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2,869,038,155,628 | arxiv | \section{Introduction}
The study of $q$-matroids, introduced by Crapo \cite{crapo1964theory}, has recently attracted renewed attention because of its link to network coding. After the reintroduction of the object by Jurrius and Pellikaan \cite{JP18} and independently that of $(q,r)$-polymatroids by Shiromoto \cite{S19}, several other papers have studied these objects, often in relation to rank metric codes. See for example \cite{BCR21,GJ20,ghorpade2021shellability,gluesing2021qpolyindep,gluesing2021qpoly,gorla2019rank,panja2019some}. \\
Roughly speaking, a $q$-analogue in combinatorics is a generalisation from sets to finite dimensional vector spaces. So a $q$-matroid is a finite dimensional vector space with a rank function defined on its subspaces, satisfying certain properties. One can also view this generalisation from the point of view of the underlying lattice: where matroids have the Boolean lattice (of sets and subsets) as their underlying structure, $q$-matroids are defined over the subspace lattice. The work of finding a $q$-analogue often comes down to writing a statement about sets in such a lattice-theoretic way that the $q$-analogue is a direct rephrasing for the subspace lattice. However, this is often not a trivial task, for two reasons. First, there might be several equivalent ways to define something over the Boolean lattice, where the $q$-analogues of these statements are not equivalent. Secondly, some statements on the Boolean lattice do not have a $q$-analogue: the subspace lattice is, contrarily to the Boolean lattice, not distributive. \\
In this paper we consider the {\em direct sum} of two $q$-matroids. An option to do this is to extend to the realm of sum-matroids \cite{panja2019some}, but we are looking for a construction that gives a $q$-matroid. This is one of the cases as mentioned above where the $q$-analogue is a lot harder than the relatively simple procedure of taking the direct sum of two classical matroids. The latter is defined as follows. Let $E_1$ and $E_2$ be disjoint sets and let $E=E_1\cup E_2$. Let $M_1=(E_1,r_1)$ and $M_2=(E_2,r_2)$ be two matroids. Then the direct sum $M_1\oplus M_2$ is a matroid with ground set $E$. For its rank function, note that we can write any $A\subseteq E$ as a disjoint union $A=A_1\sqcup A_2$ with $A_1\subseteq E_1$ and $A_2\subseteq E_2$. The rank function of the direct sum $M_1\oplus M_2$ is now given by $r(A)=r_1(A_1)+r_2(A_2)$. \\
If we try to mimic this procedure in the $q$-analogue, we run into trouble quite fast. Let $E_1$ and $E_2$ be disjoint subspaces and let $E=E_1\oplus E_2$. If we consider a subspace $A\subseteq E$, it might be that we cannot write it as a direct sum $A_1\oplus A_2$, with $A_1\subseteq E_1$ and $A_2\subseteq E_2$. In fact, most of the subspaces of $E$ can not be written in this way. Our goal is to define a rank function for these subspaces. \\
A naive try is to define a rank function in the $q$-analogue for all spaces $A\subseteq E$ that can be written as $A_1\oplus A_2$, and hope that the axioms for the rank function take care of the rest of the spaces. However, as we show with an example in Section \ref{ExDim4}, this procedure does not give us a unique direct sum. As a byproduct of this example, we find the smallest non-representable $q$-matroid. \\
Our solution for the direct sum of $q$-matroids is the following. We will first define the notion of {\em matroid union} for $q$-matroids in Section \ref{MatUn}. This notion is dual to matroid intersection, that we consider in Section \ref{IntDual}. Then we show in Section \ref{DirSum} that the direct sum of a $q$-matroid and a loop can be defined. Finally, we define the direct sum of two $q$-matroids by first adding loops to get two $q$-matroids on the same ground space, and then taking their matroid union. \\
To motivate this definition we show that this construction has several desirable properties. First of all, it generalises our naive attempt in Section \ref{ExDim4}. Also, taking the dual of a direct sum is isomorphic to first taking duals and then taking the direct sum. Lastly, restriction and contraction to $E_1$ and $E_2$ give back one of the original $q$-matroids. \\
We finish this paper by briefly considering what it would mean for a $q$-matroid to be connected (Section \ref{Connect}). As one might assume from the difficulty of the direct sum, this is also not an easy endeavour. We outline the problems that appear when trying to make a $q$-analogue of some of the several equivalent definitions of connectedness in classical matroids. \\
At the end of this paper (Appendix \ref{qcatalogue}) we give a catalogue of small $q$-matroids. In the paper, we will often refer to examples from this catalogue. Since the study of $q$-matroids is a relatively new one, we hope this catalogue to be useful for others learning about $q$-matroids.
\section{Preliminaries}\label{PrelimSec}
Following the notation of \cite{BCR21} we denote by $n$ a fixed positive integer and by $E$ a fixed $n$-dimensional vector space over an arbitrary field $\mathbb{F}.$ The notation $\mathcal{L}(E)$ indicates the \textbf{lattice of subspaces} of $E$. For any $A,B\in\mathcal{L}(E)$ with $A\subseteq B$ we denote by $[A,B]$ the interval between $A$ and $B$, that is, the lattice of all subspaces $X$ with $A\subseteq X\subseteq B$. For $A\subseteq E$ we use the notation $\mathcal{L}(A)$ to denote the interval $[\{0\},A]$. For more background on lattices, see for example Birkhoff \cite{birkhoff}. \\
We use the following definition of a $q$-matroids.
\begin{Definition}\label{rankfunction}
A $q$-matroid $M$ is a pair $(E,r)$ where $r$ is an integer-valued function defined on the subspaces of $E$ with the following properties:
\begin{itemize}
\item[(R1)] For every subspace $A\in \mathcal{L}(E)$, $0\leq r(A) \leq \dim A$.
\item[(R2)] For all subspaces $A\subseteq B \in \mathcal{L}(E)$, $r(A)\leq r(B)$.
\item[(R3)] For all $A,B$, $r(A+ B)+r(A\cap B)\leq r(A)+r(B)$.
\end{itemize}
The function $r$ is called the {\bf rank function} of the $q$-matroid.
\end{Definition}
Sometimes, we will need to deal with the rank functions of more than one $q$-matroid at a time, say $M,M'$, with ground spaces $E$, $E'$, respectively. In order to distinguish them (and emphasize the $q$-matroid in which we are computing the rank), we will write $r(M;A)$ for the rank in $M$ of a subspace $A\subseteq E$ and $r(M';A')$ for the rank in $M'$ of a subspace $A'\subseteq E'$. For a $q$-matroid $M$ with ground space $E$, we use $r(M)$ as notation for $r(M;E)$. \\
A way to visualise a $q$-matroid is by taking the Hasse diagram of the underlying subspace lattice and colour all the covers: red if the rank goes up and green if the rank stays the same. This is done in Appendix \ref{qcatalogue}. More properties of this bi-colouring can be found in \cite{BCJ17}.
There are several important subspaces in $q$-matroids.
\begin{Definition}
Let $(E,r)$ be a $q$-matroid. A subspace $A$ of $E$ is called an {\bf independent} space of $(E,r)$ if \[r(A)=\dim A.\]
An independent subspace that is maximal with respect to inclusion is called a \textbf{basis}.
A subspace that is not an independent space of $(E,r)$ is called a {\bf dependent space} of the $q$-matroid $(E,r)$.
We call $C \in \mathcal{L}(E)$ a \textbf{circuit} if it is itself a dependent space and every proper subspace of $C$ is independent.
A {\bf spanning space} of the $q$-matroid $(E,r)$ is a subspace $S$ such that $r(S)=r(E)$.
A subspace $A$ of a $q$-matroid $(E,r)$ is called a \textbf{flat} if for all $1$-dimensional subspaces $x \in \mathcal{L}(E)$ such that $x\nsubseteq A$ we have \[r(A+x)>r(A).\]
A subspace $H$ is called a $\textbf{hyperplane}$ if it is a maximal proper flat, i.e., if $H \neq E$ and the only flat that properly contains $H$ is $E$.
\end{Definition}
A $q$-matroid can be equivalently defined by its independent spaces, bases, circuits, spanning spaces, flats and hyperplanes. See \cite{BCR21} for an overview of these cryptomorphic definitions. We will explicitly use the axioms for circuits:
\begin{Definition}\label{circuit-axioms}
Let $\mathcal{C}\subseteq\mathcal{L}(E)$. We
define the following {\bf circuit axioms}.
\begin{itemize}
\item[(C1)] $\{0\}\notin\mathcal{C}$.
\item[(C2)] For all $C_1,C_2\in\mathcal{C}$, if $C_1\subseteq C_2$, then $C_1=C_2$.
\item[(C3)] For distinct $C_1,C_2 \in \mathcal{C}$ and any $X\in \mathcal{L}(E)$ of codimension $1$ there is a circuit $C_3 \subseteq \mathcal{C}$ such that $C_3 \subseteq (C_1+C_2)\cap X$.
\end{itemize}
If $\mathcal{C}$ satisfies the circuit axioms (C1)-(C3), we say that $(E,\mathcal{C})$ is a collection of {\bf circuits}.
\end{Definition}
Recall that a lattice isomorphism between a pair of lattices $(\mathcal{L}_1,\leq_1,\vee_1,\wedge_1)$ and $(\mathcal{L}_2,\leq_2,\vee_2,\wedge_2)$ is a bijective function $\varphi:\mathcal{L}_1\longrightarrow\mathcal{L}_2$ that is order-preserving and preserves the meet and join, that is, for all $x,y\in\mathcal{L}_1$ we have that $\varphi(x\wedge_1 y)=\varphi(x)\wedge_2\varphi(y)$ and $\varphi(x\vee_1 y)=\varphi(x)\vee_2\varphi(y)$. A lattice anti-isomorphism between a pair of lattices is a bijective function $\psi:\mathcal{L}_1\longrightarrow\mathcal{L}_2$ that is order-reversing and interchanges the meet and join, that is, for all $x,y\in\mathcal{L}_1$ we have that $\psi(x\wedge_1 y)=\psi(x)\vee_2\psi(y)$ and $\psi(x\vee_1 y)=\psi(x)\wedge_2\psi(y)$.
We hence define a notion of equivalence and duality between $q$-polymatroids.
\begin{Definition}
Let $E_1,E_2$ be vector spaces over the same field $\mathbb{F}$. Let $M_1=(E_1,r_1)$ and $M_2=(E_2,r_2)$ be $q$-matroids. We say that $M_1$ and $M_2$ are \textbf{lattice-equivalent} or \textbf{isomorphic} if there exists a lattice isomorphism $\varphi:\mathcal{L}(E_1)\longrightarrow \mathcal{L}(E_2)$ such that $r_1(A)=r_2(\varphi(A))$ for all $A\subseteq E_1$. In this case we write $M_1 \cong M_2$.
\end{Definition}
Fix an anti-isomorphism $\perp:\mathcal{L}(E)\longrightarrow\mathcal{L}(E)$ that is an involution. For any subspace $X \in \mathcal{L}(E)$ we denote by $X^\perp$ the \textbf{dual} of $X$ in $E$ with respect to $\perp$. Note that since an anti-isomorphism preserves the length of intervals, we have for any $X\leq\mathcal{L}(E)$ that $\dim(X^\perp)=\dim(E)-\dim(X)$. \\
From a lattice point of view we can rewrite $B=B_1\oplus B_2$ as $B=B_1\vee B_2$ and $B_1\wedge B_2=0$. Since $\perp$ is an anti-isomorphism of $\mathcal{L}(E)$, we have that $B^\perp=B_1^\perp\wedge B_2^\perp$ and $B_1^\perp\vee B_2^\perp=1$.
Important operations on $q$-matroids are restriction, contraction and duality. We give a short summary here and refer to \cite{BCIR21,JP18} for details.
\begin{Definition}\label{defdual}
Let $M=(E,r)$ be a $q$-matroid. Then $M^*=(E,r^*)$ is also a $q$-matroid, called the \textbf{dual $q$-matroid}, with rank function
\[ r^*(A)=\dim(A)-r(E)+r(A^\perp). \]
The subspace $B$ is a basis of $M$ if and only if $B^\perp$ is a basis of $M^*$. From bi-colouring point of view, we get the dual $q$-matroid by turning the Hasse diagram upside down and interchange all red and green covers.
\end{Definition}
\begin{Definition}\label{restr}
Let $M=(E,r)$ be a $q$-matroid. The \textbf{restriction} of $M$ to a subspace $X$ is the $q$-matroid $M|_X$ with ground space $X$ and rank function $r_{M|_X}(A)=r_M(A)$. In other words, it is the $q$-matroid we get when only regarding the interval $[0,X]$ in $\mathcal{L}(E)$. \\
The \textbf{contraction} of $M$ of a subspace $X$ is the $q$-matroid $M/X$ with ground space $E/X$ and rank function $r_{M/X}(A)=r_M(A)-r_M(X)$. In other words, it is the $q$-matroid we get when only regarding the interval $[X,1]$ in $\mathcal{L}(E)$.
\end{Definition}
\begin{Theorem}\label{DualRestrContr}
Restriction and contraction are dual operations, that is, $M^*/X\cong M|_{X^\perp}$ and $(M/X)^* \cong M^*|_{X^\perp}$.
\end{Theorem}
Finally, we will define what it means for a $q$-matroid to be representable.
\begin{Definition}
Let $M=(E,r)$ be a $q$-matroid of rank $k$ over a field $K$. Let $A\subseteq E$ and let $Y$ be a matrix with column space $A$. We say that $M$ is \textbf{representable} if there exists a $k\times n$ matrix $G$ over an extension field $L/K$ such that $r(A)$ is equal to the matrix rank of $GY$ over $L$.
\end{Definition}
\section{Intuitive try for the direct sum}\label{ExDim4}
As stated in the introduction, the $q$-analogue of the direct sum is not straightforward. Let $E=E_1\oplus E_2$ be a direct sum of subspaces and let $A\subseteq E$. Then we cannot, in general, decompose $A\subseteq E$ as $A=A_1\oplus A_2$ with $A_1\subseteq E_1$ and $A_2\subseteq E_2$. With other cryptomorphic definitions of $q$-matroids, through the likes of independent spaces, bases or circuits, we run into similar problems. \\
In this section we explore if we can define the rank function of a direct sum of $q$-matroids by simply defining $r(A)=r_1(A_1)+r_2(A_2)$ for all $A$ that can be written as $A=A_1\oplus A_2$, and hoping that the rank axioms will take care of the rest of the subspaces. (Spoiler alert: it will not work.)
\subsection{First definition and properties}\label{FirstDef}
Let us make our first trial to define the direct sum. We start with the next definition, mimicking the classical case. We also prove some basic properties following from this definition. The definition will turn out to be not unique, but our final definition will satisfy the properties below.
\begin{Definition}\label{def-directsum1}
Let $M_1=(E_1,r_1)$ and $M_2=(E_2,r_2)$ be two $q$-matroids on trivially intersecting ground spaces. We define a $q$-matroid $M=(E,r)$ on the ground space $E=E_1\oplus E_2$ as a direct sum of $M_1$ and $M_2$ if it satisfies the following:
\begin{itemize}
\item the minors $M|_{E_1}$ and $M/E_2$ (that is, the intervals $[0,E_1]$ and $[E_2,1]$ in $M$) are both isomorphic to $M_1$,
\item the minors $M|_{E_2}$ and $M/E_1$ (that is, the intervals $[0,E_2]$ and $[E_1,1]$ in $M$) are both isomorphic to $M_2$.
\end{itemize}
\end{Definition}
In particular, it follows from this construction that the rank of $M$ is the sum of the ranks of $M_1$ and $M_2$. The next theorem shows that this definition is equivalent to what we recognise as the $q$-analogue of the definition of direct sum in the classical case.
\begin{Theorem}\label{RangoSomma}
Let $M_1=(E_1,r_1)$ and $M_2=(E_2,r_2)$ be two $q$-matroids on trivially intersecting ground spaces. We define a $q$-matroid $M=(E,r)$ on the ground space $E=E_1\oplus E_2$. Then it satisfies the properties of Definition \ref{def-directsum1} if and only if for each $A\subseteq E_1$ and $B\subseteq E_2$ it holds $r(A+B)=r_1(A)+r_2(B)$.
\end{Theorem}
\begin{proof}
First, assume $M$ satisfies the properties of Definition \ref{def-directsum1}. Note that for all $A\subseteq E_1$ we have $r(A)=r(M;A)=r(M|_{E_1};A)=r_1(A)$ and similarly, for all $B\subseteq E_2$ we have $r(B)=r_2(B)$. So we need to show that $r(A+B)=r(A)+r(B)$. We prove this by applying semimodularity multiple times. First we apply it to $A$ and $B$. Since $A\cap B=\mathbf{0}$, we have $r(A\cap B)=0$ and (r3) gives us
\[ r(A+B)\leq r(A)+r(B). \]
Now we apply (r3) to $E_1$ and $A+B$. Note that by construction of the direct sum, $r(E_1+B)=r(E_1)+r(B)$.
\begin{align*}
r(E_1)+r(A+B) & \geq r(E_1+(A+B))+r(E_1\cap(A+B)) \\
& = r(B+E_1)+r(A) \\
& = r(B)+r(E_1)+r(A)
\end{align*}
This implies that
\[ r(A+B)\geq r(A)+r(B). \]
Combining the two inequalities gives the desired equality: $r(A+B)=r(A)+r(B)$. \\
For the other implication, suppose that $r(A+B)=r_1(A)+r_2(B)$. The two conditions in Definition \ref{def-directsum1} are symmetric, so we only need to prove the first one. We show that the rank function on $M|_{E_1}$ is equal to the rank function on $M_1$. Let $A\subseteq E_1$. Then
\[ r(M|_{E_1};A)=r(M;A)=r_1(A)+r_2(0)=r_1(A). \]
Now for $M/E_2$, let $C\subseteq E$ such that $E_2\subseteq C$. Then we can write $C=A+E_2$ with $A\subseteq E_1$. Then
\[ r(M/E_2;C/E_2)=r(M;C)-r(M;E_2)=r_1(A)+r_2(E_2)-r(M;E_2)=r_1(A). \]
It follows that $M|_{E_1}$ and $M/E_2$ are both isomorphic to $M_1$.
\end{proof}
As mentioned, the classical case of this last theorem is exactly the definition of the rank in the direct sum of matroids. This implies that Definition \ref{def-directsum1}, when applied to the classical case, completely determines the direct sum. We will see in the next section that this is not the case in the $q$-analogue. \\
We will close this section with some small results that show that Definition \ref{def-directsum1} implies the rank of all spaces of dimension and codimension $1$. Note that the next results only depend on Definition \ref{def-directsum1}, with the exception of Lemma \ref{allredontop}.
\begin{Proposition}\label{prop-noloops}
Let $M$ be a $q$-matroid satisfying the properties of Definition \ref{def-directsum1}. Suppose $M_1$ and $M_2$ do not have any loops. Then $M$ does not have any loops.
\end{Proposition}
\begin{proof}
Suppose, towards a contraction, that there is a loop $\ell$ in $M$. By construction, $\ell$ is not in $E_1$ or in $E_2$. First we apply the semimodular inequality to $E_1$ and $\ell$:
\begin{align*}
r(E_1+\ell)+r(E_1\cap\ell) & \leq r(E_1)+r(\ell) \\
r(E_1+\ell)+0 & \leq r(E_1)+0
\end{align*}
hence $r(E_1+\ell)=r(E_1)$. Now we apply the semimodular inequality to $E_1+\ell$ and $E_2$. Note that $(E_1+\ell)\cap E_2$ is a $1$-dimensional subspace in $E_2$ that has, again by construction, rank $1$.
\begin{align*}
r((E_1+\ell)+ E_2)+r((E_1+\ell)\cap E_2) & \leq r(E_1+\ell)+r(E_2) \\
r(E_1+E_2)+1 & \leq r(E_1)+r(E_2)
\end{align*}
and this is a contradiction. So there are no loops in $M$.
\end{proof}
\begin{Corollary}\label{LoopSum}
Let $M$ be a $q$-matroid satisfying the properties of Definition \ref{def-directsum1}. The loop space of $M$ is a direct sum of the loop space of $M_1$ and the loop space of $M_2$.
\end{Corollary}
\begin{proof}
Let $L_1$ be the loop space of $M_1$ and $L_2$ the loop space of $M_2$. Since loops come in subspaces \cite[Lemma 11]{JP18}, $L_1\oplus L_2$ in $E$ only contains loops. Applying Proposition \ref{prop-noloops} to $E_1/L_1$ and $E_2/L_2$, we get that there are no other loops.
\end{proof}
In particular, since we know exactly what are the loops of the direct sum, we know that all other $1$-dimensional spaces have rank $1$. Dually, we can derive a similar result for the codimension-$1$ spaces.
The next Lemma holds for all $q$-matroids. It is the dual of the statement that loops come in subspaces.
\begin{Lemma}\label{allredontop}
Let $M=(E,r)$ be a $q$-matroid. All spaces of codimension $1$ in $E$ of rank $r(M)-1$ contain a certain subspace $H$, and the spaces $A$ in the interval $[H,1]$ are exactly all the subspaces such that $r(E)-r(A)=\dim E-\dim A$.
\end{Lemma}
\begin{proof}
Let $H_1$ and $H_2$ be of codimension $1$ in $E$ and let $r(H_1)=r(H_2)=r(M)-1$. Then semimodularity gives that
\[ r(H_1\cap H_2)\leq r(H_1)+r(H_2)-r(H_1+H_2)=r(M)-1+r(M)-1-r(M)=r(M)-2 \]
hence $r(E)-r(H_1\cap H_2)=2=\dim E-\dim(H_1\cap H_2)$. So by induction, intersecting two spaces such that $r(E)-r(A)=\dim E-\dim A$ gives again a space such that this holds. This proves the lemma.
\end{proof}
The next result is the dual of Proposition \ref{prop-noloops}.
\begin{Proposition}\label{prop-nocoloops}
Let $M$ be a $q$-matroid satisfying the properties of Definition \ref{def-directsum1}. Suppose $M_1$ and $M_2$ do not have any codimension $1$ spaces of rank $r(M_1)-1$ and $r(M_2)-1$, respectively. Then $M$ does not have any codimension $1$ spaces of rank $r(M)-1$.
\end{Proposition}
\begin{proof}
Suppose, towards a contraction, that there is a codimension $1$ space $H$ of rank $r(M)-1$ in $M$. By construction, $H$ does not contain $E_1$ or $E_2$. So $E_1\cap H$ is of codimension $1$ in $E_1$, and by construction it has rank $r(E_1)-1$. \\
Now we apply the semimodular inequality to $E_1\cap H$ and $E_2$. Note that $(E_1\cap H)+ E_2$ is a codimension $1$ subspace of $E$ containing $E_2$ so it has, again by construction, rank $r(E_1)+r(E_2)$.
\begin{align*}
r((E_1\cap H)+ E_2)+r((E_1\cap H)\cap E_2) & \leq r(E_1\cap H)+r(E_2) \\
r(E_1)+r(E_2)+0 & \leq r(E_1)-1+r(E_2)
\end{align*}
and this is a contradiction. So there are no codimension $1$ spaces in of rank $r(M)$ in $M$.
\end{proof}
\subsection{Non-uniqueness of the first definition}\label{subsec-NotUnique}
In this section we show by example that Definition \ref{def-directsum1} does not uniquely define the direct sum of $q$-matroids. \\
Let $E=\mathbb{F}_2^4$ and let $M_1=M_2=U_{1,2}$. We will attempt to construct the direct sum $M=M_1\oplus M_2$. We assume that it has the properties from Definition \ref{def-directsum1}. So the $q$-matroid $M$ has at least two circuits: $E_1$ and $E_2$. Our goal is to determine $M$ completely. Note that Theorem \ref{RangoSomma} defines the rank for all subspaces of $E$ that can be written as a direct sum of a subspace of $E_1$ and a subspace of $E_2$. \\
All $1$-dimensional spaces in $E$ have rank $1$ because of Proposition \ref{prop-noloops} and by Proposition \ref{prop-nocoloops} all $3$-dimensional spaces in $E$ have rank $2$. This means that what is left to do is to decide for all $2$-dimensional spaces if they have rank $1$ or rank $2$, that is, whether they are a circuit or an independent space. We use the next lemma for this.
\begin{Lemma}\label{CircuitIntersection}
Let $M=U_{1,2}\oplus U_{1,2}$ as in Definition \ref{def-directsum1}. Let $C_1$ and $C_2$ be circuits of dimension $2$. Then $\dim(C_1\cap C_2)\neq1$.
\end{Lemma}
\begin{proof}
If $C_1=C_2$, the result is clear. So let $C_1\neq C_2$. Suppose, towards a contradiction, that $\dim(C_1\cap C_2)=1$. Then $\dim(C_1+C_2)=3$. Now apply semimodularity to $C_1$ and $C_2$.
\begin{align*}
r(C_1+C_2)+r(C_1\cap C_2) & \leq r(C_1)+r(C_2) \\
2+1 & \leq 1+1
\end{align*}
This is a contradiction, hence $\dim(C_1\cap C_2)\neq1$.
\end{proof}
This means that every $2$-dimensional space that intersects with either $E_1$ or $E_2$ is independent. A counting argument shows that there are only six $2$-dimensional spaces that have nonempty intersection with both $E_1$ and $E_2$. \\
Denote by $A$, $B$, $C$, $D$, $F$, $G$ the six $2$-spaces of unknown rank. Do this in a way such that $\{E_1,E_2,A,B,C\}$ and $\{E_1,E_2,D,F,G\}$ form a \emph{spread} in $E$ (a spread is a set of subspaces of the same dimension that intersect trivially and cover the whole space \cite{segre}). The two spreads are isomorphic, in the sense that a change of basis of $E$ maps one to the other. Since $A$, $B$, and $C$ all intersect $D$, $F$, and $G$, deciding that at least one of $\{A,B,C\}$ is a circuit means $\{D,F,G\}$ are all independent, and vice versa. So, without loss of generality, we have completely determined the matroid $M$ if we have found which of the three $2$-dimensional spaces $A$, $B$, and $C$ are circuits and this implies that $D,F,G$ are all independent.
\begin{Lemma}\label{3ContainsSpread}
Every $3$-dimensional space $T$ contains an element of the spread \[\{E_1,E_2,A,B,C\}.\]
\end{Lemma}
\begin{proof}
This can be done via a counting argument and the pigeon hole principle. $T$ intersects all spread elements in dimension at least $1$, since $\dim E=4$. All $1$-dimensional subspaces of $E$ are by definition contained in exactly one spread element. There are five spread elements and seven $1$-dimensional subspaces in $T$, so there has to be a spread element that contains at least two $1$-dimensional subspaces of $T$, and hence intersects it in dimension $2$. But that means the whole spread element is contained in $T$.
\end{proof}
If $A$, $B$, and $C$ are all circuits, there are no other circuits because of Lemma \ref{CircuitIntersection} and axiom (C2). If not all of $A$, $B$, and $C$ are circuits, there have to be circuits of dimension $3$. These will be all the $3$-dimensional spaces that do not contain a circuit of dimension $2$. These circuits do, however, contain an element of the spread, by Lemma \ref{3ContainsSpread}. \\
We check the circuit axioms for this construction. (C1) and (C2) are clear. For (C3), notice that the sum of every pair of circuits is equal to $E$. Thus it is sufficient to show that every $3$-space contains a circuit. This is true by construction: a 3-space either contains a $2$-dimensional circuit, or it is a circuit itself. \\
We have seen that no matter what we decide for the independence of $A$, $B$, and $C$, we always get a $q$-matroid. This means that the properties of the direct sum as in Definition \ref{def-directsum1} are not enough to determine the direct sum completely: we can make a $q$-matroid with $2$, $3$, $4$ or $5$ circuits that all satisfy this definition.
\subsection{A small non-representable $q$-matroid}
As a byproduct of the example in the previous section, we find a non-representable $q$-matroid in dimension $4$. The existence of non-representable $q$-(poly)matroids was established and discussed in \cite{gluesing2021qpolyindep}. However, the example here is not included in their construction and it is also the smallest possible non-representable $q$-matroid. In the classical case, the smallest non-representable matroid is of size $8$ and rank $4$ (the V\'amos matroid). For $q$-matroids it is smaller: dimension $4$ and rank $2$.
\begin{Proposition}\label{MatrixAssoc}
Let $M$ be a representable $q$-matroid over $\mathbb{F}_2$ of rank $2$ and dimension $4$, with two circuits of dimension $2$ and no loops. Then the matrix representing $M$ has the shape
$$G:=\left[\begin{array}{cccc}
1 & \alpha & 0 & 0 \\
0 & 0 & 1 & \beta
\end{array}\right],$$
with $\alpha,\beta \in \mathbb{F}_{2^m}\setminus\mathbb{F}_2$.
\end{Proposition}
\begin{proof}
Since $M$ has rank $2$ and dimension $4$, the shape of the matrix is
\[ G:=\left[\begin{array}{cccc}
x_1 & x_2 & x_3 & x_4 \\
y_1 & y_2 & y_3 & y_4
\end{array}\right], \]
with all entries in $\mathbb{F}_{2^m}$.
Without loss of generality we apply row reduction and get $x_1=1, y_1=0$.
Since there are no loops, the columns of $G$ cannot be all zero.
Consider now the two circuits. They are, without loss of generality, $E_1:=\langle(1,0,0,0),(0,1,0,0) \rangle$ and
$E_2:=\langle(0,0,1,0),(0,0,0,1) \rangle$.
We have for $E_1$
$$\left[\begin{array}{cccc}
1 & x_2 & x_3 & x_4 \\
0 & y_2 & y_3 & y_4
\end{array}\right] \cdot \left[\begin{array}{cc}
1 & 0 \\
0 & 1\\
0 & 0 \\
0 & 0
\end{array}\right] = \left[\begin{array}{cc}
1 & x_2 \\
0 & y_2
\end{array}\right],$$
whose rank must be one, leading to $y_2=0$.
Similarly, for $E_2$ we have
$$\left[\begin{array}{cccc}
1 & x_2 & x_3 & x_4 \\
0 & y_2 & y_3 & y_4
\end{array}\right] \cdot \left[\begin{array}{cc}
0 & 0 \\
0 & 0\\
1 & 0 \\
0 & 1
\end{array}\right] = \left[\begin{array}{cc}
x_3 & x_4 \\
y_3 & y_4
\end{array}\right],$$
whose rank must be one, leading to the fact that $(x_3,x_4)$ and $(y_3,y_4)$ are scalar multiples. By row reduction we can conclude that $x_3=x_4=0$ and the absence of loops implies that $x_2,y_3,y_4\neq 0$. We can finally set, again by row reduction, $y_3=1$.
Note that column operations over the ground field $\mathbb{F}_2$ give an isomorphic $q$-matroid, so we have that $x_2$ and $y_4$ are elements of $\mathbb{F}_{2^m}$ but not of $\mathbb{F}_2$.
\end{proof}
\begin{Theorem}\label{NonReprDim4}
If the $q$-matroid considered before is representable, it cannot have $4$ circuits of dimension $2$. This gives an example of a non-representable $q$-matroid.
\end{Theorem}
\begin{proof}
We know the representation is of the form
\[ \left[\begin{array}{cccc}
1 & \alpha & 0 & 0 \\
0 & 0 & 1 & \beta
\end{array}\right], \]
with $\alpha,\beta \in \mathbb{F}_{2^m}\setminus\mathbb{F}_2$ by Proposition \ref{MatrixAssoc}.
Consider
\[ \left[\begin{array}{cccc}
1 & \alpha & 0 & 0 \\
0 & 0 & 1 & \beta
\end{array}\right] \cdot
\left[\begin{array}{cc}
a_0 & b_0 \\
a_1 & b_1 \\
a_2 & b_2 \\
a_3 & b_3
\end{array}\right] =
\left[\begin{array}{cc}
a_0+a_1\alpha & b_0+b_1\alpha \\
a_2+a_3\beta & b_2+b_3\beta
\end{array}\right].
\]
In order to have a circuit of dimension $2$, the determinant of this $2\times 2$ matrix should be zero. In particular, we need to have proportional columns.
This automatically tells us that $a_0=a_1=0$ implies $b_0=b_1=0$, and that $a_2=a_3=0$ implies $b_2=b_3=0$.
These two cases correspond to the two circuits $E_1$ and $E_2$ from Proposition \ref{MatrixAssoc}. By a computer search, we found the determinants of all $2$-dimensional spaces $A,B,C,D,F,G$ from Section \ref{subsec-NotUnique}. They are the following:
\begin{itemize}
\item[$A$:] $\alpha+\beta$
\item[$B$:] $\alpha\beta +\alpha+1$
\item[$C$:] $\alpha \beta+\beta+1$
\item[$D$:] $\alpha \beta+\beta+ \alpha$
\item[$F$:] $\alpha \beta+1$
\item[$G$:] $\alpha+\beta +1$
\end{itemize}
Now, it is easy to see that if $A$ and $B$ vanish, then $C$ vanishes as well, and the same goes for $D$, $F$ and $G$. We already saw that circuits appear either in $\{A,B,C\}$ or $\{D,E,F\}$ and the other spaces are independent.
Therefore, the alternatives we have are:
\begin{itemize}
\item none of the six determinants above vanishes, so $E_1$ and $E_2$ are the only circuits of dimension $2$;
\item one determinant vanishes, so we have three circuits of dimension $2$;
\item the determinants of all the elements in a spread vanish, leading to five circuits of dimension $2$ (that are all circuits in the $q$-matroid). \qedhere
\end{itemize}
\end{proof}
\begin{Corollary}
The $q$-matroid over $\mathbb{F}_2$ of rank $2$ and dimension $4$ with four circuits, as described in Theorem \ref{NonReprDim4}, is the smallest non-representable $q$-matroid.
\end{Corollary}
\begin{proof}
See the appendix for a list of all $q$-matroids with a ground space of dimension at most $3$. All of these are representable. Hence, the $q$-matroid from Theorem \ref{NonReprDim4} is the smallest non-representable $q$-matroid.
\end{proof}
\begin{example}\label{SomeExamplesRepresentables}
Consider the finite field $\mathbb{F}_8$ and a primitive element $\alpha$. We give some examples of $q$-matroids of dimension $4$ and rank $2$, arising from our construction of $U_{1,2}\oplus U_{1,2}$, distinguishing them by the number of their circuits:
\begin{itemize}
\item $\left[\begin{array}{cccc}
1 & \alpha & 0 & 0 \\
0 & 0 & 1 & \alpha^5
\end{array}\right]$ represents a $q$-matroid with two circuits of dimension $2$;
\item $\left[\begin{array}{cccc}
1 & \alpha & 0 & 0 \\
0 & 0 & 1 & \alpha^3
\end{array}\right]$ represents a $q$-matroid with three circuits of dimension $2$;
\item $\left[\begin{array}{cccc}
1 & \alpha & 0 & 0 \\
0 & 0 & 1 & \alpha
\end{array}\right]$ represents a $q$-matroid with five circuits of dimension $2$.
\end{itemize}
\end{example}
\begin{Remark}
Example \ref{SomeExamplesRepresentables} above also tells us something about the direct sum of two representable $q$-matroids. Suppose we have two representable $q$-matroids $M_1$ and $M_2$ over the same field $K$. Suppose these $q$-matroids are representable by matrices $G_1$ and $G_2$ over an extension field $L$ of $K$. One would expect the direct sum $M_1\oplus M_2$ to be representable by
\[ \left[\begin{array}{cc} G_1 & 0 \\ 0 & G_2 \end{array}\right]. \]
However, this construction is not uniquely defined, in the sense that it depends on the representations of $M_1$ and $M_2$. Over $\mathbb{F}_8$, we can represent the $q$-matroid $U_{1,2}$ as either $\left[\begin{array}{cc} 1 & \alpha \end{array}\right]$ or $\left[\begin{array}{cc} 1 & \alpha^3 \end{array}\right]$ or $\left[\begin{array}{cc} 1 & \alpha^5 \end{array}\right]$. Taking the first of these three as $G_1$ and all of them as different options for $G_2$ gives precisely the $q$-matroids of Example \ref{SomeExamplesRepresentables}. These are not isomorphic $q$-matroids.
\end{Remark}
\section{Submodular functions and $q$-(poly)matroids}
Our goal is to define the direct sum of $q$-matroids in terms of matroid union. Before we can define that, we need some background on integer-valued increasing submodular functions. A function $f$ on the subspaces of $E$ is submodular if the following hold for all $A,B\subseteq E$:
\[ f(A+B)+f(A\cap B) \leq f(A)+f(B). \]
Such function can be viewed as the rank function of a $q$-polymatroid, and we refer to \cite{gluesing2021qpolyindep} for an extension of, and some overlap with, the results presented here.
The following Proposition and Corollary are the $q$-analogues of Proposition 11.1.1 and Corollary 11.1.2 in \cite{oxley}.
\begin{Proposition}\label{CircuitsOfMf}
Let $f$ be an integer-valued increasing submodular function on the subspaces of a finite-dimensional vector space $E$. Let
\[ \mathcal{C}(f)=\{C\subseteq R:C\text{ is minimal and non-empty such that }f(C)<\dim(C)\}. \]
Then $\mathcal{C}(f)$ is the collection of circuits of a $q$-matroid $M(f)=(E,\mathcal{C}(f))$.
\end{Proposition}
\begin{proof}
We prove that $\mathcal{C}(f)$ satisfies the circuit axioms from Definition \ref{circuit-axioms}. The axiom (C1) holds by definition and, by minimality, we have (C2).\\
Let us now prove (C3). Let $C_1 \neq C_2$ be two elements of $\mathcal{C}(f)$ and let $X$ be a codimension $1$ space containing neither $C_1$ nor $C_2$ (otherwise the assertion holds void).
We have $C_i \cap X \subseteq C_i$ for $i=1,2$. Therefore,
$C_i \cap X \notin \mathcal{C}(f)$ by (C2) and so $\dim(C_i \cap X) \leq f(C_i \cap X)$ for $i=1,2$. Since $f$ is increasing,
\[
\dim(C_i \cap X) \leq f(C_i \cap X) \leq f(C_i) \leq \dim(C_i)
\]
and thus $\dim(C_i\cap X)=\dim(C_i)-1$ and $\dim(C_i)-1 =f(C_i)$.
Since $f$ is increasing, is suffices to show $f((C_1+C_2)\cap X)<\dim((C_1+C_2)\cap X)$, because then $(C_1+C_2)\cap X$ contains a circuit. Now $f$ is increasing and submodular, so
\[
f((C_1+C_2)\cap X)\leq f(C_1+C_2)
\leq f(C_1)+f(C_2)-f(C_1 \cap C_2)\]
and because $C_1 \cap C_2 \subsetneq C_i$ for $i=1,2$, by minimality, $f(C_1 \cap C_2) \geq \dim(C_1 \cap C_2)$. Finally,
\begin{align*}
f((C_1+C_2)\cap X) & \leq f(C_1)+f(C_2)-f(C_1 \cap C_2) \\
& = \dim(C_1)+\dim(C_2)-2-f(C_1 \cap C_2) \\
& \leq \dim(C_1+C_2)-2 \\
& =\dim((C_1+C_2)\cap X)-1.
\end{align*}
This shows that $M(f)$ is a $q$-matroid defined by its circuits $\mathcal{C}(f)$.
\end{proof}
The following is a direct result of the definition of $\mathcal{C}(f)$ and the fact that every proper subspace of a circuit is independent.
\begin{Corollary}\label{IndepInMf}
A subspace $I\subseteq E$ is independent in $M(f)$ if and only if $\dim(I')\leq f(I')$ for all nontrivial subspaces $I'$ of $I$.
\end{Corollary}
The next theorem is the $q$-analogue of \cite[Chapter 8.1 Theorem 2]{welsh1976matroid}. We point out that Theorem \ref{thm-FunctionToMatroid} and Propsition \ref{IndepOfPoly} ware already proven in \cite[Theorem 3.9]{gluesing2021qpolyindep}, but with the mimum taken over the subspaces of $A$ instead of all spaces in $E$. See also Remark \ref{r-incl-only}.
\begin{Theorem}\label{thm-FunctionToMatroid}
Let $f$ be a non-negative integer-valued increasing submodular function on the subspaces of $E$ with $f(0)=0$. Then
\[ r(A)=\min_{X\subseteq E}\{f(X)+\dim(A)-\dim(A\cap X)\} \]
is the rank function of a $q$-matroid.
\end{Theorem}
\begin{proof}
We will prove that the function $r$ satisfies the rank axioms. It is clear that $r$ is integer-valued. It is non-negative because both $f(A)$ and $\dim(A)-\dim(A\cap X)$ are non-negative. By taking $X=\{0\}$ in the definition, we get $f(\{0\})+\dim(A)-\dim(\{0\})=\dim(A)$ and therefore $r(A)\leq\dim(A)$. This proves (r1). \\
In order to prove (r2), let $A\subseteq B\subseteq E$. Then for any $X\subseteq E$, we have that $\dim(B)-\dim(A)\geq\dim(B\cap X)-\dim(A\cap X)$. It follows that
\[ f(X)+\dim(A)-\dim(A\cap X) \leq f(X)+\dim(A)-\dim(B\cap X) \]
for all $X\subseteq E$ and thus $r(A)\leq r(B)$. The proof of (r3) is rather technical, but essentially a lot of rewriting. We first claim that
\begin{align*}
& \dim(A)-\dim(A\cap X)+\dim(B)-\dim(B\cap Y) \geq \\
& \quad \dim(A+B)-\dim((A+B)\cap(X+Y))+\dim(A\cap B)-\dim((A\cap B)\cap(X\cap Y)).
\end{align*}
This statement will be used later on in the proof. By using that
\[ \dim(A)+\dim(B)=\dim(A+B)+\dim(A\cap B)\]
and multiplying by $-1$ we can rewrite our claim as
\[ \dim(A\cap X)+\dim(B\cap Y) \leq \dim((A+B)\cap(X+Y))+\dim((A\cap B)\cap(X\cap Y)). \]
Using the modular equality again, we get
\begin{align*}
\dim((A\cap B)\cap(X\cap Y)) &= \dim((A\cap X)\cap(B\cap Y)) \\
& = \dim(A\cap X)+\dim(B\cap Y)-\dim((A\cap X)+(B\cap Y))
\end{align*}
and thus our claim is equivalent to
\[ \dim((A\cap X)+(B\cap Y))\leq\dim((A+B)\cap(X+Y)).\]
To prove this, it is enough to show the inclusion of vector spaces $(A\cap X)+(B\cap A)\subseteq(A+B)\cap(X+Y)$. Let $\mathbf{a}\in A\cap X$ and $\mathbf{b}\in B\cap Y$ be two nonzero vectors. Then $\mathbf{a}+\mathbf{b}\in(A\cap X)+(B\cap Y)$. We prove that $\mathbf{a}+\mathbf{b}\in(A+B)\cap(X+Y)$. Because $\mathbf{a}\in A$ also $\mathbf{a}\in A+B$ and because $\mathbf{a}\in X$ also $\mathbf{a}\in X+Y$. So $\mathbf{a}\in(A+B)\cap(X+Y)$. By a similar reasoning, $\mathbf{b}\in(A+B)\cap(X+Y)$. So $\mathbf{a}+\mathbf{b}\in(A+B)\cap(X+Y)$ as was to be shown. This finishes the proof of our claim. \\
We can now get back to proving axiom (r3). In the third step we use the claim together with the submodularity of $f$. In the fourth step we set $U=X+Y$ and $V=X\cap Y$. This will not produce all possible $U,V\subseteq E$, so the minimum is at least as big as the minimum over all $U,V\subseteq E$.
\begin{align*}
\lefteqn{r(A)+r(B)} \\
& = \min_{X\subseteq E}\{f(X)+\dim(A)-\dim(A\cap X)\} + \min_{Y\subseteq E}\{f(Y)+\dim(B)-\dim(B\cap Y)\} \\
& = \min_{X,Y\subseteq E}\{f(X)+f(Y)+\dim(A)-\dim(A\cap X)+\dim(B)-\dim(B\cap Y)\} \\
& \geq \min_{X,Y\subseteq E}\{ f(X+Y)+f(X\cap Y)+\dim(A+B)-\dim((A+B)\cap(X+Y)) \\
& \qquad +\dim(A\cap B)-\dim((A\cap B)\cap(X\cap Y))\} \\
& \geq \min_{U,V\subseteq E}\{f(U)+f(V)+\dim(A+B)-\dim((A+B)\cap U) \\
& \qquad +\dim(A\cap B)-\dim((A\cap B)\cap V)\} \\
& = \min_{U\subseteq E}\{f(U)+\dim(A+B)-\dim((A+B)\cap U)\} \\
& \qquad + \min_{V\subseteq E}\{f(V)+\dim(A\cap B)-\dim((A\cap B)\cap V)\} \\
& = r(A+B)+r(A\cap B).
\end{align*}
So the rank function $r$ satisfies all rank axioms (r1),(r2),(r3).
\end{proof}
\begin{Remark}\label{r-incl-only}
Note that the minimum in Theorem \ref{thm-FunctionToMatroid} is taken over all subspaces of $E$. This is convenient for some of the proofs, but not strictly necessary. Let $X\subseteq E$ and let $X'=A\cap X$. Then
\[ f(X')+\dim(A)-\dim(A\cap X') \leq f(X)+\dim(A)-\dim(A\cap X) \]
because $f(X')\leq f(X)$ and $\dim(A\cap X')=\dim(A\cap X)=\dim(X')$. This means that the minimum over all subspaces $X\subseteq E$ is the same as the minimum taken only over the subspaces $X'\subseteq A$. This makes calculating the rank function a lot faster in practice.
\end{Remark}
The next proposition shows that the $q$-matroids from Corollary \ref{IndepInMf} and Theorem \ref{thm-FunctionToMatroid} are the same.
\begin{Proposition}\label{IndepOfPoly}
Let $f$ be a non-negative integer-valued increasing submodular function with $f(0)=0$. Let $M(f)$ be the corresponding $q$-matroid as defined in Corollary \ref{IndepInMf} with independent spaces $\mathcal{I}$. Let $r$ be the rank function as defined in Theorem \ref{thm-FunctionToMatroid}. Then both give the same $q$-matroid because $r(I)=\dim(I)$ for all $I\in\mathcal{I}$.
\end{Proposition}
\begin{proof}
We have to prove that $r(I)=\dim(I)$ iff $\dim(I')\leq f(I')$ for all nontrivial subspace $I'$ of $I$. Note that since $f(0)=0$, this holds for all subspaces $I'$ of $I$, also the trivial one. (Note that Proposition \ref{CircuitsOfMf} does not require $f(0)=0$, but Theorem \ref{thm-FunctionToMatroid} does.)
From the remark before we have that
\[ r(I)=\min_{I'\subseteq I}\{f(I')+\dim(I)-\dim(I')\}. \]
As already proven in Theorem \ref{thm-FunctionToMatroid}, $r(I)\leq\dim(I)$. For the other inequality, the following are equivalent:
\begin{align*}
I\in\mathcal{I}(M(f)) &\Leftrightarrow f(I')\geq\dim(I') \text{ for all }I'\subseteq I \\
&\Leftrightarrow f(I')+\dim(I)-\dim(I')\geq\dim(I) \text{ for all }I'\subseteq I \\
&\Leftrightarrow r(I)\geq\dim(I)
\end{align*}
This proves that $r(I)=\dim(I)$.
\end{proof}
\section{Matroid union}\label{MatUn}
In this section we define the $q$-analogue of matroid union by means of its rank function and we show what are the independent spaces.
\begin{Definition}\label{DefUnione}
Let $M_1$ and $M_2$ be two $q$-matroids on the same ground space $E$, with rank functions $r_1$ and $r_2$, respectively. Then the \emph{matroid union} $M_1\vee M_2$ is defined by the rank function
\[ r(A)=\min_{X\subseteq A}\{r_1(X)+r_2(X)+\dim A-\dim X\} \]
\end{Definition}
\begin{Theorem}\label{Union_q-Matr}
Let $M_1$ and $M_2$ be two $q$-matroids on the same ground space $E$, with rank functions $r_1$ and $r_2$, respectively. Then the matroid union $M_1\vee M_2$ is a $q$-matroid.
\end{Theorem}
\begin{proof}
For all $A\subseteq E$, define a function $f(A)=r_1(A)+r_2(A)$. We claim that $f$ is a non-negative integer-valued submodular function on the subspaces of $E$ with $f(0)=0$. \\
Note that $r_1$ and $r_2$ are non-negative integer valued submodular functions on the subspaces of $E$ with $r_1(\{0\})=r_2(\{0\})=0$. It follows directly that $f$ is a non-negative integer-valued function on the subspaces of $E$. It is increasing, because for all $A\subseteq B\subseteq E$ we have
\[ f(A)= r_1(A)+r_2(A)\leq r_1(B)+r_2(B)=f(B). \]
Furthermore, $f$ is submodular, because for all $A,B\subseteq E$ we have
\begin{align*}
f(A+B)+f(A\cap B) & = r_1(A+B)+r_2(A+B)+r_1(A\cap B)+r_2(A\cap B) \\
& \leq r_1(A)+r_1(B)+r_2(A)+r_2(B) \\
& = f(A)+f(B).
\end{align*}
Now we apply Theorem \ref{thm-FunctionToMatroid} and Remark \ref{r-incl-only} to the function $f$: this shows that the function $r$ of Definition \ref{DefUnione} is indeed the rank function of a $q$-matroid $(E,r)$.
\end{proof}
We gather some important properties of the matroid union.
\begin{Remark}\label{dependsOnCoordinates}
The matroid union is not always invariant under coordinatisation. That is, if $\varphi:\mathcal{L}(E)\longrightarrow\mathcal{L}(E)$ is a lattice isomorphism, then it is direct from the definition that $\varphi(M_1)\vee\varphi(M_2)=\varphi(M_1\vee M_2)$. However, $M_1\vee M_2$ is not necessarily isomorphic to $\varphi(M_1)\vee M_2$. We illustrate this with a small example. \\
Let $M_1$ and $M_2$ both be isomorphic to the mixed diamond, see \ref{qcd2}. That is: $\dim(E)=2$, $r(E)=1$ and $r(A)=1$ for all $1$-dimensional spaces except one loop. Suppose the loop is at the same coordinates for both $M_1$ and $M_2$, call this subspace $\ell$. Then the rank of $M_1\vee M_2$ is one, as we will show. Consider all $X\subseteq E$. If $\dim(X)=0$ or $\dim(X)=2$ then the expression inside the minimum of Definition \ref{DefUnione} is equal to $2$. If $\dim(X)=1$ we have to distinguish between $\ell$ and the any other space. If $X=\ell$ the expression is $0+0+2-1=1$, otherwise it is $1+1+2-1=3$. Therefore, $r(E)=1$. \\
Consider now the case where the loop of $M_1$ is $\ell_1$ and the loop of $M_2$ is $\ell_2$, with $\ell_1\neq\ell_2$. Then the calculations are as before for $\dim(X)=0$ or $\dim(X)=2$. For $\dim(X)=1$ and $X\neq\ell_1,\ell_2$ we get $1+1+2-1=3$. If $X=\ell_1$ we get $0+1+2-1=2$, and similarly for $X=\ell_2$ we get $1+0+2-1=2$. So $r(E)=2$. \\
This example illustrates that we have to be careful to define $M_1$ and $M_2$ precisely, not just up to isomorphism.
\end{Remark}
We prove two straightforward lemmas concerning the matroid union.
\begin{Lemma}\label{AddGreenDoesNothing}
Let $M_1$ and $M_2$ be two $q$-matroids on the same ground space $E$ and let $r(M_2)=0$. Then $M_1\vee M_2=M_1$ and in particular, $M\vee U_{0,n}=M$.
\end{Lemma}
\begin{proof}
The rank function of the matroid union is equal to
\[ r(A)=\min_{X\subseteq A}\{r_1(X)+0+\dim A-\dim X\}. \]
We see that the minimum is attained when $\dim A=\dim X$, and in that case $r(A)=r_1(A)$. So $M_1\vee M_2=M_1$.
\end{proof}
\begin{Lemma}\label{IndipUnionFromM1M2}
Let $M_1$ and $M_2$ be $q$-matroids on the same ground space $E$. Let $I$ be independent in both $M_1$ and $M_2$. Then $I$ is independent in $M_1\vee M_2$.
\end{Lemma}
\begin{proof}
We have that $r(M_1;I)=r(M_2;I)=\dim I$ by definition. Also, all subspaces of $I$ are independent. This means that
\begin{align*}
r(M_1\vee M_2;I) &= \min_{X\subseteq I}\{r(M_1;X)+r(M_2;X)+\dim I-\dim X\} \\
&= \min_{X\subseteq I}\{\dim X+\dim X+\dim I-\dim X\} \\
&= \dim I.
\end{align*}
We conclude that $I$ is independent in $M_1\vee M_2$.
\end{proof}
The independent spaces of the matroid union can be found in the following way.
\begin{Theorem}\label{indipUnion}
Let $M_1=(E,\mathcal{I}_1)$ and $M_2=(E,\mathcal{I}_2)$ be two $q$-matroids defined by their independent spaces. Then $I\subseteq E$ is an independent space of the matroid union $M_1\vee M_2$ if and only if for all nontrivial $J\subseteq I$ there exist $I_1\in\mathcal{I}_1$ and $I_2\in\mathcal{I}_2$ such that $J=I_1\oplus I_2$. We notate the collection of independent spaces of $M_1\vee M_2$ by $\mathcal{I}$.
\end{Theorem}
\begin{proof}
Let $f(A)=r_1(A)+r_2(A)$, as in the proof of Theorem \ref{Union_q-Matr}. According to Corollary \ref{IndepInMf} and Proposition \ref{IndepOfPoly}, we know that the independent spaces of the matroid union are exactly those $I\subseteq E$ such that for all nontrivial subspaces $J\subseteq I$ we have $\dim J\leq f(J)$. \\
First, let $I\subseteq E$ such that all nontrivial subspace $J\subseteq I$ can be written as $J=I_1\oplus I_2$ with $I_1\in\mathcal{I}_1$ and $I_2\in\mathcal{I}_2$. We need to prove that $I$ is independent in $M_1\vee M_2$, that is, for all $J\subseteq I$ it holds that $\dim J\leq f(J)$. This follows from
\[ \dim(J)=\dim(I_1)+\dim(I_2) = r_1(I_1)+r_2(I_2) \leq r_1(J)+r_2(J)=f(J), \] the inequality coming from the axiom (r2). \\
For the other implication, let $I$ be independent in $M_1\vee M_2$. We need to show that we can write all $I$ as $I=I_1\oplus I_2$ with $I_1\in\mathcal{I}_1$ and $I_2\in\mathcal{I}_2$. Because all subspaces of an independent space are independent, this proves the statement. \\
First, note that if $I$ is independent in $M_1\vee M_2$, then its rank is equal to its dimension: $\dim I=r(I)=\min_{X\subseteq I}\{r_1(X)+r_2(X)+\dim I-\dim X\}$. Therefore, for each $X \subseteq I$ it holds
\[ r_1(X)+r_2(X)-\dim X\geq 0. \]
We will proceed by mathematical induction on the dimension of $I$. If $\dim I=0$ then $I=0$ and we can write $0=0\oplus 0$ where $0\in\mathcal{I}_1$ and $0\in\mathcal{I}_2$. If $\dim I=1$, then $r_1(I)+r_2(I)\geq1$ so $I$ is independent in at least one of $M_1$ and $M_2$. Without loss of generality, let $I$ be independent in $M_1$, then we can write $I=I\oplus0$ with $I\in\mathcal{I}_1$ and $0\in\mathcal{I}_2$. \\
Let $I$ be independent in $M_1\vee M_2$ with $\dim I=h+1$. Let $J\subseteq I$ with $\dim J=h$ and $J=J_1\oplus J_2$ for some $J_1\in\mathcal{I}_1$ and $J_2\in\mathcal{I}_2$. We will show that for any $1$-dimensional $x\subseteq I-J$ either $J_1\oplus x\in\mathcal{I}_1$ or $J_2\oplus x\in\mathcal{I}_2$. \\
Assume that for all $x\subseteq I-J$ the space $J_1\oplus x$ is dependent in $M_1$. This implies that $r_1(J_1\oplus x)=r_1(J_1)$ and by \cite[Prop. 7]{JP18} we have $r_1(I)=r(J_1)=\dim J_1$. Since $I\subseteq I$ we have the following equivalent statements:
\begin{align*}
r_1(I)+r_2(I)-\dim I &\geq 0 \\
\dim J_1+r_2(I)-(\dim J_1+\dim J_2+1) &\geq 0 \\
r_2(I) &\geq \dim J_2+1
\end{align*}
and hence $r_2(I)\geq 1$. This implies that not all $x\subseteq I-J$ can be loops in $M_2$, because if they were, by semimodularity this would imply $r_2(I)=0$. So assume $x\subseteq I-J$ with $r_2(x)=1$. Then, by applying semimodularity in $M_2$ again, we get
\begin{align*}
r_2(J_2+x)+r_2(J_2\cap x) &\geq r_2(J_2)+r_2(x) \\
r_2(J_2\oplus x)+0 &\geq \dim J_2+1
\end{align*}
and it follows that $J_2\oplus x$ is independent in $M_2$. This gives the decomposition $I=J_1\oplus (J_2\oplus x)$ with $J_1\in\mathcal{I}_1$ and $J_2\oplus x\in\mathcal{I}_2$.
\end{proof}
\begin{Remark}
We want to point out that Theorem \ref{indipUnion} is indeed a $q$-analogue of the classical case. There the independent sets of the matroid union are defined by
\[ \mathcal{I}=\{I_1\cup I_2:I_1\in\mathcal{I}_1,I_2\in\mathcal{I}_2\}. \]
First of all, note that the union can be rewritten as a disjoint union. Let $I=J_1\cup J_2$ with $J_1\in\mathcal{I}_1$ and $J_2\in\mathcal{I}_2$. Take $I_1=J_1$ and $I_2=J_2-J_1$, then $I=I_1\sqcup I_2$. This procedure does not create a unique $I_1$ and $I_2$, there is a lot of choice involved. However, it does imply that every independent set $I$ of the matroid union is of the form $I=I_1\sqcup I_2$, and conversely, every $I=I_1\sqcup I_2$ is independent in the matroid union.
In the classical case, if $I=I_1\sqcup I_2$ then for all $\bar{J}\subseteq I$ we can write directly $\bar{J}=\bar{J}_1\sqcup \bar{J}_2$ with $\bar{J}_1=\bar{J}\cap I_1$ and $\bar{J}_2=\bar{J}\cap I_2$. Since $\bar{J}_1\subseteq I_1$ and $\bar{J}_2\subseteq I_2$, these are independent. This reasoning does not hold in the $q$-analogue (see also the Introduction), which is why we specifically have to state it in the definition. For a counterexample, see the example in Remark \ref{dependsOnCoordinates}: if $\ell_1=\ell_2=\ell$ we can write $E=I_1\oplus I_2$ for some $1$-dimensional $I_1$ and $I_2$ that are not equal to $\ell$, but we cannot write $\ell$ as the direct sum of independent spaces of $M_1$ and $M_2$.
\end{Remark}
\section{Matroid intersection and duality} \label{IntDual}
We complete our study of the matroid union for $q$-matroids by defining the dual operation, that is matroid intersection. We follow \cite[p.123]{welsh1976matroid}.
\begin{Definition}
Let $M_1$ and $M_2$ be $q$-matroids on the same ground space $E$ with collection of spanning spaces $\mathcal{S}(M_1)$ and $\mathcal{S}(M_2)$. Define the \emph{$q$-matroid intersection} of $M_1$ and $M_2$ by its spanning spaces:
\[ \mathcal{S}(M_1\wedge M_2)=\{S_1\cap S_2: S_1\in\mathcal{S}(M_1), S_2\in\mathcal{S}(M_2)\}. \]
\end{Definition}
We need to prove that it is a $q$-matroid. This can be done by checking the axioms for spanning spaces, but we can also do this by proving a more general result:
\begin{Theorem}\label{thm-intersection}
Let $M_1$ and $M_2$ be $q$-matroids on the same ground space $E$. Then
\[ M_1\wedge M_2=(M_1^*\vee M_2^*)^*. \]
\end{Theorem}
\begin{proof}
From \cite{BCR21} we know that the orthogonal complements of the spanning spaces of a $q$-matroid $M$ are the independent spaces of the dual $q$-matroid $M^*$. So we have to prove that the orthogonal complements of $\mathcal{S}(M_1\wedge M_2)$ are the independent spaces of $M_1^*\vee M_2^*$. \\
First, start with $S\in\mathcal{S}(M_1\wedge M_2)$. Then we can write $S=S_1\cap S_2$ for $S_1\in\mathcal{S}(M_1)$ and $S_2\in\mathcal{S}(M_2)$. Let $S'\supseteq S$ be a superspace of $S$. Then we can write $S'=T_1\cap T_2$ for $T_1\supseteq S_1$ and $T_2\supseteq S_2$. Note that $T_1$ and $T_2$ are also spanning spaces of $M_1$ and $M_2$, respectively. Now we take orthogonal complements.
The orthogonal complement is $S^\perp=S_1^\perp+S_2^\perp$. Now we can write $S_1^\perp=I_1^*$ with $I_1^*$ independent in $M_1^*$, and similarly, $S_2^\perp=I_2^*$ is independent in $M_2^*$. We need to prove that $I_1^*+I_2^*$ is independent in $M_1^*\vee M_2^*$, that is, we have to show that all $I'\subseteq I_1^*+I_2^*$ can be written as $J_1\oplus J_2$ with $J_1$ independent in $M_1^*$ and $J_2$ independent in $M_2^*$. Note that all $I'\subseteq I_1^*+I_2^*$ can be written as $I'=(S')^\perp$, with $S'=T_1\cap T_2$ as above. If we take orthogonal complements of $T_1$ and $T_2$, we get independent spaces of $M_1^*$ and $M_2^*$. So we can write $I'=J_1\oplus J_2$. (We can always make the sum a direct sum by taking a subspace of $J_1$ if necessary.) We conclude that $I_1^*+I_2^*$ is independent in $M_1^*\vee M_2^*$.
For the opposite inclusion, start with an independent space $I$ of $M_1^*\vee M_2^*$. Then by Theorem \ref{indipUnion} we can write $I=I_1+I_2$ with $I_1$ independent in $M_1^*$ and $I_2$ independent in $M_2^*$. Taking orthogonal complements gives that $I^\perp=I_1^\perp\cap I_2^\perp=S_1\cap S_2$ for spanning spaces $S_1$ in $M_1$ and $S_2$ in $M_2$. This implies that $I^\perp$ is in $\mathcal{S}(M_1\wedge M_2)$.
Since $(M_1^*\vee M_2^*)^*$ is a $q$-matroid, this shows that $M_1\wedge M_2$ is a $q$-matroid as well.
\end{proof}
We have the following corollary on intersection, union, and restriction and contraction.
\begin{Corollary}\label{DualUnionIntersection}
Let $M_1$ and $M_2$ be $q$-matroids on the same ground space $E$. Then, for $T\subseteq E$,
\[ (M_1\vee M_2)|_T = M_1|_T\vee M_2|_T \]
and also
\[ (M_1\wedge M_2)/T \cong (M_1/T)\wedge(M_2/T). \]
\end{Corollary}
\begin{proof}
The first part of the statement follows directly from Definition \ref{DefUnione} of the matroid union and the fact that for the rank function of the restriction is $r_{M|_T}(A)=r(A)$. The second statement follows from the first by applying Theorem \ref{thm-intersection}, use $(M/T)^* \cong M^*|_{T^\perp}$ from Theorem \ref{DualRestrContr}, and then applying Theorem \ref{thm-intersection} again.
\begin{align*}
(M_1/T)\wedge(M_2/T) & = ((M_1/T)^*\vee(M_2/T)^*)^* \\
& \cong ((M_1^*|_{T^\perp})\vee(M_2^*|_{T^\perp}))^* \\
& = ((M_1^*\vee M_2^*)|_{T^\perp})^* \\
& \cong (M_1^*\vee M_2^*)^*/T \\
& = (M_1\wedge M_2)/T. \qedhere
\end{align*}
\end{proof}
We finish this section with the dual of Lemma \ref{AddGreenDoesNothing}.
\begin{Lemma}\label{IntersectAllRed}
Let $M$ be a $q$-matroid. Then $M= M\wedge U_{n,n}$.
\end{Lemma}
\begin{proof}
Applying Lemma \ref{AddGreenDoesNothing} to $M^*$ gives that $M^*= M^*\vee U_{0,n}=(M\wedge U_{n,n})^*$. Dualising both sides gives the desired result.
\end{proof}
\section{The direct sum}\label{DirSum}
In this section we will define the direct sum of two $q$-matroids. The idea will be to first add loops to $M_1$ and $M_2$, so they are on the same ground space, and then taking their matroid union. In the classical case, we can also write the direct sum like this: the idea from this construction comes from \cite[Proposition 7.6.13 part 2]{white}.
\subsection{Defining the direct sum} \label{AdL}
The next definition explains how to ``add a loop'' to a $q$-matroid.
\begin{Definition}\label{AggiungiLoop}
Let $M=(E,r)$ be a $q$-matroid. Then the direct sum of $M$ and a loop $\ell$ is denoted by $M'=M\oplus \ell$ and constructed in the following way. Let $E'=E+\ell$. Then for every $A'\subseteq E'$ we can write $A'+\ell=A\oplus\ell$ for a unique $A\subseteq E$. Then $r'(A')=r(A)$.
\end{Definition}
\begin{Remark}\label{caseSplitting}
The definition above divides the subspaces of $E'$ into three different kinds.
\begin{itemize}
\item If $A'\subseteq E$ then $A'=A$ and $\dim A'=\dim A$.
\item If $A'\supseteq \ell$ then $A=A'\cap E$ and $\dim A=\dim A'-1$.
\item If $A'$ is not contained in $E$ and does not contain $\ell$, then $\dim A'=\dim A$. There is a diamond with bottom $A'\cap A\subseteq E$, top $A'+\ell$ and with $A$ and $A'$ in between.
\end{itemize}
\end{Remark}
This construction is well defined, in the sense that it gives a $q$-matroid, as the next theorem shows.
\begin{Theorem}\label{AddLoop}
The direct sum $M'=M\oplus\ell$ as defined above is a $q$-matroid, that is, the rank function $r'$ satisfies (r1),(r2),(r3).
\end{Theorem}
\begin{proof}
(r1) Since $r(A)\geq0$ we have $r'(A')\geq0$ as well. We get that $r'(A')=r(A)\leq\dim A\leq\dim A'$ by Remark \ref{caseSplitting}.
(r2)
Let $A'\subseteq B'$. Since
$A=(A'+l)\cap E$,
$B=(B'+l)\cap E$ and
$A'+l \subseteq B'+l$, we have that $A \subseteq B$. Therefore $r'(A')=r(A)\leq r(B)=r'(B')$.
For (r3) let $A',B'\subseteq E'$. We first claim that $(A'+B')+\ell=(A+B)+\ell$ and $(A'\cap B')+\ell=(A\cap B)+\ell$, because this implies that
\begin{align*}
r'(A'+B') + r'(A'\cap B') & = r(A+B) + r(A\cap B) \\
& \leq r(A)+r(B) \\
& = r'(A')+r'(B').
\end{align*}
Now let us prove the claims. For addition, we see that
\[ (A'+B')+\ell=(A'+\ell)+(B'+\ell)=(A+\ell)+(B+\ell)=(A+B)+\ell. \]
For intersection we distinguish three cases depending on whether $A'$ and $B'$ contain $\ell$.
\begin{itemize}
\item Let $\ell\not\subseteq A',B'$. Then $(A'\cap B')+\ell=(A'+\ell)\cap(B'+\ell)=(A+\ell)\cap(B+\ell)=(A\cap B)+\ell$.
\item Let $\ell\subseteq A',B'$. Then $(A'\cap B')+\ell=A'\cap B'=(A+\ell)\cap(B+\ell)=(A\cap B)+\ell$.
\item Let $\ell\subseteq A'$ and $\ell\not\subseteq B'$. Then $(A'\cap B')+\ell=((A'\cap E)\cap B')+\ell=((A'\cap E)+\ell)\cap(B'+\ell)=(A+\ell)\cap(B+\ell)=(A\cap B)+\ell$.
\end{itemize}
The function $r'$ satisfies the axioms (r1),(r2),(r3), hence $M$ is a $q$-matroid.
\end{proof}
We combine the adding of loops and the matroid union to define the direct sum.
\begin{Definition}\label{DirSumWithUnion}
Let $M_1=(E_1,r_1)$ and $M_2=(E_2,r_2)$ be two $q$-matroids on trivially intersecting ground spaces. Let $n_1=\dim E_1$ and $n_2=\dim E_2$. We construct the direct sum $M_1\oplus M_2$ as follows.
\begin{itemize}
\item Let $E=E_1\oplus E_2$. This will be the ground space of $M$. By slight abuse of notation, we denote by $E_i$ both the ground space of $M_i$ and the embedding of $E_i$ in $E$.
\item In the lattice $\mathcal{L}(E)$ we have that the intervals $[0,E_1]$ and $[E_2,1]$ are isomorphic to $\mathcal{L}(E_1)$, and the intervals $[0,E_2]$ and $[E_1,1]$ are isomorphic to $\mathcal{L}(E_2)$. Fix the involution $\perp$ such that $E_1^\perp=E_2$.
\item Add $n_2$ times a loop to $M_1$, using Theorem \ref{AddLoop}. This gives the $q$-matroid $M_1'$ on ground space $E$. Assume that $M_1'|_{E_1}\cong M_1$ and $M_1'|_{E_2}\cong U_{0,n_2}$.
\item Add $n_1$ times a loop to $M_2$, using again Theorem \ref{AddLoop}. This gives the $q$-matroid $M_2'$ on ground space $E$. Assume that $M_2'|_{E_1}\cong U_{0,n_1}$ and $M_2'|_{E_2}\cong M_2$.
\end{itemize}
Now the direct sum is defined as $M_1\oplus M_2=M_1'\vee M_2'$, with the matroid union as in Theorem \ref{Union_q-Matr}.
\end{Definition}
Note that this procedure is well-defined, since we already showed that adding loops and taking the matroid union are well-defined constructions. We do, however, have to show that this procedure always defines the same $q$-matroid up to isomorphism, since it was observed in Remark \ref{dependsOnCoordinates} that matroid union is not invariant under coordinatisation.
\begin{Theorem}\label{IsomorphicIsLinearAlgebra}
Let $M_1=(E_1,r_1)$ and $M_2=(E_2,r_2)$ be two $q$-matroids on trivially intersecting ground spaces and let $M=M_1\oplus M_2$ be their direct sum as constructed in Definition \ref{DirSumWithUnion}. Let $\varphi_i$ be a lattice-isomorphism of $\mathcal{L}(E_i)$ for $i=1,2$. Then there is an isomorphism $\psi$ of $\mathcal{L}(E)$ such that $\varphi_1(M_1)\oplus\varphi_2(M_2)=\psi(M)$.
\end{Theorem}
\begin{proof}
Let $\psi$ be an isomorphism on $\mathcal{L}(E)$ such that $\psi|_{E_1}=\varphi_1$ and $\psi|_{E_2}=\varphi_2$. We can construct $\psi$ by its images of $n_1+n_2$ linearly independent $1$-dimensional spaces: we find these by taking the image under $\varphi_1$ of $n_1$ linearly independent $1$-spaces in $E_1$ and the image under $\varphi_2$ of $n_2$ linearly independent $1$-spaces in $E_2$. \\
Let $A\subseteq E$ and let $B\subseteq E_1$ such that $A+E_2=B\oplus E_2$. This means that $B=(A+E_2)\cap E_1$. Now we have that
\begin{align*}
(\psi(A)+E_2)\cap E_1 &= (\psi(A)+\psi(E_2))\cap E_1 \\
&= \psi(A+E_2)\cap E_1 \\
&= \psi(B\oplus E_2) \cap E_1 \\
&= (\psi(B)\oplus\psi(E_2))\cap E_1 \\
&= (\varphi_1(B)\oplus E_2)\cap E_1 \\
&= \varphi_1(B) \\
&= \varphi_1((A+E_2)\cap E_1).
\end{align*}
The rank function of $\varphi_1(M_1)'$ is equal to
\begin{align*}
r(\varphi_1(M_1)';A) & = r(\varphi_1(M_1);(A+E_2)\cap E_1) \\
& = r(M_1;\varphi_1((A+E_2)\cap E_1)) \\
& = r(M_1; (\psi(A)+E_2)\cap E_1) \\
& = r(M_1';\psi(A) ).
\end{align*}
We have a similar argument for $M_2$ and $\varphi_2$. Combining these gives that
\begin{align*}
\lefteqn{r(\varphi_1(M_1)\oplus\varphi_2(M_2);A)} \\
& = r(\varphi_1(M_1)'\vee\varphi_2(M_2)';A) \\
&= \min_{X\subseteq E}\{ r(\varphi_1(M_1)';X)+r(\varphi_2(M_2)';X)+\dim A-\dim(A\cap X)\} \\
&= \min_{\psi(X)\subseteq E}\{ r(M_1';\psi(X))+r(M_2';\psi(X))+\dim\psi(A)-\dim(\psi(A)\cap \psi(X))\} \\
&= r(M;\psi(A)) = r(\psi(M);A).
\end{align*}
This proves the theorem.
\end{proof}
We prove a lemma that will make the calculations in the next section easier.
\begin{Lemma}\label{l-RankOfSum}
For two $q$-matroids $M_1$ and $M_2$ it holds that
\[r(M_1\oplus M_2)=r(M_1)+r(M_2).\]
\end{Lemma}
\begin{proof}
By applying Definitions \ref{AggiungiLoop} and \ref{DirSumWithUnion}, we get that
\[ r(M_1\oplus M_2)=r(M_1'\vee M_2';E)=\min_{X\subseteq E}\{ r(M_1';X)+r(M_2';X)+\dim E-\dim X\}. \]
If we take $X=E$, we get that
\[ r(M_1';X)+r(M_2';X)+\dim E-\dim X = r(M_1';E)=r(M_2';E)=r(M_1)+r(M_2). \]
Now let $Y_1\subseteq E_1$ such that $X+E_2=Y_1\oplus E_2$. Then $r(M_1';X)=r(M_1;Y_1)$. Similarly, let $Y_2\subseteq E_2$ such that $X+E_1=Y_2\oplus E_1$ so $r(M_2';X)=r(M_2;Y_2)$. We have that $\dim(Y_1)=\dim(X)-\dim(X\cap E_2)$ and $\dim(Y_2)=\dim(X)-\dim(X\cap E_1)$. Note that, by local semimodularity, $r(M_1;Y_1)\geq r(M_1;E_1)-\dim(E_1)+\dim(Y_1)$ and similarly $r(M_2;Y_2)\geq r(M_2;E_2)-\dim(E_2)+\dim(Y_2)$. All together this gives
\begin{align*}
\lefteqn{r(M_1';X)+r(M_2';X)+\dim E-\dim X} \\
&= r(M_1';X)+r(M_2';X)+\dim E-\dim X \\
&= r(M_1;Y_1)+r(M_2;Y_2)+\dim E-\dim X \\
&\geq r(M_1;E_1)-\dim(E_1)+\dim(Y_1) \\
& \quad +r(M_2;E_2)-\dim(E_2)+\dim(Y_2)+\dim E-\dim X \\
&= r(M_1)+r(M_2)-\dim(X)+\dim(Y_1)+\dim(Y_2) \\
&= r(M_1)+r(M_2)-\dim(X)+\dim(X)-\dim(X\cap E_2)+\dim(X)-\dim(X\cap E_1) \\
&= r(M_1)+r(M_2)+\dim(X)-\dim(X\cap E_2)-\dim(X\cap E_1) \\
&\geq r(M_1)+r(M_2).
\end{align*}
This means that the minimum $\min_{X\subseteq E}\{ r(M_1';X)+r(M_2';X)+\dim E-\dim X\}$ is attained by $X=E$ and $r(M_1\oplus M_2)=r(M_1)+r(M_2)$.
\end{proof}
\subsection{Examples of the direct sum}
To get some feeling for this construction, we analyse some small examples. We refer to the Appendix for an overview of small $q$-matroids. \\
We start with the easiest examples possible, with $n_1=n_2=1$.
\begin{Example}
Let $M_1=M_2=U_{0,1}$. This is the sum of two loops. In fact, we could just use Theorem \ref{AddLoop} here, without Definition \ref{DirSumWithUnion}, but we do the whole procedure for clarity. For $M_1'=M_1\oplus\ell$, let $E_1=\langle(1,0)\rangle$. Then by Theorem \ref{AddLoop}, $M_1'$ is a $q$-matroid of rank $0$, so all its subspaces have rank zero. In fact, $M_1'\cong U_{0,2}$. Let $E_2=\langle(0,1)\rangle$. We also have that $M_2'\cong U_{0,2}$. Applying Theorem \ref{Union_q-Matr} we find that $M_1'\vee M_2'\cong U_{0,2}$. \\
Let $M_1=U_{0,1}$ and $M_2=U_{1,1}$. Then $M_1'=U_{0,2}$ as argued above. For $M_2'$, let $E_2=\langle(0,1)\rangle$ and apply Theorem \ref{AddLoop}. By construction, $r(\{0\})=0$. In dimension $1$ we have $r(\langle01\rangle)=r(E_2)=r_2(E_2)=1$, $r(\ell)=r_2(\{0\})=0$, and for all other spaces $A$ of dimension $1$ we have $r(A)=r_2(E_2)=1$. These are the three cases in Remark \ref{caseSplitting}. Note that $M_2'=U_{1,2}$. Finally, we have $r(E)=r_2(E_2)=1$. By Lemma \ref{AddGreenDoesNothing}, $M_1 \oplus M_2=M_1' \vee M_2' =M_2'$. \\
The last case to consider is $M_1=M_2=U_{1,1}$. We have seen that $M_1'=M_2'=U_{1,2}$. To get $M_1'\vee M_2'$, we first see that $r(\{0\})=0$. In dimension $1$, we have that $r(\langle(0,1)\rangle)=\min\{ 0+0+1-0, 1+0+1-1 \}=1$. For $r(\langle(1,0)\rangle)$ we get the same but in a different order, so the rank is again $1$. For a $1$-dimensional space not equal to $E_1$ or $E_2$ we get $r(A)=\min\{0+0+1-0, 1+1+1-1 \}=1$. Finally, for $E$ we get $r(E)=\min\{0+0+2-0,1+0+2-1,0+1+2-1,1+1+2-1,1+1+2-0\}=2$. So, $M_1\oplus M_2=U_{2,2}$.
\end{Example}
Note that it follows from this example that $U_{1,2}$ is connected: it cannot be written as a direct sum.
\begin{Example}\label{ex-PrimePlusRed}
We calculate the $q$-matroid $P_1^*$ (see Section \ref{P1star}), it is the sum of a prime diamond (see Section \ref{qcd2}) and an independent 1-dimensional space, that is, $M_1=U_{1,2}$ and $M_2=U_{1,1}$. Let $E_1=\langle (0,0,1),(0,1,0)\rangle$ and $E_2=\langle (1,0,0)\rangle$. We first have to make $M_1'$ and $M_2'$. \\
For $M_1'$ we take $\ell=E_2=\langle(1,0,0)\rangle$. We have that $r_1'(0)=0$ and $r_1'(E)=r(M_1)=1$. For a $1$-dimensional space inside $E_1$, the rank is $1$, while $r_1'(\ell)=0$. For any other $1$-dimensional space $A$, $r_1'(A)=r_1(A')$ for $A'\subseteq E_1$, so $r_1'(A)=1$. For the $2$-dimensional spaces $A$, $r_1'(E_1)=1$. If $\ell\subseteq A$, $r_1'(A)=r_1(A\cap E_1)=1$. For the other $2$-dimensional spaces we have $r_1'(A)=r_1(E_1)=1$. Together, we find that $M_1'$ is the $q$-matroid $P_1$ in the Section \ref{P1}. \\
For $M_2'$ we have to add a loop twice to $U_{1,1}$. The first loop gives the mixed diamond, as explained in the previous example. The second one gives a $q$-matroid isomorphic to $P_2$ (see Section \ref{P2}). \\
Now we take the union. We have $r(0)=0$ and also $r(E)=2$ by Lemma \ref{l-RankOfSum}. \\ There are three types of $1$-dimensional spaces, as well as three types of $2$-dimensional spaces. Let $\dim A=1$. If $A\subseteq E_1$ then $r(A)=\min\{0+0+1-0, 1+0+1-1\}=1$. \\If $A=E_2$ then $r(A)=\min\{0+0+1-1,0+1+1-1\}=1$. For the other $1$-dimensional spaces $A$, $r(A)=\min\{0+0+1-0,1+1+1-1\}=1$. Now let $\dim A=2$. If $A=E_1$ then $r(A)=\min\{0+0+2-0,1+0+2-1,1+0+2-2\}=1$. For the other 2-dimensional spaces $A$, note that any $1$-dimensional space has rank $1$ in either $M_1'$ or in $M_2'$, contributing $1+0+2-1=0+1+2-1=2$ to the minimum. The zero space also contributes $0+0+2-0=2$, and the space itself gives $1+1+2-1=2$. So $r(A)=2$. \\
In total, we see that $U_{1,2}\oplus U_{1,1}\cong P_1^*$ .
\end{Example}
\subsection{Properties of the direct sum}
We will now show that the direct sum as defined here has some desirable properties. All of these results are also true for the classical case, motivating the `correctness' of the definition of the direct sum presented in the previous section. Further support of the definition is provided by \cite{GJ}, where it is shown that the direct sum is the coproduct in the category of $q$-matroids and linear weak maps.
\begin{Theorem}
Let $M_1$ and $M_2$ be two $q$-matroids with ground spaces $E_1$ and $E_2$, respectively. Let their direct sum be as defined in Definition \ref{DirSumWithUnion}. Then for any $A\subseteq E$ of the form $A=A_1\oplus A_2$ with $A_1\subseteq E_1$ and $A_2\subseteq E_2$ it holds that $r(M_1\oplus M_2;A)=r(M_1;A_1)+r(M_2;A_2)$.
\end{Theorem}
\begin{proof}
By definition of the direct sum we have that
\[ r(M_1\oplus M_2;A)=\min_{X\subseteq A}\{r(M_1';X)+r(M_2';X)+\dim A-\dim X\}. \]
We will show that the minimum is attained for $X=A$. First, note that $A+E_2=A_1\oplus E_2$ and $A+E_1=A_2\oplus E_1$. Then taking $X=A$ inside the minimum gives
\[ r(M_1';A)+r(M_2';A)+\dim A-\dim A=r(M_1;A_1)+r(M_2;A_2). \]
We have left to show that for any $X\subseteq A$, the quantity inside the minimum is at least $r(M_1;A_1)+r(M_2;A_2)$. To see this, take $B_1\subseteq E_1$ and $B_2\subseteq E_2$ such that $X+E_2=B_1\oplus E_2$ and $X+E_1=B_2\oplus E_1$. \\
For the dimension of $B_1$, we have that $\dim B_1=\dim(X+E_2)-\dim E_2=\dim X-\dim(X\cap E_2)$. Furthermore, $B_1\subseteq A_1$ and thus by local semimodularity, $r(M_1;A_1)-\dim A_1\leq r(M_1;B_1)-\dim B_1$. Similar results hold for $B_2$. Finally, note that $\dim B_1+\dim B_2\leq\dim X$. \\
Combining this, we get that
\begin{align*}
\lefteqn{r(M_1';X)+r(M_2';X)+\dim A-\dim X} \\
&= r(M_1;B_1)+r(M_2;B_2)+\dim A-\dim X \\
&\geq r(M_1;A_1)-\dim A_1+\dim B_1+r(M_2;A_2)-\dim A_2+\dim B_2+\dim A-\dim X \\
&\geq r(M_1;A_1)+r(M_2;A_2).
\end{align*}
This completes the proof that $r(M_1\oplus M_2;A)=r(M_1;A_1)+r(M_2;A_2)$.
\end{proof}
From Theorem \ref{RangoSomma} the following is now immediate.
\begin{Corollary}\label{MinorTheorem}
Let $M_1$ and $M_2$ be two $q$-matroids with ground spaces $E_1$ and $E_2$, respectively. Then their direct sum, as defined in Definition \ref{DirSumWithUnion}, satisfies the properties of Definition \ref{def-directsum1}.
\end{Corollary}
Note that this implies that also the rest of the results in Section \ref{FirstDef} hold for our definition \ref{DirSumWithUnion} of the direct sum. Another desirable property of our definition of the direct sum is that the dual of the direct sum is the direct sum of the duals. \\
In order to prove that direct sum commutes with duality, we need to define duality on $E_1$, $E_2$, and $E$ in a compatible way.
\begin{Definition}
Let $E=E_1\oplus E_2$ and let $\perp$ be an anti-isomorphism on $\mathcal{L}(E)$ such that $E_1^\perp=E_2$. Define an anti-isomorphism $\perp\!\!(E_1)$ on $E_1$ by
\[ A^{\perp(E_1)}:=(A+E_2)^\perp=A^\perp\cap E_2^\perp=A^\perp\cap E_1. \]
Similarly, we define the anti-isomorphism $A^{\perp(E_2)}=A^\perp\cap E_2$.
\end{Definition}
The map $\perp\!\!(E_1)$ (and, similarly, $\perp\!\!(E_2)$) is indeed an anti-isomorphism, because it is the concatenation of the isomorphism $[0,E_1]\to[E_2,E]$ given by $A\mapsto A\oplus E_2$ and the anti-isomorphism $\perp$ restricted to $[E_2,E]\to[0,E_1]$.
\begin{Theorem}\label{thm-DualDirect}
Let $M_1$ and $M_2$ be $q$-matroids on ground spaces $E_1$ and $E_2$, respectively. Then we have that $(M_1\oplus M_2)^*=M_1^*\oplus M_2^*$.
\end{Theorem}
\begin{proof}
Let $B$ be a basis of $M_1\oplus M_2$. We will prove that $B^\perp$ is a basis of $M_1^*\oplus M_2^*$. First, note that by Lemma \ref{l-RankOfSum} we have
\begin{align*}
r(M_1^*\oplus M_2^*) &= r(M_1^*)+r(M_2^*) \\
&= \dim E_1-r(M_1)+\dim E_2-r(M_2) \\
&= \dim E-r(M_1\oplus M_2) \\
&= \dim B^\perp.
\end{align*}
This means that if we show that $B^\perp$ is independent in $M_1^*\oplus M_2^*$, it is also a basis. The rank of $B^\perp$ in $M_1^*\oplus M_2^*$ is given by
\[ r(M_1^*\oplus M_2^*)=\min_{X\subseteq E}\{r((M_1^*)';X)+r((M_2^*)';X)+\dim B^\perp-\dim(B^\perp\cap X)\}. \]
We want this to be equal to $\dim B^\perp$, hence we need to show for all $X\subseteq E$ that
\[ \dim(B^\perp\cap X)\leq r((M_1^*)';X)+r((M_2^*)';X). \]
Equality is attained by taking $X=0$. Now note that
\begin{align*}
((X+E_2)\cap E_1)^{\perp(E_1)} & = ((X+E_2)\cap E_1)^\perp\cap E_1 \\
&= ((X+E_2)^\perp + E_2)\cap E_1 \\
&= ((X^\perp\cap E_1)+E_2)\cap E_1 \\
&= X^\perp \cap E_1
\end{align*}
because for a space in $E_1$, first adding $E_2$ and then intersecting with $E_1$ is giving the same space we start with. With this in mind, we can rewrite one of the rank functions:
\begin{align*}
r((M_1^*)';X) &= r(M_1^*;(X+E_2)\cap E_1) \\
&= r(M_1;((X+E_2)\cap E_1)^{\perp(E_1)}) +\dim((X+E_2)\cap E_1)-r(M_1;E_1) \\
&= r(M_1;X^\perp \cap E_1) +\dim E-\dim(X^\perp\cap E_1)-r(M_1;E_1).
\end{align*}
We have a similar result for $r((M_2^*)';X)$. Applying this yields
\begin{align*}
\lefteqn{r((M_1^*)';X)+r((M_2^*)';X)} \\
&= r(M_1;X^\perp \cap E_1) +\dim E -\dim(X^\perp\cap E_1)-r(M_1;E_1) \\
& \quad + r(M_2;X^\perp \cap E_2) +\dim E-\dim(X^\perp\cap E_2)-r(M_2;E_2) \\
&\geq r(M_1;X^\perp \cap E_1)+r(M_2;X^\perp \cap E_2)+\dim E-r(M_1\oplus M_2) \\
&= r(M_1;X^\perp \cap E_1)+r(M_2;X^\perp \cap E_2)+\dim B^\perp \\
&\geq \dim B^\perp \\
& \geq \dim(B^\perp\cap X)
\end{align*}
as was to be shown. So $B^\perp$ is independent in $M_1^*\oplus M_2^*$, hence a basis, and we have proven that $(M_1\oplus M_2)^*=M_1^*\oplus M_2^*$.
\end{proof}
In the last example we will answer the question started in Section \ref{ExDim4} about the direct sum of two copies of $U_{1,2}$. This direct sum is now uniquely defined.
\begin{Example}\label{2U12}
Let $M_1=M_2=U_{1,2}$. We will compute $M:=M_1 \oplus M_2$. This $q$-matroid is defined as $M=U_{1,2}'\vee U_{1,2}'$. \\
Let us coordinatize the ground space of $M_1$ as
$E_1=\langle (1,0,0,0), (0,1,0,0) \rangle$ and that of $M_2$ as $E_2=\langle (0,0,1,0), (0,0,0,1) \rangle$. Let $E=E_1\oplus E_2$. \\
We first compute $U_{1,2}'$. Since $n_1=n_2=2$, we need to add two loops to $U_{1,2}$ via Definition \ref{AggiungiLoop}. This gives a $q$-matroid with ground space $E$ and $r(A)=1$ for each $A\subseteq E$, unless $A\subseteq E_2$, then $r(A)=0$. \\
To determine $M=U_{1,2}'\vee U_{1,2}'$ we use Lemma \ref{l-RankOfSum} to get $r(M)=2$. By Proposition \ref{prop-noloops}, $M$ does not have any loops. So it suffices to decide for every $2$-dimensional space $A$ whether it is a basis or a circuit. First, note that
\[r(A)=\min_{X \subseteq A} \{r_1(X)+r_2(X)+\dim(A)-\dim(X)\}
=\min_{X \subseteq A} \{r_1(X)+r_2(X)+2-\dim(X)\}.\]
We distinguish between different types of $2$-spaces, depending on their intersection with $E_1$ and $E_2$.
\begin{itemize}
\item For $A=E_1=E_2$ we have $r(A)=1$ by Corollary \ref{MinorTheorem}.
\item Let $A\cap E_1 =A \cap E_2=\{0\}$, then
\begin{itemize}
\item if $\dim(X)=0$ then $r_1(X)+r_2(X)+2-\dim(X)=2$;
\item if $\dim(X)=1$ then $r_1(X)+r_2(X)+2-\dim(X)=3$;
\item if $\dim(X)=2$ then $r_1(X)+r_2(X)+2-\dim(X)=2$;
\end{itemize}
so we conclude that $r(A)=2$.
\item In the case $\dim(A\cap E_2)=1$ and $A\cap E_1 =\{0\}$ (or vice versa) we have
\begin{itemize}
\item if $\dim(X)=0$ then $r_1(X)+r_2(X)+2-\dim(X)=2$;
\item if $\dim(X)=1$ then $r_1(X)+r_2(X)+2-\dim(X)=3$ if $X$ is not contained in $E_2$, and $r_1(X)+r_2(X)+2-\dim(X)=1$ otherwise;
\item if $\dim(X)=2$ then $r_1(X)+r_2(X)+2-\dim(X)=2$;
\end{itemize}
so we conclude that $r(A)=2$.
\item Finally, if $\dim(A\cap E_2)=\dim(A\cap E_1)=1$ we have
that
\begin{itemize}
\item if $\dim(X)=0$ then $r_1(X)+r_2(X)+2-\dim(X)=2$;
\item if $\dim(X)=1$ then $r_1(X)+r_2(X)+2-\dim(X)=3$ if $X$ is not contained in $E_1$ nor in $E_2$, and $r_1(X)+r_2(X)+2-\dim(X)=2$ otherwise;
\item if $\dim(X)=2$ then $r_1(X)+r_2(X)+2-\dim(X)=2$;
\end{itemize}
so we conclude that $r(A)=2$.
\end{itemize}
We see that all $2$-spaces except $E_1$ and $E_2$ are basis. Since we have $E_1=E_2^\perp$, it follows that this $q$-matroid is self-dual. Because $U_{1,2}^*=U_{1,2}$, this example is in agreement with Theorem \ref{thm-DualDirect}.
\end{Example}
\section{Connectedness}\label{Connect}
In the classical case, every matroid is the direct sum of its connected components. It therefor makes sense to consider the notion of connectedness in the study of the direct sum of $q$-matroids. In this final section we collect some thoughts and examples concerning a possible $q$-analogue of connectedness. We will not be able to define the concept, but we hope to argue why it is not straightforward and give some possible paths for further investigation. \\
To define connectedness in classical matroids, we use the following relation on the elements of a matroid $M=(E,r)$.
\begin{quote}
Two elements $x,y\in E$ are related if either $x=y$ or if there is a circuit of $M$ that contains both $x$ and $y$.
\end{quote}
This relation is in fact an equivalence relation \cite[Theorem 3.46]{gordonmcnulty}. We call a matroid connected if it has only one equivalence class under this relation. If there are multiple equivalence classes $E_1,\ldots,E_k$ then we can write
\[ M=M|_{E_1}\oplus\cdots\oplus M|_{E_k}. \]
We will discuss some attempts to find a $q$-analogue of this equivalence relation. Note that we are looking for an equivalence relation on the $1$-dimensional spaces of $E$.
\subsection{First attempt}
The first obvious $q$-analogue for the relation is the following:
\begin{Definition}\label{def-failed1}
Two $1$-dimensional spaces $x,y\subseteq E$ are related if either $x=y$ or if there is a circuit of $M$ that contains both $x$ and $y$.
\end{Definition}
However, this is not an equivalence relation, because it is not transitive. Look at the matroid $P_1$ from the catalogue (Section \ref{P1}). The spaces $\langle(0,1,0)\rangle$ and $\langle(0,0,1)\rangle$ are in a circuit, and also $\langle (0,0,1)\rangle$ and $\langle(1,1,0)\rangle$ are in a circuit, but $\langle(0,1,0)\rangle$ and $\langle(1,1,0)\rangle$ are not in a circuit.
\subsection{Alternative attempt}
Assume we have a $q$-matroid $M=(E,r)$ with $\mathcal{H}$ its family of hyperplanes.
\begin{Definition}\label{relation_hyperplanes}
Let $x$ and $y$ be two $1$-dimensional spaces in $E$. We say $x$ and $y$ are related if $x=y$ or if there is a hyperplane $H\in\mathcal{H}$ such that $x,y\not\subseteq H$. We call this relation $R$.
\end{Definition}
\begin{Remark}
For classical matroids, consider the following relations:
\begin{itemize}
\item $x$ and $y$ are related if $x=y$ or if there is a circuit containing both $x$ and $y$.
\item $x$ and $y$ are related if $x=y$ or if there is a hyperplane containing neither $x$ nor $y$.
\end{itemize}
It is a well established result for classical matroids (see for example \cite[Theorem 3.36]{gordonmcnulty}) that the first relation is an equivalence relation. It is also a classical result \cite[Theorem 3.48]{gordonmcnulty} that both relations give the same equivalence classes. However, the $q$-analogues of these two relations are \emph{not} equivalent. Being in a circuit is equivalent to being in the orthogonal complement of a hyperplane, not being outside a hyperplane. So the relation defined in this subsection is not equivalent to the relation in the previous subsection. In fact, Definition \ref{relation_hyperplanes} is an equivalence relation, as the next theorem shows.
\end{Remark}
\begin{Theorem}
The relation $R$ from Definition \ref{relation_hyperplanes} is an equivalence relation.
\end{Theorem}
\begin{proof}
We follow the proof of \cite[Proposition 3.36]{gordonmcnulty}, replacing circuits with hyperplanes and reversing inclusion. $R$ is clearly reflective and symmetric. So we only have to prove it is transitive. We will frequently use the following hyperplane axiom \cite{BCR21}:
\begin{itemize}
\item[(H3')] If $H_1,H_2\in\mathcal{H}$ with $y\not\subseteq H_1,H_2$ and $x\subseteq H_2$, $x\not\subseteq H_1$, then there is an $H_3\in\mathcal{H}$ such that $(H_1\cap H_2)+y\subseteq H_3$ and $x\not\subseteq H_3$.
\end{itemize}
Let $x,y,z$ be $1$-dimensional spaces in $E$. Let $x,y\not\subseteq H_1$ and $y,z\not\subseteq H_2$. We have to show there exists a hyperplane $H'$ not containing $x$ and $z$. If $x\not\subseteq H_2$ or $z\not\subseteq H_1$, we are done, so suppose $x\subseteq H_2$ and $z\subseteq H_1$. We will use induction on $\dim H_1-\dim(H_1\cap H_2)$. \\
Suppose $\dim H_1-\dim(H_1\cap H_2)=1$, then we can write $H_1$ as $(H_1\cap H_2)+z$. Applying (H3') yields an $H'\in\mathcal{H}$ such that $(H_1\cap H_2)+y\subseteq H'$ and $x\not\subseteq H'$. We need to have that $z\not\subseteq H'$, because otherwise $H_1\subsetneq H'$ and this violates axiom (H2). So $H'$ is a hyperplane not containing $x$ and $z$, as requested. \\
Now suppose $\dim H_1-\dim(H_1\cap H_2)=n>1$ and assume that $H'$ exists for all pairs of hyperplanes such that $\dim H_1-\dim(H_1\cap H_2)<n$. We will use (H3') twice to find a hyperplane $H_4\in\mathcal{H}$ such that $\dim H_1-\dim(H_1\cap H_4)<\dim H_1-\dim(H_1\cap H_2)$ and such that $x\subseteq H_4$, $x\not\subseteq H_1$ and $z\subseteq H_1$, $z\not\subseteq H_4$. Then we can apply the induction hypothesis to $H_1$ and $H_4$.
\[ \includegraphics[width=.8\textwidth]{hyperplanes.pdf} \]
First we apply (H3') to $H_1$ and $H_2$. This gives $H_3\in\mathcal{H}$ such that $(H_1\cap H_2)+y\subseteq H_3$ and $x\not\subseteq H_3$. If $z\not\subseteq H_3$ we are done, so let $z\subseteq H_3$. However, there is a $1$-dimensional space $z^*\subseteq H_1$, $z^*\not\subseteq H_2$ such that $z^*\not\subseteq H_3$: if not, $H_1\subsetneq H_3$ and this violates axiom (H2). \\
Now we apply (H3') again, to $H_2$ and $H_3$ with $z^*\not\subseteq H_2,H_3$ and $z\subseteq H_3$, $z\not\subseteq H_2$. This gives $H_4\in\mathcal{H}$ such that $(H_2\cap H_3)+z^*\subseteq H_4$ and $z\not\subseteq H_4$. If $x\not\subseteq H_4$ we are done, so let $x\subseteq H_4$. \\
By construction (see picture) we have that $(H_1\cap H_2)\subseteq(H_1\cap H_4)$. This inclusion is strict, because $z^*\subseteq H_1,H_4$ but $z^*\not\subseteq H_2$. This means we have $\dim H_1-\dim(H_1\cap H_4)<\dim H_1-\dim(H_1\cap H_4)$. By the induction hypothesis, we can now find an $H'\in\mathcal{H}$ such that $x,z\not\subseteq H'$. \\
This proves that the relation $R$ is transitive, and hence an equivalence relation.
\end{proof}
The good news is that we have found a relation that is in fact an equivalence relation. The bad news is that it does not work like we want to. The uniform $q$-matroids $U_{0,3}$ and $U_{3,3}$ only have one equivalence class, where we would want that $U_{0,3}$ is the sum of three copies of $U_{0,1}$ and $U_{3,3}$ is the sum of three copies of $U_{1,1}$. Also the $q$-matroid $P_1^*$ (Section \ref{P1star}) in the catalog has only one equivalence class, where we constructed it in Example \ref{ex-PrimePlusRed} as the direct sum $U_{1,1}\oplus U_{1,2}$. $P_1$ on the other hand (the dual of $P_1^*$) has more than one equivalence class: a signal that this attempt for an equivalence relation does not play nice with duality.
\subsection{Towards a well-defined definition}
As we saw, Definition \ref{def-failed1} is in general not an equivalence relation. However, in some $q$-matroids it is an equivalence relation. From our examples, we think the following statements could be true.
\begin{Conjecture}
The relation of Definition \ref{def-failed1} is an equivalence relation in at least one of $M$ and $M^*$.
\end{Conjecture}
\begin{Conjecture}
Let $M$ be a $q$-matroid with circuits $\mathcal{C}$ and cocircuits $\mathcal{C}^*$. Suppose $\dim(C\cap C^*)\neq1$ for all $C\in\mathcal{C}$ and $C^*\in\mathcal{C}^*$. Then Definition \ref{def-failed1} is an equivalence relation.
\end{Conjecture}
Both conjectures are of course true in the classical case. To see this for the last conjecture, note that it can be proven that the intersection between a circuit and a cocircuit can never be a single element. See for example \cite[Proposition 2.1.11]{oxley}. The $q$-analogue of this statement is not true in general: see for example the $q$-matroid $P_1^*$ of Section \ref{P1star}. It has one circuit, $\langle(0,1,0),(0,0,1)\rangle$, that intersects in dimension $1$ with the cocircuit $\langle(1,1,0),(0,0,1)\rangle$. \\
We welcome any further hints towards a better understanding of the $q$-analogues of the direct sum, connectedness, and their relation.
\section*{Acknowledgement}
We would like to express our gratitude to Heide Gluesing-Luerssen and Benjamin Jany for carefully reading the paper. Their questions and remarks have lead to significant corrections and clarifications in the manuscript.
\bibliographystyle{abbrv}
|
2,869,038,155,629 | arxiv | \section{Introduction}
Monogamy relations of quantum entanglement are important feature of quantum physics that play an important role in quantum information and quantum communication. Monogamy relations confine the entanglement of a quantum system with the other (sub)systems, thus they are closely related to quantum information processing tasks such as security analysis of quantum key distribution \cite{MP}.
The monogamy relation for a three-qubit state $\rho_{A_1A_2A_3}$ is defined \cite{MK} as
$$\mathcal{E}(\rho_{A_1|A_2A_3})\geq \mathcal{E}(\rho_{A_1A_2}) +\mathcal{E}(\rho_{A_1A_3}),$$
where $\mathcal{E}$ is a bipartite entanglement measure, $\rho_{A_1A_2}$ and $\rho_{A_1A_3}$ are the reduced density matrices of $\rho_{A_1A_2A_3}$. The monogamy relation was generalized to multiqubit quantum systems, high-dimensional quantum systems in
general settings \cite{ZXN,JZX,jll,012329,gy1,gy2,jin1,jin2, SC, RWF, ZYS}.
The first polygamy relation of entanglement was established in \cite{gg} for some three-qubit system as the inequality ${E_a}_{A_1|A_2A_3}\leq {E_a}_{A_1A_2} +{E_a}_{A_1A_3}$ where ${E_a}_{A_1|A_2A_3}$ is the assisted entanglement \cite{gg} between $A_1$ and $A_2A_3$ and later generalized to some multiqubit systems in \cite{jsb,jin3}. General polygamy inequalities of multipartite entanglement were also given in \cite{062328,295303,jsb,042332} in terms of entanglement of assistance. While it is
known that the monogamy relation does not hold for all quantum systems, it is also shown \cite{GG} that
any monotonic bipartite measure is monogamous on pure tripartite states.
It turns out that a generalized monogamy relation always holds for any quantum system. In \cite{JZX, jll,ZXN}, multiqubit monogamy relations have been demonstrated for the $x$th power of the entanglement of
formation ($x\geq\sqrt{2}$) and the concurrence ($x\geq2$), which opened new direction to study the monogamy relation. Similar general polygamy relation
has been shown for R\'ennyi-$\alpha$ entanglement \cite{GYG}. Monogamy relations for quantum steering have also been shown in \cite{hqy,mko,jk1,jk2,jk3}.
Moreover, polygamy inequalities were given in terms of the
$\alpha$th ($0\leq\alpha\leq 1$) power of square of convex-roof extended negativity (SCREN) and the entanglement of assistance \cite{j012334, 042332}.
\iffalse
Whereas the monogamy of entanglement shows the restricted sharability of multipartite quantum entanglement, the distribution of entanglement in multipartite quantum systems was shown to have a dually monogamous property.
Based on concurrence of assistance \cite{qic}, the polygamy of entanglement provides a lower bound for the distribution of bipartite entanglement in a multipartite system \cite{jmp}.
Monogamy and polygamy of entanglement can restrict the possible correlations between the authorized users and the eavesdroppers, thus tightening the security bounds in quantum cryptography \cite{MP}.
\fi
\iffalse
Whereas the monogamy of entanglement shows the restricted sharability of multipartite quantum entanglement, the distribution of entanglement in multipartite quantum systems was shown to have a dually monogamous property.
Based on concurrence of assistance \cite{qic}, the polygamy of entanglement provides a lower bound for the distribution of bipartite entanglement in a multipartite system \cite{jmp}.
Monogamy and polygamy of entanglement can restrict possible correlations between the authorized users and the eavesdroppers, thus tightening the security bounds in quantum cryptography \cite{MP}.
\fi
An important feature for this generalized monogamy relation for $\alpha$th power of the measure is its transitivity in the sense that other power of the measure satisfies a weighted monogamous relation. Recently, the authors in \cite{JFQ} provided a class of monogamy and polygamy relations of the $\alpha$th $(0\leq\alpha\leq r,r\geq2)$ and the $\beta$th $(\beta\geq s,0\leq s\leq1)$ powers for any quantum correlation. Applying the monogamy relations in \cite{JFQ} to quantum correlations like squared convex-roof extended negativity, entanglement of formation and concurrence one can get tighter monogamy inequalities than those given in \cite{zhu}.
Similarly applying the bounds in \cite {JFQ}
to specific quantum correlations such as the concurrence of assistance, square of convex-roof extended negativity of assistance (SCRENoA), entanglement of assistance,
corresponding polygamy relations were obtained, which are complementary to the existing ones \cite{jin3,062328,295303,jsb,042332} with different regions of parameter $\beta$. In \cite{ZJZ1,ZJZ2}, the authors gave another set of monogamy relations for $(0\leq\alpha\leq \frac{r}{2},r\geq2)$ and polygamy relations for $(\beta\geq s,0\leq s\leq1)$, and we note that the bound is stronger than \cite{JFQ} in monogamy case but weaker in polygamy case.
One realizes that the monogamy and polygamy relations given in \cite{JFQ, ZJZ1,ZJZ2} were obtained by bounding the function $(1+t)^x$ by various estimates. In this paper, we revisit the function $(1+t)^x$ and give a unified and improved method to estimate
its upper and lower bounds, which then lead to new improved monogamy and polygamy relations stronger than some of the recent strong ones in both situations.
For instance, we will rigorously show the monogamy and polygamy relations of quantum correlations for the cases $(0\leq\alpha\leq r,r\geq2)$ and $(\beta\geq s,0\leq s\leq1)$ are tighter than those given in \cite{JFQ, ZJZ1,ZJZ2} all together. We also use the concurrence and SCRENoA as examples to demonstrate how our bounds have improved
previously available strong bounds.
\section{Monogamy relations of quantum correlations}
Let $\rho$ be a density matrix on a multipartite quantum system $\bigotimes_{i=1}^nA_i$, and let
$\mathcal{Q}$ be a measure of quantum correlation for any bipartite (sub)system. If $\mathcal{Q}$ satisfies \cite{ARA} the inequality
\begin{eqnarray}\label{q}
&&\mathcal{Q}(\rho_{A_1|A_2,\cdots,A_{n}})\nonumber\\
&&\geq\mathcal{Q}(\rho_{A_1A_2})+\mathcal{Q}(\rho_{A_1A_3})+\cdots+\mathcal{Q}(\rho_{A_1A_{n}}),
\end{eqnarray}
$\mathcal Q$ is said to be {\it monogamous}, where $\rho_{A_1A_i}$, $i=2,...,n$, are the reduced density matrices of $\rho$. For simplicity, we denote $\mathcal{Q}(\rho_{A_1A_i})$ by $\mathcal{Q}_{A_1A_i}$, and $\mathcal{Q}(\rho_{A_1|A_2,\cdots,A_{n}})$ by $\mathcal{Q}_{A_1|A_2,\cdots,A_{n}}$. It is known that some quantum measures obey the monogamous relation \cite{AKE,SSS} for certain quantum states, while
there are quantum measures which do not satisfy the monogamy relation \cite{GLGP, RPAK}.
In Ref. \cite{SPAU,TJ,YKM}, the authors have proved that there exists $r\in \mathbb R~(r\geq2)$ such that the $r$th power of any measure $\mathcal{Q}$ satisfies the following generalized monogamy relation for arbitrary dimensional tripartite state \cite{SPAU}:
\begin{eqnarray}\label{aqq}
\mathcal{Q}^r_{A_1|A_2A_3}\geq\mathcal{Q}^r_{A_1A_2}+\mathcal{Q}^r_{A_1A_3}.
\end{eqnarray}
Assuming \eqref{aqq}, we would like to
prove that other power of $\mathcal Q$ also obeys a weighted monogamy relation. First of all, using the inequality $(1+t)^{x} \geq 1+t^{x}$ for $x \geq 1,0 \leq t \leq 1$ one can easily
derive the following generalized polygamy relation for the $n$-partite case,
$$
\mathcal{Q}_{A_1 \mid A_{2}, \ldots, A_{n}}^{r} \geq \sum_{i=2}^{n} \mathcal{Q}_{A_1 A_{i}}^{r}
$$
\iffalse
The concurrence of a bipartite pure state $|\psi\rangle_{A B}$ is given by \cite{U,RBCHJ,AF},
$$
C\left(|\psi\rangle_{A B}\right)=\sqrt{2\left[1-\operatorname{Tr}\left(\rho_{A}^{2}\right)\right]},
$$
where $\rho_{A}$ is reduced density matrix $\rho_{A}=\operatorname{Tr}_{B}\left(|\psi\rangle_{A B}\langle\psi|\right)$. The concurrence of mixed states $\rho=\sum_{i} p_{i}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|, p_{i} \geq 0, \sum_{i} p_{i}=1$, is given by the convex roof construction,
$$
C\left(\rho_{A B}\right)=\min _{\left\{p_{i},\left|\psi_{i}\right\rangle\right\}} \sum_{i} p_{i} C\left(\left|\psi_{i}\right\rangle\right)
$$
For $n$-qubit quantum states, the concurrence satisfies \cite{ZF}
$$
C_{A_1 \mid A_{2}\ldots A_{n}}^{\alpha} \geq C_{A_1 A_{2}}^{\alpha}+\ldots+C_{A_1 A_{n}}^{\alpha}
$$
for $\alpha \geq 2$, where $C_{A_1 \mid A_{2} \ldots A_{n}}$ is the concurrence of $\rho$ under bipartite partition $A_1 \mid A_{2} \ldots A_{n}$, and $C_{A_1 A_{i}}, i=$ $2 \ldots, n$, is the concurrence of the reduced states $\rho_{A_1 A_{i}}$.
\fi
We now try to generalize the monogamy relation to other powers. Let's start with a useful lemma.
\begin{lemma}\label{lem:1} Let $a\geq 1$ be a real number. Then for $t\geq a\geq 1$, the function $(1+t)^x$ satisfies the following inequality
\begin{equation}
(1+t)^x\geq (1+a)^{x-1}+(1+\frac{1}{a})^{x-1}t^x,
\end{equation}
where $0<x\leq 1$.
\end{lemma}
\begin{proof}.
Consider $g(x,y)=(1+y)^x-(1+{a})^{x-1}y^x$, $0<y\leq \frac{1}{a}$, $0\leq x\leq 1$. Then
$$\frac{\partial g}{\partial y}=xy^{x-1}\left((1+\frac{1}{y})^{x-1}-(1+a)^{x-1}\right).$$
Let $h(x,y)=(1+\frac{1}{y})^{x-1}$, $0<y\leq \frac{1}{a}$, $0\leq x\leq 1$. Since $h(x,y)$ is an increasing function of $y$, we have $h(x,y)\leq h(x,\frac{1}{a})=(1+{a})^{x-1}$.
Thus $\frac{\partial g}{\partial y}\leq 0$ and $g(x,y)$ is decreasing with respect to $y$. Therefore we have $g(x,y)\geq g(x,\frac{1}{a})=(1+\frac{1}{a})^{x-1}$. Subsequently
\begin{equation*}
g(x,\frac{1}{t})=\frac{(1+t)^x}{t^x}-\frac{(1+{a})^{x-1}}{t^x}\geq (1+\frac{1}{a})^{x-1}
\end{equation*}
for $t\geq a$. Then we have
\begin{equation*}
(1+t)^x\geq(1+{a})^{x-1}+(1+\frac{1}{a})^{x-1}t^x,
\end{equation*}
for $t\geq a$.
\iffalse
Similarly, consider ${g}(x,y)=(1+y)^x-2^{x-1}y^x$, $y\geq 1$, $0\leq x\leq 1$.
Then
$$\frac{\partial g}{\partial y}=(1+y)^x\frac{x}{1+y}-2^{x-1}y^x\frac{x}{y}=xy^{x-1}\left((1+\frac{1}{y})^{x-1}-2^{x-1}\right).$$
Let $h(x,y)=(1+\frac{1}{y})^{x-1}$, $y\geq 1$, $0\leq x\leq 1$. Since $h(x,y)$ is an increasing function of $y$, we have $h(x,y)\geq h(x,1)=2^{x-1}$.
Thus $\frac{\partial g}{\partial y}\geq 0$ and $g(x,y)$ is increasing with respect to $y$. Therefore we have $g(x,y)\geq g(x,1)=2^{x-1}$.
Thus we can easily get that
\begin{equation*}
g(x,\frac{1}{t})=\frac{(1+t)^x}{t^x}-\frac{2^{x-1}}{t^x}\geq 2^{x-1}
\end{equation*}
for $0<t\leq 1$. Then we have
\begin{equation*}
(1+t)^x\geq (1+\frac{1}{a})^{x-1}+\frac{(1+a)^x-(1+\frac{1}{a})^{x-1}}{a^x}t^x,
\end{equation*}
for $0<t\leq 1$.
\fi
\end{proof}
\begin{remark}\label{rem:1}~~
The proof of Lemma \ref{lem:1} implies that for $0\leq x\leq 1$, $t\geq a\geq 1$ and $f(x)\geq (1+{a})^{x-1} $, we have
\begin{equation*}
(1+t)^x\geq f(x)+\frac{(1+a)^x-f(x)}{a^x}t^x.
\end{equation*}
It is not hard to see that for $0\leq x\leq 1$, $t\geq a\geq 1$ and for $f(x)\geq (1+{a})^{x-1}$,
$$(1+{a})^{x-1}+(1+\frac{1}{a})^{x-1}t^x-
\left[f(x)+\frac{(1+a)^x-f(x)}{a^x}t^x\right]\geq 0.$$
In fact,
\begin{equation*}
\begin{aligned}
{ LHS}
&=(1+{a})^{x-1}-f(x)+\Big(\frac{a(1+a)^{x-1}}{a^x}t^{x}-\frac{(1+a)(1+a)^{x-1}-f(x)}{a^x}t^{x}\Big)\\
&=(1+{a})^{x-1}-f(x)+\frac{f(x)-(1+{a})^{x-1}}{a^x}t^{x}\\
&=(\frac{t^x}{a^x}-1)(f(x)-(1+{a})^{x-1})\geq 0.
\end{aligned}
\end{equation*}
\end{remark}
\begin{remark} \label{rem:2}
In \cite[Lemma 1]{JFQ}, the authors have given a lower bound of $(1+t)^x$ for $0\leq x\leq 1$ and $t\geq a\geq 1$:
\begin{equation*}
(1+t)^{x} \geq 1+\frac{(1+a)^{x}-1}{a^{x}} t^{x}.
\end{equation*}
This is a special case of our result in Remark \ref{rem:1}. In fact, let $f(x)=1\geq (1+{a})^{x-1}$ for $0\leq x\leq 1$ in Remark \ref{rem:1}, then the inequality
descends to theirs. Therefore our lower bound of $(1+t)^x$ is better than that of \cite{JFQ}, consequently
any monogamy relations based on Lemma \ref{lem:1} are better than those given in \cite{JFQ} based on Lemma 1 of \cite{JFQ}.
\end{remark}
\begin{remark}\label{rem:ZJZ1}
In \cite[Lemma 1]{ZJZ1}, the authors gave another lower bound of $(1+t)^x$ for $0\leq x\leq \frac{1}{2}$ and $t\geq a\geq 1$
\begin{equation*}
(1+t)^{x} \geq p^{x}+\frac{(1+a)^{x}-p^{x}}{a^{x}} t^{x},
\end{equation*}
where $\frac{1}{2}\leq p\leq 1$.
This is a also special case of our Remark \ref{rem:1} where $f(x)=p^{x}$ for $0\leq x\leq \frac{1}{2}$ and $t\geq a\geq 1$.
Since $(1+a)^{x-1}\leq p^x$ for $0\leq x\leq \frac{1}{2}$ and $\frac{1}{2}\leq p\leq 1$,
therefore our lower bound of $(1+t)^x$ for $0\leq x\leq \frac{1}{2}$ and $t\geq a\geq 1$ is stronger than that given in \cite{ZJZ1}.
Naturally our monogamy relations based on Lemma \ref{lem:1} will outperform those given in \cite{ZJZ1} based on Lemma 1 of \cite{ZJZ1}.
\end{remark}
\begin{remark}\label{rem:ZJZ2}
In \cite[Lemma 1]{ZJZ2}, the following lower bound was given: $(1+t)^x$ for $0\leq x\leq \frac{1}{2}$ and $t\geq a\geq 1$
\begin{equation*}
(1+t)^{x} \geq (\frac{1}{2})^{x}+\frac{(1+a)^{x}-(\frac{1}{2})^{x}}{a^{x}} t^{x},
\end{equation*}
which is the special case $p=\frac{1}{2}$ of \cite{ZJZ1}.
therefore our lower bound of $(1+t)^x$ for $0\leq x\leq \frac{1}{2}$ and $t\geq a\geq 1$ is better than the one given in \cite{ZJZ2}.
Thus, our monodamy relations based on Lemma \ref{lem:1} are better than the ones given in \cite{ZJZ2} based on \cite[Lemma 1]{ZJZ2}.
\end{remark}
\begin{lemma}\label{lem:2}
Let $p_i$ $(i=1,\cdots, n)$ be nonnegative numbers arranged as $p_{(1)}\geq p_{(2)}\geq ...\geq p_{(n)}$
for a permutation $(1)(2)\cdots (n)$ of $12\cdots n$.
If $p_{(i)}\geq a p_{(i+1)}$ for $i=1,...,n-1$,
we have
\begin{equation}\label{eq:3}
\left(\sum_{i=1}^n p_i\right)^x\geq (1+{a})^{x-1}\sum_{i=1}^n \left((1+\frac{1}{a})^{x-1}\right)^{n-i}p_{(i)}^x,
\end{equation}
for $0\leq x\leq 1$.
\iffalse
and
\begin{equation}\label{eq:4}
\left(\sum_{i=1}^n p_i\right)^x\leq \sum_{j=2}^{n}2^{(n-j+1)(x-1)}p_{(j)}^x+2^{(n-1)(x-1)}p_{(1)}^x,
\end{equation}
for $x\geq 1$,
\fi
\end{lemma}
\begin{proof}
For $0\leq x\leq 1$, if $p_{(n)}>0$ using Lemma \ref{lem:1} we have
\begin{equation*}
\begin{aligned}
\left(\sum_{i=1}^n p_i\right)^x&=p_{(n)}^{x}\left(1+\frac{p_{(1)}+...+p_{(n-1)}}{p_{(n)}}\right)^x\\
&\geq (1+{a})^{x-1}p_{(n)}^x+(1+\frac{1}{a})^{x-1}(p_{(1)}+...+p_{(n-1)})^x\\
&\geq...\\
&\geq (1+{a})^{x-1}\sum_{i=1}^n \left((1+\frac{1}{a})^{x-1}\right)^{n-i}p_{(i)}^x.
\end{aligned}
\end{equation*}
The other cases can be easily checked by Lemma \ref{lem:1}.
\end{proof}
\begin{theorem}\label{thm:1}
For any tripartite mixed state $\rho_{A_1A_2A_3}$, let $\mathcal{Q}$ be a (bipartite) quantum measure satisfying the generalized monogamy relation \eqref{aqq} for $r\geq 2$
and $\mathcal{Q}_{A_1A_3}^{r}\geq a\mathcal{Q}_{A_1A_2}^{r}$ for some $a$, then
we have
\begin{equation}
\mathcal{Q}^{\alpha}_{A_1|A_2A_3}\geq (1+{a})^{\frac{\alpha}{r}-1}\mathcal{Q}_{A_1A_2}^{\alpha}+(1+\frac{1}{a})^{\frac{\alpha}{r}-1}\mathcal{Q}_{A_1A_3}^{\alpha}.
\end{equation}
for $0\leq \alpha\leq r$.
\end{theorem}
\begin{proof} It follows from Lemma \ref{lem:1} that
\begin{equation}
\begin{aligned}
\mathcal{Q}_{A_1|A_2A_3}^{\alpha}&=\left(\mathcal{Q}^{r}_{A_1|A_2A_3}\right)^{\frac{\alpha}{r}}\geq \left(\mathcal{Q}_{A_1A_2}^{r}+\mathcal Q_{A_1A_3}^{r}\right)^{\frac{\alpha}{r}}\\
&=\mathcal{Q}_{A_1A_2}^{\alpha}\left(1+\frac{\mathcal{Q}_{A_1A_3}^{r}}{\mathcal{Q}_{A_1A_2}^{r}}\right)^{\frac{\alpha}{r}}\\
&\geq(1+{a})^{\frac{\alpha}{r}-1}\mathcal{Q}_{A_1A_2}^{\alpha}+(1+\frac{1}{a})^{\frac{\alpha}{r}-1}\mathcal{Q}_{A_1A_3}^{\alpha}.
\end{aligned}
\end{equation}
\end{proof}
\begin{theorem}\label{thm:2}
For any n-partite quantum state $\rho_{A_1A_2...A_n}$, let $\mathcal{Q}$ be a (bipartite) quantum measure satisfying the generalized monogamy relation \eqref{aqq} for $r\geq 2$.
Arrange $\mathcal{Q}_{(1)}\geq \mathcal{Q}_{(2)}\geq...\geq \mathcal{Q}_{(n-1)}$ with $\mathcal{Q}_{(j)}\in\{\mathcal{Q}_{A_1A_i}|i=2,...,n\}, j=1,...,n-1$. If for some $a$, $\mathcal{Q}^r_{(i)}\geq a \mathcal{Q}^r_{(i+1)}$ for $i=1,...,n-2$, then we have
\begin{equation}
\mathcal{Q}^{\alpha}_{A_1|A_2...A_n}\geq(1+{a})^{\frac{\alpha}{r}-1}\sum_{i=1}^{n-1} \left((1+\frac{1}{a})^{\frac{\alpha}{r}-1}\right)^{n-1-i}\mathcal{Q}_{(i)}^{{\alpha}}
\end{equation}
for $0\leq \alpha\leq r$.
\end{theorem}
\begin{proof}
By Lemma \ref{lem:2}, we have
\begin{equation*}
\begin{aligned}
\mathcal{Q}^{\alpha}_{A_1|A_2...A_n}&=\left(\mathcal{Q}^{r}_{A_1|A_2...A_n}\right)^{\frac{\alpha}{r}}
\geq \left(\mathcal{Q}_{A_1A_2}^{r}+...+\mathcal{Q}_{A_1A_n}^{r}\right)^{\frac{\alpha}{r}}\\
&\geq(1+{a})^{\frac{\alpha}{r}-1}\sum_{i=1}^{n-1} \left((1+\frac{1}{a})^{\frac{\alpha}{r}-1}\right)^{n-1-i}\mathcal{Q}_{(i)}^{{\alpha}},
\end{aligned}
\end{equation*}
\end{proof}
The general monogamy relations work for any quantum correlation measure such as concurrence, negativity, entanglement of formation etc.
Thus our theorems produce
tighter weighted monogamy relations than the existing ones (cf. \cite{JFQ,zhu,ZJZ1,ZJZ2}). Moreover, the new weighted monogamy relations also can be used for
the Tsallis-$q$ entanglement and R\'enyi-$q$ entanglement measures, and they also
outperform some of the recently found monogamy relations in \cite{jll,jin3,slh}.
In the following, we use the concurrence as an example to show the advantage of our monogamy relations.
For a bipartite pure state $\rho=|\psi\rangle_{AB}\in{H}_A\otimes {H}_B$, the concurrence is defined \cite{AU,PR,SA} by $C(|\psi\rangle_{AB})=\sqrt{{2\left[1-\mathrm{Tr}(\rho_A^2)\right]}}$,
where $\rho_A$ is the reduced density matrix. For a mixed state $\rho_{AB}$ the concurrence is given by the convex roof extension
$C(\rho_{AB})=\min_{\{p_i,|\psi_i\rangle\}}\sum_ip_iC(|\psi_i\rangle)$,
where the minimum is taken over all possible decompositions of $\rho_{AB}=\sum\limits_{i}p_i|\psi_i\rangle\langle\psi_i|$, with $p_i\geq0$, $\sum\limits_{i}p_i=1$ and $|\psi_i\rangle\in {H}_A\otimes {H}_B$.
\iffalse
For an $n$-qubit state $\rho_{A_1A_2\cdots A_{n}}\in {H}_A\otimes {H}_{A_2}\otimes\cdots\otimes {H}_{A_{n}}$, if $C(\rho_{A_1B_i})\geq C(\rho_{A|B_{i+1}\cdots B_{N-1}})$ for $i=1, 2, \cdots, N-2$, $N\geq 4$, the concurrence satisfies \cite{jll},
\begin{eqnarray}\label{mo2}
&&C^\alpha(\rho_{A|B_1B_2\cdots B_{N-1}})\geq \sum_{j=1}^{N-1}(2^\frac{\alpha}{2}-1)^{j-1}C^\alpha(\rho_{AB_j}),
\end{eqnarray}
for $\alpha\geq2$.
For any $2\otimes2\otimes2^{N-2}$ tripartite mixed state $\rho_{ABC}$, if $C(\rho_{AC})\geq C(\rho_{AB})$, the concurrence satisfies \cite{zhu}
\begin{eqnarray}\label{mo3}
C^\alpha(\rho_{A|BC})\geq C^\alpha(\rho_{AB})+(2^\frac{\alpha}{\gamma}-1)C^\alpha(\rho_{AC})
\end{eqnarray}
for $0\leq\alpha\leq \gamma$ and $\gamma\geq 2$.
\fi
For convenience, we write $C_{A_1A_i}=C(\rho_{A_1A_i})$ and $C_{A_1|A_2,\cdots,A_{n}}=C(\rho_{A_1|A_2\cdots A_{n}})$. The following conclusions are easily seen by
the similar method as in the proof of Theorem 1.
\begin{corollary} Let $C$ be the concurrence satisfying the generalized monogamy relation \eqref{aqq} for $r\geq 2$.
For any 3-qubit mixed state $\rho_{A_1A_2A_3}$, if $C_{A_1A_3}^{r}\geq aC_{A_1A_2}^{r}$ for some $a$, then
we have
\begin{equation}
C^{\alpha}_{A_1|A_2A_3}\geq (1+{a})^{\frac{\alpha}{r}-1}C_{A_1A_2}^{\alpha}+(1+\frac{1}{a})^{\frac{\alpha}{r}-1}C_{A_1A_3}^{\alpha}.
\end{equation}
for $0\leq \alpha\leq r$.
\end{corollary}
\begin{corollary} Let $C$ be the concurrence satisfying the generalized monogamy relation \eqref{aqq} for $r\geq 2$.
For any n-qubit quantum state $\rho_{A_1A_2...A_n}$, arrange
$C_{(1)}\geq C_{(2)}\geq...\geq C_{(n-1)}$ with $C_{(j)}\in\{C_{A_1A_i}|i=2,...,n\}, j=1,...,n-1$. If for some $a$, $C^r_{(i)}\geq a C^r_{(i+1)}$ for $i=1,...,n-2$, then we have
\begin{equation}
C^{\alpha}_{A_1|A_2...A_n}\geq(1+{a})^{\frac{\alpha}{r}-1}\sum_{i=1}^{n-1} \left((1+\frac{1}{a})^{\frac{\alpha}{r}-1}\right)^{n-1-i}C_{(i)}^{{\alpha}}
\end{equation}
for $0\leq \alpha\leq r$.
\end{corollary}
\begin{example}
Consider the following three-qubit state $|\psi\rangle$ with generalized Schmidt decomposition \cite{AA,XH},
$$
|\psi\rangle=\lambda_{0}|000\rangle+\lambda_{1} e^{i \varphi}|100\rangle+\lambda_{2}|101\rangle+\lambda_{3}|110\rangle+\lambda_{4}|111\rangle,
$$
where $\lambda_{i} \geq 0$ and $\sum_{i=0}^{4} \lambda_{i}^{2}=1$. Then $C_{A_1 \mid A_2 A_3}=2 \lambda_{0} \sqrt{\lambda_{2}^{2}+\lambda_{3}^{2}+\lambda_{4}^{2}}, C_{A_1 A_2}=$ $2 \lambda_{0} \lambda_{2}$, and $C_{A_1 A_3}=2 \lambda_{0} \lambda_{3}$.
Set $\lambda_{0}=\lambda_3=\frac{1}{2}, \lambda_{1}=\lambda_{2}=\lambda_{4}=\frac{\sqrt{6}}{6}$.
We have
$C_{A_1 \mid A_2A_3}=\frac{\sqrt{21}}{6}, C_{A_1 A_2}=\frac{\sqrt{6}}{6}, C_{A_1A_3}=\frac{1}{2}$. Set $a=\sqrt{6}/2$, then
the lower bound of $C_{A_1 \mid A_2A_3}^{\alpha}$ given in \cite{JFQ} is
\begin{equation*}
C_{A_1A_2}^{\alpha}+\frac{(1+a)^{\frac{\alpha}{r}}-1}{a^{\frac{\alpha}{r}}} C_{A_1 A_3}^{\alpha}=\left(\frac{\sqrt{6}}{6}\right)^{\alpha}+\frac{(1+\frac{\sqrt{6}}{2})^{\frac{\alpha}{r}}-1}{(\frac{\sqrt{6}}{2})^{\frac{\alpha}{r}}}\left(\frac{1}{2}\right)^{\alpha}=Z_1,
\end{equation*}
the lower bound of $C_{A_1 \mid A_2A_3}^{\alpha}$ given in \cite{ZJZ1,ZJZ2} is
\begin{equation*}
C_{A_1A_2}^{\alpha}p^{\frac{\alpha}{r}}+\frac{(1+a)^{\frac{\alpha}{r}}-p^{\frac{\alpha}{r}}}{a^{\frac{\alpha}{r}}} C_{A_1 A_3}^{\alpha}=\left(\frac{\sqrt{6}}{6}\right)^{\alpha}(\frac{1}{2})^{\frac{\alpha}{r}}+\frac{(1+\frac{\sqrt{6}}{2})^{\frac{\alpha}{r}}-(\frac{1}{2})^{\frac{\alpha}{r}}}{(\frac{\sqrt{6}}{2})^{\frac{\alpha}{r}}}\left(\frac{1}{2}\right)^{\alpha}=Z_2,
\end{equation*}
with $p=\frac{1}{2}$ and $\alpha\leq \frac{r}{2}$.
While our bound is
\begin{equation*}
(1+{a})^{\frac{\alpha}{r}-1}C_{A_1A_2}^{\alpha}+(1+\frac{1}{a})^{\frac{\alpha}{r}-1}C_{A_1A_3}^{\alpha}=(1+\frac{\sqrt{6}}{2})^{\frac{\alpha}{r}-1}\left(\frac{\sqrt{6}}{6}\right)^{\alpha}+(1+\frac{2}{\sqrt{6}})^{\frac{\alpha}{r}-1}\left(\frac{1}{2}\right)^{\alpha}=Z_3
\end{equation*}
Fig. 1 charts the graphs of the three bounds and the figure clearly shows that our result is better than those in \cite{JFQ, ZJZ1, ZJZ2} for $0 \leq \alpha \leq 1$ and $r \geq 2$.
\begin{figure}[!htb]
\centerline{\includegraphics[width=0.6\textwidth]{fig1.eps}}
\renewcommand{\figurename}{Fig.}
\caption{The $z$-axis shows the concurrence as a function of $\alpha, r$. The blue, green and red surfaces represent our lower bound, the lower bound from \cite{ZJZ1,ZJZ2} and the lower bound from \cite{JFQ} respectively.}
\end{figure}
\end{example}
\section{Polygamy relations for general quantum correlations}
In \cite{jinzx}, the authors proved that for arbitrary dimensional tripartite states, there exists $0 \leq s \leq 1$ such that any quantum correlation measure $\mathcal{Q}$ satisfies the following polygamy relation:
\begin{equation}\label{poly}
\mathcal{Q}_{A \mid B C}^{s} \leq \mathcal{Q}_{A B}^{s}+\mathcal{Q}_{A C}^{s}.
\end{equation}
Using the similar method as Lemma \ref{lem:1} and Lemma \ref{lem:2}, we can prove the following Lemma \ref{lem:4} and Lemma \ref{lem:5}.
\begin{lemma}\label{lem:4}
For $t\geq a\geq 1$, $x\geq 1$ we have,
\begin{equation}
(1+t)^x\leq (1+{a})^{x-1}+(1+\frac{1}{a})^{x-1}t^x.
\end{equation}
\end{lemma}
\begin{lemma}\label{lem:5}
For nonnegative numbers $p_i$, $i=1,\cdots, n$, rearrange them in descending order: $p_{(1)}\geq p_{(2)}\geq ...\geq p_{(n)}$
where $p_{(i)}\in \{p_j|j=1,\cdots, n\}$. If $p_{(i)}\geq a p_{(i+1)}$ for $i=1,...,n-1$ and $a$, then
we have
\begin{equation}
\left(\sum_{i=1}^n p_i\right)^x\leq (1+{a})^{x-1}\sum_{i=1}^n \left((1+\frac{1}{a})^{x-1}\right)^{n-i}p_{(i)}^x,
\end{equation}
for $x\geq 1$.
\end{lemma}
\begin{remark}\label{rem:3}~~
Similar argument as Remark \ref{rem:1} implies that
for the case $x\geq 1$, $t\geq a\geq 1$ and $f(x)\leq (1+{a})^{x-1} $, we have
\begin{equation*}
(1+t)^x\leq f(x)+\frac{(1+a)^x-f(x)}{a^x}t^x.
\end{equation*}
We can easily check that for $ x\geq 1$, $t\geq a\geq 1$ and $f(x)\leq (1+{a})^{x-1}$,
$$(1+{a})^{x-1}+(1+\frac{1}{a})^{x-1}t^x-
\left[f(x)+\frac{(1+a)^x-f(x)}{a^x}t^x\right]\leq 0.$$
\end{remark}
\begin{remark}\label{rem:4}
In Lemma 2 of \cite{JFQ}, the authors gave an upper bound of $(1+t)^x$ for $x\geq 1$ and $t\geq a\geq 1$
\begin{equation*}
(1+t)^{x} \leq 1+\frac{(1+a)^{x}-1}{a^{x}} t^{x}.
\end{equation*}
Actually this is a special case of $f(x)=1\leq (1+{a})^{x-1}$ for $ x\geq 1$ in Remark \ref{rem:3},
therefore our upper bound of $(1+t)^x$ is better than the one given in \cite{JFQ}.
Consequently our polygamy relations based on Lemma \ref{lem:4} are better than those given in \cite{JFQ} based on Lemma 2 of \cite{JFQ}.
\end{remark}
\begin{remark}\label{rem:Jing's paper1}
In \cite[Lemma 2]{ZJZ1}, the authors also gave an upper bound of $(1+t)^x$ for $x\geq 1$, $t\geq a\geq 1$ and $0<q\leq 1$
\begin{equation*}
(1+t)^{x} \leq q^{x}+\frac{(1+a)^{x}-q^{x}}{a^{x}} t^{x},
\end{equation*}
Actually this is the special cases of $f(x)=q^x\leq (1+{a})^{x-1}$ for $ x\geq 1$ in Remark \ref{rem:3},
therefore our upper bound of $(1+t)^x$ is better than the one given in \cite{ZJZ1}.
Thus, our polygamy relations based on Lemma \ref{lem:4} are better than those given in \cite{ZJZ1} based on Lemma 2 of \cite{ZJZ1}.
\end{remark}
\begin{remark}\label{rem:Jing's paper2}
In \cite[Lemma 1]{ZJZ2}, an upper bound of $(1+t)^x$ for $x\geq 1$, $t\geq a\geq 1$ was given:
\begin{equation*}
(1+t)^{x} \leq (\frac{1}{2})^{x}+\frac{(1+a)^{x}-(\frac{1}{2})^{x}}{a^{x}} t^{x},
\end{equation*}
Again this is a special case of $f(x)=(\frac{1}{2})^{x}\leq (1+{a})^{x-1}$ for $ x\geq 1$ in Remark \ref{rem:3},
therefore our upper bound of $(1+t)^x$ is better than the one given in \cite{ZJZ2}. Naturally
our polygamy relations based on Lemma \ref{lem:4} are stronger than those of \cite{ZJZ2} based on Lemma 1 of \cite{ZJZ2}.
\end{remark}
\begin{theorem} Let $\mathcal{Q}$ be a bipartite measure satisfying the generalized polygamy relation \eqref{poly} for $0 \leq s \leq 1$.
Suppose $\mathcal{Q}_{A_1A_3}^{s} \geq a \mathcal{Q}_{A_1 A_2}^{s}$
for $a\geq 1$ on any tripartite state $\rho_{A B C} \in$ $H_{A_1} \otimes H_{A_2} \otimes H_{A_3}$, then the quantum correlation measure $\mathcal{Q}$ satisfies
$$
\mathcal{Q}_{A_1 \mid A_2 A_3}^{\beta} \leq (1+{a})^{\frac{\beta}{s}-1}\mathcal{Q}_{A_1 A_2}^{\beta}+(1+\frac{1}{a})^{\frac{\beta}{s}-1} \mathcal{Q}_{A_1 A_3}^{\beta}
$$
for $\beta \geq s$.
\end{theorem}
\begin{theorem} Let $\rho$ be a state on the multipartite system $A_1A_2...A_n$. Let $\mathcal{Q}$ be a bipartite measure satisfying the generalized polygamy relation \eqref{poly} for $0 \leq s \leq 1$. Set $\mathcal{Q}_{(1)}\geq \mathcal{Q}_{(2)}\geq...\geq \mathcal{Q}_{(n-1)}$ with $\mathcal{Q}_{(j)}\in\{\mathcal{Q}_{A_1A_i}|i=2,...,n\}, j=1,...,n-1$. If
$\mathcal{Q}_{(i)}^{s}\geq a \mathcal{Q}_{(i+1)}^s$ for $a$ and $i=1,...,n-2$, then we have
\begin{equation}
\mathcal{Q}^{\beta}_{A_1|A_2...A_n}\leq(1+{a})^{\frac{\beta}{s}-1}\sum_{i=1}^{n-1} \left((1+\frac{1}{a})^{\frac{\beta}{s}-1}\right)^{n-1-i}\mathcal{Q}_{(i)}^{{\beta}}
\end{equation}
for $\beta\geq s$.
\end{theorem}
\begin{proof}
Since $\mathcal{Q}_{A \mid B C}^{s} \leq \mathcal{Q}_{A B}^{s}+\mathcal{Q}_{A C}^{s}$, we have
\begin{equation}
\mathcal{Q}^{s}_{A_1|A_2...A_n}\leq \mathcal{Q}^{s}_{A_1|A_2}+\mathcal{Q}^{s}_{A_1|A_3...A_n}\leq\cdots\leq \sum_{i=2}^n\mathcal{Q}^{s}_{A_1|A_i}=\sum_{j=1}^{n-1}\mathcal{Q}^{s}_{(j)}.
\end{equation}
By Lemma \ref{lem:5} we have
\begin{equation}
\begin{aligned}
\mathcal{Q}^{\beta}_{A_1|A_2...A_n}&=\left(\mathcal{Q}^{s}_{A_1|A_2...A_n}\right)^{\frac{\beta}{s}}\leq \left(\sum_{j=1}^{n-1}\mathcal{Q}^{s}_{(j)}\right)^{\frac{\beta}{s}}\\
&\leq(1+{a})^{\frac{\beta}{s}-1}\sum_{i=1}^{n-1} \left((1+\frac{1}{a})^{\frac{\beta}{s}-1}\right)^{n-1-i}\mathcal{Q}_{(i)}^{{\beta}}
\end{aligned}
\end{equation}
\end{proof}
The general monogamy relations can be applied to any quantum correlation measure such as the concurrence of assistance, square of convex-roof extended negativity of assistance (SCRENoA), and entanglement of assistance etc. Correspondingly new class of (weighted) polygamy relations are obtained. In the following, we take the SCRENoA as an example.
The negativity of bipartite state $\rho_{A_1A_2}$ is defined by \cite{GRF}:
$N(\rho_{A_1A_2})=(||\rho_{A_1A_2}^{T_{A_{1}}}||-1)/2$,
where $\rho_{A_1A_2}^{T_{A_1}}$ is the partial transposition with respect to the subsystem $A_1$ and $||X||=\mathrm{Tr}\sqrt{XX^\dag}$ is the trace norm of $X$.
For convennience, we use the following definition of negativity, $ N(\rho_{A_1A_2})=||\rho_{A_1A_2}^{T_{A_{1}}}||-1$.
For any bipartite pure state $|\psi\rangle_{A_1A_2}$, the negativity $ N(\rho_{A_1A_2})$ is given by
$$N(|\psi\rangle_{A_1A_2})=2\sum_{i<j}\sqrt{\lambda_i\lambda_j}=(\mathrm{Tr}\sqrt{\rho_{A_1}})^2-1,$$
where $\lambda_i$ are the eigenvalues for the reduced density matrix $\rho_A$ of $|\psi\rangle_{A_1A_2}$. For a mixed state $\rho_{A_1A_2}$, the square of convex-roof extended negativity (SCREN) is defined by
$N_{sc}(\rho_{A_1A_2})=[\mathrm{min}\sum_ip_iN(|\psi_i\rangle_{A_1A_2})]^2$,
where the minimum is taken over all possible pure state decompositions $\{p_i,~|\psi_i\rangle_{A_1A_2}\}$ of $\rho_{A_1A_2}$. The SCRENoA is then defined by $N_{sc}^a(\rho_{A_1A_2})=[\mathrm{max}\sum_ip_iN(|\psi_i\rangle_{A_1A_2})]^2$, where the maximum is taken over all possible pure state decompositions $\{p_i,~|\psi_i\rangle_{A_1A_2}\}$ of $\rho_{A_1A_2}$. For convenience, we denote ${N_a}_{A_1A_i}=N_{sc}^a(\rho_{A_1A_i})$ the SCRENoA of $\rho_{A_1A_i}$ and ${N_a}_{A_1|A_2\cdots A_{n}}=N^a_{sc}(|\psi\rangle_{A_1|A_2\cdots A_{n}})$.
\iffalse
In \cite{j012334} it has been shown that
${N_a}_{A|B_1\cdots B_{N-1}}\leq \sum_{j=1}^{N-1}{N_a}_{AB_j}.$
It is further improved that for $0\leq\beta\leq1$ \cite{jin3},
\begin{eqnarray}\label{n5}
{N_a}^\beta_{A|B_1\cdots B_{N-1}}\leq\sum_{j=1}^{N-1} (2^\beta-1)^j{N_a}^\beta_{AB_j}.
\end{eqnarray}
\fi
\begin{corollary} Let $s\in (0, 1)$ be the fixed number so that the SCRENoA satisfying the generalized polygamy relation \eqref{poly}.
Suppose ${N_a}^s_{A_1A_3}\geq a{N_a}^s_{A_1A_2}$
for $a\geq 1$ on a $2\otimes2\otimes2^{N-2}$ tripartite mixed state $\rho$, then the SCRENoA satisfies
\begin{eqnarray}\label{co31}
{N_a}^\beta_{A_1|A_2A_3}\leq(1+{a})^{\frac{\beta}{s}-1}{N_a}^\beta_{A_1A_2}+(1+\frac{1}{a})^{\frac{\beta}{s}-1} {N_a}^\beta_{A_1A_3}
\end{eqnarray}
for $\beta\geq \delta$.
\end{corollary}
By induction, the following result is immediate for a multiqubit quantum state $\rho_{A_1A_2\cdots A_{n}}$.
\begin{corollary} Let $s\in (0, 1)$ be the fixed number so that the SCRENoA satisfying the generalized polygamy relation \eqref{poly}.
Let $N_{a(1)}\geq N_{a(2)}\geq...\geq N_{a(n-1)}$ be a reordering of $N_{aA_1A_j}$, $j=2,...,n$.
If $N_{a(i)}^{s}\geq a N_{a(i+1)}^s$ for $a$ and $i=1,...,n-2$, then we have
\begin{equation}
N^{\beta}_{aA_1|A_2...A_n}\leq(1+{a})^{\frac{\beta}{s}-1}\sum_{i=1}^{n-1} \left((1+\frac{1}{a})^{\frac{\beta}{s}-1}\right)^{n-1-i}N_{a(i)}^{{\beta}}
\end{equation}
for $\beta\geq s$.
\end{corollary}
\begin{example} Let us consider the three-qubit generlized $W$-class state,
\begin{eqnarray}\label{W}
|W\rangle_{A_1A_2A_3}=\frac{1}{2}(|100\rangle+|010\rangle)+\frac{\sqrt{2}}{2}|001\rangle.
\end{eqnarray}
Then ${N_a}_{A_1|A_2A_3}=\frac{3}{4}$, ${N_a}_{A_1A_2}=\frac{1}{4},~{N_a}_{A_1A_3}=\frac{1}{2}$.
Let $a=2^{0.6}$, then the upper bound given in \cite{JFQ} is $$W_1=(\frac{1}{4})^\beta+\frac{(1+2^{0.6})^\frac{\beta}{s}-1}{(2^{0.6})^{\frac{\beta}{s}}}(\frac{1}{2})^\beta,$$
the upper bound given in \cite{ZJZ1,ZJZ2} is
$$W_2=(\frac{1}{2})^{\frac{\beta}{s}}(\frac{1}{4})^\beta+\frac{(1+2^{0.6})^\frac{\beta}{s}-(\frac{1}{2})^{\frac{\beta}{s}}}{(2^{0.6})^{\frac{\beta}{s}}}(\frac{1}{2})^\beta,$$
while our upper bound is $$W_3=(1+2^{0.6})^{\frac{\beta}{s}-1}(\frac{1}{4})^\beta+(1+2^{-0.6})^{\frac{\beta}{s}-1} (\frac{1}{2})^\beta.$$
Fig. 2 charts our bound together with other bounds, and Fig. 3 and Fig. 4 show the comparison of our bound with those given in \cite{JFQ,ZJZ1,ZJZ2}. Our bound is found to be stronger than the other two.
\end{example}
\begin{figure}[H]
\centerline{\includegraphics[width=0.6\textwidth]{fig2.eps}}
\renewcommand{\figurename}{Fig.}
\caption{The z-axis presents SCRENoA for the state $|W\rangle_{A_1A_2A_3}$ as a function of $\beta, s$. The red, green and blue surfaces chart
our upper bound, the upper bound of \cite{JFQ} and the upper bound of \cite{ZJZ1,ZJZ2} respectively.}
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=0.6\textwidth]{fig3.eps}}
\renewcommand{\figurename}{Fig.}
\caption{The surface depicts our upper bound of SCRENoA minus that given by \cite{JFQ}.}
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=0.6\textwidth]{fig4.eps}}
\renewcommand{\figurename}{Fig.}
\caption{The surface depicts our upper bound of SCRENoA minus that of \cite{ZJZ1,ZJZ2}.}
\end{figure}
\bigskip
\section{Conclusion}
Monogamy relations reveal special properties of correlations in terms of inequalities satisfied by various quantum measurements of the subsystems.
In this paper, we have examined the physical meanings and mathematical formulations related to monogamy and polygamy relations
in multipartite quantum systems. By grossly generalizing a technical inequality for the function $(1+t)^x$, we have obtained general stronger weighted
monogamy and polygamy relations for any quantum measurement such as concurrence, negativity, entanglement of formation etc. as well as
the Tsallis-$q$ entanglement and R\'enyi-$q$ entanglement measures. We have shown rigorously that our bounds outperform some of the strong bounds found recently
in a unified manner, notably that our results are not only stronger for monogamy relations but also polygamy relations. We have also used the concurrence and the SCRENoA
(square of convex-roof extended negativity of assistance) to show that our bounds are indeed better than the
recently available bounds through detailed examples and charts in both situations.
\bigskip
\noindent{\bf Acknowledgments}
This work is partially supported by Simons Foundation grant no. 523868
and National Natural Science Foundation of China grant no. 12126351.
\bigskip
\noindent\textbf {Data Availability Statements} All data generated or analysed during this study are included in the paper.
\bigskip
|
2,869,038,155,630 | arxiv | \section{Metric spaces}
\label{metric spaces}
\setcounter{equation}{0}
A \emph{metric space} is a set $M$ equipped with a function
$d(x, y)$ defined for $x, y \in M$ such that $d(x, y)$ is a
nonnegative real number that is equal to $0$ exactly when $x = y$,
\begin{equation}
d(y, x) = d(x, y)
\end{equation}
for every $x, y \in M$, and
\begin{equation}
d(x, z) \le d(x, y) + d(y, z)
\end{equation}
for every $x, y, z \in M$, which is known as the \emph{triangle inequality}.
Remember that the \emph{absolute value} of a real number $r$
is denoted $|r|$ and equal to $r$ when $r \ge 0$ and to $-r$ when $r
\le 0$. It is easy to check that
\begin{equation}
|r + t| \le |r| + |t|
\end{equation}
and
\begin{equation}
|r \, t| = |r| \, |t|
\end{equation}
for any pair of real numbers $r$, $t$. The standard metric on the
real line ${\bf R}$ is given by $|r - t|$, which is the first main
example of a metric space.
If $(M, d(x, y))$ is a metric space, then $d(x, y)$ is called
the \emph{distance function} or \emph{metric} on $M$. For each $x \in
M$ and $r > 0$, the \emph{open ball} in $M$ with center $x$ and radius
$r$ is
\begin{equation}
B(x, r) = \{y \in M : d(x, y) < r\}.
\end{equation}
Similarly, the \emph{closed ball} with center $x$ and radius $r \ge 0$ is
\begin{equation}
\overline{B}(x, r) = \{y \in M : d(x, y) \le r\}.
\end{equation}
Thus
\begin{equation}
B(x, r) \subseteq \overline{B}(x, r) \subseteq B(x, t)
\end{equation}
when $r < t$.
Let $a$, $b$ be real numbers with $a < b$. The \emph{open
interval} $(a, b)$ in ${\bf R}$ is defined by
\begin{equation}
(a, b) = \{r \in {\bf R} : a < r < b\},
\end{equation}
and the \emph{closed interval} $[a, b]$ is defined by
\begin{equation}
[a, b] = \{r \in {\bf R} : a \le r \le b\}.
\end{equation}
One may also allow $a = b$ for the latter. The \emph{length} of these
intervals is $b - a$. Note that open and closed balls in the real
line with respect to the standard metric are open and closed
intervals.
\section{A little calculus}
\label{calculus}
\setcounter{equation}{0}
Suppose that $a$, $b$ are real numbers with $a < b$, and that
$f(x)$ is a continuous real-valued function on the closed interval
$[a, b]$ in the real line. The \emph{extreme value theorem} states
that there are elements $p$, $q$ of $[a, b]$ at which $f$ attains its
maximum and minimum, which is to say that
\begin{equation}
f(q) \le f(x) \le f(p)
\end{equation}
for every $x \in [a, b]$. This works as well for continuous
real-valued functions on compact subsets of metric spaces, or even
topological spaces. If $p$ or $q$ is in the open interval $(a, b)$
and $f$ is differentiable there, then the derivative $f'(p)$ or
$f'(q)$ is equal to $0$.
Suppose that $f(x)$ is differentiable at every point in $(a,
b)$. If $f(a) = f(b) = 0$, then \emph{Rolle's theorem} states that
$f'(x) = 0$ for some $x \in (a, b)$. This is because the maximum or
minimum of $f$ on $[a, b]$ is attained on $(a, b)$, or $f(x) = 0$ for
every $x \in [a, b]$. No matter the values of $f(a)$, $f(b)$, the
\emph{mean value theorem} says that there is an $x \in (a, b)$ such that
\begin{equation}
f'(x) = \frac{f(b) - f(a)}{b - a}.
\end{equation}
This follows from Rolle's theorem applied to $f - \phi$, where
$\phi(x) = \alpha \, x + \beta$ and $\alpha, \beta \in {\bf R}$ are
chosen so that $\phi(a) = f(a)$, $\phi(b) = f(b)$.
Of course, the derivative of a constant function is $0$, and
the mean value theorem implies that a continuous function $f$ on $[a,
b]$ is constant if the derivative of $f$ exists and is equal to $0$ at
every point in $(a, b)$. If $f$ is monotone increasing on $[a, b]$,
in the sense that $f(x) \le f(y)$ when $a \le x \le y \le b$, then
$f'(x) \ge 0$ for every $x \in (a, b)$ at which $f$ is differentiable.
Conversely, if $f$ is continuous on $[a, b]$, differentiable on $(a,
b)$, and $f'(x) \ge 0$ for each $x \in (a, b)$, then $f$ is monotone
increasing on $[a, b]$, by the mean value theorem. If $f'(x) > 0$ for
every $x \in (a, b)$, then $f$ is strictly increasing on $[a, b]$, in
the sense that $f(w) < f(y)$ when $a \le w < y \le b$. However, the
derivative of a strictly increasing function may be equal to $0$, as
when $f(x) = x^3$.
\section{Norms on ${\bf R}^n$}
\label{norms}
\setcounter{equation}{0}
Let $n$ be a positive integer, and let ${\bf R}^n$ be the
space of $n$-tuples of real numbers. This means that an element $x$
of ${\bf R}^n$ is of the form $x = (x_1, \ldots, x_n)$, where the
coordinates $x_1, \ldots, x_n$ of $x$ are real numbers. Addition and
scalar multiplication on ${\bf R}^n$ are defined coordinatewise in the
usual way, so that ${\bf R}^n$ becomes a finite-dimensional vector
space over the real numbers.
A \emph{norm} on ${\bf R}^n$ is a function $N(x)$ such that
$N(x)$ is a nonnegative real number for every $x \in {\bf R}^n$ which
is equal to $0$ exactly when $x = 0$,
\begin{equation}
\label{N(r x) = |r| N(x)}
N(r \, x) = |r| \, N(x)
\end{equation}
for every $r \in {\bf R}$ and $x \in {\bf R}^n$, and
\begin{equation}
\label{N(x + y) le N(x) + N(y)}
N(x + y) \le N(x) + N(y)
\end{equation}
for every $x, y \in {\bf R}^n$. If $N$ is a norm on ${\bf R}^n$, then
\begin{equation}
d_N(x, y) = N(x - y)
\end{equation}
is a metric on ${\bf R}^n$.
For example, the absolute value function is a norm on ${\bf
R}$, for which the corresponding metric is the standard metric on the
real line. The standard Euclidean norm on ${\bf R}^n$ is defined by
\begin{equation}
|x| = \Big(\sum_{j = 1}^n x_j^2 \Big)^{1/2},
\end{equation}
and the corresponding metric is the standard Euclidean metric on ${\bf R}^n$.
It is not so obvious that this satisfies the triangle inequality, and hence
is a norm, and we shall discuss a proof of this fact in Section \ref{convex
sets}.
One can check directly that
\begin{equation}
\|x\|_1 = \sum_{j = 1}^n |x_j|
\end{equation}
and
\begin{equation}
\|x\|_\infty = \max(|x_1|, \ldots, |x_n|)
\end{equation}
are norms on ${\bf R}^n$. We shall see in Section \ref{convex sets} that
\begin{equation}
\|x\|_p = \Big(\sum_{j = 1}^n |x_j|^p \Big)^{1/p}
\end{equation}
is a norm when $p \ge 1$, which includes the Euclidean norm as a
special case.
\section{Convex functions}
\label{convex functions}
\setcounter{equation}{0}
A real-valued function $f(x)$ on the real line is said to be
\emph{convex} if
\begin{equation}
\label{convexity}
f(t \, x + (1 - t) \, y) \le t \, f(x) + (1 - t) \, f(y)
\end{equation}
for every $x, y \in {\bf R}$ and $t \in [0, 1]$. This is equivalent
to
\begin{equation}
\label{convexity, 2}
\frac{f(w) - f(x)}{w - x} \le \frac{f(y) - f(w)}{y - w}
\end{equation}
for every $x, w, y \in {\bf R}$ such that $x < w < y$. Applying this
condition twice, we get that
\begin{equation}
\label{convexity, 3}
\frac{f(w) - f(x)}{w - x} \le \frac{f(z) - f(y)}{z - y}
\end{equation}
when $x < w < y < z$. As another refinement of (\ref{convexity, 2}),
one can use (\ref{convexity}) to show that
\begin{equation}
\label{convexity, 4}
\frac{f(w) - f(x)}{w - x} \le \frac{f(y) - f(x)}{y - x}
\le \frac{f(y) - f(w)}{y - w}
\end{equation}
when $x < w < y$.
If $f$ is differentiable and $f'$ is monotone increasing, then
the mean value theorem implies (\ref{convexity, 2}) and hence that $f$
is convex. Conversely, (\ref{convexity, 3}) implies that the
derivative of $f$ is monotone increasing when $f$ is differentiable.
Actually, one can show that the right and left derivatives $f_+'(x)$,
$f_-'(x)$ exist for each $x \in {\bf R}$ when $f$ is convex, and
satisfy
\begin{equation}
f_-'(x) \le f_+'(x)
\end{equation}
and
\begin{equation}
f_+'(x) \le f_-'(y)
\end{equation}
when $x < y$. One can also show that these conditions characterize
convexity, using analogues of Rolle's theorem and the mean value
theorem for functions with one-sided derivatives.
A function $f : {\bf R} \to {\bf R}$ is \emph{strictly convex} if
\begin{equation}
f(t \, x + (1 - t) \, y) < t \, f(x) + (1 - t) \, f(y)
\end{equation}
when $x \ne y$ and $0 < t < 1$. This corresponds to strict inequality
in (\ref{convexity, 2}), (\ref{convexity, 3}), and (\ref{convexity,
4}) as well. If $f$ is differentiable on ${\bf R}$, then $f$ is
strictly convex if and only if $f'$ is strictly increasing.
Otherwise, strict convexity can be characterized in terms of one-sided
derivatives by the requirement that
\begin{equation}
f_+'(x) < f_-'(y)
\end{equation}
when $x < y$. Alternatively, if a convex function $f$ on ${\bf R}$ is
not strictly convex, then $f$ is equal to an affine function on an
interval of positive length.
For example, consider $f(r) = |r|^p$, $p > 0$. If $p = 1$,
then $f(r) = |r|$ is convex but not strictly convex on ${\bf R}$. If
$p = 2$, then $f(r) = r^2$ is twice-differentiable, $f''(r) = 2$, and
$f$ is strictly convex. If $p > 2$, then $f$ is twice-differentiable
on ${\bf R}$, $f''(r) > 0$ when $r \ne 0$, $f''(0) = 0$, and $f$ is
strictly convex because $f'$ is strictly increasing. If $1 < p < 2$,
then $f$ is differentiable on ${\bf R}$, twice-differentiable on ${\bf
R} \backslash \{0\}$, $f''(r) > 0$ when $r \ne 0$, and again $f$ is
strictly convex since $f'$ is strictly increasing. If $0 < p < 1$,
then $f$ is twice-differentiable on ${\bf R} \backslash \{0\}$,
$f''(r) < 0$ when $r \ne 0$, and $f$ is not convex.
\section{Convex sets}
\label{convex sets}
\setcounter{equation}{0}
A set $E \subseteq {\bf R}^n$ is said to be \emph{convex} if
\begin{equation}
t \, x + (1 - t) \, y \in E
\end{equation}
for every $x, y \in E$ and $t \in (0, 1)$. For example, open and
closed balls associated to metrics defined by norms on ${\bf R}^n$ are
convex.
Conversely, suppose that $N(x)$ is a nonnegative real-valued
function on ${\bf R}^n$ such that $N(x) > 0$ when $x \ne 0$ and the
homogeneity condition (\ref{N(r x) = |r| N(x)}) holds for all $x \in
{\bf R}^n$ and $r \in {\bf R}$. If the closed unit ball
\begin{equation}
\label{B_N}
B_N = \{x \in {\bf R}^n : N(x) \le 1\}
\end{equation}
is convex, then $N$ satisfies the triangle inequality (\ref{N(x + y)
le N(x) + N(y)}) and hence is a norm. Let $x, y \in {\bf R}^n$ be
given, and let us check (\ref{N(x + y) le N(x) + N(y)}). We may
suppose that $x, y \ne 0$, since the inequality is trivial when $x =
0$ or $y = 0$. Put
\begin{equation}
x' = \frac{x}{N(x)}, \quad y' = \frac{y}{N(y)},
\end{equation}
so that $N(x') = N(y') = 1$. By hypothesis,
\begin{equation}
N(t \, x' + (1 - t) \, y') \le 1
\end{equation}
when $0 \le t \le 1$. Applying this with
\begin{equation}
t = \frac{N(x)}{N(x) + N(y)},
\end{equation}
we get (\ref{N(x + y) le N(x) + N(y)}), as desired.
For example, suppose that $N(x) = \|x\|_p$, $1 < p < \infty$.
Let $x, y \in {\bf R}^n$ with $\|x\|_p, \|y\|_p \le 1$ be given, so that
\begin{equation}
\sum_{j = 1}^n |x_j|^p, \ \sum_{j = 1}^n |y_j|^p \le 1.
\end{equation}
We would like to show that
\begin{equation}
\|t \, x + (1 - t) \, y\|_p \le 1
\end{equation}
when $0 \le t \le 1$, which is the same as
\begin{equation}
\sum_{j = 1}^n |t \, x_j + (1 - t) \, y_j|^p \le 1.
\end{equation}
The convexity of $|r|^p$ on ${\bf R}$ implies that
\begin{equation}
|t \, x_j + (1 - t) \, y_j|^p \le t |x_j|^p + (1 - t) \, |y_j|^p
\end{equation}
for each $j$, and the desired inequality follows by summing this over
$j$.
A norm $N$ on ${\bf R}^n$ is said to be \emph{strictly convex}
if the unit ball $B_N$ is strictly convex in the sense that
\begin{equation}
N(t \, x + (1 - t) \, y) < 1
\end{equation}
when $x, y \in {\bf R}^n$, $N(x) = N(y) = 1$, $x \ne y$, and $0 < t <
1$. It is easy to see that the absolute value function is strictly
convex as a norm on ${\bf R}$, if not as a general function as in the
previous section. One can also check that $\|x\|_p$ is a strictly
convex norm on ${\bf R}^n$ when $p > 1$, using the strict convexity of
$|r|^p$ on ${\bf R}$ and computations as in the preceding paragraph.
However, $\|x\|_1$ and $\|x\|_\infty$ are not strictly convex norms on
${\bf R}^n$ when $n \ge 2$.
\section{A little more calculus}
\label{a little more calculus}
\setcounter{equation}{0}
Let $f$ be a continuous real-valued function on a closed
interval $[a, b]$, $a < b$. The integral
\begin{equation}
\int_a^b f(t) \, dt
\end{equation}
can be defined in the usual way as a limit of finite sums. The
convergence of the finite sums to the integral uses the fact that
continuous functions on $[a, b]$ are actually uniformly continuous.
It is well known that continuous functions on compact subsets of any
metric space are uniformly continuous.
Consider the indefinite integral
\begin{equation}
F(x) = \int_a^x f(t) \, dt.
\end{equation}
This defines a continuous function on $[a, b]$ which is differentiable
on $(a, b)$ and satisfies $F'(x) = f(x)$. Similarly, $F$ has
one-sided derivatives at the endpoints $a$, $b$ that satisfy the same
condition. If another differentiable function on $(a, b)$ has
derivative $f$, then the difference of $F$ and this function is
constant, by the mean value theorem.
Clearly $F$ is monotone increasing on $[a, b]$ if $f \ge 0$ on
the whole interval. If $f > 0$ on $[a, b]$, then $F$ is strictly
increasing. The same conclusion holds if $f \ge 0$ on $[a, b]$ and $f
> 0$ at some point in any nontrivial subinterval of $[a, b]$.
Equivalently, if $f \ge 0$ on $[a, b]$, and if $F$ is not strictly
increasing on $[a, b]$, then $f = 0$ at every point in a nontrivial
subinterval.
Suppose that $f' \ge 0$ on $(a, b)$, or simply that $f$ is
monotone increasing on $[a, b]$. This implies that
\begin{equation}
f(x) \le \frac{F(y) - F(x)}{y - x} \le f(y)
\end{equation}
when $a \le x < y \le b$. In particular,
\begin{equation}
\frac{F(w) - F(x)}{w - x} \le \frac{F(y) - F(w)}{y - w}
\end{equation}
when $a \le x < w < y \le b$. If $f$ is strictly increasing, then
these inequalities are strict as well.
\section{Supremum and infimum}
\label{supremum and infimum}
\setcounter{equation}{0}
A real number $b$ is said to be an \emph{upper bound} for a
set $A \subseteq {\bf R}$ if $a \le b$ for every $a \in A$. We say
that $b_1 \in {\bf R}$ is the \emph{least upper bound} or
\emph{supremum} of $A$ if $b_1$ is an upper bound for $A$ and $b_1 \le
b$ for every upper bound $b$ of $A$. If $b_2 \in {\bf R}$ also
satisfies these two conditions, then $b_1 \le b_2$ and $b_2 \le b_1$,
and hence $b_1 = b_2$. Thus the supremum of $A$ is unique when it
exists, in which case it is denoted $\sup A$. The \emph{completeness
property} of the real line states that every nonempty set with an
upper bound has a least upper bound.
More precisely, this is completeness with respect to the
ordering on the real line, which can be defined for other ordered
sets. There is also completeness for metric spaces, which means that
every Cauchy sequence converges. Both forms of completeness hold on
the real line, and are basically equivalent to each other in this
particular situation. However, the two notions are distinct, because
they can be applied in different circumstances. There are
completeness conditions concerning the existence of solutions of
ordinary differential equations as well, which may be related to
completeness for an associated metric space.
Similarly, a real number $c$ is said to be a \emph{lower bound}
for $A \subseteq {\bf R}$ if $c \le a$ for every $a \in A$, and $c_1
\in {\bf R}$ is the \emph{greatest lower bound} or \emph{infimum} of
$A$ if $c_1$ is a lower bound for $A$ and $c \le c_1$ for every lower
bound $c$ of $A$. This is unique when it exists for the same reasons
as before, and is denoted $\inf A$. It follows from completeness that
a nonempty set $A \subseteq {\bf R}$ with a lower bound has a greatest
lower bound, which can be characterized as the supremum of the set of
lower bounds of $A$. Alternatively, the infimum of $A$ is equal to
the negative of the supremum of $-A = \{-a : a \in A\}$.
\section{Bounded sets}
\label{bounded sets}
\setcounter{equation}{0}
Let $(M, d(x, y))$ be a metric space. A set $E \subseteq M$
is said to be \emph{bounded} if there is a $p \in M$ and an $r \ge 0$
such that
\begin{equation}
d(p, x) \le r
\end{equation}
for every $x \in E$. This implies that for every $q \in M$ there is a
$t \ge 0$ such that $d(q, x) \le t$ for every $x \in E$, by taking $t
= r + d(p, q)$.
Equivalently, $E \subseteq M$ is bounded if the set of
distances $d(x, y)$ for $x, y \in E$ has an upper bound in ${\bf R}$.
If $E$ is nonempty and bounded, then the \emph{diameter} of $E$ is defined by
\begin{equation}
\mathop{\rm diam} E = \sup \{d(x, y) : x, y \in E\}.
\end{equation}
The diameter of the empty set may be interpreted as $0$.
If $E_1 \subseteq E_2 \subseteq M$ and $E_2$ is bounded, then
$E_1$ is bounded, and
\begin{equation}
\mathop{\rm diam} E_1 \le \mathop{\rm diam} E_2.
\end{equation}
The union of two bounded subsets of $M$ is also bounded, but the diameter of
the union may be much larger than the sum of the diameters of the two subsets.
Suppose that $M$ is ${\bf R}^n$ equipped with a norm $N$ and
its associated metric $d_N(x, y)$. The \emph{convex hull}
$\widehat{E}$ of a set $E \subseteq {\bf R}^n$ consists of all convex
combinations of elements of $E$. More precisely, $\widehat{E}$ is the
set of all finite sums of the form
\begin{equation}
\sum_{i = 1}^k r_i \, x(i),
\end{equation}
where $k$ is a positive integer, $r_1, \ldots, r_k$ are
nonnegative real numbers such that
\begin{equation}
\sum_{i = 1}^k r_k = 1,
\end{equation}
and $x(1), \ldots, x(k)$ are elements of $E$. It is well known that
one can take $k = n + 1$ here, but we shall not need this fact. By
construction, $\widehat{E}$ is a convex set in ${\bf R}^n$ that
contains $E$. Moreover, $\widehat{E}$ is the smallest such set, in
the sense that $\widehat{E}$ is contained in any convex set in ${\bf
R}^n$ that contains. If $E$ is bounded, so that $E$ is contained in a
ball, then $\widehat{E}$ is contained in the same ball, and hence
$\widehat{E}$ is bounded. Let us check that
\begin{equation}
{\mathop{\rm diam}}_N \widehat{E} \le {\mathop{\rm diam}}_N E,
\end{equation}
where the subscript $N$ indicates that the diameter uses the norm $N$.
Let
\begin{equation}
\xi = \sum_{i = 1}^k r_i \, x(i), \quad \eta = \sum_{j = 1}^l t_j \, y(j)
\end{equation}
be arbitrary elements of $\widehat{E}$, as before. Thus
\begin{equation}
\xi - \eta = \sum_{i = 1}^k \sum_{j = 1}^l r_i \, t_j \, (x(i) - y(j)),
\end{equation}
and therefore
\begin{equation}
N(\xi - \eta) \le \sum_{i = 1}^k \sum_{j = 1}^l r_i \, t_j \, N(x(i) - y(j)),
\end{equation}
by the properties of norms. This implies that
\begin{equation}
N(\xi - \eta) \le \max \{N(x(i) - y(j)) : 1 \le i \le k, \, 1 \le j \le l\},
\end{equation}
and consequently $N(\xi - \eta) \le {\mathop{\rm diam}}_N E$, as desired.
\section{Lipschitz mappings}
\label{lipschitz mappings}
\setcounter{equation}{0}
Let $(M_1, d_1(x, y))$ and $(M_2, d_2(u, v))$ be metric
spaces. A mapping $f : M_1 \to M_2$ is said to be \emph{Lipschitz} if
\begin{equation}
d_2(f(x), f(y)) \le C \, d_1(x, y)
\end{equation}
for some $C \ge 0$ and all $x, y \in M$. More precisely, this means
that $f$ is Lipschitz of order $1$, and we shall discuss other
Lipschitz conditions later. One can also say that $f$ is
$C$-Lipschitz or $C$-Lipschitz of order $1$ to mention the constant
$C$ explicitly.
Thus $f$ is $C$-Lipschitz with $C = 0$ if and only if $f$ is
constant. Note that Lipschitz mappings are uniformly continuous.
Suppose that $(M_3, d_3(w, z))$ is another metric space, and that $f_1
: M_1 \to M_2$ and $f_2 : M_2 \to M_3$ are Lipschitz mappings with
constants $C_1$, $C_2$, respectively. The composition $f_2 \circ f_1$
is the mapping from $M_1$ to $M_2$ defined by
\begin{equation}
(f_2 \circ f_1)(x) = f_2(f_1(x)),
\end{equation}
and it is easy to check that this is Lipschitz with constant equal to
the product of $C_1$ and $C_2$.
If $f : M_1 \to M_2$ is $C$-Lipschitz and $E \subseteq M_1$
is bounded, then
\begin{equation}
f(E) = \{f(x) : x \in E\}
\end{equation}
is bounded in $M_2$, and
\begin{equation}
{\mathop{\rm diam}}_2 f(E) \le C \, {\mathop{\rm diam}}_1 E.
\end{equation}
Here the subscripts indicate in which metric space the diameter is
taken. This is easy to verify, directly from the definitions, and
suggests another way to look at the composition of Lipschitz mappings,
as in the previous paragraph.
\section{Real-valued functions}
\label{real functions}
\setcounter{equation}{0}
Let $f$ be a real-valued function on an open interval $(a, b)$
in the real line. If $f$ is $C$-Lipschitz with respect to the
standard metric on the domain and range, then
\begin{equation}
|f'(x)| \le C
\end{equation}
at every point $x \in (a, b)$ at which $f$ is differentiable, by
definition of the derivative. Conversely, if $f$ is differentiable
and satisfies this condition everywhere on $(a, b)$, then $f$ is
$C$-Lipschitz, by the mean value theorem.
Now let $(M, d(x, y))$ be a metric space. A function $f : M
\to {\bf R}$ is $C$-Lipschitz with respect to the standard metric on
${\bf R}$ if and only if
\begin{equation}
f(x) \le f(y) + C \, d(x, y)
\end{equation}
for every $x, y \in M$. This follows easily from the definitions. In
particular, $f_p(x) = d(p, x)$ is $1$-Lipschitz for every $p \in M$.
If $A \subseteq M$, $A \ne \emptyset$, and $x \in M$, then put
\begin{equation}
\mathop{\rm dist}(x, A) = \inf \{d(x, a) : a \in A\}.
\end{equation}
For each $x, y \in M$ and $a \in A$,
\begin{equation}
\mathop{\rm dist}(x, A) \le d(x, a) \le d(x, y) + d(y, a),
\end{equation}
and therefore
\begin{equation}
\mathop{\rm dist}(x, A) \le \mathop{\rm dist}(y, A) + d(x, y).
\end{equation}
This shows that $\mathop{\rm dist}(x, A)$ is $1$-Lipschitz on $M$.
Suppose that $f_1, f_2 : M \to {\bf R}$ are Lipschitz with
constants $C_1$, $C_2$, respectively. For any $r_1, r_2 \in {\bf R}$,
$r_1 \, f_1 + r_2 \, f_2$ is Lipschitz with constant $|r_1| \, C_1 +
|r_2| \, C_2$. Suppose also that $f_1$, $f_2$ are bounded on $M$, with
\begin{equation}
|f_1(x)| \le k_1, \quad |f_2(x)| \le k_2
\end{equation}
for some $k_1, k_2 \ge 0$ and every $x \in M$. Because
\begin{eqnarray}
\lefteqn{f_1(x) \, f_2(x) - f_1(y) \, f_2(y)} \\
& & = (f_1(x) - f_1(y)) \, f_2(x) + f_1(y) \, (f_2(x) - f_2(y)) \nonumber
\end{eqnarray}
for every $x, y \in M$, $f_1 \, f_2$ is Lipschitz on $M$ with constant
$k_2 \, C_1 + k_1 \, C_2$.
\section{${\bf R}^n$-valued functions}
\label{R^n functions}
\setcounter{equation}{0}
Let $N$ be a norm on ${\bf R}^n$. Thus $N$ is $1$-Lipschitz
as a real-valued function on ${\bf R}^n$ with the metric $d_N(x, y)$
associated to $N$, as in the previous section. One can also show that
$N$ is bounded by a constant multiple of the standard Euclidean norm
on ${\bf R}^n$. This uses the finite-dimensionality of ${\bf R}^n$ in
an essential way, and it implies that $N$ is Lipschitz with respect to
the standard metric on ${\bf R}^n$.
Suppose that $f$ is a continuous ${\bf R}^n$-valued function
on a closed interval $[a, b]$ in the real line. As an extension of
the triangle inequality for $N$,
\begin{equation}
N\Big(\int_a^b f(t) \, dt\Big) \le \int_a^b N(f(t)) \, dt.
\end{equation}
Indeed, the analogous statement for the finite sums follows from the
triangle inequality for $N$. The integral of $f$ can be approximated
by finite sums, and continuity of $N$ as in the preceding paragraph
can be employed to pass to the limit. Alternatively, one can use
duality, as follows. For any linear functional $\phi : {\bf R}^n \to
{\bf R}$,
\begin{equation}
\phi\Big(\int_a^b f(t) \, dt\Big) = \int_a^b \phi(f(t)) \, dt.
\end{equation}
If $|\phi(w)| \le N(w)$ for every $w \in {\bf R}^n$, then we get that
\begin{equation}
\biggl|\int_a^b \phi(f(t)) \, dt\biggr| \le \int_a^b N(f(t)) \, dt.
\end{equation}
A famous theorem states that for each $v \in {\bf R}^n$ there is such
a $\phi$ with $\phi(v) = N(v)$, which permits one to estimate the norm
of the integral. We shall not discuss the proof of this here, but one
can take $\phi(w)$ to be the standard inner product of $w$ with $v /
|v|$ when $v \ne 0$ and $N$ is the Euclidean norm on ${\bf R}^n$, and
there are also explicit expressions for $\phi$ when $N(w) = \|w\|_p$,
$1 \le p \le \infty$.
Suppose now that $F : [a, b] \to {\bf R}^n$ is $C$-Lipschitz
with respect to the standard metric on ${\bf R}$ and the metric $d_N$
on ${\bf R}^n$. If $F$ is differentiable at a point $x \in (a, b)$,
then $N(F'(x)) \le C$. This follows from the definition of the
derivative, as in the real-valued case. Conversely, if $F$ is
continuously differentiable on $[a, b]$ and $N(F') \le C$, then one
can use the fundamental theorem of calculus and the integral form of
the triangle inequality to show that that $F$ is $C$-Lipschitz with
respect to $N$. One can use duality to get the same conclusion when
$F$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with
$N(F') \le C$, by applying the mean value theorem to $\phi \circ F$
for linear functionals $\phi : {\bf R}^n \to {\bf R}$.
Let $(M, d(x, y))$ be a metric space, and let $F = (F_1,
\ldots, F_n)$ be a mapping from $M$ into ${\bf R}^n$. If ${\bf R}^n$
is equipped with the norm $\|w\|_\infty$, then it is easy to see that
$F$ is $C$-Lipschitz if and only if $F_1, \ldots, F_n$ are
$C$-Lipschitz as real-valued functions on $M$. Of course, one can
estimate Lipschitz conditions for $F$ in terms of Lipschitz conditions
for $F_1, \ldots, F_n$ for other norms on ${\bf R}^n$, and vice-versa,
but the relationship between the constants is normally not quite as
simple as for the norm $\|w\|_\infty$.
\section{Bounded variation}
\label{bounded variation}
\setcounter{equation}{0}
Let $f$ be a real-valued function on a closed interval $[a,
b]$. A \emph{partition} of $[a, b]$ is a finite sequence $\{t_j\}_{j
= 0}^n$ of real numbers such that
\begin{equation}
a = t_0 < t_1 < \cdots < t_n = b.
\end{equation}
For each partition $\mathcal{P} = \{t_j\}_{j = 0}^n$ of $[a, b]$, consider
\begin{equation}
V_\mathcal{P}(f) = \sum_{j = 1}^n |f(t_j) - f(t_{j - 1})|.
\end{equation}
This measures the variation of $f$ on the partition $\mathcal{P}$. We
say that $f$ has \emph{bounded variation} on $[a, b]$ if there is an
upper bound for $V_\mathcal{P}(f)$ over all partitions $\mathcal{P}$
of $[a, b]$. In this case, the \emph{total variation} $V_a^b(f)$ of
$f$ on $[a, b]$ is defined by
\begin{equation}
\label{V_a^b(f)}
V_a^b(f) = \sup \{ V_\mathcal{P} : \mathcal{P}
\hbox{ is a partition of } [a, b]\}.
\end{equation}
Thus $V_a^b(f) = 0$ if and only if $f$ is constant on $[a, b]$.
Using the partition that consists of only $a$, $b$, we get that
\begin{equation}
|f(b) - f(a)| \le V_a^b(f).
\end{equation}
If $f$ is monotone increasing on $[a, b]$, then
\begin{equation}
V_\mathcal{P}(f) = f(b) - f(a)
\end{equation}
for every partition $\mathcal{P}$ of $[a, b]$. Hence $f$ has bounded
variation on $[a, b]$, and
\begin{equation}
V_a^b(f) = f(b) - f(a).
\end{equation}
Conversely, if $f$ has bounded variation on $[a, b]$ and
\begin{equation}
V_a^b(f) = |f(b) - f(a)|,
\end{equation}
then $f$ is either monotone increasing or decreasing on $[a, b]$.
If $f$ is $C$-Lipschitz on $[a, b]$, then
\begin{equation}
V_\mathcal{P}(f) \le C \, (b - a)
\end{equation}
for every partition $\mathcal{P}$ of $[a, b]$. Hence $f$ has bounded
variation on $[a, b]$, and
\begin{equation}
V_a^b(f) \le C \, (b - a).
\end{equation}
If $\phi : {\bf R} \to {\bf R}$ is $C$-Lipschitz, then
\begin{equation}
V_\mathcal{P}(\phi \circ f) \le C \, V_\mathcal{P}(f)
\end{equation}
for every $f : [a, b] \to {\bf R}$ and partition $\mathcal{P}$ of $[a, b]$.
If $f$ has bounded variation on $[a, b]$, then it follows that $\phi
\circ f$ has bounded variation on $[a, b]$, and
\begin{equation}
V_a^b(\phi \circ f) \le C \, V_a^b(f).
\end{equation}
Let $f_1, f_2 : [a, b] \to {\bf R}$ and $r_1, r_2 \in {\bf R}$
be given. For any partition $\mathcal{P}$ of $[a, b]$,
\begin{equation}
V_\mathcal{P}(r_1 \, f_1 + r_2 \, f_2)
\le |r_1| \, V_\mathcal{P}(f_1) + |r_2| \, V_\mathcal{P}(f_2).
\end{equation}
If $f_1$, $f_2$ have bounded variation on $[a, b]$, then it follows
that $r_1 \, f_1 + r_2 \, f_2$ also has bounded variation, with
\begin{equation}
V_a^b(r_1 \, f_1 + r_2 \, f_2) \le |r_1| \, V_a^b(f_1) + |r_2| \, V_a^b(f_2).
\end{equation}
Suppose that $f_1$, $f_2$ are bounded on $[a, b]$, so that
\begin{equation}
|f_1(x)| \le k_1, \quad |f_2(x)| \le k_2
\end{equation}
for some $k_1, k_2 \ge 0$ and every $x \in [a, b]$. It is easy to
check that
\begin{equation}
V_\mathcal{P}(f_1 \, f_2) \le k_2 \, V_\mathcal{P}(f_1)
+ k_1 \, V_\mathcal{P}(f_2)
\end{equation}
for every partition $\mathcal{P}$ of $[a, b]$. If $f_1$, $f_2$ have
bounded variation on $[a, b]$, then $f_1 \, f_2$ has bounded variation, and
\begin{equation}
V_a^b(f_1 \, f_2) \le k_2 \, V_a^b(f_1) + k_1 \, V_a^b(f_2).
\end{equation}
This is analogous to the earlier estimate for the Lipschitz constant
of the product of bounded Lipschitz functions, and to the Leibniz rule
for differentiating the product of two functions.
Suppose that $a_1$, $b_1$ are real numbers such that $a \le
a_1 \le b_1 \le b$. If $f$ has bounded variation on $[a, b]$, then
$f$ has bounded variation on $[a_1, b_1]$, and
\begin{equation}
V_{a_1}^{b_1}(f) \le V_a^b(f).
\end{equation}
This is because every partition of $[a_1, b_1]$ can be extended to a
partition of $[a, b]$. In particular, $f$ is bounded on $[a, b]$ when
it has bounded variation.
A partition $\mathcal{P}'$ of $[a, b]$ is said to be a
\emph{refinement} of a partition $\mathcal{P}$ of $[a, b]$ if
$\mathcal{P}'$ contains all of the terms in $\mathcal{P}$.
In this case, one can check that
\begin{equation}
V_\mathcal{P}(f) \le V_{\mathcal{P}'}(f)
\end{equation}
for every $f : [a, b] \to {\bf R}$, using the triangle inequality.
Also, any finite collection of partitions of $[a, b]$ has a common
refinement.
Suppose that $f$ has bounded variation on $[a, b]$, and that
$x \in (a, b)$. Thus the restrictions of $f$ to $[a, x]$ and to $[x,
b]$ have bounded variation, and moreover
\begin{equation}
V_a^x(f) + V_x^b(f) = V_a^b(f).
\end{equation}
Indeed, any partitions $\mathcal{P}_1$, $\mathcal{P}_2$ of $[a, x]$,
$[x, b]$, respectively, can be combined to get a partition
$\mathcal{P}_3$ of $[a, b]$ for which
\begin{equation}
V_{\mathcal{P}_1}(f) + V_{\mathcal{P}_2}(f) = V_{\mathcal{P}_3}(f),
\end{equation}
which implies that $V_a^x(f) + V_x^b(f) \le V_a^b(f)$. To get the
opposite inequality, note that every partition of $[a, b]$ can be
refined if necessary to contain $x$, and hence to be a combination of
partitions of $[a, x]$ and $[x, b]$. The same argument shows that $f$
has bounded variation on $[a, b]$ if it has bounded variation on $[a,
x]$ and on $[x, b]$.
Suppose that $f$ is continuously differentiable on $[a, b]$.
If $a \le r \le t \le b$, then
\begin{equation}
|f(t) - f(r)| = \biggl|\int_r^t f'(\xi) \, d\xi\biggr|
\le \int_r^t |f'(\xi)| \, d\xi.
\end{equation}
This implies that
\begin{equation}
V_\mathcal{P}(f) \le \int_a^b |f'(\xi)| \, d\xi
\end{equation}
for every partition $\mathcal{P}$ of $[a, b]$. One can show that
\begin{equation}
V_a^b(f) = \int_a^b |f'(\xi)| \, d\xi,
\end{equation}
using very fine partitions $\mathcal{P}$ of $[a, b]$.
For each $r \in {\bf R}$, put $r_+ = r$ when $r \ge 0$ and
$r_+ = 0$ when $r \le 0$, and $r_- = -r$ when $r \le 0$ and $r-_ = 0$
when $r \ge 0$, so that
\begin{equation}
r_+ - r_- = r, \quad r_+ + r_- = |r|.
\end{equation}
Given $f : [a, b] \to {\bf R}$ and a partition $\mathcal{P} =
\{t_j\}_{j = 0}^n$ of $[a, b]$, put
\begin{equation}
P_\mathcal{P}(f) = \sum_{j = 1}^n (f(t_j) - f(t_{j - 1}))_+
\end{equation}
and
\begin{equation}
N_\mathcal{P}(f) = \sum_{j = 1}^n (f(t_j) - f(t_{j - 1}))_-.
\end{equation}
Thus
\begin{equation}
P_\mathcal{P}(f) + N_\mathcal{P}(f) = V_\mathcal{P}(f)
\end{equation}
and
\begin{equation}
P_\mathcal{P}(f) - N_\mathcal{P}(f) = f(b) - f(a).
\end{equation}
Suppose that $f$ has bounded variation on $[a, b]$, and put
\begin{equation}
P_a^b(f) = \sup \{P_\mathcal{P}(f) : \mathcal{P}
\hbox{ is a partition of } [a, b]\}
\end{equation}
and
\begin{equation}
N_a^b(f) = \sup \{N_\mathcal{P}(f) : \mathcal{P}
\hbox{ is a partition of } [a, b]\}.
\end{equation}
One can check that
\begin{equation}
P_a^b(f) + N_a^b(f) = V_a^b(f)
\end{equation}
and
\begin{equation}
P_a^b(f) - N_a^b(f) = f(b) - f(a).
\end{equation}
Similarly,
\begin{equation}
P_a^x(f) - N_a^x(f) = f(x) - f(a)
\end{equation}
when $a \le x \le b$. This implies that $f$ can be expressed as the
difference of two monotone increasing functions on $[a, b]$, since
$P_a^x(f)$, $N_a^x(f)$ are monotone increasing in $x$.
Functions of bounded variation do not have to be continuous,
but they can only have jump discontinuities. More precisely, if $f$
has bounded variation on $[a, b]$, then $f$ has one-sided limits from
both sides at every point in $(a, b)$, and from the right and left
sides at $a$, $b$, respectively. This follows from the analogous
statement for monotone functions and the fact that a function of
bounded variation can be expressed in terms of monotone functions, and
it can also be shown more directly.
\section{Lengths of paths}
\label{lengths of paths}
\setcounter{equation}{0}
Let $(M, d(x, y))$ be a metric space, let $a$, $b$ be real
numbers with $a \le b$, and let $f$ be a function on $[a, b]$ with
values in $M$. For each partition $\mathcal{P} = \{t_j\}_{j = 0}^n$
of $M$, consider
\begin{equation}
\Lambda_\mathcal{P}(f) = \sum_{j = 1}^n d(f(t_j), f(t_{j - 1})).
\end{equation}
This is the same as the variation $V_\mathcal{P}(f)$ of $f$ on
$\mathcal{P}$ when $M$ is the real line with the standard metric.
If there is an upper bound for $\Lambda_\mathcal{P}$ over all
partitions $\mathcal{P}$ of $[a, b]$, then we say that the path $f :
[a, b] \to M$ has finite length, and the length of the path is defined by
\begin{equation}
\Lambda_a^b(f) = \sup \{\Lambda_\mathcal{P} : \mathcal{P}
\hbox{ is a partition of } [a, b]\}.
\end{equation}
This is the same as the total variation $V_a^b(f)$ of $f$ when $M =
{\bf R}$. As in the previous case, $\Lambda_a^b(f) = 0$ if and only
if $f$ is constant. If $a \le r \le t \le b$, then
\begin{equation}
d(f(r), f(t)) \le \Lambda_\mathcal{P}(f)
\end{equation}
for any partition $\mathcal{P}$ of $[a, b]$ that contains $r$, $t$.
Hence $f([a, b])$ is a bounded set in $M$ when $f : [a, b] \to M$ has
finite length, with
\begin{equation}
\mathop{\rm diam} f([a, b]) \le \Lambda_a^b(f).
\end{equation}
Of course, $f([a, b])$ is a compact set in $M$ when $f$ is continuous,
and therefore bounded. If $f$ is continuous, then $f([a, b])$ is also
a connected set in $M$.
If $f : [a, b] \to M$ is $C$-Lipschitz, then $f$ has finite
length, and
\begin{equation}
\Lambda_a^b(f) \le C \, (b - a).
\end{equation}
Let $(\widetilde{M}, \widetilde{d}(u, v))$ be another metric space,
and suppose that $\phi : M \to \widetilde{M}$ is $C$-Lipschitz. For
any $f : [a, b] \to M$ and partition $\mathcal{P}$ of $[a, b]$,
\begin{equation}
\widetilde{\Lambda}_\mathcal{P}(\phi \circ f) \le C \, \Lambda_\mathcal{P}(f),
\end{equation}
where $\widetilde{\Lambda}$ is the analogous quantity for $\widetilde{M}$.
If $f : [a, b] \to M$ has finite length, then $\phi \circ f : [a, b]
\to \widetilde{M}$ does too, and
\begin{equation}
\widetilde{\Lambda}_a^b(\phi \circ f) \le C \, \Lambda_a^b(f).
\end{equation}
In particular, if $f$ has finite length and $\phi : M \to {\bf R}$ is
Lipschitz, then $\phi \circ f$ has bounded variation.
If $f : [a, b] \to M$ has finite length and $a \le a_1 \le b_1
\le b$, then the restriction of $f$ to $[a_1, b_1]$ has finite length, and
\begin{equation}
\Lambda_{a_1}^{b_1}(f) \le \Lambda_a^b(f).
\end{equation}
If $\mathcal{P}$, $\mathcal{P}'$ are partitions of $[a, b]$ and
$\mathcal{P}'$ is a refinement of $\mathcal{P}$, then
\begin{equation}
\Lambda_\mathcal{P}(f) \le \Lambda_{\mathcal{P}'}(f)
\end{equation}
for any $f : [a, b] \to M$, as in the case of real-valued functions in
the previous section. As before, one can use this to show that
\begin{equation}
\Lambda_a^x(f) + \Lambda_x^b(f) = \Lambda_a^b(f)
\end{equation}
for every $x \in (a, b)$ when $f$ has finite length. If $a \le x <
b$, then
\begin{equation}
\lim_{y \to x+} \Lambda_a^y(f)
\end{equation}
exists, because $\Lambda_a^x(f)$ is monotone increasing in $x$, and hence
\begin{equation}
\lim_{y \to x+} \sup \{\Lambda_w^y(f) : x < w \le y\} = 0.
\end{equation}
This implies that
\begin{equation}
\lim_{y \to x+} \mathop{\rm diam} f((x, y]) = 0,
\end{equation}
since
\begin{eqnarray}
\mathop{\rm diam} f((x, y]) & = & \sup \{\mathop{\rm diam} f([w, y]) : x < w \le y\} \\
& \le & \sup \{\Lambda_w^y(f) : x < w \le y\}. \nonumber
\end{eqnarray}
If $M$ is complete, then it follows that $f$ has a limit from the
right at $x$, and similarly there is a limit from the left when $a < x
\le b$.
Suppose now that $M$ is ${\bf R}^n$, equipped with a norm $N$,
and thus the metric $d_N(x, y)$ associated to $N$ too. If $f_1, f_2 :
[a, b] \to {\bf R}^n$ have finite length and $r_1, r_2 \in {\bf R}$,
then $r_1 \, f_1 + r_2 \, f_2$ has finite length, and
\begin{equation}
\Lambda_a^b(r_1 \, f_1 + r_2 \, f_2)
\le |r_1| \, \Lambda_a^b(f_1) + |r_2| \, \Lambda_a^b(f_2).
\end{equation}
This is similar to the case of real-valued functions, and one can also
treat the product of a real-valued function and an ${\bf R}^n$-valued
function on $[a, b]$ in the same way as before. If $f : [a, b] \to
{\bf R}^n$ is continuously differentiable, then one can show that $f$
has finite length and that
\begin{equation}
\Lambda_a^b(f) = \int_a^b N(f'(\xi)) \, d\xi,
\end{equation}
in practically the same way as before. It can be interesting to
consider integral norms
\begin{equation}
\Big(\int_a^b N(f'(\xi))^p \, d\xi\Big)^{1/p}
\end{equation}
as well, $1 \le p < \infty$. The $p = \infty$ case corresponds to the
maximum of $N(f')$ on $[a, b]$. This integral norm is especially
interesting when $p = 2$ and $N$ is the standard Euclidean norm on
${\bf R}^n$. For other $p$, there is some simplification when $N(v) =
\|v\|_p$. If $N(v) = \|v\|_1$, then the length of any path of finite
length in ${\bf R}^n$ is equal to the sum of the total variations of
the coordinates of the path. This uses the fact that any finite
collection of partitions of $[a, b]$ has a common refinement, so that
independent partitions for the coordinate functions are equivalent to
using the same partition for the whole path.
\section{Snowflake metrics}
\label{snowflakes}
\setcounter{equation}{0}
Let $\alpha$ be a positive real number, with $\alpha < 1$.
For any pair of nonnegative real numbers $u$, $v$,
\begin{equation}
\label{u^alpha, v^alpha}
(u + v)^\alpha \le u^\alpha + v^\alpha.
\end{equation}
To see this, observe that
\begin{equation}
\max(u, v) \le (u^\alpha + v^\alpha)^{1/\alpha},
\end{equation}
and hence
\begin{equation}
u + v \le \max(u, v)^{1 - \alpha} \, (u^\alpha + v^\alpha)
\le (u^\alpha + v^\alpha)^{1/\alpha}.
\end{equation}
Note that the inequality is strict in (\ref{u^alpha, v^alpha}) when $u, v > 0$.
If $(M, d(x, y))$ is a metric space, then it follows from
(\ref{u^alpha, v^alpha}) that $d(x, y)^\alpha$ is also a metric on
$M$. This does not change the topology of $M$, but it does change the
geometry. Many familiar examples of snowflake curves in the plane
have approximately this type of geometry, for instance.
Suppose that $f : [a, b] \to M$ is a continuous path with
finite length with respect to $d(x, y)^\alpha$. This means that
\begin{equation}
\sum_{j = 1}^n d(f(t_j), f(t_{j - 1}))^\alpha \le A
\end{equation}
for some $A \ge 0$ and every partition $\{t_j\}_{j = 0}^n$ of $[a, b]$.
Let $\epsilon > 0$ be given. By continuity and compactness, $f$
is uniformly continuous, and so there is a $\delta > 0$ such that
\begin{equation}
d(f(r), f(w)) < \epsilon
\end{equation}
for every $r, w \in [a, b]$ such that $|r - w| < \delta$. Hence
\begin{equation}
\sum_{j = 1}^n d(f(t_j), f(t_{j - 1})) \le \epsilon^{1 - \alpha} \, A
\end{equation}
when $t_j - t_{j - 1} < \delta$ for each $j = 1, \ldots, n$. Every
partition of $[a, b]$ has a refinement with this property, which
implies that the length $\Lambda_a^b(f)$ of $f$ with respect to $d(x,
y)$ satisfies
\begin{equation}
\Lambda_a^b(f) \le \epsilon^{1 - \alpha} \, A.
\end{equation}
Thus $\Lambda_a^b(f) = 0$, since $\epsilon > 0$ is arbitrary, and $f$
must be constant.
\section{H\"older continuity}
\label{holder}
\setcounter{equation}{0}
Let $(M_1, d_1(x, y))$ and $(M_2, d_2(w, z))$ be metric
spaces. A mapping $f : M_1 \to M_2$ is said to be \emph{H\"older
continuous} of order $\alpha$, $0 < \alpha < 1$, if
\begin{equation}
d_2(f(x), f(y)) \le C \, d_1(x, y)^\alpha
\end{equation}
for some $C \ge 0$ and every $x, y \in M_1$. One might also say that
$f$ is Lipschitz of order $\alpha$ in this case, but it will be
convenient to refer to this as a Lipschitz condition when $\alpha = 1$
and H\"older continuity when $0 < \alpha < 1$. Similar names are
sometimes used for other related conditions as well.
As in the previous section, $d_1(x, y)^\alpha$ is a metric on
$M_1$, and therefore $f$ is H\"older continuous of order $\alpha$ with
respect to $d_1(x, y)$ if and only if $f$ is Lipschitz with respect to
$d_1(x, y)^\alpha$. Thus many basic properties of H\"older continuous
mappings follow from the corresponding statements for Lipschitz
mappings. In particular,
\begin{equation}
f_p(x) = d_1(p, x)^\alpha
\end{equation}
is a real-valued H\"older continuous function of order $\alpha$ on
$M_1$ with $C = 1$ for each $p \in M_1$.
Let $(M, d(x, y))$ be a metric space, and consider the case of
a continuous path $f : [a, b] \to M$. If $f$ is $C$-Lipschitz, then
$f$ is H\"older continuous of order $\alpha$ for each $\alpha \in (0,
1)$ with constant $C \, (b - a)^{1 - \alpha}$. Of course, $f$ also
has finite length $\le C \, (b - a)$ when $f$ is $C$-Lipschitz.
However, continuous paths of finite length need not be H\"older
continuous of any positive order, and there are couterexamples already
for monotone increasing real-valued functions. Similarly, H\"older
continuous paths may not have finite length.
Suppose that $f : [a, b] \to M$ is H\"older continuous of
order $\alpha$ with constant $C > 0$. For each $\rho > 0$, $f([a,
b])$ is contained in the union of $O(\rho^{-1/\alpha})$ subsets of $M$
with diameter $\le \rho$, because $[a, b]$ is the union of
$O(\rho^{-1/\alpha})$ subintervals of length $\le
(\rho/C)^{1/\alpha}$. This implies that the Minkowski dimension of
$f([a, b])$ is $\le 1/\alpha$, and hence the Hausdorff dimension of
$f([a, b])$ is $\le 1/\alpha$ too. This is an analogue for H\"older
continuous paths of the fact that Lipschitz paths have finite length.
\section{Coverings}
\label{coverings}
\setcounter{equation}{0}
If $[a, b], [a_1, b_1], \ldots, [a_n, b_n]$ are closed
intervals in the real line such that
\begin{equation}
[a, b] \subseteq \bigcup_{j = 1}^n [a_j, b_j],
\end{equation}
then
\begin{equation}
b - a \le \sum_{j = 1}^n (b_j - a_j).
\end{equation}
More generally, if $E_1, \ldots, E_n$ are bounded subsets of ${\bf R}$
such that
\begin{equation}
[a, b] \subseteq \bigcup_{j = 1}^n E_j,
\end{equation}
then
\begin{equation}
b - a \le \sum_{j = 1}^n \mathop{\rm diam} E_j.
\end{equation}
Indeed, each $E_j$ is contained in a closed interval of the same diameter.
Let $(M, d(x, y))$ be a metric space, and let $A, E_1, \ldots, E_n$
be bounded subsets of $M$ such that
\begin{equation}
A \subseteq \bigcup_{j = 1}^n E_j.
\end{equation}
If $A$ is connected, then
\begin{equation}
\mathop{\rm diam} A \le \sum_{j = 1}^n \mathop{\rm diam} E_j.
\end{equation}
To see this, remember first that continuous mappings send connected
sets to connected sets. If $f : M \to {\bf R}$ is continuous, then
$f(A)$ is an interval in the real line, which may be open, or closed,
or half-open and half-closed. At any rate,
\begin{equation}
{\mathop{\rm diam}}_{\bf R} f(A) \le \sum_{j = 1}^n {\mathop{\rm diam}}_{\bf R} f(E_j),
\end{equation}
where the subscripts indicate that these are diameters in ${\bf R}$.
If $f$ is $C$-Lipschitz, then
\begin{equation}
{\mathop{\rm diam}}_{\bf R} f(A) \le C \, \sum_{j = 1}^n \mathop{\rm diam} E_j.
\end{equation}
The desired estimate follows by applying this to $1$-Lipschitz
functions of the form $f_p(x) = d(p, x)$, $p \in A$.
The hypothesis that $A$ be connected is essential here. If
$A$ is a finite set with at least two elements, then $\mathop{\rm diam} A > 0$,
but $A$ is contained in the union of finitely many sets with one
element and thus diameter $0$. Cantor's middle-thirds set in the real
line has diameter equal to $1$, and is contained in the union of
$2^\ell$ intervals of length $3^{-\ell}$ for each $\ell \ge 1$. A
compact set $A \subseteq {\bf R}$ has Lebesgue measure $0$ exactly if
for each $\epsilon > 0$ there are finitely many bounded sets $E_1,
\ldots, E_n \subseteq {\bf R}$ such that $A \subseteq \bigcup_{j =
1}^n E_j$ and $\sum_{j = 1}^n \mathop{\rm diam} E_j < \epsilon$.
In particular, if $A \subseteq M$ is a bounded connected set
and $\rho > 0$, then $A$ is not covered by fewer than $\mathop{\rm diam} A / \rho$
bounded subsets of $M$ of diameter $\le \rho$. Depending on the
situation, many more of these subsets may be required. If $M$ is
${\bf R}^n$ equipped with the standard metric, for example, then a
bounded set $A$ can be covered by $O(\rho^{-n})$ sets of diameter $\le
\rho$. One needs at least a positive multiple of $\rho^{-n}$ such
sets when $A$ has nonempty interior, because otherwise the
$n$-dimensional volume of $A$ would be too small.
\section{Domains in ${\bf R}^n$}
\label{domains, R^n}
\setcounter{equation}{0}
A set $U$ in a metric space $M$ is an open set if for every $p
\in U$ there is an $r > 0$ such that $B(p, r) \subseteq U$. Any norm
$N$ on ${\bf R}^n$ determines the same open sets as the standard
metric. This is because $N$ is less than or equal to a constant times
the standard norm, and conversely the standard norm is less than or
equal to a constant times $N$. The first statement can be checked
directly by expressing any element of ${\bf R}^n$ as a linear
combination of the standard basis for ${\bf R}^n$ and using the
triangle inequality. As mentioned previously, this and the triangle
inequality imply that $N$ is continuous with respect to the standard
norm. Hence the minimum of $N$ is attained on the standard unit
sphere in ${\bf R}^n$, since the latter is compact. The standard norm
times the minimum of $N$ on the unit sphere is less than or equal to
$N$ on all of ${\bf R}^n$, by homogeneity, which implies the second
statement. For explicit norms like $\|w\|_p$, $1 \le p \le \infty$,
the comparison with the standard norm can be verified directly.
Suppose that $U$ is a connected open set in ${\bf R}^n$, which
is to say that $U$ is not the union of two disjoint nonempty open
sets. It is well known that $U$ is then pathwise-connected, so that
for every $p, q \in U$ there is a continuous mapping $f : [a, b] \to
U$ such that $f(a) = p$ and $f(b) = q$. More precisely, one can even
take $f$ to be piecewise-affine on $[a, b]$. In particular, $f$ then
has finite length.
However, it is not clear how small the length of $f$ can be.
Of course, the length of $f$ is at least the distance between $p$ and
$q$. If $U$ is convex, then one can take $f$ to be affine, and the
length of $f$ is equal to the distance between $p$ and $q$.
Otherwise, the length of $f$ may have to be quite large compared to
the distance between $p$ and $q$. It is easy to give examples where
this happens in the plane. For instance, there may be elements of $U$
on opposite sides of the boundary locally. The boundary of $U$ might
also be complicated, with spirals or other obstacles.
Even if the boundary of $U$ is complicated, it may be that $U$
behaves well in terms of lengths of paths. For example, if $U$ is the
region in the plane bounded by the von Koch snowflake, then every pair
of elements of $U$ can be connected by a path of length bounded by a
constant multiple of the distance between them. The main idea is for
the path to avoid the boundary as much as possible, without going too
far away. There can also be relatively small parts of the boundary
that only cause minor detours for paths in the domain.
\section{Lipschitz graphs}
\label{lipschitz graphs}
\setcounter{equation}{0}
Let $k, l, n$ be positive integers such that $k + l = n$, and
let us identify ${\bf R}^n$ with ${\bf R}^k \times {\bf R}^l$, so that
an element $x$ of ${\bf R}^n$ may be expressed as $(x', x'')$, where
$x' \in {\bf R}^k$ and $x'' \in {\bf R}^l$. Also let $A : {\bf R}^k
\to {\bf R}^l$ be a continuous mapping, and consider its graph
\begin{equation}
\{(x', x'') \in {\bf R}^n : x'' = A(x')\}.
\end{equation}
This is a nice $k$-dimensional topological submanifold of ${\bf R}^n$.
If $k = n - 1$, then this hypersurface has two complementary
components $U_+$, $U_-$ consisting of $(x', x'') \in {\bf R}^n$ such
that $x'' > A(x')$ and $x'' < A(x')$, respectively. If $k < n$, then
the complement of the graph in ${\bf R}^n$ is connected. For any $k$,
\begin{equation}
(x', x'') \mapsto (x', x'' + A(x'))
\end{equation}
defines a homeomorphism on ${\bf R}^n$ that sends the $k$-plane $x'' =
0$ to the graph of $A$. If $f(t)$ is a continuous path in ${\bf R}^k$
parameterized by an interval $[a, b]$, then $\widehat{f}(t) = (f(t),
A(f(t))$ is a continuous path in the graph of $A$. The graph of $A$
is itself a curve in ${\bf R}^n$ when $k = 1$.
Suppose that $A$ is Lipschitz. If $f$ has finite length, then
$\widehat{f}$ does too, and the length of $\widehat{f}$ is bounded by
a constant multiple of the length of $f$. If the Lipschitz constant
of $A$ is small, then this constant multiple is close to $1$.
Using affine paths in ${\bf R}^k$, we get that every pair of elements
of the graph of $A$ can be connected by a continuous path in the graph
of $A$ of finite length bounded by a constant multiple of the distance
between them, where the constant multiple is close to $1$ when $A$
has small Lipschitz constant.
A $k$-dimensional $C^1$ submanifold of ${\bf R}^n$ is locally
the same as the graph of a continuously-differentiable mapping on
${\bf R}^k$ with respect to a suitable choice of coordinate axes. By
rotating the axes so that ${\bf R}^k$ is parallel to the tangent plane
of the submanifold at a particular point, the submanifold can be
represented near the point as the graph of a function with small
Lipschitz constant. The Lipschitz constant tends to $0$ as one
approaches the point in question. Thus distances on $C^1$
submanifolds are approximately the same as the infimum of lengths of
paths on the submanifold locally.
\section{Real analysis}
\label{real analysis}
\setcounter{equation}{0}
For the sake of simplicity, we have so far avoided referring
to Lebesgue integrals and measure. However, this more sophisticated
theory can be quite convenient in the present context. Let us mention
some of the key points.
A basic fact is that a monotone real-valued function on an
open interval in the real line is differentiable ``almost
everywhere'', which is to say on the complement of a set of Lebesgue
measure $0$. Thus additional hypotheses of differentiability are
sometimes superfluous. Unfortunately, even continuous monotone
functions cannot necessarily be recoved from their almost everywhere
derivative, as in the fundamental theorem of calculus, without an
extra condition of ``absolute continuity''. Indeed, there are
examples of nonconstant continuous monotone increasing functions with
derivative equal to $0$ almost everywhere.
It follows that a real-valued function of bounded variation on
an interval in the real line is differentiable almost everywhere,
since it can be expressed as the difference of two monotone increasing
functions. In particular, a real-valued Lipschitz function on an
interval is differentiable almost everywhere. Lipschitz functions are
absolutely continuous, and so there is a version of the fundamental
theorem of calculus for them. As corollaries of this fact, a
Lipschitz function $f$ on an interval is constant if $f'(x) = 0$
almost everywhere, $f$ is monotone increasing if $f'(x) \ge 0$ almost
everywhere, and $f$ is $C$-Lipschitz if $|f'(x)| \le C$ almost
everywhere.
At the same time, bounded variation and Lipschitz conditions
have natural extensions involving metric spaces, as we have seen. The
composition of a path of finite length in a metric space with a
real-valued Lipschitz function on the metric space is a function of
bounded variation, which is Lipschitz when the path is. There are
also a lot of real-valued Lipschitz functions on any metric space.
Even on ${\bf R}^n$, there are a lot of nice functions that are
Lipschitz and not continuously differentiable, such as the distance to
a point or to a set.
|
2,869,038,155,631 | arxiv | \section{Results}
\begin{figure}[htbp]
\centerline{\scalebox{.4}{\includegraphics{hancockdynfig2_red.eps}}}
\caption{(Color online) The real part of the optical conductivity of YbInCu${_{4} }$\ at various temperatures.}
\label{fig:x0}
\end{figure}
Figure \ref{fig:x0} shows the frequency-dependent reflectivity and infrared conductivity $\sigma_{1}(\omega)$\ of YbInCu${_{4} }$\ at temperatures below and above the $T_v\simeq42K$ phase transition temperature. At high temperature (250\hspace{.05cm}$K$), $\sigma_{1}(\omega)$\ consists of a narrow free-carrier (Drude-like) contribution clearly seen in the far-infrared spectrum, due to the presence of mobile carriers. This interpretation is consistent with Hall\cite{figueroa98} and resistivity measurements\cite{sarrao2}. $\omega\simeq6000cm^{-1}$ marks a clear onset of a set of strong interband transitions extending upward into the visible spectral range.
At the lowest temperature (20\hspace{.05cm}$K$), a significant decrease in conductivity in the interband region ($\omega>4000\hspace{.05cm} cm^{-1}$) accompanies a substantial increase in the conductivity below 4000\hspace{.05cm}${cm^{-1} }$\ in the form of a well-defined peak centered around 2000\hspace{.05cm}${cm^{-1} }$. The development of this peak is highly correlated with the phase transition (${T_{v} }$$=$42\hspace{.05cm}$K$) and is a prominent and essential feature of the low-$T$ phase of YbInCu${_{4} }$. The connection between this peak and the physics of hybridization is an important theme of the present work.
\begin{figure}[htbp]
\centerline{\scalebox{.4}{\includegraphics{hancockdynfig3_red.eps}}}
\caption{(Color online) The real part of the optical conductivity of YbIn${_{.7}}$Ag${_{.3}}$Cu${_{4} }$\ at various temperatures.}
\label{fig:x3}
\end{figure}
\begin{figure}[htbp]
\centerline{\scalebox{.4}{\includegraphics{hancockdynfig4_red.eps}}}
\caption{(Color online) The real part of the optical conductivity of YbAgCu${_{4} }$\ at various temperatures.}
\label{fig:x1}
\end{figure}
Figure \ref{fig:x3} shows optical data for the $x$=0.3 system. Trends similar to those in low temperature YbInCu${_{4} }$\ are apparent, with a depletion of weight around 8000\hspace{.05cm}${cm^{-1} }$, accommodated by a replenishing at lower frequencies. Vestiges of the 2000${cm^{-1} }$\ peak persist at temperatures as high as 300\hspace{.05cm}$K$, an energy scale much higher than the extrapolated value of the phase transition temperature for this composition ${T_{v} }$(\twiddle100$K$), but still lower than the Kondo scale appropriate to the low temperature phase ${T_{K} }$\twiddle360$K$.
Figure \ref{fig:x1} shows the similar plots for YbAgCu${_{4} }$. The behavior displayed by the conductivity shows behavior more typical of a heavy fermion system, with moderate temperature dependence and spectral weight which is approximately conserved below 0.5\hspace{.05cm}$eV$. There is a continuous evolution of the 2000\hspace{.05cm}${cm^{-1} }$\ feature, activated by crossing the valence transition at low $x$, and the more familiar phenomenology found in the low frequency response of the heavy fermi system YbAgCu${_{4} }$.
Figure \ref{fig:allx} profiles the $x$\ dependence of $\sigma_{1}(\omega)$\ at low temperatures. The 2,000\hspace{.05cm}${cm^{-1} }$\ peak in YbInCu${_{4} }$\ undergoes a complex shifting behavior as $x$\ is increased, blueshifting slightly when $x$=0.3, then redshifting upon further doping, reaching a minimum peak frequency when $x$=0.75, before blueshifting again as $x$\ continues to 1. The strength of this feature is also influenced by $x$\ in a nontrivial way, discussed further below.
At higher frequency, the large hump feature centered on 11,000\hspace{.05cm}${cm^{-1} }$\ in YbInCu${_{4} }$\ monotonically blueshifts and decreases in overall strength as $x$\ is increased. Further, inflection points around 6000\hspace{.05cm}${cm^{-1} }$\ and 9000\hspace{.05cm}${cm^{-1} }$\ redshift slightly upon doping to the $x$=0.3 and $x$=0.5 systems. These inflection points are not discernable for $x$=0.75, but reappear at low frequency (3000\hspace{.05cm}${cm^{-1} }$\ and 5500\hspace{.05cm}${cm^{-1} }$) in the $x$=1 system.
The $x$-dependent profiling of the low temperature conductivity is an important part of our experimental results, allowing clear identification of systematic changes of the low temperature electrodynamics as a function of an external control parameter. The discussion begins with identification of systematic trends in the 2000\hspace{.05cm}${cm^{-1} }$\ feature followed by a discussion of interband features in a later section.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{hancockdynfig5_red.eps}
\caption{(Color online) Infrared conductivity $\sigma_{1}(\omega)$\ versus $\omega$ at 20\hspace{.05cm}$K$\ for the five $x$\ values studied. Also shown are the components of a Lorentzian-Drude fit as described in the text. The dashed vertical lines are a guide to the eye and correspond to the center frequencies of the fit components when $x$=0.}
\label{fig:allx}
\end{center}
\end{figure}
\section{Results, $x$\ dependence}
\label{sec:lo}
\begin{figure}
\centerline{\scalebox{.5}{\includegraphics{hancockdynfig6_red.eps}}}
\caption{(Color online) (a) The characteristic frequencies, ${\omega_{pk}}$ (triangles) and ${\omega_{th}}$ (boxes), and (b) the Kondo temperature as a function of $x$. (c) The spectral weights, ${n(4000\hspace{.05cm} cm^{-1})}$ (boxes) and ${n(6000\hspace{.05cm} cm^{-1})}$ (circles) and the results of a combined Drude/Lorentz fitting (triangles) of the low frequency conductivity. (d) The low temperature susceptibility (open circles) and Sommerfeld coefficient (solid circles) as a function of $x$. ${T_{K} }$\ in (b) is related to $\chi(0)$ in (d). Spectral weights are computed assuming a band mass of 4$m_e$. }\label{fig:ntw}
\end{figure}
In this section, we focus on the $x$\ dependent changes of the 2000\hspace{.05cm}${cm^{-1} }$\ feature in order to explore the significance of this behavior to the correlated electron physics of hybridization and the periodic Anderson model. First, we quantify the $x$\ dependent trends of Figure \ref{fig:allx} and draw important conclusions through the comparison to previously published thermodynamic data. The inferred relationships are then explored below in sections \ref{sec:kg} and \ref{sec:pam} where we consider an interpretation based on the periodic Anderson model.
Figure \ref{fig:ntw}a shows the frequency of the 2000\hspace{.05cm}${cm^{-1} }$\ feature versus $x$\ as determined in two ways. The black triangles mark the frequency of the peak in $\sigma_{1}(\omega)$\ (also marked in Figure \ref{fig:allx}). Alternatively, a threshold frequency can be extracted from a fit of the conductivity to a calculation based on the low energy dispersion of the periodic Anderson model (PAM), discussed below in Section \ref{sec:kg}.
In addition to examining the frequency of the peak as a function of $x$, we can also look at the strength of the 2000\hspace{.05cm}${cm^{-1} }$\ feature. We quantify this characteristic through the spectral weight, defined as the integrated intensity of $\sigma_{1}(\omega)$\ over a low frequency interval:
\begin{equation} n(\omega)=\frac{2 m}{\pi
e^{2}}\int_{0^{+}}^{\omega}\sigma_1(\omega^{\prime})\mathrm{d}\omega^{\prime}
\label{eqn3:sw} \end{equation}
where $m$ represents a bare band mass. The lower limit is chosen to be nonzero ($0^+=50\hspace{.05cm} cm ^{-1}$) in order to exclude from the strength estimate the comparatively minute contribution of the free carrier (Drude) response. The upper limit of integration is chosen to encompass the 2000\hspace{.05cm}${cm^{-1} }$\ peak without including the $x$-dependence of the high frequency interband contributions. Neither integration limit is critical; in fact a lower limit of 0 and upper limits between anywhere between 3000\hspace{.05cm}${cm^{-1} }$ and 8000\hspace{.05cm}${cm^{-1} }$\ produce similar $x$\ dependence. $n(4000\hspace{.05cm} cm^{-1})$ and $n(6000\hspace{.05cm} cm^{-1})$ are shown in Figure \ref{fig:ntw}c.
As an alternative to this simple integral calculation of the strength, we can fit the complex conductivity ($\sigma=\sigma_1+i\sigma_2$) with a sum of Lorentzian and Drude response functions\cite{dressel,wooten}:
\begin{equation}
\sigma(\omega)=\sum_j\frac{\omega_{P,j}^2}{4\pi}\frac{\omega}{i(\omega_{j}^2-\omega^2)+\omega\Gamma_j}.
\label{eq:lo}
\end{equation}
The constituents of the fits include: one narrow Drude (D1, $\Gamma\sim10-40\hspace{.05cm} cm^{-1}$) contribution to represent the free carrier peak, a wide Drude (D2, $\Gamma>800\hspace{.05cm} cm^{-1}$), and a Lorentz oscillator (L1) in the vicinity of the 2000\hspace{.05cm}${cm^{-1} }$\ feature which, as we discuss below, relate to the Kondo resonance. There are in addition two Lorentz oscillators (L2 and L3, around 7,300\hspace{.05cm}${cm^{-1} }$\ and 11,000\hspace{.05cm}${cm^{-1} }$\ for $x$=0) to represent the infrared interband conductivity, and two wide Lorentz oscillators at ultraviolet frequencies ($\omega>30,000\hspace{.05cm} cm^{-1}$) representing the conductivity in that range. These fit components are labeled in Figure \ref{fig:allx}a.
Previous work\cite{tahvildar} has explicitly demonstrated that the calculated lineshape of the optical signature of the Kondo resonance is intrinsically non-Lorentzian, and furthermore demonstrated the viability of fits which combine Drude and Lorentz terms to represent the Kondo resonance. The combination of the contributions D2 and L1 reasonably fit the conductivity in the range of the 2000\hspace{.05cm}${cm^{-1} }$\ peak. The strength from that combination is shown by the triangles in Figure \ref{fig:ntw}c. This determination of the strength exhibits an $x$\ dependence similar to the simpler integral representations of the strength. This makes one confident that the ${n}$ versus $x$\ dependence shown here is an essential characteristic of the data, and independent of any of the detailed choices we have made in the analysis. $n$ determined by the methods discussed here is presented in Figure \ref{fig:ntw}c.
It is interesting to compare the $x$-dependent trends inferred from previously published thermodynamic measurements with those from our optical data. Figure \ref{fig:ntw}b shows the Kondo temperature of low-$T$ phase YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$\ as deduced by Cornelius\cite{cornelius97} \textit{et al}\ from fitting the measured magnetic susceptibility to the numerically calculated result of the $j=\frac{7}{2}$ Coqblin-Schreiffer model\cite{cornelius97,rajan}. Figure \ref{fig:ntw}d shows that the trend in the $x$\ dependence of $\chi(T=0)$\footnote{In the $N$-fold degenerate Anderson impurity model, the value of the low temperature susceptibility is related\cite{hewson} simply to the Kondo temperature ${T_{K} }$\ by $\chi(0)=(g\mu_B)^2j(j+1)w_N/3k_BT_K$, where $w_N$ is given by Equation \ref{eqn:wn}.} generally agrees with the corresponding trend in the Sommerfeld coefficient $\gamma$, implying a Wilson ratio within 10\%\ of the value ($\mathcal{R}=\frac{8}{7}$) expected for a $j=\frac{7}{2}$ Anderson impurity\cite{hewson,cornelius97}. Thus the ${T_{K} }$\ values inferred from the susceptibility analysis reasonably represent the effective energy scale relevant to the onset of strong-coupling Kondo physics within the low temperature phase.
The complicated $x$\ dependence of ${T_{K} }$\ is not understood, and may be the result of an interplay of band structure, chemical pressure, screening, disorder, and other many-body effects. While resolving the detailed cause of this complex $x$\ dependence presents a subject for future work, out focus here is the relationship between the $x$\ dependence of ${T_{K} }$\ and that of the optical data; the similar form of the $x$\ dependent electrodynamic (Figures \ref{fig:ntw}a and \ref{fig:ntw}c) and thermodynamic quantities (Figures \ref{fig:ntw}b and \ref{fig:ntw}d) indicates a common source. We show below that this is rooted in the strongly correlated electron physics of hybridization.
\section{Modeling, PAM Dispersion}
\label{sec:pam}
\begin{figure} \centerline{ \scalebox{.4}{\includegraphics{hancockdynfig7_red.eps}}}
\caption{(Color online) The PAM dispersion relations. Vertical arrows indicate possible optical transitions. The horizontal dashed lines represent $E_{F}$\ and $\tilde{\epsilon_f}$. The light diagonal line represents the unrenormalized dispersion of the conduction carriers.}
\label{fig:qpd}
\end{figure}
We can make progress toward eliciting the relationships suggested in Figures \ref{fig:ntw} by examining an interpretation of the 2000\hspace{.05cm}${cm^{-1} }$\ feature in the context of the periodic Anderson model (PAM). Complementary to the work of other authors\cite{tahvildar,jarrell95,vidhyadhiraja03,georges96,zlatic01,freericks03}, which focus the rigorous techniques of many-body theory directly toward the underlying Hamiltonian, we will use a simplified approach based on effective low energy (near-$E_{F}$) PAM dispersion relations\cite{edwards,hewson}. These dispersion relations ($\epsilon^+$ and $\epsilon^-$ in Figure \ref{fig:qpd}) provide the basis for a simple unifying picture in which much of the low energy phenomenology of heavy fermion materials can be viewed, including the mass enhancement, aspects of magnetism, and transport measurements\cite{millis87a,tannous}. In Figure \ref{fig:qpd}, the vertical extent of the plot is of order 1\hspace{.05cm}$eV$\ and the (singly occupied) ${f}$-electron level is below the bottom plot boundary. The light dashed lines indicate the bare (unhybridized) conduction electron dispersion.
In a system with no hybridization, the conduction electrons are the dominant influence on the transport properties such as thermopower and resistivity, while at the same time provide a temperature-independent Pauli-paramagnetic contribution to the magnetic susceptibility. The ${f}$-electrons, on the other hand, are localized and as a result contribute very little to the transport properties, but play a major role in magnetism, contributing a Curie $1/T$ term to the susceptibility. Inclusion of the hybridization and on-site Coulomb repulsion terms complicates this independent particle picture considerably and the new eigenstates become nontrivial admixtures of the states of pure ${f}$\ and conduction electron character.
The PAM quasiparticle dispersion relations provide a venue through which to explore the effects of nonzero hybridization on the phase space of excitations. At energies far from the chemical potential, the upper and lower bands, ${\epsilon^{+}}$ and ${\epsilon^{-}}$, follow closely the unrenormalized free carrier dispersion. There the main effect of the interaction and hybridization is to provide a channel for relaxation of the conduction states, \textit{i.e.} a broadening of the spectral function along the dispersion curves. At lower energies, these relaxation effects are reduced by the phase-space-constraining presence of a filled Fermi sea, however, renormalization also opens the Fermi surface ($k_F^{(bare)}\rightarrow k_F$ and the bands flatten) to accommodate the ${f}$-electron weight projected up to the Fermi level. This reorganization of the bands in the vicinity of the Fermi level is due to many-body interactions, the strength of which is characterized by the parameter ${\tilde{V}}$. The resultant narrow peak in the density of states, called the Kondo, or Abrikosov-Suhl, resonance\footnote{Strictly speaking, there appears a peak in the $DOS$ in the impurity case. The periodic system differs in that an additional \textit{indirect} gap within this peak is present\cite{georges96}.} is central to the understanding of heavy fermion and mixed-valent phenomenology\cite{degiorgi99,hewson, jarrell95}.
From the point of view of optical probes, a key effect of renormalization on the optical response is to create the possibility for vertical transitions from filled states below $E_{F}$, across a \textit{direct} gap, and into unoccupied levels above $E_{F}$, as illustrated by the vertical arrows of Figure \ref{fig:qpd}. The threshold for these transitions (short arrow) occurs at a frequency ${\omega=2\tilde{V}}$, where ${\tilde{V}}$ is the hybridization strength renormalized by the on-site ${f}$-electron repulsion. This energy scales with the Kondo temperature as\cite{coleman, coxcomm,millis87a,millis87b,grewe84}
\begin{equation}\tilde{V}=\sqrt{T_{K} B}\label{eq:tkb}\end{equation}
where ${B}$ is a model parameter related to the conduction electron bandwidth.
At threshold, the nesting condition for the upper and lower bands is met ($\nabla_{\textbf{k}}\epsilon^+=\nabla_{\textbf{k}}\epsilon^-$), leading to a very high joint density of states for vertical quasiparticle transitions and hence a strong peak in the conductivity.
At frequencies larger than the threshold frequency there are two distinct
contributions to the conductivity: one originating from levels
inside the unrenormalized Fermi surface (${k_{<}}$ below); the other from the states
occupied as a result of renormalization, \textit{i.e.}, outside the unrenormalized
Fermi surface (${k_{>}}$ below). An example of two such transitions with the same frequency $\omega$ are indicated by dashed arrows in Figure \ref{fig:qpd}. Transitions involving both of these sets of quasiparticles are important and must be counted independently in the determination of the total optical conductivity, as discussed further below.
In addition to the threshold frequency, another, higher frequency scale appears which is relevant to the electrodynamic response. This higher frequency scale corresponds to the vertical transition (long arrow, Figure \ref{fig:qpd}) which occurs from states \textit{on} the Fermi surface (\textit{i.e.} the locus of points which divides the set of occupied and unoccupied \textbf{k} states). Vertical transitions involving higher \textbf{k} states cannot occur because both initial and final states are unoccupied when $k>k_F$, and hence one expects a drop in the conductivity at this frequency. For a linearly dispersing conduction band, direct calculation reveals that the frequency of the last allowed transition is equal to
\begin{equation}
\label{eqn:OFS}
\Omega_{FS}=\frac{\tilde{V}^{2}+\tilde{\epsilon_{f}}^{2}}{\tilde{\epsilon_{f}}}.
\end{equation}
The identification of $\tilde{\epsilon_{f}}$ with ${T_{K} }$\ (discussed further below), together with Equation \ref{eq:tkb} implies that this scale is of order the conduction electron bandwidth, $B$. For high energy transitions, band edge final states can be reached and the linear approximation to the conduction band is likely to become poor. Quasiparticle transitions in this frequency range may be influenced by the details of the underlying band structure.
\section{Modeling, Kubo-Greenwood Analysis}
\label{sec:kg}
With these considerations of the phase space for optical transitions in mind, we can proceed with an analysis using the Kubo-Greenwood formula\cite{dressel}:
\begin{equation}
\sigma_{1}(\omega) = \frac{\pi e^{2}}{m^{2}\omega}
\sum_{\ell,\ell^{\prime}}JDOS_{\ell,\ell^{\prime}}(\omega)|\textbf{p}_{\ell,\ell^{\prime}}|^{2}
\label{eq:conduct1}
\end{equation}
where ${|\textbf{p}_{\ell,\ell^{\prime}}|}$ denotes the dipole matrix element
connecting electronic bands ${\ell}$ and ${\ell^{\prime}}$, and
${JDOS_{\ell,\ell^{\prime}}(\omega)}$ is the corresponding joint density of states. Applying this formula to hybridizing quasiparticles (as though they were electrons) allows an exploration of the phenomena of the mid-infrared conductivity in the context of the PAM. In that case, the two relevant bands are $\epsilon^{+}$ and $\epsilon^{-}$, which in this model approach leads to:
\begin{equation}
\sigma_{pam}(\omega) = \frac{e^{2}}{4 \pi^{2} m^{2}\omega}
\int_{\Delta\epsilon=\omega}\frac{dS}{|\nabla_\textbf{k}(\epsilon^{+}-\epsilon^{-})|}|\textbf{p}_{+,-}|^{2}
\label{eq:conduct2}
\end{equation}
where ${|\textbf{p}_{+,-}|}$ is the matrix element for transitions between the two bands under consideration.
In the case of a spherical Fermi surface, the integrand is constant and equation (\ref{eq:conduct2}) simplifies to
\begin{equation}
\sigma_{pam}(\omega) = \frac{e^{2}}{4 \pi^{2} m^{2}\omega}
\sum_{k'=k_< ,k_>}\frac{4 \pi k^{2}|\textbf{p}_{+,-}|^{2}}{|\partial_{k}(\epsilon^{+}-\epsilon^{-})|}\Bigg{|}_{k=k'}
\label{eq:conduct3}
\end{equation}
It is useful to consider the case of constant optical matrix elements ${|\textbf{p}_{+,-}|}$, meaning that every filled \textbf{k} state transits to the corresponding upper band state with equal probability. One can now obtain a model lineshape from Equation \ref{eq:conduct3} using the explicit PAM dispersion relations\cite{grewe84,hewson,millis87a}
\begin{equation}
\label{eqn:epm}
\epsilon^{\pm}=\frac{E_{F}+\tilde{\epsilon_{f}}+\epsilon_{\textbf{k}}\pm \sqrt{(E_{F}+\tilde{\epsilon_{f}}-\epsilon_{\textbf{k}})^{2}+4 \tilde{V}^{2}} }{2},
\end{equation}
where $E_F$ is the Fermi level, $\tilde{V}$ is the renormalized hybridization strength and ${\tilde{\epsilon_{f}}}$ is the ${f}$-level position renormalized by on-site ${f}$-electron repulsion. This latter quantity defines the scale of the low-energy physics and is commonly identified with the impurity Kondo temperature, ${T_{K} }$\ (discussed further below).
Putting Equation \ref{eqn:epm} into \ref{eq:conduct3}, the conductivity takes the form:
\begin{equation}
\sigma_{pam}(\omega) = \frac{2 e^2|\textbf{p}_{+,-}|^{2}}{m^2 |\nabla \epsilon_{k_F}|^3\pi}\frac{(E_F+\tilde{\epsilon_f})^2+\omega^2-4\tilde{V}^2}{\sqrt{\omega^2-4\tilde{V}^2}}
\label{eq:conduct4}
\end{equation}
for $2\tilde{V}<\omega<\frac{\tilde{V}^{2}+\tilde{\epsilon_{f}}^{2}}{\tilde{\epsilon_{f}}}$ and
\begin{equation}
\sigma_{pam}(\omega) = \frac{e^2|\textbf{p}_{+,-}|^{2}}{m^2 |\nabla \epsilon_{k_F}|^3\pi}\frac{(E_F+\tilde{\epsilon_f}-\sqrt{\omega^2-4\tilde{V}^2})^2}{\sqrt{\omega^2-4\tilde{V}^2}}
\label{eq:conduct5}
\end{equation}
for $\omega>\frac{\tilde{V}^{2}+\tilde{\epsilon_{f}}^{2}}{\tilde{\epsilon_{f}}}$. Figure \ref{fig:fitplot}a shows this lineshape for two sets of $\tilde{V}$ and $\tilde{\epsilon_{f}}$ values.
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{hancockdynfig8_red.eps}
\caption{(Color online) (a) The idealized conductivity (Equations \ref{eq:conduct4} and \ref{eq:conduct5}) for $\tilde{V}=93\hspace{.05cm} meV$ and $\tilde{\epsilon_f}=9\hspace{.05cm} meV$ (solid) and $\tilde{V}=40\hspace{.05cm} meV$ and $\tilde{\epsilon_f}=2.5\hspace{.05cm} meV$ (dashed). Inset contrasts the lineshapes derived with constant matrix elements $|\textbf{p}_{+,-}|$ (Equations \ref{eq:conduct4} and \ref{eq:conduct5}), and those with the coherence factors discussed in the text.
(b) The same idealized conductivity curves as (a), Lorenztian broadened with widths $\Delta_0=0.16\hspace{.05cm} eV$ and $\Delta_{0.75}=0.125\hspace{.05cm} eV$ for comparison to the measured conductivity of YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$\ with $x$=0 and $x$=0.75, respectively.}
\label{fig:fitplot}
\end{center}
\end{figure}
The rather idealized lineshape generated by these considerations is extremely sharp and a meaningful comparison with the data requires addressing the effects of relaxation, which were thus far neglected in our treatment. To this end, we convolute this idealized lineshape with a Lorentzian function, keeping the half width $\Delta$ as an adjustable parameter when performing fits to the measured conductivity. Examples of fits produced using this procedure are shown in Figure \ref{fig:fitplot}b.
The broadening parameter $\Delta$ addresses the finite width of the spectral function along the dispersion curves of Figure \ref{fig:qpd}, and therefore provides a measure of the statistical time over which a typical quasiparticle decays. The numerical values for $\Delta$ obtained from our fits range from 0.12\hspace{.05cm}$eV$\ to 0.17\hspace{.05cm}$eV$, with an associated time scale for decay in the range $\tau=\hbar/\Delta$=5.4\hspace{.05cm}$ps$ to 3.8\hspace{.05cm}$ps$, respectively. These lifetimes estimates are in good agreement with the quasiparticle lifetime of members of this class of materials (YbAgCu${_{4} }$), as measured directly by Demsar \textit{et al}\ in pump-probe experiments\cite{demsar} of electron-hole relaxation lifetime. The extraction of this parameter from the conductivity data is a meaningful consistency check on the method developed here.
We now take a moment to consider the possible influence of \textbf{k}-dependent matrix elements associated with the composite nature of the hybridized quasiparticles on the electrodynamic response.
The hybridizing quasiparticles are composite admixtures of excitations with both ${f}$\ and conduction electron character. The regions of the dispersion which are flatter correspond to quasiparticles with a large amplitude of ${f}$\ admixture whereas regions which follow more closely the bare conduction dispersion are dominated by conduction character. Generally speaking transitions among the bare states are not all equally probable (\textit{i.e.} $|\textbf{p}_{cc}|\neq|\textbf{p}_{cf}|\neq|\textbf{p}_{ff}|$). Thus one expects that the optical transition rate for quasiparticles may exhibit some dependence on \textbf{k}, which goes beyond our earlier assumptions.
One can obtain a relatively simple model with \textbf{k} dependence by assuming transitions among bare states only occur between conduction electron initial and final states ($|\textbf{p}_{cc}|\neq0, |\textbf{p}_{cf}|=|\textbf{p}_{ff}|=0$), and using that to calculate transition rates between the hybridized quasiparticle bands. To model the non-constant admixture of states, we use the coherence factors of the resonant level model\cite{tannous,coxcomm} (aka Fano-Anderson\cite{mahan}),
\begin{eqnarray}
\label{eqn:uv}
u_{\textbf{k},\sigma}=\frac{1}{\sqrt{1+(\frac{\tilde{V}}{\tilde{\epsilon_f}-\epsilon^+})^2}},&
v_{\textbf{k},\sigma}=\frac{1}{\sqrt{1+(\frac{\tilde{V}}{\tilde{\epsilon_f}-\epsilon^-})^2}}.
\end{eqnarray}
The approach follows as before however the $JDOS$ integral (Eqn. \ref{eq:conduct2}) picks up a factor $u_{\textbf{k},\sigma}^2v_{\textbf{k},\sigma}^2=\tilde{V}^2/\omega^2$ associated with these coherence factors. One thus obtains a model conductivity similar to Equations \ref{eq:conduct4} and \ref{eq:conduct5}, but with $|\textbf{p}_{+,-}|^2$ replaced by $|\textbf{p}_{cc}|^2\tilde{V}^2/\omega^2$. The most significant effect of the inclusion of coherence factors is that the conductivity at high frequency should fall to zero much more quickly than in the constant matrix element case\cite{coxcomm}. The line shapes with and without coherence factors are contrasted in the inset of Figure \ref{fig:fitplot}.
This approach may be too nuanced because in a general mixed valent system, bare ${f}$-to-conduction state transitions can be appreciable, and in fact are expected to be important in YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$. This is because the conduction band states are derived primarily from Cu-In-Ag $p$ and $d$ orbitals\cite{antonov}, whereas the ${f}$\ electrons sit on Yb sites. This physical displacement between the underlying orbital states is manifest in the banded states through nonvanishing dipole transition matrix elements, $|\textbf{p}_{fc}|$\cite{coxcomm}. Thus, in YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$, transitions involving the flatter portions of the quasiparticle dispersion, which are dominated by ${f}$-like character, \textit{can} be expected to provide a considerable contribution to the optical strength, thus we feel that Equations \ref{eq:conduct4} and \ref{eq:conduct5} are more applicable to YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$.
\section{Discussion, ${T_{K} }$\ Scaling}
\label{sec:fs}
\begin{figure}
\centerline{\scalebox{.5}{\includegraphics{hancockdynfig9_red.eps}}}
\caption{(Color online) Plot of ${\omega_{pk}}$ and ${\omega_{th}}$ versus ${\sqrt{T_{K}}}$. The dotted lines represent Eq. (\ref{eq:tkb}) and the slopes are ${1.6\hspace{.05cm}eV^{\frac{1}{2}}}$ and ${1.1\hspace{.05cm}eV^{\frac{1}{2}}}$.} \label{fig:freqscale}
\end{figure}
We now analyze the $x$\ dependent frequency of the 2000\hspace{.05cm}${cm^{-1} }$\ feature (Figure \ref{fig:ntw}a), and its relationship to the Kondo temperature ${T_{K} }$\ (Figure \ref{fig:ntw}b). Figure \ref{fig:freqscale} shows $\omega_{pk}$, the maximum of the conductivity, and the threshold frequency $\omega_{th}$, determined from the fit described above, plotted versus the square root of the Kondo temperature. The complex $x$\ dependence of $\omega_{th}$ and $\omega_{pk}$ (Figure \ref{fig:ntw}) simplifies considerably when we plot these quantities as a function of ${T_{K} }$\ (Figure \ref{fig:freqscale}). The emergence of a functional relationship between these quantities implies that the frequency of the 2000\hspace{.05cm}${cm^{-1} }$\ peak is controlled by the same physics that underlies the thermodynamic behavior. The square root dependence is evidence that hybridization physics plays a dominant role.
With that in mind, the modeling developed in the previous section can be used to extract an estimate for the band parameter $B$ (Equation \ref{eq:tkb}) by associating the threshold frequency, $\omega_{th}$, with its PAM value, $2\tilde{V}$. The slope of the line through the $\omega_{th}$ values in Figure \ref{fig:freqscale}, together with Equation \ref{eq:tkb}, directly gives $B=0.30\hspace{.05cm} eV$. This value of $B$ reflects the rate at which the frequency increases with ${T_{K} }$. This can be compared with expectations based on density of states, as well as the rate at which the strength decreases with ${T_{K} }$, as discussed below.
\begin{figure}
\begin{center}
\includegraphics[width=3.2in]{hancockdynfig10_red.eps}
\caption{Scaling relations for the strength of the 2000\hspace{.05cm}${cm^{-1} }$\ feature with ${T_{K} }$. Dark circles represent $n(4000\hspace{.05cm} cm^{-1})$ and open circles represent $n(6000\hspace{.05cm} cm^{-1})$
(a) shows the result of fitting the measured dependence with Equation \ref{eqn:ntknocf2}, and (b) shows the same data fit with Equation \ref{eqn:ntkcf}.}
\label{fig:ntk}
\end{center}
\end{figure}
Within the approach developed in Section \ref{sec:kg} the relationship between ${T_{K} }$\ and the strength of the 2000\hspace{.05cm}${cm^{-1} }$\ feature ($n$) can be addressed. We can obtain a closed-form result from our model calculation if we set the two sphere areas in (\ref{eq:conduct3}) equal to ${4\pi k_{F}^{2}}$. This approximation\footnote{In regards to the ${T_{K} }$\ dependence, this approximation is equivalent to assuming that $E_{F}$\ is the largest energy scale in the problem.} avoids the effects of bare band structure details while including the influence of the strong cusp at ${2\tilde{V}}$, which originates from the many-body physics of the PAM. Solving for the wavevectors ${k_{>}}$ and ${k_{<}}$ using the condition ${\epsilon^{+}-\epsilon^{-}=\omega}$, and substituting the result into \ref{eq:conduct3}, we obtain a model strength:
\begin{eqnarray}
n_{pam} & = & \frac{2m}{\pi e^2}\int_{2\tilde{V}}^{\Omega_{FS}}\sigma_{pam}(\omega) d\omega \\
& \simeq & \frac{4|\textbf{p}_{+,-}|^{2} k_{F}}{\pi^2}\ln\Big(\frac{\tilde{V}}{\tilde{\epsilon_{f}}}\Big)
\label{eqn:ntknocf}.
\end{eqnarray}
This is essentially the area under the curves of Figure \ref{fig:fitplot}a. Using Equation \ref{eq:tkb} and introducing the parameter $c$, defined by $T_K=c\hspace{.05cm} \tilde{\epsilon_f}$, we can express $n_{pam}$ in terms of ${T_{K} }$:
\begin{eqnarray}
n_{pam} & \simeq & \frac{4|\textbf{p}_{+,-}|^{2} k_{F}}{\pi^2}\ln\Big(c\sqrt{\frac{B}{T_{K}}}\Big)\label{eqn:ntknocf2}.
\end{eqnarray}
Figure \ref{fig:ntk}a shows a least squares fit of this logarithmic scaling relationship to the data.
The factor $c$ relates the renormalized ${f}$\ level position at low energies and the Kondo temperature, ${T_{K} }$. The value of $c$ is unambiguous in the Fermi liquid theory of the $N(=2j+1)$-fold degenerate Anderson impurity model where,
\begin{equation}
\label{ }
\tilde{\epsilon_f}=\frac{T_K}{c}=k_BT_L\frac{N^2\sin(\pi/N)\cos(\pi/N)}{\pi(N-1)}
\end{equation}
and\footnote{$T_L$ here is a common alternative definition of the Kondo temperature\cite{rajan,cornelius97} and is related to the Kondo temperature $T_K$ by\cite{hewson} Equation \ref{eqn:tktl}.}
\begin{equation}
\label{eqn:tktl}
T_K=w_NT_L.
\end{equation}
$w_N$ is the generalized Wilson number\cite{hewson} given by
\begin{equation}
\label{eqn:wn}
w_N=\frac{e^{1+C-\frac{3}{2N}}}{2\pi\Gamma(1+\frac{1}{N})}.
\end{equation}
For the $j=\frac{7}{2}$ moment of Yb, $N=8$ and $c\simeq0.66$.
Using this value of $c\simeq0.66$ and fitting the measured result to Equation \ref{eqn:ntknocf} gives $B=0.35\hspace{.05cm} eV$ and 0.45\hspace{.05cm}$eV$\ for $n(4000\hspace{.05cm} cm^{-1})$ and $n(6000\hspace{.05cm} cm^{-1})$ (Eq. 1), respectively. We consider these to be a reasonable agreement given the simplicity of our approach. These values of $B$, which are determined by the amount that $n$ goes up when ${T_{K} }$\ goes down, are similar in size to our previous estimate of the parameter $B=0.30\hspace{.05cm} eV$ determined from the amount that $\omega_{th}$ goes down as ${T_{K} }$\ goes down. Therefore, in addition to the agreement of experiment and theory regarding the \textit{direction} of the ${T_{K} }$\ dependence of $n$ and $\omega_{th}$, the \textit{sensitivity} of the dependence of $n$ and $\omega_{th}$ on ${T_{K} }$\ are in reasonable agreement \textit{with each other}. In addition, the numerical values for $B$ are reasonable bandwidths for the InAgCu $d$-orbital derived band states\cite{ibach,antonov} of YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$. Thus we conclude that the observed dependences of both $n$ and $\omega_{th}$ on ${T_{K} }$\ is consistent with the predictions of the periodic Anderson model in magnitude as well as direction.
An alternative strength estimate can be made using the coherence factor model of the conductivity, introduced in detail above. In that case, we replace $|\textbf{p}_{+,-}|^2$ by $|\textbf{p}_{cc}|^2\tilde{V}^2/\omega^2$ in Equation \ref{eq:conduct2} and repeat the steps which produced Equation \ref{eqn:ntknocf2}, giving a theoretical strength
\begin{eqnarray}
n_{pam} & = & \frac{|\textbf{p}_{cc}|^{2} k_{F}}{\pi^2}\frac{c^2B-T_K}{c^2B+T_K}.
\label{eqn:ntkcf}
\end{eqnarray}
Again using $c=0.66$, fits to the data would yield $B$ estimates $0.17\hspace{.05cm} eV$ and 0.19\hspace{.05cm}$eV$, for $n(4000\hspace{.05cm} cm^{-1})$ and $n(6000\hspace{.05cm} cm^{-1})$, respectively. These values are in order-of-magnitude agreement with the band parameter estimates made above. This functional form is fit for comparison to the data in Figure \ref{fig:ntk}b. However, as discussed at the end of section \ref{sec:kg}, the ${f}$-to-conduction electron transitions are made allowed in this system and so the calculation which includes these, represented by Equation \ref{eqn:ntknocf} is probably more appropriate for YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$.
\section{Discussion, High Frequency Interband Transitions}
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{hancockdynfig11_red.eps}
\caption{(Color online) (a) The center frequencies of the transitions L2 and L3 from (Figure 5), and (b) the strengths of the corresponding Lorentzian fit components are shown as a function of doping, $x$. A band-structure picture which we use to interpret the trends in (a) and (b) is shown for (c) YbInCu${_{4} }$\ and (d) YbAgCu${_{4} }$. The dashed lines illustrate \textbf{k}-conserving transitions which we associate with L1, L2 and L3,
respectively. (L1 is the Kondo resonance excitation.)}
\label{fig:ib}
\end{center}
\end{figure}
We now consider the $x$-dependent trends in the high frequency ($\omega>6000\hspace{.05cm} cm^{-1}$) conductivity. We will use the language introduced in Figure \ref{fig:allx} (Section \ref{sec:lo}) associated with Lorentzian fitting of the conductivity data, focussing our attention on the features L2 and L3. In Figures \ref{fig:ib}a and \ref{fig:ib}b we show the center frequency and strength of these high energy excitations as a function of $x$ . This measured $x$-dependence can inform
our understanding of the nature of the underlying states associated with these transitions and the density of
states near $E_{F}$ .
In a textbook picture of metals and semiconductors, an important effect of doping is to add or remove electrons from a set of band states, thereby influencing the position of $E_{F}$. In YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$, increasing $x$\ from 0 to 1 corresponds to the net removal of 2 electrons (per formula unit) from the system, hence we expect the Fermi level to move downward in energy as $x$\ is increased. Relativistic band structure calculations addressing these changes have been carried out by Antonov \textit{et al}\cite{antonov}. In this scenario, optical features involving transitions from filled states just below $E_{F}$\ tend to weaken and move upward as $x$\ is increased and these states are emptied\footnote{Similarly, optical features involving final states just above $E_{F}$\ may strengthen as $x$\ is increased and new final states become available.}. The dashed arrow of Figure \ref{fig:ib}c illustrates one such transition, which we identify with L3. In this scenario, the width of the feature when $x$=0 suggests bandwidths of order \twiddle1\hspace{.05cm}$eV$, and the threshold for these transitions, approximately 7000\hspace{.05cm}${cm^{-1} }$\ ($\sim$ 0.9\hspace{.05cm}$eV$) when $x$=0, gives an indication of the overall energy position relative to $E_{F}$. Furthermore, the amount of shift with doping implies that the density of states in the near-$E_{F}$\ region of the band structure is approximately 2/0.58\hspace{.05cm}$eV$\twiddle3.5$e^{-}eV^{-1}$/f.u. Both of these numbers are quite reasonable for conduction bands in rare earth and transition metal systems\cite{ibach}.
With this rigid band interpretation as a backdrop describing the salient relationships between L3 and $x$, the correlated electron effects discussed in previous sections occur in addition. The renormalization discussed there dresses these bare states and the renormalized region of the band structure, including the Kondo resonance, tracks $E_{F}$, which moves downward with increasing $x$.
We now turn to the systematics of the feature L2. This feature redshifts with $x$\ by an amount similar in magnitude to the shift of L3 (Figure \ref{fig:ib}a). However, the narrowness and nearly $x$-independent strength of L2 does not lend as easily to a simple band interpretation. When considering an identification of the component L2, we point out the significant temperature dependence in the frequency region associated with L2, as shown before in Figures \ref{fig:x0}, \ref{fig:x3} and \ref{fig:x1}. The temperature dependent interplay of spectral weight contained in the frequency intervals of L1 and L2 naturally lead one to speculate that perhaps the same correlated electron physics controlling L1 may also be relevant to L2. One is thus led toward the question of whether the presence of the feature L2 represents a further phenomenon associated with hybridization physics.
We noted in Section \ref{sec:pam} that in addition to the strong peak in the PAM conductivity associated with renormalized band nesting, a second feature could appear at higher frequencies associated with the initial state energy crossing $E_{F}$. In a linearly dispersing band model, this change occurs around the conduction electron bandwidth frequency (Equation \ref{eqn:OFS}). In a more realistic bandstructure, the filling fraction and band curvature details could influence the frequency and magnitude of this conductivity change. In particular, if the transition final states are band edge states, then significant shifting with doping concentration could result.
With these considerations, it is reasonable to suggest that L2 represents a conductivity feature arising from Fermi surface quasiparticles. The narrowness can then be attributed to the very long lifetime of the initial state (because it occurs on the Fermi surface), and weak $x$\ dependence of the strength arises because the energy structures responsible for the associated transitions track the $x$\ dependent Fermi level, as opposed to becoming filled or depleted with $x$. The culmination of the identifications suggested in this section are presented at a light and heavy doping concentration in Figures \ref{fig:ib}c and \ref{fig:ib}d, respectively.
\section{Discussion, The Phase transition of Y\lowercase{b}I\lowercase{n}C\lowercase{u}$_4$}
The physical mechanism of the phase transition in the lightly doped system remains elusive, however, progress has been made in understanding aspects of the phase transition.
In previous sections, we have analyzed in detail the low temperature conductivity in terms of PAM renormalizations of few-band models. We now discuss the how this picture identifying optical transitions may be related to the interesting phenomenology displayed by YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$\ including the phase transition at the low $x$.
The Kondo volume collapse\cite{allen1} (KVC) model, which describes the complex interplay of the system volume, hybridization, Kondo temperature, and ${f}$\ level occupation, seems adequate to describe the valence transition in elemental Ce\cite{allen1,haule}. This model, however, seems insufficient to quantitatively describe the phase transition in YbInCu${_{4} }$, as evidenced mainly by the smallness of the volume change at the transition. Some authors\cite{sarrao3,cornelius97,figueroa98} have argued that a quasigap, or region of low density of states, exists in the bare band structure located just above the Fermi level in YbInCu${_{4} }$, and is important to the phase transition. This description is qualitative, but addresses the change in carrier density as well as the changes related to Kondo physics.
In this approach, the presence of a quasigap makes the Kondo temperature very sensitive to the placement of the Fermi level and can help induce the phase transition in manner akin to the KVC model, but with the importance placed on the density of states dependence, rather than the volume.
It is of interest to ask what consequences a quasigap scenario could have for the optical data in YbInCu${_{4} }$, where the conductivity in the range of 8000\hspace{.05cm}${cm^{-1} }$\ (L2 in the fits) drastically displaces to lower frequency forming the Kondo resonance excitation (L1) discussed above. In the picture outlined in Figure \ref{fig:ib}c, an upward shift in the Fermi level off of the bare band edge and into the quasigap could have drastic consequences for the optical response, possibly forcing the contribution L2 to shift to very high frequency, as the renormalized portion of the upper band is forced across the gap in response to a small shift in $E_{F}$. This possible interpretation of the temperature dependence indicates that further theoretical work directed toward investigating the generic physics of the PAM and the fate of the Kondo scenario in the context of rapidly varying band structure is needed to understand the complex temperature dependence of YbInCu${_{4} }$.
Theoretical work on the Falicov-Kimball model\cite{zlatic01,freericks03,fkrmp} attempts to make a quantitative connection with experiment using a particular many-body model which features a first-order phase transition. Aspects of this modeling seem promising, in particular the prediction of substantial temperature dependence of the high frequency optical conductivity, however this model does not yet explicitly include the effects of hybridization. It is possible that a minimal model that describes the first order phase transition and also includes the Kondo physics may require and extension of the periodic Anderson model to include Falicov-Kimball-type interaction.
\section{Conclusions}
Our results indicate that ${T_{K} }$\ scaling is present in the low temperature finite-frequency dynamics of YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$\ and can be addressed in the context of local-moment models. Furthermore, our data provide numerical estimates of key parameters necessary for the construction of a minimal theoretical model of the valence transition as well as pointing out salient features of the underlying bandstructure. Further work may be directed toward greater understanding of this low-$T$ scaling behavior as well as unexplained temperature dependent behavior of lightly doped YbIn${_{1-x}}$Ag${_{x}}$Cu${_{4} }$.
\begin{acknowledgments}
The authors have greatly benefitted from discussions with D. N. Basov, A. L. Cornelius, D. L. Cox, P. A. Lee, B. S. Shastry, and A. P. Young. We also gratefully acknowledge S. L. Hoobler and Y. W. Rodriguez for technical assistance. Work at UCSC supported by NSF Grant Number DMR-0071949. ZF acknowledges support of NSF Grant Number DMR-0203214.
\end{acknowledgments}
|
2,869,038,155,632 | arxiv | \section{Introduction}
Many chemical reactions occur in a medium, e.g., in a solvent.
Explicitly taking this environment into account in super-molecular calculations dramatically increases
the computational cost for most quantum mechanical methods.
However, as most chemical properties are local a so-called \textit{focused} model\cite{born1920,kirkwood1934,onsager1936,kirkwood1938} can be applied.
In such a model, a pre-defined region, denoted the quantum mechanical (QM) region, is described by
an accurate electronic structure method, whereas interactions with the remaining system are described through an effective operator.
Focused models have
the additional advantage that the interpretation of the results obtained is kept simple due to the explicit system--environment separation.
In focused models,
the most widespread approximation for the environment is a dielectric continuum, where the environment is characterized solely
by a dielectric constant.\cite{tomasi2005}
Accordingly, continuum models can capture bulk effects of a solvent, but are less meaningful for structured environments
with specific interactions such as hydrogen bonding.
A more general approach is to consider a discrete environment where all molecules are modeled
explicitly.
One realization of explicit models relies on a classical description of the environment,
representing it by point charges. The most widely applied models employ partial charges derived from a
molecular force field, as done in the quantum-mechanics molecular mechanics (QM/MM) hybrid models\cite{warshel1976,senn2009}.
These schemes usually
allow only the electron density of the QM system to be polarized.
Whereas the QM system will also polarize the environment, this is normally neglected in QM/MM calculations.
Various methods that include the environment polarization have been proposed (see, for instance, Refs.\ \citenum{slip10,defu11}).
However, some of them turn out to be computationally more demanding compared to their purely electrostatic counterparts.
This is somewhat surprising as the polarizable methods are in fact closer to the spirit of the first QM/MM scheme by Warshel and Levitt\cite{warshel1976}.
One option, that has been explored within Density Functional Theory (DFT), is a full quantum mechanical description with frozen parts in the
environment through an effective potential. This is known as Frozen Density Embedding
(FDE)\cite{senatore1986,cortona1991,wesolowski1993}.
In FDE, polarization can be included by an iterative procedure known as freeze-and-thaw cycles\cite{wesolowski1996}
in which the role of system and environment is interchanged for the subsystems until convergence is reached.
The FDE method is not restricted to DFT and
the embedded subsystem can also be described by a wave function
which is usually denoted WFT-in-DFT\cite{govind1998,govind1999,manby2012,hoefner2013,csaba2013,csaba2014,dresselhaus2015,hoefner2016}.
WFT-in-WFT embedding schemes have also been investigated, e.g.,~by Chan and co-workers\cite{knizia2012,knizia2013,wout16},
by Scuseria and co-workers\cite{scus14,scus14b}, and by Fromager and co-workers\cite{fromager2015,from16}.
\textit{Polarizable Embedding}\cite{soderhjelm2009,olsen2010,olsen2010b,sneskov2011,schwabe2011,list2013,hedegaard2013a,hedegaard2014,hedegaard2015a} (PE)
relies on a different strategy, namely on atom-centered polarizabilities to represents the fragments in the environment.
Contrary to other methods of this kind\cite{thompson1995,thompson1996,gao1997,jensen2003a,jensen2003b,yu2005},
PE employs both high-order multipoles and polarizabilities obtained from QM calculations, which makes the
PE potential easily customized to a specific environment.
For locally excited states embedded in an environment, most studies apply some variant of time-dependent DFT which is based on response theory.
Here, we instead aim at a state-specific \textit{ab initio} approach. The state-specific approach
features aspects that are more appealing than methods based on response theory.
This was shown\cite{corni2005,cammi2005} by comparing the
two approaches to the analytical solution for a four-state model (in a continuum solvation model). The two methods were shown
to have different formal expressions for excitation energies under inclusion of a polarizable environment. In particular, the linear response
approach fails to account for differences of the dipole moments in ground and excited states \cite{corni2005}.
Yet, state-specific approaches have been dormant in this field for a long time. The interest was recently
revived in studies on solvated systems with the FDE scheme coupled to a
Quantum Monte Carlo wave function description for embedded system.\cite{csaba2013,csaba2014}
In state-specific excited state optimizations, the treatment of static correlation effects is of high importance
as excited states typically display large static correlation effects.
One of the most capable methods to recover static correlation is the density matrix
renormalization group (DMRG) algorithm\cite{white1992b,white1993,ors_springer,chan2008a,marti2010b,chan2011,marti2011,wouters2014rev,kurashige2014,szalay2015,yanai2015,garnet16}.
Our group recently reported
the implementation of an efficient second-generation DMRG program\cite{keller2014,dolfi2014,keller2015,knecht2016,keller2016} which
relies entirely on matrix product operators.
In this paper, we describe the coupling of this DMRG program with a PE scheme.
This approach supplements our recently described coupling of DMRG
to the FDE scheme\cite{dresselhaus2015}.
As an active-space method, DMRG relies on the selection of a proper orbital space from frontier molecular orbitals \cite{stein16a,stein16b}.
As a consequence, dynamical electron correlation must be considered either {\it a posteriori},
for instance by perturbation theory\cite{yana11,kurashige2014-cupt2,sharm14,soko16,guo16,knecht2016,roem16,wout16b} (diagonalize-and-then-perturb\cite{shavitt2002}),
or {\it a priori} from the outset (in a perturb-and-then-diagonalize approach\cite{shavitt2002}). We recently
investigated the latter option for DMRG\cite{hedegaard2015b} by employing a range-separated Hamiltonian\cite{savinbook} that recovers dynamical correlation
through DFT by short-range (sr) density functionals (DMRG--srDFT)
in close analogy to the MCSCF--srDFT ansatz\cite{fromager2007,fromager2009,stoyanova2013}.
The long-range part is then described by a DMRG wave function \textit{ansatz}.
Herein, we also report the extension of our PE-DMRG approach to PE-DMRG--srDFT.
This paper is organized as follows: In Section \ref{theory} we outline DMRG and PE theories as well as the
required extensions for PE-DMRG to accommodate short-range functionals.
Implementational aspects are described in Section \ref{implementation_details} and computational details in Section \ref{compmet}.
We then proceed with applications of PE-DMRG and PE-DMRG-srDFT in Section \ref{results} that focus on photoexcitation of water
and the retinalidyne chromophore in the channelrhodopsin protein.
Finally, conclusions and an outlook are given in Section \ref{conclusion}.
\section{Theory} \label{theory}
\subsection{The Complete-Active-Space Method} \label{CAS}
In this paper, we generally work in Hartree atomic units unless otherwise noted.
We focus first on an isolated system in state $\alpha$ with an energy $E^{\rm iso}_{\alpha}$ defined as
\begin{equation}
E^{\rm iso}_{\alpha} = \langle \Psi_{\alpha} \vert \hat{H}^{\rm iso}\vert \Psi_{\alpha} \rangle = \sum_{pq}h_{pq}D^{\alpha}_{pq}
+ \frac{1}{2}\sum_{pqrs}g_{pqrs}P^{\alpha}_{pqrs} + {V}_{\text{nn}},
\label{E_iso}
\end{equation}
where $V_{\text{nn}}$ is the nuclear repulsion potential energy and $ h_{pq} $ and $g_{pqrs}$ are the usual one- and two-electron
integrals over molecular orbitals $\phi_p (\bm{r})$, respectively. Orbital indices $p,q,r,s$ denote general
spatial orbitals, $i,j,k,l$ inactive (doubly occupied) orbitals, and $u,v,x,y$
active (partially occupied) orbitals\cite{roos1980b,siegbahn1981}.
The system is described through an electronic wave function
$\vert\Psi_{\alpha}\rangle$ for the $\alpha$-th (electronic) state; $\bm{D}^{\alpha} = \{ D^{\alpha}_{pq}\}$ and $\bm{P}^{\alpha}= \{ P^{\alpha}_{pqrs}\}$ are the corresponding
one- and two-electron reduced density matrices (1-RDM and 2-RDM, respectively),
\begin{align}
D^{\alpha}_{pq} & = \langle \Psi_{\alpha} \vert \hat{E}_{pq}\vert \Psi_{\alpha} \rangle , \label{one-and-two-electron-dens-1} \\
P^{\alpha}_{pqrs} & = \langle \Psi_{\alpha} \vert \hat{e}_{pqrs}\vert \Psi_{\alpha} \rangle . \label{one-and-two-electron-dens-2}
\end{align}
The operator $\hat{E}_{pq}$ is defined as
\begin{align}
\hat{E}_{pq} = \hat{E}^{\uparrow}_{pq} + \hat{E}^{\downarrow}_{pq}
= \hat{a}^{\dagger}_{p\uparrow}\hat{a}_{q\uparrow} + \hat{a}^{\dagger}_{p\downarrow}\hat{a}_{q\downarrow} , \label{E_pq}
\end{align}
where $\hat{a}^{\dagger}_{p}$ and $\hat{a}_{p}$ are creation and annihilation operators\cite{helgaker2004}, respectively,
defined for orbital $\phi_p(\bm{r})$ with spin-up ($\uparrow$) and spin-down ($\downarrow$) quantum numbers.
The operator $\hat{e}_{pqrs}$ then reads
\begin{equation}
\hat{e}_{pqrs} = \hat{E}_{pq}\hat{E}_{rs} - \hat{E}_{ps}\delta_{qr} . \label{e_pqrs}
\end{equation}
All complete active space methods divide the Hamiltonian of the isolated system into active (``A'') and inactive (``I'') parts
\begin{equation}
\hat{H}^{\rm iso} = \hat{H}^{\rm iso}_{I} + \hat{H}^{\rm iso}_{A},
\label{Hamilton_CAS_iso}
\end{equation}
with
\begin{align}
\hat{H}^{\rm iso}_{I} &= \frac{1}{2}\sum_{ij}\bigl(h_{ij} + f^{I}_{ij} \bigr)\hat{E}_{ij} + V_{\rm nn} \label{H_inactive}
\end{align}
and
\begin{align}
\hat{H}^{\rm iso}_{A} &= \sum_{uv}f^{I}_{uv}\hat{E}_{uv} + \frac{1}{2}\sum_{uvxy}g_{uvxy}\hat{e}_{uvxy}, \label{H_active}
\end{align}
where $f^{I}_{pq}$ denotes an element of the inactive Fock matrix
\begin{equation}
f^{I}_{pq} = h_{pq} + \sum_{k}\bigl(2 g_{pqkk} - g_{pkqk} \bigr) \label{Fock_inact} .
\end{equation}
Accordingly, the complete-active-space configuration-interaction (CAS-CI) energy is then given as a sum of an inactive energy, $E^{\rm iso}_{I}$,
and an active energy, $E^{\rm iso}_{A}$:
\begin{align}
E^{\text{iso}}_{\alpha} =
\langle\Psi_{\alpha}\vert \hat{H}^{\rm iso}_{I}\vert \Psi_{\alpha}\rangle + \langle\Psi_{\alpha}\vert \hat{H}^{\rm iso}_{A}\vert \Psi_{\alpha}\rangle
= E^{\rm iso}_{I,\alpha} + E^{\rm iso}_{A,\alpha} , \label{CAS-CI-energy}
\end{align}
where
\begin{align}
E^{\rm iso}_{I,\alpha} & = \frac{1}{2}\sum_{ij}\bigl(h_{ij} + f^{I}_{ij}\bigr) D^{I,\alpha}_{ij} + V_{\text{nn}}
= \sum_{i}\bigl(h_{ii} + f^{I}_{ii}\bigr) + V_{\text{nn}} , \label{inact-energy}
\end{align}
and
\begin{align}
E^{\rm iso}_{A,\alpha} & = \sum_{uv}f^{I}_{uv}D^{A,\alpha}_{uv} + \frac{1}{2}\sum_{uvxy}g_{uvxy}P^{A,\alpha}_{uvxy} \label{act-energy} .
\end{align}
Note that we keep the $\alpha$ state index for the inactive part in order to emphasize its implicit dependence on the choice and type of inactive
orbitals.
In Eqs.~\eqref{inact-energy} and \eqref{act-energy} we have divided the 1-RDM into an inactive part
$\bm{D}^{I,\alpha} = \{ D^{I,\alpha}_{ij}\} = \{ 2 \delta_{ij}\}$,
and an active part, $\bm{D}^{A,\alpha} = \{ D^{A,\alpha}_{uv}\}$. In the following subsection, we discuss how the operators
from Eqs.~\eqref{E_pq} and \eqref{e_pqrs} are constructed
in a matrix-product-operator (MPO) based DMRG algorithm and how the (active) 1-RDM is obtained.
\subsection{DMRG with Matrix Product Operators} \label{CAS_MPS_MPO_DMRG}
For the DMRG algorithm, the orbitals in the CAS are arranged on a linear \textit{lattice}, each orbital defining a \textit{site}.
For a lattice of length $L$ (i.e., for a CAS with $L$ orbitals) the DMRG wave function can be written in the MPS formalism as\cite{rommer1997}
\begin{equation}
\vert \Psi_{\alpha}\rangle =
\sum_{\bm{\sigma}}M^{\sigma_1}_{\alpha} M^{\sigma_2}_{\alpha} \cdots M^{\sigma_L}_{\alpha}\vert \bm{\sigma}\rangle \label{MPS_state}.
\end{equation}
The (site-) matrices $M^{\sigma_l}_{\alpha}$ in Eq.~\eqref{MPS_state}, defined for each site $l$, multiply to yield a CI coefficient
($M^{\sigma_1}_{\alpha}$ and $M^{\sigma_L}_{\alpha}$ are vectors). $\vert\bm{\sigma}\rangle = \vert \sigma_1 , \sigma_2, \ldots,\sigma_L \rangle $
denotes an occupation number vector and
on each site, four \textit{local} states ranging from doubly occupied to empty,
$\vert\sigma_l \rangle = \bigl(\vert\hspace{-1.0mm}\uparrow\downarrow\rangle, \vert\hspace{-1.0mm}\uparrow\rangle, \vert\hspace{-1.0mm}\downarrow\rangle,\vert 0\rangle\bigr)$,
are defined. The DMRG algorithm optimizes the site matrices iteratively.
The MPS formalism can be transferred to operators yielding the {Matrix Product Operators} \cite{mcclloch2007,crosswhite2008}.
Our DMRG implementation employs an MPO form for all operators\cite{dolfi2014,keller2014,keller2015}. Accordingly, an operator $\mathcal{\hat{W}}$ will be
of the form
\begin{equation}
\hat{\mathcal{W}} = \sum_{\bm{\sigma}\bm{\sigma}'}W_{\bm{\sigma}\bm{\sigma}'}\vert\bm{\sigma}\rangle\langle\bm{\sigma}'\vert , \label{mpo_1}
\end{equation}
where $W_{\bm{\sigma}\bm{\sigma}'}$ is given by
\begin{equation}
W_{\bm{\sigma}\bm{\sigma}'} = \sum_{b_1,\cdots b_{L-1}}W^{\sigma_1 ,\sigma'_1 }_{1 b_1}\cdots W^{\sigma_{l} ,\sigma'_l }_{b_{l-1} b_{l}} \cdots
W^{\sigma_{L} ,\sigma'_L }_{b_{L-1}1} , \label{mpo_2}
\end{equation}
so that an expectation value reads (see Refs.~\citenum{keller2015} and \citenum{schollwock2011} for details)
\begin{align}
\langle\Psi_{\alpha}\vert \hat{\mathcal{W}}\vert \Psi_{\alpha}\rangle
= & \sum_{\sigma_{L}\sigma'_L}\sum_{b_{L-1}}M^{\sigma_L \dagger }_{\alpha} W^{\sigma_{L}\sigma'_L}_{b_{L-1} 1 }\Bigl(
\cdots \sum_{\sigma_2 \sigma'_2}\sum_{b_1}M^{\sigma_2 \dagger }_{\alpha} W^{\sigma_{2}\sigma'_2}_{b_{1} b_2 } \notag \\
& \cdot \Bigl( \sum_{\sigma_1 \sigma'_1}M^{\sigma_1 \dagger }_{\alpha} W^{\sigma_{1}\sigma'_1}_{1 b_{1} } M^{\sigma'_1 }_{\alpha}\Bigr)
M^{\sigma'_2 }_{\alpha} \cdots \Bigr) M^{\sigma'_L }_{\alpha} \label{mpo_mps_2} .
\end{align}
For the active part of the 1-RDM associated with $\hat{E}^{\uparrow}_{uv}$, we find
\begin{align}
D^{\uparrow\alpha}_{uv} = & \langle\Psi_{\alpha}\vert \hat{E}^{\uparrow}_{uv}\vert \Psi_{\alpha}\rangle \notag \\
= & \sum_{\sigma_{L}\sigma'_L}\sum_{b_{L-1}}M^{\sigma_L \dagger }_{\alpha} \mathbb{I}^{\sigma_{L}\sigma'_L}_{b_{L-1} 1 }\Bigl(
\cdots \Bigl(\sum_{\sigma_v \sigma'_v}\sum_{b_{v-1}}M^{\sigma_v \dagger }_{\alpha} \mathbb{D}^{\sigma_{v}\sigma'_v \uparrow}_{b_{v-1} b_v } \cdots
\Bigl(\sum_{\sigma_u \sigma'_u}\sum_{b_{u-1}}M^{\sigma_u \dagger }_{\alpha} \mathbb{D}^{\sigma_{u}\sigma'_u \uparrow}_{b_{u-1} b_u } \notag \\
& \cdots
\sum_{\sigma_2 \sigma'_2}\sum_{b_1}M^{\sigma_2 \dagger }_{\alpha} \mathbb{I}^{\sigma_{2}\sigma'_2}_{b_{1} b_2 }
\cdot \Bigl( \sum_{\sigma_1 \sigma'_1}M^{\sigma_1 \dagger }_{\alpha} \mathbb{I}^{\sigma_{1}\sigma'_1}_{1 b_{1} } M^{\sigma'_1 }_{\alpha}\Bigr)
M^{\sigma'_2 \dagger }_{\alpha} \cdots \Bigr)M^{\sigma'_u \dagger }_{\alpha} \cdots\Bigr)M^{\sigma'_v \dagger }_{\alpha}
\cdots \Bigr) M^{\sigma'_L \dagger }_{\alpha} , \label{D_uv_MPO}
\end{align}
where $\mathbb{I}^{\sigma_{i}\sigma'_{i}}$ denotes unit matrices.
Note that we have not explicitly accounted for the fermionic anti-commutation of the creation and annihilation operators in Eqs.~\eqref{D_uv_MPO}.
As explained in Ref.~\citenum{keller2015}, this can be done by multiplying the matrix representations
of the creation and annihilation operators with a 4$\times$4 \textit{fill matrix} related to a Jordan-Wigner transformation.
\subsection{\label{theorya}Polarizable Embedding}\label{PE_scheme}
We now consider the energy, $E^{\rm pe}$, associated with the interaction of environment and embedded system.
We will, in general, have
both electrostatic and polarization contributions, $E^{\rm es}_{\alpha}$ and $E^{\rm pol}_{\alpha}$, respectively,
\begin{equation}
E^{\rm pe}_{\alpha} = E^{\rm es}_{\alpha} + E^{\rm pol}_{\alpha} \label{PE_energy}.
\end{equation}
The PE scheme\cite{soderhjelm2009,olsen2010} divides the environment into (molecular) fragments.
We associate to each of these fragments a set of electrostatic multipoles, usually
localized at the atomic centers\cite{gagliardi2004}. A set of
charges, $q (\bm{r}_{s'})$, dipole moments $\mu_{\alpha}(\bm{r}_{s'})$, quadrupole moments $ Q_{\alpha\beta}(\bm{r}_{s'})$ etc.,
are in this way defined for each center, $s'$, in the environment (the prime distinguishes environment centers from orbital indices).
In addition, the PE scheme allows for a polarization of the environment
by defining a set of localized polarizabilities $\{\bm{\alpha}_{s'} \}$ on each atomic center of the environment.
We obtain both the localized multipoles and polarizabilities from a quantum chemical calculation (usually within DFT).
$E^{\text{es}}_{\alpha}$ is therefore given by
\begin{equation}
E^{\text{es}}_{\alpha} = \langle\Psi_{\alpha}\vert \hat{V}^{\text{es}}\vert\Psi_{\alpha}\rangle + E^{\text{mul}}, \label{esenergy}
\end{equation}
where $E^{\text{mul}}$ contains the
interaction of all multipoles in the environment.
The operator $\hat{V}^{\text{es}}$ describes the interaction of all electrons in the QM region with the environment,
\begin{equation}
\hat{V}^{\rm es} = \sum_{s'}\left( E^{\text{es}}_{\text{n},s'} + \sum_{pq}V_{pq,s'}\hat{E}_{pq} \right) , \label{V_es_operator}
\end{equation}
where the nuclear part, $E^{\text{es}}_{\text{n},s'}$ can simply be added to the electronic part.
$V_{pq,s'}$ are the electrostatic potential integrals for center $s'$ defined through a multipole expansion\cite{olsen2010,list2013,hedegaard2013a}
\begin{equation}
V_{pq,s'} = \sum_{|k|=0}\frac{(-1)^{|k|}}{k!}t^{(k)}_{pq,s'}M^{(k)}_{s'} \label{multipole_exp} .
\end{equation}
The multipole expansion in Eq.~\eqref{multipole_exp} is described through the composite index $k$
collecting $(k_x , k_y, k_z)$ in such a way that $k! = k_x ! k_y ! k_z !$ and $|k| = k_x + k_y + k_z$. In this notation,
the $t^{(k)}_{pq,s'}$ in Eq.~\eqref{multipole_exp} are the integrals over the $T^{(k)}_{s'}$
interaction operators\cite{buckingham1967,stone2002} of order $k$
\begin{equation}
t^{(k)}_{pq,s'} = -\langle \phi_p \vert T^{(k)}_{s'}\vert \phi_q \rangle ,
\label{t_k}
\end{equation}
and the $M^{(k)}_{s'}$ are multipoles of order $k$ at center $s'$.
The composite-index notation is detailed in Refs.~\citenum{olsen2010,olsen2011,hedegaard2013a} where also
the explicit form of the $T^{(k)}_{s'}$ is given.
The energy associated with polarization, $E^{\text{pol}}_{\alpha}$, is given by
\begin{equation}
E^{\text{pol}}_{\alpha} = -\frac{1}{2}\bm{\mu}_{\text{ind}} \langle\Psi_{\alpha}\vert\hat{\bm{F}}\vert\Psi_{\alpha}\rangle ,
\end{equation}
where $\bm{\mu}_{\text{ind}}$ collects all dipole moments and $\hat{\bm{F}}$ all field-strength operators
induced locally at each center.
For a system with $s'_{\text{tot}}$ \textit{polarizable} sites in the environment, we then have
\begin{align}
\bm{\mu}_{\text{ind}} & = \bigl(\bm{\mu}^{\text{ind}}_1 , \bm{\mu}^{\text{ind}}_2, \ldots, \bm{\mu}^{\text{ind}}_{s'_{\text{tot}}} \bigr)^{T},\\
\bm{\hat{F}} & = \bigl(\bm{\hat{F}}_1 , \bm{\hat{F}}_2, \ldots, \bm{\hat{F}}_{s'_{\text{tot}}} \bigr)^{T} .
\end{align}
Note that centers within the same fragment are not allowed to polarize
each other and hence, the number of polarizable centers will be equal or less than the number of total centers in the environment.
On the atomic center $s'$, $\bm{\hat{F}}_{s'}$ will have contributions from the multipoles ($\bm{F}^{\text{es}}_{s'}$) as well as
from the nuclei ($\bm{F}^{\text{n}}_{s'}$) and electrons ($\bm{F}^{\text{e}}_{s'}$) in the QM region,
\begin{equation}
\bm{\hat{F}}_{s'} = \bm{F}^{\text{es}}_{s'} + \bm{F}^{\text{n}}_{s'} + \hat{\bm{F}}^{\text{e}}_{s'}. \label{total_field}
\end{equation}
The two latter operators are defined as
\begin{align}
\bm{F}^{\text{n}}_{s'} & = \sum_{M} Z_{M}\bm{T}^{(1)}_{M s'} = \sum_{M}\frac{Z_{M}(\bm{R}_{M} - \bm{r}_{s'})}{|\bm{R}_{M} - \bm{r}_{s'}|^{3}}
\end{align}
and
\begin{align}
\bm{\hat{F}}^{\text{e}}_{s'} & = -\sum_{pq}\bm{t}^{(1)}_{pq,s'} \hat{E}_{pq}
= \sum_{pq} \left\langle\phi_p\left| \left(\frac{\bm{r}_{} -
\bm{r}_{s'}}{|\bm{r}_{} - \bm{r}_{s'}|^{3}} \right)\right| \phi_q \right\rangle \hat{E}_{pq} \notag
= - \sum_{pq}\bm{F}^{\rm e}_{pq,s'}\hat{E}_{pq}
\end{align}
where $Z_M$ and $\bm{R}_M$ are charge numbers and coordinates of the nuclei in the QM region, respectively. $\bm{r}$ is an electronic coordinate
and $\bm{r}_{s'}$ the coordinates of the point-polarizability centers.
The dipole moment, $\bm{\mu}^{\text{ind}}_{s'}$, induced at atomic center $s'$ depends on the
polarizability on that center, $\bm{\alpha}_{s'}$, and on the locally induced field strength, $\bm{F}_{s'}$,
calculated as the observable over the operator given in Eq.~\eqref{total_field}. However, it also depends on the induced dipole moments of the other sites, $\bm{\mu}_{\text{ind},s''}$,
which can be accounted for by the dipole tensors, $\bm{T}^{(2)}_{s's''}$, given in Ref.\ \citenum{applequist1972},
\begin{align}
\bm{\mu}_{\text{ind},s'} = \bm{\alpha}_{s'}\bm{F}_{s'} - \bm{\alpha}_{s'}\sum_{\substack{s'=1 \\ s''\neq s'}}\bm{T}^{(2)}_{s's''}\bm{\mu}^{\rm ind}_{s''} .
\label{induced_dipoles}
\end{align}
Apart from being applied in QM/MM with polarizable force fields\cite{thompson1995,thompson1996,gao1997},
this expression can be found in the context of the classical description of molecular properties,
e.g., in the work of Thole\cite{thole1981} and Applequist\cite{applequist1972,applequist1977} and
earlier of Mortensen\cite{mortensen1968} and Silberstein\cite{silberstein1917a,silberstein1917b,silberstein1917c}.
The induced dipole moments are usually obtained by re-writing Eq.~\eqref{induced_dipoles} into a matrix equation\cite{applequist1972},
\begin{equation}
\bm{\mu}_{\text{ind}} = \bm{R}\bm{F} ,
\end{equation}
where $\bm{R}$ is an inverted (super) matrix with $\bm{\alpha}^{-1}_{s'}$ tensors on the diagonal and (negative) dipole
tensors, $-\bm{T}^{(2)}_{s's''}$, as off-diagonal elements (see, e.g., Refs.~\citenum{jensen2003a,olsen2010,hedegaard2013a}).
The energy due to polarization in the environment can then be written as
\begin{align}
E^{\text{pol}}_{\alpha} = -\frac{1}{2} \langle\Psi_{\alpha}\vert \bm{\hat{F}}\vert \Psi_{\alpha}\rangle^T \bm{R} \langle\Psi_{\alpha} \vert\bm{\hat{F}}\vert \Psi_{\alpha}\rangle . \label{inducedenergy_2}
\end{align}
Following Ref.~\citenum{list2013}, Eqs.~\eqref{esenergy} and \eqref{inducedenergy_2} can be combined to yield the total energy as
\begin{align}
E_{\alpha} = \langle\Psi_{\alpha}\vert \hat{H}^{\rm iso}\vert\Psi_{\alpha}\rangle + \langle\Psi_{\alpha}\vert \hat{V}^{\rm es}\vert \Psi_{\alpha}\rangle
-\frac{1}{2} \langle\Psi_{\alpha}\vert \bm{\hat{F}}\vert \Psi_{\alpha}\rangle^T \bm{R} \langle\Psi_{\alpha} \vert\bm{\hat{F}}\vert \Psi_{\alpha}\rangle + E^{\text{mul}} .
\label{total_energy}
\end{align}
A solution for the active system can be obtained by a linear variation in the wave function parameters
in Eq.~\eqref{total_energy}, corresponding to optimizing the pseudo-energy
\begin{align}
\mathcal{E}_{\alpha} = \langle\Psi_{\alpha}\vert \hat{H}^{\text{iso}} + \hat{V}^{\text{pe}} \vert\Psi_{\alpha}\rangle,
\label{pseudo-energy}
\end{align}
where we defined the effective PE operator
\begin{equation}
\hat{V}^{\rm pe} = \hat{V}^{\rm es} - \langle\Psi_{\alpha}\vert \bm{\hat{F}}\vert \Psi_{\alpha}\rangle^T \bm{R}\bm{\hat{F}}^{\rm e} \label{V_pe} .
\end{equation}
We can write Eq.~\eqref{V_pe} in terms of individual contributions as
\begin{align}
\hat{V}^{\rm pe}
& = \hat{V}^{\rm es} - \langle\Psi_{\alpha}\vert \bm{\hat{F}}^{\rm e} + \bm{F}^{\rm n} +\bm{F}^{\rm es} \vert \Psi_{\alpha}\rangle^T \bm{R}\bm{\hat{F}}^{\rm e} \notag \\
& = \hat{V}^{\rm es} - \sum_{s' \in P}\left(\langle\Psi_{\alpha}\vert \bm{\hat{\mu}}^{\rm e}_{\text{ind},s'}\vert\Psi_{\alpha}\rangle
+ \bm{\mu}^{\rm n}_{\text{ind},s'} +\bm{\mu}^{\rm es}_{\text{ind},s'}\right) \bm{\hat{F}}^{\rm e}_{s'} ,
\end{align}
where $\hat{V}^{\rm pe}$ depends on $\bm{D}^{\alpha}$ in Eq.~\eqref{one-and-two-electron-dens-1} through
\begin{align}
\bm{\mu}^{\rm e}_{\text{ind},s'}[\bm{D}^{\alpha}] \equiv \langle \Psi_{\alpha}\vert \bm{\hat{\mu}}^{\rm e}_{\text{ind},s'}\vert \Psi_{\alpha}\rangle
= \sum_{s''}\sum_{rs} \bm{R}_{s'',s'} \bm{F}^{\rm e}_{pq,s'} D^{\alpha}_{rs}. \label{mu_dens_dep}
\end{align}
The expectation value of $\hat{V}^{\rm pe}$ now reads
\begin{align}
\mathcal{E}^{\rm pe}_{\alpha} = &\langle\Psi_{\alpha}\vert\hat{V}^{\rm pe}\vert \Psi_{\alpha}\rangle
\equiv \mathcal{E}^{\rm es}_{\alpha} + \mathcal{E}^{\rm pol}_{\text{es},\alpha}
+ \mathcal{E}^{\text{pol}}_{\text{n},\alpha} + \mathcal{E}^{\rm pol}_{\text{e},\alpha}[\bm{D}^{\alpha}] \notag \\
=& \sum_{s'}\left(E^{\rm es}_{\text{n},s'} + \sum_{pq}V^{\rm es}_{pq,s'}D^{\alpha}_{pq}\right) \notag \\
& - \sum_{s'}\sum_{pq} \bigl(\bm{\mu}^{\rm es}_{\text{ind},s'}\bigr)^T \bm{F}^{\rm e}_{pq}D^{\alpha}_{pq}
- \sum_{s'}\sum_{pq} \bigl(\bm{\mu}^{\rm n}_{\text{ind},s'}\bigr)^T \bm{F}^{\rm e}_{pq,s'}D^{\alpha}_{pq} \notag \\
& - \sum_{s'}\sum_{pq} \bigl(\bm{\mu}^{\rm e}_{\text{ind},s'}[\bm{D}^{\alpha}]\bigr)^T \bm{F}^{\rm e}_{pq,s'}D^{\alpha}_{pq} . \label{pe_exp_val}
\end{align}
The energy expression in Eq.~\eqref{pe_exp_val} is not linear in the 1-RDMs, as can be directly seen by inserting
the expression for $\bm{\mu}^{\text{ind}}_{\text{e},s'}[\bm{D}^{\alpha}]$ into the last term in Eq.~\eqref{pe_exp_val}
\begin{equation}
\mathcal{E}^{\text{pol}}_{\text{e},\alpha}[\bm{D}^{\alpha}] = - \sum_{s'}\bigl(\bm{\mu}^{\rm e}_{\text{ind},s'}[\bm{D}^{\alpha}]\bigr)^T \bm{F}^{\rm e}_{pq,s'}D^{\alpha}_{pq} =
- \sum_{s's''}\sum_{pqrs}(\bm{F}^{\rm e}_{rs,s'} )^T \bm{R}_{s''s'} \bm{F}^{\rm e}_{pq,s'} D^{\alpha}_{rs} D^{\alpha}_{pq} . \label{e_pol_exp_val}
\end{equation}
However, the optimization of a standard CAS-CI wave function assumes a linear relationship of the energy on the 1-RDMs.
We have previously applied a work-around to this problem when adding a (non-linear) short-range DFT potential\cite{hedegaard2015b} to CAS-CI
or DMRG-CI following Pedersen\cite{pedersenphd2004}.
This will be elaborated in Section \ref{implementation_details}. As we will distinguish between 1-RDMs for the inactive and active parts,
we introduce the notation
\begin{align}
\mathcal{E}^{\text{pol}}_{\text{e},X,\alpha}[\bm{D}^{Y,\alpha}]
= - \sum_{s'}\bigl(\bm{\mu}^{\rm e}_{\text{ind},s'}[\bm{D}^{Y,\alpha}]\bigr)^T \bm{F}^{\rm e}_{pq,s'}D^{X,\alpha}_{pq} , \label{e_pol_master}
\end{align}
where $X$ and $Y$ indicate the origins ('$I$' or '$A$') of the 1-RDMs.
\subsection{Accommodation of Short-Range Density Functionals} \label{CAS_srDFT}
PE-DMRG with a range-separated Hamiltonian does not create any additional coupling of the short-range DFT functional and the PE operators. The
only difference between PE-DMRG and PE-DMRG--srDFT is their different expressions for $\hat{H}^{\text{iso}}$. To extend PE-DMRG to PE-DMRG--srDFT it
is therefore sufficient to implement Eq.~\eqref{Hamilton_CAS_iso} in a range-separated form\cite{fromager2007,fromager2009}
\begin{align}
\hat{H}^{\rm iso} \rightarrow \hat{H}^{\rm iso}_{\rm srDFT} = \hat{H}^{\rm iso,lr} + \hat{V}^{\rm sr}_{\rm Hxc}[\rho] ,
\label{H_iso_srDFT}
\end{align}
$\hat{H}^{\rm iso,lr}$ is the long-range Hamiltonian which in this work is evaluated as an MPO (cf.~Eq.~\eqref{mpo_2}) with
long-range two-electron integrals separated on the basis
of the error function (see, e.g., Eqs.~(14) and (15) in Ref.~\citenum{hedegaard2015b}).
The contribution from DFT is introduced through the short-range Hartree--exchange--correlation
potential, $\hat{V}^{\rm sr}_{\rm Hxc}[\rho]$,
and the pseudo-energy corresponding to Eq.~\eqref{pseudo-energy} then reads
\begin{align}
\mathcal{E}^{\rm srDFT}_{\alpha} = \langle\Psi^{\rm lr}_{\alpha}\vert \hat{H}^{\rm iso}_{\rm srDFT} + \hat{V}^{\text{pe}} \vert\Psi^{\rm lr}_{\alpha}\rangle,
\label{pseudo_srdft_energy}
\end{align}
where the integrand is the effective PE-DMRG--srDFT Hamiltonian.
The (long-range) DMRG wave function that diagonalizes $ \hat{H}^{\rm iso,lr}$ is
denoted $\vert\Psi^{\rm lr}_{\alpha}\rangle$.
\section{Implementation} \label{implementation_details}
The implementation details described in this section will mainly be concerned with the coupling of the PE model with DMRG; the implementation of the additional terms
for PE-DMRG--srDFT only concerns terms due to the introduction of the range-separated Hamiltonian in Eq.~\eqref{pseudo_srdft_energy}\cite{hedegaard2015b},
which amount to the evaluation of the short-range Hartree--exchange--correlation potential, $\hat{V}^{\rm sr}_{\rm Hxc}[\rho]$,
in $\langle\Psi^{\rm lr}_{\alpha}\vert \hat{H}^{\rm iso}_{\rm srDFT}\vert \Psi^{\rm lr}_{\alpha} \rangle$. A full account of the evaluation of this term was given
for DMRG--srDFT in Ref.~\citenum{hedegaard2015b}. We will in fact employ the same technique as in Ref.~\citenum{hedegaard2015b} to evaluate the
non-linar part of the PE operator and we therefore introduce the deviation, $\Delta D^{\alpha}_{pq}$,
\begin{align}
\Delta D^{\alpha}_{pq} = D^{\alpha}_{pq} - D^{\text{ref},\alpha}_{pq},
\label{delta_D_pq_1}
\end{align}
from some (fixed) reference density matrix, $\bm{D}^{\text{ref},\alpha} = \{ D^{\text{ref},\alpha}_{pq}\}$,
\begin{align}
D^{\text{ref},\alpha}_{pq} = D^{I,\alpha}_{ij} + D^{\text{ref},A,\alpha}_{uv},
\label{fixation}
\end{align}
so that the active-part 1-RDM elements can be written as
\begin{equation}
D^{\text{A},\alpha}_{uv} = D^{\text{ref,A},\alpha}_{uv} + \Delta D^{\text{A},\alpha}_{uv} . \label{delta_D_pq_2}
\end{equation}
Note that the linearization in Eq.~\eqref{delta_D_pq_1} is an approximation only required to obtain a self-consistent scheme.
The final (converged) state does not involve other approximations than those inherent to PE, DMRG, and srDFT.
As we seek to divide contributions into active and inactive parts, as in Eqs.~\eqref{inact-energy} and \eqref{act-energy},
\begin{equation}
\mathcal{E}^{\rm pe}_{\alpha} = \mathcal{E}^{\rm pe}_{I,\alpha} + \mathcal{E}^{\rm pe}_{A,\alpha} ,
\end{equation}
we obtain with Eq.~\eqref{delta_D_pq_1} for the non-linear part in Eq.~\eqref{pe_exp_val}
\begin{align}
\mathcal{E}^{\rm pe}_{I,\alpha} & = \mathcal{E}^{\rm es}_{I,\alpha} + \mathcal{E}^{\rm pol}_{\text{es},I,\alpha} + \mathcal{E}^{\rm pol}_{\text{n},I,\alpha} +
\mathcal{E}^{\rm pol}_{\text{e},\text{ref},\alpha}[\bm{D}^{\text{ref},\alpha}] ,
\label{E_pe_I_1}
\\
\mathcal{E}^{\rm pe}_{A,\alpha} & = \mathcal{E}^{\rm es}_{A,\alpha} + \mathcal{E}^{\rm pol}_{\text{es},A,\alpha} + \mathcal{E}^{\rm pol}_{\text{n},A,\alpha}
+ 2\Delta \mathcal{E}^{\rm pol}_{\text{e},A,\alpha}[\bm{D}^{\text{ref},A,\alpha}] .
\label{E_pe_A_1}
\end{align}
The pseudo-energies $\mathcal{E}^{\rm es}_{A,\alpha}$, $\mathcal{E}^{\rm pol}_{\text{es},A,\alpha}$, $\mathcal{E}^{\rm pol}_{\text{n},A,\alpha}$
and the corresponding inactive parts are given as in Eq.~\eqref{pe_exp_val}, replacing
$D^{\alpha}_{pq}\rightarrow D^{A,\alpha}_{uv}$ and $D^{\alpha}_{pq}\rightarrow D^{I,\alpha}_{ij}$. The terms
$\mathcal{E}^{\rm pol}_{\text{e},\text{ref},\alpha}[\bm{D}^{\text{ref},\alpha}]$ and $\Delta\mathcal{E}^{\rm pol,\text{ref}}_{\text{e},A,\alpha}[\bm{D}^{\text{ref},A,\alpha}]$
associated with inactive and active parts, respectively, are defined
in accordance with Eq.~\eqref{e_pol_master} and therefore given as
\begin{equation}
\mathcal{E}^{\text{pol},\alpha}_{\text{e},\text{ref},\alpha}[\bm{D}^{\text{ref},\alpha}]
= -\sum_{s'\in P}\sum_{pq} \bigl(\bm{\mu}^{\rm ind}_{\text{e},s'}[\bm{D}^{\text{ref},\alpha}]\bigr)^T \bm{F}^{\rm e}_{pq,s'}D^{\text{ref},\alpha}_{pq}
\label{E_pol_e_I}
\end{equation}
and
\begin{equation}
\Delta \mathcal{E}^{\text{pol}}_{\text{e},A,\alpha}[\bm{D}^{\text{ref},A,\alpha}] =
-\sum_{s'\in P}\sum_{uv} \bigl(\bm{\mu}^{\rm ind}_{\text{e},s'}[\bm{D}^{\text{ref}, A,\alpha}]\bigr)^T \bm{F}^{\rm e}_{uv,s'} \Delta D^{A,\alpha}_{uv} .
\label{E_pol_e_A_1}
\end{equation}
With Eq.~\eqref{delta_D_pq_2} and the reference density matrix being fixed by Eq.\ (\ref{fixation}),
we can rewrite Eqs.~\eqref{E_pe_I_1} and \eqref{E_pe_A_1} to become
\begin{align}
\mathcal{E}^{\rm pe}_{I,\alpha} &= \mathcal{E}^{\rm es}_{I,\alpha} + \mathcal{E }^{\rm pol}_{\text{es},I,\alpha} + \mathcal{E}^{\rm pol}_{\text{n},I,\alpha}
+ \mathcal{E}^{\rm pol}_{\text{e},I,\alpha}[\bm{D}^{I,\alpha}] - \mathcal{E}^{\rm pol}_{\text{e},\text{ref},A,\alpha}[\bm{D}^{\text{ref},A,\alpha}] , \label{E_pe_I}
\end{align}
and
\begin{align}
\mathcal{E}^{\rm pe}_{A,\alpha} &= \mathcal{E}^{\rm es}_{A,\alpha} + \mathcal{E}^{\rm pol}_{\text{es},A,\alpha} + \mathcal{E}^{\rm pol}_{\text{n},A,\alpha}
+ 2\bigl(\mathcal{E}^{\rm pol}_{\text{e},A,\alpha}[\bm{D}^{I,\alpha}] + \mathcal{E}^{\rm pol}_{\text{e},A,\alpha}[\bm{D}^{\text{ref},A,\alpha}] \bigr), \label{E_pe_A}
\end{align}
where the $\mathcal{E}^{\text{pol}}_{\text{e},\alpha}[\bm{D}^{\alpha}]$ terms in Eqs.~\eqref{E_pe_I} and \eqref{E_pe_A}
are defined according to the notation in Eq.~\eqref{e_pol_master}. These last two equations are the operational expressions implemented.
For the extension to PE-DMRG--srDFT, from Eq.~\eqref{pseudo_srdft_energy} it follows that we obtain expressions that are equilvalent to
Eqs.~\eqref{E_pe_I} and \eqref{E_pe_A} from $\langle\Psi^{\rm lr}_{\alpha}\vert \hat{V}^{\rm pe}\vert \Psi^{\rm lr}_{\alpha}\rangle$. In
addition, we obtain a term from $\hat{H}^{\rm iso}_{\rm srDFT}$ due to the short-range Hartree--exchange--correlation potential, $\hat{V}^{\rm sr}_{\rm Hxc}[\rho]$.
As shown in Ref.~\citenum{hedegaard2015b}, these terms can be treated similarly to the effective PE operator in Eq.~\eqref{V_pe}, i.e.,~by
the method of linearization in Eq.~\eqref{delta_D_pq_1}. The resulting terms are given in Eqs.~(37) and (39) of Ref.~\citenum{hedegaard2015b}.
The theory described above was implemented in a development version of the {\sc Dalton} program\cite{DALTON2016}.
This version of {\sc Dalton} includes an interface to the release version of the PE Library\cite{pelib}, {\sc pelib}, and an interface
to our MPO-based DMRG program, {\sc QCMaquis}\cite{keller2014,dolfi2014,keller2015,knecht2016,keller2016}.
\section{Computational methodology}\label{compmet}
The PE potentials constructed in this work represent a water cluster of 127 water molecules.
The potentials were obtained with a python script that automates the extraction of coordinates for each fragment (here a water molecule)
and sets up the calculation of localized multipoles and polarizabilities.\cite{jmothesis}
The latter are obtained with the LoProp procedure\cite{gagliardi2004} as implemented in {\sc Molcas}\cite{aquilante2010}.
The PE potentials include multipoles up to quadrupoles and anisotropic polarizabilities, which will
be denoted M2P2. In addition, we have constructed potentials including only charges (M0) and
charges, dipoles, and quadrupoles (M2). The LoProp procedure requires specially constructed basis sets of which we used
A-6-31GPG (M2P2) and A-AUG-CC-PVTZ (M2P2, M2 and M0).\cite{gagliardi2004}
The coordinates of the water cluster were taken from the supporting information of Ref.~\citenum{jacob2006} and
have already served in a range of previous works\cite{jacob2006,jensen2003a,jensen2003b,kongsted2002}. Therefore, they represent an excellent benchmark for our approach.
The structure for the water cluster can be seen as a representative structure for an ensemble of structures.
It was originally obtained in Ref.~\citenum{kongsted2002}
by averaging 8000 molecular dynamics (MD) trajectories into one effective structure,
where each trajectory describes a box of 128 water molecules. The MD was performed\cite{kongsted2002} with
a polarizable force field and periodic boundary conditions in which the individual trajectories spanned 20 picoseconds, starting from different velocities.
The QM water molecule and a minimal solvent shell with only three nearest neighbors involved in hydrogen bonding
are shown in Figure \ref{christoph_h2o_full}A, whereas the full
cluster is shown in Figure \ref{christoph_h2o_full}B. We also constructed a system that exhibits multiconfigurational character to a certain degree by elongating
one of the water {H--O} bonds as shown in Figure \ref{christoph_h2o_full}C.
\begin{figure}[tbh!]
\centering
\includegraphics[scale=0.40]{figure1}
\caption{Water cluster from Ref.~\citenum{kongsted2002}. The QM region is shown in ball-and-stick representation.
(A) Cut-out of the single water molecule taken as the QM region including three nearest-neighbor water molecules.
(B) Full structure. (C) Zoom in on the QM water molecule where one of the bonds was elongated by a factor of 3/2.
\label{christoph_h2o_full}}
\end{figure}
As an example of a QM subsystem requiring a large active space in a structured environment, we chose the
the retinal chromophore in rhodopsin. This system has become a benchmark system for advanced electronic structure methods
in recent years (see, for instance, Refs.\ \citenum{ferre2003,andruniow2004,cembran2005,roth05,coto06,roth07,frut07,stra08,scha11,wein11,roth13,roth14,roth14b,vals15,luk-15}).
For the channelrhodopsin protein, we employ the M2P2 potential constructed in
Ref.~\citenum{hedegaard2015a}, which assumes a QM region comprised of the retinalidyne chromophore and a small part of the lysine (lys296) side chain
that links the chromophore to the protein (depicted below in Figure \ref{mut_inf_gs_dmrg-20-27_m2p2}, middle).
All DMRG calculations for the QM parts were carried out with {\sc QCMaquis}\cite{keller2014,dolfi2014,keller2015,knecht2016,keller2016}
in a Hartree--Fock (HF) or HF--srDFT molecular orbital basis. The corresponding PE-DMRG and PE-DMRG--srDFT calculations employed
PE-HF and PE-HF--srDFT molecular orbital bases, respectively. For water, the DMRG calculations employed
512 renormalized block states $m$, whereas for the retinalidyne chromophore this number was increased to 1024 renormalized block states.
The importance of diffuse functions in the first excitation of the water molecule was previously highlighted\cite{jacob2006}. We chose a
6-31++G* basis set\cite{hehre1972,harihara1973,clark1983}, meaning that DMRG(10,30)[512] corresponds
to the active space of a full configuration interaction (FCI) calculation.
However, although this small basis set will not yield accurate results, it is a suitable choice here as it
permits us to carry out a FCI-type DMRG calculation that does not require further consideration of dynamical correlation effects.
For the retinylidene chromophore, we investigated the multiconfigurational nature of the ground and first excited states in terms of
single-orbital entropies\cite{legeza2003a} and mutual information\cite{legeza2006,rissler2006} evaluated for the frontier orbitals.
For this analysis, the DMRG(20,27)[1024] active space encompassed
the $\pi$-system and higher lying $\sigma$/$\sigma^{*}$ orbitals. The same active space was employed in the DMRG(20,27)[1024]--srDFT
calculations. The latter calculations
were performed with the tailored short-range PBE functional by Goll, Werner, and Stoll\cite{goll2005}, and we denote these
calculations DMRG(20,27)[1024]--srPBE in the following.
All calculations on retinylidene employed a 6-31G* basis set because this very small basis had already been chosen for
studies on related rhodopsins\cite{andruniow2004,altun2009} and hence makes a comparison possible.
Despite the fact that this basis set is very small, previous calculations on channelrhodopsin and a related rhodopsin showed that
the $S_0 \rightarrow S_1$ excitation is not too sensitive
to an increasing basis set size.\cite{altun2008,sneskov2013}
For the water systems we have in addition to the PE-DMRG calculations, also carried out PE-TDDFT calculations with
four different density functionals, namely BLYP\cite{becke1988,lee1988}, B3LYP\cite{becke1988,lee1988,becke1993}, PBE\cite{perdew1996b},
and PBE0\cite{adamo1999}. Due to the Rydberg character of the excitation, we investigated the effect of a range-separated functional,
for which we chose the long-range corrected CAM-B3LYP functional.\cite{yanai2004} The effect is, in this case, about 0.2 eV compared to B3LYP and hence comparatively small.
Although solvent shifts and absolute excitation energies differ between the functionals, the obtained trends between different PE potentials are similar.
We therefore chose B3LYP as representative for the DFT results and only detailed B3LYP data are given explicitly;
results obtained with the other functionals are only shown in the figures.
\section{Results and discussion}\label{results}
Within a state-specific approach, we investigated the effect of the environment on the first electronic excitation ($1^{1}A_1 \rightarrow 1^{1}B_1 $) of a water molecule.
The experimental value for this excitation is known in both gas and condensed phase. In the gas phase, the excitation energy
is approximately 7.4 eV whereas it is blue-shifted to about 8.2 eV in condensed phase, yielding a solvent shift of about 0.8 eV. A discussion
of the experimental results can be found in the paper by Christiansen et al.\cite{christiansen2000}.
In the first subsection (Section \ref{small_environment}), we consider the small solvent shell of three water molecules as the environment
that is to be represented by a PE potential (Figure \ref{christoph_h2o_full}A). In the second
subsection (Section \ref{large_environment}), we extend the environment so that all 127 water molecules are represented by a PE potential (Figure \ref{christoph_h2o_full}B).
To highlight the differences between our PE-DMRG approach and PE-TDDFT for multiconfigurational cases,
we consider in Section \ref{stretched_OH_bond} the solute with the elongated {O--H} bond (Figure \ref{christoph_h2o_full}C)
where the multireference character is incorporated already in the electronic ground state.
Although we also compare with experimental results,
it should be emphasized that results obtained with only one structural configuration must be considered with care. For an accurate comparison,
one must employ a stochastically meaningful number of snapshots. However, our main purpose is to compare the various embedding potentials
for state-specific approaches within the PE approach.
Afterwards we consider the retinylidene chromophore within its host channelrhodopsin protein (Section \ref{channelrhodopsin}).
For this system, the protein environment is known to cause a large blue-shift of
the first excitation energy\cite{sneskov2013}. DMRG studies of chromophores within proteins are rare
(see, e.g., Ref.~\citenum{sharma2014b}
where a point-charge description of the environment was invoked for a iron-sulfur cluster).
\subsection{Water Molecule Embedded in a Minimal Solvent Shell}\label{small_environment}
The state-specific excitation energies obtained with PE-DMRG and the predicted solvent
shifts are given in Table \ref{h2o-pe-dmrg-2} (TDDFT an PE-TDDFT results with B3LYP are also given in this table).
\begin{figure}[tbh!]
\centering
\includegraphics[scale=0.43]{figure2a}
\hspace{0.5cm}
\includegraphics[scale=0.43]{figure2b}
\caption{Excitation energies and solvent shifts for the environment comprised of three water molecules, see Figure \ref{christoph_h2o_full}A.
The column for the A-AUG-CC-PVTZ basis set collects contributions from charges (M0), charges, dipoles and quadrupoles (M2), and
charges, dipoles, quadrupoles, and polarizabilities (M2P2).
The experimental result is shown as a red, horizontal line. \label{trimer}}
\end{figure}
\begin{table}
\caption{Excitation energies and solvent shifts (in eV) for {H$_2$O} in PE potentials representing three environment water molecules.
\label{h2o-pe-dmrg-2}}
\begin{tabular}{lcccc}
\hline \hline \\[-1.5ex]
Potential & \multicolumn{2}{c}{Exc. Energy} & \multicolumn{2}{c}{Solvent shift} \\
\hline \\[-1.0ex ]
& ~~DMRG(10,30)[512]~~ & ~~B3LYP~~ & ~~DMRG(10,30)[512]~~ & ~~B3LYP ~~ \\[0.5ex]
Isolated & 7.46 & 6.83 & - & - \\[0.5ex]
M0$^a$ & 7.99 & 7.31 & 0.53 & 0.48 \\[0.5ex]
M2$^a$ & 7.89 & 7.18 & 0.43 & 0.35 \\[0.5ex]
M2P2$^b$ & 7.90 & 7.34 & 0.44 & 0.51 \\[0.5ex]
M2P2$^a$ & 7.95 & 7.42 & 0.49 & 0.59 \\[0.5ex]
\hline \hline
\end{tabular} \\
\noindent {\footnotesize $^a$Obtained with A-AUG-PTVZ/B3LYP} \\
\noindent {\footnotesize $^b$Obtained with A-6-31PGP/B3LYP }
\end{table}
Starting with the result for the isolated {H$_2$O} molecule, we find that the DMRG(10,30)[512] excitation energy
is in good agreement with the experimental value of 7.4 eV.
As one would have expect from the significant Rydberg character of the excitation\cite{jacob2006},
the excitation energies from TDDFT and PE-TDDFT are underestimated.
We note that this can be partially corrected by asymptotically corrected functionals, although we find that the effect is rather small (see Section \ref{compmet}).
In fact, the result for the isolated {H$_2$O} molecule from Ref.~\citenum{jacob2006}
(7.76 eV) is slightly overestimated with the SAOP potential\cite{schipper2000} in combination with a Slater-type basis set including diffuse functions.
For the embedding calculations, the state-specific calculation with the most accurate potential yields
an excitation energy of 7.95 eV and thereby a solvent shift of 0.49 eV (see Table \ref{h2o-pe-dmrg-2}).
The experimental solvent shift is therefore underestimated by about a factor two.
By comparing the M0 entry with the M2 and M2P2 entries in Table \ref{h2o-pe-dmrg-2}, we see that M0 alone produces a solvent shift of 0.44 eV and hence
the effect from the charges is by far the largest. The additional
effect from higher-order multipoles (M2) amounts to 0.10 eV and is in the opposite direction, thereby reducing the total shift slightly.
Increasing the accuracy by including polarizabilities has little effect for this small environment, but (as will be
apparent in the next subsection) becomes nonnegligible when considering a larger environment.
Regarding the PE-TDDFT results, the absolute excitation energies and solvent shifts of
all density functionals are compared to the PE-DMRG(10,30)[512] result in Figure \ref{trimer}.
The excitation energies and solvent shifts show the same trends
as the state-specific PE-DMRG
results with respect to the applied potential. Yet, the PE-TDDFT excitation energies and
solvent shifts show a larger variation with respect to how the environment potentials are obtained than our PE-DMRG results.
As expected, the absolute excitation energies are too low and the solvent shifts are predicted lower than the experimental one.
The latter was also observed in Ref.~\citenum{jacob2006} for a small environment comprising only two water molecules.
\subsection{Water Molecule Embedded in Larger Environment}\label{large_environment}
We now consider the larger environment of 127 water molecules (Figure \ref{christoph_h2o_full}B).
The excitation energies and solvent shifts for PE-DMRG and PE-TDDFT (with B3LYP) are given in Table \ref{h2o-pe-dmrg-full}.
\begin{table}
\caption{Excitation energies and solvent shifts (in eV) for a solvated system of {H$_2$O}
in PE potentials representing 127 environment water molecules.
\label{h2o-pe-dmrg-full}}
\begin{tabular}{lcccc}
\hline \hline \\[-1.5ex]
Potential & \multicolumn{2}{c}{Exc. Energy} & \multicolumn{2}{c}{Solvent shift} \\
\hline \\[-1.0ex ]
& ~~DMRG(10,30)[512]~~ & ~~B3LYP~~ & ~~DMRG(10,30)[512]~~ & ~~B3LYP ~~ \\[0.5ex]
M0$^a$ & 8.21 & 7.53 & 0.75 & 0.70 \\[0.5ex]
M2$^a$ & 8.04 & 7.32 & 0.58 & 0.49 \\[0.5ex]
M2P2$^b$ & 8.17 & 7.65 & 0.71 & 0.82 \\[0.5ex]
M2P2$^a$ & 8.36 & 7.89 & 0.90 & 1.06 \\[0.5ex]
\hline \hline
\end{tabular} \\
\noindent {\footnotesize $^a$Obtained with A-AUG-PTVZ/B3LYP} \\
\noindent {\footnotesize $^b$Obtained with A-6-31PGP/B3LYP }
\end{table}
\begin{figure}[tbh!]
\centering
\includegraphics[scale=0.43]{figure3a}
\hspace{0.5cm}
\includegraphics[scale=0.43]{figure3b}
\caption{Excitation energies and solvent shifts for the full water cluster in Figure \ref{christoph_h2o_full}B.
The column for the A-AUG-CC-PVTZ basis set collects contributions from charges (M0), charges, dipoles and quadrupoles (M2), and
charges, dipoles, quadrupoles and polarizabilities (M2P2).
The experimental result is shown as a red, horizontal line. \label{full_system}}
\end{figure}
Compared to the small system described in Section \ref{small_environment}, the excitation energies and solvent shifts increase.
For PE-DMRG with the large A-AUG-PVTZ basis set,
the charges alone give an excitation energy that is almost on top of the experimental value, which must be considered fortuitous.
From Table \ref{h2o-pe-dmrg-full} (see also Figure \ref{full_system}) we see that
including higher-order multipoles lowers the excitation energy (by 0.17 eV), while polarization again counteracts by raising the excitation energy (by 0.32 eV).
Hence, our most accurate DMRG result overestimates the excitation energy compared to experiment, albeit by a small margin.
Compared to the small environment (Section \ref{small_environment}), the effect of polarization gains importance.
Considering that the DMRG excitation energies obtained with the two potentials including polarization (M2P2) differ by 0.19 eV shows
that accurate potentials require fairly large basis sets.
This was also concluded in a DFT study\cite{olsen2015b} of amino acids in their ground state.
In this regard, it must be considered fortuitous, as for the case with M0 charges above,
that the DMRG(10,30)[512] result with the M2P2 potential
constructed with A-6-31GPG/B3LYP is very close to the experimental value.
As can be seen from Table \ref{h2o-pe-dmrg-full} and Figure \ref{full_system}, PE-TDDFT overestimates the effect of
polarization compared to PE-DMRG (and also with respect to the experimental result),
leading to an overall overestimation of the solvent shift. PE-DMRG also overestimates the experimental shift, but the overestimation is reduced, compared to PE-TDDFT.
Such an overestimation was also found within a FDE-TDDFT scheme for this particular system\cite{jacob2006}.
With the exception of the
PBE0 functional, the DFT results underestimate the absolute excitation energies compared to PE-DMRG (and also compared to the experimental result).
The large scatter in the excitation energies and solvent shifts from the different density functionals point to the value of
DFT-independent data provided by wavefunction methods (note, however, that DMRG results in general lack dynamic-correlation contributions, which
is of no importance here as our DMRG calculations are actually FCI calculations in the one-electron basis sets chosen).
\subsection{Water Molecule with Stretched O-H Bond Embedded in Larger Environment}\label{stretched_OH_bond}
The system shown in Figure \ref{christoph_h2o_full}C was constructed by elongating one bond of the QM water molecule by a factor 1.5 (to $1.435$ {\AA}).
Thereby, the natural occupation number for the orbital involved in the {O--H} bond decreases from 1.97 to 1.93 (for the ground state), indicating
increased multiconfigurational character.
The excitation energies and solvent shifts are given in Table \ref{h2o-pe-dmrg-full_1-5}.
Both PE-DMRG(10,30)[512] and PE-TDDFT (B3LYP) results show that the excited state is destabilized more than the ground state upon bond elongation. Hence,
the excitation energy is significantly lowered compared to the system with bond length derived from MD simulations (Table \ref{h2o-pe-dmrg-full}).
This conclusion also holds for results obtained with the other functionals (cf.~\ref{full_system_1-5}, left).
\begin{table}
\caption{Excitation energies and solvent shifts (in eV) for a solvated system of {H$_2$O} with stretched {O--H} bond
in PE potentials representing 127 environment water molecules.
\label{h2o-pe-dmrg-full_1-5}}
\begin{tabular}{lcccc}
\hline \hline \\[-1.5ex]
Potential & \multicolumn{2}{c}{Exc. Energy} & \multicolumn{2}{c}{Solvent shift} \\
\hline \\[-1.0ex ]
& ~~DMRG(10,30)[512]~~ & ~~B3LYP~~ & ~~DMRG(10,30)[512]~~ & ~~B3LYP ~~ \\[0.5ex]
Isolated & 4.14 & 4.25 & - & - \\[0.5ex]
M0$^a$ & 5.07 & 5.01 & 0.93 & 0.76 \\[0.5ex]
M2$^a$ & 5.01 & 4.92 & 0.87 & 0.67 \\[0.5ex]
M2P2$^b$ & 5.55 & 5.56 & 1.41 & 1.31 \\[0.5ex]
M2P2$^a$ & 5.54 & 5.62 & 1.40 & 1.37 \\[0.5ex]
\hline \hline
\end{tabular} \\
\noindent {\footnotesize $^a$Obtained with A-AUG-PTVZ/B3LYP} \\
\noindent {\footnotesize $^b$Obtained with A-6-31PGP/B3LYP }
\end{table}
\begin{figure}[tbh!]
\centering
\includegraphics[scale=0.43]{figure4a}
\hspace{0.5cm}
\includegraphics[scale=0.43]{figure4b}
\caption{Excitation energies and solvent shifts for the full water cluster in Figure \ref{christoph_h2o_full}C.
The column for the A-AUG-CC-PVTZ basis set collects contributions from charges (M0), charges, dipoles and quadrupoles (M2), and
charges, dipoles, quadrupoles and polarizabilities (M2P2).
\label{full_system_1-5}}
\end{figure}
In particular, the influence of polarization is increased for the system with elongated {O--H} bond. Naturally, this is not only due to bond elongation, but also partly
due to the decreased distance between the hydrogen atom from the elongated {O--H} bond and environment water molecules (see Figure \ref{christoph_h2o_full}). In such cases, it
can have some effect to include quantum mechanical corrections in the PE scheme, as suggested in a recent paper\cite{olsen2015a}. Our current PE-DMRG method
does not include such correction schemes, but the approach from Ref.~\citenum{olsen2015a} can be straightforwardly extended to a DMRG wave function.
We finally note that, as expected for a system that displays multiconfigurational character, the PE-TDDFT results in Figure \ref{full_system_1-5}
show a large spread of absolute excitation energies, with a clustering of results for pure density functionals (BLYP and PBE) and
hybrid density functionals (B3LYP and PBE0).
\subsection{Retinylidene in Channelrhodopsin \label{channelrhodopsin}}
Before considering the shift of the $S_{1}\rightarrow S_0$ excitation energy due to the channelrhodopsin protein, we analyzed the electron correlation
mapped by the entanglement measures, i.e., by single-orbital entropies\cite{legeza2003a} and mutual information\cite{legeza2006,rissler2006}.
These orbital-based entities have become popular as descriptors to classify
multiconfigurational character.\cite{boguslawski2012b,boguslawski2014,stein16a,stein16b}
Figure \ref{mut_inf_gs_dmrg-20-27_m2p2} shows the PE-DMRG entanglement plots of the ground-state $S_0$ (left) and the first excited state $S_1$ (right).
The entanglement plots indicate multiconfigurational character already in the ground-state with large mutual information particular between the $\pi$ and $\pi^{*}$ type orbitals
(orbital numbers 5,7--13,16,17).
Both single-orbital entropies and mutual information increase (as expected) for the first excited state. This emphasizes that for state-specific optimization
of retinalidyne chromophores, multiconfigurational methods are required.
\begin{figure}[ht] \label{ fig7}
\includegraphics[width=.40\linewidth]{figure5a}
\hspace{0.5cm}
\includegraphics[width=.10\linewidth]{figure5b}
\includegraphics[width=.40\linewidth]{figure5c}
\caption{Entanglement plots from PE-DMRG(20,27) calculations on the retinylidene chromophore (middle) showing the change upon
vertical excitation from the $S_0$ ground state
(left) to the first $S_1$ excited state (right). The magnitude of the single-orbital entropies are encoded in the size of the red circles while the
magnitude of mutual information is ecoded in color and strength of the connecting lines.
the thicker and darker the connecting line between two orbitals, the larger is their mutual information.
The $\pi$ orbitals are labeled 5,7--10 and the $\pi^{*}$ orbitals 11--13,16,17.\label{mut_inf_gs_dmrg-20-27_m2p2}}
\end{figure}
The shifts of the $S_{1}\rightarrow S_0$ excitation energy caused by the protein in the PE-DMRG and PE-DMRG--srDFT calculations are
compiled in Table \ref{pe-dmrg-srdft}. For comparison, we include in this table also results from the literature obtained with PE-TDDFT\cite{sneskov2013},
PE-CC2\cite{sneskov2013} and PE-TDCAS(6,6)--srDFT\cite{hedegaard2015a}. Before discussing the results, we should stress that PE-CC2,
PE-TDDFT, and PE-TDCAS--srDFT are all based on response theory, and hence cover the effect of the environment in a fundamentally different way compared
to state-specific PE-DMRG and PE-DMRG--srPBE. Furthermore,
the PE-TDCAS(6,6)--srPBE method also includes orbital optimization whereas
the present PE-DMRG and PE-DMRG--srPBE results are based on PE-HF or PE-HF--srPBE orbitals (for the ground state).
\begin{table}
\caption{Shifts of the first $S_{0}\rightarrow S_{1}$ excitation of the retinalidyne chromophore due to the channelrhodopsin protein (in eV). The shifts are taken with
respect to an isolated-molecule calculation of retinalidyne in its structure adopted within the protein. The strain exerted on the structure by the protein was previously found to have
an effect of 0.05 eV on the $S_{0}\rightarrow S_{1}$ excitation\cite{sneskov2013}.
We recall that 'TD' refers to linear-response calculations.
\label{pe-dmrg-srdft}}
\centering
\begin{tabular}{lc}
\hline \hline \\[-1.5ex]
Method & Shift \\
\hline \\[-1.0ex ]
PE-DMRG[1024](20,27) & 1.01 \\[0.5ex]
PE-DMRG[1024](20,27)--srPBE$^a$ & 0.52 \\[0.5ex]
PE-CC2\cite{sneskov2013} & 0.89 \\[0.5ex]
PE-TDDFT\cite{sneskov2013} & 0.49 \\[0.5ex]
PE-TDCAS(6,6)-srPBE\cite{hedegaard2015a} & 0.69 \\[0.5ex]
\hline \\[0.5ex]
exp.\ \cite{ande05,niel06,ritt08,kato12} & 0.58--0.70 \\[0.5ex]
\hline \hline
\end{tabular}
$^a$ short-range PBE functional as defined in Ref.~\citenum{goll2005} \\
$^b$ short-range PBE functional as defined in Ref.~\citenum{fromager2007}
\end{table}
The excitation energy shift due to the protein calculated with PE-DMRG[1024](20,27) and PE-CC2 are similar, namely 1.01 eV and 0.89 eV, respectively.
However, PE-DMRG[1024](20,27) lacks most of the dynamic correlation and srPBE then reduces this shift to 0.52 eV, which is
comparable to the shifts calculated with TDDFT\cite{sneskov2013}
(with the CAM-B3LYP\cite{yanai2004} functional) and TDCAS(6,6)--srPBE\cite{hedegaard2015a}.
We may compare the calculated shifts with experimental values of 0.58--0.70 eV taken from gas-phase \cite{ande05,niel06} and in-protein
\cite{ritt08,kato12} measurements, which show that almost all theoretical results listed in Table \ref{pe-dmrg-srdft} are in this range.
Only the pure PE-DMRG is slightly off, which indicates the importance of dynamic correlation. However, we also emphasize that
the orbitals were not optimized for the PE-DMRG wave function, which will be worst for the excited state. Moreover, we note that all
calculated results were obtained for a single structure and that was the one in the protein.
The absolute excitation energy from DMRG[1024](20,27) is overestimated in vacuum (3.17 eV), and also in the protein (4.18 eV).
The experimental gas-phase peak maximum is between 2.00 eV and 2.03 eV \cite{ande05,niel06}, whereas it is shifted to 2.58 and 2.70 eV
in the protein \cite{ritt08,kato12}.
Only srDFT captures the neglected dynamical correlation in such a way that for DMRG--srPBE and PE-DMRG--srPBE we finally obtained
improved excitation energies of 2.75 eV in vacuum and 3.27 eV in the protein, respectively.
For comparison, CC2 and TDDFT yield excitation energies of 2.11 eV and 2.42 eV in vacuum and 3.0 eV and 2.91 eV in the protein, respectively.
The PE-DMRG--srPBE result could be significantly improved by orbital optimization, because, as noted before, the HF-type orbitals in all DMRG
calculations are certainly not adequate for the description of the excited state and hence lead to a larger deviation from experiment.
\section{Conclusion}\label{conclusion}
We presented a coupling of our MPO-based DMRG program with the Polarizable Embedding scheme that explicitly accounts for large
environments using atom-centered multipoles and polarizabilities.
With this PE-DMRG scheme we investigated a well-studied system, namely the excitation energy and associated solvent shift of the
first excitation ($1^{1}A_1 \rightarrow 1^{1}B_1 $) of a water molecule embedded in a water environment. Ground and excited state energies
were obtained from state-specific DMRG(10,30)[512] and PE-DMRG(10,30)[512] with an active space including all orbitals.
We also compared PE-TDDFT and state-specific PE-DMRG with respect to the excitation energy and solvent shift for a range
of embedding potentials of increasing accuracy. The applied potentials were denoted M0 (charges),
M2 (charges, dipoles and quadrupoles), and M2P2 (charges, dipoles, quadrupoles, and polarizabilities). For a small environment, the
addition of polarization is less important, whereas both the absolute excitation energy and solvent shift are affected significantly for a larger solvation shell.
The differences are somewhat larger for PE-TDDFT, compared to the state-specific PE-DMRG results. Further, the PE-TDDFT results show that the differences between
the M0, M2 and M2P2 PE potentials also depend on the density functional.
The functional dependence is even more pronounced when the water molecule described by TDDFT has a stretched {O--H} bond.
We also showed how the PE-DMRG scheme can be extended to a range-separated PE-DMRG--srDFT hybrid scheme\cite{hedegaard2015b} that accounts for dynamical correlation
when the QM system is too large to include all orbitals in the active space.
With this method we investigated the blue-shift induced by a channelrhodopsin protein
on the $S_{0}\rightarrow S_{1}$ excitation of the protein's chromophore,
a retinalidyne Schiff base. In accordance with previous studies, the protein effect was found to be large. The inclusion of dynamical
correlation through a srDFT functional reduced the absolute excitation energy of $S_{0}\rightarrow S_{1}$ as compared to PE-DMRG
(where they were considerably overestimated).
It is currently not known to what extend incorporation of quantum effects in the interaction between QM region and PE potential will affect the results obtained here.
This was done (for TDDFT) in a recent extension\cite{olsen2015a} of the PE scheme and these developments can be straightforwardly included in the method(s) presented here.
\section*{Acknowledgements}
E.D.H. thanks the Villum Kann Rasmussen Foundation for a postdoctoral fellowship.
This work has been financially supported by ETH Z{\"u}rich and the Schweizer Nationalfonds (SNF project 200020\_156598).
\newcommand{\Aa}[0]{Aa}
\providecommand{\refin}[1]{\\ \textbf{Referenced in:} #1}
|
2,869,038,155,633 | arxiv | \section{Introduction}
NOMA has been recognized as a promising candidate for 5G communication systems\cite{Noma_Performace}. In contrast with conventional OMA, e.g., time-division multiple access (TDMA), NOMA serves multiple users simultaneously via power domain division. Early literature on NOMA has mainly focused on the improvement of SE. For example, in \cite{NOMA_SISO_DING}, the authors analyzed the ergodic sum rate and the outage performance of a single-input single-output (SISO) NOMA system with randomly deployed users. In \cite{UserPairing}, the impact of user pairing on two-user SISO NOMA systems was considered. Besides, the power allocation among users in a SISO NOMA system was investigated in \cite{Fairness} from the perspective of user fairness.
In addition to SE, EE has recently drawn significant attention since the information and communication technology (ICT) accounts for around 5\% of the entire world energy consumption \cite{Pseudo-concavity}, which is becoming one of the major social and economical concerns worldwide. Currently, only a few works have studied NOMA from the perspective of EE. In \cite{EE2}, the EE optimization was performed in a fading multiple-input multiple-output (MIMO) NOMA system. However, the number of users is limited and fixed as two in \cite{EE2}, which greatly restrains the application of NOMA.
Motivated by the aforementioned observations, in this correspondence, we study the EE optimization in a downlink SISO NOMA system with multiple users, where each user has its own quality of service (QoS) requirement guaranteed by a minimum required data rate.
We first determine the minimum transmitting power that is able to support the required data rate for each user. Then an energy-efficient power allocation strategy is proposed to maximize the EE by solving a non-convex fractional programming problem. This optimization is further decoupled into two concatenate subproblems and solved one by one:
1) a non-convex multivariate optimization problem that is solved in closed form;
2) a strict pseudo-concave univariate optimization problem that is solved by the bisection method.
Our numerical results show that NOMA has superior EE performance compared with conventional OMA.
\section{System Model}
Consider a downlink transmission scenario wherein one single-antenna BS simultaneously serves $K$ single-antenna users. The channel from the BS to the $k$-th user, $1\leq k\leq K$, is modeled as $h_k=g_kd_k^{-\frac{\alpha}{2}}$, where $g_k$ is the Rayleigh fading coefficient, $d_k$ is the distance between the BS and the $k$-th user, and $\alpha$ is the path loss exponent. The instantaneous channel state information (CSI) of all users is known at the BS. Without loss of generality, we assume that the channel gains are sorted in the ascending order, i.e., $0<\left|h_1\right|^2\leq\left|h_2\right|^2...\leq\left|h_K\right|^2$.
According to the principle of NOMA\cite{Noma_Performace,NOMA_SISO_DING}, the BS broadcasts the superposition of $K$ signals to its $K$ users via power domain division. We denote $P$ as the total power available at the BS, $a_k$ as the $k$-th user's power allocation coefficient, which is defined as the ratio of the transmitting power for the $k$-th user's message to the total power $P$.
At receivers, successive interference cancellation (SIC) is used to eliminate the multi-user interference. Specifically, the $k$-th user first decodes the $i$-th user's message, $i<k$, and then removes this message from its received signal, in the order $i=1,2,...,k-1$; the messages for the $i$-th user, $i>k$, are treated as noise\cite{NOMA_SISO_DING}.
The achievable rate of the $k$-th user $R_k$ and the achievable sum rate of the system $R$ are given by
\begin{align}
&R_k=\log_2\left(1+\frac{P\left|h_k\right|^2a_k}{P\left|h_k\right|^2\sum_{i=k+1}^Ka_i+\sigma^2}\right),\label{Rm}\\
&R = \sum\nolimits_{k=1}^KR_k\label{R},
\end{align}
respectively\cite{NOMA_SISO_DING}, where $\sigma^2$ is the power of the additive noise.
\section{Problem Formulation}
As done in \cite{EE2,Pseudo-concavity}, the EE is defined as the ratio of the achievable sum rate of the system to the total power consumption, which is given by
$\textrm{EE}\triangleq\frac{R}{P_t+P_c}$,
where $P_t \triangleq \sum\nolimits_{k=1}^Ka_kP$ is the actually consumed transmitting power and $P_c$ is the constant power consumption of circuits.
Our design is based on providing QoS guarantees for all users. Each user has a minimum required data rate, denoted as $R_k^{\textrm{Min}}$ for $1\leq k\leq K$, i.e.,
\begin{equation}\label{RmQm}
R_k \geq R_k^{\textrm{Min}}, ~~~1\leq k\leq K,
\end{equation}
which can be further transformed into
\begin{equation}\label{RmQm2}
a_k\geq A_k\left(\sum\nolimits_{i=k+1}^{K}a_i+\frac{\sigma^2}{P\left|h_{k}\right|^2}\right),~1\leq k\leq K,
\end{equation}
where $A_k\triangleq2^{R_k^{\textrm{Min}}}-1$. Thereby, the EE maximization problem is formulated as
\begin{subequations}\label{EE_MV_Original}
\begin{align}
&\max_{P_t,a_k,1\leq k\leq K}~\textrm{EE}\\%\frac{R}{P_t+P_c} \\
&~~~~~~\textrm{s.t.}~~~~~~~P_t\leq P~~~\textrm{and}~~~P_t=\sum\nolimits_{k=1}^Ka_kP,\label{constraintP}\\
&~~~~~~~~~~~~~~~~(\ref{RmQm2})\label{constraintQ}.
\end{align}
\end{subequations}
Due to the minimum data rate constraints in (\ref{constraintQ}), problem (\ref{EE_MV_Original}) might be infeasible when the total power $P$ is not sufficiently large. Accordingly, there must exist a minimum transmitting power $P_{\textrm{Min}}$ that satisfies all users' data rate requirements and then problem (\ref{EE_MV_Original}) is feasible only under the condition $P\geq P_{\textrm{Min}}$. Thereby, it is important to firstly establish the feasible range of $P$, the derivation of which is discussed as follows.
\subsection{Minimum Required Transmitting Power $P_{\textrm{Min}}$}
Denote $P_k$ as the power allocated to the $k$-th user's message, then the problem of figuring out $P_{\textrm{Min}}$ is formulated as
\begin{subequations}\label{minp}
\begin{align}
&P_{\textrm{Min}}\triangleq\min_{P_k,1\leq k\leq K}~~\sum\nolimits_{k=1}^KP_k\label{Ob RmQm2} \\
&\textrm{s.t.}~P_k\geq A_k\left(\sum\nolimits_{i=k+1}^{K}P_i+\frac{\sigma^2}{\left|h_{k}\right|^2}\right),~~~1\leq k\leq K,\label{C RmQm2}
\end{align}
\end{subequations}
where (\ref{C RmQm2}) comes from the minimum data rate constraints in (\ref{RmQm2}). Problem (\ref{minp}) is solved by the following theorem.
\begin{theorem}\label{theo1}
The optimal solution to problem (\ref{minp}), denoted by $\{P_k^{\textrm{Min}}\}_{k=1}^{K}$, is given as
\begin{equation}\label{calculate Pkmin}
P_k^{\textrm{Min}}=A_k\left(\sum\nolimits_{i=k+1}^{K}P_i^{\textrm{Min}}+\frac{\sigma^2}{\left|h_{k}\right|^2}\right),~~~1\leq k\leq K.
\end{equation}
\end{theorem}
\begin{IEEEproof}
It can be seen that problem (\ref{minp}) is convex, thus the following Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for its optimal solution:
\begin{align}
&1+\sum\nolimits_{i=1}^{k-1}\mu_iA_i =\mu_k,~~~~~~~~~~~~~~~~~~~~~~1\leq k\leq K,\label{KKT_Pmin}\\
&\mu_k\left[A_k\left(\sum_{i=k+1}^{K}P_i+\frac{\sigma^2}{\left|h_{k}\right|^2}\right)-P_k\right]=0,~1\leq k\leq K,\\
&\mu_k\geq 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1\leq k\leq K,\label{mu_original_Pmin}
\end{align}
where $\{\mu_k\}_{k=1}^K$ are the Lagrange multipliers for the constraints in (\ref{C RmQm2}).
According to (\ref{KKT_Pmin}), we have $\mu_k> 0$ for $1\leq k\leq K$, because $\{A_k\}_{k=1}^{K}$ and $\{\mu_k\}_{k=1}^K$ are all nonnegative numbers. This indicates that the constraints in (\ref{C RmQm2}) are all satisfied at equality. Further, by setting the constraints in (\ref{C RmQm2}) to be active for $1\leq k\leq K$, the closed-form expressions of $\{P_k^{\textrm{Min}}\}_{k=1}^{K}$ are given by (\ref{calculate Pkmin}). Specifically, $\{P_k^{\textrm{Min}}\}_{k=1}^{K}$ are calculated sequentially in the order $k=K,K-1,...,1$. Then the proof is complete.
\end{IEEEproof}
According to Theorem 1, with the instantaneous CSI, $\{P_k^{\textrm{Min}}\}_{k=1}^{K}$ are calculated in the order $k=K,K-1,...,1$ by using (\ref{calculate Pkmin}). Afterwards, $P_{\textrm{Min}}=\sum\nolimits_{k=1}^KP_k^{\textrm{Min}}$ can be used as a threshold to verify whether $P$ is large enough to meet the constraint on data rate for each user.
\section{Energy Efficiency Maximization}\label{EEOpt}
In this section, we solve problem (\ref{EE_MV_Original}) under the condition $P\geq P_{\textrm{Min}}$, which guarantees the feasibility of problem (\ref{EE_MV_Original}).
Substituting (\ref{Rm}) into (\ref{R}), we first reformulate the achievable sum rate $R$ as follows:
\begin{equation} \label{R_expanded}
\begin{aligned}
&R=\log_2\left(P\left|h_{1}\right|^2\sum\nolimits_{i=1}^Ka_i+\sigma^2\right)\\
&~~~+\sum\nolimits_{k=1}^{K-1}\left[\log_2\left(P\left|h_{k+1}\right|^2\sum\nolimits_{i=k+1}^Ka_i+\sigma^2\right)\right.\\
&~~~~~~~~~~~~~~~~\left.-\log_2\left(P\left|h_k\right|^2\sum\nolimits_{i=k+1}^Ka_i+\sigma^2\right)\right].
\end{aligned}
\end{equation}
For notational simplicity, we further define
\begin{subequations}\label{Simplicity}
\begin{align}
&C_k\triangleq P\left|h_{k}\right|^2,~~~~~~1\leq k\leq K,\\
&\theta\triangleq\sum\nolimits_{i=1}^{K}a_i=\frac{P_t}{P},\label{definetheta}\\
&x_k\triangleq\sum\nolimits_{i=k+1}^Ka_i,~~1\leq k\leq K-1,\label{tm}\\
&F_k(x_k)\triangleq\log_2\left(C_{k+1}x_k+\sigma^2\right)-\log_2\left(C_{k}x_k+\sigma^2\right).\label{J_Fun}
\end{align}
\end{subequations}
By using these notations, $R$ in (\ref{R_expanded}) is recast as
\begin{equation}\label{R_J_Fun}
R=\log_2\left(C_1\theta+\sigma^2\right) + \sum\nolimits_{k=1}^{K-1}F_k(x_k),
\end{equation}
and the original problem (\ref{EE_MV_Original}) is rewritten as
\begin{subequations}\label{EE_MV_Theta}
\begin{align}
&\max_{\theta,a_k,1\leq k\leq K}~~\frac{\log_2\left(C_1\theta+\sigma^2\right) + \sum\nolimits_{k=1}^{K-1}F_k(x_k)}{\theta P+P_c}\label{EE_theta} \\
&~~~~~\textrm{s.t.}~~~~~~~~\theta\leq1~~~\textrm{and}~~~\theta=\sum\nolimits_{k=1}^{K}a_k,\\
&~~~~~~~~~~~~~~~~(\ref{RmQm2}).
\end{align}
\end{subequations}
Here, we emphasize that $\theta$ is the ratio of the actually consumed transmitting power $P_t$ to the total power available at the BS $P$. In particular, $\theta$ might be less than one for maximizing the EE.
Problem (\ref{EE_MV_Theta}) can be further decoupled into two concatenate subproblems as follows.
\begin{subequations}\label{EE_MV_Theta_Decoupled}
\begin{align}
&\max_{\theta}~~\frac{\log_2\left(C_1\theta+\sigma^2\right) + \max\limits_{a_k,1\leq k\leq K}\sum\nolimits_{k=1}^{K-1}F_k(x_k)}{\theta P+P_c}\\
&~~~\textrm{s.t.}~~\theta\leq1~~~\textrm{and}~~~\theta=\sum\nolimits_{k=1}^{K}a_k,\label{CosP}\\
&~~~~~~~~(\ref{RmQm2}).\label{17c}
\end{align}
\end{subequations}
The inner optimization problem is performed over arguments $\{a_k\}_{k=1}^K$ by taking $\theta$ as a constant, the solution of which is a function of $\theta$. Afterwards, the outer optimization problem is taken over $\theta$. These two subproblems are sequentially solved in subsections \ref{sub_Scheme} and \ref{sub_O_Theta}, respectively.
To be specific, in subsection \ref{sub_Scheme}, taking $\theta$ as a constant, we propose a power allocation strategy to solve the inner optimization problem and meanwhile obtain closed-form expressions for the optimal power allocation coefficients $\{a^*_k\left(\theta\right)\}_{k=1}^K$.
In subsection \ref{sub_O_Theta}, we prove that the outer optimization problem is a strict pseudo-concave optimization problem with respect to (w.r.t) the unique argument $\theta$, and then the bisection method is applied to find the optimal $\theta^*$ that maximizes the EE.
\subsection{Optimal Power Allocation Strategy}\label{sub_Scheme}
By fixing $\theta$ in the feasible range $\frac{P_{\textrm{Min}}}{P}\leq\theta\leq1$, the constraint $\theta\leq1$ in (\ref{CosP}) can be eliminated and then the inner optimization problem in (\ref{EE_MV_Theta_Decoupled}) is rewritten as
\begin{subequations}\label{EE_MV_Fixed_Theta}
\begin{align}
&\max_{a_k,1\leq k\leq K}~~\sum\nolimits_{k=1}^{K-1}F_k(x_k)\label{Ob_EE_Fixed_Theta}\\
&~~~~\textrm{s.t.}~~~~~~~\sum\nolimits_{k=1}^{K}a_k =\theta~~~\textrm{and}~~~(\ref{RmQm2}).\label{constraintALLTheta}
\end{align}
\end{subequations}
\begin{remark}
Actually, by regarding $\theta$ as a constant in $\frac{P_{\textrm{Min}}}{P}\leq\theta\leq1$, the nature of the inner optimization problem (\ref{EE_MV_Fixed_Theta}) is to maximize the EE subject to the constraint that the transmitting power should exactly be $\theta P$.
\end{remark}
From (\ref{EE_MV_Fixed_Theta}), we can see that the objective function in (\ref{Ob_EE_Fixed_Theta}) is the summation of $K-1$ non-convex subfunctions sharing similar forms. Based on this observation, we propose an optimization algorithm to solve (\ref{EE_MV_Fixed_Theta}), which can be elaborated in two steps as follows. \emph{Step 1}: we individually maximize each subfunction $F_k(x_k)$ subject to the constraints in (\ref{constraintALLTheta}). \emph{Step 2}: we demonstrate that the optimal solution set of each maximization problem possesses a \emph{unique common solution}. Namely, we can find a unique solution that simultaneously maximizes $F_k(x_k)$ for $1\leq k\leq K-1$ with all the constraints in (\ref{constraintALLTheta}) satisfied. Thereby, this unique solution is the optimal solution to problem (\ref{EE_MV_Fixed_Theta}).
Mathematically, denoting $\Phi_k$ as the optimal solution set for maximizing $F_k(x_k)$ subject to the constraints in (\ref{constraintALLTheta}), we will show
\begin{equation}
\Phi_{1}\cap\Phi_{2}\cap...\cap\Phi_{K-1} = \{\{a^*_i(\theta)\}_{i=1}^K\},
\end{equation}
where $\{a_i^{*}(\theta)\}_{i=1}^K$ is the unique common solution of the $K-1$ optimization problems.
\emph{Step 1}: we now solve these $K-1$ optimization problems. Firstly, the first-order derivative of $F_k(x_k)$ w.r.t $x_k$ is given as
\begin{equation}\label{lemma}
\frac{dF_k(x_k)}{dx_k}=\frac{\left(C_{k+1}-C_{k}\right)\sigma^2}{\ln2\left(C_{k+1}x_k+\sigma^2\right)\left(C_{k}x_k+\sigma^2\right)}\geq0,
\end{equation}
which demonstrates that $F_k(x_k)$ is a monotonically increasing function of $x_k$. Therefore, maximizing $F_k(x_k)$ is equivalent to maximizing $x_k$. As a result, we can uniformly formulate the aforementioned $K-1$ optimization problems as
\begin{subequations}\label{op_uniform}
\begin{align}
&\max_{a_k, 1\leq k\leq K}~~x_{K_0}\\
&~~~~\textrm{s.t.}~~~~~\sum\nolimits_{k=1}^{K}a_k =\theta, \label{gamma_condition3}\\
&~~~~~~~~~~~~~~(\ref{RmQm2}),\label{gamma_condition1}
\end{align}
\end{subequations}
where $1\leq K_0\leq K-1$ is the index for the $K-1$ optimization problems. Problem (\ref{op_uniform}) is solved by the following proposition.
\begin{proposition}
Problem (\ref{op_uniform}) is solved when the constraints in (\ref{gamma_condition1}) are active for $1\leq k\leq K_0$, and the closed-form expressions of $\{a_k\}_{k=1}^{K_0}$ and $x_{K_0}$ are given by
\begin{subequations}\label{calculate gammatm}
\begin{align}
&a_k=D_k\left(\theta-\sum\nolimits_{i=1}^{k-1}a_i\right)+\frac{D_k\sigma^2}{P\left|h_k\right|^2},~~ 1\leq k\leq K_0,\label{calculate gamma}\\
&x_{K_0}=\theta-\sum\nolimits_{k=1}^{K_0}a_k,\label{calculate tm}
\end{align}
\end{subequations}
respectively, where $D_k=A_k/2^{R_k^{\textrm{Min}}}$.
\end{proposition}
\begin{IEEEproof}
Please see Appendix A.
\end{IEEEproof}
\emph{Step 2}: based on Proposition 1, the following theorem further gives a closed-form expression for the unique solution to problem (\ref{EE_MV_Fixed_Theta}).
\begin{theorem}\label{theo2}
The optimal power allocation coefficients $\{a^*_k\left(\theta\right)\}_{k=1}^K$ that maximize the objective function in (\ref{Ob_EE_Fixed_Theta}), are given by
\begin{equation} \label{gammaMaxS}
a_k^{*}(\theta)=
\begin{cases} D_k\left(\theta-\sum\nolimits_{i=1}^{k-1}a_i^{*}(\theta)\right)+\frac{D_k\sigma^2}{P\left|h_k\right|^2},&k\neq K,\\
\theta - \sum_{i=1}^{K-1}a_i^{*}(\theta),&k=K.
\end{cases}
\end{equation}
\end{theorem}
\begin{IEEEproof}
According to (\ref{calculate gamma}) in Proposition 1, arguments $\{a_k\}_{k=1}^{K_0}$ are uniquely and sequentially determined in the order $k=1,2,...,K_0$ for maximizing $x_{K_0}$. This implies that more power allocation coefficients will be determined when $K_0$ increases. Namely, the size of the optimal solution set of problem (\ref{op_uniform}), i.e., $\Phi_{K_0}$, becomes smaller as $K_0$ increases, which can be characterized by
\begin{subequations}
\begin{align}
&\mathfrak{}\Phi_{1}\supset\Phi_{2}\supset...\supset\Phi_{K-1},\label{setrelation1}\\
&\Phi_{1}\cap\Phi_{2}\cap...\cap\Phi_{K-1} = \Phi_{K-1}\label{setrelation2}.
\end{align}
\end{subequations}
Accordingly, $\Phi_{K-1}$ is the optimal solution set that simultaneously maximizes $F_{K_0}(x_{K_0})$ for $1\leq K_0\leq K-1$, consequently solving problem (\ref{EE_MV_Fixed_Theta}).
By setting $K_0$ to $K-1$ in (\ref{calculate gammatm}), the first $K-1$ optimal arguments $\{a_k^{*}(\theta)\}_{k=1}^{K-1}$ are uniquely and sequentially determined in the order $k=1,2,...,K-1$ by using (\ref{calculate gamma}). Further, we have $a_K^{*}(\theta)=\theta-\sum_{k=1}^{K-1}a_k^{*}(\theta)$ from (\ref{gamma_condition3}). As a result, the closed-form expressions of $\{a_k^{*}(\theta)\}_{k=1}^K$ that maximize the objective function in (\ref{Ob_EE_Fixed_Theta}), are given by (\ref{gammaMaxS}). Then the proof is complete.
\end{IEEEproof}
From Theorem 2, we find that the inner optimization problem (\ref{EE_MV_Fixed_Theta}) is solved when the minimum data rate constraints in (\ref{RmQm2}) are active for $1\leq k\leq K-1$, which implies that the optimal power allocation strategy is to use the extra power $\left(\theta P-P_{\textrm{Min}}\right)$ only for increasing the $K$-th user's data rate. This is because, the $K$-th user has the largest channel gain and it achieves the highest data rate among all users with the same amount of power. Namely, the $K$-th user can use power more efficiently than the other users do.
As a result, when the transmitting power is fixed as $\theta P$, the nature of maximizing the EE is to enlarge the data rate of the user with the largest channel gain as much as possible.
However, this does not signify that the extra power $\left(\theta P-P_{\textrm{Min}}\right)$ should be totally allocated to the $K$-th user, since its signal also interferes with the other $K-1$ users. More explicitly, the following corollary further reveals the essence of the proposed optimal power allocation strategy.
\begin{corollary}
$\{\frac{da^*_k(\theta)}{d\theta}\}_{k=1}^K$ are positive constants.
\end{corollary}
\begin{IEEEproof}
Please see Appendix B.
\end{IEEEproof}
Corollary 1 means that the power of the $k$-th user's signal $a^*_k(\theta)P$ increases linearly as $\theta$ increases. This implies that the extra power $\left(\theta P-P_{\textrm{Min}}\right)$ is allocated to the $k$-th user with the constant proportion $\frac{da^*_k(\theta)}{d\theta}$.
\subsection{Optimal Transmitting Power $\theta^* P$ for Maximizing the \emph{EE}}\label{sub_O_Theta}
In the previous subsection, the inner optimization problem (\ref{EE_MV_Fixed_Theta}) is solved with the closed-form solution in (\ref{gammaMaxS}), of which $\theta$ is the unique argument. Consequently, the outer optimization problem in (\ref{EE_MV_Theta_Decoupled}) is transformed into a univariate optimization problem w.r.t $\theta$, which is given by
\begin{subequations}\label{EE_SV}
\begin{align}
&\max_{\theta}~~\frac{\log_2\left(C_1\theta+\sigma^2\right) + \sum\nolimits_{k=1}^{K-1}F_k(x_k^*(\theta))}{\theta P+P_c}\label{EE_SV_Final}\\
&~~\textrm{s.t.}~~~P_{\textrm{Min}}\leq\theta P\leq P,\label{Feasible_Range}
\end{align}
\end{subequations}
where $x_k^*(\theta)=\sum_{i=k+1}^Ka_i^*(\theta)=\theta-\sum_{i=1}^ka_i^*(\theta)$ and the constraint in (\ref{Feasible_Range}) indicates the feasible range of $\theta$.
\begin{theorem}\label{theo3}
Denote the objective function in (\ref{EE_SV_Final}) as $\textrm{EE}(\theta)$, then EE$\left(\theta\right)$ is a strict pseudo-concave function w.r.t $\theta$.
\end{theorem}
\begin{IEEEproof}
It can be easily verified that the second-order derivative of $F_k(x_k)$ w.r.t $x_k$ is non-positive, which indicates that $F_k(x_k)$ is a concave function of $x_k$ for $1\leq k \leq K-1$. Based on this property, we further conclude that $F_k(x_k^*(\theta))$ is a concave function of $\theta$. This is because $\{a^{*}_i(\theta)\}_{i=1}^{K}$ and $\{x_k^*(\theta)\}_{k=1}^{K-1}$ are all affine mappings according to their linear expressions, which preserves the convexity of $F_k(x_k^*(\theta))$ w.r.t $\theta$. Moreover, it can be easily verified that $\log_2\left(C_1\theta+\sigma^2\right)$ is a strict concave function of $\theta$. As a result, the numerator of EE$\left(\theta\right)$, which is the summation of $\sum\nolimits_{k=1}^{K-1}F_k(x_k^*(\theta))$ and $\log_2\left(C_1\theta+\sigma^2\right)$, must be a strict concave function w.r.t $\theta$, since the convexity is preserved by the addition operation. By now, we have proved that EE$\left(\theta\right)$ has a strict concave numerator and an affine denominator, which ensures that EE$\left(\theta\right)$ is a strict pseudo-concave function w.r.t $\theta$ \cite[Proposition 6]{Pseudo-concavity}.
\end{IEEEproof}
According to Theorem 3, EE($\theta$) is a strict pseudo-concave function of $\theta$ and thus admits a unique maximizer which is the unique root of the equation $\frac{d\textrm{EE}\left(\theta\right)}{d\theta}=0$ \cite[Proposition 5]{Pseudo-concavity}. The expression of $\frac{d\textrm{EE}\left(\theta\right)}{d\theta}$ is given by (\ref{dEE}) at the top of the next page. Then, the bisection method\footnote{Problem (\ref{EE_SV}) can be also solved by Dinkelbach's algorithm or Charnes-Cooper Transform (see, e.g., \cite{EEProgram} and references therein).} can be applied to find out $\theta^*$ that maximizes EE($\theta$) with polynomial complexity.
\begin{figure*}[!t]
\normalsize
\begin{equation}\label{dEE}
\begin{split}
\frac{d\textrm{EE}\left(\theta\right)}{d\theta}=
&\frac{\frac{1}{\ln2}
\left(\frac{C_K\frac{da_K^*(\theta)}{d\theta}}{\sigma^2+C_Ka^{*}_K(\theta)}\right)\left(\theta P+P_c\right)-\left[\log_2\left(C_1\theta+\sigma^2\right) + \sum\nolimits_{k=1}^{K-1}F_k(x_k^*(\theta))\right]P}{\left(\theta P+P_c\right)^2}
\end{split}
\end{equation}
\hrulefill
\vspace*{-6pt}
\end{figure*}
\section{Simulation Results}
In this section, we numerically evaluate the proposed energy-efficient power allocation strategy, which is labeled as ``EEPA''.
Besides, another strategy that uses full power $P$ for maximizing the SE of the system is also presented, which is labeled as ``MaxSE''. This ``MaxSE'' strategy is actually the solution of the inner optimization problem (\ref{EE_MV_Fixed_Theta}) with $\theta=1$.
For the comparison between NOMA and conventional OMA, we use a TDMA system as a baseline, where the time slots with equal duration are individually allocated to users and the transmit power is fixed, of which the maximum EE is obtained via exhausted search on the transmit power.
We solve problem (\ref{EE_MV_Original}) for 10,000 times with random channel realizations. The parameter setting is: $\{g_k\}_{k=1}^K\sim\mathcal{CN}(0,1)$, $\alpha=3$, $\sigma^2=-70$ dBm and $P_c=30$ dBm. In particular, when the total power $P$ is not large enough for guaranteeing all users' minimum required date rates, the BS will not send messages and the EE is set to zero for this case.
\begin{figure}[!t]
\vspace{-1.0em}
\centering
\includegraphics[height = 5.2cm, width = 6.4cm]{EE_P}
\vspace{-1.0em}
\caption{Average EE (bits/Joule/Hz) versus total power available at the BS $P$ (dBm). $R_k^{\textrm{Min}}=1$ bits/s/Hz and $d_k=80$ m, where $1\leq k\leq K$.}
\label{fig:EE_P}
\end{figure}
Fig. \ref{fig:EE_P} depicts the average EE versus $P$. We can see that there exists a ``Green Point'' at which the maximum EE is achieved by both ``EEPA'' and ``MaxSE'' strategies. When $P$ is smaller than the Green Point's corresponding power on the horizontal axis, the increase of SE will simultaneously bring an increase of EE. But when $P$ is larger, using full power $P$ is not optimal from the perspective of EE. Besides, NOMA is superior to OMA in terms of EE, and the performance gains of NOMA become more significant as $K$ increases. This is because when more users are simultaneously served, higher diversity gains and higher SE can be achieved.
\begin{figure}[!t]
\centering
\vspace{-1.0em}
\includegraphics[height = 5.2cm, width = 6.4cm]{EE_QoS}
\vspace{-1.0em}
\caption{Average EE versus minimum required data rate $R^{\textrm{Min}}$ for different numbers of users, where $\{R_k^{\textrm{Min}}\}_{k=1}^K = R^{\textrm{Min}}$ and $P$=20 dBm.}
\label{fig:EE_QoS}
\vspace{-1.0em}
\end{figure}
By setting $\{R_k^{\textrm{Min}}\}_{k=1}^K$ to the same value, denoted by $R^{\textrm{Min}}$, Fig. \ref{fig:EE_QoS} shows the average EE versus $R^{\textrm{Min}}$. We can see that as $R^{\textrm{Min}}$ increases, it is more difficult to achieve a high EE. This is because, the increase of $R^{\textrm{Min}}$ requires the BS to allocate more power to the users with worse channel conditions, which consequentially degrades the EE performance. It can be further seen that as $R^{\textrm{Min}}$ becomes very large, the EE approaches zero faster for NOMA. This is because $P$ is not large enough for satisfying the highly demanding data rate requirements and then the BS does not send messages, which implies that NOMA is more suitable for low-rate communications and less robust for the increase of data rate requirements in comparison with conventional OMA.
\begin{figure}[!t]
\centering
\vspace{-1.0em}
\includegraphics[height = 5.2cm, width = 6.4cm]{EE_P_D}
\vspace{-1.0em}
\caption{Average EE (bits/Joule/Hz) versus total power available at the BS $P$ (dBm), for different cases of user locations.
\protect\\ Case 1: $d_1$=60m, $d_2$=50m, $d_3$=40m, $(d_1+d_2+d_3)/3=50$m.
\protect\\ Case 2: $d_1$=70m, $d_2$=55m, $d_3$=40m, $(d_1+d_2+d_3)/3=55$m.
\protect\\ Case 3: $d_1$=60m, $d_2$=55m, $d_3$=50m, $(d_1+d_2+d_3)/3=55$m.
\protect\\ Case 4: $d_1$=80m, $d_2$=80m, $d_3$=80m, $(d_1+d_2+d_3)/3=80$m.}
\vspace{-1.0em}
\label{fig:EE_Pd}
\end{figure}
Fig. \ref{fig:EE_Pd} investigates the influence of user locations on the EE. First of all, there is no doubt that the system must have a low EE when all users locate far from the BS (shown as case 4). More importantly, we can see: 1) case 1 and case 2 have very close EE; 2) case 2 outperforms case 3 although they have an equal average user distance. These observations imply that the EE performance is mainly determined by the user with the closest distance to the BS, since this user is most likely to have the largest channel gain so as to use energy most efficiently, which validates our analysis in subsection \ref{sub_Scheme}.
\section{Conclusion}
In this correspondence, we have studied the EE optimization in a SISO NOMA system where multiple users have their own data rate requirements. An energy-efficient power allocation strategy has been proposed to maximize the EE. Our numerical results have shown that NOMA has superior EE performance compared with conventional OMA. This is because, in NOMA, multiple users are simultaneously served via power domain division, which makes energy be more efficiently used.
\appendices
\section{Proof of Proposition 1}
Since problem (\ref{op_uniform}) is convex, the following KKT conditions are necessary and sufficient for the optimality of problem (\ref{op_uniform}):
\begin{align}
&\hspace{-0.05in}\lambda =
\begin{cases}
\mu_k-\sum_{i=1}^{k-1}\mu_iA_i, &~~~~1\leq k \leq K_0,\\
\mu_k-\sum_{i=1}^{k-1}\mu_iA_i+1, &~~~~K_0+1\leq k\leq K,
\end{cases}\label{KKT_lambda}\\
&\hspace{-0.05in}\mu_k\left[A_k\left(\sum_{i=k+1}^{K}a_i+\frac{\sigma^2}{P\left|h_{k}\right|^2}\right)-a_k\right]=0,~1\leq k\leq K,\hspace{-0.05in}\\
&\hspace{-0.05in}\mu_k\geq 0,~~~1\leq k\leq K,\label{mu_original}
\end{align}
where $\lambda$ and $\{\mu_k\}_{k=1}^K$ are the Lagrange multipliers for constraints (\ref{gamma_condition3}) and (\ref{gamma_condition1}), respectively.
In the following, we prove that $\{\mu_k\}_{k=1}^{K_0}$ are positive numbers, which is equivalent to that the constraints in (\ref{gamma_condition1}) are active for $1\leq k\leq K_0$.
\emph{Firstly}, we demonstrate $\mu_1>0$ by contradiction: suppose $\mu_1=0$, then we have $\lambda=\mu_1=0$ by setting $k=1$ in (\ref{KKT_lambda}). Accordingly, for $1\leq k\leq K_0$ in (\ref{KKT_lambda}), we can further obtain $\mu_k=\sum_{i=1}^{k-1}\mu_iA_i$, which indicates that $\{\mu_k\}_{k=1}^{K_0}$ are all zeros, since $\mu_k=0$ can be calculated in the order $k=2,3,...,K_0$.
However, by setting $k=K_0+1$ in (\ref{KKT_lambda}), we have
\begin{equation}\label{v_t_lambda}
\mu_1=\lambda=\mu_{K_0+1}-\sum\nolimits_{i=1}^{K_0}\mu_iA_i+1=\mu_{K_0+1}+1>0,
\end{equation}
which contradicts to the assumption that $\mu_1=0$. As a result, we have proved that $\lambda=\mu_1>0$.
\emph{Afterwards}, for $2\leq k\leq K_0$ in (\ref{KKT_lambda}), we have $\mu_k=\sum\nolimits_{i=1}^{k-1}\mu_iA_i + \lambda$ , which indicates that $\mu_k>0$ for $2\leq k\leq K_0$.
Thereby, constraints (\ref{gamma_condition1}) must be active for $1\leq k\leq K_0$.
We set constraints (\ref{gamma_condition1}) to be active for $1\leq k\leq K_0$ and replace $\sum_{i=k+1}^{K}a_i$ by $(\theta-\sum_{i=1}^{k}a_i)$ in (\ref{gamma_condition1}), then the closed-form expressions of $\{a_k\}_{k=1}^{K_0}$ and $x_{K_0}$ are derived and given by (\ref{calculate gamma}) and (\ref{calculate tm}), respectively. Specifically, $\{a_k\}_{k=1}^{K_0}$ are calculated sequentially in the order $k=1,2,...,K_0$.
\section{Proof of Corollary 1}
Firstly, $\frac{da^*_k(\theta)}{d\theta}$ can be derived from (\ref{gammaMaxS}):
\begin{equation} \label{dgammaOpt}
\frac{da^*_k(\theta)}{d\theta} =
\begin{cases}
D_k\left(1-\sum_{i=1}^{k-1}\frac{da^*_i(\theta)}{d\theta}\right), &k\neq K,\\ 1-\sum_{i=1}^{K-1}\frac{da^*_i(\theta)}{d\theta}, &k=K.
\end{cases}
\end{equation}
It can be seen that $0\leq\sum_{i=1}^{k-1}\frac{da^*_i(\theta)}{d\theta}<1$ is a sufficient condition for $\frac{da^*_k(\theta)}{d\theta}>0$ due to $0\leq D_k<1$ for $1\leq k\leq K$. In the following, we use Mathematical Induction to prove
\begin{equation}\label{MI}
0\leq\sum\nolimits_{i=1}^{k-1}\frac{da^*_k(\theta)}{d\theta}<1,~~~1\leq k\leq K.
\end{equation}
It is obvious that (\ref{MI}) holds when $k=1$. When $k=N+1$,
\begin{align}
&\sum_{i=1}^{N}\frac{da^*_i(\theta)}{d\theta}=\sum_{i=1}^{N-1}\frac{da^*_i(\theta)}{d\theta}+D_N\left(1-\sum_{i=1}^{N-1}\frac{da^*_i(\theta)}{d\theta}\right)\nonumber\\
&~~~~~~~~~~~~~~=(1-D_N)\sum\nolimits_{i=1}^{N-1}\frac{da^*_i(\theta)}{d\theta}+D_N.\label{MIK}
\end{align}
By using the induction hypothesis that (\ref{MI}) holds when $k=N$, i.e., $0\leq\sum_{i=1}^{N-1}\frac{da^*_i(\theta)}{d\theta}<1$, we have $0\leq\sum_{i=1}^{N}\frac{da^*_i(\theta)}{d\theta}<1$. Thereby, we have proved $\frac{da^*_k(\theta)}{d\theta}>0$. Besides, $\{\frac{da^*_k(\theta)}{d\theta}\}_{k=1}^K$ are calculated sequentially in the order $k=1,2,...,K$.
|
2,869,038,155,634 | arxiv | \section{Introduction}
\label{Intro}
The \emph{demixing} problem involves disentangling two (or more) high-dimensional vectors from their linear superposition~\cite{mccoyTropp2014,mccoy2014convexity,soltani2016fast,SoltaniHegde_Asilomar,SoltaniHegde_Globalsip}. In statistical learning applications involving parameter estimation, such superpositions can be used to model situations when there is some ambiguity in the parameters (e.g., the true parameters can be treated as ``ground truth'' + ``outliers'') or when there is some existing prior knowledge that the true parameter vector is a superposition of two components. Mathematically, suppose that the parameter vector is given by $\beta =\Phi\theta_1 + \Psi \theta_2$ where $\beta,\theta_1, \theta_2\in\mathbb{R}^p$ and $\Phi, \Psi$ are orthonormal bases.
If a linear observation model is assumed, then given samples $y\in\mathbb{R}^n$ and a design matrix $X \in \mathbb{R}^{n \times p}$, the goal is to recover the parameter vector $\beta$ that minimizes a loss function $\mathcal{L}(X,y; \beta)$. We focus on the sample-poor regime where the dimension far exceeds the number of samples; this regime has received significant attention from the machine learning and signal processing communities in recent years~\cite{negahban2009unified,CandesCS}.
However, fitting the observations according to a linear model can be restrictive. One way to ease this restriction is to assume a \emph{nonlinear} observation model:
\begin{align}\label{nonlindex}
y = g(X\beta) + e =g(X(\Phi\theta_1 + \Psi \theta_2) )+ e,
\end{align}
where $g$ denotes a nonlinear \textit{link} function and $e$ denotes observation noise. This is akin to the \emph{Generalized Linear Model} (GLM) and \emph{Single Index Model} (SIM) commonly used in statistics~\cite{kakade2011}. Here, the problem is to estimate $w$ and $z$ from the observations $y$ with as few samples as possible.
The above estimation problem is challenging in several different aspects: (i) there is a basic identifiability of issue of obtaining $\theta_1$ and $\theta_2$ even with perfect knowledge of $\beta$; (ii) there is a second identifiability issue arising from the nontrivial null-space of the design matrix (since $n \ll p$); and (iii) the nonlinear nature of $g$, as well as the presence of noise $e$ can further confound recovery.
Standard techniques to overcome each of these challenges are well-known. By and large, these techniques all make some type of \emph{sparseness} assumption on the components $\theta_1$ and $\theta_2$ \cite{CandesCS}; some type of \emph{incoherence} assumption on the bases $\Phi$ and $\Psi$ \cite{elad2005simultaneous,donoho2006stable}; some type of \emph{restricted strong convexity} (RSC)
~\cite{negahban2009unified}; and some type of \emph{Lipschitz} (\emph{restricted strong smoothness} (RSS)) assumptions on the link function $g$~\cite{yang2015sparse}. See section \ref{Perm} for details.
In this short paper, we demonstrate an algorithm that stably estimate the components $\theta_1$ and $\theta_2$ under general \emph{structured sparsity} assumptions on these components. Structured sparsity assumptions are useful in applications where the support patterns (i.e., the coordinates of the nonzero entries) belong to certain restricted families (for example, the support is assumed to be \emph{group-sparse} \cite{huang2010benefit}). It is known that such assumptions can significantly reduce the required number of samples for estimating the parameter vectors, compared to generic sparsity assumptions~\cite{modelcs,SPINIT,approxIT}.
We note that demixing approaches in high dimensions with structured sparsity assumptions have appeared before in the literature~\cite{mccoyTropp2014,mccoy2014convexity,rao2014forward}. However, our method differs from these earlier works in a few different aspects. The majority of these methods involve solving a convex relaxation problem; in contrast, our algorithm is manifestly \emph{non-convex}. Despite this feature, for certain types of structured superposition models our method provably recovers the components given merely $n = \mathcal{O}(s)$ samples; moreover, our methods achieve a fast (linear) convergence rate, and also exhibits fast (near-linear) per-iteration complexity for certain types of structured models. Moreover, these earlier methods have not explicitly addressed the nonlinear observation model (with the exception of \cite{plan2014high}). We show that under certain smoothness assumptions on $g$, the performance of our method matches (in terms of asymptotics) the best possible sample-complexity.
\section{Preliminaries}
\label{Perm}
Let $\|.\|_q$ denote the $\ell_q$-norm of a vector. Denote the spectral norm of the matrix $X$ as $\|X\|$. Denote the true parameter vector, $\theta = [\theta_1^T \ \theta_2^T ]^T\in\mathbb{R}^{2p}$ as the vector obtaining by stacking the true and unknown coefficient vectors, $\theta_1, \theta_2$. For simplicity of exposition, we suppose that components $\theta_1$ and $\theta_2$ have block sparsity with sparsity $s$ and block size $b$~\cite{modelcs} (Analogous approaches apply for other structured sparsity models.)
The problem~\eqref{nonlindex} is inherently unidentifiable and to resolve this issue, we need to assume that the coefficient vectors $\theta_1, \theta_2$ are distinguishable from each other. This issue is characterized by a notion of incoherence of the components $\theta_1, \theta_2$~\cite{SoltaniHegde_Globalsip}.
\begin{definition}\label{incoherence}
The bases $\Phi$ and $\Psi$ are called $\varepsilon$-incoherent if
$
\varepsilon = \sup_{\substack{\|u\|_0\leq s,\ \|v\|_0\leq s \\ \|u\|_2 = 1,\ \|v\|_2 = 1}}|\langle{\Phi u, \Psi v}\rangle|.
$
\end{definition}
For the analysis of our proposed algorithm we need the following standard definition~\cite{negahban2009unified}:
\begin{definition}\label{rssrsc}
$f : \mathbb{R}^{2p} \rightarrow \mathbb{R}$ satisfies \new{Structured} \textit{Restricted Strong Convexity/Smoothness \new{(SRSC/SRSS)} }if:
\begin{align*}
m_{4s}\leq\|\nabla^2_{\xi} f(t)\|\leq M_{4s},\ \ t\in\mathbb{R}^{2p},
\end{align*}
where $\xi = \textrm{supp}(t_1)\cup \textrm{supp}(t_2)$, for all $t_i\in\mathbb{R}^{2p}$ such that \new{$t_i$ belongs to $(2s,b)$ block-sparse vectors}
for $i=1,2$, and $m_{4s}$ and $M_{4s}$ are (respectively) the \new{SRSC and SRSS} constants. Also $\nabla^2_{\xi} f(t)$ denotes a $4s\times 4s$ sub-matrix of the Hessian matrix $\nabla^2 f(t)$ comprised of row/column indices indexed by $\xi$.
\end{definition}
Also, we assume that the derivative of the link function is strictly bounded either within a positive interval, or within a negative interval
\section{Algorithm and main theory}
In this section, we describe our algorithm which we call it \emph{Structured Demixing with Hard Thresholding} (STRUCT-DHT) and our main theory. To solve demixing problem in~\eqref{nonlindex}, we consider the minimization of a special loss function $F(t)$ following~\cite{SoltaniHegde_Globalsip}:
\begin{equation} \label{optprob}
\begin{aligned} \underset{t \in \mathbb{R}^{2p}}{\text{min}}
\ \ F(t) &= \frac{1}{m}\sum_{i=1}^m \Theta(x_i^T\Gamma t) - y_i x_i^T\Gamma t \\
& \text{s.\ t.} \ \ t\in\mathcal{D}
\end{aligned}
\end{equation}
where $\Theta(x) = \int_{-\infty}^{x} g(u)du$ denotes as the integral of the link function $g$, $\Gamma = [\Phi \ \Psi]$, $x_i$ is the $i^{th}$ row of the design matrix $X$ and $\mathcal{D}$ denotes the set of length-$2p$ vectors formed by stacking a pair of $(s,b)$ block-sparse vectors. The objective function in~\eqref{optprob} is motivated by the single index model in statistics; for details, see~\cite{SoltaniHegde_Globalsip}. To approximately solve~\eqref{optprob}, we propose STRUCT-DHT which is detailed as Algorithm \ref{algHTM}.
\begin{algorithm}[t]
\caption{Structured Demixing with Hard Thresholding (STRUCT-DHT)
\label{algHTM}
}
\begin{algorithmic}
\State \textbf{Inputs:} Bases $\Phi$ and $\Psi$, design matrix $X$, link function $g$, observation $y$, sparsity $s$, step size $\eta'$.
\State \textbf{Outputs:} Estimates $\widehat{\beta}=\Phi\widehat{\theta_1} + \Psi\widehat{\theta_2}$, $\widehat{\theta_1}$, $\widehat{\theta_2}$
\State\textbf{Initialization:}
\State$\left(\beta^0, \theta_1^0, \theta_2^0\right)\leftarrow\textsc{random initialization}$
\State$k \leftarrow 0$
\While{$k\leq N$}
\State $t^k \leftarrow [ \theta_1^k ; \theta_2^k ]$\quad\quad\{Forming constituent vector\}
\State $t_1^k\leftarrow\frac{1}{m}\Phi^TX^T(g(X\beta^k) - y)$
\State$t_2^k\leftarrow\frac{1}{m}\Psi^TX^T(g(X\beta^k) - y)$
\State$\nabla F^k \leftarrow [ t_1^k ; t_2^k ]$
\quad\quad\{Forming gradient\}
\State${\tilde{t}}^k = t^k - \eta'\nabla F^k$
\quad\{Gradient update\}
\State$[ \theta_1^k ; \theta_2^k ]\leftarrow\mathcal{P}_{s;s}\left(\tilde{t}^k\right)$
\quad\{Projection\}
\State$\beta^k\leftarrow\Phi \theta_1^k + \Psi \theta_2^k$\quad\{Estimating $\widehat{x}$\}
\State$k\leftarrow k+1$
\EndWhile
\State\textbf{Return:} $\left(\widehat{\theta_1}, \widehat{\theta_2}\right)\leftarrow \left(\theta_1^N, \theta_2^N\right)$
\end{algorithmic}
\end{algorithm}
At a high level, \textsc{STRUCT-DHT} tries to minimize loss function defined in~\eqref{optprob} (tailored to $g$) between the observed samples $y$ and the predicted responses $X\Gamma \widehat{t}$, where $\widehat{t} = [\widehat{\theta}_1; \ \widehat{\theta}_2]$ is the estimate of the parameter vector after $N$ iterations. The algorithm proceeds by iteratively updating the current estimate of $\widehat{t}$ based on a gradient update rule followed by (myopic) \emph{hard thresholding} of the residual onto the set of $s$-sparse vectors in the span of $\Phi$ and $\Psi$. Here, we consider a version of \textsc{DHT}~\cite{SoltaniHegde_Globalsip} which is applicable for the case that coefficient vectors $\theta_1$ and $\theta_2$ have block sparsity. For this setting, we replace the hard thresholding step, $\mathcal{P}_{s;s}$ by component-wise block-hard thresholding~\cite{modelcs}. Specifically, $\mathcal{P}_{s;s}(\tilde{t}^k)$ projects the vector $\tilde{t}^k\in\mathbb{R}^{2p}$ onto the set of concatenated $(s,b)$ block-sparse vectors by projecting the first and the second half of $\tilde{t}^k$ separately.
Now, we provide our main theorem supporting the convergence analysis and sample complexity (required number of observations for successful estimation of $\theta_1, \theta_2$) of \textsc{STRUCT-DHT}.
\begin{theorem}
\label{mainThConvergence}
Consider the observation model~\eqref{nonlindex} with all the assumption and definitions mentioned in the section~\ref{Perm}. Suppose that the corresponding objective function $F$ satisfies the \new{Structured SRSS/SRSC} properties with constants $M_{6s}$ and $m_{6s}$
such that $1\leq\frac{M_{6s}}{m_{6s}}\leq\frac{2}{\sqrt{3}}$ . Choose a step size parameter $\eta'$ with $\frac{0.5}{M_{6s}}<\eta^{\prime}<\frac{1.5}{m_{6s}}$.
Then, \textsc{DHT} outputs a sequence of estimates $(\theta_1^k, \theta_1^k)$ ($t^{k+1} = [\theta_1^k; \theta_1^k]$) such that the estimation error of the parameter vector satisfies the following upper bound (in expectation) for any $k\geq 1$:
\begin{align}
\label{eq:linconverge}
\|t^{k+1} - \theta\|_2\leq\left(2q\right)^k\|t^0-\theta\|_2 + C\tau\sqrt{\frac{s}{m}},
\end{align}
where $q = 2\sqrt{1+{\eta^{\prime}}^2M_{6s}^2-2\eta^{\prime} m_{6s}}$ and $C>0$ is a constant that depends on the step size $\eta^{\prime}$ and the convergence rate $q$. Here, $\theta$ denotes the true parameter vector defined in section~\ref{Perm}
\end{theorem}
\begin{proof}[Proof sketch]
The proof follows the technique used to prove Theorem 4.6 in~\cite{soltani2016fast}. The main steps are as follows.
Let $b'\in\mathbb{R}^{2p} =[b_1';b_2']= t^k - \eta'\nabla F(t^k)$, $b = t^k - \eta'\nabla_J F(t^k)$ where $J = \text{supp}(t^k)\cup \text{supp}(t^{k+1})\cup \text{supp}(\theta)$ and $b_1', \ b_2'\in\mathbb{R}^{p}$ (Here, $\theta = [\theta_1;\theta_2]$ denotes the true parameter vector). Also define $\ t^{k+1} = \mathcal{P}_{s;s}(b') = [\mathcal{P}_s(b'_1); \mathcal{P}_s(b'_2)]$. Now, by the triangle inequality, we have:
$\|t^{k+1} - \theta\|_2\leq \|t^{k+1} - b\|_2 + \|b- \theta\|_2$.
The proof is completed by showing that $\|t^{k+1} - b\|_2\leq 2\|b - \theta\|_2$.
Finally, we use the Khintchine inequality~\cite{vershynin2010introduction} to bound the expectation of the $\ell_2$-norm of the restricted gradient function, $\nabla F(\theta)$ (evaluated at the true parameter vector $\theta$) with respect to the support set $J$).
\end{proof}
\begin{figure}
\begin{center}
\begingroup
\setlength{\tabcolsep}{.1pt}
\renewcommand{\arraystretch}{.1}
\begin{tabular}{cc}
\includegraphics[trim = 8mm 58mm 15mm 30mm, clip, width=0.45\linewidth]{ProbRecRE.pdf}&
\includegraphics[trim = 8mm 58mm 15mm 30mm, clip, width=0.45\linewidth]{ReErr.pdf}\\
(a) & (b)
\end{tabular}
\endgroup
\end{center}
\caption{\small{\emph{Comparison of \textsc{DHT} with structured sparsity with other algorithms. (a) Probability of recovery in terms of normalized error. (b) Normalized error between $\widehat{\beta} = \Phi \widehat{\theta_1} + \Psi \widehat{\theta_2}$ and true $\beta$.}}}
\label{fig:ComparisonSyn}
\end{figure}
Inequality~\eqref{eq:linconverge} indicates the linear convergence behavior of our proposed algorithm. Specifically, in the noiseless scenario to achieve $\kappa$-accuracy in estimating the parameter vector $\widehat{t} = [\widehat{\theta}_1; \ \widehat{\theta}_2]$, \textsc{Struct-DHT} only requires $\log\left(\frac{1}{\kappa}\right)$ iterations.
We also have the following theorem regarding the sample complexity of Alg.\ \ref{algHTM}:
\begin{theorem}
If the rows of $X$ are independent subgaussian random vectors~\cite{vershynin2010introduction}, then the required number of samples for successful estimation of the components, $n$ is given by $\mathcal{O}\left(\frac{s}{b}\log\frac{p}{s}\right)$. Furthermore, if $b = {\Omega}\left(\log\frac{p}{s}\right)$, then the sample complexity of our proposed algorithm is given by $n = \mathcal{O}(s)$, which is asymptotically optimal.
\end{theorem}
\begin{proof}
The proof is similar to the proof of Theorem 4.8 in~\cite{soltani2016fast} where we derived upper bounds on the sample complexity by proving the RSC/RSS for the objective function $F$. Here, the steps are essentially the same as in~\cite{soltani2016fast}, except that we need to compute union bound over the set of $(s,b)$ block-sparse vectors. This set is considerably smaller than the set of \emph{all} sparse vectors and results in an asymptotic gain in sample complexity.
\end{proof}
The big-Oh constant hides dependencies on various parameters, including the coherence parameter $\varepsilon$, as well as the upper bound and lower bounds on the derivative of the link function $g$.
\section{Numerical results}
To show the efficacy of \textsc{Struct-DHT} for demixing components with structured sparsity, we numerically compare \textsc{Struct-DHT} with ordinary \textsc{DHT} (which does \emph{not} leverage structured sparsity), and also with an adaptation of a convex formulation described in~\cite{yang2015sparse} that we call \emph{Demixing with Soft Thresholding} (\textsc{DST}). We first generate true components $\theta_1$ and $\theta_2$ with length $p = 2^{16}$ with nonzeros grouped in blocks with length $b = 16$ and total sparsity $s = 656$. The nonzero (active) blocks are randomly chosen from a uniform distribution over all possible blocks.
We construct a design (observation) matrix following the construction of~\cite{krahmer2011new}. Finally, we use a (shifted) sigmoid link function given by $g(x) = \frac{1-e^{-x}}{1 + e^{-x}}$ to generate the observations $y$. Fig~\ref{fig:ComparisonSyn} shows the the performance of the three algorithms with different number of samples averaged over $10$ Monte Carlo trials. In Fig~\ref{fig:ComparisonSyn}(a), we plot the probability of successful recovery, defined as the fraction of trials where the normalized error is less than 0.05. Fig~\ref{fig:ComparisonSyn}(b) just shows the normalized estimation error for these algorithms. As we can see, \textsc{Struct-DHT} shows much better sample complexity (the required number of samples for obtaining small relative error) as compared to \textsc{DHT} and \textsc{DST}.
\bibliographystyle{unsrt}
|
2,869,038,155,635 | arxiv | \section{Introduction}
The minimal leptogenesis scenario \cite{FY} is based on the seesaw (type I) mechanism, consisting
of the Standard Model (SM) extended by 2 or 3 right-handed (RH) Majorana neutrinos with hierarchical masses. The
lightest RH neutrino, produced by thermal scattering after inflation, decays through out-of-equilibrium processes that violate lepton number, C and CP symmetries. These processes induce a dynamical production of a lepton asymmetry, which can be converted into a
baryon asymmetry through
$(B+L)$-violating sphaleron interactions.
In this context, several studies
\cite{leptogen,towards,pedes} investigated this possible explanation of the baryon asymmetry of the Universe (BAU)
and the different analyses have led to
constraints on neutrino physics from the requirement of a successfull leptogenesis.
For instance, a lower bound on the reheating temperature
$T_{\rm RH}$ (see e.g. \cite{towards,towardsbis,DI}) and an upper bound on
the absolute scale of light neutrino masses (for example in \cite{bound}) have been derived.
These studies have been performed using the so called ``one-flavour state" approximation.
It is becoming well known that the explanation of the BAU from a successfull thermal leptogenesis
has to be revisited when the mass of the decaying right-handed neutrino that produces the lepton asymmetry is $M_{N_{1}} \lesssim 10^{12}$ GeV \cite{flav,zeno}. In this case, the Yukawa couplings of the charged leptons affect the dynamics of the Boltzmann equations (BE) and one cannot use the usual one-state-dominance approximation (one flavour), but the ``flavoured" dynamical equations in the derivation of the BAU must be taken into account~ \cite{flav}-\cite{zeno}.
\section{Flavours in leptogenesis}
We consider the SM Lagrangian with three heavy Majorana right-handed neutrinos $N_i$, $i=1,2,3$. The RH neutrinos couple to the left-handed (LH) ones through the complex Yukawa coupling matrix, $\lambda$. The small neutrino masses generated by the seesaw (type I) mechanism are given by~:
\begin{eqnarray}
m_{\nu}=v^2\, U^{T}\,\lambda^{T}\,{\cal M }^{-1}\,\lambda\,U\ ,
\end{eqnarray}
where $U$ is the PMNS mixing matrix, $v$ is the vacuum expectation value (vev) of the Higgs field ($v\simeq 174$ GeV), and ${\cal M}$ is the $3\times 3$ diagonal Majorana mass matrix.
We assume a hierarchical spectrum for the right handed neutrino, $M_{N_{3}}\gg M_{N_{2}} \gg M_{N_{1}}$, and consider that the lepton asymmetry is produced by the decay of the lightest RH neutrino $N_1$. Then, in the context of leptogenesis, the baryon asymmetry is obtained by partial conversion of the leptonic asymmetry via sphaleron interactions. With the correct treatment of flavoured leptogenesis, this conversion reads~\cite{nardi}~:
\begin{eqnarray}
Y_{\cal B} \simeq \frac{12}{37} \sum_{\alpha} Y_{\Delta \alpha}\ .
\end{eqnarray}
$Y_{\Delta \alpha}$ is the $B/3-L_{\alpha}$ asymmetry in the lepton flavour $\alpha$, which is conserved by sphaleron interactions and transmitted to a baryonic asymmetry $Y_{\cal B}$
\noindent Defining the variable $z=M_1/T$, the BE governing the abundance of RH neutrinos $Y_{N_1}$, and the asymmetry $Y_{\Delta _\alpha}$ are \cite{matters}~:
\begin{eqnarray}
\label{n1}
Y_{N_1}^{\prime}(z)&=&- \kappa \left(D(z)+S(z)\right) \left(Y_{N_1}(z)-Y_{N_1}^{eq}(z)\right) \ ,\\
Y_{\Delta \alpha}^{\prime}(z)&=&-\epsilon_{\alpha} \kappa \left(D(z)+S(z)\right) \left(Y_{N_1}(z)-Y_{N_1}^{eq}(z)\right)-\kappa_{\alpha} W(z) \sum_{\beta} A'_{\alpha \beta} Y_{\Delta \beta}(z)\ ,
\label{n2}\end{eqnarray}
where $Y_{N_1}^{eq}$ is the thermal population of the lightest RH neutrino $N_1$
\begin{eqnarray}
Y_{N_1}^{eq}(z)\simeq\frac{45 \, \zeta(3)}{ 2\pi^4 g_{\star}}\frac{3}{4} \, \,z^{2} K_{2}(z)\ ,
\end{eqnarray}
with $g_{\star}=106.75$ in the SM. The CP-asymmetry generated by $N_1$ in the flavour $\alpha$ is given
by~\cite{issues}:
\begin{eqnarray}
\epsilon_{\alpha} & =& \frac{\Gamma_{N1\: \ell_{\alpha}}-\Gamma_{N1 \:\bar{\ell}_{\alpha}}}{\sum_{\alpha}\left(\Gamma_{N1\: \ell_{\alpha}}+\Gamma_{N1 \:\bar{\ell}_{\alpha}}\right) } \nonumber \\ &=&
\frac{1}{(8\pi)}\frac{1}{ [\lambda \lambda^{\dagger}]_{11}}
\sum_{j} {\rm Im}\, \left\{ ( \lambda_{1 \alpha} ) (\lambda
\lambda^{\dagger})_{1j}
(\lambda^{*}_{j \alpha}) \right\}
g\left(\frac{M_{j}^2}{M_{1}^{2}}\right)\, ,
\end{eqnarray}
where $g$ is the usual loop function~\cite{roulet}. \\
In eqs. (\ref{n1},\ref{n2}),
the wash-out parameters have been factorized out, and are defined as follows:
\begin{eqnarray}
\label{washoutfactor}
\kappa_{\alpha}&\equiv&\frac{\Gamma_{N1\: \ell_{\alpha}}}{H(M_{N_{1}})}=\lambda_{1\,\alpha}\lambda_{1\,\alpha}^{\star}\frac{ v^{2}}{M_{1} m^*} \equiv\frac{\tilde{m_\alpha}}{ m^*}\ ,\\
\kappa&=&\sum_{\alpha} \kappa_{\alpha}\equiv \frac{\tilde{m}}{ m^*}\, ,
\end{eqnarray}
with $m^*$ the equilibrium neutrino mass, $m^*\simeq 1.08\times10^{-3} \,{\rm eV}$. These parameters exhibit the
out-of-equilibrium condition on the decay of the right-handed neutrino: the decay process is out-of-equilibrium when the decay rate is slower than the Hubble expansion rate at the temperature $M_{N_{1}}$: $\Gamma \lesssim H(M_{N_{1}})$ and thus $\kappa \lesssim 1$.
The regime where $\kappa \ll 1 $ is called the weak wash-out regime, in which case the inverse reactions involving the thermal scatters are rather slow and do not efficiently wash-out the lepton asymmetry. On the contrary, $\kappa \gg 1$ is the strong wash-out regime, where the inverse reactions strongly wash-out the asymmetry. Depending on the value of this wash-out parameter, analytical approximations of the production and wash-out terms allow us to derive semi-analytical formulae for the baryon asymmetry, as will be see in the next section.
The processes we take into account in eqs. (\ref{n1},\ref{n2}) are decays and inverse decays labeled $D(z)$, and $\Delta L=1$ scattering, $S(z)$. Notice that in eq. (\ref{n2}), CP violation in scattering is also taken into account, and thus $S(z)$ further contributes to the production of the $Y_{\Delta _\alpha}$ asymmetry. In this study we neglect $\Delta L=2$ scatterings (except the real intermediate states already substracted) that are negligible as $M_{N_1}/10^{14}$ GeV $\ll 0.1\times\kappa$ \cite{matters}, as well as well as scatterings involving gauge bosons.
The thermally averaged decay rate is given by:
\begin{eqnarray}
D(z)=z \frac{K_1(z)}{K_2(z)}\ ,
\end{eqnarray}
where $K_n(z)$ are the modified Bessel functions of the $2^\text{nd}$ kind.
The Higgs-mediated scatterings in the $s-$ and $t-$channel contribute to the production of the asymmetry, as well as to the wash-out term. Their effects can be parametrized by two functions $f_1(z)$ and $f_2(z)$:
\begin{eqnarray}
f_1(z)&=& \frac{S(z)+D(z)}{ D(z)}\simeq \frac{0.1}{ z^2}\left(\frac {15} {8} +z\right) \left[ 1+a_h(z)z^2\log{\left(1+ \frac{0.1}{a_h(z) z}\right)}\right]\, ,
\label{f1}\\
f_2(z)&\simeq& \frac{0.1}{ z^2}\left(\frac {15} {8} +z\right) \left[ \mu(\kappa)+a_h(z)z^2\log{\left(1+ \frac{0.1}{ a_h(z) z}\right)}\right] \, ,\label{f2}
\end{eqnarray}
where $a_h(z) \simeq \log{(\frac{M_1}{M_h(T)})}\simeq \log{(\frac{z}{0.4})}$ and $\mu (\kappa)\simeq 1$ ($2/3$) in the case of weak (strong) wash-out regime. The wash-out term $W(z)$ contains a part from the inverse decay and a part from scatterings \cite{pedes} and is given by:
\begin{eqnarray}
W(z)=W_{ID}(z) f_2(z)\, , \quad
W_{ID}(z)=\frac{1}{4} z^{3} K_{1}(z) \ .
\end{eqnarray}
The function $f_1(z)$ ($f_2(z)$) parametrizes the effect of the scatterings in the production (wash-out) factor, and the r.h.s. of eqs.(\ref{f1},\ref{f2}) comes from high-temperature approximations of the reduced-cross sections, when the scattering effects are fully relevant. In the low temperature regime, scatterings become negligible, so the functions $f_1$ and $f_2$ tend to unity.
The matrix $A_{\alpha \beta}$~\footnote{For convenience we use $A'=-A$ in the BE so that the diagonal elements are positive.} depends on which charged lepton interactions are in thermal equilibrium, and parametrizes the conversion of the leptonic asymmetry into a $B/3-L_{\alpha}$ asymmetry according to $Y_{\alpha}=\sum_{\beta} A_{\alpha \beta} Y_{\Delta \beta}$. If the temperature at which leptogenesis occurs, $M_1$, is below $10^{9}$ GeV, interactions involving charged $\mu$ and $\tau$ couplings are fast compared to the Hubble expansion rate, and are therefore in equilibrium. Thus, $\mu$ and $\tau$ flavours have to be treated separately and so the electron flavour is also distinguishable. Then one has \cite{matters}:
\begin{eqnarray}
A=-A'= \left( \begin{array}{ccc}
-151/179 & 20/179 & 20/179 \\
25/358 & -344/537 & 14/537 \\
25/358 & 14/537 & -344/537
\end{array} \right) \ .
\end{eqnarray}
For $M_1$ between $10^{9}$ GeV and $10^{12}$ GeV, only the charged $\tau$ Yukawa interactions are in equilibrium. The interactions involving the $e$ and $\mu$ flavours are slower than the expansion rate, so that those flavours are indistinguishable, and the decay of $N_1$ will generate a $Y_{ e+ \mu}$ asymmetry. The ``flavoured" asymmetries are then reduced to ($Y_{\tau}$-$Y_{e+\mu}$), and the $B-L\leftrightarrow L$ conversion matrix reads :
\begin{eqnarray}
A=-A'=\left( \begin{array}{cc}
-417/589 & 120/589 \\
30/589 & -390/589 \\
\end{array} \right)\ .
\end{eqnarray}
In a recent work~\cite{zeno}, it has been argued that the interaction rates involving the charged Yukawa couplings should be fast compared to the interactions involving the decaying $N_1$ in order to have sufficient time to project the produced lepton asymmetry onto flavour-space. It has been derived that $M_{N_{1}}$ should be below $5\times 10^{11}$ GeV for the tau-Yukawa to be in equilibrium, hence projecting the lepton asymmetry on the ($Y_{\tau}$-$Y_{e+\mu}$) space. This point will be discussed in section 4.\\A formal solution of eq. (\ref{n2}) for the $B/3-L_\alpha$ asymmetry is given by:
\begin{eqnarray}
\label{solform}
Y_{\Delta \alpha}(z)&=&-\epsilon_{\alpha} \kappa \int_{z_{in}}^{z} dx D(x)f_1(x) \Delta N_{1}(x) e^{-\kappa_{\alpha} A'_{\alpha \alpha} \int_{x}^{z} dy W(y)} \\
&-& \kappa_{\alpha} \sum_{\beta \neq \alpha} A'_{\alpha \beta} \int_{z_{in}}^{z} dx W(x) Y_{\Delta \beta}(x) e^{-\kappa_{\alpha} A'_{\alpha \alpha} \int_{x}^{z} dy W(y) } \, , \nonumber
\end{eqnarray} where $\Delta N_{1}(z)=\left(Y_{N_1}(z)-Y_{N_1}^{eq}(z)\right)$ is the departure from thermal equilibrium.
The first term in eq. (\ref{solform}) had been estimated for a vanishing initial $N_1$ abundance, $N_{1}(z_{in})=0$, and for an $N_1$ abundance initially at thermal equilibrium \footnote{We will refer to the case of a vanishing initial abundance as a dynamical case, as the population of $N_1$ is created dynamically by thermal processes. The case of a $N_1$ initially at thermal equilibrium will be refered as thermal case, even if it recquires a non-thermal production mechanism.} $N_{1}(z_{in})=N_{1}(z_{in})^{eq}$, in Refs.~\cite{pedes,matters,BdB}.
The second term, which has been neglected in these previous studies, is responsible for the interdependency of the flavours through the off-diagonal matrix elements $A'_{\alpha \beta}$.
This term drives a new contribution to the $B/3-L_{\alpha}$ asymmetry and can be relevant. In some cases the latter is in fact the dominant contribution, as we will later see.\\
We parametrize, as usual, the final asymmetry $Y_{\Delta \alpha}$ in terms of efficiency factors that contain all the dependency on the wash-out factors $\kappa$, $\kappa_\alpha$. Those effiencies $\eta_{\alpha}$ are defined by:
\begin{eqnarray}
\label{eta}
Y_{\Delta \alpha} &\equiv& - \epsilon_{\alpha}\: \eta_{\alpha}\,Y_{N_1}^{eq}(T \gg M_{N_1})\nonumber \\
&\simeq& -3.9\times10^{-3}\,\epsilon_{\alpha} \left( \eta_{\alpha}^{d}+\eta_{\alpha}^{nd} \right) \ .
\end{eqnarray}
The first term $\eta_{\alpha}^{d}$ has been derived in \cite{matters,BdB}, and its expression will be presented in the next section. The second term, $\eta_{\alpha}^{nd}$, arises from the non-diagonal conversion of a leptonic flavour, say $L_{\beta}$, into the $B/3-L_{\alpha}$ direction and its effect is also studied in the next section. It is clear from eq. (\ref{solform}) that the efficiency $\eta_{\alpha}$ of the process depends on the individual wash-out parameter $\kappa_{\alpha}$, but weighted by the factor $A'_{\alpha \alpha}$ arising from the $B-L \leftrightarrow L$ conversion. We then define $\tilde{\kappa}_{\alpha} \equiv A'_{\alpha \alpha} \kappa_{\alpha}$ (and consequently $\tilde{\kappa} =\sum_{\alpha} \tilde{\kappa}_{\alpha}$ ) as the ``real" wash-out parameter, and thus $\eta_{\alpha}=\eta(\tilde{\kappa}_{\alpha})$.
\\ The baryon asymmetry reads:
\begin{eqnarray}
Y_{\cal B}=\frac{12}{37} \sum_{\alpha} Y_{\Delta \alpha}\simeq -1.26\times10^{-3}\,\sum_{\alpha}\epsilon_{\alpha}\: \eta_{\alpha} \ ,\quad \text{with}\quad \eta_{\alpha}=\eta_{\alpha}^{d} + \eta_{\alpha}^{nd}\ .
\end{eqnarray}
The baryon asymmetry is, as the lepton asymmetry, the sum of the diagonal term proportional to $A_{\alpha \alpha}\simeq1$ and of the off-diagonal term proportional to $A_{\alpha \beta,\ \beta\neq \alpha} \simeq 1/10$. The latter contribution will be negligible for the baryon asymmetry, but will strongly modify the individual lepton asymmetries.
\section{Efficiency factors}
Here we proceed to numerically solve the BE (eqs. (\ref{n1},\ref{n2})) for different configurations of the individual CP asymmetries and wash-out factors, for distinct flavour ``alignments".
\subsection{Study of the efficiency $\eta^d$}
In this part, we neglect the off-diagonal part in the last term of eq. (\ref{n2}) and solve the BE (eqs. (\ref{n1},\ref{n2})). Depending on the initial conditions and on the strenght of the wash-out, several analytical approximations can be derived for $\eta^{d}$ (eq. (\ref{eta})).
\subsubsection{Vanishing initial $N_1$ abundance}
In the case where the population of $N_1$ is dynamically generated, i.e. $N_{1}(z_{in})=0$, one can derive the expression for $\eta^{d}$ in different wash-out regimes~\cite{matters}:
\begin{itemize}
\item all flavours in the strong wash-out case: $\kappa_{\alpha} \gg 1$
\begin{equation}
\eta^{d}(\tilde{\kappa}_{\alpha}) \simeq 3.5 \: \left( \frac{1}{6 \tilde{\kappa}_{\alpha}} \right)^{1.16}\ , \label{fort}
\end{equation}
\item all flavours in the weak wash-out case: $\kappa_{\alpha} \ll 1$\ ,
\begin{equation}
\eta^{d}(\tilde{\kappa}_{\alpha}) \simeq \, \: 1.4 \: \tilde{\kappa}_{\alpha}\ \tilde{\kappa}\: \ . \label{faible}
\end{equation}
However, within the choosen range of wash-out parameter, we find that the efficiency is better fitted with $\eta^{d}(\tilde{\kappa}_{\alpha}) \simeq \: 0.4 \: \tilde{\kappa}_{\alpha}\ \sqrt{\tilde{\kappa}} $ , as can be seen in fig.\ref{figure1}.
\item some flavours in strong $\kappa_{\alpha} \gg 1$ and some others in weak $\kappa_{\beta} \ll 1$ wash-out regimes.
In this case the efficiency for the flavour $\alpha$ is given by eq. (\ref{fort}) and the efficiency for the flavour $\beta$ is given by:
\begin{equation}
\eta^{d}(\tilde{\kappa}_{\beta}) \simeq \: 0.3 \:\tilde{\kappa}_{\beta}\ . \label{mid}
\end{equation}
\end{itemize}
The efficiency $\eta^{d}$ is then obtained by simple interpolation between these three generic cases.
For the sake of illustration, we choose 3 representative cases:
\begin{itemize}
\item Case a): all the wash-out parameters $\kappa_{\alpha}$ are equal.
\item Case b): some flavours (e.g. $\beta$) are weakly washed-out with $\kappa_{\beta}=5\times 10^{-2}$.
\item Case c): some flavours ($\beta$) are stronlgy washed-out with $\kappa_{\beta}=30$.
\end{itemize}
We checked the validity of those expressions in the considered range of wash-out parameters, $\kappa_{\alpha}$ between $10^{-2}$ and $10^{2}$.
It is interesting to notice the dependence of $\eta_{\alpha}^{d}$ on the total wash-out parameter $\kappa$ in eq.~(\ref{faible}).
From this term, a flavour $\alpha$ that is weakly washed-out will be sensitive to the wash-out of the other flavours implying that
there is a correlation of the flavours. This can be seen in figure \ref{figure1}, where we represent the efficiency of a given flavour ($\alpha$) as a function of the respective wash-out parameter $\kappa_{\alpha}$, for the three representative cases a), b) and c) discussed above. The democratic scenario, case a), which is similar to the one-flavour approximation, is represented in red and the misaligned cases b) and c) are represented in blue and green, respectively. For comparison, we also represent the analitycal estimates (dashed lines) of the efficiencies, eqs. (\ref{fort}-\ref{mid}).
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.25]{figure1.eps}}
\caption{\small Efficiency $\eta^{d}(\tilde{\kappa}_{\alpha})$ in the dynamical case, for specific values of
$\kappa_{\beta}$, $\beta\neq\alpha$. Case a) (red curves):~$\kappa_{\beta}~=~\kappa_{\alpha}$~;~case b) (blue curves):~$\kappa_{\beta}=5\times 10^{-2}$;~case c) (green curves):~$\kappa_{\beta}=30$. In each case, the solid lines represent the numerical computation and the dashed ones the results of the analytical approximation. In the left panel, the upper (dotted) curve corresponds to the analytical estimates of the efficiency in the weak wash-out regime eq.~(\ref{faible}).}
\label{figure1}
\end{figure}
We see that the agreement between the numerical and the analytical results is very good.\\
Firstly, in the case where the right handed neutrino population is created by inverse decays and scatterings, the wash-out factor which parametrizes the strenght of those thermal production, has to be non-negligible in order to produce a sufficient amount of $N_1$. Therefore, the efficiency is maximized in this dynamical case for $\kappa_{\alpha} \simeq 1$ with $\eta_{dyn.}^{max}\simeq 0.2$. \\
Secondly, and this is the main point here, we see that the efficiency of the process when the flavour $\alpha$ is weakly washed-out does indeed depend on the strenght of the wash-out of the other flavours. For example, for $\kappa_{\alpha}\simeq 5\times 10^{-2}$ (and assuming $A'_{\alpha \alpha}\simeq A'_{\beta \beta} \simeq 1$), we roughly have $\eta_{\alpha}^\text{b)}\simeq \eta_{\alpha}^\text{b)} \simeq3\times 10^{-3}$ and $\eta_{\alpha}^\text{c)}\simeq 1.5\times 10^{-2}$.
The enhancement $\eta^\text{c)}$ comes from the fact that $\kappa \simeq 10$ but still $\kappa_{\alpha}\lesssim 1$, and the eq.(\ref{mid}) applies. For cases $a)$ and $b)$, eq.(\ref{faible}) applies, and we have an extra supression from the factor $\kappa \simeq 10^{-1}$. Looking at the strong wash-out regime, $\kappa_{\alpha} > 1 $ , we see that the efficiency of the flavour $\alpha$ does not depend on the wash-out of the other flavours.
\subsubsection{Thermal initial abundance}
In the case where the population of $N_1$ is initially in thermal equilibrium, $N_{1}(z_{in})=N_{1}(z_{in})^{eq}$, the computation of the efficiencies is modified. Following \cite{pedes,BdB}, one has for the efficiency
factors:
\begin{eqnarray}
\label{eqth}
\eta^{d}(\tilde{\kappa}_{\alpha})&\simeq&\frac{2}{\tilde{\kappa}_{\alpha}\,z_{B}(\tilde{\kappa}_{\alpha}) f_{1}(z_{B}(\tilde{\kappa}_{\alpha})) }\:\left(1-e^{-\frac{1}{2} \, \left[ {\tilde{\kappa}_{\alpha}\,z_{B}(\tilde{\kappa}_{\alpha})\, f_{1}(z_{B}(\tilde{\kappa}_{\alpha}))}\right]}\right) \ ,
\end{eqnarray}
where
\begin{eqnarray}
z_{B}(\tilde{\kappa})&\simeq&2+4 \ \tilde{\kappa}^{\, 0.13} \ e^{-2.5\, / \,\tilde{\kappa}}\ .
\end{eqnarray}
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.25]{figure2.eps}}
\caption{\small Efficiency $\eta^{d}(\tilde{\kappa}_{\alpha})$ in the case of a thermal initial $N_1$ abundance, for specific values of
$\kappa_{\beta}$, $\beta\neq\alpha$. Line and color code as in figure \ref{figure1}.}
\label{grapheTeff}
\end{figure}
The study of the thermal case is shown in figure \ref{grapheTeff}, where we again consider the 3 cases a), b) and c), with the same colour code as in figure \ref{figure1}. The difference from the case of a vanishing initial $N_1$ is striking in the weak wash-out regime: besides the obvious behaviour of $\eta$ that is maximized for small wash-out, $\eta_{ther.}^{max}\simeq 1$ for $\kappa_{\alpha} \ll 1$, we see that the effect of flavour is negligible, except for $\kappa_{\alpha}\simeq1$, where a small distinction of the cases a), b) and c) is possible. In the strong wash-out regime, the individual efficiencies do not depend on the wash-out of other flavours, as in the dynamical case. Furthermore, in this strong wash-out regime, there is no distinction between the thermal and the dynamical cases, as can be seen in the left panel of figure \ref{figure4} which is included in the discussion of the next section. We also see that the agreement between the analytical result for the efficiency given in eq. (\ref{eqth}) and the numerical result is very good.
\subsection{Study of $\eta^{nd}$}
Now we study the effect of the non-diagonal terms $A_{\alpha \beta}\ , \alpha \neq \beta$, that exhibit the interplay between different flavours. This interplay can be seen in the last term of eq. (\ref{solform}):
\begin{eqnarray}
\label{depen}
-Y_{\Delta \alpha}^{nd}=3.9\times 10^{-3}\,\epsilon_{\alpha} \eta_{\alpha}^{nd}&=&\kappa_{\alpha} \sum_{\beta \neq \alpha} A'_{\alpha \beta} \int_{z_{in}}^{z} dx\,W(x) Y_{\Delta \beta}(x)\,e^{-\kappa_{\alpha} A'_{\alpha \alpha} \int_{x}^{z} dy W(y) }\ .
\end{eqnarray}
One can find an approximate expression for this term: considering that the variations of $Y_{\Delta _\beta}(x)$ are negligible compared to the rest of the integrand, one can approximate it to its final value $Y_{\Delta _\beta}(x)\simeq Y_{\Delta _\beta}(\infty)$, and can thus be factorized out from the integral. Another approximation is to consider that $Y_{\Delta _\beta}$ is mainly generated by the diagonal part, that is we neglect the effects of $\mathcal{O}(A_{nd}^{2})$ that are corrections of the order of a few percent. We obtain:
\begin{eqnarray}
3.9\times10^{-3}\,\epsilon_{\alpha}\eta_{\alpha}^{nd}&\simeq&\kappa_{\alpha}\, \sum_{\beta \neq \alpha}\, A'_{\alpha \beta}\, Y_{\Delta _\beta}^{d}\:f_{c}( \tilde{\kappa}_{\alpha} )\, ,
\end{eqnarray}
where
$\tilde{\kappa}_{\alpha}=\kappa_{\alpha} A'_{\alpha \alpha}$ and $f_c(\tilde{\kappa}_{\alpha})$ is given by:
\begin{eqnarray}
\label{offdiag}
f_{c}(\tilde{\kappa}_{\alpha})&=& \int_{z_{in}}^{\infty} dx \ W(x)\ e^{-\tilde{\kappa}_{\alpha} \int_{x}^{\infty} dy W(y) }\\
&\simeq&1.3\,\frac{1}{1+0.8\times\tilde{\kappa}_{\alpha}^{1.17}}\ .
\end{eqnarray}
The total efficiency of a given flavour is the sum of the contribution from the diagonal part of $A$ which, contains slight contamination from the other flavours, and from the non-diagonal part, that will be responsible, as we will see, for a huge modification of the total efficiency, in some cases becoming dominant compared to the diagonal contribution,
\begin{eqnarray}
\eta(\tilde{\kappa}_{\alpha})=\eta^{d}(\tilde{\kappa}_{\alpha})+\kappa_{\alpha}\:f_{c}( \tilde{\kappa}_{\alpha} )\, \sum_{\beta \neq \alpha}\, A_{\alpha \beta}\, \frac{\epsilon_{\beta}}{\epsilon_{\alpha}}\:\eta^{d}(\tilde{\kappa_{\beta}})\, .
\end{eqnarray}
The effect of the non-diagonal part on the asymmetry $Y_{\Delta_{\alpha}}$ depends on the wash-out and on the CP asymmetries of the different flavours. For example, if we consider the flavour $\alpha$, the asymmetry produced by the diagonal part proportional to $ \eta^{d}$ depends on the strenght $\tilde{\kappa}_{\alpha}$.
If $\tilde{\kappa}_{\alpha} \gg 1$, then $Y_{\Delta_{\alpha}}$ is strongly washed-out and therefore is too small, as can be seen in figure \ref{figure1}: for $\kappa_{\alpha}\gtrsim 100$ we have $\eta^{d}(\tilde{\kappa}_{\alpha})\lesssim 10^{-3}$. Now consider the non-diagonal part: it depends on the strenght of $\tilde\kappa_{\beta}$. If the wash-out of the flavour $\beta$ is weak (or even mild), then $\eta^{d}(\tilde\kappa_{\beta})$ will be close to its maximal value, and then one has in this case:
\begin{eqnarray}
\eta(\tilde{\kappa}_{\alpha})&\propto& \frac{\epsilon_{\beta}}{\epsilon_{\alpha}}\:
\eta^{d}_{\beta \, \text{max}}\times\left(10^{-1}-10^{-2}\right) \nonumber \ ,
\end{eqnarray}
the value of $\eta_\text{max}$ depending on the inital conditions.
Typically, this effect can, in favourable situations, drive up the asymmetry $Y_{\Delta_{\alpha}}$ by one or two orders of magnitude as can be seen in the different panels of figure \ref{grapheMeff}. There we represent the efficiency factor of a given asymmetry $\eta_{\alpha}$ as a function of the wash-out parameter $\kappa_{\alpha}$, for the different alignments of the flavours.
\begin{figure}[ht]
\begin{center}
\hspace{-1.5cm}
\includegraphics[scale=0.25]{figure3haut.eps} \\
\hspace{-2cm}
\includegraphics[scale=0.25]{figure3bas.eps}
\end{center}
\vspace{-0.9cm}
\caption{}
{\small
Influence of the other flavours on the efficiency $\eta(\tilde{\kappa}_{\alpha})$. Upper panels: case of a vanishing initial abundance. Lower panels: case of a thermal initial abundance. The curves represent different inputs for the wash-out parameters $\kappa_{\beta}$, $\beta\neq\alpha$: for the case a) (red curves) we have a democratic scenario where $\kappa_{\beta}=\kappa_{\alpha}$, while for the case b) (blue curves) $\kappa_{\beta}=5\times 10^{-2}$ and for the case c) (green curves) $\kappa_{\beta}=30$. The solid lines represent the numerical computation and the dashed ones the results of the analytical approximation.
}
\label{grapheMeff}
\end{figure}
We clearly see that the off-diagonal terms of $A$ modify the efficiency $\eta_{\alpha}$ in the strong-wash-out case $\kappa_{\alpha}\gg1$. Another consequence of this dependence of $\eta^{nd}_{\alpha}$ on the wash-out of the other flavours is illustrated in figure \ref{figure4}.
\begin{figure}[ht]
\hspace{-1.5cm}
\begin{center}
\includegraphics[scale=0.3]{figure4.eps}
\end{center}
\vspace{-0.8cm}
\caption{}
{\small
Influence of the other flavours on the efficiency $\eta(\tilde{\kappa}_{\alpha})$. The left panel represents the diagonal contribution to the efficiency, whereas the right panel represents total, the diagonal and non-diagonal contributions. We represent the case of a zero and thermal initial $N_1$ abundance, for the three cases a), b), c). Line and color code as in figure \ref{figure1}.
}
\label{figure4}
\end{figure}
We see that when we neglect the off-diagonal terms there is no distinction in the strong wash-out regime between the dynamical and the thermal case. That is, in the strong wash-out regime, the efficiency is independent of the thermal history of the decaying $N_1$. However, when we include the off-diagonal terms, we see that for the case c), the two efficiencies do not coincide anymore in the strong wash-out regime, as each $\eta(\tilde{\kappa}_{\alpha})$ is related to $\eta^{d}(\tilde{\kappa}_{\beta})$, the latter strongly depending on whether the $N_1$ was initially at thermal equilibrium or had a vanishing abundance. This clearly illustrates the effect of flavours in leptogenesis.
\\
The above discussion is based on a strong approximation ($Y_{\Delta _\beta}(x)\simeq Y_{\Delta _\beta}(\infty)$), which was made in order to quantify the off-diagonal effect in eq.~(\ref{depen}).
However this excessively naive approximation does not describe all wash-out and CP asymmetry configurations. Furthermore,
the dynamics of the flavours $\beta\neq \alpha$ is neglected and in particular, in the case of a strongly washed-out flavour $\beta$, this approximation cannot be used.
We thus solve the BE, eqs. (\ref{n1}), (\ref{n2}), including the off-diagonal terms of the $A$ matrix and
illustrate in figure \ref{figure5} contours of the ratio $Y_{e+\mu}/Y_{\tau}$ (absolute value) in logarithmic scales, for the thermal and dynamical cases, as function of the ratio of the flavoured CP asymmetries $\epsilon_{e+\mu}/\epsilon_{\tau}$ and of the flavoured wash-out parameters $\kappa_{e+\mu}/\kappa_{\tau}$. At first sight, we see that $Y_{e+\mu} \geq (\leq) Y_{\tau}$ for $\epsilon_{e+\mu} \geq (\leq) \epsilon_{\tau}$.
Looking in more detail, we see that the wash-out parameters influence the former statement. For example, in the thermal case, for $\kappa_{e+\mu}/\kappa_{\tau}\simeq 0.1$ and $\epsilon_{e+\mu}/\epsilon_{\tau}\simeq 0.5$, we have $Y_{e+\mu}\sim 2 Y_{\tau}$.
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.25]{YsurYtFINAL.eps}}
\caption{\small Contour plots of the ratio of the individual flavoured asymmetries, plotted as a function of the ratio of wash-out parameter versus the ratio of CP asymmetries. We place ourselves in a 2 flavour case with $M_{N_1}=5\times10^{9} \,{\rm GeV}$, $\epsilon_{e+\mu}+\epsilon_{\tau}=10^{-6}$ and $\kappa_{e+\mu}+\kappa_{\tau}=10$. The left (right) panel stands for the dynamical (thermal) case. On both graphs, the black area denotes the equality $Y_{e+\mu}=Y_{\tau}$. Moving to the left of this countour, $Y_{e+\mu}<Y_{\tau}$ whereas to the right we have $Y_{e+\mu}>Y_{\tau}$.}
\label{figure5}
\end{figure}
\subsection{The baryon asymmetry}
Summing-up the contribution from the diagonal and off-diagonal parts of the $A$ matrix, we finally obtain the baryon asymmetry:
\begin{eqnarray}
Y_{\cal B} &=&\frac{12}{37} \sum_{\alpha} Y_{\Delta {\alpha}}\simeq -1.26 \times 10^{-3} \sum_{\alpha} \epsilon_{\alpha}\,\eta_{\alpha} \, .
\end{eqnarray}
For $\epsilon\equiv \sum_{\alpha} \epsilon_{\alpha}\neq 0$, one can define an efficiency for the baryon asymmetry $\eta_B$ such that:
\begin{eqnarray}
Y_{\cal B} &=&-1.26\times10^{-3} \epsilon \, \eta_{ B} \, .
\end{eqnarray}
The baryon asymmetry will be the sum of the individual lepton asymmetries. Therefore, the baryon asymmetry, and hence the efficiency $\eta_{B}$, will depend on the alignment of the flavours. We illustrate this point in figure \ref{yb}, where the efficiency $\eta_{B}$ is plotted as a function of the wash-out parameter $\kappa_{\alpha}$, for cases a), b) and c). The solid lines represent the diagonal contributions, and the dashed ones represent both diagonal and non-diagonal contributions. As the baryonic efficiency can be defined only for $\epsilon \neq 0$ and is strongly dependent on the flavoured CP violation $\epsilon_{\alpha}$, we set $\epsilon_{\alpha}=8\times10^{-7}$ for all flavours.
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.3]{figure6old.eps}}
\caption{Effects of the off-diagonal terms of the $A$ matrix on the total baryon efficiency $\eta_{\cal B}$.}
\label{yb}
\end{figure}
First, we consider only the diagonal contributions. As expected, we see that the efficiency strongly depends on the flavour alignment. The democratic scenario (case a), in red) shows the one-flavour approximation-like behaviour. The flavours play a full role in the misaligned case, and especially in case b) (in blue), where we have $\kappa_{\beta}=5\times10^{-2}$. Then the efficiency for the flavour(s) $\beta$ is (are) close to its (their) maximum, depending on the thermal history of $N_1$. If the flavour $\alpha$ is strongly washed-out, the baryon asymmetry is mainly composed of the flavour(s) $\beta$, and is thus weakly washed-out, even if the total wash-out is strong. This is the very effect of taking into account the flavours in leptogenesis. We also see that the baryonic efficiency, as well as the leptonic one, depend on the initial abundance of $N_1$ in the strong wash-out regime.\\
Now, let us consider the dashed lines, which represent the sum of the diagonal and off-diagonal terms of $A$. We see that the off-diagonal terms account for percent corrections. A more detailed analysis of their effect on $Y_{B}$ is shown in fig.\ref{figureYb}, where we represent contours of $Y_{B}^{\text{ total}}/Y_{B}^{\text{diagonal}}$ in the interesting two flavour-case. We choose the individual CP asymmetries to be equal, $\epsilon_{e+\mu}=\epsilon_{\tau}(=10^{-6}$ but the actual value is not relevant). We see that the off-diagonal terms affect the baryon asymmetry up to $40\%$ in the dynamical case in both ways (increasing and decreasing). On the other hand, the thermal case is only enhanced, up to the same amount. Effects of the off-diagonal terms are particularly important in the strong wash-out regime.
\begin{figure}[ht]
\hspace{-0.6cm}
\includegraphics[scale=0.4]{GAllcomp.eps}
\vspace{-0.8cm}
\caption{}
{\small
Effects of the off-diagonal terms on the baryon asymmetry: contour plot of $Y_{B}^{\text {total}}/Y_{B}^{\text {diag.}}$ for fixed CP asymmetries $\epsilon_{e+\mu}=\epsilon_{\tau}$ and varying wash-out parameters. The left (right) panel stands for the case of a vanishing (thermal) initial $N_{1}$ abundance.
}
\label{figureYb}
\end{figure}
In some particular cases the non-diagonal terms also have a relevant role, namely in the democratic scenario, where $\tilde{\kappa}_{\alpha}$ are equal, and where the total CP asymmetry is zero $\sum_{\alpha} \epsilon_{\alpha}=0$. Then ignoring the $\mathcal{O}(A)$ effect will lead to a vanishing baryonic asymmetry, as $\sum_{\alpha} \epsilon_{\alpha}\: \eta_{\tilde{\kappa}_{\alpha}}^{d}=0$ \cite{lownrjCPV}. On the contrary, taking into account those effects avoids the latter cancellation. For example, in the case of three distinguishable flavours with the specific alignment in which $\tilde{\kappa}_{\alpha}=\tilde{\kappa}_{\beta}=\tilde{\kappa}_{\delta}$ and $\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{\delta}\equiv\epsilon_{\alpha}(1+b+d)=0 $, one finds for the baryon asymmetry :
\begin{eqnarray}
Y_{\cal B}=-1.26\times10^{-3}\,\epsilon_{\alpha}\:\eta^{d}(\tilde{\kappa}_{\alpha})\left(\sum_{\alpha_{1}\neq \alpha}\kappa_{\alpha_{1}}\:f_{c}(\tilde{\kappa}_{\alpha_{1}}) A_{\alpha_{1}\,\alpha}+b \sum_{\beta_{1}\neq \beta}\kappa_{\beta_{1}}\:f_{c}(\tilde{\kappa}_{\beta_{1}}) A_{\beta_{1}\,\beta}+d \sum_{\delta_{1}\neq \delta}\kappa_{\delta_{1}}\:f_{c}(\tilde{\kappa}_{\delta_{1}}) A_{\delta_{1}\,\delta} \right) \ .\nonumber
\end{eqnarray}
In order to maximize the above function, we take $\alpha=\mu$, $\beta=e$ and $\delta=\tau$ for a positive value of $b$ (in the case of a negative value of $b$, $\alpha=\mu$, $\beta=\tau$ and $\delta=e$). We then get $ Y_{\cal B}\simeq\epsilon_{\mu}\:\eta^{d}(\tilde{\kappa}_{\mu})\:f_{c}(\tilde{\kappa}_{\mu})\:\kappa_{\mu} \times4.5\:\times 10^{-5}\left(1+2 b\:\right) $,
which is non-zero for $b\neq -1/2$.
\section{Lower bound on $M_{N_{1}}$ }
When the flavours are taken into account in leptogenesis, the modification of the asymmetry, combined with the change in the efficiency factor may have an impact on the lower bound of the mass of ${N_{1}}$. From the bound on each individual CP asymmetry~\cite{issues},
\begin{eqnarray}
\epsilon_{\alpha}\lesssim \frac{3 M_{N_{1}} m_\text{max}}{16 \pi v^{2}} \sqrt{\frac{\kappa_{\alpha}}{\kappa}}\ ,
\end{eqnarray}
one has
\begin{eqnarray}
\vert Y_{\cal B} \vert &\simeq& 1.26\times 10^{-3}\sum_{\alpha}\,\epsilon_{\alpha}\:\eta_{\alpha} \nonumber \\
&\lesssim&1.26\times10^{-3}\,\frac{3\,M_{N_{1}}\,m_\text{max}}{16 \pi v^{2}}\sum_{\alpha} \sqrt{\frac{\kappa_{\alpha}}{\kappa}}\:\eta_{\alpha}\, ,
\end{eqnarray}
from which a lower bound on $M_{N_1}$ is derived,
\begin{eqnarray}
\label{boundM1}
M_{N_1}&\gtrsim &\frac{16 \pi }{3\times 1.26\times10^{-3}}\frac{v^2}{m_\text{max}} \frac{\vert Y_{\cal B} \vert }{\sum_{\alpha} \sqrt{\frac{\kappa_{\alpha}}{\kappa}}\:\eta_{\alpha}} \\
M_{N_1} &\gtrsim& 7.1\times10^{8} \,{\rm GeV} \left( \frac{m_{\rm{atm}}}{m_\text{max}} \right)\,\left| \frac{Y_{\cal B}}{Y_{\cal B}^{CMB}} \right| \frac{1}{\sum_{\alpha} \sqrt{\frac{\kappa_{\alpha}}{\kappa}}\:\eta_{\alpha}}\, .
\end{eqnarray}
Since the lower bound on $M_{N_1}$ is inversely proportional to the efficiency $\eta_{\alpha}$, it will therefore depend on the thermal history of the decaying right-handed neutrino. In the case where $N_1$ are produced by scatterings, the efficiency is maximized for a wash-out $\kappa_{\alpha}\simeq 1$, where $\eta_{\alpha}\simeq 0.2$. In the case where $N_1$ are non-thermally produced, the efficiency is maximized to $1$ for a very weak wash-out $\kappa_{\alpha} \ll 1$. The lower bound will depend on the alignement of flavours, and in the democratic case one has:
\begin{eqnarray}
\label{bound}
M_{N_1}\gtrsim \left\lbrace
\begin{array}{l}
4.1\times10^{8} \,{\rm GeV}\: \ {\rm in\: the\: thermal\: case}\\
2.5\times10^{9} \,{\rm GeV} \:\ {\rm in\: the\: dynamical\: case} \, .
\end{array} \right .
\end{eqnarray}
This bound is close to the one derived in the ``one flavour approximation", where $M_{N_1} \gtrsim 2.1\times10^{8} \,{\rm GeV}$ in the thermal case and $M_{N_1} \gtrsim 1.05\times10^{9} \,{\rm GeV}$ in the dynamical one~\cite{bound}. Besides flavour effects, the difference between the lower bounds of the flavoured and unflavoured cases comes from a different factor in the $B-L\leftrightarrow B$ conversion. Indeed, in the one flavour dominance, the Davidson-Ibarra bound reads \cite{DI}:
\begin{eqnarray}
\epsilon \leq \frac{3}{8 \pi} \frac{M_{N_{1}} m_{max}}{v^{2}} \ ,
\end{eqnarray}
and the conversion from sphalerons is $ Y_{B}=28/51 Y_{L} $. Therefore the lower bound on $M_{N_{1}}$ in the unflavoured case is $\sim$ $1/2\times12/37\times 51/28 \sim 0.3$ times the flavoured one. \\
An implication of this bound resides in the well-known conflict between the reheating temperature and leptogenesis. Indeed, $T_{RH}$ should be above $M_{N_1}$ in order to avoid the erasing of the lepton asymmetry. In view of the above estimates, $T_{RH} \lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$} 4\times 10^{9(8)} \,{\rm GeV}$ in the dynamical (thermal) case. In this case, the inclusion of flavours does not really help.
In the regime of strong wash-out, $\kappa\gg1$, where the effective neutrino mass is close to the mass inferred from atmospheric oscillations, the situation changes. In the one-flavour approximation, the efficiency $\eta(\kappa)\propto \kappa^{-1}$, therefore $M_{N_1}^{\rm min}\propto \kappa$, and increases with the wash-out, and so does the reheating temperature. In this strong wash-out regime, we roughly estimate $T_{RH}\geq M_{N_1}/10 \simeq 9\times10^{8} \,{\rm GeV} \: \kappa$, $\kappa\gg1$~\cite{pedes}. Including flavours, one generically has non-alignment of the individual asymmetries, and one can have a strong total wash-out, with some weakly washed-out flavours. If for example (see fig. \ref{M1m1}), $\kappa\simeq m_{\text{atm}}/m^{*}\simeq 45$ with $\kappa_{\beta}\simeq 40$ but $\kappa_{\alpha}\simeq 5$, one has $M_{N_1}^{\rm min}\simeq 2\times10^{10} \,{\rm GeV}$ for the flavoured case and $8\times 10^{10} \,{\rm GeV}$ for the unflavoured case. Hence, including flavours, the reheating temperature in this strong wash-out regime is lowered by a factor $\simeq 4$, and one roughly has $T_{RH} \geq 2\times10^{9} \,{\rm GeV}$.
\subsection{Numerical results}
We numerically solve the set of coupled BE (eqs.(\ref{n1}\ref{n2})) and obtain the allowed parameter space. The input parameters of the BE are the Yukawa couplings $\lambda$, which define the wash-out parameters and the CP asymmetries. These have been built using the Casas-Ibarra parametrization \cite{CI}. In this parametrization the matrix $\lambda$ reads:
\begin{equation}
\lambda=M_{N}^{1/2}\,R\, m^{1/2} U^{\dagger } v^{-1} , \label{yukawa}
\end{equation}
$U$ being the PMNS matrix parametrized by three angles and three phases. The solar and atmospheric angles
are well measured, whereas the $\theta_{13}$ angle (Chooz angle) is only upper-constrained. The three CP violating phases are yet undetermined. The matrix $R$ is a $3\times3$ orthogonal matrix depending on three complex angles. We impose a normal hierarchical spectrum, both for the light and for the heavy neutrino sectors, so that the only free parameters that enter in the mass matrix are the lightest neutrino masses $m_{1}$ and $M_{N_1}$. Therefore, we have 12 independent variables that are not experimentally determined.
In our numerical computation, we scan over the whole parameter space by randomly choosing the free parameters. Moreover, we impose a perturbative limit $\lambda_{33} \lesssim y_{t}$ ($y_{t}$ being the top Yukawa coupling) that will upper-constraint $M_{N_{1}}$, leading to the upper bound $M_{N_1}\lesssim 4\times10^{11} \,{\rm GeV}$.
From \cite{zeno} and in order for the flavour to be relevant in leptogenesis, an upper bound on $M_{N_{1}}$ can be derived, namely $ M_{N_1}\lesssim 5.8\times10^{11} \,{\rm GeV}$. Here, we only consider cases were the interactions involving the charged lepton Yukawas are faster than the inverse decay, so that flavours are fully relevant. Finally, by imposing the obtained baryon asymmetry to be in the experimental range \footnote{The experimental bounds come from cosmological constraints: the lower bound corresponds to the Big-Bang nucleosynthesis (BBN) constraint on the light species abundance, and the upper-bound from cosmic background radiation (CBR) constraints \cite{BBN}. The WMAP~\cite{wmap} constraints are included therein.}, $5.2\times10^{-10} \lesssim 7.04\times Y_{\cal B} \lesssim 7.2\times10^{-10}$, we represent in figure \ref{M1m1} the ($M_{N_{1}}-\tilde{m}$) parameter space allowed by the requirement of a sucessfull leptogenesis.
\begin{figure}[htb]
\hspace{0.4cm}{
\includegraphics[scale=0.4]{figure6haut.eps}}\\
\hspace{0.4cm}{ \includegraphics[scale=0.4]{figure6bas.eps}}
\vspace{-0.8cm}
\caption{}
{\small
Successfull leptogenesis: bound on $M_1$, in the case of a zero (thermal) initial $N_1$ abundance for the up (down) panels. The left panels show the allowed ($M_{N_1}$-$\tilde{m}$) parameter space for the one-flavour approximation, whereas the right panels stand for the case when lepton flavours are taken into account. The vertical lines represent $\sqrt{\Delta m^{2}_\text{atm}}$ (in blue) and $\sqrt{\Delta m^{2}_\text{sol}}$ (in green).
}
\label{M1m1}
\end{figure}
Many remarks are in order. Firstly, the lower bound numerically derived confirms the one given in eq. (\ref{bound}), and we notice that the ``one flavour approximation" (left panels in figure \ref{M1m1}) lowers the bound compared to the correct flavour treatment (right panels). Secondly, comparing the dynamical and thermal cases, that is the up and down panels in figure \ref{M1m1}, we see that $\tilde{m}$ can take much smaller values in the case of a thermal initial abundance. As this mass encodes the out-of-equilibrium condition, cf. eq. (\ref{washoutfactor}), this means that in the thermal case, leptogenesis occurs even for extremely small values of $\tilde m$,
while in the dynamical case, this is not possible.
Finally, comparing the left and right panels, one clearly sees the effect of flavour in the ``re-opening" of the parameter space for higher values of $\tilde{m}$. Indeed, by introducing the lepton flavour asymmetries, we relax the one-flavour approximation that corresponds either to a common behaviour of the individual asymmetries, or to the dominance of one flavour. Now, other configurations are allowed, and the misaligned ones widen the parameter space. The mass $\tilde{m}$ is related to the total wash-out, that is, to the sum of each individual wash-out. In the one-flavour approximation, a high value of $\tilde{m}$ corresponds to a strong wash-out and is disfavored by leptogenesis. On the contrary we observe (cf. figure \ref{yb}) that even if the total wash-out is strong, we can still have flavours that are weakly washed-out, hence dominating the baryon asymmetry and allowing a successfull leptogenesis. Thus, by the inclusion of flavour in leptogenesis no upper-bound on $\tilde{m}$ can be derived.\\ Notice that for $\tilde{m} (m_{1}) \gtrsim \text{atm}$ in figure \ref{M1m1}(\ref{M1m1f}), the points drop below the upper-bound $M_{N_1}\simeq 5\times 10^{11} \,{\rm GeV}$. Indeed, as $m_1\gtrsim \text{atm}$ , $m_{max}\simeq m_{1}\simeq m_{2}\simeq m_{3}$, and the upper-bound on $M_{N_1}$ scales as $1/m_{1}$, c.f eq. (\ref{boundM1}). \\
In the one flavour approximation, a bound on the light neutrino mass was derived in \cite{bound}, and this no longer holds when flavours are accounted for \cite{issues}.
However, in \cite{FlavOsc} a bound on the neutrino mass scale of about $2 \,{\rm eV}$ is derived in the flavoured leptogenesis context in the strong wash-out regime and hierarchical wash-out factors $1\ll\kappa_\alpha \ll \kappa_\beta$ and equal CP-asymmetries.
In this work, we impose $m_{\nu}$ to be lighter than the cosmological bound $\sum m_{\nu} \lesssim 1 \,{\rm eV}$ and we do not explore configurations leading to higher $m_{\nu}$.
This can be seen in figure \ref{M1m1f}, which represents the allowed parameter space ($M_{N_{1}}$-$m_1$ ) in different cases. The black points are the result when flavours are included, whereas red ones represent the one-flavour approximation. We clearly see that the cosmological bound is saturated when flavours are considered, and this does not occur in the one-flavour approximation.
In figure \ref{M1m1f}, for $m_1$ above $m_{\text{atm}}$ , the solutions have in general a specific flavour alignement: the flavoured CP-asymmetries are almost equal $\epsilon_{e+\mu}\sim \epsilon_{\tau}$ and the individual wash-out factors are hierarchical $1\ll\kappa_\alpha\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$} 10 \kappa_\beta$ and the total wash-out is strong. It is well known that such configurations of wash-out parameters achieve sucessfull leptogenesis in the flavoured case whereas the unflavoured one fails. For such specific configurations, effect of the off-diagonal terms of the $A$-matrix on $Y_{B}$ is maximisal (c.f fig \ref{figureYb}) but nevertheless is only a correction without important impact.
\begin{figure}[ht]
\hspace{0.2cm}
\includegraphics[scale=0.4]{figure7.eps}
\vspace{-0.8cm}
\caption{}
{\small
Successfull leptogenesis: $M_1$-$m_{1}$ space, in the dynamical case (left panel) and in the thermal case (right panel). The vertical lines represent $\sqrt{\Delta m^{2}_\text{atm}}$ (in blue) and $\sqrt{\Delta m^{2}_\text{sol}}$ (in green).
}
\label{M1m1f}
\end{figure}
\section{Conclusion}
The behaviour of individual lepton asymmetries in the case of vanishing initial $N_1$ abundance has been analysed in~\cite{matters}. In this study we give semi-analytical results including fine-tuning corrections that depend on flavour alignment. We extend the study to the case of $N_1$ initially in thermal equilibrium, and confirm that in this case, when off-diagonal entries of the conversion $B/3-L_{\alpha}\leftrightarrow L$ are neglected, the efficiency factor for a given flavour is independent of the wash-out of other flavours.\\ Independently of the thermal history of the decaying right-handed neutrino, we observed that misalignment of flavours can greatly enhance the baryon asymmetry, when compared to the one-flavour approximation, for an identical wash-out strenght.We also include off-diagonal entries of the $B/3-L_{\alpha}\leftrightarrow L$ conversion that couple flavours among themselves.
Even if this inclusion only modifies the baryon asymmetry by a few percent, thus allowing to safely disregard these terms for $Y_{B}$ computation, we nevertheless stressed that the individual lepton asymmetries are very sensitive to such interdependencies.
Finally, we studied the lower bound on the $N_1$ mass and the leptogenesis allowed parameter space. We confirm the lower bound to be $\sim 4\times 10^{8 (9)}$ GeV for a thermal (vanishing) initial $N_1$ abundance. We have also shown that the parameter space is enlarged, as the flavour (mis)alignment allows for higher values of the wash-out (or equivalently of $\tilde{m}$).
\section*{Acknowledgements}
The authors wish to thank S.~Davidson for enlightening discussions and A.M. Teixeira for usefull comments and for reading the manuscript. This project is partly supported by the ANR project NEUPAC.
|
2,869,038,155,636 | arxiv | \section*{Preliminaries}
Let $X$ denote a set and $x \in X$ denote its elements. For notational convenience random variables are not distinguished---probability measures on $X$ are denoted $P(X)$. The Cartesian product is denoted $\times$, and for any object $V_i = V_{i1} \times \ldots \times V_{in}$, $\overline{V_i}$ shall denote the family of component sets of $V_i$, $\overline{V_i} = \{ V_{i1}, \ldots, V_{in} \}$. The cardinality of $X$ is denoted $|X|$. The powerset is denoted $\mathcal{P}$. Herein it has two uses. Frequently, in order to express input-output conditions for a learning system we will only use its input-output representation $S:D \times X \to Y$. In contexts where $S \subset \times \{A, D, \Theta, H, X, Y\}$, we use $(d, x, y) \in \mathcal{P}(S)$ to make reference to the input-output representation. Also, the subset of the powerset of a powerset $K \subset \mathcal{P}(\mathcal{P}(D \cup \Theta))$ is used to denote that $K$ can be $\subset D$, $\subset \Theta$, or $\subset D \times \Theta$, etc., i.e., to make reference to ordered pairs. Often, we make reference to $d \in D$ to say a particular set of data $d$ from the larger set $D$. Additionally, for a system $S \subset X \times Y$, when we discuss $x \in X$ or $y \in Y$ it is assumed that $(x, y) \in S$ unless stated otherwise. This is to save the reader from the pedantry of Mesarovician abstract systems theory.
\section{Introduction}
Transfer learning, unlike classical learning, does not assume that the training and operating environments are the same, and, as such, is fundamental to the development of real-world learning systems. In transfer learning, knowledge from various \emph{source} sample spaces and associated probability distributions is \emph{transferred} to a particular \emph{target} sample space and probability distribution. Transfer learning enables learning in environments where data is limited. Perhaps more importantly, it allows learning systems to propagate their knowledge forward through distributional changes.
Mechanisms for knowledge transfer are a bottleneck in the deployment of learning systems. Learning in identically distributed settings has been the focus of learning theory and machine learning research for decades, however, such settings represent a minority of use cases. In real-world settings, distributions and sample spaces vary between systems and evolve over time. Transfer learning addresses such differences by sharing knowledge between learning systems, thus offering a theory principally based on distributional difference, and thereby a path towards the majority of use cases.
Existing transfer learning frameworks are incomplete from a systems theoretic perspective. They focus on domain and task, and neglect perspectives offered by explicitly considering system structure and behavior. Mesarovician systems theory can be used as a super-structure for learning to top-down model transfer learning, and although existing transfer learning frameworks may better reflect and classify the literature, the resulting systems theoretic framework offers a more rigorous foundation better suited for system design and analysis.
Mesarovician systems theory is a set-theoretic meta-theory concerned with the characterization and categorization of systems. A system is defined as a relation on sets and mathematical structure is sequentially added to those sets, their elements, or the relation among them to formalize phenomena of interest. By taking a top-down, systems approach to framing transfer learning, instead of using a bottom-up survey of the field, we naturally arrive at a framework for modeling transfer learning without necessarily referencing solution methods. This allows for general considerations of transfer learning systems, and is fundamental to the understanding of transfer learning as a mathematical construct.
We provide a novel definition of transfer learning systems, dichotomize transfer learning in terms of structure and behavior, and formalize notions of negative transfer, transferability, transfer distance, and transfer roughness in subsequent elaborations. First we review transfer learning and Mesarovician abstract systems theory in Section 2. We then define learning systems and discuss their relationship to abstract systems theory and empirical risk minimization in Section 3. Using this definition, transfer learning systems are defined and studied in Sections 4 and 5. We conclude with a synopsis and remarks in Section 6.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{framework-comparison-mindmap-3.png}
\caption{Existing frameworks interpret the informal definition of transfer learning given by DARPA in terms of domain $\mathcal{D}$ and task $\mathcal{T}$. In contrast, we use structure and behavior, which provide a more formal basis for elaboration.}
\label{fig:frameworks}
\end{figure*}
\section{Background}
In the following we review transfer learning and make explicit the principal differences between existing frameworks and ours. Then, pertinent Mesarovician abstract systems theory is introduced. A supplemental glossary of Mesarovician terms can be found in the Appendix.
\subsection{Transfer Learning}
DARPA describes transfer learning as ``the ability of a system to recognize and apply knowledge and skills learned in previous tasks to novel tasks'' in Broad Agency Announcement (BAA) 05-29. The previous tasks are referred to as source tasks and the novel task is referred to as the target task. Thus, transfer learning seeks to transfer knowledge from some source learning systems to a target learning system.
Existing frameworks focus on a dichotomy between \emph{domain} $\mathcal{D}$ and \emph{task} $\mathcal{T}$. The domain $\mathcal{D}$ consists of the input space $X$ and its marginal distribution $P(X)$. The task $\mathcal{T}$ consists of the output space $Y$ and its posterior distribution $P(Y|X)$. The seminal transfer learning survey frames transfer learning in terms of an inequality of domains $\mathcal{D}$ and tasks $\mathcal{T}$ \cite{pan2009survey}. Therein, Pan and Yang define transfer learning as follows.
\begin{definition}{\emph{Transfer learning}.} \\
Given a source domain $\mathcal{D}_S$ and task $\mathcal{T}_S$ and a target domain $\mathcal{D}_T$ and task $\mathcal{T}_T$, transfer learning aims to improve the learning of $P(Y_T|X_T)$ in the target using knowledge in $\mathcal{D}_S$ and $\mathcal{T}_S$, where $\mathcal{D}_S \neq \mathcal{D}_T$ or $\mathcal{T}_S \neq \mathcal{T}_T$.
\end{definition}
Pan and Yang continue by defining \emph{inductive transfer} as the case where the source and target tasks are not equal, $\mathcal{T}_S \neq \mathcal{T}_T$, and \emph{transductive transfer} as the case where the source and target domains are not equal but their tasks are, $\mathcal{D}_S \neq \mathcal{D}_T \land \mathcal{T}_S = \mathcal{T}_T$. They use these two notions, and their sub-classes, to categorize the transfer learning literature and its affinity for related fields of study. Alternative frameworks use notions of \emph{homogeneous} and \emph{heterogeneous} transfer, which correspond to the cases where the sample spaces of the source and target domains $X$ and tasks $Y$ are or are not equal, respectively \cite{weiss2016survey}.
While these formalisms describe the literature well, they are not rich enough to maintain formalism in the elaboration of their respective frameworks. For example, Pan and Yang address what, how, and when to transfer in a largely informal manner, making reference to inductive and transductive transfer as guideposts, but ultimately resorting to verbal descriptions \cite{pan2009survey}. In contrast, instead of starting with domain $\mathcal{D}$ and task $\mathcal{T}$ as the fundamental notions of transfer learning, we use structure and behavior---two concepts with deep general systems meaning, define transfer learning as a relation on systems, and carry formalism through into subsequent elaboration. The principal difference between existing frameworks and ours is depicted in Figure \ref{fig:frameworks}.
Importantly, despite our formalism, we maintain a general systems level of abstraction, in contrast to purely learning theoretical frameworks for transfer learning \cite{kuzborskij2013stability}. As such, we compare our general framework with those of Pan and Yang \cite{pan2009survey} and Weiss et. al \cite{weiss2016survey}. We greatly expand on previous, initial efforts in this direction \cite{cody2019systems, cody2020motivating}.
\begin{figure}[b]
\centering
\includegraphics[width=8cm]{ast-block.png}
\caption{AST is a minimally formal framework. In modeling learning, learning theory brings formalism to AST, and machine learning specifies the detailed model.}
\label{fig:ast-block}
\end{figure}
\subsection{Abstract Systems Theory}
Mesarovician abstract systems theory (AST) is a general systems theory that adopts the formal minimalist world-view \cite{mesarovic1989abstract, dori2019system}. AST is developed top-down, with the goal of giving a verbal description a parsimonious yet precise mathematical definition. Mathematical structure is added as needed to specify systems properties of interest. This facilitates working at multiple levels of abstraction within the same framework, where mathematical specifications can be added without restructuring the framework. In modeling, it is used as an intermediate step between informal reasoning and detailed mathematics by formalizing block-diagrams with little to no loss of generality, see Figure \ref{fig:ast-block}. Apparently this generality limits its deductive powers, but, in return, it helps uncover fundamental mathematical structure related to the general characterization and categorization of phenomena.
We will now review the AST definitions of a system, input-output system, and goal-seeking system, and the related notions of system structure and behavior. Additional details can be found in the Appendix.
In AST, a system is defined as a relation on component sets. When those sets can be partitioned, the system is called an input-output system. Systems and input-output systems are defined as follows.
\begin{definition}{\emph{System}.} \\
A (general) system is a relation on non-empty (abstract) sets,
$$S \subset \times \{ V_{i} : i \in I \}$$
where $\times$ denotes the Cartesian product and $I$ is the index set. A component set $V_{i}$ is referred to as a system object.
\end{definition}
\begin{definition}{\emph{Input-Output Systems}.} \\
Consider a system $S$, where $S \subset \times \{ V_{i} : i \in I \}$. Let $I_{x} \subset I$ and $I_{y} \subset I$ be a partition of $I$, i.e., $I_{x} \cap I_{y} = \emptyset$, $I_{x} \cup I_{y} = I$. The set $X = \times \{ V_{i} : i \in I_{x} \}$ is termed the input object and $Y = \times \{ V_{i} : i \in I_{y} \}$ is termed the output object. The system is then
$$S \subset X \times Y$$
and is referred to as an input-output system. If $S$ is a function $S:X \to Y$, it is referred to as a function-type system.
\end{definition}
AST is developed by adding structure to the component sets and the relation among them. Input-output systems with an internal feedback mechanism are referred to as goal-seeking (or cybernetic) systems. The internal feedback of goal-seeking systems is specified by a pair of consistency relations $G$ and $E$ which formalize the notions of goal and seeking, respectively. Figure \ref{fig:i-o-system} depicts input-output and goal-seeking systems. Goal-seeking systems are defined as follows.
\begin{definition}{\emph{Goal-Seeking Systems}.} \\
A system $S:X \to Y$ has a goal-seeking representation if there exists a pair of maps
\begin{gather*}
S_G:X \times Y \to \Theta \\
S_F:\Theta \times X \to Y
\end{gather*}
and another pair
\begin{gather*}
G: \Theta \times X \times Y \to V \\
E: X \times Y \times V \to \Theta
\end{gather*}
such that
\begin{gather*}
(x, y) \in S \leftrightarrow (\exists \theta) [(\theta, x, y) \in S_F \wedge (x, y, \theta) \in S_G] \\
(x, y, G(\theta, x, y), \theta) \in E \leftrightarrow (x, y, \theta) \in S_G
\end{gather*}
where
$$x \in X, y \in Y, \theta \in \Theta.$$
$S_G$ is termed the goal-seeking system and $S_F$ the functional system. $G$ and $E$ are termed the goal and seeking relations, and $V$ the value.
\label{def:gs}
\end{definition}
System structure and behavior are focal in Mesarovician characterizations of systems. System structure refers to the mathematical structure of a system's component sets and the relations among them. For example, there may be algebraic structure related to the specification of the relation, e.g. the linearity of a relationship between two component sets. System behaviors, in contrast, are properties or descriptions paired with systems. For example, consider a system $S:X \to Y$ and a map $S \to \{ stable, neutral, unstable \}$. A linear increasing function and an increasing power function may both be considered behaviorally unstable, but clearly their structures are different \cite{mesarovic1989abstract}.
Similarity of systems is a fundamental notion, and it can be expressed well in structural and behavioral terms. Structural similarity describes the \emph{homomorphism} between two systems' structures. Herein, in accord with category theory, a map from one system to another is termed a morphism, and homomorphism specifies the morphism to be onto. Homomorphism is formally defined as follows.
\begin{definition}{\emph{Homomorphism}.} \\
An input-output system $S \subset X \times Y$ is homomorphic to $S' \subset \times X' \times Y'$ if there exists a pair of maps,
\begin{align*}
\varrho:X \to X', \vartheta:Y \to Y'
\end{align*}
such that for all $x\in X$, $x'\in X'$, and $y\in Y$, $y'\in Y'$, $\varrho(x)=x'$ and $\vartheta(y)=y'$.
\end{definition}
\noindent Behavioral similarity, in contrast, describes the \emph{proximity} or \emph{distance} between two systems' behavior. As in AST generally, we use structure and behavior as the primary apparatus for elaborating on our formulation of transfer learning systems. Refer to the Appendix for additional details on structure, behavior, and similarity.
\begin{figure}[t]
\centering
\includegraphics[]{i-o-system.png}
\caption{Input-output systems (left) and goal-seeking systems (right).}
\label{fig:i-o-system}
\end{figure}
\section{Learning Systems}
We follow Mesarovic's top-down process to sequentially construct a learning system $S$. Learning is a relation on data and hypotheses. To the extent that a scientific approach is taken, those hypotheses are explanations of initial-final condition pairs \cite{popper2005logic}. Otherwise put, we are concerned with learning as function estimation. We additionally note that learning algorithms use data to select those hypotheses and that the data is a sample of input-output pairs \cite{vapnik2013nature}. Such a learning system can be formally defined as follows.
\begin{definition}{\emph{(Input-Output) Learning System}.} \\
A learning system $S$ is a relation
$$S \subset \times \{A, D, \Theta, H, X, Y \}$$
such that
\begin{gather*}
D \subset X \times Y, A:D \to \Theta, H:\Theta \times X \to Y \\
(d, x, y) \in \mathcal{P}(S) \leftrightarrow (\exists \theta) [(\theta, x, y) \in H \wedge (d, \theta) \in A]
\end{gather*}
where
$$x \in X, y \in Y, d \in D, \theta \in \Theta.$$
The algorithm $A$, data $D$, parameters $\Theta$, hypotheses $H$, input $X$, and output $Y$ are the component sets of $S$, and learning is specified in the relation among them.
\label{def:ls}
\end{definition}
The above definition of learning formalizes learning as a cascade connection of two input-output systems: an inductive system $S_I \subset \times \{ A, D, \Theta \}$ responsible for inducing hypotheses from data, and a functional system $S_F \subset \times \{ \Theta, H, X, Y \}$, i.e. the induced hypothesis. $S_I$ and $S_F$ are coupled by the parameter $\Theta$. Learning is hardly a purely input-output process, however. To address this, we must specify the goal-seeking nature of $S_I$, and, more particularly, of $A$.
$A$ is goal-seeking in that it makes use of a \emph{goal} relation $G: D \times \Theta \to V$ that assigns a value $v \in V$ to data-parameter pairs, and a \emph{seeking} relation $E: V \times D \to \Theta$ that assigns parameter $\theta \in \Theta$ to data-value pairs. These consistency relations $G$ and $E$ specify $A$, but not by decomposition; i.e., in general, $G$ and $E$ cannot be composed to form $A$. The definition of a learning system can be extended as follows.
\begin{definition}{\emph{(Goal-Seeking) Learning System}.} \\
A learning system $S$ is a relation
$$S \subset \times \{ A, D, \Theta, G, E, H, X, Y \}$$
such that
\begin{gather*}
D \subset X \times Y, A:D \to \Theta, H:\Theta \times X \to Y \\
(d, x, y) \in \mathcal{P}(S) \leftrightarrow (\exists \theta) [(\theta, x, y) \in H \wedge (d, \theta) \in A] \\
G: D \times \Theta \to V, E: V \times D \to \Theta \\
(d, G(\theta, d), \theta) \in E \leftrightarrow (d, \theta) \in A
\end{gather*}
where
$$x \in X, y \in Y, d \in D, \theta \in \Theta.$$
The algorithm $A$, data $D$, parameters $\Theta$, consistency relations $G$ and $E$, hypotheses $H$, input $X$, and output $Y$ are the component sets of $S$, and learning is specified in the relation among them.
\label{def:gsls}
\end{definition}
\begin{figure}[t]
\centering
\includegraphics[]{learning-system.png}
\caption{Learning systems are a cascade connection of the inductive system $S_I$ and the induced hypothesis $S_F$. $S_I$ is goal-seeking.}
\label{fig:learning-systems}
\end{figure}
Learning systems are depicted in Figure \ref{fig:learning-systems}. These systems theoretic definitions of learning have an affinity to learning theoretic constructions. Consider empirical risk minimization (ERM), where empirical measures of risk are minimized to determine the optimal hypothesis for a given sample \cite{vapnik2013nature}. Apparently, ERM specifies $G$ to be a measure of risk calculated on the basis of a sample drawn independently according to a probability measure on the approximated function $f:X \to Y$ and specifies $E$ to be a minimization of $G$ over $\Theta$.
We have demonstrated how our definition of a learning system anchors our framework to both AST and ERM. We posit these definitions not as universal truths, but rather as constructions that anchor our framing of transfer learning to systems and learning theory. We abstain from further elaboration on these definitions, however, proofs of the above propositions can be found in the Appendix. In the following, we leave $G$ and $E$ implicit, only making reference to $f$ and related probability measures.
\begin{exmp}{\emph{Learning in an Unmanned Aerial Vehicle.}} \\
Consider an unmanned aerial vehicle (UAV) with a learning system $S$ for path planning. $H$ is a function from sensor data $X$, e.g., from accelerometers, cameras, and radar, to flight paths $Y$. $D$, then, consists of sets of sensor-path pairs. If $S$ is a support-vector machine (SVM), then $H$ is a set of half-spaces parameterized by $\Theta$ and $A$ is a convex optimization routine\cite{suykens1999least}.
\label{ex:learning-systems}
\end{exmp}
\section{Transfer Learning Systems}
Transfer learning is conventionally framed as a problem of sharing knowledge from source domains and tasks to a target domain and task. We propose an alternative approach. We formulate transfer learning top-down in reference to the source and target learning systems, and then dichotomize subsequent analysis not by domain and task, but rather by structure, described primarily by the $X \times Y$ space, and behavior, described primarily by probability measures on the estimated function $f:X \to Y$.
A transfer learning system is a relation on the source and target systems that combines knowledge from the source with data from the target and uses the result to select a hypothesis that estimates the target learning task $f_T$. We define it formally as follows.
\begin{definition}{\emph{Transfer Learning System}.} \\
Given source and target learning systems $S_S$ and $S_T$
\begin{gather*}
S_S \subset \times \{ A_S, D_S, \Theta_S, H_S, X_S, Y_S \} \\
S_T \subset \times \{ A_T, D_T, \Theta_T, H_T, X_T, Y_T \}
\end{gather*}
a transfer learning system $S_{Tr}$ is a relation on the component sets of the source and target systems $S_{Tr} \subset \overline{S_S} \times \overline{S_T}$ such that
$$K_S \subset D_S \times \Theta_S, D \subset D_T \times K_S$$
and
\begin{gather*}
A_{Tr}: D \to \Theta_{Tr}, H_{Tr}: \Theta_{Tr} \times X_T \to Y_T \\
(d, x_T, y_T) \in \mathcal{P}(S_{Tr}) \leftrightarrow \\
(\exists \theta_{Tr})[(\theta_{Tr}, x_T, y_T) \in H_{Tr} \land (d, \theta_{Tr}) \in A_{Tr}]
\end{gather*}
where
$$x_T \in X_T, y_T \in Y_T, d \in D, \theta_{Tr} \in \Theta_{Tr}.$$
The nature of source knowledge $K_S$\footnote{Here, we define the transferred knowledge $K_S$ to be $D_S$ and $\Theta_S$, the source data and parameters, following convention \cite{pan2009survey}. In general, however, source knowledge $K_S \subset \mathcal{P}(\mathcal{P}(\overline{S_S}))$.}, the transfer learning algorithm $A_{Tr}$, hypotheses $H_{Tr}$, and parameters $\Theta_{Tr}$ specify transfer learning as a relation on $\overline{S_S}$ and $\overline{S_T}$.
\label{def:tl}
\end{definition}
Trivial transfer occurs when the structure and behavior of $S_S$ and $S_T$ are the same, or, otherwise put, when transfer learning reduces to classical, identically distributed learning. Transfer is non-trivial when there is a structural difference $X_S \times Y_S \neq X_T \times Y_T$ or a behavioral difference $P(X_S) \neq P(X_T) \lor P(Y_S|X_S) \neq P(Y_T|X_T)$ between the source $S_S$ and target $S_T$. If the posterior distributions $P(Y|X)$ and marginal distributions $P(X)$ are equal between the source and target systems, then transfer is trivial. Non-trivial transfer is implied when $X_S \times Y_S \neq X_T \times Y_T$.
\begin{prop}
\emph{$S_{Tr}$ in Definition \ref{def:tl} is a learning system as defined in Definition \ref{def:ls}.}\\
\emph{Proof:}
As stated in Definition \ref{def:tl}, a transfer learning system is a relation $S_{Tr} \subset \overline{S_S} \times \overline{S_T}$. More particularly, it is a relation $S_{Tr} \subset (D_S \times \Theta_S) \times (D_T \times X_T \times Y_T)$, and has a function-type representation $S_{Tr}: D_S \times \Theta_S \times D_T \times X_T \to Y_T$. Its inductive system is the relation $A_{Tr}:D \to \Theta_{Tr}$, where $D \subset D_S \times \Theta_S \times D_T$. And its functional system is the relation $H_{Tr}: \Theta_{Tr} \times X_T \to Y_T$. Thus, we can restate $S_{Tr}$ as a relation
$$S_{Tr} \subset \times \{A_{Tr}, D, \Theta_{Tr}, H_{Tr}, X_T, Y_T\}$$
and since by Definition \ref{def:tl}
\begin{gather*}
(d, x_T, y_T) \in \mathcal{P}(S_{Tr}) \leftrightarrow \\
(\exists \theta_{Tr})[(\theta_{Tr}, x_T, y_T) \in H_{Tr} \land (d, \theta_{Tr}) \in A_{Tr}]
\end{gather*}
where
$$x_T \in X_T, y_T \in Y_T, d \in D, \theta_{Tr} \in \Theta_{Tr},$$
we have that $S_{Tr}$ is an input-output learning system as in Definition $\ref{def:ls}$.
\end{prop}
Transfer learning systems are distinguished from general learning systems by the selection and transfer of $K_S$, and its relation to $D_T$ by way of $D \subset K_S \times D_T$ and its associated operator $K_S \times D_T \to D$. In cases where $\{A_{Tr}, \Theta_{Tr}, H_{Tr}\} \leftrightarrow \{A_T, \Theta_T, H_T\}$, e.g., as is possible when transfer learning consists of pooling samples with identical supports, the additional input $K_S$ is all that distinguishes $S_{Tr}$ from $S_T$. Classical and transfer learning systems are depicted in Figure \ref{fig:tl-system}.
As we will see, however, this is no small distinction, as it allows for consideration of learning across differing system structures and behaviors. But before we elaborate on the richness of structural and behavioral considerations, first, in the following subsections, we interpret existing frameworks in terms of structure and behavior and define preliminary notions related to generalization in transfer learning.
\begin{figure}[t]
\centering
\includegraphics[]{tl-1.png}
\caption{Transfer learning systems $S_{Tr}$ are a relation $K_S \times D_T \times X_T \to Y_T$, while the target system $S_T$ is a relation $D_T \times X_T \to Y_T$.}
\label{fig:tl-system}
\end{figure}
\begin{exmp}{\emph{Transfer Learning in UAVs.}} \\
Consider UAVs with learning systems $S_S$ and $S_T$ defined according to Example \ref{ex:learning-systems} and a transfer learning system $S_{Tr} \subset \overline{S_S} \times \overline{S_T}$. If $S_{Tr}$ is also a SVM, then $H_{Tr}$ are also half-spaces parameterized by $\Theta_{Tr}$. If $K_S \subset D_S \times \Theta_S$, $\Theta_S$ can provide an initial estimate for $\Theta_{Tr}$, and $D_S$ can be pooled with $D_T$ to update this estimate. $A_{Tr}$, in distinction to $A_T$, must facilitate this initialization and pooling.
\label{ex:transfer}
\end{exmp}
\subsection{Comparison to Existing Frameworks}
Using Definition \ref{def:tl}, the central notions of existing frameworks can be immediately defined in terms of structural and behavioral inequalities. Homogeneous transfer specifies structural equality of the source and target sample spaces, $X_S \times Y_S = X_T \times Y_T$, and heterogeneous transfer specifies otherwise. Domain adaptation, co-variate shift, and prior shift are all examples of homogeneous transfer \cite{jiang2008literature, pan2009survey, csurka2017comprehensive}. Transductive and inductive transfer entail more nuanced specifications.
Recall, inductive transfer specifies that $\mathcal{T}_S \neq \mathcal{T}_T$ and transductive transfer specifies that $\mathcal{D}_S \neq \mathcal{D}_T \land \mathcal{T}_S = \mathcal{T}_T$, where $\mathcal{D}=\{P(X), X\}$ and $\mathcal{T}=\{P(Y|X), Y\}$. Technically, transductive transfer occurs if $X_S \neq X_T$ or if $P(X_S) \neq P(X_T)$. However, if $X_S \neq X_T$, then it is common for $P(Y_S|X_S) \neq P(Y_T|X_T)$ because the input set conditioning the posterior has changed, and thus it is likely that $\mathcal{T_S} \neq \mathcal{T_T}$. To that extent, in the main, transductive transfer specifies a difference between input behavior while output behavior remains equal. Inductive transfer, on the other hand, is more vague, and merely specifies that there is a structural difference in the outputs, $Y_S \neq Y_T$, or a behavioral difference in the posteriors, $P(Y_S|X_S) \neq P(Y_S|X_T)$. Note, this behavioral difference in the posteriors can be induced by a structural difference in the inputs as previously mentioned, and is implied by a structural difference in the outputs.
In short, the homogeneous-heterogeneous dichotomy neglects behavior and the transductive-inductive framing muddles the distinction between structure and behavior. While frameworks based on either cover the literature well, they only provide high-level formalisms which are difficult to carry through into general, formal characterizations of transfer learning systems. In contrast, Definition \ref{def:tl} provides a formalism that can be used to define transfer learning approaches and auxiliary topics in generalization.
\subsection{Transfer Approaches}
Consider how the seminal framework informally classifies transfer learning algorithms \cite{pan2009survey}. Three main approaches are identified: `instance transfer', `parameter transfer', and `feature-representation transfer'. While the transductive or inductive nature of a transfer learning system gives insight into which approaches are available, the approaches cannot be formalized in those terms, or in terms of domain $\mathcal{D}$ and task $\mathcal{T}$ for that matter, because they are a specification on the inductive system $S_I \subset \times \{A_{Tr}, D_{Tr}, \Theta_{Tr}\}$, whereas the former are specifications on the functional system $S_F \subset \times \{\Theta_{Tr}, H_{Tr}, X_{Tr}, Y_{Tr}\}$.
With the additional formalism of Definition \ref{def:tl}, these transfer approaches can be formalized using system structure. First, note that differently structured data $D$ leads to different approaches. Consider the categories of transfer learning systems corresponding to the various cases where $D \subset \mathcal{P}(\mathcal{P}(D_T \cup D_S \cup \Theta_S))$. Instance and parameter transfer correspond to transferring knowledge in terms of $D_S$ and $\Theta_S$, respectively, and can be formally defined as follows.
\begin{definition}{\emph{Instance Transfer}.} \\
A transfer learning system $S_{Tr}$ is an instance transfer learning system if $K_S \subset D_S$, i.e., if
$$\mathcal{A}_{Tr}: D \to \Theta_{Tr} \iff \mathcal{A}_{Tr}: D_S \times D_T \to \Theta_{Tr}.$$
\end{definition}
\begin{definition}{\emph{Parameter Transfer}.} \\
A transfer learning system $S_{Tr}$ is a parameter transfer learning system if $K_S \subset \Theta_S$, i.e., if
$$\mathcal{A}_{Tr}: D \to \Theta_{Tr} \iff \mathcal{A}_{Tr}: \Theta_S \times D_T \to \Theta_{Tr}.$$
\end{definition}
Feature-representation transfer, in contrast, specifies that learning involves transformations on $\overline{S_T}$, $K_S$, or both. It can be defined formally as follows.
\begin{definition}{\emph{Feature-Representation Transfer}.} \\
Consider a transfer learning system $S_{Tr}$ and a learning system $S_L$, termed the latent learning system. Note, $S_{Tr}$ and $S_L$ can be represented as function-type systems,
\begin{gather*}
S_{Tr}: D \times X_T \to Y_T \\
S_L: D_L \times X_L \to Y_L.
\end{gather*}
$S_{Tr}$ is a feature-representation transfer learning system if there exist maps
$$m_D:D \to D_L, m_{X_T}:X_T \to X_L, m_{Y_L}:Y_L \to Y_T$$
such that
\begin{gather*}
\forall (d, x_T, y_T) \in (S_{Tr}) \\
S_{Tr}(d, x_T) \leftrightarrow m_{Y_L}(S_L(m_D(d), m_{X_T}(x_T)))
\end{gather*}
where
$$d \in D, x_T \in X_T, y_T \in Y_T.$$
In other words, $S_{Tr}$ is a feature-representation transfer learning system if transfer learning involves transforming to and from a latent system where learning occurs.
\end{definition}
\begin{prop}
\emph{Learning in $S_S$, $S_T$, and $S_L$.} \\
Consider a case of feature-representation transfer where $K_S \subset D_S$. Let $m_{D_T}:D_T \to D_L$ and $m_{D_S}:D_S \to D_L$. Then, $m_D \iff (m_{D_T}, m_{D_S})$. Recall $D_i \subset X_i \times Y_i$. If $m_{D_T}$ is the identity and $m_{D_S}$ is not, then $X_T \times Y_T = X_L \times Y_L$---learning occurs in the target sample space. If $m_{D_S}$ is the identity and $m_{D_T}$ is not, then $X_S \times Y_S = X_L \times Y_L$---learning occurs in the source sample space. If $m_D$ is the identity, then $X_S \times Y_S = X_T \times Y_T = X_L \times Y_L$, i.e., $S_{Tr}$ involves homogeneous transfer. If neither $m_{D_T}$ or $m_{D_S}$ are the identity, then learning occurs in a latent sample space $X_L \times Y_L$ that is unequal to $X_T \times Y_T$ and $X_S \times Y_S$.
\end{prop}
In feature-representation transfer, data $D \subset D_T \times K_S$ is mapped to a latent system $S_L$ where learning occurs. By way of $m_D:D \to D_L$, feature-representation transfer involves relating the source and target input-output spaces to a latent space $X_L \times Y_L$. Learning can occur in $X_L \times Y_L$, and, using $m_{Y_L}$, the output can be given in terms of the target output $Y_T$. Similarly, the target can be mapped onto the source, $X_L \times Y_L = X_S \times Y_S$, where learning can occur given $m_{Y_L}$, or the source can be mapped onto the target, $X_L \times Y_L = X_T \times Y_T$.
Figure \ref{fig:latent-learning} depicts these three cases of morphisms using a commutative diagram. As the individual maps that compose these morhpisms become more dislike identities and partial, feature-representation transfer becomes more difficult. We will discuss this further in our elaboration on structural considerations. Additionally note, even if $X_S \times Y_S = X_T \times Y_T$, feature-representation transfer may still be used to better relate source and target behavior.
\begin{table}[t]
\centering
\ra{1.3}
\begin{tabular}{@{}ll@{}}
\toprule
Transfer Approach & Algorithm Structure\\
\midrule
Instance & $A_{Tr}:D_T \times D_S \to \Theta_{Tr}$\\
Parameter & $A_{Tr}:D_T\ \times \Theta_S \to \Theta_{Tr}$\\
Instance \& Parameter & $A_{Tr}:D_T \times D_S \times \Theta_S \to \Theta_{Tr}$\\
Feature-Representation & $A_{Tr}: m_D(D) \to \Theta_{Tr}$ \\
\bottomrule
\\
\end{tabular}
\caption{Structural differences between transfer approaches.}
\label{table:approaches}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[]{latent-learning23.png}
\caption{Morphisms in feature representation learning. Learning in the target sample space requires a morphism from that of the source, as shown in red. Learning in the source sample space requires a morphism from that of the target, as shown in blue, and a map from the source output to the target output, as shown by the dashed blue arrow. And learning in a latent sample space requires morphisms from both the source and target sample spaces to that of the latent system, as shown in green, and a map from the latent output to the target output, as shown by the dashed green arrow. As discussed in Section 5, the nature of these morphisms affects the difficulty of transfer.}
\label{fig:latent-learning}
\end{figure*}
Instance, parameter, and feature-based approaches are shown in terms of their specification on transfer learning algorithms $A_{Tr}$ in Table \ref{table:approaches}. Another general notion in transfer learning is \emph{n}-shot transfer. It can be defined as follows.
\begin{definition}{\emph{N-shot Transfer}.} \\
A transfer learning system $S_{Tr}$ with target data $d_T \in D_T$ is referred to as a n-shot transfer learning system if $|d_T| = n$. Zero-shot transfer occurs if $A_{Tr}: D \to \Theta_{Tr} \iff A_{Tr}: K_S \to \Theta_{Tr}$.
\end{definition}
\noindent Machine learning is often concerned with few-shot learners---transfer learning systems that can generalize with only a few samples from the target. We will discuss generalization in transfer learning in the following subsection, but first, to get a sense of how we formalize instance, parameter, and feature-representation transfer, consider how a few canonical transfer learning algorithms are modeled by our framework.
Transfer component analysis uses a modified principal component analysis approach to project the source and target data into a relatable latent space \cite{pan2010domain}, i.e., it is an instance approach in that $D_S$ is used in $A_{Tr}$ and a feature-representation approach in that $X_S$ and $X_T$ are projected into a latent $X_L$. Constraining parameters to be within a range of those of the source, as in hierarchical Bayesian and regularization approaches, is parameter transfer \cite{evgeniou2004regularized, schwaighofer2005learning}. Deep learning approaches often involve parameter transfer in that the weights $\Theta_S$ of the source network are shared and frozen in the target, or otherwise used to initialize $\Theta_T$ \cite{bengio2012deep}. Other deep learning approaches also involve instance transfer to increase sample size, such as those that use generative adversarial networks \cite{sankaranarayanan2018generate}. When the source and target data must first be transformed before the data can be related, they are also feature-representation approaches, as in joint adaptation networks \cite{long2017deep}.
By formalizing the canonical classes of transfer approaches, we are better able to understand them in terms of their general requirements on $S_{Tr}$, particularly on $S_I$, and more particularly on $A_{Tr}$ and $D$. The informal use of these classes by existing frameworks, wherein a solution method's dominant nature sorts it into a particular class, does well to organize the literature. Our formalisms can cloud these scholarly distinctions, as shown in the case of deep learning where a single method can belong to all three classes, however, they give a basis for defining formal categories of transfer learning systems $S_{Tr}$ in terms of their inductive systems $S_I$.
\subsection{Generalization in Transfer Learning}
Generalization is, perhaps, the ultimate aim of learning. It is the ability for the learned hypothesis to approximate $f$ out-of-sample, i.e., on samples not seen in training. Generalization as a goal for learning systems is implicit in $A$ when a measure of error $\epsilon$ between $h(\theta)$ and $f$ specifies $G$, such as in ERM. Herein, we define it as follows.
\begin{definition}{\emph{Generalization.}} \\
Given a learning system $S$ and data $d \in D$, generalization is the ability for a learned hypothesis $h(\theta)$ to estimate learning task $f:X \to Y$, on samples $(x, y) \notin d$.
\end{definition}
In moving from the classical, identically distributed learning setting to transfer learning, we move from generalizing to a new sample from the same system, to generalizing to a new sample from a different system. In classical learning, for a learning system $S$, the estimated function $f$ is specified by $P(Y|X)$ and data $D$ are drawn from a related joint $P(X,Y)$. In transfer learning, however, the $X \times Y$ space and probability measures specifying $f$ and $D$ vary between $S_S$ and $S_T$.
In classical learning, given a learning system $S$, data $d \in D$, a measure of error $\epsilon: H(\Theta) \times f \to \mathbb{R}$, and a threshold on error $\epsilon^* \in \mathbb{R}$, we generalize if
$$\epsilon(H(A(d)), f) \leq \epsilon^*.$$
That, is, if the measure of error between the learned hypothesis and the function it estimates is below a threshold. In practice, since $f$ is not known, error is empirically estimated using samples $(x, y) \in X \times Y$ such that $(x, y) \notin d$.
In transfer learning, given $S_{Tr}$ and data $d \in D$, we generalize if
$$\underbrace{\epsilon(H_{Tr}(A_{Tr}(d)), f_T)}_{\epsilon_T} \leq \epsilon^*.$$
If $\epsilon_T$ is smaller without any transferred knowledge from $S_S$ than with, transfer from $S_S$ to $S_T$ is said to result in negative transfer. Negative transfer is defined in accord with Wang et. al as follows.
\begin{definition}{\emph{Negative Transfer}.} \\
Consider a transfer learning system $S_{Tr}$. Recall $D \subset D_T \times K_S$. Let $d \in D$ and $d_T \in D_T$. Given a measure of error $\epsilon: H(\Theta) \times f \to \mathbb{R}$, negative transfer is said to occur if
$$\epsilon(H_T(A_T(d_T)), f_T) < \epsilon(H_{Tr}(A_{Tr}(d)), f_T),$$
that is, if the error in estimating $f_T$ is higher with the transferred knowledge than without it.
\label{def:nt}
\end{definition}
\noindent As Wang et. al note, negative transfer can arise from behavioral dissimilarity between the source and target \cite{wang2019characterizing}. In general, it can arise from structural dissimilarity as well.
Because generalization in transfer learning considers generalization across systems, as opposed to generalization within a given system, naturally, it is concerned with the set of systems to and from which transfer learning can generalize. Using $\epsilon_T$ and $\epsilon^*$, we can describe these sets as neighborhoods of systems \emph{to} which we can transfer and generalize,
$$ \underbrace{\{ S_T | S_S, \epsilon_T \leq \epsilon^* \}}_{\text{Neighborhood of Targets } S_T}$$
and neighborhoods of systems \emph{from} which we can transfer and generalize,
$$ \underbrace{\{ S_S | S_T, \epsilon_T \leq \epsilon^* \}}_{\text{Neighborhood of Sources } S_S}.$$
Noting Definition \ref{def:nt}, if $\epsilon^* = \epsilon(H_T(A_T(d_T)), f_T)$, these neighborhoods are those systems to and from which transfer is positive.
The size of these neighborhoods describes the transferability of a learning system in terms of the number of systems it can transfer to or from and generalize. To the extent that cardinality gives a good description of size\footnote{Cardinality counts arbitrarily close systems as different, and it may be preferable to define a measure of equivalence, and consider the cardinality of the neighborhoods after the equivalence relation is applied.}, transferability can be defined formally as follows.
\begin{definition}{\emph{Transferability}.} \\
Consider a target learning system $S_T$ and a source learning system $S_S$. Given a measure of error $\epsilon_T: H_{Tr}(\Theta_{Tr}) \times f_T \to \mathbb{R}$ and a threshold on error $\epsilon^* \in \mathbb{R}$, the transferability of a source is the cardinality of the neighborhood of target systems $S_T$ to which it can transfer and generalize,
$$|\{S_T| S_S, \epsilon_T \leq \epsilon^*\}|,$$
and the transferability of a target is the cardinality of the neighborhood of source systems $S_S$ from which we can transfer and generalize,
$$|\{S_S|S_T, \epsilon_T \leq \epsilon^*\}|.$$
These cardinalities are termed the source-transferability and target-transferability, respectively.
\end{definition}
\noindent Note, this defines transferability as an attribute of a particular system---not an attribute of a source-target pairing.
Our interest in transferability as an aim of transfer learning systems echoes a growing interest of the machine learning community in a notion of \emph{generalist} learning systems \cite{kolesnikov2019big, huang2019gpipe, tschannen2020self}. Put informally, generalists are learning systems which can generalize to many tasks with few samples. Using our formalism, these systems can be described as learning systems with high source-transferability. More particularly, they can be defined as follows.
\begin{definition}{\emph{Generalist Learning Systems}.} \\
A generalist learning system $S_S$ is a system that can transfer to at least $t$ target systems $S_T$ with data $d_T \in D_T$ and generalize with at most $n$ target samples $(x_T, y_T) \in X_T \times Y_T.$ That is, they are systems $S_S$ where
$$|\{S_T | S_S, |d_T| \leq n, \epsilon_T < \epsilon^*\}| \geq t$$
\end{definition}
\noindent Generalists are sources $S_S$ that can $n$-shot transfer learn to $t$ or more targets $S_T$. Generalists are typically studied in the context of deep learning for computer vision, where a single network is tasked with few-shot learning a variety of visual tasks, e.g., classification, object detection, and segmentation, in a variety of environments \cite{kolesnikov2019big}.
In the following, we go beyond existing frameworks to explore notions of transferability---and thereby generalization, transfer roughness, and transfer distance in the context of structure and behavior. In doing so, we demonstrate the mathematical depth of Definition \ref{def:tl}. We show that not only does it allow for immediate, formal consideration of surface-level phenomena covered by existing frameworks, but moreover, it allows for a considerable amount of modeling to be done at the general level, i.e., without reference to solution methods, in following with the spirit of AST depicted in Figure \ref{fig:ast-block}.
\section{Structure and Behavior in Transfer Learning}
To the extent that generalization in transfer learning is concerned with sets of systems, it is concerned with how those sets can be expressed in terms of those systems' structures and behaviors. In the following subsections, we discuss how structural and behavioral equality and, moreover, similarity relate to the difficulty of transfer learning. Equalities between $S_S$ and $S_T$ give a basic sense of the setting and what solution methods are available. Similarities between $S_S$ and $S_T$ are a richer means for elaboration, and can give a sense of the likelihood of generalization.
Learning systems are concerned with estimating functions $f:X \to Y$. As transfer learning is concerned with sharing knowledge used to estimate a source function $f_S:X_S \to Y_S$ to help estimate a target function $f_T:X_T \to Y_T$, naturally, the input-output spaces of the source $X_S \times Y_S$ and target $X_T \times Y_T$ are the principal interest of structural considerations. Similarly, the principal interest of behavioral considerations are the probability measures which specify $f_S$ and $f_T$, and, correspondingly, $D_S$ and $D_T$.
\subsection{Structural Considerations}
For source and target systems $S_S$ and $S_T$ we have the following possible equalities between system structures:
\begin{align*}
X_S = X_T, Y_S = Y_T, \\
X_S \neq X_T, Y_S = Y_T, \\
X_S = X_T, Y_S \neq Y_T, \\
X_S \neq X_T, Y_S \neq Y_T.
\end{align*}
The first case $X_S \times Y_S = X_T \times Y_T$ specifies transfer as homogeneous---all others specify heterogeneous transfer. This is the extent of discussion of structure in the existing frameworks \cite{pan2009survey, weiss2016survey}. We elaborate further.
To do so, we extend past structural equality to notions of structural similarity. Recall, structural similarity is a question of the structural homomorphism between two systems. As is common in category theory, we define a morphism as simply a map between systems, and define an onto map between systems as a homomorphism. We can investigate homomorphism in reference to a morphism $m: S_S \to S_T$. First, note that we can quantify structural similarity using equivalence classes. Let $m_x:X_S \to X_T$ and $m_y:Y_S \to Y_T$ such that $m \leftrightarrow (m_x, m_y)$. And let $S_S/m$, $X_S/m_x$, and $Y_S/m_y$ be the equivalence classes of $S_S$, $X_S$, and $Y_S$ with respect to $m$, $m_x$, and $m_y$, respectively.
Consider the two sets of relations
\begin{equation*}
\begin{split}
w &: S_S \to S_S/m \\
w_x &:X_S \to X_S/m_x \\
w_y &: Y_S \to Y_S/m_y
\end{split}
\qquad
\begin{split}
z &: S_S/m \to S_T \\
z_x &: X_S/m \to X_T \\
z_y &: Y_S/m \to Y_T
\end{split}
\end{equation*}
Relation $w$ maps the source $S_S$ to its equivalence class $S_S/m$ and relation $z$ maps $S_S/m$ to the target $S_T$, as depicted by the commutative diagram shown in Figure \ref{fig:roughness}. That is,
$$S_S \xrightarrow[(w_x, w_y)]{} S_S/m \xrightarrow[(z_x, z_y)]{} S_T $$
The equivalence class $S_S/m$ describes the `roughness' of the structural similarity from $S_S$ to $S_T$. Its cardinality quantifies the `surjective-ness' of $m:S_S \to S_T$. The greater the difference between $|S_S|$ and $|S_S/m|$, the more structurally dissimilar $S_S$ and $S_T$ are. However, in the large, structural similarity is not measurable in the same way as behavioral similarity.
\begin{figure}
\centering
\includegraphics{roughness.png}
\caption{A commutative diagram depicting how equivalence classes can describe roughness.}
\label{fig:roughness}
\end{figure}
The homomorphism between $S_S$ and $S_T$ is better investigated in terms of the properties of $m$, such as whether it is injective, surjective, invertible, etc. For example, partial morphisms from $X_S \times Y_S$ to $X_T \times Y_T$ are associated with partial transfer \cite{cao2018partial}. When the partial morphism is surjective, only a subset of the source is transferred to the target. When the partial morphism is injective, the source transfers to only a subset of the target. Also, structural similarity can be expressed using category theory, where the structural similarity between two systems can be studied with respect to the categories of systems to which they belong. To describe structural similarity in a broad sense, we define \emph{transfer roughness} as follows.
\begin{definition}{\emph{Transfer Roughness}.} \\
Transfer roughness describes the structural homomorphism from the source system $S_S$ to the target system $S_T$. When $S_S$ and $S_T$ are isomorphic, transfer roughness is minimal or otherwise non-existent. When roughness exists, it is defined by its properties, and thus there is no clear notion of maximal roughness.
\end{definition}
The structure of the source relative to that of the target determines the roughness of transfer. Structures can be too dissimilar to transfer no matter what the behavior. Homomorphisms are onto and thus structure preserving, and, as such, it is a reasonable principle to characterize structural transferability in terms of the set of homomorphisms shared between the source and target. The supporting intuition is that either the source must map onto the target or they must both map onto some shared latent system, if not fully, at least in some aspect. Otherwise information in the source is lost when transferring to the target.
Let $\mathcal{H}(X, Y)$ denote the set of all structures homomorphic to $X \times Y$. The set of homomorphic structures between $S_S$ and $S_T$ is given by,
$$\mathcal{H}(X_S, Y_S) \cap \mathcal{H}(X_T, Y_T).$$
In transfer learning, we are specifically interested in using knowledge from $S_S$ to help learn $f_T$. Thus, not all elements of this intersection are valid structures for transfer learning, only those whose output can be mapped to $Y_T$. This set of valid structures can be expressed as,
$$\mathcal{V} = \{X \times Y \in \mathcal{H}(X_S, Y_S) \cap \mathcal{H}(X_T, Y_T) | \exists m_y:Y \to Y_T\}.$$
Apparently not all elements of $\mathcal{V}$ will be useful structures for estimating $f_T$, however, those that are useful, presuming structural homomorphism is necessary, will be in $\mathcal{V}$.
If we define $\mathcal{V}'$ to be the subset of $\mathcal{V}$ where transfer learning generalizes, i.e., the homomorphic structures where $\epsilon_T < \epsilon^*$, transferability can be defined in structural terms as follows.
\begin{definition}{\emph{Structural Transferability}.} \\
Consider a target learning system $S_T$ and a source learning system $S_S$. The structural transferability of a source $S_S$ is,
$$|\{S_T|S_S, \exists (X \times Y) \in \mathcal{V}'(S_S, S_T)\}|,$$
and the structural transferability of a target is,
$$|\{S_S|S_T, \exists (X \times Y) \in \mathcal{V}'(S_S, S_T)\}|.$$
\end{definition}
\noindent In other words, structural transferability concerns the set of systems that share a useful homomorphism with $S_S$ and $S_T$. While in practice $\mathcal{V}$ and $\mathcal{V}'$ are difficult to determine, they provide a theoretical basis for considering whether transfer learning is structurally possible between two systems and the structural invariance of the usefulness of transferred knowledge, respectively.
The relation $\mathcal{V}' \subset \mathcal{V}$ is particularly difficult. Ordering structural usefulness by homomorphism alone is difficult because of the vagueness of how homomorphism can be measured. The more isomorphism there is between $S_S$ and $S_T$, the more the question of usefulness shifts to the behavior. There, the error $\epsilon$ provides the ordering\footnote{$\epsilon$ is a transfer distance between posteriors specifying $h(\theta)$ and $f$.} and the threshold $\epsilon^*$ provides the partition. Structural similarity provides no clear parallel.
It is true that if no homomorphism exists between $S_S$ and $S_T$, they are from different categories. While functors can be used to map between categories, they necessarily distort transferred knowledge because they must add or remove structure to do so. Homomorphisms between systems, in contrast, are structure preserving. And so perhaps a partial order between homomorphic and non-homomorphic systems is justified. But this ordering is hardly granular. A more formal digression on this topic is beyond the scope of this paper, but well within the scope of AST\cite{mesarovic1989abstract}.
\begin{exmp}{\emph{Transfer Roughness in UAVs.}} \\
Consider $S_S$, $S_T$, and $S_{Tr}$ defined according to Example \ref{ex:transfer}. From Example \ref{ex:learning-systems} $X_S \times Y_S = X_T \times Y_T$, so $S_{Tr}$ involves homogeneous transfer. But, if $X_T$ did not include radar, transfer would be heterogeneous. Similarly so if $Y_S$ described paths up to 100 meters in length and $Y_T$ paths up to 10 meters. In either case, $X_S \times Y_S$ can map onto $X_T \times Y_T$, but $X_T \times Y_T$ cannot map onto $X_S \times Y_T$. Thus, transfer from $S_T$ to $S_S$ is rougher than transfer from $S_S$ to $S_T$.
\end{exmp}
\subsection{Behavioral Considerations}
In transfer learning, the primary behaviors of interest are $P(X)$ and $P(Y|X)$ from the domain $\mathcal{D}$ and task $\mathcal{T}$, respectively, and the joint distribution they form,
$$P(X,Y) = P(X) P(Y|X).$$
It is important to realize that $P(X_S, Y_S) \neq P(X_T, Y_T)$ only implies that $P(X_S) \neq P(X_T) \lor P(Y_S|X_S) \neq P(Y_T|X_T)$. That is, the posteriors $P(Y|X)$ can still be equal when the joints $P(X, Y)$ are not if the marginals $P(X)$ offset the difference, and vice versa. In the main, these behavioral equalities make absolute statements on the inductive or transductive nature of a transfer learning system. Behavioral similarities, in contrast, have the richness to make statements on the likelihood of generalization, and, thereby, on transferability.
In AST, behavior is a topological-type concept and, accordingly, behavioral similarity is akin to a generalized metric. However, because in transfer learning we are concerned primarily with behaviors which are probability measures, behavioral similarity between $S_S$ and $S_T$ takes the form of distributional divergences. In our elaboration of behavioral similarity we focus on a notion of \emph{transfer distance}. Transfer distance is the abstract distance knowledge must traverse to be transferred from one system to another. We consider it to be a measure on the input spaces $X_S \times X_T$, output spaces $Y_S \times Y_T$, or input-output spaces $(X_S \times Y_S) \times (X_T \times Y_T)$---more specifically, as a measure on probability measures over those spaces. It can be defined formally as follows.
\begin{definition}{\emph{Transfer Distance}.} \\
Let $S_S$ and $S_T$ be source and target learning systems. Let $Z_i$ be a non-empty element of $\mathcal{P}(X_i \cup Y_i)$. Transfer distance $\delta_T$ is a measure
$$\delta_T:P(Z_S) \times P(Z_T) \to \mathbb{R}$$
of distance between the probability measures $P(Z_i)$ related to the estimated functions $f_i:X_i \to Y_i$ of $S_S$ and $S_T$.
\end{definition}
In practice, transfer distances are often given by $f$-divergences \cite{ditzler2011hellinger}, such as KL-divergence or the Hellinger distance, Wasserstein distances \cite{shen2017wasserstein}, and maximum mean discrepancy \cite{pan2008transfer, long2017deep, jiang2015integration}. Others use generative adversarial networks, a deep learning distribution modeling technique, to estimate divergence \cite{tzeng2015simultaneous, ganin2016domain}. Commonly, these distances are used to calculate divergence-based components of loss functions. Herein, we consider transfer distance's more general use in characterizing transfer learning systems.
In heterogeneous transfer, transfer distances can be used after feature-representation transfer has given the probability measures of interest the same support. Transfer distances between measures with different support are not widely considered in existing machine learning literature. However, the assumptions of homogeneous transfer and domain adaptation, i.e., $X_S \times Y_S = X_T \times Y_T$, allow for a rich theory of the role of transfer distance in determining the upper-bound on error.
Upper-bounds on $\epsilon_T$ have been given in terms of statistical divergence \cite{blitzer2008learning}, $H$-divergence \cite{ben2010theory}, Rademacher complexity \cite{mohri2009rademacher}, and integral probability metrics \cite{zhang2012generalization}, among others. Despite their differences, central to most is a transfer distance $\delta_T: P(X_S) \times P(X_T) \to \mathbb{R}$ that concerns the closeness of input behavior and a term $C$ that concerns the complexity of estimating $f_T$. These bounds roughly generalize to the form,
\begin{equation}
\epsilon_T \leq \epsilon_S + \delta_T + C
\label{eq:inequality}
\end{equation}
where $\epsilon_T$ and $\epsilon_S$ are the errors in $S_T$ and $S_S$, $\delta_T$ is the transfer distance, and $C$ is a constant term. $C$ is often expressed in terms of sample sizes, e.g., $|D_S|$ and $|D_T|$, capacity, e.g., the VC-dimension of $H_T$ \cite{ben2010theory}, and information complexity, e.g., the Rademacher complexity of $D_T$ \cite{mohri2009rademacher}. Note, closeness and complexity are often not as separable as suggested by Inequality \ref{eq:inequality}.
To the extent that Inequality \ref{eq:inequality} holds, we can describe transferability in terms of transfer distance. Generalization in transfer learning occurs if $\epsilon_T \leq \epsilon^*$, and since $\epsilon_T \leq \epsilon_S + \delta_T + C$, $\epsilon_S + \delta_T + C \leq \epsilon^* \implies \epsilon_T \leq \epsilon^*$. Thus, transferability can be defined in behavioral terms as follows.
\begin{definition}{\emph{Behavioral Transferability}.} \\
Consider a target learning system $S_T$ and a source learning system $S_S$. The behavioral transferability of a source $S_S$ is,
$$|\{S_T|S_S, \epsilon_S + \delta_T + C < \epsilon^*\}|,$$
and the behavioral transferability of a target is,
$$|\{S_S|S_T, \epsilon_S + \delta_T + C < \epsilon^*\}|.$$
\end{definition}
\noindent For $S_S$ with similar $\epsilon_S$ and $S_T$ with similar $C$, given a threshold on distance $\delta^* \in \mathbb{R}$, behavioral transferability can be expressed entirely in terms of transfer distance:
$$|\{S_T | S_S, \delta_T < \delta^*\}| \text{ and } |\{S_S | S_T, \delta_T < \delta^*\}|.$$
Of course, specific bounds on $\epsilon_T$ with specific distances $\delta_T$ from the literature can be substituted in the stead of Inequality \ref{eq:inequality}. Also note, we are assuming $X_S \times Y_S = X_T \times Y_T$. When $X_S \times Y_S \neq X_T \times Y_T$, transfer distance is a measure between probability measures with different supports, and while an upper-bound like Inequality \ref{eq:inequality} may be appropriate, it is not supported by existing literature. In such cases it is important to consider structural similarity.
\begin{exmp}{\emph{Transfer Distance in UAVs.}} \\
Consider $S_S$, $S_T$, and $S_{Tr}$ defined according to Example \ref{ex:transfer}. Let source $S_S$ be associated with a desert biome and $S_T$ a jungle biome. When comparing $P(X_T)$ to $P(X_S)$, increased foliage in $S_T$ suggests accelerometer readings with higher variance, camera images with different hue, saturation, and luminance, and radar readings with more obstacles. Similarly, increased foliage may also mean paths in $P(Y_T|X_T)$ must compensate more for uncertainty than those in $P(Y_S|X_S)$. In contrast, foliage is more similar between the desert and tundra, thus, transfer distance is likely larger from the desert to the jungle than from the desert to the tundra.
\end{exmp}
\subsection{Remarks}
In summary, structure and behavior provide a means of elaborating deeply on transfer learning systems, just as they do for systems writ large. Structural considerations center on the structural relatability of $S_S$ and $S_T$ and the usefulness of the related structures $X \times Y$ for transfer learning. Behavioral considerations center on the behavioral closeness of $S_S$ and $S_T$ and the complexity of learning $f_T$. These concerns provide guideposts for the design and analysis of transfer learning systems. While the joint consideration of structure and behavior is necessary for a complete perspective on transfer learning systems, herein, in following with broader systems theory, we advocate that their joint consideration ought to come from viewing structure and behavior as parts of a whole---instead of approaching their joint consideration directly by neglecting notions of structure and behavior entirely, as is advocated implicitly by the existing frameworks pervasive use of domain $\mathcal{D}$ and task $\mathcal{T}$.
\section{Conclusion}
Our framework synthesizes systems theoretic notions of structure and behavior with key concepts in transfer learning. These include homogeneous and heterogeneous transfer, domain adaptation, inductive and transductive transfer, negative transfer, and more. In subsequent elaborations, we provide formal descriptions of transferability, transfer roughness, and transfer distance, all in reference to structure and behavior.
This systems perspective places emphasis on different aspects of transfer learning than existing frameworks. When we take behavior to be represented by a posterior or joint distribution, we arrive at constructs similar to existing theory. More distinctly, when we introduce structure, and study it in isolation, we arrive at notions of roughness, homomorphism, and category neglected in existing literature.
The presented framework offers a formal approach for modeling learning. The focal points of our theory are in aspects central to the general characterization and categorization of transfer learning as a mathematical construct, not aspects central to scholarship. This strengthens the literature by contributing a framework that is more closely rooted to engineering design and analysis than existing frameworks. Because our framework is pointedly anchored to concepts from existing surveys, practitioners should face little difficulty in the simultaneous use of both. Taken together, practitioners have a modeling framework and a reference guide to the literature.
Herein, we have modeled transfer learning as a subsystem. Transfer learning systems can be connected component-wise to the systems within which they are embedded. Subsequently, deductions can be made regarding the design and operation of systems and their learning subsystems with the interrelationships between them taken into account. In this way, we contribute a formal systems theory of transfer learning to the growing body of engineering-centric frameworks for machine learning.
Real-world systems need transfer learning, and, correspondingly, engineering frameworks to guide its application. The presented framework offers a Mesarovician foundation.
\section{Appendix}
\subsection{Mesarovician Glossary}
\begin{definition}{\emph{System Behavior}.} \\
System behaviors are properties or descriptions paired with systems. For example, consider a system $S:X \to Y$ and a map $S \to \{ stable, neutral, unstable \}$ or from $S \to P(X,Y)$. System behavior is a topological-type concept in the sense that it pairs systems with elements of sets of behaviors.
\end{definition}
\begin{definition}{\emph{Behavioral Similarity}.} \\
Behavioral similarity describes the `proximity' between two systems' behavior. To the extent that behavior can be described topologically, behavioral similarity can be expressed in terms of generalized metrics (topological `distance'), metrics and pseudo-metrics (measure theoretic `distance'), and statistical divergences (probability/information theoretic `distance'), depending on the nature of the topology.
\end{definition}
\begin{definition}{\emph{System Structure}.} \\
System structure is the mathematical structure of a system's component sets and the relations among them. For example, there may be algebraic structure, e.g. the linearity of a relationship between two component sets, related to the definition of the relation.
\end{definition}
\begin{definition}{\emph{Structural Similarity}.} \\
Structural similarity describes the homomorphism between two systems' structures. It is described in reference to a relation $m:S_1 \to S_2$, termed a morphism. The equivalence class $S_1/m$ describes the `roughness' of the structural similarity between $S_1$ and $S_2$. Its cardinality gives a quantity to the `surjective-ness' of $m:S_1 \to S_2$. However, in the large, structural similarity is not measurable in the same way as behavioral similarity. The homomorphism is better studied using properties of $m$.
\end{definition}
\begin{definition}{\emph{Cascade Connection}.} \\
Let $\circ: \overline{S} \times \overline{S} \to \overline{S}$ be such that $S_1 \circ S_2 = S_3$, where,
\begin{gather*}
S_1 \subset X_1 \times (Y_1 \times (Z_1)), S_2 \subset (X_2 \times Z_2) \times Y_2 \\
S_3 \subset (X_1 \times X_2) \times (Y_1 \times Y_2), Z_1 = Z_2 = Z
\end{gather*}
and,
\begin{gather*}
((x_1, x_2), (y_1, y_2)) \in S_3 \leftrightarrow \\
(\exists z) ((x_1, (y_1, z)) \in S_1 \wedge ((x_2, z), y_2) \in S_2)
\end{gather*}
$\circ$ is termed the cascade (connecting) operator.
\end{definition}
\subsection{Learning Systems}
\begin{prop}
\emph{$S$ in Definition \ref{def:ls} is a cascade connection of two input-output systems.} \\
\emph{Proof:}
Recall $S \subset \times \{A, D, \Theta, H, X, Y\}$. First we will show $A$ and $H$ to be input-output systems. First note that $A \subset \times \{ D, \Theta \}$. Noting $D \subset X \times Y$, apparently $D \cap \Theta = \emptyset$ and $D \cup \Theta = \overline{A}$. Similarly, $H \subset \times \{ \Theta, X, Y \}$. Letting $X' = \{ X, \Theta \}$, apparently $X' \cap Y = \emptyset$ and $X' \cup Y = \overline{H}$. Therefore, by definition, $A$ and $H$ are input-output systems. Let $S_C:D \times X \to Y$. Apparently, for $d \in D, x \in X, y \in Y, \theta \in \Theta$, $((d, x), y) \in S_C \leftrightarrow \exists \theta ((d, \theta) \in A \wedge (\theta, x, y) \in H$. Therefore, $S_C: A \circ H$. Lastly, note $S_C$ is a function-type representation of $S$, where $A$, $H$, and $\Theta$ are left as specifications on relations, not included as component sets.
\end{prop}
\begin{prop}
\emph{$S$ in Definition \ref{def:gsls} is a goal-seeking system.} \\
\emph{Proof:}
Goal-seeking is characterized by the consistency relations $(G, E)$ and by the internal feedback of $X \times Y$ into $S_G$. Note $D \subset X \times Y$ satisfies internal feedback. The consistency relations $(G, E)$ in Definition \ref{def:gs} and \ref{def:gsls} can be shown to be isomorphic by substituting $D \subset X \times Y$ into consistency relations $G$ and $E$ in Definition \ref{def:gs} and $(x, y) \in d$ into their constraints. Thus, by definition, $S$ in Definition \ref{def:gsls} is a goal-seeking system, where $S_G$ is the inductive system $A$ and $S_F$ is the functional system $H$.
\end{prop}
\begin{prop}
\emph{Empirical risk minimization is a special case of a learning system as defined in Definition \ref{def:gsls}.} \\
\emph{Proof:}
A learning system given by Definition \ref{def:gsls} is an empirical risk minimization learning system if (1) $D$ is a sample of $l$ independent and identically distributed observations sampled according to an unknown distribution $P(X, Y)$, and (2) $A$ selects $\theta \in \Theta$ by minimizing the empirical risk $R_{emp}$, calculated on the basis of $D$, over $\theta \in \Theta$. Otherwise put, ERM is a learning system $S \subset \times \{A, D, \Theta, G, E, H, X, Y\}$ where $G(D, \theta) = R_{emp}(D, \theta) = \frac{1}{l}\sum\limits_{i=1}^l L(y_i, h(x_i, \theta))$ and $E = \min_{\theta \in \Theta} G(D, \theta)$, where $L$ is a loss function.
\end{prop}
\bibliographystyle{IEEEtran}
\section*{Preliminaries}
Let $X$ denote a set and $x \in X$ denote its elements. For notational convenience random variables are not distinguished---probability measures on $X$ are denoted $P(X)$. The Cartesian product is denoted $\times$, and for any object $V_i = V_{i1} \times \ldots \times V_{in}$, $\overline{V_i}$ shall denote the family of component sets of $V_i$, $\overline{V_i} = \{ V_{i1}, \ldots, V_{in} \}$. The cardinality of $X$ is denoted $|X|$. The powerset is denoted $\mathcal{P}$. Herein it has two uses. Frequently, in order to express input-output conditions for a learning system we will only use its input-output representation $S:D \times X \to Y$. In contexts where $S \subset \times \{A, D, \Theta, H, X, Y\}$, we use $(d, x, y) \in \mathcal{P}(S)$ to make reference to the input-output representation. Also, the subset of the powerset of a powerset $K \subset \mathcal{P}(\mathcal{P}(D \cup \Theta))$ is used to denote that $K$ can be $\subset D$, $\subset \Theta$, or $\subset D \times \Theta$, etc., i.e., to make reference to ordered pairs. Often, we make reference to $d \in D$ to say a particular set of data $d$ from the larger set $D$. Additionally, for a system $S \subset X \times Y$, when we discuss $x \in X$ or $y \in Y$ it is assumed that $(x, y) \in S$ unless stated otherwise. This is to save the reader from the pedantry of Mesarovician abstract systems theory.
\section{Introduction}
Transfer learning, unlike classical learning, does not assume that the training and operating environments are the same, and, as such, is fundamental to the development of real-world learning systems. In transfer learning, knowledge from various \emph{source} sample spaces and associated probability distributions is \emph{transferred} to a particular \emph{target} sample space and probability distribution. Transfer learning enables learning in environments where data is limited. Perhaps more importantly, it allows learning systems to propagate their knowledge forward through distributional changes.
Mechanisms for knowledge transfer are a bottleneck in the deployment of learning systems. Learning in identically distributed settings has been the focus of learning theory and machine learning research for decades, however, such settings represent a minority of use cases. In real-world settings, distributions and sample spaces vary between systems and evolve over time. Transfer learning addresses such differences by sharing knowledge between learning systems, thus offering a theory principally based on distributional difference, and thereby a path towards the majority of use cases.
Existing transfer learning frameworks are incomplete from a systems theoretic perspective. They focus on domain and task, and neglect perspectives offered by explicitly considering system structure and behavior. Mesarovician systems theory can be used as a super-structure for learning to top-down model transfer learning, and although existing transfer learning frameworks may better reflect and classify the literature, the resulting systems theoretic framework offers a more rigorous foundation better suited for system design and analysis.
Mesarovician systems theory is a set-theoretic meta-theory concerned with the characterization and categorization of systems. A system is defined as a relation on sets and mathematical structure is sequentially added to those sets, their elements, or the relation among them to formalize phenomena of interest. By taking a top-down, systems approach to framing transfer learning, instead of using a bottom-up survey of the field, we naturally arrive at a framework for modeling transfer learning without necessarily referencing solution methods. This allows for general considerations of transfer learning systems, and is fundamental to the understanding of transfer learning as a mathematical construct.
We provide a novel definition of transfer learning systems, dichotomize transfer learning in terms of structure and behavior, and formalize notions of negative transfer, transferability, transfer distance, and transfer roughness in subsequent elaborations. First we review transfer learning and Mesarovician abstract systems theory in Section 2. We then define learning systems and discuss their relationship to abstract systems theory and empirical risk minimization in Section 3. Using this definition, transfer learning systems are defined and studied in Sections 4 and 5. We conclude with a synopsis and remarks in Section 6.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{framework-comparison-mindmap-3.png}
\caption{Existing frameworks interpret the informal definition of transfer learning given by DARPA in terms of domain $\mathcal{D}$ and task $\mathcal{T}$. In contrast, we use structure and behavior, which provide a more formal basis for elaboration.}
\label{fig:frameworks}
\end{figure*}
\section{Background}
In the following we review transfer learning and make explicit the principal differences between existing frameworks and ours. Then, pertinent Mesarovician abstract systems theory is introduced. A supplemental glossary of Mesarovician terms can be found in the Appendix.
\subsection{Transfer Learning}
DARPA describes transfer learning as ``the ability of a system to recognize and apply knowledge and skills learned in previous tasks to novel tasks'' in Broad Agency Announcement (BAA) 05-29. The previous tasks are referred to as source tasks and the novel task is referred to as the target task. Thus, transfer learning seeks to transfer knowledge from some source learning systems to a target learning system.
Existing frameworks focus on a dichotomy between \emph{domain} $\mathcal{D}$ and \emph{task} $\mathcal{T}$. The domain $\mathcal{D}$ consists of the input space $X$ and its marginal distribution $P(X)$. The task $\mathcal{T}$ consists of the output space $Y$ and its posterior distribution $P(Y|X)$. The seminal transfer learning survey frames transfer learning in terms of an inequality of domains $\mathcal{D}$ and tasks $\mathcal{T}$ \cite{pan2009survey}. Therein, Pan and Yang define transfer learning as follows.
\begin{definition}{\emph{Transfer learning}.} \\
Given a source domain $\mathcal{D}_S$ and task $\mathcal{T}_S$ and a target domain $\mathcal{D}_T$ and task $\mathcal{T}_T$, transfer learning aims to improve the learning of $P(Y_T|X_T)$ in the target using knowledge in $\mathcal{D}_S$ and $\mathcal{T}_S$, where $\mathcal{D}_S \neq \mathcal{D}_T$ or $\mathcal{T}_S \neq \mathcal{T}_T$.
\end{definition}
Pan and Yang continue by defining \emph{inductive transfer} as the case where the source and target tasks are not equal, $\mathcal{T}_S \neq \mathcal{T}_T$, and \emph{transductive transfer} as the case where the source and target domains are not equal but their tasks are, $\mathcal{D}_S \neq \mathcal{D}_T \land \mathcal{T}_S = \mathcal{T}_T$. They use these two notions, and their sub-classes, to categorize the transfer learning literature and its affinity for related fields of study. Alternative frameworks use notions of \emph{homogeneous} and \emph{heterogeneous} transfer, which correspond to the cases where the sample spaces of the source and target domains $X$ and tasks $Y$ are or are not equal, respectively \cite{weiss2016survey}.
While these formalisms describe the literature well, they are not rich enough to maintain formalism in the elaboration of their respective frameworks. For example, Pan and Yang address what, how, and when to transfer in a largely informal manner, making reference to inductive and transductive transfer as guideposts, but ultimately resorting to verbal descriptions \cite{pan2009survey}. In contrast, instead of starting with domain $\mathcal{D}$ and task $\mathcal{T}$ as the fundamental notions of transfer learning, we use structure and behavior---two concepts with deep general systems meaning, define transfer learning as a relation on systems, and carry formalism through into subsequent elaboration. The principal difference between existing frameworks and ours is depicted in Figure \ref{fig:frameworks}.
Importantly, despite our formalism, we maintain a general systems level of abstraction, in contrast to purely learning theoretical frameworks for transfer learning \cite{kuzborskij2013stability}. As such, we compare our general framework with those of Pan and Yang \cite{pan2009survey} and Weiss et. al \cite{weiss2016survey}. We greatly expand on previous, initial efforts in this direction \cite{cody2019systems, cody2020motivating}.
\begin{figure}[b]
\centering
\includegraphics[width=8cm]{ast-block.png}
\caption{AST is a minimally formal framework. In modeling learning, learning theory brings formalism to AST, and machine learning specifies the detailed model.}
\label{fig:ast-block}
\end{figure}
\subsection{Abstract Systems Theory}
Mesarovician abstract systems theory (AST) is a general systems theory that adopts the formal minimalist world-view \cite{mesarovic1989abstract, dori2019system}. AST is developed top-down, with the goal of giving a verbal description a parsimonious yet precise mathematical definition. Mathematical structure is added as needed to specify systems properties of interest. This facilitates working at multiple levels of abstraction within the same framework, where mathematical specifications can be added without restructuring the framework. In modeling, it is used as an intermediate step between informal reasoning and detailed mathematics by formalizing block-diagrams with little to no loss of generality, see Figure \ref{fig:ast-block}. Apparently this generality limits its deductive powers, but, in return, it helps uncover fundamental mathematical structure related to the general characterization and categorization of phenomena.
We will now review the AST definitions of a system, input-output system, and goal-seeking system, and the related notions of system structure and behavior. Additional details can be found in the Appendix.
In AST, a system is defined as a relation on component sets. When those sets can be partitioned, the system is called an input-output system. Systems and input-output systems are defined as follows.
\begin{definition}{\emph{System}.} \\
A (general) system is a relation on non-empty (abstract) sets,
$$S \subset \times \{ V_{i} : i \in I \}$$
where $\times$ denotes the Cartesian product and $I$ is the index set. A component set $V_{i}$ is referred to as a system object.
\end{definition}
\begin{definition}{\emph{Input-Output Systems}.} \\
Consider a system $S$, where $S \subset \times \{ V_{i} : i \in I \}$. Let $I_{x} \subset I$ and $I_{y} \subset I$ be a partition of $I$, i.e., $I_{x} \cap I_{y} = \emptyset$, $I_{x} \cup I_{y} = I$. The set $X = \times \{ V_{i} : i \in I_{x} \}$ is termed the input object and $Y = \times \{ V_{i} : i \in I_{y} \}$ is termed the output object. The system is then
$$S \subset X \times Y$$
and is referred to as an input-output system. If $S$ is a function $S:X \to Y$, it is referred to as a function-type system.
\end{definition}
AST is developed by adding structure to the component sets and the relation among them. Input-output systems with an internal feedback mechanism are referred to as goal-seeking (or cybernetic) systems. The internal feedback of goal-seeking systems is specified by a pair of consistency relations $G$ and $E$ which formalize the notions of goal and seeking, respectively. Figure \ref{fig:i-o-system} depicts input-output and goal-seeking systems. Goal-seeking systems are defined as follows.
\begin{definition}{\emph{Goal-Seeking Systems}.} \\
A system $S:X \to Y$ has a goal-seeking representation if there exists a pair of maps
\begin{gather*}
S_G:X \times Y \to \Theta \\
S_F:\Theta \times X \to Y
\end{gather*}
and another pair
\begin{gather*}
G: \Theta \times X \times Y \to V \\
E: X \times Y \times V \to \Theta
\end{gather*}
such that
\begin{gather*}
(x, y) \in S \leftrightarrow (\exists \theta) [(\theta, x, y) \in S_F \wedge (x, y, \theta) \in S_G] \\
(x, y, G(\theta, x, y), \theta) \in E \leftrightarrow (x, y, \theta) \in S_G
\end{gather*}
where
$$x \in X, y \in Y, \theta \in \Theta.$$
$S_G$ is termed the goal-seeking system and $S_F$ the functional system. $G$ and $E$ are termed the goal and seeking relations, and $V$ the value.
\label{def:gs}
\end{definition}
System structure and behavior are focal in Mesarovician characterizations of systems. System structure refers to the mathematical structure of a system's component sets and the relations among them. For example, there may be algebraic structure related to the specification of the relation, e.g. the linearity of a relationship between two component sets. System behaviors, in contrast, are properties or descriptions paired with systems. For example, consider a system $S:X \to Y$ and a map $S \to \{ stable, neutral, unstable \}$. A linear increasing function and an increasing power function may both be considered behaviorally unstable, but clearly their structures are different \cite{mesarovic1989abstract}.
Similarity of systems is a fundamental notion, and it can be expressed well in structural and behavioral terms. Structural similarity describes the \emph{homomorphism} between two systems' structures. Herein, in accord with category theory, a map from one system to another is termed a morphism, and homomorphism specifies the morphism to be onto. Homomorphism is formally defined as follows.
\begin{definition}{\emph{Homomorphism}.} \\
An input-output system $S \subset X \times Y$ is homomorphic to $S' \subset \times X' \times Y'$ if there exists a pair of maps,
\begin{align*}
\varrho:X \to X', \vartheta:Y \to Y'
\end{align*}
such that for all $x\in X$, $x'\in X'$, and $y\in Y$, $y'\in Y'$, $\varrho(x)=x'$ and $\vartheta(y)=y'$.
\end{definition}
\noindent Behavioral similarity, in contrast, describes the \emph{proximity} or \emph{distance} between two systems' behavior. As in AST generally, we use structure and behavior as the primary apparatus for elaborating on our formulation of transfer learning systems. Refer to the Appendix for additional details on structure, behavior, and similarity.
\begin{figure}[t]
\centering
\includegraphics[]{i-o-system.png}
\caption{Input-output systems (left) and goal-seeking systems (right).}
\label{fig:i-o-system}
\end{figure}
\section{Learning Systems}
We follow Mesarovic's top-down process to sequentially construct a learning system $S$. Learning is a relation on data and hypotheses. To the extent that a scientific approach is taken, those hypotheses are explanations of initial-final condition pairs \cite{popper2005logic}. Otherwise put, we are concerned with learning as function estimation. We additionally note that learning algorithms use data to select those hypotheses and that the data is a sample of input-output pairs \cite{vapnik2013nature}. Such a learning system can be formally defined as follows.
\begin{definition}{\emph{(Input-Output) Learning System}.} \\
A learning system $S$ is a relation
$$S \subset \times \{A, D, \Theta, H, X, Y \}$$
such that
\begin{gather*}
D \subset X \times Y, A:D \to \Theta, H:\Theta \times X \to Y \\
(d, x, y) \in \mathcal{P}(S) \leftrightarrow (\exists \theta) [(\theta, x, y) \in H \wedge (d, \theta) \in A]
\end{gather*}
where
$$x \in X, y \in Y, d \in D, \theta \in \Theta.$$
The algorithm $A$, data $D$, parameters $\Theta$, hypotheses $H$, input $X$, and output $Y$ are the component sets of $S$, and learning is specified in the relation among them.
\label{def:ls}
\end{definition}
The above definition of learning formalizes learning as a cascade connection of two input-output systems: an inductive system $S_I \subset \times \{ A, D, \Theta \}$ responsible for inducing hypotheses from data, and a functional system $S_F \subset \times \{ \Theta, H, X, Y \}$, i.e. the induced hypothesis. $S_I$ and $S_F$ are coupled by the parameter $\Theta$. Learning is hardly a purely input-output process, however. To address this, we must specify the goal-seeking nature of $S_I$, and, more particularly, of $A$.
$A$ is goal-seeking in that it makes use of a \emph{goal} relation $G: D \times \Theta \to V$ that assigns a value $v \in V$ to data-parameter pairs, and a \emph{seeking} relation $E: V \times D \to \Theta$ that assigns parameter $\theta \in \Theta$ to data-value pairs. These consistency relations $G$ and $E$ specify $A$, but not by decomposition; i.e., in general, $G$ and $E$ cannot be composed to form $A$. The definition of a learning system can be extended as follows.
\begin{definition}{\emph{(Goal-Seeking) Learning System}.} \\
A learning system $S$ is a relation
$$S \subset \times \{ A, D, \Theta, G, E, H, X, Y \}$$
such that
\begin{gather*}
D \subset X \times Y, A:D \to \Theta, H:\Theta \times X \to Y \\
(d, x, y) \in \mathcal{P}(S) \leftrightarrow (\exists \theta) [(\theta, x, y) \in H \wedge (d, \theta) \in A] \\
G: D \times \Theta \to V, E: V \times D \to \Theta \\
(d, G(\theta, d), \theta) \in E \leftrightarrow (d, \theta) \in A
\end{gather*}
where
$$x \in X, y \in Y, d \in D, \theta \in \Theta.$$
The algorithm $A$, data $D$, parameters $\Theta$, consistency relations $G$ and $E$, hypotheses $H$, input $X$, and output $Y$ are the component sets of $S$, and learning is specified in the relation among them.
\label{def:gsls}
\end{definition}
\begin{figure}[t]
\centering
\includegraphics[]{learning-system.png}
\caption{Learning systems are a cascade connection of the inductive system $S_I$ and the induced hypothesis $S_F$. $S_I$ is goal-seeking.}
\label{fig:learning-systems}
\end{figure}
Learning systems are depicted in Figure \ref{fig:learning-systems}. These systems theoretic definitions of learning have an affinity to learning theoretic constructions. Consider empirical risk minimization (ERM), where empirical measures of risk are minimized to determine the optimal hypothesis for a given sample \cite{vapnik2013nature}. Apparently, ERM specifies $G$ to be a measure of risk calculated on the basis of a sample drawn independently according to a probability measure on the approximated function $f:X \to Y$ and specifies $E$ to be a minimization of $G$ over $\Theta$.
We have demonstrated how our definition of a learning system anchors our framework to both AST and ERM. We posit these definitions not as universal truths, but rather as constructions that anchor our framing of transfer learning to systems and learning theory. We abstain from further elaboration on these definitions, however, proofs of the above propositions can be found in the Appendix. In the following, we leave $G$ and $E$ implicit, only making reference to $f$ and related probability measures.
\begin{exmp}{\emph{Learning in an Unmanned Aerial Vehicle.}} \\
Consider an unmanned aerial vehicle (UAV) with a learning system $S$ for path planning. $H$ is a function from sensor data $X$, e.g., from accelerometers, cameras, and radar, to flight paths $Y$. $D$, then, consists of sets of sensor-path pairs. If $S$ is a support-vector machine (SVM), then $H$ is a set of half-spaces parameterized by $\Theta$ and $A$ is a convex optimization routine\cite{suykens1999least}.
\label{ex:learning-systems}
\end{exmp}
\section{Transfer Learning Systems}
Transfer learning is conventionally framed as a problem of sharing knowledge from source domains and tasks to a target domain and task. We propose an alternative approach. We formulate transfer learning top-down in reference to the source and target learning systems, and then dichotomize subsequent analysis not by domain and task, but rather by structure, described primarily by the $X \times Y$ space, and behavior, described primarily by probability measures on the estimated function $f:X \to Y$.
A transfer learning system is a relation on the source and target systems that combines knowledge from the source with data from the target and uses the result to select a hypothesis that estimates the target learning task $f_T$. We define it formally as follows.
\begin{definition}{\emph{Transfer Learning System}.} \\
Given source and target learning systems $S_S$ and $S_T$
\begin{gather*}
S_S \subset \times \{ A_S, D_S, \Theta_S, H_S, X_S, Y_S \} \\
S_T \subset \times \{ A_T, D_T, \Theta_T, H_T, X_T, Y_T \}
\end{gather*}
a transfer learning system $S_{Tr}$ is a relation on the component sets of the source and target systems $S_{Tr} \subset \overline{S_S} \times \overline{S_T}$ such that
$$K_S \subset D_S \times \Theta_S, D \subset D_T \times K_S$$
and
\begin{gather*}
A_{Tr}: D \to \Theta_{Tr}, H_{Tr}: \Theta_{Tr} \times X_T \to Y_T \\
(d, x_T, y_T) \in \mathcal{P}(S_{Tr}) \leftrightarrow \\
(\exists \theta_{Tr})[(\theta_{Tr}, x_T, y_T) \in H_{Tr} \land (d, \theta_{Tr}) \in A_{Tr}]
\end{gather*}
where
$$x_T \in X_T, y_T \in Y_T, d \in D, \theta_{Tr} \in \Theta_{Tr}.$$
The nature of source knowledge $K_S$\footnote{Here, we define the transferred knowledge $K_S$ to be $D_S$ and $\Theta_S$, the source data and parameters, following convention \cite{pan2009survey}. In general, however, source knowledge $K_S \subset \mathcal{P}(\mathcal{P}(\overline{S_S}))$.}, the transfer learning algorithm $A_{Tr}$, hypotheses $H_{Tr}$, and parameters $\Theta_{Tr}$ specify transfer learning as a relation on $\overline{S_S}$ and $\overline{S_T}$.
\label{def:tl}
\end{definition}
Trivial transfer occurs when the structure and behavior of $S_S$ and $S_T$ are the same, or, otherwise put, when transfer learning reduces to classical, identically distributed learning. Transfer is non-trivial when there is a structural difference $X_S \times Y_S \neq X_T \times Y_T$ or a behavioral difference $P(X_S) \neq P(X_T) \lor P(Y_S|X_S) \neq P(Y_T|X_T)$ between the source $S_S$ and target $S_T$. If the posterior distributions $P(Y|X)$ and marginal distributions $P(X)$ are equal between the source and target systems, then transfer is trivial. Non-trivial transfer is implied when $X_S \times Y_S \neq X_T \times Y_T$.
\begin{prop}
\emph{$S_{Tr}$ in Definition \ref{def:tl} is a learning system as defined in Definition \ref{def:ls}.}\\
\emph{Proof:}
As stated in Definition \ref{def:tl}, a transfer learning system is a relation $S_{Tr} \subset \overline{S_S} \times \overline{S_T}$. More particularly, it is a relation $S_{Tr} \subset (D_S \times \Theta_S) \times (D_T \times X_T \times Y_T)$, and has a function-type representation $S_{Tr}: D_S \times \Theta_S \times D_T \times X_T \to Y_T$. Its inductive system is the relation $A_{Tr}:D \to \Theta_{Tr}$, where $D \subset D_S \times \Theta_S \times D_T$. And its functional system is the relation $H_{Tr}: \Theta_{Tr} \times X_T \to Y_T$. Thus, we can restate $S_{Tr}$ as a relation
$$S_{Tr} \subset \times \{A_{Tr}, D, \Theta_{Tr}, H_{Tr}, X_T, Y_T\}$$
and since by Definition \ref{def:tl}
\begin{gather*}
(d, x_T, y_T) \in \mathcal{P}(S_{Tr}) \leftrightarrow \\
(\exists \theta_{Tr})[(\theta_{Tr}, x_T, y_T) \in H_{Tr} \land (d, \theta_{Tr}) \in A_{Tr}]
\end{gather*}
where
$$x_T \in X_T, y_T \in Y_T, d \in D, \theta_{Tr} \in \Theta_{Tr},$$
we have that $S_{Tr}$ is an input-output learning system as in Definition $\ref{def:ls}$.
\end{prop}
Transfer learning systems are distinguished from general learning systems by the selection and transfer of $K_S$, and its relation to $D_T$ by way of $D \subset K_S \times D_T$ and its associated operator $K_S \times D_T \to D$. In cases where $\{A_{Tr}, \Theta_{Tr}, H_{Tr}\} \leftrightarrow \{A_T, \Theta_T, H_T\}$, e.g., as is possible when transfer learning consists of pooling samples with identical supports, the additional input $K_S$ is all that distinguishes $S_{Tr}$ from $S_T$. Classical and transfer learning systems are depicted in Figure \ref{fig:tl-system}.
As we will see, however, this is no small distinction, as it allows for consideration of learning across differing system structures and behaviors. But before we elaborate on the richness of structural and behavioral considerations, first, in the following subsections, we interpret existing frameworks in terms of structure and behavior and define preliminary notions related to generalization in transfer learning.
\begin{figure}[t]
\centering
\includegraphics[]{tl-1.png}
\caption{Transfer learning systems $S_{Tr}$ are a relation $K_S \times D_T \times X_T \to Y_T$, while the target system $S_T$ is a relation $D_T \times X_T \to Y_T$.}
\label{fig:tl-system}
\end{figure}
\begin{exmp}{\emph{Transfer Learning in UAVs.}} \\
Consider UAVs with learning systems $S_S$ and $S_T$ defined according to Example \ref{ex:learning-systems} and a transfer learning system $S_{Tr} \subset \overline{S_S} \times \overline{S_T}$. If $S_{Tr}$ is also a SVM, then $H_{Tr}$ are also half-spaces parameterized by $\Theta_{Tr}$. If $K_S \subset D_S \times \Theta_S$, $\Theta_S$ can provide an initial estimate for $\Theta_{Tr}$, and $D_S$ can be pooled with $D_T$ to update this estimate. $A_{Tr}$, in distinction to $A_T$, must facilitate this initialization and pooling.
\label{ex:transfer}
\end{exmp}
\subsection{Comparison to Existing Frameworks}
Using Definition \ref{def:tl}, the central notions of existing frameworks can be immediately defined in terms of structural and behavioral inequalities. Homogeneous transfer specifies structural equality of the source and target sample spaces, $X_S \times Y_S = X_T \times Y_T$, and heterogeneous transfer specifies otherwise. Domain adaptation, co-variate shift, and prior shift are all examples of homogeneous transfer \cite{jiang2008literature, pan2009survey, csurka2017comprehensive}. Transductive and inductive transfer entail more nuanced specifications.
Recall, inductive transfer specifies that $\mathcal{T}_S \neq \mathcal{T}_T$ and transductive transfer specifies that $\mathcal{D}_S \neq \mathcal{D}_T \land \mathcal{T}_S = \mathcal{T}_T$, where $\mathcal{D}=\{P(X), X\}$ and $\mathcal{T}=\{P(Y|X), Y\}$. Technically, transductive transfer occurs if $X_S \neq X_T$ or if $P(X_S) \neq P(X_T)$. However, if $X_S \neq X_T$, then it is common for $P(Y_S|X_S) \neq P(Y_T|X_T)$ because the input set conditioning the posterior has changed, and thus it is likely that $\mathcal{T_S} \neq \mathcal{T_T}$. To that extent, in the main, transductive transfer specifies a difference between input behavior while output behavior remains equal. Inductive transfer, on the other hand, is more vague, and merely specifies that there is a structural difference in the outputs, $Y_S \neq Y_T$, or a behavioral difference in the posteriors, $P(Y_S|X_S) \neq P(Y_S|X_T)$. Note, this behavioral difference in the posteriors can be induced by a structural difference in the inputs as previously mentioned, and is implied by a structural difference in the outputs.
In short, the homogeneous-heterogeneous dichotomy neglects behavior and the transductive-inductive framing muddles the distinction between structure and behavior. While frameworks based on either cover the literature well, they only provide high-level formalisms which are difficult to carry through into general, formal characterizations of transfer learning systems. In contrast, Definition \ref{def:tl} provides a formalism that can be used to define transfer learning approaches and auxiliary topics in generalization.
\subsection{Transfer Approaches}
Consider how the seminal framework informally classifies transfer learning algorithms \cite{pan2009survey}. Three main approaches are identified: `instance transfer', `parameter transfer', and `feature-representation transfer'. While the transductive or inductive nature of a transfer learning system gives insight into which approaches are available, the approaches cannot be formalized in those terms, or in terms of domain $\mathcal{D}$ and task $\mathcal{T}$ for that matter, because they are a specification on the inductive system $S_I \subset \times \{A_{Tr}, D_{Tr}, \Theta_{Tr}\}$, whereas the former are specifications on the functional system $S_F \subset \times \{\Theta_{Tr}, H_{Tr}, X_{Tr}, Y_{Tr}\}$.
With the additional formalism of Definition \ref{def:tl}, these transfer approaches can be formalized using system structure. First, note that differently structured data $D$ leads to different approaches. Consider the categories of transfer learning systems corresponding to the various cases where $D \subset \mathcal{P}(\mathcal{P}(D_T \cup D_S \cup \Theta_S))$. Instance and parameter transfer correspond to transferring knowledge in terms of $D_S$ and $\Theta_S$, respectively, and can be formally defined as follows.
\begin{definition}{\emph{Instance Transfer}.} \\
A transfer learning system $S_{Tr}$ is an instance transfer learning system if $K_S \subset D_S$, i.e., if
$$\mathcal{A}_{Tr}: D \to \Theta_{Tr} \iff \mathcal{A}_{Tr}: D_S \times D_T \to \Theta_{Tr}.$$
\end{definition}
\begin{definition}{\emph{Parameter Transfer}.} \\
A transfer learning system $S_{Tr}$ is a parameter transfer learning system if $K_S \subset \Theta_S$, i.e., if
$$\mathcal{A}_{Tr}: D \to \Theta_{Tr} \iff \mathcal{A}_{Tr}: \Theta_S \times D_T \to \Theta_{Tr}.$$
\end{definition}
Feature-representation transfer, in contrast, specifies that learning involves transformations on $\overline{S_T}$, $K_S$, or both. It can be defined formally as follows.
\begin{definition}{\emph{Feature-Representation Transfer}.} \\
Consider a transfer learning system $S_{Tr}$ and a learning system $S_L$, termed the latent learning system. Note, $S_{Tr}$ and $S_L$ can be represented as function-type systems,
\begin{gather*}
S_{Tr}: D \times X_T \to Y_T \\
S_L: D_L \times X_L \to Y_L.
\end{gather*}
$S_{Tr}$ is a feature-representation transfer learning system if there exist maps
$$m_D:D \to D_L, m_{X_T}:X_T \to X_L, m_{Y_L}:Y_L \to Y_T$$
such that
\begin{gather*}
\forall (d, x_T, y_T) \in (S_{Tr}) \\
S_{Tr}(d, x_T) \leftrightarrow m_{Y_L}(S_L(m_D(d), m_{X_T}(x_T)))
\end{gather*}
where
$$d \in D, x_T \in X_T, y_T \in Y_T.$$
In other words, $S_{Tr}$ is a feature-representation transfer learning system if transfer learning involves transforming to and from a latent system where learning occurs.
\end{definition}
\begin{prop}
\emph{Learning in $S_S$, $S_T$, and $S_L$.} \\
Consider a case of feature-representation transfer where $K_S \subset D_S$. Let $m_{D_T}:D_T \to D_L$ and $m_{D_S}:D_S \to D_L$. Then, $m_D \iff (m_{D_T}, m_{D_S})$. Recall $D_i \subset X_i \times Y_i$. If $m_{D_T}$ is the identity and $m_{D_S}$ is not, then $X_T \times Y_T = X_L \times Y_L$---learning occurs in the target sample space. If $m_{D_S}$ is the identity and $m_{D_T}$ is not, then $X_S \times Y_S = X_L \times Y_L$---learning occurs in the source sample space. If $m_D$ is the identity, then $X_S \times Y_S = X_T \times Y_T = X_L \times Y_L$, i.e., $S_{Tr}$ involves homogeneous transfer. If neither $m_{D_T}$ or $m_{D_S}$ are the identity, then learning occurs in a latent sample space $X_L \times Y_L$ that is unequal to $X_T \times Y_T$ and $X_S \times Y_S$.
\end{prop}
In feature-representation transfer, data $D \subset D_T \times K_S$ is mapped to a latent system $S_L$ where learning occurs. By way of $m_D:D \to D_L$, feature-representation transfer involves relating the source and target input-output spaces to a latent space $X_L \times Y_L$. Learning can occur in $X_L \times Y_L$, and, using $m_{Y_L}$, the output can be given in terms of the target output $Y_T$. Similarly, the target can be mapped onto the source, $X_L \times Y_L = X_S \times Y_S$, where learning can occur given $m_{Y_L}$, or the source can be mapped onto the target, $X_L \times Y_L = X_T \times Y_T$.
Figure \ref{fig:latent-learning} depicts these three cases of morphisms using a commutative diagram. As the individual maps that compose these morhpisms become more dislike identities and partial, feature-representation transfer becomes more difficult. We will discuss this further in our elaboration on structural considerations. Additionally note, even if $X_S \times Y_S = X_T \times Y_T$, feature-representation transfer may still be used to better relate source and target behavior.
\begin{table}[t]
\centering
\ra{1.3}
\begin{tabular}{@{}ll@{}}
\toprule
Transfer Approach & Algorithm Structure\\
\midrule
Instance & $A_{Tr}:D_T \times D_S \to \Theta_{Tr}$\\
Parameter & $A_{Tr}:D_T\ \times \Theta_S \to \Theta_{Tr}$\\
Instance \& Parameter & $A_{Tr}:D_T \times D_S \times \Theta_S \to \Theta_{Tr}$\\
Feature-Representation & $A_{Tr}: m_D(D) \to \Theta_{Tr}$ \\
\bottomrule
\\
\end{tabular}
\caption{Structural differences between transfer approaches.}
\label{table:approaches}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[]{latent-learning23.png}
\caption{Morphisms in feature representation learning. Learning in the target sample space requires a morphism from that of the source, as shown in red. Learning in the source sample space requires a morphism from that of the target, as shown in blue, and a map from the source output to the target output, as shown by the dashed blue arrow. And learning in a latent sample space requires morphisms from both the source and target sample spaces to that of the latent system, as shown in green, and a map from the latent output to the target output, as shown by the dashed green arrow. As discussed in Section 5, the nature of these morphisms affects the difficulty of transfer.}
\label{fig:latent-learning}
\end{figure*}
Instance, parameter, and feature-based approaches are shown in terms of their specification on transfer learning algorithms $A_{Tr}$ in Table \ref{table:approaches}. Another general notion in transfer learning is \emph{n}-shot transfer. It can be defined as follows.
\begin{definition}{\emph{N-shot Transfer}.} \\
A transfer learning system $S_{Tr}$ with target data $d_T \in D_T$ is referred to as a n-shot transfer learning system if $|d_T| = n$. Zero-shot transfer occurs if $A_{Tr}: D \to \Theta_{Tr} \iff A_{Tr}: K_S \to \Theta_{Tr}$.
\end{definition}
\noindent Machine learning is often concerned with few-shot learners---transfer learning systems that can generalize with only a few samples from the target. We will discuss generalization in transfer learning in the following subsection, but first, to get a sense of how we formalize instance, parameter, and feature-representation transfer, consider how a few canonical transfer learning algorithms are modeled by our framework.
Transfer component analysis uses a modified principal component analysis approach to project the source and target data into a relatable latent space \cite{pan2010domain}, i.e., it is an instance approach in that $D_S$ is used in $A_{Tr}$ and a feature-representation approach in that $X_S$ and $X_T$ are projected into a latent $X_L$. Constraining parameters to be within a range of those of the source, as in hierarchical Bayesian and regularization approaches, is parameter transfer \cite{evgeniou2004regularized, schwaighofer2005learning}. Deep learning approaches often involve parameter transfer in that the weights $\Theta_S$ of the source network are shared and frozen in the target, or otherwise used to initialize $\Theta_T$ \cite{bengio2012deep}. Other deep learning approaches also involve instance transfer to increase sample size, such as those that use generative adversarial networks \cite{sankaranarayanan2018generate}. When the source and target data must first be transformed before the data can be related, they are also feature-representation approaches, as in joint adaptation networks \cite{long2017deep}.
By formalizing the canonical classes of transfer approaches, we are better able to understand them in terms of their general requirements on $S_{Tr}$, particularly on $S_I$, and more particularly on $A_{Tr}$ and $D$. The informal use of these classes by existing frameworks, wherein a solution method's dominant nature sorts it into a particular class, does well to organize the literature. Our formalisms can cloud these scholarly distinctions, as shown in the case of deep learning where a single method can belong to all three classes, however, they give a basis for defining formal categories of transfer learning systems $S_{Tr}$ in terms of their inductive systems $S_I$.
\subsection{Generalization in Transfer Learning}
Generalization is, perhaps, the ultimate aim of learning. It is the ability for the learned hypothesis to approximate $f$ out-of-sample, i.e., on samples not seen in training. Generalization as a goal for learning systems is implicit in $A$ when a measure of error $\epsilon$ between $h(\theta)$ and $f$ specifies $G$, such as in ERM. Herein, we define it as follows.
\begin{definition}{\emph{Generalization.}} \\
Given a learning system $S$ and data $d \in D$, generalization is the ability for a learned hypothesis $h(\theta)$ to estimate learning task $f:X \to Y$, on samples $(x, y) \notin d$.
\end{definition}
In moving from the classical, identically distributed learning setting to transfer learning, we move from generalizing to a new sample from the same system, to generalizing to a new sample from a different system. In classical learning, for a learning system $S$, the estimated function $f$ is specified by $P(Y|X)$ and data $D$ are drawn from a related joint $P(X,Y)$. In transfer learning, however, the $X \times Y$ space and probability measures specifying $f$ and $D$ vary between $S_S$ and $S_T$.
In classical learning, given a learning system $S$, data $d \in D$, a measure of error $\epsilon: H(\Theta) \times f \to \mathbb{R}$, and a threshold on error $\epsilon^* \in \mathbb{R}$, we generalize if
$$\epsilon(H(A(d)), f) \leq \epsilon^*.$$
That, is, if the measure of error between the learned hypothesis and the function it estimates is below a threshold. In practice, since $f$ is not known, error is empirically estimated using samples $(x, y) \in X \times Y$ such that $(x, y) \notin d$.
In transfer learning, given $S_{Tr}$ and data $d \in D$, we generalize if
$$\underbrace{\epsilon(H_{Tr}(A_{Tr}(d)), f_T)}_{\epsilon_T} \leq \epsilon^*.$$
If $\epsilon_T$ is smaller without any transferred knowledge from $S_S$ than with, transfer from $S_S$ to $S_T$ is said to result in negative transfer. Negative transfer is defined in accord with Wang et. al as follows.
\begin{definition}{\emph{Negative Transfer}.} \\
Consider a transfer learning system $S_{Tr}$. Recall $D \subset D_T \times K_S$. Let $d \in D$ and $d_T \in D_T$. Given a measure of error $\epsilon: H(\Theta) \times f \to \mathbb{R}$, negative transfer is said to occur if
$$\epsilon(H_T(A_T(d_T)), f_T) < \epsilon(H_{Tr}(A_{Tr}(d)), f_T),$$
that is, if the error in estimating $f_T$ is higher with the transferred knowledge than without it.
\label{def:nt}
\end{definition}
\noindent As Wang et. al note, negative transfer can arise from behavioral dissimilarity between the source and target \cite{wang2019characterizing}. In general, it can arise from structural dissimilarity as well.
Because generalization in transfer learning considers generalization across systems, as opposed to generalization within a given system, naturally, it is concerned with the set of systems to and from which transfer learning can generalize. Using $\epsilon_T$ and $\epsilon^*$, we can describe these sets as neighborhoods of systems \emph{to} which we can transfer and generalize,
$$ \underbrace{\{ S_T | S_S, \epsilon_T \leq \epsilon^* \}}_{\text{Neighborhood of Targets } S_T}$$
and neighborhoods of systems \emph{from} which we can transfer and generalize,
$$ \underbrace{\{ S_S | S_T, \epsilon_T \leq \epsilon^* \}}_{\text{Neighborhood of Sources } S_S}.$$
Noting Definition \ref{def:nt}, if $\epsilon^* = \epsilon(H_T(A_T(d_T)), f_T)$, these neighborhoods are those systems to and from which transfer is positive.
The size of these neighborhoods describes the transferability of a learning system in terms of the number of systems it can transfer to or from and generalize. To the extent that cardinality gives a good description of size\footnote{Cardinality counts arbitrarily close systems as different, and it may be preferable to define a measure of equivalence, and consider the cardinality of the neighborhoods after the equivalence relation is applied.}, transferability can be defined formally as follows.
\begin{definition}{\emph{Transferability}.} \\
Consider a target learning system $S_T$ and a source learning system $S_S$. Given a measure of error $\epsilon_T: H_{Tr}(\Theta_{Tr}) \times f_T \to \mathbb{R}$ and a threshold on error $\epsilon^* \in \mathbb{R}$, the transferability of a source is the cardinality of the neighborhood of target systems $S_T$ to which it can transfer and generalize,
$$|\{S_T| S_S, \epsilon_T \leq \epsilon^*\}|,$$
and the transferability of a target is the cardinality of the neighborhood of source systems $S_S$ from which we can transfer and generalize,
$$|\{S_S|S_T, \epsilon_T \leq \epsilon^*\}|.$$
These cardinalities are termed the source-transferability and target-transferability, respectively.
\end{definition}
\noindent Note, this defines transferability as an attribute of a particular system---not an attribute of a source-target pairing.
Our interest in transferability as an aim of transfer learning systems echoes a growing interest of the machine learning community in a notion of \emph{generalist} learning systems \cite{kolesnikov2019big, huang2019gpipe, tschannen2020self}. Put informally, generalists are learning systems which can generalize to many tasks with few samples. Using our formalism, these systems can be described as learning systems with high source-transferability. More particularly, they can be defined as follows.
\begin{definition}{\emph{Generalist Learning Systems}.} \\
A generalist learning system $S_S$ is a system that can transfer to at least $t$ target systems $S_T$ with data $d_T \in D_T$ and generalize with at most $n$ target samples $(x_T, y_T) \in X_T \times Y_T.$ That is, they are systems $S_S$ where
$$|\{S_T | S_S, |d_T| \leq n, \epsilon_T < \epsilon^*\}| \geq t$$
\end{definition}
\noindent Generalists are sources $S_S$ that can $n$-shot transfer learn to $t$ or more targets $S_T$. Generalists are typically studied in the context of deep learning for computer vision, where a single network is tasked with few-shot learning a variety of visual tasks, e.g., classification, object detection, and segmentation, in a variety of environments \cite{kolesnikov2019big}.
In the following, we go beyond existing frameworks to explore notions of transferability---and thereby generalization, transfer roughness, and transfer distance in the context of structure and behavior. In doing so, we demonstrate the mathematical depth of Definition \ref{def:tl}. We show that not only does it allow for immediate, formal consideration of surface-level phenomena covered by existing frameworks, but moreover, it allows for a considerable amount of modeling to be done at the general level, i.e., without reference to solution methods, in following with the spirit of AST depicted in Figure \ref{fig:ast-block}.
\section{Structure and Behavior in Transfer Learning}
To the extent that generalization in transfer learning is concerned with sets of systems, it is concerned with how those sets can be expressed in terms of those systems' structures and behaviors. In the following subsections, we discuss how structural and behavioral equality and, moreover, similarity relate to the difficulty of transfer learning. Equalities between $S_S$ and $S_T$ give a basic sense of the setting and what solution methods are available. Similarities between $S_S$ and $S_T$ are a richer means for elaboration, and can give a sense of the likelihood of generalization.
Learning systems are concerned with estimating functions $f:X \to Y$. As transfer learning is concerned with sharing knowledge used to estimate a source function $f_S:X_S \to Y_S$ to help estimate a target function $f_T:X_T \to Y_T$, naturally, the input-output spaces of the source $X_S \times Y_S$ and target $X_T \times Y_T$ are the principal interest of structural considerations. Similarly, the principal interest of behavioral considerations are the probability measures which specify $f_S$ and $f_T$, and, correspondingly, $D_S$ and $D_T$.
\subsection{Structural Considerations}
For source and target systems $S_S$ and $S_T$ we have the following possible equalities between system structures:
\begin{align*}
X_S = X_T, Y_S = Y_T, \\
X_S \neq X_T, Y_S = Y_T, \\
X_S = X_T, Y_S \neq Y_T, \\
X_S \neq X_T, Y_S \neq Y_T.
\end{align*}
The first case $X_S \times Y_S = X_T \times Y_T$ specifies transfer as homogeneous---all others specify heterogeneous transfer. This is the extent of discussion of structure in the existing frameworks \cite{pan2009survey, weiss2016survey}. We elaborate further.
To do so, we extend past structural equality to notions of structural similarity. Recall, structural similarity is a question of the structural homomorphism between two systems. As is common in category theory, we define a morphism as simply a map between systems, and define an onto map between systems as a homomorphism. We can investigate homomorphism in reference to a morphism $m: S_S \to S_T$. First, note that we can quantify structural similarity using equivalence classes. Let $m_x:X_S \to X_T$ and $m_y:Y_S \to Y_T$ such that $m \leftrightarrow (m_x, m_y)$. And let $S_S/m$, $X_S/m_x$, and $Y_S/m_y$ be the equivalence classes of $S_S$, $X_S$, and $Y_S$ with respect to $m$, $m_x$, and $m_y$, respectively.
Consider the two sets of relations
\begin{equation*}
\begin{split}
w &: S_S \to S_S/m \\
w_x &:X_S \to X_S/m_x \\
w_y &: Y_S \to Y_S/m_y
\end{split}
\qquad
\begin{split}
z &: S_S/m \to S_T \\
z_x &: X_S/m \to X_T \\
z_y &: Y_S/m \to Y_T
\end{split}
\end{equation*}
Relation $w$ maps the source $S_S$ to its equivalence class $S_S/m$ and relation $z$ maps $S_S/m$ to the target $S_T$, as depicted by the commutative diagram shown in Figure \ref{fig:roughness}. That is,
$$S_S \xrightarrow[(w_x, w_y)]{} S_S/m \xrightarrow[(z_x, z_y)]{} S_T $$
The equivalence class $S_S/m$ describes the `roughness' of the structural similarity from $S_S$ to $S_T$. Its cardinality quantifies the `surjective-ness' of $m:S_S \to S_T$. The greater the difference between $|S_S|$ and $|S_S/m|$, the more structurally dissimilar $S_S$ and $S_T$ are. However, in the large, structural similarity is not measurable in the same way as behavioral similarity.
\begin{figure}
\centering
\includegraphics{roughness.png}
\caption{A commutative diagram depicting how equivalence classes can describe roughness.}
\label{fig:roughness}
\end{figure}
The homomorphism between $S_S$ and $S_T$ is better investigated in terms of the properties of $m$, such as whether it is injective, surjective, invertible, etc. For example, partial morphisms from $X_S \times Y_S$ to $X_T \times Y_T$ are associated with partial transfer \cite{cao2018partial}. When the partial morphism is surjective, only a subset of the source is transferred to the target. When the partial morphism is injective, the source transfers to only a subset of the target. Also, structural similarity can be expressed using category theory, where the structural similarity between two systems can be studied with respect to the categories of systems to which they belong. To describe structural similarity in a broad sense, we define \emph{transfer roughness} as follows.
\begin{definition}{\emph{Transfer Roughness}.} \\
Transfer roughness describes the structural homomorphism from the source system $S_S$ to the target system $S_T$. When $S_S$ and $S_T$ are isomorphic, transfer roughness is minimal or otherwise non-existent. When roughness exists, it is defined by its properties, and thus there is no clear notion of maximal roughness.
\end{definition}
The structure of the source relative to that of the target determines the roughness of transfer. Structures can be too dissimilar to transfer no matter what the behavior. Homomorphisms are onto and thus structure preserving, and, as such, it is a reasonable principle to characterize structural transferability in terms of the set of homomorphisms shared between the source and target. The supporting intuition is that either the source must map onto the target or they must both map onto some shared latent system, if not fully, at least in some aspect. Otherwise information in the source is lost when transferring to the target.
Let $\mathcal{H}(X, Y)$ denote the set of all structures homomorphic to $X \times Y$. The set of homomorphic structures between $S_S$ and $S_T$ is given by,
$$\mathcal{H}(X_S, Y_S) \cap \mathcal{H}(X_T, Y_T).$$
In transfer learning, we are specifically interested in using knowledge from $S_S$ to help learn $f_T$. Thus, not all elements of this intersection are valid structures for transfer learning, only those whose output can be mapped to $Y_T$. This set of valid structures can be expressed as,
$$\mathcal{V} = \{X \times Y \in \mathcal{H}(X_S, Y_S) \cap \mathcal{H}(X_T, Y_T) | \exists m_y:Y \to Y_T\}.$$
Apparently not all elements of $\mathcal{V}$ will be useful structures for estimating $f_T$, however, those that are useful, presuming structural homomorphism is necessary, will be in $\mathcal{V}$.
If we define $\mathcal{V}'$ to be the subset of $\mathcal{V}$ where transfer learning generalizes, i.e., the homomorphic structures where $\epsilon_T < \epsilon^*$, transferability can be defined in structural terms as follows.
\begin{definition}{\emph{Structural Transferability}.} \\
Consider a target learning system $S_T$ and a source learning system $S_S$. The structural transferability of a source $S_S$ is,
$$|\{S_T|S_S, \exists (X \times Y) \in \mathcal{V}'(S_S, S_T)\}|,$$
and the structural transferability of a target is,
$$|\{S_S|S_T, \exists (X \times Y) \in \mathcal{V}'(S_S, S_T)\}|.$$
\end{definition}
\noindent In other words, structural transferability concerns the set of systems that share a useful homomorphism with $S_S$ and $S_T$. While in practice $\mathcal{V}$ and $\mathcal{V}'$ are difficult to determine, they provide a theoretical basis for considering whether transfer learning is structurally possible between two systems and the structural invariance of the usefulness of transferred knowledge, respectively.
The relation $\mathcal{V}' \subset \mathcal{V}$ is particularly difficult. Ordering structural usefulness by homomorphism alone is difficult because of the vagueness of how homomorphism can be measured. The more isomorphism there is between $S_S$ and $S_T$, the more the question of usefulness shifts to the behavior. There, the error $\epsilon$ provides the ordering\footnote{$\epsilon$ is a transfer distance between posteriors specifying $h(\theta)$ and $f$.} and the threshold $\epsilon^*$ provides the partition. Structural similarity provides no clear parallel.
It is true that if no homomorphism exists between $S_S$ and $S_T$, they are from different categories. While functors can be used to map between categories, they necessarily distort transferred knowledge because they must add or remove structure to do so. Homomorphisms between systems, in contrast, are structure preserving. And so perhaps a partial order between homomorphic and non-homomorphic systems is justified. But this ordering is hardly granular. A more formal digression on this topic is beyond the scope of this paper, but well within the scope of AST\cite{mesarovic1989abstract}.
\begin{exmp}{\emph{Transfer Roughness in UAVs.}} \\
Consider $S_S$, $S_T$, and $S_{Tr}$ defined according to Example \ref{ex:transfer}. From Example \ref{ex:learning-systems} $X_S \times Y_S = X_T \times Y_T$, so $S_{Tr}$ involves homogeneous transfer. But, if $X_T$ did not include radar, transfer would be heterogeneous. Similarly so if $Y_S$ described paths up to 100 meters in length and $Y_T$ paths up to 10 meters. In either case, $X_S \times Y_S$ can map onto $X_T \times Y_T$, but $X_T \times Y_T$ cannot map onto $X_S \times Y_T$. Thus, transfer from $S_T$ to $S_S$ is rougher than transfer from $S_S$ to $S_T$.
\end{exmp}
\subsection{Behavioral Considerations}
In transfer learning, the primary behaviors of interest are $P(X)$ and $P(Y|X)$ from the domain $\mathcal{D}$ and task $\mathcal{T}$, respectively, and the joint distribution they form,
$$P(X,Y) = P(X) P(Y|X).$$
It is important to realize that $P(X_S, Y_S) \neq P(X_T, Y_T)$ only implies that $P(X_S) \neq P(X_T) \lor P(Y_S|X_S) \neq P(Y_T|X_T)$. That is, the posteriors $P(Y|X)$ can still be equal when the joints $P(X, Y)$ are not if the marginals $P(X)$ offset the difference, and vice versa. In the main, these behavioral equalities make absolute statements on the inductive or transductive nature of a transfer learning system. Behavioral similarities, in contrast, have the richness to make statements on the likelihood of generalization, and, thereby, on transferability.
In AST, behavior is a topological-type concept and, accordingly, behavioral similarity is akin to a generalized metric. However, because in transfer learning we are concerned primarily with behaviors which are probability measures, behavioral similarity between $S_S$ and $S_T$ takes the form of distributional divergences. In our elaboration of behavioral similarity we focus on a notion of \emph{transfer distance}. Transfer distance is the abstract distance knowledge must traverse to be transferred from one system to another. We consider it to be a measure on the input spaces $X_S \times X_T$, output spaces $Y_S \times Y_T$, or input-output spaces $(X_S \times Y_S) \times (X_T \times Y_T)$---more specifically, as a measure on probability measures over those spaces. It can be defined formally as follows.
\begin{definition}{\emph{Transfer Distance}.} \\
Let $S_S$ and $S_T$ be source and target learning systems. Let $Z_i$ be a non-empty element of $\mathcal{P}(X_i \cup Y_i)$. Transfer distance $\delta_T$ is a measure
$$\delta_T:P(Z_S) \times P(Z_T) \to \mathbb{R}$$
of distance between the probability measures $P(Z_i)$ related to the estimated functions $f_i:X_i \to Y_i$ of $S_S$ and $S_T$.
\end{definition}
In practice, transfer distances are often given by $f$-divergences \cite{ditzler2011hellinger}, such as KL-divergence or the Hellinger distance, Wasserstein distances \cite{shen2017wasserstein}, and maximum mean discrepancy \cite{pan2008transfer, long2017deep, jiang2015integration}. Others use generative adversarial networks, a deep learning distribution modeling technique, to estimate divergence \cite{tzeng2015simultaneous, ganin2016domain}. Commonly, these distances are used to calculate divergence-based components of loss functions. Herein, we consider transfer distance's more general use in characterizing transfer learning systems.
In heterogeneous transfer, transfer distances can be used after feature-representation transfer has given the probability measures of interest the same support. Transfer distances between measures with different support are not widely considered in existing machine learning literature. However, the assumptions of homogeneous transfer and domain adaptation, i.e., $X_S \times Y_S = X_T \times Y_T$, allow for a rich theory of the role of transfer distance in determining the upper-bound on error.
Upper-bounds on $\epsilon_T$ have been given in terms of statistical divergence \cite{blitzer2008learning}, $H$-divergence \cite{ben2010theory}, Rademacher complexity \cite{mohri2009rademacher}, and integral probability metrics \cite{zhang2012generalization}, among others. Despite their differences, central to most is a transfer distance $\delta_T: P(X_S) \times P(X_T) \to \mathbb{R}$ that concerns the closeness of input behavior and a term $C$ that concerns the complexity of estimating $f_T$. These bounds roughly generalize to the form,
\begin{equation}
\epsilon_T \leq \epsilon_S + \delta_T + C
\label{eq:inequality}
\end{equation}
where $\epsilon_T$ and $\epsilon_S$ are the errors in $S_T$ and $S_S$, $\delta_T$ is the transfer distance, and $C$ is a constant term. $C$ is often expressed in terms of sample sizes, e.g., $|D_S|$ and $|D_T|$, capacity, e.g., the VC-dimension of $H_T$ \cite{ben2010theory}, and information complexity, e.g., the Rademacher complexity of $D_T$ \cite{mohri2009rademacher}. Note, closeness and complexity are often not as separable as suggested by Inequality \ref{eq:inequality}.
To the extent that Inequality \ref{eq:inequality} holds, we can describe transferability in terms of transfer distance. Generalization in transfer learning occurs if $\epsilon_T \leq \epsilon^*$, and since $\epsilon_T \leq \epsilon_S + \delta_T + C$, $\epsilon_S + \delta_T + C \leq \epsilon^* \implies \epsilon_T \leq \epsilon^*$. Thus, transferability can be defined in behavioral terms as follows.
\begin{definition}{\emph{Behavioral Transferability}.} \\
Consider a target learning system $S_T$ and a source learning system $S_S$. The behavioral transferability of a source $S_S$ is,
$$|\{S_T|S_S, \epsilon_S + \delta_T + C < \epsilon^*\}|,$$
and the behavioral transferability of a target is,
$$|\{S_S|S_T, \epsilon_S + \delta_T + C < \epsilon^*\}|.$$
\end{definition}
\noindent For $S_S$ with similar $\epsilon_S$ and $S_T$ with similar $C$, given a threshold on distance $\delta^* \in \mathbb{R}$, behavioral transferability can be expressed entirely in terms of transfer distance:
$$|\{S_T | S_S, \delta_T < \delta^*\}| \text{ and } |\{S_S | S_T, \delta_T < \delta^*\}|.$$
Of course, specific bounds on $\epsilon_T$ with specific distances $\delta_T$ from the literature can be substituted in the stead of Inequality \ref{eq:inequality}. Also note, we are assuming $X_S \times Y_S = X_T \times Y_T$. When $X_S \times Y_S \neq X_T \times Y_T$, transfer distance is a measure between probability measures with different supports, and while an upper-bound like Inequality \ref{eq:inequality} may be appropriate, it is not supported by existing literature. In such cases it is important to consider structural similarity.
\begin{exmp}{\emph{Transfer Distance in UAVs.}} \\
Consider $S_S$, $S_T$, and $S_{Tr}$ defined according to Example \ref{ex:transfer}. Let source $S_S$ be associated with a desert biome and $S_T$ a jungle biome. When comparing $P(X_T)$ to $P(X_S)$, increased foliage in $S_T$ suggests accelerometer readings with higher variance, camera images with different hue, saturation, and luminance, and radar readings with more obstacles. Similarly, increased foliage may also mean paths in $P(Y_T|X_T)$ must compensate more for uncertainty than those in $P(Y_S|X_S)$. In contrast, foliage is more similar between the desert and tundra, thus, transfer distance is likely larger from the desert to the jungle than from the desert to the tundra.
\end{exmp}
\subsection{Remarks}
In summary, structure and behavior provide a means of elaborating deeply on transfer learning systems, just as they do for systems writ large. Structural considerations center on the structural relatability of $S_S$ and $S_T$ and the usefulness of the related structures $X \times Y$ for transfer learning. Behavioral considerations center on the behavioral closeness of $S_S$ and $S_T$ and the complexity of learning $f_T$. These concerns provide guideposts for the design and analysis of transfer learning systems. While the joint consideration of structure and behavior is necessary for a complete perspective on transfer learning systems, herein, in following with broader systems theory, we advocate that their joint consideration ought to come from viewing structure and behavior as parts of a whole---instead of approaching their joint consideration directly by neglecting notions of structure and behavior entirely, as is advocated implicitly by the existing frameworks pervasive use of domain $\mathcal{D}$ and task $\mathcal{T}$.
\section{Conclusion}
Our framework synthesizes systems theoretic notions of structure and behavior with key concepts in transfer learning. These include homogeneous and heterogeneous transfer, domain adaptation, inductive and transductive transfer, negative transfer, and more. In subsequent elaborations, we provide formal descriptions of transferability, transfer roughness, and transfer distance, all in reference to structure and behavior.
This systems perspective places emphasis on different aspects of transfer learning than existing frameworks. When we take behavior to be represented by a posterior or joint distribution, we arrive at constructs similar to existing theory. More distinctly, when we introduce structure, and study it in isolation, we arrive at notions of roughness, homomorphism, and category neglected in existing literature.
The presented framework offers a formal approach for modeling learning. The focal points of our theory are in aspects central to the general characterization and categorization of transfer learning as a mathematical construct, not aspects central to scholarship. This strengthens the literature by contributing a framework that is more closely rooted to engineering design and analysis than existing frameworks. Because our framework is pointedly anchored to concepts from existing surveys, practitioners should face little difficulty in the simultaneous use of both. Taken together, practitioners have a modeling framework and a reference guide to the literature.
Herein, we have modeled transfer learning as a subsystem. Transfer learning systems can be connected component-wise to the systems within which they are embedded. Subsequently, deductions can be made regarding the design and operation of systems and their learning subsystems with the interrelationships between them taken into account. In this way, we contribute a formal systems theory of transfer learning to the growing body of engineering-centric frameworks for machine learning.
Real-world systems need transfer learning, and, correspondingly, engineering frameworks to guide its application. The presented framework offers a Mesarovician foundation.
\section{Appendix}
\subsection{Mesarovician Glossary}
\begin{definition}{\emph{System Behavior}.} \\
System behaviors are properties or descriptions paired with systems. For example, consider a system $S:X \to Y$ and a map $S \to \{ stable, neutral, unstable \}$ or from $S \to P(X,Y)$. System behavior is a topological-type concept in the sense that it pairs systems with elements of sets of behaviors.
\end{definition}
\begin{definition}{\emph{Behavioral Similarity}.} \\
Behavioral similarity describes the `proximity' between two systems' behavior. To the extent that behavior can be described topologically, behavioral similarity can be expressed in terms of generalized metrics (topological `distance'), metrics and pseudo-metrics (measure theoretic `distance'), and statistical divergences (probability/information theoretic `distance'), depending on the nature of the topology.
\end{definition}
\begin{definition}{\emph{System Structure}.} \\
System structure is the mathematical structure of a system's component sets and the relations among them. For example, there may be algebraic structure, e.g. the linearity of a relationship between two component sets, related to the definition of the relation.
\end{definition}
\begin{definition}{\emph{Structural Similarity}.} \\
Structural similarity describes the homomorphism between two systems' structures. It is described in reference to a relation $m:S_1 \to S_2$, termed a morphism. The equivalence class $S_1/m$ describes the `roughness' of the structural similarity between $S_1$ and $S_2$. Its cardinality gives a quantity to the `surjective-ness' of $m:S_1 \to S_2$. However, in the large, structural similarity is not measurable in the same way as behavioral similarity. The homomorphism is better studied using properties of $m$.
\end{definition}
\begin{definition}{\emph{Cascade Connection}.} \\
Let $\circ: \overline{S} \times \overline{S} \to \overline{S}$ be such that $S_1 \circ S_2 = S_3$, where,
\begin{gather*}
S_1 \subset X_1 \times (Y_1 \times (Z_1)), S_2 \subset (X_2 \times Z_2) \times Y_2 \\
S_3 \subset (X_1 \times X_2) \times (Y_1 \times Y_2), Z_1 = Z_2 = Z
\end{gather*}
and,
\begin{gather*}
((x_1, x_2), (y_1, y_2)) \in S_3 \leftrightarrow \\
(\exists z) ((x_1, (y_1, z)) \in S_1 \wedge ((x_2, z), y_2) \in S_2)
\end{gather*}
$\circ$ is termed the cascade (connecting) operator.
\end{definition}
\subsection{Learning Systems}
\begin{prop}
\emph{$S$ in Definition \ref{def:ls} is a cascade connection of two input-output systems.} \\
\emph{Proof:}
Recall $S \subset \times \{A, D, \Theta, H, X, Y\}$. First we will show $A$ and $H$ to be input-output systems. First note that $A \subset \times \{ D, \Theta \}$. Noting $D \subset X \times Y$, apparently $D \cap \Theta = \emptyset$ and $D \cup \Theta = \overline{A}$. Similarly, $H \subset \times \{ \Theta, X, Y \}$. Letting $X' = \{ X, \Theta \}$, apparently $X' \cap Y = \emptyset$ and $X' \cup Y = \overline{H}$. Therefore, by definition, $A$ and $H$ are input-output systems. Let $S_C:D \times X \to Y$. Apparently, for $d \in D, x \in X, y \in Y, \theta \in \Theta$, $((d, x), y) \in S_C \leftrightarrow \exists \theta ((d, \theta) \in A \wedge (\theta, x, y) \in H$. Therefore, $S_C: A \circ H$. Lastly, note $S_C$ is a function-type representation of $S$, where $A$, $H$, and $\Theta$ are left as specifications on relations, not included as component sets.
\end{prop}
\begin{prop}
\emph{$S$ in Definition \ref{def:gsls} is a goal-seeking system.} \\
\emph{Proof:}
Goal-seeking is characterized by the consistency relations $(G, E)$ and by the internal feedback of $X \times Y$ into $S_G$. Note $D \subset X \times Y$ satisfies internal feedback. The consistency relations $(G, E)$ in Definition \ref{def:gs} and \ref{def:gsls} can be shown to be isomorphic by substituting $D \subset X \times Y$ into consistency relations $G$ and $E$ in Definition \ref{def:gs} and $(x, y) \in d$ into their constraints. Thus, by definition, $S$ in Definition \ref{def:gsls} is a goal-seeking system, where $S_G$ is the inductive system $A$ and $S_F$ is the functional system $H$.
\end{prop}
\begin{prop}
\emph{Empirical risk minimization is a special case of a learning system as defined in Definition \ref{def:gsls}.} \\
\emph{Proof:}
A learning system given by Definition \ref{def:gsls} is an empirical risk minimization learning system if (1) $D$ is a sample of $l$ independent and identically distributed observations sampled according to an unknown distribution $P(X, Y)$, and (2) $A$ selects $\theta \in \Theta$ by minimizing the empirical risk $R_{emp}$, calculated on the basis of $D$, over $\theta \in \Theta$. Otherwise put, ERM is a learning system $S \subset \times \{A, D, \Theta, G, E, H, X, Y\}$ where $G(D, \theta) = R_{emp}(D, \theta) = \frac{1}{l}\sum\limits_{i=1}^l L(y_i, h(x_i, \theta))$ and $E = \min_{\theta \in \Theta} G(D, \theta)$, where $L$ is a loss function.
\end{prop}
\bibliographystyle{IEEEtran}
|
2,869,038,155,637 | arxiv | \section{Introduction}\label{sec:introduction}
Sound synthesizers have been widely used in music production since the late 50s. Because of their inner complexity, many musicians and producers polish presets' parameters until they reach the desired sound. This procedure is time-consuming and sometimes results in failed attempts to achieve a desired sound.
Much research has been done in the area of automating the generation of these sounds through the aid of machine learning and neural networks. Common approaches included directly generating the waveform in the time domain \cite{donahue_adversarial_2019} or predicting synthesis parameters based on hand-picked analysis features \cite{Blaauw2017}. In their 2020 paper on Differentiable Digital Signal Processing (DDSP)\cite{engel_ddsp_2020}, Engel et al.\ proposed a novel approach to neural audio synthesis. Rather than generating signals directly in the time or frequency domain, DDSP offers a complete end-to-end toolbox consisting of a synthesizer based on Spectral Modeling Synthesis (SMS) \cite{serra_spectral_1990}, and an autoencoder neural network architecture that takes care of both extracting analysis features and predicting synthesis parameters.
The authors of the DDSP paper released a public demonstration of "tone transfer"\footnote{\url{https://sites.research.google/tonetransfer}, last accessed on 2020-11-30}, allowing the user to upload their own recordings, select from a list of models trained on various instruments and "transfer" their recorded melodies to the sound of a trumpet, a violin etc.
\begin{figure}[bthp]
\centering
\includegraphics[width=0.99\columnwidth]{Images/DDSP_Synth_GUI.jpg}
\caption{Our real-time DDSP Synthesizer GUI.}
\label{fig:ddsp_gui}
\end{figure}
We implemented the DDSP back-end as a virtual instrument playable in real-time. Figure \ref{fig:ddsp_gui} shows the GUI of our synthesizer. This paper documents the background, our requirement-driven design and implementation approach, including model compenents and training, the GUI design, and user experience evaluation. The structure of this paper follows these main topics in order.
Besides our contribution to the real-time neural audio synthesis, we release our real-time MATLAB and JUCE implementations at \url{https://github.com/SMC704/juce-ddsp} and \url{https://github.com/SMC704/matlab-ddsp}, respectively.
\section{Background}\label{sec:background}
In addition to the DDSP paper \cite{engel_ddsp_2020}, our work is inspired by the commercially produced additive synthesizer called \emph{Razor} by Native Instruments\cite{native-instruments}. Razor's core consists of a powerful additive synthesizer and features various modulation options for manipulating the sound output. What is especially interesting about Razor is that every modulation option (e.g. filters, stereo imaging, reverbs and delays) is actually modulating individual partial harmonics (non-integer multiples of the fundamental frequency) in the additive synthesis engine. Furthermore, Razor enables musicians and producers to intuitively control partials via different parameters while relying on a visual representation of partial manipulation. We therefore focused on the harmonic and the stochastic components of the DDSP.
\begin{comment}
\begin{figure}[htp]
\centering
\includegraphics[width=0.9\columnwidth]{SMC704 DDSP/Images/ddsp_autoencoder.png}
\caption{Autoencoder architecture from the DDSP Library \cite{engel_ddsp_2020}
\label{fig:ddsp_autoencoder}}
\end{figure}
\end{comment}
\subsection{Harmonic Oscillator / Additive Synthesizer}\label{subsubsec:additive_synth}
The additive synthesizer is the main core of the whole synthesis and is responsible for generating all the harmonic components of the reconstructed sound. The output is characterized by the sum of several harmonic integer multiples of the fundamental frequency $f_0$:
\begin{equation}
f_k = k \cdot f_0(n) .
\label{eq:kth_harmonic}
\end {equation}
In order to generate the harmonics, we can implement $k$ oscillators in the discrete time:
\begin{equation}
x(n) = \sum_{k = 1}^{K} A_k(n) \cdot sin(\phi_k(n)),
\label{eq:add_synth}
\end {equation} where $A_k(n)$ is the time-varying amplitude of the $k_{th}$ sinusoidal component and $\phi_k(n)$ is its instantaneous phase. $\phi_k(n)$ is obtained using equation \ref{eq:phase}.
\begin{equation}
\phi_k(n) = 2\pi \sum_{m = 0}^{n} f_k(m) + \phi_{0,k}.
\label{eq:phase}
\end {equation}
The only two parameters necessary to control the synthesizer are the frequency $f_0(n)$ and the harmonic amplitudes $A_k(n)$. These are retrieved directly from the input sound using the encoder contained in the autoencoder network. As reported in \cite{engel_ddsp_2020}, the network outputs are scaled and normalized to fall within an interpretable value range for the synthesizer
\subsection{Filtered Noise / Subtractive Synthesizer}\label{subsubsec:ddsp_subtractive_synth}
The subtractive synthesis is used to recreate the non-harmonic part of natural sounds. The parameters necessary to obtain a frequency-domain transfer function of a linear time-variant finite impulse response (LTV-FIR) filter are retrieved from the neural network in frames that are subsets of the input signal. The corresponding impulse responses (IRs) are calculated and a windowing function is applied. The windowed IRs are then convolved with white noise via transformation to and multiplication in the frequency domain.
\subsection{Reverb}\label{subsubsec:reverb}
In addition to the SMS model, the sound is also given a sense of space using a reverberation algorithm performed in the frequency domain. Thus, the operation of convolution between the impulse response of the reverb and the synthesized signal is a more efficient multiplication.
\begin{comment}
\subsubsection{Instrument-specific models}\label{subsubsec:instrument_models}
The Google Magenta Research team has trained four different models to test the DDSP autoencoder: flute, saxophone, trumpet and violin. The audio file used for the latter training has been taken from the Musopen database\footnote{\url{https://musopen.org/music/13574-violin-partita-no-1-bwv-1002/}, last accessed 2020-12-02} while for the other instruments they used the NSynth dataset \cite{nsynth2017}.
The supervised models obtained from the DDSP autoencoder significantly outperformed other state-of-the-art models by several quantitative metrics, demonstrating higher accuracy in tasks such as $f_0$-resynthesis while requiring a smaller number of parameters.
The neural network architecture of the DDSP library has been implemented in TensorFlow, an open-source machine learning library developed by the Google Brain team \cite{tensorflow2015-whitepaper}.
\subsection{Other Related Work}\label{subsec:other_related_work}
In addition to the DDSP library synthesis method, we explored various other synthesizers and synthesis methods. By looking into other ways of working with synthesis and synth-parameter estimation we gained knowledge of possibilities and limitations that the DDSP library offers in relation to other synthesizer approaches. In this section we briefly elaborate on the 3 synth concepts that we drew the most inspiration from in our concept development phase: \emph{FlowSynth, NSynth} and \emph{Razor}.
\end{comment}
\begin{comment}
\emph{FlowSynth} is an intuitive synthesizer-controller which aims to solve the issue of complex synthesizers having \emph{“highly non-linear relationships between the parameters and the resulting audio.”}\cite{Esling2019}. FlowSynth aims to make it easier for non-expert users to explore complex software synthesizers by implementing a machine learning based preset exploration method. Esling et al. argues that FlowSynth facilitates a way of \emph{“finding an organized latent audio space that represents the capabilities of a synthesizer, while constructing an invertible mapping to the space of its parameters.”}\cite{Esling2019}. The controller works as a tool that can be trained on specific existing synthesizers' parameters to learn which sounds it is capable of creating (‘audio space’). It is then able to recreate sounds using the existing synthesizer parameters (‘parameter space’).
Another source of inspiration for our project is \emph{NSynth} (Neural Synthesizer) which, instead of featuring parameters such as oscillators and wavetables, relies on deep neural networks to generate sounds \cite{nsynth2017}. NSynth aims to achieve quality instrument sounds by training models on large datasets of musical notes which are sampled from individual instruments. Therefore, gathering a large dataset with around 300k samples from 1000 different instruments which they call “The NSynth Dataset” is another goal for the Google Magenta team \cite{nsynth2017}. The team has released The NSynth Dataset to contribute to the research and development in the audio machine learning community.
Finally, we drew inspiration from a commercially produced additive synthesizer called \emph{Razor} by Native Instruments\cite{native-instruments}. Razor's core consists of a powerful additive synthesizer and features various modulation options for manipulating the sound output. What is especially interesting about Razor is that every modulation option (e.g. filters, stereo imaging, reverbs and delays) is actually modulating individual partial harmonics (non-integer multiples of the fundamental frequency) in the additive synthesis engine. Furthermore, Razor enables musicians and producers to intuitively control partials via different parameters while relying on a visual representation of partial manipulation.
\section{Analysis}\label{sec:Analysis}
In this section we will introduce the initial research path conducted around the DDSP library as well as the research question that drove our project from the beginning.
\subsection{Exploration of the problem space}\label{subsec:exploration_problem_space}
During the first phase of our research, the team analyzed and collected several DDSP application ideas trying to evaluate the possible issues linked to the development of each of them. A great effort has been put into the formulation of new concepts and associated questions about feasibility, technical and technological challenges, usability and musicality. Each idea has been evaluated using the SWOT analysis \cite{leigh2009swot}, an investigation tool widely employed to detect and analyze four different key aspects of projects: strengths, weaknesses, opportunities and threats.
One of the first ideas investigated is the embedded platform microcontroller application inspired by recent research published in the DAFx-20 Proceedings conducted by Esling et al. \cite{esling_diet_2020}. In this paper, a \textit{structured trimming} has been performed on deep learning models applied to audio applications, obtaining light models for generative audio. Later, due to the monophonic constraint imposed by the architecture of the DDSP model, we took into account the challenge of a polyphonic implementation of this library. Finally, we considered using DDSP to develop a synthesizer, examining two different proposals. At the end of this research process, these concepts were polished and combined together. The first was focused on the design of a synthesizer based on DDSP with semantic macro-controls to help both beginners and experts reaching the wanted result in a fast and efficient way. The second application was inspired by Yee-King et al. \cite{yee-king_automatic_2018} and involved the design of a synthesizer based on sound matching, i.e. an instrument able to reconstruct a target sound adjusting the DDSP components.
To sum up, we found the design of a synthesizer based on sound-matching with macro-controls to be a perfect challenge for this project. It is not only focused on a new application of DDSP, but also introduces new features such as macro controls for the synthesis and real-time implementation.
\end{comment}
\subsection{Research question}\label{subsec:research_question_statement}
Based on this background
we have formulated the following research question: How can we develop a playable software instrument, based on the DDSP library, that would: a) allow customization of model-estimated synth parameters through top-level macro controls, b) enable existing workflow-integration in Digital Audio Workstations (DAWs), and c) facilitate a simple approach for beginners without limiting usability for expert music producers?
\begin{comment}
\section{Design Requirements}\label{sec:design_requirements}
For the purpose of requirements management and evaluation we created a set of software design requirements for our software application which are based on the approach of
\subsection{Requirements generation - Approach}\label{subsubsec:requirements_generation_approach}
Pandey et al.'s Requirement Engineering approach aims to discover quality requirements for software development and argues that \emph{“A well-formed requirement is a statement of system functionality that satisfies customer needs.”}\cite{Pandey2010}. As our project revolves around development of a software music production tool which is intended to be used by musicians and music producers, it is crucial for the success of our product that it meets user needs to a certain degree. Therefore, we have followed the overall structure of Pandeys et al.'s Requirement Engineering’s phases but mainly focused on elements from the two first phases: \emph{Requirement elicitation and development} and \emph{Documentation of requirements} because these phases are very relevant in formulating meaningful design requirements\cite{Pandey2010}.
\subsection{Requirements elicitation and development}\label{subsubsec:requirements_elicitation_and_development}
Pandey et al. suggests to begin with finding ‘objectives’ (raw requirements) for the system from different stakeholder’s viewpoint, e.g. the customers, users, constraints, marketing, etc.\cite{Pandey2010}. We started by gathering objectives for our application focusing mainly on the user’s perspective since our synthesizer is supposed to be used on an experimentation level by musicians and music producers who are interested in new technologies for the music industry. Thus, we do not take other stakeholders' viewpoint into consideration, as this would be out of the scope of this paper. We aim to come up with solutions to these objectives through the design of our software application. As this project aims to explore technical possibilities of implementing DDSP in an instrument-like context, instead of collecting user data through surveys or interviews, we based the objectives for the synthesizer on our research of e.g. popular music-focused YouTubers' opinions\cite{neely_2020, huang_2020}, state-of-the-art music production tools \cite{Esling2019, nsynth2017, native-instruments} as well as our own experience with software synthesizers and virtual instruments.
We found that DDSP is not currently implemented to work in real-time\cite{engel_ddsp_2020} and we believe that this new feature will provide interesting new application opportunities. Especially in the context of music composition and music production, a real-time implementation, in the form of a software synthesizer plugin, could potentially provide a new tool for generating unique sounds and gaining inspiration. For this to be adopted by music producers, we want to develop a plugin-based instrument because our goal is that musicians and producers are able to integrate our application conveniently into their everyday workflows in different DAWs. To maintain the opportunity for different workflows and composition methods, our application will have both MIDI and Line input methods, from which the application will retrieve the fundamental frequency and amplitude information. We want the users to be able to choose from different instrument-specific models (\ref{subsubsec:instrument_models}), which will control the parameters of the additive synth, subtractive synth and the reverb module in real-time to generate the desired instrument-specific output sound. If this was the only user-input that the application had, it would be rather limited in creative expressivity. Therefore, it must be possible for the user to alter parameters of the application's modules (such as harmonics/noise mix, amplitude of individual partials, reverb length etc.) in an easy user-friendly way, after a model was used to estimate them.
\subsection{Documentation of requirements}\label{subsubsec:documentation_of_requirements}
Documentation of requirements is according to Pandey et al. a matter of describing \emph{“[…] the behaviour of the system or software to be developed.”}\cite{Pandey2010}. In table \ref{tab:design_requirements}, each of the user objectives is listed with a solution to assure that a specific user need is met by the software design. The
, and present them on Table \ref{tab:objectives}.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.9\columnwidth}{|l|X|}
\hline
\textbf{Obj. \#} & \textbf{User objectives} \\
\hline
1 & Provide a new playable instrument for unique sound generation and inspiration \\
\hline
2 & Conveniently integrate into existing workflows \\
\hline
3 & Adapt to different composition methods \\
\hline
4 & Easy fast unique sound generation \\
\hline
5 & Customizability of generated sounds \\
\hline
\end{tabularx}
\end{center}
\caption{List of objectives from a user-centered viewpoint.}
\label{tab:objectives}
\end{table}
\end{comment}
To sum up the design requirements, we aim to build a software instrument plugin that is playable in real-time. The instrument must support different composition techniques, thus having a line and MIDI input mode. The instrument must include at least four pre-trained models which serve the purpose of estimating synthesizer parameters to output a desired sound. Finally, the instrument must include graphical user interface components that provide intuitive controls for the manipulation of synthesizer and effect parameters.
\section{Design \& Implementation}\label{sec:implementation}
Based on this research question, we have identified five user needs \cite{Pandey2010}, and matched them with a solution, reformulating them as a concrete measurable design requirement. The design requirements are thus documented on Table \ref{tab:design_requirements}.
\begin{table*}[ht]
\begin{center}
\begin{tabularx}{0.99\columnwidth}{|l|X|X|X|}
\hline
\textbf{\#} & \textbf{User Obj.} & \textbf{Solution} & \textbf{Design Requirement} \\
\hline
1 & Provide a new playable instrument for unique sound generation and inspiration & Real-time implementation & \emph{Must work in real-time as a playable software instrument.} \\ \hline
2 & Conveniently integrate into existing workflows & Plugin format application & \emph{Must be implemented as a software plugin.}\\
\hline
3 & Adapt to different composition methods & Allow line and MIDI input & \emph{Must allow switching between Line and MIDI input.} \\
\hline
4 & Easy fast unique sound generation & Choose models for sound generation & \emph{Must implement at least four pre-trained models.} \\
\hline
5 & Convenient customizability of sounds & Tweakable parameters that effects the audio output & \emph{Must include GUI components for intuitive manipulation of synth and effects parameters.} \\ \hline
\end{tabularx}
\end{center}
\caption{Documentation of Design Requirements}
\label{tab:design_requirements}
\end{table*}
\subsection{Architecture overview}\label{subsec:architecture_overview}
\begin{figure*}[htp]
\centering
\includegraphics[width=0.99\columnwidth]{Images/Architecture.png}
\caption{Schematic overview of the project architecture.
\label{fig:architecture_scheme}}
\end{figure*}
To meet our criteria of creating a real-time software instrument, we decided to build the plugin in C++ using the JUCE application framework\footnote{\url{https://juce.com/}, last accessed on 2020-12-15}. With JUCE, we had a multi-platform supported audio plugin template that was handling MIDI and audio inputs and outputs. This allowed us to mainly focus on the audio processing and GUI.
Creating a real-time implementation of the non-real-time DDSP library posed some immediate challenges. To analyze and understand these challenges we decided to start by doing a direct translation of the additive and subtractive synthesizers from the DDSP library into MATLAB. The synthesizers could then be changed into real-time implementations and tested. In order to use our MATLAB implementation in the JUCE framework, we used inbuilt MATLAB tools to generate C++ code.
We transformed the autoencoder models pretrained by Google into models that could be used to estimate synthesizer parameters directly from our plugin's user input.
A general overview of this architecture can be seen in figure \ref{fig:architecture_scheme}.
The following sections will discuss each component in more detail.
\subsubsection{Synth in MATLAB}\label{subsubsec:synth_MATLAB}
MATLAB’s environment and visualization tools gave us access to quick prototyping and testing. This allowed us to do the implementation over multiple iterations. We tested our synthesizers' compatibility with the predicted parameters from the DDSP models by invoking the encoders and decoders in isolation through MATLAB's Python interface.
At first we implemented the non-real-time synthesis algorithms of the DDSP library. Then the synthesizers were changed to real-time, i.e., synthesizing a single frame at a time.
Using the MATLAB Audio Test Bench, we could then test the functionality of the synthesizer components and parameters with real-time audio and varying sample rate and buffer size.
The last iterations consisted of optimizing the code with the constraints of real-time audio processing on CPUs.
\subsubsection{MATLAB to C++}\label{subsubsec:MATLAB_to_C++}
Using the MATLAB coder tool\footnote{\url{https://se.mathworks.com/products/matlab-coder.html}, last accessed on 2020-12-15} we were able to generate C++ functions from the MATLAB code. For the simplest integration between the generated C++ functions and the JUCE plugin we chose to limit the function inputs and outputs to built-in and derived C++ data types. This required our MATLAB functions to have fixed-sized inputs and outputs. We decided on a maximum input/output size of 4096 double-precision floating point numbers, this being the maximum buffer size the plugin would be able to work with.
A helper file was created to ensure code consistency, allowing the user and MATLAB coder to verify the functions with different inputs. Having this setup made it easy to go back to the MATLAB code and generate updated C++ functions without breaking the JUCE plugin.
\subsubsection{TensorFlow in C++}\label{subsubsec:Tensorflow_in_C++}
Running the DDSP TensorFlow implementation in a real-time audio application is a heavy computational challenge. Moving from TensorFlow in Python to the TensorFlow C API\footnote{\url{https://www.tensorflow.org/install/lang_c}, last accessed on 2020-12-15} allowed us to integrate the models into the C++ codebase. By moving the TensorFlow computations to a separate thread, we can load the models, set the inputs, run the parameter estimation and save the outputs, without experiencing buffer underruns in the main audio processing thread.
\subsubsection{Input signals}\label{subsubsec:input_signals}
The DDSP autoencoder needs the input values \emph{fundamental frequency} ($f_0$) and \emph{loudness} ($ld$). Since we allow both MIDI and line-in audio, two separate implementations are needed to calculate these values. Functions for this were created in MATLAB, but in the C++ implementation we chose to use the implementation of the YIN pitch tracking algorithm \cite{yin2002} from the C library Aubio\cite{aubio}, since it yielded more precise results.
\subsection{Training models}\label{subsec:training_models}
\subsubsection{Pre-trained models}\label{subsubsec:pre-trained_models}
Next to the \emph{tone transfer} website mentioned in the introduction, the authors of the DDSP paper also published a Jupyter Notebook Demo on Google Colab called \emph{timbre transfer.}\footnote{\url{https://colab.research.google.com/github/magenta/ddsp/blob/master/ddsp/colab/demos/timbre_transfer.ipynb}, last accessed on 2020-12-15}
We accessed the available checkpoint files for violin, flute, tenor saxophone and trumpet from this notebook for our real-time implementation of the timbre transfer. However, we were not immediately able to use them in the JUCE plugin. The DDSP models are trained using TensorFlow's \emph{eager execution mode}, while the TensorFlow C API is constructed around \emph{graph mode}. Additionally, since we required the models to be controllable by MIDI input, we needed direct access to the decoder part of the model instead of supplying audio to the encoder.
The \texttt{convert\_models.py} script from the Python folder of the plugin code repository deals with these requirements by loading the eager model from the downloaded checkpoint file, constructing a graph-based model only containing the decoder and then copying all weights from the old model to the new one. The resulting checkpoint now contains a graph that can be loaded by the TensorFlow C API.
\subsubsection{Custom models}\label{subsubsec:our_models}
In order to make use of the DDSP training library and extend the synthesizer with additional models, we created four custom models trained on:
\begin{itemize}
\item Bass sounds of the Moog One, Moog Minimoog and Moog Minitaur synthesizers
\item Studio recordings of Middle Eastern instruments, the Hammered Dulcimer and Santoor
\item Studio recordings of a Handpan (also known as Hang Drum)
\item Nature field recordings of birds chirping
\end{itemize}
For training we used the official DDSP (version 0.14.0) Jupyter notebook on Google Colab called \emph{train autoencoder}\footnote{\url{https://colab.research.google.com/github/magenta/ddsp/blob/master/ddsp/colab/demos/train_autoencoder.ipynb}, last accessed on 2020-12-15} which allows training on a Google Cloud GPU using own data. We chose the recordings listed above in order to obtain interesting sounds that differ from the more traditional pre-trained DDSP models. According to the recommendations of the DDSP authors given in the notebook, trained models perform best using recordings of a single, monophonic sound source, in one acoustic environment, in .wav or .mp3 format with a total duration of 10 to 20 minutes. Since the DDSP Autoencoder is conditioned on the loudness $A$ and the fundamental frequency $f_0$, i.e., the model learns to associate different synthesizer configurations to specific value pairs of $(A, f_0)$, training on multiple instruments, acoustic environments or polyphonic sounds prevents the autoencoder to learn a unified representation. However, these thereby introduced artifacts can also be used in a musical context, that is why we decided to challenge the autoencoder with less conform training data and eventually achieved interesting timbres.
The training process is performed as follows. The first step is comprised of data generation and pre-processing of the training data. The raw audio is split into short parts of a few seconds, each analyzed on the specified features, i.e., the fundamental frequency and loudness, and finally saved in the TensorFlow \emph{TFRecord} format. The fundamental frequency is thereby estimated by using the state-of-the-art pitch tracking technique, called \emph{CREPE} by Kim et al. \cite{kim2018crepe} that applies a deep convolutional neural network on time-domain audio. The second step is the actual training, using a Python based configuration framework for dependency injection by Google, called \emph{Gin}\footnote{\url{https://github.com/google/gin-config}, last accessed on 2020-12-15}. In this way, all available training hyperparameters can be defined in a gin config file that is passed to the training function. The training process does not include any optimization techniques, such as a hyperparameter search or early stopping, the authors just recommend in the code documentation to train for 5,000 to 30,000 steps until a spectral loss of about 4.5-5 is reached for an optimal learning representation without overfitting. The third and last step is a short evaluation based on resynthesis. That means, a training sample is randomly picked, passed to the autoencoder that encodes and decodes, i.e., reconstructs the input sample based on the learned features.
We successfully conducted training of all four models and validated their performance in the previously mentioned timbre transfer demo. While validation using the DDSP library went smoothly and showed musically interesting results, we ran into issues during inference using the TensorFlow C API within our plugin. We monitored a much higher loudness of the custom models compared to the pre-trained models, resulting in a distorted, clipping sound. Furthermore, we detected a constant harmonic distribution independent of the incoming pitch and loudness while the pre-trained models adapt harmonics and frequency response according to these inputs. The overall experience with the training script provided by the DDSP authors is that it works without problems for standard parameters, but as soon as own hyperparameters within the gin framework are chosen, a lot of side-effects appear. For the mentioned reasons, integrating and possibly adapting the custom-trained models to make them work in the DDSP synthesizer will be a part of future work.
\subsubsection{Real-time implementation of the models}\label{subsubsec:realtime_implementation_of_the_models}
The DDSP non-real-time implementation synthesizes several frames before processing them into one output. Reading through the DDSP code base we experienced the number of frames (time steps) to be defined by the size of the input audio and a hop size defined by constants in the gin config file of the selected pre-trained model.
\begin{comment}
\begin{equation}
\text{hop\ size} = \frac{\text{number\ of\ samples\ in\ training}}{\text{number\ of\ time\ steps\ in\ training}}
\label{eq:hop_size}
\end{equation}
\begin{equation}
\text{time\ steps} = \frac{\text{size\ of\ audio\ input}}{\text{hop\ size}}
\label{eq:time_steps}
\end{equation}
\end{comment}
For our real-time implementation we wanted to calculate one frame with a size of the input buffer each time the buffer is ready.
Given the static nature of our TensorFlow model implementation we were not able to change the number of time steps on the run. Therefore, we set the number of time steps to one. Each run of the TensorFlow model would then return a set of values for one time step, independent of the buffer size.
\subsection{Additive synthesizer}\label{subsec:additive_synthesizer}
The implementation of the additive synthesizer can be found in the \texttt{additive.m} MATLAB code file. During the development of the DDSP synthesizer we went from a re-implementation of the DDSP equivalent to an adapted real-time optimized version with additional parameters for high-level control. While the original DDSP library provides two different implementations of the additive synthesis, the harmonic and sinusoidal approach, this work focuses on the harmonic synthesis that models a signal by adding only integer multiples of the fundamental frequency.
In the following, the initial implementation as well as the main modifications in its final state are clarified.
As already explained in \ref{subsubsec:additive_synth}, the additive synthesizer models audio using a bank of harmonic sinusoidal oscillators. The synthesis algorithm takes amplitudes, harmonic distribution and fundamental frequencies for a specified number of frames as input and computes the sample-wise audio signal as output. The harmonic distribution provides frame-wise amplitudes of the harmonics. The additive synthesis as implemented in the DDSP library is performed in two main steps:
\begin{itemize}
\item Translation of neural network outputs to the parameter space of the synthesizer controls
\item Computing the output signal from synthesizer controls
\end{itemize}
In order to make the output of the neural network usable for controlling the synthesizer, it needs to be transformed accordingly. In detail, that means the amplitudes are scaled and the harmonic distribution is scaled, bandlimited and normalized while the fundamental frequencies remain unchanged. Bandlimiting the harmonic distribution means removing the harmonics that exceed Nyquist in order to avoid artifacts.
After retrieving valid synthesizer controls, the harmonic synthesis is performed. Since the DDSP approach works frame-based while the output needs to be delivered sample-based, the synthesizer controls need to be upsampled. This is done by linearly interpolating the frequency envelopes and windowing the amplitude envelopes by using 50\% overlapping Hann windows. Having calculated all controls on a sample basis, the signal can be synthesized by accumulative summation of the corresponding phases, i.e., adding the calculated sinusoids together, sample by sample.
The following changes were made to optimize the algorithm for a real-time application and to add additional high-level control for the synthesis.
\begin{itemize}
\item Since the frame-based calculation was computationally too heavy, we adapted the code so that the input is always one frame (equivalent to the buffer size) and all computations are sample-based. Therefore, no resampling or windowing is needed.
\item Each time the function is called, the phases of all harmonics are saved and returned along with the signal and added as offset in the next call to avoid artifacts caused by phase jumps.
\item In order to be able to optionally introduce non-harmonic partials to the signal, a stretch parameter was added that transforms the distance between the integer multiples while maintaining the fundamental frequency. An additional shift parameter adds the functionality to modify the fundamental frequency from one octave below to one octave above the current pitch in a continuous scale.
\end{itemize}
\subsection{Subtractive synthesizer}\label{subsec:matlab_subtractive_synth}
This component is responsible for the non-harmonic parts of instrument sounds, such as the audible non-pitched flow of air that accompanies the harmonic part of a flute sound. Our implementation, which can be found in the \texttt{subtractive.m} MATLAB code file, generates a frame of random noise and then filters it according to a given frequency response.
The function's parameters are the frame length (number of samples), noise color (see below) and the frequency response, which is given as a vector of $N$ magnitudes $m_0,\ldots,m_{N-1}$, where $m_0$ corresponds to the DC component and $m_i$ to frequency $f_{\text{nyquist}} / (N - i)$ with $f_{\text{nyquist}} = f_s / 2$ and samplerate $f_s$.
While we started with a direct re-implementation of the DDSP FilteredNoise approach described in \ref{subsubsec:ddsp_subtractive_synth}, we made the following adaptations over the course of the project:
\begin{itemize}
\item Simplified filtering calculation. The DDSP synthesizer processes multiple frames at once. For the sake of a real-time implementation, we removed the step of calculating the impulse response for each frame and applying a windowing function. Instead, we simply perform a Fourier transform on the generated noise and multiply the result with the filter magnitude response that the model predicted for the single current frame.
\item Noise color. We provide functionality to shape the frequency distribution of the generated noise. Noise color generally refers to the frequency $f$ being emphasized proportionally to $1/f^{\alpha}$ for some exponent $\alpha$ \cite{kasdin_1995}. $\alpha < 1$ results in higher frequencies becoming more prominent, while $\alpha > 1$ increases the energy of the lower frequencies. Uniform white noise is achieved by setting $\alpha = 1$.
\end{itemize}
\subsection{Graphical User Interface}\label{subsec:GUI}
After the development of all the features of our synthesizer, we focused our attention on designing an interface with high-level controls for the additive and the subtractive synthesis, the reverb, the modulation and the models. Our process started from a list of all the parameters we wanted to manipulate. We also looked for some inspiration from well-known VST synthesizers, comparing them in terms of usability and trying to understand what their best interaction features were. Later we organized the controls of our synthesizer in different modules and displayed them in a rectangular interface, trying to find a layout that was pleasant but also respectful of the instrument's architecture logic. In table \ref{tab:GUI_features}, we list all the controls for each module of our synthesizer. Because of the particular choice of a graphic control for the harmonics' amplitude, the team opted for a spectrogram representing the output of our plugin. In this way, the user is able to clearly see which harmonics are being played.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.9\columnwidth}{|l|X|}
\hline
\textbf{Module} & \textbf{Feature controls}\\
\hline
Input selector & MIDI/line selector\\
\hline
\multirow{8}{50pt}{Models selector} & Violin\\
& Flute\\
& Saxophone\\
& Trumpet\\
& Moog Bass (not included)\\
& Dulcimer (not included)\\
& Handpan (not included)\\
& Chirps (not included)\\
\hline
\multirow{4}{50pt}{Additive synthesis} & Graphic harmonics editor\\
& $f_0$ shift\\
& Harmonics stretching\\
& Global amplitude\\
\hline
\multirow{2}{50pt}{Subtractive synthesis} & Noise color\\
& Global amplitude\\
\hline
\multirow{3}{50pt}{Modulation} & Modulation rate\\
& Delay control\\
& Amount\\
\hline
\multirow{3}{50pt}{Reverb} & Dry/wet mix\\
& Size\\
& Glow\\
\hline
Output & Master gain\\
\hline
Spectrogram & Clear visualization of the output\\
\hline
\end{tabularx}
\end{center}
\caption{List of GUI's features}
\label{tab:GUI_features}
\end{table}
Once we defined the layout and the parameters that we wanted to control, we moved to the software development in JUCE. In order to customize the appearance of knobs, we used the "Custom LookandFeel" objects while we designed ad hoc images for the buttons and background texture using a vector graphics software. Figure \ref{fig:ddsp_gui} previously presented the GUI of our synthesizer.
\subsection{Plugin setup}\label{subsec:plugin_setup}
The synthesizer ended up being built as a standalone executable and a DAW plugin using Steinberg’s VST3 format.
Using JUCE’s AudioProcessorValueTreeState class we are exposing the different controllable parameters to the DAW, allowing control and automation of the plugin. Using this class we will also be able to easily store and read plugin states, enabling generation of presets, though this has not been implemented yet.
The synthesizer is configured to load the models from a given path with subfolders containing the individual models, as well as configuration files containing key-value pairs such as number of harmonics and scaling values.
\section{Evaluation}\label{sec:evaluation}
In order to understand the strengths and weaknesses of our product to improve it, we designed an evaluation strategy for both User Experience (UX) and sound output.
Our target users are musicians and music producers. Accordingly, we shared a release of our VST plugin with selected sound engineers, musicians and producers to collect opinions and user insights. Moreover, we designed two different questionnaires and asked participants to evaluate the UX and the sound accuracy of our software. The DDSP Synthesizer as well as the two questionnaires have been distributed online and the participants received an email with all the indications to properly conduct the test.
In the next two sub-sections we will describe each evaluation in detail, including approach, desired outcome, survey design and results.
\subsection{User Experience Evaluation}\label{subsec:ux_evaluation}
\subsubsection{Approach}\label{subsubsec:ux_approach}
The aim of this evaluation is to collect feedback about the user interface from people with experience on synthesizers and music production. One of the goals of our project was to design a simple and efficient interface able to control several parameters with a single gesture without giving up functionality in the pursuit of simplicity.
After a trial period where the participants had the chance to familiarize themselves with the software, we asked them to compile a form.
\subsubsection{Survey structure}\label{subsubsec:ux_survey_structure}
Google Forms was chosen as a platform because of its simplicity and wide spread. We designed the survey with different sections to group the questions by theme. We included an experiment in order to ask each participant to load and perform some changes to a model and export the result in an audio file. In this way, we are sure that every participant had at least used and interacted with the plugin for a while. Moreover we are able to compare each audio export to understand if some of the instructions were not clear or if the UX itself was not effective.
Four usage questions have been asked to collect information about the user's DAW and for how much time they used the plugin. In the next sections we asked the participants to report their experience during the experiment and evaluate the user interface rating 9 different statements with a Likert-scale, a widely used bipolar symmetric scaling method in questionnaires. In this way, users were able to express their agreement/disagreement related to each sentence. Furthermore, we asked 4 open questions to let the participants express their opinion about the overall UX. Finally we added 8 questions to locate demographics and musical-related personal experiences. Table \ref{tab:ux_survey} summarizes the content of each section.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.9\columnwidth}{|l|X|X|}
\hline
\textbf{\#} & \textbf{Section} & \textbf{Content}\\
\hline
1 & Introduction & Aim of the questionnaire\\
\hline
2 & Experiment & Task instructions\\
\hline
3 & Usage & 4 mixed questions\\
\hline
4 & UX evaluation & 9 Likert scale evaluations\\
\hline
5 & UX experience & 4 open questions\\
\hline
5 & Demographics & 8 mixed questions\\
\hline
\end{tabularx}
\end{center}
\caption{Content of the UX survey}
\label{tab:ux_survey}
\end{table}
\subsubsection{Expected results}\label{subsubsec:ux_expected_results}
In this section we want to express a projection of the feedback regarding the User Experience. Considering that the software is still under development, we are expecting reports about compatibility issues with different DAWs as well as some stability problems. Moreover, because of the VST's instability in the first release, it is possible that some users will not be able to conduct the small experiment that requires the plugin to be embedded in a DAW track. Considering the whole interface, one of the main points of our design requirements was the simplicity
and thus our hope is to facilitate the user's interaction.
\subsubsection{Results}\label{subsubsec:ux_results}
We received 7 answers. Five participants identified as males, 1 female and one preferred not to say. The age average is 28.57 years (STD 8.42). Six of them declared that sound production is their hobby while one said music production is related to their job. The mean experience in the music production field is 7.43 years (STD 4.87).
Six users do not have experience with machine learning VST plugins and only one of them does not know if she/he ever used one. Each user spent an average of 23.57 minutes using our synthesizer (STD 17.74). We suppose that some mistake has been made reporting the usage time for at least one user. In table \ref{tab:used_daw} we report the number of user tests per different software environment.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.7\columnwidth}{|l|X|}
\hline
\textbf{\# users} & \textbf{Environment} \\
\hline
3 & Reaper\\
\hline
2 & Ableton Live\\
\hline
1 & Cubase\\
\hline
1 & Standalone version\\
\hline
\end{tabularx}
\end{center}
\caption{List of used DAWs in the evaluation.}
\label{tab:used_daw}
\end{table}
In general, the experiment
has been rated a medium difficult task with a mean rating of 3.43 in a scale from 1 to 5 being 1 "easy to accomplish" and 5 "hard to accomplish". In figure \ref{fig:ux_likert_results} we summarize the answers obtained from the questions with an associated likert scale. The users were asked to rate each sentence from 1 to 5 with 1 corresponding to "strongly disagree" and 5 to "strongly agree". We can observe that the graphical user interface has been really appreciated with a 4.43 mean value while the interface's controls seem not to let the participants easily reach the wanted results. The other statements reported in the likert section obtained a medium rating between 3 and 3.86 which might mean that the GUI is in general appreciated.
\begin{figure*}[htp]
\centering
\includegraphics[width= 0.99\columnwidth]{Images/Likert_Graph.jpg}
\caption{User experience evaluation - Likert scale}
\label{fig:ux_likert_results}
\end{figure*}
As expected, some of the participants encountered difficulties in the installation procedure of the VST3 plugin in both Windows and macOS environments while the standalone version seems to be more stable. Furthermore, three users reported an unsatisfactory audio result related to the presets obtained from models. Here we report part of one of the feedback: "\textit{[...] It's possible to get some cool sounds but the default sound when you just start it is not so nice.}". On the other hand, the audio input feature was appreciated: "\textit{[...] I think the audio input feature has a lot of potential and I caught myself experimenting with this a lot and loosing track of time.}". Two participants reported that the possible interaction with the interface for the additive synthesizer was not immediate to spot and they realized its features after a while. For this reason they suggest a graphical indication to guide the user to the interaction with the harmonic sliders.
A significant outcome is the unexpected audio results that participants reported. Even though they described output sounds as "awkward", they highlighted the new creative way of producing unexpected sounds, finding the whole synthesizer experience engaging.
\begin{comment}
\subsection{Sound Accuracy Evaluation}\label{subsec:soundaccuracey_evaluation}
\subsubsection{Approach}\label{subsubsec:sound_accuracy_approach}
In this part of the evaluation we aim to test the sound quality of the output of our real-time software synthesizer in comparison to the Google Magenta DDSP 'tone transfer' implementation\footnote{\url{https://sites.research.google/tonetransfer}, last accessed on 2020-12-17}. More specifically, we test for the perceived timbre transfer accuracy in the output audio from both systems. As opposed to section \ref{subsec:ux_evaluation}, this section focuses solely on evaluation of the output audio and does not consider any user-interactable parameters for sound generation, i.e. all audio sources used in this part of the evaluation are purely model-generated for the sake of comparability.
We approached the sound accuracy evaluation with a focus on timbre transfer capability because, instead of testing for sound similarity between our implementation and Google Magenta, we believe that testing timbre transfer capability will provide more useful insights on the overall perceived sound recreation of our plugin (while this approach also implicitly compares audio output similarity).
To conduct this evaluation, we created a set of audio samples of a piano playing in different octaves. We inputted our sample-set to both Google Magenta's implementation as well as our synthesizer to generate an output-set of transferred audio files, with a transferred timbre. We created multiple output-sets, one for each of the following models: violin saxophone, flute and trumpet. By using piano-samples that are played in different octaves, we are able to test each of the model’s capability of reproducing timbre of different frequency content. Furthermore, as opposed to Google Magenta's implementation, our synthesizer does not feature automatic modification of the input signals frequency and loudness to minimize the difference between the input signal and the training data. Therefore, we modified the input signal manually (changing octaves of the piano-sample) before inputting it into our synthesizer and Google Magenta's implementation.
\subsubsection{Listening test tool: ‘webMUSHRA’}\label{subsubsec:test_tool_mushra}
To construct a suitable platform for conducting audio-listening tests we have used the ‘webMUSHRA’ web framework\cite{Schoeffler2018}. The webMUSHRA framework is designed to help researchers and experimenters conduct valid, convenient tests or evaluations that revolve around listening tests and audio files. Many existing survey tools do not allow for convenient online listening test environments and therefore we chose to use webMUSHRA. As stated by Schoeffler et. al. \emph{“[…] there exist only minor differences between listening tests carried out in an laboratory environment and those carried out over the Internet, if the experiment was properly designed.”}\cite{Schoeffler2018}.
The main feature of webMUSHRA that we used to build this survey is called a MUSHRA (Multi-Stimulus Test with Hidden Reference and Anchor), which can be used to evaluate auditory systems\cite{Schoeffler2018}. This test environment is suitable for conducting our evaluation of timbre reproduction capability of our synthesizer implementation and Google Magentas implementation.
The MUSHRA structure is suitable because it features a reference audio source as well as multiple ‘conditions’ where some of the conditions are so called ‘anchors’ which are \emph{“[…] low-pass-filtered versions of the reference stimulus and were introduced to make the ratings of different assessors and labs more comparable.”}\cite{Schoeffler2018}. i.e. anchors are included to make the survey results more valid and valuable. Even though this is the ideal scenario, we decided to exclude the anchor conditions in our evaluation because, from our pilot-tests, we discovered that the anchor conditions confused participants and therefore would affect the results validity negatively. For conditions in our survey we are using the output sets generated from our synthesizer implementation and Google Magenta's implementation. We are creating a MUSHRA for each of the model output-sets to evaluate each models capability of timbre reproduction.
\subsubsection{Survey structure}\label{subsubsec:mushra_survey_structure}
The survey structure is constructed of 5 sections. Sections 1 through 4 is dedicated to one model respectively thus testing 4 different models (see appendix \ref{subsec:mushra_survey_content}). The 5th section is concerned with demographics of participants because we want to consider e.g. participants experience working with audio, to evaluate whether they are proficient with timbre recognition etc.
\begin{figure}[htp]
\centering
\includegraphics[width= 0.9\columnwidth]{SMC704 DDSP/Images/mushra_1anchor.png}
\caption{MUSHRA user interface}
\label{fig:mushra_ui}
\end{figure}
For each section (model), participants are presented with a reference sample (which is the raw input sample) as well as three conditions. The three conditions include:
\begin{enumerate}
\item Output audio from Magenta implementation
\item Output audio from Plugin Synth
\item Reference (input sample)
\end{enumerate}
For each condition, participants are asked to rate how accurate they think the timbre of the reference is recreated in the condition audio source. This input is translated through one vertical slider per condition which features a rating scale from "bad" transfer to "excellent" transfer of timbre and is represented by a value from 0-100 (see Figure \ref{fig:mushra_ui}).
By including a third reference as an additional condition, we aim to facilitate a way to validate participants survey-answers. This is a way to validate that participants understood the questions and actually listened to the auditory stimuli to evaluate the timbre transfer accuracy. For illustration purposes, the labels of each condition are visible in figure \ref{fig:mushra_ui} (underlined in red), but these labels will of course be hidden to participants completing the survey thus facilitating a ‘blind’ listening-test where participants have to rate each condition exclusively based on auditory stimuli. Then it becomes obvious if participants e.g. rate the reference condition as bad timbre transfer , then we are able to conclude that the validity of this particular participants answers are relatively low. Even though, we are only interested in the rating of our synthesizer implementation and Google Magenta's implementation, it is important to include the reference for the purpose of validating participants answers.
\subsubsection{Expected Results}\label{subsubsec:mushra_expectations}
In this subsection we describe our expected results of the Sound Accuracy Evaluation as well as possible reasons for those assumptions. Given that our synthesizer, including the pitch detection, is running in real-time, the timbre recreation might be perceived as less accurate in comparison to Google Magenta's implementation which is not running in real-time. Furthermore, the Google Magenta implementation features modification options of the input signal to make sure that the incoming signal is sounding as close to the training data as possible (to optimize the resulting output). Our implementation does not have this feature, so instead we experimented with input signals with different frequency content and found the optimal input frequency. Taking these factors into consideration, our expectations is that our implementation will not have as accurate timbre transfer as the Google Magenta implementation, but we expect it to receive a decent score for its timbre transfer ability.
\subsubsection{Results}\label{subsubsec:mushra_results}
We conducted the evaluation via the webMUSHRA-framework and received 11 answers from participants. Via our demographics section of the evaluation we know that there were 8 males, 2 females and 1 identifying as other. The participants age is varying from 22 to 57 with an average of age 30.5. Amongst the participants, are 7 working with audio at a hobby level and 2 are working with audio in relation to their job while 2 has no experience working with audio. Additionally, 9 participants report experience using a software-synthesizer, thus the majority of participants are aware of the sound reproduction capabilities of a software synthesizer.
The main takeaway from the MUSHRA evaluation is to compare the perceived timbre transfer capability of our plugins output in comparison to Google Magenta's implementation. In figure \ref{fig:mushra_resultsbar}, we illustrate this relationship of perceived timber transfer capabilities rated by the participants.
\begin{figure}[htp]
\centering
\includegraphics[width= 1\columnwidth]{SMC704 DDSP/Images/mushra_results_barchart.jpg}
\caption{MUSHRA timbre transfer capability ratings
\label{fig:mushra_resultsbar}}
\end{figure}
The illustration in figure \ref{fig:mushra_resultsbar} is showing the mean of all participant-ratings for each model evaluated. To emphasize the comparison of outputs, the blue bars represent the synthesizer output and the red bars represent the Magenta-implementation output. Furthermore, the bars are grouped according to which model (e.g. the violin) that were used to re-synthesize the reference sample.
From the illustration in figure \ref{fig:mushra_resultsbar} it is obvious that the Magenta-implementation received ratings with averages in the ‘good’ to ‘excellent’ (60-100) timbre transfer categories while our synthesizer received ratings with averages in the ‘fair’ (40-60) category.
The results are varying slightly across the different models. The Magenta implementation’s re-synthesis scored almost the same on average for the Violin and flute model (76.6 and 76.1) but the average of the Trumpet and Saxophone models are slightly higher with ratings of 79.2 and 80.5 respectively.
Looking at our synthesizers timbre transfer score, the Flute and Saxophone models scored 52.3 and 49.2 while the Trumpet and Violin scored slightly higher with ratings of 54.8 and 56.5. It is interesting that the model that is perceived as transferring the timbre most accurately is, for Magenta, the Saxophone model with a rating of 80.5 and for our synthesizer, the Violin model with a score of 56.5.
Since the Magenta implementation scored considerably higher across all models in this evaluation, this means that our synthesizer implementation suffers re-synthesis accuracy compared to the non-real-time web-based Magenta implementation.
Keep in mind, this evaluation is based on 10 answers and therefore might not be entirely representative.
By including the reference as a Condition, we are able to validate the answers of the evaluation. The average of the ratings for each model were 97.9 or 100, which we consider a valid test (see appendix \ref{subsec:mushra_survey_validation}).
\end{comment}
\subsection{Design Requirements Validation}\label{subsec:designrequirements_validation}
To validate if our synthesizer implementation met our design requirements we conducted two evaluations, which were presented in the previous sections.
Design requirement 1: \emph{‘Must work in real-time as a playable software instrument’} and 2: \emph{‘Must be implemented as a software plugin’} has been met as our software synthesizer plugin features models that are implemented to be playable in real-time (elaborated in \ref{subsubsec:realtime_implementation_of_the_models}). We confirmed that these requirements were met by running user-tests where participants had to install the VST3 plugin as well as play it as an instrument in real-time \ref{subsubsec:ux_results}. Design requirement 3: \emph{‘Must allow switching between Line and MIDI input’} and 4: \emph{‘Must implement at least 4 pre-trained models’} was met and addressed in the user-tests as well. Participants reported that switching between Line and MIDI input makes sense, and that they especially enjoy the Line-input functionality \ref{subsubsec:ux_results}. The four pre-trained models that we implemented in our synthesizer were violin, flute, saxophone and trumpet. The timbre transfer capabilities of these models in context of our implementation, were evaluated in section \ref{subsubsec:mushra_results}. Furthermore, we trained additional models but those were not included in this version of the synthesizer as stated in section \ref{subsubsec:our_models}. Lastly, the fifth design requirement \emph{‘Must include graphical UI components for intuitive manipulation of synth parameters’} was partially met with our implementation of the individual partials control GUI, however as reported in section \ref{subsubsec:ux_results} the participants found the general UI to be visually appealing and self-explanatory, but the graphical interface for controlling the harmonic distribution can be improved by making its functionality more obvious to the user. Additionally, implementation of the "modulation" section of our synthesizer would contribute to fulfilling this design requirement as participants reported a lack of controls to achieve a desired output sound \ref{subsubsec:ux_results}. The current state of our synthesizer sufficiently meets two of our design requirements while partially meeting the other three.
\section{Discussion}\label{sec:discussion}
\subsection{Synthesizer controls}
From the Likert evaluation (see figure \ref{fig:ux_likert_results}), we can conclude that users found the interface to be visually appealing and for the most part intuitive to understand, with the exception of the graphical control over the harmonic distribution. However, there is also an indication that the users desired more control over the sound and did not find themselves being able to reach the expression they were trying to achieve. We attribute this sentiment partially to some of the initially planned modules not being implemented in the distributed product. To address this, in the future we would like to fully implement the reverb and modulation modules, and support MIDI Polyphonic Expression (MPE) features.
\begin{comment}
\begin{itemize}
\item The reverb module and its user controls
\item The modulator and the ability to connect it to both the additive and the subtractive synthesizer
\item Controls that allow the user to shape the magnitude response of the filter used in the subtractive synthesizer
\item Additional explanation and a way to reset the harmonic distribution interface
\item Support for MIDI Polyphonic Expression (MPE) features such as Aftertouch
\end{itemize}
\end{comment}
\subsection{Real-time timbre transfer}
We found the quality of the timbre transfer in our real-time implementation below that of the demonstrations published by the Magenta team. Our converted models preserve some characteristics of the original ones, such as wind noises in the flute model, but do not accurately reproduce the timbre overall. We confirmed that on the level of a single frame, our models produce the same output as their original counterparts; will investigate and improve the quality in the future. Additionally, we would like to further investigate why we were unable to perform the timbre transfer with models that we trained both within the framework provided by Magenta, and within custom environments.
A recently released realtime reimplementation of DDSP in PyTorch\footnote{\url{https://github.com/acids-ircam/ddsp_pytorch}} provides a possibly more seamless way of interfacing with DDSP models in C++ that proved compatible with our plugin and JUCE. Extending that API to allow the user some control over the synthesis parameters seems a promising avenue to improve the sound quality of our timbre transfer.
\begin{comment}
so we propose the following potential reasons for the quality mismatch that we would like to
\begin{itemize}
\item Frame-based tracking of the input. Where the timbre transfer demo uses a deep convolutional neural network that operates on the entire input, our project makes use of the Aubio implementation of the YIN algorithm \cite{yin2002}, considering only up to 4096 samples at a time to track the input audio and convert it into f0 and loudness data. This leads to a loss of information and accuracy when it comes to pitch and note onset detection, and it would stand to reason that less accurate input data leads to less accurate output from the models.
\item The models were originally trained on receiving the loudness and pitch information for seconds of audio at a time, and generate the synthesizer controls using a recurrent neural network, meaning that future output depends on previous inputs to a degree. We suspect that a lot of that information is lost through our implementation decision to generate the audio independently from frame to frame. We plan to experiment with keeping internal buffers of previous input to remedy this, keeping an eye of the impact larger input sizes to the models might have on real-time performance.
\end{itemize}
\end{comment}
\subsection{Distribution as a VST3 plugin}
When it came to distributing our project to users, we encountered some difficulties in packaging the required libraries and model files together with the generated VST3 plugin. Some of the DAWs that users tested on, like Ableton or Reaper, did not recognize the plugin or experienced stability issues during its usage.
Although the core functionality could still be accessed via the standalone application generated by JUCE, the project was designed first and foremost as a plugin. Functionality like handling of external audio sources and wet/dry mixing was expected to be handled by the host DAW. Users who had to resort to the standalone when their DAW did not recognize or stably run the plugin reported those features as missing.
Thus, we would like to improve the distribution process in the future, ensuring that the project can be seamlessly installed as a plugin in multiple DAWs on Windows and macOS.
\section{Conclusion}\label{sec:conclusion}
In this paper, we presented an approach to integrate the DDSP library into a real-time plugin and standalone application using the JUCE framework. We succeeded in implementing a synthesizer playable based on pure user input. While we were generally able to use the output from pre-trained models to control the DDSP backend, further research is needed to match the sound quality of these real-time models to that of the offline timbre transfer examples provided by the DDSP authors.
\section{Introduction}\label{sec:introduction}
Sound synthesizers have been widely used in music production since the late 50s. Because of their inner complexity, many musicians and producers polish presets' parameters until they reach the desired sound. This procedure is time-consuming and sometimes results in failed attempts to achieve a desired sound.
Much research has been done in the area of automating the generation of these sounds through the aid of machine learning and neural networks. Common approaches included directly generating the waveform in the time domain \cite{donahue_adversarial_2019} or predicting synthesis parameters based on hand-picked analysis features \cite{Blaauw2017}. In their 2020 paper on Differentiable Digital Signal Processing (DDSP)\cite{engel_ddsp_2020}, Engel et al.\ proposed a novel approach to neural audio synthesis. Rather than generating signals directly in the time or frequency domain, DDSP offers a complete end-to-end toolbox consisting of a synthesizer based on Spectral Modeling Synthesis (SMS) \cite{serra_spectral_1990}, and an autoencoder neural network architecture that takes care of both extracting analysis features and predicting synthesis parameters.
The authors of the DDSP paper released a public demonstration of "tone transfer"\footnote{\url{https://sites.research.google/tonetransfer}, last accessed on 2020-11-30}, allowing the user to upload their own recordings, select from a list of models trained on various instruments and "transfer" their recorded melodies to the sound of a trumpet, a violin etc.
\begin{figure}[bthp]
\centering
\includegraphics[width=0.99\columnwidth]{Images/DDSP_Synth_GUI.jpg}
\caption{Our real-time DDSP Synthesizer GUI.}
\label{fig:ddsp_gui}
\end{figure}
We implemented the DDSP back-end as a virtual instrument playable in real-time. Figure \ref{fig:ddsp_gui} shows the GUI of our synthesizer. This paper documents the background, our requirement-driven design and implementation approach, including model compenents and training, the GUI design, and user experience evaluation. The structure of this paper follows these main topics in order.
Besides our contribution to the real-time neural audio synthesis, we release our real-time MATLAB and JUCE implementations at \url{https://github.com/SMC704/juce-ddsp} and \url{https://github.com/SMC704/matlab-ddsp}, respectively.
\section{Background}\label{sec:background}
In addition to the DDSP paper \cite{engel_ddsp_2020}, our work is inspired by the commercially produced additive synthesizer called \emph{Razor} by Native Instruments\cite{native-instruments}. Razor's core consists of a powerful additive synthesizer and features various modulation options for manipulating the sound output. What is especially interesting about Razor is that every modulation option (e.g. filters, stereo imaging, reverbs and delays) is actually modulating individual partial harmonics (non-integer multiples of the fundamental frequency) in the additive synthesis engine. Furthermore, Razor enables musicians and producers to intuitively control partials via different parameters while relying on a visual representation of partial manipulation. We therefore focused on the harmonic and the stochastic components of the DDSP.
\begin{comment}
\begin{figure}[htp]
\centering
\includegraphics[width=0.9\columnwidth]{SMC704 DDSP/Images/ddsp_autoencoder.png}
\caption{Autoencoder architecture from the DDSP Library \cite{engel_ddsp_2020}
\label{fig:ddsp_autoencoder}}
\end{figure}
\end{comment}
\subsection{Harmonic Oscillator / Additive Synthesizer}\label{subsubsec:additive_synth}
The additive synthesizer is the main core of the whole synthesis and is responsible for generating all the harmonic components of the reconstructed sound. The output is characterized by the sum of several harmonic integer multiples of the fundamental frequency $f_0$:
\begin{equation}
f_k = k \cdot f_0(n) .
\label{eq:kth_harmonic}
\end {equation}
In order to generate the harmonics, we can implement $k$ oscillators in the discrete time:
\begin{equation}
x(n) = \sum_{k = 1}^{K} A_k(n) \cdot sin(\phi_k(n)),
\label{eq:add_synth}
\end {equation} where $A_k(n)$ is the time-varying amplitude of the $k_{th}$ sinusoidal component and $\phi_k(n)$ is its instantaneous phase. $\phi_k(n)$ is obtained using equation \ref{eq:phase}.
\begin{equation}
\phi_k(n) = 2\pi \sum_{m = 0}^{n} f_k(m) + \phi_{0,k}.
\label{eq:phase}
\end {equation}
The only two parameters necessary to control the synthesizer are the frequency $f_0(n)$ and the harmonic amplitudes $A_k(n)$. These are retrieved directly from the input sound using the encoder contained in the autoencoder network. As reported in \cite{engel_ddsp_2020}, the network outputs are scaled and normalized to fall within an interpretable value range for the synthesizer
\subsection{Filtered Noise / Subtractive Synthesizer}\label{subsubsec:ddsp_subtractive_synth}
The subtractive synthesis is used to recreate the non-harmonic part of natural sounds. The parameters necessary to obtain a frequency-domain transfer function of a linear time-variant finite impulse response (LTV-FIR) filter are retrieved from the neural network in frames that are subsets of the input signal. The corresponding impulse responses (IRs) are calculated and a windowing function is applied. The windowed IRs are then convolved with white noise via transformation to and multiplication in the frequency domain.
\subsection{Reverb}\label{subsubsec:reverb}
In addition to the SMS model, the sound is also given a sense of space using a reverberation algorithm performed in the frequency domain. Thus, the operation of convolution between the impulse response of the reverb and the synthesized signal is a more efficient multiplication.
\begin{comment}
\subsubsection{Instrument-specific models}\label{subsubsec:instrument_models}
The Google Magenta Research team has trained four different models to test the DDSP autoencoder: flute, saxophone, trumpet and violin. The audio file used for the latter training has been taken from the Musopen database\footnote{\url{https://musopen.org/music/13574-violin-partita-no-1-bwv-1002/}, last accessed 2020-12-02} while for the other instruments they used the NSynth dataset \cite{nsynth2017}.
The supervised models obtained from the DDSP autoencoder significantly outperformed other state-of-the-art models by several quantitative metrics, demonstrating higher accuracy in tasks such as $f_0$-resynthesis while requiring a smaller number of parameters.
The neural network architecture of the DDSP library has been implemented in TensorFlow, an open-source machine learning library developed by the Google Brain team \cite{tensorflow2015-whitepaper}.
\subsection{Other Related Work}\label{subsec:other_related_work}
In addition to the DDSP library synthesis method, we explored various other synthesizers and synthesis methods. By looking into other ways of working with synthesis and synth-parameter estimation we gained knowledge of possibilities and limitations that the DDSP library offers in relation to other synthesizer approaches. In this section we briefly elaborate on the 3 synth concepts that we drew the most inspiration from in our concept development phase: \emph{FlowSynth, NSynth} and \emph{Razor}.
\end{comment}
\begin{comment}
\emph{FlowSynth} is an intuitive synthesizer-controller which aims to solve the issue of complex synthesizers having \emph{“highly non-linear relationships between the parameters and the resulting audio.”}\cite{Esling2019}. FlowSynth aims to make it easier for non-expert users to explore complex software synthesizers by implementing a machine learning based preset exploration method. Esling et al. argues that FlowSynth facilitates a way of \emph{“finding an organized latent audio space that represents the capabilities of a synthesizer, while constructing an invertible mapping to the space of its parameters.”}\cite{Esling2019}. The controller works as a tool that can be trained on specific existing synthesizers' parameters to learn which sounds it is capable of creating (‘audio space’). It is then able to recreate sounds using the existing synthesizer parameters (‘parameter space’).
Another source of inspiration for our project is \emph{NSynth} (Neural Synthesizer) which, instead of featuring parameters such as oscillators and wavetables, relies on deep neural networks to generate sounds \cite{nsynth2017}. NSynth aims to achieve quality instrument sounds by training models on large datasets of musical notes which are sampled from individual instruments. Therefore, gathering a large dataset with around 300k samples from 1000 different instruments which they call “The NSynth Dataset” is another goal for the Google Magenta team \cite{nsynth2017}. The team has released The NSynth Dataset to contribute to the research and development in the audio machine learning community.
Finally, we drew inspiration from a commercially produced additive synthesizer called \emph{Razor} by Native Instruments\cite{native-instruments}. Razor's core consists of a powerful additive synthesizer and features various modulation options for manipulating the sound output. What is especially interesting about Razor is that every modulation option (e.g. filters, stereo imaging, reverbs and delays) is actually modulating individual partial harmonics (non-integer multiples of the fundamental frequency) in the additive synthesis engine. Furthermore, Razor enables musicians and producers to intuitively control partials via different parameters while relying on a visual representation of partial manipulation.
\section{Analysis}\label{sec:Analysis}
In this section we will introduce the initial research path conducted around the DDSP library as well as the research question that drove our project from the beginning.
\subsection{Exploration of the problem space}\label{subsec:exploration_problem_space}
During the first phase of our research, the team analyzed and collected several DDSP application ideas trying to evaluate the possible issues linked to the development of each of them. A great effort has been put into the formulation of new concepts and associated questions about feasibility, technical and technological challenges, usability and musicality. Each idea has been evaluated using the SWOT analysis \cite{leigh2009swot}, an investigation tool widely employed to detect and analyze four different key aspects of projects: strengths, weaknesses, opportunities and threats.
One of the first ideas investigated is the embedded platform microcontroller application inspired by recent research published in the DAFx-20 Proceedings conducted by Esling et al. \cite{esling_diet_2020}. In this paper, a \textit{structured trimming} has been performed on deep learning models applied to audio applications, obtaining light models for generative audio. Later, due to the monophonic constraint imposed by the architecture of the DDSP model, we took into account the challenge of a polyphonic implementation of this library. Finally, we considered using DDSP to develop a synthesizer, examining two different proposals. At the end of this research process, these concepts were polished and combined together. The first was focused on the design of a synthesizer based on DDSP with semantic macro-controls to help both beginners and experts reaching the wanted result in a fast and efficient way. The second application was inspired by Yee-King et al. \cite{yee-king_automatic_2018} and involved the design of a synthesizer based on sound matching, i.e. an instrument able to reconstruct a target sound adjusting the DDSP components.
To sum up, we found the design of a synthesizer based on sound-matching with macro-controls to be a perfect challenge for this project. It is not only focused on a new application of DDSP, but also introduces new features such as macro controls for the synthesis and real-time implementation.
\end{comment}
\subsection{Research question}\label{subsec:research_question_statement}
Based on this background
we have formulated the following research question: How can we develop a playable software instrument, based on the DDSP library, that would: a) allow customization of model-estimated synth parameters through top-level macro controls, b) enable existing workflow-integration in Digital Audio Workstations (DAWs), and c) facilitate a simple approach for beginners without limiting usability for expert music producers?
\begin{comment}
\section{Design Requirements}\label{sec:design_requirements}
For the purpose of requirements management and evaluation we created a set of software design requirements for our software application which are based on the approach of
\subsection{Requirements generation - Approach}\label{subsubsec:requirements_generation_approach}
Pandey et al.'s Requirement Engineering approach aims to discover quality requirements for software development and argues that \emph{“A well-formed requirement is a statement of system functionality that satisfies customer needs.”}\cite{Pandey2010}. As our project revolves around development of a software music production tool which is intended to be used by musicians and music producers, it is crucial for the success of our product that it meets user needs to a certain degree. Therefore, we have followed the overall structure of Pandeys et al.'s Requirement Engineering’s phases but mainly focused on elements from the two first phases: \emph{Requirement elicitation and development} and \emph{Documentation of requirements} because these phases are very relevant in formulating meaningful design requirements\cite{Pandey2010}.
\subsection{Requirements elicitation and development}\label{subsubsec:requirements_elicitation_and_development}
Pandey et al. suggests to begin with finding ‘objectives’ (raw requirements) for the system from different stakeholder’s viewpoint, e.g. the customers, users, constraints, marketing, etc.\cite{Pandey2010}. We started by gathering objectives for our application focusing mainly on the user’s perspective since our synthesizer is supposed to be used on an experimentation level by musicians and music producers who are interested in new technologies for the music industry. Thus, we do not take other stakeholders' viewpoint into consideration, as this would be out of the scope of this paper. We aim to come up with solutions to these objectives through the design of our software application. As this project aims to explore technical possibilities of implementing DDSP in an instrument-like context, instead of collecting user data through surveys or interviews, we based the objectives for the synthesizer on our research of e.g. popular music-focused YouTubers' opinions\cite{neely_2020, huang_2020}, state-of-the-art music production tools \cite{Esling2019, nsynth2017, native-instruments} as well as our own experience with software synthesizers and virtual instruments.
We found that DDSP is not currently implemented to work in real-time\cite{engel_ddsp_2020} and we believe that this new feature will provide interesting new application opportunities. Especially in the context of music composition and music production, a real-time implementation, in the form of a software synthesizer plugin, could potentially provide a new tool for generating unique sounds and gaining inspiration. For this to be adopted by music producers, we want to develop a plugin-based instrument because our goal is that musicians and producers are able to integrate our application conveniently into their everyday workflows in different DAWs. To maintain the opportunity for different workflows and composition methods, our application will have both MIDI and Line input methods, from which the application will retrieve the fundamental frequency and amplitude information. We want the users to be able to choose from different instrument-specific models (\ref{subsubsec:instrument_models}), which will control the parameters of the additive synth, subtractive synth and the reverb module in real-time to generate the desired instrument-specific output sound. If this was the only user-input that the application had, it would be rather limited in creative expressivity. Therefore, it must be possible for the user to alter parameters of the application's modules (such as harmonics/noise mix, amplitude of individual partials, reverb length etc.) in an easy user-friendly way, after a model was used to estimate them.
\subsection{Documentation of requirements}\label{subsubsec:documentation_of_requirements}
Documentation of requirements is according to Pandey et al. a matter of describing \emph{“[…] the behaviour of the system or software to be developed.”}\cite{Pandey2010}. In table \ref{tab:design_requirements}, each of the user objectives is listed with a solution to assure that a specific user need is met by the software design. The
, and present them on Table \ref{tab:objectives}.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.9\columnwidth}{|l|X|}
\hline
\textbf{Obj. \#} & \textbf{User objectives} \\
\hline
1 & Provide a new playable instrument for unique sound generation and inspiration \\
\hline
2 & Conveniently integrate into existing workflows \\
\hline
3 & Adapt to different composition methods \\
\hline
4 & Easy fast unique sound generation \\
\hline
5 & Customizability of generated sounds \\
\hline
\end{tabularx}
\end{center}
\caption{List of objectives from a user-centered viewpoint.}
\label{tab:objectives}
\end{table}
\end{comment}
To sum up the design requirements, we aim to build a software instrument plugin that is playable in real-time. The instrument must support different composition techniques, thus having a line and MIDI input mode. The instrument must include at least four pre-trained models which serve the purpose of estimating synthesizer parameters to output a desired sound. Finally, the instrument must include graphical user interface components that provide intuitive controls for the manipulation of synthesizer and effect parameters.
\section{Design \& Implementation}\label{sec:implementation}
Based on this research question, we have identified five user needs \cite{Pandey2010}, and matched them with a solution, reformulating them as a concrete measurable design requirement. The design requirements are thus documented on Table \ref{tab:design_requirements}.
\begin{table*}[ht]
\begin{center}
\begin{tabularx}{0.99\columnwidth}{|l|X|X|X|}
\hline
\textbf{\#} & \textbf{User Obj.} & \textbf{Solution} & \textbf{Design Requirement} \\
\hline
1 & Provide a new playable instrument for unique sound generation and inspiration & Real-time implementation & \emph{Must work in real-time as a playable software instrument.} \\ \hline
2 & Conveniently integrate into existing workflows & Plugin format application & \emph{Must be implemented as a software plugin.}\\
\hline
3 & Adapt to different composition methods & Allow line and MIDI input & \emph{Must allow switching between Line and MIDI input.} \\
\hline
4 & Easy fast unique sound generation & Choose models for sound generation & \emph{Must implement at least four pre-trained models.} \\
\hline
5 & Convenient customizability of sounds & Tweakable parameters that effects the audio output & \emph{Must include GUI components for intuitive manipulation of synth and effects parameters.} \\ \hline
\end{tabularx}
\end{center}
\caption{Documentation of Design Requirements}
\label{tab:design_requirements}
\end{table*}
\subsection{Architecture overview}\label{subsec:architecture_overview}
\begin{figure*}[htp]
\centering
\includegraphics[width=0.99\columnwidth]{Images/Architecture.png}
\caption{Schematic overview of the project architecture.
\label{fig:architecture_scheme}}
\end{figure*}
To meet our criteria of creating a real-time software instrument, we decided to build the plugin in C++ using the JUCE application framework\footnote{\url{https://juce.com/}, last accessed on 2020-12-15}. With JUCE, we had a multi-platform supported audio plugin template that was handling MIDI and audio inputs and outputs. This allowed us to mainly focus on the audio processing and GUI.
Creating a real-time implementation of the non-real-time DDSP library posed some immediate challenges. To analyze and understand these challenges we decided to start by doing a direct translation of the additive and subtractive synthesizers from the DDSP library into MATLAB. The synthesizers could then be changed into real-time implementations and tested. In order to use our MATLAB implementation in the JUCE framework, we used inbuilt MATLAB tools to generate C++ code.
We transformed the autoencoder models pretrained by Google into models that could be used to estimate synthesizer parameters directly from our plugin's user input.
A general overview of this architecture can be seen in figure \ref{fig:architecture_scheme}.
The following sections will discuss each component in more detail.
\subsubsection{Synth in MATLAB}\label{subsubsec:synth_MATLAB}
MATLAB’s environment and visualization tools gave us access to quick prototyping and testing. This allowed us to do the implementation over multiple iterations. We tested our synthesizers' compatibility with the predicted parameters from the DDSP models by invoking the encoders and decoders in isolation through MATLAB's Python interface.
At first we implemented the non-real-time synthesis algorithms of the DDSP library. Then the synthesizers were changed to real-time, i.e., synthesizing a single frame at a time.
Using the MATLAB Audio Test Bench, we could then test the functionality of the synthesizer components and parameters with real-time audio and varying sample rate and buffer size.
The last iterations consisted of optimizing the code with the constraints of real-time audio processing on CPUs.
\subsubsection{MATLAB to C++}\label{subsubsec:MATLAB_to_C++}
Using the MATLAB coder tool\footnote{\url{https://se.mathworks.com/products/matlab-coder.html}, last accessed on 2020-12-15} we were able to generate C++ functions from the MATLAB code. For the simplest integration between the generated C++ functions and the JUCE plugin we chose to limit the function inputs and outputs to built-in and derived C++ data types. This required our MATLAB functions to have fixed-sized inputs and outputs. We decided on a maximum input/output size of 4096 double-precision floating point numbers, this being the maximum buffer size the plugin would be able to work with.
A helper file was created to ensure code consistency, allowing the user and MATLAB coder to verify the functions with different inputs. Having this setup made it easy to go back to the MATLAB code and generate updated C++ functions without breaking the JUCE plugin.
\subsubsection{TensorFlow in C++}\label{subsubsec:Tensorflow_in_C++}
Running the DDSP TensorFlow implementation in a real-time audio application is a heavy computational challenge. Moving from TensorFlow in Python to the TensorFlow C API\footnote{\url{https://www.tensorflow.org/install/lang_c}, last accessed on 2020-12-15} allowed us to integrate the models into the C++ codebase. By moving the TensorFlow computations to a separate thread, we can load the models, set the inputs, run the parameter estimation and save the outputs, without experiencing buffer underruns in the main audio processing thread.
\subsubsection{Input signals}\label{subsubsec:input_signals}
The DDSP autoencoder needs the input values \emph{fundamental frequency} ($f_0$) and \emph{loudness} ($ld$). Since we allow both MIDI and line-in audio, two separate implementations are needed to calculate these values. Functions for this were created in MATLAB, but in the C++ implementation we chose to use the implementation of the YIN pitch tracking algorithm \cite{yin2002} from the C library Aubio\cite{aubio}, since it yielded more precise results.
\subsection{Training models}\label{subsec:training_models}
\subsubsection{Pre-trained models}\label{subsubsec:pre-trained_models}
Next to the \emph{tone transfer} website mentioned in the introduction, the authors of the DDSP paper also published a Jupyter Notebook Demo on Google Colab called \emph{timbre transfer.}\footnote{\url{https://colab.research.google.com/github/magenta/ddsp/blob/master/ddsp/colab/demos/timbre_transfer.ipynb}, last accessed on 2020-12-15}
We accessed the available checkpoint files for violin, flute, tenor saxophone and trumpet from this notebook for our real-time implementation of the timbre transfer. However, we were not immediately able to use them in the JUCE plugin. The DDSP models are trained using TensorFlow's \emph{eager execution mode}, while the TensorFlow C API is constructed around \emph{graph mode}. Additionally, since we required the models to be controllable by MIDI input, we needed direct access to the decoder part of the model instead of supplying audio to the encoder.
The \texttt{convert\_models.py} script from the Python folder of the plugin code repository deals with these requirements by loading the eager model from the downloaded checkpoint file, constructing a graph-based model only containing the decoder and then copying all weights from the old model to the new one. The resulting checkpoint now contains a graph that can be loaded by the TensorFlow C API.
\subsubsection{Custom models}\label{subsubsec:our_models}
In order to make use of the DDSP training library and extend the synthesizer with additional models, we created four custom models trained on:
\begin{itemize}
\item Bass sounds of the Moog One, Moog Minimoog and Moog Minitaur synthesizers
\item Studio recordings of Middle Eastern instruments, the Hammered Dulcimer and Santoor
\item Studio recordings of a Handpan (also known as Hang Drum)
\item Nature field recordings of birds chirping
\end{itemize}
For training we used the official DDSP (version 0.14.0) Jupyter notebook on Google Colab called \emph{train autoencoder}\footnote{\url{https://colab.research.google.com/github/magenta/ddsp/blob/master/ddsp/colab/demos/train_autoencoder.ipynb}, last accessed on 2020-12-15} which allows training on a Google Cloud GPU using own data. We chose the recordings listed above in order to obtain interesting sounds that differ from the more traditional pre-trained DDSP models. According to the recommendations of the DDSP authors given in the notebook, trained models perform best using recordings of a single, monophonic sound source, in one acoustic environment, in .wav or .mp3 format with a total duration of 10 to 20 minutes. Since the DDSP Autoencoder is conditioned on the loudness $A$ and the fundamental frequency $f_0$, i.e., the model learns to associate different synthesizer configurations to specific value pairs of $(A, f_0)$, training on multiple instruments, acoustic environments or polyphonic sounds prevents the autoencoder to learn a unified representation. However, these thereby introduced artifacts can also be used in a musical context, that is why we decided to challenge the autoencoder with less conform training data and eventually achieved interesting timbres.
The training process is performed as follows. The first step is comprised of data generation and pre-processing of the training data. The raw audio is split into short parts of a few seconds, each analyzed on the specified features, i.e., the fundamental frequency and loudness, and finally saved in the TensorFlow \emph{TFRecord} format. The fundamental frequency is thereby estimated by using the state-of-the-art pitch tracking technique, called \emph{CREPE} by Kim et al. \cite{kim2018crepe} that applies a deep convolutional neural network on time-domain audio. The second step is the actual training, using a Python based configuration framework for dependency injection by Google, called \emph{Gin}\footnote{\url{https://github.com/google/gin-config}, last accessed on 2020-12-15}. In this way, all available training hyperparameters can be defined in a gin config file that is passed to the training function. The training process does not include any optimization techniques, such as a hyperparameter search or early stopping, the authors just recommend in the code documentation to train for 5,000 to 30,000 steps until a spectral loss of about 4.5-5 is reached for an optimal learning representation without overfitting. The third and last step is a short evaluation based on resynthesis. That means, a training sample is randomly picked, passed to the autoencoder that encodes and decodes, i.e., reconstructs the input sample based on the learned features.
We successfully conducted training of all four models and validated their performance in the previously mentioned timbre transfer demo. While validation using the DDSP library went smoothly and showed musically interesting results, we ran into issues during inference using the TensorFlow C API within our plugin. We monitored a much higher loudness of the custom models compared to the pre-trained models, resulting in a distorted, clipping sound. Furthermore, we detected a constant harmonic distribution independent of the incoming pitch and loudness while the pre-trained models adapt harmonics and frequency response according to these inputs. The overall experience with the training script provided by the DDSP authors is that it works without problems for standard parameters, but as soon as own hyperparameters within the gin framework are chosen, a lot of side-effects appear. For the mentioned reasons, integrating and possibly adapting the custom-trained models to make them work in the DDSP synthesizer will be a part of future work.
\subsubsection{Real-time implementation of the models}\label{subsubsec:realtime_implementation_of_the_models}
The DDSP non-real-time implementation synthesizes several frames before processing them into one output. Reading through the DDSP code base we experienced the number of frames (time steps) to be defined by the size of the input audio and a hop size defined by constants in the gin config file of the selected pre-trained model.
\begin{comment}
\begin{equation}
\text{hop\ size} = \frac{\text{number\ of\ samples\ in\ training}}{\text{number\ of\ time\ steps\ in\ training}}
\label{eq:hop_size}
\end{equation}
\begin{equation}
\text{time\ steps} = \frac{\text{size\ of\ audio\ input}}{\text{hop\ size}}
\label{eq:time_steps}
\end{equation}
\end{comment}
For our real-time implementation we wanted to calculate one frame with a size of the input buffer each time the buffer is ready.
Given the static nature of our TensorFlow model implementation we were not able to change the number of time steps on the run. Therefore, we set the number of time steps to one. Each run of the TensorFlow model would then return a set of values for one time step, independent of the buffer size.
\subsection{Additive synthesizer}\label{subsec:additive_synthesizer}
The implementation of the additive synthesizer can be found in the \texttt{additive.m} MATLAB code file. During the development of the DDSP synthesizer we went from a re-implementation of the DDSP equivalent to an adapted real-time optimized version with additional parameters for high-level control. While the original DDSP library provides two different implementations of the additive synthesis, the harmonic and sinusoidal approach, this work focuses on the harmonic synthesis that models a signal by adding only integer multiples of the fundamental frequency.
In the following, the initial implementation as well as the main modifications in its final state are clarified.
As already explained in \ref{subsubsec:additive_synth}, the additive synthesizer models audio using a bank of harmonic sinusoidal oscillators. The synthesis algorithm takes amplitudes, harmonic distribution and fundamental frequencies for a specified number of frames as input and computes the sample-wise audio signal as output. The harmonic distribution provides frame-wise amplitudes of the harmonics. The additive synthesis as implemented in the DDSP library is performed in two main steps:
\begin{itemize}
\item Translation of neural network outputs to the parameter space of the synthesizer controls
\item Computing the output signal from synthesizer controls
\end{itemize}
In order to make the output of the neural network usable for controlling the synthesizer, it needs to be transformed accordingly. In detail, that means the amplitudes are scaled and the harmonic distribution is scaled, bandlimited and normalized while the fundamental frequencies remain unchanged. Bandlimiting the harmonic distribution means removing the harmonics that exceed Nyquist in order to avoid artifacts.
After retrieving valid synthesizer controls, the harmonic synthesis is performed. Since the DDSP approach works frame-based while the output needs to be delivered sample-based, the synthesizer controls need to be upsampled. This is done by linearly interpolating the frequency envelopes and windowing the amplitude envelopes by using 50\% overlapping Hann windows. Having calculated all controls on a sample basis, the signal can be synthesized by accumulative summation of the corresponding phases, i.e., adding the calculated sinusoids together, sample by sample.
The following changes were made to optimize the algorithm for a real-time application and to add additional high-level control for the synthesis.
\begin{itemize}
\item Since the frame-based calculation was computationally too heavy, we adapted the code so that the input is always one frame (equivalent to the buffer size) and all computations are sample-based. Therefore, no resampling or windowing is needed.
\item Each time the function is called, the phases of all harmonics are saved and returned along with the signal and added as offset in the next call to avoid artifacts caused by phase jumps.
\item In order to be able to optionally introduce non-harmonic partials to the signal, a stretch parameter was added that transforms the distance between the integer multiples while maintaining the fundamental frequency. An additional shift parameter adds the functionality to modify the fundamental frequency from one octave below to one octave above the current pitch in a continuous scale.
\end{itemize}
\subsection{Subtractive synthesizer}\label{subsec:matlab_subtractive_synth}
This component is responsible for the non-harmonic parts of instrument sounds, such as the audible non-pitched flow of air that accompanies the harmonic part of a flute sound. Our implementation, which can be found in the \texttt{subtractive.m} MATLAB code file, generates a frame of random noise and then filters it according to a given frequency response.
The function's parameters are the frame length (number of samples), noise color (see below) and the frequency response, which is given as a vector of $N$ magnitudes $m_0,\ldots,m_{N-1}$, where $m_0$ corresponds to the DC component and $m_i$ to frequency $f_{\text{nyquist}} / (N - i)$ with $f_{\text{nyquist}} = f_s / 2$ and samplerate $f_s$.
While we started with a direct re-implementation of the DDSP FilteredNoise approach described in \ref{subsubsec:ddsp_subtractive_synth}, we made the following adaptations over the course of the project:
\begin{itemize}
\item Simplified filtering calculation. The DDSP synthesizer processes multiple frames at once. For the sake of a real-time implementation, we removed the step of calculating the impulse response for each frame and applying a windowing function. Instead, we simply perform a Fourier transform on the generated noise and multiply the result with the filter magnitude response that the model predicted for the single current frame.
\item Noise color. We provide functionality to shape the frequency distribution of the generated noise. Noise color generally refers to the frequency $f$ being emphasized proportionally to $1/f^{\alpha}$ for some exponent $\alpha$ \cite{kasdin_1995}. $\alpha < 1$ results in higher frequencies becoming more prominent, while $\alpha > 1$ increases the energy of the lower frequencies. Uniform white noise is achieved by setting $\alpha = 1$.
\end{itemize}
\subsection{Graphical User Interface}\label{subsec:GUI}
After the development of all the features of our synthesizer, we focused our attention on designing an interface with high-level controls for the additive and the subtractive synthesis, the reverb, the modulation and the models. Our process started from a list of all the parameters we wanted to manipulate. We also looked for some inspiration from well-known VST synthesizers, comparing them in terms of usability and trying to understand what their best interaction features were. Later we organized the controls of our synthesizer in different modules and displayed them in a rectangular interface, trying to find a layout that was pleasant but also respectful of the instrument's architecture logic. In table \ref{tab:GUI_features}, we list all the controls for each module of our synthesizer. Because of the particular choice of a graphic control for the harmonics' amplitude, the team opted for a spectrogram representing the output of our plugin. In this way, the user is able to clearly see which harmonics are being played.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.9\columnwidth}{|l|X|}
\hline
\textbf{Module} & \textbf{Feature controls}\\
\hline
Input selector & MIDI/line selector\\
\hline
\multirow{8}{50pt}{Models selector} & Violin\\
& Flute\\
& Saxophone\\
& Trumpet\\
& Moog Bass (not included)\\
& Dulcimer (not included)\\
& Handpan (not included)\\
& Chirps (not included)\\
\hline
\multirow{4}{50pt}{Additive synthesis} & Graphic harmonics editor\\
& $f_0$ shift\\
& Harmonics stretching\\
& Global amplitude\\
\hline
\multirow{2}{50pt}{Subtractive synthesis} & Noise color\\
& Global amplitude\\
\hline
\multirow{3}{50pt}{Modulation} & Modulation rate\\
& Delay control\\
& Amount\\
\hline
\multirow{3}{50pt}{Reverb} & Dry/wet mix\\
& Size\\
& Glow\\
\hline
Output & Master gain\\
\hline
Spectrogram & Clear visualization of the output\\
\hline
\end{tabularx}
\end{center}
\caption{List of GUI's features}
\label{tab:GUI_features}
\end{table}
Once we defined the layout and the parameters that we wanted to control, we moved to the software development in JUCE. In order to customize the appearance of knobs, we used the "Custom LookandFeel" objects while we designed ad hoc images for the buttons and background texture using a vector graphics software. Figure \ref{fig:ddsp_gui} previously presented the GUI of our synthesizer.
\subsection{Plugin setup}\label{subsec:plugin_setup}
The synthesizer ended up being built as a standalone executable and a DAW plugin using Steinberg’s VST3 format.
Using JUCE’s AudioProcessorValueTreeState class we are exposing the different controllable parameters to the DAW, allowing control and automation of the plugin. Using this class we will also be able to easily store and read plugin states, enabling generation of presets, though this has not been implemented yet.
The synthesizer is configured to load the models from a given path with subfolders containing the individual models, as well as configuration files containing key-value pairs such as number of harmonics and scaling values.
\section{Evaluation}\label{sec:evaluation}
In order to understand the strengths and weaknesses of our product to improve it, we designed an evaluation strategy for both User Experience (UX) and sound output.
Our target users are musicians and music producers. Accordingly, we shared a release of our VST plugin with selected sound engineers, musicians and producers to collect opinions and user insights. Moreover, we designed two different questionnaires and asked participants to evaluate the UX and the sound accuracy of our software. The DDSP Synthesizer as well as the two questionnaires have been distributed online and the participants received an email with all the indications to properly conduct the test.
In the next two sub-sections we will describe each evaluation in detail, including approach, desired outcome, survey design and results.
\subsection{User Experience Evaluation}\label{subsec:ux_evaluation}
\subsubsection{Approach}\label{subsubsec:ux_approach}
The aim of this evaluation is to collect feedback about the user interface from people with experience on synthesizers and music production. One of the goals of our project was to design a simple and efficient interface able to control several parameters with a single gesture without giving up functionality in the pursuit of simplicity.
After a trial period where the participants had the chance to familiarize themselves with the software, we asked them to compile a form.
\subsubsection{Survey structure}\label{subsubsec:ux_survey_structure}
Google Forms was chosen as a platform because of its simplicity and wide spread. We designed the survey with different sections to group the questions by theme. We included an experiment in order to ask each participant to load and perform some changes to a model and export the result in an audio file. In this way, we are sure that every participant had at least used and interacted with the plugin for a while. Moreover we are able to compare each audio export to understand if some of the instructions were not clear or if the UX itself was not effective.
Four usage questions have been asked to collect information about the user's DAW and for how much time they used the plugin. In the next sections we asked the participants to report their experience during the experiment and evaluate the user interface rating 9 different statements with a Likert-scale, a widely used bipolar symmetric scaling method in questionnaires. In this way, users were able to express their agreement/disagreement related to each sentence. Furthermore, we asked 4 open questions to let the participants express their opinion about the overall UX. Finally we added 8 questions to locate demographics and musical-related personal experiences. Table \ref{tab:ux_survey} summarizes the content of each section.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.9\columnwidth}{|l|X|X|}
\hline
\textbf{\#} & \textbf{Section} & \textbf{Content}\\
\hline
1 & Introduction & Aim of the questionnaire\\
\hline
2 & Experiment & Task instructions\\
\hline
3 & Usage & 4 mixed questions\\
\hline
4 & UX evaluation & 9 Likert scale evaluations\\
\hline
5 & UX experience & 4 open questions\\
\hline
5 & Demographics & 8 mixed questions\\
\hline
\end{tabularx}
\end{center}
\caption{Content of the UX survey}
\label{tab:ux_survey}
\end{table}
\subsubsection{Expected results}\label{subsubsec:ux_expected_results}
In this section we want to express a projection of the feedback regarding the User Experience. Considering that the software is still under development, we are expecting reports about compatibility issues with different DAWs as well as some stability problems. Moreover, because of the VST's instability in the first release, it is possible that some users will not be able to conduct the small experiment that requires the plugin to be embedded in a DAW track. Considering the whole interface, one of the main points of our design requirements was the simplicity
and thus our hope is to facilitate the user's interaction.
\subsubsection{Results}\label{subsubsec:ux_results}
We received 7 answers. Five participants identified as males, 1 female and one preferred not to say. The age average is 28.57 years (STD 8.42). Six of them declared that sound production is their hobby while one said music production is related to their job. The mean experience in the music production field is 7.43 years (STD 4.87).
Six users do not have experience with machine learning VST plugins and only one of them does not know if she/he ever used one. Each user spent an average of 23.57 minutes using our synthesizer (STD 17.74). We suppose that some mistake has been made reporting the usage time for at least one user. In table \ref{tab:used_daw} we report the number of user tests per different software environment.
\begin{table}[htp]
\begin{center}
\begin{tabularx}{0.7\columnwidth}{|l|X|}
\hline
\textbf{\# users} & \textbf{Environment} \\
\hline
3 & Reaper\\
\hline
2 & Ableton Live\\
\hline
1 & Cubase\\
\hline
1 & Standalone version\\
\hline
\end{tabularx}
\end{center}
\caption{List of used DAWs in the evaluation.}
\label{tab:used_daw}
\end{table}
In general, the experiment
has been rated a medium difficult task with a mean rating of 3.43 in a scale from 1 to 5 being 1 "easy to accomplish" and 5 "hard to accomplish". In figure \ref{fig:ux_likert_results} we summarize the answers obtained from the questions with an associated likert scale. The users were asked to rate each sentence from 1 to 5 with 1 corresponding to "strongly disagree" and 5 to "strongly agree". We can observe that the graphical user interface has been really appreciated with a 4.43 mean value while the interface's controls seem not to let the participants easily reach the wanted results. The other statements reported in the likert section obtained a medium rating between 3 and 3.86 which might mean that the GUI is in general appreciated.
\begin{figure*}[htp]
\centering
\includegraphics[width= 0.99\columnwidth]{Images/Likert_Graph.jpg}
\caption{User experience evaluation - Likert scale}
\label{fig:ux_likert_results}
\end{figure*}
As expected, some of the participants encountered difficulties in the installation procedure of the VST3 plugin in both Windows and macOS environments while the standalone version seems to be more stable. Furthermore, three users reported an unsatisfactory audio result related to the presets obtained from models. Here we report part of one of the feedback: "\textit{[...] It's possible to get some cool sounds but the default sound when you just start it is not so nice.}". On the other hand, the audio input feature was appreciated: "\textit{[...] I think the audio input feature has a lot of potential and I caught myself experimenting with this a lot and loosing track of time.}". Two participants reported that the possible interaction with the interface for the additive synthesizer was not immediate to spot and they realized its features after a while. For this reason they suggest a graphical indication to guide the user to the interaction with the harmonic sliders.
A significant outcome is the unexpected audio results that participants reported. Even though they described output sounds as "awkward", they highlighted the new creative way of producing unexpected sounds, finding the whole synthesizer experience engaging.
\begin{comment}
\subsection{Sound Accuracy Evaluation}\label{subsec:soundaccuracey_evaluation}
\subsubsection{Approach}\label{subsubsec:sound_accuracy_approach}
In this part of the evaluation we aim to test the sound quality of the output of our real-time software synthesizer in comparison to the Google Magenta DDSP 'tone transfer' implementation\footnote{\url{https://sites.research.google/tonetransfer}, last accessed on 2020-12-17}. More specifically, we test for the perceived timbre transfer accuracy in the output audio from both systems. As opposed to section \ref{subsec:ux_evaluation}, this section focuses solely on evaluation of the output audio and does not consider any user-interactable parameters for sound generation, i.e. all audio sources used in this part of the evaluation are purely model-generated for the sake of comparability.
We approached the sound accuracy evaluation with a focus on timbre transfer capability because, instead of testing for sound similarity between our implementation and Google Magenta, we believe that testing timbre transfer capability will provide more useful insights on the overall perceived sound recreation of our plugin (while this approach also implicitly compares audio output similarity).
To conduct this evaluation, we created a set of audio samples of a piano playing in different octaves. We inputted our sample-set to both Google Magenta's implementation as well as our synthesizer to generate an output-set of transferred audio files, with a transferred timbre. We created multiple output-sets, one for each of the following models: violin saxophone, flute and trumpet. By using piano-samples that are played in different octaves, we are able to test each of the model’s capability of reproducing timbre of different frequency content. Furthermore, as opposed to Google Magenta's implementation, our synthesizer does not feature automatic modification of the input signals frequency and loudness to minimize the difference between the input signal and the training data. Therefore, we modified the input signal manually (changing octaves of the piano-sample) before inputting it into our synthesizer and Google Magenta's implementation.
\subsubsection{Listening test tool: ‘webMUSHRA’}\label{subsubsec:test_tool_mushra}
To construct a suitable platform for conducting audio-listening tests we have used the ‘webMUSHRA’ web framework\cite{Schoeffler2018}. The webMUSHRA framework is designed to help researchers and experimenters conduct valid, convenient tests or evaluations that revolve around listening tests and audio files. Many existing survey tools do not allow for convenient online listening test environments and therefore we chose to use webMUSHRA. As stated by Schoeffler et. al. \emph{“[…] there exist only minor differences between listening tests carried out in an laboratory environment and those carried out over the Internet, if the experiment was properly designed.”}\cite{Schoeffler2018}.
The main feature of webMUSHRA that we used to build this survey is called a MUSHRA (Multi-Stimulus Test with Hidden Reference and Anchor), which can be used to evaluate auditory systems\cite{Schoeffler2018}. This test environment is suitable for conducting our evaluation of timbre reproduction capability of our synthesizer implementation and Google Magentas implementation.
The MUSHRA structure is suitable because it features a reference audio source as well as multiple ‘conditions’ where some of the conditions are so called ‘anchors’ which are \emph{“[…] low-pass-filtered versions of the reference stimulus and were introduced to make the ratings of different assessors and labs more comparable.”}\cite{Schoeffler2018}. i.e. anchors are included to make the survey results more valid and valuable. Even though this is the ideal scenario, we decided to exclude the anchor conditions in our evaluation because, from our pilot-tests, we discovered that the anchor conditions confused participants and therefore would affect the results validity negatively. For conditions in our survey we are using the output sets generated from our synthesizer implementation and Google Magenta's implementation. We are creating a MUSHRA for each of the model output-sets to evaluate each models capability of timbre reproduction.
\subsubsection{Survey structure}\label{subsubsec:mushra_survey_structure}
The survey structure is constructed of 5 sections. Sections 1 through 4 is dedicated to one model respectively thus testing 4 different models (see appendix \ref{subsec:mushra_survey_content}). The 5th section is concerned with demographics of participants because we want to consider e.g. participants experience working with audio, to evaluate whether they are proficient with timbre recognition etc.
\begin{figure}[htp]
\centering
\includegraphics[width= 0.9\columnwidth]{SMC704 DDSP/Images/mushra_1anchor.png}
\caption{MUSHRA user interface}
\label{fig:mushra_ui}
\end{figure}
For each section (model), participants are presented with a reference sample (which is the raw input sample) as well as three conditions. The three conditions include:
\begin{enumerate}
\item Output audio from Magenta implementation
\item Output audio from Plugin Synth
\item Reference (input sample)
\end{enumerate}
For each condition, participants are asked to rate how accurate they think the timbre of the reference is recreated in the condition audio source. This input is translated through one vertical slider per condition which features a rating scale from "bad" transfer to "excellent" transfer of timbre and is represented by a value from 0-100 (see Figure \ref{fig:mushra_ui}).
By including a third reference as an additional condition, we aim to facilitate a way to validate participants survey-answers. This is a way to validate that participants understood the questions and actually listened to the auditory stimuli to evaluate the timbre transfer accuracy. For illustration purposes, the labels of each condition are visible in figure \ref{fig:mushra_ui} (underlined in red), but these labels will of course be hidden to participants completing the survey thus facilitating a ‘blind’ listening-test where participants have to rate each condition exclusively based on auditory stimuli. Then it becomes obvious if participants e.g. rate the reference condition as bad timbre transfer , then we are able to conclude that the validity of this particular participants answers are relatively low. Even though, we are only interested in the rating of our synthesizer implementation and Google Magenta's implementation, it is important to include the reference for the purpose of validating participants answers.
\subsubsection{Expected Results}\label{subsubsec:mushra_expectations}
In this subsection we describe our expected results of the Sound Accuracy Evaluation as well as possible reasons for those assumptions. Given that our synthesizer, including the pitch detection, is running in real-time, the timbre recreation might be perceived as less accurate in comparison to Google Magenta's implementation which is not running in real-time. Furthermore, the Google Magenta implementation features modification options of the input signal to make sure that the incoming signal is sounding as close to the training data as possible (to optimize the resulting output). Our implementation does not have this feature, so instead we experimented with input signals with different frequency content and found the optimal input frequency. Taking these factors into consideration, our expectations is that our implementation will not have as accurate timbre transfer as the Google Magenta implementation, but we expect it to receive a decent score for its timbre transfer ability.
\subsubsection{Results}\label{subsubsec:mushra_results}
We conducted the evaluation via the webMUSHRA-framework and received 11 answers from participants. Via our demographics section of the evaluation we know that there were 8 males, 2 females and 1 identifying as other. The participants age is varying from 22 to 57 with an average of age 30.5. Amongst the participants, are 7 working with audio at a hobby level and 2 are working with audio in relation to their job while 2 has no experience working with audio. Additionally, 9 participants report experience using a software-synthesizer, thus the majority of participants are aware of the sound reproduction capabilities of a software synthesizer.
The main takeaway from the MUSHRA evaluation is to compare the perceived timbre transfer capability of our plugins output in comparison to Google Magenta's implementation. In figure \ref{fig:mushra_resultsbar}, we illustrate this relationship of perceived timber transfer capabilities rated by the participants.
\begin{figure}[htp]
\centering
\includegraphics[width= 1\columnwidth]{SMC704 DDSP/Images/mushra_results_barchart.jpg}
\caption{MUSHRA timbre transfer capability ratings
\label{fig:mushra_resultsbar}}
\end{figure}
The illustration in figure \ref{fig:mushra_resultsbar} is showing the mean of all participant-ratings for each model evaluated. To emphasize the comparison of outputs, the blue bars represent the synthesizer output and the red bars represent the Magenta-implementation output. Furthermore, the bars are grouped according to which model (e.g. the violin) that were used to re-synthesize the reference sample.
From the illustration in figure \ref{fig:mushra_resultsbar} it is obvious that the Magenta-implementation received ratings with averages in the ‘good’ to ‘excellent’ (60-100) timbre transfer categories while our synthesizer received ratings with averages in the ‘fair’ (40-60) category.
The results are varying slightly across the different models. The Magenta implementation’s re-synthesis scored almost the same on average for the Violin and flute model (76.6 and 76.1) but the average of the Trumpet and Saxophone models are slightly higher with ratings of 79.2 and 80.5 respectively.
Looking at our synthesizers timbre transfer score, the Flute and Saxophone models scored 52.3 and 49.2 while the Trumpet and Violin scored slightly higher with ratings of 54.8 and 56.5. It is interesting that the model that is perceived as transferring the timbre most accurately is, for Magenta, the Saxophone model with a rating of 80.5 and for our synthesizer, the Violin model with a score of 56.5.
Since the Magenta implementation scored considerably higher across all models in this evaluation, this means that our synthesizer implementation suffers re-synthesis accuracy compared to the non-real-time web-based Magenta implementation.
Keep in mind, this evaluation is based on 10 answers and therefore might not be entirely representative.
By including the reference as a Condition, we are able to validate the answers of the evaluation. The average of the ratings for each model were 97.9 or 100, which we consider a valid test (see appendix \ref{subsec:mushra_survey_validation}).
\end{comment}
\subsection{Design Requirements Validation}\label{subsec:designrequirements_validation}
To validate if our synthesizer implementation met our design requirements we conducted two evaluations, which were presented in the previous sections.
Design requirement 1: \emph{‘Must work in real-time as a playable software instrument’} and 2: \emph{‘Must be implemented as a software plugin’} has been met as our software synthesizer plugin features models that are implemented to be playable in real-time (elaborated in \ref{subsubsec:realtime_implementation_of_the_models}). We confirmed that these requirements were met by running user-tests where participants had to install the VST3 plugin as well as play it as an instrument in real-time \ref{subsubsec:ux_results}. Design requirement 3: \emph{‘Must allow switching between Line and MIDI input’} and 4: \emph{‘Must implement at least 4 pre-trained models’} was met and addressed in the user-tests as well. Participants reported that switching between Line and MIDI input makes sense, and that they especially enjoy the Line-input functionality \ref{subsubsec:ux_results}. The four pre-trained models that we implemented in our synthesizer were violin, flute, saxophone and trumpet. The timbre transfer capabilities of these models in context of our implementation, were evaluated in section \ref{subsubsec:mushra_results}. Furthermore, we trained additional models but those were not included in this version of the synthesizer as stated in section \ref{subsubsec:our_models}. Lastly, the fifth design requirement \emph{‘Must include graphical UI components for intuitive manipulation of synth parameters’} was partially met with our implementation of the individual partials control GUI, however as reported in section \ref{subsubsec:ux_results} the participants found the general UI to be visually appealing and self-explanatory, but the graphical interface for controlling the harmonic distribution can be improved by making its functionality more obvious to the user. Additionally, implementation of the "modulation" section of our synthesizer would contribute to fulfilling this design requirement as participants reported a lack of controls to achieve a desired output sound \ref{subsubsec:ux_results}. The current state of our synthesizer sufficiently meets two of our design requirements while partially meeting the other three.
\section{Discussion}\label{sec:discussion}
\subsection{Synthesizer controls}
From the Likert evaluation (see figure \ref{fig:ux_likert_results}), we can conclude that users found the interface to be visually appealing and for the most part intuitive to understand, with the exception of the graphical control over the harmonic distribution. However, there is also an indication that the users desired more control over the sound and did not find themselves being able to reach the expression they were trying to achieve. We attribute this sentiment partially to some of the initially planned modules not being implemented in the distributed product. To address this, in the future we would like to fully implement the reverb and modulation modules, and support MIDI Polyphonic Expression (MPE) features.
\begin{comment}
\begin{itemize}
\item The reverb module and its user controls
\item The modulator and the ability to connect it to both the additive and the subtractive synthesizer
\item Controls that allow the user to shape the magnitude response of the filter used in the subtractive synthesizer
\item Additional explanation and a way to reset the harmonic distribution interface
\item Support for MIDI Polyphonic Expression (MPE) features such as Aftertouch
\end{itemize}
\end{comment}
\subsection{Real-time timbre transfer}
We found the quality of the timbre transfer in our real-time implementation below that of the demonstrations published by the Magenta team. Our converted models preserve some characteristics of the original ones, such as wind noises in the flute model, but do not accurately reproduce the timbre overall. We confirmed that on the level of a single frame, our models produce the same output as their original counterparts; will investigate and improve the quality in the future. Additionally, we would like to further investigate why we were unable to perform the timbre transfer with models that we trained both within the framework provided by Magenta, and within custom environments.
A recently released realtime reimplementation of DDSP in PyTorch\footnote{\url{https://github.com/acids-ircam/ddsp_pytorch}} provides a possibly more seamless way of interfacing with DDSP models in C++ that proved compatible with our plugin and JUCE. Extending that API to allow the user some control over the synthesis parameters seems a promising avenue to improve the sound quality of our timbre transfer.
\begin{comment}
so we propose the following potential reasons for the quality mismatch that we would like to
\begin{itemize}
\item Frame-based tracking of the input. Where the timbre transfer demo uses a deep convolutional neural network that operates on the entire input, our project makes use of the Aubio implementation of the YIN algorithm \cite{yin2002}, considering only up to 4096 samples at a time to track the input audio and convert it into f0 and loudness data. This leads to a loss of information and accuracy when it comes to pitch and note onset detection, and it would stand to reason that less accurate input data leads to less accurate output from the models.
\item The models were originally trained on receiving the loudness and pitch information for seconds of audio at a time, and generate the synthesizer controls using a recurrent neural network, meaning that future output depends on previous inputs to a degree. We suspect that a lot of that information is lost through our implementation decision to generate the audio independently from frame to frame. We plan to experiment with keeping internal buffers of previous input to remedy this, keeping an eye of the impact larger input sizes to the models might have on real-time performance.
\end{itemize}
\end{comment}
\subsection{Distribution as a VST3 plugin}
When it came to distributing our project to users, we encountered some difficulties in packaging the required libraries and model files together with the generated VST3 plugin. Some of the DAWs that users tested on, like Ableton or Reaper, did not recognize the plugin or experienced stability issues during its usage.
Although the core functionality could still be accessed via the standalone application generated by JUCE, the project was designed first and foremost as a plugin. Functionality like handling of external audio sources and wet/dry mixing was expected to be handled by the host DAW. Users who had to resort to the standalone when their DAW did not recognize or stably run the plugin reported those features as missing.
Thus, we would like to improve the distribution process in the future, ensuring that the project can be seamlessly installed as a plugin in multiple DAWs on Windows and macOS.
\section{Conclusion}\label{sec:conclusion}
In this paper, we presented an approach to integrate the DDSP library into a real-time plugin and standalone application using the JUCE framework. We succeeded in implementing a synthesizer playable based on pure user input. While we were generally able to use the output from pre-trained models to control the DDSP backend, further research is needed to match the sound quality of these real-time models to that of the offline timbre transfer examples provided by the DDSP authors.
|
2,869,038,155,638 | arxiv | \section{Introduction}
Sustainability is a core component in modern wireless communications, and for the upcoming sixth generation (6G) systems, an energy efficiency (EE) of up to 1 Terabit/Joule is anticipated \cite{white}. Such ultra-high EE is quite challenging with very large active antenna arrays using a large number (i.e. in the order of hundreds) of power-demanding radio-frequency (RF) chains for their operation.
\par On the other hand, intelligent reflective surfaces (IRSs) are envisioned to be an attractive solution for energy-efficient communications. IRSs are nearly-passive planar surfaces capable of tweaking the wireless environment by means of smart reflections of impinging signals \cite{jian2022reconfigurable}. Each surface consists of a large number of small unit cells (UCs), which can be digitally configured to introduce phase and/or amplitude manipulations on impinging electromagnetic waves. The UCs at the IRS are nearly passive components, and they do not require power-hungry RF chains to provide signal reflections.
\par In principle, the IRS is similar to a multi-antenna amplify-and-forward relay with two main differences. The first one is that IRSs can provide almost instant signal reflections without introducing large delays as it is the case with active relaying, and the second main difference is that IRSs are nearly passive devices that cannot provide active power amplifications, and are therefore highly energy-efficient. For a detailed comparison between relays and IRSs, we refer the reader to the works in \cite{huang2019reconfigurable, bjornson2019intelligent} and the references therein.
\par Recently, few works have demonstrated that hybrid relaying networks (HRNs) amalgamating both relays and IRSs can bring about large improvements in terms of achievable rates and/or total transmit powers \cite{abdullah2020hybrid, abdullah2020optimization, Obeed, yildirim2021hybrid}. In particular, the idea of HRNs was first reported in \cite{abdullah2020hybrid}, where it was demonstrated that a cooperative network comprising both an IRS and a single-antenna half-duplex (HD) decode-and-forward (DF) relay can achieve a large rate improvement compared to utilizing only an IRS (i.e. without a relay), given that the number of UCs and/or the transmit power are/is limited. Furthermore, the work in \cite{kang2021irs} investigated the location and deployment strategy of IRSs in HRNs. Finally, the work in \cite{doubleRIS} investigated the number of relays and transmission strategy for maximum rate performance in double-IRS assisted networks, where the signal is subject to reflections from two spatially separated IRSs.
\par In this work, we investigate the performance of hybrid and non-hybrid relaying schemes in terms of required transmit powers and EE performance. Unlike previous works on HRNs \cite{abdullah2020hybrid, abdullah2020optimization, Obeed, yildirim2021hybrid, kang2021irs, doubleRIS}, the required overhead to estimate the channel state information (CSI) is taken into account when the reflective beamforming design (RBD) at the IRS is carried out based on instantaneous CSI (iCSI) as well as statistical CSI (sCSI) models, and for both low- and high-mobility scenarios. For the EE, the power consumption model corresponding to both iCSI- and sCSI-based RBD is formulated, and the EE is evaluated for a wide range of targeted rate thresholds. \par The rest of this paper is organized as follows. In Section~\ref{system model}, we present the system model of different relaying schemes. The achievable rates and IRS optimization are tackled in Section~\ref{RBD}. Transmit powers and EE performance are investigated in Section~\ref{EE}. Numerical evaluations and discussions appear in Section \ref{results}. Concluding remarks are given in Section \ref{conclusions}.
\par \textit{Notations}: Matrices and vectors are represented by boldface uppercase and lowercase letters, respectively. The conjugate, transpose, and Hermitian transpose of a vector $\boldsymbol v$ are denoted by $\boldsymbol v^\ast$, $\boldsymbol v^T$, and $\boldsymbol v^H$, respectively. The ($i,j$)th entry of $\boldsymbol V$ is denoted by $[\boldsymbol V]_{i,j}$, while the $n$th entry of $\boldsymbol v$ is $[\boldsymbol v]_n$. The $N\times N$ identity matrix is $\boldsymbol I_N$, while $\boldsymbol 0_{N}$ and $\boldsymbol 1_N$ are vectors of length $N$ with entries of all $0$'s and $1$'s, respectively. The absolute, expected, and trace operators are expressed as $|\cdot|$, $\mathbb E \{\cdot\}$, and $\text{tr}(\cdot)$, respectively. Moreover, $\angle{(v)}$ denotes the phase of a complex number $v$. Finally, $\boldsymbol V = \text{diag}\{\boldsymbol v\}$ is a diagonal matrix whose diagonal are the elements of $\boldsymbol v$.
\section{System Model} \label{system model}
We consider a network where a source node ($\mathrm S$), aims to transmit data to a destination node ($\mathrm D$), with the help of either an HD-DF relay ($\mathrm R$), an IRS ($\mathrm I$), or both (see Fig. \ref{Fig1}). The source, destination, and relay are each equipped with a single isotropic radiating element, while the IRS has $M$ reflective UCs. A two-dimensional (2D) square array is adopted for the IRS such that $M = M_d^2$ with $M_d$ being the number of UCs per dimension. Spatial correlation at the IRS is taken into account through the correlation matrix $\boldsymbol R$, whose ($n,k$)th entry is \cite{bjornson2020rayleigh}:
\begin{equation} \label{R}
\small
\left[\boldsymbol R\right]_{n,k} = \mathrm{sinc} \left( \frac{2\left\|\boldsymbol u_{n} - \boldsymbol u_{k} \right\|}{\lambda}\right), \ \ \ \ \ \ \forall \{n,k\} \in \mathcal M,
\end{equation}where $\left\|\boldsymbol u_{n} - \boldsymbol u_{k} \right\|$ is the distance between the $n$th and $k$th UCs at the IRS, $\lambda$ is the carrier wavelength, and $\mathcal M = \{1, 2, \ldots, M\}$ is a set containing the indices of all UCs.
\par Moreover, direct links exist between all nodes except between the source and destination due to blockage, which justifies the deployment of the relay and/or IRS. Block fading is adopted such that the response of channels remains constant within the duration of transmitting one data frame, but changes independently from one frame to another. \par In the following, we formulate the expressions of received
signals and the corresponding signal-to-noise ratios (SNRs)
for the relay-assisted, IRS-assisted, and HRN cases.
\subsection{Relay-Assisted Scenario}
In this case, only the relay is utilized to facilitate the communication, which takes place over two phases.
\subsubsection{First-hop} During this phase, $\mathrm S$ transmits a block of data to $\mathrm R$,\footnote{The indices of different data frames and information symbols are dropped for the sake of simplicity, and without having any impact on the analysis or the results presented in this work.} and the received signal at the latter is given as
\begin{equation}
\small
y_{1\mathrm{R}} = \sqrt{P_1} h_{\mathrm{SR}}~ s + w_{1},
\end{equation}where the subscripts in $y_{1\mathrm R}$ indicate that this is the first phase of the transmission for the relay-assisted scenario, $P_1$ is the transmit power in Watts during the first phase, $s$ is the information symbol satisfying $\mathbb E\{\left|s\right|^2\} = 1$, and $w_1 \sim \mathcal N_{\mathbb C} (0, \sigma^2) $ is the additive white Gaussian noise (AWGN) at the relay. In addition, $h_{\mathrm{SR}}\in\mathbb C$ is the channel coefficient between the source and the relay expressed as $h_{\mathrm{SR}} = \sqrt{\rho_{\mathrm{SR}}} {g}_{\mathrm{SR}}$ with $\rho_{\mathrm{SR}}$ being the channel variance, while $g_{\mathrm{SR}} \sim \mathcal N_{\mathbb C} (0,1)$ accounts for the Rayleigh distributed small-scale flat-fading component. \\ The instantaneous received SNR at the relay is thus given as:
\begin{equation}
\small
\gamma_{1\mathrm R} = \frac{P_1}{\sigma^2} \left| h_{\mathrm {SR}}\right|^2.
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=1.75]{Fig1.eps}
\caption{The considered system in a scattering environment.}
\label{Fig1}
\end{figure}
\subsubsection{Second-hop} Here, $\mathrm R$ re-transmits the signal to $\mathrm D$ after performing decoding and re-encoding on $s$. Assuming a successful decoding at $\mathrm R$, the received signal at $\mathrm D$ during the second transmission phase for the relay-assisted scenario, denoted as $y_{2\mathrm R}$, is given as:
\begin{equation}
\small
y_{2\mathrm R} = \sqrt{P_2} h_{\mathrm{RD}}~s + w_2.
\end{equation}where $P_2$ is the transmit power from $\mathrm R$, $h_{\mathrm{ RD}} = \sqrt{\rho_{\mathrm{RD}}} g_{\mathrm {RD}}$ represents the channel response with $\rho_{\mathrm{RD}}$ being the variance, while $g_{\mathrm {RD}}\sim\mathcal N_{\mathbb C} (0,1)$ accounts for the Rayleigh distributed flat-fading component, and $w_2 \sim \mathcal N_{\mathbb C} (0,\sigma^2)$ is the AWGN at~$\mathrm D$. Therefore, the instantaneous received SNR at $\mathrm D$ is:
\begin{equation}
\small
\gamma_{2\mathrm R} = \frac{P_2}{\sigma^2} \left| h_{\mathrm {RD}}\right|^2.
\end{equation}We next shift our attention to the IRS-assisted case.
\subsection{IRS-Assisted Scenario}
In this scenario, only the IRS assists the communication between $\mathrm S$ and $\mathrm D$. A single-phase transmission is sufficient, and the received signal at $\mathrm D$ is:
\begin{equation}
y_{\mathrm {IRS}} = \sqrt{P} (\boldsymbol h_{\mathrm{ID}}^T\boldsymbol \Theta \boldsymbol h_{\mathrm {SI}})~s + w,
\end{equation}
where $P$ is the transmit power from the source for the IRS-assisted case, $\boldsymbol h_{\mathrm{ID}} \in \mathbb C^{M\times 1}$ and $\boldsymbol h_{\mathrm{SI}}\in \mathbb C^{M\times 1}$ are the channel coefficients between $\mathrm I\rightarrow \mathrm D$ and $\mathrm S\rightarrow \mathrm I$. For $x\in\{\mathrm {ID,~ SI}\}$, we have $\boldsymbol h_x = \sqrt{\rho_x} \boldsymbol R ^{\frac{1}{2}}{\boldsymbol g}_x$ with $\rho_x$ being the channel variance between $\{\mathrm S, \mathrm D\}$ and each UC at the IRS, $\boldsymbol R$ is the correlation matrix given in (\ref{R}), and ${\boldsymbol g}_x \sim \mathcal N_{\mathbb C} (\boldsymbol 0_M, \boldsymbol I_M)$ is a vector with independent and identically distributed (i.i.d) Rayleigh fading components. Also, $w\sim\mathcal N_{\mathbb C}(0,\sigma^2)$ is the AWGN at $\mathrm D$, while $\boldsymbol \Theta \in \mathbb C^{M\times M}$ controls the response of each UC at the IRS for the IRS-assisted transmission, and it can be expressed as:
\begin{equation}
\small
\boldsymbol \Theta = \text{diag}\left\{\left[\mu_{1} e^{\jmath \left[{\boldsymbol {\theta}}\right]_1}, \mu_{2} e^{\jmath \left[ {\boldsymbol {\theta}}\right]_2}, \cdots, \mu_{M} e^{\jmath \left[{\boldsymbol {\theta}}\right]_M}\right]\right\},
\end{equation}where $\mu_{m} \in [0,1]$ and $[\boldsymbol \theta]_m \in [0, 2\pi]$ are the reflection amplitude and phase of $m$th ($m\in\mathcal M$) UC at the IRS.\\ The instantaneous received SNR at the destination for IRS-assisted transmission is given as:
\begin{equation} \label {gamma_IRS}
\small
\gamma_{\mathrm{IRS}} = \frac{P}{\sigma^2} \left|\boldsymbol h_{\mathrm{ID}}^T\boldsymbol \Theta \boldsymbol h_{\mathrm {SI}}\right|^2.
\end{equation}Next, we introduce the system model for the HRN.
\subsection{HRN scenario}
In this case, both the HD relay and IRS contribute to the communication between the source and destination \cite{abdullah2020hybrid}, and the data transmission requires two phases.
\subsubsection{First-hop}
During this phase, the source transmits its signal to the relay through the direct link and the link via the IRS. The received signal at the relay is given as:
{\small \begin{align}
y_{1\mathrm H} = \sqrt{P_1} \left(h_{\mathrm{SR}} + \boldsymbol h_{\mathrm{I}\mathrm{R}}^T \boldsymbol \Theta_1 \boldsymbol h_{\mathrm S\mathrm I}\right) s + w_{1},
\end{align}}where the subscripts in $y_{1\mathrm H}$ reflects the first phase of the HRN scenario, $\boldsymbol h_{\mathrm{IR}} \in \mathbb C^{M\times 1}$ is a vector containing the channel coefficients between $\mathrm I\rightarrow \mathrm R$ given as $\boldsymbol h_{\mathrm{IR}} = \sqrt{\rho_{\mathrm{IR}}} \boldsymbol R ^{\frac{1}{2}}{\boldsymbol g}_{\mathrm{IR}}$ with $\rho_{\mathrm{IR}}$ being the channel variance, and ${\boldsymbol g}_{\mathrm{IR}} \sim \mathcal N_{\mathbb C} (\boldsymbol 0_M, \boldsymbol I_M)$ is the Rayleigh distributed flat-fading channel vector with i.i.d entries. The diagonal matrix $\boldsymbol \Theta_1 \in \mathbb C^{M\times M}$ controls the response of each UC at the IRS during the first phase, such that $\left[\boldsymbol \Theta_1\right]_{m,m} = \mu_{1,m} e^{\jmath [\boldsymbol \theta_1]_m}$, where $\mu_{1,m} \in [0,1]$ and $[\boldsymbol \theta_1]_m \in [0, 2\pi]$ are the reflection amplitude and phase of $m$th ($m\in\mathcal M$) UC at the IRS.\\ The instantaneous received SNR at the relay is given as:
\begin{equation}
\small
\gamma_{1\mathrm H} = \frac{P_1}{\sigma^2} \left|h_{\mathrm{SR}} + \boldsymbol h_{\mathrm{I}\mathrm{R}}^T \boldsymbol \Theta_1 \boldsymbol h_{\mathrm S\mathrm I}\right|^2.
\end{equation}
\subsubsection{Second-hop} During this phase, the relay broadcasts the signal to the destination through the direct link and reflections from the IRS. The received signal at the destination is:
{\small \begin{align}
y_{2\mathrm H} = \sqrt{P_2} \left(h_{\mathrm{RD}} + \boldsymbol h_{\mathrm{I}\mathrm{D}}^T \boldsymbol \Theta_2 \boldsymbol h_{\mathrm R\mathrm I}\right) s + w_{2},
\end{align}}where $\boldsymbol h_{\mathrm{RI}} = \sqrt{\rho_{\mathrm{RI}}} \boldsymbol R ^{\frac{1}{2}}{\boldsymbol g}_{\mathrm{RI}} \in \mathbb C^{M\times 1}$ is a vector containing the channel coefficients between $\mathrm R\rightarrow \mathrm I$ with $\rho_{\mathrm{RI}}$ being the channel variance, and ${\boldsymbol g}_{\mathrm{RI}} \sim \mathcal N_{\mathbb C} (\boldsymbol 0_M, \boldsymbol I_M)$ is the Rayleigh distributed flat-fading channel vector with i.i.d entries. Also, $\boldsymbol \Theta_2 \in \mathbb C^{M\times M}$ is the IRS reflection matrix during the second transmission phase, such that $\left[\boldsymbol \Theta_2\right]_{m,m} = \mu_{2,m} e^{\jmath [\boldsymbol \theta_2]_m}$, where $\mu_{2,m} \in [0,1]$ and $[\boldsymbol \theta_2]_m \in [0, 2\pi]$ are the reflection amplitude and phase of $m$th ($m\in\mathcal M$) UC at the IRS during the second transmission phase.\\ The instantaneous received SNR at the destination is:
\begin{equation}
\small
\gamma_{2\mathrm H} = \frac{P_2}{\sigma^2} \left|h_{\mathrm{RD}} + \boldsymbol h_{\mathrm{I}\mathrm{D}}^T \boldsymbol \Theta_2 \boldsymbol h_{\mathrm R\mathrm I}\right|^2.
\end{equation}In the next section, we formulate the achievable rate expressions under both iCSI- and sCSI-based RBD of the IRS.
\section{Achievable Rates and RBD} \label{RBD}
In this section, we formulate the achievable rate expressions for the three different relaying schemes. We take into account the amount of training required to estimate the channels under both iCSI- and sCSI-based RBD.
\par We denote the length of the coherence interval (in samples) by $\tau_c$, while $L\ge 1$ denotes the number of samples (i.e., pilot signals) utilized to estimate the channel response of a single link.\footnote{Here, we focus on the impact of required training for channel estimation on the rate and hence, the EE performance, while a perfect estimation accuracy with $L$ pilot signals is assumed throughout this work. Nonetheless, the impact of estimation errors on HRNs was investigated in our previous work in \cite{abdullah2020hybrid}.} A frame-based transmission is assumed, where the frame length is aligned with the coherence interval. Furthermore, each frame contains $\tau_p = LT$ pilot symbols with $T$ being the number of channel links to be estimated, followed by $\tau = \tau_c-\tau_p-\tau_g$ data symbols.\footnote{The parameter $\tau_g$ reflects a time gap between pilot and data symbols, which is required only for networks utilizing the IRS. During this period, the receiving node performs the RBD of the IRS, based on the estimated channels, and feedback the optimized phase-shifts to the IRS controller.}
\subsection{Relay-Assisted Scenario}
In this case, and since $\{\mathrm {S, R, D}\}$ are each equipped with a single antenna, only $L$ samples are required to estimate the channel per hop. During the first transmission phase, $\mathrm S$ transmits $L$ pilots to $\mathrm R$ at the start of each coherence interval, and the latter utilizes the received samples to estimate the channel between its antenna and the source (i.e. estimates $h_{\mathrm{SR}}$) and recover the original signal. Similarly, in the second phase, $\mathrm R$ transmits $L$ pilots to $\mathrm D$ for the latter to estimate $h_{\mathrm{RD}}$ and perform the decoding operation. \\ Therefore, the achievable rate for the relay-assisted network with an HD-DF relay is given as:
\begin{equation} \label{rate_r}
\small
\mathcal R_{\mathrm R} = \eta_{\mathrm R} \min \Big\{ \log_2\left(1 + \gamma_{1\mathrm R}\right), \log_2\left(1 + \gamma_{2\mathrm R}\right) \Big\},
\end{equation}where $\eta_{\mathrm R} = \frac{\tau_c - L}{2\tau_c}$ and the division over two is the result of the HD transmission.
\subsection{IRS-Assisted Scenario}
In this case, the RBD at the IRS can be carried out based on either iCSI or sCSI models. In the following, we tackle each case separately.
\subsubsection{iCSI-based RBD}
When the phase shifts at the IRS are reconfigured at each coherence interval, each of the $M$ sub-links of the cascaded channel needs to be estimated (i.e. $[\boldsymbol h_{\mathrm {ID}}]_1 [\boldsymbol h_{\mathrm {SI}}]_1, \cdots, [\boldsymbol h_{\mathrm {ID}}]_M [\boldsymbol h_{\mathrm {SI}}]_M$). As such, at the start of each coherence interval, $LM$ pilots need to be transmitted from $\mathrm S$ to $\mathrm D$ through the IRS to estimate all $M$ channel links at $\mathrm D$, which will then inform the IRS control unit of the optimized phase-shift values through a dedicated control channel.\footnote{Note that it is possible to reduce the amount of training even under iCSI-based RBD by equipping the IRS with estimation capabilites as in \cite{taha2021enabling}. In addition, strong spatial correlation or channel sparsity can also result in reduced pilot signaling. However, here we focus on a general scenario that does not rely on any channel conditions and/or IRS capabilities.} \\ In such a case, the optimal phase-shift of $m$th UC is $[\bar{\boldsymbol \theta}^\star]_{m} = - \angle\left([\boldsymbol h_{\mathrm{ID}}]_m [\boldsymbol h_{\mathrm{SI} }]_m\right)$ \cite{abdullah2020hybrid}, where the bar notation is used to indicate that the RBD is carried out based on iCSI. Assuming that all UCs have the same reflection amplitude of $\mu$, the maximum received SNR is:
\begin{equation}
\small
\bar{\gamma}_{\mathrm {IRS}} = \frac{P}{\sigma^2}\Big(\mu\sum_{m\in \mathcal M}\Big|[\boldsymbol h_{\mathrm{ID}}]_m [\boldsymbol h_{\mathrm{SI} }]_m\Big|\Big)^2.
\end{equation}
\subsubsection{sCSI-based RBD}
To reduce the amount of training, one can optimize the response of all UCs based on the statistical CSI model, which is independent of the instantaneous channels' realizations and varies very slowly in practice \cite{demir2022channel}. In this case, the RBD is carried out to maximize the ergodic SNR $\mathbb{E} \left\{\gamma_{\mathrm{IRS}}\right\}$ given in (\ref{gamma_IRS}). Assuming that each UC has a reflection amplitude of $\mu$, an optimal configuration of the IRS can be obtained by setting $\hat{\boldsymbol \Theta}^{\star} = \mu{\boldsymbol I_M}$ (see Appendix A for details), where the hat notation indicates that an sCSI model is adopted for the RBD. Then, the overall cascaded channel $(\boldsymbol h_{\mathrm {ID}}^T \hat{\boldsymbol \Theta}^\star \boldsymbol h_{\mathrm{SI}} = \mu \boldsymbol h_{\mathrm {ID}}^T \boldsymbol h_{\mathrm{SI}})$ can be treated as a single link, and hence, only $L$ pilot samples are required for channel estimation (CE) at the start of each coherence interval. \\ Accordingly, the received SNR under sCSI-based RBD is:
\begin{equation}
\small
\hat{\gamma}_{\mathrm{IRS}} = \frac{P}{\sigma^2} \Big| \mu\boldsymbol h_{\mathrm {ID}}^T \boldsymbol h_{\mathrm{SI}}\Big|^2.
\end{equation}\par It follows that the achievable rate for the IRS-assisted scenario can be expressed as:
\begin{equation}
\small
{\mathcal R}_{\mathrm{IRS}} = \tilde{\eta}_{\mathrm{IRS}} \log_2\Big(1+\tilde{\gamma}_{\mathrm {IRS}}\Big),
\end{equation}where $\tilde{\gamma}_{\mathrm{IRS}}\in\{\bar{\gamma}_{\mathrm{IRS}}, \hat{\gamma}_{\mathrm{IRS}}\}$ and $\tilde{\eta}_{\mathrm{IRS}}\in\{\bar{\eta}_{\mathrm{IRS}}, \hat{\eta}_{\mathrm{IRS}}\}$, depending on whether the RBD is carried out based on iCSI or sCSI. Regarding the parameter $\tilde{\eta}_{\mathrm{IRS}}$, we have $\bar{\eta}_{\mathrm{IRS}} = \frac{\tau_c - LM - \tau_g}{\tau_c}$, while $\hat{\eta}_{\mathrm{IRS}} = \frac{\tau_c - L}{\tau_c}$.
\subsection{HRN Scenario}
For the HRN, one can also adopt either the iCSI or the sCSI based RBD. In each case, the CE is performed at $\mathrm R$ and $\mathrm D$, during the first and the second hops, respectively.
\subsubsection{iCSI-based RBD}
In this case, and for each of the two transmission phases, the $M$ links through the IRS as well as the direct link need to be estimated. Therefore, $LM+L$ pilot samples are required at the start of each coherence interval.
\par The phase-shifts are adjusted to maximize the instantaneous SNRs at $\mathrm R$ and $\mathrm D$, during the first and second transmission phases, respectively. The optimal phase response of the $m$th UC ($m\in\mathcal M$) during the first phase is $[\bar{\boldsymbol \theta}^\star_1]_{m} = \angle({h_\mathrm{SR}}) - \angle\left([\boldsymbol h_{\mathrm{IR}}]_m [\boldsymbol h_{\mathrm{SI} }]_m\right)$, while during the second phase of transmission we have $[\bar{\boldsymbol \theta}^\star_2]_{m} = \angle({h_\mathrm{RD}}) - \angle\left([\boldsymbol h_{\mathrm{ID}}]_m [\boldsymbol h_{\mathrm{RI} }]_m\right)$. Assuming a fixed reflection amplitude of $\mu$ at each UC, the corresponding SNRs at $\mathrm R$ and $\mathrm D$ are expressed as follows \cite{abdullah2020hybrid}
\begin{subequations}
\begin{equation}
\small
\bar{\gamma}_{1\mathrm H} = \frac{P_1}{\sigma^2} \Big( \left|h_{\mathrm SR}\right| + \mu \sum_{m\in \mathcal M}\left|\left[\boldsymbol h_{\mathrm {IR}}\right]_m \left[\boldsymbol h_{\mathrm {SI}}\right]_m \right| \Big)^2,
\end{equation}
\begin{equation}
\small
\bar{\gamma}_{2\mathrm H} = \frac{P_2}{\sigma^2} \Big( \left|h_{\mathrm RD}\right| + \mu \sum_{m\in \mathcal M}\left|\left[\boldsymbol h_{\mathrm {ID}}\right]_m \left[\boldsymbol h_{\mathrm {RI}}\right]_m \right| \Big)^2.
\end{equation}
\end{subequations}
\subsubsection{sCSI-based RBD} Here, the phase optimization is carried out to maximize the ergodic SNRs, and the optimal RBD during both transmission phases can be obtained as $\hat{\boldsymbol \Theta}_i = \mu \boldsymbol I_M~(i\in\{1,2\})$ (see Appendix B for details). The overall effective channel between $\mathrm S$ and $\mathrm R$ $\small \left(h_{\mathrm{SR}} + \mu \boldsymbol h_{\mathrm{I}\mathrm{R}}^T \boldsymbol h_{\mathrm S\mathrm I}\right)$ is treated as a single channel link, and the same applies for the second transmission phase between $\mathrm R$ and $\mathrm D$. Therefore, one only needs $L$ pilot samples per transmission phase to estimate the overall channel link at the start of each coherence interval.
\\ The corresponding SNRs under sCSI-based RBD are:
\begin{subequations}
\begin{equation}
\small
\hat{\gamma}_{1\mathrm H} = \frac{P_1}{\sigma^2} \Big|h_{\mathrm SR} + \mu \boldsymbol h_{\mathrm{I}\mathrm{R}}^T \boldsymbol h_{\mathrm S\mathrm I} \Big|^2,
\end{equation}
\begin{equation}
\small
\hat{\gamma}_{2\mathrm H} = \frac{P_2}{\sigma^2} \Big|h_{\mathrm RD} + \mu \boldsymbol h_{\mathrm{I}\mathrm{D}}^T \boldsymbol h_{\mathrm R\mathrm I} \Big|^2.
\end{equation}
\end{subequations}
\par Therefore, the achievable rate of the HRN with an HD-DF relay can be given as:
\begin{equation} \label{rate_H}
\small
{\mathcal R}_{\mathrm H} = \tilde{\eta}_{\mathrm H} \min \Big\{ \log_2\left(1 + \tilde{\gamma}_{1\mathrm H}\right), \log_2\left(1 + \tilde{\gamma}_{2\mathrm H}\right) \Big\},
\end{equation}where $\tilde{\gamma}_{i\mathrm H}\in\{\bar{\gamma}_{i\mathrm H}, \hat{\gamma}_{i\mathrm H}\}$ ($i\in\{1,2\}$) and $\tilde{\eta}_{\mathrm H}\in\{\bar{\eta}_{\mathrm H}, \hat{\eta}_{\mathrm H}\}$, such that $\bar{\eta}_{\mathrm H} = \frac{\tau_c - (LM+L)-\tau_g}{2\tau_c}$ and $\hat{\eta}_{\mathrm H} = \frac{\tau_c - L}{2\tau_c}$.
\section{Transmit Power Levels and EE Performance} \label{EE}
Here, we study the EE performance of the different relaying schemes. First, we obtain the minimum required transmit powers to achieve a given data rate threshold of $R_{{th}}$ at the destination. Then, the EE is evaluated based on the required power levels. In general, the EE can be defined as follows \cite{bjornson2019intelligent}:
\begin{equation}
\small
\mathrm{EE} = \frac{R_{{th}}} {{P}_{\text{total}}/B},
\end{equation}where $B$ is the bandwidth and $P_{\text{total}}$ is the total power consumption of the considered communication network.
\subsection{Relay-Assisted Scenario}
Let us define $\beta_{\mathrm{SR}} = |h_{\mathrm{SR}}|^2$ and $\beta_{\mathrm{RD}} = |h_{\mathrm{RD}}|^2$. Then, under a transmit power constraint of ${\small P = \frac{P_1 + P_2}{2}}$, it follows that the maximum achievable rate, which is obtained when $\gamma_{1\mathrm R} = \gamma_{2\mathrm R}$ (see Eq.(\ref{rate_r})), is ${\small \mathcal R_{\mathrm R}^\star = \eta_{\mathrm R}\log_2\left(1 + \frac{2P\beta_{\mathrm{SR}} \beta_{\mathrm{RD}} }{(\beta_{\mathrm{SR}} + \beta_{\mathrm{RD}})\sigma^2}\right)}$. Therefore, the required transmit power for the relay-assisted scenario to achieve a rate of $R_{{th}}$ is:
\begin{equation}
\small
P_{\mathrm R} = \Big(2^{\frac{R_{{th}}}{\eta_{\mathrm R}}}-1\Big) \frac{\left(\beta_{\mathrm{SR}} + \beta_{\mathrm{RD}}\right)\sigma^2 }{2\beta_{\mathrm{SR}} \beta_{\mathrm{RD}}},
\end{equation}and the total power consumption for the relay-assisted case is:
\begin{equation} \label{P_R}
\small
P_{\text{total}}^{\mathrm R} = \frac{P_{\mathrm R}}{\zeta} + \frac{1}{2} \overline{p}_{\mathrm S} + \frac{1}{2} \overline{p}_{\mathrm D} + \overline{p}_{\mathrm R},
\end{equation}where $\zeta \in (0,1]$ is the power amplifier efficiency, while $\overline{p}_{\mathrm S},~ \overline{p}_{\mathrm R}$, and $\overline{p}_{\mathrm D}$ are the hardware-dissipated power at the source, relay, and destination, respectively. The division over two of $\{\overline{p}_{\mathrm S}, \overline{p}_{\mathrm D}\}$ in (\ref{P_R}) is due to the fact that the source and destination are only active for half of the transmission time.
\subsection{IRS-Assisted Scenario} Let the channel gains under iCSI- and sCSI-based RBD for the IRS case be $\small \bar{\beta}_{\mathrm{IRS}}=\left(\mu\sum_{m\in \mathcal M}\Big|[\boldsymbol h_{\mathrm{ID}}]_m [\boldsymbol h_{\mathrm{SI} }]_m\Big|\right)^2$ and $\small \hat{\beta}_{\mathrm{IRS}} = \left|\mu \boldsymbol h_{\mathrm{ID}}^T \boldsymbol h_{\mathrm{SI}}\right|^2$, respectively. Then, the required transmit power to achieve a rate of $R_{{th}}$ is:
\begin{equation}
\small
{P}_{\mathrm{IRS}} = \left(2^{\frac{R_{{th}}}{\tilde{\eta}_{\mathrm{IRS}}}}-1\right) \frac{\sigma^2}{\tilde{\beta}_{\mathrm{IRS}}}
\end{equation}with $\tilde{\beta}_{\mathrm{IRS}} \in \{\bar{\beta}_{\mathrm{IRS}}, \hat{\beta}_{\mathrm{IRS}}\}$, depending on the RBD criterion.
\par The total power consumption of the IRS case is given as:
\begin{equation}
\small
P_{\text{total}}^{\mathrm{IRS}} = \frac{P_{\mathrm{IRS}}}{\zeta} + \overline{p}_{\mathrm S} + \overline{p}_{\mathrm D} + M\left(\overline p_{\text{st}} + \overline{p}_{\text{dyn}}\right)
\end{equation}with $\overline p_{\text{st}}$ and $\overline{p}_{\text{dyn}}$ being the static and dynamic power dissipation at the IRS, respectively. In particular, $\overline p_{\text{st}}$ is the static power that the IRS consumes just for being connected to an energy source, while $\overline p_{\text{dyn}}$ is the power consumed due to the reconfiguration of UCs \cite{report}. Note that when sCSI-based RBD is carried out, $\overline p_{\text{dyn}}$ is equal to zero since the UCs are not reconfigured at each coherence interval.
\subsection{HRN Scenario}
We define ${\small \bar{\beta}_{1\mathrm H} = \left( \left|h_{\mathrm SR}\right| + \mu \sum_{m\in \mathcal M}\left|\left[\boldsymbol h_{\mathrm {IR}}\right]_m \left[\boldsymbol h_{\mathrm {SI}}\right]_m \right| \right)^2}$ and ${\small \bar \beta_{2\mathrm H} = \left( \left|h_{\mathrm RD}\right| + \mu \sum_{m\mathcal M}\left|\left[\boldsymbol h_{\mathrm {ID}}\right]_m \left[\boldsymbol h_{\mathrm {RI}}\right]_m \right| \right)^2}$ as the effective channel gains during the first and second transmission phases for the HRN with iCSI-based RBD. Similarly, under sCSI-based RBD, we define ${\small \hat \beta_{1\mathrm H} = \left|h_{\mathrm SR} + \mu \boldsymbol h_{\mathrm {IR}}^T \boldsymbol h_{\mathrm {SI}} \right|^2}$ and ${\small \hat \beta_{2\mathrm H} = \left|h_{\mathrm RD} + \mu \boldsymbol h_{\mathrm {ID}}^T \boldsymbol h_{\mathrm {RI}} \right|^2}$. Then, to achieve optimal received SNRs, one should optimize the transmit powers such that $\tilde \gamma_{1\mathrm H} = \tilde \gamma_{2\mathrm H}$ (see Eq. (\ref{rate_H})). Therefore, under a transmit power constraint of $P = \frac{P_1 + P_2}{2}$, the maximum achievable rate with optimal transmit powers for the HRN is ${\small \mathcal R_{\mathrm H}^\star = \tilde \eta_{\mathrm H} \log_2\left(1 + \frac{2P\tilde\beta_{1\mathrm H}\tilde\beta_{2\mathrm H}}{(\tilde\beta_{1\mathrm H} + \tilde\beta_{2\mathrm H})\sigma^2}\right)}$, where $\tilde \beta_{i\mathrm H} \in\{\bar \beta_{i\mathrm H}, \hat \beta_{i\mathrm H}\}$ and $i\in\{1, 2\}$.
\\ To achieve a rate of $R_{{th}}$, the required transmit power is:
\begin{equation}
\small
P_{\mathrm H} = \left( 2^{ \frac{R_{{th}}}{\tilde \eta_{\mathrm H}} } -1\right) \frac{(\tilde \beta_{1\mathrm H} + \tilde \beta_{2\mathrm H})\sigma^2}{2\tilde \beta_{1\mathrm H}\tilde \beta_{2\mathrm H}},
\end{equation}and the total power consumed in such a case is given as:
\begin{equation}
\small
P_{\text{total}}^{\mathrm H} = \frac{P_{\mathrm H}}{\zeta} + \frac{1}{2} \overline{p}_{\mathrm S} + \frac{1}{2} \overline{p}_{\mathrm D} + \overline{p}_{\mathrm R} + M\left(\overline p_{\text{st}} + \overline{p}_{\text{dyn}}\right).
\end{equation}
\begin{figure*}[t]
\centering
\subfigure[\footnotesize $\tau_c = 10^4$ samples, ${R}_{th} = 3$ bits/s/Hz.]{%
\label{Fig2a}%
\includegraphics[width=.311\linewidth]{Fig2a.eps}
}\hspace{0.31cm}
\subfigure[\footnotesize $\tau_c = 10^3$ samples, ${R}_{th} = 3$ bits/s/Hz.]{%
\label{Fig2b}%
\includegraphics[width=.311\linewidth]{Fig2b.eps}
}\hspace{0.31cm}
\subfigure[\footnotesize $\tau_c = 10^4$ samples, $M = 144$.]{%
\label{Fig2c}%
\includegraphics[width=.311\linewidth]{Fig2c.eps}
}%
\caption{Performance comparison among different relaying schemes. The legend in (a) applies to all figures.}\label{fig:exp1}
\end{figure*}
\section{Results and Discussions} \label{results}
We start by introducing the locations of different communication nodes. In particular, a three-dimensional network setup was considered, with the $xyz-$coordinates of the source, relay, and destination being fixed at $(0,~0,~0), ~(100,~0,~0)$, and $(200,~0,~0)$, respectively, all in meters. Regarding the location of the IRS,\footnote{We highlight that for IRS-assisted scenarios, higher channel gains are obtained when the IRS is closest to the source or destination as in such cases the double path-loss is minimal. In contrast, when dealing with HRNs, the IRS should ideally be in a close proximity to the relay\cite{abdullah2020hybrid, abdullah2020optimization, kang2021irs}, and it is well known that under identical channel characteristics of the two sub-links, the relay provides the highest performance enhancement when located in the middle between the two end nodes of the network.} it was located near the relay at $(100,~2,~8)$~meters when dealing with an HRN, while for the IRS-assisted system, we evaluate the performance under two different scenarios. \textbf{Scenario~1}: The IRS is located between the two end nodes at $(100,~2,~8)$~meters similar to the HRN case, and \textbf{Scenario~2}: The IRS is located near the source at $(0,~2,~8)$~meters.
\par The channel variance between any two nodes $i$ and $j$ was modelled as $\rho_{ij}~[\mathrm{dB}] = 10\log_{10} (\frac{d_{ij}}{d_0})^{-\alpha} - 20$, where $d_{ij}$ is the distance between the two nodes, $d_0 = 1~$m is the reference distance, and $\alpha$ is the path-loss exponent. In particular, a path-loss exponent of $3$ was set for all links that involve the IRS, while a path-loss exponent of $3.7$ was selected for channel links between the relay and both the source and destination.\footnote{The justification of different path-losses is that IRSs can be mounted on the facades of tall buildings, and thereby experiencing a better link quality than relays who can be cooperative users in a dense urban network.} In addition, the carrier frequency is $1.9$~GHz \cite{demir2022channel}, $\sigma^2 = -107~\mathrm{dBm}$, $L = 1$, $\tau_g = M$, $\mu = 0.9$, $B = 10$~MHz, $\zeta = 0.5$, $\overline{p}_{\mathrm S} = \overline{p}_{\mathrm D} = \overline{p}_{\mathrm R} = 100$~mW \cite{bjornson2019intelligent}, $\overline{p}_{\text{dyn}} = 5$~mW \cite{bjornson2019intelligent}, and $\overline{p}_{\text{st}}=1$~mW. Finally, the IRS element spacing (i.e. the distance between two adjacent UCs located on the same row/column of the IRS surface) is $\lambda/8$.
\par Fig.~\ref{Fig2a} illustrates the required transmit powers of different relaying schemes to achieve a rate threshold of $3$ bits/s/Hz with $\tau_c = 10^4$ samples, which is a typical value for low-mobility scenarios \cite{demir2022channel}. The results show that an HRN with iCSI-based RBD can achieve superior power savings compared to all other schemes. As the number of UCs at the IRS increases, the power savings of the HRN become larger compared to the relay-assisted case, while the opposite holds when it comes to the comparison with the IRS-assisted cases. This is in line with the findings of \cite{abdullah2020hybrid} which stated that an IRS with a very large number of UCs outperforms an HRN with both an HD relay and an IRS. Moreover, for an IRS-assisted network, if the IRS is located within a close proximity of the source, higher power savings can be achieved. This is due to the fact that the double path-loss of an IRS-assisted network worsen as the location of the IRS moves toward the middle point between the two transceiving end nodes.
\par Fig. \ref{Fig2b} presents a comparison of the transmit powers under a fast-changing environment with $\tau_c = 1000$ samples. The size of the IRS plays a crucial role in such scenarios if the RBD is carried out based on the iCSI. In particular, the achievable rate of an IRS-assisted network could suffer from a rate penalty of $\bar{\eta}_{\text{IRS}}\approx 0.5$ for $M = 256$, which means that about $50\%$ of the transmission time is allocated for the CE and RBD phase. Such a challenge becomes even worse for the HRN with an HD relay, as for the same number of $256$ UCs, the parameter $\bar{\eta}_{\text{H}}\approx 0.25$ means that about $75\%$ of the achievable rate is wasted as a result of the CE with RBD phase and also the HD operation mode. Therefore, and as shown in Fig. \ref{Fig2b}, the HRN with iCSI-based RBD is the most affected when dealing with large IRSs. On the contrary, the sCSI-based RBD cases show improved performance as the number of UCs increases. This is due to the fact that the amount of pilot samples required do not increase with the number of UCs when sCSI is adopted to carry out the IRS-phase configuration.
\par Finally, Fig. \ref{Fig2c} demonstrates the EE performance. We observe that when the targeted rate is low, the relay-assisted case is by far the most efficient choice. In contrast, for high targeted rates, the HRN is the most energy-efficient system if the IRS was located near the relay, while an optimally placed IRS shows higher efficiency compared to the HRN. Interestingly, an IRS with sCSI-based RBD achieves the highest EE at medium targeted rate thresholds (between $5.3$ and $7.5$ bits/s/Hz), which demonstrates that although the sCSI-based RBD requires higher transmit powers compared to iCSI-based RBD (as shown in Fig.~\ref{Fig2a} and Fig.~\ref{Fig2b}), it can be more energy efficient due to a lower overall power consumption.
\par From all the above, one can identify the scenarios where HRNs can be efficiently utilized. For example, if the transmit power of a communication device is limited (such as a mobile user or even an IoT device), then HRNs can help achieving higher power savings compared to non-hybrid schemes, while ensuring a targeted rate threshold as shown in Fig. \ref{Fig2a}. Also, when both the transmitter and receiver are far away from the IRS, then incorporating a cooperative relay device that is close to the IRS can lead to better EE when the targeted rate is high as shown in Fig.~\ref{Fig2c}.
\section{Concluding Remarks} \label{conclusions}
We thoroughly investigated the power requirements and EE performance of hybrid and non-hybrid relaying networks under both iCSI- and sCSI-based RBD models. We highlighted the role of various parameters on the power and EE performance, such as the RBD models, effects of high mobility, the number of UCs, the targeted rate, as well as the IRS placement.
\section*{Acknowledgment}
This work was supported by the Luxembourg National Research Fund (FNR) through the CORE Project under Grant RISOTTI C20/IS/14773976.
\section*{Appendix A}
The ergodic SNR in (\ref{gamma_IRS}) with correlated Rayleigh fading is:
{\small \begin{eqnarray}
\mathbb E\{\gamma_{\mathrm{IRS}}\} & = & \frac{P}{\sigma^2} \mathbb E \Big\{\boldsymbol h_{\mathrm {ID}}^T\boldsymbol \Theta \boldsymbol h_{\mathrm {SI}}\boldsymbol h_{\mathrm {SI}}^H\boldsymbol \Theta^H \boldsymbol h_{\mathrm {ID}}^\ast\Big\} \nonumber \\ & = & \frac{P}{\sigma^2} \mathbb E \Big\{\boldsymbol h_{\mathrm {ID}}^T\boldsymbol \Theta \mathbb E\{\boldsymbol h_{\mathrm {SI}}\boldsymbol h_{\mathrm {SI}}^H\} \boldsymbol \Theta^H \boldsymbol h_{\mathrm {ID}}^\ast\Big\} \nonumber \\ & = & \frac{P}{\sigma^2} \rho_{\mathrm{SI}} \mathbb E \Big\{\boldsymbol h_{\mathrm{ID}}^T\boldsymbol \Theta \boldsymbol R \boldsymbol \Theta^H \boldsymbol h_{\mathrm{ID}}^\ast\Big\} \nonumber \\ & = & \frac{P}{\sigma^2} \rho_{\mathrm{SI}} \text{tr}\Big(\mathbb E \Big\{\boldsymbol h_{\mathrm{ID}}^\ast\boldsymbol h_{\mathrm{ID}}^T \Big\} \boldsymbol \Theta \boldsymbol R \boldsymbol \Theta^H \Big) \nonumber \\ & = & \frac{P}{\sigma^2} \rho_{\mathrm{ID}} \rho_{\mathrm{SI}} \text{tr}\big(\boldsymbol R \boldsymbol \Theta \boldsymbol R \boldsymbol \Theta^H \big).
\end{eqnarray}}Then, from \cite[Theorem 2]{abdullah2022impact}, the optimal solution must satisfy $\boldsymbol \Theta^\star = \text{diag}\{\exp\left(\jmath c\boldsymbol 1_M\right)\}$, with $c$ being any real number. For a reflection amplitude of $\mu$, and by letting $c = 0$, we obtain $\boldsymbol \Theta^\star = \mu\boldsymbol I_M$ as an optimal solution under sCSI-based RBD.
\section*{Appendix B}
The ergodic SNR for HRN during the first phase is:
{\small \begin{eqnarray}
\mathbb E\left\{\gamma_{1\mathrm H}\right\} & = & \mathbb E \left\{\frac{P_1}{\sigma^2} \left|h_{\mathrm{SR}} + \boldsymbol h_{\mathrm{I}\mathrm{R}}^T \boldsymbol \Theta_1 \boldsymbol h_{\mathrm S\mathrm I}\right|^2 \right\}\nonumber \\ & \stackrel{\mathrm{a}}{=} & \frac{P_1}{\sigma^2} \Big(\mathbb E \left\{\left|h_{\mathrm{SR}} \right|^2 \right\} + \mathbb E \Big\{\left| \boldsymbol h_{\mathrm{I}\mathrm{R}}^T \boldsymbol \Theta_1 \boldsymbol h_{\mathrm S\mathrm I}\right|^2 \Big\} \Big) \nonumber \\ & = & \frac{P_1}{\sigma^2} \left(\rho_{\mathrm{SR}} + \rho_{\mathrm{IR}}\rho_{\mathrm{SI}} \text{tr}\big(\boldsymbol R \boldsymbol \Theta_1 \boldsymbol R \boldsymbol \Theta^H_1 \big)\right),
\end{eqnarray}}where equality ($\mathrm a$) holds due to the statistical independence of the direct and reflected channels. Then, from \cite[Theorem~2]{abdullah2022impact}, and for a reflection amplitude of $\mu$, we obtain the solution $\boldsymbol \Theta_1^\star = \mu\boldsymbol I_M$. The phase-shift matrix during the second transmission phase can be obtained following similar steps.
\bibliographystyle{IEEEtran}
|
2,869,038,155,639 | arxiv | \section{Introduction}
\label{001}
\nopagebreak
The observation of solar and very-long-baseline reactor neutrino oscillations due to the squared-mass difference
$ \Delta{m}^{2}_{\text{SOL}} = ( 7.59 \pm 0.21 ) \times 10^{-5} \, \text{eV}^{2} $
\protect\cite{0801.4589}
and the observation of atmospheric and long-baseline accelerator neutrino oscillations due to the squared-mass difference
$ \Delta{m}^{2}_{\text{ATM}} = 2.74 {}^{+0.44}_{-0.26} \times 10^{-3} \, \text{eV}^{2} $
\protect\cite{Adamson:2007gu}
give very robust evidence of three-neutrino mixing
(for reviews of the theory and phenomenology of neutrino mixing, see
Refs.~\protect\cite{Bilenky:1978nj,Bilenky:1987ty,hep-ph/9812360,hep-ph/0202058,hep-ph/0310238,hep-ph/0405172,hep-ph/0506083,hep-ph/0606054,Giunti-Kim-2007}).
There are, however,
some anomalies in the data of neutrino experiments which could
be interpreted as indications of exotic neutrino physics beyond three-neutrino mixing:
the LSND anomaly \protect\cite{hep-ex/0104049},
the Gallium radioactive source experiments anomaly \protect\cite{nucl-ex/0512041},
and the MiniBooNE low-energy anomaly
\protect\cite{0704.1500}.
In this paper we consider the anomaly observed
in the Gallium radioactive source experiments
\protect\cite{Anselmann:1995ar,Hampel:1998fc-Cr-51,Abdurashitov:1996dp,hep-ph/9803418,nucl-ex/0512041},
in which the Gallium solar neutrino detectors
GALLEX \protect\cite{Hampel:1998xg} and SAGE \protect\cite{nucl-ex/0509031}
were tested by measuring the electron neutrino flux
produced by intense artificial radioactive sources
placed inside the detectors.
The Gallium radioactive source experiments
measured a number of events smaller than expected.
This deficit
can be interpreted\footnote{
Another possible explanation is that
the theoretical
cross section of the Gallium detection process
has been overestimated \protect\cite{nucl-ex/0512041,hep-ph/0605186}.
}
as an indication of the disappearance of electron neutrinos
due to neutrino oscillations
\protect\cite{Laveder:2007zz,hep-ph/0610352,0707.4593}.
Under this hypothesis,
we analyze the data of the Gallium radioactive source experiments in the
effective framework of two-neutrino mixing,
which describes neutrino oscillations due to a $ \Delta{m}^{2} $
that is much larger than the solar and atmospheric ones
(see Refs.~\protect\cite{hep-ph/9812360,hep-ph/0202058,Giunti-Kim-2007}).
We also study the compatibility of this interpretation of
the Gallium radioactive source experiments anomaly
with the data of the Bugey \protect\cite{Declais:1995su} and Chooz \protect\cite{hep-ex/0301017}
reactor short-baseline antineutrino disappearance experiments.
\section{Gallium}
\label{002}
\nopagebreak
The GALLEX \protect\cite{Hampel:1998xg} and SAGE \protect\cite{nucl-ex/0509031}
solar neutrino detectors
(see
Refs.~\protect\cite{Bilenky:1978nj,Bilenky:1987ty,hep-ph/9812360,hep-ph/0202058,hep-ph/0310238,hep-ph/0405172,hep-ph/0506083,hep-ph/0606054,Giunti-Kim-2007})
have been tested
in so-called
"Gallium radioactive source experiments"
which consist in the detection of electron neutrinos
produced by intense artificial ${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ radioactive sources
placed inside the detectors.
The radioactive nuclei
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$
decay through electron capture
($ e^{-} + {}^{51}\text{Cr} \to {}^{51}\text{V} + \nu_{e} $
and
$ e^{-} + {}^{37}\text{Ar} \to {}^{37}\text{Cl} + \nu_{e} $)
emitting $\nu_{e}$ lines with the energies and branching ratios listed in Tab.~\ref{004}.
These neutrinos
were detected through the same reaction used for the detection of solar neutrinos
\protect\cite{Kuzmin-Ga-65}:
\begin{equation}
\nu_{e} + {}^{71}\text{Ga} \to {}^{71}\text{Ge} + e^{-}
\,,
\label{003}
\end{equation}
which has the low neutrino energy threshold
$ E_{\nu}^{\text{th}}({}^{71}\text{Ga}) = 0.233 \, \text{MeV} $.
The cross sections of the $\nu_{e}$ lines emitted in
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$
decay interpolated from Tab.~II of Ref.~\protect\cite{Bahcall-Ga-97}
are listed in Tab.~\ref{004}.
\begin{table}[t!]
\begin{center}
\begin{tabular}{l|cccc|cc}
&
\multicolumn{4}{c|}{${}^{51}\text{Cr}$}
&
\multicolumn{2}{c}{${}^{37}\text{Ar}$}
\\
\hline
$E_{\nu}\,[\text{keV}]$ & $ 747 $ & $ 752 $ & $ 427 $ & $ 432 $ & $ 811$ & $ 813$ \\
B.R. & $0.8163$ & $0.0849$ & $0.0895$ & $0.0093$ & $0.902$ & $0.098$ \\
$\sigma\,[10^{-46}\,\text{cm}^{2}]$ & $ 60.8 $ & $ 61.5 $ & $ 26.7 $ & $ 27.1 $ & $ 70.1$ & $70.3 $ \\
\hline
\end{tabular}
\caption{ \label{004}
Energies ($E_{\nu}$), branching ratios (B.R.) and Gallium cross sections ($\sigma$)
of the $\nu_{e}$ lines emitted in
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$
decay through electron capture.
The cross sections are interpolated from Tab.~II of Ref.~\protect\cite{Bahcall-Ga-97}.
}
\end{center}
\end{table}
The ratios $R$ of measured and predicted ${}^{71}\text{Ge}$
production rates in the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments\footnote{
As explained in Ref.~\protect\cite{nucl-ex/0512041},
the values of $R$ in Tab.~\ref{006} for the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments
are different from those published in Refs.~\protect\cite{Anselmann:1995ar,Hampel:1998fc-Cr-51},
because of an improved reanalysis of the data.
Similar results have been published recently in a PhD thesis \cite{Kaether:2007zz}
and discussed at the Neutrino 2008 Conference \cite{Hahn-Nu2008}:
$ R(\text{Cr1}) = 0.997 \pm 0.11 $
and
$ R(\text{Cr2}) = 0.807 {}^{+0.11}_{-0.10} $
in a standard rise-time analysis;
$ R(\text{Cr1}) = 0.953 \pm 0.11 $
and
$ R(\text{Cr2}) = 0.812 {}^{+0.10}_{-0.11} $
in a pulse-shape analysis.
We have verified that our results are stable against such small changes of the data.
},
Cr1 \protect\cite{Anselmann:1995ar} and Cr2 \protect\cite{Hampel:1998fc-Cr-51},
and
the SAGE
${}^{51}\text{Cr}$ \protect\cite{Abdurashitov:1996dp,hep-ph/9803418} and ${}^{37}\text{Ar}$ \protect\cite{nucl-ex/0512041} radioactive source experiments,
as reported in Ref.~\protect\cite{nucl-ex/0512041},
are listed in Tab.~\ref{006}.
Since the weighted average,
\protect\cite{nucl-ex/0512041}
\begin{equation}
R_{\text{Ga}}
=
0.88 \pm 0.05
\,,
\label{005}
\end{equation}
is smaller than unity by more than $2\sigma$,
it can be interpreted as an indication of the disappearance of electron neutrinos
due to neutrino oscillations
\protect\cite{Laveder:2007zz,hep-ph/0610352,0707.4593}.
The $\chi^2$ in the absence of oscillation is $8.19$ for 4 degrees of freedom,
corresponding to a 8.5\% goodness-of-fit\footnote{
The goodness-of-fit is the probability to obtain a worse fit under the assumption
that the model under consideration is correct (see Ref.~\cite{PDG-2006}).
It is the standard statistic used for the estimation of the quality of a fit
obtained with the least-squares method,
assuming the validity of the approximation in which
$\chi^{2}_{\text{min}}$ has a $\chi^2$ distribution with
$ \text{NDF} = N_{\text{D}} - N_{\text{P}} $ degrees of freedom,
where
$N_{\text{D}}$ is the number of data points and $N_{\text{P}}$ is the number of fitted parameters.
The fit is usually considered to be acceptable if the goodness-of-fit is larger than about 1\%.
}, as shown in Tab.~\ref{010}.
Therefore,
a fluctuation of the data in the case of no oscillations cannot be excluded.
However,
since from a physical point of view it is interesting to explore possible indications
of non-standard physics,
in the following we consider the case of neutrino oscillations.
\begin{table}[t!]
\begin{center}
\begin{tabular}{l|cc|cc}
&
\multicolumn{2}{c|}{GALLEX}
&
\multicolumn{2}{c}{SAGE}
\\
\hline
&
Cr1
&
Cr2
&
${}^{51}\text{Cr}$
&
${}^{37}\text{Ar}$
\\
\hline
$R$ & $ 1.00 \pm 0.10 $ & $ 0.81 \pm 0.10 $ & $ 0.95 \pm 0.12 $ & $ 0.79 \pm 0.10 $ \\
radius [m] & \multicolumn{2}{c|}{$1.9$} & \multicolumn{2}{c}{$0.7$} \\
height [m] & \multicolumn{2}{c|}{$5.0$} & \multicolumn{2}{c}{$1.47$} \\
source height [m] & $2.7$ & $2.38$ & \multicolumn{2}{c}{$0.72$} \\
\hline
\end{tabular}
\caption{ \label{006}
Ratios $R$ of measured and predicted ${}^{71}\text{Ge}$
production rates in the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments,
Cr1 \protect\cite{Anselmann:1995ar} and Cr2 \protect\cite{Hampel:1998fc-Cr-51},
and
the SAGE
${}^{51}\text{Cr}$ \protect\cite{Abdurashitov:1996dp,hep-ph/9803418} and ${}^{37}\text{Ar}$ \protect\cite{nucl-ex/0512041} radioactive source experiments,
as reported in Ref.~\protect\cite{nucl-ex/0512041}.
We give also the radii and heights of the GALLEX and SAGE cylindrical detectors
and the heights from the base of the detectors at which the radioactive sources were placed along the axes of the detectors.
}
\end{center}
\end{table}
In the effective framework of two-neutrino oscillations,
which is appropriate in the case of short-baseline oscillations generated by a squared-mass difference
much larger than
$ \Delta{m}^{2}_{\text{SOL}} $
and
$ \Delta{m}^{2}_{\text{ATM}} $
(see
Refs.~\protect\cite{hep-ph/9812360,Giunti-Kim-2007}),
the survival probability of electron neutrinos and antineutrinos
with energy $E_{\nu}$ at a distance $L$ from the source
is given by\footnote{
The symmetry under CPT transformations,
which is a characteristic of all relativistic local quantum field theories,
implies that the survival probabilities of neutrinos and antineutrinos are equal
(see Ref.~\protect\cite{Giunti-Kim-2007}).
}
\begin{equation}
P_{\boss{\nu}{e}\to\boss{\nu}{e}}(L,E_{\nu})
=
1 - \sin^{2}2\vartheta \, \sin^{2}\left( \frac{ \Delta{m}^{2} L }{ 4 E_{\nu} } \right)
\,,
\label{007}
\end{equation}
where $\vartheta$ is the mixing angle and $\Delta{m}^{2}$ is the squared-mass difference.
The fit of the data gives information on the values of the mixing parameters $\sin^{2}2\vartheta$ and $\Delta{m}^{2}$.
In our calculation, the theoretical value of
the ratio $R$ of the predicted ${}^{71}\text{Ge}$
production rates in each of the Gallium radioactive source experiments
in the cases of presence and absence of neutrino oscillations
is given by
\begin{equation}
R
=
\frac
{ \int \text{d}V \, L^{-2} \sum_{i} (\text{B.R.})_{i} \, \sigma_{i} \, P_{\nu_{e}\to\nu_{e}}(L,E_{\nu,i}) }
{ \sum_{i} (\text{B.R.})_{i} \, \sigma_{i} \int \text{d}V \, L^{-2} }
\,,
\label{008}
\end{equation}
where $i$ is the index of the $\nu_{e}$ lines emitted in
${}^{51}\text{Cr}$ or ${}^{37}\text{Ar}$,
which are listed in Tab.~\ref{004}.
The measured ratios are listed in Tab.~\ref{006},
together with the dimensions of the detectors,
which we approximate as cylindrical,
and the height from the base of each detector at which the radioactive sources were placed along the axis of the respective detector.
We averaged the neutrino path length $L$ with a Monte Carlo integration over the volume $V$ of each cylindrical detector.
\begin{figure}[t!]
\begin{center}
\setlength{\tabcolsep}{1pt}
\begin{tabular}{cc}
\includegraphics*[bb=25 147 572 702, width=0.49\textwidth]{fig/gus-g1-upl.eps}
&
\includegraphics*[bb=25 147 572 702, width=0.49\textwidth]{fig/gus-g2-cnt.eps}
\\
\includegraphics*[bb=25 147 572 702, width=0.49\textwidth]{fig/gus-s1-upl.eps}
&
\includegraphics*[bb=25 147 572 702, width=0.49\textwidth]{fig/gus-s2-cnt.eps}
\end{tabular}
\caption{ \label{023}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane obtained
from the fits of the results of
the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments,
Cr1 and Cr2,
and
the SAGE
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ radioactive source experiments.
The curves in the GALLEX Cr1 and SAGE ${}^{51}\text{Cr}$ plots exclude the region on the right.
In the GALLEX Cr2 and SAGE ${}^{37}\text{Ar}$ plots,
the pairs of $1\sigma$ and $2\sigma$ curves delimit allowed regions,
whereas the $3\sigma$ curves exclude the region on the right.
}
\end{center}
\end{figure}
\begin{table}[t!]
\begin{center}
\begin{tabular}{cccccccc}
&
&
Ga
&
Bu
&
Ga+Bu
&
Bu+Ch
&
Ga+Ch
&
Ga+Bu+Ch
\\
\hline
& $\chi^{2}_{\text{min}}$ & $ 8.19 $ & $ 50.94 $ & $ 59.13 $ & $ 51.00 $ & $ 8.26 $ & $ 59.19 $ \\
No Osc. & NDF & $ 4 $ & $ 55 $ & $ 59 $ & $ 56 $ & $ 5 $ & $ 60 $ \\
& GoF & $ 0.085 $ & $ 0.63 $ & $ 0.47 $ & $ 0.66 $ & $ 0.14 $ & $ 0.51 $ \\
\hline & $\chi^{2}_{\text{min}}$ & $ 2.91 $ & $ 47.97 $ & $ 53.87 $ & $ 48.63 $ & $ 6.60 $ & $ 54.80 $ \\
& NDF & $ 2 $ & $ 53 $ & $ 57 $ & $ 54 $ & $ 3 $ & $ 58 $ \\
Osc. & GoF & $ 0.23 $ & $ 0.67 $ & $ 0.59 $ & $ 0.68 $ & $ 0.086 $ & $ 0.60 $ \\
& $\sin^{2}2\vartheta_{\text{bf}} $ & $ 0.22 $ & $ 0.048 $ & $ 0.062 $ & $ 0.041 $ & $ 0.08 $ & $ 0.054 $ \\
& $\Delta{m}^{2}_{\text{bf}}\,[\text{eV}^{2}]$ & $ 1.98 $ & $ 1.85 $ & $ 1.85 $ & $ 1.85 $ & $ 1.72 $ & $ 1.85 $ \\
\hline & $\Delta\chi^{2}_{\text{min}}$ &&& $ 2.98 $ & $ 0.59 $ & $ 3.63 $ & $ 3.85 $ \\
PG & NDF &&& $ 2 $ & $ 1 $ & $ 1 $ & $ 3 $ \\
& GoF &&& $ 0.23 $ & $ 0.44 $ & $ 0.057 $ & $ 0.28 $ \\
\hline
\end{tabular}
\caption{ \label{010}
Values of
$\chi^{2}_{\text{min}}$,
number of degrees of freedom (NDF) and
goodness-of-fit (GoF)
for the fit of different combinations of
the results of the Gallium radioactive source experiments and the
Bugey and Chooz reactor experiments.
The first three lines correspond to the case of no oscillations (No Osc.).
The following five lines,
including the best-fit values of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$,
correspond to the case of oscillations (Osc.).
The last three lines describe the parameter goodness-of-fit (PG) \protect\cite{hep-ph/0304176}.
}
\end{center}
\end{table}
In the separate analysis of the result of each Gallium radioactive source experiment
in terms of neutrino oscillations,
the two mixing parameters cannot be determined
through a least-squares analysis from one data point.
Therefore,
we adopt a Bayesian approach,
as done in Ref.~\protect\cite{hep-ph/9411414},
considering $R$ as a random variable with a uniform prior probability distribution
between zero and one.
Then,
if $R_{\text{obs}}$ is the observed value of $R$,
the normalized posterior probability distribution of $R$ is given by
\begin{equation}
p(R|R_{\text{obs}})
=
\frac
{ p(R_{\text{obs}}|R) }
{ \int_{0}^{1} \text{d}R \, p(R_{\text{obs}}|R) }
\,.
\label{009}
\end{equation}
Here, $p(R_{\text{obs}}|R)$ is the sampling distribution of $R_{\text{obs}}$ given $R$,
which we assume to be Gaussian
with standard deviation equal to the experimental uncertainty.
The allowed interval of $R$ with a given Bayesian Confidence Level
is given by the Highest Posterior Density interval with integrated probability equal
to the Confidence Level.
Figure~\ref{023} shows
the resulting allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane.
One can see that the first GALLEX source experiment (Cr1)
and the ${}^{51}\text{Cr}$ SAGE source experiment,
in which the measured rate is within $1\sigma$ from unity,
imply only upper limits for the mixing parameters.
On the other hand,
the analyses of the second GALLEX source experiment (Cr2)
and the ${}^{37}\text{Ar}$ SAGE source experiment
give $2\sigma$ allowed bands,
which have a large overlap for $ \Delta{m}^{2} \gtrsim 1 \, \text{eV}^{2} $.
Let us now discuss the combined fit of the four Gallium source experiments.
Since there are enough data points to determine the two mixing parameters
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$,
we abandon the Bayesian approach in favor of a standard
frequentist least-squares fit.
This method is
based on a global minimization of the $\chi^{2}$ in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane and the calculation
of the Confidence Level contours corresponding to a
$\Delta\chi^{2}$ with two degrees of freedom:
$\Delta\chi^{2}=2.30,6.18,11.83$ for
68.27\% ($1\sigma$), 95.45\% ($2\sigma$) and 99.73\% ($3\sigma$) C.L., respectively
(see Ref.~\cite{PDG-2006}).
The result of the combined least-squares analysis of the four Gallium source experiments
is shown in Fig.~\ref{024}.
One can see that there is an allowed region in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane at $1\sigma$
for
$ \Delta{m}^{2} \gtrsim 0.6 \, \text{eV}^{2} $
and
$ 0.08 \lesssim \sin^{2}2\vartheta \lesssim 0.4 $.
The values of
$\chi^{2}_{\text{min}}$,
the number of degrees of freedom (NDF),
the goodness-of-fit
(GoF)
and
the best-fit values of the mixing parameters are given in Tab.~\ref{010}.
The value of the the goodness-of-fit (23\%) shows that the fit is acceptable.
Table~\ref{011} shows the allowed ranges of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the corresponding marginal $\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{\text{min}}$ in Fig.~\ref{024}.
The presence of $2\sigma$ lower limits for $\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
in spite of the absence of a $2\sigma$ lower limit in
the $\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane in Fig.~\ref{024}
is an effect due to the statistical analysis:
for one parameter $2\sigma$ corresponds to $\Delta\chi^{2}=4$,
whereas for two parameters it corresponds to $\Delta\chi^{2}=6.18$.
Hence,
it is fair to conclude that there is an indication of a possible neutrino disappearance
due to neutrino oscillations with
$\sin^{2}2\vartheta \gtrsim 0.03$
and
$\Delta{m}^{2} \gtrsim 0.1 \, \text{eV}^{2}$
at a confidence level between one and two sigmas
($ \sim 70 - 90 \% \, \text{C.L.} $).
\begin{table}[t!]
\begin{center}
\begin{tabular}{clccc}
Parameter
&
\null \hfill C.L. \hfill \null
&
Ga
&
Bu
&
Ga+Bu
\\
\hline
& 68.27\% ($1\sigma$) & $ 0.12 - 0.33 $ & $ 0.021 - 0.075 $ & $ 0.035 - 0.087 $ \\
$\sin^{2}2\vartheta$ & 95.45\% ($2\sigma$) & $ > 0.028 $ & $ - $ & $ 0.007 - 0.19 $ \\
& 99.73\% ($3\sigma$) & $ - $ & $ - $ & $ - $ \\
\hline
& 68.27\% ($1\sigma$) & $ > 0.85 $ & $ 1.77 - 1.91 $ & $ 1.79 - 1.91 $ \\
$\Delta{m}^{2}\,[\text{eV}^{2}]$ & 95.45\% ($2\sigma$) & $ > 0.079 $ & $ - $ & $ > 0.77 $ \\
& 99.73\% ($3\sigma$) & $ - $ & $ - $ & $ - $ \\
\hline
\end{tabular}
\caption{ \label{011}
Allowed ranges of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
from the combined fit of the results of Gallium radioactive source experiments,
from the fit of the results of the Bugey reactor experiment,
and
from the combined fit.
The dash indicates the absence of limits.
}
\end{center}
\end{table}
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=23 144 572 704, width=\textwidth]{fig/chi-ga-cnt.eps}
\caption{ \label{024}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
combined fit of the results of
the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments
and
the SAGE
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ radioactive source experiments.
The best-fit point corresponding to $\chi^2_{\text{min}}$ is indicated by a cross.
}
\end{center}
\end{figure}
\section{Bugey}
\label{012}
\nopagebreak
The disappearance of electron antineutrinos have been investigated
by several reactor neutrino experiments at different baselines
(see Refs.~\protect\cite{hep-ph/0107277,Giunti-Kim-2007}).
Since,
according to Eq.~(\ref{007}),
the survival probabilities of neutrinos and antineutrinos are equal,
the interpretation of the results of Gallium radioactive source experiments
in terms of electron neutrino disappearance can be compared directly with the
results of reactor neutrino experiments.
In this section we consider the results of the reactor short-baseline Bugey experiment \protect\cite{Declais:1995su},
which put the most stringent constraints on the
disappearance of electron antineutrinos due to
$ \Delta{m}^{2} \gtrsim 0.1 \, \text{eV}^{2} $.
Reactor neutrino experiments detect electron antineutrinos
through the inverse neutron decay process
\begin{equation}
\bar\nu_{e} + p \to n + e^{+}
\,.
\label{013}
\end{equation}
The neutrino energy $E_{\nu}$ and the positron kinetic energy $T_{e}$ are related by
\begin{equation}
E_{\nu} = T_{e} + T_{n} + m_{e} + m_{n} - m_{p} \simeq T_{e} + 1.8 \, \text{MeV}
\,,
\label{014}
\end{equation}
where $T_{n}$ is the negligibly small recoil kinetic energy of the neutron.
In the Bugey experiment the survival probability of electron antineutrinos
was measured at three source-detector distances:
$ L_{j} = 15, 40, 95 \, \text{m} $, for $j=1,2,3$, respectively.
We use the ratio of observed and expected
(in the case of no oscillation)
positron spectra given in Fig.~17 of Ref.~\protect\cite{Declais:1995su},
in which there are
$ N_{j} = 25, 25, 10$
energy bins.
We analyze the data with the following $\chi^{2}$,
taken from Ref.~\protect\cite{Declais:1995su}:
\begin{equation}
\chi^{2}
=
\sum_{j=1}^{3}
\left\{
\sum_{i=1}^{N_{j}}
\dfrac{ \left[ \left( A a_{j} + b \left( E_{ji} - E_{0} \right) \right) R_{ji}^{\text{the}} - R_{ji}^{\text{exp}} \right]^{2} }{ \sigma_{ji}^{2} }
+
\dfrac{ \left( a_{j} - 1 \right)^{2} }{ \sigma_{a_{j}}^{2} }
\right\}
+
\dfrac{ \left( A - 1 \right)^{2} }{ \sigma_{A}^{2} }
+
\dfrac{ b^{2} }{ \sigma_{b}^{2} }
\,,
\label{015}
\end{equation}
where
$E_{ji}$ is the central energy of
the $i\text{th}$ bin in the positron kinetic energy spectrum measured at the $L_{j}$ source-detector distance,
$R_{ji}^{\text{exp}}$ and $R_{ji}^{\text{the}}$
are, respectively, the corresponding measured and calculated ratios.
The uncertainties $\sigma_{ji}$ include the statistical uncertainty of each bin
and a 1\% systematic uncertainty added in quadrature,
which takes into account the uncertainty of the spectrum calculation
(with a total of about 5\% uncorrelated systematic uncertainty over 25 bins).
The coefficients
$ \left( A a_{j} + b \left( E_{ji} - E_{0} \right) \right) $,
with $E_{0} = 1 \, \text{MeV}$,
were introduced in Ref.~\protect\cite{Declais:1995su}
in order to take into account the systematic uncertainty of the positron energy calibration.
The value of $\chi^{2}$ as a function of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
is calculated by minimizing Eq.~(\ref{015}) with respect to the five parameters
$A$, $a_{j}$ ($j=1,2,3$), $b$, which have, respectively,
uncertainties
$\sigma_{A} = 0.048$,
$\sigma_{a_{j}} = 0.014$
$\sigma_{b} = 0.02 \, \text{MeV}^{-1}$ \protect\cite{Declais:1995su}.
Following Ref.~\protect\cite{hep-ph/0102252},
we approximate
the neutrino flux, the detection cross section and the detection efficiency
as constants in each energy bin.
Then,
$R_{ji}^{\text{the}}$ is given by
\begin{equation}
R_{ji}^{\text{the}}
=
\frac
{
\int \text{d}L
\,
L^{-2}
\int_{E_{ji}-\Delta{E_{j}}/2}^{E_{ji}+\Delta{E_{j}}/2} \text{d}E
\int_{-\infty}^{+\infty} \text{d}T_{e}
\,
F(E,T_{e})
\,
P_{\bar\nu_{e}\to\bar\nu_{e}}(L,E_{\nu}) }
{
\Delta{E_{j}}
\int \text{d}L \, L^{-2}
}
\,.
\label{016}
\end{equation}
Here
$T_{e}$ and $E_{\nu}$ are, respectively, the positron kinetic energy and the neutrino energy,
related by Eq.~(\ref{014}),
whereas $E$ is the measured positron kinetic energy,
which is connected to $T_{e}$ by the energy resolution function of the detector $F(E,T_{e})$.
We considered a Gaussian energy resolution function with standard deviation
$0.252\sqrt{E/4.2\text{MeV}}\,\text{MeV}$
\protect\cite{Declais:1995su}.
The quantities $ \Delta{E_{j}} $ are the widths of the energy bins in each detector.
The integration over the neutrino path length $L$ is performed by a Monte Carlo
which takes into account the geometries of
the reactor and of the detectors and their relative positions \protect\cite{Declais-2008}.
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=25 147 564 702, width=\textwidth]{fig/chi-bu-ras.eps}
\caption{ \label{025}
90\% C.L. exclusion curves in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
obtained from a raster-scan analysis of Bugey data (solid line)
and from a standard global least-squares fit (dashed line).
}
\end{center}
\end{figure}
With this method we obtained the 90\% C.L. raster-scan\footnote{
In the raster-scan method,
$\chi^{2}_{\text{min}}$ is found for each fixed value of $\Delta{m}^{2}$.
The corresponding upper limit for $\sin^{2}2\vartheta$ is calculated
as the value of $\sin^{2}2\vartheta$ for which
the cumulative distribution function of
$\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{\text{min}}$, which has one degree of freedom,
is equal to the Confidence Level
($\Delta\chi^{2}=2.71$ for 90\% C.L.).
}
exclusion curve
shown in Fig.~\ref{025},
which is similar to the original 90\% C.L. raster-scan Bugey exclusion curve in Ref.~\protect\cite{Declais:1995su}.
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=23 144 572 704, width=\textwidth]{fig/chi-bu-cnt.eps}
\caption{ \label{026}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
least-squares analysis of Bugey data.
The best-fit point corresponding to $\chi^2_{\text{min}}$ is indicated by a cross.
}
\end{center}
\end{figure}
Let us emphasize that the raster-scan method is statistically weak,
because it does not have proper coverage \protect\cite{physics/9711021}.
We presented in Fig.~\ref{025} the raster-scan exclusion curve only to show by comparison
with the analogous figure in Ref.~\protect\cite{Declais:1995su} that
our analysis of the Bugey data is acceptable.
The dashed line in Fig.~\ref{025} shows the 90\% C.L. Bugey exclusion curve
obtained with the standard least-squares method,
which we adopted also in the previous Fig.~\ref{024} and the following Figs.~\ref{026}--\ref{031}.
From Fig.~\ref{025} one can see that the 90\% C.L. raster-scan exclusion curve
overcovers for all values of $\Delta{m}^2$,
except for small intervals around
$ \Delta{m}^2 \simeq 0.9 \, \text{eV}^2 $
and
$ \Delta{m}^2 \simeq 1.9 \, \text{eV}^2 $.
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=28 142 571 700, width=0.8\textwidth]{fig/chi-bu-hst.eps}
\caption{ \label{027}
Best fit of Bugey data (points with error bars \protect\cite{Declais:1995su}).
The three panels show the ratio $R$ of observed and expected
(in the case of no oscillation) event rates
at the
three source-detector distances in the Bugey experiment
as functions of the measured positron kinetic energy $E$ (see Eq.~(\ref{016})).
In each panel,
the solid and dashed histograms correspond, respectively, to the best-fit values of
$ \left( A a_{j} + b \left( E_{ji} - E_{0} \right) \right) R_{ji}^{\text{the}} $
and
$ R_{ji}^{\text{the}} $
(see Eq.~(\ref{015})).
}
\end{center}
\end{figure}
Figure~\ref{026}
shows the allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
the marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
least-squares analysis of Bugey data.
The value and location in the $\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
of the minimum of the $\chi^{2}$,
the number of degrees of freedom (NDF) and the goodness-of-fit (GoF) are given in Tab.~\ref{010}.
The fit is satisfactory,
since the goodness-of-fit is 67\%.
The best-fit value of the oscillation parameters
and the small $1\sigma$ allowed regions in Fig.~\ref{026}
are in favor of neutrino oscillations.
However,
the $2\sigma$ and $3\sigma$ contours in Fig.~\ref{026}
provide only upper limits to neutrino oscillations.
Also,
the value of the $\chi^{2}$ in the case of absence of oscillations
and the corresponding goodness-of-fit
(63\%)
do not allow us to exclude the absence of oscillations.
The reason of the hint in favor
of neutrino oscillations given by the Bugey data
is illustrated in Fig.~\ref{027},
where
the histogram relative to the best fit is shown against
the Bugey $R_{ji}^{\text{exp}}$'s.
With the help of the histogram,
one can see that there is a weak hint of oscillations.
The $1\sigma$ allowed regions in Fig.~\ref{026} have very narrow
$\Delta{m}^{2}$ ranges around
$0.9\,\text{eV}^{2}$,
$1.85\,\text{eV}^{2}$, and
$3\,\text{eV}^{2}$,
because slight shifts of $\Delta{m}^{2}$ from these optimal values
spoil the agreement with the data of the histogram in Fig.~\ref{027}.
Table~\ref{011} shows the marginal allowed ranges of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the corresponding $\Delta\chi^{2}$'s in Fig.~\ref{026}.
One can see that there is a hint of neutrino oscillations
with
$0.02 \lesssim \sin^{2}2\vartheta \lesssim 0.08$
and
$\Delta{m}^{2} \approx 1.8 \, \text{eV}^{2}$.
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=23 144 572 704, width=\textwidth]{fig/chi-gabu-cnt.eps}
\caption{ \label{028}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
combined fit of the results of
the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments,
the SAGE
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ radioactive source experiments
and the Bugey reactor experiment.
The best-fit point corresponding to $\chi^2_{\text{min}}$ is indicated by a cross.
}
\end{center}
\end{figure}
From a comparison of Figs.~\ref{024} and \ref{026}
one can see that the allowed regions
of the Gallium radioactive source experiments and the Bugey experiment
are marginally compatible
for
$ \sin^{2} 2\vartheta \sim 0.1 $
and
$ \Delta{m}^{2} \gtrsim 1 \, \text{eV}^{2} $.
Figure~\ref{028} shows the allowed regions obtained from the combined fit.
Since the Bugey data are statistically dominant,
the curves in Fig.~\ref{028} are not very different from those in Fig.~\ref{026},
which have been obtained from the fit of the Bugey data alone.
The inclusion of the Gallium data has the effect
of eliminating the $1\sigma$ allowed region at
$ \Delta{m}^{2} \approx 0.9 \, \text{eV}^{2} $
and
of disfavoring at $1\sigma$ values of $ \sin^{2} 2\vartheta $
smaller than about $2\times10^{-2}$.
The value and location of $\chi^{2}_{\text{min}}$,
the number of degrees of freedom and the goodness-of-fit are listed in Tab.~\ref{010}.
One can see that the Gallium data do not spoil the good fit of the Bugey data.
Indeed,
the value of the parameter goodness-of-fit\footnote{
The value of $(\Delta\chi^{2}_{\text{min}})_{\text{A+B}}$
corresponding to the parameter goodness-of-fit of two experiments A and B
is given by
$ (\chi^{2}_{\text{min}})_{\text{A+B}} - [ (\chi^{2}_{\text{min}})_{\text{A}} + (\chi^{2}_{\text{min}})_{\text{B}} ] $.
It has a $\chi^2$ distribution with number of degrees of freedom
$ \text{NDF} = P_{\text{A}} + P_{\text{B}} - P_{\text{A}+\text{B}} $,
where $P_{\text{A}}$, $P_{\text{B}}$ and $P_{\text{A}+\text{B}}$ are, respectively,
the number of parameters in the fits of A, B and A+B data
\protect\cite{hep-ph/0304176}.
}
\protect\cite{hep-ph/0304176}
reported in Tab.~\ref{010} shows that the Bugey and Gallium data are compatible
under the hypothesis of neutrino oscillations.
The marginal allowed ranges of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the corresponding $\Delta\chi^{2}$'s in Fig.~\ref{028}
are given in Tab.~\ref{011}.
\section{Chooz}
\label{017}
\nopagebreak
In this section we consider the result of the long-baseline reactor neutrino experiment
Chooz \protect\cite{hep-ex/0301017},
which gives limits on neutrino oscillations which are comparable with those of the Bugey experiment for
$ \Delta{m}^{2} \gtrsim 2 \, \text{eV}^{2} $.
In the Chooz experiment
the ratio of the number of observed events and that expected in the absence of neutrino oscillations
is
\begin{equation}
R_{\text{Chooz}}
=
1.01 \pm 0.04
\,.
\label{018}
\end{equation}
The value of this ratio puts a constraint on the disappearance of
electron (anti)neutrinos with energies in the MeV range
at distances smaller than about 1 km.
This corresponds to a constraint on $ \sin^{2} 2\vartheta $
for $ \Delta{m}^{2} \gtrsim 10^{-3} \, \text{eV}^{2} $.
In the range of sensitivity of the Gallium radioactive source experiments,
$ \Delta{m}^{2} \gtrsim 10^{-1} \, \text{eV}^{2} $
(see Figs.~\ref{024}),
the oscillation length of reactor antineutrinos
is much shorter than the Chooz source-detector distance.
In this case, the Chooz experiment is only sensitive to the averaged survival probability
\begin{equation}
\langle P_{\boss{\nu}{e}\to\boss{\nu}{e}} \rangle
=
1 - \frac{1}{2} \, \sin^{2}2\vartheta
\,.
\label{019}
\end{equation}
Therefore, the Chooz result in Eq.~(\ref{018})
can be combined\footnote{
In our figures we considered $\Delta{m}^{2}$
in the range $10^{-2}-10^{2}\,\text{eV}^2$.
For simplicity, we neglected the small
$\Delta{m}^{2}$ dependence of the CHOOZ exclusion curve
for $\Delta{m}^{2}\lesssim4\times10^{-2}\,\text{eV}^2$
(see Fig.~55 of Ref.~\protect\cite{hep-ex/0301017}).
}
with the results of the Gallium radioactive source experiments
simply by considering it as a measurement of $\sin^{2}2\vartheta$:
in the Bayesian approach of Eq.~(\ref{009})
\begin{equation}
\sin^{2}2\vartheta
<
0.071 ,\,
0.15 ,\,
0.23
\,,
\label{020}
\end{equation}
at
68.27\% ($1\sigma$),
95.45\% ($2\sigma$),
99.73\% ($3\sigma$) Bayesian Confidence Level, respectively.
\begin{table}[t!]
\begin{center}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{clccc}
Parameter
&
\null \hfill C.L. \hfill \null
&
Bu+Ch
&
Ga+Ch
&
Ga+Bu+Ch
\\
\hline
& 68.27\% ($1\sigma$) & $ 0.012 - 0.067 $ & $ 0.017 - 0.14 $ & $ 0.028 - 0.078 $ \\
$\sin^{2}2\vartheta$ & 95.45\% ($2\sigma$) & $ < 0.096 $ & $ < 0.20 $ & $ 0.002 - 0.12 $ \\
& 99.73\% ($3\sigma$) & $ < 0.18 $ & $ < 0.26 $ & $ < 0.18 $ \\
\hline
& 68.27\% ($1\sigma$) & $ 0.83 - 1.92 $ & $ > 0.62 $ & $ 1.78 - 1.91 $ \\
$\Delta{m}^{2}\,[\text{eV}^{2}]$ & 95.45\% ($2\sigma$) & $ - $ & $ - $ & $ > 0.74 $ \\
& 99.73\% ($3\sigma$) & $ - $ & $ - $ & $ - $ \\
\hline
\end{tabular}
\caption{ \label{021}
Allowed ranges of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
from the combined fit of the results of
the Bugey and Chooz reactor experiments,
the Gallium radioactive source and Chooz reactor experiments,
and
the Gallium radioactive source and the Bugey and Chooz reactor experiments.
The dash indicates the absence of limits.
}
\end{center}
\end{table}
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=23 144 572 704, width=\textwidth]{fig/chi-buch-cnt.eps}
\caption{ \label{029}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
combined fit of the results of
the Bugey and Chooz reactor experiments.
The best-fit point corresponding to $\chi^2_{\text{min}}$ is indicated by a cross.
}
\end{center}
\end{figure}
First,
we performed a combined frequentist least-squares analysis of the Bugey and Chooz data,
which yielded the allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
shown in Fig.~\ref{029},
the best fit values of the mixing parameters reported in Tab.~\ref{010},
and the marginal allowed ranges listed in Tab.~\ref{021}.
One can see that the addition of the Chooz result to the
Bugey data analysis has the effect of improving slightly
the upper limit on $\sin^{2}2\vartheta$ for $\Delta{m}^{2} \gtrsim 3 \, \text{eV}^{2}$
and that of excluding values of $\sin^{2}2\vartheta$ larger than about 0.1
for $\Delta{m}^{2} \lesssim 3 \times 10^{-2} \, \text{eV}^{2}$,
where Bugey is not sensitive.
In the intermediate range of $\Delta{m}^{2}$,
where Bugey is sensitive to the oscillations,
the addition of the Chooz result weakens the hint in favor of oscillations
given by the Bugey data: the $1\sigma$ allowed regions in Fig.~\ref{026}
are stretched towards small values of $\sin^{2}2\vartheta$ in Fig.~\ref{029}.
However,
the best-fit value of the mixing parameters remain unchanged,
because of the dominance of the Bugey data.
From Tab.~\ref{010},
one can see that
the parameter goodness-of-fit implies that Bugey and Chooz results are compatible under the hypothesis of
neutrino oscillations,
but the goodness-of-fit obtained in the case of no oscillations do not allow us to exclude this possibility.
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=23 144 572 704, width=\textwidth]{fig/chi-gach-cnt.eps}
\caption{ \label{030}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
combined fit of the results of
the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments,
the SAGE
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ radioactive source experiments
and the Chooz reactor experiment.
The best-fit point corresponding to $\chi^2_{\text{min}}$ is indicated by a cross.
}
\end{center}
\end{figure}
From the comparison of Eq.~(\ref{020}) and Fig.~\ref{024},
one can see that the results of the Chooz and the Gallium radioactive source experiments
are compatible only at the $2\sigma$ level.
In fact the parameter goodness-of-fit reported in Tab.~\ref{010} shows a tension between
Gallium and Chooz data under the hypothesis of
neutrino oscillations.
Figure~\ref{030} shows
the allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane obtained with the combined least-squares fit of Gallium and Chooz data.
The values of
$\chi^{2}_{\text{min}}$ and Goodness of Fit
and
the best-fit values of the mixing parameters are given in Tab.~\ref{010}.
It is clear that the combined fit is not good,
since the results of Chooz and the Gallium radioactive source experiments
are in contradiction regarding neutrino disappearance.
The marginal allowed ranges of
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$ in Tab.~\ref{021}
are of little interest,
since the minima of the corresponding $\Delta\chi^{2}$'s in Fig.~\ref{030}
are very shallow,
except for the upper bound on $\sin^{2}2\vartheta$
driven by Chooz data.
As one can see from the allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane in Fig.~\ref{030},
the Chooz bound on $\sin^{2}2\vartheta$ in Eq.~(\ref{020})
is weakened by the results of the Gallium radioactive source experiments
in a significant way only for
$\Delta{m}^{2} \gtrsim 10^{-1} \, \text{eV}^{2}$
at the $1\sigma$ level.
\begin{figure}[t!]
\begin{center}
\includegraphics*[bb=23 144 572 704, width=\textwidth]{fig/chi-gabuch-cnt.eps}
\caption{ \label{031}
Allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
and
marginal $\Delta\chi^{2}$'s
for
$\sin^{2}2\vartheta$ and $\Delta{m}^{2}$
obtained from the
combined fit of the results of
the two GALLEX ${}^{51}\text{Cr}$ radioactive source experiments,
the SAGE
${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ radioactive source experiments
and the Bugey and Chooz reactor experiments.
The best-fit point corresponding to $\chi^2_{\text{min}}$ is indicated by a cross.
}
\end{center}
\end{figure}
Finally,
we performed a combined fit of the results of Bugey, Chooz, and Gallium data.
The resulting allowed regions in the
$\sin^{2}2\vartheta$--$\Delta{m}^{2}$ plane
are shown in Fig.~\ref{031}.
The best fit values and the marginal allowed ranges of the mixing parameters
are listed, respectively, in Tabs.~\ref{010} and \ref{021}.
One can see that the Gallium and Chooz data tend to compensate each other,
leading to results which are similar to those obtained in the analysis of Bugey data alone.
The value of the parameter goodness-of-fit reported in Tab.~\ref{010} does not allow us
to exclude the compatibility of the Bugey, Chooz and Gallium data under the hypothesis of neutrino oscillations.
Also the goodness-of-fit obtained in the case of no oscillations,
given in Tab.~\ref{010}, is acceptable.
Therefore,
we can only conclude that
the combined analysis of all the experimental data that we have considered
is compatible both with the case of no oscillations
and with the hint in favor of neutrino oscillations
with
$0.02 \lesssim \sin^{2}2\vartheta \lesssim 0.08$
and
$\Delta{m}^{2} \approx 1.8 \, \text{eV}^{2}$
found in the analysis of Bugey data.
\section{Conclusions}
\label{022}
\nopagebreak
We interpreted the deficit
observed in the Gallium radioactive source experiments
as a possible indication of
the disappearance of electron neutrinos.
We have analyzed the data
in the effective framework of two-neutrino mixing,
which describes neutrino oscillations due to a $ \Delta{m}^{2} $
that is much larger than the solar and atmospheric ones.
We found that
there is an indication of electron neutrino disappearance
due to neutrino oscillations with
$\sin^{2}2\vartheta \gtrsim 0.03$
and
$\Delta{m}^{2} \gtrsim 0.1 \, \text{eV}^{2}$.
We have also studied the compatibility of the data of the Gallium radioactive source experiments
with the data of the Bugey and Chooz reactor short-baseline antineutrino disappearance experiments
in the same effective framework of two-neutrino mixing,
in which the disappearance of neutrinos and antineutrinos are equal.
We found that the Bugey data present a hint of neutrino oscillations
with
$0.02 \lesssim \sin^{2}2\vartheta \lesssim 0.08$
and
$\Delta{m}^{2} \approx 1.8 \, \text{eV}^{2}$,
which is compatible with the region of the mixing parameters
allowed by the analysis of the data of the Gallium radioactive source experiments.
We have also performed combined analyses
of the Bugey and Chooz data,
of the Gallium and Bugey data,
of the Gallium and Chooz data,
which show that
the Bugey and Chooz data are compatible,
the Gallium and Bugey data are compatible, and
the Gallium and Chooz data are marginally compatible.
The weak indication in favor of neutrino oscillations
found in the analysis of the Bugey data persists in the combined analyses
of the Bugey data with the Gallium and Chooz data.
However,
we cannot exclude the absence of oscillations.
From a physical point of view,
a hint in favor of short-baseline neutrino oscillations
generated by $\Delta{m}^{2} \gtrsim 0.1 \, \text{eV}^{2}$
is extremely interesting.
This squared-mass difference
is too large to be compatible with the three-neutrino mixing scheme
inferred from the observation of neutrino oscillations in
solar, very-long-baseline reactor, atmospheric and long-baseline accelerator experiments,
in which there are only two independent squared-mass differences,
$ \Delta{m}^{2}_{\text{SOL}} \approx 8 \times 10^{-5} \, \text{eV}^{2} $
and
$ \Delta{m}^{2}_{\text{ATM}} \approx 3 \times 10^{-3} \, \text{eV}^{2} $.
Therefore,
the results of our analysis indicate the possible existence of at least one
light sterile neutrino $\nu_{s}$
(see Refs.~\protect\cite{hep-ph/9812360,hep-ph/0202058,Giunti-Kim-2007}).
We think that it is very important to explore
this intriguing hint of
new physics beyond the Standard Model.
As already discussed in Ref.~\cite{0707.4593},
short-baseline $\boss{\nu}{e}\to\boss{\nu}{s}$ transitions
have an influence on the interpretation of all experiments
with an initial $\boss{\nu}{e}$ beam.
In the existing
solar and atmospheric neutrino experiments the survival probability of $\boss{\nu}{e}$ is
the averaged one in Eq.~(\ref{019}).
However, the uncertainties of the experimental data and our knowledge of the initial flux
do not allow us to exclude $\boss{\nu}{e}\to\boss{\nu}{s}$ transitions at the level of about 20\%
in the case of solar neutrinos
\cite{hep-ph/0406294}
and about 30\%
(see Ref.~\cite{Giunti-Kim-2007})
in the case of atmospheric neutrinos.
Future experiments which are well suited for finding small
$\boss{\nu}{e}\to\boss{\nu}{s}$ transitions
are those with a source producing a $\boss{\nu}{e}$ flux
which is known with high accuracy.
Since sterile neutrinos are invisible,
$\boss{\nu}{e}\to\boss{\nu}{s}$ transitions
can be revealed either by measuring a disappearance of
$\boss{\nu}{e}$'s without $\bos{\mu}$ or $\bos{\tau}$ production in the detection process
or by measuring a disappearance of
$\boss{\nu}{e}$'s due to oscillations with a squared-mass difference
much larger than
$ \Delta{m}^{2}_{\text{SOL}} $
and
$ \Delta{m}^{2}_{\text{ATM}} $.
We are aware of the following possibilities:
Beta-Beam experiments \protect\cite{Zucchelli:2002sa}
which have a pure $\nu_{e}$ or $\bar\nu_{e}$ beam from nuclear decay
(see the reviews in Refs.~\protect\cite{physics/0411123,hep-ph/0605033});
Neutrino Factory experiments
in which the beam is composed of
$\nu_{e}$ and $\bar\nu_{\mu}$,
from $\mu^{+}$ decay,
or
$\bar\nu_{e}$ and $\nu_{\mu}$,
from $\mu^{-}$ decay
(see the review in Ref.~\protect\cite{hep-ph/0210192,physics/0411123});
Mossbauer neutrino experiments,
with a $\bar\nu_{e}$ beam
produced in recoiless nuclear decay
and detected in recoiless nuclear antineutrino capture
\protect\cite{hep-ph/0601079};
the LENS detector
\protect\cite{Raghavan:1997ad,LENS-2002}
with an artificial Megacurie $\nu_{e}$ source
\protect\cite{Grieb:2006mp}.
Let us also notice the very interesting possibility to reveal
the existence of sterile neutrinos in the flux of
high-energy astrophysical neutrinos after their passage through the Earth
by measuring the peculiar matter effects \cite{hep-ph/0302039,0709.1937}.
\section*{Acknowledgments}
\nopagebreak
We would like to express our gratitude to Y. Declais for giving us detailed information
on the Bugey experiment.
M.A. Acero would like to thank the International Doctorate on AstroParticle Physics
(IDAPP) for financial support.
C. Giunti would like to thank the Department of Theoretical Physics of the University of Torino
for hospitality and support.
\raggedright
|
2,869,038,155,640 | arxiv | \section{Introduction}
\label{cintro}
A long--standing problem in astrophysics is a result of our inability to determine the three-dimensional structure of distant objects.
This limitation has often inhibited our understanding of the internal structure of even relatively well--defined and isolated astronomical objects, such as molecular cloud cores.
Assuming that such an object is spherically symmetric, or has some other simple geometry, often permits us to describe the object's internal structure using one or more radial profile functions.
Such radial profiles are frequently used to examine or model the physics and chemistry which govern such objects.
It is relatively safe to assume that stars and planets are spherical or that a spiral galaxy has a disk and a bulge.
These assumptions become problematic when studying objects without obvious symmetry.
Molecular cloud cores exhibit a wide variety of shapes that very rarely resemble any simple geometry.
Hence, determining their internal structure while using a geometric assumption will always yield some bias in any derived radial profile.
In this paper we describe a technique which may be used to obtain limited, yet useful information about an object's radial profile function without making any assumptions about the object's shape, orientation, or the nature of the radial profile function. This is done using a single two-dimensional column density map as the entire available data on the source.
A variety of such techniques have been used to determine the radial density distribution in molecular cloud cores (also referred to as dense cores) in studies over more than three decades. Early work employed $^{13}$CO\ emission \citep{Dickman1983, Arquilla1985}. Optical extinction was utilized by \citet{Cernicharo1985} to determine the density distribution within a number of dark clouds in Taurus. These techniques were recognized to have weaknesses (inability to trace high column density regions for optical extinction, and variable abundance due to e.g. freezeout for carbon monoxide). Subsequent efforts have largely moved to measument of stellar reddening in the near-infrared, allowing accurate probing of the extinction to much greater columns \citep{Alves2001a, Alves2001b}. \citet{Kandori2005} employed measurement of infrared colors and stellar densities to obtain the density structure of 10 dense cores. The infrared color excess technique was utlized by \citet{Pineda2010} to derive the density distribution in cloud cores in Taurus using 2MASS data. Continuum emission may also be used to study the temperature, and density distribution of dust in the ISM. With Herschel data, \citet{Launhardt13} were able to probe the dust within 12 molecular cloud cores. \citet{Stutz10} used both dust extinction as well as emission to model the column density and temperature distribution of CB244. To introduce this new technique, we will constrain ourselves to dust extinction as it is simpler, and temperature-independent.
In some of the previous work, a singe power law radial density profile was fitted to the data \citep{Arquilla1985, Cernicharo1985}, with exponents typically between 1 and 2 found. Other studies used a Bonner--Ebert Sphere \citep{Bonnor1956, Ebert1955} to model the density profile, which characteristically has a flat density profile in the central region transitioning to a $r^{-2}$ radial dependence towards the edge of the core \citet{Dickman1983, Alves2001a, Alves2001b, Kandori2005}. A function with a similar form gave a good fit to the data of \citet{Pineda2010}.
Our technique improves on previous methods by eliminating any geometric assumptions, as well as any a priori assumptions about the nature of the radial profile function.
There are certain limitations to the technique as well as criteria which must be fulfilled.
These are discussed in detail in Section \ref{cassumptions}.
The most important limitation and constraints may be summarized as follows.
\begin{itemize}
\item Since a two dimensional projection cannot uniquely define a three-dimensional object without additional information, it is impossible to obtain absolute values for the radial profile function without additional information or assumptions. It is however possible to obtain the form of the function which differs from the original profile by two unknown, geometry-dependent scalars.
\item The internal structure of the object in question must be describable using a radial profile function.
\end{itemize}
In theory the technique may be used to study spectral line emission, absorption, continuum emission, extinction, etc.
It may be applied to any object provided it is consistent with the assumptions described in Section \ref{cassumptions}. We will demonstrate that the technique is useful even in cases where only a portion of an object exhibits contour self-similarity.
To illustrate and validate the technique we have chosen to apply it to maps of the dust extinction in molecular cloud core column density maps derived from 2MASS data on stellar reddening.
This paper is formulated so as to introduce a novel methodology by presenting an analytical derivation, testing it against simulated data, and finally applying it to real data.
Section \ref{cassumptions} describes the initial assumptions which must be fulfilled in order for the technique to be applicable to a given object.
The assumptions yield critical relationships which illustrate key aspects of this technique.
Section \ref{cderivation} derives the technique analytically using two different methods.
Section \ref{cnumeric} applies the technique to a set of simulated data designed to test its validity as well as to expose its performance under a variety of circumstances.
Section \ref{creal} discusses the use of 2MASS dust extinction maps and applies the technique to several clouds.
We make a comparison with previous methods for measuring radial profiles in Section \ref{ccomparison}.
We discuss the results and the performance of this new technique in Section \ref{cdiscussion}.
\section{Assumptions}
\label{cassumptions}
The goal of this research is to extract the maximum available information regarding the internal volume density structure of an object using a single column density map observed from one line of sight direction, while making the fewest possible assumptions.
We show how it is possible under certain conditions to obtain the form of an object's volume density profile function without assuming a specific geometry, or making any assumptions about the function that governs the radial density profile.
To this end it is necessary to detail the assumptions used in this work.
The method described here only relies on the three assumptions below which are made for all cases.
\begin{description}
\item[Assumption 1:] The object studied must be optically thin in whatever observable quantity is being measured in the sense that
\begin{equation}
\label{csimplen}
N(y,z) = \int_{-\infty}^{\infty} n(x,y,z) dx \>\> ,
\end{equation}
where $N(y,z)$ is the measured column density at position $(y,z)$ and $n(x,y,z)$ represents the volume density at position $(x,y,z)$.
Throughout this paper, the x axis is arbitrarily chosen to represent the line of sight direction.
\item[Assumption 2:] The volume density of the object can be entirely characterized using a single function that describes the volume density profile.
\end{description}
The following can be considered to follow from assumption 2.
\begin{description}
\item[Assumption 2a:] Any object which satisfies assumption 2 must be described using two functions;
One describes the cloud's geometry, while the second describes its radial volume density profile.
\end{description}
We define the object's shape using a core function
\begin{equation}
a(\alpha, \theta)=a_c f_c(\alpha,\theta) \>\> ,
\end{equation}
where $a(\alpha,\theta)$ has units of length and describes the size of the object's core along each direction originating from the object's center. $a_c$ is a constant with units of length, and $f_c(\alpha,\theta)$ is a dimensionless function which scales the core radius along each $(\alpha,\theta)$ to produce a shape for the object.
In the case of a sphere, $f_c(\alpha,\theta)=1$ while $a_c$ represents the radius.
Spherical coordinates are chosen here to emphasize the fact that the core function depends only on direction from the object's center, and not on distance.
When working with arbitrary shapes it is convenient to define a new, dimensionless parameter $r_{rc}$ which is equal to the ratio between the distance from some point $(x,y,z)$ to the object's center, and the core radius $a(\alpha,\theta)$ along the same direction.
With the object's center located at the origin of the coordinate system $(0,0,0)$,
\begin{equation}
\label{corirrc}
r_{rc}(x,y,z)= \frac{\sqrt{x^2 + y^2 + z^2}}{a(\alpha,\theta)} \>\> .
\end{equation}
For a sphere, $r_{rc}(x,y,z)=\sqrt{x^2 + y^2 + z^2}/a_c$.
We commonly refer to the surface described by $r_{rc}=1$ as the core.
In order to fully describe the geometry of an object which meets assumption 2, $f_c(\alpha,\theta)$ must describe a closed surface such that a vector from the object's center along any direction will cross the surface exactly once.
This permits the definition of a radial volume density function that is dependent on $r_{rc}$ and governs the volume density distribution of the entire object. We define
\begin{equation}
\label{corin}
n(x,y,z) = n_0 f_n(r_{rc}) \>\> ,
\end{equation}
where $n(x,y,z)$ represents the volume density at position $(x,y,z)$, and $f_n(r_{rc})$ is a dimensionless function that governs the radial volume density profile.
$n_0$ is a constant representing the volume density where $f_n(r_{rc})=1$.
$r_{rc}(x,y,z)$ and $n(x,y,z)$ can fully characterize any object which satisfies assumption 2.
Neither $r_{rc}(x,y,z)$, nor $n(x,y,z)$ can ever be fully determined from a single column density map using only one observable quantity without additional information, since a column density map in and of itself can not uniquely define a three-dimensional object.
It is possible to determine the function $f_n$ as well as certain properties of $a(\alpha,\theta)$ by taking advantage of the self-similarity imposed on the object by assumption 2.
Any object which satisfies assumption 2 satisfies the implied assumptions below.
\begin{figure}
\includegraphics[scale=.75]{cfig1.eps}
\caption{(Left) A three-dimensional representation of surfaces with three specific values of $r_{rc}$ for an arbitrary object which satisfies assumption 2. Each surface shares the same shape and orientation, while differing only in scale. (Right) Projections of the three surfaces along the line of sight (LOS). Each surface is characterized by a specific value of $r_{rc}$, its projected Area ($A$), and its volume density ($n$) as in Equation \ref{corin}. Projected areas have the same shape and orientation, while differing only in scale. Since the object is assumed to be optically thin, the projected column densities from each surface add linearly to produce the total column density.
\label{cfig1}}
\end{figure}
\begin{description}
\item[Assumption 2b:] Specific values of $r_{rc}$ describe three-dimensional surfaces of equal volume density. The left panel of Figure \ref{cfig1} illustrates three such surfaces belonging to an arbitrary object, and having three distinct values of $r_{rc}$. All volume density surfaces share the same shape, orientation, and center position. The only differences between surfaces of different $r_{rc}$ are in their sizes, and volume densities.
\item[Assumption 2c:] Each volume density surface, when projected onto a plane perpendicular to the line of sight produces a two-dimensional boundary whose area ($A$) is directly proportional to $r_{rc}^2$. The right panel of Figure \ref{cfig1} describes such projections of three surfaces with independent values of $r_{rc}$. All such surface boundaries are identical except in their size and the corresponding volume densities they represent.
\end{description}
Self-similarity between different volume density surfaces is a critical aspect of assumptions 2b and 2c. Aside from certain constants, the only parameters which differentiate the projected boundaries of different volume density surfaces are their areas (expressable in terms of $r_{rc}^2$), and the corresponding volume density (determined by $f_n(r_{rc})$). Therefore there must be a relationship between the area of each projected boundary and its volume density which is dependent on $f_n(r_{rc})$, but is, aside from some constants, independent of $f_c(\alpha,\theta)$. \emph{$f_c(\alpha,\theta)$ determines the shape and orientation which are identical for each surface, while the relationship between the projected area and volume density of each surface is governed by the radial density profile function ($f_n(r_{rc})$)}.
The observable column density map is a superposition of all the volume density surfaces projected onto a plane perpendicular to our line of sight. The column density map should thus exhibit the same self-similarity seen among the individual volume density surfaces. If assumptions 1 and 2 hold for a given object, then the following must be valid as well
\begin{description}
\item[Assumption 3:] Comparing the column densities and areas of different column density contours should yield a relationship which, aside from some constants, is independent of the object's geometry.
\end{description}
Assumption 3 is confirmed analytically by equation \ref{cNarea} in Section \ref{canalyticder}. The following section shows how the function $f_n(r_{rc})$ may be derived using that relationship.
\section{Derivation}
\label{cderivation}
No truly general proof that applies to all possible shapes is evident at this time. Therefore it is necessary to restrict this analytic derivation to those geometries which can be described by a quadratic definition of $r_{rc}$. Geometries which do not conform to equation \ref{crrc} are tested numerically in Section \ref{cnumeric}. A useful form for $r_{rc}$ is
\begin{equation}
\label{crrc}
r_{rc}(x,y,z)=\sqrt{x^2 a + x b(y,z) + c(y,z)} \>\> ,
\end{equation}
where $a$ is a constant and $b(y,z)$ and $c(y,z)$ are any functions that conform to assumption 2. The above quadratic representation, while not universal, can describe a wide variety of geometries encountered in nature including triaxial ellipsoids. No specific values for $a,b(y,z),$ and $c(y,z)$ are invoked in the following derivation except where noted for purposes of illustration. In such cases, a spheroid model with axial ratio $\alpha$, inclined by an angle $i$ through a rotation about the y axis will be used. A spheroid is chosen because it is mathematically tractable yet versatile enough to demonstrate changes in shape and orientation by varying $\alpha$ and $i$ respectively. Such a spheroid may be described by the relationships
\begin{eqnarray}
\label{cadef}
a=\frac{\omega^2}{a_c^2 \alpha^2} \>\> , \ \omega^2=\alpha^2 cos^2(i) + sin^2(i) \>\> , \\
\label{cbdef}
b(z)=\frac{2 z cos(i) sin(i) (1-\alpha^2)}{a_c^2 \alpha^2} \>\> , \\
\label{ccdef}
c(y,z)=\frac{y^2 \alpha^2 + z^2(\alpha^2 sin^2(i) + cos^2(i))}{a_c^2 \alpha^2} \>\> .
\end{eqnarray}
Values of $r_{rc}$ describe individual surfaces of fixed volume density. Since the line of sight is chosen to be along the x axis it is useful to express the x positions of each surface with a specific $r_{rc}$ as
\begin{equation}
x_{r_{rc},\pm}(y,z)=\frac{-b(y,z) \pm \sqrt{b(y,z)^2 - 4 a c(y,z) + 4 a r_{rc}^2}}{2 a} \>\> .
\end{equation}
Defining a new function
\begin{equation}
E(y,z) = \frac{4 a c(y,z) - b(y,z)^2}{4 a} \>\> ,
\end{equation}
yields
\begin{equation}
\label{cxpos}
x_{r_{rc},\pm}(y,z)=\frac{-b(y,z)}{2 a} \pm \frac{\sqrt{r_{rc}^2 - E(y,z)}}{\sqrt{a}} \>\> .
\end{equation}
$x_{r_{rc},\pm}(y,z)$ denotes the two line of sight ($x$) positions for a surface defined by a particular value of $r_{rc}$ at sky position $(y,z)$. Functions $a$, $b(y,z)$, and $c(y,z)$ are defined by the geometry of the object in question. Equations \ref{cadef}, \ref{cbdef}, and \ref{ccdef} describe the appropriate functions representing a spheroid with axial ratio $\alpha$ and inclination $i$. $E(y,z)$ is a function that is entirely dependent on the object's shape; the following section discusses its conceptual meaning further.
\subsection{Discrete Derivation}
\label{cdiscretesec}
It is possible to define an object as a discrete series of shells, each of which is defined as the region between an inner ($r_{rc,i}$) and an outer ($r_{rc,o}$) surface with an average volume density ($n_{i,o}$) within the shell. The depth along the x axis of each such shell at different (y,z) positions will vary according to
\begin{equation}
\label{cdori}
D(y,z)_{i,o} = (x_{r_{rc,o},+}(y,z) - x_{r_{rc,o},-}(y,z)) - (x_{r_{rc,i},+}(y,z) - x_{r_{rc,i},-}(y,z)) \>\> ,
\end{equation}
where $D(y,z)_{i,o}$ is the total depth along the line of sight at position $(y,z)$ for the shell made up of two surfaces defined by $r_{rc,i}$ and $r_{rc,o}$. Substituting equation \ref{cxpos} into equation \ref{cdori} yields
\begin{equation}
\label{cdinc}
D(y,z)_{i,o}= \frac{2}{\sqrt{a}} \left(\sqrt{r_{rc,o}^2 - E(y,z)} - \sqrt{r_{rc,i}^2 - E(y,z)}\right) \>\> .
\end{equation}
Equation \ref{cdinc} describes the depth of each shell, however this representation is of limited use since $r_{rc}$ is not an observable quantity. Similarly, $E(y,z)$ is a function that is directly dependent on the object's unknown shape. Equation \ref{cdinc} must be put in terms of observable quantities: the observed column density, and the area within each column density contour.
Each surface as described by equation \ref{cxpos}, when projected onto the line of sight, produces a closed boundary composed of those $(y,z)$ positions where $x_{r_{rc},+}(y,z) = x_{r_{rc},-}(y,z)$. In view of Equation \ref{cxpos} the projected boundaries of each shell are defined by
\begin{equation}
\label{ceboundary}
E(y,z)_{boundary} = r_{rc}^2 \>\> .
\end{equation}
Solving for $E(y,z)$ using the spheroid model above yields a familiar relation,
\begin{equation}
\label{cespheroid}
E(y,z)_{spheroid} = \frac{1}{a_c^2} \left( \frac{z^2}{\omega^2} + {y^2} \right) \>\> ,
\end{equation}
which is a simple ellipse that results from projecting a three dimensional spheroid surface onto a two dimensional plane. Equation \ref{ceboundary} makes clear that each contour of equal $E(y,z)$ corresponds to the boundary of a particular volume density surface, with a specific value of $r_{rc}$ and possesses a unique projected area. In general, the projected area of a surface of a particular $r_{rc}$ can be expressed as
\begin{equation}
\label{careaeq}
A=\epsilon r_{rc}^2 \>\> ,
\end{equation}
where $\epsilon$ is a geometry-dependent unknown constant($\epsilon_{sphere}=\pi a_c^2, \epsilon_{spheroid}=\omega \pi a_c^2$). All positions with equal $E(y,z)$ correspond to the projected boundary of a surface with $r_{rc,c}$ with corresponding area $A_c$. The additional subscript c denotes a specific contour.
Therefore, equation \ref{cdinc} can be reformed in terms of areas as
\begin{equation}
\label{cd}
D_{i,o,c} =\frac{2}{\sqrt{a\epsilon}} \left(\sqrt{A_o - A_c} - \sqrt{A_i - A_c}\right) \>\> ,
\end{equation}
where $D_{i,o,c}$ represents the depth along the line of sight of the shell between surfaces defined by $r_{rc}=r_{rc,i}$ and $r_{rc}=r_{rc,o}$ at all positions defined by the contour formed by the projected boundary of the surface defined by $r_{rc,c}$. The observed column density can then be defined as
\begin{equation}
\label{cncori}
N_{c}= \sum_{k=c}^{\infty} n_{k,k+1} D_{k,k+1,c} = \sum_{k=c}^{\infty} n_{k,k+1} \frac{2}{\sqrt{a\epsilon}} \left(\sqrt{A_{k+1} - A_c} - \sqrt{A_k - A_c}\right)\>\> .
\end{equation}
$N_{c}$ represents the column density at all positions $(y,z)$ defined by the projected boundary of the $r_{rc,c}$ surface. $n_{k,k+1}$ represents the mean volume density within the shell whose surfaces are defined by $r_{rc,k}$ and $r_{rc,k+1}$. The column density and area are observable quantities, however $n_{k,k+1}/(a\epsilon)$ are unknowns. The relationship between column density ($N$) and area ($A$) can be obtained through contouring the observed map, yielding a discrete series of contour column densities and associated areas. Using such data it should be possible to obtain information on the quantity $n_{k,k+1}/(a\epsilon)$.
It is useful to define two new variables which will represent the derived volume density profile function.
\begin{eqnarray}
\label{cr'n'}
r'= \sqrt{\frac{A}{\pi}} = \sqrt{\frac{\epsilon}{\pi}} r_{rc} \>\> , \\
\label{cn'}
n'(r')= n(r_{rc}) \sqrt{\frac{\pi}{a\epsilon}} = n_0 \sqrt{\frac{\pi}{a\epsilon}} f_n\left(r'\sqrt{\frac{\pi}{\epsilon}}\right) \>\> ,
\end{eqnarray}
permitting equation \ref{cncori} to be rewritten as
\begin{equation}
\label{cdiscfinal}
N_{c} = \frac{2}{\sqrt{\pi}} \sum_{k=c}^{\infty} n'_{k,k+1} \left(\sqrt{A_{k+1} - A_c} - \sqrt{A_k - A_c}\right)\>\> .
\end{equation}
The observed column density map thus yields a series of contours denoted by their column density ($N_{c}$) and area ($A_{c}$). Beginning with the outermost contour with the largest area, and moving recursively inward it is possible to derive a series of $n'_{c,c+1}$ measurements for the object using equation \ref{cdiscfinal}. Equation \ref{cr'n'} yields a series of $r'_{c}$ measurements derived from the contour Areas ($A_c$), yielding $n'(r')$. Equation \ref{cn'} shows that $n'(r')$ is related to $n(r_{rc})$ and $f_n(r_{rc})$ through a series of constants ($\epsilon, n_0, a$) that are all unknown. Knowledge of the object's geometry would yield values for $\epsilon$ and $a$ allowing the determination of $n_0$ and the full definition of the object's radial volume density profile $n(r_{rc})$. Conversely, knowledge of $n_0$ could yield information on the object's geometry.
Without such a priori knowledge there are limits to the information which may be obtained from a single column density map, however $f_n(r_{rc})$ can be determined to within 2 unknown scalars so as to obtain the form of the volume density profile function. The nature of those two scalars (G and $\chi$) is best elucidated through a non-discrete derivation as discussed in the following two sections. This is done without assuming a specific geometry for the object, or the nature of $f_n(r_{rc})$. This derivation is dependent on obtaining valid $N_c$ vs. $A_c$ measurements from the column density map which may be a non-trivial process when working with real data. Methods for obtaining such measurements are discussed in Section \ref{creal} along with examples of the derivation applied to simulated data.
\subsection{Analytic Derivation using Gaussians and Attenuated Power Laws}
\label{canalyticder}
Equation \ref{cdiscfinal} is useful for deriving $n'(r')$ from real data, and is used in all practical examples in this paper with both simulated and real data. However, it does not necessarily give the most insight into the problem. Any practical application of this theorem requires a strict understanding of the relation between $n'(r')$ and $n(r_{rc})$ with respect to the two scalars which separate them. To this end an analytic derivation is invoked in this section which is equivalent to that in Section \ref{cdiscretesec}, yet is qualitatively different in that it illustrates different aspects of the derived $n'(r')$ function. This derivation does not invoke discreteness, but instead uses integration. The integrals prohibit the use of a truly general form for $f_n(r_{rc})$, thus two radial density profiles are invoked for illustrative purposes along with the same spheroid geometry from Section \ref{cdiscretesec}. A gaussian and an attenuated power law are selected as mathematically tractable profiles that are frequently observed in nature. They may be described as
\begin{equation}
\label{cngnp}
n_g(x,y,z) = n_0 e^{-\frac{r_{rc}(x,y,z)^2}{2}}, n_p(x,y,z)=n_0 (r_{rc}(x,y,z)^2 + 1)^{\frac{-\gamma}{2}} \>\> ,
\end{equation}
where $n_g$ and $n_p$ represent the gaussian and attenuated power law functions respectively. $\gamma$ is a constant greater than 1. This attenuated power law function can be viewed as a form of the well-studied Type IV Pareto distribution. It is inspired by, and represents a more generalized form of the King profile \citep{King62}. The King profile was also used by \cite{King62,Dapp09}, and \cite{Pineda2010} when addressing the problem of density distributions within molecular cloud cores, however they each utilized geometric assumptions which we do not invoke.
If Assumption 1 holds then the observed column density map for each profile can be written as
\begin{equation}
N_g(y,z)= \int_{-\infty}^{\infty} n_g(x,y,z) dx,\ N_p(y,z)= \int_{-\infty}^{\infty} n_p(x,y,z) dx \>\> .
\end{equation}
Since specific radial density profile functions are used, it is possible to directly perform each integral, yielding
\begin{equation}
\label{cNdirect}
N_g(y,z)=n_0 \sqrt{\frac{2 \pi}{a}} e^{\frac{-1}{2}E(y,z)},\ N_p(y,z)=n_0 \sqrt{\frac{2\pi}{a}} \sqrt{2\pi} {\gamma-2 \choose \frac{\gamma-2}{2}} \left(E(y,z) +1\right)^{\frac{-\gamma+1}{2}} \>\> ,
\end{equation}
where ${\gamma-2 \choose \frac{\gamma-2}{2}}$ is the binomial coefficient.
Similarly to Section \ref{cdiscretesec}, the preceding equation may be used to express the column density in terms of the area covered by each column density contour resulting in
\begin{equation}
\label{cNarea}
N_g(A)=n_0 \sqrt{\frac{2 \pi}{a}} e^{\frac{-A}{2\epsilon}},\ N_p(A)=n_0 \sqrt{\frac{2\pi}{a}} \sqrt{2\pi} {\gamma-2 \choose \frac{\gamma-2}{2}} \left(\frac{A}{\epsilon} +1\right)^{\frac{-\gamma+1}{2}} \>\> .
\end{equation}
Since no specific shape has yet been invoked, Equation \ref{cNarea} verifies Assumption 3 by showing that the relationship between column density and the area of its contours is, aside from some constants ($\epsilon, a, n_0$), independent of the object's geometry.
Alternatively, using equations \ref{cNdirect} and \ref{cxpos} along with the relation
\begin{equation}
\frac{d x_{r_{rc}}}{dr_{rc}} = \frac{r_{rc}}{\sqrt{a(r_{rc}^2 - E(y,z))}}
\end{equation}
yields the following expression for the column density which is equivalent to the derivation in section \ref{cdiscretesec}
\begin{equation}
\label{cNrrc}
N(r_{rc}) = \int_{r_{rc}}^{\infty} n(r_{rc,e}) \frac{2 r_{rc,e}}{\sqrt{a(r_{rc,e}^2 - r_{rc}^2)}} dr_{rc,e} \>\> .
\end{equation}
Equation \ref{cNrrc} specifies the observed column density at all positions $(y,z)$ described by the projected boundary of the shell defined by $r_{rc}$. $r_{rc,e}$ is the integration variable, where the e subscript denotes that the integration is performed over all surfaces exterior to $r_{rc}$. Solving equation \ref{cNrrc} for the gaussian and attenuated power law profiles yields
\begin{equation}
\label{cNrrc2}
N_g(r_{rc}) = n_0 \sqrt{\frac{2 \pi}{a}} e^{-\frac{r_{rc}^2}{2}}, N_p(r_{rc})=n_0 \sqrt{\frac{2\pi}{a}} \sqrt{2\pi} {\gamma-2 \choose \frac{\gamma-2}{2}} \left(r_{rc}^2 +1\right)^{\frac{-\gamma+1}{2}} \>\> .
\end{equation}
Equation \ref{cNrrc2}, when converted to Areas using Equation \ref{careaeq}, is identical to Equation \ref{cNarea}, thus confirming that the discrete derivation in Section \ref{cdiscretesec} is equivalent to integrating the volume density along the line of sight.
\subsection{Understanding $n'(r')$}
Deriving $n'(r')$ through the method defined above yields a function with the same form as the original $n(r_{rc})$. It is important to understand the relation between the derived and actual density profile functions. This relationship may be defined as
\begin{equation}
\label{cGepsilon}
n'(r')=G n(r_{rc}) = G n\left(\frac{r'}{\chi}\right) \>\> ,
\end{equation}
where $G$ and $\chi$ are unknown constants. Applying the above method to the spheroid model with gaussian and power-law profiles yields derived volume density functions described by
\begin{equation}
\label{cnprimegnprimep}
n'_g (r') = G n_0 e^{\frac{-r_{rc}^2}{2 \chi^2}},\ n'_p(r') = G n_0 \left( \frac{r_{rc}^2}{\chi^2} + 1 \right) ^{\frac{-\gamma}{2}} \>\> .
\end{equation}
Equations \ref{cr'n'} and \ref{cGepsilon} in conjunction with Equation \ref{cnprimegnprimep} show that for a spheroid model,
\begin{equation}
\label{cGchi}
G = \sqrt{\frac{\pi}{a \epsilon}} = \frac{\alpha}{\omega^{\frac{3}{2}}}, \chi = \sqrt{\frac{\epsilon}{\pi}} =\sqrt{\omega} a_c \>\> .
\end{equation}
$G$ is dimensionless, while $\chi$ has dimensions of length. It is important to note that $G$ and $\chi$ are completely geometry dependent and thus identical in both the power law and gaussian cases. Neither parameter can be fully determined by the method described here without knowledge of the object's geometry, further data, or assumptions. As evidenced by Equation \ref{cGchi}, $G$ and $\chi$ are not independent quantities due to their dependence on $\omega$. Aside from scalars $G$ and $\chi$, the derived $n'(r')$, and the original $n(r_{rc})$ are identical. For a sphere, $G = 1$ and $\chi = a_c$. These scalars contain all of the unknown geometric information about the observed object. We may derive $n'(r')$ from an observed column density map, however this function will differ from the object's volume density profile ($n(r_{rc})$) by the two unknown scalars $G$ and $\chi$. The form of $n'(r')$ will however be identical to $n(r_{rc})$ regardless of the two scalars. If an object's geometry is known, $G$ and $\chi$ may be calculated (values for the spheroid are shown in Equation \ref{cGchi}). In the most general terms, $G$ may be viewed as the ratio between the depth and the width of an object along the line of sight, though this interpretation will rarely be strictly true. If $G$ is greater than 1, then the object is deeper than it is wide, and the derived $n'_0$ will be greater than the actual $n_0$.
\section{Examples using simulated data}
\label{cnumeric}
Numerical models using simulated data can validate the technique described in Section \ref{cderivation}, as well as illustrate the behaviour of $G$ and $\chi$ under various conditions. Models of several objects are constructed using known geometries and volume density profiles in order to create simulated column density maps. These maps are then used to derive $n'(r')$ which are finally compared to the original (known) volume density functions used to construct the model.
\subsection{Model Construction and Analysis}
\begin{figure}
\includegraphics[scale=.75]{cfig2.eps}
\caption{A model object with spherical geometry ($a=1/a_c^2, b=0, c=(y^2 + z^2)/a_c^2$), and a radial volume density profile described by a gaussian ($n(r_{rc}) = n_0 e^{-r_{rc}^2/2} $). a) A simulated column density map with sample contours. Gaussian noise is added equivalent to 1\% of the maximum column density. $Y$ and $Z$ coordinates are represented in units of $a_c$. b) Column Density ($N$) and corresponding Area ($A$) for each contour (not displayed) used in the analysis. c) A contour diagnostic plot for the object, as described in Section \ref{cnumeric}. d) The derived volume density profile function ($n'(r')$). Black points represent the values derived from each $N$ and $A$ contour pair. The red line represents the original function $n(r_{rc})$ used by the model as scaled by $G$ and $\chi$.
\label{cfig2}}
\end{figure}
Figure \ref{cfig2} illustrates how such a model is constructed and analyzed using the simplest case of a sphere with a gaussian volume density profile and minimal noise. To produce a column density map such as in Figure \ref{cfig2}a it is necessary to first choose a geometry (in this case a sphere) and a volume density profile function (in this case a gaussian) and construct a three-dimensional array whose elements represent the object's volume density. This array is then integrated along the line of sight to produce a column density map. Normally distributed noise with mean zero and a certain standard deviation (1\% of the maximum column density in the case of Figure \ref{cfig2}a) is then added to the column density map. Selected contour levels are drawn for illustration purposes to produce a map as in Figure \ref{cfig2}a. Such column density maps are the only source of information for further model analysis, as knowledge of model scalars such as $a_c$ and $n_0$ is used only for the purposes of scaling the plots.
Many column density contours (not drawn) are measured on the map in order to produce a plot of column density ($N$) versus contour area ($A$) as in Figure \ref{cfig2}b. It is impossible to properly sample the whole range of column densities without a priori knowledge of the volume density profile function. We found it most appropriate to measure the same number of contours as the number of pixels that span the object, and to space them equally in column density. This choice often results in oversampling, as discussed in Section \ref{cuncertainty}, but has been experimentally found to be the most useful.
The technique described here requires implicitly that all column density contours exhibit self-similarity, sharing the same shape, orientation, and center position.
The suitability of an object to this analysis technique may be verified by comparing the contours.
Similarly, it is necessary to remove from consideration any contours which are created by noise in the column density map.
Figure \ref{cfig2}c illustrates how these requirements are satisfied. Each contour is scaled to the same size, translated to the same center position, and then plotted so as to overlap as in Figure \ref{cfig2}c.
The innermost third of the contours with the smallest area are colored red, while the outermost third are colored blue, and intermediate contours are colored green.
In the case of Figure \ref{cfig2}c the simulated noise is quite low and thus only the innermost (smallest) contours display any deviations from a circle.
These variations are due to the small number of pixels within the smallest contours.
This representation is useful in that any deviations from a single contour shape and orientation may be easily identified.
A simple method for numerically filtering out noise-induced contours from consideration is to compare the geometric centers of each contour to the geometric center of mass for the object from the column density map. Any contours whose centers exceed some small distance from the center of mass are excluded. Figure \ref{cfig2}c represents the object's center of mass as the dashed cross in the center. The center positions of each contour are plotted in relation to the center of mass with the green, red, and blue colors representing the contours with the smallest, intermediate, and largest areas respectively. The solid black circle represents the radius used to filter out questionable contours. The square represents the relative position and scale of the pixel from the column density map which contains the object's center of mass. This diagnostic plot is useful in determining how well a given object complies with assumption 2, as well as which contours are suitable for analysis.
Once all unsuitable column density contours are removed from consideration it is possible to apply Equations \ref{cr'n'} and \ref{cdiscfinal} recursively to the $N$ and $A$ pairs in order to derive $n'(r')$ as in Figure \ref{cfig2}d. Since modeled data is used here it is possible to directly determine the values of $G$ and $\chi$ as well as to scale $r'$ and $n'(r')$ using the known values of $a_c$ and $n_0$ as in Figure \ref{cfig2}d.
\subsection{Tests Using Various Geometries and Profile Functions}
\label{csimulatedgeometries}
Figure \ref{cfig2} verifies that the derived $n'(r')$ has the same form as the original $n(r_{rc})$ function to a very high degree for the low-noise sphere with a gaussian profile function.
As expected, $G=1$ and $\chi = a_c$.
The technique described in Section \ref{cderivation} should apply to any profile function, as well as to any geometries which fulfill the requirements described in Section \ref{cassumptions}.
To that end, Figures \ref{cfig3}a-b, \ref{cfig3}c-d, and \ref{cfig4}a-b present three cases beyond the simple gaussian sphere in Figure \ref{cfig2}.
Figures \ref{cfig3}a-b, and \ref{cfig3}c-d represent the spheroid defined in Section \ref{cderivation} with two different orientations, while Figure \ref{cfig4}a-b represents a tri-axial ellipsoid.
The derived and original volume density profile function forms agree to a great extent, verifying that the technique is valid for tri-axial ellipsoids of any shape and orientation.
Several other geometries (not shown) which satisfy the assumptions in Section \ref{cassumptions} were tested and were all shown to produce valid results.
\begin{figure}
\includegraphics[scale=.75]{cfig3.eps}
\caption{a) Simulated column density map of a prolate spheroid as described by Equations \ref{cadef}, \ref{cbdef}, and \ref{ccdef} with $\alpha = 2$, $i = 0^o$, and 3\% noise added. b) Actual (red line) and derived (black dots) volume density profile for the object in a. An attenuated power-law as in Equation \ref{cngnp} with $\gamma = 3$ is used to construct the object in a and b. c) Simulated column density map of an object using the same geometry as in a, except that the object is rotated by $90^o$ about the y axis and 5\% noise is added. d) Actual (red line) and derived (black dots) volume density profile for the object in c. The radial volume density profile used in c and d is given by $n(r_{rc})= n_0 e^{-r_{rc}^2/2} (r_{rc}^2 + 1)^{\frac{-\gamma}{2}}$ with $\gamma = 1.5$.
\label{cfig3}}
\end{figure}
\begin{figure}
\includegraphics[scale=.75]{cfig4.eps}
\caption{a) Simulated column density map of a triaxial ellipsoid with axis dimensions of 0.5,1, and 2 $a_c$ and 10\% noise added. b) Actual (red line) and derived (black dots) volume density profile for the object in a. A triple gaussian volume density profile function is used to construct the object in a and b. c) Simulated column density map of an object with nonuniform $a$ (Equation \ref{cadef}). d) Actual (red line) and derived (black dots) volume density profile for the object in c. Since contour self-similarity is not present throughout, the object is not expected to be adequately modelled by this technique. The radial volume density profile used in c and d is described by the same triple gaussian as in a and b.
\label{cfig4}}
\end{figure}
The derivation in Section \ref{cderivation} requires that $a$ be a constant (Equation \ref{crrc}).
Any geometry which involves a definition of $a$ that is dependent on spatial coordinates $y$ and $z$ results in an object which does not conform to the assumptions in Section \ref{cassumptions}, and thus is not suitable to the analysis presented here.
This is due to the variation in the depth of each shell resulting from an inhomogeneous $a(y,z)$ as in Equation \ref{cd}.
If the object has a non-constant value of $a(y,z)$ then the relationship between the area and depth of each shell is no longer a constant, and the technique described by Equation \ref{cncori} fails as $a$ would be dependent on the shell number $k$.
It is possible to determine from the column density map whether the object in question has a geometry which is dependent on a constant value of $a$.
Equation \ref{careaeq} shows that the projected area of each shell is dependent on $\epsilon$.
From the definition of the spheroid it can be shown that $\epsilon_{spheroid}=\omega \pi a_c^2 = \sqrt{a} a_c \alpha \pi a_c^2$.
If $a$ is nonuniform then so is $\epsilon$, meaning that the relationship between a shell's area and $r_{rc}^2$ is no longer constant.
This implies that the projected boundary of each shell, and thus each column density contour, has a different shape.
\emph{Inhomogeneities in contour shape invalidate this technique}.
Figures \ref{cfig4}c-d show such a geometric shape which utilizes the same quadratic definition for $r_{rc}$ as in Equation \ref{crrc} where
\begin{eqnarray}
\beta=0.5, \gamma=0.5 \>\> , \\
a(y) = 1 + \gamma y^2 \>\> , \\
b(y) = -2 \beta^2 \gamma y^2 \>\> , \\
c(y,z) = (1 + \beta^2 \gamma) y^2 + z^2 \>\> .
\end{eqnarray}
In this case, $a$ is not a constant, the object does not produce self-similar contours, and our technique fails to reproduce the correct radial profile as seen in Figure \ref{cfig4}d.
It is important to note however, that the inner-most and outer-most regions (where there contours are self-similar) do correctly reproduce the original profile function.
Fortunately, the influence of each individual contour of the overall density profile is limited.
Each contour only affects contours interior to itself, and has the greatest influence on adjacent contours.
Thus, in our derivation, as one moves from the outermost shells to the innermost, a change in the value of $\epsilon$ will only begin to have an effect at the contour where the change first occurs, and its influence will decline as we move further into the interior.
We refer to this change in contour shape as an $\epsilon$ discontinuity.
Assuming only one such discontinuity occurs within a given map (so that there are only two contour shapes present), the derived volume density profile should still be correct in the region outside of the discontinuity.
The discontinuity will invalidate the derived profile interior to itself, but as its influence weakens the innermost region of the derived profile may still be accurate, as in Figure \ref{cfig4}d.
This phenomenon is frequently present when dealing with real data, and is illustrated further in Section \ref{creal}, but it only prohibits us from accurately obtaining $n'(r')$ in those portions of the profile where $\epsilon$ discontinuities occur. $\epsilon$ discontinuities are simple to identify visually through plots such as Figure 2c, and numerically by calculating the values of $\epsilon$ for each contour.
\subsection{Understanding Uncertainties and Distinguishing Real Features From Noise and Systematic Effects}
\label{cuncertainty}
Sections \ref{cderivation} and \ref{cnumeric} have proven that our technique can accurately derive the form of the radial volume density profile under idealized conditions.
Such circumstances rarely occur in nature so it is important to be able to distinguish real data from noise and systematic effects.
Ideally, one would assign an uncertainty to each measurement in the derived profile to determine which points are likely to represent real measurements.
However, a serious shortcoming of our method here is that we are unable to assign such uncertainties.
Many methods were tried, but the root problem is that we are unable to properly assign an accurate uncertainty to the $N$ vs. $A$ measurements for the contours.
Even assuming that all contours ultimately have the same shape, systematic noise affects them in ways that are difficult to quantify.
Each of the above simulated data sets, are made with different levels of systematic noise in order to illustrate its effects on the solutions.
Figure \ref{cfig2} has minimal noise equivalent to $1\%$ of the peak intensity, resulting in a near perfect derived profile.
Figures \ref{cfig3}a-b contain $3\%$ noise which is sufficient to add some irregularity to the shapes of the observed contours.
Those irregularities are most evident in the smallest contours, and are seen as slight offsets in the innermost regions of the radial density profile.
The $5\%$ noise map in Figures \ref{cfig3}c-d shows a new phenomenon.
Here, the noise is such that the innermost contour is broken into two pieces which are unusable.
As a result, the derived profile has no measurements interior to $r'= 0.3 a_0$.
Further, there are larger irregularities in the derived profile.
These are caused by individual pixels with a significantly different signal compared to their surroundings.
Contours will tend to bend around such pixels, until a certain column density threshold is reached and the contours snap onto the other side of the pixel.
This phenomenon may be recognized in that all the irregularities will have roughly the same width in the derived profile, corresponding roughly to the width of each pixel in the map as evident in Figure \ref{cfig3}d.
Figures \ref{cfig4}a-b show a more difficult case in which the noise is $10\%$ of the peak signal.
The object in the map can be difficult to discern under such conditions.
The systematic noise will prevent the construction of the innermost contours, but may also prevent the formation of contours in other regions leading to gaps in the derived profile such as in Figure \ref{cfig4}b. It is important to note that the gaps did not prevent the derivation of the correct profile in the regions where contours could be formed.
Changes in contour shape introduce a bias to the resulting profiles, as seen in Figure \ref{cfig4}c-d.
No reliable method for removing such bias is apparent.
However, the bias seems to be localized to only those regions of the derived profile immediately interior to the $\epsilon$ discontinuity. As a result, the innermost portion of the derived density profile in Figure \ref{cfig4}d agrees well with the original profile. How these $\epsilon$ discontinuities manifest themselves is best illustrated with real data as seen in the following section.
\section{Derivation of Molecular Cloud Core Volume Density Profiles Using Real Data}
\label{creal}
Star formation theory abounds with open questions, many of which are related to the process by which dense cores within molecular clouds collapse.
Of particular interest is the balance between forces which induce collapse, such as gravity and external pressure, and support mechanisms such as thermal pressure, turbulence, angular momentum, magnetic fields, etc.
A cloud core's density distribution is central to understanding this balance.
Thus measuring both the gas and dust components to obtain the distribution of the total proton density is of critical importance, representing a long-standing and active field of study.
The basic model of an equilibrium mass distribution \citep{Bonnor1956, Ebert1955} assumes an isothermal sphere bounded by some external pressure.
Even such a simple model yields powerful insights, such as that radial volume density profiles in molecular cloud cores should resemble power laws with an approximate $n\ \alpha \ r^{-2}$ relationship between radius and volume density at the edge of the core, with a weaker dependence on radius towards the center of the core.
Several recent studies have utilized Bonnor-Ebert spheres, or some derivative thereof, such as \cite{Evans2001}, \cite{Alves2001a}, and \cite{Teixeira2005}.
As mentioned above, some studies \citep[e.g.][]{Cernicharo1985, Arquilla1985} have fitted a power law to the radial density profile. While often satisfactory, there is considerable variation in the value of the exponent among different cloud cores, with $n \propto r^{-1~{\rm to}~-3}$ found in these studies.
Precise measurements of the exponents in cloud cores speak to the significance of the support mechanisms.
Studies employing a variety of techniques have convincingly demonstrated that clouds in different evolutionary states exhibit different density distributions \citep[with selected examples being][] {Ward-Thompson94, Kainulainen07, Liu2012, Wu2012}.
\cite{Liu2012} made a strong case for the urgent need of investigations of density distributions and support mechanisms in pre-stellar cores in light of new data from Planck.
We have chosen to demonstrate our technique using total proton counts in molecular clouds due to their well-known power-law nature which may be used to validate the technique.
By not assuming a geometry, we remove a significant source of bias present in previous observations.
The relatively small sample size here is used for demonstration purposes, while a more focused study will be presented in forthcoming publications.
It is assumed under most circumstances that gas and dust are fairly well mixed in the diffuse ISM and in molecular clouds.
Some molecular species, such as $H_{2}$ or $^{13}$CO are generally good tracers of the total proton count in molecular clouds. However, $H_{2}$ cannot be directly measured in clouds unless through absorption against a background source.
$^{13}$CO requires us to determine its excitation temperature to obtain column densities.
Further, carbon monoxide has been shown to freeze onto dust grains at higher densities \citep{Kramer1999, Tafalla2002, Bergin2002, Pineda2010}.
While not without its limitations, dust provides a well-tested, proven alternative.
Estimating total proton column densities through stellar reddening avoids the need to determine temperatures and may be used ubiquitously throughout a cloud assuming sufficient background stars are visible.
\cite{Goodman09} compared dust extinction, Near-Infrared Emission, and $^{13}$CO emission as probes of the total proton content of dense clouds.
After a detailed examination they concluded that dust extinction provided the simplest and most reliable probe.
Methods for deriving total proton column densities through stellar reddening data are well--developed \citep[e.g.][]{Lada94, Lombardi01,Dobashi11} and can be readily employed using infrared data from the 2MASS sky survey \citep{2MASS}.
The 2MASS data in conjunction with these methods provide a widely accessible, and comparatively uncontroversial method for obtaining total proton column density maps.
The data for two of the sources, which might actually be described as either dense clouds or cloud cores, (B133, and L466) were derived by the authors from the raw 2MASS stellar reddening catalog using an implementation of the NICER method \citep{Lombardi01, Chapman2009}.
Maps for the other clouds (L1765, L1709, B5, NGC1333) were obtained from the Perseus and Ophiuchus final extinction maps as part of the COMPLETE survey that also utilize the NICER method \citep{Lombardi01}.
L1765, and L1709 are part of the Ophiuchus Complex, while B5 and NGC1333 are part of the Perseus complex. Each of these four clouds is in a region with substantial background extinction and is accompanied by neighboring features. For these, a 3 arcminute beam size was used. B133 and L466 are more isolated, with comparatively little background extinction or neighboring features. The maps for B133 and L466 employ a 1 arcminute beam.
\begin{figure}
\includegraphics[scale=1.0]{cfig5.eps}
\caption{a) L1756 Column Density map. Darker pixels represent greater column density. Sample contours are drawn, with each color representing an individual contour group. b) $N$ vs. $A$ plot derived from contours actually used in the derivation, color groups correspond to the coloring in a. The fitted line(s) correspond to simple power-law fits for each color-group with the exponent(s) printed in the legend, and are only drawn through segments which are believed to be trustworthy. c) Contour Diagnostic Plot similar to that in Figure \ref{cfig2}a. d) Derived volume density profile. The fitted line(s) correspond to simple power-law fits for each color-group with the exponent(s) printed in the legend, and are only drawn through segments which are believed to be trustworthy.
\label{cfig5}}
\end{figure}
The scheme used in Figure \ref{cfig5} is used to describe each of the clouds in this study.
Figure \ref{cfig5}a represents the column density map for L1756, with some sample contours added.
Column density contours are applied throughout the map and filtered as described in Section \ref{cnumeric} to produce the column density vs. area ($N$ vs. $A$)plot in Figure \ref{cfig5}b.
It is apparent that there are three different behaviors in the $N$ vs. $A$ plot, therefore each measurement has been coded with a symbol and color.
This color and symbol scheme is applied throughout the whole Figure.
The colors of the contours drawn on the column density map correspond to the same column density levels as the colors in the $N$ vs. $A$ plot.
Figure \ref{cfig5}c is a diagnostic plot similar to Figure \ref{cfig2}c.
The derived volume density profile is depicted in Figure \ref{cfig5}d.
In contrast to the simulated data in Figure \ref{cfig2}d, $n_0$ and $a_0$ for the real cloud are not known, and thus the plot is scaled in terms of $n'$ and $r'$.
Three regions are evident in L1765 with red crosses representing the highest column density contours, blue representing the outer-most contours, and green those in between.
The changes in behavior between the three groups are in fact characterized by two $\epsilon$ discontinuities.
In the case of L1765, only the green data are believed to be trustworthy.
The red group represents the inner core. In the column density map and the contour diagnostic plot, the red contours do not appear to be self-similar.
The contours may in fact be self-similar, but there are too few pixels to properly define their shape, and thus they are not reliable.
Experimentation with real data reveals that the smallest contours must have an area greater than approximately 25 Nyquist-sampled pixels to be sufficiently well-defined and thus be usable.
With L1765, these innermost contours are displayed as an example; they are removed from consideration in the other clouds. The map shows green contours centered around the main cloud, as well as separate contours along the secondary clump.
The blue contours however encircle the secondary clump as well.
This technique cannot correctly function where there are two cores.
As a result the blue contours and the related data cannot be trusted.
Only the green contours around the main cloud are trusted.
The green group in L1765 seems to exhibit a very good power law with slope $-0.3$ in the $N$ vs. $A$ plot, and the fitted line is drawn in green in Figure \ref{cfig5}b.
From Section \ref{cderivation} it is expected that if the $N$ vs. $A$ exhibits a power-law behavior, then so should the derived volume density profile.
Figure \ref{cfig5}d indeed shows that the green region's volume density profile function follows a power law with slope $-2.47$.
We are thus able to determine the form of the volume density profile in the intermediate region of the cloud (the green group), where the contours are well-defined, self-similar, and include only one core or clump.
A reasonable concern is how can the green region be trusted when the red and blue are not. The red measurements are interior to the green ones, and thus have no effect at all on the green measurements.
Equation \ref{cd} reveals that the depth of each shell is roughly constant in the region interior to the shell, and thus so is its contribution to all interior shells in the derived profile.
The primary contribution of the blue measurements is to add an approximately constant volume density to the interior green and red measurements during the derivation process.
However, that constant contribution is irrelevant unless we know the specific geometry of the cloud.
Only the changes in the shell depth near the edge of each shell can alter the form of the derived interior profile.
Therefore, only the few inner-most points in the derived profile interior to the $\epsilon$ discontinuity are affected.
Our derived profile yields values for $n'$ and $r'$ that are scaled by unknown geometry-dependent constants.
Based on Figure \ref{cfig5}d, we cannot say that at a radius of $0.9$ pc, the volume density within L1765 is equal to $500\ cm^{-3}$, nor is that the goal of this research.
We can however say with significant confidence that in those regions of the main cloud encompassed by the green contours in Fig \ref{cfig5}a, or approximately the middle third of the cloud radially, the volume density profile is governed by a power law with exponent $-2.47$.
If, and only if, the cloud is assumed to be approximately spherical, $n'$ and $r'$ may actually represent similar values to the real $n$ and $r$.
Without geometry information however, we can only be certain of the profile's form within the trusted region.
The fact that we observed a strict power-law within a molecular cloud is in line with previous observations made through other methods.
It is both encouraging and disconcerting that the derived profile follows a power law quite so well. It is encouraging to see that a real map, with real data, will produce an orderly volume density profile function (in the green region) and that a power-law is observed as it has been by with previous studies. However, it is necessary to make certain that the power-law is not a systematic effect of our technique or of the data itself. We therefore examine additional clouds to determine their behavior, and verify the validity of our technique.
\begin{figure}
\includegraphics[scale=1.]{cfig6.eps}
\caption{L1709 analyzed as in Figure \ref{cfig5}
\label{cfig6}}
\end{figure}
Figure \ref{cfig6} represents L1709. In this case, the region interior to the red group is not shown as those contours have too small an area to be useful. The green contours however, are not trustworthy as they are influenced by the secondary peak at the edge of the map.
The contours for the red group in Figure \ref{cfig6}c exhibit a remarkable self-similarity in shape and center position even though they vary in area from 0.3 to 0.7 square parsecs.
Similarly to L1765, the derived profile exhibits a strong power-law with a slope of $-2.43$.
\begin{figure}
\includegraphics[scale=1.]{cfig7.eps}
\caption{B5 analyzed as in Figure \ref{cfig5}
\label{cfig7}}
\end{figure}
The analysis for B5 is depicted in Figure \ref{cfig7}. The blue and green regions are not trustworthy in this case because they encompass two secondary clumps in the top, and bottom right regions of the map. The red contours do not exhibit quite the same level of self-similarity as found in L1756 and L1709 due to the distension in the bottom right region. As a result, even the red contours may be somewhat suspect, however the distension corresponds to a variance of less than $5\%$ in the value of $\epsilon$ among the red contours. The red region corresponds to a power law with slope of $-2.15$.
\begin{figure}
\includegraphics[scale=0.65]{cfig8.eps}
\caption{L466 analyzed as in Figure \ref{cfig5}
\label{cfig8}}
\end{figure}
The cloud cores examined so far have all belonged to the Ophiuchus and Perseus complexes and have been surrounded by neighboring clumps which prevented us from measuring the density profiles in the outermost regions of the clouds. Furthermore, they have all exhibited a very similar power-law behavior, while originating from the same data source, which raises concerns that perhaps the way they were gridded, or the reduction method, may somehow be influencing the results. Hence, we located two cloud cores (L466, and B133) which are isolated, and employed an independent data reduction, gridding the maps to 1 arcminute beams.
Figure \ref{cfig8} represents L466. In this case, there are no adjoining clumps, or background extinction. Thus we were able to utilize a much wider range of contours in the exterior regions of the cloud. Here, both the red and blue regions are trustworthy, while the green region corresponds to an $\epsilon$ discontinuity and is not trustworthy. Here, the red region exhibits the same behavior as the previous clouds with a power-law slope of $-2.21$. The outermost, diffuse region of the cloud also follows a power-law, but with a slope of $-4.64$.
\begin{figure}
\includegraphics[scale=0.65]{cfig9.eps}
\caption{B133 analyzed as in Figure \ref{cfig5}
\label{cfig9}}
\end{figure}
Figure \ref{cfig9} reveals that in B133 we can measure the density profile in the outermost region of the cloud. Similarly to the case of L466, there seem to be two power laws present in the red and blue regions with slopes of $-1.81$, and $-4.02$. The green region is again the site of an $\epsilon$ discontinuity (significantly larger than in L466) and cannot be trusted.
What is the meaning of the two power-laws in the two clouds (L466 and B133) where we have been able to confidently measure the radial profile in the diffuse region of the cloud? How can two distinct power laws exist within the same cloud? This kind of discontinuity can be troubling. Previous researchers have noted that attenuated power-laws can well describe such clouds, while exhibiting different localized power laws in individual regions \citep{Pineda2010}.
It may be possible that an attenuated power law, such as that used in Equation \ref{cngnp} may accurately represent these clouds.
There is insufficient data to fit $n_p$ to the derived profile from L466 and B133 since the innermost region is still unmeasured and $n_p$ has three free parameters ($n_0$, $\gamma$, and $a_c$).
Using three free parameters it is not possible to derive a constrained fit for L466 and B133.
However, the total mass of the cloud may be used to reduce this to a two-parameter problem utilizing the relationship
\begin{equation}
M = 4 \pi \int_0^{\infty} n(r) r^2 dr= 4 \pi n_0 \int_0^{\infty} \left(1+ \frac{r^2}{a_c^2}\right)^{\frac{-\gamma}{2}} r^2 dr \>\> ,
\end{equation}
which may be integrated using the $_2F_1$ Hyper-Geometric function to yield
\begin{equation}
\label{cmasseqn}
M=\pi^{\frac{3}{2}} n_0 a_c^3 \left( \frac{\Gamma(\gamma/2 -1)}{\Gamma(2)}\right) \>\> ,
\end{equation}
where $\Gamma$ represents the Gamma function ($\Gamma(a+1)=a!$).
Equation \ref{cmasseqn} permits us to turn the attenuated power-law fit into a two parameter problem using a cloud mass measured from the column density map.
$\gamma$ and $n'_0$ seemed the most appropriate free parameters to use. It can be shown using the derivation in Section \ref{cderivation} that it is appropriate to use $n'_0$ and $a'_c$ along with the cloud mass even though geometric information is entangled in those parameters.
\begin{figure}
\includegraphics[scale=0.65]{cfig10.eps}
\caption{The results of the attenuated power-law fit for L466. The green region is ignored in the calculations. a) Image of the fit residuals on a map as a function of $\gamma$ and $n'_0$. The cross represents the position of the best fit with the lowest residual. The black contour represents the $1\sigma$ uncertainty for the fit. The blue and red contours represent the uncertainties considering masses respectively $10\%$ lower, and higher than the measured value. The straight lines represent positions where $a'_c$ equals 0.1, 0.2, and 0.3 parsecs in a clockwise order. They represent the likely size of the cloud's core assuming the attenuated power-law is a correct fit. b) The derived density profile (points) along with the fitted attenuated power-law (solid line) and the $1\sigma$ uncertainty (dashed lines).
\label{cfig10}}
\end{figure}
\begin{figure}
\includegraphics[scale=0.65]{cfig11.eps}
\caption{The results of the attenuated power-law fit for B133 presented in the same form as in Figure \ref{cfig10}.
\label{cfig11}}
\end{figure}
Figures \ref{cfig10} and \ref{cfig11} present the results of such fits. The left panel in each figure represents the residuals map. The best fit for values of $n'_0$ and $\gamma$ is represented by a cross. The contours represent the $1\sigma$ uncertainties calculated from the residuals and are not necessarily symmetric. There may be some uncertainty in determining the masses of these clouds from the column density maps, due to background extinction, biases, uncertainty in the dust to gas ratio, distance estimates, and ambiguity in defining the edges of each cloud. These uncertainties may combine for several tens of percent in some cases. The blue and red contours correspond to alternate solutions with masses $10\%$ lower, and higher from the measured value to illustrate the effect. Underestimating a cloud's mass will result in a lower modelled value of $\gamma$. While mass constrains the value of $a'_c$ for any given $n'_0$ and $\gamma$, lines where $a'_c$ equals 0.1, 0.2, and 0.3 parsecs are drawn with the 0.1 parsec line closest to the vertical axis. The best-fit curves (solid line) are plotted in the right panels over the derived profiles, with the $1\sigma$ uncertainty bounds marked by dashed lines. The uncertainty bounds are not symmetric as they are not necessarily gaussian and are calculated using all possible solutions, not just the best fit.
In both clouds, the exterior (blue) and interior (red) regions can be very well fit by an attenuated power law. It is a peculiarity of the attenuated power-law model that there may be a great deal of uncertainty in the values of $\gamma$ which, in combination with $a'_c$, and $n'_0$ can produce very similar results. The greatest uncertainty in the fit occurs in the innermost core of each cloud, as expected. There is nothing in the data which would suggest that the innermost regions of the clouds follow the fitted profile, as there is no data there. However, these fits do show that an attenuated power-law could explain why two different regions within the same cloud could appear to follow very different localized power-laws.
\section{Previous Methods and Practices}
\label{ccomparison}
This technique may be applied to many fields of study.
In the previous section we examined the total proton volume density radial profiles in molecular cloud cores.
To our knowledge, no one has previously applied a geometry--independent method for measuring such volume density radial profiles.
The imposition of geometric assumptions has been especially problematic
since molecular clouds rarely resemble simple geometries.
In fact, most studies involving the internal structure of molecular clouds have limited themselves to studying the distribution of the observed column densities.
Due to the desire to have more direct information on the internal structure of these clouds, various methods have been used to get volume density estimates despite their inherent limitations.
The simplest and most common method involves estimating the object's shape from a column density map to arrive at an educated guess for its depth along one or more lines of sight \citep{Liu2012,Wu2012}.
This method is usually used only along a single line of sight, typically the center, due to the uncertainties in estimating an object's three-dimensional shape.
This method yields only the mean volume density along the line of sight, and is directly influenced by geometric assumptions.
It yields no information on how volume density varies within a cloud.
Understanding the internal structure of clouds allows us to determine the relevant physics and chemistry.
Many studies have sought to do this by comparing an assumed cloud geometry to an observed column density map in order to produce a best-fit radial profile function.
Spheres and ellipsoids are most commonly assumed.
Early examples are the Bonnor-Ebert Sphere \citep{Bonnor1956, Ebert1955} or the study by \cite{Arquilla1985}. More recent examples of this methodology are found in \cite{Pineda2010}, and \cite{Dapp09}.
The primary advantage of this method is that it yields a volume density radial profile.
However, it is often difficult to reconcile the idealized geometric assumptions with the actual objects studied.
\cite{Alves2001a} found that B68 seemed to provide an excellent fit to the Bonnor-Ebert Sphere in their survey.
Any deviations from the assumed geometry manifest themselves in the derived radial profiles as bias in ways that are often unpredictable, and are thus frequently ignored or simply misinterpreted as uncertainties.
We avoid such biases by discarding assumptions on the object's shape.
As a result, the afore-mentioned $\epsilon$ discontinuities become readily apparent and allow us to avoid regions where the self-similarity assumption fails.
\section{Conclusions}
\label{cdiscussion}
This paper has presented a novel new method for determining the forms of the radial volume density profiles of objects such as molecular cloud cores without making assumptions about their geometry. While the method has been applied here only to dust extinction maps of molecular clouds, it is highly generalized and may be applied to any objects, and any observable quantities that satisfy the assumptions in Section \ref{cassumptions}. Those assumptions may be briefly summarized as requiring that the object can be described using a single radial profile function as well as the validity of Equation \ref{csimplen}. As such, this method may be widely useful in a number of fields. The method relies on using only a column density map, which necessarily cannot uniquely define a three-dimensional object whose geometry is unknown. It is a fortunate mathematical peculiarity that makes this method possible in that all of the object's geometric information, and no information about the form of the radial profile function are embedded into two dependent scalars ($G$ and $\chi$). These constants scale the derived $n'(r')$ profile to the cloud's original function $n(r_{rc})$. Values for $G$ and $\chi$ can only be determined with knowledge of an object's geometry. However, the form of the derived radial profile can be derived accurately independent of geometry within the bounds of the assumptions in Section \ref{cassumptions}.
Those methods which rely on geometric assumptions necessarily introduce an often significant, yet difficult to predict bias due to deviations from an idealized geometry. Our method yields the maximum amount of information attainable without the introduction of such a bias.
Our method is limited in several ways, the chief of which is the presence of $\epsilon$ discontinuities which arise due to variations in contour shapes as a result of the failure of assumption 2. Figure \ref{cfig4}b
a cloud core for which self-similarity is satisfied only its the outer-most and central regions
Regions of the derived profile immediately interior to the $\epsilon$ discontinuities are affected, while exterior regions and those sufficiently far to the interior of the discontinuities still maintain their form.
If one were to assume the simulated object in Figure \ref{cfig4}a were a sphere, the $\epsilon$ discontinuities would still be presented, but manifested in a less obvious, and more unpredictable manner. Despite its limitations, our method presents the best option for discerning the form of the radial profile in those situations where it is suitable for use.
Section \ref{cderivation} presents an analytic derivation from basic principles, while Section \ref{cnumeric} bolsters the derivation through tests using simulated data. All the clouds studied here exhibited similar behavior with power laws present in each. All clouds exhibited power-laws with similar slopes ranging from $-1.8$ to $-2.5$ in their middle regions. We believe that these power-laws are not artifacts of our method since analytic derivation, and numeric simulations show that the technique should be capable of properly deriving any kind of radial profile function.
We chose to demonstrate the technique using 2MASS data on molecular cloud cores.
The total proton density is of particular interest as it contains information on support mechanisms as well the formation rates of molecules such as H$_2$, or $^{13}$CO.
While we did not display the data for all clouds studied in this paper, it seems that quite often there are regions where the cloud's column density contours exhibit remarkable self-similarity accompanied by sharp changes in contour shape ($\epsilon$ discontinuities) between regions.
Our method shows that there does not appear to be a gradual change in the derived local power laws, but rather sudden shifts where the interior profile may follow a power law of $~2$, followed by an $\epsilon$ discontinuity, and a much steeper power law of $~4$ in the exterior regions. That this sudden break accompanies a stark shift in contour shapes is intriguing. With only two isolated clouds, there is insufficient data with which to draw general conclusions, and that is beyond the scope of demonstration of the technique.
If it can be proven that this is a common characteristic of isolated molecular cloud cores and is not some kind of artifact of the column density maps or the technique for deriving radial profiles, then is there a real effect which produces a sudden change in the density behavior of these clouds. One possibility may be that the properties of the dust particles change at lower densities, thus producing a sharper drop in observed extinction, or one of the cloud's support mechanisms ceases to be effective at a certain point leading to a steeper drop in density in the exterior. It may also be possible that the interior of the cloud is undergoing gravitational collapse while the exterior is not.
We can say at the present time that we have no reason to believe that this change in local power laws is due to biases within the data or our reduction technique within the bounds of the limitations discussed throughout the paper as we have tested the technique using a variety of geometries, profile forms, beam widths, and reduction techniques. There is nothing in the analytical derivation suggesting that such a phenomenon should be produced as a side-effect. Furthermore, while previous studies have found that the localized power law seems to change within different regions of individual cloud cores, these studies may not have been able to discern how sharp the change is due to their use of geometric assumptions which obscure the manifestation of $\epsilon$ discontinuities.
\section{Acknowledgements}
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
We wish to thank Jorge Pineda for useful discussions.
This research was carried out in part at the Jet Propulsion Laboratory operated for NASA by the California Institute of Technology.
We thank the Hayden Planetarium, and Rebecca Oppenheimer in particular for generously providing a conducive environment in which a portion of this research was carried out. We appreciate the very insightful comments and suggestions from the anonymous reviewer that significantly improved this paper.
|
2,869,038,155,641 | arxiv | \section{Introduction}
Substantial progress has been accomplished in this last decade on our
theoretical understanding of the acceleration of particles at
relativistic shocks, revealing in more than one place crucial
differences with Fermi acceleration at non-relativistic shock
waves. For instance, Gallant \& Achterberg (1999), Achterberg et
al. (2001) have emphasized the strong anisotropy of the cosmic ray
population propagating upstream, which is directly related to the fact
that the relativistic shock wave is always trailing right behind the
accelerated particles. These particles are confined into a beam of
opening angle $\theta\,\lesssim\,1/\Gamma_{\rm sh}$ (with $\Gamma_{\rm
sh}$ the Lorentz factor of the shock wave in the upstream frame) and
are overtaken by the shock wave on a timescale $r_{\rm L}/\Gamma_{\rm
sh}$, with $r_{\rm L}$ the typical Larmor radius of these particles
in the background magnetic field. One consequence of the above is to
restrict the energy gain per
up$\,\rightarrow\,$down$\,\rightarrow\,$up cycle, $\Delta E/E$, to a
factor of order unity. Early Monte Carlo numerical experiments
nonetheless observed efficient Fermi acceleration, with a generic
spectral index $s=2.2-2.3$ in the ultra-relativistic limit (Bednarz \&
Ostrowski 1998, Achterberg et al. 2001, Lemoine \& Pelletier 2003,
Ellison \& Double 2004), in agreement with semi-analytical studies
(Kirk et al. 2000) and analytical calculations (Keshet \& Waxman
2005). This value of the spectral index is however restricted to the
assumption of isotropic turbulence both upstream and downstream of the
shock (Niemiec \& Ostrowski 2004; Lemoine \& Revenu 2006), whereas the
shock crossing conditions imply a mostly perpendicular magnetic field
downstream, which severely limits the possibility of downstream
scattering. Furthermore, it was later stressed by Niemiec \& Ostrowski
(2006) and Lemoine, Pelletier \& Revenu (2006) that these early
studies implicitly ignored the correlation between the upstream and
downstream particle trajectories during a cycle. In particular, the
former numerical study demonstrated that Fermi acceleration became
inefficient if the proper shock crossing conditions were applied to
the background magnetic field. This result was demonstrated
analytically in the latter study, concluding that Fermi acceleration
could only proceed if strong turbulence ($\delta B \gg B$) existed on
a scale much smaller than the typical larmor radius. The addition of
turbulence on large scales $\,\gg\,r_{\rm L}$ does not help in this
respect, as the particle then experiences a roughly coherent field on
the short length scales that it probes during its cycle. Further
studies by Niemiec, Ostrowski \& Pohl (2006) have confirmed that Fermi
acceleration proceeds if short scale turbulence is excited to high
levels, either downstream or upstream. The detailed conditions under
which Fermi acceleration can proceed have been discussed analytically
in Pelletier, Lemoine \& Marcowith (2009); they are found to agree
with the numerical results of Niemiec, Ostrowski \& Pohl (2006).
Amplification of magnetic fields on short spatial scales thus appears
to be an essential ingredient in Fermi processes at ultra-relativistic
shock waves. Quite interestingly, strong amplification has been
inferred from the synchrotron interpretation of gamma-ray burst
afterglows, downstream at the level of $\delta B/B \,\gtrsim\,
10^4-10^5$ (Waxman 1997; see Piran 2005 for a review), and upstream
with $\delta B/B \,\gtrsim\, 10^2-10^3$ (Li \& Waxman 2006), assuming
an upstream magnetic field typical of the interstellar
medium. Understanding the mechanism by which the magnetic field gets
amplified is crucial to our understanding to relativistic Fermi
acceleration, since the nature of this short scale turbulence will
eventually determine the nature of scattering, hence the spectral
index and the acceleration timescale.
Concerning the amplification of the downstream magnetic field, the
Weibel two stream instability operating in the shock transition layer
has been considered as a prime suspect (Gruzinov \& Waxman 1999,
Medvedev \& Loeb 1999, Wiersma \& Achterberg 2004, Achterberg \&
Wiersma 2007, Achterberg, Wiersma \& Norman 2007; Lyubarsky \& Eichler
2006). Several questions nevertheless remain open. For instance,
Hededal \& Nishikawa (2005) and Spitkovsky (2005) have observed, by
the means of numerical simulations that this instability gets quenched
when the magnetization of the upstream field becomes sufficiently
large. On analytical grounds, Wiersma \& Achterberg (2004), Achterberg
\& Wiersma (2007), and Lyubarsky \& Eichler (2006) have argued that it
saturates at a level too low to explain the gamma-ray burst
afterglow. The long term evolution of the generated turbulence also
remains an open question, although Medvedev et al. (2005) claim to see
the merging of current filaments into larger filaments through
dedicated numerical experiments.
Regarding upstream instabilities, the relativistic generalization of
the non-resonant Bell instability has been investigated by
Milosavljevi\'c \& Nakar (2006) and Reville, Kirk \& Duffy (2006) in the
case of parallel shock waves. However, ultra-relativistic shock waves
are generically superluminal, with an essentially transverse magnetic
field in the shock front. For this latter case, Pelletier, Lemoine \&
Marcowith (2009) have shown that the equivalent of the Bell
non-resonant instability excites magnetosonic compressive modes and
saturates at a moderate level $\delta B/ B\,\sim\,1$ in the frame of
the linear theory.
In recent years, particle-in-cell (PIC) simulations have become a key
tool in the investigation of these various issues. Such simulations go
(by construction) beyond the test particle approximation and may
therefore probe the wave -- particle relationship, which is central to
all of the above issues. Of course, such benefice comes at the price
of numerical limitations of the simulations, both in terms of
dimensionality and of dynamic range, which in turns impact on the mass
ratios accessible to the computation. Nonetheless, early PIC
simulations have been able to simulate the interpenetration of
relativistic flows and to study the development of two stream
instabilities at early times, see e.g. Silva et al. (2003),
Frederiksen et al. (2004), Hededal et al. (2004), Dieckmann (2005),
Dieckmann, Drury \& Shukla (2006), Dieckmann, Shukla \& Drury (2006),
Nishikawa et al. (2006), Nishikawa et al. (2007) and Frederiksen \&
Dieckmann (2008) for unmagnetized colliding plasma shells, and
Nishikawa et al. (2003), Dieckmann, Eliasson \& Shukla (2004a, b),
Nishikawa et al. (2005) and Hededal \& Nishikawa (2005) for studies of
the magnetized case. The formation of the shock itself has been
observed for both electron-positron and electron-proton plasmas thanks
to recent simulations that were able to carry the integration on to
longer timescales, see e.g. Spitkovsky (2005), Kato (2007), Chang et
al. (2008), Dieckmann, Shukla \& Drury (2008), Spitkovsky (2008a, b),
Keshet et al. (2009). All of the above studies use different
techniques for the numerical integration, and varying parameters
(dimensions, composition, mass ratios, density ratios of the colliding
plasmas and relative Lorentz factors) in order to examine different
aspects of the instabilities to various degrees of accuracy and over
different timescales.
Several of these studies have reported hints for particle acceleration
through non Fermi processes (Dieckmann, Eliasson \& Shukla 2004b;
Frederiksen et al. 2004; Hededal et al. 2004; Hededal \& Nishikawa
2005; Nishikawa et al. 2005; Dieckmann, Shukla \& Drury 2006,
2008). Concrete evidence for Fermi acceleration, i.e. particles
bouncing back and forth across the shock wave has come with the recent
simulations of Spitkovsky (2008b), and was studied in more details for
both magnetized and unmagnetized shock waves in Sironi \& Spitkovsky
(2009). In particular, this latter study has demonstrated the
inefficiency of Fermi acceleration at high upstream magnetization in
the superluminal case, along with the absence of amplification of the
magnetic field (thus in full agreement with the calculations of
Lemoine, Pelletier \& Revenu 2006). This result is particularly
interesting, because it suggests that the magnetization of the
upstream plasma, in limiting the length of the precursor, may hamper
the growth of small scale magnetic fields, and therefore inhibit Fermi
cycles. Finally, the long term simulations of Keshet et al. (2009)
have also observed a steady development of turbulence upstream of the
shock wave, suggesting that as time proceeds, particles are
accelerated to higher and higher energies and may thus stream further
ahead of the shock wave. We will discuss this issue as well at the end
of the present work.
The main objective of this paper is to undertake a systematic study of
micro-instabilities in the upstream medium of a relativistic shock
wave. We should emphasize that we assume the shock structure to exist
and we concentrate our study on the shock transition region where the
incoming upstream plasma collides with the shock reflected and shock
accelerated ions that are moving towards upstream infinity. Therefore,
care should be taken when confronting the present results to the above
numerical simulations which reproduce the collision of two neutral
plasma flows in order to study the development of instabilities that
eventually lead to the formation of the shock (through the
thermalization of the electron and ion populations). The physical
set-up that we have in mind matches best that obtained in the
simulations of shock formation and particle acceleration described in
Spitkovsky (2008b), Keshet et al. (2009) and Sironi \& Spitkovsky
(2009), or that simulated in Dieckmann, Eliasson and Shukla (2004a, b)
and Frederiksen \& Dieckmann (2008), or that studied in Medvedev \&
Zakutnyaya (2008). Our approach also rests on the following
observation, namely that in the ultra-relativistic limit, the
accelerated (or the reflected) particle population essentially behaves
as an {\em unmagnetized cold beam of Lorentz factor $\sim\,\Gamma_{\rm
sh}^2$}.
In the present paper, we assume the beam to be carrying a weak current
and in so doing, we neglect electromagnetic current instabilities. We
will nevertheless include in our summary of instabilities the
relativistic generalization of the Bell current instability (Bell
2004), since it has been studied in detail in several recent studies
(Milosavljevi\'c \& Nakar 2006; Reville, Kirk \& Duffy 2006). The
instability triggered by the cosmic-ray current in the case of oblique
shock waves has also been discussed in the relativistic regime in
Pelletier, Lemoine \& Marcowith (2009). Note also that in the case of
pair plasmas, electromagnetic current instabilities do not take place
as the beam remains neutral.
The layout of the present paper is as follows. We examine the
instabilities triggered by this beam, considering in turn the cases of
an unmagnetized upstream plasma (Section~3) and that of a magnetized
plasma (Section~4). In Section~5, we discuss the intermediate limit
and construct a phase diagram indicating which instability prevails as
a function of shock Lorentz factor and magnetization level. We then
discuss the possibility of Fermi acceleration in the generated
turbulence and apply these results to the case of gamma-ray bursts
shock waves and pulsar winds. We will recover the trend announced
above, namely that a magnetized upstream medium inhibits the growth of
the magnetic field hence particle acceleration. In Section~2, we first
discuss the general structure of a collisionless shock, in the case of
a electron--proton plasma with a quasi perpendicular mean field,
borrowing from analyses in the non-relativistic limit.
\section{General considerations}
\subsection{On the configuration of a relativistic collisionless shock wave}
A collisionless shock is built with the reflection of a fraction of
incoming particles at some barrier, generally of electrostatic or
magnetic nature. Let us sketch the general picture, borrowing from
model of non-relativistic collisionless electro-ion shocks (see
e.g. Treumann \& Jaroschek 2008a, b for recent reviews). In an
electron--proton- plasma carrying an oblique magnetic field, one
expects a barrier of both electrostatic and magnetic nature to
rise. Because the magnetic field is frozen in most part of the plasma,
its transverse component is amplified by the velocity decrease. This
in itself forms a magnetic barrier which can reflect back a fraction
of the incoming protons. Similarly, the increase of electron density
together with the approach of the electron population towards
statistical equilibrium is concomittant with the rise of an
electrostatic potential such that $e\Phi \simeq T_e \log (n/n_{\rm
u\vert\rm sh})$ ($n$ is the local density in the front frame, and
$n_{\rm u\vert\rm sh}$ the upstream incoming density viewed in the
front frame). The electron temperature is expected to grow to a value
comparable to, but likely different from that of protons, which
reaches $T_p \,\sim\, \left(\Gamma_{\rm sh}-1\right)m_pc^2$. The
electrostatic barrier thus allows the reflection of a significant part
of the incoming protons since $e \Phi \,\sim\, \left(\Gamma_{\rm
sh}-1\right)m_pc^2$. Although it reflects a fraction of protons, it
favors the transmission of electrons that would otherwise be reflected
by the magnetic barrier. The reflection of a fraction of the protons
ensures the matter flux preservation against the mass density increase
downstream. However because the magnetic field is almost transverse,
an intense electric field $E \,=\,\beta_{\rm sh} B$ energizes these
reflected protons such that they eventually cross the
barrier. Interactions between the different streams of protons are
then expected to generate a turbulent heating of the proton
population, which takes place mostly in the so-called ``foot''
region. This foot region extends from the barrier upstream over a
length scale (in the shock front frame, as indicated by the
$_{\vert\rm sh}$ subscript) $\ell_{\rm F\vert\rm sh} = r_{\rm L\vert
sh}$, where $r_{\rm L\vert\rm sh}$ denotes the Larmor radius of the
reflected protons.
Entropy production in the shock transition region comes from two
independent anomalous (caused by collisionless effects) heating
processes for electrons and ions. The three ion beams in the foot
(incoming, reflected in the foot and accelerated) interact through the
``modified two stream instability'', which seemingly constitutes the
main thermalisation process of the ion population. A careful
description of these anomalous heating processes certainly requires an
appropriate kinetic description. For the time being, we note that the
growth of the ion temperature develops on a length scale $\ell_{\rm
F}$. The temperature of the electrons rather grows on a very short
scale scale $\ell_{\rm R}\,\ll\,\ell_{\rm F}$ which defines the
``ramp'' of the shock. In non-relativistic shocks, electrons reach a
temperature larger than ions; however we do not know yet whether this
is the case in relativistic shocks. These electrons also experience
heating in the convection electric field. Moreover, due to the strong
gradient of magnetic field, an intense transverse electric current is
concentrated, inducing anomalous heat transfer through the
ramp. Probably an anomalous diffusion of electron temperature occurs
that smoothes out the temperature profile; however it has not been
identified in relativistic shocks. Electron heating is described by
Ohm's law in the direction of the convection electric field (in the
$\mathbf{x\times B}$ direction, taken to be $\mathbf{z}$):
\begin{equation}
\label{eq:jh}
\beta_x B + E = \frac{\eta c}{4\pi} \frac{{\rm d}B}{{\rm d}x}\ ,
\end{equation}
with $\beta_x<0$ in the shock front frame, $E\,=\,\beta_{\rm sh}B_{\rm
u}$, $B_{\rm u}$ denoting the background magnetic field at
infinity. The magnetic field profile can be obtained by prescribing a
velocity profile going from $-\beta_{\rm sh}c\sim -c$ to $\simeq-c/3$
over a distance much larger than $\ell_{\rm R}$. The profile displays
a ramp at scale $\ell_{\rm R}$ followed by an overshoot before
reaching the asymptotic value $3B_{\rm u}$. The above result indicates that
the relevant scale for $\ell_{\rm R}$ is the relativistic resistive
length:
\begin{equation}
\ell_{\rm R} \sim \frac{\eta c}{4\pi} = \delta_{\rm e}
\frac{\nu_{\rm eff}}{\omega_{\rm pe}}\ .
\end{equation}
This is a very short scale not larger than the electron inertial
length $\delta_{\rm e}\,\equiv\,c/\omega_{\rm pe}$ even when the anomalous
resistivity is so strong that the effective collision frequency
$\nu_{\rm eff}$ is of order $\omega_{\rm pe}$. This scale thus represents the
growth scale of three major quantities, namely, the potential, the
magnetic field and the electron temperature. It is of interest to
point out that this scale always remains much smaller than the foot
scale. Indeed, even if $\delta_{\rm e}$ is estimated with ultra-relativistic
electrons of relativistic mass $\Gamma_{\rm sh} m_p$,
i.e. $\delta_{\rm e}\,=\, \left[\Gamma_{\rm sh} m_p c^2/ (4\pi n_{e\vert\rm
sh} e^2)\right]^{1/2}$, it remains smaller than the foot length,
since
\begin{equation}
\label{eq:2}
\frac{\delta_{\rm e}}{\ell_{\rm F\vert\rm sh}}\,=\, \left(\frac{B_{\vert\rm
sh}^2}
{4\pi n_{e\vert\rm u} \Gamma_{\rm sh}^2 m_p c^2}\right)^{1/2} \,\ll\,1 \ ,
\end{equation}
using the value of $\ell_{\rm F\vert\rm sh}$ for particles with typical
energy $\Gamma_{\rm sh}m_pc^2$ in the shock front. The last inequality
in the above equation is a natural requirement for a strong shock.
The downstream flow results from the mixing of the flow of first
crossing ions (adiabatically slowed down) with the flow of transmitted
ions after reflection. All the ingredients of a shock are then
realized.
In the case of an electron-positron plasma, when a magnetic field is
considered, no electrostatic barrier rises, only the magnetic barrier
appears. However, if the mean magnetic field is negligible, a barrier
can rise only through the excitation of waves, as demonstrated by the
PIC simulations discussed above.
The structure is thus described by two scales $\ell_{\rm R}$ and
$\ell_{\rm F}$ and three small parameters: $\xi_{\rm cr}$, the
fraction of thermal energy density behind the shock converted into
cosmic ray energy, $\sigma_B$ the ratio of magnetic energy density
over the incoming energy density and $1/ \Gamma_{\rm sh}$.
\subsection{Particle motion}
As mentioned above, there are three particle populations in the foot:
the cold incoming particles, the reflected protons, and the
accelerated particle population which has undergone at least one
up$\,\rightarrow\,$down$\,\rightarrow\,$up cycle. This latter
population arrives upstream with a typical Lorentz factor
$\Gamma_{\rm b}\,\sim\,\Gamma_{\rm sh}^2$, with a typical relative spread
of order unity. The second population of reflected protons also
carries an energy $\simeq\Gamma_{\rm sh}^2m_pc^2$, since these
particles have performed a Fermi-like cycle, albeit in the front
rather than downstream. Therefore one can treat these two populations
as a single beam. From the point of view of the instabilities, one can
approximate this beam as cold, with momentum distribution $\propto
\delta\left(p_x-\Gamma_{\rm
sh}^2m_pc\right)\delta\left(p_\perp\right)$. Indeed, the
instabilities are governed by the beam velocity, the dispersion of
which remains very small, being of order
$\Delta\beta_{\rm b}\,\sim\,-(2/\Gamma_{\rm b}^2)\Delta\Gamma_{\rm b}/\Gamma_{\rm b}$.
In order to verify this, one writes the susceptibility of the beam,
assuming as above that it is unmagnetized on the scale of the
instabilities (Melrose 1986):
\begin{eqnarray}
\chi^{\rm b}_{ij}&\,=\,&-\frac{4\pi e^2}{m_p \omega^2}\int \frac{{\rm
d}^3p}{\gamma}\,
f_{\rm b}(\mathbf{p})\nonumber\\ &&\,\,\,\times\left[\delta_{ij}+ \frac{k_i
c\beta_{j}+ k_jc \beta_{i}}{\omega- \mathbf{k}\cdot
\bfs{\beta}c} + \frac{(k^2c^2-\omega^2)\beta_{i}\beta_{j}}{\left(\omega -
\mathbf{k}\cdot\bfs{\beta}c\right)^2}\right] \ ,\label{eq:beams}
\end{eqnarray}
with $p=\beta\gamma m_pc$ and $f_{\rm b}(\mathbf{p})$ the distribution
function of the beam. Since the velocity distribution of the beam is
essentially delta like, one may then indeed approximate the above beam
susceptibility with that of a cold beam; the difference amounts to a
redefinition of the beam plasma frequency by a factor of order unity.
Another crucial length scale in our study is the length scale of the
precursor. As discussed above, this length scale $\ell_{\rm F\vert
sh}=r_{\rm L\vert\rm sh}$ in the front shock in the case of a
magnetized shock wave. In the upstream frame, this can be rewritten
as:
\begin{equation}
\label{eq:lfb}
\ell_{\rm F \vert\rm u} \,\simeq\, \frac{r_{\rm L\vert\rm u}}{\Gamma_{\rm sh}^3}
\,=\,
\frac{c}{\omega_{\rm ci} \Gamma_{\rm sh} \sin \theta_B} \,\quad
(B_{\rm u}\neq 0)\ .
\end{equation}
We assume that the field is almost perpendicular in the front frame,
but in the upstream comoving frame we consider its obliquity (angle
$\theta_B$ with respect to the shock normal), assuming that $\sin
\theta_B > 1/\Gamma_{\rm sh}$. The particular case of a parallel shock
wave for which $\Gamma_{\rm sh}\sin \theta_B < 1$ is discussed in
Section~\ref{sec:parshock}; there it will be shown that a fraction
$\left(1-\Gamma_{\rm sh}\theta_B\right)^2$ of the particles that
return upstream may actually propagate to upstream infinity in the
limit of a fully coherent upstream magnetic field, while
Eq.~(\ref{eq:lfb}) remains correct for the rest of the accelerated
particle population. The size of the precursor for the particles that
escape away is eventually given by the level of turbulence ahead of
the shock wave.
In the case of an unmagnetized shock wave, the size of the precursor
is determined by the length traveled by the reflected protons in the
self-generated short scale turbulence. Neglecting for simplicity the
influence of the short scale upstream electric fields (we will see in
Section~\ref{sec:Fermi} that this does not affect the following
result), this length scale can be written (Milosavljevi\'c \& Nakar
2006; Pelletier, Lemoine \& Marcowith 2009):
\begin{equation}
\label{eq:lfu}
\ell_{\rm F\vert\rm u}\,\simeq\, \frac{r_{\rm L\vert\rm u}^2}{\Gamma_{\rm
sh}^4\ell_{\rm c}}
\,\simeq\,\frac{c^2}{\omega_{\rm ci}^2\ell_{\rm c}}\ ,
\end{equation}
where $\ell_{\rm c}$ represents the typical scale of short scale
magnetic fluctuations. Whether one or the other formula applies
depends on several possible situations and outcomes: if the shock is
magnetized and one considers the first generation of cosmic rays, one
should use Eq.~(\ref{eq:lfb}); if the shock is magnetized and one
assumes that a stationary state has developed with strong
self-generated turbulence, one should use Eq.~(\ref{eq:lfu});
obviously, if the development of the turbulence cannot take place, one
should rather use Eq.~(\ref{eq:lfb}); finally, for an unmagnetized
shock, Eq.~(\ref{eq:lfu}) applies. In the following, we discuss the
turbulence growth rate for these different cases.
There seems to be a consensus according to which magnetic fluctuations
have to be tremendously amplified through the generation of cosmic
rays upstream in order for Fermi acceleration to proceed. A fraction
$\xi_{\rm cr}$ of the incoming energy is converted into cosmic rays
and a fraction of this cosmic rays energy is converted into
electromagnetic fluctuations, which add up to a fraction $\xi_{\rm
em}$ of the incoming energy. This process is expected to develop
such that the generation of cosmic rays allows the generation of
electromagnetic waves that in turn, through more intense scattering,
allows further cosmic ray acceleration and so on until some saturation
occurs. We write the quantities $\xi_{\rm cr}$ and $\xi_{\rm em}$ as:
\begin{equation}
\label{eq:defxi}
\xi_{\rm cr}\,\equiv\, \frac{P_{\rm cr}}{\Gamma_{\rm sh}^2n_{\rm u}m_pc^2}\ ,\quad
\xi_{\rm em}\,\equiv\, \frac{U_{\rm em}}{\Gamma_{\rm sh}^2 n_{\rm u}m_pc^2}\ ,
\end{equation}
with $\xi_{\rm em}\,<\,\xi_{\rm cr}$. We approximate the beam
pressure with that of the cosmic rays, i.e. $P_{\rm cr}\,\approx\,
\Gamma_{\rm sh}n_{{\rm b}\vert\rm sh}m_p c^2$ for the first generation of
accelerated particles, as expressed in the shock front frame. The
electromagnetic energy density is written $U_{\rm em}$ in the same
frame, as usual.
Unless otherwise noted, our discussion takes place in the upstream
rest frame in what follows.
\section{Upstream instabilities in the absence of a mean magnetic field}
When the ambient magnetic field can be neglected or is absent, the
reflected particles and the fraction of particles that participate to
the first Fermi cycle constitute a relativistic cold beam that
pervades the ambient plasma and trigger three major
micro-instabilities. One is the two stream electrostatic instability,
which amplifies the electrostatic Langmuir field through a
\v{C}erenkov resonant interaction $\omega - \mathbf{k \cdot v}_{\rm b} =
0$, with
$\mathbf{k}\parallel\mathbf{E}\parallel\mathbf{v}_{\rm b}$. Another is the
Weibel instability, with
$\mathbf{k}\parallel\mathbf{v}_{\rm b}\perp\mathbf{E}$ and its analog
filamentation instability, with
$\mathbf{k}\perp\mathbf{v}_{\rm b}\parallel\mathbf{E}$ (Bret, Firpo \&
Deutsch 2004, 2005a, 2005b; see also Bret 2009 for a recent
compilation). These two instabilities are non-resonant and mostly
electromagnetic with a low phase velocity so that the magnetic
component of the wave is dominant. It is thus particularly relevant
for developing particle scattering. Finally, these authors have also
discovered an oblique resonance which grows faster than the above
two. It is mostly longitudinal (see further below) but $\mathbf{k}$ is
neither perpendicular nor parallel to the beam. These growth rates are
easily recovered as follows.
For a cold beam, Eq.~(\ref{eq:beams}) gives the following
susceptibility:
\begin{equation}
\label{eq:chib}
\chi^{\rm b}_{ij} = -\frac{\omega_{{\rm
p}{\rm b}}^2}{\omega^2}\Biggl[\delta_{ij} + \frac{k_i c\beta_{{\rm b}j}+
k_jc \beta_{{\rm b}i}}{\omega- \mathbf{k}\cdot \bfs{\beta}_{\rm b}c} +
\frac{(k^2c^2-\omega^2)\beta_{{\rm b}i}\beta_{{\rm b}j}}{\left(\omega -
\mathbf{k}\cdot\bfs{\beta}_{\rm b}c\right)^2}\Biggr] \ .
\end{equation}
The beam propagates with velocity
\mbox{\boldmath{$\beta$}}$_{\rm b}c=\left(1-1/\Gamma_{\rm b}^2\right)^{1/2}\mathbf{x}$;
the relativistic beam plasma frequency (in the upstream frame) is
given by:
\begin{equation}
\omega_{{\rm p}{\rm b}}\,\equiv\, \left(\frac{4\pi n_{{\rm b}\vert\rm u}e^2}
{\Gamma_{\rm b}m_p}\right)^{1/2}\ ,
\end{equation}
recalling $\Gamma_{\rm b}\,\simeq\,\Gamma_{\rm sh}^2$. One can solve the
dispersion relation, including the beam response, to first order in
$\chi^{\rm b}$ since its contribution is of order:
\begin{equation}
\left(\frac{\omega_{{\rm p}{\rm b}}}{\omega_{\rm
pe}}\right)^2\,=\,\frac{m_e}{m_p}\xi_{\rm cr}\,\ll\,1 \ .
\label{eq:oo}
\end{equation}
\subsubsection{Weibel/filamentation instability}
Consider now a mode with $k_y=0$, but $k_x\neq0$, $k_z\neq0$. The
dispersion relation, including the beam response can be written as
follows, to first order in $\chi^{\rm b}_{ij}$:
\begin{eqnarray}
& & \left(\omega^2-\omega_{\rm
p}^2-k^2c^2-\chi^{\rm b}_{yy}\omega^2\right)\nonumber\\ & &
\times\Biggl[\left(\omega^2-\omega_{\rm p}^2-k_z^2c^2 +
\chi^{\rm b}_{xx}\right)\left(\omega^2-\omega_{\rm
p}^2-k_x^2c^2+\chi^{\rm b}_{zz}\right)\nonumber\\ & &
\,\,\,\,-\left(k_xk_zc^2 +
\chi^{\rm b}_{xz}\omega^2\right)^2\Biggr]\,=\,0 \ ,
\end{eqnarray}
with $\omega_{\rm p}^2\,\equiv\,\omega_{\rm pi}^2+\omega_{\rm pe}^2$. In the limit
$k_x\rightarrow 0$, one recovers the filamentation (Weibel like)
instability by developing the above dispersion relation to first order
in $\chi^{\rm b}$, with:
\begin{equation}
\omega^2\,=\,-\omega_{{\rm p}{\rm b}}^2\frac{k^2c^2}{\omega_{\rm p}^2+k^2c^2}\ .
\end{equation}
It saturates at a growth rate ${\cal I}(\omega_{\rm We.})\,
\simeq\,\omega_{{\rm p}{\rm b}}$ in the limit $kc\,\gg\,\omega_{\rm p}$.
\subsubsection{\v{C}erenkov resonance with oblique electrostatic modes}
In the other limit $k_z\rightarrow 0$, one can simplify the dispersion
relation for electrostatic modes down to:
\begin{equation}
\omega^2 - \omega_{\rm p}^2 + \chi_{xx}^{\rm b}\omega^2\,\simeq\,0\ .\label{eq:long1}
\end{equation}
Then, the two stream instability resonance condition between the
Langmuir modes and the beam reads:
\begin{equation}
\omega\,=\,\omega_{\rm p}\left(1+\delta\right)\,=\,
\beta_{\rm b}k_x c\left(1+\delta\right)\ ,\label{eq:res1}
\end{equation}
with by assumption $\vert\delta\vert\,\ll\,1$. After insertion into
Eq.~(\ref{eq:long1}), this yields:
\begin{equation}
\delta^3\,=\,\frac{\omega_{{\rm p}{\rm b}}^2}{2\Gamma_{\rm b}^2\omega_{\rm p}^2}\ ,
\end{equation}
hence a growth rate:
\begin{equation}
{\cal I}(\omega)\,\simeq\,\frac{\sqrt{3}}{2^{4/3}}
\left(\frac{\omega_{{\rm p}{\rm b}}^2\omega_{\rm
p}}{\Gamma_{\rm b}^2}\right)^{1/3}\ .\label{eq:long}
\end{equation}
One should note that the \v{C}erenkov resonance can only take place
with plasma modes with phase velocity smaller than $c$ (refraction
index $kc/\omega(k)>1$), hence transverse modes are excluded in this
respect.
The oblique mode, with $k_z\neq0$ and a resonance as above yields a
growth rate that is larger by a factor $\Gamma_{\rm b}^{2/3}$ than
the two stream rate given in Eq.~(\ref{eq:long}) for $k_z=0$ (Bret,
Firpo \& Deutsch 2004, 2005a, b). This can be understood as
follows. The instability arises from the $xx$ component of the beam
susceptibility tensor, which dominates over the other components at
the resonance [see Eq.~(\ref{eq:chib})], and which reads:
\begin{equation}
\chi_{xx}^{\rm b}\,=\,-\frac{\omega_{{\rm p}{\rm b}}^2}{\omega^2}
\frac{\omega^2/\Gamma_{\rm b}^2 + \beta_{\rm b}^2k_z^2c^2}
{\left(\omega-\beta_{\rm b}k_x c\right)^2}\ .
\end{equation}
This component is suppressed by $1/\Gamma_{\rm b}^2$ when $k_z=0$,
which explains the factor appearing in the r.h.s. of
Eq.~(\ref{eq:long}). For $k_z\neq0$ however, the algebra is more
cumbersome. Nevertheless, proceeding as above, with the resonance
condition Eq.~(\ref{eq:res1}), one obtains in the limit
$\delta\,\ll\,1$ and $\beta_{\rm b}\simeq 1$:
\begin{equation}
\delta^3 \,\simeq\, \frac{\omega_{p{\rm b}}^2}{\omega_{\rm p}^2}\frac{k_z^2}{2k^2}\ .
\end{equation}
In the limit $k_z\,\gg\,k_x\,\simeq\,\omega_{\rm p}/c$, one recovers the
growth rate of the oblique mode:
\begin{equation}
{\cal I}(\omega)\,\simeq\,\frac{\sqrt{3}}{2^{4/3}}
\left(\omega_{{\rm p}{\rm b}}^2\omega_{\rm p}\right)^{1/3}\ .\label{eq:obl}
\end{equation}
This mode obviously grows faster than the previous two.
Obviously, the mode is quasi-longitudinal, since resonance takes place
with the electrostatic modes. However it also comprises a small
electromagnetic component, $\left\vert B_y\right\vert/\left\vert
E_z\right\vert\,\approx\,2\left\vert\delta\right\vert$, as can be
seen by solving for the eigenmode, using the full dispersion relation
including the beam contribution.
\section{Instabilities in the presence of a mean field}
As before, we look for an instability of the upstream plasma waves,
triggered by the beam of accelerated (and shock reflected)
particles. At non-relativistic shocks, one usually considers an
interaction at the Larmor resonance. However this cannot be relevant
in the ultra-relativistic case, because the interaction must develop
on a distance scale $\lesssim \ell_{\rm F}$ which is itself much
shorter than the Larmor radius.
The particular case of a relativistic
parallel shock wave will be briefly discussed thereafter. Note
finally that for the frequently valid condition $\beta_{\rm A}
\Gamma_{\rm sh} \sin \theta_B \ll 1$, the precursor has a length much
larger than the minimum scale for MHD description ($\ell_{\rm MHD}/
\ell_{\rm F\vert\rm u} = \beta_{\rm A} \Gamma_{\rm sh} \sin \theta_B $),
which justifies the resonance between the beam and the MHD modes.
\subsection{Oblique magnetic field}
In order to excite fast waves of frequency higher than the Larmor
frequency, we consider again the \v{C}erenkov resonance between the
non-magnetized beam and the magnetized plasma waves: $\omega -
\mathbf{k \cdot v}_{\rm b} = 0$. Let us recall that for a
ultra-relativistic beam, the velocity distribution is strongly peaked
at $v_{\rm b}\sim c$, even if the dispersion in Lorentz factor of the beam
is significant. We also discuss the possibility of generating the
magnetic field through a (non-resonant) Weibel (filamentation)
instability with $k_x=0$.
\subsubsection{Weibel -- filamentation instability}
This instability taking place in the shock transition layer between
the unshocked plasma and the shocked plasma has been discussed in
detail in the waterbag approximation for an unmagnetized plasma
(Medvedev \& Loeb 1999; Wiersma \& Achterberg 2004; Lyubarsky \&
Eichler 2006; Achterberg \& Wiersma 2007, Achterberg, Wiersma \&
Norman 2007). As we now argue, the Weibel instability can also proceed
in the regime of unmagnetized proton -- magnetized plasma electrons at
smaller frequencies, corresponding to the range $\omega_{\rm
ci}\,\ll\,\omega\,\ll\,\omega_{\rm ce}$ (see also Achterberg \&
Wiersma 2007). Again, we should stress that we consider a pure ion
beam (reflected and accelerated particles), whereas most above studies
consider two neutral interpenetrating plasmas.
To simplify the algebra, we write down the dispersion relation in a
frame in which the $(\mathbf{x},\mathbf{z})$ plane has been rotated in
such a way as to align $\mathbf B$ with the third axis, denoted
$\mathbf{z}_B$; $\mathbf y$ remains the second axis $\mathbf{y}_B$. To
simplify further a cumbersome algebra, we consider a wavenumber
$\mathbf{k}\parallel\mathbf{y}_B$, perpendicular to both the beam
motion and the magnetic field. The plasma di-electric tensor is
written in this $B$ frame as:
\begin{eqnarray}
\Lambda_{ij\vert B} \,=\,\left(\begin{array}{ccc}
\varepsilon_1-\eta^2 & i\varepsilon_2 & 0 \\ -i\varepsilon_2 &
\varepsilon_1 & 0 \\0 & 0 &
\varepsilon_{\parallel}-\eta^2\end{array}\right)\ ,\label{eq:plasmadiel}
\end{eqnarray}
with the following usual definitions (for $\omega_{\rm
ci}\,\ll\omega\,\ll\omega_{\rm ce}$):
\begin{equation}
\varepsilon_1\,\simeq\,1 - \frac{\omega_{\rm
pi}^2}{\omega^2}+\frac{\omega_{\rm pe}^2}{\omega_{\rm
ce}^2}\ ,\,\,\,\, \varepsilon_2\,\simeq\, \frac{\omega_{\rm
pe}^2}{\omega\omega_{\rm ce}}\ ,\,\,\,\,
\varepsilon_\parallel\,\simeq\,1-\frac{\omega_{\rm
p}^2}{\omega^2}\ .\label{eq:dielcomp-int}
\end{equation}
and $\eta\,\equiv\,kc/\omega$. One needs to rotate the beam
susceptibility tensor to this $B$ frame. The quantity of interest will
turn out to be the $3-3$ component
$\chi^{\rm b}_{z_Bz_B}\,=\,\cos^2\theta_B\chi^{\rm b}_{xx} +
\sin^2\theta_B\chi^{\rm b}_{zz}$. To first order in $\chi^{\rm b}$, the
dispersion relation indeed has the solution:
\begin{equation}
\varepsilon_\parallel - \eta^2 + \cos^2\theta_B\chi^{\rm b}_{xx} +
\sin^2\theta_B\chi^{\rm b}_{zz}\,=\,0 \ .
\end{equation}
Given the dependence of $\chi^{\rm b}_{xx}$ on $\omega$, this is a quartic
equation which admits the solution leading to the Weibel
(filamentation) instability:
\begin{equation}
\omega^2\,\simeq\,-\omega_{{\rm p}{\rm b}}^2\cos^2\theta_B
\frac{k^2c^2}{\omega_{\rm p}^2 + k^2c^2}\ .\label{eq:wb}
\end{equation}
As in the unmagnetized case, it saturates at a growth rate $\simeq
\omega_{{\rm p}{\rm b}}\cos\theta_B$ (up to the angular dependence on $B$). Note
that in the limit $\cos\theta_B\rightarrow 0$, this instability does
not disappear. In order to see this, one has to consider the other
branch of the dispersion relation, for $\cos\theta_B=0$,
$\mathbf{k}=k_z\mathbf{z}$:
\begin{equation}
\left(\epsilon_1-\eta^2 + \chi_{xx}^{\rm b}\right)
\left(\epsilon_1-\eta^2+ \chi_{yy}^{\rm b}\right) -\epsilon_2^2\,=\,0\ .
\end{equation}
One of the roots corresponds to the Whistler mode and the other to the
Weibel unstable mode with $\omega^2\,\simeq\,-\omega_{{\rm p}{\rm b}}^2$.
The above thus shows that fast waves can be excited by the
relativistic stream in the intermediate range between MHD and electron
dynamics, i.e. with unmagnetized plasma ions but magnetized
electrons. The typical length scale of these waves for which maximal
growth occurs is obviously the electron inertial scale $\delta_{\rm e}
\equiv c/\omega_{\rm p}$ as before.
\subsubsection{\v{C}erenkov resonance with longitudinal modes}
The previous discussion of the \v{C}erenkov instability with
electrostatic modes can be generalized to the magnetized plasma limit
by considering those modes with $\mathbf{k}\parallel\mathbf{B}$, which
do not feel the magnetic field (see Lyubarsky 2002 for a discussion of
this instability in the case of pulsar magnetospheres). Rewriting the
above plasma di-electric tensor for a wavenumber parallel to the
magnetic field, it is straightforward to see that the dispersion
relation admits the longitudinal branch given by:
\begin{equation}
\varepsilon_\parallel + \cos^2\theta_B\chi^{\rm b}_{xx} +
\sin^2\theta_B\chi^{\rm b}_{zz}\,=\,0\ .
\end{equation}
In order to avoid confusion, it may be useful to stress that the
previous {\em oblique} denomination refers to the angle between the
wavenumber and the beam direction, while the present term {\em
longitudinal} here refers to the parallel nature of $\mathbf{k}$ and
$\mathbf{B}$. At the \v{C}erenkov resonance $\omega=\omega_{\rm
p}(1+\delta)=\beta_{\rm b}k_xc(1+\delta)$, with
$\vert\delta\vert\ll1$, one has $\vert\chi^{\rm
b}_{xx}\vert\,\gg\,\vert\chi^{\rm b}_{zz}\vert$, therefore one can
obtain the following approximate solution for the growth rate:
\begin{equation}
{\cal I}(\omega)\,\simeq\,\frac{\sqrt{3}}{2^{4/3}} \left[\omega_{{\rm
p}{\rm b}}^2\omega_{\rm p}\cos^2\theta_B\left(\frac{1}{\Gamma_{\rm b}^2} +
\frac{k^2c^2\sin^2\theta_B}{\omega_{\rm
p}^2}\right)\right]^{1/3}\ .\label{eq:Blong}
\end{equation}
Recalling that $k_x=k\cos\theta_B=\omega_p/c$, and that $\Gamma_{\rm
b}^2=\Gamma_{\rm sh}^4$, one can neglect in all generality the first
term in the parenthesis, so that
${\cal I}(\omega)\,\simeq\,\left(\omega_{{\rm p}{\rm b}}^2\omega_{\rm
p}\sin^2\theta_B\right)^{1/3}$.
\subsubsection{Resonant instability with Alfv\'en modes}
Turning now to resonant instabilities with Alfv\'en waves, we consider
a wavector in the $(\mathbf{x},\mathbf{z})$ plane. The resonance
condition for Alfv\'en modes reads: $\beta_{\rm b}k_x \,\simeq\,\beta_{\rm
A}k \cos\theta_k$, where $\theta_k$ represents the angle between the
wavenumber and the magnetic field direction. Since $\beta_{\rm
A}\,<\,1$, this implies $k_x\,\ll\,k$, therefore the wavenumber is
mostly aligned along $\mathbf{z}$ and $\theta_k\,\simeq\,\pi/2 -
\theta_B$.
The plasma dielectric tensor now reads (we omitted negligible
contributions in $\sin^2 \theta_k$):
\begin{eqnarray}
\Lambda_{ij\vert B}
\,=\,\left(\begin{array}{ccc}
\varepsilon_1-\eta^2\cos^2\theta_k & i\varepsilon_2 &
\eta\cos\theta_k\sin\theta_k \\
-i\varepsilon_2 & \varepsilon_1 -\eta^2& 0 \\
\eta\cos\theta_k\sin\theta_k & 0 &
\varepsilon_{\parallel}-\eta^2\sin^2\theta_k\end{array}\right)\ ,
\label{eq:diel-B}
\end{eqnarray}
with ($\omega\,\ll\,\omega_{\rm ci}$):
\begin{equation}
\varepsilon_1\,\simeq\,\frac{1}{\beta_A^2} \ ,\,\,
\varepsilon_2\,\simeq\,0\ ,\,\,
\varepsilon_\parallel\,\simeq\, - \frac{\omega_{\rm p}^2}{\omega^2}\ .
\end{equation}
The beam susceptibility can be approximated accurately by neglecting
all components in front of $\chi^{\rm b}_{xx}$, which dominates at the
resonance, as explained above. The relevant components then are:
\begin{eqnarray}
\chi^{\rm b}_{x_Bx_B}&\,\simeq\,& \sin^2\theta_B\chi^{\rm b}_{xx}\ ,\,\,
\chi^{\rm b}_{z_Bz_B}\,\simeq\, \cos^2\theta_B\chi^{\rm b}_{xx}\ ,\nonumber\\
\chi^{\rm b}_{x_Bz_B}&\,=\,&\chi^{\rm b}_{z_Bx_B}\,\simeq\,
\sin\theta_B\cos\theta_B\chi^{\rm b}_{xx}\ .\label{eq:chib-app}
\end{eqnarray}
The dispersion relation then takes the form:
\begin{eqnarray}
&&\left(\frac{\omega^2}{\beta_A^2} - k^2c^2\cos^2\theta_k\right)
\left(\omega_{\rm p}^2 + k^2c^2\sin^2\theta_k\right)\nonumber \\
&& + k^4c^4\sin^2\theta_k\cos^2\theta_k - \omega^4A_{xx}\chi^{\rm b}_{xx}\,=\,0\ ,
\end{eqnarray}
where $A_{xx}\,\simeq\, -\sin^2\theta_B\omega_{\rm p}^2/\omega^2$ in the
limit $k \delta_{\rm e} \,\ll\,1$. Writing down the resonance condition
$\omega\,=\,\beta_A k\cos\theta_k
c\left(1+\delta\right)\,=\,\beta_{\rm b}k_xc\left(1+\delta\right)$,
with $\vert\delta\vert\,\ll\,1$ as before, one obtains the growth
rate:
\begin{equation}
{\cal I}(\omega)\,\simeq\,\frac{\sqrt{3}}{2^{4/3}}\,\left(\omega_{{\rm p}{\rm b}}^2
\beta_{\rm A}kc\cos\theta_k\right)^{1/3}\ ,
\end{equation}
where we approximated $k_z\,\simeq\,k$; recall furthermore that
$\cos\theta_k\,\simeq\,\sin\theta_B$. This instability disappears in
the limit of a parallel shock wave as one can no longer satisfy the
\v{C}erenkov resonance condition.
One should stress that the above perturbative treatment remains valid
as long as the condition $\vert\delta\vert\,\ll\,1$, which amounts to
$\xi_{\rm cr}\,\ll\,\beta_{\rm A}^2$ at maximum wave growth rate
($kc=\omega_{\rm ci}$). Therefore Alfv\'en growth is limited to
strongly magnetized shock waves only.
In the continuity of right Alfv\'en waves (the left modes being
absorbed at the ion-cyclotron resonance), there are Whistler waves for
quasi parallel propagation (with respect to the mean field), that are
electromagnetic waves with a dominant magnetic component. For quasi
perpendicular propagation, there are the ionic extraordinary modes,
which have frequencies between the ion-cyclotron frequency and the
low-hybrid frequency (obtained for large refraction index) and which
are mostly electrostatic with a weaker electromagnetic component. For
scattering purpose, the whistler waves are the most interesting in
this intermediate range; they are actually excited in the foot of
non-relativistic collisionless shocks in space plasmas. But for
pre-heating purposes, the extraordinary ionic modes are more
interesting (they are actually used for additional heating in
tokamaks). Let us now discuss these in turn.
\subsubsection{Resonant instability with Whistler waves}\label{sec:whistler}
We proceed as before, using the plasma di-electric tensor
Eq.~(\ref{eq:diel-B}) in the range
$\omega_{\rm ci}\,\ll\,\omega\,\ll\,\omega_{\rm ce}$ with the components given
in Eq.~(\ref{eq:dielcomp-int}). The Whistler branch of the dispersion
relation reads, to first order in the beam response $\chi^{\rm b}$
approximated by Eq.~(\ref{eq:chib-app}):
\begin{equation}
\left(\epsilon_1-\eta^2\cos^2\theta_k+\chi^{\rm
b}_{xx}\sin^2\theta_B\right)
\left(\epsilon_1 - \eta^2\right)
-\epsilon_2^2\,=\,0\ .
\end{equation}
When the beam response is absent, one recovers the dispersion relation
for oblique Whistler waves:
\begin{equation}
\omega_{\rm Wh.}^2\,\simeq\,\frac{\omega_{\rm ce}^2}
{\omega_{\rm pe}^4}k^4c^4\cos^2\theta_k\ .
\end{equation}
Introducing the resonance $\omega=\omega_{\rm
Wh.}\left(1+\delta\right)\,=\,\beta_{\rm b}k_x
c\left(1+\delta\right)$, with $\vert\delta\vert\,\ll\,1$, we obtain
the growth rate:
\begin{equation}
{\cal I}(\omega)\,\simeq\,\frac{\sqrt{3}}{2^{4/3}}
\left(\omega_{{\rm p}{\rm b}}^2\omega_{\rm Wh.}\right)^{1/3}\ .
\end{equation}
In the latter equation, we again approximated $k_z\,\simeq\,k$, since
the resonance condition implies $k_x\,\ll\,k$ (therefore
$\cos\theta_k\,\simeq\,\sin\theta_B$). The instability disappears in
the limit of a parallel shock wave as well, because the resonance
condition cannot be satisfied. Maximum growth occurs here as well for
$k\,\simeq\,c/\omega_{\rm pe}\,\simeq\,c/\omega_{\rm p}$, i.e. at the electron
inertial scale $\delta_{\rm e}$, however the excitation range extends to the
proton inertial scale $\delta_{\rm i}$ where it matches with the Alfv\'en
wave instability.
As before, the perturbative treatment remains valid as long as
$\vert\delta\vert\,\ll\,1$, which amounts to $\xi_{\rm
cr}\,\ll\,(m_p/m_e)^2\beta_{\rm A}^2$. This condition is more easily
satisfied that the corresponding one for amplification of Alfv\'en
waves. It will be discussed in more detail in
Section~\ref{sec:disclim}.
\subsubsection{Resonant instability with extraordinary modes}
At MHD scales, the extraordinary ionic modes (that propagate with wave
vectors almost perpendicular to the magnetic field) assimilate to
magneto-sonic modes. These modes has been shown to be unstable when
there is a net electric charge carried by the cosmic rays (Pelletier,
Lemoine \& Marcowith 2009). The obtained growth rates are increasing
with wave numbers indicating an instability that reaches its maximum
growth at scales shorter than the MHD range. Let us therefore discuss
how this instability extends to sub-MHD scales.
Let us first discuss the ionic (lower hybrid) branch,
$\omega\,<\,\omega_{\rm lh}$, with $\omega_{\rm
lh}\equiv\sqrt{\omega_{\rm ci}\omega_{\rm ce}}$. In the following,
we assume for simplicity $\omega_{\rm ce}\,\ll\,\omega_{\rm pe}$,
i.e. a weakly magnetized plasma. In the $B$ frame, in which
$\mathbf{B}$ is along $\mathbf{z}_B$ and the beam propagates in the
$(\mathbf{x},\mathbf{z})$ plane, take
$\mathbf{k}\parallel\mathbf{y}_B$, with a small component $k_{x_B}$,
i.e. in the $(x,z)$ plane but perpendicular to $B$. The dispersion
relation to zeroth order in $\chi^{\rm b}$ reads:
\begin{equation}
\eta^2\,=\,\frac{\epsilon_1^2-\epsilon_2^2}{\epsilon_1}\ .
\end{equation}
with (since $\omega\,<\,\omega_{\rm lh}\,\ll\,\omega_{\rm ce}$):
\begin{equation}
\frac{\epsilon_1^2-\epsilon_2^2}{\epsilon_1}\,\simeq\,
\frac{\omega_{\rm ce}^2}{\omega_{\rm ci}^2\omega_{\rm pe}^2}
\frac{\omega^2\omega_{\rm ci}^2-\left(\omega_{\rm ci}^2+\omega_{\rm pi}^2\right)^2}
{\omega^2-\omega_{\rm lh}^2}\ ,
\end{equation}
hence
\begin{eqnarray}
\frac{\epsilon_1^2-\epsilon_2^2}{\epsilon_1} &\,\simeq\,&
\frac{\omega_{\rm pi}^2}{\omega_{\rm ci}^2}\quad (\omega\,\ll\,\omega_{\rm ci})\ ,
\nonumber\\
\frac{\epsilon_1^2-\epsilon_2^2}{\epsilon_1} &\,\simeq\,&
\frac{\omega_{\rm pe}^2}{\omega_{\rm lh}^2-\omega^2}
\quad(\omega_{\rm ci}\,\ll\omega\,\ll\omega_{\rm lh})\ .
\end{eqnarray}
At $\omega\,\ll\omega_{\rm ci}$, this gives the fast magnetosonic
branch with $\omega_{\rm H}\,\simeq\,\beta_{\rm A}kc$, while at
$\omega_{\rm ci}\,\ll\omega\,\ll\,\omega_{\rm lh}$, $\omega_{\rm
H}\,\sim\,\omega_{\rm lh}kc/\sqrt{k^2c^2+\omega_{\rm pe}^2}$. We
define:
\begin{equation}
{\cal D}(k,\omega)\,\equiv\,\frac{\epsilon_1^2-\epsilon_2^2}{\epsilon_1} - \eta^2\ .
\end{equation}
so that:
\begin{eqnarray}
\omega^2\frac{\partial}{\partial \omega^2}{\cal
D}(k,\omega)\,\simeq\, \eta^2
\quad (\omega\,\ll\,\omega_{\rm ci})\ ,
\nonumber\\
\omega^2\frac{\partial}{\partial \omega^2}{\cal
D}(k,\omega)\,\simeq\,
\eta^2 \frac{\omega_{\rm lh}^2}{\omega_{\rm lh}^2-\omega^2} \quad
(\omega_{\rm ci}\,\ll\omega\,\ll\omega_{\rm lh})\ .
\end{eqnarray}
Including the beam response, the dispersion relation becomes:
\begin{equation}
\epsilon_1^2 - \epsilon_2^2 - \epsilon_1\eta^2 +
\left(\epsilon_1-\eta_{x_B}^2\right)\sin^2\theta_B\chi^{\rm b}_{xx}\,=\,0\ .
\end{equation}
We neglect the term $\eta_{x_B}^2\,\ll \eta^2$ in front of
$\epsilon_1\,\sim\,1/\beta_{\rm A}^2$ (at
$\omega\,\ll\,\omega_{\rm ci}$). At the resonance $\omega\,=\,\omega_{\rm
H}(1+\delta)$, with $\omega_{\rm H}$ the solution of ${\cal
D}(k,\omega_{\rm H})\,=\,0$, one finds:
\begin{equation}
\delta^3\,\simeq\,\frac{1}{2}\frac{\omega_{{\rm
p}{\rm b}}^2\sin^2\theta_B}{\omega_{\rm H}^2}
\left[\omega^2\frac{\partial}{\partial\omega^2}{\cal
D}(k,\omega)\right]^{-1} \frac{k_y^2c^2}{\omega_{\rm
H}^2}\ . \label{eq:gH}
\end{equation}
The growth rate for \v{C}erenkov resonance with the lower hybrid
extraordinary mode thus reads:
\begin{eqnarray}
{\cal I}(\omega_{\rm LX})&\,\simeq\,&\frac{\sqrt{3}}{2^{4/3}}
\left(\omega_{{\rm p}{\rm b}}^2\sin^2\theta_B\frac{k_y^2}{k^2}\beta_{\rm
A}kc\right)^{1/3} \quad (\omega\,\ll\,\omega_{\rm
ci})\ ,\nonumber\\ {\cal I}(\omega_{\rm
LX})&\,\simeq\,&\frac{\sqrt{3}}{2^{4/3}} \left[\omega_{{\rm
p}{\rm b}}^2\sin^2\theta_B\frac{k_y^2}{k^2}\frac{\omega_{\rm
lh}\omega_{\rm pe}^2kc} {\left(k^2c^2+\omega_{\rm
pe}^2\right)^{3/2}}\right]^{1/3} \nonumber\\ & &
\quad\quad\quad\quad (\omega_{\rm ci}\,\ll\,\omega\,\ll\,\omega_{\rm
lh})\ .
\end{eqnarray}
In the limit of magnetosonic modes, $\omega \, \ll \, \omega_{\rm
ci}$, one recovers the same growth rate as for Alfv\'en waves; note
that $\beta_{\rm A}kc\,\ll\,\omega_{\rm ci}$ implies
$k\,\ll\,\omega_{\rm pi}/c$. At smaller scales, one finds that the
growth rate reaches its maximum at $k\,\simeq\,\omega_{\rm pe}/c$ with
${\cal I}(\omega_{\rm LX})\,\sim\,(\omega_{{\rm
p}{\rm b}}^2\sin^2\theta_B\omega_{\rm lh})^{1/3}$. We can expect this
instability to provide efficient heating of the protons in the foot.
Turning to the electronic (upper hybrid) modes, around
$\omega\,\sim\,\omega_{\rm pe}$, one obtains:
\begin{eqnarray}
\frac{\epsilon_1^2-\epsilon_2^2}{\epsilon_1}&\,\simeq\,&
\frac{(\omega^2-\omega_x^2)(\omega^2-\omega_z^2)}
{\omega_{\rm pe}^2(\omega^2-\omega_{\rm uh}^2)}\ ,
\end{eqnarray}
with $\omega_x\,\simeq\,\omega_{\rm pe}-\omega_{\rm ce}/2$,
$\omega_z\,\simeq\,\omega_{\rm pe}+\omega_{\rm ce}/2$ and
$\omega_{\rm uh}\,\equiv\,\left(\omega_{p}^2+\omega_{\rm ce}^2\right)^{1/2}$.
The dispersion relation takes the same form ${\cal D}(k,\omega)=0$, but now:
\begin{equation}
\frac{\partial}{\partial\omega^2}{\cal D}(k,\omega)\,\simeq\,
\eta^2\left(\frac{\omega^2}{\omega^2-\omega_x^2}+
\frac{\omega^2}{\omega^2-\omega_z^2} -
\frac{\omega^2}{\omega^2-\omega_{\rm uh}^2}+1 \right)\ .
\end{equation}
The growth rate can be written in the same algebraic form as
(\ref{eq:gH}). It vanishes in both limits $\omega\rightarrow\omega_x$
and $\omega\rightarrow\omega_z$, while for
$\omega\,\simeq\,\omega_{\rm pe}$, giving $\eta\,\simeq\,1$, one
obtains:
\begin{equation}
{\cal I}(\omega_{\rm
UX})\,\simeq\,\frac{\sqrt{3}}{2^{4/3}}\left(\omega_{{\rm
p}{\rm b}}^2\sin^2\theta_B \omega_{\rm pe}\frac{\omega_{\rm
ce}^2}{\omega_{\rm pe}^2}\frac{k_y^2}{k^2}\right)^{1/3}\ .
\end{equation}
It vanishes in the limit $\omega_{\rm ce}/\omega_{\rm
pe}\rightarrow0$, in which limit the electronic extraordinary branch
actually disappears.
Being electrostatic in nature, these waves participate mostly to the
heating process in the shock foot or precursor. However their
scattering efficiency is comparable to the magnetic perturbations as
will be seen further on.
\subsection{The particular case of a parallel magnetic field}\label{sec:bell}
When the magnetic field is almost parallel, i.e. $\theta_B < 1/
\Gamma_{\rm sh}$, the relativistic Bell non-resonant instability (Bell
2004, 2005) can develop (e.g. Milosavljevi\'c \& Nakar 2006; Reville,
Kirk \& Duffy 2006). This instability is triggered by the charge
current carried by the cosmic rays in the precursor, which induces a
return current in the plasma, thereby destabilizing non-resonant waves
of wavelength shorter than the typical Larmor radius, the cosmic rays
being unresponsive to the excitation of the waves. The growth rate of
this instability in the upstream frame is (Reville, Kirk \& Duffy
2006):
\begin{equation}
{\cal I}\left(\omega_{\rm Bell}\right)\,\simeq\,
\frac{\beta_{\rm b}n_{{\rm b}\vert\rm u}}{n_{\rm u}}\omega_{\rm pi}\ ,
\end{equation}
and growth is maximal at the scale $k_c\,\simeq\,{\cal I}(\omega_{\rm
Bell})/(\beta_{\rm A}c)$.
One can then verify that, under quite general assumptions, this growth
rate is larger than the growth rate of the Weibel instability, since
the ratio of these two is given by:
\begin{equation}
\frac{{\cal I}\left(\omega_{\rm Bell}\right)}{{\cal I}\left(\omega_{\rm
We.}\right)}
\,\simeq\,\Gamma_{\rm sh}^2\xi_{\rm cr}^{1/2}\ .
\end{equation}
One must emphasize however that the Bell instability is quenched when
the growth rate exceeds the ion cyclotron frequency see Couch,
Milosavljevic \& Nakar (2008), Riquelme \& Spitkovsky (2009), Ohira et
al. (2009). This limitation will be made clear in
Section~\ref{sec:parshock}.
\section{Limitations of the instabilities}\label{sec:disclim}
Using the growth rates derived previously, we can now delimit the
conditions under which the various instabilities become effective, and
which one dominates. We then discuss the limit between unmagnetized
and magnetized shock waves, from the point of view of these upstream
instabilities.
\subsection{Superluminal shock waves}
In this Section, we discuss the generic case of relativistic
superluminal shock waves, taking $\sin^2\theta_B\,\sim\,1$. Unless
otherwise noted, we assume an $e-p$ plasma; we will discuss how the
results are modified in the limit of a pair plasma at the end of this
discussion. The more particular case of relativistic parallel shock
waves is treated further below.
We start by introducing the two parameters $X$ and $Y$ defined as
follows:
\begin{eqnarray}
X&\,\equiv\,& \Gamma_{\rm sh}\frac{m_e}{m_p}\ ,\nonumber\\
Y &\,\equiv\,& \Gamma_{\rm sh}^4\frac{B_{\rm u\vert\rm u}^2}
{4\pi n_{{\rm b}\vert\rm u}m_pc^2}\,=\,\Gamma_{\rm sh}^2\sigma_{\rm u}\xi_{\rm cr}^{-1}\ .
\end{eqnarray}
The upstream magnetization parameter $\sigma_{\rm u}$ also corresponds
to the Alfv\'en velocity squared of the upstream plasma (in units of
$c^2$). If the field is fully perpendicular, the shock crossing
conditions imply $B_{\rm d\vert d,\perp}\,\simeq\, B_{\rm u\vert\rm
u,\perp}\Gamma_{\rm sh}\sqrt{8}$, and for the enthalpy $h_{\rm
d\vert d}\,\simeq\, (8/3)\Gamma_{\rm sh}^2h_{\rm u\vert\rm u}$ (for
a cold upstream plasma, see Blandford \& McKee 1976), so that
$\sigma_{\rm d}\,\simeq\,3\sigma_{\rm u}\sin^2\theta_B$. If the
magnetic field is mostly parallel, meaning
$\sin\theta_B\,\leq\,1/\Gamma_{\rm sh}$, then $\sigma_{\rm
d}\,\sim\,(3/8)\Gamma_{\rm sh}^{-2}\sigma_{\rm u}$.
Let us first compare the growth rates of the instabilities obtained in
the magnetized case; the unmagnetized case (in particular the oblique
mode) will be discussed thereafter. We carry out this comparison at
the wavenumber where the growth rates reach their maximum, namely $k
\,\sim\,\omega_{\rm pe}/c$. The ratio of the Weibel to Whistler
instability growth rates is given by:
\begin{equation}
\frac{{\cal I}\left(\omega_{\rm We.}\right)}{{\cal I}\left(\omega_{\rm Wh.}\right)}\,=\,
\left(\frac{X}{Y}\right)^{1/6}\ ,
\end{equation}
hence the Weibel instability will dominate over the Whistler
\v{C}erenkov resonant instability whenever $Y\,\ll\,X$. The fastest
mode however corresponds to the \v{C}erenkov resonance with the
longitudinal modes along the magnetic field, since the ratio of the
growth rates of this mode to the Weibel mode is
$(m_p/m_e)^{1/6}\xi_{\rm cr}^{-1/6}$, which is always greater than
one.
Since the \v{C}erenkov resonant instabilities for the Whistler and
Alfv\'en waves scale in a similar way with the eigenfrequencies of the
resonant plasma modes, it is straightforward to see that Whistler
waves will always grow faster than the Alfv\'en waves.
Concerning the extraordinary modes, one finds that ${\cal I}(\omega_{\rm
Wh.})/{\cal I}(\omega_{\rm LX})\,\sim\,(m_p/m_e)^{1/6}$ on the ionic
(lower hybrid) branch, while ${\cal I}(\omega_{\rm Wh.})/{\cal I}(\omega_{\rm
UX})\,\sim\,(\omega_{\rm pe}/\omega_{\rm ce})^{1/3}$ on the
electronic (upper hybrid) branch. Therefore the growth of these modes
is always sub-dominant with respect to that of Whistler and Weibel
modes. Since the growth rates of the Alfv\'en and extraordinary modes
are always smaller than that of the Whistler modes, we discard the
former in the following.
Additional constraints can be obtained as follows. First of all, the
above derivation of the instabilities has assumed the beam to be
unmagnetized, i.e. that the growth time be much shorter than the
Larmor time of the beam particles. This condition is always easily
satisfied, since it reads: $Y\,\ll\,\Gamma_{\rm sh}^6$ for the Weibel
instability, $Y\,\ll\,\Gamma_{\rm sh}^6\xi_{\rm
cr}^{-1/3}(m_e/m_p)^{-1/3}$ for the longitudinal mode and
$Y\,\ll\,\Gamma_{\rm sh}^8m_p/m_e$ for the Whistler \v{C}erenkov
resonant mode. One can explicit the dependence of $Y$ on the shock
parameters in order to verify this; for the Weibel instability, the
condition amounts to $\xi_{\rm cr}\,\gg\,\Gamma_{\rm
sh}^{-4}\sigma_{\rm u}$, which is indeed easily verified at large
Lorentz factors.
Concerning the \v{C}erenkov resonant modes, one must also require that
$\vert\delta\vert\,<\,1$ in order for the perturbative treatment to be
apply. In the case of longitudinal modes, this is automatically
satisfied since $\omega_{{\rm p}{\rm b}}<\omega_{\rm p}$. However, for
Whistler modes of smaller eigenfrequency, this implies a non-trivial
constraint $\omega_{{\rm p}{\rm b}}<\omega_{\rm Wh.}$ which can be
translated into $Y\,\gg\,X^2$ for $\omega_{\rm Wh.}=\omega_{\rm ce}$.
Further bounds can be obtained by requiring that the background
protons are non-magnetized in the case of the Weibel instability,
which requires ${\cal I}\left(\omega\right)\,\gg\,\omega_{\rm ci}$. This
condition is however superseded by the requirement that the growth can
occur on the precursor length scale, since $\ell_{\rm F}/c\sim
(\Gamma_{\rm sh}\omega_{\rm ci})^{-1}$ [see Eq.~(\ref{eq:lfb})]. At
this stage, it is important to point out a fundamental difference
between the \v{C}erenkov resonant instabilities and the Weibel /
filamentation instabilities. The former have, by definition of the
resonance, a phase velocity along the shock normal which, to zeroth
order in $\vert\delta\vert$ exceeds the shock velocity, while the
latter have vanishing phase velocity along $\mathbf{x}$. Therefore the
timescale available for the growth of these non-resonant waves is the
crossing time of the precursor: they are sourced at a typical distance
$\ell_{\rm F}$ away from the shock, then advected downstream on this
timescale. Regarding the resonant modes, their phase velocity along
$\mathbf{x}$ is $\beta_{\phi,x}=\beta_{\rm b}(1 + \delta_R)$, with
$\delta_R={\cal R}(\delta)$. Since $\delta_{\rm R}<0$ for the resonant
modes, one must consider three possible cases: (i)
$\beta_{\phi,x}<\beta_{\rm sh}$, in which case the mode is advected
away on a timescale $\ell_{\rm F}/c$ as for the non-resonant modes;
(ii) $\beta_{\phi,x}>\beta_{\rm sh}$, in which case the mode
propagates forward, but exits the precursor (where it is sourced) on a
similar timescale; and (iii) $\beta_{\phi,x}\,\simeq\,\beta_{\rm sh}$,
in which case the mode can be excited on a timescale
$\simeq\,c^{-1}\ell_{\rm F}/(\beta_{\rm sh}-\beta_{\phi,x})$ and where
the divergence corresponds to the situation of a mode surfing on the
shock precursor. However condition (i) appears to be the most likely,
as least in the ultra-relativistic limit, for it amounts to
$2\Gamma_{\rm sh}^2\vert\delta_{\rm R}\vert\,\gg\,1$. Indeed, all
resonant instabilities have a growth rate $\sim (\omega_{\rm p
{\rm b}}^2\omega)^{1/3}$ where $\omega$ is the eigenfrequency of the
resonant mode (an exception is the upper hybrid mode for which the
growth rate is smaller by $(\omega_{\rm ce}/\omega_{\rm pe})^{2/3}$,
in which case the following condition is even stronger), therefore the
condition $2\Gamma_{\rm sh}^2\vert\delta_{\rm R}\vert\,\gg\,1$ can be
rewritten as $\omega/\omega_{\rm pe}\,\ll\,7 (\Gamma_{\rm
sh}/10)^3(\xi_{\rm cr}/0.1)^{1/2}$, which is generically
satisfied. This means that the phase velocity of the resonant modes,
when corrected by the effect of the beam becomes smaller than the
shock front velocity, so that these modes are advected on a timescale
$\sim\ell_{\rm F}/c$ and transmitted downstream, after all. For the
purpose of magnetic field amplification downstream and particle
acceleration, this is certainly noteworthy, as such true plasma
eigenmodes (Whistler, Alfv\'en, extraordinary and electrostatic
longitudinal or oblique modes) can be expected to have a longer
lifetime than the Weibel modes.
The modes thus grow on the precursor crossing timescale if
${\cal I}\left(\omega\right)\ell_{\rm F}/c\,\gg\,1$, which can be recast as
$Y\,\ll\,1$ for the Weibel instability, $Y\,\ll\,\xi_{rm
cr}^{-1/3}(m_e/m_p)^{-1/3}$ and $XY\,\ll\,1$ for the \v{C}erenkov
resonant Whistler mode. Henceforth, we use the parameter
$G\,\equiv\,(\omega_{\rm p{\rm b}}/\omega_{\rm p})^{-2/3}=\xi_{\rm
cr}^{-1/3}(m_e/m_p)^{-1/3}\,>\,1$.
In short, we find that the various instabilities discussed here are
more likely quenched by advection rather than by saturation. In
Section~\ref{sec:appl}, we provide several concrete estimates for
cases of astrophysical interest and it will be found that this limit
is indeed quite stringent.
Finally, one must also require that the growth rate of the
\v{C}erenkov resonant instabilities does not exceed the proper
eigenfrequency of the mode. For the Whistler modes, as discussed at
the end of Section~\ref{sec:whistler}, this implies $Y\,\gg\,X^2$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig_1.ps}
\caption{Instability diagram for superluminal relativistic shock
waves (assuming $\sin^2\theta_B\sim 1$): in abscissa,
$X\,\equiv\,\Gamma_{\rm sh}m_e/m_p$, in ordinates $Y \,\equiv\,
\Gamma_{\rm sh}^4B_{\rm u}^2/ \left(4\pi n_{{\rm b}\vert\rm
u}m_pc^2\right)$. The parameter $G\,=\,\xi_{\rm
cr}^{-1/3}(m_e/m_p)^{-1/3}>1$. The axes are plotted in log-log
on arbitrary scale. The main result is summarized by the thick
solid line, which indicates the maximum value of $Y(X)$ which
allows electromagnetic waves to grow. The other lines indicate the
boundaries of the regions of growth of the various instabilities,
as indicated. The hierarchy of growth rates, from largest to
smallest is as follows: oblique and longitudinal, then Whistler
and/or Weibel. The long dashed line separates the regions in
which the growth of Whistler or Weibel modes is faster: for values
of $Y(X)$ larger than the long dashed line, Whistler modes grow
faster. The growth rates of the oblique mode and the longitudinal
mode are comparable. Unlike the longitudinal mode, the oblique
instability is limited by the assumption of unmagnetization (see
main text), but at the same time, it applies to a larger
wavenumber phase space. The regions for Alfv\'en and
extraordinary modes are not indicated (see main text).
\label{fig:XY_perp}}
\end{figure}
In Section~3, we have also examined the growth rates in the absence of
a mean magnetic field, and concluded that the oblique mode of Bret,
Firpo \& Deutsch (2004, 2005a, b) was by far the fastest. This
instability is very similar to the \v{C}erenkov resonance with the
longitudinal modes propagating along the magnetic field and indeed the
growth rates only differ by $\sin^{2/3}\theta_B$, see
Eqs.~(\ref{eq:obl}),(\ref{eq:Blong}). The difference lies in the
degree of magnetization of the ambient plasma: while the oblique mode
is limited to the unmagnetized limit, the longitudinal mode does not
suffer from such constraint; the oblique mode however covers a larger
fraction of the wavenumber phase space than the longitudinal mode.
With respect to the oblique mode instability, the shock can be
described as unmagnetized as long as the background electrons and
protons remain unmagnetized on the timescale of the instability; of
course, one must also require that the instability has time to grow on
the length scale of the precursor. Note that the latter condition also
implies that the beam can be considered as unmagnetized over the
instability growth timescale, which is another necessary
condition. For the oblique modes, those conditions amount to:
\begin{eqnarray}
{\cal I}\left(\omega_{\rm obl.}\right)\,\gg\,\omega_{\rm ce} &
\Leftrightarrow & Y\,\ll\,G
X^2\label{eq:obllimmag}\ ,\\ {\cal I}\left(\omega_{\rm
obl.}\right)\,\gg\,c/\ell_{\rm F} & \Leftrightarrow &
Y\,\ll\,G
\ ,\label{eq:obllimadv}
\end{eqnarray}
with $G\,=\,\xi_{\rm cr}^{-1/3}(m_e/m_p)^{-1/3}\,>\,1$ as above.
Provided the above two conditions are satisfied, the oblique mode
dominates over the Weibel and Whistler \v{C}erenkov instability growth
rates, just as the longitudinal mode. The \v{C}erenkov resonant
instability with Whistler waves dominates over the oblique modes when
$X\,\lesssim\,G^{-1/3}$ and $G
X^{2}\,\lesssim\,Y\,\lesssim\,1/X$. The \v{C}erenkov resonant
instability with Whistler waves dominates over the longitudinal
modes when $X\,\lesssim\,G^{-1}$ and
$G\,\lesssim\,Y\,\lesssim\,1/X$. For $X\,\lesssim\,G^{-1}$ and
$Y\,\gtrsim\,X^{-1}$, or for $G^{-1}\,\lesssim\,X$ and $Y\,\gtrsim\,
G$ neither of the above instabilities can grow. For reference,
$X\,\lesssim\,G^{-1}$ corresponds to $\Gamma_{\rm
sh}\,\lesssim\,150 \xi_{\rm cr}^{1/3}$. The above regions can be
summarized in the $X-Y$ plane as in Fig.~\ref{fig:XY_perp}, which
delimit the domains in which the various instabilities can grow, and
which of these instabilities dominates in each case.
From the above discussion, the case of a pair shock is easily obtained
by taking $m_p/m_e\,\rightarrow\,1$, by restricting oneself to the
study of the oblique and Weibel modes, and by considering only the
right part of Fig.~\ref{fig:XY_perp} with $X>1$, since $X=\Gamma_{\rm
sh}$ for a pair shock. One sees that, irrespectively of $\Gamma_{\rm
sh}$, the oblique and longitudinal modes can grow if
$Y\,\lesssim\,\xi_{cr}^{-1/3}$ and the Weibel mode grows on the
precursor timescale if $Y\,\lesssim\,1$.
In this respect, it is instructive to compare the present results with
the latest simulations of Sironi \& Spitkovsky (2009). These authors
find that the growth of instabilities is quenched when the
magnetization $\sigma_{\rm u}\,\gtrsim\,0.03$ for a perpendicular (or
oblique) pair shock with $\Gamma_{\rm sh}\,\simeq\,20$. This
corresponds to $X\,\simeq\,20$ and $Y\,\simeq\,10\xi_{\rm
cr}^{-1}(\sigma_{\rm u}/0.03)$. Our results indicate that indeed, at
this high level of magnetisation, both Weibel and oblique/longitudinal
instabilities are quenched by advection. Note that these simulations
do not exclude that the instabilities are quenched even at lower
magnetisations. Our calculations thus bring to light the following
point of interest. One should not infer from the simulations of Sironi
\& Spitkovsky (2009) that superluminal shock waves cannot lead to
magnetic field amplification. This conclusion entirely depends on the
level of magnetization. It would therefore be interesting to extend
the PIC simulations down to weakly magnetized shocks with $\sigma_{\rm
u}\,\sim\, 3\times 10^{-3}\xi_{\rm cr}^{2/3}(\Gamma_{\rm
sh}/20)^{-2}$ in order to probe the limit at which the oblique mode
can grow.
\subsection{Parallel shock waves}\label{sec:parshock}
In the ultra-relativistic limit, parallel shock waves are non-generic;
however, they may lead more easily to particle acceleration than
superluminal shock waves (since the argument discussed in Lemoine,
Pelletier \& Revenu 2006 no longer applies) and consequently provide
interesting observational signatures. One can extend the above
discussion to the case of parallel shock waves as follows.
First of all, the main limitation of the instabilities, that is due to
the precursor crossing timescale disappears in the limit $\Gamma_{\rm
sh}\sin\theta_B\rightarrow 0$ as a fraction
\begin{equation}
p\,\equiv\,\frac{1-\beta_{\rm sh} \cos \theta_B - \frac{1}{\Gamma_{\rm
sh}} \sin \theta_B}{1-\beta_{\rm sh}}\,\simeq \,(1-\Gamma_{\rm sh}
\theta_B)^2
\end{equation}
of the particles can propagate to upstream infinity (at least in the
limit of a fully coherent magnetic field). This can be seen as
follows. Particles cross the shock wave back toward downstream once
their angle cosine with the shock normal becomes smaller than
$\beta_{\rm sh}$. However, when $\Gamma_{\rm sh}\sin\theta_B<1$,
there exists a cone ${\cal C}_B$ around the magnetic field direction,
of opening angle $\theta_B - {\rm acos}(\beta_{\rm sh})$, that never
intersects the cone ${\cal C}$ defined around the shock normal with
opening angle ${\rm acos}(\beta_{\rm sh})$ . Because the pitch angle
of the particles with respect to the magnetic field direction is
conserved, particles that enter toward upstream in this cone ${\cal
C}_B$ never recross the shock toward downstream (up to the
influence of the turbulence). The fraction of particles that enter in
this cone is approximately given by the ratio of the solid angles,
i.e. the factor $(1-\Gamma_{\rm sh}\theta_B)^2$ quoted
before. Depending on the value of $\Gamma_{\rm sh}\theta_B$, this
fraction can be substantial and these particles can excite plasma
waves up to large distances from the shock. Of course, the actual
precursor length remains finite as a result of the influence of large
and short scale turbulence. In the following, we will simply discard
these advection constraints in order to avoid introducing new
parameters.
We also choose to discuss the limitations as a function of $X$ and
$Y$, but at a fixed value of $\Gamma_{\rm sh}\sin\theta_B\,<\,1$. The
fact that $X\,\propto\,\Gamma_{\rm sh}$ and that $\Gamma_{\rm
sh}\sin\theta_B$ is fixed modifies slightly the limitations derived
previously. The main limitation for the oblique mode is the
non-magnetization condition, ${\cal I}(\omega_{\rm obl.})\,\gg\,\omega_{\rm
ce}$ which can be rewritten $Y\,\ll\, G X^2$ as before. For the
Weibel mode, the condition of non-magnetization of the protons,
${\cal I}(\omega_{\rm We.})\,\gg\,\omega_{\rm ci}$ now reads
$Y\,\ll\,(m_p/m_e)^2X^2$.
The \v{C}erenkov resonant mode with longitudinal waves does not suffer
any constraint in this parallel configuration, but the growth rate now
becomes significantly smaller than that of the oblique mode (when the
latter applies), by a factor $\sin^{2/3}\theta_B$.
Concerning the Whistler modes, the condition $\vert\delta\vert\,<\,1$
now leads to $Y\,\gg\,(m_e/m_p)^{-2}(\Gamma_{\rm
sh}\sin\theta_B)^{-2}X^4$. One recovers the condition expressed in
the case of superluminal shock waves for $\sin\theta_B^2\,\sim\,1$, as
one should. At a fixed value of $\Gamma_{\rm sh}\sin\theta_B$ however,
the condition appears slightly different. Note that in the particular
case of parallel shock waves, there is another non-trivial condition
for these Whistler modes, which is related to the fact that the
eigenfrequency $\omega_{\rm Wh.}\,\simeq\,\omega_{\rm ce}\sin\theta_B$
(at maximal growth) should exceed $\omega_{\rm ci}$, in order for the
Whistler branch to apply. This translates into $X\,\ll\,\Gamma_{\rm
sh}\sin\theta_B\,<\,1$.
Finally, the Bell non-resonant instability requires the background
protons to be magnetized, as noted in Section~\ref{sec:bell}, which
corresponds to $Y\,\gg\,\xi_{\rm cr}(m_p/m_e)^6X^6$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig_2.ps}
\caption{Same as Fig.~\ref{fig:XY_perp} for the case of a parallel
relativistic shock wave. The regions of growth are drawn at a fixed
value of $\Gamma_{\rm sh}\sin\theta_B$ (here taken to be 0.3). The
axes are plotted in log-log on arbitrary scale. The longitudinal
mode can grow in all parameter space (in the idealized limit of a
fully coherent upstream magnetic field) due to the divergence of the
precursor length (see main text).
\label{fig:XY_par}}
\end{figure}
These various conditions are expressed in Fig.~\ref{fig:XY_par}, which
is the analog of Fig.~\ref{fig:XY_perp} for parallel shock waves. Here
as well, the limit of a pair shock can be obtained simply by taking
the limit $m_e/m_p\rightarrow 1$ and discarding the Whistler branch as
well as the Bell instability (which requires a net current to exist
upstream).
As for the oblique shock wave, it is instructive to compare the
present results to the simulations of Sironi \& Spitkovsky (2009), who
find in particular that instabilities can be triggered in parallel
shocks for a magnetisation $\sigma_{\rm SS}=0.1$. Here $\sigma_{\rm
SS}$ corresponds to the definition of the magnetisation given in
Sironi \& Spitkovsky (2009), or to our defintion of downstream
magnetisation up to a factor $3/4$, the latter factor of $3/4$
representing the difference between enthalpy and energy density for a
relativistic gas. For a parallel shock wave, this thus corresponds to
an upstream magnetisation $\sigma_{\rm u}\,=\,0.05\Gamma_{\rm sh}^2$,
hence to $Y\,\simeq\,0.05\Gamma_{\rm sh}^4\xi_{\rm
cr}^{-1}\,\simeq\,10^4\xi_{\rm cr}^{-1}$ for $X\,\simeq\,20$. One
may note that at this large level of magnetisation, one has
$\omega_{\rm ce}>\omega_{\rm pe}$. One can check immediately, using
the above, that neither the oblique nor the Weibel instabilities can
grow, as their respective non-magnetisation conditions are not
satisfied. The growth of instabilities observed in the simulations of
Sironi \& Spitkovsky (2009) is thus likely due to the \v{C}erenkov
resonance with the longitudinal modes that propagate along the
magnetic field, which does not suffer from these constraints.
The above also allows to understand the abrupt transition as a
function of shock obliquity: when $\sin\theta_B\rightarrow
1/\Gamma_{\rm sh}$, the growth of fluctuations is suddenly inhibited
and so is Fermi acceleration. Indeed, as $\sin\theta_B\rightarrow
1/\Gamma_{\rm sh}$, the fraction of particles that can escape to
upstream infinity vanishes, hence the precursor length now rapidly
decreases to the value given by Eq.~(\ref{eq:lfb}). This prevents the
growth of fluctuations at high magnetisation levels, as discussed
above for superluminal shock waves, and consequently this prevents
successful Fermi acceleration (Lemoine, Pelletier \& Revenu 2006).
\section{Triggering Fermi acceleration}\label{sec:Fermi}
It is important to underline that Fig.~\ref{fig:XY_perp} indicates
whether instabilities triggered by the first generation of cosmic rays
returning upstream have time to grow or not. If these instabilities
cannot be triggered by the first generation, meaning if the shock wave
characteristics are such that $(X,Y)$ lie above the thick solid line
of Fig.~\ref{fig:XY_perp}, then instabilities cannot be triggered,
either upstream or downstream (at least in the frame of our
approach). Consequently Fermi cycles will not develop, at least for
superluminal shock waves, in accordance with the arguments of Lemoine,
Pelletier \& Revenu 2006, Pelletier, Lemoine \& Marcowith 2009 and
with the simulations of Niemiec, Ostrowski \& Pohl 2006.
If, however, the initial values of $X$ and $Y$ are such that
instabilities can develop, Fig.~\ref{fig:XY_perp} suggest that these
instabilities will develop upstream and be transferred
downstream. Fermi cycles may then develop provided the appropriate
conditions discussed in Lemoine, Pelletier \& Revenu (2006) and
Pelletier, Lemoine \& Marcowith (2009) are satisfied. These conditions
have been discussed under the assumption of isotropic short scale
magnetic turbulence, and we restrict ourselves to this assumption in
the present work as well. It would certainly be interesting to
generalize this discussion to more realistic turbulence
configurations, as in Hededal et al. (2004), Dieckmann, Drury \&
Shukla (2006) for instance. However, this clearly becomes more model
dependent in terms of turbulence configuration and for this reason, we
postpone such a study to future work.
Let us discuss first the case of upstream turbulence. When particles
are scattered off short scale $\ell_{\rm c}$, but intense magnetic
fluctuations, the scattering frequency of a relativistic particle of
momentum $p$ is
\begin{equation}
\label{NUS}
\nu_{\rm s} \,\sim\, c \frac{e^2\langle \delta B^2 \rangle
}{p^2}\ell_{\rm c}\ .
\end{equation}
Since the oblique mode dominates over the Whistler and Weibel waves
over most of the parameter space, one cannot ignore the influence of
short scale electrostatic fields. These electrostatic waves lead to a
second order Fermi process in the upstream medium, with a concomittant
pitch angle scattering. Indeed, the particle scatters against random
electric fields $\pm E_\parallel$ along the shock normal ($\mathbf{x}$
direction), gaining momentum $\Delta p_\parallel\,\simeq\, \pm
eE_\parallel \Delta t$, with $\Delta t\,\simeq\,\omega_{\rm p}^{-1}$
at each interaction, and similarly in the perpendicular direction. The
initial pitch angle of the particle (with respect to the shock normal)
$\theta\,\ll\,1$ in the upstream frame, and the particle is overtaken
by the shock wave whenever $\theta\,\gtrsim\,1/\Gamma_{\rm sh}$
(Achterberg et al. 2001). This pitch angle diffuses according to:
\begin{equation}
\frac{\langle\Delta \theta^2\rangle}{\Delta
t}\,\simeq\, \frac{\langle\Delta p^2\rangle}{p^2 \Delta t}
\,\simeq\, e^2 \frac{E_\perp^2 +
2\theta^2E_\parallel^2}{p_\parallel^2}\tau_{\rm c} \ ,
\end{equation}
for a correlation time $\tau_{\rm c}=\ell_{\rm c}/c \sim \omega_{\rm
pe}^{-1}$. Therefore we obtain a scattering rate similar to the
previous one (\ref{NUS}) in which the magnetic field fluctuation is
replaced by the electric field fluctuation:
\begin{equation}
\label{eq:diffrate}
\nu_{\rm s}' \sim c \frac{e^2\langle \delta E^2 \rangle}{p^2}
\ell_{\rm c}\ .
\end{equation}
This correspondence justifies that we treat the short scale electric
and magnetic fields on a similar footing and consider the total
electromagnetic energy content. A conversion of a fraction of the
energy of the beam into magnetic or electrostatic fluctuations is
expected with $\xi_{\rm em} < \xi_{\rm cr}$, with typically $\xi_{\rm
cr} \sim 10^{-1}$ and $\xi_{\rm em} \sim 10^{-2}-10^{-1}$ (Spitkovsky
2008a). Scattering in the short scale electromagnetic turbulence will
govern the scattering process if it leads to $\langle\Delta
p^2\rangle/p^2\,\sim\,1/\Gamma_{\rm sh}^2$ on a timescale $r_{{\rm
L}\vert B}/(\Gamma_{\rm sh}c)$, with $r_{{\rm L}\vert B}$ the
Larmor radius of first generation cosmic rays as measured upstream
relatively to the background magnetic field (see the corresponding
discussion in Pelletier, Lemoine \& Marcowith 2009). If this short
scale turbulence governs the scattering process, then Fermi
acceleration will operate. Assuming $\ell_{\rm c}=c/\omega_{\rm pe}$,
this condition amounts to:
\begin{equation}
\xi_{\rm em} > \Gamma_{\rm sh} \left(\frac{m_p}{m_e}\right)^{1/2}
\sigma_{\rm u}^{1/2} \ .
\label{eq:fup1}
\end{equation}
Using the fact that $\xi_{\rm em}\,<\,\xi_{\rm cr}$, this constraint
can be rewritten as a bound on $\sigma_{\rm u}$:
\begin{equation}
\sigma_{\rm u}\,\ll\, \xi_{\rm cr}^2 \frac{m_e}{m_p}\Gamma_{\rm
sh}^{-2}\ .\label{eq:fup}
\end{equation}
This limit is very stringent indeed; in terms of our above parameters,
it can rewritten as $Y\,\ll\,X\xi_{\rm cr}/\Gamma_{\rm sh}$. We will
discuss the applicability of this inequality in concrete cases in the
following sub-section.
If this condition is not verified, the background unamplified magnetic
field remains the main agent of particle scattering upstream. In this
case, Fermi acceleration cycles can develop only if short scale
turbulence govern the scattering downstream of the shock wave. As
discussed in Pelletier, Lemoine \& Marcowith (2009), this requires:
\begin{equation}
\ell_{\rm c\vert d}\,<\,r_{\rm L\vert d}\,<\,\frac{\delta B_{\vert\rm
d}}{B_{\vert\rm d}}\ell_{\rm c\vert d}\ ,\label{eq:fdown}
\end{equation}
where all quantities should be evaluated in the downstream rest frame,
and $r_{\rm L\vert d}$ refers to the Larmor radius of the accelerated
particles in this frame. This double inequality amounts to requiring
that $\ell_{\rm c\vert d}/c \,<\ \tau_{\rm s} \,<\ \tau_{\rm L,0}$,
i.e. that the scattering time $\tau_{\rm s}=\nu_{\rm s}^{-1}$ be
shorter than the Larmor time in the mean field $\tau_{\rm L,0}$ in
order to break the inhibition constraint of the mean field that tends
to drag the particles in the downstream flow. The scattering must also
develop in a special regime where the correlation time $\ell_{\rm
c\vert d}/c$ is shorter than the Larmor time. Regarding $\ell_{\rm
c\vert d}$, two main spatial scales are to be envisaged: the
previous upstream electron skin depth, if one assumes that the typical
scale of transverse fluctuations is preserved through shock crossing,
and the downstream electron skin depth, if reorganization takes place
through shock crossing. Assuming a typical electron temperature $\sim
\Gamma_{\rm sh}m_p c^2$ behind the shock, and accounting for shock
compression of the electron density, this latter scale can actually be
written as $c\,\omega_{\rm pi}^{-1}$ ($\omega_{\rm pi}$ the {\em
upstream} ion plasma frequency), a factor $43$ larger than the
previous one. One should also envisage the possibility that the
turbulence spectrum evolves to larger scales with time (Medvedev et
al. 2005; Lemoine \& Revenu 2006; Katz, Keshet \& Waxman 2007) but we
will not do so here. Let us consider the above two possibilities in
turn.
If $\ell_{\rm c\vert d}\,=\,c/\omega_{\rm pe}$ (upstream electron skin
depth), then the first inequality in Eq.~(\ref{eq:fdown}) can be
rewritten as $\xi_{\rm em}\,<\,m_p/m_e$ and is therefore always
satisfied. The second inequality amounts to $\sigma_{\rm d}\,<\,
(m_e/m_p)\xi_{\rm em}^2$, hence $Y\,<\,\Gamma_{\rm sh}X \xi_{\rm
em}^2/\xi_{\rm cr}$. This latter inequality is much more
stringent. If satisfied, it means that the downstream short scale
turbulence governs the scattering process, in particular it allows the
particle to escape its orbit around the shock compressed background
magnetic field on a timescale smaller than the Larmor time in this
field. This is a necessary condition for successful Fermi cycles.
If $\ell_{\rm c\vert d}\,=\,c\,\omega_{\rm pi}^{-1}$ (equivalently,
the downstream electron skin depth), then the first inequality in
Eq.~(\ref{eq:fdown}) becomes $\xi_{\rm em}\,<\,1$, which is always
true. The second inequality reads $\sigma_{\rm d}\,<\,\xi_{\rm em}^2$
(or $Y\,<\,\Gamma_{\rm sh}^2\xi_{\rm em}^2/\xi_{\rm cr}$). We will
summarize the two above two possible cases for $\ell_{\rm c\vert d}$
and parameterize the uncertainty on $\ell_{\rm c\vert d}$ by writing
the condition as:
\begin{equation}
\sigma_{\rm d}\, \ll\, \sigma_{*} \equiv \kappa\, \xi_{\rm em}^2\ ,
\label{eq:fdown2}
\end{equation}
with $\ell_{\rm c\vert d}\,=\,\kappa c/\omega_{pi}$ and
$m_e/m_p\,\lesssim\,\kappa\,\lesssim\,1$. One should however recall
that the typical scale of electromagnetic fluctuations could evolve
with the distance to the shock front, as envisaged in Medvedev et
al. (2005), Lemoine \& Revenu (2006) and Katz, Keshet \& Waxman
(2007). This amounts to making $\kappa$ be a growing function of the
energy taking values larger than 1. The above result clearly reveals
the need for dedicated PIC simulations of shock wave at moderate
magnetisation, with realistic proton to mass ratio and geometry in
order to reduce this large uncertainty on $\kappa$ and determine the
precise conditions under which Fermi acceleration can take place.
To summarize this discussion, we obtain the following conditions for
successful Fermi acceleration. If Eq.~(\ref{eq:fup}) is satisfied [or,
to be more accurate, Eq.~(\ref{eq:fup1})], then Fermi acceleration
will operate, because the short scale fluctuations produced upstream
are sufficiently intense to govern the scattering. In this case, it is
important to stress that Eq.~(\ref{eq:lfb}), which defines the length
of the precursor, no longer applies. It should be replaced by
Eq.~(\ref{eq:lfu}), which is larger. Physically, the precursor widens,
giving more time for the fluctuations to grow, thus reaching a higher
efficiency in terms of $\xi_{\rm em}/\xi_{\rm cr}$. If
Eq.~(\ref{eq:fup}) is not satisfied, e.g. because the upstream
magnetization is not small enough, particles gyrate in the background
magnetic field before experiencing the short scale turbulence. Then
Fermi acceleration will operate if Eq.~(\ref{eq:fdown2}) is verified.
Consequently, a sufficient condition for the development of Fermi
cycles is $\xi_{em} > \left(\sigma_{\rm crit}/\kappa\right)^{1/2}$, or
equivalently $\sigma_* > \sigma_{\rm crit}$, where $\sigma_{\rm crit}$
is the maximum value of the upstream magnetization that allows
turbulence to grow upstream and then be transferred downstream. As
shown previously for a superluminal configuration, $\sigma_{\rm crit}
= \Gamma_{\rm sh}^{-3}\xi_{\rm cr} m_p/m_e$ for an electron-proton
plasma in the realistic case where $\Gamma_s \lesssim 150\xi_{\rm
cr}^{1/3}$ so that the transition is governed by the excitation of
whistler waves; or $\sigma_{\rm crit}=\Gamma_{\rm
sh}^{-2}\xi_{cr}^{2/3}(m_p/m_e)^{1/3}$ for an electron-positron
plasma, with the development of the oblique two stream instability.
The spectral index and the maximal energy remain to be determined
however. In this respect, we note that Eq.~(\ref{eq:fdown}) provides
an upper bound for this maximal energy: $\epsilon_{\rm max} \simeq
\Gamma_{\rm sh} m_pc^2 (\sigma_*/\sigma_{\rm u})^{1/2}$ in the front
frame.
The more likely development of the Fermi process is thus hybrid, in
the sense that it is of drift type upstream and of diffusive type
downstream. As Fermi cycles develop, particles are accelerated beyond
the energy $\Gamma_{\rm sh}^2m_pc^2$ considered here for the first
generation. Although they are less numerous, they stream farther ahead
of the shock and are therefore liable to induce stronger
amplification. One can only speculate about these issues, since the
spectral index depends strongly on the assumption made on the shape of
the turbulence spectra, upstream as well as downstream. In particular,
if the magnetic field amplified downstream through the Weibel
instability decays on scales of order of tens or hundreds of electron
inertial lengths $\delta_{\rm e}$, the particles will likely escape
towards downstream because of the lack of scattering agents, thereby
cutting off the Fermi process prematurely. Nevertheless, assuming for
the sake of discussion that Fermi cycles develop with a spectral index
$s\,\sim\,2-3$, the number density of cosmic rays streaming upstream
scales as $n_{{\rm b}\vert\rm u}(>p_*)\,\propto\, (p_*/p_0)^{1-s}$,
with $p_0\,\sim\,\Gamma_{\rm sh}^2m_pc^2$. The beam plasma frequency,
which controls the growth rates of the instabilities, $\omega_{{\rm
p}*}(>p_*)\,\propto\,(p_*/p_0)^{-s/2}$, whereas the precursor
length $\ell_{\rm F\vert\rm u}(>p_*)\,\propto\,(p_*/p_0)$. Since the
growth rates of the resonant instabilities which develop upstream
scale as $\omega_{{\rm p}*}^{2/3}$, $s<3$ would guarantee that the
growth factor of the instabilities triggered by these high energy
particles exceeds that for the first generation. These findings seem
in agreement with the numerical simulations of Keshet et al. (2009)
and Sironi \& Spitkovsky (2009) who observe wave growth farther from
the shock from high energy particles, as time increases.
\subsection{Applications}\label{sec:appl}
It is interesting to situate the relativistic shock waves of physical
interest in the above diagram. Here we consider three proto-typical
cases: a pulsar wind, a gamma-ray burst external shock waves expanding
in the interstellar medium, and a gamma-ray burst external shock wave
propagating along a density gradient in a Wolf-Rayet wind. We find the
following:
\begin{itemize}
\item Pulsar winds: with $\Gamma\,\simeq\,10^6$ and $\sigma_{\rm
u}\,\simeq\,0.01$, one finds $(X,Y)\,\sim\,(500,10^{10}\xi_{\rm
cr}^{-1})$; the level of magnetization is thus so high that no wave
can grow, either upstream or downstream. Fermi acceleration should
consequently be inhibited.
\item Gamma-ray burst external shock waves expanding in the
interstellar medium: for $\Gamma\,\simeq\,300$ and $\sigma_{\rm
u}\,\sim\,10^{-9}$ (i.e. $B\,\sim\,3\,\mu$G), one finds
$(X,Y)\,\sim\,(0.1,10^{-5}\xi_{\rm cr}^{-1})$. Wave growth should be
efficient both usptream and downstream. Concerning Fermi
acceleration, Eq.~(\ref{eq:fup}) amounts to $Y\,<\,\xi_{\rm
cr}m_e/m_p$. It can thus be only marginally satisfied. However,
Eq.~(\ref{eq:fdown2}) is most likely satisfied, so that Fermi
acceleration should develop, even in the early afterglow phase when
$\Gamma_{\rm sh}\,\sim\,300$.
\item Gamma-ray burst external shock waves propagating along a density
gradient in a Wolf-Rayet wind: taking a surface magnetic field of
$1000\,G$ for a $10R_\odot$ Wolf-Rayet progenitor, the magnetization
at distances of $10^{17}\,$cm is $\sigma_{\rm u}\,\sim\,10^{-4}$
(Crowther 2007). This gives $(X,Y)\,\sim\,(0.1, \xi_{\rm
cr}^{-1})$. Growth may or may not occur in this case, depending on
the precise values of $\Gamma_{\rm sh}$, $\sigma_{\rm u}$ and
$\xi_{\rm cr}$. In detail, the condition for Weibel growth
$Y\,\lesssim\,1$ is likely not verified for the above fiducial
values, but could be verified in less magnetized winds and at later
stages of evolution, with a smaller value of $\Gamma_{\rm sh}$. The
condition for growth of Whistler waves, $Y\,\lesssim\,1/X$, may be
satisfied if $\xi_{\rm cr}\,\gtrsim\,0.1$ and it is likely to be
more easily verified at smaller values of $\Gamma_{\rm sh}$ and
$\sigma_{\rm u}$. Finally, the (most stringent) condition for growth
of the oblique mode, Eq.~(\ref{eq:obllimmag}), is likely not
verified in the initial stages with $\Gamma_{\rm sh}\simeq300$ and
the above fiducial value of $\sigma_{\rm u}$, but would be verified
if $\sigma_{\rm u}$ were smaller.
However, Eq.~(\ref{eq:fup}) cannot be satisfied in this case, meaning
that the orbit of the particle upstream is governed by the wind
magnetic field, not by the amplified short scale component. Regarding
the bound Eq.~(\ref{eq:fdown2}), it can be satisfied, depending on the
values of the wind magnetisation and most particularly on the value of
$\kappa$. The possibility of Fermi acceleration thus remains open in
this case. More work is necessary to understand the properties of
downstream turbulence in order to determine whether particle can
eventually be accelerated.
\end{itemize}
\subsection{Further considerations}
It is important to emphasize that we do not understand yet the
structure of a relativistic shock front in detail. In the previous
section we have assumed that the shock front is structured like a
non-relativistic front and just extended the non-relativistic
results. Since MHD compressive instability and extraordinary ionic
modes can be excited, we cannot exclude that the foot be full of
relativistically hot protons and electrons of similar temperature
$\bar \gamma m_pc^2$ with $1\, \ll \, \bar \gamma\,\leq \,\Gamma_s$.
In that case the plasma
response would be different, because the intermediate whistler range
(and also extraordinary range) would disappear so that the plasma
would behave like a relativistic pair plasma. Then, the relevant
instabilities are the Weibel and oblique modes (in the unmagnetized
approximation). The length of the precursor and the Weibel growth rate
remain unchanged, hence the domain of growth of the Weibel instability
also remains unchanged. The growth rate of the oblique mode is however
reduced because the background plasma frequency is smaller by a ratio
$(\gamma m_p/m_e)^{1/2}$. Therefore the condition of growth on the
advection timescale now reads $Y\,\ll\,\xi_{\rm cr}^{-1/3} \Gamma_{\rm
sh}^{1/3} X^{-1/3} (\gamma m_p/m_e)^{-1/3}$. The ratio of the growth
rates of the oblique mode to the Weibel mode can be written as
$(\gamma \xi_{\rm cr})^{-1/6}$, hence the Weibel instability becomes
the dominant mode if $\gamma\,\gg\,\xi_{\rm cr}^{-1}$.
In the downstream plasma, the magnetic fluctuations generated by the
Weibel instability are expected to disappear rapidly because they do
not correspond to plasma modes. Whistler and other resonant eigenmodes
(when they are excited) are however transmitted and although they are
not excited downstream, their damping is weak. When Fermi cycles
develop, they create ``inverted'' distribution downstream, that should
produce a maser effect.
Tangled magnetic field carried by the upstream flow are very
compressed downstream and thus opposite polarization field lines come
close together. This should produce magnetic reconnections in an
unusual regime where protons and electrons have a similar relativistic
mass of order $\Gamma_{\rm sh} m_pc^2$. Such a regime of reconnection
deserves a specific investigation with appropriate numerical
simulations. Despite magnetic dissipation, reconnections would
probably create a chaotic flow that favors diffusion of particles from
downstream to upstream. \\
\section{Conclusions}
In this work, we have carried out a detailed study of the
micro-instabilities at play in the precursor of a ultra-relativistic
shock wave. The main limitation for the growth of these waves is
related to the length of precursor, which is itself related to the
level of magnetisation in the upstream plasma (where magnetisation
refers to the background field, not the shock generated short scale
fields). Nevertheless, we have found electronic and ionic
instabilities that grow sufficiently fast in the precursor of a
relativistic shock. The fastest growing instabilities are due to the
\v{C}erenkov resonance between the beam of accelerated (and shock
reflected protons) and the upstream plasma Whistler waves and
electrostatic modes. The Weibel instability, which is non-resonant by
essence, is also excited, but its growth is generally superseded by
that of the previous modes. The strongest amplification occurs on very
short spatial scales $\sim\delta_{\rm e}$, the electron skin depth in
the upstream plasma. Our results are summarized in
Fig.~\ref{fig:XY_perp} for the generic case of relativistic
superluminal shock waves, which delimits the domains in which
electromagnetic modes are excited in terms of shock Lorentz factor and
upstream magnetisation and defines the critical value $\sigma_{crit}$
of the magnetization below which Fermi process can operate. Figure
\ref{fig:XY_par} presents the corresponding limitations for the case
of parallel shock waves; in this case, the growth of instabilities is
made much easier by the divergence of the precursor length for a
fraction of the particles returning upstream. Our results explain some
features of recent PIC simulations of relativistic pair shocks of
various geometries and magnetisation levels.
We have discussed the conditions under which Fermi acceleration can
proceed superluminal shock waves once a significant fraction of the
cosmic ray energy has been dumped into these short scale
electromagnetic fluctuations. Fermi acceleration can operate if the
upstream magnetisation ($\sigma_{\rm u}$) or downstream magnetisation
($\sigma_{\rm d}$) are low enough for the shock generated turbulence
to govern the scattering of particles. This is the second condition
that states the required level of electromagnetic energy density
versus the magnetization. This requires either $\sigma_{\rm u}\,\ll\,
\xi_{\rm em}^2 (m_e / m_p)\Gamma_{\rm sh}^{-2}$ (for upstream
scattering), which is however difficult to fulfill or $\sigma_{\rm
d}\,\ll\, \kappa\, \xi_{\rm em}^2$ (for downstream scattering, which
is easily fulfilled with a low level of turbulence); $\xi_{\rm em}$
indicates the fraction of incoming energy transferred into
electromagnetic fluctuations, with $\xi_{\rm
em}\,\sim\,10^{-2}-10^{-1}$ generally indicated by PIC simulations,
and $\kappa$ is a fudge factor that encaptures our ignorance of the
transfer of electromagnetic modes excited upstream through the shock,
$m_e/m_p\,\lesssim\,\kappa\,\lesssim\,1$ (and it may be even larger
depending on the particle energy if the scale of the electromagnetic
flucutations evolves with the distance to the shock). We emphasize the
need for PIC simulations with realistic geometry, realistic proton to
electron mass ratios and moderate magnetisation (of order of the
above) in order to lift this uncertainty on $\kappa$ and to determine
the precise conditions under which Fermi acceleration can take place.
This limitation also places a strict upper bound on the maximum energy
that is achievable through the Fermi process, namely $\epsilon_{\rm
max} \simeq \Gamma_{\rm sh}m_p c^2 (\sigma_{\rm crit}/\sigma_{\rm
u})^{1/2}$, with $\sigma_{\rm crit}$ the maximum magnetisation that
allows waves to grow, and $\sigma_{\rm u}$ the upstream background
magnetisation (see discussion after Eq.~(\ref{eq:fdown2}). Beyond this
intrinsic limit, the scattering time indeed becomes longer than the
Larmor time in the mean field downstream, so that the particle is
advected downstream by the mean field and Fermi cycles end.
We have also applied our calculations to several cases of
astrophysical interest. In practice, we find that terminal shocks of
pulsar winds have a magnetisation level that is too high to allow for
the amplification of short scale electromagnetic fields, so that
particle acceleration must be inhibited. We have found that gamma-ray
burst external shock waves propagating into a typical interstellar
medium should lead to strong amplification of the magnetic field and
to Fermi cycles, even at high Lorentz factor. The energies reached by
the suprathermal electrons can easily explain the afterglow emission
through jitter radiation (Medvedev 2000). However, if the shock wave
propagates in a stellar wind, the upstream magnetisation may be too
large to allow for particle acceleration, eventhough magnetic field
amplification should take place.
\section*{Acknowledgments}
We warmly acknowledge A. Marcowith for collaboration at an earlier
stage of this work and for a careful reading of the manuscript. We
also thank the referee for a detailed reading of the manuscript and
for useful suggestions. One of us (G.P.) acknowledges fruitful
discussions with J. Arons, A. Bell, L. Drury, J. Kirk, Y. Lyubarsky,
J. Niemiec, M. Ostrowski, B. Reville, H. V\"olk, A. Spitkovsky and
L. Sironi. We also thanks an anonymous referee for his accurate and
detailed analysis of our work.
|
2,869,038,155,642 | arxiv | \section{Introduction}
Chest X-rays are the most common imaging modality read by radiologists in hospitals and tele-radiology practices today. With advances in artificial intelligence, there is now the promise of obtaining automated preliminary reads that can expedite clinical workflows, improve accuracy and reduce overall costs. Current automated report generation methods are based on image captioning approaches of computer vision \cite{vinyals2015show,xu2015show} and use an encoder-decoder architecture where a Convolutional Neural Network (CNN) is used to encode images into a set of semantic topics \cite{jing2017automatic} or limited findings \cite{jiebo} and a Recurrent Neural Network (RNN) decoder or a hierarchical LSTM generates the most likely sentence given the topics \cite{harzig2019addressing,rennie2017self,krause2017hierarchical,wang2018tienet,li2018hybrid,jing2017automatic}. Other approaches have leveraged template sentences to aid in paraphrasing and report generation \cite{li2018hybrid,gale2018producing,li2019knowledge}. Recent approaches have also emphasized the role of clinical accuracy measured loosely through clinical correlation between disease states in the objective functions \cite{szolovits}.
Despite the progress, the quality of reports generated by current approaches is not yet clinically acceptable as they do not ensure the correct detection of a comprehensive set of findings nor the description of their clinical attributes such as laterality, anatomical location, severity, etc. The emphasis is usually more on the report language generation than the visual detection of findings. In this paper we take a new approach in which we train deep learning networks on a large number of detailed finding labels that represent an in-depth and comprehensive characterization of findings in chest X-ray images. An initial set of core findings label vocabulary were derived through a multi-year chest X-ray lexicon building effort involving several radiologists and clinical experts. The detailed finding labels were then automatically derived from their associated radiology reports through a concept detection and phrasal grouping algorithm that associates detailed characterization modifiers with the initially identified core findings using natural language analysis. The resulting labels with large image support were used to train a novel deep learning network based on feature pyramids. Given a new chest X-ray image, the joint occurrence of detailed finding labels is predicted as a pattern vector from the learned model and is matched against a pre-assembled database of label patterns and their associated reports. Finally, the retrieved report is post-processed to remove mentioned findings whose evidence is absent in the predicted label pattern.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth,clip]{figures/combined1.png}
\end{center}
\caption{Illustration of the finer description labels for capturing the essence of reports.}
\label{reportexample1}
\end{figure}
\section{Describing images through fine finding labels (FFL)}
Consider the chest X-ray image shown in Figure~\ref{reportexample1}a. Its associated report is shown in Figure~\ref{reportexample1}b. In order to automatically produce such sentences from analyzing images, we need image labels that cover not only the core finding, such as opacity, but also its laterality, location, size, severity, appearance, etc. Specifically, a full description of the finding can be denoted by a fine finding label (FFL) as
\begin{equation}
F_{i}=<T_{i}|N_{i}|C_{i}|M_{i}^{*}>
\end{equation}
\noindent where $F_{i}$ is the FFL label, $T_{i}$ is the finding type, $N_{i}=yes|no$ indicates a positive or negative finding (i.e is present versus absent), $C_{i}$ is the core finding itself, and $M_{i}$ are one or more of the possible finding modifiers. The finding types in chest X-rays are adequately covered by six major categories namely, anatomical findings, tubes and lines and their placements, external devices, viewpoint-related issues, and implied diseases associated with findings. The vocabulary for core findings as well as possible modifiers was semi-automatically assembled through a multi-year chest X-ray lexicon development process in which a team of 4 clinicians including 3 radiologists, iteratively searched through the best practice literature such as Fleishner Society guidelines \cite{hansell2008fleischner} and used every day use terms to expand the vocabulary by examining a large dataset of 220,000 radiology reports in a vocabulary building tool \cite{dla} addressing abbreviations, misspellings, semantical equivalence and ontological relationships. Currently, the lexicon consists of over 11000 unique terms covering the space of 78 core findings and 9 modifiers and represents the largest set of core findings assembled so far. The set of modifiers associated with each core finding also depends on the finding type and the FFL label syntax captures these for various finding types.
The FFL labels capture the essence of a report adequately as can be seen in Figure~\ref{reportexample1}c and comparing with the actual report in Figure~\ref{reportexample1}b. Further, if the FFL labels are similar, a similarity is also implied in the associated reports. Figure~\ref{reportexample1}d-g show examples of similar reports all of which are characterized by similar FFL patterns. Thus if we can infer the FFL labels from the visual appearance of findings in chest X-ray images, we can expect to generate an adequate report directly from the labels.
\subsection{Extraction of FFL Labels from reports}
The algorithm for extracting FFL labels from sentences in reports consists of 4 steps, namely, (a) core finding and modifier detection, (b) phrasal grouping, (c) negation sense detection, (d) pattern completion.
The vocabulary of core findings from lexicon and their synonyms were used to detect core concepts in sentences of reports using the vocabulary-driven concept extraction algorithm described in \cite{guo2017efficient}. To associate modifiers with relevant core findings, we used a natural language parser called the ESG parser \cite{esg} which performed word tokenization and morpholexical analysis to create a dependency parse tree for the words in a sentence as shown in Figure~\ref{dependency}. The initial grouping of words is supplied directly by the parse tree such as the grouping of terms 'alveolar' and 'consolidation' into one term 'alveolar consolidation' shown in Figure \ref{dependency}. Further phrasal grouping is done by clustering the lemmas using word identifiers specified in the dependency tree. For this, a connected component algorithm is used on the word positions in slots, skipping over unknowns (marked with u in tuples). This allows all modifiers present within a phrasal group containing a core finding to be automatically associated with the finding. For example, the modifier 'stable' is associated with the core finding 'alveolar consolidation' in Figure~\ref{dependency}. The modifiers in phrasal groups that do not contain a core finding are associated with the adjacent phrasal groups that contain a core finding.
To determine if a core finding is a positive or negative finding (e.g. "no pneumothorax"), we use a two-step approach that combines language structuring and vocabulary-based negation detection as described in \cite{guo2017efficient}. The negation pattern detection algorithm iteratively identifies words within the scope of negation by iteratively expanding neighborhood of seed negation terms by traversing the dependency parse tree of a sentence. The details are described in \cite{guo2017efficient}.
The last step completes the FFL pattern using a priori knowledge captured in the lexicon for the associated anatomical locations of findings when these are not specified in the sentence itself as shown in Figure~\ref{dependency} where the term "alveoli" is inserted from the knowledge of the location of the finding 'alveolar consolidation'. Thus the final FFL label produced may show more information than the original sentence from which it was extracted. In addition, the name of the core finding may be ontologically rolled up to the core findings as seen in Figure~\ref{reportexample1} for 'emphysema'.
\begin{figure}
\begin{center}
\includegraphics[width=0.90\linewidth]{figures/phrasalgroups.png}
\end{center}
\caption{Illustration of the dependency parse tree and phrasal grouping.}
\label{dependency}
\end{figure}
The FFL label extraction algorithm was applied to all sentences from a collection of 232,964 reports derived from MIMIC-4 \cite{mimic-4} and NIH \cite{wang2017chestx} datasets, to generate all possible FFL patterns from the Findings and Impression sections of reports. A total of 203,938 sentences were processed resulting in 102,135 FFL labels. By retaining only those labels with at least 100 image support, a total of 457 FFL labels were selected. As shown in the Results section, the label extraction process is highly accurate, so that spot check clinical validation is sufficient for use in image labeling. Since the FFL labels were seeded by clinically selected core findings, nearly 83\% of all FFL labels extracted could be mapped into their nearest counterpart in the 457 FFL Label set. Thus the set of 457 labels were found sufficient to cover a wide variety in spoken sentences and were used as labels for building deep learning models. Of these, 78 were the original core labels (called the CFL labels) given by clinicians, and the remaining were finer description labels with modifiers extracted automatically.
\subsection{Learning FFL labels from images}
The learning of FFL labels from chest X-rays is a fine-grained classification problem for which single networks used for computer vision problems may not yield the best performance, particularly since large training sets are still difficult to obtain. The work in \cite{Conference:Nguyen:ISCAS2018} shows that concatenating different ImageNet-pretrained features from different networks can improve classification on microscopic images. Following this idea, we combine the ImageNet-pretrained features from different models through the Feature Pyramid Network in \cite{Conference:Lin:CVPR2017}. This forms the multi-model feature pyramid which combines the features in multiple scales. The VGGNet (16 layers) \cite{Journal:Simonyan:arXiv2014} and ResNet (50 layers) \cite{Conference:He:CVPR2016} are used as the feature extractors. As nature images and chest X-rays are in different domains, low-level features are used. From the VGGNet, the feature maps with 128, 256, and 512 channels are used, which are concatenated with the feature maps from the ResNet of the same spatial sizes which have 256, 512, and 1024 feature channels.
We propose dilated blocks to learn the high-level features from the extracted ImageNet features. Each dilated block is composed of dilated convolutions for multi-scale features \cite{Journal:Yu:arXiv2015}, a skip connection of identity mapping to improve convergence \cite{Conference:He:ECCV2016}, and spatial dropout to reduce overfitting. Group normalization (16 groups) \cite{Conference:Wu:ECCV2018} whose performance is independent of the training batch size is used with ReLU. Dilated blocks with different feature channels are cascaded with maxpooling to learning more abstract features. Instead of global average pooling, second-order pooling is used, which is proven to be effective for fine-grained classification \cite{Conference:Yu:ECCV2018}. Second-order pooling maps the features to a higher-dimensional space where they can be more separable. Following \cite{Conference:Yu:ECCV2018}, the second-order pooling is implemented as a 1$\times$1 convolution followed by global square pooling (Figure~\ref{network}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth]{figures/network.pdf}
\end{center}
\caption{Illustration of the custom deep learning network developed for large number of label recognition problem.}
\label{network}
\end{figure}
Image augmentation with rigid transformations is used to avoid overfitting. As most of an image should be included, we limit the augmentation to rotation ($\pm$\ang{10}) and shifting ($\pm${10}\%). The probability of an image to be transformed is 80\%. The optimizer Nadam is used with a learning rate of 2$\times$10$^{-6}$, a batch size of 48, and 20 epochs. To ensure efficient learning, we developed two instances of this network, one for the core finding labels (CFL labels) and the other for detailed FFL labels that have a support of at least 100 images for training to exploit the mutually reinforcing nature of the coarse-fine labels.
Due to the variability in the size of the dataset per FFL label, the AUC per FFL label is not always a good indicator for precision on a per image level as it is dominated by the negative examples. To ensure we report as few irrelevant findings while still detecting all critical findings within an image, we select operating points on the ROC curves per label based on optimizing the F1 score, a well-known measure of accuracy, as
\begin{equation}
L(\theta)=-ln(\frac{1}{n}\sum_{i=1}^{n}F1_{i}(\theta))
\end{equation}
\subsection{FFL Pattern-Report Database Creation}
Using the FFL label detection algorithm, we can describe a report (its relevant sections) as a binary pattern vector $P=\{I_{P}(F_{j})\}$ where $I_{P}(F_{j})=1$ if the FFL label $F_{j}\in \hat{F}$ is present in the report and zero otherwise. Here $\hat{F}$ is the set of FFL labels used in training the deep learning models. During the database creation process, we collect all reports characterized by the same binary pattern vector, and rank them based on the support provided by their constituent sentences. Let $\hat{R_{P}}={r_{s}}$ be the collection of reports spanned by a pattern vector $P$. Then
\begin{equation}
Rank(r_{s})=\sum_{j=1}^{M_{s}}h(s_{j})
\label{rank}
\end{equation}
where $M_{s}$ is the number of relevant sentences in report $r_{s}$ spanned by one or more of the FFL labels in the pattern $P$. Here $h(s_{j})$ is given by
$h(s_{j})=\frac{\mbox{Number of reports }r_{i}\mbox{ that contain }s_{j}}{|\hat{R_{P}}|}$
The highest ranked reports are then stored as associated reports with the binary pattern vectors in a database.
\subsection{Report assembly}
The overall report generation workflow is illustrated in Figure~\ref{overall}. An image is fed to the two deep learning networks built for CFL and FFL patterns and their predictions thresholded using the image-based precision-recall F1-score for optimization. The resulting pattern vectors are combined to result in the consolidated FFL pattern vector $Q=\{I_{Q}(F_{j})\}$. The best matching reports are then derived from the semantically nearest pattern vectors in the database. The semantic distance between the query FFL bit pattern vector $Q$ and a matching pattern vector from the database $P$ is given by
\begin{equation}
d(Q,P)=\frac{\sqrt{\sum_{l=1}^{|\hat{F}|}w_{l}(I_{P}(F_{l})-I_{Q}(F_{l}))^{2}}}{|\hat{F}|}
\label{nearest}
\end{equation}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\linewidth]{figures/cleanerpicture.png}
\end{center}
\caption{Illustration of quality of reports generated by different methods.}
\label{resultexample}
\end{figure}
where $w_{l}$ is the weight associated with the FFL label $F_{l}$. A criticality rank for each core findings on a scale of 1 to 10 was supplied by the clinicians which was normalized and used to weigh the clinical importance of a finding during matching. Once the matching FFL pattern is determined, the highest ranked report as given by Equation~\ref{rank} associated with the FFL pattern is retrieved as the best matching report. Finally, we drop all sentences from the retrieved report whose evidence cannot be found in the FFL label pattern of the query thus achieving the variety needed in returned reports per query. Although with 457 FFL labels, the number of possible binary patterns would be large ($2^{457}$), due to the sparseness of 5-7 findings per report, the actual number of distinct binary patterns in the database of over 232,000 reports was only 924 patterns corresponding to 5246 distinct sentences in the precomputed ranked list across all patterns. Thus the lookup per predicted pattern is a fairly trivial operation which is O(1) with indexing and takes less than 5 msec.
\section{Results}
We collected datasets from three separate sources, namely, the MIMIC \cite{mimic-4} dataset of over 220,000 reports with associated frontal images, 2964 unique Indiana reports \cite{indiana} with associated images and a set of 10,000 NIH \cite{wang2017chestx} released images re-read by our team of radiologists to produce a total of 232,964 image-report pairs for our experiments.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Dataset & FFL Label Phases& Train& Validate& Test & Average AUC & Weighted \\
\hline
MIMIC-4 + NIH &CFL labels& 249,286&35,822& 70,932&0.805 & 0.841\\
\hline
MIMIC-4+NIH &FFL labels& 75,613 &10,615& 20,941 &0.729 & 0.716 \\
\hline
\end{tabular}
\caption{Illustration of the datasets and performance of fine grained classification model for CFL and FFL labels (last column is average of AUCs weighted by the number of samples per each category). }
\label{datasets}
\end{center}
\end{table}
\textbf{Evaluating FFL label extraction accuracy: }We evaluated the accuracy of FFL label extraction by noting the number of findings missed and overcalled (which included errors in negation sense detection) as well the correctness and completeness of association of modifiers with the relevant core findings. The result of evaluation for the Indiana dataset \cite{indiana} by our clinicians is shown in Table~\ref{fllevaluation}. As can be seen, the FFL label extraction is highly accurate in terms of the coverage of findings with around 3\% error mostly due to negation sense detection. Further, the association of modifiers to core findings given by the phrasal grouping algorithm is also accurate with over 99\% precision and recall.
\textbf{FFL Label Prediction from Deep Learning: }The training, validation and test image datasets used for building the CFL and FFL models used MIMIC-4 and NIH datasets as shown in Table~\ref{datasets}. The AUC averaged for all CFL labels and FFL labels is also shown in that table. In addition, using the F1-score-based optimization, the mean average image-based precision for CFL labels was 73.67\% while the recall was 70.6\%.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{figures/Fig5.png}
\end{center}
\caption{Illustration of the report generation algorithm.}
\label{overall}
\end{figure}
\begin{table}
\begin{center}
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
reports & relevant & FFL patterns & missed & overcall & incorrect association & missed\\
analyzed & sentences & extracted & findings & (negated findings) & of modifiers & modifiers \\
\hline
2964 & 3046 & 5245 & 0 & 168 & 49 & 11
\\
\hline
\end{tabular}
\end{scriptsize}
\end{center}
\caption{The accuracy of FFL label extraction from reports.}
\label{fllevaluation}
\end{table}
\textbf{Evaluation of report generation: } Due to the ontological mapping used to abstract the description of findings, the match produced from our approach is at a more semantic level rather than lexical in comparison to other approaches. Figure~\ref{resultexample} shows the reports manually and automatically produced by our approach and a comparative approach implemented from a visual attention-based captioning model \cite{vinyals2015show}.
We compared the performance of our algorithm with several state-of-the-art baselines from recent literature \cite{vinyals2015show,multimodalrecurrent,li2019knowledge,szolovits,jing2017automatic,jiebo}. These included a range of approaches from visual attention-based captioning \cite{vinyals2015show}, knowledge-driven report generation \cite{li2019knowledge}, clinically accurate report generation \cite{szolovits}, to a strawman approach using a set of template sentences manually chosen by clinicians for the FFL labels instead of the nearest report selection algorithm described earlier. Although we have tested our algorithm for very large number of images from the combined MIMIC-NIH data, for purposes of comparison, we show the results on the same Indiana test dataset that has been used most commonly by other algorithms as reported in \cite{jing2017automatic}. The resulting performance using the popular scoring metrics is shown in Table~\ref{fflresults} showing that our algorithm outperforms other approaches in all the established scoring metrics.
\textbf{Conclusions: }We presented an explainable AI approach to semantically correct radiology report generation. The results show superior performance both because of the detailed descriptive nature of labels, and due to a statistically informed report retrieval process that ensures a semantic match.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Methods &BLEU-1 &BLEU-2 &BLEU-3 &BLEU-4 &METEOR &ROUGE-L\\
\hline
Vis-Att \cite{vinyals2015show} &0.39 &0.25 &0.16 &0.11 &0.16 &0.32\\
\hline
MM-Att \cite{multimodalrecurrent} &0.46 &0.35 &0.27 &0.19 &0.27 &0.36\\
\hline
KERP \cite{li2019knowledge} &0.48 &0.32 &0.22 &0.16 &- &0.33\\
\hline
Template-based &0.28 &0.29 &0.32 &0.27 &0.35 &0.34\\
\hline
Clinical Accurate \cite{szolovits} &0.35 &0.22 &0.15 &0.1 &- &0.45\\
\hline
Co-Att \cite{jing2017automatic} &0.51 &0.39 &0.30 &0.25 &0.21 &0.44\\
\hline
Jiebo Luo \cite{jiebo} &0.53 &0.37 &0.31 &0.25 &0.34 &0.45\\
\hline
CFL-only & 0.49 & 0.39 & 0.36 &0.32 &0.48 &0.52 \\
\hline
FFL+CFL -based (ours) &\bf{0.56} &\bf{0.51} &\bf{0.5 }&\bf{0.49} &\bf{0.55} &\bf{0.58}\\
\hline
\end{tabular}
\caption{Comparative performance of report generation by various methods.}
\label{fflresults}
\end{center}
\end{table}
{\small
\bibliographystyle{splncs03}
|
2,869,038,155,643 | arxiv | \section{Introduction}
Inflation has established itself as a key idea in our understanding of the universe \cite{inflation}. A single stage of accelerated expansion can explain in one shot why the universe is homogeneous and isotropic, spatially flat, what is the origin of the entropy in the universe and why primordial perturbations are adiabatic, Gaussian and nearly scale invariant. For this reason, due to its central role in modern cosmology, it is important to ascertain whether there are alternatives to explain some of the inflationary predictions \cite{alternatives}, or whether the universe inevitably had to undergo an inflationary stage \cite{non-alternatives}.
The generation of an adiabatic, Gaussian and scale invariant spectrum of super-horizon perturbations arguably is the most important success of inflation. Indeed, one can reason that it is natural for the universe to begin in a symmetric state, thus accounting for its homogeneity and isotropy. But in an expanding universe with a monotonically decreasing Hubble parameter, there seems to be no way to causally explain the origin of super-horizon perturbations other than inflation, regardless of their spectral index.
In previous works, Eugene Lim and I explored alternative ways of seeding a scale invariant spectrum of primordial perturbations by relaxing some of the assumptions made in our descriptions of the origin of structure. In particular, in \cite{Armendariz-Picon:2003ht} we studied whether a contracting sound horizon can lead to the seeding of perturbations, and in \cite{Armendariz-Picon:2003ku} I considered the impact of a time-varying scalar field mass on the primordial spectrum. It turns out that in both cases, the generation of scale invariant super-horizon sized primordial perturbations still requires the existence of an inflationary stage, although some of the constraints on its properties can be significantly relaxed.
This work is to some extent a natural continuation of these previous attempts. Our models on the origin of structure are based on extrapolations of low-energy theories to extremely high momenta. However, as we go back in time in the history of the universe, we expect physical laws to change as typical energies increase. In particular, the dispersion relations used to describe the evolution of structure should receive corrections suppressed by a cut-off scale $M$, which become important at high energies. A similar phenomenon occurs in crystal lattices; as the wave vector of an electron approaches a Bragg plane, its energy deviates from the conventional quadratic behavior.
In this work, we explore whether modified dispersion relations containing higher powers of the spatial momentum $k$, can lead to exactly or nearly scale invariant primordial spectra in a non-inflating universe. Ironically, similar dispersion relations have been also considered to study the robustness of inflationary predictions. Specifically, they have been used to analyze whether ``trans-Planckian" physics may significantly affect the spectrum of primordial perturbations seeded during inflation \cite{Martin:2003kp}. Our goal here is quite different. We would like to determine whether modified dispersion relations may lead to nearly scale invariant spectra in circumstances where we do not expect them to be so.
\section{Modified Dispersion Relations}
Consider an expanding, homogeneous, isotropic and spatially flat universe, $ds^2=a^2(\eta)(-d\eta^2+d\vec{x}^2).$ Let us assume for simplicity that the universe power-law expands,
\begin{equation}\label{eq:a}
a(\eta)=\left(\frac{\eta}{\eta_T}\right)^p,
\end{equation}
where we have arbitrarily set to one the value of the scale factor at an arbitrary time $\eta_T$ we shall later define. The universe accelerates for $-\infty<p<-1$ and decelerates for $1/2<p<\infty$, where we have assumed that the dominant energy condition is satisfied. Concretely, a universe with effective equation of state parameter $w$ has
\begin{equation}
p=\frac{2}{1+3w}.
\end{equation}
In Fourier space, the equation of motion of a massless scalar field $\varphi$ in such a universe is
\begin{equation}\label{eq:motion}
v_k''+\left(\omega^2(k,\eta)-\frac{a''}{a}\right)v_k=0,
\end{equation}
where $v_k= a\cdot \varphi_k$, and a prime denotes a derivative with respect to conformal time. In this work we assume that the dispersion relation of the scalar field is
\begin{equation}\label{eq:dispersion}
\omega^2(k,\eta)=\frac{1 }{M^{2n-2}}\frac{k^{2n}}{a^{2m}},
\end{equation}
where $n$ and $m$ are constant parameters, and $M$ is a mass scale (recall that we have set $a_T=1$.) Conventionally, the dispersion relation of a scalar field is $\omega^2=k^2$, that is, $n=1$ and $m=0$. Fluids with a non-constant speed of sound yield a dispersion relation with $n=1$ and $m\neq 0$. We discuss in Section \ref{sec:realizations} how dispersion relations with different values of $n$ and $m$ \cite{Brandenberger:2000wr} can arise from generally covariant scalar field actions. In the meantime, the reader can think of equation (\ref{eq:dispersion}) as a phenomenological description of new physics at momenta $k/a$ that at time $\eta_T$ are above the cut-off $M$ \cite{Unruh:1994je,Corley:1996ar}.
\subsection{Initial conditions}
The solution of equation (\ref{eq:motion}) with dispersion relation (\ref{eq:dispersion}) in a universe that expands according to equation (\ref{eq:a}) is
\begin{equation}\label{eq:solution}
v_k(\eta)=\sqrt{\eta}\left[C_1\, H_\nu^{(1)}\left(\frac{\omega\cdot \eta}{mp-1}\right)+
C_2 \,H_\nu ^{(2)}\left(\frac{\omega \cdot\eta}{mp-1}\right)\right],
\end{equation}
where the $H_\nu$ are the Hankel functions of the first and second kind, and
\begin{equation}
\nu=\frac{1}{2}\left|\frac{2p-1}{mp-1}\right|.
\end{equation}
Note that the equation of motion (\ref{eq:motion}) is invariant under $\eta\to -\eta$. Thus, in order to obtain a solution during inflation (where conformal time is negative), we can simply replace $\eta$ by $-\eta$. Because the differential equation (\ref{eq:motion}) is second order, its general solution contains two arbitrary integration constants, $C_1$ and $C_2$. Their values are determined by requiring that perturbations be in the adiabatic vacuum at early times. Essentially, this is the only way we know of to specify initial conditions.
The notion of an adiabatic vacuum exists only if the equation of motion admits the approximate solution
\begin{equation}\label{eq:vacuum}
v_k\approx \frac{1}{\sqrt{2\,\omega(k,\eta)}}
\exp\left(-i\int^\eta \omega(k,\tilde{\eta}) \, d\tilde{\eta}\right),
\end{equation}
where $\omega$ is a possibly time-dependent frequency \cite{BirrellDavies}. This in fact resembles the vacuum mode functions of a field in Minkowski space. If such an approximate solution exists, perturbations are defined to be in the adiabatic vacuum if they match the approximate solution (\ref{eq:vacuum}) at early times.
It can be verified that in the limit $\omega \, \eta \gg 1$, the solution (\ref{eq:solution}) reproduces the adiabatic vacuum (\ref{eq:vacuum}) if
\begin{equation}\label{eq:initial}
C_1=\frac{1}{2}\sqrt{\frac{\pi}{mp-1}}, \quad C_2=0.
\end{equation}
Hence, what remains to be verified is whether $\omega\,\eta \to \infty$ at early times. Substituting equation (\ref{eq:a}) into (\ref{eq:dispersion}) we find that $\omega\, \eta\gg 1$ amounts to
\begin{equation}
\left(\frac{H}{H_T}\right)^{\frac{mp-1}{p+1}} k^n \, H_T^{-1}\gg M^{n-1},
\end{equation}
where $H_T$ is the value of the Hubble parameter $H$ at the arbitrary time $\eta_T$. It is hence clear that the adiabatic condition is satisfied at early times (when $H\to \infty$) if
\begin{equation}\label{eq:condition}
(1+p) (mp-1)>0.
\end{equation}
Therefore, for a conventional dispersion relation, $n=1$ and $m=0$, we can specify vacuum initial conditions only when ${p<-1}$, that is, during inflation. However, if we drop this assumption, we find that an adiabatic solution at early times exists even if $p>1/2$, that is, even if the universe decelerates. Note that if condition (\ref{eq:condition}) is satisfied, $\omega^2\cdot \eta^2$ decreases as the universe expands.
\subsection{Horizon Crossing}
It is generally argued that only during a stage of inflation it is possible to seed primordial perturbations, regardless of their spectrum. The argument goes as follows. The physical length of a mode is $a/k$, and the Hubble radius is $H^{-1}$. Only during a stage of inflation does the length of a mode increase faster than the Hubble radius, that is, only during a stage of inflation can modes begin sub-horizon sized and later become super-horizon sized. In other words, $k^2 \eta^2$ decreases only if the universe accelerates.
Our previous analysis shows the limits of the previous argument (see also \cite{Armendariz-Picon:2003ht}). Going back to the equation of motion (\ref{eq:motion}) one realizes that what really matters is whether $\omega^2$ grows faster or slower than than $a''/a$. Only if $\omega^2\gg a''/a$ at early times there is a chance that we can find solutions that match the adiabatic vacuum (\ref{eq:vacuum}). We call modes for which $\omega^2>a''/a$ ``short-wavelength" modes, and modes for which $\omega^2<a''/a$ ``long-wavelength" modes. Typically, $a''/a \sim 1/\eta^2$, so a transition between the short and long-wavelength regimes occurs when $\omega^2 \eta^2$ decreases, irrespective of the universe expansion \cite{Armendariz-Picon:2003ht}. Moreover, in some (degenerate) cases even this conclusion does not apply. In a radiation dominated universe, $a''/a$ equals zero, and hence all modes (both sub-horizon and super-horizon) are short-wavelength.
Nevertheless, the existence of a short-wavelength regime is not sufficient for the existence of an adiabatic regime, which is the regime needed to specify meaningful initial conditions. Equation (\ref{eq:vacuum}) typically is an approximate solution of the equations of motion only if the adiabatic condition
$\omega'/\omega^2\ll 1$ holds \cite{BirrellDavies}. This equation is formally different from the short-wavelength condition $\omega^2\gg a''/a$, although in practice both are equivalent. Because $\omega'/\omega \sim 1/\eta$ and $a''/a\sim 1/\eta^2$, short-wavelength modes are generally adiabatic. Therefore, the only condition we have to impose on any model for the origin of structure is that $\omega^2 \, \eta^2$ be monotonically decreasing. Since the period of oscillation of the perturbations (in conformal time) is $1/\omega$, what this means is that the crucial requirement for the causal generation of perturbations is freeze-out ($\omega\, \eta > 1\to\omega\, \eta < 1$), rather than horizon-crossing ($k\, \eta>1 \to k \, \eta< 1$).
\subsection{The Power Spectrum}
The solution (\ref{eq:solution}) of the equation of motion (\ref{eq:motion}) with adiabatic vacuum initial conditions (\ref{eq:initial}) is
\begin{equation}\label{eq:exact}
v_k=\frac{1}{2}\sqrt{\frac{\pi \eta}{mp-1}} H^{(1)}_\nu\left(\omega(k,\eta)\cdot \eta\right).
\end{equation}
With this exact solution in hand, we are ready to calculate the primordial spectrum of field perturbations. Because, by assumption, $\varphi$ is in a vacuum state, its expectation value vanishes, $\langle \varphi \rangle=0$. The two-point function of the field is characterized by the power spectrum
\begin{equation}
\mathcal{P}_{\varphi}=\frac{k^3}{2\pi^2} |\varphi_k|^2,
\end{equation}
which is also a measure of the mean squared inhomogeneities of the field on comoving length-scales $1/k.$ Using the relation $v_k= a\, \varphi_k$ and equation (\ref{eq:exact})
we find that at late times ($\omega \, \eta\ll1$) the Fourier mode $\varphi_k$ remains constant for ${\text{sgn}[(2p-1)/(mp-1)]=1}$. In this case, the spectrum is given by
\begin{equation}\label{eq:power}
\mathcal{P}_\varphi= A^2 \cdot
\left(\frac{k}{k_T}\right)^{n_s-1},
\end{equation}
where the spectral index is
\begin{equation}\label{eq:index}
n_s-1= 3-n \left|\frac{2p-1}{mp-1}\right|,
\end{equation}
and the squared amplitude equals
\begin{equation}
A^2=\frac{\Gamma^2(\nu)}{4\pi^3}\left(\frac{1}{2}\frac{p}{mp-1}\right)^{1-2\nu}H_T^2\cdot
\left(\frac{H_T}{M}\right)^{-2(n-1)\nu}.
\end{equation}
Note that, in the last equations, $k_T$ is the mode that crosses the Hubble radius at time $\eta_T$, $k_T=H_T$. Since, by construction, the equation of motion of the field $\varphi$ is linear, and because its modes are in a vacuum state, the fluctuations in $\varphi$ are Gaussian.
The primordial spectrum seeded during a stage of de Sitter inflation ($p=-1$) happens to be scale invariant for any value of $n$ if $m=n-1$. This includes the conventional dispersion relation, $n=1$ and $m=0$, and the dispersion relation of the ghost condensate \cite{Arkani-Hamed:2003uz}, $n=2$ and $m=1$. But as we see, it is also possible to seed an exactly or nearly scale invariant spectrum of primordial perturbations during a non-inflationary stage of expansion. In particular, many combinations of parameters lead to $n_s\approx 1$ in a non-accelerating universe ($p>1/2$), given that the only restriction on the model parameters so far is $mp>1$, from equation (\ref{eq:condition}). And even for $n=1$ the spectrum seeded during a non-inflationary phase can be scale-invariant if the effective speed of sound is time-dependent and decreasing ($m>0$), although this realization requires a superluminal \emph{effective} speed of sound \cite{Armendariz-Picon:2003ht}.
To conclude this section, let us note that current observations actually disfavor an exact scale-invariant spectrum \cite{Spergel:2006hy} (see however \cite{Magueijo:2006we}). This does not pose any particular problem in our approach. Although in the realizations we discuss below $n$ has to be an integer, $p$ and $m$ can be arbitrary real numbers. In this case, just as for inflation, departures from scale invariance arise from, e.g., small departures of $p$ from integer values.
\subsection{The scale M}
Within a classical description of the universe, nothing prevents us from taking the early-time limit $\eta\to 0$. However, we expect quantum effects at Planckian energy densities to invalidate our semi-classical description. Hence, rather than requiring that modes be adiabatic as $\eta\to 0$, we should require that cosmologically relevant modes be in the adiabatic vacuum at the Planckian time at the earliest.
Let us assume that the seeding stage (\ref{eq:a}) is followed by the conventional period of radiation domination, and let us assume that the transition occurs at time $\eta_T$. Requiring the adiabatic condition $\omega \, \eta\gg1 $ to hold at the Planck time for a mode of the size of our present horizon, we arrive at the upper bound
\begin{equation}\label{eq:long}
\left(\frac{M}{M_P}\right)^{n-1}\ll 10^{-29 n}
\left(\frac{H_T}{M_P}\right)^{n/2-p(m+1)/(p+1)},
\end{equation}
where $M_P=G^{-1/2}=10^{19}$~GeV is the Planck mass. We shall later argue that the perturbations in the field $\varphi$ are converted into adiabatic perturbations at the time of the transition between the seeding stage and radiation domination. Because in our calculation of the spectrum we assumed that the relevant modes are long-wavelength, we have to make sure that this condition is satisfied at the time of the transition. Conservatively requiring than five decades in $k$ above our present horizon be long-wavelength at that time we arrive then at the lower bound
\begin{equation}\label{eq:short}
\left(\frac{H_T}{M_P}\right)^{n/2-1}\ll \left(\frac{M}{M_P}\right)^{n-1} 10^{24n}.
\end{equation}
Note that the universe has to be radiation dominated by the time of nucleosynthesis at the latest, which implies the additional constraint $H_T >10^{-44} M_P$.
Conditions (\ref{eq:long}) and (\ref{eq:short}) can be simultaneously satisfied with parameters that lead to a nearly scale invariant spectrum. Set for instance $p\approx 2$, $n=1$ and $m=1$. In this case, the mass scale $M$ does not enter the equations. Choose in addition $H_T=10^{-40} M_P$, which amounts to a temperature at the transition $T_T\sim100$ MeV. All the required conditions are then met. If the parameters are instead, say, $p\approx 1$, $n=3$ and $m=2$, the spectrum is again nearly scale invariant, and equations (\ref{eq:long}) and (\ref{eq:short}) are satisfied with $M=10^{-17}$ eV and $H_T=10^{-40} M_P$. Note the tiny value of the scale $M$, which is a reflection of the horizon problem; in a non-accelerating universe, our current horizon has a size much larger than the Hubble radius at early times. However, the value of this parameter should be interpreted with care, as the scale $M$ is only defined modulo an appropriately normalized time-dependent function (see below).
\section{Realizations}\label{sec:realizations}
We still have to answer whether the dispersion relation (\ref{eq:dispersion}) follows from any generally covariant scalar field theory. What we have to explain is $i)$ the origin of the higher powers in spatial momentum $k$, and $ii)$ why these powers are not accompanied by corresponding powers of the scale factor $a$. As we shall see, ingredient $i)$ arises in any low-energy effective theory, while ingredient $ii)$ follows from couplings between $\varphi$ and background scalars and tensors.
\subsection{Effective Field Theories}
Let us begin by addressing the origin of the higher powers of spatial momenta in equation (\ref{eq:dispersion}). Suppose we deal with a low energy effective action with cut-off scale $M$. The action is expected to contain all possible terms compatible with the symmetries of the theory. Generically no symmetry forbids a term
\begin{equation}\label{eq:higher-order}
L_\mathrm{corr}=\frac{1}{2 M^{2n-2}}\varphi \, \Box^n \varphi,
\end{equation}
where $\Box=\nabla^\mu \nabla_\mu$, so we expect the action to contain this operator, in addition to many other similar ones. Upon variation of the action, this term leads to corrections to the equation of motion, $\Box\, \varphi+M^{2-2n}\Box^n \varphi=0$. In a spatially flat Friedman-Robertson-Walker universe with metric (\ref{eq:a}) these corrections have the structure
\begin{equation}
\Box^n \varphi=\frac{1}{a^{2n+1}}\sum_{p+2q+s=2n} c^n{}_{pq} \cdot
\eta^{-p} \, k^{2q} \, v_k^{(s)},
\end{equation}
where $c^n{}_{pq}$ are dimensionless coefficients, a superscript $(s)$ denotes $s$ derivatives with respect to conformal time and we have reintroduced $v_k=a\,\varphi_k$.
Although the equation of motion then contains the desired term, proportional to $k^{2n}$, it includes in addition many others that actually dominate over the gradients. Compare for instance $k^{2n}\,v_k$ with $(k^{2n-2}/\eta^2) v_k$. The first one dominates when $k^2>\eta^{-2},$ that is, for sub-horizon modes. But without inflation, cosmologically relevant scales are well outside the horizon in the past! Hence, higher-dimensional operators like the one in equation (\ref{eq:higher-order}) do not lead to the dispersion relation (\ref{eq:dispersion}).
\subsection{Spatial Projectors}
Upon closer inspection of the structure of the dispersion relation (\ref{eq:dispersion}) one realizes that the key point is the presence of higher-order spatial derivatives in the equation of motion, but the absence at the same time of higher-order time derivatives. This structure naturally appears in the presence of appropriate second rank background tensors \cite{Jacobson:2000gw}.
Imagine a spacetime endowed with a symmetric tensor $h_{\mu\nu}$ that projects onto the space orthogonal to a timelike vector field $u^\mu$. By definition, the projector $h_{\mu\nu}$ satisfies $h_{\mu\nu}h^\nu{}_\rho=h_{\mu\rho}$ and $h_{\mu\nu}u^\nu=0$. In our context, the reader can think of the vector field $u^\mu$ as the four velocity of observers for which the universe appears to be isotropic. In conformal time coordinates, this vector has components $u^\mu=(a^{-1},0,0,0)$, so the projection tensor is
$h_{\mu\nu}=a^2\cdot \mathrm{diag}(0,1,1,1)$. If this is the case, it then follows that, acting on a scalar,
\begin{equation}
D_\mu = h_\mu{}^\nu \nabla_\nu
\end{equation}
is orthogonal to $u^\mu$ and has only spatial components, i.~e., is a spatial derivative. On the other hand
\begin{equation}
d_\mu = (\delta_\mu{}^\nu-h_\mu{}^\nu) \nabla_\nu
\end{equation}
has vanishing spatial components and hence represents a time derivative. Therefore, by combining these two generally covariant derivatives, we can obtain any combination of time and spatial derivatives. In particular,
\begin{equation}\label{eq:spatial}
\varphi \cdot (D_\mu D^\mu)^n \varphi=(-1)^n
\frac{k^{2n}}{a^{2n}}\varphi_k^2.
\end{equation}
\subsection{Time functions}
Note that in equation (\ref{eq:spatial}) every factor of $k$ is accompanied by a corresponding factor of the scale factor $a$. In order to justify the presence of different powers of $a$ in the dispersion relation (\ref{eq:dispersion}), a non-vanishing background scalar $r$ is sufficient. Because of the symmetry of the spacetime, this scalar can depend only on time, $r=r(\eta)$, so this is why we shall call $r$ a time function. At this point let us simply assume that $r=a^{2(n-m-1)}$. We shall justify this functional dependence below; what matters here is that at time $\eta_T$, $r=1$. Given equation (\ref{eq:a}), it follows that the Lagrangian
\begin{equation}\label{eq:L}
L= -\frac{1}{2}d_\mu \varphi\, d^\mu\varphi -\frac{(-1)^n}{2 M^{2n-2}} \cdot r \cdot
\varphi \cdot (D_{\mu}D^{\mu})^n \varphi
\end{equation}
yields the equation of motion (\ref{eq:motion}) with dispersion relation given by (\ref{eq:dispersion}), provided we assume that the fluctuations in $r$, $g_{\mu\nu}$ and $h_{\mu\nu}$ are negligible; in other words, provided that the universe is highly homogeneous and isotropic. Notice that $n=m+1$ leads to a simpler model, since in that case $r$ is a constant. In the following, we present possible different origins of the spatial tensor $h_{\mu\nu}$ and the time function $r$. Similar constructions of the projection tensor involving vector fields \cite{Jacobson:2000gw} are also possible.
\paragraph{Non-minimal gravitational couplings}
The existence of an appropriate tensor $h_{\mu\nu}$ is quite natural in a Friedman-Robertson-Walker universe, and does not imply the breaking of any spacetime symmetry. Consider for instance
\begin{equation}
h_{\mu\nu}=\frac{3(p-1)}{p+1} \frac{R_{\mu\nu}}{R}
+\frac{3g_{\mu\nu}}{2(p+1)} \;\text{and} \;
r= \left(\frac{R}{R_T}\right)^{-\frac{p(n-m-1)}{p+1}}
\end{equation}
when $p\neq 1$. The reader can readily verify that the tensor $h_{\mu\nu}$ and the scalar $r$ have the required properties in a universe whose scale factor evolves according to equation (\ref{eq:a}). This choice of $r$ defines $\eta_T$ to be the time at which the the Ricci scalar equals the free parameter $R_T$. The presence of the additional curvature terms however leads to modifications in the gravitational equations. Inverse powers of the Ricci scalar have been also postulated as explanations for cosmic acceleration in \cite{Carroll:2003wy}. Note that the scalar curvature does not necessarily vanish during radiation-domination, say, due to the presence of dark matter particles.
\paragraph{Couplings to other fields}
Even though in our analysis all we had to assume is the appropriate form of the scalar field equation of motion, regardless of the dynamics of gravity, it might be phenomenologically desirable to restrict ourselves to actions where gravity is described by general relativity. If $\chi$ is a homogeneous time-evolving scalar field in a FRW universe, then, the
projector
\begin{equation}
h_{\mu\nu}=g_{\mu\nu}-\frac{\partial_\mu\chi \,
\partial_\nu \chi}{\partial_\alpha \chi \partial^\alpha \chi}
\end{equation}
also has the required properties. In addition, if the scalar field $\chi$ is rolling down an exponential potential
\begin{equation}\label{eq:chi}
L_\chi=-\frac{1}{2}\partial_\mu \chi \partial^\mu \chi - V_0 \exp\left(-\sqrt{\frac{16\pi (1+p)}{p}}\frac{\chi}{M_P}\right),
\end{equation}
the time-keeping function
\begin{equation}
r=\exp\left[(n-m-1)\sqrt{\frac{16\pi p}{1+p}}\frac{\chi-\chi_T}{M_P}\right]
\end{equation}
leads to the desired power of the scale factor in the dispersion relation.\footnote{Since $\varphi$ vanishes classically, $\varphi_\mathrm{cl}\equiv \langle\varphi\rangle=0$, the terms in the Lagrangian (\ref{eq:L}) do not alter the $\chi$ equation of motion.} In this case, $\eta_T$ is the time at which the scalar field $\chi$ equals $\chi_T$.
\section{Metric Perturbations}
The mechanism we have presented generates a nearly scale invariant spectrum of perturbations in a subdominant scalar field $\varphi$, that is, it leads to \emph{isocurvature} perturbations. Observations however suggest that primordial perturbations are not only nearly scale invariant, but also \emph{adiabatic} and Gaussian. Hence, we still have to explain how these isocurvature perturbations are converted into adiabatic ones. We shall discuss this conversion in the context of the realization of our model where the spatial projectors arises from an evolving and homogeneous scalar field $\chi$.
Imagine that the early universe is dominated by a scalar field $\chi$ with an exponential potential, as in equation (\ref{eq:chi}). Assume that this potential has a minimum at $\chi_\mathrm{min}$, that is, the potential deviates from an exponential around $\chi\sim \chi_\mathrm{min}.$ Once the field reaches this minimum, it will start to oscillate and decay into the fields $\psi$ it couples to, thus reheating the universe. Now, assume that the couplings of $\chi$ to $\psi$ are not constants, but depend on the fluctuating field $\varphi$ \cite{Dvali:2003em},
\begin{equation}
L_\mathrm{int}=-\lambda_0 \cdot\left(1+f \frac{\varphi}{M}\right)\chi \bar{\psi}{\psi},
\end{equation}
where $\lambda_0$ and $f$ are two dimensionless constants. Because the reheating temperature depends on the coupling of $\chi$ to matter, the fluctuations in $\varphi$ lead then to fluctuations in the reheating temperature and the Newtonian potential $\Phi$ \cite{Dvali:2003em, Armendariz-Picon:2003ku},
\begin{equation}
\Phi\sim \frac{\delta T}{T}\sim f \frac{\varphi}{M}.
\end{equation}
Combining this equation with our results in equation (\ref{eq:power}) we find that in cases where the spectrum of $\varphi$ is nearly scale invariant, the decay of $\chi$ results again in a nearly scale independent spectrum of adiabatic perturbations with squared amplitude
\begin{equation}
\mathcal{P}_\Phi\sim f^2 \cdot \left(\frac{H_T}{M}\right)^{\frac{3-n}{n}}.
\end{equation}
These perturbations are nearly Gaussian ($f_{NL}$ equals a few), provided that the decay rate of $\chi$ at the end of the seeding stage is not too high \cite{Zaldarriaga:2003my}.
Cosmic microwave background measurements \cite{Spergel:2006hy} require $\mathcal{P}_\Phi\sim 10^{-10}$, which fixes the parameter $f$ for given $H_T$ and $M$. Hence, the amplitude of metric perturbations is not uniquely determined by the amplitude of the perturbations in $\varphi$. For instance, for $n=3$ the amplitude of primordial perturbations is just fixed by $f$, and observations hence require $f\sim 10^{-5}$.
\section{Homogeneity}
Some inflationary models not only explain the origin of a nearly scale invariant spectrum of primordial perturbations, but also why the universe is homogeneous, isotropic and spatially flat.
Whereas our scenario has nothing to say about the spatial geometry of the universe, it can indeed partially address the homogeneity of space.
To that purpose, consider the energy density of an inhomogeneous field $\varphi$ in an expanding universe. The energy momentum tensor of the scalar field has been calculated in \cite{Lemoine:2001ar}. In the case of a scalar field with action (\ref{eq:L}) the energy density due to the inhomogeneities in $\varphi$ is
\begin{equation}
\rho_\mathrm{inh}\approx \frac{1}{2}\frac{\varphi_k'^2}{a^2}+\frac{1}{2}\frac{\omega^2 }{a^2} \varphi_k^2.
\end{equation}
We would like to compare the energy density of these inhomogeneities with the total energy density of the universe, $\rho_\mathrm{tot} \sim M_P^2/ (a^2 \eta^2)$. We have seen that as the universe expands, modes leave the short-wavelength regime and become long-wavelength, that is $\omega \, \eta$ decreases. In the limit $\omega \, \eta\ll 1$, the solution of the equation of motion is $\varphi_k =const$. Therefore, the ratio of energy densities approaches
\begin{equation}
\frac{\rho_\mathrm{inh}}{\rho_\mathrm{tot}}\sim \omega^2\, \eta^2 \, \frac{\varphi_k^2}{M_P^2}.
\end{equation}
and, hence, just as during inflation, decreases. Clearly, this argument is somewhat heuristic, but it nevertheless suggests that inhomogeneities are ``washed-out" by the expansion even if the universe does not inflate.
\section{Conclusions}
In this article we have described a way to seed a nearly scale invariant spectrum of adiabatic primordial perturbations in a non-inflating universe. The key element is the modified dispersion relation of a matter component in the universe. No epoch of contraction is required, and the Hubble parameter does not have to increase at any time during the past history of the universe. For concreteness, we have illustrated this idea with a scalar field, but other realizations of this mechanism might be also possible. In this scalar field context, the modified dispersion relations arises from rather non-generic couplings of the scalar field to background scalars and tensors. Typically, these couplings have to be suppressed by a tiny scale $M$, a reflection of the horizon problem. Because the spectrum of primordial perturbations is seeded during a non-inflationary stage in an otherwise perfectly homogeneous universe, no gravitational waves are produced. Perturbations are also expected to be non-Gaussian, with an $f_{NL}$ of a few, and thus potentially detectable in the near future. The mechanism does not address the spatial flatness of the universe, but might have the potential to tackle the homogeneity problem.
It is natural to compare our scenario with the inflationary benchmark. Certainly, \emph{some} inflationary models still provide the simplest explanation for many of the features of our universe. But this work should be interpreted from a different perspective anyway. At this stage, it merely shows that the causal generation of the primordial perturbations we observe in our universe does not necessarily require a stage of inflation or an unconventional expansion history.
\begin{acknowledgments}
The author thanks the Yukawa Institute for Theoretical Physics and the Max-Planck-Institut f\"ur Gravitationsphysik for their kind hospitality. I am also indebted to Shinji Mukohyama for useful conversations and remarks. This work was supported in part by the NSF under grant PHY-0604760.
\end{acknowledgments}
|
2,869,038,155,644 | arxiv | \section{Introduction}
\label{sec:intro}
The mapping class group $\Mod(S_g)$ of a closed, orientable surface $S_g$ of genus $g$ is the group of homotopy classes of orientation-preserving homeomorphisms of $S_g$. The extended mapping class group $\MCG(S_g)$ is the group of homotopy classes of all homeomorphisms of $S_g$.
\p{Normal subgroups.} The main result of this paper, Theorem~\ref{main:normal}, gives a general condition for a normal subgroup of $\Mod(S_g)$ or $\MCG(S_g)$ to have automorphism group and abstract commensurator group isomorphic to $\MCG(S_g)$. Previously this result was only known for very specific subgroups, namely, the Torelli group and its variants. Our general condition, which is that the normal subgroup contains an element of ``small'' support, is easy to verify and applies to most natural normal subgroups, including the Torelli groups and their variants, as well as many others.
Farb has posed the problem of computing the abstract commensurators for various subgroups of $\Mod(S_g)$ \cite[Problem 2.2]{farb}. Our theorem solves this problem in many cases. It also addresses the so-called metaconjecture of Ivanov; see below.
An observation of L. Chen further implies that each normal subgroup with an element of small support is unique in that no other normal subgroup of $\MCG(S_g)$ is isomorphic to it; see Corollary~\ref{cor:chen}. So, for example, the terms of the Johnson filtration, the terms of the Magnus filtration, the level $m$ congruence subgroups together form a collection of pairwise non-isomorphic subgroups of the mapping class group.
Two further applications of our theorem are restrictions on the isomorphism types of subgroups of mapping class groups. For example, if $g \neq g'$ a normal subgroup of $\Mod(S_g)$ with an element of small support cannot be isomorphic to a normal subgroup of $\Mod(S_{g'})$ with an element of small support; see Corollary~\ref{cor:notiso}. Also, a normal subgroup of $\Mod(S_g)$ with an element of small support cannot be isomorphic to a surface group or a right-angled Artin group; see Corollary~\ref{cor:raag}. The key idea for both of these applications is to use the automorphism group as an invariant of the isomorphism class of a group.
Our results suggest a dichotomy for the normal subgroups of mapping class groups, namely, into those that have automorphism group isomorphic to the extended mapping class group and those that do not contain elements of small support; see Conjecture~\ref{conj:dichotomy} below. As discussed by Farb \cite{farb}, a traditional classification theorem for normal subgroups of mapping class groups, in the form of a complete list of isomorphism types, is almost certainly out of reach. However, our conjecture provides a new framework for a coarser classification of normal subgroups of mapping class groups.
\p{Simplicial complexes.} We prove our main result about normal subgroups of mapping class groups by reducing it to a problem about automorphisms of simplicial complexes. To this end, we consider simplicial complexes whose vertices correspond to connected subsurfaces of the ambient surface and whose edges correspond to disjointness. Our main theorem about automorphisms of simplicial complexes, Theorem~\ref{main:complex}, gives general conditions for such a simplicial complex to have automorphism group isomorphic to the extended mapping class group.
Our result applies to many natural simplicial complexes associated to a surface, including some that were already known to have automorphism group the extended mapping class group. Our theorem is the first to address infinitely many complexes with a single argument, and indeed it applies to a wide class.
\p{Ivanov's metaconjecture} Our work in this paper has its origins in the seminal work of N. V. Ivanov \cite{ivanov}. He proved that for $g$ at least 3, the automorphism group of the complex of curves $\C(S_g)$ is isomorphic to $\MCG(S_g)$ (see also \cite{korkmaz,luo}). As one application, he proved that the automorphism group of $\Mod(S_g)$---and also the abstract commensurator group of $\Mod(S_g)$---is isomorphic to $\MCG(S_g)$ (cf. \cite{mccarthy,tchangang}). Ivanov's work inspired a number of theorems of the following form:
\begin{enumerate}
\item the automorphism group of some particular simplicial complex associated to a surface $S$ is isomorphic to $\MCG(S)$, and
\item the automorphism group of some particular normal subgroup of $\Mod(S)$ is isomorphic to $\MCG(S)$.
\end{enumerate}
Many are found among the 96 (and counting) citations on MathSciNet for the aforementioned paper of Ivanov. In response, Ivanov posed the following \cite{ivanov15}.
\begin{metaconjecture}
Every object naturally associated to a surface $S$ and having a sufficiently rich structure has $\MCG(S)$ as its group of automorphisms. Moreover, this can be proved by a reduction to the theorem about the automorphisms of $\C(S)$.
\end{metaconjecture}
There are many results supporting Ivanov's metaconjecture, some quite classical, going back to the work of Dehn \cite[Paper 8]{dehn} and Nielsen \cite{nielsen} in the 1920s. Also, in the 1930s Teichm\"uller's showed that the group of automorphisms of the universal curve over Teichm\"uller space is the extended mapping class group \cite{TeichTrans,teich}; this theorem was put into a more general framework by Grothendieck \cite[Theorem 3.1]{grothendieck}. For an overview of other related results, see our survey paper \cite{survey} or the one by McCarthy--Papadopoulos \cite{mcpap}. In our survey we suggest a generalization of Ivanov's metaconjecture, from surfaces to other spaces.
Our results may be viewed as a resolution of Ivanov's metaconjecture for a wide class of normal subgroups of $\Mod(S_g)$ and a wide class of simplicial complexes associated to $S_g$. Ivanov's metaconjecture is deliberately vague: the terms ``object,'' ``naturally,'' and ``sufficiently rich'' are left open to interpretation. In this paper we formulate his metaconjecture into two precise statements about normal subgroups and simplicial complexes (the objects at hand) by finding appropriate notions of sufficient richness in each case.
Our work follows Ivanov in the sense that we reduce our problem about normal subgroups of the mapping class group to a problem about automorphisms of simplicial complexes. In his work, as well as in the other subsequent works, there is a single group being considered, and a single simplicial complex. A central challenge in this paper, and one of the main departures from Ivanov's work, is that we consider many groups all at the same time, each requiring its own simplicial complex. As such, we need to prove that all of these simplicial complexes have automorphism group isomorphic to the extended mapping class group. Also, when it comes to the normal subgroups we consider we do not have full information about which elements are in, and are not in, our given subgroups; we are only given the information that there is a single element whose support is small (in the precise sense defined below).
\subsection{Results on normal subgroups}
In order to state our main theorem about normal subgroups of the mapping class group, Theorem~\ref{main:normal}, we require several definitions.
\p{Small components.} Let $f \in \Mod(S_g)$ be a pure mapping class. Briefly, this means that $f$ is a product of partial pseudo-Anosov mapping classes and Dehn twists, all with disjoint supports; see Section~\ref{section:normal} for details. We will define a certain measure of complexity $\hat g(f)$ for $f$. First, a \emph{region} in $S_g$ is a compact, connected subsurface with no boundary component homotopic to a point in $S_g$. Next, a \emph{fitting region} for $f$ is a region $Q$ in $S_g$ so that some Nielsen--Thurston component of (some representative of) $f$ has support that is non-peripheral in $Q$ and so that the other Nielsen--Thurston components of $f$ have support disjoint from $Q$. Finally we define $\hat g(f)$ to be the smallest number $k$ so that there is a region of $S_g$ that has genus $k$ and connected (possibly empty) boundary and that contains a fitting region for $f$.
We will say that a pure element $f$ of $\Mod(S_g)$ has a \emph{small component} if $\hat g(f) < g/3$. The main hypothesis of Theorem~\ref{main:normal} is that the normal subgroup under consideration has a nontrivial element with a small component.
For example, if $f$ has a partial pseudo-Anosov Nielsen--Thurston component whose support is contained in a region of genus $k$ with connected boundary then $\hat g(f)$ is at most $k$. Also if the entire support of $f$ is contained in a region of genus $k$ with connected boundary then $\hat g(f)$ is at most $k+1$ (for instance if $f$ is a Dehn twist about a separating curve of genus $k$ then $\hat g(f) = k+1$).
For a subgroup $N$ of $\MCG(S_g)$, we define $\hat g(N)$ to be the minimum of $\hat g (f)$ for nontrivial pure $f$ in $N$ (by default pure elements lie in $\Mod(S_g)$). See Section~\ref{section:normal} for the definitions of pure elements and Nielsen--Thurston components.
\p{Abstract commensurators.} The \emph{abstract commensurator group} of a group $G$ is the group of equivalence classes of isomorphisms between finite-index subgroups of $G$, where two isomorphisms are equivalent if they agree on some finite-index subgroup of $G$.
\p{Natural maps} Let $N$ be a normal subgroup of $\Mod(S_g)$. There is a natural homomorphism $\Mod(S_g) \to \Aut N$ where $f \in \Mod(S_g)$ maps to the element of $\Aut N$ given by conjugation by $f$; there is a similar map if $N$ is normal in $\MCG(S_g)$. Also, there is a natural homomorphism $\Aut N \to \Comm N$ where an automorphism maps to its equivalence class.
There is one more natural homomorphism in the statement of Theorem~\ref{main:normal}. If $N$ is a subgroup of $\MCG(S_g)$, there is a map $\MCG(S_g) \to \Hom(N,\MCG(S_g))$ where $f \in \MCG(S_g)$ maps to the homomorphism taking $n$ to $fnf^{-1}$. If for each $f \in \MCG(S_g)$ there is a finite-index subgroup $H$ of $N$ so that $fHf^{-1}$ has finite index in $N$ then we may regard the map $\MCG(S_g) \to \Hom(N,\MCG(S_g))$ as a map $\MCG(S_g) \to \Comm N$. When this map exists, we call it the natural map $\MCG(S_g) \to \Comm N$.
\p{Statement of our main theorem about normal subgroups.} Our first main theorem describes the automorphism group and the abstract commensurator groups of all normal subgroups of $\Mod(S_g)$ and $\MCG(S_g)$ that contain elements of small support. So each of these subgroups remembers the structure of the full mapping class group.
\begin{theorem}
\label{main:normal} Let $N$ be a normal subgroup of either $\MCG(S_g)$ or $\Mod(S_g)$ with $g \geq 3 \hat g(N) + 1$.
\begin{enumerate}
\item\label{normal emcg} If $N$ is normal in $\MCG(S_g)$ then the natural maps
\[ \MCG(S_g) \to \Aut N \to \Comm N \]
are isomorphisms.
\item\label{normal mcg} If $N$ is normal in $\Mod(S_g)$ but not in $\MCG(S_g)$ then there is a natural map $\Comm N \to \MCG(S_g)$, the natural maps
\[ \Mod(S_g) \to \Aut N \to \Comm N \to \MCG(S_g) \]
are injective, the first map is an isomorphism, and the composition is the inclusion. In particular $\Comm N$ is isomorphic to either $\Mod(S_g)$ or to $\MCG(S_g)$. In the first case the second map is an isomorphism. In the second case the inverse of the isomorphism $\Comm N \to \MCG(S_g)$ is the natural map $\MCG(S_g) \to \Comm N$.
\end{enumerate}
\end{theorem}
Most of the well-studied normal subgroups of the mapping class group---for instance the Torelli group and the terms of the Johnson filtration---are normal in the extended mapping class group, and so the first statement of Theorem~\ref{main:normal} applies. We expect that there are subgroups $N$ that are normal in $\Mod(S_g)$ but not $\MCG(S_g)$ and that satisfy $\Comm N \cong \MCG(S_g)$. Examples in the case $g=1$ were explained to us by Jones \cite{jones}.
By Lemma~\ref{comm trick} below, any normal subgroup of $\Mod(S_g)$ or $\MCG(S_g)$ that has a pure element with a small component also has a pure element whose entire support is small, meaning that the support is contained as a nonperipheral subsurface in a subsurface of $S_g$ with connected boundary and genus $k < g/3$. Therefore, the hypothesis on $N$ in Theorem~\ref{main:normal} is equivalent to the hypothesis that $N$ has a nontrivial element with small support.
The hypothesis of small supports in Theorem~\ref{main:normal} is certainly not optimal. Indeed, if we take $N=\Mod(S_g)$, then Theorem~\ref{main:normal}\eqref{normal emcg} implies that $\Aut \Mod(S_g)$ is isomorphic to $\MCG(S_g)$ when $g \geq 4$. On the other hand, Ivanov already proved this result for $g \geq 3$.
\p{Exotic normal subgroups.} One might hope that all normal subgroups of $\Mod(S_g)$ have automorphism group $\MCG(S_g)$, in other words that the hypothesis on $\hat g(N)$ in Theorem~\ref{main:normal} is not necessary. However, this is certainly not the case: Dahmani, Guirardel, and Osin \cite{dgo} proved that there are normal subgroups of $\Mod(S_g)$ isomorphic to infinitely generated free groups; see also the recent work of Clay, Mangahas, and the second author \cite{cmm}. Each nontrivial element in the Dahmani--Guirardel--Osin subgroups is pseudo-Anosov. The hypotheses of Theorem~\ref{main:normal} exactly rule out this type of example, as $\hat g(f) = g$ for any pseudo-Anosov $f$.
\p{Prior results.} Our Theorem~\ref{main:normal} recovers many previously known results. After Ivanov's original work, Farb and Ivanov \cite{farbivanovannounce,farbivanov} proved that the automorphism group and the abstract commensurator group of the Torelli subgroup of $\Mod(S_g)$ is isomorphic to $\MCG(S_g)$, and the authors of this paper proved \cite{kg} that the automorphism group and the abstract commensurator group of the Johnson kernel, an infinite index subgroup of the Torelli group, is isomorphic to $\MCG(S_g)$. Bridson, Pettet, and Souto \cite{bps} then announced the following result: every normal subgroup of the extended mapping class group that is contained in the Torelli group and has the property that each subsurface of Euler characteristic $-2$ supports a non-abelian free subgroup has automorphism group and abstract commensurator group isomorphic to $\MCG(S_g)$. In particular for $g \geq 4$ this applies to every term of the Johnson filtration of $\Mod(S_g)$. The Johnson filtration is an infinite sequence of nested normal subgroups of $\Mod(S_g)$ whose intersection is the trivial subgroup; the first two groups in the sequence are the Torelli group and the Johnson kernel. Theorem~\ref{main:normal} implies each of the above results.
\p{Applications and examples} Many natural subgroups of $\Mod(S_g)$ and $\MCG(S_g)$ come in families, meaning that there is one normal subgroup $N_g$ for each $g$. Also, it is often the case that $\hat g(N_g)$ does not depend on $g$, and so Theorem~\ref{main:normal} applies to all members of the family once $g$ is large enough.
As one example the Torelli group $\I(S_g)$ is the normal subgroup of $\MCG(S_g)$ defined as the kernel of the action of $\Mod(S_g)$ on $H_1(S_g;\Z)$. The Johnson kernel $\K(S_g)$ is the infinite-index subgroup of $\I(S_g)$ generated by Dehn twists about separating curves. For all $g$ we have $\hat g(\I(S_g)) = \hat g(\K(S_g)) = 2$. Theorem~\ref{main:normal}\eqref{normal emcg} applies to both, thus recovering our earlier result and the result of Farb and Ivanov for $g \geq 7$.
Similarly, Theorem~\ref{main:normal}\eqref{normal emcg} applies to the $k$th term $N_g^k$ of the Johnson filtration, which is the kernel of the action (by outer automorphisms) of $\Mod(S_g)$ on $\pi/\pi_k$ where $\pi=\pi_1(S_g)$ and $\pi_k$ is the $k$th term of its lower central series. We have $\hat g(N_g^k) = 2$ \cite[Proof of Theorem 5.10]{farb}. In particular for $g \geq 7$ our theorem recovers the results announced by Bridson--Pettet--Souto.
Beyond this, the terms of the derived series for the Torelli group, the terms of the lower central series of the Torelli group, the kernel of the Chillingworth homomorphism, and the kernel of the Birman--Craggs--Johnson homomorphism each have $\hat g = 2$ and so Theorem~\ref{main:normal} applies for $g \geq 7$.
A further application of our theorem is to the Magnus filtration of the Torelli group, defined by McNeill \cite{taylor}. The $k$th term $M_g^k$ is the subgroup of $\Mod(S_g)$ acting trivially on $\pi/\pi_k'$ where $\pi_k'$ is the $k$th term of the lower central series of $[\pi,\pi]$. The first term $M_g^1$ is the kernel of the Magnus representation of $\Mod(S_g)$, defined in the 1930s by Magnus \cite{magnus}. McNeill proves that $\hat g(M_g^k) \leq 3$ for all $g \geq 3$ and $k \geq 1$ \cite[Lemma 5.2]{taylor}, and so Theorem~\ref{main:normal}\eqref{normal emcg} applies for $g \geq 10$. (McNeill discusses surfaces with boundary, but the capping homomorphism to $\Mod(S_g)$ respects the Magnus filtration.)
One may readily construct many other examples of normal subgroups satisfying the hypotheses of Theorem~\ref{main:normal}, for instance the group generated by $k$th powers of Dehn twists ($\hat g = 1$), the group generated by $k$th powers of Dehn twists about separating curves ($\hat g = 2$), the terms of the lower central series of the Torelli group ($\hat g = 2$), the normal closure of any partial pseudo-Anosov element supported on a torus with one boundary component ($\hat g = 1$), and the normal closure of any multitwist ($\hat g \leq 2$). In the last case, to make an example with $\hat g = 2$ we should choose the support to be a pants decomposition where each curve is nonseparating.
Any normal subgroup of $\Mod(S_g)$ or $\MCG(S_g)$ containing one of the above groups automatically satisfies the hypothesis of Theorem~\ref{main:normal}. For instance, if $N$ is the kernel of the \textrm{SU}(2)-TQFT representations of the mapping class group (see e.g. Funar \cite{funartqft}), then $N$ contains the group generated by $k$th powers of Dehn twists, and hence Theorem~\ref{main:normal} applies for $g \geq 4$. The same applies to the subgroup of $\Mod(S_g)$ generated by the $k$th powers of all elements.
Finally, any normal subgroup of $\Mod(S_g)$ or $\MCG(S_g)$ that has finite index in a group satisfying the hypothesis of Theorem~\ref{main:normal}. This includes, for example, the level $m$ congruence subgroups of $\Mod(S_g)$ and also the congruence subgroups defined by Ivanov via characteristic covers of surfaces \cite[Problem 1]{ivanov15}.
\p{Chen's corollary} Ivanov--McCarthy proved that any injective map $\MCG(S_g) \to \MCG(S_g)$ is an inner automorphism \cite[Theorem 1]{im}. As observed by Chen \cite{lei}, this theorem has the following corollary: if $N$ is a normal subgroup of $\MCG(S_g)$ where the natural map $\MCG(S_g) \to \Aut N$ is an isomorphism then $N$ is unique in the sense that every normal subgroup of $\MCG(S_g)$ isomorphic to $N$ is equal to $N$. This applies, for example, to the Torelli group, as well as all of the other subgroups discussed above.
Indeed, suppose that $M$ is a normal subgroup of $\MCG(S_g)$ isomorphic to $N$. Consider the composition
\[
\Xi : \MCG(S_g) \stackrel{}{\to} \Aut M \stackrel{} \to \Aut N \stackrel{}{\to} \MCG(S_g),
\]
where the first map is the natural map given by conjugation, the second map is the isomorphism induced by any isomorphism $M \to N$, and the third map is the inverse of the natural map $\MCG(S_g) \to \Aut N$, which is an isomorphism by assumption. All of the maps are injective (cf. Lemma~\ref{inj}) and hence the composition $\Xi$ is an injective map from $\MCG(S_g)$ to itself. By the Ivanov--McCarthy result, $\Xi$ is an inner automorphism of $\MCG(S_g)$. From the definitions of the three maps we observe that
\[
M \mapsto \Inn M \mapsto \Inn N \mapsto N
\]
and so $\Xi(M) = N$. Since $\Xi$ is inner and $M$ is normal it follows that $M=N$, as desired.
Combining Chen's corollary with our main theorem we obtain the following corollary of Theorem~\ref{main:normal}.
\begin{cor}
\label{cor:chen}
Suppose $N$ is a normal subgroup of $\Mod(S_g)$ with $g \geq 3 \hat g(N) + 1$. Any normal subgroup of $\Mod(S_g)$ isomorphic to $N$ is equal to $N$.
\end{cor}
Here is a sample application of Corollary~\ref{cor:chen}. Fix some $g \geq 4$. For each natural number $k$ let $\T_k$ denote the subgroup of $\Mod(S_g)$ generated by all $k$th powers of Dehn twists. For $k < \ell$ the subgroups $\T_k$ and $\T_{\ell}$ are not equal, since $\T_\ell$ lies in the level $\ell$ congruence subgroup of $\Mod(S_g)$ and $\T_k$ does not. Thus by Corollary~\ref{cor:chen} the subgroup $\T_k$ is not isomorphic to $\T_\ell$. So $\T_1,\T_2,\dots$ is an infinite sequence of pairwise non-isomorphic subgroups of $\Mod(S_g)$. Similarly, the terms $\N_k(S_g)$ of the Johnson filtration are all non-isomorphic and each such term is not isomorphic to any $\T_\ell$, etc.
The reader should compare Corollary~\ref{cor:chen} with the theorem of Akin, which states that the point-pushing subgroup $\pi_1(S_g)$ is unique among normal subgroups of $\Mod(S_{g,1})$ in the same sense. Akin's theorem is not implied by our corollary since our Theorem~\ref{main:normal} does not apply to punctured surfaces. McLeay \cite{alan,alan2} has proved an analogue of Theorem~\ref{main:normal} for punctured surfaces; however Akin's group does not satisfy the hypotheses there.
\p{Application: non-commensurability of normal subgroups in different mapping class groups} One kind of application of Theorem~\ref{main:normal} is to show that certain normal subgroups of $\Mod(S_g)$ cannot be isomorphic to, or even commensurable to, certain normal subgroups of $\Mod(S_{g'})$ with $g \neq g'$. Specifically we have the following corollary of Theorem~\ref{main:normal}.
\begin{cor}
\label{cor:notiso}
Suppose $N$ and $N'$ are normal subgroups of $\Mod(S_g)$ and $\Mod(S_{g'})$, respectively, with $3\hat g(N)+1 < g$ and $3 \hat g(N') +1 < g'$. If $g \neq g'$ then $N$ is not abstractly commensurable to $N'$. In particular, $N$ and $N'$ are not isomorphic.
\end{cor}
Indeed, consider $N$ and $N'$ as in the corollary. By Theorem~\ref{main:normal} we have that $\Aut N \cong \MCG(S_g)$ and $\Aut N' \cong \MCG(S_{g'})$. Since $\MCG(S_g)$ is not isomorphic to $\MCG(S_{g'})$ when $g \neq g'$ (consider, for instance, the rank of a maximal abelian subgroup), it follows that $N$ and $N'$ are not isomorphic. Moreover, since $\Comm N$ is isomorphic to $\Mod(S_g)$ or $\MCG(S_g)$ and $\Comm N'$ is isomorphic to $\Mod(S_{g'})$ or $\MCG(S_{g'})$ then, since the abstract commensurator group is an invariant of the abstract commensurability class, it similarly follows that $N$ is not commensurable to $N'$ (again use the ranks of maximal abelian subgroups).
To illustrate Corollary~\ref{cor:notiso}, consider the following normal subgroups of $\Mod(S_g)$ and $\Mod(S_{2g})$. Let $N$ be the normal closure in $\Mod(S_g)$ of a partial pseudo-Anosov element supported on a torus with one boundary component and let $N'$ be the normal closure in $\Mod(S_{2g})$ of a partial pseudo-Anosov element supported on a subsurface of genus two with one boundary component. If $g \geq 4$ then by Corollary~\ref{cor:notiso}, the groups $N$ and $N'$ are not abstractly commensurable.
We do not know a proof of the non-commensurability of such subgroups that is independent of Theorem~\ref{main:normal}. For example, the groups $N$ and $N'$ above cannot be distinguished by their virtual cohomological dimensions or the maximal ranks of their abelian subgroups in any obvious way (both invariants are at least $g$ for both $N$ and $N'$, but their exact values seem hard to compute).
\p{Application: an obstruction theorem for normal subgroups} Another kind of application of Theorem~\ref{main:normal} is to rule out isomorphism types for certain normal subgroups of the mapping class group. For example, if $N \trianglelefteq \Mod(S_g)$ contains a nontrivial pure element with a small component then $N$ cannot be isomorphic to---or even abstractly commensurable to---any group whose automorphism group or abstract commensurator group is not isomorphic to $\Mod(S_g)$ or to $\MCG(S_g)$.
There are many classes of groups where no member of the class has both its automorphism group and its abstract commensurator group isomorphic to a mapping class group of a closed surface. For example, if $G = \pi_1(S_h)$ with $h \geq 2$ then $\Aut G$ is isomorphic to $\MCG(S_{h,1})$, the extended mapping class group of a punctured surface. The group $\MCG(S_{h,1})$ is not isomorphic to any $\Mod(S_g)$ or $\MCG(S_g)$ (the rank of a maximal abelian subgroup is divisible by 3 in the closed case and not in the punctured case), and so $\Aut G$ is not isomorphic to any $\MCG(S_g)$ (even more, $\Comm G$ is quite large).
The same holds for all right-angled Artin groups. The abstract commensurator group of an abelian right-angled Artin group is isomorphic to $\GL_n(\Q)$ for some $n$ (and anyway there are no infinite abelian normal subgroups of $\Mod(S_g)$). Also, the abstract commensurator group of any non-abelian right-angled Artin group contains arbitrarily large finite groups, and $\MCG(S_g)$ does not have this property; see \cite{CLM}.
We summarize the above discussion with the following corollary.
\begin{cor}
\label{cor:raag}
If $G$ is a group with $\Aut(G)$ or $\Comm(G)$ not isomorphic to $\Mod(S_g)$ or to $\MCG(S_g)$, and $N$ is a normal subgroup of $\Mod(S_g)$ with $g \geq 3 \hat g(N) + 1$, then $N$ is not isomorphic to $G$ and further $N$ is not abstractly commensurable to $G$. In particular, this applies when $G$ is any surface group or right-angled Artin group.
\end{cor}
It was, for instance, a folk conjecture that the normal subgroup $T_g^k$ of $\Mod(S_g)$ generated by all $k$th powers of Dehn twists is a right-angled Artin group \cite{funar}, but this is false for $g \geq 4$ since $\hat g(T_g^k) = 1$.
As a consequence of Corollary~\ref{cor:raag}, we see that all normal right-angled Artin subgroups of $\Mod(S_g)$ and all surface subgroups of $\Mod(S_g)$ must be like the Dahmani--Guirardel--Osin examples in that the support of every nontrivial Nielsen--Thurston component of every element must be large. In this direction, Clay, Mangahas, and the second author of this paper have produced normal right-angled Artin groups of $\Mod(S_g)$ where the support of each element is large (but not all pseudo-Anosov as in the Dahmani--Guirardel--Osin examples) \cite{cmm}.
\p{A conjectural sharpening of our theorem.} As mentioned, the hypothesis $\hat g(N) < g/3$ in Theorem~\ref{main:normal} is not optimal. We conjecture that the $g/3$ can be improved to $g/2$.
\begin{conjecture}
If $N$ is a normal subgroup of $\MCG(S_g)$ with $g \geq 2 \hat g(N) + 1$ then the natural maps
\[ \MCG(S_g) \to \Aut N \to \Comm N \]
are isomorphisms.
\end{conjecture}
Even better, we expect that one can replace the hypothesis $\hat g(N) < g/2$ with the hypothesis that $N$ contains a nontrivial pure element with a component whose support takes up less than half of $S_g$ in the sense that it is homeomorphic to a proper subsurface of its complement.
\p{A conjectural dichotomy.} Combining our Theorem~\ref{main:normal} with the exotic subgroups produced by Dahmani--Guirardel--Osin and those constructed by Clay, Mangahas, and the second author, we are led to a conjectural dichotomy for normal subgroups of the mapping class group.
\begin{conjecture}
\label{conj:dichotomy}
Let $N$ be a normal subgroup of either $\Mod(S_g)$ or $\MCG(S_g)$. Then either $\Aut N \cong \MCG(S_g)$ or $N$ contains an infinitely generated right-angled Artin group with finite index.
\end{conjecture}
A further conjecture of Clay, Mangahas, and the second author \cite{cmm} is that if $N$ is a normal, right-angled Artin subgroup of $\Mod(S_g)$ then $N$ is isomorphic to a free product of groups from the following list:
\[ F_\infty \, , \qquad \displaystyle\mathop{\mbox{\Huge{$\ast$}}}_\infty \left(F_\infty \times F_\infty\right)\, , \qquad \displaystyle\mathop{\mbox{\Huge{$\ast$}}}_\infty \left(F_\infty \times \Z\right)\, ,\quad \text{ and } \quad \displaystyle\mathop{\mbox{\Huge{$\ast$}}}_\infty \left(F_\infty \times F_\infty \times \Z\right).\]
\p{Mapping class groups versus lattices} Ivanov's original motivation for the study of the automorphism group of the complex of curves stems from the analogous work about lattices and arithmetic groups. For example, the fundamental theorem of projective geometry states that for $k$ a field and $n \geq 3$ the automorphism group of the Tits building for $k^n$ (the poset of nontrivial proper subspaces) is the group of projective semilinear automorphisms of $k^n$ (see \cite{ftpg}).
Ivanov's work on abstract commensurators also is inspired by the theory of lattices. By the work of Margulis, an irreducible lattice in a connected semisimple noncompact Lie group with finite center is arithmetic if and only if it has infinite index in its abstract commensurator \cite{margulis}. As observed by Ivanov \cite{ivanov}, this implies that mapping class groups are not arithmetic as (for most surfaces) mapping class groups have finite index in their abstract commensurator. Since arithmetic groups are not normal subgroups of their abstract commensurators, our Theorem~\ref{main:normal} gives a new point of contrast between arithmetic groups and normal subgroups of mapping class groups.
\subsection{Results on complexes of regions}\label{sec:complex}
Our next goal is to state our results about automorphisms of simplicial complexes associated to a surface. We begin by describing a class of simplicial complexes first defined by McCarthy and Papadopoulos \cite{mcpap}.
\p{Complexes of regions.} By a \emph{subsurface} of a compact surface $S$ we will always mean a compact subsurface $R$ where no component of $\partial R$ is homotopic to a point in $S$. And (as above) a \emph{region} is a connected, non-peripheral subsurface. A \emph{complex of regions} for $S$ is any nonempty simplicial flag complex that has vertices corresponding to homotopy classes of regions in $S$ and edges corresponding to vertices with disjoint representatives and that admits an action of $\MCG(S)$.
For the purposes of this paper, the difference between a graph and a flag complex is purely cosmetic, since the automorphism group of a flag complex is completely determined by the 1-skeleton. In other words, we could replace complexes of regions with graphs of regions without affecting the theory. The only difference is that we will use the term ``simplex'' instead of ``clique'' etc. On the other hand, it might be an interesting problem to understand the topological properties of the complexes of regions as defined.
Let $\R(S)$ be the set of $\MCG(S)$-orbits of homotopy classes of regions in $S$. For any subset $A$ of $\R(S)$ we denote the associated complex of regions by $\C_A(S)$. One can recover traditional complexes of curves in this context by using an annulus as a proxy for a curve: if $A$ consists of all orbits of annuli, then $\C_A(S)$ is isomorphic to the usual complex of curves.
\p{Prior results} There are several examples of complexes of regions that have been shown to have automorphism group isomorphic to the extended mapping class group: the complex of nonseparating curves by Irmak \cite{irmak}, the complex of separating curves by the authors of this paper \cite{kg}, the truncated complex of domains by McCarthy and Papadopoulos \cite{mcpap}, the arc complex by Irmak and McCarthy and by Disarlo \cite{irmakmac,disarlo}, the arc and curve complex by Korkmaz and Papadopoulos \cite{korkpap}, and the complex of strongly separating curves by Bowditch \cite{bowditch}.
There are also more general theorems characterizing isomorphisms---and injective maps---between different complexes; see for instance the work of Aramayona \cite{aramayona}, Aramayona--Leininger \cite{al}, Bavard--Dowdall--Rafi \cite{bdr}, Birman--Broaddus--Menasco \cite{bbm}, Hern\'andez \cite{jhh2,jhh3}, Irmak \cite{irmak1,irmak2}, and Shackleton \cite{shack}.
\p{Pathologies} In spite of all of the aforementioned positive results, there are many natural complexes of regions on which $\MCG(S)$ acts but where the full group of automorphisms is much larger than $\MCG(S_g)$. There are two immediate problems:
\begin{enumerate}
\item $\C_A(S_g)$ might be disconnected, and
\item $\C_A(S_g)$ might admit an \emph{exchange automorphism}, that is, an automorphism that interchanges two vertices and fixes all others.
\end{enumerate}
Typically, a disconnected complex of regions has automorphism group larger than $\MCG(S_g)$. For instance, if $\C_A(S_g)$ has infinitely many isomorphic components (like the complex of curves for the torus or the complex of nonseparating curves for the multi-punctured torus) then the automorphism group contains an infinite permutation group.
Also, an element of $\MCG(S_g)$ cannot act on $\C_A(S_g)$ by an exchange automorphism. Indeed, if a mapping class fixes all but finitely many vertices of $\C_A(S_g)$ then it must be the identity (cf. Lemma~\ref{inj} below). McCarthy and Papadopoulos were the first to address the issue of exchange automorphisms; they showed that the complex of domains $\C_{\R(S)}(S)$ admits exchange automorphisms when $S$ has more than one boundary component.
We would like to rule out these two types of pathologies. First we will list two situations that give rise to exchange automorphisms---holes and corks---and later in Section~\ref{sec:suff rich} we will prove that all exchange automorphisms arise in this way.
\p{Holes and corks} Let $\C_A(S_g)$ be a complex of regions. First, we say that a vertex $v$ of $\C_A(S_g)$ has a \emph{hole} if a representative region $R$ has a complementary region $Q$ with the property that no vertex of $\C_A(S_g)$ is represented by a subsurface of $Q$ (we refer to $Q$ as the hole). Note that annular vertices cannot have holes. Indeed, if $R$ represents an annular vertex, then there is an annulus parallel to $R$ in every complementary region.
Next, we say a vertex $v$ of $\C_A(S_g)$ is a \emph{cork} if (1) $v$ is represented by an annulus $A$, (2) one complementary region $R$ of $A$ represents a vertex $w$ of $\C_A(S_g)$, and (3) no proper, non-peripheral subsurface of $R$ represents a vertex of $\C_A(S_g)$. Any such pair $\{v,w\}$ will be referred to as a \emph{cork pair}.
\begin{figure}
\includegraphics[scale=.15]{bowlegged}
\hspace*{.5in}
\includegraphics[scale=.15]{cork}
\caption{\emph{Left:} vertices with holes; \emph{Right:} a cork pair}
\label{fig:bow}
\end{figure}
An example of a complex of regions that has a vertex with a hole is the complex whose vertices correspond to bow-legged pairs of pants, that is, pairs of pants that are embedded in such a way that two of the boundary components are parallel in the surface (the hole is the annulus between these two boundary components). Any two vertices with representatives contained in the same handle (torus with one boundary component) can be exchanged by an automorphism of this complex; see the left-hand side of Figure~\ref{fig:bow}.
An example of a complex of regions with a cork is the complex of regions that includes all regions except the nonseparating annuli and the bow-legged pairs of pants; in this case the corks are the separating annuli that cut off a handle. These vertices can be exchanged with the vertices corresponding to the handles they cut off.
In Section~\ref{sec:suff rich} below we will show that a connected complex of regions admits an exchange automorphism if and only if it has a hole or a cork (meaning that it has a vertex with a hole or a vertex that is a cork).
\p{Statement of the main theorem about complexes} For a subsurface $R$ of $S_g$ we define $\bar g(R)$ to be the smallest number $k$ so that $R$ is contained in a subsurface of $S_g$ of genus $k$ with connected boundary (we allow for the possibility that $\bar g(R) = g$). We define $\bar g(A)$ to be the minimum of $\bar g(R)$ where $R$ represents an element of $A$. The definitions of $\hat g(f)$ above and $\bar g(R)$ here are similar in spirit, although an important difference is that in the present case $R$ is not required to be non-peripheral in the subsurface of genus $k$. As such, we use different notations to avoid confusion.
\begin{theorem}
\label{main:complex}
Let $\C_A(S_g)$ be a complex of regions that is connected and has no holes or corks and assume that $g \geq 3\bar g(A) + 1$. Then the natural map
\[ \MCG(S_g) \to \Aut \C_A(S_g) \]
is an isomorphism.
\end{theorem}
Again the hypothesis on $g$ is not sharp. In the case where $\C_A(S_g)$ is the complex of curves, Theorem~\ref{main:complex} says that $\Aut \C_A(S_g) \cong \MCG(S_g)$ when $g \geq 4$, while on the other hand this isomorphism is known to hold for $g \geq 3$.
\p{Applications.} Theorem~\ref{main:complex} is the first result to address infinitely many distinct complexes with a unified argument. It covers many of the previously studied examples of simplicial complexes with automorphism group $\MCG(S_g)$, such as the complex of curves, the complex of separating curves, the complex of nonseparating curves, and the Bridson--Pettet--Souto complex of four-holed spheres and two-holed tori. It is also easy to construct new examples, such as the complex of handles, the complex of separating curves of odd genus, and the complex of nonseparating seven-holed tori, etc.
\p{Automorphisms in the presence of holes and corks.} In the case where $\C_A(S_g)$ has a hole or a cork, we can still describe its automorphism group. Let $\Exchange \C_A(S_g)$ denote the normal subgroup of $\Aut \C_A(S_g)$ generated by all exchange automorphisms.
\begin{theorem}
\label{ex thm}
Let $g \geq 3$. Let $\C_A(S_g)$ be a complex of regions that is connected and satisfies $g \geq 3 \bar g(A)+1$. Then
\[
\Aut \C_A(S_g) \cong \Exchange \C_A(S_g) \rtimes \MCG(S_g).
\]
\end{theorem}
We will prove Theorem~\ref{ex thm} in Section~\ref{sec:suff rich}. McCarthy and Papadopoulos proved Theorem~\ref{ex thm} in the case where $\C_A(S_g)$ is the complex of domains \cite[Theorem 8.9]{mcpap}.
\p{A conjectural sharpening of the theorem.} We conjecture that the condition on $\hat g(A)$ in Theorem~\ref{main:complex} is not necessary.
\begin{conjecture}
\label{crconj}
Let $\C_A(S_g)$ be a complex of regions that is connected and has no holes or corks. Then the natural map
\[ \MCG(S_g) \to \Aut \C_A(S_g) \]
is an isomorphism.
\end{conjecture}
\p{Further possible generalizations} There are other ways that one might extend our Theorem~\ref{main:complex}, for instance by generalizing the definition of a complex of regions. There are several examples of simplicial complexes that do not satisfy our definition of a complex of regions but still have automorphism group isomorphic to the extended mapping class group. For example, the systolic complex of curves, studied by Schmutz--Schaller \cite{pss} has edges that do not correspond to disjointness. The Torelli complex, studied by Farb and Ivanov \cite{farbivanov} (see also \cite{kg}), has vertices corresponding to disconnected subsurfaces. And the pants complex, studied by the second author \cite{pants}, has both deficiencies: its vertices correspond to disconnected subsurfaces and its edges do not correspond to disjointness. On the other hand, all of these complexes have automorphism group the extended mapping class group.
Other natural directions are to study the analogs for punctured surfaces, surfaces of infinite type, and other manifolds. Work in these directions has already been done by McLeay \cite{alan,alan2} and Scott \cite{shane}.
\subsection{Plan of the paper} We now give a summary of the remaining five sections of the paper. Along the way, we explain how the various sections fit together to prove our main results.
\p{Exchange automorphisms} As mentioned, Section~\ref{sec:suff rich} is devoted to the classification of exchange automorphisms and to the determination of the automorphism group of a connected complex with exchange automorphisms. More precisely, we prove Theorem~\ref{theorem:suff rich} below, which states that all exchange automorphisms arise from holes and corks, and we prove Theorem~\ref{ex thm} above, which gives a semi-direct product decomposition of the group of automorphisms into the extended mapping class group and the group of exchange automorphisms.
\p{Complexes of separating curves} In Section~\ref{sec:sep} we extend previous work of the authors on the complex of separating curves. This section is the only part of the paper that closely parallels earlier work in the subject.
Recall that the complex of curves $\C(S_g)$ is the simplicial flag complex with vertices corresponding to homotopy classes of simple closed curves in $S_g$ and edges connecting vertices with disjoint representatives. Vertices of $\C(S_g)$ can be separating or nonseparating, meaning that they have a representative that is such.
The \emph{genus} of a separating curve is the minimum of the genera of the two complementary regions of $S_g$. Let $\C_k(S_g)$ be the subcomplex of $\C(S_g)$ spanned by all vertices represented by separating curves of genus at least $k$. The first ingredient in our proof of Theorem~\ref{main:complex} is the following.
\begin{theorem}
\label{theorem:sep k}
Let $k \geq 1$ and let $g \geq 3k+1$. The natural map
\[ \MCG(S_g) \to \Aut(\C_k(S_g)) \]
is an isomorphism.
\end{theorem}
We prove Theorem~\ref{theorem:sep k} in Section~\ref{sec:sep}. The proof proceeds by induction on $k$. The base case is $k=1$, in which case $\C_k(S_g)$ is the complex of separating curves. This case was proved in our earlier work \cite{kg,kgadd}.
The bounds on genus in Theorems~\ref{main:normal} and~\ref{main:complex} are derived from the bound on $g$ in Theorem~\ref{theorem:sep k}. If the bound here can be improved, one would obtain improved versions of those theorems.
\p{Complex of dividing sets} In Section~\ref{sec:div} we apply Theorem~\ref{theorem:sep k} in order to determine the automorphism group of a different complex, which is of a different nature and requires a specialized set of tools and techniques. To state the theorem, we require some definitions.
A \emph{dividing set} in $S_g$ is a disjoint union of essential simple closed curves that divides $S_g$ into exactly two regions in such a way that each curve lies in the boundary of both regions. We allow for the possibility that one of the two regions associated to a dividing set is an annulus. We say that two dividing sets are \emph{nested} if one is contained entirely in a single region defined by the other.
Let $\D(S_g)$ denote the set of $\MCG(S_g)$-orbits of isotopy classes of dividing sets in $S_g$. For any $D \subseteq \D(S_g)$ we define $\C_D(S_g)$ to be the abstract simplicial flag complex whose vertices correspond to isotopy classes of dividing sets in $S_g$ representing elements of $D$ and whose vertices are connected by an edge when they have nested representatives.
We define a partial order on $\D(S_g)$ as follows: we say that $a \preceq b$ if $a$ and $b$ have nested representatives $A$ and $B$ and, of the two regions of $S_g$ complementary to $B$, the dividing set $A$ lies in one with minimal genus.
For any set $X$ with a partial order, an \emph{upper set} is a subset $Y \subseteq X$ with the property that $y \in Y$ and $y \preceq z$ implies $z \in Y$.
Finally, we define $\check g(D)$ to be the minimum of the genera of the separating curves corresponding to elements of $D$.
\begin{theorem}
\label{theorem:multi k}
Fix a nonempty upper set $D \subseteq \D(S_g)$ so that $\C_D(S_g)$ is connected and assume that $g \geq 3 \check g(D)+1$. Then the natural map
\[ \MCG(S_g) \to \Aut \C_D(S_g) \]
is an isomorphism.
\end{theorem}
We prove Theorem~\ref{theorem:multi k} in Section~\ref{sec:div}. The first observation is that since $D$ is an upper set the complex $\C_D(S_g)$ has a subcomplex isomorphic to $\C_{\check g(D)}(S_g)$. The proof then proceeds by showing that an automorphism of $\C_{D}(S_g)$ induces an automorphism of $\C_{\check g(D)}(S_g)$ and then applying Theorem~\ref{theorem:sep k}.
The proof of Theorem~\ref{theorem:multi k} is more subtle than the previous theorems about automorphisms of curve complexes. The first major distinction is that edges in $\C_D(S_g)$ do not correspond to disjointness, and so the usual arguments do not apply. On top of this, the hypotheses of the theorem do not specify which dividing sets do, and do not, correspond to vertices of $\C_D(S_g)$; they only specify that $D$ is an upper set (compare this with the hypotheses of our Theorem~\ref{main:normal} about normal subgroups of $\Mod(S_g)$).
\p{Complexes of regions} In Section~\ref{sec:reg} we derive Theorem~\ref{main:complex} from Theorem~\ref{theorem:multi k}. The starting point is a correspondence
\[
\left\{ \text{maximal joins in } \C_A(S_g) \right\} \longleftrightarrow \left\{ \text{vertices of } \C_D(S_g) \right\}.
\]
We use this correspondence to show that an automorphism of $\C_A(S_g)$ induces an automorphism of some $\C_D(S_g)$, and then apply Theorem~\ref{theorem:multi k}. The main work is in showing that the induced map $\Aut \C_A(S_g) \to \Aut \C_D(S_g)$ is injective. Again a difficulty is that we do not have an explicit list of vertices of the complex.
\p{Normal subgroups} In Section~\ref{section:normal} we prove Theorem~\ref{main:normal} using Theorem~\ref{main:complex}. The core idea is that to a normal subgroup $N$ of $\Mod(S_g)$ or $\MCG(S_g)$ we associate a complex of regions whose vertices correspond to the supports of certain subgroups of $N$. To this end, we introduce the notion of a basic subgroup of $N$, which is a non-abelian subgroup of $N$ whose centralizer in $N$ is maximal among non-abelian subgroups of $N$.
We consider the complex of regions whose vertices are the supports of the basic subgroups of $N$. Since $N$ has a pure element with a small component, the complex of regions has a small vertex. This construction enables us to extract the necessary topological data from $N$ without knowing any specific information about its elements. After possibly modifying the complex of regions so that it satisfies the other hypotheses of Theorem~\ref{main:complex}, we then show that an automorphism of $N$ gives rise to an automorphism of the complex of regions and then apply Theorem~\ref{main:complex}.
The proof then proceeds by analyzing separately the case where $N$ is normal in $\MCG(S_g)$ and the (harder) case where $N$ is normal in $\Mod(S_g)$.
\p{Bird's-eye view of the proof} One interpretation of our proofs of Theorems~\ref{main:normal} and~\ref{main:complex} is that there is a sequence of maps
\begin{align*}
\Aut N \to \Aut &\ \C_A(S_g) \to \Aut \C_{D}(S_g) \to \Aut \C_k(S_g) \to \Aut \C_{k-1}(S_g) \to \\
& \cdots \to \Aut \C_1(S_g) \to \Aut \C(S_g) \to \MCG(S_g)
\end{align*}
and that the appropriate compositions are inverse to the natural maps $\MCG(S_g) \to \Aut N$ and $\MCG(S_g) \to \Aut \C_A(S_g)$.
Therefore, not only does our main theorem validate Ivanov's metaconjecture, but by going through Ivanov's original theorem the proof does as well.
\p{Acknowledgments} This project was begun in conjunction with the Mathematical Research Communities program on Geometric Group Theory in 2013. We are grateful to the American Mathematical Society, the organizers, and the participants there, in particular Matthew Durham and Brian Mann, for helpful conversations and inspiration. We would like to thank Justin Lanier, Alan McLeay, and Shane Scott for their comments on earlier drafts, and Martin Bridson, Lei Chen, Tom Church, Luis Paris, Alexandra Pettet, Juan Souto, Masaaki Suzuki, and Daniel Studenmud for helpful conversations. We would like to thank Kevin Wortman for detailed discussions about arithmetic groups. We are also grateful to Joan Birman, Benson Farb, and Andrew Putman for their encouragement on this project. Finally we would like to express our deepest gratitude to Kathleen Margalit and Brendan Owens for their support throughout this project.
\section{Exchange automorphisms}
\label{sec:suff rich}
Before we proceed to the proofs of our main theorems, we prove two theorems that clarify the role that exchange automorphisms play in the theory of automorphisms of complexes of regions. Theorem~\ref{theorem:suff rich} gives a complete characterization of exchange automorphisms in complexes of regions. Theorem~\ref{ex thm} then describes the automorphism group of a connected complex of regions that has exchange automorphisms.
\subsection{Characterization of exchange automorphisms}\label{sec:sr} To state the first theorem about exchange automorphisms, we require a definition. Let $\C_A(S_g)$ be a complex of regions, and let $v$ be a non-annular vertex. Let $R$ be a representative of $v$ and let $Q_1,\dots,Q_n$ be the set of complementary regions of $R$ that do not contain representatives of vertices of $\C_A(S_g)$ (the $Q_i$ are the holes of $v$). The \emph{filling} of $v$ is the homotopy class of regions represented by $R' = R \cup Q_1 \cup \cdots \cup Q_n$. When $v$ has a hole (so $n > 0$), there are infinitely many vertices of $\C_A(S_g)$ with the same filling; these are the translates of $v$ under the elements of $\MCG(S_g)$ that preserve $R'$. For convenience, we define the filling of an annular vertex $v$ to be $v$ itself.
\begin{theorem}
\label{theorem:suff rich}
Let $g \geq 3$. Let $\C_A(S_g)$ be a complex of regions with no isolated vertices or edges. Then $\C_A(S_g)$ admits an exchange automorphism if and only if it has a hole or a cork. Moreover, two vertices can be interchanged by an exchange automorphism if and only if they are non-annular vertices with equal fillings or they form a cork pair.
\end{theorem}
\begin{proof}
The first statement follows from the second statement and the fact that if a vertex has a hole then there is another vertex (in fact infinitely many) with the same filling. Thus, it suffices to prove the latter statement.
Suppose first that $v$ and $w$ form a cork pair in $\C_A(S_g)$, and say that $v$ is the annular vertex in the pair. Clearly any vertex connected by an edge to $w$ must also be connected to $v$. If there were a vertex of $\C_A(S_g)$ that was connected to $v$ but not $w$, then it would be represented by a subsurface of a representative of $w$. By the definition of a cork, no such vertex exists. Thus, the stars of $v$ and $w$ in $\C_A(S_g)$ are equal and so the two vertices can be interchanged by an exchange automorphism.
Now suppose that $v$ is a vertex with a hole and that $v$ and $w$ have equal fillings. Say that $R$ is a representative of the filling. By the definition of a filling, any vertex of $\C_A(S_g)$ that is connected to $v$ by an edge must be represented in the complement of $R$. But then this vertex is also connected to $w$ by an edge. It then follows that $v$ and $w$ have equal links and can be interchanged by an exchange automorphism.
For the other direction of the theorem, we will show that two vertices of $\C_A(S_g)$ either have equal fillings, form a cork pair, or cannot be interchanged by an exchange automorphism. Let $v$ and $w$ be two vertices of $\C_A(S_g)$ and say they are represented by regions $P$ and $Q$. First we treat the case where $v$ and $w$ are connected by an edge, so $P$ and $Q$ are disjoint. There are three subcases.
The first subcase is where there is a component of the boundary of $P$ that is not parallel to the boundary of $Q$. In this case there is a region $R$ of $S_g$ that contains $P$ as a proper, non-peripheral subsurface and is disjoint from $Q$. From this it follows that there are $\MCG(S_g)$-translates of $v$ that are connected by an edge to $w$ but not to $v$. Therefore $v$ and $w$ cannot be exchanged.
The second subcase is where $P$ and $Q$ are complementary regions in $S_g$. If there is a vertex of $\C_A(S_g)$ corresponding to a proper, non-peripheral subsurface of either $P$ or $Q$ then this vertex would be connected by an edge to one of $v$ or $w$ but not the other, and we would have that $v$ and $w$ were not exchangeable. So we may assume there is no such vertex. It follows that $P$ and $Q$ are homeomorphic. We may also assume that there is a vertex $u$ of $\C_A(S_g)$ corresponding to a component of the boundary of $P$, for otherwise $v$ and $w$ would span an isolated edge in $\C_A(S_g)$. If $u$ is a nonseparating annular vertex then since there are no vertices of $\C_A(S_g)$ represented by proper, non-peripheral subsurfaces of $P$ or $Q$, it follows that $P$ and $Q$ are pairs of pants and so $g=2$, a contradiction. If $u$ is a separating annular vertex, then $u$ and $v$ form a cork pair.
The third and final subcase is where $P$ has connected boundary and $Q$ is an annulus parallel to the boundary of $P$. If there is a vertex $u$ of $\C_A(S_g)$ represented by a proper, essential subsurface of $P$ then $v$ and $w$ cannot be exchanged ($u$ is connected by an edge to $w$ but not $v$). If there is no such vertex $u$ then $v$ and $w$ form a cork pair.
We now proceed to the case where $v$ and $w$ are not connected by an edge. Here there are two subcases, according to whether or not one of the two vertices is annular. Again let $P$ and $Q$ be representatives of $v$ and $w$; in this case, $P$ and $Q$ have essential intersection.
The first subcase is where $P$ is annular. The complement of $P$ in $S_g$ consists of either one or two regions; the region $Q$ has essential intersection with each such region. Since $v$ is not an isolated vertex of $\C_A(S_g)$, there is a vertex $u$ of $\C_A(S_g)$ represented in a region of $S_g$ complementary to $P$. Some $\MCG(S_g)$-translate of $u$ is connected by an edge to $v$ but not $w$, and so $v$ and $w$ are not exchangeable.
The second and final subcase is where neither $P$ or $Q$ is annular. Since $v$ and $w$ are distinct, there must be (after possibly renaming the vertices) a complementary region $R$ of $P$ that has essential intersection with $Q$. If we assume that $v$ and $w$ are exchangeable, then $R$ must represent a hole for $P$ (otherwise we would find a vertex connected by an edge to $v$ but not $w$). It follows that after filling $v$ and $w$, they do not intersect each other's complementary regions. In other words, $v$ and $w$ have equal fillings. This completes the proof.
\end{proof}
\subsection{Automorphism groups in the presence of exchange automorphisms} In this section we prove Theorem~\ref{ex thm} which describes the group of automorphisms of a general complex of regions, possibly with holes and corks.
We will require a sequence of lemmas about fillings (these lemmas will also be used in Section~\ref{section:normal}). In what follows, the \emph{filling} of a complex of regions $\C_A(S_g)$ is the complex of regions $\C_{\bar A}(S_g)$ obtained by replacing each vertex of $\C_A(S_g)$ with its filling.
\begin{lemma}
\label{filling}
Let $\C_A(S_g)$ be a complex of regions. Then its filling $\C_{\bar A}(S_g)$ has no holes.
\end{lemma}
\begin{proof}
Let $R$ be a non-annular region of $S_g$ representing a vertex of $\C_{\bar A}(S_g)$. Since the vertices of $\C_{\bar A}(S_g)$ are fillings of vertices of $\C_A(S_g)$, it follows that there is a non-annular region $Q$ in $R$ that represents a vertex of $\C_A(S_g)$ and so that the $R$-vertex of $\C_{\bar A}(S_g)$ is the filling of the $Q$-vertex of $\C_A(S_g)$. Denote the complementary regions of $Q$ in $S_g$ by $P_1,\dots,P_n$ and say that $P_{m+1},\dots,P_n$ are the complementary regions corresponding to holes (i.e. the regions that do not support any vertex of $\C_A(S_g)$). Then $R$ is represented by the union of $Q$ with $P_{m+1} \cup \cdots \cup P_n$ and so the complementary regions to $R$ are $P_{1},\dots,P_n$. By assumption each of these regions supports a vertex of $\C_A(S_g)$.
We must show that each of $P_1,\dots,P_n$ supports a vertex of $\C_{\bar A}(S_g)$. Let $Q_{1}$ be a region in $P_{1}$ representing a vertex of $\C_A(S_g)$. If $Q_1$ is annular then the filling of the $Q_1$-vertex is itself and there is nothing to do. So we may assume $Q_1$ is not annular. Since the complement of $P_1$ is connected, there is a single complementary region of $Q_{1}$ containing the complement of $P_1$. Since this complementary region contains $Q$ it cannot represent a hole for $Q_{1}$. Thus the filling of the $Q_{1}$-vertex is contained in $P_1$ as desired. The same argument applies to $P_2,\dots,P_m$, and we have finished showing that $\C_{\bar A}(S_g)$ has no holes.
\end{proof}
\begin{lemma}
\label{small filling}
Let $\C_A(S_g)$ be a complex of regions with a small vertex. Then its filling $\C_{\bar A}(S_g)$ has a small vertex.
\end{lemma}
\begin{proof}
Let $v$ be a small vertex of $\C_A(S_g)$. Let $R$ be a subsurface of genus less than $g/3$ and with connected boundary that contains a representative of $v$. Let $Q$ denote the region complementary to $R$. Note that $Q$ lies in a single complementary region for (a representative of) $v$. Also note that $Q$ does not lie in a hole for $v$ since there are $\MCG(S_g)$-translates of $v$ that are represented in $Q$. Therefore the filling of $v$ is represented in $R$. The lemma follows.
\end{proof}
We are ready now for the proof of Theorem~\ref{ex thm}. In the proof we will refer to the set of vertices of $\C_A(S_g)$ with a given filling as an \emph{equal filling set}. As mentioned, if an equal filling set has more than one element, then it has infinitely many.
\begin{proof}[Proof of Theorem~\ref{ex thm}]
Let $\C_{\bar A}(S_g)$ be the filling of $\C_A(S_g)$ and let $\C_{\bar A'}(S_g)$ be the complex of regions obtained from $\C_{\bar A}(S_g)$ by removing all corks.
The proof is divided into two parts. The first part is to show that $\C_{\bar A'}(S_g)$ satisfies the hypotheses of Theorem~\ref{main:complex} and hence has automorphism group $\MCG(S_g)$. The second part to show that there is a split short exact sequence
\[
1 \to \Exchange \C_A(S_g) \to \Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g) \to 1.
\]
For the first part of the proof there are four steps, namely, to show that $\C_{\bar A'}(S_g)$ is connected, that is has a small vertex, that it has no corks, and that it has no holes. We treat these four steps in order.
The first step is to show that $\C_{\bar A'}(S_g)$ is connected. There are natural maps
\[
\C_A^{(0)}(S_g) \to \C_{\bar A}^{(0)}(S_g) \to \C_{\bar A'}^{(0)}(S_g)
\]
defined as follows. Under the first map each vertex of $\C_A(S_g)$ is sent to its filling. Under the second map, each vertex that is not a cork is sent to itself and each cork is assigned to have the same image as the vertex for which is it a cork.
We would like to show that both maps extend to simplicial maps of $\C_A(S_g)$ and $\C_{\bar A}(S_g)$, respectively. For the first map, assume that $v$ and $w$ are vertices of $\C_A(S_g)$ that are connected by an edge. By the definition of a hole, neither $v$ nor $w$ can lie in a hole for the other, and it follows that each is disjoint from the other's filling. Thus, the images of $v$ and $w$ are connected by an edge, and so the map indeed extends.
For the second map, let us assume that $v$ and $w$ are vertices of $\C_{\bar A}(S_g)$ that are connected by an edge. The only nontrivial case is where $v$ is a cork. In this case $w$, being disjoint from $v$, either is equal to or is connected by an edge to the vertex that forms a cork pair with $v$, as desired. Since the image of a connected complex under a simplicial map is connected, we have thus proven that $\C_{\bar A'}(S_g)$ is connected.
The second step is to show that $\C_{\bar A'}(S_g)$ has a small vertex. By assumption $\C_A(S_g)$ has a small vertex, and hence by Lemma~\ref{small filling} the filling $\C_{\bar A}(S_g)$ has a small vertex; call it $v$. If $v$ does not represent a cork in $\C_{\bar A}(S_g)$ then it survives in $\C_{\bar A'}(S_g)$ and is the desired small vertex. If $v$ does represent a cork in $\C_{\bar A}(S_g)$, then there is a vertex $w$ of $\C_{\bar A}(S_g)$ that it forms a cork pair with. This $w$ is represented in $R$ and is the desired small vertex of $\C_{\bar A'}(S_g)$.
The third step is to show that $\C_{\bar A'}(S_g)$ has no corks. Suppose for contradiction that $\C_{\bar A'}(S_g)$ had a cork pair corresponding to the non-annular region $R$ and the annular region $Q$. All regions representing vertices of $\C_{\bar A'}(S_g)$ also represent vertices of the intermediate complex $\C_{\bar A}(S_g)$; in particular $R$ and $Q$ do. By the definition of a cork pair, there is no vertex of $\C_{\bar A'}(S_g)$ represented by a non-peripheral, proper subsurface of $R$. But since $R$ and $Q$ do not represent a cork pair for $\C_{\bar A}(S_g)$ (otherwise the $Q$-vertex would have been removed), there must be a vertex of $\C_{\bar A}(S_g)$ represented by a non-peripheral, proper subsurface $P$ of $R$. Since $P$ does not represent a vertex of $\C_{\bar A'}(S_g)$, it must represent a cork in $\C_{\bar A}(S_g)$. But then there is a region $P'$ so that the $P$- and $P'$-vertices form a cork pair in $\C_{\bar A}(S_g)$. The region $P'$ must be contained in $R$, for otherwise the annular region $Q$ would prevent $P$ from being a cork. The region $P'$ is further proper and non-peripheral in $R$, contradicting our assumption about $R$. This completes the proof that $\C_{\bar A'}(S_g)$ has no corks.
The fourth step is to show that $\C_{\bar A'}(S_g)$ has no holes. Suppose that $R$ is a non-annular region of $S_g$ representing a vertex of $\C_{\bar A'}(S_g)$. Then $R$ also represents a vertex of $\C_{\bar A}(S_g)$. Let $Q$ be a region of $S_g$ complementary to $R$. We would like to show that $Q$ does not represent a hole for the $R$-vertex of $\C_{\bar A'}(S_g)$. By Lemma~\ref{filling} the complex $\C_{\bar A}(S_g)$ has no holes. Thus there is a region $P$ in $Q$ representing a vertex of $\C_{\bar A}(S_g)$. If $P$ does not represent a cork for $\C_{\bar A}(S_g)$ then $P$ also represents a vertex of $\C_{\bar A'}(S_g)$ and we are done. Now suppose that $P$ does represent a cork for $\C_{\bar A}(S_g)$. If the $R$-vertex of $\C_{\bar A}(S_g)$ forms a cork pair with the $P$-vertex, then since $\C_{\bar A'}(S_g)$ has a small vertex the genus of $R$ is less than $g/3$. As such, the complementary region $Q$ contains a $\MCG(S_g)$-translate of the $R$-vertex of $\C_{\bar A'}(S_g)$ and again we are done. Finally, we are in the situation where $P$ represents a cork for $\C_{\bar A}(S_g)$ but $R$ does not represent the other vertex in the cork pair; say that $P'$ represents the other vertex of the cork pair. By the definition of a cork, $R$ is not contained in $P'$. The annulus $P$ only has two complementary regions, one of which is $P'$. So it must be that $P'$ is the complementary region to $P$ not containing $R$. It follows that $P'$ is contained in $Q$. Hence $Q$ does not represent a hole for the $R$-vertex of $\C_{\bar A'}(S_g)$ and so $\C_{\bar A'}(S_g)$ has no holes.
We have shown that $\C_{\bar A'}(S_g)$ is connected, has a small vertex, and has no holes or corks. By Theorem~\ref{main:complex} we have
\[
\Aut \C_{\bar A'}(S_g) \cong \MCG(S_g).
\]
We now proceed to the second part of the proof, which is to show that there is a short exact sequence
\[
1 \to \Exchange \C_A(S_g) \to \Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g) \to 1.
\]
We treat three steps in turn, namely, to show that there is a well-defined map $\Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g)$, to show that this map has a right inverse, and then to show that the kernel is $\Exchange \C_A(S_g)$.
We begin with the first step, which is to show that there is a well-defined map $\Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g)$. Above we described simplicial maps
\[
\C_A(S_g) \to \C_{\bar A}(S_g) \to \C_{\bar A'}(S_g).
\]
We would like to show that these simplicial maps induce well-defined maps
\[
\Aut \C_A(S_g) \to \Aut \C_{\bar A}(S_g) \to \Aut \C_{\bar A'}(S_g).
\]
For the first map we need to show that the image of an equal filling set under an automorphism of $\C_A(S_g)$ is another equal filling set. But this is true by Theorem~\ref{theorem:suff rich}, which implies that a collection of vertices of $\C_A(S_g)$ is an equal filling set if and only if it is either a singleton or an infinite set on which $\Exchange \C_A(S_g)$ acts transitively.
For the second map we need to show that $\Aut \C_{\bar A}(S_g)$ preserves the set of cork pairs. But again by Theorem~\ref{theorem:suff rich} a collection of vertices of $\C_{\bar A}(S_g)$ is a cork pair if and only if it is a set with two elements upon which $\Exchange \C_{\bar A}(S_g)$ acts transitively.
We will now show that the map $\Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g)$ has a right inverse. We already showed in the first part of the proof that the map $\MCG(S_g) \to \Aut \C_{\bar A'}(S_g)$ is an isomorphism and so the natural homomoprhism $\MCG(S_g) \to \Aut \C_A(S_g)$ is a candidate for a right inverse. Because the processes of filling vertices and removing corks both commute with the action of $\MCG(S_g)$, this indeed gives the desired right inverse.
To complete the second part of the proof---and hence the theorem---it remains to show that the kernel of the composition $\Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g)$ is indeed the group $\Exchange \C_A(S_g)$. The kernel of the first map $\Aut \C_A(S_g) \to \Aut \C_{\bar A}(S_g)$ is clearly the subgroup of $\Exchange \C_A(S_g)$ generated by the permutations of equal filling sets. Since $\C_{\bar A}(S_g)$ has no holes the kernel of the second map $\Aut \C_{\bar A}(S_g) \to \Aut \C_{\bar A'}(S_g)$ is equal to the subgroup of $\Exchange \C_{\bar A}(S_g)$ generated by cork pair swaps and by Theorem~\ref{theorem:suff rich} this is all of $\Exchange \C_{\bar A}(S_g)$.
The statement that the kernel of $\Aut \C_A(S_g) \to \Aut \C_{\bar A'}(S_g)$ is $\Exchange \C_A(S_g)$ now amounts to the following statement: if $v$ and $w$ are vertices of $\C_A(S_g)$ whose images $\bar v$ and $\bar w$ in $C_{\bar A}(S_g)$ form a cork pair---say that $\bar v$ is the cork---then either $v$ and $w$ form a cork pair or they are not exchangeable. Since $\bar v$ is annular, it follows that $v$ is represented by the same annulus. Also, the filling of $w$ corresponds to $\bar w$. If $w$ and $\bar w$ are represented by the same region, then $v$ and $w$ form a cork. If not, then $w$ has a nontrivial filling, and there are infinitely many other vertices of $\C_A(S_g)$ with the same filling. Each of these vertices is connected to $v$ by an edge but not to $w$. It follows that $v$ and $w$ are not exchangeable. This completes the proof.
\end{proof}
\section{Complexes of separating curves}
\label{sec:sep}
In this section we prove Theorem~\ref{theorem:sep k}. The ideas in this section build on our earlier work. Indeed, the main tool in this section is a certain configuration of curves called a sharing pair, which was introduced in our previous paper \cite{kg}. Our approach here puts our earlier work into a more conceptual framework, allowing for the more general argument.
As discussed in the introduction, Theorem~\ref{theorem:sep k} is proven by induction with the base case $k=1$ (again the base case was proved in our earlier work \cite{kg,kgadd}). The main point of the inductive step is to show that there is a map
\begin{align*}
\Aut \C_k(S_g) &\to \Aut \C_{k-1}(S_g) \\
\phi &\mapsto \hat \phi
\end{align*}
so that each element $\phi$ of $\Aut \C_k(S_g)$ is the restriction to $\C_k(S_g)$ of its image $\hat \phi$ (we regard $\C_k(S_g)$ as a subcomplex of $\C_{k-1}(S_g)$). Given an automorphism $\phi$ of $\C_k(S_g)$, we thus need to specify the action of $\hat \phi$ on the vertices of $\C_{k-1}(S_g)$ of genus $k-1$.
\begin{figure}
\begin{minipage}{5in}
\centering
$\vcenter{\hbox{
\labellist
\small\hair 2pt
\pinlabel {$k-1$} [ ] at 110 30
\endlabellist
\includegraphics[scale=.35]{sp}
}
}$
\hspace*{.2in}
$\vcenter{\hbox{\includegraphics[scale=.35]{sp3d}}}$
\end{minipage}
\caption{\emph{Left:} A sharing pair of genus $k$. \emph{Right:} Another view of a sharing pair of genus three}
\label{fig:sp}
\end{figure}
To this end, we define a \emph{sharing pair of genus $k$} as a pair of vertices of $\C_k(S_g)$ that both have genus $k$ and are configured as in Figure~\ref{fig:sp}. Specifically, if we choose representative curves in minimal position, then the two curves intersect in two points and cut the surface into four regions, each with one boundary component, and the genera of the regions are $g-k-1$, $k-1$, 1, and 1. The key point is that (as long as $g > 2k$) a sharing pair specifies a unique separating curve of genus $k-1$, namely, the boundary of the region of genus $k-1$ defined by the sharing pair.
The main steps are to show for any automorphism $\phi$ of $\C_k(S_g)$ that
\begin{enumerate}
\item if two vertices of $\C_k(S_g)$ form a sharing pair their $\phi$-images do, and
\item if two sharing pairs in $\C_k(S_g)$ specify the same vertex of $\C_{k-1}(S_g)$ then their $\phi$-images do.
\end{enumerate}
Given these properties, we can define an automorphism $\hat \phi$ of $\C_{k-1}(S_g)$, whereby the action of $\hat \phi$ on a vertex of genus $k-1$ is dictated by the action of $\phi$ on sharing pairs.
Here is the outline for the section. First we begin with two preliminaries, dealing with basic separation properties (Lemma~\ref{lemma:joins}) and with subsurface projections (Lemma~\ref{lemma:vertex}). Then we prove the two properties above (Lemmas~\ref{lemma:sharing} and Lemma~\ref{lemma:sp graph connected}) before finally finishing the proof.
\p{Separation properties.} We begin with some basic characterizations. If $z$ is a vertex of $\C_k(S_g)$ of genus $h$, then any representative of $z$ divides $S_g$ into two regions, one of genus $h$ and one of genus $g-h$. Both regions are well defined up to isotopy and refer to them as the \emph{sides} of $z$. If $a$ is any other vertex connected to $z$ by an edge, then $a$ lies on one side of $z$; this side is well-defined.
The \emph{join} of two simplicial complexes $X$ and $Y$ is the simplicial complex $X \ast Y$ whose simplices are the disjoint unions of the simplices in $X$ and $Y$.
\begin{lemma}
\label{lemma:joins}
Let $k \geq 1$. Let $\phi \in \Aut \C_{k}(S_g)$, let $z$ be a vertex of $\C_k(S_g)$, and let $a$ and $b$ be a vertices connected by edges to $z$. Then
\begin{enumerate}
\item $z$ and $\phi(z)$ have the same genus,
\item\label{small side} the sides of $z$ and $\phi(z)$ corresponding to $a$ and $\phi(a)$ have the same genus, and
\item $a$ and $b$ lie on the same side of $z$ if and only if $\phi(a)$ and $\phi(b)$ lie on the same side of $\phi(z)$.
\end{enumerate}
\end{lemma}
\begin{proof}
The link of $z$ in $\C_k(S_g)$ is the join of two subcomplexes, namely, the subcomplexes corresponding to the two sides of $z$. Moreover, if we write the link of $z$ as the join of two subcomplexes, then those two subcomplexes must be the ones corresponding to the two sides. This is a consequence of the fact that if $v$ and $w$ are two vertices of $\C_{k}(S_g)$ lying on the same side of $z$ then there is another vertex $x$ in the link of $z$ that is not connected by an edge to either $v$ or $w$ in $\C_{k}(S_g)$.
Say that the genus of $z$ is $h$. The three statements of the lemma follow from the previous paragraph and the fact that the subcomplexes corresponding to the two sides of $z$ have dimensions $h-k-1$ and $g-h+k+1$.
\end{proof}
\p{Projections} Before showing that automorphisms preserve sharing pairs, we need to introduce one further tool: subsurface projection maps. This idea was introduced by Masur and Minsky in their work on the geometry of the complex of curves \cite{masurminsky}.
Let $z$ be a vertex of $\C_k(S_g)$ whose genus is strictly less than $g/2$. Let $R$ be a region of $S_g$ whose boundary represents $z$ and whose genus is equal to that of $z$. Let $v$ be a vertex of $\C_k(S_g)$ that is not in the star of $z$ (that is, $v$ intersects $z$). The projection $\pi_z(v)$ is a collection of homotopy classes of disjoint arcs in $R$ defined as follows: we choose a representative of $v$ that lies in minimal position with $\partial R$, take the intersection of this representative with $R$, and identify parallel arcs to a single arc.
The projection $\pi_z(v)$ is a well-defined collection of homotopy classes of arcs in $R$. We say that $\pi_z(v)$ is a \emph{nonseparating arc} if a representative has a single component and is nonseparating. Next, we say that $\pi_z(v)$ and $\pi_z(w)$ are \emph{unlinked} if they have representatives that are disjoint and whose endpoints on $\partial R$ alternate. Finally we say that $\pi_z(v)$ and $\pi_z(w)$ form a \emph{handle pair} if they have representatives that are distinct nonseparating arcs and so that the subsurface of $R$ filled by these projections is a surface of genus one with two boundary components.
\begin{lemma}
\label{lemma:vertex}
Let $k \geq 2$. Let $z$ be a vertex of $\C_k(S_g)$ of genus $h$ where $k < h < g/2$. Let $u$, $v$, and $w$ be vertices of $\C_k(S_g)$ that are not in the star of $z$, and suppose that $u$ is connected to $v$ by an edge.
\begin{enumerate}
\item If $\pi_z(v)$ is a nonseparating arc then $\pi_{\phi(z)}(\phi(v))$ is as well;
\item if $\pi_z(v)$ and $\pi_z(w)$ are distinct nonseparating arcs then $\pi_{\phi(z)}(\phi(v))$ and $\pi_{\phi(z)}(\phi(w))$ are distinct nonseparating arcs;
\item if $\pi_z(v)$ and $\pi_z(w)$ form a handle pair then $\pi_{\phi(z)}(\phi(v))$ and $\pi_{\phi(z)}(\phi(w))$ form a handle pair.
\item if $\pi_z(u)$ and $\pi_z(v)$ are unlinked nonseparating arcs then $\pi_{\phi(z)}(\phi(u))$ and $\pi_{\phi(z)}(\phi(v))$ are unlinked nonseparating arcs;
\end{enumerate}
\end{lemma}
\begin{proof}
We fix a region $R$ of genus $h$ whose boundary represents $z$. For the first statement, we claim that $\pi_z(v)$ is a nonseparating arc if and only if $\C_k(S_g)$ has more than one vertex of genus $h-1$ that lies on the genus $h$ side of $z$ and is connected by an edge to $v$. The first statement will follow from the claim and Lemma~\ref{lemma:joins}.
For the forward direction, assume that $\pi_z(v)$ is a single nonseparating arc. If we cut $R$ along a representative of the arc $\pi_z(v)$ the resulting surface has genus $h-1$ and two boundary components. There are infinitely many isotopy classes of simple closed curves in the cut surface that separate the cut surface into a pair of pants and a surface of genus $h-1$ with one boundary component. Each of these corresponds to a vertex of genus $h-1$ in $\C_k(S_g)$ that is connected by an edge to $v$.
For the other direction, there are two cases: either $\pi_z(v)$ contains the homotopy class of a separating arc or $\pi_z(v)$ contains more than one homotopy class of nonseparating arcs. In the first case, if we divide a representative of $R$ along such a separating arc, we obtain two surfaces of genus $h_1$ and $h_2$ with $h_2 > h_1 > 0$ and $h_1+h_2=h$. In particular, $h_i \leq h-1$. Any vertex of $\C_k(S_g)$ that has a representative in $R$ and is connected to $v$ by an edge must lie in one of these subsurfaces. If $h_1 =1$ then there is a unique such vertex; otherwise, there is no such vertex. For the second case, we note that if we cut $R$ along two disjoint nonseparating arcs we either obtain a surface of genus $h-1$ with a single boundary component, we obtain a surface of genus less than $h-1$, or we obtain two surfaces, each with two boundary components and genus less than $h-1$. So either there is a single vertex of genus $h-1$ as in the claim or there are none.
The second statement follows from Lemma~\ref{lemma:joins} and the first statement, since $\pi_z(v)$ and $\pi_z(w)$ are determined by the vertices of genus $h-1$ that lie on the genus $h$ side of $z$ and are disjoint from $v_1$ and $v_2$, respectively.
We now proceed to the third statement. Two distinct, nonseparating projections $\pi_z(v)$ and $\pi_z(w)$ form a handle pair if and only if there exists a vertex of genus $h-1$ in $\C_k(S_g)$ that lies on the genus $h$ side of $z$ and is connected by edges to both $v$ and $w$ in $\C_k(S_g)$. The third statement then follows from Lemma~\ref{lemma:joins}.
Finally we prove the fourth statement. Since $u$ and $v$ are connected by an edge, their projections are disjoint. Also, by the first statement we know that the projections are nonseparating if and only if their images are. It remains to characterize linking and unlinking for disjoint nonseparating projections. But disjoint nonseparating projections $\pi_z(u)$ and $\pi_z(v)$ are linked if and only if they form a handle pair, and so an application of the third statement completes the proof.
\end{proof}
\p{Sharing pairs} We now show that automorphisms of $\C_k(S_g)$ preserve sharing pairs. As discussed at the start of this section, this is the main tool used to extend an automorphism $\phi$ of $\C_k(S_g)$ to an automorphism $\hat \phi$ of $\C_{k-1}(S_g)$.
\begin{lemma}
\label{lemma:sharing}
Let $k \geq 2$, let $g \geq 3k+1$, and let $a$ and $b$ be two vertices of $\C_k(S_g)$ that form a sharing pair of genus $k$. If $\phi$ is an automorphism of $\C_k(S_g)$, then $\phi(a)$ and $\phi(b)$ form a sharing pair of genus $k$.
\end{lemma}
\begin{figure}
\labellist
\small\hair 2pt
\pinlabel {$a$} [ ] at 170 185
\pinlabel {$b$} [ ] at 290 185
\pinlabel {$z$} [ ] at 315 185
\pinlabel {$k-1$} [ ] at 90 65
\pinlabel {$k-1$} [ ] at 585 125
\pinlabel {$k-1$} [ ] at 585 15
\pinlabel {$P$} [ ] at 585 180
\pinlabel {$Q$} [ ] at 585 70
\endlabellist
\includegraphics[scale=.525]{sp_soup}
\caption{Curves and arcs used to characterize a sharing pair}
\label{fig:sp soup}
\end{figure}
\begin{proof}
We will show that two vertices $a$ and $b$ of $\C_k(S_g)$ form a sharing pair of genus $k$ if and only if there are vertices $x_1$, $x_2$, $y_1$, $y_2$, and $z$ of $\C_k(S_g)$ with the following properties:
\begin{enumerate}
\item\label{condition:genus} the genus of $z$ is $k+1$;
\item\label{condition:small} $a$ and $b$ are vertices of genus $k$ lying on the genus $k+1$ side of $z$;
\item\label{condition:penta} each $x_i$ is connected by edges to $a$, $y_1$, and $y_2$ but not to $b$;
\item\label{condition:pentb} each $y_i$ is connected by edges to $b$, $x_1$, and $x_2$ but not to $a$;
\item\label{condition:handle} $\pi_z(x_1)$ and $\pi_z(x_2)$ form a handle pair and $\pi_z(y_1)$ and $\pi_z(y_2)$ form a handle pair; and
\item\label{condition:unlink} each $\pi_z(x_i)$ is unlinked with each $\pi_z(y_j)$.
\end{enumerate}
(We have implicitly used the fact that $z$ has only one side of genus $k+1$, but this is implied by the conditions on $g$ and $k$.) Lemmas~\ref{lemma:joins} and ~\ref{lemma:vertex} together imply that $a$, $b$, the $x_i$, the $y_i$, and $z$ satisfy the given conditions if and only if their $\phi$-images do. Therefore, it remains to prove that the existence of such $a$, $b$, the $x_i$, the $y_i$, and $z$ is equivalent to the condition that $a$ and $b$ form a sharing pair.
The forward direction is given by explicit construction; refer to Figure~\ref{fig:sp soup}. There is a unique configuration for $a$ and $b$ as in the figure. The curve $z$ is shown. For each $i$ the curve $x_i$ is obtained by taking the boundary of a regular neighborhood of the union of the region $P$ with one of the arcs in the picture with endpoints on $P$ (the order of the arcs does not matter). Similarly each $y_i$ is obtained as the boundary of a regular neighborhood of the union of the region $Q$ with one of the arcs with endpoints on $Q$.
We now proceed to the other implication. Assume that $a$, $b$, $x_1$, $x_2$, $y_1$, $y_2$, and $z$ satisfy the conditions in the claim. Let $R$ be a region of $S_g$ representing the genus $k+1$ side of $z$, as per property \eqref{condition:genus}. By property~\eqref{condition:small}, the vertices $a$ and $b$ have representatives in $R$. Let $\bar R$ denote the closed surface obtained by collapsing the boundary of $R$ to a marked point. Each of $a$ and $b$ separates $\bar R$ into a surface of genus $k$ with one boundary component and a surface of genus one with one boundary component and one marked point.
By property~\eqref{condition:handle}, the vertices $x_1$ and $x_2$ give rise to a pair of nonseparating arcs in $\bar R$ based at the marked point, and these arcs fill a subsurface $Q_x$ of $\bar R$ homeomorphic to a surface of genus one with one boundary component and one marked point. Similarly, the $y_i$ give a pair of nonseparating arcs in $\bar R$ that fill a subsurface $Q_y$ with the same properties.
Since there is only one separating curve of genus $k$ disjoint from $Q_x$, namely the boundary of $Q_x$, property~\eqref{condition:penta} implies that $a$ is represented by the boundary of $Q_x$. Similarly, property~\eqref{condition:pentb} implies that $b$ is represented by the boundary of $Q_y$. Our main goal at this point is to show that the geometric intersection number $i(a,b)$ is 2, and so we have reduced this to a problem about the $x_i$ and $y_i$.
If we consider a small closed disk around the marked point of $\bar R$, the $x_i$-arcs and $y_i$-arcs give a collection of eight disjoint arcs connecting the marked point to the boundary of the disk, and since no triple among the $x_i$ and $y_i$ have pairwise nontrivial intersection, the eight arcs in the disk come in a well-defined cyclic order (depending only on the $x_i$, the $y_i$, and $z$). Any homotopically distinct based simple loops in $Q_x$ must cross transversely at the base point (this follows from the identification of the set of oriented nonseparating simple closed curves in the punctured torus with the primitive elements of $\Z^2$). It follows that the two $x_1$-arcs and the two $x_2$-arcs alternate in the cyclic order. Then, since the $x_i$-arcs fill $Q_x$, it follows from property~\eqref{condition:unlink} that in the cyclic order the four $x_i$-arcs in the disk appear in order, followed by the four $y_i$-arcs.
Since the $x_i$-arcs fill $Q_x$ and since the $x_i$ are disjoint from the $y_i$ it follows from the previous paragraph that if we take the $y_i$-arcs in $\bar R$ and intersect them all with $Q_x$ we obtain a set of four parallel arcs connecting the marked point to the boundary. As $Q_y$ is obtained from a regular neighborhood of the $y_i$-arcs, it follows that the intersection of $Q_x$ with $Q_y$ is a disk. Since $a$ and $b$ are identified with the boundary components of $Q_x$ and $Q_y$, it follows that $i(a,b)=2$. There is only one possible configuration for two separating curves of genus $k$ in $R$ with intersection number two, and so $a$ and $b$ form a sharing pair, as desired.
\end{proof}
\p{Sharing triples} As discussed at the beginning of the section, Lemma~\ref{lemma:sharing} suggests a method for extending an automorphism of $\C_k(S_g)$ to an automorphism of $\C_{k-1}(S_g)$: the image of a vertex of genus $k-1$ is determined by the image of a corresponding sharing pair. We need to show that this rule is well defined. We will use sharing triples to address this issue.
\begin{figure}
\includegraphics[scale=.35]{triple}
\caption{A sharing triple of genus three}
\label{fig:triple}
\end{figure}
We say that three vertices of $\C_k(S_g)$ form a \emph{sharing triple of genus $k$} if they are configured as in Figure~\ref{fig:triple}. Specifically, each pair of vertices in a sharing triple forms a sharing pair for the same vertex of $\C_{k-1}(S_g)$. It is also true that if three vertices of $\C_k(S_g)$ pairwise form sharing pairs for the same vertex of $\C_{k-1}(S_g)$ then they form a sharing triple but we will not use this.
The next lemma tells us that if two sharing pairs belong to the same sharing triple, then their images under any automorphism of $\C_k(S_g)$ represent the same vertex of genus $k-1$. Lemma~\ref{lemma:sp graph connected} below then tells us that if two arbitrary sharing pairs represent the same vertex of genus $k-1$, then their images under any automorphism of $\C_k(S_g)$ also represent the same vertex.
\begin{lemma}
\label{lemma:triples}
Let $k \geq 2$, let $g \geq 3k+1$, and let $\phi$ be an automorphism of $\C_k(S_g)$. If $a$, $b$, and $c$ are three vertices of $\C_k(S_g)$ that form a sharing triple of genus $k$ then $\phi(a)$, $\phi(b)$, and $\phi(c)$ form a sharing triple of genus $k$.
\end{lemma}
\begin{proof}
We claim that three vertices $a$, $b$, and $c$ form a sharing triple of genus $k$ if and only if there are vertices $z$ and $d$ of $\C_k(S_g)$ so that the following conditions hold:
\begin{enumerate}
\item the genus of $z$ is $k+2$,
\item $a$, $b$, and $c$ all lie on a genus $k+2$ side of $z$,
\item any two of $a$, $b$, and $c$ form a sharing pair of genus $k$,
\item any vertex of $\C_k(S_g)$ that is connected by an edge to each of $a$, $b$, and $c$ must also be connected by an edge to $z$, and
\item $d$ forms a sharing pair with $c$ and is connected by edges to $a$ and $b$.
\end{enumerate}
(In the special case where $k=2$ and $g=7$ the first condition should be interpreted as saying that the genus of $z$ is 3.) The lemma follows from the claim plus Lemmas~\ref{lemma:joins} and~\ref{lemma:sharing}. The last condition in the claim is included only to rule out one configuration in the case $k=2$. On the other hand for any $k \geq 2$ there are fake sharing triples that satisfy the first three conditions but not the fourth.
\begin{figure}
\labellist
\small\hair 2pt
\pinlabel {$a$} [ ] at 370 70
\pinlabel {$b$} [ ] at 345 305
\pinlabel {$c$} [ ] at 285 70
\pinlabel {$z$} [ ] at 240 305
\pinlabel {$d$} [ ] at 115 70
\endlabellist
\includegraphics[scale=.35]{zd}
\caption{The vertices $z$ and $d$ in the characterization of sharing triples.}
\label{fig:sp2}
\end{figure}
The forward direction of the claim is straightforward. The construction of the vertices $z$ and $d$ is indicated in Figure~\ref{fig:sp2}. Note that the vertex $d$ exists because $g \geq 2k+1$.
For the other direction, assume that $a$, $b$, and $c$ are three vertices of $\C_k(S_g)$ that satisfy the conditions of the claim. We must show that $a$, $b$, and $c$ form a sharing triple of genus $k$.
Choose a vertex $z$ as in the claim, and choose a representative curve in $S_g$. Let $R$ be the region of $S_g$ that is determined by this curve and contains representatives of $a$, $b$, and $c$. Choose representative curves for $a$ and $b$ in $R$. Since $a$ and $b$ form a sharing pair, we can assume that these curves are configured as in Figure~\ref{fig:sp}.
There is a region $Q$ of $R$ that lies between the boundary of $R$ and the union of the $a$-curve and the $b$-curve. This region $Q$ is homeomorphic to a surface of genus one with two boundary components; one boundary component is the $z$-curve and the other boundary component is made up of one arc of the $a$-curve and one arc of the $b$-curve.
Choose a curve in $R$ representing $c$, and take this curve to be in minimal position with the $a$-curve and the $b$-curve, and so that there are no triple intersections. Since $c$ lies on the same side of $z$ as $a$ and $b$ and is not connected by an edge to $a$ or $b$, the intersection of the $c$-curve with $Q$ is a collection of disjoint arcs, each starting and ending on the boundary component coming from $a$ and $b$. Since two curves in a sharing pair intersect in two points, there are at most two arcs. We may further assume that no $c$-arc in $Q$ is peripheral, for in this case we can push this arc out of $Q$ without increasing the number of intersections of the $c$-curve with either the $a$-curve or the $b$-curve.
\begin{figure}
\includegraphics[scale=.4]{q}
\caption{Three possibilities for the intersection of $c$ with $Q$ in the proof of Lemma~\ref{lemma:triples}}
\label{fig:q}
\end{figure}
By the fourth condition of the claim, the $c$-arcs in $Q$ must have the following property: if we cut $Q$ along the $c$-arcs the component containing the $z$-curve must be an annulus. We thus have the following possibilities:
\begin{enumerate}
\item there are two $c$-arcs in $Q$ that are not homotopic and nonseparating,
\item there are two $c$-arcs in $Q$ that are parallel and separating, or
\item there is a single $c$-arc in $Q$ that is separating;
\end{enumerate}
see Figure~\ref{fig:q}.
The first case can be ruled out because in this case the $c$-curve is nonseparating in $S_g$ (we can find a curve in $Q$---hence $S_g$---that intersects it in one point).
The second case can be ruled out as follows. First, any vertex $v$ of $\C_k(S_g)$ that is connected by edges to $a$ and $b$ but not $c$ is necessarily not connected by an edge to $z$. This is simply because $v$ has genus $k$ and the genus of $Q$, the region between $z$ and $a \cup b$, is only one. It follows that the vertex $d$ from the fifth condition of the claim must intersect $z$. But in the second case any curve that is disjoint from $a$ and $b$ but not $z$ must intersect $c$ in at least four points. This contradicts the assertion that $c$ and $d$ form a sharing pair.
We now consider the third case, where there is a single separating $c$-arc in $Q$. In this case the $c$-arc in $Q$ is configured like the $c$-arc on the outside of $a$ and $b$ in Figure~\ref{fig:sp2}. It remains to determine the configuration of $c$ in the union of the interiors of $a$ and $b$ in Figure~\ref{fig:sp2}. The $c$-arcs in the genus $k$ sides of the $a$- and $b$-curves are separating and must cut off a subsurface of genus $k-1$ in each region. The only possibility then is that $c$ is configured exactly as in Figure~\ref{fig:sp2} and hence forms a sharing triple with $a$ and $b$.
\end{proof}
\p{The sharing pair graph} Assume $g \geq 2k$ and let $y$ be a vertex of $\C_{k-1}(S_g)$. We define the \emph{sharing pair graph} for $y$ as the graph whose vertices correspond to sharing pairs of genus $k$ that specify $y$ and whose edges correspond to sharing pairs $\{a,b\}$ and $\{b,c\}$ with the property that $\{a,b,c\}$ is a sharing triple.
Let $\Mod(S_g,y)$ denote the subgroup of $\Mod(S_g)$ consisting of elements represented by homeomorphisms that act by the identity on the region of $S_g$ corresponding to the genus $k-1$ side of $y$. Note that $\Mod(S_g,y)$ acts on the sharing pair graph for $y$ and acts transitively on the vertices.
\p{Putman's trick} If $G$ is a group with a generating set $\{g_i\}$ and $G$ acts on a graph $X$ with base point $v$ and $G$ acts transitively on the vertices of $X$ then $X$ is connected if for each $i$ the vertices $g_i \cdot v$ and $v$ lie in the same component of $X$. As Putman explains \cite{putman}, this method is useful in the theory of mapping class groups because one can often choose the $v$ and the $g_i$ so that most of the $g_i \cdot v$ are equal to $v$ and the other $g_i \cdot v$ are very close to $v$. We refer to this method as Putman's trick. We will presently apply it to the action of $\Mod(S_g,y)$ on the sharing pair graph for $y$.
\begin{lemma}
\label{lemma:sp graph connected}
Let $k \geq 2$ and let $g \geq 2k$. Let $y$ be a vertex of $\C_{k-1}(S_g)$. The sharing pair graph for $y$ is connected.
\end{lemma}
\begin{proof}
Let $y$ be the vertex of $\C_{k-1}(S_g)$ shown in Figure~\ref{fig:hump}. We enlist the sharing pair $\{a,b\}$ shown in Figure~\ref{fig:hump} to act as a base point $v$ for the sharing pair graph for $y$. As mentioned, the group $\Mod(S_g,y)$ acts transitively on the vertices of this graph.
Denote $g-k+1$ by $g'$. The group $\Mod(S_g,y)$ is isomorphic to the mapping class group of the region of $S_g$ that lies on the genus $g'$ side of the $y$-curve. As such it is generated by the Dehn twists about the curves $\{d_0,\dots,d_{2g'}\}$ indicated in Figure~\ref{fig:hump} for the case where $g=k+4$; see \cite[Theorem 1]{johnson}.
\begin{figure}
\labellist
\small\hair 2pt
\pinlabel {$b$} [ ] at 260 38
\pinlabel {$a$} [ ] at 260 110
\pinlabel {$y$} [ ] at 185 120
\pinlabel {$d_0$} [ ] at 410 -10
\pinlabel {$d_1$} [ ] at 290 -10
\pinlabel {$d_2$} [ ] at 293 48
\pinlabel {$d_3$} [ ] at 350 50
\pinlabel {$d_4$} [ ] at 410 68
\pinlabel {$d_5$} [ ] at 455 70
\pinlabel {$d_6$} [ ] at 545 78
\pinlabel {$d_7$} [ ] at 465 116
\pinlabel {$d_8$} [ ] at 410 135
\pinlabel {$d_9$} [ ] at 365 132
\pinlabel {$d_{10}$} [ ] at 290 155
\endlabellist
\includegraphics[scale=.6]{hump}
\caption{Curves $d_i$ whose Dehn twists generate $\Mod(S_g,y)$}
\label{fig:hump}
\end{figure}
By Putman's trick, it is enough to show that each $T_{d_i}(v)$ lies in the same connected component of the sharing pair graph as $v$. We have arranged things so that when $d_i \notin \{d_3,d_{2g'-1}\}$ we have $T_{v_i} \cdot v = v$ and there is nothing to check. It remains to check that $T_{d_3} \cdot v$ and $T_{d_{2g'-1}} \cdot v$ lie in the same component as $v$. The configuration $\{a,b,d_3\}$ is equivalent to the configuration $\{a,b,d_{2g'-1}\}$ and so it suffices to treat the case of $d_3$.
\begin{figure}
\labellist
\small\hair 2pt
\pinlabel {$a$} [ ] at 210 70
\pinlabel {$b$} [ ] at 380 55
\pinlabel {$c$} [ ] at 480 200
\pinlabel {$d_3$} [ ] at 120 65
\endlabellist
\includegraphics[scale=.4]{d3}
\caption{The curves $a$, $b$, $c$, and $d_3$}
\label{fig:d3}
\end{figure}
There exists a vertex $c$ of $\C_k(S_g)$ so that $\{a,b,c\}$ forms a sharing triple of genus $k$ for $y$ and so that $i(c,d_3)=0$; this is clear from the point of view of Figure~\ref{fig:d3}. We also see from Figure~\ref{fig:hump} that $i(b,d_3)=0$. Since $\{a,b,c\}$ is a sharing triple, its $T_{d_3}$-image $\{T_{d_3}(a),b,c\}$ is a sharing triple as well. Thus, $T_{d_3} \cdot v = \{T_{d_3}(a),T_{d_3}(b)\} = \{T_{d_3}(a),b\}$ is connected by an edge to $w=\{b,c\}$. Since $v=\{a,b\}$ is also connected to $w$, we are done.
\end{proof}
\p{Finishing the proof} We are almost ready to finish the proof of Theorem~\ref{theorem:sep k}, which states that for $k \geq 2$ and $g \geq 3k+1$ the natural map $\MCG(S_g) \to \Aut \C_k(S_g)$ is an isomorphism.
\begin{lemma}
\label{inj}
Let $g \geq 3$ and let $X$ be one of the complexes $\C_k(S_g)$, $\C_D(S_g)$, or $\C_A(S_g)$, where $k \geq 1$, $D \subseteq \D(S_g)$ is nonempty, or $A \subseteq \R(S_g)$ is nonempty. Then the natural map $\MCG(S_g) \to \Aut X$ is injective and the image contains no exchange automorphisms.
\end{lemma}
\begin{proof}
A vertex of $X$ represents a point in $\PMF(S_g)$, the space of projective measured foliations on $S_g$. Indeed, for $\C_k(S_g)$ and $\C_D(S_g)$ we take the usual inclusion of the set of multicurves in $S_g$ into $\PMF(S_g)$ \cite[Section 5.4]{flp}, and for $\C_A(S_g)$ we first take the boundary of the corresponding subsurface and then take the usual inclusion of the set of multicurves into $\PMF(S_g)$. The action of $\MCG(S_g)$ on $X$ clearly agrees with the action on $\PMF(S_g)$.
The action of $\MCG(S_g)$ on $\PMF(S_g)$ is continuous and minimal, meaning the orbit of every point is dense; see \cite[Theorem 6.19]{flp}. It follows that if an element of $\MCG(S_g)$ acts trivially on $X$ then it acts trivially on $\PMF(S_g)$. But the kernel of the action of $\MCG(S_g)$ on $\PMF(S_g)$ is trivial for $g \geq 3$ (see \cite[Proof of Theorem 3.10]{primer}) and so the first statement follows. The second statement also follows, since the complement of two points in a dense subset of $\PMF(S_g)$ is still dense.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:sep k}]
Fix $g \geq 4$. We would like to show that for each $k$ with $1 \leq k < g/3$ the automorphism group of $\C_k(S_g)$ is isomorphic to $\MCG(S_g)$. We proceed by induction on $k$. The case $k=1$ was proven in our earlier paper \cite{kg}. So suppose $1 < k < g/3$, and assume that the natural map $\MCG(S_g) \to \Aut \C_{k-1}(S_g)$ is an isomorphism. By Lemma~\ref{inj} it suffices to show that the natural map $\MCG(S_g) \to \Aut \C_k(S_g)$ is surjective. Let $\phi \in \Aut \C_{k}(S_g)$. By Lemmas~\ref{lemma:sharing} and~\ref{lemma:sp graph connected} there is a well-defined automorphism $\hat \phi$ of the 0-skeleton of $\C_{k-1}(S_g)$ that agrees with $\phi$ on the 0-skeleton of $\C_k(S_g)$. Specifically, if $v$ is a vertex of $\C_{k-1}(S_g)$ that does not lie in $\C_k(S_g)$ then $\hat \phi(v)$ is the vertex of $\C_{k-1}(S_g)$ determined by the $\phi$-image of any sharing pair for $v$.
We claim that $\hat \phi$ extends to the 1-skeleton of $\C_{k-1}(S_g)$. To this end, let $v$ and $w$ be vertices of $\C_{k-1}(S_g)$. If $v$ and $w$ both lie in the subcomplex $\C_k(S_g)$ of $\C_{k-1}(S_g)$, there is nothing to show since the restriction of $\hat \phi$ to $\C_k(S_g)$ is equal to $\phi$. Next suppose that neither $v$ nor $w$ lies in $\C_k(S_g)$; in other words $v$ and $w$ both have genus $k-1$. In this case since $g \geq 3k+1 \geq 2k+2$ we can find sharing pairs for $v$ and $w$ that are disjoint, and from this---and the fact that two disjoint separating curves of genus $k$ cut off disjoint regions of genus $k$---the result follows. The final case is where $v$ lies in $\C_k(S_g)$ but $w$ does not. If $v$ has genus $k$ and $w$ lies on the genus $k$ side of $v$ then $v$ lies in a sharing pair for $w$, and the claim follows. In all other cases, we can choose a sharing pair for $w$ that is disjoint from $v$ and again the claim follows.
By induction there exists some $f \in \MCG(S_g)$ whose image in $\Aut \C_{k-1}(S_g)$ is precisely $\hat \phi$. Since the restriction of $\hat \phi$ to $\C_{k}(S_g)$ is equal to $\phi$, it follows that the image of $f$ in $\Aut \C_{k}(S_g)$ is $\phi$, as desired.
\end{proof}
\section{Complexes of dividing sets}
\label{sec:div}
The goal of this section is to prove Theorem~\ref{theorem:multi k}, which states that whenever $D$ is an upper set in $\D(S_g)$ with $g \geq 3\check{g}(D)+1$, the group of automorphisms of $\C_D(S_g)$ is isomorphic to $\MCG(S_g)$.
We will say that a vertex of $\C_D(S_g)$ has \emph{type $(k,n)$} if one of the regions of $S_g$ determined by a representative of $v$ is a surface of genus $k$ with $n$ boundary components. Because a dividing set determines two complementary regions of $S_g$, a vertex of type $(k,n)$ is also a vertex of type $(g-k-n+1,n)$.
Next, we say that a dividing set has...
\begin{enumerate}
\item \emph{type N} if it is of type $(0,2)$,
\item \emph{type S} if it is of type $(k,1)$ for some $k$, and
\item \emph{type M} otherwise
\end{enumerate}
(N, S, and M are for nonseparating curve, separating curve, and multicurve). Vertices of type N and S correspond to vertices of the complex of curves. Also, a dividing set is nested with a vertex of type N or S if and only if it is disjoint. Therefore, the subgraph of $\C_D(S_g)$ spanned by vertices of type N and S is isomorphic to a subgraph of $\C(S_g)$. The main idea of the proof is to show that the type of a vertex is invariant under automorphisms of $\D(S_g)$, and so an automorphism of $\D(S_g)$ induces an automorphism of either the complex of curves or---in the absence of vertices of type N---an automorphism of the complex of separating curves $\C_{\check g(D)}(S_g)$.
The only upper set in $\D(S_g)$ containing vertices of type N is the entire set $\D(S_g)$ itself. We deal with this case first since it is especially easy, and also because in the absence of vertices of type N a simplex in $\C_D(S_g)$ has a normal form representative that is unique (Lemma~\ref{lemma:div set normal form} below).
\subsection{Complexes with vertices of type N}
Before dispensing with the case where $D = \D(S_g)$ (equivalently, where $D$ contains an element of type N), we require one new idea.
\p{Subordinacy} We say that a vertex $v$ of $\C_{\D(S_g)}(S_g)$ is \emph{subordinate} to a vertex $w$ if $v$ has a representative that is homotopic to a subset of a representative of $w$ (the homotopy might take distinct components to a single component). We make the following observations.
\begin{enumerate}
\item If $v$ and $w$ are distinct vertices of $\C_{\D(S_g)}(S_g)$ with $v$ subordinate to $w$ then $v$ has type N and $w$ has type M.
\item If $v$ is a vertex of $\C_{\D(S_g)}(S_g)$ that has type N, then there exists another vertex $w$ so that $v$ is subordinate to $w$.
\item If $w$ is a vertex of $\C_{\D(S_g)}(S_g)$ that has type M then there is another vertex $v$ subordinate to $w$.
\end{enumerate}
In other words, in $\C_{\D(S_g)}(S_g)$ the type of a given vertex is completely determined by the subordinacy relation.
\begin{lemma}
\label{subord}
Suppose that $g \geq 1$ and let $\phi$ be an automorphism of $\C_{\D(S_g)}$. If $v$ and $w$ are vertices of $\C_{\D(S_g)}$ with $v$ subordinate to $w$ then $\phi(v)$ is subordinate to $\phi(w)$.
\end{lemma}
\begin{proof}
We make the following claim: a vertex $v$ is subordinate to a vertex $w$ if and only if the star of $w$ is contained in the star of $v$. Since automorphisms preserve stars, the lemma will follow from this.
For the forward direction of the claim, suppose that $v$ is subordinate to $w$. As above, $v$ has type N. Thus a vertex $u$ is connected to $v$ by an edge if and only if $u$ and $v$ have disjoint representatives. Therefore, if a vertex $u$ is not connected to $v$ by an edge, then $u$ is not connected to $w$ by an edge (it intersects the component of $w$ corresponding to $v$); thus the star of $w$ is contained in the star of $v$.
Now suppose that $v$ is not subordinate to $w$. If $v$ and $w$ are not joined by an edge, then $w$ is not contained in the star of $v$ and we are done. If $v$ and $w$ are joined by an edge, then they have disjoint, nested representatives. Since $v$ is not subordinate to $w$, the representative of $v$ has a component that is not homotopic to any component of $w$, that is, $v$ is represented by a non-peripheral dividing set in one of the two regions of $S_g$ determined by the representative of $w$. It follows that there is a $\MCG(S_g)$-translate $v'$ of $v$ that is connected by an edge to $w$ and has essential intersection with $v$, so $v'$ is not connected by an edge to $v$. This completes the proof.
\end{proof}
\p{Finishing the proof in the easy case} The next proposition constitutes the special case of Theorem~\ref{theorem:multi k} in the case where $D = \D(S_g)$.
\begin{prop}
\label{prop:cd easy case}
Suppose that $g \geq 3$. The natural map
\[ \MCG(S_g) \to \Aut \C_{\D(S_g)}(S_g) \]
is an isomorphism.
\end{prop}
\begin{proof}[Proof of Proposition~\ref{prop:cd easy case}]
We already explained that the type of a vertex in $\C_{\D(S_g)}(S_g)$ is completely determined by the subordinacy relation. It follows from this and from Lemma~\ref{subord} that the vertices of type N, S, and M form three characteristic subsets of $\C_{\D(S_g)}(S_g)$ (meaning each subset is preserved by automorphisms). Identifying $\C(S_g)$ with the subcomplex of $\C_{\D(S_g)}(S_g)$ spanned by vertices of type N and S, we thus obtain a homomorphism
\[ \Aut \C_{\D(S_g)}(S_g) \to \Aut \C(S_g) \]
given by restriction. This map is injective because each vertex of type M is determined by the vertices of type N that are subordinate to it. Thus the composition
\[ \Aut \C_{\D(S_g)}(S_g) \to \Aut \C(S_g) \stackrel{\cong}{\to} \MCG(S_g) \]
is injective. It remains to check that the map $\MCG(S_g) \to \Aut \C_{\D(S_g)}(S_g)$ is a right inverse. But this is true because the composition $\MCG(S_g) \to \Aut \C_{\D(S_g)}(S_g) \to \Aut \C(S_g)$ is equal to the natural map $\MCG(S_g) \to \Aut \C(S_g)$.
\end{proof}
\subsection{Complexes without vertices of type N}
To prove Theorem~\ref{theorem:multi k}, it remains to deal with the case where $\C_D(S_g)$ has no vertices of type N. For this case, there are three technical ingredients. The first ingredient, which takes advantage of the assumption that there are no vertices of type N, is that every simplex of $\C_D(S_g)$ has a normal form representative that is unique up to isotopy in $S_g$ (Lemma~\ref{lemma:div set normal form}). The second ingredient is the notion of a linear simplex in $\C_D(S_g)$, a special type of simplex in $\C_D(S_g)$. The third ingredient is the idea of a exceptional edge, a certain type of configuration that only can involve vertices of type M. Along the way we also give a topological characterization of upper sets in $\D(S_g)$. Then we determine $\Aut \C_D(S_g)$ by showing that automorphisms of $\C_D(S_g)$ preserve the set of vertices of type S.
\p{Normal form representatives.} Let $\sigma$ be a simplex of $\C_D(S_g)$ with vertices $v_0,\dots,v_n$. A \emph{normal form} representative for $\sigma$ in $S_g$ is a collection of pairwise nested multicurves $m_0, \dots, m_n$ where $m_i$ represents $v_i$.
\begin{lemma}\label{lemma:div set normal form}
Suppose that $D \subseteq \D(S_g)$ contains no elements of type N. Every simplex of $\C_D(S_g)$ has a normal form representative, unique up to isotopy of $S_g$.
\end{lemma}
\begin{proof}
Let $\sigma$ be a simplex in $\C_D(S_g)$ with vertices $v_0, \dots, v_n$. Choose representatives $m_i$ for the $v_i$ that are pairwise disjoint. Such a collection $\cup m_i$ exists and is unique up to isotopy of $S_g$ and reordering the curves in each parallel family; see, e.g., \cite[Section 1.2.4]{primer}. It remains to show there is a unique choice of ordering of each parallel family so that the resulting representatives of the $v_i$ are pairwise nested, hence are in normal form.
We first deal with the case where $\sigma=\{v_0,v_1\}$ is an edge and then use this to prove the general case. By the definition of $\C_D(S_g)$, the vertices $v_0$ and $v_1$ have nested representatives $m_0$ and $m_1$. Such representatives are unique up to ambient isotopy in $S_g$ and reordering the connected components that come in parallel families. Finally, there is a unique way to order a given pair of parallel curves: since $m_0$ has at least one component $c$ that is not parallel to any component of $m_1$, we must order the curves so that the $m_0$-curves lie on the side of $m_1$ containing $c$.
Now let $\sigma$ be an arbitrary simplex of dimension greater than one and again choose disjoint representatives $m_i$. Consider one particular parallel family of curves in $\cup m_i$, and let $A$ be a (nonseparating) annulus in $S_g$ containing this family. Arbitrarily name the two boundary components of $A$ as the left and right boundary components.
For any $m_i$ and $m_j$ with components in $A$, we choose a normal form representative of the edge spanned by $v_i$ and $v_j$ (possibly modifying $m_i$ and $m_j$ in the process). We can then declare $v_i$ to be to the left of $v_j$ in $A$ if the $m_i$-curve is closer to the left boundary of $A$ than the $m_j$-curve in $A$. We need to show that this gives a total ordering on the set of vertices of $\sigma$ incident to $A$, that is, we need to show that the given relation is transitive.
Suppose that $v_i$ is to the left of $v_j$ in $A$ and $v_j$ is to the left of $v_k$ in $A$. This means that there are nested representatives $m_i$ and $m_j$ of $v_i$ and $v_j$ so that $m_i$ is to the left of $m_j$ in $A$, and nested representatives $m_j'$ and $m_k$ of $v_j$ and $v_k$ so that $m_j'$ is to the left of $m_k$ in $A$. By the uniqueness mentioned earlier, we can apply an ambient isotopy to $m_j'$ and $m_k$ so that $m_j'$ is equal to $m_j$. In particular, $m_j$---hence $m_i$---is to the left of $m_k$ in $A$, as desired.
After rearranging each parallel family of curves according to the above ordering, the resulting representative of $\sigma$ is a normal form representative, since a representative of a simplex is in normal form if and only if it restricts to a normal form representative for each edge. This completes the proof.
\end{proof}
\p{A characterization of upper sets.} Assume that $\C_D(S_g)$ has no vertices of type N. Let $\sigma = \{v_0,\dots,v_n\}$ be a simplex of $\C_D(S_g)$. As a consequence of Lemma~\ref{lemma:div set normal form} it makes sense to say that a vertex $v_j$ lies \emph{between} $v_i$ and $v_k$ in $S_g$; specifically, this means that for any normal form representative $\{m_0,\dots,m_n\}$, the dividing sets $m_i$ and $m_k$ lie on opposite sides of $m_j$. Equivalently, we can say that $v_i$ and $v_j$ lie \emph{on the same side} of $v_k$.
We say that a subset $D$ of $\D(S_g)$ is \emph{closed under separations} if whenever $u$ and $w$ are homotopy classes of dividing sets representing elements of $D$ and $v$ is a homotopy class of dividing sets lying between $u$ and $w$ we have that $v$ represents an element of $D$.
\begin{lemma}
\label{upper set}
Let $g \geq 0$. Let $D \subseteq \D(S_g)$ be a subset with no element of type N. Then $D$ is an upper set if and only if it is closed under separations.
\end{lemma}
\begin{proof}
Let $D \subseteq \D(S_g)$. Suppose first that $D$ is an upper set. Let $u$ and $w$ be homotopy classes of dividing sets representing elements of $D$ and so that $v$ lies between $u$ and $w$; for concreteness we imagine that $u$ lies to the left of $v$ and $w$ lies to the right. We will show that $v$ is either larger than $u$ or larger than $w$ in the partial order on $\D(S_g)$. Suppose that $v$ is not larger than $w$. Then the regions of $S_g$ to the left and right of $v$ have genus $g_1$ and $g_2$, respectively, with $g_1 < g_2$. It follows that the regions of $S_g$ to the left and right of $u$ have genus $h_1$ and $h_2$, respectively, with $h_1 \leq g_1$ and $h_2 \geq g_2$. It follows that $u \preceq v$. Thus $v$ represents an element of $D$ and we have shown that $D$ is closed under separations.
For the other direction suppose that $D$ is closed under separations. Let $u$ and $v$ be homotopy classes of dividing sets with $u \in D$ and with $u \preceq v$. We would like to show that $v$ represents an element of $D$. Say that $u$ lies to the left of $v$. Since $u \preceq v$, there is a $\MCG(S_g)$-translate $w$ of $u$ that lies to the right of $v$. This $w$ represents the same element of $D$ as $u$. Since $D$ is closed under separations, $v$ represents an element of $D$, as desired.
\end{proof}
\p{Linear simplices.} We say that a simplex $\sigma$ of $\C_D(S_g)$ is \emph{linear} if we can label the vertices of $\sigma$ by $v_0,\dots,v_n$ in such a way that $v_j$ lies between $v_i$ and $v_k$ whenever $i < j < k$. We can think of a linear simplex as a sequence of cobordisms between 1-manifolds, starting and ending with the empty manifold. We will refer to the vertices $v_0$ and $v_n$ as \emph{extreme} vertices.
\begin{lemma}
\label{lemma:linear}
Let $g \geq 0$. Suppose that $D \subseteq \D(S_g)$ contains no elements of type N. Let $D \subseteq \D(S_g)$, let $\sigma$ be a linear simplex of $\C_D(S_g)$ with ordered vertices $v_0,\dots,v_n$ and let $\phi$ be an automorphism of $\C_D(S_g)$. Then $\phi(\sigma)$ is a linear simplex of $\C_D(S_g)$ with ordered vertices $\phi(v_0),\dots,\phi(v_n)$.
\end{lemma}
\begin{proof}
Let $u$, $v$, and $w$ be three vertices of an arbitrary simplex of $\C_D(S_g)$. We will show that $v$ lies between $u$ and $w$ if and only if the link of $v$ in $\C_D(S_g)$ is contained in the union of the links of $u$ and $w$; in other words, if a vertex $z$ is not connected by an edge to either $u$ or $w$ then it is not connected by an edge to $v$ either. This will imply the lemma.
First suppose that $v$ lies between $u$ and $w$ in $S_g$. If $z$ is any vertex of $\C_D(S_g)$ connected by an edge to $v$, then $z$ is also connected by an edge to one of $u$ and $w$, specifically, the one lying on the opposite side of $v$ from $z$. This completes the forward direction of the claim.
For the other direction, suppose $v$ does not lie between $u$ and $w$. Let $\{m_u,m_v,m_w\}$ be a normal form representative for the simplex $\{u,v,w\}$. Both $m_u$ and $m_w$ must lie on the same side of $m_v$ and each must have a component that is not homotopic into $m_v$. We can thus find a vertex $u'$ of $\C_D(S_g)$ in the $\MCG(S_g)$-orbit of $u$ that is connected to $v$ but not to $u$ or $w$ (because $u'$ intersects $u$ and $w$ nontrivially). This completes the proof.
\end{proof}
\begin{figure}
\includegraphics[scale=.5]{linear}
\caption{A specific type of linear simplex}
\label{fig:linear}
\end{figure}
\begin{lemma}
\label{lemma:four}
Suppose that $D \subseteq \D(S_g)$ is an upper set that contains no elements of type N and suppose that $g \geq \max\{3 \check g(D) + 1,5\}$. If $v$ is a vertex of $\C_D(S_g)$ of type S then $v$ is an extreme vertex of linear simplex of $\C_D(S_g)$ with five vertices.
\end{lemma}
\begin{proof}
There is a linear simplex $\tau = \{v_0,\dots,v_n\}$ in $\C_D(S_g)$ where $n$ is even, where each $v_i$ with $i$ even is of type $S$, and where $\tau$ is maximal with respect to these properties. The simplex $\tau$ is unique up to the action of $\MCG(S_g)$; see Figure~\ref{fig:linear}. Note that the existence of $\tau$ uses the assumption that $D$ is an upper set.
We claim that $n \geq 6$. Indeed, since $\tau$ is maximal, both $v_0$ and $v_n$ have genus $\check g(D)$. So the region between $v_0$ and $v_n$ has genus $g-2\check g(D)$. Thus $\tau$ has $g-2\check g(D)+1$ vertices of type S and $g-2\check g(D)$ vertices of type $M$ (each with two components), for a total of $2g-4\check g(D)+1$ vertices, so $n=2g-4\check g(D)$. Since $g \geq 3 \check g(D) + 1$ we have $n \geq 2\check g(D) + 2$. It immediately follows that $n \geq 6$ when $\check g(D) \geq 2$. When $\check g(D) = 1$ it follows from the assumption that $g \geq 5$ and the equality $n=2g-4\check g(D)$ that $n \geq 6$.
Every vertex of type S lies in the orbit of one of the vertices $v_i$ with $i$ even and $0 \leq i \leq n/2$. And so it suffices to show that each such $v_i$ satisfies the statement of the lemma. But since $n \geq 6$ it is the case that for all $i$ even with $0 \leq i \leq n/2$ we have $i + 4 \leq n$ and so $\{v_i,\dots,v_{i+4}\}$ is the desired linear simplex.
\end{proof}
\begin{figure}
\labellist
\small\hair 2pt
\pinlabel {$v$} [ ] at 198 -10
\pinlabel {$w$} [ ] at 246 -10
\endlabellist
\includegraphics[scale=.45]{exceptional}
\caption{A typical exceptional edge}
\label{fig:example}
\end{figure}
\p{Exceptional edges} We now explain the third tool required for the proof of Theorem~\ref{theorem:multi k}. We say that vertices $v_1$ and $v_2$ of $\C_D(S_g)$ form an \emph{exceptional edge} if they are connected by an edge in $\C_D(S_g)$ and---after taking a normal form representative for the edge---the subsurface between $v_1$ and $v_2$ is the disjoint union of some number of annuli and at least two pairs of pants. Note that in an exceptional edge both vertices must have type M. An example of an exceptional edge is shown in Figure~\ref{fig:example}.
\begin{lemma}
\label{lemma:tight edges}
Suppose $D \subseteq \D(S_g)$ is an upper set that contains no elements of type N. The image of an exceptional edge under an automorphism of $\C_D(S_g)$ is an exceptional edge.
\end{lemma}
\begin{proof}
We will show that an edge in $\C_D(S_g)$ is exceptional if and only if its link is the join of a nonempty finite graph with some other (possibly empty) graph (see Section~\ref{sec:sep} for the definition of a join). Since the latter property is clearly preserved by automorphisms, the lemma will follow.
Let $\{v_1,v_2\}$ be an edge in $\C_D(S_g)$. Normal form representatives for $v_1$ and $v_2$ divide $S_g$ into three regions, $L$, $C$, and $R$ (for left, center, and right). Say that $C$ is the region lying between $v_1$ and $v_2$. The regions $L$ and $R$ are connected but $C$ may not be. We think of $v_1$ as lying to the left of $v_2$ so that $L$ is bounded by $v_1$.
Suppose first that $\{v_1,v_2\}$ is an exceptional edge, so the subsurface $C$ is the disjoint union of $n \geq 2$ pairs of pants $P_1,\dots,P_n$ and some number of annuli. There are $2^n$ vertices of the star of $\{v_1,v_2\}$ supported in $C$: for each $i$ we make a choice between the left side of $P_i$ or the right and for each annulus we include the core curve. If we choose the left side of each pair of pants, we obtain $v_1$ and if we choose the right side in each case we obtain $v_2$. Thus, there are $2^n-2>0$ vertices of the link of $\{v_1,v_2\}$ supported in $C$; call this set of vertices $F$. Since all other vertices of the link of $\{v_1,v_2\}$ are supported in $L$ or $R$, the link of $\{v_1,v_2\}$ is the join of $F$ with the graph spanned by those vertices.
Now suppose that $\{v_1,v_2\}$ is an edge of $\C_D(S_g)$ whose link is the join of a finite graph $F$ with some other (possibly empty) graph. Assume that $\{v_1,v_2\}$ is not an exceptional edge. We must show that $F$ is empty. We will repeatedly use the following fact: to show that a vertex $w$ is not contained in $F$ it is enough to show that there are infinitely many vertices of the link of the edge $\{v_1,v_2\}$ to which $w$ is not connected by an edge.
Suppose that $w$ is a vertex of the link of $\{v_1,v_2\}$ represented in $L$ (or $R$). Then $w$ must have at least one component that is not peripheral in $L$ (or $R$). But then there are infinitely many $\MCG(S_g)$-translates of $w$ that lie in the link of $\{v_1,v_2\}$ and have nonzero intersection with $w$. As above, this implies $w$ does not lie in $F$. Applying the same argument to the region $C$, we see that all elements of $F$ must be represented by dividing sets that are peripheral in $C$.
If $C$ has fewer than two components that are not annuli then the only vertices of the star of $\{v_1,v_2\}$ realized as peripheral dividing sets in $C$ are $v_1$ and $v_2$ and we have succeeded in showing that $F$ is empty. So assume that $C$ has at least two components that are not annuli.
Let $w$ be a vertex of $F$. Again, we may assume that $w$ is peripheral in $C$. Since $\{v_1,v_2\}$ is not exceptional, there is some component $C_0$ of $C$ that is not an annulus or a pair of pants. Since $C_0$ is not an annulus or a pair of pants there is a vertex $u$ of the link of $\{v_1,v_2\}$ so that $u$ is supported in $C$ and some component of a representative of $u$ is non-peripheral in $C_0$. In $C_0$ it makes sense to say that $u$ lies to the left or right of $w$. Say it is to the left. This means that in $C_0$ the vertex $w$ is parallel to the $v_2$-side (the right-hand side) of $C_0$.
Since $w$ is not equal to $v_1$ or $v_2$, it must have some component that lies on the $v_1$-side (the left-hand side) of some non-annular component $C_1$ of $C$. After possibly changing $u$ we may assume that the intersection of $u$ with $C_1$ is the collection of curves parallel to the $v_2$-side (the right-hand side) of $C_1$. By construction this $u$ is not connected to $w$ by an edge. Also, since $u$ has a component in the interior of $C_0$ there are infinitely many $\MCG(S_g)$-translates of $u$ that are not connected to $w$ by an edge. We have thus shown that $w$ cannot lie in $F$, and so $F$ is empty.
\end{proof}
We already said that the two vertices in an exceptional edge must be of type M. The next lemma is a partial converse to this statement.
\begin{lemma}
\label{lemma:char}
Suppose that $D \subseteq \D(S_g)$ contains no elements of type N. Let $D \subseteq \D(S_g)$ be an upper set with $g \geq 3 \check g (D) + 1$. Let $v$ be a vertex of $\C_D(S_g)$ that is of type M and is an extreme vertex of a linear simplex $\sigma$ with five vertices. Then $v$ lies in an exceptional edge.
\end{lemma}
\begin{proof}
Denote the ordered vertices of $\sigma$ by $v=v_0,\dots,v_4$. Choose a normal form representative for $\sigma$ and let $Q$ denote the region of $S_g$ lying between the dividing sets representing $v_0$ and $v_4$. We will think of $Q$ as lying to the right of $v$.
We must find a vertex $w$ so that $v$ and $w$ form an exceptional edge. To do this, we will find two disjoint pairs of pants $P_1$ and $P_2$ that lie in $Q$ and are adjacent to $v$; the vertex $w$ is then obtained from $v$ by replacing the left-hand side of each $P_i$ with the right-hand side. It follows from Lemma~\ref{upper set} that $w$ will indeed represent a vertex of $\C_D(S_g)$.
We can think of $Q$ as the union of four cobordisms connecting the vertices of $\sigma$. As such we have $\chi(Q) \leq -4$. We can assume without loss of generality that $\chi(Q) = -4$: since $D$ is an upper set, we can by Lemma~\ref{upper set} replace $\sigma$ (if necessary) by a linear simplex with five vertices where each of the four corresponding cobordisms is the disjoint union of one pair of pants with some number of annuli.
Let $Q_0$ denote the union of the non-annular components of $Q$. If $Q_0$ is not connected, then we can find the desired $w$ by choosing pair of pants in two different components of $Q_0$.
\begin{figure}
\includegraphics[scale=.25]{w}
\caption{\emph{Top:} finding $w$ when $Q$ is a sphere with six boundary components; \emph{Bottom:} finding $w$ when $Q$ is a torus with four boundary components}
\label{fig:w}
\end{figure}
Now suppose that $Q_0$ is connected. There are three cases for $Q_0$: a surface of genus zero with six boundary components, a surface of genus one with four boundary components, or a surface of genus two with two boundary components. If $Q_0$ has more than one boundary component parallel to $v$ then we can easily find the desired pairs of pants $P_1$ and $P_2$, hence the desired vertex $w$; see Figure~\ref{fig:w}. (Note that we do note need to consider here the case where $Q_0$ has genus two since $Q_0$ does not have all of its boundary components parallel to $v$.)
So it remains to consider the case where $Q_0$ is one of the three surfaces described in the previous paragraph and $Q_0$ has a single component of its boundary parallel to $v$. We treat this case by reducing to the previous cases.
Say that $v$ has type $(k,n)$ and that the region $R$ of $S_g$ that is determined by $v$ and does not contain $Q_0$ has genus $k$. If $Q_0$ is a sphere with six boundary components, then $v_4$ has type $(k,n+4)$. Since $n \geq 2$, we can form another vertex of type $(k,n+4)$ by gluing a sphere with five boundary components to one component of the boundary of $R$ and a pair of pants to another component of the boundary of $R$ (the new dividing set is the boundary of the union of $R$ and the two additional spheres). This new dividing set represents a vertex $v_4'$ of $\C_D(S_g)$ since it has the same type as $v_4$. Also, since the region between $v$ and $v_4'$ has more than one component with negative Euler characteristic, we have reduced to a previous case.
Similarly, if $Q_0$ is a surface of genus one with four boundary components, then $v_4$ has type $(k+1,n+2)$. In this case we can obtain a vertex $v_4'$ of the same type by gluing a sphere with six boundary components to $R$ along two of the six boundary components. Again, this is a case we have already dealt with.
Finally if $Q_0$ is a surface of genus two with two boundary components, then $v_4$ has type $(k+2,n)$, and in this case $v_4'$ is obtained by gluing a sphere with six boundary components to $R$ along three of its boundary components, another case we have previously treated.
\end{proof}
\p{Finishing the proof} For the proof of Theorem~\ref{theorem:multi k}, we say that a vertex $v$ of $\C_D(S_g)$ is \emph{1-sided} if all vertices in the link of $v$ lie to one side of $v$. Similarly, we say that $v$ is \emph{2-sided} if there are vertices of the link of $v$ lying on different sides of $v$.
\begin{proof}[Proof of Theorem~\ref{theorem:multi k}]
By Proposition~\ref{prop:cd easy case} we may assume that $\C_D(S_g)$ does not contain vertices of type N. And by Lemma~\ref{inj}, it suffices to show that the natural map $\MCG(S_g) \to \Aut \C_D(S_g)$ is surjective. Let $\phi \in \Aut \C_D(S_g)$.
We claim that $\phi$ preserves the set of vertices of type S. First assume that $g \geq 5$. By Lemma~\ref{lemma:four}, if a vertex of $\C_D(S_g)$ is not an extreme vertex of some linear simplex with five vertices then it is a vertex of type M. The set of extreme vertices of linear simplices with five vertices is preserved by automorphisms (Lemma~\ref{lemma:linear}). Therefore, it suffices to show that if a vertex of type M is an extreme vertex of a linear simplex with five vertices then its image under $\phi$ is a vertex of type M. Let $v$ be such a vertex. By Lemma~\ref{lemma:char}, $v$ is a vertex of an exceptional edge. So by Lemma~\ref{lemma:tight edges} the vertex $\phi(v)$ is a vertex of an exceptional edge. Thus $\phi(v)$ is of type M, giving the claim in the case $g \geq 5$.
It remains to prove the claim in the case where $g=4$ and $\check g(D) = 1$. The first step is to show that automorphisms of $\C_D(S_4)$ preserve the set of vertices of type S that have genus one, that is, the ones of type $(1,1)$. The only vertices than can be extreme vertices of linear simplices with five vertices are those of type $(0,3)$ and of type $(1,1)$. By the argument of the previous paragraph, these two types are each preserved (although type $(0,3)$ may not be present in the complex).
The second step for the proof of the claim in the $g=4$ case is to show that automorphisms of $\C_D(S_4)$ preserve the set of vertices of type S that have genus two. These vertices are distinguished by the following property: if we have a simplex with five vertices, four of which are vertices of type S that have genus one, then the fifth vertex must be a vertex of type S that has genus two. This completes the proof of the claim.
The claim implies that $\phi$ restricts to an automorphism $\bar \phi$ of $\C_{\check g(D)}(S_g)$, regarded as a subcomplex of $\C_D(S_g)$. By Theorem~\ref{theorem:sep k}, there exists some $f \in \MCG(S_g)$ so that the automorphism of $\C_{\check g(D)}(S_g)$ induced by $f$ is $\bar \phi$. We would like to show that the automorphism of $\C_D(S_g)$ induced by $f$ is $\phi$.
To do this, it suffices to show that an automorphism of $\C_D(S_g)$ that restricts to the identity on the subcomplex $\C_{\check g(D)}(S_g)$ must itself be the identity. We proceed by induction on distance in $\C_D(S_g)$ from $\C_{\check g(D)}(S_g)$; since $\C_D(S_g)$ is connected by assumption, this will prove the theorem.
We assume for induction that $\phi$ restricts to the identity on all vertices of $\C_D(S_g)$ that have distance at most $k$ from $\C_{\check g(D)}(S_g)$. The base case $k=0$ is true by assumption.
We perform the inductive step in two stages, first for 1-sided vertices and then for 2-sided vertices. Let $v$ be a 1-sided vertex of $\C_D(S_g)$ that has distance $k+1$ from $\C_{\check g(D)}(S_g)$. Let $w$ be any vertex of $\C_D(S_g)$ that is connected by an edge to $v$ and has distance $k$ from $\C_{\check g(D)}(S_g)$. There are $\MCG(S_g)$-translates of $w$ that fill the region of $S_g$ that lies to one side of $v$ and contains $w$. Since $\C_{\check g(D)}(S_g)$ is invariant under the action of $\MCG(S_g)$, all of these vertices have distance $k$ from $\C_{\check g(D)}(S_g)$ and so by the inductive hypothesis they are fixed by $\phi$. As $v$ is 1-sided, it is the unique vertex of $\C_D(S_g)$ connected by edges to all of these translates of $w$ and so $\phi$ must fix $v$ as well.
Now let $v$ be a 2-sided vertex of $\C_D(S_g)$ that has distance $k+1$ from $\C_{\check g(D)}(S_g)$. Let $w$ be any vertex of $\C_D(S_g)$ that is connected by an edge to $v$ and has distance $k$ from $\C_{\check g(D)}(S_g)$. The vertex $w$ lies to one side of $v$, and the $w$-side of $v$ is filled by $\MCG(S_g)$-translates of $w$. As above, all of these vertices have distance $k$ from $\C_{\check g(D)}(S_g)$ and so are fixed by $\phi$. Let $u$ be any 1-sided vertex of $\C_D(S_g)$ that lies on the other side of $v$. Note that the distance from $u$ to $\C_{\check g(D)}(S_g)$ is at most $k+1$ since $u$ is connected by an edge to $w$. By the previous paragraph, $u$ is fixed by $\phi$. Moreover, there are $\MCG(S_g)$-translates of $u$ that fill the $u$-side of $v$, and all of these vertices are also fixed by $\phi$. The vertex $v$ is the unique vertex connected by edges to all of these vertices, and so $v$ is also fixed by $\phi$, and we are done.
\end{proof}
\section{Complexes of regions}
\label{sec:reg}
In this section we apply our theorem about dividing sets (Theorem~\ref{theorem:multi k}) in order to prove Theorem~\ref{main:complex}, which states that if $\C_A(S_g)$ is a connected complex of regions with no holes or corks and with a small vertex then the automorphism group of $\C_A(S_g)$ is isomorphic to $\MCG(S_g)$.
The basic idea of the proof of Theorem~\ref{main:complex} is to relate maximal joins in $\C_A(S_g)$ to dividing sets in $S_g$ (Lemma~\ref{lemma:bijection}). This relationship is somewhat strained in the case where $\C_A(S_g)$ has nonseparating annular vertices. So similar to Section~\ref{sec:div} we will break this off as a special case.
We say that a vertex of $\C_A(S_g)$ has...
\begin{enumerate}
\item \emph{type N} if it is represented by a nonseparating annulus,
\item \emph{type S} if it is represented by a separating annulus, and
\item \emph{type P} if it is represented by a nonseparating pair of pants.
\end{enumerate}
There is a partial order on the set of vertices of $\C_A(S_g)$ whereby $a \preceq b$ if the link of $b$ is contained in the link of $a$. We say that a vertex is \emph{$\preceq$-minimal} if it is minimal with respect to this ordering.
\subsection{Complexes with vertices of type N}
In this section, we prove Theorem~\ref{main:complex} in the special case where $\C_A(S_g)$ has vertices corresponding to nonseparating annuli.
\begin{lemma}
\label{nonsep ann}
Let $\C_A(S_g)$ be a complex of regions with vertices of type N.
\begin{enumerate}
\item Any automorphism of $\C_A(S_g)$ preserves the set of vertices of type N.
\item Any automorphism of $\C_A(S_g)$ preserves the set of vertices of type P.
\end{enumerate}
\end{lemma}
\begin{proof}
We claim that a vertex of $\C_A(S_g)$ is of type N if and only if
\begin{enumerate}
\item it is a $\preceq$-minimal vertex of $\C_A(S_g)$ and
\item its link in $\C_A(S_g)$ is not a join.
\end{enumerate}
The first statement of the lemma follows from the claim since an automorphism of $\C_A(S_g)$ preserves these two properties.
For the forward direction, assume $v$ is of type N. The vertex $v$ is $\preceq$-minimal because a representative of $v$ contains no proper subsurfaces. Also, the link of $v$ is not a join because for any two vertices of the link of $v$ we can find a $\MCG(S_g)$-translate of $v$ that is not connected by an edge to either.
For the other direction of the claim, assume that $v$ is not of type N. Let $R$ be a representative of $v$. If $R$ is separating then each complementary region to $R$ supports at least one vertex of type N, and so the link of $v$ is a join. So we may assume that $R$ is nonseparating. If the boundary of $R$ is connected, then $v$ is not $\preceq$-minimal, as there are vertices of type N that are smaller in the partial order. So we may further assume that $R$ has disconnected boundary. If $R$ is the complementary region to a nonseparating annulus, then clearly $v$ is not $\preceq$-minimal. The remaining case is that $R$ is nonseparating with disconnected boundary and no two components of the boundary are parallel. In this case, the link of $v$ is a join. Indeed, any vertex of type N corresponding to a component of the boundary of $R$ is a cone point. This completes the proof of the claim and hence the first statement of the lemma.
For the second statement, we claim that a vertex $v$ of $\C_A(S_g)$ is of type P if and only if there is a triangle $\sigma = \{w_1,w_2,w_3\}$ so that each $w_i$ is of type N and so that $v$ and $\sigma$ have equal stars. The second statement follows from this and the first statement.
For the forward direction, let $v$ be a vertex of type P. We take $\sigma$ to be the triangle corresponding to the boundary of a representative of $v$. Since the only proper regions in this pair of pants are those corresponding to the vertices of $\sigma$, it follows that $\sigma$ and $v$ have equal stars.
For the other direction, assume we have a vertex $v$ and a triangle $\sigma$ as in the claim. Let $Q$ and $R$ be representatives of $\sigma$ and $v$. Since $v$ and $\sigma$ have equal stars, we can take $Q$ and $R$ to be disjoint. Also, each component of $Q$ must be parallel to the boundary of $R$, for otherwise we could find a vertex in the star of $v$ but not $\sigma$. Conversely, each component of the boundary of $R$ must be parallel to some component of $Q$, for otherwise we could find a vertex in the star of $\sigma$ but not $v$. Thus, $R$ has exactly three boundary components, each of which is a nonseparating curve in $S_g$. It follows that $R$ is nonseparating. If $R$ is not a pair of pants, then we could find a vertex of type N that is in the star of $\sigma$ but not $v$. This completes the proof of the claim, hence the lemma.
\end{proof}
\p{Finishing the proof in the easier case} We are now ready to prove Theorem~\ref{main:complex} in the case where $\C_A(S_g)$ contains vertices of type N. In the proof we say that a region of $S_g$ is of type P if it is a nonseparating pair of pants (we make the distinction between a region and a vertex here because $\C_A(S_g)$ may or may not have vertices of type P).
Let $\N(S_g)$ denote the \emph{complex of nonseparating curves} for $S_g$, that is, the subcomplex of the complex of curves $\C(S_g)$ spanned by vertices of type N.
\begin{prop}
\label{prop:ca easy}
Let $g \geq 3$ and suppose that $\C_A(S_g)$ contains vertices of type N. Then the natural map
\[ \MCG(S_g) \to \Aut \C_A(S_g) \]
is an isomorphism.
\end{prop}
\begin{proof}
By Lemma~\ref{inj} the map $\Psi : \MCG(S_g) \to \Aut \C_A(S_g)$ is injective. So it suffices to show that $\Psi$ is surjective. We will construct a right inverse.
The subcomplex of $\C_A(S_g)$ spanned by the vertices of type N is naturally isomorphic to the complex of nonseparating curves $\N(S_g)$ and so we may regard $\N(S_g)$ as a subcomplex of $\C_A(S_g)$. By Lemma~\ref{nonsep ann}, there is a map
\[
\Aut \C_A(S_g) \to \Aut \N(S_g)
\]
given by restriction. Also Irmak proved that the natural map
\[
\MCG(S_g) \to \Aut \N(S_g)
\]
is an isomorphism \cite{irmak}. Consider the composition
\[
\Omega: \Aut \C_A(S_g) \to \Aut \N(S_g) \stackrel{\cong}{\to} \MCG(S_g).
\]
We would like to show that $\Psi \circ \Omega$ is the identity. Let $\phi \in \Aut \C_A(S_g)$. Then $\Omega(\phi)$ is a mapping class $f$ whose action on $\N(S_g)$ agrees with the restriction of $\phi$. To show that $\Psi \circ \Omega(\phi) = \phi$ we must show that $\phi$ is determined by its restriction, or, that any element of $\Aut \C_A(S_g)$ restricting to the identity in $\Aut \N(S_g)$ must itself be the identity.
So suppose $\phi \in \Aut \C_A(S_g)$ restricts to the identity in $\Aut \N(S_g)$. Let $v$ be a vertex of $\C_A(S_g)$ that is not of type N. We must show that $\phi(v)=v$.
First suppose that $v$ is of type P. By the second statement of Lemma~\ref{nonsep ann}, $\phi(v)$ is also of type P. But since $g \geq 3$ such a vertex is clearly determined by the vertices of type N in its link, and so we are done in this case.
Now suppose that $v$ is not of type P. Again by Lemma~\ref{nonsep ann} the same is true for $\phi(v)$. Let $R$ be a representative of $v$. Each complementary region to $R$ is nonseparating. If a complementary region is not of type P, then it is filled by vertices of type N. If a complementary region is of type P, then its three boundary components all correspond to vertices of type N. Thus, if we take the subsurface of $S_g$ filled by vertices of type N in the link of $v$, then $R$ is the unique complementary region that is not of type P. Thus $\phi(v)$ is represented in $R$. In other words, the support of $v$ cannot become larger under $\phi$. Since $\phi$ is invertible, we conclude that the support of $v$ cannot become smaller under $\phi$ either. This means that $\phi(v)$ is represented by $R$, that is, $\phi(v)=v$, as desired.
\end{proof}
\subsection{Complexes without vertices of type N}
To complete the proof of Theorem~\ref{main:complex} it remains to treat the case where $\C_A(S_g)$ does not have any vertices of type N.
\p{From regions to dividing sets.} Let $A \subseteq \R(S_g)$ and let $D \subseteq \D(S_g)$. We say that vertices of $\C_A(S_g)$ and $\C_D(S_g)$ are \emph{compatible} if they have disjoint representatives. Also, we say that an element $d$ of $\D(S_g)$ is \emph{compatible} with $A$ if both regions of $S_g$ determined by a representative of $d$ contain representatives of elements of $A$ (the representatives are allowed to be peripheral). We then define
\[ \delta A = \{ d \in \D(S_g) \mid d \text{ is compatible with } A \}. \]
For any $A$, the set $\delta A$ is clearly closed under separations and so by Lemma~\ref{upper set} it is an upper set.
Most of our work here is devoted to showing that there is a well-defined map
\[ \Aut \C_A(S_g) \to \Aut \C_{\delta A} (S_g), \]
so that we may apply Theorem~\ref{theorem:multi k}.
We now define a function
\[ \Phi : \{ \text{vertices of } \C_{\delta A}(S_g) \} \to \{ \text{subcomplexes of } \C_A(S_g) \}. \]
For a vertex $v$ of $\C_{\delta A}(S_g)$, the image $\Phi(v)$ is defined to be the full subcomplex of $\C_A(S_g)$ spanned by all vertices that are compatible with $v$.
\p{Two join decompositions.} For a vertex $v$ of $\C_{\delta A}(S_g)$, we will now define two different join decompositions of $\Phi(v)$, one topological and one combinatorial. In the next lemma we will show that the two decompositions are the same.
First, we define the \emph{left/right decomposition} $V_L \ast V_M \ast V_R$ of $\Phi(v)$ as follows. A representative of $v$ divides $S_g$ into two complementary regions, which we arbitrarily label as $L$ and $R$ (for left and right). The complexes $V_L$ and $V_R$ are the subcomplexes of $\C_A(S_g)$ spanned by the vertices corresponding to non-peripheral subsurfaces of $L$ and $R$. The complex $V_M$ is the subcomplex spanned by the vertices corresponding to annuli parallel to a representative of $v$; since $\C_A(S_g)$ does not have vertices of type N, the complex $V_M$ is either empty or is a single vertex of type S. It is clear that $V_L \ast V_M \ast V_R$ is a join decomposition of $\Phi(v)$.
Next, the \emph{complete join decomposition} of $\Phi(v)$ is a decomposition $V_1 \ast \cdots \ast V_n$ with the property that no $V_i$ can be decomposed as a join. We have $n \leq 3g-3$ since there is an upper bound of $3g-4$ to the dimension of a simplex in $\C_A(S_g)$.
We say that a subcomplex of a simplicial complex is a \emph{maximal join} if it is a join that is not contained in any other join.
\begin{lemma}
\label{lemma:bijection}
Let $\C_A(S_g)$ be a complex of regions without vertices of type N and without isolated vertices.
\begin{enumerate}
\item The map $\Phi$ gives a bijection
\[ \Phi : \{ \text{vertices of } \C_{\delta A}(S_g) \} \to \{ \text{maximal joins in } \C_A(S_g) \}. \]
\item For each vertex $v$ of $\C_{\delta A}(S_g)$ the left/right decomposition of $\Phi(v)$ is the same as the complete join decomposition.
\end{enumerate}
\end{lemma}
\begin{proof}
We start with the first statement. First we will show that each $\Phi(v)$ is a maximal join and then we will show that each maximal join is in the image of $\Phi$ (it is clear that $\Phi$ is injective).
Let $v$ be a vertex of $\C_{\delta A}(S_g)$. We will begin by showing that the left/right decomposition of $\Phi(v)$ is a nontrivial join decomposition. Assume for contradiction that the left/right decomposition of $\Phi(v)$ is a trivial join. By the definition of $\delta A$ there are vertices of $\C_A(S_g)$ corresponding to (possibly peripheral) subsurfaces of both sides of $v$, and so the assumption implies that $\Phi(v)$ consists only of vertices corresponding to components of $v$. Since $\C_A(S_g)$ has no nonseparating annular vertices, it follows that $v$ is represented by a separating curve (it is of type S). It follows that $\Phi(v)$ consists of a single vertex, and this vertex is isolated in $\C_A(S_g)$, the desired contradiction.
Now that we know $\Phi(v)$ is a join, we would like to show that it is maximal. Let $V_1 \ast V_2$ be a join in $\C_A(S_g)$ that properly contains $\Phi(v)$. We would like to show that one of the $V_i$ is trivial. First note that $V_L$ must be contained in one of $V_1$ or $V_2$; this is because for any two vertices of $V_L$ there is a third vertex of $V_L$ that is not connected by an edge to either. Similar for $V_R$. Since $V_M$ has at most one vertex (as mentioned above), it also must be contained in $V_1$ or $V_2$. Suppose now for contradiction that $w$ is a vertex of $V_1 \ast V_2$ that does not lie in $\Phi(v)$. If $V_L$ is nonempty, then $w$ has nonempty intersection with some vertex of $V_L$, and similarly for $V_M$ and $V_R$. Thus $w$ must be contained in the same $V_i$ as each nonempty element of the set $\{V_L,V_M,V_R\}$. Since $w$ was arbitrary, it follows that one of the $V_i$ is empty, as desired.
We now must show that each maximal join in $\C_A(S_g)$ lies in the image of $\Phi$. Let $X$ be a maximal join in $\C_A(S_g)$. Let $V_1 \ast \cdots \ast V_n$ be the complete join decomposition of $X$. For each $i$ let $R_i$ be the subsurface of $S_g$ determined by $V_i$, that is, $R_i$ is the smallest subsurface containing a representative of each vertex of $V_i$.
We claim that:
\begin{enumerate}
\item the $R_i$ are connected,
\item the $R_i$ are disjoint, and
\item at least one $R_i$ is not an annulus.
\end{enumerate}
The $R_i$ are connected because otherwise the corresponding $V_i$ could be written as the join of the complexes corresponding to the components. If the $R_i$ were not disjoint, then since each $R_i$ is filled by the vertices of the corresponding $V_i$ we could find vertices of $V_i$ and $V_j$ that had essential intersection. For the last statement, suppose for contradiction that each $R_i$ were an annulus. Consider the subgraph of $\C_A(S_g)$ spanned by the vertex corresponding to $R_1$ and the vertices represented in the complement of $R_1$. This graph is a join (it is a cone on the $R_1$-vertex) and it clearly contains $X$. It is strictly larger than $X$ because it contains $\MCG(S_g)$-translates of $R_2$ that do not lie in $X$. This is a contradiction and the claim is proven.
Suppose then that $R_1$ is non-annular. We claim that $R_1$ is nonseparating. Suppose to the contrary that $R_1$ has complementary regions $P_1,\dots,P_k$ with $k \geq 2$. Since $n \geq 2$, there is an $R_i$ contained in some $P_j$, say $P_1$. The complement of $P_1$ is a region $Q$ containing $R_1$ as a proper subsurface. Consider the subcomplex of $\C_A(S_g)$ spanned by the vertices represented in either $Q$ or $P_1$. This is the join of the subcomplexes corresponding to $Q$ and to $P_1$, and it is a nontrivial join since both are nonempty by assumption. Moreover this subcomplex properly contains $X$ since there are $\MCG(S_g)$-translates of vertices of $X$ that are not contained in any $R_i$. This is a contradiction, and we conclude that $R_1$ is indeed nonseparating.
Since $R_1$ is nonseparating, its boundary is a dividing set. Also, since $X$ is a nontrivial join, this dividing set represents a vertex $v$ of $\C_{\delta A} (S_g)$. Since $R_1$ is compatible with $v$, we must have that $\Phi(v)$ contains $X$. Since $X$ is maximal, it follows that $\Phi(v)$ is equal to $X$, and so $X$ is in the image of $\Phi$. This completes the proof of the first statement of the lemma.
Since we chose $R_1$ to be an arbitrary non-annular $R_i$, it follows that $v$ corresponds to the boundary of each non-annular $R_i$. In particular, there are at most two non-annular $R_i$ and if there are two then they are complementary in $S_g$. In other words, it makes sense to think of the non-annular $R_i$ as the left and right sides of $v$. The second statement of the lemma follows.
\end{proof}
\p{Compatible subcomplexes.} For us, the main consequence of Lemma~\ref{lemma:bijection} is that an automorphism of $\C_A(S_g)$ induces an automorphism of the 0-skeleton of $\C_{\delta A}(S_g)$: the image of a vertex of $\C_{\delta A}(S_g)$ is determined by the image of the corresponding maximal join in $\C_A(S_g)$. The next lemma tells us that this automorphism extends to the 1-skeleton of $\C_{\delta A}(S_g)$. We say that a subcomplex $V$ of a complex is \emph{compatible} with a subcomplex $W$ if $V$ can be written as $V_1 \ast V_2$ with $V_1$ nonempty and $V_1 \subseteq W$.
\begin{lemma}
\label{lemma:ca edges}
Suppose $\C_A(S_g)$ has no isolated vertices. Let $v$ and $w$ be vertices of $\C_{\delta A}(S_g)$. Then $v$ and $w$ are connected by an edge in $\C_{\delta A}(S_g)$ if and only if $\Phi(v)$ is compatible with $\Phi(w)$. In particular, compatibility of maximal joins in $\C_A(S_g)$ is a symmetric relation.
\end{lemma}
\begin{proof}
Denote the left/right decompositions of $\Phi(v)$ and $\Phi(w)$ by $V_L \ast V_M \ast V_R$ and $W_L \ast W_M \ast W_R$. By Lemma~\ref{lemma:bijection} these are both nontrivial join decompositions.
Suppose first that $v$ and $w$ are connected by an edge in $\C_{\delta A}(S_g)$. This means that there are representatives of $v$ and $w$ in $S_g$ that are nested dividing sets. We can choose the left and right subsurfaces of $S_g$ associated to $v$ and $w$ in a compatible way and so that $v$ lies to the left of $w$. It then follows from the definition of the left/right decomposition that $V_L \ast V_M$ is contained in $W_L$. Since the left/right decomposition is the same as the complete join decomposition (Lemma~\ref{lemma:bijection}) it must be that $V_L \ast V_M$ is nonempty. So $\Phi(v)$ is compatible with $\Phi(w)$, as desired.
For the other direction, suppose that $v$ and $w$ are not connected by an edge in $\C_{\delta A}(S_g)$. Any pair of representatives for $v$ and $w$ are dividing sets that are not nested. Thus each complementary subsurface for the $v$-dividing set has essential intersection with each complementary subsurface for $w$. It follows that no component of the left/right decomposition of $\Phi(v)$ can be contained in $\Phi(w)$. But by Lemma~\ref{lemma:bijection} the left/right decomposition is the same as the complete join decomposition and so $\Phi(v)$ is not compatible with $\Phi(w)$, as desired.
\end{proof}
\p{Finishing the proof.} We are almost ready to prove Theorem~\ref{main:complex}.
\begin{lemma}
\label{lemma:sr punchline}
Let $A$ be a subset of $\R(S_g)$ so that $\C_A(S_g)$ is connected and has no holes.
\begin{enumerate}
\item\label{bound} Let $R$ be a representative of a vertex of $\C_A(S_g)$; then $\partial R$ represents a simplex in $\C_{\delta A}(S_g)$.
\item The complex $\C_{\delta A}(S_g)$ is connected.
\end{enumerate}
\end{lemma}
\begin{proof}
We begin with the first statement. Clearly $\partial R$ is a disjoint union of dividing sets; the individual dividing sets are in bijection with the regions of $S_g$ complementary to $R$. If $R$ is not an annulus, then the statement follows from the no holes condition. If $R$ is an annulus, then the statement is automatic since there are representatives of the $R$-vertex on both sides of $\partial R$. This completes the proof of the first statement. (In the case where $R$ is a separating annulus, $\partial R$ is two parallel separating curves that both represent the same vertex of $\C_{\delta A}(S_g)$.)
We now proceed to the second statement. Let $v$ and $w$ be vertices of $\C_{\delta A}(S_g)$. We would like to find a path between $v$ and $w$ in $\C_{\delta A}(S_g)$. By the definition of $\delta A$ we can choose a vertices $a$ and $b$ of $\C_A(S_g)$ so that $v$ is compatible with $a$ and $w$ is compatible with $b$. Let $a=a_0,\dots,a_n=b$ be a path in $\C_A(S_g)$. By the first statement of the lemma the boundary of each $a_i$ represents a simplex $\delta a_i$ of $\C_{\delta A}(S_g)$. The vertices $v$ and $w$ are connected by edges to $\delta a_0$ and $\delta a_n$, respectively. Moreover, since each $a_i$ corresponds to a connected subsurface of $S_g$, each component of $\delta a_{i+1}$ lies in a single region of $S_g$ determined by any given component of $\delta a_i$; this is to say that each $\delta a_i \cup \delta a_{i+1}$ is also a simplex of $\C_{\delta A}(S_g)$. The second statement now follows, as the desired path between $v$ and $w$ lies in the sequence of simplices $v,\delta a_0,\dots, \delta a_n,w$.
\end{proof}
In the proof of Theorem~\ref{main:complex} we will use the partial order on vertices of $\C_A(S_g)$ defined above. Also, we say that a vertex $v$ of $\C_A(S_g)$ is \emph{1-sided} if for any representative of $v$ in $S_g$ there is exactly one complementary region that contains a representative of a vertex of the link of $v$ in $\C_A(S_g)$.
\begin{proof}[Proof of Theorem~\ref{main:complex}]
As in the statement of the theorem, we have a connected complex of regions $\C_A(S_g)$ with a small vertex and no holes or corks. By Proposition~\ref{prop:ca easy} we may further assume that $\C_A(S_g)$ has no vertices of type N. Let $\eta$ denote the natural map $\MCG(S_g) \to \Aut \C_A(S_g)$. We would like to show that $\eta$ is an isomorphism. By Lemma~\ref{inj}, $\eta$ is injective. It remains to show that $\eta$ is surjective.
By Lemmas~\ref{lemma:bijection} and~\ref{lemma:ca edges} there is a well-defined map
\[
\delta : \Aut \C_A(S_g) \to \Aut \C_{\delta A}(S_g);
\]
for any $\phi \in \Aut \C_A(S_g)$ the image under $\delta \phi$ of a dividing set is determined by the image under $\phi$ of the corresponding maximal join. The main step in the proof is to show that $\delta$ is injective.
Let $\phi$ be an automorphism of $\C_A(S_g)$ and suppose that $\delta\phi$ is the identity. We would like to show that $\phi$ is the identity. To this end, let $v$ be a vertex of $\C_A(S_g)$. We would like to show that $\phi(v)=v$. We proceed in three cases.
\begin{enumerate}
\item the case where $v$ is a 1-sided annular vertex,
\item the case where $v$ is a non-annular, $\preceq$-minimal, 1-sided vertex, and
\item the general case.
\end{enumerate}
\medskip \noindent {\em Case 1.} First assume that $v$ is a 1-sided annular vertex. A representative of $v$ divides $S_g$ into two regions; let $R$ be a region of smallest genus, and let $Q$ be the other region. We would like to show that there is a vertex of $\C_{\delta A}(S_g)$ represented by a multicurve in $Q$ that is not peripheral in $Q$ (in other words, this vertex should not be the dividing set that is parallel to $v$). Since $\C_A(S_g)$ is connected there must be some vertex $w$ of $\C_A(S_g)$ represented by a subsurface of $Q$. If $w$ is annular then it is not parallel to $v$ and so the vertex of $\C_{\delta A}(S_g)$ parallel to $w$ is the desired vertex. If $w$ is not annular and is not represented by the entire region $Q$ then we can apply Lemma~\ref{lemma:sr punchline}\eqref{bound} in order to find the desired vertex of $\C_{\delta A}(S_g)$. So the only remaining possibility is that $w$ corresponds to $Q$ and no other subsurface of $Q$ represents a vertex of $\C_A(S_g)$. But this pair would be a cork pair, a contradiction.
Since there is one vertex of $\C_{\delta A}(S_g)$ corresponding to a non-peripheral dividing set in $Q$, it follows that $Q$ is filled by representatives of vertices of $\C_{\delta A}(S_g)$. By assumption, all of these vertices are fixed by $\delta\phi$. By the definition of $\delta\phi$, the vertex $\phi(v)$ must be disjoint from all of these vertices of $\C_{\delta A}(S_g)$, and hence $\phi(v)$ must be represented by a subsurface of $R$. Since $v$ is 1-sided it must be that $\phi(v)=v$, as desired.
\medskip \noindent {\em Case 2.} Next assume that $v$ is non-annular, $\preceq$-minimal, and 1-sided. The argument is very similar to the previous case. Let $R$ be a representative of $v$. Since $\C_A(S_g)$ has no holes and $v$ is 1-sided, the complement of $R$ in $S_g$ is connected; denote this region by $Q$. We again would like to show that there is a vertex of $\C_{\delta A}(S_g)$ represented by a dividing set in $Q$ that is not peripheral in $Q$. Let $w$ be a vertex of $\C_A(S_g)$ in the link of $v$, so $w$ is represented by a subsurface of $Q$. If $w$ corresponds to a separating annulus that is not peripheral in $Q$ then the vertex of $\C_{\delta A}(S_g)$ parallel to $w$ is the desired vertex. If $w$ corresponds to an annulus parallel to the boundary of $Q$, then since $\C_A(S_g)$ has no corks there must be a vertex of $\C_A(S_g)$ represented by a subsurface of $R$. And since $\C_A(S_g)$ has no holes there must be a further vertex of $\C_A(S_g)$ that is represented by a subsurface of $R$ and that is smaller than $v$ in the partial order, a contradiction. If $w$ is a non-annular vertex corresponding to a proper subsurface of $Q$ then again we apply Lemma~\ref{lemma:sr punchline}\eqref{bound} to find the desired vertex of $\C_{\delta A}(S_g)$. So the only remaining case to consider is where $w$ corresponds to $Q$ and no other vertex of $\C_A(S_g)$ corresponds to a subsurface of $Q$. By the minimality of $v$ it follows that $Q$ and $R$ are homeomorphic and hence that $v$ and $w$ span an isolated edge in $\C_A(S_g)$, a contradiction.
Again, we must have that $Q$ is filled by vertices of $\C_{\delta A}(S_g)$ and so again $\phi(v)$ must correspond to a subsurface of $R$. By the minimality of $v$ and the assumption that $\C_A(S_g)$ has no holes or corks, it must be that $\phi(v)=v$.
\medskip \noindent {\em Case 3.} We now attack the case of an arbitrary vertex $v$ of $\C_A(S_g)$. Let $Q$ be a region of $S_g$ that is complementary to some representative of $v$. We would like to analyze the set of vertices of the link of $v$ represented in $Q$. Since we already dealt with the case of 1-sided annular vertices and since $\C_A(S_g)$ has no holes we may assume that this set of vertices is nonempty.
We put a partial order on this set of vertices of the link of $v$ represented in $Q$. For any such vertex $u$, we write $\hat u$ for the smallest nonseparating region of $Q$ that contains a representative of $u$. Then we say $u \leq w$ if $\hat u \subseteq \hat w$. Since there are finitely many vertices of $\C_A(S_g)$ represented in $Q$ up to homeomorphism of $Q$ the set of $\leq$-minimal elements is nonempty. We would like to show that each $\leq$-minimal element is either a 1-sided annular vertex or a non-annular, $\preceq$-minimal, 1-sided vertex.
Let $u$ be a vertex of the link of $v$ that is represented in $Q$. Assume that $u$ is neither a 1-sided annular vertex or a non-annular, $\preceq$-minimal, 1-sided vertex. We must show that $u$ is not $\leq$-minimal.
Suppose first that $u$ is non-annular and 2-sided. Let $P$ be a complementary region to $u$ that does not contain the boundary of $Q$ (there is such a region since the complement of $Q$ is connected). Since $\C_A(S_g)$ has no holes, there exists a vertex $w$ of $\C_A(S_g)$ supported in $P$. We have $w < u$, as desired.
Next suppose that $u$ is non-annular, 1-sided vertex that is not $\preceq$-minimal. Since $u$ is non-annular and $\C_A(S_g)$ has no holes, $u$ is represented by a nonseparating subsurface $P$ of $S_g$. Since $u$ is not $\preceq$-minimal and is 1-sided and since $\C_A(S_g)$ has no holes, it again follows that there is a vertex $w$ with $w < u$.
Finally suppose that $u$ is annular and 2-sided. Let $P$ denote the region of $Q$ that is determined by $u$ and does not contain the boundary of $Q$. Since $\C_A(S_g)$ has no corks, there is a vertex $w$ of $\C_A(S_g)$ represented by a proper subsurface of $P$. If we do not have $w < u$ then $w$ is non-annular and has one boundary component parallel to $u$. Since $\C_A(S_g)$ has no holes, there is a vertex $x$ of $\C_A(S_g)$ with $x < u$ and $u$ is again not $\leq$-minimal. We have succeeded in characterizing the $\leq$-minimal elements.
We will now show that one of the following conditions holds:
\begin{enumerate}
\item $Q$ is filled by the 1-sided annular vertices and non-annular, $\preceq$-minimal, 1-sided vertices of $\C_A(S_g)$ represented in $Q$, or
\item the boundary of $Q$ is parallel to a 1-sided annular vertex of $\C_A(S_g)$ and no non-peripheral subsurfaces of $Q$ represent vertices of $\C_A(S_g)$.
\end{enumerate}
If there is a vertex of $\C_A(S_g)$ that is $\leq$-minimal and is non-peripheral in $Q$ then by our classification of $\leq$-minimal elements, we are in the first case. If all $\leq$-minimal elements are peripheral in $Q$, then since there are no vertices of type N it must be that the boundary of $Q$ is connected. Then since there are no holes or corks we must be in the second case.
We are finally ready to complete the proof that $\phi(v)=v$, and hence that $\delta$ is injective. From our analysis of the vertices of $\C_A(S_g)$ represented in the complementary regions for $v$, we conclude that $v$ is completely determined as follows: it is the unique vertex of $\C_A(S_g)$ represented by a region that is complementary to the supports of the 1-sided annular vertices and non-annular, $\preceq$-minimal, 1-sided vertices of $\C_A(S_g)$ that lie in the link of $v$. Indeed, no other complementary region contains a representative of a vertex of $\C_A(S_g)$ and---since $\phi$ is invertible---$\phi(v)$ cannot correspond to a proper subregion of $v$. Since we already showed that 1-sided annular vertices and non-annular, $\preceq$-minimal, 1-sided vertices are fixed by $\phi$, it follows that $\phi(v)=v$. We have thus proven $\delta$ is injective.
\medskip
Having shown that $\delta$ is injective, we now proceed to complete the proof of the theorem. Since $\bar g(A) < g/3$, it follows that $\check g(\delta A) < g/3$. Also it follows from Lemma~\ref{lemma:sr punchline} and the assumption that $\C_A(S_g)$ is connected that $\C_{\delta A}(S_g)$ is connected. As we already mentioned, Lemma~\ref{upper set} implies that $\delta A$ is an upper set. Thus by Theorem~\ref{theorem:multi k} the natural map $\MCG(S_g) \to \Aut \C_{\delta A}(S_g)$ is an isomorphism. We thus have the following diagram:
\[
\xymatrix{
\Aut \C_A(S_g)\ \ar@{^{(}->}[rr]^\delta & & \ \Aut \C_{\delta A}(S_g) \\
& \MCG(S_g) \ar[ur]_{\cong} \ar@{_{(}->}[ul]^{\eta} &
}
\]
It follows from the definition of $\delta$ that the diagram is commutative. It follows then from the injectivity of $\delta$ and $\eta$ that both are isomorphisms. This completes the proof of the theorem.
\end{proof}
\section{Normal subgroups}
\label{section:normal}
In this section we prove Theorem~\ref{main:normal}, which describes the automorphism group and the abstract commensurator group of a normal subgroup of $\Mod(S_g)$ or $\MCG(S_g)$ that has a pure element with a small component. We will define a complex of regions where the vertices correspond to the supports of certain subgroups of $G$, called basic subgroups. Theorem~\ref{main:normal} will then be derived from Theorem~\ref{main:complex}, our theorem about automorphisms of complexes of regions.
It might seem more intuitive to relate regions of $S_g$ to elements as opposed to subgroups. One immediate advantage of using subgroups is that typically the centralizer of a subgroup has its support disjoint from that of the subgroup; this is not true for individual elements. We were inspired to take the subgroup approach after reading a paper by Hensel \cite{Hensel}, although this idea already appears in the work of Ivanov, cf. \cite[Section 7.20]{ivanovshiny}.
\subsection{Nielsen--Thurston normal forms and pure elements}
Before getting to the proof of Theorem~\ref{main:normal}, we must recall some ideas related to the Nielsen--Thurston classification for elements of $\Mod(S_g)$. For basics on this theory, including the definition of a pseudo-Anosov mapping class, see the book by Farb and the second author of this paper \cite[Chapter 13]{primer}.
For a compact surface $R$ with marked points, let $\PMod(R)$ denote the subgroup of $\Mod(R)$ consisting of elements that induce the trivial permutation of the marked points (for us the mapping class group of a surface with boundary is the group of homotopy classes of orientation-preserving homeomorphisms that restrict to the identity on the boundary, so the components of the boundary are automatically not permuted).
A \emph{partial pseudo-Anosov} element of $\Mod(S_g)$ is the image of a pseudo-Anosov element of $\PMod(R)$ under the map $\PMod(R) \to \Mod(S_g)$ induced by the inclusion of a region $R$ of $S_g$; here $R$ has no marked points and we allow $R=S_g$. By the work of Thurston and Birman--Lubotzky--McCarthy \cite{blm} the region $R$---the \emph{support}---is canonically defined up to isotopy. Similarly, the support of a power of a Dehn twist is well defined.
Following Ivanov \cite{ivanovshiny}, a mapping class $f \in \MCG(S_g)$ is \emph{pure} if it is equal to a product $f_1 \cdots f_k$ where
\begin{enumerate}
\item each $f_i$ is a partial pseudo-Anosov element or a power of a Dehn twist,
\item for $i \neq j$ the supports of $f_i$ and $f_j$ have disjoint representatives, and
\item for $i \neq j$ the support of $f_i$ is not homotopic into the support of $f_j$.
\end{enumerate}
The $f_i$ are called the \emph{Nielsen--Thurston components}, or simply \emph{components}, of $f$. Each component can be characterized as a \emph{pseudo-Anosov component} or a \emph{Dehn twist component}. Note that the third condition is vacuous if $f_i$ is a pseudo-Anosov component. Also note that pure elements of $\MCG(S_g)$ lie in $\Mod(S_g)$.
Under these definitions, the Nielsen--Thurston components of a pure mapping class are not canonical when the supports of pseudo-Anosov components $f_i$ and $f_j$ have a pair of parallel boundary components. Indeed, if the homotopy class of these boundary components is $c$ then we may replace $f_i$ and $f_j$ with $f_iT_c$ and $f_jT_c^{-1}$. Still, the support of a pure element is a well-defined subsurface, invariant under taking nontrivial powers.
A subgroup of $\Mod(S_g)$ is \emph{pure} if each element is pure. Ivanov proved that there is a subgroup of finite index in $\Mod(S_g)$ that is pure \cite[Corollary 1.8]{ivanovshiny}.
The support of a pure subgroup of $\Mod(S_g)$ is again a well-defined subsurface of $S_g$ (sometimes called the active subsurface; cf. \cite{Mosher}). The support of a pure subgroup is invariant under passage to finite-index subgroups.
If $R$ is a component of the support of the pure subgroup $H$, then there is a well-defined \emph{reduction homomorphism}
\[
H \to \PMod(R^\circ),
\]
where $R^\circ$ is the surface obtained from $R$ by collapsing each component of the boundary to a marked point. By definition, the image is an irreducible subgroup of $\PMod(R^\circ)$. (The `P' in $\PMod$ also stands for pure, but in general $\PMod(R)$ is not pure in the sense of Ivanov.)
If we have two partial pseudo-Anosov elements with equal support $R$, we say that the elements are \emph{independent} if their corresponding pseudo-Anosov foliations are distinct from each other. If a partial pseudo-Anosov mapping class $f$ has support $R$ then its image under the reduction homomorphism $\langle f \rangle \to \Mod(R^\circ)$ is pseudo-Anosov, and so we can say that two partial pseudo-Anosov elements with support $R$ are independent if their reductions are independent pseudo-Anosov elements.
We will repeatedly use the following fact, usually without mention.
\begin{fact}
\label{commute fact}
Two elements of a pure subgroup of $\Mod(S_g)$ commute if and only if
\begin{enumerate}
\item the supports of their components are pairwise disjoint or equal and
\item in the case where two pseudo-Anosov components have equal support $R$ the components are dependent.
\end{enumerate}
In particular if two pure elements commute then all of their nontrivial powers commute and if two pure elements do not commute then all of their nontrivial powers fail to commute.
\end{fact}
Fact~\ref{commute fact} follows from the fact that commuting elements have compatible canonical reduction systems \cite[Lemma 3.1(1)]{blm} and the fact that in a pure subgroup the centralizer of a pseudo-Anosov element is cyclic \cite[Lemma 5.10]{ivanovshiny}.
\begin{lemma}
\label{irred}
Let $H$ be a non-abelian pure subgroup of $\Mod(S_g)$. Then there is a component $R$ of the support of $H$ so that the reduction homomorphism $H \to \PMod(R^\circ)$
has non-abelian image. For any such $R$, the centralizer of $H$ is supported in the complement of $R$.
\end{lemma}
\begin{proof}
For the first statement, let $f$ and $h$ be two elements of $H$ that do not commute. By Fact~\ref{commute fact} they have components whose supports $R_1$ and $R_2$ have essential intersection and if $R_1=R_2$ then these components are independent partial pseudo-Anosov elements. The regions $R_1$ and $R_2$ must lie in the same component $R$ of the support of $H$. The images of $R_1$ and $R_2$ in $R^\circ$ still have essential intersection and if $R_1=R_2$ then the images of the chosen components of $f$ and $h$ are still independent. By Fact~\ref{commute fact} the images of $f$ and $h$ in $\PMod(R^\circ)$ do not commute.
We now prove the second statement. Let $R$ be as in the statement. By the definition of a component of the support of $H$, the image of $H$ under the reduction homomorphism is irreducible; since the image is also non-abelian it contains a pseudo-Anosov element \cite[Theorem 5.9]{ivanovshiny}. Since (the image of) $H$ is not abelian, it follows that its image in $\Mod(R^\circ)$ contains two independent pseudo-Anosov elements (conjugate the first one by any element that does not commute with it). The preimages in $H$ of these pseudo-Anosov elements of $\Mod(R^\circ)$ are two elements of $H$ that have independent partial pseudo-Anosov components with support $R$. The second statement now follows from Fact~\ref{commute fact}.
\end{proof}
\subsection{The commutator trick}
Fix $g$ and let $N$ be some fixed pure, normal subgroup of $\Mod(S_g)$ that contains a pure element with a small component. Also let $G$ be a subgroup of $N$ of finite index.
We would like to define a complex of regions where the vertices correspond to the supports of certain subgroups of $N$. One of the basic difficulties we need to overcome is that a typical element (or subgroup) of $N$ has disconnected support, but in a complex of regions the vertices must correspond to connected subsurfaces. Further, if an element with multiple Nielsen--Thurston components lies in $N$ then the individual components may or may not lie in $N$. The next lemma deals with this problem. The key point is that if $N$ contains an element $f$ so that one component of the support of $f$ is a non-annular region $R$, then there is a different element $f'$ of $N$---not equal to a component of $f$---whose support is $R$. The element $f'$ is obtained as a commutator of $f$ with an appropriately chosen element of $\Mod(S_g)$.
\begin{lemma}
\label{comm trick}
Let $g \geq 0$, let $N$ be a pure, normal subgroup of $\Mod(S_g)$ and let $G$ be a finite-index subgroup of $N$.
Let $f$ be an element of $G$ and let $R$ be a region of $S_g$ so that some component of $f$ has support contained as a non-peripheral subsurface of $R$ and all other components of $f$ have support that is either contained in or is disjoint from $R$. Let $J$ be the subgroup of $G$ consisting of all elements supported in $R$. Then
\begin{enumerate}
\item\label{comm trick 1} $J$ is not abelian,
\item\label{comm trick 2} $J$ contains an element with support $R$, and
\item\label{comm trick 3} the centralizer $C_G(J)$ is supported in the complement of $R$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first step is to show that $J$ contains a nontrivial element $j$ whose support is a non-peripheral subsurface of $R$. To this end, consider the reduction homomorphism $G \to \PMod(R^\circ)$. Since the support of $f$ is not peripheral in $R$, the image of $f$ is nontrivial. Also, since $f$ is pure it follows that the image of $f$ is not central in $\PMod(R^\circ)$ (cf. \cite[page 77]{primer}). Thus there is an $\bar h \in \PMod(R^\circ)$ that does not commute with the image of $f$. Let $h$ be any element of the preimage of $\bar h$ in $\PMod(R)$ (the reduction homomorphism $\PMod(R) \to \PMod(R^\circ)$ is surjective). As above we may identify $h$ as an element of $\Mod(S_g)$. Since $N$ is normal in $\MCG(S_g)$, the commutator $[f,h]$ lies in $N$. By construction $[f,h]$ is supported in $R$. Let $j$ be a nontrivial power of $[f,h]$ that lies in $G$. The kernel of the map $\PMod(R) \to \PMod(R^\circ)$ is precisely the set of elements with peripheral support, so $j$ is the desired element.
The second step is to show that $J$ contains two independent partial pseudo-Anosov elements with support $R$. All three statements of the lemma follow from this and Fact~\ref{commute fact}. Let $j$ be the element found in the first step. Any conjugate of $j$ by an element of $\Mod(S_g)$ has a power in $G$. It follows that the image of $J$ under the reduction homomorphism $\PMod(R) \to \PMod(R^\circ)$ is irreducible and not abelian (apply Fact~\ref{commute fact}). Any such subgroup contains two independent pseudo-Anosov elements (irreducible subgroups contain pseudo-Anosov elements \cite[Theorem 5.9]{ivanovshiny} and any nontrivial conjugate of a pseudo-Anosov element is an independent pseudo-Anosov element and has a power in the image of $J$). Any preimages of these elements in $J$ are the desired elements and the proof is complete.
\end{proof}
\subsection{Basic subgroups}
As in the previous section let $N$ be a pure, normal subgroup of $\Mod(S_g)$ that contains a pure element with a small component, and let $G$ be a finite-index subgroup of $N$. We now define basic subgroups of $N$. We will show that these subgroups have connected supports, and so they can used to build a complex of regions for $N$.
We define a strict partial order on subgroups of $G$ by the following rule:
\[
H \prec H' \ \ \text{ if } \ \ C_G(H') \subsetneq C_G(H).
\]
(by a strict partial order we mean a binary relation that is irreflexive---meaning that no element is related to itself---and transitive). A subgroup of $G$ is \emph{basic} if among non-abelian subgroups of $G$ it is minimal with respect to this strict partial order; specifically $B$ is basic if there is no non-abelian subgroup $B'$ of $G$ with $B' \preceq B$.
Consider for example the case where $N$ is the Torelli group $\I(S_g)$. Let $R$ be a sphere with four separating boundary components in $S_g$. Then the subgroup $B$ of $N$ consisting of elements that are supported in $R$ is a basic subgroup. Indeed, it is not abelian since it contains Dehn twists about curves that intersect and it is minimal because all proper subsurfaces of $R$ have abelian mapping class group.
\begin{lemma}
\label{lemma:basic}
Let $g \geq 0$, let $N$ be a pure, normal subgroup of $\Mod(S_g)$ that contains a pure element with a small component, and let $G$ be a finite-index subgroup of $N$.
\begin{enumerate}
\item\label{basic region} The support of a basic subgroup of $G$ is a non-annular region of $S_g$.
\item\label{basic passage} If $B$ is a basic subgroup of $N$ then $B \cap G$ is a basic subgroup of $G$; similarly, any basic subgroup of $G$ is also a basic subgroup of $N$.
\item\label{basic small} $N$ contains a basic subgroup with small support.
\item\label{basic action} $\MCG(S_g)$ acts on the set of supports of basic subgroups of $G$.
\end{enumerate}
\end{lemma}
\begin{proof}
We begin with the first statement. Let $H$ be a basic subgroup of $G$. Since $H$ is not abelian, the support of $H$ is clearly not empty and not an annulus. It is also easy to see that the support of $H$ is proper. Indeed if the support of $H$ were $S_g$, then by Lemma~\ref{irred} the centralizer $C_G(H)$ would be trivial. On the other hand, since $N$---hence $G$---contains an element $f$ with a small component, we can apply Lemma~\ref{comm trick} to $f$ and a small region $Q$ in order to produce a non-abelian subgroup $J$ with nontrivial centralizer $C_G(J)$ (since $Q$ is small there is an $h \in \Mod(S_g)$ so that $h J h^{-1} \cap G$ lies in $C_G(J)$). Any such $J$ would be strictly smaller than $H$ in the strict partial order.
Suppose for contradiction that the support of $H$ is not connected. By Lemma~\ref{irred} there is a component $R$ of the support of $H$ so that the reduction homomorphism $H \to \PMod(R^\circ)$ has non-abelian image. In particular, there must be an element $h$ of $H$ with a Nielsen-Thurston component whose support is non-peripheral in $R$. By Lemma~\ref{comm trick}\eqref{comm trick 1}, the subgroup
\[ J = \{ h \in G \mid \Supp(h) \subseteq R \}. \]
of $G$ is not abelian.
We will show that $J \prec H$. By Lemma~\ref{irred} the centralizer $C_G(H)$ of $H$ is supported in the complement of $R$. It follows that $C_G(H) \subseteq C_G(J)$. We must now produce an element of $C_G(J) \setminus C_G(H)$. Let $h$ be an element of $H$ whose support is not contained in $R$. Let $Q$ be a non-annular region of $S_g$ that is disjoint from $R$, that contains a component of the support of $h$ as a non-peripheral subsurface, and is disjoint from all other components of the support of $h$. Applying Lemma~\ref{comm trick}\eqref{comm trick 1} to $h$ and $Q$ we find the desired element of $C_G(J) \setminus C_G(H)$. Thus $J \prec H$, a contradiction. This completes the proof of the first statement.
The second statement has two parts. For the first part, let $B$ be a basic subgroup of $N$. We would like to show that $B \cap G$ is a basic subgroup of $G$. It follows from Fact~\ref{commute fact} that $B \cap G$ is non-abelian, and it follows from the first statement and Lemma~\ref{irred} that the centralizer of $B \cap G$ in $G$ consists of all elements whose support lies outside the support of $B \cap G$. Thus, if $B \cap G$ were not basic in $G$ there would be a non-abelian subgroup $H$ of $G$ whose centralizer contains the centralizer of $B \cap G$ and also contains at least one extra element. The support of this extra element would have to intersect the support of $B \cap G$ and hence by the first statement and Lemma~\ref{irred} the support of $H$ would be a proper subsurface of the support of $B \cap G$. Since $B \cap G$ has finite index in $B$ the latter is equal to the support of $B$ and we can conclude that $H$ is smaller than $B$ in the strict partial order on subgroups of $N$, contradicting the assumption that $B$ was basic in $N$.
For the second part of the second statement, let $B$ be a basic subgroup of $G$. We would like to show that $B$ is basic in $N$. Suppose $B'$ is a non-abelian subgroup of $N$ that is smaller than $B$ in the strict partial order on subgroups of $N$. Consider the subgroup $B' \cap G$ of $G$. Again by Fact~\ref{commute fact} the subgroup $B' \cap G$ is not abelian. Since $B' \prec B$ (in $N$) the centralizer of $B$ in $N$ is strictly contained in the centralizer of $B'$ in $N$. Thus the centralizer of $B$ in $G$ is contained in the centralizer of $B'$ in $G$. Since the containment is strict there is an element $f$ of $N$ that lies in the centralizer in $N$ of $B'$ but not of $B$. Some power of $f$ lies in $G$. It then follows from Fact~\ref{commute fact} that this power of $f$ lies in the centralizer of $B'$ in $G$ but not the centralizer of $B$ in $G$, contradicting the assumption that $B$ is basic in $G$.
We now treat the third statement. By assumption, $N$---hence $G$---contains a nontrivial pure element $f$ with a small component $f_1$.
Let $R_1$ be a fitting region for $f$ corresponding to $f_1$, as per the definition of $\hat g(f)$ in Section~\ref{sec:complex}; the component $f_1$ and the region $R_1$ satisfy the hypotheses of Lemma~\ref{comm trick}. There is a region $Q$ of $S_g$ that has genus less than $g/3$, has connected boundary, and contains $R_1$. We define
\[
J_{R_1} = \{ h \in G \mid \Supp h \subseteq R_1 \}.
\]
It follows from parts \eqref{comm trick 1} and \eqref{comm trick 3} of Lemma~\ref{comm trick} that $J_{R_1}$ is not abelian and that $C_G(J_{R_1})$ is the set of elements of $G$ with support in the complement of $R_1$. We would like to show that $J_{R_1}$ contains a basic subgroup. If $J_{R_1}$ itself is not minimal, then it contains a non-abelian subgroup $J_{R_1}'$ with $J_{R_1}' \prec J_{R_1}$. It follows that $C_G(J_{R_1}')$ contains an element whose support intersects $R_1$. By Lemma~\ref{irred} the support of $J_{R_1}'$ has a component that is a proper subsurface $R_2$ of $R_1$. If $J_{R_2}$ is the subgroup of $G$ consisting of all elements with support in $R_2$ we still have $J_{R_2} \prec J_{R_1}$. If $J_{R_2}$ is not minimal we can repeat the process. Since the Euler characteristics of the $R_i$ are strictly increasing and negative, the process must eventually terminate at a basic subgroup $H$ with support contained in $Q$.
For the fourth statement, suppose that $R$ is the support of a basic subgroup $B$ in $G$ and let $f \in \MCG(S_g)$. We would like to show that $f(R)$ is the support of a basic subgroup of $G$. Since $\Mod(S_g)$ acts transitively on each $\MCG(S_g)$-orbit of regions in $S_g$, we may assume without loss of generality that $f$ lies in $\Mod(S_g)$.
Let $J_R$ denote the subgroup of $G$ consisting of all elements with support in $R$. Let $B' = (f J_R f^{-1}) \cap G$. As $N$ is normal in $\Mod(S_g)$ and $G$ has finite index in $N$ the subgroup $B'$ has finite index in $f J_R f^{-1}$. Since the support of a pure subgroup of $\Mod(S_g)$ is invariant under taking finite index subgroups, the support of $B'$ is $f(R)$. We would like to show that $B'$ is basic. It follows from Fact~\ref{commute fact} that $B'$ is not abelian. Finally we must show that $B'$ is minimal. By part \eqref{comm trick 3} of Lemma~\ref{comm trick} the support of $J_R$ is $R$ (not a proper subsurface of $R$) and so the support of $f J_R f^{-1}$, hence $B'$, is $f(R)$. Since $B'$ is not abelian it then follows from Fact~\ref{commute fact} that $C_G(B')$ consists of exactly the elements of $G$ with support outside $f(R)$. If there were a subgroup $B''$ of $G$ with $B'' \prec B'$ then there would be an element of $C_G(B'')$ whose support has essential intersection with $f(R)$ and so the support of $B''$ would be contained in $f(R)$. But then---again using Fact~\ref{commute fact}---the subgroup $(f^{-1}B''f) \cap G$ would be strictly smaller than $B$ in the strict partial order, contradicting the minimality of $B$.
\end{proof}
\subsection{The complex}
Again, let $N$ be some fixed pure, normal subgroup of $\Mod(S_g)$ that contains a pure element with a small component. We are now ready to construct the desired complex of regions $\C_N(S_g)$ for $N$. By statements~\eqref{basic region} and~\eqref{basic action} of Lemma~\ref{lemma:basic} there is a complex of regions $\C_N^\sharp(S_g)$ whose set of vertices is in bijection with the set of supports of basic subgroups of $N$. One point to note here is that there are many basic subgroups of $N$ corresponding to a given vertex of $\C_N^\sharp(S_g)$.
By Lemma~\ref{lemma:basic}\eqref{basic region} the complex of regions $\C_N^\sharp(S_g)$ has no annular vertices and so it has no corks. Also by Lemma~\ref{lemma:basic}\eqref{basic small} it has a small vertex. On the other hand, $\C_N^\sharp(S_g)$ does not necessarily satisfy the other hypotheses of Theorem~\ref{main:complex}: it may have holes and it may be disconnected.
To illustrate the first point, we again consider the above example where $N=\I(S_g)$. Again, let $R$ be a four-holed sphere where each component of the boundary is separating in $S_g$. Also assume that one of the complementary regions $Q$ is a handle. We already explained why there is a basic subgroup with support $R$, and so $R$ represents a vertex of $\C_N^\sharp(S_g)$. But in this particular case $R$ represents a vertex of $\C_N^\sharp(S_g)$ with a hole. Indeed, the subgroup of $\I(S_g)$ consisting of elements supported in $Q$ is cyclic (it is generated by the Dehn twist about the boundary of $Q$) and so there are no basic subgroups of $N$ supported in $Q$. Thus $Q$ represents a hole for the $R$-vertex.
We can also imagine an example where $\C_N^\sharp(S_g)$ is not connected. Let $N$ be the normal closure in $\MCG(S_g)$ of two elements $f$ and $h$ of $\Mod(S_g)$, where $f$ is a partial pseudo-Anosov element supported on a handle $Q$ and $h$ is a partial pseudo-Anosov element supported on a subsurface $R$ of genus zero with $g+1$ boundary components. Using Lemma~\ref{comm trick} we can find non-abelian subgroups of $N$ with supports $Q$ and $R$. For the typical choices of $f$ and $h$ we would expect these subgroups to be basic, and so $Q$ and $R$ will represent vertices $v$ and $w$ of $\C_N^\sharp(S_g)$. It is also possible that all vertices of $\C_N^\sharp(S_g)$ lie in the orbit of $v$ and $w$. Since no vertex in the orbit of $v$ is connected to $w$ the complex $\C_N^\sharp(S_g)$ is disconnected in this case.
In light of these issues, we now set about modifying $\C_N^\sharp(S_g)$ so that it satisfies all of the hypotheses of Theorem~\ref{main:complex}. First, let $\C_N^\flat(S_g)$ be the filling of $\C_N^\sharp(S_g)$ (cf. Section~\ref{sec:sr}). By Lemma~\ref{filling}, the complex $\C_N^\flat(S_g)$ has no holes and by Lemma~\ref{small filling} it has a small vertex. Since the filling of a non-annular vertex is non-annular, and since $\C_N^\sharp(S_g)$ has no annular vertices, $\C_N^\flat(S_g)$ has no corks.
In summary, the complex $\C_N^\flat(S_g)$ has no holes or corks and it has a small vertex, but it might be disconnected. We have the following fact, which is a straightforward application of the Putman trick from the proof of Lemma~\ref{lemma:sp graph connected}.
\begin{lemma}
\label{small component}
Let $\C_A(S_g)$ be a complex of regions. The small vertices of $\C_A(S_g)$ all lie in the same connected component of $\C_A(S_g)$.
\end{lemma}
Finally, we may define $\C_N(S_g)$ as the connected component of $\C_N^\flat(S_g)$ containing the small vertices. Clearly $\C_N(S_g)$ is connected and has a small vertex. Also since $\C_N^\flat(S_g)$ has no annular vertices, $\C_N(S_g)$ has no corks. To check that $\C_N(S_g)$ satisfies the hypotheses of Theorem~\ref{main:complex}, it remains to check that $\C_N(S_g)$ has no holes.
Suppose that $R$ is a region of $S_g$ that represents a vertex of $\C_N(S_g)$. Then $R$ also represents a vertex of $\C_N^\flat(S_g)$. Since the latter has no holes, each complementary region of $R$ supports a vertex of $\C_N^\flat(S_g)$. Each of these vertices clearly lies in the connected component of $\C_N^\flat(S_g)$ containing the $R$-vertex and so they all correspond to vertices of $\C_N(S_g)$. So $\C_N(S_g)$ has no holes. We thus have the following consequence of Theorem~\ref{main:complex}.
\begin{prop}
\label{prop:defining cn}
Let $N$ be a pure, normal subgroup of $\Mod(S_g)$ that contains a pure element with a small component. Then the natural map
\[
\MCG(S_g) \to \Aut \C_N(S_g)
\]
is an isomorphism.
\end{prop}
\subsection{Action of the commensurator groups on the complex}
For the complex of regions $\C_N(S_g)$ to be useful, we would like to know that an automorphism---or an abstract commensurator---of $N$ gives rise to an automorphism of $\C_N(S_g)$. In order to obtain a well-defined action, we must deal with the issue that there are many basic subgroups of $N$ giving rise to the same vertex of $\C_N(S_g)$.
In what follows, we will denote by $v_B$ the vertex of $\C_N(S_g)$ arising from the basic subgroup $B$ of $N$. As mentioned, we may have two basic subgroups $B$ and $B'$ with $v_B = v_{B'}$. Also, for $G$ a finite-index subgroup of $N$ and $B$ a basic subgroup of $G$ we define the \emph{basic centralizer} of $B$ in $G$ to be the subgroup of $G$ generated by the basic subgroups of $G$ in the centralizer of $B$; we denote this group by $BC_G(H)$.
\begin{lemma}
\label{lemma:bc}
Let $N$ be a pure, normal subgroup of $\Mod(S_g)$ that contains an element with a small component. Let $G$ be a subgroup of $N$ of finite index. Let $h \in \MCG(S_g)$ and let $h_\star$ denote its image under the natural map $\MCG(S_g) \to \Aut \C_N(S_g)$. Let $B$ and $B'$ be two basic subgroups of $G$. Then
\begin{enumerate}
\item\label{bc v} $v_B = v_{B'}$ if and only if $BC_G(B) = BC_G(B')$,
\item\label{bc wd} $v_B = v_{B'}$ if $B'$ is a finite index subgroup of $B$,
\item\label{bc e} $v_B$ is connected by an edge to $v_{B'}$ if and only if $B' \leqslant BC_G(B)$, and
\item\label{bc act} $h_\star(v_B) = v_{hBh^{-1}}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $R$ denote the support of $B$. By Lemma~\ref{lemma:basic}\eqref{basic region}, the subsurface $R$ is a non-annular region of $S_g$. Denote by $P_1,\dots,P_m$ the complementary regions that do contain the supports of other basic subgroups of $G$ and denote by $Q_1,\dots,Q_n$ the complementary regions that do not. By the definition of $\C_N(S_g)$, the vertex $v_B$ is represented by the union of $R$ with the $Q_i$; call this region $R'$. By Lemma~\ref{irred}, we have that $BC_G(B)$ is the subgroup of $G$ generated by the basic subgroups of $G$ with support in the complement of $R'$. All statements of the lemma follow (for the second statement apply Fact~\ref{commute fact}).
\end{proof}
\begin{prop}
\label{prop:comm to aut}
Let $N$ be a pure, normal subgroup of $\Mod(S_g)$ that contains an element with a small component. There is a map
\[
\Comm N \to \Aut \C_N(S_g)
\]
defined as follows: if $\alpha : G_1 \to G_2$ is an isomorphism between finite-index subgroups of $N$ and $[\alpha]_\star$ is the image in $\Aut \C_N(S_g)$ of the equivalence class of $\alpha$, then for any basic subgroup $B$ of $N$ we have
\[
[\alpha]_\star (v_B) = v_{\alpha(B \cap G_1)}.
\]
\end{prop}
\begin{proof}
Our first objective is to show that the formula given in the statement of the proposition makes sense. Let $\alpha : G_1 \to G_2$ be an isomorphism between finite-index subgroups of $N$ and let $B$ be a basic subgroup of $N$. By Lemma~\ref{lemma:basic}\eqref{basic passage}, the group $B \cap G_1$ is a basic subgroup of $G_1$. Since $\alpha$ is an isomorphism from $G_1$ to $G_2$ it follows that $\alpha(B \cap G_1)$ is a basic subgroup of $G_2$. Again by Lemma~\ref{lemma:basic}\eqref{basic passage} the group $\alpha(B \cap G_1)$ is a basic subgroup of $N$. Thus $v_{\alpha(B \cap G_1)}$ is indeed a vertex of $\C_N(S_g)$.
Next we must show that the formula in the statement gives a well-defined action of $\Comm N$ on the set of vertices of $\C_N(S_g)$. There are two issues, namely, that an element of $\Comm N$ has many representatives and also that there are many basic subgroups of $N$ giving rise to the same vertex of $\C_N(S_g)$.
Let $\alpha : G_1 \to G_2$ be an isomorphism between finite-index subgroups of $N$ and let $B$ be a basic subgroup of $N$. Let $\alpha' : G_1' \to G_2'$ be another isomorphism of finite-index subgroups of $N$ that represents the same element of $\Comm N$ as $\alpha$. We must show that
\[
v_{\alpha(B \cap G_1)} = v_{\alpha'(B \cap G_1')}.
\]
Since $\alpha$ and $\alpha'$ agree on a finite-index subgroup of $N$ it suffices to treat the case where $G_1'$ is a finite-index subgroup of $G_1$ and $\alpha|(B \cap G_1') = \alpha'|(B \cap G_1')$. In this case $B \cap G_1'$ has finite index in $B \cap G_1$ and so $\alpha'(B \cap G_1')=\alpha(B \cap G_1')$ has finite index in $\alpha(B \cap G_1)$. It follows that the supports of $\alpha'(B \cap G_1')$ and $\alpha(B \cap G_1)$ are equal, which is to say that $v_{\alpha(B \cap G_1)} = v_{\alpha'(B \cap G_1')}$, as desired.
To deal with the second ambiguity, suppose that $B'$ is another basic subgroup of $N$ with $v_B = v_{B'}$. With $\alpha$ as above, we must show that
\[
v_{\alpha(B \cap G_1)} = v_{\alpha(B' \cap G_1)}.
\]
By Lemma~\ref{irred} the centralizer of a basic subgroup is invariant under passage to finite-index subgroups. It follows from this and Lemma~\ref{lemma:basic}\eqref{basic passage} that $v_{B \cap G_1} = v_{B' \cap G_1}$. It further follows from Lemma~\ref{lemma:bc}\eqref{bc v} that $BC_{G_1}(B \cap G_1)$ is equal to $BC_{G_1}(B' \cap G_1)$. As basic centralizers are preserved by isomorphisms, we have that $BC_{G_2}(\alpha(B \cap G_1))$ is equal to $BC_{G_2}(\alpha(B' \cap G_1))$. Again by Lemma~\ref{lemma:bc}\eqref{bc v} we have the desired equality $v_{\alpha(B \cap G_1)} = v_{\alpha(B' \cap G_1)}$.
Having now shown that $[\alpha]$ induces a well-defined permutation of the set of vertices of $\C_N(S_g)$, it remains to check that this permutation preserves the set of edges. To this end, we claim that if $B$ and $B'$ are basic subgroups of $N$ then $v_B$ and $v_{B'}$ are connected by an edge if and only if $B$ and $B'$ commute. The subgroups $B$ and $B'$ commute if and only if the subgroups $B \cap G_1$ and $B' \cap G_1$ commute, and the latter holds if and only if $\alpha(B \cap G_1)$ and $\alpha(B' \cap G_1)$ commute. It then follows from Lemma~\ref{lemma:bc}\eqref{bc e} that the given permutation of vertices extends to an automorphism of $\C_N(S_g)$.
\end{proof}
\subsection{Proof of the theorem}
\label{sec:finishing}
We are almost ready to prove Theorem~\ref{main:normal}. Let us first introduce some notation. For $f \in \MCG(S_g)$ denote by $\alpha_f$ the automorphism of $\MCG(S_g)$ given by conjugation by $f$, that is, $\alpha_f(h) = fhf^{-1}$ for all $h \in \MCG(S_g)$. If $f$ lies in the normalizer of $N$ then we may consider $\alpha_f$ as an element of $\Aut N$ (technically, the restriction of $\alpha_f$ to $N$ gives an element of $\Aut N$). Similarly, if there is a restriction of $\alpha_f$ that is an isomorphism between finite-index subgroups of $N$ then we may regard $[\alpha_f]$ as an element of $\Comm N$ (this is an abuse of notation: we should more properly write $[\bar \alpha_f]$ where $\bar \alpha_f$ is the restriction).
\begin{proof}[Proof of Theorem~\ref{main:normal}]
For simplicity we first deal with the case where $N$ is normal in $\MCG(S_g)$ (this is the first statement of Theorem~\ref{main:normal}). Let $P$ be a pure normal subgroup of finite index in $\MCG(S_g)$. We will begin by describing a sequence of five maps $\Phi_1,\dots,\Phi_5$ as follows:
\begin{align*}
\MCG(S_g) \stackrel{\Phi_1}{\to}& \Aut N \stackrel{\Phi_2}{\to} \Comm N \stackrel{\Phi_3}{\to} \Comm N \cap P \\
& \stackrel{\Phi_4}{\to} \Aut \C_{N \cap P}(S_g) \stackrel{\Phi_5}{\to} \MCG(S_g).
\end{align*}
Here are the definitions of the maps:
\begin{itemize}
\item $\Phi_1$ is the conjugation map, that is, $\Phi_1(f)= \alpha_f$,
\item $\Phi_2$ maps an element of $\Aut N$ to its equivalence class in $\Comm N$,
\item $\Phi_3$ maps the equivalence class of an isomorphism between finite index subgroups of $N$ to the equivalence class of any restriction that is an isomorphism between finite-index subgroups of $N \cap P$,
\item $\Phi_4$ is the map from Proposition~\ref{prop:comm to aut}, and
\item $\Phi_5$ is the isomorphism from Proposition~\ref{prop:defining cn}.
\end{itemize}
To prove the theorem in the case where $N$ is normal in $\MCG(S_g)$ we will show that $\Phi_1$, $\Phi_2$, $\Phi_3$, $\Phi_4$, and $\Phi_5$ are all injective and that the composition
\[
\Phi_5 \circ \Phi_4 \circ \Phi_3 \circ \Phi_2 \circ \Phi_1
\]
is the identity. The injectivity of the $\Phi_i$ and the surjectivity of the composition together imply that the $\Phi_i$ are surjective, and hence are isomorphisms. That $\Phi_1$ and $\Phi_2$ are isomorphisms is the content of the first statement of Theorem~\ref{main:normal}. The map $\Phi_5 \circ \Phi_4 \circ \Phi_3$ is the natural map $\Comm N \to \MCG(S_g)$ from the statement of the theorem.
We begin by showing that the $\Phi_i$ are injective. The map $\Phi_1$ is injective by an argument similar the one given in Lemma~\ref{inj}; indeed, if $f \in \MCG(S_g)$ commutes with $h \in N$ then $f$ fixes the canonical reduction system of $h$. Any element of $N$ with a small component has a nonempty canonical reduction system, and as in the proof of Lemma~\ref{inj} the orbit under $\MCG(S_g)$ of this canonical reduction system is dense in $\PMF(S_g)$. Since $N$ is normal in $\MCG(S_g)$ the canonical reduction system for $khk^{-1}$ is the $k$-image of the canonical reduction system for $h$, the injectivity follows.
We now show that $\Phi_2$ is injective. Let $\alpha \in \Aut N$ be an element of the kernel. Let $f$ be an element of $N$. We would like to show that $\alpha(f)=f$. Let $h$ be a pseudo-Anosov element of $N$ (all infinite normal subgroups of $\Mod(S_g)$ contain such elements). Since $\Phi_2(\alpha)$ is the identity, there is a finite-index subgroup $G$ of $N$ so that $\alpha|G$ is the identity. There is an $m > 0$ so that $h^{m}$ and $(fhf^{-1})^{m}$ lie in $G$ and so we have $\alpha(h^{m})=h^{m}$ and $\alpha((fhf^{-1})^{m}) = (fhf^{-1})^{m}$. We have
\begin{align*}
f h^m f^{-1} = (fhf^{-1})^m = \alpha((fhf^{-1})^m) = \alpha(f) \alpha(h^m) \alpha(f)^{-1} = \alpha(f) h^m \alpha(f)^{-1}.
\end{align*}
In other words $f^{-1}\alpha(f)$ commutes with $h^m$ and so $f^{-1}\alpha(f)$ fixes the point in $\PMF(S_g)$ corresponding to the unstable foliation of $h$. Since $h$ was arbitrary, $f^{-1}\alpha(f)$ fixes all the points in $\PMF(S_g)$ corresponding to the unstable foliations of pseudo-Anosov elements of $N$. But since $N$ is normal in $\Mod(S_g)$, these points are dense in $\PMF(S_g)$. As in the proof of Lemma~\ref{inj} we conclude that $f^{-1}\alpha(f)$ is the identity, which is to say that $\alpha(f)=f$. Thus, $\alpha$ is the identity, as desired.
The map $\Phi_3$ is an isomorphism since a finite-index subgroup of $N \cap P$ also has finite index in $N$.
Next, we will show that $\Phi_4$ is injective. To this end, fix some isomorphism $\alpha : G_1 \to G_2$ representing an element $[\alpha]$ of $\Comm N \cap P$. As in Proposition~\ref{prop:comm to aut} denote the image of $[\alpha]$ in $\Aut \C_{N \cap P}(S_g)$ by $\alpha_\star$. Assume that $\alpha_\star$ is the identity. We must show that $[\alpha]$ is the identity. We will show that in fact $\alpha$ is the identity (in particular $G_1=G_2$). So let $h \in G_1$. We would like to show that $\alpha(h)=h$. Let $h_\star$ and $\alpha(h)_\star$ denote the images of $h$ and $\alpha(h)$ under natural map $\MCG(S_g) \to \Aut \C_{N \cap P}(S_g)$. Since the latter is injective (Lemma~\ref{inj}) it suffices to show that $h_\star = \alpha(h)_\star$.
So let $B$ be an arbitrary basic subgroup of $N$. By Lemma~\ref{lemma:basic}\eqref{basic passage} and Lemma~\ref{lemma:bc}\eqref{bc wd} we may assume without loss of generality that $B$ is contained in $G_1$. We have
\begin{align*}
h_\star (v_B) &= v_{hBh^{-1}} = \alpha_\star v_{hBh^{-1}} = v_{\alpha(hBh^{-1})} = v_{\alpha(h)\alpha(B)\alpha(h)^{-1}} \\
&\quad = \alpha(h)_\star v_{\alpha(B)} = \alpha(h)_\star \alpha_\star v_{B} = \alpha(h)_\star v_{B}.
\end{align*}
In order, the equalities use Lemma~\ref{lemma:bc}\eqref{bc act}, the assumption that $\alpha_\star$ is the identity, Proposition~\ref{prop:comm to aut}, the assumptions that $B$ and $h$ both lie in $G_1$, again Lemma~\ref{lemma:bc}\eqref{bc act}, again Proposition~\ref{prop:comm to aut}, and again the assumption that $\alpha_\star$ is the identity. It follows that $h_\star = \alpha(h)_\star$ and so $\alpha(h)=h$, as desired.
The fifth and final map $\Phi_5 : \Aut \C_{N \cap P}(S_g) \to \MCG(S_g)$ is an isomorphism by Proposition~\ref{prop:defining cn}; in particular it is injective.
As in Theorem~\ref{main:complex} the isomorphism $\Phi_5$ is the inverse of the natural map $\MCG(S_g) \to \Aut \C_{N \cap P}(S_g)$. It follows that the composition $\Phi_5 \circ \Phi_4 \circ \Phi_3 \circ \Phi_2 \circ \Phi_1$ is the identity. Indeed, given $f \in
\Mod(S_g)$ the image in $\Comm N \cap P$ is the element given by conjugation by $f$. Thus the image in $\Aut \C_{N \cap P}(S_g)$ is $f_\star$ and so the image in $\MCG(S_g)$ must again be $f$. This completes the proof of the first statement of the theorem.
We now prove the second statement of the theorem. Assume that $N$ is normal in $\Mod(S_g)$ but not in $\MCG(S_g)$. Again let $P$ be a pure normal subgroup of finite index in $\MCG(S_g)$. We will consider a collection of homomorphisms $\Phi_i$ analogous to the ones from the proof of the first statement:
\[
\xymatrix{
& \MCG(S_g) \ar@{-->}[ddd]^{\Phi_6} & \\
\Aut \C_{N \cap P}(S_g) \ar[ur]^{\Phi_5}_{\cong} & & \Mod(S_g) \ar@{_{(}->}[ul] \ar[d]^{\Phi_1} \\
\Comm N \cap P \ar[u]^{\Phi_4} & & \Aut N \ar[dl]^{\Phi_2} \\
& \Comm N \ar[ul]^{\Phi_3}_{\cong} &
}
\]
The maps $\Phi_1,\dots,\Phi_5$ are all defined in the same way as in the first case, except that the domain of $\Phi_1$ has changed. The map $\Phi_6$ is the natural map $\MCG(S_g) \to \Comm N$ from the statement of the theorem. It maps $f \in \MCG(S_g)$ to the element of $\Comm N$ given by conjugation by $f$. For $f$ not in $\Mod(S_g)$ there may be no subgroup of finite index $H$ in $N$ so that $fHf^{-1}$ has finite index in $N$; if such $f$ exists the map $\Phi_6$ is not well defined.\footnote{One is tempted to think that since $N$ is normal in $\Mod(S_g)$ and since $\Mod(S_g)$ has index two in $\MCG(S_g)$, there is a subgroup of finite index in $N$ that is normal in $\MCG(S_g)$. If this were true it would imply that $\Phi_6$ is always well defined. However, it is not true. Consider for example the group $A = \Z^2 \rtimes \Z/2$ where $\Z/2$ acts on $\Z^2$ by interchanging the two factors; the subgroup $N$ of $A$ corresponding to the first factor of $\Z^2$ is normal in $\Z^2$ but there is a conjugate of $N$ in $A$ with which $N$ has trivial intersection.}
By the same arguments as in the proof of the first statement of the theorem, the maps $\Phi_1$, $\Phi_2$, $\Phi_3$, $\Phi_4$, and $\Phi_5$ are all injective and the composition
\[
\Phi_5 \circ \Phi_4 \circ \Phi_3 \circ \Phi_2 \circ \Phi_1
\]
is the inclusion map.
We consider two cases, according to whether the image of $\Phi_5 \circ \Phi_4 \circ \Phi_3$ is $\Mod(S_g)$ or $\MCG(S_g)$. Let us first assume that the image is $\Mod(S_g)$. In this case $\Comm N$ is isomorphic to $\Mod(S_g)$ under the natural map $\Phi_5 \circ \Phi_4 \circ \Phi_3$. Further $\Phi_1$ and $\Phi_2$ are isomorphisms since $\Phi_1$, $\Phi_2$, and $\Phi_5 \circ \Phi_4 \circ \Phi_3$ are injective and their composition is the inclusion, as desired.
We now proceed to the case where the image of $\Phi_5 \circ \Phi_4 \circ \Phi_3$ is $\MCG(S_g)$. Since we already know that $\Phi_1$, $\Phi_2$, and $\Phi_5 \circ \Phi_4 \circ \Phi_3$ are injective, and that their composition is the inclusion, the only remaining statements to prove are that $\Phi_6$ is the inverse to $\Phi_5 \circ \Phi_4 \circ \Phi_3$ and that $\Phi_1$ is surjective, hence an isomorphism. We treat each of these statements in turn.
We first show that $\Phi_6$ is a left inverse to $\Phi_5 \circ \Phi_4 \circ \Phi_3$ (hence is the inverse). The proof of this statement follows along similar lines as in the proof of the injectivity of $\Phi_4$. We fix some isomorphism $\alpha : G_1 \to G_2$ representing an element $[\alpha]$ of $\Comm N \cap P$ (which under $\Phi_3$ is canonically isomorphic to $\Comm N$). As above we denote the $\Phi_4$-image of $[\alpha]$ in $\Aut \C_{N \cap P}(S_g)$ by $\alpha_\star$. Assume that $\alpha_\star$ maps to $f \in \MCG(S_g)$ under $\Phi_5$. We would like to show that $[\alpha]$ is given by the restriction of the conjugation map $\alpha_f$. This is the same as saying that $[\alpha]=\Phi_6 \circ \Phi_5 \circ \Phi_4 \circ \Phi_3 ([\alpha])$, as desired.
Let $h \in G_1$. We would like to show that $\alpha(h)=fhf^{-1}$. For a mapping class $j \in \MCG(S_g)$ denote by $j_\star$ the image of $j$ under natural map $\Phi_5^{-1} : \MCG(S_g) \to \Aut \C_{N \cap P}(S_g)$ (so by definition $\alpha_\star=f_\star$). Since $\Phi_5^{-1}$ is injective (Lemma~\ref{inj}) it suffices to show that $(fhf^{-1})_\star = \alpha(h)_\star$, or $(fh)_\star = (\alpha(h)f)_\star$. Let $B$ be an arbitrary basic subgroup of $N$. Again we may assume without loss of generality that $B$ is contained in $G_1$. We have
\begin{align*}
(fh)_\star v_B &= f_\star(h_\star v_B) = \alpha_\star(h_\star v_B) = \alpha_\star v_{hBh^{-1}} = v_{\alpha(hBh^{-1})} \\ &= v_{\alpha(h)\alpha(B)\alpha(h^{-1})} = \alpha(h)_\star v_{\alpha(B)} = \alpha(h)_\star \alpha_\star (v_B) \\ &= \alpha(h)_\star f_\star v_B = (\alpha(h)f)_\star v_B.
\end{align*}
In order, the equalities use the fact that $\Phi_5^{-1}$ is a homomorphism, the assumption that $f_\star= \alpha_\star$, Lemma~\ref{lemma:bc}\eqref{bc act}, Proposition~\ref{prop:comm to aut}, the assumptions that $B$ and $h$ both lie in $G_1$, again Lemma~\ref{lemma:bc}\eqref{bc act}, again Proposition~\ref{prop:comm to aut}, again the assumption that $f_\star= \alpha_\star$, and again the fact that $\Phi_5^{-1}$ is a homomorphism. It follows that $(fh)_\star = (\alpha(h)f)_\star$ and so $\alpha(h) = fhf^{-1}$, as desired.
We have proven that the left inverse of the natural map $\Phi_5 \circ \Phi_4 \circ \Phi_3 : \Comm N \to \MCG(S_g)$ is the natural map $\Phi_6 : \MCG(S_g) \to \Comm N$, as in the statement of the theorem.
To complete the proof of the theorem, it remains to show that $\Phi_1$ is an isomorphism, in other words that $\Phi_1$ is surjective. Let $\alpha \in \Aut N$. As usual, denote $\Phi_2(\alpha)$ by $[\alpha]$. Let $f$ denote $\Phi_5 \circ \Phi_4 \circ \Phi_3([\alpha])$. Since $\Phi_6$ is the inverse of $\Phi_5 \circ \Phi_4 \circ \Phi_3$, we have $\Phi_6(f) = [\alpha]$. In other words $[\alpha] = [\alpha_f]$. To show that $\alpha$ lies in the image of $\Phi_1$ we will show that $\alpha = \alpha_f$. The proof will be similar to the proof of the injectivity of $\Phi_2$.
Let $n \in N$. We would like to show that $\alpha(n) = fnf^{-1}$. Let $h$ be a pseudo-Anosov element of $N$. Since $[\alpha] = [\alpha_f]$, there is a finite-index subgroup $G$ of $N$ so that $\alpha|G = \alpha_f|G$. There is an $m > 0$ so that $h^{m}$ and $nh^mn^{-1}$ lie in $G$. So $\alpha(h^{m})=fh^{m}f^{-1}$ and $\alpha(nh^mn^{-1}) = fnh^mn^{-1}f^{-1}$. We have
\begin{align*}
fnh^mn^{-1}f^{-1} = \alpha(nh^mn^{-1}) = \alpha(n)\alpha(h^m)\alpha(n)^{-1} = \alpha(n)fh^mf^{-1}\alpha(n)^{-1}.
\end{align*}
From the equality of the first and last expressions we deduce that
\[
f^{-1}\alpha(n)^{-1}fnh^m = h^mf^{-1}\alpha(n)^{-1}fn,
\]
in other words that $f^{-1}\alpha(n)^{-1}fn$ commutes with $h^m$. Since $h$ was arbitrary, it follows as in the proof of the injectivity of $\Phi_2$ that $f^{-1}\alpha(n)^{-1}fn$ is the identity, which is to say that $\alpha(n) = f nf^{-1}$, as desired. This completes the proof of the theorem.
\end{proof}
\bibliographystyle{plain}
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2,869,038,155,645 | arxiv | \section{Introduction}
\subsection{Background and history}
Let $\L g$ be a real Lie algebra. A function $\zeta: \mathfrak{g} \rightarrow \mathbb{R}$ is called a \emph{Lie quasi-state} if
\[
\zeta(aX + bY) = a\zeta(X) + b\zeta(Y).
\]
for all $a, b \in \mathbb{R}$ and every pair $(X, Y)$ of \emph{commuting} elements in $\mathfrak{g}$. In other words, $\zeta$ is linear on abelian subalgebras of $\mathfrak{g}$.
We refer the reader to \cite{E14, EP09} for an overview of the history of the notion of Lie quasi-states. Roughly speaking, Lie quasi-states (or closely related notions) arose more or less independently in three different contexts: In connection with the foundations of quantum mechanics (see e.g. \cite{Gl57, Gleason2, Gleason3}), in symplectic topology (see the recent survey \cite{E14} for references) and in the study of quasimorphisms on finite (see e.g. \cite{GuichardetWigner, BuMo, Shtern, Surface, BSH, BSH2, Calegari}) and infinite-dimensional (see e.g. \cite{EP1, EPBiran, BS, Usher}) Lie groups.
One of the basic theorems in the mathematical foundations of quantum mechanics is Gleason's theorem on rigidity of frame functions \cite{Gl57, Gleason2}. Although Lie quasi-states do not feature explicitly in Gleason' work, his result is essentially equivalent to the statement that every locally bounded Lie quasi-state on $\L u(n)$ is linear, provided $n \geq 3$. (See the introductions of \cite{EPZ} and \cite{EP09} for a discussion of this equivalence.) This can be seen as the first major non-trivial result concerning Lie quasi-states.
In symplectic topology, Lie quasi-states constructed from Floer homology and spectral invariants (as in \cite{EP1, EPBiran, Usher}) have recently become an important tool in studying the displacability of subsets of symplectic manifolds under Hamiltonian diffeomorphisms (see \cite{EP2}). Entov's recent ICM address \cite{E14} gives an overview of these developments and an extensive list of references. In most of these symplectic applications, the Lie quasi-states considered arise as directional derivatives of continuous quasimorphisms on the corresponding infinite-dimensional Lie groups.
Both in the quantum-mechanical and the symplectic setting the focus is naturally on Lie quasi-states on infinite-dimensional Lie algebras. A systematic analysis of Lie quasi-states on finite-dimensional Lie algebras was initiated only recently by Entov and Polterovich in \cite{EP09}. Even if one is primarily interested in the infinite-dimensional case, such an analysis is relevant in order to understand the behaviour of Lie quasi-states along finite-dimensional subalgebras. However, to the best of our knowledge \cite{EP09} remains the only paper so far which concerns Lie quasi-states on finite-dimensional Lie algebras.
The purpose of the present paper is to extend some of the results from \cite{EP09} to larger classes of finite-dimensional Lie algebras and to obtain a clearer picture about Lie quasi-states on general finite-dimensional Lie algebras through some key examples. Our main focus will be on Ad-invariant Lie quasi-states, since these are comparatively easy to handle and at the same time the most relevant ones in applications. One particular goal of this article is to understand their connection with homogeneous quasimorphisms on finite-dimensional Lie groups.
\subsection{Integrable Lie quasi-states and homogeneous quasimorphisms}
From now on we assume that all Lie algebras are real and finite-dimensional. We denote by $Q(\mathfrak{g})$ the space of Lie quasi-states on $\mathfrak{g}$ and by $\mathcal Q(\mathfrak{g}) \subset Q(\mathfrak{g})$ the subspace of continuous Lie quasi-states on $\mathfrak{g}$ respectively. Note that the adjoint group associated to the Lie algebra $\mathfrak{g}$ acts on these spaces by $g.\zeta(X) = \zeta({\rm Ad}(g)^{-1}(X))$.
The notion of a Lie quasi-state on a Lie algebra $\L g$ has a global counterpart on the level of Lie groups. Indeed, given a Lie group $G$ (not necessarily connected) we can consider \emph{integrated Lie quasi-states}, i.e. Borel measurable functions $f: G \rightarrow \mathbb{R}$ such that
\begin{equation*}
f(gh) = f(g) + f(h).
\end{equation*}
for all pairs $(g,h)$ of commuting elements in $G$. Any integrated Lie quasi-state $f$ on a Lie group $G$ induces a Lie quasi-state $\zeta$ on the Lie algebra $\mathfrak{g}$ on $G$, given by the \emph{directional derivative}
\[
\zeta(X) =f(\exp_G(X)), \quad \textrm{for all $X \in \mathfrak{g}$},
\]
where $\exp_G$ denotes the exponential map from $\mathfrak{g}$ to $G$.
We note that $\zeta$ is continuous provided $f$ is. If $\zeta$ is the directional derivative of an integrated Lie quasi-state $f$, then we say that $\zeta$ is \emph{integrable} and we say that $\zeta$ \emph{integrates} to $f$. Typical examples of integrated Lie quasi-states are given by homogeneous (continuous) quasimorphisms, i.e. continuous functions $f: G\rightarrow \mathbb{R}$ which satisfy
\[
D(f) := \sup_{g,h \in G} |f(gh)-f(g)-f(h)| < \infty,
\]
and $f(g^n) = n \cdot f(g)$ for all $n \in \mathbb Z$. We note that if $g$ and
$h$ in $G$ commute, then
\[
n \cdot f(gh) = f((gh)^n) = f(g^n h^n) - f(g^n) - f(h^n) + n \cdot (f(g) + f(h))
\]
for all $n$, which readily implies that $f(gh) = f(g) + f(h)$ upon dividing with
$n$ and letting $n$ tend to infinity. In particular, every homogeneous (continuous) quasimorphism is a conjugation-invariant integrated Lie quasi-state on $G$, and thus its directional derivative gives rise to an {\rm Ad}-invariant
(continuous) Lie quasi-states on $\mathfrak{g}$.
It turns out that homogeneous quasimorphisms on connected Lie groups are rare. On a solvable Lie group every homogeneous quasimorphism is in fact a homomorphism, and a simple Lie group admits a (non-trivial) homogeneous quasimorphism if and only if it has infinite center, in which case there is a unique (non-trivial) homogeneous quasimorphism up
to multiples (see e.g. \cite{BSH2}).
\subsection{Lie quasi-state rigidity}
In the sequel we shall denote by $Q_{\rm Ad}(\mathfrak{g})$ and $Q_{\rm int}(\mathfrak{g})$ the spaces of Ad-invariant and integrable Lie quasi-states respectively. We also denote by $Q_{\rm qm}(\mathfrak{g})$ the space of Lie quasi-states which are directional derivatives of homogeneous quasimorphisms on some Lie group $G$ with Lie algebra $\mathfrak{g}$. We then use the notations $\mathcal Q_{\rm Ad}(\mathfrak{g})$, $\mathcal Q_{\rm int}(\mathfrak{g})$, $\mathcal Q_{\rm qm}(\mathfrak{g})$ for the corresponding subspaces of continuous quasi-states. We write $\mathfrak{g}^*$ for the space of linear functionals on $\mathfrak{g}$ and ${\rm Hom}(\mathfrak{g}, \mathbb{R}) \subset \mathfrak{g}^*$ for the subspace of Lie algebra homomorphisms from $\mathfrak{g}$ to $\mathbb{R}$. Then we have a chain of inclusions
\[
{\rm Hom}(\mathfrak{g}, \mathbb{R}) \subset Q_{\rm qm}(\mathfrak{g}) = \mathcal Q_{\rm qm}(\mathfrak{g}) \subset \mathcal Q_{\rm Ad}(\mathfrak{g}) \cap \mathcal Q_{\rm int}(\mathfrak{g}) \subset \mathcal Q_{\rm Ad}(\mathfrak{g}) \subset \mathcal Q(\mathfrak{g}).
\]
It follows from the results in \cite{BuMo} that the first inclusion is an equality for all Lie algebras not containing a simple Hermitian Lie algebra, and it is of codimension one for simple Hermitian Lie algebras (see also \cite{BSH} for a more detailed discussion). In particular, $Q_{\rm qm}(\mathfrak{g})$ is always finite-dimensional, whereas all of the larger spaces listed above can be infinite-dimensional. We say that a Lie algebra $\mathfrak{g}$ is \emph{quasi-state rigid}, or just \emph{rigid} for short, if
\[
\mathcal Q_{\rm qm}(\mathfrak{g}) =\mathcal Q_{\rm Ad}(\mathfrak{g}).
\]
This implies in particular, that every {\rm Ad}-invariant Lie quasi-state is integrable and that $\mathcal Q_{\rm Ad}(\mathfrak{g})$ is finite-dimensional. For Lie algebras not containing a simple Hermitian Lie algebra it is moreover equivalent to showing that every {\rm Ad}-invariant Lie quasi-state is linear, hence a homomorphism. In the sequel we will refer to the problem of classifying all rigid Lie algebras as the \emph{(quasi-state)} \emph{rigidity problem}, and this problem will be the main concern of the present paper.
\subsection{Statements of main results}
We recall that every finite-dimensional Lie algebra $\mathfrak{g}$ is the semidirect product of a semisimple Lie subalgebra and a maximal solvable ideal, called the solvable radical of $\mathfrak{g}$. It is therefore a common strategy in the study of Lie algebras to separately analyze a problem for semisimple (or, slightly more generally, reductive) and solvable Lie algebras and then to attack the general `mixed' case by means of semi-direct products. This is also the approach which we will take towards the quasi-state rigidity problem here.
The reductive case is by far the simplest one, since the fine structure of reductive Lie algebras is very well understood. It was already established in \cite[Theorem 4.2]{EP09} that for every Hermitian simple Lie algebra $\mathfrak{g}$ we have
\[
Q_{\rm Ad}(\L g) = {\mathcal Q}_{\rm Ad}(\L g)
={\mathcal Q}_{\rm qm}(\L g).
\]
By \cite[Theorem 4.1]{EP09} the same holds for any compact Lie algebra $\mathfrak{g}$ (which is automatically reductive). In this paper, we strengthen this
result as follows:
\begin{theorem}\label{MainTheorem1}
For every reductive Lie algebra $\mathfrak{g}$ we have
\[Q_{\rm Ad}(\L g) = {\mathcal Q}_{\rm Ad}(\L g)
={\mathcal Q}_{\rm qm}(\L g).\]
In particular, every reductive Lie algebra is rigid and every Ad-invariant Lie quasi-state on a non-Hermitian simple Lie algebra (such as $\L {sl}_n(\mathbb{R})$, $n \geq 3$) is trivial.
\end{theorem}
This settles the quasi-state rigidity problem in the reductive setting completely. However, we stress that the classification of
\textit{non}-Ad-invariant quasi-states on reductive Lie algebras
remains open. There are some important partial results towards such a classification. For example, if $n \geq 3$ and $\mathfrak{g}$ denotes either the Lie algebra $\mathfrak{g} = \L u(n)$ or the Lie algebra $\mathfrak{g} = \L {sp}(2n)$, then
\begin{equation}\label{EPGl}
{\mathcal Q}(\L g) ={\mathcal Q}_{\rm qm}(\L g)+\mathfrak{g}^*,
\end{equation}
where in the $\L u(n)$ case the first summand on the right hand side is actually trivial. For $\mathfrak{g} = \L u(n)$ this follows from a classical theorem of Gleason (\cite{Gl57}, see also the introduction of
\cite{EP09}) and for $\mathfrak{g} = \L {sp}(2n)$ this is one of the main results of \cite{EP09}. Currently we do not know whether \eqref{EPGl} can be extended to more general compact Lie algebras, let alone reductive Lie algebras. If such a general result exists, then it has to evoke some kind of higher rank assumption, since \eqref{EPGl} fails for $\L{sp}(2)$ and $\L u(2)$.
Let us mention in passing that there is also a global version of Theorem \ref{MainTheorem1} which can be stated as follows (see Theorem \ref{GlobalThm} below):
\begin{theorem} Let $G$ be a connected reductive Lie group. Then every conjugation-invariant integrated Lie quasi-state on $G$ is a homogeneous quasimorphism.
\end{theorem}
The rigidity problem for solvable Lie algebras is more subtle than in the reductive case. The smallest example of a non-rigid solvable Lie algebra is provided by the three-dimensional Heisenberg algebra $\L h_3$, which can be characterized as the unique non-rigid Lie algebra of dimension $\leq 3$ (see Theorem \ref{ThmLow}). We actually show that $\mathcal Q_{Ad}(\L h_3)$ is infinite-dimensional by providing an explicit parametrization of all Ad-invariant continuous Lie quasi-states on $\L h_3$ (see Corollary \ref{CorH3}). It then follows from general principles that $\mathcal Q_{Ad}(\L g)$ is also infinite-dimensional for every Lie algebra $\mathfrak{g}$ which surjects onto $\L h_3$. In particular, such a Lie algebra can never be rigid. One can ask whether this is the only obstruction to rigidity, at least for solvable Lie algebras:
\begin{question}\label{Question} Let $\mathfrak{g}$ be a solvable Lie algebra which does not surject onto $\L h_3$. Does it follow that $\mathfrak{g}$ is rigid?
\end{question}
This question is motivated by the following partial results. Consider the class of Lie algebras $\mathfrak{g}$ which are almost abelian in the in the sense that $\mathfrak{g}$ admits a codimension one abelian ideal $V$ such that the extension
\[
0 \rightarrow V \rightarrow \mathfrak{g} \rightarrow \mathbb{R} \rightarrow 0
\]
splits as a semidirect product. Such Lie algebras are automatically (two-step) solvable, but not necessarily nilpotent. Examples include the Lie algebras of the $(ax+b)$-group, of the three-dimensional SOL group and of the $1$-dimensional unitary motion group $\mathbb{C} \rtimes U(1)$, as well as the Lie algebra $\L h_3$. For Lie algebras in this class, the answer to Question \ref{Question} is positive:
\begin{theorem}\label{MainTheorem2}
Let $\mathfrak{g}$ be a Lie algebra which splits over a codimension one abelian ideal. Then $\mathfrak{g}$ is rigid if and only if it does not surject onto the three-dimensional Heisenberg algebra $\L h_3$.
\end{theorem}
Given the special role of the three-dimensional Heisenberg algebra in this theorem, it is natural to ask about rigidity of the higher-dimensional Heisenberg algebras. Interestingly enough, it was already established in \cite[Prop. 2.13]{EP09} that these algebras are always rigid. In fact, on these algebras every Lie quasi-state (neither assumed Ad-invariant nor even continuous) is linear. Thus in order to find potential counterexamples to Question \ref{Question} one has to look elsewhere (and to go beyond dimension $3$), and it it currently not clear to us, what would be a natural class of solvable Lie algebras to consider.
The example of the Heisenberg algebra should also serve as a warning concerning the study of the rigidity problem for general (i.e. neither solvable nor reductive) Lie algebras. Here one is immediately confronted with the following problem:
\begin{question}\label{Question2}
Let $\L g_1$ be a rigid solvable Lie algebra and $\L g_2$ be a semisimple (hence rigid) Lie algebra acting on $\L g_1$ by automorphisms. Is the semi-direct product $\L g_1 \rtimes \L g_2$ rigid?
\end{question}
If the assumption semisimple is replaced by reductive the answer to this question is no, since $\L h_3 = \mathbb{R}^2 \rtimes \mathbb{R}$ is even a semi-direct product of abelian subalgebras. For the moment we have no systematic way to deal with Question \ref{Question2}.
However, we do understand some specific, but important, examples:
\begin{theorem}\label{MainTheorem3} Let $\mathfrak{g}_n$ be the $n$-dimensional unitary motion algebra, i.e. the semidirect product of the Lie algebras $\L{u}(n)$ and $\mathbb{C}^n$ with respect to the standard representation of $\L{u}(n)$ on $\mathbb{C}^n$. Then the following hold:
\begin{enumerate}[(i)]
\item $\mathfrak{g}_n$ is rigid for all $n \geq 1$.
\item If $n \geq 3$, then every every continuous quasi-state on $\mathfrak{g}$ is linear.
\item If $n \leq 2$, then the space of continuous quasi-states on $\mathfrak{g}_n$ is infinite-dimensional.
\end{enumerate}
\end{theorem}
The proof of Theorem \ref{MainTheorem3} is close in spirit to the proofs
of Gleason and of Entov-Polterovich of the rigidity of the Lie algebras
$\L u(n)$ and $\L{sp}(n)$ respectively in that it depends on the analysis of
the values of a given Lie quasi-state at certain elements related to rank one projections. Extending Theorem \ref{MainTheorem3} even to the Euclidean motion algebras $\mathbb{R}^n \rtimes \L{o}(n)$ would require a deeper
understanding of (non-Ad-invariant) Lie quasi-states on $\L{o}(n)$.
\subsection{Organization of the paper}
The paper consists of four main sections and an appendix. In Section \ref{sec:reductive} we discuss additive Jordan decompositions
of simple Lie algebras and we show how these decompositions can be used
to prove Theorem \ref{MainTheorem1}. In Section \ref{sec:semidirect} we provide an explicit formula for Lie quasi-states on solvable Lie algebras which split over an abelian ideal of
codimension $1$. We then use this formula to derive Theorem \ref{MainTheorem2}.
In Section \ref{sec:unitary} we extend some arguments of Gleason and
Entov-Polterovich to the setting of unitary motion Lie algebras and
establish Theorem \ref{MainTheorem3}. In the appendix we study a class of generalized frame functions which appear in the proof of Theorem \ref{MainTheorem3}
\subsection{Acknowledgments}
The authors would like to express their gratitude to Michael Entov
for bringing the problem of classifying Lie quasi-states on finite-dimensional
Lie algebras to their attention. They are also indebted to Leonid Polterovich for comments on an earlier version of this paper and in particular for challenging them to extend Theorem \ref{MainTheorem3} to the present form. The authors are grateful to the anonymous referee for a careful reading of the manuscript which lead to several improvements and corrections. In particular, the referee pointed out a crucial mistake in one of the computations in an earlier version of this paper, and thereby saved us from publishing a wrong classification statement. The present paper is an outgrowth of discussions between the authors which took place at Chalmers University of Technology in Gothenburg in September 2014. The second author would
like to thank the Guest Research Program of Chalmers and the Department of Mathematical Sciences at Chalmers for their hospitality.
\section{The reductive case}
\label{sec:reductive}
Recall that a Lie algebra $\mathfrak{g}$ is \emph{reductive} if it decomposes as a direct sum of abelian and simple Lie algebras. The study of (Ad-invariant) Lie quasi-states on such Lie algebras can be immediately reduced to the case of simple Lie algebras by means of the following observation:
\begin{lemma}\label{Trivialities} If $\L g = \L g_1 \oplus \L g_2$ is a direct sum of Lie algebras, then $Q(g) $ decomposes as $ Q(\mathfrak{g}) =Q(\mathfrak{g}_1) \oplus Q(\mathfrak{g}_2)$, and the subspaces ${Q}_{\rm qm}(\L g)$, ${Q}_{\rm Ad}(\L g)$, ${Q}(\L g)$, ${\mathcal Q}_{\rm qm}(\L g)$, ${\mathcal Q}_{\rm Ad}(\L g)$, ${\mathcal Q}(\L g)$ decompose accordingly. If $\mathfrak{g}$ is abelian then all these spaces coincide with $\mathfrak{g}^*$.
\end{lemma}
\subsection{Additive Jordan decompositions of simple Lie algebras}
Thus in order to establish Theorem \ref{MainTheorem1} it suffices to consider the case of a finite-dimensional simple real Lie algebra $\mathfrak{g}$. Such Lie algebras come in two different flavors, \emph{Hermitian} and \emph{non-Hermitian}. More precisely, let $\theta$ be a Cartan involution on $\mathfrak{g}$ and let $\mathfrak{k}<\mathfrak{g}$ be the fixed point algebra of $\theta$. Then $\mathfrak{k}$ is reductive, i.e. can be written as $\mathfrak{k} = \L {z(k)} \oplus \mathfrak{k}_{ss}$ where $\mathfrak{k}_{ss}$ is semisimple and where $\L{z(k)}$ denotes the centre of $\mathfrak{k}$, and moreover $\dim \L{z(k)} \leq 1$. Now $\mathfrak{g}$ is Hermitian if $\dim \L{z(k)} =1$ and non-Hermitian otherwise.
A key feature of simple Lie algebras is that they are always algebraic, i.e. their associated adjoint groups are not only Lie groups, but in fact algebraic groups. Now simple algebraic groups admit a (multiplicative) Jordan decomposition, and this induces an additive Jordan decomposition of the corresponding Lie algebras. We will now describe this decomposition explicitly in our setting.
We first recall that given the pair $(\mathfrak{g}, \mathfrak{k})$, there exists an Iwasawa decomposition of the form
\[
\mathfrak{g} = \mathfrak{k} \oplus \L a \oplus \L n = \L {z(k)} \oplus \mathfrak{k}_{ss} \oplus \L a \oplus \L n,
\]
such that $\L a < \L g$ is an abelian subalgebra with $\theta|_{\L a} \equiv -1$ and maximal with respect to these two properties and $\L n$ consists of {\rm ad}-nilpotent elements \cite[Prop. 6.43]{Knapp}. Now the additive refined Jordan decomposition of $\mathfrak{g}$ can be stated as follows \cite[Cor. 2.5]{Borel}:
\begin{lemma}\label{AdditiveRJC} Let $\mathfrak{g}$ be a simple Lie algebra with Iwasawa decomposition $\mathfrak{g} =\L {z(k)} \oplus \mathfrak{k}_{ss} \oplus \L a \oplus \L n$ and let $G$ be the $1$-connected Lie group with Lie algebra $\mathfrak{g}$. Then for every $X \in \mathfrak{g}$ there exist unique elements $X_c, X_k, X_a, X_n \in \mathfrak{g}$ and (in general non-unique) elements $Y_c \in \L{z(k)}, Y_k \in \mathfrak{k}_{ss}$, $Y_a \in \L a$, $Y_n \in \L n$ such that the following hold:
\begin{enumerate}[(i)]
\item $X = X_c+X_k + X_a + X_n$.
\item $X_c, X_k, X_a, X_n$ commute pairwise.
\item $X_j \in {\rm Ad(G)}(Y_j)$ for $j \in \{c, k,a,n\}$.
\end{enumerate}
\end{lemma}
Note that the elements $Y_c, Y_k, Y_a, Y_n$ are determined up to the action of ${\rm Ad}(G)$. In fact, the element $Y_c$ is uniquely determined, as can be seen as follows: Assume $Y_c, Y_c'$ were elements of $\L{z(k)}$ and ${\rm Ad}(g)(Y_c) = Y_c'$. Since $\dim \L{z(k)} \leq 1$ we have $Y_c = \lambda Y_c'$ for some $\lambda \in \mathbb{R}$. In particular, if $f: G\rightarrow \mathbb{R}$ is conjugation-invariant then $f(\exp(Y_c)) = f(\exp(Y_c'))$. If $\lambda \neq 1$ we would deduce that every continuous conjugation-invariant homogeneous function on $G$ would have to vanish on $Z(K)$; however, a non-trivial such function is given by the Guichardet-Wigner quasimorphism \cite{GuichardetWigner}. We deduce that $Y_c$ is uniquely determined by $X$ and refer to $Y_c$ as the \emph{central elliptic part} of $X$.
We need one other basic structural property of semisimple Lie algebras. Recall from \cite[Prop. 14.31]{FultonHarris} that given an irreducible abstract root system $\Sigma$ spanning a vector space $V$, the action of the Weyl group $W$ of $\Sigma$ on $V$ is irreducible. It follows that the space $V^W$ of $W$-invariants is trivial, and this conclusion extends to reducible root systems. In particular, if $\L a$ is the Cartan subalgebra of a semisimple Lie algebra and $W$ the corresponding Weyl group, then $(\L a^*)^W = \{0\}$.
\subsection{Ad-invariant Lie quasi-states on simple Lie algebras}
We can now prove the following strengthening of Theorem \ref{MainTheorem1}.
\begin{theorem}\label{MainTheorem1Ext} Let $\mathfrak{g}$ be a simple real Lie algebra and $\zeta\in {{Q}}_{\rm Ad}(\mathfrak{g})$. Then there exists a linear functional $\alpha \in \L{z(k)}^*$ such that for every $X \in \mathfrak{g}$ with central elliptic part $Y_c \in \L z(\mathfrak{k})$ we have
\begin{equation}\label{GWFormula}\zeta(X) = \alpha(Y_c).\end{equation}
In particular,
\[
\dim {{Q}}_{\rm Ad}(\mathfrak{g}) =\left\{\begin{array}{ll} 1,& \mathfrak{g} \: \textrm{ Hermitian},\\ 0, & \mathfrak{g} \: \textrm{ non-Hermitian},\end{array}\right.
\]
and thus ${Q}_{\rm Ad}(\L g) = {\mathcal Q}_{\rm Ad}(\L g)
={\mathcal Q}_{\rm qm}(\L g)$.
\end{theorem}
\begin{proof}
Let $\zeta$ be an \rm{Ad-} invariant quasi-state on $\mathfrak{g}$. By the first two parts of Lemma \ref{AdditiveRJC} we have
\[\zeta(X) = \zeta(X_c) + \zeta(X_k)+\zeta(X_a)+\zeta(X_n).\]
Since $\zeta$ is \rm{Ad-} invariant the last part of that lemma yields $\zeta(X_j) =\zeta(Y_j)$ for $j \in \{k,a,n\}$, whence
\begin{equation}\label{JordanApplication}
\zeta(X) = \zeta(Y_c)+\zeta(Y_k)+\zeta(Y_a)+\zeta(Y_n).
\end{equation}
Since $\L{z(k)}$ is abelian the restriction $\alpha := \zeta|_\L{z(k)}$ is a linear functional. It thus remains to show only that $\zeta$ vanishes on $\L a$, $\L n$ and $\L k_{ss}$.
Since $\L a$ is abelian, $\zeta|_{\L a} \in \L a^*$ is linear, and since $\zeta$ is Ad-invariant, $\zeta|_{\L a}$ is invariant under the adjoint action of the Weyl group $W := N_K(\L a)/Z_K(\L a)$. Now $(\L a^*)^W=\{0\}$ by the remark at the end of the last subsection, and hence $\zeta|_{\L a} = 0$.
Now let $X \in \mathfrak{g}$ be an arbitrary nilpotent element. By the Jacobson-Morozov theorem (see e.g. \cite[Thm. 10.3]{Knapp}), there exists an embedding $\phi: \L{sl}_2(\mathbb{R}) \hookrightarrow \mathfrak{g}$ such that $X = \varphi(e)$, where
\[
e = \left(\begin{matrix} 0 & 1\\ 0& 0 \end{matrix}\right).
\]
Now $e$ is conjugate to $2e$ inside $\L{sl}_2(\mathbb{R})$, since
\[
\left(\begin{matrix} \sqrt 2 & 0\\ 0& \frac{1}{\sqrt 2} \end{matrix}\right)\cdot\left(\begin{matrix} 0 & 1\\ 0& 0 \end{matrix}\right)\cdot\left(\begin{matrix} \frac{1}{\sqrt 2} & 0\\ 0& \sqrt 2 \end{matrix}\right) = \left(\begin{matrix} 0 & 2\\ 0& 0 \end{matrix}\right),
\]
and hence $X = \phi(e)$ is conjugate to $2X = \phi(2e)$ inside $\L g$. Thus invariance of $\zeta$ implies $\zeta(X) = \zeta(2X) = 2\cdot \zeta(X)$, whence $\zeta(X) = 0$. This shows that $\zeta$ vanishes on all nilpotent elements of $\mathfrak{g}$, and in particular on $\L n$.
The fact that $\zeta$ vanishes on $\mathfrak{k}_{ss}$ was already observed in \cite{EP09}. We repeat the argument here for completeness: Since $\mathfrak{k}_{ss}$ is compact, every Ad-orbit intersects any given maximal toral subalgebra $\L t < \L k_{ss}$, whence $\zeta|_{\mathfrak{k}_{ss}}$ is determined by $\zeta|_{\L t}$. Since $\L t$ is abelian we have $\zeta|_{\L t}\in \L t^*$ and moreover $\zeta|_{\L t}$ is invariant under the adjoint action of $W_{\mathfrak{k}} := N_K(\L t)/Z_K(\L t)$. Since $\mathfrak{k}_{ss}$ is semisimple we have $(\L t^*)^{W_{\mathfrak{k}}} = \{0\}$ (again by the remark at the end of the last subsection) and thus $\zeta|_{\L t} = 0$ and consequently also $\zeta|_{\mathfrak{k}_{ss}} = 0$. The theorem follows.
\end{proof}
\begin{remark}
\begin{enumerate}
\item The most substantial ingredient in the proof of Theorem \ref{MainTheorem1Ext} is the vanishing of $\zeta$ along nilpotent elements, which we deduced from the Jacobson-Morozov theorem. If one is willing to assume continuity of $\zeta$ then one can give a more elementary proof of this result which does not invoke the Jacobson-Morozov theorem. Instead one uses the fact that
for every $X \in \L n$ there exists a sequence $(g_n)$ in $G$ such that \[\lim_{n \rightarrow \infty}{\rm Ad}(g_n)(X) = 0.\] Assuming continuity of $\zeta$ this is enough to deduce that
\[
0 = \zeta(\lim_{n \rightarrow \infty}{\rm Ad}(g_n)(X)) = \lim_{n \rightarrow \infty}\zeta({\rm Ad}(g_n)(X)) = \lim_{n \rightarrow \infty}\zeta(X) = \zeta(X).
\]
\item In the case of a simple Hermitian Lie algebra, Theorem \ref{MainTheorem1Ext} gives an explicit formula for all Ad-invariant Lie quasi-states on $\mathfrak{g}$. Now let $G$ denote the simply-connected Lie group associated with $\mathfrak{g}$ and $Z(K) \cong \mathbb{R}$ denote the analytic subgroup of $G$ with Lie algebra $\L{z(k)}$. Then for every measurable homomorphism $\chi: Z(K) \rightarrow \mathbb{R}$ there is a unique homogeneous quasimorphism $f$ on $G$ subject to the normalisation $f|_{Z(K)} = \chi$, and by Theorem \ref{MainTheorem1Ext} we have
\[
f(\exp(X)) = \chi(\exp(Y_c)).
\]
A global version of this formula was first pointed out in \cite{Surface}.\\
\item The classification of Lie quasi-states on $\mathfrak{g}$ does not by itself provide a classification of integrated Lie quasi-states on $G$, since there is no general argument which would ensure that two integrated Lie quasi-states with the same directional derivatives coincide. (Such an argument would work e.g. if the Lie quasi-states in question were of class $C^1$, but in that case they are necessarily linear anyway.) It therefore requires additional effort to prove the following global version of Theorem \ref{MainTheorem1Ext}.
\end{enumerate}
\end{remark}
\begin{theorem}\label{GlobalThm} Let $G$ be a connected reductive Lie group and $f: G \rightarrow \mathbb{R}$ a conjugation-invariant integrated Lie quasi-state. Then $f$ is a homogeneous quasimorphism.
\end{theorem}
\begin{proof} We first observe that we may assume that $G$ is simply-connected. Indeed, if $G$ is arbitrary and $\widetilde{G}$ is its universal covering group, then every conjugation-invariant integrated Lie quasi-state on $G$ lifts to $\widetilde{G}$, and if this can be shown to be a homogeneous quasimorphism, then it descends to a homogeneous quasimorphism on $G$, see \cite{BSH2}. Now a simply-connected reductive Lie group $G$ is a product of abelian and simple factors. Since every integrated Lie quasi-state on an abelian group is a homomorphism, it suffices to prove the theorem for a general simply-connected simple Lie group $G$.
In this case, we consider the Lie algebra $\mathfrak{g}$ of $G$ and and fix an Iwasawa decomposition $\mathfrak{g} =\L k \oplus \L a \oplus \L n$. Let $K$ be a maximal {\rm Ad}-compact subgroup of $G$ with Lie algebra $\L k$ and let $A$ and $N$ be the analytic subgroups associated with $\L a$ and $\L n$. Since $G$ is connected and $K$ is a deformation retract of $G$, also $K$ is connected. Since $K$ is compact-times-abelian and $A$ and $N$ are nilpotent and all three are connected the restricted exponential functions $\L k \rightarrow K$, $\L a \rightarrow A$ and $\L n \rightarrow N$ are all onto. It thus follows from Theorem \ref{MainTheorem1Ext} that there exists a homogeneous quasimorphism $\phi: G \rightarrow \mathbb{R}$ such that $f_0 := f-\phi$ vanishes on $K$, $A$ and $N$. It suffices to show that $f_0 \equiv 0$.
To this end we first observe that since $Z(G) \subset K$ we have $f_0|_{Z(G)}\equiv 0$, and hence $f$ factors through a quasimorphism on $G_0 := {\rm Ad}(G) = G/Z(G)$ which vanishes on the images $K_0$, $A_0$ and $N_0$ of $K$, $A$ and $N$ in $G_0$. The group $G_0$ (unlike $G$) is automatically algebraic, whence admits a multiplicative Jordan decomposition: Every $g \in G$ can be written as a product of elements $g_k$, $g_a$ and $g_n$ which pairwise commute and are conjugate to elements in $K_0$, $A_0$ and $N_0$ respectively. It follows that $f_0(g) = f_0(g_k) + f_0(g_a)+f_0(g_n) = 0$, which finishes the proof.
\end{proof}
\section{Semidirect products and solvable examples}\label{SecAlmostAbelian}
\label{sec:semidirect}
\subsection{Semidirect products and normalized Lie quasi-states}\label{SemidirectGeneral}
The goal of this section is to establish Theorem \ref{MainTheorem2} concerning Ad-invariant Lie quasi-states on certain solvable Lie algebras. Before we turn to the specific setting of that theorem we collect a few general facts about semidirect product algebras of the form $\mathfrak{g} = V \rtimes \mathfrak{h}$, where $V\lhd \mathfrak{g}$ is an abelian ideal and $\mathfrak{h} < \mathfrak{g}$ is a complementary subalgebra. We denote by $\rho: \mathfrak{h} \rightarrow \mathfrak{gl}(V)$ the restriction of the adjoint action of $\mathfrak{h}$ to $V$; then $\rho$ is a representation of $\mathfrak{h}$ and $\mathfrak{g} = \mathfrak{g}(V, \mathfrak{h}, \rho)$ is uniquely determined by the triple $(V, \mathfrak{h}, \rho)$.
Note that if $\alpha \in V^*$ is any linear functional, then the map $(v, X) \mapsto \alpha(v)$ defines a linear functional, hence a Lie quasi-state on $\mathfrak{g}$. Similarly, if $\zeta \in Q(\mathfrak{h})$ is a Lie quasi-state, then so is the map $(v, X)\mapsto \zeta(X)$. This is because the projection onto the second coordinate is a Lie algebra homomorphism and thus $[(v,X), (w, Y)] = 0$ implies $[X,Y] = 0$. In particular, we can consider $V^*$ and $Q(\mathfrak{h})$ as subspaces of $Q(\mathfrak{g})$.
We define subspaces $Q_0(\mathfrak{g}) \subset Q(\mathfrak{g})$ and $\mathcal Q_0(\mathfrak{g}) \subset \mathcal Q(\mathfrak{g})$ by
\[
Q_{0}(\mathfrak{g}) = \big\{\zeta \in Q(\mathfrak{g})\mid \forall v \in V, X \in \mathfrak{h}:\;\zeta(v,0) = \zeta(0,X)=0\big\}.
\]
and $\mathcal Q_0(\mathfrak{g}) := Q_{0}(\mathfrak{g}) \cap \mathcal Q(\mathfrak{g})$. We observe:
\begin{lemma}\label{SemidirectLemma}
Let $\mathfrak{g} = \mathfrak{g}(V, \mathfrak{h}, \rho)$ as above. Then $Q(\mathfrak{g})$ can be written as the internal direct sum
\begin{equation}\label{NormalizedQS}
Q(\mathfrak{g}) = Q_{0}(\mathfrak{g}) \oplus V^* \oplus Q(\mathfrak{h}),\end{equation}
and similarly we have
\[\mathcal Q(\mathfrak{g}) = \mathcal Q_{0}(\mathfrak{g}) \oplus V^* \oplus \mathcal Q(\mathfrak{h}).\]
\end{lemma}
\begin{proof} Given $\zeta \in Q(\mathfrak{g})$, the restrictions $\zeta|_{V \times \{0\}}$ and $\zeta|_{\{0\} \times \L h}$ are Lie quasi-states. Now $V \times \{0\} \cong V$ is abelian, hence $\alpha(v) :=\zeta(v,0) \in V^*$. Similarly, $\psi(X) := \zeta(0,X)$ defines a Lie quasi-state on $\mathfrak{h}$. Now let
\[
\zeta_0(v,X) := \zeta(v,X) - \alpha(v) - \psi(X).
\]
Then $\zeta_0 \in Q_{0}(\mathfrak{g})$ and $\zeta = \zeta_0+\alpha+\psi$, whence $Q(\mathfrak{g}) = Q_{0}(\mathfrak{g})+V^* +Q(\mathfrak{h})$. Since the pairwise intersections of $Q_{0}(\mathfrak{g})$, $V^*$ and $Q(\mathfrak{h})$ (considered as subspaces of $Q(\mathfrak{g})$) are trivial, the sum is direct. Finally, if $\zeta$ is continuous, then so are $\zeta_0, \alpha$ and $\psi$ in the above decomposition.
\end{proof}
In the sequel we will refer to an element $\zeta \in Q_{0}(\mathfrak{g})$ as a \emph{normalized Lie quasi-state} on $\mathfrak{g}$. We record the following property of continuous normalized quasi-states for later use:
\begin{lemma}\label{Sublinearity}
Let $\zeta \in \mathcal Q_{0}(\mathfrak{g})$ be a continuous normalized Lie quasi-state and let $X \in \mathfrak{h}$. Then the function $\zeta_X: V \rightarrow \mathbb{R}$ given by
$\zeta_X(v):= \zeta(v,X)$ is sublinear in the sense that
\[
\lim_{v \rightarrow \infty} \frac{\zeta_X(v)}{\|v\|} \rightarrow 0
\]
for any norm on $V$.
\end{lemma}
\begin{proof} Since $\zeta$ is continuous, it is uniformly continuous on the product of unit balls. Since it is moreover normalized, we can compute
\[
\lim_{v \rightarrow \infty} \frac{\zeta_X(v)}{\|v\|} = \lim_{v \rightarrow \infty} \frac{\zeta(v, X)}{\|v\|} = \lim_{v \rightarrow \infty} \zeta(v/\|v\|, X/\|v\|) = \lim_{v \rightarrow \infty} \zeta(v/\|v\|, 0)=0.
\]
\end{proof}
\subsection{Classification of Lie quasi-states when $\mathfrak{h}$ is one-dimensional}
We now specialize to our main case of interest which is given by $\mathfrak{h} = \mathbb{R}$, i.e. $\mathfrak{g} = V \rtimes \mathbb{R}$ with $V \lhd \mathfrak{g}$ abelian. (We will return to the more general situation in the next section.) In this situation we will adopt the following notation: We denote by $\phi \in {\rm End}(V)$ the restriction of the adjoint action of the generator $1 \in \mathbb{R}$ to $V$ so that
\[
[(v,s), (w,t)] = (s\phi(w)-t\phi(v),0).
\]
We then write $\mathfrak{g} = \mathfrak{g}_\phi$ and refer to $\phi$ as the \emph{underlying endomorphism} of $\mathfrak{g}$. Note that if $\phi = 0$ then $\mathfrak{g}_\phi = V\oplus \mathbb{R}$ is abelian and thus $Q(\mathfrak{g}) = \mathfrak{g}^*$. Thus we are going to assume from now on that $\phi \neq 0$.
Given a Lie quasi-state $\zeta \in {Q}(\mathfrak{g})$ we refer to the linear functional $\zeta_V \in V^*$ given by
\[
\zeta_V(v):=\zeta(v,0), \quad \textrm{for all $v \in V$ },
\]
as the \emph{canonical character} of $\zeta$. The following theorem provides a full classification of Lie quasi-states on $\mathfrak{g}$. We will denote by $U$ the kernel and by $W$ the image of the endomorphism $\phi$ underlying $\mathfrak{g}$.
\begin{theorem}\label{SLTheorem}
A function $\zeta: \mathfrak{g} \rightarrow \mathbb{R}$ is a Lie quasi-state of canonical character $\alpha \in V^*$ if and only if there exists a function $c: W \rightarrow \mathbb{R}$ such that
\begin{equation}\label{QSFormulaSemidirect1}
\zeta(v,t) = \left\{\begin{array}{ll}c(\phi(v)/t)\cdot t + \alpha(v) & t \neq 0\\ \alpha(v)& t= 0\end{array}\right.
\end{equation}
The quasi-state $\zeta$ given by \eqref{QSFormulaSemidirect1} is continuous if and only if the function $c$ is continuous and sublinear.
\end{theorem}
\begin{proof} Assume first that $\zeta \in Q_{0}(\mathfrak{g})$ is normalized. Then $\zeta(v,0) = 0$ for every $v \in V$ and if $\widetilde{c}: V \rightarrow \mathbb{R}$ is given by $\widetilde{c}(v) := \zeta(v,1)$, then for every $v \in V$ and $t \neq 0$ we have
\[
\zeta(v,t) = \widetilde{c}(v/t)\cdot t.
\]
Moreover, since $\zeta$ is normalized we have $\widetilde{c}(0) = \zeta(0,1) = 0$. If $u \in U = \ker(\phi)$, then $(u, 0)$ is central in $\mathfrak{g}_\phi$, hence
\[
\widetilde{c}(v+u) = \zeta(v+u, 1) = \zeta((v,1)+(u,0)) = \zeta(v,1)+\zeta(u,0) = \zeta(v,1) = \widetilde{c}(v).
\]
It follows that $\widetilde{c}$ descends to a function on $V/U$. Since the latter space is isomorphic to $W$ via $\phi$, we conclude that there exists a function $c: W \rightarrow \mathbb{R}$ with $c(0) = 0$ such that
\begin{equation}\label{QSCandidate}
\zeta(v,t) = \left\{\begin{array}{ll}c(\phi(v)/t)\cdot t & t \neq 0\\ 0 & t= 0.\end{array}\right.
\end{equation}
Conversely, assume that $\zeta: \mathfrak{g} \rightarrow \mathbb{R}$ is given by \eqref{QSCandidate} for some function $c: W \rightarrow \mathbb{R}$ with $c(0) = 0$. Then, by definition, $\zeta(v, 0) = \zeta(0,t) = 0$ for all $v\in V$ and $t\in \mathbb{R}$. Now assume that $(u, s)$ and $(v,t)$ commute, i.e. $s\phi(v) = t\phi(u)$. We claim that $\zeta(u,s) + \zeta(v,t) = \zeta(u+v, s+t)$. There are several cases:
\begin{enumerate}[\textsc{Case} 1.]
\item If $0 \not \in \{s,t, s+t\}$ then we have
\[
\frac{\phi(u)}{s} = \frac{\phi(v)}{t} = \frac{\phi(u+v)}{s+t},
\]
and we deduce that
\begin{eqnarray*}
\zeta(u,s)+\zeta(v,t) &=& c(\phi(u)/s) \cdot s + c(\phi(v)/t)\cdot t = c(\phi(u+v)/(s+t)) \cdot s + c(\phi(u+v)/(s+t)) \cdot t\\
&=& c(\phi(u+v)/(s+t)) \cdot (s+t) = \zeta(u+v, s+t).
\end{eqnarray*}
\item If $s = 0$, then the condition $s\phi(v) = t\phi(u)$ implies either $t=0$ or $\phi(u) = 0$. In the former case we have
$\zeta(u,s)+\zeta(v,t) = 0+0 = 0 = \zeta(u+v, s+t)$, and in the latter case we have
\[
\zeta(u,s)+\zeta(v,t) = 0+ c(\phi(v)/t)\cdot t = c((0+\phi(v))/(0+t)) \cdot (0+t) = c((\phi(u)+\phi(v))/(s+t)) \cdot (s+t) =\zeta(u+v, s+t).
\]
\item If $t = 0$ then we can argue as in \textsc{Case 2}, since the roles of $s$ and $t$ are symmetric.
\item It remains to deal with the case $s=-t \neq 0$. In this case,
\[
\frac{\phi(u)}{s} = \frac{\phi(v)}{t},
\]
and thus
\begin{eqnarray*}
\zeta(u,s)+\zeta(v,t) &=& c(\phi(u)/s) \cdot s + c(\phi(v)/t)\cdot t = c(\phi(u)/s) \cdot s + c(\phi(u)/s)\cdot t \\
&=& c(\phi(u)/s) \cdot (s+t) = 0 = \zeta(u+v, 0) = \zeta(u+v, s+t).
\end{eqnarray*}
\end{enumerate}
It follows that $\zeta$ is a normalized Lie quasi-state. We have thus shown that the normalized Lie quasi-states on $\mathfrak{g}$ are exactly the functions of the form \eqref{QSCandidate}, where $c: W \rightarrow \mathbb{R}$ is an arbitrary function with $c(0) = 0$.
This finishes the classification of normalized Lie quasi-states on $\mathfrak{g}$. As for general Lie quasi-states, by Lemma \ref{SemidirectLemma} these are of the form $\zeta(v,t) = \zeta_0(v,t)+\alpha(v)+ c_0\cdot t$, where $\zeta_0 \in Q_0(\mathfrak{g})$ is normalized, $\alpha \in V^*$ denotes the central character of $\zeta$ and $c_0 \in \mathbb{R}$ is given by $c_0 = \zeta(0,1)$. Since $\zeta_0$ is of the form \eqref{QSCandidate} for some $c: W \rightarrow \mathbb{R}$ with $c(0) = 0$, it follows that $\zeta$ is of the form described in the statement of the theorem.
As for continuity, if the quasi-state $\zeta$ given by \eqref{QSFormulaSemidirect1} is continuous, then in particular the map $v \mapsto c(\phi(v)) = \zeta(v, 1)$ is continuous, whence $c$ is continuous. Moreover, continuity of $\zeta$ at $t = 0$ implies sublinearity of $c$ by Lemma \ref{Sublinearity}. Conversely, if $c$ is continuous, then $\zeta$ is obviously continuous on $\mathfrak{g} \setminus(V \times \{0\})$, and if $c$ is moreover sublinear, then $\zeta$ is continuous on all of $\mathfrak{g}$.
\end{proof}
In the sequel we write $\zeta_{\alpha, c}$ for the Lie quasi-state on $\mathfrak{g}$ given by equation \eqref{QSFormulaSemidirect1}. If we denote by $C_{sl}(W) < C(W)$ the space of sub-linear continuous functions on $W$ then we have:
\begin{corollary} If $\mathfrak{g} = \mathfrak{g}_\phi$ for some $\phi \neq 0$ and $W = {\rm im}(\phi)$ then the map
\[
V^*\oplus C_{sl}(W) \rightarrow \mathcal Q(\mathfrak{g}), \quad (\alpha, c)\mapsto \zeta_{\alpha,c}.
\]
is an isomorphism and thus ${\mathcal Q}(\mathfrak{g})$ is infinite-dimensional.
\end{corollary}
\subsection{The Ad-invariant case}
We keep the notation of the previous subsection, in particular $V$ is an abelian Lie algebra, $\phi \in {\rm End}(V) \setminus\{0\}$ and
$\L g = \L g_{\phi} = V \rtimes \mathbb{R}$ with ${\rm ad}(1)|_V = \phi$. Moreover, we abbreviate $W := {\rm im}(\phi)$ and given $\alpha \in V^*$ and $c \in C_{sl}(W)$ we denote by $\zeta_{\alpha, c}$ the Lie quasi-state on $\mathfrak{g}$ given by \eqref{QSFormulaSemidirect1}. The following theorem classifies those quasi-states $\zeta_{\alpha,c}$ which are invariant under the action of the adjoint group of $\mathfrak{g}$.
\begin{theorem}\label{SolvableAdInv}
Let $\mathfrak{g} = \mathfrak{g}_\phi$ as above. Given $\alpha \in V^*$ and $c \in C_{sl}(W)$, the Lie quasi-state $\zeta_{\alpha, c}$ is ${\rm Ad}$-invariant if and only if the following hold:
\begin{enumerate}[(i)]
\item $\im \phi \subset \ker \alpha$.
\item $c: W \rightarrow \mathbb{R}$ descends to a function $c : W/\im \phi^2 \to \mathbb{R}$.
\end{enumerate}
\end{theorem}
\begin{proof} The function $\zeta_{\alpha,c}$ is ${\rm Ad}$-invariant if and only if for all $(w,s), (v,t) \in \mathfrak{g}$ we have
\begin{equation}\label{AdInvNew}
\zeta_{\alpha, c}(\exp(\ad(w,s))(v,t)) = \zeta_{\alpha, c}(v,t)
\end{equation}
In fact, by continuity of $\zeta_{\alpha, c}$ we may assume that $s \neq 0 \neq t$. Note that
\[
\ad(w,s)(v,t) = \left(\begin{matrix} s\phi & -\phi(w)\\0&0\end{matrix}\right) \cdot \left(\begin{matrix}v\\ t\end{matrix}\right),
\]
and hence for all $k >0$,
\[
\ad(w,s)^k(v,t) = \left(\begin{matrix} s\phi & -\phi(w)\\0&0\end{matrix}\right)^k \cdot \left(\begin{matrix}v\\ t\end{matrix}\right) = \left(\begin{matrix} s^k\phi^k & -s^{k-1}\phi^k(w)\\0&0\end{matrix}\right) \cdot \left(\begin{matrix}v\\ t\end{matrix}\right) =
\left(\begin{matrix}s^k\phi^k(v)-ts^{k-1}\phi^k(w)\\0 \end{matrix}\right).
\]
We deduce that
\begin{eqnarray*}
\exp(\ad(w,s))(v,t) &=& \left(\begin{matrix} 1+\sum_{k=1}^\infty \frac{1}{k!} s^k\phi^k & -\sum_{k=1}^\infty \frac{1}{k!} s^{k-1}\phi^k(w)\\0&1\end{matrix}\right) \cdot \left(\begin{matrix}v\\ t\end{matrix}\right) \\
&=& \left(\begin{matrix} \exp(s\phi) & \frac{1}{s}(\exp(s\phi)w - w)\\0&1\end{matrix}\right) \cdot \left(\begin{matrix}v\\ t\end{matrix}\right) \\
&=& \Big(\exp(s\phi)v - \frac{t}{s}\big(\exp(s\phi)w-w\big),t\Big),
\end{eqnarray*}
and thus the condition \eqref{AdInvNew} amounts to
\begin{eqnarray}\label{AdInvNew2}
&&c\Big(\frac{\exp(s\phi)\phi(v)}{t} - \frac{1}{s}\Big(\exp(s\phi)\phi(w)-\phi(w)\Big)\Big)\cdot t + \alpha(\exp(s\phi)v) - t \cdot \alpha\Big(\frac{1}{s}\Big(\exp(s\phi)w-w\Big)\Big) \\
&=& \nonumber c\Big(\frac{\phi(v)}{t}\Big)\cdot t + \alpha(v).
\end{eqnarray}
We now show that \eqref{AdInvNew2} implies (i) and (ii) above. Specialize \eqref{AdInvNew2} to $t:=1$ and then let $s \rightarrow 0$. Since
\[
\lim_{s \to 0} \frac{\exp(s\phi)u - u}{s} = \phi(u) \quad \textrm{for all $u \in V$},
\]
we obtain
\begin{equation}\label{AdInvNew3}
c\big(\phi(v) - \phi^2(w)\big) - \alpha(\phi(w)) = c(\phi(v)).
\end{equation}
Replacing $w$ by $rw$ for some $r > 0$ and dividing by $r$ we obtain
\[
\frac 1 r \cdot c\big(\phi(v) - r\cdot \phi^2(w)\big) - \alpha(\phi(w)) = \frac{c(\phi(v))}{r}.
\]
If $\phi^2(w) \neq 0$, then $\frac 1 r \|\phi(v) - r\cdot \phi^2(w)\| \rightarrow \|\phi^2(w)\|$ and thus sublinearity of $c$ implies that the first term converges to $0$ as $r \rightarrow \infty$. This in term implies $\alpha(\phi(w)) = 0$. If, on the other hand, $\phi^2(w) = 0$, then \eqref{AdInvNew3} simplifies to $\alpha(\phi(w)) = \alpha(v)$ for all $v \in V$ and choosing $v =0$ yields $\alpha(\phi(w)) = 0$ also in this case. We have thus established (i). This in turn allows us to rewrite \eqref{AdInvNew3} as
\begin{equation}\label{AdInvNew4}
c\big(\phi(v) - \phi^2(w)\big)= c(\phi(v)).
\end{equation}
which is (ii). Hence we have seen that Ad-invariance implies both (i) and (ii).
Towards the converse implication, observe that
\[
\exp(s\phi)\phi(v) \in \phi(v) + \im \phi^2
\]
and similarly $\exp(s\phi)\phi(w) \in \phi(w) + \im \phi^2$. It follows that
\[\frac{\exp(s\phi)\phi(v)}{t} - \frac{1}{s}\Big(\exp(s\phi)\phi(w)-\phi(w)\Big) \in \frac{\phi(v)}{t} + \im \phi^2
\]
and thus
\[
c\left(\frac{\exp(s\phi)\phi(v)}{t} - \frac{1}{s}\Big(\exp(s\phi)\phi(w)-\phi(w)\Big)\right)= c\left(\frac{\phi(v)}{t} \right).
\]
for any $c$ satisfying (ii). Thus if (ii) holds, then the Ad-invariance condition \eqref{AdInvNew2} simplifies into
\[\alpha\left(\exp(s\phi)(v) -\frac{t}{s}\Big(\exp(s\phi)w-w\Big)\right)=\alpha(v).\]
Since
\[
\exp(s\phi)(v) -\frac{t}{s}\Big(\exp(s\phi)w-w\Big) \in v + \im \phi,
\]
this follows from (i), and hence (i) and (ii) together imply Ad-invariance.
\end{proof}
The theorem allows us to determine whether the algebra $\mathfrak{g}_\phi$ is Lie quasi-state rigid. We can summarize the result as follows:
\begin{corollary}\label{SplittingNew} The following are equivalent for the Lie algebra $\mathfrak{g} = \mathfrak{g}_\phi$:
\begin{enumerate}[(i)]
\item $\mathfrak{g}_\phi$ is not Lie quasi-state rigid.
\item $\mathcal Q_{Ad}(\mathfrak{g})$ is infinite-dimensional (and, in fact, of uncountable dimension).
\item $\im \phi^2 \subsetneq \im \phi$.
\item There exists a $\phi$-invariant splitting $V = V_0 \oplus V_1$ with $n := \dim V_0 \geq 2$ such that in some suitable basis of $V_0$ the restriction $\phi|_{V_0}$ is represented by the matrix
\begin{equation}\label{phin}
\phi_n = \left(
\begin{matrix}
0&1& &&&&\\
&0&1&&&\\
&&\ddots&\ddots&&\\
&&&0&1\\
&&&&0
\end{matrix}
\right)
\end{equation}
\item $\mathfrak{g}_\phi$ surjects onto the three-dimensional Heisenberg algebra $\L h_3$.
\end{enumerate}
\end{corollary}
Note that the equivalence (i)$\Leftrightarrow$(v) is precisely Theorem \ref{MainTheorem2} from the introduction. The proof of Corollary \ref{SplittingNew} will make use of some basic functoriality properties of Lie quasi-states which we list in the following lemma:
\begin{lemma}\label{QuotientMaps} Let $p: \L g \rightarrow \L q$ be a surjective Lie algebra homomorphism and let $\zeta \in \mathcal Q(\L q)$ be a quasi-state.
\begin{enumerate}[(i)]
\item The functions $p^*\zeta: \mathfrak{g} \rightarrow \mathbb{R}$ given by
\[
p^*\zeta(X) := \zeta(p(X))
\]
is a Lie quasi-state on $\L g$.
\item If $\zeta$ is Ad-invariant, then so is $p^*\zeta$.
\item If $\zeta \in Q_{qm}(\L q)$, then $p^*\zeta \in Q_{qm}(\mathfrak{g})$
\end{enumerate}
\end{lemma}
\begin{proof} (i) If $X, Y \in \mathfrak{g}$ with $[X, Y] = 0$, then $[p(X), p(Y)] = p([X, Y]) = 0$ and thus
\[
p^*\zeta(X+Y) = \zeta(p(X) + p(Y)) = \zeta(p(X)) + \zeta(p(Y)) = p^*\zeta(X) + p^*\zeta(Y),
\]
and $\mathbb{R}$-linearity is obvious.
(ii) Assume that $\zeta$ is Ad-invariant. Then for all $X, Y \in \mathfrak{g}$ we have
\begin{eqnarray*}
p(\Ad(\exp(X))(Y)) &=& p(\exp(\ad(X))(Y)) \\
&=&p\left( \sum_{k=0}^\infty \frac{1}{k!} \ad(X)^k(Y)\right)\\
&=&\sum_{k=0}^\infty \frac{1}{k!} \ad(p(X))^k(p(Y))\\
&=&\exp(\ad(p(X))(p(Y))\\
&=&\Ad(\exp(p(X)))(p(Y))
\end{eqnarray*}
and thus
\[
p^*\zeta(\Ad(\exp(X))(Y)) = \zeta(p(\Ad(\exp(X)(Y)) ) =\zeta(\Ad(\exp(p(X)))(p(Y))) = \zeta(p(Y)) = p^*\zeta(Y),
\]
which implies Ad-invariance of $p^*\zeta$.
(iii) Denote by $Q$ and $G$ the $1$-connected Lie groups associated with $\L q$ and $\L g$ respectively and denote by $\widehat{p}: G \rightarrow Q$ the lift of $p$. If $\zeta \in Q_{qm}(\L q)$ is the directional derivative of a homogeneous quasimorphism $f: Q \rightarrow \mathbb{R}$, then
\[
p^*\zeta = p^*(f \circ \exp) = f\circ \exp \circ p = f \circ \widehat{p} \circ \exp = \widehat{p}^*f \circ \exp,
\]
whence $p^*\zeta$ is the directional derivative of $\widehat{p}^*f$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{SplittingNew}] We first show the equivalence of the first three conditions:
(i) $\Rightarrow$ (iii): Assume $\im(\phi^2) = \im(\phi)$ and let $\zeta_{\alpha, c} \in \mathcal Q_{Ad}(\mathfrak{g})$. Theorem \ref{SolvableAdInv} then implies that $c$ is constant, whence $\zeta_{\alpha, c}$ is linear. This show that $\mathfrak{g}$ is Lie quasi-state rigid.
(iii) $\Rightarrow$ (ii): If (iii) holds, then $\dim(\im \phi/\im \phi^2)>0$, whence $C_{sl}(\im \phi/\im \phi^2)$ is of uncountable dimension, and the map $c\mapsto \zeta_{0, c}$ defines a linear embedding of $C_{sl}(\im \phi/\im \phi^2)$ into $\mathcal Q_{\rm Ad}(\mathfrak{g}_\phi)$.
(ii) $\Rightarrow$ (i): We already observed in the introduction that if $\mathfrak{g}$ is quasi-state rigid, then $\dim \mathcal Q_{\rm Ad}(\mathfrak{g}_\phi) < \infty$.
On the other hand, the equivalence (iii) $\Leftrightarrow$ (iv) is an immediate consequence of the real Jordan decomposition of $\phi$. Namely, $\im \phi^2 \subsetneq \im \phi$ if and only if $\phi$ contains a nilpotent Jordan block of size $\geq 2$. We have thus established equivalence of the first four properties.
Before we turn to condition (v) we observe that if $\phi_2 \in {\rm End}(\mathbb{R}^2)$ is given by
\[
\phi_2 = \left(\begin{matrix} 0 & 1\\ 0 &0 \end{matrix}\right),
\]
then $\mathfrak{g}_{\phi_2}$ is precisely the three-dimensional Heisenberg algebra $\L h_3$. It thus follows from the implication (iv)$\Rightarrow$(ii), that $\mathcal Q_{Ad}(\L h_3)$ is infinite-dimensional. Now we can establish the remaining implications:
(v) $\Rightarrow$(ii): Assume $p: \mathfrak{g} \rightarrow \L h_3$ is a surjection. By Lemma \ref{QuotientMaps}, $p$ induces a map $p^*: \mathcal Q_{Ad}(\L h_3) \rightarrow \mathcal Q_{Ad}(\L g)$, and since $p$ is surjective, this map is injective. It follows that $\dim \mathcal Q_{Ad}(\L g) \geq \dim \mathcal Q_{Ad}(\L h_3) = \infty$.
(iv) $\Rightarrow$ (v): Assume that $\mathfrak{g}_\phi$ satisfies (iv) and let $V_0$, $V_1$ be as described in (iv). Then $V_1 \lhd \mathfrak{g}_\phi$ is an ideal, and $\mathfrak{g}_\phi/V_1 \cong \mathfrak{g}_{\phi_n}$, where $\phi_n \in {\rm End}(\mathbb{R}^n)$ is given as in (iv). By definition the Lie algebra $\mathfrak{g}_{\phi_n}$ has a basis of the form $(X, Y_1, \dots, Y_n)$ with bracket relations
\[
[X, Y_1] = Y_2,\; \dots, \; [X, Y_{n-1}] = Y_n,\; [X, Y_n] = 0,\; [Y_i, Y_j] = 0\quad(i,j = 1, \dots, n).
\]
In this basis the center $\L z(\mathfrak{g}_{\phi_n})$ of $\mathfrak{g}_{\phi_n}$ is given by $\L z(\mathfrak{g}_{\phi_n}) = \mathbb{R} \cdot Y_n$, and we have $\mathfrak{g}_{\phi_n}/\L z(\mathfrak{g}_{\phi_n}) = \mathfrak{g}_{\phi_{n-1}}$. Thus if $\mathfrak{g}_{\phi}$ satisfies (iv), then we have a chain of surjections
\[
\mathfrak{g}_{\phi} \rightarrow \mathfrak{g}_{\phi_n} \rightarrow \mathfrak{g}_{\phi_{n-1}} \rightarrow \dots \rightarrow \mathfrak{g}_{\phi_3} \rightarrow \mathfrak{g}_{\phi_2} = \L h_3,
\]
and thus $\mathfrak{g}_\phi$ surjects onto $\L h_3$. This finishes the proof.
\end{proof}
\begin{remark}
Recall that if the space of homogeneous quasimorphisms modulo homomorphisms on some group $G$ is infinite-dimensional, then its dimension is at least the cardinality of the continuum. Indeed, this follows from the fact that the second bounded cohomology of a group is a Banach space with respect to the Gromov norm (and thus any Hamel basis has either finite or uncountable cardinality). This should be compared to the equivalence (i)$\Leftrightarrow$(ii) above, which allows the even stronger conclusion that the space $\mathcal Q_{\rm Ad}(\mathfrak{g}_\phi)/{\rm Hom}(\mathfrak{g}_\phi, \mathbb{R})$ is of uncountable dimension as soon as it is non-trivial. We do not know whether for an arbitrary Lie algebra $\mathfrak{g}$ the dimension of $\mathcal Q_{\rm Ad}(\mathfrak{g})/{\rm Hom}(\mathfrak{g}, \mathbb{R})$ can be countable-dimensional. For a solvable Lie algebra $\mathfrak{g}$ we do not even know whether it can be finite-dimensional without being $0$.
\end{remark}
\subsection{Low-dimensional examples}\label{SolvableExample}
Let us summarize what the results obtained so far imply for low-dimensional Lie algebras. Every one-dimensional Lie algebra is abelian and thus rigid. Every non-abelian two-dimensional Lie algebra is isomorphic to the Lie algebra of the $(ax+b)$-group. This Lie algebra can be realized as $\mathfrak{g}_{\phi}$, where $\phi={\rm Id}_{\mathbb{R}}$, and thus is rigid by the criterion from Corollary \ref{SplittingNew}.(iii).
Now assume that $\mathfrak{g}$ is a three-dimensional real Lie algebra. If $\mathfrak{g}$ contains a simple subalgebra, then it is actually simple (namely $\mathfrak{g}$ is one of the two three-dimensional simple Lie algebras $\L{sl}_2(\mathbb{R})$ or $\L{su}(2)$), and thus rigid. Otherwise $\mathfrak{g}$ is a solvable $3$-dimensional Lie algebra. We now recall the classification of such Lie algebras.
In order to determine a three-dimensional Lie algebra $\mathfrak{g}$ it suffices to compute the commutators $[X_3, X_1]$, $[X_3, X_2]$ and $[X_1, X_2]$ for some basis $(X_1, X_2, X_3)$ of $\mathfrak{g}$. The following table thus determines four families of three-dimensional Lie algebras:
\begin{center}
\begin{tabular}{ l | c | c | c }
Name of the Lie algebra in \cite{deGraaf} & $[X_3, X_1]$ & $[X_3, X_2]$&$[X_1, X_2]$ \\
\hline
$L^1$ & $0$ & $0$ & $0$ \\
$L^2$ & $X_1$ & $X_2$ & $0$ \\
$L^3_a$ ($a \in \mathbb{R}$)& $X_2$ & $aX_1+X_2$ & $0$ \\
$L^4_a$ ($a \in \mathbb{R}$) & $X_2$ & $aX_1$ & $0$
\end{tabular}
\end{center}
According to \cite{deGraaf}, every three-dimensional solvable Lie algebra is isomorphic to a Lie algebra in one of these families. Moreover, Lie algebras from different families are never isomorphic, Lie algebras of the form $L^3_a$ are pairwise non-isomorphic and $L^4_a = L^4_b$ if and only if $a = \lambda \cdot b$ for some $\lambda >0$. The Lie algebra $L^1$ is abelian, and in particular rigid. If $\mathfrak{g} \in \{L^2, L^3_a, L^4_a\}$, then the subspace $V = {\rm span}(X_1, X_2)$ is a $2$-dimensional abelian ideal of $\mathfrak{g}$, for which the short exact sequence
\[
0 \rightarrow V \rightarrow \mathfrak{g} \rightarrow \mathbb{R} \rightarrow 0
\]
splits. It follows that $\mathfrak{g}$ is of the form $\mathfrak{g} = \mathfrak{g}_{\phi}$, where $\phi$ is given by
\[
\phi^{(2)} = \left(\begin{matrix} 1&0\\0&1\end{matrix}\right), \quad \phi^{(3,a)} = \left(\begin{matrix} 0&1\\a&1\end{matrix}\right), \quad \phi^{(4,a)} = \left(\begin{matrix} 0&1\\a&0\end{matrix}\right),
\]
according to whether $\mathfrak{g}$ is equal to $L^2$, $L^3_a$ or $L^4_a$. Note that the algebra $L^4_0$ is isomorphic to the three-dimensional Heisenberg algebra $\L h_3$. It it immediate from the criterion in Corollary \ref{SplittingNew}.(v) that $\L h_3$ is the only non-rigid three-dimensional Lie algebra of the form $\mathfrak{g}_\phi$. Since, by the above classification, every solvable Lie algebra of dimension $3$ is of the form $\mathfrak{g}_\phi$ for some $\phi$, it follows that $\L h_3$ is also the unique non-rigid solvable three-dimensional Lie algebra. Combining this with our previous observations we have proved:
\begin{theorem}\label{ThmLow}
A Lie algebra of dimension $\leq 3$ is rigid if and only if it is not isomorphic to the three-dimensional Heisenberg algebra $\L h_3$.
\end{theorem}
For example, the Lie algebras of the three-dimensional SOL group and of the $1$-dimensional unitary motion group $\mathbb{C} \rtimes U(1)$ are rigid.
\subsection{The three-dimensional Heisenberg algebra}
Since the three-dimensional Heisenberg algebra $\L h_3$ plays such a prominent role in our classification, let us describe the space of Ad-invariant Lie quasi-states on $\L h_3$ explicitly. Here we will think of $\L h_3$ as the Lie algebra of strictly upper triangular $3 \times 3$-matrices.
\begin{corollary}\label{CorH3} Let $a \in \mathbb{R}$ and let $c$ be a sublinear continuous function on $\mathbb{R}$. Then the function
\[
\zeta_{\alpha, c}\left(\left(\begin{matrix}0&x&z\\0&0&y\\0&0&0\end{matrix} \right)\right) = c(y/x) \cdot x + a \cdot y \quad(x \in \mathbb{R}^\times, y, z \in \mathbb{R})
\]
extends continuously to an Ad-invariant quasi state $\zeta_{\alpha, c}: \L h_3 \rightarrow \mathbb{R}$. Moreover, every continuous Ad-invariant quasi-state on $\L h_3$ is of this form.
\end{corollary}
\begin{proof} Let $(X, Y, Z)$ be the basis of $\L h_3$ given by the elementary matrices $X = E_{1,2}$, $Y = E_{2,3}$ and $Z= E_{1, 3}$ so that
\[
[X, Y] = Z, \; [X, Z] = 0, \; [Y, Z] = 0.
\]
Then $V := {\rm span}(Y, Z)\lhd \mathfrak{g}$ is a $2$-dimensional ideal and $\mathfrak{g} = \mathfrak{g}_\phi$, where $\phi \in {\rm End}(V)$ is given by $\phi(Y) = Z$, $\phi(Z) = 0$. In particular, $\ker \phi = \im \phi = \mathbb{R} \cdot Z$ and $\im \phi^2 = 0$. It then follows from Theorem \ref{SolvableAdInv} that the Ad-invariant Lie quasi-states on $\L h_3$ are of the form $\zeta=\zeta_{\alpha, c}$, where $\alpha$ is a linear functional on $V$ which vanishes on $\mathbb{R}\cdot Z$ and $c$ is a sublinear continuous function on $\mathbb{R} \cdot Z$. Let $a \in \mathbb{R}$ such that $\alpha(yY+zZ) = a \cdot y$ and identify $c$ with a sublinear continuous function on $\mathbb{R}$ via the isomorphism $\lambda \mapsto \lambda \cdot Z$. Then we obtain for all $x \neq 0$,
\[
\zeta_{\alpha, c}(xX+yY+zZ) = c(\phi(yY+zZ)/x) \cdot x + \alpha(yY+zZ) = c(y/x) \cdot x + a \cdot y,
\]
which finishes the proof.
\end{proof}
\section{Unitary motion algebras}
\label{sec:unitary}
In this section we shall consider the family of unitary motion algebras $\L g_n := \mathbb{C}^n \rtimes \L{u}(n)$. We have already seen in Subsection \ref{SolvableExample} that the Lie algebra $\L g_1$ is rigid and that $\mathcal Q(\mathfrak{g}_1)$ is infinite-dimensional. For the rest of this section we will thus assume $n \geq 2$, unless otherwise mentioned.
\subsection{Normalized Lie quasi-states and generalized frame function}
Recall from Subsection \ref{SemidirectGeneral} that a Lie quasi-state $\zeta: \mathfrak{g}_n \rightarrow \mathbb{R}$ is called \emph{normalized} provided
\[
\zeta(v,0) = \zeta(0, X) = 0
\]
for all $v \in \mathbb{C}^n$ and $X \in \L u(n)$. We are going to show that for $n \geq 2$ every normalized continuous Lie quasi-state is actually trivial, thereby reducing the study of continuous Lie quasi-states on $\L g_n$ to the separate study of Lie quasi-states on $\mathbb{C}^n$ and $\L u(n)$. Recall that in \cite{Gl57}, Gleason established linearity of Lie quasi-states on $\L u(n)$, $n \geq 3$, by relating them to so-called frame functions and showing that such frame functions are necessarily of a special form. We will follow a similar strategy and relate normalized Lie quasi-states on $\mathfrak{g}_n$ to generalized frame functions in the sense of the following definition:
\begin{definition} A function $f: \mathbb{C}^n \rightarrow \mathbb{R}$ is called a \emph{generalized frame function} if it satisfies $f(0) = 0$ and for every pair $(u,v)$ of orthogonal vectors in $\mathbb{C}^n \setminus\{0\}$ we have
\[
f(u+v) = f(u)+f(v).
\]
\end{definition}
Obviously, every homomorphism $f: \mathbb{C}^n \rightarrow \mathbb{R}$ is a generalized frame function, in particular there exist generalized frame function of linear growth in arbitrary dimensions. However, as we establish in Theorem \ref{FFTrivial} in the appendix, there are no frame functions of \emph{sublinear} growth for any $n \geq 2$. Here a function $f: \mathbb{C}^n \rightarrow \mathbb{R}$ is called sublinear provided
\[
\lim_{v \rightarrow \infty} \frac{f(v)}{\|v\|} = 0.
\]
The following propositions links generalized frame functions to normalized Lie quasi-states on $\L g_n$.
\begin{proposition}\label{FrameSL} Let $n \geq 2$ and let $\zeta: \mathfrak{g}_n \rightarrow \mathbb{R}$ be a normalized Lie quasi-state. Then the following hold:
\begin{enumerate}
\item The function $f_\zeta: \mathbb{C}^n \rightarrow \mathbb{R}$ given by
\[
f_\zeta(w) := \zeta(-iw, i\cdot{\bf 1})
\]
is a generalized frame function.
\item If $\zeta$ is continuous, then $f_\zeta$ is continuous and sublinear.
\item $\zeta$ is uniquely determined by $f_\zeta$. In particular, $f_\zeta = 0$ implies $\zeta = 0$.
\end{enumerate}
\end{proposition}
\begin{proof} For the proof we introduce the following notation: Firstly, we
denote by $\langle \cdot, \cdot \rangle$ an inner product on $\mathbb{C}^n$ with respect to which elements of $\L u(n)$ are skew-Hermitian. We use the convention that $\langle \cdot, \cdot \rangle$ is linear in the first argument and anti-linear in the second argument. Given a vector $v \in \mathbb{C}^n\setminus\{0\}$ we define $P_v \in \L u(n)$ by
\[
P_v(w) = i\frac{\langle w,v \rangle}{\langle v, v\rangle} v,
\]
so that $-iP_v$ is the orthogonal projection onto the line spanned by $v$. Note that $P_v = P_w$ if and only if $v =\lambda w$ for some $\lambda \in \mathbb{C}^\times$ and that for every orthogonal basis $(v_1, \dots, v_n)$ we have
\begin{equation}\label{ProjectorSum}
\sum_{j=1}^nP_{v_j} = i\cdot {\bf 1}.\end{equation}
We claim that for all $w \in \mathbb{C}^n \setminus\{0\}$ we have
\begin{equation}\label{EqProjector}
f_\zeta(w) = \zeta(-iw, P_w).
\end{equation}
Observe first that if $Q \in \L u(n)$ satisfies $Qv = 0$ for some $v \in \mathbb{C}^n$, then for all $P \in \L u(n)$ that commutes with $Q$ we have $[(v, P), (0, Q)] = 0$ and hence
\[
\zeta(v, P+Q) = \zeta(v, P) + \zeta(0, Q) = \zeta(v,P).
\]
In particular, if $u \in \mathbb{C}^n \setminus\{0\}$ we can apply this to $u:= -iv$, $P:= P_v$ and $Q := i\cdot {\bf 1}-P_v$ to obtain
\[
f_\zeta(v) = \zeta(-iv, i\cdot{\bf 1}) = \zeta(-iv, P_v+Q) = \zeta(-iv, P_v),
\]
which establishes \eqref{EqProjector}. We now use this to establish (i)-(iii):
(i) If $u,v \in \mathbb{C}^n \setminus\{0\}$ are orthogonal and $Q := i \bf{\bf 1} - P_u - P_v$, then the elements $(-iu, P_u)$, $(-iv, P_v)$ and $(0, Q)$ pairwise commute, and we deduce with \eqref{EqProjector} that
\[
f_\zeta(u+v) = \zeta(-iu-iv, P_u+P_v+Q) = \zeta(-iu, P_u)+\zeta(-iv, P_v)+\zeta(0,Q) = \zeta(-iu, P_u)+\zeta(-iv, P_v) =f_\zeta(u) + f_\zeta(v).
\]
Since $\zeta$ is normalized we also have $f_\zeta(0) = 0$, and thus $f_\zeta$ is indeed a generalized frame function.
(ii) Continuity of $f_\zeta$ is immediate from continuity of $\zeta$, and sublinearity follows from Lemma \ref{Sublinearity}.
(iii) Let $w \in \mathbb{C}^n$ and let $X \in \L u(n)$. We are going to express $\zeta(w,X)$ in terms of the function $f:=f_\zeta$. For this let $(v_1, \dots, v_n)$ be an orthonormal basis of eigenvectors of $X$ with corresponding (purely imaginary) eigenvalues $(i\lambda_1, \dots, i\lambda_n)$. Note that $X \in \L u(n)$ implies that $\lambda_j \in \mathbb{R}$, so we can order them by decreasing absolute value and assume that $|\lambda_1| \geq \dots \geq |\lambda_l| > \lambda_{l+1} = \dots = \lambda_n = 0$,
for some index $l = l(X)$. If we wish to further emphasize the dependence on $X$ we write $\lambda_j(X)$ and $v_j(X)$ instead of $\lambda_j$ and $v$.
Now let $w \in \mathbb{C}^n$ and set $w_j := P_{v_j}(w)$. We observe that by \eqref{ProjectorSum},
\[
X.w = X\left(-i\cdot \sum_{j=1}^nP_{v_j}(w)\right) = \sum_{j=1}^n \lambda_j P_{v_j}(w).
\]
and
\[
w=-i \sum_{j=1}^n w_j.
\]
We thus obtain
\[
(w, X) = \sum_{j=1}^n (-iw_j, \lambda_jP_{v_j})
\]
Since for $j \neq k$ we have $\lambda_jP_{v_j}(-iw_k) = 0 =\lambda_kP_{v_k}(-iw_j)$ the summands on the right hand side commute, and thus
\begin{equation}\label{EqZetawX}
\zeta(w, X) = \sum_{j=1}^n \zeta(-iw_j, \lambda_jP_{v_j})= \sum_{j=1}^l \zeta(-iw_j, \lambda_jP_{v_j})
\end{equation}
Now assume that $j \leq l$. If $w_j \neq 0$, then $w_j/\lambda_j$ is proportional to $v_j$, whence $P_{w_j/\lambda_j} = P_{v_j}$. In this case we thus have
\begin{eqnarray*}
\zeta(-iw_j, \lambda_jP_{v_j}) &=& \lambda_j \cdot \zeta(-iw_j/\lambda_j, P_{v_j})\\
&=& \lambda_j \cdot \zeta(-iw_j/\lambda_j, P_{w_j/\lambda_j})\\
&=& \lambda_j \cdot f(w_j/\lambda_j)\\
&=& \lambda_j \cdot f(P_{v_j}(w)/\lambda_j),
\end{eqnarray*}
i.e.
\begin{equation}\label{wjEq}
\zeta(-iw_j, \lambda_jP_{v_j})=\lambda_j \cdot f(P_{v_j}(w)/\lambda_j).
\end{equation}
If $w_j = 0$ then \eqref{wjEq} still holds, since both sides of the equation are $0$ by our normalisation. Plugging \eqref{wjEq} into \eqref{EqZetawX} we obtain
\begin{equation*}\label{QSfromFF}
\zeta(w,X) = \sum_{j=1}^{l(X)} \lambda_j(X) \cdot f_\zeta(P_{v_j(X)}(w)/\lambda_j(X)).
\end{equation*}
This shows in particular that $\zeta$ is uniquely determined by $f_\zeta$.
\end{proof}
Combining this with the aforementioned Theorem \ref{FFTrivial} we deduce:
\begin{corollary}\label{NCLQSTrivial} Let $n \geq 2$. Then every normalized continuous Lie quasi-state $\zeta: \mathfrak{g}_n \rightarrow \mathbb{R}$ is trivial.
\end{corollary}
\begin{proof} If $\zeta$ is any continuous quasi-state on $\mathfrak{g}_n$, then $f_\zeta$ is a sublinear generalized frame function by Proposition \ref{FrameSL}. We thus deduce from Theorem \ref{FFTrivial} that $f_\zeta \equiv 0$, which in turn implies $\zeta \equiv 0$ by invoking Proposition \ref{FrameSL} again.
\end{proof}
\subsection{The case $n \geq 3$}
Recall from Lemma \ref{SemidirectLemma} that the space of continuous Lie quasi-states on $\L g_n$ is given by
\begin{equation*}\label{UMADec}
\mathcal Q(\L g_n) = \mathcal Q_{0}(\L g_n) \oplus (\mathbb{C}^n)^* \oplus \mathcal Q(\L u(n)), \end{equation*}
where $\mathcal Q_{0}(\L g_n)$ denotes the subspace of normalised continuous Lie quasi-states. We have just established in Corollary \ref{NCLQSTrivial} that this space is trivial, whence
\begin{equation}\label{NoMixedQS}
\mathcal Q(\L g_n) =(\mathbb{C}^n)^* \oplus \mathcal Q(\L u(n)),
\end{equation}
i.e. every continuous quasi-state on $\L g_n$ decomposes into a sum of continuous quasi-states on $\mathbb{C}^n$ and $\L u(n)$. If $n \geq 3$, then every continuous Lie quasi-state on $\L u(n)$ is linear by Gleason's theorem, and Lie quasi-states on $\mathbb{C}^n$ are linear anyway. We deduce:
\begin{theorem}\label{unStrongRigidity} For $n \geq 3$ every continuous Lie quasi-state on $\mathfrak{g}_n$ is linear.
\end{theorem}
Note that the theorem does not hold for $n \in \{1, 2\}$. In both cases the space of continuos Lie quasi-states on $\mathfrak{g}_n$ is infinite-dimensional. For $n=1$ this was established in Subsection \ref{SolvableExample}, whereas for $n=2$ it follows from the fact that $\mathcal Q(\L u(2))$ is infinite-dimensional.
\subsection{Rigidity of unitary motion algebras}
Theorem \ref{unStrongRigidity} implies in particular that the Lie algebras $\L g_n$ are rigid for all $n \geq 3$. Moreover, we have already established rigidity for $n=1$ in Subsection \ref{SolvableExample}. The following theorem deals with the remaining case $n=2$.
\begin{theorem}\label{MotionAd}
The Lie algebra $\L g_n = \mathbb{C}^n \rtimes \L u(n)$ is rigid for all $n \geq 1$. Moreover, $\dim {\mathcal Q}_{\rm Ad}(\mathfrak{g}_n)=1$ for all $n \geq 1$ and every continuous ${\rm Ad}$-invariant Lie quasi-state on $\L g_n$ is of the form
\[
\zeta(v,X) = i\lambda \cdot {\rm tr}(X)
\]
for some $\lambda \in \mathbb{R}$.
\end{theorem}
\begin{proof} Observe first that every Lie algebra homomorphism (i.e. every Ad-invariant \emph{linear} Lie quasi-state) $\mathfrak{g}_n \rightarrow \mathbb{R}$ factors through $\L u(n)$, and every Lie algebra homomorphism $\L u(n) \rightarrow \mathbb{R}$ is of the form $X \mapsto i\lambda \cdot {\rm tr}(X)$ for some $\lambda \in \mathbb{R}$. The latter is obvious for $n=1$ and follows from the decomposition $\L u(n) = \L{su}(n) \oplus \mathbb{R} \cdot i {\bf 1}$ for $n \geq 2$, since then $\L{su}(n)$ is simple. We conclude that the second statement of the theorem follows from the rigidity statement. In view of our previous results it thus only remains to establish rigidity for $\mathfrak{g}_2$; the following argument works actually for all $n \geq 2$.
Let $n \geq 2$ and $\zeta: \mathfrak{g}_n \rightarrow \mathbb{R}$ be a continuous Ad-invariant Lie quasi-state. By Subsection \ref{UMADec} there exist $\psi \in (\mathbb{C}^n)^*$ and $\zeta_{\L u(n)} \in \mathcal Q(\L u(n))$ such that
\[
\zeta(v,X) = \psi(v) + \zeta_{\L u(n)}(X).
\]
Now since $\zeta$ is ${\rm Ad}$-invariant, we have
\[
\zeta(v,X) = \zeta({\rm Ad}(k)(v, X)) = \zeta(k.v, {\rm Ad}(k)(X))
\]
for all $(v,X) \in \mathfrak{g}_n$ and all $k \in U(n)$, which we can write out as
\[
\psi(v) + \zeta_{\L u(n)}(X) = \psi(k.v) +\zeta_{\L u(n)}({\rm Ad}(k)(X)).
\]
Setting $X = 0$, respectively $v=0$, we obtain that $\psi$ is radial and that $\zeta_{\L u(n)}$ is an Ad-invariant Lie quasi-state. Now every radial linear functional is trivial, and since $\L u(n)$ is reductive, we deduce from Theorem \ref{MainTheorem1} that every Ad-invariant Lie quasi-state on $\L u(n)$ is linear. It follows that $\zeta$ itself is linear, and thus $\mathfrak{g}_n$ is rigid for $n \geq 2$.
\end{proof}
\newpage
|
2,869,038,155,646 | arxiv | \section{Introduction}
The ability to measure a qubit with high fidelity is of great importance in quantum computation \cite{Divincenzo2000,Knill2005}
and metrology \cite{Blatt2008,Giovannetti2011},
as well as in measurement-based feedback control \cite{Vijay2012,Sayrin2011,Cook2007,Yamamoto2008,Cramer2016,Ofek2016}
and computation \cite{Raussendorf2003,Gross2007,Briegel2009}.
Experimentally, much progress has been made in recent years toward realizing high-fidelity qubit measurement. High-fidelity single-shot measurements have been demonstrated in a wide variety of physical systems, including nitrogen-vacancy centers \cite{Robledo2011,Shields2015,DAnjou2016}, superconducting circuits \cite{Reed2010,Jeffrey2014,Walter2017}, and quantum dots \cite{Barthel2009,Harvey-Collardnodate,Nakajima2017}. Qubit relaxation is often a limiting factor in such experiments, and in systems with longer qubit lifetimes higher readout fidelities are possible. Indeed, in trapped ions---known for their long coherence times---readout fidelities in excess of $99.9\%$~\cite{Noek2013,Harty2014} and even $99.99\%$~\cite{Myerson2008,Burrell2010} have been demonstrated experimentally.
While this experimental progress is encouraging, strategies to further improve qubit readout fidelity are of great interest. One such strategy involves coupling the primary qubit to an ancillary readout qubit. Measurements are performed by mapping the system's state onto the ancilla, whose state is then read out. These measurements are said to be quantum non-demolition (QND) if the system's measurement eigenstates are unaffected by the ancilla readout procedure. QND measurements are necessarily repeatable, and the overall measurement fidelity can be improved by repeating measurements to suppress individual measurement infidelity (\hyperref[fig:QNDcircuit]{Fig.~\ref{fig:QNDcircuit}}). Highly QND readouts have already been realized in trapped-ion systems \cite{Hume2007,Wolf2016}, nitrogen vacancy centers \cite{Jiang2009,Neumann2010,Cramer2016}, and circuit QED systems~\cite{Johnson2010, Peaudecerf2014,Sun2014, Saira2014,Ofek2016,Lupascu2007, Vijay2011}.
\begin{figure}[bp]
\begin{center}
\includegraphics[width=\columnwidth]{QNDcircuit2}
\caption{Repeated QND readouts. Contributions to overall measurement infidelity from ancilla preparation/readout errors and noise can be exponentially suppressed by repeating measurements. }
\label{fig:QNDcircuit}
\end{center}
\end{figure}
For a qubit encoded in a two-level system, the fidelity of such repeated readout procedures is fundamentally limited by the fact that there exist single errors, such as relaxation of the excited state to the ground state, that can destroy the information in the qubit. This fundamental limit can be overcome, however, by robustly encoding the information within a larger Hilbert space, so that single errors leave states distinguishable.
The combination of repeated QND measurements and robust encoding thus enables one to overcome limits imposed by both individual measurement infidelity and qubit relaxation.
In this work we propose a robust readout scheme for bosonic systems in the dispersive coupling regime---a class of systems where information can be both encoded robustly and read out in a QND way. The infinite-dimensional Hilbert space of a single bosonic mode (quantum harmonic oscillator) provides room to encode information and protect it from errors~\cite{Cochrane1999, Gottesman2001, Michael2016}, while the mode's dispersive coupling to an ancillary quantum system enables repeated QND readout~\cite{Brune1992,Blais2004,Wallraff2004,Wallraff2005,Sun2014}. We show explicitly how the combination of these two techniques allows one to simultaneously suppress the contributions to readout infidelity from qubit relaxation and individual measurement noise to higher order, potentially yielding orders-of-magnitude improvement in readout fidelity.
This scheme is applicable to a variety of systems where bosonic modes, typically in the form of photons or phonons, are naturally available. Circuit QED systems, where the strong dispersive regime is experimentally accessible~\cite{Wallraff2004,Schuster2007,Boissonneault2009}, provide one example. Other examples include optomechanical systems, where dispersive couplings necessary for QND readout have been demonstrated~\cite{Jayich2008, Thompson2008}, and in principle also nanomechanical systems~\cite{LaHaye2009, OConnell2010} or circuit quantum acoustodynamic systems~\cite{Manenti2017,Chu2017}, provided sufficiently strong couplings and long mode lifetimes can be engineered.
More broadly, this scheme can be applied in any system where a bosonic mode has a strong dispersive coupling to an ancilla, so it can even be applied to more exotic systems, e.g.~quantum magnonics, where strong dispersive couplings were recently demonstrated~\cite{Lachance-Quirion2017}.
This article is organized as follows. In \hyperref[decay-twolevel]{Sec.~\ref{decay-twolevel}}, we use a simple Fock state encoding to introduce the robust readout scheme for a lossy bosonic mode dispersively coupled to a two-level readout ancilla. This encoding serves as a straightforward example of an encoding suited to robust readout, and its analysis (Secs.~II-V) is intended to make the ideas underlying the robust readout scheme abundantly clear. With this encoding, we explicitly compute the readout infidelity and show that contributions from relaxation and individual measurement noise are suppressed.
In \hyperref[decayexcite-twolevel]{Sec.~\ref{decayexcite-twolevel}} we generalize the readout scheme so that contributions to the infidelity from spontaneous heating are also suppressed.
In \hyperref[decayexcite-manylevel]{Sec.~\ref{decayexcite-manylevel}} we show how, given a readout ancilla with more than two levels, readout fidelity can be significantly improved by using a maximum likelihood estimate as opposed to simple majority voting.
In \hyperref[InfoTheory]{Sec.~\ref{InfoTheory}}, we consider the robust readout scheme from the perspective of classical information theory and place a lower bound on readout infidelity. This concludes our analysis of the Fock state encoding.
We consider other encodings in \hyperref[Encodings]{Sec.~\ref{Encodings}}, which contains the main results of this article. We identify criteria on encodings that are sufficient for robust, ancilla-assisted readout of a qubit encoded in a lossy bosonic mode, and as examples,
we explicitly show that these criteria are satisfied by cat codes and binomial codes. We approximate the readout fidelity for both codes.
\section{Robust readout of a qubit encoded in a lossy bosonic mode }
\label{decay-twolevel}
\subsection{Robust readout scheme}
Let a bit of quantum information be encoded in a bosonic mode as $\ket{\psi}_B = \alpha \ket{0}_B + \beta \ket{1}_B$, where $\ket{0}_B$ and $\ket{1}_B$ are the ``logical'' states in the mode's Hilbert space that we seek to distinguish with maximal fidelity. Readout of this qubit (henceforth referred to as the bosonic qubit) is performed by repeatedly mapping its state onto a two-level ancillary quantum system, whose state is subsequently measured. The mapping and ancilla readout processes are assumed to be QND so that they can be repeated without disturbing the bosonic qubit. This assumption is justified in the dispersive coupling regime, as will be shown explicitly.
We refer to the process of mapping the bosonic qubit onto the ancilla, followed by ancilla readout, as a level-1 readout. Each level-1 readout yields one classical bit of information (the ancilla is either found to be in $\ket{g}$ or $\ket{e}$). In our scheme, $N$ repeated level-1 readouts are performed, and their outcomes are collectively analyzed, e.g.~with majority voting, to yield a single bit of classical information (the bosonic qubit is determined to be in either $\ket{0}_B$ or $\ket{1}_B$). We refer to the entire procedure---performing $N$ level-1 readouts and combining the results---as a level-2 readout. This scheme is shown schematically in \hyperref[fig:setup]{Fig.~\ref{fig:setup}(a)}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{decay_twolevel_setup6}
\caption{Robust readout scheme. (a) Quantum circuit for the readout scheme. The state of the bosonic qubit is read out through repeated QND mappings of its state onto an ancilla. (b) The bosonic mode and mapping procedure. Fock states in the bosonic mode decay with rates proportional to their excitation number. All excited states are mapped to the excited state of the two-level readout ancilla. (c) Schematic of a circuit QED system. The state of a microwave cavity mode can be read out via a dispersive coupling to a transmon qubit. The transmon is measured via its coupling to some other device, typically a resonator or cavity. Realistic parameter values for this architecture, c.f.~Refs.~\cite{Heeres2017,Chou2018,Axline2018}, are shown in the table (see \hyperref[decay-twolevelB]{Sec.~\ref{decay-twolevelB}} for parameter definitions). }
\label{fig:setup}
\end{center}
\end{figure}
We now define the logical states, the specific mapping required for this scheme, and the relaxation properties of the bosonic mode, all of which are summarized in \hyperref[fig:setup]{Fig.~\ref{fig:setup}(b)}. The logical states encoding the bosonic qubit are chosen to be the Fock states
\begin{align}
\label{Fock_logical_states}
\ket{0}_B &= \ket{0} \nonumber \\
\ket{1}_B &= \ket{L},
\end{align}
for positive integer $L$.
We make three remarks on this choice of encoding. First, the reason that we begin by considering this ``Fock code'' is that the analysis of its readout fidelity is straightforward, so the code serves as an instructive example. Second, we note that the Fock code has previously been used in quantum information processing applications. For example, the initialization~\cite{Heeres2015, Heeres2017, Chou2018,Axline2018,Hofheinz2008,Hofheinz2009} and manipulation~\cite{Chou2018} of qubits with this encoding have been demonstrated experimentally. Third, the Fock code is a quantum error-detecting code, capable of detecting excitation loss errors for $L>1$. Thus, this code could be useful in a concatenated encoding scheme, for example, since the ability to detect errors at one level enables more efficient correction of errors at the next level of encoding~\cite{Knill2005}. Other possible choices of the logical states, including quantum error-correcting codes like the cat and binomial codes, are considered in \hyperref[Encodings]{Sec.~\ref{Encodings}}.
In the dispersive coupling regime, the logical states (\hyperref[Fock_logical_states]{\ref{Fock_logical_states}}) can be distinguished through a measurement procedure that is QND. In this work we consider projective measurements and define QND as follows. A projective measurement can be described by a collection of measurement operators $\{\hat{M}_k\}$ that constitute a complete set of orthogonal projectors, satisfying $\hat{M}_k^2 = \hat{M}_k$ and $\sum_k \hat{M}_k = 1$. Such a measurement is QND if
\begin{equation}
\label{qnd_criterion}
\left[\hat{M}_k, \hat H(t) \right] = 0,
\end{equation}
for all $k$ and $t$, where $\hat{H}(t)$ is an operator describing the ancilla preparation, its coupling to the bosonic mode, and the ancilla readout. In the robust readout scheme, the level-1 measurements are defined by operators $\hat M_0$ and $\hat M_1$ that act on the Hilbert space of the bosonic mode
\begin{align}
\label{Eqn:decay_meas_ops}
\hat{M}_0 &= \ket{0}\bra{0} \nonumber \\
\hat{M}_1 &= \ket{1}\bra{1} + \ldots + \ket{L}\bra{L}.
\end{align}
For a bosonic mode dispersively coupled to a two-level ancilla, QND measurements are possible because these operators commute with the dispersive coupling Hamiltonian,
\begin{equation}
H_{DC}/\hbar = -\chi \,\hat{a}^\dagger \hat{a} \ket{e}\bra{e},
\end{equation}
where $\ket{g}$ and $\ket{e}$ denote the basis states of the ancilla, and $\hat a$ is the bosonic annihilation operator. Similar QND measurements have already been demonstrated experimentally in circuit QED systems \cite{Johnson2010}.
During the mapping process, the bosonic state $\ket{0}$ ($\ket{L}$) is mapped to the ancilla state $\ket{g}$ ($\ket{e}$), while all intermediate Fock states $\ket{0<n<L}$ are mapped to the ancilla state $\ket{e}$. Experimentally, this mapping can be realized by initializing the ancilla in the ground state, then utilizing the dispersive coupling to apply a collection of selective pulses \cite{Schuster2007, Johnson2010, Krastanov2015, Heeres2015, Reagor2016} at frequencies $(\omega_{ge} - k\chi)$ for $k = 0, 1, \ldots L$, where $\omega_{ge}$ is the bare frequency of the ancilla qubit. These pulses, which can be applied simultaneously, flip the ancilla to the excited state only if the bosonic mode state is $\ket{1}$, $\ket{2}$, ..., or $\ket{L}$. As a simpler alternative, a single selective pulse can be applied at $\omega_{ge}$ to flip the qubit conditioned on whether the bosonic mode is in $\ket{0}$. The only difference between this latter procedure and the mapping in~\hyperref[fig:setup]{Fig.~\ref{fig:setup}(b)} is that the roles of the ancilla states are reversed---a trivial change in bookkeeping.
Because readouts are frequently limited by qubit lifetime, we consider a bosonic mode that is subject to spontaneous relaxation. Specifically, the decay rate of a Fock state $\ket{n}$ to $\ket{n-1}$ is given by $n \kappa_\downarrow$, where the factor of $n$ is due to bosonic enhancement. Transitions between non-adjacent Fock states are suppressed by selection rules, and excitations will be considered later in \hyperref[decayexcite-twolevel]{Sec.~\ref{decayexcite-twolevel}}.
As a figure of merit for this readout scheme, the readout fidelity $\mathcal{F}$ is defined as~\cite{Gambetta2007,Walter2017}
\begin{equation}
\mathcal{F} = 1 - P(0_B|1_B) - P(1_B|0_B),
\label{eqn:fidelity}
\end{equation}
where $P(i|j)$ is the probability of the level-2 readout yielding $i$ when the initial state of the bosonic qubit was $j$, for $i,j \in \{ \ket{0}_B,\ket{1}_B \}$. $\mathcal{F}$ varies continuously from 0, for readouts which yield no information about the initial state, to 1, for perfect readouts. In the robust readout scheme, both $P(0_B|1_B)$ and $P(1_B|0_B)$ are suppressed by increasing $L$ and $N$, as is shown quantitatively in the following sections.
Finally, to make the following analysis more concrete, in \hyperref[fig:setup]{Fig.~\ref{fig:setup}(c)} we show an example of a real system where the robust readout scheme can be applied---a circuit QED system. In this system, a microwave cavity mode (the bosonic mode) dispersively couples to a transmon qubit (the ancilla), and this coupling can be used to perform repeated QND measurements of the cavity state~\cite{Johnson2010, Peaudecerf2014,Sun2014,Ofek2016}. For a qubit stored in the cavity mode with a suitable encoding (e.g.~with the Fock, cat, or binomial codes), it will be shown that contributions to readout infidelity from cavity decay, mapping errors, and transmon readout errors can all be suppressed to higher order with this scheme.
\subsection{Discrete model of the robust readout scheme}
\label{decay-twolevelB}
A Hidden Markov Model (HMM) is used to model the robust readout scheme of \hyperref[fig:setup]{Fig.~\ref{fig:setup}}. A HMM is a Markov chain where, instead of being able to observe a system's state directly, the only information about the system is provided by a series of noisy emissions. HMMs have been previously used as effective models of qubit readout \cite{Dreau2013,Gammelmark2013,Ng2014,Wolk2015}. In our case, a discrete model (\hyperref[fig:setup]{Fig.~\ref{fig:HMM}}) is used where each level-1 readout is modeled as a possible transition, representing the bosonic qubit's decay, followed by a noisy emission, representing the mapping and the readout of the ancilla.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{HMM_v4}
\caption{Hidden Markov Model for the robust readout scheme. (a) Markov chain and emissions. At each step of the HMM the bosonic system transitions to a new state and releases an emission. Here, $B_n$ denotes the bosonic mode state after step $n$, and $A_n$ denotes the $n^{th}$ ancilla measurement outcome. (b) Transition and emission probabilities. Transitions and emissions are shown diagrammatically for the case $L=2$, where the matrix elements along the arrows are the associated probabilities. Bolded arrows indicate the intended mappings. }
\label{fig:HMM}
\end{center}
\end{figure}
The model is parameterized by transition probabilities $T_{ij}$ and emission probabilities $E_{ij}$. The transition probability $T_{ij}$ is defined to be the probability that the bosonic state $\ket{i}$ transitions to $\ket{j}$ during a single level-1 readout, with $i, j \in \{0, 1, \ldots, L \}$. The emission probability $E_{ij}$ is the probability that the bosonic system, having transitioned to state $\ket{i}$, with $i,\in \{0, 1, \ldots, L \}$, is read out as ancilla state $\ket{g}$ for $j=0$, or $\ket{e}$ for $j=1$. The emission probabilities are defined in terms of the probability $\delta$ that an error occurs during the mapping and readout processes which causes the ancilla readout to be misleading
\begin{equation}
E_{ij}=
\begin{cases}
1-\delta,& \text{if } i=j=0 \text{ or } i>0,j=1,\\
\delta,& \text{otherwise. }
\end{cases}
\end{equation}
In cases where different Fock states have different probabilities of producing misleading ancilla readouts, taking $\delta$ to be the largest of these probabilities will yield a conservative estimate of readout fidelity.
Explicit expressions for transition probabilities $T_{ij}$ are derived from the bosonic decay rates. Consider a population of quantum harmonic oscillators, with $p_i(t)$ of the oscillators in Fock state $\ket{i}$ at time $t$. The system of differential equations describing the time evolution of the populations is
\begin{equation}
\dot{p}_i(t) = \sum_{j=0}^L (K_\downarrow)_{ij}\,p_j(t),
\end{equation}
where $(K_\downarrow)_{ij}$ is the transition rate from state $\ket{j}$ to $\ket{i}$. For bosonic systems,
\begin{equation}
(K_\downarrow)_{ij} = \begin{cases}
-j\,\kappa_{\downarrow}, & i = j \\
\,\,\,\,\,j\, \kappa_{\downarrow}, &i = j-1\\
\,\,\,\,\,0, & \text{otherwise}.
\end{cases}
\end{equation}
This system has the solution $\mathbf{p}(t) = e^{K_\downarrow t}\,\mathbf{p}(0)$.
The transition probabilities for a level-1 readout taking time $\tau$ are thus obtained by explicitly computing the matrix elements of $e^{K_\downarrow \tau}$,
\begin{equation}
T_{ij}(\tau)=(e^{K_\downarrow \tau})_{ji} = {{i}\choose{j}}\left(e^{\kappa_\downarrow \tau}-1\right)^{i-j} e^{-i\kappa_{\downarrow}\tau}.
\end{equation}
As an aside, we note that both $\tau$ and $\delta$ can depend implicitly on the strength of the dispersive coupling $\chi$. For example, larger coupling strengths can enable faster or more selective pulses. The values of $\tau$ and $\delta$ given in \hyperref[fig:setup]{Fig.~\ref{fig:setup}(c)} are estimated from the given $\chi$ value based on such considerations. In order to keep the following discussion general, however, we do not assume a particular functional dependence of either of these parameters on $\chi$.
To provide intuition as to why increasing the number of levels $L$ can improve the readout fidelity, we calculate the expected value of the time $\tau_0$ which it takes initial state $\ket{L}$ to decay to $\ket{0}$,
\begin{align}
\left< \tau_0\right> = \int_0^\infty d\tau\, \tau\, \frac{d\, }{d\tau} T_{L0}(\tau) =\frac{1}{\kappa_\downarrow} \sum_{n=1}^L \frac{1}{n}.
\end{align}
Because $\tau_0$ grows with $L$, so too does the effective signal lifetime, thereby improving readout fidelity. Indeed, the effective lifetime diverges with $L$, though there are diminishing returns in using higher levels because the divergence is only logarithmic. Interestingly, it should be noted that using higher-level encodings can improve readout fidelity even in the absence of an increase in effective signal lifetime \cite{DAnjou2017}.
\subsection{Readout infidelity in the discrete model}
Using the HMM, we calculate the infidelity $1-\mathcal{F}$ of the robust readout scheme in terms of the ``experimental'' parameters $\delta$ and $\kappa_{\downarrow}\tau$. This infidelity depends on how the level-2 measurement outcomes are determined. We consider two approaches: simple majority voting and a maximum likelihood estimate (MLE).
In majority voting, each level-2 measurement outcome is determined by tallying the $N$ level-1 measurement outcomes, with ancilla readouts of $\ket{g}$ ($\ket{e}$) counted as votes for initial state $\ket{0}$ ($\ket{L}$). In the MLE, which is the statistically optimal approach, the known values of the transition and emission matrix elements are used to calculate which initial state was more likely to have produced a series of observed ancilla readouts. Explicitly, the likelihood $\lambda_{\mathbf{a}}(i)$ that a discrete set of ancilla readouts $a_n \in \{g,e \}$, for $n \in \{1, \ldots, N \}$, was produced with initial state $\ket{i}$ is
\begin{equation}
\lambda_{\mathbf{a}}(i) = \sum_{j_1,\ldots,j_N} T_{i,j_1} E_{j_1,a_1} \ldots T_{j_{N-1},j_N}E_{j_N,a_N},
\end{equation}
which is efficiently calculable in $O(NL^2)$ operations \cite{Press2007}.
The outcome of a level-2 measurement is then decided by determining which of the two initial states was more likely to have produced the emissions, i.e.~by comparing $\lambda_{\mathbf{a}}(0)$ and $\lambda_{\mathbf{a}}(L)$.
For both majority voting and the MLE classification strategies, the infidelity is given exactly as a function of the likelihoods
\begin{align}
1- \mathcal{F} &= \sum_{\mathbf{a} \in \mathcal{A}_0 }\lambda_{\mathbf{a}}(L) + \sum_{\mathbf{a'} \in \mathcal{A}_L }\lambda_{\mathbf{a'}}(0).
\end{align}
where $\mathcal{A}_0$ ($\mathcal{A}_L$) is the set of ancilla readout vectors $\mathbf{a}$ which are classified as initial state $\ket{0}$ ($\ket{L}$). Whether a given $\mathbf{a}$ falls in either $\mathcal{A}_0$ or $\mathcal{A}_L$ depends on the classification strategy. By definition, the MLE chooses the sets $\mathcal{A}_0$ and $\mathcal{A}_L$ to be those which minimize the infidelity.
Plots of the infidelity as a function of $N$ are shown in \hyperref[fig:decay]{Fig.~\ref{fig:decay}} for both majority voting and the MLE. The values of $\kappa\tau$ and $\delta$ used in the figure are the same as those given for the circuit QED system in \hyperref[fig:setup]{Fig.~\ref{fig:setup}(c)}, so the infidelities shown in the main panel are realistically attainable. Notably, the minimum infidelity attained by both majority voting and the MLE decreases by over an order of magnitude as $L$ increases from 1 to 2. Indeed, the inset shows that increasing $L$ can lead to multiple orders of magnitude improvement. It is also clear that the MLE can dramatically outperform majority voting as $N$ increases. This discrepancy is due to decays: majority voting weights all votes equally, even those that are recorded long after initial state $\ket{L}$ is likely to have decayed to $\ket{0}$. The minimal infidelities attained by the two methods, however, are not significantly different, meaning that simple majority voting is a near-optimal strategy until decays begin to play a significant role.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{decayEXACT_v4}
\caption{Infidelity of the robust readout scheme. The infidelity is plotted as a function of the number of measurements $N$ for $L=1$ and 2, with the parameter choices $\delta = 2\%$ and $\kappa_{\downarrow}\tau = 1\%$. The solid lines denote the approximate majority voting infidelity~(\ref{eqn:approx}), while circles and squares respectively denote exact calculations for the majority voting and MLE schemes. Inset: the minimum attainable infidelity is plotted as a function of $L$.}
\label{fig:decay}
\end{center}
\end{figure}
To compute the exact infidelity, it is necessary to enumerate all possible combinations of $N$ level-1 readouts and to compute the likelihoods of each, a computation which takes $O(NL^2\times2^N)$ operations. To provide a more accessible means of quickly estimating the readout infidelity, and to elucidate its scaling, we derive a simple approximation for the infidelity in the majority voting scheme. The approximation depends on a small number of general experimental parameters: the level-1 readout error probability $\delta$, the decay rate of the bosonic system $\kappa_{\downarrow}$, and the level-1 readout time~$\tau$.
There are two dominant processes which are most likely to fool the majority voting. The first is sufficiently quick decay of the initial state $\ket{L}$ to $\ket{0}$, but with no level-1 readout errors occurring. The second is a sufficient number of level-1 readout errors occurring so as to fool the voting, but with no decays occurring. All other processes which fool the voting, such as combinations of decays and level-1 readout errors, have probabilities that are higher order in the parameters $\delta$ or $\kappa_\downarrow \tau$. We approximate the probabilities of incorrectly identifying initial states by neglecting the contributions of these higher-order processes,
\begin{subequations}
\begin{align}
P(0|L) &\approx T_{LL}\left(N\tau\right) \times \sum_{k = \ceil{N/2}}^L {{N}\choose{k}} \delta^k (1-\delta)^{N-k} \nonumber \\[0.1cm]
&+ T_{L0}\left(\ceil{N/2}\tau\right)\times(1-\delta)^N \\[0.1cm]
P(L|0) &\approx \sum_{k = \ceil{N/2}}^L {{N}\choose{k}} \delta^k (1-\delta)^{N-k},
\end{align}
\end{subequations}
where $\ceil{\cdot}$ denotes the ceiling function.
Expanding to lowest order in $\delta$ and $\kappa_\downarrow \tau$ gives
\begin{align}
\label{eqn:approx}
1-\mathcal{F} & = P(0|L) + P(L|0) \nonumber \\[0.2cm]
&\approx \, 2{{N}\choose{\ceil{N/2}}} \delta^{\ceil{N/2}} + \left( \ceil{N/2} \kappa_\downarrow \tau\right)^{L} .
\end{align}
This approximation is valid when both $N\delta \ll 1$ and $N\kappa_{\downarrow} \tau\ll1$ so that higher order terms can be neglected. This approximation is plotted along with the exact result in \hyperref[fig:decay]{Fig.~\ref{fig:decay}}, where the two agree well because the approximation is valid in the regime shown.
\hyperref[eqn:approx]{Eqn.~\ref{eqn:approx}} elucidates the benefit of combining robust encoding with repeated measurement. In two-level systems, such as trapped ions, the fidelity is limited by $\kappa_\downarrow\tau$ because $L = 1$ is fixed. On the other hand, in multi-level systems where repetitive QND readouts are not possible, the fidelity is limited by $\delta$ because $N=1$ is fixed. For bosonic systems in the dispersive coupling regime, however, one has the freedom to increase both $L$ and $N$. Thus, both terms contributing to the infidelity are suppressed to higher order, and readout is no longer theoretically limited by either individual measurement errors or relaxation. This is the strength of the robust readout scheme.
\section{Robust readout with both relaxation and heating}
\label{decayexcite-twolevel}
We now consider the case where the bosonic mode is subject to heating, defined here as a nonzero excitation rate $\kappa_\uparrow$. Without modification, the readout fidelity of the above scheme would be limited by the probability of the initial state $\ket{0}$ spontaneously exciting to $\ket{1}$, a process which is first order in $\kappa_\uparrow\tau$. In this section, we generalize the scheme so that contributions to the infidelity from heating are also suppressed to higher orders.
The modified readout scheme is shown in \hyperref[fig:decayexcite_setup]{Fig.~\ref{fig:decayexcite_setup}}, where the excitation rate\footnote{In order to study the fidelity with a finite HMM, we truncate the Hilbert space to the first $L+1$ Fock states, taking the heating rate from $\ket{L}$ to $\ket{L+1}$ to be 0. It is safe to neglect the additional levels when $\kappa_{\uparrow}\tau \ll \kappa_{\downarrow}\tau \ll 1$.} between the adjacent Fock states $\ket{n}$ and $\ket{n+1}$ is $n\kappa_{\uparrow}$.
To account for this heating, we define a threshold state $\ket{m}$ such that the mapping from the bosonic mode to the ancilla is
\begin{equation}
\ket{n} \rightarrow
\begin{cases}
\ket{g}, & n\leq m \\
\ket{e}, &n>m .
\end{cases}
\end{equation}
This mapping can be implemented by initializing the ancilla in the ground state, then applying selective pulses at frequencies $(\omega_{ge} - k\chi)$ for $k = m+1,m+2, \ldots, L$. These pulses flip the ancilla from $\ket{g}$ to $\ket{e}$ only if the bosonic mode state is $\ket{m+1}$, $\ket{m+2}$, ..., or $\ket{L}$. The level-1 readouts are then described by the measurement operators
\begin{align}
\hat{M}_0 &= \,\,\, \sum_{k=0}^m\,\,\,\, \ket{k}\bra{k} \nonumber \\
\hat{M}_1 &= \sum_{k=m+1}^L \ket{k}\bra{k} .
\end{align}
For $m>0$, the contribution to the infidelity from heating of the initial state $\ket{0}$ will thus be suppressed to higher order in $\kappa_\uparrow$ because multiple excitations are required for $\ket{0}$ to heat to a state which is mapped to ancilla state $\ket{e}$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{decayexcite_twolevel_setup2}
\caption{Robust readout scheme for relaxation and heating. Decays and excitations occur between adjacent Fock states with rates proportional to the excitation number. All Fock states $\ket{n>m}$ are mapped to the excited state of the two-level ancilla. }
\label{fig:decayexcite_setup}
\end{center}
\end{figure}
As in the previous section, this scheme is quantitatively analyzed with a HMM. The emission probabilities $E_{ij}$ are similarly defined in terms of the level-1 readout error probability $\delta$ as
\begin{equation}
E_{ij}=
\begin{cases}
1-\delta,& \text{if } i\leq m,j=0 \text{ or } i>m,j=1,\\
\delta,& \text{otherwise.}
\end{cases}
\end{equation}
The transition probabilities $T_{ij}$ are calculated as functions of the decay and excitation rates. The system of differential equations describing the time-evolution of the Fock state populations is
\begin{equation}
\dot{p}_i(t) = \sum_{j=0}^L (K_\downarrow +K_\uparrow)_{ij}\,p_j(t),
\end{equation}
where $K_\uparrow$ has matrix elements
\begin{equation}
(K_\uparrow)_{ij} = \begin{cases}
-(j+1)\,\kappa_{\uparrow}, & i = j < L \\
\,\,\,\,\,(j+1)\, \kappa_{\uparrow}, &i = j+1 \\
\,\,\,\,\,0, & \text{otherwise}.
\end{cases}
\end{equation}
The transition probabilities are then given as a function of the level-1 readout time $\tau$,
\begin{equation}
T_{ij}(\tau)=\left[e^{( K_{\downarrow}+ K_{\uparrow})\tau}\right]_{ji}.
\end{equation}
Exact calculations of the infidelity proceed as in the previous section. We also approximate the infidelity by again considering only the dominant error processes, now including the probability that initial state $\ket{0}$ heats to $\ket{m+1}$, with no level-1 readout errors occurring. With this additional term, the level-2 readout error probabilities are approximately given by
\begin{subequations}
\begin{align}
P(0|L) &\approx T_{LL}(N \tau) \sum_{k = \ceil{N/2}}^L {{N}\choose{k}} \delta^k (1-\delta)^{N-k} \nonumber \\[0.1cm]
&+ (1-\delta)^N\left(e^{K_\downarrow \ceil{N/2}\tau}\right)_{m,L} \\[0.1cm]
P(L|0) &\approx T_{00}(N\tau) \sum_{k = \ceil{N/2}}^L {{N}\choose{k}} \delta^k (1-\delta)^{N-k}\nonumber \\[0.1cm]
&+ (1-\delta)^N\left(e^{K_\uparrow \ceil{N/2}\tau}\right)_{m+1,L}.
\end{align}
\end{subequations}
To lowest order in $\delta$, $\kappa_\downarrow \tau$, and $\kappa_\uparrow \tau$, the infidelity is
\begin{align}
\label{heatinfidelity}
1-\mathcal{F} &\approx {{L}\choose{m}}\left( \left\lceil \frac{N}{2}\right\rceil \kappa_\downarrow \tau\right)^{L-m}
+ \left(\left\lceil \frac{N}{2}\right\rceil \kappa_\uparrow \tau\right)^{m+1} \nonumber \\[0.1cm]
&+2{{N}\choose{\ceil{N/2}}} \delta^{\ceil{N/2}} .
\end{align}
It is clear that, within this approximation, all contributions to the infidelity are suppressed to higher orders in $\kappa_\downarrow\tau$, $\kappa_\uparrow\tau $, and $\delta$, by increasing $L$, $m$, and $N$, respectively.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{decayexciteEXACT_v4}
\caption{
Infidelity of the robust readout scheme with both relaxation and heating. The infidelity is plotted as a function of the number of measurements $N$ for $L=1,2$ and 3, with the parameter choices $\delta = 2\%$, $\kappa_{\downarrow}\tau = 1\%$, and $\kappa_{\uparrow}\tau= 0.5\%$. Inset: the minimum attainable infidelity is plotted as a function of $L$. For $m=0$ (red) the infidelity asymptotes to a finite value, but for optimal $m$ (black) it continues to decrease. }
\label{fig:decayexciteinset}
\end{center}
\end{figure}
Plots of the infidelity with both majority voting and the MLE are shown in \hyperref[fig:decayexciteinset]{Fig.~\ref{fig:decayexciteinset}}. Though the heating rates $\kappa_\uparrow$ of physical systems are typically much smaller than the decay rate $\kappa_\downarrow$ (e.g.~\cite{Chen2016}), the two are chosen to be comparable in the plot so that the importance of the threshold state is apparent. For the parameters shown in the figure, $m=0$ is the optimal choice of the threshold for $L\leq2$, but at $L=3$ the optimal choice is $m=1$. In the inset, the minimum majority voting infidelity is plotted as a function of $L$ for both fixed $m=0$ (red) and the optimal choice of $m$ (black). It is clear that without increasing $m$ the readout infidelity is limited by the first-order heating process, but when $m$ is allowed to increase it is again possible to improve readout fidelity by orders of magnitude. We also note that here again the optimal MLE and majority voting infidelities do not differ significantly.
\section{Robust readout with a multi-level ancilla}
\label{decayexcite-manylevel}
There exist experimental systems where a bosonic mode can be dispersively coupled to an ancilla with more than two levels. Circuit QED systems provide one example; the higher excited states of a superconducting transmon qubit have been populated and measured in experiment \cite{Bianchetti2010,Peterer2015}. We now consider a version of the robust readout scheme applicable to such systems and show that the use of a multi-level ancilla can lead to significant improvements in readout fidelity when the MLE is used.
The readout scheme for this case is shown in \hyperref[fig:decayexciteL_setup]{Fig.~\ref{fig:decayexciteL_setup}}. As before, nonzero decay and excitation rates are assumed, but in this case the level-1 measurement operators are
\begin{equation}
\{\hat{M}_k = \ket{k}\bra{k}, \text{ for $k=0,1,\ldots,L$}\}.
\end{equation}
The threshold state $\ket{m}$ is used only to determine which of the $L+1$ possible ancilla state readouts are counted as votes for initial bosonic state $\ket{0}$ or $\ket{L}$ in the majority voting scheme. It plays no role in the MLE.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{decayexcite_Llevel_setup2}
\caption{Robust readout scheme for multi-level ancilla. (a) Schematic description. Both decays and excitations occur between adjacent bosonic mode Fock states. Each Fock state is mapped to a unqiue ancilla state. In the majority voting all ancilla readouts of $\ket{n>m}$ are counted as votes for $\ket{L}$, while readouts of $\ket{n\leq m}$ are counted as votes for $\ket{0}$. (b) Mapping circuit. Fourier gates on the ancilla, in combination with evolution under the dispersive coupling, implement the mapping used in the QND measurement. }
\label{fig:decayexciteL_setup}
\end{center}
\end{figure}
A circuit that uses the dispersive coupling to implement the mapping from the bosonic mode to the ancilla is shown in \hyperref[fig:decayexciteL_setup]{Fig.~\ref{fig:decayexciteL_setup}(b)} \cite{Li2017}. The ancilla is initialized in the ground state, and a Fourier gate $\hat{F}_{L+1}$ maps this state to an even superposition of the first $L+1$ Fock states. For a bosonic mode dispersively coupled to an $(L+1)$-level ancilla, the coupling Hamiltonian is
\begin{equation}
\hat{H}_{DC}/\hbar = -\sum_{j=0}^{L} j\, \chi \ket{j}\bra{j}\hat{a}^\dagger \hat{a},
\end{equation}
where $\ket{j}$ are the ancilla states. The bosonic mode and ancilla are allowed to evolve under this coupling for a time $t = 2\pi/(L+1)\chi$, implementing the unitary
\begin{equation}
\hat{U}_{DC} = e^{i \frac{2\pi j}{L+1} \ket{j}\bra{j}\hat{a}^\dagger \hat{a} },
\end{equation}
after which the application of the gate $\hat{F}_{L+1}^\dagger$ completes the mapping of the bosonic mode's excitation number onto the ancilla. With this mapping, the measurement procedure is QND because the measurement operators $\hat{M}_k$ commute with the dispersive coupling.
As a practical matter, we note that, since the number of excitations in the bosonic mode is not known \textit{a priori}, the dispersive coupling causes an unknown shift of the ancilla transition frequencies. However, this unknown frequency shift does not pose a barrier to implementing the Fourier gates in \hyperref[fig:decayexciteL_setup]{Fig.~\ref{fig:decayexciteL_setup}(b)}. If we can drive the ancilla with strength $\Omega$ much larger the dispersive coupling $\chi$, the standard control pulse has a small error decreasing with the driving strength as $(L\chi/\Omega)^2$. Moreover, dispersive coupling induced ancilla gate errors can be further suppressed to even higher order using composite pulses~\cite{Vandersypen2005} or numerically optimized control pulses~\cite{Khaneja2005, de_Fouquieres2011}.
The HMM transition probabilities $T_{ij}$ are the same as in the previous section, but it is necessary to redefine the emission probabilities $E_{ij}$ to incorporate the $L+1$ possible ancilla readouts. We define the emission matrix elements
\begin{align}
\label{emissionelements}
E_{ij}=
\begin{cases}
(1-\delta),& \text{for } i = j \\
\delta/L,& \text{otherwise. }
\end{cases}
\end{align}
This choice\footnote{Note that with this definition $\delta$ is no longer the probability of obtaining a misleading readout. As a result, expressions involving $\delta$ in this section are not directly comparable to those in previous sections.} is made so that $\delta$ remains an easily measurable parameter: given the ability to reliably prepare an initial Fock state, $(1-\delta)$ is measurable as the probability that the state is correctly read out as the corresponding ancilla state.
\begin{figure}[bp]
\begin{center}
\includegraphics[width=\columnwidth]{decayexciteLEXACT_v4}
\caption{ Infidelity of the robust readout scheme with multi-level ancilla. The infidelity is plotted with the parameter choices $\delta = 2\%$, $\kappa_{\downarrow}\tau = 1\%$, and $\kappa_{\uparrow}\tau= 0.5\%$. The dashed line is a lower bound on the fidelity determined through information-theoretic considerations (see \hyperref[InfoTheory]{Sec.~V}). }
\label{fig:decayexciteL}
\end{center}
\end{figure}
As before, the infidelity of the level-2 readout for both the majority voting and MLE is exactly calculable with the HMM. We also approximate the infidelity for the majority voting scheme:
\begin{align}
&1-\mathcal{F} \approx {{L}\choose{m}}\left( \left\lceil \frac{N}{2}\right\rceil \kappa_\downarrow \tau\right)^{L-m}
+ \left(\left\lceil \frac{N}{2}\right\rceil \kappa_\uparrow \tau\right)^{m+1} \nonumber \\[0.1cm]
&+{{N}\choose{\ceil{N/2}}}\left[ \left(\frac{(m+1)}{L}\delta\right)^{\left\lceil \frac{N}{2}\right\rceil}+\left(\frac{(L-m)}{L}\delta\right)^{\left\lceil \frac{N}{2}\right\rceil} \right].
\end{align}
Representative infidelities are plotted in \hyperref[fig:decayexciteL]{Fig.~\ref{fig:decayexciteL}}. The most salient feature of the plot is the discrepancy between the minimum infidelities attained by the majority voting and the MLE. Whereas in the previous cases the two were not found to differ significantly, here the MLE is a clearly superior strategy. This discrepancy is due to the fact that the majority voting uses only binary information (votes for $\ket{0}$ or $\ket{L}$) to classify the $N$ level-1 outcomes. In contrast, the MLE can take any of the $L+1$ possible ancilla readouts as input and thus extracts more information from each level-1 readout. With this additional information, the MLE is able to more accurately determine the initial state. We further explore an information-theoretic description of the robust readout scheme in the next section.
\section{information-theoretic description }
\label{InfoTheory}
In this section, we consider the fidelity of the robust readout scheme from the perspective of classical information theory. The initial state of the bosonic mode constitutes one bit\footnote{In this section, all logarithms are base 2. } of information, and it is the goal of the robust readout scheme to extract as much of this information as possible. By quantifying the amount of information extracted, it is possible to place a general lower bound on the readout infidelity.
We treat the initial state of the bosonic mode as a classical discrete random variable $B$ and suppose that initial states $\ket{0}$ and $\ket{L}$ are equally likely,
\begin{equation}
p_B(b) = \frac{1}{2},
\end{equation}
where $b \in \{0,L\}$ is a realization of $B$.
Similarly, we treat the series of $N$ ancilla readouts as a discrete random variable $A$. The conditional probability distribution of $A$ given $B$ is given by the likelihood
\begin{align}
&p_{A|B}(\mathbf{a}|b) = \lambda_{\mathbf{a}}(b) \nonumber\\
= &\sum_{j_1, \ldots j_N} T_{bj_1}E_{j_1a_1}\ldots T_{j_{N-1}j_N}E_{j_N a_N},
\end{align}
where $\mathbf{a}$, an $N$-vector whose components are the ancilla measurement outcomes, is a realization of $A$. (For a two-level ancilla, $a_i \in \{0,1\}$, while for an $(L+1)$-level ancilla $a_i \in \{0,1,\ldots,L\}$.) We also calculate the remaining distributions in terms of the likelihoods: the joint probability distribution for $A$ and $B$,
\begin{equation}
p_{AB}(\mathbf{a},b) = \frac{1}{2}\lambda_{\mathbf{a}}(b);
\end{equation}
the marginal probability distribution for $A$,
\begin{align}
p_A(\mathbf{a}) = \frac{\lambda_{\mathbf{a}}(0) + \lambda_{\mathbf{a}}(L) }{2};
\end{align}
and the conditional probability distribution of $B$ given $A$,
\begin{equation}
p_{B|A}(b|\mathbf{a}) = \frac{\lambda_{\mathbf{a}}(b)}{\lambda_{\mathbf{a}}(0) + \lambda_{\mathbf{a}}(L)}.
\end{equation}
The bosonic mode's initial state contains one bit of information, as quantified by the entropy $H$ of random variable $B$,
\begin{equation}
H(B) = -\sum_b p_B(b)\log(p_B(b)) = 1.
\end{equation}
The goal of the robust readout scheme is to indirectly extract as much of this information as possible through random variable $A$. The conditional entropy
\begin{equation}
H(B|A) = - \sum_{\mathbf{a},b}p_{AB}(\mathbf{a},b) \log(p_{B|A}(b|\mathbf{a})),
\end{equation}
quantifies the amount of uncertainty in $B$ given $A$, and it follows that the mutual information
\begin{equation}
I(A;B) = H(B) - H(B|A)
\end{equation}
quantifies the amount of information extracted through the robust readout procedure.
These quantities are used to bound the readout fidelity. Consider a classification process where one attempts to determine $B$ from $A$. Let $\hat{B}(A)$ be the guessed value of $B$. The probability of an incorrect assignment $P(\hat{B}(A) \neq B) \equiv p_e$ is related to the conditional entropy through \textit{Fano's inequality},
\begin{equation}
H(B|A) \leq H_2(p_e) + p_e \log(|\mathcal{B}|-1).
\end{equation}
Here, $\mathcal{B}$ is the support of random variable $B$, and $H_2$ is the binary entropy,
\begin{equation}
H_2(p_e) = -p_e \log(p_e) - (1-p_e) \log(1-p_e).
\end{equation}
Thus, Fano's inequality places a lower bound on the infidelity of the robust readout scheme $(1-\mathcal{F}) = 2\,p_e$, and the bound is calculable in terms of the relaxation and heating probabilities, $\kappa_\downarrow\tau$ and $\kappa_\uparrow\tau$, and the level-1 readout error probability $\delta$.
This lower bound is shown in \hyperref[fig:decayexciteL]{Fig.~\ref{fig:decayexciteL}} for the case of $L=3$. This bound behaves similarly to the MLE, since the MLE is the optimal classification strategy. Despite the fact that the bound is not saturated, it is clear from the figure that classical information theory provides a reasonable alternative perspective from which the fidelity of the robust readout scheme can be understood.
For completeness, we show why the MLE does not attain the bound. The bound is saturated only if $H(B|A) = H_2(p_e)$, since $|\mathcal{B}|=2$. Equivalently, this condition may be written as $H(E|A) = H(E)$, where $E$ is the discrete random variable
\begin{equation}
E = \begin{cases}
1, & \hat{B} \neq B\\
0, & \hat{B} = B.
\end{cases}
\end{equation}
Qualitatively, $H(E|A) = H(E)$ holds when $A$ does not provide any information about whether a classification error will happen, i.e.~when classification errors are equally likely for all realizations of $A$. This property does not generally hold for the robust readout scheme since typically $P(0|L) \neq P(L|0)$. This is a consequence of the asymmetry between relaxation and heating rates, which enables one to be more confident in a correct classification for some sequences of ancilla readouts over others.
\section{Robust readout for bosonic encodings}
\label{Encodings}
Given a qubit stored in a bosonic mode as $\ket{\psi}_B = \alpha \ket{0}_B + \beta \ket{1}_B$, we have thus far only considered readout using the Fock state encoding
\begin{align}
\label{FockEncoding}
\ket{0}_B &= \ket{0} \nonumber \\
\ket{1}_B &= \ket{L}.
\end{align}
This choice was made for simplicity---with this encoding the readout fidelity can be computed classically. While this error-detecting code could be useful in a concatenated architecture~\cite{Knill2005}, it may not be ideal for more general applications. Thus, in this section we consider alternate encodings. We develop a set of sufficient encoding criteria for the robust readout procedure to be applicable, show how these criteria are satisfied by cat codes and binomial codes, and approximate the majority voting readout fidelity for both encodings.
\subsection{Encoding criteria}
For a qubit encoded in a lossy bosonic mode as $\ket{\psi}_B = \alpha \ket{0}_B + \beta \ket{1}_B$,
we identify three encoding criteria that are sufficient for robust, ancilla-assisted readout in the $\{\ket{0}_B, \ket{1}_B \}$ basis.
\textit{Criterion 1:} Encodings must be robust against excitation loss so that a single loss error cannot destroy all information about the initial state. Explicitly, when subject to $k$ excitation losses, let the logical states $\ket{0}_B$ and $\ket{1}_B$ be respectively mapped to error states $\ket{E_0^{k}}$ and $\ket{E_1^{k}}$.
The encoding is said to be robust against $d$ excitation losses if
\begin{equation}
\left\langle E_0^{k} | E_1^{\ell} \right\rangle = 0,\, \text{ for } k \text{ and }\ell \in \{0,1,\ldots, d\},
\end{equation}
where $\ket{E_0^{0}}$ $(\ket{E_1^{0}})$ denotes $\ket{0}_B$ ($\ket{1}_B$).
For example, the Fock state encoding (\ref{FockEncoding}) is robust against $d= L-1$ excitation losses.
We note that this criterion is less stringent than the Knill-Laflamme conditions for quantum error correction \cite{Knill1997} because we only need to protect a bit of $\textit{classical}$ information.
\textit{Criterion 2:} The two logical states and their corresponding error states must be distinguishable through an ancilla readout procedure that is QND. For a projective measurement described by $\{ \hat M_k\}$ that is capable of distinguishing these states, the measurement is QND if
\begin{equation}
\left[\hat{H}(t), \hat{M_k} \right] = 0\, \text{ for all $k$},
\end{equation}
where $\hat H(t)$ is the Hamiltonian describing the readout procedure. The satisfaction of this criterion enables repeated readouts. As an example, a measurement described by the operators
\begin{align}
\hat{M}_0 &= \ket{0}_B \bra{0}_B + \ket{E_0^1} \bra{E_0^1} + \ldots + \ket{E_0^d} \bra{E_0^d} \nonumber \\
\hat{M}_1 &= \ket{1}_B \bra{1}_B + \ket{E_1^1} \bra{E_1^1} + \ldots + \ket{E_1^d} \bra{E_1^d},
\end{align}
is capable of distinguishing the logical states and their corresponding error states, and it is QND if both $\hat M_0$ and $\hat M_1$ commute with $\hat H(t)$.
For the two-level ancilla readout procedure of \hyperref[decay-twolevel]{Sec.~II}, the measurement operators (\ref{Eqn:decay_meas_ops}) commute with the dispersive coupling Hamiltonian, thereby satisfying this criterion.
\textit{Criterion 3:} Ancilla errors must not induce damaging changes in the bosonic mode's state. Let possible ancilla errors be described by a set of jump operators $\{\hat{J}_\ell\}$. For an ancilla error occurring at time $t$ during a level-1 readout, the evolution of the combined system is described by the operator
\begin{align}
\hat{J}_\ell '(t) =& \mathcal{T} e^{- \frac{i}{\hbar} \int_t^\tau \hat{H}(t') dt' }\,
\hat{J}_\ell\, \mathcal{T}e^{- \frac{i}{\hbar} \int_0^t \hat{H}(t') dt'},
\end{align}
where $\mathcal{T}$ denotes time-ordering.
We must have
\begin{equation}
\left[\hat{J}_\ell '(t), \hat{M_k} \right] = 0, \text{ for all $k$ and $\ell$},
\end{equation}
so that ancilla jumps do not affect measurement outcomes by altering the bosonic mode state.
More concretely, for a $d$-level ancilla we consider the possible ancilla errors
\begin{equation}
\hat{J} \in \left\{ \ket{n}\bra{m}, \text{ for $n\neq m$ and $n,m \leq d$ } \right\},
\end{equation}
corresponding to spontaneous transitions of the ancilla state. In the dispersive coupling regime, such jumps induce dephasing of the bosonic mode that can be modeled as applications of the operator $\hat{J}'\sim\hat{n}$ and its higher powers \cite{Michael2016,Reagor2016}. Therefore, we must have $[\hat{n},\hat{M}_k] = 0$ for this criterion to be satisfied, lest readout fidelity be limited by the probability of spontaneous ancilla transitions.
These three criteria are satisfied by the Fock state encoding (\ref{FockEncoding}). We now show explicitly that the criteria are also satisfied by cat codes and binomial codes, and we approximate the fidelity of the robust readout scheme for both types of codes.
\subsection{Cat codes}
Cat codes \cite{Cochrane1999,Leghtas2013, Mirrahimi2014, Li2017} are quantum error correcting codes designed to protect against excitation loss. Quantum error correction with cat codes has recently reached the break-even point where the lifetime of encoded qubits exceeds the lifetimes of all constituent components \cite{Ofek2016}. The codewords are formed from equal superpositions of coherent states. Let the state $\ket{C_\alpha^{n}}$ be defined as a superposition of $2L$ coherent states evenly distributed around a circle in the bosonic mode's phase space
\begin{align}
\ket{C_\alpha^{n}} &= \frac{1}{2L\sqrt{N_\alpha^n}} \sum_{k=0}^{2L-1} e^{-ikn\pi/L} \ket{e^{ik\pi/L}\alpha},
\end{align}
where $N_\alpha^n$ is a normalization factor \cite{Li2017}.
These sates can be expressed in terms of Fock states as
\begin{align}
\ket{C_\alpha^{n}} &=
\frac{1}{\sqrt{N_\alpha^n}}\sum_{m=0}^{\infty}\frac{e^{-|\alpha|^2/2} \alpha^{n+2mL}}{\sqrt{(n+2mL)!}}\ket{n+2mL}_F
\end{align}
where the subscript $F$ is used in this section to distinguish Fock states from coherent states. It is important to note that $\ket{C_\alpha^{n}}$ is a superposition of Fock states which all have the same excitation number $n$ modulo $2L$. We define the logical states
\begin{align}
\ket{0}_B &= \ket{C_\alpha^L} \nonumber\\
\ket{1}_B &= \ket{C_\alpha^{2L}}.
\end{align}
\textit{Criterion 1.} After $k$ excitation loss events, the state $\ket{C_\alpha^{n}}$ is mapped to $\ket{C_\alpha^{n-k}}$. The cat codes are robust against $L-1$ excitation loss events since
\begin{equation}
\left\langle C_\alpha^{L-k} | C_\alpha^{2L-\ell} \right\rangle = 0, \text{ for $k,\ell \leq L-1$}.
\end{equation}
\textit{Criterion 2.} The cat code logical states and their corresponding error states can be distinguished by measurement of the excitation number modulo $2L$. This measurement can be described by the set of measurement operators $\{\hat{M}_k, \text{ $k = 0, \ldots, 2L -1$}\}$, where
\begin{equation}
\label{Eqn:cat_ops}
\hat{M}_k = \sum_{m = 0}^\infty \ket{k + 2Lm }\bra{k + 2Lm }.
\end{equation}
This measurement can be implemented using the dispersive coupling $\hat{H}_{DC}$ with a procedure similar to the one shown in \hyperref[fig:decayexciteL_setup]{Fig.~\ref{fig:decayexciteL_setup}(b)}. Using Fourier gates on the $2L$-level ancilla, in combination with evolution under the dispersive coupling, implement the unitary
\begin{equation}
\hat{U} = \hat F_{2L}^\dagger e^{i \frac{2\pi j}{2L}\ket{j}\bra{j} \hat a ^\dagger \hat a}\hat F_{2L},
\end{equation}
which maps the bosonic mode's excitation number modulo $2L$ onto the ancilla. This measurement process is QND because
$\left[\hat{H}_{DC}, \hat{M_k} \right] = 0$ for all $k$.
\textit{Criterion 3.} Spontaneous ancilla transitions during the readout process do not induce damaging changes in the bosonic mode's state because the measurement operators $\hat{M}_k$ commute with dephasing errors $\hat{n}$ for all $k$.
\textit{Fidelity.} To approximate the fidelity of the majority voting scheme we consider the two processes most likely to fool the voting: (1) sufficient level-1 readout errors with no excitation loss events, and (2) $L$ excitation loss events occurring sufficiently quickly with no level-1 readout errors. The probability of process (1) can be computed in terms of $\delta$, the probability of obtaining a misleading level-1 readout, as in the previous sections. To compute the probability of process (2), we first note that the Kraus operator-sum representation for the lossy bosonic channel \cite{Chuang1997} is
\begin{equation}
\mathcal{L}(\hat\rho) = \sum_{k=0}^\infty \hat A_k\, \hat \rho\, \hat A_k^\dagger,
\end{equation}
where
\begin{equation}
\hat A_k = \sqrt{\frac{(1-e^{-\kappa_\downarrow t})^k}{k!}} e^{-\kappa_\downarrow t \hat n/2} \hat a^k
\end{equation}
is the Kraus operator corresponding to $k$ excitation losses. The probability of process (2) is the probability of initial state $\ket{C_\alpha ^n}$ suffering $L$ excitation loss events in a time $\ceil{N/2}\tau$, which is approximately given by
\begin{align}
\left\langle \hat A_L^\dagger \hat A_L \right\rangle &\approx \frac{(\ceil{N/2}\kappa_\downarrow \tau)^L}{L!}
\left\langle \hat a^{\dagger L} \hat a^L \right\rangle \nonumber \\
&\approx \frac{1}{L!} \left(|\alpha|^2\ceil{N/2}\kappa_\downarrow\tau \right)^L.
\end{align}
To lowest order in $\delta$ and $\kappa_\downarrow\tau$, the cat code readout fidelity $\mathcal{F}_{\text{cat}}$ is thus given by
\begin{equation}
\mathcal{F}_{\text{cat}} \approx 1 - 2{{N}\choose{\ceil{N/2}}}\delta^{\ceil{N/2}} - \frac{2}{L!} \left(|\alpha|^2\ceil{N/2}\kappa_\downarrow\tau \right)^L.
\end{equation}
Within this approximation it is clear that both error terms are suppressed to higher order. The contribution from individual measurement infidelity is suppressed by increasing $N$, and the contribution from excitation loss is suppressed by increasing the number of coherent states comprising the cat state---analogous to increasing the excitation number used in the Fock state encoding.
\subsection{Binomial codes}
Binomial codes \cite{Michael2016} are a new class of quantum error correcting codes that can protect against excitation loss and gain errors as well as dephasing errors. The codewords are formed from superpositions of Fock states weighted with binomial coefficients
\begin{align}
\label{Eqn:binomial_codes}
\ket{0}_B &= \frac{1}{\sqrt{2^{M-1}}}\sum_{p \text{ even}}^{[0,M]}\sqrt{{M}\choose{p}}\ket{pL} \nonumber \\
\ket{1}_B &= \frac{1}{\sqrt{2^{M-1}}}\sum_{p \text{ odd}}^{[0,M]}\sqrt{{M}\choose{p}}\ket{pL},
\end{align}
where $M$ and $L$ are positive integers, and the range of the index $p$ is from $0$ to $M$.
\textit{Criterion 1.} The error state $\ket{E_0^k}$ is a superposition of Fock states with excitation number $L-k$ mod $2L$, while error state $\ket{E_1^\ell}$ is a superposition with excitation number $2L-\ell$ mod $2L$. Therefore, the binomial codes are robust against $L-1$ excitation loss events since $\left\langle E_0^{k} | E_1^{\ell} \right\rangle = 0$ for $k$ and $\ell$ between 0 and $d$.
\textit{Criterion 2.} The binomial code logical states and corresponding error states can be distinguished by measuring the excitation number modulo $2L$. This measurement (\ref{Eqn:cat_ops}) is the same as that considered for cat codes, and it is QND by the same argument.
\textit{Criterion 3.} Spontaneous ancilla transitions during the readout process do not induce damaging changes in the bosonic mode's state by the same argument as for cat codes.
\textit{Fidelity.} We approximate the fidelity of a majority voting scheme by considering the two processes most likely to fool the voting. The argument here proceeds analogously to the one given for cat codes, except that the probability of process (2) is different for binomial codes.
The probability that one of the initial states (\ref{Eqn:binomial_codes}) suffers $L$ excitation loss events in a time $\ceil{N/2}\tau$ is approximately given by
\begin{align}
\left\langle \hat A_L^\dagger \hat A_L \right\rangle &\approx \frac{(\ceil{N/2}\kappa_\downarrow \tau)^L}{L!}
\left\langle \hat a^{\dagger L} \hat a^L \right\rangle \nonumber \\
&\approx \frac{1}{L!} \left( \frac{LM}{2} \ceil{N/2}\kappa_\downarrow\tau \right)^L.
\end{align}
To lowest order in $\delta$ and $\kappa_\downarrow\tau$, the binomial code readout fidelity $\mathcal{F}_{\text{bin}}$ is then given by
\begin{equation}
\mathcal{F}_{\text{bin}} \approx 1 - 2{{N}\choose{\ceil{N/2}}}\delta^{\ceil{N/2}} - \frac{2}{L!} \left( \frac{LM}{2} \ceil{N/2}\kappa_\downarrow\tau \right)^L.
\end{equation}
As with the cat codes, it is clear that both error terms are suppressed to higher orders.
\section{Conclusions}
We have shown how the combination of robust encoding and repeated QND measurements constitutes a powerful means of improving qubit readout fidelity. Robust encodings allow one to suppress contributions to the infidelity from relaxation, and repeated QND measurements allow one to suppress contributions from individual measurement infidelity. For bosonic systems in the dispersive coupling regime, these two techniques are simultaneously applicable.
Strong dispersive couplings have already been experimentally demonstrated in circuit QED systems~\cite{Wallraff2004, Schuster2007,Boissonneault2009}, meaning the robust readout scheme can be readily applied, potentially yielding orders of magnitude improvement in readout fidelity. In principle, the scheme could also be applied to optomechanical~\cite{Jayich2008, Thompson2008}, nanomechanical~\cite{LaHaye2009, OConnell2010}, circuit quantum acoustodynamic~\cite{Manenti2017,Chu2017}, or quantum magnonics systems~\cite{Tabuchi2015, Tabuchi2016, Lachance-Quirion2017}.
In this work we have not only studied the fidelity of the scheme for a simple Fock state encoding, but we have also provided general criteria that characterize other applicable encodings. We have shown that both cat codes and binomial codes can be read out robustly, thereby providing examples of quantum error correcting codes where the robust readout scheme is applicable. Ultra-high-fidelity logical state readout would be of great practical use in a number of applications where measurement fidelity is prioritized, including gate teleportation, entanglement purification, and modular quantum computation.
\section{Acknowledgements}
We thank K.~Noh for helpful discussions. We acknowledge support from the ARL-CDQI, ARO (Grants No. W911NF-14-1-0011 and No. W911NF-14-1- 0563), ARO MURI (Grant No. W911NF-16-1-0349), NSF (Grants No. DMR-1609326 and No. DGE-1122492), AFOSR MURI (Grants No. FA9550-14-1-0052 and No. FA9550-15-1-0015), the Alfred P. Sloan Foundation (Grant No. BR2013-049), and the Packard Foundation (Grant No. 2013-39273).
\normalbaselines
|
2,869,038,155,647 | arxiv | \section{Introduction}
\label{sec:intro}
There is general belief that quantum effects dominate in a final stage of some physical phenomena to avoid undesired results predicted by classical theories. In the context of gravitational physics, one may say that the chronology protection conjecture is a manifestation of such belief stated in a verifiable form~\cite{Hawking:1991nk}. This conjecture asserts that the emergence of a closed timelike curve, which can be thought of as a natural time machine and lead to paradoxes, would be prevented by the backreaction of explosive particle creation.
A wormhole of spacetime has many interesting properties but is unfavorable in the sense that its existence leads to the emergence of closed timelike curves~\cite{Morris:1988tu}. It is not surprising if semiclassical effects act in the direction of preventing the dynamical formation of wormholes, even if it is allowed in classical frameworks.
The formation of a wormhole can happen in the merger of disconnected two spaces (see the right panel of Fig.~\ref{fg:tr}, and see Ref.~\cite{Maeda:2008bh} for a simple exact solution to the Einstein equation representing such a wormhole formation). Quantum fields living in such a spacetime will undergo the drastic change of their environments. For example, the spacetime merger gives rise to the sudden increase of spatial volume and the sudden change of boundary conditions for the quantum fields near the merger point. What we are concerned in this paper is whether or not these sudden changes excite the vacuum of quantum fields and the backreactions to the spacetime play crucial roles.
\begin{figure}[b]
\begin{center}
\begin{minipage}[c]{0.8\textwidth}
\linespread{1}
\begin{center}
\includegraphics[height=3.5cm]{fig1_trousers.eps}
\caption{{Schematic pictures of spacetime splitting (left) and
merger (right). Quantum fields living in these spacetimes undergo the
drastic change of environments such as boundary conditions.
}}
\label{fg:tr}
\end{center}
\end{minipage}
\end{center}
\end{figure}
Particle creation in the spacetime merger has not been studied extensively in the literature, as far as the present authors know (but see \cite{Braunstein:1996aj}). On the other hand, the particle creation in some situations analogous to the spacetime splitting (see the left panel of Fig.~\ref{fg:tr}) has been investigated in the literature. A pioneering work is that of Anderson and DeWitt~\cite{Anderson:1986ww}. They considered the dynamical change of spatial topology, $S^1 \to S^1 + S^1$ (consider the cut of the boundary of splitting spacetime in Fig.~\ref{fg:tr} by constant-time planes), and the vacuum excitation of a test scalar field in such a background. They found that an explosive flux of created particles is emitted from the `crotch', suggesting that the backreaction prevents the spacetime from bifurcating. This conclusion was supported by~\cite{Manogue}.
In this paper, we consider a test scalar field in a 1D Dirichlet
cavity. We assume that a both-sided Dirichlet wall can appear and
disappear instantaneously at the center of cavity, which mimics the
sudden splitting and merger of spacetime, respectively.
As we will see, when the central wall suddenly appears, the initial vacuum is highly
excited to result in a strong flux in an almost same way as the
Anderson-DeWitt analysis. Namely, the flux contains a delta function squared multiplied by a logarithmically diverging factor. On the other hand, when the central wall suddenly disappears, the initial vacuum is also highly excited to result in a diverging flux, although it does not contain the delta function squared.
As we will see, the total number of created particles diverges both in
the appearance and disappearance cases. This means that two ground
states defined when the Dirichlet wall is absent and present are
orthogonal. Namely, the vacuum structures are completely different
before and after the appearance and disappearance.
Therefore, we expect that the diverging flux appears in the both cases.
In fact, we will see that this is the case. We will discuss this point in Conclusion again.
Before starting our analysis, we note that the vacuum excitation by time-varying boundary conditions, as those considered in this paper, is not only interesting from the gravitational physics point of view, but also one of hot topics in the fundamental studies of relativistic quantum field theory. As predicted by Moore in 1970~\cite{Moore}, the non-inertial motion of cavity boundary induces the emission of photons, which is called the dynamic Casimir effect. Although the rapid acceleration of boundary, of which speed has to be comparable with the speed of light, was experimental challenge, the boundary motion turned out to be effectively realized by modulating the electromagnetic properties of boundary with high frequencies. Then, the particle creation was recently observed in a laboratory experiment using superconducting circuit~\cite{nature}. It is beyond the scope of the present paper to list the references before or after this experimental breakthrough. See, e.g. \cite{Brown:2015yma}, and references therein for recent developments and updated interests such as the quantum information theory and black-hole firewall problem.
The organization of this paper is as follows. In Sec.~\ref{sec:app}, we investigate the vacuum excitation due to the sudden appearance of the both-sided Dirichlet boundary. While the obtained result in this section is up to our expectations, we can prepare for the succeeding investigations and test the validity of our formulation. In Sec.~\ref{sec:disapp}, we investigate the vacuum excitation due to the sudden disappearance of the Dirichlet boundary. Most parts of the analysis proceed in parallel with those in Sec.~\ref{sec:app}, but the results are different. In Sec.~\ref{sec:smooth}, we discuss the vacuum excitation by the smooth appearance and disappearance of Dirichlet wall adopting the formulation in Ref.~\cite{Brown:2015yma}, and compare its instantaneous limit with the results in Secs.~\ref{sec:app} and \ref{sec:disapp}. Section~\ref{sec:conc} is devoted to a conclusion. The proof and derivation of several equations are given in Appendices. In particular, the vacuum expectation values of energy-momentum tensor in both appearance and disappearance cases, which are main results of this paper, are reproduced in Appendix~\ref{sec:green} using the Green functions, rather than the Bogoliubov transformation used in the text. We use the natural unit in which $c=\hbar=1$.
\section{Sudden appearance of a Dirichlet boundary}
\label{sec:app}
\subsection{Classical behaviors of a massless scalar field}
\begin{figure}[bht]
\begin{center}
\begin{minipage}[c]{0.8\textwidth}
\linespread{1}
\begin{center}
\includegraphics[height=6cm]{fig2_appearance.eps}
\caption{The sudden appearance of a Dirichlet boundary in a 1D
cavity. The scalar field obeys the Dirichlet boundary conditions at the
both ends $(x=\pm \frac{L}{2})$ for $-\infty<t<\infty$ and at the
center $(x=0)$ for $t>0$. The null lines $z_\pm := t \pm x = 0 \; (t>0)$ and the spatial configurations of mode
functions $g_m$ and $f_n^{(\gamma)} \; (m,n \in {\bf N},
\gamma \in \{1,2 \})$ are schematically depicted.
}
\label{fg:bc1}
\end{center}
\end{minipage}
\end{center}
\end{figure}
We consider a massless scalar field confined in a 1D cavity, which obeys the following equation of motion
\be
(-\pd_t^2 + \pd_x^2 ) \phi (t,x)=0,
\;\;\;
-\infty<t<\infty,
\;\;\;
-\frac{L}{2} \leq x \leq \frac{L}{2}.
\label{eom}
\ee
We impose the Dirichlet boundary conditions at the both ends
\be
\phi (t,\pm \frac{L}{2}) = 0,
\;\;\;
-\infty<t<\infty.
\label{BC_end}
\ee
In addition, we impose another Dirichlet boundary condition at the center after $t=0$,
\be
\phi (t, 0) = 0,
\;\;\;
t > 0,
\label{BC_App}
\ee
in order to model the sudden appearance of a perfect mirror in the cavity (see Fig.~\ref{fg:bc1}).
Preparing for the quantization of the scalar field, we shall find appropriate positive-energy mode functions before and after $t=0$.
In the past asymptotic region $t \to -\infty$, the following $\{
g_m \} \; ( m \in {\bf N} )$ constitute a set of positive-energy mode functions,
\be
g_m(t,x)
=
\frac{1}{\sqrt{m\pi}} e^{ -i q_m t} \times
\begin{cases}
\cos q_m x & (m: \mbox{odd}) \\
\sin q_m x & (m: \mbox{even})
\end{cases},
\;\;\;
q_m := \frac{m\pi}{L},
\;\;\;
-\frac{L}{2} \leq x \leq \frac{L}{2},
\label{g}
\ee
which satisfy the orthonormal conditions
\be
\langle g_m, g_{m'} \rangle =- \langle g_m^\ast, g_{m'}^\ast \rangle = \delta_{mm'},
\;\;\;
\langle g_m,g_{m'}^\ast \rangle =0.
\ee
Here, the asterisk denotes the complex conjugate and $ \langle \phi,\psi \rangle :=i\int_{-L/2}^{L/2} (\phi^\ast \pd_t \psi - \pd_t \phi^\ast \psi) dx $ is the Klein-Gordon inner product, evaluated on a spacelike curve $t={\rm const.}$~\cite{Birrell:1982ix}. This inner product is conserved, namely independent of time, whenever both $\phi$ and $\psi$ are solutions to the equation of motion~\eqref{eom} and vanish on every boundary. Note that the above expression of $g_m$, Eq.~\eqref{g}, is valid only before the appearance of the Dirichlet wall.
In the future asymptotic region $t \to +\infty$, the following $\{
f_n^{(\gamma)} \}\; (\gamma \in \{1,2\},\; n \in {\bf N})$ constitute a set of positive-energy mode functions,
\begin{align}
\begin{split}
&
f^{(1)}_n (t,x)
=
\begin{cases}
\displaystyle 0 & \displaystyle ( -\frac{L}{2} \leq x < 0 )\\
\displaystyle \frac{1}{\sqrt{n\pi}} e^{-i p_n t} \sin p_n x & \displaystyle ( 0 \leq x \leq \frac{L}{2} )
\end{cases},
\\
&
f^{(2)}_n (t,x)
=
\begin{cases}
\displaystyle -\frac{1}{\sqrt{n\pi}} e^{ -i p_n t } \sin p_n x
& \displaystyle ( -\frac{L}{2} \leq x < 0 ) \\
\displaystyle 0 & \displaystyle ( 0 \leq x \leq \frac{L}{2} )
\end{cases},
\;\;\;
p_n := \frac{2n\pi}{L},
\label{f}
\end{split}
\end{align}
which satisfy the orthonormal conditions
\be
\langle f_{n}^{(\gamma)}, f_{n'}^{(\gamma')} \rangle
=
- \langle f_{n}^{(\gamma)\ast}, f_{n'}^{(\gamma')\ast} \rangle
=
\delta_{\gamma\gamma'} \delta_{nn'},
\;\;\;
\langle f_{n}^{(\gamma)}, f_{n'}^{(\gamma')\ast} \rangle = 0.
\ee
Noted that the above expression of $f_m^{(\gamma)}$, Eq.~\eqref{f}, is valid only after the appearance of the Dirichlet wall.
We consider the expansion of $g_m$ by $f_n^{(\gamma)}$,
\be
g_m
=
\sum_{\gamma=1}^2 \sum_{n=1}^\infty
(
\rho_{mn}^{(\gamma)} f_n^{(\gamma)}
+
\sigma_{mn}^{(\gamma)} f_n^{(\gamma)\ast}
),
\label{gBYf}
\ee
where the expansion coefficients, called the Bogoliubov coefficients, are evaluated as
\be
\rho_{mn}^{(\gamma)}
=
\langle f_n^{(\gamma)}, g_m \rangle,
\;\;\;
\sigma_{mn}^{(\gamma)}
=
- \langle f_n^{(\gamma)\ast}, g_m \rangle.
\label{bogo}
\ee
We take $t=0$ as the spacelike curve on which the inner products in Eq.~\eqref{bogo} are evaluated. Substituting Eqs.~\eqref{g} and \eqref{f} into Eq.~\eqref{bogo}, we obtain
\be
\rho_{mn}^{(\gamma)}
=
\begin{cases}
\displaystyle \frac{2}{(2n-m)\pi} \sqrt{ \frac{n}{m} } & (m:\mbox{odd}) \\
\displaystyle \frac{ (-1)^{\gamma-1} }{\sqrt{2}}\delta_{m,2n} & (m:\mbox{even})
\end{cases},
\;\;\;
\sigma_{mn}^{(\gamma)}
=
\begin{cases}
\displaystyle \frac{2}{(2n+m)\pi} \sqrt{ \frac{n}{m} } & (m:\mbox{odd}) \\
\displaystyle 0 & (m:\mbox{even})
\end{cases}.
\label{rho_sigma}
\ee
Here, we note that the expansion \eqref{gBYf} for odd $m$ is valid
everywhere except for $x=0$ (namely, almost everywhere in a mathematical sense).
This is because the mode functions $g_m$ for odd $m$ can take non-zero values at $x=0$ while the mode functions $f_n$ are always zero by the boundary conditions. We will look into the implication of Eq.~\eqref{gBYf} in Conclusion, comparing with corresponding relation~\eqref{f_g} in the disappearance case.
\subsection{Quantization of the scalar field}
The canonical quantization of the scalar field is implemented by expanding the field operator $ {\bm \phi} $ by two set of mode functions, $\{ g_m \}$ or $\{ f_n^{(\gamma)} \}$, as
\begin{align}
{\bm \phi}
&=
\sum_{m=1}^\infty
(
{\bm b}_m g_m + {\bm b}_m^\dagger g_m^\ast
)
\label{phi_g}
\\
&=
\sum_{\gamma=1}^2 \sum_{n=1}^\infty
(
{\bm a}_n^{(\gamma)} f_n^{(\gamma)}
+
{\bm a}_n^{(\gamma) \dagger } f_n^{(\gamma) \ast }
),
\label{phi_f}
\end{align}
and imposing the commutation relations on the expansion coefficients
\begin{align}
[ {\bm b}_m, {\bm b}_{m'}^{\dagger} ]
=
\delta_{mm'},
\;\;\;
&
[ {\bm b}_m, {\bm b}_{m'} ]
=
0,
\label{CR_b}
\\
[ {\bm a}_n^{(\gamma)}, {\bm a}_{n'}^{(\gamma')\dagger} ]
=
\delta_{\gamma \gamma'} \delta_{nn'},
\;\;\;
&
[ {\bm a}_n^{(\gamma)}, {\bm a}_{n'}^{(\gamma')} ]
=
0.
\label{CR_a}
\end{align}
Then, ${\bm b}_m$ and ${\bm a}_n^{ (\gamma) }$ are interpreted as annihilation operators, and ${\bm b}_m^\dagger$ and ${\bm a}_n^{(\gamma) \dagger }$ creation operators.
Substituting Eq.~\eqref{gBYf} into Eq.~\eqref{phi_g}, and comparing it with Eq.~\eqref{phi_f}, one obtains
\be
{\bm a}_n^{(\gamma)}
=
\sum_{m=1}^\infty
(
\rho_{mn}^{(\gamma)} {\bm b}_m
+
\sigma_{mn}^{(\gamma) \ast} {\bm b}_m^\dagger
).
\label{aBYb}
\ee
Substituting Eq.~\eqref{aBYb} into Eq.~\eqref{CR_a}, and using Eq.~\eqref{CR_b}, one finds that the following conditions must hold for the two quantizations, Eqs.~\eqref{phi_g} and \eqref{phi_f}, to be consistent.
\begin{align}
\sum_{m=1}^\infty
(
\rho_{mn}^{(\gamma)} \rho_{mn'}^{(\gamma') \ast}
-
\sigma_{mn}^{(\gamma)\ast} \sigma_{mn'}^{(\gamma')}
)
&=
\delta_{\gamma \gamma'} \delta_{nn'},
\label{Consis1}
\\
\sum_{m=1}^\infty
(
\rho_{mn}^{(\gamma)} \sigma_{mn'}^{(\gamma') \ast}
-
\sigma_{mn}^{(\gamma)\ast} \rho_{mn'}^{(\gamma')}
)
&=
0.
\label{Consis2}
\end{align}
Hereafter, we call these relations unitarity relations. In Appendix~\ref{sec:con1}, we prove that the Bogoliubov coefficients given by Eq.~\eqref{rho_sigma} satisfy these unitarity relations.
Since we are interested in the particle creation caused by the appearance of boundary, we assume that the quantum field is in the vacuum state $| 0_g \rangle$ in which any particle corresponding to $g_m$ does not exist. Such a vacuum is characterized by
\be
{\bm b}_m | 0_g \rangle =0,
\;\;\;
\langle 0_g | 0_g \rangle =1,
\;\;\;
\forall m \in {\bf N}.
\label{vac1}
\ee
\subsection{Spectrum and energy-momentum density}
The vacuum $| 0_g \rangle$ contains no particle corresponding to $g_m$ but can contain particles corresponding to $f_n^{(\gamma)}$. This is examined by calculating the vacuum expectation value of particle-number operator,
\begin{align}
\langle 0_g | {\bm a}_n^{(\gamma)\dagger} {\bm a}_n^{(\gamma)} | 0_g \rangle
=
\sum_{m=1}^\infty | \sigma_{mn}^{(\gamma)} |^2
=
\frac{4n}{\pi^2}
\sum_{\substack{ m=1 \\ m:{\rm odd}}}^\infty
\frac{1}{ m (m+2n)^2 }.
\label{NumDen2}
\end{align}
While this is finite, its summation over $n $ and $\gamma $, i.e.\ the total number of created particles, diverges.
This implies that the Fock space representation associated with ${\bm
b}_{m}$ is unitarily inequivalent to that associated with ${\bm a}_n^{(\gamma)}$~\cite{Wald:1995yp,Rodriguez-Vazquez:2014hka}.
In order to see directly what happens, we calculate the vacuum
expectation value of the energy-momentum tensorial operator, which is
given for the massless scalar field by $ {\bm T}_{\mu\nu} = \pd_\mu {\bm
\phi} \pd_\nu {\bm \phi} - \frac12 \eta_{\mu\nu} (\pd {\bm
\phi})^2$. Here, $\eta_{\mu\nu}={\rm Diag}.\ (-1,1)$ is the two-dimensional
Minkowski metric. If one introduces the double null coordinates, the non-zero components of the energy-momentum operator are
\be
{\bm T}_{\pm \pm} = ( \pd_\pm {\bm \phi} )^2,
\;\;\;
z_{\pm} := t \pm x.
\label{tpm}
\ee
Note that the energy density and momentum density (or energy-flux density) in the original Cartesian coordinates are given by $ {\bm T}^{tt} = {\bm T}_{--} + {\bm T}_{++}$ and ${\bm T}^{tx} = {\bm T}_{--} - {\bm T}_{++}$, respectively.
Substituting Eq.~\eqref{phi_g} into Eq.~\eqref{tpm}, and using
Eq.~\eqref{g}, we obtain the vacuum expectation value before the
appearance of the Dirichlet boundary,
\begin{align}
\langle 0_g | {\bm T}_{\pm\pm} | 0_g \rangle_{t<0}
=
\sum_{m=1}^\infty | \pd_\pm g_m |^2
=
\frac{\pi}{4L^2} \sum_{m=1}^\infty m.
\label{Tab_02}
\end{align}
This is clearly divergent but can be renormalized by
standard procedures~\cite{Birrell:1982ix}.
If we remove the ultraviolet divergence caused by the vacuum energy of
the Minkowski space (namely $L\to\infty$), then we obtain well-known
finite result as
\begin{equation}
\langle 0_g | {\bm T}_{\pm\pm} | 0_g \rangle_{{\rm ren}, t<0} = -
\frac{\pi}{48L^2}.
\label{eq:simple_Casimir}
\end{equation}
Such a negative energy is called the Casimir energy. In the ordinary 3D electromagnetic case, this kind of negative energy gives rise to an attractive force between two parallel neutral plates put in vacuum~\cite{Casimir}.
What we are most interested in is the same quantity after the appearance
of the Dirichlet boundary. Substituting Eq.~\eqref{phi_f} into Eq.~\eqref{tpm}, and then using Eq.~\eqref{aBYb}, we obtain
\begin{align}
&
\langle 0_g| {\bm T}_{\pm\pm} |0_g \rangle_{t>0}
\nn
\\
&=
\sum_{\gamma=1}^2 \sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty \sum_{n=1}^\infty \sum_{n'=1}^\infty
[
( \rho_{mn}^{(\gamma)} \sigma_{mn'}^{(\gamma)} + \rho_{mn'}^{(\gamma)} \sigma_{mn}^{(\gamma)} )
{\rm Re}
( \pd_\pm f_n^{(\gamma)} \pd_\pm f_{n'}^{(\gamma)} )
+
( \rho_{mn}^{(\gamma)} \rho_{mn'}^{(\gamma)} + \sigma_{mn}^{(\gamma)} \sigma_{mn'}^{(\gamma)} )
{\rm Re}
( \pd_\pm f_n^{(\gamma)} \pd_\pm f_{n'}^{(\gamma)\ast} )
]
\nn
\\
&+
\sum_{\gamma=1}^2 \sum_{ \substack{ m=2 \\ m: {\rm even} } }^\infty \sum_{n=1}^\infty \sum_{n'=1}^\infty
\rho_{mn}^{(\gamma)} \rho_{mn'}^{(\gamma)} {\rm Re}
( \pd_\pm f_n^{(\gamma)} \pd_\pm f_{n'}^{(\gamma)\ast} ).
\label{TAB_01}
\end{align}
To derive Eq.~\eqref{TAB_01}, we symmetrize the dummy indices $n$ and $n'$. In addition, we use the facts that $ \sigma_{mn}^{(\gamma)} $ vanishes for even $m$, and $\pd_\pm f_n^{(1)}$ and $\pd_\pm f_{n'}^{(2)}$ have no common support, i.e.\ $ \pd_\pm f_n^{(\gamma)} \pd_\pm f_{n'}^{(\gamma')} \propto \delta_{\gamma \gamma'}$.
Substituting explicit form of Bogoliubov coefficients~\eqref{rho_sigma} and mode functions~\eqref{f} into Eq.~\eqref{TAB_01}, we obtain
\begin{align}
&
\langle 0_g| {\bm T}_{\pm\pm} |0_g \rangle_{t>0}
\nn
\\
&=
\frac{1}{\pi L^2} \sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty
\left(
\frac{1}{4m}
[
4 \sum_{n=1}^\infty \cos ( \frac{2n\pi}{L} z_\pm )
+
m^2 \sum_{n=1}^\infty \frac{ \cos ( \frac{2n\pi}{L} z_\pm ) }{ n^2-(m/2)^2 }
]^2
+
m
[
\sum_{n=1}^\infty \frac{ n \sin ( \frac{ 2n\pi }{L} z_\pm ) }{ n^2-(m/2)^2 }
]^2
\right)
+
\frac{\pi}{4L^2} \sum_{ \substack{ m=2 \\ m: {\rm even} } }^\infty m.
\label{TAB_02}
\end{align}
This is an even function of $z_\pm$ with period $L$, as it is invariant
under reflection $z_\pm \to - z_\pm$ and translation $z_\pm \to z_\pm +
L$. Therefore, it is sufficient to calculate it in $0 \leq z_\pm < L$,
and then generalize the obtained expression to the one valid in the entire domain appropriately.
The first and second summations over $n$ in Eq.~\eqref{TAB_02} can be calculated to give
\begin{align}
&
\langle 0_g| {\bm T}_{\pm\pm} |0_g \rangle_{t>0}
\nn
\\
&=
\frac{1}{\pi L^2} \sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty
\left(
\frac{1}{4m}
[
4 L^2 \delta (z_\pm)^2 + m^2 \pi^2 \sin^2 ( \frac{m\pi}{L} z_\pm )
]
+
m
[
\sum_{n=1}^\infty \frac{ n \sin ( \frac{ 2n\pi }{L} z_\pm ) }{ n^2-(m/2)^2 }
]^2
\right)
+
\frac{\pi}{4L^2} \sum_{ \substack{ m=2 \\ m: {\rm even} } }^\infty m,
\label{TAB_03}
\end{align}
which is valid in $ 0 \leq z_\pm < L $. Here, we have used the following formulas,
\begin{align}
\sum_{k=1}^\infty \cos( \frac{ 2k \pi}{a} y )
&=
-\frac12 + \frac{a}{2} \sum_{\ell = -\infty}^\infty \delta( y- \ell a ),
\;\;\;
( - \infty < y < \infty ),
\\
\sum_{k=1}^\infty
\frac{ \cos ky }{ k^2-a^2 }
&=
-\frac{\pi}{2a} \cos[ a(\pi - y) ] {\rm cosec} (a \pi )+\frac{1}{2a^2},
\;\;\;
(0 \leq y \leq 2\pi),
\end{align}
where $\delta$ represents the Dirac delta function. See Ref.~\cite[p.~730]{maru} for the second formula.
For $z_\pm = 0$, from Eq.~\eqref{TAB_03}, we have
\be
\langle 0_g| {\bm T}_{\pm\pm} |0_g \rangle_{t>0}
=
\frac{ 1 }{\pi}
\sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty
\frac{ \delta(0)^2 }{m}
+
\frac{\pi}{4L^2} \sum_{ \substack{ m=2 \\ m: {\rm even} } }^\infty m,
\;\;\;
(z_\pm = 0).
\label{TAB_04}
\ee
For $ 0 < z_\pm < L$, the summation over $n$ in Eq.~\eqref{TAB_03} can be calculated to give
\be
\langle 0_g| {\bm T}_{\pm\pm} |0_g \rangle_{t>0}
=
\frac{\pi}{4L^2} \sum_{m=1}^\infty m,
\;\;\;
(0<z_\pm < L),
\label{TAB_05}
\ee
using the following formula~\cite[p.~730]{maru}
\begin{align}
\sum_{k=1}^\infty
\frac{ k \sin ky }{ k^2-a^2 }
&=
\frac{\pi}{2} \sin[ a(\pi - y) ] {\rm cosec} (a \pi ),
\;\;\;
(0 <y < 2\pi).
\end{align}
Extending the domain of Eqs.~\eqref{TAB_04} and \eqref{TAB_05} to the entire domain periodically, we obtain
\be
\langle 0_g| {\bm T}_{\pm\pm} |0_g \rangle_{t>0}
=
\frac{ 1 }{\pi}
\sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty
\frac{ 1 }{m}
\sum_{\ell=-\infty}^\infty \delta( z_\pm - \ell L )^2
+
\begin{cases}
\displaystyle
\frac{\pi}{4L^2} \sum_{ \substack{ m=2 \\ m: {\rm even} } }^\infty m
&
(z_\pm = \ell L, \; \ell \in {\bf Z} ) \\
\displaystyle
\frac{\pi}{4L^2} \sum_{m=1}^\infty m
&
({\rm otherwise})
\end{cases}.
\label{Tab_03}
\ee
From the above result, we immediately see that an infinitely strong energy flux which
behaves as the delta function squared multiplied by a logarithmically
divergent factor emanates from the appearance point of the Dirichlet
boundary.
Note that the delta function squared means that not only the energy
density but also the total energy emitted diverge.
Such a divergent flux suggests that its backreaction to the spacetime
and boundary is not ignorable. See Fig.~\ref{fg:VEV} for 3D plots of the
energy density and momentum density with cutoff.
We have to pay attention also to the second term on the right-hand side of Eq.~\eqref{Tab_03}. Everywhere except the null lines emanating from the appearance
point, this divergent summation has the same form as Eq.~\eqref{Tab_02}.
Hence, by the same renormalization procedure, we obtain a finite result for $t>0$ which is the same as Eq.~\eqref{eq:simple_Casimir} for $t<0$. On the other hand, on the null lines, the summation is different from the previous one. Such a different divergence may yield a non-renormalizable ultraviolet divergence.
Indeed, we present another derivation of Eq.~\eqref{Tab_03} by the
Green-function method in Appendix~\ref{sec:green} and evaluate
``a renormalized energy-momentum tensor'' by the point-splitting method.
The result gives Eq.~(\ref{eq:simple_Casimir}) for $t<0$ but
the following for $t>0$:
\be
\langle 0_g | {\bm T}_{\pm\pm} | 0_g \rangle_{{\rm ren}, t>0}=
\frac{ 1 }{\pi}
\sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty
\frac{ 1 }{m}
\sum_{\ell=-\infty}^\infty \delta( z_\pm - \ell L )^2
+
\begin{cases}
\displaystyle
- \frac{\pi}{24L^2}
+ \lim_{\Delta z_\pm \to 0} \frac{1}{8\pi\Delta z_\pm^2}
& (z_\pm = \ell L, \; \ell \in {\bf Z}) \\
\displaystyle
- \frac{\pi}{48L^2} & ({\rm otherwise}) \\
\end{cases} .
\label{Tab_03_ren}
\ee
Thus, in the coincidence limit $\Delta z_\pm \to 0$, an ultraviolet divergence like $(\Delta z_\pm)^{-2}$ remains on the null lines.
The divergence like $(\Delta z_{\pm})^{-2}$ suggests that the contribution to the total energy from this term also diverges.
\begin{figure}[bht]
\begin{center}
\begin{minipage}[c]{0.8\textwidth}
\linespread{1}
\begin{center}
\setlength{\tabcolsep}{ 0 pt }
\begin{tabular}{ cc }
\includegraphics[height=6.5cm]{fig3a_e-density.eps} &
\includegraphics[height=6.5cm]{fig3b_m-density.eps} \\
\end{tabular}
\caption{The vacuum expectation values of energy density $\langle 0_g | ( {\bm T}_{--} + {\bm T}_{++} ) | 0_g \rangle_{t>0}$ (left) and momentum density $ \langle 0_g | ({\bm T}_{--} - {\bm T}_{++}) | 0_g \rangle_{t>0} $ (right) with cutoff from which the Casimir contribution is subtracted. We set $L=1$ and summation over modes in Eq.~\eqref{TAB_02} is taken until $n = m = 14$. The exact results without cutoff are given by Eq.~\eqref{Tab_03}.
}
\label{fg:VEV}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\section{Sudden disappearance of a Dirichlet boundary}
\label{sec:disapp}
In this section, we consider the sudden disappearance of the Dirichlet boundary (see Fig.~\ref{fg:bc2}). Since the situation is a kind of time reversal of that in the previous section, most parts of calculation can be reused in this section.
\begin{figure}[bth]
\begin{center}
\begin{minipage}[c]{0.8\textwidth}
\linespread{1}
\begin{center}
\includegraphics[height=6cm]{fig4_disappearance.eps}
\caption{The sudden disappearance of a Dirichlet boundary in a 1D
cavity. The scalar field obeys the Dirichlet boundary conditions at the
both ends $(x=\pm \frac{L}{2})$ for $-\infty<t<\infty$ and at the
center $(x=0)$ for $t < 0$. The null lines $z_\pm := t \pm x = 0 \; (t>0)$ and the spatial configurations of mode functions $g_m$ and $f_n^{(\gamma)} \; (m,n \in {\bf N}, \gamma \in \{1,2 \})$ are schematically depicted.
}
\label{fg:bc2}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\subsection{Classical behaviors and quantization of the massless scalar field}
In addition to the Dirichlet boundary conditions at the both ends~\eqref{BC_end}, the scalar field obeys the Dirichlet boundary condition at the center before $t=0$,
\be
\phi (t,0)=0, \;\;\; t<0.
\label{BC_DisApp}
\ee
Then, the positive-energy mode functions in the asymptotic regions $t \to -\infty$ and $t \to \infty$ are given by Eqs.~\eqref{f} and \eqref{g}, respectively.
We expand $f_n^{(\gamma)}$ by $g_m$,
\be
f_n^{(\gamma)}
=
\sum_{m=1}^\infty ( \alpha_{nm}^{(\gamma)} g_m + \beta_{nm}^{(\gamma)} g_m^\ast ).
\label{f_g}
\ee
Here, the expansion coefficients are given by
\be
\alpha_{nm}^{(\gamma)}
=
\langle g_m, f_n^{(\gamma)} \rangle
=
\rho_{mn}^{(\gamma)\ast},
\;\;\;
\beta_{nm}^{(\gamma)}
=
- \langle g_m^\ast, f_n^{(\gamma)} \rangle
=
-\sigma_{mn}^{(\gamma)},
\label{alpha_beta}
\ee
where $\rho_{mn}^{(\gamma)}$ and $\sigma_{mn}^{(\gamma)}$ are given by Eq.~\eqref{rho_sigma}.
The quantization of the scalar field is again implemented by
Eqs.~\eqref{phi_g}--\eqref{CR_a}. Substituting Eq.~\eqref{f_g} into
Eq.~\eqref{phi_f} and comparing it with \eqref{phi_g}, we obtain
\be
{\bm b}_m
=
\sum_{\gamma=1}^2 \sum_{n=1}^\infty
(
\alpha_{nm}^{(\gamma)} {\bm a}^{(\gamma)}_n
+
\beta_{nm}^{(\gamma) \ast } {\bm a}^{(\gamma)\dagger}_n
).
\label{bBYa}
\ee
Substituting Eq.~\eqref{bBYa} into Eq.~\eqref{CR_b}, and using Eq.~\eqref{CR_a}, one finds that the following conditions must hold for the two quantizations, Eqs.~\eqref{phi_g} and \eqref{phi_f}, to be consistent.
\begin{align}
\sum_{\gamma=1}^2 \sum_{n=1}^\infty
(
\alpha_{nm}^{(\gamma)} \alpha_{nm'}^{(\gamma)\ast}
-
\beta_{nm}^{(\gamma)\ast} \beta_{nm'}^{(\gamma)}
)
&=
\delta_{mm'},
\label{Consis3}
\\
\sum_{\gamma=1}^2 \sum_{n=1}^\infty
(
\alpha_{nm}^{(\gamma)} \beta_{nm'}^{(\gamma)\ast}
-
\beta_{nm}^{(\gamma)\ast} \alpha_{nm'}^{(\gamma)}
)
&=
0.
\label{Consis4}
\end{align}
It is shown in Appendix~\ref{sec:con2} that these unitarity relations indeed hold for the Bogoliubov coefficients given by Eq.~\eqref{alpha_beta}.
Since we are interested in the particle creation due to the disappearance of boundary, we assume that the quantum field is in the vacuum state $| 0_f \rangle$ in which any particle corresponding to $f_n^{(\gamma)}$ does not exist. Such a vacuum is characterized by
\be
{\bm a}_n^{(\gamma)} |0_f \rangle = 0,
\;\;\;
\langle 0_f | 0_f \rangle =1,
\;\;\;
\forall \gamma \in \{ 1,2 \},
\;\;\;
\forall n \in {\bf N}.
\label{vac2}
\ee
\subsection{Spectrum and energy-momentum density}
The vacuum $| 0_f \rangle$ contains no particle corresponding to $f_n^{(\gamma)}$ but can contain particles corresponding to $g_m$. This is examined by calculating the vacuum expectation value of particle-number operator,
\begin{align}
\langle 0_f | {\bm b}_m^\dagger {\bm b}_m | 0_f \rangle
=
\sum_{\gamma=1}^2 \sum_{n=1}^\infty | \beta_{nm}^{(\gamma)} |^2
=
\begin{cases}
\displaystyle \frac{2}{m\pi^2} \sum_{n=1}^\infty \frac{n}{(n+m/2)^2} & (m:\mbox{odd}) \\
\displaystyle 0 & (m:\mbox{even})
\end{cases}.
\label{NumDen4}
\end{align}
This is logarithmically divergent for odd $m$. Therefore, the total
number of created particles, i.e.\ the summation over $m \in {\bf N}$ of
Eq.~\eqref{NumDen4}, also diverges. This implies again that the Fock
space representation associated with ${\bm a}_n^{(\gamma)}$ is unitarily
inequivalent to that associated with ${\bm b}_{m}$~\cite{Wald:1995yp,Rodriguez-Vazquez:2014hka}.
Substituting Eq.~\eqref{phi_f} into Eq.~\eqref{tpm}, and using
Eq.~\eqref{f}, we obtain the vacuum expectation value of the
energy-momentum tensor before the disappearance of the Dirichlet boundary,
\begin{align}
\langle 0_f | {\bm T}_{\pm \pm} |0_f \rangle_{t<0}
=
\sum_{\gamma=1}^2 \sum_{n=1}^\infty | \pd_{\pm} f_n^{(\gamma)} |^2
=
\frac{\pi}{L^2} \sum_{n=1}^\infty n.
\label{Tab_12}
\end{align}
We can renormalize this by standard procedures again to obtain a finite
result,
\begin{equation}
\langle 0_f | {\bm T}_{\pm \pm} |0_f \rangle_{{\rm ren}, t<0} = - \frac{\pi}{12L^2}.
\label{eq:simple_Casimir_2}
\end{equation}
What we are most interested in is the same quantity after the disappearance of the Dirichlet boundary. Substituting Eq.~\eqref{phi_g} into Eq.~\eqref{tpm}, and then using Eq.~\eqref{bBYa}, such a quantity is obtained as
\begin{align}
&\langle 0_f |{\bm T}_{\pm\pm}| 0_f \rangle_{t>0}
\nn
\\
&=
\sum_{\gamma=1}^2
\sum_{n=1}^\infty
\sum_{ \substack{ m=1 \\ m:{\rm odd} } }^\infty
\sum_{ \substack{ m'=1 \\ m':{\rm odd} } }^\infty
[
(
\alpha_{nm}^{(\gamma)} \beta_{nm'}^{(\gamma)}
+
\alpha_{nm'}^{(\gamma)} \beta_{nm}^{(\gamma)}
) {\rm Re} (\pd_\pm g_m \pd_\pm g_{m'})
+
(
\alpha_{nm}^{(\gamma)} \alpha_{nm'}^{(\gamma)}
+
\beta_{nm'}^{(\gamma)} \beta_{nm}^{(\gamma)}
) {\rm Re} (\pd_\pm g_m \pd_\pm g^\ast_{m'})
]
\nn
\\
&+
\sum_{\gamma=1}^2
\sum_{n=1}^\infty
\sum_{ \substack{ m=2 \\ m:{\rm even} } }^\infty
\sum_{ \substack{ m'=2 \\ m': {\rm even} } }^\infty
\alpha_{nm}^{(\gamma)} \alpha_{nm'}^{(\gamma)}
{\rm Re} (\pd_\pm g_m \pd_\pm g^\ast_{m'}).
\label{TAB_11}
\end{align}
To derive Eq.~\eqref{TAB_11}, we symmetrize the dummy indices $m$ and $m'$. In addition, we use the facts that $ \beta_{nm}^{(\gamma)} $ vanishes for even $m$, and implicitly use a few properties of Bogoliubov coefficients~\eqref{alpha_beta} such as the $\gamma$-dependence.
Using the explicit form of Bogoliubov coefficients and mode functions, Eqs.~\eqref{alpha_beta}, \eqref{rho_sigma}, and \eqref{g}, equation \eqref{TAB_11} is written as
\begin{align}
\langle 0_f |{\bm T}_{\pm\pm}| 0_f \rangle_{t>0}
=
\frac{8}{\pi L^2}
\sum_{n=1}^\infty
\left(
4n^3
[
\sum_{ \substack{ m=1 \\ m: {\rm odd} } }^\infty
\frac{ \cos ( \frac{m\pi}{L} z_\pm ) }{ m^2-(2n)^2 }
]^2
+
n
[
\sum_{ \substack{ m=1 \\ m: {\rm odd} } }^\infty
\frac{ m \sin ( \frac{m\pi}{L} z_\pm ) }{ m^2-(2n)^2 }
]^2
\right)
+ \frac{\pi}{2L^2}
\sum_{n=1}^\infty n.
\label{TAB_12}
\end{align}
This is an even function of $z_\pm$ with period $L$, as it is invariant under reflection $z_\pm \to - z_\pm$ and translation $z_\pm \to z_\pm + L$. Therefore, it is sufficient to calculate it in $0 \leq z_\pm < L$, and then generalize the obtained expression to one valid in the entire domain appropriately.
The first summation over odd $m$ in Eq.~\eqref{TAB_12} can be calculated to give
\begin{align}
\langle 0_f |{\bm T}_{\pm\pm}| 0_f \rangle_{t>0}
=
\frac{8}{\pi L^2}
\sum_{n=1}^\infty
n \left(
\frac{\pi^2}{16} \sin^2 ( \frac{2n\pi}{L} z_\pm )
+
[
\sum_{ \substack{ m=1 \\ m: {\rm odd} } }^\infty
\frac{ m \sin ( \frac{m\pi}{L} z_\pm ) }{ m^2-(2n)^2 }
]^2
\right)
+ \frac{\pi}{2L^2}
\sum_{n=1}^\infty n,
\label{TAB_13}
\end{align}
which is valid in $0 \leq z_\pm < L$. Here, we have used the following formula~\cite[p.\ 733]{maru},
\begin{align}
\sum_{k=0}^\infty
\frac{ \cos [(2k+1)y] }{(2k+1)^2 -a^2}
=
\frac{\pi}{4a} \sin[ \frac{a}{2}(\pi - 2y) ] \sec (\frac{a\pi}{2}),
\;\;\;
( 0 \leq y \leq \pi ).
\label{maru_typo1}
\end{align}
It is noted here that there are typos in Ref.~\cite[p.\ 733]{maru} about formulas~\eqref{maru_typo1} and \eqref{maru_typo2} (see below).
For $z_\pm=0$, from Eq.~\eqref{TAB_13}, we have
\be
\langle 0_f |{\bm T}_{\pm\pm}| 0_f \rangle_{t>0}
=
\frac{\pi}{2L^2}
\sum_{n=1}^\infty n,
\;\;\;
(z_\pm = 0).
\label{TAB_14}
\ee
For $ 0 < z_\pm < L$, we find that the summation over odd $m$ in Eq.~\eqref{TAB_13} can be calculated to give
\be
\langle 0_f |{\bm T}_{\pm\pm}| 0_f \rangle_{t>0}
=
\frac{\pi}{L^2}
\sum_{n=1}^\infty n,
\;\;\;
(0 < z_\pm < L),
\label{TAB_15}
\ee
using the following formula~\cite[p.\ 733]{maru},
\begin{align}
\sum_{k=0}^\infty
\frac{ (2k+1) \sin [(2k+1)y] }{(2k+1)^2 -a^2}
&=
\frac{\pi}{4} \cos[ \frac{a}{2}(\pi - 2y) ] \sec (\frac{a\pi}{2}),
\;\;\;
( 0 < y < \pi ).
\label{maru_typo2}
\end{align}
Extending the domain of Eqs.~\eqref{TAB_14} and \eqref{TAB_15} to the entire domain periodically,
we obtain
\be
\langle 0_f | {\bm T}_{\pm \pm} |0_f \rangle_{t>0}
=
\begin{cases}
\displaystyle \frac{\pi}{2L^2} \sum_{n=1}^\infty n & (z_\pm = \ell L, \; \ell \in {\bf Z} )\\
\displaystyle \frac{\pi}{L^2} \sum_{n=1}^\infty n & ({\rm otherwise})\\
\end{cases} .
\label{Tab_13}
\ee
Although we have no term proportional to the delta function squared in contrast to the appearance case, we have to pay attention again to the diverging summations in Eq.~\eqref{Tab_13}. On the null lines, the energy-momentum tensor \eqref{Tab_13} takes the different form from Eq.~\eqref{Tab_12}. Indeed, the renormalized energy-momentum tensor is given by Eq.~(\ref{eq:simple_Casimir_2}) for $t<0$ but by the following for $t>0$:
\be
\langle 0_f | {\bm T}_{\pm\pm} | 0_f \rangle_{{\rm ren}, t>0}=
\begin{cases}
\displaystyle
- \frac{\pi}{24L^2}
+ \lim_{\Delta z_\pm \to 0} \frac{1}{8\pi\Delta z_\pm^2}
& (z_\pm = \ell L, \; \ell \in {\bf Z}) \\
\displaystyle
- \frac{\pi}{12L^2}
& ({\rm otherwise}) \\
\end{cases},
\label{Tab_14}
\ee
which diverges on the null lines in the coincidence limit $\Delta z_\pm \to 0$. See Appendix~\ref{sec:green} for the derivation of this result using the Green functions.
The above ultraviolet divergence on the null lines seems to play a physically
significant role as follows. After the Dirichlet wall disappears and the cavity size becomes
$L$ for $t>0$, the ambient Casimir energy density remains the same as the energy density with the cavity size $L/2$ for $t<0$. This means that the amount of energy for $t>0$ would be lower than that of the ground state with the cavity size $L$ if the divergence on the null lines was not taken into account. Thus, it is expected that this divergent flux would compensate for the shortage of energy in the cavity. This expectation holds if the total energy radiated on the null lines diverges due to the term proportional to $(\Delta
z_{\pm})^{-2}$.
\section{Instantaneous limit of smooth appearance and disappearance}
\label{sec:smooth}
As we have seen, the sudden appearance and disappearance of the
Dirichlet wall will cause the different behaviors of ultraviolet
divergence for the energy-momentum tensor.
For understanding their origins, it would be helpful to compare results
in smooth appearance and disappearance models.
In Ref.~\cite{Brown:2015yma}, the authors investigated the vacuum
excitation by the smooth appearance of a both-sided Dirichlet wall in
$1+1$ dimensional Minkowski spacetime and its instantaneous limit.
Therefore, following their formulation, we will see that the divergent
behaviors are quite similar to those observed in Secs.~\ref{sec:app} and
\ref{sec:disapp} after taking a certain limit of smooth appearance and
disappearance models.
Let us briefly review the formulation and result of
Ref.~\cite{Brown:2015yma} in Sec.~\ref{BL}. Then, in Sec.~\ref{SA}, we
will show that the divergent behavior such as the delta function
squared and the ultraviolet divergence on the null lines appear by
taking the instantaneous limit of its model. Also, in Sec.~\ref{SD}, we generalize the formulation
to the disappearance case and consider its instantaneous limit.
In this section, for simplicity, we leave the existence of the cavity boundaries out of consideration because we are only interested in the ultraviolet divergent behavior of the energy-momentum tensor independent of cavity size $L$.
In addition, we focus on the even-parity modes of the scalar field
because the odd-parity modes are irrelevant to existence or absence of
the Dirichlet wall at $x=0$.
\subsection{Smooth-appearance model}
\label{BL}
In Ref.~\cite{Brown:2015yma}, for analyzing a smooth appearance of the
Dirichlet wall, the authors introduce the $\delta$-function potential with a smooth time-dependent coefficient into the Klein-Gordon equation of motion,
\begin{equation}
\left[\partial^2_t - \partial^2_x +
\frac{2\cot(\theta(t))}{ {\cal L} }\delta(x)\right]\phi = 0 ,
\label{eq:KG_pot}
\end{equation}
where ${\cal L}$ is a positive constant. They assume that function $\theta$ in the coefficient is given by
\begin{equation}
\theta(t) = \arctan\left(\frac{1+e^{-\lambda t}}{\lambda {\cal L} }\right),
\label{theta1}
\end{equation}
where $\lambda$ is a positive constant. This choice of $\theta$ corresponds to the following time-dependent boundary condition at $x=0$,
\be
\pd_x \phi(t,x)|_{x=0+} = \frac{ \lambda }{ 1+e^{- \lambda t} } \phi(t,x) |_{x=0+},
\label{SA_BC}
\ee
where we have used $\phi(t,x)$ is an even function with respect to $x=0$. Note that boundary condition~\eqref{SA_BC} is obtained by
integrating Eq.~\eqref{eq:KG_pot} over an infinitesimal interval across $x=0$ and substituting Eq.~\eqref{theta1} into it. One can see that boundary condition~\eqref{SA_BC} reduces to the Neumann one in the asymptotically past $t \to -\infty$ and to the Dirichlet one in the asymptotically future $t \to +\infty$ as long as $\lambda$ is sufficiently large. $\lambda^{-1}$ represents the time scale of appearance process, and therefore the limit of $\lambda \to \infty$ corresponds to the instantaneous limit of smooth appearance of a Dirichlet wall.
A set of positive-energy mode function $ \{ U_k \} \; ( k>0) $ is written in the following form
\be
U_k (z_-,z_+) = \frac{1}{\sqrt{8\pi k}} \left[e^{-ik z_+} + E_k( z_- )\right].
\label{eq:Uk}
\ee
Substituting Eq.~\eqref{eq:Uk} into Eq.~\eqref{SA_BC} and imposing the regularity at $t \to \infty$, one obtains
\be
E_k ( z_- ) = \frac{e^{-ik z_- }}{1+e^{\lambda z_-}}
\left(1- \frac{\lambda + ik}{\lambda - ik} e^{\lambda z_-}\right).
\label{eq:Ek}
\ee
When (the even sector of) field operator ${\bm \phi}$ is expanded as
\begin{equation}
{\bm \phi}
=
\int^\infty_0 dk
({\bm a}_k U_k + {\bm a}_k^\dagger U_k^\ast) ,
\end{equation}
a non-vanishing component of the renormalized energy-momentum tensor is given by
\begin{equation}
\langle 0 | {\bm T}_{--} | 0 \rangle_{\rm ren} = \int_\mu^\infty \frac{dk}{8\pi k}
\left(|E'_k(z_-)|^2 - k^2\right) ,
\label{BL-vev}
\end{equation}
where $|0 \rangle$ denotes an ordinary Minkowski vacuum, which is annihilated by all right-propagating and left-propagating positive-energy modes, and $\mu$ is an infrared cutoff introduced by hand. The second term ($-k^2$) in the integrand of Eq.~\eqref{BL-vev} is the subtraction term for the vacuum expectation value in the Minkowski spacetime.
Substituting Eq.~\eqref{eq:Ek} into Eq.~\eqref{BL-vev}, one obtains
\be
\langle 0 | {\bm T}_{--} | 0 \rangle_{\rm ren}
=
\frac{ \lambda^2 \ln [ 1+(\lambda/\mu)^2 ] }{ 64\pi \cosh^4 ( \lambda z_-/2 ) }.
\ee
It is clear that in the limit of $\lambda \to \infty$ this quantity diverges on the null line $z_-=0$ and vanishes on $z_- \neq 0$. In order to estimate the strength of divergence on $z_-=0$, the above expression is rewritten as
\be
\langle 0 | {\bm T}_{--} | 0 \rangle_{\rm ren}
=
\frac{ \lambda \ln [ 1+(\lambda/\mu)^2 ] }{ 24\pi }
\delta_\lambda ( z_- ),
\;\;\;
\delta_\lambda (z_-) := \frac{ 3\lambda }{ 8 \cosh^4( \lambda z_-/2 ) }.
\ee
Taking into account that $ \lim_{\lambda \to \infty} \delta_\lambda (z_-) = \delta (z_-) $ (see Eq.~\eqref{acosh^k} below), Ref.~\cite{Brown:2015yma} concluded that the divergence on the null line $z_-=0$ of $ \langle 0 | {\bm T}_{--} | 0 \rangle_{\rm ren} $ in the instantaneous limit is too strong to have a distributional limit.
\subsection{Smooth appearance and instantaneous limit}
\label{SA}
Now, we will see that the divergent behavior similar to that observed in Sec.~\ref{sec:app} can be obtained by taking an instantaneous limit of the above smooth appearance model.
Note that Ref.~\cite{Brown:2015yma} takes the instantaneous limit
$\lambda \to \infty$ after computing the momentum integration in
Eq.~\eqref{BL-vev}. Instead, we take the instantaneous limit before the
momentum integration. Using Eq.~\eqref{eq:Ek}, we compute the integrand
in Eq.~\eqref{BL-vev} in an instantaneous regime $ k /\lambda \ll 1
\footnote{
As we mentioned, in the current model, we should take sufficiently large
$\lambda$ to realize the Dirichlet boundary condition at the asymptotic future.
In fact, the asymptotic form becomes
$E_k(z_-) \sim - e^{-ikz_-} (\lambda - i k)/(\lambda + i k)$ as
$t\to\infty$ rather than $E_k(z_-) \sim - e^{-ikz_-}$.
Therefore, $k/\lambda \ll 1$ must be kept even if momentum $k$ becomes large.
}
as
\begin{equation}
|E'_k(z_-)|^2 - k^2 = \frac{\lambda^2}{4\cosh^4(\lambda z_-/2)}
- \frac{ k^2 }{ 4\cosh^4( \lambda z_- /2) }
\big[ 1+ \mathcal{O}(\frac{ k^2 }{ \lambda^2 } ) \big],
\end{equation}
where $ \mathcal{O}(\frac{ k^2 }{ \lambda^2 } ) $-term has no dependence on $z_-$.
Taking the limit $\lambda \to \infty$ of the above, we have
\begin{equation}
|E'_k(z_-)|^2 -k^2 = 4 \delta (z_-)^2 -
\begin{cases}
\displaystyle
\frac{k^2}{4}
& (z_- = 0) \\
\displaystyle
0
& ({\rm otherwise}) \\
\end{cases},
\;\;\;
( \lambda \to \infty),
\label{BL-integrand}
\end{equation}
where we have used the following mathematical relations
\be
&&
\lim_{a \to \infty} \frac{ a }{ \cosh^k ( a y )} = \frac{ 2^k[ (k-2)!! ]^2 }{ (2k-2)!! } \delta (y),
\;\;\;
k=2,4,6,\cdots,
\label{acosh^k}
\\
&&
\lim_{a \to \infty} \frac{ 1 }{ \cosh^k (ay) }
=
\begin{cases}
\displaystyle
1
& (y = 0) \\
\displaystyle
0
& ( y \neq 0 ) \\
\end{cases},
\;\;\;
k=1,2,3,\cdots.
\ee
Substituting Eq.~\eqref{BL-integrand} into Eq.~\eqref{BL-vev}, we have
\be
\langle 0 | {\bm T}_{--} | 0 \rangle_{\rm ren}
=
\frac{ \delta(z_-)^2 }{ 2\pi } \int_\mu^\infty \frac{dk}{k}
-
\begin{cases}
\displaystyle
\frac{1}{32\pi} \int_\mu^\infty dk k
& (z_- = 0) \\
\displaystyle
0
& ({\rm otherwise}) \\
\end{cases},
\;\;\;
( \lambda \to \infty).
\ee
Thus, we have obtained the delta function squared multiplied by a logarithmically divergent factor and the ultraviolet divergence that exists only on the null line $z_-=0$. Note that divergent integral $\int_\mu^\infty dk k$ turns out to be proportional to $-(\Delta z_-)^{-2}$ if one adopts the point-splitting regularization.
\subsection{Smooth disappearance and instantaneous limit}
\label{SD}
Here, let us generalize the argument of smooth appearance of the Dirichlet wall in Ref.~\cite{Brown:2015yma} to the smooth disappearance of Dirichlet wall. Then, we will consider its instantaneous limit.
In order to model the disappearance of Dirichlet wall in the formulation, we consider the time reversal $t \to -t$ of smoothing function \eqref{theta1} as
\be
\theta(t)
=
\arctan \left( \frac{ 1+e^{\lambda t} }{ \lambda {\cal L} } \right),
\;\;\;
(\lambda >0),
\label{theta2}
\ee
which corresponds to the following time-dependent boundary condition at the center,
\be
\pd_x \phi(t,x)|_{x=0+} = \frac{ \lambda }{ 1+e^{\lambda t} } \phi(t,x) |_{x=0+}.
\label{SD_BC}
\ee
Substituting the ansatz of mode function~\eqref{eq:Uk} into Eq.~\eqref{SD_BC} and imposing the regularity at $ t \to -\infty$, we obtain
\be
E_k (z_-)
=
-e^{ -ik z_- }
- \frac{ 2ik }{ \lambda -i k } (1+e^{ \lambda z_- })
{}_2 F_1 \left( 1,1- i \frac{k}{\lambda}, 2- i \frac{k}{\lambda} ; -e^{ \lambda z_- } \right) e^{-ik z_-},
\label{eq:Ek2}
\ee
where ${}_2 F_1$ is the hypergeometric function.
Using Eq.~\eqref{eq:Ek2}, we compute the integrand in Eq.~\eqref{BL-vev} in the instantaneous regime $ k /\lambda \ll 1$,
\be
| E_k' (z_-) |^2 -k^2
=
-4
e^{-\lambda z_-} \ln ( 1+e^{\lambda z_-} ) \cdot
[ 1-e^{-\lambda z_-} \ln ( 1+e^{\lambda z_-} ) ] k^2 + {\cal O} (\frac{ k^3 }{ \lambda^3 } ) .
\ee
If we take the instantaneous limit $\lambda \to \infty$ of the above, we obtain
\be
| E_k' (z_-) |^2 -k^2
=
\begin{cases}
- 4 (1-\ln 2) \ln 2 \cdot k^2& (z_- = 0) \\
0 & (\mbox{otherwise}) \\
\end{cases},
\;\;\;
( \lambda \to \infty ),
\label{BL-integrand2}
\ee
using the following,
\be
\lim_{ a \to \infty } e^{-ay} \ln ( 1+e^{ay} )
=
\begin{cases}
1 & (y<0) \\
\ln 2 & (y=0) \\
0 & (y>0) \\
\end{cases}.
\ee
Substituting Eq.~\eqref{BL-integrand2} into Eq.~\eqref{BL-vev}, we obtain
\be
\langle 0 | {\bm T}_{--} | 0 \rangle_{\rm ren}
=
\begin{cases}
\displaystyle -\frac{ (1-\ln 2) \ln 2}{ 2\pi }\int_\mu^\infty dk k & (z_-=0) \\
0 & (\mbox{otherwise}) \\
\end{cases},
\;\;\;
( \lambda \to \infty ).
\ee
From the above expression, we observe that the term of delta function squared is absent, and only the ultraviolet divergence that exists only on the null line $z_-=0$ appears. Thus, we have obtained the divergent energy-momentum tensor of which main features are the same as those in Sec.~\ref{sec:disapp}.
\section{Conclusion}
\label{sec:conc}
We have investigated the vacuum excitation of a massless
Klein-Gordon scalar field due to the sudden appearance and disappearance
of a both-sided Dirichlet wall in a 1D cavity.
For the sudden appearance of the Dirichlet wall, we found that the
vacuum is highly excited to result in the infinitely strong flux given
by Eq.~\eqref{Tab_03_ren}. This result suggests that the backreaction to the background spacetime and boundary cannot be ignored. In other words, the background spacetime is forced to be dynamical and/or the instantaneous insertion of the Dirichlet wall itself is prohibited by the quantum field. We note that the result is quite similar to those in
the investigation of the topology change~\cite{Anderson:1986ww,Manogue}
and the strong curvature singularity~\cite{Ishibashi:2002ac}, although the boundary condition in the present work is different from those in the papers.
Also for the sudden disappearance of the Dirichlet wall, we
found that the vacuum is highly excited to result in the infinitely strong flux given by Eq.~\eqref{Tab_14}. In contrast to the appearance case, the renormalized energy-momentum tensor does not contain the term proportional to the delta function squared, although it contains the diverging term proportional to $(\Delta z_{\pm})^{-2}$.
The infinite flux is what we expect from the viewpoint of the number of created particles as mentioned in Introduction, while the lack of the delta function squared is not.
Let us mention the divergence of the renormalized energy-momentum tensor appearing both in the sudden appearance and disappearance cases. We have seen that the standard procedure of the point-splitting regularization gives a finite value of the renormalized energy-momentum tensor at the spacetime points not on the null lines which emanate from the transition point, while it is divergent on the null lines. We have interpreted this result as the diverging flux on the null lines for $t>0$. While there seems no ambiguity in this straightforward interpretation, it is more convincing if such a peculiar divergence on null lines appears as the result of an instantaneous limit of finite-time appearance and disappearance of the Dirichlet wall. Therefore, using the formulation in Ref.~\cite{Brown:2015yma}, which estimates the particle creation by a smoothly appearing Dirichlet wall, we have shown in Sec.~\ref{sec:smooth} that the $(\Delta z_\pm)^{-2}$-type divergence appears on the null lines after taking an instantaneous limit for the appearance and disappearance cases, although the discussion is restricted only to the infinite cavity limit $(L \to \infty)$.
The discrepancy between the appearance and disappearance cases seems to
stem from the different behaviors of two sets of mode functions,
$\{f_n^{(\gamma)}\}$ and $\{ g_m \}$, which define distinct vacua $|0_f \rangle$ and $ |0_g \rangle $, respectively.
First, let us see the behavior of $f_n^{(\gamma)}$. While $f_n^{(\gamma)}$ is given by
Eq.~\eqref{f} for $t<0$, it is expressed as Eq.~\eqref{f_g} for
$t>0$. Here, the point is that $ f_n^{(\gamma)} $ given by
Eqs.~\eqref{f} and \eqref{f_g} coincide in the limit of $t \to 0$, which
implies that $f_n^{(\gamma)}$ is continuous at $t=0$. Next, let us see
the behavior of $ g_m $. While $g_m$ is given by Eq.~\eqref{g} for
$t<0$, $g_m$ is expressed as Eq.~\eqref{gBYf} for $t>0$. In this case,
$g_m$ given by Eqs.~\eqref{g} and \eqref{gBYf} do not coincide in the
limit of $t \to 0$ at every point of $[-L/2, L/2]$. Namely, when $m$ is
odd, while $\lim_{t \to -0} g_m(t,0) \neq 0$ from Eq.~\eqref{g},
$\lim_{t \to +0} g_m(t,0) = 0$ from Eq.~\eqref{gBYf} (note that the
right-hand side of Eq.~\eqref{gBYf} consists only of sine functions),
which implies the discontinuity of $g_m \; (m \in {\rm odd})$ at
$t=0$. We conjecture that the existence of such a discontinuity of mode
functions is the origin of the square of the delta function in the appearance case.
Given the results in this paper, we have many things to examine. In
particular, it is important to prove (or disprove) that
the present result
is not an artifact of simplification and idealization adopted in our analysis (i.e., equal lengths of left and right regions, low dimensionality, scalar field, and so on). The generalizations of this work in this direction will be indispensable to understand how much the semiclassical effects play crucial roles in the gravitational phenomena such as the spacetime connection and disconnection.
\subsection*{Acknowledgments}
The authors would like to thank an anonymous referee for suggesting us to compare the result in this paper with that of Ref.~\cite{Brown:2015yma}, which deepened our understanding about the current topic. UM would like to thank H.\ Maeda, A.\ Ishibashi, and H.\ Okamoto for useful discussions. This work was supported by JSPS KAKENHI Grant
Numbers 26400282 (TH) and 15K05086 (UM).
|
2,869,038,155,648 | arxiv | \section{\label{sec:level1}Introduction}
The problem of water wave transformation in a canal of a variable
cross-section is one of the classic problems of theoretical and
applied hydrodynamics. It has been studied in many books, reports,
and journal papers starting from the first edition (1879) of the
famous monograph by H. Lamb, {\em Hydrodynamics} (see the last
lifetime publication \citep{Lamb-1932}). In particular, the
coefficients of transformation of long linear waves in a canal of
a rectangular cross-section with an abrupt change of geometrical
parameters (width and depth) were presented. The transmission and
reflection coefficients were found as functions of depth ratio $X
= h_2/h_1$ and width ratio $Y = b_2/b_1$, where $h_1$ and $b_1$
are the canal depth and width at that side from which the incident
wave arrives, and $h_2$ and $b_2$ are the corresponding canal
parameters at the opposite side where the transmitted wave goes to
(see Fig. \ref{f01}). The parameters $X$ and $Y$ can be both less
than 1, and greater than 1. As explained in Ref.
\citep{Lamb-1932}, the canal cross-section can vary smoothly, but
if the wavelengths of all scattered waves are much greater than
the characteristic scale of variation of the canal cross-section,
then the canal model with the abrupt change of parameters is
valid.
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{Fig01.pdf}
\vspace*{-3.0cm}%
\caption{(Color online). Sketch of a canal consisting of two
sections of different rectangular cross-sections. The wave number
of incident wave is ${\bf k}_i$, and the wave number of
transmitted wave is ${\bf k}_t$ (a reflected wave is not shown).
Water flow $U$ is co-directed with the $x$-axis.}
\label{f01}%
\end{figure}
The Lamb model has been further generalised for waves of arbitrary
wavelengths and applied to many practical problems. One of the
typical applications of such a model is in the problem of oceanic
wave transformation in the shelf zone; the numerous references can
be found in the books and reviews \citep{Massel-1989, Dingem-1997,
Kurkin-2015}. In such applications the canal width is assumed to
be either constant or infinitely long and only the water depth
abruptly changes.
A similar problem was studied also in application to internal
waves, but analytical results were obtained only for the
transformation coefficients of long waves in a two-layer fluid
\citep{GrimPelTal-2008}, whereas for waves of arbitrary wavelength
only the numerical results were obtained and the approximative
formulae were suggested \citep{ChurSemStep-2015}.
All aforementioned problems of wave transformation were studied
for cases when there is no background current. However, there are
many situations when there is a flow over an underwater step or in
the canals or rivers with variable cross-sections. The presence of
a current can dramatically affect the transformation coefficients
due to the specific wave-current interaction (see, e.g., Ref.
\citep{Belibassakis-2011} and references therein). The amplitudes
and energies of reflected and transmitted waves can significantly
exceed the amplitude and energy of an incident wave. Such
over-reflection and over-transmission phenomena are known in
hydrodynamics and plasma physics (see, e.g., Ref.
\citep{Jones-1968}); the wave energy in such cases can be
extracted from the mean flow. Apparently, due to complexity of
wave scattering problem in the presence of a background flow, no
results were obtained thus far even for a relatively weak flow and
small flow variation in a canal. There are, however, a number of
works devoted to wave-current interactions and, in particular,
wave scattering in spatially varying flows mainly on deep water
(see, for instance, Refs. \cite{Smith-1975, StiasDagan-79,
TrulMei-1993, Belibassakis-2011} and references therein). In Ref.
\citep{Belibassakis-2011} the authors considered the surface wave
scattering in two-dimensional geometry in $(x, y)$-plane for the
various models of underwater obstacles and currents including
vortices. In particular, they studied numerically wave passage
over an underwater step in the shoaling zone in the presence of a
current. However, the transformation coefficients were not
obtained even in the plane geometry.
Here we study the problem of long wave scattering analytically for
all possible configurations of the background flow and incident
wave (downstream and upstream propagation) in the narrowing or
widening canal (accelerating or decelerating flow) for the
subcritical, transcritical, and supercritical regimes when the
current speed is less or greater than the typical wave speed $c_0
= \sqrt{gh}$ in calm water in the corresponding canal section ($g$
is the acceleration due to gravity, and $h$ is the canal depth).
Because we consider a limiting model case of very long waves when
the variation of canal geometry is abrupt, the wave blocking
phenomenon here has a specific character of reflection. Such a
phenomenon has been studied in shallow-water limit in Ref.
\cite{Smith-1975}, but transformation coefficients were not
obtained.
Notice also that in the last decade the problem of wave-current
interaction in water with a spatially varying flow has attracted a
great deal of attention from researchers due to application to the
modelling of Hawking's radiation emitted by evaporating black
holes \citep{Unruh-1981} (see also Refs. \citep{Jacobson-1991,
Unruh-1995, Faccio-2013}). Recent experiments in a water tank
\citep{Euve-2016} have confirmed the main features of the Hawking
radiation; however many interesting and important issues are still
under investigation. In particular, it is topical to calculate the
transformation coefficients of all possible modes generated in the
process of incident mode conversion in the spatially varying flow.
Several papers have been devoted to this problem both for the
subcritical \citep{CoutWein-2016, RobMichPar-2016} and
transcritical \citep{CoutParFin-2012, Robert-2012} flows. However,
in all these papers the influence of wave dispersion was
important, whereas there is no dispersion in the problem of black
hole radiation. Our results for the dispersionless wave
transformation can shed light on the problem of mode conversion in
the relatively simple model considered in this paper.
\section{\label{sec:level2}Problem statement and dispersion relation}
Consider a long surface gravity wave propagating on the background
current in a canal consisting of two portions of different
cross-section each as shown in Fig. \ref{f01}. A similar problem
with a minor modification can be considered for internal waves in
two-layer fluid, but we focus here on the simplest model to gain
an insight in the complex problem of wave-current interaction. We
assume that both the canal width and depth abruptly change at the
same place, at the juncture of two canal portions. The current is
assumed to be uniform across the canal cross-section and flows
from left to right accelerating, if the canal cross-section
decreases, or decelerating, if it increases. In the presence of a
current the water surface does not remain plane even if the canal
depth is unchanged, but the width changes. According to the
Bernoulli law, when the current accelerates due to the canal
narrowing, the pressure in the water decreases and, as a result,
the level of the free surface reduces. Therefore, asymptotically,
when $x \to \infty$, the portion of canal cross-section occupied
by water is $S_2 = b_2h_2$. A similar variation in the water
surface occurs in any case when the current accelerates due to
decrease of the canal cross-section in general; this is shown
schematically in Fig. \ref{f02} (this figure is presented not in
scale, just for the sake of a vivid explanation of the wave
scattering, whereas in fact, we consider periodic waves with the
wavelengths much greater than the fluid depth).
\begin{figure}[h]
\centering
\includegraphics[width=15cm]{Fig02.pdf}
\vspace*{-7.0cm}%
\caption{(Color online). The side view of a flow in a canal with a
variable cross-section. Wave 1 schematically represents an
incident wave, wave 2 -- a reflected wave, and wave 3 -- a
transmitted wave. The water surface slightly lowers when the
background flow increases as shown schematically by thin line.}
\label{f02}%
\end{figure}
The relationship between the water depth $h_2$, which
asymptotically onsets at the infinity, and variations of canal
width and depth at the juncture point is nontrivial. In
particular, even in the case when the canal width is unchanged,
and the canal cross-section changes only due to the presence of a
bottom step of a height $d$, the water depth $h_2$ at the infinity
is not equal to the difference $h_1 - d$ (see, e.g., Ref.
\citep{GazizMaklak-2004.}). As shown in the cited paper, variation
of a free surface due to increase of water flow is smooth even in
the case of abruptly changed depth, but in the long-wave
approximation it can be considered as abrupt. In any case, we will
parameterize the formulas for the transformation coefficients in
terms of the real depth ratio at plus and minus infinity $X =
h_2/h_1$ and canal width aspect ratio $Y = b_2/b_1$. The long-wave
approximation allows us to neglect the dispersion assuming that
the wavelength $\lambda$ of any wave participating in the
scattering is much greater than the canal depth $h$ in the
corresponding section.
In the linear approximation the main set of hydrodynamic equations
for shallow-water waves in a perfect incompressible fluid is (see,
e.g., Ref. \citep{Lamb-1932}):
\begin{eqnarray}
\frac{\partial u}{\partial t} + U\frac{\partial u}{\partial x}
&=& -g\frac{\partial \eta}{\partial x}, \label{LinEurEq} \\%
\frac{\partial \eta}{\partial t} + U\frac{\partial \eta}{\partial
x} &=& -h\frac{\partial u}{\partial x}. \label{LinMassCons} %
\end{eqnarray}
Here $u(x, t)$ is a wave induced perturbation of a horizontal
velocity, $U$ is the velocity of background flow which is equal to
$U_1$ at minus infinity and $U_2$ at plus infinity, $\eta(x, t)$
is the perturbation of a free surface due to the wave motion, and
$h$ is the canal depth which is equal to $h_1$ at minus infinity
and $h_2$ at plus infinity -- see Fig. \ref{f02}.
For the incident harmonic wave of the form $\sim
\mathrm{e}^{\mathrm{i}(\omega t - kx)}$ co-propagating with the
background flow we obtain from Eq. (\ref{LinMassCons})%
\begin{equation}
\label{Eq01}%
\left(\omega - U_1k_i\right)\eta_i = h_1k_iu_i,%
\end{equation}
where index $i$ pertains to incident wave (in what follows indices
$t$ and $r$ will be used for the transmitted and reflected waves
respectively).
Combining this with Eq. (\ref{LinEurEq}), we derive the
dispersion relation for the incident wave%
\begin{equation}
\label{DispRelIn}%
\omega = \left(U_1 + c_{01}\right)k_i,%
\end{equation}
where $c_{01} = \sqrt{gh_1}$.
Similarly for the transmitted wave we have $\left(\omega -
U_2k_t\right)\eta_t = h_2k_tu_t$ and the dispersion relation
$\omega = \left(U_2 + c_{02}\right)k_t$, where $c_{02} =
\sqrt{gh_2}$. Notice that the wave frequency remains unchanged in
the process of wave transformation in a stationary, but spatially
varying medium. Then, equating the frequencies for the incident
and transmitted waves, we obtain $k_t/k_i = \left(U_1 +
c_{01}\right)/\left(U_2 + c_{02}\right)$.
From the mass conservation for the background flow we have
$U_1h_1b_1 = U_2h_2b_2$ or $U_1/U_2 = XY$. Using this
relationship, we obtain for the wave number of the transmitted
wave
\begin{equation}
\label{WaveNumTr}%
\frac{k_t}{k_i} = XY\frac{1 + \mathrm{Fr}}{X^{3/2}Y + \mathrm{Fr}},%
\end{equation}
where $\mathrm{Fr} = U_1/c_{01}$ is the Froude number.
\begin{figure}[b]
\centering
\includegraphics[width=15cm]{Fig03.pdf}
\vspace*{-3.5cm}%
\caption{(Color online). The dependence of wave number ratio on
the depth drop $X = h_2/h_1$ for different Froude numbers and $Y =
1$. Line 1 pertains to the reference case when $\mathrm{Fr} = 0$,
lines 2 and $2'$ -- to $\mathrm{Fr} = 0.1$, lines 3 and $3'$ -- to
$\mathrm{Fr} = 0.5$, line 4 and $4'$ -- to $\mathrm{Fr} = 1$.
Dashed vertical lines 5 and 6 show the boundaries between the
subcritical and supercritical regimes in the downstream domain for
$\mathrm{Fr} = 0.1$, line 5, and $\mathrm{Fr} = 0.5$, line 6.}
\label{f03}%
\end{figure}
The relationship between the wave numbers of incident and
transmitted waves as functions of the depth drop $X$ is shown in
Fig. \ref{f03} for several values of $\mathrm{Fr}$ and $Y = 1$. As
one can see, the ratio of wave numbers $k_t/k_i$ non-monotonically
depends on $X$; it has a maximum at $X_m =
\left(2\mathrm{Fr}/Y\right)^{2/3}$. The maximum value
$\left(k_t/k_i\right)_{max} = \sqrt[3]{4Y}\left(1 +
\mathrm{Fr}\right)/\left(3\sqrt[3]{\mathrm{Fr}}\right)$ is also a
non-monotonic function of the Froude number; it has a minimum at
$\mathrm{Fr} = 0.5$ where $\left(k_t/k_i\right)_{max} =
\sqrt[3]{Y}$. In the limiting case, when there is no current
($\mathrm{Fr} = 0$), $k_t/k_i = X^{-1/2}$ independently of $Y$
(see line 1 in Fig. \ref{f03}). The current with the Froude number
$\mathrm{Fr} < 1$ remains subcritical in the downstream domain, if
$X > \left(\mathrm{Fr}/Y\right)^{2/3}$. Otherwise it becomes
supercritical. Dashed lines 5 and 6 in Fig. \ref{f03} show the
boundaries between the subcritical and supercritical regimes in
the downstream domains for two values of the Froude number,
$\mathrm{Fr} = 0.1$ and $\mathrm{Fr} = 0.5$ respectively.
For the upstream propagating reflected wave the harmonic
dependencies of free surface and velocity perturbations are
$\{\eta, u\} \sim \mathrm{e}^{\mathrm{i}(\omega t + k_rx)}$. Then
from Eq. (\ref{LinMassCons}) we obtain $\left(\omega +
U_1k_r\right)\eta_r = -h_1k_ru_r$, and combining this with Eq.
(\ref{LinEurEq}), we derive the dispersion relation for
the reflected wave with $k_r < 0$%
\begin{equation}
\label{DispRelRef}%
\omega = \left(c_{01} - U_1\right)|k_r|.%
\end{equation}
Equating the frequencies of the incident and reflected waves, we
obtain from the dispersion relations the relationship between the
wave numbers:
\begin{equation}
\label{WaveNumRef}%
\frac{|k_r|}{k_i} = \frac{1 + \mathrm{Fr}}{1 - \mathrm{Fr}}.%
\end{equation}
Notice that the ratio of wave numbers $|k_r|/k_i$ depends only on
$\mathrm{Fr}$, but does not depend on $X$ and $Y$.
The dispersion relations for long surface waves on a constant
current are shown in Fig. \ref{f04}. Lines 1 and 2 show the
dispersion dependencies for the downstream and upstream
propagating waves, respectively, in the upstream domain, if the
background current is subcritical, i.e., when $\mathrm{Fr} < 1$.
Lines 3 and 4 show the dispersion dependencies for the downstream
and upstream propagating waves, respectively, which can
potentially exist in the downstream domain, if the background
current remains subcritical in this domain too, i.e. when
$U_2/c_{02} \equiv \mathrm{Fr}/\left(X^{3/2}Y\right) < 1$. If
there is a source generating an incident wave of frequency
$\omega$ and wave number $k_i$ at minus infinity, then after
scattering at the canal juncture the reflected wave appears in the
upstream domain with the same frequency and wave number $k_r$.
Dashed horizontal line 7 in Fig. \ref{f04} shows the given
frequency $\omega$. In the downstream domain with a subcritical
flow the incident wave generates only one transmitted wave with
the wave number $k_t$.
\begin{figure}[h]
\centering
\includegraphics[width=15cm]{Fig04.pdf}
\vspace*{-4.0cm}%
\caption{(Color online). Qualitative sketch of dispersion lines
for long surface waves on a uniform background flow in a canal.
For details see the text.}
\label{f04}%
\end{figure}
If the flow in one of the domains becomes faster and faster so
that $\mathrm{Fr} \to 1_-$, then the dispersion line corresponding
to the upstream propagating waves tilts to the negative portion of
horizontal axis $k$ in Fig. \ref{f04} (cf. lines 2 and 4), and its
intersection with the horizontal dashed line 7 shifts to the minus
infinity. In the case of a supercritical flow, $\mathrm{Fr}
> 1$, the dispersion line corresponding to the upstream
propagating waves is line 6 in Fig. \ref{f04}. Its intersection
with the horizontal dashed line 7 originates at the plus infinity
(as the continuation of the intersection point of line 4 with line
7 disappeared at the minus infinity) and moves to the left when
the flow velocity increases. The speeds of such waves in a calm
water are smaller than the speed of a current, therefore despite
the waves propagate counter current, the current traps them and
pulls downstream. In the immovable laboratory coordinate frame
they look like waves propagating to the right jointly with the
current. As shown in Refs. \citep{StepFabr-1989, FabrStep-1998,
MaiRusStep-2016A}, such waves possess a negative energy. This
means that the total energy of a medium when waves are excited is
less then the energy of a medium without waves. Obviously, this
can occur only in the non-equilibrium media, for example, in
hydrodynamical flows possessing kinetic energy. In the equilibrium
media, wave excitation makes the total energy greater than the
energy of the non-perturbed media (more detailed discussion of the
negative energy concept one can find in the citations presented
above and references therein). In Appendix \ref{appA} we present
the direct calculation of wave energy for the dispersionless case
considered here and show when it become negative.
With the help of dispersion relations, the links between the
perturbations of fluid velocity and free surface in the incident,
reflected and transmitted waves can be presented as
\begin{equation}
\label{Rel-u&eta}%
u_i = c_{01}\eta_i/h_1; \quad u_r = -c_{01}\eta_r/h_1; \quad u_t =
c_{02}\eta_t/h_2.%
\end{equation}
Using these relationships, we calculate in the next sections the
transformation coefficients for all possible flow regimes and
wave-current configurations.
\section{\label{sec:level3}Subcritical flow in both the upstream and downstream domains}
\subsection{\label{sec:level31}Downstream propagating incident wave}
Consider first the case when the current is co-directed with the
$x$-axis (see Fig. \ref{f02}) and the incident wave travels in the
same direction. Then, the transmitted wave is also co-directed
with the current, but the reflected wave travels against the
current. We assume that the current is subcritical in both left
domain and right domains, i.e. its speed $U_1 < c_{01}$ and $U_2 <
c_{02}$. This can be presented alternatively in terms of the
Froude number and canal specific ratios, viz $\mathrm{Fr} < 1$ and
$\mathrm{Fr} < X^{3/2}Y$.
To derive the transformation coefficients, we use the boundary
conditions at the juncture point $x = 0$. These conditions
physically imply the continuity of pressure and continuity of
horizontal mass flux induced by a surface wave. The total pressure
in the moving fluid consists of hydrostatic pressure $\rho g (h +
\eta)$ and kinetic pressure $\rho(U + u)^2/2$. The condition of
pressure continuity in the linear approximation reduces to
\begin{equation}
\label{PresCont}%
g\eta_1 + U_1u_1 = g\eta_2 + U_2u_2,%
\end{equation}
where indices 1 and 2 pertain to the left and right domains
respectively far enough from the juncture point $x = 0$. In the
left domain we have $\{\eta_1, u_1\} = \{\eta_i + \eta_r, u_i +
u_r\}$, whereas in the right domain $\{\eta_2, u_2\} = \{\eta_t,
u_t\}$.
Using the relationships between $u_{i,r,t}$ and $\eta_{i,r,t}$ as
per Eq. (\ref{Rel-u&eta}) and assuming that the incident wave has
a unit amplitude in terms of $\eta$, we obtain from Eq.
(\ref{PresCont})
\begin{equation}
\label{PresCont1}%
g\left(1 + R_{\eta}\right) + U_1\frac{c_{01}}{h_1}\left(1 -
R_{\eta}\right) = gT_{\eta} + U_2\frac{c_{02}}{h_2}T_{\eta},%
\end{equation}
where $R_{\eta}$ and $T_{\eta}$ are amplitudes of reflected and
transmitted waves respectively. In the dimensionless form this
equations reads
\begin{equation}
\label{PresCont2}%
1 + \mathrm{Fr} + \left( 1 - \mathrm{Fr}\right)R_{\eta} =
T_{\eta}\left( 1 + \frac{\mathrm{Fr}}{X^{3/2}Y}\right).
\end{equation}
The condition of mass flux continuity leads to the equation
\begin{equation}
\label{MassFluxCont}%
\rho b_1\left(h_1 + \eta_1\right)\left(U_1 + u_1\right) = \rho
b_2\left(h_2 + \eta_2\right)\left(U_2 + u_2\right).%
\end{equation}
In the linear approximation and dimensionless form this gives:
\begin{equation}
\label{MassFluxCont1}%
1 + \mathrm{Fr} - \left(1 - \mathrm{Fr}\right)R_{\eta} =
T_{\eta}\sqrt{X}Y\left(1 + \frac{\mathrm{Fr}}{X^{3/2}Y}\right).
\end{equation}
After that we derive the transformation coefficients $R_{\eta}$
and $T_{\eta}$ from Eqs. (\ref{PresCont2}) and
(\ref{MassFluxCont1}):
\begin{equation}
\label{TransCoef1}%
R_{\eta} = \frac{1 + \mathrm{Fr}}{1 - \mathrm{Fr}}\frac{1 -
\sqrt{X}Y}{1 + \sqrt{X}Y}, \quad T_{\eta} = \frac{1 +
\mathrm{Fr}}{X^{3/2}Y + \mathrm{Fr}} \frac{2X^{3/2}Y}{1 +
\sqrt{X}Y}.
\end{equation}
These formulas naturally reduce to the well-known Lamb formulas
\citep{Lamb-1932} when $\mathrm{Fr} \to 0$. Graphics of $T_{\eta}$
and $R_{\eta}$ as functions of depth drop $X$ are shown in Fig.
\ref{f05} for the particular value of Froude number $\mathrm{Fr} =
0.5$ and $Y = 1$.
\begin{figure}[h]
\centering
\includegraphics[width=15cm]{Fig05.pdf}
\vspace*{-3.5cm}%
\caption{(Color online). The transformation coefficients of
surface waves on a uniform subcritical current in a canal with
flat walls, $Y = 1$, as functions of the depth drop $X$. Line 1
for $T_\eta$ and line $1'$ for $R_\eta$ pertain to the reference
case given by the Lamb formulas with $\mathrm{Fr} = 0$; lines 2
(for $T_\eta$) and $2'$ (for $R_\eta$) pertain to the flow with
$\mathrm{Fr} = 0.5$.}
\label{f05}%
\end{figure}
As follows from the formula for $R_\eta$, the reflection
coefficient increases uniformly in absolute value, when the Froude
number increases from 0 to 1, provided that $\sqrt{X}Y \ne 1$. It
is important to notice that the reflectionless propagation can
occur in the case, when $\sqrt{X}Y = 1$, whereas neither $X$, nor
$Y$ are equal to one. The transmission coefficient in this case
$T_\eta = \left(1 + \mathrm{Fr}\right)/\left(1 +
Y^2\mathrm{Fr}\right) \ne 1$ in general, except the case when
$\mathrm{Fr} = 0$. The reflection coefficient is negative when
$\sqrt{X}Y > 1$, which means that the reflected wave is in
anti-phase with respect to the incident wave.
The dependence of $T_\eta$ on the Froude number is more
complicated and non-monotonic in $X$. However, in general
$T_{\eta} \to 0$ in two limiting cases, when $X \to 0$, then
$T_\eta \approx 2X^{3/2}Y\left(1 + 1/\mathrm{Fr}\right)$, and when
$X \to \infty$, then $T_\eta \approx 2\left(1 +
\mathrm{Fr}\right)/ \left(\sqrt{X}Y\right)$ (see Fig. \ref{f05}).
It is appropriate to mention here the nature of singularity of the
reflection coefficient $R_\eta$ and wave number $k_r$ of the
reflected wave as per Eq. (\ref{WaveNumRef}) when $\mathrm{Fr} \to
1$. In such case, the dispersion line 2 in Fig. \ref{f04}
approaches negative half-axis of $k$, and the point of
intersection of line 2 with the dashed horizontal line 7 shifts to
the minus infinity, i.e. $k_r \to -\infty$, and the wavelength of
reflected wave $\lambda_r = 2\pi/|k_r| \to 0$. Thus, we see that
when $\mathrm{Fr} \to 1$, then the amplitude of the reflected wave
$R_\eta$ infinitely increases, and its wavelength vanishes. It
will be shown below that the wave energy flux associated with the
reflected wave remains finite even when $\mathrm{Fr} = 1$.
The results obtained for the transformation coefficients are in
consistency with the wave energy flux conservation in an
inhomogeneous stationary moving fluid (see, e.g., Ref.
\citep{LongHig-1996}), $W \equiv V_gE = \mathrm{const.}$, where
$V_g \equiv d\omega/dk$ is the group speed in the moving fluid,
and $E$ is the density of wave energy. In the case of long waves
in shallow water we have $\left(V_g\right)_{1,2} =
\left(c_{0}\right)_{1,2} \pm U_{1,2}$. As shown in Appendix
\ref{appA} (see also Refs. \citep{Dysthe-2004, MaiRusStep-2016A}),
the period-averaged energy density in the long-wave limit is $E =
gA^2b\left(1 \pm \mathrm{Fr}\right)/2$, where $A$ is the amplitude
of free surface perturbation, $b$ is the canal width, sign plus
pertains to waves co-propagating with the background flow, and
sign minus -- to waves propagating against the flow. Taking into
account that the energy fluxes in the incident and transmitted
waves are directed to the right, and the energy flux in the
reflected wave is directed to the left, we obtain
\begin{equation}
\label{EnFluxCons}%
\left(1 + \mathrm{Fr}\right)^2 - \left(1 -
\mathrm{Fr}\right)^2R_{\eta}^2 = \sqrt{X}Y \left(1 +
\frac{\mathrm{Fr}}{X^{3/2}Y}\right)^2 T_{\eta}^2,
\end{equation}
where the factor $\sqrt{X}Y$ accounts for the change of the
cross-sectional area of the canal.
Substituting here the expressions for the transformation
coefficients Eq. (\ref{TransCoef1}), we confirm that Eq.
(\ref{EnFluxCons}) reduces to the identity. Notice that the second
term in the left-hand side of Eq. (\ref{EnFluxCons}), which
represents the energy flux induced by the reflected wave, remains
finite even at $\mathrm{Fr} = 1$.
The gain of energy densities in the reflected and transmitted
waves can be presented as the ratios $E_r/E_i$ and $E_t/E_i$.
Using the formulas for the transformation coefficients and
expression for the wave energy in a moving fluid (see above), we
obtain
\begin{equation}
\label{EnergyGain}%
\frac{E_r}{E_{i}} = \frac{1 + \mathrm{Fr}}{1 -
\mathrm{Fr}}\left(\frac{1 - \sqrt{X}Y}{1 + \sqrt{X}Y}\right)^2,
\quad \frac{E_t}{E_{i}} = \frac{4Y}{\left(1 +
\sqrt{X}Y\right)^2}\frac{1 + \mathrm{Fr}}{1 +
\mathrm{Fr}/X^{3/2}Y}.
\end{equation}
\begin{figure}[b]
\centering
\includegraphics[width=15cm]{Fig06.pdf}
\vspace*{-3.5cm}%
\caption{(Color online). The gain of energy density in the
transmitted wave for several Froude numbers and $Y = 1$ as
functions of the depth drop $X$. Line 1 pertains to the reference
case when $\mathrm{Fr} = 0$; lines 2 and 3 pertain to the
downstream propagating waves in the subcritical flows with
$\mathrm{Fr} = 0.1$ and 0.5 respectively; and lines 5 and 6
pertain to the upstream propagating waves in the same flows. Line
4 shows the typical dependence of energy density gain in the
upstream propagating reflected wave with $\mathrm{Fr} = 0.5$.
Lines 7 and 8 show the boundaries of subcritical regimes for
$\mathrm{Fr} = 0.1$ and 0.5 respectively.}
\label{f06}%
\end{figure}
As follows from the first of these expressions, the density of
wave energy in the reflected wave is enhanced uniformly by the
current at any Froude number ranging from 0 to 1 regardless of $X$
and $Y$, whereas the density of wave energy in the transmitted
wave can be slightly enhanced by the current only if $X^{3/2}Y >
1$; otherwise, it is less than that in the incident wave. Figure
\ref{f06} illustrates the gain of energy density in the
transmitted wave for several Froude numbers and $Y = 1$. Line 4 in
that figure shows the typical dependence of $E_r/E_i$ on $X$ for
$\mathrm{Fr} = 0.5$ and $Y = 1$. When $\mathrm{Fr} \to 1$ the gain
of wave energy in the reflected wave infinitely increases within
the framework of a linear model considered here (in reality the
nonlinear, viscous, or dispersive effects can restrict infinite
growth). In this case the typical over-reflection phenomenon
\citep{Jones-1968} occurs in the scattering of downstream
propagating wave, when the energy density in the reflected wave
becomes greater than the energy density in the incident wave. This
can occur due to the wave energy extraction from the mean flow.
\subsection{\label{sec:level32}Upstream propagating incident wave}
Consider now the case when the current is still co-directed with
the $x$ axis (see Fig. \ref{f02}) and the incident wave travels in
the opposite direction from plus infinity. Then, the transmitted
wave in the left domain propagates counter current, and the
reflected wave in the right domain is co-directed with the
current. In the dispersion diagram shown in Fig. \ref{f04} the
incident wave now corresponds to the intersection of line 2 with
the dashed horizontal line 7 (with the wave number $k_r$ replaced
by $k_i$), the reflected wave corresponds to intersection of line
1 with line 7 (with the wave number $k_i$ replaced by $k_r$), and
the transmitted wave corresponds to the intersection of line 4
with line 7 (not visible in the figure).
To derive the transformation coefficients, we use the same
boundary conditions at the juncture point $x = 0$ and after simple
manipulations similar to those presented in the previous
subsection we obtain essentially the same formulas for the wave
numbers of transmitted and reflected waves as in Eqs.
(\ref{WaveNumTr}) and (\ref{WaveNumRef}), as well as the
transformation coefficients as in Eqs. (\ref{TransCoef1}) with the
only difference that the sign of the Froude number should be
changed everywhere to the opposite, $\mathrm{Fr} \to
-\mathrm{Fr}$. However, the change of sign in the Froude number
leads to singularities in both the wave number of the transmitted
wave and the transmission coefficient. Therefore for the wave
numbers of scattered waves we obtain:
\begin{equation}
\label{WaveNumb0.22}%
\frac{k_r}{k_i} = \frac{1 - \mathrm{Fr}}{1 + \mathrm{Fr}}, \quad
\frac{k_t}{k_i} = XY\frac{1 - \mathrm{Fr}}{X^{3/2}Y -
\mathrm{Fr}}.%
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=15cm]{Fig07.pdf}
\vspace*{-3.5cm}%
\caption{(Color online). The dependences of normalized wave
numbers of transmitted waves on the depth drop $X$ for $Y = 1$ and
several particular values of the Froude number. Line 1 pertains to
the reference case when there is no flow ($\mathrm{Fr} = 0$);
other lines pertain to the subcritical cases (line 2 --
$\mathrm{Fr} = 0.1$; line 3 -- $\mathrm{Fr} = 0.5$) and
supercritical cases (line $2'$ -- $\mathrm{Fr} = 0.1$; line $3'$
-- $\mathrm{Fr} = 0.5$). Dashed vertical lines 4 and 5 show the
boundaries between the subcritical and supercritical cases for
$\mathrm{Fr} = 0.1$ and 0.5, respectively.}
\label{f07}%
\end{figure}
In Fig. \ref{f07}, lines 1 -- 3 show the dependencies of
normalized wave numbers of transmitted waves on the depth drop $X$
for $Y = 1$ and several particular values of the Froude number.
Line 1 pertains to the reference case studied by \citet{Lamb-1932}
when there is no flow ($\mathrm{Fr} = 0$). As one can see, when
the depth drop decreases and approaches the critical value, $X \to
X_c = \left(\mathrm{Fr}/Y\right)^{2/3}$, the wave number of the
transmitted wave becomes infinitely big (and the corresponding
wavelength vanishes). This means that the current in the left
domain becomes very strong and supercritical; the transmitted wave
cannot propagate against it and the blocking phenomenon occurs
(see, e.g., Refs. \citep{BasTal-1977, MaiRusStep-2016B} and
references therein).
The transformation coefficients for this case are
\begin{equation}
\label{TransCoef0.22}%
R_{\eta} = \frac{1 - \mathrm{Fr}}{1 + \mathrm{Fr}}\frac{1 -
\sqrt{X}Y}{1 + \sqrt{X}Y}, \quad T_{\eta} = \frac{1 -
\mathrm{Fr}}{X^{3/2}Y - \mathrm{Fr}} \frac{2X^{3/2}Y}{1 +
\sqrt{X}Y}.
\end{equation}
They are as shown in Fig. \ref{f08} in the domains where the
subcritical regime occurs, $X > \left(\mathrm{Fr}/Y\right)^{2/3}$
as the functions of depth drop $X$ for $Y = 1$ and two values of
the Froude number. When depth drop decreases and approaches the
critical value $X_c$, the transmission coefficient infinitely
increases, and the over-transmission phenomenon occurs. However,
it can be readily shown that the energy flux remains finite, and
the law of energy flux conservation Eq. (\ref{EnFluxCons}) with
$\mathrm{Fr} \to -\mathrm{Fr}$ holds true in this case too.
\begin{figure}[h]
\centering
\includegraphics[width=15cm]{Fig08.pdf}
\vspace*{-4.0cm}%
\caption{(Color online). The transformation coefficients for the
upstream propagating incident waves in a canal with flat walls, $Y
= 1$, as functions of depth drop $X$. Line 1 for $T_\eta$ and line
$1'$ for $R_\eta$ pertain to the reference case when $\mathrm{Fr}
= 0$; lines 2 (for $T_\eta$) and $2'$ (for $R_\eta$) pertain to
$\mathrm{Fr} = 0.1$, and lines 3 (for $T_\eta$) and $3'$ (for
$R_\eta$) pertain to $\mathrm{Fr} = 0.5$.}
\label{f08}%
\end{figure}
The gain of energy densities in the reflected and transmitted
waves follows from Eq. (\ref{EnergyGain}) if we replace
$\mathrm{Fr}$ by $-\mathrm{Fr}$ (see lines 4 and 5 in Fig.
\ref{f06}):
\begin{equation}
\label{EnergyGain0.22}%
\frac{E_r}{E_{i}} = \frac{1 - \mathrm{Fr}}{1 +
\mathrm{Fr}}\left(\frac{1 - \sqrt{X}Y}{1 + \sqrt{X}Y}\right)^2,
\quad \frac{E_t}{E_{i}} = \frac{4Y}{\left(1 +
\sqrt{X}Y\right)^2}\frac{1 - \mathrm{Fr}}{1 -
\mathrm{Fr}/X^{3/2}Y}.
\end{equation}
The presence of a subcritical current leads to uniform decrease of
wave energy density in the reflected wave regardless of $X$ and
$Y$. Moreover, the wave density in this wave vanishes when
$\mathrm{Fr} \to 1$. However, in the transmitted wave the density
of wave energy quickly increases when $X \to X_c$ being greater
than $X_c$ (see lines 5 and 6 in Fig. \ref{f06}). Thus, the
typical over-transmission phenomenon occurs in the scattering of
upstream propagating wave (cf. with the over-reflection phenomenon
described at the end of the previous subsection).
\section{\label{sec:level4}Subcritical flow in the upstream domain,
but supercritical in the downstream domain}
In such a case an incident wave can propagate only along the
current. In the downstream domain where the current is
supercritical no one wave can propagate against it. Therefore, we
consider here a scattering of only a downstream propagating
incident wave which arrives from minus infinity in Fig. \ref{f01}.
We assume that the Froude number and geometric parameters of a
canal are such that $X^{3/2}Y < \mathrm{Fr} < 1$.
In the upstream domain two waves of frequency $\omega$ can
propagate in the subcritical flow. One of them is an incident wave
with the unit amplitude and wave number $k_i = \omega/(c_{01} +
U_1)$ and another one is the reflected wave with the amplitude
$R_\eta$ and wave number $k_r = \omega/(c_{01} - U_1)$. In the
downstream domain two waves can exist too. One of them is the
transmitted wave of positive energy with the amplitude $T_p$ and
wave number $k_{t1} = \omega/(U_2 + c_{02})$ and another one is
the transmitted wave of negative energy (see the Appendix) with
the amplitude $T_n$ and wave number $k_{t2} = \omega/(U_2 -
c_{02})$.
The relationships between the wave numbers of scattered waves
follows from the frequency conservation. For the transmitted wave
of positive energy and reflected wave we obtain the same formulas
as in Eqs. (\ref{WaveNumTr}) and (\ref{WaveNumRef}), whereas for
the transmitted wave of negative energy we obtain
\begin{equation}
\label{WaveNumTr2}%
\frac{k_{t2}}{k_i} = XY\frac{\mathrm{Fr} + 1}{\mathrm{Fr} - X^{3/2}Y}.%
\end{equation}
As follows from this formula, the wave number $k_{t2}$ infinitely
increases when $X \to X_c$ being less than $X_c$. The dependencies
of $k_{t1}/{k_i}$ are shown in Fig. \ref{f03} by lines $2'$, $3'$,
and $4'$ for $\mathrm{Fr} = 0.1, \; 0.5$, and 1, respectively,
whereas the dependencies of $k_{t2}/{k_i}$ are shown in Fig.
\ref{f07} by lines $2'$ and $3'$ for $\mathrm{Fr} = 0.1$ and 0.5
respectively.
To find the transformation coefficients we use the same boundary
conditions as in Eqs. (\ref{PresCont1}) and (\ref{MassFluxCont}),
but now they provide the following set of equations:
\begin{eqnarray}
1 + \mathrm{Fr} + \left( 1 - \mathrm{Fr}\right)R_{\eta} &=&
T_{p}\left( 1 + \frac{\mathrm{Fr}}{X^{3/2}Y}\right) + T_{n}\left(1
- \frac{\mathrm{Fr}}{X^{3/2}Y}\right), \label{PCond1} \\%
1 + \mathrm{Fr} - \left( 1 - \mathrm{Fr}\right)R_{\eta} &=&
\sqrt{X}Y\left[T_{p}\left( 1 + \frac{\mathrm{Fr}}{X^{3/2}Y}\right)
- T_{n}\left(1 - \frac{\mathrm{Fr}}{X^{3/2}Y}\right)\right].
\label{PCond2}%
\end{eqnarray}
This set relates three unknown quantities, $R_\eta$, $T_p$, and
$T_n$. We can express, for example, amplitudes of transmitted
waves $T_p$ and $T_n$ in terms of unit amplitude of incident wave
and amplitude of reflected wave $R_\eta$:
\begin{eqnarray}
T_p &=& \frac{X}{2\left(X^{3/2}Y +
\mathrm{Fr}\right)}\left[\left(1 +
\mathrm{Fr}\right)\left(\sqrt{X}Y + 1\right) + \left(1 -
\mathrm{Fr}\right)\left(\sqrt{X}Y - 1\right)R_\eta\right],
\label{Sub-superT1} \\%
T_n &=& \frac{X}{2\left(X^{3/2}Y -
\mathrm{Fr}\right)}\left[\left(1 +
\mathrm{Fr}\right)\left(\sqrt{X}Y - 1\right) + \left(1 -
\mathrm{Fr}\right)\left(\sqrt{X}Y + 1\right)R_\eta\right],
\label{Sub-superT2}%
\end{eqnarray}
whereas the reflection coefficient $R_\eta$ remains unknown.
It can be noticed a particular case when the background flow
could, probably, spontaneously generate waves to the both sides of
a juncture where the background flow abruptly changes from the
subcritical to supercritical value. Bearing in mind that the
transformation coefficients are normalized on the amplitude of an
incident wave, $R_\eta \equiv A_r/A_i$, $T_p \equiv A_p/A_i$, $T_n
\equiv A_n/A_i$, and considering a limit when $A_i \to 0$, we
obtain from Eqs.~(\ref{Sub-superT1}) and (\ref{Sub-superT2}):
\begin{equation}
\label{SpontGen}%
\frac{A_r}{A_p} = \frac{2}{X\left(1 -
\mathrm{Fr}\right)}\frac{X^{3/2} + \mathrm{Fr}}{\sqrt{X}Y - 1},
\quad \frac{A_n}{A_p} = \frac{\sqrt{X}Y + 1}{\sqrt{X}Y -
1}\frac{X^{3/2} + \mathrm{Fr}}{X^{3/2} - \mathrm{Fr}}.%
\end{equation}
The conservation of wave energy flux in general is
\begin{equation}
\label{EnFluxCons0}%
\left(1 + \mathrm{Fr}\right)^2 - \left(1 -
\mathrm{Fr}\right)^2R_\eta^2 = \frac{1}{X^{5/2}Y}\left[
\left(X^{3/2}Y + \mathrm{Fr}\right)^2T_p^2 - \left(X^{3/2}Y -
\mathrm{Fr}\right)^2T_n^2\right].
\end{equation}
After substitution here of the transmission coefficients Eqs.
(\ref{Sub-superT1}) and (\ref{Sub-superT2}) we obtain the identity
regardless of $R_\eta$. In the case of spontaneous wave generation
when there is no incident wave, Eq.~(\ref{EnFluxCons0}) turns to
the identity too after its re-normalization and substitution of
Eqs. (\ref{SpontGen}). This resembles a spontaneous wave
generation due to Hawking's effect \citep{Unruh-1981, Unruh-1995,
Faccio-2013}) at the horizon of an evaporating black hole, when a
positive energy wave propagates towards our space (the upstream
propagating wave $A_r$ in our case), whereas a negative energy
wave together with a positive energy wave propagates towards the
black hole (the downstream propagating waves $A_n$ and $A_p$).
Thus, within the model with an abrupt change of canal
cross-section the complete solution for the wave scattering cannot
be obtained in general. One needs to discard from the
approximation when the current speed abruptly increases at the
juncture and consider a smooth current transition from one value
$U_1$ to another one $U_2$ (this problem was recently studied in
Ref. \citep{ChurErStep-2017}).
\section{\label{sec:level5}Supercritical flow in both the upstream
and downstream domains}
Now let us consider a wave scattering in the case when the flow is
supercritical both in upstream and downstream domain, $U_1 >
c_{01}$ and $U_2 > c_{02}$. In terms of the Froude number we have
$\mathrm{Fr} > 1$ and $\mathrm{Fr} > X^{3/2}Y$. It is clear that
in such a situation, similar to the previous subsection, only a
downstream propagating incident wave can be considered.
In the upstream supercritical flow there is no reflected wave. In
the dispersion diagram of Fig. \ref{f04} the downstream
propagating incident wave of frequency $\omega$ can be either the
wave on the intersection of line 5 with the dashed horizontal
line, or on the intersection of line 6 with the dashed horizontal
line (the intersection point is off the figure), or even both. The
former wave is the wave of positive energy and has the wave number
$k_{i1} = \omega/(U_1 + c_{01})$, whereas the latter is the wave
of negative energy (see the Appendix) and has the wave number
$k_{i2} = \omega/(U_1 - c_{01})$.
In the downstream domain where we assume that the flow is
supercritical too, two waves appear as the result of scattering of
incident waves. As in the upstream domain, one of the transmitted
waves has positive energy and the wave number $k_{t1} =
\omega/(U_2 + c_{02})$, and the other has negative energy and the
wave number $k_{t2} = \omega/(U_2 - c_{02})$.
Let us assume that there is a wavemaker at minus infinity that
generates a sinusoidal surface perturbation of frequency $\omega$.
Then, two waves of positive and negative energies with the
amplitudes $A_p$ and $A_n$, respectively, can jointly propagate.
In the process of wave scattering at the canal juncture two
transmitted waves with opposite energies will appear with the
amplitudes $T_p$ and $T_n$. Their amplitudes can be found from the
boundary conditions Eqs. (\ref{PresCont1}) and
(\ref{MassFluxCont}). Then, after simple manipulations similar to
those in Secs. \ref{sec:level3} and \ref{sec:level4} we obtain:
\begin{eqnarray}
T_p &=& \frac{X}{2\left(X^{3/2}Y +
\mathrm{Fr}\right)}\left[\left(\mathrm{Fr +
1}\right)\left(\sqrt{X}Y + 1\right)A_p - \left(\mathrm{Fr} -
1\right)\left(\sqrt{X}Y - 1\right)A_n\right],
\label{Super-superT1} \\%
T_n &=& \frac{X}{2\left(X^{3/2}Y -
\mathrm{Fr}\right)}\left[\left(\mathrm{Fr +
1}\right)\left(\sqrt{X}Y - 1\right)A_p - \left(\mathrm{Fr} -
1\right)\left(\sqrt{X}Y + 1\right)A_n\right].
\label{Super-superT2}%
\end{eqnarray}
At certain relationships between the amplitudes $A_p$ and $A_n$ it
may happen that there is only one transmitted wave, either of
positive energy ($T_n = 0$), when
\begin{equation}
\label{ZeroNEWave}%
A_n = A_p\frac{\mathrm{Fr} + 1}{\mathrm{Fr} - 1}\frac{\sqrt{X}Y -
1}{\sqrt{X}Y + 1},
\end{equation}
or of negative energy ($T_p = 0$), when
\begin{equation}
\label{ZeroPEWave}%
A_n = A_p\frac{\mathrm{Fr} + 1}{\mathrm{Fr} - 1}\frac{\sqrt{X}Y +
1}{\sqrt{X}Y - 1}.
\end{equation}
From the law of wave energy flux conservation we obtain
\begin{equation}
\label{EnFluxCons1}%
\left(\mathrm{Fr} + 1\right)^2A_p^2 - \left(\mathrm{Fr} -
1\right)^2A_n^2 = \sqrt{X}Y\left[
\left(\frac{\mathrm{Fr}}{X^{3/2}Y} + 1\right)^2T_p^2 -
\left(\frac{\mathrm{Fr}}{X^{3/2}Y} - 1\right)^2T_n^2\right].
\end{equation}
Substituting here the expressions for $T_p$ and $T_n$ as per Eqs.
(\ref{Super-superT1}) and (\ref{Super-superT2}), we see that Eq.
(\ref{EnFluxCons1}) becomes an identity regardless of amplitudes
of incoming waves $A_p$ and $A_n$, including the cases when they
are related by Eqs.~(\ref{ZeroNEWave}) or (\ref{ZeroPEWave}). In
the particular cases one of the incident waves can be suppressed,
ether the wave of negative energy or wave of positive energy. In
the former case we set $A_n = 0$ and $A_p = 1$, and in the latter
case we set $A_p = 0$ and $A_n = 1$.
When there is only one incident wave of {\it positive energy} with
the amplitude $A_p = 1$ and there is no wave of negative energy
($A_n = 0$), then the transmission coefficients Eqs.
(\ref{Super-superT1}) and (\ref{Super-superT2}) reduce to
\begin{equation}
\label{Super-superT3}%
T_p = \frac{X}{2}\frac{\mathrm{Fr} + 1}{\mathrm{Fr} +
X^{3/2}Y}\left(1 + \sqrt{X}Y\right), \quad T_n =
\frac{X}{2}\frac{\mathrm{Fr} + 1}{\mathrm{Fr} - X^{3/2}Y}\left(1 -
\sqrt{X}Y\right).
\end{equation}
Recall that these formulas are valid for supercritical flows when
$\mathrm{Fr} > 1$ and $\mathrm{Fr} > X^{3/2}Y$. In the limiting
case when $X \to 0$ and $Y = \mathrm{const.}$, we obtain
\begin{equation}
\label{Super-superT4}%
T_p \approx T_n \approx X\frac{\mathrm{Fr} + 1}{\mathrm{2Fr}}.
\end{equation}
In another limiting case when $X^{3/2}Y \to \mathrm{Fr}$ the
transmission coefficient for the positive energy wave remains
constant, whereas the transmission coefficient for the negative
energy wave within the framework of linear theory goes to plus or
minus infinity depending on the value of $Y$. Figure \ref{f09}(a)
illustrates the transmission coefficients $T_p$ and $T_n$ as
functions of $X$ for $Y = 1$ and two particular values of the
Froude number.
\begin{figure}[h]
\centering
\includegraphics[width=16cm]{Fig09.pdf}
\vspace*{-4.5cm}%
\caption{(Color online). The transmission coefficients for the
downstream propagating incident waves of positive energy (frame a)
and negative energy (frame b) in a canal with the flat walls, $Y =
1$, as functions of the depth drop $X$. Line 1 for $T_p$ and line
$1'$ for $T_n$ pertain to $\mathrm{Fr} = 1.5$, and lines 2 (for
$T_p$) and $2'$ (for $T_n$) pertain to $\mathrm{Fr} = 2.5$. Data
for lines 1 and 2 in frame (b) were multiplied by a factor of ten
to make the graphics clearly visible.}
\label{f09}%
\end{figure}
When there is only one incident wave of {\it negative energy} with
the amplitude $A_n = 1$ and there is no wave of positive energy
($A_p = 0$), then the transmission coefficients Eqs.
(\ref{Super-superT1}) and (\ref{Super-superT2}) reduce to
\begin{equation}
\label{Super-superT5}%
T_p = \frac{X}{2}\frac{\mathrm{Fr} - 1}{\mathrm{Fr} +
X^{3/2}Y}\left(1 - \sqrt{X}Y\right), \quad T_n =
\frac{X}{2}\frac{\mathrm{Fr} - 1}{\mathrm{Fr} - X^{3/2}Y}\left(1 +
\sqrt{X}Y\right).
\end{equation}
In the limiting case when $X \to 0$, and $Y = \mathrm{const.}$, we
obtain
\begin{equation}
\label{Super-superT6}%
T_p \approx T_n \approx X\frac{\mathrm{Fr} - 1}{2\mathrm{Fr}}.
\end{equation}
In another limiting case when $X^{3/2}Y \to \mathrm{Fr}$, the
transmission coefficient for the positive energy wave remains
finite, whereas, the transmission coefficient for the negative
energy wave within the framework of linear theory goes to plus
infinity. Figure \ref{f09}(b) shows the transmission coefficients
$T_p$ and $T_n$ as functions of $X$ for $Y = 1$ for two particular
values of the Froude number.
\section{\label{sec:level6}Supercritical flow in the upstream and
subcritical in the downstream domain}
Let us consider, at last, the case when the flow is supercritical
in the upstream domain, where $U_1 > c_{01}$, but due to canal
widening becomes subcritical in the downstream domain, where $U_2
< c_{02}$. Thus, the flow is decelerating and in terms of the
Froude number we have $1 < \mathrm{Fr} < X^{3/2}Y$. Assume first
that the incident wave propagates downstream.
\subsection{\label{sec:level61}Downstream propagating incident wave}
As was mentioned in the previous section, two waves with the
amplitudes $A_p$ and $A_n$ can propagate simultaneously from minus
infinity, if they are generated by the same wavemaker with the
frequency $\omega$. In the downstream domain potentially two waves
of positive energy can exist, but only one of them propagating
downstream can appear as the transmitted wave with the amplitude
$T_\eta$ as the result of wave scattering at the juncture.
The amplitudes of scattered waves can be found from the boundary
conditions Eqs. (\ref{PresCont1}) and (\ref{MassFluxCont}). This
gives, after simple manipulations:
\begin{eqnarray}
\left(1 + \mathrm{Fr}\right)A_p + \left(1 - \mathrm{Fr}\right)A_n
&=& T_{\eta}\left(1 + \frac{\mathrm{Fr}}{X^{3/2}Y}\right), \label{PCond51} \\%
\left(1 + \mathrm{Fr}\right)A_p - \left(1 - \mathrm{Fr}\right)A_n
&=& \sqrt{X}YT_{\eta}\left( 1 +
\frac{\mathrm{Fr}}{X^{3/2}Y}\right). \label{PCond52}%
\end{eqnarray}
This set of equations provides a unique solution for the
transmission coefficient $T_\eta$ only in the case when the
amplitudes of incoming waves are related:
\begin{equation}
\label{RelApAn}%
A_n = \frac{1 + \mathrm{Fr}}{1 - \mathrm{Fr}}\frac{1 -
\sqrt{X}Y}{1 + \sqrt{X}Y}A_p, \quad T_\eta = \frac{1 +
\mathrm{Fr}}{X^{3/2}Y + \mathrm{Fr}}\frac{2X^{3/2}Y}{1 +
\sqrt{X}Y}A_p.
\end{equation}
If one of the incident waves is absent ($A_n = 0$ or $A_p = 0$) or
amplitudes of incoming waves are not related by Eq.
(\ref{RelApAn}), then the set of Eqs. (\ref{PCond51}) and
(\ref{PCond52}) is inconsistent. In such cases the problem of wave
scattering in the canal does not have a solution within the
framework of a model with a sharp change of the cross-section.
If the amplitudes of incident waves $A_p$ and $A_n$ are related by
Eq. (\ref{RelApAn}), then the conservation of wave energy flux
holds and takes the form
\begin{equation}
\label{EnFluxCons51}%
\left(\mathrm{Fr} + 1\right)^2A_p^2 - \left(\mathrm{Fr} -
1\right)^2A_n^2 = \sqrt{X}Y \left(\frac{\mathrm{Fr}}{X^{3/2}Y} +
1\right)^2T_\eta^2.
\end{equation}
Substituting here $A_n$ and $T_\eta$ from Eq. (\ref{RelApAn}), we
see that it becomes just the identity.
\subsection{\label{sec:level62}Upstream propagating incident wave}
For the incident wave arriving from the plus infinity and
propagating upstream in the subcritical domain of the flow, the
problem of wave scattering within the model with a sharp change of
a current is undefined. The incoming wave cannot penetrate from
the domain with a subcritical flow into the domain with a
supercritical flow, therefore one can say that formally the
reflection coefficient in this case $R_\eta = 1$, and the
transmission coefficients $T_\eta = 0$. However such a problem
should be considered within a more complicated model with a smooth
transcritical flow; this has been done in Ref.
\citep{ChurErStep-2017}.
\section{\label{sec:level7}Conclusion}
In this paper within the linear approximation we have studied a
scattering of long surface waves at the canal juncture when its
width and depth abruptly change at a certain place. We have
calculated the transformation coefficients for the reflected and
transmitted waves in the presence of a background flow whose speed
changes from $U_1$ to $U_2$ in accordance with the mass flux
conservation. The calculated coefficients represent the
effectiveness of the conversion of the incident wave into the
other wave modes -- reflected and transmitted of either positive
or negative energy. Our consideration generalizes the classical
problem studied by Ref. \cite{Lamb-1932} when the background flow
is absent. It was assumed that the characteristic scale of current
variation in space is much less than the wavelengths of scattered
waves. Such a simplified model allows one to gain insight into the
complex problem of wave-current interaction and find the
conditions for the over-reflection and over-transmission of water
waves. We have analyzed all possible orientations of the incident
wave with respect to flow and studied all possible regimes of
water flow (subcritical, supercritical, and transcritical).
In the study of the subcritical and supercritical flows (see Secs.
\ref{sec:level3} and \ref{sec:level5}) we have succeeded in
calculating the transmission and reflection coefficients in the
explicit forms as functions of the depth drop $X = h_2/h_1$,
specific width ratio $Y = b_2/b_1$, and Froude number
$\mathrm{Fr}$. Based on these, the conditions for the
over-reflection and over-transmission have been found in terms of
the relationships between the Froude number and canal geometric
parameters $X$ and $Y$. It appears that it is not possible to do
the same for the transcritical flows, at least within the
framework of the simplified model considered in this paper (see
Secs. \ref{sec:level4} and \ref{sec:level6}). The reason for that
is in the critical point where $\mathrm{Fr} = 1$ which appears in
the smooth transient domain between two portions of a canal with
the different cross-sections. The transition through the critical
point is a rather complex problem which was recently studied on
the basis of a model with a continuously varying flow speed in a
duct of smoothly varying width \citep{ChurErStep-2017}. The
summary of results obtained is presented in
Table I.\\
Table I. The summary of considered cases. A cocurrent propagating
incident waves is denoted by $k_i \uparrow\uparrow U$, whereas a
countercurrent propagating incident waves is denoted by $k_i
\downarrow\uparrow U$. The acronyms PEW and NEW pertain to
positive and negative energy waves, correspondingly.\\
\noindent\begin{tabular}{|l|c|c|c|} \hline
\multicolumn{4}{|c|}{\textbf{I. Subcritical
flow in the upstream and downstream domains}}\\
\hline %
\;${\bf k}_i$, ${\bf U}$ & {\bf Reflect. coeff.} & {\bf Transmiss. coeff.} & {\bf Peculiarity of a scattering} \\
\hline %
$k_i \uparrow\uparrow U$ & $R_\eta$ see Eq.~(\ref{TransCoef1})
& $T_\eta$ see Eq.~(\ref{TransCoef1}) & Regular scattering \\
\hline %
$k_i \downarrow\uparrow U$ & $R_\eta$ see
Eq.~(\ref{TransCoef0.22}) & $T_\eta$ see Eq.~(\ref{TransCoef0.22}) & Regular scattering \\
\hline %
\multicolumn{4}{|c|}{\textbf{II. Subcritical flow in the upstream domain and supercritical in the}}\\
\multicolumn{4}{|c|}{\textbf{downstream domain. PEW and NEW appear downstream.}}\\
\hline%
\;${\bf k}_i$, ${\bf U}$ & {\bf Reflect. coeff.} & {\bf Transmiss. coeff.} & {\bf Peculiarity of a scattering} \\
\hline %
$k_i \uparrow\uparrow U$ & $R_{\eta}$ is undetermined, &
$T_{p}$ see Eq.~(\ref{Sub-superT1}) & Undefined problem statement, \\
& according to \citep{ChurErStep-2017}, $R_{\eta} = 1$ & $T_{n}$ see Eq.~(\ref{Sub-superT2}) & according to \citep{ChurErStep-2017}, $T_p = -T_n = 1$ \\
\hline%
$k_i \downarrow\uparrow U$ & \multicolumn{3}{c|}{Impossible situation} \\
\hline \multicolumn{4}{|c|}{\textbf{III. Supercritical flow in the upstream and downstream domains}}\\
\hline%
\;${\bf k}_i$, ${\bf U}$ & {\bf Reflect. coeff.} & {\bf Transmiss. coeff.} & {\bf Peculiarity of a scattering} \\
\hline %
$k_i \uparrow\uparrow U$ & No reflected wave &
$T_{p}$ see Eq.~(\ref{Super-superT1}) & Incident wave can be PEW or NEW,\\
& & $T_{n}$ see Eq.~(\ref{Super-superT2}) & or both. See Eqs.~(\ref{Super-superT3}), (\ref{Super-superT5}).\\
\hline%
$k_i \downarrow\uparrow U$ & \multicolumn{3}{c|}{Impossible situation} \\
\hline \multicolumn{4}{|c|}{\textbf{IV. Supercritical flow in the upstream and}}\\
\multicolumn{4}{|c|}{\textbf{subcritical in the downstream domain}}\\
\hline%
\;${\bf k}_i$, ${\bf U}$ & {\bf Reflect. coeff.} & {\bf Transmiss. coeff.} & {\bf Peculiarity of a scattering} \\
\hline %
$k_i \uparrow\uparrow U$ & No reflected wave & $T_{\eta}$ provided that & Over-determined problem if \\
& & $A_n \sim A_p$, Eq.~(\ref{RelApAn}) & there is only one incident wave \\
\hline%
$k_i \downarrow\uparrow U$ & Formally $R_\eta = 1$ & Formally $T_\eta = 0$ & See Ref. \citep{ChurErStep-2017}\\
\hline
\end{tabular}\\
\bigskip
The problem studied can be further generalized for waves of
arbitrary length taking into account the effect of dispersion.
Similar works in this direction were published recently for
relatively smooth current variation in the canal with the
finite-length bottom obstacles \citep{RobMichPar-2016,
CoutWein-2016}. It is worthwhile to notice that in the dispersive
case for purely gravity waves there is always one wave of negative
energy for which the flow is supercritical. This negative energy
mode smoothly transforms into the dispersionless mode when the
flow increases. In such cases two other upstream propagating modes
disappear, and the dispersion relations reduces to one of
considered in this paper. It will be a challenge to compare the
theoretical results obtained in this paper with the numerical and
experimental data; this may be a matter of future study.
\begin{acknowledgments}
This work was initiated when one of the authors (Y.S.) was the
invited Visiting Professor at the Institut Pprime, Universit{\'e}
de Poitiers in August--October, 2016. Y.S. is very grateful to the
University and Region Poitou-Charentes for the invitation and
financial support during his visit. Y.S. also acknowledges the
funding of this study from the State task program in the sphere of
scientific activity of the Ministry of Education and Science of
the Russian Federation (Project No. 5.1246.2017/4.6), and G.R.
acknowledges the funding from the ANR Grant HARALAB No.
ANR-15-CE30-0017-04. The research of A.E. was supported by the
Australian Government Research Training Program Scholarship.
The authors are thankful to Florent Michel, Renaud Parentani,
Thomas Philbin, and Scott Robertson for useful discussions.
\end{acknowledgments}
|
2,869,038,155,649 | arxiv | \section{Introduction}
High-throughput assays are transforming the study of biology, and are generating a rich,
complex and diverse collection of high-dimensional data sets. Joint analyses combining data from different studies and technologies are crucial to improve accuracy of conclusions and to produce generalizable knowledge.
Most measurements from high-throughput experiments display variation arising
from both biological and artifactual sources.
Within a study, effects driven by unique issues with the experimental conditions of a specific laboratory or technology can be so large to surpass the biological signal for many biological features \citep{aach2000}. In gene expression, for example, large systematic differences arising from different laboratories or technological platforms have been long recognized \citep{irizarry2003,shi2006,kerr2007}. Systematic collections of gene expression data, collected with technologies that have evolved over time, are widely available, as exemplified by the breast cancer datasets that motivate our work, described in Section~2.
A strength of multi-study analyses is that, generally, genuine biological signal is more likely than spurious signal to be present in multiple studies, particularly when studies are collected from biologically similar populations. Thus, multi-study analyses offer the opportunity to learn replicable features shared among multiple studies.
Discovering these features is, broadly speaking, more valuable than discovering signal in a single study. Joint analyses of multiple genomic datasets have begun more than a decade ago, they are now increasingly common, and can be highly successful \citep{Rhodes:2002ko,Huttenhower:2006wq,Gao:2014tj,Pharoah:2013tm,Riester:2014ga,Ciriello:2013js}. Many such analyses focus on identifying parameters that relate biological features measured at high throughput to phenotypes. These effects can be replicable, though signal extraction across studies can be challenging \citep{Garrett-Mayer2008}.
An important goal in high-dimensional data analysis is the unsupervised identification of latent components or factors. Despite the importance of this goal, the development of formal statistical approaches for unsupervised multi-study analyses is relatively unexplored.
In applications, joint unsupervised analyses of high-throughput biological studies often proceed by pooling all the data. Despite their success,
these studies rely critically on simplified methods of analysis to capture common signal. For example \cite{wang2011} and \cite{edefonti2012} stack all studies and then perform standard analyses, such as factor analysis (FA) or Principal Component Analysis (PCA). The results will capture some common features,
but the information about study-specific components will likely be lost, and ignoring it could compromise the accuracy of the common factors found.
Alternatively, it is also common to analyze each study separately and then heuristically explore common structures from the results \citep{Hayes2006}.
Co-Inertia Analysis (CIA) \citep{dray2003} explores the common structure of two different sets of variables by first separately performing dimension reduction on each set to estimate factor scores, and then investigating the correlation between these factors. Multiple Co-Inertia Analysis (MCIA) is a generalization of CIA to more than two data sets, which projects different studies into a common hyperspace \citep{meng2014}. Multiple Factor analysis (MFA) \citep{abdi2013} is an extension of PCA and consists of three steps. The first step applies PCA to each study. In the second step, each data set is normalized by dividing by the first singular value of the covariance matrix. In the third step, these normalized data are stacked by row creating a single data set to which PCA is then applied.
In practice, there is a need to automatically and rigorously model across studies the common signal that can reliably be identified, while at the same time modeling study-specific variation.
A methodological tool for this task is Multi-Study Factor Analysis (MSFA), recently introduced in \citet{DV2016}. Inspired by models used in the social sciences, MSFA extends FA to the joint analysis of multiple studies,
separately estimating signal reproducibly shared across multiple studies
from study-specific components arising from artifactual and population-specific sources of variation. This dual goal clearly sets MSFA aside from
earlier applications of FA to gene expression studies, such as
\citet{carvalho2008}, \citet{friguet2009}, \citet{blum2010}, or \citet{runcie2013}.
The MSFA methodology in \citet{DV2016} is limited to settings where enough samples are available in each study, and no sparsity is expected or necessary. This is because
model parameters are estimated by maximum likelihood (MLE) and model selection is performed
by standard information criteria. In high-throughput biology, the sample size routinely exceeds the number of variables, and it is essential to employ regularization through priors or penalties.
In this paper we introduce a Bayesian generalization of Multi-study factor analysis.
Bayesian approaches naturally provide helpful regularization, and offer further advantages, discussed later. We leverage the sparse Bayesian infinite factor model, and generalize the multiplicative gamma prior of \citet{bhattacharya2011} to the MSFA setting, to induce sparsity on each loading matrix. We then sample from the posterior distribution via MCMC, without any ex-ante constraints on the loading matrices. This avoids the order dependence induced by the often-used assumption of a lower-triangular form of the loading matrices \citep{geweke1996, lopes2004}, which was employed by the original MSFA proposal. Although useful inferences can be obtained with careful implementation of the constraint, removing it makes the application of FA much simpler and general. We regard this to be an important advantage of our proposal.
Our prior and parametrization also facilitate inference on the covariance matrices and precision matrices of the observed variables. These are often important goals. An important example is inference on gene networks, often implemented by first estimating the covariance matrix through FA \citep{zhao2014, gao2016}. Through the estimation of common factors implied by the decomposition of the covariance matrix described in \S \ref{sub:def}, the approach we propose allows to detect a common network across the studies, and also to recover the study-specific contributions to gene networks.
The original implementation of the sparse Bayesian infinite factor model \citet{bhattacharya2011} truncates the dimension of the loading matrices at a fixed value. In MSFA, this point is even more important, since our model introduces $(S+1)$ loading matrices if there are $S$ studies. We suggest a pragmatic approach, where the number of dimensions is chosen based on a simple eigenvalue decomposition of covariance matrices obtained as output of the MCMC sampling from the posterior. The specific choice of prior makes the choice of the dimension less critical than would alternative approaches, as we discuss later.
A further strength of our proposal is the recovery of the loading matrices, which are not estimated in \citet{bhattacharya2011}.
We leverage the recently proposed Orthogonal Procrustes (OP) method, introduced in \cite{assmann2016}. OP performs an ex-post recovery of the
estimated loadings by processing the MCMC output, after fitting the model without any restrictions. The method provides a satisfactory solution to the rotation invariance of FA.
Our results show that the good properties of OP can be generalized to our multiple study setting.
The plan of the paper is as follows. Section~2 describes the data. Section~3 introduces the Bayesian Multi-study factor analysis (BMSFA) framework, describes our prior, our extension of OP, and our procedure for choosing the number of shared and study-specific factors. Section~4 presents extensive simulation studies, providing evidence on the performance of BMSFA and comparing it with standard methods. We also investigate determining the truncation level for latent
factors. Section~\ref{sec:CS} applies BMSFA
to the breast cancer data described in Section~2. Section~5 contains a discussion.
\section{The Breast Cancer Data sets}
\label{sub:BCdata}
Breast cancer is both a clinically diverse and a genetically heterogeneous disease \citep{perou2000, planey2016}. The complex nature of breast cancer has been clarified by classifying breast cancer into subtypes using gene expression measurements from tumor samples. Reliably identifying these subtypes has the potential of driving personalized patient treatment regimens \citep{masuda2013} and risk prediction
models \citep{parker2009}. Several groups \citep{sorlie2001, sotiriou2003, hu2006, planey2016} have focused on finding replicable gene expression patterns across different studies, to better classify breast carcinomas into distinct subtypes.
A very valuable statistical approach is unsupervised clustering using different microarrays that query the same set of genes \citep{perou2000, sorlie2001, sorlie2003, castro2016}. A challenge is to characterize the extent to which variation in gene expression, and the resulting subtypes, are stable across different studies \citep{Hayes2006}. When different microarray studies are considered together, one is likely to encounter significant and unknown sources of study-to-study heterogeneity \citep{simon2009, bernau2014}. These sources
include differences in design, hidden biases, technologies used for
measurements, batch effects, and also variation in the populations studied ---for example, differences in treatment or disease stage and severity. Quantifying these heterogeneities
and dissecting their impact on the replicability of patterns is essential.
A typical bioinformatics analysis pipeline would attempt to remove
variation attributable to experimental artifacts before further analysis.
If information on batches of other relevant experimental factors is available, their effects can be addressed \citep{draghici2007systems}.
For example, \cite{sorlie2001} use the SAM (significance analysis of microarrays) algorithm to detect genes not influenced by batch effect, and then use this set of genes to perform unsupervised cluster analysis.
In general, it is challenging to fully remove artifactual effects, particularly if they are related to unobserved confounders rather than known batches or factors \citep{draghici2007systems}.
The joint analysis of multiple studies offers the opportunity to understand replicable variation across different studies. The overarching goal of this work is to improve the identification of a stable and replicable signal by simultaneously modeling both the components of variation shared across studies, and those that are study-specific. The latter could include artifacts and batch effects that were not addressed by the study specific preprocessing, as well as biological signal that may hard to replicate or genuinely unique to a study. An example of the latter would be the gene expression signature resulting from the administration of a treatment that is used in one study only.
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{l c c c c l}
\hline\hline
Study & Adjuvant Therapy & N & N: ER$+$ & 3Q survival & Reference\\ [0.5ex]
\hline
CAL & Chemo, hormonal & 118 & 75 & 42 & \cite{chin2006} \\
MAINZ & none & 200 & 162 & 120 & \cite{schmidt2008}\\
MSK & combination & 99 & 57 & 76 & \cite{minn2005}\\
EXPO & hormonal & 517 & 325 & 126 & \cite{symmans2010}\\
TRANSBIG & none & 198 & 134 & 143 & \cite{desmedt2007}\\
UNT & none & 133 & 86 & 151 & \cite{sotiriou2006}\\
VDX & none & 344 & 209 & 44 & \cite{minn2007}\\
\hline
\end{tabular}
\caption{\it The seven data sets considered in the illustration and their characteristics. N is the total number of samples; N: ER$+$ is the number of Estrogen Receptor positive patients. 3Q survival is the third quartile of the survival function for all patients in the study.}
\label{tab: dataset}
\end{table}
In our case study, we consider a systematic collection of publicly available breast cancer microarray studies compiled by \cite{haibe2012}.
Table~\ref{tab: dataset} provides an overview of the studies, the corresponding references, sample size, Estrogen Receptor (ER) status prevalence, and survival time. Additional details about these studies, their preprocessing, curation, criteria for inclusion, and public availability are described in \cite{haibe2012}. Four of these studies only include patients who did not receive hormone therapy or chemotherapy. Within the Affymetrix technology, genes can be represented by multiple probe-sets. Our analysis considers, for each gene, only the probe-set with maximum mean \citep{miller2011}. As in \cite{bernau2014}, we only consider we only consider genes measured in all the seven studies and focus on the 50\% of genes with higher variance.
\section{A Bayesian Framework for multi-study analysis}
This section provides details of our model, in four parts:
\begin{itemize}
\item[i)] Definition of the multi-study factor model sampling distribution;
\item[ii)] Choice of the multiplicative gamma prior \citep{bhattacharya2011}, with shrinkage priors for the loading matrices to incorporate sparsity. Posterior sampling
is carried out by Gibbs sampling, without any constraints on the model parameters;
\item[iii)] Choice of truncation level for the latent
factor dimensions, determined by a suitable singular value decomposition;
\item[iv)] Recovery of the loading matrices, performed by the OP approach.
\end{itemize}
\subsection{Model definition}
\label{sub:def}
We consider $S$ studies, each with the same $P$ genomic variables.
Study $s$, $s=1, \dots, S$, has $n_s$ subjects and
$P$-dimensional data vector $\mathbf{x}_{is}$, $i=1,\ldots,n_s$, centered at its sample mean. Our sampling distribution follows the multi-study factor model \citep{DV2016}. The variables in study $s$ are decomposed into $K$ factors shared among all studies, and
$J_s$ further factors specific to study $s$, as follows:
\begin{equation}
\mathbf{x}_{is} = \boldsymbol{\Phi} \mathbf{f}_{is} + \boldsymbol{\Lambda}_s \mathbf{l}_{is} + \mathbf{e}_{is} \, .
\label{eqn:MFA}
\end{equation}
Here $\mathbf{f}_{is} \sim N_k(\mathbf{0}, \mathbf{I}_{k})$ are the \textit{shared} latent factors, $\boldsymbol{\Phi}$ is their $P \times K$ loading matrix; $\mathbf{l}_{is} \sim N_{j_s}(\mathbf{0}, \mathbf{I}_{j_s})$ are the \textit{study-specific} latent factors and
$\boldsymbol{\Lambda}_{s},\; s=1,\dots, S$ are the corresponding $P \times J_s$ loading matrices; lastly, $\mathbf{e}_{is}$ is the $p \times 1$ Gaussian error vector with covariance $\mathbf{\boldsymbol{\Psi}}_{s} = \rm{diag} (\psi_{\textit{s}_1}^2, \dots, \psi_{\textit{s}_p}^2)$.
The resulting marginal distribution of $\mathbf{x}_{is}$ is a multivariate normal with mean vector $\mathbf{0}$ and covariance matrix
$\boldsymbol{\Sigma}_s = \boldsymbol{\Phi} \boldsymbol{\Phi}^\top+\boldsymbol{\Lambda}_s\boldsymbol{\Lambda}_s^\top+\boldsymbol{\Psi}_s$.
The covariance matrix of study $s$ can be rewritten as
\begin{equation}
\boldsymbol{\Sigma}_s = \boldsymbol{\Sigma}_{\Phi}+ \boldsymbol{\Sigma}_{\Lambda_s}+\boldsymbol{\Psi}_s,
\label{eqn:sigma_mfa}
\end{equation}
where $\boldsymbol{\Sigma}_{\Phi}= \boldsymbol{\Phi} \boldsymbol{\Phi}^\top$ is the covariance of the shared factors, and $\boldsymbol{\Sigma}_{\Lambda_s} = \boldsymbol{\Lambda}_s\boldsymbol{\Lambda}_s^\top$ is the
covariance of the study-specific factors. A straightforward implication of (\ref{eqn:sigma_mfa}) is
that $\boldsymbol{\Sigma}_{\Phi}$ and $\boldsymbol{\Sigma}_{\Lambda_s}$ describe the variability of the $P$ variables in study $s$
that can be interpreted as shared across studies and specific to study $s$, respectively.
The decomposition of $\boldsymbol{\Sigma}_s$ is not unique, as there are infinite possibilities to represent it because $\boldsymbol{\Phi}^* = \boldsymbol{\Phi}\mathbf{Q}$ and $\boldsymbol{\Lambda}_s^* = \boldsymbol{\Lambda}_s \mathbf{Q}_s$ both satisfy (\ref{eqn:sigma_mfa}) for any two orthogonal matrices $\mathbf{Q}$ and $\mathbf{Q}_s$. MSFA identifies the parameters by imposing constraints on the two factor loadings matrices, such as the lower triangular constraint used in Factor Analysis (FA) \citep{geweke1996, lopes2004} .
This constraint generates an order dependence among the variables. Thus,
as noted by \citet{carvalho2008},
the choice of the first $K+J_S$ variables becomes an important modeling choice.
Several approaches focus on the estimation of covariance matrix \citep{bhattacharya2011} or precision matrix \citep{gao2014, zhao2014}. These methods do not require identifiability of the loading matrix. Our approach is also based on this concept: we focus on the estimation of the common variation $\boldsymbol{\Sigma}_{\Phi}$ shared among the studies and the variation specific to each study $\boldsymbol{\Sigma}_{\Lambda_s}$. The two matrices $\boldsymbol{\Sigma}_{\Phi}$ and $\boldsymbol{\Sigma}_{\Lambda_s}$ are only assumed to be positive semidefinite normal matrices, i.e. symmetric matrices with a subset of positive non-null eigenvalues.
\subsection{ The multiplicative gamma shrinkage prior}
\label{sub:sbif}
We adapt a shrinkage prior from \cite{bhattacharya2011} for both the common and study-specific factor loadings.
The shrinkage priors favor sparsity by removing some entries of the loading matrix.
When an element is close to zero, the variable corresponding to the row does not contribute to the common or study-specific latent factor corresponding to the column. In the genomic context, this sparsity
models the biological reality that only a subset of the genes represented in a cell's transcriptome is participating in a specific biological function \citep{tegner2003}.
Another important property of the \cite{bhattacharya2011} prior is that the shrinkage towards zero increasing with the column index of the loading matrix.
Our extension of the multiplicative gamma shrinkage prior to the multiple study setting is as follows.
The prior for the elements of the shared factor loading matrix $\boldsymbol{\Phi}$
is
$$
\phi_{pk} \mid \omega_{pk}, \tau_{k} \sim N(0,\omega^{-1}_{pk} \tau^{-1}_k), \quad p=1,\dots,P, \; k=1,\dots, \infty,
$$
$$
\omega_{pk} \sim
\Gamma \left( \frac{\nu}{2},\frac{\nu}{2} \right)
\;\;\;\;\;\; \tau_{k} = \prod_{l=1}^k \delta_l \;\;\;\;\;\; \delta_1 \sim \Gamma (a_1,1) \;\;\;\;\;\; \delta_l \sim \Gamma (a_2,1),\;\;\;l \geq 2
$$
where $\delta_l \; (l=1,2,\dots)$ are independent, $\tau_k$ is the global shrinkage parameter for the $k$-th column and $\omega_{pk}$ is the local shrinkage for the element $p$ in column $k$.
We then replicate this scheme to specify
the prior for the elements of the study-specific factor loading matrix $\boldsymbol{\Lambda}_s$:
$$
\lambda_{pj_s} \mid \omega^s_{pj_s}, \tau^s_{j_s} \sim N(0,\omega^{s^{-1}}_{pj_s} \tau^{s^{-1}}_{j_s}), \quad p=1,\dots,P, \; j_s=1,\dots, \infty, \; \mbox{ and } s=1,\dots,S,
$$
$$
\omega_{pj_s} \sim
\Gamma \left( \frac{\nu^s}{2},\frac{\nu^s}{2} \right)
\;\;\;\;\;\; \tau^s_{j_s} = \prod_{l=1}^{j_s} \delta^s_l \;\;\;\;\;\; \delta^s_1 \sim \Gamma (a^s_1,1) \;\;\;\;\;\; \delta^s_l \sim \Gamma (a^s_2,1),\;\;\;l \geq 2
$$
where $\delta^s_l (l=1,2, \dots)$ are independent, $\tau^s_{j_s}$ is the global shrinkage parameter for the $j_s$ column and $\omega^s_{pj_s}$ is the local shrinkage for the element $p$ in column $j_s$.
For each of the error variances $\psi_{ps},\,\,p =1,\dots,P$ we assume an inverse
gamma prior
$
\psi_{ps}^{-1} \sim \Gamma (a_{\psi},b_{\psi}).
$
This choice, made also by \citet{bhattacharya2011}, is common
in standard FA \citep{lopes2004, gao2013, rovckova2016}.
Sampling from the posterior distribution of the model parameters
is carried out by Gibbs sampling. Details are in Supplementary Materials.
\subsection{Choosing the number of latent factors}
\label{sub:nfac}
In practical applications, the number of important latent factors is likely to be small compared to the number of variables
$P$. As suggested by \cite{bhattacharya2011}, the effective number of factors would be small when data are sparse.
Our approach circumvents the need for pre-specifying the latent dimension since the shrinkage prior gives positive mass to an infinite number of them. However, we need a proper computational strategy for choosing accurate truncation levels $K$ and $J_s$, $s=1,\ldots,S$. Ideally, we would like to retain the relevant factors discarding the redundant ones.
An analogous task for FA is addressed in \citet{bhattacharya2011} who truncate the number of factors to a finite value, usually far smaller than the number of variables $P$.
This truncation level is chosen by checking the columns of the estimated loading matrix, to assess which ones are formed entirely by elements of negligible size. The fact that the shrinkage implied by the prior increases in later columns greatly simplifies
this task, compared to what required by alternative shrinkage priors such as the spike and slab~\citep{carvalho2008}. We use the same idea, though computational details differ.
Our practical method to assess the numbers of shared factors $K$ and study-specific factors $J_s$ is based on singular value decomposition (SVD) and proceeds as follows.
Starting from a considerable number of shared and study-specific factors, we seek $K \ll P$ and $J_s \ll P$. In the MSFA model, this implies that
the two matrices $\boldsymbol{\Sigma}_{\Phi}$ and $\boldsymbol{\Sigma}_{\Lambda_s}$ are singular, with ranks $K$ and $J_s$, respectively. Since these matrices are symmetric, they have $K$ and $J_s$ non-null eigenvalues.
Based on this, we compute the eigenvalues $\nu_1, \ldots, \nu_P$ of $\widehat{\boldsymbol{\Sigma}}_{\Phi}$, with $\nu_p \geq 0$, $p=1,\ldots,P$,
ordered in decreasing size. We then choose $K$ as the number of eigenvalues
larger than a pre-specified positive threshold, to achieve
$
\mathbf{U} \, \mathbf{N}_K \mathbf{U}^\top \doteq \widehat{\boldsymbol{\Sigma}}_{\Phi} \, ,
$
where $\mathbf{N}_K={\rm diag}(\nu_1, \ldots, \nu_K)$, and the columns of $\mathbf{U}$, of size $P \times K$, are given by $K$ (normalized) eigenvectors of $\widehat{\boldsymbol{\Sigma}}_{\Phi}$. We proceed in the same way for $J_s$, $s=1,\ldots,S$.
\subsection{Recovering loading matrices}
\label{sub:iden}
The method of \S \ref{sub:def}-\ref{sub:nfac} provides a practical route to the estimation of $\boldsymbol{\Sigma}_{\Phi}$ and $\boldsymbol{\Sigma}_{\Lambda_s}$, but in many applications recovery of the loading matrices is also useful.
Recently \cite{assmann2016} solved the identification issue in the context of FA by first generating an MCMC sample without any constraints, and then filtering out the possible effect of orthogonal rotations. They solve an Orthogonal Procrustes (OP) problem \citep{gower2004} by building a sequence of orthogonal matrices defined from the MCMC output.
Here we extend this procedure to BMSFA.
When the model parameters are not constrained, the Gibbs sampler is said to be orthogonally mixed \citep{assmann2016}, as each chain may produce
different orthogonal transformations (represented by the matrices $\mathbf{Q}$ and $\mathbf{Q}_s$) for the factor loadings $\boldsymbol{\Phi}^*$ and $\boldsymbol{\Lambda}_s^*$. Starting from
a sequence of $R$ draws from the posterior distribution of $\boldsymbol{\Phi} (\boldsymbol{\Phi}^1, \dots, \boldsymbol{\Phi}^R)$, the OP algorithm circumvents this problem by estimation the loading matrices via the following constrained optimization:
\begin{equation}
\left\{\left\{ \tilde\mathbf{Q} \right\}_{r=1}^{R}, \tilde\boldsymbol{\Phi}^*\right\} = \underset{\mathbf{Q}^{(r)}, \boldsymbol{\Phi}^*}{\mbox{argmin }} \sum_{r=1}^{R} L_Q \left( \boldsymbol{\Phi}^*, \boldsymbol{\Phi}^{(r)} \mathbf{Q}^{(r)} \right) \,\, \mbox{ s.t. } \, \mathbf{Q}^{(r)} \mathbf{Q}^{(r)^\top} = \mathbf{I}_K, \, r=1, \dots,R
\label{eqn:qr}
\end{equation}
where $L_Q$ is the loss function
\begin{equation*}
L_Q \left( \boldsymbol{\Phi}^*, \boldsymbol{\Phi}^{(r)} \mathbf{Q}^{(r)} \right) = {\rm tr} \left\{ \left( \boldsymbol{\Phi}^{(r)} \mathbf{Q}^{(r)} - \boldsymbol{\Phi}^* \right)^\top \left( \boldsymbol{\Phi}^{(r)} \mathbf{Q}^{(r)} - \boldsymbol{\Phi}^* \right) \right\}.
\label{eqn:L_q}
\end{equation*}
The optimization is carried out by iterating two steps:
\begin{enumerate}
\item Minimize equation (\ref{eqn:qr}), for a given $\boldsymbol{\Phi}^*$ by computing the SVD of $\boldsymbol{\Sigma}_{\Phi^*} = \boldsymbol{\Phi}^{(r)} \boldsymbol{\Phi}^{*\top}$ and setting $ \tilde{\mathbf{Q}}^{(r)} = \mathbf{U}_r \mathbf{V}_r$, where $ \mathbf{U}_r$ and $\mathbf{V}_r$ are the two orthogonal matrices obtained by the SVD at MCMC iteration $r$,
\item Compute
$
\tilde{\boldsymbol{\Phi}}^{*(r)} = \frac{1}{R} \sum_{r=1}^{R} \boldsymbol{\Phi}^{(r)} \tilde{\mathbf{Q}}^{(r)}.
$
\end{enumerate}
The algorithm is then iterated using the updated value of $\tilde{\boldsymbol{\Phi}}^{*}$ in place of $\boldsymbol{\Phi}^*$. The search stops when subsequent estimates of $\boldsymbol{\Phi}$ are close enough.
This algorithm requires a starting value for $\boldsymbol{\Phi}^*$. \cite{assmann2016} suggests the last iteration of the Gibbs sampler as initial value for $\boldsymbol{\Phi}^*$. The same
procedure can be applied to each of the study-specific loading matrices.
This algorithm provides an approximate solution to identifiability, since the posterior distribution of the loading matrices is only known in approximate form. Yet, \cite{assmann2016}
show that it can be quite effective.
The OP procedure is iterative in nature. However, we verified that typically the first iteration is sufficient
to get close to the final estimate. Since the OP algorithm is computationally
demanding, the one-step version is recommendable.
All the results of this paper
have been obtained with a single iteration of the OP algorithm.
This point will be further examined for our setting in the following section.
\section{Simulation Results}
\label{sub:simulation}
In this section we use simulation experiments to assess BMSFA's ability to recover common and study-specific latent dimensions, by itself and in comparison to standard FA applied to the merged datasets.
We generate 50 datasets from the distributions specified in Table~\ref{tab:simX}. We fixed $\boldsymbol{\Phi}$, $\boldsymbol{\Lambda}_s$ and $\boldsymbol{\Psi}_s$ and thus $\boldsymbol{\Sigma}_s$.
\begin{table}[!t]
\scriptsize
\parbox{.45\linewidth}{
\begin{tabular}{c}
\hline \hline
$\mathbf{X}_s \sim \mbox{MVN}\left( \mathbf{0}, \boldsymbol{\Sigma}_s \right)$ \\
$\boldsymbol{\Sigma}_s = \boldsymbol{\Phi} \boldsymbol{\Phi}^\top+\boldsymbol{\Lambda}_s\boldsymbol{\Lambda}_s^\top+\boldsymbol{\Psi}_s$\\
fixed $\boldsymbol{\Phi}$ and $\boldsymbol{\Lambda}_s$: sparse matrices with $\approx$ 80 \% of zeros \\
fixed $\boldsymbol{\Phi}$ and $\boldsymbol{\Lambda}_s$: non zero elements drawn once from U$ (-1,1)$\\
fixed $\boldsymbol{\Psi}_s$: diagonal elements drawn once from U$(0,1)$\\
\hline \hline
\end{tabular}
\caption{\it \small Distributions used to generate observations in study $s$, for simulation experiments. }
\label{tab:simX}
}
\hfill \parbox{.45\linewidth}{\begin{tabular}{c}
\hline \hline
Common factor loadings : $\omega_{pk} \sim \Gamma\left(\frac{\nu =3}{2}, \frac{\nu =3}{2}\right)$ \\
Study-Specific factor loadings : $\omega_{pj_s} \sim \Gamma\left(\frac{\nu^s =3}{2}, \frac{\nu^s =3}{2}\right)$\\
$\delta_1 \sim \Gamma(a_1 = 2.1,1)$ and $\delta_l \sim \Gamma(a_2 = 3.1,1)$ with $l\geq 2$\\
$\delta_1^s \sim \Gamma(a_1^s = 2.1,1)$ and $\delta_l^s \sim \Gamma(a_2^s = 3.1,1)$ with $l\geq 2$\\
$\boldsymbol{\Psi}_s^{-1} \sim \Gamma ( a_\psi = 1, b_\psi = 0.3)$\\
\hline \hline
\end{tabular}
\caption{\it \small Prior distributions used in the simulation experiments and real data analysis.}
\label{tab:priSim}
}
\end{table}
We consider four scenarios differing in the number of studies, study sample sizes, and covariance structure (see Figure~\ref{fig: sig}). Scenarios~1 and~2 are similar to \cite{zhao2014}: $n_s$ is chosen to be smaller than $P$ to mimic large $P$ and small $n_s$ conditions while operating with a manageable set of variables for visualization and summarization. In the Scenario~3 we wish to model a situation where not all the studies have $P\gg n$. Moreover, in this scenario, study-specific factor loadings are large. The motivation behind this scenario is to investigate if our method recovers the shared biological signal in the presence of large study-specific or batch effects, and if it can isolate these sources. In Scenario~4 we closely mimic the data in Table~\ref{tab: dataset}, choosing $S=7$ and matching the sample sizes to those of Table~\ref{tab: dataset}.
Moreover, in Scenarios 1, 2 and 4 we randomly allocate the zeros in each column of $\boldsymbol{\Phi}$ and $\boldsymbol{\Lambda}_S$ (Table~\ref{tab:simX}), while in Scenario 3, we allocate zeros matching the central panel in the third row of Figure~\ref{fig: sig}.
\begin{figure}[!b]
\centering
\includegraphics[width=15cm]{heatSIG+RV.pdf}
\caption{\it \small Covariance matrices $\boldsymbol{\Sigma}_{\Phi}$ and their Bayesian estimates in four simulation scenarios. The right column shows the boxplots of RV coefficient between the true and the estimated $\boldsymbol{\Sigma}_{\Phi}$.}
\label{fig: sig}
\end{figure}
We run the Gibbs sampler for 15000 iterations with a burn-in of 5000 iterations. We set priors as in Table~\ref{tab:priSim}.
We first evaluate, for fixed latent dimension $K$ and $J_s$, BMSFA's ability
to recover the covariance component $\boldsymbol{\Sigma}_\Phi$ determined by the shared factors, as well as the shared factors' loadings $\boldsymbol{\Phi}$. For one randomly selected simulation dataset,
Figure~\ref{fig: sig} compares the true and estimated elements of $\boldsymbol{\Sigma}_\Phi$. We also present a summary of the analyses of 50 datasets.
To quantify the similarity between $\boldsymbol{\Sigma}_\Phi^{true}$ and $\widehat{\boldsymbol{\Sigma}}_\Phi$ we use the RV coefficient (Robert and Escouffer, 1976)\cite{} of similarity of two $P \times P$ matrices $\boldsymbol{\Sigma}_1$ and $\boldsymbol{\Sigma}_2$:
$$
RV (\mathbf{S}_1, \mathbf{S}_2) = \frac{tr ((\boldsymbol{\Sigma}_1 \boldsymbol{\Sigma}_2^\top) (\boldsymbol{\Sigma}_1 \boldsymbol{\Sigma}_2^\top)}{tr(\boldsymbol{\Sigma}_1 \boldsymbol{\Sigma}_1^\top)^2 tr (\boldsymbol{\Sigma}_2 \boldsymbol{\Sigma}_2^\top)^2}.
$$
RV varies in $[0, 1]$. The closer RV is to 1 the more similar the two matrices are.
\citet{smilde2008} argue that the RV coefficient can overestimate similarity between data sets
in high-dimensions, and propose a modified version that addresses this problem. We use it in Scenario~4, though differences will not be pronounced. The red boxplots in the right column of Figure~\ref{fig: sig} show the RV distributions across 50 simulations in our four scenarios.
Figure~\ref{fig: phipro} presents a similar analysis comparing the true factor loadings to their estimates obtained through posterior sampling and the OP procedure. The correlations between true and estimated values in both Figures~\ref{fig: sig}~and~\ref{fig: phipro}) are very high, suggesting that our estimands are well identified and our sampling approaches are appropriate.
\begin{figure}[!h]
\centering
\resizebox{1\textwidth }{!}{ %
\includegraphics[width=0.3cm]{procrust_simula.pdf}
}
\caption{\it Heatmap of the true (left) and estimated (center) shared factor loadings $\boldsymbol{\Phi}$ in the four scenarios of Figure~\ref{fig: sig}. In Scenario 4 we only show common factor loadings $\geq 0.5$. The right column displays boxplots of correlations between the true and estimated common factor loadings over 50 datasets for each scenario.}
\label{fig: phipro}
\end{figure}
Next we compare BMSFA to a Bayesian FA, using the same prior distribution. For Bayesian FA, we stacked all studies into a single dataset, ignoring that samples originate from distinct studies. The RV coefficients for BMSFA are systematically greater than FA's (Figure~\ref{fig: sig}, right column), demonstrating that BMSFA recovers shared factors better than a merged analysis. In Scenario~3 the gap is more pronounced, as study-specific factor loadings are large. In most simulations, FA captures study-specific effects that are not actually shared. BMSFA recovers the shared signal better. Also, the distribution of BMSFA's RV coefficient is narrower than FA's. This comparison illustrates that BMSFA identifies the shared signal across the studies and improves its estimation compared to standard Bayesian FA. Moreover, the BMSFA estimations are more efficient compared to the FA estimation, due to the beneficial effects of removing the study-specific components that lack cross-study reproducibility.
So far we took $K$, the number of shared factors, and $J_s$'s, the numbers of study-specific factors, to be known. We next focus on the latent dimensions calculated via SVD of matrices $\boldsymbol{\Sigma}_\Phi$ and $\boldsymbol{\Sigma}_{\Lambda_s}$, as described earlier, and using an eigenvalue threshold of 0.05. The simple adaptive method described in \S \ref{sub:nfac} for latent factor selection, common $K$ and specific $J_s$, proved to be extremely robust respect to the choice of this threshold. Conclusion with a threshold of 0.1 was the same. We choose a lower value as are more concerned to lose important shared biological factors than to include additional shared factors.
\begin{figure}[!t]
\centering
\resizebox{1\textwidth }{!}{ %
\includegraphics[width=1.6cm]{scelta.pdf}
}
\caption{\it Dimensions of shared and study specific factors in the four scenarios. Model selection procedure for the shared $K$ and the study-specific $J_s$ latent dimension via SVD of $\boldsymbol{\Sigma}_\Phi$ and $\boldsymbol{\Sigma}_{\Lambda_s}$. The true dimensions are visualized by the dashed lines.}
\label{fig: scelta}
\end{figure}
Figure~\ref{fig: scelta} shows the results obtained by fitting the model for 50 different data sets generated from the BMSFA with $K = 3$ in the four different scenarios. The vertical lines show the 50 estimated latent dimensions in each data set. Our method consistently selects the right dimensions for both the shared and the study-specific factors.
The simulation analysis highlights the merit of our method in a variety of scenarios, with improved performance over FA in terms of covariance matrices estimation in multi-study settings, estimation of the reproducible signal across studies, and identification issue for the factor loading matrix.
\section{Breast Cancer Case Study}\label{sec:CS}
The aim of this analysis is to identify shared common factors describing the common correlation structure across the 7 breast cancer microarray studies listed in Table~\ref{tab: dataset}. Recovering shared gene co-expression patterns from different high-throughput studies is important to identify replicable genetic regulation. This case study considers a relatively well understood area of cancer biology and provides a realistic positive control for the BMSFA methodology.
We consider genes measured in all studies and remove the $50\%$ of genes with the lowest variance. We use the prior of Table~\ref{tab:priSim}.
Our method chooses a shared latent dimension of $K=8$, through the SVD of $\boldsymbol{\Sigma}_\Phi$. We first summarize and visualize the shared co-expression patterns via a co-expression network (Figure~\ref{fig: cluster}) built on $\boldsymbol{\Sigma}_{\Phi}$, and thus representing all studies. A gene co-expression network is an undirected graph. Each node corresponds to a gene and each edge represents a high co-expression between genes. The importance of genes in a cluster is represented by the node size.
\begin{figure}[!t]
\centering
\includegraphics[height=14.26cm, trim={0cm 5.2cm 0cm 5.4cm},clip]{7study_geneco.pdf}
\caption{\it \small Shared gene co-expression network across the 7 studies of Table~\ref{tab: dataset}. We include edges between two genes if the corresponding element in the shared part of the covariance matrix is greater than 0.5. Edges in blue (orange) represent positive (negative) associations.
}
\label{fig: cluster}
\end{figure}
Our analysis identifies five larger clusters. Co-expressed genes tend to be members of the same, highly plausible, biological pathways. All clusters are associated with biological processes known for explaining heterogeneity of expression across breast cancers, lending credibility to BMSFA.
The first cluster is driven by expression of the estrogen receptor (ESR1), which historically is one of the earliest cancer biomarkers to have been discovered, and plays a crucial role in the biology and treatment of breast cancer \citep{jordan2007, robinson2013}. High dimensional expression pattern are found in \cite{sorlie2001}. Many studies have shown the relation of ESR1 with growth of cancer \citep{osborne2001, iorio2005, toy2013}. Levels of ESR1 expression are associated with different outcomes \citep{ross2012, theodorou2013}. Three other genes stand out: GATA3, XBP1 and FOXA1. These are ESR1-cooperating transcription factors altered in breast tumors \citep{lacroix2004, theodorou2013}. In breast cancer cell, many studies revealed strong and positive association of GATA3, XBP1 and FOXA1 with ESR1 \citep{hoch1999, sotiriou2003, sorlie2003, lacroix2004, lai2013, theodorou2013}.
The second cluster is related to the cell cycle. One of the most important genes in this cluster is CCNB1, which encodes cyclin B. Cyclins are prime cell cycle regulators. Many analyses found a common pattern of overexpression of the mitotic cyclins A and B and their dependent kinase in the tumor cell of breast cancer \citep{keyomarsi1993, lin2000, basso2002}. Two other important genes in this cluster are CDK1, a kinase dependent on cyclins, and CDC20, a gene related to the metaphase and anaphase of cell cycle.
All genes in the third cluster are related to regulation of the immune response. The CD genes are important for the immune system pathway and the HLA genes are a crucial element for immune function.
The fourth cluster includes several genes expressed by the connective tissue, including collagen genes (COL1A1, COL1A2, COL3A1, COL5A1, COL5A2, COL10A1, COL11A1), previously associated with stromal cells \citep{ross2000, ioachim2002}. Of note are also ADAM, a protease related to the degradation of the connective tissue, and smooth muscle cell marker TAGLN, also previously found to play a role in breast cancer.
Finally, all the RP genes in the fifth cluster codify the ribosome, which synthesizes proteins. Dysregulation of Ribosome function is related to tumor progression in breast cancer \citep{belin2009}.
To further explore the patterns found in the shared gene co-expression network, we estimate the shared factor loadings after the post-process procrustes algorithm.
The heatmap in Figure~\ref{fig: heatmap} depicts the
estimates of the shared factor loadings
that can be identified reproducibly across the studies.
\begin{figure}[!t]
\centering
\includegraphics[height=5.26cm]{phi_BC1.jpg}
\caption{\it \small Heatmap of the estimated shared factor loadings obtained with BMSFA across the 7 studies in Table~\ref{tab: dataset}. We only show common factor loadings $\geq 0.5$.}
\label{fig: heatmap}
\end{figure}
To extract biological insight from the shared factors, we explore whether specific gene sets are enriched among the loadings using Gene Set Enrichment
Analysis (GSEA) \citep{Mootha2003}. We used the package \texttt{RTopper} in \texttt{R} in \texttt{Bioconductor}, following the method of \citet{tyekucheva2011} and considering all the gene sets representing pathways from {\tt reactome.org}. The resulting analysis shows concordant results with the pathways obtained with the shared gene co-expression network, further suggesting that we identify genuine biological signal. The first shared factor is significantly enriched with the ``Cell communication" and ``Cell cycle" pathways. The second factor is associated with the Immune system pathway and all the sub-pathway included in it, namely the ``Adaptive Immune System", ``Innate Immune System" and ``Cytokine Signaling in Immune System". Factor 3 shows a significant association with cell cycle, namely with the pathway ``Cell cycle", ``Cell cycle mitotic", ``Cell cycle checkpoints", ``Regulation of mitotic cell cycle". The shared 5, 6 and 7 factors have protein production ``Transport of ribonucleoproteins into the host nucleus", ``Protein folding", ``Mitochondrial protein import", ``Metabolism of proteins", ``NRIF signals cell death from the nucleus". Finally, factor 8 is related to the ER pathway, ``ER phagosome pathway", ``Interferon signaling", and ``Interferon alpha beta signaling".
An important feature of BMSFA in this case study is regularization of the common factor loadings. To illustrate this in more detail, we conclude this section comparing BMSFA to the MSFA which uses MLE for parameter estimation. The data consists of 63 genes in the Immune System Pathway. Their loadings are compared in Figure~\ref{fig: MLEbayes}.
\begin{figure}[!t]
\centering
\includegraphics[width=5.2cm]{plotMLE1.pdf}
\includegraphics[width=5.2cm]{plotMLE2.pdf}
\caption{\it Left: Comparison between the first two estimated common factor loadings with MLE and BMSFA. Right: Comparison between the first two estimated factor loadings with MLE, followed by varimax rotation, and BMSFA.}
\label{fig: MLEbayes}
\end{figure}
BMSFA regularizes common factor loadings by shrinking small and moderate MLE loadings to zero while systematically amplifying larger MLE loadings (Figure~\ref{fig: MLEbayes}, left panel). This regularization behavior is somewhat unique to this setting, as it is far more common for regularization to only result in shrinkage. Here, the prior helps the posterior perform a factor rotation method which results in more sparse factors. To further illustrate we rotate the loadings obtained with the MLE using the varimax rotation \citep{kaiser1958} and we compare it with the BMSFA (Figure~\ref{fig: MLEbayes} right panel). The BMSFA loadings are far more similar to the varimax rotated MLE (correlation 0.97) than the original MLE (correlation 0.7).
\section{Discussion}
In this paper we propose a general Bayesian framework for the unsupervised analysis of high-dimensional biological data across multiple studies, building upon~\cite{DV2016}. We address the unmet need to rigorously model replicable signal across studies, while at the same time capturing study-specific variation. Our approach is not limited by $P\ll n$ and, in addition to replicability, shows considerable promise in modeling sparsity and enhancing interpretability via rotation-like shrinkage. Building on \cite{bhattacharya2011} we propose a computationally efficient MCMC algorithm.
The work in this paper is motivated by identifying replicable signal in unsupervised genomic applications. The results of our analysis of breast cancer gene expression across seven studies identified shared gene patterns, which we also represented via clusters in a co-expression network. Both factors and clusters are clearly related to well-known breast cancer pathways. Our analytic tools allows investigators to focus on the replicable shared signal, after properly accounting for, and separating, the influence of study-specific variation. While we focused on the shared signal, study-specific loadings can also be examined directly.
BMSFA may have broad applicability in a wide variety of genomic platforms, including microarrays,
RNA-seq, SNP-based genome-wide association studies, proteomics, metabolomics, and epigenomics. Relevance is also immediate in other fields of biomedical
research, such as those generated by exposome studies, Electronic Medical Record (EMR), or
high dimensional epidemiological data. In dietary pattern analysis, it is important to find replicable dietary patterns in different populations \citep{castello2016}.
Our analysis could be applied to check if there are shared dietary patterns across different populations and to detect the study-specific dietary patterns of a particular population. In this field generally, it is common to apply a varimax rotation to factor loading matrix, for a better interpretation. Specifically, the interpretation of a factor relies on loadings. The interpretation of the model is simplified if more of the loadings are shrunk towards zero and the factor is defined by few large loadings. In the frequentist analysis, this is possible by rotation methods, such as varimax. In our representation, the BMSFA embeds this step giving an immediate representation of the two sparse factor loading matrices through the shrinkage prior, as shown in Section~\ref{sec:CS}.
Our Bayesian non-parametric approach offers more flexibility in the choice of the dimensionality of shared latent factors. Moreover, we provide shrinkage of the latent factor loadings, enhancing the role of the variables that are most important in each factor.
To address the choice of model dimension, we developed, building on \cite{bhattacharya2011}, a practical procedure based on separate SVD of the shared covariance part and the study-specific covariance parts. The choice of the number of factors remains an important open problem. The most common method for choosing latent dimension fits the factor model for different choices of $K$ and compares them using selection criteria such as BIC. This approach presents many problems especially in a $p \gg n$ setting where MLE is not duable. \cite{lopes2004} proposed a reversible jump MCMC to estimate the number of factors in standard FA, but this method is also often computationally intensive. \citet{bhattacharya2011} developed an interesting adaptive scheme that dynamically changes the dimension of the latent factors as the Gibbs sampling progresses. In our approach, we develop a practical approach where we have a balance between retaining important factors and removing the redundant ones.
We also address identification. Identifiability remains a challenge in standard FA. In the Bayesian approach, constraints were proposed to tackle this issue, such as that of a block lower triangular matrix \citep{lopes2004, carvalho2008}. As \cite{carvalho2008} noticed, in this constraint different ordering of variables could lead to different conclusions. In our work, we adopt a procrustes algorithm and demonstrate through a series of simulation analyses that this method applied to the BMSFA is effective. \citet{rovckova2016} solves this problems in a Bayesian context by rotating the factor loadings matrix with the varimax rotation \citep{kaiser1958}. We also compared the BMSFA estimates after the procrustes algorithm with the MLE after rotating the common factor loadings. The resulting analysis are close, demonstrating that the prior we adopt works similarly to a rotation.
We hope BMSFA will encourage joint analyses of multiple high-throughput studies in biology, and contribute to alleviating the current challenges in replicability of unsupervised analyses in this fields and across data science.
\clearpage
\bibliographystyle{apa}
|
2,869,038,155,650 | arxiv | \section{Introduction and Preliminaries}
The notion of fractal interpolation function (FIF in what follows) has proved to be an attractive strategy to produce interpolants and approximants for a wide class of problems. The basic setting of FIF as defined by Barnsley \cite{MF1} stems from the concept of iterated function system (IFS), one of the most popular methods of generating fractals; see, for instance, \cite{Hut}. The book \cite{PRM2} and the monograph \cite{PRM1} are good references for fractal functions and related areas. The theory of fractal interpolation is an active research topic in the field of fractal approximation theory, as shown, for example in \cite{BL,CK,DCL,MAN2,Ri,S, WY}. In what follows, we shall hint at the technical details concerning the notion of FIF; the readers are referred to \cite{MF1} for more details.
\par
Let $\mathbb{N}_r$ denote the set of first $r$ natural numbers. As is customary, we shall denote by $\mathcal{C}(I)$ the Banach space of all real-valued continuous
functions on a closed bounded interval $I$, endowed with the supremum norm.
\par
Let $\{(x_i,y_i): i=0,1,2,...,N\}$, $N \ge 2$ be a data set with strictly increasing abscissae. Set $I=[x_0,x_N]$ and $I_i= [x_{i-1},x_i]$ for $i \in \mathbb{N}_N$. For every $i \in \mathbb{N}_N$, suppose that $L_i: I \to I_i$ is a contractive increasing homeomorphism and $F_i: I \times \mathbb{R} \to \mathbb{R} $ is a map that is continuous, contractive with respect to the second variable and such that
$$F_i(x_0,y_0) = y_{i-1}, \quad F_i(x_N,y_N)= y_i.$$
For $i \in \mathbb{N}_N$, define
$$W_i(x,y) = \Big(L_i(x),F_i(x,y) \Big) ~~\forall ~~(x,y) \in I \times \mathbb{R}.$$
The system $\{I \times \mathbb{R}: W_i, i \in \mathbb{N}_N\}$ is an IFS and it has a unique attractor which is the graph of a continuous function $g: I \to \mathbb{R}$ such that $g(x_i)=y_i$ for every $i=0,1,\dots, N$ and
$$g(x) = F_i \Big (L_i^{-1}(x),g\big(L_i^{-1}(x)\big) \Big) ~~\forall ~~x \in I_i.$$
The function $g$ is called a FIF \cite{MF1} and it has the property that its graph is self-referential, that is, the graph is a union of transformed copies of itself. The main
difference with the classical interpolants resides in the definition by the aforementioned functional equation endowing self-referentiality to the interpolant $g$.
\par
Navascu\'{e}s explored the idea of fractal interpolation further to associate a class of fractal functions with a prescribed function $f$ in $\mathcal{C}(I)$ as follows \cite{MAN1}. Consider a partition $\Delta:=\{x_0,x_1, \dots,x_N\}$ of $I=[x_0,x_N]$ such that $x_0<x_1<\dots< x_N$. For $i \in \mathbb{N}_N$, let
$$L_i(x)=a_ix+b_i, \quad F_i(x,y) = \alpha_iy + f \big( L_i(x) \big)-\alpha_i b(x),$$
where $b \neq f $ is such that
$$b(x_0) = f(x_0), \quad b(x_N) = f(x_N),$$ and $\alpha:= (\alpha_1, \alpha_2, \dots, \alpha_N) \in (-1,1)^N$. The corresponding FIF denoted by $f_{\Delta,b}^\alpha$ or simply as $f^\alpha$ (for notational convenience) is called an \emph{$\alpha$-fractal function} and it satisfies the self-referential equation
$$f_{\Delta,b}^\alpha(x) = f(x) + \alpha_i (f_{\Delta,b}^{\alpha}- b)\big(L_i^{-1}(x)\big) ~ ~ ~~\forall~~ x \in I_i,~~ i \in \mathbb{N}_N.$$
The function, or rather the class of functions, $f_{\Delta,b}^\alpha$ may be treated as fractal perturbation of the original function $f$, termed the \emph{germ function} or \emph{seed function}. Note that the perturbation process involves three elements: the \emph{partition} $\Delta$ of the domain $I$, function $b$ that is referred to as the \emph{base function} and vectorial parameter $\alpha$ termed \emph{scaling vector}. Taking advantage of the scaling vector, fractal interpolation is more robust than the classical piecewise interpolation.
\par
As was observed by Navascu\'{e}s \cite{MAN2}, a particular interesting case arises if one chooses the base function $b=Lf$, where $L: \mathcal{C}(I) \to \mathcal{C}(I)$ is a bounded linear map. With fixed choices of $\Delta$, $\alpha$ and $L$, one can define an operator $\mathcal{F}_{\Delta,L}^\alpha$ denoted for simplicity as $\mathcal{F}^\alpha$ that assigns $f_{\Delta,L}^\alpha$ to $f$:
\begin{equation}\label{eq0}
\mathcal{F}_{\Delta,L}^\alpha: \mathcal{C}(I) \to \mathcal{C}(I), \quad \mathcal{F}_{\Delta,L}^\alpha(f)= f_{\Delta,L}^\alpha,
\end{equation}
referred to as the \emph{$\alpha$-fractal operator.} Let $$|\alpha|_\infty:= \max \big\{ |\alpha_i|: i \in \mathbb{N}_N\big\}.$$ The following properties of the $\alpha$-fractal operator $\mathcal{F}_{\Delta,L}^\alpha$ are well-known; see, for instance, \cite{MAN2,Rational}.
\begin{theorem}\textup{\cite[Theorem 2.2]{Rational}}. \label{prelthm} Let $Id$ be the identity operator on $\mathcal{C}(I)$.
\begin{enumerate}
\item The fractal operator $\mathcal{F}_{\Delta,L}^{\alpha}: \mathcal{C}(I) \to \mathcal{C}(I)$ is a bounded linear map. Further, the operator norms satisfy the following inequalities $$ \|Id-\mathcal{F}_{\Delta,L}^{\alpha}\| \leq \frac{| \alpha|_{\infty} }{1-| \alpha|_{\infty}} \| Id - L\|,\quad \| \mathcal{F}_{\Delta,L}^{\alpha}\| \leq 1+ \frac{| \alpha|_{\infty} }{1-| \alpha|_{\infty}} \| Id - L\|.$$
\item For $| \alpha |_{\infty} < \|L\|^{-1}$, $\mathcal{F}_{\Delta,L}^{\alpha}$ is bounded below. In particular, $\mathcal{F}_{\Delta,L}^{\alpha}$ is an injective map.
\item For $| \alpha|_{\infty} < \|L\|^{-1}$, the fractal operator $\mathcal{F}_{\Delta,L}^{\alpha}$ is not a compact operator.
\item If $| \alpha|_{\infty} < \big(1 + \|Id - L\|\big)^{-1}$, then $\mathcal{F}_{\Delta,L}^{\alpha}$ is a topological isomorphism (i.e., a bijective bounded linear map with a bounded inverse). Moreover, $$\| (\mathcal{F}_{\Delta,L}^{\alpha})^{-1}\| \leq \dfrac{1+|\alpha|_{\infty} }{1-|\alpha|_{\infty} \|L\|}.$$
\end{enumerate}
\end{theorem}
\par Navascu\'{e}s and coworkers approached the construction of new classes of functions in $\mathcal{C}(I)$ by taking the image of the popular approximation classes of functions such as polynomials, trigonometric and rational functions under the fractal operator $\mathcal{F}_{\Delta,L}^\alpha$ \cite{MAN1,MAN3,Rational}. These new functions defined as perturbations of the classical may preserve properties of the latter or display new characteristics such as non-smoothness or quasi-random behavior. These fractal maps tends to bridge the gap between the smoothness of the classical mathematical objects and the pseudo-randomness of the experimental data, breaking in this way their apparent contradiction \cite{MAN4}. Motivated and influenced by the aforementioned works, in this article we define the class of \emph{fractal rational trigonometric functions} and study some approximation aspects of the same. In this way, we approach the classical problems of periodicity and approximation from a fractal viewpoint. Besides providing the motivation for our researches reported herein, the works in \cite{MAN1,MAN3,Rational} also offered us an
array of basic tools which we have modified and adapted.
\par In Theorem \ref{error1a} we provide an upper bound for the fractal rational trigonometric minimax error, that is, the smallest error in approximating a prescribed continuous function by a fractal rational trigonometric function in the uniform norm. Our approach to the fractal minimax error also points out that the upper bound for the minimax error in approximating a continuous function by a fractal rational function as announced in \textup{\cite[Theorem 4.3]{Rational}}
does not hold. The second author regrets to inform that this error may invalidate some results given as corollaries of \textup{\cite[Theorem 4.3]{Rational}}. However, as stated in the analogous theorem in this paper (see, Theorem \ref{error1a}), there is a way to fix this mistake by including an additional term. The incorrect arguments in \textup{\cite[Theorem 4.3]{Rational}} is subsequently carried over almost verbatim in \textup{\cite[Theorem 3.9]{Vij2}}. However, we note that the result stated in \textup{\cite[Theorem 3.9]{Vij2}} remains valid and provide a correct proof for it.
\par
The following upper bound for the uniform distance between the original function $f$ and its fractal version $f_{\Delta,b}^\alpha$ can be obtained; see, for instance, \cite{MAN1,MAN2}:
\begin{equation} \label{eq1}
\|f_{\Delta,b}^\alpha-f\|_\infty \le \frac{|\alpha|_\infty}{1-|\alpha|_\infty} \|f-b\|_\infty.
\end{equation}
The previous estimate reveals that by choosing the scaling vector $\alpha$ such that $|\alpha|_\infty$ is close enough to zero or by selecting the base function $b$ near to $f$, the perturbed fractal function $f_{\Delta,b}^\alpha$ can be made sufficiently close to the original seed function $f$. In particular, if $\alpha^m \in \mathbb{R}^N$, $|\alpha^m|_\infty<1$ and $\alpha^m \to 0$ as $m \to \infty$, then $f_{\Delta,b}^{\alpha^m} \to f$ uniformly as $m \to \infty$.
\par Since FIFs do not possess a closed form expression, standard methods such as the Taylor
series analysis, Cauchy remainder form, and Peano kernel theorem (see, for instance, \cite{PJD}) may not be easily
adapted for the convergence analysis of fractal interpolants and approximants. Instead, in the literature, the closeness of a fractal approximant $f_{\Delta,b}^\alpha$ (which is perturbation of a classical approximant $f$) to the original function $\Phi$ is established using the closeness of $f$ to $\Phi$ via the following triangle inequality:
\begin{equation} \label{eq2}
\|\Phi -f_{\Delta,b}^{\alpha}\|_\infty \le \|\Phi-f\|_\infty + \|f_{\Delta,b}^{\alpha}- f\|_\infty.
\end{equation}
The second term in the right hand side of the Inequality (\ref{eq2}) can be bounded via (\ref{eq1}) to conclude that for the scaling vector $\alpha$ with small enough value of $|\alpha|_\infty$, the error in approximating $\Phi$ with $f_{\Delta,b}^\alpha$ is small, whenever $f$ is a good approximant to $\Phi$. As various fractal interpolants studied in the literature can be realized as fractal perturbation of their classical counterparts, a similar comment holds for their convergence (see, for example, \cite{CK,MAN5}). Note that the scaling vector $\alpha$ has the most influence on the fractal dimension of the graph of $f_{\Delta,b}^\alpha$ and hence on the ``roughness" of the function $f_{\Delta,b}^\alpha$. For instance, we have the following proposition given in \cite{Akhtar}.
\begin{theorem}\textup{\cite[Corollary 3.1]{Akhtar}}.\label{Akhtar}
Let $f$ and $b$ be Lipschitz continuous functions defined on $I$ with $b(x_0)=f(x_0)$ and $b(x_N)=f(x_N).$ Let $\Delta=\{x_0,x_1,\dots , x_N\}$ be a partition of $I=[x_0,x_N]$ satisfying $x_0<x_1< \dots < x_N$ and $ \alpha =(\alpha_1,\alpha_2, \dots ,\alpha_{N}) \in (-1,1)^N.$ If the data points $ \{(x_i, f(x_i)): i =0,1 \dots, N\}$ are not collinear, then the graph of the $\alpha$-fractal function $f_{\Delta,b}^{\alpha}$ denoted by
$G_{f_{\Delta,b}^\alpha}$ has the box dimension
\begin{equation*}
\dim_B(G_{f_{\Delta,b}^\alpha})=
\begin{cases} D, \text{ if $\sum_{i=1}^{N} |\alpha_i| > 1$}\\
1, \text{ otherwise ,}
\end{cases}
\end{equation*}
where $D$ is the solution of $\sum_{i=1}^{N} |\alpha_i|a_i^{D-1}=1.$
\end{theorem}
Therefore, it appears that the roughness in the constructed fractal interpolant (approximant) and the convergence (closeness) to the original function may not be simultaneously achieved.
\par
One can circumvent this by exploring the choices of other parameters in the construction of the fractal function $f_{\Delta,b}^\alpha$. For instance, a look back at the estimate in (\ref{eq1}) should convince the reader that with any permissible choice of $\alpha$, the fractal function $f_{\Delta,b}^\alpha$ is close to the seed function $f$, provided the base function $b$ is close enough to $f$. In particular, if $(b_n)_{n \in \mathbb{N}}$ is a sequence of base functions satisfying $b_n(x_0)=f(x_0)$ and $b_n(x_N)=f(x_N)$ for all $n\in \mathbb{N}$ and $b_n \to f$ as $n \to \infty$, then $f_{\Delta,b_n}^\alpha \to f$ uniformly as $n \to \infty.$ For instance, one can take $b_n= B_n(f)$, where $B_n: \mathcal{C}(I) \to \mathcal{C}(I)$ defined by
$$B_nf (x) = \sum_{k=0}^n f\Big(x_0+ \frac{k}{n}(x_N-x_0)\Big) {n \choose k} \frac{(x-x_0)^k (x_N-x)^{n-k}}{(x_N-x_0)^n},$$
is the classical Bernstein operator. This simple but noteworthy observation was exploited to define what is called \emph{Bernstein $\alpha$-fractal functions} corresponding to $f$, denoted by $f_{\Delta, B_n(f)}^\alpha =f_n^\alpha$ \cite{Vij1}. Theorem $2$ in reference \cite{Vij1}, which reads as follows, contains an error in the statement.
\begin{theorem}\textup{\cite[Theorem 2]{Vij1}}.
Let $\Delta=\{x_0,x_1,\dots , x_N\}$ be a partition of $I=[x_0,x_N]$ satisfying $x_0<x_1< \dots < x_N$ and $ \alpha =(\alpha_1,\alpha_2, \dots ,\alpha_{N}) \in (-1,1)^N.$ For an irregular function $f \in \mathcal{C}(I)$ if all the Bernstein $\alpha$-fractal functions in the sequence $(f_n^\alpha)_{n \in \mathbb{N}}$ are obtained with a fixed choice of scaling vector $\alpha$ whose components satisfy $\sum_{i=1}^N |\alpha_i| >1$, then all the Bernstein $\alpha$-fractal functions in the sequence $(f_n^\alpha)_{n \in \mathbb{N}}$ have the same fractal dimension and $f_n^\alpha \to f$ uniformly as $n \to \infty.$
\end{theorem}
The author claims that the above theorem follows from Theorem \ref{Akhtar}, but overlooked that Theorem \ref{Akhtar} needs additional assumptions, for instance, the seed function $f$ has to be Lipschitz. We shall refine this theorem by inserting the required additional condition. While the original proof holds with this additional assumption, we also provide a revised proof that shows that the assumption is not needed. We take the opportunity to observe that similar adjustments must be made to the identical results appeared elsewhere; see, for instance, \textup{\cite[Theorem 2.3]{Vij2}} and \textup{\cite[Theorem 2]{Vij3}}.
\par
Let the partition $\Delta$ and scale vector $\alpha$ be fixed. If a suitable sequence $(b_n)_{n \in \mathbb{N}}$ is used in the place of a single base function $b$, then corresponding to a fixed $f \in \mathcal{C}(I)$ we obtain a family of fractal functions $\{f_{\Delta,b_n}^\alpha:n \in \mathbb{N}\}$. In this case, corresponding to the fractal operator $\mathcal{F}_{\Delta,b}^\alpha=\mathcal{F}^\alpha$ in Equation (\ref{eq0}) one obtains a multi-valued operator or a set-valued operator
\begin{equation}\label{eq3}
\mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I), \quad \mathcal{F}^\alpha(f)= \{f^\alpha_{\Delta,b_n}: n\in \mathbb{N}\}.
\end{equation}
In \textup{\cite[Theorem 3]{Vij3}} the author attempts to prove that the aforementioned set-valued operator is linear and bounded. It seems that in the proof, the operator is treated as a single-valued operator. This is the case, for instance, for each fixed $n \in \mathbb{N}$. However, to the best of our knowledge the linearity and boundedness of a multi-valued operator need to be approached in a different way. A \emph{closed convex process} is treated as a set-valued analogue of a continuous (bounded) linear operator in the sense that a closed convex processes enjoy almost all properties of continuous linear operators, including the open mapping theorem, closed graph theorem and uniform boundedness principle; see, for instance, \cite{Aubin}. We prove that the multi-valued fractal operator in (\ref{eq3}) is a \emph{process} and \emph{Lipschitz} but not \emph{linear}, where linearity is interpreted in an appropriate sense. Similarly, in \textup{\cite[Theorem 4]{Vij3}}, with a suitable assumption on the scaling vector, the author attempts to prove that the multi-valued operator $\mathcal{F}^\alpha$ in (\ref{eq3}) is bounded below, but not compact. But, in the ``proof" of this theorem, $\mathcal{F}^\alpha$ is treated as a single-valued operator. A possible explanation of this could be that the author deals, or rather intends to deal, with the single-valued operator $\mathcal{F}_n^\alpha: \mathcal{C}(I) \to \mathcal{C}(I)$ defined by $\mathcal{F}_n^\alpha(f)=f^\alpha_{\Delta,b_n}$ for each fixed $n \in \mathbb{N}$. Another observation worth noting regarding \textup{\cite[Theorem 4]{Vij3}} is that in case one intends to handle the single-valued operator $\mathcal{F}_n^\alpha: \mathcal{C}(I) \to \mathcal{C}(I)$, thanks to items (2) and (3) of Theorem \ref{prelthm} above, the assumptions on the scaling vector made in \textup{\cite[Theorem 4]{Vij3}} can be dropped. We collect all these refinements that act as a corrigendum to \cite{Vij1,Vij2,Vij3} in the last section.
\section{Fractal Rational Trigonometric Functions}
We consider here the space of $2\pi$-periodic continuous functions $$ \mathcal{C}(2\pi)= \big\{f:[-\pi,\pi] \rightarrow \mathbb{R}; f ~ \text{is ~continuous}, f(-\pi)=f(\pi)\big\}.$$ Let $\mathfrak{T}_m(2 \pi)$ be the set of trigonometric polynomials of degree at most $m.$ Recall that $\mathfrak{T}_m(2 \pi)$ is linearly spanned by the set $$\big\{1, \sin x,\cos x,\sin 2x,\cos 2x,\dots , \sin mx,\cos mx\big\}.$$ In fact, this family constitutes a basis for $\mathfrak{T}_m(2 \pi)$ and this system is orthogonal with respect to the standard inner product
$$ \langle f, g \rangle := \int_{- \pi }^\pi f(x) g(x) \mathrm{d}x.$$
Let $\Delta: -\pi=x_0<x_1< \dots < x_N=\pi $ be a partition of the interval $I=[-\pi,\pi].$
The following class of functions is introduced in \cite{MAN3}.
\begin{definition}\textup{\cite[Definition 4.1]{MAN3}}.\label{basicdef1}
Let $m$ be a nonnegative integer. We define the set of $\alpha-$fractal trigonometric polynomials of degree at most $m$ denoted by $\mathfrak{T}_m^{\alpha}(2 \pi)$ as $\mathcal{F}_{\Delta,L}^{\alpha}\big(\mathfrak{T}_m(2 \pi)\big)$, where $\mathcal{F}_{\Delta,L}^\alpha= \mathcal{F}^\alpha$ is the (single-valued) $\alpha$-fractal operator defined in (\ref{eq0}). An element in $\mathfrak{T}_m^\alpha(2 \pi)$ is referred to as an $\alpha$-fractal trigonometric polynomial or simply as a fractal trigonometric polynomial. Further, the set of all $\alpha$-fractal trigonometric polynomials is defined as $\mathfrak{T}^\alpha (2 \pi)= \cup_{m}\mathfrak{T}_m^{\alpha}(2 \pi)$.
\end{definition}
Similar to the class of trigonometric polynomials, one can define rational trigonometric functions in $\mathcal{C}(2 \pi)$ as follows.
For $m, n \in \mathbb{N}\cup \{0\}$, let $$\mathfrak{R}_{mn}(2 \pi):= \Big\{t=\frac{p}{q}: p \in \mathfrak{T}_m, q \in \mathfrak{T}_n~\text{and}~ q>0 ~ \text{on} ~ [-\pi, \pi] \Big\},$$ the set of all real-valued rational trigonometric functions of type $(m,n)$ and $$ \mathfrak{R}(2\pi)= \cup_{m,n} \mathfrak{R}_{mn}(2\pi).$$
Following the construction of fractal versions of classical functions such as polynomials, trigonometric functions and rational functions \cite{MAN2,MAN3,Rational}, in the upcoming definition we apply the fractal operator to map the class of rational trigonometric functions to its fractal counterpart.
\begin{definition}
For $m, n \in \mathbb{N}\cup \{0\}$ we define the class of \emph{$\alpha-$fractal rational trigonometric functions} of type $(m,n)$ denoted by $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ as the image of $\mathfrak{R}_{mn}(2 \pi)$ under the fractal operator $\mathcal{F}_{\Delta,L}^{\alpha}$. That is,
$$\mathfrak{R}_{mn}^\alpha(2\pi):= \mathcal{F}_{\Delta,L}^{\alpha}\big(\mathfrak{R}_{mn}(2 \pi)\big).$$
Further, we let
$$\mathfrak{R}^{\alpha}(2\pi)= \mathcal{F}_{\Delta,L}^{\alpha}\big(\mathfrak{R}(2\pi)\big),$$ the set of all $\alpha-$fractal rational trigonometric functions.
\end{definition}
\begin{remark}
We can also define a new class of $\alpha-$fractal rational trigonometric functions in the following way $$\mathfrak{S}^{\alpha}_{mn}(2\pi)= \Big\{\frac{p^{\alpha}}{q^{\alpha}}: p^{\alpha} \in \mathfrak{T}_m^{\alpha}, q^{\alpha} \in \mathfrak{T}_n^{\alpha}~\text{and}~ q^{\alpha}>0 ~ \text{on} ~ I \Big\}.$$
For suitable choices of the scale vector, one can obtain $q^{\alpha}> 0$ on $I$ whenever so is $q$ \cite{PV2}. A difference in the two classes of fractal functions $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ and $\mathfrak{S}^{\alpha}_{mn}(2\pi)$ defined above is the following. It is evident that a function $r^\alpha$ in $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ satisfies the self-referential equation of the form
$$r_{\Delta,L}^\alpha(x) = r(x) + \alpha_i (r_{\Delta,L}^{\alpha}- Lr)\big(L_i^{-1}(x)\big) ~ ~ ~~\forall~~ x \in I_i,~~ i \in \mathbb{N}_N,$$
and its graph is the attractor of an IFS whereas it is not certain if a similar self-referentiality is applicable for a function in $\mathfrak{S}^{\alpha}_{mn}(2\pi)$. Please consult also Section \ref{nonselfssec} of this article.
\end{remark}
\begin{remark}
To obtain a more general class of FIFs, the constant scaling factors $\alpha_i \in (-1,1)$ can be replaced by functions $\alpha_i \in \mathcal{C}(I)$ such that
$ \|\alpha_i\|_\infty< 1$ for all $i \in \mathbb{N}_N$ \cite{WY}. Correspondingly, for a given $f \in \mathcal{C}(I)$, with $\alpha:= (\alpha_1, \alpha_2, \dots, \alpha_N)$, we can define an $\alpha$-fractal function $f_{\Delta,b}^\alpha=f^\alpha$ satisfying the self-referential equation
$$f^\alpha(x) = f(x) + \alpha_i(L_i^{-1}(x)) (f^{\alpha}- b)\big(L_i^{-1}(x)\big) ~ ~ ~~\forall~~ x \in I_i,~~ i \in \mathbb{N}_N.$$
In the sequel, we shall use constant scaling factors but we remark here that most of our results can be applied to the setting of variable scaling factors as well.
\end{remark}
\begin{example} Let $p(x)=27 \sum_{k=0}^{2}\sin(x_{k3})J_3^2(x -x_{k3})$ and $q(x)=19 + 8 \cos(3x)$ for $x \in [0,1] $. Here $x_{k3}=\frac{2k\pi}{3} $ for $k=0,1,2,$ and $J_3$ is the Jackson function (see, Section \ref{AEB}). The rational trigonometric function $r(x)=\dfrac{p(x)}{q(x)}$ is plotted in Fig. \ref{figFTR}(a). We consider the partition $\Delta: 0 < \frac{1}{10}<\frac{2}{10}< \dots < \frac{9}{10}<1$ and scale vector $\alpha$ with components $\alpha_i=0.9$ for $i \in \mathbb{N}_{10}$. Figs. \ref{figFTR}(b)-(c) correspond to the fractal rational trigonometric function $r_{\Delta,L}^\alpha$ with $L$ defined by
\begin{enumerate}[(i)]
\item $Lf= \nu f$, where $\nu(x)= 1+x(x-1)$
\item $Lf= f\circ \varphi$, where $\varphi(x)= x^3$.
\end{enumerate}
Fig. \ref{figFTR}(d) depicts two graphs one (red color) corresponds to the self-referential rational trigonometric function $r_{\Delta,L}^\alpha$ with parameters as in Fig.\ref{figFTR}(b) and the other (blue color) corresponds to $\dfrac{p_{\Delta,L}^{\alpha}(x)}{q_{\Delta,L}^{\alpha}(x)} $, where the fractal functions $p^\alpha$ and $q^\alpha$ are constructed with same $\Delta$ and $\alpha$ as before, and $L$ as in item (1) above.
\end{example}
\begin{figure}[h!]
\begin{center}
\begin{minipage}{0.47\textwidth}
\epsfig{file=FTR_orig.eps,scale=0.17}
\centering{(a)}
\end{minipage}\hfill
\begin{minipage}{0.47\textwidth}
\epsfig{file=FTR_class_1.eps,scale=0.17}
\centering{(b)}
\end{minipage}\\
\begin{minipage}{0.47\textwidth}
\epsfig{file = FTR_nbase.eps,scale=0.17}
\centering{(c)}
\end{minipage}\hfill
\begin{minipage}{0.47\textwidth}
\epsfig{file=FTR_both.eps,scale=0.17}
\centering{(d)}
\end{minipage}
\caption{A rational trigonometric function and its fractal versions.}\label{figFTR}
\end{center}
\end{figure}
\subsection{Weierstrass-type theorems}
In the following theorems we show that a given $f \in \mathcal{C}(2 \pi)$ can be uniformly well-approximated by a fractal rational trigonometric function. The idea is to apply Inequality (\ref{eq2}) to find suitable parameters that provide a close enough fractal perturbation $t_{\Delta,L}^\alpha$ of a rational trigonometric function $t$ that well approximates $f$. This basic idea is not claimed to be new, and is, in fact, explored in various contexts scattered in the fractal approximation literature (see, for instance, \cite{MAN1,Rational,Vij1}).
\begin{theorem}\label{WLT1}
Let $f \in \mathcal{C}(2\pi)$ and $\epsilon>0.$ Suppose that the partition $\Delta:= \{x_0,x_1, \dots,x_N: x_0< x_1< \dots <x_N\}$ of the interval $I=[-\pi,\pi]$, and the bounded linear operator $L:\mathcal{C}(2\pi)\rightarrow \mathcal{C}(2\pi),$ $L \ne Id$ satisfying $(Lf)(x_0)=f(x_0),(Lf)(x_N)=f(x_N)$ are arbitrary, but fixed. Then there exists a scale vector $\alpha=\alpha(\epsilon)$ in $(-1,1)^N$, $\alpha \neq 0$
and an $\alpha$-fractal rational trigonometric function $t_{\Delta,L}^{\alpha}$ such that
$$ \|f- t_{\Delta,L}^{\alpha}\|_{\infty} <\epsilon .$$
\end{theorem}
\begin{proof}
Let $\epsilon >0$ be given. By the Stone-Weierstrass theorem, there exists a rational trigonometric function $ t \in \mathcal{R}(2\pi)$ such that $$ \|f- t\|_{\infty} <\frac{\epsilon}{2} .$$ For a partition $ \Delta:= \{x_0,x_1, \dots,x_N: x_0< x_1< \dots <x_N\} $ of $I=[x_0,x_N]=[-\pi,\pi]$ and for a bounded linear operator $L:\mathcal{C}(2\pi)\rightarrow \mathcal{C}(2\pi),$ $L \ne I_d$ satisfying $(Lf)(x_0)=f(x_0),(Lf)(x_N)=f(x_N),$ select $ \alpha \in (-1,1)^{N}$, $\alpha \neq 0$ such that $$ |\alpha |_{\infty} <\frac{\frac{\epsilon}{2}}{\frac{\epsilon}{2}+\|Id-L\| \|t\|_{\infty}} .$$
Then we have
\begin{equation*}
\begin{aligned}
\|f- t_{\Delta,L}^{\alpha}\|_{\infty} & \leq \|f- t\|_{\infty}+\|t- t_{\Delta,L}^{\alpha}\|_{\infty}\\
& \leq \|f- t\|_{\infty}+ \frac{|\alpha |_{\infty}}{1-|\alpha |_{\infty}}\|Id- L\| \|t\|_{\infty}\\
& < \frac{\epsilon}{2} + \frac{\epsilon}{2}\\
& = \epsilon,
\end{aligned}
\end{equation*}
completing the proof.
\end{proof}
\begin{theorem}\label{WLT2}
Let $f \in \mathcal{C}(2\pi)$ and $\epsilon>0. $ Let the partition $\Delta:= \{x_0,x_1, \dots,x_N: x_0< x_1< \dots <x_N\}$ of the interval of $I=[x_0,x_N]=[-\pi,\pi]$ and scale vector $ \alpha \in (-1,1)^N$ be arbitrary but fixed. Then, there exists a bounded linear operator $L: \mathcal{C}(2\pi)\rightarrow \mathcal{C}(2\pi),$ $L \ne I_d$ satisfying $(Lf)(x_0)=f(x_0),(Lf)(x_N)=f(x_N)$ and an
$\alpha$-fractal rational trigonometric function $t_{\Delta,L}^{\alpha}$ such that
$$ \|f- t_{\Delta,L}^{\alpha}\|_{\infty} < \epsilon .$$
\end{theorem}
\begin{proof}
Let $\epsilon >0$ be given. By the Stone-Weierstrass theorem, there exists a rational trigonometric function $ t \in \mathcal{C}(2\pi)$ such that $$ \|f- t\|_{\infty} <\frac{\epsilon}{2} .$$ Choose a partition $ \Delta:= \{x_0,x_1, \dots,x_N: x_0< x_1< \dots <x_N\} $ of $I=[x_0,x_N]=[-\pi,\pi]$ and a scale vector $ 0 \ne \alpha \in (-1,1)^{N}$ satisfying $ |\alpha|_{\infty} <1.$ Now let us consider a bounded linear operator $L:\mathcal{C}(2\pi)\rightarrow \mathcal{C}(2\pi),$ $L \ne Id$ satisfying $(Lf)(x_0)=f(x_0),(Lf)(x_N)=f(x_N),$ such that $$ \|Id-L\| <\frac{1-|\alpha |_{\infty}}{ |\alpha |_{\infty} \|t\|_{\infty}} \frac{\epsilon}{2} .$$
Then we have
\begin{equation*}
\begin{aligned}
\|f- t_{\Delta,L}^{\alpha}\|_{\infty} & \leq \|f- t\|_{\infty}+\|t- t_{\Delta,L}^{\alpha}\|_{\infty}\\
& \leq \|f- t\|_{\infty}+ \frac{|\alpha |_{\infty}}{1-|\alpha|_{\infty}}\|Id- L\| \|t\|_{\infty}\\
& < \frac{\epsilon}{2} + \frac{\epsilon}{2}\\
&=\epsilon,
\end{aligned}
\end{equation*}
and this completes the proof.
\end{proof}
\begin{remark}
Let $f \in \mathcal{C}(2\pi)$. The above theorems, in particular, assert the following.
\begin{enumerate}
\item Let $\alpha^m \in \mathbb{R}^N$, $|\alpha^m|_\infty<1$ and $\alpha^m \to 0$ as $m \to \infty$. Then there exists a sequence of fractal rational trigonometric functions $\big(t_{\triangle, L}^{\alpha^m}\big)$ which converges to $f$ uniformly.
\item Let $(L_n)_{n \in \mathbb{N}}$ be a sequence of bounded linear operators on $\mathcal{C}(2\pi)$ satisfying $ L_n(g) \to g $ for each $g \in \mathcal{C}(2\pi)$. Then
there exists a sequence of fractal rational trigonometric functions $\big(t_{\triangle, L_n}^{\alpha}\big)$ which converges to $f$ uniformly. For instance, one can work with Bernstein operators corresponding to $f$.
\end{enumerate}
\end{remark}
\begin{theorem} \label{newthm1}
Let $\mathfrak{R}_{\Delta, B_n}^{\alpha}(2 \pi)=\mathcal{F}_{\Delta, B_n}^{\alpha} \big( \mathfrak{R} (2 \pi) \big)$ be the class of all
$\alpha$-fractal rational trigonometric functions with a fixed choice of the scale vector $\alpha$, partition $\Delta$ and Bernstein operator $B_n$.
The set $ \bigcup_{n \in \mathbb{N}} \mathfrak{R}_{\Delta, B_n}^{\alpha}(2 \pi)$ is dense in $\mathcal{C}(2\pi).$
\end{theorem}
\begin{proof}
The proof is immediate from item (2) of the previous remark.
\end{proof}
The following theorem demonstrates that a continuous non-negative function on a compact interval can be uniformly well-approximated by a non-negative $\alpha$-fractal rational trigonometric function. A similar result in the setting of $\alpha$-fractal rational function can be consulted in \textup{\cite[Theorem 3.5]{Rational}}. Although the proof is patterned after \textup{\cite[Theorem 3.5]{Rational}}, the difference lies in the fact that in the following theorem, the scale vector $\alpha$ is arbitrary, except that $|\alpha|_\infty<1$.
\begin{theorem}
Let $f \in \mathcal{C}(2\pi)$ be such that $f(x) \geq 0$ for all $x \in I=[-\pi,\pi].$ Then for any $\epsilon >0 ,$ and for any $\alpha \in (-1,1)^N$, there exists a nonnegative $\alpha$-fractal rational trigonometric function $ t_{\Delta,L}^{\alpha}$ such that $ \|f- t_{\Delta,L}^{\alpha}\|_{\infty} <\epsilon .$ A similar result holds for a continuous non-positive function.
\end{theorem}
\begin{proof}
Let $\epsilon >0 $ and $ f \in \mathcal{C}(2\pi)$ be such that $f(x) \geq 0$ for all $x \in I.$ We assume further that the operator $L$ used in the construction of $f_{\Delta,L}^\alpha$ fixes the constant function $1$ defined by $ 1(x)=1 $ for all $x \in I.$ That is, $L(1)=1.$ For instance, note that the Bernstein operators $B_n$ fixes the function $f(x)=1$. Assume $ \alpha \in (-1,1)^{N}.$ From the self-referential equation for $f_{\Delta,L}^\alpha$, we obtain
$$ \|f_{\Delta,L}^{\alpha}- f\|_{\infty} \leq |\alpha |_{\infty}\|f_{\Delta,L}^{\alpha}- Lf\|_{\infty}.$$ For $f=1$, the above inequality gives $ \|f_{\Delta,L}^{\alpha}- 1\|_{\infty} \leq |\alpha |_{\infty}\|f_{\Delta,L}^{\alpha}- 1\|_{\infty}$ and this gives $\|f_{\Delta,L}^{\alpha}- 1\|_{\infty} = 0 .$ Therefore, $f_{\Delta,L}^\alpha=1$, that is $\mathcal{F}_{\Delta,L}^{\alpha}(1)=1.$\\
For $\epsilon >0 $, $\alpha \in (-1,1)^N$ and $ f \in \mathcal{C}(2\pi).$ In view of Theorem \ref{WLT2}, there exists a rational trigonometric function $t_{\Delta,L}^\alpha$ such that $$ \|f- t_{\Delta,L}^{\alpha}\|_{\infty} <\frac{\epsilon}{2}, ~ \text{ where}~ \mathcal{F}_{\Delta,L}^{\alpha}(t)=t_{\Delta,L}^{\alpha}.$$ Define
$ r_{\Delta,L}^{\alpha}(x)=t_{\Delta,L}^{\alpha}(x)+ \frac{\epsilon}{2} $ for all $x \in I .$ Since $1$ is a fixed point of $\mathcal{F}_{\Delta,L}^{\alpha},$ $$ r_{\Delta,L}^{\alpha}(x)=t_{\Delta,L}^{\alpha}(x)+ \frac{\epsilon}{2}1(x) = t_{\Delta,L}^{\alpha}(x)+ \frac{\epsilon}{2}1^{\alpha}(x).$$
Further, since $\mathcal{F}_{\Delta,L}^{\alpha}$ is a linear operator $$ r_{\Delta,L}^{\alpha} = t_{\Delta,L}^{\alpha}+ \frac{\epsilon}{2}1^{\alpha}= \mathcal{F}_{\Delta,L}^{\alpha}(t+\frac{\epsilon}{2} 1).$$
The above equation tells that $ r_{\Delta,L}^{\alpha}$ is a fractal rational trigonometric polynomial. Further, we have $$ r_{\Delta,L}^{\alpha}(x)=t_{\Delta,L}^{\alpha}(x)+ \frac{\epsilon}{2} = t_{\Delta,L}^{\alpha}(x)+ \frac{\epsilon}{2}-f(x)+f(x) \geq f(x)+ \frac{\epsilon}{2} - \| t_{\Delta,L}^{\alpha}- f \|_{\infty} \geq 0. $$
Moreover, we obtain
$$\| f-r_{\Delta,L}^{\alpha} \|_{\infty}\leq \| f-t_{\Delta,L}^{\alpha} \|_{\infty}+\| t_{\Delta,L}^{\alpha}-r_{\Delta,L}^{\alpha} \|_{\infty}< \frac{\epsilon}{2}+\frac{\epsilon}{2} = \epsilon $$ and hence the proof.
\end{proof}
\begin{remark}
An analogous result can be proved for $\alpha$-fractal rational function, which can be treated as an improvement to \textup{\cite[Theorem 3.5]{Rational}}
in the sense that the scale vector $\alpha$ is arbitrary.
\end{remark}
\section{Best Approximation Property of $\mathfrak{R}^{\alpha}_{mn}(2\pi)$}
\begin{definition} \textup{\cite[p. 372]{DVpai}}.
Let $(X,\|.\|)$ be a normed linear space over $\mathbb{K},$ the field of real or complex numbers. Given a nonempty set $V \subset X$ and an element $x \in X ,$ distance from $V$ to $x$ is defined as $$d(x,V)=\inf\{\|x-y\| : y\in V\}.$$ An element $v(x) \in V$ such that $\|x- v(x)\|= d(x,V)$ if it exists is called a \emph{best approximant} to $x$ from $V.$ A subset $V$ of $X$ is called \emph{proximinal (proximal or existence set)} if for each $x \in X$ a best approximant $v(x) \in V$ of $x$ exists.
\end{definition}
We recall a well-known fact (see, for example, \cite{DVpai,Cheney}) that
\begin{theorem}\textup{\cite[p. 20]{Cheney}}.\label{BATthm1}
Let $X$ be a normed linear space and $E$ be a finite dimensional subspace of $X$. Then $E$ is proximinal in $X$, that is, for each $x$ in $X$, a best approximant from $E$ to $x$ exists.
\end{theorem}
\begin{remark}
Since $\mathfrak{T}_m$ and consequently
$\mathfrak{T}_m^{\alpha}$ is a finite dimensional subspace of $\mathcal{C}(2\pi)$, it follows that for each $f \in \mathcal{C}(2\pi)$, the best approximant $t^\alpha_m (f)$ from $\mathfrak{T}_m^{\alpha}$ to $f$ exists. That is,
$$ \| f- t^\alpha_m (f)\|_\infty = \inf\big\{\|f - t^{\alpha}_m \|_{\infty}:t^{\alpha}_m \in \mathcal{T}^{\alpha}_m \big\}.$$
\end{remark}
\begin{definition}\textup{\cite[p. 71]{DVpai}}.
If $V$ is a proximinal subset of $X$ and the best approximant $v^*(x)$ for each $x \in X$ is unique, then we can define a map $P_V:X \rightarrow V $ by $ P_V(x)=v^*(x).$ This map is called as \emph{a best approximation operator}.
In general, best approximant to $x$ from $V$ is not unique, therefore, $ P_V(x)$ is the set of all best approximants to $x$ from $V.$ The set valued map $ P_V:X \rightrightarrows V$ is called the \emph{metric projection} supported on $V.$
\end{definition}
In this section, we shall establish that $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ is a proximinal subset of $\mathcal{C}(2\pi),$ i.e., for each $f \in \mathcal{C}(2\pi),$ there exists an element $r_*^{\alpha} \in \mathfrak{R}^{\alpha}_{mn}(2\pi)$ such that $$\|f- r_*^{\alpha}\|_{\infty} =\text{dist} \big(f, \mathfrak{R}^{\alpha}_{mn}(2\pi)\big) :=\inf\big\{\|f-r^{\alpha}\|_{\infty} : r^{\alpha} \in \mathfrak{R}^{\alpha}_{mn}(2\pi) \big\}.$$ Note that $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ is not a linear subspace of $\mathcal{C}(2\pi)$ and hence in contrast to the case $\mathfrak{T}_m^{\alpha} (2 \pi)$, Theorem \ref{BATthm1} cannot be applied to infer that $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ is proximinal. First let us record the following definition and lemma.
\begin{definition}\textup{\cite[p. 376]{DVpai}}.
Let $Y$ be a non-empty subset of a normed linear space $X.$ Then $Y$ ia said to be approximately compact if for every $x \in X,$ each sequence $ (y_n) \subseteq Y$ such that $\| x-y_n\| \rightarrow d(x,Y),$ has a subsequence convergent in $Y.$
\end{definition}
\begin{lemma}\textup{\cite[p. 156]{Cheney}}. Let $P$ and $Q$ be two non-zero trigonometric polynomials with real coefficients such that $|P(\theta)|\leq |Q(\theta)|$ for all real $\theta$. If $Q$ has a real zero, then there exist non-zero trigonometric polynomials $P^*$ and $Q^*$ with real coefficients such that $\deg(P^*) < \deg(P) , \deg(Q^*) < \deg(Q)$ and $P^*Q=PQ^*.$
\end{lemma}
\begin{theorem}\label{proximal}
If the fractal operator $\mathcal{F}_{\Delta,L}^{\alpha}$ is bounded below, then for each $f \in \mathcal{C}(2\pi)$ there exists a fractal rational trigonometric function $r_*^{\alpha} \in \mathfrak{R}^{\alpha}_{mn}(2\pi)$ such that $\|f- r_*^{\alpha}\|_{\infty} = \text{dist} \big(f, \mathfrak{R}^{\alpha}_{mn}(2\pi) \big)$. In particular, $\mathfrak{R}^{\alpha}_{mn}(2\pi) $ is approximately compact.
\end{theorem}
\begin{proof}
Let $ d = \text{dist} (f, \mathfrak{R}^{\alpha}_{mn}(2\pi) ).$ By the definition of infimum, we get a sequence $r^{\alpha}_k$ in $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ such that $$\|f- r^{\alpha}_k\|_{\infty} < d+ \frac{1}{k}, ~ ~k=1,2,\dots .$$ It follows that $$ \|r^{\alpha}_k\|_{\infty} \leq \|r^{\alpha}_k - f\|_{\infty} + \| f\|_{\infty} \leq d+1+\| f\|_{\infty},~ ~ k=1,2,\dots .$$
Let $ r^{\alpha}_k = \mathcal{F}_{\Delta,L}^{\alpha}(r_k),$ where $r_k=\dfrac{p_k}{q_k}$, $ p_k \in \mathfrak{T}_m$, $q_k \in \mathfrak{T}_n$, $\|q_k\|_{\infty}=1$ and $q_k(x) > 0$ on $I .$ Since the fractal operator is bounded below, there exists $C>0$ such that $$ C \|f\|_{\infty} \leq \|\mathcal{F}_{\Delta,L}^{\alpha}(f)\|_{\infty} ~~ \forall~~ f \in \mathcal{C}(I).$$ Therefore
$$ \|r_k\|_{\infty} \leq \frac{1}{C}\|\mathcal{F}_{\Delta,L}^{\alpha}(r_k)\|_{\infty}= \frac{1}{C}\|r^{\alpha}_k\|_{\infty} \leq \frac{1}{C}\big( d+1+\|f\|_{\infty}\big):= K.$$
Since $\mathfrak{T}_m, \mathfrak{T}_n$ are finite dimensional spaces and $$|p_k(x)|=|q_k(x)||r_k(x)| \leq \|q_k\|_{\infty}\|r_k\|_{\infty} \leq A,$$ the pairs $(p_k,q_k)$ lie in the compact sets defined by the inequalities $\|p\|_{\infty} \leq A$ and $\|q\|_{\infty} =1 .$ We may assume, by passing to a subsequence if necessary, that $p_k \rightarrow p$ and $q_k \rightarrow q.$ Clearly, $\|q\|_{\infty}=1;$ whence using the Haar condition there can be at most $2n$ zeros for $q$. At the points that are not zeros of $q,$ $\dfrac{p(x)}{q(x)}$ is well defined, and we have $\dfrac{p_k(x)}{q_k(x)}\rightarrow \dfrac{p(x)}{q(x)}.$ Therefore at points in $I$ where $q$ does not vanish,
$$\dfrac{|p(x)|}{|q(x)|} \leq A, \quad |p(x)| \leq A|q(x)|.$$ Since there are only finite number of zeros for $q,$ by continuity, the last inequality holds for all $x \in I.$ One can apply previous lemma (perhaps repeatedly) to obtain other trigonometric polynomials $p_*$ and $q_*$ such that $\deg(p_*) < \deg(p)$ , $\deg(q_*) < \deg(q)$, $q_*(x)>0$ on $ I$ and $p_*(x)q(x)= p(x)q_*(x).$ The resulting element $r_*:=\dfrac{p_*}{q_*}$ is in $\mathfrak{R}_{mn}(2 \pi).$ As $ r_k \rightarrow r_*$ uniformly and $ \mathcal{F}_{\Delta,L}^{\alpha}$ is a bounded linear map, we get $ r_k^{\alpha} \rightarrow r_*^{\alpha}$ and hence $f-r_k^{\alpha} \rightarrow f-r_*^{\alpha}.$ By the continuity of norm, we have, $\|f-r_k^{\alpha}\|_{\infty} \rightarrow \|f-r_*^{\alpha}\|_{\infty}.$ Therefore, $\|f-r_*^{\alpha}\|_{\infty}=d.$
\end{proof}
\begin{remark}
Approach in the previous proof is identical to the one used in \textup{\cite[Theorem 4.1]{Rational}} for proving the proximality of the class of fractal rational functions in $\mathcal{C}(I)$ except for a few lines at the end. However, we included a expanded rendition of the arguments for the sake of completeness and record.
\end{remark}
\begin{remark}
It is known that (Cf. Theorem \ref{prelthm}) for $|\alpha |_{\infty}< \|L\|^{-1}$, the fractal operator $\mathcal{F}_{\Delta,L}^{\alpha}$ is bounded below. Therefore, for $|\alpha |_{\infty}< \|L\|^{-1}$, $\mathfrak{R}^{\alpha}_{mn}(2\pi) $ is a proximinal approximately compact subset of $\mathcal{C}(2\pi)$.
\end{remark}
In general, best approximant from $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ to $f \in \mathcal{C}(2\pi)$ may not be unique. For $f \in \mathcal{C}(2\pi),$ let us write
$$P_{\mathfrak{R}^{\alpha}_{mn}(2\pi)}(f)= \Big\{r^\alpha \in \mathfrak{R}^{\alpha}_{mn}(2\pi): \|f-r^\alpha\|_\infty= \text{dist}\big(f, \mathfrak{R}^{\alpha}_{mn}(2\pi)\big) \Big\}.$$
\begin{theorem}
If the fractal operator $\mathcal{F}_{\Delta,L}^{\alpha}$ is bounded below, then the set-valued map $P_{\mathfrak{R}^{\alpha}_{mn}(2\pi)}: \mathcal{C}(2\pi) \rightrightarrows \mathfrak{R}^{\alpha}_{mn}(2\pi)$ supported on the nonempty proximal subset $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ is upper semicontinuous and closed.
\end{theorem}
\begin{proof}
By Theorem \ref{proximal}, $\mathfrak{R}^{\alpha}_{mn}(2\pi)$ is a nonempty approximately compact subset of the normed linear space $\mathcal{C}(2\pi)$.
Therefore the multi-valued map $P_{\mathfrak{R}^{\alpha}_{mn}(2\pi)}: \mathcal{C}(2\pi) \rightrightarrows \mathfrak{R}^{\alpha}_{mn}(2\pi)$ is upper semicontinuous and its values are compact. This follows by a result that if $X$ is a normed linear space and $V$ is a nonempty approximately compact subset of $X$, then the metric projection set-valued function $P_V : X \rightrightarrows V$ is upper semicontinuous and its values are compact (see, for instance, \textup{\cite[p. 440]{DVpai}}). The set-valued map $P_{\mathfrak{R}^{\alpha}_{mn}(2\pi)} $ is closed follows from the fact that if $X$ is a topological space, $Y$ is a Hausdorff space and $T:X \rightrightarrows Y$ is upper semicontinuous with compact values, then $T$ is closed (see, for instance, \textup{\cite[p. 434]{DVpai}}).
\end{proof}
\begin{remark}
Taking $n=0$, $ \mathfrak{R}^{\alpha}_{m0}(2\pi)= \mathfrak{T}^{\alpha}_m(2\pi) .$ Therefore the above theorems and remark are valid for the class of fractal trigonometric polynomials as well. This observation serves as an addendum to the researches in \cite{MAN3}.
\end{remark}
\subsection{Approximation Error Bound}\label{AEB}
Define $$ E^{\alpha}_{mn}(f;[-\pi,\pi]):=\inf\big\{\|f - r^{\alpha} \|_{\infty}:r^{\alpha} \in \mathfrak{R}^{\alpha}_{mn}(2\pi) \big\}$$ and $$ E_{mn}(f;[-\pi,\pi]):=\inf\big\{\|f - r \|_{\infty}:r \in \mathfrak{R}_{mn}(2\pi)\big \}.$$
\begin{theorem} \label{error1a}
Let $f \in \mathcal{C}( 2 \pi)$. Then,
$$ E^{\alpha}_{mn}(f;[-\pi,\pi])\leq \frac{1+|\alpha|_\infty(\|Id-L\|-1)}{1-|\alpha|_\infty} E_{mn}(f;[-\pi,\pi])+ \frac{|\alpha |_{\infty}}{1-|\alpha |_{\infty}}\|Id-L\|~\|f\|_\infty.$$
\end{theorem}
\begin{proof}
Let $f \in \mathcal{C}(2 \pi)$ and $r_*$ be a best approximant to $f$ from $\mathfrak{R}_{mn}(2 \pi)$. We have
\begin{equation*}
\begin{aligned}
E^{\alpha}_{mn}(f;[-\pi,\pi]) & \leq \|f - r^{\alpha}_* \|_{\infty}\\
& \leq \|f - r_* \|_{\infty} + \|r_* - r^{\alpha}_* \|_{\infty} \\
& = E_{mn}(f;[-\pi,\pi])+\|r_* - r^{\alpha}_* \|_{\infty}\\
& \leq E_{mn}(f;[-\pi,\pi])+\frac{|\alpha |_{\infty}}{1-|\alpha |_{\infty}}\|Id-L\|~\|r_*\|_\infty\\
& \leq E_{mn}(f;[-\pi,\pi])+\frac{|\alpha |_{\infty}}{1-|\alpha |_{\infty}}\|Id-L\|~ \big(\|f-r^*\|_\infty+\|f\|_\infty \big),
\end{aligned}
\end{equation*}
and hence the theorem.
\end{proof}
Let $n \in \mathbb{N}$ and $x_{kn}= \frac{2k\pi}{n}$, $k=0,1,\dots,n-1$. Let $f$ be an arbitrary $2\pi$-periodic continuous function. From
\cite{AKVarma}, we recall a sequence of positive linear interpolating operators $\wedge_n$, $n=1,2,\dots$, which map $\mathcal{C}(2\pi)$ into the set of rational trigonometric functions of order $ \leq 2n-2$ defined by
$$\wedge_n(f,x)= \frac{\sum_{k=0}^{n-1}f(x_{kn})J^2_n(x-x_{kn})}{1-((n^2 -1)/3n^2)(1-\cos(nx))},$$
where $J_n$ are Jackson functions
$$J_n(x)= \Big( \frac{\sin(nx/2)}{n~\sin(x/2)}\Big)^2.$$
Let us recall also that the modulus of continuity of a bounded function $f$ on the compact interval $I$ is defined by
$$\omega_f (\delta) = \omega_f \big(I; \delta \big):= \sup\Big\{|f(x)-f(y)|: x, y \in I, |x-y| \le \delta \Big\}.$$
\begin{theorem}(\cite{AKVarma}).\label{error2}
Let $f \in \mathcal{C}(2\pi)$. Then, for $n=2,3,4,\dots ,$ and for all $x,$ $$ |f(x)- \wedge_n(f,x)| \leq 2\omega_f\Big( \frac{\pi \sqrt{3}}{n}\Big).$$
\end{theorem}
Theorems \ref{error1a}-\ref{error2} now dictate
\begin{theorem}
Let $f \in \mathcal{C}(2\pi)$ with modulus of continuity $ \omega_f(\delta)$. Then, $$ E^{\alpha}_{nn}(f;[-\pi,\pi])\leq \frac{1+|\alpha|_\infty(\|Id-L\|-1)}{1-|\alpha|_\infty} 2\omega_f\Big( \frac{2 \pi \sqrt{3}}{n+2}\Big) + \frac{|\alpha |_{\infty}}{1-|\alpha |_{\infty}}\|Id-L\|~\|f\|_\infty.$$
\end{theorem}
\section{Some Comments and Corrections}
This section aims to provide corrections and comments to some results scattered in the literature that are based on the concept of $\alpha$-fractal functions.
\subsection{On minimax error}\label{SCC1}
Let us begin by noting that a result similar to Theorem \ref{error1a} in the previous section is announced in \textup{\cite[Theorem 4.3]{Rational}} to compare fractal rational minimax error with classical rational minimax error. With the notation
\begin{equation*}
\begin{split}
\mathcal{R}_{mn}(I) :=&~ \Big \{r=\frac{p}{q}: p \in \mathcal{P}_m(I), q \in \mathcal{P}_n(I); q >0 ~\text{on}~ I \Big \}, \\ \mathcal{R}_{mn}^\alpha (I) :=&~ \mathcal{F}_{\Delta, L}^\alpha \big ( \mathcal{R}_{mn} (I)\big),
\end{split}
\end{equation*}
where $\mathcal{P}_k(I)$ is the space of all algebraic polynomials of degree at most $k$, the authors claim that
\begin{theorem}\textup{\cite[Theorem 4.3]{Rational}} \label{Thmamend1}.
For any $f \in \mathcal{C}(I )$, $$\text{dist}\big(f,\mathcal{R}_{mn}^\alpha(I)\big) \le \text{dist}\big(f,\mathcal{R}_{mn}(I)\big).$$
\end{theorem}
However, the proof of the above theorem as mentioned in \cite{Rational} is inaccurate. The inaccuracy comes from the fact that the proof uses:
\begin{equation*}
\begin{split}
& \inf \{ \|f-\mathcal{F}^\alpha(r)\|_\infty: r \in \mathcal{R}_{mn}(I)\}\\ \le &~ \inf\big\{\|f-r\|_\infty + \|r- \mathcal{F}^\alpha(r)\|_\infty: r \in \mathcal{R}_{mn}(I) \big\}\\
\le &~ \inf \big\{ \|f-r\|_\infty: r \in \mathcal{R}_{mn}(I)\big \} + \inf\big\{ \|r-\mathcal{F}^\alpha(r)\|_\infty: r \in \mathcal{R}_{mn}(I)\big \},
\end{split}
\end{equation*}
which is not true. If $A=\{x_\beta: \beta \in \Lambda\}$, $B=\{y_\beta: \beta \in \Lambda\}$ are subsets of $\mathbb{R}$ and $C=\{x_\beta+y_\beta: \beta \in \Lambda\}$, then it is easy to see that $\inf C \ge \inf A + \inf B$. However, in general, $\inf C \le \inf A + \inf B$ is not true. Let us recall also that, in fact, $\inf(A+B) = \inf(A) + \inf(B)$ holds, however here $C \neq A+B$. P. Viswanathan regrets for this careless mistake and would like to mention that it was also observed and pointed out by Prof. Navascu\'{e}s in a different context during some personal communications.
Our result in Theorem \ref{error1a} suggests that at this point Theorems \ref{Thmamend1} above can be corrected by supplying suitable additional terms. For instance, with notation as in \cite{Rational}, we have
\begin{theorem} \label{Thmamend3}
For any $f \in \mathcal{C}(I )$,
\begin{equation*}
\begin{split}
\text{dist}\big(f,\mathcal{R}_{mn}^\alpha(I)\big) \le &~ \text{dist}\big(f,\mathcal{R}_{mn}(I)\big) \\ &~+ \frac{|\alpha|_\infty}{1-|\alpha|_\infty} \|Id-L\| \Big(\text{dist}\big(f,\mathcal{R}_{mn}(I)\big) + \|f\|_\infty\Big).
\end{split}
\end{equation*}
\end{theorem}
The same incorrect arguments in the proof of Theorem \ref{Thmamend1} is repeated recently for the class of Bernstein $\alpha$-fractal rational functions in \cite{Vij2}. We note that the theorem remains valid and supply a correct proof for it. Let us recall the following notation as in \cite{Vij2}. For a fixed partition $\Delta$ and scale vector $\alpha$
\begin{equation*}
\begin{split}
\mathcal{R}_{l,m}(I) :=&~ \Big \{r=\frac{p}{q}: p \in \mathcal{P}_l(I), q \in \mathcal{P}_m(I); q >0 ~\text{on}~ I \Big \},\\ \mathcal{R}_{l,m}^\alpha (I) :=&~ \big\{ \mathcal{F}^\alpha_{\Delta, B_n}(r): r \in \mathcal{R}_{l,m}(I), n \in \mathbb{N}\big\}.
\end{split}
\end{equation*}
\begin{theorem}\textup{\cite[Theorem 3.9]{Vij2}}. \label{Thmamend2}
For any $f \in \mathcal{C}(I ),$
$$\text{dist}\big(f,\mathcal{R}_{l,m}^\alpha(I)\big) \le \text{dist}\big(f,\mathcal{R}_{l,m}(I)\big).$$
\end{theorem}
\begin{proof}
Let $r^*$ be the unique best approximant to $f \in \mathcal{C}(I )$ from $\mathcal{R}_{l,m}(I)$, that is, $\|f-r^*\|_\infty = \text{dist}\big(f,\mathcal{R}_{l,m}(I)\big)$ (see, for instance, \textup{\cite[p. 164]{Cheney}}).
Using item (1) in Theorem \ref{prelthm} we have
\begin{equation*}
\begin{split}
\text{dist}\big(f,\mathcal{R}_{l,m}^\alpha(I)\big) \le &~ \|f- \mathcal{F}^\alpha_{\Delta, B_n}(r^*)\|_\infty \\
\le &~ \|f-r^*\|_\infty + \|r^*-\mathcal{F}^\alpha_{\Delta, B_n}(r^*)\|_\infty\\
\le &~ \text{dist}\big(f,\mathcal{R}_{l,m}(I)\big)+ \frac{|\alpha|_\infty}{1-|\alpha|_\infty}\|Id- B_n\| \|r^*\|_\infty
\end{split}
\end{equation*}
Since the above estimate holds for all $n \in \mathbb{N}$ and $\|Id-B_n\| \to 0$ as $n \to \infty$, we infer that
$\text{dist}\big(f,\mathcal{R}_{l,m}^\alpha(I)\big) \le \text{dist}\big(f,\mathcal{R}_{l,m}(I)\big)$ and thus the proof.
\end{proof}
\begin{remark}
In view of Theorems \ref{Thmamend3} -\ref{Thmamend2} it appears that the approximation class $\mathcal{R}_{m,n}^\alpha(I)$
of Bernstein fractal rational functions introduced in \cite{Vij2} is ``better" than the class $\mathcal{R}_{mn}^\alpha(I)$ of fractal rational functions that made its debut in \cite{Rational}. In this regard, let us note that corresponding to a fixed rational function $r$ of order $(m,n)$, there exists a unique fractal rational function $r_{\Delta,L}^\alpha$ whereas there exist a sequence of Bernstein fractal rational functions $\big(r_{\Delta,B_n}^\alpha\big)$ converging to $r$.
\end{remark}
\subsection{On Bernstein $\alpha$-fractal functions}
As mentioned in the introductory section, Inequality (\ref{eq1}) should convince the reader that the fractal function $f_{\Delta,b}^\alpha$ can be made close to the seed function $f$ by taking the parameter map $b$ close to $f$. In \cite{Vij1,Vij2,Vij3}, the author uses this simple observation effectively by selecting $b=B_n(f)$, the Bernstein polynomials for $f$ to introduce what is called Bernstein $\alpha$-fractal functions. In particular, using a result by Akhtar et. al. (see Theorem \ref{Akhtar}), the following claim is made in \cite{Vij1,Vij2,Vij3}.
\begin{theorem} \textup{\cite[Theorem 2]{Vij1}},\textup{\cite[Theorem 2.3]{Vij2}}, \textup{\cite[Theorem 2]{Vij3}}.
Let $f \in \mathcal{C}(I).$ Let $\Delta=\{x_0,x_1,\dots , x_N\}$ be a partition of $I=[x_0,x_N]$ satisfying $x_0<x_1< \dots < x_N$ and $ \alpha =(\alpha_1,\alpha_2, \dots ,\alpha_{N}) \in (-1,1)^N.$ If the $\alpha$-fractal functions in the sequence $\big(f^{\alpha}_{\Delta,B_n}\big)_{n=1}^{\infty}$ are obtained with same fixed choice of scaling vector $\alpha$ whose components satisfy the condition $\sum_{i=1}^{N} |\alpha_i | > 1,$ then all the $\alpha$-fractal functions in the sequence $\big(f_{\Delta,B_n}^{\alpha}\big)_{n=1}^{\infty}$ have the same fractal dimension $D \in (1,2)$ and $\lim_{n \to \infty}f_{\Delta, B_n}^{\alpha}=f.$
\end{theorem}
Theorem \ref{Akhtar} has the hypothesis that the seed function $f$ and base function $b$ are Lipschitz continuous and data points sampled from $f$ are not collinear. The Bernstein $\alpha$-fractal functions use Bernstein polynomials as base function which obviously are Lipschitz. However, to apply Theorem \ref{Akhtar} other hypotheses are to be taken care, and a possible refinement to the above theorem could be
\begin{theorem}
Let $f: I \to \mathbb{R}$ be a Lipschitz continuous function and $\Delta=\{x_0,x_1,\dots , x_N: x_0<x_1< \dots < x_N\}$ be a partition of $I=[x_0,x_N]$ such that the data set $\big\{\big(x_i,f(x_i)\big): i=0,1,\dots,N\big\}$ is not collinear. Let $ \alpha =(\alpha_1,\alpha_2, \dots ,\alpha_{N}) \in (-1,1)^N$ be a fixed vector such that $\sum_{i=1}^{N} |\alpha_i | > 1.$ Then the graphs of the Bernstein $\alpha$-fractal functions $f_{\Delta, B_n}^{\alpha}$, $n \in \mathbb{N}$ have the same box dimension
$D$ given by the formula in Theorem \ref{Akhtar} and the sequence $ \Big(f_{\Delta, B_n}^{\alpha}\Big)_{n=1}^{\infty}$ converges to $f$ uniformly.
\end{theorem}
Now we shall prove that the assumption of Lipschitz continuity of $f$ can be dropped, if one intends only to obtain a sequence of $\alpha$-fractal functions converging uniformly to $f$ with additional property that the graphs of all functions in this sequence have the same box dimension.
\begin{theorem}
Let $f \in \mathcal{C}(I)$ and $\Delta=\{x_0,x_1,\dots , x_N: x_0<x_1< \dots < x_N\}$ be a partition of $I=[x_0,x_N]$. Let $ \alpha =(\alpha_1,\alpha_2, \dots ,\alpha_{N}) \in (-1,1)^N$ be a fixed vector. Then there exists a sequence of $\alpha$-fractal functions with graphs having same box dimension that converges uniformly to $f$.
\end{theorem}
\begin{proof}
Let $f \in \mathcal{C}(I).$ Without loss of generality assume that the points in $\big\{\big(x_i,f(x_i)\big): i=0,1,\dots,N\big\}$ obtained by sampling $f$ are not collinear. Consider the sequence of Bernstein polynomials $(p_n)_{n \in \mathbb{N}}$ that converges to $f$ uniformly. That is, $p_n = B_n(f)$ for $n \in \mathbb{N}$, where $B_n$ is the $n$-th Bernstein operator. Fix a partition $\Delta$ and a scale vector $\alpha$. For each fixed $n \in \mathbb{N}$, find the $\alpha$-fractal function $(p_n)_{\Delta, B_n(p_n)}^\alpha$ corresponding to $p_n$ by choosing the base function as $B_n(p_n)$. We have
\begin{equation*}
\begin{split}
\| f- (p_n)_{\Delta, B_n(p_n)}^\alpha\| _\infty \le &~ \|f-p_n \|_\infty + \|p_n - (p_n)_{\Delta, B_n(p_n)}^\alpha\|_\infty \\
\le &~ \|f -p_n \|_\infty + \frac{|\alpha|_\infty}{1-|\alpha|_\infty} \| p_n - B_n(p_n)\|_\infty\\
\le &~ \|f -p_n \|_\infty + \frac{|\alpha|_\infty}{1-|\alpha|_\infty} \|Id-B_n \| \|p_n \|_\infty \\
= &~ \|f-p_n\|_\infty + \frac{|\alpha|_\infty}{1-|\alpha|_\infty} \|Id-B_n \| \|B_n(f)\|_\infty \\
\le &~ \|f-p_n\|_\infty + \frac{|\alpha|_\infty}{1-|\alpha|_\infty} \|Id-B_n \| \|f\|_\infty.
\end{split}
\end{equation*}
The above estimate shows that $(p_n)_{\Delta, B_n(p_n)}^\alpha \to f$ uniformly as $ n \to \infty.$ For each fixed $n \in \mathbb{N}$, the germ function $p_n$ and base function $B_n(p_n)$ are Lipschitz and the set of data points $\{(x_i, p_n(x_i): I=0,1,\dots, N\}$ is not collinear. Therefore, by the formula in Theorem \ref{Akhtar}, the box dimensions of the graphs of $(p_n)_{\Delta, B_n(p_n)}^\alpha$, which depend only on the partition and scaling vector, are all same.
\end{proof}
\subsection{On non-self-referential Bernstein fractal rational functions}\label{nonselfssec} The class of fractal rational functions studied in detail in \cite{Rational} is defined as the image of the set of rational functions under the bounded linear map $\mathcal{F}_{\Delta, L}^\alpha$. However, it is hinted in \textup{\cite[Remark 3.2]{Rational}} that a class of fractal rational functions can also be defined by considering the quotients of suitable fractal polynomials.
Let us recall these two classes of fractal rational functions. Following notation in \cite{Rational}, let $\mathcal{P}_k (I)$ be the set of polynomials of degree less than or equal to $k$ defined on $I$,
\begin{equation*}
\begin{split}
&\mathcal{R}_{mn}(I) :=\Big\{\frac{p}{q}: p \in \mathcal{P}_m(I), q \in \mathcal{P}_n(I);~~ q>0~~ \text{on}~~I \Big\},\\
& \mathcal{P}_k ^\alpha (I) := \mathcal{F}_{\Delta, L}^\alpha\big(\mathcal{P}_k (I)\big),\\ & \mathcal{R}_{mn}^\alpha(I):= \mathcal{F}_{\Delta, L}^\alpha \big(\mathcal{R}_{mn}(I)\big),\\
& \mathcal{K}_{mn}^\alpha(I):= \Big\{\frac{p^\alpha}{q^\alpha}: p \in \mathcal{P}_m^\alpha(I), q \in \mathcal{P}_n^\alpha(I);~~ q^\alpha>0~~ \text{on}~~I \Big\}.
\end{split}
\end{equation*}
In \cite{Vij2}, by taking $L$ as the sequence of Bernstein operators, two classes of Bernstein rational functions are considered, which we shall denote again by $\mathcal{R}_{mn}^\alpha(I)$ and $\mathcal{K}_{mn}^\alpha(I)$. The author claims that an element $\varphi=\dfrac{p^\alpha}{q^\alpha}\in \mathcal{K}_{mn}^\alpha(I)$ is \emph{non-self-referential} function as its graph does not correspond to any IFS \textup{\cite[Section 4]{Vij2}}. To this end, we observe the following.
\par We know \big(Cf. item (4) Theorem \ref{prelthm}\big) that for the scaling vector $\alpha \in (-1,1)^{N}$ satisfying $|\alpha|_\infty < \big(1 + \|Id - L\|\big)^{-1}$, the operator $\mathcal{F}^\alpha: \mathcal{C}(I) \to \mathcal{C}(I)$ is invertible. Consequently, for $\alpha$ satisfying the aforementioned condition and $\varphi := \dfrac{p^\alpha}{q^\alpha} \in \mathcal{K}_{mn}^\alpha(I)\subseteq \mathcal{C}(I)$, there exists
$g \in \mathcal{C}(I)$ such that $\varphi = \mathcal{F}^\alpha (g)=g^\alpha$. By the very construction of the $\alpha$-fractal function, the graph of $g^\alpha$, and consequently the graph of $\varphi$, is the attractor of an IFS. In fact, $\varphi$ satisfies the self-referential equation
$$ g_{\Delta,b}^\alpha(x) = g(x) + \alpha_i (g_{\Delta,b}^{\alpha}- b)\big(L_i^{-1}(x)\big) ~ ~ ~~\forall~~ x \in I_i,~~ i \in \mathbb{N}_N.$$
Therefore, it is perhaps an abuse of terminology to regard functions in $\mathcal{K}_{mn}^\alpha(I)$ as non-self-referential for all permissible choice of $\alpha$. Let us remark that the question whether $\varphi \in \mathcal{K}_{mn}^\alpha(I)$ is equal to $\mathcal{F}^\alpha (r)$ for some rational function $r$ remains unsettled.
\subsection{On multi-valued fractal operator}
The definition of Bernstein $\alpha$-fractal function $f_{\Delta, B_n(f)}^\alpha$ corresponding to each $f \in \mathcal{C}(I)$ provides
\begin{enumerate}
\item a sequence of single-valued operators $(\mathcal{F}_{\Delta, B_n}^\alpha)_{n \in \mathbb{N}}$ defined by $$\mathcal{F}_{\Delta, B_n}^\alpha: \mathcal{C}(I) \to \mathcal{C}(I); \quad \mathcal{F}_{\Delta, B_n}^\alpha(f)= f_{\Delta, B_n(f)}^\alpha~~\text{for ~each}~~n \in \mathbb{N}.$$
\item a multi-valued operator $\mathcal{F}^\alpha : \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ defined by
$$ \mathcal{F}^\alpha (f) = \{f_{\Delta, B_n(f)}^\alpha: n \in \mathbb{N}\}.$$
\end{enumerate}
It appears that in \cite{Vij3}, the author switches between the sequence of single-valued operators and multi-valued operator in items (1) -(2) above without making a clear distinction between them. For instance, it is claimed that
\begin{theorem}\textup{\cite[Theorem 3]{Vij3}}. \label{thmmulval}
Let $\mathcal{C}(I)$ be endowed with the supnorm. The multi $\alpha$-operator $\mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ defined by
$\mathcal{F}^\alpha (f) = f_{\Delta, B_n(f)}^\alpha= f_n^\alpha$ is linear and bounded.
\end{theorem}
A careful examination of the proof of the above theorem indicates that, the author treats $\mathcal{F}^\alpha$ as an ordinary function, but not as a set-valued map. In what follows, we shed some light on this. Since for $\alpha =0$, $\mathcal{F}_{\Delta, B_n}^\alpha(f) =f$ for all $f \in \mathcal{C}(I)$ and $n \in \mathbb{N}$, we consider the case $\alpha \neq 0.$
\par
Since the Bernstein operator $B_n: \mathcal{C}(I) \to \mathcal{C}(I)$, $n \in \mathbb{N}$ is a bounded linear operator, by Theorem \ref{prelthm}, it is straightforward to see that $\mathcal{F}_{\Delta, B_n}^\alpha$ is a bounded linear operator for each $n \in \mathbb{N}$. However, as mentioned in the introductory section the linearity and boundedness of the multi-valued operator is dealt with a slightly different approach \cite{Aubin}. A suitable question to ask would be whether the multi-valued operator $\mathcal{F}^\alpha$ is a closed convex process. We provide a partial answer to this question and arguments to conclude that $\mathcal{F}^\alpha$ is not ``linear", contradicting Theorem \ref{thmmulval}. As a prelude, let us recall a few definitions and results.
\begin{definition}(\cite{Aubin}).
Let $X$ and $Y$ be two real normed linear spaces over $\mathbb{R}$. For a set-valued map $T$ from $X$ to $Y$ denoted by $T: X \rightrightarrows Y$, the
domain of $T$ is defined by
$\text{Dom}(T):= \{x \in X: T(x) \neq \emptyset\}.$ Then $T: X \rightrightarrows Y$ is
\begin{enumerate}
\item \emph{convex} if for all $x_1, x_2 \in \text{Dom}(T)$ and for all $\lambda \in [0,1],$ $$\lambda T(x_1)+(1-\lambda)T(x_2) \subseteq T\big(\lambda x_1+(1-\lambda)x_2\big).$$
\item \emph{process} if for all $~x \in X$ and for all $\lambda > 0$, $$\lambda T(x)= T(\lambda x) ~\text{and}~ 0 \in T(0).$$
\item \emph{linear} if for all $x_1, x_2 \in \text{Dom}(T)$ and for all $\beta, \gamma \in \mathbb{R},$ $$\beta T(x_1)+ \gamma T(x_2) \subseteq T\big(\beta x_1+\gamma x_2\big).$$
\item \emph{closed} if graph of $T$ $$G_T:= \big\{(x,y)\in X \times Y: y \in T(x) \big\}$$ is closed.
\item \emph{Lipschitz} if there exists a constant $l >0$ such that for all $x_1, x_2 \in \text{Dom}(T)$
$$T(x_1) \subseteq T(x_2) + l \|x_1-x_2\| U_Y,$$
where $U_Y$ is the closed unit ball in $Y$.
\end{enumerate}
\end{definition}
\begin{theorem}\textup{\cite[Corollary 1.4]{DS}}. \label{Multhm2}
Let $X$ and $Y$ be real vector spaces and $\mathcal{P}_0(Y)$ be the collection of all nonempty subsets of $Y$. If a set-valued map $T: X \to \mathcal{P}_0(Y)$ is linear and $T(0)= \{0\}$, then $T$ is single-valued.
\end{theorem}
\begin{theorem}\textup{\cite[Corollary 2.1]{DS}}.\label{Multhm2a}
Let $X$ and $Y$ be real vector spaces and $\mathcal{P}_0(Y)$ be the collection of all nonempty subset of $Y$. If a set-valued map $T: X \to \mathcal{P}_0(Y)$ is such that $T(x_0)$ is singleton for some $x_0\in X$, then $T: X \to \mathcal{P}_0(Y)$ is convex if and only if $T$ is single-valued and affine.
\end{theorem}
\begin{theorem}\label{Multhm3}
The set-valued map $: \mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ defined by $$\mathcal{F}^\alpha(f) =\{f^{\alpha}_{\Delta,B_n(f)}: n \in \mathbb{N} \}$$ is a Lipschitz process.
\end{theorem}
\begin{proof}
Let $f \in \mathcal{C}(I)$ and $\lambda > 0.$ Since for each fixed $n \in \mathbb{N}$, the operator defined by $\mathcal{F}_{\Delta, B_n}^\alpha(f)= f_{\Delta, B_n(f)}^\alpha$ is linear
\begin{equation*}
\begin{aligned}
\mathcal{F}^\alpha (\lambda f) = &\{(\lambda f)^{\alpha}_{\Delta,B_n}:n \in \mathbb{N} \}\\
=&\{ \lambda f^{\alpha}_{\Delta,B_n}:n \in \mathbb{N}\}\\
= &\lambda\{f^{\alpha}_{\triangle,B_n}:n \in \mathbb{N} \}\\
=& \lambda \mathcal{F}^\alpha(f).
\end{aligned}
\end{equation*}
Further, since for each $n \in \mathbb{N}$, $\mathcal{F}_{\Delta, B_n}^\alpha$ is a linear operator, it follows that for
$\mathcal{F}_{\Delta, B_n}^\alpha(0) = 0_{\Delta, B_n(0)}^\alpha=0$. Thus, $\mathcal{F}^\alpha (0) = \{0\},$ and consequently $\mathcal{F}^\alpha$ is a process. Let $f,g \in \mathcal{C}(I).$
Using the functional equation for the $\alpha$-fractal function we have
$$ f^{\alpha}_{\Delta,B_m}(x)= f(x)+\alpha_i \big(f^{\alpha}_{\Delta,B_m}- B_m(f)\big)\circ L_i^{-1}(x) ~~\forall~~ x \in I_i,~~ i \in \mathbb{N}_N,$$
and
$$ g^{\alpha}_{\Delta,B_m}(x)= g(x)+\alpha_i \big(g^{\alpha}_{\Delta,B_m}- B_m(g)\big)\circ L_i^{-1}(x) ~~\forall~~ x~~ \in I_i,~~ i \in \mathbb{N}_N.$$
Therefore,
\begin{equation*}
\begin{aligned}
f^{\alpha}_{\Delta,B_m}(x)- g^{\alpha}_{\Delta,B_m}(x) =& f(x)- g(x)+\alpha_i(f^{\alpha}_{\Delta,B_m}- g^{\alpha}_{\Delta,B_m})\circ L_i^{-1}(x)\\&+\alpha_i \big(B_m(g)- B_m(f)\big)\circ L_i^{-1}(x),
\end{aligned}
\end{equation*}
for all $x \in I_i,~~ i \in \mathbb{N}_N.$
Further, we deduce
$$\|f^{\alpha}_{\Delta,B_m}- g^{\alpha}_{\Delta,B_m}\|_{\infty} \le \frac{1+|\alpha|_{\infty}\|B_m\|}{1- |\alpha|_{\infty}} \|f-g\|_{\infty}.$$
Since $\|B_m \| \le 1 ,$ we get
$$\|f^{\alpha}_{\Delta,B_m}- g^{\alpha}_{\Delta,B_m}\|_{\infty} \le \frac{1+|\alpha|_{\infty}}{1- |\alpha|_{\infty}} \|f-g\|_{\infty}.$$
Choosing $l= \dfrac{1+ |\alpha|_{\infty}}{1-|\alpha|_{\infty}}, $ we have
$$ \mathcal{F}^\alpha(g) \subseteq \mathcal{F}^\alpha(f) +l~ \|f-g\|_{\infty}U_{\mathcal{C}(I)},$$
establishing that $\mathcal{F}^\alpha$ is a Lipschitz set-valued map.
\end{proof}
\begin{remark}
For the set-valued map $: \mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ defined by $\mathcal{F}^\alpha(f) =\{f^{\alpha}_{\Delta,B_n(f)}: n \in \mathbb{N} \}$, we have the following.
\begin{enumerate}
\item as observed in the previous theorem, $\mathcal{F}^\alpha (0) = \{0\}.$
\item $ \mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ is not single-valued. This follows by observing that for a scale vector $\alpha \neq 0$, $b \neq c$ implies $f_{\Delta,b}^\alpha \neq f_{\Delta,c}^\alpha$.
\end{enumerate}
\end{remark}
In view of the previous remark, Theorems \ref{Multhm2}-\ref{Multhm2a} provide
\begin{theorem}
The multi-valued operator $ \mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ is not convex, and hence, in particular, not linear.
\end{theorem}
Similarly, the following theorem allegedly reports that the multi-valued operator $\mathcal{F}^\alpha: \mathcal{C}(I) \rightrightarrows \mathcal{C}(I)$ is ``bounded below" and ``non-compact" whereas in the proof author deals actually with the single-valued operator $\mathcal{F}_{\Delta,B_n}^\alpha$ for each fixed $n \in \mathbb{N}$.
\begin{theorem}\textup{\cite[Theorem 4]{Vij3}}. Let $\theta=\max \{\|B_1\|, \|B_2\|,\dots ,\|B_{N_0}\|, 1+\epsilon\}.$ If $|\alpha|_{\infty} < \frac{1}{\theta},$ then $ \mathcal{F}^{\alpha}$ is bounded below and not compact.
\end{theorem}
Our search for the set-valued analogues of the property of being bounded below and compactness of a linear operator came up emptyhanded.
Let us further remark that in the case where one intends to work only with the single-valued fractal operator $\mathcal{F}_{\Delta,B_n}^\alpha$, by Theorem \ref{prelthm}, it follows that it is bounded below and not compact for any choice of the scaling vector $\alpha$ with $|\alpha|_\infty<1$. This is because of the fact that since $\|B_n\| \le 1$, the condition $|\alpha|_\infty < \|B_n\|^{-1}$ is automatically satisfied for the scaling vector $\alpha$ with $|\alpha|_\infty<1$. Therefore, one may replace the above theorem with the following, which is immediate from Theorem \ref{prelthm}.
\begin{theorem}
If $|\alpha|_\infty<1$, then for each $n \in \mathbb{N}$, the (single-valued) fractal operator $\mathcal{F}_{\Delta, B_n}^\alpha: \mathcal{C}(I)\to \mathcal{C}(I)$ is bounded below and not compact.
\end{theorem}
\subsection*{Acknowledgements}
The first author thanks the University Grants
Commission (UGC), India for financial support in the form of a Junior Research Fellowship.
|
2,869,038,155,651 | arxiv | \section{Introduction}
The scintillation that a photonic quantum state experience as it propagates through a turbulent atmosphere is a topic of considerable importance for free-space quantum communication. The evolution of the quantum state in this scenario can be considered, using a single phase screen (SPS) model \cite{paterson}, provided that the scintillation remains weak. Although the SPS model is used in most of the work that has been done in this field \cite{sr,qturb4,qturb3,pors,malik,toddbrun,leonhard}, a more accurate multiple phase screen (MPS) approach has been proposed recently \cite{ipe,iperr,lindb,notrunc}. The MPS approach is based on the principle of infinitesimal propagation, which allows one to derive an equation for the evolution of the quantum state, called the infinitesimal propagation equation (IPE). The IPE is a first-order differential equation with respect to the propagation distance, which can be solved \cite{notrunc} to obtain an expression for the density matrix of the quantum state at arbitrary propagation distances and under arbitrary turbulence conditions.
However, the derivation of the IPE employs a Markov approximation to obtain the expression for the differential equation. In this approximation it is assumed that the medium is delta-correlated with itself along the propagation direction. For the derivation of the first-order differential equation of the IPE, one effectively assumes that the infinitesimal propagation step size is larger than the intrinsic scale, which in this case is the outer scale of the turbulence. To obtain the differential equation, one takes the limit where the step size goes to zero. On the other hand, the outer scale is assumed to go to infinity, allowing one to use the Kolmogorov turbulence model. This, seems to be a clear contradiction without a suitable justification. To some extent, the fact that the refractive index fluctuations are very small and thus allows light to propagate over long distances with minimal effect, mitigates this contradictory relationship between the step size and the outer scale. Still, our understanding of the evolution of photonic quantum states in turbulence would clearly benefit from a non-Markovian approach.
The Markov approximation is deeply ingrained in the work that has been done in the propagation of classical light through turbulence. Right from the start the assumption is made that the medium is delta-correlated along the propagation direction (see for instance \cite{scintbook}) and that for this reason the refractive index power spectral density can be treated as a two-dimensional function by setting the coordinate for the third dimension to zero. Thence, the theory is developed for all aspects of optical fields in turbulence, both within weak and strong fluctuation scenarios. Although the resulting theory seems to predict the behavior of classical light in turbulence adequately for the applications and conditions under consideration, one cannot currently say whether such a Markov approximation would be adequate for the evolution of quantum light in turbulence.
It is important to note that, although the system under investigation here deals with the evolution of a quantum state, it should not be confused with a non-Markovian quantum process. The latter concerns a situation where a system interacts with an environment such that the process needs to be described as an interacting quantum theory, formulated in terms of quantum mechanics. Such non-Markovian quantum processes are in general quite complex (see for instance \cite{petru}). In contrast, the non-Markovianity that one encounters in the evolution of a quantum state through turbulence is of a simpler nature. The process is linear --- there is no interaction --- and therefore it does not have a quantum bath that acts as the environment and interacts with the system. In the case of light propagating through turbulence, the effect of the medium is simply a continuous modulation process that extends over the propagation distance.
In this paper we consider a non-Markovian approach to study the evolution of photonic quantum states propagating through turbulence. We provide the derivation of a non-Markovian IPE, which takes the form of a second-order differential equation. The resulting equation has a form that does not in general have a solution. For this reason one needs to apply some simplifications or approximations to solve the differential equation. Here, we'll show two such approaches. In the first approach we assume that the turbulence is weak, which allows one to perform a perturbative expansion of the solution for the differential equation. The weak turbulence conditions can be considered as complimentary to the SPS model, which implies strong turbulence conditions \cite{notrunc}. The second approach is to modify the functional form of the differential equation. The resulting differential equation then does have a solution. Here, we'll only consider the single-photon case for this approach.
The paper is organized as follows. In Sec.~\ref{agter}, we provide a brief review of background material, followed by a discussion of the approach that we'll use to obtain a non-Markovian equation in Sec.~\ref{niemarkov}. The derivation of the non-Markovian IPE is shown in detail in Sec.~\ref{derive}. We provide the two different approaches to find solutions for the non-Markovian IPE in Secs.~\ref{oplos} and \ref{modi}, respectively. In Sec.~\ref{disc} we discuss some pertinent aspects of these solutions, followed by some conclusions in Sec.~\ref{concl}.
\section{Background}
\label{agter}
\subsection{Notation}
The discussions in this paper include both two-dimensional functions (such as the phase functions) and three-dimensional functions (such as the refractive index fluctuations). For this reason we need to define both two-dimensional and three-dimensional vectors to represent coordinate vectors. The two-dimensional coordinate vectors are always defined in the transverse plane, perpendicular to the propagation direction, the latter being the $z$-direction. For position coordinates, the two-dimensional position vector is denoted by a bold small ${\bf x}$, while the three-dimensional position vector is denoted by a bold capital ${\bf X}$. In the Fourier domain we prefer to work with spatial frequency vectors. The two-dimensional spatial frequency vector is denoted by a bold small ${\bf a}$, while the three-dimensional spatial frequency vector is denoted by a bold capital ${\bf A}$. Occasionally, we will also use the three-dimensional propagation vector, denoted by a bold capital ${\bf K}=2\pi{\bf A}$. The small $k$ is used to represent the wavenumber, which is not equal to $|{\bf K}|$.
During the analysis we'll obtain expressions for density matrices in terms of different sets of coordinates. Instead of denoting all these density matrices by the same symbol $\rho$, we rather avoid possible confusion by using different symbols $H$, $G$, etc. to represent the density matrices, depending on their arguments. We only use $\rho$ to represent the density matrix in generic discussions.
\subsection{Scintillation}
\label{spsagter}
For a thin enough slab of the turbulent medium, one can represent the scintillation process as a phase modulation. The phase functions that represent the turbulent medium in such a modulation process are random functions taken from an ensemble of such functions. Each one is obtained from an element of the ensemble of refractive index fluctuations $\delta n({\bf X})$, by an integration along the direction of propagation --- the $z$-direction. The phase functions are therefore defined by
\begin{equation}
\theta({\bf x}) = k \int_0^z \delta n({\bf X})\ {\rm d}z ,
\label{faseint}
\end{equation}
where $k$ is the wavenumber, given as $k=2\pi/\lambda$ in terms of the wavelength $\lambda$.
In the calculations of the evolution process, one often finds ensemble averages over phase functions, which give rise to the phase structure function in the following way
\begin{equation}
{\cal E}\!\left\{\exp\left[i\theta({\bf x}_1)-i\theta({\bf x}_2)\right]\right\} = \exp\left[-\frac{1}{2} D_{\theta} (\Delta {\bf x}) \right] ,
\label{expavg}
\end{equation}
where $\Delta {\bf x} = {\bf x}_1-{\bf x}_2$. Here
\begin{equation}
D_{\theta} (\Delta {\bf x}) = {\cal E}\!\left\{\right[\theta({\bf x}_1)-\theta({\bf x}_2)\left]^2\right\} ,
\label{strukt}
\end{equation}
is the phase structure function, which is related to the phase autocorrelation function
\begin{equation}
D_{\theta} (\Delta {\bf x}) = 2 B_{\theta} \left(0\right) - 2 B_{\theta} (\Delta {\bf x}) ,
\label{dtnabt}
\end{equation}
The phase autocorrelation function is given by
\begin{equation}
B_{\theta} (\Delta {\bf x}) = {\cal E}\!\left\{\theta({\bf x}_1)\theta({\bf x}_2)\right\} .
\label{kordeft}
\end{equation}
It is also referred to as a covariance function, because these random functions are assumed to have zero mean.
A similar relationship exists between the refractive index structure function and the refractive index autocorrelation function
\begin{equation}
D_n (\Delta {\bf X}) = 2 B_n \left(0\right) - 2 B_n (\Delta {\bf X}) ,
\label{dnnabn}
\end{equation}
where $\Delta {\bf X} = {\bf X}_1-{\bf X}_2$. The refractive index autocorrelation function is defined as
\begin{equation}
B_n (\Delta {\bf X}) = {\cal E}\!\left\{\delta n({\bf X}_1)\delta n({\bf X}_2)\right\} ,
\label{kordefn}
\end{equation}
and the refractive index structure function in the Kolmogorov theory is given by
\begin{equation}
D_n(\Delta {\bf X}) = C_n^2 |\Delta {\bf X}|^{2/3} .
\label{dndef}
\end{equation}
Using Eqs.~(\ref{faseint}) and (\ref{kordeft}), we express the two-dimensional phase autocorrelation function in terms of the three-dimensional refractive index autocorrelation function:
\begin{eqnarray}
B_{\theta} (\Delta {\bf x}) & = & k^2\!\int_0^z\!\!\int_0^z\!{\cal E}\!\left\{\delta n({\bf X}_1)\delta n({\bf X}_2)\right\} {\rm d}z_1 {\rm d}z_2 \nonumber \\
& = & k^2 \int_0^z \int_0^z B_n (\Delta {\bf X})\ {\rm d}z_1\ {\rm d}z_2 .
\label{kortnan}
\end{eqnarray}
Autocorrelation functions are related to power spectral density functions by the Wiener-Kinchine theorem \cite{statopt}. For the refractive index autocorrelation function we have
\begin{equation}
B_n ({\bf X}) = \int \Phi_n({\bf K}) \exp(-i 2\pi {\bf A}\cdot{\bf X})\ {\rm d}^3 a ,
\label{wkn}
\end{equation}
where $\Phi_n({\bf K})$ is the refractive index power spectral density, which, in the Kolmogorov theory, reads \cite{scintbook}
\begin{equation}
\Phi_n ({\bf K}) = 0.033 (2\pi)^3 C_n^2 |{\bf K}|^{-11/3} ,
\label{klmgrv}
\end{equation}
where $C_n^2$ is the refractive index structure constant and the extra $(2\pi)^3$ factor is due to a difference in the definition of the Fourier transform \cite{iperr}. For the phase autocorrelation function we have
\begin{equation}
B_{\theta} ({\bf x}) = \int \Phi_{\theta}({\bf a}) \exp(-i 2\pi {\bf a}\cdot{\bf x})\ {\rm d}^2 a ,
\label{wkt}
\end{equation}
where $\Phi_{\theta}({\bf a})$ is the phase power spectral density. Using Eqs.~(\ref{kortnan}), (\ref{wkn}) and (\ref{wkt}), one can express the phase autocorrelation function in terms of the refractive index power spectral density, which is given by
\begin{eqnarray}
\Phi_{\theta} ({\bf a}) & = & k^2 \int \int_{z_0}^{z} \int_{z_0}^{z} \exp[-i 2\pi (z_1-z_2) c] \nonumber \\
& & \times \Phi_n({\bf K})\ {\rm d} z_2\ {\rm d} z_1\ {\rm d} c .
\label{verwant}
\end{eqnarray}
The integrals over $z$ indicate that the refractive index fluctuations over the entire propagation path up to $z$ contribute to the behavior at $z$.
\subsection{Multiple phase screens}
\label{mpsagter}
The infinitesimal propagation principle, which allows a multiple-phase-screen approach, follows from considering the change in the photonic state due to an infinitesimal propagation through the medium. The operation of such an infinitesimal propagation on the density operator can be expressed by
\begin{equation}
\hat{\rho}(z) \rightarrow \hat{\rho}(z+\delta z) = dU \hat{\rho}(z) dU^{\dag} ,
\label{infquop}
\end{equation}
where $dU$ is a unitary operator representing the infinitesimal propagation through the turbulent medium. When the density operator is expressed as a density matrix in terms of some arbitrary discrete basis $\ket{m}$, the output density matrix elements, after the infinitesimal propagation, are given by
\begin{equation}
\rho_{mn}(z+\delta z) = \sum_{pq} \bra{m}dU\ket{p} \rho_{pq}(z) \bra{q}dU^{\dag}\ket{n} .
\label{infpsom}
\end{equation}
Using the paraxial wave equation in an inhomogeneous medium, given by \cite{scintbook}
\begin{equation}
\nabla_T^2 g({\bf X}) - i 2k\partial_z g({\bf X}) + 2k^2 \delta n({\bf X}) g({\bf X}) = 0 ,
\label{eomturb}
\end{equation}
where $g({\bf X})$ is the scalar electric field and $\delta n({\bf X})$ is the refractive index fluctuations, one can show that \cite{lindb}
\begin{equation}
\bra{m}dU\ket{p} = \delta_{mp} + i \delta z\ {\cal P}_{mp} + \delta z\ {\cal L}_{mp} ,
\label{infudef}
\end{equation}
where
\begin{equation}
{\cal P}_{mp}(z) \triangleq {2\pi^2 \over k} \int |{\bf a}|^2 G_m^*({\bf a},z) G_p({\bf a},z)\ {{\rm d}^2 a}
\label{kindef}
\end{equation}
and
\begin{equation}
{\cal L}_{mp}(z) \triangleq - i k \iint G_m^*({\bf a},z) N({\bf a}-{\bf a}',z) G_p({\bf a}',z)\ {{\rm d}^2 a}\ {{\rm d}^2 a'} .
\label{dispdef}
\end{equation}
Here, $G_m({\bf a},z)$ and $N({\bf a},z)$ represent the two-dimensional transverse Fourier transforms of $g_m({\bf X})=\braket{x}{m}$ and $\delta n({\bf X})$, respectively.
The infinitesimal propagation of the density operator then leads to the following equation for each element in the ensemble \cite{lindb}
\begin{eqnarray}
\rho_{mn}(z_0+\delta z) & = & \rho_{mn}(z_0) + i \delta z \left[ {\cal P}, \rho(z_0) \right]_{mn} \nonumber \\
& & + \delta z \sum_p \left[ {\cal L}_{mp}(z_0) \rho_{pn}(z_0) \right. \nonumber \\
& & \left. + \rho_{mp}(z_0) {\cal L}_{pn}^{\dag}(z_0) \right] .
\label{lb1}
\end{eqnarray}
The right-hand side of Eq.~(\ref{lb1}) can be represented by an integral over a small range of $z$ to replace the factor of $\delta z$. If one were to compute the ensemble average of Eq.~(\ref{lb1}), the dissipative term (sum over $p$) would vanish, because ${\cal E}\!\{N\}=0$. One needs an expression with terms that are second-order in $N$ before computing the ensemble averages to have nonzero dissipative terms. The result of such ensemble averages would then contain autocorrelation functions of $N({\bf a},z)$.
\subsection{Markov approximation}
\label{markov}
The Markov approximation enters at the point where one computes the autocorrelation function of $N({\bf a},z)$
\begin{equation}
\Gamma_0({\bf a}_1,{\bf a}_2,z_1,z_2) = {\cal E}\!\{N({\bf a}_1,z_1) N^*({\bf a}_2,z_2)\} .
\label{nkor}
\end{equation}
One can model $N({\bf a},z)$ as
\begin{equation}
N({\bf a},z) = \int \left[ {\Phi_n({\bf K})\over\Delta^3} \right]^{1/2} \tilde{\chi}({\bf A}) \exp(-i 2\pi c z)\ {\rm d} c ,
\label{spek3d}
\end{equation}
where $\Delta$ is a correlation length in the frequency domain, $c$ is the `$z$-component' of ${\bf A}$ and $\tilde{\chi}({\bf A})$ is a normally distributed, delta-correlated, random complex function, with a zero mean. Hence,
\begin{equation}
{\cal E}\!\{\tilde{\chi}({\bf A}_1)\tilde{\chi}^*({\bf A}_2)\} = \Delta^3 \delta_3({\bf A}_1-{\bf A}_2) .
\label{deltkor}
\end{equation}
Since $\delta n({\bf X})$ is a real-valued function, the random complex function also obeys $\tilde{\chi}^*({\bf A})=\tilde{\chi}(-{\bf A})$.
With the aid of Eq.~(\ref{spek3d}) we write Eq.~(\ref{nkor}) as
\begin{eqnarray}
\Gamma_0({\bf a}_1,{\bf a}_2,z_1,z_2) & = & \delta_2({\bf a}_1-{\bf a}_2) \int \exp[-i 2\pi (z_1-z_2) c_1] \nonumber \\
& & \times \Phi_n({\bf K}_1)\ {\rm d} c_1 .
\label{nkorc}
\end{eqnarray}
In the Markov approximation it is assumed that only the values of the field and the medium at $z$ contribute to the behavior at $z$. This assumption implies that the refractive index fluctuations are delta-correlated along the $z$-direction. The result is that one can substitute $k_z=0$ ($c=0$) in $\Phi_n({\bf K})$. Making this substitution and evaluating the integrals in Eq.~(\ref{verwant}), one arrives at a simpler relationship given by
\begin{equation}
\Phi_{\theta} ({\bf a}) = z k^2 \Phi_n(2\pi{\bf a},0) .
\label{verwantm}
\end{equation}
The simpler expression for $\Phi_{\theta} ({\bf a})$ can in turn be used to simplify the model for $N$:
\begin{equation}
N({\bf a},z) = \tilde{\chi}({\bf a}) \left[ \frac{\Phi_{\theta}({\bf a})}{\Delta^2} \right]^{1/2} ,
\label{spek2d}
\end{equation}
where $\tilde{\chi}({\bf a})$ is now a two-dimensional random function, but other than that has the same properties as $\tilde{\chi}({\bf A})$.
The Markov approximation is introduced into Eq.~(\ref{nkorc}) by setting $k_z=0$ in $\Phi_n({\bf K})$, which gives
\begin{equation}
\Gamma_1({\bf a}_1,{\bf a}_2,z_0,z) \approx \frac{\delta z}{2} \delta_2({\bf a}_1-{\bf a}_2) \Phi_n(2\pi{\bf a}_1,0) ,
\label{gam2}
\end{equation}
where $\delta z=z-z_0$. The factor of $\delta z$ leads to a first-order differential equation --- the Markovian IPE \cite{lindb}.
\section{Non-Markovian approach}
\label{niemarkov}
For the non-Markovian approach, we proceed without setting $k_z=0$ in $\Phi_n({\bf K})$. As a result, the integrals in Eq.~(\ref{verwant}) need to be evaluated by using an explicit expression for $\Phi_n({\bf K})$. On the other hand, one can exploit the fact that $\delta z$ is small for infinitesimal propagations. This allows one to expand Eq.~(\ref{nkorc}) up to leading order in $\delta z$. As a result we have
\begin{equation}
\Gamma_1({\bf a}_1,{\bf a}_2,z_0,z) \approx {\delta z^2\over 2} \delta_2 ({\bf a}_1-{\bf a}_2) \int \Phi_n({\bf K}_1)\ {\rm d} c_1 .
\label{gam3}
\end{equation}
The factor of $\delta z^2$ (instead of just $\delta z$) suggests that the non-Markovian equation could be a second-order differential equation.
In the derivation in Sec.~\ref{derive} and Appendix~\ref{eenaf}, we'll find that $z_1=z_2=z$. Thus, the correlation function in Eq.~(\ref{nkor}) or (\ref{nkorc}) becomes independent of $z$, so that
\begin{eqnarray}
\Gamma_0({\bf a}_1,{\bf a}_2) & = & {\cal E}\!\{N({\bf a}_1,z) N^*({\bf a}_2,z)\} \nonumber \\
& = & \delta_2({\bf a}_1-{\bf a}_2) \Phi_1({\bf a}_1) ,
\label{nmkor}
\end{eqnarray}
where
\begin{equation}
\Phi_1({\bf a}_1) \triangleq \int \Phi_n({\bf K}_1)\ {\rm d} c_1 .
\label{phi1def}
\end{equation}
Master equations for non-Markovian systems (for example, the Nakajima-Zwanzig equation \cite{naka,zwanz}) in general have the form
\begin{equation}
\partial_z \rho(z) = \int_{z_0}^{z} K(z,z') \rho(z')\ {\rm d} z ,
\label{gennm}
\end{equation}
where $K(z,z')$ is a super-operator that represents the memory in the system. Taking another derivative with respect to $z$ on both sides,
\begin{equation}
\partial_z^2 \rho(z) = K(z,z) \rho(z) + \int_{z_0}^{z} \left[ \partial_z K(z,z') \right] \rho(z')\ {\rm d} z ,
\label{gennm2}
\end{equation}
one finds that the right-hand side still contains an integral over $z$. It is therefore not in general possible to describe non-Markovian systems in terms of a pure second-order differential equation, having no integrals over $z$. If, however, $K(z,z')=K(z')$ in Eq.~(\ref{gennm}), one would obtain
\begin{equation}
\partial_z^2 \rho(z) = K(z) \rho(z) ,
\label{gennm3}
\end{equation}
which does not contain an integral over $z$.
In the particular case under consideration, it is possible to obtain a pure second-order differential equation, having no integrals over $z$. Consider Eq.~(\ref{lb1}), expressed as a first-order differential equation
\begin{eqnarray}
\partial_z \rho_{mn}(z) & = & i \left[ {\cal P}(z), \rho(z) \right]_{mn} + \sum_p \left[ {\cal L}_{mp}(z) \rho_{pn}(z) \right. \nonumber \\
& & \left. + \rho_{mp}(z) {\cal L}_{pn}^{\dag}(z) \right] .
\label{lb2}
\end{eqnarray}
If one differentiates Eq.~(\ref{lb2}) on both sides with respect to $z$, replaces the resulting first derivatives $\partial_z \rho(z)$ again by Eq.~(\ref{lb2}) and computes the ensemble average, by taking into account that ${\cal E}\!\{N\} = {\cal E}\!\{\partial_z N\} = 0$, one obtains a result that reads
\begin{eqnarray}
\partial_z^2 \rho_{mn}(z) & = & i \left[ \partial_z {\cal P}(z), \rho(z) \right]_{mn} - \left[ {\cal P}(z), \left[ {\cal P}(z), \rho(z) \right] \right]_{mn} \nonumber \\
& & + \sum_{p,q} {\cal E}\!\left\{ 2 {\cal L}_{mp}(z) \rho_{pq}(z) {\cal L}_{qn}^{\dag}(z) \right. \nonumber \\
& & - {\cal L}_{mp}^{\dag}(z) {\cal L}_{pq}(z) \rho_{qn}(z) \nonumber \\
& & \left. - \rho_{mp}(z) {\cal L}_{pq}^{\dag}(z) {\cal L}_{qn}(z) \right\} .
\label{lb3}
\end{eqnarray}
Here we used the fact that the ${\cal L}$'s are anti-hermitian: ${\cal L}_{mn}^{\dag}=-{\cal L}_{mn}$. We note that Eq.~(\ref{lb3}) does not contain any integrals over $z$.
Using Eqs.~(\ref{dispdef}), (\ref{spek2d}) and (\ref{nmkor}), we compute the ensemble average over the ${\cal L}_{pq}$'s. The result is \cite{lindb}
\begin{eqnarray}
\Lambda_{mnpq} & \triangleq & {\cal E}\!\{{\cal L}_{mp}(z) {\cal L}_{qn}^{\dag}(z)\} \nonumber \\
& = & k^2 \iiint G_m^*({\bf a}_1+{\bf a}_2,z) G_p({\bf a}_2,z) G_q^*({\bf a}_3,z) \nonumber \\
& & \times G_n({\bf a}_3+{\bf a}_1,z) \Phi_1({\bf a}_1)\ {{\rm d}^2 a_1}\ {{\rm d}^2 a_2}\ {{\rm d}^2 a_3} \nonumber \\
& = & k^2 \int W_{mp}({\bf a},z) W_{nq}^*({\bf a},z) \Phi_1({\bf a})\ {{\rm d}^2 a} ,
\label{verll0}
\end{eqnarray}
where
\begin{equation}
W_{ab}({\bf a},z) \triangleq \int G_a^*({\bf a}'+{\bf a},z) G_b({\bf a}',z)\ {{\rm d}^2 a'} .
\label{wdef}
\end{equation}
When two of the indices on the ${\cal L}_{pq}$'s are contracted, one can use the orthogonality and completeness conditions of the modal basis to show that \cite{lindb}
\begin{equation}
\sum_p \Lambda_{mnpp} = \delta_{mn} k^2 \int \Phi_1({\bf a})\ {{\rm d}^2 a} \triangleq \delta_{mn} \Lambda_T .
\label{verll2}
\end{equation}
Substituting Eqs.~(\ref{verll0}) and (\ref{verll2}) into Eq.~(\ref{lb3}), we obtain
\begin{eqnarray}
\partial_z^2 \rho_{mn}(z) & = & i \left[ \partial_z {\cal P}(z), \rho(z) \right]_{mn} - \left[ {\cal P}(z), \left[ {\cal P}(z), \rho(z) \right] \right]_{mn} \nonumber \\
& & + 2 k^2 \int \sum_{p,q} W_{mp}({\bf a},z) \rho_{pq}(z) W_{qn}^{\dag}({\bf a},z) \nonumber \\
& & \times \Phi_1({\bf a})\ {{\rm d}^2 a} - 2 \Lambda_T \rho_{mn}(z) .
\label{lb4}
\end{eqnarray}
The result in Eq.~(\ref{lb4}) is a general expression for the non-Markovian IPE in an arbitrary discrete basis for a single photon propagating through turbulence.
Below, we'll repeat this derivation in detail, but we'll perform the derivation in the plane wave basis (Fourier domain), which is more beneficial for the purpose of finding solutions for the differential equation \cite{notrunc}.
\section{The non-Markovian IPE}
\label{derive}
In the transverse Fourier domain, the paraxial wave equation in an inhomogeneous medium is given by
\begin{equation}
\partial_z G({\bf a},z) = i\pi\lambda |{\bf a}|^2 G({\bf a},z) - i k N({\bf a},z) \star G({\bf a},z) ,
\label{transgft}
\end{equation}
where $\star$ represents convolution. The first term on the right-hand side of Eq.~(\ref{transgft}) represents free-space propagation and the second term produces distortions due to the effect of the medium.
It is convenient to work in a `rotating' frame in which the free-space term is removed. This is done by using
\begin{equation}
G({\bf a},z) = F({\bf a},z) \exp\left(i\pi\lambda z |{\bf a}|^2\right) ,
\label{irrot}
\end{equation}
to convert the paraxial wave equation in Eq.~(\ref{transgft}) into
\begin{eqnarray}
\partial_z F({\bf a},z) & = & - i k \int N({\bf a}-{\bf u},z) F({\bf u},z) \nonumber \\
& & \times \exp\left[-i\pi\lambda z \left(|{\bf a}|^2-|{\bf u}|^2\right) \right]\ {\rm d}^2 u .
\label{transgft0}
\end{eqnarray}
To derive a non-Markovian IPE for a single-photon input state from Eq.~(\ref{transgft0}), we assume that the input is a single-photon pure state in the plane wave basis, given (in the rotating frame) by
\begin{equation}
R({\bf a}_1,{\bf a}_2,z) = F({\bf a}_1,z) F^*({\bf a}_2,z) .
\label{enkfot}
\end{equation}
The derivation of the non-Markovian IPE for the single-photon input state in Eq.~(\ref{enkfot}) is shown in Appendix~\ref{eenaf}. The result is
\begin{eqnarray}
\partial_z^2 R({\bf a}_1,{\bf a}_2,z) & = & 2 k^2 \int \left\{ R({\bf a}_1-{\bf u},{\bf a}_2-{\bf u},z) \right. \nonumber \\
& & \times \exp\left[-i 2\pi\lambda z \left({\bf a}_1-{\bf a}_2\right)\cdot{\bf u} \right] \nonumber \\
& & \left. - R({\bf a}_1,{\bf a}_2,z) \right\} \Phi_1({\bf u})\ {\rm d}^2 u .
\label{dveenfot}
\end{eqnarray}
Although it has an integral over the Fourier variables ${\bf u}$, the non-Markovian IPE is a second-order differential equation without any integrals over $z$.
\begin{widetext}
The expression, equivalent to Eq.~(\ref{dveenfot}), for the two-photon states is given by
\begin{eqnarray}
\partial_z^2 R({\bf a}_1,{\bf a}_2,{\bf a}_3,{\bf a}_4,z) & = & 2 k^2 \int \left\{ R({\bf a}_1-{\bf u},{\bf a}_2-{\bf u},{\bf a}_3,{\bf a}_4,z) \exp\left[-i 2\pi\lambda z \left({\bf a}_1-{\bf a}_2\right)\cdot{\bf u} \right] \right. \nonumber \\
& & + R({\bf a}_1,{\bf a}_2,{\bf a}_3-{\bf u},{\bf a}_4-{\bf u},z) \exp\left[-i 2\pi\lambda z \left({\bf a}_3-{\bf a}_4\right)\cdot{\bf u} \right] \nonumber \\
& & \left. - 2 R({\bf a}_1,{\bf a}_2,{\bf a}_3,{\bf a}_4,z) \right\} \Phi_1({\bf u})\ {\rm d}^2 u .
\label{dvtweefot}
\end{eqnarray}
\end{widetext}
To aid the solution of the non-Markovian IPE we cast it in a form that decouples the $z$-dependence from the Fourier variables. This is done in a similar way as in \cite{notrunc}, by performing the following steps.
First, we redefine the Fourier variables (spatial frequencies) in terms of sums and differences, defined by
\begin{eqnarray}
{\bf a}_1 & = & {\bf a}+\frac{1}{2}{\bf a}_d \\
{\bf a}_2 & = & {\bf a}-\frac{1}{2}{\bf a}_d .
\end{eqnarray}
The state is then also refined
\begin{eqnarray}
R({\bf a}_1,{\bf a}_2,z) & = & R({\bf a}+{\bf a}_d/2,{\bf a}-{\bf a}_d/2,z) \nonumber \\
& \triangleq & S({\bf a},{\bf a}_d,z) .
\end{eqnarray}
The expression in Eq.~(\ref{dveenfot}) then becomes
\begin{eqnarray}
\partial_z^2 S({\bf a},{\bf a}_d,z) & = & 2 k^2 \int \left[ S({\bf a}-{\bf u},{\bf a}_d,z) \exp\left(-i 2\pi\lambda z {\bf a}_d\cdot{\bf u} \right) \right. \nonumber \\
& & \left. - S({\bf a},{\bf a}_d,z) \right] \Phi_1({\bf u})\ {\rm d}^2 u .
\label{dv7}
\end{eqnarray}
The next step is to perform an inverse Fourier transform with respect to the sum coordinates:
\begin{equation}
H({\bf x},{\bf a}_d,z) = \int S({\bf a},{\bf a}_d,z) \exp(-i 2\pi {\bf a}\cdot {\bf x})\ {\rm d}^2 a .
\end{equation}
Equation~(\ref{dv7}) then reads
\begin{equation}
\partial_z^2 H({\bf x},{\bf a}_d,z) = - 2 k^2 Q(\lambda z {\bf a}_d+{\bf x}) H({\bf x},{\bf a}_d,z) ,
\label{dv9}
\end{equation}
where
\begin{equation}
Q({\bf x}) \triangleq \int \left[ 1-\exp(-i 2\pi {\bf x}\cdot{\bf u}) \right] \Phi_1({\bf u})\ {\rm d}^2 u .
\label{qint}
\end{equation}
By combining the integral in Eq.~(\ref{qint}) with the definition in Eq.~(\ref{phi1def}), we find that $Q({\bf x})$ is related to the refractive index structure function, with $z=0$
\begin{equation}
Q({\bf x}) = \frac{1}{2} D_n ({\bf x},0) = \frac{1}{2} C_n^2 |{\bf x}|^{2/3} .
\label{qint2}
\end{equation}
Using the expression in Eq.~(\ref{qint2}), we obtain an expression for the single-photon non-Markovian IPE, given by
\begin{equation}
\partial_z^2 H(z) = - k^2 C_n^2 H(z) |\lambda z {\bf a}_d+{\bf x}|^{2/3} .
\label{nmipe}
\end{equation}
where $H(z)\equiv H({\bf x},{\bf a}_d,z)$. The equivalent expression for the two-photon case is
\begin{eqnarray}
\partial_z^2 H(z) & = & - k^2 C_n^2 H(z) \left( |\lambda z {\bf a}_d+{\bf x}_1|^{2/3} \right. \nonumber \\
& & \left. + |\lambda z {\bf b}_d+{\bf x}_2|^{2/3} \right) ,
\label{nm2ipe}
\end{eqnarray}
where $H(z)\equiv H({\bf x}_1,{\bf a}_d,{\bf x}_2,{\bf b}_d,z)$.
\section{Perturbative solution}
\label{oplos}
For the first method to solve the differential equation in Eq.~(\ref{nmipe}), we assume that the turbulence is weak enough [$C_n$ in Eq.~(\ref{nmipe}) is small enough] to allow a perturbative approach. This approach has the benefit that it can be generalized to the two-photon case, but first we'll consider the single-photon case.
\subsection{Single-photon state}
Consider a second-order differential equation given by Eq.~(\ref{gennm3}), but with the coupling constant $g$, which is proportional to the turbulence strength, made explicit
\begin{equation}
\partial_z^2 \rho(z) = g K(z) \rho(z) .
\label{gennm4}
\end{equation}
Expand the solution as an asymptotic series in $g$
\begin{equation}
\rho(z) = \rho_0(z) + g \rho_1(z) + g^2 \rho_2(z) + .... ,
\label{pertrho}
\end{equation}
and substitute it back into Eq.~(\ref{gennm4}). Setting g=0, one obtains the zeroth-order perturbation
\begin{equation}
\partial_z^2 \rho_0(z) = 0 .
\label{pert0}
\end{equation}
Its solution must satisfy the initial conditions.
The two initial conditions for the second-order differential equation in Eq.~(\ref{gennm4}) can be stated as follows:
\begin{enumerate}
\item the initial rate of change of the state is zero
\begin{equation}
\left.\partial_z \rho(z) \right|_{z=0}=0 ,
\label{randv1}
\end{equation}
and
\item the state at $z=0$ is given by the input state
\begin{equation}
\rho(0) = \rho_{\rm in} .
\label{randv2}
\end{equation}
\end{enumerate}
The solution of Eq.~(\ref{pert0}) that satisfies these initial conditions is
\begin{equation}
\rho_0(z) = \rho_{\rm in} .
\label{popl0}
\end{equation}
The first-order perturbation is obtained by taking a derivative with respect to $g$ before setting $g=0$. The resulting equation
\begin{equation}
\partial_z^2 \rho_1 (z) = K(z) \rho_0(z) = K(z) \rho_{\rm in} ,
\label{pert1}
\end{equation}
has a solution satisfying the initial conditions, given by
\begin{equation}
\rho_1(z) = \rho_{\rm in} \int_0^z \int_0^{z_2} K(z_1)\ {\rm d} z_1\ {\rm d} z_2 .
\label{popl1}
\end{equation}
Here we'll only go to sub-leading order in $g$. Therefore, our total solution, obtained from Eqs.~(\ref{popl0}) and (\ref{popl1}), is
\begin{equation}
\rho(z) = \rho_{\rm in} \left[ 1+\int_0^z \int_0^{z_2} K(z_1)\ {\rm d} z_1\ {\rm d} z_2 \right] ,
\label{pertopl}
\end{equation}
where we reabsorbed $g$ into $K(z)$.
To obtain an explicit expression for Eq.~(\ref{pertopl}), one needs to evaluate the double $z$-integration of $K(z)$. The expression for $K(z)$ for the single-photon case, according to Eq.~(\ref{nmipe}), is
\begin{equation}
K(z) = - k^2 C_n^2 \left(|\lambda z {\bf a}_d+{\bf x}|^2\right)^{1/3} .
\label{kdef}
\end{equation}
The solution in Eq.~(\ref{pertopl}) can thus be expressed as
\begin{equation}
\rho(z) = \rho_{\rm in} \left[ 1 - k^2 C_n^2 \int_0^z \int_0^{z_2} P(z)^{1/3}\ {\rm d} z_1\ {\rm d} z_2 \right]
\label{pertopln}
\end{equation}
where
\begin{equation}
P(z) = |\lambda z {\bf a}_d+{\bf x}|^2 =\lambda^2 z^2 |{\bf a}_d|^2 + 2\lambda z ({\bf a}_d\cdot{\bf x})+|{\bf x}|^2 .
\label{pol0}
\end{equation}
The evaluation of the integrations over $z$ in Eq.~(\ref{pertopln}) is briefly discussed in Appendix~\ref{strukint}. The result, in the $H$-notation of Eq.~(\ref{nmipe}), is given by
\begin{widetext}
\begin{eqnarray}
H({\bf x},{\bf a}_d,z) & = & H_{\rm in}({\bf x},{\bf a}_d) \left\{ 1 + \frac{3}{8} \frac{k^2 C_n^2 \left[\left(|{\bf a}_d|^2 \lambda z + {\bf a}_d\cdot{\bf x}\right)^2+\left({\bf a}_d\times{\bf x}\right)^2\right]^{4/3}-|{\bf a}_d|^{8/3} |{\bf x}|^{8/3}}{\lambda^2 |{\bf a}_d|^{14/3}} \right. \nonumber \\
& & + \frac{k^2 C_n^2 \left({\bf a}_d\cdot{\bf x}\right) \left(|{\bf a}_d|^2 \lambda z + {\bf a}_d\cdot{\bf x}\right) \left|{\bf a}_d\times{\bf x}\right|^{2/3}}{\lambda^2 |{\bf a}_d|^{14/3}} {_2{\rm F}_1} \left[\left(\frac{-1}{3},\frac{1}{2}\right),\left(\frac{3}{2}\right),\frac{-\left({\bf a}_d\cdot{\bf x}\right)^2}{\left({\bf a}_d\times{\bf x}\right)^2}\right] \nonumber \\
& & \left. - \frac{k^2 C_n^2 \left(|{\bf a}_d|^2 \lambda z + {\bf a}_d\cdot{\bf x}\right)^2 \left|{\bf a}_d\times{\bf x}\right|^{2/3}}{\lambda^2 |{\bf a}_d|^{14/3}} {_2{\rm F}_1} \left[\left(\frac{-1}{3},\frac{1}{2}\right),\left(\frac{3}{2}\right),\frac{-\left(|{\bf a}_d|^2 \lambda z + {\bf a}_d\cdot{\bf x}\right)^2}{\left({\bf a}_d\times{\bf x}\right)^2}\right] \right\} ,
\label{pertnaz}
\end{eqnarray}
\end{widetext}
where $H_{\rm in}({\bf x},{\bf a}_d)$ is the input state, ${_2{\rm F}_1}$ denotes a hyper-geometrical function and where we used the identity
\begin{equation}
({\bf A}\cdot{\bf B})^2+|{\bf A}\times{\bf B}|^2 = |{\bf A}|^2 |{\bf B}|^2 .
\label{vekid}
\end{equation}
The expression in Eq.~(\ref{pertnaz}) depends on a mixture of Fourier and position domain coordinates. It is preferable to obtain an expression that only depends on position domain coordinates. The expression in Eq.~(\ref{pertnaz}) has the form
\begin{equation}
H({\bf x},{\bf a}_d,z) = H_{\rm in}({\bf x},{\bf a}_d) T({\bf x},{\bf a}_d,z) .
\label{genopl0}
\end{equation}
where the function $T(\cdot)$ is given by the part in curly brackets in Eq.~(\ref{pertnaz}).
The general approach to obtain a position domain expression for the solution of the non-Markovian IPE, is to perform the steps of Sec.~\ref{derive} in reverse, keeping the expressions in terms of sum and difference coordinates all the way through. Starting from Eq.~(\ref{genopl0}), we first convert the expression completely to the Fourier domain by performing Fourier transforms with respect to ${\bf x}$. Then we add the free-space phase factor and perform an inverse Fourier transform on all coordinates to obtain the position space expression. Finally, one may simplify the expression by redefining the integration variables, using for instance ${\bf u}_s\rightarrow \lambda z {\bf a}$ and/or ${\bf u}_d\rightarrow {\bf x}_d-{\bf u}$. The resulting position domain expression reads
\begin{eqnarray}
G({\bf x}_s,{\bf x}_d,z) & = & \int G_{\rm in}({\bf x}_s-\lambda z {\bf a},{\bf u}) T \left({\bf u},\frac{{\bf x}_d-{\bf u}}{\lambda z},z\right) \nonumber \\
& & \times \exp \left[ -i 2\pi {\bf a}\cdot \left( {\bf x}_d-{\bf u} \right) \right]\ {\rm d}^2 a\ {\rm d}^2 u ,
\label{xopl}
\end{eqnarray}
in terms of the sums and differences of the position coordinates
\begin{eqnarray}
{\bf x}_s & = & \frac{1}{2} \left({\bf x}_1+{\bf x}_2\right) \\
{\bf x}_d & = & {\bf x}_1-{\bf x}_2 .
\end{eqnarray}
In Eq.~(\ref{xopl}), $T(\cdot)$, which is the same function that appears in Eq.~(\ref{genopl0}), serves as a kernel function for the propagation process. We'll refer to it as the (non-Markovian) turbulence propagation kernel.
Comparing the arguments of the turbulence propagation kernels in Eq.~(\ref{genopl0}) and (\ref{xopl}), one finds that the position domain expression requires the replacements ${\bf x}\rightarrow {\bf u}$ and ${\bf a}_d\rightarrow ({\bf x}_d-{\bf u})/\lambda z$. This leads to the following replacements for the quantities appearing in Eq.~(\ref{pertnaz})
\begin{eqnarray}
|{\bf x}| & \rightarrow & |{\bf u}| \nonumber \\
|{\bf a}_d| & \rightarrow & \frac{|{\bf x}_d-{\bf u}|}{\lambda z} \nonumber \\
({\bf a}_d\cdot{\bf x}) & \rightarrow & \frac{\left({\bf x}_d\cdot{\bf u}-|{\bf u}|^2\right)}{\lambda z} .
\label{verv0}
\end{eqnarray}
We use these replacements to obtain a position domain expression for the turbulence propagation kernel
\begin{widetext}
\begin{eqnarray}
T \left({\bf u},\frac{{\bf x}_d-{\bf u}}{\lambda z},z\right) & = & 1 + \frac{g t^2}{w_0^{2/3}|{\bf x}_d-{\bf u}|^{14/3}} \left\{ \frac{3}{8} \left( |{\bf x}_d|^{8/3}-|{\bf u}|^{8/3}\right) |{\bf x}_d-{\bf u}|^{8/3} \right. \nonumber \\
& & + \left({\bf x}_d\cdot{\bf u}-|{\bf u}|^2\right) \left(|{\bf x}_d|^2-{\bf x}_d\cdot{\bf u}\right) |{\bf x}_d\times{\bf u}|^{2/3} {_2{\rm F}_1} \left[\left(\frac{-1}{3},\frac{1}{2}\right),\left(\frac{3}{2}\right),\frac{-\left({\bf x}_d\cdot{\bf u}-|{\bf u}|^2\right)^2}{|{\bf x}_d\times{\bf u}|^2}\right] \nonumber \\
& & \left. - \left(|{\bf x}_d|^2-{\bf x}_d\cdot{\bf u}\right)^2 |{\bf x}_d\times{\bf u}|^{2/3} {_2{\rm F}_1} \left[\left(\frac{-1}{3},\frac{1}{2}\right),\left(\frac{3}{2}\right),\frac{-\left(|{\bf x}_d|^2-{\bf x}_d\cdot{\bf u}\right)^2}{|{\bf x}_d\times{\bf u}|^2}\right] \right\} ,
\label{kernel0}
\end{eqnarray}
\end{widetext}
where
\begin{equation}
t \triangleq \frac{\lambda z}{\pi w_0^2} ,
\label{tnaz}
\end{equation}
is a normalized propagation distance and
\begin{equation}
g \triangleq \frac{4{\cal T}}{\Theta^4} ,
\label{gdef}
\end{equation}
is a dimensionless coupling constant. The expression of this coupling constant is obtained by considering the complete expression Eq.~(\ref{pertopln}) in terms of dimensionless quantities. The details of this analysis is provided in Appendix \ref{koppel}.
Notice that in Eq.~(\ref{kernel0}) the propagation distance only appears together with the coupling constant in front of the dissipative term and not anywhere inside the curly brackets. In fact, there are no dimension parameters inside the curly brackets, only the difference in position coordinates ${\bf x}_d={\bf x}_1-{\bf x}_2$ and the integration variables ${\bf u}$. However, some $z$ dependence also enters via the arguments of the input state in Eq.~(\ref{xopl}).
\subsection{Two photon state}
The result in Eq.~(\ref{kernel0}), together with Eq.~(\ref{xopl}), represents a perturbative solution for the single-photon differential equation given in Eq.~(\ref{nmipe}). One can generalize this solution to the two-photon case. The general perturbative solution for the two-photon case, analogous to Eq.~(\ref{pertopl}), is
\begin{eqnarray}
\rho(z) & = & \rho_{\rm in} \left[ 1+\int_0^z \int_0^{z_2} K_1(z_1)\ {\rm d} z_1\ {\rm d} z_2 \right. \nonumber \\
& & \left. + \int_0^z \int_0^{z_2} K_2(z_1)\ {\rm d} z_1\ {\rm d} z_2 \right] ,
\label{pert2opl}
\end{eqnarray}
where $K_1(z)$ and $K_2(z)$ are associated with the two photons, respectively.
The single-photon turbulence propagation kernel, given in Eq.~(\ref{kernel0}), has the form
\begin{equation}
T \left({\bf u},\frac{{\bf x}_d-{\bf u}}{\lambda z},z\right) = 1 + g W({\bf u},{\bf x}_d,z) ,
\label{trubkern1}
\end{equation}
where $W(\cdot)$ is given by the part in Eq.~(\ref{kernel0}) that is multiplied by $g$. For the two-photon case, the form of the expression of the turbulence propagation kernel, simply becomes
\begin{eqnarray}
T_2 \left({\bf u}_1,{\bf x}_{1d},{\bf u}_2,{\bf x}_{2d},z\right) & = & 1 + g W({\bf u}_1,{\bf x}_{1d},z) \nonumber \\
& & + g W({\bf u}_2,{\bf x}_{2d},z) ,
\label{trubkern2}
\end{eqnarray}
where $W(\cdot)$ is the same function as in Eq.~(\ref{trubkern1}).
\section{Modified differential equation}
\label{modi}
Here we consider an alternative approach to solve the differential equation in Eq.~(\ref{nmipe}). The idea is that, although the differential equation in Eq.~(\ref{gennm3}) does not have a solution in general, it does have solutions when $K(z)$ has a particular functional form. In what follows, we'll consider one such example.
The differential equation in Eq.~(\ref{nmipe}) can be written as
\begin{equation}
\partial_z^2 \rho(z) = - k^2 C_n^2 P(z)^{1/3} \rho(z) ,
\label{gennm5}
\end{equation}
where $P(z)$ is given in Eq.~(\ref{pol0}). With the aid of Eq.~(\ref{vekid}), one can express $P(z)$ as
\begin{equation}
P(z) = \frac{\left[z\lambda|{\bf a}_d|^2 + ({\bf a}_d\cdot{\bf x})\right]^2 + |{\bf a}_d\times{\bf x}|^2}{|{\bf a}_d|^2} .
\label{pol1}
\end{equation}
A special case that does allow a solution for Eq.~(\ref{nmipe}), is when the cross-product term in Eq.~(\ref{pol1}) is neglected. The differential equation then has the form
\begin{equation}
\partial_z^2 \rho(z) = - \alpha^2 (z+\zeta)^{2/3} \rho(z) ,
\label{dvf}
\end{equation}
where
\begin{eqnarray}
\alpha & = & \frac{2\pi |{\bf a}_d|^{1/3} \sqrt{C_n^2}}{\lambda^{2/3}} \label{alphadef} \\
\zeta & = & \frac{({\bf a}_d\cdot{\bf x})}{\lambda|{\bf a}_d|^2} . \label{z0def}
\end{eqnarray}
The differential equation in Eq.~(\ref{dvf}) has the solution,
\begin{eqnarray}
\rho(z) & = & C_1 \sqrt{z+\zeta}\ {\rm J}_{3/8}\left[\frac{3\alpha}{4} (z+\zeta)^{4/3} \right] \nonumber \\
& & + C_2 \sqrt{z+\zeta}\ {\rm Y}_{3/8}\left[\frac{3\alpha}{4} (z+\zeta)^{4/3} \right] ,
\label{genopl}
\end{eqnarray}
where ${\rm J}_{\nu}$ and ${\rm Y}_{\nu}$ are Bessel functions of the first and second kind, respectively, and $C_1$ and $C_2$ are constant to be determined by the initial conditions, given in Eqs.~(\ref{randv1}) and (\ref{randv2}).
Applying the first initial condition Eq.~(\ref{randv1}), one finds that the constants much have the forms
\begin{eqnarray}
C_1 & = & C_0 {\rm Y}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) \label{f1def} \\
C_2 & = & - C_0 {\rm J}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) , \label{f2def}
\end{eqnarray}
where $C_0$ is a constant, common to both $C_1$ and $C_2$. Substituting Eqs.~(\ref{f1def}) and (\ref{f2def}) into Eq.~(\ref{genopl}), one obtains an interim expression for the solution, given by
\begin{eqnarray}
\rho(z) & = & C_0 \sqrt{z+\zeta} \nonumber \\
& & \times \left\{ {\rm Y}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) {\rm J}_{3/8}\left[\frac{3\alpha}{4} (z+\zeta)^{4/3} \right] \right. \nonumber \\
& & \left. - {\rm J}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) {\rm Y}_{3/8}\left[\frac{3\alpha}{4} (z+\zeta)^{4/3} \right] \right\} . \nonumber \\
\label{opl1}
\end{eqnarray}
Now we apply the second initial condition Eq.~(\ref{randv2}) to the expression in Eq.~(\ref{opl1}) to obtain
\begin{eqnarray}
\rho(0) = \rho_{\rm in} & = & C_0 \sqrt{\zeta} \left[ {\rm Y}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) {\rm J}_{3/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) \right. \nonumber \\
& & \left. - {\rm J}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) {\rm Y}_{3/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) \right] \nonumber \\
& = & \frac{8 C_0}{3\pi \zeta^{5/6} \sqrt{\alpha}} ,
\label{randv3}
\end{eqnarray}
where we used the Wronskian
\begin{equation}
{\rm J}_{\nu+1}(z) {\rm Y}_{\nu}(z)-{\rm Y}_{\nu+1}(z) {\rm J}_{\nu}(z) = \frac{2}{\pi z} ,
\label{wronsk}
\end{equation}
to obtain the last expression in Eq.~(\ref{randv3}). It gives a relationship between $C_0$ and $\rho_{\rm in}$, which is then used to replace $C_0$ in Eq.~(\ref{opl1}). The resulting solution reads
\begin{eqnarray}
\rho(z) & = & \frac{3\pi}{8} \rho_0 \zeta^{5/6} \sqrt{\alpha} \sqrt{z+\zeta} \nonumber \\
& & \times \left\{ {\rm Y}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) {\rm J}_{3/8}\left[\frac{3\alpha}{4} (z+\zeta)^{4/3} \right] \right. \nonumber \\
& & \left. - {\rm J}_{-5/8}\left(\frac{3\alpha}{4} \zeta^{4/3} \right) {\rm Y}_{3/8}\left[\frac{3\alpha}{4} (z+\zeta)^{4/3} \right] \right\} . \nonumber \\
\label{opl2}
\end{eqnarray}
The expression for the solution of the simplified differential equation that satisfies the initial conditions, is obtained from Eq.~(\ref{opl2}) by substituting Eqs.~(\ref{alphadef}) and (\ref{z0def}), into it. We express the result
\begin{eqnarray}
H({\bf x},{\bf a}_d,z) & = & \frac{\pi}{2} \beta H_{\rm in}({\bf x},{\bf a}_d) \left(z\lambda|{\bf a}_d|^2+{\bf a}_d\cdot{\bf x}\right)^{1/2} \nonumber \\
& & \times \frac{\left({\bf a}_d\cdot{\bf x}\right)^{5/6}}{|{\bf a}_d|^{7/3}}\left\{ {\rm Y}_{-5/8}\left[\frac{\beta \left({\bf a}_d\cdot{\bf x}\right)^{4/3}}{|{\bf a}_d|^{7/3}}\right] \right. \nonumber \\
& & \times {\rm J}_{3/8}\left[\frac{\beta \left(z\lambda|{\bf a}_d|^2+{\bf a}_d\cdot{\bf x}\right)^{4/3}}{|{\bf a}_d|^{7/3}} \right] \nonumber \\
& & - {\rm J}_{-5/8}\left[\frac{\beta \left({\bf a}_d\cdot{\bf x}\right)^{4/3}}{|{\bf a}_d|^{7/3}}\right] \nonumber \\
& & \left. \times {\rm Y}_{3/8}\left[\frac{\beta \left(z\lambda|{\bf a}_d|^2+{\bf a}_d\cdot{\bf x}\right)^{4/3}}{|{\bf a}_d|^{7/3}} \right] \right\} ,
\label{opl3}
\end{eqnarray}
in terms of the $H$-notation of Eq.~(\ref{nmipe}). Here $H_{\rm in}({\bf x},{\bf a}_d)$ is the input state and
\begin{equation}
\beta \triangleq \frac{3\pi\sqrt{C_n^2}}{2\lambda^2} = \frac{3\sqrt{g}}{4\pi w_0^{7/3}} .
\label{betadef}
\end{equation}
To convert the expression in Eq.~(\ref{opl3}) to the position domain, we use an approach that is similar to the one followed to obtain Eq.~(\ref{xopl}) from Eq.~(\ref{genopl0}). However, here we find it more convenient to perform a shift in the integration variables
\begin{equation}
{\bf u} \rightarrow {\bf x}_d-{\bf u} ,
\label{skuif}
\end{equation}
with the result that Eq.~(\ref{xopl}) becomes
\begin{eqnarray}
G({\bf x}_s,{\bf x}_d,z) & = & \int G_0({\bf x}_s-\lambda z {\bf a},{\bf x}_d-{\bf u}) T \left({\bf x}_d-{\bf u},\frac{{\bf u}}{\lambda z},z\right) \nonumber \\
& & \times \exp \left( i 2\pi {\bf a}\cdot {\bf u} \right)\ {\rm d}^2 a\ {\rm d}^2 u .
\label{xopl1}
\end{eqnarray}
The resulting replacements in the arguments of the propagation kernel $T(\cdot)$ in this case are ${\bf x}\rightarrow {\bf x}_d-{\bf u}$ and ${\bf a}_d\rightarrow {\bf u}/\lambda z$, leading to the following replacements
\begin{eqnarray}
|{\bf a}_d| & \rightarrow & \frac{|{\bf u}|}{\lambda z} \nonumber \\
({\bf a}_d\cdot{\bf x}) & \rightarrow & \frac{\left({\bf x}_d\cdot{\bf u}-|{\bf u}|^2\right)}{\lambda z} .
\label{verv1}
\end{eqnarray}
Applying these replacements to the expression in Eq.~(\ref{opl3}), we obtain the following position domain expression for the single-photon turbulence propagation kernel
\begin{eqnarray}
T \left({\bf x}_d-{\bf u},\frac{{\bf u}}{\lambda z},z\right) & = & \frac{\pi\lambda z\beta}{2}
\frac{\left({\bf u}\cdot{\bf x}_d\right)^{1/2}}{|{\bf u}|^{7/3}} \left({\bf u}\cdot{\bf x}_d-|{\bf u}|^2\right)^{5/6} \nonumber \\
& & \times \left\{ {\rm Y}_{-5/8}\left[\frac{z\lambda\beta \left({\bf u}\cdot{\bf x}_d-|{\bf u}|^2\right)^{4/3}}{|{\bf u}|^{7/3}}\right] \right. \nonumber \\
& & \times {\rm J}_{3/8}\left[\frac{z\lambda\beta \left({\bf u}\cdot{\bf x}_d\right)^{4/3}}{|{\bf u}|^{7/3}} \right] \nonumber \\
& & - {\rm J}_{-5/8}\left[\frac{z\lambda\beta \left({\bf u}\cdot{\bf x}_d-|{\bf u}|^2\right)^{4/3}}{|{\bf u}|^{7/3}}\right] \nonumber \\
& & \left. \times {\rm Y}_{3/8}\left[\frac{z\lambda\beta \left({\bf u}\cdot{\bf x}_d\right)^{4/3}}{|{\bf u}|^{7/3}} \right] \right\} .
\label{kernel1}
\end{eqnarray}
Unfortunately, the result given in Eq.~(\ref{kernel1}) cannot be directly generalized to a two-photon case, as in the perturbative case above. The reason is that, when the simplification that was applied to Eq.~(\ref{nmipe}) to give Eq.~(\ref{dvf}), is applied to Eq.~(\ref{nm2ipe}), the resulting differential equation is not solvable.
\section{Discussion}
\label{disc}
Here we only consider the MPS approach. Although the SPS model appears to follow from a Markovian approach, the leading contribution in the non-Markovian approach gives the same expression for the SPS model. The conclusions that one can derive from the SPS model are therefore applicable regardless of whether one considers a Markovian or non-Markovian approach.
One of the pertinent aspects the SPS model is that it gives the behavior of the state in terms of a single dimensionless parameter ${\cal W}=w_0/r_0$, where $w_0$ is the optical beam waist radius and $r_0$ is the Fried parameter \cite{fried}. The relationship between the Rytov variance \cite{scintbook}, which quantifies scintillation strength, and ${\cal W}$ indicates that, for a constant ${\cal W}$, the scintillation strength increases with propagation distance. The SPS model is only valid under weak scintillation conditions. Therefore, it breaks down when the propagation distance becomes too large. In the context of the evolution of an entangled quantum state propagating through turbulence, one finds that the SPS model can only describe this evolution correctly for the entire duration of a nonzero entanglement, if the turbulence is strong enough to complete this evolution over a relatively short propagation distance.
As a result, one can conclude that the SPS model provides a tool to study quantum state evolution under strong turbulence conditions \cite{notrunc}. What is needed then is another model that can provide a tool to study quantum state evolution under weak turbulence conditions. For the Markovian approach, such a tool was presented in the form of the Markovian IPE \cite{ipe,notrunc}. Here we provide such a tool for the non-Markovian approach, where we exploit the weakness of the turbulence to obtain a perturbative solution.
We also provide another solution for the non-Markovian IPE that does not assume weak turbulence. This is obtained by modifying the differential equation for the non-Markovian IPE. The resulting modified differential equation only works to the single-photon case. Its solution cannot be generalized to the two-photon case, because the simplification that is used does not render a readily solvable differential equation in the two-photon case. Nevertheless, it is not inconceivable that one may be able to find a simplification that can be applied to the two-photon differential equation which would allow solutions. The resulting expressions would in general be even more complex than those that we obtained here.
The modification that is applied assumes that the cross-product between particular coordinate vectors gives a vanishing contribution to final result. This assumption depends on the particular input optical field. For instance, if the input optical field is a Gaussian beam, then the expectation value of this cross-product is zero.
\section{Conclusions}
\label{concl}
The propagation of a photonic quantum state through a turbulent atmosphere is considered in terms of a non-Markovian approach. This is done in contrast to the existing Markovian methods that have been proposed before. We derive a non-Markovian IPE, which takes the form of a second-order differential equation with respect to the propagation distance. The non-Markovian IPE contains no integrations over the propagation distance. The form of this second-order differential equation does not allow immediate solutions.
To solve the non-Markovian IPE, we follow two different approaches. The first is to assume the turbulence is weak enough to allow a perturbative analysis. This approach gives a solution that contains hyper-geometrical functions. Although we obtain the solution for the single-photon case, it can be generalized to the two-photon case.
The second approach is to apply a particular simplification to the form of the differential equation. The resulting simplified differential equation can be solved to give a solution in terms of Bessel functions of fractional order. It only applies to the single-photon case.
|
2,869,038,155,652 | arxiv | \section{Introduction}
Peculiar velocity can be a powerful probe of cosmology. On one hand, peculiar velocity causes redshift space distortion \cite{Kaiser1987}, thus to get cosmological information from galaxy surveys, one needs to model the peculiar velocity as well. On the other hand, redshift space distortion causes anisotropy, which gives rise to higher order multipoles in the correlation function/power spectrum \cite{Hamilton1992}, and hence a useful signal. Peculiar velocity can also help distinguish general relativity from modified gravity, e.g.~\cite{SongPercival_2009, TaruyaKoyama_etal2014}. The redshift space distortion has been used to constrain the growth rate and testing gravity from some recent galaxy surveys \cite{GuzzoPierleoni_etal2008,BlakeKazin_2011,ReidSamushia_etal2012,BeutlerBlake_etal2012,SamushiaReid_etal2012,delaTorre_etal2013,OkaSaito_etal2014}. In future the peculiar velocity surveys can also be fruitful \cite{KodaBlake_etal2014}.
As in galaxy surveys, only galaxies are observable, not the dark matter, it is important to understand if the velocity of the galaxy is biased with respect to that of the underlying dark matter or not. As galaxies are hosted in halos, and halos are simpler than galaxies because they are only governed by the gravitational physics, in this paper, we take studying the halo bias as a step towards understanding the galaxy bias. Recently there have been some indications from the measurements using halos at low redshifts that velocity bias may be non-negligible, although the quantitative measurement is still hard \cite{EliaLudlowPorciani2012,ChanScoccimarroSheth2012,BaldaufDesjacquesSeljak2014,JenningsBaughHatt2014,ZhengZhangJing2014b}. Velocity measurement is nontrivial because it requires the velocity field of the tracers to be weighted by volume, and it is easy to mistakenly get the density weighted velocity, i.e. momentum, instead of velocity \cite{Scoccimarro2004}. When the number density of the tracers is low, it suffers from numerical sampling artifact, see e.g.~\cite{Juszkiewiczetal1995,ZhangZhengJing2014,ZhengZhangJing2014a, ZhengZhangJing2014b}. One way out is to use momentum instead, e.g.~in \cite{OkumuraSeljak_etal2012,BaldaufDesjacquesSeljak2014}. Unlike the velocity field, however, there is an additional complication that in momentum the galaxy density bias is involved as well.
On the theory side, in the usual fluid approximation for dark matter and galaxy, even if the initial velocity field of the galaxy differs from that of the dark matter, i.e.~there is initial velocity bias, large scale gravitational evolution will naturally drive the galaxy velocity field to that of the dark matter \cite{EliaKulkarnietal2011,ChanScoccimarroSheth2012}. On the other hand, the peak model predicts that the velocity bias persists and remains constant at late time \cite{DesjacquesCrocceetal2014}. This result seems to be favoured by the recent simulation results, which suggest that the halo velocity bias at late time is non-negligible at $k \sim 0.15 \, \mathrm{Mpc}^{-1} \, h $. Ref.~\cite{BaldaufDesjacquesSeljak2014} argued that the force on the halo has to be ``biased'' in order for the coupled-fluid approach to agree with peak model result, although no further justification was given. Ref.~\cite{BiagettiDesjacquesKehagiasRiotto2014b} tried to derive the peak theory results using the distribution function approach. Here we take a different approach. A halo is a composite object consisting of a collection of particles. The position of the halo is defined by the position of the center of mass (CM) of its constituent particles. Thus the force acting on the CM position of the halo should be averaged over its constituent particles. In this way, we give a physical origin for the ``biased'' force on the halo. We will show that the halo profile correction naturally gives rise to the leading $k^2$ correction to the velocity bias and it does not decay away.
We note that our approach also has rather different interpretation for the generation of velocity bias from that in \cite{DesjacquesSheth2010,DesjacquesCrocceetal2014, BaldaufDesjacquesSeljak2014,BiagettiDesjacquesKehagiasRiotto2014b}. In peak theory, although the smoothing window is an important ingredient, the window function is usually assumed to be static. Sometimes, the attention is focused on the discrete peak ``points'', which have the same velocity as the dark matter locally, it was argued that the velocity bias is a ``statistical'' effect. In our model, the velocity bias physically arises from the fact that halos are finite-sized objects, not point particles, and it also highlights the dynamical nature of window.
On the other hand, our approach may not be mutually exclusive with the peak model approach. In the modelling of halos starting from the Lagrangian space, one defines halos with window function and the smoothing scale is fixed to be the Lagrangian szie when they are transformed to the Eulerian space. Even if our velocity bias contribution is not the dominant one seen in simulations, the profile correction effects should be taken into account in the calculations as well.
This paper is organized as follows. As we will show that the halo profile gives rise to the velocity bias correction, to set the stage, we will first review the evolution of the halo profile using the spherical collapse model in Sec.~\ref{sec:spherical_collpase} and the numerical halo profile is measured from simulation and compared with the spherical collapse model in Sec.~\ref{sec:Measurements}. In Sec.~\ref{sec:LinearBiasComputation} we compute the correction to the linear velocity and density bias due to the halo profile, and the second order corrections are presented in Sec.~\ref{sec:SecondBiasComputation}. We conclude in Sec.~\ref{sec:Conclusions}.
\section{ Halo Profile Evolution}
\label{sec:halo_profile}
Halo profile is often used in the context of halo model for modelling the dark matter power spectrum \cite{Seljak2000, PeacockSmith2000,Scoccimarroetal2001,CooraySheth}. In this case, the virialized halo profile, such as the NFW profile \cite{NFW1996} is often used. However, we will follow the proto-halo from its infancy to the final virialized stage in modelling the bias evolution. To this end, we will first review the evolution of a halo using the spherical collapse (SC) model. We will then construct proto-halos at various redshifts and measure the profile evolution in numerical simulations. The results are compared with the SC model. To our knowledge, this is the first systematic measurements of the proto-halo profile evolution.
\subsection{Profile evolution from SC model}
\label{sec:spherical_collpase}
A simple analytic model for halo evolution is given by the SC model \cite{GunnGott1972} (see also \cite{Peebles1980,Padmanabhan1993, MoBoschWhite2010}). Suppose that the initial fluctuations are spherically symmetric about some point in position space. To avoid shell crossing, we assume that the radial profile is non-increasing as the distance from the center increases. We will consider the matter dominated universe as the resultant equation can be integrated analytically, and also the more realistic $\Lambda$CDM model. Under the Newtonian approximation, the equation of motion for a mass shell at a distance $r$ from the center is given by
\begin{equation}
\label{eq:EoM_sphericalcollapse}
\frac{d^2 r }{ d t^2 } = -\frac{G M(r)}{ r^2 } + \frac{\Lambda }{3 } r ,
\end{equation}
where both $r$ and $t$ are the \textit{physical} distance and time, and $G$ is the gravitational constant, $ M(r)$ is the total mass inside the mass shell and $\Lambda $ is the cosmological constant.
Integrating Eq.~\ref{eq:EoM_sphericalcollapse} once, we obtain the first integral of motion
\begin{equation}
\label{eq:1stintg_sphericalcollapse}
\frac{1 }{2}\Big( \frac{dr}{ dt } \Big)^2 - \frac{GM}{ r } - \frac{\Lambda r^2 }{ 6} = E,
\end{equation}
where the total energy $E$ is a constant of integration.
We will solve Eq.~\ref{eq:EoM_sphericalcollapse} numerically when $\Lambda \neq 0 $. When $\Lambda=0$, Eq.~\ref{eq:1stintg_sphericalcollapse} can be further integrated analytically, and the solution can be expressed in the form of a cycloid solution
\begin{eqnarray}
\label{eq:r_cycloid}
r &=& A ( 1 - \cos \theta ), \\
\label{eq:t_cycloid}
t + T &=& B ( \theta - \sin \theta ) , \\
\label{eq:AB_cycloid}
A^3 &=& GM B^2,
\end{eqnarray}
where $A$, $B$ and $T$ are constants. The parameter $ \theta $, also called the development angle, runs from 0 to $ 2 \pi $. When $\theta $ is close to 0, the overdensity inside the mass shell is small, the mass shell essentially follows the Hubble expansion. The mass shell reaches maximum $r_{\rm m} $ at $\theta= \pi $. Beyond that the mass shell overcomes the Hubble expansion and turns around. At $\theta = 2 \pi $, the shell collapses to a point according to Eq.~\ref{eq:r_cycloid}. However, it was argued that during the rapid infall the potential varies quickly, the particles no longer follow the energy conserving orbits, instead the energy available to the particles is widened, and the system reaches virial equilibrium \cite{LyndenBell1967, BindoniSecco2008}. This procoess is called ``violent relaxation'' \cite{LyndenBell1967}. From virial equilibrium, one finds that the virial size $r_{\rm v} $ is related to $r_{\rm m} $ as
\begin{equation}
\label{eq:rvir_rm_half}
r_{\rm v} = \frac{ r_{\rm m} }{2}.
\end{equation}
The virial size is often given in terms of the virial density, $\Delta_{\rm v}$, as
\begin{equation}
\label{eq:rv_Deltav}
r_{\rm v} = \Big( \frac{ M }{ \frac{4 \pi}{3} \bar{\rho}_{\rm m} \Delta_{\rm v} } \Big)^{ \frac{1}{3} },
\end{equation}
where $M$ is the mass of the halo and $\bar{\rho}_{\rm m} $ is the comoving density of matter. For EdS universe, $ \Delta_{\rm v} $ is equal to 178. When $\Lambda \neq 0 $, a fitting formula for the virial density, $\Delta_{\rm v}$, is given in \cite{BryanNorman1998}. For the flat $\Lambda $CDM with $\Omega_{\rm m} =0.25 $ adopted in this paper, $\Delta_{\rm v}$ for a halo virailzed at $z=0 $ is 380. In practice other values of $\Delta_{\rm v} $ are often adopted, such as 200 and 500. We will use $\Delta_{\rm v} = 500$ as we will see later on it gives a good description of our simulation data. Note that for non-EdS universe, in Eq.~\ref{eq:rv_Deltav}, the critical density is often used instead of $ \bar{\rho}_{\rm m} $ to define $r_{\rm v}$.
To solve Eq.~\ref{eq:EoM_sphericalcollapse}, the initial conditions that the initial overdensity is obtained by extrapolating the collapse threshold from the present time to the initial time using the linear growth factor and zero initial peculiar velocity is often assumed. However, the zero initial peculiar velocity condition excites both the growing mode and the decaying mode. It can be shown that the linear amplitude of the perturbation is reduced by a factor of $3/5$ \cite{Peebles1980,Padmanabhan1993,Scoccimarro98}. This is equivalent to setting up the initial condition incorrectly and transient effects are induced. To get the right final amplitude, one quick fix is to increase the initial perturbation by a factor of $5/3 $ to compensate the loss to the decaying mode. As we set up the initial conditions at not very high redshifts, the transient effects are not negligible. A better approach is to set the initial peculiar velocity such that the decaying mode vanishes. Thus we will use the initial conditions
\begin{eqnarray}
\label{eq:rstar_SC}
r_* & =& \Big( \frac{ 3 M }{ 4 \pi \bar{\rho}_{\rm m} } \Big)^{ \frac{1}{ 3 } } \frac{ a_* }{ ( 1 + \bar{\delta}_*)^{\frac{1 }{3 } } }, \\
\label{eq:rdotstar_SC}
\dot{r}_* &=& H_* r_* \Big( 1 - \frac{1 }{ 3 } \bar{ \delta}_* \Big) ,
\end{eqnarray}
In this paper, we use ``*'' to denote a quantity at some initial time. Thus $a_*$, $H_*$ and $ \bar{ \delta }_* $ are the scale factor, Hubble parameter and the average density contrast inside the spherical shell at the initial time.
Using these initial conditions, we can write the coefficents $A$ and $B$ in Eq.~\ref{eq:r_cycloid} and \ref{eq:t_cycloid} as
\begin{eqnarray}
A &=& \frac{r_* }{ 2} \frac{ \Omega_{\rm m}^* ( 1 + \bar{\delta}_* ) }{ \Omega_{\rm m }^* ( 1 + \bar{ \delta}_* ) - \big( 1 - \frac{1 }{ 3 } \bar{\delta}_* \big)^2 } , \\
B &=& \frac{1 }{ 2 } \frac{ \Omega_{\rm m}^* ( 1 + \bar{\delta}_* )}{
H_* \Big[ \Omega_{\rm m}^* ( 1 + \bar{ \delta}_* ) - \big( 1 - \frac{1 }{3 } \bar{\delta}_* \big)^2 \Big]^{ \frac{3}{2 }} },
\end{eqnarray}
where $ \Omega_{\rm m}^* $ is the density parameter of matter at the initial time. We first note that $A$ and hence $r$ is proportional to $r_*$. On the other hand, the collapse history given by $ t $ is independent of $r_*$, and it depends only on the matter inside through $\Omega_{\rm m}^*$ and $\bar{\delta}_* $.
In Eq.~\ref{eq:EoM_sphericalcollapse}, after dividing by $r_*$, one can easily see that the collapse history is independent of $r_*$ in $\Lambda $CDM model. Thus given $\Omega_{\rm m}$ and $\Omega_{\Lambda}$, the collapse history depends only on $\bar{\delta}_* $.
In Fig.~\ref{fig:SC_r_EdS_OCDM_LCDM}, we show the evolution of the profile as a function of $a$ for three different cosmological models: EdS, Open CDM with $\Omega_{\rm m}=0.25 $ and $\Lambda$CDM with $\Omega_{\rm m}=0.25 $ and $\Omega_{ \Lambda }=0.75 $. Note that in this paper $\Lambda$CDM always refers to this model. The mass of the halo is chosen to be $ 2 \times 10^{13} \, M_{\odot} {h}^{-1} $. The collapse threshold at $z=0$ is set to be $\delta_{\rm c} =1.68 $ and extrapolated to the initial time using the linear growth factor for the corresponding cosmology. The initial conditions are set using Eq.~\ref{eq:rstar_SC} and \ref{eq:rdotstar_SC}. Nonetheless, we still find that we need to choose $a_*$ to be sufficiently small ($ a_* = 0.01 $ here) to reduce the effects of transients. For example when $a_* =0.02$ is chosen instead, we find that the collapse epochs are increased by a few per cent compared to the ones shown. We emphasize that this is because we set up the initial conditions using linear theory, and the transients can be further suppressed using higher order perturbation theory. This is analogous to setting up initial conditions in simulations using 2LPT \cite{Scoccimarro98, CroccePeublasetal2006}. Although these cosmologies are rather different, the final collapse epochs are very similar as long as the correct linear growth factor is used to set $\bar{\delta}_*$. In other words the collapse threshold is insensitive to the cosmological model \cite{EkeColeFrenk1996}. However, the intermediate stages of collapse are quite different among these models. Thus this suggests that we need to solve the model explicitly in order to follow the evolution of the halo profile accurately. We also indicate in Fig.~\ref{fig:SC_r_EdS_OCDM_LCDM} the virial size after the collapse. For EdS and OCDM, it is computed using Eq.~\ref{eq:rvir_rm_half}, while for $\Lambda $CDM we use Eq.~\ref{eq:rv_Deltav} with $\Delta_{\rm v} = 500$.
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{SC_r_EdS_OCDM_LCDM.pdf}
\caption{ The evolution of the halo profile in three different cosmological models EdS (blue), OCDM with $\Omega_{\rm m}=0.25 $ (red) and $\Lambda$CDM with $\Omega_{\rm m}=0.25 $ and $\Omega_{\rm m}=0.75 $ (green). The virial size is indicated as a dot after the collapse. }
\label{fig:SC_r_EdS_OCDM_LCDM}
\end{figure}
To illustrate the evolution of the profile, we further assume that the initial halo profile is described by a top-hat profile, and in Fourier space, it reads
\begin{equation}
W_{\rm TH}( \kappa ) = \frac{ 3 }{ \kappa^3 } ( \sin \kappa - \kappa \cos \kappa ).
\end{equation}
Top-hat profile is a good approximation at high redshifts. In SC with top-hat perturbation, the top-hat shape is preserved during evolution. The only part that changes is the width of the window. On the other hand, the Eulerian virialized spherical halo profile is well described by the NFW profile \cite{NFW1996}. This means that a top-hat window of perturbation cannot evolve to the NFW-like profile. We will see in Sec.~\ref{sec:Measurements} that the halo profile measured from simulation goes from one resembling top-hat to an NFW-like profile as the redshift decreases.
In Fig.~\ref{fig:WTH_evolution}, we show the evolution of $W_{\rm TH}(k x) $ for a series of values of the comoving size $x$ at different time $a$. We start with $x_*=4.07 \, \mathrm{Mpc} \, h^{-1} $ at $z_*=99$, and the size of the spherical shell is then evolved according to Eq.~\ref{eq:EoM_sphericalcollapse}. We have adopted a flat $\Lambda$CDM model with $\Omega_{\Lambda } = 0.75 $, and $\delta_{\rm c} =1.68 $. Note that although the physical size $r$ first expands and then collapses as in Fig.~\ref{fig:SC_r_EdS_OCDM_LCDM}, the comoving size $x$ is always decreasing as the effect of expansion is removed. At $a=1$, the shell has not fully collpased yet. As we will see later on, we are mainly interested in the low $k$ part of the window, thus to a very good approximation, the window is essentially 1 up to $k \sim 3 \, \mathrm{Mpc}^{-1} \, h $ at the present time. In this plot we have not substituted the halo size at $a=1$ with the virial size. Most of time in the paper, the sudden change during virialization does not matter as it occurs almost instantaneous for our purpose. From now on, the SC model refers the one obtained by evolving an initially top-hat perturbation using Eq.~\ref{eq:EoM_sphericalcollapse}.
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{WTH_evolution.pdf}
\caption{ The evolution of the top-hat window, $W_{\rm TH} ( kx )$ for a suite of scale factor $a$. The comoving size $x$ of the spherical shell is computed using the SC model. }
\label{fig:WTH_evolution}
\end{figure}
\subsection{Measurement of the profile evolution from simulations }
\label{sec:Measurements}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{rho_r_zE0_zLMany_bpt156_Oriana.pdf}
\caption{ The halo profile at $z=0$, 0.34, 0.73, 0.97, 1.5, and 49. The Eulerian profile is at $z=0$, and the proto-halo profiles are obtained by tracing the particles in the Eulerian halo to higher redshifts, $ z_{\rm L}$. Results from halos of mass ranging from $6.9 \times 10^{13} \, M_{\odot} {h}^{-1} $ to $1.3 \times 10^{15} \, M_{\odot} {h}^{-1} $ are shown. The density is normalized with respect to the mean comoving density of matter, $\bar{\rho}_{\rm m} $ and the radial distance $r$ is normalized with respect to the Eulerian virial size of the halo, $r_{\rm v } $. The results are from the Oriana simulations. }
\label{fig:rho_r_zE0_zLMany_bpt156_Oriana}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{rho_k_norm_zE0_zLMany_bpt156_Oriana.pdf}
\caption{ Same as Fig.~\ref{fig:rho_r_zE0_zLMany_bpt156_Oriana}, except in Fourier space. The halo profile is normalized such that it approaches 1 as $k$ tends to 0. }
\label{fig:rho_k_norm_zE0_zLMany_bpt156_Oriana}
\end{figure*}
In this section, we will measure the evolution of halo profile from $N$-body simulation. Starting from Eulerian halos, such as those at $z=0$, we trace the particles in the Eulerian halo back in time to construct the proto-halos at earlier times. The position of the proto-halo at redshift $z$ is defined by the CM of its constituent particles at redshift $z$. We will consider proto-halos at various redshifts.
Before presenting the numerical results we would like to first outline the details of the $N$-body simulation used here. We shall use the Oriana and Carmen simulations in the LasDamas project. In these simulations, a flat $\Lambda$CDM model with the cosmological parameters, $\Omega_{\rm m} =0.25 $, $\Omega_{\Lambda} =0.75$ and $\sigma_8 =0.8 $ are adopted. The transfer function is output from CMBFAST \cite{CMBFAST}. The initial conditions are Gaussian with spectral index being 1. The initial displacement fields are created using 2LPT \cite{CroccePeublasetal2006} at $z=49$. The simulations are evolved using the code Gadget2 \cite{Gadget2}. In the Oriana simulations, there are $1280^3$ particles in a cubic box of size 2400 $ \, \mathrm{Mpc} \, h^{-1} $, while for Carmen simulation there are $1120^3$ particles in a box of size $1000 \, \mathrm{Mpc} \, h^{-1} $. Thus the particle masses are $4.57 \times 10^{11} $ and $4.94 \times 10^{10} \, M_{\odot} {h}^{-1} $ for Oriana and Carmen respectively. We shall use five realizations for Oriana and seven for Carmen. The halos are obtained using Friend-of-Friend halo finder. For Oriana, the linking length $b=0.156$ is used, while $b=0.2$ for Carmen. To resolve the halo better, we use halos with at least 150 particles. Although the Carmen simulations have better mass resolution than Oriana, we find that their results are quite similar. To avoid redundancy, most of the time, we only show results from Oriana.
In Fig.~\ref{fig:rho_r_zE0_zLMany_bpt156_Oriana}, we show the halo profile at redshifts, $z=0$, 0.34, 0.73, 0.97, 1.5, and 49. The halo profile is obtained by stacking the halos in the same mass bin together and spherically averaged to get the spherically symmetric profile. In this plot, the Eulerian halo is at $z=0$ and the proto-halos at higher redshifts are constructed from the Eulerian ones. We find that when the size of halos of different masses normalized by their corresponding virial size, they coincide well with each other. We computed $r_{\rm v}$ using $\Delta_{\rm v} =500$ although this is immaterial to our purpose here.
Note that for redshift $z=0$, the halos are in fact in the Eulerian space. The virialized spherical Eulerian halo profile at low redshift is well fitted by the NFW profile. However, the halos used to construct this profile are carefully selected, see e.g.~\cite{NFW1996,NetoGaoBettetal2007}. These halos are constructed using spherical overdensity finder and they are chosen to be spherically symmetric and in a relaxed state without signatures of recent mergers. Here we use all the halos obtained from the halo finder without further screening. We find that our Eulerian profile is reasonably well fitted by the NFW profile, but we also find that the profile close to the virial radius drops faster than $r^{-3}$, the scaling of the NFW profile near virial radius. Using the NFW profile, one can show that the Eulerian profile is approximately universal in the variable $r/r_{\rm v} $ for different masses because of the fact that the concentration only weakly depends on the mass of the halo \cite{CooraySheth}.
The deviation of the proto-halo profile from the NFW profile increases as the redshift increases. At $z=49$, the proto-halo profile corresponds to the one in the initial condition of the simulation. Theoretically, the Lagrangian profile is often assumed to be a top-hat. In \cite{ChanShethScoccimarro2015}, it is found to be in between a Gasussian and a top-hat. More precisely, in Fourier space, the Lagrangian profile is well fitted by a product of a Gaussian and a top-hat window. We note that as $z$ increases, there are large deviations in the profile at small $r$ among different halo masses.
As there is no universal halo profile that can fit the proto-halo profile at various redshifts well, we shall use the numerical profile directly. The profile is Fourier transformed numerically. In the case of the NFW profile, the integration is cut-off at the virial radius $r_{\rm v }$ \cite{CooraySheth}. In our case, for proto-halos at intermediate redshifts, i.e.~in between $z=0$ and $z=49$, it is not clear what the cut-off size should be. Nonetheless, our profile drops rapidly for $ r$ greater than a few $r_{\rm v }$, we can take $r$ to be infinity, and the results are unaffected. As the mass of the proto-halo is conserved, we expect the low-$k$ part of the profile at different redshifts to be the same. Numerically, however, this is not always achieved. In each set of simulation, we find that the fractional deviation of the $\rho(k=0) $ across redshifts decreases as the mass of the halo increases. For example, for the lowest mass halo used for Oriana, the fractional deviation of $\rho(k=0) $ is within 10\%. This is one of the indications that we should use halos with large number of particles. From now on, we simply normalize the profile so that it is 1 at low $k$. In Fig.~\ref{fig:rho_k_norm_zE0_zLMany_bpt156_Oriana}, we show the Fourier transform of the halo profile for a selection of halo masses. As the redshift $z_{\rm L}$ increases, the size of the proto-halo increases, the window in Fourier space decreases and the low-$k$ plateau shrinks. We also note that there are oscillations in the Fourier transform of the window. It is more visible as the mass of the halo increases because the oscillations are pushed to lower $k$. They are also more prominant as $ z_{\rm L} $ increases because the halo profile is more top-hat-like, and hence the wiggles are stronger.
To show the scale-dependence of the window function, in Fig.~\ref{fig:W_1_zE0_zLMany_bpt156_Oriana}, we plot $ | W - 1 | $ obtained from simulations and the SC model. Absolute value is taken because the $k^2$ correction is negative. We have introduced the time variable
$y=\ln D $, where $D$ is the linear growth factor (defined by Eq.~\ref{eq:DM_grothfactor}). It is normalized such that $y=0 $ at $z_*=49$. As we will see in next section, this time variable is convenient. First at $y=0$, SC model agrees with data well except for the highest mass bin shown ($ 3.7 \times 10^{15} \, M_{\odot} {h}^{-1} $). Unfortunately, we have no simulation data available in the range $0 \lesssim y \lesssim 3 $, although we expect that the overdensity is still in the expansion stage ($y \lesssim 3 $), the SC model should work reasonably well. When the region turns around and collapses, we expect the SC model to fail to describe the simulation data accurately. In fact, during the turn-around and collapse phase, the SC results are larger than the simulation data. We also note that for various values of $k$ shown, the agreement between the SC model and the data is qualitatively smiliar. In Fig.~\ref{fig:W_1_zE0_zLMany_bpt2_Carmen} we show the corresponding results obtained using the Carmen simulations. In this plot, the Eulerian halos are at $z=0$ and proto-halos are constructed at $z=0.13 $, 0.52, 0.97 and 49. Although Carmen has better mass resolution, the results are quite similar to those obtained from Oriana. Overall, the agreement between the simulation results and SC model is reasonable.
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{W_1_zE0_zLMany_bpt156_Oriana.pdf}
\caption{ The function $|W-1|$ as a function of $y$, obtained from simulations (filled circles) and SC model (solid line). Results from four mass bins of mass $6.9 \times 10^{13}$, $2.0 \times 10^{14}$, $6.1 \times 10^{14}$, and $3.7 \times 10^{15} \, M_{\odot} {h}^{-1} $ are shown (from left to right). For each mass bin, $| W(k,y)-1 |$ at six different $k$'s are plotted. The data is from Oriana. }
\label{fig:W_1_zE0_zLMany_bpt156_Oriana}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{W_1_zE0_zLMany_bpt2_Carmen.pdf}
\caption{ Similar to Fig.~\ref{fig:W_1_zE0_zLMany_bpt156_Oriana}, except for Carmen. }
\label{fig:W_1_zE0_zLMany_bpt2_Carmen}
\end{figure*}
The SC model only works qualitatively at late time. After all, in SC the halo profile shape does not change, but for real halos the halo profile shape does change. In \cite{Engineer_etal2000}, a modified SC model that tries to overcome the jump at the final virialization stage was proposed. The model joins smoothly to the final virial scale at the expense of two additional free parameters. The modified model is valid only when the density is high as it is an expansion in $1/\delta$. Nonetheless, using these additional parameters, one may get a profile evolution history, especially the part from turn-around to collapse, to agree with the simulation results better. Another potential way to improve the modelling is to use ellipsoidal model \cite{Peebles1980,BondMyers1996, ShenAbelMoSheth2006,MoBoschWhite2010}. For example, the halo mass function motivated by the ellipsoidal collapse improves the agreement with simulation \cite{ShethMoTormen2001} compared to the spherical Press-Schechter one. The halo collapse threshold is also better modelled by the ellipsoidal collapse model \cite{RobertsonKravtsovTinkerZentner2009}. However, as the halo profile considered here is spherically averaged, one still need to average over the ellipsoidal profile to get the spherically symmetric one. On the data side, we hope to get the data to fill the gap in between 0 and 3 in future. As we see the model does not work very well, in practice it will be useful to come up with a parametrized form for the evolution of the profile. Also the paramertized form of halo can be used to improve the halo model. In the standard halo model one assumes that all the matter exists within halos, and the virialized halo profile is used, such as the NFW profile for halos \cite{CooraySheth}. However at higher redshift, virialized halos are rare, and this assumption is not justified. One can improve the halo model using the proto-halo profile instead.
\section{Bias with profile corrections}
\label{sec:BiasComputations}
We shall apply the fluid approximation to model the evolution of the dark matter and the galaxy field. The fluid approximation enable one to derive the nonlocal bias parameters \cite{Fry1996,ChanScoccimarroSheth2012,Baldaufetal2012} which results in better modelling of the halo power spectrum and bispectrum \cite{ChanScoccimarroSheth2012,Baldaufetal2012,SaitoBaldauf_etal2014,BiagettiDesjacquesKehagiasRiotto2014a} and halo 3-point function \cite{BelHoffmannGaztanagn2015}. In this paper we use halo and galaxy interchangeably. For dark matter, we will use the standard perturbation theory (SPT) results (see \cite{PTreview} for a review). In this framework, the evolution of the density contrast of the galaxy, $\delta_{\rm g} $, and its velocity divergence $\theta_{\rm g} $ are governed by the continuity equation and the Euler equation
\begin{widetext}
\begin{eqnarray}
\label{eq:continuity_0}
\frac{\partial \delta_{\rm g} }{ \partial \tau } + \theta_{\rm g} & = & - \int d^3 k_1 d^3 k_2 \delta_{\rm D} (\mathbf{k} - \mathbf{k}_{12} ) \alpha( \mathbf{k}_1, \mathbf{k}_2 ) \theta_{\rm g}( \mathbf{k}_1 ) \delta_{\rm g}( \mathbf{k}_2 ) , \\
\label{eq:Euler_0}
\frac{ \partial \theta_{\rm g} }{ \partial \tau } + \mathcal{H} \theta_{\rm g} + \frac{ 3 }{ 2 } \mathcal{H}^2 \Omega_{\rm m} W \delta &=& - \int d^3 k_1 d^3 k_2 \delta_{\rm D} (\mathbf{k} - \mathbf{k}_{12} ) \beta( \mathbf{k}_1, \mathbf{k}_2 ) \theta_{\rm g}( \mathbf{k}_1 ) \delta_{\rm g}( \mathbf{k}_2 ),
\end{eqnarray}
where $ \tau $ is the conformal time, $\mathcal{H}$ is the conformal Hubble parameter $d \ln a / d \tau $, $\mathbf{k}_{12}$ denotes $ \mathbf{k}_1 + \mathbf{k}_2 $, and $\alpha $ and $\beta $ are the coupling kernels
\begin{equation}
\alpha( \mathbf{k}_1, \mathbf{k}_2 ) = \frac{ \mathbf{k}_{12} \cdot \mathbf{k}_1 }{ k_1^2 }, \quad \beta( \mathbf{k}_1, \mathbf{k}_2 ) = \frac{ k_{12}^2 \mathbf{k}_1 \cdot \mathbf{k}_2 }{ 2 k_1^2 k_2^2 }.
\end{equation}
Here $\Omega_{\rm m} $ is the density parameter of matter.
Eq.~\ref{eq:continuity_0} and \ref{eq:Euler_0} are similar to the fluid equations widely adopted for modeling the evolution of multiple components \cite{SomogyiSmith2010, EliaKulkarnietal2011,BernardeauVandeRijtVernizzi2012,ChanScoccimarroSheth2012}, except with the window function $W$, which is central to the results in this paper. We also note that in \cite{BaldaufDesjacquesSeljak2014}, a similar modification of the Euler equation was proposed, in which the authors argued the forced for halos should be biased. However, the physical origin of this modification and its form are quite different from that in \cite{BaldaufDesjacquesSeljak2014}. Another important difference from \cite{BaldaufDesjacquesSeljak2014,BiagettiDesjacquesKehagiasRiotto2014b} is that we do not impose the peak constraint in the evolution equations. The proto-halos after initial identification, they simply evolve following Eq.~\ref{eq:continuity_0} and \ref{eq:Euler_0}.
The introduction of $W$ is to model the fact that although $\delta_{\rm g}$ denotes the density contrast of the spatial distribution of the CM of the halos, each individual halo consists of a collection of particles. Thus the force on the CM of the halo should be the average force acting on all the individual particles in the halo. Hence in real space the effective source of the gravitational force for a finite-sized object is $W*\delta$, instead of only $\delta $ at the CM position of the object. In Fourier space, it is given by the product between $ W $ and $\delta$ thanks to the convolution theorem. This window function describes the profile of the object. We will use the window function/profile studied in Sec.~\ref{sec:halo_profile}.
To simplify Eq.~\ref{eq:continuity_0} and \ref{eq:Euler_0} further, we introduce the new time variable $y=\ln D $ where $D$ is the linear growth factor for the dark matter satisfying the equation
\begin{equation}
\label{eq:DM_grothfactor}
\frac{d ^2 D }{ d \tau^2 } + \mathcal{H} \frac{d D }{d \tau } - \frac{3 }{2} \mathcal{H}^2 \Omega_{\rm m} D = 0.
\end{equation}
We note that $f^2 \approx \Omega_{\rm m} $, with $f = d \ln D / d \ln a $ is a very good approximation for the epoch that we are interested in \cite{SCFFHM98}. Using this approximation Eq.~\ref{eq:continuity_0} and \ref{eq:Euler_0} can be written as
\begin{eqnarray}
\label{eq:continuity_1}
\frac{ \partial \delta_{\rm g} }{ \partial y } - \tilde{ \theta_{\rm g}} & =& \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} )
\alpha( \mathbf{k}_1 , \mathbf{k}_2 ) \tilde{ \theta_{\rm g}} ( \mathbf{k}_1 ) \delta_{\rm g} ( \mathbf{k}_2 ) , \\
\label{eq:Euler_1}
\frac{ \partial \tilde{ \theta_{\rm g}} }{ \partial y } + \frac{ 1 }{2 } \tilde{ \theta_{\rm g}} - \frac{3 }{2 } W \delta & = & \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} )
\beta( \mathbf{k}_1 , \mathbf{k}_2 ) \tilde{ \theta_{\rm g}} ( \mathbf{k}_1 ) \tilde{ \theta_{\rm g}} ( \mathbf{k}_2 ) ,
\end{eqnarray}
where $\tilde{ \theta_{\rm g}}$ denotes $ \theta_{\rm g} / ( - f \mathcal{H} )$. In the rest of the paper, we shall abuse the notation and simply use $\theta $ and $ \theta_{\rm g} $ to denote $ \theta / ( - f \mathcal{H} )$ and $ \theta_{\rm g} / ( - f \mathcal{H} )$ respectively.
In the following subsections, we will solve Eq.~\ref{eq:continuity_1} and \ref{eq:Euler_1} to linear and second order respectively to reveal the effects of the window function on the bias parameters. In \cite{ChanScoccimarroSheth2012}, the continuity and Euler equation of the galaxy field together with the other two equations for dark matter were written in a concise form, and hence a general perturbative solution was obtained using the transient formalism \cite{Scoccimarro98}, thanks to the fact that the coefficients of the equations are not explicitly time-dependent. However, $W$ is time-dependent as we see in Sec.~\ref{sec:halo_profile}. Here we will solve Eq.~\ref{eq:continuity_1} and \ref{eq:Euler_1} directly.
\end{widetext}
\subsection{ Linear biases }
\label{sec:LinearBiasComputation}
We start from the linearized version of Eq.~\ref{eq:continuity_1} and \ref{eq:Euler_1}
\begin{eqnarray}
\label{eq:continuity_deltag_linear}
\partial_y \delta_{\rm g}^{(1)} &= & \theta_{\rm g}^{(1)} , \\
\label{eq:Euler_deltag_linear}
\partial_y \theta_{\rm g}^{(1)} + \frac{ 1 }{ 2 } \theta_{\rm g}^{(1)} &=& \frac{ 3 }{ 2 } W \delta^{(1)} ,
\end{eqnarray}
where the superscript (1) emphasizes that the field is linear. We have suppressed the explicit $ \mathbf{k}$-dependence.
\subsubsection{Velocity}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{bv_tot_evolve.pdf}
\caption{ Evolution of the linear velocity bias with static window (dashed) and that with window given by SC (solid). Three redshifts are shown $z=5$ (blue), 1 (red) and 0 (green). The case with static window is almost constant (thus some of the curves are covered by the blue curves), while the evolving window case gives decaying, but not negligible $b_{\rm v} $. }
\label{fig:bv_tot_evolve}
\end{figure}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{j3o2_zE0_zLMany_bpt156_Oriana.pdf}
\caption{ The integrand $ | j_{\frac{3 }{2 }} | $ as a function of $y$ for various values of $k$, obtained from simulation data (filled circles). The results from the SC (solid) and the static window (dashed) are also shown. The simulation data is from Oriana. }
\label{fig:j3o2_zE0_zLMany_bpt156_Oriana}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{W1vsU_zE0_zLMany_bpt156_Oriana.pdf}
\caption{ The comparison of the contribution to $k^2$ from $|W-1| $ (blue) and $\frac{3 }{ 2} e^{- \frac{3 y}{ 2} } | J_{ \frac{3}{2} } |$ (red), which is the main contribution to $k^2$ correction in the velocity bias in Eq.~\ref{eq:bw_in_w-1}. The results from simulations (symbol) and SC (solid line) are shown. The simulation data is from Oriana. For $W-1$ from SC, we have used $r_{\rm v} $ computed with $\Delta_{\rm v}=500 $. The quantities are evaluated are $z=0$. }
\label{fig:W1vsU_zE0_zLMany_bpt156_Oriana}
\end{figure*}
We will work on the Euler equation first because it depends only on $ \theta_{\rm g}^{(1)} $. Integrating Eq.~\ref{eq:Euler_deltag_linear} in terms of $W$ and $\delta^{(1)} $, we get
\begin{equation}
\label{eq:theta_g_lin}
\theta_{\rm g}^{(1)} (y) = \theta_{\rm g}^{*(1)} e^{- \frac{y }{ 2 } } + \frac{ 3 }{ 2 } \int_0^y dy' W(y') \delta^{(1)} (y') e^{ -\frac{ 1 }{ 2 }( y - y' ) }.
\end{equation}
As we mentioned before, in this paper, we use star ``*'' to denote a quantity at some initial time. Thus $ \theta_{\rm g}^{*(1)} $ is the velocity divergence of the galaxy at the initial time. For convenience, we define the function $I_n$ as
\begin{equation}
I_n(y) = \int_0^y d y' W(y') e^{ n y' },
\end{equation}
thus we have
\begin{equation}
\label{eq:thetag_1}
\theta_{\rm g}^{(1)}(y) = \theta_{\rm g}^{*(1)} e^{- \frac{y }{ 2 } } + \frac{ 3 }{ 2 } \delta_*^{(1)} e^{- \frac{ y}{ 2} } I_{\frac{3 }{2} },
\end{equation}
where we have used the SPT result
\begin{equation}
\delta^{(1)}(y) = \delta_*^{(1)} e^y.
\end{equation}
Note that we have normalized the linear growth factor to be 1 at the initial time so that $y_* =0 $.
The linear velocity bias $ b_{\rm v} $ then is given by \footnote{The linear bias parameters are defined differently from that in peak theory \cite{DesjacquesSheth2010}, where a smoothing window function is divided by. For example, the $b_{\rm v}$ defined here is equal $\tilde{b}_{\rm v} W_{\rm s} $, where $W_{\rm s} $ is a smoothing window and only $\tilde{b}_{\rm v} $ is the called the velocity bias in \cite{DesjacquesSheth2010}. Our more ``direct'' definition is closer to the standard treatment, where the window is not explicitly written down but its effects will be included in $b_{\rm v}$. }
\begin{eqnarray}
\label{eq:bvlin_W}
b_{\rm v} (y) & \equiv & \frac{ \theta_{\rm g}^{(1)}(y) }{\theta^{(1)} (y) } \nonumber \\
&=& b_{\rm v} ^* e^{ - \frac{ 3 y }{ 2 } } + \frac{ 3 }{ 2 } e^{ - \frac{ 3 }{2 } y } I_{ \frac{ 3 }{ 2} } ,
\end{eqnarray}
where we have used
\begin{equation}
\theta^{(1)}(y) = \theta_*^{(1)} e^y = \delta_*^{(1)} e^y .
\end{equation}
Correspondingly the initial linear velocity bias $ b_{\rm v} ^* $ is defined as
\begin{equation}
b_{\rm v} ^* \equiv \frac{ \theta_{\rm g}^{*(1)} }{ \theta_*^{(1)} }.
\end{equation}
In Fourier space, the window function approaches 1 at low $k$, thus it is convenient to express the integral in Eq.~\ref{eq:bvlin_W} in terms of $W-1 $. Hence we have instead
\begin{equation}
\label{eq:bw_in_w-1}
b_{\rm v} = 1 + ( b_{\rm v} ^* - 1 ) e^{-\frac{ 3 y}{ 2 } } + \frac{ 3 }{ 2 } e^{-\frac{ 3 y}{ 2 } } J_{ \frac{ 3 }{ 2 } }(y),
\end{equation}
where $J_n$ denotes the integral
\begin{equation}
J_n (y) = \int_0^y d y' [ W(y') - 1 ] e^{n y' }.
\end{equation}
The advantage of introducing $ J_n $ is that it gives at least $k^2$ order correction, thus it represents the genuine halo profile correction. The first two terms in Eq.~\ref{eq:bw_in_w-1} are the velocity bias evolution obtained in \cite{ChanScoccimarroSheth2012}, and the last term is new, which arises from the halo profile. In the limit of large $y$, the profile correction does not vanish, instead $ b_{\rm v} $ tends to $1 + \frac{ 3 }{ 2 } e^{-\frac{ 3 y}{ 2 } } J_{ \frac{ 3 }{ 2 } }(y) $. If we assume that the window is static, we get $W$.
In Fig.~\ref{fig:bv_tot_evolve}, we plot $ b_{\rm v} $ at $z=5$, 1 and 0. To set the initial condition, $b_{\rm v}^* $ at $z_*=49 $, we borrow the results from peak theory \cite{DesjacquesSheth2010}
\begin{equation}
\label{eq:bvs_peak}
b_{\rm v} ^* = \Big( 1 - \frac{ s_0 }{ s_1 } k^2 \Big) W_{\rm G}( k R_{\rm G}),
\end{equation}
where $W_{\rm G} $ denotes the Gaussian window function and $s_n $ is the spectral moment defined as
\begin{equation}
s_n = 4 \pi \int dk k^{ 2( n+1) } P(k) W_{\rm G}^2( kR_{\rm G} ) .
\end{equation}
Let's clarify the reason that we set the initial conditions using the peak theory even though we dispute about its prediction at late time. Various studies, e.g.~\cite{EliaLudlowPorciani2012,ChanShethScoccimarro2015} show that the density and velocity cross power spectrum between halo and matter in the \textit{Lagrangian} space can be well fitted by the functional form motivated by the peak theory. Thus one may think that here we only use an established fact from simulations. However we argue that the subsequent evolution can be modelled by the simple fluid approximation augmented with the window function.
In the plot, as an example we will consider halo of mass $2 \times 10^{13} \, M_{\odot} {h}^{-1} $. We map the top-hat window size to the Gaussian window size using the relation $ R_{\rm G} = R_{\rm TH} / \sqrt{5} $. We compare the case when the window is static, with the window size given by the Lagrangian size and the case in which the window size is evolved by the SC model. The difference in treatment is only in $J_{\frac{3}{2} } $ in Eq.~\ref{eq:bw_in_w-1}. We note that when the window is static, the resultant $b_{\rm v} $ is almost constant over time. However, when the window is evolved by the SC model, $b_{\rm v} $ decays over time, but it is not negligible at late time.
Although we have used Eq.~\ref{eq:bvs_peak}, as in the initial condition, the contributions from the initial scale-dependent part is small compared to the ones due to $J_n$, even if we have used $ b_{\rm v} ^* =1 $, we find that the results are quite similar to that in Fig.~\ref{fig:bv_tot_evolve}. This highlights that the scale dependence of $ b_{\rm v} $ is mainly driven by the late time halo profile.
In the literature, the velocity bias is often associated with $k^2 $ correction to both the density and velocity biases, e.g. these $k^2 $ corrections can be derived from the peak model \cite{DesjacquesSheth2010}. In \cite{DesjacquesCrocceetal2014}, Zel'dovich approximation was used to displace the peaks to the Eulerian space, and they found that the velocity bias remains constant over time. The numerical measurement seemed to be in favour of the peak model result \cite{BaldaufDesjacquesSeljak2014}. Here we show that taking into account that halos are composite objects there is significant $k^2$-correction to the velocity bias and it does not decay away over time. When the static window is used, we also find that $ b_{\rm v}$ reduces to $W$ in the long term limit. However, when the evolving SC model is applied, the velocity bias is not constant, as can be seen from Fig.~\ref{fig:bv_tot_evolve}. Even in the static window limit, our result ($W$) is still different from \cite{DesjacquesCrocceetal2014}, which gets $W b_{\rm v}^{\rm pk }$ instead. The reason for this difference is that we do not impose the peak constraint in proto-halo evolution \cite{BiagettiDesjacquesKehagiasRiotto2014b}. These differences can be used to differentiate these two models.
Alternatively we can express the linear bias parameters in terms of the time derivative of the profile. Integrating the integral in Eq.~\ref{eq:bvlin_W} by parts, $ b_{\rm v} $ can be written as
\begin{eqnarray}
\label{eq:bvlin_Wp}
b_{\rm v}( y) &=& W(y) + [ b_{\rm v}^* - W(0) ] e^{ - \frac{ 3 }{ 2 } y } \nonumber \\
&-& \int_0^y d y' W'(y') e^{ - \frac{ 3 }{ 2 } (y - y') } ,
\end{eqnarray}
where $W'$ denotes $ \partial W/ \partial y $. This form shows that there are two contributions to the $k^2$ correction from the profile, one from $W$ and another from $W'$. The contribution from $W$ is simply the trivial smoothing. However, numerically performing derivatives on sparse data can lead to noisy results. Thus we will only use the form in terms of $J_n$, such as Eq.~\ref{eq:bw_in_w-1}.
As the velocity bias is mainly generated by $J_{\frac{3}{2}} $ in Eq.~\ref{eq:bw_in_w-1}, to gain insight into which part of the integral of $J_{\frac{3}{2}} $ contributes most, we plot the integrand of $J_{\frac{3}{2}}$, $j_{\frac{3}{2} }$
\begin{equation}
j_{\frac{3}{2}} ( y) = e^{ \frac{3 }{2 } y } \Big( W(y) -1 \Big)
\end{equation}
in Fig.~\ref{fig:j3o2_zE0_zLMany_bpt156_Oriana}. Again, in the range $ 0 <y \lesssim 3 $, there are no data available. We also show the prediction from the SC model. SC predicts that the contribution to the results in that range is small, while the contribution around $y\sim 3$ is the largest. However, we note that the SC results often overshoots in this range. We also show the results obtained with the static window. Static window approximation is good for $y\lesssim 2 $, but it overestimates the results for $y \gtrsim 3 $.
As both the window function $W$ and the time integral $J_{\frac{3}{2} }$ contribute to leading $k^2$ correction, we would like to compare the magnitude of these terms. In Fig.~\ref{fig:W1vsU_zE0_zLMany_bpt156_Oriana}, we compare $k^2$ contributions from $| W-1 |$ and $\frac{3 }{ 2 } e^{- \frac{3 y}{2} } | J_{ \frac{3}{2}} | $ using both the numerical results and the SC model. The results are for $z=0$. The SC model gives quite good description of the results from the data. In particular, the value of $\frac{3 }{ 2 } e^{- \frac{3 y}{2} } | J_{ \frac{3}{2}} | $ from numerical window and the SC model agrees quite well.
For the SC result for $ |W-1| $, we have compared a few prescriptions for the size of the window. Although at $z=0$ the size has not yet collapsed exactly to zero, using such a value gives the magnitude of $W-1$ much smaller than the simulation results. We have tried using Eq.~\ref{eq:rv_Deltav} with $\Delta_{\rm v} =200 $, 380 and 500, and $\Delta_{\rm v} = 500 $ gives the best agreement with simulations. In Fig.~\ref{fig:W1vsU_zE0_zLMany_bpt156_Oriana} we have shown the results obtained using $\Delta_{\rm v} = 500 $. In passing, if we simply use Eq.~\ref{eq:rvir_rm_half}, which is strictly only for matter-dominated universe, we get the results very similar to those from $\Delta_{\rm v} = 380 $. We have cross-checked the results using Carmen, and they are consistent with those from Oriana.
We note that recently there are reports of measurements of velocity bias at late time \cite{BaldaufDesjacquesSeljak2014,JenningsBaughHatt2014,ZhengZhangJing2014b}. In \cite{BaldaufDesjacquesSeljak2014}, the momentum was measured, and they found that the model with $b_{\rm pk} $ arised due to the peak constraint seems to fit the data better at high redshift such as $z=20$ than the evolution model without it. At such high redshifts, the effect of the profile evolution is small as can be seen from Fig.~\ref{fig:bv_tot_evolve}. Thus if confirmed, this would shows that the profile correction would not be the dominant effect of the velocity bias seen in simulations.
In the study of \cite{ZhengZhangJing2014b}, velocity bias was measured with sampling bias correction applied. They found that the velocity bias at $k\sim 0.08 \, \mathrm{Mpc}^{-1} \, h $ is slightly positive, with $ b_{\rm v} \sim 1.01 $. In our model, velocity bias can only be negative in the mildly nonlinear regime. However, as the number density of halos is low and halos are more inhomogeneously distributed in Eulerian space, thus it is hard to get an accurate volume weighted measurement. It is not clear that these measurements are free of artifacts. Thus we will keep these in mind and hope to report our own numerical comparison in future.
\subsubsection{Density}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{b1_tot_evolve.pdf}
\caption{ Evolution of the linear density bias with static window (dashed) and evolving window given by the SC (solid). Three redshifts are shown: $z=5 $ (blue), 1 (red) and 0 (green). }
\label{fig:b1_tot_evolve}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{b1_tot_evolve_constant_bstar.pdf}
\caption{ Same as Fig.~\ref{fig:b1_tot_evolve}, except the scale independent initial conditions $b_1^* = b_{\nu} $ and $ b_{\rm v} ^*=1 $ are assumed. }
\label{fig:b1_tot_evolve_constant_bstar}
\end{figure}
We now turn to the density bias. Plugging Eq.~\ref{eq:theta_g_lin} into Eq.~\ref{eq:continuity_deltag_linear}, we have
\begin{eqnarray}
\label{eq:deltag_step1}
\delta_{\rm g}^{(1)} (y) &=& \delta_{\rm g}^{*(1)} + 2 \theta_{\rm g}^{*(1)} (1 - e^{ - \frac{y }{ 2 } } ) \nonumber \\
&+& \frac{3 }{ 2 } \delta_*^{(1)} \int_0^y d y' e^{ - \frac{ y' }{ 2 } } I_{\frac{ 3 } { 2 }} (y') .
\end{eqnarray}
It is useful to note that
\begin{equation}
\label{eq:int_e_I}
\int_0^y dy' e^{a y'} I_b(y') = \frac{ 1 }{ a } [ e^{ay} I_b ( y) - I_{a+b} (y)].
\end{equation}
Using Eq.~\ref{eq:int_e_I}, we can simplify Eq.~\ref{eq:deltag_step1} to
\begin{equation}
\label{eq:deltag_1}
\delta_{\rm g}^{(1)}(y) = \delta_{\rm g}^{*(1)} + 2 \theta_{\rm g}^{*(1)} (1 - e^{ - \frac{y }{ 2 } } ) - 3 \delta_*^{(1)} [ e^{ - \frac{ y }{ 2 } } I_{\frac{3 }{ 2 }} - I_1 ].
\end{equation}
Thus the linear density bias is given by
\begin{eqnarray}
b_1 (y) &\equiv & \frac{ \delta_{\rm g}^{(1)} (y) }{ \delta^{(1)}(y) } \nonumber \\
&=& b_1^* e^{-y} + 2 b_{\rm v} ^* e^{-y} ( 1 -e^{ - \frac{ y }{ 2 } } ) \nonumber \\
& & \quad - 3 e^{-y} ( e^{ - \frac{ y }{ 2 } } I_{\frac{ 3 }{ 2 }} - I_1 ) ,
\end{eqnarray}
where $b_1^* $ is defined as
\begin{equation}
b_1^* \equiv \frac{ \delta_{\rm g}^{*(1)} }{ \delta_*^{(1)} }.
\end{equation}
Or in terms of $J_n$ using $ I_n = J_n + (e^{ny} -1)/n $, we have
\begin{eqnarray}
\label{eq:b1_w_1}
b_1(y) &=& 1 + ( b_1^* + 2 b_{\rm v} ^* - 3 ) e^{-y} + 2 (1 - b_{\rm v} ^* ) e^{ -\frac{3 y}{2} } \nonumber \\
&+& 3 e^{-y} J_1 - 3 e^{ - \frac{ 3y }{2} } J_{\frac{ 3 }{ 2 } }.
\end{eqnarray}
The first line in Eq.~\ref{eq:b1_w_1} is the same as the time evolution of linear density bias obtained in \cite{ChanScoccimarroSheth2012}, while the second line results from the halo profile correction. Unlike the decaying terms in the first line, they do not decay away. In the long term limit, $b_1$ reduces to $ 1 + 3 e^{-y} J_1 - 3 e^{ - \frac{ 3y }{2} } J_{\frac{ 3 }{ 2 } } $. If we assume that the window is static, we get $b_1=W$ in the long term limit.
In Fig.~\ref{fig:b1_tot_evolve}, the evolution of the linear density bias is plotted. Again we use the form of the initial condition motivated by the peak theory \cite{DesjacquesSheth2010}
\begin{equation}
\label{eq:b1s_peak}
b_1^* = ( b_{\nu} + b_{ \zeta } k^2 ) W_{\rm G}( k R_{\rm G}).
\end{equation}
We take $R_{\rm G} $ corresponding to halo of mass $2 \times 10^{13} \, M_{\odot} {h}^{-1} $. Instead of using the peak theory results, we take $ b_{\nu}= 15.9$ and $ b_{ \zeta } = 40.0 \, ( \, \mathrm{Mpc} \, h^{-1} )^2 $, which are obatined from measurement of the initial cross power spectrum \cite{ChanShethScoccimarro2015}. Both the results from the static window and SC evolving window are shown, however, the differences are very small.
The initial condition term $ ( b_1^* + 2 b_{\rm v} ^* - 3 ) e^{- y } $, especially due to $b_1^* $, is important at low $k$. In fact, at low $ k$, it gives the decay of linear bias \cite{Fry1996} . The term due solely to $ b_{\rm v} ^*$, $2( 1- b_{\rm v} ^*) e^{-3 y /2} $ is negligible in the whole range of $k$ shown. The sum of the two profile correction terms, $J_1$ and $J_{\frac{3}{2}} $ gives small overall correction. That is also the reason why the static and evolving window gives almost identical results. The reason that the profile correction term $J_n$ gives much more significant effect for $b_{\rm v} $ than for $b_1$ is that unlike the case of $ b_{\rm v} $, $b_1 $ at late time is still dominated by the reminant effect of $b_1^* $ because the magnitude of $b_1^* $ is much larger than that of $ b_{\rm v} ^* $.
Unlike the case $ b_{\rm v} $, the magnitude of the scale-dependent part of $b_1^*$ in Eq.~\ref{eq:b1s_peak} is significant compared to other contributions. To highlight its effect, we plot the results when the initial bias is scale-independent, i.e.~$b_1^* = b_{\nu} $ and $ b_{\rm v} ^*=1$ in Fig.~\ref{fig:b1_tot_evolve_constant_bstar}. This plot shows that the bump in Fig.~\ref{fig:b1_tot_evolve} around $k\sim 0.7 \, \mathrm{Mpc}^{-1} \, h $ is due to the large magnitude of the initial $ b_{ \zeta } k^2 $ term. The low $k$ plateau is due to $ b_{\nu}$ from initial condition and the scale-dependent transition comes from $J_1$ and $J_{\frac{3}{2}} $ terms.
\begin{widetext}
\subsection{Second order biases }
\label{sec:SecondBiasComputation}
To second order, Eq.~\ref{eq:continuity_1} and \ref{eq:Euler_1} become
\begin{eqnarray}
\label{eq:continuity_g}
\frac{ \partial \delta_{\rm g}^{(2)} }{ \partial y } - \theta_{\rm g}^{(2)} & =& \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} )
\alpha( \mathbf{k}_1 , \mathbf{k}_2 ) \theta_{\rm g}^{(1)} ( \mathbf{k}_1 ) \delta_{\rm g}^{(1)} ( \mathbf{k}_2 ) , \\
\label{eq:Euler_g}
\frac{ \partial \theta_{\rm g}^{(2)} }{ \partial y } + \frac{ 1 }{2 } \theta_{\rm g}^{(2)} - \frac{3 }{2 } W \delta^{ (2)} & = & \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} )
\beta( \mathbf{k}_1 , \mathbf{k}_2 ) \theta_{\rm g}^{(1)} ( \mathbf{k}_1 ) \theta_{\rm g}^{(1)} ( \mathbf{k}_2 ) .
\end{eqnarray}
We shall solve Eq.~\ref{eq:continuity_g} and \ref{eq:Euler_g} perturbatively to obtain $\delta_{\rm g}^{(2)} $ and $\theta_{\rm g}^{(2)}$.
\subsubsection{Velocity}
We will start from Eq.~\ref{eq:Euler_g} to compute $\theta_{\rm g}^{(2) }$ first. Using the dark matter SPT result
\begin{equation}
\delta^{ (2) } (y ) = e^{2y} \delta^{ (2) }_*.
\end{equation}
and Eq.~\ref{eq:thetag_1}, we can integrate Eq.~\ref{eq:Euler_g} to get
\begin{eqnarray}
\theta_{\rm g}^{(2) } ( y ) & = & \theta_{\rm g}^{* (2) } e^{- \frac{ y }{ 2 }} + \frac{ 3 }{ 2 } \delta_*^{(2)} e^{-\frac{ y }{ 2 }} I_{\frac{ 5 }{ 2 } } (y ) + \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \beta( \mathbf{k}_1 , \mathbf{k}_2 ) \nonumber \\
& \times & e^{- \frac{ y }{ 2 } } \Big\{ 2 ( 1 - e^{ - \frac{y }{2 } } ) \theta_{\rm g}^{*(1)}( \mathbf{k}_1 ) \theta_{\rm g}^{*(1)} ( \mathbf{k}_2 ) - \frac{ 3}{2} \big[ ( e^{ - \frac{ y }{ 2 } } I_{ \frac{ 3 }{ 2 } }( \mathbf{k}_2 ) - I_1( \mathbf{k}_2 ) ) \theta_{\rm g}^{*(1)} ( \mathbf{k}_1 ) \delta_*^{(1)} ( \mathbf{k}_2 ) + ( \mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ) \big] \nonumber \\
&+& \frac{9 }{4 } \int_0^y d y' e^{ - \frac{ y' }{ 2 } } I_{\frac{3}{ 2}} (y', \mathbf{k}_1) I_{\frac{3}{ 2}} (y', \mathbf{k}_2) \delta_*^{(1)}(\mathbf{k}_1 ) \delta_*^{(1)}(\mathbf{k}_2 ) \Big\}.
\end{eqnarray}
Replacing $ \theta_{\rm g}^{* (1) } $ by $ b_{\rm v}^* \delta_*^{(1)} $ and extrapolating $\delta_*^{(1)} $ to the present time, we have
\begin{eqnarray}
\label{eq:theta_g_2}
\theta_{\rm g}^{(2) } ( y ) & = & \theta_{\rm g}^{* (2) } e^{- \frac{ y }{ 2 }} + \frac{ 3 }{ 2 } \delta^{(2)}(y) e^{-\frac{ 5 y }{ 2 }} I_{\frac{ 5 }{ 2 } } (y ) + \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) K_{ \theta 2 } ( \mathbf{k}_1, \mathbf{k}_2 ) \delta^{(1)}( \mathbf{k}_1 ) \delta^{(1)}( \mathbf{k}_2 ),
\end{eqnarray}
where
$ K_{ \theta 2 } $ is given by
\begin{eqnarray}
K_{ \theta 2 } ( \mathbf{k}_1, \mathbf{k}_2 ) &=& e^{ - \frac{ 5 y }{ 2 } } \beta( \mathbf{k}_1, \mathbf{k}_2 ) \Big\{ 2( 1 - e^{- \frac{ y}{ 2 } } ) b_{\rm v }^* (\mathbf{k}_1 ) b_{\rm v }^* (\mathbf{k}_2 )
- \frac{3}{2} \big[ ( e^{ - \frac{ y }{ 2 } } I_{ \frac{ 3 }{ 2 } }( \mathbf{k}_2 ) - I_1( \mathbf{k_2} ) ) b_{\rm v }^* (\mathbf{k}_1 ) + (\mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ) \big] \nonumber \\
& +& \frac{9 }{4 } \int_0^y d y' e^{ - \frac{ y' }{ 2 } } I_{\frac{3}{ 2}} (y', \mathbf{k}_1) I_{\frac{3}{ 2}} (y', \mathbf{k}_2)
\Big\} ,
\end{eqnarray}
where $ ( \mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ) $ is a shorthand for a similar term obtained with $\mathbf{k}_1 $ and $\mathbf{k}_2 $ interchanged. To be general, we allow the initial linear biases to be scale-dependent. Note that $ K_{ \theta 2 } $ is already symmetric in $\mathbf{k}_1$ and $\mathbf{k}_2$, so symmetrization is not required.
In terms of $J_n$, $\theta_{\rm g}^{(2)}$ can be expressed as
\begin{equation}
\label{eq:theta_g_2_J_final}
\theta_{\rm g}^{(2) } ( y ) = \theta_{\rm g}^{* (2) } e^{- \frac{ y }{ 2 }} + \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \mathcal{K}_{ \theta 2 } ( \mathbf{k}_1, \mathbf{k}_2 ) \delta^{(1)}( \mathbf{k}_1 ) \delta^{(1)}( \mathbf{k}_2 ),
\end{equation}
where
$ \mathcal{K}_{ \theta 2 } $ is given by
\begin{equation}
\mathcal{K}_{ \theta 2 }= T_{F} + T_{G} + T_{ b_{\rm v} ^* } + T_{JJ} + T_J .
\end{equation}
The five types of terms $ T_{F}$, $T_{G}$, $ T_{ b_{\rm v} ^* }$, $ T_{JJ}$, and $ T_J$ are given by
\begin{eqnarray}
T_G &=& ( 1 - e^{- \frac{ 5 y }{ 2 } } ) G_2( \mathbf{k}_1, \mathbf{k}_2 ) , \\
T_F &=& \frac{3}{ 2 } e^{- \frac{ 5 y }{ 2 } } J_{ \frac{ 5 }{ 2} } (\mathbf{k} )F_2( \mathbf{k}_1, \mathbf{k}_2 ) , \\
T_{ b_{\rm v} ^* } &=& \beta( \mathbf{k}_1, \mathbf{k}_2 ) \Big\{ 2 e^{ - \frac{ 5 y }{ 2 } } ( 1 - e^{- \frac{ y}{ 2 } } ) b_{\rm v }^* (\mathbf{k}_1 ) b_{\rm v }^* (\mathbf{k}_2 )
+ e^{ - \frac{ 5 y }{ 2 } } ( e^{y} + 2 e^{ - \frac{ y }{ 2 } } - 3 ) \big( b_{\rm v }^* (\mathbf{k}_1 ) + b_{\rm v }^* (\mathbf{k}_2 ) \big) \nonumber \\
&-& 2 ( e^{ - \frac{3 y}{ 2 } } - 2 e^{ - \frac{ 5y}{ 2 } } + e^{-3y} )
- \frac{3}{2} e^{ - 3 y } \Big[ \big( J_{ \frac{ 3 }{2 } }( \mathbf{k}_2 ) - J_1 ( \mathbf{k}_2 ) \big) b_{\rm v }^* (\mathbf{k}_1 ) + ( \mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ) \Big] \Big\} , \\
T_{JJ} &=& \beta( \mathbf{k}_1, \mathbf{k}_2 ) e^{ - \frac{ 5 y }{ 2 } } \int_0^y d y' e^{ - \frac{ y' }{ 2 } } \frac{9 }{ 4 } J_{\frac{3}{ 2}} (y', \mathbf{k}_1 ) J_{\frac{3}{ 2}} ( y', \mathbf{k}_2 ), \\
T_J &=& \beta( \mathbf{k}_1, \mathbf{k}_2 ) e^{ - \frac{ 5 y }{ 2 } } \int_0^y d y' e^{ - \frac{ y' }{ 2 } } \frac{3}{2} ( e^{ \frac{3y'}{2} } - 1 ) \big( J_{\frac{3}{ 2}} (y', \mathbf{k}_1 ) + J_{\frac{3}{ 2}} (y', \mathbf{k}_2 ) \big) ,
\end{eqnarray}
where $\mathbf{k} = \mathbf{k}_{12} $, and $F_2$ and $G_2$ represent the coupling kernels
\begin{equation}
F_2 ( \mathbf{k}_1, \mathbf{k}_2 ) = \frac{5 }{7} + \frac{1 }{2} \mu \big( \frac{k_1 }{ k_2 } + \frac{k_2 }{ k_1 } \big) + \frac{2 }{7} \mu^2, \quad \quad
G_2 ( \mathbf{k}_1, \mathbf{k}_2 ) = \frac{3 }{7} + \frac{1 }{2} \mu \big( \frac{k_1 }{ k_2 } + \frac{k_2 }{ k_1 } \big) + \frac{4 }{7} \mu^2,
\end{equation}
with $ \mu =\hat{ \mathbf{k}}_1 \cdot \hat{ \mathbf{k}}_2$.
As a cross-check, we pause to consider the limit $ b_{\rm v} ^* =1 $ and $J_n=0$. Then $\mathcal{K}_{\theta 2}$ reduces to $ (1 -e^{ - \frac{5y}{ 2} }) G_2 $. Note that in this limit $ \theta_{\rm g}^{* (2) } e^{- \frac{ y }{ 2 }} = \theta^{* (2) } e^{- \frac{ y }{ 2 }} = \theta^{ (2) } e^{- \frac{5 y }{ 2 }} $, thus Eq.~\ref{eq:theta_g_2_J_final} reduces to $\theta^{(2)} $ because the galaxy field reduces to the dark matter field. On the other hand, in the long term limit $y \rightarrow \infty$, the transient terms vanish, in particular those arising from $ b_{\rm v} ^* $, and we end up with $G_2 + T_F + T_{JJ} + T_J $.
In Fig.~\ref{fig:theta2g_tot_evolve}, we show the evolution of the kernel $\mathcal{K}_{\theta 2 } $. In this plot, we have used the same parameters as those in the previous section and have set $k_1 = k_2 $ and $\mu = - 1/2$, which corresponds to the equilateral triangle configuration. We have compared the case with the static window and the one with SC evolving window, and find that the high $k$ corrections are quite different. In particular, the magnitude of the static one decreases while the evolving one increases over time. When $ b_{\rm v} ^* $ is assumed to be scale-independent instead, the results are similar to Fig.~\ref{fig:theta2g_tot_evolve}, thus we do not show it here.
We now look at the individual components of $\mathcal{K}_{\theta 2 } $ in details in this example. At low $k$, the only non-vanishing component is $T_G$ and it is (almost) constant for the reshifts shown. The term $T_F$ gives negative $k^2$ correction and its magnitude is large among all the high $k$ correction terms. The term $T_J$ and $T_{JJ} $ are of opposite signs, but the magnitude of $T_J $ is slightly larger. In particular, as the leading correction from $T_{JJ}$ is of $k^4$, compared to $T_{J}$, it is unimportant for $k\lesssim 0.6 \, \mathrm{Mpc}^{-1} \, h $. $T_{F}$ and $T_J$ are the largest scale-dependent correction terms, they are of similar magnitude but of opposite signs. The term with $ b_{\rm v} ^* $, $T_{ b_{\rm v} ^*} $ gives small negative contribution, which is negligible compared to the other correction terms. In fact for $ k \lesssim 0.6 \, \mathrm{Mpc}^{-1} \, h $, the kernel $\mathcal{K}_{\theta 2 } $ is well captured by the sum $T_F + T_G + T_J $. This is similar to $ b_{\rm v} $, for which the term solely due to $ b_{\rm v} ^* $ is negligible at late time (even at $z\sim 5$), and the dominant correction term comes from the $J_n$-term.
\subsubsection{Density}
We now compute $\delta_{\rm g}^{(2)} $. Integrating Eq.~\ref{eq:continuity_g} in terms of $\theta_{\rm g}^{(1) }$, $\theta_{\rm g}^{(2) }$, and $\delta_{\rm g}^{(1)} $ yields
\begin{equation}
\label{eq:deltag_2_general}
\delta_{\rm g}^{(2)} = \delta_{\rm g}^{*(2)} + \int_0^y d y' \theta_{\rm g}^{(2)}(y') + \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \alpha( \mathbf{k}_1 , \mathbf{k}_2 ) \int_0^y dy' \theta_{\rm g}^{(1)} ( \mathbf{k}_1 ) \delta_{\rm g}^{(1)} ( \mathbf{k}_2 ) .
\end{equation}
Using Eq.~\ref{eq:theta_g_2}, replacing $ \delta_{\rm g}^{* (1) } $ by $ b_{\rm v}^* \delta_*^{(1)} $ and extrapolating $\delta_*^{(1)} $ to the present time, we have
\begin{eqnarray}
\label{eq:int_theta2}
\int_0^y dy' \theta_{\rm g}^{(2)}(y') & = & 2 \theta_{\rm g}^{*(2)} ( 1 - e^{ - \frac{y}{2} } ) - 3 \delta^{(2)} [e^{- \frac{ 5y }{2 } } I_{\frac{ 5 }{2}} -e^{-2y} I_2 ] + \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \beta( \mathbf{k}_1 , \mathbf{k}_2 ) \nonumber \\
& \times & e^{-2y} \Big \{ 2( 1 - 2 e^{- \frac{y }{2} } + e^{-y} ) b_{\rm v}^*( \mathbf{k}_1 )b_{\rm v}^*( \mathbf{k}_2 ) + \frac{3}{2} \Big [ \big( e^{-y} I_{\frac{ 3 }{ 2 } }(\mathbf{k}_2) - 2 e^{ - \frac{ y }{ 2 } } I_1( \mathbf{k}_2 ) + I_{ \frac{ 1 }{ 2 } }( \mathbf{k}_2) \big) b_{\rm v}^*( \mathbf{k}_1 ) + ( \mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ) \Big] \nonumber \\
&-& \frac{ 9 }{2} \int_0^y dy' ( e^{ -\frac{ y}{ 2} } - e^{ -\frac{ y' }{ 2} } ) e^{ -\frac{ y' }{ 2} } I_{\frac{ 3 }{ 2} } (y', \mathbf{k}_1 ) I_{\frac{ 3 }{ 2} } (y', \mathbf{k}_2 ) \Big\} \delta^{(1)} ( \mathbf{k}_1 ) \delta^{(1)} ( \mathbf{k}_2 ) .
\end{eqnarray}
Making use of Eq.~\ref{eq:thetag_1} and \ref{eq:deltag_1}, we can compute the second integral in Eq.~\ref{eq:deltag_2_general} to get
\begin{eqnarray}
\label{eq:alpha_theta_delta}
&& \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \alpha( \mathbf{k}_1, \mathbf{k}_2 ) \int_0^y dy' \theta_{\rm g}^{(1)} ( y', \mathbf{k}_1 ) \delta_{\rm g}^{(1)} ( y', \mathbf{k}_2 ) \nonumber \\
& = & \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) e^{-2y} \alpha( \mathbf{k}_1, \mathbf{k}_2 ) \Big\{
2 b_{\rm v}^* (\mathbf{k}_1 ) b_{1}^* (\mathbf{k}_2 ) ( 1 - e^{ - \frac{ y }{2 } } )
+ 2 b_{\rm v}^* (\mathbf{k}_1 ) b_{\rm v}^* (\mathbf{k}_2 ) (1 -2 e^{ - \frac{ y }{ 2 } } + e^{ -y } ) \nonumber \\
&-& 3 b_1^*( \mathbf{k}_2 ) \big( e^{ - \frac{ y }{ 2 } } I_{\frac{ 3}{ 2 }}( \mathbf{k}_1 ) - I_1 ( \mathbf{k}_1 ) \big)
+ 3 b_{\rm v}^* ( \mathbf{k}_1 ) \big( e^{-y} I_{ \frac{ 3 }{ 2 } } ( \mathbf{k}_2 ) - 2 e^{- \frac{ y }{ 2 }} I_1 ( \mathbf{k}_2 ) + I_{ \frac{ 1 }{ 2 } } ( \mathbf{k}_2 ) \big)
+ 3 b_{\rm v}^* ( \mathbf{k}_2 ) \big[ ( e^{-y} - 2 e^{- \frac{ y }{ 2 }} ) I_{\frac{ 3 }{ 2 }} ( \mathbf{k}_1 ) \nonumber \\
& +& 2 I_1 ( \mathbf{k}_1 ) - I_{\frac{ 1}{ 2 }} ( \mathbf{k}_1 ) \big]
- \frac{ 9 }{ 2 } \int_0^y d y' e^{- \frac{ y' }{ 2 } } I_{\frac{3}{ 2} }(y', \mathbf{k}_1 ) [ e^{- \frac{ y'}{ 2 } } I_{\frac{3}{2} }(y', \mathbf{k}_2) - I_1(y' , \mathbf{k}_2) ]
\Big\} \delta^{(1)}( \mathbf{k}_1 ) \delta^{(1)}( \mathbf{k}_2 ).
\end{eqnarray}
Therefore, we have
\begin{equation}
\delta_{\rm g}^{(2)} = \delta_{\rm g}^{*(2)} + 2 \theta_{\rm g}^{*(2)} ( 1 - e^{ - \frac{y}{2} } ) - 3 \delta^{(2)} [e^{- \frac{ 5y }{2 } } I_{\frac{ 5 }{2}} -e^{-2y} I_2 ] + \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) K_{\delta2}( \mathbf{k}_1, \mathbf{k}_2) \delta^{(1)} (\mathbf{k}_1 ) \delta^{(1)}(\mathbf{k}_2 ) ,
\end{equation}
where $ K_{\delta2} $ is given by
\begin{eqnarray}
K_{\delta2} ( \mathbf{k}_1, \mathbf{k}_2 )
&=& e^{-2 y} \Big\{
2 ( e^{-y} - 2 e^{ - \frac{ y }{ 2 } } + 1 ) \big( \beta(\mathbf{k_1}, \mathbf{k}_2 ) + \alpha (\mathbf{k_1}, \mathbf{k}_2 ) \big) b_{\rm v} ^*( \mathbf{k}_1 ) b_{\rm v} ^*( \mathbf{k}_2 ) \nonumber \\
&+& 2 ( 1 - e^{ - \frac{ y }{ 2 } } ) b_{\rm v} ^* ( \mathbf{k}_1 ) b_1^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 )
+ 3 \big[ e^{-y} I_{\frac{ 3 }{ 2 } } ( \mathbf{k}_2 ) - 2 e^{ - \frac{ y }{ 2 } } I_1 ( \mathbf{k}_2 ) + I_{ \frac{ 1 }{ 2 } } ( \mathbf{k}_2 ) \big] b_{\rm v} ^* ( \mathbf{k}_1 ) ( \alpha(\mathbf{k_1}, \mathbf{k}_2 ) + \beta(\mathbf{k_1}, \mathbf{k}_2 ) ) \nonumber \\
&+& 3 \big[ ( e^{-y} - 2 e^{ - \frac{y }{ 2 } } ) I_{\frac{3 } { 2 }} ( \mathbf{k}_1 ) + 2 I_1 ( \mathbf{k}_1 ) - I_{ \frac{1 }{2 } } ( \mathbf{k}_1 ) \big] b_{\rm v} ^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 )
- 3 \big[ e^{ - \frac{ y }{2 } } I_{\frac{ 3 }{ 2 } } ( \mathbf{k}_1 ) - I_1 (\mathbf{k}_1 ) \big] b_1^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 ) \nonumber \\
&-& \frac{ 9 }{ 2 } \int_0^y d y' e^{- \frac{ y' }{ 2 } } I_{\frac{ 3 }{ 2 } }(y', \mathbf{k}_1 ) \Big[ ( e^{ - \frac{ y }{2 } } - e^{ - \frac{ y' }{2 } } ) I_{\frac{ 3 }{ 2 } }(y', \mathbf{k}_2 ) \beta(\mathbf{k_1}, \mathbf{k}_2 ) + \Big( e^{ - \frac{ y' }{2 } } I_{\frac{ 3 }{ 2 } }(y', \mathbf{k}_2) - I_1(y',\mathbf{k}_2) \Big) \alpha (\mathbf{k_1}, \mathbf{k}_2 ) \Big]
\Big\} . \nonumber \\
\end{eqnarray}
In terms of $J_n$, $\delta_{\rm g}^{(2)}$ can be expressed as
\begin{equation}
\label{eq:deltag_final}
\delta_{\rm g}^{(2)} = \delta_{\rm g}^{*(2)} + 2 \theta_{\rm g}^{*(2)} ( 1 - e^{ - \frac{y}{2} } )
+ \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} )\mathcal{ K}_{\delta2}( \mathbf{k}_1, \mathbf{k}_2) \delta^{(1)}(\mathbf{k}_1 ) \delta^{(1)}(\mathbf{k}_2 ) ,
\end{equation}
where $ K_{\delta2} $ is given by
\begin{eqnarray}
\label{eq:Kmathcal_final}
\mathcal{K}_{\delta2} ( \mathbf{k}_1, \mathbf{k}_2 )
&=& e^{-2 y} \Big\{
[ \frac{ 3 }{ 10 } ( e^{2y} + 4 e^{- \frac{ y }{ 2 } } - 5 ) - 3 \big( e^{- \frac{ y }{ 2 } } J_{\frac{ 5 }{ 2 } }( \mathbf{k} ) - J_2 (\mathbf{k}) \big) ] F_2( \mathbf{k}_1, \mathbf{k}_2 ) \nonumber \\
&+& 2( e^{-y} - 2 e^{ - \frac{ y}{ 2 } } + 1 ) \big( \alpha (\mathbf{k_1}, \mathbf{k}_2) + \beta (\mathbf{k_1}, \mathbf{k}_2) \big) b_{\rm v} ^*( \mathbf{k}_1 ) b_{\rm v} ^*( \mathbf{k}_2 )
+ 2 ( 1 - e^{ - \frac{ y }{ 2 } } ) b_{\rm v} ^* ( \mathbf{k}_1 ) b_1^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 )
\nonumber \\
&+& \big[ 2 ( e^{ \frac{ y}{ 2 } } - 3 +3 e^{- \frac{ y }{2 }} - e^{-y} )
+ 3 \big( e^{-y}J_{\frac{ 3}{2 }} (\mathbf{k}_2) - 2 e^{- \frac{y}{2} } J_1( \mathbf{k}_2 )+ J_{\frac{ 1 }{ 2 }}( \mathbf{k}_2 ) \big) \big] b_{\rm v} ^* ( \mathbf{k}_1 ) \big( \alpha(\mathbf{k_1}, \mathbf{k}_2 ) + \beta(\mathbf{k_1}, \mathbf{k}_2 ) \big) \nonumber \\
&+& \big[ 2 \big( e^y - 2 e^{\frac{ y}{ 2 }} + 2 e^{ - \frac{ y }{ 2 } } - e^{-y} \big) + 3 \big( (e^{-y} - 2 e^{- \frac{ y }{ 2 } } ) J_{ \frac{ 3 }{ 2 } }( \mathbf{k}_1 ) + 2 J_1 ( \mathbf{k}_1 ) - J_{ \frac{1}{2} } ( \mathbf{k}_1 ) \big) \big] b_{\rm v} ^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 ) \nonumber \\
&+ & \big[ e^y - 3 + 2 e^{ - \frac{ y }{ 2 } } - 3 \big( e^{- \frac{ y }{ 2 } } J_{ \frac{ 3 }{ 2 }}( \mathbf{k}_1 ) - J_1 ( \mathbf{k}_1 ) \big) \big] b_1^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 )
+ A_1 + A_2 + A_3
\Big\},
\end{eqnarray}
where $A_1$, $A_2$, and $A_3$ represent
\begin{eqnarray}
A_1 &= &- \frac{ 9 }{ 2 } \int_0^y dy' e^{ - \frac{ y' }{ 2 } } J_{ \frac{ 3 }{ 2 } }( y', \mathbf{k}_1 ) [ ( e^{- \frac{y }{ 2 }} - e^{- \frac{y' }{2 } } ) J_{\frac{3 }{ 2} } ( y', \mathbf{k}_2 ) \beta(\mathbf{k}_1, \mathbf{k}_2 ) + \big( e^{- \frac{y' }{2 } } J_{ \frac{ 3 }{ 2 } } ( y', \mathbf{k}_2 ) - J_1 ( y', \mathbf{k}_2 ) \big) \alpha(\mathbf{k}_1, \mathbf{k}_2 ) ], \\
A_2 & =& - 3 \int_0^y dy' e^{ - \frac{ y' }{ 2 } } \Big\{ 2 ( e^{- \frac{y }{ 2 }} - e^{- \frac{y' }{2 } } ) ( e^{ \frac{ 3y' }{ 2 } } - 1 ) J_{\frac{3}{2}}(y', \mathbf{k}_1) \beta( \mathbf{k}_1, \mathbf{k}_2 ) \nonumber \\
&+& \Big[ \big( \frac{3}{2} - \frac{e^{y' } }{ 2 } - e^{ - \frac{y' }{ 2} } \big) J_{\frac{ 3 }{2 } }(y', \mathbf{k}_1 ) + ( e^{ y' } - e^{ - \frac{ y' }{ 2 } } ) J_{\frac{ 3 }{ 2 } } (y', \mathbf{k}_2 ) - ( e^{ \frac{3 y' }{ 2 } } - 1 ) J_1 ( y', \mathbf{k}_2 ) \Big] \alpha( \mathbf{k}_1, \mathbf{k}_2 ) \Big\} , \\
A_3 &=& \frac{ 1 }{ 10 } e^{-y} ( e^{ \frac{ y}{ 2} } - 1 )^4 [ 5 ( 2 + e^{ \frac{ y}{ 2} } )^2 \alpha( \mathbf{k}_1, \mathbf{k}_2 ) + 2 ( e^y + 4 e^{ \frac{ y}{ 2} } + 10 ) \beta ( \mathbf{k}_1, \mathbf{k}_2 ) ] .
\end{eqnarray}
When the limit $b_1^* =1 $, $ b_{\rm v} ^* = 1 $ and $J_n=0$ are taken, $\delta_{\rm g}^{(2)} $ in Eq.~\ref{eq:deltag_final} reduces to $\delta^{(2)} $ as the galaxy field becomes the dark matter field. In the large $y$ limit, we have
\begin{equation}
\mathcal{K}_{\delta 2 } = \Big[ \frac{3}{10} - 3 e^{ -2 y } \Big( e^{ -\frac{ y }{ 2 } } J_{\frac{5}{2}}(\mathbf{k}) -J_2(\mathbf{k}) \Big) \Big] F_2 ( \mathbf{k}_1, \mathbf{k}_2 ) + e^{ -2 y } ( A_1 + A_2 + A_3 ).
\end{equation}
As only the symmetric part of the kernel $\mathcal{K}_{\delta 2 } $ contributes to the integral in Eq.~\ref{eq:deltag_final}, we need to symmetrize the kernel as
\begin{equation}
\mathcal{K}_{\delta 2}^{\rm s} (\mathbf{k}_1, \mathbf{k}_2 ) = \frac{1}{2} \Big( \mathcal{K}_{\delta 2} (\mathbf{k}_1, \mathbf{k}_2 ) + \mathcal{K}_{\delta 2} (\mathbf{k}_2, \mathbf{k}_1 ) \Big) .
\end{equation}
However, to reduce the length of the formulas, we do not explicitly symmetrize them. However, in the final results, we always use the symmetrized kernel.
Similar to that in Ref.~\cite{ChanScoccimarroSheth2012}, we define the terms that deviate from the local biasing prescription as nonlocal terms. Thus at second order, the nonlocal terms are defined as
\begin{equation}
\chi_{\rm nonloc}^{(2)} = \delta_{\rm g}^{(2)} - ( b_1 \delta^{(2)} + \frac{b_2 }{2 } (\delta^{(1)} )^2 ).
\end{equation}
At second order, the nonlocal terms are induced by the initial linear bias, and the initial second order biases do not generate new terms \cite{ChanScoccimarroSheth2012}. For convenience we consider
\begin{equation}
\label{eq:deltag2_nonloc}
\delta_{\rm g}^{(2)} - b_1 \delta^{(2)} = \delta_{\rm g}^{*(2)} + 2 \theta_{\rm g}^{*(2)} ( 1 - e^{ - \frac{y}{2} } )
+ \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \chi ( \mathbf{k}_1, \mathbf{k}_2) \delta(\mathbf{k}_1 ) \delta(\mathbf{k}_2 )
\end{equation}
where $ \chi_{\delta2} $ is given by
\begin{eqnarray}
\label{eq:chi_delta2}
\chi ( \mathbf{k}_1, \mathbf{k}_2)
&=& e^{-2 y} \Big\{
\Big[ - \frac{ 7 }{ 10 } e^{2y} - \big ( b_1^*(\mathbf{k}) + 2 b_{\rm v} ^*( \mathbf{k}) - 3 \big) e^y - 2 \big(1 - b_{\rm v} ^*( \mathbf{k} ) \big) e^{\frac{ y}{ 2}} \nonumber \\
&+& \frac{ 6 }{ 5 } e^{ - \frac{ y}{ 2} } -\frac{ 3 }{ 2 } - 3 e^{ - \frac{ y}{ 2} } J_{\frac{5 }{2 }}(\mathbf{k} ) + 3 J_2(\mathbf{k} ) - 3 e^y J_1(\mathbf{k}) + 3 e^{\frac{y}{2} } J_{\frac{ 3}{ 2 }} (\mathbf{k}) \Big]F_2( \mathbf{k}_1 , \mathbf{k}_2 ) \nonumber \\
&+& 2( e^{-y} - 2 e^{ - \frac{ y}{ 2 } } + 1 ) \big( \alpha (\mathbf{k_1}, \mathbf{k}_2) + \beta (\mathbf{k_1}, \mathbf{k}_2) \big) b_{\rm v} ^*( \mathbf{k}_1 ) b_{\rm v} ^*( \mathbf{k}_2 )
+ 2 ( 1 - e^{ - \frac{ y }{ 2 } } ) b_{\rm v} ^* ( \mathbf{k}_1 ) b_1^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 )
\nonumber \\
&+& \big[ 2 ( e^{ \frac{ y}{ 2 } } - 3 +3 e^{- \frac{ y }{2 }} - e^{-y} )
+ 3 \big( e^{-y}J_{\frac{ 3}{2 }} (\mathbf{k}_2) - 2 e^{- \frac{y}{2} } J_1( \mathbf{k}_2 )+ J_{\frac{ 1 }{ 2 }}( \mathbf{k}_2 ) \big) \big] b_{\rm v} ^* ( \mathbf{k}_1 ) \big( \alpha(\mathbf{k_1}, \mathbf{k}_2 ) + \beta(\mathbf{k_1}, \mathbf{k}_2 ) \big) \nonumber \\
&+& \big[ 2 \big( e^y - 2 e^{\frac{ y}{ 2 }} + 2 e^{ - \frac{ y }{ 2 } } - e^{-y} \big) + 3 \big( (e^{-y} - 2 e^{- \frac{ y }{ 2 } } ) J_{ \frac{ 3 }{ 2 } }( \mathbf{k}_1 ) + 2 J_1 ( \mathbf{k}_1 ) - J_{ \frac{1}{2} } ( \mathbf{k}_1 ) \big) \big] b_{\rm v} ^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 ) \nonumber \\
&+ & \big[ e^y - 3 + 2 e^{ - \frac{ y }{ 2 } } - 3 \big( e^{- \frac{ y }{ 2 } } J_{ \frac{ 3 }{ 2 }}( \mathbf{k}_1 ) - J_1 ( \mathbf{k}_1 ) \big) \big] b_1^* ( \mathbf{k}_2 ) \alpha (\mathbf{k_1}, \mathbf{k}_2 )
+ A_1 + A_2 + A_3
\Big\},
\end{eqnarray}
In Eq.~\ref{eq:deltag2_nonloc}, we have used $b_1 $ given by Eq.~\ref{eq:b1_w_1}. The initial second order biases are hidden in $\delta_{\rm g}^{*(2) } $.
Suppose that the initial conditions are given by
\begin{equation}
\delta_{\rm g}^{*(2)} = \frac{ b_{2}^* }{ 2 } ( \delta^{(1)}_*)^2 + b_{1}^* \delta^{(2)}_* , \quad b_{\rm v} ^* = 1, \quad \theta_{\rm g}^{*(2)} = \theta^{(2)}_*,
\end{equation}
where $ b_{2 }^* $ and $b_{1}^* $ are scale-independent. In other words, we suppose that the initial density biases are local in Lagrangian space, and there is no initial velocity bias. If we also neglect all the $J_n$ terms, then Eq.~\ref{eq:deltag2_nonloc} is simplified substantially and we end up with \cite{ChanScoccimarroSheth2012}
\begin{equation}
\delta_{\rm g}^{(2)} - b_1 \delta^{(2)} = \frac{ b_2 }{ 2 } ( \delta^{(1)})^2 + \gamma_2 \mathcal{G}_2 ,
\end{equation}
where $ b_2 $ and $\gamma_2$ are given by
\begin{equation}
b_2 = b_2^* e^{-2y}, \quad \quad \gamma_2 = \frac{2}{ 7} ( b_1^* - 1) e^{-2y} ( e^{y} - 1 ),
\end{equation}
and $\mathcal{G}_2 $ denotes
\begin{equation}
\mathcal{G}_2 (\mathbf{k}) = \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) ( \mu^2 -1 ) \delta^{(1)}(\mathbf{k}_1 ) \delta^{(1)}(\mathbf{k}_2 ).
\end{equation}
In particular, because there is no velocity bias as the dipole term vanishes.
We now consider the correction to these results due to initial scale-dependent biases and the corrections arising from the profile corrections. For the initial conditions, we will assume that $ b_{\rm v} ^* $ and $b_1^*$ are given by Eq.~\ref{eq:bvs_peak} and \ref{eq:b1s_peak} respectively, and for simplicity $b_2^*$ is a constant and $\theta_{\rm g}^{*(2)} = \theta_*^{(2)} $.
We first define the scale-dependent parameters of the initial biases
\begin{equation}
\epsilon_{1}^* (k) = b_1^*(k) - b_{\nu} , \quad \quad \epsilon_{\rm v}^* ( k) = b_{\rm v} ^*(k) - 1 .
\end{equation}
Then we have
\begin{equation}
\delta_{\rm g}^{(2)} - b_1 \delta^{(2)} - \frac{ b_2 }{ 2 } (\delta^{(1)} )^2 - \gamma_2 \mathcal{G}_2 = \int d^3 k_1 d^3 k_2 \delta_{\rm D} ( \mathbf{k}- \mathbf{k}_{12} ) \psi(\mathbf{k}_1, \mathbf{k}_2 ) \delta^{(1)}(\mathbf{k}_1 ) \delta^{(1)}(\mathbf{k}_2 ),
\end{equation}
where the kernel $ \psi $ is defined as
\begin{equation}
\psi( \mathbf{k}_1, \mathbf{k}_2 ) = T_{ \epsilon_{1}^* } + T_{ \epsilon_{\rm v}^* } + T_J ,
\end{equation}
with various terms given by
\begin{eqnarray}
T_{ \epsilon_{1}^* } &=& \frac{ 1}{2} e^{-2 y } \big[ - e^{y} \epsilon_{1}^* ( \mathbf{k} ) F_2( \mathbf{k}_1, \mathbf{k}_2 ) + ( e^y -1 ) \epsilon_{1}^* ( \mathbf{k}_2 ) \alpha( \mathbf{k_1}, \mathbf{k_2} ) \big] + (\mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ), \\
T_{ \epsilon_{\rm v}^* } &=& \frac{ 1}{2} e^{-2y} \Big\{ 2 ( e^{-y} - 2 e^{ - \frac{ y }{ 2 } } + 1 ) ( \alpha( \mathbf{k}_1, \mathbf{k}_2 ) + \beta( \mathbf{k}_1, \mathbf{k}_2 ) ) \big( \epsilon_{\rm v}^* ( \mathbf{k}_1 ) + \epsilon_{\rm v}^* ( \mathbf{k}_2 ) + \epsilon_{\rm v}^* ( \mathbf{k}_1 ) \epsilon_{\rm v}^* ( \mathbf{k}_2 ) \big) \nonumber \\
&+& 2( e^{ \frac{y }{ 2 } } - e^{y} ) \epsilon_{\rm v}^* (\mathbf{k} ) F_2 (\mathbf{k}_1 , \mathbf{k}_2 ) + 2 (1 - e^{ - \frac{ y }{ 2 } } ) \epsilon_{\rm v}^* ( \mathbf{k}_1 ) \big( b_{\nu} + \epsilon_{1}^* (\mathbf{k}_2) \big) \alpha( \mathbf{k}_1 , \mathbf{k}_2 ) \nonumber \\
&+& 2( e^{\frac{ y }{ 2 }} - 3 + 3 e^{- \frac{ y }{ 2 } } - e^{ -y } ) \epsilon_{\rm v}^* ( \mathbf{k}_1 ) \big( \alpha( \mathbf{k}_1 , \mathbf{k}_2 ) + \beta( \mathbf{k}_1 , \mathbf{k}_2 ) \big) \nonumber \\
&+& 2 ( e^y - 2 e^{ \frac{y }{2 } } + 2 e^{ - \frac{y }{2 } } - e^{-y} ) \epsilon_{\rm v}^* ( \mathbf{k}_2 ) \alpha( \mathbf{k}_1 , \mathbf{k}_2 ) \Big\} + (\mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ), \\
T_J &=& \frac{ 1}{2} e^{-2y} \Big\{ \big[ -3 e^{ - \frac{ y }{ 2 }} J_{ \frac{ 5 }{ 2 }} ( \mathbf{k} ) + 3 J_2 ( \mathbf{k} ) - 3 e^y J_1 ( \mathbf{k} ) + 3 e^{ \frac{y }{ 2 } } J_{ \frac{ 3 }{ 2 } } ( \mathbf{k} ) \big] F_2 ( \mathbf{k}_1, \mathbf{k}_2 ) \nonumber \\
&+& 3 \big[ e^{-y} J_{ \frac{3 }{ 2 } }( \mathbf{k}_2 ) - 2 e^{- \frac{ y }{2 } } J_1( \mathbf{k}_2 ) + J_{\frac{1}{2}} ( \mathbf{k}_2 ) \big] b_{\rm v} ^*( \mathbf{k}_1 ) \big( \alpha( \mathbf{k}_1, \mathbf{k}_2 ) + \beta ( \mathbf{k}_1 , \mathbf{k}_2 ) \big) \nonumber \\
&+ & 3 \big[ ( e^{-y} - 2 e^{- \frac{ y }{ 2}} ) J_{ \frac{ 3 }{2 }} ( \mathbf{k}_1 ) + 2 J_1( \mathbf{k}_1 ) - J_{ \frac{ 1}{ 2 } } ( \mathbf{k}_1 ) \big] b_{\rm v} ^* (\mathbf{k}_2 ) \alpha ( \mathbf{k}_1 , \mathbf{k}_2 ) \nonumber \\
& -& 3 ( e^{ - \frac{ y }{ 2 } } J_{ \frac{ 3}{ 2 } } ( \mathbf{k}_1 ) - J_1 ( \mathbf{k}_1 ) ) b_1^*( \mathbf{k}_2 ) \alpha( \mathbf{k}_1, \mathbf{k}_2 ) + A_1 + A_2 \Big\} + (\mathbf{k}_1 \leftrightarrow \mathbf{k}_2 ) .
\end{eqnarray}
In Fig.~\ref{fig:delta2g_tot_evolve}, we show the evolution of the kernel $ \psi$ and its components at $ z=5$, 1 and 0 respectively. The parameters used are the same as those in the previous section and we have set $k_1 = k_2$ and $\mu= - 1/2$. Similar to the case of $b_1$, there is no noticeable difference between the case with static window and the evolving one. All these contributions peaks around $ k \sim 0.7 - 0.9 \, \mathrm{Mpc}^{-1} \, h $. In this case, $T_{ \epsilon_1^* } $ and $T_{ \epsilon_{\rm v}^* } $ are of similar magnitude but opposite signs, so they roughly cancel each other. Thus the net contribution is mainly given by $T_{J}$. The overall contribution of $ \psi$ decays over time.
Again to highlight the effects of the initial scale-dependent bias, we show the case when the initial condition is scale-independent, i.e.~$b_1^* = b_{\nu}$ and $ b_{\rm v} ^* = 1 $ are assumed in Fig.~\ref{fig:delta2g_tot_evolve_constant_bstar}. We find that the bump around $k\sim 1 \, \mathrm{Mpc}^{-1} \, h $ Fig.~\ref{fig:delta2g_tot_evolve} is no longer present, instead there is smooth transition from $k\sim 0.2$ to $1 \, \mathrm{Mpc}^{-1} \, h $.
The second order kernel will contribtute to the tree level bispectrum. As a quick check of the importance of the correction term, we compare the kernel $\psi $ with the nonlocal term kernel $\gamma_2 ( \mu^2 -1 ) $ in Fig.~\ref{fig:psi_G2_ratio}. The parameters used are the same as the previous ones. From Fig.~\ref{fig:psi_G2_ratio}, we can see the $k^2$ correction at low $k$. However, the $k^2$ correction term starts to surpass the nonlocal term at $k \sim 0.4 \, \mathrm{Mpc}^{-1} \, h $. Hence one should also find the signature of the $k^2$ correction term in the halo bispectrum at $k \gtrsim 0.2 \, \mathrm{Mpc}^{-1} \, h $.
\end{widetext}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{theta2g_tot_evolve.pdf}
\caption{ The evolution of the kernel of $\theta_{\rm g}^{(2)} $ for the case with static window (dashed) and the SC evolving one (solid). The parameters $k_1 = k_2 $ and $\mu =- 1/2$ are used. Three redshifts are shown, $z=5$ (blue), 1 (red) and 0 (green). }
\label{fig:theta2g_tot_evolve}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{delta2g_tot_evolve.pdf}
\caption{ The evolution of the kernel of $\psi $ for the case with static window (dashed) and the SC evolving one (solid). However, these two cases are indistinguishable. The parameters are set such that $k_1 = k_2 $ and $\mu = -1/2$. Three redshifts are shown, $z=5$ (blue), 1 (red) and 0 (green). }
\label{fig:delta2g_tot_evolve}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{delta2g_tot_evolve_constant_bstar.pdf}
\caption{ Same as Fig.~\ref{fig:delta2g_tot_evolve}, except with scale-independent initial conditions $b_1^* = b_{\nu} $ and $ b_{\rm v} ^* =1 $. }
\label{fig:delta2g_tot_evolve_constant_bstar}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{psi_G2_ratio.pdf}
\caption{ The ratio between the kernel $\psi$ and $\gamma_2 ( \mu^2 -1 ) $, the kernel of the nonlocal term $\mathcal{G}_2$. The parameters $k_1 = k_2 $ and $\mu =- 1/2$ are used. Three redshifts are shown, $z=5$ (blue), 1 (red) and 0 (green). Solid line for positive value and dotted line for negative one. }
\label{fig:psi_G2_ratio}
\end{figure}
\section{Conclusions}
\label{sec:Conclusions}
Recent measurements of the velocity bias suggest that the velocity bias of the halos is non-negligible at the weakly nonlinear regime $k \sim 0.1 \, \mathrm{Mpc}^{-1} \, h $ at late time. In the time evolution model of the halo field, previously it was shown that it leads to decay of the initial velocity bias using the point particle approximation so that it becomes negligible at late time. On the other hand, the peak model gives constant velocity bias over time. Thus the measurement seems to be in favour of the peak model result.
It is often assumed that halos are point particles and focus only on their center of mass. Here we argue that as halos consist of a collection of particles, the force acting on its CM of the halo should be the force averaged over its constituent particles instead of only the force at the position of the CM. To take into account of the halo profile, we introduce a window function in the Euler equation. We find that the window function leads to non-negligible $k^2$ correction to the linear velocity bias. While the initial $k^2 $ velocity bias decays away, the correction due to profile correction does not. In contrast, in the peak model, the imposition of the peak constraint leads to an extra constant scale-dependent bias. This difference can be used to distinguish these models. The profile correction also gives $k^2$ correction to the second order velocity kernel. For the density bias, the effect of the profile correction is not important at low $k$ because the magnitude of the initial scale-dependent density bias is large in the peak model. Thus even at low $z$, the magnitude of the terms due to $ b_1^*$ are the most important ones. Nonetheless, this implies that the $k^2$ correction is non-negligible for $k\gtrsim 0.2 \, \mathrm{Mpc}^{-1} \, h $, especially for bispecturm.
Since the window function is dynamical, we model it using the spherical collapse model. We also measure the evolution of the halo profile by constructing proto-halos at different redshifts. To our knowledge this is the first systematic numerical study of the evolution of the proto-halo profile. We find that the proto-halo profile evolves from a top-hat-like profile to an NFW profile. We find reasonable agreement between the spherical collapse and the numerical results. On the theory side, one may improve the modelling using ellipsoidal collapse model instead. Computationally, it would be useful to come up with a parameterization for the halo profile at various epochs.
Our work has highlighted the importance of halo profile and its evolution on bias. In theories such as the excursion set theory and peak model, window function are used to define halos in Lagrangian space. They are often assumed to be static and the window size given by the Lagrangian size even when they are transformed to the Eulerian space. The idea of profile evolution can be easily applied to these models as well.
In our model, the effect of the window function correction is most apparent in the velocity bias. Although there are some existing measurements of velocity bias, it is still hard because it is prone to sampling artifacts. We hope to report the comparison of our model with velocity bias measurement in future.
\section*{Acknowledgement}
I am grateful to Rom\'an Scoccimarro for suggesting that the halo profile evolution as the source of velocity bias and motivating me to look at the effects of halo profile evolution analytically and numerically. He also made numerous valuable suggestions to this paper. I thank Andreas Berlind, Matteo Biagetti and Ravi Sheth for useful discussions. I also thank Vincent Desjacques and Ravi Sheth for comments on the draft of the paper. I thank LasDamas project \footnote{\url{http://lss.phy.vanderbilt.edu/lasdamas}} for the simulations used in the work. The simulations were run using a Teragrid allocation and some RPI and NYU computing resources were also used. This work is supported by the Swiss National Science Foundation.
|
2,869,038,155,653 | arxiv | \section{Introduction}
Peak demand is the time when consumer demand for electricity is at its highest. In distribution systems, the installed power transfer capacity must be greater than the expected annual peak demand. However, in reality, it has become common for forecast electricity demand to exceed the distribution network's supply capacity in the near, as peak demand continues to increase \cite{AEMO2}.
In response to this, traditionally, distribution network service providers (DNSPs) have sometimes invested in new generation in substations (e.g. diesel generators) to accommodate the growing peak demand within their operation areas. Specifically, diesel generators make sense for very `peaky' peaks, i.e. seasonal/weather driven, characterized by high peak-to-average load profiles. However, this is an expensive method considering that the diesel costs are uncertain due to the relatively high capital cost and price uncertainty of liquid fuels.
Residential PV-battery systems have become an attractive alternative for effectively providing peak demand support, reducing the risk of supply falling short of demand. One of the key drivers to this development is the falling cost of PV-battery systems with the fast advancement of the technology \cite{AEMO3}. Meanwhile, the benefits of residential PV-battery systems are well recognized. First, battery systems store surplus PV generation, and utilize the stored energy in the evening for peak-load support \cite{tonkoski2012impact}. Second, they reduce network power losses and help with voltage regulation \cite{ma2019novel}. Although the technology is currently expensive to implement, it has already been shown effective in reducing diesel generation \cite{scott2019network}, and thus providing opportunities to replace the costly diesel generator investment, as well as to defer expensive network augmentation \cite{Clean,Evan}.
However, with the increasing uncertainties in distribution network investment and operation in the electricity system, DNSPs confront a great challenge in determining efficient and well-timed investment in residential PV-battery systems. In Australia and some other jurisdictions, asset ring-fencing regulations mean that these PV-battery systems cannot be owned by (regulated) DNSPs if they are also used for supplying energy and services to contestable markets. However, to promote the use of this technology, DNSPs may provide incentives to cover a large proportion of PV-battery procurement and installation cost, and in this way, they can effectively invest in these assets.
Given this context, the decision rule of the traditional \textit{net present value} (NPV) analysis is to undertake the investment immediately if the NPV is positive and reject those with a negative NPV, regardless of how future uncertainties will unfold \cite{vlaovic2013advantages}. However, this method ignores the \textit{real options} in an irreversible distribution network investment, such as the options to \textit{defer}, \textit{expand}, \textit{contract} and \textit{abandon}. Each of these options is a contingent decision that the investor has the flexibility to make based on the realization of future uncertainties. This highlights the importance and value of managerial flexibility, which is not captured in standard NPV assessments.
In contrast, in this paper we use \textit{real options valuations} (ROV) to identify the options embedded in an investment, and hence the possible flexible investment directions in the light of various uncertainties. Specifically, ROV takes into account the value of managerial flexibility of making contingent decisions upon the realizations of future uncertainties, typically captured by \textit{Monte Carlo} (MC) simulations. Given this, we reduce the exposure to risk, by considering management's right, but not obligation, to make contingent decisions \cite{kodukula2006project}.
However, there are generally several interacting options embedded in an investment. The most common interaction happens when a subsequent option becomes available after the prerequisite option is executed. In this case, the subsequent option provides additional future contingency that affects the investor's decisions.
Examples include system expansion, relocation, re-contracting and abandonment which often appear only after the option to invest has been executed. Properly incorporating these subsequent options into the valuation of the prerequisite option may lead to a different investment strategy, and therefore provide additional value to the investment. Nonetheless, there is currently limited literature that presents a valuation framework to explicitly consider the interdependency among the options available in a distribution network investment.
\subsection{Contributions}
Against this background, this work proposes a multi-stage valuation framework that uses ROV to evaluate an investment with compound options under multiple future uncertainties. To demonstrate the characteristics of the framework, we apply it to assess the economic benefits and costs accrued to a DNSP for providing incentives to network customers to purchase PV-battery systems.
By doing so, we remove the need to install additional substation diesel generators.
Given this, we consider two interacting options in this investment, which are:
\begin{itemize}
\item The option to \textit{defer} the PV-battery investment in the first decision period; and
\item The option to \textit{expand} this investment in the second decision period.
\end{itemize}
The option to expand is a subsequent option, which is only considered after the option to defer is executed. Given this, the value of the deferral option is also dependent on this subsequent option. The proposed framework derives the optimal investment strategy subject to these options by quantitatively taking into account the uncertainties and the flexibility of making decisions contingent to unfolding information.
The uncertainties considered in this work are:
\begin{itemize}
\item Growing peak power demand which decides the capacity of the PV-battery investment, as well as diesel generator operational cost;
\item Varying diesel price that determines the cost of diesel procurement; and
\item The declining cost of PV-battery technology.
\end{itemize}
These uncertainties are chosen as the state variables to the ROV because their future projections affect the behaviour of the cash flow, and the stochastic variations provide opportunities for a DNSP to increase the investment value by making decisions contingent on their realizations.
In order to capture the value of the options, and determine the optimal investment strategy within a pre-identified decision period, the framework overcomes two main challenges that have not been addressed in the existing literature: (i) the need to enable ROV on the application of PV-battery hybrid system investment and determine the optimal investment timing given multiple interacting options available, under multiple uncertainties, and (ii) integrate multiple investments (diesel generator and PV-battery systems) within one ROV process to raise the potential for greater option values. Thus, the major contributions of the proposed framework are summarized as follows:
\begin{itemize}
\item We define the economic benefits of the PV-battery investment as the cost saving from replacing the expensive diesel generation investment to enable the application of ROV;
\item We use ROV to determine the optimal strategy that subjects to multiple interacting options available in the application of PV-battery system investment; and
\item By combining the investments in diesel generator and PV-battery systems, we integrate the uncertainties involved in both investments within one valuation process to provide greater opportunity values.
\end{itemize}
By doing so, we demonstrate the process of deriving the optimal strategy for a distribution network investment that has multiple interacting options. Although ROV based on the LSMC method has already been used in the evaluation of network investments, this is the first time that the usefulness of these techniques has been shown for assessing the investment value of customer-owned PV-battery systems in particular, and for distribution investments in general.
Our results show that, from the traditional NPV analysis, the PV-battery investment is abandoned immediately because the NPV is negative. However, this decision is changed after considering the options embedded in the investment. Specifically, by considering the option to expand in the future, the proposed ROV framework suggests to delay the investment and thereby increase the value of the investment. In addition to this, we observe that changing the drift parameter and volatility of the state variables can have significant impacts on the distribution of optimal strategies.
\subsection{Literature review}
In this subsection we review existing literature for: (i) ROV in transmission and distribution network investments, and (ii)~the methods to calculate the value of real options.
In recent years, ROV has been frequently applied for valuation of distribution and transmission network investments, including transmission network expansion \cite{salazar2007decision, pringles2015real}, distributed generation \cite{zou2017optimal}, and renewable generation including hydro-power \cite{linnerud2014investment}, wind generation \cite{boomsma2012renewable, bockman2008investment, eryilmaz2016does,ritzenhofen2016optimal,boomsma2015market}, solar generation \cite{martinez2013assessment, zhang2016real, gahrooei2016timing, zhang2017optimal, eissa2017lobatto} and large scale battery storage systems \cite{ma2019estimating}. A common feature of these studies is that they determine the value to execute one or several \textit{independent} options embedded in a network investment in the presence of uncertain electricity market conditions or regulatory policies, and hence the optimal investment timing.
However, in practice, a distribution network investment often involves multiple \textit{interacting} (i.e.~not independent) options which actively engage with each other to produce a greater investment value. For this setting, the authors in \cite{loncar2017compound} determine the value of multiple options, including the option to invest, followed by the subsequent options (expand, re-power, contract and abandon), in a wind farm investment. However, the investment timing for the subsequent options are fixed (i.e.~a European call option) to reduce the size of the problem.
Currently, the method to properly value American compound options, with flexible execution timing, has not been presented in the existing literature.
More importantly, we have found that the topic of addressing investment in hybrid renewable generation systems, such as PV-battery, using ROV has not been presented, which drives our research direction.
Given this context, we need to decide the most suitable approach to address the problems above.
The \textit{partial differential equations} (PDE) approach \cite{de2006flexibility, macdougall2015value} based on the research of Black, Scholes and Merton, and \textit{binomial lattice} \cite{eryilmaz2016does, boomsma2015market, eissa2017lobatto} have been widely applied to calculate the value of real options. However, the PDE approach can be used to incorporate only one uncertainty, or at most two correlated ones. This is not the case in the electricity market where multiple uncertainties exist. Meanwhile, PDE is designed for European-type of option analysis, in which the investment can only be taken on a specific future date \cite{schachter2016critical}. On the other hand, lattice models use backward induction, and the prior path of the underlying variable is unknown at the time computations are made, making it impossible to incorporate multiple interacting options with numerous state variables \cite{schachter2016critical}. Thus, these methods are not suitable for distribution network investments where there are multiple sources of uncertainty and interacting options.
In contrast, \textit{binomial tree} models can handle the cases with compound options under multiple uncertainties better, by adding additional decision nodes \cite{loncar2017compound}. However, their computation complexity grows exponentially with the increasing number of decision nodes, constraining their use to complex but smaller-sized problems.
In light of this, the \textit{least square Monte Carlo} (LSMC) method \cite{longstaff2001valuing} is applied to determine the value of real options, and hence derive the optimal investment strategy \cite{pringles2015real, zou2017optimal, linnerud2014investment, boomsma2012renewable, zhang2016real, zhang2017optimal, blanco2011real}. This method allows us to incorporate many sources of uncertainty, and accurately capture the flexibility in delaying investments using MC simulations. For example, the authors in \cite{blanco2011real} applied this method to value flexible AC transmission systems devices in transmission networks, by modeling both demand and fuel costs as stochastic processes, and providing the optimal timing for the option to install, locate and remove the asset.
For these reasons, the LSMC method is used in our work.
The rest of the paper is organized as follows. Section \ref{section2} explains the LSMC approach which is used for valuing the managerial flexibility. Section \ref{section3} describes the costs and benefits analysis of the PV-battery investment, and presents the stochastic modeling of future uncertainties including power demand growth, varying diesel fuel price, and the declining cost of PV-battery systems. The outcomes are evaluated in Section \ref{section4}. Finally, Section \ref{section5} draws conclusions.
\section{Real Options Theory} \label{section2}
The traditional NPV method fails to appraise an investment under uncertainties and in the presence of contingency, as it considers managerial flexibility as a passive factor and attains only deterministic investment decisions. In contrast, ROV recognizes the benefits of contingency and includes this as an active entity in its calculation, which potentially changes the value of the investment \cite{kodukula2006project}.
For multi-stage compound options valuation, we need to determine the value of the subsequent options (the option to expand), and incorporate this value when valuing the deferral option. In this case, the option to defer is to be executed within a 5-year decision period, $t \in \mathcal{T}_\mathrm{inv}$, while the option to expand is considered in the next 5-year decision period, $t \in \mathcal{T}_\mathrm{exp}$. The optimal investment strategy is extracted from this process. The mathematical formulation for solving compound options via the LSMC method is detailed in this section.
\subsection{ROV with the LSMC method}
To determine the optimal investment strategy subject to the compound options, the first task is to apply the LSMC method to calculate the value of the subsequent option to expand the investment, assuming that the investment has already been carried out. Specifically, this method combines a forward-looking model for incorporating uncertainties, and a backward recursion (least square regression) for determining the value of an option \cite{schachter2016critical}. The option can be executed any year from the initial year $t_0$ to the maturity $T_{\mathrm{m}}$. This time-span is divided in to $n \in \mathcal{N}$ intervals, whose length is 1 year.
We assume that there are $h\in \! \mathcal{H}$ options within an investment, thus, we use $h$ and $h\! +\! 1$ to represent the options to defer and expand, respectively. Given this, the investment value in year ${t}$ considering the option to expand is denoted as ${F_{h+1}(t_{n},{X}_{t_{n}})}$, where ${{X}_{t_{n}}}$ is the state variable of the investment, including the growing power demand, varying diesel fuel price, and the declining PV-battery cost. Hence, the future discounted value of the subsequent investment option can be expressed as follows:
\begin{equation} \label{ROVeq1}
F_{h+1}(t_{0},{X}_{t_{0}}) = \max_{\tau \in [t_{0}, {T}_{\mathrm{m}}]}\left\{e^{-r(\tau-t_{0})}E^{*}_{t_{0}}\left[\Pi_{h+1}(\tau,X_{\tau})\right]\right\},
\end{equation}
\noindent where $\tau$ is the optimal stopping time on each MC path, $\Pi_{h+1}(\tau,X_{\tau})$ is the payoff for expanding the project, ${{E}^{*}_{t_{n}}\left[\Pi_{h+1}(\tau,X_{\tau})\right]}$ is the expectation on the information available at ${t_{n}}$, and ${r}$ is the risk neutral discount rate.
LSMC integrates MC simulations with least square regression to accurately estimate the option value. The dynamic features of state variables are simulated by generating a set, ${\Omega}$, of MC realization paths by means of \textit{geometric Brownian motion}. Then, we estimate the continuation value, denoted ${\Phi(t_{n},\omega,X_{t_{n}})}$, which is essentially an estimate of the investment value of the next time step, given a realization $\omega \in \Omega$. This value represents the value of continuing to wait for the realization of future random variables at each time along the ${\omega^{\mathrm{th}}}$ path. By comparing ${\Phi(t_{n},\omega,X_{t_{n}})}$ to the value of investing immediately, the optimal stopping time along each MC path is found. The process for calculating the optimal stopping times is described by the \textit{Bellman's principle of optimality}, which is expressed as follows:
\begin{equation}\label{ROVeq2}
F_{h+1}(t_{n},{X}_{t_{n}}) = \max\left\{\Pi_{h+1}(t_{n},X_{t_{n}}),\Phi_{h+1}(t_{n},X_{t_{n}})\right\},
\end{equation}
\noindent where
\begin{equation}\label{ROVeq3}
\Phi_{h+1}(t_{n},X_{t_{n}}) = E^{*}_{t_{n}}\left[\sum_{i=n+1}^{N}e^{-r(t_{i}-t_{n})} \Pi_{h+1}(t_{n},t_{i},\tau)\right].
\end{equation}
In more detail, the least square regression estimates the continuation value by regressing the discounted future investment values on a linear combination of a group of basis functions of the current state variables. Each group of the basis functions represents the payoff trajectory within a time interval, and we use these basis functions to estimate the payoff at $t\! + \! 1$ (continuation value). This process has been employed in many recent studies, which generally apply simple powers of the state variable ${{X}_t}$ as basis functions \cite{zou2017optimal}, \cite{blanco2011real}. Given this, we define the orthonormal basis of the ${j^\mathrm{th}}$ state variable as ${L_{j}}$. The optimal coefficients, ${\hat{\phi}_{j}}$, for the basis functions are obtained using \eqref{ROVeq4}.
\begin{equation}\label{ROVeq4}
\begin{split}
\begin{aligned}
\hat{\phi}_{j}(t) & = \underset{\phi_{j}(t)}{\operatorname{argmin}} \biggl\{\sum_{i=n+1}^{N} e^{-r(t_{i}-t_{n})} \Pi_{h+1}\left(t_{i},X_{t_{i}}(\omega)\right) \\ & \quad - \sum_{j=1}^{J}\phi_{j}(t)L_{j}\left(X_{t_{n}}(\omega)\right)\biggl\}^{2}.
\end{aligned}
\end{split}
\end{equation}
The continuation value for each MC path is thus calculated by feeding the optimal coefficient ${\hat{\phi}_{j}}$ into the linear combination of the basis functions, that is:
\begin{equation}\label{ROVeq5}
\Phi_{h+1}\left(t,X_{t_{n}}(\omega)\right) = \sum^{J}_{j=1} \hat{\phi}_{j}(t_{n})L_{j}\left(X_{t_{n}}(\omega)\right).
\end{equation}
The option value is maximized along each path if the option is executed as soon as the payoff exceeds the continuation value. The optimal stopping time along each of the in-the-money paths is determined by applying Bellman's principle of optimality, given by \eqref{ROVeq2} recursively from maturity ${{T}_{\mathrm{exp}}}$ to ${{t}_{0}}$. If the decision rule holds true at ${t_{n}}$, the stopping time ${\tau_{\omega}}$ along the ${{\omega}^{\mathrm{th}}}$ path will be updated to ${t_{n}}$, that is:
\begin{equation}\label{ROVeq6}
\text{if} \ \ \Phi_{h+1}\left(t_{n},X_{t_{n}}(\omega)\right) \leq \Pi_{h+1}\left(\tau,X_{t_{n}}(\omega)\right), \ \ \text{then} \ \ \tau(\omega) = t_{n}.
\end{equation}
The optimal stopping time of each MC path ${\tau_{\omega}}$ forms a unique optimal stopping time matrix, which includes the earliest investment timing for all in-the-money MC paths. Using this matrix, the option value at $t_{0}$ that considers managerial flexibility and future uncertainty is determined by the following equation:
\begin{equation}\label{ROVeq7}
\begin{split}
\begin{aligned}
F_{h+1}(t_{0},{X}_{t_{0}}) &= \frac{1}{|\Omega|} \sum\limits_{\omega\in\Omega}{ e^{-r\tau(\omega)}\Pi_{h+1}\left(\tau(\omega),X_{\tau(\omega)}(\omega)\right) }.
\end{aligned}
\end{split}
\end{equation}
Given the value of the subsequent option, to calculate the value of the option to defer the PV-battery investment, the Bellman's principle of optimality described by \eqref{ROVeq2} becomes:
\begin{equation}\label{ROVeq8}
\begin{split}
\begin{aligned}
F_{h}(t_{n},{X}_{t_{n}}) &= \max\left\{\Pi_{h}(t_{n},X_{t_{n}})+F_{h+1}(t_{n},X_{t_{n}}), \Phi_{h}(t_{n},X_{t_{n}})\right\},
\end{aligned}
\end{split}
\end{equation}
\noindent where
\begin{equation}\label{ROVeq81}
\Phi_{h}(t_{n},X_{t_{n}}) = E^{*}_{t_{n}}\left[\sum_{i=n+1}^{N}e^{-r(t_{i}-t_{n})} \sum_{l=h}^{H} \Pi_{l}(t_{n},t_{i},\tau)\right].
\end{equation}
For the deferral option value, the continuation value, $\Phi_{h}(t,X_{t})$ is compared to the payoff of the investment, $\Pi_{h}(t,X_{t})$, plus the value of the option to expand, $F_{h+1}(t,X_{t})$. Thus, the stopping time ${\tau_{\omega}}$ along the ${{\omega}^{\mathrm{th}}}$ path is updated to ${t}$, if:
\begin{equation}\label{ROVeq9}
\begin{split}
\begin{aligned}
\Phi_{h}\left(t_{n},X_{t_{n}}(\omega)\right) \leq \left(\Pi_{h}\left(t_{n},X_{t_{n}}(\omega)\right) +F_{h+1}\left(t_{n},X_{t_{n}}(\omega)\right)\right).
\end{aligned}
\end{split}
\end{equation}
We use the updated decision rule in the LSMC method to calculate the deferral option value, and hence the optimal investment strategy. The accuracy of the estimation grows as the number of simulation paths and basis functions increases. The presented method can be applied to multiple subsequent options, where the payoff of the $(h+n)^{\mathrm{th}}$ option needs to be incorporated when calculating the value of the $(h+n-1)^{\mathrm{th}}$ option in \eqref{ROVeq8}, \eqref{ROVeq81} and \eqref{ROVeq9} within the LSMC approach.
\section{Costs and Benefits Analysis} \label{section3}
To carry out the financial assessment of the PV-battery investment, we need to determine the cost and payoff for carrying out and expanding the investment, respectively. These values are used as the inputs to the ROV to value the compound options, and therefore the corresponding optimal investment strategy. The valuation algorithm is described in Algorithm \ref{alg1}, and discussed in more detail below.
Specifically, we calculate the cost of the investment via the traditional NPV method, and hence the payoff ($\Pi_{h,t,\omega}$) from replacing the diesel generator investment with the PV-battery investment during the first 5-year decision period, and the payoff ($\Pi_{h+1,t,\omega}$) from expanding this investment in the next 5-year decision period. This period is chosen so that the DNSP can fully capture the benefit of delaying the PV-battery investment, while carrying out the appropriate investment for peak demand support within a relatively short time-span. The detailed formulation is described in this section. Then, we demonstrate the modelling of future uncertainties including (i) growing power demand, (ii) varying diesel fuel price, and (iii) declining cost of the PV-battery technology.
\subsection{Payoff Calculation}
The energy delivered through the transformer that is over the thermal limit for each day between 2014 and 2017 is calculated from the aggregated historical electricity usage data from the Top Ryde substation (\SI{132}{\kilo\volt}/\SI{33}{\kilo\volt}), which has a thermal limit of \SI{35}{MVA}. These data, provided by Ausgrid\footnote{a DNSP in New South Wales, Australia; see
\url{https://www.ausgrid.com.au/Industry/Innovation-and-research/Data-to-share/Distribution-zone-substation-data}.}, have a 15-minute resolution. The future growth in power demand that is over the thermal limit is then simulated. We take the average across the simulated data for each future decision year, and use it as the aggregated installed capacity for the PV systems and diesel generator, denoted as $E^{\mathrm{Cap}}_{t,\omega}$. The battery size is decided based on the PV size. In Australia, \SI{2}{kWh} of battery is typically used per \SI{1}{kW} of PV installed.
The future costs of the PV-battery investment ($c^{\mathrm{PV}}_{t,\omega}$) and diesel generator investment ($c^{\mathrm{DG}}_{t,\omega}$) need to be discounted via the traditional NPV method as follows before calculating the payoffs:
\begin{equation} \label{eq3}
\begin{aligned}
NPV = \sum_{t=1}^{T}{\frac{c_t}{(1+r)^t}}.
\end{aligned}
\end{equation}
\noindent where $r$ is the risk-free discount rate\footnote{In our work, we assume that the future costs are risk free. Thus, we have fixed this value to a constant (0.06).}.
We assume that the DNSP is responsible for 70\% of PV-battery procurement (the rest is paid by the customers), and it is obliged to cover the cost of electricity usage over the thermal limit and is not covered by the additional generation capacity provided by PV-battery systems (denoted as ${c^{\mathrm{g}}_{t,\omega}}$). This cost occurs when the PV-battery investment is delayed or the peak demand exceeds the installed generation capacity. Thus, ${\Pi_{h,t,\omega}}$, including ${c^{\mathrm{g}}_{t,\omega}}$ is given by:
\begin{equation} \label{eq1}
\begin{aligned}
\Pi_{h,t,\omega} = \left(c^{\mathrm{PV}}_{t,\omega} - c^{\mathrm{DG}}\right) E^{\mathrm{Cap}}_{t,\omega} - c^{\mathrm{OM}} + c^{\mathrm{g}}_{t,\omega},
\end{aligned}
\end{equation}
\noindent where $c^{\mathrm{PV}}_{t,\omega}$ and $c^{\mathrm{DG}}$ are the costs per \SI{1}{kW} for the same aggregated capacity $E^{\mathrm{Cap}}_{t,\omega}$, $c^{\mathrm{OM}}_{h,t,\omega}$ is the maintenance and operation cost, while $c^{\mathrm{g}}_{t,\omega}$ is given by:
\begin{equation} \label{eq2}
\begin{aligned}
c^{\mathrm{g}}_{t,\omega} = c^{\mathrm{f}}_{t,\omega}\Delta{E^{\mathrm{g}}_{t,\omega}},
\end{aligned}
\end{equation}
\noindent where $c^{\mathrm{f}}_{t,\omega}$ is the diesel price, and $\Delta{E^{\mathrm{g}}_{t,\omega}}$ is the electricity usage over the thermal limit and fails to be covered by PV-battery systems.
\begin{table}[t]
\centering\small
\caption{Cost specifications.}
\begin{tabular}{cccc}
\hline
Items & Cost\\
PV-battery system, $c^{\mathrm{PV}}_{t,\omega}$ & Risk neutral valuation \\
Diesel generator, $c^{\mathrm{DG}}$ & {\$}{600}/kW \cite{el2005integrated} \\
Peak demand, $E^{\mathrm{Cap}}_{t,\omega}$ & GBM \\
Operation and maintenance, $c^{\mathrm{OM}}$ & {\$}{100}{k/year} \cite{el2005integrated} \\
Diesel fuel, $c^{\mathrm{f}}_{t,\omega}$ & Mean-reverting process \\
\hline
\end{tabular}
\label{T1}
\end{table}
The aggregated capacity of the PV-battery systems installed in the first decision period does not cover the power demand growth in the future, which leaves an open question as to whether the expansion is necessary. If the option to expand is abandoned, the future power demand growth will be covered by installing an additional diesel generator.
In more detail, the payoff for expanding the PV-battery investment, ${\Pi_{h+1,t,\omega}}$, assuming that the investment has already been executed, is the difference between the cost of installing additional PV-battery systems to cover the peak demand growth after the first decision period, and the cost of expanding the diesel generation investment for the same amount of aggregated generation capacity, $E^{\mathrm{Cap}}_{h+1,t,\omega}$; that is:
\begin{equation} \label{eq_payoff2}
\begin{aligned}
\Pi_{h+1,t,\omega} = \left(c^{\mathrm{PV}}_{t,\omega} - c^{\mathrm{DG}}\right) E^{\mathrm{Cap}}_{h+1,t,\omega} - c^{\mathrm{OM}}_{h+1,t,\omega} + c^{\mathrm{g}}_{t,\omega}.
\end{aligned}
\end{equation}
In our work, we assume that the DNSP is in charge of electricity distribution, network planning, monitoring and maintenance. The electricity bills generated from the diesel generator are allocated to this entity, and these are the primary source of cash inflow in this investment. Cash outflows in this investment include system procurement, installation, maintenance and operation costs for the generator.
\begin{algorithm}[t]
\footnotesize
\caption{Real options valuation algorithm}
\begin{algorithmic}[1]
\STATE Generate $\Omega$ realization paths of $\Delta{E^\mathrm{Cap}}$, $c^{\mathrm{f}}$ and $c^{\mathrm{PV}}$.
\FOR{$\omega$ = 1:$\Omega$}
\FOR{year = 6:10}
\STATE Calculate $\Pi_{h+1}$.
\ENDFOR
\ENDFOR
\FOR{$t$ = $T_{\mathrm{exp}}$:-1:1}
\STATE Calculate $\Phi_{h+1}$ via LSMC for all in-the-money paths.
\FOR{$\omega$ = 1:$\Omega$}
\IF{${\Pi} >= \Phi_{h+1}(t,X_{t}(\omega))$}
\STATE Update $\tau$ = t.
\ENDIF
\ENDFOR
\ENDFOR
\STATE Calculate the $F_{h+1}$ using \eqref{ROVeq7}.
\STATE Repeat the steps 3 to 6 to calculate $\Pi_{h}$ from Years 1 to 5
\FOR{$t$ = $T_{\mathrm{inv}}$:-1:1}
\STATE Calculate $\Phi_{h}$ via LSMC for all in-the-money paths.
\FOR{$\omega$ = 1:$\Omega$}
\STATE Update the decision rule to \eqref{ROVeq9}.
\IF{$\Pi + F_{h+1}>= \Phi_{\mathrm{h}}(t,X_{t}(\omega))$}
\STATE Update $\tau$ = t.
\ENDIF
\ENDFOR
\ENDFOR
\STATE Calculate $F_{h}$ and extract the optimal investment strategy.
\end{algorithmic}
\label{alg1}
\end{algorithm}
\subsection{Modelling of Future Uncertainties} \label{SectionGBM}
In this subsection we simulate the random variables used in the payoff calculation, including growing power demand that exceeds the thermal limit ($\Delta{E^{\mathrm{Cap}}_{t,\omega}}$), cost of diesel fuel ($c^{\mathrm{f}}_{t,\omega}$), and the declining cost of PV-battery technology ($c^{\mathrm{PV}}_{t,\omega}$). Specifically, the growing peak demand that exceeds the thermal limit ($\Delta{E^{\mathrm{Cap}}_{t,\omega}}$) is governed by the GBM and simulated using MC analysis. GBM is used in this case as we assume that the stochastic peak demand evolution over time can be captured by a GBM process. This assumption relies on the fact that the increments of process in the GBM show the Markov property, which assumes that any future change is independent from the previous values, while the variable remains positive throughout the process \cite{fleten2007optimal}. The GBM for the growing peak demand is therefore described mathematically as follows:
\begin{equation} \label{eq4}
\begin{aligned}
dS^{\mathrm{d}}_{t} = \mu S^{\mathrm{d}}_{t}dt + \sigma S^{\mathrm{d}}_{t} d W_{t},
\end{aligned}
\end{equation}
\noindent where ${S^{\mathrm{d}}_{t}}$ describes the sought stochastic process of power demand, ${\mu}$ is the percentage drift that describes the rate of growth in the aggregated peak demand that is over the thermal limit, ${\sigma}$ is the percentage volatility of the data, and ${W_{t}}$ is the Wiener process that describes the stochastic component. Thus, the discretization recursion formula is given by:
\begin{equation} \label{eq5}
\begin{aligned}
S^{\mathrm{d}}_{t+\Delta{t}} = S^{\mathrm{d}}_{t}e^{\left(\mu-\frac{\sigma^2}{2}\right)\Delta{t}+\sigma dW_{t}}.
\end{aligned}
\end{equation}
\begin{figure}[t]
\centering
\begin{framed}
\includegraphics[width=7cm,keepaspectratio]{Demand.pdf}%
\end{framed}
\vspace{-8pt}
\caption{Distribution of simulated average monthly growth in future peak demand that is over the transformer thermal limit via GBM (drift = 1.5\% and volatility = 9.8\%)}
\label{fig11}
\end{figure}
\begin{figure}[t]
\centering
\begin{framed}
\includegraphics[width=7cm,keepaspectratio]{Dieselprice.pdf}%
\end{framed}
\vspace{-8pt}
\caption{Distribution of simulated future diesel price via a mean reverting process (speed of reversion = 5\%, reversion level = 2.6, volatility = 4.7\%)}
\label{fig22}
\end{figure}
\begin{figure}[t]
\centering
\begin{framed}
\includegraphics[width=7cm,keepaspectratio]{PV_battery.pdf}%
\end{framed}
\vspace{-8pt}
\caption{Distribution of simulated future cost of PV-battery system via risk neutral valuation (risk-free rate = 0.06, volatility = 9\%)}
\label{fig33}
\end{figure}
The simulation of future power demand takes the historical electricity usage data provided by Ausgrid. The future power demand is simulated for 10,000 MC paths.
On the other hand, diesel fuel price, like all the other bulk commodities such as oil, gas and metal tends to conform to its long-term mean, with stochastic shocks when unforeseen "events" occur. Given this we used a mean-reverting process to simulate the varying diesel fuel prices; that is:
\begin{equation} \label{eqMRP1}
\begin{aligned}
dS^{\mathrm{f}}_{t} = \beta_{\mathrm{f}}(\hat{S}^{\mathrm{f}} - S^{\mathrm{f}}_t)dt + \sigma S^{\mathrm{f}}_{t} d W_{t},
\end{aligned}
\end{equation}
\noindent where ${S^{\mathrm{f}}_{t}}$ describes the sought stochastic process for the cost of diesel fuel, ${\beta_{\mathrm{f}}}$ is the speed of reversion to the mean, and $\hat{S}^{\mathrm{f}}$ is the reversion level. Thus, the discretization recursion formula is given by:
\begin{equation} \label{eqMRP2}
\begin{aligned}
S^{\mathrm{f}}_{t+\Delta{t}} = e^{-\beta_{\mathrm{f}}\Delta t}(S^{\mathrm{f}}_t - \hat{S}^{\mathrm{f}}) + \hat{S}^{\mathrm{f}} + \sigma \epsilon \sqrt{(1- e^{-2\beta_{\mathrm{f}} \Delta t })/2\beta_{\mathrm{f}}}.
\end{aligned}
\end{equation}
In the LSMC approach, we assume that the sample paths of costs of assets (PV-battery systems) over the relevant time horizon are simulated according to the risk-neutral measure\footnote{This is because these costs depend crucially on their risk as investors typically demand more profit for bearing more risk. It is difficult to adjust the simulated expected values based on an investor's preference, and therefore, risk neutral measure is used.}. Given this, we first describe the evolution of the cost using GBM:
\begin{equation} \label{eqRNM}
\begin{aligned}
dS^{\mathrm{PV}}_{t} = \mu S^{\mathrm{PV}}_{t}dt + \sigma S^{\mathrm{PV}}_{t} d W_{t},
\end{aligned}
\end{equation}
\noindent where $S^{\mathrm{PV}}_{t}$ describes the sought stochastic process for the cost of PV-battery systems. Then, we define $dW_t = d\hat{W}_t - \frac{\mu - r}{\sigma} dt$, where $d\hat{W}_t$ is the Brownian motion under risk neutral measure and $r$ is the risk-free rate. Substituting this to \eqref{eqRNM} yields the risk neutral valuation:
\begin{equation} \label{eqRNM2}
\begin{aligned}
dS^{\mathrm{PV}}_{t} = r S^{\mathrm{PV}}_{t}dt + \sigma S^{\mathrm{PV}}_{t} d \hat{W}_{t}.
\end{aligned}
\end{equation}
Thus, the discretization formula becomes:
\begin{equation} \label{eqRNM3}
\begin{aligned}
S^{\mathrm{PV}}_{t+\Delta{t}} = S^{\mathrm{PV}}_{t}e^{\left(r-\frac{\sigma^2}{2}\right)\Delta{t}+\sigma d\hat{W}_{t}}.
\end{aligned}
\end{equation}
The historical PV and battery price and diesel price data are obtained from \cite{battery} and \cite{bureau}, respectively. For brevity, in this work, only 6 years of the simulated future data are shown. It is observed that the electricity usage that is over the thermal limits, $\Delta{E^{\mathrm{Cap}}_{t,\omega}}$, and the cost of diesel fuel $c^{\mathrm{f}}_{t,\omega}$ increase gradually in the future, as illustrated in Fig.~\ref{fig11} and Fig.~\ref{fig22}, respectively, while the cost of PV-battery $c^{\mathrm{PV}}_{t,\omega}$ continues to decrease (Fig.~\ref{fig33}).
\section{Case Studies} \label{section4}
In this section, we demonstrate that the proposed ROV framework can be used to evaluate a distribution network investment with interacting options under multiple uncertainties. Specifically, we value the PV-battery investment considering two interacting options, which are (i) the option to defer in the first 5-year decision period, and (ii) the option to expand the investment in the next 5-year decision period. The subsequent option for the investment expansion can only be considered after the investment has been carried out. The option values are calculated via the LSMC approach described in Section \ref{section2}. As previously described, the expansion of the PV-battery investment aims to cover the further growth in the aggregated peak power demand after the first decision period.
The study period is 10 years, including the two consecutive 5-year decision periods, and the annual risk-neutral discount rate is assumed to be \SI{6}{\%}. The DG investment and PV-battery investment are thereafter referred to as P1 and P2, respectively.
\subsection{Diesel Generator Cash Flow Analysis}
The NPV for executing P1 in Year 1 is calculated in this subsection. The capital cost of diesel generator is {\$}{600} per \SI{1}{kW} \cite{el2005integrated}, while the future electricity generation cost is dependent on the increase in the aggregated peak demand and diesel fuel price simulated using the GBM in Section \ref{section3}.
The average NPV of P1 decreases from {--\$}{900}{k} in Year 1 to greater than {--\$}{2}{M} by the end of the 10-year study period, as illustrated by Fig.~\ref{fig2}, left, this is due to the expenses in maintenance and electricity generation. Specifically, the magnitude of the outliers outside the boxes, especially below, increases significantly with respect to time, showing the increasing risks for large investment costs in the case of fast-growing peak demand and diesel fuel cost. We use the NPV of P1 in Year 10 in \eqref{eq1} to calculate the payoff, $\Pi_{h,t,\omega}$, of P2.
\subsection{PV-Battery Cash Flow Analysis}
Compared to the NPV of P1, the average NPV of P2 in Year 1 is roughly {--\$}{1.6}{M}, which decreases to just under {--\$}{2}{M} in Year 10, assuming P2 is executed in Year 1, as shown in Fig.~\ref{fig2}, right. The reduction in the NPV is less pronounced because PV-battery systems rely on cost-free solar power, and only the maintenance cost is committed to this investment during the study period.
Based on the traditional NPV analysis, P2 is not profitable as the NPV in all MC paths are negative, as shown in Fig.~\ref{fig2}, right, and hence this investment is abandoned. However, using the ROV, we create an opportunity to make profits of this investment by postponing it to a time when the uncertainties turn favourable, and the trade-off to this is the cost to cover the electricity usage that is over the transformer thermal limit, given that neither P1 nor P2 is implemented until the optimal investment timing.
\begin{figure}[t]%
\centering
\includegraphics[width=4cm,keepaspectratio]{DG_NPV.pdf}
\qquad
\includegraphics[width=4cm,keepaspectratio]{PV_NPV.pdf}
\caption{Costs for executing the diesel generator investment (left) and the PV-battery investment in Year 1 (right)}%
\label{fig2}%
\end{figure}
\begin{figure}[t]%
\centering
\includegraphics[width=4cm,keepaspectratio]{Payoff_INV.pdf}
\qquad
\includegraphics[width=4cm,keepaspectratio]{Payoff_EXP.pdf}
\caption{Payoffs for executing the PV-battery investment in each of the decision years (left), and for expanding the investment in each of the decision years (right)}%
\label{fig3}%
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=7cm,keepaspectratio]{OIS.pdf}
\caption{Frequency distribution of optimal investment timing for the option to defer the investment in the first decision period, and the option to expand the investment in the second decision period (benchmark).}
\label{fig4}
\end{figure}
The future payoffs of investing and expanding the PV-battery investment needed for the LSMC approach are shown in Fig.~\ref{fig3}. Specifically, these costs are calculated using \eqref{eq1} and \eqref{eq_payoff2}, respectively, where the cost parameters are shown in Table~.\ref{T1} and the simulation of the state variables is described in Section \ref{SectionGBM}. Observe that the payoffs for both investing in and expanding in the future gradually increase throughout the decision periods due to the declining cost of the advancing technology. Thus, there exists an opportunity to make the investment profitable in the future via ROV.
\begin{table*}[]
\footnotesize
\caption{ROV results under different scenarios.}
\begin{tabular}{ccccccccccc}
\hline
Scenario & Description & Year 1 (\%) & Year 2 (\%) & Year 3 (\%) & Year 4 (\%) & Year 5 (\%) & Standard NPV (k\$) & ROV (k\$) & Flexible NPV (k\$) \\
S1 & Benchmark & 1.2 & 1.2 & 2.3 & 3.4 & 90 & --240 & 660 & 420 \\
S2 & $\mu_{\mathrm{d}} = 3\%$ & 8.7 & 3.1 & 2.9 & 5.4 & 77 & --110 & 740 & 630 \\
S3 & $\sigma_{\mathrm{d}} = 20\%$ & 4.5 & 4.8 & 5.5 & 4.3 & 80 & --240 & 600 & 360 \\
S4 & $\beta_{\mathrm{f}} = 15\%$ & 2.2 & 0.5 & 0.3 & 0.4 & 97 & --230 & 690 & 460 \\
S5 & $\sigma_{\mathrm{f}} = 20\%$ & 0.3 & 6.1 & 4.8 & 5.0 & 84 & --250 & 640 & 390 \\
S6 & $\sigma_{\mathrm{pv}} = 20\%$ & 1.7 & 1.6 & 1.7 & 1.9 & 92 & --240 & 720 & 480 \\
\hline
\end{tabular}
\label{T2}
\end{table*}
\begin{figure*}[t]%
\centering
\includegraphics[width=4cm,keepaspectratio]{one11.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{two11.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{three11.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{four11.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{one22.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{two22.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{three22.pdf}%
\qquad
\includegraphics[width=4cm,keepaspectratio]{four22.pdf}%
\caption{Impact on optimal investment timing with respect to changing drift parameter and volatility of the state variables.}
\label{fig5}%
\end{figure*}
\subsection{Real Options Valuation}
The opportunity values provided by future uncertainties are calculated using the proposed ROV framework. Specifically, the ROV suggests to delay the investment and wait for the market conditions to turn favourable if the deferral option value is positive and greater than the payoff calculated via the NPV method, and the optimal investment timing is when the payoff exceeds this option value. The flexible investment value is equal to the sum of the option value and the payoff calculated by the NPV analysis.
The LSMC approach for calculating the value of the compound options and the corresponding optimal investment strategies are detailed in Section \ref{section2}: (i) use the payoff (${\Pi_{h+1,t,\omega}}$) of expanding P2 to calculate the continuation value, ${\Phi_{h+1,t,\omega}}$, via least square regression, (ii) compare the payoff with continuation value for each year during the decision period using \eqref{ROVeq6}, the optimal timing to execute the investment on the $\omega^{\mathrm{th}}$ path is when the payoff is greater than the continuation value for the first time, (iii) calculate the investment value considering managerial flexibility using \eqref{ROVeq7}, and (iv) incorporate the value of the option to expand, $F_{h+1}$, in the LSMC method to determine the deferral option value, $F_{h}$, and the optimal investment strategy. The options are abandoned in the case of a negative payoff to reduce the computation time.
The average payoff of the investment from the standard NPV analysis is {--\$}{240}{k}, as observed in Fig.~\ref{fig3}, Investment. The value of the compound options calculated using \eqref{ROVeq7} is {\$}{660}{k}. Therefore, the flexible investment value in this case equals {\$}{420}{k} ({\$}{660}{k} + (-- {\$}{240}{k})), as indicated in Table.~\ref{T2}, Benchmark. In this scenario, the drift parameter and volatility of the state variables are given in Table.~\ref{T1}. Based on the standard NPV approach, the investment is abandoned immediately because the payoff is negative. However, ROV sees potential hidden within the state variables, and discovers a possible additional benefit of {\$}{660}{k} if the investment is postponed to a later year. To predict the optimal investment timing, the LSMC computes the optimal stopping timing (${\tau_{\omega}}$) for each MC path, from this we extract the frequency distribution of ${\tau_{\omega}}$, for the deferral option, as shown in Fig.~\ref{fig4}, in red. The optimal timing of the investment is indicated by the greatest frequency. Thus, P2 is delayed to Year 5 (90\%), despite the fact that the DNSP needs to pay for the additional grid supply that covers the future growth in peak demand, $c^{\mathrm{g}}_{t,\omega}$, incurred by postponing P2.
On the other hand, the option to expand calculated using \eqref{ROVeq7} in the second decision period is worth {\$}{330}{k}, while the payoff is only {\$}{30}{k} in Year 6, as seen in Fig.~\ref{fig3}, Expansion. In this case, the investment value is already positive before considering the flexibility, thus, based on the NPV analysis, the investment is expanded immediately. However, the ROV sees the potential for the market conditions to become favourable in the future, and hence suggests to delay the expansion. Specifically, based on the frequency distribution extracted from LSMC, as shown in Fig.~\ref{fig4}, blue, the option to expand is to be executed in Year 5 in the second decision period, with a 81\% possibility.
If the deferral option is considered independent of the option to expand, the deferral option value is reduced from {\$}{660}{k} to only {\$}{350}{k}, while the optimal investment timing is kept in Year 5 with a slight decrease in the frequency (84\%), compared to Fig.~\ref{fig4}. Eliminating the option to expand in this calculation means that there is less managerial flexibility considered, leading to a smaller option value. This result shows that properly establish the relation between interacting options leads to a greater option value, and hence significantly increases the investment value.
To summarize, the ROV framework accounts for the opportunity values from executing options under future uncertainties, and properly considers the interaction between these options. By doing so, it reverses the decisions drawn from the traditional NPV analysis. Specifically, the fast-declining PV-battery cost is the main driver to the deferral of the investment. This case serves as the benchmark in our work.
\subsection{Sensitivity Analysis}
The optimal investment strategy can be affected by altering the state variables, including growing power demand, varying diesel fuel price and the declining costs of PV-battery systems. Thus, it is useful to evaluate the sensitivity of the option value and optimal investment timing subject to changing scenarios characterized by these variables. The ROV results are summarized in Table.~\ref{T2}.
Overall speaking, increasing the growth rate of future peak demand (S2) leads to a greater operational cost for the diesel generator, and thus a greater standard NPV if the PV-battery investment is executed in Year 1. Meanwhile, this also means a greater deferral option value, because the payoff increases further by delaying the investment. On the other hand, increasing the volatility of future peak demand shifts the optimal investment timing for some paths to early years, as seen in Table.~\ref{T2}, S3. Nevertheless, the option value rises due to the increasing potential for a greater payoff in the future, compared to the benchmark. Increasing the mean reversion speed $\beta_{\mathrm{f}}$ to 15\% means diesel fuel price will reach the reversion level faster. As a result, the market becomes more confident to defer to the investment as the probability to execute the investment in Year 5 nearly reaches 100\%.
In more detail, we use the example of peak demand growth to demonstrate how changing drift parameter and volatility affect the optimal investment timing. Specifically, the optimal investment timing is kept in Year 5, while the frequency decreases from 97\% to just under 50\% when $\mu_{\mathrm{d}}$ rises from $1\%$ to $5\%$, as shown in Fig.~\ref{fig5}, column one, top. As the demand grows faster, greater payoffs are expected. Thus, there is a greater probability for the payoff to exceed the continuation value in early investment years, and we are less confident to execute the investment in Year 5. Further, we observe an increasing deferral option value as there is larger profit in the future with a greater demand growth rate (Fig.~\ref{fig5}, column one, bottom). Similarly, the value of the option to expand rises with the same increase in $\mu_{\mathrm{d}}$, while the investment frequency in Year 5 is decreased, as shown in Fig.~\ref{fig5}, column two.
In contrast, as we increase the volatility of demand ($\sigma_{\mathrm{d}}$) from 10\% to 30\%, greater payoffs are expected, so are the amount of negative payoff cash flows (due to the increasing uncertainties). Given that all negative cash flows are discarded by the LSMC approach, overall we observe a subtle increasing trend for the option value (Fig.~\ref{fig5}, column three, bottom). However, even though both the NPV and option value are increasing, the probability to execute P2 has been reduced from 90\% to 77\% in Year 5 (Fig.~\ref{fig5}, column three, top) as there is a greater possibility for the payoffs to become negative. Due to the same reasons, similar changes are observed in the case of expanding the investment, as shown in Fig.~\ref{fig5}, column four.
Therefore, the optimal investment timing and the option value react differently to different values of drift parameter and volatility of the state variables. Interestingly, a greater deferral option value does not necessarily mean that the investment should be further delayed. Our results show that the optimal investment timing depends on how payoff and continuation value interact with each other. Meanwhile, it is possible for the option value to decrease as we increase future uncertainties, because there will be a greater number of payoffs becoming negative as time moves forward. These payoffs are discarded when calculating the option value. The sensitivity analysis shows how ROV responds to uncertain events in electricity market, and hence increases the robustness of the decision making process. In general, it is up to the DNSP to check how ROV responds to changing state variables, re-evaluate the investment, and make contingent decisions as time moves forward.
\section{Conclusions} \label{section5}
This work proposed a ROV framework that evaluates a distribution network investment that embeds compound options, and derives the optimal investment strategy under uncertain market environment. We demonstrated the characteristics of the framework by determining the optimal strategy for the investment in residential PV-battery systems, and calculating the values of the option to expand, and then the deferral option via the LSMC approach. Through the proposed ROV framework, we demonstrated that delaying the PV-battery investment/expansion to a later year when future uncertainties have turned favourable increases the investment value and mitigates the risk of financial losses. Meanwhile, the framework has shown that the investment value can be significantly increased when the interaction between the real options are properly considered. More importantly, the framework allows the managerial flexibility to optimally respond to uncertain events characterized by the underlying state variables. Thus, the proposed ROV framework can be employed in the future to value a distribution network investment with multiple interacting options, and derive the corresponding optimal investment strategy. In future research, we expect to include network uncertainties including the size and location of PV-battery systems, load behaviour in the ROV framework. Capturing value of the opportunities presented by these uncertainties requires solving the power flow problem within the MC analysis underpinned by the ROV.
\section*{References}
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2,869,038,155,654 | arxiv | \section{#1}}
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\preprint{}
\title{\center Linking Entanglement and Discrete Anomaly}
\author{Ling-Yan Hung${}^{1,2,3}$\ ,~Yong-Shi Wu${}^{3,1,2,4}$\ ,~Yang Zhou${}^1$\\
${}^1$ Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China\\
${}^2$ Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China\\
${}^3$ State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China\\
${}^4$ Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah, 84112, U.S.A\\
{\tt E-mails : [email protected], [email protected], [email protected]}
}
\abstract{In $3d$ Chern-Simons theory, there is a discrete one-form symmetry, whose symmetry group is isomorphic to the center of the gauge group. We study the 't Hooft anomaly associated to this discrete one-form symmetry in theories with generic gauge groups, $A,B,C,D$-types. We propose to detect the discrete anomaly by computing the Hopf state entanglement in the subspace spanned by the symmetry generators and develop a systematical way based on the truncated modular S matrix. We check our proposal for many examples.}
\begin{document}
\pagestyle{plain} \setcounter{page}{1}
\newcounter{bean}
\baselineskip16pt
\section{Introduction}
The goal of this paper is to establish a relation between discrete global anomaly and multi-boundary entanglement in topological quantum field theory. The former has been used to classify symmetry-protected topological phases~\cite{Chen:2011pg,Hung:2012nf}. The anomalies associated to discrete symmetries are also very useful to detect phases of QCD in $3d$ and $4d$~\cite{Gaiotto:2017yup,Komargodski:2017keh,Gomis:2017ixy,Gaiotto:2017tne}. It is well known that 't Hooft anomaly is powerful to constrain infrared dynamics. On the other hand quantum entanglement is a natural quantity to characterize the long-range physics. Moreover, both the entanglement and anomaly are pure quantum effects without classical analogies. Therefore one might expect to build up some precise connection between the two. In this paper we move a small initial step along this direction. Namely we have established a concrete relation between the quantum entanglement and the discrete anomaly in certain field theories. In particular the simplicity of topological field theories allows us to focus on the global nature of both the entanglement and the anomaly without worrying about the local interactions.
There has been a lot of interest to study entanglement in topological field theories. The most popular observable is the topological entanglement entropy~\cite{Levin:2006zz,Dong:2008ft,Kitaev:2005dm}, which is associated to partitions in a single boundary of a Euclidean $d$-dimensional manifold where the topological theories are living on. It is an interesting open question to understand the possible patterns of entanglement that can arise in field theory. One might expect that the entanglement encoded in wavefunctions is always a result of interactions through local Hamiltonians. In fact, entanglement is a property encoding physics beyond locality.
Another way to study entanglement in field theories is to think about entanglement between multi-boundaries. The computation becomes relatively simple in topological quantum field theories. In Euclidean path integral formalism, the $d$-dimensional bulk path integral can be used to prepare the entanglement between different disconnected regions for the boundary state.
The multi-boundary entanglement was first studied in AdS$_3$/CFT$_2$~\cite{Balasubramanian:2014hda,Marolf:2015vma}, later in $3d$ Chern-Simons theory where the entanglement structure of a link state was nicely connected to the framing-independent link invariants~\cite{Salton:2016qpp,Balasubramanian:2016sro}. Recently the study of entanglement structure was further carried out in Chern-Simons theories with generic gauge groups~\cite{Dwivedi:2017rnj}. It is therefore desirable to extract more physical information out of the large amount of entanglement data, which is one of the motivations of this paper.
In the works~\cite{Salton:2016qpp,Balasubramanian:2016sro,Dwivedi:2017rnj} mentioned above, the Chern-Simons theory was placed on a 3-manifold with a boundary consisting of $n$ topologically linked tori. The Hilbert space is the $n$-fold tensor product $\mathcal{H}^{\otimes n}$, where $\mathcal{H}$ is the Hilbert space of Chern-Simons theory on a torus. In this paper, instead of studying the wavefunction in the entire $\mathcal{H}$ space consisting of all integrable representations of the affine Lie algebra, we focus on a subspace spanned by the symmetry generators of a discrete 1-form symmetry. We propose that the reduced density matrix on such a subspace captures all the information about the anomaly of this discrete symmetry.
{\it Note added:} While this paper was in completion, \cite{Numasawa:2017crf} appeared, where the invariant boundary state conditions in WZW models coincide with our $3d$ anomaly free conditions.
\section{Review of Anomaly of Discrete Symmetry}
Anomaly was originally discovered as the violation of a classical symmetry at the quantum level, namely the ABJ anomaly~\cite{Bell:1969ts,Adler:1969gk}. This can be interpreted as a 't Hooft anomaly. The famous axial anomaly is an example of a 't Hooft anomaly for $U(1)_{em}\times U(1)_A$~\cite{Kapustin:2014lwa}. For a global symmetry $G$, the obstruction to be promoted to a gauge symmetry is called 't Hooft anomaly~\cite{tHooft:1979rat}. These anomalies are preserved along renormalization group flows. If $G$ is a connected Lie group, the 't Hooft anomaly is tightly constrained by Wess-Zumino consistency conditions~\cite{Wess:1971yu}. Furthermore, the mechanism of anomaly inflow implies that a 't Hooft anomaly in $d$ space-time dimensions must be classified by possible Chern-Simons actions in $d+1$ dimensions. For finite $G$, Wess-Zumino consistency condition does not apply, but we expect that the inflow mechanism still works. The topological actions in $d+1$ dimensions are classified by elements of the group $H^{d+2}(BG, Z)$~\cite{Dijkgraaf:1989pz}. For finite $G$, we instead have $H^{d+1}(BG,U(1))$, which also classifies bosonic SPT phase with global symmetry $G$ in $d+1$ dimensions~\cite{Chen:2011pg,Hung:2012nf}.
Another way to see the anomaly is from the commutation relation of the generators of $G$ by noting that the symmetry $G$ is only realized projectively~\cite{Kapustin:2014lwa,Vishwanath:2012tq}. We will see from an example in later discussions that the generators $x$ and $y$ of $Z_k\times Z_k$ do not commute. That is, the symmetry of the theory is a central extension of $Z_k\times Z_k$.
For continuous symmetries, the 't Hooft anomalies are not affected by the RG flows and therefore constrain the infrared dynamics. When the symmetries are spontaneously broken, the anomalies give rise to Wess-Zumino-Witten terms in the low energy effective action. This is usually called 't Hooft anomaly matching. Since there is no massless Goldstone particles for the spontaneous breaking of discrete symmetries, one can only keep track of the remaining unbroken symmetries in the IR and the UV and impose anomaly matching.
A higher ($q$-form) symmetry is a global symmetry under which the charged operators are of space-time dimension $q$ and the generators have co-dimension $q+1$ in space-time~\cite{Gaiotto:2014kfa}. This can be contrasted with ordinary symmetries, where $q=0$. The properties of ordinary symmetries can be readily generalized to accommodate higher symmetries. In particular, we will focus on the associated 't Hooft anomalies.
$3d$ Chern-Simons theory is the simplest example where a higher form symmetry has a 't Hooft anomaly. We will closely follow~\cite{Gaiotto:2014kfa} and review in this section the anomalies they studied.
\subsection{$U(1)_k$}
$U(1)_k$ Chern-Simons theory has a one-form $Z_k$ symmetry with the generators given by the Wilson loops
\begin{equation}
U_g[W]=\exp\left(in\oint_W A\right)\ , \quad g=e^{i2\pi n/k}\ ,
\end{equation} with $n=0,1,\dots, k-1$. The charged operators are the same Wilson loops with the acting rule
\begin{equation}\label{symmetrydef}
U_g[W]\,U_{g'}[V] = e^{2\pi i mn\over k} U_{g'}[V]\ ,\quad g=e^{2\pi i n/k}\,,g'=e^{2\pi i m/k}\ ,
\end{equation} where $W$ is a circle around $V$.\footnote{Here we mean the linking number is $1$.} It is clear that the generators are charged under themselves, which was interpreted as the appearance of a't Hooft anomaly associated to $Z_k$.
It is of interest to note that in the construction of SET via field extension say in \cite{Hung:2012nf} that embeds the topological order in a larger topological order, it is required that the set of generators of the global symmetries correspond to Bosonic sectors in the extended topological theory, with trivial mutual statistics. This readily matches with the criterion for an anomaly free symmetry as described above.
\subsection{$SU(2)_k$}
Another example is $SU(2)_k$ Chern Simons theory. There is a one-form $Z_2$ symmetry generated by the Wilson loop associated with the $SU(2)$ representation with $j=k/2$. The charged operators are the various Wilson loops and the acting rule is
\begin{equation}
U_g[W] C_{j'}[V]=(-1)^{2{j'}}C_{j'}[V],\quad g=e^{{2\pi i\over k}j}=-1\ ,
\end{equation} with $j'=0,{1\over 2},\dots,{k\over 2}$.
This is consistent with the expectation value of two linked loops in the representations $j$ and $j'$ in $S^3$. It is
\begin{equation}
S_{jj'} = \sqrt{2\over k+2}\sin\left(\pi(2j+1)(2j'+1)\over k+2\right)\ .
\end{equation} For $j={k\over 2}$ one has
\begin{equation}
S_{{k\over 2}j'}=(-1)^{2j'}S_{0j'}\ .
\end{equation} Now let us consider the action of generators on themselves
\begin{equation}
U_g[W]U_j[V]=(-1)^{2j}U_j[V], \quad j={k\over 2}\ .
\end{equation}
For odd $k$ there is a 't Hooft anomaly of $Z_2$ symmetry.
Notice that this $Z_2$ symmetry coincides with the center of the $SU(2)$ gauge group.
\subsection{$SU(N)_k$}
Similarly one can consider $SU(N)_k$ Chern-Simons theory. This theory has a $Z_N$ one-form symmetry. The generator $U_1$ is a Wilson loop with $k$ boxes in a symmetric representation, the other group elements $U_n$ are labeled by rectangular Young Tableaux with $nk$ boxes. Again focus on the acting rule of the generators on themselves
\begin{equation}
U_1[W]U_1[V] = e^{-2\pi i k\over N}U_1[V]\ ,
\end{equation} which indicates that for general $k$ there is a 't Hooft anomaly except that $k$ is a multiple of $N$.
Our observation is that the above anomaly can be directly seen from the linking entanglement, which we will illustrate below.
\section{Entanglement as measure of anomaly}
Consider a 2-component link (such as Hopf link) inside $S^3$ in Chern-Simons theory with gauge group $G$ at level $k$, cut along a tube neighborhood of the link, then there are {\it inside} part and {\it outside} part, the {\it solid torus} and {\it link complement}.
The {\it inside} path integral over solid torus with a Wilson loop insertion, which runs over all integrable representations of affine algebra $G_k$, provides a base of a Hilbert space. Since there are two {\it inside} parts, the total Hilbert space is naturally given by the tensor product of the two. The wave function of the state defined by the {\it outside} path integral in the basis just constructed is precisely the usual link invariant~\cite{Witten:1988hf}. By tracing out one of the sub Hilbert spaces, one can compute the entanglement encoded in this wavefunction (so called multi-boundary entanglement). This is what we call linking entanglement for short. In the following subsection we briefly review the set up in~\cite{Balasubramanian:2016sro}. All the other parts of this section are devoted to the study of the precise relation between linking entanglement and discrete anomaly in different theories.
\subsection{Linking entanglement}
Consider Chern-Simons theory with gauge group $G$ at level $k$, with action defined on 3-manifold
\begin{equation}
S_{CS}[A] = {k\over 4\pi} \int_M \mbox{Tr} \left(A\wedge dA + {2\over 3} A\wedge A\wedge A\right)\ .
\end{equation} Consider states defined on $n$ copies of $T^2$, $\Sigma_n$. Then the Hilbert space is the $n$-fold tensor product $\mathcal{H}^{\otimes n}$ as mentioned in the introduction. We construct states by performing the Euclidean path integral of the theory on $M$ with boundary $\partial M=\Sigma_n$. There could be many options for $M$ to satisfy this boundary condition, but we choose the link complement of a $n$-component link in $S^3$ as our $M$ following~\cite{Balasubramanian:2016sro}. Then the path integral will produce a state
\begin{equation}
|L\rangle \in \mathcal{H}^{\otimes n}\ .
\end{equation}
To be specific, let us assign a basis for the Hilbert space $\mathcal{H}$ and write
\begin{equation}\label{outstate}
|L\rangle = \sum_{j_1,\dots,j_n} C_{L}(j_1,\dots,j_n) | j_1,\dots,j_n\rangle\ ,
\end{equation} where $\{|j\rangle\}$ can be prepared as the path integral of a solid torus with a Wilson line in the integrable representations $R_j$ inserted along the non-contractible circle. As such, one can write the coefficient in (\ref{outstate})
\begin{equation}
C_{L}(j_1,\dots,j_n) = \langle j_1,\dots,j_n|L\rangle\ .
\end{equation} This actually means that if we assign in $S^3$ the $n$-component link with Wilson lines in the conjugate representation $R^*_{j_i}$ for the $i$-th component, the coefficient in (\ref{outstate}) is nothing but the expectation value of $n$ Wilson lines
\begin{equation}
C_{L}(j_1,\dots,j_n) = \left\langle W_{R^*_{j_1}}(L_1)\dots W_{R^*_{j_n}}(L_n)\right\rangle_{S^3}\ .
\end{equation}
By the above construction, we assign a quantum entanglement structure to a link in three-sphere. With the wavefunction $C_{L}(j_1,\dots,j_n)$ at hand and also the total Hilbert space factorized, one can explore the entanglement structure sufficiently.
For instance, one can bi-partition a link $L$ as $L=L^m|L^{n-m}$. The reduced density matrix can be evaluated
\begin{equation}
\rho = {1\over \langle L|L\rangle}\mbox{Tr}_{L_1,\dots,L_m}|L\rangle\langle L|\ .
\end{equation} And the entanglement entropy is given by
\begin{equation}
S = -\mbox{Tr}_{L_{m+1},\dots,L_n}\left(\rho\ln\rho\right)\ .
\end{equation}
In most cases below, we will focus on Hopf link, where the wave function of the link state is given by the modular S matrix
\begin{equation}
C_{\text{Hopf}} (j_1,j_2)= S_{j_1,j_2}\ .
\end{equation}
\subsection{$U(1)_k$}
We consider a 2-component link with linking number $n$, for $U(1)_k$ Chern-Simons, the entanglement entropy constructed as above is given by~\cite{Balasubramanian:2016sro}
\begin{equation}\label{AbEE}
S_{EE} = \ln\left({k\over \mathrm{gcd}(k,n)}\right)\ .
\end{equation}
In the case of Hopf link $n=1$, the entanglement entropy is $\ln k$. We interpret the non-vanishing of $S_{EE}$ as the signal of the 't Hooft anomaly of $Z_k$, given that the symmetry actions on operators is defined as (\ref{symmetrydef}). Notice that in the Abelian case the generators of the $Z_k$ symmetry span the whole Hilbert space. One can generalize the symmetry definition (\ref{symmetrydef}) to the case with linking number $n$ between $W$ and $V$, then the faithful symmetry becomes $Z_{k/\mathrm{gcd}(k,n)}$. The rank of the group precisely matches with the entanglement entropy (\ref{AbEE}).
\subsection{$SU(2)_k$}
In the case of $SU(2)_k$ Chern-Simons theory, the generators of $Z_2$ (Wilson loops) are the spin $0$ and spin ${k\over2}$ representations. We will first review the computation of the entanglement entropy for the Hopf link state.
As mentioned before the state defined by the path integral on a 3-manifold with linked torus boundaries has the wave function
\begin{equation}
S_{j_1j_2}(k)=\sqrt{2\over k+2} \sin\left({\pi(2j_1+1)(2j_2+1)\over k+2}\right)\ ,
\end{equation} where $j_1$ and $j_2$ span $0,{1\over 2},\dots,{k\over 2}$. The reduced density matrix after integrating out $\mathcal{H}_2$ is
\begin{equation}
\rho_{j_1j'_1}(k) = {SS^\dagger\over \mathrm{tr} SS^\dagger} = {1\over k+1}\mathrm{diag}(1,1,\dots,1)\ ,
\end{equation} where ${1\over k+1}$ is the normalization factor since the dimension is $k+1$. To gain information of the anomaly associated to the $Z_2$ symmetry, we instead consider a $2\times 2$ small $D(k)$ matrix constructed from $S_{j_1j_2}(k)$ with $j_1,j_2$ span $0,{k\over 2}$:
\begin{equation}
D_{11}(k)=S_{00}=\sqrt{2\over k+2}\sin\left(\pi\over k+2\right),\quad D_{12}(k)=S_{0{k\over 2}}=\sqrt{2\over k+2}\sin\left({\pi(k+1)\over k+2}\right),
\end{equation}
\begin{equation}
D_{21}(k)=S_{{k\over 2}0}=\sqrt{2\over k+2}\sin\left({\pi(k+1)\over k+2}\right),\quad D_{22}(k)=S_{{k\over 2}{k\over 2}}=\sqrt{2\over k+2}\sin\left({\pi(k+1)^2\over k+2}\right).
\end{equation}
The reduced density matrix after tracing out $\widetilde{\mathcal{H}}_2$ is~\footnote{$\widetilde{\mathcal{H}}_2$ is the sub Hilbert space which spans representations $0,{k\over 2}$.}
\begin{equation}
\rho_{2\times 2}(k)={DD^\dagger\over \mathrm{tr} DD^\dagger}\ .
\end{equation} The von Neumann entropy of $\rho_{2\times 2}(k)$ is
\begin{equation}
S =-\mathrm{tr}\left(\rho_{2\times 2}\ln\rho_{2\times 2}\right)= \mathrm{log}\,2\ ,
\end{equation} only if $k$ is odd and otherwise vanishes. We illustrate $D(100)$ and $D(101)$ as follows
\begin{equation}
D(100) = \left(
\begin{array}{cc}
\frac{\sin \left(\frac{\pi }{102}\right)}{\sqrt{51}} & \frac{\sin \left(\frac{\pi }{102}\right)}{\sqrt{51}} \\
\frac{\sin \left(\frac{\pi }{102}\right)}{\sqrt{51}} & \frac{\sin \left(\frac{\pi }{102}\right)}{\sqrt{51}} \\
\end{array}
\right);\quad D(101) = \left(
\begin{array}{cc}
\sqrt{\frac{2}{103}} \sin \left(\frac{\pi }{103}\right) & \sqrt{\frac{2}{103}} \sin \left(\frac{\pi }{103}\right) \\
\sqrt{\frac{2}{103}} \sin \left(\frac{\pi }{103}\right) & -\sqrt{\frac{2}{103}} \sin \left(\frac{\pi }{103}\right) \\
\end{array}
\right)\ .
\end{equation}
In the following we will consider more general CS theories of A,B,C and D types.
\subsection{$A_{N-1, N\geq2}$}
We will first illustrate the process to find the anomaly free condition for $SU(N)_k$ Chern-Simons theory. $SU(N)_k$ Chern-Simons theory has a $Z_N$ global symmetry. The integrable representations are labeled by a set of non-negative integers, $\hat a = [a_0, a_1,\dots,a_{N-1}]$, where $(a_1,\dots,a_{N-1})$ are Dynkin labels. The integrability condition reads
\begin{equation}
\phi_1 a_1 +\dots+ \phi_{N-1} a_{N-1} \leq k\ ,
\end{equation} where $(\phi_1,\dots,\phi_{N-1})=(1,\dots,1)$ are the comarks of $su(N)$ algebra and $k$ is the Chern-Simons level. Suppose $R_1$ and $R_2$ are two representations of $SU(N)$. The element $S_{R_1,R_2}$ of the modular S matrix is given by
\begin{equation}
S_{R_1,R_2} = (-i)^{N(N-1)\over 2}{N^{-{1\over 2}}\over (k+N)^{N-1\over 2}} \det \left[M_{R_1,R_2}\right]\ ,
\end{equation} where the matrix $M_{R_1,R_2}$ is a $N\times N$ matrix whose element is defined as
\begin{equation}
M_{R_1,R_2}[i,j]= \exp \left[{2i\pi \phi_{R_1}[i]\phi_{R_2}[j]\over k+N}\right]\ .
\end{equation} The indexed function $\phi$ has a total of $N$ components and its $i$-th component for a representation $R$ is defined as
\begin{equation}
\phi_R[i]=\ell[i] - i -{\ell\over N}+{N+1\over 2}\ ,\quad i=1,\dots,N\ ,
\end{equation} where $\ell[i]$ is the number of boxes in the $i$-th row of the reduced Young diagram of representation $R$ and $\ell$ is the total number of boxes ($\ell[N]=0$).
One can take the reduction of the large modular S matrix to a small $N\times N$ matrix, where the row and column of the small matrix $D_{N\times N}$ spans the representations generating the $Z_N$ symmetry. Then one can evaluate the von Neumann entropy of a reduced density matrix constructed from the small $D$ matrix and find that
\begin{equation}
S = -\mathrm{tr}\left(\rho_{N\times N}\ln\rho_{N\times N}\right) = 0\ ,
\end{equation} only if $k$ is a multiple of $N$ and otherwise finite. This means that only when $k$ is a multiple of $N$, the theory is anomaly free. Here we should emphasize that the truncation method here is only needed for $SU(N\geq 2)_k$ theory and the $U(1)_k$ theory as our first example is beyond this pattern. There we do not need to truncate a modular matrix to diagnose the anomaly because the symmetry $Z_k$ spans the whole Hilbert space. We illustrate some examples of $D_{N\times N}(k)$ and $\rho_{N\times N}(k)$ as follows
\begin{equation}\label{example1}
D_{4\times 4}(k=2) = {1\over 2\sqrt{6}}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
\end{array}
\right)\ ,\quad \rho_{4\times 4} (k=2) = \left(
\begin{array}{cccc}
\frac{1}{4} & 0 & \frac{1}{4} & 0 \\
0 & \frac{1}{4} & 0 & \frac{1}{4} \\
\frac{1}{4} & 0 & \frac{1}{4} & 0 \\
0 & \frac{1}{4} & 0 & \frac{1}{4} \\
\end{array}
\right) ;
\end{equation}
\begin{equation}
D_{4\times 4}(k=3) = \text{c}\times\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & i & -1 & -i \\
1 & -1 & 1 & -1 \\
1 & -i & -1 & i \\
\end{array}
\right)\ ,\quad \rho_{4\times 4} (k=3) = \left(
\begin{array}{cccc}
\frac{1}{4} & 0 & 0 & 0 \\
0 & \frac{1}{4} & 0 & 0 \\
0 & 0 & \frac{1}{4} & 0 \\
0 & 0 & 0 & \frac{1}{4} \\
\end{array}
\right) ;
\end{equation}
\begin{equation}
\rho_{3\times 3}(k=1)=
\left(
\begin{array}{ccc}
\frac{1}{3} & 0 & 0 \\
0 & \frac{1}{3} & 0 \\
0 & 0 & \frac{1}{3} \\
\end{array}
\right);
\quad\rho_{3\times 3}(k=2) = \left(
\begin{array}{ccc}
\frac{1}{3} & 0 & 0 \\
0 & \frac{1}{3} & 0 \\
0 & 0 & \frac{1}{3} \\
\end{array}
\right)\ .
\end{equation}
Notice that the 1-form symmetry group of $SU(N)_k$ Chern-Simons theory coincides with the center of the gauge group. This is because the outer automorphism group of the affine Lie algebra is isomorphic to the center group.\footnote{For an explicit description of outer-automorphism groups of affine Lie algebras and the proof of this isomorphism, see for instance Section 14.2.1 and Section 14.2.3 in the CFT book by Francesco, Mathieu and S\'en\'echal.} Therefore one can generalize the discussion for $A$-type gauge group to other $B,C,D,E$ types. This will give conditions to justify whether there is 't Hooft anomaly for one-form symmetries in other types of theories. Below we illustrate the details for $B_N$, $C_N$ and $D_N$ theories.
\subsection{$B_{N\geq 3}$}
We will illustrate the process to find the anomaly free condition for $SO(2N+1)_k$ Chern-Simons theory. $SO(2N+1)_k$ Chern-Simons theory has a $Z_2$ global symmetry. The integrable representations are labeled by a set of non-negative integers, $\hat a = [a_0, a_1,\dots,a_{N}]$, where $(a_1,\dots,a_{N})$ are Dynkin labels. The integrability condition reads
\begin{equation}
\phi_1a_1 +\phi_2a_2+\dots+ \phi_{N} a_{N} \leq k\ ,
\end{equation} where $(\phi_1,\phi_2,\dots,\phi_{N-1},\phi_N)=(1,2,\dots,2,1)$ are the comarks of $so(2N+1)$ algebra and $k$ is the Chern-Simons level. Suppose $R_1$ and $R_2$ are two representations of $SO(2N+1)$ with Dynkin labels $R_1=[a_1,\dots,a_N]$ and $R_2=[b_1,\dots,b_N]$. The element $S_{R_1,R_2}$ of the modular S matrix is given by
\begin{equation}
S_{R_1,R_2} = (-1)^{N(N-1)\over 2}{2^{N-1}\over (k+2N-1)^{N\over 2}} \det \left[M_{R_1,R_2}\right]\ ,
\end{equation} where the matrix $M_{R_1,R_2}$ is $N\times N$ matrix whose element is defined as
\begin{equation}
M_{R_1,R_2}[i,j]= \sin \left({2\pi \phi_{R_1}[i]\phi_{R_2}[j]\over k+2N-1}\right)\ .
\end{equation} The indexed function $\phi$ has a total $N$ components and its $i$-th component for a representation $R$ is defined as
\begin{equation}
\phi_R[i]=\ell_i - i +{2N+1\over 2}\ ,\quad i=1,\dots,N\ ,
\end{equation} where $\ell_i$ is determined by Dynkin labels
\begin{equation}
\ell_i = \sum_{n=i}^{N-1} a_n + {a_N\over 2}\ ,\quad \ell_N = {a_N\over 2}\ .
\end{equation}
One can take the reduction of the large modular S matrix to a small $2\times 2$ matrix, where the row and column of the small matrix span the representations generating $Z_2$ symmetry. Then one obtains a reduced density matrix from the small $D$ matrix and find that
\begin{equation}
S=-\mathrm{tr}\left(\rho_{2\times 2}\ln\rho_{2\times 2}\right) =0\ ,
\end{equation} for any $N\geq 3$ and any $k$. This means that the theories are always anomaly free.
\subsection{$C_{N\geq 2}$}
We will illustrate the process to find the anomaly free condition for $Sp(2N)_k$ Chern-Simons theory. $Sp(2N)_k$ Chern-Simons theory has a $Z_2$ global symmetry. The integrable representations are labeled by a set of non-negative integers, $\hat a = [a_0, a_1,\dots,a_N]$, where $(a_1,\dots,a_N)$ are Dynkin labels. The integrability condition reads
\begin{equation}
\phi_1a_1 +\phi_2a_2+\dots+ \phi_{N} a_{N} \leq k\ ,
\end{equation} where $(\phi_1,\phi_2,\dots,\phi_N)=(1,1,\dots,1)$ are the comarks of $sp(2N)$ algebra and $k$ is the Chern-Simons level. Suppose $R_1$ and $R_2$ are two representations of $Sp(2N)$ with Dynkin labels $R_1=[a_1,\dots,a_N]$ and $R_2=[b_1,\dots,b_N]$. The element $S_{R_1,R_2}$ of the modular S matrix is given by
\begin{equation}
S_{R_1,R_2} = (-1)^{N(N-1)\over 2}\left({2\over k+N+1}\right)^{N\over 2} \det \left[M_{R_1,R_2}\right]\ ,
\end{equation} where the matrix $M_{R_1,R_2}$ is $N\times N$ matrix whose element is defined as
\begin{equation}
M_{R_1,R_2}[i,j]= \sin \left({\pi \phi_{R_1}[i]\phi_{R_2}[j]\over k+N+1}\right)\ .
\end{equation} The indexed function $\phi$ has total $N$ components and its $i$-th component for a representation $R$ is defined as
\begin{equation}
\phi_R[i]=\ell_i - i +N+1\ ,\quad i=1,\dots,N\ ,
\end{equation} where $\ell_i$ is determined by Dynkin labels
\begin{equation}
\ell_i = \sum_{n=i}^N a_n\ .
\end{equation}
One can take the reduction of the large modular S matrix to a small $2\times 2$ matrix, where the row and column of the small matrix span the representations generating the $Z_2$ symmetry. Then one can obtain a reduced density matrix from the small $D$ matrix and find that
\begin{equation}
S=-\mathrm{tr}\left(\rho_{2\times 2}\ln\rho_{2\times 2}\right)> 0\ ,
\end{equation} only if $N$ is odd and also $k$ is odd, but vanishes otherwise. This means that there is a 't Hooft anomaly only for the case where both $N$ and $k$ are odd.
\subsection{$D_{N\geq 4}$}
We will illustrate the process to find the anomaly free condition for $SO(2N)_k$ Chern-Simons theory. $SO(2N)_k$ Chern-Simons theory has a $Z_4$ global symmetry when $N$ is odd and a $Z_2\times Z_2$ global symmetry when $N$ is even. The integrable representations are labeled by a set of non-negative integers, $\hat a = [a_0, a_1,\dots,a_N]$, where $(a_1,\dots,a_N)$ are Dynkin labels. The integrability condition reads
\begin{equation}
\phi_1a_1 +\phi_2a_2+\dots+ \phi_{N} a_{N} \leq k\ ,
\end{equation} where $(\phi_1,\phi_2,\dots,\phi_{N-2},\phi_{N-1},\phi_N)=(1,2,\dots,2,1,1)$ are the comarks of $so(2N)$ algebra and $k$ is the Chern-Simons level. Suppose $R_1$ and $R_2$ are two representations of $SO(2N)$ with Dynkin labels $R_1=[a_1,\dots,a_N]$ and $R_2=[b_1,\dots,b_N]$. The element $S_{R_1,R_2}$ of modular S matrix is given by
\begin{equation}
S_{R_1,R_2} = (-1)^{N(N-1)\over 2}{2^{N-2}\over (k+2N-2)^{N\over 2}} \left(\det \left[M_{R_1,R_2}\right]+i^N\det\left[G_{R_1,R_2}\right]\right)\ ,
\end{equation} where the matrix $M_{R_1,R_2}$ and $G_{R_1,R_2}$ are $N\times N$ matrix whose element is defined as
\begin{equation}
M_{R_1,R_2}[i,j]= \cos \left({2\pi \phi_{R_1}[i]\phi_{R_2}[j]\over k+2N-2}\right)\ ,\quad G_{R_1,R_2}[i,j]=\sin \left({2\pi \phi_{R_1}[i]\phi_{R_2}[j]\over k+2N-2}\right)\ .
\end{equation} The indexed function $\phi$ has total $N$ components and its $i$-th component for a representation $R$ is defined as
\begin{equation}
\phi_R[i]=\ell_i - i +N\ ,\quad i=1,\dots,N\ ,
\end{equation} where $\ell_i$ is determined by Dynkin labels
\begin{equation}
\ell_i = \sum_{n=i}^{N-2} a_n + {a_N+a_{N-1}\over 2}\ ,\quad \ell_{N-1} = {a_N+a_{N-1}\over 2}\ ,\quad\ell_N = {a_N-a_{N-1}\over 2}\ .
\end{equation}
One can take the reduction of the large modular S matrix to a small $4\times 4$ matrix, where the row and column of the small matrix span the representations generating $Z_4$ or $Z_2\times Z_2$ symmetry. Then one can obtain a reduced density matrix from the small $D$ matrix and find that
\begin{equation}
S=-\mathrm{tr}\left(\rho_{4\times 4}\ln\rho_{4\times 4}\right)= 0\ ,
\end{equation} only if $k$ is even when $N$ is even or $k=4\mathbb{Z}$ when $N$ is odd. This means that the theories are anomaly free only for these two cases. We illustrate some examples of $D_{4\times 4}(N,k)$ matrices and $\rho_{4\times 4}(N,k)$ matrices as follows
\begin{equation}
D_{4\times 4}(N=5,k=1) = \left(
\begin{array}{cccc}
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & \frac{i}{2} & -\frac{i}{2} & -\frac{1}{2} \\
\frac{1}{2} & -\frac{i}{2} & \frac{i}{2} & -\frac{1}{2} \\
\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
\end{array}
\right); \quad D_{4\times 4}(N=5,k=4)={2-\sqrt{3}\over 24}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}
\right);
\end{equation}
\begin{equation}\label{example2}
D_{4\times 4}(N=5,k=2) ={1\over 2\sqrt{10}}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 \\
1 & -1 & -1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}
\right)\ ,\quad \rho_{4\times 4}(N=5,k=2) = \left(
\begin{array}{cccc}
\frac{1}{4} & 0 & 0 & \frac{1}{4} \\
0 & \frac{1}{4} & \frac{1}{4} & 0 \\
0 & \frac{1}{4} & \frac{1}{4} & 0 \\
\frac{1}{4} & 0 & 0 & \frac{1}{4} \\
\end{array}
\right);
\end{equation}
\begin{equation}
D_{4\times 4}(N=4,k=3)=\frac{1}{9} \left(2 \sin \left(\frac{\pi }{18}\right)+\cos \left(\frac{\pi }{9}\right)-\cos \left(\frac{2 \pi }{9}\right)\right)\times\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1 \\
\end{array}
\right)\ ,
\end{equation}
\begin{equation}
\rho_{4\times 4}(N=4,k=3)=\left(
\begin{array}{cccc}
\frac{1}{4} & 0 & 0 & 0 \\
0 & \frac{1}{4} & 0 & 0 \\
0 & 0 & \frac{1}{4} & 0 \\
0 & 0 & 0 & \frac{1}{4} \\
\end{array}
\right);\quad D_{4\times 4}(N=4,k=2) ={1\over 4\sqrt{2}}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}
\right)\ .
\end{equation}
\begin{equation}
D_{4\times 4}(N=6,k=2)={1\over 4\sqrt{3}}\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}
\right)\ ,\quad D_{4\times 4}(N=6,k=4)= c\times \left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}
\right)\ .
\end{equation}
\section{Discussion}
Now we discuss what the entropy we have constructed actually counts. As noted in some examples we have discussed, the entropy actually counts how many symmetries are anomalous. More precisely, the entropy is
\begin{equation}
S = \log \mathcal{D}\ ,
\end{equation} where $\mathcal{D}$ is the rank of the anomalous group. Let us see a few examples. In the case of $SU(4)_{k=2}$ CS theory (\ref{example1}), the one-form symmetry is $Z_4$, but the entropy is $S = \log 2$ and the eigenvalues of the $4\times 4$ reduced density matrix is $({1\over 2},{1\over 2},0,0)$. This is because only a subgroup $Z_2$ of $Z_4$ is anomalous. The same thing happens for $SO(10)_{k=2}$ CS theory (\ref{example2}). One can therefore examine which subgroup of the global symmetry is anomalous by looking at the entropy.
Another physical meaning of $\mathcal{D}$ in $U(1)_k$ theory is the ground state degeneracy. Let us explain by looking at the $U(1)_k$ CS quantized on a torus. We refer to~\cite{Witten:2015aoa} for recent reviews.
The gauge invariant operators are Wilson loops. There are two different cycles so there will be two different Wilson loop operators $W_a$ and $W_b$. From the quantization condition, $W_{a,b}$ must obey the algebra
\begin{equation}\label{commurel}
W_{a}W_{b}=e^{2\pi i \over k}W_{b}W_{a}\ .
\end{equation}
$W_{a}$ and $W_{b}$ are the generators of a symmetry $Z_k\times Z_k$. This commutation relation allows us to see the anomaly since $Z_k\times Z_k$ is only realized projectively. The actual symmetry is a central extension of $Z_k\times Z_k$. This interpretation is useful to distinguish the anomaly we are discussing from other types of anomaly (such as mixed anomalies).
One can immediately see that (\ref{commurel}) can not be realized on a single vacuum~\cite{Wen:1990zza,Sato:2006de}. The minimal representation has dimension $k$ , with the acting rules
\begin{equation}
W_a|j\rangle =e^{{2\pi i\over k}j}|j\rangle, \quad W_b|j\rangle = | j+1\rangle\ ,\quad j=0,1,\dots, k-1\ .
\end{equation}
So the ground state degeneracy is $k$, which matches with the entropy we have constructed (\ref{AbEE}), $S=\log k$. It is interesting to ask which degrees of freedom could respect the 't Hooft anomaly (of one-form symmetries) in non-abelian CS theories.
\section*{Acknowledgement}
We are grateful for helpful discussions with Siddharth Dwivedi and Adar Sharon.
|
2,869,038,155,655 | arxiv | \section{Introduction}
\label{sec:intro}
A prediction of the strength of future solar cycles is of interest
because a) expected future levels of solar activity have implications
for space missions and for foreseeing potential hazards due to `space
weather', and b) it can be used to test theoretical models of the solar
cycle, if the prediction is based upon a physical approach. Existing
attempts to predict solar activity levels can be broadly divided into
two groups: 1) extrapolation models, deriving a prediction from a purely
mathematical analysis of the past records of solar activity, for
instance by harmonic analysis \citep[e.g.,][]{Rigozo:etal:2001,
Echer:etal:2004} or by using concepts from nonlinear dynamics
\citep[e.g.,][]{Sello:2001}, and 2) precursor models based upon
correlations between certain measured quantities in the declining phase
of a cycle and the strength of the next cycle
\citep[e.g.,][]{Hathaway:etal:1999, Schatten:2003}. While the overall
success of these models in predicting the {\em future} has been rather
limited (e.g., see Figure~14.2 in Wilson 1994 and Figure~6 of Lantos \&
Richard 1998), a number of methods have demonstrated considerable skill
in `predicting' the strength of the past solar cycles
\citep[e.g.][]{Hathaway:etal:1999}. In particular, some methods based on
the level of geomagnetic variations (as measured, e.g., by the $aa$ or
$A_p$ indices) a few years before and around sunspot minimum provide
high correlation coefficients of up to 0.97 between the respective
predictor and past maxima of the sunspot record
\citep[e.g.,][]{Legrand:Simon:1981,Layden:etal:1991,Thompson:1993,
Lantos:Richard:1998, Hathaway:etal:1999}
Since the recurrent geomagnetic variations in the late phases of a cycle
are presumably dominated by the fast solar wind streams from the
equatorward expanding polar coronal holes of the Sun, some relation to
the strength of the polar magnetic field during that phase has been
suggested. As noted by, e.g., \citet{Schatten:etal:1978} and
\citep{Layden:etal:1991}, such a relationship would be generally
consistent with the Babcock-Leighton type of dynamo models for the solar
cycle \citep{Babcock:1961, Leighton:1969, Giovanelli:1985, Wang:Sheeley:1991,
Choudhuri:etal:1995, Durney:1995, Durney:1996, Dikpati:Charbonneau:1999,
Kueker:etal:2001}. Such models assume that the differential rotation
winds up a dipolar global poloidal field which is thought to dominate
the large-scale surface field distribution around solar minimum and
creates sub-surface toroidal fields, whose later eruption causes the
magnetic activity of the following cycle. The larger the global dipole
moment is, the stronger the toroidal field becomes and, presumably, the
stronger the next cycle will be. In the course of the cycle, the global
dipole field is reversed and an opposite-polarity field built up, owing
to the preferential poleward drift of the follower polarity parts of
solar active regions and the diffusion and cancellation of
preceding-polarity flux over the equator. This occurs because there is a
(statistically) systematic tilt angle of the bipolar magnetic regions
with respect to the equator (Joy's law). The important point is that the
mechanism which provides the poloidal field for the next cycle operates
in the near-surface layers and thus is directly accessible to
observation while the old cycle is still ongoing. If quantitatively
understood this could provide a physical basis for prediction and,
at the same time, a testbed for dynamo models.
The first attempt to actually use a theoretical dynamo model for
predicting future solar activity levels was presented recently by
\citet{Dikpati:etal:2006} and \citet{Dikpati:Gilman:2006}, in the
following referred to as the DDG model. They considered a
Babcock-Leighton type flux-transport dynamo with solar-like differential
rotation, advection of the magnetic field by a prescribed meridional
circulation pattern (poleward near the surface, equatorward in the lower
convection zone) and two sources for the poloidal field: one near the
bottom of the convection zone and one resulting from active-region tilt
near the surface. Parametrizing the near-surface source with the
observed record of sunspot areas since 1874 and making simple
assumptions concerning the tilt angle and the equatorward migration of
the source latitude (corresponding to the butterfly diagram), they took
the amount of flux of the low-latitude toroidal field generated by the
model as a measure of predicted solar activity. Their model is able to
reproduce the variation of the amplitudes of cycles 16--23 remarkably
well, with correlation coefficients reaching up to nearly 0.99,
depending on the model parameters.
The success of the DGG model suggests that it may have captured the
essential ingredients of the solar dynamo mechanism and that the dynamo
is of the Babcock-Leighton flux-transport type. This would be an
enormous advance in our understanding and, therefore, the evidence must
be evaluated thoroughly and critically. We have therefore set out to
answer the following questions: 1) Which are the key physical features
that underly the predictive skill of the the DGG model? 2) How much of
the predictive skill is already contained in the surface transport part?
3) Could the predictive skill be affected by correlations in the sunspot
record (used as input data for the model) that are not specific to any
particular dynamo model?
In order approach answers to the first two questions, we have carried
out an exploratory study with a simple surface flux transport
model. Two-dimensional versions of such models have successfully
reproduced the observed evolution of the solar surface field
\citep[e.g.,][]{Wang:etal:1989, Wang:etal:1989b,
Ballegooijen:etal:1998a, Schrijver:2001, Durrant:etal:2004,
Baumann:etal:2004}. For simplicity (and consistent with the surface part
of the DGG model) we have considered azimuthally averaged quantities in
a one-dimensional flux transport model and determined the evolution of
the latitude-dependent surface field on the basis of observational input
data (sunspot groups or sunspot numbers). If the generation of the
poloidal field for the next cycle actually takes place near the surface
in the fashion exhibited by the Babcock-Leighton approach, we should be
able to find relevant predictors from such results. We have therefore
compared quantities related to the reversal and built-up of the global
dipole field, particularly the amount of flux diffusing and cancelling
over the equator, with the activity of the following cycle.
In order to address the third question, we have also studied whether the
predictive skill of the precursor-type models (including our own and
possibly also DDG model) could be affected by the overlap of consecutive
cycles and the resulting shift of the epoch of activity minimum
depending on the strength and the rise time of the following cycle.
The paper is organized as follows. We describe the flux transport model
in Sect.~\ref{sec:ftm} and define the quantities considered as
predictors in Sect.~\ref{sec:predictors}. Results based upon
observational datasets for sunspot areas and sunspot numbers are shown
in Sect.~\ref{sec:results}. We first closely follow the procedure used
by DDG and compare with their results. Then we evaluate the sensitivity
of the results to parameter variations and to modifications of the model
assumptions concerning the input data, particularly the latitude
distribution of emerging flux. In Sect.~\ref{sec:origin} we consider
series of synthetic cycles with random amplitudes in order to study the
effect of the amplitude-dependent shift of the activity minimum epoch on
prediction methods. Sect.~\ref{sec:discussion} contains a discussion of
the results and gives our conclusions.
\section{A simple flux transport model}
\label{sec:ftm}
We model the evolution of the (azimuthally averaged) component of the
field normal to the solar surface, $B_r$, as a function of latitude,
$\lambda$, and time, $t$. We assume that the surface flux is passively
advected by a meridional flow, $v(\lambda)$, and by supergranulation,
the latter being described as a turbulent diffusion process with an
effective diffusivity $\eta$. In addition, we have a source term,
$S(\lambda,t)$, describing the emergence of new bipolar magnetic regions
at the surface, and an exponential decay term with a characteristic time
scale $\tau$, which roughly mimics 3-D effects
\citep{Schrijver:etal:2002, Baumann:etal:2006}. Accordingly, our simple
1-D surface transport equation is written as
\begin{eqnarray}
\frac{\partial B_r}{\partial t} &=&
\frac{1}{R_{\odot}\cos\lambda}\frac{\partial}{\partial \lambda}
\bigg[ v(\lambda) B_r \cos\lambda \bigg] +
\nonumber\\ \noalign{\vspace{0.2cm}}
&+& \frac{\eta}{R_{\odot}^2\cos\lambda}
\frac{\partial}{\partial \lambda}
\left( \sin\lambda\frac{\partial B_r}{\partial\lambda}\right)
- \frac{B_r}{\tau} + S(\lambda,t)\,.
\label{eq:transport}
\end{eqnarray}
For simplicity, we take $B_r$ to be antisymmetric with respect to
the equator $(\lambda=0$) and consider only the Northern hemisphere,
thus ignoring North-South asymmetries.
We solve this 1D linear advection-diffusion problem by a fully implicit
finite difference scheme with 900 grid cells per hemisphere. Since the
bipolar regions described by the source term typically have a size of a
few degrees, all features occurring in the simulation are very well
resolved. In fact, tests with only 90 grid points give essentially the
same results. The timestep of the numerical integration is set
according to the Courant-Friedrichs-Levy condition on the basis of the
maximum meridional flow speed. Values typically are of the order of a
day.
\section{Predictors}
\label{sec:predictors}
Which quantities provided by the surface flux transport model could be
considered as potential predictors for the strength of the next cycle?
In the framework of Babcock-Leighton dynamos, one is tempted to use the
maximum value of the reversed polar field built up after solar maximum
as an indicator for the strength of the poloidal field from which the
toroidal field of the next cycle is being generated by differential
rotation \citep{Schatten:etal:1978}. In fact,
\citep{Svalgaard:etal:2005} have reported a correlation between observed
polar fields and sunspot activity of the subsequent cycle, but regular
measurements of the polar fields exist only since about 1970 and still
are rather uncertain and noisy \citep[see,
e.g.,][]{Arge:etal:2002}. Furthermore, \citet{Layden:etal:1991} have
tried to extend the temporal basis by considering potential proxies for
the polar field such as the shape of the polar corona, coronal holes, or
polar faculae, but found no clear evidence of a `predictive' skill.
Even in the framework of the Babcock-Leighton scenario, it is not so
clear whether the maximum polar field strength necessarily represents
the relevant quantity that determines the strength of the next cycle. In
a flux transport dynamo, the amount of poloidal magnetic flux reaching
deep into the convection zone (the global dipole) is relevant for the
generation of the toroidal field emerging during the next cycle. The
actual polar surface field, on the other hand, contains also more
superficial flux which is still topologically connected with the
corresponding preceding-polarity flux on the same
hemisphere. Furthermore, high-latitude flux emergence could possibly
also contribute to the polar field \citep{Durrant:etal:2002}. The
reversal and built-up of the global dipole field relevant for a
flux-transport dynamo is determined by the amount of preceding-polarity
flux that diffuses across the equatorial plane and reconnects with the
opposite-polarity preceding flux on the other hemisphere. This is nicely
illustrated by Figure~10 of \citet{Dikpati:Gilman:2006}. We therefore
consider the amount of flux crossing the equator (per unit time, or
time-integrated) as the potentially most relevant precursor for the
strength of the subsequent cycle in the framework of a Babcock-Leighton
dynamo. It can easily be calculated in surface flux transport
simulations and it also lends itself to observational determination
\citep{Durrant:etal:2004}.
Note that this predictor differs from what is used by DGG, namely,
the low-latitude toroidal flux at the bottom of the convection zone,
which is not present in our model. However, in a linear Babcock-Leighton
type flux transport dynamo, the generated toroidal field is directly
related to the poloidal flux diffusing over the equator. Thus there is a
physical link between the two models, although we do not claim that our
model provides a direct test of the DGG model.
\section{Results of the flux transport model}
\label{sec:results}
We have considered two sets of simulation runs with our flux transport
model. In the first set, the results of which are presented in
Sect.~\ref{subsec:Dikpati}, we have followed as closely as possible the
procedures in the DDG model in order to study whether part
of the predictive skill of their model is already contained in the
surface flux transport. The second set of simulations addresses the
dependence of the predictive skill on the parameters of the flux
transport model as well as on the assumptions and procedures concerning
the emergence latitudes of tilted bipolar regions.
\subsection{Input model according to Dikpati et al.(2006)}
\label{subsec:Dikpati}
The DDG model \citep{Dikpati:etal:2006} is described in some more
detail in \citet{Dikpati:Gilman:2006}. Excluding parameters of the
dynamo model which are of no concern here, the procedures can be
described as follows:
\begin{enumerate}
\item The surface source term for the poloidal magnetic field (more
precisely, for the azimuthal component of the vector potential) is
taken proportional to the sunspot areas given in the combined RGO/SOON
observational record provided by D. Hathaway
(NASA/MSFC)\footnote{http://solarscience.msfc.nasa.gov/greenwch.shtml}.
\item The individual solar cycles are stretched or
compressed in time to fit with a mean cycle length of 10.75 yr.
\item The source term has a Gaussian latitudinal profile with a fixed
full width at half maximum (FWHM) of $6^{\circ}$, migrating equatorward
between $35^{\circ}$ and 5$^{\circ}$ between cycle minima.
\item The magnetic diffusivity in the near-surface layers has a value of
about 300$\,\,\rm km^2\,s^{-1}$.
\item The poleward meridional flow at the surface has the form
$v(\lambda)=v_0\,\sin(2\lambda)$ with $v_0=14.5\,\rm m\,s^{-1}$.
\end{enumerate}
We have adapted our surface transport model as far as possible to these
parameters and procedures. We describe the source term (in our case,
written for $B_r$) by two Gaussians of opposite sign with a FWHM of
6$^{\circ}$, centered at $\lambda_0(t)\pm 0.\dgr5$, where $\lambda_0(t)$
parametrizes the equatorward drift of the flux emergence latitude
between $35^{\circ}$ and $5^{\circ}$ as a linear function of cycle phase. The
amplitude of the Gaussians as a function of time is taken to be
proportional to the monthly averages of the sunspot area from the
RGO/SOON dataset, smoothed by a 6-month boxcar running mean. For the
decay time, $\tau$, we use a value of 5.7 years \citep[roughly
corresponding to a volume diffusivity of about 100$\,\rm km^2\,s^{-1}$,
see][]{Baumann:etal:2006}.
The epochs of the sunspot minima have been taken from the National
Geophysical Data Center
(NOOA/NGDC)\footnote{ftp://ftp.ngdc.noaa.gov/STP/SOLAR\_DATA/SUNSPOT\_NUMBERS/maxmin.new}.
Since cycle 23 is still ongoing, we have considered three possible
values for the epoch of the upcoming sunspot minimum, namely 2007.0,
2007.5 and 2008.0, respectively. Since the RGO/SOON data are available
only until 10/2005, we have substituted the missing data between 11/2005
and the assumed minimum epoch by data for the corresponding phase of
cycle 22.
Figure~\ref{fig:Dikpati_reference} shows the time evolution (for the
period covered by the RGO/SOON data) of various quantities and the
corresponding correlation diagrams. The top panels give the RGO/SOON
sunspot areas and the correlations between the maximum of a cycle with
that of the preceding cycle; obviously, the amplitude of the foregoing
cycle is a poor predictor (correlation coefficient $r=0.47$). The second
panel from the top shows the polar field as resulting from the flux
transport simulation. It peaks typically around sunspot
minimum and largely reflects the level of activity of the {\em ongoing}
cycle, a common result of surface flux transport models
\citep{Schrijver:etal:2002, Wang:etal:2002b, Wang:etal:2005,
Baumann:etal:2006}. Consequently, the predictive power is poor
($r=0.35$).
The third panel shows the amount of magnetic flux diffusing over the
equator per unit time, which we henceforth denote by $\Phi$, for
simplicity. We consider this predictor to be related to the build-up of
the global poloidal field relevant for the dynamo. We see that this
quantity shows a reasonable predictive skill with $r=0.90$. In
particular, the strong drop of activity from cycle 19 and 20 is
correctly anticipated. On the other hand, the prediction for the
relatively high cycle 21 is quite bad. Finally, the bottom panel gives
the dipole component of the simulated surface field, which mainly
results from the accumulated flux diffusing over the equator. It shows a
predictive power that is comparable to that of $\Phi$ ($r=0.83$).
As an alternative to correlating the maxima, we have also considered
time-integrated quantities, namely, cycle-integrated sunspot area and
magnetic flux diffused over the equator as well as polar field and
dipole component of the surface field integrated between polarity
reversals. All integrals have been determined for unstretched cycles,
i.e., keeping the actual cycle lengths. The correlation coefficients of
the integrated sunspot area of the subsequent cycle with the integrated
flux over the equator ($r=0.92$) and the dipole component ($r=0.82$) are
similar to the corresponding values for the maxima. Interestingly, the
correlations coefficients for the integrated polar field and the sunspot
area of the preceding cycle are much larger: 0.79 and 0.78,
respectively. At least part of this increase comes from the better
`prediction' of cycle 20, whose lower amplitude is largely compensated
by a its longer duration in the integrals. This result indicates that
all such correlations should be considered with great caution because
they can be sensitive to the precise definition of the quantities
considered and their real significance is difficult to evaluate.
The results presented in Figure~\ref{fig:Dikpati_reference} show that a
significant part of the predictive skill contained in the DDG model is
covered by surface quantities resulting from a simple flux transport
model, although we do not quite reach such high correlation coefficients
when using their model parameters. The fact that the magnetic flux
diffusing over the equator appears to be a fairly good predictor would
be consistent with its putatively important role played in a
Babcock-Leighton dynamo. What is relevant for the amplitude $\Phi$?
Most importantly, it is the amount of flux emerging near to the equator
in the late phases of a cycle. Since the distance to the equator is
small and since the poleward meridional flow accelerates away from the
equator, low-latitude bipolar magnetic regions contribute most to
$\Phi$. Figure~\ref{fig:blowup} illustrates the importance of
low-latitude emergence during the declining phase of the cycle for the
predictor $\Phi$. Given are the observed sunspot area (solid) and the
flux diffusing over the equator, $\Phi$ (dashed), for cycles 18, 19, and
20. In addition, the dotted lines (with a separate axis on the right
hand side) show the linear progression of the centroid of the emergence
latitude assumed in the model. The maxima of $\Phi$ are shifted by about
3--4 years with respect to the preceding sunspot maxima and occur about
2--3 years before the subsequent minima. Since the $\Phi$ curve
essentially represents a convolution of the sunspot area curve with the
latitude sensitivity of the flux diffusing over the equator (which is
heavily weighted towards the low latitudes), its amplitude is determined
by the sunspot area and the emergence latitude at a given time. The
sunspot area is much larger during the time interval when the emergence
latitude ranges between about $15^{\circ}$ and $5^{\circ}$ in the case of cycle
18 than for cycle 19. Consequently, the prediction for cycle 19 is much
higher than that for cycle 20. This result already indicates that the
predictor could be rather sensitive to the definition of the source
latitudes in the model.
A note concerning the prediction for cycle 24: the results shown in
Figure~\ref{fig:Dikpati_reference} are based on assuming 2007.0 for the
epoch of the upcoming sunspot minimum. This yields a predicted maximum
sunspot number of about 160 for cycle 24. However, a later minimum would
lead to significantly lower predictions: about 130 for a 2007.5 minimum,
and about 110 for a minimum at 2008.0. The first two values are
consistent with the DDG prediction of a cycle 24 amplitude exceeding
that of cycle 23 by 30--50\%, while the third case would predict an
activity similar to that of cycle 23. All these results are affected in
an unknown way by the substitution of data from cycle 22 for unknown
cycle 23 data. In any case, we see that the predictions rather
sensitively depend on the assumed epoch of the upcoming activity minimum
- the significance of this result will become clear when we consider the
origin of the predictive skill in Sect.~\ref{sec:origin}.
\subsection{Modifications of the procedure}
\label{subsec:modifications}
We have seen that the amount of preceding-polarity flux diffusing over
the equator appears to represent a reasonably good predictor for the
next cycle. In this section we will evaluate a) how strongly this result
depends on parameter choices in our flux transport model and b) whether
the predictive skill can be improved by using more of the observational
information contained in the available sunspot data. It turns out that
the correlation coefficients for the maxima of the polar field are
always small, while the results for the dipole component of the surface
field and for the flux crossing the equator, $\Phi$, are similar to each
other in most cases. Therefore, in what follows we restrict ourselves to
showing results for the latter quantity.
\subsubsection*{Dependence on model parameters}
\clearpage
\begin{deluxetable}{cl|c|ccc }
\tabletypesize{\small}
\tablewidth{0pt}
\tablenum{1}
\tablecolumns{6}
\tablecaption{Correlation coefficients
between the predictor, $\Phi$, and the strength of the
following activity cycle \label{tab:ccs}}
\tablehead{
\colhead{\ \ } & \colhead{variable} & \colhead{reference} &
\colhead{$a$} & \colhead{$b$} & \colhead{$c$}}
\startdata
1. & meridional flow velocity, $v_0\,(\,\rm m\,s^{-1})$
& 14.5 & 10. & 20. & \\
& & {\it 0.90} & {\it 0.88} & {\it 0.85} & \\[6pt]
2. & meridional flow profile
& DDG & SB2006 & & \\
& & {\it 0.90} & {\it 0.82} & & \\[6pt]
3. & magnetic diffusivity, $\eta\,(\,\rm km^2\,s^{-1})$
& 300. & 100. & 200. & 600. \\
& & {\it 0.90} & {\it 0.70} & {\it 0.79} & {\it 0.76} \\[6pt]
4. & decay time, $\tau$ (yr)
& 5.6 & 1.7 & 27.8 & \\
& & {\it 0.90} & {\it 0.89} & {\it 0.90} & \\[6pt]
5. & tilt factor in source amplitude
& none & $\sin\lambda$ & & \\
& & {\it 0.90} & {\it 0.88} & & \\[6pt]
6. & Gaussian width, FWHM (deg)
& 6. & 3. & 12. & \\
& & {\it 0.90} & {\it 0.85} & {\it 0.66} & \\[6pt]
7. & Latitude shift of Gaussians (deg)
& 1. & 2. & 3. & 4. \\
& & {\it 0.90} & {\it 0.90} & {\it 0.90} & {\it 0.91} \\[6pt]
8. & Latitude range of emerging flux (deg)
& 5--35 & 1--31 & 5--25 & 5--40 \\
& & {\it 0.90} & {\it 0.80} & {\it 0.74} & {\it 0.82} \\[6pt]
9. & stretching of cycles
& yes & no & & \\
& & {\it 0.90} & {\it 0.91} & & \\[6pt]
\enddata
\tablecomments{The first column indicates the variable or procedure of
the model being varied. The subsequent columns show the results for
the reference case (Sect.~\ref{subsec:Dikpati}) and the variations of
the model. In each case, the upper number gives the value of the
parameter under consideration and the lower number (in italics) the gives
corresponding correlation coefficient.}
\end{deluxetable}
\clearpage
As a first step, we consider the effect of parameter variations in our
model on the correlation coefficient between the maximum of $\Phi$, the
amount of flux diffusing over the equator, and the activity maximum of
the subsequent cycle. The basic procedure for including the emerging
flux as a source in the model is the same as in the reference case
discussed in the preceding section. We have varied the following
parameters:
\begin{enumerate}
\item {\em Peak meridional velocity, $v_0$.}
\item {\em Latitude profile of the meridional flow, $v(\lambda)$.} As an
alternative to the profile used by DDG, we consider a
semi-empirical profile adapted to helioseismic measurements,
$v(\lambda)=1.6\sin(2\lambda)\,\exp[\pi(1-\vert\lambda\vert/90^{\circ})]$,
in units of $\,\rm m\,s^{-1}$ \citep{Schuessler:Baumann:2006}.
\item {\em Turbulent magnetic diffusivity, $\eta$.} The value of $600\,\rm km^2\,s^{-1}$
favored by surface flux-transport models
\citep[e.g.,][]{Durrant:Wilson:2003, Baumann:etal:2004} is a
factor of 2 larger than the surface value considered by DDG. We
have also considered smaller values of $\eta$.
\item {\em Decay time, $\tau$.} The results of \citet{Schrijver:etal:2002}
and \citet{Baumann:etal:2006} favor a value between 5 and 10
years.
\item {\em Tilt angle factor in the source amplitude.} DDG have taken
the amplitude of their source term proportional to the sunspot
area as a proxy of the amount of emerging flux, thus implicitly
assuming that the average tilt angle of the emerging bipolar
regions is independent of latitude. In reality, the average tilt
angle is roughly proportional to $\sin\lambda$
\citep[e.g.,][]{Howard:1991}. This can be included into the model
by multiplying the source term by $\sin\lambda$. The linearity of
the flux transport model permits us to ignore the factor of
proportionality since it represents only a scaling factor that
does not affect the correlation results.
\item {\em FWHM of the Gaussians in the source term.} DDG have assumed a
fixed latitudinal width of $6^{\circ}$ for their source term,
corresponding to a constant width of the activity belt in the
course of the cycle.
\item {\em Latitude shift between the Gaussians.} As long as this shift is
small compared to the width of the Gaussian, the sum of the two
Gaussians (of opposite sign) used in our source term is
proportional to the derivative of the Gaussian with respect to
latitude and the width of the resulting source term is only determined
by the FWHM of the individual Gaussians. The effect of moving away
from this limit can be evaluated by increasing the shift from
$1^{\circ}$ (reference case) to larger values.
\item {\em Range of emergence latitudes.} The procedure of DDG assumes that
the source progresses through a fixed latitude interval in the
course of each cycle. The boundaries of this interval are
adjustable parameters.
\item {\em Stretching of cycles.} In order to keep their dynamo model in phase
with the source term derived from the actual sunspot data, DDG had
to stretch (or compress) the lengths of the individual cycles to a
fixed value of 10.75 years. Our simple model does not require such
stretching, so that the actual cycle length can be used.
\end{enumerate}
Table~\ref{tab:ccs} gives the correlation coefficients between the
maximum of $\Phi$ and the maximum sunspot area of the subsequent cycle
for various parameter changes, following the sequence of the preceding
list. The correlation coefficients are printed in italic numbers in the
second row for each case. As reference we take the case presented in
Sect.~\ref{subsec:Dikpati}, which follows closely the DDG procedures.
We see that the most sensitive parameters are the magnetic diffusivity
and the latitudinal width of the source term, which is not surprising in
view of the importance of these parameters with respect to our
predictor, $\Phi$. The corresponding correlation coefficients based on
the cycle-integrated quantities are very similar to those shown in
Table~\ref{tab:ccs} and thus need not to be discussed any further.
Altogether, it is clear from the table that, as long as we keep the
basic procedures used by DDG, the correlation coefficients are not
strongly dependent on the parameters, with values around $r=0.90$ being
common. We can easily obtain even higher values by performing some fine
tuning: for instance, for $\eta=200\,\rm km^2\,s^{-1}$ and a latitude range of
$5^{\circ}$--$25^{\circ}$ for the source term, we find a correlation coefficient
of 0.95 between the maxima of $\Phi$ and the sunspot area. However, such
a procedure is dubious for obvious reasons and we have not tried to
further `optimize' the values.
\subsubsection*{Dependence on input data set}
The cross-calibration between the RGO and the SOON data to obtain a
consistent data set spanning the whole time period from 1874 on is not
trivial. \citet{Balmaceda:etal:2005} have recently considered an
additional sunspot data set, the ``Russian books'', compiled from data
obtained at observatories in the former USSR. Applying a careful
cross-calibration, they have bridged the RGO and the SOON data by parts
of this data set and thus obtained a consistent record of sunspot
areas. In order to evaluate the dependence of the predictive skill on
the input data set, we have taken these data as input for our flux
transport model. In addition, we also have used the monthly sunspot
numbers as provided by the Solar Influences Data Analysis Center
(SIDC)\footnote{http://sidc.oma.ce} at the Royal Observatory of
Belgium. Figure~\ref{fig:Dikpati_datasets} shows the quantity $\Phi$
obtained with the surface flux transport model using the three data sets
for cycles 12-23. All other parameters are the same as those in
Sect.~\ref{subsec:Dikpati}. As apparent from
Figure~\ref{fig:Dikpati_datasets}, there are no big differences between
the results for the various datasets. In all cases, the drop from cycle
19 to cycle 20 is predicted well while the prediction for cycle 21 is
much too low. The correlation coefficients with the activity maxima of
the next cycle are $r=0.87$ for the dataset of
\citet{Balmaceda:etal:2005} and $r=0.73$ for the sunspot numbers, to be
compared with $r=0.90$ for the reference case. The corresponding values
for cycle-integrated quantities are 0.83, 0.80, and 0.92.
The satisfactory predictive performance of the model with input from
monthly sunspot numbers suggests that we can use this dataset to include
more cycles. The result for cycles 1--23 is shown in
Figure~\ref{fig:Dikpati_sunspot_number}. The correlation between the
maxima of the flux diffusing over the equator and the maxima of the
sunspot number of the subsequent cycle is $r=0.80$. The two curves are
maximally correlated for a forward time shift of the $\Phi$ curve by
6.8~yr.
\subsubsection*{The relevance of the declining phase}
In order to illustrate the importance of the activity level during the
declining phase, we consider as a predictor the average sunspot activity
three years before the minima of the (unstretched) historical
cycles. The result based on the sunspot areas for cycles 12--23 are
shown in Figure~\ref{fig:Precursor_Laura}, while sunspot numbers for
cycles 1--23 are considered in Figure~\ref{fig:Precursor_SSN}. In both
figures, the predictor is shown by the diamond-shaped symbols. In order
to illustrate the predictive skill, we have marked the corresponding
activity level (multiplied by a factor 3) by circles at the times of the
following activity maxima and connected them by dashed lines. The
general trends and the significant drops of activity after cycles 4, 11,
and 19 are reproduced in both cases. The corresponding correlation
coefficients are $r=0.84$ for Figure~\ref{fig:Precursor_Laura} and
$r=0.89$ for Figure~\ref{fig:Precursor_SSN}. The values for using the
activity level 2 years and 4 years before minimum are $r=0.89$ and
$r=0.83$, respectively, for the sunspot numbers of cycles 1--23.
Consequently, the predictive skill of this very simple precursor is
comparable to the corresponding cases using the predictor $\Phi$ from
the flux transport model. Together with the considerations presented in
connection with Figure~\ref{fig:blowup} this indicates that the level of
activity in the declining phase of a cycle is underlying the predictive
skill of our flux transport model when the source term is specified
according to the DGG model.
\subsubsection*{Changes of the source model}
Since the existing sunspot data since 1874 do actually contain the areas
and coordinates of each individual observed sunspot group, one can use
this information to make the source term more realistic. We have done so
in two steps, both based on the cross-calibrated sunspot group data set
provided by \citet{Balmaceda:etal:2005}.
1) In the first step, we change from the schematically prescribed fixed
range and drift rate of the emergence latitudes to the average
observed actual latitudes of the sunspot groups. This is done in the
following way (cf. Figure~\ref{fig:data_emerg_lat}). We average
the latitudes of the individual sunspot groups that appeared within
one month, weighted by group area. Since a sunspot group typically
appears more than once in the data, we consider a group only at the
day of its maximum area in order to avoid multiple counting. When
cycles overlap around solar minimum, sunspot groups near the equator
are attributed to the old cycle while those appearing in higher
latitudes are considered to be part of the new cycle (red and blue
dotted areas in Figure~\ref{fig:data_emerg_lat}). For each cycle,
the resulting monthly averages of the emergence latitudes (green
curves in Figure~\ref{fig:data_emerg_lat}), are fitted to a
parabola (black lines). This procedure leads to a observationally
based time-latitude profiles of flux emergence for each cycle, also
allowing for the overlapping of cycles. Inspection of these parabolic
profiles in Figure~\ref{fig:data_emerg_lat} shows that, for most
cycles, they deviate strongly from the linear profiles between
$35\deg$ and $5\deg$ assumed in the reference case. In fact, they are
better represented by a linear profile between $25\deg$ and $5\deg$.
We have run the flux transport model with these emergence profiles,
considering separately the sunspot areas for each cycle in the source
amplitudes, so that overlapping cycles are properly accounted for.
Also, the tilt-angle factor $\sin\lambda$ has been included in the
source amplitudes. All other parameters are as in the reference case
discussed in
Sect.~\ref{subsec:Dikpati}. Figure~\ref{fig:predict_emerg_lat} shows
the resulting time evolution of the predictor $\Phi$ together with
the observed (total) sunspot areas. Although the general trend of the
cycle amplitudes is roughly reproduced, the value of $r=0.75$ for the
correlation coefficient between the maxima shows that the predictive
skill is significantly diminished with respect to the reference case.
If we avoid the parabolic fit and directly use the monthly weighted
averages (green curve in Figure~\ref{fig:data_emerg_lat}), the
correlation coefficient drops to $r=0.43$. Likewise, omitting the
monthly averages of the emergence latitudes and using a area-weighted
direct parabolic fit through the emergence latitudes of the
individual sunspot groups (red and blue dots in
Figure~\ref{fig:data_emerg_lat}), yields $r=0.45$. In both cases, the
predictive skill is almost completely lost.
This leaves us with the surprising result that the predictive skill
is strongly dimished or even almost lost when more detailed
observational data are used for prescribing the source in the flux
transport model. Moreover, the skill depends critically on the way
that the emergence latitudes of the individual sunspot groups are
averaged. These results become less puzzling when we consider in
Figure~\ref{fig:data_emerg_lat} the broad distribution of the actual
emergence latitudes for any given time: since the contribution to the
predictor $\Phi$ depends sensitively on the emergence latitude, in the
sense that near-equator emergence is much more strongly weighted,
variations in the averaging procedure that lead to small changes
in the low-latitude part of the averaged emergence latitudes used in
the flux-transport model can have a strong impact on the predictor
quantity, $\Phi$.
2) We can avoid the sensitivity of the results with respect to the
averaging procedure of the emergence latitude by directly considering
the contribution to the source term of each individual sunspot group,
so that no averaging is required. Each sunspot group in the data is
identified with a bipolar magnetic region that is introduced into the
flux transport simulation at its recorded latitude. The magnetic flux
content of the region is
assumed to be proportional to its maximum sunspot area. The
orientation of the magnetic polarities follows Hale's laws and
thus alternates from cycle to cycle. Taking the
latitude separation of the two polarities of the region to be small
compared to their diameter, which itself is assumed to be
proportional to the square root of the sunspot area, we choose for
the FWHM of the two Gaussians describing the radial field source
\begin{equation}
{\rm FWHM}= \left( {A_{\rm s}}\over{A_{\rm s,max}} \right) ^{1/2}
\times 5 ^{\circ}\,,
\label{eq:FWHM}
\end{equation}
where $A_{\rm s}$ is the area of the sunspot group and $A_{\rm s,max}$
the value for the largest sunspot groups in the dataset. For these
groups, we assume a latitude extension of $5\degr$, roughly
corresponding to half the size of the largest groups. It turns out that
the the results are rather insensitive to the exact value of this
parameter (see below). To account for the latitude dependence of the
tilt angle \citep{Howard:1991}, the amplitude of the corresponding
source term is multiplied by a factor $\sin\lambda$.
Figure~\ref{fig:actual_emerg_lat} shows the prediction results. It is
obvious that the predictor, $\Phi$, now mainly reflects the strength of
the ongoing cycle and thus provides very low predictive
capability. Accordingly, the correlation coefficient between the maxima
of $\Phi$ and the sunspot area of the subsequent cycle is low: $r=0.33$
(the corresponding values for maximum latitude extensions of $10\deg$
and $2.5\deg$ are $r=0.27$ and $r=0.34$, respectively. The origin of
this result is in the high sensitivity of $\Phi$ to bipolar regions
emerging in low latitudes (cf. Figure~\ref{fig:blowup}). Although the
tilt angle goes to zero, $\Phi$ is still dominated by these low-latitude
emergences. Since we have a broad distribution of emergence latitudes at
any given time, there is nearly always some flux emergence in low
latitudes (cf. Figure~\ref{fig:data_emerg_lat}. Because the amount of
this low-latitude flux emergence is mainly determined by the overall
strength of the {\it ongoing} cycle, the quantity $\Phi$ is no longer
dominated by the late phase as in the reference case, which did not take
into account the broad latitude range of emerging flux. Consequently,
the predictor mainly reflects the ongoing cycle, so that the predictive
skill of the model is a nearly completely lost. Note that this version
of the model makes the most direct use of the actual data.
A result pointing in this direction appears already in
our parameter study (see Table~\ref{tab:ccs}): when doubling the
latitudinal width of the source term (to 12 degrees FWHM of the
corresponding Gaussian), the correlation coefficient dropped from 0.90
to 0.66. When we impose a narrow latitudinal extension of the source
term, low-latitude emergence occurs exclusively in the declining phase
of a solar cycle, i.e., during a few years before sunspot minimum. The
more activity (flux emergence) during the this phase, the more flux
diffuses over the equator, and the higher is the amplitude of our
predictor (cf. Figure~\ref{fig:blowup}).
\section{What is the origin of the predictive skill?}
\label{sec:origin}
We have seen that our predictor $\Phi$ determined with the flux
transport model provides reasonable predictive skill in the case of a
narrow latitudinal width and a prescribed fixed latitude range of the
source term and we have shown that this is related to the activity level
in the declining phase of the cycle. However, the predictive skill of
$\Phi$ is strongly dimished or even nearly completely lost when averages
of the actually observed observed emergence latitudes are considered or
if the individual emergence latitudes of sunspot groups are used
directly. This result casts serious doubt upon connecting the predictive
skill in the standard case with the Babcock-Leighton dynamo scheme and
with the dipole strength of the surface field. But how else can we
understand the unquestionable correlation between the activity level in
the declining phase of a cycle and the strength of the next cycle?
We suggest that in fact no direct physical link between the surface
manifestations of the old and the new cycle is required to understand
the predictive skill of the activity level in the declining phase. All
that is needed are two well-established properties of the sunspot
record: 1) the temporal overlapping of cycles with high-latitude spots
of the new cycle already appearing when the old cycle is still in
progression in low latitudes \citep[e.g.,][]{Harvey:1992a} and 2) the
relation between the rise time of a cycle towards its maximum and its
strength: stronger cycles rise faster towards sunspot maximum
\citep[known as the Waldmeier effect, e.g.][]{Waldmeier:1935}. When
considering latitude-integrated quantities like the sunspot number or
area, the combined effect of both properties leads to a systematic shift
of the minimum epochs between two cycles of different strength: the
minimum occurs earlier if the following cycle is stronger than the
preceding cycle and later for a weaker following cycle. This effect
explains the empirical statistical relationship between cycle length and
amplitude: strong cycles tend to be preceded by short cycles
\citep[e.g.,][]{Hathaway:etal:1999, Hathaway:etal:2002,
Solanki:etal:2002b}.
Figure~\ref{fig:idea} illustrates how the overlapping of cycles with
amplitude-dependent rise time leads to a predictive skill of the
activity level during the declining phase of the preceding cycle. In
this figure we have considered time profiles of the activity cycles
according to a prescribed functional form
\citep{Li:1999,Hathaway:etal:1994}, which reproduces both the rise and
decay parts of a cycle, including the Waldmeier effect:
\begin{equation}
f(t) = \frac{a(t-t_0)^3}{e^{(t-t_0)/b}-c}\,,
\label{eq:Li_f}
\end{equation}
where $t_0$ denotes the starting time of the cycle. The parameter
$b=1.128\,$yr and the relation between $c$ and $a$
of the form
\begin{equation}
c = -1.104\,10^{-4}\,a^2 + 0.24666\,a - 123.593
\label{eq:Li_c}
\end{equation}
have been determined by \citet{Li:1999} on the basis of sunspot area
data since 1876. By fitting the two parameters, $t_0$ and $a$, the
cycles 12--22 are well reproduced by the functional form given by
equation (\ref{eq:Li_f}). The left panel of Figure~\ref{fig:idea} shows
the summed activity levels of a cycle followed by a stronger cycle
(solid curve) or by a weaker cycle (dashed curve), respectively, the
follower cycle starting 11 years after the first cycle in both cases.
The individual cycles profiles are indicated by the dotted lines. The
faster rise of the stronger following cycle leads to a earlier `sunspot
minimum' than in the case of a weaker follower (marked in the figure by
`M1' and `M2', respectively), the time shift being about one year.
Since observationally a sunspot cycle is defined as the time between
adjacent minima, the activity in the declining phase of the first cycle,
(i.e., in a fixed time interval relative to the respective solar minimum
epoch) is considerably larger when the follower cycle is stronger than
when it it weaker \citep{Hathaway:etal:1999}. In our example, for
instance, the activity level three years before the respective minimum
is about a factor of two larger for the stronger following cycle than
for the weaker follower (marked in the figure by `P1' and `P2'),
corresponding to the relative strength of the two follower
cycles. Consequently, the activity level at time P1/P2 can be used as a
predictor for the amplitude of the following cycles, but it can only be
determined after the epoch of the minimum is known; before, only upper
limits can be given.
The left panel of Fig.~\ref{fig:idea} shows that the above defined
predictor P1/P2 (activity level three years before cycle minimum) is
only weakly affected by the strength of the first cycle. This results
from the fact that the declining phase of a cycle largely represents an
exponential decay, irrespective of the cycle strength.
Fig.~\ref{fig:idea} also suggests that the shift of sunspot minima
depending on the amplitudes of overlapping cycles leads to a correlation
of the activity level at sunspot minimum with the amplitude of the
following cycle. In fact, \citet{Hathaway:etal:2002} find a correlation
coefficient of $r=0.72$ between these quantities for the historical
sunspot record.
As a consequence, simply two well-known properties of the solar cycles,
the Waldmeier effect and the overlapping, explain the predictive skill
of the activity level in the declining phase of a cycle. This explanation also
carries over to the skill of the our predictor from the flux transport
model, the amount of flux diffusing over the equator: assuming for the
source term a linear profile of emergence latitude versus time between
two sunspot minima and an amplitude proportional to the instantaneous
activity level, leads to a stronger source in low latitudes (declining
phase of the cycle) for an earlier minimum, which, in turn, results from
a stronger following cycle. Through the dependence of the sunspot
minimum shift on the amplitude of the next cycle,
information about the strength of the following cycle propagates into
our predictor, completely independent of whether there is a physical
connection between surface manifestations of the cycles or not.
No further `memory' of the system is required.
We demonstrate the effect of the minimum shift on the prediction with a
precursor method by considering series of overlapping synthetic cycles
with profiles according to equation~(\ref{eq:Li_f}). The cycle
amplitudes form a random sequence with a flat probability density
between 500 and 3000 microhemispheres for the maximum sunspot area, so
that the strengths of subsequent cycles are completely independent of
each other. The cycles start at regular intervals of $10.75\,$yr;
because of their overlapping, the cycle length defined by the time
between subsequent activity minima varies. As an example,
Figure~\ref{fig:synthetic} shows the analog to
Figure~\ref{fig:Precursor_SSN} for one realization of 23 random
synthetic cycles. Statistics of this synthetic series, such as the
correlation of the minima with the amplitudes of the subsequent cycle
($r=0.60$) and the correlation between the length of a cycle and the
amplitude of the next cycle ($r=-0.82$) are roughly consistent with the
properties of the actual sunspot record \citep[exhibiting values of 0.72
and $-0.69$, respectively, see][]{Hathaway:etal:1999}, which is
sufficient for the purpose of illustration. Using the activity level 3
years before the activity minima as predictor, we find about the same
skill for the synthetic random series (correlation coefficient $r=0.84$)
as for the actual data ($r=0.89$, see Figure~\ref{fig:Precursor_SSN}).
This demonstrates that the amplitude-dependent overlapping of cycles
affects the timing of the activity minima in such a way that information
from cycle $n+1$ can be picked up by considering the activity in the
declining phase of cycle $n$, {\em but only after the epoch of the
minimum is known}. As we have seen, this effect allows us to `predict' a
random sequence of cycle amplitudes without any relation between
subsequent cycles apart from overlapping.
For 1000 random series of 8 cycles each, Figure~\ref{fig:synthetic_ccs}
shows the cumulative probability distribution for the correlation
coefficient between the activity level three years before minimum and
the subsequent maximum. The median value corresponds to $r=0.83$, i.e.,
in 50\% of the cases the correlation coefficient is larger than this
value; it is larger than 0.95 in 5\% of the cases. These results with
random sequences of cycle amplitudes indicate that correlation
coefficients in the range of 0.8--0.9, which are regularly found with
precursor methods, may be easily be explained by the minimum shift
effect alone and require no further physical connection of the surface
quantities between subsequent cycles. This does not mean that such a
connection may not exist, only that correlations of that size do not
compel us to assume such a connection.
\section{Discussion and conclusions}
\label{sec:discussion}
The DDG model is able to reproduce the amplitudes of cycles 16--23 with
impressively large correlation coefficients (up to $r\simeq 0.99$) and
we have seen that our very much simpler model exhibits somewhat less (up
to $r\simeq 0.95$), but still considerable predictive skill, provided
that we closely follow the DDG treatment of the surface source term. In
fact, for a Babcock-Leighton type flux-transport dynamo, our predictor,
the flux diffusing over the equator determines the strength of the
reversed global dipole field from which differential rotation generates
the toroidal flux for the next activity cycle. Therefore, it is tempting
to consider these correlations as indicating that such a type of dynamo
model in fact represents the engine underlying the solar activity cycle.
However, at least in the case of our model, the predictive skill is
nearly completely lost when we use more of the available observational
information about flux emergence in the photosphere, particularly
concerning the emergence latitudes of bipolar magnetic regions. If
we allow for the overlapping of cycles and use the full
information provided by the butterfly diagram, the predictor largely
reflects the strength of the ongoing cycle and exhibits almost no
relation to the amplitude of the next cycle.
We have shown that this puzzling result, i.e., the schematic source
providing predictive skill while the source based on actual data showing
none, can be understood by the strong dependence of the schematic source
on the epochs of the sunspot minima. The overlapping of cycles whose
rise time is anticorrelated their amplitude (the Waldmeier effect)
naturally leads to a time shift of the minima that is strongly related
to the strength of the following cycle, thus affecting the strength of
the schematic source for low-latitude emergence in the declining phase of a
cycle. We have demonstrated the importance of this effect by showing
that such propagation of amplitude information from the next cycle into
the minimum epochs allows us to `predict' synthetic cycles with random
amplitudes with about the same skill as for the actual sequence of
sunspot cycles.
The amplitude-dependent minimum shift explains the results obtained with
our flux transport model. Moreover, it could account for the (partial)
success of a number of precursor methods \citep[for a summary,
see][]{Hathaway:etal:1999}. This includes the correlation of the sunspot
maximum with the activity level during the preceding minimum and the
anticorrelation with the length of the preceding cycle
\citep{Wilson:etal:1998,Solanki:etal:2002b}, which directly result from
the amplitude-related shift of the minimum epoch (see Figure
\ref{fig:idea}). Likewise, the anticorrelation of the skewness of the
cycle profile with the amplitude of the subsequent cycle
\citep{Lantos:2006} can be understood by the minimum shift: a weak
follower cycle is preceded by a late minimum, so that the foregoing
cycle has a longer decay time and thus becomes more asymmetric, and vice
versa.
The amplitude-dependent minimum shift also explains the predictive skill
of some geomagnetic precursors, such as the minimum level of the {\em
aa} index \citep{Ohl:1966}. Other precursor methods are not so easily
reduced to the minimum effect. For instance, when considering cycles
12-22 in retrospect, the method of \citet{Thompson:1993} yields a
coefficient of 0.97 between the number geomagnetically disturbed days
($Ap>25$) during cycle $n$ with the sum of the sunspot number maxima of
cycle $n$ and $n+1$ \citep{Hathaway:etal:1999}. For cycle 23, however,
this method predicted a maximum (yearly) sunspot number of about 160,
more than 30\% too large. It is therefore unclear whether this method
has a real physical basis or if the good correlation in the past is just
fortuitous, the method being just one of a large number of other
possibilities and happened to provide a large correlation coefficient
for the existing data. Some methods depending on a ad-hoc recipes for
the selection or division of geomagnetic data during the declining phase
of the sunspot cycle into an `activity-related' and an `interplanetary'
component \citep{Legrand:Simon:1981,Feynman:1982} have also been shown
to provide high correlations for past cycles
\citep[e.g.,][]{Hathaway:Wilson:2006}. They may capture early
manifestations of the (extended) new cycle in middle to high latitudes,
for instance, in the form of ephemeral active regions
\citep{Harvey:1992a}, but the physical connection remains to be
understood. In any case, it appears to be the early influence of the
{\em new} cycle which makes a prediction possible.
In summary, we have shown that the predictive skill of many precursor
methods and that of our simple flux transport model with a schematic
source can be reduced to the time shift of solar minima depending on the
strength of the next cycle. Therefore, information about the next cycle
is available (in statistical sense, of course) at the time when the
solar minimum is clearly identified -- not always an easy and
straightforward task, see \citet{Harvey:White:1999}. Concerning our
physical understanding of the solar cycle and the dynamo mechanism, the
crucial point is that the predictive skill of such precursors does
neither require nor imply a physical relation between surface
manifestations of subsequent cycles -- we have shown that such methods
can be applied to for random sequences of cycles with fully independent
amplitudes.
In any case, one should not be misled by high correlation coefficients
for reproducing the past because most methods can be adjusted once the
actual numbers are known. At best, precursor methods could indicate a
trend for the next cycle, but we cannot expect to fully capture the
intrinsic variability of the solar dynamo process with such simple
recipes. Our work gives no indication that flux-transport models of the
surface field could improve the predictions, quite the contrary: our
model shows no predictive skill when the actually observed emergence
latitudes are used. There is no reason to assume that a more
sophisticated 2D surface flux transport model would lead to a different
result. The predictive skill arising in the case of a schematic
prescription of the emergence latitudes results from the shift of the
minimum epoch depending on the strength of the {\em next} cycle, i.e.,
on information from the cycle to be predicted. We can tune the
parameters of the model to obtain correlation coefficients with past
cycles of $r\simeq 0.95$, but much simpler precursors (like the
activity level 3 years before minimum) also reach values around 0.90.
While lacking the sub-surface transport and the generation of the
toroidal field, our flux-transport approach is conceptually similar to
the model of \citet{Dikpati:etal:2006} and
\citep{Dikpati:Gilman:2006}. We conjecture that the amplitude-dependent
minimum shift should have an impact on the predictive skill of their
model as well. This could be easily tested by replacing the schematic
latitude dependence of the source by the actual emergence latitudes for
each sunspot group, thus avoiding the minimum shift effect.
\acknowledgments {Extended discussions with Mausumi Dikpati and Peter
Gilman about the DGG model are gratefully acknowledged. Laura Balmaceda
kindly put to our disposal her cross-calibrated dataset of sunspot
group areas. Helpful comments by an anonymous referee led to a
substantial improvement of the presentation in various parts of the
paper.}
|
2,869,038,155,656 | arxiv | \section{Introduction}
\label{sec1}
In preceding papers of ours (see mainly \cite{conc1,adv}) the arguments have been proved leading to the conclusion that the hidden-variable (HV) theory should be preferred to the Copenhagen quantum mechanics. The results based on these arguments have been presented to the conference held in Trieste (January 2008) and published in corresponding Proceedings \cite{ffp9} (similar results being contained also in Ref. \cite{hradec}). In the present paper the more systematic explanation of the whole approach will be provided.
The new results may be shortly summarized: Two different (Copenhagen and ensemble) interpretations of quantum-mechanical model discussed earlier have represented in fact two divers theories differing in important assumptions. The ensemble alternative being identical practically with mere Schr\"{o}dinger equation (and equivalent to hidden-variable theory) refuses in principle physically interpreted superposition principle and requires to extend corresponding Hilbert space to be in full harmony with time-dependent solutions of the equation proposed by Schr\"{o}dinger \cite{schr}. All previous criticisms (Pauli, Susskind and Glogover, Einstein) often discussed have been removed in such a case. And no logical contradictions and paradoxes are involved more in the description of matter reality. It has been also proved that the basic solutions (represented always by one Hamilonian eigenfunction) of Schr\"{o}dinger equation lead to the same results as classical physics. However, some classically possible states do not seem to correspond to basic Schr\"{o}dinger solutions, which concerns the existence of discrete states. It is also possible to say that the microscopic physics may provide now the same ontological picture as it is known from classical physics.
Main assumptions the Copenhagen quantum mechanics has been based on will be introduced in Sec. \ref{sec2} and the difference between two earlier often discussed interpretation alternatives will be explained. Critical comments of Pauli \cite{pauli} and others discussed during the second half of the 20th century will be explained in Sec. \ref{sec3}. The necessity of extended Hilbert space and time irreversibility will be then handled in Sec. \ref{sec4}. The story of EPR experiment will be described and explained in Sec. \ref{sec5}. And in Sec. \ref{sec6} the physical meanings of different limits of Bell's operator will be shown. The possibility of applying the HV (hidden-variable) theory to whole physical reality will be discussed in Sec. \ref{sec7}. The results and consequences of the experiment with three polarizers will be presented in Sec. \ref{sec8}.
The actual relation between the Schr\"{o}dinger equation and classical physics will be then shown in Sec. \ref{sec9}. The structure of Hilbert space corresponding to the HV theory will be discussed in Sec. \ref{sec10}.
Some concluding remarks will be introduced in Sec. \ref{sec11}.
\section { Copenhagen quantum mechanics and basic assumptions }
\label{sec2}
To understand the whole problem it is necessary to start with main assumptions on which the Copenhagen quantum mechanics is based. Leaving aside some technical assumptions it is possible to introduce four following ones:
- first of all it is the validity of time-dependent Schr\"{o}dinger equation \cite{schr}
\begin{equation}
i\hbar\frac{\partial}{\partial t}\psi(x,t)=H\psi(x,t), \;\;\;\;
H=-\frac{\hbar^2}{2m}\triangle + V(x) \label{schr}
\end{equation}
where Hamiltonian $H$ represents the total (kinetic and potential) energy of a given physical system and $x$ represents the coordinates of all matter objects;
- the evolution of a physical system is described by function $\psi(x,t)$ and all physical quantities may be expressed as the expected values of corresponding operators:
\begin{equation}
A(t) = \int\psi^*(x,t)A_{op}\psi(x,t)dx
\end{equation}
where $A_{op}$ and functions $\psi(x,t)$ may be represented by operators and vectors in a suitable Hilbert space;
- in the case of Copenhagen quantum mechanics it has been required for the corresponding Hilbert space to be spanned on one set of Hamiltonian eigenfunctions $\psi_E(x)$:
\begin{equation}
H\psi_E(x) = E\psi_E(x);
\end{equation}
- and in addition to, the mathematical superposition principle valid in any Hilbert space has been interpreted in physical sense, i.e., any superposition of two vectors has represented again another pure (basic) physical state.
It has been spoken often about two different interpretation alternatives of the quantum-mechanical mathematical model: orthodox (or Copenhagen) and statistical (or ensemble). However, it has been never sufficiently stressed that both these alternatives have not concerned identical model and have corresponded to the different sets of basic assumptions. While the Copenhagen alternative has involved all four preceding assumptions the ensemble alternative (denoted usually also as HV theory) has corresponded to the first two assumptions only (being equivalent in principle to the mere Schr\"{o}dinger equation). And it is necessary to speak about two different theories, differing significantly in their properties as well as in assumptions.
The Copenhagen alternative has been often denoted as supported by different experimental data. However, in all such cases only the assumption of the mere Schr\"{o}dinger equation (i.e., of the HV theory) has been tested; without the last two assumptions (forming Copenhagen alternative) having been actually involved. All four assumptions have been involved practically only in interpreting the EPR experiment, which will be discussed in Sec. \ref{sec5}.
As to the HV theory the Hilbert space must be chosen according to corresponding physical system; it must be always extended (at least doubled) in comparison to the Hilbert space required by the third assumption. In such a suitably extended Hilbert space the critical comments of Pauli and also of Susskind and Glogover are not more valid (see the next section).
\section { Critical comments of Pauli and others }
\label{sec3}
The Copenhagen quantum mechanics was accepted as valid by physical community even if it exhibited some logical and ontological paradoxes and some critical comments were brought against its full regularity.
Already in 1933 Pauli \cite{pauli} showed that under the validity of all assumptions introduced in Sec. \ref{sec2} it was necessary for the corresponding Hamiltonian to possess continuous energy spectrum from $-\infty$ to $+\infty$, which disagreed with the fact that the energy was defined as positive quantity, or at least limited always from below.
In 1964 Susskind and Glogover \cite{suss} showed then that exponential phase operator was not unitary, which indicated that the given Hilbert space was not fully complete to represent a corresponding physical system quite regularly. Many attempts have been done to solve these deficiencies during the 20th century. The reason of having been unsuccessful may be seen in the fact that practically in all cases both the shortages were regarded and being solved as one common problem.
The corresponding solution seems to have been formulated only recently (see Refs. \cite{kulo1,kulo2}) when it has been shown that it is necessary to remove two mentioned shortages one after the other. As to the simple system of two free colliding particles the criticism of Pauli may be removed if the Hilbert space required by the third assumption has been doubled as proposed by Lax and Phillips in 1967 (see \cite{lax1,lax2}); it consists then at least of two mutually orthogonal subspaces ($\mathcal{H} = \Delta^- \oplus \Delta^+$); each of them being spanned on one basis of Hamiltonian eigenfunctions (see also Sec. \ref{sec4}).
As to the non-unitarity of exponential phase operator it is necessary to mention that this operator was discussed for the first time by Dirac \cite{dirac} in the case of linear harmonic oscillator (i.e., $V(x)=kq^2$). Annihilation and creation operators
\vspace{-4mm}
\begin{equation}
a \;=\; p - im\omega q, \;\;\; a^{\dag} \;=\; p+im\omega q,
\;\;\;\;\: \omega=\sqrt{\frac{k}{m}}
\end{equation}
\vspace{-1mm}
were introduced, fulfilling the relations
\begin{equation}
[H,a] \;=\; -\omega a\,, \;\;\; [H, a^\dag] \;=\; \omega a^{\dag}\,.
\end{equation}
It was then possible to define operator
\begin{equation}
\mathcal{E}\,=\, (a a^\dag+1)^{1/2}a\:,
\;\;\; \mathcal{E}^\dag \;=\; a^\dag (aa^\dag +1)^{1/2})
\end{equation}
fulfilling the relations
\begin{equation}
[H,\mathcal{E}] \;=\; -\omega \mathcal{E}\;, \;\;\;
[H,\mathcal{E}^\dag] \;=\; +\omega \mathcal{E}^\dag
\end{equation}
and representing exponential phase operator defined as
\begin{equation}
\mathcal{E} \;=\;e^{-i\omega\Phi} \label{exph}
\vspace{-2mm}
\end{equation}
where $\Phi$ is the phase.
And it was shown later by Susskind and Glogower \cite{suss} that in the Hilbert space corresponding to the third assumption the operator $\mathcal{E}$ was not unitary, but only isometric, as it held
$\;\mathcal{E}^\dag\mathcal{E}\:u_{1/2} \,\equiv\, 0$. The unitarity condition has not been fulfilled for the state vector corresponding to the minimum-energy (or vacuum) state.
The Hilbert space (extended to solve Pauli's problem) should be further doubled and formed by combining two mutually orthogonal subspaces corresponding to systems with opposite angular momentums.
They should be bound together by the added action of exponential phase operator, linking together the corresponding vacuum states as it was proposed already by Fain \cite{fajn}; see also \cite{kulo1,kulo2}.
In any case, the given non-unitarity of exponential phase operator has represented always a less substantial problem than the criticism of Pauli, as it has concerned the completeness of the Hilbert space and not any actual discrepancy.
Very important criticism was delivered, however, by Einstein and collaborators \cite{einst} who proposed the so called EPR Gedankenexperiment to argue that some action at macroscopic distance between microscopic particles should exist in the Copenhagen model. The EPR problem has been repeatedly discussed, which has been accompanied by some important mistakes leading to false conclusions The whole story and contemporary solution of the given problem will be described in Sec. \ref{sec5}.
However, before ending the present section another criticism should be yet added. It concerns the existence of discrete states in Schr\"{o}dinger equation when the two last assumptions are added. It is evident that in such a case all mathematical superpositions should represent mutually equivalent physical states and practically no quantized (discrete) states should exist in experimental reality. The given problem has been removed in the HV theory as only eigenstates belonging to Hamiltonian eigenvalues represent now "pure" states and any superposition of theirs represents statistical mixture.
It means that the structure of the Hilbert space in the HV theory differs significantly from that required in the case of the Copenhagen quantum mechanics. It consists always of mutually orthogonal subspaces in the dependence on the type of a corresponding physical system (see Sec. \ref{sec4} and eventually also Sec. \ref{sec10}). The time flow is then described as irreversible, which will be demonstrated on the physical situation corresponding to the evolution of two-particle system being described in the next section.
\section { Hidden-variable theory and irreversible time flow }
\label{sec4}
The HV theory differs from the Copenhagen quantum mechanics also in one further important point; the time evolution is not more reversible, which follows simply from the Schr\"{o}dinger equation and the extended Hilbert space.
Let us suppose now that the given physical system consists of two particles. It may be described in its center mass system (CMS) with the help of Hamiltonian
\vspace{-2mm}
\begin{equation}
H \;=\; \frac{p^2}{2m} \;+\; V(x) \label{ham}
\end{equation}
where $m$ is reduced mass, $p$ - momentum of one particle, $V(x)$ is mutual potential between the given particle pair and $x$ represents the positions (mutual distance) of particles. The Hilbert space $\mathcal H$ corresponding to the HV theory consists then of two subspaces:
\begin{equation}
\mathcal{H} \;\equiv\; \{\Delta^-,\Delta^+\}
\end{equation}
that are mutually related by evolution operator
\begin{equation}
U(t) \;=\; e^{-iHt} \;\; (t \ge 0).
\end{equation}
It holds, e.g., (see \cite{lax1,lax2})
\begin{equation}
\mathcal{H} \;=\; \overline{\Sigma_t U(t)\Delta^-}
\;=\;\overline{\Sigma_t U(-t)\Delta^+}.
\end{equation}
Individual subspaces $\Delta^-$ and $\Delta^+$ are spanned on one set of
Hamiltonian eigenfunctions in usual way. However, the superposition principle cannot be applied to in the physical sense. The basic physical states are represented by Hamiltonian eigenstates only; their superpositions represent statistical combinations of basic states. Any $t$-dependent
function $\psi(x,t)$ obtained by solving Schr\"{o}dinger equation may be then
represented by a trajectory characterized by given initial
conditions.
In case of continuous Hamiltonian spectrum (two free particles) any point on such a trajectory may be characterized by expectation values of the operator
\vspace{-2mm}
$$ R\;=\;\frac{1}{2}\{{\bf p}.{\bf q}\}, \;\;\; \langle i[H,R]\rangle >0 $$
where ${\bf q}$ and ${\bf p}$ are coordinates and momentum
components of one particle in CMS. The states belonging to
$\Delta^-$ are incoming states, and those of $\Delta^+$ - outgoing
states (independently of chosen coordinate system). The evolution goes always in one direction from "in"-subspace to "out"-subspace (for more details see \cite{lok98}).
As these two different kinds of states represent quite different experimental situations it is necessary to separate "in" and "out" states into
two mutually orthogonal subspaces. It is then also possible to join an additional orthogonal subspace that might represent corresponding resonance formed in a particle collision (or an unstable particle decaying into the given particle pair), i.e.
\begin{equation}
\mathcal{H} \;\equiv\; \{\Delta^-\oplus\Theta\oplus\Delta^+\}\; ;
\end{equation}
see also Ref. \cite{alda} where the corresponding extended Hilbert space was derived independently as the consequence of exact exponential decay law.
It is only necessary to define the action of evolution operator
between $\Theta$ and other subspaces in agreement with evolution
defined already in individual $\Delta^\pm$-subspaces.
The evolution goes in one direction, at least from the global point of
view; evolution processes between internal states of $\Theta$ may be
rather chaotic. However, global trajectories tend always in
one direction; see the scheme in Fig. 1.
\vspace{5mm}
\begin{figure}[htb]
\begin{center}
\hspace*{-0.1cm} $\Delta^{(-)}\; \hspace{1.54cm} |
\hspace{0.8cm} \;\Delta^{(+)}$ \\
\hspace*{-0.1cm} { $\langle R\rangle <0 \hspace{1.33cm} |
\hspace{0.8cm}\langle R\rangle >0$ \\
\hspace*{0.1cm} { "in" } \hspace{0.1cm} $\searrow
\hspace{0.4cm} \longrightarrow |
\hspace{0.8cm} { "out" } \longrightarrow$ \\
\hspace*{.05cm} $ \_\_\_\_\_\_\_ \_\_\_\_\_\_\_\_\_\_\_\_\_\_|
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ $ \\
\hspace*{0.7cm}$| \hspace{1.6cm} \nearrow \hspace{1.15cm} | $ \\
\hspace*{0.64cm}$| \hspace{1.4cm} { \Theta}\hspace{1.7cm}| $ \\
\hspace*{0.8cm}$|\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_| $ }
\caption { \it { Scheme of the Hilbert space (for a two-particle system)
extended according to original proposal of Lax and Phillips; three mutually orthogonal subspaces and irreversible time evolution. } }
\end{center}
\end{figure}
The subspace $\Theta$ may represent a resonance (or generally an unstable object) decaying into the given particle pair. The unstable object may be, of course, formed also in the decays of heavier unstable objects. In such a case the decay process may be represented always as the transition from $\Theta$ to $\Delta^+$.
In the case of discrete Hamiltonian spectrum (e.g., harmonic
oscillator) the wave function has similar irreversible t-dependent form.
However, the evolution is periodical as a rule. The evolution may be again characterized by trajectories
corresponding to different initial conditions.
The corresponding Hilbert space consists then of infinite number of identical pairs of mutually orthogonal subspaces belonging subsequently to two intervals of phase: $(k.\pi, (k+1).\pi)$ and $((k+1).\pi,(k+2).\pi)$; individual states being characterized by expectation values of phase $\Phi$ lying in principle in the interval from $-\infty$ to $+\infty$ (see also \cite{lok98}).
Such an extended model enables to represent the time evolution described by Schr\"{o}dinger equation in the Hilbert space in full agreement with actual behavior of physical systems. It removes fully the criticism of Pauli; however, it does not give any answer to the criticism of Susskind and Glogover. To this goal some other extension of Hilbert space would be necessary as introduced in Sec. \ref{sec3}.
As to the irreversibility of time evolution it corresponds to the behavior of real objects in microscopic world similarly as it has been in macroscopic one. There is not practically any greater physical gap between these two regions.
\section{ EPR experiment and HV theory}
\label{sec5}
In the early years of quantum mechanics only the Copenhagen alternative was taken as valid since the physical community was influenced strongly by the "proof" of von Neumann \cite{vonne} that any other (hidden) parameters were excluded by the standard quantum-mechanical mathematical model and that the given quantum-mechanical model represented complete description of microscopic world. It remained without any attention that already in 1935 Grete Herrmann \cite{gher} showed that the approach of von Neumann was practically a "circle proof".
It was undoubtedly the main reason why also the criticism of Einstein and his collaborators \cite{einst} was refused. Einstein proposed the known EPR Gedankenexperiment and argued in 1935 that the quantum mechanics was not a complete theory to describe fully a physical system.
Bohr \cite{bohr} opposed strongly; having stated that the Copenhagen model corresponded fully to microscopic reality. And physical community accepted practically his standpoint.
The argument that a hidden variable was contained already in Schr\"{o}dinger equation was published by D. Bohm \cite{bohm} in 1952; however, it was accepted seriously by a very small number of the then physicists. Only the approach of J. Bell \cite{bell} met with greater attention, as also some formulas were presented that seemed to be able to bring the decision between the two discussed (Copenhagen and ensemble) alternatives of the quantum-mechanical model on experimental basis.
However, in broad physical community there has not been any interest to change generally accepted paradigm. Any greater doubts about the standard quantum theory have not been evoked, which was influenced from the very beginning in principle by the mentioned mistaking proof of von Neumann.
Nevertheless, the approach initiated by Bell influenced decisively the whole further story of the EPR experiment. The original Gedankenexperiment was modified to be possible to perform it. And the corresponding experiments started to be prepared. However, two other mistaking arguments played a decisive role in the then solution of the given problem.
One argument followed from the statement of Belinfante \cite{belin} that the Copenhagen alternative and the HV theory had to give mutually different predictions, or in other words, that the prediction of HV theory had to differ significantly from the Malus law (approximately measured dependence of light transmission in the case of two polarizers). That was not, however, true as the given statement was based on mistaking interchange of transmission probabilities through a polarizer pair and one polarizer; a more detailed explanation having been given, e.g., in Refs. \cite{lk02,contr}. In fact, the approximate Malus law (as measured) has been fully consistent with the HV theory.
The other mistaking argument was then involved in the already mentioned formula of Bell, which will be explained to a greater detail in the next section. Now the actual story of the EPR experiment will be described.
The experimentally feasible EPR experiment consisted in the measurement of coincidence transmission probabilities of two photons with opposite spins and running in opposite directions through two polarizers:
\[ <---|^{\beta}---o---|^{\alpha}---> \]
where $\alpha$ and $\beta$ are deviations of individual polarizer axes from a common zero position.
According to Bell \cite{bell} any four transmission probabilities $a_j, b_j \;(j=1,2)$ corresponding to two different orientations of both the polarizers (4 different combinations) should have fulfilled the following condition
\[ B = a_1b_1+a_2b_1+a_1b_2-a_2b_2 \leq 2 \]
if the HV theory had hold (which has not been, however, true as it will be shown in Sec. \ref{sec6}). For the Copenhagen alternative this upper limit should have been higher.
The then situation was described to a greater detail in already mentioned book of
Belinfante \cite{belin}.
The main series of corresponding EPR experiments was being performed
since 1971. It was finished practically in 1982 by the
following conclusion (see Aspect et al. \cite{asp}):
- Bell's inequalities have been violated; and the hidden-variable
alternative seemed, therefore, to be refused;
- measured values have been practically in agreement with
quantum-mechanical predictions (approximate Malus law having been obtained).
At that time the results were interpreted as the victory of Copenhagen
quantum mechanics. However, it was seen soon that practically
nothing from earlier problems was solved. And the discussions
concerning EPR experiments have continued. In fact none of both the results of Aspect's experiment has led to refusal of HV theory as the corresponding conclusion has been based on the mentioned mistakes.
As to the latter result of Aspect et al. obtained in the given EPR experiment (approximate Malus law) we have already mentioned the mistake of Belinfante and referred to its detailed explanation in Ref. \cite{contr}. The approximate Malus law may correspond fully to both the theoretical alternatives and no such an argument against the HV theory exists.
More detailed explanation is needed, of course, as to the first experimental result. Bell started in principle from the HV approach. However, to derive the given inequalities he had to interchange some probabilities, which seemed to be natural at the first sight only. In fact, it represented a strong assumption that corresponded to the passage from the hidden-variable concept to the classical one. The given inequalities were derived, of course, in other ways, too; see, e.g., Ref. \cite{clau}. Similar assumptions were involved, however, in all these approaches (see \cite{lk98}). Bell's limit corresponded to classical physics and not to HV theory, which will be discussed and explained to a greater detail in the next section.
However, at the end of this section it is necessary to mention also the recent delayed-choice experiment \cite {jac} trying to bring a new support for Copenhagen alternative. It follows from our theoretical analysis that no further experiment of EPR type may influence our conclusions concerning the decisive preference of the hidden-variable theory.
\section{ Bell's operator and different inequalities }
\label{sec6}
It is evident from the preceding that Bell's combination of
coincidence probabilities may exhibit different limits according
to basic assumptions concerning the individual processes. We will discuss this problem now in the language of the so called
Bell operator obtained by substituting individual probabilities by basic operators representing individual measurement acts (see, e.g., \cite{hill}). According to chosen assumptions three different limits may be obtained.
The Bell operator $B$ may be represented in the Hilbert space
\begin{equation}
{\mathcal H}\;=\; {\mathcal H}_a \otimes {\mathcal H}_b
\label{tens}
\end{equation}
where the subspaces ${\mathcal H}_a$ and ${\mathcal H}_b$
correspond to individual measuring devices (polarizers) in the
coincidence arrangement. It is then possible to introduce the
operator
\begin{equation}
B\;=\;a_1b_1+a_1b_2+a_2b_1-a_2b_2
\end{equation}
where $a_j$ and $b_k$ are now operators acting in subspaces ${\mathcal
H}_a$ and ${\mathcal H}_b$ and corresponding to different
measurements in individual polarizers. It holds for the
expectation values of these operators \cite{hill}
$$ 0\;\leq\; |\langle a_j\rangle|, \;|\langle b_k\rangle|\; \leq \;1\, . $$
The expectation values $|\langle B\rangle|$ of the Bell operator
may then possess different upper limits according to the mutual
commutation relations of the operators $a_j$ and $b_k$.
If it holds
$$ [a_j,b_k]\neq 0, \;\;[a_1,a_2]\neq 0, \;\;[b_1,b_2]\neq 0 $$
one can obtain by a rough estimate \cite{tsil}
$$ \langle BB^+\rangle \leq 16 \;\; \mathrm {or} \;\;\langle B\rangle \leq 4 \;. $$
However, after more detailed calculation one obtains (see
\cite{revz})
\begin{equation}
\langle BB^+\rangle \leq 12, \;\;\;\;|\langle B\rangle| \leq 2{\sqrt 3}\; . \label{one}
\end{equation}
If
$$ [a_j,b_k]= 0\;,\;\;\mathrm {and}\;\;\;[a_1,a_2]\neq 0,\;[b_1,b_2]\neq 0\,, $$
it holds
\begin{equation}
\langle BB^+\rangle \leq 8,
\;\;\;\;|\langle B\rangle| \leq 2{\sqrt 2} \;. \label{two}
\end{equation}
And finally, if all operators $a_j$ and $b_k$ commute mutually one
obtains
\begin{equation}
\langle BB^+\rangle \leq 4,
\;\;\;\;|\langle B\rangle| \leq 2 \; ; \label{three}
\end{equation}
the same limit being obtained also if at least the operators
belonging to one of subspaces ${\mathcal H}_a$ or ${\mathcal
H}_b$ commute mutually and with all other operators \cite{revz}.
It has been, therefore, derived that there are in principle three different limiting bounds for Bell's combination of coincidence probability measurements. In the following we will attempt to correlate them to individual physical alternatives:
(i) In contradistinction to hitherto common opinion the last limit
(\ref{three}) corresponds to the conditions of classical physics.
(ii) The limit (\ref{two}) represents the properties of the HV alternative. There are not, of course, any hidden parameters; all physical parameters are standard ones; only some of them being statistically distributed in corresponding initial states.
(iii) As to the limit (\ref{one}) it represents the case when the
results of both the measuring devices may influence mutually one
another; it would be the case of the orthodox (Copenhagen) quantum mechanics.
It is only the classical limit that has been excluded by experimental EPR data. As to the HV alternative it does not contradict
the experimental results and may be brought to agreement with
experimentally established coincidence polarization data
(obtained, e.g., by Aspect et al. \cite{asp}). It is, of course, also the
Copenhagen quantum mechanics that has not been excluded by EPR experiment results. Here the previously discussed internal discrepancies must be taken into account.
\section{ HV theory and physical reality }
\label{sec7}
It follows from the preceding that there are not any objections against the HV theory as to the description of microscopic world. However, the introduced arguments indicate that this theory should be not only preferred to the standard quantum mechanics but also used in describing all physical reality, as it has been considered recently by A. Legget \cite{legg,legg2}.
He has asked whether the every day world may be described with the help of the same physical model as microscopic objects.
However, he has not taken into account the significant difference between two alternatives of quantum-theoretical model: statistical (or HV theory) and Copenhagen. Consequently, he has not been able to get a positive answer.
His question has been answered, of course, positively in our paper, as the former alternative, i.e. HV theoretical approach, differs from the classical physics only in the existence of discrete energy spectrum in closed physical systems (see also Sec. \ref{sec9}).
The HV theory may be applied, therefore, practically also to the description of standard macroscopic processes, since in the case of discrete spectrum the differences between individual energy values should remain quite immeasurable. However, quite recent analyses concerning the cavity optomechanical systems seem to provide the way to much stronger support for the HV theory to be denoted as generally valid; see e.g. \cite{thom,bhatt}. The behavior analogous to the excitations of single molecules may be expected for pairs of membranes in vibrational states \cite{bhatt}. However, it is not possible to interpret such states as the phenomenon of entanglement (see Ref. \cite{hart}) as this phenomenon might be bound only to the Copenhagen quantum mechanics that contains internal contradictions. The application of the HV theory might be probably very helpful in the solution of corresponding problems of optomechanics.
It is also the famous problem of Schr\"{o}dinger cat (representing a macroscopic closed physical system) that may be seen in a quite new light now. There are two basic states ("eigenstates"), i.e., living and dead cat, and any superposition of these states correspond to their statistical combination.
\section{ Light transmission through three polarizers }
\label{sec8}
In the preceding we have discussed the internal discrepancies and unphysical aspects of the Copenhagen quantum mechanics and demonstrated that they might be removed by passing to the HV theory. At the same time the criticism of Einstein may be removed, too. Any objections against HV theory do not exist.
In the case of the EPR experiment with polarized photons both the quantum theories may give practically the same predictions for coincidence transmission, even if the physical interpretations of polarization and transmission mechanisms are quite different. And one may ask whether different predictions may exist for another arrangement of polarizers. Already some time ago we have attempted to analyze the transmission of light through three polarizers:
\[ o---|---|^{\alpha}---|^{\beta}---> \]
where individual polarizers have been differently oriented; $\alpha$ and $\beta$ denoting axis deviations of the second and third polarizers from the first one. According to the Copenhagen quantum-mechanical model it should hold for corresponding light transmission probability
\begin{equation}
P(\alpha,\beta)\,=\,\cos^2\!\alpha\;cos^2(\alpha-\beta). \label{Cqm}
\end{equation}
And it was possible to expect measurable deviations in the case of the HV theory.
The corresponding experiment was performed and the results were published in 1993-4; see \cite{krasa1,krasa2}. For a given angle $\alpha$ the angle $\beta$ was always established, so as the total light transmission be minimal. Fundamental deviations from the given quantum-mechanical formula (\ref{Cqm}) have been found, as it may be seen from Figs. 2 and 3.
The mentioned angle pairs (giving the minimum light transmission for given $\alpha$) are shown in Fig. 2. And the corresponding experimental values of transmitted light are represented by dashed line in Fig. 3 (experimental points taken from \cite{krasa2}). The quantum-mechanical predictions for the given angle pairs are then represented by full line; the position of this line being shifted in vertical
direction somewhat arbitrarily as the values of the so called "imperfectness" of given polarizers were not available. In any case the standard quantum-mechanical prediction requires maximums at the positions where the experiment exhibits deep minimums. The difference against the Copenhagen quantum mechanics was not accented in Ref. \cite{krasa2} where the experimental results were published for the first time as we were afraid reasonably that the paper would not have been accepted for publication in such a case. We have only mentioned that similar characteristics may be obtained in the framework of classical theory of Stokes.
And it is possible to conclude that the given experimental data represent further falsification of Copenhagen quantum mechanics, while good agreement may be evidently obtained in the framework of HV theory as a series of free parameters for the polarization description is available.
\begin{figure}[t!]
\begin{center}
\includegraphics*[scale=.40, angle=-90]{prevod.eps}
\caption { \it { Pairs of angles $\,\alpha$ and $\beta\,$ corresponding to minimal transmission values for chosen $\alpha$;
pair values used for the measurement shown in Fig. 3. } }
\end{center}
\vspace{4mm}
\begin{center}
\includegraphics*[scale=.40, angle=-90]{prevoc.eps}
\caption { \it { Light transmission through three polarizers (for
$\,\alpha\,-\,\beta\,$ pairs shown in Fig. 2); experimental data -
points on dashed line; quantum-mechanical (orthodox) prediction -
full line. } }
\end{center}
\end{figure}
\section{ Schr\"{o}dinger equation and classical physics }
\label{sec9}
To complete our analysis it is necessary to discuss the actual relation between the Schr\"{o}dinger equation and classical physics, yet. E. Schr\"{o}dinger \cite{schr} was successful with his equation when he showed that for particles exhibiting inertial motion the identical behavior with classical physics was obtained.
Let us return now, therefore, to the time-dependent Schr\"{o}dinger equation
\begin{equation}
i\hbar\frac{\partial}{\partial t}\psi(x,t)=H\psi(x,t) \label{tsch}
\end{equation}
where the complex function $\psi(x,t)$ is expressed as
\begin{equation}
\psi(x,t) \;=\; \lambda(x,t)\, e^{\frac{i}{\hbar}\Phi(x,t)} \label{psi}
\end{equation}
and both the functions $\lambda(x,t)$ and $\Phi(x,t)$ are real. Let us limit to time-independent potential $V(x)$ (see Eq. (\ref{schr})). Eq. (\ref{tsch}) may be substituted by two equations for two real functions (see D. Bohm \cite{bohm})
\begin{eqnarray}
\frac{(\nabla \Phi)^2}{2m}\,+\, V(x)\,+\,V_q(x,t)
&=& -\,\partial_t\,\Phi \;, \label{hamj} \\
\triangle\Phi \,+\,2(\nabla\,\Phi)(\nabla\,lg\,\lambda) &=&
-2m\;\partial_t\, lg\,\lambda \label{ham2}
\end{eqnarray}
where
\begin{equation}
V_q(x,t)\,=\,-\frac{\hbar^2}{2m}\frac{\triangle\lambda}{\lambda}
\end{equation}
has been denoted as quantum potential.
Eq. (\ref{hamj}) resembles Hamilton-Jacobi equation
\begin{equation}
\frac{1}{2m}(\nabla S(x,t))^2 + V(x) = -{\partial_t S(x,t)} \label{haja}
\end{equation}
where $S(x,t)$ has been replaced by $\Phi(x,t)$ and the quantum potential $V_q(x,t)$ has been added. $S(x,t)$ is Hamilton principal function, from which the momentum values may be derived:
\[ p(x,t)=\nabla S(x,t). \]
For inertial motion it holds
\[ V_q(x,t)\,=\, V(x)\,=\,0 \]
and
\[ \Phi(x,t)\,=\, S(x,t); \]
the phase being identical with Hamilton principal function in such a case.
Let us assume now
\[ V(x)\; \neq \; 0 \]
and let us limit to basic solutions corresponding to different values of energy (Hamiltonian eigenvalues) and fulfilling the conditions
\[ \psi^{(E)}(x,t)\,=\,\psi_E(x)e^{-iEt},\;\;\; H\psi_E(x)\,=\,E\psi_E(x). \]
In such a case $V_q(x)\neq 0$ is independent of $t$ and $\Phi(x,t)$ and $S(x,t)$ are mutually different; $V_q(x)$ representing the numerical measure of such difference.
There is not, however, any difference between the physical results of Schr\"{o}dinger equation and classical physics; see also \cite{adv}. All basic solutions of Schr\"{o}dinger equation are fully equivalent to classical solutions corresponding to the same energy. However, it does not hold in opposite direction. For some solutions of Hamilton equation the corresponding counterparts in the Schr\"{o}dinger equation (or in the HV theory) do not seem to exist in the case of discrete spectrum.
In the past when the existence of quantum potential was assumed to represent decisive physical difference between Schr\"{o}dinger equation and classical physics there were done some attempts to interpret it as the consequence of Brown motions of individual microscopic objects. Our result is, however, in full agreement with results of Ioanidou \cite{ioan} and Hoyer \cite{hoyer} who have shown that Schr\"{o}dinger equation may be derived if classical physics is combined with a kind of statistical distribution.
The advantage of the Schr\"{o}dinger equation consists then in obtaining a complete statistical result in one solution if a statistical distribution of initial basic states is given. Consequently, the Schr\"{o}dinger equation is very suitable if some initial parameters (e.g., the impact parameter in collision processes and, consequently, also interaction energy values) are not exactly defined and only their statistical distributions may be established, as it occurs in collision measurement approaches.
\section{ HV theory and Hilbert space }
\label{sec10}
It follows from the preceding that individual solutions of Schr\"{o}dinger equation may be truly represented in the Hilbert space that is extended (i.e., at least doubled) in comparison to the third assumption introduced in Sec. \ref{sec2}; and when the fourth assumption is refused, too. The evolution of a physical system is then characterized by a trajectory that represents irreversible behavior. Exact solutions may be derived, of course, practically in the case of a system consisting of two stable particles. However, the representation of time evolution in a corresponding Hilbert space may be very helpful also in the case of more complex physical systems.
Our considerations will be based on the analysis of a two-particle system as it has been described in Sec. 4. Such a scheme may represent a basic structure for describing the evolution of any more complex physical system even if it involves objects that are not stable. A resonance may be formed in the collision of two simpler particles and also the decay of an unstable object (resonance) may be interpreted at least in the first approximation as the transition to two-particle system even if any of the arising particles may be unstable. In such a case the Hilbert space must be, of course, more complex consisting not only of orthogonal sums but also of tensor products of simpler subspaces.
It is clear that all subspaces in one orthogonal sum ($\Delta^+,\Delta^-,\Theta$) must correspond to the same quantum numbers that must be conserved during the whole evolution. One two-particle system at a given time may be then represented in $\Delta^\pm$ by one vector. However, even the stable objects exhibit usually some internal structures and internal evolutions that might influence the transition probabilities to other subspaces in collision processes. It may be taken into account by substituting, e.g., $\Delta ^-$ by the tensor product of Hilbert subspaces $\Delta ^-\otimes{\mathcal P}_1\otimes{\mathcal P}_2$ where the two latter subspaces represent main properties of individual objects.
It would be, of course, difficult to allocate a more general Schr\"{o}dinger equation to such a physical system. In fact, it is hardly possible to describe the detailed evolution of corresponding collision processes when the detailed internal structures and evolutions of individual particles are not known. It is possible to characterize the influence of some characteristics only by establishing some probability distributions in transmission processes. And in such a case the representations of ${\mathcal P}^\pm$ may consist in characterizing them by some basis vectors corresponding to different classes of particle properties that may change during the time evolution.
And it is evident that even rather complicated processes might be represented correspondingly (at least at not very high energies).
The same holds before all also for the subspace $\Theta$ representing a resonance (or an unstable object); it may be suitable to represent it, e.g., by vector basis corresponding to individual decay channels (see \cite{lk69}), where the frequencies of individual basic states may be derived from experimental transmission data.
\section{ Conclusion }
\label{sec11}
The physics of microscopic world in the 20th century has been represented by the Copenhagen quantum mechanics with its logical paradoxes and contradictions. The given theory was taken as valid during the whole century even if Pauli called the attention to one important contradiction already in 1933: the Hamiltonian had to possess the continuous spectrum in the whole interval of all real numbers $(-\infty,+\infty)$ while the actual energy value must be practically positive. Other critical arguments including also the disagreement with experimental data (light transmission through three polarizers) have been then summarized in the preceding.
It has been shown that all known critical points can be removed when one passes from the Copenhagen quantum mechanics to the HV theory that is based practically on mere Schr\"{o}dinger equation, while earlier additional assumptions (see Sec. \ref{sec2}) are abandoned and the Hilbert space is adapted to actual time-dependent Schr\"{o}dinger solutions. E.g., for the system of two particles it has been necessary to extend the Hilbert space so as to consist at least of two mutually orthogonal subspaces (see Sec. \ref{sec4}).
The given passage to the HV theory has solved practically all known problems; the mistaking statements having been repaired. It has been also the EPR experiment that should be (and may be) interpreted as it was required by Einstein.
It has been shown also that the Schr\"{o}dinger equation or the HV theory is practically equivalent to classical physics with the only exception, concerning the existence of discrete bound states.
The HV theory may be then applied in principle to the description of microscopic as well as macroscopic objects. There is practically no gap between these two physical regions. It is possible also to say that the question put by A. Legget (see Ref. \cite{legg2}) has been answered in positive way only if one has passed from Copenhagen quantum mechanics to HV theory (see also Sec. \ref{sec7}).
{\footnotesize
|
2,869,038,155,657 | arxiv | \section{Introduction}
The achievement of producing Bose--Einstein condenstates with ultracold dilute gases of atoms has seen a wealth of theoretical and experimental activity. One enticing prospect of Bose-Einstein condensates is that they may allow for a better understanding of the interface between classical and quantum mechnics, through the possibility of macroscopic
Schr\"odinger cat states \cite{zoller} and macroscopic quantum tunneling \cite{leggett}. Another intriguing field of study is the chemistry of Bose-Einstein condensates, where the atomic constituents may form molecules through Feshbach resonances
\cite{feshbach} or photoassociation \cite{photo}. A novel feature of a molecular Bose--Einstein condensate is that the atomic and molecular states can exist as a superposition
\cite{donley}, providing a chemical analogue of a Schr\"odinger cat state. In cases where the molecules are heteronuclear, the presence of a permanent electric dipole moment also opens the possibility for manipulating the condensate through electrostatic forces \cite{stwalley}.
Since systems of Bose--Einstein condensates exist at ultracold temperatures, it is to be expected that significant insights into their behaviour can be obtained from studying their ground-state properties.
From a general theoretical perspective there has been substantial progress in the understanding of quantum (i.e. ground-state) phases in many-body quantum systems, due largely to a cross fertilisation of ideas between the condensed matter theory and the quantum information theory communities. Much of this study has explored the relationship between entanglement and quantum criticality \cite{on,oaff,vlrk}.
However other characterisations of quantum criticality have been sought too \cite{vbkos,hamma}. Recently the notion of wavefunction overlaps (also known as the {\it fidelity}), which is again common in quantum information theory, has been applied to the study of quantum phase transitions \cite{zanardi,huanandjp}. An advantage of this approach is its universality, as it can be applied to any system independent of the choice of decomposition into subsystems.
With the above points in mind here we analyse a simple, yet non-trivial, three-mode model describing a heteronuclear molecular condensate. Two modes are associated with two distinguishable atomic constituents, which can combine to form a molecule represented by the third mode. Besides the interaction describing the interconversion of atoms and molecules, the Hamiltonian contains terms which are linear in the mode number operators (corresponding to external fields) and terms which are second-order in the mode number operators (corresponding to scattering interactions between atoms and molecules).
We mainly concern ourselves with the ground-state properties of the model, with the aim of identifying the ground-state phases. We avoid taking the thermodynamic limit and restrict our analysis to finite systems, for reasons which will be discussed later. This in turn presents challenges in rigourously identifying quantum phases, since for finite systems there are no singularities in physical quantities such as the ground state energy and its derivatives. However several recent works have addressed the issues of quantum phases in finite systems \cite{iz,dhl,adgv,leviatan}. We mention that traditional techniques of renormalisation group methods are not applicable to the model under consideration, due to the low number of degrees of freedom. Neither is the concept of symmetry breaking, as the model does not admit global symmetries, nor long-range order, as the model is in essence zero-dimensional.
We start our analysis with a semi-classical treatment, following the approach of \cite{fitonel}. Since the model with which we are dealing is integrable, the semi-classical many-body system can be reduced to a problem with a single degree of freedom. We study the phase space of this system, in particular determining the fixed points. It is found that for certain coupling parameters bifurcations of the fixed points occur, and we can determine a parameter space diagram which classifies the fixed points. An unexpected result is that the boundaries between the regions in parameter space are extremely sensitive on whether the number of constituent atoms is equal or not. Specifically, when the number of constituent atoms is equal (i.e. the {\it atomic imbalance} is zero) there is a spontaneous appearance of additional boundaries in the parameter space, some of which can be identified with bifurcations of the global minimum of the classical Hamiltonian.
We next investigate the extent to which the classical behaviour influences the ground-state properties of the quantum system. Our first goal in the full quantum analysis is to derive an exact Bethe ansatz solution for the model. We use the Bethe ansatz solution to map the spectrum of the Hamiltonian into that of a one-body
Schr\"odinger equation in one-dimensional. An advantage of this method is that it allows for an analysis of the finite system, following the ideas of \cite{dhl}, as the mapping to the one-body Schr\"odinger equation is not dependent on taking the thermodynamic limit of the original many-body system. The results of the analysis of the associated Schr\"odinger equation are in general agreement with the results obtained from the semi-classical treatment, supporting the picture of an additional phase boundary when the atomic imbalance is zero.
However, due to quantum fluctuations, the emergence of the phase boundary is smooth rather than spontaneous. This property is apparent from a study of ground-state expectation values and quantum dynamics.
In order to simply characterise the ground-state phases for the finite system, we finally define the notion of a {\it quantum phase pre-transition} in terms of wavefunction overlaps. Specifically, a quantum phase pre-transition is identified with each coupling for which the incremental ground-state wavefunction overlap is a local minimum. We numerically calculate these for several cases and discuss these results in relation to the semi-classical and quantum analyses which have been described above. The results confirm the emergence of a quantum phase boundary in the limit of zero atomic imbalance.
\section{The model}
We consider a general three-mode Hamiltonian describing a heteronuclear molecular Bose--Einstein condensate with two distinct
species of atoms, labelled by $a$ and $b$, which can combine to produce a molecule labelled by $c$. We introduce canonical creation and annihilation operators $\{a,\,b,\,c,\,a^{\dagger},\,b^{\dagger},\,c^{\dagger}\}$ satisfying the usual commutation relations $[a,\,a^\dagger]=I$ etc., which represent
the three degrees of freedom in the model. The Hamiltonian reads \cite{jzrg}
\begin{eqnarray}
H&=&U_{aa}N_a^2 + U_{bb}N_b^2 +U_{cc}N_c^2+U_{ab}N_aN_b+U_{ac}N_aN_c+U_{bc}N_bN_c \nonumber \\
&+& \mu_aN_a +\mu_bN_b+\mu_cN_c + \Omega(a^{\dagger}b^{\dagger}c +c^{\dagger}ba).
\label{ham}
\end{eqnarray}
The parameters $U_{ij}$ describe S-wave scattering,
$\mu_i$ are external potentials and $\Omega$ is the amplitude for interconversion of atoms and molecules.
We remark that in the limit
$U_{aa}=U_{bb}=U_{cc}=U_{ab}=U_{ac}=U_{bc}=0$, equation
(\ref{ham}) is the Hamiltonian studied in \cite{wb,wt}
in the context of quantum optics. In the latter stages of the manuscript we will
study this limiting case in some detail.
The Hamiltonian acts on the Fock space spanned by the (unnormalised) vectors
\begin{equation}
\left|n_a;n_b;n_c\right>
={(a^\dagger)^{n_a}(b^\dagger)^{n_b}(c^\dagger)^{n_c}}\left|0\right>
\label{states}
\end{equation}
where $\left|0\right>$ is the Fock vacuum. We then have
$$N_a \left|n_a;n_b;n_c\right> = n_a \left|n_a;n_b;n_c\right> $$
etc., where $N_a=a^{\dagger}a$, $N_b=b^{\dagger}b$ and $N_c=c^{\dagger}c$.
The Hamiltonian commutes
with $J=N_a-N_b$ and the total atom
number $N=N_a+N_b+2N_c$. We refer to $J$ as the atomic imbalance and introduce $k=J/N,\,k\in[-1,1]$
as the fractional atomic imbalance.
As there are three degrees of freedom and three conserved operators, the system is integrable. This fact will allow us to analyse the model in some depth. Below we begin with a semi-classical analogue of the model, and determine the fixed points of the system.
\section{Semi-classical analysis} \label{sca}
Let $N_j,\,\phi_j,\,j=a,\,b,\,c$ be
quantum variables satisfying the canonical relations
$$[\phi_j,\,\phi_k]=[N_j,\,N_k]=0,~~~~~[N_j,\,\phi_k]=i\delta_{jk}I.$$
We make a change of variables from the operators $\{j,\,j^\dagger |\,j=a,\,b,\,c\}$ to a number-phase representation via
$$j=\exp(i\phi_j)\sqrt{N_j} \;\;\;\;\;\;\;j=a,\,b,\,c$$
such that the canonical commutation relations are preserved.
We now make a further change of variables
$$ z=\frac{1}{N}(N_a+N_b-2N_c),$$
$$\phi=\frac{N}{4}(\phi_a+\phi_b-\phi_c),$$
such that $z$ and $\phi$ are canonically conjugate variables; i.e.
$$[z,\,\phi]=iI. $$
For large $N$ we can now approximate the (rescaled) Hamiltonian by
\begin{eqnarray}
H=\lambda z^2 + 2(\alpha-\lambda)z +\lambda -2\alpha + \beta
+\sqrt{2(1-z)(z+c_+)(z+c_-)} \cos\left(\frac{4\phi}{N}\right)
\label{ham2}
\end{eqnarray}
with
\begin{eqnarray*}
\lambda &=& \frac{\sqrt{2N}}{\Omega}\left(\frac{U_{aa}}{4}+
\frac{U_{bb}}{4}+\frac{U_{cc}}{4}+ \frac{U_{ab}}{4}-\frac{U_{ac}}{4}-\frac{U_{bc}}{4}\right) \\
\alpha &=&\frac{\sqrt{2N}}{\Omega}\left( \frac{1+k}{2}U_{aa}
+\frac{1-k}{2}U_{bb}+\frac{1}{2}U_{ab} -\frac{1+k}{4}U_{ac}-\frac{1-k}{4}U_{bc}+\frac{1}{2N}(\mu_a+\mu_b-\mu_c)\right) \\
\beta &=& \frac{\sqrt{2N}}{\Omega}\left((1+k)^2U_{aa}+(1-k)^2U_{bb}+(1-k^2)U_{ab}+\frac{2}{N}((1+k)\mu_a+(1-k)\mu_b) \right)
\end{eqnarray*}
where $c_{\pm}=1\pm 2k $.
Since $N$ and $k$ are conserved, we treat them as constant.
We now regard (\ref{ham2}) as a classical Hamiltonian and
investigate the fixed points of the system. The first step is to
derive Hamilton's equations of motion yielding
\begin{eqnarray*}
\frac{dz}{dt}=\frac{\partial H}{\partial \phi}&=& - \frac{4}{N}\sqrt{2(1-z)(z+c_+)(z+c_-)} \sin\left(\frac{4\phi}{N}\right), \label{de1} \\
-\frac{d\phi}{dt}=\frac{\partial H}{\partial z} &=&2\lambda z +2\alpha-2\lambda +\frac{(1-z)(2z+2)-(z+c_+)(z+c_-)}{\sqrt{2(1-z)(z+c_+)(z+c_-)}} \cos\left(\frac{4\phi}{N}\right).
\label{de2}
\end{eqnarray*}
The fixed points of the system are determined by the condition
\begin{equation}
\frac{\partial H}{\partial \phi}=\frac{\partial H}{\partial z}=0.
\label{fixed}
\end{equation}
Due to periodicity of the solutions, below we restrict to $\phi\in[0,\,N\pi/2)$. It is necessary to treat the cases of
$k\neq 0$ and $k=0$ separately, and without loss of generality we assume $k\geq 0$.
\subsection{Case I: $k\neq 0$}
Define the functions
\begin{eqnarray}
f(z)&=& \lambda z+\alpha-\lambda \label{f} \\
g(z)&=& \frac{(z-1)(2z+2)+ (z+c_+)(z+c_-)}{2\sqrt{2(1-z)(z+c_+)(z+c_-)}} \label{g}
\end{eqnarray}
Note that the domain of $g(z)$ is $z \in [2k-1,1]$, and $g(z)$ is divergent at $z=2k-1$ and $z=1$.
For $k\neq 0$, we then have the following classification of solutions for (\ref{fixed}):
\begin{itemize}
\item $\phi=0$, and $z$ is a solution of
\begin{equation}
f(z)=g(z)
\label{sol2}
\end{equation}
which can admit one, two or three solutions.
\item $\phi={N\pi}/4$, and $z$ is a solution of
\begin{equation}
f(z)=-g(z)
\label{sol1}
\end{equation}
which can admit one, two or three solutions.
\end{itemize}
\begin{figure}[ht]
\begin{center}
\epsfig{file=figures/Fig1.eps,scale=0.4} \\
\end{center}
\caption{On the left, graphical solution of (\ref{sol2}) with $k=0.8$. Depending on the values of $\lambda$ and $\alpha$, there may be one, two or three solutions.
On the right, graphical solution of (\ref{ssol2}) with $k=0$. Depending on the values of $\lambda$ and $\alpha$, there may be zero, one or two solutions.}
\label{curve2}
\vspace{1.00cm}
\end{figure}
A graphical representation of possible types of solutions for $\theta=0$ is given in Fig. \ref{curve2}.
From the equations (\ref{sol2}, \ref{sol1}) we can determine there are fixed point bifurcations for
certain choices of the coupling parameters. These bifurcations allow us to divide the coupling parameter space into
different regions. To construct this diagram, we observe that bifurcations occur when $f$ is the tangent line to $g_{\pm}$; i.e. for values of $\lambda,\,\alpha$ such that
\begin{eqnarray}
\lambda&=&\pm\left.\frac{dg}{dz}\right|_{z_0} \label{boundarya} \\
f(z_0)&=& \pm g(z_0) \label{boundaryb}
\end{eqnarray}
for some $z_0$. This requirement determines the boundaries in parameter space, which are depicted in Fig.
\ref{curvekdiff}
\begin{figure}[ht]
\begin{center}
\epsfig{file=figures/Fig2.eps,height=6cm,angle=0}
\end{center}
\caption{Parameter space diagram identifying the different types of solutions for
equation (\ref{fixed}), for
$k=0.002,\,0.02,\,0.2$. In each case
the diagram is divided into regions
$A$ (one solution for $z$ when $\phi = 0$ and one solution when $\phi=N\pi/4$ ),
$B$ (three solutions for $z$ when $\phi = 0$ and one solution when $\phi=N\pi/4$) and
$C$ (one solution for $z$ when $\phi = 0$ and three solutions when $\phi=N\pi/4$).
The boundary separating the regions is given by solutions to the equations (\ref{boundarya},\ref{boundaryb}).
}
\label{curvekdiff}
\end{figure}
\subsection{ Case II: $k=0$}
Next we consider the case $k=0$ for which the function $g(z)$ has substantially different properties. Setting
$c_+=c_-=1$ into (\ref{g}), we find that $g(z)$ reduces to
\begin{equation}
g(z)= \frac{1-3z}{2\sqrt{2(1-z)}}
\end{equation}
Here we observe that $g(z)$ is divergent at $z=1$, but finite at $z=-1$. This property affects the types of solutions for (\ref{fixed}). Specifically, we now have the following classifications of solutions for $k=0$:
\begin{itemize}
\item $\phi=0$, and $z$ is a solution of
\begin{equation}
f(z)=g(z)
\label{ssol2}
\end{equation}
which can admit zero, one or two solutions.
\item $\phi={N\pi}/4$, and $z$ is a solution of
\begin{equation}
f(z)=-g(z)
\label{ssol1}
\end{equation}
which can admit zero, one or two solutions.
\item $z=-1$ and $\phi$ is a solution of
\begin{equation}
\cos\left(\frac{4\phi}{N}\right)=-2\lambda+{\alpha}
\label{ssol3}
\end{equation}
which can admit zero, one or two solutions.
\end{itemize}
A graphical representation of possible types of solutions for $\theta=0$ is given in Fig. \ref{curve2}.
Because $g(-1)$ is finite, for this case there can be either zero, one or two solutions. As in the $k\neq 0$ case, we can determine the region boundaries in parameter space from equations (\ref{boundarya},\ref{boundaryb}). Moreover, because of the existence of solutions of the form given by (\ref{ssol3}) for $k=0$, which do not have an analogue for $k\neq 0$, we see the appearance of new boundaries given by the conditions $\lambda=(\alpha\pm 1)/2$ for all values of $\alpha$.
The boundaries in parameter space are depicted in Fig. \ref{parametro1}.
\begin{figure}[h]
\begin{center}
\epsfig{file=figures/Fig3.eps,height=6cm,angle=0}
\end{center}
\caption{Parameter space diagram identifying the different types of solutions for
equation (\ref{fixed}) when $k=0$.
In region I there is no solution for $z$ when $\phi = 0$, and one solution for
$z$ when $\phi = {N\pi}/{4}$. In region II there are two solutions for $z$ when $\phi = 0$, and one solution for
$z$ when $\phi = {N\pi}/{4}$. In region III exists one solution for $z$ when $\phi = 0$, one solution for
$z$ when $\phi = {N\pi}/{4}$, and two solutions for $\phi$ when $z=-1$. In region IV there is one solution for $z$
when $\phi = 0$, and no solution for $z$ when $\phi = {N\pi}/{4}$. In region V there is one solution for $z$ when $\phi=0$, and two solutions for $z$ when $\phi = {N\pi}/{4}$. The boundary separating regions II and III
is given by $\lambda=(\alpha+1)/2$, while the equation $\lambda =(\alpha-1)/2$ separates the regions III and IV.
The boundary between regions I and II has been obtained numerically.
}
\label{parametro1}
\end{figure}
To help visualise the classical dynamics, it is useful to plot the level curves of the Hamiltonian (\ref{ham2}).
Since the fixed point bifurcations change the topology of the level curves, qualitative differences can be observed
between each of the regions. The results are shown respectively in Fig. \ref{levelcurve2} for $k=0.2$ and
Fig. \ref{levelcurve1} for $k=0$, where
for clarity we show $4\phi/N\in[-2\pi,\,2\pi]$.
\begin{figure}[h]
\begin{center}
\begin{tabular}{cccc}
\epsfig{file=figures/l10alm5.eps,width=3.5cm,height=5cm,angle=0}&
\epsfig{file=figures/l10al4.eps,width=3.5cm,height=5cm,angle=0} &
\epsfig{file=figures/l10al12.eps,width=3.5cm,height=5cm,angle=0}&
\epsfig{file=figures/l10al16.eps,width=3.5cm,height=5cm,angle=0} \\
(a) & (b) & (c) & (d)\\
\end{tabular}
\end{center}
\caption{Level curves of the Hamiltonian (\ref{ham2}) for $k=0.2$, where the dark regions indicate lower values than the light regions. Figures (a) and (d) correspond to region A while
Figures (b), and (c) correspond to region B. The parameter values are: (a) $\lambda=10,\,\alpha=-5$; (b) $\lambda=10,\,\alpha=4$;
(c) $\lambda=10,\,\alpha=12$ and (d) $\lambda=10,\,\alpha=16$.
In region A, there is a maximal point at $\phi=0$ and minima at
$4\phi/N=\pm\pi$. Two additional fixed points, a saddle and a maximum, occur in region
B at $\phi=0$.
}
\label{levelcurve2}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cccc}
\epsfig{file=figures/reg1k0.eps,width=3.5cm,height=5cm,angle=0}&
\epsfig{file=figures/reg2k0.eps,width=3.5cm,height=5cm,angle=0}&
\epsfig{file=figures/reg3k0.eps,width=3.5cm,height=5cm,angle=0}&
\epsfig{file=figures/reg4k0.eps,width=3.5cm,height=5cm,angle=0} \\
(I)& (II) & (III)& (IV) \\
\end{tabular}
\end{center}
\caption{Level curves of the Hamiltonian (\ref{ham2}) for $k=0$, showing the typical behaviour for regions I, II,
III and IV. The dark regions indicate lower values than the light regions.
The parameter values are $\lambda=1.0,\,\alpha=-2.0$ for region I, $\lambda=2.0,\,\alpha=2.0$ for region II, $\lambda=0.5,\,\alpha=0.5$ for region III
and $\lambda=0.5,\,\alpha=3.0$ for region IV.
In region I there are local minima for $4\phi/N=\pm\pi$.
Besides the minima at $4\phi/N=\pm\pi$,
two additional fixed points (a maximum and a saddle point)
are apparent in region II occurring at $\phi=0$. In region III there are minima at $4\phi/N=\pm\pi$
and for $\phi=0$ just one fixed point, a maximum. There are also saddle points for when $z=-1$.
In region IV just one fixed point, a maximum, occurs for $\phi=0$, which always has $z<1$. In contrast the global
minimum occurs for $z=-1$.
}
\label{levelcurve1}
\end{figure}
Hereafter we will focus most attention on the case where $\lambda=0$, so the model has one effective coupling parameter, $\alpha$. From Figs.
\ref{curvekdiff}, \ref{parametro1}, it can be seen that for this submanifold there are no bifurcations when the atomic imbalance is non-zero, with bifurcations occuring at $\alpha=\pm 1$ when the atomic imbalance is zero. For the case when the atomic imbalance is non-zero, the global minimum of the classical Hamiltonian (\ref{ham2}) occurs when $\phi=N\pi/4$ and $z$ is the unique solution of (\ref{sol1}). In particular, for the solution $z\in[2k-1,1]$ $dz/d\alpha$ is a continuous function of $\alpha$.
When the atomic imbalance is zero and $\alpha>1$, the global minimum of the classical Hamiltonian (\ref{ham2}) always occurs at the phase space boundary $z=-1$ with $\phi$ arbitrary. At $\alpha=1$ a bifurcation occurs, and for $\alpha$ slightly less than 1 two saddle points arise for $z=-1$ with $\phi$ given by solution to (\ref{ssol3}) and a new global minimum emerges corresponding to $\phi=N\pi/4$ with $z$ the unique solution of (\ref{ssol1}). In this case $dz/d\alpha$ is discontinuous at $\alpha=1$.
In the following sections we will conduct an analysis of the quantum Hamiltonian (\ref{ham}). In particular we will establish that the bifurcation occuring at $(\alpha,\lambda)=(1,0)$ when the atomic imbalance is zero can be seen to influence the ground-state properties of the quantum system. In the context of the quantum system we will refer to the boundaries in Figs. \ref{curvekdiff},
\ref{parametro1} as {\it threshold couplings}. We avoid using the terminology {\it quantum phase transition} as the analysis is conducted for finite particle number, not in the thermodynamic limit. The reason for not taking the thermodynamic limit is that the quantities $\lambda$ and $\alpha$ are dependent on $N$. Additionally, in the thermodynamic limit $N\rightarrow\infty$ with $k$ finite the semi-classical results predict qualitative differences between the cases $k=0$ and $k\neq 0$. However if $N$ is odd then we cannot have $k=0$, raising technical issues about whether the limit is convergent. Consequently we only consider the case of finite particle number. To deal with the subtleties of the finite size of the system we will formally define a quantum phase {\it pre-transition} in Sect. \ref{wo}.
\section{Exact solution of the quantum Hamiltonian}
We now turn our attention to a quantum mechanical treatment of the model, to investigate the nature of the additional
threshold couplings when the atomic imbalance is zero. First we derive an exact Bethe ansatz solution of the model, and then use this to map the spectrum of the Hamiltonian (\ref{ham}) into the spectrum of a one-body Schr\"odinger operator.
\subsection{Energy eigenvalues as roots of a polynomial equation}
We rewrite the system Hamiltonian in a compact form as
\begin{equation}
H=U+\Omega(a^{\dagger}b^{\dagger}c+c^{\dagger}ba)
\end{equation}
where the operator $U$ is a function of the number operators:
\begin{align*}
U&=U_{aa}N_{a}^{2}+U_{bb}N_{b}^{2}+U_{cc}N_{c}^{2}+U_{ab}N_{a}N_{b}+U_{ac}N_{a}N_{c}+U_{bc}N_{b}N_{c}\\
&\qquad+\mu_{a}N_{a}+\mu_{b}N_{b}+\mu_cN_c.
\end{align*}
Since the operators $N$ and $k=J/N$ are conserved we fix these and without loss of generality consider cases where
$k\geq 0$. This restricts the Hilbert space to a subspace of dimension $(m+1)$ spanned by the vectors
\begin{equation}
\left|{l-j};{m-j};{j}\right>
\label{basis}
\end{equation}
where we have defined
\begin{align}
l=\frac{N(1+k)}{2},\nonumber\\
m=\frac{N(1-k)}{2}\nonumber
\end{align}
such that $l+m=N$.
We then look for eigenstates of (\ref{ham}) of the form
\begin{equation}
\label{eq:gstate}
|\psi\rangle=\sum_{j=0}^{m}\rho_j \left|{l-j};{m-j};{j}\right>
\end{equation}
Since the basis states (\ref{basis}) are eigenstates of each of the number operators they are eigenstates
of the operator $U$ so we can define the quantities ${\mathcal U}_j$ through
\begin{equation}
U \left|{l-j};{m-j};{j}\right>
={\mathcal U}_j \left|{l-j};{m-j};{j}\right>.
\end{equation}
The Hamiltonian acts on the general state (\ref{eq:gstate}) as
\begin{align}
H |\psi\rangle &=\sum_{j=1}^{m-1} ({\mathcal U}_{j}\rho_j + \Omega((j+1)\rho_{j+1} + (l+1-j)(m+1-j)\rho_{j-1}))
\left|l-j;m-j;j\right> \nonumber\\
& \qquad +({\mathcal U}_0 \rho_0 + \Omega\rho_1)\left|l;m;0\right> +
({\mathcal U}_m \rho_m + \Omega \rho_{m-1}(l-m+1)) \left|l-m;0;m\right> \label{Hside}
\end{align}
Requiring that (\ref{eq:gstate}) is an eigenstate of the Hamiltonian with energy eigenvalue $E$
leads to the following recursion relations that must be satisfied by coefficients $\rho_j$:
\begin{subequations}
\label{recrels}
\begin{align}
\Omega\rho_1+{\mathcal U}_0\rho_0&=E\rho_0,\label{rho0} \\
\Omega((j+1)\rho_{j+1}+(l+1-j)(m+1-j)\rho_{j-1})+{\mathcal U}_j\rho_j&=E\rho_j,
\label{rhoj} \\
{\mathcal U}_m\rho_m +\Omega(l-m+1)\rho_{m-1} &= E \rho_m, \label{rhom}
\end{align}
\end{subequations}
where $1<j<m-1$ in (\ref{rhoj}).
As the normalisation of the state (\ref{eq:gstate}) can be chosen arbitrarily, we have the freedom to choose
$\rho_0=1$. The recursion relation (\ref{rhoj}) then shows that $\rho_j$ is a polynomial in $E$ of order $j$.
The constraint (\ref{rhom}) is thus a polynomial in $E$ of order $(m+1)$, whose roots are the energy eigenvalues of
(\ref{ham}). Since the number of roots is the same as the dimension of the subspace spanned by the vectors (\ref{basis}),
all energy eigenvalues are given by the roots of (\ref{rhom}).
\subsection{Bethe ansatz solution and mapping to a Schr\"odinger equation}
With the above implicit form for the energy eigenvalues we are able to map the energy spectrum into that
of a one-dimensional Schr\"odinger equation. We start by mapping the energy eigenstates to polynomial solutions
of a particular second-order ordinary differential equation (ODE) and then utilise a change of variables such that the differential equation takes the form of the
Schr\"odinger equation.
Each eigenstate of the system (\ref{eq:gstate}) can be represented by an $m^{\text{th}}$ order polynomial with coefficients $\rho_j$ ($j=1,2,..m.$). For a particular energy, we can then construct an ODE for $G(u)$ such that the polynomial coefficients must satisfy the recursion relations of (\ref{recrels}). Below we outline the details of this construction.
Consider a general second-order ODE eigenvalue problem satisfied by an $m^{\text{th}}$ polynomial function $G(u)$:
\begin{equation}
a(u)G^{''}+b(u)G'+c(u)G=EG
\label{ODEform}
\end{equation}
First we write the polynomial $G(u)$ with roots $\{u_p\}_{p=1}^m$ in the factorised form
\begin{equation*}
G(u)=\prod_{p=1}^{m}(u-u_{p})
\end{equation*}
such that
\begin{eqnarray*}
G'(u)&=&\sum_{p=1}^m\prod_{q\neq p}^m(u-u_q), \\
G''(u)&=&\sum_{p=1}^m\sum^m_{q\neq p}\prod^m_{\substack{r\neq p\\r\neq q}}(u-u_r).
\end{eqnarray*}
Evaluating (\ref{ODEform}) at the root $u_q$ leads to the Bethe ansatz equations
\begin{equation}
\label{BAeqns}
\frac{b(u_q)}{a(u_q)}=\sum_{p\neq q}^{m}\frac{2}{u_p-u_q},\hspace{1cm}q=1,2,...,m.
\end{equation}
Hence, the roots of the polynomial must satisfy the system of coupled equations (\ref{BAeqns}) if $G(u)$ is a solution to (\ref{ODEform}).
We can map the solutions of (\ref{ODEform}) with eigenvalue $E$ to solutions of a Schr\"odinger equation
\begin{equation}
\label{shro}
\frac{-d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x)
\end{equation}
with the same eigenvalues,
by mapping the polynomial solution of (\ref{ODEform}) to a wavefunction of (\ref{shro}) via
\begin{equation*}
\psi(x)=e^{f(x)}{G(u(x))}.
\end{equation*}
Substituting into the Schr\"odinger equation gives the following relations to be satisfied
\begin{subequations}
\label{mapp}
\begin{align}
a(u)&=-\left(\frac{du}{dx}\right)^2 \label{a}\\
b(u)&=-\frac{d^2u}{dx^2}-2\frac{du}{dx}\frac{df}{dx}\label{b}\\
c(u)&=V(x)-\frac{d^2f}{dx^2}-\left(\frac{df}{dx}\right)^2\label{c}
\end{align}
\end{subequations}
In view of the above discussions, we now formulate the Bethe ansatz solution for (\ref{ham}) and the associated
mapping to a Schr\"odinger equation.
To simplify the notation, we define
\begin{equation*}
{\mathcal U}_j = A(m-j)(m-j-1)+ B(m-j)+ C
\end{equation*}
where
\begin{align*}
A&=U_{aa} +U_{bb} +U_{cc} +U_{ab}-U_{ac}-U_{bc}\\
B&=(1+2l-2m)U_{aa}+U_{bb} +(1-2m)U_{cc} +(1+l-m)U_{ab}\\
&~~~~~~+(2m-l-1)U_{ac} +(m-1)U_{bc}+\mu_a+\mu_b-\mu_c \\
C&=(1-m)^2U_{aa}+m(l-m)U_{ac} +m^2 U_{cc}+(m-l)\mu_a +m \mu_c.\\
\end{align*}
The polynomial defined as
\begin{equation}
G(u)=\sum_{j=0}^{m}\rho_{j}u^{m-j},
\end{equation}
with the $\rho_j$ satisfying (\ref{rho0},\ref{rhoj},\ref{rhom}), is a solution to the following differential equation
\begin{equation}
(Au^2+\Omega u)G^{''}+(Bu+\Omega(l-m+1-u^2))G^{'}+(\Omega mu+C)G=EG.
\label{ODEG}
\end{equation}
The roots of $G(u)$ are solutions of the Bethe ansatz equations
\begin{equation}
\frac{\Omega(l-m+1-u^2_q)+Bu_q}{u_q(\Omega+Au_q)}=\sum^m_{p\neq q}\frac{2}{u_p-u_q},\hspace{1cm}q=1,2,...,m.
\end{equation}
We can also derive an expression for the energy eigenvalues of the model in terms of the roots $u_q$. Consider the leading order expansions
\begin{align*}
G(u)&=u^m- u^{m-1} \sum_{q=1}^{m} u_{q} +...\\
G'(u)&=m u^{m-1}-(m-1)u^{m-2}\sum_{q=1}^m u_{q}+...\\
G''(u)&=m(m-1)u^{m-2}-(m-1)(m-2)u^{m-3} \sum_{q=1}^m u_{q} +...\\
\end{align*}
We substitute these expressions into (\ref{ODEG}) and equate terms of order $m$ to arrive at the following expression for the energy eigenvalues of the system
\begin{equation}
E=A m(m-1)+ Bm+C-\Omega \sum_{q=1}^{m}u_{q}
\end{equation}
Next we determine the explicit form of the Schr\"odinger equation.
Comparing (\ref{ODEG}) to (\ref{ODEform}) gives
\begin{align*}
a(u)&=Au^2 + \Omega u\\
b(u)&=(l-m+1-u^2)\Omega+Bu\\
c(u)&=mu\Omega+C
\end{align*}
Using (\ref{a},\ref{b},\ref{c}) we may perform the mapping to the Schr\"odinger equation by choosing
\begin{align*}
\frac{du}{dx}&=\pm\sqrt{-Au^2-\Omega u}\\
\end{align*}
Integrating this expression (with a convenient choice for the constant of integration) gives
\begin{align}
u=\frac{\Omega}{2A}(\cos({\sqrt{A}x})-1)
\end{align}
We also find that
\begin{align*}
\frac{df}{dx}&=\frac{\Omega^2}{4A^{\frac{3}{2}}}\sin({\sqrt{A}x})+\left(\sqrt{A}(l-m+1)-\frac{B}{2\sqrt{A}}-\frac{\Omega^2}{2A^{3/2}}\right)\csc(\sqrt{A}x)\\
&\qquad+\left(\frac{-\sqrt{A}}{2}+\frac{\Omega^2}{2A^{{3}/{2}}}+\frac{B}{2\sqrt{A}}\right)\cot{(\sqrt{A}x)}
\end{align*}
So the wavefunction
\begin{equation}
\Psi(x)=\exp\left(f(x)\right)\prod_{p=1}^{m}\left(\frac{\Omega}{2A}(\cos({\sqrt{A}x})-1)-u_p\right)
\label{wf}
\end{equation}
satisfies the Schr\"odinger equation (\ref{shro}) with potential
\begin{align*}
V(x)&=mu\Omega+C+\frac{d^2f}{dx^2}+\left(\frac{df}{dx}\right)^2\\
&=\left(C+\frac{\Omega^2}{2A}(l-2m+2)-\frac{\Omega^4}{2A^3}-\frac{A}{4}
+B\left(\frac{1}{2}-\frac{3\Omega^2}{4A^2}-\frac{B}{4A}\right)\right)\\
&\quad+\frac{\Omega^4}{16A^3}\sin^2(\sqrt{A}x)+\frac{\Omega^2}{2A}\left(m+\frac{\Omega^2}{2A^2}+\frac{B}{2A}\right)
\cos({\sqrt{A}x})\\
&\quad+\left(\frac{3A}{4}+A(l-m+1)^2+\frac{\Omega^2}{A}(l-m+1)+\frac{\Omega^4}{2A^3}-\frac{\Omega^2}{A}
+B\left(\frac{B}{2A}+\frac{\Omega^2}{A^2}-1\right)\right) \\
&\qquad\qquad \times \csc^2({\sqrt{A}x})\\
&\quad+\left(\left(\frac{\Omega^2}{A}+B\right)(l-m+2)-2A(l-m+1)-\frac{\Omega^4}{2A^3}
-B\left(\frac{B}{2A}+\frac{\Omega^2}{A^2}\right)\right)\\
&\qquad\qquad \times \cot(\sqrt{A}x)\csc(\sqrt{A}x).
\end{align*}
The above potential is an example of a quasi-exactly solvable potential \cite{quasi}, whereby a finite number of eigenstates of the form (\ref{wf}) can be constructed. The concept of mapping the spectrum of many-body systems into those of one-body Schr\"odinger equations has been discussed in detail in \cite{uz}.
\section{Analysis in the no scattering limit}
In this section we now conduct a deeper analysis of the Hamiltonian in the no scattering limit where
$U_{jk}=0$ for all $j,k=a,b,c.$ In this limit the model simplifies substantially, yet remains sufficiently non-trivial to enable us to gain an understanding of the quantum behaviour through the Schr\"odinger equation mapping, ground-state expectation values and quantum dynamics. Specifically the no scattering limit corresponds to the coupling $\lambda=0$ in the semi-classical analysis of Sect. \ref{sca}. With reference to Fig. \ref{parametro1} there are two threshold couplings in the case of zero atomic imbalance. One occurs at $(\alpha,\lambda)=(1,0)$, signifying the bifurcation of the global minimum of the Hamiltonian, while the other occurs at
$(\alpha,\lambda)=(-1,0)$, signifying the bifurcation of the global maximum. In contrast there are no bifurcations along the line $\lambda=0$ in Fig. \ref{curvekdiff}. We focus our attention to the coupling $(\alpha,\lambda)=(1,0)$ as the bifurcation of the fixed point in phase space is associated with the ground state of the quantum system.
\subsection{Schr\"odinger equation mapping} \label{sem}
For small values of $A$, we can take series expansions for the trigonometric functions in the potential $V(x)$ and wavefunction $\Psi(x)$.
Then taking the limit $A\rightarrow 0$ (corresponding to $\lambda=0$ for the analogous classical system (\ref{ham2})) the Schr\"odinger potential becomes
\begin{eqnarray}
V(x)&=&C-\frac{B}{2}(N+1)+ \left(J^2-\frac{1}{4}\right)x^{-2} \nonumber \\
&&\qquad + \left( \frac{B^2}{16} - \frac{ \Omega^2}{8}(N+2) \right)x^2
+ \frac{\Omega^2B}{32}x^4 +\frac{\Omega^4}{256}x^6
\label{semigeneralpotential}
\end{eqnarray}
where we now parametrise the system in terms of the variable $J=l-m$ and $N=l+m$.
Now consider a simple subclass of the general Hamiltonian (\ref{ham})
\begin{equation}
H=\mu N_c +\Omega(a^{\dagger}b^{\dagger}c+c^{\dagger}ba)
\label{hamres}
\end{equation}
We have mapped the general model to a Schr\"odinger equation in the previous section.
The above case (\ref{hamres}) corresponds to $A=0$, $B=-\mu$ and $C=m\mu$ in equation (\ref{semigeneralpotential}). The energy eigenstates map to solutions of the Schr\"odinger equation with potential
\begin{align}
\label{genpot}
V(x)&=\frac{\mu(N+1)}{2}+ \left(J^2-\frac{1}{4}\right) x^{-2}\nonumber\\
&\qquad + \frac{1}{16}\left(\mu^2-2\Omega^2 (N+2)\right) x^2 -\frac{\mu \Omega^2}{32} x^4+ \frac{\Omega^4}{256} x^6.
\end{align}
The associated wavefunction is given by
\begin{equation}
\Psi(x)=x^{(J+1/2)}\exp\left(\frac{-\Omega^2 x^4}{64}+\frac{\mu x^2}{8}\right)\prod_{p=1}^{m}\left(\frac{-\Omega x^2}{4}-u_p\right)
\label{wf1}
\end{equation}
with energy eigenvalues
\begin{equation}
E= -\Omega \sum_{q=1}^{(N-J)/2}u_{q}
\end{equation}
where the $\{u_q\}$ are solutions to the Bethe ansatz equations
\begin{equation}
\frac{(J+1)}{u_q}-u_q-\frac{\mu}{\Omega}=\sum_{p\neq q}^{(N-J)/2}\frac{2}{u_p-u_q},\hspace{1cm}q=1,2,...,(N-J)/2.
\end{equation}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=7cm,height=7cm]{figures/imbalancepaper.eps}
&\quad &
\includegraphics[width=7cm,height=7cm]{figures/noimbalance.eps}
\\
(a) &\quad & (b) \\
\end{tabular}
\end{center}
\caption{ The potential $V(x)$ given by (\ref{genpot}). (a) $N=501$ and atomic imbalance $J=1$. The potential is bounded from below, with the inset showing
$V\rightarrow\infty $ as $x\rightarrow 0$. Varying the coupling parameter $\alpha$ across the threshold value
$\alpha=1$, it is apparent there is no bifurcation of the potential minimum. (b) $N=500$ and atomic imbalance $J=0$. The potential is not bounded below, with the inset showing $V\rightarrow-\infty $ as $x\rightarrow 0$.
As the coupling parameter $\alpha$ is varied across the threshold value $\alpha=1$, it is apparent there is a bifurcation with the formation of a local minimum and a local maximum for $\alpha<1$. }
\label{potentials}
\end{figure}
The semi-classical analysis predicts a threshold coupling at $\alpha=1$ when the atomic imbalance is zero where for the case under consideration
$\alpha=-\mu/(\Omega\sqrt{2N})$. When the atomic imbalance is non-zero there is no predicted threshold coupling.
Fig. \ref{potentials} (a) depicts the potential (\ref{genpot}) for $N=501,\, J=1$ and various values of $\alpha$ close to the threshold value $\alpha=1$.
It can be seen that the potential has a single minimum for all $\alpha$. In contrast, Fig. \ref{potentials} (b) shows (\ref{genpot}) for $N=500$ and $J=0$. For this case the potential is not bounded from below and there is a bifurcation for $\alpha\approx 1$. For the model (\ref{hamres}), the predictions of the semi-classical analysis conducted in Sect. \ref{sca} of a threshold coupling at $\alpha\approx 1$ are consistent with qualitative differences of the associated Sch\"odinger equation.
Now we examine bifurcations of the critical points of the potential (\ref{genpot}) in more detail. Consider the general class of potentials
\begin{eqnarray}
V(x)&=& {\mathcal A} x^{-2}+{\mathcal B} x^2+{\mathcal C} x^4 +{\mathcal D} x^6 \label{generic}
\end{eqnarray}
where ${\mathcal C},\,{\mathcal D}$ are assumed to be positive and no constraints are placed on ${\mathcal A}$ nor ${\mathcal B}$. In particular we wish to determine when the condition
\begin{eqnarray}
\frac{dV}{dx}=\frac{d^2V}{dx^2}=0 \label{condition}
\end{eqnarray}
can be met. Since the potential is a symmetric function, we restrict to non-negative values of $x$. When ${\mathcal A}=0$ it is straightforward to deduce that, for any values of ${\mathcal C}$ and ${\mathcal D}$,
(\ref{condition}) holds at $x=0$ when ${\mathcal B}=0$.
For non-zero values of ${\mathcal A}$ we find
\begin{eqnarray}
\frac{dV}{dx}&=& -2{\mathcal A} x^{-3}+2{\mathcal B} x +4{\mathcal C} x^3 +6{\mathcal D} x^5 \label{dvdx} \\
\frac{d^2V}{dx^2}&=& 6{\mathcal A} x^{-4} +2{\mathcal B} +12 {\mathcal C} x^2 +30{\mathcal D} x^4 \label{d2vdx2}
\end{eqnarray}
We set both (\ref{dvdx}) and (\ref{d2vdx2}) to zero and take particular linear combinations to obtain the following relations:
\begin{eqnarray}
\frac{1}{8}\frac{d^2V}{dx^2}+\frac{3}{8x}\frac{dV}{dx}&=& {\mathcal B} + 3 {\mathcal C} x^2 +6 {\mathcal D} x^4 =0\label{bcd} \\
\frac{x^4}{8}\frac{d^2V}{dx^2}-\frac{5x^3}{8}\frac{dV}{dx}&=& 2 {\mathcal A} - {\mathcal B} x^4 -{\mathcal C} x^6=0 \label{abc}
\end{eqnarray}
Note that eq. (\ref{bcd}) is independent of ${\mathcal A}$, and has solutions
\begin{eqnarray}
x^2&=&\frac{-3{\mathcal C}\pm\sqrt{9{\mathcal C}^2-24{\mathcal B}{\mathcal D}}}{12{\mathcal D}}. \label{x}
\end{eqnarray}
We take the positive square root in (\ref{x}) and impose ${\mathcal B}<0$, ensuring $x^2>0$.
Treating ${\mathcal B}$ as a small parameter such that
\begin{eqnarray}
\left|{\mathcal B}\right|\ll \frac{{\mathcal C}^2}{{\mathcal D}} \label{muchless}
\end{eqnarray}
yields
\begin{eqnarray}
x^2\approx-\frac{{\mathcal B}}{3{\mathcal C}}. \label{z}
\end{eqnarray}
Next we substitute (\ref{z}) into (\ref{abc}) and solve for ${\mathcal B}$:
\begin{equation}
{\mathcal B}=3({\mathcal A}{\mathcal C}^2)^{1/3}. \label{B}
\end{equation}
Since ${\mathcal B}$ is negative such a solution only exists when ${\mathcal A}$ is negative.
Matching the co-efficients between (\ref{genpot}) and (\ref{generic}) gives
\begin{eqnarray*}
{\mathcal A}&=& J^2-\frac{1}{4} \\
{\mathcal B}&=& \frac{1}{16}\left(\mu^2-2\Omega^2(N+2)\right) \\
{\mathcal C}&=& -\frac{\mu \Omega^2}{32}\\
{\mathcal D}&=& \frac{\Omega^4}{256}
\end{eqnarray*}
When $\alpha\approx 1$, or equivalently $\mu\approx -\Omega\sqrt{2N}$, we satisfy the requirements ${\mathcal C},\,{\mathcal D}>0$ and
(\ref{muchless}). A bifurcation of the potential only occurs when the atomic imbalance $J$ is zero, as ${\mathcal A}$ must be negative. Using
(\ref{B}) we then find $\mu$ satisfies
$$3\mu^{2/3}\Omega^{4/3}+\mu^2=2\Omega^2(N+2). $$
From this expression we determine the leading quantum correction to the semi-classical result for the threshold coupling
for (\ref{hamres}):
$$\mu\approx-\Omega (2(N+2))^{1/2}+\frac{3\Omega}{2}(2(N+2))^{-1/6}. $$
\subsection{Ground-state expectation values and quantum dynamics}
\begin{figure}[ht]
\begin{center}
\epsfig{file=figures/correlationfunction.eps,scale=0.45}
\end{center}
\caption{Ground-state expectation values of the (scaled) molecular number operator $N_c$ as a function of the coupling $\mu$, for the Hamiltonian
(\ref{hamres}). Results shown correspond to $\Omega=1$ and both zero and non-zero atomic imbalance. The inset shows the first derivative of the expectation values with respect to the coupling $\mu$. While there are quantitative differences there is no significant qualitative change between the case of zero and minimal non-zero atomic imbalance.}
\label{expectationvalue}
\end{figure}
Next we examine the behaviour of the ground state of (\ref{hamres}) as the threshold coupling $\alpha=1$ is crossed. From the semi-classical analysis we have found that the global minimum for $\alpha>1$ and $J=0$ occurs at $z=-1$ in phase space. In the Hilbert space of states this corresponds to $|0;0;N/2 \rangle$. It is then appropriate to compute
the gound-state expectation value $\langle N_c\rangle$ for the quantum system as the coupling is varied. Results are shown in Fig. \ref{expectationvalue}. In general agreement with the semi-classical result, it can be seen that the expectation value $\langle N_c\rangle/N$ is close to unity
when $-\mu>\Omega\sqrt{2N}$ for the case of zero atomic imbalance. When $-\mu<\Omega\sqrt{2N}$ the expectation value decreases. The figure also shows that when the imbalance $J=1$ the results are qualitatively similar. However from the predictions of both the semi-classical analysis of Sect. \ref{sca} and the the associated one-body Schr\"odinger potential of Sect. \ref{sem} we do not obtain any prediction about the change in the ground-state properties when the imbalance is non-zero. Further increase in the value of $J$ (not shown) does not indicate any dramatic change in the qualitative features of $\left<2 N_c\right>/N$ as a function of $J$. As mentioned previously, because we are studying a finite system changes in the ground-state properties are smooth as $J$ is varied. If we instead look at the quantum dynamics as the threshold coupling is crossed, qualitative differences are more apparent.
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc}
\epsfig{file=figures/Fig7a.eps,width=7cm,height=8cm,angle=0} &
\epsfig{file=figures/Fig7b.eps,width=7cm,height=8cm,angle=0} \\
(a) & (b) \\
\end{tabular}
\end{center}
\caption{Time evolution of the expectation value of $z$ for the Hamiltonian (\ref{hamres}) with $N=500$. The cases shown are, from top to bottom, $\alpha=0.9,\,0.95,\,1,\,1.05,\,1.1$. (a) $J=0$ and initial state $|0;0;250\rangle$. The oscillations are largely irregular with significantly decreasing amplitude as the point at $\alpha=1$ is crossed. This point corresponds to the boundary at $(\alpha,\lambda)=(1,0)$ between regions $III$ and $IV$ as shown in Fig. \ref{parametro1}.
(b) $J=10$ with initial state
$|10;0;245\rangle$. The oscillations display collapse and revival behaviour with smoothly decreasing amplitude as the point at $\alpha=1$ is crossed, indicative of the fact there is no boundary at $(\alpha,\lambda)=(1,0)$ in Fig. \ref{curvekdiff}.}
\label{quantumdynamics}
\end{figure}
In general the time evolution of any state is given
by $|\Psi(t) \rangle = U(t)|\phi \rangle$,
where $U(t)$ is the temporal evolution operator $\displaystyle U(t)=\sum_{j=0}^{m}|j\rangle \langle j|\exp(-i E_{j} t)$,
$|j\rangle$ is an eigenstate with energy $E_{j}$ and $|\phi \rangle$ represents
the initial state with $N=N_a+N_b+2N_c$.
We adopt the method of directly diagonalising the Hamiltonian (\ref{hamres}) as done in \cite{fitonel},
and compute the expectation value of $z(t)$ through
$$
\langle z(t)\rangle=\frac{1}{N}\langle \Psi (t)|N_a+N_b-2N_c|\Psi (t)\rangle.
$$
For a fixed atomic imbalance $J$ we will use the initial state configuration $|J;0;(N-J)/2 \rangle$. When $J=0$ this state correspond to $z=-1$ in phase space, which is a fixed point when $\alpha>1$. We thus expect that in this case
$\langle z(t)\rangle$ will not vary significantly in time (i.e. the system is {\it localised}).
On the other hand when $J\neq 0$ the state
$|J;0;(N-J)/2 \rangle$ does not correspond to a fixed point.
We therefore compare the two cases of the quantum dynamics, with atomic imbalance $J=0$ and $J\neq 0$, as the value $\alpha=1$ is crossed.
We fix the parameter $\Omega=1$ and use $\mu$ as the variable coupling parameter.
Results of the expectation value for $z$ are shown in Fig. \ref{quantumdynamics} for the cases of zero and non-zero atomic imbalance. The qualitative difference are quite apparent. In the case of zero atomic imbalance ($k=0$), Fig. \ref{quantumdynamics} (a), we find that for $\alpha<1$ there are irregular oscillations in $z$. By comparison the dynamics in Fig. \ref{quantumdynamics} (b) for non-zero imbalance ($k=0.02$) show a collapse and revival of oscillations. As the coupling parameter $\alpha$ is increased across the threshold value at $\alpha=1$, the transition to localised oscillations is much sharper in case (a) compared to case (b). Note in particular the vertical scales in (a) and (b) are not the same.
We remark that the nature of the dynamics for $\alpha>1$ {\it does} change smoothly from localisation to delocalisation over the intermediate values $0<k<0.02$ (not shown).
Taking the thermodynamic limit $N\rightarrow \infty$ does not aid in the analysis. For the Hamiltonian (\ref{hamres}) the condition for localisation of oscillations for zero atomic imbalance, $\alpha> 1$, is equivalent to $-\mu>\Omega\sqrt{2N}$. Hence for fixed $-\mu>0$ and $\Omega>0$ this condition imposes an upper bound on $N$ for which localisation occurs.
\section{Wavefunction overlaps} \label{wo}
\begin{figure}[ht]
\begin{center}
\epsfig{file=figures/iodi.eps,scale=0.25,angle=-90}
\end{center}
\caption{Ground-state wavefunction overlaps for the Hamiltonian (\ref{hamres}) with $N=1000$ and various values of $\Delta$. The value of the local minimum at $\alpha\approx 1$ is a decreasing function of $\Delta$, asymptotically approaching zero.}
\label{gsoverlaps1}
\end{figure}
In order to gain a better insight into the effect of the threshold couplings for the quantum system, in our final analysis we adopt the method of wavefunction overlaps \cite{zanardi,huanandjp}.
If a system admits a quantum phase transition, then two states belonging to different phases of the same system are distinguishable. If states are distinguishable they must be orthogonal~\cite{nc} and consequently the wave function overlaps vanish. For systems which exhibit a quantum phase transition in the thermodynamic limit, the wavefunction overlaps between states in different phases go to zero in this limit. The occurrence of a minimum in the incremental ground-state wavefunction overlap in a finite system is then a precursor for a quantum phase transition in the thermodynamic limit. Thus for finite systems we identify quantum phase {\it pre-transitions} at couplings for which the incremental wavefunction overlap is (locally) minimal.
\begin{figure}[ht]
\begin{center}
\begin{tabular}{ccc}
\epsfig{file=figures/molec_BEC_replot_Figure12_b.eps,scale=0.25,angle=-90} &\quad&
\epsfig{file=figures/molec_BEC_Fig8_replotted.eps,scale=0.25,angle=-90} \\
(a) &\quad & (b) \\
\end{tabular}
\end{center}
\caption{ (a) Ground-state wavefunction overlaps of the Hamiltonian (\ref{hamres}), for $N=500,1000,1500$. The solid lines correspond to cases when the atomic imbalance is zero, while the dashed lines illustrate the behaviour for the fractional imbalance
$k=0.02$. Two general properties that can be observed at the pre-transition coupling $\alpha\approx 1$ are (i) the minimum value decreases with increasing $N$; (ii) for fixed $N$, the value of the minimum is lower for $k=0$ compared to $k\neq 0$.
(b) Ground-state wavefunction overlaps of the Hamiltonian (\ref{hamres}) for $N=1000$ and different values of $\lambda$.
The solid lines correspond to cases when the atomic imbalance is zero, while the dashed lines illustrate the behaviour for the fractional imbalance $k=0.02$. The locations of the minima fit the line of threshold couplings given by $\lambda=(\alpha-1)/2$ as predicted by the semi-classical analysis.}
\label{gsoverlaps2}
\end{figure}
We now formally define
a quantum phase pre-transition in terms of ground-state wavefunction overlaps. Let $H(\delta)$ denote a generic Hamiltonian depending on a coupling parameter $\delta$. Assuming the ground state of the system is non-degenerate,
let $\left|\Psi(\delta)\right>$ denote the unique normalised ground state. For fixed small $\Delta$ we define the function
$W_{\Delta}(\delta) $ by
\begin{eqnarray*}
W_{\Delta}(\delta) = \left|\left<\Psi(\delta(1-\Delta))|\Psi(\delta(1+\Delta))\right>\right|
\end{eqnarray*}
which is symmetric in $\Delta$, bounded between 0 and 1, and satisfies $W_0(\delta)=1$.
Generically, $W_{\Delta}(\delta)$ is a decreasing function of $\Delta$. Fig. \ref{gsoverlaps1} shows the behaviour of the wavefunction overlaps for the Hamiltonian (\ref{hamres}) with $N=1000$, and different values of $\Delta$. It is clear that there is a distinct dip in the quantity $W_{\Delta}(\alpha)$ near the threshold coupling $\alpha=1$. The choice for
$\Delta$ affects the magnitude of the minimum, which can be made arbitrarily small. In Fig. \ref{gsoverlaps1} when $\Delta=0.05$, representing a coupling change of about
$5\%$ on either side of the threshold coupling, the ground states are essentially orthogonal. However the value of $\alpha$ at which the minimum occurs is largely independent of $\Delta$.
For a given $\Delta$ we say that there is a quantum phase pre-transition at $\delta_c$ if $W_{\Delta}(\delta)$, treated as a single-variable function of $\delta$, is locally minimal at $\delta_c$.
We have computed the wavefunction overlaps for several cases with both zero and non-zero atomic imbalance. Fig. \ref{gsoverlaps2} (a) shows the behaviour of $W_{\Delta}(\alpha)$ with $\lambda=0,\,\Delta=0.01$ and varying $N$ for both $k=0$ and $k=0.02$. It is clear the minimum value of $W_{\Delta}(\alpha)$, which determines the quantum phase pre-transition, is at $\alpha\approx 1$. The distinction between the predicted threshold coupling and the observed pre-transition coupling is that the pre-transition coupling also occurs for $k\neq 0$, although for fixed $N$ the value of the minimum is lower for
$k=0$ compared to $k=0.02$. In all instances the value of the mimimum decreases with increasing $N$. Fig. \ref{gsoverlaps2} (b) shows similar results for fixed $N=1000$ and varying $\lambda$. In this latter case we see that the occurences of the mimima, determining the pre-transition couplings, fit well with the predicted boundary of threshold couplings given by $\lambda=(\alpha-1)/2$ (cf. Fig. \ref{parametro1}). However we again observe pre-transitions couplings for $k=0.02$ which were not predicted in Sect. \ref{sca}.
In the above cases the minimum of $W_{\Delta}(\alpha)$ is substantially more pronounced for zero imbalance compared to non-zero imbalance. We can see that although quantum phase pre-transitions as defined exist for $k\neq 0$, the distiguishability of the two phases is more reliable in the limit as $k$ goes to zero. In the analyses of Sect. \ref{sca} and Sect. \ref{sem} qualitative differences are only found precisely when $k=0$. We interpret these results as the emergence of quantum phase boundaries at $k=0$.
\section{Discussion}
We have studied the ground-state phases of a three-mode model describing a heteronuclear molecular Bose--Einstein condensate, through a variety of techniques. Using a semi-classical analysis we were able to determine threshold couplings associated with fixed point bifurcations in phase space. We then derived the exact Bethe ansatz solution for the system and discussed how the spectrum of the Hamiltonian maps into that of an associated one-body Schr\"odinger equation. It was shown that in the particular subcase of no scattering interactions the threshold coupling for the global minimum in phase space for zero atomic imbalance was consistent with the existence of a bifurcation of the potential of the Schr\"odinger equation. For the non-zero imbalance case where the semi-classical results do not predict any threshold coupling, it was found that there was no bifurcation of the Schr\"odinger potential. These results suggested that the ground-state properties of the model are sensitive to whether the atomic imbalance was zero or non-zero. However due to the finite nature of the system, such a transition is not associated with a discontinuity that is defined in the thermodynamic limit.
We then introduced the notion of a quantum phase pre-transition for finite systems, defined in terms of ground-state wavefunction overlaps. Applying this idea to the model under consideration we argued that a quantum phase boundary emerges in the limit as the atomic imbalance goes to zero.
For future work it would be useful to investigate the ground-state entanglement properties of the Hamiltonian. Not only would this be of interest in the study of the behaviour of the entanglement at the quantum phase boundary as $k$ approaches zero, but the model is also a simple example of a strictly tripartite system which may offer insights into the role of three-way entanglement in the description of quantum phases.
\section*{Acknowledgements}
E.C.M. thanks CAPES-Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'{\i}vel Superior for financial support.
A.T. and A.F. thank FAPERGS-Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado do Rio Grande do Sul for financial
support. A.F also acknowledges support from PRONEX under contract CNPq 66.2002/1998-99.
M.D., J.L. and N.O. are funded by the Australian Research Council through the Discovery Projects DP0557949 and DP0663773.
|
2,869,038,155,658 | arxiv | \section{Introduction}\label{sec:introduction}
\IEEEPARstart{E}{llipse} detection is a fundamental technique in image processing field and plays an indispensable role in shape detection and geometric measurement.
Actually, ellipse detector can be utilized to handle various real-world problems.
In PCB inspection field, one basic function of the defect detection machine is to precisely as well as fast locate the circular pads or holes. Moreover, accurate measurement of circular control points and elliptic fiducial markers is helpful to homography estimation and camera calibration~\cite{huang2016homography,calvet2016detection,heikkila2000geometric}, and some irregular objects could be fitted as ellipses to simplify the shape structure for efficient mathematical modeling \cite{crocco2016structure,da2010fitting,bai2009splitting,kothari2009automated}.
However, to our best knowledge, there exist few robust, stable, efficient and accurate ellipse detector algorithms to universally handle the ellipse detection problem in real-world images, which may have the presence of cluttered edges, motion blur, illumination, occlusion, noise and so on. The major reason is that an ellipse involves five parameters rather than that a circle just needs three, which results in detecting ellipse both efficiently and accurately to be a tough problem. Recently, convolutional neural network (CNN) \cite{lecun2015deep} is revolutionizing objection detection field, like Mask R-CNN \cite{he2017mask} and YOLO \cite{redmon2018yolov3}. Deep learning based methods can provide image proposals which contain oval objects for image pre-processing while they are still inappropriate to directly handle ellipse detection due to the issues of limited segmentation accuracy and expensive manual annotation. In most real-world applications, the practical requirements with regard to higher location accuracy and faster speed make ellipse detection problem even challenging.\par
\begin{figure*}[!t]
\centering
\includegraphics[width=\hsize]{pic//1.pdf}
\caption{A comparison of various ellipse detection methods\protect\footnotemark[1]. The processing time of various methods is counted on the same computer with Intel Core i7-7500U 2.7GHz CPU and 8 GB memory. Except that the methods marked with (*) are implemented in C++, the remaining methods are in MATLAB. (a) the origin image is with the resolution of $720 \times 435$; (b) shows the ground true ellipses; (c) RHT \cite{mclaughlin1998randomized} can detect the ellipses while easily generating duplicates; (d) most ellipses can be located by Prasad et al. method \cite{prasad2012edge} at the cost of long running time; (e) and (f) show that the methods proposed by Fornaciari et al. \cite{fornaciari2014fast} and Qi et al. \cite{jia2017fast} are very fast. However, both methods cause either missing detections or false positives; (g) ELSDc \cite{patraucean2017joint} can jointly detect ellipses, arcs, and line segments while suffering from a long time; (h) our proposed method could accurately detect the ellipses with competitive running time, which reveals its high-quality detection performance.}
\label{fig1}
\end{figure*}
Existing commonly used methods for ellipse detection can
be briefly grouped into two categories: 1) \text{\it{Hough Transform}}; 2) \text{\it{Edge Following}}.\par
Hough Transform (HT) has been widely used for detecting geometric primitives such as line segment (or LS for short), circle and ellipse~\cite{Duda1972Use}. The basic idea of HT for ellipse detection is voting arbitrary edge pixels into 5D parameter space. The local peak will occur when the corresponding bin of accumulator exceeds a threshold of votes, which implies for detecting an ellipse. But it is almost impractical to directly apply HT in practice due to the heavy computation burden and excessive consumption of memory. To alleviate these issues, considerable improved methods are put forward. Probabilistic Hough Transform (PHT) randomly selects a small subset of the edge points which is used as input for HT~\cite{kiryati1991probabilistic}, but large-scale attempts are taken to find the points all sharing a common ellipse and it leads to inferior performance when substantial noise exists. Yuen et al. decomposed the 5D parameter space by finding the ellipse center using some geometric properties like colinearity and symmetry on the first stage and then finding the remaining three parameters on the second stage~\cite{yuen1989detecting,tsuji1978detection}. Instead of transforming each edge point into a 5D parameter space, Xu et al.~\cite{xu1990RHT} proposed a Randomized Hough Transform (RHT) to detect curves, which randomly chooses five edge pixels each time and maps them into a point of the ellipse parameter space. McLaughlin et al. \cite{mclaughlin1998randomized} combined the aforementioned two-stage decomposition method and RHT at the aim of reducing the computation time and improving the detection performance compared with the standard HT, which becomes a baseline of ellipse detection method in the literature (Fig.~\ref{fig1}(c)). However, it is still not efficient enough in practice and always generates false detections due to the lack of novel validation strategies. Despite the simplicity of HT, HT based ellipse detection methods suffer from the following legacy problems: First, it is vulnerable in front of substantial image noise and complicated real-world background due to false peaks; Second, it requires much effort to tune the model parameters, e.g. bin size and peak threshold.\par
\footnotetext[1]{best viewed in color\label{footnote1}}
The second well-known family of ellipse detection methods is edge following, in which the connectivity between edge pixels, convexity of arc segments and geometric constraints are used. The general idea of edge following always starts from computing the \text{\it{binary edge}} map and corresponding gradients acquired by Canny or Sobel detector \cite{canny1986computational, plataniotis2013color} and then refining the arc segments from the \text{\it{binary edge}} for the ellipse fitting.\par
Many of edge following methods use line segments (LSs), which are extracted from the binary edges, as an intermediary to find the arc segments. The approach proposed by Kim et al. \cite{kim2002fast} merges the very short LSs to represent arc segments, where the arc fitting algorithms are frequently called. \cite{mai2008hierarchical} shares similar ideas with \cite{kim2002fast} to extract short LSs from the edge map while the difference lies in linking the LSs as well as the LS's edge points to form arc segments by using simple preset adjacency threshold and proper curvature condition. This method further iteratively groups two arc segments and applies the Random Sample Consensus (RANSAC) to the arc segment groups to recover the ellipse models. Although \cite{mai2008hierarchical} tries to promote the ellipse detector's robustness by iterative grouping and RANSAC, the massive missing detections (FNs) and false positives (FPs) appear.
The method proposed by Chia et al.~\cite{chia2011split} improves the framework illustrated in~\cite{mai2008hierarchical}, but a more complicated fragments merging and grouping procedures were employed. The merging of arc fragments is formalized as an alignment problem, where an alignment function is defined to score the rationality of merging, and a total cost function is built to incrementally search the optimal paired arc segments for grouping. Though the complex and iterative mathematical optimization boosts the detection performance to some extent, \cite{chia2011split} shows slow speed in the real-world images as reported in \cite{prasad2012edge,patraucean2017joint}. The ellipse detector proposed by Prasad et al. \cite{prasad2012edge} incorporates the edge curvature and convexity to extract smooth edge contours and performs a 2D HT to rank the edge contours in a group by the relationship scores for the better generation of ellipse hypotheses. But it also suffers from a long computation time, as shown in Fig.~\ref{fig1}(d).\par
Another stream of edge following methods tries to extract arc segments from binary edge directly and prunes straight edges for the purpose of fast detection speed. The ellipse detector proposed by Fornaciari et al. \cite{fornaciari2014fast} assigns a bounding box for each arc, removes the straight edges and determines the convexity of the arc by comparing the areas of region under and over the arc. In addition, this method accelerates the detection process by utilizing the property of that ellipse center should be colinear to the midpoints of parallel chords. However, it raises the detection speed at the cost of localization accuracy and robustness (Fig.~\ref{fig1}(e)). Recently, the method presented by Qi et al. \cite{jia2017fast} inherits \cite{fornaciari2014fast} and uses the similar convexity classification approach, but the difference lies in that \cite{jia2017fast} filters straight edges efficiently by calculating the edge connected component's characteristic number, which is a kind of projective invariant being able to distinguish the lines and conic curves within images. \cite{jia2017fast} is fast and yet prone to generate duplicates due to the absence of novel clustering (Fig.~\ref{fig1}(f)). In addition, both \cite{fornaciari2014fast} and \cite{jia2017fast} require at least three arc segments to recover the ellipse model, which might disable the algorithms when handling the incomplete ellipses.\par
Some researchers generalize the LS detection method to be a multi-functional detector which can jointly detect the LS and elliptic arcs. ELSDc proposed by P{\u{a}}tr{\u{a}}ucean et al. \cite{patraucean2017joint} uses an improved LSD \cite{grompone2010lsd} version for detecting LS, and then iteratively searches the remaining LSs from the start and end points of the detected LS. Eventually, both LS detection and grouping tasks are established simultaneously. Notably, ELSDc stands out other methods by detecting LSs from the \text{\it{greyscale image}} instead of \text{\it{binary edge}} such that abundant gradient and geometric cues can be fully exploited.
ELSDc and our proposed method are both based on LSD \cite{grompone2010lsd} for LS detection from the \text{\it{greyscale image}}, but they are fundamentally different from the generated LS type, ellipse candidates generation and validation strategies. Our method merely generates the arc-support LSs and do not chain them in the LS generation step. Moreover, ELSDc fits and validates the locally grouped LSs, omitting the global situation, which may be prone to produce the false positives (Fig.~\ref{fig1}(g)).\par
Arc-support LS is our previous work as introduced in \cite{Lu2017Circle}, each pair of which is successfully used for circle detection. However, it cannot handle the ellipse detection scheme since an arbitrary ellipse cannot be determined by two paired LSs. Admittedly, ellipse detection owns much higher complexity and requires more geometric cues. For example, the continuity feature, which is neglected in \cite{Lu2017Circle}, can be fully embodied in the arc-support group and is important in ellipse detection. Therefore, the careful arc-support groups forming, complicated geometric constraints, accurate ellipse generation and clustering, and novel validation strategy accustomed to ellipse detection are required, which will be addressed in this paper.\par
The main research purpose of this paper is to propose a high-quality ellipse detection method to handle the long-standing issue that cannot detect ellipses both accurately and efficiently in ellipse detection field. To that end, for the first time, we take the advantage of arc-support LSs for ellipse detection. The arc-support groups are formed by robustly linking the consecutive arc-support LSs which share similar geometric properties in point statistics level. Each arc-support group will be measured and assigned a saliency score. Secondly, we generate the initial ellipse set by two complementary approaches both locally and globally. The superposition principle of ellipse fitting and the novel geometric constraints, which are polarity constraint, region restriction and adaptive inliers criterion, are employed to consolidate the proposed method's accuracy and efficiency. Thirdly, we decompose the 5D ellipse parameter space into three subspaces according to ellipse center, orientation and semi-axes and perform three-stage efficient clustering. Finally, the candidates which pass the rigorous and effective verification will be refined by fitting again.\par
The rest of this paper is organized as follows. Section \ref{sec:Preliminary} introduces the preliminaries about arc-support LS and superposition principle of ellipse fitting. Section \ref{sec:ellipse detection} presents the high-quality ellipse detection framework, as a four-stage detection procedure: arc-support groups forming, initial ellipse set generation, clustering, and candidate verification. Section \ref{sec:complexity} analyzes the computation complexity of the proposed ellipse detection algorithm. Experimental results, as well as the accuracy and efficiency detection performance of the proposed method, are detailed in Section \ref{sec:experimental results}. Section \ref{sec:conclusion} concludes the paper.
\section{Preliminary}\label{sec:Preliminary}
In this section, the arc-support LS and its appendant properties are introduced as the basic geometric primitives for ellipse detection. Then we develop a superposition principle of fast ellipse fitting, which will save running time for the ellipse generation.
\subsection{Arc-support LS}
\begin{figure}[!t]
\centering
\includegraphics[width=\hsize]{pic//2.pdf}
\caption{Level-line angle and two types of LS. (a) the level-line angle is acquired by clockwise rotating the gradient angle $90^{\circ}$; (b) greyscale image; (c) straight and arc-support LSs generated from (b).}
\label{fig2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=\hsize]{pic//3.pdf}
\caption{Features of arc-support LS. (a) the overall gradient direction in the local greyscale area is same as arc-support direction and the three main angles in the corresponding level-line map change anticlockwise; (b) the conter-example of (a).}
\label{fig3}
\end{figure}
In image processing, LS mainly derives from two situations, as shown in Fig. \ref{fig2}. The first type LS comes from the support region where points share nearly the same level-line angle and overall distribute straight. Another type of LS derives from the arc-support region whose distribution changes like a curve. Thus, we call the LS approximated from arc-support region as ``arc-support LS''. Arc-support LS is built on top of LSD \cite{grompone2010lsd} as it is superior to other methods due to its efficiency and false control ability. The corresponding extraction procedures can be found in \cite{Lu2017Circle}. With the help of arc-support LS, the straight LS can be pruned while the arc geometric cues remain. Hereon, some properties of arc-support LS critical for ellipse detection are detailed.
\subsubsection{Arc-support Direction}
Different from conventional LS, arc-support LS carries the nature of convexity, standing for the ellipse center direction of an elliptic arc, namely the arc-support direction, as shown in Fig. \ref{fig3}. Assume that the two terminals of the circumscribed rectangle of the support region are $A$ and $B$ and the centroid is $C$. Thus the main angle of the support region is denoted as $\angle \overrightarrow{AB}$ and can be set to
\begin{equation}
\label{eq1}
\text{arctan}\left(\frac{{\sum\nolimits_{{p_i} \in \text{Region}} {\text{sin} (\text{level-line angle}({p_i}))} }}{{\sum\nolimits_{{p_i} \in \text{Region}} {\text{cos} (\text{level-line angle}({p_i}))} }}\right).
\end{equation}
Analogously, the main angles of two subregions $\angle \overrightarrow{AC}$ and $\angle \overrightarrow{CB}$ can be obtained according to Eq. (\ref{eq1}). Therefore, the arc-support direction can be set by anticlockwise (or clockwise) rotating $\angle \overrightarrow{AB}$ by $90^{\circ}$ if $\angle \overrightarrow{AC}$, $\angle \overrightarrow{AB}$ and $\angle \overrightarrow{CB}$ change in the anticlockwise (or clockwise) direction and have an angle interval at least $T_{ai}$ in \{$\angle \overrightarrow{AC}$,$\angle \overrightarrow{AB}$ \} and \{$\angle \overrightarrow{AB}$,$\angle \overrightarrow{CB}$ \}.
\subsubsection{Polarity of Arc-support LS}
In the greyscale image, the overall gradient direction in the local area indicates the tendency of illumination variation. After the careful observation, there exist two situations between elliptic arc's overall gradient direction and arc-support direction. We define the polarity of an arc-support LS, namely $Pol_{L}$, is positive ($+1$) if the corresponding gradient direction and are-support direction are consistent, otherwise is negative ($-1$). A fast decision to the polarity of an arc-support LS is by judging the rotation direction of the main angles $\angle \overrightarrow{AC}$, $\angle \overrightarrow{AB}$ and $\angle \overrightarrow{CB}$, as shown in Fig. \ref{fig3}(a) and Fig. \ref{fig3}(b).
\subsection{Superposition Principle of Ellipse Fitting}
Ellipse fitting is very important in ellipse detection since it directly affects the quality of detected ellipse. Least-squares based ellipse fitting methods focus on minimizing the residue between points and ellipse \cite{rosin1993note,gander1994least,fitzgibbon1999direct}. As the constraint of ellipse fitting problem is quadratic, it usually leads to unsatisfactory efficiency along with the iterative procedure. Therefore, Fitzgibbon et al. \cite{fitzgibbon1999direct} proposed a non-iterative algorithm by solving the positive eigenvector of eigensystem. And we develop the superposition principle on the basis of \cite{fitzgibbon1999direct} due to its efficiency. Suppose that there are $n$ data points in the set $\Gamma_1 = \{ {p_1},{p_2}, \cdots ,{p_n}\}$, $p_{i}=\{ x_{i},y_{i} \}$. We first calculates $\Gamma_1$'s scatter matrix $\text{\bf{S}} =
\text{\bf{D}}^{\text{T}}\text{\bf{D}}$, and {\bf{D}} is denoted as
\begin{equation}
\label{eq2}
\text{\bf{D}} = {\left[ {\begin{array}{*{20}{c}}
{{x_1}^2}&{{x_1}{y_1}}&{{y_1}^2}&{{x_1}}&{{y_1}}&1\\
{{x_2}^2}&{{x_2}{y_2}}&{{y_2}^2}&{{x_2}}&{{y_2}}&1\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
{{x_n}^2}&{{x_n}{y_n}}&{{y_n}^2}&{{x_n}}&{{y_n}}&1
\end{array}} \right]_{n\times6}}.
\end{equation}
Then by solving the generalized eigensystem $\text{\bf{S}}^{-1}\text{\bf{C}}$, where {\bf{C}} is the constant constraint matrix
\begin{equation}\label{eq:constant matrix}
\text{\bf{C}} = {\left[ {\begin{array}{*{20}{c}}
{\rm{0}}&{\rm{0}}&{{\rm{ - 1}}}& \cdots &{\rm{0}}\\
{\rm{0}}&{\rm{2}}&{\rm{0}}&{}&{}\\
{{\rm{ - 1}}}&{\rm{0}}&{\rm{0}}&{}& \vdots \\
\vdots &{}&{}& \ddots &{}\\
{\rm{0}}&{}& \cdots &{}&{\rm{0}}
\end{array}} \right]_{{6\times6}}},
\end{equation}
the obtained eigenvector with positive eigenvalue is the desired fitted ellipse to $\Gamma_1$.\par
In practical ellipse detection process, it always needs to attempt fitting extensive different combinations of point sets for finding the most suitable fitted ellipse. Assuming $\Gamma_1$ has already been computed to fit an ellipse and after that several additional point sets belonging to the same ellipse are newly discovered, which are denoted by $\Gamma_2, \Gamma_3, \cdots, \Gamma_k$, an efficient computation approach to fit the new ellipse should be based on the previous computation results. Denote the design matrix and scatter matrix of $\Gamma_i$ as $\text{\bf{D}}(\Gamma_{i})$ and $\text{\bf{S}}(\Gamma_{i})$, respectively. Thus the design matrix $\text{\bf{D}}_{c}$ of the combination of $k$ point sets $\Gamma_1, \Gamma_2, \cdots, \Gamma_k$ can be written as
\begin{equation}
\label{eq3}
{\text{\bf{D}}_{c}} = \left[ {\begin{array}{*{20}{c}}
{\text{\bf{D}}({\Gamma_{1}})}\\
\vdots \\
{\text{\bf{D}}({\Gamma_{k}})}
\end{array}} \right],
\end{equation}
and the corresponding scatter matrix $\text{\bf{S}}_{c}$ is
\begin{equation}
\begin{aligned
\label{eq4}
{\text{\bf{S}}_c} &= \text{\bf{D}}_c^\text{T}{\text{\bf{D}}_c} = \text{\bf{D}}{({\Gamma_{1}})^\text{T}}\text{\bf D}({\Gamma_{1}}) + \cdots + \text{\bf D}{({\Gamma_{k}})^\text{T}}\text{\bf D}({\Gamma_{k}}) \\
&= \text{\bf S}({\Gamma_{1}}) + (\text{\bf S}({\Gamma_{2}}) + \cdots + \text{\bf S}({\Gamma_{k}})).
\end{aligned}
\end{equation}
Eq. \ref{eq4} indicates that the scatter matrix of any combinatorial point sets equals the summation of the scatter matrix of each point set, which casts light on the feasibility of calculating the scatter matrix of each group merely once. The above superposition feature can cut computation time down when fitting one or more sets into an ellipse.
\section{High-quality Ellipse Detection}\label{sec:ellipse detection}
In this section, a high-quality ellipse detection is proposed by introducing the arc-support LSs. The overall procedure consists of: (1) arc-support groups forming, (2) initial ellipse set generation, (3) clustering, and (4) candidate verification. The arc-support group collects the consecutive arc-support LSs belonging to the same curve, which can avoid the disturbance of the useless straight LSs. In the initial ellipse set generation step, accuracy and efficiency keep pace with the aid of fast ellipse fitting and effective geometric constraints. The efficient clustering and rigorous verification further facilitate the high detection performance of the proposed detector. An overall detection example of our method is demonstrated in Fig. \ref{fig4}.
\begin{figure}[!tb]
\centering
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//origin.jpg}
\label{fig4:a}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//arcSupportLSs.jpg}
\label{fig4:b}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//arcSupportGroups.jpg}
\label{fig4:c}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//initialEllipses.jpg}
\label{fig4:d}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//candidates.jpg}
\label{fig4:e}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//results.jpg}
\label{fig4:f}
}
\caption{Pipeline illustration of the proposed ellipse detection. (a) origin image; (b) 42 extracted arc-support LSs; (c) 20 arc-support groups; (d) 13 initial ellipses; (e) 10 ellipse candidates after clustering; (f) 2 qualified detections after verification and refinement.}
\label{fig4}
\end{figure}
\subsection{Arc-support Groups forming}
\subsubsection{Robust Linking and Groups Forming}
\begin{figure}[!bt]
\centering
\includegraphics[width=\hsize]{pic//4.pdf}
\caption{The arc-support LSs are linked to form an arc-support group in point statistics level based on continuity and convexity.}
\label{fig5}
\end{figure}
Since an elliptic curve may consist of several arc-support LSs, we can link the discovered arc-support LSs to form a group. Any two consecutive arc-support LSs that will be linked should meet the continuity and convexity conditions. For continuity condition, the proximity between the head of an arc-support LS to the tail of another one should be close enough. For convexity condition, the linked LSs should change in the same direction either clockwise or anticlockwise.
Note that a support pixel's level-line angle should be within tolerance $\alpha$ with the support region's main angle, therefore, the angle deviation between two consecutive arc-support LSs should be less than $2\alpha$. To avoid incorrect LSs linking in the existence of noise, we count the number of support points of each next LS within a local statistical area near the terminal of current LS ($k$th LS's number of support points is represented as $\sum p_{i}^{L_{k}}$), and create a histogram for choosing the LS with maximum votes to link with current LS, as shown in Fig.~\ref{fig5}. Iteratively, the linked arc-support LSs which share the similar geometric properties are called as ``arc-support group''. Algorithm~\ref{algo1} details the arc-support LSs linking and groups forming process.\par
\begin{algorithm}[!b]
\caption{ Arc-support groups forming.}
\label{algo1}
\begin{algorithmic}[1]
\Require
Arc-support line segment set, $T_l$;
Arc-support regions that generate line segments, $T_r$;
Angle tolerance, $\alpha$;
Status where line segment used, $S$;
\Ensure
Arc-support groups, $G$;
\State Initialize groups $G = \varnothing$;
\label{code:alg1:initialization}
\Repeat
\State Choose an arc-support line segment $l_i$ which satisfies $S(l_i) \neq used$ from $T_l$;
\State Set the arc-support groups searched from the head and tail of $l_i$ as $g_{\text{head}}$ = $\varnothing$, $g_{\text{tail}}$ = $\varnothing$;
\State Set $l_i$ as the seed of line segment $l_s$;
\Repeat
\State Searching consecutive arc-support line segments at the head end of $l_s$;
\State Rule out the searched line segments which are $used$ and beyond $2\alpha$ angle deviation to $l_s$;
\State Determine statistical area at the head end of $l_s$;
\State Acquire the line segment $l_k$ with highest point votes by using $T_r$;
\State Update $g_{\text{head}}$ = $g_{\text{head}}\cup L_{{k}}$, $S(l_k) = used$, $l_s$ = $l_k$;
\Until{$l_s$ is $\varnothing$}
\State Set $l_i$ as the seed of line segment $l_s$ again;
\State $g_{\text{tail}}$ can be obtained by repeating the above searching process at the tail of $l_s$;
\State Combine the searched arc-support groups $g_{\text{head}}$ = $\{L_{h1},\cdots,L_{hn}\}$ and $g_{\text{tail}}$ = $\{L_{t1},\cdots,L_{tn}\}$ as $g$ = $\{L_{tn},\cdots,L_{t1},L_{\text{i}},L_{h1},\cdots,L_{hn}\}$;
\State Update $G$ = $G\cup g$;
\State Update $S(l_i) = used$;
\Until{every arc-support line segment is traversed}\\
\Return $G$;
\end{algorithmic}
\end{algorithm}
\subsubsection{Spanning Angle Measurement for Each Group}
Each arc-support group which is composed of several arc-support LSs is essentially the polygonal approximation of a curve. If the $i$th group contains $n$ arc-support LSs, it will have $n-1$ angle intervals derived from every two consecutive arc-support LSs. Supposing that the angle interval sequence of $i$th group is $\{\theta^{i}_{1}, \theta^{i}_{2}, \cdots, \theta^{i}_{n-1}\}$, therefore, the spanning angle of $i$th group is $\sum\limits_{j = 1}^{n-1} {\theta _j^i}$. If an arc-support group is more salient to an ellipse, its spanning angle will be larger. Therefore, we have $S^{i}\propto (\sum\limits_{j = 1}^{n-1} {\theta _j^i})/{360^{\circ}}$,
where $S^{i}$ is the saliency score of the $i$th group. For convenience, the proportionality coefficient is set to $1$, and thus the range of $S^{i}$ is $[0,1]$.\par
As shown in Fig.~\ref{fig4:b}, only arc-support LSs are retained while the straight LSs are filtered. With the procedure of robust arc-support LSs linking, the LSs that belong to the same curves are pooled into arc-support groups (Fig.~\ref{fig4:c}).
\subsection{Initial Ellipse Set Generation}
Considering the fact that an arc-support group might contain all arc-support LSs of a curve, or merely a separate arc-support LS, therefore, we use two complementary methods to generate the initial ellipse set. First, from the local perspective, the arc-support LS group with relatively high saliency score is probably the dominant component of the polygonal approximation of an ellipse. A simple and effective manner is to individually fit the arc-support LS group to an ellipse completely relying on the threshold $T_{ss}$ such that the salient ellipse, for instance, the one with spanning angle close to $360^{\circ}$, can be picked out precedently. Second, from the global perspective, we search all the valid pairs of arc-support groups globally to reconstruct the latent ellipses on the image, one advantage of which is dealing with the troublesome situation of the arc-support groups for a common ellipse but far apart. All the valid pairs of groups should satisfy three criteria below: (1) polarity constraint; (2) region restriction; (3) adaptive inliers criterion. In addition, to avoid the overwhelming fitting process, the superposition principle is adopted during ellipse fitting for time efficiency.\par
\subsubsection{Polarity Constraint}
After observing the image regions around an elliptical edge, the inner of an ellipse is always either brighter or darker than the peripheral, where brighter means that the polarity of the arc-support LSs is positive and darker means the one is negative. If all arc-support LSs come from the same ellipse, generally, their polarity should be the same, too. Thus, the first criterion is that the polarity of the paired arc-support LS groups should be congruous. In certain cases, it may only require to detect the oval objects which are brighter (or darker) than background. Following our approach, the ellipse with specified polarity can be easily recognized, which will be detailed in Section \uppercase\expandafter{\romannumeral5}. E. As for the case of the arc-support groups derived from common ellipse while owning different polarity, e.g. an ellipse sharing different backgrounds, it still could be successfully detected as finding the most salient pair of arc-support groups with same polarity for ellipse fitting.\par
\subsubsection{Region Restriction}
Actually, most of the arc-support groups do not contribute to building a valid pair because of the high probability that two groups are from different ellipses or curves. Therefore, an early decision before fitting ellipse is essential.
\begin{figure}[!tb]
\centering
\includegraphics[width=0.8\hsize]{pic//8.pdf}
\caption{Region restriction for a pair of two arc-support groups.}
\label{fig6}
\end{figure}
As shown in Fig.~\ref{fig6}, groups $\{g_{1},g_{3}\}$ and $\{g_{1},g_{4}\}$ should not be paired. The reason is that group $g_{3}$ is out of $g_{1}$'s valid region, which is along with the arc-support direction of every arc-support LS in $g_{1}$. Although $g_{4}$ is in the valid region of $g_{1}$, $g_{1}$ is not in the valid region of $g_{4}$. Consequently, if two groups are paired, they should locate in the mutual valid regions. In Fig.~\ref{fig6}, the start point of group $g_{i}$ is denoted as $P_{s}^{g_{i}}$ and the end point is $P_{e}^{g_{i}}$; the start arc-support LS is $L_{s}^{g_{i}}$ and the end arc-support LS is $L_{e}^{g_{i}}$; the midpoint of $P_{s}^{g_{i}}$ and $P_{e}^{g_{i}}$ is $M_{i}$. Then, we have
\begin{equation}
\label{eq6}
\begin{aligned}
\left\{
\begin{array}{c}
\overrightarrow{ARC_{L_{s}^{g_{i}}}}\cdot\overrightarrow{P_{s}^{g_{i}}P_{e}^{g_{j}}} > \rho_{d} \\
\overrightarrow{ARC_{L_{e}^{g_{i}}}}\cdot\overrightarrow{P_{e}^{g_{i}}P_{s}^{g_{j}}} > \rho_{d} \\
(-Pol)\cdot Dir_{\bot}(\overrightarrow{P_{s}^{g_{i}}P_{e}^{g_{i}}})\cdot \overrightarrow{M_{i}M_{j}} > \rho_{d}
\end{array}
\right.
\end{aligned}
\end{equation}
where the $\overrightarrow{ARC_{L}}$ is the normalized arc-support direction vector of $L$. $Dir_{\bot}(\overrightarrow{v})$ represents the normalized vector acquired by rotating $\overrightarrow{v}$ clockwise $90^{\circ}$. The $Pol$ equals to $+1$ when the polarity of the pair $\{g_{1},g_{2}\}$ is positive, otherwise is $-1$. $\rho_{d}$ is distance threshold. Meanwhile, $(i,j)$ runs over $(1,2)$ and $(2,1)$.\par
\subsubsection{Adaptive Inliers Criterion}
For the pair $\{g_{1},g_{2}\}$ that passes the validations of polarity constraint and region restriction, an ellipse fitting against the endpoints of arc-support LSs in the pair will be implemented immediately. Assuming that the fitted ellipse is $e$, and the normal vector of edge point $P$ to $e$ is $\nabla e(P)$ (its direction points to the exterior of the ellipse),
we define edge point $P_{i}$ as a support inlier to $e$ if meeting both the distance tolerance $\epsilon$ and normal tolerance $\alpha$, namely the Rosin approximation distance \cite{rosin1998ellipse} from $P_{i}$ to $e$ should be less than $\epsilon$, and the absolute angle difference between $\nabla e(P_{i})$ and $-Pol_{e}\cdot Grad(P_{i})$ is less than $\alpha$, where $Pol_{e}$ is the polarity of $e$ and $Grad(P_{i})$ is image gradient of $P_{i}$. From the geometric perspective, we can approximate the arc length by the number of edge pixels. Therefore, the number of support inliers made up each arc-support LS in pair $\{g_{1},g_{2}\}$ should be greater than the corresponding LS's length, namely
\begin{equation}
\label{eq7}
\# \{p_i : p_i \in \text{SI}(L_{j})\} > \text{Length}(L_{j}),
\end{equation}
where $\text{SI}(L_{j})$ represents the support inliers set of arc-support LS $L_{j}$, $j = 1,2,\cdots,N_{g_{1}}+N_{g_{2}}$. $N_{g_{1}}$ and $N_{g_{2}}$ are the number of arc-support LSs in $g_{1}$ and $g_{2}$, respectively. In this step, the length of LS is an adaptive threshold for the corresponding support inliers. If an ellipse $e$ satisfies the adaptive inliers criterion, we will fit the support inliers to produce an initial ellipse. Eventually, all the valid pairs of arc-support groups are transformed to the initial ellipse set.
\subsection{Ellipse Clustering}
Considering the existence of duplicates in the initial ellipse set, an efficient clustering method is extremely important to trim them down, which should not only maintain the isolated points but also suppress the non-maximum. To that end, we develop a hierarchical clustering method based on mean shift~\cite{cheng1995mean}, which decomposes the 5D ellipse parameter space clustering problem into three low and cascaded dimensional space clustering problems (centers, orientations and semi-axes).\par
Assume that the initial ellipse set is $E^{\text{init}}$ and the element number of $E^{\text{init}}$ is $N^{\text{init}}$. We have
\begin{equation}
\label{eq8}
E^{\text{init}} = \bigcup_{1 \le i \le N^{\text{init}}}e_{i}
\end{equation}
where $e_{i} = \{(x,y)_{i}, \varphi_{i}, (a,b)_{i}\}$. Meanwhile, $(x,y)_{i}$, $\varphi_{i}$ and $(a,b)_{i}$ are the center, orientation and semi-axes of the initial ellipse $e_{i}$, respectively.\par
Firstly, our method clusters the ellipse centers of $E^{\text{init}}$ based on mean shift with limited iterations. It then produces $n^{\text{c}}$ elliptic cluster centers $(x,y)_{1}^{c}$, $(x,y)_{2}^{c}$, $\cdots$, $(x,y)_{n^{\text{c}}}^{c}$. If $e_{i}$ is the nearest to $(x,y)_{k}^{c}$, we will add $e_{i}$ to set $\Omega_{k}$. Therefore, $E^{\text{init}}$ can be divided into $n^{\text{c}}$ partitions. The $k$th ($1\le k \le n^{\text{c}}$) partition is represented as
\begin{equation}
\label{eq9}
\Omega_{k} = \{ e_{i} \parallel (x,y)_{i} \in e_{i} \; \text{and} \; (x,y)_{i}\rightarrow (x,y)_{k}^{c} \}.
\end{equation}
$(x,y)_{i} \rightarrow (x,y)_{k}^{c}$ means the Euclidean distance between data point $(x,y)_{i}$ and cluster center $(x,y)_{k}^{c}$ is least among all the cluster centers.\par
Secondly, each initial ellipse subset $\Omega_{k}$ is clustered with respect to their orientations. So $n^{\varphi}_{k}$ orientation cluster centers are generated, which are $\varphi^{c}_{1}, \varphi^{c}_{2}, \cdots, \varphi^{c}_{n^{\varphi}_{k}}$. In a similar way, $\Omega_{k}$ can be divided into $n^{\varphi}_{k}$ subsets. And the $s$th ($1\le s \le n^{\varphi}_{k}$) subset $\Omega_{k,s}$ corresponding to the orientation cluster center $\varphi^{c}_{s}$ is
\begin{equation}
\label{eq10}
\Omega_{k,s} = \{ e_{i} \parallel e_{i} \in \Omega_{k}, \varphi_{i} \in e_{i} \; \text{and} \; \varphi_{i}\rightarrow \varphi_{s}^{c} \}.
\end{equation}
\par
Finally, we implement the clustering step based on semi-axes for each initial ellipse subset $\Omega_{k,s}$ and therefore, $n^{\text{axes}}_{k,s}$ cluster centers are produced: $(a,b)^{c}_{1}$, $(a,b)^{c}_{2}$, $\cdots$, $(a,b)^{c}_{n^{\text{axes}}_{k,s}}$. Eventually, each mode of the combination $\{(x,y)^{c}_{k}, \varphi^{c}_{s}, (a,b)^{c}_{t}\}$ ($1\le t \le n^{\text{axes}}_{k,s}$) is the initial ellipse clustering result, namely the ellipse candidate. The ellipse candidate set $E^{c}$ can be described as
\begin{equation}
\label{eq11}
E^{c} = \bigcup_{k,s,t}\{ (x,y)^{c}_{k}, \varphi^{c}_{s}, (a,b)^{c}_{t} \}
\end{equation}
and the number of ellipse candidates $N^{c}$ is
\begin{equation}
\label{eq12}
{N^{c}} = \sum\limits_{k} {\sum\limits_{s}{n_{k,{\rm{ }}s}^{\text{axes}}} }.
\end{equation}
\par
As a result, the purer ellipse candidates can be generated after applying the hierarchical clustering approach to initial ellipse set. The benefits of such ellipse parameter space decomposition are two folds.
First, we distinguish the different geometrical significance of ellipse center, orientation, and semi-axes for hierarchical clustering since it is difficult to assign an accurate distance measure between two ellipses directly. Second, our clustering method is conveniently implemented and its computation complexity is quadratic (details can be found in Section \uppercase\expandafter{\romannumeral4}), which vastly outperforms than the direct clustering behavior in 5D space.
\subsection{Ellipse Candidate Verification}
The learned ellipse candidates cover almost all possible ellipses existing in the image while keeping nearly few duplicates, which saves much verification time and guarantees good recalls due to their pure quantity and high saliency. In this section, to further ensure the quality of the detected ellipses, we conduct the ellipse candidate verification that incorporates the stringent regulations for goodness measurement and elliptic geometric properties for refinement.
\subsubsection{Goodness Measurement}
The study has shown positive correlation between the number of support inliers on an ellipse and corresponding perimeter $\mathcal{B}$~\cite{kulpa1979properties}. Although the precise numerical formula of $\mathcal{B}$ has not yet been revealed, in practice, it can use $\mathcal{B} \approx \pi[\frac{3}{2}(a+b)-\sqrt{ab}]$ for approximation where $a$ and $b$ are the semi-major axis and semi-minor axis respectively \cite{zhang2016consistency}. In addition, the larger the angular coverage $\mathcal{C}$ of the elliptic connected component of support inliers becomes, the more salient the ellipse candidate will be. Based on these facts, we employ the following evaluation regarding both the number of support inliers and the angular coverage. The support inliers should satisfy the distance tolerance $\frac{\epsilon}{2}$ and the normal tolerance $\alpha$. Therefore, the ``goodness'' can be formulated as
\begin{equation}
\label{eq13}
\text{Goodness}(e)=\sqrt{\frac{\# \{p_i : p_i \in \text{SI}(e)\}}{\mathcal{B}} \cdot \frac{\mathcal{C}}{360^{\circ}}}
\end{equation}
where $\text{SI}(e)$ represents the support inliers of ellipse $e$. Note that we tighten the distance tolerance $\epsilon$ to $\frac{\epsilon}{2}$ to rule out the noise, and thus the goodness measurement is more credible. Finally, we apply a pseudo descending order (in linear time) to the ellipse candidates according to their goodness scores and preferentially pick out the candidates for subsequent validation.
\subsubsection{Verification and Fitting Again}
In verification, the proposed method continues to loose the distance tolerance $\frac{\epsilon}{2}$ to $\epsilon$, and validate the candidate individually against: (1) the number of support inliers; (2) the angular coverage of ellipse. We use a ratio threshold $T_{r}$ and expect that there are $T_{r}\mathcal{B}$ support inliers on the ellipse. In the meanwhile, our method only accepts the ellipse whose angular coverage is at least $T_{ac}$ degrees.\par
Given sufficient support inliers, we may have better ellipse results. Recall that we have found the support inliers with respect to each candidate in the validation step. If a candidate generates the true ellipse (or TP for short), its support inliers should be more sufficient than the previous one. This motivates us to fit ellipse again, which improves the overall accuracy and shows self-calibrated ability. Notably, the time complexity of the additional fitting only relates to the number of final detected ellipses, which will not bring on significant changes in the running time.
\section{Complexity of The Algorithm}\label{sec:complexity}
Assuming that the image size is $N \times N$, therefore, the arc-support LS extraction approach has an $O(N^{2})$ complexity equivalent to LSD \cite{grompone2010lsd}. In the procedure of the arc-support groups forming, the computational complexity is $O(N_{L})$, where $N_{L}$ denotes the number of arc-support LS. If there are $N_{G}$ groups, spanning angle measurement of groups has the complexity of $O(N_{G})$. In the initial ellipse set generation step, the computational complexity has upper bound $O(N_{G}^{2})$ in the worst case, which may rarely occur due to the three novel geometric constraints. In the ellipse clustering step, the running time has the complexity of $O(1+n^{\text{c}}+\sum \limits_{k}n^{\varphi}_{k})$. Noting that after clustering, the number of cluster centers is less than or equal to the number of original data points, which means that $n^{\text{c}} \leq N^{\text{init}}$ and $\sum \limits_{k}n^{\varphi}_{k} \leq N^{\text{init}}$. Note that the initial ellipses' number $N^{\text{init}}$ is of $O(N_{G}^{2})$, and thus the process of initial ellipse clustering step has the computational complexity of $O(2N_{G}^{2})$. In the validation step, the computation time relates to the number of candidates $N^{c}$. Eventually, the computational complexity of the proposed method is upper bounded by $O( N^{2} + N_{L} + 3N_{G}^{2} + N^{c})$, which reveals that the ellipse detection complexity is as fast as quadratic in $N$ and $N_{G}$.
\section{Experimental Results}\label{sec:experimental results}
In this section, extensive and detailed experiments are implemented to demonstrate the high-quality ellipse detection performance of the proposed method compared to the existing state-of-the-art methods.\par
\subsection{Experiments Setup}
\begin{table*}[!t]
\centering
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\caption{Ellipse Detection Results on Three Public Real-world Datasets Compared with the Popular and State-of-the-art Methods.}
\label{tab1}
\begin{tabular}{cccccccc}
\toprule[1pt]
Dataset &Index & RHT \cite{mclaughlin1998randomized} & ELSDc \cite{patraucean2017joint} & Prasad \cite{prasad2012edge} & Fornaciari \cite{fornaciari2014fast} & Qi \cite{jia2017fast} & Our \\\midrule[1pt
\multirow{4}*{Traffic Sign Dataset} &Precision & 0.0706 & 0.0429 & 0.1401 & 0.4545 & 0.5814 & \textbf{\textbf{0.9110}} \\%\cline{2-8}
&Recall & 0.5149 & 0.6933 & 0.5665 & 0.7277 & 0.7324 & \textbf{0.8811} \\%\cline{2-8}
&F-measure & 0.1242 & 0.0808 & 0.2246 & 0.5596 & 0.6482 & \textbf{0.8958} \\%\cline{2-8}
&Time/s & 1.2875 & 4.6006 & 9.9918 & 0.2875 & \textbf{0.1488} & 0.5791 \\\midrule[0.5pt]
\multirow{4}*{Prasad Dataset} &Precision & 0.1941 & 0.0922 & 0.2360 & 0.7039 & 0.7161 & \textbf{0.7523} \\
&Recall & 0.2479 & 0.2940 & 0.3145 & 0.2154 & 0.2393 & \textbf{0.3504} \\
&F-measure & 0.2177 & 0.1403 & 0.2697 & 0.3298 & 0.3587 & \textbf{0.4781} \\
&Time/s & 0.2011 & 1.4348 & 3.9011 & 0.0939 & \textbf{0.0529} & 0.1685 \\\midrule[0.5pt]
\multirow{4}*{\tabincell{c}{PCB Dataset}}&Precision&0.4165 &0.4168 &0.7645 &0.8933 & 0.9393 &\textbf{0.9715}\\
&Recall & \textbf{0.9500} & 0.9154 & 0.7115 & 0.8692 & 0.8923 & 0.9192 \\
&F-measure & 0.5791 & 0.5728 & 0.7371 & 0.8811 & 0.9152 & \textbf{0.9447} \\
&Time/s & 0.2271 & 3.4830 & 1.0758 & 0.0980 & \textbf{0.0738} & 0.1693
\\\bottomrule[1pt]
\end{tabular}
\end{table*}
\subsubsection{Model Parameters}
Our ellipse detection method mainly involves seven parameters, which are discussed as follows
(1) $T_{ai}$ is the angle interval of each subregion to the support region, which is used in arc-support LS extraction. It indicates the least curvature degree of the support region that generates an arc-support LS. Obviously, the LS derived from the straight edge will be filtered since its angle interval is less than $T_{ai}$. We fix $T_{ai}$ to $2.25^{\circ}$ as it performs well in experiments. (2) Angle tolerance $\alpha$ is used in the cases when evaluating the angle deviation of a geometric primitive, e.g. the point's level-line angle or gradient angle, to the corresponding reference angle. $\alpha$ can be empirically set to $22.5^\circ$ which yields the best results for thousands of images~\cite{grompone2010lsd,burns1986extracting}. (3) Saliency score threshold $T_{ss}$ is used in the initial ellipse set generation. Admittedly, we can fit any group as long as its saliency score is higher than zero because of the negligible computational cost. However, it is unnecessary because we will find all the valid pairs of arc-support groups for generating the initial ellipses. Therefore, we set $T_{ss}$ to $0.25$. (4) Distance threshold $\rho_{d}$ is used in region restriction, which can be minus for dealing with some extreme cases. Thus $\rho_{d}$ is fixed to $-3\epsilon$. (5) The distance tolerance $\epsilon$ is used to recover the inliers to arc-support LS or ellipse. If $\epsilon$ becomes more restrictive, the inliers will be purer, even in presence of spurious edges or noise. As our algorithm aims at high localization accuracy, we are able to set $\epsilon$ to $2$ pixels. (6) $T_{r}$ is the ratio of support inliers on an ellipse and (7) $T_{ac}$ is the elliptic angular coverage threshold. They both are used in the ellipse verification. $T_{r}$ and $T_{ac}$ are easily tuned due to their geometric significance. Since most of the true ellipses in real-world images have a degree of completeness, we set $T_{r} = 0.6$ and $T_{ac} = 165^{\circ}$ in default.\par
Notably, we merely open \text{\it{two}} external adjustable parameters $T_r$ and $T_{ac}$ after regarding the other five parameters as intrinsic parameters since they can be empirically fixed and work fairly well, which enables our ellipse detection algorithm to be easily used.
\subsubsection{Evaluation Metrics}
We employ the following metrics for evaluations: (1) \textbf{ precision}, (2) \textbf{recall}, (3) \textbf{F-measure}. The precision = TPs$/$(TPs + FPs), recall = TPs$/$(TPs + FNs) and F-measure = $2/$(precision${}^{-1}$ + recall${}^{-1}$). A detected ellipse is regarded as a TP if its overlap area ratio to the corresponding ground true ellipse is larger than $D_{0}$. And we set $D_{0}$ to $0.8$ throughout all experiments as did in \cite{prasad2012edge,fornaciari2014fast, jia2017fast}.
\subsubsection{Compared Methods}
We mainly select five most popular and competitive algorithms for quantitative and qualitative comparisons, which are RHT \cite{mclaughlin1998randomized}, ELSDc \cite{patraucean2017joint} and the methods proposed by Prasad et al. \cite{prasad2012edge}, Fornaciari et al. \cite{fornaciari2014fast}, and Qi et al. \cite{jia2017fast}. RHT is the most popular in the literature and often used as the baseline method. ELSDc is robust and achieves precise location accuracy amongst state-of-the-art methods. Prasad method combines HT and the techniques of edge following, which reflects very well detection performance. The methods proposed by Fornaciari et al. and Qi et al. are very efficient and obtain extremely high F-measure scores on public datasets. We compare our method\footnote[2]{The source code and more detection examples of our method can be found at https://github.com/AlanLuSun/High-quality-ellipse-detection.\label{footnote2}} to the aforementioned ellipse detectors as they are the most popular or state-of-the-art detectors existing in ellipse detection field.\par
For the fair comparison purpose, we adopt the source codes of ELSDc \cite{patraucean2017joint}, Prasad et al. \cite{prasad2012edge}, Fornaciari et al.~\cite{fornaciari2014fast}, and Qi et al. \cite{jia2017fast} which are available online, and reimplement RHT \cite{mclaughlin1998randomized} according to the original paper. RHT, Prasad method and our method are run in MATLAB while the remaining three methods are in C++. All experiments are performed with default parameters and on the same computer with Intel Core i7-7500U 2.7GHz CPU and 8 GB memory.
\subsubsection{Datasets}
To test the competitive ellipse detectors, three public challenging real-world datasets are utilized:
\begin{itemize}
\item \textbf{Traffic sign dataset}. As a portion of Dataset $\sharp2$ which is created by Fornaciari et al. \cite{fornaciari2014fast}, it contains 273 images with various ellipses that are projected by round traffic signs at different real-life scenarios. These images derive from the frames of several videos captured by smartphone,
suffering from the blurry and varying lighting conditions by motion and autofocus.
\item \textbf{Prasad dataset}. A complex real-world image dataset, employed before by many well-known ellipse detectors in their experiments~\cite{prasad2012edge,fornaciari2014fast, patraucean2017joint, jia2017fast}. Note that Prasad dataset consists of 198 images which are complex enough due to the unpredictable conditions and substantial disturbances.
\item \textbf{PCB dataset}. PCB (Printed Circuit Board) dataset \cite{Lu2017Circle} includes 100 industrial PCB images with various disturbances and each image contains at least one circular or elliptic shape. All PCB images are labeled manually and precisely.
\end{itemize}
\par
\subsection{Experiments on Real-world Datasets}
\begin{figure*}[!tb]
\centering
\includegraphics[width=0.9\hsize]{pic//threedatasets.pdf}
\caption{Ellipse detection examples on three real-world datasets. The first and second columns are the input images and ground truth (GT). The input images of the first three rows are from traffic sign dataset while the second and third three-row images are from Prasad dataset and PCB dataset.}
\label{fig7}
\end{figure*}
\begin{figure*}[!tb]
\centering
\subfigure[]{\includegraphics[width=0.3\textwidth]{pic//curves//Traffic-Precision.pdf
\label{fig8:Traffic-P}}
\subfigure[]{\includegraphics[width=0.3\textwidth]{pic//curves//Traffic-Recall.pdf
\label{fig8:Traffic-R}}
\subfigure[]{\includegraphics[width=0.3\textwidth]{pic//curves//Traffic-FMeasure.pdf
\label{fig8:Traffic-F}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Prasad-Precision.pdf
\label{fig8:Prasad-P}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Prasad-Recall.pdf
\label{fig8:Prasad-R}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Prasad-FMeasure.pdf
\label{fig8:Prasad-F}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Industrial-Precision.pdf
\label{fig8:Industrial-P}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Industrial-Recall.pdf
\label{fig8:Industrial-R}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Industrial-FMeasure.pdf
\label{fig8:Industrial-F}}
\caption{Ellipse detection performance of our method against other methods by varying overlap ratio $D_{0}$ from $0.65$ to $0.95$ at the step of $0.05$ on three real-world datasets. The three rows reveal the metric indexes of precision, recall and F-measure on traffic sign dataset, Prasad dataset and PCB dataset respectively.}
\label{fig8}
\end{figure*}
The detailed ellipse detection results on three real-world datasets are shown in Table \ref{tab1}. The highest scores in precision, recall, F-measure and cost time are stressed in boldface. As we can see, our method achieves the best F-measure scores 0.8959, 0.4781 and 0.9447 across traffic sign dataset, Prasad dataset and PCB image dataset respectively, which highlights its extraordinary overall ellipse detection ability. Moreover, our method holds the winner almost in all precision and recall indexes in three real-world datasets except the recall in PCB dataset which is acquired by RHT \cite{mclaughlin1998randomized}. The higher precision indicates that our ellipse detector is more rigorous to reject false positives and the ellipses reported by our method own larger possibility to truly exist. The precision and recall of our method keep pace, consequently, yielding the larger F-measure scores.
The iterative random search helps RHT \cite{mclaughlin1998randomized} raise the recall, however, its precision and F-measure scores are unsatisfactory due to the lack of novel validations. ELSDc \cite{patraucean2017joint} tries to detect ellipses and line segments simultaneously. However, the side-effects are also brought into such as the limited ellipse detection performance and long computation time in real-world images. Actually, ELSDc \cite{patraucean2017joint} are more suitable to deal with PCB images as its performance gets promoted vastly in PCB dataset. Similar to ELSDc \cite{patraucean2017joint}, Prasad method \cite{prasad2012edge} also suffers from long computation time due to the heavy detection procedures.
From Table \ref{tab1}, it can observe that Qi et al. method \cite{jia2017fast} which is developed on the basis of Fornaciari et al. method \cite{fornaciari2014fast} could achieve second best overall ellipse detection results across three real-world datasets. Moreover, Qi et al. method \cite{jia2017fast} is able to consume quite small running time. The reasons behind this are mainly due to its usage of projective invariant for effectively pruning straight edges, fast arc selection strategy, and simple clustering \cite{jia2017fast, Prasad2010Clustering}. In contrast to Qi et al. method accelerating detection speed at the risk of generating duplicates surrounding a common ground truth, our method employs a more useful and yet relatively more time-consuming hierarchical clustering method for ellipse candidates, which could reduce the false positives significantly. Although our method implemented in MATLAB would further slow down the ellipse detection speed, it could achieve competitive running time compared with the methods proposed by Qi et al. \cite{jia2017fast} and Fornaciari et al. \cite{fornaciari2014fast} which are in C++.\par
Some ellipse detection examples on three real-world datasets are shown in Fig. \ref{fig7}. The images of rows (1) to (3) are from the traffic sign dataset, where exist the disturbances of illumination, varied eccentricity and extremely close concentric ellipses. The second three-row images are from Prasad dataset which is the most complicated among three real-world datasets due to noise, occlusions and various backgrounds. The left three images are from PCB dataset, which is with the substantial Gaussian white noise and blur. As a result, both accurately and efficiently detecting ellipses in such images is difficult to an ellipse detector. As illustrated in Fig. \ref{fig7}, RHT \cite{mclaughlin1998randomized} tends to detect every possible ellipse while generates many false positives. The detection performance of RHT gets worse especially in the images with substantial noise and textures. Although ellipse detectors proposed by Fornaciari et al. \cite{fornaciari2014fast} and Qi et al. \cite{jia2017fast} are very efficient, both methods report many ellipse duplicates as well as resulting in poor location accuracy, as revealed in the images of (2) to (5) rows. In contrary, Prasad method \cite{prasad2012edge} and ELSDc \cite{patraucean2017joint} could relatively more accurately locate the ellipses. The relating concerns of both Prasad method and ELSDc are the massive missing detections for ground truth and modeling small contours as ellipses, which worsens the overall ellipse detection performance as the F-measure scores shown in Table \ref{tab1}. Unlike other ellipse detectors being tough to balance the issues between accuracy and efficiency, our method can both efficiently and accurately detect the ellipses and is robust to noise and textures. Especially, our method also performs well in handling the incomplete and occluded ellipses, as shown in the last column of Fig. \ref{fig7}. The elements guarantee our method's accuracy mainly due to the false control ability of arc-support LSs, novel verification criteria and self-calibrated refinement. And the reasons for the good efficiency are attributed to arc-support LSs alleviating the disturbance of straight LSs and effective initial ellipse generation aided with polarity constraint, region restriction and adaptive inliers criterion. \par
In order to comprehensively evaluate the ellipse detection performance of the compared methods, we vary the overlap ratio threshold $D_{0}$ from $0.65$ to $0.95$ at the step of $0.05$, the higher of which indicates the stricter a detected ellipse being regarded as true positive. The corresponding results are shown in Fig. \ref{fig8}. Again, our method achieves the best overall ellipse detection performance among three real-world datasets as the F-measure curves are above those of compared methods, which accords with the before performance analysis. It is evident that our method shows high-quality ellipse detection performance.
\subsection{Localization Accuracy and Efficiency Analysis}
\begin{table}[!tb]
\centering
\caption{The Mean Overlap Ratio (MOR) of Correctly Detected Ellipses on Three Real-world Datasets.}
\label{tab2}
\begin{tabular}{cccc}
\toprule[1pt]
MOR & Traffic Sign & Parasad Dataset & PCB Dataset \\\midrule[1pt]
RHT \cite{mclaughlin1998randomized} & 0.9080 & 0.8963 & 0.9459 \\
ELSDc \cite{patraucean2017joint} & 0.9229 & 0.8818 & 0.9352 \\
Prasad \cite{prasad2012edge} & 0.9226 & 0.9144 & \textbf{0.9603} \\
Fornaciari \cite{fornaciari2014fast} & 0.9274 & 0.9080 & 0.9442 \\
Qi \cite{jia2017fast} & 0.9239 & 0.9047 & 0.9428 \\
Our & \textbf{0.9383} & \textbf{0.9291} & 0.9574 \\\bottomrule[1pt]
\end{tabular}
\end{table}
Localization accuracy is a critical index to testify whether an ellipse detector to be high-quality or not. To this end, we compute each ellipse detection method's mean overlap ratio (MOR) of correctly detected ellipses on three real-world datasets and the results are shown in Table \ref{tab2}. Our method achieves the best MOR scores in traffic sign dataset and Prasad dataset and second highest MOR $0.9574$ in PCB dataset compared to $0.9603$ which is acquired by Prasad method \cite{prasad2012edge}. The higher MOR indicates that our method aims at the high localization accuracy and does not rest content with picking out the true positives, which stands the proposed method out the compared methods. Actually, accurate localization could favor an ellipse detector to distinguish the very closed ellipses. An accurate ellipse detection example of our method is shown in Fig. \ref{fig9}. There are eight ground truth in the input image and the average distance of each two concentric ellipses is $\Delta 4.18_{-1.08}^{+0.78}$ pixels. Although the ellipses are so close, our method still can successfully locate each individual and report high overlap ratio, as shown in Fig. \ref{fig9:accuracy3} and Fig. \ref{fig9:accuracy4}.\par
\begin{figure}[!tb]
\centering
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//1.jpg
\label{fig9:accuracy1}}
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//2.pdf
\label{fig9:accuracy2}}
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//3.jpg
\label{fig9:accuracy3}}
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//4.pdf
\label{fig9:accuracy4}}
\caption{An illustration example of the proposed method with high localization accuracy. (a) input image; (b) ground truth; (c) eight detected ellipses; (d) the overlap ratio of each detected ellipse with ground truth.}
\label{fig9}
\end{figure}
\begin{figure}[!tb]
\centering
\includegraphics[width=0.9\hsize]{pic//computation_complexity.pdf}
\caption{Computation time with regard to the number of pixels of real-world images.}
\label{fig10}
\end{figure}
In order to verify the quadratic complexity of the proposed ellipse detector in image longer length $N$ and arc-support groups $N_{G}$, we record the computation time of 100 different real-world images, the sizes of which range from 46 x 51 to 4600 x 5100. The correlation between computation time and the number of pixels is shown in Fig. \ref{fig10}. The scatters are general in linear distribution and the ellipse computation time almost linearly increases with the number of pixels, which indicates that our method is quadratic in $N$ ($N>N_{G}$ in most of the images). Admittedly, the ellipse detector should own larger time complexity than line segment or circle detector. Our method is still efficient and can handle the real-world images in quadratic time complexity which is superior to most of existing ellipse detectors.
\subsection{Robustness to Parameters Setting and Ellipse Variations}
The angular coverage $T_{ac}$ and the ratio of support inliers $T_r$ are two extrinsic parameters of the proposed ellipse detection method. Firstly, $T_{ac}$ and $T_{r}$ have geometric significance, which enables us easy to tune when applied in the real application. Secondly, both parameters are insensitive and have robustness in a wide setting range. To validate the robustness to tunable parameters, we select PCB dataset as the testset and perform quantitative experiments. We first freeze $T_r$ as $0.6$ which is the default parameter and vary the elliptic angular coverage $T_{ac}$ from $105^\circ$ to $225^\circ$ at the step of $10^\circ$. Then the curves of precision, recall and F-measure according to the experimental results are plotted, as shown in Fig. \ref{fig11:AngleVarying}. Similarly, the ratio of support inliers $T_r$ is changed from $0.4$ to $0.8$ at the step of $0.05$ and the angular coverage $T_{ac}$ are fixed to the default parameter $165^\circ$. The corresponding ellipse detection performance is shown in Fig. \ref{fig11:TrVarying}. As the angular coverage $T_{ac}$ and ratio of support inliers $T_r$ rise, the recall tends to decline as the detected ellipses are more likely to be rejected due to the stricter requirements. However, the precision gets boosted since the detected ellipses are purer. Notably, the F-measure curves in both Fig. \ref{fig11:AngleVarying} and Fig. \ref{fig11:TrVarying} are relatively smooth and little fluctuating in a wide range, which reveals the robustness of the proposed method to different parameter settings.\par
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//parameterSensitivity//AngleVarying.pdf
\label{fig11:AngleVarying}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//parameterSensitivity//TrVarying.pdf
\label{fig11:TrVarying}}
\caption{Ellipse detection performance of the proposed method in PCB image dataset with varying angular coverage and ratio of support inliers. a) the elliptic angular coverage are set from $105^\circ$ $\sim$ $225^\circ$ at step of $10^\circ$ with fixed ratio of support inliers 0.6; b) the ratio of support inliers ranges from 0.4 $\sim$ 0.8 at step of 0.05 while the angular coverage is $165^\circ$.}
\label{fig11}
\end{figure}
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//SemiAxisAxisRatio.pdf
\label{fig12:a}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//OrientationAxisRatio.pdf
\label{fig12:b}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//AngularCoverageAxisRatio.pdf
\label{fig12:c}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//SemiAxisAxisRatio2.pdf
\label{fig12:d}}
\caption{Extensive detection results subject to various ellipse variations. The horizontal axis is the ratio of semi-minor axis to semi-major one, which ranges from 0.01 to 1 at the step 0.01. The vertical axes of (a), (b) and (c) are the semi-major axis length, ellipse orientation and angular coverage of ellipse arc. (d) shows the effects after upscaling the synthetic images two times.}
\label{fig12}
\end{figure}
In order to investigate the robustness of our method to different ellipse variations such as ellipse size, orientation and incompleteness, three synthetic datasets are prepared. The first dataset includes 10000 images, in which the semi-major axis of the ellipse is varied from $1\sim 100$ pixels at the step of 1 pixel and the axes ratio ranges from 0.01 to 1 at the step of 0.01. To evaluate the influence of orientation, we build the second dataset by rotating the ellipse from $-88^\circ$ to $90^\circ$ at the step of $2^{\circ}$. For each orientation, the major-axis is fixed to 100 pixels and the axes ratio changes from $0.01\sim 1$ at the step of 0.01, which totally results in 9000 images. Actually, a high-quality ellipse detector should accurately detect the incomplete ellipses, namely elliptic arcs. Therefore, the third dataset is built and consists of 12000 images, where the angular coverage of ellipse varies from $3^\circ$ to $360^\circ$ at the step of $3^\circ$ and the axes ratio ranges from $0.01 \sim 1$ at the step of $0.01$. Each synthetic image contains an ellipse and is with the size of 250 x 250.\par
The effects of ellipse variations on our ellipse detector are shown in Fig. \ref{fig12}, where the white region indicates the corresponding ellipses could be correctly detected while the black region means the detection failures. Firstly, in Fig. \ref{fig12:a}, our method has wide successful detection area and could detect the small ellipse with the semi-major axis of about 20 pixels and axes ratio of 0.25. The extremely oblate and small ellipses are failed to detect by our method as well as by the methods proposed by Fornaciari et al. \cite{fornaciari2014fast} and Qi et al. \cite{jia2017fast}. Secondly, the black region distributed vertically in Fig. \ref{fig12:b}, which indicates that the ellipse detection performance is invariant to ellipse orientation. Our method is robust to the orientation as it is a basic nature of high-quality ellipse detector. Thirdly, our method can successfully detect the elliptic arc with angular coverage of about $165^\circ$ since our parameter $T_{ac}$ is acquiescently set to $165^\circ$, as shown in Fig. \ref{fig12:c}. This result reveals that our method is able to tackle the incomplete ellipses and detect the specified elliptic arc with assigned angular coverage. Finally, our ellipse detection performance gets improved after upscaling the image size two times since the black region shrinks in Fig. \ref{fig12:d}, which provides a feasible approach to boost detection performance of the proposed ellipse detector.
\subsection{Polarity-specific Ellipse Detection}
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//25.jpg
\label{fig13:a}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//positive.jpg
\label{fig13:b}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//negative.jpg
\label{fig13:c}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//allellipses.jpg
\label{fig13:d}}
\caption{An example of polarity-specific ellipse detection. (a) origin image, 993 x 595; (b) detection for the ellipse whose polarity is positive; (c) detection for the negative polarity ellipse; (d) detecting all ellipses in the image.}
\label{fig13}
\vspace{-3pt}
\end{figure}
Recall that the polarity of an ellipse is positive if the corresponding inside adjacent area of the boundary is brighter than outside, otherwise is negative. Actually, our method is able to detect the polarity-specific ellipses because we only need to retain the arc-support LSs with the corresponding polarity for generating the initial ellipse set. As shown in Fig.~\ref{fig13:a}, the black elliptic ring belts and white ring belts are concentric and adjacent. Each ring belt will generate two different ellipses with positive or negative polarity. In Fig.~\ref{fig13:b}, the concentric ellipses with positive polarity are successfully detected by our method and they are highlighted in blue color. In Fig.~\ref{fig13}(c), the detected ellipses in yellow are all with negative polarity. Naturally, if we use all the arc-support LSs for ellipse detection, the target is to detect all potential ellipses in the image, as the detected red ellipses in Fig.~\ref{fig13}(d). The information of polarity of arc-support LS is greatly important and useful, which not only contributes to reducing the computation time for searching all the valid paired arc-support groups but also helps to detect the polarity-specific ellipses in the certain case.
\section{Conclusion}\label{sec:conclusion}
In this paper, we propose a high-quality ellipse detection method by introducing the arc-support LSs, which aims at both accurately and efficiently detecting ellipses in real-world images. To this end, our method follows a four-stage ellipse detection framework: arc-support groups forming, initial ellipse set generation, clustering, and candidate verification. With the help of arc-support LSs, straight LSs are filtered and the abundant geometric features such as overall gradient direction of the local area, arc-support direction and polarity can be thoroughly exploited. The robust forming of arc-support groups, the adoption of the superposition principle of ellipse fitting and the efficient generation of initial ellipse set with three novel geometric constraints guarantee the overall efficiency of the proposed method. Moreover, the rigorous ellipse verification defend the high localization accuracy and robustness as well as rejecting the false positives. The self-calibrated refinement facilitates higher accuracy. The quantitative experiments compared with existing novel methods evidently demonstrate that our method could well balance the relationship between accuracy and efficiency, and achieves the high-quality ellipse detection performance.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
\section{Introduction}\label{sec:introduction}
\IEEEPARstart{E}{llipse} detection is a fundamental technique in image processing field and plays an indispensable role in shape detection and geometric measurement.
Actually, ellipse detector can be utilized to handle various real-world problems.
In PCB inspection field, one basic function of the defect detection machine is to precisely as well as fast locate the circular pads or holes. Moreover, accurate measurement of circular control points and elliptic fiducial markers is helpful to homography estimation and camera calibration~\cite{huang2016homography,calvet2016detection,heikkila2000geometric}, and some irregular objects could be fitted as ellipses to simplify the shape structure for efficient mathematical modeling \cite{crocco2016structure,da2010fitting,bai2009splitting,kothari2009automated}.
However, to our best knowledge, there exist few robust, stable, efficient and accurate ellipse detector algorithms to universally handle the ellipse detection problem in real-world images, which may have the presence of cluttered edges, motion blur, illumination, occlusion, noise and so on. The major reason is that an ellipse involves five parameters rather than that a circle just needs three, which results in detecting ellipse both efficiently and accurately to be a tough problem. Recently, convolutional neural network (CNN) \cite{lecun2015deep} is revolutionizing objection detection field, like Mask R-CNN \cite{he2017mask} and YOLO \cite{redmon2018yolov3}. Deep learning based methods can provide image proposals which contain oval objects for image pre-processing while they are still inappropriate to directly handle ellipse detection due to the issues of limited segmentation accuracy and expensive manual annotation. In most real-world applications, the practical requirements with regard to higher location accuracy and faster speed make ellipse detection problem even challenging.\par
\begin{figure*}[!t]
\centering
\includegraphics[width=\hsize]{pic//1.pdf}
\caption{A comparison of various ellipse detection methods\protect\footnotemark[1]. The processing time of various methods is counted on the same computer with Intel Core i7-7500U 2.7GHz CPU and 8 GB memory. Except that the methods marked with (*) are implemented in C++, the remaining methods are in MATLAB. (a) the origin image is with the resolution of $720 \times 435$; (b) shows the ground true ellipses; (c) RHT \cite{mclaughlin1998randomized} can detect the ellipses while easily generating duplicates; (d) most ellipses can be located by Prasad et al. method \cite{prasad2012edge} at the cost of long running time; (e) and (f) show that the methods proposed by Fornaciari et al. \cite{fornaciari2014fast} and Qi et al. \cite{jia2017fast} are very fast. However, both methods cause either missing detections or false positives; (g) ELSDc \cite{patraucean2017joint} can jointly detect ellipses, arcs, and line segments while suffering from a long time; (h) our proposed method could accurately detect the ellipses with competitive running time, which reveals its high-quality detection performance.}
\label{fig1}
\end{figure*}
Existing commonly used methods for ellipse detection can
be briefly grouped into two categories: 1) \text{\it{Hough Transform}}; 2) \text{\it{Edge Following}}.\par
Hough Transform (HT) has been widely used for detecting geometric primitives such as line segment (or LS for short), circle and ellipse~\cite{Duda1972Use}. The basic idea of HT for ellipse detection is voting arbitrary edge pixels into 5D parameter space. The local peak will occur when the corresponding bin of accumulator exceeds a threshold of votes, which implies for detecting an ellipse. But it is almost impractical to directly apply HT in practice due to the heavy computation burden and excessive consumption of memory. To alleviate these issues, considerable improved methods are put forward. Probabilistic Hough Transform (PHT) randomly selects a small subset of the edge points which is used as input for HT~\cite{kiryati1991probabilistic}, but large-scale attempts are taken to find the points all sharing a common ellipse and it leads to inferior performance when substantial noise exists. Yuen et al. decomposed the 5D parameter space by finding the ellipse center using some geometric properties like colinearity and symmetry on the first stage and then finding the remaining three parameters on the second stage~\cite{yuen1989detecting,tsuji1978detection}. Instead of transforming each edge point into a 5D parameter space, Xu et al.~\cite{xu1990RHT} proposed a Randomized Hough Transform (RHT) to detect curves, which randomly chooses five edge pixels each time and maps them into a point of the ellipse parameter space. McLaughlin et al. \cite{mclaughlin1998randomized} combined the aforementioned two-stage decomposition method and RHT at the aim of reducing the computation time and improving the detection performance compared with the standard HT, which becomes a baseline of ellipse detection method in the literature (Fig.~\ref{fig1}(c)). However, it is still not efficient enough in practice and always generates false detections due to the lack of novel validation strategies. Despite the simplicity of HT, HT based ellipse detection methods suffer from the following legacy problems: First, it is vulnerable in front of substantial image noise and complicated real-world background due to false peaks; Second, it requires much effort to tune the model parameters, e.g. bin size and peak threshold.\par
\footnotetext[1]{best viewed in color\label{footnote1}}
The second well-known family of ellipse detection methods is edge following, in which the connectivity between edge pixels, convexity of arc segments and geometric constraints are used. The general idea of edge following always starts from computing the \text{\it{binary edge}} map and corresponding gradients acquired by Canny or Sobel detector \cite{canny1986computational, plataniotis2013color} and then refining the arc segments from the \text{\it{binary edge}} for the ellipse fitting.\par
Many of edge following methods use line segments (LSs), which are extracted from the binary edges, as an intermediary to find the arc segments. The approach proposed by Kim et al. \cite{kim2002fast} merges the very short LSs to represent arc segments, where the arc fitting algorithms are frequently called. \cite{mai2008hierarchical} shares similar ideas with \cite{kim2002fast} to extract short LSs from the edge map while the difference lies in linking the LSs as well as the LS's edge points to form arc segments by using simple preset adjacency threshold and proper curvature condition. This method further iteratively groups two arc segments and applies the Random Sample Consensus (RANSAC) to the arc segment groups to recover the ellipse models. Although \cite{mai2008hierarchical} tries to promote the ellipse detector's robustness by iterative grouping and RANSAC, the massive missing detections (FNs) and false positives (FPs) appear.
The method proposed by Chia et al.~\cite{chia2011split} improves the framework illustrated in~\cite{mai2008hierarchical}, but a more complicated fragments merging and grouping procedures were employed. The merging of arc fragments is formalized as an alignment problem, where an alignment function is defined to score the rationality of merging, and a total cost function is built to incrementally search the optimal paired arc segments for grouping. Though the complex and iterative mathematical optimization boosts the detection performance to some extent, \cite{chia2011split} shows slow speed in the real-world images as reported in \cite{prasad2012edge,patraucean2017joint}. The ellipse detector proposed by Prasad et al. \cite{prasad2012edge} incorporates the edge curvature and convexity to extract smooth edge contours and performs a 2D HT to rank the edge contours in a group by the relationship scores for the better generation of ellipse hypotheses. But it also suffers from a long computation time, as shown in Fig.~\ref{fig1}(d).\par
Another stream of edge following methods tries to extract arc segments from binary edge directly and prunes straight edges for the purpose of fast detection speed. The ellipse detector proposed by Fornaciari et al. \cite{fornaciari2014fast} assigns a bounding box for each arc, removes the straight edges and determines the convexity of the arc by comparing the areas of region under and over the arc. In addition, this method accelerates the detection process by utilizing the property of that ellipse center should be colinear to the midpoints of parallel chords. However, it raises the detection speed at the cost of localization accuracy and robustness (Fig.~\ref{fig1}(e)). Recently, the method presented by Qi et al. \cite{jia2017fast} inherits \cite{fornaciari2014fast} and uses the similar convexity classification approach, but the difference lies in that \cite{jia2017fast} filters straight edges efficiently by calculating the edge connected component's characteristic number, which is a kind of projective invariant being able to distinguish the lines and conic curves within images. \cite{jia2017fast} is fast and yet prone to generate duplicates due to the absence of novel clustering (Fig.~\ref{fig1}(f)). In addition, both \cite{fornaciari2014fast} and \cite{jia2017fast} require at least three arc segments to recover the ellipse model, which might disable the algorithms when handling the incomplete ellipses.\par
Some researchers generalize the LS detection method to be a multi-functional detector which can jointly detect the LS and elliptic arcs. ELSDc proposed by P{\u{a}}tr{\u{a}}ucean et al. \cite{patraucean2017joint} uses an improved LSD \cite{grompone2010lsd} version for detecting LS, and then iteratively searches the remaining LSs from the start and end points of the detected LS. Eventually, both LS detection and grouping tasks are established simultaneously. Notably, ELSDc stands out other methods by detecting LSs from the \text{\it{greyscale image}} instead of \text{\it{binary edge}} such that abundant gradient and geometric cues can be fully exploited.
ELSDc and our proposed method are both based on LSD \cite{grompone2010lsd} for LS detection from the \text{\it{greyscale image}}, but they are fundamentally different from the generated LS type, ellipse candidates generation and validation strategies. Our method merely generates the arc-support LSs and do not chain them in the LS generation step. Moreover, ELSDc fits and validates the locally grouped LSs, omitting the global situation, which may be prone to produce the false positives (Fig.~\ref{fig1}(g)).\par
Arc-support LS is our previous work as introduced in \cite{Lu2017Circle}, each pair of which is successfully used for circle detection. However, it cannot handle the ellipse detection scheme since an arbitrary ellipse cannot be determined by two paired LSs. Admittedly, ellipse detection owns much higher complexity and requires more geometric cues. For example, the continuity feature, which is neglected in \cite{Lu2017Circle}, can be fully embodied in the arc-support group and is important in ellipse detection. Therefore, the careful arc-support groups forming, complicated geometric constraints, accurate ellipse generation and clustering, and novel validation strategy accustomed to ellipse detection are required, which will be addressed in this paper.\par
The main research purpose of this paper is to propose a high-quality ellipse detection method to handle the long-standing issue that cannot detect ellipses both accurately and efficiently in ellipse detection field. To that end, for the first time, we take the advantage of arc-support LSs for ellipse detection. The arc-support groups are formed by robustly linking the consecutive arc-support LSs which share similar geometric properties in point statistics level. Each arc-support group will be measured and assigned a saliency score. Secondly, we generate the initial ellipse set by two complementary approaches both locally and globally. The superposition principle of ellipse fitting and the novel geometric constraints, which are polarity constraint, region restriction and adaptive inliers criterion, are employed to consolidate the proposed method's accuracy and efficiency. Thirdly, we decompose the 5D ellipse parameter space into three subspaces according to ellipse center, orientation and semi-axes and perform three-stage efficient clustering. Finally, the candidates which pass the rigorous and effective verification will be refined by fitting again.\par
The rest of this paper is organized as follows. Section \ref{sec:Preliminary} introduces the preliminaries about arc-support LS and superposition principle of ellipse fitting. Section \ref{sec:ellipse detection} presents the high-quality ellipse detection framework, as a four-stage detection procedure: arc-support groups forming, initial ellipse set generation, clustering, and candidate verification. Section \ref{sec:complexity} analyzes the computation complexity of the proposed ellipse detection algorithm. Experimental results, as well as the accuracy and efficiency detection performance of the proposed method, are detailed in Section \ref{sec:experimental results}. Section \ref{sec:conclusion} concludes the paper.
\section{Preliminary}\label{sec:Preliminary}
In this section, the arc-support LS and its appendant properties are introduced as the basic geometric primitives for ellipse detection. Then we develop a superposition principle of fast ellipse fitting, which will save running time for the ellipse generation.
\subsection{Arc-support LS}
\begin{figure}[!t]
\centering
\includegraphics[width=\hsize]{pic//2.pdf}
\caption{Level-line angle and two types of LS. (a) the level-line angle is acquired by clockwise rotating the gradient angle $90^{\circ}$; (b) greyscale image; (c) straight and arc-support LSs generated from (b).}
\label{fig2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=\hsize]{pic//3.pdf}
\caption{Features of arc-support LS. (a) the overall gradient direction in the local greyscale area is same as arc-support direction and the three main angles in the corresponding level-line map change anticlockwise; (b) the conter-example of (a).}
\label{fig3}
\end{figure}
In image processing, LS mainly derives from two situations, as shown in Fig. \ref{fig2}. The first type LS comes from the support region where points share nearly the same level-line angle and overall distribute straight. Another type of LS derives from the arc-support region whose distribution changes like a curve. Thus, we call the LS approximated from arc-support region as ``arc-support LS''. Arc-support LS is built on top of LSD \cite{grompone2010lsd} as it is superior to other methods due to its efficiency and false control ability. The corresponding extraction procedures can be found in \cite{Lu2017Circle}. With the help of arc-support LS, the straight LS can be pruned while the arc geometric cues remain. Hereon, some properties of arc-support LS critical for ellipse detection are detailed.
\subsubsection{Arc-support Direction}
Different from conventional LS, arc-support LS carries the nature of convexity, standing for the ellipse center direction of an elliptic arc, namely the arc-support direction, as shown in Fig. \ref{fig3}. Assume that the two terminals of the circumscribed rectangle of the support region are $A$ and $B$ and the centroid is $C$. Thus the main angle of the support region is denoted as $\angle \overrightarrow{AB}$ and can be set to
\begin{equation}
\label{eq1}
\text{arctan}\left(\frac{{\sum\nolimits_{{p_i} \in \text{Region}} {\text{sin} (\text{level-line angle}({p_i}))} }}{{\sum\nolimits_{{p_i} \in \text{Region}} {\text{cos} (\text{level-line angle}({p_i}))} }}\right).
\end{equation}
Analogously, the main angles of two subregions $\angle \overrightarrow{AC}$ and $\angle \overrightarrow{CB}$ can be obtained according to Eq. (\ref{eq1}). Therefore, the arc-support direction can be set by anticlockwise (or clockwise) rotating $\angle \overrightarrow{AB}$ by $90^{\circ}$ if $\angle \overrightarrow{AC}$, $\angle \overrightarrow{AB}$ and $\angle \overrightarrow{CB}$ change in the anticlockwise (or clockwise) direction and have an angle interval at least $T_{ai}$ in \{$\angle \overrightarrow{AC}$,$\angle \overrightarrow{AB}$ \} and \{$\angle \overrightarrow{AB}$,$\angle \overrightarrow{CB}$ \}.
\subsubsection{Polarity of Arc-support LS}
In the greyscale image, the overall gradient direction in the local area indicates the tendency of illumination variation. After the careful observation, there exist two situations between elliptic arc's overall gradient direction and arc-support direction. We define the polarity of an arc-support LS, namely $Pol_{L}$, is positive ($+1$) if the corresponding gradient direction and are-support direction are consistent, otherwise is negative ($-1$). A fast decision to the polarity of an arc-support LS is by judging the rotation direction of the main angles $\angle \overrightarrow{AC}$, $\angle \overrightarrow{AB}$ and $\angle \overrightarrow{CB}$, as shown in Fig. \ref{fig3}(a) and Fig. \ref{fig3}(b).
\subsection{Superposition Principle of Ellipse Fitting}
Ellipse fitting is very important in ellipse detection since it directly affects the quality of detected ellipse. Least-squares based ellipse fitting methods focus on minimizing the residue between points and ellipse \cite{rosin1993note,gander1994least,fitzgibbon1999direct}. As the constraint of ellipse fitting problem is quadratic, it usually leads to unsatisfactory efficiency along with the iterative procedure. Therefore, Fitzgibbon et al. \cite{fitzgibbon1999direct} proposed a non-iterative algorithm by solving the positive eigenvector of eigensystem. And we develop the superposition principle on the basis of \cite{fitzgibbon1999direct} due to its efficiency. Suppose that there are $n$ data points in the set $\Gamma_1 = \{ {p_1},{p_2}, \cdots ,{p_n}\}$, $p_{i}=\{ x_{i},y_{i} \}$. We first calculates $\Gamma_1$'s scatter matrix $\text{\bf{S}} =
\text{\bf{D}}^{\text{T}}\text{\bf{D}}$, and {\bf{D}} is denoted as
\begin{equation}
\label{eq2}
\text{\bf{D}} = {\left[ {\begin{array}{*{20}{c}}
{{x_1}^2}&{{x_1}{y_1}}&{{y_1}^2}&{{x_1}}&{{y_1}}&1\\
{{x_2}^2}&{{x_2}{y_2}}&{{y_2}^2}&{{x_2}}&{{y_2}}&1\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
{{x_n}^2}&{{x_n}{y_n}}&{{y_n}^2}&{{x_n}}&{{y_n}}&1
\end{array}} \right]_{n\times6}}.
\end{equation}
Then by solving the generalized eigensystem $\text{\bf{S}}^{-1}\text{\bf{C}}$, where {\bf{C}} is the constant constraint matrix
\begin{equation}\label{eq:constant matrix}
\text{\bf{C}} = {\left[ {\begin{array}{*{20}{c}}
{\rm{0}}&{\rm{0}}&{{\rm{ - 1}}}& \cdots &{\rm{0}}\\
{\rm{0}}&{\rm{2}}&{\rm{0}}&{}&{}\\
{{\rm{ - 1}}}&{\rm{0}}&{\rm{0}}&{}& \vdots \\
\vdots &{}&{}& \ddots &{}\\
{\rm{0}}&{}& \cdots &{}&{\rm{0}}
\end{array}} \right]_{{6\times6}}},
\end{equation}
the obtained eigenvector with positive eigenvalue is the desired fitted ellipse to $\Gamma_1$.\par
In practical ellipse detection process, it always needs to attempt fitting extensive different combinations of point sets for finding the most suitable fitted ellipse. Assuming $\Gamma_1$ has already been computed to fit an ellipse and after that several additional point sets belonging to the same ellipse are newly discovered, which are denoted by $\Gamma_2, \Gamma_3, \cdots, \Gamma_k$, an efficient computation approach to fit the new ellipse should be based on the previous computation results. Denote the design matrix and scatter matrix of $\Gamma_i$ as $\text{\bf{D}}(\Gamma_{i})$ and $\text{\bf{S}}(\Gamma_{i})$, respectively. Thus the design matrix $\text{\bf{D}}_{c}$ of the combination of $k$ point sets $\Gamma_1, \Gamma_2, \cdots, \Gamma_k$ can be written as
\begin{equation}
\label{eq3}
{\text{\bf{D}}_{c}} = \left[ {\begin{array}{*{20}{c}}
{\text{\bf{D}}({\Gamma_{1}})}\\
\vdots \\
{\text{\bf{D}}({\Gamma_{k}})}
\end{array}} \right],
\end{equation}
and the corresponding scatter matrix $\text{\bf{S}}_{c}$ is
\begin{equation}
\begin{aligned
\label{eq4}
{\text{\bf{S}}_c} &= \text{\bf{D}}_c^\text{T}{\text{\bf{D}}_c} = \text{\bf{D}}{({\Gamma_{1}})^\text{T}}\text{\bf D}({\Gamma_{1}}) + \cdots + \text{\bf D}{({\Gamma_{k}})^\text{T}}\text{\bf D}({\Gamma_{k}}) \\
&= \text{\bf S}({\Gamma_{1}}) + (\text{\bf S}({\Gamma_{2}}) + \cdots + \text{\bf S}({\Gamma_{k}})).
\end{aligned}
\end{equation}
Eq. \ref{eq4} indicates that the scatter matrix of any combinatorial point sets equals the summation of the scatter matrix of each point set, which casts light on the feasibility of calculating the scatter matrix of each group merely once. The above superposition feature can cut computation time down when fitting one or more sets into an ellipse.
\section{High-quality Ellipse Detection}\label{sec:ellipse detection}
In this section, a high-quality ellipse detection is proposed by introducing the arc-support LSs. The overall procedure consists of: (1) arc-support groups forming, (2) initial ellipse set generation, (3) clustering, and (4) candidate verification. The arc-support group collects the consecutive arc-support LSs belonging to the same curve, which can avoid the disturbance of the useless straight LSs. In the initial ellipse set generation step, accuracy and efficiency keep pace with the aid of fast ellipse fitting and effective geometric constraints. The efficient clustering and rigorous verification further facilitate the high detection performance of the proposed detector. An overall detection example of our method is demonstrated in Fig. \ref{fig4}.
\begin{figure}[!tb]
\centering
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//origin.jpg}
\label{fig4:a}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//arcSupportLSs.jpg}
\label{fig4:b}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//arcSupportGroups.jpg}
\label{fig4:c}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//initialEllipses.jpg}
\label{fig4:d}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//candidates.jpg}
\label{fig4:e}
}
\subfigure[]{
\includegraphics[width=0.28\hsize]{pic//flowDiagrams//results.jpg}
\label{fig4:f}
}
\caption{Pipeline illustration of the proposed ellipse detection. (a) origin image; (b) 42 extracted arc-support LSs; (c) 20 arc-support groups; (d) 13 initial ellipses; (e) 10 ellipse candidates after clustering; (f) 2 qualified detections after verification and refinement.}
\label{fig4}
\end{figure}
\subsection{Arc-support Groups forming}
\subsubsection{Robust Linking and Groups Forming}
\begin{figure}[!bt]
\centering
\includegraphics[width=\hsize]{pic//4.pdf}
\caption{The arc-support LSs are linked to form an arc-support group in point statistics level based on continuity and convexity.}
\label{fig5}
\end{figure}
Since an elliptic curve may consist of several arc-support LSs, we can link the discovered arc-support LSs to form a group. Any two consecutive arc-support LSs that will be linked should meet the continuity and convexity conditions. For continuity condition, the proximity between the head of an arc-support LS to the tail of another one should be close enough. For convexity condition, the linked LSs should change in the same direction either clockwise or anticlockwise.
Note that a support pixel's level-line angle should be within tolerance $\alpha$ with the support region's main angle, therefore, the angle deviation between two consecutive arc-support LSs should be less than $2\alpha$. To avoid incorrect LSs linking in the existence of noise, we count the number of support points of each next LS within a local statistical area near the terminal of current LS ($k$th LS's number of support points is represented as $\sum p_{i}^{L_{k}}$), and create a histogram for choosing the LS with maximum votes to link with current LS, as shown in Fig.~\ref{fig5}. Iteratively, the linked arc-support LSs which share the similar geometric properties are called as ``arc-support group''. Algorithm~\ref{algo1} details the arc-support LSs linking and groups forming process.\par
\begin{algorithm}[!b]
\caption{ Arc-support groups forming.}
\label{algo1}
\begin{algorithmic}[1]
\Require
Arc-support line segment set, $T_l$;
Arc-support regions that generate line segments, $T_r$;
Angle tolerance, $\alpha$;
Status where line segment used, $S$;
\Ensure
Arc-support groups, $G$;
\State Initialize groups $G = \varnothing$;
\label{code:alg1:initialization}
\Repeat
\State Choose an arc-support line segment $l_i$ which satisfies $S(l_i) \neq used$ from $T_l$;
\State Set the arc-support groups searched from the head and tail of $l_i$ as $g_{\text{head}}$ = $\varnothing$, $g_{\text{tail}}$ = $\varnothing$;
\State Set $l_i$ as the seed of line segment $l_s$;
\Repeat
\State Searching consecutive arc-support line segments at the head end of $l_s$;
\State Rule out the searched line segments which are $used$ and beyond $2\alpha$ angle deviation to $l_s$;
\State Determine statistical area at the head end of $l_s$;
\State Acquire the line segment $l_k$ with highest point votes by using $T_r$;
\State Update $g_{\text{head}}$ = $g_{\text{head}}\cup L_{{k}}$, $S(l_k) = used$, $l_s$ = $l_k$;
\Until{$l_s$ is $\varnothing$}
\State Set $l_i$ as the seed of line segment $l_s$ again;
\State $g_{\text{tail}}$ can be obtained by repeating the above searching process at the tail of $l_s$;
\State Combine the searched arc-support groups $g_{\text{head}}$ = $\{L_{h1},\cdots,L_{hn}\}$ and $g_{\text{tail}}$ = $\{L_{t1},\cdots,L_{tn}\}$ as $g$ = $\{L_{tn},\cdots,L_{t1},L_{\text{i}},L_{h1},\cdots,L_{hn}\}$;
\State Update $G$ = $G\cup g$;
\State Update $S(l_i) = used$;
\Until{every arc-support line segment is traversed}\\
\Return $G$;
\end{algorithmic}
\end{algorithm}
\subsubsection{Spanning Angle Measurement for Each Group}
Each arc-support group which is composed of several arc-support LSs is essentially the polygonal approximation of a curve. If the $i$th group contains $n$ arc-support LSs, it will have $n-1$ angle intervals derived from every two consecutive arc-support LSs. Supposing that the angle interval sequence of $i$th group is $\{\theta^{i}_{1}, \theta^{i}_{2}, \cdots, \theta^{i}_{n-1}\}$, therefore, the spanning angle of $i$th group is $\sum\limits_{j = 1}^{n-1} {\theta _j^i}$. If an arc-support group is more salient to an ellipse, its spanning angle will be larger. Therefore, we have $S^{i}\propto (\sum\limits_{j = 1}^{n-1} {\theta _j^i})/{360^{\circ}}$,
where $S^{i}$ is the saliency score of the $i$th group. For convenience, the proportionality coefficient is set to $1$, and thus the range of $S^{i}$ is $[0,1]$.\par
As shown in Fig.~\ref{fig4:b}, only arc-support LSs are retained while the straight LSs are filtered. With the procedure of robust arc-support LSs linking, the LSs that belong to the same curves are pooled into arc-support groups (Fig.~\ref{fig4:c}).
\subsection{Initial Ellipse Set Generation}
Considering the fact that an arc-support group might contain all arc-support LSs of a curve, or merely a separate arc-support LS, therefore, we use two complementary methods to generate the initial ellipse set. First, from the local perspective, the arc-support LS group with relatively high saliency score is probably the dominant component of the polygonal approximation of an ellipse. A simple and effective manner is to individually fit the arc-support LS group to an ellipse completely relying on the threshold $T_{ss}$ such that the salient ellipse, for instance, the one with spanning angle close to $360^{\circ}$, can be picked out precedently. Second, from the global perspective, we search all the valid pairs of arc-support groups globally to reconstruct the latent ellipses on the image, one advantage of which is dealing with the troublesome situation of the arc-support groups for a common ellipse but far apart. All the valid pairs of groups should satisfy three criteria below: (1) polarity constraint; (2) region restriction; (3) adaptive inliers criterion. In addition, to avoid the overwhelming fitting process, the superposition principle is adopted during ellipse fitting for time efficiency.\par
\subsubsection{Polarity Constraint}
After observing the image regions around an elliptical edge, the inner of an ellipse is always either brighter or darker than the peripheral, where brighter means that the polarity of the arc-support LSs is positive and darker means the one is negative. If all arc-support LSs come from the same ellipse, generally, their polarity should be the same, too. Thus, the first criterion is that the polarity of the paired arc-support LS groups should be congruous. In certain cases, it may only require to detect the oval objects which are brighter (or darker) than background. Following our approach, the ellipse with specified polarity can be easily recognized, which will be detailed in Section \uppercase\expandafter{\romannumeral5}. E. As for the case of the arc-support groups derived from common ellipse while owning different polarity, e.g. an ellipse sharing different backgrounds, it still could be successfully detected as finding the most salient pair of arc-support groups with same polarity for ellipse fitting.\par
\subsubsection{Region Restriction}
Actually, most of the arc-support groups do not contribute to building a valid pair because of the high probability that two groups are from different ellipses or curves. Therefore, an early decision before fitting ellipse is essential.
\begin{figure}[!tb]
\centering
\includegraphics[width=0.8\hsize]{pic//8.pdf}
\caption{Region restriction for a pair of two arc-support groups.}
\label{fig6}
\end{figure}
As shown in Fig.~\ref{fig6}, groups $\{g_{1},g_{3}\}$ and $\{g_{1},g_{4}\}$ should not be paired. The reason is that group $g_{3}$ is out of $g_{1}$'s valid region, which is along with the arc-support direction of every arc-support LS in $g_{1}$. Although $g_{4}$ is in the valid region of $g_{1}$, $g_{1}$ is not in the valid region of $g_{4}$. Consequently, if two groups are paired, they should locate in the mutual valid regions. In Fig.~\ref{fig6}, the start point of group $g_{i}$ is denoted as $P_{s}^{g_{i}}$ and the end point is $P_{e}^{g_{i}}$; the start arc-support LS is $L_{s}^{g_{i}}$ and the end arc-support LS is $L_{e}^{g_{i}}$; the midpoint of $P_{s}^{g_{i}}$ and $P_{e}^{g_{i}}$ is $M_{i}$. Then, we have
\begin{equation}
\label{eq6}
\begin{aligned}
\left\{
\begin{array}{c}
\overrightarrow{ARC_{L_{s}^{g_{i}}}}\cdot\overrightarrow{P_{s}^{g_{i}}P_{e}^{g_{j}}} > \rho_{d} \\
\overrightarrow{ARC_{L_{e}^{g_{i}}}}\cdot\overrightarrow{P_{e}^{g_{i}}P_{s}^{g_{j}}} > \rho_{d} \\
(-Pol)\cdot Dir_{\bot}(\overrightarrow{P_{s}^{g_{i}}P_{e}^{g_{i}}})\cdot \overrightarrow{M_{i}M_{j}} > \rho_{d}
\end{array}
\right.
\end{aligned}
\end{equation}
where the $\overrightarrow{ARC_{L}}$ is the normalized arc-support direction vector of $L$. $Dir_{\bot}(\overrightarrow{v})$ represents the normalized vector acquired by rotating $\overrightarrow{v}$ clockwise $90^{\circ}$. The $Pol$ equals to $+1$ when the polarity of the pair $\{g_{1},g_{2}\}$ is positive, otherwise is $-1$. $\rho_{d}$ is distance threshold. Meanwhile, $(i,j)$ runs over $(1,2)$ and $(2,1)$.\par
\subsubsection{Adaptive Inliers Criterion}
For the pair $\{g_{1},g_{2}\}$ that passes the validations of polarity constraint and region restriction, an ellipse fitting against the endpoints of arc-support LSs in the pair will be implemented immediately. Assuming that the fitted ellipse is $e$, and the normal vector of edge point $P$ to $e$ is $\nabla e(P)$ (its direction points to the exterior of the ellipse),
we define edge point $P_{i}$ as a support inlier to $e$ if meeting both the distance tolerance $\epsilon$ and normal tolerance $\alpha$, namely the Rosin approximation distance \cite{rosin1998ellipse} from $P_{i}$ to $e$ should be less than $\epsilon$, and the absolute angle difference between $\nabla e(P_{i})$ and $-Pol_{e}\cdot Grad(P_{i})$ is less than $\alpha$, where $Pol_{e}$ is the polarity of $e$ and $Grad(P_{i})$ is image gradient of $P_{i}$. From the geometric perspective, we can approximate the arc length by the number of edge pixels. Therefore, the number of support inliers made up each arc-support LS in pair $\{g_{1},g_{2}\}$ should be greater than the corresponding LS's length, namely
\begin{equation}
\label{eq7}
\# \{p_i : p_i \in \text{SI}(L_{j})\} > \text{Length}(L_{j}),
\end{equation}
where $\text{SI}(L_{j})$ represents the support inliers set of arc-support LS $L_{j}$, $j = 1,2,\cdots,N_{g_{1}}+N_{g_{2}}$. $N_{g_{1}}$ and $N_{g_{2}}$ are the number of arc-support LSs in $g_{1}$ and $g_{2}$, respectively. In this step, the length of LS is an adaptive threshold for the corresponding support inliers. If an ellipse $e$ satisfies the adaptive inliers criterion, we will fit the support inliers to produce an initial ellipse. Eventually, all the valid pairs of arc-support groups are transformed to the initial ellipse set.
\subsection{Ellipse Clustering}
Considering the existence of duplicates in the initial ellipse set, an efficient clustering method is extremely important to trim them down, which should not only maintain the isolated points but also suppress the non-maximum. To that end, we develop a hierarchical clustering method based on mean shift~\cite{cheng1995mean}, which decomposes the 5D ellipse parameter space clustering problem into three low and cascaded dimensional space clustering problems (centers, orientations and semi-axes).\par
Assume that the initial ellipse set is $E^{\text{init}}$ and the element number of $E^{\text{init}}$ is $N^{\text{init}}$. We have
\begin{equation}
\label{eq8}
E^{\text{init}} = \bigcup_{1 \le i \le N^{\text{init}}}e_{i}
\end{equation}
where $e_{i} = \{(x,y)_{i}, \varphi_{i}, (a,b)_{i}\}$. Meanwhile, $(x,y)_{i}$, $\varphi_{i}$ and $(a,b)_{i}$ are the center, orientation and semi-axes of the initial ellipse $e_{i}$, respectively.\par
Firstly, our method clusters the ellipse centers of $E^{\text{init}}$ based on mean shift with limited iterations. It then produces $n^{\text{c}}$ elliptic cluster centers $(x,y)_{1}^{c}$, $(x,y)_{2}^{c}$, $\cdots$, $(x,y)_{n^{\text{c}}}^{c}$. If $e_{i}$ is the nearest to $(x,y)_{k}^{c}$, we will add $e_{i}$ to set $\Omega_{k}$. Therefore, $E^{\text{init}}$ can be divided into $n^{\text{c}}$ partitions. The $k$th ($1\le k \le n^{\text{c}}$) partition is represented as
\begin{equation}
\label{eq9}
\Omega_{k} = \{ e_{i} \parallel (x,y)_{i} \in e_{i} \; \text{and} \; (x,y)_{i}\rightarrow (x,y)_{k}^{c} \}.
\end{equation}
$(x,y)_{i} \rightarrow (x,y)_{k}^{c}$ means the Euclidean distance between data point $(x,y)_{i}$ and cluster center $(x,y)_{k}^{c}$ is least among all the cluster centers.\par
Secondly, each initial ellipse subset $\Omega_{k}$ is clustered with respect to their orientations. So $n^{\varphi}_{k}$ orientation cluster centers are generated, which are $\varphi^{c}_{1}, \varphi^{c}_{2}, \cdots, \varphi^{c}_{n^{\varphi}_{k}}$. In a similar way, $\Omega_{k}$ can be divided into $n^{\varphi}_{k}$ subsets. And the $s$th ($1\le s \le n^{\varphi}_{k}$) subset $\Omega_{k,s}$ corresponding to the orientation cluster center $\varphi^{c}_{s}$ is
\begin{equation}
\label{eq10}
\Omega_{k,s} = \{ e_{i} \parallel e_{i} \in \Omega_{k}, \varphi_{i} \in e_{i} \; \text{and} \; \varphi_{i}\rightarrow \varphi_{s}^{c} \}.
\end{equation}
\par
Finally, we implement the clustering step based on semi-axes for each initial ellipse subset $\Omega_{k,s}$ and therefore, $n^{\text{axes}}_{k,s}$ cluster centers are produced: $(a,b)^{c}_{1}$, $(a,b)^{c}_{2}$, $\cdots$, $(a,b)^{c}_{n^{\text{axes}}_{k,s}}$. Eventually, each mode of the combination $\{(x,y)^{c}_{k}, \varphi^{c}_{s}, (a,b)^{c}_{t}\}$ ($1\le t \le n^{\text{axes}}_{k,s}$) is the initial ellipse clustering result, namely the ellipse candidate. The ellipse candidate set $E^{c}$ can be described as
\begin{equation}
\label{eq11}
E^{c} = \bigcup_{k,s,t}\{ (x,y)^{c}_{k}, \varphi^{c}_{s}, (a,b)^{c}_{t} \}
\end{equation}
and the number of ellipse candidates $N^{c}$ is
\begin{equation}
\label{eq12}
{N^{c}} = \sum\limits_{k} {\sum\limits_{s}{n_{k,{\rm{ }}s}^{\text{axes}}} }.
\end{equation}
\par
As a result, the purer ellipse candidates can be generated after applying the hierarchical clustering approach to initial ellipse set. The benefits of such ellipse parameter space decomposition are two folds.
First, we distinguish the different geometrical significance of ellipse center, orientation, and semi-axes for hierarchical clustering since it is difficult to assign an accurate distance measure between two ellipses directly. Second, our clustering method is conveniently implemented and its computation complexity is quadratic (details can be found in Section \uppercase\expandafter{\romannumeral4}), which vastly outperforms than the direct clustering behavior in 5D space.
\subsection{Ellipse Candidate Verification}
The learned ellipse candidates cover almost all possible ellipses existing in the image while keeping nearly few duplicates, which saves much verification time and guarantees good recalls due to their pure quantity and high saliency. In this section, to further ensure the quality of the detected ellipses, we conduct the ellipse candidate verification that incorporates the stringent regulations for goodness measurement and elliptic geometric properties for refinement.
\subsubsection{Goodness Measurement}
The study has shown positive correlation between the number of support inliers on an ellipse and corresponding perimeter $\mathcal{B}$~\cite{kulpa1979properties}. Although the precise numerical formula of $\mathcal{B}$ has not yet been revealed, in practice, it can use $\mathcal{B} \approx \pi[\frac{3}{2}(a+b)-\sqrt{ab}]$ for approximation where $a$ and $b$ are the semi-major axis and semi-minor axis respectively \cite{zhang2016consistency}. In addition, the larger the angular coverage $\mathcal{C}$ of the elliptic connected component of support inliers becomes, the more salient the ellipse candidate will be. Based on these facts, we employ the following evaluation regarding both the number of support inliers and the angular coverage. The support inliers should satisfy the distance tolerance $\frac{\epsilon}{2}$ and the normal tolerance $\alpha$. Therefore, the ``goodness'' can be formulated as
\begin{equation}
\label{eq13}
\text{Goodness}(e)=\sqrt{\frac{\# \{p_i : p_i \in \text{SI}(e)\}}{\mathcal{B}} \cdot \frac{\mathcal{C}}{360^{\circ}}}
\end{equation}
where $\text{SI}(e)$ represents the support inliers of ellipse $e$. Note that we tighten the distance tolerance $\epsilon$ to $\frac{\epsilon}{2}$ to rule out the noise, and thus the goodness measurement is more credible. Finally, we apply a pseudo descending order (in linear time) to the ellipse candidates according to their goodness scores and preferentially pick out the candidates for subsequent validation.
\subsubsection{Verification and Fitting Again}
In verification, the proposed method continues to loose the distance tolerance $\frac{\epsilon}{2}$ to $\epsilon$, and validate the candidate individually against: (1) the number of support inliers; (2) the angular coverage of ellipse. We use a ratio threshold $T_{r}$ and expect that there are $T_{r}\mathcal{B}$ support inliers on the ellipse. In the meanwhile, our method only accepts the ellipse whose angular coverage is at least $T_{ac}$ degrees.\par
Given sufficient support inliers, we may have better ellipse results. Recall that we have found the support inliers with respect to each candidate in the validation step. If a candidate generates the true ellipse (or TP for short), its support inliers should be more sufficient than the previous one. This motivates us to fit ellipse again, which improves the overall accuracy and shows self-calibrated ability. Notably, the time complexity of the additional fitting only relates to the number of final detected ellipses, which will not bring on significant changes in the running time.
\section{Complexity of The Algorithm}\label{sec:complexity}
Assuming that the image size is $N \times N$, therefore, the arc-support LS extraction approach has an $O(N^{2})$ complexity equivalent to LSD \cite{grompone2010lsd}. In the procedure of the arc-support groups forming, the computational complexity is $O(N_{L})$, where $N_{L}$ denotes the number of arc-support LS. If there are $N_{G}$ groups, spanning angle measurement of groups has the complexity of $O(N_{G})$. In the initial ellipse set generation step, the computational complexity has upper bound $O(N_{G}^{2})$ in the worst case, which may rarely occur due to the three novel geometric constraints. In the ellipse clustering step, the running time has the complexity of $O(1+n^{\text{c}}+\sum \limits_{k}n^{\varphi}_{k})$. Noting that after clustering, the number of cluster centers is less than or equal to the number of original data points, which means that $n^{\text{c}} \leq N^{\text{init}}$ and $\sum \limits_{k}n^{\varphi}_{k} \leq N^{\text{init}}$. Note that the initial ellipses' number $N^{\text{init}}$ is of $O(N_{G}^{2})$, and thus the process of initial ellipse clustering step has the computational complexity of $O(2N_{G}^{2})$. In the validation step, the computation time relates to the number of candidates $N^{c}$. Eventually, the computational complexity of the proposed method is upper bounded by $O( N^{2} + N_{L} + 3N_{G}^{2} + N^{c})$, which reveals that the ellipse detection complexity is as fast as quadratic in $N$ and $N_{G}$.
\section{Experimental Results}\label{sec:experimental results}
In this section, extensive and detailed experiments are implemented to demonstrate the high-quality ellipse detection performance of the proposed method compared to the existing state-of-the-art methods.\par
\subsection{Experiments Setup}
\begin{table*}[!t]
\centering
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\caption{Ellipse Detection Results on Three Public Real-world Datasets Compared with the Popular and State-of-the-art Methods.}
\label{tab1}
\begin{tabular}{cccccccc}
\toprule[1pt]
Dataset &Index & RHT \cite{mclaughlin1998randomized} & ELSDc \cite{patraucean2017joint} & Prasad \cite{prasad2012edge} & Fornaciari \cite{fornaciari2014fast} & Qi \cite{jia2017fast} & Our \\\midrule[1pt
\multirow{4}*{Traffic Sign Dataset} &Precision & 0.0706 & 0.0429 & 0.1401 & 0.4545 & 0.5814 & \textbf{\textbf{0.9110}} \\%\cline{2-8}
&Recall & 0.5149 & 0.6933 & 0.5665 & 0.7277 & 0.7324 & \textbf{0.8811} \\%\cline{2-8}
&F-measure & 0.1242 & 0.0808 & 0.2246 & 0.5596 & 0.6482 & \textbf{0.8958} \\%\cline{2-8}
&Time/s & 1.2875 & 4.6006 & 9.9918 & 0.2875 & \textbf{0.1488} & 0.5791 \\\midrule[0.5pt]
\multirow{4}*{Prasad Dataset} &Precision & 0.1941 & 0.0922 & 0.2360 & 0.7039 & 0.7161 & \textbf{0.7523} \\
&Recall & 0.2479 & 0.2940 & 0.3145 & 0.2154 & 0.2393 & \textbf{0.3504} \\
&F-measure & 0.2177 & 0.1403 & 0.2697 & 0.3298 & 0.3587 & \textbf{0.4781} \\
&Time/s & 0.2011 & 1.4348 & 3.9011 & 0.0939 & \textbf{0.0529} & 0.1685 \\\midrule[0.5pt]
\multirow{4}*{\tabincell{c}{PCB Dataset}}&Precision&0.4165 &0.4168 &0.7645 &0.8933 & 0.9393 &\textbf{0.9715}\\
&Recall & \textbf{0.9500} & 0.9154 & 0.7115 & 0.8692 & 0.8923 & 0.9192 \\
&F-measure & 0.5791 & 0.5728 & 0.7371 & 0.8811 & 0.9152 & \textbf{0.9447} \\
&Time/s & 0.2271 & 3.4830 & 1.0758 & 0.0980 & \textbf{0.0738} & 0.1693
\\\bottomrule[1pt]
\end{tabular}
\end{table*}
\subsubsection{Model Parameters}
Our ellipse detection method mainly involves seven parameters, which are discussed as follows
(1) $T_{ai}$ is the angle interval of each subregion to the support region, which is used in arc-support LS extraction. It indicates the least curvature degree of the support region that generates an arc-support LS. Obviously, the LS derived from the straight edge will be filtered since its angle interval is less than $T_{ai}$. We fix $T_{ai}$ to $2.25^{\circ}$ as it performs well in experiments. (2) Angle tolerance $\alpha$ is used in the cases when evaluating the angle deviation of a geometric primitive, e.g. the point's level-line angle or gradient angle, to the corresponding reference angle. $\alpha$ can be empirically set to $22.5^\circ$ which yields the best results for thousands of images~\cite{grompone2010lsd,burns1986extracting}. (3) Saliency score threshold $T_{ss}$ is used in the initial ellipse set generation. Admittedly, we can fit any group as long as its saliency score is higher than zero because of the negligible computational cost. However, it is unnecessary because we will find all the valid pairs of arc-support groups for generating the initial ellipses. Therefore, we set $T_{ss}$ to $0.25$. (4) Distance threshold $\rho_{d}$ is used in region restriction, which can be minus for dealing with some extreme cases. Thus $\rho_{d}$ is fixed to $-3\epsilon$. (5) The distance tolerance $\epsilon$ is used to recover the inliers to arc-support LS or ellipse. If $\epsilon$ becomes more restrictive, the inliers will be purer, even in presence of spurious edges or noise. As our algorithm aims at high localization accuracy, we are able to set $\epsilon$ to $2$ pixels. (6) $T_{r}$ is the ratio of support inliers on an ellipse and (7) $T_{ac}$ is the elliptic angular coverage threshold. They both are used in the ellipse verification. $T_{r}$ and $T_{ac}$ are easily tuned due to their geometric significance. Since most of the true ellipses in real-world images have a degree of completeness, we set $T_{r} = 0.6$ and $T_{ac} = 165^{\circ}$ in default.\par
Notably, we merely open \text{\it{two}} external adjustable parameters $T_r$ and $T_{ac}$ after regarding the other five parameters as intrinsic parameters since they can be empirically fixed and work fairly well, which enables our ellipse detection algorithm to be easily used.
\subsubsection{Evaluation Metrics}
We employ the following metrics for evaluations: (1) \textbf{ precision}, (2) \textbf{recall}, (3) \textbf{F-measure}. The precision = TPs$/$(TPs + FPs), recall = TPs$/$(TPs + FNs) and F-measure = $2/$(precision${}^{-1}$ + recall${}^{-1}$). A detected ellipse is regarded as a TP if its overlap area ratio to the corresponding ground true ellipse is larger than $D_{0}$. And we set $D_{0}$ to $0.8$ throughout all experiments as did in \cite{prasad2012edge,fornaciari2014fast, jia2017fast}.
\subsubsection{Compared Methods}
We mainly select five most popular and competitive algorithms for quantitative and qualitative comparisons, which are RHT \cite{mclaughlin1998randomized}, ELSDc \cite{patraucean2017joint} and the methods proposed by Prasad et al. \cite{prasad2012edge}, Fornaciari et al. \cite{fornaciari2014fast}, and Qi et al. \cite{jia2017fast}. RHT is the most popular in the literature and often used as the baseline method. ELSDc is robust and achieves precise location accuracy amongst state-of-the-art methods. Prasad method combines HT and the techniques of edge following, which reflects very well detection performance. The methods proposed by Fornaciari et al. and Qi et al. are very efficient and obtain extremely high F-measure scores on public datasets. We compare our method\footnote[2]{The source code and more detection examples of our method can be found at https://github.com/AlanLuSun/High-quality-ellipse-detection.\label{footnote2}} to the aforementioned ellipse detectors as they are the most popular or state-of-the-art detectors existing in ellipse detection field.\par
For the fair comparison purpose, we adopt the source codes of ELSDc \cite{patraucean2017joint}, Prasad et al. \cite{prasad2012edge}, Fornaciari et al.~\cite{fornaciari2014fast}, and Qi et al. \cite{jia2017fast} which are available online, and reimplement RHT \cite{mclaughlin1998randomized} according to the original paper. RHT, Prasad method and our method are run in MATLAB while the remaining three methods are in C++. All experiments are performed with default parameters and on the same computer with Intel Core i7-7500U 2.7GHz CPU and 8 GB memory.
\subsubsection{Datasets}
To test the competitive ellipse detectors, three public challenging real-world datasets are utilized:
\begin{itemize}
\item \textbf{Traffic sign dataset}. As a portion of Dataset $\sharp2$ which is created by Fornaciari et al. \cite{fornaciari2014fast}, it contains 273 images with various ellipses that are projected by round traffic signs at different real-life scenarios. These images derive from the frames of several videos captured by smartphone,
suffering from the blurry and varying lighting conditions by motion and autofocus.
\item \textbf{Prasad dataset}. A complex real-world image dataset, employed before by many well-known ellipse detectors in their experiments~\cite{prasad2012edge,fornaciari2014fast, patraucean2017joint, jia2017fast}. Note that Prasad dataset consists of 198 images which are complex enough due to the unpredictable conditions and substantial disturbances.
\item \textbf{PCB dataset}. PCB (Printed Circuit Board) dataset \cite{Lu2017Circle} includes 100 industrial PCB images with various disturbances and each image contains at least one circular or elliptic shape. All PCB images are labeled manually and precisely.
\end{itemize}
\par
\subsection{Experiments on Real-world Datasets}
\begin{figure*}[!tb]
\centering
\includegraphics[width=0.9\hsize]{pic//threedatasets.pdf}
\caption{Ellipse detection examples on three real-world datasets. The first and second columns are the input images and ground truth (GT). The input images of the first three rows are from traffic sign dataset while the second and third three-row images are from Prasad dataset and PCB dataset.}
\label{fig7}
\end{figure*}
\begin{figure*}[!tb]
\centering
\subfigure[]{\includegraphics[width=0.3\textwidth]{pic//curves//Traffic-Precision.pdf
\label{fig8:Traffic-P}}
\subfigure[]{\includegraphics[width=0.3\textwidth]{pic//curves//Traffic-Recall.pdf
\label{fig8:Traffic-R}}
\subfigure[]{\includegraphics[width=0.3\textwidth]{pic//curves//Traffic-FMeasure.pdf
\label{fig8:Traffic-F}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Prasad-Precision.pdf
\label{fig8:Prasad-P}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Prasad-Recall.pdf
\label{fig8:Prasad-R}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Prasad-FMeasure.pdf
\label{fig8:Prasad-F}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Industrial-Precision.pdf
\label{fig8:Industrial-P}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Industrial-Recall.pdf
\label{fig8:Industrial-R}}
\subfigure[]{\includegraphics[width=0.3\hsize]{pic//curves//Industrial-FMeasure.pdf
\label{fig8:Industrial-F}}
\caption{Ellipse detection performance of our method against other methods by varying overlap ratio $D_{0}$ from $0.65$ to $0.95$ at the step of $0.05$ on three real-world datasets. The three rows reveal the metric indexes of precision, recall and F-measure on traffic sign dataset, Prasad dataset and PCB dataset respectively.}
\label{fig8}
\end{figure*}
The detailed ellipse detection results on three real-world datasets are shown in Table \ref{tab1}. The highest scores in precision, recall, F-measure and cost time are stressed in boldface. As we can see, our method achieves the best F-measure scores 0.8959, 0.4781 and 0.9447 across traffic sign dataset, Prasad dataset and PCB image dataset respectively, which highlights its extraordinary overall ellipse detection ability. Moreover, our method holds the winner almost in all precision and recall indexes in three real-world datasets except the recall in PCB dataset which is acquired by RHT \cite{mclaughlin1998randomized}. The higher precision indicates that our ellipse detector is more rigorous to reject false positives and the ellipses reported by our method own larger possibility to truly exist. The precision and recall of our method keep pace, consequently, yielding the larger F-measure scores.
The iterative random search helps RHT \cite{mclaughlin1998randomized} raise the recall, however, its precision and F-measure scores are unsatisfactory due to the lack of novel validations. ELSDc \cite{patraucean2017joint} tries to detect ellipses and line segments simultaneously. However, the side-effects are also brought into such as the limited ellipse detection performance and long computation time in real-world images. Actually, ELSDc \cite{patraucean2017joint} are more suitable to deal with PCB images as its performance gets promoted vastly in PCB dataset. Similar to ELSDc \cite{patraucean2017joint}, Prasad method \cite{prasad2012edge} also suffers from long computation time due to the heavy detection procedures.
From Table \ref{tab1}, it can observe that Qi et al. method \cite{jia2017fast} which is developed on the basis of Fornaciari et al. method \cite{fornaciari2014fast} could achieve second best overall ellipse detection results across three real-world datasets. Moreover, Qi et al. method \cite{jia2017fast} is able to consume quite small running time. The reasons behind this are mainly due to its usage of projective invariant for effectively pruning straight edges, fast arc selection strategy, and simple clustering \cite{jia2017fast, Prasad2010Clustering}. In contrast to Qi et al. method accelerating detection speed at the risk of generating duplicates surrounding a common ground truth, our method employs a more useful and yet relatively more time-consuming hierarchical clustering method for ellipse candidates, which could reduce the false positives significantly. Although our method implemented in MATLAB would further slow down the ellipse detection speed, it could achieve competitive running time compared with the methods proposed by Qi et al. \cite{jia2017fast} and Fornaciari et al. \cite{fornaciari2014fast} which are in C++.\par
Some ellipse detection examples on three real-world datasets are shown in Fig. \ref{fig7}. The images of rows (1) to (3) are from the traffic sign dataset, where exist the disturbances of illumination, varied eccentricity and extremely close concentric ellipses. The second three-row images are from Prasad dataset which is the most complicated among three real-world datasets due to noise, occlusions and various backgrounds. The left three images are from PCB dataset, which is with the substantial Gaussian white noise and blur. As a result, both accurately and efficiently detecting ellipses in such images is difficult to an ellipse detector. As illustrated in Fig. \ref{fig7}, RHT \cite{mclaughlin1998randomized} tends to detect every possible ellipse while generates many false positives. The detection performance of RHT gets worse especially in the images with substantial noise and textures. Although ellipse detectors proposed by Fornaciari et al. \cite{fornaciari2014fast} and Qi et al. \cite{jia2017fast} are very efficient, both methods report many ellipse duplicates as well as resulting in poor location accuracy, as revealed in the images of (2) to (5) rows. In contrary, Prasad method \cite{prasad2012edge} and ELSDc \cite{patraucean2017joint} could relatively more accurately locate the ellipses. The relating concerns of both Prasad method and ELSDc are the massive missing detections for ground truth and modeling small contours as ellipses, which worsens the overall ellipse detection performance as the F-measure scores shown in Table \ref{tab1}. Unlike other ellipse detectors being tough to balance the issues between accuracy and efficiency, our method can both efficiently and accurately detect the ellipses and is robust to noise and textures. Especially, our method also performs well in handling the incomplete and occluded ellipses, as shown in the last column of Fig. \ref{fig7}. The elements guarantee our method's accuracy mainly due to the false control ability of arc-support LSs, novel verification criteria and self-calibrated refinement. And the reasons for the good efficiency are attributed to arc-support LSs alleviating the disturbance of straight LSs and effective initial ellipse generation aided with polarity constraint, region restriction and adaptive inliers criterion. \par
In order to comprehensively evaluate the ellipse detection performance of the compared methods, we vary the overlap ratio threshold $D_{0}$ from $0.65$ to $0.95$ at the step of $0.05$, the higher of which indicates the stricter a detected ellipse being regarded as true positive. The corresponding results are shown in Fig. \ref{fig8}. Again, our method achieves the best overall ellipse detection performance among three real-world datasets as the F-measure curves are above those of compared methods, which accords with the before performance analysis. It is evident that our method shows high-quality ellipse detection performance.
\subsection{Localization Accuracy and Efficiency Analysis}
\begin{table}[!tb]
\centering
\caption{The Mean Overlap Ratio (MOR) of Correctly Detected Ellipses on Three Real-world Datasets.}
\label{tab2}
\begin{tabular}{cccc}
\toprule[1pt]
MOR & Traffic Sign & Parasad Dataset & PCB Dataset \\\midrule[1pt]
RHT \cite{mclaughlin1998randomized} & 0.9080 & 0.8963 & 0.9459 \\
ELSDc \cite{patraucean2017joint} & 0.9229 & 0.8818 & 0.9352 \\
Prasad \cite{prasad2012edge} & 0.9226 & 0.9144 & \textbf{0.9603} \\
Fornaciari \cite{fornaciari2014fast} & 0.9274 & 0.9080 & 0.9442 \\
Qi \cite{jia2017fast} & 0.9239 & 0.9047 & 0.9428 \\
Our & \textbf{0.9383} & \textbf{0.9291} & 0.9574 \\\bottomrule[1pt]
\end{tabular}
\end{table}
Localization accuracy is a critical index to testify whether an ellipse detector to be high-quality or not. To this end, we compute each ellipse detection method's mean overlap ratio (MOR) of correctly detected ellipses on three real-world datasets and the results are shown in Table \ref{tab2}. Our method achieves the best MOR scores in traffic sign dataset and Prasad dataset and second highest MOR $0.9574$ in PCB dataset compared to $0.9603$ which is acquired by Prasad method \cite{prasad2012edge}. The higher MOR indicates that our method aims at the high localization accuracy and does not rest content with picking out the true positives, which stands the proposed method out the compared methods. Actually, accurate localization could favor an ellipse detector to distinguish the very closed ellipses. An accurate ellipse detection example of our method is shown in Fig. \ref{fig9}. There are eight ground truth in the input image and the average distance of each two concentric ellipses is $\Delta 4.18_{-1.08}^{+0.78}$ pixels. Although the ellipses are so close, our method still can successfully locate each individual and report high overlap ratio, as shown in Fig. \ref{fig9:accuracy3} and Fig. \ref{fig9:accuracy4}.\par
\begin{figure}[!tb]
\centering
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//1.jpg
\label{fig9:accuracy1}}
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//2.pdf
\label{fig9:accuracy2}}
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//3.jpg
\label{fig9:accuracy3}}
\subfigure[]{\includegraphics[width=0.45\hsize]{pic//accuracyNew//4.pdf
\label{fig9:accuracy4}}
\caption{An illustration example of the proposed method with high localization accuracy. (a) input image; (b) ground truth; (c) eight detected ellipses; (d) the overlap ratio of each detected ellipse with ground truth.}
\label{fig9}
\end{figure}
\begin{figure}[!tb]
\centering
\includegraphics[width=0.9\hsize]{pic//computation_complexity.pdf}
\caption{Computation time with regard to the number of pixels of real-world images.}
\label{fig10}
\end{figure}
In order to verify the quadratic complexity of the proposed ellipse detector in image longer length $N$ and arc-support groups $N_{G}$, we record the computation time of 100 different real-world images, the sizes of which range from 46 x 51 to 4600 x 5100. The correlation between computation time and the number of pixels is shown in Fig. \ref{fig10}. The scatters are general in linear distribution and the ellipse computation time almost linearly increases with the number of pixels, which indicates that our method is quadratic in $N$ ($N>N_{G}$ in most of the images). Admittedly, the ellipse detector should own larger time complexity than line segment or circle detector. Our method is still efficient and can handle the real-world images in quadratic time complexity which is superior to most of existing ellipse detectors.
\subsection{Robustness to Parameters Setting and Ellipse Variations}
The angular coverage $T_{ac}$ and the ratio of support inliers $T_r$ are two extrinsic parameters of the proposed ellipse detection method. Firstly, $T_{ac}$ and $T_{r}$ have geometric significance, which enables us easy to tune when applied in the real application. Secondly, both parameters are insensitive and have robustness in a wide setting range. To validate the robustness to tunable parameters, we select PCB dataset as the testset and perform quantitative experiments. We first freeze $T_r$ as $0.6$ which is the default parameter and vary the elliptic angular coverage $T_{ac}$ from $105^\circ$ to $225^\circ$ at the step of $10^\circ$. Then the curves of precision, recall and F-measure according to the experimental results are plotted, as shown in Fig. \ref{fig11:AngleVarying}. Similarly, the ratio of support inliers $T_r$ is changed from $0.4$ to $0.8$ at the step of $0.05$ and the angular coverage $T_{ac}$ are fixed to the default parameter $165^\circ$. The corresponding ellipse detection performance is shown in Fig. \ref{fig11:TrVarying}. As the angular coverage $T_{ac}$ and ratio of support inliers $T_r$ rise, the recall tends to decline as the detected ellipses are more likely to be rejected due to the stricter requirements. However, the precision gets boosted since the detected ellipses are purer. Notably, the F-measure curves in both Fig. \ref{fig11:AngleVarying} and Fig. \ref{fig11:TrVarying} are relatively smooth and little fluctuating in a wide range, which reveals the robustness of the proposed method to different parameter settings.\par
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//parameterSensitivity//AngleVarying.pdf
\label{fig11:AngleVarying}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//parameterSensitivity//TrVarying.pdf
\label{fig11:TrVarying}}
\caption{Ellipse detection performance of the proposed method in PCB image dataset with varying angular coverage and ratio of support inliers. a) the elliptic angular coverage are set from $105^\circ$ $\sim$ $225^\circ$ at step of $10^\circ$ with fixed ratio of support inliers 0.6; b) the ratio of support inliers ranges from 0.4 $\sim$ 0.8 at step of 0.05 while the angular coverage is $165^\circ$.}
\label{fig11}
\end{figure}
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//SemiAxisAxisRatio.pdf
\label{fig12:a}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//OrientationAxisRatio.pdf
\label{fig12:b}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//AngularCoverageAxisRatio.pdf
\label{fig12:c}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//ellipseVariations//SemiAxisAxisRatio2.pdf
\label{fig12:d}}
\caption{Extensive detection results subject to various ellipse variations. The horizontal axis is the ratio of semi-minor axis to semi-major one, which ranges from 0.01 to 1 at the step 0.01. The vertical axes of (a), (b) and (c) are the semi-major axis length, ellipse orientation and angular coverage of ellipse arc. (d) shows the effects after upscaling the synthetic images two times.}
\label{fig12}
\end{figure}
In order to investigate the robustness of our method to different ellipse variations such as ellipse size, orientation and incompleteness, three synthetic datasets are prepared. The first dataset includes 10000 images, in which the semi-major axis of the ellipse is varied from $1\sim 100$ pixels at the step of 1 pixel and the axes ratio ranges from 0.01 to 1 at the step of 0.01. To evaluate the influence of orientation, we build the second dataset by rotating the ellipse from $-88^\circ$ to $90^\circ$ at the step of $2^{\circ}$. For each orientation, the major-axis is fixed to 100 pixels and the axes ratio changes from $0.01\sim 1$ at the step of 0.01, which totally results in 9000 images. Actually, a high-quality ellipse detector should accurately detect the incomplete ellipses, namely elliptic arcs. Therefore, the third dataset is built and consists of 12000 images, where the angular coverage of ellipse varies from $3^\circ$ to $360^\circ$ at the step of $3^\circ$ and the axes ratio ranges from $0.01 \sim 1$ at the step of $0.01$. Each synthetic image contains an ellipse and is with the size of 250 x 250.\par
The effects of ellipse variations on our ellipse detector are shown in Fig. \ref{fig12}, where the white region indicates the corresponding ellipses could be correctly detected while the black region means the detection failures. Firstly, in Fig. \ref{fig12:a}, our method has wide successful detection area and could detect the small ellipse with the semi-major axis of about 20 pixels and axes ratio of 0.25. The extremely oblate and small ellipses are failed to detect by our method as well as by the methods proposed by Fornaciari et al. \cite{fornaciari2014fast} and Qi et al. \cite{jia2017fast}. Secondly, the black region distributed vertically in Fig. \ref{fig12:b}, which indicates that the ellipse detection performance is invariant to ellipse orientation. Our method is robust to the orientation as it is a basic nature of high-quality ellipse detector. Thirdly, our method can successfully detect the elliptic arc with angular coverage of about $165^\circ$ since our parameter $T_{ac}$ is acquiescently set to $165^\circ$, as shown in Fig. \ref{fig12:c}. This result reveals that our method is able to tackle the incomplete ellipses and detect the specified elliptic arc with assigned angular coverage. Finally, our ellipse detection performance gets improved after upscaling the image size two times since the black region shrinks in Fig. \ref{fig12:d}, which provides a feasible approach to boost detection performance of the proposed ellipse detector.
\subsection{Polarity-specific Ellipse Detection}
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//25.jpg
\label{fig13:a}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//positive.jpg
\label{fig13:b}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//negative.jpg
\label{fig13:c}}
\subfigure[]{\includegraphics[width=0.48\hsize]{pic//specifiedPolarity//allellipses.jpg
\label{fig13:d}}
\caption{An example of polarity-specific ellipse detection. (a) origin image, 993 x 595; (b) detection for the ellipse whose polarity is positive; (c) detection for the negative polarity ellipse; (d) detecting all ellipses in the image.}
\label{fig13}
\vspace{-3pt}
\end{figure}
Recall that the polarity of an ellipse is positive if the corresponding inside adjacent area of the boundary is brighter than outside, otherwise is negative. Actually, our method is able to detect the polarity-specific ellipses because we only need to retain the arc-support LSs with the corresponding polarity for generating the initial ellipse set. As shown in Fig.~\ref{fig13:a}, the black elliptic ring belts and white ring belts are concentric and adjacent. Each ring belt will generate two different ellipses with positive or negative polarity. In Fig.~\ref{fig13:b}, the concentric ellipses with positive polarity are successfully detected by our method and they are highlighted in blue color. In Fig.~\ref{fig13}(c), the detected ellipses in yellow are all with negative polarity. Naturally, if we use all the arc-support LSs for ellipse detection, the target is to detect all potential ellipses in the image, as the detected red ellipses in Fig.~\ref{fig13}(d). The information of polarity of arc-support LS is greatly important and useful, which not only contributes to reducing the computation time for searching all the valid paired arc-support groups but also helps to detect the polarity-specific ellipses in the certain case.
\section{Conclusion}\label{sec:conclusion}
In this paper, we propose a high-quality ellipse detection method by introducing the arc-support LSs, which aims at both accurately and efficiently detecting ellipses in real-world images. To this end, our method follows a four-stage ellipse detection framework: arc-support groups forming, initial ellipse set generation, clustering, and candidate verification. With the help of arc-support LSs, straight LSs are filtered and the abundant geometric features such as overall gradient direction of the local area, arc-support direction and polarity can be thoroughly exploited. The robust forming of arc-support groups, the adoption of the superposition principle of ellipse fitting and the efficient generation of initial ellipse set with three novel geometric constraints guarantee the overall efficiency of the proposed method. Moreover, the rigorous ellipse verification defend the high localization accuracy and robustness as well as rejecting the false positives. The self-calibrated refinement facilitates higher accuracy. The quantitative experiments compared with existing novel methods evidently demonstrate that our method could well balance the relationship between accuracy and efficiency, and achieves the high-quality ellipse detection performance.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
2,869,038,155,659 | arxiv | \section{Introduction}
Large deviations are used extensively in Physics (thermodynamics, statistical mechanics) as well as in Mathematics (information theory, stochastic analysis, mathematical finance)
to estimate the exponential decay of probability measures of rare events.
Varadhan~\cite{Varadhan}, Schilder~\cite{Schilder}, Freidlin and Wentzell~\cite{FWBook} proved,
in different degrees of generality, large deviations principles (in~$\mathbb{R}^n$ and on path space)
for solutions of stochastic differential equations with small noise,
and the monographs by Dembo and Zeitouni~\cite{DZ} and Deuschel-Stroock~\cite{DBookS}
provide a precise account of those advances (at least up to the mid-1990s).
In the past decade, this set of techniques and results has been adopted by the mathematical finance community:
finite-dimensional large deviations (in the sense of G\"artner-Ellis) have been used
to prove small-and large-time asymptotics of implied volatility in affine models ~\cite{FJL, JKM},
sample-path LDP (\`a la Freidlin-Wentzell~\cite{FWBook}) have proved efficient to determine importance sampling changes of probability~\cite{GT16,GR08, Rob10},
and heat kernel expansions (following Ben Arous~\cite{Benarous} and Bismut~\cite{B84}),
have led to a general understanding of small-time and tail behaviour of
multi-dimensional diffusions~\cite{BO1, BO2, DFJV1, DFJV2}.
These asymptotics have overall provided a deeper understanding of the behaviour of models,
and, ultimately, allow for better calibration of real data;
a general overview can be found in~\cite{Pham}.
Motivated by financial applications, we derive here asymptotic small-time and tail behaviours
of the solution to a generalised version of the Stein-Stein stochastic volatility model,
originally proposed in~\cite{SZ99,SS}.
We in particular consider two important (in light of the recent trends in the literature proposed models) extensions:
(i) the SDE driving the instantaneous volatility process is started from a random distribution;
this so-called `randomised' type of models was recently proposed in~\cite{JR16, JS17, M15},
in particular to understand the behaviour of the so-called `forward volatility';
(ii) the volatility process is driven by a fractional Brownian motion.
Fractional stochastic volatility models, originally proposed by Comte and Renault~\cite{ComteRenault, ComteCR} with $H\in (1/2,1)$, have recently been extended to the case $H \in (0,1/2)$,
and a recent flourishing activity in this area~\cite{ALV07, BFG16, Fukasawa, GJR14, JMM17}
has established these models as the go-to standards for estimation and calibration.
The original motivation behind randomisation of the initial starting point is rooted in financial practice, where only the initial value of the stock price process is observed directly and the instantaneous value of volatility is subject to calibration.
The effect of randomisation of the initial volatility on the implied volatility surface was explored by Jacquier and Roome~\cite{JR15} in a simple `random environment' setting, where the volatility component was assumed to follow CEV dynamics.
Their results give an impetus both on the theoretical and the practical level: they solve a practical modelling problem in a simple tractable setting and at the same time raise awareness for the potential prowess of applying random evolution equations for financial modelling.
In this paper we follow up on this direction and blend more involved approaches (proposed by~\cite{M00, MNS}) from the literature around random environment and random evolution equations into our financial model,
where randomness also appears in the drift and diffusion coefficients of the process.
On the practical level, independently from the results of~\cite{JR15},
Mechkov~\cite{M15} goes a step further in endorsing the idea of randomising the initial volatility and makes a strong case to move away from modelling hidden variables (such as stochastic volatility) in the traditional way.
He argues that starting the volatility from a fixed starting point heavily underestimates the effect of the hidden variable on the slope of the implied volatility smile,
and therefore `hot start' volatility models (with random starting point) significantly outperform traditional ones altogether.
Indeed, both randomised models~\cite{JR15,M15} produce the desired explosion in the smile at short maturities.
Jacquier and Shi~\cite{JS17} develop this further by providing a precise link between the rate of explosion of implied volatilities on the short end and the tail distribution of the initial distribution of the volatility process in a `randomised' Heston model.
These outputs confirm that stochastic volatility models with random starting point constitute a class of counterexamples to the long-standing belief formulated by Gatheral~\cite[Chapter 5]{Gat06},
that jumps in the stock price process are needed to produce steep short-dated implied volatility skews.
Another example of broadly different design was provided by Caravenna and Corbetta~\cite{CC17}.
In their `multiscaling' model, the stock price process is continuous,
while the volatility process has (carefully designed) jumps,
and steepness of the smile is achieved with a heavy-tail distribution of the small-time distribution of the volatility.
Rough fractional volatility models (with continuous volatility paths) have recently been proposed,
and are able to capture the volatility skew \cite{ALV07,BFG16, BFGHS17, EFR16, FZ16, Fukasawa, GJR14, GuliViens, JPS17}.
In this paper we analyse the combined effect of a rough fractional Brownian driver
(with Hurst parameter $H\in (0,1)$) in the volatility and a random starting point.
We quantify how the tail behaviour (parametrised by a scaling coefficient $b>0$) of the random starting point modulates the rate of explosion in the implied volatility in the presence of rough fractional volatility.
Finally, in a specific simplified setting we highlight how our model blends naturally into the setting of forward-start options in stochastic volatility models,
whose asymptotic properties have been studied in~\cite{JR16}.
In proving our results, we improve the large deviations literature on both SDEs with random starting points and fractional SDEs.
In Section~\ref{Sec:Notations}, we recall some concepts that will be used in the paper and set the notations.
We also introduce the model~\eqref{eq:model}, and the main assumptions on its dynamics and on the initial random starting point.
Section~\ref{Sec:Mainresults} collects the main large deviations estimates in different regimes:
tail behaviour (Section~\ref{Sec:Tails}), and small-time behaviour (Section~\ref{Sec:Smalltime}).
In each case, we present two different scenarios consisting of an appropriately rescaled fractional model (Theorems~\ref{Th:tails_frac_X} and~\ref{Th:LDP_ST_X})
and a simplified diffusive model (Theorems~\ref{thm:ext_MNSLDP} and~\ref{Th_rdm_new})
with more restrictive conditions on the random starting point (allowing for simpler large deviations rate functions).
Section~\ref{sec:applications} displays applications to implied volatility asymptotics (Corollaries~\ref{Cor:ImpliedvolTails} and~\ref{Cor:ImpliedvolST}),
and presents an application to forward-start options.
Proofs can be found in Appendix.
\section{Set up and notations}\label{Sec:Notations}
As outlined in the introduction, we prove pathwise large deviations
for a two-dimensional system generalising the Stein-Stein model~\cite{SZ99, SS}, with random initial datum.
In particular, via suitable rescaling, we determine the small-time and the large-tail behaviours of the system.
Before delving into the core of the paper, let us recall some useful facts about large deviations and Gaussian processes, which shall also serve as setting the notations for the rest of the paper.
Unless otherwise stated, we always work on a finite time horizon,
say $[0,1]$ without loss of generality, which we denote by~$\mathcal{T}$, and we write $\mathcal{T}^*:=\mathcal{T}\setminus\{0\}$.
We let $\mathcal{C}:=\mathcal{C}(\mathcal{T}, \mathbb{R})$ be the space of continuous functions from~$\mathcal{T}$ to~$\mathbb{R}$
and $\mathcal{C}_b^2$ the space of twice differentiable functions on~$\mathcal{T}$
with bounded partial derivatives up to the second order.
We write $X^\varepsilon\sim\mathrm{LDP}(h_\varepsilon, I)$ when the sequence~$(X^\varepsilon)_{\varepsilon>0}$
satisfies a large deviations principle (Definition~\ref{def:LDP}) on~$\mathcal{C}$,
as~$\varepsilon$ tends to zero with good rate function~$I$ and speed~$h_{\varepsilon}$, where~$h_{\varepsilon}$ denotes a function satisfying $\lim_{\varepsilon \downarrow 0} h_{\varepsilon}=0$.
For a random variable~$X$, we denote by $\mathrm{supp}(X)$ its support.
\subsection{Large deviations and fractional Brownian motion}
We shall use~\cite[Chapter 1.2]{DZ} and~\cite[Chapter 1]{DZ} as our guides through large deviations.
Given a topological space~$\mathcal{X}$ and the completed Borel $\sigma$-field~$\mathcal{B}_{\mathcal{X}}$ corresponding to~$\mathcal{X}$, for any $A \in \mathcal{B}_{\mathcal{X}}$, we denote by~$\mathring{A}$ and~$\overline{A}$
respectively its interior and closure, and consider a sequence~$(X^\varepsilon)_{\varepsilon>0}$ on $(\mathcal{X}, \mathcal{B}_{\mathcal{X}})$.
\begin{definition}
A (good) rate function is a lower semi-continuous mapping $I: \mathcal{X} \rightarrow [0, \infty]$
such that the level sets $\left\{ x: I(x) \le z \right\}$ are closed (compact) subsets of~$\mathcal{X}$ for any $z\geq 0$.
\end{definition}
\begin{definition}\label{def:LDP}
The sequence $(X^{\varepsilon})_{\varepsilon > 0}$ satisfies a large deviations principle (LDP)
on $(\mathcal{X}, \mathcal{B}_{\mathcal{X}})$
as~$\varepsilon$ tends to zero,
with speed~$h_\varepsilon$, and rate function~$I$, if for any Borel subset $A \subset \mathcal{X}$,
the following inequalities hold:
\begin{equation}
-\inf_{A^o} \ I(\varphi)
\le \liminf_{\varepsilon \downarrow 0} h_\varepsilon \log \mathbb{P} (X^{\varepsilon} \in A)
\le \limsup_{\varepsilon \downarrow 0} h_\varepsilon \log \mathbb{P} (X^{\varepsilon} \in A)
\le - \inf_{\overline{A}} I(\varphi).
\end{equation}
\end{definition}
A particularly convenient tool to prove large deviations is the so-called exponential equivalence,
which we recall from~\cite[Definition 4.2.10]{DZ} as follows:
\begin{definition}\label{def:ExpEquiv}
On a metric space $(\mathcal{Y},d)$, two $\mathcal{Y}$-valued sequences ${(X^{\varepsilon})}_{\varepsilon > 0}$ and
${(\widetilde{X}^{\varepsilon})}_{\varepsilon > 0}$ are called exponentially equivalent
(with speed~$h_\varepsilon$) if there exist probability spaces $(\Omega, \mathcal{B}_\varepsilon, \mathbb{P}_\varepsilon)_{\varepsilon>0}$ such that
for any $\varepsilon>0$, $\mathbb{P}^{\varepsilon}$ is the joint law and,
for each $\delta >0$, the set $\left\{ \omega: (\widetilde{X}^{\varepsilon},X^{\varepsilon}) \in \Gamma_{\delta} \right\}$ is $\mathcal{B}_{\varepsilon}$-measurable, and
\begin{equation*}
\limsup_{\varepsilon \downarrow 0} h_\varepsilon \log \mathbb{P}^{\varepsilon}\left(\left\{ (\tilde{y},y): d(\tilde{y},y) > \delta \right\}\right)=- \infty.
\end{equation*}
\end{definition}
\begin{theorem}
Let ${(X^{\varepsilon})}_{\varepsilon > 0}$ and ${(\widetilde{X}^{\varepsilon})}_{\varepsilon > 0}$ be two
exponentially equivalent sequences (with speed~$h_\varepsilon$) on some metric space.
If $(X^{\varepsilon})\sim\mathrm{LDP}(h_\varepsilon, \Lambda^X)$ for some good rate function~$\Lambda^X$,
then $(\widetilde{X}^{\varepsilon})\sim\mathrm{LDP}(h_\varepsilon, \Lambda^X)$.
\end{theorem}
On a real, separable Banach space $(\mathcal{E}, \|\cdot\|)$,
we denote by $\mathcal{B}$ the associated Borel sigma field.
Letting~$\mathcal{E}^*$ denote the topological dual of~$\mathcal{E}$, we define a Gaussian measure as follows:
\begin{definition}
A Gaussian measure~$\mu$ on $(\mathcal{E}, \|\cdot\|)$ is such that every $\vartheta^* \in\mathcal{E}^*$,
when viewed as a random variable via the dual pairing
$\vartheta\mapsto\langle \vartheta^*, \vartheta\rangle_{\mathcal{E}^*\mathcal{E}}$,
is a real Gaussian random variable on $(\mathcal{E}, \mathcal{B}, \mu)$.
\end{definition}
We associate a Gaussian process to a Gaussian measure in the usual way~\cite[Section 3.2]{CT06}.
Particular examples of Gaussian processes, crucial for the rest of the paper,
include standard Brownian motion on the time interval~$\mathcal{T}$,
where $\mathcal{E}=\mathcal{C}$ equipped with the supremum norm and with the topology of uniform convergence,
and~$\mathcal{E}^*$ is the space of signed measures on~$\mathcal{T}$.
In fact, this construction applies to all (centered) continuous Gaussian processes,
which are uniquely characterised by their covariance operator.
A fractional Brownian motion $W^H$ with Hurst parameter $H \in (0,1)$
is such a Gaussian process, starting from zero,
with covariance
$$
\left\langle W^H_t, W^H_s\right\rangle = \frac{1}{2} \left( {|t|}^{2H} + {|s|}^{2H} - {|t-s|}^{2H}\right), \qquad \textrm{for any } s,t \in \mathcal{T}.
$$
Of primary importance in understanding small-noise behaviours of Gaussian systems
is the concept of reproducing kernel Hilbert spaces (RKHS),
which we recall following~\cite[Definition 3.3]{CT06}:
\begin{definition}
Let~$\mu$ be a Gaussian measure on~$\mathcal{E}$ and define the map
$\mathcal{R} : \mathcal{E}^* \rightarrow \mathcal{E}$ by $\mathcal{R}x^* := \int_\mathcal{E} \langle x^*, x \rangle x \mu(\mathrm{d} x)$.
The RKHS~$\mathcal{H}_{\mu}$ of~$\mu$ is the completion of the image $\mathcal{R}\mathcal{E}^*$ for the norm
$\|\mathcal{R}x^*\|_{\mathcal{H}_{\mu}} := \left(\langle x^*, \mathcal{R}x^*\rangle\right)^{1/2}$,
for all $x^*\in\mathcal{E}^*$.
\end{definition}
To characterise the RKHS of fractional Brownian motion,
the usual tool is its Volterra representation~\cite{NVV99}
\begin{equation}\label{eq:VolterraW}
W^H_t = \int_0^t K^H(t,s) \mathrm{d} B_s,
\end{equation}
which holds almost surely for all $t\in\mathcal{T}$,
where~$B$ is a standard Brownian motion generating the same filtration as~$W^H$,
and $K^H$ is the Volterra kernel defined, for any $s, t \in \mathcal{T}$ with $0<s<t$,
by~\cite[Theorem 5.2]{NVV99}
\begin{equation}\label{eq:VolterraK}
K^H(t,s) =
\left\{
\begin{array}{ll}
\displaystyle \frac{\kappa_H}{s^{H_{-}}}
\left[ \left(t(t-s)\right)^{H_{-}} - H_{-}
\int_{s}^{t} \frac{(u-s)^{H_{-}}}{u^{1-H_{-}}} \mathrm{d} u\right], & \displaystyle \text{if }H<\frac{1}{2},\\
\displaystyle \frac{\kappa_H H_{-}}{s^{H_{-}}}
\int_{s}^{t}\frac{u^{H_{-}}\mathrm{d} u}{|u-s|^{1-H_{-}}}, & \displaystyle \text{if }H>\frac{1}{2},\\
1, & \displaystyle \text{if }H=\frac{1}{2},\end{array}
\right.
\end{equation}
with $H_{\pm} := H\pm\frac{1}{2}$
and
$\displaystyle
\kappa_H := {\left(\frac{2H \Gamma(1-H_{-})}{\Gamma(H_{+})\Gamma(2-2H)}\right)}^{1/2}$.
For $t\in\mathcal{T}^*$, the map $K^H(t,\cdot)$ is square integrable around the origin,
and the reproducing kernel Hilbert space of the fractional Brownian motion is given by
$$
\mathcal{H}_{K^H} := \left\{ \int_{0}^{t} K^H(t,s) f(s) \mathrm{d} s, t\in\mathcal{T} : f \in \mathrm{L}^2(\Tt)\right\}
$$
with inner product
$$
\left\langle \int_{0}^{\cdot} K^H(\cdot,s) f_1(s) \mathrm{d} s, \int_{0}^{\cdot} K^H(\cdot,s) f_2(s) \mathrm{d} s \right\rangle_{\mathcal{H}_{K^H}} := {\langle f_1,f_2 \rangle }_{\mathrm{L}^2(\Tt)}.
$$
The notation~$\mathcal{H}_{K^H}$, emphasising the link with the underlying kernel,
will be useful later (in Definition~\ref{def:RKHS}) for more general kernels.
In particular, the RKHS associated to (standard) Brownian motion ($H=1/2$)
is the Cameron-Martin space, and corresponds to the space of absolutely continuous functions starting at zero,
with square integrable derivatives.
In other words, for any $H \in (0,1)\setminus\{1/2\}$,
the identity $\mathcal{H}_{K^{H}}=\mathbb{K}_{K^H}\mathrm{L}^2(\Tt)$
holds, where $\mathbb{K}_{K^H}: \mathrm{L}^2(\Tt)\ni f \mapsto \int_{0}^{t} K^H(t,s) f(s) \mathrm{d} s$.
This characterisation motivates the following definition:
\begin{definition}\label{def:RKHS}
For any strictly positive function~$\Phi:\mathbb{R}_+^2\to\mathbb{R}$ such that $\Phi(t, \cdot) \in L^2(\mathcal{T})$
for any $t\in\mathcal{T}$,
the corresponding RKHS is defined as
\begin{align}\label{eq:RKHSgeneral}
\mathcal{H}_{\Phi} := \left\{ \int_{0}^{t} \Phi(t,s) f(s) \mathrm{d} s, t\in\mathcal{T}: f \in \mathrm{L}^2(\Tt)\right\},
\end{align}
with inner product
$$
\left\langle \int_{0}^{\cdot} \Phi(\cdot,s) f_1(s) \mathrm{d} s, \int_{0}^{\cdot} \Phi(\cdot,s) f_2(s) \mathrm{d} s \right\rangle_{\mathcal{H}_\Phi} := {\langle f_1,f_2 \rangle }_{\mathrm{L}^2(\Tt)}.
$$
\end{definition}
Reproducing Kernel Hilbert Spaces, together with their inner products,
turn out to provide the right spaces to characterise large deviations rate functions.
In particular, for a given Gaussian Volterra process of the form
$\int_{0}^{\cdot}\Phi(\cdot,s)\mathrm{d} B_s$, for some Volterra kernel~$\Phi$,
it follows from~\cite[Theorem 3.4.5]{DBookS}
that the sequence
$(\varepsilon \int_{0}^{\cdot}\Phi(\cdot,s)\mathrm{d} B_s)_{\varepsilon>0}$
satisfies a large deviations principle with speed~$\varepsilon^2$ and rate function
\begin{equation}\label{eq:LDPVolterrageneral}
\Lambda_{\Phi}(\varphi) = \left\{
\begin{array}{ll}
\displaystyle \frac{1}{2}\|\varphi\|_{\mathcal{H}_{\Phi}}, & \text{if }\varphi \in \mathcal{H}_{\Phi},\\
+ \infty, & \text{otherwise}.
\end{array}
\right.
\end{equation}
An obviously special role is played by the standard Brownian motion $H=1/2$,
and we shall adopt the simplified notation~$\mathcal{H}$ (the classical Cameron-Martin space)
and~$\Lambda$ in place of $\mathcal{H}_{K^{1/2}}$ and $\Lambda_{K^{1/2}}$.
\subsection{Setting and assumptions}
The particular system we are interested in is
\begin{equation}\label{eq:model}
\left\{
\begin{array}{ll}
\displaystyle
\mathrm{d} X_t = - \frac{1}{2}\sigma(Y_t)^2\mathrm{d} t + \sigma(Y_t) \left(\rho\mathrm{d} B_t + \bar{\rho} \mathrm{d} B^{\perp}_t\right),
& X_0 = 0, \\
\mathrm{d} Y_t = (\lambda + \beta Y_t) \mathrm{d} t + \xi \mathrm{d} W^H_t, & Y_0 \sim \Theta,
\end{array}
\right.
\end{equation}
where $W^H$ is a fractional Brownian motion, with Hurst parameter $H \in (0,1)$,
$(B,B^\perp)$ is a two-dimensional standard Brownian motion,
$\beta<0$, $\lambda, \xi>0$, $\rho \in (-1,1)$, $\bar{\rho} := \sqrt{1-\rho^2}$,
and~$\Theta$ is a square-integrable continuous random variable.
In order to guarantee existence and uniqueness of a strong solution,
we further assume~\cite[Theorem 3.1.3]{M08}
that~$\sigma$ is Lipschitz continuous,
satisfies the growth condition $|\sigma(y)| \le C(1+|y|)$ for $y \in \mathbb{R}$,
and is differentiable with locally H\"older continuous derivative.
In order to prove our main results below, we make the following technical assumptions:
\noindent \textbf{Assumption $\mathcal{A}_b$}:
There exist a measurable function $\widetilde{\sigma}:\mathbb{R}\to\mathbb{R}$,
with locally H\"older continuous derivative, such that~$\widetilde{\sigma}$ is Lipschitz continuous,
satisfying the growth condition~$|\widetilde{\sigma}(y)| \le C(1+|y|)$ for $y \in \mathbb{R}$,
as well as a constant $b>0$ such that the scaling property $\lim_{\varepsilon \downarrow 0} \varepsilon^b \sigma(y/\varepsilon^b) = \widetilde{\sigma}(y)$ holds true uniformly on~$\mathbb{R}$.
\section{Main results}\label{Sec:Mainresults}
Centrepiece of our analysis are large deviations estimates for suitably rescaled versions of~\eqref{eq:model}.
The first rescaling (presented in Section~\ref{Sec:Tails}) is tailored
to the analysis of the tail behaviour of~\eqref{eq:model},
while the second rescaling (Section~\ref{Sec:Smalltime}) is bespoke to its short-time asymptotic properties.
In addition to these asymptotic results in the general fractional case (Theorems~\ref{Th:tails_frac_X} and~\ref{Th:LDP_ST_X}),
we present two special simplified diffusive cases,
where particularly tractable rate functions can be obtained (Theorems~\ref{thm:ext_MNSLDP}
and~\ref{Th_rdm_new} respectively).
For this we impose stronger conditions (Assumption~\ref{ass:MNS}) on the random starting point.
This allows us to establish, following~\cite{MNS} in Section~\ref{app:MNS}, an exponential equivalence
between~\eqref{eq:model} and an analogous process with fixed starting point.
In Section~\ref{Sec:Mellouk} we construct a third
rescaling~\eqref{eq:SystemEps} inspired by Mellouk~\cite{M00} in the short-time diffusive case under the assumption that the support of the random starting point is bounded.
We shall work with rescaled versions $Y^\varepsilon$ of the process~$Y$ in~~\eqref{eq:model}
(see~\eqref{eq:fracSSRandom} and~\eqref{eq:SmalltimeModel} for specific examples), together with a function~$h_\varepsilon$
describing the speed of the large deviations estimates, for which we introduce the following assumptions:
\noindent \textbf{Assumption $\mathcal{A}_b'$}:
There exists a family of continuous functions $(\sigma_n)_{n \ge 0}$ on $\mathbb{R}$ such that
\begin{itemize}
\item[(i)] $(\sigma_n)_{n\geq 0}$ converges uniformly to $\widetilde{\sigma}$ on~$\mathbb{R}$;
\item[(ii)] for all $\delta >0$,
$\lim\limits_{n \uparrow \infty} \limsup\limits_{\varepsilon \downarrow 0} h_{\varepsilon} \log \mathbb{P} \left(|\sigma_n(Y^\varepsilon) - \widetilde{\sigma}(Y^\varepsilon) \ge \delta| \right) = -\infty$.
\end{itemize}
\noindent \textbf{Assumption $\mathcal{A}_{b}^{\Theta}$} (tail behaviour of~$\Theta$):
The limit
$\limsup\limits_{\varepsilon \downarrow 0} h_\varepsilon \log\mathbb{P}(\varepsilon^b |\Theta|>1) = -\infty$ holds.\\
In Assumptions $\mathcal{A}_b'$ and $\mathcal{A}_{b}^{\Theta}$ above, the large deviations speed~$h_\varepsilon$
takes the value~$\varepsilon^{2b}$ in Section~\ref{Sec:Tails}
and~$\varepsilon^{4H+2b}$ in Section~\ref{Sec:Smalltime} respectively.
The constant~$b$ (which may vary below)
plays an essential role in subsequent large deviations estimates,
via exponential equivalence techniques (Definition~\ref{def:ExpEquiv}).
Assumptions~$\mathcal{A}_b$ and~$\mathcal{A}_b'$ are naturally satisfied in the fractional Stein-Stein case
(where $\sigma(y)\equiv y$), but imposing them allow us to state our results for more general~$\sigma$,
in particular when using extended Contraction Principles~\cite[Proposition~2.3]{MNP92}.
\subsection{Tail behaviour}\label{Sec:Tails}
\subsubsection{The general case}
For $b,\varepsilon>0$, introduce the rescaling
$(X^{\varepsilon}, Y^{\varepsilon}) := (\varepsilon^{2b}X, \varepsilon^b Y)$, so that~\eqref{eq:model} becomes
\begin{equation}\label{eq:fracSSRandom}
\left\{
\begin{array}{ll}
\displaystyle
\mathrm{d} X^{\varepsilon}_t = - \frac{\varepsilon^{2b}}{2} \sigma\left(\frac{Y^{\varepsilon}_t}{\varepsilon^b}\right)^2 \mathrm{d} t + \varepsilon^{2b} \sigma\left(\frac{Y^{\varepsilon}_t}{\varepsilon^b}\right) (\rho\mathrm{d} B_t + \bar{\rho} \mathrm{d} B^{\perp}_t), &X^{\varepsilon}_0=0, \\
\mathrm{d} Y^{\varepsilon}_t = \left( \varepsilon^b \lambda + \beta Y^{\varepsilon}_t \right) \mathrm{d} t + \varepsilon^b \xi \mathrm{d} W^H_t, &Y^{\varepsilon}_0 \sim \varepsilon^b \Theta.
\end{array}
\right.
\end{equation}
The particular rescaling considered here is perfectly suited for tail behaviour,
as large deviations provide estimates for $\mathbb{P}(X^\varepsilon\geq 1) = \mathbb{P}(X \geq \varepsilon^{-{2b}})$.
Our main result is as follows, and is proved in Appendix~\ref{Sec:Proof1}:
\begin{theorem}\label{Th:tails_frac_X}
For any $H \in (0,1)$, the following hold:
\begin{enumerate}[(i)]
\item for any $b>0$ such that~$\mathcal{A}_{b}^{\Theta}$ holds, $Y^{\varepsilon} \sim \mathrm{LDP} \left(\varepsilon^{2b}, \Lambda_{F^{H}} \right)$,
with~$\Lambda_{F^{H}}$ in~\eqref{eq:LDPVolterrageneral}
and~$F^{H}$ in Lemma~\ref{lem:rep_fBm};
\item
for any $b \ge \frac{1}{2}$ such that Assumptions~$\mathcal{A}_b$, $\mathcal{A}_b'$, $\mathcal{A}_{b}^{\Theta}$ hold,
$X^{\varepsilon}\sim\mathrm{LDP}(\varepsilon^{2b}, \widetilde{\Lambda})$,
with~$\widetilde{\Lambda}$ in~\eqref{eq:grf_X_tails}.
\end{enumerate}
\end{theorem}
The proof of the theorem, developed later, requires a precise analysis of
the Reproducing Kernel Hilbert Spaces of the processes under consideration,
and we first state the following two key ingredients
(proved in Appendices~\ref{App:Lem_F} and~\ref{sec:proof_RKHS_tails}),
which are also of independent interest:
\begin{lemma}\label{lem:rep_fBm}
For any $H\in (0,1)$ and $\beta>0$, there exists a standard Brownian motion~$Z$,
such that
\begin{equation}\label{eq:RepfOU}
\xi \int_{0}^{t} \mathrm{e}^{\beta (t-s)} \mathrm{d} W^H_s = \int_{0}^{t} F^{H}(t,s) \mathrm{d} Z_s,
\end{equation}
holds almost surely for $t\in\mathcal{T}$, where $F^{H}:\mathcal{T}\times\mathcal{T}\to\mathbb{R}$ is defined for $0<s<t$,
with~$\kappa_H$ in~\eqref{eq:VolterraK}, as
\begin{equation*}
F^{H}(t,s) := \left\{
\begin{array}{ll}
\displaystyle
\frac{\xi\kappa_H}{s^{H_{-}}}\left[[t(t-s)]^{H_{-}}
+ \int_{s}^{t}\left\{\frac{1-2H}{2u} + \beta \right\}
[u(u-s)]^{H_{-}} \mathrm{e}^{\beta (t-u)} \mathrm{d} u \right],
& \displaystyle\text{if }H <\frac{1}{2},\\
\displaystyle\frac{\xi\kappa_H H_{-}}{s^{H_{-}}} \int_s^t \frac{u^{H_{-}}\mathrm{e}^{\beta (t-u)} \mathrm{d} u}{(u-s)^{1-H_{-}}},
& \displaystyle\text{if }H >\frac{1}{2},\\
\displaystyle\xi\mathrm{e}^{\beta (t-s)}, & \displaystyle\text{if } H = \frac{1}{2}.
\end{array}
\right.
\end{equation*}
\end{lemma}
The case $\beta=0$ (and $\xi =1$) is excluded since, in that case,
the lemma boils down to the classical Volterra representation
of the fractional Brownian motion~\eqref{eq:VolterraW} and the function~$F^{H}$ is nothing else that~$K^H$ given in~\eqref{eq:VolterraK}.
For $H\in (\frac{1}{2},1)$, the expression for~$F^{H}$ is in agreement with~\cite[Definition 2.1]{YLX08},
but, for $H \in(0,\frac{1}{2})$,
it corrects the slightly erroneous expression therein.
This function~$F^H$ allows us to fully characterise the following RKHS,
with proof postponed to Section~\ref{sec:proof_RKHS_tails}:
\begin{proposition}\label{prop:RKHS_tails}
For any $H \in (0,1)$, $\beta>0$, the space~$\mathcal{H}_{F^{H}}$
is the RKHS of the Gaussian process
$\xi\int_{0}^{\cdot} \mathrm{e}^{\beta (\cdot-s)} \mathrm{d} W^H_s$.
\end{proposition}
\subsubsection{The Millet-Nualart-Sanz approach}\label{app:MNS}
In~\cite{MNS}, Millet, Nualart and Sanz consider a perturbed stochastic differential equation of the form
\begin{equation}\label{eq:MNSsystem}
\mathrm{d} \mathrm{X}^{\varepsilon}_t = b(\varepsilon,\mathrm{X}^{\varepsilon}_t)\mathrm{d} t + \varepsilon a(\mathrm{X}^{\varepsilon}_t)\mathrm{d} \mathrm{W}_t.
\end{equation}
Here, for any~$\varepsilon>0$, the functions
$b(\varepsilon,\cdot):\mathbb{R}^n\to\mathbb{R}^n$ and $a: \mathbb{R}^n \rightarrow \mathcal{M}_{(n,d)}(\mathbb{R})$
are bounded Borel measurable and uniformly Lipschitz,
$b(\varepsilon,\cdot)$ converges uniformly to a function~$b(\cdot)$ as~$\varepsilon$ tends to zero,
$\mathrm{W}$ is a $d$-dimensional Brownian motion,
and~$\mathrm{X}^\varepsilon_0$ is an $\mathbb{R}^n$-valued square-integrable random variable.
Existence and uniqueness of a strong solution can be found in~\cite[Chapter 5, Theorem 2.1]{Friedman}.
Following classical large deviations steps, consider,
for any $\varphi\in\mathcal{H}$, the controlled ordinary differential equation on~$\mathcal{T}$:
\begin{equation}\label{eq:ControlODE}
\dot{\psi}_t = a(g_t) \dot{\varphi}_t + b(\psi_t),
\end{equation}
the solution flow of which, starting from~$\mathrm{x}_0\in\mathbb{R}^n$ is denoted by~$\mathcal{S}_{\mathrm{x}_0}(\varphi)$.
Millet, Nualart and Sanz~\cite{MNS} proved a large deviations principle~\cite[Theorem 4.1]{MNS}
for the sequence~$(\mathrm{X}^\varepsilon)_{\varepsilon>0}$ under the following assumption:
\begin{assumption}\label{ass:MNS}
Both~$a(\cdot)$ and~$b(\cdot)$ belong to~$\mathcal{C}_b^2$, and there exists $\mathrm{x}_0 \in \mathbb{R}^n$ such that, for any $\delta >0$,
\begin{equation}\label{assump1}
\limsup_{\varepsilon \downarrow 0} \varepsilon^2 \log \mathbb{P} \left(|\mathrm{X}^\varepsilon_0-\mathrm{x}_0| > \delta \right) = - \infty.
\end{equation}
\end{assumption}
\begin{theorem}\label{thm:MNSLDP}
Under Assumption~\ref{ass:MNS},
$(\mathrm{X}^{\varepsilon})_{\varepsilon >0}\sim\mathrm{LDP}(\varepsilon^2, I)$ with
$I(\psi) = \inf\{\Lambda(\varphi): \varphi \in \mathcal{H}, \psi=\mathcal{S}_{\mathrm{x}_0}(\varphi)\}$.
\end{theorem}
Condition~\eqref{assump1} is an exponential equivalence property
between the initial random variable~$\mathrm{X}^\varepsilon_0$
and the constant~$\mathrm{x}_0$, and ensures that large deviations are preserved under exponentially small perturbations
of the starting point.
Therefore, in the standard diffusion case $H = \frac{1}{2}$,
it is possible to obtain a similar result to Theorem~\ref{Th:tails_frac_X} with a simplified rate function
(albeit with slightly more restrictions on the starting point),
by using the approach considered by Millet-Nualart-Sanz in~\cite{MNS}.
For this, we rewrite~\eqref{eq:MNSsystem} to correspond to~\eqref{eq:fracSSRandom}, albeit with stronger assumptions on the coefficients, with
$\mathrm{W}:=(W_1, W_2)'$ a Brownian motion,
$\mathrm{X}^\varepsilon = (X^\varepsilon, Y^\varepsilon)$, $H=1/2$,
$\varepsilon\to\varepsilon^b$,
$\mathrm{X}_0^\varepsilon = (0, \varepsilon \Theta)$, and $\bar{\rho} := \sqrt{1-\rho^2}$,
$$
b(\varepsilon,\mathrm{X}^{\varepsilon}_t) =
\begin{pmatrix}
- \frac{1}{2} \widetilde{\sigma}(Y^{\varepsilon}_t)^2\\
\varepsilon \lambda + \beta Y^{\varepsilon}_t
\end{pmatrix}
\qquad\text{and}\qquad
a(\mathrm{X}^{\varepsilon}_t) =
\begin{pmatrix}
\bar{\rho}\widetilde{\sigma}(Y^{\varepsilon}_t) & \rho\widetilde{\sigma}(Y^{\varepsilon}_t)\\
0 & \xi
\end{pmatrix}.
$$
The correlation between the two components of $\mathrm{W}$ is explicitly represented in the diffusion matrix~$a$.
\begin{theorem}~\label{thm:ext_MNSLDP}
Under Assumption~\ref{ass:MNS},
the solution~$\mathrm{X}^{\varepsilon}$ to~\eqref{eq:MNSsystem}
satisfies $(\mathrm{X}^{\varepsilon})_{\varepsilon >0}\sim\mathrm{LDP}(\varepsilon^2, \mathbf{I})$ with
$\mathbf{I}(\chi) = \inf \left\{ \Lambda(\varphi) : \varphi \in \mathcal{H}, \chi = \mathcal{S}_{\mathrm{x}_0}(\Psi^{\rho}(\varphi))\right\}$
and
$\Psi^{\rho}:\mathbb{R}^2\ni\mathrm{z}\mapsto
\begin{pmatrix}
\bar{\rho} & \rho\\
0 & 1
\end{pmatrix}
\mathrm{z},
$.
\end{theorem}
\begin{proof}
The proof of Theorem~\ref{thm:MNSLDP} relies
first on proving a large deviations principle
for the flow~$\mathcal{S}_{\mathrm{x}_0}$ using Schilder's theorem~\cite[Theorem 5.2.3]{DZ},
then on extending this LDP to the original system.
One can easily extend it to include a correlation parameter $\rho \in (-1,1)$,
the main difference being the rate function.
Indeed, since $W^\perp_2$ and $W_2$ are independent,
Schilder's theorem yields that $\varepsilon(W^\perp_2, W_2)' \sim \mathrm{LDP}(\varepsilon^2, \Lambda)$.
Since the map~$\Psi^{\rho}$ is continuous on $(\mathcal{C}, \|\cdot\|_{\infty})$
and $\varepsilon\mathrm{W}' =\Psi^{\rho}(\varepsilon(W^\perp_2, W_2)')$,
the theorem follows from the Contraction Principle giving an LDP for~$\varepsilon\mathrm{W}'$
as~$\varepsilon$ tends to zero with speed~$\varepsilon^2$ and good rate function
\begin{equation}\label{eq:ext_grf}
\Lambda^{\rho}(\psi):=
\inf \left\{ \Lambda(\varphi): \varphi\in\mathcal{H}, \psi=\Psi^{\rho}(\varphi) \right\}.
\end{equation}
\end{proof}
\subsection{Small-time behaviour}\label{Sec:Smalltime}
We now tackle the small-time behaviour of the process~\eqref{eq:model}.
Under the general set of assumptions
$\mathcal{A}_b$, $\mathcal{A}'_b$, $\mathcal{A}_{b}^{\Theta}$, we need to introduce a particular rescaling,
both in time and in space in order to observe some weak convergence.
This is different from the classical It\^o diffusion case (with fixed starting point),
where solutions of such SDEs generally converge in small time.
In the It\^o case, though, if the distribution of the starting point has compact support,
we show in Section~\ref{Sec:Mellouk} that space rescaling is not required any longer.
\subsubsection{The general case}\label{sec:GenCase}
With the rescaling
$(X^{\varepsilon}_t, Y^{\varepsilon}_t) := (\varepsilon^{2H+2b-1} X_{\varepsilon^2 t}, \varepsilon^b Y_{\varepsilon^2 t})$,
with $b>0$, \eqref{eq:model} becomes
\begin{equation}\label{eq:SmalltimeModel}
\left\{
\begin{array}{ll}
\displaystyle
\mathrm{d} X^{\varepsilon}_t = - \frac{\varepsilon^{2H+1+2b}}{2} \sigma\left(\frac{Y^{\varepsilon}_t}{\varepsilon^b}\right)^2 \mathrm{d} t + \varepsilon^{2H+2b} \sigma\left(\frac{Y^{\varepsilon}_t}{\varepsilon^b}\right) \left(\rho\mathrm{d} B_t + \bar{\rho} \mathrm{d} B^{\perp}_t\right), &X^{\varepsilon}_0=0, \\
\mathrm{d} Y^{\varepsilon}_t = \left( \varepsilon^{b+2} \lambda + \beta \varepsilon^2 Y^{\varepsilon}_t \right) \mathrm{d} t + \varepsilon^{2H+b} \xi \mathrm{d} W^H_t, &Y^{\varepsilon}_0 \sim \varepsilon^b \Theta.
\end{array}
\right.
\end{equation}
Our main result is as follows:
\begin{theorem}\label{Th:LDP_ST_X}
For any $H \in (0,1)$,
\begin{enumerate}[(i)]
\item for any $b>0$ such that~$\mathcal{A}_{b}^{\Theta}$ holds
$Y^{\varepsilon}\sim\mathrm{LDP}\left(\varepsilon^{4H+2b}, \Lambda_{G^H_0}\right)$, with~$\Lambda_{G^H_0}$
as in~\eqref{eq:LDPVolterrageneral};
\item
if $b\ge \frac{1}{2}-2H$ such that $\mathcal{A}_b$, $\mathcal{A}'_b$, $\mathcal{A}_{b}^{\Theta}$ hold,
$X^\varepsilon\sim\mathrm{LDP}(\varepsilon^{4H+2b}, \mathtt{I})$, with~$\mathtt{I}$ defined in~\eqref{eq:grf_X_ST}.
\end{enumerate}
\end{theorem}
The proof of~(i) is similar to that of Theorem~\ref{Th:tails_frac_X}(i)
and relies on proving LDP for an auxiliary process, defined in~\eqref{eq:auxi},
exponentially equivalent to the original (rescaled) process~$Y^{\varepsilon}$.
The proof of~(ii) is more involved and postponed to Appendix~\ref{sec:Th:LDP_ST_X}.
In order to state the following key result, define, for any $\varepsilon>0$, $G^H_{\varepsilon}$
as the function~$F^{H}$ in Lemma~\ref{lem:rep_fBm},
replacing~$\beta$ by~$\beta \varepsilon^2$, and for $s,t\in\mathcal{T}$ with $0<s<t$, its pointwise limit
\begin{equation*}
G^H_0 (t,s) := \lim_{\varepsilon \downarrow 0} G^H_\varepsilon (t,s) =
\left\{
\begin{array}{ll}
\displaystyle \frac{\xi\kappa_H H_{-}}{s^{H_{-}}} \int_s^t \frac{u^{H_{-}}}{(u-s)^{1-H_{-}}} \mathrm{d} u,
& \displaystyle \text{for } H \in \left(\frac{1}{2}, 1\right),\\
\displaystyle \frac{\xi\kappa_H}{s^{H_{-}}}\left({(t(t-s))}^{H_{-}}
- H_{-} \int_{s}^{t} {(u-s)}^{H_{-}} u^{H_{-} - 1} \mathrm{d} u \right), & \displaystyle \text{for } H \in \left(0, \frac{1}{2}\right), \\
\xi, & \displaystyle \text{for }H=\frac{1}{2}.
\end{array}
\right.
\end{equation*}
The following proposition is similar to Proposition~\ref{prop:RKHS_tails},
as $G^H_0(t,\cdot) \in \mathrm{L}^2(\Tt)$ and for all $0<s<t$, $G^H_0(t,s) >0$, and its proof is omitted.
\begin{proposition}\label{Pp:RKHS_ST}
For any $H \in (0,1)$, the space $\mathcal{H}_{G^H_0}$
is the RKHS of the Gaussian process
$\int_0^{\cdot} G^H_0 (\cdot,s) \mathrm{d} Z_s$.
\end{proposition}
\subsubsection{Small-time asymptotics for bounded support in the diffusion case}\label{Sec:Mellouk}
In the standard case $H=\frac{1}{2}$, the rescaling in the previous subsection is not really `natural',
in the sense that small-time weak convergence usually holds for It\^o diffusions
without space rescaling.
In this case, using an approach introduced by Bezuidenhout~\cite{B87} and further developed
by Mellouk~\cite{M00},
we can obtain simpler large deviations estimates if the support of the initial datum~$\Theta$ is bounded.
A simplified version of Mellouk considers, for any $\varepsilon>0$, the system, on~$\mathcal{T}$,
\begin{equation}\label{eq_ree}
\mathrm{d} \mathrm{X}^\varepsilon_t = b(\mathrm{X}^\varepsilon_t, \mathrm{Z})\mathrm{d} t + \varepsilon a(\mathrm{X}^\varepsilon_t,\mathrm{Z})\mathrm{d} \mathrm{W}_t,
\qquad \mathrm{X}^\varepsilon_0 = \mathrm{x}_{0} \in \mathbb{R}^n,
\end{equation}
where $b:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ and
$a: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathcal{M}_{(n,d)}$ are bounded Borel measurable,
uniformly Lipschitz continuous,
$\mathrm{Z}$ is a random variable with compact support on~$\mathbb{R}^m$
and~$\mathrm{W}$ a $d$-dimensional standard Brownian motion, independent of~$\mathrm{Z}$.
The main result of the paper is a large deviations principle on $\mathcal{C}^\alpha(\mathcal{T},\mathbb{R}^n)$,
the space of $\alpha$-H\"older continuous functions,
for $0\le\alpha< \frac{1}{2}$, for the sequence $(\mathrm{X}^\varepsilon)_{\varepsilon >0}$,
under the following assumptions:
\begin{assumption}\
\label{assM}
\begin{itemize}
\item[(H1)] $b(\cdot,\cdot)$ is jointly measurable on $\mathbb{R}^n \times \mathbb{R}^m$ and there exists $C>0$ such that, for all $\mathrm{x},\mathrm{x}' \in \mathbb{R}^n$, $\mathrm{z},\mathrm{z}' \in \mathbb{R}^m$,
$$
|b(\mathrm{x},\mathrm{z})| \le C(1+|\mathrm{x}|)
\qquad\text{and}\qquad
|b(\mathrm{x},\mathrm{z}) - b(\mathrm{x}',\mathrm{z}')| \le C(|\mathrm{x}-\mathrm{x}|+|\mathrm{z}-\mathrm{z}'|).
$$
\item[(H2)] $a(\cdot,\cdot)$ is jointly measurable on $\mathbb{R}^n \times \mathbb{R}^m$ and there exists $C>0$ such that, for all $\mathrm{x},\mathrm{x}' \in \mathbb{R}^n$, $\mathrm{z},\mathrm{z}' \in \mathbb{R}^m$,
$$
\|a(\mathrm{x},\mathrm{z})\| \le C
\qquad\text{and}\qquad
\|a(\mathrm{x},\mathrm{z}) - a(\mathrm{x}',\mathrm{z}')\| \le C (|\mathrm{x}-\mathrm{x}'|+|\mathrm{z}-\mathrm{z}'|).
$$
\end{itemize}
\end{assumption}
For $f \in \mathcal{H}$, $u \in \mathrm{supp}(\mathrm{Z})$ and $\mathrm{x}_0 \in \mathbb{R}^n$,
let $\mathcal{S}_{\mathrm{x}_0}(f,u)$ denote the unique solution to the controlled ODE
$g_t = \mathrm{x}_0 + \int_{0}^{t} b(g_s,u_s)\mathrm{d} s + \int_{0}^{t} a(g_s,u_s) \dot{f}_s \mathrm{d} s$,
for $t \in \mathcal{T}$.
Let us now introduce the following definition:
\begin{definition}\label{def:LSC}
Let $\alpha\in [0,\frac{1}{2})$ and~$\mathcal{B}_a$ be the ball of radius~$a$ in the $\alpha$-H\"older norm.
The lower semi-continuous regularisation~$\breve{I}:\mathcal{C}^\alpha(\mathcal{T},\mathbb{R}^m)\to\mathcal{C}^\alpha(\mathcal{T},\mathbb{R}^m)$
of a functional~$I:\mathcal{C}^\alpha(\mathcal{T},\mathbb{R}^m)\to\mathcal{C}^\alpha(\mathcal{T},\mathbb{R}^m)$ is defined as
$$
\breve{I}(\psi) := \lim_{a \downarrow 0} \inf\left\{I(\varphi): \varphi\in \mathcal{B}_{a}(\psi)\right\}.
$$
\end{definition}
\begin{theorem}[Theorem 2.1 in~\cite{M00}]\label{Th_Mell}
Under Assumption~\ref{assM},
$(\mathrm{X}^\varepsilon)_{\varepsilon >0}\sim\mathrm{LDP}(\varepsilon^2, \breve{I}^\alpha)$,
where (with $\Lambda$ in~\eqref{eq:LDPVolterrageneral})
$$
I^\alpha(\psi) := \inf \left\{\Lambda(\varphi): \varphi \in \mathcal{H}, \mathcal{S}_{\mathrm{x}_0}(\varphi,u) = \psi,
\text{ for some }u \in \mathrm{supp}(\mathrm{Z}) \right\}.
$$
\end{theorem}
Coming back to our model, the rescaling
$(X^{\varepsilon}_t, Y^{\varepsilon}_t) := (X_{\varepsilon^2 t}, Y_{\varepsilon^2 t})$,
equivalent to that of Section~\ref{sec:GenCase} with $b=0$ and $H=\frac{1}{2}$,
the small-noise system~\eqref{eq:model}, under Assumption~\textbf{$\mathcal{A}_b$}, becomes,
similarly to Section~\ref{sec:GenCase},
\begin{equation}\label{eq:SystemEps}
\left\{
\begin{array}{ll}
\displaystyle
\mathrm{d} X^{\varepsilon}_t = - \frac{\varepsilon^{2}}{2} \widetilde{\sigma}(Y^{\varepsilon}_t)^2 \mathrm{d} t + \varepsilon \widetilde{\sigma}(Y^{\varepsilon}_t) \left(\rho\mathrm{d} B_t + \bar{\rho} \mathrm{d} B^{\perp}_t\right), &X^{\varepsilon}_0=0, \\
\mathrm{d} Y^{\varepsilon}_t = \left(\varepsilon^{2} \lambda + \beta \varepsilon^2 Y^{\varepsilon}_t \right)\mathrm{d} t + \varepsilon \xi \mathrm{d} B_t, &Y^{\varepsilon}_0 \sim \Theta,
\end{array}
\right.
\end{equation}
with~$B$ a standard Brownian motion.
Subtracting the initial random datum~$\mathrm{X}^\varepsilon_0 = \mathrm{X}_0 = (0,\Theta)'$, this system can be expressed in the form~\eqref{eq_ree} with $\mathrm{X}^\varepsilon = (X^\varepsilon, Y^\varepsilon)$,
\begin{equation}\label{eq:SystemEpsCoef}
b(\varepsilon,\mathrm{X}^\varepsilon,\mathrm{X}_0) =
\varepsilon^{2}
\begin{pmatrix}
\displaystyle - \frac{1}{2} \widetilde{\sigma}(Y^{\varepsilon}_t + \Theta)^2\\
\displaystyle \lambda + \beta (Y^{\varepsilon}_t + \Theta)
\end{pmatrix}
\qquad\text{and}\qquad
a(\mathrm{X}^\varepsilon,\mathrm{X}_0) =
\begin{pmatrix}
\rho \widetilde{\sigma}(Y^{\varepsilon}_t+\Theta) & \bar{\rho}\widetilde{\sigma}(Y^{\varepsilon}_t+\Theta)\\
\xi & 0
\end{pmatrix},
\end{equation}
and note that~$b(\varepsilon,\cdot,\cdot)$ converges to the null map as~$\varepsilon$ tends to zero.
The assumptions imposed in~\cite{M00} on the drift and diffusion coefficients are clearly satisfied here.
While Mellouk allows the drift and diffusion to depend explicitly on external random factors,
we can write our setting (dependence on a random starting point) into this framework.
The large deviations estimate for the sequence $(\mathrm{X}^\varepsilon)_{\varepsilon>0} = (X^\varepsilon, Y^\varepsilon)_{\varepsilon >0}$
thus obtained is stronger than that in the previous section, as it holds on $\mathcal{C}^\alpha(\mathcal{T},\mathbb{R}^n)$,
for any $0\le\alpha< \frac{1}{2}$.
Note that the mild conditions on the coefficients~\cite[($H_0$)-($H_2$)]{M00} are easily satisfied in our case, so the only additional assumption is the boundedness of the support on~$\Theta$.
We remark here that~\cite{M00} is not directly applicable to the current setting
but has to be extended to include $\varepsilon$-dependence in the drift, and we do so following
the Azencott~\cite{A80}'s inspired approach developed by Peithmann~\cite[Subsection 2.2.1]{P07}.
In order to state LDP (proved in Section~\ref{sec:Proof-Th_rdm_new}), we impose the following assumption:
\begin{assumption}\label{ass:UniqueSol}
For $u \in \mathrm{supp}(\mathrm{X}_0)$ and $\varphi\in \mathcal{H}$,
the ODE $\psi_t = \int_{0}^{t} a(\psi_s,u_s) \dot{\varphi}_s \mathrm{d} s$
has a unique solution on~$\mathcal{T}$, denoted by $\mathcal{S}_{0}(\varphi,u)$.
\end{assumption}
\begin{theorem}\label{Th_rdm_new}
Under Assumption~\ref{ass:UniqueSol}, if~$\Theta$ has compact support, then
$(\mathrm{X}^\varepsilon)_{\varepsilon >0}\sim\mathrm{LDP}(\varepsilon^2,\breve{I}^\alpha)$, with
$$
I^\alpha(\psi) := \inf \left\{ \Lambda^{\rho}(\varphi): \varphi\in \mathcal{H},
\text{ such that } \mathcal{S}_{0}(\varphi,u) = \psi, \text{ for some }u \in \mathrm{supp}(\mathrm{X}_0) \right\},
$$
with $\Lambda^{\rho}$ defined in~\eqref{eq:ext_grf}.
\end{theorem}
\section{Applications to Implied volatility asymptotics}\label{sec:applications}
As announced in the introduction, we unify here two branches of research,
both aimed at reproducing the steepness of the implied volatility surface on the short end via models with continuous paths.
While there are now numerous outputs~\cite{ALV07,BFG16,BFGHS17,FZ16,Fukasawa,GJR14,JPS17}
in the literature confirming that a fractional driving noise (with Hurst exponent $H<1/2$)
in the volatility leads to the observed steepness of the smile,
recent results~\cite{JR15,JS17} reproduce this effect by randomising the initial volatility in classical diffusive models.
In this section we demonstrate how to modulate the two effects with respect to one another.
In the Black-Scholes-Merton model, the price of a European Call option is $\mathcal{C}_{BS}(t,\mathrm{e}^k, \Sigma)$,
with associated volatility~$\Sigma$.
Considering a market with observed Call option prices $\mathcal{C}_{obs}(t, \mathrm{e}^k)$,
with maturity~$t$ and strike~$\mathrm{e}^k$, we denote by $\Sigma_{t}(k)$ the implied volatility,
defined as the unique non-negative solution to $\mathcal{C}_{BS}(t, \mathrm{e}^k, \Sigma_{t}(k)) = \mathcal{C}_{obs}(t, \mathrm{e}^k)$.
\subsection{General fractional case}
From Theorems~\ref{Th:tails_frac_X} and~\ref{Th:LDP_ST_X},
we can deduce the asymptotic behaviour of the implied volatility
for large strikes and for small maturities.
We state those below, and postpone the proofs to Appendices~\ref{sec:Proof_IVTails} and~\ref{sec:Proof_IVST}.
\begin{corollary}(Large-strike implied volatility asymptotics)\label{Cor:ImpliedvolTails}
For any $H \in (0,1)$ and any $b \geq 1/2$ such that Theorem~\ref{Th:tails_frac_X}
holds, we have the following large-strike asymptotic estimates of the implied volatility:
$$
\lim_{k\uparrow \infty} \frac{\Sigma^2_t(k)t}{k}
= \frac{1}{2}\left(\inf_{y \ge 1} \widetilde{\Lambda}(\phi)|_{\phi_t =y}\right)^{-1}
\quad \textrm{with } \widetilde{\Lambda} \textrm{ as in~\eqref{eq:grf_X_tails}}, \text{ and for any } t \in \mathcal{T}.
$$
\end{corollary}
Similarly, from Theorem~\ref{Th:LDP_ST_X} we can deduce the asymptotic behaviour of the implied volatility when time becomes small.
The following Corollary generalises~\cite[Corollary 4.10]{FZ16}.
\begin{corollary}[Small-time Implied volatility asymptotics]\label{Cor:ImpliedvolST}
For any $H \in (0,1)$ and any $b \geq 1/2-2H$ such that Theorem~\ref{Th:LDP_ST_X} holds,
the following small-time estimate is true for any $k\ne 0$:
\begin{equation}\label{eq:IV_ST}
\lim_{t\downarrow 0} t^b \Sigma^2_t\left(t^{1/2-H-b}k\right)
= \frac{k^2}{2}\left(\inf_{y \ge k} \mathtt{I}(\phi)|_{\phi_{1} =y}\right)^{-1},
\quad \textrm{with } \mathtt{I} \textrm{ as in~\eqref{eq:grf_X_ST}}.
\end{equation}
\end{corollary}
This implies that the implied volatility explodes with rate~$t^{-b}$.
For $b=0$, it is identical to~\cite[Formula (26)]{FZ16}.
\subsection{Refined asymptotic results in the special diffusive case from Sections~\ref{app:MNS} and~\ref{Sec:Mellouk}}
\label{Ch3_rdm_future}
\subsubsection{Large-strike asymptotics}
We consider here a specific case of a multidimensional diffusion, as we are only interested
in studying its tail asymptotics.
Let~$\mathrm{X}^{\zeta}:=(X^{\zeta, 1},\cdots, X^{\zeta,n})$ be the unique strong solution in~$\mathbb{R}^n$ to
$$
\mathrm{d} \mathrm{X}_{t}^{\zeta} = \tilde{b}(\mathrm{X}_{t}^{\zeta})\mathrm{d} t + a(\mathrm{X}_{t}^{\zeta}) \mathrm{d} W_t,
\qquad \mathrm{X}_{0}^{\zeta} = \zeta,
$$
for some $d$-dimensional standard Brownian motion and some square integrable random variable~$\zeta$,
and with
$\tilde{b}:\mathbb{R}^n\to\mathbb{R}^n$ and $a: \mathbb{R}^{n} \rightarrow \mathcal{M}_{(n,d)}(\mathbb{R})$.
Consider the following scaling assumption:
\begin{assumption}\label{assu:Tails}
There exist $b_1,\cdots,b_n>0$ with $b_1=2$ such that
$\mathrm{X}^{\varepsilon,\zeta}:=\left(\varepsilon^{b_1}X^{\zeta,1},\cdots,\varepsilon^{b_n}X^{\zeta,n}\right)$
satisfies
\begin{equation}\label{eq:ScalingTails}
\mathrm{d} \mathrm{X}_{t}^{\varepsilon,\zeta} = \varepsilon \tilde{b}(\varepsilon,\mathrm{X}_{t}^{\varepsilon,\zeta})\mathrm{d} t
+ \varepsilon a(\mathrm{X}_{t}^{\varepsilon,\zeta}) \mathrm{d} W_t,
\qquad \mathrm{X}_{0}^{\zeta} = \left(\varepsilon^{b_1}\zeta^{(1)},\cdots,\varepsilon^{b_n}\zeta^{(n)}\right).
\end{equation}
Furthermore,
$\varepsilon \tilde{b}(\varepsilon, \cdot/\varepsilon)$ converges uniformly to some function~$b(\cdot)$ as~$\varepsilon$ tends to zero.
\end{assumption}
We can then state the main result about tail asymptotics:
\begin{proposition}\label{prop:RSP}
Under Assumption~\ref{assu:Tails}, if there exists~$\mathrm{x}_0 \in \mathbb{R}^n$ such that for any $\delta >0$,
$\limsup_{\varepsilon \downarrow 0} \varepsilon^2 \log \mathbb{P}(|\zeta-\mathrm{x}_0|>\delta)=-\infty$,
and if the triple $(\mathrm{x}_0, b(\cdot), a(\cdot))$ satisfies Assumption~\ref{ass:MNS}, then
$$
\lim_{\varepsilon \downarrow 0} \varepsilon^2 \log \mathbb{P}(\varepsilon X^{\zeta,1}_t \ge 1)
= \lim_{\varepsilon \downarrow 0} \varepsilon^2 \log \mathbb{P}(\varepsilon X^{\mathrm{x}_0,1}_t \ge 1),
\qquad\text{for any }t\in\mathcal{T}.
$$
\end{proposition}
The scaling from Assumption~\ref{assu:Tails} may be odd at first, but reflects the fact that
components of stochastic models may each act on different scales.
Consider for example the Ornstein-Uhlenbeck process, solution to
\begin{equation*}
\begin{array}{rll}
\mathrm{d} X_t & = -\frac{1}{2}Y_t^2 \mathrm{d} t + Y_t \mathrm{d} W_t, & X_0 = \zeta,\\
\mathrm{d} Y_t & = (\lambda + \beta Y_t) \mathrm{d} t + \xi \mathrm{d} B_t, & Y_0 = y_0>0,
\end{array}
\end{equation*}
where $W$ and $B$ are two correlated Brownian motions.
The rescaling $(X^\varepsilon,Y^\varepsilon) := (\varepsilon^2 X, \varepsilon Y)$
(corresponding to $b_1=2$ and $b_2=1$) yields
\begin{equation*}
\begin{array}{rll}
\mathrm{d} X_t^{\varepsilon} & = -\frac{1}{2}(Y_t^{\varepsilon})^2 \mathrm{d} t + \varepsilon Y_t^{\varepsilon} \mathrm{d} W_t, & X_0^{\varepsilon} = \varepsilon^2\zeta,\\
\mathrm{d} Y_t^{\varepsilon} & = (\varepsilon\lambda + \beta Y_t^{\varepsilon}) \mathrm{d} t + \varepsilon\xi \mathrm{d} B_t, & Y_0^{\varepsilon} = \varepsilon y_0>0,
\end{array}
\end{equation*}
namely~\eqref{eq:ScalingTails}, and the assumptions are satisfied.
The proof of Proposition~\ref{prop:RSP} is postponed to Appendix~\ref{sec:Proof_prop_RSP}.
The assumption of the random initial condition $ \mathrm{X}^\zeta_0 = \zeta$ being $\mathcal{F}_0$-measurable distribution can be relaxed.
Indeed, the filtration $\mathcal{F}$ is the filtration generated by the $d$-dimensional Brownian motion~$\mathrm{W}$. Instead, one could work on a filtration $\mathcal{F}' := {(\mathcal{F}'_t)}_{t\in\mathcal{T}^*}$ generated by
$\mathcal{F}'_t := \sigma(\left\{\mathrm{W}_u, u\le t\right\} \cup \left\{\zeta\right\})$, for all $t\in\mathcal{T}^*$.
Then the random initial point~$X^\zeta_0$ has a $\mathcal{F}'_0$-measurable distribution
and the results above still hold, in particular Theorem~\ref{thm:MNSLDP} and Proposition~\ref{prop:RSP},
on the new filtered probability space $(\Omega, \mathcal{F}', {(\mathcal{F}')}_{t\in\mathcal{T}}, \mathbb{P})$.
In the context of implied volatility asymptotics, this result has the following meaning:
\begin{corollary}
\label{AsympIV}
The wings of the smile are independent of the starting point ($\zeta$ or $\mathrm{x}_0$).
\end{corollary}
\begin{proof}
Gao and Lee~\cite{GL14} show that asymptotic behaviour of the implied volatility can be directly inferred
from comparing tail probabilities to those of the Black-Scholes model.
It is straightforward to see that the scaling of Proposition~\ref{prop:RSP} is the same in Black-Scholes,
and the corollary follows immediately.
\end{proof}
\subsubsection{Small-time asymptotics for the `forward' Stein-Stein model}
We are interested in a `forward' process, as defined by Jacquier and Roome~\cite{JR15}
in the context of forward-start European options:
$$
\mathbb{E} {\left( \frac{S_{t+\tau}}{S_t} - \mathrm{e}^k\right)}^{+}
= \mathbb{E} {\left( \mathrm{e}^{X_{t+\tau}-X_t} - \mathrm{e}^k\right)}^{+}
=: \mathbb{E} {\left(\mathrm{e}^{X_{\tau}^{(t)}} - \mathrm{e}^k\right)}^{+},
$$
with ${(X_\tau^{(t)})}_{\tau \ge 0}$ the so-called `forward' process, defined path-wise by
$X_\tau^{(t)} := X_{t+\tau} - X_t$, for some fixed $t>0$, and for all $\tau\geq 0$.
The 'forward' process ${(X_\tau^{(t)})}_{\tau \ge 0}$ then satisfies the following stochastic differential equation:
\begin{equation}\label{eq:rSS}
\left\{
\begin{array}{ll}
\displaystyle \mathrm{d} X_\tau^{(t)} = -\frac{1}{2} (Y^{(t)}_\tau)^2 \mathrm{d} \tau + Y^{(t)}_\tau \mathrm{d} W_{1,\tau}, & X_0^{(t)} = 0,\\
\displaystyle \mathrm{d} Y^{(t)}_\tau = (\lambda + \beta Y^{(t)}_\tau)\mathrm{d} \tau + \xi \mathrm{d} W_{2,\tau}, & Y^{(t)}_0 ~\sim \sigma_t.
\end{array}
\right.
\end{equation}
The stochastic differential equation for $(X_\tau^{(t)} , Y_\tau^{(t)})_{\tau\geq 0}$
is the same as that for $(X_t , Y_t)_{t\geq 0}$,
albeit with an initial random distribution $(\delta_0,\sigma_t)$,
where~$\sigma_t$ is Gaussian
with mean $\mathrm{e}^{\beta t}(\sigma_0 + \frac{\lambda}{\beta}) - \frac{\lambda}{\beta}$
and variance $\frac{\xi^2}{2\beta}(\mathrm{e}^{2\beta t}-1)$.
We now apply the results of Section~\ref{Sec:Mellouk} to obtain small-time asymptotics for a version of the Stein-Stein `forward' model
with a generalised random starting point.
\begin{proposition}
With the scaling $(X^\varepsilon_\tau, Y^\varepsilon_\tau) := (X^{(t)}_{\varepsilon^2 \tau}, Y^{(t)}_{\varepsilon^2 \tau})$ for $\varepsilon,t >0$, the randomised Stein-Stein rescaled model~\eqref{eq:rSS}
is the same as~\eqref{eq:SystemEps} with coefficients given in~\eqref{eq:SystemEpsCoef},
with $\widetilde{\sigma}(y)\equiv y$, and Theorem~\ref{Th_rdm_new} applies.
\end{proposition}
We can translate this result into forward implied volatility asymptotics directly using~\cite[Theorem 2.4]{JF09},
and refer to this very paper for a precise definition of the forward implied volatility~$\Sigma_{t,\tau}$:
\begin{corollary}
The small-time forward smile reads
$\displaystyle \lim_{\tau \downarrow 0} \Sigma^2_{t,\tau}(k)
= \frac{k^2}{2}\left(\inf_{y \ge k} \breve{I}^\alpha (\phi)|_{\phi_{1}=y}\right)^{-1}$,
with~$\breve{I}^\alpha$ in Theorem~\ref{Th_rdm_new}.
\end{corollary}
|
2,869,038,155,660 | arxiv | \section{Introduction}
\label{Introduction}
At large momentum scale, quantum chromodynamics (QCD) predicts asymptotic freedom \cite{J-Gross,D-Politzer} or a remarkable weakening in the running strong coupling. Accordingly, phase transition takes place from hadrons in which quarks and gluons are conjectured to remain confined (at low temperature and density) to quark-gluon plasma (QGP) \cite{N. Cabibbo,Collins}, in which quarks and gluons become deconfined (at high temperature and density) \cite{Rischke:2003mt}. Furthermore, at low temperature, the QCD chiral symmetry is spontaneously broken; $SU(N_f )_L \times SU(N_f )_R \rightarrow SU(N_f )_V$. In this limit, the chiral condensate
remains finite below the critical temperature ($T_c$). The broken chiral symmetry is restored at high temperatures. The finite quark masses explicitly break QCD chiral symmetry.
Nambu-Jona-Lasinio (NJL) model \cite{Nambu:1961tp} describes well the hadronic degrees of freedom. Polyakov Nambu-Jona-Lasinio (PNJL) model \cite{Fukushima:2003fw,Ratti:2005jh,Fukushima:2008wg} takes into consideration the quark dynamics \cite{Hatsuda:1994pi} and has been utilized to study the QCD phase-diagram \cite{Asakawa:1989bq,Fujii:2003bz}. Also, linear-$\sigma$ model (LSM) \cite{Gell Mann:1960} can be used in mapping out the QCD phase-diagram.
Many studies have been performed on LSM like O(4) LSM \cite{Gell Mann:1960} at vanishing temperature, O(4) LSM at finite temperature \cite{Lenaghan:1999si, Petropoulos:1998gt} and $SU(N_f)_R \times SU(N_f)_L$ LSM for $N_f=2$, $3$ and $4$ quark flavors \cite{l, Hu:1974qb, Schechter:1975ju, Geddes:1979nd}. In order to obtain reliable results, extended LSM to PLSM ] can be utilized, in which information about the confining glue sector of the theory is included in form of Polyakov-loop potential. The latter can be extracted from pure Yang-Mills lattice simulations \cite{Polyakov:1978vu, Susskind:1979up, Svetitsky:1982gs,Svetitsky:1985ye}. Also, the Polyakov linear sigma-model (PLSM), and Polyakov quark meson model (PQM) \cite{Schaefer:2007pw,Kahara:2008yg,Schaefer:2008ax} deliver reliable results. Furthermore, the quasi-particle model (QPM) \cite{kmpf1,qp18b} was suggested to reproduce the lattice QCD calculations \cite{QPMqcd1,QPMqcd2}, in which two types of actions are implemented; the lattice QCD simulations utilizing the standard Wilson action and the ones with renormalization-improved action.
In the present work, we integrate the gluonic sector of QPM to LSM \cite{Tawfik:2014bna} (QLSM) in order to reproduce the recent lattice QCD calculations \cite{Borsanyi:2013bia}. In QLSM \cite{Tawfik:2014bna}, the Polyakov contributions to the gluonic interactions and to the confinement-deconfinement phase transition are entirely excluded. Instead we just add the gluonic part of QPM. Therefore, the quark masses should be very heavy. We shall comment on this, later on. In section \ref{sec:Results}, we outline the QLSM results. They are similar to that of PNJL. This might be interpreted due the very heavy quark masses implemented in both models. Similar approach has been introduced in \cite{MAS2006}. The authors described in inclusion of gluonic Polyakov loop, which is assumed to generate a large gauge invariance and lead to a remarkable modification in hadron thermodynamics. A quite remarkable bridging between PNJL model quantum and local Polyakov loop and HRG model has been introduced \cite{AMS2014}. A large suppression of the thermal effects has been reported and it was concluded that the center symmetry breaking becomes exponentially small with increasing the massed of constituent quarks. In other words, the chiral symmetry restoration becomes exponentially small with increasing the pion mass.
The hadron resonance gas (HRG) model gives a good description for the thermal and dense evolution of various thermodynamic quantities in the hadronic matter \cite{Karsch:2003vd,Karsch:2003zq,Redlich:2004gp,Tawfik:2004sw,Tawfik:2004vv,Tawfik:2006yq,Tawfik:2010uh,Tawfik:2010pt,Tawfik:2012zz}. Also, it has been successfully utilized to characterizing the conditions deriving the chemical freeze-out at finite densities~\cite{Tawfik:2012si}. In light of this, the HRG model can be well used in calculating the thermal and dense dependence of quark-antiquark condensate \cite{Tawfik:2005qh}. The HRG grand canonical ensemble includes two important features \cite{Tawfik:2004sw}; the kinetic energies and the summation over all degrees of freedom and energies of the resonances. On other hand, it is known that the formation of resonances can only be achieved through strong interactions~\cite{Hagedorn:1965st}; {\it resonances (fireballs) are composed of further resonances (fireballs), which in turn consist of resonances (fireballs) and so on}. In other words, the contributions of the hadron resonances to the partition function are the same as that of free (non-interacting) particles with an effective mass. At temperatures comparable to the resonance half-width, the effective mass approaches the physical one \cite{Tawfik:2004sw}. Thus, at high temperatures, the strong interactions are conjectured to be taken into consideration through the inclusion of heavy resonances. It is found that the hadron resonances with masses up to $2\;$GeV are representing suitable constituents for the partition function ~\cite{Karsch:2003vd,Karsch:2003zq,Redlich:2004gp,Tawfik:2004sw,Tawfik:2004vv,Tawfik:2006yq,Tawfik:2010uh,Tawfik:2010pt,Tawfik:2012zz}. Such a way, the singularity expected at the Hagedorn temperature~\cite{Karsch:2003zq,Karsch:2003vd} can be avoided and the strong interactions are assumed to be taken into consideration. Nevertheless, validity of the HRG model is limited to the temperatures below the critical one, $T_c$.
In the present paper, we review PLSM, QLSM, PNJL and HRG with respect to their descriptions for the chiral phase-transition. We analyse the chiral order-parameter $M(T)$, the normalized net-strange condensate $\Delta_{q,s}(T)$ and the chiral phase-diagram and compare the results with the lattice QCD \cite{LQCD1,Schmidt:2010ss,Borsanyi:2010zi}. The present work is organized as follows. In section \ref{Model}, we introduce the different approaches SU(3) PLSM \cite{Tawfik:2014uka} (section \ref{PLSM}), QLSM (section \ref{LSM+QPM}), PNJL (section \ref{PNJL}) and HRG (section \ref{HRG:main}). The corresponding mean field approximations are also outlined. Section \ref{sec:Results} is devoted to the results. The conclusions and outlook shall be given in section \ref{sec:conclusion}.
\section{SU(3) effective models}
\label{Model}
\subsection{Polyakov Linear Sigma-Model (PLSM)}
\label{PLSM}
As discussed in Ref. \cite{Tawfik:2014uka,Tawfik:2014bna}. the Lagrangian of LSM with $N_f =3$ quark flavors and $N_c=3$ (for quarks, only) color degrees and with quarks coupled to Polyakov loop dynamics was introduced in \cite{Schaefer:2008ax,Mao:2010},
\begin{eqnarray}
\mathcal{L}=\mathcal{L}_{chiral}-\mathbf{\mathcal{U}}(\phi, \phi^*, T). \label{plsmI}
\end{eqnarray}
where the chiral part of the Lagrangian of the SU(3)$_{l}\times$ SU(3)$_{R}$ symmetric linear sigma-model Lagrangian with $N_f =3$ is \cite{Lenaghan,Schaefer:2008hk} $\mathcal{L}_{chiral}=\mathcal{L}_q+\mathcal{L}_m$. The first term is the fermionic part, Eq. (\ref{lfermion1}) with a flavor-blind Yukawa coupling $g$ of the quarks. The second term is the mesonic contribution, Eq. (\ref{lmeson1})
\begin{eqnarray}
\mathcal{L}_q &=& \sum_f \overline{\psi}_f(i \gamma^{\mu}
D_{\mu}-gT_a(\sigma_a+i \gamma_5 \pi_a))\psi_f, \label{lfermion1} \\
\mathcal{L}_m &=&
\mathrm{Tr}(\partial_{\mu}\Phi^{\dag}\partial^{\mu}\Phi-m^2
\Phi^{\dag} \Phi)-\lambda_1 [\mathrm{Tr}(\Phi^{\dag} \Phi)]^2
-\lambda_2 \mathrm{Tr}(\Phi^{\dag}
\Phi)^2+c[\mathrm{Det}(\Phi)+\mathrm{Det}(\Phi^{\dag})]
+\mathrm{Tr}[H(\Phi+\Phi^{\dag})]. \label{lmeson1}
\end{eqnarray}
The summation $\sum_f$ runs over the three flavors (f=1, 2, 3 for the three quarks u, d, s). The flavor-blind Yukawa coupling $g$ should couple the quarks to the mesons. The coupling of the quarks to the Euclidean gauge field $A_{\mu}=\delta_{\mu 0}A_0$ is given via the covariant derivative $D_{\mu}=\partial_{\mu}-i A_{\mu}$ \cite{Polyakov:1978vu,Susskind}. In Eq. (\ref{lmeson1}), $\Phi$ is a complex $3 \times 3$ matrix which depends on the $\sigma_a$ and $\pi_a$ \cite{Schaefer:2008hk}, where $\gamma^{\mu}$ are Dirac $\gamma$ matrices, $\sigma_a$ are the scalar mesons and $\pi_a$ are the pseudoscalar mesons.
\begin{eqnarray}
\Phi= T_a \phi _{a} =T_a(\sigma_a+i\pi_a),\label{Phi}
\end{eqnarray}
where $T_a=\lambda_a/2$ with $a=0, \cdots, 8$ are the nine generators of the U(3) symmetry group and $\lambda_a$ are the eight Gell-Mann matrices \cite{Gell Mann:1960}. The chiral symmetry is explicitly broken through
\begin{eqnarray}
H = T_a h_a.\label{H}
\end{eqnarray}
which is a $3 \times 3$ matrix with nine parameters $h_a$. Three finite condensates $\bar{\sigma_0}$, $\bar{\sigma_3}$ and $\bar{\sigma_8}$ are possible, because the finite values of vacuum expectation of $\Phi$ and $\bar{\Phi}$ are conjectured to carry the vacuum quantum numbers and the diagonal components of the explicit symmetry breaking term, $h_a$, where $h_0 \neq 0$, $h_3=0$ and $h_8 \neq 0$, and squared tree level mass of the mesonic fields $m^2$, two possible coupling constants $\lambda_1$ and $\lambda_2$, Yukawa coupling $g$ and a cubic coupling constant $c$ can be estimated as follows. $c=4807.84~$MeV, $h_1=(120.73)^3~$MeV$^3$, $h_s=(336.41)^3~$MeV$^3$, $m^2=-(306.26)^2$MeV$^2$, $\lambda _1=13.48$ and $\lambda _3=46.48$ and $g=6.5$.
In order to get a good analysis it is more convenient to convert the condensates $\sigma_0$ and $\sigma_8$ into a pure non-strange $\sigma_x$ and strange $\sigma_y$ condensates \cite{Kovacs:2006}
\begin{eqnarray}
\label{sigms}
\left( {\begin{array}{c}
\sigma _x \\
\sigma _y
\end{array}}
\right)=\frac{1}{\sqrt{3}}
\left({\begin{array}{cc}
\sqrt{2} & 1 \\
1 & -\sqrt{2}
\end{array}}\right)
\left({ \begin{array}{c}
\sigma _0 \\
\sigma _8
\end{array}}
\right).
\end{eqnarray}
The second term in Eq. (\ref{plsmI}) $\mathbf{\mathcal{U}}(\phi, \phi^*, T)$ represents Polyakov-loop effective potential \cite{Polyakov:1978vu}, which agrees well with the non-perturbative lattice QCD simulations and should have $Z(3)$ center symmetry as pure gauge QCD Lagrangian does \cite{Ratti:2005jh,Schaefer:2007pw}. In the present work, we use the potential $U(\phi, \phi^{*},T)$ as a polynomial expansion in $\phi $ and $ \phi^{*}$ \cite{Ratti:2005jh,Roessner:2007,Schaefer:2007d,Fukushima:2008wg}
\begin{eqnarray}
\frac{\mathbf{\mathcal{U}}(\phi, \phi^*, T)}{T^4}=-\frac{b_2(T)}{2}|\phi|^2-\frac{b_3
}{6}(\phi^3+\phi^{*3})+\frac{b_4}{4}(|\phi|^2)^2, \label{UloopI}
\end{eqnarray}
where
\begin{eqnarray}
b_2(T)=a_0+a_1\left(\frac{T_0}{T}\right)+a_2\left(\frac{T_0}{T}\right)^2+a_3\left(\frac{T_0}{T}\right)^3.
\end{eqnarray}
In order to reproduce pure gauge lattice QCD thermodynamics and the behavior of the Polyakov loop as a function of
temperature, we use the parameters $a_0=6. 75$, $a_1=-1. 95$, $a_2=2. 625$, $a_3=-7. 44$, $b_3=0.75$ and $b_4=7.5$. For a much better agreement with the lattice QCD results, the deconfinement temperature $T_0$ in pure gauge sector is fixed at $270$ MeV.
\subsubsection{Polyakov Linear Sigma-Model (PLSM) in Mean Field Approximation}
\label{PLSM:main}
In thermal equilibrium the grand partition function can be defined by using a path integral over the quark, anti-quark and meson fields
\begin{eqnarray} \label{MFAEQ}
\mathcal{Z} &=& \mathrm{Tr \,exp}[-(\hat{\mathcal{H}}-\sum_{f=u, d, s}
\mu_f \hat{\mathcal{N}}_f)/T]
= \int\prod_a \mathcal{D} \sigma_a \mathcal{D} \pi_a \int
\mathcal{D}\psi \mathcal{D} \bar{\psi} \mathrm{exp} \left[ \int_x
(\mathcal{L}+\sum_{f=u, d, s} \mu_f \bar{\psi}_f \gamma^0 \psi_f )
\right],
\end{eqnarray}
where $\int_x\equiv i \int^{1/T}_0 dt \int_V d^3x$ and $V$ is the volume of the system. $\mu_f$ is the chemical potential for $f=(u, d, s)$. We consider symmetric quark matter and define a uniform blind chemical potential $\mu_f \equiv \mu_{u, d}=\mu_s$. Then, we evaluate the partition function in the mean field approximation \cite{Schaefer:2008hk,Scavenius:2000qd}. We can use standard methods \cite{Kapusta:2006pm} in order to calculate the integration. This gives the effective potential for the mesons.
We define the thermodynamic potential density of PLSM as
\begin{eqnarray}
\Omega(T, \mu)=\frac{-T \mathrm{ln}
\mathcal{Z}}{V}=U(\sigma_x, \sigma_y)+\mathbf{\mathcal{U}}(\phi, \phi^*, T)+\Omega_{\bar{\psi}
\psi}. \label{potentialI}
\end{eqnarray}
Assuming degenerate light quarks, i.e. $q\equiv u, d$, the quarks and anti-quarks contribution potential is given as \cite{Mao:2010}
\begin{eqnarray} \label{qqpotioI}
\Omega_{\bar{\psi} \psi} &=& -2 T N_q \int \frac{d^3\vec{p}}{(2
\pi)^3} \{ \mathrm{ln} [ 1+3(\phi+\phi^* e^{-(E_q-\mu)/T})\times
e^{-(E_q-\mu)/T}+e^{-3 (E_q-\mu)/T}] \nonumber \\&& +\mathrm{ln} [
1+3(\phi^*+\phi e^{-(E_q+\mu)/T})\times e^{-(E_q+\mu)/T}+e^{-3
(E_q+\mu)/T}] \} \nonumber \\&& -2 T N_s \int \frac{d^3\vec{p}}{(2
\pi)^3} \{ \mathrm{ln} [ 1+3(\phi+\phi^* e^{-(E_s-\mu)/T})\times
e^{-(E_s-\mu)/T}+e^{-3 (E_s-\mu)/T}] \nonumber \\&& +\mathrm{ln} [
1+3(\phi^*+\phi e^{-(E_s+\mu)/T})\times e^{-(E_s+\mu)/T}+e^{-3
(E_s+\mu)/T}] \},
\end{eqnarray}
where $N_q=2$, $N_s=1$, and the valence quark and antiquark energy for light and strange quark, $E_q=\sqrt{\vec{p}^2+m_q^2}$ and $E_s=\sqrt{\vec{p}^2+m_s^2}$, respectively. Also, as per \cite{Kovacs:2006} the light quark sector, Eq. (\ref{sqmass}), decouples from the strange quark sector ($m_s$) and light quark mass $m_q$ gets simplified in this new basis to
\begin{eqnarray}
m_q &=& g \frac{\sigma_x}{2}, \label{qmass} \\
m_s &=& g \frac{\sigma_y}{\sqrt{2}}. \label{sqmass}
\end{eqnarray}
The purely mesonic potential is given as
\begin{eqnarray}
U(\sigma_x, \sigma_y) &=& \frac{m^2}{2} (\sigma^2_x+\sigma^2_y)-h_x \sigma_x-h_y \sigma_y-\frac{c}{2\sqrt{2}} \sigma^2_x \sigma_y + \frac{\lambda_1}{2} \sigma^2_x \sigma^2_y
+ \frac{1}{8} (2 \lambda_1 +\lambda_2)\sigma^4_x + \frac{1}{4} (\lambda_1+\lambda_2)\sigma^4_y. \label{UpotioI}
\end{eqnarray}
We notice that the sum in Eqs. (\ref{UloopI}), (\ref{qqpotioI}) and (\ref{UpotioI}) give the thermodynamic potential density similar to Eq. (\ref{potentialI}), which has seven parameters $m^2, h_x, h_y, \lambda_1, \lambda_2, c$ and $ g$, two unknown condensates $\sigma_x$ and $ \sigma_y$ and two order parameters for the deconfinement $\phi$ and $\phi^*$. The six parameters $m^2, h_x, h_y, \lambda_1, \lambda_2 $ and $ c$ are fixed in the vacuum by six experimentally known quantities \cite{Schaefer:2008hk}. In order to evaluate the unknown parameters $\sigma_x$, $ \sigma_y$, $\phi$ and $\phi^*$, we minimize the thermodynamic potential, Eq. (\ref{potentialI}), with respective to $\sigma_x$, $ \sigma_y$, $\phi$ and $\phi^*$, respectively. Doing this, we obtain a set of four equations of motion
\begin{eqnarray}\label{cond1}
\frac{\partial \Omega}{\partial \sigma_x}= \frac{\partial
\Omega}{\partial \sigma_y}= \frac{\partial \Omega}{\partial
\phi}= \frac{\partial \Omega}{\partial \phi^*}\mid_{min} =0,
\end{eqnarray}
where {\it min} means $\sigma_x=\bar{\sigma_x}, \sigma_y=\bar{\sigma_y}, \phi=\bar{\phi}$ and $\phi^*=\bar{\phi^*}$ are the global minimum.
\begin{eqnarray}
\Omega_{PLSM} &=& {\cal {U}}(\phi,\bar \phi,T)+\frac{m^2}{2} (\sigma^2_x+\sigma^2_y)-h_x
\sigma_x-h_y \sigma_y-\frac{c}{2\sqrt{2}} \sigma^2_x \sigma_y + \frac{\lambda_1}{2} \sigma^2_x \sigma^2_y
+ \frac{1}{8} (2 \lambda_1 +\lambda_2)\sigma^4_x + \frac{1}{4} (\lambda_1+\lambda_2)\sigma^4_y \nonumber \\
&-&
2 T N_q \int \frac{d^3\vec{p}}{(2 \pi)^3} \{ \mathrm{ln} [ 1+3(\phi+\phi^* e^{-(E_q-\mu)/T})\times
e^{-(E_q-\mu)/T}+e^{-3 (E_q-\mu)/T}] \nonumber \\
&& + \mathrm{ln} [1+3(\phi^*+\phi e^{-(E_q+\mu)/T})\times e^{-(E_q+\mu)/T}+e^{-3 (E_q+\mu)/T}] \} \nonumber \\
&-& 2 T N_s \int \frac{d^3\vec{p}}{(2 \pi)^3} \{ \mathrm{ln} [ 1+3(\phi+\phi^* e^{-(E_s-\mu)/T})\times
e^{-(E_s-\mu)/T}+e^{-3 (E_s-\mu)/T}] \nonumber \\
&& + \mathrm{ln} [1+3(\phi^*+\phi e^{-(E_s+\mu)/T})\times e^{-(E_s+\mu)/T}+e^{-3 (E_s+\mu)/T}] \}, \label{plsm:omega}
\end{eqnarray}
Accordingly, the chiral order parameter can be deduced as
\begin{eqnarray}
M_{PLSM} &=& m_s \dfrac{\langle \bar{\psi}\psi \rangle_{PLSM}}{T^{4}} = \frac{m_s}{T^{4}} \dfrac{\partial \Omega_{PLSM}}{\partial m_l}.
\label{Eq:M}
\end{eqnarray}
\subsection{Linear Sigma-Model and Quasi-Particle Sector (QLSM) }
\label{LSM+QPM}
When the Polyakov contributions to the gluonic interactions and to the confinement-deconfinement phase transition are entirely excluded, the Lagrangian of LSM with $N_f=3$ quark flavors and $N_c=3$ (for quarks, only) color degrees of freedom, where the quarks couple to the Polyakov loop dynamics, has been introduced in Ref. \cite{Schaefer:2008ax,Mao:2010},
\begin{eqnarray}
\mathcal{L}=\mathcal{L}_{chiral}-\mathbf{\mathcal{U}}(\phi, \phi^*, T). \label{plsm}
\end{eqnarray}
The main original proposal of the present work is the modification of Eq. (\ref{plsm})
\begin{eqnarray}
\mathcal{L}=\mathcal{L}_{chiral}-\mathbf{\mathcal{U}_g}(T,\mu), \label{newplsm}
\end{eqnarray}
where the chiral part of the Lagrangian $\mathcal{L}_{chiral}=\mathcal{L}_q+\mathcal{L}_m$ is of SU(3)$_{L}\times$ SU(3)$_{R}$ symmetry \cite{Lenaghan,Schaefer:2008hk}. Instead of $\mathbf{\mathcal{U}}(\phi, \phi^*, T)$, the gluonic potential $\mathbf{\mathcal{U}_g}(T,\mu)$, which is similar to the gluonic sector of the quasi-particle model, is inserted, (review Eq. (\ref{Eq:Ug})). The Lagrangian with $N_f =3$ consists of two parts; fermionic and mesonic contributions, Eqs. (\ref{lfermion1}) and (\ref{lmeson1}), respectively.
Some details about the quasi-particle model are in order. The model gives a good phenomenological description for lattice QCD simulation and treats the interacting massless quarks and gluons as non-interacting massive quasi-particles \cite{qusim1}. The corresponding degrees of freedom are treated in a similar way as the electrons in condensed matter theory \cite{paul}, i.e. the interaction with the medium provides the quasi-particles with dynamical masses. Consequently, most of the interactions can be taken into account. When confronting it to the lattice QCD calculations, the free parameters can be fixed. The pressure at finite $T$ and $\mu$ is given as
\begin{eqnarray}
p &=& \sum_{i=q,g}\, p_i - B(T,\mu), \qquad\qquad
p_i = \frac{g_i}{6\, \pi^2} \int_0^{\infty} k^4\, dk\, \frac{1}{E_i(k)} \left[f_i^+(k)+f_i^-(k)\right], \label{Eq:pg}
\end{eqnarray}
where the function $B(T,\mu)$ stands for bag pressure at finite $T$ and $\mu$ which can be determined by thermodynamic self-consistency and $\partial p/\partial \Pi_a=0$; the stability of $p$ with respect to the self-energies ($\Pi_a$) and the distribution function for bosons and fermions, $\pm$ respectively is given as
\begin{eqnarray} \label{Eq:fi}
f_i^{\pm}(k) &=& \frac{1}{\exp\left(\frac{E_i(k)\mp \mu}{T}\right)\pm 1}.
\end{eqnarray}
The quasi-particle dispersion relation can be approximated by the asymptotic mass shell expression near the light cone \cite{kmpf1,qp18b},
\begin{eqnarray} \label{Eq:Ei}
E_i^2(k) &=& k^2 + m_i^2(T,\mu)=k^2 + \Pi_i(k; T,\mu) + (x_i\, T)^2,
\end{eqnarray}
where $\Pi_i(k; T,\mu)$ is the self-energy at finite $T$ and $\mu$ and $x_i^2$ is a factor taking into account the mass scaling as used in the lattice QCD simulations. In other words, $x_i^2$ was useful when the lattice QCD simulations have been performed with quark masses heaver than the physical ones. In the present work, the gluon self-energies $\Pi_g(k; T,\mu)$ are relevant \cite{Bluhm}
\begin{eqnarray} \label{pi:Ei}
\Pi_g(k; T,\mu)=\left(\left[3+\dfrac{N_f}{2}\right] T^2 + \dfrac{3}{2 \pi ^2} \sum_f \mu_{f}^2\right) \dfrac{G^2}{6},
\end{eqnarray}
where the effective coupling $G$ at vanishing chemical potential is given as,
\begin{eqnarray} \label{G:Ei}
G^2(T) &=& \left\{ \begin{array}{ll}
G^2_{\text{2loop}}(T), & T\geq T_c, \\
& \\
G^2_{\text{2loop}}(T)+b\left(1-\frac{T}{T_c}\right), & T< T_c.\end{array}
\right.,
\end{eqnarray}
And the two-loop effective coupling $ G^2_{\text{2loop}}(T) $ reads \cite{kmpf1}
\begin{eqnarray}
G^2_{\text{2loop}}(T) &=& \frac{16\, \pi^2}{\beta_0 \ln(\xi^2)} \left[1-2 \frac{\beta_1}{\beta_0^2} \frac{\ln(\ln(\xi^2))}{\ln(\xi^2)}\right], \qquad \xi = \lambda \frac{T-T_s}{T_c}, \label{Gloop:Ei}
\end{eqnarray}
and $T_s$ is a regulator at $T_c$. For The parameter $\lambda$ is used to adjust the scale as found in lattice QCD simulations. These two parameters are not very crucial in the present calculations. The regulator and scale are controlled by the condensates ($\sigma_x$ and $\sigma_y$) and the order parameters ($\phi$ and $\phi^*$), which are given as function of temperature and baryon chemical potential. The $\beta$ function \cite{betaf} depends on the QCD coupling $G$, $\beta=\partial\, G/(\partial\, \ln(\Delta_{\mu}))$, with $\Delta_{\mu}$ is the energy scale. It is obvious that the QCD coupling in Eqs. (\ref{G:Ei}) and (\ref{Gloop:Ei}) is given in $T$-dependence, only. In calculating $\beta=\partial\, G/(\partial\, \ln(\Delta_{\mu}))$ at finite $\mu$ it is apparently needed to extend $G$ to be $\mu$-dependent, as well. The two-loop perturbation estimation for $\beta$ functions gives
\begin{eqnarray}
\beta_0 &=& \frac{1}{3} \left(11\, n_c - 2\, n_f\right), \\
\beta_1 &=& \frac{1}{6} \left(34\, n_c^2 - 13\, n_f\, n_c + 3\, \frac{n_f}{n_c}\right).
\end{eqnarray}
\subsubsection{Linear Sigma-Model and Quasi-Particle Sector (QLSM) in Mean Field Approximation}
\label{LSM+QPM:main}
As in section \ref{PLSM:main} and Eq. (\ref{MFAEQ}), we derive the thermodynamic potential density in the mean field approximation. This consists of three parts: mesonic and quasi-gluonic potentials in additional to the quark potential
\begin{eqnarray}
\Omega(T, \mu)=\frac{-T\, \ln (Z)}{V}=U(\sigma_x,\, \sigma_y) + U_{g}(T,\, \mu) + \Omega_{\bar{\psi}
\psi}. \label{potential}
\end{eqnarray}
\begin{itemize}
\item First, the quark potential part \cite{Schaefer:2008hk}
\begin{equation}
\label{eq:quark_pot}
\Omega_{\bar{q}q}(T,\mu_f) = d_q\, T \sum_{f=u,d,s}
\int\limits_0^\infty \! \frac{d^3 k}{(2\pi)^3} { \ln \left[1- n_{q,f}(T,\mu_f)\right] + \ln \left[1-n_{\bar{q},f}(T,\mu_f)\right] }
\end{equation}
It is obvious that $\Omega_{\bar{\psi}\psi}$ is equivalent to $\Omega_{\bar{q}q}$.
The occupation quark/aniquarks numbers read,
\begin{equation}
n_{q|\bar{q},f}\left(T,\mu_{f}\right)=\frac{1}{1+\exp\left[(E_{q,f}\pm \mu_f)/T\right]},
\end{equation}
and antiquarks $n_{\bar q,f}(T,\mu_{f}) \equiv n_{q,f} (T,-\mu_{f})$, respectively. The number of internal quark degrees of freedom is denoted by $d_q=2$ and $N_{c}=6$ (for quarks and antiquarks). The energies are given as
\begin{equation}
E_{q,f}= \sqrt{k^2 + m_f^2},
\end{equation}
with the quark masses $m_f$ which is related to $m_q$ and $m_s$ for $u$-, and $d$- and $s$-quark, respectively.
As given, the latter are proportional to the $\sigma$-fields
\begin{eqnarray}
m_q &=& g \frac{\sigma_x}{2}, \label{qmass2} \\
m_s &=& g \frac{\sigma_y}{\sqrt{2}} \label{sqmass2},
\end{eqnarray}
where the Yukawa coupling $g=8.3$. The symbols for the chiral condensates, $\sigma_x$ and $\sigma_y$ for light- and strange-quarks, respectively, are kept as in literature.
\item Second, the purely mesonic potential part reads
\begin{eqnarray}
U(\sigma_x, \sigma_y) &=& \frac{m^2}{2} \left(\sigma^2_x+\sigma^2_y\right)-h_x
\sigma_x-h_y \sigma_y-\frac{c}{2\sqrt{2}} \sigma^2_x \sigma_y
+\frac{\lambda_1}{2} \sigma^2_x \sigma^2_y
+ \frac{\left(2 \lambda_1 + \lambda_2\right)\sigma^4_x}{8} + \frac{\left(\lambda_1+\lambda_2\right)\sigma^4_y}{4}. \label{Upotio} \hspace*{8mm}
\end{eqnarray}
\item Third, the quasi-gluonic potential part is constructed from Eqs. (\ref{Eq:Ei}), (\ref{Eq:fi}) and (\ref{Eq:pg})
\begin{eqnarray}
U_g &=& -\frac{d_g}{6\, \pi^2} \int_0^{\infty} k^4\, dk \frac{1}{E_i} \left[\frac{1}{\exp\left(\frac{E_i - \mu}{T}\right)- 1} +
\frac{1}{\exp\left(\frac{E_i + \mu}{T}\right)- 1}\right]. \label{Eq:Ug}
\end{eqnarray}
In Eq. (\ref{Eq:Ug}), the degeneracy factor $ d_g = 8$ and two parameters $\lambda$ and $T_s$, which were given in Eq. (\ref{Gloop:Ei}), should be fixed in order to reproduce the lattice QCD calculations. Here, we find that $\lambda=2.0 $ and $T_s=0.0~$MeV give excellent results.
When adding the three potentials given in Eqs. (\ref{Eq:Ug}), (\ref{Upotio}) and (\ref{eq:quark_pot}), the thermodynamics and chiral phase translation can be analysed. The resulting potential $\Omega_{QLSM}$ can be used to determine the normalized net-strange condensate and chiral order parameter, Eq. (\ref{Eq:M}).
\begin{eqnarray}
\Omega_{QLSM}&=&\frac{m^2}{2} (\sigma^2_x+\sigma^2_y) - h_x\, \sigma_x-h_y \sigma_y-\frac{c}{2\sqrt{2}} \sigma^2_x \sigma_y + \frac{\lambda_1}{2} \sigma^2_x \sigma^2_y + \frac{1}{8} (2 \lambda_1 + \lambda_2)\sigma^4_x + \frac{1}{4} (\lambda_1+\lambda_2)\sigma^4_y \nonumber \\
&-& T \frac{d_q}{2 \pi^2} \int_0^{\infty}k^2\, dk \left[2\ln (1-f_q^-(T,\mu ))+2\ln (1-f_q^+(T,\mu ))
+ \ln (1-f_s^{-}(T,\mu ))+\ln (1-f_s^{+}(T,\mu ))\right] \nonumber \\
&-& 3\, \pi ^2\, d_g \int_0^{\infty}k^4\, dk \left[ \left(e^{\frac{\omega _g(T,\mu )}{T}}-1\right) \omega _g(T,\mu )\right]^{-1}, \label{Pq}
\end{eqnarray}
where,
\begin{eqnarray}
f_q^{\pm} (T,\mu) &=& \frac{1}{e^{\frac{\sqrt{\frac{1}{4} g^2 \sigma _x(T,\mu ){}^2+k^2}\pm \mu }{T}}+1}, \label{var1}\\
f_s^{\pm} (T,\mu) &=& \frac{1}{e^{\frac{\sqrt{\frac{1}{2} g^2 \sigma _y(T,\mu ){}^2+k^2}\pm \mu }{T}}+1}, \label{var2}\\
\omega _g(T,\mu ) &=& \left[k^2+\frac{8\, \pi ^2 \left(\frac{9}{2\, \pi ^2}\, \mu+\left(\frac{N_f}{2}+3\right) T^2\right) \left(1-\frac{3 \left(34\, N_c^2-13\, N_c\, N_f+3\, \frac{N_f}{N_c}\right) \ln \left(\ln ^2\left(\xi^2\right)\right)}{(11\, N_c-2\, N_f)^2 \ln^2\left(\xi^2\right)}\right)}{(11\, N_c-2\, N_f) \ln ^2\left(\xi^2\right)}\right]^{1/2}, \label{var3}
\end{eqnarray}
$\xi$ is function of $T$ and the quark masses should be very heavy. The QLSM results are similar to that of PNJL, section \ref{sec:Results}. This might be interpreted due the very heavy quark masses implemented in both models.
\subsection{Polyakov Nambu-Jona-Lassinio (PNJL) Model}
\label{PNJL}
The Lagrangian of PNJL reads \cite{OHP2006,Abhijit}
\begin {align}
{\cal L} &= {\sum_{f=u,d,s}}{\bar\psi_f}\gamma_\mu i D^\mu
{\psi_f}-\sum_f m_{f}{\bar\psi_f}{\psi_f}
+\sum_f \mu \gamma_0{\bar \psi_f}{\psi_f}\nonumber\\
&+{\frac {g_S} {2}} \sum_{a=0,\ldots,8} [({\bar\psi} \lambda^a {\psi})^2+
({\bar\psi} i\gamma_5\lambda^a {\psi})^2]
-{g_D} [det{\bar\psi_f}{P_L}{\psi_{f^\prime}}+det{\bar\psi_f}
{P_R}{\psi_{f^\prime}}]\nonumber\\
&-{\cal {U}}(\phi[A],\bar \phi[A],T)
\label{lag}
\end {align}
where the matrices $P_{L,R}=(1\pm \gamma_5)/2$ are chiral projectors, ${\cal {U}}(\phi[A],\bar \phi[A],T)$ is the Polyakov loop potential (Landau-Ginzburg type potential) and $D^\mu=\partial^\mu-i{A_4}\delta_{\mu 4}$ stand for gauge field interactions. The mass of a particular flavor is denoted by $m_f$, where $f=u,d,s$. The two coupling constants $g_D$ and $g_S$, $\lambda^a$ are Gell-Mann matrices \cite{Gell Mann:1960} and $\gamma_\mu$ are Dirac $\gamma$ matrices. The model is not renormalizable so that we have to use three-momentum cut-off regulator $\Lambda$ in order to keep quark loops finite.
The Polyakov loop potential is given by \cite{ratti},
\begin{equation}
\frac {{\cal U}(\phi, \bar \phi, T)}{ {T^4}}=-\frac {{b_2}(T)}{ 2}
{\bar \phi}\phi-\frac {b_3}{ 6}(\phi^3 + \bar \phi^3)
+\frac {b_4}{ 4}{(\bar\phi \phi)}^2
\end{equation}
with
\begin {eqnarray}
\phi &=& \frac{Tr_c L}{N_c}, \qquad
{\bar \phi} = \frac{Tr_c L^\dagger}{N_c}, \qquad
{b_2}(T) = a_0+{a_1}\left(\frac { {T_0}}{ T}\right)+{a_2}\left(\frac {{T_0}}{ T}\right)^2+
{a_3}\left(\frac {{T_0}}{T}\right)^3, \nonumber
\end {eqnarray}
$b_3$ and $b_4$ being constants and we choose the following fitting values for the potential parameters,
$a_0=6. 75$, $a_1=-1. 95$, $a_2=2. 625$, $a_3=-7. 44$, $b_3 = 0.75$ $b_4=7.5$ and $T_0=187$~MeV. These are adjusted to the pure gauge lattice data such that the equation of state and the Polyakov-loop expectation values are reproduced.
\subsubsection{Polyakov Nambu-Jona-Lassinio (PNJL) Model in Mean Field Approximation}
\label{NJL:main}
The thermodynamic potential density of PNJL is defined as
\begin {align} \label{PNJLpotuntial}
\Omega &= {\cal {U}}[\phi,\bar \phi,T]+2{g_S}{\sum_{f=u,d,s}}
{\sigma_f^2}-\frac {{g_D} }{2}{\sigma_u}
{\sigma_d}{\sigma_s}-6{\sum_f}{\int_{0}^{\Lambda}}
\frac {{d^3p}}{{(2\pi)}^3} E_{pf}\Theta {(\Lambda-{ |\vec p|})}\nonumber \\
&-2{\sum_f}T{\int_0^\infty}\frac {{d^3p}}{{(2\pi)}^3}
\ln\left[1+3\left(\phi+{\bar \phi}e^{\frac {-(E_{pf}-\mu)}{ T}}\right)
e^{\frac {-(E_{pf}-\mu)}{ T}}+e^{\frac {-3(E_{pf}-\mu)}{ T}}\right]
\nonumber\\
&-2{\sum_f}T{\int_0^\infty}\frac {{d^3p}}{{(2\pi)}^3}
\ln\left[1+3\left({\bar \phi}+{ \phi}e^{\frac {-(E_{pf}+\mu)}{ T}}\right)
e^{\frac {-(E_{pf}+\mu)}{ T}}+e^{ \frac {-3(E_{pf}+\mu)}{ T}}\right],
\end {align}
where $E_{pf}=\sqrt {p^2+M^2_f}$ is the single quasi-particle energy,
$\sigma_f^2=\sigma_u^2+\sigma_d^2+\sigma_s^2$ and from isospin symmetry, $\sigma_q=\sigma_u=\sigma_d$. In the above integrals, the vacuum integral has a cutoff $\Lambda$
whereas the medium dependent integrals have been extended to infinity. By the self-consistent gap equation, the quark mass can be estimated,
\begin {equation}
M_f =m_f - 2 g_S\, \sigma_f + \frac {{g_D}}{2}\, \sigma_{f+1}\; \sigma_{f+2},
\end {equation}
where $\sigma_f=\langle{\bar \psi_f} \psi_f\rangle$ denotes the chiral condensate of quark with flavor $f$ and other parameters are listed out in Tab. \ref{table2PNL} \cite{OHP2006,Abhijit}. For isospin symmetry, we define the light and strange-quark masses as
\begin {eqnarray}
M_s &=& m_s - 2 g_S\, \sigma_s + \frac {{g_D}}{2}\; \sigma_{q}^2, \\
M_q &=& m_q - 2 g_S\, \sigma_q + \frac {{g_D}}{2}\; \sigma_{q}\; \sigma_{s}.
\end {eqnarray}
Here, we notice the strong dependence on the $\sigma$-fields.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$ m_u$ [MeV] & $m_s$ [MeV]&$ \Lambda $ [MeV] & $g_S \Lambda^2 $&$ g_D \Lambda^5 $\\
\hline
$ 5.5 $&$ 134.758 $&$ 631.357 $&$ 3.664 $&$ 74.636$ \\
\hline
\end{tabular}
\caption{Parameters of the SU(3) PNJL model. \label{table2PNL}}
\end{center}
\end{table}
Now, we have all the PNJL Model parameters except $\sigma_{q}, \sigma_{s}, \phi$ and $\bar \phi$, which can be estimated from minimizing the thermodynamic potential, Eq. (\ref{PNJLpotuntial}), with respective to $ \sigma_{q}, \sigma_{s}, \phi$ and $\bar \phi$, respectively. Doing this, we obtain a set of four equations of motion
\begin{eqnarray}\label{cond1}
\frac{\partial \Omega}{\partial \sigma_x}= \frac{\partial \Omega}{\partial \sigma_y}= \frac{\partial \Omega}{\partial \phi}= \frac{\partial \Omega}{\partial {\bar \phi}}\mid_{min} =0.
\end{eqnarray}
Then, the potential of the PNJL model reads
\begin {align}
\Omega _{PNJL} &= {\cal {U}}[\phi,\bar \phi,T]+2{g_S}{\sum_{f=u,d,s}}
{\sigma_f^2}-\frac {{g_D} }{2}{\sigma_u}
{\sigma_d}{\sigma_s}-6{\sum_f}{\int_{0}^{\Lambda}}
\frac {{d^3p}}{{(2\pi)}^3} E_{pf}\Theta {(\Lambda-{ |\vec p|})}\nonumber \\
&-2{\sum_f}T{\int_0^\infty}\frac {{d^3p}}{{(2\pi)}^3}
\ln\left[1+3\left(\phi+{\bar \phi}e^{\frac {-(E_{pf}-\mu)}{T}}\right)
e^{\frac {-(E_{pf}-\mu)}{T}}+e^{\frac {-3(E_{pf}-\mu)}{T}}\right]
\nonumber\\
&-2{\sum_f}T{\int_0^\infty}\frac {{d^3p}}{{(2\pi)}^3}
\ln\left[1+3\left({\bar \phi}+{ \phi}e^{\frac {-(E_{pf}+\mu)}{T}}\right)
e^{\frac {-(E_{pf}+\mu)}{T}}+e^{ \frac {-3(E_{pf}+\mu)}{T}}\right]. \label{PNJLpotuntial}
\end {align}
Having completed the introduction of both PLSM and PNJL, it is in order now to discuss central $Z(3)$ symmetry related to the Polyakov loop. It has been shown that the SU(3) color-singlet has $Z(3)$ symmetry through the normalized character in the fundamental representation of SU(3), $\Phi(\theta_1,\theta_2)$. This becomes equivalent to an ensemble of Polyakov loop \cite{IAMRG}. Furthermore, it was concluded that $\Phi(\theta_1,\theta_2)$ can taken as an order parameter for color-confinement to color-deconfinement phase transition, i.e. the center symmetry is spontaneously broken at high temperatures.
Ref \cite{MHOB:2012} introduced an attempt to resolve some incongruities within NJL and PNJL. It was argued that by integrating corresponding extremum conditions, the thermodynamic potential is directly obtained, where the integration constant
can be fixed from Stefan-Boltzmann law. Keeping the regulator finite at finite temperature and chemical potential is the main advantage of this approach.
\subsection{Hadron Resonance Gas (HRG) Model}
\label{HRG:main}
Treating hadron resonances as a free (non-interacting) gas~\cite{Karsch:2003vd,Karsch:2003zq,Redlich:2004gp,Tawfik:2004sw,Tawfik:2004vv} is conjectured to give an accurate estimation for the thermodynamic pressure below $T_c$. It has been shown that thermodynamics of strongly interacting system can also be approximated as an ideal gas composed of hadron resonances with masses $\le 2~$GeV ~\cite{Tawfik:2004sw,Vunog}, i.e. confined QCD matter (hadrons) is well modelled as a non-interacting gas of resonances. The grand canonical partition function reads
\begin{eqnarray}
Z(T, \mu, V) &=& \bf{Tr} \left[ \exp^{\frac{\mu\, N-H}{T}}\right],
\end{eqnarray}
where $T$ ($\mu$) is temperature (chemical potential). The Hamiltonian ($H$) is given as the kinetic energies of the relativistic Fermi and Bose particles.
The main motivation of using $H$ is that
\begin{itemize}
\item it contains all relevant degrees of freedom of confined, {\it strongly interacting} QCD matter,
\item it {\it implicitly} includes interactions, especially the ones leading to formation of resonances and
\item it gives a quite satisfactory description of the particle production in heavy-ion collisions.
\end{itemize}
With these assumptions, the thermodynamics is resulted from {\it single-particle partition} functions $Z_i^1$
\begin{eqnarray}
\ln Z(T, \mu_i ,V) &=& \sum_i\pm \frac{V g_i}{2\pi^2}\int_0^{\infty} k^2\, \ln\left\{1 \pm \exp\left[\frac{\mu_i -\varepsilon_i(k)}{T}\right]\right\}\, d k, \label{eq:lnz1}
\end{eqnarray}
where $\varepsilon_i(k)=(k^2+ m_i^2)^{1/2}$ is the $i-$th particle dispersion relation, $g_i$ is
spin-isospin degeneracy factor and $\pm$ stands for bosons and fermions, respectively.
For hadron resonances which not yet measured, experimentally, a parametrization for a total spectral weight has been proposed \cite{brnt} as a recent estimation for Hagedorn mass spectrum \cite{hgdrn}. In the present work, we merely include known (measured) hadron resonances with mass $\leq 2~$GeV. This mass cut-off is assumed to define the validity of HRG in modelling the hadronic phase. Resonances with heavier masses diverge all thermodynamic quantities at the Hagedorn temperature~\cite{Karsch:2003zq, Karsch:2003vd}.
Very recently, it has been shown that indeed the viral expansion is a reliable way to include hadron resonances, because the phase shift is a directly-accessible quantity in experiments \cite{BGB2015}. For instance, for accurate isospin-averaged observables, the scalar-isoscalar $f_0(500)$ ($\sigma$ meson) resonance and scalar $K^*(800)$ should be not be included in the HRG model.
The HRG model has been used in calculating the higher-order moments of the particle multiplicity, in which a grand canonical partition function of an ideal gas with experimentally-observed states up to a certain mass cut-off is utilized \cite{Tawfik:2012si}. The HRG model has been successfully utilized in characterizing two different conditions generating the chemical freeze-out at finite densities, namely constant normalized-entropy density $s/T^3=7$ \cite{sT3p1,sT3p2,Tawfik:2005qn,Tawfik:2004ss}, constant product of kurtosis and variance $\kappa\, \sigma^2=0$ \cite{Tawfik:2013dba} and constant trace-anomaly $(\epsilon-3 p)/T^4=7/2$ \cite{Tawfik:2013eua}. As introduced in \cite{Tawfik:2004ss}, the third freeze-out conditions, which is characterized by constant $s/T^3$ is accompanied by constant $s/n$.
Our HRG model was used to study the possible differences between the behavior of light
$\langle\bar{q}q\rangle=\langle\bar{u}u\rangle=\langle\bar{d}d\rangle$ and strange $\langle\bar{s}s\rangle$ quark-antiquark condensates in hadron phase. The contribution to the pressure due to a particle of mass $m_h$, baryon charge $B$, isospin $I_3$, strangeness $S$, and degeneracy $g$ is given by
\begin{eqnarray}
\Delta p=\frac{g\, m_h^2\, T^2}{2\pi^2} \, \sum_{n=1}^\infty \,\frac{(-\eta)^{n+1}}{n^2} \,\exp\left(n\frac{B\mu_B - I_3\mu_I - S \mu_S}{T} \right) \, K_2\left(n\frac{m_h}{T} \right), \label{p1}
\end{eqnarray}
where $K_n(x)$ is the modified Bessel function. In hadrons, the isospin is an almost exact symmetry.
The quark-antiquark condensates are given by the derivative of Eq. (\ref{p1}) with respect to the constituent quark masses
\begin{eqnarray}
\langle\bar{q}q\rangle &=&\langle\bar{q}q\rangle_0+ \sum_h \frac{\partial m_h}{\partial m_q}
\frac{\partial \Delta p}{\partial m_h}, \nonumber \\
\langle\bar{s}s\rangle &=&\langle\bar{s}s\rangle_0+ \sum_h \frac{\partial m_h}{\partial m_s} \frac{\partial \Delta
p}{\partial m_h}, \label{qqHRG}
\end{eqnarray}
where $\langle\bar{q}q\rangle_0$ and $\langle\bar{s}s\rangle_0$ are light and strange quark-antiquark condensates in vacuum, respectively \cite{Tawfik:2005qh}. It was found that at small chemical potential the strange quark-antiquark condensate is larger that the light one. At large chemical potential, such difference gradually diminishes.
Some authors still prefer to take into account repulsive ({\it electromagnetic}) van der Waals interactions in order to compensate the strong interactions in hadron matter \cite{Tawfik:2013eua}. Accordingly, each resonance constituent is allowed to have an {\it eigen}-volume. Thus, such total volume should be subtracted from the fireball volume or that of the heat bath. Also, considerable modifications in thermodynamics of hadron gas including energy, entropy and number densities are likely. The hard-core radius of hadron nuclei can be related to the multiplicity fluctuations.
About ten years ago, Tawfik derived $S$-matrix for the HRG model \cite{Tawfik:2004sw}, which describes the scattering processes in the thermodynamical system~\cite{Dashen:1969mb}. Accordingly, Eq.~(\ref{eq:lnz1}) can be written as an expansion of the fugacity term
\begin{eqnarray}
\ln\, Z^{(int)}(V,T,\mu) &=& \ln\, Z^{(id)}(V,T,\mu) + \sum_{\nu=2}^{\infty} a_{\nu}(T) \exp(\mu_{\nu}/T). \label{eq:p2}
\end{eqnarray}
where $a_{\nu}(T)$ are the virial coefficients and the subscript $\nu$ refers to the order of multiple-particle interactions.
\begin{eqnarray}
a_{\nu}(T) &=& \frac{g_r}{2\pi^3} \int_{M_{\nu}}^{\infty}dw\; \exp\{-\varepsilon_r(w)/T\}\;
\sum_l(2l+1)\frac{\partial}{\partial w}\delta_l(w). \label{eq:p3}
\end{eqnarray}
The sum runs over the spatial waves. The phase shift $\delta_l(w)$ of two-body inelastic interactions, for instance, depends on the resonance half-width $\Gamma_r$, spin and mass of produced resonances,
\begin{eqnarray}
\ln\, Z^{(int)}(V,T,\mu) &=& \ln\, Z^{(id)}(V,T,\mu) + \frac{g_r}{2\pi^3}\int_{M_{\nu}}^{\infty}dw \frac{\Gamma_r\; \exp\{(-\varepsilon_r(w)+\mu_r)/T\}}{(M_r-w)^2+\left(\frac{\Gamma_r}{2}\right)^2}. \label{eq:p4}
\end{eqnarray}
In Eq.~(\ref{eq:p4}), by replacing $\mu$ by $-\mu$, the anti-particles are taken into consideration. For a narrow width and/or at low temperature, the virial term decreases so that the {\it non-relativistic} ideal partition function of hadron resonances with effective masses $M_{\nu}$ is obtained. This means that, the resonance contributions to the partition function are the same as that of massive {\it free} resonances. At temperatures comparable to $\Gamma_r$, the effective mass approaches the physical one. Thus, we conclude that at high temperatures, the strong interactions are taken into consideration via heavy resonances, Eq.~(\ref{eq:lnz1}), i.e. Hagedorn picture. We therefore utilise the grand canonical partition function, Eq.~(\ref{eq:p2}), without any corrections.
In order to verify this picture, Tawfik checked the ability of HRG with finite-volumed constituents in reproducing lattice QCD thermodynamics \cite{Tawfik:2013eua}. At radius $r>0.2~$fm, the disagreement becomes obvious and increases with increasing $r$. At high temperatures, the resulting thermodynamics becomes {\it non}-physical. It was concluded that the excluded volume seems to be practically irrelevant. It has e negligible effect, at $r\leq 0.2~$fm. On the other hand, a remarkable deviation from the lattice QCD calculations appears, especially when the radius $r$ become large.
In the present work, the chiral parameters, $M(T)$ and $\Delta_ {q,s}(T)$, see section \ref{sec:Results}, are extracted from HRG assuming fully and partially chemical non-equilibrium \cite{Tawfik:2014dha}. There is no difference, when $\gamma_S=1.0$ and when it is allowed to have values different than unity, where $\gamma_q$ and $\gamma_S$ refer to non-equilibrium treatment or occupation factors for light and strange quarks, respectively. These two parameters enter Eq. (\ref{eq:lnz1}) after raising them to exponents reflecting the light and strange quarks contents of $i$-th hadron. They are identical to the fugacity factor and therefore are multiplied to the exponential function.
\section{Results}
\label{sec:Results}
A systematic comparison between PLSM, PNJL, QLSM and HRG is presented. It intends to calculate two chiral quantities, the order parameter $M(T)$ and the normalized net strange non-strange condensate $\Delta_{q,s}(T)$. The results shall be confronted to the lattice QCD simulations \citep{LQCD1,Schmidt:2010ss,Borsanyi:2010zi}. The comparison with the lattice should signal which model is close to the lattice and on other hand offers differentiation between the SU(3) effective models, themselves.
\subsection{Chiral order-parameter $M(T)$}
\label{subsec:Results1}
\begin{figure}[htb]
\includegraphics[width=8.cm,angle=-90]{PLSM-M.eps}
\label{sec:QCDcomp}
\caption{The thermal behaviour of the dimensionless chiral order-parameter, $M$, calculated as function of temperature from the four SU(3) effective models, PLSM (solid curve), QLSM (dotted curve), PNJL (double-dotted curve) and HRG (dash-double-dotted curve) and compared with the lattice QCD calculations (solid circles) \cite{LQCD1} and (open circles) \cite{Schmidt:2010ss} at $m_l/m_s=0.037$.
\label{fig:M}
}
\end{figure}
The chiral order-parameter $M(T)$ was originated in lattice techniques \cite{Mqcd}. The latter calculates dimensionless quantities in units of lattice spacing rather than physical units. The lattice spacing can then be converted into the physical units. $M(T)$ relates the light quark condensate to the strange quark mass
\begin{eqnarray}
M(T) &=& m_s\, \dfrac{\langle\bar{\psi}\psi\rangle_{l}}{T^{4}}, \label{Eq:M}
\end{eqnarray}
where $m_s$ is the strange quark physical mass which fixed here to $138~MeV$ in order to get the ratio of light and strange quark masses $m_l/m_s=0.037$. Also, we notice that the dimensionless $M(T)$ depends on the thermal behavior of the light quark condensate $\langle\bar{\psi}\, \psi_{l}\rangle$. In lattice QCD, the chiral condensate remains finite. But it contains contributions which would diverge in the continuum limit. Therefore, it requires renormalization, in particular an additive and multiplicative renormalization. In order to remove - at least - the multiplicative renomalization factor, we take into consideration Eq. (\ref{Eq:M}) as a definition for the order parameter. The light quark condensate itself can be calculated from the potential, PLSM: Eq. (\ref{plsm:omega}),
QLSM: Eq. (\ref{Pq}),
PNJL: Eq. (\ref{PNJLpotuntial}) and
HRG: Eq. (\ref{qqHRG}).
Accordingly, we estimate Eq. (\ref{Eq:M}) from the four models and then compare them with the lattice QCD calculations, Fig. \ref{fig:M}. We find that this chiral order-parameter in the SU(3) effective models and first-principle lattice QCD simulations \cite{LQCD1,Schmidt:2010ss} rapidly decreases with increasing $T$.
Comparing with the lattice QCD \cite{LQCD1}, the best agreement is found with PLSM, but PLSM underestimates the recent lattice QCD \cite{Schmidt:2010ss}. In fact, the lattice calculations \cite{Schmidt:2010ss} lay on top of all curved from the SU(3) effective models. This might be originated to the specific configurations of the lattices and the actions implemented in each simulations, section \ref{sec:QCDcomp}. The other effective models lay below the two sets of lattice calculations.
The four models PLSM, QLSM, PNJL, HRG and different sets of lattice calculations have different critical temperatures. In Tab. \ref{tab:1}, we list out the critical temperatures corresponding to each order parameter. In determining the pseudocritical temperatures, $T_{\chi}$, different criteria are implemented. They are not only quite unorthodox but they are distinguishable from each other. Further details shall be elaborated in Section \ref{subsec:Results2}.
\begin{table}
\begin{center}
\begin{tabular}{||c||c|c||}
\hline\hline
& $T_{\chi}$ [MeV] & Order Parameter \\ \hline\hline
PLSM & $164$ & crossing of ($\sigma_x$, $\sigma_y$) and ($\phi$, $\bar{\phi}$) \\ \hline
QLSM & $200$ & ($\sigma_x$, $\sigma_y$) and largest fluctuation in $m_2/\mu^2$ \\ \hline
PNJL & $217$ & crossing of ($\sigma_x$, $\sigma_y$) and ($\phi$, $\bar{\phi}$) \\ \hline
HRG & $184$ & vanishing $\langle \bar{\psi}\psi\rangle$-condensate \\ \hline
LQCD \cite{LQCD1} & $156$ at $m_s/m_l=0.037$ & sudden drop in $M(T)$ and $\Delta_{l,s}(T)$ \\ \hline
LQCD \cite{Schmidt:2010ss} & $165-170$ & sudden drop in $M(T)$ and $\Delta_{l,s}(T)$ \\ \hline
LQCD \cite{Borsanyi:2010zi} & $165$ & sudden drop in $M(T)$ and $\Delta_{l,s}(T)$ \\ \hline
\end{tabular}
\caption{The pseudocritical temperatures $T_{\chi}$ as calculated from PLSM, QLSM, PNJL, HRG and the different sets of lattice QCD calculations. \label{tab:1} }
\end{center}
\end{table}
\subsubsection{A short comparison between the two sets of lattice QCD calculations}
\label{sec:compLQCD}
Refs. \cite{LQCD1,Schmidt:2010ss} presented results for $2+1$ quark flavors, where all systematics are controlled, the quark masses are set to their physical values and the continuum extrapolation is carried out. Larger lattices and a Symanzik improved gauge besides a stout-link improved staggered fermion action are implemented. Depending on the exact definition of the observables, the remnant of the chiral transition is obtained at $T_c=150$~MeV. Extending these results, the transition temperature was also determined for small non-vanishing baryonic chemical potentials. At high temperatures, the lattice pressure is found $\sim 30\%$ lower than the Stefan-Boltzmann limit.
Ref. \cite{Borsanyi:2010zi} used $2+1$ quark flavors with physical strange quark mass and almost physical light quark masses. The calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, a good agreement between results obtained with these two $O(a^2)$ improved staggered fermion discretization schemes is found. At high temperatures, the lattice pressure is $\sim 14\%$ lower than the Stefan-Boltzmann limit.
From this short comparison, we find that:
\begin{itemize}
\item \cite{LQCD1,Schmidt:2010ss} implement Symanzik improved gauge and stout-link improved staggered fermion action. The resulting pressure is found $\sim 30\%$ lower than the Stefan-Boltzmann limit.
\item \cite{Borsanyi:2010zi} uses improved staggered fermion actions; the asqtad and p4 actions. The resulting pressure is $\sim 14\%$ lower than the Stefan-Boltzmann limit.
\end{itemize}
\subsubsection{Couplings in PLSM}
In the effective models, the parameters, especially the couplings, are very crucial for the outcome of the calculations. One of the motivations for the present work is the failure of PLSM \cite{Tawfik:2014uka} in reproducing the lattice QCD results \cite{LQCD1,Schmidt:2010ss,Borsanyi:2010zi} even with large coupling $g$. In Ref. \cite{Tawfik:2014uka}, $g$ ranges between $6.5$ and $10.5$. The first value was enough to reproduce the lattice QCD calculations, PRD80, 014504 (2009) and PLB730, 99 (2014). Increasing $g$ to $10.5$ does not enable PLSM to reproduce the other lattice simulations \cite{LQCD1,Schmidt:2010ss,Borsanyi:2010zi}. Furthermore, through fitting with lattice QCD calculations and experiments, the parameters of PLSM can be estimated. This was described in details in Ref. \cite{Tawfik:2014uka,Schaefer:2008hk}.
Scope of the present script is the regeneration for the lattice QCD calculations \cite{LQCD1,Schmidt:2010ss,Borsanyi:2010zi}. In the present work, we tackle this problem through comparison with various effective models. In doing this, we have modified LSM and present systematic analysis for two order parameters. We have added to LSM the gluonic sector of the quasi-particle model. This is the essential original proposal of the present script. Thus, waiving details about PLSM itself is though as legitimated. But for a complete list of the PLSM parameters, the readers are kindly advised to consult \cite{Tawfik:2014uka,Schaefer:2008hk}.
\subsection{Normalized net-strange condensate $\Delta_{q,s}(T)$}
\begin{figure}[htb]
\includegraphics[width=8cm,angle=-90]{PLSM-delta.eps}
\caption{The thermal dependence of $\Delta_{q,s}$ calculated from PLSM (solid curve), QLSM (dotted curve), PNJL (double-dotted curve) and HRG (dash-double-dotted curve) and compared with the lattice QCD calculations (solid circles) \cite{LQCD1} and (open circles) \citep{Borsanyi:2010zi} at $m_l/m_s=0.037$. \label{fig:Delta}
}
\end{figure}
Another dimensionless quantity shows the difference between non-strange and strange condensates.
\begin{eqnarray}
\Delta_ {q,s}(T) &=& \dfrac{\langle\bar{q}q\rangle - \dfrac{m_q}{m_s} \langle\bar{s}s\rangle}{\langle\bar{q}q\rangle_{0} - \dfrac{m_q}{m_s} \langle\bar{s}s\rangle_{0}},\label{Eq:Delta}
\end{eqnarray}
where $\langle\bar{q}q\rangle$ ($\langle\bar{s}s\rangle$) are non-strange (strange) condensates, and $m_q$ ($m_s$) are non-strange (strange) masses. Using Ward identities and Gell-Mann-Oakes-Renner relation, expression (\ref{Eq:Delta}) might be given in terms of pion and kaon masses and their decay constants \cite{GMOR}. Accordingly, the final results might be scaled but their thermal behavior remains unchanged.
The lattice QCD calculations for $\Delta_{q,s}(T)$ (solid circles) \cite{LQCD1} and (open circles) \citep{Borsanyi:2010zi} are compared with the calculations from PLSM (solid curve), QLSM (dotted curve), PNJL (double-dotted curve) and HRG (dash-double-dotted curve) in Fig. \ref{fig:Delta}.
It is obvious that PLSM agrees with the lattice results \cite{Borsanyi:2010zi} at low and also at high temperatures. Its ability to reproduce the other set of lattice results \cite{Schmidt:2010ss} is limited to the high temperatures. This might be originated in the difference between the two sets of lattice QCD simulations, section \ref{sec:compLQCD}. The HRG model agrees well with this lattice calculations \cite{Schmidt:2010ss}. It is apparent that such agreement is limited to temperatures below the critical value due to the limited applicability of the HRG model. The remaining two models PNJL and QLSM show qualitative thermal behavior, as that from the other effective models and lattice calculations, which can be described by large plateau at low temperatures, around the critical temperature the values of $\Delta_{q,s}(T)$ decrease rapidly and at high temperature, $\Delta_{q,s}(T)$ vanishes but very slowly. Both models are closer to \cite{Borsanyi:2010zi} rather than to \cite{Schmidt:2010ss}
Both Figs. \ref{fig:M} and \ref{fig:Delta} show the PNJL model and HRG model describe much better the LQCD data for the magnetization and normalized net strange condensate, respectively, than for the chiral condensate. One should bear in mind that the magnetizations have been simulated in a different lattice that the one for the net strange condensate. Unfortunately, both quantities are not available from the same lattice simulation.
\subsection{QCD chiral phase-diagram}
\label{subsec:Results2}
For mapping out the QCD chiral phase-diagram, various approaches are available. From PLSM and PNJL, as they possess two order parameters; one for strange and one for non-strange chiral condensates, hints about QCD chiral phase transition can be analysed. Furthermore, PLSM and PNJL possess deconfinement order parameter because of the Polyakov loop potential. Therefore, from strange and non-strange chiral condensates, a dimensionless quantity reflecting the difference between both condensates, $\Delta_{q,s}(T)$, can be deduced as function of temperature at fixed baryon chemical potential. Apparently, this signals the QCD chiral phase-transition. At the same value of baryon chemical potential, we can also deduce the deconfinement order-parameter as function of temperature. At a fixed baryon chemical potential, the thermal dependence of these two quantities intersect with each other at a characterizing point representing the phase transition. When repeating this procedure at different values of the baryon chemical potentials, we get a set of points representing the QCD phase-diagram. The results are given in Fig. \ref{fig:C-PD}, as solid curve for PLSM and dotted curve for PNJL.
For the QCD chiral phase-diagram from QLSM, we implement another method. As no Polyakov loop potential is included, the QCD chiral phase-diagram is characterized by the higher-order moments of particle multiplicity \cite{Tawfik:2014bna}, which are assumed to highlight various types of fluctuations in $T$ and $\mu$. Therefore, we utilize the possible fluctuations accompanying normalized second-order moment \cite{Tawfik:2014bna} in mapping out the QCD chiral phase-transition. The problematic of determining pseudocritical temperature from the second moment has been discussed in Ref. \cite{Karsch2009a}. Accordingly, we observe that the peaks corresponding to different temperatures are conjectured to be characterized by different values of the baryon chemical potentials, where the QCD chiral phase-transition is conjectured to occur. We analyse this dependence at different values of the temperature $T$. Then, we follow the scheme to determine $T$ and $\mu$, which is characterized by maximum $m_2/\mu^2$, where $m_2$ is the second-order moment of the particle multiplicity. The results are illustrated in Fig. \ref{fig:C-PD}, as dash-dotted curve.
For the HRG model, we map out the QCD chiral phase-diagram by utilizing the quark-antiquark condensate as order parameter \cite{Tawfik:2005qh}. It is assumed that the thermal dependence of the quark-antiquark condensate remains finite in the hadronic phase, but vanishes at temperature higher than the critical chiral-temperature. The results are given in Fig. \ref{fig:C-PD}, as well, as double-dotted curve.
We can now shortly summarize the methods implemented to determine the pseuodcritical temperatures:
\begin{itemize}
\item PLSM and PNJL: due to chiral and deconfinement phase transitions for light and strange quarks, $\Delta_{q,s}(T)$ is determined as function of temperature at a fixed baryon chemical potential, $\mu$. This signals the QCD chiral phase-transition. At the same value of $\mu$, the deconfinement order-parameter can be studied as function of temperature, as well. Then, the thermal dependence of these two quantities is conjectured to intersect with each other at a characterizing point. When repeating this procedure for different values of $\mu$, a set of points of pseudocritical temperatures $T_{\chi}$ and $\mu$ can be deduced.
\item QLSM: the normalized second-order moment of particle multiplicity is implemented in mapping out the QCD chiral phase-transition. Peaks corresponding to different temperatures are conjectured to be characterized by different values of $T_{\chi}$ and $\mu$.
\item HRG: the quark-antiquark condensates are implemented as order parameters. At vanishing and finite $\mu$, the thermal dependence of the quark-antiquark condensate remains finite in the hadronic phase and vanishes at temperature higher than the critical chiral-temperature, $T_{\chi}$.
\end{itemize}
We observe that the chiral boundary from PLSM (solid curve) is positioned within the upper band of the lattice QCD calculations \cite{KarschA,LQCDA} and agrees well with the freeze-out results deduced from the STAR BES measurements (symbols) \cite{Tawfik:2013bza}. The temperatures calculated from the HRG model by using the quark-antiquark condensate as the order parameter (double-dotted curve) \cite{Tawfik:2005qh} is higher than the chiral temperatures from the PLSM and the freeze-out temperatures calculated in the lattice QCD (band) and from the STAR BES measurements (symbols). Despite this difference, the corresponding $T$-$\mu$ sets are very similar to that of the PLSM. The results from PNJL and QLSM are higher than that from the HRG model.
\begin{figure}[htb]
\includegraphics[width=8cm,angle=-90]{ChiralPT_LSM_NJL_LQCD1.eps}
\caption{The PLSM $T$-$\mu$ chiral phase-diagram (lines with points), with which the freeze-out parameters deduced from lattice the QCD calculations \cite{KarschA,LQCDA} (band) and that from different the thermal models \cite{Tawfik:2013bza,SHMUrQM} (symbols) are compared.
\label{fig:C-PD}}
\end{figure}
\section{Conclusions and outlook}
\label{sec:conclusion}
In the present work, we report on a systematic comparison between PLSM, PNJL, QLSM and HRG in generating the chiral quantities, order parameter $M(T)$ and normalized net strange and non-strange condensates $\Delta_{q,s}(T)$. Furthermore, we confront the results deduced from the four effective models to the recent lattice QCD calculations in order to distinguish between the models and to interpret the first-principle lattice QCD calculations.
For the order parameter $M(T)$, the best agreement is found with PLSM, while the recent lattice QCD \cite{Schmidt:2010ss} lay on top of all curves. This might be understood from the lattice configurations and the actions implemented in the simulations. The other effective models lay below the two sets of the lattice calculations. We notice that the effective PLSM, QLSM, PNJL and HRG and the different sets of the lattice calculations have different critical temperatures, Tab. \ref{tab:1}.
For the normalized net strange and non-strange condensates $\Delta_{q,s}(T)$, PLSM again gives an excellent agreement with the lattice results \cite{Borsanyi:2010zi} at low and high temperatures. But its ability to reproduce the lattice simulations \cite{Schmidt:2010ss} is limited to high temperatures. This might be originated in the difference between the two sets of lattice QCD simulations. Furthermore, we find that the HRG model agrees well with this the lattice QCD calculations \cite{Schmidt:2010ss}. It is apparent that this is restricted to temperatures below the critical value. The effective models PNJL and QLSM show the same qualitative thermal behavior. There is a large plateau at low temperatures. Around the critical temperature the values of $\Delta_{q,s}(T)$ decrease, rapidly. At high temperature, $\Delta_{q,s}(T)$ vanishes but very slowly. The effective models PNJL and QLSM are closer to \cite{Borsanyi:2010zi} rather than to \cite{Schmidt:2010ss}.
In light of this, we conclude that the PLSM reproduces $M(T)$ and $\Delta_{q,s}(T)$, well. The HRG model is able to reproduce $\Delta_{q,s}(T)$, while PNJL and QLSM seem to fail. These features and differences are present in the chiral phase-diagram, Fig. \ref{fig:C-PD}, as well.
In section \ref{subsec:Results2}, we have introduced the various order parameters used in the different models in order to deduce $T$ and $\mu$ of the QCD chiral phase-transition. The strange and non-strange chiral condensates and the Polyakov loop potentials are utilized in PLSM and PNJL. The thermal dependence of these two quantities are assumed to intersect with each other at a characterizing point representing the QCD chiral phase-transition. For QLSM, no Polyakov loop potential is included in, therefore, the chiral phase-diagram is characterized by the higher-order moments of the particle multiplicity. The possible fluctuations accompanying the normalized second-order moment are assumed to map out the QCD chiral phase-transition. For the HRG model, we utilize the quark-antiquark condensates as order parameter.
Again, we find that the PLSM chiral boundary (solid curve) is located within the upper band of the lattice QCD calculations and agrees well with the freeze-out results deduced from the experiments and the thermal models (symbols). It is obvious that the chiral temperature calculated from the HRG model is larger than that from the PLSM. This is also larger than the freeze-out temperatures calculated in the lattice QCD (band) and from the experiments and the thermal models (symbols). Despite this difference, the corresponding $T$ and $\mu$ sets are very similar to that from the PLSM. This might be explained as follows. The $T$ and $\mu$ are calculated using different order parameters; in HRG vanishing quark-antiquark condensate but in the PLSM crossing (equalling) the chiral condensates and the Polyakov loop potential. The latter assumes that the two phase transitions; the chiral and the deconfinement, occur at the same temperature. The earlier deals with the chiral phase-transition independent on the confinement-deconfinement one.
The results from the two model PNJL and QLSM show the same qualitative behavior. The chiral temperatures are higher than that from the PLSM and HRG. This might be interpreted due the heavy quark masses implemented in both models.
Any model comparison with lattice results should span as much as possible of the parameter space. Even with the narrow parameter space explored in the present paper, we would like to highlight that the results are limited this. But, with reference to previous work \cite{Tawfik:2014uka}, the parameters alone are not able to explain the diversity with the results in this study. We have to attack essential components of LSM and integrate gluonic sector taken from the quasi-particle models.
\section*{Acknowledgement}
The present work was supported by the World Laboratory for Cosmology And Particle Physics (WLCAPP) http://wlcapp.net/.
The authors are very grateful to the anonymous referee for his/her very constructive comments, suggestions and even criticisms, which helped a lot in improving the manuscript!
|
2,869,038,155,661 | arxiv | \section*{Abstract}
{\bf
Universal quantum gates lie at the heart of designing quantum computer. We construct two compact quantum circuits to implement post-selected controlled-phase-flip (CPF) gate and Toffoli gate with linear optics assisted by one and two single photons, respectively. The current existing maximum success probability of 1/4 for linear optical CPF gate is achieved by resorting to an ancillary single photon rather than an entangled photon pair or two single photons. Remarkably, our Toffoli gate is accomplished with current maximum success probability of 1/30 without using additional entangled photon pairs and the standard decomposition-based approach. Linear optical implementations of the presented two universal gates are feasible under current technology and provide a potential application in large-scale optical quantum computing.
}
\vspace{10pt}
\noindent\rule{\textwidth}{1pt}
\tableofcontents\thispagestyle{fancy}
\noindent\rule{\textwidth}{1pt}
\vspace{10pt}
\section{Introduction}\label{sec1}
Quantum computing \cite{book} has the remarkable potential to dramatically surpass its classical counterpart on solving certain complex tasks in terms of the
processing speed or resource overhead. Universal quantum gates are crucial building blocks in quantum circuit model \cite{Barenco,circuits,Fredkin-Liu1,Fredkin-Liu2,Nikolaeva}, quantum algorithms \cite{Grover,Shor,Bharti}, quantum simulations \cite{Georgescu}, and quantum communication \cite{Pan2}. Photon is generally viewed as one of the promising candidates for flying and solid-state quantum computing owing to its outstanding low decoherence, high-speed transmission, natural information carrier, flexible single-qubit manipulations, and available atom-like qubit interconnector \cite{computing0,computing}. Strong interactions between individual photons are the key resources for nontrivial multi-photon quantum gates, and the prohibited photon-photon interactions can be remedied efficiently by using linear optics \cite{KLM} or solid-state media \cite{QD,NV,atom}. Unfortunately, solid-state platforms are challenged by inefficiency and imperfection in experiments. The probabilistic character of universal quantum gates with linear optics is unavoidable. Hence, minimizing the quantum resources required to implement quantum gates with higher success probability is a central problem of linear optical quantum computing, and tremendous efforts have been made on it \cite{CNOT-KLM,CNOT-Bell-1/4-1,CNOT-Bell-1/4-2,CNOT-Bell-1/4-3,CNOT-Bell-1/4-4,CNOT-Bell-1/4-5,CNOT-1/9-2,CNOT-1/9-4,CNOT-1/9-5,CNOT-1/9-3,CNOT-1/9-6,CNOT-1/9-7,CNOT-1/8-1,CNOT-1/8-2,CNOT-1/16,CNOT-SR}.
Controlled phase flip (CPF) gate or its equivalent controlled-NOT (CNOT) gate is the most quintessential universal quantum gate \cite{book}. CNOT gates together with single-qubit rotations are sufficient to implement any quantum computation \cite{Barenco}. Nowadays, CNOT gate has been experimentally demonstrated in several physical systems \cite{CNOT-superconducting,CNOT-atom,CNOT-ion}.
The KLM scheme \cite{KLM} is served as a stepping stone for implementing CPF gate with a sheer number of linear optics, large and good quantum memory, and giant interferometer phase stable. Various improved works were later proposed both in theory and experiment \cite{CNOT-KLM,CNOT-Bell-1/4-1,CNOT-Bell-1/4-2,CNOT-Bell-1/4-3,CNOT-Bell-1/4-4,CNOT-Bell-1/4-5,CNOT-1/9-2,CNOT-1/9-4,CNOT-1/9-5,CNOT-1/9-3,CNOT-1/9-6,CNOT-1/9-7,CNOT-1/8-1,CNOT-1/8-2,CNOT-1/16,CNOT-SR}.
So far, it has been demonstrated that CPF gate can be completed with a success probability of 1/9, which is the existing maximum value achievable without ancillary photons \cite{CNOT-1/9-2,CNOT-1/9-4,CNOT-1/9-5,CNOT-1/9-3,CNOT-1/9-6,CNOT-1/9-7},
and the success probability can be improved to 1/8 via two additional independent single photons \cite{CNOT-1/8-1,CNOT-1/8-2}.
The current existing highest success probability of 1/4 for a CNOT gate has been achieved assisted by a necessary entangled photon pair \cite{CNOT-Bell-1/4-1,CNOT-Bell-1/4-2,CNOT-Bell-1/4-3,CNOT-Bell-1/4-4,CNOT-Bell-1/4-5}. Deterministic generation of entangled photon pairs based on spontaneous parametric down-conversion remains a key technical obstacle in experiments due to multi-photon emissions and probabilistic properties \cite{SPDC}.
Toffoli gate supplemented with Hadamard gate can simulate any multi-qubit quantum computing \cite{book}. Toffoli gate is also served as an essential part in quantum factoring algorithm \cite{Shor1}, quantum search algorithm \cite{Grover1}, quantum half-adder \cite{adder}, quantum error correction \cite{correction1}, and quantum fault tolerance \cite{tolerant1}, etc. Much attention has been paid to the realization of Toffoli gate \cite{Toffoli-ion,Toffoli-superconducting,Toffoli-atom}.
It has been confirmed theoretically that the optimal cost of a Toffoli gate is six CNOT gates \cite{Barenco} or five two-qubit entangling gates \cite{five}. Such synthesis might be helpful to design complex quantum gate, but it makes the gate further susceptible to the environmental noise and increases the time scale of the system.
Without using the standard decomposition-based approach, early in 2006, Fiur\'{a}\v{s}ek \cite{T-Fiurasek} first showed a three-photon polarization Toffoli gate with a success probability of 0.75\% (approximately 1/133) using linear optics. Using higher-dimensional Hilbert spaces, Ralph \emph{et al.} \cite{T-PRA} improved the success probability of a linear optical Toffoli gate to 1/72 in 2007, and this interesting probabilistic scheme was later experimentally demonstrated in 2009 \cite{T-NatPhy}. Recently, Liu \emph{et al.} \cite{T-Liu} further enhanced the success probability of the Toffoli gate to 1/64. Additionally, in 2022, Li \emph{et al.} \cite{T-path} experimentally demonstrated a path-based three-photon Toffoli gate with a success probability of 1/72. Many hybrid multiple degrees of freedom (DOFs) probabilistic and deterministic Toffoli gates were also reported in recent years \cite{hybrid1,hybrid2,hybrid3,hybrid4,hybrid5}.
In this paper, we propose two compact quantum circuits to implement post-selected CPF gate and Toffoli gate in the coincidence basis using solely polarizing beam splitters (PBSs), half-wave plates (HWPs), beam splitters (BSs), and single-photon detectors.
Assisted by one and two independent single photons, our CPF and Toffoli gates are accomplished respectively when exactly one photon appears in each output mode.
Our schemes are appealing for higher success probabilities and less quantum resource requirements.
The existing highest success probability of a linear optical CPF gate 1/4 is achieved resorting to an auxiliary single photon in our scheme rather than an auxiliary entangled photon pair \cite{CNOT-Bell-1/4-1,CNOT-Bell-1/4-2,CNOT-Bell-1/4-3,CNOT-Bell-1/4-4,CNOT-Bell-1/4-5}.
The presented CPF gate also beats the ones with the success probability of 1/8 assisted by two single photons \cite{CNOT-1/8-1,CNOT-1/8-2}, and the ones with 1/9 without auxiliary photons \cite{CNOT-1/9-2,CNOT-1/9-4,CNOT-1/9-5,CNOT-1/9-3,CNOT-1/9-6,CNOT-1/9-7}.
In addition, the average success probability of our Toffoli gate is high to 1/30, which far exceeds all previous results for the same works \cite{T-Fiurasek,T-PRA,T-NatPhy,T-Liu}.
\section{Post-selected CPF gate with linear optics} \label{sec2}
It is well-known that CPF gate introduces a $\pi$ phase shift when the first qubit and the second qubit are both $|1\rangle$, and the rest remains unchanged. We encode the gate qubit in two polarization DOFs of a single photon, i.e., the horizontally polarized photon $|H\rangle=|0\rangle$ and vertically polarized photon $|V\rangle=|1\rangle$, respectively.
\begin{figure} [!h]
\begin{center}
\includegraphics[width=9.5 cm,angle=0]{CPF.eps}
\caption{ Schematic diagram of a post-selected CPF gate. The gate is completed in case each spatial modes 1, 3 and 4 contain exactly one photon. PBS, a polarizing beam splitter, transmits the $H$-polarized photon and reflects the $V$-polarized photon. HWP$^{22.5^\circ}$, a half-wave plate setting at $22.5^\circ$, results in $|H\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$ and
$|V\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle)$. $D_{H_3}$ and $D_{V_3}$ stand for the single photon detectors. Feed-forward operation $\sigma_z=|H\rangle \langle H| -|V\rangle \langle V|$ is applied when $D_{V_3}$ is triggered. } \label{CPF}
\end{center}
\end{figure}
The scheme described in Fig. \ref{CPF} can complete a polarization-based CPF gate in the following three steps.
First, the two gate photons and one auxiliary photon in the states
\begin{eqnarray} \label{eq1}
\begin{split}
|\phi\rangle_{c_{in}}=\alpha_1|H\rangle_{c_{in}}+\beta_1|V\rangle_{c_{in}},
\end{split}
\end{eqnarray}
\begin{eqnarray} \label{eq2}
\begin{split}
|\phi\rangle_{t_{in}}=\alpha_2|H\rangle_{t_{in}}+\beta_2|V\rangle_{t_{in}},
\end{split}
\end{eqnarray}
\begin{eqnarray} \label{eq3}
\begin{split}
|\phi\rangle_a=\frac{1}{\sqrt{2}}(|H\rangle_a+|V\rangle_a),
\end{split}
\end{eqnarray}
are injected into the spatial modes $c_{in}$, $t_{in}$, and $a$, respectively. Here coefficients $\alpha_1$, $\beta_1$, $\alpha_2$, $\beta_2$ satisfy the conditions $|\alpha_1|^2+|\beta_1|^2=1$ and $|\alpha_2|^2+|\beta_2|^2=1$. The subscripts denote the spatial modes of photons (also named photon's paths).
The photons emitted from spatial modes $c_{in}$ and $a$ are fed into PBS$_1$, simultaneously. PBS$_1$ transforms the state of the whole system from $|\Phi_1\rangle =|\phi\rangle_{c_{in}}\otimes|\phi\rangle_a\otimes|\phi\rangle_{t_{in}}$ to
\begin{eqnarray} \label{eq4}
\begin{split}
|\Phi_2\rangle =&\frac{1}{\sqrt{2}} \big(\alpha_1|H\rangle_1|H\rangle_2+\beta_1|H\rangle_1|V\rangle_1+\alpha_1|H\rangle_2|V\rangle_2+\beta_1|V\rangle_1|V\rangle_2\big)\\
&\otimes\big(\alpha_2|H\rangle_{t_{in}}+\beta_2|V\rangle_{t_{in}}\big).
\end{split}
\end{eqnarray}
Based on Eq. (\ref{eq4}), one can see that PBS$_1$ can complete a parity-check measurement on the polarization photons by choosing the instance in which each of the spatial mode contains exactly one photon in post-selection principle, and then the system would be changed into the state
\begin{eqnarray} \label{eq5}
\begin{split}
|\Phi_3\rangle =\big(\alpha_1|H\rangle_1|H\rangle_2+\beta_1|V\rangle_1|V\rangle_2 \big)
\otimes\big(\alpha_2|H\rangle_{t_{in}}+\beta_2|V\rangle_{t_{in}} \big),
\end{split}
\end{eqnarray}
with a probability of 1/2. While the instance in which each spatial mode involves two photons or none photon indicates the gate operation fails.
Second, as shown in Fig. \ref{CPF}, before and after the photons from modes 2 and $t_{in}$ pass through PBS$_2$ simultaneously, two polarization Hadamard operations are performed on them by using HWP$_1^{22.5^\circ}$ and HWP$_2^{22.5^\circ}$, respectively. Here half-wave plate oriented at 22.5$^\circ$ (HWP$^{22.5^\circ}$) completes the transformations
\begin{eqnarray} \label{eq6}
\begin{split}
&|H\rangle\xrightarrow{\text{HWP}^{22.5^\circ}}\frac{1}{\sqrt{2}}\big(|H\rangle+|V\rangle\big), \quad
&|V\rangle\xrightarrow{\text{HWP}^{22.5^\circ}}\frac{1}{\sqrt{2}}\big(|H\rangle-|V\rangle\big).
\end{split}
\end{eqnarray}
Operations $\text{HWP}_1^{22.5^\circ} \rightarrow \text{PBS}_2 \rightarrow \text{HWP}_2^{22.5^\circ}$ transform the state $|\Phi_3\rangle$ into
\begin{eqnarray} \label{eq7}
\begin{split}
|\Phi_4\rangle =&\frac{1}{2}\big( \alpha_1\alpha_2|H\rangle_1|H\rangle_4+\alpha_1\beta_2|H\rangle_1|V\rangle_4
+\beta_1\alpha_2|V\rangle_1|H\rangle_4-\beta_1\beta_2|V\rangle_1|V\rangle_4 \big)|H\rangle_3\\&
+\frac{1}{2}\big( \alpha_1\alpha_2|H\rangle_1|H\rangle_4-\alpha_1\beta_2|H\rangle_1|V\rangle_4
+\beta_1\alpha_2|V\rangle_1|H\rangle_4+\beta_1\beta_2|V\rangle_1|V\rangle_4\big)|V\rangle_3\\&
+\frac{1}{2\sqrt{2}} (\alpha_1\alpha_2|H\rangle_1-\beta_1\alpha_2|V\rangle_1)(|H\rangle_3+|V\rangle_3)(|H\rangle_3-|V\rangle_3)\\&
+\frac{1}{\sqrt{2}}\big( \alpha_1\beta_2|H\rangle_1+\beta_1\beta_2|V\rangle_1\big)|H\rangle_4|V\rangle_4.
\end{split}
\end{eqnarray}
We choose the case where exactly one photon in each of the spatial modes 3 and 4, and then the system would be in a normalization state
\begin{eqnarray} \label{eq8}
\begin{split}
|\Phi_5\rangle =&\frac{1}{\sqrt{2}}\big( \alpha_1\alpha_2|H\rangle_1|H\rangle_4+\alpha_1\beta_2|H\rangle_1|V\rangle_4
+\beta_1\alpha_2|V\rangle_1|H\rangle_4-\beta_1\beta_2|V\rangle_1|V\rangle_4 \big)|H\rangle_3\\&
+\frac{1}{\sqrt{2}}\big( \alpha_1\alpha_2|H\rangle_1|H\rangle_4-\alpha_1\beta_2|H\rangle_1|V\rangle_4
+\beta_1\alpha_2|V\rangle_1|H\rangle_4+\beta_1\beta_2|V\rangle_1|V\rangle_4\big)|V\rangle_3,
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.
Finally, the photon emitted from spatial mode $a$ will be detected by using PBS$_3$ and single-photon detectors $D_{H_3}$ and $D_{V_3}$.
Based on Eq. (\ref{eq8}), one can see that when $D_{H_3}$ is fired, the photons emitted from $c_{out}$ and $t_{out}$ kept are in the state
\begin{eqnarray} \label{eq9}
\begin{split}
|\Phi_6\rangle =\alpha_1\alpha_2|H\rangle_1|H\rangle_4+\alpha_1\beta_2|H\rangle_1|V\rangle_4
+\beta_1\alpha_2|V\rangle_1|H\rangle_4-\beta_1\beta_2|V\rangle_1|V\rangle_4,
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{4}\times\frac{1}{2}$. And then, the performance of CPF gate is completed.
When $D_{V_3}$ is fired, the system will collapse into the state
\begin{eqnarray} \label{eq10}
\begin{split}
|\Phi_6'\rangle =\alpha_1\alpha_2|H\rangle_1|H\rangle_4-\alpha_1\beta_2|H\rangle_1|V\rangle_4
+\beta_1\alpha_2|V\rangle_1|H\rangle_4+\beta_1\beta_2|V\rangle_1|V\rangle_4,
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{4}\times\frac{1}{2}$. It is easily to convert Eq. (\ref{eq10}) to Eq. (\ref{eq9}) by applying a feed-forward $\sigma_z$ operation on the photon emitted from spatial mode 4, which can be achieved by an $\text{HWP}^{0^\circ}$.
Putting all the pieces together, one can see that the quantum circuit shown in Fig. \ref{CPF} completed a CPF operation conditional on exactly one photon in each of the output spatial modes. The total success probability of the presented gate can reach the current existing best result $\frac{1}{8}+\frac{1}{8}=\frac{1}{4}$, and only one additional single photon is required.
\section{Post-selected Toffoli gate with linear optics} \label{sec3}
\begin{figure} [t
\begin{center}
\includegraphics[width=14.7 cm,angle=0]{Toffoli.eps}
\caption{Schematic diagram of a post-selected Toffoli gate. The gate works successfully conditioned on exactly one photon in each spatial modes 6, 12, $c_{1out}$, $c_{2out}$, and $t_{out}$. BS, a 50:50 beam splitter, results in $|H\rangle_5\rightarrow\frac{1}{\sqrt{2}}\big(|H\rangle_{5'}+|H\rangle_{5''}\big)$ and
$|V\rangle_5\rightarrow\frac{1}{\sqrt{2}}\big(|V\rangle_{5'}+|V\rangle_{5''}\big)$. HWP$^{67.5^\circ}$ completes the transformations $|H\rangle\rightarrow\frac{1}{\sqrt{2}}(-|H\rangle+|V\rangle)$ and
$|V\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$. Pauli operation $\sigma_x=|H\rangle \langle V|+ |V\rangle \langle H|$ is applied when $D_{V_6}$ or
$D_{H_{12}}$ is triggered, which is achieved by an HWP$^{45^\circ}$.} \label{Toffoli}
\end{center}
\end{figure}
Toffoli gate flips the state of the target qubit if the two controlled qubits both are in $|1\rangle$, and has no effect otherwise.
Figure \ref{Toffoli} depicts a scheme for implementing a Toffoli gate with an average success probability of 1/30 in the linear optical system.
Suppose two controlled photons, one target photon, and two auxiliary photons are initially prepared in the following states
\begin{eqnarray} \label{eq11}
\begin{split}
|\psi\rangle_{c_{1in}}=\alpha_1|H\rangle_{c_{1in}}+\beta_1|V\rangle_{c_{1in}},
\end{split}
\end{eqnarray}
\begin{eqnarray} \label{eq12}
\begin{split}
|\psi\rangle_{c_{2in}}=\alpha_2|H\rangle_{c_{2in}}+\beta_2|V\rangle_{c_{2in}},
\end{split}
\end{eqnarray}
\begin{eqnarray} \label{eq13}
\begin{split}
|\psi\rangle_{t_{in}}=\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}},
\end{split}
\end{eqnarray}
\begin{eqnarray} \label{eq14}
\begin{split}
|\psi\rangle_{a_1}=\frac{1}{\sqrt{2}}(|H\rangle_{a_1}+|V\rangle_{a_1}),
\end{split}
\end{eqnarray}
\begin{eqnarray} \label{eq15}
\begin{split}
|\psi\rangle_{a_2}=\frac{1}{\sqrt{2}}(|H\rangle_{a_2}+|V\rangle_{a_2}),
\end{split}
\end{eqnarray}
where $|\alpha_1|^2+|\beta_1|^2=1$, $|\alpha_2|^2+|\beta_2|^2=1$, and $|\alpha_3|^2+|\beta_3|^2=1$.
In the first step, we employ PBS$_1$ to complete the parity-check measurement on the first controlled photon (emitted from spatial $c_{1in}$) and the first additional photon (emitted from spatial $a_1$), and choose the instance in which each outing mode contains exactly one photon. And then, PBS$_1$ converts the whole system from the initial state $|\Psi_0\rangle =|\psi\rangle_{c_{1in}}\otimes|\psi\rangle_{a_1}\otimes|\psi\rangle_{c_{2in}}\otimes|\psi\rangle_{a_2}\otimes|\psi\rangle_{t_{in}}$ to
\begin{eqnarray} \label{eq16}
\begin{split}
|\Psi_1\rangle =&\frac{1}{\sqrt{2}} \big(\alpha_1|H\rangle_1|H\rangle_2+\beta_1|V\rangle_1|V\rangle_2 \big)\otimes\big(\alpha_2|H\rangle_{c_{2in}}+\beta_2|V\rangle_{c_{2in}}\big)\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big) \\
&\otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big),
\end{split}
\end{eqnarray}
with a probability of 1/2.
In the second step, PBS$_2$ transmits $H_{c_{2in}}$-polarized component to PBS$_8$ and reflects $V_{c_{2in}}$-polarized component to spatial mode 3 for mixing with the components emitted from spatial mode 2 at PBS$_3$. After the photons emitted from spatial modes 2 and 3 experience the block composed of four HWP$^{22.5^\circ}$s and PBS$_3$, we choose the instance in which each of spatial modes 5 and 6 contains exactly one photon, and then the system will become the normalization state
\begin{eqnarray} \label{eq17}
\begin{split}
|\Psi_2\rangle =&\frac{1}{2\sqrt{2}} \big[ \big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4|H\rangle_6+\beta_1\alpha_2|V\rangle_1|H\rangle_4|H\rangle_6 +\alpha_1\alpha_2|H\rangle_1|H\rangle_4|V\rangle_6 \\&
+\beta_1\alpha_2|V\rangle_1|H\rangle_4|V\rangle_6+\alpha_1\alpha_2|H\rangle_1|H\rangle_4|H\rangle_5-\beta_1\alpha_2|V\rangle_1|H\rangle_4|H\rangle_5 \\&-\alpha_1\alpha_2|H\rangle_1|H\rangle_4|V\rangle_5+\beta_1\alpha_2|V\rangle_1|H\rangle_4|V\rangle_5\big)
+\frac{1}{2} \big(\alpha_1\beta_2|H\rangle_1|V\rangle_5|H\rangle_6 \\
&+\beta_1\beta_2|V\rangle_1|H\rangle_5|H\rangle_6
+\alpha_1\beta_2|H\rangle_1|H\rangle_5|V\rangle_6+\beta_1\beta_2|V\rangle_1|V\rangle_5|V\rangle_6 \big)\big]\\
&\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big)\otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big),
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}$.
In order to complete the Toffoli gate with unity fidelity in principle, we next reduce the amplitude of the photon emitted from spatial mode 5 to half by using a 50:50 beam splitter (BS). The unitary transformations of the BS can be described as
\begin{eqnarray} \label{eq18}
\begin{split}
&|H\rangle_5\xrightarrow{\text{BS}}\frac{1}{\sqrt{2}}\big(|H\rangle_{5'}+|H\rangle_{5''}\big), \quad
&|V\rangle_5\xrightarrow{\text{BS}}\frac{1}{\sqrt{2}}\big(|V\rangle_{5'}+|V\rangle_{5''}\big).
\end{split}
\end{eqnarray}
That is, BS yields the state
\begin{eqnarray} \label{eq19}
\begin{split}
|\Psi_3\rangle =&\frac{1}{2\sqrt{2}} \big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4+\beta_1\alpha_2|V\rangle_1|H\rangle_4 +\alpha_1\beta_2|H\rangle_1|V\rangle_{5'}\\&+\beta_1\beta_2|V\rangle_1|H\rangle_{5'}+\alpha_1\beta_2|H\rangle_1|V\rangle_{5''}+\beta_1\beta_2|V\rangle_1|H\rangle_{5''}\big)
\\&\otimes|H\rangle_6\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big) \otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big)\\
&+\frac{1}{2\sqrt{2}}\big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4+\beta_1\alpha_2|V\rangle_1|H\rangle_4 +\alpha_1\beta_2|H\rangle_1|H\rangle_{5'}\\
&+\beta_1\beta_2|V\rangle_1|V\rangle_{5'}+\alpha_1\beta_2|H\rangle_1|H\rangle_{5''}+\beta_1\beta_2|V\rangle_1|V\rangle_{5''} \big)\\
&\otimes|V\rangle_6\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big)\otimes \big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big) \\
%
%
&+\frac{1}{4} \big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4|H\rangle_{5'}-\alpha_1\alpha_2|H\rangle_1|H\rangle_4|V\rangle_{5'}-\beta_1\alpha_2|V\rangle_1|H\rangle_4|H\rangle_{5'}\\
&+\beta_1\alpha_2|V\rangle_1|H\rangle_4|V\rangle_{5'}+\alpha_1\alpha_2|H\rangle_1|H\rangle_4|H\rangle_{5''}-\alpha_1\alpha_2|H\rangle_1|H\rangle_4|V\rangle_{5''}\\&-\beta_1\alpha_2|V\rangle_1|H\rangle_4|H\rangle_{5''}
+\beta_1\alpha_2|V\rangle_1|H\rangle_4|V\rangle_{5''}\big)\\&\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big) \otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big).
\end{split}
\end{eqnarray}
If $D_{H_6}$ is triggered, the photon emitted from spatial mode $5'$ is led to PBS$_5$ to mix with the photon emitted from mode $a_2$, Eq. (\ref{eq19}) will collapse into the state
\begin{eqnarray} \label{eq20}
\begin{split}
|\Psi_4\rangle =&\frac{1}{\sqrt{2}} \big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4+\alpha_1\beta_2|H\rangle_1|V\rangle_{5'}+\beta_1\alpha_2|V\rangle_1|H\rangle_4+\beta_1\beta_2|V\rangle_1|H\rangle_{5'} \big)\\
&\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big) \otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big), \\
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times \frac{1}{4}$. If $D_{V_6}$ is triggered, the photon emitted from spatial mode $5'$ will be applied a feedback $\sigma_x$ operation to convert the state
\begin{eqnarray} \label{eq20.5}
\begin{split}
|\Psi'_4\rangle =&\frac{1}{\sqrt{2}} \big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4+\alpha_1\beta_2|H\rangle_1|H\rangle_{5'} +\beta_1\alpha_2|V\rangle_1|H\rangle_4+\beta_1\beta_2|V\rangle_1|V\rangle_{5'} \big)\\
&\otimes\big(|H\rangle_{a_2}+|V\rangle_{a_2}\big) \otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big), \\
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times \frac{1}{4}$ to Eq. (\ref{eq20}). The $\sigma_x$ operation can be achieved easily by an HWP$^{45^\circ}$ setting in mode $5'$.
In the third step, after PBS$_5$ completes the parity-check measurement on the photons emitted from spatial modes $5'$ and $a_2$, the instance in which the spatial mode 8 involves exactly one photon is chosen, and then the system will be in the following normalization state
\begin{eqnarray} \label{eq21}
\begin{split}
|\Psi_5\rangle =& \big(\alpha_1\alpha_2|H\rangle_1|H\rangle_4|V\rangle_8+\alpha_1\beta_2|H\rangle_1|V\rangle_7|V\rangle_8 +\beta_1\alpha_2|V\rangle_1|H\rangle_4|V\rangle_8\\
&+\beta_1\beta_2|V\rangle_1|H\rangle_7|H\rangle_8 \big) \otimes\big(\alpha_3|H\rangle_{t_{in}}+\beta_3|V\rangle_{t_{in}}\big), \\
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times (\frac{1}{4}+\frac{1}{4})\times \frac{1}{2}$.
In the fourth step, the photons emitted from spatial modes 7 and 4 pass through HWP$^{67.5^\circ}$ and HWP$^{22.5^\circ}$ respectively, and then are fed into PBS$_8$. Before and after the photons emitted from spatial modes 8 and $t_{in}$ mix at PBS$_6$, four HWP$^{22.5^\circ}$s are performed on them. The even-parity is chosen for the polarized photons in modes 11 and 12 after PBS$_6$. Therefore, before $D_{H_{12}}$ or $D_{V_{12}}$ is fired, these elements induce the outing photons in the state
\begin{eqnarray} \label{eq22}
\begin{split}
|\Psi_6\rangle =&\frac{1}{2} \big(\gamma_1|H\rangle_1|H\rangle_9|H\rangle_{11}+\gamma_2|H\rangle_1|H\rangle_9|V\rangle_{11} +\gamma_3|H\rangle_1|V\rangle_9|H\rangle_{11}\\&+\gamma_4|H\rangle_1|V\rangle_9|V\rangle_{11}+\gamma_5|V\rangle_1|H\rangle_9|H\rangle_{11}+\gamma_6|V\rangle_1|H\rangle_9|V\rangle_{11} \\&+\gamma_7|V\rangle_1|V\rangle_9|V\rangle_{11}+\gamma_8|V\rangle_1|V\rangle_9|H\rangle_{11}\big) \otimes |V\rangle_{12} \\
&+\frac{1}{2} \big(\gamma_1|H\rangle_1|H\rangle_9|V\rangle_{11}+\gamma_2|H\rangle_1|H\rangle_9|H\rangle_{11} +\gamma_3|H\rangle_1|V\rangle_9|V\rangle_{11}\\
&+\gamma_4|H\rangle_1|V\rangle_9|H\rangle_{11}+\gamma_5|V\rangle_1|H\rangle_9|V\rangle_{11}+\gamma_6|V\rangle_1|H\rangle_9|H\rangle_{11} \\&+\gamma_7|V\rangle_1|V\rangle_9|H\rangle_{11}+\gamma_8|V\rangle_1|V\rangle_9|V\rangle_{11}\big) \otimes |H\rangle_{12} \\
&+\frac{1}{2} \big(\gamma_1|H\rangle_1|V\rangle_{10}|H\rangle_{11}+\gamma_2|H\rangle_1|V\rangle_{10}|V\rangle_{11} +\gamma_3|H\rangle_1|H\rangle_{10}|H\rangle_{11}\\
&+\gamma_4|H\rangle_1|H\rangle_{10}|V\rangle_{11}+\gamma_5|V\rangle_1|V\rangle_{10}|H\rangle_{11}+\gamma_6|V\rangle_1|V\rangle_{10}|V\rangle_{11} \\
& -\gamma_7|V\rangle_1|H\rangle_{10}|V\rangle_{11}-\gamma_8|V\rangle_1|H\rangle_{10}|H\rangle_{11}\big)\otimes |V\rangle_{12} \\
&+\frac{1}{2} \big(\gamma_1|H\rangle_1|V\rangle_{10}|V\rangle_{11}+\gamma_2|H\rangle_1|V\rangle_{10}|H\rangle_{11} +\gamma_3|H\rangle_1|H\rangle_{10}|V\rangle_{11}\\
&+\gamma_4|H\rangle_1|H\rangle_{10}|H\rangle_{11}+\gamma_5|V\rangle_1|V\rangle_{10}|V\rangle_{11}+\gamma_6|V\rangle_1|V\rangle_{10}|H\rangle_{11} \\
&-\gamma_7|V\rangle_1|H\rangle_{10}|H\rangle_{11}-\gamma_8|V\rangle_1|H\rangle_{10}|V\rangle_{11}\big) \otimes |H\rangle_{12}, \\
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times (\frac{1}{4}+\frac{1}{4})\times \frac{1}{2}\times \frac{1}{2}$. Here, for simplicity, the coefficients are written as $\gamma_1=\alpha_1\alpha_2\alpha_3$, $\gamma_2=\alpha_1\alpha_2\beta_3$, $\gamma_3=\alpha_1\beta_2\alpha_3$, $\gamma_4=\alpha_1\beta_2\beta_3$, $\gamma_5=\beta_1\alpha_2\alpha_3$, $\gamma_6=\beta_1\alpha_2\beta_3$, $\gamma_7=\beta_1\beta_2\alpha_3$, and $\gamma_8=\beta_1\beta_2\beta_3$.
Half-wave plate HWP$^{67.5^\circ}$ induces the transformations
\begin{eqnarray} \label{eq23}
\begin{split}
&|H\rangle\xrightarrow{\text{HWP}^{67.5^\circ}}\frac{1}{\sqrt{2}}\big(|V\rangle-|H\rangle\big), \quad
&|V\rangle\xrightarrow{\text{HWP}^{67.5^\circ}}\frac{1}{\sqrt{2}}\big(|H\rangle+|V\rangle\big).
\end{split}
\end{eqnarray}
Based on Eq. (\ref{eq22}), one can see that when $D_{V_{12}}$ is fired, the outing photons from the spatial modes $1$, $9$, and $11$ kept are in the state
\begin{eqnarray} \label{eq24}
\begin{split}
|\Psi_7\rangle =&\gamma_1|H\rangle_1|H\rangle_9|H\rangle_{11}+\gamma_2|H\rangle_1|H\rangle_9|V\rangle_{11} +\gamma_3|H\rangle_1|V\rangle_9|H\rangle_{11}+\gamma_4|H\rangle_1|V\rangle_9|V\rangle_{11}\\
&+\gamma_5|V\rangle_1|H\rangle_9|H\rangle_{11}+\gamma_6|V\rangle_1|H\rangle_9|V\rangle_{11} +\gamma_7|V\rangle_1|V\rangle_9|V\rangle_{11}+\gamma_8|V\rangle_1|V\rangle_9|H\rangle_{11},
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times (\frac{1}{4}+\frac{1}{4})\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{4}$. That is, the probabilistic Toffoli gate is completed.
\begin{figure} [!h]
\begin{center}
\includegraphics[width=8.5 cm,angle=0]{Psuccess.eps}
\caption{The relations between the success probability of Toffoli gate ($P_\text{Toff}$) and parameter $\alpha_2$. The blue dashed curve indicates $P_\text{Toff}=\frac{1+\alpha_2^2}{64}$ without recycling the photons in mode $5''$, while the red solid curve indicates $P_\text{Toff}=\frac{(1+\alpha_2^2)(2-\alpha_2^2)}{64}$ with recycling the photons in mode $5''$.} \label{PToff}
\end{center}
\end{figure}
\begin{table}[!h]
\centering \caption{Measurement outcomes and corresponding feed-forward operations in mode $5'$ or 11 for realizing a Toffoli gate. $I_2$ is an identity operation and $\sigma_x$ is a Pauli $X$ operation, which can be realized by an HWP setting at $45^\circ$.}
\begin{tabular}{ccccccc
\hline \hline
\multicolumn {2}{c}{Measurement} & \qquad\qquad & \multicolumn {2}{c}{Feed-forward} & \qquad\qquad & Achieved \\ \cline{1-2} \cline{4-5}
Detector & Detector & \quad & mode $5'$ & mode 11 & \qquad\qquad &result \\
\hline
$D_{H_6}$ & $D_{V_{12}}$ & \quad & $I_2$ & $I_2$ & & Toffoli \\
$D_{H_6}$ & $D_{H_{12}}$ & \quad & $I_2$ & $\sigma_x$ & & Toffoli \\
$D_{V_6}$ & $D_{V_{12}}$ & \quad & $\sigma_x$ & $I_2$ & & Toffoli \\
$D_{V_6}$ & $D_{H_{12}}$ & \quad & $\sigma_x$ & $\sigma_x$ & & Toffoli \\
\hline \hline
\end{tabular}\label{table1}
\end{table}
When $D_{H_{12}}$ is fired, the outing photons from the spatial modes $1$, $9$, and $11$ kept are in the state
\begin{eqnarray} \label{eq25}
\begin{split}
|\Psi_7'\rangle =&\gamma_1|H\rangle_1|H\rangle_9|V\rangle_{11}+\gamma_2|H\rangle_1|H\rangle_9|H\rangle_{11}+\gamma_3|H\rangle_1|V\rangle_9|V\rangle_{11}+\gamma_4|H\rangle_1|V\rangle_9|H\rangle_{11}\\
&+\gamma_5|V\rangle_1|H\rangle_9|V\rangle_{11}+\gamma_6|V\rangle_1|H\rangle_9|H\rangle_{11} +\gamma_7|V\rangle_1|V\rangle_9|H\rangle_{11}+\gamma_8|V\rangle_1|V\rangle_9|V\rangle_{11},
\end{split}
\end{eqnarray}
with a probability of $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times (\frac{1}{4}+\frac{1}{4})\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{4}$. To complete the Toffoli gate, a feed-forward $\sigma_x$ operation is applied to photon in spatial mode 11. The outcomes of measurement and corresponding
feed-forward operations for completing Toffoli gate are summarized in Tab. \ref{table1}. When $D_{10}$ is fired, it means that the scheme fails.
The quantum circuit shown in Fig. \ref{Toffoli} completed a post-selection Toffoli gate with a success probability of $P_\text{Toff}=\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}= \frac{1+\alpha_2^2}{64}$.
Alternately, the success probability can be further improved to $\frac{1}{2}\times\frac{1+\alpha_2^2}{2}\times\frac{2-\alpha_2^2}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}= \frac{(1+\alpha_2^2)(2-\alpha_2^2)}{64}$ by recycling the photons emitted from spatial mode $5''$ into the PBS$_5$ because the state has the same form as the state shown in Eq. (\ref{eq20}).
In Fig. \ref{PToff}, we can see that the minimum $P_\text{Toff}$ without recycling is approximately $\frac{1}{64}$, the maximum $P_\text{Toff}$ without recycling reaches approximately $\frac{1}{32}$, and the average $P_\text{Toff}$ without recycling is $\frac{1}{2} \int_{-1}^{1}\frac{1+\alpha_2^2}{64}d\alpha_2=\frac{1}{48}$. In contrast, with recycling one, the minimum $P_\text{Toff}$ is approximately $\frac{1}{32}$, the maximum $P_\text{Toff}$ can reach $\frac{9}{256}$, and the average $P_\text{Toff}$ is $\frac{1}{30}$. The success probability of our Toffoli gate is much higher than previous works \cite{T-Fiurasek,T-PRA,T-NatPhy,T-Liu}.
\begin{table}[!h]
\centering\caption{A comparison of proposed post-selected CPF gate with linear optics and previous schemes.}
\begin{tabular}{lcccccc}
\hline \hline
Proposed & & Ancillary & &Success & & Achieved \\
schemes & & photons & &probability & & results \\
\hline
Refs. \cite{CNOT-Bell-1/4-1,CNOT-Bell-1/4-2,CNOT-Bell-1/4-3,CNOT-Bell-1/4-4,CNOT-Bell-1/4-5} & & A Bell-state & & 1/4 & & CNOT \\
Refs. \cite{CNOT-1/9-2,CNOT-1/9-4,CNOT-1/9-5} & & No & & 1/9 & & CPF \\
Refs. \cite{CNOT-1/9-3,CNOT-1/9-6,CNOT-1/9-7} & & No & & 1/9 & & CNOT \\
Refs. \cite{CNOT-1/8-1,CNOT-1/8-2} & & Two single photons & & 1/8 & & CNOT \\
Ref. \cite{CNOT-1/16} & & Two single photons & & 1/16 & & CNOT \\
Ref. \cite{T-Liu} & & No & & 1/8 & & CNOT \\
This work & & A single photon & & 1/4 & & CPF \\
\hline \hline
\end{tabular}\label{compareCPF}
\end{table}
\begin{table}[!h]
\centering\caption{A comparison of proposed post-selected Toffoli gate with linear optics and previous schemes.}
\begin{tabular}{lcccccc}
\hline \hline
Proposed & \quad\; & Ancillary & \quad\; &Success \\
schemes & \quad\; & photons & \quad\; &probability \\
\hline
Fiur\'{a}\v{s}ek \cite{T-Fiurasek} & & No & & 1/133 \\
Ralph \emph{et al.} \cite{T-PRA} & & No & & 1/72 \\
Ralph \emph{et al.} \cite{T-PRA} & & Two Bell-states & & 1/32 \\
Lanyon \emph{et al.} \cite{T-NatPhy} & & No & & 1/72 \\
Liu \emph{et al.} \cite{T-Liu} & & No & & 1/64 \\
This work & & Two single photons & & 1/30 \\
\hline \hline
\end{tabular}\label{compareToffoli}
\end{table}
\section{Conclusion} \label{sec4}
We have proposed two schemes to implement post-selected CPF and Toffoli gates in the coincidence basis by solely using linear optics. The comparisons between our proposed CPF gate and Toffoli gate and previous schemes are presented in Tab. \ref{compareCPF} and Tab. \ref{compareToffoli}, respectively. A maximally entangled photon pair \cite{CNOT-Bell-1/4-1,CNOT-Bell-1/4-2,CNOT-Bell-1/4-3,CNOT-Bell-1/4-4,CNOT-Bell-1/4-5} (or two single photons \cite{CNOT-1/8-1,CNOT-1/8-2}) is necessary for implementing a CNOT gate with the current existing maximum success probability of 1/4 (or 1/8).
Only one auxiliary single photon is introduced to accomplish our CPF gate with the success probability of 1/4. In addition, our approach to implement Toffoli gate is much more efficient than the synthesis one. Assisted by two independent single photons, our Toffoli gate is constructed with the current maximum success probability of 1/30. The presented two architectures provide an alternative insight into probabilistic quantum gates using linear optical elements and suggest that they maybe have various applications in photonic quantum information processing.
\section*{Acknowledgments}
This work was supported by the National Natural Science Foundation of China under Grant No. 11604012, the Fundamental Research Funds for the Central Universities under Grants FRF-TP-19-011A3.
\bibliographystyle{SciPost_bibstyle}
|
2,869,038,155,662 | arxiv | \section*{Introduction}
Rotations of objects characterize one of the fundamental properties of the dynamics and can be measured by various optical techniques.
It is however significantly limited on the nanoscale owing to the difficulty of acquiring directional information under the optical diffraction limit.
For characteristic dimensions above the sub-micron scale, optical microscopy provides ways to observe the rotational motion of small particles,
such as capturing the anisotropic morphology of particles
\cite{han2006brownian,cheng2003rotational,romodina2016detection},
polarization-sensitive optical detection of metallic nanorods~\cite{ruijgrok2011brownian,xiao2011imaging},
and extracting rotational information based on the detailed analysis of the 3D-translation dynamics~\cite{isojima2016direct}.
On the other hand, at the atomic scale, fluorescence depolarization spectroscopy provides information
on the rotational Brownian motion (diffusion) of ensembles of molecules~\cite{mann2003fluorescence}.
However, detecting rotational motion on the nanoscale, which lies between the atomic and the micron scales,
has not been hitherto well explored.
The difficulty of detecting rotational motion on the nanoscale arises from two major technical challenges.
First, it is difficult to capture the shapes of nanoparticles by optical means due to their sizes being smaller than the optical diffraction limit.
Emission (either fluorescence or scattering) from nanoparticles is treated as coming from a point light source and cannot provide the directional information.
Second, the timescale of the rotational diffusion of nanoparticles is quite fast with a high dynamic range; for example, in water,
it can vary from millisecond to microsecond for particle diameters from 100 nm to 10 nm,
which is several orders of magnitude faster than that of micron-scale particles (1 \si{\um} gives 1.45 Hz)~\cite{berg1993random}.
Most of the image-based optical techniques cannot provide such high-frequency detection.
Thus, the detection of the rotational motion of nanoparticles has been elusive.
A new approach that could access the rotational motion of single nanoparticles is to exploit the electron spins of
nitrogen vacancy (NV) centers in nanodiamonds.
Nanodiamonds can be used as very stable fluorescnece nano-light emitters when incorporating NV centers~\cite{mochalin2012properties}.
The intensity of the NV fluorescence is electron-spin-dependent and can be affected by the nanoscale local environment of the nanodiamonds,
such as magnetic fields, electric fields, and temperature~\cite{Doherty20131,schirhagl2014nitrogen}, which allows for quantum-enhanced nanoscale sensing~\cite{maze2008nanoscale,knowles2014observing,fujiwara2016manipulation}.
Applications of NV centers now extend to 3D-orientation tracking of nanoparticles~\cite{geiselmann2013three,Andrich2014,Horowitz2012} and nanoscale thermometry in living cells~\cite{kucsko2013nanometre,simpson2017non}.
The electron spins of NV centers are in principle able to sense the rotational motion of the host nanodiamonds,
because the random walk of the spin precession angle is accumulated as the geometric phase fluctuation of the NV quantum system.
The geometric phase fluctuation leads to dephasing of the electron spin coherence and thus broadens the electron spin resonance
(ESR) line of NV centers in continuous-wave (CW) ESR detection~\cite{maclaurin2013nanoscale,maclaurin2010masterthesis,Ledbetter2012,Ajoy2012}.
Here, we report the linewidth broadening of the ESR lines of single NV centers by the rotational diffusion of the host nanodiamonds.
Single nanodiamonds incorporating single NV centers are slowly detached from the host substrates in an aqueous buffer solution.
Continuous optical measurement of the NV centers during the detachment process allows for measuring the ESR spectra of the same NV centers when the nanodiamonds are either fixed to the substrate or fluctuating during the detachment.
The ESR line is clearly broadened by 1.8 MHz (full width at half maximum, FWHM) by the nanodiamond fluctuation.
The observed broadening shows good agreement with the diffusion constant of nanodiamonds with a diameter of 11.4 nm, which is derived from the Einstein--Smoluchowski relation.
Our findings can provide a new method to measure the rotational motion of single nanoparticles and enable the exploration of nano-scale fluid mechanics.
\section*{Results}
\begin{figure}[t!]
\centering
\includegraphics[width=120mm]{fig1-8.eps}
\caption{(a) Schematic drawing of a single nanodiamond gradually detached from a coverslip it was attached to. The nanodiamond orientation (NV axis) is fluctuating due to the rotational Brownian motion. (b) Close up of the central part of the experimental setup. The nanodiamond is placed in a home-made perfusion chamber that simultaneously allows exchange of the solution and the optical experiment. A thin copper wire is fed through into the chamber for the spin excitation. Single NV centers hosted in nanodiamonds are excited by 532-nm laser light and are observed with red-shifted fluorescence collected through the same microscope objective. (c) Energy diagram of NV centers. The main optical transition occurs between the ground state ($\ket{A^3}$) and the excited state ($\ket{E^3}$). Microwave excites the electron spin from $\ket{0}$ to $\ket{\pm1}$ in the sub-states of $\ket{A^3}$, followed by spin-conserving optical transitions and intersystem crossing from $\ket{\pm1}$ states in $\ket{E^3}$ to the lower singlet state, ($\ket{A^1}$). The population in $\ket{A^1}$ is nonradiatively relaxed to the spin $\ket{0}$ state of the triplet ground state, which causes a decrease of the fluorescence intensity.}
\label{fig1}
\end{figure}
\subsection*{Confocal fluorescence microscopy of singe NV centers in nanodiamonds in aqueous buffer solutions}
Figure~\ref{fig1}(a) shows a schematic drawing,
which depicts nanodiamonds that are about to be detached from the coverslip.
We use commercially available nanodiamonds with a median size of 25 nm (Microdiamant, MSY0-0.05).
A droplet of the nanodiamond suspension is spin-coated on a coverslip.
The uniform nanodiamond distribution on the coverslip is confirmed by
atomic force microscopy (see Supplementary Fig. S1).
We then fabricate a home-made perfusion chamber on the coverslip,
which simultaneously allows liquid exchange and optical observation as
shown in Fig.~\ref{fig1}(b) (see Methods and Supplementary Information for the details).
Distilled water is sent into the perfusion chamber to immerse nanodiamonds in liquid, followed by subsequent optical and ESR measurements.
The total volume of the tube line and the perfusion chamber is 490 \si{\uL}.
The liquid flow keeps running with the flow rate of 80 $\si{\uL} \cdot \si{\min}^{-1}$ in the subsequent experiments
unless specifically mentioned.
\begin{figure}[b!]
\centering
\includegraphics[width=120mm]{fig2-4.eps}
\caption{(a) Confocal fluorescence scanning image of nanodiamonds deposited on a coverslip and immersed in water. The fluorescent nanodiamond indicated by the dashed circle is to be detached. (b) Scanning image of the same region after the nanodiamond is detached. (c) The second-order photon correlation histogram and (d) fluorescence spectrum of the nanodiamond, when it is fixed on the coverslip in water. }
\label{fig2}
\end{figure}
Figure~\ref{fig2}(a) shows a confocal fluorescence scanning image of nanodiamonds fixed on a coverslip under the flow of distilled water.
There are isolated nanodiamonds showing fluorescence signals, most of which are ascribed to NV centers.
Figures~\ref{fig2}(c) and (d) show the second-order photon correlation histogram and fluorescence spectrum of the nanodiamond indicated by the dashed circle, respectively.
The photon-correlation histogram shows an antibunching dip with $g^{(2)}(0) = 0.16$ (the time origin $t_0 = 47$ ns) and a bunching shoulder at $t \sim 100$ ns due to the population exchange with the nearby metastable state~\cite{berthel2015photophysics} (see Fig.~\ref{fig1}(c)),
which clearly indicates incorporation of a single negatively charged NV$^{-}$ center.
The temporal profile of the photon correlation data can be fitted with an equation reported elsewhere\cite{berthel2015photophysics},
which yields $\tau_1 = 11$ ns and $\tau_2 = 166$ ns.
The fluorescence spectrum is another clear signature of the presence of NV$^{-}$ centers;
the zero-phonon line is observed at 634 nm, accompanied by a broad phonon sideband up to 750 nm
\cite{zhao2012effect,zhao2012suppression}.
We then replace the water with a buffer solution of pH = 7.5 and measure the ESR spectra.
The pH of the solution is changed stepwise (7.5 $\to$ 8.2 $\to$ 9.1 $\to$ 9.9) by adding a carbonate buffer solution.
After pH = 9.9 is reached, it is brought back to pH = 9.1 by adding HCl.
A single process of changing $\Delta {\rm pH} \sim 1$ takes about an hour that includes
adjusting the pH of the reservoir, liquid circulation, and optical--ESR characterization.
The total time to change from water to pH = 9.1 (the final pH in this experiment) is about 6 hours.
When the pH is changed back to pH = 9.1, the fluorescence from the nanodiamond starts to fluctuate,
as the nanodiamond is about to detach from the coverslip.
During the time of the nanodiamond fluctuation of several minutes, we are able to measure the ESR spectrum, which will be described in the next section.
The nanodiamond finally moves away due to the continuous flow in the perfusion chamber.
Figure~\ref{fig2}(b) shows a confocal scanning image of the same region as imaged in Fig.~\ref{fig2}(a)
after the nanodiamond is completely detached.
There is no more prominent fluorescent spot remaining inside the dashed circle,
while other fluorescent nanodiamonds are still located at the same positions.
There remain residues that show very small fluorescence of 32 kcps; this is about 5 times smaller than the average fluorescence count of single NV centers detected in water in our laboratory setup ($\sim$150 kcps).
We therefore confirm that only the nanodiamond indicated by the dashed circle is removed during the pH change. Note that there are some new fluorescent blurred spots emerged in Fig.~\ref{fig2}(b)
(see Supplementary Fig. S2 for the details).
These spots were created by the green laser excitation when we stopped the liquid flow during the experiment.
Without the continuous liquid flow, the green laser excitation gradually generates such fluorescent spots
(the fluorescence eventually grows much brighter than the single NV fluorescence if the liquid flow is stopped for a long time).
This is probably because nanodiamonds detached from other locations (beyond the imaging region)
are accumulated around the laser spot due to the strong green laser excitation (optical forces, laser heating, etc.) ~\cite{nishimura2014control}.
This phenomenon is more prominent in more acidic pH buffer solutions, which may be related to the zeta potential of nanodiamonds,
as nanodiamonds show lower negative zeta-potentials for more acidic pH~\cite{williams2010size,petit2015probing,gines2017positive}.
\subsection*{ESR measurements on single NV centers}
\begin{figure}[b!]
\centering
\includegraphics[width=70mm]{fig3-6.eps}
\caption{(a) Schematic illustration of the gated photon counting for the ESR measurements. The APD detection is gated for microwave irradiation ON and OFF.
The APD gate width is 200 \si{\us} common to the both gates,
giving $\Delta I_{\rm PL} = I_{\rm PL}^{\rm ON} / I_{\rm PL}^{\rm OFF}$.
The repetition rate of the gating (including laser off time of 100 $\si{\us}$) is 2 kHz.
The 532-nm green laser irradiation is continuously on during these gating periods.
532: 532-nm green laser pulse. MW: microwave pulse. Sig: photon counting while the microwave is ON.
Ref: photon counting while the microwave is OFF.
(b) ESR spectra of the nanodiamond indicated by the dashed circle (Fig.~\ref{fig2}) when it is fixed on the coverslip and (c) is fluctuating. The blue solid lines in (b) and (c) are the Lorentzian fits to the data, and the red dashed lines in (b) are the 2-peak components of the fitting. The olive dashed line in (c) is the reproduced curve for the minor peak, which is almost buried in the noise,
calculated by taking account of the reduction of the main peak area.}
\label{fig3}
\end{figure}
We measure ESR spectra of the nanodiamonds throughout the course of the pH change.
Figure~\ref{fig3}(a) shows a pulse sequence used to obtain the ESR spectra of single NV centers in nanodiamonds.
The microwave excitation and APD detection are gated with a common gate width of 200 $\si{\us}$ and a repetition rate of 2 kHz,
in order to extract spin-dependent signals out of the fluorescence fluctuation
due to the environmental noise such as defocusing of the laser spot, heating by the microwave irradiation, and the nanodiamond fluctuation (see Methods)~\cite{Horowitz2012}.
The fluorescence intensities with/without the microwave excitation ($ I_{\rm PL}^{\rm ON}$ / $I_{\rm PL}^{\rm OFF}$) are measured, and their ratio $\Delta I_{\rm PL} = I_{\rm PL}^{\rm ON} / I_{\rm PL}^{\rm OFF}$ (ESR contrast) is plotted as a function of the microwave frequency.
The external magnetic field is not applied in this experiment.
Figure~\ref{fig3}(b) shows the ESR spectrum of a single NV center when the nanodiamond is fixed to the coverslip
in distilled water (Fig.~\ref{fig2}(a)).
The ESR spectrum is composed of two Lorentzian peaks.
Curve fitting with a two-peak Lorentzian profile determines that
the major fluorescence peak is located at 2.8784(1) GHz with a linewidth of 12.2(4) MHz (FWHM), and
another associated minor peak is located at 2.9036(11) GHz with a linewidth of 12.1(40) MHz (FWHM).
These two peaks are the result of intrinsic splitting of the magnetic sublevels of $\ket{\pm 1}$ spin states by lattice strain
in the nanodiamond,
which is often observed in nanodiamond NV centers~\cite{tisler2009fluorescence}.
We note that the presented errors are from the curve fitting.
Figure~\ref{fig3}(c) shows the ESR spectrum of the single NV center while the fluorescence from the nanodiamond
is fluctuating in the final buffer solution at pH = 9.1.
The major peak is clearly broadened, and the other minor peak is weakened.
We fit the data with a single Lorentzian profile, since the intensity of the minor associated peak is comparable to the noise level,
which makes curve fitting difficult (the peak still exists at around 2.905 GHz).
The major peak is located at 2.8800(3) GHz with a FWHM linewidth of 14.0(9) MHz.
The linewidth is broadened by 1.8(9) MHz compared with that of the fixed configuration (Fig.~\ref{fig3}(b)).
We note that the peak area of the major peak is decreased by 36.5 \% compared to that shown in Fig.~\ref{fig3}(b).
This decrease buries the minor peak under the noise, which justifies the use of a single Lorentzian profile for the curve fitting (we discuss this issue in Discussion).
\subsection*{Broadening of the ESR peak linewidth}
The observed linewidth broadening comes from the rotational Brownian motion of the host nanodiamonds.
Fast rotation of nanodiamonds adds a geometric phase to the time evolution of the NV spin system
as theoretically studied in Refs.~\citenum{maclaurin2010masterthesis,maclaurin2013nanoscale}.
When the spin measurement time is sufficiently long compared with the rotational diffusion constant of the nanodiamonds,
the final ESR signal is averaged over the entire ensemble of initial orientations and rotational trajectories.
The random fluctuation of the geometric phase creates an additional decay channel for the quantum superposition,
which modifies the spin coherence time as $T_2^\ast \to T_2^\ast + k_d^{-1}$.
Here, $k_d$ is the rotational diffusion constant of nanodiamonds and, according to the Einstein-–Smoluchowski relation, is given by
\begin{equation}
k_d \left( = \frac{\Delta \Gamma}{2} \right) = \frac{k_B T}{8 \pi (d/2)^3 \eta},
\end{equation}
where $\Delta \Gamma$, $k_B$, $T$, $d$, and $\eta$ are the observed linewidth broadening in the ESR spectrum,
Boltzmann constant, temperature, diameter of the nanoparticles, and viscosity of the surrounding medium, respectively.
Note that $\Delta \Gamma = 2k_d$, considering the time--frequency transformation relation of
the exponential decay and the Lorentzian line shape.
In the present case, the surrounding medium is a carbonate buffer-based solution
that has a viscosity of $\eta = 0.97$ m\si{\Pa} at room temperature (T = 293 K) determined by a viscotester (Toki Sangyo, RE100L).
The spin measurement time is 200 \si{\us}, sufficiently longer than the rotational diffusion time of
the nanodiamonds, as it corresponds to the diffusion constant of nanoparticles with a diameter of 66 nm (5 kHz).
The observed linewidth broadening is hence calculated to be the rotational diffusion constant of
nanoparticles with a diameter of $d$ = 11.4(2) \si{\nm}.
The size statistics of our nanodiamonds exhibit a mean diameter of 30 nm with a distribution ranging from 10 to 50 nm
on the basis of the AFM topography image (Supplementary Fig.~S1).
The observed linewidth is within the range of the particle distribution,
thus indicating that it comes from the rotational Brownian motion.
\section*{Discussion}
In this paper, we reported the linewidth broadening of the ESR peaks of single NV centers
due to the rotational Brownian motion of the host nanodiamonds.
We found that the ESR peak was broadened by 1.8 MHz (FWHM) that corresponds to the rotational diffusion constant of the nanodiamonds with a diameter of 11.4 nm.
While the results clearly demonstrate the effect of the rotational Brownian motion on the ESR peaks of NV centers,
there still remain questions that need to be addressed to fully understand the presented results.
\subsubsection*{Other possible factors contributing to the linewidth broadening}
Laser heating of the nanodiamonds by the 532-nm green laser may affect the linewidth broadening as it can change the local viscosity.
The local temperature in the present situation can rise up to, for example, 90 $^\circ {\rm C}$
(slightly lower than the boiling point of water), which reduces the viscosity of water to almost one third of the original value~\cite{korson1969viscosity}
(1.0 m\si{\Pa} at 20 $^\circ {\rm C}$ $\to$ 0.31 m\si{\Pa} at 90 $^\circ {\rm C}$).
The change of the local viscosity may lead to overestimation of the linewidth broadening by a factor of 3, giving the
corresponding particle size of $d$ = 16.4(3) \si{\nm}.
Such temperature change, however, might not occur in the present situation,
since we did not observe the peak shift that corresponds to this temperature change of -0.7 MHz (the temperature change causes the peak shift of -74 kHz$\cdot {\rm K}^{-1}$~\cite{acosta2010temperature}).
Note that the microwave power and the laser intensity can affect the linewidth, which we describe in the following subsection.
The pH change does not cause such a drastic change of the linewidth as observed here. We have investigated the effect of the pH on the electron spin properties of single NV centers in the same nanodiamond samples.
The variation of the linewidth over the pH change from 4 to 11 is always smaller than 0.6 MHz~\cite{tracking2018fujiwara}.
It is interesting to note that the linewidth broadening of the ESR peaks has been observed in optically trapped nanodiamonds with a relatively large diameter of 74 nm~\cite{geiselmann2013three}.
The nanodiamonds used in Ref.~\citenum{geiselmann2013three} have a mean particle size of 74 nm based on the dynamic light scattering data
and are trapped in a viscous solution (glycerin:water = 5:1, viscosity 132 m\si{\Pa} at 20 $^\circ {\rm C}$~\cite{cheng2008formula}),
which gives $k_D$ of $\sim$ 24 Hz.
A detectable broadening on the kHz level was confirmed by statistically comparing the ESR spectra of the trapped nanodiamonds with those of the fixed ones (Fig. S9 in Ref.~\citenum{geiselmann2013three}) and qualitatively ascribed to the precession of the NV axis.
Further quantitative evaluation would be important to reveal the detailed mechanism of the linewidth broadening.
\subsubsection*{Reduced peak area during the fluctuation}
The peak area of the major peak in Fig.~\ref{fig3}(c) is decreased by 36.5 \% compared to that in Fig.~\ref{fig3}(b).
This peak reduction results in the associated minor peak being buried under the noise level,
which makes it impossible to fit with the original double Lorentzian peak profile.
By applying the same change (36.5\%-peak-area reduction and 1.6-MHz-frequency shift) to the fitting parameters of the minor peak in Fig.~\ref{fig3}(b) and keeping other parameters fixed (see Supplementary Table~S1), we reproduce the simulated curve for the minor peak as shown in Fig.~\ref{fig3}(c) (dashed olive line).
The reproduced minor peak is smaller than the noise,
which justifies the use of a single Lorentzian profile for the curve fitting.
Both the microwave power and the laser excitation power can affect the peak area,
as the ESR linewidth exhibits corresponding dependences~\cite{Lesik2011}.
We measured the dependence of the ESR peaks on both the microwave power and
laser power for nanodiamonds fixed to a coverslip in water (see Supplementary Fig. S3) to
estimate the effect of these parameters on the reduction of the peak area (and linewidth broadening).
In the present situation, the linewidth can be broadened by $\pm$ 0.2 MHz by a typical microwave power change of 1 \% and
$\pm$ 0.2 MHz by a typical laser intensity fluctuation of 5--10 \%.
The observed linewidth broadening and the peak area reduction therefore do not originate from the power change of the
microwave and the laser, confirming the origin of the linewidth broadening as the rotation Brownian motion of the nanodiamonds.
\subsubsection*{Future perspectives of rotational-motion sensing}
Despite of these unresolved issues,
the present observation suggests attractive applications of this rotational-motion quantum sensing to various fields like nanofluidics or
biological sensing.
On the nanoscale, the persistent photostability of NV centers, together with the present rotational-motion sensing, will make nanodiamonds indispensable tools to investigate nanoscale fluid mechanics.
The classical experimental tool to visualize nanofluids is organic-molecular fluorescent probes~\cite{fornander2016visualizing,sinton2004microscale} that can also be used for fluorescence depolarization spectroscopy~\cite{smith2015review}.
This method, however, suffers from bleaching of dyes and can only be used for a short period of time and in some specific conditions at specific pH and temperature range.
Furthermore, the nanoscale volume restricts the number of fluorescent molecules, thereby shortening the observation time further.
In contrast, fluorescent nanodiamonds can provide long-term tracking in various pHs and temperature ranges with excellent fluorescence stability.
Recent advancement of fabrication technology has enabled the incorporation of NV centers in nanodiamonds smaller than 5 nm~\cite{bradac2010observation,tisler2009fluorescence}.
One could insert such ultra-small nanodiamonds into structures with a size of tens of nanometers.
It is also possible to access the translational Brownian motion at the same time with measuring the rotational Brownian motion
by a wide-field imaging technique.
Combining NV-fluorescent nanodiamonds with walking protein motors would be interesting because NV centers provide
information on the 3D protein motion (rotation or torsion in addition to the translational motion)~\cite{isojima2016direct}.
Our NV sensing technique can thus provide a way for extracting full information on the Brownian motion of protein motors in fluorescence microscopes.
\section*{Methods}
\subsection*{Sample preparation}
A commercially available nanodiamond suspension (Microdiamant, MSY 0-0.05, median particle size: 25 nm) was purified by centrifugation and was dispersed in pure water.
A small droplet of the suspension was spin-coated on a cleaned coverslip.
A 25-\si{\um}-thin copper wire was placed on the coverslip as a microwave linear antenna, and both ends were soldered to SMA connectors.
An acrylic spacer with a height of about 4 mm with inlet and outlet tubes was then glued
on top of the sample using a UV-curing resin.
It was sealed with a glass plate to make a perfusion chamber.
The spin-coated samples were raster scanned with an atomic force microscope (Bruker, Edge) to obtain topography images.
The peak heights of the distributed nanodiamonds were measured to extract the particle size distributions.
The nanodiamonds were detached from the substrate by changing the pH of the buffer solution stepwise in the perfusion chamber.
We first sent distilled water to the chamber.
A flow of pH-buffer solution (0.1 M sodium carbonate with 5\% HCl) was next created.
During the optical excitation, the continuous flow of these solutions with a rate of 80 $\si{\uL} \cdot {\rm min}^{-1}$ was maintained to prevent photothermal accumulation of nanodiamonds.
\subsection*{Optical measurements}
The perfusion chamber was mounted on a 3-axis piezo stage in a home-built confocal fluorescence microscope.
A continuous-wave 532 nm laser was used for the excitation with an excitation intensity of 94 kW$\cdot {\rm cm ^{-2}}$ (250 \si{\uW}).
An oil-immersion microscope objective with a numerical aperture of 1.4
was used both for the excitation and the fluorescence collection.
The NV fluorescence was filtered by a dichroic beam splitter (Semrock, FF560-FDi01)
and a long-pass filter (Semrock, BLP01-561R) to remove the residual green laser scattering.
It was then coupled to an optical fiber acting as a pinhole (Thorlabs, 1550HP, core diameter $\sim 10$ \si{\um}).
The fiber-coupled fluorescence was finally guided into a Hanbury-Brown-Twiss (HBT) setup consisting of two APDs (Perkin Elmer SPCM AQRH-14) and a 50:50 beam splitter or connected to a spectrometer equipped with a liquid-nitrogen-cooled CCD camera (Princeton, LNCCD).
By scanning the sample with the piezo stage, we were able to obtain the fluorescence scanning images.
A time-correlated single-photon counting module (PicoQuant, TimeHarp-260) was used to obtain second-order photon correlation histograms.
\subsection*{ESR measurements}
Microwave generated from a source (Rohde \& Schwarz, SMB100A) was boosted by 45 dB with an amplifier (Mini-circuit, ZHL-16W-43+) and was fed to the microwave linear antenna in the perfusion chamber.
The microwave excitation power was 35 dBm (3.2 W).
To extract the ESR spectra from the fluorescence fluctuation of the nanodiamonds,
the APD detection was gated for microwave irradiation ON and OFF states
by using an RF switch (Mini-circuit, ZYSWA-2-50DR-S) and a bit pattern generator (Spincore, PBESR-PRO-300).
The gate width was 200 \si{\us} common for both gates, followed by a laser shut-off time of 100 $\si{\us}$,
giving $I_{PL}^{ON}$ and $I_{PL}^{OFF}$.
The repetition rate of the gating pulses was 2 kHz.
No external magnetic field was applied.
|
2,869,038,155,663 | arxiv | \section{Quantum Measurements}
The chief ingredients for a quantum measurement on a quantum system are i) an appropriate apparatus, with well defined
pointer
states $P_i$ (these, in the present folklore, are to be determined by suitable apparatus decoherence processes), and an
appropriate
measurement interaction ${\cal M}$ between the system and the apparatus. The latter is determined by the observable of the
system to be measured. A point to be emphasised is that the \emph{same} measurement interaction can be used both for
the strong,
projective measurements, as well as for the so called \emph{weak measurements}. For example, for qubit measurements, this
can be taken to be (A,S are for apparatus and system,respectively, and $P_i$ are the pointer-states of the apparatus):
\begin{eqnarray}
\label{eq:measint}
& &|P_i\rangle_A\otimes|\uparrow\rangle_S\xrightarrow{{\cal M}}\,|P_{i+1}\rangle_A\otimes|\uparrow\rangle_S\nonumber\\
& &|P_i\rangle_A\otimes|\downarrow\rangle_S\xrightarrow{{\cal M}}\,|P_{i-1}\rangle_A\otimes|\downarrow\rangle_S
\end{eqnarray}
This is sybolically depicted in Figure.(\ref{fig:measint0}) where the central line denotes the pointer state $P_i$,
and those flanking it denote
$P_{i\pm 1}$.
\begin{figure}[htp!]
\centering
\includegraphics[width=1.5in]{measint0.eps}
\caption{Measurement interaction of eqn.(\ref{eq:measint}).}
\label{fig:measint0}
\end{figure}
\subsection{Projective measurements}
We now discuss the so called projective or strong measurements. For this, the initial state of the apparatus is taken to be a \emph{single}
pointer state, say, $P_0$. The same measurement interaction discussed above now reads:
\begin{equation}
|P_0\rangle_A\otimes|\pm\rangle_S\xrightarrow{{\cal M}}\,|P_\pm\rangle_A\otimes|\pm\rangle_S
\end{equation}
in an obvious relabelling of states. Henceforth we shall drop the $\otimes$.
If the initial state of the system is taken to be:
\begin{equation}
\label{eq:sysini}
|\psi\rangle = \alpha|\uparrow\rangle+\beta|\downarrow\rangle\quad\quad |\alpha|^2+|\beta|^2=1
\end{equation}
and the initial state of the apparatus-system complex
is taken to be $|\psi\rangle\,|P_0\rangle$, the \emph{post-measurement-interaction state} of the composite is given by
\begin{equation}
|P_0\rangle\,|\psi\rangle\xrightarrow\,|\Psi\rangle_{SA}=\alpha|P_+\rangle|\uparrow\rangle+\beta|P_-\rangle|\downarrow\rangle
\end{equation}
As is well known, this is an entangled state and does not correspond to the expected state after a definite measurement
outcome.
The current folklore is that \emph{environmental decoherence} reduces the density matrix $\rho_{SA}$ of this
pure state to the mixed state, which by construction, is diagonal in the pointer-states bases:
\begin{equation}
\rho_{SA}\xrightarrow{decoh}\,|\alpha|^2|P_+\rangle\langle P_+|\,|\uparrow\rangle\langle \uparrow|+|\beta|^2\,|P_-\rangle\langle P_-||\downarrow\rangle\langle \downarrow|
\end{equation}
The system itself can be efficiently characterized by its \emph{reduced density matrix}:
\begin{equation}
\rho_{red} = |\alpha|^2|\uparrow\rangle\langle \uparrow|+|\beta|^2|\downarrow\rangle\langle \downarrow|\quad\quad
\end{equation}
The so called \emph{Purity} of this mixed state, defined as $tr_A\,\rho_{SA}^2$, is given by $ 1-2|\alpha|^2|\beta|^2$.
This is generically far from a purity value of unity. It should be appreciated that decoherence, however, does not explain the measurement
process on an event by event basis.
With each outcome, the system is irretrievably altered. The pointer position $+1$ occurs with probability $|\alpha|^2$,
while the outcome $-1$ occurs with probability $|\beta|^2$. The mean pointer position is $|\alpha|^2-|\beta|^2$. The
variance is the standard uncertainty associated with the state $|\psi\rangle$, and the error in the result of M
measurements falls off as $\frac{1}{{\sqrt M}}$.
\subsection{Weak measurements}
Now we turn to the so called \emph{weak measurements}. To demystify the hopelessly large hype(and many wrong statements), we
consider a highly idealised example which nevertheless contains the essential features of this very interesting new category
of measurements introduced by Aharonov and his collaborators \cite{aharonovorig} (for a detailed exposition of many aspects
of weak measurements see \cite{nori}). The initial state of the apparatus is now taken to be a very
broad superposition of pointer states with equal weights and no relative phases:
\begin{equation}
|A\rangle = \frac{1}{\sqrt{N}}\,\sum_{i=1}^{i=N}\,|P_i\rangle
\end{equation}
In some of the current literature, even this very very broad state is treated as a pointer state, with its
centroid identified as the corresponding \emph{pointer position}. It is quite meaningless to take this position.
Introducing the apparatus state
\begin{equation}
|{\bar A}\rangle = \frac{1}{\sqrt{N-2}}\,\sum_{i=2}^{i=N-1}\,|P_i\rangle
\end{equation}
one sees that the measurement interaction of eqn(\ref{eq:measint}) leads in this case to
\begin{equation}
|A\rangle|\uparrow\rangle\,\rightarrow\: \{\sqrt{\frac{N-2}{N}}\,|{\bar A}\rangle+\frac{1}{\sqrt{N}}\,
(|P_N\rangle+|P_{N+1}\rangle)\}|\uparrow\rangle
\end{equation}
\begin{equation}
|A\rangle|\downarrow\rangle\,\rightarrow\: \{\sqrt{\frac{N-2}{N}}\,|{\bar A}\rangle+\frac{1}{\sqrt{N}}\,
(|P_0\rangle+|P_{1}\rangle)\}|\downarrow\rangle
\end{equation}
This is depicted in Figure.(\ref{fig:measint}).
\begin{figure}[htp!]
\centering
\includegraphics[width=1.5in]{measint.eps}
\caption{A weak measurement}
\label{fig:measint}
\end{figure}
If the initial state of the apparatus and system is taken to be $|A\rangle\otimes|\psi\rangle$, with $|\psi\rangle$ as given by eqn.(\ref{eq:sysini}),
the \emph{post-measurement-interaction} composite state is now given by:
\begin{eqnarray}
\label{eq:postmeasweakexample}
& &\sqrt{\frac{N-2}{N}}\,|{\bar A}\rangle|\psi\rangle+\frac{\alpha}{\sqrt{N}}\,(|P_N\rangle+|P_{N+1}\rangle)|\uparrow\rangle\nonumber\\
& &+
\frac{\beta}{\sqrt{N}}\,(|P_0\rangle+|P_1\rangle)|\downarrow\rangle
\end{eqnarray}
The \emph{post-decoherence} system-apparatus mixed state, which is by construction diagonal in $P_i$ (the incorrectness of
treating the initial apparatus state $|A\rangle$ becomes evident here), is easily worked out to be:
\begin{eqnarray}
\label{eq:postdecohweakexample}
& &\frac{N-2}{N}\,\sum_{i=2}^{i=N-1}|P_i\rangle\langle P_i||\psi\rangle\langle \psi|\nonumber\\
&+&\frac{|\alpha|^2}{N}(|P_N\rangle\langle P_N|+|P_{N+1}\rangle\langle P_{N+1}|)|\uparrow\rangle\langle \uparrow|\nonumber\\
&+&\frac{|\beta|^2}{N}(|P_0\rangle\langle P_0|+|P_{1}\rangle\langle P_{1}|)|\downarrow\rangle\langle \downarrow|
\end{eqnarray}
The post-measurement reduced density matrix of the system is obtained by tracing over the apparatus state-space:
\begin{equation}
\label{eq:redrhoweakexample}
\rho^{weak}_{red} = |\psi\rangle\langle \psi|-\frac{2}{N}(\alpha\beta^*|\uparrow\rangle\langle \downarrow|+\alpha^*\beta|\downarrow\rangle
\langle \uparrow|)
\end{equation}
The purity of this reduced density matrix is
\begin{equation}
\label{eq:weakpurity}
{\cal P}_{weak} = 1-\frac{8}{N}|\alpha|^2|\beta|^2
\end{equation}
When $N >> 1$, this post-measurement purity can be arbitrarily close to the unit purity of the system state before
measurement. In this sense, the weak measurements appear to be highly \emph{non-invasive}, but there is more to
invasiveness than just this measure.
A number of important properties attributed to weak measurements in general can be gleaned from this highly idealized
example. From eqn.(\ref{eq:postmeasweakexample}), it follows that with probability $1-2/N$, the system is not changed at
all(extreme weakness). It is also important to observe that this 'weakness' has nothing to do with the strength of the
measurement interaction. Rather, it is completely controlled by N, the \emph{width} of the initial apparatus state.
While with most measurement outcomes, there is no change of the system, the \emph{information} obtained about the system
by these outcomes is also zero. This follows from the fact that the probabilities for these outcomes has \emph{no}
dependence on the initial state. On the other hand, the outcomes $i=N, N+1$ occur with the very low probability
$\frac{|\alpha|^2}{N}$ and likewise, $i=0,1$ with probability $\frac{|\beta|^2}{N}$. For these outcomes, the system is
irretrievably changed exactly as in projective measurements! These probabilities being dependent on the system state, these
outcomes give full information!
Let us now calculate the mean pointer position ${\bar i}$ and the associated variance. Elementary calculations give this
to be $(N+1)/2$ before measurement, and, $(N+1)/2+|\alpha|^2-|\beta|^2$. Therefore the shift in the mean pointer position
is exactly the expectation value of the observable, as in the projective measurements. The variance in the pointer
positions is now dramatically different. Before measurements it is $(N^2-1)/12$ while after measurements, it is still
essentially this, but shifted by a tiny system-dependent part:
\begin{equation}
\label{eq:weakexamplevariance}
(\Delta i)^2_{pre} = \frac{N^2-1}{12}\quad\quad (\Delta i)^2_{post} = (\Delta i)^2_{pre}+(\Delta S)^2_{\psi}
\end{equation}
The results for the mean and variance are exactly the same as for the most generic weak measurements \cite{nori}.
In this elementary example, the deviations of pointer outcomes can trivially be much larger than the eigenvalues
of the observable in question. There is no big mystery that needs some special understanding. Another noteworthy
feature is that since for pointer outcomes, the system is mostly not
an eigenstate of the observable (in the example, this happens only when the outcomes are $i=0,1,N,N+1$), there is no
\emph{value} of the observable associated with the value of the pointer outcome, unlike the case in projective measurements.
Too much has been made of this starting from the title of the first paper on weak measurements \cite{aharonovorig}.
Now we turn out to a standard treatment of weak measurements. The
\emph{Pointer variable} is taken to be $p$ the momentum. The \emph{Pointer states} are taken to be the momentum eigenstates $|p\rangle$.
In practice, these are taken to be narrow gaussian wave packets in momentum representation.
As seen in our extreme example, the initial apparatus state for weak measurements should be a \emph{very broad superposition} of
pointer states i.e
\begin{equation}
\label{eq:appstategen}
|A\rangle = {\bar N}_p\,\int\,dp\,e^{-\frac{p^2}{2\Delta_p^2}}\,|p\rangle \quad\quad {\bar N}_p = (\pi\Delta_p^2)^{-1/4}
\end{equation}
with $\Delta_p >> 1$.
The \emph{measurement interaction} is taken to be $e^{-iQA}$ where $A$ is the observable that is being measured, and Q the
variable conjugate to momentum. As in the von Neumann model, this is taken to be impulsive, acting exactly at the time of
measurement. For simplicity, we take the observable A to have the discrete, non-degenerate spectrum
$a_i, |a_i\rangle$.
The initial system state is taken to be:
\begin{equation}
\label{eq:syststategen}
|\psi\rangle = \sum_i\,\alpha_i\,|a_i\rangle\quad\quad \sum_i\,|\alpha_i|^2=1
\end{equation}
The \emph{post-measurement-interaction} state of system and apparatus is then given by:
\begin{eqnarray}
|\Psi\rangle_{SA,weak} &=& {\bar N}_p\,\sum_i\,\alpha_i\,\int\,dp\,e^{-\frac{p^2}{2\Delta_p^2}}|p+a_i\rangle|a_i\rangle\nonumber\\
&=&\,\int\,dp\,N(p,\{\alpha\})|p\rangle|\psi_p\rangle
\end{eqnarray}
Where
\begin{eqnarray}
N(p,\{\alpha\}) &=& {\bar N}_p\,\sqrt{\sum_i\,|\alpha_i|^2\,e^{-\frac{(p-a_i)^2}{\Delta_p^2}}}\nonumber\\
|\psi_p\rangle &=& \frac{{\bar N}_p}{N(p,\{\alpha\})}\,\sum_i\,\alpha_i\,e^{-\frac{(p-a_i)^2}{2\Delta_p^2}}|a_i\rangle
\end{eqnarray}
Hence, weak measurements can be viewed as the so called Positive Operator Valued Measurements(POVM) with
measurement operators:
\begin{equation}
\label{eq:weakpovm}
M_p = {\bar N}_p\,\sum\,e^{-\frac{(p-a_i)^2}{2\Delta_p^2}}|a_i\rangle\langle a_i|
\end{equation}
The \emph{post-decoherence} mixed state of the system and apparatus is easily calculated to be:
\begin{equation}
\rho^{post-decoh}_{SA} = \int\,dp\,|N(p,\{\alpha\})|^2\, |p\rangle\langle p||\psi_p\rangle\langle \psi_p|
\end{equation}
The probability distribution for the pointer outcomes is given by $|N(p,\{\alpha\}|^2$. As the eigenvalues $a_i$ are
bounded, this distribution, when $p|a_i| << \Delta_p^2$, is well approximated by
\begin{equation}
\label{eq:lowpapprox}
|N(p,\{\alpha\}|^2\simeq\quad {\bar N}_p^2\,e^{-\frac{P^2}{\Delta_p^2}}+\ldots
\end{equation}
In this case
\begin{equation}
\label{eq:lowpstate}
|\psi_p\rangle\simeq\,|\psi\rangle+\ldots
\end{equation}
where the dots represent small corrections. One once again observes the same features encountered in the example, namely,
that for most of the outcomes the state changes very little(in the example, that change was zero while in
the more
realistic cases, as here, it is small). But precisely for those cases, the probability of outcome is either
independent, or nearly independent, of the system
state and no information can be obtained about the system state. Nevertheless, as in the example, the mean pointer position
has full information about the state(provided a \emph{complete} set of weak measurements are performed).
The average outcome and its variance can be calculated exactly:
\begin{equation}
\langle p \rangle = \sum_i\,|\alpha_i|^2\,a_i = \langle A \rangle_\psi\quad (\Delta p)^2 = \frac{\Delta_p^2}{2}+(\Delta A)^2_\psi
\end{equation}
These necessarily have to be \emph{ensemble} measurements.
The errors in weak measurements are very large because $\Delta_p >> 1$. These are to be reduced \emph{statistically}.
It is instructive to compute the \emph{reduced} density matrix of the system:
\begin{equation}
\label{eq:weakredrho}
\rho^{red,weak}_{sys} = |\psi\rangle\langle \psi| -\frac{1}{4\Delta_p^2}\sum_{i,j}\,\alpha_i\alpha_j^*\,(a_i-a_j)^2|a_i\rangle\langle a_j|
\end{equation}
\section{Weak Measurements and Leggett-Garg Inequalities}
We saw in eqn.(\ref{eq:weakredrho}) that the reduced density matrix after a weak measurement is \emph{practically} the same as the initial pure density matrix.
In this sense, the weak measurements can be said to be \emph{non-invasive}.
Non-invasive measurements have been emphasized in a variety of contexts.
The most notable of these has been the \emph{Leggett-Garg} inequalities \cite{agarg,dhome,mahesh}.
A typical experimental setup
consists of four series of measurements on
identical initial states.
In each series, some quantity $Q(t)$ is measured at two instants of time. In the first, measurements are done at $t_1$ and $t_2$;
in the second, at $t_2$ and $t_3$, in the third at $t_3$ and $t_4$, and finally in the fourth at $t_1$ and $t_4$. It is to be noted
that $t_1 < t_2 < t_3 < t_4$.
The first measurement in each series is required to be \emph{non-invasive}, as then the second measurement can be
\emph{construed} to have also been made on the \emph{same state} as the initial one.
Thus a total of 8 measurements of which 4 have to be non-invasive.
The natural question is whether weak measurements can be used to achieve this?
The answer to this hinges on the accuracy of measurements(errors) as well as the \emph{available resources}, the apparent
non-invasiveness of weak measurements notwithstanding.
An obvious resource to be considered is the \emph{ensemble size} of the initial state. Let this be M identical copies.
If we consider using weak measurements to provide the required non-invasive measurements, it will be necessary to divide
M into 4 equal subensembles of $M/4$ copies each, and use one for each series of measurements.
The statistical error in the resulting weak measurements will be $\epsilon_w=\frac{\Delta_p}{\sqrt{2}}\frac{1}{\sqrt{M/4}}$.
It should be remembered that for the second measurement in each series, the state will not be exactly the same as the
original state. Depending on $\Delta_p$, this could be an important factor to reckon with in practical implementations.
Since the second measurement does not have to be non-invasive, it can even be done with strong measurements, which, for
the same ensemble size would yield an error substantially lowered by a factor $\frac{\sqrt{2}(\Delta A)_\psi}{\Delta_p}$.
The error analysis of LG-inequalities would be more complicated then.
Let us estimate the ensemble size that would yield the same error $\epsilon_w$ but now done with strong measurements.
The relation between statistical error and ensemble size for strong measurements is $\epsilon_s = \frac{(\Delta A)_\psi}{\sqrt{M_s}}$,
where $M_s$ is the relevant ensemble size.
Therefore the ensemble size for strong measurements with the same error as in the weak measurements is:
\begin{equation}
M_s = \frac{(\Delta A)_\psi^2}{\epsilon_w^2}=\frac{M}{2}\cdot\frac{(\Delta A)_\psi^2}{\Delta_p^2}
\end{equation}
The idea now is to divide the original resource into 8 equal subensembles and use each of them to perform the total of 8 measurements
required.
Altogether 8 strong measurements need to be done and the total ensemble size required is $M\cdot\frac{4(\Delta A)_\psi^2}{\Delta_p^2}$
Hence it follows that as long as $\frac{(\Delta A)_\psi}{\Delta_p} << 1/2$, the ensemble size required for strong version of checking LG inequalities is \emph{much smaller} than what was required for the weak version of the same!
Furthermore, in the strong version the states used for all the 8 measurements are \emph{exactly identical}!
In summary, if $\Delta_p$ is very large, one can test the LG-inequalities with much smaller resources using strong measurements.
If $\Delta_p$ is not so large, the weak measurements are no longer non-invasive.
Either way, there is no case for invoking weak measurements to test the LG-inequalities. Similar considerations for
determination of so called trajectories will be taken up elsewhere.
\section{Repeated Weak Measurements On a Single Copy}
One of the most \emph{surprising} and \emph{shocking} facets of the Copenhagen view of quantum mechanics is what
one may call the \emph{demise of the individual} (for a detailed exposition see \cite{ndhonto}).
More precisely, that view predicated that no information can be obtained about the unknown state of a \emph{single copy}.
This is a trivial consequence if one uses projective or strong measurements.
This is so as the first measurement randomly results in an eigenstate and all subsequent measurements have no bearing on the
original unknown state.
Weak measurements offer a \emph{superficial hope} that it may be possible to determine the unknown state of a single copy.
The basis for that hope is that each weak measurement, with high probability, very weakly alters the system state while giving
some information about the original state.
Consider the following \emph{schema} for repeated weak measurements on a single copy. (i) Perform a weak measurement
of observable A on a single copy of an unknown state $|\psi\rangle$.
Let the apparatus outcome be, say, $p_1$.
Consequently, the system state at this stage is $|\psi_{p_1}\rangle$.
(ii) Restore the apparatus to the same state before the first weak measurement.
(iii) Perform weak measurement of A in the new system state $|\psi_{p_1}\rangle$.
(iv) Repeat.
The crucial question is whether the statistics of outcomes $p_1,p_2,...,p_N$ have anything to say about the
original unknown $|\psi\rangle$? The naive argument would be that since at each step one gathers some \emph{information}
about the original unknown state, although very little, with sufficiently large repetitions one ought to gather enough
information to determine the original state. The question will be answered in the negative here. The details can be found in \cite{ndhweakrepeat}. Alter and Yamamoto \cite{orly,orlybook} had
in fact analysed a very similar problem in the context of repeated QND measurements long ago, but issues of degradation of the state
as well connections
to strong measurements were not considered by them.
The probability $P^{(1)}(p_1)$ of the first outcome $p_1$ is given by:
\begin{equation}
P^{(1)}(p_1) = |N(p_1,\{\alpha\}|^2
\end{equation}
The system state after this outcome is $|\psi_{p_1}\rangle$.
It is useful to describe this state as one with \emph{changed values} of $\{\alpha\}$:
\begin{equation}
\alpha_i^{(1)} = \frac{{\bar N}_p}{N(p_1,\{\alpha\})}\:e^{-\frac{(p_1-a_i)^2}{2\Delta_p^2}}
\end{equation}
The probability $P(p_2)$ of the second outcome $p_2$ is, therefore:
\begin{eqnarray}
& &P^{(2)}(p_2) = |N(p_2,\{\alpha^{(1)}\}|^2\nonumber\\
&=&\frac{{\bar N}_p^4}{N(p_1,\{\alpha\})^2}\,
\sum_i\,|\alpha_i|^2\,e^{-\frac{(p_1-a_i)^2}{\Delta_p^2}}\cdot e^{-\frac{(p_2-a_i)^2}{\Delta_p^2}}
\end{eqnarray}
But this is the \emph{conditional probability} $P(p_2|p_1)$ for obtaining $p_2$ \emph{given} that the first outcome was $p_1$.
The \emph{joint probability} distribution $P(p_1,p_2)$ is given by Bayes theorem to be $P(p_1)\cdot P(p_2|p_1)$:
\begin{equation}
P(p_1,p_2) = {\bar N}_p^4\,
\sum_i\,|\alpha_i|^2\,e^{-\frac{(p_1-a_i)^2}{\Delta_p^2}}\cdot e^{-\frac{(p_2-a_i)^2}{\Delta_p^2}}
\end{equation}
The state of the system when the outcomes are $p_1,p_2$ is:
\begin{equation}
|\psi(p_1,p_2)\rangle = \frac{\sum_i\,\prod_{j=1}^2\,e^{-\frac{(p_j-a_i)^2}{\Delta_p^2}}\alpha_i\,|a_i\rangle}{\sqrt{\sum_i\,
\prod_j\,|\alpha_i|^2\,e^{-\frac{(p_j-a_i)^2}{\Delta_p^2}}}}
\end{equation}
These readily generalize to the case of M repeated measurements:
\begin{eqnarray}
P(p_1,p_2,\ldots,p_M) &=& ({\bar N}_P^2)^M\,\sum_i\,|\alpha_i|^2\,\prod_{j=1}^M\,e^{-\frac{(p_j-a_i)^2}{\Delta_p^2}}\nonumber\\
|\psi(p_1,p_2,..,p_M)\rangle &=& \frac{\sum_i\,\prod_{j=1}^M\,e^{-\frac{(p_j-a_i)^2}{\Delta_p^2}}\alpha_i\,|a_i\rangle}{\sqrt{\sum_i\,
\prod_j\,|\alpha_i|^2\,e^{-\frac{(p_j-a_i)^2}{\Delta_p^2}}}}
\end{eqnarray}
These equations codify \emph{all} the information that can be obtained by repeated weak measurements on a single copy of an unknown state.
The joint probability distribution is not \emph{factorisable} as the outcomes are \emph{not mutually independent}, but it is still of
the so called \emph{separable} form.
The average $y_M$ of the M outcomes is $\sum_i\,|\alpha_i|^2\,a_i = \langle A \rangle_\psi$!
Does this mean we have obtained the same information in a weak measurement on a single copy what
could only be obtained by ensemble measurements of the strong kind?
It is necessary to look into the distribution function $P(y_M)$ for such an average.
Recall that in ensemble measurements this takes the form (Central Limit Theorem):
\begin{equation}
P(y_M)_{ensemble} = {\tilde N}\,e^{-\frac{M(y_M-\mu)^2}{\Delta^2}}
\end{equation}
In ensemble measurements too, the sequence of outcomes in \emph{a particular realization} will be different, and \emph{unpredictable}.
But the average obtained in any particular realisation converges to the true average as
$M\rightarrow\,\infty$.
Now it turns out that the story is entirely different for repeated weak measurements on a single copy!
The distribution function $P(y_M)$:
\begin{eqnarray}
P(y_M)&=& \sqrt{\frac{M}{\pi\Delta_p^2}}\,\sum_i\,|\alpha_i|^2\,e^{-\frac{M(y_M-a_i)^2}{\Delta_p^2}}\nonumber\\
&\rightarrow&
\sum_i\,|\alpha_i|^2\,\delta(y_M-a_i)
\end{eqnarray}
The distribution of $y_M$ is no longer peaked at the true average with errors decreasing as $M^{-1/2}$.
Instead, it is a weighted sum of distributions that increasingly peak around the eigenvalues as $\Delta_p$ increases.
In the limiting case, averages over a particular realisation will be eigenvalues occurring with probability $|\alpha_i|^2$, exactly
as in the case of strong measurements.
Hence averages over any particular realisation do not give any information about the initial state.
To substantiate this picture further, one can investigate the average value of the post-measurement system reduced density
matrix:
\begin{equation}
\langle \rho^{red} \rangle = \rho - \sum_{i,j}\,\alpha_i\alpha_j^*\,(1-e^{-\frac{M(a_i-a_j)^2}{4\Delta_p^2}})|a_i\rangle\langle a_j|
\end{equation}
Therefore as M becomes larger and larger, there is significant change in the system state.
In the limit $M\rightarrow\,\infty$, the \emph{off-diagonal} parts of the density matrix get completely quenched, as
in \emph{decoherence}!
In that limit, the density matrix becomes diagonal in the eigenstate(of A) basis:
\begin{equation}
\langle \rho^{red} \rangle\,\rightarrow\: \sum_i\,|\alpha_i|^2\,|a_i\rangle\langle a_i|
\end{equation}
This is exactly the post-measurement system state in the case of strong measurements.
It should be noted that this decoherence in eigenstate basis has nothing to do with the environmental decoherence
in the pointer state basis of the apparatus.
It is entirely due to the large number of repeated weak measurements.
Such an effect had also been noted by Gurvitz in 1997 \cite{gurwitz}.
We can view the distance between the initial $\rho$ and the average post-measurement reduced density matrix
$\langle \rho^{red} \rangle$, according to some reasonable distance measure, as a measure of the \emph{disturbance}
caused by the repeated weak measurements on the single copy.
For example, ${\cal D} = 1-tr\,\rho\,\langle \rho^{red}\rangle$ is one such distance measure.
The statistical error $\epsilon = \frac{\Delta_p}{\sqrt{2M}}$.
Then one gets the \emph{error-disturbance} relation:
\begin{eqnarray}
{\cal D}(\epsilon)&=& \sum_{i,j}\:|\alpha_i|^2|\alpha_j|^2(1-e^{-\frac{(a_i-a_j)^2}{8\epsilon^2}})\nonumber\\
&\rightarrow&
\sum_i\,|\alpha_i|^2(1-|\alpha_i|^2)
\end{eqnarray}
Reducing errors can only be at the cost of increasing invasiveness! It should be noted that this error-disturbance relation bears
no obvious relation to the ones being discussed by Ozawa \cite{ozawa}.
\section{Weak value coordinates and optimal weak value measurements}
This section is based on the works \cite{ndhsaicoord} and \cite{ndhsaioptimal}.
Consider the projection operators ${\cal P}_\pm$ for the eigenstates $|\pm\rangle$ of, say, $S_z$.
Let the preselected state be $|\psi =\alpha |+\rangle\,+\,\beta|-\rangle$, with $|\alpha|^2+|\beta|^2=1$.
Let the post-selected state be $|b\rangle$.
If $w_\pm$ are the weak values of ${\cal P}_\pm$
\begin{equation}
w_\pm=\frac{\langle b||\pm\rangle\langle \pm||\psi\rangle}{\langle b|\psi\rangle} \quad\quad w_+\,+\,w_-=1
\end{equation}
The idea of \emph{weak value tomography}($b_\pm=\langle b|\pm\rangle$):
\begin{equation}
\alpha = \frac{\frac{w_+}{b_+}}{\sqrt{|\frac{w_+}{b_+}|^2+|\frac{w_-}{b_-}|^2}}\quad\quad
\beta = \frac{\frac{w_-}{b_-}}{\sqrt{(\frac{w_+}{b_+})^2+(\frac{w_-}{b_-})^2}}
\end{equation}
Thus experimentally determining a single complex weak value ($w_+$ or $w_-$) suffices to determine the state.
$w_+=\frac{1}{2}+w_z$ and $w_-=\frac{1}{2}-w_z$, where $w_z$ is the weak value of $S_z$.
Thus it suffices to measure the weak value of a single observable to determine the state as against conventional tomography
which would require the \emph{expectation values} of \emph{two} independent observables and a \emph{sign}!
At this stage, the fact that $Re\: w, Im\: w$ are \emph{unbounded} becomes crucial.
It indicates that the real and imaginary parts of weak values provide a \emph{stereographic projection} of the \emph{Riemann sphere}.
The \emph{metric} on the state space can be introduced through the line element
\begin{equation}
dl^2 = 2\,tr\,d\rho\,d\rho
\end{equation}
For example, if the pure state density matrix is parametrised as
\begin{equation}
\rho = \frac{I}{2} +\langle S_x \rangle\,\sigma_x +\langle S_y \rangle\,\sigma_y +\langle S_z \rangle\,\sigma_z
\end{equation}
with
\begin{equation}
\langle S_x \rangle^2+ \langle S_y \rangle^2+ \langle S_z \rangle^2 = \frac{1}{4}
\end{equation}
The line element becomes
\begin{equation}
dl^2 = 4\{(dS_x)^2+(dS_y)^2+(dS_z)^2\}
\end{equation}
This is just the metric on a sphere.
The most general form of the line element is
\begin{equation}
dl^2 = g_{ww}\,dw^2\,+g_{{\bar w}{\bar w}}\,d{\bar w}^2\,+g_{w{\bar w}}\,dw\,d{\bar w}
\end{equation}
Explicit evaluation yields
\begin{equation}
g_{w{\bar w}}=\frac{4}{|b_+|^2|b_-|^2}\,\frac{1}{\sqrt{|\frac{w_+}{b_+}|^2+\frac{w_-}{b_-}|^2}}
\end{equation}
with $g_{ww} = g_{{\bar w}{\bar w}}=0$.
Therefore, the weak value coordinates have the nice feature that they are \emph{conformal}!
In terms of $Re\:w_+=x,Im\:w_+=y$, the line element can be rewritten as
\begin{equation}
dl^2 = \frac{4|b_+|^2|b_-|^2\:(dx^2+dy^2)}{\{x^2+y^2+x(|b_-|^2-|b_+|^2)+\frac{1}{4}\}^2}
\end{equation}
The volume(area) element of the state space is then
\begin{equation}
dA = \frac{4|b_+|^2|b_-|^2\:dx\,dy}{\{x^2+y^2+x(|b_-|^2-|b_+|^2)+\frac{1}{4}\}^2}
\end{equation}
The total volume of $\rho$-space is correctly reproduced as \emph{4$\pi$}(area of unit sphere).
Now another remarkable feature of weak measurements comes into play i.e the measurement errors in both x and y are the
\emph{same}, and are \emph{state-independent}.
The common statistical error is $\Delta_s = \frac{\Delta_p}{\sqrt{2M}}$.
This is in contrast to strong measurements.
Following Wootters and Fields, the \emph{error volume} is
\begin{equation}
(\Delta A)_{err} = \frac{16\,\Delta_s^2\,|b_+|^2|b_-|^2\:dx\,dy}{\{x^2+y^2+x(|b_-|^2-|b_+|^2)+\frac{1}{4}\}^2}
\end{equation}
As noted by Wootters and Fields in the case of standard tomography, this is \emph{state-dependent}, and it is not possible to
optimise it.
We follow them and optimise the error volume \emph{averaged over state space}.
The state averaged error volume can easily be worked out:
\begin{equation}
\langle (\Delta A)_{err} \rangle = \frac{16\,\Delta_s^2}{|b_+|^2|b_-|^2}
\end{equation}
Since $\Delta_s$ has no dependence on the post-selected state $|b\rangle$, it is straight forward to optimise this.
The solution is $|b_+|^2=|b_-|^2=\frac{1}{2}$.
In other words \emph{weak value measurements are optimal in the sense of minimizing state averaged error volume when the
post-selected states are MUB with respect to the eigenstates of the observable measured}.
Extension to spin-1 and higher spin values is under investigation.
|
2,869,038,155,664 | arxiv | \section{Introduction}
Pulsars are very efficient particle accelerators as witnessed by they broad band electromagnetic spectrum from radio \citep{manchester_australia_2005} up to very high-energy, GeV \citep{abdo_second_2013}, and sometimes TeV emission like for the Crab \citep{ansoldi_teraelectronvolt_2016} and Vela \citep{djannati-atai_h.e.s.s._2017}. Particle acceleration and therefore radiation is rooted to the fast rotation of a strongly magnetized neutron star. Rotational kinetic energy is converted into radiation by curvature, synchrotron and inverse Compton emission leading to the stellar braking accounted by the spin-down rate derived from the period~$P$ and its derivative~$\dot{P}$. However, where exactly within the magnetosphere those mechanisms occur is still unclear. Undoubtedly particles flow at very high Lorentz factor from the star to the interstellar medium, shaping the pulsar wind as a ballerina similar to the solar wind. The global magnetosphere electrodynamics is intimately related to the motion of these particles and their subsequent radiation. Some localized regions are prone to efficient conversion of the rotational kinetic energy into acceleration and radiation but where and how remains to be self-consistently determined from global magnetosphere simulations including dissipation.
The simplest approach to find such solutions starts with force-free regime (FFE) where an ideal plasma is considered, neglecting particle inertia and temperature, meaning that the electric field $\mathbf{E}$ is orthogonal to the magnetic field $\mathbf{B}$, $\mathbf{E} \cdot \mathbf{B}=0$ and the electric field is weaker than the magnetic field in normalized units, meaning $E<c\,B$ where $c$ is the speed of light.
In this picture, the Poynting flux is conserved because the electromagnetic field does no work on the plasma via the electric current~$\mathbf{j}$, meaning $\mathbf{j} \cdot \mathbf{E}=0$.
Strictly speaking, such magnetospheres are invisible because no photons are produces.
Numerical simulations have been pioneered by \cite{contopoulos_axisymmetric_1999} for the axisymmetric rotator that was extended to an oblique rotator by \cite{spitkovsky_time-dependent_2006} and retrieved by other authors, whether only aligned \citep{timokhin_force-free_2006, cao_spectral_2016, komissarov_simulations_2006, parfrey_introducing_2012, chen_electrodynamics_2014} or oblique \citep{petri_pulsar_2012, kalapotharakos_three-dimensional_2009, tchekhovskoy_three-dimensional_2016}. See also different solutions not requiring a current sheet like for instance in \cite{lovelace_jets_2006}.
The aforementioned fluid description offers a good starting point to understand the global electric circuit made of charge and current densities. However, it neglects some fundamental kinetic aspects required to self-consistently include single particle acceleration as well as radiation feedback. As kinetic simulations are much more demanding than fluid models, this approach was only scarcely investigated in the last century. Let us mention \cite{krause-polstorff_electrosphere_1985} who computed axisymmetric dead pulsar magnetospheres called electrospheres. Due to the axisymmetry of the problem they used rings of charges instead of point particles. Later with the advent on computational power, \cite{smith_numerical_2001} showed with slightly more sophisticated simulations that a fully field magnetosphere is unstable and collapse to an electrosphere. The first full three-dimensional electrosphere was constructed by \cite{mcdonald_investigations_2009} using an electromagnetic Particle in Cell (PIC) code. They neglect pair creation and therefore did not add any particle injection process. Eventually \cite{philippov_ab_2014} computed the first two-dimensional axisymmetric pulsar magnetosphere for an aligned rotator by permanently injecting particle supposed to be released from the surface, avoiding to end to an electrosphere configuration \citep{petri_global_2002}. Depending on the volume injection rate, they were able to find any equilibrium between the force-free and the fully charge separated state. \cite{chen_electrodynamics_2014} improved this model by adding a prescription for the pair creation, putting a threshold on the lepton Lorentz factor. Following the same lines \cite{cerutti_particle_2015} assumed particle injection only from the vicinity of the stellar surface. \cite{belyaev_dissipation_2015} injected particles from regions where a parallel electric field exists. The first full three dimensional PIC simulations of a pulsar magnetosphere were performed by \cite{philippov_ab_2015}. For an aligned rotator \cite{philippov_ab_2015-1} also included general-relativistic corrections with frame-dragging. Soon after some observational signature predictions were added to compute light curves and spectra emanating from curvature and or synchrotron radiation like for instance \cite{cerutti_modelling_2016} who then included polarization \citep{cerutti_polarized_2016}. This PIC simulations were then extended to the striped wind well outside the light-cylinder to study its dissipation \citep{cerutti_dissipation_2017, cerutti_dissipation_2020}. The oblique magnetosphere with radiation and general-relativistic correction was eventually computed by \cite{philippov_ab-initio_2018}. Several other groups performed similar simulations like \cite{brambilla_electronpositron_2018} or \cite{kalapotharakos_fermi_2017} and \cite{kalapotharakos_three-dimensional_2018} who tried to explicitly connect their simulation results to gamma-ray observations. Alternatively, more simply test particle trajectories can be explored within a fluid code, see for instance \cite{brambilla_testing_2015}.
Even if the PIC approach is now mature to include several ingredients like pair creation and its subsequent radiative signature, its main flaw resides in its inability to simulate neutron star magnetospheres with realistic stellar magnetic field strengths and rotation periods. For instance the Larmor radius is 10 to 15 orders of magnitude smaller than the light-cylinder radius, putting stringent constraints on the time step than cannot be fulfilled with current computational technology. It is therefore difficult to connect straightforwardly the microphysics dynamics induced by the gyro motion to the dynamics of the global magnetosphere although the time and spatial scale hierarchy is maintained in current PIC simulations. Consequently, a fluid description, as the one we employ in this paper, remains a valuable tool to explore the dynamics of pulsar magnetospheres. A hybrid approach using a particle kinetic description wherever necessary and a fluid model elsewhere would represent a good compromise. Recent developments indeed combine the PIC technique to the MHD evolution as for instance performed by \cite{marle_magnetic_2018} for particle shock acceleration or by \cite{bai_magnetohydrodynamic-particle--cell_2015} for investigation of cosmic rays interaction with a thermal plasma. Such hybrid modules are also implemented in available MHD codes like PLUTO \citep{mignone_particle_2018}. This new trend highlights the need to pursue our ongoing effort on improving plasma fluid models jointly with particle and Vlasov techniques by adding more physics on macro and micro scales simultaneously.
Returning to a simpler fluid and not particle description of the magnetosphere, the next step requires a proper treatment of dissipation with radiation. Some resistive simulations have been performed but with a resistivity not always based on pure physical grounds or with some arbitrariness leading to no unique prescription \citep{li_resistive_2012, kalapotharakos_toward_2012, gruzinov_strong-field_2008}. Although the best view would be a full kinetic description including, acceleration and radiation is conceptually possible, we believe it better at this stage to use a fluid description of the plasma by deriving an Ohm's law according to the radiation reaction limit and derived in for instance \cite{mestel_stellar_1999}.
\cite{contopoulos_are_2016}, build on this work and found radiative magnetospheric solution for oblique rotators however in a simplified manner. Recently a fully radiative solution has been computed for aligned rotators by \cite{petri_radiative_2020} and extended to oblique rotators \cite{petri_electrodynamics_2020} assuming that force-free dynamics holds inside the light-cylinder. \cite{cao_three-dimensional_2020} found similar solutions but allowing also possible dissipation within the light cylinder. Recently, \cite{cao_pulsar_2022} computed high resolution radiative magnetosphere solutions including some test particle dynamics in order to predict synchrotron spectra and light curves.
In the radiation reaction limit, the equation of motion is solved for a single particle in a stationary regime where the Lorentz force is counterbalanced by a so called radiative friction for ultra-relativistic speeds. Assuming that the particle moves at exactly the speed of light leads to a unique solution for the velocity vector given some decades ago by \cite{mestel_stellar_1999}. This expression is sometimes also called Aristotelian electrodynamics. Electrons and positrons, although possessing the same electric drift motion because being independent of the particle charge, will move in opposite direction with respect to the perpendicular plane. Their charge density will generate a current leading to a one parameter family of current prescription reminded in the next section. Therefore, it should be clear that radiative simulations as those we present in this paper include single particle dynamics concretized through the electric current density prescription.
The radiation reaction approximation solves the single particle equation of motion in the ultra-relativistic regime where the speed is exactly equal to the speed of light. The solution to the Lorentz force leads to the Aristotelian dynamics, the velocity being only a function of the local electric and magnetic field. However in order to avoid complication due to finite masses, as in the force-free case, we neglect their inertia. In this picture, particles only contribute to the charge and current density required to solve Maxwell equations. The derived current density possesses a component along the electric field and leads naturally to a kind of resistivity.
A full description of the plasma in this limit requires knowledge of its lepton content, separating the contribution from the electrons and the positrons. This pair multiplicity factor~$\kappa$ serves this goal and is the only free parameter in this radiative Ohm law. A fully self-consistent picture must however add pair-production but this small scale physics is still difficult to reconcile with the global scale of the magnetosphere. No self-consistent simulations have been performed so far taking all the ingredients self-consistently into account. Nevertheless, non thermal acceleration and radiative feedback is included in the present work thanks to the Aristotelian dynamics.
Because the radiation reaction limit regime relies on assumptions note always met within the magnetosphere, it is worth keeping in mind several caveats of our approach. The velocity field in Aristotelian electrodynamics is derived in the limit of significant radiative friction in the Lorentz force, reaching on a short time scale an asymptotic regime of exact balance between electric field acceleration and radiation damping. While this regime could be achieved in many places within the magnetosphere, there exist localized regions where such intense radiation damping is not effective due to negligible radiation reaction. Indeed, in the vicinity of polar caps, pair production efficiently screens the electric field component parallel to the magnetic field and particles do not experience the radiation reaction force \citep{timokhin_current_2013}. Moreover radiation damping involves ultra-relativistic particles with very large Lorentz factors that fails to be produced at several places except maybe for a tiny population of highly energetic particles. The outcome is a complex particle distribution function resembling more to a power law than to a mono-energetic population we take in this paper. As will be shown, Aristotelian dynamics allows for large regions where the electric field~$E$ dominates the magnetic field~$B$, i.e. where $E > c\,B$. However, recent studies showed that this should only occur as a transition stage to a magnetically dominated regime where $E < c\,B$ \citep{li_fast_2021}, see also \cite{beskin_radio_2018}. Moreover, our treatment neglects magnetic reconnection, especially within the current sheet of the striped wind outside the light cylinder, although it has been observed in PIC and MHD simulations. Our scheme represents a simplified two stage process where dissipation is directly converted into radiation.
In this paper, we compute oblique pulsar magnetospheres in the radiation reaction limit taking into account the exact dissipative current containing the electric drift component as well as the components aligned with the electric and magnetic field. In Sec.~\ref{sec:Modele}, we describe the model of our radiative magnetosphere and the prescription for Ohm's law derived from the radiation reaction regime. Some examples of magnetic topologies for an aligned and an orthogonal rotator are presented in Sec.~\ref{sec:LigneChamp} for the ideal FFE field and for the radiative magnetospheres. Next, in Sec.~\ref{sec:Luminosite} we compute the spin-down luminosity extracted from these models and compare it with previous works. The importance of dissipation is pointed out in Sec.~\ref{sec:Dissipation}. The importance of the parallel electric field component is stressed in Sec.~\ref{sec:parallel_E} Influences on the polar cap shape and size is explored in Sec.~\ref{sec:Calottes}. Sky maps and light-curves are presented in Sec.~\ref{sec:emission}. Conclusions are drawn in Sec.~\ref{sec:Conclusion}.
\section{Magnetospheric model}
\label{sec:Modele}
In this section, we present the underlying model to compute radiative pulsar magnetospheres starting from Maxwell equations and then explaining the electric current prescription.
\subsection{Maxwell equations}
In our models, the plasma only furnishes the required charge~$\rho_{\rm e}$ and current~$\mathbf{j}$ densities to evolve Maxwell equations written in standard MKSA units as
\begin{subequations}
\begin{align}
\label{eq:Maxwell1}
\mathbf{\nabla}\cdot \mathbf B & = 0 \\
\label{eq:Maxwell2}
\mathbf{\nabla} \times \mathbf E & = - \frac{\partial \mathbf B}{\partial t} \\
\label{eq:Maxwell3}
\mathbf{\nabla}\cdot \mathbf E & = \frac{\rho_{\rm e}}{\varepsilon_0} \\
\label{eq:Maxwell4}
\mathbf{\nabla} \times \mathbf B & = \mu_0 \, \mathbf j + \frac{1}{c^2} \, \frac{\partial \mathbf E}{\partial t} .
\end{align}
\end{subequations}
Apart from the obvious boundary conditions on the stellar surface, the current density $\mathbf{j}$ is the only unknown of the problem. Once fixed according to a given plasma model, Maxwell equations can be solved numerically, leading to a magnetosphere solution. So let us describe the possibilities for this current.
\subsection{Current prescription}
\subsubsection{Force-free limit}
The simplest model corresponds to an ideal plasma with infinite conductivity, leading to the force-free prescription as
\begin{equation}
\label{eq:J_ideal}
\mathbf j = \rho_{\rm e} \, \frac{\mathbf{E}\wedge \mathbf{B}}{B^2} + \frac{\mathbf{B} \cdot \mathbf{\nabla} \times \mathbf{B} / \mu_0 - \varepsilon_0 \, \mathbf{E} \cdot \mathbf{\nabla} \times \mathbf{E}}{B^2} \, \mathbf{B} .
\end{equation}
By construction, this current does not work on particles since $\mathbf{j} \cdot \mathbf{E} = 0$. All the rotational kinetic energy goes into the Poynting flux of the low frequency large amplitude electromagnetic wave. For completeness and comparison with other models shown in this paper, we compute again some force-free magnetospheres.
\subsubsection{Radiative solution}
If some emission is taken into account, for instance like in the radiation reaction limit of ultra-relativistic particles, the velocity of particles is fully determined by the local electromagnetic field configuration. Indeed, the friction caused by a radiative term can be seen as an isotropic emission of photons in the particle rest frame and at a rate controlled by the radiated power $\mathcal{P} \geq 0$ such that the balance between Lorentz force and radiative friction is
\begin{equation}\label{eq:balance_lorentz_radiation}
q \, ( \mathbf{E} + \mathbf{v} \wedge \mathbf{B}) = \frac{\mathcal{P}}{c^2} \, \mathbf{v} .
\end{equation}
A justification and argumentation about the validity of this balance can be found in \cite{mestel_axisymmetric_1985} starting from the Lorentz-Abraham-Dirac equation.
This equation is solved explicitly with respect to the velocity vector $\mathbf{v}$ and given for positive charges as $\mathbf{v}_+$ and negative charges as $\mathbf{v}_-$ according to
\begin{equation}
\label{eq:VRR}
\mathbf{v}_\pm = \frac{\mathbf{E} \wedge \mathbf{B} \pm ( E_0 \, \mathbf{E} / c + c \, B_0 \, \mathbf{B})}{E_0^2/c^2+B^2} .
\end{equation}
This expression only assumes that particles move exactly at the speed of light.
$E_0$ and $B_0$ are the strength of the electric and magnetic field deduced from the electromagnetic invariants and satisfying $\mathcal{I}_1 = \bmath E^2 - c^2 \, \bmath B^2 = E_0^2 - c^2 \, B_0^2$ and $\mathcal{I}_2 = c \, \bmath E \cdot \bmath B = c\,E_0 \, B_0$. Explicitly solving for $E_0\geq0$ and $B_0$ we find
\begin{subequations}
\label{eq:E0B0}
\begin{align}
E_0^2 & = \frac{1}{2} \, (\mathcal{I}_1 + \sqrt{\mathcal{I}_1^2 + 4 \, \mathcal{I}_2^2 }) \\
c\,B_0 & = \textrm{sign} (\mathcal{I}_2) \, \sqrt{E_0^2 - \mathcal{I}_1} .
\end{align}
\end{subequations}
$E_0$ and $B_0$ are interpreted as the electric and magnetic field strength in a frame where electric and magnetic field are parallel to each other.
The radiated power is then simply
\begin{equation}\label{eq:puissance_rayonnee}
\mathcal{P} = |q| \, E_0 \, c \geq 0 .
\end{equation}
Therefore within physical constants, $E_0$ is a direct measure of the radiated power.
Single particles are therefore evolved according to eq.~\eqref{eq:VRR}. It corresponds to the exact solution of the Lorentz equation of motion for charges subject to a friction and moving at exactly the speed of light. The particle inertia has been neglected and all species with the same sign of charge possess the same velocity vector irrespective of their charge to mass ratio~$q/m$ as long as their sign does not change.
From the velocity expression in \eqref{eq:VRR} we can derive the electric current associated to this particle flow. The detailed derivation is given by \cite{petri_theory_2016}, and the associated radiative current density~$\mathbf{j}$ with minimal assumption is explained in \cite{petri_radiative_2020}. The final expression reduces to
\begin{equation}
\label{eq:J_rad}
\mathbf j = \rho_{\rm e} \, \frac{\mathbf E \wedge \mathbf B}{E_0^2/c^2 + B^2} + ( |\rho_{\rm e}| \, + 2\,\kappa \, n_0 \, e) \, \frac{E_0 \, \mathbf E/c^2 + B_0 \, \mathbf B}{E_0^2/c^2 + B^2}
\end{equation}
where $\kappa$ is the pair multiplicity.
The background particle density number is depicted by $n_0$ and varies from point to point within the magnetosphere. For a self-consistent picture, this density should be constrained by the pair production rate. However such task is out of the scope of the present study. In order to go further, we replace $n_0$ by $|\rho_{\rm e}/q|$ and therefore is directly connected to the electric field via Maxwell-Gauss equation. Consequently
\begin{equation}\label{eq:happa_n0}
|\rho_{\rm e}| \, + 2\,\kappa \, n_0 \, e = |\rho_{\rm e}| \, (1 + 2\,\kappa)
\end{equation}
but actually this factor could be any function in the most general situation.
We stress that in our simulations, individual particles follow the velocity given in eq.\eqref{eq:VRR}. Each particle possesses its own velocity depending solely on the local electromagnetic field configuration. Therefore particles are present indirectly in the simulations with an analytical expression for the velocity, the Aristotelian dynamics. Nevertheless we do not follow individual particle trajectory because this would also require some assumption about the particle injection rate and its spatial dependence. To avoid such arbitrariness, we fixed the local charge density to the Gauss-Maxwell expectations \eqref{eq:Maxwell3}.
We observe a difference between positive and negative charges because they move in opposite direction with respect to the electric and magnetic field direction. Radiation feedback is taken into account by a friction term in the Lorentz force, opposite to the velocity, which is proportional to the radiative power $\mathcal{P}$, see eq.\eqref{eq:balance_lorentz_radiation}. We do not use any electron-ion plasma in thermal equilibrium, rather an electron-positron plasma, although the difference between ions and positrons is anecdotal because the particle mass does not intervene in the ultra-relativistic regime, like photons. We also do not have to worry about the fluid motion because particle fill the whole space with a charge density given by Gauss law. This procedure is very similar to its force-free avatar.
In the radiative regime, dissipation of electromagnetic field is controlled by the electric field strength~$E_0$ as measured in the frame where $\vec{E}$ and $\vec{B}$ are parallel because
\begin{equation}
\label{eq:jscalaireE}
\mathbf{j} \cdot \mathbf{E} = |\rho_e| \, ( 1 + 2 \, \kappa ) \, c \, E_0 \geq 0 .
\end{equation}
Expressed in terms of the radiated power we find
\begin{equation}\label{eq:dissipation}
\mathbf{j} \cdot \mathbf{E} = n_0 \, ( 1 + 2 \, \kappa ) \, \mathcal{P} .
\end{equation}
where $n_0 = |\rho_e/q|$ represents the particle density number required for the minimalistic model of a totally charge separated plasma.
We do not expect reconnection to play any role in the dissipation of the magnetic energy. The radiative Ohm's law behaves as a resistive term. FFE is therefore broken when switching to the radiative solution. All the losses funnel into the particle velocity component along the magnetic field, making particles moving approximately at the speed of light, therefore copiously radiating energy and momentum. The only requirement is that kinetic energy losses being compensated by the electric field work. The balance equation \eqref{eq:balance_lorentz_radiation} connects the radiative losses to the Lorentz force in a stationary state. In this picture we neglect particle inertia compared to the electromagnetic energy and radiative losses. To make an analogy with FFE, particle not only produce the required charge and current density but now they also produce some emission. Their dissipation rate is completely controlled by the radiative term and no reconnection is observed.
In the force-free limit, $E_0$ vanishes and dissipation disappears.
In the minimalistic view, the current~\eqref{eq:J_rad} is imposed only where necessary that is in regions where the condition $E<c\,B$ is violated whether inside or outside the light-cylinder. We call this model the radiative solution (RAD). We could also allow for less dissipation, for instance outside the light-cylinder, in the spirit of the force-free inside/dissipative outside approach of \cite{kalapotharakos_fermi_2017}. We call it force-free inside/radiative outside (FIRO).
\subsubsection{Force-free inside/radiative outside}
Inside the light-cylinder, the corotating electric field~$E$ remains less than $c\,B$. It is therefore always possible to set force-free conditions in this region. However, outside the light-cylinder, the $E$ field can easily surpass the $B$ field strength. In such cases, we can artificially decrease the $E$ strength as in the force-free model. However, as a less stringent method and more realistically, we let the system evolve by adding dissipation not requiring any condition on the $E$ field as allowed by the radiative current prescribed previously in Eq.~\eqref{eq:J_rad}. Therefore in a last regime, we enforce a force-free inside radiative outside (FIRO) model, allowing force-free conditions inside and radiative dissipation outside the light-cylinder.
\subsubsection{Numerical setup}
We performed several sets of runs with the aforementioned three regimes leading to a priori different magnetosphere models. The neutron star radius is set to $R/r_{\rm L}=0.3$ and the outer boundary of the simulation sphere is located at $7\,r_{\rm L}$ where the light-cylinder is defined by $r_{\rm L}=c/\Omega$. This allows us to clearly compute the base of the striped wind on almost one wavelength. The pair multiplicity is set by the user, we chose $\kappa=\{0,1,2\}$.
The pair multiplicity~$\kappa$ must always be a positive integer. It quantifies the deviation from a purely charge separated plasma. Indeed, in the minimalistic regime, a fully charge separated plasma requires $\kappa=0$. If some weak pair production occurs within the plasma we chose low values such as $\kappa=1,2,5$ or any small integer. For large multiplicities $\kappa \gg 1$, we will show that the solution tends quickly to the force-free magnetosphere when $\kappa$ augments. With $\kappa=2$ the magnetosphere configurations becomes already indistinguishable from the FFE case. The multiplicity $\kappa$ is intimately related to the pair production efficiency within the magnetosphere. PIC simulations have shown that the pair injection process, rate and location, crucially determines the outcome, tending either to a charge separated plasma forming an electrosphere or to an almost neutral plasma leading to a force-free and completely filled magnetosphere. The pair multiplicity remains so far largely unconstrained by observations. However, detailed numerical simulations of pair cascades around the polar caps performed by \cite{timokhin_time-dependent_2010, timokhin_current_2013} showed that values up to $\kappa=10^5$ can be expected. Such high multiplicity leads to an almost perfect force-free regime. Nevertheless, the current prescription containing $\kappa$ as the only free parameter in our model could be supplemented by the freedom in the background charge density $\rho_0$, possibly differing from the standard corotating prescription given by $\rho_{\rm e} \neq \rho_0$. This change in the background dynamics dramatically impacts the magnetosphere electrodynamics. Having no way to constrain this density $\rho_0$ we kept minimal assumption by imposing $\rho_0 = \rho_{\rm e}$.
The pulsar obliquity is denoted by the angle~$\rchi$. We implemented absorbing outer boundary conditions, meaning that the solution becomes unrealistic at distances $r>5\,r_{\rm L}$. In the following sections, we derive important quantities related to the pulsar electrodynamics such as its electromagnetic field structure, its spin-down losses, the work done on the plasma and the observational outcome relying on the polar cap shape, their light-curves and the slot gap/striped wind emission properties. A numerical grid of $N_r\times N_\theta \times N_\varphi = 257\times32\times64$ was sufficient for obtaining accurate solutions in all cases.
The adopted resolution stems from a convergence study of the spin down luminosity. Acceptable accuracy is reached whenever the luminosity at the light-cylinder has converged within 1\%.
We ran simulations for an orthogonal rotator with several radial and latitudinal grid points from the lowest resolution of $65\times16\times32$ to the highest resolution of $257\times128\times256$. The spin-down luminosity is plotted in Fig.\ref{fig:convergence} and shows that a resolution of $257\times32\times64$ or even $129\times32\times64$ is already sufficient for acceptable accuracy.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{convergence_r0.3_ro7_cfl0.5_a90_ba7_alp0.1_o3.png}
\caption{Convergence study for the spin down luminosity~$L$ normalized to the vacuum luminosity~$L_\perp$ of an orthogonal rotator depending on the grid resolution with $N_r$ radial points and $N_\theta$ latitudinal points.}
\label{fig:convergence}
\end{figure}
This seemingly low resolution is actually due to the very low dissipation of our fully pseudo-spectral method compared to finite volume or finite difference methods. The grid resolution can be much coarser for spectral methods, especially if the solution is continuous.
Eventually, as a check of our algorithm, including filtering, de-aliasing, absorbing boundary layers and resolution, we computed vacuum solutions comparing our results with expectations from the \cite{deutsch_electromagnetic_1955} solution.
\section{Magnetic field lines}
\label{sec:LigneChamp}
Electrodynamics of neutron stars relies heavily on its electromagnetic field. We therefore start by showing the magnetic field structure. A full 3D picture being difficult to visualize on a sheet of paper, we restrict ourself to the geometry of magnetic field lines in the meridional plane for an aligned rotator and in the equatorial plane for an orthogonal rotator. Such lines are shown in Fig.~\ref{fig:ligne_champ_j} for the FFE limit and different radiative regimes for an aligned rotator on the left panel and an orthogonal rotator on the right panel. Because the pair multiplicity factor only weakly impacts on the geometry, we only show the cases with $\kappa=0$. Compared to the cases~$\kappa \in \{1,2\}$ we have not found any significant changes, therefore they are not shown in Fig.~\ref{fig:ligne_champ_j}. Inside the light-cylinder, shown as a black dashed line on the left panel and as a circle on the right panel, the magnetic field of the FFE and radiative cases are very similar, whatever the pair multiplicity.
As expected the radiative solutions close more field lines along the equator outside the light-cylinder.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{lignes_champ_B_xz_a0.png} &
\includegraphics[width=\columnwidth]{lignes_champ_B_xy_a90.png}
\end{tabular}
\caption{Magnetic field lines for an aligned rotator, left panel, and for an orthogonal rotator, right panel, in the force-free limit (FFE) in blue, a minimalist radiative magnetosphere (RAD) in red and a FIRO magnetosphere in green, both latter with $\kappa=0$.
\label{fig:ligne_champ_j}}
\end{figure*}
Next we diagnose quantitatively the effect of a radiative magnetosphere by computing relevant physical quantities such as the spin down luminosity and the work done on the plasma.
\section{Poynting flux}
\label{sec:Luminosite}
A radiative magnetosphere has the interesting property to allow conversion of the Poynting flux into particle acceleration and radiation accounting for the feedback of this current onto the electromagnetic field in a self-consistent manner. In this section, we report the efficiency of Poynting flux decrease depending on the model and on the pair multiplicity. The electromagnetic flux must be compared to the reference situation of a force-free magnetosphere for which no dissipation is expected by construction. However, such solutions develop current sheets that are strongest for an aligned rotator. Numerically such discontinuities are tricky to handle especially for a spectral method where the Gibbs phenomenon easily arises. Some artificial dissipation must be introduced to avoid strong oscillations. However for radiative models, the current prescription naturally leads to some dissipation controlled by a physical parameter like resistivity or radiation damping.
For purely electromagnetic interactions, energy is shared between three quantities, the electromagnetic energy density~$u$ defined by
\begin{equation}
\label{eq:DensiteElectromagnetic}
u = \frac{\varepsilon_0 \, E^2}{2} + \frac{B^2}{2\,\mu_0}
\end{equation}
the Poynting flux defined by
\begin{equation}
\label{eq:FluxPoynting}
\mathbf{S} = \frac{\mathbf{E} \wedge \mathbf{B}}{\mu_0}
\end{equation}
and the work done on the plasma represented by current density interacting with the electric field at a rate
\begin{equation}\label{eq:Dissipation_j.E}
\mathcal{D} = \mathbf{j} \cdot \mathbf{E} .
\end{equation}
Particle are assumed to have zero inertia in these simulations. Their velocity is governed by Aristotelian dynamics according to the local electromagnetic field eq.~\eqref{eq:VRR}. In this respect, the lost energy is directly converted into radiation because of this zero lepton mass limit. The strength of the radiative feedback depends on the $E_0$ field which is proportional to the radiated power as shown decades ago by \cite{mestel_axisymmetric_1985}. Magnetic energy is dissipated not via reconnection but via radiation damping, impacting the particle velocity and leading to eq.~\eqref{eq:VRR}.
The dissipative term $\mathcal{D}$ vanishes for a force-free plasma and in the radiation reaction limit it reduces to expression~\eqref{eq:jscalaireE}.
The energy conservation law then reads
\begin{equation}
\label{eq:ConservationEnergie}
\frac{\partial u}{\partial t} + \mathbf{\nabla}\cdot \mathbf{S} + \mathbf{j} \cdot \mathbf{E} = 0 .
\end{equation}
In a stationary state, the electromagnetic energy density~$u$ remains unchanged. Without dissipation, the Poynting flux across a closed surface is conserved but with for instance radiative losses energy flows into the plasma. From the conservation law Eq.~\eqref{eq:ConservationEnergie} integrated within a sphere~$\Sigma$ of radius~$r$ we get
\begin{equation}
\label{eq:Travail}
\iint_\Sigma \mathbf{S} \cdot \mathbf{e}_{\rm r} \, d\Sigma = - \iiint_V \mathbf{j} \cdot \mathbf{E} \, dV
\end{equation}
where $\mathbf{S} \cdot \mathbf{e}_{\rm r} = S_{\rm r}$ is the radial component of the Poynting flux, $d\Sigma$ a surface element on the sphere and $dV$ a volume element inside the sphere~$\Sigma$.
The radial evolution of the Poynting flux is shown in Fig.~\ref{fig:Luminosite} for several models, the force-free (FFE), the force-free inside/radiative outside prescription (FIRO) and the minimalistic radiative approach (RAD). As a check, the vacuum solution (VAC) is also shown to estimate the numerical dissipation. The luminosity is normalized with respect to the vacuum point dipole orthogonal rotator
\begin{equation}\label{eq:spindown_dipole_vide}
L_{\rm vac} = \frac{8\,\upi}{3\,\mu_0\,c^3} \, \Omega^4 \, B^2 \, R^6
\end{equation}
such that the flux plotted is $\ell = L/L_{\rm vac}$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{luminosite_r0.3_ro7_n129_nt32_np64_cfl0.5_ba7_alp0.1_o3.png}
\caption{Radial decrease of the Poynting flux depending on the model. FFE is shown in blue, RAD in red and FIRO in green, both latter for $\kappa=0$. For reference, the vacuum case is also shown in yellow.}
\label{fig:Luminosite}
\end{figure}
The angular dependence on the obliquity~$\rchi$ is the same for all regimes, see Fig.~\ref{fig:Spindown}. All the plasma filled fits are well approximated by
\begin{equation}\label{eq:Lapprox}
L / L_{\rm vac} \approx 1.3 + 1.5 \, \sin^2\rchi .
\end{equation}
As a check, for the vacuum case we get
\begin{equation}\label{eq:LapproxVide}
L / L_{\rm vac} \approx 0.96 \, \sin^2\rchi .
\end{equation}
We observe an important dissipation of the Poynting flux for the FIRO case. In order to better localize this dissipative effect within the magnetosphere, we show in the next section the work done on the plasma for an aligned and an orthogonal case.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{spindown_r0.3_ro7_n129_nt32_np64_cfl0.5_ba7_alp0.1_o3.png}
\caption{The Poynting flux crossing the light-cylinder for oblique rotators in vacuum (VAC) in yellow, force-free (FFE) in blue, RAD in red and FIRO in green. The solid lines shows the best fit.\label{fig:luminosite_regime}}
\label{fig:Spindown}
\end{figure}
The sensitivity to the pair multiplicity~$\kappa$ is only weakly perceptible because the prescription for particle injection according to the local electric field via Maxwell-Gauss law already tends to the FFE limit for low multiplicities. What effectively controls the spin down losses and the magnetosphere solution, either closer to an electrosphere or to the FFE regime is the particle density number. PIC simulations have also shown that the injection procedure is critical for the final outcome of the simulation.
Fig.~\ref{fig:comparaison} shows the different spin down luminosities at the light cylinder depending on the plasma model. The variation in luminosities remains very small except although the radiative solution seems slight more dissipative. The sensitivity to the pair multiplicity is also only weakly perceptible. Most of the dissipation occurs outside the light-cylinder as shown in the next section.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{comparaison_r0.3_ro7_n129_nt32_np64_cfl0.5_ba7_alp0.1_o3.png}
\caption{Comparison of spin down luminosities at the light cylinder depending on the plasma model. The integer within the bracket denotes the pair multiplicity factor~$(\kappa)$.}
\label{fig:comparaison}
\end{figure}
The spin down luminosity is relatively insensitive to the dissipation term because the particle injection scheme follows the force-free scheme by dropping particles where the Maxwell-Gauss law imposes it. Therefore the radiative model tends quickly to a nearly FFE state, even with a low to moderate multiplicity factor~$\kappa$.
\section{Dissipation}
\label{sec:Dissipation}
Conservation of the total energy implies that some electromagnetic energy went into particle acceleration and radiation. The radial decrease in the Poynting flux~$L$ indicates a sink of electromagnetic energy imputed to the presence of a non ideal plasma. The location where this conversion arises is important for the prediction of observational signatures such as radio and gamma-ray light-curves and spectra. Within a spherical shell of radius~$r$, this dissipation is given by the opposite of the Poynting flux radial derivative as
\begin{equation}
\label{eq:DeriveeTravail}
W_E = \iint_\Sigma \mathbf{j} \cdot \mathbf{E} \, d\Sigma = - \frac{dL}{dr} .
\end{equation}
Fig.~\ref{fig:dLsdr} shows how fast dissipation occurs outside the light-cylinder depending on the radius.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{dLsdr_r0.3_ro7_n129_nt32_np64_cfl0.5_ba7_alp0.1_o3_firo.png} &
\includegraphics[width=\columnwidth]{dLsdr_r0.3_ro7_n129_nt32_np64_cfl0.5_ba7_alp0.1_o3_rad.png}
\end{tabular}
\caption{Efficiency of the dissipation according to the integral in Eq.~\eqref{eq:DeriveeTravail} for the FIRO model on the left panel and for the RAD model on the right panel. The pair multiplicity is shown in the legend with different colours. For each multiplicity~$\kappa$, the dissipation is shown for all obliquities in the same colour.}
\label{fig:dLsdr}
\end{figure*}
For FFE magnetospheres, dissipation vanishes but because of our numerical filtering procedure and of grid size effects, a small residual work is observed. In the FIRO model, dissipation starts at the light cylinder, increasing to a maximum in the interval $[1,2]\,r_{\rm L}$ and then decreases slowly. In the RAD regime, the dissipation sets in with a delay, increasing significantly only beyond a radius $r \gtrsim 2\,r_{\rm L}$. This delayed dissipation impacts on the radio time lag of gamma-rays photons with respect to radio photons. We expect to observe a decrease in this time lag for radio loud gamma-ray pulsar, helping to better jointly fit radio and gamma-ray light-curves, as done in \cite{petri_young_2021}.
In order to better localize the radiative regions where the particle dynamics is the most important, we show maps of the work done locally on the plasma by computing the power defined in Eq.~\eqref{eq:jscalaireE} for aligned and orthogonal rotators. Enlightening cases are shown for FIRO and RAD models in Fig.~\ref{fig:dissipation_aligne} for an aligned rotator and in fig.~\ref{fig:dissipation_orthogonal} for an orthogonal rotator. The Poynting flux flows into the plasma outside the light-cylinder in the vicinity of the current sheet of the striped wind. With increasing distance, the power sharply decreases by two orders of magnitude at the outer boundary because of the decreasing electric field and current density. The thickness of this dissipative region is about $0.2\,r_{\rm L}$.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{dissipation_aligne_r0.3_ro7_n257_nt64_np1_cfl0.5_ba7_alp0.1_o3_j2_ka0.png} &
\includegraphics[width=\columnwidth]{dissipation_aligne_r0.3_ro7_n257_nt64_np1_cfl0.5_ba7_alp0.1_o3_j3_ka0.png}
\end{tabular}
\caption{Work done on the plasma for $\kappa=0$ as given by Eq.~(\ref{eq:jscalaireE}), for an aligned rotator with the FIRO model on the left panel and the RAD model on the right panel.}
\label{fig:dissipation_aligne}
\end{figure*}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{dissipation_perp_r0.3_ro7_n257_nt64_np128_cfl0.5_ba7_alp0.1_o3_j2_ka0.png} &
\includegraphics[width=\columnwidth]{dissipation_perp_r0.3_ro7_n257_nt64_np128_cfl0.5_ba7_alp0.1_o3_j3_ka0.png}
\end{tabular}
\caption{Work done on the plasma for $\kappa=0$ as given by Eq.~(\ref{eq:jscalaireE}), for an orthogonal rotator with the FIRO model on the left panel and the RAD model on the right panel.
}
\label{fig:dissipation_orthogonal}
\end{figure*}
This dissipation layers are the privileged places where high-energy radiation is produced and detected as pulsed gamma-ray emission. This fact is supported by the investigation of radio loud young gamma-ray pulsars for which the rotating vector model is consistent with gamma-ray light-curve fitted for more than a dozen of pulsars \citep{petri_young_2021}.
The acceleration and radiation processes are implicitly implemented by the Aristotelian velocity dynamics. Particles move at the speed of light because of zero inertia approximation and the radiated power is controlled by the $E_0$ field. Particles in the simulations are present but they only contribute to the charge and current density as in the FFE approximation.
In the closed field zone, within the light-cylinder, for the FIRO model, we found a dissipation rate $\mathcal{D}$ which is less than $10^{-4}$ or even $10^{-5}$. Such small values are almost zero from a numerical point of view. The plasma really remains force-free as it should within the numerical error of the algorithm.
The connection between dissipation layers and radiation zones has been confirmed by kinetic simulations such as \cite{philippov_ab-initio_2018} and \cite{chen_filling_2020}.
The current sheet thickness is governed by the local physics, the breakdown of the FFE conditions. It is controlled by the radiative term and not by numerical dissipation which has been checked to remain negligible compared to the dissipation introduced by the radiative Ohm law.
\section{Parallel electric field}
\label{sec:parallel_E}
In order to quantify the presence of a parallel electric field~$E_\parallel$, we plot some maps of the strength of $E_\parallel = \mathbf{E} \cdot \mathbf{B} / B = E_0\,B_0/B$ in the observer frame. This component of the electric field could be responsible for particle acceleration and therefore represents a good indicator of the deviation from force-free conditions. Actually, significant values of $E_\parallel$ are coincident with the dissipation maps shown in the previous section, in Fig.~\ref{fig:dissipation_aligne} and \ref{fig:dissipation_orthogonal}.
Fig.~\ref{fig:E_para_aligne} shows two $E_\parallel$ maps for an aligned rotator with $\kappa=0$, for the FIRO model on the left panel and for the RAD model on the right panel.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{E_para_aligne_r0.3_ro7_n257_nt64_np1_cfl0.5_ba7_alp0.1_o3_j2_ka0.png} &
\includegraphics[width=\columnwidth]{E_para_aligne_r0.3_ro7_n257_nt64_np1_cfl0.5_ba7_alp0.1_o3_j3_ka0.png}
\end{tabular}
\caption{Absolute value of the parallel electric field~$E_\parallel$ for an aligned rotator, right panel, in the minimalist radiative magnetosphere (RAD) in red and a FIRO magnetosphere in green, both with $\kappa=0$.}
\label{fig:E_para_aligne}
\end{figure*}
Fig.~\ref{fig:E_para} shows the same quantities for an orthogonal rotator.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{E_para_0.3_ro7_n257_nt64_np128_cfl0.5_ba7_alp0.1_o3_j2_ka0.png} &
\includegraphics[width=\columnwidth]{E_para_0.3_ro7_n257_nt64_np128_cfl0.5_ba7_alp0.1_o3_j3_ka0.png}
\end{tabular}
\caption{Same as fig.~\ref{fig:E_para_aligne} but for an orthogonal rotator.}
\label{fig:E_para}
\end{figure*}
Particles do not follow field lines any more due to dissipation. They are efficiently accelerated by the electric field~$\mathbf{E}$ along the magnetic field~$\mathbf{B}$ in regions where dissipation is maximal.
Particles are evolved implicitly according to eq.\eqref{eq:VRR}. They follow trajectories imposed by the local electromagnetic field $\mathbf{(E,B)}$ and possess a component along $\mathbf{E}$. This electric field aligned acceleration captures non ideal effects deviating from the pure force-free picture. $\mathbf{E}$ can be larger than $\mathbf{c\, B}$ and there exist a parallel electric field component $E_\parallel$ responsible for these non ideal effects.
There are no strong FFE violation because the particle injection scheme resembles very much to the force-free scheme and therefore the radiative model is still able to tends to a nearly FFE state. The spin down is not much affected but the electric field and the corresponding dissipation rate term eq.\eqref{eq:Dissipation_j.E} are very sensitive to the radiative mechanism however only in very localized areas where $E>c\,B$.
\section{Polar caps}
\label{sec:Calottes}
As a preparation for the investigation of the radio and gamma-ray light-curves of radio loud gamma-ray pulsars detected by Fermi/LAT and related to our radiative magnetosphere, we compute the shape of the polar cap in the different plasma regimes, comparing them to the force-free limit. For better readability with different obliquities $\rchi$, the origin of the plots corresponds to the location of the magnetic north pole, the axes are defined locally for each obliquity by performing a rotation from the rotation axis to the magnetic axis.
Illustrative examples are shown in fig.~\ref{fig:polar_cap} for the polar cap rim with $\rchi = \{15\degr,45\degr,75\degr\}$, in vacuum (VAC), force-free (FFE), FIRO and RAD regimes. The vacuum polar cap shapes computed from our pseudo-spectral simulations are shown in orange solid line and checked against those polar caps found from the exact analytical Deutsch solution and shown in dashed blue lines. The agreement between both contours is excellent and gives us confidence about our results for plasma fields magnetospheres. The polar caps for the FFE, FIRO and RAD regimes are also shown in fig.~\ref{fig:polar_cap}, respectively in blue, green and red solid line. We have not noticed any significant change in these caps and their rims almost overlap whatever the regime.
Nevertheless, compared to vacuum, the area of these polar caps is larger than in the vacuum case because magnetic field lines open up due to the magnetospheric current. The presence of the plasma inflates the caps.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{polar_cap.png}
\caption{Polar cap geometry for a rotator with obliquity $\rchi=\{15\degr,45\degr,75\degr\}$ in vacuum (VAC), force-free (FFE) or in the radiation reaction limit, respectively FIRO and RAD. The blue dashed line corresponds to the polar cap computed from the exact Deutsch solution.}
\label{fig:polar_cap}
\end{figure*}
We conclude that the impact of radiation on the polar cap shape remains rather weak when compared to the force-free solution. Even if the ideal plasma approximation does not produce any parallel electric fields that could accelerate particles and therefore does not produce any radiation, from a geometrical point of view, it nevertheless offers a reasonably faithful picture of the electromagnetic field of a dissipative magnetosphere, wherever the radiative dissipation occurs, everywhere outside the light-cylinder or only where required by local conditions imposed by the electromagnetic field. We stress that within a fluid model, as the one employed in our study, the acceleration mechanism cannot produce power law distribution functions. It only heats up particles by keeping them thermal. This acceleration is accounted for by the parallel electric fields~$E_\parallel$ component for a resistive plasma for which magnetic energy is dissipated into particle kinetic energy. This requires an Ohm's law like eq.\eqref{eq:J_rad} which allows the presence of a parallel electric field~$E_\parallel$. Injecting test particles into this configuration would offer a good compromise between a fully kinetic simulation and a fluid approximation, leading to possible non thermal acceleration and building power law distribution functions for those particles.
The observation that force-free and dissipative magnetospheres look rather similar relies mainly on the fact that the particle injection scheme derives from Maxwell-Gauss law, sowing particles locally at the rate imposed by the electric field. In such an approach, a fully charge separated plasma present in the whole magnetosphere deviates only slightly from the force-free counterpart. This conclusion is not an artefact from the grid resolution which could impact on the physical dissipative term but a natural consequence of the lepton injection method. Replacing the charge density derived from Maxwell-Gauss by another explicit spatio-temporal injection dependence would lead to more effective dissipation. But this is at the expense of adding more arbitrariness into the model that we wanted to avoid in the present investigation.
Small changes on the stellar surface are amplified at the light-cylinder and beyond, therefore we could expect a significant change in the multi-wavelength light-curve predictions. This last point connecting our simulations to observations is touched in the next section through comparison of sky maps in the radio and the gamma-ray band.
\section{Pulsed emission}
\label{sec:emission}
Dissipative magnetospheres are necessary to produce some radiation as detected by a distant observer. The location and geometry of the emission regions strongly imprint on the multi-wavelength light-curves and the phase-resolved spectra. In this last section, we compute sky maps and light-curves for the aforementioned models, highlighting the differences expected depending on the plasma regime, ideal or radiative.
We consider the three main emission regions to be, first the polar cap for radio photons, second the slot gap for high-energy gamma-ray photons and third the striped wind model for high and very high-energy gamma-rays up to the TeV range. For radio photons, we assume a polar cap model with emissivity shaped by a Gaussian function centred on the magnetic axis. For gamma-ray photons, we assume a striped wind model with emissivity starting at the light-cylinder and focused along the current sheet or a slot gap extending from the surface (meaning here 0.3~$r_{\rm L}$) up to the light-cylinder. Some more details about these emission models can be found in \cite{petri_general-relativistic_2018}.
In our model, the size of the radio cone emission is controlled by the polar cap rims computed in the previous section. We assume that photon are produced with an altitude in the interval $[0.3,0.4]\,r_{\rm L}$. In this region close to the stellar surface, the electric field remains weak compared to the magnetic field and particles follow almost magnetic field lines. Photons are therefore shoot in a direction tangential to the magnetic field lines, including retardation and aberration effects. Moreover, we implemented a Gaussian profile centred along the magnetic moment axis with a typical width equal to the cone supported by the last open field lines. We adopted this picture in order to mimic the true radio profiles observed in many pulsars.
Gamma-ray photons are produced in the equatorial current within the striped wind, outside the light-cylinder. Because the wind is expanding almost radially, these photons are emitted in the radial direction. The emissivity is maximal at the centre of the current sheet and decreases following another Gaussian shape when deviating from this sheet. The emission zone extend from $1r_{\rm L}$ to $3r_{\rm L}$. Much more details can be found in \cite{benli_constraining_2021} and \cite{petri_young_2021} and references therein.
In order to probe the radio emission mechanism from the polar cap and the gamma-ray emission from higher altitudes, kinetic physics is required to capture the non-ideal electric field, the gap formation, and the pair production. However such study is out of the present scope. Attempts to better understanding the radio emission generation have been pursued by for instance \cite{philippov_origin_2020} and for the gamma-ray emission by for instance \cite{kalapotharakos_fermi_2017}. Here we are only interested in the radio and high energy pulse profiles implied by the geometrical configuration of radiative magnetospheres. We occult the detailed energetics of individual particles, radiation and electromagnetic interactions.
\subsection{Sky maps}
We start by reckoning a full set of light-curves in radio and high-energy, following the three emission regions. The combined polar cap/striped sky maps are summarized in Fig.~\ref{fig:carte_complete_vent} for $\rchi = \{15\degr,45\degr,75\degr\}$ and the combined polar cap/slot gap is summarized in Fig.~\ref{fig:carte_complete_cavite} for the same obliquities. It is also instructive to show the expectations from the Deutsch vacuum solution. The observer line of sight inclination~$\zeta$ varies from 0\degr to 180\degr. Each column in this plot depicts a particular model. From left to right column we have successively the vacuum (VAC), the force-free (FFE), the FFE inside/radiative outside (FIRO), and the radiative (RAD) regimes. Each line represents a different obliquity given from top to bottom by $\rchi = \{15\degr,45\degr,75\degr\}$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{carte_complete_vent.png}
\caption{Sky maps of the ideal and radiative models compared to the vacuum case. From left to right column: vacuum (VAC), force-free (FFE), FFE inside/radiative outside (FIRO), and radiative (RAD) on the right. Each line represents a different obliquity given from top to bottom by $\rchi = \{15\degr,45\degr,75\degr\}$. The gamma-ray emission assumes a striped wind model.}
\label{fig:carte_complete_vent}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{carte_complete_cavite.png}
\caption{Same as fig.~\ref{fig:carte_complete_vent} but for the gamma-ray emission assumes a slot gap model.}
\label{fig:carte_complete_cavite}
\end{figure}
Interestingly, the force-free maps are very similar to the radiative maps.
The FFE case and radiative runs are different in their details. Radiative models clearly show some regions where the electric field exceeds the magnetic field $E>c\,B$ and some dissipative regions where $\mathcal{D}$ is non negligible. Whereas the global magnetosphere is not drastically impacted by the local dissipative terms, the sky maps are.
Even the vacuum solution already produces maps resembling the plasma filled cases, especially when looking close to the equator ($\zeta \approx 90\degr$). Inspecting more carefully Fig.~\ref{fig:carte_complete_vent} and Fig.~\ref{fig:carte_complete_cavite}, we conclude that the actual plasma regime only weakly impacts on the sky maps, whether in radio or in gamma-rays. From a geometrical point of view, when investigating light-curve shapes, the force-free limit allows for an accurate study of pulse profile without adding any free parameter into the game. The geometric dependence on $\rchi$ and $\zeta$ is already faithfully reproduced in FFE. However, when energetic considerations come into play, the radiative models will generate very different phase-resolved spectra and multi-wavelength light curves because of the varying parallel electric field acting on the particle dynamics. This requires deeper investigation of particle acceleration and radiation that we leave for future work.
\subsection{Light curves}
As a typical example of different light curves constructed from these models, we plot an atlas of gamma-ray light curves in Figure~\ref{fig:atlasgamma} for $\rchi=\{15\degr,45\degr,75\degr\}$ and in steps of 10\degr for $\zeta\in[0\degr,90\degr]$, according to the striped wind and the slot gap model in the force-free limit.
\begin{figure*}
\centering
\includegraphics[width=0.95\linewidth]{atlas_gamma.png}
\caption{Atlas of striped wind and slot gap light-curves for $\rchi = \{15\degr,45\degr,75\degr\}$ and $\zeta\in[0\degr,90\degr]$ in steps of 10\degr, see the inset in the format $\{\rchi, \zeta\}$. The first, third and fifth column are for the striped wind whereas the second, fourth and sixth column are for the slot gap.}
\label{fig:atlasgamma}
\end{figure*}
For the plasma filled models, we only notice a variation of several percent in phase lag between the light curves. To a large extent, the double peak gamma-ray separation remains insensitive to the model used. Discrepancies in phase are difficult to detect in real gamma-ray observations. However, more importantly are the variations in the peak maximal intensity between the models, notably the reversal of the dominant peak, leading or trailing with respect to radio, when switching from FFE/FIRO to RAD model, see Fig.~\ref{fig:courbe_lumiere} with $\{\rchi,\zeta\}=\{45\degr,40\degr\}$ on the left panel and $\{\rchi,\zeta\}=\{75\degr,60\degr\}$ on the right panel.
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.45\columnwidth]{courbe_lumiere_a45_z40.png} &
\includegraphics[width=0.45\columnwidth]{courbe_lumiere_a75_z60.png}
\end{tabular}
\caption{Examples of gamma-ray light curves extracted from the sky maps shown in figure~\ref{fig:carte_complete_vent} and \ref{fig:carte_complete_cavite} with the geometry given in the inset. On the top panel the results for the striped wind and on the bottom panel for the slot gap.}
\label{fig:courbe_lumiere}
\end{figure}
The slot gap model produces much wider profiles, broader than what is observed by Fermi/LAT such that we favour the striped wind to explain GeV light-curves although for some pulsar showing a kind of plateau emission the slot gap would better fit the observations. As a conclusion, we think that the majority of the gamma-ray pulsars fall into the striped wind emission model except for some outliers possibly also dominantly emitting in the slot gap sites.
\section{Conclusions}
\label{sec:Conclusion}
Finding self-consistent dissipative pulsar magnetospheres is important to localize possible sites for their broadband pulsed emission. The underlying processes of particle acceleration and radiation remain to be properly identified. In this paper, we constructed self-consistent radiative pulsar magnetospheres in the radiation reaction regime where ultra-relativistic particles flow around the neutron star. We introduced partially and minimalistic radiative models, demonstrating that the global electromagnetic topology remains mainly insensitive to the actual dissipation regime. In any case, most of the radiation occurs in the current sheet outside the light-cylinder but close to it.
Although different models produce marginal differences in the gamma-rays and radio light-curves, we believed that current observations especially in high-energy are not sensitive enough to disentangle for instance the minimalistic from the partially radiative magnetosphere. For better comparison with observations, the computation of phase-resolved spectra and multi-wavelength light-curves will undoubtedly help to segregate between competing dissipative models like the radiative or resistive magnetospheres introduced in the literature. The energetic of the magnetosphere model will leave the degeneracy contrary to a pure geometrical study of light-curves. As phase-resolved spectra are available in radio and gamma-rays, we plan to compute multi-wavelength light-curves and spectra based on the above magnetosphere models.
The physics of radiative magnetospheres requires a more detailed investigation of the central role of particle injection and its crucial impacts on the magnetosphere energetics before to confront to the observations. Some more ingredients are required before investigating the gamma-ray light curves of individual pulsars.
A large amount of work has been carried out to identify these effects via in particle-in-cell simulations, taking the feedback between particle acceleration and radiation self-consistently into account. Relativistic magnetic reconnection is supposed to play a key role in the dissipation of the Poynting flux channelling into particle acceleration and emitting synchrotron photons as demonstrated by \cite{cerutti_modelling_2016} and \cite{philippov_ab-initio_2018}. The particle production assumption, either from the surface or from the light-cylinder or from the whole magnetosphere volume decides what on the global solution found in numerical simulations. \cite{chen_filling_2020} presented simulations not requiring efficient pair production in the vicinity of the light cylinder and found quasi-periodic solution able to explain the cone versus and core emission characteristics of radio pulses. Previously \cite{chen_electrodynamics_2014} already found significant curvature and synchrotron photon production around the current sheet of an aligned rotator. These results are able to reproduce the Fermi gamma-ray pulsar observations. In the same vain, \cite{kalapotharakos_fermi_2017} computed test particle trajectories and the associated radiation and particle efficiency in the so called FIDO model assuming curvature emission in the radiation reaction, in a similar way to the present study. Based on gamma-ray pulsar luminosity, they found a positive correlation between the pair multiplicity factor and the pulsar spin down.
Nevertheless almost all these simulations rely on particle injection prescriptions which are still not fully resolved from a physical point of view. As the magnetospheric solution tend to depends crucially on the injection rate, it is not yet clear how all these processes operate. Moreover, the Lorentz factors obtained are still many orders of magnitude below realistic values expected from observations. Our approach represents a good alternative to tackle this important issue of very high Lorentz factor within the magnetosphere.
\section*{Acknowledgements}
I am grateful to the referee for helpful comments and suggestions.
This work has been supported by the CEFIPRA grant IFC/F5904-B/2018 and ANR-20-CE31-0010. We acknowledge the High Performance Computing center of the University of Strasbourg for supporting this work by providing scientific support and access to computing resources.
\section*{Data availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
|
2,869,038,155,665 | arxiv | \section{Introduction}
Topological groupoids are extensively used in dynamics, topology,
non-commutative geometry, and
$C^*$-algebras,
see~\cite{haefl:foliations,paterson:gr,renault:groupoids}.
With recent results on topological full groups (see~\cite{matui:etale,juschenkomonod,YNS})
new applications of groupoids to group theory were
discovered.
Our paper studies growth and complexity for \'etale groupoids
with applications to the theory of growth and Gelfand-Kirillov
dimension of algebras. We give examples of groupoids
whose convolution algebras (over an arbitrary field) have prescribed growth.
In particular, we give first examples of simple algebras of quadratic
growth over finite fields and simple algebras of Gelfand-Kirillov
dimension 2 that do not have quadratic growth.
A \emph{groupoid} $\mathfrak{G}$ is the set of isomorphisms of a small category,
i.e., a set $\mathfrak{G}$ with partially defined multiplication
and everywhere defined operation of taking inverse satisfying the
following axioms:
\begin{enumerate}
\item If the products $ab$ and $bc$ are defined, then $(ab)c$ and $a(bc)$ are
defined and are equal.
\item The products $a^{-1}a$ and $bb^{-1}$ are always defined and
satisfy $abb^{-1}=a$ and $a^{-1}ab=b$ whenever the product $ab$ is defined.
\end{enumerate}
It follows from the axioms that $(a^{-1})^{-1}=a$ and that a product
$ab$ is defined if and only if $bb^{-1}=a^{-1}a$.
The elements of the form $aa^{-1}$ are called \emph{units} of the
groupoid. We call $\mathsf{o}(g)=g^{-1}g$ and $\mathsf{t}(g)=gg^{-1}$ the \emph{origin}
and the \emph{target} of the element $g\in\mathfrak{G}$.
A \emph{topological groupoid} is a groupoid together with topology
such that multiplication and taking inverse are continuous.
It is called \emph{\'etale} if every element has a basis of neighborhoods
consisting of \emph{bisections}, i.e., sets $F$ such that
$\mathsf{o}:F\longrightarrow\mathsf{o}(F)$ and $\mathsf{t}:F\longrightarrow\mathsf{t}(F)$ are homeomorphisms.
For example, if $G$ is a discrete group acting (from the left)
by homeomorphisms on a topological space
$\mathcal{X}$, then the topological space $G\times\mathcal{X}$ has a natural structure of an \'etale
groupoid with respect to the multiplication
\[(g_1, g_2(x))\cdot (g_2, x)=(g_1g_2, x).\]
In some sense \'etale groupoids are generalization of actions of
discrete groups on topological spaces.
We consider two growth functions for an \'etale
groupoid $\mathfrak{G}$ with compact totally disconnected
space of units. The first one is the most straightforward and classical: growth of
fibers of the origin map. If $S$ is an open compact generating set of
$\mathfrak{G}$ then, for a given unit $x$, we can consider the growth function $\gamma_S(r,
x)$ equal to the
number of groupoid elements with origin in $x$ that can be expressed
as a product of at most $n$ elements of $S\cup S^{-1}$. This notion of
growth of a groupoid has appeared in many situations, especially in
amenability theory for topological groupoids,
see~\cite{kaim,delaroche_renault}.
See also Theorem~\ref{th:nofree} of our paper,
where for a class of groupoids we show how sub-exponential growth
implies absence of free subgroups in the topological full group of the
groupoid.
This notion of growth does not capture full complexity of a groupoid
precisely because it is ``fiberwise''. Therefore, we introduce the
second growth function: complexity of the groupoid. Let $\mathcal{S}$
be a finite covering by open bisections of an open
compact generating set $S$ of $\mathfrak{G}$. For a given natural number $r$
and units $x, y\in\mathfrak{G}^{(0)}$ we write $x\sim_r y$ if for any two
products $S_1S_2\ldots S_n$ and $R_1R_2\ldots R_m$ of elements of
$\mathcal{S}\cup\mathcal{S}^{-1}$ such that $n, m\le r$ we have $S_1S_2\ldots
S_nx=R_1R_2\ldots R_mx$ if and only if $S_1S_2\ldots S_ny=R_1R_2\ldots
R_my$. In other words, $x\sim_r y$ if and only if balls of radius $r$
with centers in $x$ and $y$
in the natural $\mathcal{S}$-labeled \emph{Cayley graphs} of
$\mathfrak{G}$ are isomorphic. Then the \emph{complexity function} $\delta(r, \mathcal{S})$ is
the number of $\sim_r$-equivalence classes of points of $\mathfrak{G}^{(0)}$.
This notion of complexity (called in this case \emph{factor complexity}, or
\emph{subword complexity}) is well known and studied for groupoids of
the action of shifts on closed
shift-invariant subsets of $X^{\mathbb{Z}}$, where $X$ is a finite
alphabet. There is an extensive literature on it,
see~\cite{cassaigneets:factorcomplexity,ferenczi:complexity}
An interesting result from the group-theoretic point of view is a
theorem of
N.~Matte~Bon~\cite{mattebon:liouville} stating
that if complexity of a subshift is
strictly sub-quadratic, then the topological full group of the
corresponding groupoid is Liouville. Here the \emph{topological full
group} of an \'etale groupoid $\mathfrak{G}$ is the group of all
$\mathfrak{G}$-bisections $A$ such that $\mathsf{o}(A)=\mathsf{t}(A)=\mathfrak{G}^{(0)}$.
It seems that complexity of groupoids in more general
\'etale groupoids has not been well studied yet. It would be
interesting to understand how complexity function (together with the
growth of fibers) is related with the properties of the topological
full group of an \'etale groupoid. For example, it would be
interesting to know if there exists a non-amenable (e.g., free) group
acting faithfully on a compact topological space so that the
corresponding groupoid of germs has sub-exponential growth and
sub-exponential complexity functions.
We relate growth and complexity of groupoids with growth of algebras
naturally associated with them. Suppose that $\mathcal{A}$ is a finitely
generated algebra with a unit over a field $\Bbbk$. Let
$V$ be the $\Bbbk$-linear span of
a finite generating set containing the unit. Denote by $V^n$ the linear
span of all products $a_1a_2\ldots a_n$ for $a_i\in V$. Then
$\mathcal{A}=\bigcup_{n=1}^\infty V^n$. \emph{Growth} of $\mathcal{A}$
is the function
\[\gamma(n)=\dim V^n.\]
It is easy to see that if $\gamma_1, \gamma_2$ are growth functions
defined using different finite generating sets, then there exists $C>1$ such
that $\gamma_1(n)\le\gamma_2(Cn)$ and $\gamma_2(n)\le\gamma_1(Cn)$.
\emph{Gelfand-Kirillov} dimension of $\mathcal{A}$ is defined as
$\limsup_{n\to\infty}\frac{\log\dim V^n}{\log n}$, which informally is
the degree of polynomial growth of the algebra. If $\mathcal{A}$ is
not finitely generated, then its Gelfand-Kirillov dimension is defined
as the supremum of the Gelfand-Kirillov dimensions of all its sub-algebras.
See the monograph~\cite{krauselenagan} for a survey of results on growth of algebras
and their Gelfand-Kirillov dimension.
It is known, see~\cite{warfield:gk}
and~\cite[Theorem~2.9]{krauselenagan}, that Gelfand-Kirillov dimension can be any
number in the set $\{0, 1\}\cup [2, \infty]$. The values in the interval
$(1, 2)$ are prohibited by a theorem of G.M.~Bergman,
see~\cite[Theorem~2.5]{krauselenagan}.
There are examples of prime algebras of
arbitrary Gelfand-Kirillov dimension $d\in [2, \infty]$, see~\cite{vishne:gk}, but
it seems that no examples of simple algebras of arbitrary Gelfand-Kirillov dimension
over finite fields were known so far.
A naturally defined \emph{convolution algebra} $\Bbbk[\mathfrak{G}]$ over
arbitrary field $\Bbbk$ is
associated with every \'etale groupoid $\mathfrak{G}$ with totally
disconnected space of units. If the groupoid $\mathfrak{G}$ is Hausdorff, then
$\Bbbk[\mathfrak{G}]$ is the convolution algebra of all continuous functions
$f:\mathfrak{G}\longrightarrow\Bbbk$ with compact support, where $\Bbbk$ is taken with the
discrete topology. Here convolution $f_1\cdot f_2$ of two functions
is the function given by the formula
\[f(g)=\sum_{g_1g_2=g}f_1(g_1)f_2(g_2).\]
In the non-Hausdorff case we follow A.~Connes~\cite{conn:foliations}
and B.~Steinberg~\cite{steinberg:groupoidapproach}, and define $\Bbbk[\mathfrak{G}]$ as the linear span of
the functions that are continuous on open compact subsets of
$\mathfrak{G}$. Equivalently, $\Bbbk[\mathfrak{G}]$ is the linear span of the
characteristic functions of open compact $\mathfrak{G}$-bisections.
Note that
the set $\mathcal{B}(\mathfrak{G})$ of all open compact $\mathfrak{G}$-bisections (together with the empty
one) is a semigroup. The algebra $\Bbbk[\mathfrak{G}]$ is isomorphic to
the quotient of the semigroup algebra of $\mathcal{B}(\mathfrak{G})$
by the ideal generated by the relations $F-(F_1+F_2)$ for
all triples $F, F_1, F_2\in\mathcal{B}(\mathfrak{G})$ such that
$F=F_1\cup F_2$ and $F_1\cap F_2=\emptyset$.
We prove the following relation between growth of groupoids and growth
of their convolution algebras.
\begin{theorem}
\label{th:main}
Let $\mathfrak{G}$ be an \'etale groupoid with compact totally disconnected
space of units. Let $\mathcal{S}$ be a finite set of open compact
$\mathfrak{G}$-bisections such that $S=\bigcup\mathcal{S}$ is a generating set of
$\mathfrak{G}$. Let $V\subset\Bbbk[\mathfrak{G}]$ be the linear span of the characteristic
functions of elements of $\mathcal{S}$. Then
\[\dim V^n\le\overline\gamma(r, \mathcal{S})\delta(r, \mathcal{S}),\]
where $\overline\gamma(r,
\mathcal{S})=\max_{x\in\mathfrak{G}^{(0)}}\gamma_S(r, x)$.
\end{theorem}
We say that a groupoid $\mathfrak{G}$ is \emph{minimal} if every $\mathfrak{G}$-orbit is
dense in $\mathfrak{G}^{(0)}$. We say that $\mathfrak{G}$ is \emph{essentially principal}
if the set of points $x$ with trivial isotropy group
is dense in $\mathfrak{G}^{(0)}$. Here the isotropy group of a point $x$ is the
set $\{g\in\mathfrak{G}\;:\;\mathsf{o}(g)=\mathsf{t}(g)=x\}$. It is known,
see~\cite{brownclarketc:simplicity}, that for a Hausdorff minimal essentially principal groupoid
$\mathfrak{G}$ with compact totally disconnected set of units the algebra
$\Bbbk[\mathfrak{G}]$ is simple. We give a proof of this fact for completeness in
Proposition~\ref{pr:simple}.
We give in Proposition~\ref{prop:expansive} a condition
(related to the classical notion of an \emph{expansive dynamical system})
ensuring that $\Bbbk[\mathfrak{G}]$ is finitely generated.
Fibers of the origin map provide us with naturally defined
$\Bbbk[\mathfrak{G}]$-modules. Namely, for a given unit $x\in\mathfrak{G}^{(0)}$ consider
the vector space $\Bbbk\mathfrak{G}_x$ of functions $\phi:\mathfrak{G}_x\longrightarrow\Bbbk$ with
finite support, where $\mathfrak{G}_x=\mathsf{o}^{-1}(x)$ is the set of elements of the
groupoid $\mathfrak{G}$ with origin in $x$. Then convolution $f\cdot \phi$ for
any $f\in\Bbbk[\mathfrak{G}]$ and $\phi\in\Bbbk\mathfrak{G}_x$ is an element of
$\Bbbk\mathfrak{G}_x$, and hence $\Bbbk\mathfrak{G}_x$ is a left $\Bbbk[\mathfrak{G}]$-module.
It is easy to prove that if the isotropy group of $x$ is trivial, then
$\Bbbk\mathfrak{G}_x$ is simple and that growth of $\Bbbk\mathfrak{G}_x$ is bounded by
$\gamma_S(x, r)$, see Proposition~\ref{pr:modules}.
As an example of applications of these results, we consider the
following family of algebras. Let $X$ be a finite alphabet, and let
$w:X\longrightarrow\mathbb{Z}$ be a bi-infinite sequence of elements of $X$.
Denote by $D_x$, for $x\in X$ the diagonal matrix $(a_{i, j})_{i,
j\in\mathbb{Z}}$ given by
\[a_{i, j}=\left\{\begin{array}{ll} 1 & \text{if $i=j$ and $w(i)=x$,}\\ 0 &
\text{otherwise.}\end{array}\right.\]
Let $T$ be the matrix $(t_{i, j})_{i, j\in\mathbb{Z}}$ of the shift given by
\[t_{i, j}=\left\{\begin{array}{ll} 1 & \text{if $i=j+1$,}\\ 0 &
\text{otherwise.}\end{array}\right.\]
Fix a field $\Bbbk$, and let $\mathcal{A}_w$ be the $\Bbbk$-algebra
generated by the matrices $D_x$, for $x\in X$, by $T$, and its transpose $T^\top$.
We say that $w$ is \emph{minimal} if for every finite subword
$(w(n), w(n+1), \ldots, w(n+k))$ there exists $R>0$ such that for any
$i\in\mathbb{Z}$ there exists $j\in\mathbb{Z}$ such that $|i-j|\le R$ and
$(w(j), w(j+1), \ldots, w(j+k))=(w(n), w(n+1), \ldots, w(n+k))$. We
say that $w$ is \emph{non-periodic} if there does not exist $p\ne 0$
such that $w(n+p)=w(n)$ for all $n\in\mathbb{Z}$.
\emph{Complexity function} $p_w(n)$ of the sequence $w\in X^{\mathbb{Z}}$ is
the number of different subwords $(w(i), w(i+1), \ldots, w(i+n-1))$ of
length $n$ in $w$.
The following theorem is a corollary of the results of our paper,
see Subsection~\ref{sss:subshifts} and Example~\ref{ex:matrices}.
\begin{theorem}
Suppose that $w\in X^{\mathbb{Z}}$ is minimal and non-periodic. Then the
algebra $\mathcal{A}_w$ is simple, and its growth $\gamma(n)$
satisfies \[C^{-1}n\cdot p_w(C^{-1}n)\le Cn\cdot p_w(Cn)\]
for some $C>1$.
\end{theorem}
We can apply now results on complexity of sequences to
construct simple algebras of various growths. For example, if $w$ is
\emph{Sturmian}, then $p_w(n)=n+1$, and hence $\mathcal{A}_w$ has quadratic growth. For
different \emph{Toeplitz} sequences we can obtain simple algebras of
arbitrary Gelfand-Kirillov dimension $d\ge 2$, or simple algebras of
growth $n\log n$, etc., see Subsection~\ref{sss:subshifts}.
Another class of examples of groupoids considered in our paper are
groupoids associated with groups acting on a rooted tree. If $G$ acts
by automorphisms on a locally finite rooted tree $T$, then it acts
by homeomorphisms on the boundary $\partial T$. One can consider the
\emph{groupoid of germs} $\mathfrak{G}$ of the action. Convolution algebras
$\Bbbk[\mathfrak{G}]$ are related to the \emph{thinned algebras} studied in~\cite{sid:ring,bartholdi:ring}.
In the case when $G$ is a \emph{contracting self-similar group},
Theorem~\ref{th:main} implies a result of L.~Bartholdi from~\cite{bartholdi:ring} giving
an estimate of Gelfand-Kirillov dimension for the thinned algebras of
contracting self-similar groups.
\section{\'Etale groupoids}
A \emph{groupoid} is a small category of isomorphisms (more precisely,
the set of its morphisms). For a groupoid
$\mathfrak{G}$, we denote by $\mathfrak{G}^{(2)}$ the set of composable pairs, i.e., the
set of pairs $(g_1, g_2)\in\mathfrak{G}\times\mathfrak{G}$ such that the product $g_1g_2$
is defined. We denote by $\mathfrak{G}^{(0)}$ the set of units of $\mathfrak{G}$, i.e.,
the set of identical isomorphisms. We also denote by $\mathsf{o},
\mathsf{t}:\mathfrak{G}\longrightarrow\mathfrak{G}^{(0)}$ the \emph{origin} and \emph{target} maps given by
\[\mathsf{o}(g)=g^{-1}g,\qquad \mathsf{t}(g)=gg^{-1}.\]
We interpret then an element $g\in\mathfrak{G}$ as an arrow from $\mathsf{o}(g)$ to
$\mathsf{t}(g)$. The product $g_1g_2$ is defined if and only if
$\mathsf{t}(g_2)=\mathsf{o}(g_1)$.
For $x\in\mathfrak{G}^{(0)}$, denote
\[\mathfrak{G}_x=\{g\in\mathfrak{G}\;:\;\mathsf{o}(g)=x\},\qquad\mathfrak{G}^x=\{g\in\mathfrak{G}\;:\;\mathsf{t}(g)=x\}.\]
The set $\mathfrak{G}_x\cap\mathfrak{G}^x$ is called the \emph{isotropy group} of $x$. A
groupoid is said to be \emph{principal} (or an equivalence relation)
if the isotropy group of every point is trivial.
Two
units $x, y\in\mathfrak{G}^{(0)}$ belong to one \emph{orbit} if there exists
$g\in\mathfrak{G}$ such that $\mathsf{o}(g)=x$ and $\mathsf{t}(g)=y$. It is easy to see that
belonging to one orbit is an equivalence relation.
A \emph{topological groupoid} is a groupoid $\mathfrak{G}$ with a topology on it
such that multiplication $\mathfrak{G}^{(2)}\longrightarrow\mathfrak{G}$ and taking inverse
$\mathfrak{G}\longrightarrow\mathfrak{G}$ are continuous maps. We do not require that $\mathfrak{G}$ is
Hausdorff, though we assume that the space of units $\mathfrak{G}^{(0)}$ is
metrizable and locally compact.
A \emph{$\mathfrak{G}$-bisection} is a subset $F\subset\mathfrak{G}$ such that the maps
$\mathsf{o}:F\longrightarrow\mathsf{o}(F)$ and $\mathsf{t}:F\longrightarrow\mathsf{t}(F)$ are homeomorphisms.
\begin{defi}
A topological groupoid $\mathfrak{G}$ is \emph{\'etale} if the set of all open
$\mathfrak{G}$-bisections is a basis of the topology of $\mathfrak{G}$.
\end{defi}
Let $\mathfrak{G}$ be an \'etale groupoid. It is easy to see that
product of two open bisections is an open bisection. It follows that
for every bisection $F$ the sets $\mathsf{o}(F)=F^{-1}F$ and $\mathsf{t}(F)=FF^{-1}$
are open, which in turn implies that $\mathfrak{G}^{(0)}$ is an open subset of $\mathfrak{G}$.
If $\mathfrak{G}$ is not Hausdorff, then there exist $g_1, g_2\in\mathfrak{G}$ that do not
have disjoint bisections. Since $\mathfrak{G}^{(0)}$ is Hausdorff,
this implies that $\mathsf{o}(g_1)=\mathsf{o}(g_2)$ and $\mathsf{t}(g_1)=\mathsf{t}(g_2)$. It
follows that the unit $x=\mathsf{o}(g_1)$ and the element $g_2^{-1}g_1$ of
the isotropy group of $x$ do not have disjoint open neighborhoods. In
particular, it means that principal \'etale groupoids are always
Hausdorff, and that an \'etale groupoid is Hausdorff if and only if
$\mathfrak{G}^{(0)}$ is a closed subset of $\mathfrak{G}$.
\begin{examp}
Let $G$ be a discrete group acting by homeomorphisms on a space $\mathcal{X}$. Then the
space $G\times\mathcal{X}$ has a natural groupoid structure with given by the
multiplication
\[(g_2, g_1(x))(g_1, x)=(g_2g_1, x).\]
This is an \'etale groupoid, since every set $\{g\}\times\mathcal{X}$ is an
open bisection. The groupoid $G\times\mathcal{X}$ is called the \emph{groupoid
of the action}, and is denoted $G\ltimes\mathcal{X}$.
\end{examp}
Our main class of groupoids will be naturally defined
quotients of the groupoids of actions,
called groupoids of germs.
\begin{examp}
Let $G$ and $\mathcal{X}$ be as in the previous example. A
\emph{germ} is an equivalence class of a pair $(g, x)\in G\times\mathcal{X}$
where $(g_1, x)$ and $(g_2, x)$ are equivalent if there exists a
neighborhood $U$ of $x$ such that the maps $g_1:U\longrightarrow\mathcal{X}$ and
$g_2:U\longrightarrow\mathcal{X}$ coincide. The set of germs is also an \'etale groupoid
with the same multiplication rule as in the previous example. We call
it \emph{groupoid of germs of the action}.
\end{examp}
The spaces of units in both groupoids are naturally identified with
the space $\mathcal{X}$ (namely, we identify the pair or the germ $(1, x)$ with
$x$). The groupoid of the action is Hausdorff if $\mathcal{X}$ is Hausdorff, since it is
homeomorphic to $G\times\mathcal{X}$. The groupoid of germs, on the other hand,
is frequently non-Hausdorff, even for a Hausdorff space $\mathcal{X}$.
If every germ of every non-trivial element of $G$ is not a unit (i.e.,
not equal to a germ of the identical homeomorphism), then
the groupoid of the action coincides with the groupoid of germs.
Many interesting examples of \'etale groupoids appear in dynamics and
topology, see~\cite{haefl:foliations,bellissardjuliensavinien,nek:hyperbolic}.
\section{Compactly generated groupoids}
For the rest of the paper, $\mathfrak{G}$ is an \'etale groupoid such that
$\mathfrak{G}^{(0)}$ is a compact totally disconnected metrizable space.
Note that then there exists a basis of topology of $\mathfrak{G}$
consisting of open compact $\mathfrak{G}$-bisections. Note that we allow compact
non-closed and compact non-Hausdorff sets, since $\mathfrak{G}$ in general is
not Hausdorff. However, if
$F$ is an open compact bisection, then $\mathsf{o}(F)$ and $\mathsf{t}(F)$ are clopen
(i.e., closed and open) and $F$ is Hausdorff.
\subsection{Cayley graphs and their growth}
\begin{defi}
A groupoid $\mathfrak{G}$ with compact totally disconnected unit space
is \emph{compactly generated} if there exists a
open compact subset $S\subset\mathfrak{G}$ such that $\mathfrak{G}=\bigcup_{n\ge 0}(S\cup S^{-1})^n$.
The set $S$ is called the \emph{generating set} of $\mathfrak{G}$.
\end{defi}
This definition is equivalent (for \'etale groupoids with compact
totally disconnected unit space) to the definition of~\cite{haefliger:compactgen}.
\begin{examp}
Let $G$ be a group acting on a Cantor set
$\mathcal{X}$. If $S$ is a finite generating set of $G$, then $S\times\mathcal{X}$ is an
open compact generating set of the groupoid $G\ltimes\mathcal{X}$. The set all
of germs of elements of $S$ is an open compact generating set of the
groupoid of germs of the action. Thus, both groupoids are compactly
generated if $G$ is finitely generated.
\end{examp}
Let $S$ be an open compact generating set of $\mathfrak{G}$. Let
$x\in\mathfrak{G}^{(0)}$. The \emph{Cayley graph} $\mathfrak{G}(x, S)$ is the directed
graph with the set of vertices $\mathfrak{G}_x$ in which we have an arrow from
$g_1$ to $g_2$ whenever there exists $s\in S$ such that $g_2=sg_1$.
We will often consider the graph $\mathfrak{G}(x, S)$ as a \emph{rooted graph} with root $x$. Morphism
$\phi:\Gamma_1\longrightarrow\Gamma_2$ of
rooted graphs is a morphism of graphs that maps the root of $\Gamma_1$
to the root of $\Gamma_2$.
Note that since $S$ can be covered
by a finite set of bisections, the degrees of vertices of the graphs
$\mathfrak{G}(x, S)$ are uniformly bounded.
\begin{examp}
Let $G$ be a finitely generated group acting on a totally disconnected
compact space $\mathcal{X}$. Let $S$ be a finite generating set of $G$, and let
$S\times\mathcal{X}$ be the corresponding generating set of the groupoid of
action $G\ltimes\mathcal{X}$. The
Cayley graphs $G\ltimes\mathcal{X}(x, S\times\mathcal{X})$ coincide then with the Cayley
graphs of $G$ (with respect to the generating set $S$).
The groupoid of germs $\mathfrak{G}$ will have smaller Cayley graphs. Let
$S'\subset\mathfrak{G}$ be the set of all germs of elements of $S$. Denote,
for $x\in\mathcal{X}$, by $G_{(x)}$ the subgroup of $G$ consisting of all
elements $g\in G$ such that there exists a neighborhood $U$ of $x$
such that $g$ fixes every point of $U$. Then $\mathfrak{G}(x, S')$ is isomorphic
to the \emph{Schreier graph} of $G$ modulo $G_{(x)}$. Its vertices are
the cosets $hG_{(x)}$, and a coset $h_1G_{(x)}$ is connected by an
arrow with $h_2G_{(x)}$ if there exists a generator $s\in S$ such that
$sh_1G_{(x)}=h_2G_{(x)}$.
\end{examp}
Cayley graphs $\mathfrak{G}(x, S)$ are closely related to the \emph{orbital
graphs}, which are defined as graphs $\Gamma(x, S)$ with the set of
vertices equal to the orbit of $x$, in which a vertex $x_1$ is
connected by an arrow to a vertex $x_2$ if there exists $g\in S$ such
that $\mathsf{o}(s)=x_1$ and $\mathsf{t}(s)=x_2$. Orbital graph $\Gamma(x, S)$ is the
quotient on the Cayley graph $\mathfrak{G}(x, S)$ by the natural right action of
the isotropy group of $x$. In particular, orbital graph and the Cayley
graph coincide if the isotropy group of $x$ is trivial.
Denote by $B_S(x, n)$ the ball of radius $n$ with center $x$ in the
graph $\mathfrak{G}(x, S)$ seen as a rooted graph (with root $x$). Let
\[\gamma_S(x, n)=|B_S(x, n)|,\qquad
\overline\gamma(n, S)=\max_{x\in\mathfrak{G}^{(0)}}\gamma_S(x, n).\]
If $S_1$ and $S_2$ are two open compact generating sets of $\mathfrak{G}$, then
there exists $m$ such that $S_2\subset\bigcup_{1\le k\le m}(S_1\cup
S_1^{-1})^k$ and $S_1\subset\bigcup_{1\le k\le m}(S_2\cup
S_2^{-1})^k$. Then $\gamma_{S_1}(x, mn)\ge\gamma_{S_2}(x,
n)$ and $\gamma_{S_2}(x, mn)\ge\gamma_{S_1}(x, n)$ for all $n$. It also follows
that $\overline\gamma(mn, S_1)\ge\overline\gamma(n, S_2)$ and
$\overline\gamma(mn, S_2)\ge\overline\gamma(n, S_1)$ for all $n$. In
other words, the \emph{growth rate} of the functions $\gamma_S(x, n)$
and $\overline\gamma(n, S)$ do not depend on the choice of $S$, if $S$
is a generating set.
Condition of polynomial growth of Cayley graphs of groupoids (or, in
the measure-theoretic category, of connected components of graphings of
equivalence relations) appear in the study of amenability of
groupoids, see~\cite{kaim,delaroche_renault}.
Here is another example of applications of the notion of growth of groupoids.
\begin{theorem}
\label{th:nofree}
Let $G$ be a finitely generated subgroup of the automorphism group of a locally finite
rooted tree $T$. Consider the groupoid of germs $\mathfrak{G}$ of the action of
$G$ on the boundary $\partial T$ of the tree. If $\gamma_S(x, n)$
has sub-exponential growth for every $x\in\partial T$, then $G$ has no
free subgroups.
\end{theorem}
\begin{proof}
By~\cite[Theorem~3.3]{nek:free}, if $G$ has a free subgroup, then either there exists a free
subgroup $F$ and a point $x\in\partial T$ such that the stabilizer of
$x$ in $F$ is trivial, or there exists a free subgroup $F$ and a point
$x\in\partial T$ such that $x$ is fixed by $F$ and every non-trivial
element $g$ of $F$ the germ $(g, x)$ is non-trivial. But both
conditions imply that the Cayley graph $\mathfrak{G}(x, S)$ has exponential growth.
\end{proof}
\subsection{Complexity}
Let $\mathcal{S}$ be a finite set of open compact
$\mathfrak{G}$-bisections such that $S=\bigcup\mathcal{S}$ is a generating set. Note that
every compact subset of $\mathfrak{G}$ can be covered by a finite number of open
compact $\mathfrak{G}$-bisections.
Denote by $\mathfrak{G}(x, \mathcal{S})$ the oriented labeled
graph with the set of vertices $\mathfrak{G}_x$ in which we have an arrow
from $g_1$ to $g_2$ labeled by $A\in\mathcal{S}$ if there exists $s\in
A$ such that $g_2=sg_1$.
The graph $\mathfrak{G}(x, \mathcal{S})$ basically
coincides with $\mathfrak{G}(x, S)$ for $S=\bigcup\mathcal{S}$. The only difference is the
labeling and that some
arrows of $\mathfrak{G}(x, S)$ become multiple arrows in $\mathfrak{G}(x,
\mathcal{S})$. In particular, the metrics induced on the sets of
vertices of graphs $\mathfrak{G}(x, S)$ and $\mathfrak{G}(x, \mathcal{S})$ coincide.
We denote by $B_{\mathcal{S}}(x, r)$ or just by $B(x, r)$ the ball of radius $r$ with center in $x$, seen
as a rooted oriented labeled graph.
We write $x\sim_r y$ if $B_{\mathcal{S}}(x, r)$ and $B_{\mathcal{S}}(y, r)$ are isomorphic.
\begin{defi}
\emph{Complexity} of $\mathcal{S}$ is the function $\delta(r,
\mathcal{S})$ equal to the number of $\sim_r$-equivalence classes.
\end{defi}
It is easy to see that $\delta(r, \mathcal{S})$ is finite for every
$r$ and $\mathcal{S}$.
\subsection{Examples}
\subsubsection{Shifts}
\label{sss:shifts}
Let $X$ be a finite alphabet containing more than one letter. Consider
the space $X^{\mathbb{Z}}$ of all bi-infinite words over $X$, i.e., maps
$w:\mathbb{Z}\longrightarrow X$. Denote by $s:X^{\mathbb{Z}}\longrightarrow X^{\mathbb{Z}}$ the shift map given by the rule
$s(w)(i)=w(i+1)$. The space $X^{\mathbb{Z}}$ is homeomorphic to the Cantor set
with respect to the direct product topology (where $X$ is
discrete).
A \emph{sub-shift} is a closed $s$-invariant subset
$\mathcal{X}\subset X^{\mathbb{Z}}$. We always assume that $\mathcal{X}$ has no isolated points.
For a sub-shift $\mathcal{X}$, consider the groupoid
$\mathfrak{S}$ of the germs of the
action of $\mathbb{Z}$ on $\mathcal{X}$ generated by the shift. It is easy to see that
all germs of non-zero powers of the shift are non-trivial,
hence the groupoid $\mathfrak{S}$ coincides with the
groupoid $\mathbb{Z}\ltimes\mathcal{X}$ of the action.
As usual, we will identify $\mathcal{X}$ with the space of units $\mathfrak{S}^{(0)}$. The
set $S=\{(s, x)\;:\;x\in\mathcal{X}\}$ is an open compact generating set of $\mathfrak{S}$. The
Cayley graphs $\mathfrak{S}(w, S)$ are isomorphic to the Cayley graph of $Z$
with respect to the generating set $\{1\}$.
If $\mathcal{X}$ is \emph{aperiodic}, i.e., if it does not contain periodic
sequences, then $\mathfrak{S}$ is principal. Note that $\mathfrak{S}$ is always Hausdorff.
For $x\in X$, denote by $S_x$ set of germs of the restriction of $s$
onto the cylindrical set $\{w\in\mathcal{X}\;:\;w(0)=x\}$. Then $\mathcal{S}=\{S_x\}_{x\in
X}$ is a covering of $S$ by disjoint clopen subsets of $S$. Then for
every $w\in\mathcal{X}$, the Cayley graph $\mathfrak{S}(w, \mathcal{S})$ basically
repeats $w$: its set of vertices is the set of germs $(s^n, w)$, $n\in
\mathbb{Z}$; for every $n$ we have an arrow from $(s^n, w)$ to $(s^{n+1}, w)$
labeled by $S_{w(n)}$.
In particular, we have
\[\delta(n, \mathcal{S})=p_{\mathcal{X}}(2n),\]
where $p_{\mathcal{X}}(k)$ denotes the
number of words of length $k$ that appear as subwords of elements of
$\mathcal{X}$.
Complexity $p_{\mathcal{X}}(n)$ of subshifts is a well studied subject,
see~\cite{kurka:topsymb,ferenczi:complexity,cassaigneets:factorcomplexity}
and references therein.
Two classes of subshifts are especially interesting for us: Sturmian and
Toeplitz subshifts.
Let $\theta\in (0, 1)$ be an irrational number, and consider
the rotation \[R_\theta:x\mapsto x+\theta\pmod{1}\] of the circle
$\mathbb{R}/\mathbb{Z}$. For a number $x\in\mathbb{R}/\mathbb{Z}$ not belonging to the
$R_\theta$-orbit of $0$, consider the
\emph{$\theta$-itinerary $I_{\theta, x}\in\{0, 1\}^{\mathbb{Z}}$} given by
\[I_{\theta, x}(n)=\left\{\begin{array}{ll} 0 & \text{if $x+n\theta\in (0,
\theta)\pmod{1}$},\\
1 & \text{if $x+n\theta\in (\theta, 1)\pmod{1}$}.\end{array}\right.\]
In other words, $I_{\theta, x}$ describes the itinerary of $x\in\mathbb{R}/\mathbb{Z}$ under
the rotation $R_\theta$ with respect to the partition
$[0, \theta), [\theta, 1)$ of the circle $\mathbb{R}/\mathbb{Z}$.
If $x$ belongs to the orbit of $0$, then we define two itineraries
$I_{\theta, x+0}=\lim_{t\to x+0}I_{\theta, t}$ and
$I_{\theta, x-0}=\lim_{t\to x-0}I_{\theta, t}$, where $t$ in the limits
belongs to the complement of the orbit of $0$.
The set $\mathcal{X}_\theta$ of all itineraries is a subshift of $\{0, 1\}^{\mathbb{Z}}$ called the
\emph{Sturmian subshift} associated with $\theta$. Informally, the space
$\mathcal{X}_\theta$ is obtained from the circle $\mathbb{R}/\mathbb{Z}$ by ``cutting'' it
along the $R_\theta$-orbit of $0$, i.e., by replacing each point
$x=n\theta$ by two copies $x+0$ and $x-0$. A basis of topology of
$\mathcal{X}_\theta$ is the set of arcs of the form $[n\theta+0,
m\theta-0]$. The shift is identified
in this model with the natural map induced by the
rotation $R_\theta$.
Complexity $p_{\mathcal{X}_\theta}(n)$ of the Sturmian subshift is equal to the number of all possible
$R_\theta$-itineraries of length $n$. Consider the set
$\{R_\theta^{-k}(\theta)\}_{k=0, 1, \ldots, n}$. It separates the
circle $\mathbb{R}/\mathbb{Z}$ into $n+1$ arcs such that two points $x, y$ have equal
length $n$ segments $\{0, \ldots, n-1\}\longrightarrow\{0, 1\}$ of their
itineraries $I_{\theta, x}$, $I_{\theta, y}$ if and only if
they belong to one arc. It follows that $p_{\mathcal{X}_\theta}(n)=n+1$. The
subshifts of the form $\mathcal{X}_\theta$ and their elements are called
\emph{Sturmian} subshifts and \emph{Sturmian} sequences.
A sequence $w:X\longrightarrow\mathbb{Z}$ is a \emph{Toeplitz} sequence if it is not
periodic and for every
$n\in\mathbb{Z}$ there exists $p\in\mathbb{N}$ such that $w(n+kp)=w(n)$ for all
$k\in\mathbb{Z}$. Complexity of Toeplitz sequences is well studied.
It is known, for example,
(see~\cite[Proposition~4.79]{kurka:topsymb}) that for any
$1\le\alpha\le\beta\le\infty$ there exists a Toeplitz subshift $\mathcal{X}$
(i.e., closure of the shift orbit of a Toeplitz sequence)
such that
\[\liminf_{n\to\infty}\frac{\ln p_{\mathcal{X}}(n)}{\ln n}=\alpha,\qquad
\limsup_{n\to\infty}\frac{\ln p_{\mathcal{X}}(n)}{\ln n}=\beta.\]
The following theorem is proved by M.~Koskas in~\cite{koskas}.
\begin{theorem}
For every rational number $p/q>1$
and every positive increasing differentiable function $f(x)$ satisfying
$f(n)=o(n^\alpha)$ for all $\alpha>0$, and $nf'(n)=o(n^\alpha)$ for
all $\alpha>0$, there exists a Toeplitz subshift $\mathcal{X}$ and two
constants $c_1, c_2>0$ satisfying
$c_1f(n)n^{p/q}\le p_{\mathcal{X}}(n)\le c_2f(n)n^{p/q}$ for all $n\in\mathbb{N}$.
\end{theorem}
\subsubsection{Groups acting on rooted trees}
Let $X$ be a finite alphabet, $|X|\ge 2$. Denote by $X^*$ the set of
all finite words (including the empty word $\varnothing$). We consider
$X^*$ as a rooted tree with root $\varnothing$ in which every word
$v\in X^*$ is connected to the words of the form $vx$ for all $x\in
X$. The \emph{boundary} of the tree is naturally identified with the
space $X^{\mathbb{N}}$ of all one-sided sequences $x_1x_2x_3\ldots$.
Every automorphism of the rooted tree $X^*$ naturally induces a
homeomorphism of $X^{\mathbb{N}}$.
Let $g$ be an automorphism of the tree $X^*$. For every $v\in X^*$
there exists a unique automorphism $g|_v$ of the tree $X^*$ such that
\[g(vw)=g(v)g|_v(w)\]
for all $w\in X^*$. We say that a group $G$ of automorphisms of $X^*$
is \emph{self-similar} if $g|_v\in G$ for every $g\in G$ and $v\in
X^*$. For every $v\in X^*$ and $w\in X^{\mathbb{N}}$ the germ $(g, vw)$
depends only on the quadruple $(v, g(v), g|_v, w)$.
\begin{examp}
\label{ex:admach}
Consider the automorphism $a$ of the binary tree $\{0, 1\}^*$ defined by the recursive rules
\[a(0w)=1w,\qquad a(1w)=0a(w).\]
It is called the \emph{adding machine}, or \emph{odometer}. The cyclic group generated by $a$ is
self-similar.
\end{examp}
\begin{examp}
\label{ex:grigorchuk}
Consider the automorphisms of $\{0, 1\}^*$ defined by the recursive rules
\[a(0w)=1w, a(1w)=0w\]
and
\begin{alignat*}{2}
b(0w)&=0a(w), & \qquad b(1w)&=1c(w),\\
c(0w)&=0a(w), &\qquad c(1w)&=1d(w),\\
d(0w)&=0w, &\qquad d(1w)&=1b(w).
\end{alignat*}
The group generated by $a, b, c, d$ is the \emph{Grigorchuk group}, see~\cite{grigorchuk:80_en}.
\end{examp}
For more examples of self-similar groups and their applications, see~\cite{nek:book}.
Let $G$ be a finitely generated self-similar group, and let $l(g)$
denote the length of an element $g\in G$ with respect to some fixed
finite generating set of $G$. The \emph{contraction coefficient} of
the group $G$ is the number
\[\lambda=\limsup_{n\to\infty}\limsup_{g\in G,
l(g)\to\infty}\max_{v\in X^n}\frac{l(g|_v)}{l(g)}.\]
The group is said to be \emph{contracting} if $\lambda<1$.
For example, the adding machine action of $\mathbb{Z}$ and the Grigorchuk
group are both contracting with contraction coefficient $\lambda=1/2$.
\begin{proposition}
\label{pr:contractingestimates}
Let $G$ be a contracting self-similar group acting on the tree
$X^*$, and let $\lambda$ be the contraction coefficient. Consider the
groupoid of germs $\mathfrak{G}$ of the action of $G$ on $X^{\mathbb{N}}$, let $S$
be a finite generating set of $G$, and let $\mathcal{S}$ be the set of
$\mathfrak{G}$-bisets of the form $\{(s, w)\;:\;w\in X^{\mathbb{N}}\}$ for $s\in S$.
Then we have
\[\limsup_{n\to\infty}\frac{\log\overline\gamma(n, \mathcal{S})}{\log n}\le
\frac{\log|X|}{-\log\lambda},\qquad
\limsup_{n\to\infty}\frac{\log\delta(n, \mathcal{S})}{\log
n}\le \frac{\log|X|}{-\log\lambda}.\]
\end{proposition}
\begin{proof}
Let $\rho$ be any number in the interval $(\lambda, 1)$. Then there
exist $n_0$, $l_0$ such that for all elements $g\in G$ such that
$l(g)>l_0$ we have $l(g|_v)\le \rho^{n_0} l(g)$ for all $v\in
X^{n_0}$. It follows that there exists a finite set $\mathcal{N}$
such that $g|_v\in\mathcal{N}$ for all $v\in X^*$ and for every $g\in
G\setminus\mathcal{N}$ we have $l(g|_v)\le \rho^{n_0} l(g)$ for all words
$v\in X^*$ of length at least $n_0$.
Then for every $g\in G$ and for every word $v\in
X^*$ of length at least $\left\lfloor\frac{\log l(g)-\log l_0}{-\log
\rho}\right\rfloor+n_0$ we
have $g|_v\in\mathcal{N}$.
Let $w=x_1x_2\ldots\in X^{\mathbb{N}}$, and denote $v=x_1x_2\ldots x_n$,
$w'=x_{n+1}x_{n+2}\ldots$ for $n=\left\lfloor\frac{\log r-\log l_0}{-\log
\rho}\right\rfloor+n_0$. Then for fixed $w$ and all $g$ such that
$l(g)\le r$, the germ $(g, w)$ depends only on
$g(v)$ and $g|_v$. There are not more than $|X|^n$ possibilities for
$g(v)$, hence the number of germs $(g, w)$ is not more than
\[|\mathcal{N}|\cdot|X|^n\le|\mathcal{N}|\exp\left(\log|X|\left(\frac{\log
r-\log l_0}{-\log\rho}+n_0\right)\right)\le
C_1r^{\frac{\log|X|}{-\log\rho}}\]
for $C_1=|\mathcal{N}|\cdot|X|^{\frac{\log
l_0}{\log\rho}+n_0}$. Consequently, for every $\rho\in (\lambda,
1)$ there exists $C_1>0$ such that
\[\overline\gamma(r, \mathcal{S})\le C_1r^{\frac{\log|X|}{-\log\rho}},\]
hence $\limsup_{r\to\infty}\frac{\log\overline\gamma(r,
\mathcal{S})}{\log r}\le\frac{\log|X|}{-\log\lambda}$.
It is enough, in order to know the ball $B_{\mathcal{S}}(w, r)$, to
know for every word $g\in G$ of length at most $2r$ whether the germ
$(g, w)$ is a unit. Let, as above, $w=vw'$, where length of $v$ is
$n=\left\lfloor\frac{\log 2r-\log
l_0}{-\log\rho}\right\rfloor+n_0$. For every $g\in G$ of
length at most $2r$ the germ $(g, w)$ is a unit if and
only if $g(v)=v$ and $(g|_v, w')$ is a unit. We have
$g|_v\in\mathcal{N}$, so
$B_{\mathcal{S}}(w, r)$ depends only on $v$ and the set
$T_{w'}=\{h\in\mathcal{N}\;:\;(h, w')\in\mathfrak{G}^{(0)}\}$. Consequently,
\[\delta(r, \mathcal{S})\le 2^{|\mathcal{N}|}\cdot |X|^n\le
C_2r^{\frac{\log|X|}{-\log\rho}},\]
where $C_2=2^{|\mathcal{N}|}|X|^{\frac{\log l_0-\log 2}{\log\rho}+n_0}$, which shows that
$\limsup_{r\to\infty}\frac{\log\delta(r, \mathcal{S})}{\log r}\le\frac{\log|X|}{-\log\lambda}$.
\end{proof}
Both estimates in Proposition~\ref{pr:contractingestimates} are not
sharp in general. For example, consider a self-similar action of
$\mathbb{Z}^2$ over the alphabet $X$ of size 5 associated with the virtual endomorphism
given by the matrix
$A=\left(\begin{array}{cc} 2 & 1\\ 1 &
3\end{array}\right)^{-1}=\left(\begin{array}{rr}3/5 & -1/5\\ -1/5 &
2/5\end{array}\right)$, see~\cite[2.9, 2.12]{nek:book}
and~\cite{neksid} for details. Note that the eigenvalues
of $A$ are $\left(\frac{5\pm\sqrt{5}}{2}\right)^{-1}\in
(0, 1)$, hence the contraction coefficient is $\lambda=\frac
2{5-\sqrt{5}}=\frac{5+\sqrt{5}}{10}$.
On the other hand $\overline\gamma(r, \mathcal{S})$ grows as a
quadratic polynomial, while $\delta(r, \mathcal{S})$ is bounded.
\section{Convolution algebras}
\subsection{Definitions}
Let $\mathfrak{G}$ be an \'etale groupoid, and let $\Bbbk$ be a
field. \emph{Support} of a function $f:\mathfrak{G}\longrightarrow\Bbbk$ is closure of the
set of points $x\in\mathfrak{G}$ such that $f(x)\ne 0$. If $f_1, f_2$ are
functions with compact support, then their \emph{convolution} is given
by the formula
\[f_1*f_2(g)=\sum_{h\in\mathfrak{G}_{\mathsf{o}(g)}}f_1(gh^{-1})f_2(h).\]
Note that since $f_2$ has compact support, the set of elements
$h\in\mathfrak{G}_{\mathsf{o}(g)}$ such that $f_2(h)\ne 0$ is finite.
It is easy to see that if $f_1, f_2$ are supported on the space of
units, then their convolution coincides with their pointwise
product. If $F_1, F_2$ are bisections, then their characteristic
functions satisfy $1_{F_1}*1_{F_2}=1_{F_1F_2}$.
The set of all functions $f:\mathfrak{G}\longrightarrow\Bbbk$ with compact support forms an
algebra over
$\Bbbk$ with respect to convolution. But this algebra is too big,
and its definition does not use the topology of $\mathfrak{G}$ much. On the other
hand, the algebra of all continuous functions (with discrete topology
on $\Bbbk$) is too small in the non-Hausdorff case. Therefore, we
adopt the next definition, following Connes~\cite{conn:foliations}, see also~\cite{paterson:gr}
and~\cite{steinberg:groupoidapproach}.
\begin{defi}
The \emph{convolution algebra} $\Bbbk[\mathfrak{G}]$ is the $\Bbbk$-algebra generated by
the characteristic functions $1_F$ of open compact $\mathfrak{G}$-bisections (with
respect to convolution).
\end{defi}
If $\mathfrak{G}$ is Hausdorff, then $\Bbbk[\mathfrak{G}]$ is the algebra of all
continuous (i.e., locally constant) functions $f:\mathfrak{G}\longrightarrow\Bbbk$, where
$\Bbbk$ has discrete topology. In the non-Hausdorff case the
algebra $\Bbbk[\mathfrak{G}]$ contains discontinuous functions
(e.g., characteristic functions of non-closed open compact
bisections).
From now on we will use the usual multiplication sign for convolution.
The unit of the algebra $\Bbbk[\mathfrak{G}]$ is the characteristic function of
$\mathfrak{G}^{(0)}$, which we will often denote just by $1$.
If $\mathfrak{G}=G\ltimes\mathcal{X}$ is the groupoid of an action, then $\Bbbk[\mathfrak{G}]$
is generated by the commutative algebra of locally
constant functions $f:\mathcal{X}\longrightarrow\Bbbk$ (with pointwise multiplication and
addition) and the group ring $\Bbbk[G]$ subject to relations
\[g^{-1}\cdot f\cdot g=f\circ g,\]
for all $f:\mathcal{X}\longrightarrow\Bbbk$ and $g\in G$, where $f\circ g:\mathcal{X}\longrightarrow\Bbbk$ is
given by $(f\circ g)(x)=f(g(x))$.
In other words, it is the \emph{cross-product} of the algebra of
functions and the group ring.
Let $\mathcal{T}\subset\mathfrak{G}^{(0)}$ be the set of units with trivial
isotropy groups. The set $\mathcal{T}$ is $\mathfrak{G}$-invariant, i.e., is a
union of $\mathfrak{G}$-orbits.
\begin{defi}
We say that $\mathfrak{G}$ is \emph{essentially principal} if the set
$\mathcal{T}$ is dense in $\mathfrak{G}^{(0)}$. It is \emph{principal} if
$\mathcal{T}=\mathfrak{G}^{(0)}$. The groupoid $\mathfrak{G}$ is said to be \emph{minimal}
if every $\mathfrak{G}$-orbit is dense in $\mathfrak{G}^{(0)}$.
\end{defi}
\begin{examp}
For every homeomorphism $g$ of a metric space $\mathcal{X}$, the set of
points $x\in\mathcal{X}$ such that $g(x)=x$ and the germ $(g, x)$ is
non-trivial is a closed nowhere dense set. It follows that if $G$ is
a countable group of homeomorphisms of $\mathcal{X}$, then groupoid of
germs of the action is essentially principal.
\end{examp}
Simplicity of essentially principal minimal groupoids is a well known fact,
see~\cite{brownclarketc:simplicity} and a $C^*$-version in~\cite[Proposition~4.6]{renault:groupoids}.
We provide a proof of the following simple proposition just for completeness.
\begin{proposition}
\label{pr:simple}
Suppose that $\mathfrak{G}$ is essentially principal and minimal.
Let $I$ be the set of functions $f\in\Bbbk[\mathfrak{G}]$ such that $f(g)=0$ for
every $g\in\mathfrak{G}$ such that $\mathsf{o}(g), \mathsf{t}(g)\in\mathcal{T}$. Then $I$ is a
two-sided ideal, and the algebra $\Bbbk[\mathfrak{G}]/I$ is simple. In
particular, if $\mathfrak{G}$ is Hausdorff, then $\Bbbk[\mathfrak{G}]$ is simple.
\end{proposition}
\begin{proof}
The fact that $I$ is a two-sided ideal follows directly from the fact
that $\mathcal{T}$ is $\mathfrak{G}$-invariant.
In order to prove simplicity of $\Bbbk[\mathfrak{G}]$ it is enough to show that
if $f\in\Bbbk[\mathfrak{G}]\setminus I$, then there exist elements $a_i,
b_i\in\Bbbk[\mathfrak{G}]$ such that $\sum_{i=1}^ka_ifb_i=1$.
If $f\in\Bbbk[\mathfrak{G}]\setminus I$, then there exists $g\in\mathfrak{G}$ such that
$\mathsf{o}(g), \mathsf{t}(g)\in\mathcal{T}$ and $f(g)\ne 0$. Let
$f=\sum_{i=1}^m\alpha_i1_{F_i}$, where $F_i$ are open compact
$\mathfrak{G}$-bisections. Let $A=\{1\le i\le m\;:\;g\in F_i\}$. Then
$f(g)=\sum_{i\in A}\alpha_i$. Since $\mathsf{o}(g)\in\mathcal{T}$, an
equality of targets $\mathsf{t}(F_i\mathsf{o}(g))=\mathsf{t}(F_j\mathsf{o}(g))$ implies the
equality $F_i\mathsf{o}(g)=F_j\mathsf{o}(g)$ of groupoid elements. It follows that
$\mathsf{t}(F_i\mathsf{o}(g))\ne\mathsf{t}(g)$ for every $i\notin A$. We can find therefore
a clopen neighborhood $U$ of $\mathsf{o}(g)$ such that $U\subset\mathsf{o}(F_i)$, $F_iU=F_jU$, for all
$i, j\in A$, $U\cap\mathsf{o}(F_j)=\emptyset$ for all $j\notin A$,
and $\mathsf{t}(F_iU)\cap\mathsf{t}(F_jU)=\emptyset$ for all $i\in A$ and $j\notin
A$. Denote $F_iU=F$ for any $i\in A$.
We have $1_{F^{-1}}f1_U=\sum_{i\in A}\alpha_i 1_U$. It follows that
$1_U=\alpha 1_{F^{-1}}f1_U$ for some $\alpha\in\Bbbk$.
The groupoid $\mathfrak{G}$ is minimal, hence for every $x\in\mathfrak{G}^{(0)}$ there
exists $h\in\mathfrak{G}$ such that $\mathsf{o}(h)=x$ and $\mathsf{t}(h)\in U$. There exists
therefore an open compact $\mathfrak{G}$-bisection $H$ such that $x\in\mathsf{o}(H)$ and
$\mathsf{t}(H)\subset U$. Then $1_{\mathsf{o}(H)}=1_{H^{-1}}1_U1_H=\alpha
1_{H^{-1}F^{-1}}f1_U1_H$. It follows that $\mathfrak{G}^{(0)}$ can be covered by
a finite collection of sets $V_i$ such that $1_{V_i}$ can be written
in the form $a_ifb_i$ for some $a, b\in\Bbbk[G]$. Note that if
$V_i'$ is a clopen subset of $V_i$, then $1_{V_i'}=1_{V_i'}1_{V_i}$,
hence we may replace the covering $\{V_i\}$ by a finite covering by disjoint
clopen sets. But in that case we have $1=\sum 1_{V_i}$.
\end{proof}
\subsection{Growth of $\Bbbk[\mathfrak{G}]$}
\begin{theorem}
\label{th:growth} Let $\mathfrak{G}$ be an \'etale groupoid with compact totally
disconnected unit space.
Let $\mathcal{S}$ be a finite set of open compact $\mathfrak{G}$-bisections. Let
$V\subset\Bbbk[\mathfrak{G}]$ be the $\Bbbk$-subspace generated by the
characteristic functions of the elements of $\mathcal{S}$.
Then
\[\dim V^n\le\overline\gamma(n, \mathcal{S})\delta(n, \mathcal{S}).\]
\end{theorem}
\begin{proof}
Fix $n$, and let $\mathcal{S}^n$ be the set of all products $S_1S_2\ldots S_n$ of length
$n$ of elements of $\mathcal{S}$. Then $V^n$ is the linear span of the characteristic functions of elements
of $\mathcal{S}^n$. Denote, for $x\in\mathfrak{G}^{(0)}$,
\[A_x=\bigcap_{F\in\mathcal{S}^n, x\in\mathsf{o}(F)}\mathsf{o}(F)\setminus\bigcup_{F\in\mathcal{S}^n, x\notin\mathsf{o}(F)}\mathsf{o}(F).\]
Since $\mathsf{o}(F)$ is clopen for every $F\in\mathcal{S}^n$, the sets $A_x$ are also clopen.
Note that for every $F\in\mathcal{S}^n$ and $x\in\mathfrak{G}^{(0)}$, either
$A_x\subset\mathsf{o}(F)$, or $A_x\cap\mathsf{o}(F)=\emptyset$.
If $F_1, F_2$ are open $\mathfrak{G}$-bisections and
$F_1\cdot x=F_2\cdot x$ for a unit $x$, then the set of points $y$
such that $F_1\cdot y=F_2\cdot y$ is equal to the intersection of
$F_1^{-1}F_2$ with $\mathfrak{G}^{(0)}$.
Since $\mathfrak{G}$ is \'etale, this set is open.
Denote by $B_x$ the set of all points $y\in A_x$ such that $F_1\cdot
x=F_2\cdot x$ implies $F_1\cdot y=F_2\cdot y$ for all $F_1,
F_2\in\mathcal{S}^n$. Then $B_x$ is open and $x\in B_x$.
Note that if $x\sim_n y$, then $A_x=A_y$, as belonging of a point $y$ to the domain of a product $S_1S_2\ldots S_n$ of elements of $\mathcal{S}$
is equivalent to the existence of a path in $\mathfrak{G}(y, \mathcal{S})$ of length $n$ starting at $y$ and labeled by the sequence $S_n, S_{n-1}, \ldots, S_1$.
Similarly, if $x\sim_n y$, then $B_x=B_y$, since an equality $F_1\cdot x=F_2\cdot x$ is equivalent to coincidence of endpoints of the paths corresponding
to the products $F_1$ and $F_2$ starting at $x$.
Let $\mathcal{B}=\{B_x\;:\;x\in\mathfrak{G}^{(0}\}$. Since $B_x=B_y$ for $x\sim_n y$, the set $\mathcal{B}$ consists of at most
$\delta(n, \mathcal{S})$ elements.
\begin{lemma}
There exists a covering
$\widetilde{\mathcal{B}}=\{\widetilde B\}_{B\in\mathcal{B}}$ of $\mathfrak{G}^{(0)}$ by
disjoint clopen sets
such that $\widetilde B\subset B$ for every $B\in\mathcal{B}$.
\end{lemma}
We allow some of the sets $\widetilde B$ to be empty.
\begin{proof}
By the Shrinking Lemma, we can find for every $B\in\mathcal{B}$ an
open set $B'\subset
B$ such that $\{B'\}_{B\in\mathcal{B}}$ is a covering of $\mathfrak{G}^{(0)}$,
and closure of $B'$ is contained in $B$. Then closure of $B'$ is
compact, and can be covered by a finite collection of clopen subsets
of $B$. Hence, after replacing $B'$ by the union of these clopen
subsets, we may assume that $B'$ are clopen. Order the set
$\mathcal{B}$ into a sequence $B_1, B_2, \ldots, B_m$, define
$\widetilde B_1=B_1'$, and inductively, $\widetilde B_i=B_i'\setminus(B_1'\cup
B_2'\cup\cdots\cup B_{i-1}')$. Then $\{\widetilde B\}_{B\in\mathcal{B}}$
satisfies the conditions of the lemma.
\end{proof}
Let $x_1, x_2, \ldots, x_m$ be a transversal of the $\sim_n$ equivalence relation, where $m=\delta(n, \mathcal{S})$.
For every $F\in\mathcal{S}^n$ and $x_i\in\mathsf{o}(F)$, consider the
restriction $F\cdot \widetilde B_{x_i}$ of
$F$ onto $\widetilde B_{x_i}$. Since $\{\widetilde B_{x_i}\}_{i=1, \ldots, m}$ is a
covering of $\mathfrak{G}^{(0)}$ by disjoint subsets, the sets
$F\cdot\widetilde B_{x_i}$ form a covering of
$F$ by disjoint subsets, and $1_F=\sum_{i=1}^m1_{F\cdot \widetilde B_{x_i}}$.
If $F_1, F_2\in\mathcal{S}^n$ and $x_i$ are such that $x_i\in\mathsf{o}(F_1)\cap\mathsf{o}(F_2)$, and $F_1\cdot x_i=F_2\cdot x_i$, then for every
$y\in\widetilde B_{x_i}$ we have $y\in\mathsf{o}(F_1)\cap\mathsf{o}(F_2)$ and $F_1\cdot y=F_2\cdot y$, hence
$F_1\cdot\widetilde B_{x_i}=F_2\cdot\widetilde B_{x_i}$. It follows that $F\cdot\widetilde B_{x_i}$ depends only on $F\cdot x_i$, and we have not more than $\gamma(n, x_i, \mathcal{S})\le
\overline\gamma(n, \mathcal{S})$ non-empty sets of the form $F\cdot\widetilde B_{x_i}$, for every given $x_i$. Hence we have at most
$\overline\gamma(n, \mathcal{S})\delta(n, \mathcal{S})$ functions of the form $1_{F\cdot x_i}$ in total, and every function $1_F$, for
$F\in\mathcal{S}^n$ is equal to the sum of a subset of these functions, which finishes the proof of the theorem.
\end{proof}
\subsection{Finite generation}
For a given finite set $\mathcal{S}$ of open compact $\mathfrak{G}$-bisections, generating $\mathfrak{G}$, denote
\[A_{x, n}=\bigcap_{F\in\mathcal{S}^n,
x\in\mathsf{o}(F)}\mathsf{o}(F)\setminus\bigcup_{F\in\mathcal{S}^n,
x\notin\mathsf{o}(F)}\mathsf{o}(F),\]
see the proof of Theorem~\ref{th:growth}.
Recall that the sets $A_{x, n}$ are clopen. It is also easy to see that two sets $A_{x, n}$ and $A_{y, n}$ are either disjoint
or coincide. Note also that $A_{x, n}\subset A_{x, m}$ if $n>m$. It follows that for any $x, y\in\mathfrak{G}^{(0)}$ and $n>m$, either $A_{x, n}\subset A_{y, m}$,
or $A_{x, n}\cap A_{y, m}=\emptyset$.
\begin{defi}
We say that $\mathcal{S}$ is \emph{expansive}
if for any two different points $x, y\in\mathfrak{G}^{(0)}$ there exists $n$ such that $A_{x, n}$ and $A_{y, n}$ are disjoint.
\end{defi}
\begin{proposition}
\label{prop:expansive}
If $\mathcal{S}$ is expansive, then the set $\{1_S\;:\;S\in\mathcal{S}\cup\mathcal{S}^{-1}\}$ generates $\Bbbk[\mathfrak{G}]$.
\end{proposition}
\begin{proof}
Let $\mathcal{A}$ be the algebra generated by the functions $1_S$ for $S\in\mathcal{S}\cup\mathcal{S}^{-1}$.
Note that $\mathsf{o}(F)=F^{-1}F$, hence $1_F\in A$ for every $F\in(\mathcal{S}\cup\mathcal{S}^{-1})^n$. Note also that $1_{A\cap B}=1_A\cdot 1_B$,
$1_{A\setminus B}=1_A\cdot(1_A-1_B)$, and $1_{A\cup B}=1_A+1_B-1_A1_B$ for every $A, B\subset\mathfrak{G}^{(0)}$. It follows that
$1_{A_{x, n}}\in \mathcal{A}$ for all $x\in\mathfrak{G}^{(0)}$ and $n$.
Let us show that for every open set $A\subset\mathfrak{G}^{(0)}$ and every $x\in A$ there exists $n$ such that $A_{x, n}\subset A$. For every $y\notin A$
there exists $n_y$ such that $A_{x, n_y}\cap A_{y, n_y}=\emptyset$. Since $\mathfrak{G}^{(0)}\setminus A$ is compact, there exists a finite
covering $A_{y_1, n_{y_1}}, A_{y_2, n_{y_2}}, \ldots, A_{y_m, n_{y_m}}$ of $\mathfrak{G}^{(0)}\setminus A$. Let $n=\max n_{y_i}$. Then $A_{x, n}\subset A$.
Let $F$ be an arbitrary open compact $\mathfrak{G}$-bisection. For every $g\in F$ there exists $n$ and $F'\in(\mathcal{S}\cup\mathcal{S}^{-1})^n$
such that $g\in F'$. There also exists $n_g$ such that $A_{\mathsf{o}(g), n_g}\subset\mathsf{o}(F)$ and $F\cdot A_{\mathsf{o}(g), n_g}=F'\cdot A_{\mathsf{o}(g), n_g}$.
We get a covering of $F$ by sets of the form $F'\cdot A_{x, m}$, where $F'\in(\mathcal{S}\cup\mathcal{S}^{-1})^n$. Since any two
sets of the form $A_{x, n}$ are either disjoint or one is a subset of the other, we can find a covering of $F$ by disjoint sets of the form
$F'\cdot A_{x, m}$ for $F'\in(\mathcal{S}\cup\mathcal{S}^{-1})^n$. This implies that $1_F\in\mathcal{A}$, which finishes the proof.
\end{proof}
\subsection{Examples}
\subsubsection{Subshifts}
\label{sss:subshifts}
Let $\mathcal{X}\subset X^{\mathbb{Z}}$ be a subshift, and let
$\mathfrak{S}$ be the groupoid of germs generated by the shift
$s:\mathcal{X}\longrightarrow\mathcal{X}$. Let, as in~\ref{sss:shifts}, $S_x=\{(s, w)\;:\;w(0)=x\}$,
$\mathcal{S}=\{S_x\}_{x\in X}$. Note that
for every word $x_1x_2\ldots x_n$ domain of the product
$S_{x_1}S_{x_2}\cdots S_{x_n}$ is the set of words $w\in\mathcal{X}$ such that
$w(0)=x_n$, $w(1)=x_{n-1}$, \ldots, $w(n-1)=x_1$. It follows that the
set $\mathcal{S}\cup\mathcal{S}^{-1}$ is expansive, and by
Proposition~\ref{prop:expansive},
$\{1_S\}_{S\in\mathcal{S}\cup\mathcal{S}^{-1}}$ is a generating set of
$\Bbbk[\mathfrak{S}]$.
Since $\mathfrak{S}$ coincides with the groupoid of the
$\mathbb{Z}$-action on $\mathcal{X}$ defined by the shift, the algebra $\Bbbk[\mathfrak{S}]$ is
the corresponding cross-product of the algebra of continuous $\Bbbk$-valued
functions with the group algebra of $\mathbb{Z}$. Every its element is
uniquely written as a Laurent polynomial $\sum a_n\cdot t^n$, where $t\in\Bbbk[\mathfrak{G}]$ is
the characteristic function of the set of germs of the shift $s:\mathcal{X}\longrightarrow\mathcal{X}$, and
$a_n$ are continuous $\Bbbk$-valued functions. Multiplication rule for such
polynomials follows from the relations $t\cdot a=b\cdot t$, where
$a, b:\mathcal{X}\longrightarrow\Bbbk$ satisfy $b(w)=a(s^{-1}(w))$ for every $w\in\mathcal{X}$.
\begin{proposition}
\label{pr:shiftgrowth}
Let $V$ be the linear span of
$\{1\}\cup\{1_S\}_{S\in\mathcal{S}\cup\mathcal{S}^{-1}}$. Then
\[\left\lfloor\frac{n}2\right\rfloor
p_{\mathcal{X}}\left(\left\lfloor\frac n2\right\rfloor\right)\le\dim V^n\le (2n+1)p_{\mathcal{X}}(2n).\]
\end{proposition}
\begin{proof}
The upper bound follows from Theorem~\ref{th:growth}. For the lower
bound note that $S_{x_1}S_{x_2}\ldots S_{x_n}$ and
$S_{y_1}S_{y_2}\ldots S_{y_m}$ are disjoint if $x_1x_2\ldots x_n\ne
y_1y_2\ldots y_m$, hence the set of characteristic functions of all
non-zero products of elements of $\mathcal{S}$ is linearly
independent, so that $\sum_{k=0}^np_{\mathcal{X}}(n)\le\dim V^n$. Since
$p_{\mathcal{X}}(n)$ is non-decreasing, we have
$\left\lfloor\frac{n}2\right\rfloor
p_{\mathcal{X}}\left(\left\lfloor\frac n2\right\rfloor\right)\le\sum_{k=0}^np_{\mathcal{X}}(n)$.
\end{proof}
Note that since the characteristic functions of the
products $S_{x_1}S_{x_2}\ldots S_{x_n}$ are linearly independent,
their linear span is a sub-algebra of $\Bbbk[\mathfrak{S}]$ isomorphic
to the semigroup algebra $\mathcal{M}_{\mathcal{X}}$
of the semigroup generated by the set
$\{S_x\;:\;x\in X\}$. It is easy to see that $\mathcal{M}_{\mathcal{X}}$ is
isomorphic to the quotient of the free associative algebra generated
by $X$ modulo the ideal generated by all words $w\in X^*$ such that
$w$ is not a subword of any element of the subshift $\mathcal{X}$. It follows
from Proposition~\ref{pr:shiftgrowth} that growths of
$\Bbbk[\mathfrak{S}]$ and $\mathcal{M}_{\mathcal{X}}$ are equivalent. Note that
the algebras $\mathcal{M}_{\mathcal{X}}$ are the original examples of algebras
of arbitrary Gelfand-Kirillov dimension, see~\cite{warfield:gk}
and~\cite[Theorem~2.9]{krauselenagan}.
\begin{examp}
Let $\mathcal{X}$ be a Sturmian subshift. It is minimal
and $p_{\mathcal{X}}(n)=n+1$, hence \[\frac{(n+1)(n+2)}2\le\dim
V^n\le 2n(2n+1),\]
so that $\Bbbk[\mathfrak{S}]$ is a quadratically growing finitely
generated algebra. Note that it is simple by Proposition~\ref{pr:simple}. This disproves
Conjecture~3.1 in~\cite{bell:simple}.
\end{examp}
\begin{examp}
It is easy to see that every Toeplitz subshift is
minimal. Consequently, known examples of Toeplitz subshifts
(see Subsection~\ref{sss:shifts}) provide us with simple finitely
generated algebras of arbitrary Gelfand-Kirillov dimension $\alpha\ge
2$, and also uncountably many different growth types of simple
finitely generated algebras of
Gelfand-Kirillov dimension two (see a question on existence of such
algebras on page 832 of~\cite{bell:dichotomy}).
\end{examp}
\subsubsection{Self-similar groups}
Let $G$ be a self-similar group of automorphisms of the tree
$X^*$. Let $\mathfrak{G}$ be the groupoid of germs of its action on the boundary
$X^{\mathbb{N}}$ of the tree. Suppose that $G$ is \emph{self-replicating},
i.e., for all $x, y\in X$ and $g\in G$ there exists $h\in G$ such that
$g(x)=y$ and $h|_x=g$. Then for all pairs of words $v, u\in X^*$ of equal
length and every $g\in G$ there exists $h\in G$ such that $h(v)=u$ and
$h|_v=g$. In other words, the transformation $vw\mapsto ug(w)$ is an
open compact $\mathfrak{G}$-bisection (more pedantically, the set of its germs
is a bisection, but we will identify a $\mathfrak{G}$ bisection $F$ with the map
$\mathsf{o}(g)\mapsto\mathsf{t}(g)$, $g\in F$).
Fix $n\ge 0$, and consider the set of all $\mathfrak{G}$-bisections of the form
$R_{u, g, v}:vw\mapsto ug(w)$ for $v, u\in X^n$ and $g\in G$. Note
that these bisections are multiplied by the rule
\begin{equation}\label{eq:Rugv}
R_{u_1, g_1, v_1}R_{u_2, g_2, v_2}=\left\{\begin{array}{ll} 0 &
\text{if $v_1\ne u_2$;}\\ R_{u_1, g_1g_2, v_2} & \text{if $v_1=u_2$}.\end{array}\right.
\end{equation}
Let $A_n$ be the formal linear span of the elements $R_{u, g, v}$ for $u, v\in X^n$ and $g\in G$. Extend
multiplication rule~\eqref{eq:Rugv} to $A_n$. It is easy to see then that $A_n$ is isomorphic to the algebra
$M_{d^n\times d^n}(\Bbbk[G])$ of matrices of size $d^n\times d^n$ over the group ring $\Bbbk[G]$.
The map $R_{u, g, v}\mapsto\sum_{x\in X} R_{ug(x), g|_x, vx}$ induces a homomorphism $A_n\mapsto A_{n+1}$ called
the \emph{matrix recursion}. More on matrix recursions for
self-similar groups
see~\cite{bgr:spec,bartholdi:ring,nek:bim,nek:cpalg,grinek:schur}.
\begin{examp}
For the adding machine action (see Example~\ref{ex:admach}) the matrix recursions replace every entry $a^n$ by $\left(\begin{array}{cc} 0 & a\\
1 & 0\end{array}\right)^n$, i.e., are induced by the map
\[a\mapsto\left(\begin{array}{cc} 0 & a\\ 1 & 0\end{array}\right).\] For
example, the image of $a$ in $A_2$ is
\[\left(\begin{array}{cccc}0 & 0 & 0 & a\\ 0 & 0 & 1 & 0\\ 1 & 0 & 0 &
0\\ 0 & 1 & 0 & 0\end{array}\right).\]
For the Grigorchuk group the matrix recursions are induced by the map
\[a\mapsto\left(\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right),\qquad
b\mapsto\left(\begin{array}{cc} a & 0\\ 0 & c\end{array}\right),\]
\[c\mapsto\left(\begin{array}{cc} a & 0\\ 0 & d\end{array}\right),\qquad
d\mapsto\left(\begin{array}{cc} 1 & 0\\ 0 & b\end{array}\right).\]
\end{examp}
\begin{proposition}
The convolution algebra $\Bbbk[\mathfrak{G}]$ of the groupoid of germs of the action of $G$ on $X^{\mathbb{N}}$ is isomorphic to the direct
limit of the matrix algebras $A_n\cong M_{d^n\times d^n}(\Bbbk[G])$ with respect to the matrix recursions.
\end{proposition}
\begin{proof}
Denote by $A_\infty$ the direct limit of the algebras $A_n$ with respect to the matrix recursions. Let $\phi:A_\infty\longrightarrow\Bbbk[\mathfrak{G}]$ be
the natural map given by $\phi(R_{u, g, v})=1_{R_{u, g, v}}$. Note
that $1_{R_{u, g, v}}=\sum_{x\in X}1_{R_{ug(x), g|_x, vx}}$, hence
the map $\phi$ is well defined. It also follows from equation~\eqref{eq:Rugv} that $\phi$ is a homomorphism of algebras. It remains to show that
$\phi$ is injective. Let $f$ be a non-zero element of $\Bbbk[\mathfrak{G}]$, and let $(g, w)\in\mathfrak{G}$ be such that $f(g, w)\ne 0$. Suppose that
$\phi(f)=\sum_{u, v\in X^n}\alpha_{u, v}R_{u, g_{u, v}, v}$ for some $\alpha_{u, v}\in\Bbbk$ and $g_{u, v}\in G$.
Denote the set
of all pairs $(u, v)$ such that $(g, w)\in R_{u, g_{u, v}, v}$ and $\alpha_{u, v}\ne 0$ by $P$.
The set $\bigcap_{(u, v)\in P}R_{u, g_{u, v}, v}$ is an open neighborhood of $(g, w)$, hence there exists a $\mathfrak{G}$-bisection
$R_{w_1, h, w_2}$ contained in $\bigcap_{(u, v)\in P}R_{u, g_{u, v}, v}$. Applying the matrix recursion, we get a representation
of $f$ as an element $\sum_{u, v\in X^{|w_1|}}\beta_{u, v}R_{u, h_{u, v}, v}\in A_{|w_1|}$ such that $(g, w)$ does not belong to
any set $R_{u, h_{u, v}, v}$, $u, v\in X^{|w_1|}$, $(u, v)\ne (w_1, w_2)$. Then $f(g, w)=\beta_{u, v}\ne 0$, hence $\phi(f)\ne 0$.
\end{proof}
As a corollary of Proposition~\ref{pr:contractingestimates}
and Theorem~\ref{th:growth} we get the following result of L.~Bartholdi~\cite{bartholdi:ring}.
\begin{proposition}
Let $G$ be a contracting self-replicating group, and let $\mathfrak{G}$ be the groupoid of germs
of its action on $X^{\mathbb{N}}$. Every finitely generated sub-algebra of $\Bbbk[\mathfrak{G}]$ has Gelfand-Kirillov dimension at most
$\frac{2\log|X|}{-\log\lambda}$, where $\lambda$ is the contraction coefficient of $G$.
\end{proposition}
The image of the group ring $\Bbbk[G]$ in $\Bbbk[\mathfrak{G}]$ is called the
\emph{thinned algebra}. It was defined in~\cite{sid:ring}, see also~\cite{bartholdi:ring}.
Let us come back to the case of the Grigorchuk group. Since its contraction coefficient is equal to $1/2$, every finitely generated sub-algebra
of $\Bbbk[\mathfrak{G}]$ has Gelfand-Kirillov dimension at most 2. It is easy to prove that it is actually equal to 2 in this case. Moreover, it
has quadratic growth, see~\cite{bartholdi:ring}.
This example is also an illustration of the non-Hausdorffness phenomenon.
The groupoid of germs of the Grigorchuk group is not Hausdorff: the germs $(b, 111\ldots)$, $(c, 111\ldots)$, $(d, 111\ldots)$,
and $(1, 111\ldots)$ do not have disjoint neighborhoods.
\begin{examp}
\label{ex:nonHausdorff}
Consider the convolution algebra $\mathbb{F}_2[\mathfrak{G}]$ for the groupoid
of germs of the Grigorchuk group over the field with two elements.
The matrix recursion for the element $b+c+d+1$ is
\[b+c+d+1\mapsto\left(\begin{array}{cc}0 & 0 \\ 0 &
b+c+d\end{array}\right).\] It follows that
$b+c+d$ is a non-trivial element of $\mathbb{F}_2[\mathfrak{G}]$ but, as a
function on $\mathfrak{G}$ is zero everywhere except for the germs of $b, c, d,
1$ at $111\ldots$,
where it is equal to 1. This shows that the ideal $I$ from
Proposition~\ref{pr:simple} is non-zero in this case, and the algebra
$\mathbb{F}_2[\mathfrak{G}]$ is not simple.
\end{examp}
\subsection{Modules $\Bbbk\mathfrak{G}_x$}
Let $\mathfrak{G}$ be an \'etale minimal groupoid. Consider the
space $\Bbbk\mathfrak{G}_x$ of maps $\phi:\mathfrak{G}_x\longrightarrow\Bbbk$ with finite support, where
$\mathfrak{G}_x=\{g\in\mathfrak{G}\;:\;\mathsf{o}(g)=x\}$. It is easy to see that for every
$\phi\in\Bbbk\mathfrak{G}_x$ and $f\in\Bbbk[\mathfrak{G}]$ the convolution
$f\cdot\phi$ is an element of
$\Bbbk\mathfrak{G}_x$, and that $\Bbbk\mathfrak{G}_x$ is a left $\Bbbk[\mathfrak{G}]$-module with
respect to the convolution.
\begin{proposition}
\label{pr:modules}
Let $\mathcal{S}$ be
an finite set of open compact $\mathfrak{G}$-bisections, and let $V\subset\Bbbk[\mathfrak{G}]$ be the
linear span of their characteristic functions and $1_{\mathfrak{G}^{(0)}}$. Then
for every $n\ge 1$ we have
\[\dim V^n\cdot\delta_x\le\gamma_{\mathcal{S}}(x, n),\]
where $\delta_x\in\Bbbk\mathfrak{G}_x$ is the characteristic function of
$x\in\mathfrak{G}_x$, and $\gamma_{\mathcal{S}}(x, n)$ is the growth of the
Cayley graph based at $x$ of the groupoid generated by the union of
the elements of $\mathcal{S}$.
If the isotropy group of $x$ is trivial, then the module $\Bbbk\mathfrak{G}_x$
is simple.
\end{proposition}
\begin{proof}
The growth estimate is obvious, since for every $g\in\mathfrak{G}_x$ and
$S\in\mathcal{S}$ we have $1_S\cdot\delta_g=\delta_{Sg}$, if
$Sg\ne\emptyset$, and $1_S\cdot\delta_g=0$ otherwise.
Let us show that $\Bbbk\mathfrak{G}_x$ is simple if the isotropy group of $x$ is
trivial. It is enough to show that for
every non-zero element
$\phi\in\Bbbk\mathfrak{G}_x$ there exist elements $f_1, f_2\in\Bbbk[\mathfrak{G}]$
such that $f_1\cdot\phi=\delta_x$ and $f_2\cdot\delta_x=\phi$.
Let $\phi\in\Bbbk\mathfrak{G}_x$, and let $\{g_1, g_2, \ldots, g_k\}$ be the
support of $\phi$. Since the isotropy group of $x$ is trivial,
$\mathsf{t}(g_i)$ are pairwise different. Let $U_1, U_2, \ldots, U_k$ be open
compact $\mathfrak{G}$-bisections such that $g_i\in U_i$ and $\mathsf{t}(U_i)$ are
disjoint. Then $\left(\sum_{i=1}^k\phi(g_i)1_{U_i}\right)\cdot
\delta_x=\phi$ and $\phi(g_1)^{-1}1_{U_1^{-1}}\phi=\delta_x$.
\end{proof}
\begin{examp}
\label{ex:matrices}
Let $X$ be a finite alphabet, and let $w\in X^{\mathbb{Z}}$ be a non-periodic
sequence such that closure $\mathcal{X}_w$ of the shift orbit of $w$ is
minimal. Let $\mathfrak{S}$
be the groupoid generated by the action of the shift on $\mathcal{X}_w$. Denote
by $T$ and $T^{-1}$ the characteristic functions of the sets of germs
of the shift and its
inverse, and for every $x\in X$, denote by $D_x$ the characteristic
function of the cylindrical set $\{w\in\mathcal{X}_w\;:\;w(0)=x\}$. Then
$\Bbbk[\mathcal{S}]$ is generated by $T, T^{-1}$ and $D_x$ for $x\in
X$. Note that we can remove one of the generators $D_x$, since
$\sum_{x\in X}D_x=1=TT^{-1}$. Consider the set
$\mathfrak{S}_w=\{(s^n, w)\;:\;n\in\mathbb{Z}\}$ and the corresponding module
$\Bbbk\mathfrak{S}_w$. Its basis as a $\Bbbk$-vector space consists of
the delta-functions $e_n=\delta_{(s^n, w)}$, $n\in\mathbb{Z}$. In this
naturally ordered basis left
multiplication by $T$ is given by the
matrix \[T=\left(\begin{array}{cccccc}\ddots & \vdots & \vdots &
\vdots & \vdots & \\ \cdots & 0 & 0 & 0 & 0 & \cdots \\ \cdots & 1 & 0 &
0 & 0 & \cdots \\ \cdots & 0 & 1 & 0 & 0 & \cdots \\
\cdots & 0 & 0 & 1 & 0 & \cdots \\
& \vdots &
\vdots &
\vdots & \vdots & \ddots\end{array}\right)=(t_{ij})_{i\in\mathbb{Z},
j\in\mathbb{Z}}\]
with the entries $t_{m, n}=\delta_{m-1, n}$. The element $T^{-1}$ is given
by the transposed matrix, and an element $D_x$ is given by the
diagonal matrix $(a_{ij})$ with entries given by the rule
\[a_{nn}=\left\{\begin{array}{ll}1 & \text{if $w(n)=x$,}\\ 0 &
\text{otherwise.}\end{array}\right.\]
It follows that the algebra $\Bbbk[\mathfrak{S}]$ is isomorphic to the
algebra generated by such matrices. For example, if $X=\{0, 1\}$, then
the algebra is generated by the matrices $T$, $T^\top$, and the
diagonal matrix with the sequence $w$ on the diagonal.
\end{examp}
\def$'${$'$}\def\ocirc#1{\ifmmode\setbox0=\hbox{$#1$}\dimen0=\ht0
\advance\dimen0 by1pt\rlap{\hbox to\wd0{\hss\raise\dimen0
\hbox{\hskip.2em$\scriptscriptstyle\circ$}\hss}}#1\else {\accent"17 #1}\fi}
\def$'${$'$}
|
2,869,038,155,666 | arxiv | \section{Introduction}\label{}
Higher Education Institutions (HEI) are operating in an ever-increasingly competitive and dynamic environment. The recently accelerated transition to online learning by many institutions has been one of the key drivers of these emerging pressures. Within this context, student retention rates have also become a prominent issue at HEI, as well as concerns around student performance in general due to the prevalence of low grades \cite{namoun2020predicting}. Considerable research effort has been invested into technological means to address these issues and the Learning Analytics (LA) field, in general, has been an important vehicle for supporting these endeavours \cite{wong2020review}.
Within LA, it is widely appreciated that predictive analytics tools that identify at-risk students hold considerable potential to address these challenges at least in part, by providing the ability for timely interventions to be initiated with at-risk learners which can result in corrective measures being undertaken by them \cite{namoun2020predicting}. However, in their survey of LA applications, \citet{hernandez2022learning} conclude that LA technologies are generally not yet widely used in this sector despite the evident potential they offer to HEI. Indeed, \citet{jang2022practical} highlight that despite the clear opportunities offered by the predictive analytics technologies, the developed tools tend to persist only as research content. The authors posit that to genuinely integrate the predictive analytics technologies into educational contexts, barriers like educators’ general distrust in the tools, as well as the lack of interpretability and visual representation of information accompanying them, need to be overcome and are perceived as being some of the biggest obstacles. \citet{rets2021exploring} also point out that learners will engage with a LA tools only if they can understand how the LA system's outputs regarding them are generated.
Predictive models themselves have over time become more complex and as a result, they have assumed 'black-box' characteristics. Their complexity ensures that it is not possible to apprehend how these models arrive at their predictions, and crucially, what aspects of the learners’ behaviours are key determinants of their prognosticated outcomes. This absence of model \textit{interpretability} (how a model works) and \textit{explainability} of their specific predictions in turn negatively affects their utility, and ultimately results in distrust by the stakeholders \cite{baneres2021predictive}.
According to \citet{villagra2017improving}, for a prediction model to truly be useful, it ought to also do more than merely classify learners into risk categories. The authors argue that the models should in addition offer interpretative characteristics from which learners can gain insights into possible causes of their learning obstacles. Transparency of this kind facilitates the development of trust by all stakeholders in general and thus increases the prospects of adoption. The greater interpretability of the predictive models enhances the tool's capabilities to support the dispensing of effective guidance towards resolution and remediation to at-risk learners. In their recent systematic literature review of predictive LA studies, \citet{namoun2020predicting} also encouraged the pursuit of research into interpretability and explainability of the predictive models where the focus should more heavily focus on developing explanatory aspects of predictions rather than the development of models that merely forecast student outcomes.
In the most ideal setting, the analytics technologies used in the educational contexts need to even go beyond interpretability and explainability properties, and should in fact embody \textit{prescriptive} analytics capabilities which leverage data-driven techniques to communicate to learners precisely what remedial actions are most likely to result in improved outcomes \cite{susnjak2022learning}. While descriptive analytics answers \textit{'what happened?'} and predictive analytics addresses \textit{'what will happen?'}, prescriptive analytics tackles \textit{'how to make it happen?'} \cite{frazzetto2019prescriptive}. The power of prescriptive analytics, therefore, lies in its ability to transform information into implementable decisions. \citet{liu2017going} highlight the importance of LA tools which lead directly to actionability. Arguably, therefore, the most beneficial and insight-rich form of analytics is found in the prescriptive data-driven outputs which generate the greatest intelligence and value \cite{lepenioti2020prescriptive}.
Prescriptive analytics frequently uses machine learning in order to suggest at the existence of possible causal relationships within the features describing learners, and consequently, recommendations can be constructed from these outputs which can be used as advice concerning which behavioural adjustments are likely to result in more desirable outcomes. By offering tailored and evidence-based recommendations that venture beyond generic advice, and are instead customised to each learner with specific and measurable goals, more effective advice can be provided to support interventions, which \citet{wong2020review} confirm in their recent review of learning analytics intervention studies. Thereby, the prospects of positively affecting both retention rates and student performances are consequently improved. Increasingly studies are emerging that have highlighted the importance of LA tools which provide insights to learners with a prescriptive component that are in the form of recommendations for guiding learners \cite{valle2021predict} and this is an emerging research frontier.
\subsection{eXplainable AI}
With the increased embedding of complex predictive models into contexts that were previously dominated by human decision-making, the need has arisen for predictive models to display a greater degree of transparency behind their mechanisms of reasoning. This is the case not least for establishing trust in them, but also from the perspective of compliance. It is increasingly becoming a legal requirement across international jurisdictions\footnote{GDPR \cite{regulation2016regulation} is one example.} that the mechanisms behind any automated decisions affecting humans be clearly explained to those affected by them. \citet{wachter2017counterfactual} also point out the necessity of being able to contest decisions made by automated systems by those concerned and to also have the opportunity to be informed with respect to the current decision-making model as to what it would need to see change in their data inputs to produce an alternative or a more desirable decision.
It is for these reasons that a relatively new research field of eXplainable Artificial Intelligence (XAI) (sometimes referred to as Interpretable Machine Learning) has emerged. Some of the key aspirations of this field are focused on developing techniques that address the high-level interpretability of opaque predictive models as well as on devising tools that enable them to interrogate and extract explanations of how the models arrive at given conclusions \cite{molnar2020interpretable}. Meanwhile, there currently exists a gathering the research interest exploring prescriptive analytics where methods are being developed that leverage data-driven approaches to generate evidence-based actionable insights \cite{lepenioti2020prescriptive}.
From a technical point of view, model interpretability tends to pertain to the tasks of making sense of a model's internals generated post-training by a machine learning algorithm. It usually concerns a level of clarity into the mechanics of a model at a \textit{global-level}, while explainability of models refers to extracting the reasoning behind the model's prediction for a specific learner in this context, which is referred to as \textit{local-level} explainability. Both perspectives on model behaviour are important and serve multiple overlapping purposes. Model interpretability affords an institution the ability to communicate to all concerned stakeholders how a predictive model works in general terms using broad brush strokes. While local-level explainability of models enables the validation of specific predictions to take place by student support teams before interventions are initiated, it also enables clear responses to be given to affected students as to exactly how and why they have been identified as being at-risk. However, clues can also be gleaned from the local-level explanations into possible remedial actions that can be suggested to the learners.
Tools and techniques supporting the goals of model transparency are reaching maturity. SHAP \cite{lundberg2017unified} is a visualisation technique that is currently recognised as being state-of-the-art in the field of XAI \cite{gramegna2021shap} for realising both global and local-level model transparency. The Anchors \cite{ribeiro2018anchors} technique has also been developed recently as a tool that imparts a high degree of local-level explainability. The benefit of Anchors is that it produces human-readable rule-based models which are succinct approximations of the behaviour of the underlying complex models. However, a more advanced approach using Counterfactuals \cite{wachter2017counterfactual} enables the analytics processes to surpass predictive capabilities, and enter into the prescriptive. This technique provides precise suggestions to the learners regarding the smallest set of adjustments they need to make in their learning behaviour for an alternative prediction to be generated.
This study demonstrates the applicability of all three of these technologies for different steps of the proposed prescriptive analytics framework.
\subsection{Research Contribution}
This study presents a novel prescriptive analytics framework for supporting LA aims of identifying and initiating both timely and effective interventions with at-risk students. The proposed framework demonstrates how both predictive and prescriptive analytics can be more fully leveraged than has previously been done. This study illustrates how effective models predicting qualification completion outcomes can be developed using machine learning and made transparent at both global and local levels to meet the needs of all stakeholders. Furthermore, this work goes on to illustrate through case study examples how prescriptive analytics tools can subsequently be utilised to automatically generate specific prescriptive feedback, that is evidence-based as well as actionable, and how the output of these methods can be converted into human-readable text.
\section{Background}\label{}
In the last two years, there have been numerous systematic literature reviews (SLR) in the field of LA and Educational Data Mining (EDM) centring on predictive themes, casting multiple perspectives onto the state of this field, and revealing where the focus of the research efforts. Indeed, these works have identified current gaps, emerging trends, and future aspirations which are brought out in the following sections.
\subsection{Recent Survey Findings}
An in-depth SLR was conducted by \citet{namoun2020predicting}, covering a total of 62 relevant studies within predictive LA, in which they focused on three areas: (1) ways in which academic performance was measured using learning outcomes and subsequently predicted, (2) the types of algorithms used to forecast student learning outcomes, and (3) which features are most impactful for predicting student outcomes. The authors found that 90\% of the studies predicted course-level outcomes, while only three studies considered predicting programme/qualification-level outcomes. As part of their findings, they urged the research community to conduct further work in predictive modelling, and more specifically, to do so at the programme level which they described presently to be in its infancy. Meanwhile, the authors called researchers to rise above simply predicting outcomes, and to also incorporate model interpretability and explainability into their studies.
\citet{albreiki2021systematic} likewise conducted an SLR consisting of 78 studies of relevant EDM literature from 2009 to 2021 concerning predicting at-risk learners of non-completion. The review indicated that only a handful of studies proposed means for generating remedial solutions consisting of feedback that learners and educators can use to address the underlying obstacles. The authors noted that future research will place greater emphasis on devising machine learning methods to predict students’ performance in general and will also augment this with automatically generated remedial actions to assist learners as early as possible.
Another up-to-date survey by \citet{xiao2022survey} examined almost 80 studies using EDM to predict students’ performance and provided insights in line with previous studies. The authors note that one of the deficiencies in this field is that very few studies have attended to explore model interpretability and have thus neglected explanations of the mechanics of predictions and the role that different features play in the predicted outputs. Indeed, the authors stress that the use of model interpretability tools ought to be one of the chief pursuits and the direction of future research in this field of study.
There were several other recently conducted reviews into LA and EDM studies. Interestingly, these did not address the issue of model interpretability and explainability, nor was there coverage of the use of prescriptive analytics tools indicating that these approaches are not yet in wide use. \citet{fahd2021application} carried out a broad meta-analysis of literature of 89 studies from 2010 to 2020. Their survey analysed the application of machine learning approaches in predicting student academic performance. Their primary focus was on considering what types of models were being used in research at HEI, and their identification of trends centred around specific matters of how to achieve better predictive accuracies using machine learning.
Similarly, \citet{batool2022educational} conducted a LA survey of some 260 studies over the last 20 years on the topic of student outcome prediction. Similar to previous reviews, they focused on highlighting the most effective features for this task and the types of algorithms and techniques used, as well as data mining tools that are most frequently applied. However, the important issues of model interpretability and prescriptive analytics were not raised.
Likewise, \citet{shafiq2022student} conducted an SLR covering 100 papers. Their focus was again on predictive analytics, looking mostly at what was being predicted, what kind of data was used, which sets of features were effective or otherwise, and what types of algorithms were explored. The focus was essentially on ways to enhance the accuracy of predictive models but there was no coverage of techniques that extend beyond mere predictions
Instead, the recommendations emphasised that more ensemble-based and clustering methods should be explored to predict the performance of students and enhance the prediction accuracy. In the same vein, \citet{tjandra2022student} conducted a comprehensive review of recent studies based on student performance prediction. The review considered features and algorithms used, accuracies attained, as well as commonly used tools. The authors concluded that there is still limited use of personal characteristics data such as psychological and social/behavioural features for developing student performance predictions and that future research should focus on including these to address dropout rates. Finally, \citet{hernandez2022learning} investigated the practices of 16 HEIs that have deployed LA projects. The authors found that they have mostly used LA technologies for student retention. These tools largely supported strategies for identifying at-risk students through predictive analytics, which served as a springboard for initiating various types of interventions.
A vast majority of the predictive models identifying at-risk students have focused on course-level outcomes rather than on programme-level completions \cite{namoun2020predicting}. Since course-level predictive models have tended to only be deployed across subsets of all courses on offer, the likelihood of at-risk students evading detection is therefore high. Thus, there is high utility in pursuing programme-level completion predictions. Though prior works in the prediction of programme completions are very rare, those that exist have generally followed an outcomes-based approach, which breaks down all constituent parts of programme-level outcome requirements into course outcomes first, and subsequently proceeds to map them all to programmes. These types of bottom-up predictive models then operate on the level of course-level outcomes and these predictions are combined to generate programme-level outcome predictions.
Examples of this approach are \cite{dandin2018attainment}, as well as \cite{bhatia2017automated} and \cite{bindra2017outcome} who likewise developed programme-completion prediction models and claimed to achieve accuracies in the upper ranges between 90\% and 95\% using various techniques from the WEKA data mining toolkit. More recently, \citet{Gupta2021} followed a similar approach in first determining course-level outcome predictions and mapping these to programmes, before determining overall outcomes. Such systems embody a great deal of complexity and appear to have been mostly research prototypes since productionisation and long-term maintenance of these approaches in a dynamic setting are questionable with respect to sustainability.
\subsection{Model Interpretability in LA}
Model interpretability is not only important for building trust, meeting compliance and extracting maximal value from predictive systems, but it is also vital for developing accurate predictive models. Feature engineering and selection tasks are more critical to the success of machine learning models than the choice of algorithm or the size of the datasets used \cite{domingos2012few}. While a wide range of features for predicting student performance have been used in literature, there is still a lack of clarity on which specific features are most effective and how they interact to influence the attainment of course and programme outcomes \cite{namoun2020predicting}. Therefore, tools and approaches that specifically illuminate the influence and the behaviour of the models in terms of the underlying features, are important. In one of the earliest studies into methods of illuminating the predictions of black-box models for learners, \citet{villagra2017improving} developed a set of proprietary graphical tools to exploit the output information and provide a meaningful guide to both learners and instructors.
\citet{dass2021predicting} develop dropout prediction models for students in a MOOC course and did strive to gain a deeper understanding of the effectiveness of the various features to support accurate predictions, as well as to predict the point in time when the students were likely to drop out.
Similarly, \citet{jang2022practical} demonstrate the usage of XAI techniques to assist in interpreting the classification results of the models which were designed to identify at-risk students. The study paid particular attention to selecting features that were relevant to stakeholders.
\subsection{Prescriptive LA Systems}
\citet{jenhani2016course} developed one of the earliest automated remedial prescriptive systems which leveraged machine learning. The system was designed in such a way that a separate classification model was trained on a remedial action dataset describing historical data based on experts’ and instructors’ actions to improve the low learning outcomes. The types of predictive outputs that the system made as suggestions for at-risk students were: revise a concept, attempt extra quizzes, solve specific practice examples, take extra assignments, etc. All these prescriptions were at a generic level. \citet{elhassan2018remedial} extended this work further to enable the system to recommend a set of remedial actions to address specific shortcomings rather than a single action as previously.
\citet{albreiki2021customized} developed predictive models for at-risk students which they combined with a customised system that enabled the instructor to set various thresholds and weights alongside the predictive models. Based on the configuration and the outputs of the predictive models, the system would then select from a predefined list of remedial actions the most suitable one.
Most recently, \citet{susnjak2022learning} proposed a student-oriented LA dashboard which apart from using descriptive and predictive analytics, also demonstrated a prototype of how to integrate interpretability and explainability aspects of the predictive models, as well as the automatic generation of feedback based on prescriptive analytics.
\subsection{Summary and Research Aims}
From this literature review, several conclusions can be drawn. The literature indicates that the use of predictive analytics within the LA field is widespread and on the rise. The literature points to an acute gap in the current research concerning the use of predictive analytics tools which enable interpretability and explainability of the models. The bulk of the research is still largely focused on devising means of more effectively predicting various outcomes. There is a general absence of XAI tools in usage. These tools have only just started to emerge in LA research even though these tools are not esoteric. Much less is there any evidence that there exists the use of data-driven prescriptive analytics which provide automated and tangible remedial advice to at-risk students. Thus the conversion of predictions into actionable insights is broadly missing from the literature at this point.
There is therefore a need and an emerging requirement to begin to implement predictive analytics systems with responsible and accountable characteristics. These LA systems ought to have embedded transparency that is enabled by an expressive set of technologies, while venturing beyond the predictive realm and ideally being augmented by prescriptive tools that assist learners in addressing their challenges. Unless an effective predictive system can be matched with the reasoning behind its decisions surrounding the underlying variables, and suggestions about concrete pathways for moving forward, these systems will only ever have constrained capabilities and limited uptake.
To that end, this work proposes a prescriptive learning analytics framework that attempts to address the outlined gaps. This work defines a step-by-step process for building predictive models and highlights ways for enhancing them with various levels of interpretability, while ultimately showing how prescriptive analytics can complement the entire process. This work demonstrates the proposed framework using programme/qualification completion predictive models and maps each step of the framework with specific technologies while showing the practicality of the framework with use case scenarios.
\section{Proposed Framework}\label{}
The proposed Prescriptive Learning Analytics Framework (PLAF) is outlined here and depicted in Figure \ref{claf}. The figure portrays the two key components, namely predictive and prescriptive phases, as well as the process flow of the various steps comprising the framework. The framework assumes that the data identification and acquisition steps have already been completed. Each step in the framework is discussed in turn.
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.4]{framework.png}
\caption{ The Prescriptive Learning Analytics Framework (PLAF) highlights each step in the process. }\label{claf}
\end{figure}
The initial Step (1) assumes that the raw data has been cleaned and preprocessed to support subsequent analyses. Here, all relevant exploratory data analyses are performed, and an investigation into the reliability of the data. This includes investigating the prevalence of missing values, together with methods to impute them if necessary. Additionally, the ability of the existing data to support deeper analyses of the subsequent phases is conducted here.
\subsubsection*{Predictive Analytics Phase}
Predictive analytics begins with the feature engineering Step (2). Here the researchers are guided by domain knowledge for creating new features from the raw data with the motivation that the derived features amplify the signal in the data which will then increase the accuracy of the machine learning models. This process is creative and involves considerable trial-and-error in conjunction with subsequent steps in the framework.
A selection of new and original raw-data features is subsequently used to develop predictive machine learning models in Step (3). Multiple algorithms representing a wide variety are used to develop competing models. This is necessary since no one algorithm outperforms the rest across all possible domains and datasets\footnote{No Free Lunch Theorem \cite{wolpert1997no}}. The competing algorithms are tuned with respect to their hyperparameters to ensure that the model training is not misspecified, thus entailing an iterative approach. The models are robustly evaluated against a variety of metrics to identify the best candidate out of the competing models. In case of insufficient model accuracy, new features may be required, thus returning to Step (2).
All models are by definition a simplification of some phenomenon. However, as machine learning algorithms become more sophisticated, the induced models in turn become uninterpretable. The subsequent steps in the predictive analytics framework aim to expose the opaque mechanics of black-box models. This is achieved by simulating the behaviour of the primary model and in effect constructing a \textit{model-of-the-model} which is a simplification of the original while having human-understandable characteristics. These second-order models are commonly referred to as \textit{surrogate} or \textit{proxy} models.
In Step (4), the selected best-performing model is evaluated for interpretability. This step has a two-fold purpose. First, during the development stage, it serves as another means for achieving model validation by the researchers. However, more important for the current context, is that in this step the global-level mechanics of the model's behaviour can be revealed. In this step, analytics tools are used that depict which features are the key drivers of predicted outcomes, and how the various feature values positively or negatively impact the final predicted outcomes. The outputs of this step are communicated to key stakeholders so that trust in the black-box models can be fostered and the predictions relied upon. If the interpretation of the models raises concerns about the validity of the models and the underlying features, then the process returns to Step (2), and new features may need to be engineered.
In Step (5), we transition towards local-level explainability of the model by interrogating it on how precisely it has reasoned to classify a specific learner with a given outcome. The information in this step becomes particularly meaningful for two reasons. First, the academic advisors who manage interventions are able to gather information and gain a deeper understanding of which factors for each learner are the contributing factors to their predicted negative outcome. This step is also critical for legal compliance aspects where affected stakeholders have the right to understand how the autonomous decision-making systems have arrived at particular conclusions about them \cite{mathrani2021perspectives}.
At this stage, the limits of what predictive modelling can offer are reached. Seeking to understand what exactly can be done to maximise better outcomes belongs within the scope of prescriptive analytics.
\subsubsection*{Prescriptive Analytics Phase}
In prescriptive analytics, the aim is to leverage techniques like machine learning models to perform counterfactual or \textit{what-if} simulations. The end goal is to use the outputs of the simulations as a basis for automatically generating evidence-based feedback to learners concerning adjustments to their learning behaviour which may result in improved outcomes. The simulations entail specifying an alternative outcome for a given learner to the one that the predictive models have arrived at. In this study's context, we are seeking to define a new target outcome of a 'successful qualification completion' for an at-risk learner. The counterfactual modelling in Step (6) takes the predictive model and uses it to uncover a set of minimal changes that would need to take place in the learner's feature values for the desired outcome to be predicted for a specific learner. Naturally, for this to be useful counterfactual modelling needs to be performed using features upon which actionable steps can be taken. This means that a set of target features need to be selected like: engagement levels with the VLE, average assignment marks and the total number of on-time assignment submissions etc. The counterfactual modelling step is flexible. New models can be developed at this stage solely for this task whereby more actionable features are used, and immutable features like demographics (eg. learner age) are removed.
Counterfactual outputs need to be configured to model changes only in the selected features upon which advisors desire to base their feedback. Additionally, each feature in this modelling process needs to be constrained within a range of values for them to be varied to generate feasible and realistic pathways to a desirable predicted outcome. Several possible pathways towards an alternative outcome for a learner can be generated and not all will be feasible. The unrealistic pathways need to be filtered and discarded.
Once the most realistic pathways have been selected, the next Step (7) entails two parts. The first requires that the engineered feature values be converted back into raw values so that they become interpretable and actionable for the learner. This step, therefore, has a dependency on Step (1) in the framework where the raw values reside and enable the engineered values can be converted back to them. The second part converts the numeric values into natural language text which can be dispensed to learners either directly via LA dashboards and emails or through conversations with academic advisors. Converting the counterfactual outputs into a human-readable format also makes the prescriptive feedback more usable for the academic advisors for the task of filtering and selecting the most suitable prescriptive feedback from all the candidate options. The actual conversion task itself can be programmed and thus automated using templates and natural language processing tools. Once the prescriptive feedback has been generated and the most suitable selected, then the intervention can be initiated - Step (8).
\section{Methodology}\label{}
\subsection{Dataset}\label{}
\paragraph{ Setting: } The data was acquired from an Australasian HEI. The data was sourced from the institution's Student Management System (SMS) and the Virtual Learning Environment (VLE) - Moodle. The dataset consisted of undergraduate students who commenced studies from 2018 through 2022 and who either completed or abandoned their studies during this period. The two categories of outcome represented an evenly balanced target variable where the total number of completed learners accounted for 52\% (3693), and those who abandoned their studies comprised 48\% (3415).
\paragraph{Predictive problem formulation: } The target prediction variable was programme/qualification completion. Given the nature of the underlying data, each student represents a data point at a given point in time denoted by the academic year. Most students are enrolled over multiple academic years of their programme, and therefore the dataset represents students as snapshots across each of their years of enrollment. For example, if a student was enrolled in a three-year bachelor's qualification with ultimately a successful completion, then the student would be represented in the dataset with three data points and each would be designated the target variable of 'completed'. As a result, the total dataset consisted of 14918 data points. The learners who completed their programme of study comprised 72\% (10736) of all data points since they enrolled in multiple academic years, while non-completion learners comprised 28\% (4182), thus making the final dataset relatively unbalanced.
Given the nature of the dataset with a learner's educational journey represented by multiple data points, the predictive problem, in this case, was both \textit{formative} and \textit{summative}. In the formative approach to the prediction of learner outcomes, students' outcomes are considered at various checkpoints of their studies of their journey towards completion. However, in the summative prediction, the learner outcomes are predicted at the end of the qualification or semester when course-level predictive models are used.
\paragraph{Features: } The features used to describe each learner at a particular point in time can be grouped into four broad categories. These categories are learner academic performance and learning behaviours, as well as the more immutable learner characteristics data. Additionally, these features were augmented with the characteristics of the programme into which they enrolled.
Once the data was prepared (Step 1), raw feature values were engineered to extract new derivative features to create more descriptive feature values - Step (2). In order to draw out more value from the raw behavioural and academic performance features such as a learner's average grade or the number of online learning resources they accessed, these were transformed and relativised to the learner's cohort. This conversion made the feature values more generic, contextually meaningful and comparable across different cohorts who study different courses/programmes that have different means and spreads across various features. To achieve this, each course-specific feature was transformed into a normalised value using z-score standardisation. This conversion converted the learner's absolute and mean values for a given feature into a form that captures the degree to which it deviates from the learner's cohort. The utility of this approach to creating course-agnostic features was underscored by \citet{ramaswami2022developing}. The z-score calculation Formula \ref{zscore} denotes $x$ as the value being converted, $\mu$ and $\sigma$ represent the mean and the standard deviation of all the values of $x$:
\begin{ceqn}\label{zscore}
\begin{align}
z-score = \frac{x-\mu}{\sigma}
\end{align}
\end{ceqn}
In essence, the value of the z-score communicates how many standard deviations a given value is from the mean. For a z-score equal to 0, it implies that the value is directly on the mean. A positive z-score signifies that the raw $x$ value is above the mean and the opposite holds for negative z-scores. The z-score values possess a greater descriptive power leading to more accurate machine learning models, but this comes with the cost of lower interpretability. All features used in the study are summarised and described in Table \ref{features}, with engineered features representing Step (2) being identified with $\bigstar$.
\begin{table}[p!]
\caption{Descriptions of features used for predictive and prescriptive modelling. $\bigstar$ denoting engineered features, while $\bigcirc$ describes features used only for prescriptive model generation and $\oplus$ denotes features used for both prescriptive model generation and prescriptive feedback generation. }\label{features}
\fontsize{7}{7.2}\selectfont
\centering
\begin{tabular} {p{0.17\textwidth}@{\extracolsep{\fill}} p{0.25\textwidth}p{0.45\textwidth} >{\centering\arraybackslash}p{0.1\textwidth} >{\centering\arraybackslash}p{0.07\textwidth}}
\toprule
Category & Feature Name & Description & Predictive Modelling & Prescriptive Modelling \\
\midrule
LEARNER CHARACTERISTICS & Basis For Admission Description & Eg. NZ Entrance, NCEA, adult admission, discretionary entrance etc. & $\checkmark$ & \\
& Has Previous Tertiary Study & yes/no & $\checkmark$ & \\
& Highest School Qualification Description & Eg. NZ University Entrance, International Baccalaureate etc. & $\checkmark$ & \\
& Current Full Time Status & full-time/part-time & $\checkmark$ & $\oplus$ \\
& Current Student Mode Numeric & On-campus or online/distance & $\checkmark$ & $\oplus$\\
& Current Prior Activity Description & What was the primary activity that the student was engaged in, in the previous year & $\checkmark$&\\
& Age Description & current student age & $\checkmark$ & $\bigcirc$ \\
& Gender & male/female/other & $\checkmark$ & \\
\cmidrule{2-5}
ACADEMIC PERFORMANCE DATA & Grade Mark Mean $\bigstar$& Student’s mean grade for current academic year courses & $\checkmark$& $\oplus$ \\
& Grade Mark Max $\bigstar$& Student’s max grade for current academic year courses & & $\bigcirc$ \\
& Grade Mark Deviation From Class Mean $\bigstar$ & Student’s Z-score calculation with respect to the average cohort grade across all courses enrolled in & $\checkmark$ & $\bigcirc$ \\
& Papers Failed For Student Academic Year $\bigstar$ & & $\checkmark$ & $\oplus$ \\
& Online Learning Has Passed Assessment Count Zscore $\bigstar$& The total number of assessments a student has passed represented as a relativised Z-score with respect to the student's current cohort values. & & $\bigcirc$ \\
& Total Qualification Percent Completed $\bigstar$ & Percent of the student's qualification completed with respect to the number of courses required & & $\oplus$ \\
& Online Learning Submitted Assignment Zscore $\bigstar$ & The average assignment mark for a student represented as a relativised Z-score with respect to the student's current cohort average. & $\checkmark$ & $\oplus$ \\
\cmidrule{2-5}
LEARNER BEHAVIOUR DATA & Papers Withdrawn For Student Academic Year $\bigstar$ & Number of paper that a student has withdrawn from in a given year. & $\checkmark$ & $\oplus$ \\
& Online Learning Pages Viewed Count Zscore $\bigstar$ & The total amount of VLE content accessed by a student represented as a relativised Z-score with respect to the student's current cohort totals
& $\checkmark$ & $\oplus$ \\
& Online Learning Quiz Taken Count Zscore $\bigstar$ & The total number of quizzes taken by a student represented as a relativised Z-score with respect to the student's current cohort totals & $\checkmark$ & $\oplus$ \\
& Online Learning Forum Post Created Count Zscore $\bigstar$ & The total number of discussion forum posts created by a student represented as a relativised Z-score with respect to the student's current cohort totals
& $\checkmark$ & $\oplus$ \\
& Online Learning Forum Post Read Count Zscore $\bigstar$ & The total number of discussion forum posts read by a student represented as a relativised Z-score with respect to the student's current cohort totals & $\checkmark$ & $\oplus$ \\
& Online Learning On Time Submission Count Zscore $\bigstar$ & The total number of on-time assignment submissions by a student represented as a relativised Z-score with respect to the student's current cohort totals & $\checkmark$ & $\oplus$ \\
\cmidrule{2-5}
PROGRAMME CHARACTERISTICS DATA & Programme Title & Name of qualification, eg. Bachelor of Arts & $\checkmark$ & \\
& Programme Credits Required & Eg. 60, 120, 360, 480. & $\checkmark$& $\oplus$ \\
& & & & \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Predictive Machine Learning}\label{}
\paragraph{ Algorithms: } A wide variety of algorithms from a broad range of machine learning families of techniques were used for the experiments to generate candidate models as part of Step (3). These consisted of Scikit-learn \cite{ScikitLearn2022} Python implementations of Random Forest (RF) \cite{breiman2001random}, K-Nearest Neighbour Regression (kNN) \cite{cover1967nearest}, Naive Bayes (NB), Support Vector Machines (SVM), Gradient Boosting (GB), Logistic Regression (LR), Decision Tree (DT) and CatBoost (CB) \cite{catboost}\footnote{ This is a non-scikit-learn catboost 1.0.6 implementation found here: https://pypi.org/project/catboost/ }. Two baseline models were used for demonstrating the predictive value of the candidate models, namely, the stratified random guessing model (Baseline 1) as well as the mode (Baseline 2).
\paragraph{Data Preparation: } In cases where there were missing values and the underlying algorithms required the presence of all values, these were replaced with zero. For algorithms that required all values to be numeric, Binary Encoding of categorical values was used which produced more concise feature sets and reduced the likelihood of overfitting\footnote{CatBoost was the only algorithm in the suite of techniques used which did not require the imputation of missing values and the encoding of categorical values into numeric data types.}.
\paragraph{Training Approach and Hyperparameter Tuning: } Training was performed in such a way as to prevent data leakage from occurring which would compromise the validity of the predictive modelling results. Given that each learner is potentially represented by several data points, the experimental design involved applying train/test splits in a manner that ensured all data points from a given learner were either in the training or the test set.
All algorithms possessing consequential hyperparameters were first tuned using Random Grid Search in conjunction with a 5-fold cross-validation approach. The best performing hyperparameters were then used on a separate 10-fold cross-validation process to collect the estimated generalisability scores. The range of hyperparameters for each algorithm and the best performing values can be seen in Table \ref{tuning}. The final predictive models were evaluated using the overall F1-measure due to the unbalanced target variable. F1-measure calculates the harmonic mean of Precision and Recall values. However, for completion, the Area Under the Curve (AUC), Accuracy, Recall and Precision metrics are also reported separately as an average value across all test folds, together with their dispersion in the form of the standard deviation.
\begin{table}[hbt]
\caption{Optimal hyperparameters resulting from random search tuning. }\label{tuning}
\begin{tabular}{>{\raggedleft}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.7\textwidth}}
\hline
Algorithms & Optimised hyperparameter values \\
\hline
SVM & kernel=rbf, gamma=0.0001, C=1000 \\
Decision Tree & max\_depth=10, criterion=entropy \\
Logistic Regression & solver=liblinear, penalty=l2, C=336 \\
CatBoost & depth=7, iterations=150, learning\_rate=0.07 \\
kNN & weights=distance, p=1, n\_neighbors=5, metric=chebyshev, leaf\_size=20 \\
Random Forest & n\_estimators=200, min\_samples\_split=2, max\_features=auto, max\_depth=9 \\
Gradient Boosting & subsample=0.85, n\_estimators=500, min\_samples\_split=0.47, min\_samples\_leaf=0.115, max\_features=sqrt, max\_depth=5, learning\_rate=0.2, criterion=mae \\
\hline
\end{tabular}
\end{table}
\subsection{Explainable AI Tools}\label{}
SHAP (SHapley Additive exPlanation) which is based on Shapley values \cite{shapley1953quota} drawn from game theory literature, was used to generate \textit{global} interpretability defined in Step (4), as well as \textit{local} model explainability required for Step (5), addressing both the "how" and "why" questions around model behaviour respectively.
In addition, the Anchors tool was also used in Step (5) alongside SHAP. Both techniques approach the generation of new \textit{surrogate models} approximating the behaviour of the original "black-box" models differently and offer complementary insights. The advantage of SHAP lies in its detailed quantification of effects that each feature and their values exert on the final prediction, while Anchors reduces a black-box model into a human-readable set of predicates resembling a degenerate decision tree.
For the prescriptive analytics phase, the Diverse Counterfactual Explanations (DiCE) \cite{mothilal2020dice} technique was used in Step (6) to simulate \textit{what-if} scenarios and generate candidate prescriptive feedback suggestions.
DiCE can generate counterfactuals for many machine learning models, approaching the task as an optimisation problem. However, counterfactual modelling is a challenge and needs to be configured in such a way that, firstly, the features used are relevant for a learner and represent learning behaviours that can be adjusted, and secondly, the suggested action must be feasible and practical \cite{poyiadzi2020face}. To that end, a new underlying machine learning model was generated using an alternative set of features comprising predominately mutable features which would support the development of actionable feedback. These features can be seen in Table \ref{features} under the Prescriptive Modelling column. However, the counterfactual modelling was further configured to exclusively rely on features upon which prescriptive advice was to be based. Table \ref{features} therefore shows features that were used for creating a new prescriptive model as denoted by $\bigcirc$, and features that were used for both modelling and for generating counterfactuals $\oplus$. Secondly, the features used for counterfactual modelling have been constrained to fall within feasible and realistic ranges. Finally, once the candidate counterfactuals were generated, these were then converted into human-readable text as indicated in Step (7).
\subsection{Framework Evaluation}\label{}
The quantitative component of the framework concerning predictive analytics is evaluated through empirical experiments that demonstrate the accuracy of the generated models, which also validate the efficacy of the engineered features (Steps 2 - 3).
The remaining parts of the framework are qualitative. Steps 4 and 5 are evaluated by assessing the suitability of the tools to visually convey the mechanics of the underlying model and to establish its reasonableness.
A case study using two hypothetical students is used to evaluate in detail the model behaviour in explaining its predictions for Step 5.
The same case study examples are then carried through to the prescriptive analytics component (Steps 6 - 7), where several prescriptive feedback suggestions are generated for each student, and subsequently converted into human-readable text for assessment of their feasibility.
\section{Results}\label{}
The efficacy of the candidate predictive models to identify learners at risk of programme/qualification non-completion is evaluated initially, and the best algorithm for this dataset is identified (Step 3). The characteristics of the best-performing predictive model are next examined to interpret its mechanics (Step 4). A case study involving two hypothetical students with non-completion predictions is then conducted to demonstrate how model explainability can be leveraged to interrogate the models about their reasoning for arriving at given predictions (Step 5) - two approaches to achieve this are presented. The application of prescriptive analytics is subsequently demonstrated using the same two hypothetical students for illustrative purposes. The capability of counterfactual modelling (Step 6) to derive potential prescriptive feedback for learners is then shown. Finally, the conversion of the prescriptive feedback into human-readable text suitable for learners (and advisors) is demonstrated (Step 7).
\subsection{Predictive Model}\label{}
Table \ref{accuracies} summarises the generalisation performances of all candidate models across several evaluation measures. The results indicate that overall, the models have achieved a high level of efficacy for predicting whether a learner would eventually complete or abandon their studies. The preferred F1-measure on the unbalanced dataset indicates that the values range from ~92\% to ~95\% and represent a significant improvement over the baseline models. The table highlights the importance of reporting the accuracies of baseline models so that the genuine value of the predictive models with respect to various forms of random guessing can be quantified. On average, the best-performing algorithm for this dataset is CatBoost, which is slightly better than Random Forest. Some of the performance advantages of CB can arguably be attributed to its inherent ability to handle both missing values and categorical data directly in contrast to the other algorithms in these experiments.
\begin{table}[hbt]
\caption{Predictive accuracies of all candidate models listed in a descending order of estimated generalisability using the F1-measure. }\label{accuracies}
\begin{tabular}{rlllll}
\hline
Algorithm & F1-measure & Accuracy & AUC & Recall & Precision \\
\hline
CatBoost & 94.5 $\pm$0.6 & 92.0 $\pm$0.7 & 88.3 $\pm$0.6 & 96.7 $\pm$0.7 & 92.5 $\pm$0.7\\
Random Forest & 93.9 $\pm$0.6 & 90.9 $\pm$0.9 & 85.6 $\pm$1.2 & 97.7 $\pm$0.5 & 90.4 $\pm$1.0\\
SVM & 93.8 $\pm$0.6 & 90.7 $\pm$0.9 & 85.8 $\pm$1.2 & 97.1 $\pm$0.7 & 90.7 $\pm$1.1\\
Logistic Regression & 93.5 $\pm$0.8 & 90.3 $\pm$1.1 & 85.5 $\pm$1.4 & 96.5 $\pm$0.8 & 90.7 $\pm$1.1\\
Decision Tree & 93.0 $\pm$0.6 & 89.6 $\pm$0.8 & 84.1 $\pm$1.3 & 96.6 $\pm$0.4 & 89.7 $\pm$1.1\\
Gradient Boosting & 92.9 $\pm$0.5 & 89.4 $\pm$0.7 & 84.4 $\pm$0.8 & 95.9 $\pm$0.8 & 90.0 $\pm$0.7\\
kNN & 92.9 $\pm$0.7 & 89.3 $\pm$0.9 & 83.5 $\pm$1.2 & 96.8 $\pm$0.5 & 89.3 $\pm$1.1\\
Naive Bayes & 91.5 $\pm$0.6 & 87.6 $\pm$0.8 & 83.2 $\pm$0.9 & 93.2 $\pm$0.7 & 89.9 $\pm$0.8\\
Baseline 2 & 83.7 $\pm$1.0 & 71.9 $\pm$1.5 & 50.0 $\pm$0.0 & 100.0 $\pm$0.0 & 71.9 $\pm$1.5\\
Baseline 1 & 72.0 $\pm$0.9 & 59.6 $\pm$1.0 & 49.6 $\pm$1.0 & 72.4 $\pm$1.7 & 71.7 $\pm$1.0\\
\hline
\end{tabular}
\end{table}
\subsection{Model Interpretability}\label{}
Having determined the efficacy of the model to identify students at risk of programme non-completion, the global-level interpretability of the best-performing CatBoost model is next examined.
Figure \ref{shapsummary} depicts SHAP's perspective of the model's dynamics. Two key components are shown. The first lists features in their order of importance, from highest to lowest in terms of the impact they exert on the eventual prediction. It can be seen that the learners' current full-time status, their prior activity with respect to the current academic year, the mean grade mark, together with the number of failed papers are the most impactful features.
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.45]{summary_plot.png}
\caption{SHAP summary plot showing the high-level behaviour of the CatBoost model. Most impactful features are shown from top to bottom. }\label{shapsummary}
\end{figure}
The second component in the figure offers an additional dimension concerning the interpretability of the model. Here we observe how an increase or decrease in feature values affects the final prediction. The colour gradients represent increasing (red) and decreasing (blue) feature values, while grey represents categorical values. The x-axis depicts SHAP values. Data points with a positive SHAP value (appearing to the right of the vertical zero line) have a positive impact on the predictions, in other words, they contribute towards driving the prediction towards programme completion predictions. Conversely, the points with a negative SHAP value (to the left of the vertical zero line) influence the prediction towards programme non-completion. The extent of the points from the vertical line signifies the magnitude of the effect that they contribute to the final prediction.
In the figure, it can be observed that the full-time study status (value 1) has a positive effect on completion predictions, while part-time (value 0), has the opposite. Learners' mean grade has an unsurprising effect; however, a nuanced interpretation can be extracted from the plot where one can deduce that lower grade averages have more of a negative predictive outcome than high grade-averages have on positive outcomes. A similar interpretation can be made regarding the number of failed papers. A large number of failed papers has more of a negative predictive effect than having no failed papers has on positive predictive outcomes.
In general, elevated assignment scores and submission counts are predictive of positive outcomes. Learner engagement with the VLE conveys a more mixed picture. VLE pages viewed and the number of online quizzes taken have a positive effect; however, forum post creation counts are ambivalent, while the reverse holds for the number of forum posts read. As the number of qualification credits increases, the effect is stronger for negative outcomes, while the reverse holds for the learner's age. Possessing previous studies (value 1) indeed has a more positive effect than otherwise (value 0), while the on-campus study mode (value 1) is associated with more negative qualification outcomes than studying online (value 0).
By considering both the feature importance ranks and how the feature values affect the final prediction, it is possible to validate the model against an expected behaviour and communicate its mechanics in a simplified form to all stakeholders. The behaviour of the examined model confirms that it is reasonable and thus valid.
\subsection{Model Explainability}\label{}
Having achieved interpretation and validation of the predictive model, the next step is to examine the model behaviour at an individual (or local) prediction level. SHAP as well as Anchors\footnote{An alternative technology for this is LIME which has some additional advantages.} are used in this step. To demonstrate this, two hypothetical students are used - Student A and Student B. Both students have been predicted by the CatBoost model with non-completion outcomes, with the probability of 97\% and 90\% respectively.
Figure \ref{shapwaterfall}(a) shows the top nine features and their values for Student A on the y-axis, rank-ordered by influence on the final prediction. Informally, the figure can be viewed and interpreted as a tug-of-war. The mid-line represented with the value of 1.823 is the expected or the average SHAP value of all the predicted data points. A final SHAP value to the left of this line represents a non-completion prediction and alternatively, the values on the right side denote positive outcome predictions. Blue bars represent the forcing effects towards negative predictions, while red the opposite. The size of the bars represents the magnitude of the forcing of the corresponding features and their values. These graphs are best interpreted from the bottom up. The topmost feature represents the final SHAP value that includes all feature contributions.
\begin{figure}[hbt]
\centering
\subfloat(a){%
\includegraphics[clip,width=0.85\columnwidth]{studentA.png}%
}
\subfloat(b){%
\includegraphics[clip,width=0.85\columnwidth]{studentB.png}%
}
\caption{SHAP force plot for Student A and B depicting the effects that their feature values have on the final prediction outcome. }\label{shapwaterfall}
\end{figure}
Figure \ref{shapwaterfall}(a) shows that the least significant 11 features collectively have an influence tending towards negative outcomes to the left of the 1.823 value mid-line. This increases with each of the features except for the student's learning behaviour for on-time assignment submissions, which has a positive effect.
Model reasoning is depicted in Figure \ref{shapwaterfall}(b) for Student B. Similar patterns are observed with the exception that this student is a returning learner (prior activity is 'university student') and that the student's mean grade is 66\%, which have a strong positive influence on the final prediction. However, in totality, the majority of the features are forcing the prediction towards negative outcomes, and this is where they ultimately settle. The utility of the SHAP tool to visually explain its reasoning to academic advisors pre-intervention as well as to other relevant stakeholders is demonstrable through these examples.
While SHAP provides a detailed visual perspective into the model mechanics, some cccasions will require that a more succinct and simplified explanation of a prediction is communicated.
This can be achieved using Anchors. Figures \ref{proxy}(a) and \ref{proxy}(b) show how the complex predictive model can be reduced to the most essential and explained in a more simplified manner for both Student A and B respectively. The figures show that the prediction model has been re-cast as a rule-based decision tree consisting of only three conditions, which result in a non-completion prediction if they all hold true. Student A and B's actual values for the three features are shown in the first column with the conditional statements and their thresholds for each feature shown in the second column.
\begin{figure}[hbt]
\centering
\subfloat(a){%
\includegraphics[clip,width=0.9\columnwidth]{studentAproxy.png}%
}
\subfloat(b){%
\includegraphics[clip,width=0.9\columnwidth]{studentBproxy.png}%
}
\caption{Proxy model explanation of predictions for Student A and B.}\label{proxy}
\end{figure}
In both cases, the Anchor surrogate model has identified and used the number of papers failed as one of the reasons for a negative prediction. For Student A, the fact that the student has a low average grade of 53\% while undertaking a commitment to a programme that is higher than the standard bachelor's qualification (360 credits), has been used as further conditions for the classification as non-completion. In the case of Student B, the fact that the student has already withdrawn from papers in the current academic year and is presently a part-time student, has created conditions for a non-completion predicted outcome. These forms of model simplifications serve as effective and suitable tools for communicating to affected learners how exactly they have come to be identified as being at-risk, thus meeting the requirements of transparency and responsible use of predictive analytics.
The generation of surrogate models as demonstrated above already hints at possible prescriptive suggestions which can be constructed from them. Indeed, there is a potential to do this, however, a more data-driven and robust method ought to be pursued which considers the interaction of the features and their effects in a more principled approach. This is where prescriptive analytics tools make their contribution.
\subsection{Prescriptive Modelling}\label{}
Counterfactual modelling is used in this step to generate a set of possible pathways for a specific learner that would lead them to a positive outcome prediction. In more precise terms, here we are looking at several possible sets of minimal adjustments to selected feature values which would result in an alternate outcome for a student who is predicted to be on track for non-completion. This type of \textit{what-if} modelling is demonstrated for Student A and B in Figure \ref{cfs}.
Figure \ref{cfs} depicts both the most concise set of features and the smallest required adjustments in their values which would be needed for the selected students to toggle their predicted outcome to a successful completion. The first column lists the selected features by the counterfactual model, and the next column shows the actual values for each of the hypothetical students, followed by three sets of counterfactuals from which automated and data-driven Prescriptive Feedback (PF) advice can be generated. In each PF set, only three feature values have been varied. The dash represents no required changes to the actual values.
\begin{figure}[hbt]
\centering
\subfloat(a){%
\includegraphics[clip,width=0.9\columnwidth]{studentA_CF.png}%
}
\subfloat(b){%
\includegraphics[clip,width=0.9\columnwidth]{studentB_CF.png}%
}
\caption{Depiction of three sets of Prescriptive Feedback (PF) options generated using counterfactuals for Student A and B. }\label{cfs}
\end{figure}
In Figure \ref{cfs}(a), it can be seen that modest increases to the grade average have been identified as a pathway to completion for Student A, together with a switch to a smaller programme of study, as well as a mixture of adjustments to the online learning engagement behaviours. In the case of Student B, Figure \ref{cfs}(b) shows multiple pathways towards completion if the study mode is changed to full-time as well as to online mode on another occasion. Pathways exist through some modifications in online learning behaviours, while no adjustment is needed to be made for the grade average of 66\%. Interestingly, the qualification percent completion feature has been identified in the case of this student as being potentially helpful. In all three PFs, it is observable that if the student succeeds in completing 8.2\% of their programme, up from the current 4.1\%, then this also suggests that positive outcomes become more likely.
From this, it becomes immediately apparent that z-score values and percentages of completion carry with them very little meaning and actionable potential on behalf of the target learners as well as for the academic advisors. It is for this reason, that the engineered features need to be converted back into original raw values which will then make them practical.
\subsection{Remedial Advice Generation}\label{}
The final step in the proposed framework performs two types of conversions. The first converts the engineered features chosen by the counterfactuals into raw values, while the second converts the candidate PFs into a natural language form which can then be dispensed to students. The example results of both types of conversions of the counterfactuals into final text-based PFs are shown in Figure \ref{text_pfs} for Student A and B.
\begin{figure}[hbt]
\centering
\subfloat(a){%
\includegraphics[clip,width=0.9\columnwidth]{studentA_PF.png}%
}
\subfloat(b){%
\includegraphics[clip,width=0.9\columnwidth]{studentB_PF.png}%
}
\caption{Depiction of the conversion of three sets of Prescriptive Feedback (PF) options into contextualised and human-readable student advice generated for Student A(a) and B(b), showing derived values from counterfactuals in bold. }\label{text_pfs}
\end{figure}
Figure \ref{text_pfs} shows that the proposed format of the final PFs to be used for remedial interventions comprises two parts. The first informs learners of their current status concerning the particular features that are to be used for PFs. This provides a context for the second part, which consists of sets of possible feedback suggestions that advisors can provide to the students. The figure shows that both the z-scores and the programme completion percentages have been converted into meaningful values which are now actionable and measurable. This underscores the dependence between Step 7 and Step 1 in the proposed framework for the conversion of engineered values to the raw values. The conversion of the counterfactuals into text-based PFs can either be automated through text-based templates or using Natural Language processing techniques. Once converted, interventions can be initiated as outlined in Step (8).
\section{Discussion}\label{}
The end goal of the proposed framework is the automated generation of evidence-based prescriptive feedback to learners who have been identified as being at-risk. The framework is generic and therefore the definition of at-risk is flexible and adaptable to any context. This study has demonstrated how this framework can be applied to learners who are at risk of programme non-completion.
In order to identify at-risk learners, the framework proposed how predictive analytics should be used to develop highly accurate predictive models, and how to use these models responsibly and with accountability. Using the models responsibly means exposing their internal mechanics and interpreting their behaviour to verify and validate them, and to be able to communicate this as plainly as possible to relevant stakeholders. This builds confidence and trust in the underlying black-box systems. Responsible use also means interrogating the models as to how they arrive at predictions for specific students which underscores accountability. The proposed framework outlines these steps and provides a clear roadmap in terms of which technologies and tools can be leveraged to support these tasks.
The proposed framework demonstrates how the leap from merely predicting outcomes to prescribing actions can be bridged for the first time in this domain using more advanced analytics. The danger in using counterfactuals in this domain is in assuming that causality is definitively established based on the fact that the prescriptive models have identified pathways to a positive outcome. This is not the case. The underlying prescriptive models rely on associations and cannot establish causality and this is a limitation.
It must be emphasised that many latent variables cannot be captured that have a significant bearing on eventual learner outcomes. The features used in this study are largely proxies. We cannot capture variables describing a learner's true level of motivation, their sense of progress, and confidence which all have a bearing on eventual outcomes amongst many other variables. However, the possibility exists that if a learner is provided with actionable and achievable data-driven prescriptive feedback, and if it is followed and attained, the possibility exists that it may have a cascading effect on a learner's sense of achievement and thus on their level of motivation, progress and overall confidence, which may then lead to positive outcomes.
Ultimately, the proposed framework demonstrates how multiple data-driven remedial advice options can be generated which can then be processed by academic advisors who are the human-in-the-middle, and they can then combine their experience and established theory of how to select the most suitable feedback suggestions for the target learners. Future work will involve evaluating the effectiveness of the proposed framework to improve student retention and qualification completion rates.
\section{Conclusion}\label{}
It has been a consistent feature of predictive Learning Analytics (LA) research targetting at-risk students, to focus exclusively on merely the predictive component. Predictive analytics is however much broader and it includes the unpacking of the internals of the predictive models' behaviour to stakeholders. It also encompasses responsible use of these automated systems which assist decision-making affecting humans. This includes the ability to interrogate the predictive models and seeks their reasoning as to how they have arrived at particular conclusions. eXplainable AI is a field that offers a suite of mature tools which enable this form of transparency that is largely absent in the current body of LA literature.
What is more, predictions and their understanding while important, only address one part of the challenge in improving retention rates and increasing successful learner outcomes. Additional approaches are needed that can provide specific and tailored remedial advice to learners that are most likely to improve their outcomes. Prescriptive analytics tools support these aims and make any analytics endeavours more complete.
This work proposes a prescriptive analytics framework that demonstrates how both transparent predictive analytics can be achieved and combined with prescriptive analytics techniques. The study develops predictive models for identifying at-risk learners of programme non-completion. This work demonstrates through case studies how transparent and responsible predictive modelling can be augmented with prescriptive
\urlstyle{same}
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2,869,038,155,667 | arxiv | \section{Introduction}
Coupled SUSY (CSusy) was introduced in \cite{coupledsusy} in connection with quantum mechanical systems in presence of some particular ladder operators. A simple example of CSusy is provided by the quantum harmonic oscillator, but other examples are discussed in \cite{coupledsusy}. Many relevant systems described in terms of ladder operators have been discussed in the literature during many years. We refer here to the monograph \cite{dong}, and to the papers \cite{lad1}-\cite{lad5}, which are just few of those written in the past three years.
This very partial list of references is sufficient to give an idea of the vitality of the topic, also in contexts which might appear quite non standard, see for instance \cite{bagbook1} and \cite{bagbook2}.
It is well known that ladder operators are not uniquely related to canonical commutation, or anti-commutation, rules. Raising and lowering operators also appear for quons, \cite{fiv,green,kar}, and in the analysis of the truncated harmonic oscillator, \cite{buc,bagchi}, just to cite two other interesting situations. As already observed, they also appear in CSusy, \cite{coupledsusy}, where they are proved to be related to the $\mathfrak{su}(1,1)$ Lie algebra.
In almost all the cases listed above, these operators are used to write an Hamiltonian of the physical system, and to deduce its dynamics out of it. The Hamiltonian, and more in general the observables of the system, are taken to be self-adjoint. This is, as it is well known, the standard approach in Quantum Mechanics, \cite{mess,merz}. However, since some decades, it has become clearer and clearer that self-adjointness of an operator is a sufficient condition for its eigenvalues to be real. But it is not also necessary. And this has physical consequences. In fact, it is trivial to construct counterexamples: it is enough to consider a non self-adjoint operator $T$ which is similar (but not unitarily equivalent) to a self-adjoint
$T_0$. This means that an invertible operator $V$ exists such that $T=VT_0V^{-1}$. Here, to make the situation simple, we assume that all the operators are bounded, $T,T_0,V,V^{-1}\in B(\mc H)$, the C*-algebra of the bounded operators over the Hilbert space $\mc H$. In this case, if $\varphi$ is an eigenstate of $T_0$ with eigenvalue $E\in\mathbb{R}$, it is clear that $V\varphi$ is different from zero, and that it is an eigenstate of $T$ with the same eigenvalue. Furthermore, if $V^{-1}\neq V^\dagger$, then $T\neq T^\dagger$. This simple consideration, with a seminal paper on the now famous Hamiltonian $H=p^2+ix^3$, \cite{ben1}, turned on a renewed interest to physicists for non self-adjoint observables\footnote{Mathematicians were already aware of the possibility of having real eigenvalues for non self-adjoint operators.}. A very rich line of research was then generated, involving people interested in the new physical consequences of this approach, and people more focused on its mathematical aspects, which are many and not so easy to deal with. For instance, even if the reality of the eigenvalues is preserved, the eigenvectors need not being orthonormal, or even being a basis in $\mc H$, also in presence of a pure point spectrum. Some references on the physical aspects of what is now usually called {\em PT} or {\em Pseudo-Hermitian} Quantum Mechanics are \cite{ben1}-\cite{bagprocpa}, while \cite{bagbookPT}-\cite{petr4} are references more mathematically oriented.
This partial list of references prove that there is a growing interest in a deeper understanding of the properties of many classes of non self-adjoint operators, and in particular of some connected {\em extended number-like operators} and their related ladder operators. This analysis has begun several years ago when people realized that, in some physically relevant situation, it is possible to factorize a certain Hamiltonian $H$, non self-adjoint, as follows: $H=A^*A$. Here $A^*$ is not the (Dirac) adjoint of $A$, $A^\dagger$, and this explains why $H\neq H^\dagger$. This property has been analyzed in details in different contexts along the years for several different systems, producing deformed versions of the standard {\em particles}: ${\mc D}$-pseudo bosons (${\mc D}$-PBs), pseudo-fermions, deformed quons, and other deformations of {\em ordinary} bosons, fermions and quons. The common feature, in all these cases, is that the raising operator of the model is not the Dirac adjoint of the lowering operator. In principle, they are unrelated, even if they usually satisfy some peculiar (anti-)commutation rules. We refer to \cite{baginbagbook,bagthmp} for some self-consistent reviews on these particular families of deformations.
In this paper we continue our analysis, starting from CSusy and deforming the related ladder operators. In this way we consider two different (but related) families of four raising and four lowering operators, acting on different vectors which are biorthogonal in pairs and connected by these operators. They are also eigenvectors of four different number-like, manifestly non self-adjoint, operators. This is essentially the content of Section \ref{sect2} where, after introducing the problem, we construct the algebraic framework where all the operators considered all along the paper live. After creating the settings, we deform CSusy and we discuss some useful aspects of the deformed $\mathfrak{su}(1,1)$ Lie algebra arising in our analysis. The supersymmetric aspects of the model are discussed in Section \ref{sectBECSusy}. In Section \ref{sectPBs} we show how ${\mc D}$-PBs produce an interesting example of our construction, while in Section \ref{sectDPBs} we consider a deformed version of ${\mc D}$-PBs, with an explicit example connected to the shifted harmonic oscillator, \cite{petr4}. Our conclusions are given in Section \ref{sectconcl}.
\section{The settings}\label{sect2}
In \cite{coupledsusy} the authors introduced the notion of {\em coupled SUSY} (CSusy), and they used it in the analysis of some physical systems with Hamiltonians written in terms of ladder operators. Roughly speaking, a CSusy arises out of two operators $a$ and $b$, acting on an Hilbert space $\mc H$, and two real non-zero numbers $\gamma$, $\delta$, with $\delta>\gamma$, satisfying the following:
\begin{equation}
a^\dagger a=bb^\dagger+\gamma1 \!\! 1, \qquad aa^\dagger=b^\dagger b+\delta1 \!\! 1.
\label{21}\end{equation}
Here $1 \!\! 1$ is the identity operator on $\mc H$. Since $a$, $b$ and their adjoints could be unbounded, it is clear that some domain conditions must be imposed to these operators. For instance, for (\ref{21}) to make sense, the range of $a$ ($b$) must be contained in the domain of $a^\dagger$ ($b^\dagger$), and viceversa. It is easy to see that (\ref{21}) extends the ordinary bosonic case. In fact, if $c$ is an operator on $\mc H$ satisfying (in the sense on unbounded operators) the canonical commutation relation $[c,c^\dagger]=1 \!\! 1$, then the equations in (\ref{21}) are satisfied taking $a=b=c$, $\delta=1$, $\gamma=-1$
The interesting result deduced in \cite{coupledsusy} is that most of the typical ladder structure {\em attached} to $c$ and $c^\dagger$, producing in particular an orthonormal basis in $\mc H$, can be recovered also using the pair $(a,b)$ in (\ref{21}). In particular, the operators
\begin{equation}
{\mathfrak K}_+=\frac{1}{\delta-\gamma}a^\dagger b^\dagger, \qquad {\mathfrak K}_-=\frac{1}{\delta-\gamma} ba, \qquad {\mathfrak K}_0=\frac{1}{\delta-\gamma}\left(a^\dagger a-\frac{\gamma}{2}\right),
\label{22}\end{equation}
satisfy the following commutation rules,
\begin{equation}
[{\mathfrak K}_0,{\mathfrak K}_\pm]=\pm{\mathfrak K}_\pm, \qquad [{\mathfrak K}_+,{\mathfrak K}_-]=-2{\mathfrak K}_0,
\label{23}\end{equation}
which are those of the $\mathfrak{su}(1,1)$ Lie algebra. Hence, ${\mathfrak K}_\pm$ act as ladder operators, while ${\mathfrak K}_0$ is connected with some Hamiltonian.
Equations (\ref{22}) show that ${\mathfrak K}_0={\mathfrak K}_0^\dagger$ and ${\mathfrak K}_+^\dagger={\mathfrak K}_-$. Our interest is focused in what happens when these conditions are lost. In particular, we are interested in considering what happens if the ladder operators extending ${\mathfrak K}_\pm$ are not one the adjoint of the other, and if ${\mathfrak K}_0\neq {\mathfrak K}_0^\dagger$. As already discussed in the Introduction, this become interesting in view of the growing interest for physical systems driven by manifestly non self-adjoint Hamiltonians, \cite{ben1}-\cite{bagbookPT}, which has produced several results both in theoretical and experimental physics, and in pure mathematics. To produce this extension, we will replace (\ref{21}) with two new equalities involving four, in general unrelated, operators $c$, $d$, $r$ and $s$, all acting on $\mc H$. However, before doing this, it is convenient to introduce an algebraic settings which can be useful to deal with our situation, creating a rather general framework.
\subsection{$O^*$-algebras}\label{sectalgebras}
Since the original proposal by R. Haag and H. Kastler in 1964, \cite{hk}, it become clear that the use of algebras of operators can be useful in the analysis of several physical systems, and in particular of those with infinite degrees of freedom. However, $C^*$ or von Neumann-algebras are sometimes not the best choice, since the operators they contain are all bounded, while many physical systems are deeply connected with operators which are not bounded. This is quite often the case in many-body theory, in quantum field theory, and in statistical mechanics, for instance, \cite{br1}-\cite{sewbook2}. But it is also true for simple quantum mechanical systems. For instance, the ladder operators used in the analysis of the quantum harmonic oscillator, and their related number operator, are all unbounded, \cite{br2,llt,araki}. To deal with these cases, in the past twenty years or so several examples of {\em unbounded operator algebras} have been introduced and studied in details. We refer to \cite{aitbook}-\cite{Inoue} for some relevant publications on this subject.
In this paper we will use a particular unbounded operator algebra, the $O^*$-algebra ${\cal L}^\dagger({\mc D})$.
Let us briefly review how ${\cal L}^\dagger({\mc D})$ can be introduced, and let us comment on why it is so relevant for us. We start with the following definition:
\begin{defn}\label{o*}Let $\mathcal{H}$ be a separable Hilbert space and $N_0$ an
unbounded, densely defined, self-adjoint operator. Let $D(N_0^k)$ be
the domain of the operator $N_0^k$, $k \ge 0$, and $\mathcal{D}$ the domain of
all the powers of $N_0$, that is, \begin{equation} \mathcal{D} = \bigcap_{k\geq 0}
D(N_0^k). \label{add1}\end{equation} This set is dense in $\mathcal{H}$. We call
$\mathcal{L}^\dagger(\mathcal{D})$ the $*$-algebra of all \textit{ closable operators}
defined on $\mathcal{D}$ which, together with their adjoints, map $\mathcal{D}$ into
itself. Here the adjoint of $X\in\mathcal{L}^\dagger(\mathcal{D})$,
$X^\dagger$, is the restriction of the adjoint of $X$ in $\mc H$ (which we also indicate with $X^\dagger$) to ${\mc D}$.
\end{defn}
In $\mathcal{D}$ the topology is defined by the following $N_0$-depending
seminorms: $$\phi \in \mathcal{D} \rightarrow \|\phi\|_n\equiv \|N_0^n\phi\|,$$
where $n \ge 0$, while the topology $\tau_0$ in $\mathcal{L}^\dagger(\mathcal{D})$ is introduced by the seminorms
$$ X\in \mathcal{L}^\dagger(\mathcal{D}) \rightarrow \|X\|^{f,k} \equiv
\max\left\{\|f(N_0)XN_0^k\|,\|N_0^kXf(N_0)\|\right\},$$ where
$k \ge 0$ and $f \in \mathcal{C}$, the set of all the positive,
bounded and continuous functions on $\mathbb{R}_+$, which are
decreasing faster than any inverse power of $x$:
$\mathcal{L}^\dagger(\mathcal{D})[\tau_0]$ is a { complete *-algebra}. This implies, in particular, that taken any
$x,y\in \mathcal{L}^\dagger(\mathcal{D})$, we can multiply them and the results, $xy$ and $yx$, both belong to $\mathcal{L}^\dagger(\mathcal{D})$, as well as their difference, the commutator $[x,y]$. Also, powers of $x$ and $y$ all belong to ${\cal L}^\dagger({\mc D})$, which therefore is a good candidate to work with, also in presence of unbounded operators. In fact, if $N_0=c^\dagger c$, where $[c,c^\dagger]=1 \!\! 1$ as above, we can prove that $c, c^\dagger\in{\cal L}^\dagger({\mc D})$. Hence $N_0\in {\cal L}^\dagger({\mc D})$ as well.
Let now $a$ and $b$ be two operators
on $\mathcal{H}$, with domains $D(a)$ and $D(b)$ respectively, $a^\dagger$ and $b^\dagger$ their adjoint, and let $\mathcal{D}$ be a dense subspace of $\mathcal{H}$
such that $a^\sharp\mathcal{D}\subseteq\mathcal{D}$ and $b^\sharp \mathcal{D} \subseteq \mathcal{D}$, where with $x^\sharp$ we indicate $x$ or $x^\dagger$. Of course, $\mathcal{D}\subseteq D(a^\sharp)$
and $\mathcal{D}\subseteq D(b^\sharp)$.
\begin{defn}\label{def21}
The operators $(a,b)$ are $\mathcal{D}$-\textit{pseudo-bosonic} if, for all $f\in\mathcal{D}$, we have
\begin{equation}\label{A1}
a\,b\,f-b\,a\,f=f.
\end{equation}
\end{defn}
By means of $a$ and $b$, a number-like operator $N=ba$ can be defined, which is manifestly non self-adjoint, with $N^\dagger$ sharing with $N$ all its eigenvalues, $n=0,1,2,3,\ldots$. This is just the begin of the story for ${\mc D}$-PBs, which have a very rich structure.
More on ${\mc D}$-PBs can be found in Section \ref{sectPBs}, while a rather complete (but not particularly recent) review is \cite{baginbagbook}. In \cite{bagrusso} it is shown that, under some mild extra condition, $a,b,N$ and their adjoint can be seen as elements of a $O^*$-algebra ${\cal L}^\dagger({\mc D})$ and, as such, we can multiply them, raise to (non-negative) integer powers, compute commutators, and so on. This is useful for what follows.
\subsection{Extending CSusy}\label{sect2.2}
We are now ready to extend significantly the definition in (\ref{21}). In doing so, non self-adjoint operators will become relevant and natural. For that we start considering an Hilbert space $\mc H$, endowed with scalar product $\langle.,.\rangle$, and with an adjoint $\dagger$ connected to $\langle.,.\rangle$: $\langle X^\dagger f,g\rangle=\langle f,Xg\rangle$, $\forall f,g\in\mc H$. Let us further consider a suitable subspace ${\mc D}\subset\mc H$, and the $^*$-algebra ${\cal L}^\dagger({\mc D})$ constructed as in \cite{aitbook}, see also Definition \ref{o*}. Here ${\mc D}$ can be constructed as in (\ref{add1}) for some operator $N_0$, or being a convenient dense subset of $\mc H$. Then
\begin{defn}\label{defecsusy}
Let $d$, $c$, $r$ and $s$ be four elements of ${\cal L}^\dagger({\mc D})$, and let $\gamma, \delta$ be two real numbers with $\delta>\gamma$. We say that $(d,c,r,s;\delta,\gamma)$ define an {\em extended coupled Susy} (ECSusy), if the following equalities are satisfied:
\begin{equation}
\left\{
\begin{array}{ll}
dc=rs+\gamma1 \!\! 1,\\
cd=sr+\delta1 \!\! 1,\\
\end{array}
\right.
\label{25}\end{equation}
\end{defn}
Here, as usual, $1 \!\! 1$ is the identity operator on $\mc H$, and the formulas above could be understood as follows: $d(cf)=r(sf)+\gamma f$ and $c(df)=s(rf)+\delta f$, for all $f\in{\mc D}$. These equalities are both well defined since, if $f\in{\mc D}$, then $cf,df,sf,rf\in{\mc D}$ as well, and, therefore, f we also have $c(df)\in{\mc D}$, and so on.
In \cite{coupledsusy} it is shown that the two equalities in (\ref{21}) are really different: one can be satisfied while the other does not hold. This is what happens, for instance, if $a=\frac{1}{\sqrt{2}}\left(\frac{d}{dx}+x\right)$ and $b=\frac{1}{\sqrt{2}}\left(\frac{d}{dx}+x\right)e^{ix}$. Of course, it is not difficult to imagine that this difference between the two equations in (\ref{21}) is strengthen further in our case, i.e. for equations (\ref{25}). Stated differently, the two equations in (\ref{25}) are really different, and for this reason have to be considered together in the following.
Let us define the following operators, still in ${\cal L}^\dagger({\mc D})$:
\begin{equation}
k_+=\frac{1}{\delta-\gamma}ds, \qquad k_-=\frac{1}{\delta-\gamma}rc, \qquad k_0=\frac{1}{\delta-\gamma}\left(dc-\frac{\gamma}{2}\,1 \!\! 1\right),
\label{26}\end{equation}
and
\begin{equation}
l_+=\frac{1}{\delta-\gamma}sd, \qquad l_-=\frac{1}{\delta-\gamma}cr, \qquad l_0=\frac{1}{\delta-\gamma}\left(sr+\frac{\delta}{2}\,1 \!\! 1\right).
\label{27}\end{equation}
Using (\ref{25}) it is easy to check that they obey the following commutation relations:
\begin{equation}
[k_0,k_\pm]=\pm k_\pm, \qquad [k_+,k_-]=-2k_0,
\label{28}\end{equation}
as well as
\begin{equation}
[l_0,l_\pm]=\pm l_\pm, \qquad [l_+,l_-]=-2l_0.
\label{29}\end{equation}
These look like the commutators in (\ref{23}), but with a big difference: $k_+$ and $l_+$ are not the adjoint of $k_-$ and $l_-$, and $k_0$ and $l_0$ are not self-adjoint. This gives us the possibility to introduce two other families of operators, $p_\alpha$ and $q_\alpha$, $\alpha=0,\pm$:
\begin{equation}
p_0=k_0^\dagger, \qquad p_\pm=k_\mp^\dagger; \qquad\qquad q_0=l_0^\dagger, \qquad q_\pm=l_\mp^\dagger.
\label{210}\end{equation}
They satisfy the same commutators in (\ref{28}) and (\ref{29}):
\begin{equation}
[p_0,p_\pm]=\pm p_\pm, \qquad [p_+,p_-]=-2p_0; \qquad [q_0,q_\pm]=\pm q_\pm, \qquad [q_+,q_-]=-2q_0.
\label{211}\end{equation}
Hence we conclude that (\ref{25}) implies the existence of four (in general) different triples of operators obeying the same commutators of an $\mathfrak{su}(1,1)$ Lie algebra, but with different relations under the adjoint operation.
\subsection{Deformed $\mathfrak{su}(1,1)$ Lie algebra: a view to its eigenstates}\label{sectdla}
In the attempt to clarify the consequences of what deduced before, in this section we will deduce the main properties of three operators, $x_\pm$ and $x_0$, in ${\cal L}^\dagger({\mc D})$, satisfying $[x_0,x_\pm]=\pm x_\pm$, and $[x_+,x_-]=-2x_0$, but with $x_+^\dagger\neq x_-$ and $x_0^\dagger\neq x_0$. In particular we will show that all the useful (for us) results valid for $\mathfrak{su}(1,1)$ can be also deduced in the present situation.
First we put, with a slight abuse of notation,
\begin{equation}
x^2=x_0^2-\frac{1}{2}(x_+x_-+x_-x_+)=x_0^2+x_0-x_-x_+=x_0^2-x_0-x_+x_-.
\label{212}\end{equation}
We call it {\em an abuse} since $x^2$ is not really the square of an operator $x$ (to be identified) and, moreover, $x^2$ in (\ref{212}) is not even positive. We use this notation since it is somehow standard. In fact, the definition in (\ref{212}) is the one usually adopted in the literature for the {\em ordinary} $\mathfrak{su}(1,1)$ Lie algebra, and we borrow this terminology from that. The operator $x^2$ commutes with each $x_\alpha$: $[x^2,x_\alpha]=0$, for $\alpha=0,\pm$, and this is the main reason why it is so relevant for us. Now, since in particular $x^2$ and $x_0$ commute, we can look for common eigenstates of these two operators. Using again the same notation adopted for ordinary $\mathfrak{su}(1,1)$, we assume the following: there exists a non zero vector $\Phi_{j,q_0}\in{\mc D}$ satisfying the following eigenvalue equations:
\begin{equation}
\left\{
\begin{array}{ll}
x^2\Phi_{j,q_0}=j(j+1)\Phi_{j,q_0},\\
x_0\Phi_{j,q_0}=q_0\Phi_{j,q_0},\\
\end{array}
\right.
\label{213}\end{equation}
for some $j$ and $q_0$. We should stress that, in principle, there is no reason a priori to assume here that $j$ and $q_0$ are real or positive. This is because, as already observed, $x^2$ is not positive or self-adjoint, and $x_0$ is not self-adjoint, too. This makes in general more complicated to describe the range of values of $j$ and $q_0$. However, some useful result can still be found, as we will see. Using the commutation rules for $(x^2,x_\alpha)$ we deduce that
\begin{equation}
\left\{
\begin{array}{ll}
x^2(x_\pm\Phi_{j,q_0})=j(j+1)(x_\pm\Phi_{j,q_0}),\\
x_0(x_\pm\Phi_{j,q_0})=(q_0\pm1)(x_\pm\Phi_{j,q_0}),\\
\end{array}
\right.
\label{214}\end{equation}
at least if $\Phi_{j,q_0}\notin \ker(x_\pm)$. This means that $x_\pm$ are ladder operators and, in particular, that $x_+$ is a raising while $x_-$ is a lowering operator. Using the same standard arguments for $\mathfrak{su}(1,1)$, we can also deduce that
\begin{equation}
\left\{
\begin{array}{ll}
x_+\Phi_{j,q_0}=(q_0-j)\Phi_{j,q_0+1},\\
x_-\Phi_{j,q_0}=(q_0+j)\Phi_{j,q_0-1}.\\
\end{array}
\right.
\label{215}\end{equation}
These equations are in agreement with the fact that, as it is easy to check,
$$
[x_0,x_-x_+]=[x_0,x_+x_-]=0.
$$
In fact, from (\ref{215}) we see that $x_0$ and $x_-x_+$ have the same eigenvectors. The same is true for $x_0$ and $x_+x_-$.
We have several possibilities:
\vspace{2mm}
{\bf Case 1:--} for some $m\in\mathbb{N}_0=\mathbb{N}\cup\{0\}$ we have $x_-^{m-1}\Phi_{j,q_0}\neq0$ and $x_-^{m}\Phi_{j,q_0}=0$. In this case the set of eigenvalues of $x_0$, $\sigma(x_0)$, is bounded below: $\sigma(x_0)=\{q_0-m+1,q_0-m+2,q_0-m+3,\ldots\}$.
\vspace{2mm}
{\bf Case 2:--} for some $k\in\mathbb{N}_0=\mathbb{N}\cup\{0\}$ we have $x_+^{k-1}\Phi_{j,q_0}\neq0$ and $x_+^{k}\Phi_{j,q_0}=0$. In this case $\sigma(x_0)$, is bounded above: $\sigma(x_0)=\{\ldots,q_0+k-3,q_0+k-2,q_0+k-1\}$.
\vspace{2mm}
{\bf Case 3:--} both conditions above are true: for some $m\in\mathbb{N}_0=\mathbb{N}\cup\{0\}$ we have $x_-^{m-1}\Phi_{j,q_0}\neq0$ and $x_-^{m}\Phi_{j,q_0}=0$, and for some $k\in\mathbb{N}_0=\mathbb{N}\cup\{0\}$ we have $x_+^{k-1}\Phi_{j,q_0}\neq0$ and $x_+^{k}\Phi_{j,q_0}=0$. In this case, of course, $\sigma(x_0)$, is bounded above and below: $\sigma(x_0)=\{q_0-m+1,q_0-m+2,\ldots,q_0+k-2,q_0+k-1\}$.
\vspace{2mm}
{\bf Case 4:--} neither Case 1, nor Case 2, hold. Then $\sigma(x_0)$ has no bound below and above.
\subsection{Back to ECSusy}\label{sectBECSusy}
We can use now the results of the previous section in the analysis of the operators introduced in Section \ref{sect2.2}. However, this will not be the only ingredient of the procedure we are going to propose. In fact, as we will see, the natural biorthonormality connected to the appearance of non self-adjoint number-like operators will play a relevant role. We first consider the operators $k_\alpha$, $\alpha=0,\pm$. As in (\ref{213}), we assume a non zero vector $\varphi_{j,q}\in{\mc D}$ exists, $j,q\in\mathbb{C}$, such that
\begin{equation}
k^2\varphi_{j,q}=j(j+1)\varphi_{j,q}, \qquad k_0\varphi_{j,q}=q\varphi_{j,q}.
\label{216}\end{equation}
Here, as in (\ref{212}), $k^2=k_0^2+k_0-k_-k_+$, for instance. The operators $k_\pm$ act on $\varphi_{j,q}$ as ladder operators:
\begin{equation}
k_+\varphi_{j,q}=(q-j)\varphi_{j,q+1}, \qquad k_-\varphi_{j,q}=(q+j)\varphi_{j,q-1},
\label{217}\end{equation} for all $\varphi_{j,q}\notin\ker(k_\pm)$. Let us now call $I_j$ the set of all the $q's$ for which $\varphi_{j,q}$ is not annihilated by at least one between $k_+$ and $k_-$: if $q\in I_j$, then $\varphi_{j,q}\notin\ker(k_+)$ or $\varphi_{j,q}\notin\ker(k_-)$, or both, and let ${\cal F}_\varphi(j):=\{\varphi_{j,q}, \,\forall q\in I_j\}$. Let then introduce ${\cal E}_j=l.s.\{\varphi_{j,q}, \, q\in I_j\}$, the linear span of the vectors in ${\cal F}_\varphi(j)$, and $\mc H_j$ the closure of ${\cal E}_j$, with respect to the norm of $\mc H$. Of course, $\mc H_j\subseteq\mc H$, for each fixed $j$. By construction, ${\cal F}_\varphi(j)$ is a basis for $\mc H_j$. Let ${\cal F}_\psi(j):=\{\psi_{j,q}, \,\forall q\in I_j\}$ be its unique biorthogonal basis, \cite{chri}. Then
\begin{equation}
\langle\varphi_{j,q},\psi_{j,r}\rangle=\delta_{q,r},
\label{218}\end{equation}
for all $q,r\in I_j$, and $l.s.\{\psi_{j,q}, \, q\in I_j\}$ is dense in $\mc H_j$. Using (\ref{210}), it is possible to check the following eigenvalue and ladder equalities:
\begin{equation}
\left\{
\begin{array}{ll}
p_0\psi_{j,q}=\overline{q}\psi_{j,q},\\
p_+\psi_{j,q}=\overline{(q+1+j)}\psi_{j,q+1},\\
p_-\psi_{j,q}=\overline{(q-1-j)}\psi_{j,q+1},\\
\end{array}
\right.
\label{219}\end{equation}
at least if $\psi_{j,q}\notin\ker(p_\pm)$.
\vspace{2mm}
{\bf Remark:--} These equations appear different from those deduced in (\ref{215}) and, in fact, this is so. In particular, the coefficients in the ladder equations are not those in (\ref{215}). The difference arises because the vectors $\{\psi_{j,q}\}$ are introduced here as the only family which is biorhonormal to the $\{\varphi_{j,q}\}$, which is more convenient, and more natural, for us. The other possibility would be to introduce, in analogy to what we have done in (\ref{216}), a family of eigenstate of $p^2$ and $p_0$, $\{\tilde\psi_{j,q}\}$, which, however, turns out to be biorthogonal, but not biorthonormal, to $\{\varphi_{j,q}\}$. In other words, the difference we have with these different procedures is in the normalization of the states: $\psi_{j,q}$ and $\tilde{\psi}_{j,q}$ are proportional to each other. This is confirmed by the fact that, from (\ref{219}), we also deduce that $p^2\psi_{j,q}=j(j+1)\psi_{j,q}$. This aspect will appear clear in the analysis of ${\mc D}$-PBs, in Section \ref{sectPBs}, where these vectors will be explicitly computed and compared.
\vspace{2mm}
The biorthonormality between ${\cal F}_\varphi(j)$ and ${\cal F}_\psi(j)$ is connected to the fact that they are eigenstates of pairs of operators related by the adjoint map, as in (\ref{210}). The same formulas show that a similar relation exists between the operators $l_\alpha$ and $q_\alpha$. Also, comparing (\ref{26}) and (\ref{27}), we can see that, for instance, $l_\pm$ are a sort of supersymmetric version of $k_\pm$, meaning with this that the various operators are all factorized, and the order of the operators in, say, $l_\alpha$ is the opposite with respect to their order in $k_\alpha$. Hence the eigenstates of $(k_0,k^2)$ are also related to those of $(l_0,l^2)$, for example, as we will show next.
We begin by noticing that, as a consequence of (\ref{26}), (\ref{27}) and (\ref{210}), we can deduce the following intertwining relations:
\begin{equation}
\left\{
\begin{array}{ll}
sk_+=l_+s,\qquad\qquad k_+d=dl_+\\
ck_-=l_-c,\qquad\qquad k_-r=rl_-\\
r^\dagger p_+=q_+r^\dagger,\qquad\quad p_+c^\dagger=c^\dagger q_+\\
d^\dagger p_-=q_-d^\dagger,\qquad\quad p_-s^\dagger=s^\dagger q_-,\\
\end{array}
\right.
\label{220}\end{equation}
as well as
\begin{equation}
\left\{
\begin{array}{ll}
l_0s=s\left(k_0+\frac{1}{2}1 \!\! 1\right)\\
l_0c=c\left(k_0-\frac{1}{2}1 \!\! 1\right)\\
rl_0=\left(k_0+\frac{1}{2}1 \!\! 1\right)r\\
dl_0=\left(k_0-\frac{1}{2}1 \!\! 1\right)d,\\
\end{array}
\right.
\label{221}\end{equation}
and their adjoint which, recalling (\ref{210}), we can write as follows:
\begin{equation}
\left\{
\begin{array}{ll}
q_0r^\dagger=r^\dagger\left(p_0+\frac{1}{2}1 \!\! 1\right)\\
q_0d^\dagger=d^\dagger\left(p_0-\frac{1}{2}1 \!\! 1\right)\\
s^\dagger q_0=\left(p_0+\frac{1}{2}1 \!\! 1\right)s^\dagger\\
c^\dagger q_0=\left(p_0-\frac{1}{2}1 \!\! 1\right)c^\dagger.\\
\end{array}
\right.
\label{222}\end{equation}
\vspace{2mm}
The equalities in (\ref{221}) and (\ref{222}) show that the eigenvalues of $l_0$ differ from those of $k_0$ by half integers, as those of $q_0$ from those of $p_0$. Indeed we have, considering a vector $\varphi_{j,q}$ with $s\varphi_{j,q}\neq0$ and $c\varphi_{j,q}\neq0$,
$$
l_0\left(s\varphi_{j,q}\right)=s\left(k_0+\frac{1}{2}1 \!\! 1\right)\varphi_{j,q}=\left(q+\frac{1}{2}\right)\left(s\varphi_{j,q}\right).
$$
as well as
$$
l_0\left(c\varphi_{j,q}\right)=\left(q-\frac{1}{2}\right)\left(c\varphi_{j,q}\right).
$$
In other words, $s\varphi_{j,q}$ and $c\varphi_{j,q}$ are both eigenstates, with different eigenvalues, of $l_0$. Hence, as in (\ref{214}), they must be connected by the ladder operators $l_\pm$. However, we do not expect that the ladder equations are exactly those in (\ref{215}), since the eigenstates of $l_0$ are introduced using SUSY rather than considering the approach described in Section \ref{sectBECSusy}. Nonetheless, the vectors which are deduced here are proportional to those we would introduce using (\ref{213}), (\ref{214}) and (\ref{215}), $\chi_{j,q'}$, where $q'$ differs from $q$ for half-integers. The same equations (\ref{221}) allow to check that, if $r\chi_{j,q'}\neq0$ and $d\chi_{j,q'}\neq0$,
$$
k_0\left(r\chi_{j,q'}\right)=\left(q'-\frac{1}{2}\right)\left(r\chi_{j,q'}\right), \qquad k_0\left(d\chi_{j,q'}\right)=\left(q'+\frac{1}{2}\right)\left(d\chi_{j,q'}\right).
$$
This situation is clearly shown in Figure \ref{fig1} in a concrete situation which is deduced deforming ${\mc D}$-PBs, see Section \ref{sectDPBs}. More concretely, what we have just deduced here is {\em the right half part} of Figure \ref{fig1}. The left half part is connected with the consequences of the intertwining relations in (\ref{222}), consequences which are completely analogous to those discussed so far for $k_0$ and $l_0$. We will go back to Figure \ref{fig1} later on, when dealing with ${\mc D}$-PBs.
\section{A detailed example: ${\mc D}$-PBs}\label{sectPBs}
In this Section we will discuss in details how ${\mc D}$-PBs, see Definition \ref{def21}, fit into the present framework. In particular, we will show that a rather rich functional structure can be constructed, and generalized even more, as we will show in Section \ref{sectDPBs}.
We start this section recalling few facts related to Definition \ref{def21}, starting with the following working assumptions, valid in several physical systems considered along the years.
\vspace{2mm}
{\bf Assumption ${\mc D}$-pb 1.--} there exists a non-zero $\varphi_{ 0}\in{\mc D}$ such that $a\,\varphi_{ 0}=0$.
\vspace{1mm}
{\bf Assumption ${\mc D}$-pb 2.--} there exists a non-zero $\Psi_{ 0}\in{\mc D}$ such that $b^\dagger\,\Psi_{ 0}=0$.
\vspace{2mm}
The invariance of ${\mc D}$ under the action of $b$ and $a^\dagger$ implies
that the vectors \begin{equation} \varphi_n:=\frac{1}{\sqrt{n!}}\,b^n\varphi_0,\qquad \Psi_n:=\frac{1}{\sqrt{n!}}\,{a^\dagger}^n\Psi_0, \label{31}\end{equation}
$n\geq0$, are well defined and they all belong to ${\mc D}$ and, as a consequence, to the domain of $a^\sharp$, $b^\sharp$ and $N^\sharp$, where $N=ba$. Let us put ${\cal F}_\Psi=\{\Psi_{ n}, \,n\geq0\}$ and
${\cal F}_\varphi=\{\varphi_{ n}, \,n\geq0\}$.
It is simple to deduce the following lowering and raising relations:
\begin{equation}
\left\{
\begin{array}{ll}
b\,\varphi_n=\sqrt{n+1}\varphi_{n+1}, \qquad\qquad\quad\,\, n\geq 0,\\
a\,\varphi_0=0,\quad a\varphi_n=\sqrt{n}\,\varphi_{n-1}, \qquad\,\, n\geq 1,\\
a^\dagger\Psi_n=\sqrt{n+1}\Psi_{n+1}, \qquad\qquad\quad\, n\geq 0,\\
b^\dagger\Psi_0=0,\quad b^\dagger\Psi_n=\sqrt{n}\,\Psi_{n-1}, \qquad n\geq 1,\\
\end{array}
\right.
\label{32}\end{equation} as well as the eigenvalue equations $N\varphi_n=n\varphi_n$ and $N^\dagger\Psi_n=n\Psi_n$, $n\geq0$. Then, if we choose the normalization of $\varphi_0$ and $\Psi_0$ in such a way $\left<\varphi_0,\Psi_0\right>=1$, we deduce that
\begin{equation} \left<\varphi_n,\Psi_m\right>=\delta_{n,m}, \label{33}\end{equation}
for all $n, m\geq0$. Hence ${\cal F}_\Psi$ and ${\cal F}_\varphi$ are biorthonormal. In \cite{baginbagbook} it is shown that, in several quantum models, ${\cal F}_\Psi$ and ${\cal F}_\varphi$ are complete in $\mc H$, but they are not bases. However, they still produce useful resolution of the identity since they are always, at least in all the systems considered so far, ${\cal G}$-quasi bases, where ${\cal G}$ is some subspace dense in $\mc H$. This means that
for all $f$ and $g$ in ${\cal G}$,
\begin{equation}
\left<f,g\right>=\sum_{n\geq0}\left<f,\varphi_n\right>\left<\Psi_n,g\right>=\sum_{n\geq0}\left<f,\Psi_n\right>\left<\varphi_n,g\right>.
\label{34}
\end{equation}
We refer to \cite{baginbagbook} for many more results and examples on ${\mc D}$-quasi bosons. Here, what is relevant for us, are the ladder properties described by (\ref{32}), and the fact that they produce a concrete, and highly non trivial, example of ECSusy.
In fact, let us fix the operators and the numbers $\gamma$ and $\delta$ in Definition \ref{defecsusy} as follows: $c=r=a$, $d=s=b$, $\delta=-\gamma=1$, where $a$ and $b$ satisfy Definition \ref{def21}. Hence the operators in (\ref{26}), (\ref{27}) and (\ref{210}) become
\begin{equation}
k_+=l_+=\frac{1}{2}b^2, \qquad k_-=l_-=\frac{1}{2}a^2, \qquad k_0=l_0=\frac{1}{2}\left(N+\frac{1}{2}1 \!\! 1\right),
\label{35}\end{equation}
where $N=ba$, and
\begin{equation}
p_+=q_+=\frac{1}{2}{a^\dagger}^2, \qquad p_-=q_-=\frac{1}{2}{b^\dagger}^2, \qquad p_0=q_0=\frac{1}{2}\left(N^\dagger+\frac{1}{2}1 \!\! 1\right).
\label{36}\end{equation}
It is clear that the four original families collapse to two. Formula (\ref{212}) produce further the following result:
\begin{equation}
k^2=p^2=-\frac{3}{16}1 \!\! 1,
\label{37}\end{equation}
which, of course, commute with all the other operators, as expected. We notice that this formula clarifies what already observed in Section \ref{sectdla}, after formula (\ref{213}): despite of their "names", $k^2$ and $p^2$ are not positive operators. Formula (\ref{213}) is based on the assumption that a non zero eigenstate of $x^2$ and $x_0$ exists. In our situation, such a vector can be easily found: in fact, if we consider the vacuum $\varphi_0$ introduced before, see Assumption ${\mc D}$-pb 1., we have $$k^2\varphi_0=-\frac{3}{16}\varphi_0, \qquad k_0\varphi_0=\frac{1}{4}\varphi_0.$$
Hence, comparing these with (\ref{213}), we have $q_0=\frac{1}{4}$ and $j(j+1)=-\frac{3}{16}$, that is $j=-\frac{1}{4}$ or $j=-\frac{3}{4}$. Because of formula (\ref{217}), and observing that $k_-\varphi_0=0$, we choose
$j=-\frac{1}{4}$ and we define
\begin{equation}
\varphi_{-\frac{1}{4},\frac{1}{4}}:=\varphi_0.
\label{38}\end{equation}
Hence we are in Case 1 of Section \ref{sectalgebras}, with $m=1$. In fact, since the spectrum of $N$ is the set $\mathbb{N}_0$, $\sigma(k_0)$ is bounded below.
If we act $m$ times with $k_+$ on $\varphi_{-\frac{1}{4},\frac{1}{4}}$, $m=1,2,3,\ldots$, formula (\ref{217}) produces
\begin{equation}
\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m)!}}{(2m-1)!!}\varphi_{2m},
\label{39}\end{equation}
where $\varphi_{2m}$ are those in (\ref{31}) and, with standard notation, $(2m-1)!!=1\cdot3\cdots(2m-3)\cdot(2m-1)$, with $0!!=(-1)!!=1$. Using (\ref{216}) and (\ref{217}), or with a direct check, we find
\begin{equation}
k_0\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{4}\right)\varphi_{-\frac{1}{4},m+\frac{1}{4}},
\label{310}\end{equation}
and
\begin{equation}
k_+\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{2}\right)\varphi_{-\frac{1}{4},m+\frac{5}{4}}, \qquad k_-\varphi_{-\frac{1}{4},m+\frac{1}{4}}=m\,\varphi_{-\frac{1}{4},m-\frac{3}{4}}.
\label{311}\end{equation}
In particular, this last equality is true only if $m\geq1$. If $m=0$ we have $k_-\varphi_{-\frac{1}{4},\frac{1}{4}}=k_-\varphi_0=0$, as already noticed.
According to Section \ref{sectBECSusy}, we can now define the set of linearly independent vectors ${\cal F}_\varphi^{(e)}\left(\frac{1}{4}\right)=\{\varphi_{-\frac{1}{4},m+\frac{1}{4}}, \, m=0,1,2,3,\ldots\}$, and the Hilbert space $\mc H_{-\frac{1}{4}}^{(e)}$, constructed by taking the closure of the linear span of its vectors. Here the suffix {\em e} stands for {\em even}, since only the vectors $\varphi_{2m}$ belong to ${\cal F}_\varphi^{(e)}\left(\frac{1}{4}\right)$. It is clear that $\mc H_{-\frac{1}{4}}^{(e)}\subset\mc H$, since all the vectors with odd index, $\varphi_{2m+1}$, which belong to $\mc H$, do not belong to $\mc H_{-\frac{1}{4}}^{(e)}$. Hence, the set
${\cal F}_\varphi^{(e)}\left(\frac{1}{4}\right)$ cannot be complete in $\mc H$, and, as a consequence, cannot be a basis for $\mc H$. Nevertheless, by construction, $\mc H_{-\frac{1}{4}}^{(e)}$ is an Hilbert space as well, and ${\cal F}_\varphi^{(e)}\left(\frac{1}{4}\right)$ is a basis for it. Then, see \cite{chri}, an unique biorthonormal basis ${\cal F}_\psi^{(e)}\left(\frac{1}{4}\right)=\{\psi_{-\frac{1}{4},m+\frac{1}{4}}, \, m=0,1,2,3,\ldots\}$ exists, such that
\begin{equation}
\langle\varphi_{-\frac{1}{4},m+\frac{1}{4}},\psi_{-\frac{1}{4},l+\frac{1}{4}}\rangle=\delta_{m,l},
\label{312}\end{equation}
where the scalar product is the one in $\mc H$, and, for each $f\in \mc H_{-\frac{1}{4}}^{(e)}$,
\begin{equation}
f=\sum_{m=0}^{\infty}\langle\varphi_{-\frac{1}{4},m+\frac{1}{4}},f\rangle\,\psi_{-\frac{1}{4},m+\frac{1}{4}}=\sum_{m=0}^{\infty}\langle\psi_{-\frac{1}{4},m+\frac{1}{4}},f\rangle\,\varphi_{-\frac{1}{4},m+\frac{1}{4}}.
\label{313}\end{equation}
From (\ref{39}) and (\ref{33}) it is clear that the vectors of this biorthonormal basis are the following:
\begin{equation}
\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{(2m-1)!!}{\sqrt{(2m)!}}\psi_{2m}.
\label{314}\end{equation}
Formulas (\ref{219}) can now be explicitly checked, and we get
\begin{equation}
p^2\psi_{-\frac{1}{4},m+\frac{1}{4}}=-\frac{3}{16}\psi_{-\frac{1}{4},m+\frac{1}{4}}, \qquad p_0\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{4}\right)\psi_{-\frac{1}{4},m+\frac{1}{4}},
\label{315}\end{equation}
together with
\begin{equation}
p_+\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+1\right)\psi_{-\frac{1}{4},m+\frac{5}{4}}, \qquad p_-\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m-\frac{1}{2}\right)\psi_{-\frac{1}{4},m-\frac{3}{4}}.
\label{316}\end{equation}
Once more, we stress that the difference between these ladder equations and those in (\ref{311}) arises because, while the $\varphi_{-\frac{1}{4},m+\frac{1}{4}}$'s are introduced using directly the deformed $\mathfrak{su}(1,1)$ algebra, the $\psi_{-\frac{1}{4},m+\frac{1}{4}}$'s are just the unique basis which is biorthonormal to ${\cal F}_\varphi^{(e)}\left(\frac{1}{4}\right)$. However, as (\ref{315}) and (\ref{316}) show, these vectors are still eigenstates of $p^2$ and $p_0$, and obey interesting ladder equations with respect to $p_\pm$, which are slightly different from those in (\ref{215}).
Let us now consider the four intertwining relations (\ref{221}). In the case of ${\mc D}$-PBs, these correspond to the following two equalities:
\begin{equation}
k_0b=b\left(k_0+\frac{1}{2}1 \!\! 1\right), \qquad k_0a=a\left(k_0-\frac{1}{2}1 \!\! 1\right).
\label{317}\end{equation}
The consequence of these equalities is widely considered in the literature: if $\rho$ is an eigenstate of $k_0$ with eigenvalue $E$, $k_0\rho=E\rho$, and if $a\rho$ and $b\rho$ are both non zero, then
$$
k_0(a\rho)=\left(E-\frac{1}{2}\right)(a\rho), \qquad k_0(b\rho)=\left(E+\frac{1}{2}\right)(b\rho),
$$
which means that $a\rho$ and $b\rho$ are both eigenstates of $k_0$, but with two shifted (and different) eigenvalues, $E\pm\frac{1}{2}$. Now, since (\ref{310}) shows that the eigenvalues related to different vectors $\varphi_{-\frac{1}{4},m+\frac{1}{4}}$ and $\varphi_{-\frac{1}{4},l+\frac{1}{4}}$ differ for integer quantities, we conclude that neither $a\varphi_{-\frac{1}{4},m+\frac{1}{4}}$, nor $b\varphi_{-\frac{1}{4},m+\frac{1}{4}}$, can still be of the same form $\varphi_{-\frac{1}{4},l+\frac{1}{4}}$, for any $l\in\mathbb{N}_0$. And, in fact, this can be explicitly checked, since
\begin{equation}
a\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\sqrt{2m}\frac{\sqrt{(2m)!}}{(2m-1)!!}\,\varphi_{2m-1}, \qquad b\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\,\varphi_{2m+1},
\label{318}\end{equation}
with the agreement that $\varphi_{-1}=0$. Let us now define
\begin{equation}
\varphi_{-\frac{1}{4},m+\frac{3}{4}}:=b\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\,\varphi_{2m+1},
\label{319}\end{equation}
for all $m\geq0$. The reason for calling this vector in this way is because $\varphi_{-\frac{1}{4},m+\frac{3}{4}}$ is an eigenstate of $k_0$ with eigenvalue $m+\frac{3}{4}$, as expected because of our previous analysis on $b\rho$:
\begin{equation}
k_0\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+\frac{3}{4}\right)\varphi_{-\frac{1}{4},m+\frac{3}{4}},
\label{320}\end{equation}
We further deduce the following raising and lowering relations:
\begin{equation}
k_+\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+\frac{1}{2}\right)\varphi_{-\frac{1}{4},m+\frac{7}{4}}, \qquad k_-\varphi_{-\frac{1}{4},m+\frac{3}{4}}=m\,\frac{2m+1}{2m-1}\,\varphi_{-\frac{1}{4},m-\frac{1}{4}},
\label{321}\end{equation}
with the agreement that $\varphi_{-\frac{1}{4},-\frac{1}{4}}=0$.
\vspace{2mm}
{\bf Remark:--} It is now possible to check explicitly that (\ref{321}) are different from (\ref{215}). The reason is that, as it is clear, we are extending here the strategy described in Section \ref{sectBECSusy}, where a single family of operators were considered. Now, in fact, we are considering 4 families of operators satisfying the same commutator rules, and we are putting all these operators together, keeping biorthonormality as our main requirement, since we think this is {\em the most natural way}, mainly for technical reasons.
For completeness, let us briefly consider what happens if we repeat for the operators $p_\alpha$ what we have done for the $k_\alpha$. In analogy with (\ref{38}), we put $\tilde\psi_{-\frac{1}{4},\frac{1}{4}}=\psi_0$, since $b^\dagger\psi_0=0$. Then, using the first equation in (\ref{215}) (rather than the second in (\ref{219})), $p_+\tilde\psi_{j,q}=(q-j)\tilde\psi_{j,q+1}$, we deduce that
$$
\tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{(2m-1)!!}{\sqrt{(2m)!}}\,\psi_{2m}=\frac{(2m)!}{((2m-1)!!)^2}\psi_{-\frac{1}{4},m+\frac{1}{4}},
$$
which shows the difference in the normalizations between the $\tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}$ and the $\psi_{-\frac{1}{4},m+\frac{1}{4}}$. Now, while $\tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}$ satisfies the analogous of formulas (\ref{215}), $\psi_{-\frac{1}{4},m+\frac{1}{4}}$ satisfies (\ref{316}), which is slightly different. On the other hand, while this last vector satisfies (\ref{312}), $\tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}$ does not. In conclusion, we prefer to keep biorthonormality of the sets we work with, rather than using (\ref{215}) several times. But this is, of course, just a matter of personal taste.
\vspace{3mm}
It is clear that, in the same way in which $a$ and $b$ map $\varphi_{-\frac{1}{4},m+\frac{1}{4}}$ into some $\varphi_{-\frac{1}{4},l+\frac{3}{4}}$, they also map these last vectors into the first ones. More explicitly, we have
\begin{equation}
a\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\sqrt{2m+1}\,\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\,\varphi_{2m}, \qquad b\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\frac{\sqrt{(2m+2)!}}{(2m-1)!!}\,\varphi_{2m+2},
\label{322}\end{equation}
for all $m\geq0$. Notice that the vectors in the RHS of these equalities are proportional to $\varphi_{-\frac{1}{4},m+\frac{1}{4}}$ and to $\varphi_{-\frac{1}{4},m+\frac{5}{4}}$, see (\ref{29}).
In analogy with what we have done before, we introduce now the set ${\cal F}_\varphi^{(o)}\left(\frac{1}{4}\right)=\{\varphi_{-\frac{1}{4},m+\frac{3}{4}}, \, m=0,1,2,3,\ldots\}$, where {\em o} stands for {\em odd}, and the Hilbert space $\mc H_{-\frac{1}{4}}^{(o)}$, constructed by taking the closure of the linear span of its vectors. It is clear that $\mc H_{-\frac{1}{4}}^{(e)}\cap\mc H_{-\frac{1}{4}}^{(o)}=\emptyset$, and that, together, ${\cal F}_\varphi\left(\frac{1}{4}\right):={\cal F}_\varphi^{(e)}\left(\frac{1}{4}\right)\cup{\cal F}_\varphi^{(o)}\left(\frac{1}{4}\right)$ is complete in $\mc H$, at least if the set ${\cal F}_\varphi$ is complete, which is always the case in all the concrete examples in the literature, so far. In particular, if the ${\mc D}$-PBs are {\em regular}, see \cite{baginbagbook}, ${\cal F}_\varphi$ and ${\cal F}_\psi$ are biorthonormal Riesz bases. Hence ${\cal F}_\varphi\left(\frac{1}{4}\right)$ is a Riesz basis as well.
Now, since ${\cal F}_\varphi^{(o)}\left(\frac{1}{4}\right)$ is a basis for $\mc H_{-\frac{1}{4}}^{(o)}$, we can introduce an unique biorthonormal basis ${\cal F}_\psi^{(o)}\left(\frac{1}{4}\right)=\{\psi_{-\frac{1}{4},m+\frac{3}{4}}, \, m=0,1,2,3,\ldots\}$, whose vectors can be easily identified using (\ref{319}) and (\ref{33}). We have
\begin{equation}
\psi_{-\frac{1}{4},m+\frac{3}{4}}=\frac{(2m-1)!!}{\sqrt{(2m+1)!}}\,\psi_{2m+1}=\frac{1}{2m+1}a^\dagger \psi_{-\frac{1}{4},m+\frac{1}{4}}.
\label{323}\end{equation}
It may be interesting to notice the difference in the normalization between $\psi_{-\frac{1}{4},m+\frac{3}{4}}$ and $\varphi_{-\frac{1}{4},m+\frac{3}{4}}$, in terms of their $m+\frac{1}{4}$ counterparts, see (\ref{319}) and (\ref{323}). This difference arises because we want to maintain biorthonormality of the vectors. In fact, with the choice in (\ref{323}) we get
\begin{equation}
\langle\varphi_{-\frac{1}{4},m+\frac{3}{4}},\psi_{-\frac{1}{4},l+\frac{3}{4}}\rangle=\delta_{m,l},
\label{324}\end{equation}
where the scalar product is the one in $\mc H$, and, for each $f\in \mc H_{-\frac{1}{4}}^{(o)}$,
\begin{equation}
f=\sum_{m=0}^{\infty}\langle\varphi_{-\frac{1}{4},m+\frac{3}{4}},f\rangle\,\psi_{-\frac{1}{4},m+\frac{3}{4}}=\sum_{m=0}^{\infty}\langle\psi_{-\frac{1}{4},m+\frac{3}{4}},f\rangle\,\varphi_{-\frac{1}{4},m+\frac{3}{4}}.
\label{325}\end{equation}
Repeating then what we have done for $\mc H^{(e)}$, we can define the set ${\cal F}_\psi^{(o)}\left(\frac{1}{4}\right)=\{\psi_{-\frac{1}{4},m+\frac{3}{4}}, \, m=0,1,2,3,\ldots\}$, and observe that
${\cal F}_\psi\left(\frac{1}{4}\right):={\cal F}_\psi^{(e)}\left(\frac{1}{4}\right)\cup{\cal F}_\psi^{(o)}\left(\frac{1}{4}\right)$ is complete in $\mc H$, or it is even a Riesz basis for $\mc H$, depending on the nature of the ${\mc D}$-PBs we are considering. More in detail, if we now introduce the families ${\cal F}_\Phi=\{\Phi_k, \, k\geq0\}$ and ${\cal F}_\xi=\{\xi_k, \, k\geq0\}$, where
$$
\Phi_k=\left\{
\begin{array}{ll}
\varphi_{-\frac{1}{4},j+\frac{1}{4}}, \qquad\quad\,\, \mbox{if } k=2j,\\
\varphi_{-\frac{1}{4},j+\frac{3}{4}}, \qquad\quad\,\, \mbox{if } k=2j+1,\\
\end{array}
\right.\quad\mbox{ and }\quad \xi_k=\left\{
\begin{array}{ll}
\psi_{-\frac{1}{4},j+\frac{1}{4}}, \qquad\quad\,\, \mbox{if } k=2j,\\
\psi_{-\frac{1}{4},j+\frac{3}{4}}, \qquad\quad\,\, \mbox{if } k=2j+1,\\
\end{array}
\right.
$$
$k\geq0$, we can check that $\langle\Phi_k,\xi_l\rangle=\delta_{k,l}$, and that, $\forall f,g\in{\mc D}$,
$$
\sum_{k=0}^{\infty}\langle f,\Phi_k\rangle\langle\xi_k,g\rangle= \sum_{k=0}^{\infty}\langle f,\varphi_k\rangle\langle\psi_k,g\rangle, \qquad \sum_{k=0}^{\infty}\langle f,\xi_k\rangle\langle\Phi_k,g\rangle= \sum_{k=0}^{\infty}\langle f,\psi_k\rangle\langle\varphi_k,g\rangle.
$$
These equalities imply that ${\cal F}_\Phi$ and ${\cal F}_\xi$ are biorthonormal, and, \cite{baginbagbook}, that they are ${\mc D}$-quasi bases
if and only if ${\cal F}_\varphi$ and ${\cal F}_\psi$ are ${\mc D}$-quasi bases. This property, useful to deduce several mathematical properties of the system, as already stated, is always true in all the physical systems where ${\mc D}$-PBs have been shown to appear so far, \cite{baginbagbook,bagthmp}.
We end this section with Tables \ref{table1} and \ref{table2} which contain several useful formulas involving all the vectors introduced in this section.
\begin{table}[h]
\begin{tabular}{||l|l||}
\hline\hline
\rule[-4mm]{0mm}{1.2cm}
$\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m)!}}{(2m-1)!!}\varphi_{2m}$ & $\varphi_{-\frac{1}{4},m+\frac{3}{4}}=b\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\varphi_{2m+1}$\\
\hline
\rule[-4mm]{0mm}{1.0cm}
$\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{(2m-1)!!}{\sqrt{(2m)!}}\psi_{2m}$ & $\psi_{-\frac{1}{4},m+\frac{3}{4}}=\frac{a^\dagger}{2m+1}\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{(2m-1)!!}{\sqrt{(2m+1)!}}\psi_{2m+1}$\\
\hline\hline
\rule[-4mm]{0mm}{1.2cm}
$a\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\sqrt{2m}\frac{\sqrt{(2m)!}}{(2m-1)!!}\varphi_{2m-1}=\frac{2m}{2m-1}\varphi_{-\frac{1}{4},m-\frac{1}{4}}$ & $a\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\sqrt{2m+1}\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\varphi_{2m}=(2m+1)\varphi_{-\frac{1}{4},m+\frac{1}{4}}$ \\
\hline
\rule[-4mm]{0mm}{1.2cm}
$b\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\varphi_{2m+1}=\varphi_{-\frac{1}{4},m+\frac{3}{4}}$ & $b\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\frac{\sqrt{(2m+2)!}}{(2m-1)!!}\varphi_{2m+2}=(2m+1)\varphi_{-\frac{1}{4},m+\frac{5}{4}}$ \\
\hline
\rule[-4mm]{0mm}{1.2cm}
$a^\dagger\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m+1)!}}{(2m+1)!!}\psi_{2m+1}=(2m+1)\psi_{-\frac{1}{4},m+\frac{3}{4}}$ & $a^\dagger\psi_{-\frac{1}{4},m+\frac{3}{4}}=\sqrt{2m+2}\frac{(2m-1)!!}{\sqrt{(2m+1)!}}\psi_{2m+2}=\frac{2m+2}{2m+1}\varphi_{-\frac{1}{4},m+\frac{5}{4}}$ \\
\hline
\rule[-4mm]{0mm}{1.2cm}
$b^\dagger\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{(2m-1)!!}{\sqrt{(2m-1)!}}\psi_{2m+1}=(2m-1)\psi_{-\frac{1}{4},m-\frac{1}{4}}$ & $b^\dagger\psi_{-\frac{1}{4},m+\frac{3}{4}}=\frac{(2m-1)!!}{\sqrt{(2m)!}}\psi_{2m}=\psi_{-\frac{1}{4},m+\frac{1}{4}}$ \\
\hline\hline
\end{tabular}
\caption{Useful formulas for ${\mc D}$-PBs, part 1}\label{table1}
\end{table}
\begin{table}[h]
\begin{tabular}{||l|l|l||}
\hline\hline
\rule[-4mm]{0mm}{1.2cm}
$k_0\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{4}\right)\varphi_{-\frac{1}{4},m+\frac{1}{4}}$&$k_+\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{2}\right)\varphi_{-\frac{1}{4},m+\frac{5}{4}}$&$k_-\varphi_{-\frac{1}{4},m+\frac{1}{4}}=m\,\varphi_{-\frac{1}{4},m-\frac{3}{4}}$\\
\hline
\rule[-4mm]{0mm}{1.2cm}
$p_0\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{4}\right)\psi_{-\frac{1}{4},m+\frac{1}{4}}$&$p_+\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+1\right)\psi_{-\frac{1}{4},m+\frac{5}{4}}$&$p_-\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m-\frac{1}{2}\right)\,\psi_{-\frac{1}{4},m-\frac{3}{4}}$\\
\hline
\rule[-4mm]{0mm}{1.2cm}
$k_0\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+\frac{3}{4}\right)\varphi_{-\frac{1}{4},m+\frac{3}{4}}$&$k_+\varphi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+\frac{1}{2}\right)\varphi_{-\frac{1}{4},m+\frac{7}{4}}$&$k_-\varphi_{-\frac{1}{4},m+\frac{3}{4}}=m\,\frac{2m+1}{2m-1}\,\varphi_{-\frac{1}{4},m-\frac{1}{4}}$\\
\hline
\rule[-4mm]{0mm}{1.2cm}
$p_0\psi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+\frac{3}{4}\right)\psi_{-\frac{1}{4},m+\frac{3}{4}}$&$p_+\psi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+1\right)\frac{2m+3}{2m+1}\psi_{-\frac{1}{4},m+\frac{7}{4}}$&$p_-\psi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m-\frac{1}{2}\right)\,\psi_{-\frac{1}{4},m-\frac{1}{4}}$\\
\hline\hline
\end{tabular}
\caption{Useful formulas for ${\mc D}$-PBs, part 2}\label{table2}
\end{table}
\subsection{Deformed ${\mc D}$-PBs}\label{sectDPBs}
In Section \ref{sectPBs} we were somehow forced to simplify the structure of ECSusy because the four original families of operators collapse into two. In this section we will sketch how to deform ${\mc D}$-PBs to find four different operators satisfying Definition \ref{defecsusy}. It will be obvious that this can be done in very different ways, each of which produces a concrete example of ECSusy.
Let $a$ and $b$ be ${\mc D}$-pseudo-bosonic in the sense of Definition \ref{def21}. We work, as always, under the conditions which ensure that they belong, together with their adjoints, to ${\cal L}^\dagger({\mc D})$, for some suitable ${\mc D}$. Let now $S, T\in{\cal L}^\dagger({\mc D})$ be two invertible operators, with $S^{-1}, T^{-1}\in{\cal L}^\dagger({\mc D})$. In the following we will assume that ${T^{-1}}^\dagger={T^{\dagger}}^{-1}$ and ${S^{-1}}^\dagger={S^{\dagger}}^{-1}$. Conditions for these to be satisfied can be found, for instance, in \cite{hiroshi}, Lemma 3.2. If we define now
\begin{equation}
c=SaT^{-1}, \qquad s=SbT^{-1}, \qquad d=TbS^{-1}, \qquad r=TaS^{-1},
\label{326}\end{equation}
it is clear that these operators, which are all in ${\cal L}^\dagger({\mc D})$, satisfy (\ref{25}) with $\delta=-\gamma=1$. Using (\ref{26}), (\ref{27}) and (\ref{210}), together with (\ref{326}), we find that
\begin{equation}
\tilde k_\alpha=Tk_\alpha T^{-1}, \qquad \tilde l_\alpha=Sk_\alpha S^{-1}, \qquad \tilde p_\alpha={T^{-1}}^\dagger p_\alpha T^{\dagger}, \qquad \tilde q_\alpha={S^{-1}}^\dagger p_\alpha S^{\dagger},
\label{327}\end{equation}
where $\alpha=0,\pm$ and where the {\em un-tilted} operators $k_\alpha$ and $p_\alpha$ are those in (\ref{35}) and (\ref{36}). Recalling now that $\varphi_{-\frac{1}{4},m+\frac{1}{4}}, \psi_{-\frac{1}{4},m+\frac{1}{4}}, \varphi_{-\frac{1}{4},m+\frac{3}{4}}, \psi_{-\frac{1}{4},m+\frac{3}{4}}\in{\mc D}$, for all $m=0,1,2,3,\ldots$, it follows that the following vectors are in ${\mc D}$ as well:
\begin{equation}
\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}}=T\varphi_{-\frac{1}{4},m+\frac{1}{4}}; \qquad \tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}={T^{-1}}^\dagger\psi_{-\frac{1}{4},m+\frac{1}{4}};
\label{328}\end{equation}
and
\begin{equation}
\tilde\chi_{-\frac{1}{4},m+\frac{3}{4}}=S\varphi_{-\frac{1}{4},m+\frac{3}{4}}; \qquad
\tilde\eta_{-\frac{1}{4},m+\frac{3}{4}}={S^{-1}}^\dagger\psi_{-\frac{1}{4},m+\frac{3}{4}}.
\label{329}\end{equation}
They are eigenstates respectively of $\tilde k_0$ and $\tilde p_0$, with eigenvalue $m+\frac{1}{4}$, and of $\tilde l_0$ and $\tilde q_0$, with eigenvalue $m+\frac{3}{4}$. Moreover, they satisfy the following ladder equations:
\begin{equation}
\left\{
\begin{array}{ll}
\tilde k_+\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+\frac{1}{2}\right)\tilde\varphi_{-\frac{1}{4},m+\frac{5}{4}}, \hspace{2.4cm} \tilde k_-\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}}=m\tilde\varphi_{-\frac{1}{4},m-\frac{3}{4}},\\
\tilde p_+\tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m+1\right)\tilde\psi_{-\frac{1}{4},m+\frac{5}{4}}, \hspace{2.4cm} \tilde p_-\tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}=\left(m-\frac{1}{2}\right)\tilde\psi_{-\frac{1}{4},m-\frac{3}{4}},\\
\tilde l_+\tilde\chi_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+\frac{1}{2}\right)\tilde\chi_{-\frac{1}{4},m+\frac{7}{4}}, \hspace{2.4cm} \tilde l_-\tilde\chi_{-\frac{1}{4},m+\frac{3}{4}}=m\,\frac{2m+1}{2m-1}\tilde\chi_{-\frac{1}{4},m-\frac{1}{4}},\\
\tilde q_+\tilde\eta_{-\frac{1}{4},m+\frac{3}{4}}=\left(m+1\right)\,\frac{2m+3}{2m+1}\tilde\eta_{-\frac{1}{4},m+\frac{7}{4}}, \hspace{1.6cm} \tilde q_-\tilde\eta_{-\frac{1}{4},m+\frac{3}{4}}=\left(m-\frac{1}{2}\right)\tilde\eta_{-\frac{1}{4},m-\frac{1}{4}},\\
\end{array}
\right.
\label{330}\end{equation}
for every $m$ for which the lowering operators do not destroy the state. Also, they are biorthonormal in pairs, meaning that
\begin{equation}
\langle\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}},\tilde\psi_{-\frac{1}{4},l+\frac{1}{4}}\rangle=\langle\tilde\chi_{-\frac{1}{4},m+\frac{3}{4}},\tilde\eta_{-\frac{1}{4},l+\frac{3}{4}}\rangle=\delta_{m,l},
\label{331}\end{equation}
for all $m,l\in\mathbb{N}_0$, while, if $S$ and $T$ are not chosen in some special way, we get, for instance, $\langle\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}},\tilde\eta_{-\frac{1}{4},l+\frac{3}{4}}\rangle\neq0$. More than this: if we repeat here the same construction leading to the definition of the families ${\cal F}_\Phi$ and ${\cal F}_\xi$, see Section \ref{sectDPBs}, we do not get ${\mc D}$-quasi bases for $\mc H$, except for special choices of $S$ and $T$. A trivial possibility is when $S$ and $T$ are different, but proportional. Another more interesting case, can be set up if the various eigenvectors have definite parity. In this case, biorthormality of the sets is ensured in both $S$ and $T$ are multiplication operators: $Sf(x)=s(x)f(x)$ and $Tf(x)=t(x)f(x)$, $f(x)\in{\mc D}$, if $s(x)$ and $t(x)$ are even functions.
The whole situation is summarized in Figure \ref{fig1}: the vertical lines describe the action of the various ladder operators for the different families of vectors. The dashed horizontal lines connect the various biorthonormal families (and, in fact, this is the meaning of {\em b.o.} over the lines). The various slanted lines represent the extension of the results in Table \ref{table1} in the present case. For instance, we can easily check that
\begin{equation}
s\,\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\tilde\chi_{-\frac{1}{4},m+\frac{3}{4}}, \qquad c\,\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{2m}{2m-1}\tilde\chi_{-\frac{1}{4},m-\frac{1}{4}},
\label{add2}\end{equation}
and so on. Incidentally we observe that the slanted lines are only meant to show which vector is mapped into which other vector, but does not give any information on the related coefficients: for instance in the two cases just mentioned in (\ref{add2}), in the first case the coefficient is just $1$, while in the second is $\frac{2m}{2m-1}$. This difference is not made explicit in the Figure. It is also useful to stress that Figure \ref{fig1} does not really refer only to the present, very particular, example of ECSusy, deduced as a deformation of ${\mc D}$-PBs, but it is absolutely general. Also, it is not hard to understand what Figure \ref{fig1} becomes in the case described in Section \ref{sectPBs}, since, for instance, $c$ and $r$ collapse (and coincide with $a$), and so on.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{diagrampbs.png}
\caption{The role of the ladder operators, of the adjoint map, and of the biorthonormality of the eigenvectors.}
\label{fig1}
\end{figure}
\subsubsection{A concrete choice of $a$, $b$, $S$ and $T$}
The example we discuss here is connected to a {\em shifted harmonic oscillator}. The relevant aspect of the example is that the shift is complex, and this produces a manifestly non self-adjoint Hamiltonian. We refer to \cite{petr4,bag2013pra} for several results on this and similar systems.
Let $c_0$ be the standard bosonic annihilation operators on ${\cal L}^2(\mathbb{R})$, $c_0=\frac{1}{\sqrt{2}}\left(x+\frac{d}{dx}\right)$, $[c_0,c_0^\dagger]=1 \!\! 1$. Since $c_0$ and $c_0^\dagger$ are in ${\cal L}^\dagger({\cal S}(\mathbb{R}))$, where ${\cal S}(\mathbb{R})$ is the Schwartz space of the test functions, this commutator is well defined in ${\cal L}^\dagger({\cal S}(\mathbb{R}))$. Also, if we call $N_0=c_0^\dagger c_0$, then $H_0=N_0+\frac{1}{2}1 \!\! 1$ is the (self-adjoint) Hamiltonian of the quantum harmonic oscillator. It is well known how its eigenvectors can be constructed: if $e_0(x)\in{\cal S}(\mathbb{R})$ satisfies $c_0\,e_0(x)=0$, then
$$
e_n(x)=\frac{1}{\sqrt{n!}}\,(c_0^\dagger)^ne_0(x)=\frac{1}{\sqrt{2^nn!}\,\pi^{1/4}}\,H_n(x)e^{-x^2/2},
$$
where $H_n(x)$ is the $n-$th Hermite polynomial, and we have $H_0e_n(x)=E_ne_n(x)$, with $E_n=n+\frac{1}{2}$, $n=0,1,2,3\ldots$.
Now, let us introduce, for a fixed $\alpha\in\mathbb{R}$, an operator $V_\alpha$ in the following way:
$$
(V_\alpha f)(x)=f(x-i\alpha),
$$
for all $f(x)\in D(V_\alpha)=\{h(x)\in{\cal L}^2(\mathbb{R}): \, h(x-i\alpha)\in{\cal L}^2(\mathbb{R})\}$. This set is dense in ${\cal L}^2(\mathbb{R})$, since it contains ${\mc D}$, the linear span of the $e_n(x)$'s, whose set ${\cal F}_e=\{e_n(x)\}$ is an orthonormal basis for ${\cal L}^2(\mathbb{R})$. Notice that ${\mc D}\subset{\cal S}(\mathbb{R})$. The inclusion ${\mc D}\subseteq D(V_\alpha)$ follows from the fact that
$$
|e_n(x-i\alpha)|^2=\frac{e^{\alpha^2}}{2^nn!\sqrt{\pi}}\,H_n(x+i\alpha)H_n(x-i\alpha)e^{-x^2},
$$
which is integrable since $H_n(x+i\alpha)H_n(x-i\alpha)$ is a (real and non negative) polynomial of degree $2n$. It is clear that $V_\alpha$ is invertible, and that, for all $f(x)\in{\mc D}$, $(V_\alpha^{-1}f)(x)=f(x+i\alpha)$. It is also easy to check that, $\forall f(x), g(x)\in{\mc D}$, $\langle V_\alpha\,f,g\rangle=\langle f,V_\alpha g\rangle$ and $\langle V_\alpha^{-1}\,f,g\rangle=\langle f,V_\alpha^{-1} g\rangle$. Also, $\langle V_\alpha\,f,V_\alpha^{-1}g\rangle=\langle f, g\rangle$. These equalities allow us to check that the two families
$$
{\cal F}_\varphi=\{\varphi_n(x)=(V_\alpha e_n)(x)=e_n(x-i\alpha)\}, \qquad {\cal F}_\psi=\{\psi_n(x)=(V_\alpha^{-1} e_n)(x)=e_n(x+i\alpha)\},
$$
are biorthonormal and ${\mc D}$ quasi-bases:
$$
\langle f,g\rangle=\sum_{n=0}^{\infty}\langle f,\varphi_n\rangle \langle \psi_n,g\rangle=\sum_{n=0}^{\infty}\langle f,\psi_n\rangle \langle \varphi_n,g\rangle,
$$
for all $f,g\in{\mc D}$. To introduce the ${\mc D}$-pseudo-bosonic ladder operators for ${\cal F}_\varphi$ we first observe that $V_\alpha c_0 V_\alpha^{-1}$ and $V_\alpha c_0^\dagger V_\alpha^{-1}$ are well defined on ${\mc D}$. In fact, taking $f(x)=\sum_{l=0}^{N}k_le_l(x)\in{\mc D}$, it follows first of all that $(V_\alpha^{-1}f)(x)=\sum_{l=0}^{N}\tilde k_lH_l(x+i\alpha)e^{-(x+i\alpha)^2/2}$, where $\tilde k_k=\frac{k_l}{2^ll!\sqrt{\pi}}$. Now, since each polynomial of degree $M$ with complex coefficients, $P_M(x)$, can be written as a linear combination of the first $M$ Hermite polynomials, $P_M(x)=\sum_{j=0}^Mc_jH_j(x)$ for some complex $\{c_j\}$, and since $\sum_{l=0}^N\tilde k_l xH_l(x+i\alpha)$ and $\sum_{l=0}^N\tilde k_l (H_l'(x+i\alpha)-(x+i\alpha)H_l(x+i\alpha))$ are both polynomials of degree $N+1$, we deduce that
$$
c_0 V_\alpha^{-1}f(x)=\sum_{l=0}^{N+1}k_l^+H_l(x)e^{-(x+i\alpha)^2/2}, \qquad c_0^\dagger V_\alpha^{-1}f(x)=\sum_{l=0}^{N+1}k_l^-H_l(x)e^{-(x+i\alpha)^2/2},
$$
for some properly chosen set of coefficients $k_l^\pm$. It is clear that these two sums do not belong to ${\mc D}$, but they both belong to $D(V_\alpha)$, and we get, for instance
$$
V_\alpha c_0 V_\alpha^{-1}f(x)=\sum_{l=0}^{N+1}k_l^+H_l(x-i\alpha)e^{-x^2/2}=\sum_{l=0}^{N+1}q_l^+H_l(x)e^{-x^2/2},
$$
for some $q_l^+$ to be identified\footnote{It is clear that this (and other) identification is not relevant for us, here. We are only interested in checking that the operators are well defined, not really in computing their action. This is done in a different way.}. This suggests to introduce $a=V_\alpha c_0 V_\alpha^{-1}$, which is densely defined, since $D(a)\supseteq {\mc D}$. Similarly, we introduce also $b=V_\alpha c_0^\dagger V_\alpha^{-1}$, and we find that $D(b)\supseteq{\mc D}$.
To the same conclusion we can arrive
using the Baker-Campbell-Hausdorff formula. In fact, the operator $V_\alpha$ coincides on ${\mc D}$ with $e^{\alpha p}$, $p$ being the momentum operator, and therefore
\begin{equation}
a=V_\alpha c_0 V_{\alpha}^{-1}=c_0-\frac{i\alpha}{\sqrt{2}}, \qquad b=V_\alpha c_0^\dagger V_{\alpha}^{-1}=c_0^\dagger-\frac{i\alpha}{\sqrt{2}}.
\label{332}\end{equation}
which are in agreement with the fact that $D(a), D(b)\supseteq{\mc D}$, and with the fact that, as we have seen before, $a,b:{\mc D}\rightarrow{\mc D}$.
It is clear that $b^\dagger\neq a$, for $\alpha\neq0$, and that $[a,b]=1 \!\! 1$. The adjoints of $a$ and $b$ are the raising and the lowering operators for ${\cal F}_\psi$, \cite{baginbagbook}.
Now, the easiest example we can construct is the one we obtain by choosing $S=V_\sigma$ and $T=V_\tau$, $\sigma,\tau>0$ fixed, in (\ref{326}). They are both invertible, and we find, for instance,
$$
c=V_{\sigma+\alpha}c_0V_{-(\tau+\alpha)}, \quad d=V_{\tau+\alpha}c_0^\dagger V_{-(\sigma+\alpha)}, \quad s=V_{\sigma+\alpha}c_0^\dagger V_{-(\tau+\alpha)}, \quad r=V_{\tau+\alpha}c_0 V_{-(\tau+\alpha)}.
$$
Using (\ref{332}) we can rewrite, say, $c$, as follows:
$$
c=V_{\sigma-\tau}\left(c_0-\frac{i(\alpha+\tau)}{\sqrt{2}}\right).
$$
Hence, taking $f(x)\in{\mc D}$ and observing that $(c_0-\frac{i(\alpha+\tau)}{\sqrt{2}})f(x)\in{\mc D}$ as well, we conclude that $cf(x)$ is well defined. Similar formulas can be established for $d$, $s$ and $r$.
We can also rewrite formulas (\ref{327}), (\ref{328}) and (\ref{329}). For instance, $\tilde k_+=V_{\alpha+\tau}\left(\frac{1}{2}{c_0^\dagger}^2\right)V_{-(\alpha+\tau)}$, while the {\em tilted} vectors look as follows:
$$
\tilde\varphi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{\sqrt{(2m)!}}{(2m-1)!!}\,e_{2m}(x-i(\alpha+\tau)); \quad \tilde\psi_{-\frac{1}{4},m+\frac{1}{4}}=\frac{(2m-1)!!}{\sqrt{(2m)!}}\,e_{2m}(x+i(\alpha+\tau));
$$
and
$$
\tilde\chi_{-\frac{1}{4},m+\frac{3}{4}}=\frac{\sqrt{(2m+1)!}}{(2m-1)!!}\,e_{2m+1}(x-i(\alpha+\sigma)); \quad \tilde\eta_{-\frac{1}{4},m+\frac{3}{4}}=\frac{(2m-1)!!}{\sqrt{(2m+1)!}}\,e_{2m+1}(x+i(\alpha+\sigma)).
$$
It may be worth remarking that, even in presence of biorthogonal sets, the basis property of these sets is not granted, in general, due to the fact that it does not even hold for ordinary pseudo-bosons. In fact, what we have always found in concrete examples considered along the years is that the different families of functions connected by our ladder operators are indeed complete in $\mc H$, and ${\cal G}$-quasi bases (for properly chosen ${\cal G}$), but not necessarily bases.
\section{Conclusions}\label{sectconcl}
In this paper we have introduced and analyzed the notion of ECSusy, deforming and extending some original ideas proposed in \cite{coupledsusy}. We have shown that, in doing this extension, how a deformed $\mathfrak{su}(1,1)$ Lie algebra can be introduced. We have also shown that this algebra, where, among other differences with the {\em standard} one, the ladder operators are not connected by the adjoint map, most of the usual results on $\mathfrak{su}(1,1)$ can be established, at least those related to the eigenstates of the number-like operator and to the ladder equations. ${\mc D}$-PBs have been considered in this perspective, and we have shown how to construct concrete examples of ECSusy. In particular, the role of biorthonormal sets, ladder operators and SUSY has been analyzed in detail.
\section*{Acknowledgements}
The author acknowledges partial support from Palermo University and from G.N.F.M. of the INdAM.
|
2,869,038,155,668 | arxiv | \section{Introduction }
Let $H$ and $K$ be two complex Hilbert spaces and $\mathcal{B}(H, K)$ be the Banach space of all bounded linear operators from $H$ into $K$. In the case $K=H$, $\mathcal{B}(H,H)$ is simply denoted by $\mathcal{B}(H) $ which is a Banach algebra. For $T\in \mathcal{B}(H,K)$, we set $\mathcal{R}(T)$ and $\mathcal{N}(T)$ for the range and the null-space of $T$, respectively. We also denote by $T^* \in \mathcal{B}(K,H)$ the adjoint operator of $T$.
The spectrum of an operator $T\in\mathcal{B}(H)$ is denoted by $\sigma(T)$ and $W(T)$ is the numerical range of $T$.
An operator $T\in \mathcal{B}(H,K)$ is a {\it partial isometry} when $T^*T$ is an orthogonal projection (or, equivalently $TT^*T = T$).
In particular $T$ is an {\it isometry} if $T^*T=I$, and {\it unitary} if $T$ is a surjective isometry.
The polar decomposition of $T\in\mathcal{B}(H)$ is given by $T=V|T|$, where $|T|=\sqrt{T^*T}$ and $V$ is an appropriate partial isometry such that $\mathcal{N}(T)=\mathcal{N}(V)$ and $\mathcal{N}(T^*)=\mathcal{N}(V^*)$.
The Aluthge transform introduced in \cite{MR1047771} as $\Delta(T)=|T|^{\frac{1}{2}}V|T|^{\frac{1}{2}}$ to extend some properties of hyponormal operators.
Later, in \cite{MR1997382}, Okubo introduced a more general notion called $\lambda-$Aluthge
transform which has also been studied in detail.
For $\lambda \in [ 0, 1]$, the $\lambda-$Aluthge transform is defined by,
$$
\Delta_{\lambda}(T)=|T|^{\lambda}V|T|^{1-\lambda}.
$$
Notice that $\Delta_0(T) =V|T|=T$, and $\Delta_1(T) = |T|V$ which is known as Duggal's
transform. It has since been studied in many different contexts and considered by a
number of authors (see for instance, \cite{ MR2013473, MR2148169, MR1780122, MR1829514, MR1971744, MR1873788} and some
of the references there). The interest of the Aluthge transform lies in the fact that
it respects many properties of the original operator. For example,
\begin{equation}
\sigma_*(\Delta_{\lambda}(T)) = \sigma_*(T), \mbox{ for every } \; \; T \in \mathcal{B}(H),
\end{equation}
where $ \sigma_*$ runs over a large family of spectra. See \cite[Theorems 1.3, 1.5]{MR1780122}.
Another important property is that $Lat(T)$, the lattice of $T$-invariant subspaces of
$H$, is nontrivial if and only if $Lat(\Delta(T))$ is nontrivial
(see \cite[Theorem 1.15]{MR1780122}).
Recently in \cite{MR3455750}, F.Bothelho, L.Moln\'ar and G.Nagy studied the linear bijective mapping on Von Neumann algebras which commutes with the $\lambda$-Aluthge transforms. They focus of bijective linear maps such that $$\Delta_{\lambda}(\Phi (T))=\Phi (\Delta_{\lambda}(T )) \mbox{ for every } T \in \mathcal{B}(H). $$
We are concerned in this paper with the more general problem of product commuting maps with the $\lambda$-Aluthge transform in the following sense,
\begin{equation}\label{c1}
\Delta_{\lambda}(\Phi (A)\Phi (B))=\Phi (\Delta_{\lambda}(A B)) \mbox{ for every } A, B\in \mathcal{B}(H),
\end{equation}
for some fixed $\lambda \in]0,1[$. \\
Our main result gives a complete description of the bijective map $\Phi :\mathcal{B}(H)\to\mathcal{B}(K)$ which satisfies Condition (\ref{c1}) and is stated as follows.
\begin{thm}\label{th1}
Let $H$ and $K$ be a complex Hilbert spaces, with $H$ of dimension greater than $2$. Let $\Phi :\mathcal{B}(H)\to\mathcal{B}(K)$ be bijective.
Then,
$\Phi $ satisfies $\left(\ref{c1}\right)$, if and only if, there exists an unitary operator $U:H\to K$ such that
$$\Phi (A)=UAU^*\quad \text{ for all} \; A\in\mathcal{B}(H).$$
\end{thm}\
\begin{rem}
$(1)$ In one dimensional, the result of Theorem \ref{th1} fails, as given in the following example: let the map $\Phi: \mathbb{C}\to\mathbb{C}$ defined by $${\Phi(z)=\left\{\begin{array}{l}\frac{1}{z}\text{ if } z\ne 0,\\ 0 \text{ if } z=0. \end{array}\right.}$$
Clearly $\Phi$ is bijective and satisfies $\left(\ref{c1}\right)$, but it is not additive.
$(2)$ The map $\Phi $ considered in our theorem is not assumed to satisfy any kind of continuity. However, an automatic continuity is obtained as a consequence.\\
\end{rem}
The proof of Theorem \ref{th1} is stated in next section. Several auxiliary results are needed for the proof and are established below.
\section{ Proof of the main theorem}
We first recall some basic notions that are used in the sequel. An operator $T\in\mathcal{B}(H)$ is normal if $T^*T=TT^*$, and is quasi-normal, if it commutes with $T^*T$ ( i.e. $TT^*T=T^*T^2$), or equivalently $|T|$ and $V$ commutes. In finite dimensional spaces every quasi-normal operator is normal. It is easy to see that if $T$ is quasi-normal, then $T^2$ is also quasi-normal, but the converse is false as shown by nonzero nilpotent operators .
Also, quasi-normal operators are exactly the fixed points of $\Delta_{\lambda}$ (see \cite[Proposition 1.10]{MR1780122}).
\begin{equation}\label{qn}
T \; \; \mbox{ quasi-normal} \; \; \iff \Delta_{\lambda}(T) = T.
\end{equation}
An idempotent self adjoint operator $P\in\mathcal{B}(H)$ is said to be an orthogonal projection. Clearly quasi-normal idempotents are orthogonal projections. \\
Two projections $P,Q\in\mathcal{B}(H)$ are said to be orthogonal if $PQ=QP=0$ and we denote $P\perp Q$.
A partial ordering between orthogonal projections is defined as follows, $$P\leq Q\: \mbox{ if } \: PQ = QP = P.$$
We start with the following lemma, which gives the "only if" part in our theorem. It has already been mentioned in other papers in the case $H=K$ (see \cite{MR2148169}, for example). We give the proof for completeness.
\begin{lem}\label{l0} Let $U:H\to K$ be an unitary operator, and $\lambda\in [0,1]$. We have the following identity
$${ \Delta_{\lambda}}(UTU^*)=U{ \Delta_{\lambda}}(T)U^*, \quad \mbox{for every} \; T\in\mathcal{B}(H).$$
\end{lem}
\begin{proof}
Let $T\in\mathcal{B}(H)$. It is easy to check
$$ \vert UTU^*\vert = U\vert T\vert U^* \quad \mbox{and} \quad \vert UTU^*\vert^{\lambda} = U\vert T\vert^{\lambda}U^*, \; \; \lambda\in [0,1].$$
Now, let $T=V\vert T\vert$ be a polar decomposition. Then
$$UTU^*=UV\vert T\vert U^*=(UVU^*)(U\vert T\vert U^*)=\tilde{V}\vert UTU^*\vert,$$
where $\tilde{V}=UVU^*$. $\tilde{V}$ is a partial isometry, $\mathcal{N}(UTU^*) = \mathcal{N}(\tilde{V})$ and hence $\tilde{V}\vert UTU^*\vert$ is the polar decomposition of $UTU^*$. This implies that :
\begin{align*}
{ \Delta_{\lambda}}(UTU^*) &= |UTU^*|^{\lambda}\tilde{V}|UTU^*|^{1-\lambda} \\
&= U|T|^{\lambda}U^* \tilde{V}U|T|^{1-\lambda}U^*\\
&= U|T|^{\lambda}V|T|^{1-\lambda}U^*\\
& = U{ \Delta_{\lambda}}(T)U^*.
\end{align*}
This completes the proof.
\end{proof}
For $x,y \in H$ , we denote by $x\otimes y$ the at most rank one operator defined by $$(x\otimes y)u=<u,y>x \: \mbox{ for } \: u\in H.$$ It is easy to show that every rank one operator has the previous form and that $x\otimes y$ is an orthogonal projection, if and only if $x=y$ and $\|x\|=1$. We have the following proposition,
\begin{pro}\label{p1} Let $x,y \in H$ be nonzero vectors. We have
$$\Delta_{\lambda}(x\otimes y)=\dfrac{<x,y>}{\|y\|^2}(y\otimes y)\: \: \mbox{ for every } \: \lambda \in ]0,1[
.$$
\end{pro}
\begin{proof}
Denote $T=x\otimes y$, then $T^*T=|T|^2=\|x\|^2(y\otimes y)=\big(\frac{\|x\|}{\|y\|}(y\otimes y)\big)^2\;\;\text{and}\;\;|T|=\sqrt{T^*T}=\frac{\|x\|}{\|y\|}(y\otimes y).$
It follows that $|T|^2=\|x\|\|y\| |T|\;\; \text{and}\;\; |T|^\gamma =(\|x\|\|y\|)^{\gamma-1}|T|$ for every $\gamma>0$.
Now, let $T=U|T|$ be the polar decomposition of $T$, we have
\begin{align*}
\Delta_{\lambda}(T)&=|T|^{\lambda}U|T|^{1-\lambda}\\&=(\|x\|\|y\|)^{{\lambda-1}}(\|x\|\|y\|)^{{-\lambda}}|T|U|T|\\&=\dfrac{1}{\|x\|\|y\|}|T|T\\&=\frac{1}{\|y\|^2}(y\otimes y)\circ (x\otimes y)=\dfrac{<x,y>}{\|y\|^2}(y\otimes y).
\end{align*}
\end{proof}
We deduce the next
\begin{cor}\label{rt}Let $R$ be a bounded linear operator on $H$ and $\lambda \in ]0,1[$. Suppose that $$\Delta_{\lambda}(RT)= \Delta_{\lambda}(TR), $$ for every rank one operator of the form $T=y\otimes y$. Then, there exists some $\alpha \in \mathbb{C}$ such that $R =\alpha I$.
\end{cor}
\begin{proof} Denote $A=R^*$. First, we claim that the linear operator $A$ satisfies the property that for every $z\in H$ we either have $Az$ is orthogonal to $z$ (calling $z$ being of the first kind) or $Az,z$ are linearly dependent (calling $z$ being of the second kind). Indeed, let $z\in H$ and $T=z\otimes z$, from the assumption and the Proposition \ref{p1}, we have
$$<Rz,z>z\otimes z=\Delta_\lambda(Rz\otimes z)=\Delta_\lambda(z\otimes Az).$$
In the case when $<Rz,z>=0$, then $z$ is of the first kind. And if $<Rz,z>\ne 0$ then $Az\ne 0$, and from the last equality it follows that
$$<Rz,z>z\otimes z=\dfrac{<Rz,z>}{\|Az\|^2}Az\otimes Az.$$
Thus $Az$ and $z$ are linearly dependent.
Now, $A$ is a scalar multiple of the identity. Indeed, on contrary assume that we have vector $x$ which is of the first kind but not of the second kind and that we have a vector $y$ which is of the second kind but not of the first kind. Then $x,y$ are linearly independent. We may assume that $Ay=y$. Set $x'=Ax$. For a real number $t$ from the unit interval and for $z_t=tx+(1-t)y$ we have $Az_t=tx'+(1-t)y$. It is clear that the equation $<Az_t,z_t>=t(1-t)\big( <x',y>+<y,x>\big)+(1-t)^2\|y\|^2=0$ has at most one solutions $t_1\in ]0,1[$. Also, with the fact that $x,y$ are linearly independent, then $Az_t,z_t$ are linearly independent for all $t\in]0,1[$ except for at most one $t\in ]0,1[$. So, for example, for small enough positive $t$ the vector $z_t$ does not of the first kind nor of the second.
This shows that either have that $Az$ is orthogonal to $z$ for all vectors $z$ or we have $Az,z$ are linearly dependent for all vectors $z$. In the first case we have that $A=0$, in the second one $A$ is a scalar multiple of the identity. In any way $A$ is a scalar multiple of the identity. Thus $R=A^*=\alpha I$ for some $\alpha \in\mathbb{C}$.
\end{proof}
The following lemma, provides a criterion for an operator to be positive through its $\lambda$-Aluthge transform. It will play a crucial role in the proof of Theorem \ref{th1}.
\begin{lem}\label{l1}
Let $T\in \mathcal{B}(H)$ be an invertible operator. The following conditions are equivalent :
\begin{enumerate}[(i)]
\item $T$ is positive;
\item for every $\lambda \in [0,1], {\Delta_{\lambda}}(T)$ is positive;
\item there exists $\lambda \in [0,1]$ such that ${\Delta_{\lambda}}(T)$ is positive.
\end{enumerate}
In particular, ${\Delta_{\lambda}}(T)=cI$ for some nonzero scalar $c$, if and only if
$T=cI$.
\end{lem}
\begin{proof}
The implications $(i)\Rightarrow (ii)\Rightarrow (iii)$ are trivial. It remains to show that $(iii)\Rightarrow (i)$. Let us consider the polar decomposition $T=U|T|$ of $T$ and assume that ${\Delta_{\lambda}}(T)$ is a positive operator. Since $T$ invertible it follows that $|T|^{1-\lambda}$ is invertible and $U$ is unitary. We claim that $U = I$. Indeed, let us denote $A=|T|^{2\lambda-1}$, we have
\begin{align*}
AU &=|T|^{2\lambda-1}U\\&=|T|^{\lambda-1}(|T|^{\lambda}U|T|^{1-\lambda})|T|^{\lambda-1}\\
&=|T|^{\lambda-1}{\Delta_{\lambda}}(T)|T|^{\lambda-1}.
\end{align*}
This follows that $AU=|T|^{\lambda-1}{\Delta_{\lambda}}(T)|T|^{\lambda-1}$ is positive. In particular it is self adjoint. Thus $AU=(AU)^*=U^*A$ and then $UAU=A$.
Therefore $(AU)^2=A^2$. It follows that $AU=A$ since $AU$ and $A$ are positive. Thus $U=I$ and this gives $T=U|T|=|T|$ is positive.
\end{proof}
\begin{rem} The assumption $T$ is invertible is necessary in the previous lemma.
Indeed, let $ T=x\otimes y$, with $x, y$ be nonzero independent vectors such that $<x,y>\ge 0$. Using proposition \ref{p1}, we get ${\Delta_{\lambda}}(T)$ is positive while $T$ is not.
\end{rem}
\begin{lem}\label{l2}
Let $T\in\mathcal{B}(H)$ be an arbitrary operator and $P\in\mathcal{B}(H)$ be an orthogonal projection. The following are equivalent :
\begin{enumerate}[(i)]
\item $\Delta_{\lambda}(TP)=T$ ;
\item $TP=PT=T\: \mbox{ and } T\: \mbox{ is quasi-normal}.$
\end{enumerate}
\end{lem}
\begin{proof} The implication $(ii)\Rightarrow (i)$ is obvious. We show the direct implication.
Consider $TP=U|TP|$ the polar decomposition of $TP$. Suppose that $\Delta_{\lambda}(TP)=T$, then
\begin{equation}
|TP|^{\lambda}U|TP|^{1-\lambda}=T \mbox{~~~ and ~~~} |TP|^{1-\lambda}U^*|TP|^{\lambda}=T^*.
\end{equation}
It follows that $$\mathcal{R}(T)\subseteq \mathcal{R}(|TP|^{\lambda})\subseteq \overline{\mathcal{R}(|TP|^2)}$$
and
$$\mathcal{R}(T^*)\subseteq \mathcal{R}(|TP|^{1-\lambda})\subseteq \overline{\mathcal{R}(|TP|^2)}.$$
In the other hand, we have $|TP|^2=PT^*TP=P|T|^2P$. Thus $\overline{\mathcal{R}(|TP|^2)}\subseteq \mathcal{R}(P)$. Hence $\mathcal{R}(T)\subset \mathcal{R}(P)$ and $\mathcal{R}(T^*)\subset \mathcal{R}(P)$. Which implies that $PT=T$ and $PT^*=T^*$. Therefore $$PT=TP=T\;\;\; \text{and}\;\; T \: \mbox{ is quasi-normal}.$$
\end{proof}
\begin{pro}\label{l5} Let $\Phi $ be a bijective map satisfying $\left(\ref{c1}\right)$. Then\\
$$\Phi (0)=0.$$
Moreover, there exists a bijective function $h:\mathbb{C}\to\mathbb{C}$ such that:
\begin{enumerate}[(i)]
\item $\Phi (\alpha I)=h(\alpha)I$ for all $\alpha \in\mathbb{C}$.
\item $h(\alpha \beta)=h(\alpha)h(\beta)$ for all $\alpha,\beta \in \mathbb{C}$.
\item $h(1)=1$ and $h(-\alpha)=-h(\alpha)$ for all $\alpha \in \mathbb{C}$.
\end{enumerate}
\end{pro}
\begin{proof} For the first assertion, since $\Phi$ is bijective, there exists $A\in \mathcal{B}(H)$ such that $\Phi (A)=0$. Therefore $\Phi (0)=\Delta_{\lambda}(\Phi (A)\Phi (0))=0$.
Let us show now that there exists a function $h : \mathbb{C}\to\mathbb{C}$ such that $\Phi (\alpha I)=h(\alpha)I$ for all $\alpha \in\mathbb{C}$.
If $\alpha=0$ the function $h$ is defined by $h(0)=0$ since $\Phi (0)=0$. Now, suppose that $\alpha$ is a nonzero scalar and denote by $R=\Phi (\alpha I)$ in particular $R\ne 0$. From $\left(\ref{c1}\right)$ it follows that
\begin{equation}
\Delta_{\lambda}(R\Phi (A))=\Phi (\Delta_{\lambda}(\alpha A))=\Delta_{\lambda}(\Phi (A\alpha I))=\Delta_{\lambda}(\Phi (A)R),
\end{equation}
for every $A\in\mathcal{B}(H)$.
Since $\Phi $ is onto, then $\Delta_{\lambda}(RT)=\Delta_{\lambda}(TR)$ for every rank one operator of the form $T=y\otimes y$ from $\mathcal{B}(K)$. Since $R=\Phi (\alpha I)$ different from zero and by Corollary \ref{rt}, there exists a nonzero scalar $h(\alpha) \in \mathbb{C}$ such that $R=\Phi (\alpha I)=h(\alpha) I$. In the other hand, $\Phi $ is bijective and its inverse $\Phi ^{-1}$ satisfies the same condition as $\Phi $. It follows that the map $h: \mathbb{C}\to \mathbb{C}$ is well defined and it is bijective.
Moreover, using again Condition $\left(\ref{c1}\right)$, we get
$$h(\alpha\beta)I=\Delta_{\lambda}(\Phi (\alpha\beta I))=\Delta_{\lambda}(\Phi (\alpha I)\Phi (\beta I))=h(\alpha)h(\beta)I,$$
for every $\alpha, \beta \in \mathbb{C}$ and therefore $h$ is multiplicative.
Since $(h(1))^2=h(1)$ and $h$ is bijective with $h(0)=0$, we obtain $h(1)=1$ . Similarly $h(-1)=-1$, thus $h(-\alpha)=h(-1)h(\alpha)=-h(\alpha)$ for all $\alpha\in\mathbb{C}$.
\end{proof}
As a direct consequence we have the following corollary :
\begin{cor}\label{cor1}
Let $\Phi :\mathcal{B}(H)\to\mathcal{B}(K)$ be a bijective map satisfying $\left(\ref{c1}\right)$. Then
\begin{enumerate}[(i)]
\item $\Phi (I)=I$.
\item $\Delta_{\lambda}\circ \Phi =\Phi \circ \Delta_{\lambda}$. In particular, $\Phi $ preserves the set of quasi-normal operators in both directions.
\item$\Phi (\alpha A)=h(\alpha)\Phi (A)$ for all $\alpha$ and $A$ quasi-normal.
\end{enumerate}
\end{cor}
The following lemma gives some properties of bijective maps satisfying $\left(\ref{c1}\right)$.
\begin{lem}\label{l6} Let $\Phi $ be a bijective map satisfying $\left(\ref{c1}\right)$. Then
\begin{enumerate}[(1)]
\item $\Phi (A^2)=(\Phi (A))^2$ for all $A$ quasi-normal.
\item $\Phi $ preserves the set of orthogonal projections.
\item $\Phi $ preserves the orthogonality between the projections ; $$P\perp Q \Leftrightarrow \Phi (P)\perp\Phi (Q).$$
\item $\Phi $ preserves the order relation on the set of orthogonal projections in the both directions ; $$Q\leq P \Leftrightarrow \Phi (Q) \leq \Phi (P).$$
\item $\Phi (P+Q)=\Phi (P)+\Phi (Q)$ for all orthogonal projections $P,Q$ such that $P\perp Q$.
\item $\Phi $ preserves the set of rank one orthogonal projections in the both directions.
\end{enumerate}
\end{lem}
\begin{proof}
$(1)$ From $\left(\ref{c1}\right)$ , we have $\Delta_{\lambda}((\Phi (A))^2)=\Phi (\Delta_{\lambda} (A^2))$ for every operator $A$.
Let $A$ be a quasi-normal operator; since $\Phi $ preserves the set of quasi-normal operators, we get
$\Phi (A), \Phi (A^2), (\Phi (A))^2 $ are quasi-normal. It follows from \eqref{qn}) that $ \Delta_{\lambda} (A^2)= A^2$ and $ \Delta_{\lambda} (\Phi (A^2))=\Phi (A^2)$. We deduce that
$$(\Phi (A))^2 = \Delta_{\lambda}((\Phi (A))^2)=\Phi (\Delta_{\lambda} (A^2))=\Phi (A^2).$$
$(2)$ Follows from the first assertion since orthogonal projections are quasi-normal.
$(3)$ Assume that $P,Q$ are orthogonal and denote $N=\Phi (P)$ and $M=\Phi (Q)$. From $\left(\ref{c1}\right)$ we have,
$\Delta_{\lambda}(MN)=\Delta_{\lambda}(NM)=0$ and using \cite[Theorem 4]{MR2392831}, we obtain
$$(MN)^{2}=MNMN=0 \mbox{~~ and~~~} (NM)^{2}=NMNM=0.$$
It follows that,
$$\|MN\|^2 = \|(MN)^*MN\|= \|NMN\|= \|(NMN)^2\|^{\frac{1}{2}}= \|NMNMN\|^{\frac{1}{2}}= 0$$
and similarly, $NM = 0$.
Finally $\Phi $ preserves the orthogonality between the projections.
$(4)$ Now, assume that $Q \leq P$, then $PQ=QP=Q$. By $\left(\ref{c1}\right)$ we have $$\Delta_{\lambda}(\Phi (Q)\Phi (P))=\Phi (Q).$$ By Lemma \ref{l2}, we get $\Phi (Q)\Phi (P)=\Phi (P)\Phi (Q)=\Phi (Q)$ since $\Phi (P)$ is an orthogonal projection. Therefore $\Phi (Q) \leq \Phi (P)$. Since $\Phi $ is bijective and its inverse satisfies the same conditions as $\Phi $, hence $\Phi $ preserves the order relation between the projections in both directions.
$(5)$ Suppose that $P,Q$ are orthogonal. We have $P \leq P+Q$ and $Q \leq P+Q$. Which gives $\Phi (P) \leq \Phi (P+Q)$ and $\Phi (Q) \leq \Phi (P+Q)$. From $\Phi (P) \perp \Phi (Q)$, it follows that $$\Phi (P)+\Phi (Q)\leq \Phi (P+Q).$$ Since $\Phi ^{-1}$ satisfies the same assumptions as $\Phi $, we have
$$ \begin{array}{lll}
\Phi (P+Q) &=& \Phi [\Phi^{-1}(\Phi(P) ) +\Phi^{-1}(\Phi(Q) )]\\
&\le & \Phi [\Phi^{-1}(\Phi(P) +\Phi(Q) )]\\
&= & \Phi (P)+\Phi (Q).
\end{array}
$$
Finally $\Phi (P+Q)=\Phi (P)+\Phi (Q)$.\\
$(6)$
Let $P=x\otimes x$ be a rank one projection. We claim that $\Phi (P)$ is a non zero minimal projection. Indeed, let $y\in K$ be an unit vector such that $y\otimes y \leq \Phi (P)$. Thus $\Phi ^{-1}(y\otimes y) \leq P$. Since $P$ is a minimal projection and $\Phi ^{-1}(y\otimes y)$ is a non zero projection, then $\Phi ^{-1}(y\otimes y)= P$. Therefore $\Phi (P)=y\otimes y$ is a rank one projection.
\end{proof}
We now prove the following lemma which is needed in the proof of our result.
\begin{lem}\label{l7}
Let $\Phi $ be a bijective map satisfying $\left(\ref{c1}\right)$. Let $P=x\otimes x, Q=x'\otimes x'$ be rank one projections such that $P\perp Q$. Then
$$\Phi (\alpha P+\beta Q)=h(\alpha)\Phi (P)+h(\beta)\Phi (Q)$$
for every $\alpha, \beta\in\mathbb{C}$.
\end{lem}
\begin{proof}
If $\alpha=0$ or $\beta=0$ the result is trivial. Suppose that $\alpha\not =0$ and
$\beta\not=0$.
Clearly $\alpha P+\beta Q$ is normal, hence $\Phi (\alpha P+\beta Q)$ is quasi-normal.
By Condition $\left(\ref{c1}\right)$ we get
\begin{eqnarray*}
\Phi (\alpha P+\beta Q)&=&\Delta_{\lambda}(\Phi (\alpha P+\beta Q))\\&=&\Phi
(\Delta_{\lambda}(\alpha P+\beta Q))\\&=&\Phi \Big(\Delta_{\lambda}\big((\alpha P+\beta
Q)
(P+Q)\big)\Big)\\&=&\Delta_{\lambda}\big(\Phi (\alpha P+\beta Q)\Phi (P+Q)\big)\\
&=&\Phi (\alpha P+\beta Q)\Phi (P+Q).
\end{eqnarray*}
Since $\Phi (P+Q)=\Phi (P)+\Phi (Q)$ is a an orthogonal projection,
hence by Lemma \ref{l2}
\begin{eqnarray*}
\Phi (\alpha P+\beta Q)&=&\Phi (\alpha P+\beta Q)(\Phi (P)+\Phi (Q))=(\Phi (P)+\Phi
(Q))\Phi (\alpha P+\beta Q)\\
&=&(\Phi (P)+\Phi (Q))\Phi (\alpha P+\beta Q)(\Phi (P)+\Phi (Q)).
\end{eqnarray*}
Denote by $T=\Phi (\alpha P+\beta Q)$. We write $\Phi (x\otimes x)=y\otimes y$ and
$\Phi (x'\otimes x')=y'\otimes y'$ with $ y\perp y'$, since $\Phi $ preserves orthogonality and rank one projections.
We have,
$$T=(y\otimes y+y'\otimes y')T(y\otimes y+y'\otimes y').$$
Hence
\begin{equation}\label{eq1}
T= <Ty,y>y\otimes y+ <Ty',y>y\otimes y'+ <Ty,y'>y'\otimes y+ <Ty',y'>y'\otimes y'.
\end{equation}
We show that $<Ty',y>=<Ty,y'>=0$ by using $\left(\ref{c1}\right)$
\begin{eqnarray*}
\Delta_{\lambda}(\Phi (\alpha P+\beta Q)\Phi (P))&=& \Phi (\Delta_{\lambda}((\alpha P+\beta Q)P))\\&=& \Phi (\alpha P)=h(\alpha)\Phi (P).
\end{eqnarray*}
In other terms, we write
$$\Delta_{\lambda}(Ty\otimes y)=\Delta_{\lambda}(y\otimes T^*y )=h(\alpha) y\otimes y.$$
Since $h(\alpha)\not=0$, then $T^*y\not=0$. By Proposition \ref{p1} follows that
$$<Ty,y>y\otimes y =\dfrac{<y,T^*y>}{\|T^*y\|^2}T^*y\otimes T^*y=h(\alpha) y\otimes y.$$
Therefore $<Ty,y>=h(\alpha)$ and $T^*y=\overline{h(\alpha)}y$. Using (\ref{eq1}) we
deduce $$T^*y=<T^*y,y>y+<T^*y,y'>y'=\overline{h(\alpha)}y.$$ It follows that
$<Ty',y>=<T^*y,y'>=0$.
By similar arguments we get $<Ty',y'>=h(\beta)$ and $<Ty,y'>=0$. Again (\ref{eq1})
implies that
$$ \Phi (\alpha P+\beta Q)=T=h(\alpha)y\otimes y+h(\beta)y'\otimes y'=h(\alpha)\Phi
(P)+h(\beta)\Phi (Q).$$
\end{proof}
Now, we are in a position to prove our main result
\bigskip
{\it \bf Proof of Theorem \ref{th1}.} The "only if" part is an immediate consequence of Lemma \ref{l0}. \\
We show the "if" part. Assume that $\Phi :\mathcal{B}(H)\to\mathcal{B}(K)$ is bijective and satisfies $\left(\ref{c1}\right)$. The proof of theorem is organized in several steps.
\begin{enumerate}[Step 1.]
\item For every $A\in\mathcal{B}(H)$, we have
\begin{equation}\label{eq}
<\Phi (A)y,y>=h(<Ax,x>) \;\;\text{for all unit vectors $x,y$ such $\Phi (x\otimes x)=y\otimes y$}.
\end{equation}
Let $x,y\in H$ be unit vectors such that $\Phi (x\otimes x)=y\otimes y$. From $\left(\ref{c1}\right)$, we obtain
\begin{eqnarray*} \Delta_{\lambda}(\Phi (A)y\otimes y)&=& \Delta_{\lambda}(\Phi (A)\Phi (x\otimes x))\\
&=& \Phi (\Delta_{\lambda}(A(x\otimes x)))\\
&=& \Phi (\Delta_{\lambda}(Ax\otimes x)).
\end{eqnarray*}
Using Proposition \ref{p1}, we get
$$<\Phi (A)y,y>y\otimes y=\Phi (<Ax,x>x\otimes x)=h(<Ax,x>)y\otimes y.$$
It follows that $$<\Phi (A)y,y>=h(<Ax,x>).$$
\item The function $h$ is additive.
Let $P=x\otimes x, Q=x'\otimes x'$ are rank one projections such that $P\bot Q$ and
$\alpha, \beta \in\mathbb{C}$. Denote by $z=\frac{1}{\sqrt{2}}(x+x')$, then $\|z\|=1$ and $\|Pz\|^2= \|Qz\|^2=\frac{1}{{2}}$. Note $z\otimes z$ is rank one projection, then there exist an unit vector $u\in K$ such that $\Phi (z\otimes z)=u \otimes u$. We take $A=\alpha P+\beta Q$ in the identity (\ref{eq}), we get that
\begin{eqnarray*} <\Phi (\alpha P+ \beta Q)u,u>&=&h(<\alpha Pz +\beta Q z,z>)\\
&=&h(\alpha \|Pz\|^2+\beta \|Qz \|^2)\\
&=&h(\frac{1}{{2}})h(\alpha+\beta).
\end{eqnarray*}
Thus
\begin{equation}\label{eqq}
<\Phi (\alpha P+ \beta Q)u,u>= h(\frac{1}{{2}})h(\alpha+\beta).
\end{equation}
In the other hand, by Lemma \ref{l7} we have
$$\Phi (\alpha P+ \beta Q)=\Phi (\alpha P)+\Phi (\beta Q)=h(\alpha)\Phi (P)+ h(\beta) \Phi (Q).$$
And therefore
\begin{eqnarray*}
<\Phi (\alpha P+\beta Q)u,u>&=&<\Phi (\alpha P)u+\Phi (\beta Q)u,u>\\&=&<\Phi (\alpha P)u,u>+<\Phi (\beta Q)u,u>\\&=& h(<\alpha Pz,z>)+h(<\beta Qz,z>)\\&=&h(\alpha \|Pz\|^2)+h(\beta \|Qz\|^2)\\&=& h(\frac{1}{2})(h(\alpha)+h(\beta)).
\end{eqnarray*}
Using (\ref{eqq}) and the preceding equality, it follows that $$h(\frac{1}{{2}})h(\alpha+\beta)=h(\frac{1}{{2}})(h(\alpha)+h(\beta)).$$
Now $h(\frac{1}{{2}})\neq 0$ gives $$h(\alpha+\beta)=h(\alpha)+h(\beta).$$
\item $h$ is continuous.
Let $\mathcal{E}$ be a bounded subset in $\mathbb{C}$ and $A\in\mathcal{B}(H)$ such that $\mathcal{E}\subset W(A)$.
By (\ref{eq}),
\begin{equation*}
h(\mathcal{E})\subset h(W(A)) = W(\Phi (A))
\end{equation*}
Now , $W(\Phi (A))$ is bounded and thus $h$ is bounded on the bounded subset. With the fact that $h$ is additive and multiplicative, it then follows that $h$ is continuous (see, for example, \cite{MR804038}). We derive that $h$ is a continuous automorphism over the complex field $\mathbb{C}$. It follows that $h$ is the identity or the complex conjugation map. \\
\item The map $\Phi $ is linear or anti-linear.
Let $y\in K$ and $x\in H$ be two unit vectors, such that $y \otimes y=\Phi ( x\otimes x)$. Let $\alpha\in \mathbb{C}$ and $A ,B\in \mathcal{B}(H)$ be arbitrary. Using (\ref{eq}), we get
\begin{eqnarray*}
<\Phi (A+B)y,y>&=&h(<(A+B)x,x>)\\&=&h(<Ax,x>+<Bx,x>)\\&=&h(<Ax,x>)+h(<Bx,x>)
\\&=&<\Phi (A)y,y>+<\Phi (B)y,y>\\&=&<(\Phi (A)+\Phi (B))y,y>,
\end{eqnarray*}
and
\begin{eqnarray*}
<\Phi (\alpha A)y,y>&=&h(<\alpha A x,x>)=h(\alpha)h(<Ax,x>)=h(\alpha)<\Phi (A)y,y>.
\end{eqnarray*}
Therefore $$<\Phi (A+B)y,y>=<(\Phi (A)+\Phi (B))y,y>\;\;\text{and} <\Phi (\alpha A)y,y>=h(\alpha)<\Phi (A)y,y>,$$
for all unit vectors $y\in K$. It follows that $\Phi (A+B)=\Phi (A)+\Phi (B)$ and $\Phi (\alpha A)=h(\alpha)\Phi (A)$ for all $A,B\in\mathcal{B}(H)$. Therefore $\Phi $ is linear or anti-linear since $h$ is the identity or the complex conjugation. \\
\item There exists an unitary operator $U\in \mathcal{B}(H,K)$, such that $\Phi (A)=UAU^*$ for every $A\in \mathcal{B}(H)$.
Let $A\in\mathcal{B}(H)$ be invertible. By $\left(\ref{c1}\right)$, we have
$$\Delta_{\lambda}(\Phi (A)\Phi (A^{-1}))=\Delta_{\lambda}(\Phi (A^{-1})\Phi (A))=\Phi (\Delta_{\lambda}(I))=I.$$
By Lemma \ref{l1}, we get that $$\Phi (A)\Phi (A^{-1})=\Phi (A^{-1})\Phi (A)=I.$$
It follows that $\Phi (A)$ is also invertible and $(\Phi (A))^{-1}=\Phi (A^{-1})$. Therefore $\Phi $ preserves the set of invertible operators. By \cite[Corollary 4.3]{MR2669430}, there exists a bounded linear and bijective operator
$V:H\to K$ such that $\Phi $ takes one of the following form
\begin{equation}\label{f1}
\Phi (A)=VAV^{-1}\;\; \; \text{for all}\;\; A\in\mathcal{B}(H)
\end{equation}
or
\begin{equation}\label{f2}
\Phi (A)=VA^*V^{-1} \;\; \; \text{for all}\;\; A\in\mathcal{B}(H)
\end{equation}
In order to complete the proof we have to show that $V$ is unitary and $\Phi $ has form (\ref{f1}).
First, we show that $V:H\to K$ in (\ref{f1}) ( or in (\ref{f2})) is necessarily unitary. Indeed, let $x\in H$ be a unit vector. We know that $x\otimes x$ is an orthogonal projection, hence $\Phi (x\otimes x)=Vx\otimes (V^{-1})^*x$ is also an orthogonal projection. It follows that $ (V^{-1})^*x=Vx$ for all unit vector $x\in H$ and then $ (V^{-1})^*=V$. Therefore $V$ is unitary.
Seeking contradiction, we suppose that (\ref{f2}) holds. Multiplying (\ref{f2}) by $V^*$ left and by $V$ right, since $\Phi$ commutes with ${\Delta_{\lambda}}$, we obtain
\begin{equation}\label{ee}
{\Delta_{\lambda }}(A^{*})= ({\Delta_{\lambda}}(A))^{*},\quad \mbox{ for every} \; A\in\mathcal{B}(H).
\end{equation}
Let us consider $A=x\otimes x'$ with $x, x'$ are unit independent vectors in $H$. $A^*=x'\otimes x$ and by Proposition \ref{p1}, we have
$$ {\Delta_{\lambda}}(A)=<x,x'>(x'\otimes x') \; \; \mbox{and} \; \; {\Delta_{\lambda}}(A^{*})=<x',x>(x\otimes x),$$
which contradicts (\ref{ee}). This completes the proof.\\
\end{enumerate}
{\bf Acknowledgments.}
I wish to thank Professor Mostafa Mbekhta for the interesting discussions as well as his useful suggestions for the improvement of this paper. Also, I thank the referee for valuable comments that helped to improve the paper, in particular the proof of Corollary 2.1.
This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).\\
{\bf References}
\bibliographystyle{abbrv}
|
2,869,038,155,669 | arxiv | \section{Introduction}
The role played by symmetry in the understanding and development of physics can hardly be overstated. It is an important part of the process of building mathematical
models of the physical world, and is often crucial for their solubility
in both classical and quantum settings.
Less widely recognised, however, is the effect that symmetry has in limiting
what is measurable.
In fact,
as is common in gauge theories, e.g., \cite{haag}, and postulated
in, e.g.,
\cite{lbm, lmb}
in the context of quantum reference frames, theoretical quantities which do not commute with a symmetry
action are unobservable, even in principle.
The tension between this apparent unobservability and the use of such quantities in the description
of real physical systems is relieved by noting that any system of interest is
a part of a larger whole - there is another system (called the environment,
ancilla, reservoir, reference frame, or apparatus) whose presence is often assumed only implicitly
and which does not appear in the formulation. It is then possible that unobservable
quantities of the system of interest may be re-interpreted as representatives of
observable {\it relative} quantities of system-plus-environment.
\par
In this paper we study the extent to which the same kind of restrictions (due to symmetry) hold in
a dynamical context. To make the problem concrete,
we must identify, in analogy to the invariance requirement
for observables, the right notion of restriction for channels, since it is not unique. This
will be discussed in detail in a future publication.
One natural example arises in the
presence of conserved quantities.
The Wigner-Araki-Yanase (WAY) theorem \cite{ew1, ay1,lb1,lb2,lov1,mi} shows that
conservation laws restrict the accuracy and repeatability properties of a class of quantum measurements.
Also in the presence of
an additive conserved quantity,
\r{A}berg \cite{ab1} has shown that any unitary dynamics can be
approximately realized by preparing ``coherent" (thus asymmetric) states of the environment.
We here emphasize the necessity of the highly asymmetric (coherent) state
(cf.\ \cite{lbm, lmb, mlb}), which requires a large environment in a sense to be described.
If the environment is not large, states cannot have
enough coherence and the possible dynamics is
restricted. In addition, if the symmetry is not Abelian,
there may be some restriction due to the uncertainty relation, because
large coherence with respect to some observable
implies small coherence with respect to its conjugate.
Another possible symmetry constraint is
covariance of the dynamics, whose relevance
to the reference frame context will be discussed
briefly in the next section.
As will be shown later, this constraint
is weaker than the one given by the presence of
conserved quantities.
In this paper, we study the possible dynamics
of the system under the
symmetry constraint
on the whole system (object system plus reference).
We consider a quantum channel on the system and study how well
this (target) channel is approximately realized by a
covariant channel
on the whole system, contingent upon a choice of state of
the environment. We derive a quantitative relation which shows
that for the covariant channel to be well approximated by
the target channel, high asymmetry/coherence is required for the
state of the environment.
As an example we apply our relation to a qubit system under
$SU(2)$ symmetry.
Such size-versus-inaccuracy trade-offs are already present in the literature in various different contexts (see, e.g., \cite{mlb,ajr1,bartfin}), and our findings are broadly in line
with other findings - that good accuracy needs large size. Specific mention must be given to \cite{taj1,taj2,taj3}, which
has already investigated the dynamical setting and some elegant bounds have been provided, particularly in the unitary case.
However, we provide a novel quantitative bound in the dynamical context.
From a technical point of view, the present paper may be regarded as a descendant of \cite{mlb}, in which
we derived a quantitative bound in approximating an arbitrary effect by a
globally invariant effect.
There we reinterpreted the issue as an
approximate
joint measurement problem of observables and employed
a quantitative uncertainty relation \cite{MiyaIma}.
In this paper, we show that a similar technique,
which we may call uncertainty relation based method,
can be applied also to the approximating channel problem.
\section{Symmetry constraints on channels}
The principle of symmetry limits the possible observables to invariant ones (see, e.g., \cite{lmb}).
There are some different generalizations
of this constraint to channels.
\par
Suppose that we have a system described by
a Hilbert space $\hi$.
By $\mathbf{B}(\hi)$
we denote the algebra of bounded operators on
$\hi$.
Throughout this paper every Hilbert space we encounter will be finite dimensional.
We assume that
a finite dimensional connected Lie group $G$
is acting on $\hi$.
$G$ is assumed to define a true smooth unitary representation on each system, denoted by
$U(g)$.
\par
One of the possible constraints on dynamics is
given by a conservation law.
Suppose that there exists a conserved
charge
$N$,
which generates a $U(1)$ action.
In the situation that the
system is closed/isolated, the possible dynamics $\Lambda$
must satisfy $\Lambda(N)=N$ (in the Heisenberg
picture).
We thus arrive at the following definition.
\begin{definition}
\emph{Invariant channels} are
defined as those $\Phi: \mathbf{B}(\mathcal{H}) \to \mathbf{B}(\mathcal{H})$ for which $\Phi(U(g)) = U(g) \text{~for all~} g \in G$,
where $U$ is an $\hi$-representation of $G$.
\end{definition}
\if
On the other hand,
suppose that we have a particle and are asked to perform a repeatable measurement
of the $z$-component of angular momentum. The dynamics may be
described by a L\"{u}ders channel
$\Lambda(X)= \sum_{m} E_m X E_m$, where
$L_z= \hbar \sum_m m E_m$ is the spectral
decomposition.
This channel implicitly assumes the existence of
a reference frame/system which specifies the $z$-axis.
If we employ another reference frame,
what we measure is $U(R) L_z U(R)^*$ with
some $R \in SO(3)$ and the corresponding
channel becomes
$\Lambda_R(X)= \hbar \sum_m U(R) E_m U(R)^*
X U(R) E_m U(R)^*$.
It satisfies $\Lambda_R(U(R)X U(R)^*)
=U(R) \Lambda (X) U(R)^*$.
That is, $\Lambda_R(Y) = U(R) \Lambda
(U(R)^* Y U(R)) U(R)^*$ holds.
If we are not informed which reference frame
is to be used, we may choose one randomly.
In this case a channel is described as
\begin{eqnarray}
\overline{\Lambda}(Y)
= \int \mu (dR) \Lambda_{R}(Y)
= \int \mu(dR) U(R) \Lambda(U(R)^*Y
U(R))U(R)^*,
\end{eqnarray}
where $\mu(\cdot)$ is (the) Haar measure on $SO(3)$.
\fi
On the other hand,
suppose that we have a system and
are asked to rotate the state around the $z$-axis
by some angle $\theta$. The desired
channel is $\Phi (X)
= e^{i S_z \theta}
X e^{-i S_z \theta}$,
where $S_z$ is the $z$-component of
angular momentum. (We work in units in which $\hbar =1$.)
This channel implicitly assumes the existence of
a reference frame/system which specifies the $z$-axis.
If we employ another reference frame,
the $z$-component of
angular momentum is represented as
$U(R) S_z U(R)^*$ with
some $R \in SO(3)$ and the corresponding
channel becomes
$\Phi_R(X)
= U(R) \Phi(U(R)^* X U(R))U(R)^*$.
If we are not informed which reference frame
is to be used, we may choose one randomly, in which
case the channel is described as
\begin{eqnarray}
\overline{\Phi}(X)
=
\int \mu(dR) \Phi_R(X),
\end{eqnarray}
where $\mu(\cdot)$ is (the) Haar measure on $SO(3)$.
$\overline{\Phi}$ is an example of a \emph{covariant channel}
which we now define.
\begin{definition}
A channel $\Lambda : \mathbf{B}(\mathcal{H}) \to \mathbf{B}(\mathcal{H})$ is called
covariant if and only if
\begin{eqnarray}
\Lambda(U(g)^* X U(g))= U(g)^* \Lambda (X) U(g)
\label{covchannel}
\end{eqnarray}
holds for all $X \in \mathbf{B}(\mathcal{H})$
and for all $g\in G$.
\end{definition}
As the next proposition shows, invariant channels form an important subclass of covariant channels.
\begin{proposition}\label{prop:p1}
Invariant channels are covariant.
\end{proposition}
\begin{proof}
We begin by using a channel $\Lambda$ to define an ``operator-valued inner product"
$\langle\langle A|B \rangle\rangle:=
\Lambda(A^*B)- \Lambda(A)^*\Lambda(B)$,
which satisfies a Cauchy-Schwarz type inequality (see \cite{janssens} and Lemma 3 in \cite{mlb}):
\begin{eqnarray}\label{eq:C-S}
\Vert \langle \langle A|B\rangle \rangle \Vert^2
\leq \Vert \langle \langle A|A\rangle \rangle\Vert
\Vert \langle \langle B |B \rangle \rangle\Vert,
\end{eqnarray}
where $|| \cdot ||$ denotes the standard operator norm in $\mathbf{B}(\mathcal{H})$.
Suppose that a unitary $U$ is a fixed point, i.e., $\Lambda(U) = U$.
Then it holds that
\begin{eqnarray*}
\langle \langle U| U \rangle \rangle
= \id - \Lambda(U)^* \Lambda(U)= 0 .
\end{eqnarray*}
Thus for such a $U$ and arbitrary $A$ we find
\begin{eqnarray*}
\langle \langle A|U\rangle \rangle =\Lambda(A^*U) -\Lambda(A^*)\Lambda(U)=
\Lambda(A^*U)-\Lambda(A)^* U=0, {~\text{by}~ \eqref{eq:C-S}}.
\end{eqnarray*}
Now Let $\Lambda: \mathbf{B}(\mathcal{H}) \to \mathbf{B}(\mathcal{H})$ be an invariant channel. Then for all $g \in G$,
\begin{eqnarray*}
\Lambda(AU(g))= \Lambda(A)\Lambda(U(g))
= \Lambda(A)U(g).
\end{eqnarray*}
Similarly $\Lambda(U(g)^* B) = U(g)^* \Lambda(B)$, and thus $\Lambda(U(g)^* A U(g))= U(g)^* \Lambda(A) U(g).$
\end{proof}
We note that there exist covariant channels
which are not invariant; for instance, for any
invariant state
$\omega_0$, the channel
$\Lambda(X) = \omega_0(X) \id$ is covariant but not invariant.
However, covariance and invariance are equivalent for unitary channels:
\begin{proposition}
Unitary covariant channels are invariant.
\end{proposition}
\begin{proof}
Let us consider a unitary (and thus automorphic), covariant channel $\Lambda : \mathbf{B}(\mathcal{H}) \to \mathbf{B}(\mathcal{H})$, i.e.,
$\Lambda(X) = V^*XV$ and $U(g)^* \Lambda (X) U(g) = \Lambda (U(g)^*XU(g))$.
Then it holds that for all $X$
\begin{eqnarray*}
\Lambda(U(g)^* XU(g))
= V^* U(g)^* V V^* X V V^* U(g) V
= U(g)^* (V^* X V) U(g).
\end{eqnarray*}
Now putting
$V^*XV=Z$, we apply $U(g) \cdot U(g)^*$ to the above equation to obtain
\begin{eqnarray*}
(U(g)V^* U(g)^* V)Z (V^* U(g) V U(g)^*)=Z,
\end{eqnarray*}
which implies
$V^*U(g) VU(g)^* = c(g) \id$ with $|c(g)|=1$ and
$V^* U(g) V = c(g) U(g)$. As the left-hand side
satisfies $V^*U(g)V V^*U(g')V =V^*U(gg') V$,
$c(g)c(g')=c(gg')$ holds for all $g, g\in G$.
Now in a neighborhood of $e\in G$,
for each element $l$ of Lie algebra the corresponding generator
$L$ exists satisfying $U(e^{ls})= e^{iL s}$ for sufficiently small $|s|$.
If we put the generator of $V^*U(e^{ls})V$ as $L'$,
it satisfies $V^*LV=L'$. It in addition satisfies $L' = L+ k\id$
for some $k\in \mathbf{R}$ as $V^*U(e^{ls})V= c(e^{ls}) U(e^{ls})$ must hold.
But as $L$ is bounded (as $\mathcal{H}$ is finite dimensional) and $\Vert L\Vert = \Vert L'\Vert $ holds,
$k=0$ is the only possible choice.
Thus we have shown that for a neighbourhood $N_e$ of $e\in G$
$V^* U(g) V= U(g)$ is satisfied.
As $G$ is connected, it is generated by
$N_e$. It implies that $c(g)=1$ for all $g\in G$.
\end{proof}
\section{The setting and results}
As we have seen in the last section,
we cannot implement (for instance) the rotation
around the $z$-axis without
using a ``correct"
reference frame. More precisely, we may
implement the right rotation but this occurs
only by chance. The averaged channel is
a covariant $\overline{\Phi}$ which is different from
the desired rotation. In the worst case,
the discrepancy is larger than the averaged case.
Thus we must have a reference frame.
Since a reference frame is also a physical system,
there should be a quantum description. Our question is
then to ask what is the condition on the
quantum reference frame so that it works
well to implement the desired channel.
In the following we formulate the problem
in a general setting.
\par
Let $G$ be a connected Lie group.
We have a system and a reference frame
described by (as always, finite dimensional) Hilbert spaces $\mathcal{H}_{\mathcal{S}}$ and $\mathcal{H}_{\mathcal{R}}$, and
on each space, $G$ has a smooth true unitary
representation $U_{\mathcal{S}}(g)$ and $U_{\mathcal{R}}(g)$.
Their composition is written as
$U(g)=U_{\mathcal{S}}(g) \otimes U_{\mathcal{R}}(g)$
which acts on $\hi=\hi_{\mathcal{S}}\otimes \hi_{\mathcal{R}}$.
Our purpose is to study how well a general channel $\Lambda: \mathbf{B}(\mathcal{H}_{\mathcal{S}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$
is approximately realized by the restriction of
a covariant channel
$\Phi: \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}})$. We view $\Phi$ as representing the ``true" transformation,
with its restriction representing the transformation with the additional system suppressed.
Therefore,
$\Phi$ satisfies
\begin{eqnarray*}
\Phi(U(g)^* X U(g))=U(g) \Phi(X) U(g)
\end{eqnarray*}
for all $g\in G$ and $X \in \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}})$.
On the level of observables, the restriction to the system
$\Gamma_{\rho_{\mathcal{R}}}: \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$ is determined by a state $\rho_{\mathcal{R}}$ on $\mathbf{B}(\mathcal{H}_{\mathcal{R}})$
and is defined by the completely positive conditional expectation
\begin{eqnarray*}
\mbox{tr}[\rho_{\mathcal{S}} \Gamma_{\rho_{\mathcal{R}}}(X)]
= \mbox{tr}[(\rho_{\mathcal{S}} \otimes \rho_{\mathcal{R}})X],
\end{eqnarray*}
which holds for all states $\rho_{\mathcal{S}}$ of the system and $X \in \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}})$.
In order to define the restriction for channels,
we use the natural inclusion $\iota: \mathbf{B}(\mathcal{H}_{\mathcal{S}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}})$, given as
\begin{eqnarray*}
\iota(A) = A\otimes \id_{\mathcal{R}}.
\end{eqnarray*}
Then the realized channel is written as
$\Phi_{\rho_{\mathcal{R}}}:=\Gamma_{\rho_{\mathcal{R}}} \circ \Phi\circ \iota: \mathbf{B}(\mathcal{H}_{\mathcal{S}})\to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$,
and we wish to quantify the discrepancy between $\Phi_{\rho_{\mathcal{R}}}$ and $\Lambda$.
As a quantity to characterize the discrepancy,
one may employ the norm difference between
two channels defined by
\begin{eqnarray*}
\Vert \Phi_{\rho_{\mathcal{R}}} - \Lambda\Vert_{\mathrm{channel}}
:= \sup_{X\in \mathbf{B}(\mathcal{H}_S),
\Vert X\Vert =1}
\Vert \Phi_{\rho_{\mathcal{R}}}(X) - \Lambda(X)\Vert.
\end{eqnarray*}
\par
For each element of the Lie algebra $\frak{g}$
of a Lie group $G$ there exists a corresponding self-adjoint operator (the generator)
acting in $\mathcal{H}_{\mathcal{S}}$.
For each $l\in \frak{g}$, there exist operators
$L_{\mathcal{S}}$ and $L_{\mathcal{R}}$ satisfying $U_{\mathcal{S}}(e^{l s})
= e^{i L_{\mathcal{S}} s}$ and $U_{\mathcal{R}}(e^{ls})= e^{i L_{\mathcal{R}} s}$
and therefore $U(e^{ls})= e^{i (L_{\mathcal{S}} \otimes \id_{\mathcal{R}} + \id_{\mathcal{S}} \otimes L_{\mathcal{R}})}$.
As unitary operators have norm $1$, we obtain
an inequality for each $U_{\mathcal{S}}(e^{l s_0})$,
\begin{eqnarray*}
\Vert \epsilon(l:s_0)
\Vert :=
\Vert
\left(\Gamma_{\rho_{\mathcal{R}}}\circ \Phi \circ \iota\right) (U_{\mathcal{S}}(e^{l s_0}))
- \Lambda(U_{\mathcal{S}}(e^{l s_0}))
\Vert
\leq
\Vert \Phi_{\rho_{\mathcal{R}}} - \Lambda\Vert_{\mathrm{channel}}.
\end{eqnarray*}
$F(\rho_0, \rho_1)$ represents the fidelity between two states $\rho_0$ and
$\rho_1$ defined by $F(\rho_0, \rho_1):= \mbox{tr}[\sqrt{\rho_0^{1/2}\rho_1
\rho_0^{1/2}}]$. This quantity is positive and equals $1$ if and only if
$\rho_0= \rho_1$ holds.
\begin{theorem}\label{th:main}
Let $L_{\mathcal{S}}$ and $L_{\mathcal{R}}$ be generators of unitary representations
of $e^{ls}\ (s\in \mathbb{R})$ on $\mathcal{H}_{\mathcal{S}}$ and $\mathcal{H}_{\mathcal{R}}$ for $l\in \frak{G}$.
Define $U_{\mathcal{S}}(l:s):=e^{i L_{\mathcal{S}} s}$ and $U_{\mathcal{R}}(l:s) = e^{i L_{\mathcal{R}} s}$.
Then, for any $s_0 \in \mathbb{R}$,
$\epsilon(l:s_0):= \left(\Gamma_{\rho_{\mathcal{R}}}\circ \Phi \circ \iota\right) (U_{\mathcal{S}}(l:s_0))
- \Lambda(U_{\mathcal{S}}(l:s_0))$ is bounded for all $s\in \mathbb{R}$ by:
\begin{align*}
& \Vert
[\Lambda(U_{\mathcal{S}}(l:s_0)), U_{\mathcal{S}}(l:s)]
\Vert
\leq
2 \Vert U_{\mathcal{S}}(l:s)- \id\Vert \Vert \epsilon(l:s_0) \Vert
\\
&
+
\left(\frac{1}{F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{R}U_{\mathcal{R}}(l:s)^*)^2}
-1\right)^{1/2}
\left(
\left(
\Vert \id_{\mathcal{S}} - \Lambda(U_{\mathcal{S}}(l:s_0))^* \Lambda(U_{\mathcal{S}}(l:s_0))\Vert
+ 2 \Vert \epsilon(l:s_0)\Vert\right)^{1/2}
\right.
\\
&
\left.
+
\left(
\Vert \id_{\mathcal{S}} - \Lambda(U_{\mathcal{S}}(l:s_0))\Lambda(U_{\mathcal{S}}(l:s_0))^*\Vert
+2 \Vert \epsilon(l:s_0)\Vert\right)^{1/2}
\right).
\end{align*}
\end{theorem}
Before proving Theorem \ref{th:main}, we present some immediate implications.
We first observe that the left hand side of the above inequality
vanishes for covariant $\Lambda$, since
\begin{eqnarray}
\Vert [\Lambda(U_{\mathcal{S}}(l:s_0)), U_{\mathcal{S}}(l:s)]\Vert
&=& \Vert U_{\mathcal{S}}(l:s)^* [\Lambda(U_{\mathcal{S}}(l:s_0)), U_{\mathcal{S}}(l:s)]\Vert
\nonumber
\\
&=&
\Vert U_{\mathcal{S}}(l:s)^* \Lambda(U_{\mathcal{S}}(l:s_0))
U_{\mathcal{S}}(l:s) - \Lambda(U_{\mathcal{S}}(l:s_0))\Vert,
\label{eq1}
\end{eqnarray}
and
\begin{eqnarray*}
U_{\mathcal{S}}(l:s)^* \Lambda(U_{\mathcal{S}}(l:s_0))
U_{\mathcal{S}}(l:s) = \Lambda( U_{\mathcal{S}}(l:s)^* U_{\mathcal{S}}(l:s_0)
U_{\mathcal{S}}(l:s))
= \Lambda(U_{\mathcal{S}}(l:s_0)).
\end{eqnarray*}
Therefore, there is no bound for approximating covariant channels
$\mathbf{B}(\mathcal{H}_{\mathcal{S}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$ by restrictions of covariant channels
$\mathbf{B}(\hi) \to \mathbf{B}(\hi)$. Indeed, any covariant $\Lambda$ can trivially
be written as the restriction of a covariant channel $\Phi$ on $\mathbf{B}(\hi)$, i.e., as
$\Phi_{\rho_{\mathcal{R}}}$ for all $\rho_{\mathcal{R}}$, by setting $\Phi = \Lambda \otimes \mbox{id}$.
If $\Lambda$ is a unitary channel, Theorem \ref{th:main} takes a much simpler form.
\begin{corollary}
Under the same assumptions as Theorem \ref{th:main}, but for unitary $\Lambda$,
it holds that
\begin{align*}
\Vert
[\Lambda(U_{\mathcal{S}}(l:s_0)), U_{\mathcal{S}}(l:s)]
\Vert \leq
&2 \Vert U_{\mathcal{S}}(l:s)- \id\Vert \Vert \epsilon(l:s_0) \Vert
\\
+& 2
\left(\frac{1}{F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{R}U_{\mathcal{R}}(l:s)^*)^2}
-1\right)^{1/2}
\Vert \epsilon(l:s_0)\Vert^{1/2}.
\end{align*}
\end{corollary}
The proof follows from the observation that if $\Lambda$ is a unitary channel (or indeed, multiplicative), then for any unitary operator $U\in \mathbf{B}(\mathcal{H}_{\mathcal{S}})$, we have $\Lambda(U)^* \Lambda(U) = \id$.
Therefore, we see that in order to make possible good agreement between $\Lambda$ and $\Phi_{\rho_{\mathcal{R}}}$, a highly ``asymmetric" reference state $\rho_{\mathcal{R}}$ is necessary, since
$F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{\mathcal{R}}U_{\mathcal{R}}(l:s)^*)$ must decrease rapidly
with respect to $|s|$ as otherwise the left-hand side of the inequality
can be large for non-covariant $\Lambda$.
\par
Furthermore, this asymmetry, or \emph{coherence} factor, can be
bounded by the ``spread" of the (symmetry) generator $L_{\mathcal{R}}$:
\begin{corollary}\label{cor:der}
In the same scenario as Theorem \ref{th:main}, it holds that
\begin{align*}
\Vert
[\Lambda(U_{\mathcal{S}}(l:s_0)), L_{\mathcal{S}}]
\Vert
\leq
& 2 \Vert L_{\mathcal{S}} \Vert \Vert \epsilon(l:s_0) \Vert \\
+
&(\Delta_{\rho_{\mathcal{R}}}L_{\mathcal{R}})
\biggl( (
\Vert \id_{\mathcal{S}} - \Lambda(U_{\mathcal{S}}(l:s_0))^* \Lambda(U_{\mathcal{S}}(l:s_0))\Vert
+ 2 \Vert \epsilon(l:s_0)\Vert )^{1/2} \\
+ &
(
\Vert \id_{\mathcal{S}} - \Lambda(U_{\mathcal{S}}(l:s_0))\Lambda(U_{\mathcal{S}}(l:s_0))^*\Vert
+2 \Vert \epsilon(l:s_0)\Vert )^{1/2}
\biggr),
\end{align*}
where $\Delta_{\rho_{\mathcal{R}}}L_{\mathcal{R}}
:= \sqrt{\mathrm{tr}[\rho_{\mathcal{R}} L_{\mathcal{R}}^2]- \mathrm{tr}[\rho_{\mathcal{R}}L_{\mathcal{R}}]^2}$
represents the standard deviation of $L_{\mathcal{R}}$ in the state $\rho_{\mathcal{R}}$.
\end{corollary}
\begin{corollary}
Under the same assumptions as Theorem \ref{th:main}, but for unitary $\Lambda$,
it holds that
\begin{eqnarray*}
&&\Vert
[\Lambda(U_{\mathcal{S}}(l:s_0)), L_{\mathcal{S}}]
\Vert
\leq
2 \Vert L_{\mathcal{S}} \Vert \Vert \epsilon(l:s_0) \Vert
+2\sqrt{2}
(\Delta_{\rho_{\mathcal{R}}}L_{\mathcal{R}})
\Vert \epsilon(l:s_0)\Vert^{1/2}.
\end{eqnarray*}
\end{corollary}
This immediately follows from Corollary
\ref{cor:der}.
The inequality is easy to interpret.
For non-covariant $\Lambda$ which
yields non-vanishing left-hand side,
$\Delta_{\rho_{\mathcal{R}}}L_{\mathcal{R}}$ must be large
to attain small $\Vert \epsilon(l:s_0)\Vert$.
Thus it implies that the reference system $\mathcal{R}$
must be large (macroscopic). This result has some qualitative similarity to
the bounds obtained in \cite{taj1,taj2}, where large size/coherence/energy fluctuation of the reference is shown to be necessary for implementing unitary dynamics.
\par
We now present proofs of Theorem \ref{th:main} and Corollary \ref{cor:der}.
To prove Theorem \ref{th:main}, we need the following lemma \cite{mlb, janssens}:
\begin{lemma}\label{th:uncertain}
Consider a channel $\Gamma:\mathbf{B}(\mathcal{H}) \to \mathbf{B}(\hik)$ for
Hilbert spaces $\hi$ and $\hik$.
If $A,B\in \mathbf{B}(\mathcal{H})$ satisfy $[A,B]=0$, then
\begin{align}\label{eq:incomprehensible_inequality}
\Vert
[\Gamma(A),\Gamma(B)]
\Vert
\leq
&\Vert
\Gamma(A^*A)-\Gamma(A)^*\Gamma(A)\Vert^{1/2}
\Vert
\Gamma(BB^*)-\Gamma(B)\Gamma(B)^*\Vert^{1/2}
\\
+
& \Vert
\Gamma(AA^*)-\Gamma(A)\Gamma(A)^*\Vert^{1/2}
\Vert
\Gamma(B^*B)-\Gamma(B)^*\Gamma(B)\Vert^{1/2}.
\end{align}
\end{lemma}
We now present the proof of Theorem \ref{th:main}.
\begin{proof}
If the state $\rho_{\mathcal{R}}$ and $s \in \mathbb{R}$ satisfy
$F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{\mathcal{R}}
U_{\mathcal{R}}(l:s)^*)=0$, the claim follows trivially, and thus we assume otherwise.
For notational simplicity, we omit the dependence on $l$ and
write $U_{\mathcal{S}}(s)$ for $U_{\mathcal{S}}(l:s)$,
$U_{\mathcal{R}}(s)$ for $U_{\mathcal{R}}(l:s)$
and $\epsilon(s_0)$ for $\epsilon(l:s_0)$.
We first write
\begin{eqnarray}\label{eq:bcom}
[\Lambda(U_{\mathcal{S}}(s_0)), U_{\mathcal{S}}(s)]
=[U_{\mathcal{S}}(s), \epsilon(s_0)] + [\Gamma_{\rho_{\mathcal{R}}}\Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}), U_{\mathcal{S}}(s)].
\end{eqnarray}
The first term on the right hand side is bounded as
\begin{eqnarray*}
\Vert [U_{\mathcal{S}}(s), \epsilon(s_0)]\Vert
= \Vert[U_{\mathcal{S}}(s)-\id, \epsilon(s_0)]\Vert
\leq 2 \Vert U_{\mathcal{S}}(s)- \id\Vert \Vert \epsilon(s_0) \Vert.
\end{eqnarray*}
To estimate the second term on the right hand side of \eqref{eq:bcom},
we introduce a purification of $\rho_{\mathcal{R}}$ to
$|\phi_{RZ}\rangle \in \mathcal{H}_{\mathcal{R}} \otimes \hi_Z$, where we choose the
purification space $\hi_Z$ to be minimal, i.e., its dimension coincides with
the rank of $\rho_{\mathcal{R}}$.
We denote $\Gamma_{|\phi_{RZ}\rangle \langle \phi_{RZ}|}
: \mathbf{B}(\mathcal{H}_{\mathcal{S}} \otimes \mathcal{H}_{\mathcal{R}} \otimes \hi_Z)
\to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$ by $\Gamma$ for simplicity.
Now, for an arbitrary operator $W_Z$ on $\hi_Z$, we have
\begin{eqnarray*}
\Gamma(U_{\mathcal{S}}(s) \otimes U_{\mathcal{R}}(s) \otimes W_Z)
= U_{\mathcal{S}}(s) \langle \phi_{RZ}| U_{\mathcal{R}}(s) \otimes W_Z|\phi_{RZ}\rangle.
\end{eqnarray*}
In the following
we denote $\Phi \otimes \mbox{id}_{Z}$ by
$\hat{\Phi}$, and in order to simplify some long expressions we will make the abbreviations $A_0 = U_{\mathcal{S}}(s_0)\otimes \id _{\mathcal{R}} \otimes \id _Z$ and $A_s = U_{\mathcal{S}}(s)\otimes U_{\mathcal{R}}(s)\otimes W_Z$ when convenient.
Thus we have, for $W_Z$ with $ \langle \phi_{RZ} |U_{\mathcal{R}}(s) \otimes W_Z|\phi_{RZ}\rangle
\neq 0$,
\begin{equation*}
[\Gamma (\hat{\Phi}(A_0)),
U_{\mathcal{S}}(s)]
=
\frac{[\Gamma(\hat{\Phi}(A_0)),
\Gamma(A_s)]}{\langle \phi_{RZ} |U_{\mathcal{R}}(s) \otimes W_Z|\phi_{RZ}\rangle }.
\label{eqn1234}
\end{equation*}
Since $\Phi$ is a covariant channel, it holds that
\begin{eqnarray*}
(U_{\mathcal{S}}(s)^* \otimes U_{\mathcal{R}}(s)^*)\Phi(U_{\mathcal{S}}(s_0)\otimes \id_{\mathcal{R}})
(U_{\mathcal{S}}(s) \otimes U_{\mathcal{R}}(s))
= \Phi (U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}).
\end{eqnarray*}
Therefore we find
\begin{eqnarray*}
[\Phi( U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}) \otimes \id_Z,
U_{\mathcal{S}}(s) \otimes U_{\mathcal{R}}(s) \otimes W_Z]=0,
\end{eqnarray*}
which enables us to apply Lemma \ref{th:uncertain}.
In the following, $W_Z$ is chosen to be unitary.
Now we bound
\begin{equation*}
\Vert [\Gamma (\hat{\Phi}(A_0)),
U_{\mathcal{S}}(s)]\Vert \\
=
\frac{\Vert [\Gamma(\hat{\Phi}(A_0)),
\Gamma(A_s)]\Vert}{| \langle \phi_{RZ} |U_{\mathcal{R}}(s) \otimes W_Z|\phi_{RZ}\rangle | }.
\end{equation*}
Then
Lemma \ref{th:uncertain} yields
the numerator of the above equation to be
bounded as
\begin{align*}
\Vert [\Gamma (\hat{\Phi}(A_0)), \Gamma(A_s)]
\Vert
\leq &\Vert \Gamma (\hat{\Phi}(A_0)^*\hat{\Phi}
(A_0))
- \Gamma( \hat{\Phi}(A_0))^*\Gamma(\hat{\Phi}(A_0))\Vert^{1/2}
\Vert
\id - \Gamma(A_s) \Gamma( A_s)^* \Vert^{1/2} \\
+ &\Vert \Gamma(\hat{\Phi}(A_0)
\hat{\Phi}(A_0)^*)
- \Gamma( \hat{\Phi}(A_0))\Gamma(\hat{\Phi}
(A_0))^* \Vert^{1/2}
\Vert
\id - \Gamma(A_s)^*
\Gamma( A_s) \Vert^{1/2}.
\end{align*}
We first estimate the norm of
\begin{equation*}
A:=
\Gamma(\hat{\Phi}(A_0)^*\hat{\Phi}
(A_0)) - \Gamma( \hat{\Phi}(A_0)^*\Gamma(\hat{\Phi}(A_0)).
\end{equation*}
Due to the two-positivity of
$\Gamma$ (i.e., $\Gamma(X^*X) \geq \Gamma(X)^*
\Gamma(X)$ for all $X$) the operator $A$ is
positive.
Furthermore applying the two-positivity of $\hat{\Phi}$,
we obtain
\begin{equation*}
\hat{\Phi}(A_0)^* \hat{\Phi}(A_0)
\leq \hat{\Phi}(A_0^*
A_0)
=\id.
\end{equation*}
Since $\Gamma$ is a positive map we find
\begin{eqnarray*}
\mathbf{0}\leq A \leq \id
- \Gamma( \hat{\Phi}(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}
\otimes \id_Z))^*\Gamma(\hat{\Phi}(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}
\otimes \id_Z)),
\end{eqnarray*}
from which we conclude
\begin{eqnarray*}
\Vert A\Vert
\leq \Vert \id
- \Gamma( \hat{\Phi}(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}
\otimes \id_Z))^*\Gamma(\hat{\Phi}(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}
\otimes \id_Z))\Vert.
\end{eqnarray*}
The term
\begin{equation*}
\Vert
\Gamma(\hat{\Phi}(A_0)
\hat{\Phi}(A_0)^*)
- \Gamma( \hat{\Phi}(A_0))\Gamma(\hat{\Phi}
(A_0))^* \Vert^{1/2}
\end{equation*}
can be treated similarly. Writing $c_{RZ} \equiv \langle \phi_{RZ} |U_{\mathcal{R}}(s) \otimes W_Z|\phi_{RZ}\rangle$, we thus obtain
\begin{multline*}
\Vert [\Gamma (\hat{\Phi}(A_0)), U_{\mathcal{S}}(s)]\Vert
\leq \frac{1}{| c_{RZ}| }
\biggl(
\Vert \id - \Gamma( \hat{\Phi}(A_0))^*\Gamma(\hat{\Phi}(A_0))\Vert^{1/2}
\Vert
\id - \Gamma(A_s)
\Gamma( A_s)^* \Vert^{1/2}\\ +
\Vert \id - \Gamma( \hat{\Phi}(A_0))\Gamma(\hat{\Phi}(A_0))^* \Vert^{1/2}
\Vert
\id - \Gamma(A_s)^*
\Gamma( A_s) \Vert^{1/2} \biggr) ,
\end{multline*}
which is bounded above by
\begin{multline*}
\frac{1}{|c_{RZ} | }
( 1- |c_{RZ}|^2)^{1/2} \biggl(
\Vert \id - \Gamma_{\rho_{\mathcal{R}}}( \Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}))^*\Gamma_{\rho_{\mathcal{R}}} (\Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}})) \Vert^{1/2}\\
+
\Vert \id - \Gamma_{\rho_{\mathcal{R}}}( \Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}))\Gamma_{\rho_{\mathcal{R}}}(\Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}))^* \Vert^{1/2}
\biggr).
\end{multline*}
We estimate
\begin{align*}
\Vert \id - &\Gamma_{\rho_{\mathcal{R}}}( \Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}))^*\Gamma_{\rho_{\mathcal{R}}} (\Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}})) \Vert \\
&=\Vert \id - \Lambda(U_{\mathcal{S}}(s_0))^* \Lambda(U_{\mathcal{S}}(s_0))
- \epsilon(s_0)^* \epsilon (s_0) - \epsilon (s_0)^* \Lambda(U_{\mathcal{S}}(s_0))-
\Lambda(U_{\mathcal{S}}(s_0))^* \epsilon (s_0)\Vert \\
&\leq \Vert \id - \Lambda(U_{\mathcal{S}}(s_0))^* \Lambda(U_{\mathcal{S}}(s_0))\Vert
+ 2 \Vert \epsilon(s_0)\Vert.
\end{align*}
Similarly we obtain
\begin{align*}
\Vert \id - &\Gamma_{\rho_{\mathcal{R}}}( \Phi(U_{\mathcal{S}}(s_0)
\otimes \id_{\mathcal{R}}))\Gamma_{\rho_{\mathcal{R}}}(\Phi(U_{\mathcal{S}}(s_0) \otimes \id_{\mathcal{R}}))^* \Vert
\\
&\leq
\Vert \id - \Lambda(U_{\mathcal{S}}(s_0))\Lambda(U_{\mathcal{S}}(s_0))^*\Vert
+2 \Vert \epsilon(s_0)\Vert.
\end{align*}
Finally, one can choose $W_Z$ so as to maximize
$|\langle \phi_{RZ}| U_{\mathcal{R}}(s) \otimes W_Z |\phi_{RZ}\rangle|$,
which coincides with $F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(s) \rho_{R}U_{\mathcal{R}}(s)^*)$ due to
Uhlmann's theorem \cite{uhl1}, thereby completing the proof.
\end{proof}
We now provide a proof of Corollary \ref{cor:der}.
\begin{proof}
Adopting the shorthand $F \equiv F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{R}U_{\mathcal{R}}(l:s)^*)$, the equality (\ref{eq1})
replaces the inequality of Theorem \ref{th:main} by,
\begin{multline*}
\Vert U_{\mathcal{S}}(l:s)^* \Lambda(U_{\mathcal{S}}(l:s_0))
U_{\mathcal{S}}(l:s) - \Lambda(U_{\mathcal{S}}(l:s_0))\Vert
\leq
2 \Vert U_{\mathcal{S}}(l:s)- \id\Vert \Vert \epsilon(l:s_0) \Vert
\\
+
\bigl(\frac{1}{F^2}
-1\bigr)^{1/2}
\left(
\left(
\Vert \id_{\mathcal{S}} - \Lambda(U_{\mathcal{S}}(l:s_0))^* \Lambda(U_{\mathcal{S}}(l:s_0))\Vert
+ 2 \Vert \epsilon(l:s_0)\Vert\right)^{1/2}
\right.
\\
\left.
+
\left(
\Vert \id_{\mathcal{S}} - \Lambda(U_{\mathcal{S}}(l:s_0))\Lambda(U_{\mathcal{S}}(l:s_0))^*\Vert
+2 \Vert \epsilon(l:s_0)\Vert\right)^{1/2}
\right).
\end{multline*}
To bound the first term on the right hand side we write
\begin{eqnarray*}
U_{\mathcal{S}}(s) = \id_{\mathcal{S}} +i \int^s_0 dt U_{\mathcal{S}}(t) L_{\mathcal{S}} ,
\end{eqnarray*}
and therefore
\begin{eqnarray*}
\Vert U_{\mathcal{S}}(s) - \id_{\mathcal{S}}\Vert
\leq |s| \Vert L_{\mathcal{S}}\Vert.
\end{eqnarray*}
For the second term, we bound $F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{\mathcal{R}}U_{\mathcal{R}}(l:s)^*)$ by choosing a purification of $\rho_{\mathcal{R}}$ as $|\phi\rangle \in \mathcal{H}_{\mathcal{R}} \otimes \hi_Z$.
Then Uhlmann's theorem states that the fidelity is written as
\begin{eqnarray*}
F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{\mathcal{R}} U_{\mathcal{R}}(l:s)^*)
= \sup_{|\phi\rangle} |\langle \phi | e^{i L_{\mathcal{R}} s} \otimes \id_Z|\phi\rangle|.
\end{eqnarray*}
For each purification $|\phi\rangle$, the Mandelstam-Tamm uncertainty
relation \cite{mt1,bu1} provides a bound for $0 \leq \Delta_{\rho_{\mathcal{R}}} L_{\mathcal{R}} \cdot s\leq \pi/2$,
\begin{eqnarray*}
|\langle \phi | e^{i L_{\mathcal{R}} s} \otimes \id_Z |\phi \rangle |
\geq \cos (\Delta_{\rho_{\mathcal{R}}} L_{\mathcal{R}} \cdot s).
\end{eqnarray*}
Thus we obtain
\begin{eqnarray*}
\left(\frac{1}{F(\rho_{\mathcal{R}}, U_{\mathcal{R}}(l:s) \rho_{R}U_{\mathcal{R}}(l:s)^*)^2}
-1\right)^{1/2}
\leq \tan (\Delta_{\rho_{\mathcal{R}}} L_{\mathcal{R}} \cdot s).
\end{eqnarray*}
We divide the both terms by $|s|$ and take $|s|\to 0$ to obtain,
\begin{align*}
\Vert[ &\Lambda(U_{\mathcal{S}}(s_0)), L_{\mathcal{S}}] \Vert
\leq 2 \Vert L_{\mathcal{S}}\Vert \Vert \epsilon(s_0)\Vert
+ \Delta_{\rho_{\mathcal{R}}} L_{\mathcal{R}}
\biggl (\Vert \id- \Lambda(U_{\mathcal{S}}(s_0))^* \Lambda(U_{\mathcal{S}}(s_0))\Vert
+ 2 \Vert \epsilon (s_0)\Vert)^{1/2}\\
&+
(\Vert \id- \Lambda(U_{\mathcal{S}}(s_0)) \Lambda(U_{\mathcal{S}}(s_0))^*\Vert
+ 2 \Vert \epsilon (s_0)\Vert)^{1/2}
\biggr ).\qedhere
\end{align*}
\end{proof}
\section{Rotational symmetry}
As an example of the general behaviour we have investigated, we consider the possible dynamics
of a qubit with Hilbert space $\mathcal{H}_{\mathcal{S}} = \mathbb{C}^2$ under $SO(3)$ symmetry, realized by a true irreducible unitary representation of
its universal covering group $SU(2)$.
Since only a trivial unitary operator proportional to $\id$
commutes with all $SU(2)$ generators (angular momenta),
one cannot change the state of the qubit in isolation
(i.e., unitarily).
The environment $\mathcal{H}_{\mathcal{R}}$ also has $SU(2)$ as a symmetry.
We denote the angular momenta of the system and the reference frame
by $s_j$ and $S_j$ $(j =x,y,z)$ respectively.
We consider an $SU(2)$-covariant channel $\Phi: \mathbf{B}(\mathcal{H}_{\mathcal{S}} \otimes \mathcal{H}_{\mathcal{R}})
\to \mathbf{B}(\mathcal{H}_{\mathcal{S}} \otimes \mathcal{H}_{\mathcal{R}})$.
The following corollary is immediately obtained
from Corollary \ref{cor:der}.
\begin{corollary}
Let $G$ be a Lie group.
For a covariant channel $\Phi:
\mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}}\otimes \mathcal{H}_{\mathcal{R}})$, its restriction $R \equiv \Gamma_{\rho_{\mathcal{R}}}\circ \Phi \circ \iota:
\mathbf{B}(\mathcal{H}_{\mathcal{S}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$,
satisfies
\begin{equation*}
\Vert[ R (U_{\mathcal{S}}(s_0)),
L_{\mathcal{S}}]\Vert
\leq (\Delta_{\rho_{\mathcal{R}}} L_{\mathcal{R}})
\biggl(
\Vert \id - R
(U_{\mathcal{S}}(s_0))^*R (U_{\mathcal{S}}(s_0))\Vert^{1/2}
+
\Vert \id - R(U_{\mathcal{S}}(s_0))
R(U_{\mathcal{S}}(s_0))^*\Vert^{1/2}
\biggr).
\end{equation*}
\end{corollary}
We apply this to
the case $G= SU(2)$ and $\mathcal{H}_{\mathcal{S}} = \mathbb{C}^2$.
For $l = s_x$,
$U_{\mathcal{S}}(s)$ is written as $U_{\mathcal{S}}(s)= e^{i s_x s} = e^{i \frac{\sigma_x}{2} s}$.
We set $s_0= \pi$ to obtain $U_{\mathcal{S}}(s_0) = i \sigma_x$.
Then the
restriction to the system $\Lambda:=\Gamma_{\rho_{\mathcal{R}}}\circ \Phi \circ \iota :
\mathbf{B}(\mathcal{H}_{\mathcal{S}}) \to \mathbf{B}(\mathcal{H}_{\mathcal{S}})$ satisfies the following three
inequalities:
\begin{align*}
&\Vert[\Lambda(\sigma_x), s_x]\Vert
\leq 2 (\Delta_{\rho_{\mathcal{R}}}S_x) \Vert \id - \Lambda(\sigma_x)^2\Vert^{1/2}\\
&\Vert[\Lambda(\sigma_y), s_y]\Vert
\leq 2 (\Delta_{\rho_{\mathcal{R}}}S_y) \Vert \id - \Lambda(\sigma_y)^2\Vert^{1/2}\\
&\Vert[\Lambda(\sigma_z), s_z]\Vert
\leq 2 (\Delta_{\rho_{\mathcal{R}}}S_z) \Vert \id - \Lambda(\sigma_z)^2\Vert^{1/2},
\end{align*}
where $s_x=\frac{1}{2}\sigma_x$, {\it etc}.
The uncertainty relations for angular momenta gives a non-trivial bound
on sums of their fluctuations. We consider
\begin{eqnarray*}
(\Delta S_x)^2 + (\Delta S_y)^2 + (\Delta S_z)^2
&=&
\langle S_x^2 + S_y^2 + S_z^2\rangle
- (\langle S_x\rangle^2 + \langle S_y\rangle^2
+\langle S_z\rangle^2)
\\
&\leq& l(l+1) - (\langle S_x\rangle^2 + \langle S_y\rangle^2
+\langle S_z\rangle^2),
\end{eqnarray*}
where $l$ is the magnitude of the largest spin of the environment.
(Note that $\mathcal{H}_{\mathcal{R}}$ is written as a direct sum of
irreducible representations of $SU(2)$ as
$\mathcal{H}_{\mathcal{R}} = \oplus_s \mathbb{C}^{2s+1}$. $l$ is the largest value of $s$ in
the summation.)
It is easy to show that
$\langle S_x\rangle^2 + \langle S_y\rangle^2 + \langle S_z\rangle^2$
is rotationally invariant.
We consider the quantity $\langle \mathbf{S}\cdot \mathbf{n}\rangle$
for $|\mathbf{n}|=1$.
This is a smooth function over the sphere and therefore has
a maximum value at a certain point.
To estimate the value of $\langle S_x\rangle^2 + \langle S_y\rangle^2
+ \langle S_z\rangle^2$, we assume
that the maximum of $\langle \mathbf{S}\cdot \mathbf{n}\rangle$
is attained at $\mathbf{n} =\mathbf{e}_z$.
By differentiating in polar coordinates, one can conclude
that this state shows $\langle S_x\rangle = \langle S_y\rangle =0$.
Thus we have $\langle S_x\rangle^2 +
\langle S_y\rangle^2 + \langle S_z\rangle^2
= \langle S_z\rangle^2$.
Using $0 \leq \langle S_z\rangle^2 \leq l^2$,
we conclude that
\begin{eqnarray*}
l \leq (\Delta S_x)^2 + (\Delta S_y)^2 + (\Delta S_z)^2\leq l(l+1).
\end{eqnarray*}
Thus we obtain the bound
\begin{align*}
&\Vert [\Lambda(\sigma_x), \sigma_x]\Vert
+\Vert [\Lambda(\sigma_y), \sigma_y]\Vert
+ \Vert [\Lambda(\sigma_z), \sigma_z]\Vert\\
&\leq 2 \sqrt{l(l+1)}
(\Vert \id- \Lambda(\sigma_x)^2\Vert
+ \Vert \id- \Lambda( \sigma_y)^2\Vert
+\Vert \id - \Lambda(\sigma_z)^2\Vert)^{1/2},
\end{align*}
where we used the Cauchy-Schwarz inequality.
One can confirm,
as expected, that any realizable non-covariant channel
is inevitably dissipative,
as non-dissipative (=unitary) dynamics
satisfies $\id = \Lambda(\sigma_x)^2
=\Lambda(\sigma_y)^2
= \Lambda(\sigma_z)^2$.
The right-hand side can be regarded as
a quantity measuring the ``dissipativity" of $\Lambda$,
while the left-hand side represents the ``magnitude'' of
dynamics.
If the environment consists of $N$ qubits, as $l =\frac{N}{2}$ holds the
term $\sqrt{l(l+1)}$ in the
right-hand side of the above inequality is proportional to $N$.
Thus for $\Lambda$ whose magnitude of dynamics
is $O(1)$, its dissipativity cannot be smaller than
$O\left(\frac{1}{N}\right)$ in the presence of
$N$ environment qubits.
\par
We employ Stokes parameterization
\cite{HeinosaariZiman}
to illustrate possible channels.
Any qubit state is written as
$\rho = \frac{1}{2}(\id_{\mathcal{S}} + \mathbf{x}\cdot \mathbf{\sigma})$
with $|\mathbf{x}|\leq 1$.
$\Lambda^*$, the dual of $\Lambda$, maps
$\rho$ to another state $\rho' = \frac{1}{2}
(\id_{\mathcal{S}} + \mathbf{y} \cdot \mathbf{\sigma})$.
This map $(1,\mathbf{x}) \mapsto (1,\mathbf{y})$ is a linear map
on $\mathbf{R}^4$ since
$\Lambda$ is self-adjoint. We denote this map by $\ ^t\tilde{T}_{\Lambda}$ with
a parameterization,
\begin{eqnarray*}
\ ^t\tilde{T}_{\Lambda}= \left(
\begin{array}{cccc}
1&0&0&0\\
t_1& t_{11}& t_{12}& t_{13}\\
t_2& t_{21}& t_{22}& t_{23}\\
t_3& t_{31}& t_{32}& t_{33}
\end{array}
\right)
=\left(
\begin{array}{cc}
1& \mathbf{0} \\
\mathbf{t} & T
\end{array}
\right),
\end{eqnarray*}
where $T$ is a $3\times 3$ matrix.
Back in the Heisenberg picture,
we obtain
\begin{eqnarray*}
\Lambda( a_0 \id_{\mathcal{S}} + \mathbf{a} \cdot \mathbf{\sigma})
= (a_0 + \mathbf{t}\cdot \mathbf{a}) \id_{\mathcal{S}}
+ (T \mathbf{a})\cdot \mathbf{\sigma}.
\end{eqnarray*}
$T$ can be written as
\begin{eqnarray*}
T= R_1 D R_2,
\end{eqnarray*}
where $R_1$ and $R_2$ are elements of $SO(3)$ and
$D$ is a diagonal matrix as,
\begin{eqnarray*}
D= \left(
\begin{array}{ccc}
\lambda_1 &0&0\\
0&\lambda_2&0\\
0&0& \lambda_3
\end{array}
\right).
\end{eqnarray*}
One can choose the coordinate system so that
$R_2= \id$ is satisfied. Thus
we will consider $T$ with form $T= RD$.
Then we obtain, for redefined $\mathbf{t}$,
\begin{eqnarray*}
\Lambda( a_0 \id_{\mathcal{S}} + \mathbf{a}\cdot \mathbf{\sigma})
= (a_0 + \mathbf{t}\cdot \mathbf{a} )\id_{\mathcal{S}}
+ \sum_{ij=1}^3 R_{ij} \lambda_j a_j \sigma_i,
\end{eqnarray*}
where $R_{ij} \in SO(3)$.
Assume that $R$ is written as a rotation around the $z$-axis, with the vector ${\bf t}= {\bf 0}$, as,
\begin{eqnarray*}
R=\left(
\begin{array}{ccc}
\cos \theta & \sin \theta &0\\
-\sin \theta & \cos \theta &0 \\
0&0& 1 \\
\end{array}
\right).
\end{eqnarray*}
Then we have
\begin{eqnarray*}
\lambda_x |\sin \theta| \leq (\Delta_{\rho_{\mathcal{R}}} S_x) \sqrt{1 -\lambda_x^2}\\
\lambda_y |\sin \theta| \leq (\Delta_{\rho_{\mathcal{R}}} S_y) \sqrt{1 -\lambda_y^2}.
\end{eqnarray*}
That is, we have a relation between the dissipative and
symmetry breaking natures.
\begin{eqnarray*}
\lambda_x^2 \leq \frac{(\Delta_{\rho_{\mathcal{R}}} S_x)^2}
{(\Delta_{\rho_{\mathcal{R}}} S_x)^2 +(\sin \theta)^2};\\
\lambda_y^2 \leq \frac{(\Delta_{\rho_{\mathcal{R}}} S_y)^2}
{(\Delta_{\rho_{\mathcal{R}}} S_y)^2 +(\sin \theta)^2}.
\end{eqnarray*}
\section{Concluding remarks}
We have seen that there is a positive lower bound on the difference between an arbitrary quantum channel and the restriction of a covariant channel, and moreover, that in order to reduce this discrepancy a large spread in the generator of the symmetry is needed in the reference system. This result bears similarities with the WAY theorem, and is in line with the relational view of quantum mechanics, wherein we interpret non-symmetric channels as representatives of their symmetric counterparts of system and reference taken together. The large spread required for good approximation of relative (symmetric) by non-relative (asymmetric) can be understood as a condition on the quality of the reference frame, in the sense of the findings of \cite{lmb} and \cite{mlb}.
As a final remark, we mention that
there is yet another symmetry condition
on channels that differs from the one employed
in this paper and arises naturally in the context of quantum reference frames.
We will return to this issue elsewhere.
\section*{Acknowledgments}
TM acknowledges financial support from JSPS (KAKENHI Grant Number 20K03732).
\section*{References}
|
2,869,038,155,670 | arxiv | \section{Introduction}
In a seminal paper, \citet{ai94} proposed an interpretation of the
instrumental variables (IV) estimand as a Local Average Treatment
Effect (LATE) \textendash{} an average effect for a subpopulation
of ``compliers'' compelled to change treatment status by an external
instrument. The plausibility and transparency of the conditions underlying
this interpretation are often cited as an argument for preferring
IV estimators to nonlinear estimators based on parametric models \citep{angristpischke2009,angrist_pischke_jep}.
On the other hand, LATE itself has been criticized as difficult to
interpret, lacking in policy relevance, and problematic for generalization
\citep{heckman_jhr_1997,deaton_2009,heckman_urzua_2010}. Adherents
of this view favor estimators motivated by joint models of treatment
choice and outcomes with structural parameters defined independently
of the instrument at hand.
This note develops some connections between IV and structural estimators
intended to clarify how the choice of estimator affects the conclusions
researchers obtain in practice. Our first result is that, in the familiar
binary instrument/binary treatment setting with imperfect compliance,
a wide array of structural ``control function'' estimators derived
from parametric threshold-crossing models yield LATE estimates numerically
identical to IV. Notably, this equivalence applies to appropriately
parameterized variants of Heckman's (\citeyear{heckman_1976,heckman_1979})
classic two-step (``Heckit'') estimator that are nominally predicated
on bivariate normality. Differences between structural and IV estimates
therefore stem in canonical cases entirely from disagreements about
the target parameter rather than from functional form assumptions.
After considering how this result extends to settings with instruments
taking multiple values, we probe its limits by examining some estimation
strategies where equivalence fails. First, we revisit a control function
estimator considered by \citet{lalonde_1986} and show that it produces
results identical to IV only under a symmetry condition on the estimated
probability of treatment. Next, we study an estimator motivated by
a selection model that violates the monotonicity condition of \citet{ai94}
and establish that it yields a LATE estimate different from IV, despite
fitting the same sample moments. Standard methods of introducing observed
covariates also break the equivalence of control function and IV estimators,
but we discuss a reweighting approach that ensures equivalence is
restored. We then consider full information maximum likelihood (FIML)
estimation of some generalizations of the textbook bivariate probit
model and show that this yields LATE estimates that coincide with
IV at interior solutions. However, FIML diverges from IV when the
likelihood is maximized on the boundary of the structural parameter
space, which serves as the basis of recent proposals for testing instrument
validity in just-identified settings \citep{huber_mellace,kitagawa_2015}.
Finally, we discuss why estimation of over-identified models generally
yields LATE estimates different from IV.
The equivalence results developed here provide a natural benchmark
for assessing the credibility of structural estimators, which typically
employ a number of over-identifying restrictions in practice. As \citet{angrist_pischke_jep}
note: ``A good structural model might tell us something about economic
mechanisms as well as causal effects. But if the information about
mechanisms is to be worth anything, the structural estimates should
line up with those derived under weaker assumptions.'' Comparing
the model-based LATEs implied by structural estimators with unrestricted
IV estimates provides a transparent assessment of how conclusions
regarding a common set of behavioral parameters are influenced by
the choice of estimator. A parsimonious structural estimator that
rationalizes a variety of IV estimates may reasonably be deemed to
have survived a ``trial by fire,'' lending some credibility to its
predictions.
\section{Two views of LATE}
We begin with a review of the LATE concept and its link to IV estimation.
Let $Y_{i}$ represent an outcome of interest for individual $i$,
with potential values $Y_{i}(1)$ and $Y_{i}(0)$ indexed against
a binary treatment $D_{i}$. Similarly, let $D_{i}(1)$ and $D_{i}(0)$
denote potential values of the treatment indexed against a binary
instrument $Z_{i}$. Realized treatments and outcomes are linked to
their potential values by the relations $D_{i}=Z_{i}D_{i}(1)+\left(1-Z_{i}\right)D_{i}(0)$
and $Y_{i}=D_{i}Y_{i}(1)+\left(1-D_{i}\right)Y_{i}(0)$. \citet{ai94}
consider instrumental variables estimation under the following assumptions:
\begin{enumerate}
\item[IA.1] Independence/Exclusion: $(Y_{i}(1),Y_{i}(0),D_{i}(1),D_{i}(0))\perp \! \! \! \perp Z_{i}$.
\item[IA.2] First Stage: $Pr\left[D_{i}=1|Z_{i}=1\right]>Pr\left[D_{i}=1|Z_{i}=0\right]$.
\item[IA.3] Monotonicity: $Pr\left[D_{i}(1)\geq D_{i}(0)\right]=1$.
\end{enumerate}
Assumption IA.1 requires the instrument to be as good as randomly
assigned and to influence outcomes only through its effect on $D_{i}$.
Assumption IA.2 requires the instrument to increase the probability
of treatment, and assumption IA.3 requires the instrument to weakly
increase treatment for all individuals.
\citet{ai94} define LATE as the average treatment effect for ``compliers''
induced into treatment by the instrument (for whom $D_{i}(1)>D_{i}(0))$.
Assumptions IA.1-IA.3 imply that the population \citet{wald_1940}
ratio identifies LATE:
\begin{center}
$\dfrac{E\left[Y_{i}|Z_{i}=1\right]-E\left[Y_{i}|Z_{i}=0\right]}{E\left[D_{i}|Z_{i}=1\right]-E\left[D_{i}|Z_{i}=0\right]}=E\left[Y_{i}(1)-Y_{i}(0)|D_{i}(1)>D_{i}(0)\right]\equiv LATE.$
\par\end{center}
Suppose we have access to an $iid$ vector of sample realizations
$\left\{ Y_{i},D_{i},Z_{i}\right\} _{i=1}^{n}$ obeying the following
condition:
\begin{condition}
\noindent \textbf{$\tfrac{1}{\sum_{i}Z_{i}}\sum_{i}Z_{i}D_{i}>\tfrac{1}{\sum_{i}(1-Z_{i})}\sum_{i}(1-Z_{i})D_{i}$.
\label{assu:1}}
\end{condition}
\noindent When IA.2 is satisfied the probability of Condition 1 being
violated approaches zero at an exponential rate in $n$. The analogy
principle suggests estimating LATE with:
\noindent
\[
\widehat{LATE}^{IV}=\dfrac{\tfrac{1}{\sum_{i}Z_{i}}\sum_{i}Z_{i}Y_{i}-\tfrac{1}{\sum_{i}(1-Z_{i})}\sum_{i}(1-Z_{i})Y_{i}}{\tfrac{1}{\sum_{i}Z_{i}}\sum_{i}Z_{i}D_{i}-\tfrac{1}{\sum_{i}(1-Z_{i})}\sum_{i}(1-Z_{i})D_{i}}.
\]
\noindent This IV estimator is well-defined under Condition \ref{assu:1},
and is consistent for $LATE$ under assumptions IA.1-IA.3 and standard
regularity conditions.
\subsection*{Threshold-crossing representation}
\citet{vytlacil_2002} showed that the LATE model can be written as
a joint model of potential outcomes and self-selection in which treatment
is determined by a latent index crossing a threshold. Suppose treatment
status is generated by the equation
\begin{center}
$D_{i}=1\left\{ \psi(Z_{i})\geq V_{i}\right\} $,
\par\end{center}
\noindent where the latent variable $V_{i}$ is independently and
identically distributed according to some continuous distribution
with cumulative distribution function $F_{V}\left(.\right):\mathbb{R}\rightarrow[0,1]$,
and $\psi\left(.\right):\{0,1\}\rightarrow\mathbb{R}$ defines instrument-dependent
thresholds below which treatment ensues. Typically $F_{V}\left(.\right)$
is treated as a structural primitive describing a stable distribution
of latent costs and benefits influencing program participation that
exists independently of a particular instrument, as in the classic
selection models of \citet{roy_1951} and \citet{heckman_1974}. We
follow \citet{heckman_vytlacil_2005} and work with the equivalent
transformed model
\begin{equation}
D_{i}=1\left\{ P(Z_{i})\geq U_{i}\right\} ,\label{eq:choice}
\end{equation}
\noindent where $U_{i}\equiv F_{V}(V_{i})$ follows a uniform distribution
and $P(Z_{i})\equiv F_{V}(\psi(Z_{i}))$ is the propensity score.
The instrument $Z_{i}$ is presumed to increase the likelihood of
treatment ($P(1)>P(0)),$ and to be independent of $U_{i}$ and potential
outcomes:
\begin{equation}
(Y_{i}(1),Y_{i}(0),U_{i})\perp \! \! \! \perp Z_{i}.\label{eq:independence}
\end{equation}
The selection model defined by (\ref{eq:choice}) and (\ref{eq:independence})
is equivalent to the treatment effects model described by assumptions
IA.1-IA.3. Equation (\ref{eq:choice}) merely translates the behavioral
responses that are permitted in the LATE model into a partition of
the unit interval. In the terminology of \citet{air96}, assumption
IA.3 implies that the population consists of compliers with $D_{i}(1)>D_{i}(0)$,
``always takers'' with $D_{i}(1)=D_{i}(0)=1$, and ``never takers''
with $D_{i}(1)=D_{i}(0)=0$. The latent variable $U_{i}$ is defined
such that always takers have $U_{i}\in[0,P(0)]$, compliers have $U_{i}\in(P(0),P(1)]$,
and never takers have $U_{i}\in(P(1),1]$. Condition (\ref{eq:independence})
implies that potential outcomes and treatment choices are independent
of the instrument and imposes no further restrictions on the joint
distribution of these quantities. It follows that we can equivalently
define $LATE=E\left[Y_{i}(1)-Y_{i}(0)|P(0)<U_{i}\leq P(1)\right]$.
Though Vytlacil's \citeyearpar{vytlacil_2002} results establish equivalence
between a non-parametric latent index model and the LATE model, the
fully non-parametric model is typically not used for estimation. Rather,
to motivate alternatives to IV estimation, it is conventional to make
additional assumptions regarding the joint distribution of the latent
cost $U_{i}$ and the potential outcomes $\left(Y_{i}(1),Y_{i}(0)\right)$.
The goal of this note is to investigate the consequences of such assumptions
for empirical work.
\section{Control function estimation}
We begin by considering estimators predicated on the existence of
a parametric ``control function'' capturing the endogeneity in the
relationship between outcomes and treatment \citep{heckman_robb,blundell_matzkin,wooldridge_2015}.
The workhorse models in this literature obey the following semi-parametric
restriction:
\begin{equation}
E\left[Y_{i}(d)|U_{i}=u\right]=\alpha_{d}+\gamma_{d}\times\left(J(u)-\mu_{J}\right),\mbox{ \ensuremath{d\in\left\{ 0,1\right\} ,}}\ u\in(0,1),\label{eq:linear}
\end{equation}
\noindent where $J(\cdot):\left(0,1\right)\rightarrow\mathbb{R}$
is a strictly increasing continuous function and $\mu_{J}\equiv E\left[J(U_{i})\right]$.
\citet{lee_selection_1983} studied this dependence structure in the
context of classic ``one-sided'' selection problems where outcomes
are only observed when $D_{i}=1$. Setting $J(\cdot)$ equal to the
inverse normal CDF yields the canonical Heckman \citeyearpar{heckman_1976,heckman_1979}
sample selection (``Heckit'') model, while choosing $J(u)=u$ yields
the linear selection model studied by \citet{olsen_1980}, and choosing
the inverse logistic CDF for $J\left(.\right)$ yields the logit selection
model considered by \citet{mroz_1987}.
Subsequent work applies versions of (\ref{eq:linear}) to policy evaluation
by modeling program participation as a ``two-sided'' sample selection
problem with coefficients indexed by the treatment state $d$. For
example, \citet{bjorklund_moffitt} build on the Heckit framework
by assuming $J(\cdot)$ is the inverse normal CDF and allowing $\alpha_{1}\neq\alpha_{0}$,
$\gamma_{1}\neq\gamma_{0}$. Likewise, the linear estimator of \citet{brinch_etal}
is a two-sided variant of Olsen's \citeyearpar{olsen_1980} approach
that imposes an identity $J(\cdot)$ function with coefficients indexed
by $d$. Interestingly, Dubin and McFadden's \citeyearpar{dubin_mcfadden}
classic multinomial selection model collapses in the binary treatment
effects case to a two-sided version of Mroz's \citeyearpar{mroz_1987}
logit model.
Assumption (\ref{eq:linear}) nullifies Vytlacil's \citeyearpar{vytlacil_2002}
equivalence result by imposing restrictions on the relationships between
mean potential outcomes of subgroups that respond differently to the
instrument $Z_{i}$. Let $\mu_{dg}$ denote the mean of $Y_{i}(d)$
for group $g\in\{at,nt,c\}$, representing always takers, never takers
and compliers. For any strictly increasing $J(\cdot)$, equation (\ref{eq:linear})
implies $sgn(\mu_{dat}-\mu_{dc})=sgn(\mu_{dc}-\mu_{dnt})$ for $d\in\{0,1\}$.
In contrast, the nonparametric model defined by assumptions IA.1-IA.3
is compatible with any arrangement of differences in mean potential
outcomes for the three subgroups. We next consider whether these additional
restrictions are consequential for estimation of LATE.
\subsection*{LATE}
When non-compliance is ``two-sided'' so that $0<P(0)<P(1)<1$, equation
(\ref{eq:linear}) implies that mean outcomes conditional on treatment
status are
\begin{center}
$E\left[Y_{i}|Z_{i},D_{i}=d\right]=\alpha_{d}+\gamma_{d}\lambda_{d}\left(P(Z_{i})\right)$,
\par\end{center}
\noindent where $\lambda_{1}(\cdot):\left(0,1\right)\rightarrow\mathbb{R}$
and $\lambda_{0}(\cdot):\left(0,1\right)\rightarrow\mathbb{R}$ are
control functions giving the means of $\left(J\left(U_{i}\right)-\mu_{J}\right)$
when $U_{i}$ is truncated from above and below at $p\in\left(0,1\right)$:
\begin{center}
$\lambda_{1}(p)=E\left[J(U_{i})-\mu_{J}|U_{i}\leq p\right],\ \lambda_{0}(p)=E\left[J(U_{i})-\mu_{J}|U_{i}>p\right]$.
\par\end{center}
\noindent \begin{flushleft}
While attention in parametric selection models often focuses on the
population average treatment effect $\alpha_{1}-\alpha_{0}$ \citep{garen_84,heckman_varieties,wooldridge_2015},
equation (\ref{eq:linear}) can also be used to compute treatment
effects for other subgroups. The average effect on compliers can be
written
\begin{equation}
LATE=\alpha_{1}-\alpha_{0}+\left(\gamma_{1}-\gamma_{0}\right)\Gamma\left(P(0),P(1)\right),\label{eq:complier_lambda}
\end{equation}
\par\end{flushleft}
\noindent where $\Gamma(p,p^{\prime})$ gives the mean of $J(U_{i})-\mu_{J}$
when $U_{i}$ lies between $p$ and $p^{\prime}>p$:
\begin{center}
$\Gamma(p,p^{\prime})=E\left[J(U_{i})-\mu_{J}|p<U_{i}\leq p^{\prime}\right]=\dfrac{p^{\prime}\lambda_{1}(p^{\prime})-p\lambda_{1}(p)}{p^{\prime}-p}$.
\par\end{center}
\noindent \begin{flushleft}
The last term in (\ref{eq:complier_lambda}) adjusts the average treatment
effect to account for non-random selection into compliance with the
instrument.
\par\end{flushleft}
\subsection*{Estimation}
To motivate control function estimation, suppose that the sample exhibits
two-sided non-compliance as follows:
\begin{condition}
\noindent $0<\sum_{i}1\{D_{i}=d\}Z_{i}<\sum_{i}1\left\{ D_{i}=d\right\} $
for $d\in\{0,1\}$.\label{assu:2}
\end{condition}
\noindent \begin{flushleft}
This condition requires at least one observation with every combination
of $Z_{i}$ and $D_{i}$. Condition \ref{assu:2} is satisfied with
probability approaching one at an exponential rate in $n$ whenever
$0<Pr[Z_{i}=1]<1$ and $0<P(z)<1$ for $z\in\{0,1\}$.
\par\end{flushleft}
Control function estimation typically proceeds in two steps, both
for computational reasons and because of the conceptual clarity of
plug-in estimation strategies \citep{heckman_1979,smith_blundell}.
Deferring a discussion of one-step estimation approaches to later
sections, we define the control function estimator as a procedure
which first fits the choice model in equation (\ref{eq:choice}) by
maximum likelihood, then builds estimates of $\lambda_{1}(\cdot)$
and $\lambda_{0}(\cdot)$ to include in second-step ordinary least
squares (OLS) regressions for each treatment category. The first step
estimates can be written
\begin{equation}
\left(\hat{P}(0),\hat{P}(1)\right)={\displaystyle \arg\max_{P(0),P(1)}\sum_{i}D_{i}\log P(Z_{i})+\sum_{i}(1-D_{i})\log\left(1-P(Z_{i})\right)}.\label{eq:mle}
\end{equation}
\noindent The second step OLS estimates are
\begin{equation}
\left(\hat{\alpha}_{d},\hat{\gamma}_{d}\right)={\displaystyle \arg\min_{\alpha_{d},\gamma_{d}}}\sum_{i}1\left\{ D_{i}=d\right\} \left[Y_{i}-\alpha_{d}-\gamma_{d}\lambda_{d}(\hat{P}(Z_{i}))\right]^{2},\ d\in\{0,1\}.\label{eq:second_step}
\end{equation}
The analogy principle then suggests the following plug-in estimator
of LATE:
\begin{center}
$\widehat{LATE}^{CF}=\left(\hat{\alpha}_{1}-\hat{\alpha}_{0}\right)+\left(\hat{\gamma}_{1}-\hat{\gamma}_{0}\right)\Gamma(\hat{P}(0),\hat{P}(1))$.
\par\end{center}
Note that when non-compliance is ``one-sided'' so that $\sum_{i}D_{i}(1-Z_{i})=0$
or $\sum_{i}(1-D_{i})Z_{i}=0$, the maximum likelihood estimates in
(\ref{eq:mle}) are not well-defined. Condition \ref{assu:2} ensures
that $\hat{P}(0)$ and $\hat{P}(1)$ exist, and that $\hat{\alpha}_{d}$
and $\hat{\gamma}_{d}$ can be computed for each value of $d$. Condition
\ref{assu:1} additionally ensures that $\hat{P}(0)<\hat{P}(1)$,
guaranteeing that $\widehat{LATE}^{CF}$ exists.
\section{\label{sec:Equivalence}Equivalence results}
Compared to $\widehat{LATE}^{IV}$, $\widehat{LATE}^{CF}$ would seem
to be highly dependent upon the functional form assumed for $J(\cdot)$
and the linearity of equation (\ref{eq:linear}). Our first result
shows that this is not the case.
\begin{thm}
\noindent If Conditions \ref{assu:1} and \ref{assu:2} hold then
$\widehat{LATE}^{CF}=\widehat{LATE}^{IV}$.\label{thm:LATE_equivalence}
\end{thm}
\noindent \textbf{Proof:} The maximum likelihood procedure in (\ref{eq:mle})
yields the empirical treatment rates $\hat{P}(z)=\tfrac{\sum_{i}1\left\{ Z_{i}=z\right\} D_{i}}{\sum_{i}1\{Z_{i}=z\}}$
for $z\in\{0,1\}$. The second-step OLS regressions can be rewritten
\noindent \begin{center}
$\left(\hat{\alpha}_{d},\hat{\gamma}_{d}\right)={\displaystyle \arg\min_{\alpha_{d},\gamma_{d}}}\sum_{i}1\left\{ D_{i}=d\right\} \left(Y_{i}-\left[\alpha_{d}+\gamma_{d}\lambda_{d}(\hat{P}(0))\right]-\gamma_{d}\left[\lambda_{d}(\hat{P}(1))-\lambda_{d}(\hat{P}(0))\right]Z_{i}\right)^{2}$.
\par\end{center}
\noindent This is a least squares fit of $Y_{i}$ on an intercept
and the indicator $Z_{i}$ in the subsample with $D_{i}=d$. Such
regressions can be estimated as long as there is two-sided non-compliance
with the instrument $Z_{i}$, which follows from Condition \ref{assu:2}.
Defining $\bar{Y}_{d}^{z}\equiv\tfrac{\sum_{i}1\left\{ D_{i}=d\right\} 1\left\{ Z_{i}=z\right\} Y_{i}}{\sum_{i}1\left\{ D_{i}=d\right\} 1\left\{ Z_{i}=z\right\} }$,
we have
\begin{center}
$\bar{Y}_{d}^{0}=\hat{\alpha}_{d}+\hat{\gamma}_{d}\lambda_{d}(\hat{P}(0))$,
$\bar{Y}_{d}^{1}-\bar{Y}_{d}^{0}=\hat{\gamma}_{d}\left[\lambda_{d}(\hat{P}(1))-\lambda_{d}(\hat{P}(0))\right]$.
\par\end{center}
\noindent Under Condition \ref{assu:1}, we have $\lambda_{d}(\hat{P}(1))\neq\lambda_{d}(\hat{P}(0))$,
and this pair of equations can be solved for $\hat{\gamma}_{d}$ and
$\hat{\alpha}_{d}$ as
\begin{center}
$\hat{\gamma}_{d}=\tfrac{\bar{Y}_{d}^{1}-\bar{Y}_{d}^{0}}{\lambda_{d}(\hat{P}(1))-\lambda_{d}(\hat{P}(0))},\ \hat{\alpha}_{d}=\tfrac{\lambda_{d}(\hat{P}(1))\bar{Y}_{d}^{0}-\lambda_{d}(\hat{P}(0))\bar{Y}_{d}^{1}}{\lambda_{d}(\hat{P}(1))-\lambda_{d}(\hat{P}(0))}$.
\par\end{center}
We can therefore rewrite the control function estimate of LATE as
\begin{center}
$\widehat{LATE}^{CF}=\left(\left[\tfrac{\lambda_{1}(\hat{P}(1))\bar{Y}_{1}^{0}-\lambda_{1}(\hat{P}(0))\bar{Y}_{1}^{1}}{\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))}\right]-\left[\tfrac{\lambda_{0}(\hat{P}(1))\bar{Y}_{0}^{0}-\lambda_{0}(\hat{P}(0))\bar{Y}_{0}^{1}}{\lambda_{0}(\hat{P}(1))-\lambda_{0}(\hat{P}(0))}\right]\right)$
\par\end{center}
\begin{center}
$+\left(\left[\tfrac{\bar{Y}_{1}^{1}-\bar{Y}_{1}^{0}}{\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))}\right]-\left[\tfrac{\bar{Y}_{0}^{1}-\bar{Y}_{0}^{0}}{\lambda_{0}(\hat{P}(1))-\lambda_{0}(\hat{P}(0))}\right]\right)\times\left(\frac{\hat{P}(1)\lambda_{1}(\hat{P}(1))-\hat{P}(0)\lambda_{1}(\hat{P}(0))}{\hat{P}(1)-\hat{P}(0)}\right)$.
\par\end{center}
\noindent Using the fact that $\lambda_{0}(p)=-\lambda_{1}(p)p/(1-p)$,
this simplifies to
\begin{center}
$\widehat{LATE}^{CF}=\dfrac{\left[\hat{P}(1)\bar{Y}_{1}^{1}+(1-\hat{P}(1))\bar{Y}_{0}^{1}\right]-\left[\hat{P}(0)\bar{Y}_{1}^{0}+(1-\hat{P}(0))\bar{Y}_{0}^{0}\right]}{\hat{P}(1)-\hat{P}(0)}$,
\par\end{center}
\noindent which is $\widehat{LATE}^{IV}$. $\blacksquare$
\begin{rem}
An immediate consequence of Theorem \ref{thm:LATE_equivalence} is
that $\widehat{LATE}^{CF}$ is also equivalent to the coefficient
on $D_{i}$ associated with a least squares fit of $Y_{i}$ to $D_{i}$
and a first stage residual $D_{i}-\hat{P}(Z_{i})$. \citet{blundell_matzkin}
attribute the first proof of the equivalence between this estimator
and IV to \citet{telser_1964}.
\end{rem}
\begin{rem}
Theorem \ref{thm:LATE_equivalence} extends the analysis of \citet{brinch_etal}
who observe that linear control function estimators produce LATE estimates
numerically equivalent to IV. The above result implies that a wide
class of non-linear control function estimators share this property.
With a binary treatment and instrument, an instrumental variables
estimate can always be viewed as the numerical output of a variety
of parametric control function estimators.
\end{rem}
\subsection*{Potential outcome means}
Corresponding equivalence results hold for estimators of other parameters
identified in the LATE framework. \citet{imbens_rubin_97} and \citet{abadie_2002}
discuss identification and estimation of the treated outcome distribution
for always takers, the untreated distribution for never takers, and
both marginal distributions for compliers. Nonparametric estimators
of the four identified marginal mean potential outcomes are given
by
\begin{center}
$\hat{\mu}_{1at}^{IV}=\bar{Y}_{1}^{0}$, $\hat{\mu}_{0nt}^{IV}=\bar{Y}_{0}^{1}$,
\par\end{center}
\begin{center}
$\hat{\mu}_{1c}^{IV}=\tfrac{\hat{P}(1)\bar{Y}_{1}^{1}-\hat{P}(0)\bar{Y}_{1}^{0}}{\hat{P}(1)-\hat{P}(0)}$,
$\hat{\mu}_{0c}^{IV}=\tfrac{(1-\hat{P}(0))\bar{Y}_{0}^{0}-(1-\hat{P}(1))\bar{Y}_{0}^{1}}{\hat{P}(1)-\hat{P}(0)}$.
\par\end{center}
The corresponding control function estimators are:
\begin{center}
$\hat{\mu}_{1at}^{CF}=\hat{\alpha}_{1}+\hat{\gamma}_{1}\lambda_{1}(\hat{P}(0))$,
$\hat{\mu}_{0nt}^{CF}=\hat{\alpha}_{0}+\hat{\gamma}_{0}\lambda_{0}(\hat{P}(1))$,
\par\end{center}
\begin{center}
$\hat{\mu}_{dc}^{CF}=\hat{\alpha}_{d}+\hat{\gamma}_{d}\Gamma(\hat{P}(0),\hat{P}(1)),\ d\in\{0,1\}$.
\par\end{center}
\noindent The following proposition shows that these two estimation
strategies produce algebraically identical results.
\begin{prop}
\noindent If Conditions \ref{assu:1} and \ref{assu:2} hold then\label{prop:po_means}
\end{prop}
\begin{center}
$\hat{\mu}_{dg}^{CF}=\hat{\mu}_{dg}^{IV}$ for $(d,g)\in\{(1,at),(0,nt),(1,c),(0,c)\}$.
\par\end{center}
\noindent \textbf{Proof:} Using the formulas from the proof of Theorem
\ref{thm:LATE_equivalence}, the control function estimate of $\mu_{1at}$
is
\begin{center}
$\hat{\mu}_{1at}^{CF}=\left(\tfrac{\lambda_{1}(\hat{P}(1))\bar{Y}_{1}^{0}-\lambda_{1}(\hat{P}(0))\bar{Y}_{1}^{1}}{\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))}\right)+\left(\tfrac{\bar{Y}_{1}^{1}-\bar{Y}_{1}^{0}}{\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))}\right)\lambda_{1}(\hat{P}(0))=\bar{Y}_{1}^{0}$,
\par\end{center}
\noindent which is $\hat{\mu}_{1at}^{IV}$. Likewise,
\begin{center}
$\hat{\mu}_{0nt}^{CF}=\left(\tfrac{\lambda_{0}(\hat{P}(1))\bar{Y}_{0}^{0}-\lambda_{0}(\hat{P}(0))\bar{Y}_{1}^{1}}{\lambda_{0}(\hat{P}(1))-\lambda_{0}(\hat{P}(0))}\right)+\left(\tfrac{\bar{Y}_{0}^{1}-\bar{Y}_{0}^{0}}{\lambda_{0}(\hat{P}(1))-\lambda_{0}(\hat{P}(0))}\right)\lambda_{0}(\hat{P}(1))=\bar{Y}_{0}^{1}$
,
\par\end{center}
\noindent which is $\hat{\mu}_{0nt}^{IV}$. The treated complier mean
estimate is
\begin{center}
$\hat{\mu}_{1c}^{CF}=\left(\tfrac{\lambda_{1}(\hat{P}(1))\bar{Y}_{1}^{0}-\lambda_{1}(\hat{P}(0))\bar{Y}_{1}^{1}}{\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))}\right)+\left(\tfrac{\bar{Y}_{1}^{1}-\bar{Y}_{1}^{0}}{\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))}\right)\times\left(\frac{\hat{P}(1)\lambda_{1}(\hat{P}(1))-\hat{P}(0)\lambda_{1}(\hat{P}(0))}{\hat{P}(1)-\hat{P}(0)}\right)$
\par\end{center}
\begin{center}
$=\tfrac{\left(\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))\right)\hat{P}(1)\bar{Y}_{1}^{1}-\left(\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))\right)\hat{P}(0)\bar{Y}_{1}^{0}}{\left(\lambda_{1}(\hat{P}(1))-\lambda_{1}(\hat{P}(0))\right)\left(\hat{P}(1)-\hat{P}(0)\right)}=\tfrac{\hat{P}(1)\bar{Y}_{1}^{1}-\hat{P}(0)\bar{Y}_{1}^{0}}{\hat{P}(1)-\hat{P}(0)}$,
\par\end{center}
\noindent which is $\hat{\mu}_{1c}^{IV}$. Noting that $\widehat{LATE}^{IV}=\hat{\mu}_{1c}^{IV}-\hat{\mu}_{0c}^{IV}$
and $\widehat{LATE}^{CF}=\hat{\mu}_{1c}^{CF}-\hat{\mu}_{0c}^{CF}$,
it then follows by Theorem \ref{thm:LATE_equivalence} that $\hat{\mu}_{0c}^{CF}=\hat{\mu}_{0c}^{IV}$.
$\blacksquare$
\section{Equivalence and extrapolation}
Proposition \ref{prop:po_means} establishes that all control function
estimators based on equation (\ref{eq:linear}) produce identical
estimates of the potential outcome means that are nonparametrically
identified in the LATE framework. Different functional form assumptions
generate different estimates of quantities that are under-identified,
however. For example, the choice of $J(\cdot)$ in equation (\ref{eq:linear})
determines the shapes of the curves that the model uses to extrapolate
from estimates of the four identified potential outcome means $(\mu_{1at},\mu_{0nt},\mu_{1c},\mu_{0c})$
to the two under-identified potential outcome means $(\mu_{0at},\mu_{1nt})$.
\begin{center}
\includegraphics[scale=1.02]{fig1}
\par\end{center}
Figures 1 and 2 illustrate this extrapolation in a hypothetical example.
The horizontal axis plots values $u$ of the unobserved treatment
cost $U_{i}$, while the vertical axis plots mean potential outcomes
$m_{d}(u)=E\left[Y_{i}(d)|U_{i}=u\right]$ as functions of this cost.
Estimates of these functions are denoted $\hat{m}_{d}(u)=\hat{\alpha}_{d}+\hat{\gamma}_{d}\times(J(u)-\mu_{J})$
and their difference $\hat{m}_{1}\left(u\right)-\hat{m}_{0}\left(u\right)$
provides an estimate of the marginal treatment effect \citep{bjorklund_moffitt,heckman_vytlacil_2005,heckman_essential_heterogeneity}
for an individual with latent cost $u$.
Assumptions IA.1-IA3 ensure two averages of $m_{d}\left(U_{i}\right)$
are identified for each potential outcome: the treated means for always
takers and compliers, and the untreated means for never takers and
compliers. The control function estimator chooses $\hat{\alpha}_{d}$
and $\hat{\gamma}_{d}$ so that averages of $\hat{m}_{d}(U_{i})$
over the relevant ranges match the corresponding nonparametric estimates
for each compliance group. The coefficient $\hat{\gamma}_{1}$ parameterizes
the difference in mean treated outcomes between compliers and always
takers, while $\hat{\gamma}_{0}$ measures the difference in mean
untreated outcomes between compliers and never takers. Several tests
of endogeneous treatment assignment (see, e.g., \citealp{angrist_2004_tehet,battistin_rettore,berentha_imbens};
and \citealp{kowalski_2016}) amount to testing whether $\left(\hat{\gamma}_{0},\hat{\gamma}_{1}\right)$
are significantly different from zero.
\begin{center}
\includegraphics[scale=0.98]{fig2}
\par\end{center}
Figure 1 depicts the results of parametric extrapolation based on
the Heckit model, while Figure 2 shows results for the linear control
function model discussed by \citet{brinch_etal}. Both models match
the same four estimated mean potential outcomes, thereby generating
identical estimates of LATE. Note that by Jensen's inequality, the
nonlinear $\hat{m}_{d}(u)$ curves in Figure 1 do not pass directly
through the group mean potential outcomes. The two models yield different
imputations for the missing potential outcomes of always takers and
never takers, and therefore also different estimates of the ATE, which
averages over all three subpopulations. This sensitivity to functional
form is intuitive: treatment effects for always and never takers are
fundamentally under-identified, an insight that has led to consideration
of bounds on these quantities \citep{manski_1990,balke_pearle,mst}.
\section{Multi-valued instruments}
Consider an instrument $Z_{i}$ taking values in $\{0,1,..,K\},$
and suppose that $0<\hat{P}\left(z-1\right)<\hat{P}\left(z\right)<1$
for $z\in\left\{ 1,2,...,K\right\} $. Let $D_{i}(z)$ denote $i$'s
treatment choice when $Z_{i}=z$. If assumptions IA.1-IA.3 hold for
every pair of instrument values, Wald ratios of the form $\frac{E\left[Y_{i}|Z_{i}=z\right]-E\left[Y_{i}|Z_{i}=z-1\right]}{E\left[D_{i}|Z_{i}=z\right]-E\left[D_{i}|Z_{i}=z-1\right]}$
identify the average treatment effect among compliers indexed by a
unit increment in the instrument, which we denote $LATE_{z}\equiv E\left[Y_{i}(1)-Y_{i}(0)|D_{i}(z)>D_{i}\left(z-1\right)\right]$.
Analog estimators of $LATE_{z}$ are given by the following pairwise
IV estimator:
\[
\widehat{LATE}_{z}^{IV}=\dfrac{\tfrac{1}{\sum_{i}1\{Z_{i}=z\}}\sum_{i}1\{Z_{i}=z\}Y_{i}-\tfrac{1}{\sum_{i}1\{Z_{i}=z-1\}}\sum_{i}1\{Z_{i}=z-1\}Y_{i}}{\tfrac{1}{\sum_{i}1\{Z_{i}=z\}}\sum_{i}1\{Z_{i}=z\}D_{i}-\tfrac{1}{\sum_{i}1\{Z_{i}=z-1\}}\sum_{i}1\{Z_{i}=z-1\}D_{i}}.
\]
From Theorem \ref{thm:LATE_equivalence}, $\widehat{LATE}_{z}^{IV}$
is numerically equivalent to the corresponding pairwise control function
estimator of $LATE_{z}$ constructed from observations with $Z_{i}\in\{z-1,z\}$.
However, to improve precision, it is common to impose additional restrictions
on the $LATE_{z}$.
Consider the following restriction on potential outcomes:
\begin{equation}
E\left[Y_{i}(d)|U_{i}\right]=\alpha_{d}+{\displaystyle \sum_{\ell=1}^{L}\gamma_{d\ell}\times\left(J(U_{i})-\mu_{J}\right)^{\ell}},\ d\in\{0,1\}.\label{eq:multi_z-1}
\end{equation}
\noindent Polynomial models of this sort have been considered by,
among others, \citet{brinch_etal} and \citet{dustmann_reverse_roy}.
Letting $\lambda_{1\ell}(p)=E\left[(J(U_{i})-\mu_{J})^{\ell}|U_{i}\leq p\right]$
and $\lambda_{0\ell}(p)=E\left[(J(U_{i})-\mu_{J})^{\ell}|U_{i}>p\right]$,
a two-step control function estimator of the parameters of equation
(\ref{eq:multi_z-1}) is
\noindent \begin{center}
$\left(\hat{\alpha}_{d},\hat{\gamma}_{d1},...,\hat{\gamma}_{dL}\right)={\displaystyle \arg\min_{\alpha_{d},\gamma_{d1},...,\gamma_{dL}}}\sum_{i}1\left\{ D_{i}=d\right\} \left[Y_{i}-\alpha_{d}-\sum_{\ell=1}^{L}\gamma_{d\ell}\lambda_{d\ell}(\hat{P}(Z_{i}))\right]^{2}$.
\par\end{center}
\noindent The resulting control function estimator of $LATE_{z}$
is then
\begin{equation}
\widehat{LATE}_{z}^{CF}=(\hat{\alpha}_{1}-\hat{\alpha}_{0})+\sum_{\ell=1}^{L}(\hat{\gamma}_{1\ell}-\hat{\gamma}_{0\ell})\Gamma_{\ell}(\hat{P}(z-1),\hat{P}(z)),\label{eq:multi_z_cf}
\end{equation}
where $\Gamma_{\ell}(p,p^{\prime})=[p^{\prime}\lambda_{1\ell}(p^{\prime})-p\lambda_{1\ell}(p)]/[p^{\prime}-p].$
The following proposition establishes that this estimator is identical
to $\widehat{LATE}_{z}^{IV}$ when $L=K$.
\begin{prop}
\noindent If Conditions 1 and 2 hold for every pair of instrument
values and the polynomial order $L$ equals $K$ then $\widehat{LATE}_{z}^{CF}=\widehat{LATE}_{z}^{IV}\ \forall z\in\{1,2,...,K\}$.\label{prop:multi_z_equivalence}
\end{prop}
\noindent \textbf{Proof:} See the Appendix. $\blacksquare$
\begin{rem}
\noindent Instrumenting $D_{i}$ with a scalar function $g(Z_{i})$
generates an IV estimate equal to a convex weighted average of the
$\widehat{LATE}_{z}^{IV}$ \citep{ai94}. From Proposition \ref{prop:multi_z_equivalence},
applying these weights to the $\widehat{LATE}_{z}^{CF}$ when $L=K$
will yield an identical result. By contrast, the set of $\widehat{LATE}_{z}^{CF}$
that result from imposing $L<K$ need not correspond to weighted averages
of the $\widehat{LATE}_{z}^{IV}$, but are likely to exhibit reduced
sampling variability.
\end{rem}
\begin{rem}
When $L<K-1$, the restriction in (\ref{eq:multi_z-1}) can be used
to motivate estimators of particular LATEs that are convex combinations
of IV estimators. In the case where $K=3$ and $L=1$, one can show
that:
\[
LATE_{2}=\left(\tfrac{\Gamma(P(1),P(2))-\Gamma(P(0),P(1))}{\Gamma(P(2),P(3))-\Gamma(P(0),P(1))}\right)LATE_{3}+\left(\tfrac{\Gamma(P(2),P(3))-\Gamma(P(1),P(2))}{\Gamma(P(2),P(3))-\Gamma(P(0),P(1))}\right)LATE_{1}.
\]
\end{rem}
\noindent This representation suggests combination estimators of the
form
\[
\widehat{LATE}{}_{2}^{\xi}=\xi\widehat{LATE}_{2}^{IV}+\left(1-\xi\right)\left[\left(\tfrac{\Gamma(\hat{P}(1),\hat{P}(2))-\Gamma(\hat{P}(0),\hat{P}(1))}{\Gamma(\hat{P}(2),\hat{P}(3))-\Gamma(\hat{P}(0),\hat{P}(1))}\right)\widehat{LATE}_{3}^{IV}+\left(\tfrac{\Gamma(\hat{P}(2),\hat{P}(3))-\Gamma(\hat{P}(1),\hat{P}(2))}{\Gamma(\hat{P}(2),\hat{P}(3))-\Gamma(\hat{P}(0),\hat{P}(1))}\right)\widehat{LATE}_{1}^{IV}\right],
\]
for $\xi\in(0,1)$. To maximize precision, one can set $\xi=[\hat{v}_{2}-\hat{v}_{12}]/[\hat{v}_{1}+\hat{v}_{2}-2\hat{v}_{12}]$
, where $\hat{v}_{1}$ and $\hat{v}_{2}$ are estimated variances
of $\widehat{LATE}_{2}^{IV}$ and the term in brackets, respectively,
and $\hat{v}_{12}$ is their covariance. By construction, $\widehat{LATE}{}_{2}^{\xi}$
provides an estimate of $LATE_{2}$ more precise than $\widehat{LATE}_{2}^{IV}$.
Though $\widehat{LATE}_{2}^{\xi}$ will tend to be less precise than
$\widehat{LATE}_{2}^{CF}$ when restriction (\ref{eq:multi_z-1})
is true, the probability limit of $\widehat{LATE}_{2}^{\xi}$ retains
an interpretation as a weighted average of causal effects for complier
subpopulations when (\ref{eq:multi_z-1}) is violated, a robustness
property emphasized elsewhere by \citet{angristpischke2009}.
\section{Equivalence failures}
Though Theorem \ref{thm:LATE_equivalence} establishes equivalence
between IV and a wide class of control function estimates of LATE,
other control function estimators fail to match IV even with a single
binary instrument. \citet{lalonde_1986} considered OLS estimation
of the following model:
\begin{equation}
Y_{i}=\alpha+\beta D_{i}+\gamma\left[D_{i}\times\left(-\tfrac{\phi(\Phi^{-1}(\hat{P}(Z_{i})))}{\hat{P}(Z_{i})}\right)+(1-D_{i})\times\left(\tfrac{\phi(\Phi^{-1}(\hat{P}(Z_{i})))}{1-\hat{P}(Z_{i})}\right)\right]+\epsilon_{i}.\label{eq:lalonde_cf}
\end{equation}
By imposing a common coefficient $\gamma$ on the Mills ratio terms
for the treatment and control groups, this specification allows for
selection on levels but rules out selection on treatment effects.
The term in brackets in equation (\ref{eq:lalonde_cf}) simplifies
to $(D_{i}-\hat{P}(Z_{i}))\times\{-\phi(\Phi^{-1}(\hat{P}(Z_{i})))/[\hat{P}(Z_{i})(1-\hat{P}(Z_{i}))]\}$.
When $\hat{P}(1)=1-\hat{P}(0)$ this term is proportional to the first
stage residual and least squares estimation of (\ref{eq:lalonde_cf})
yields an estimate of $\beta$ numerically identical to IV. This is
a finite sample analogue of Heckman and Vytlacil's \citeyearpar{heckman_vytlacil_nber2000}
observation (elaborated upon in \citealp{angrist_2004_tehet}) that
LATE equals ATE when both the first stage and the error distribution
are symmetric. When $\hat{P}(1)\neq1-\hat{P}(0)$, however, the control
function in equation (\ref{eq:lalonde_cf}) differs from the first
stage residual and the estimate of $\beta$ will not match IV.
\begin{rem}
\noindent When $\hat{P}(1)=1-\hat{P}(0)$, the ATE estimate $\hat{\alpha}_{1}-\hat{\alpha}_{0}$
from a control function estimator of the form given in (\ref{eq:second_step})
coincides with IV whenever $J(U_{i})$ is presumed to follow a symmetric
distribution.
\end{rem}
\subsection*{Moments and monotonicity}
Theorem \ref{thm:LATE_equivalence} relied upon the fact that equation
(\ref{eq:linear}) includes enough free parameters to allow the control
function estimator to match the sample mean of $Y_{i}$ for every
combination of $D_{i}$ and $Z_{i}$. One might be tempted to conclude
that any structural estimator that fits these moments will produce
a corresponding LATE estimate equal to IV. We now show that this is
not the case.
Suppose that treatment status is generated by a heterogeneous threshold
crossing model:
\begin{equation}
D_{i}=1\left\{ \kappa+\delta_{i}Z_{i}\geq U_{i}\right\} ,\label{eq:humpty}
\end{equation}
\noindent where $U_{i}$ is uniformly distributed and the random coefficient
$\delta_{i}$ is a mixture taking values in $\{-\eta,\eta\}$ for
some known positive constant $\eta$. Define $\upsilon\equiv Pr\left[\delta_{i}=\eta\right]$,
and suppose that $\delta_{i}$ is independent of $(Y_{i}(1),Y_{i}(0),U_{i},Z_{i})$.
Note that this model does not admit a representation of the form of
equation (\ref{eq:choice}) as it allows $D_{i}(1)<D_{i}(0)$.
Model (\ref{eq:humpty}) has two unknown parameters, $\kappa$ and
$\upsilon$, and can therefore rationalize the two observed choice
probabilities by choosing $\hat{\kappa}=\hat{P}(0)$ and $\hat{\upsilon}=(\eta+\hat{P}(1)-\hat{P}(0))/2\eta$.
Equations (\ref{eq:linear}) and (\ref{eq:humpty}) imply
\begin{center}
$E\left[Y_{i}|D_{i}=d,Z_{i}\right]=\alpha_{d}+\gamma_{d}\times\left[\upsilon\lambda_{d}\left(\kappa+\eta Z_{i}\right)+(1-\upsilon)\lambda_{d}(\kappa-\eta Z_{i})\right]$.
\par\end{center}
\noindent As before, we can use $\hat{\kappa}$ and $\hat{\upsilon}$
to construct control functions to include in a second-step regression,
producing estimates $\hat{\alpha}_{d}$ and $\hat{\gamma}_{d}$ that
exactly fit $\bar{Y}_{d}^{1}$ and $\bar{Y}_{d}^{0}$.
Though this estimator matches all choice probabilities and conditional
mean outcomes, it produces an estimate of LATE different from IV.
The model's implied LATE is
\begin{center}
$E\left[Y_{i}(1)-Y_{i}(0)|D_{i}(1)>D_{i}(0)\right]=(\alpha_{1}-\alpha_{0})+(\gamma_{1}-\gamma_{0})\times E\left[J(U_{i})-\mu_{J}|\delta_{i}=\eta,\kappa<U_{i}\leq\kappa+\eta\right]$.
\par\end{center}
\noindent The corresponding control function estimator of this quantity
is
\begin{equation}
\widehat{LATE}^{*}=\left(\hat{\alpha}_{1}-\hat{\alpha}_{0}\right)+\left(\hat{\gamma}_{1}-\hat{\gamma}_{0}\right)\times\left(\tfrac{(\hat{\kappa}+\eta)\lambda_{1}(\hat{\kappa}+\eta)-\hat{\kappa}\lambda_{1}(\hat{\kappa})}{\eta}\right).\label{eq:fake_late}
\end{equation}
\noindent It is straightforward to verify that $\widehat{LATE}^{*}$
is not equal to $\widehat{LATE}^{IV}$. Equivalence fails here because
the selection model implies the presence of ``defiers'' with $D_{i}(1)<D_{i}(0)$.
IV does not identify LATE when there are defiers; hence, the model
suggests using a different function of the data to estimate the LATE.
\subsection*{Covariates}
\noindent It is common to condition on a vector of covariates $X_{i}$
either to account for possible violations of the exclusion restriction
or to increase precision. Theorem \ref{thm:LATE_equivalence} implies
that IV and control function estimates of LATE coincide if computed
separately for each value of the covariates, but this may be impractical
or impossible when $X_{i}$ can take on many values.
A standard approach to introducing covariates is to enter them additively
into the potential outcomes model (see, e.g., \citealp{cornelissen_review,kline_walters_2016};
and \citealp{brinch_etal}). Suppose treatment choice is given by
$D_{i}=1\{P(X_{i},Z_{i})\geq U_{i}\}$ with $U_{i}$ independent of
$(X_{i},Z_{i})$, and assume
\begin{equation}
E\left[Y_{i}(d)|U_{i},X_{i}\right]=\alpha_{d}+\gamma_{d}\times(J(U_{i})-\mu_{J})+X_{i}^{\prime}\tau,\ d\in\{0,1\}.\label{eq:Xlinear}
\end{equation}
Letting $\hat{P}(X_{i},Z_{i})$ denote an estimate of $Pr[D_{i}=1|X_{i},Z_{i}]$,
the control function estimates for this model are
\begin{equation}
\left(\hat{\alpha}_{1},\hat{\gamma}_{1},\hat{\alpha}_{0},\hat{\gamma}_{0},\hat{\tau}\right)={\displaystyle {\displaystyle \arg\min_{\alpha_{1},\gamma_{1},\alpha_{0},\gamma_{0},\tau}}\sum_{i}\sum_{d\in\{0,1\}}1\left\{ D_{i}=d\right\} \left[Y_{i}-\alpha_{d}-\gamma_{d}\lambda_{d}(\hat{P}(X_{i},Z_{i}))-X_{i}'\tau\right]^{2}}.\label{eq:cf_covs}
\end{equation}
To ease exposition, we will study the special case of a single binary
covariate $X_{i}\in\{0,1\}$. Define $LATE(x)\equiv E[Y_{i}(1)-Y_{i}(0)|P(x,0)<U_{i}\leq P(x,1),X_{i}=x]$
as the average treatment effect for compliers with $X_{i}=x$, and
let $\hat{\alpha}_{d}(x)$ and $\hat{\gamma}_{d}(x)$ denote estimates
from unrestricted control function estimation among the observations
with $X_{i}=x$. The additive separability restriction in (\ref{eq:Xlinear})
suggests the following two estimators of $LATE(1)$:
\[
\widehat{LATE}_{x}^{CF}(1)=\left(\hat{\alpha}_{1}(x)-\hat{\alpha}_{0}(x)\right)+\left(\hat{\gamma}_{1}(x)-\hat{\gamma}_{0}(x)\right)\Gamma(\hat{P}(1,0),\hat{P}(1,1)),\ x\in\{0,1\}.
\]
\noindent By Theorem \ref{thm:LATE_equivalence} $\widehat{LATE}_{1}^{CF}(1)$
is a Wald estimate for the $X_{i}=1$ sample. $\widehat{LATE}_{0}^{CF}(1)$
gives an estimated effect for compliers with $X_{i}=1$ based upon
control function estimates for observations with $X_{i}=0$. The following
proposition describes the relationship between these two estimators
and the restricted estimator of $LATE\left(1\right)$ based upon (\ref{eq:cf_covs}).
\begin{prop}
Suppose Conditions \ref{assu:1} and \ref{assu:2} hold for each value
of $X_{i}\in\{0,1\}$ and let $\widehat{LATE}_{r}^{CF}(1)=(\hat{\alpha}_{1}-\hat{\alpha}_{0})+(\hat{\gamma}_{1}-\hat{\gamma}_{0})\Gamma(\hat{P}(1,0),\hat{P}(1,1))$
denote an estimate of $LATE(1)$ based on (\ref{eq:cf_covs}). Then\label{prop:cov_formula}
\end{prop}
\begin{center}
$\widehat{LATE}_{r}^{CF}(1)=w\widehat{LATE}_{1}^{CF}(1)+(1-w)\widehat{LATE}_{0}^{CF}(1)+b_{1}\left(\hat{\gamma}_{1}(1)-\hat{\gamma}_{1}(0)\right)+b_{0}\left(\hat{\gamma}_{0}(1)-\hat{\gamma}_{0}(0)\right)$.
\par\end{center}
\noindent \emph{The coefficients $w$, $b_{1}$, and $b_{0}$ depend
only on the joint empirical distribution of $D_{i}$, $X_{i}$, and
$\hat{P}(X_{i},Z_{i})$.}
\noindent \textbf{Proof:} See the Appendix. $\blacksquare$
\begin{rem}
Proposition \ref{prop:cov_formula} demonstrates that control function
estimation under additive separability gives a linear combination
of covariate-specific estimates plus terms that equal zero when the
separability restrictions hold exactly in the sample. One can show
that the coefficient $w$ need not lie between 0 and 1. By contrast,
two-stage least squares estimation of a linear model with an additive
binary covariate using all interactions of $X_{i}$ and $Z_{i}$ as
instruments generates a weighted average of covariate-specific IV
estimates \citep{angristpischke2009}.
\end{rem}
\begin{rem}
Consider the following extension of equation (\ref{eq:Xlinear}):
\end{rem}
\begin{center}
$E\left[Y_{i}(d)|U_{i},X_{i}\right]=\alpha_{d}+\gamma_{d}\times(J(U_{i})-\mu_{J})+X_{i}^{\prime}\tau_{dc}+1\{U_{i}\leq P(X_{i},0)\}X_{i}^{\prime}\tau_{at}+1\{U_{i}>P(X_{i},1)\}X_{i}^{\prime}\tau_{nt},\ d\in\{0,1\}$.
\par\end{center}
\noindent This equation allows different coefficients on $X_{i}$
for always takers, never takers, and compliers by interacting $X_{i}$
with indicators for thresholds of $U_{i}$, and also allows the complier
coefficients to differ for treated and untreated outcomes. When $X_{i}$
includes a mutually exclusive and exhaustive set of indicator variables
and $\hat{P}(X_{i},Z_{i})$ equals the sample mean of $D_{i}$ for
each $(X_{i},Z_{i})$, control function estimation of this model produces
the same estimate of $E[Y_{i}|X_{i},D_{i},D_{i}(1)>D_{i}(0)]$ as
the semi-parametric procedure of \citet{abadie_2003}. Otherwise the
estimates may differ even asymptotically as the control function estimator
employs a different set of approximation weights when the model is
misspecified.
\begin{rem}
A convenient means of adjusting for covariates that maintains the
numerical equivalence of IV and control function estimates is to weight
each observation by $\omega_{i}=Z_{i}/\hat{e}(X_{i})+(1-Z_{i})/(1-\hat{e}(X_{i}))$
where $\hat{e}(x)\in\left(0,1\right)$ is a first step estimate of
$Pr\left[Z_{i}=1|X_{i}=x\right]$. It is straightforward to show that
the $\omega_{i}-$weighted IV and control function estimates of the
unconditional LATE will be identical, regardless of the propensity
score estimator $\hat{e}(X_{i})$ employed. See \citet{hull_jmp}
for a recent application of this approach to covariate adjustment
of a selection model.
\end{rem}
\section{Maximum likelihood}
A fully parametric alternative to two-step control function estimation
is to specify a joint distribution for the model's unobservables and
estimate the parameters in one step via full information maximum likelihood
(FIML). Consider a model that combines (\ref{eq:choice}) and (\ref{eq:independence})
with the distributional assumption
\begin{equation}
Y_{i}(d)|U_{i}\sim F_{Y|U}\left(y|U_{i};\theta_{d}\right),\label{eq:parametric}
\end{equation}
\noindent where $F_{Y|U}(y|u;\theta)$ is a conditional CDF indexed
by a finite dimensional parameter vector $\theta$. For example, a
fully parametric version of the Heckit model is $Y_{i}(d)|U_{i}\sim N\left(\alpha_{d}+\gamma_{d}\Phi^{-1}(U_{i}),\sigma_{d}^{2}\right)$.
Since the marginal distribution of $U_{i}$ is also known, this model
provides a complete description of the joint distribution of $(Y_{i}(d),U_{i})$.
FIML exploits this distributional knowledge, estimating the model's
parameters as
\begin{equation}
\begin{aligned}\left(\hat{P}(0)^{ML},\hat{P}(1)^{ML},\hat{\theta}_{0}^{ML},\hat{\theta}_{1}^{ML}\right)=\arg & {\displaystyle \max_{(P(0),P(1),\theta_{0},\theta_{1})}}{\displaystyle \sum_{i}}D_{i}\log\left(\int_{0}^{P(Z_{i})}f_{Y|U}\left(Y_{i}|u;\theta_{1}\right)du\right)\\
& +{\displaystyle \sum_{i}}(1-D_{i})\log\left(\int_{P(Z_{i})}^{1}f_{Y|U}(Y_{i}|u;\theta_{0})du\right),
\end{aligned}
\label{eq:ml}
\end{equation}
\noindent where $f_{Y|U}(\cdot|u;\theta_{d})\equiv dF_{Y|U}\left(.|u;\theta_{d}\right)$
denotes the density (or probability mass function) of $Y_{i}(d)$
given $U_{i}=u$. The corresponding FIML estimates of treated and
untreated complier means are
\begin{center}
$\hat{\mu}_{dc}^{ML}=\dfrac{\int_{\hat{P}(0)^{ML}}^{\hat{P}(1)^{ML}}\int_{-\infty}^{\infty}yf_{Y|U}(y|u;\hat{\theta}_{d}^{ML})dydu}{\hat{P}(1)^{ML}-\hat{P}(0)^{ML}}$,
\par\end{center}
\noindent and the FIML estimate of LATE is $\widehat{LATE}^{ML}=\hat{\mu}_{1c}^{ML}-\hat{\mu}_{0c}^{ML}$.
\subsection*{Binary outcomes}
We illustrate the relationship between FIML and IV estimates of LATE
with the special case of a binary $Y_{i}$. A parametric model for
this setting is given by
\begin{equation}
\begin{aligned}Y_{i}(d)= & 1\left\{ \alpha_{d}\geq\epsilon_{id}\right\} ,\\
\epsilon_{id}|U_{i} & \sim F_{\epsilon|U}\left(\epsilon|U_{i};\rho_{d}\right),
\end{aligned}
\label{eq:ml_binary}
\end{equation}
\noindent where $F_{\epsilon|U}(\epsilon|u;\rho)$ is a conditional
CDF characterized by the single parameter $\rho$. Equations (\ref{eq:choice})
and (\ref{eq:ml_binary}) include six parameters, which matches the
number of observed linearly independent probabilities (two values
of $Pr\left[D_{i}=1|Z_{i}\right]$, and four values of $Pr\left[Y_{i}=1|D_{i},Z_{i}\right]$).
The model is therefore ``saturated'' in the sense that a model with
more parameters would be under-identified.
The following result establishes the conditions under which maximum
likelihood estimates of complier means (and therefore LATE) coincide
with IV.
\begin{prop}
\noindent Consider the model defined by (\ref{eq:choice}), (\ref{eq:independence})
and (\ref{eq:ml_binary}). Suppose that Conditions \ref{assu:1} and
\ref{assu:2} hold, and that the maximum likelihood problem (\ref{eq:ml})
has a unique solution. Then $\hat{\mu}_{dc}^{ML}=\hat{\mu}_{dc}^{IV}$
for $d\in\{0,1\}$ if and only if $\hat{\mu}_{dc}^{IV}\in[0,1]$ for
$d\in\{0,1\}$.\label{prop:MLE_equivalence}
\end{prop}
\noindent \textbf{Proof:} See the Appendix. $\blacksquare$
\begin{rem}
The intuition for Proposition \ref{prop:MLE_equivalence} is that
the maximum likelihood estimation problem can be rewritten in terms
of the six identified parameters of the LATE model: $(\mu_{1at},\mu_{0nt},\mu_{1c},\mu_{0c},\pi_{at},\pi_{c})$,
where $\pi_{g}$ is the population share of group $g$. Unlike the
IV and control function estimators, the FIML estimator accounts for
the binary nature of $Y_{i}(d)$ by constraining all probabilities
to lie in the unit interval. When these constraints do not bind the
FIML estimates coincide with nonparametric IV estimates, but the estimates
differ when the nonparametric approach produces complier mean potential
outcomes outside the logically possible bounds. Logical violations
of this sort have been proposed elsewhere as a sign of failure of
instrument validity \citep{balke_pearle,imbens_rubin_97,huber_mellace,kitagawa_2015}.
\end{rem}
\begin{rem}
A simple ``limited information'' approach to maximum likelihood
estimation is to estimate $P\left(0\right)$ and $P\left(1\right)$
in a first step and then maximize the plug-in conditional log-likelihood
function
\end{rem}
\begin{center}
${\displaystyle \sum_{i}}D_{i}\log\left(\int_{0}^{\hat{P}(Z_{i})}f_{Y|U}\left(Y_{i}|u;\theta_{1}\right)du\right)+{\displaystyle \sum_{i}}(1-D_{i})\log\left(\int_{\hat{P}(Z_{i})}^{1}f_{Y|U}(Y_{i}|u;\theta_{0})du\right)$
\par\end{center}
\noindent with respect to $\left(\theta_{0},\theta_{1}\right)$ in
a second stage. One can show that applying this less efficient estimator
to a saturated model will produce an estimate of LATE equivalent to
IV under Conditions \ref{assu:1} and \ref{assu:2}. This broader
domain of equivalence results from some cross-equation parameter restrictions
being ignored by the two-step procedure. For example, the FIML estimator
may choose an estimate of $\pi_{c}$ other than $\hat{P}(1)-\hat{P}(0)$
in order to enforce the constraint that $(\mu_{1c},\mu_{0c})\in[0,1]^{2}$.
\subsection*{Overidentified models}
Equivalence of FIML and IV estimates at interior solutions in our
binary example follows from the fact that the model satisfies monotonicity
and includes enough parameters to match all observed choice probabilities.
Similar arguments apply to FIML estimators of sufficiently flexible
models for multi-valued outcomes. When the model includes fewer parameters
than observed choice probabilities, overidentification ensues. For
example, the standard bivariate probit model is a special case of
(\ref{eq:ml_binary}) that uses a normal distribution for $F_{\epsilon|U}(\cdot)$
and imposes $\epsilon_{i1}=\epsilon_{i0}$ and therefore $\rho_{1}=\rho_{0}$
(see \citealp{greene_text}). Hence, only five parameters are available
to rationalize six linearly independent probabilities.
Maximum likelihood estimation of this more parsimonious model may
yield an estimate of LATE that differs from IV even at interior solutions.
This divergence stems from the model's overidentifying restrictions
which, if correct, may yield efficiency gains but if wrong can compromise
consistency. Though maximum likelihood estimation of misspecified
models yields a global best approximation to the choice probabilities
\citep{white_1982_mle}, there is no guarantee that it will deliver
a particularly good approximation to the LATE.
\section{Model evaluation}
In practice researchers often estimate selection models that impose
additive separability assumptions on exogenous covariates, combine
multiple instruments, and employ additional smoothness restrictions
that break the algebraic equivalence of structural LATE estimates
with IV. The equivalence results developed above provide a useful
conceptual benchmark for assessing the performance of structural models
in such applications. An estimator derived from a properly specified
model of treatment assignment and potential outcomes should come close
to matching a nonparametric IV estimate of the same parameter. Significant
divergence between these estimates would signal that the restrictions
imposed by the structural model are violated.
Figure 3 shows an example of this approach to model assessment from
Kline and Walters' \citeyearpar{kline_walters_2016} reanalysis of
the Head Start Impact Study (HSIS) \textendash{} a randomized experiment
with two-sided non-compliance \citep{puma_hsis}. On the vertical
axis are non-parametric IV estimates of the LATE associated with participating
in the Head Start program relative to a next best alternative for
various subgroups in the HSIS defined by experimental sites and baseline
child and parent characteristics. On the horizontal axis are two-step
control function estimates of the same parameters derived from a heavily
over-identified selection model involving multiple endogenous variables,
baseline covariates, and excluded instruments. Had this model been
saturated, all of the points would lie on the 45 degree line. In fact,
a Wald test indicates these deviations from the 45 degree line cannot
be distinguished from noise at conventional significance levels, suggesting
that the approximating model is not too far from the truth.
\begin{center}
\includegraphics[scale=0.77]{fig2_v2}
\par\end{center}
Passing a specification test does not obviate the fundamental identification
issues inherent in interpolation and extrapolation exercises. As philosophers
of science have long argued, however, models that survive empirical
scrutiny deserve greater consideration then those that do not \citep{popper_1959,lakatos_1976}.
Demonstrating that a tightly restricted model yields a good fit to
IV estimates not only bolsters the credibility of the model's counterfactual
predictions, but serves to clarify what the estimated structural parameters
have to say about the effects of a research design as implemented.
Here the control function estimates reveal that Head Start had very
different effects on different sorts of complying households, a finding
rationalized by estimated heterogeneity in both patterns of selection
into treatment and potential outcome distributions.
\section{Conclusion}
This paper shows that two-step control function estimators of LATE
derived from a wide class of parametric selection models coincide
with the instrumental variables estimator. Control function and IV
estimates of mean potential outcomes for compliers, always takers,
and never takers are also equivalent. While many parametric estimators
produce the same estimate of LATE, different parameterizations can
produce dramatically different estimates of population average treatment
effects and other under-identified quantities. The sensitivity of
average treatment effect estimates to the choice of functional form
may be the source of the folk wisdom that structural estimators are
less robust than instrumental variables estimators. Our results show
that this view confuses robustness for a given target parameter with
the choice of target parameter.
Structural estimators that impose overidentifying restrictions may
generate LATE estimates different from IV. Reporting the LATEs implied
by such estimators facilitates comparisons with unrestricted IV estimates
and is analogous to the standard practice of reporting average marginal
effects in binary choice models \citep{wooldridge_text}. Such comparisons
provide a convenient tool for assessing the behavioral restrictions
imposed by structural models. Model-based estimators that cannot rationalize
unrestricted IV estimates of LATE are unlikely to fare much better
at extrapolating to fundamentally under-identified quantities. On
the other hand, a tightly constrained structural estimator that fits
a collection of disparate IV estimates enjoys some degree of validation
that bolsters the credibility of its counterfactual predictions.
\bibliographystyle{ecta}
|
2,869,038,155,671 | arxiv | \section*{Results}
\subsection*{Electron band lying just above $E_F$}
Figure 1(a) shows an intensity plot of $E$ vs. $k$ (energy vs. momentum) measured at 25 K ($> T_c$ = 14.5 K) along $\Gamma$-$X$ line in the Brillouin zone of FeTe$_{0.6}$Se$_{0.4}$ after dividing by the Fermi-Dirac (FD) function broadened with the Gaussian corresponding to the experimental energy resolution. We employed three different methods to determine the band dispersions: a second derivative map with respect to energy, fitting to the energy distribution curves (EDCs), and fitting to the momentum distribution curves (MDCs)~(see Supplementary Information). Three hole bands can be clearly recognized in the second derivative map.
Band-structure calculations based on density functional theory (DFT) were carried out for the parent FeTe and are overlaid as solid lines in Fig. 1(a) after a suitable energy shift and rescaling which are ascribed to renormalization effects, as is known from earlier work~\cite{Chen2010PRB,Tamai2010PRL,Nakayama2010PRL,Lubashevsky2012NP}. The calculation details and complete band structure are discussed in the Supplementary Information and the energy shifts and rescaling are listed in Table I. The calculated band structure and orbital characters are also consistent with known results~\cite{Subedi2008PRB,Miyake2010JPSJ} and were confirmed by measuring the linear polarization dependence of spectral intensities~(see Supplementary Information). However, in contrast to the DFT calculations which predict existence of three hole FSs around the $\Gamma$ point, we find that the band top of the two dominantly $xz/yz$-orbital derived bands are located around 15 meV below $E_F$, i.e., these bands sink below $E_F$, and only one hole band originating in the $x^2-y^2$ orbital crosses $E_F$. From the degeneracy of the two $xz/yz$ bands, we conclude that $k_z$ $\sim$ 0 in the reduced Brillouin zone for the present laser ARPES measurements.
The open circles in Fig. 1(a) show band dispersions deduced from the peak positions of the second derivative spectra shown in Fig. S3. The second derivative map after dividing by the FD function shown in Fig. S3(b) clearly shows that the dispersion around the $\Gamma$ point is electron-like. The origin of this electronic dispersion is presumably another dominantly $xz/yz$-orbital derived band, which is located just above $E_F$ for the DFT results for the parent FeTe\cite{note_DFT}. For checking the dispersions above $E_F$, Fig. 1(b) shows the band dispersions in a narrow energy window near $E_F$, and band dispersions deduced from fits to the EDCs are overlaid.
The fits to the EDCs, obtained after dividing by the FD function, are shown in Fig. 1(c). It is clear that there are two bands above $E_F$ at the $\Gamma$-point, which get merged around $k$ $\sim$ 0.07 {\AA} and then again separate out into two bands around $k$ $\sim$ 0.1 {\AA}. Figure 1(d) shows the fits with the component Lorentzian functions for these three cases.
We also performed measurements with another sample at higher temperatures of 35 K and 50 K in addition to 25 K as shown in Fig. S7. The peak positions are consistent with those shown in Fig. 1.
In the occupied states below $E_F$, the degenerate band top of the $xz/yz$ hole bands are positioned at $\sim$ 15 meV below $E_F$. The band top of the $E_F$-crossing $x^2-y^2$ band is located at least above $\sim$ 6.5 meV from $E_F$ at the $\Gamma$-point. Most interestingly, we do find the expected electron band existing just above $E_F$ at the $\Gamma$-point, with the band bottom located at $\sim$ 0.7 $\pm$ 0.2 meV above $E_F$ (Fig. 1(b)). This electron band has been missed in all earlier studies of the momentum resolved electronic structure of Fe(Te,Se)~\cite{Chen2010PRB,Tamai2010PRL,Nakayama2010PRL,Lubashevsky2012NP}.
We note that the $x^2-y^2$ hole band and the electron-like band just above $E_F$ may be hybridized due to spin-orbit interactions~\cite{note_SOI}, for example, and result in a wing-shaped dispersion. However, even if these two bands are hybridized and merge to a single band, this does not affect our conclusions. Since the nature of the conducting carriers being electron-like or hole-like is determined by the gradient of the band dispersion ($\partial E/\partial k$), the carriers at $k_F$ and the thermally-excited carriers at the $\Gamma$ point will be hole-like and electron-like, respectively.
Also, the details of electron band at the higher energy region above $E_F$ are not relevant to superconductivity. Only the positions of the top of hole band and the bottom of the electron band are important, and they can be evaluated rather clearly from the MDCs, of which line shape is not affect by dividing by the FD function~(see Supplementary Information).
\subsection*{Sharp superconducting coherence peak in the electron band just above $E_F$}
Figures 2(a) and 2(b) show the energy distribution curves (EDCs) after dividing by the FD functions corresponding to each temperature along $\Gamma$-$X$ line at $T$ = 25 K (above $T_c$) and at $T$ = 2.5 K (below $T_c$), respectively. The open circles in Fig. 2(a) mark the normal-state band dispersions obtained from the second derivative spectra shown in Fig. S3. In Fig. 2(b), we can clearly see that the superconducting coherence peaks emerge below $T_c$ for the hole band. The small circles in Fig. 2(b) mark the positions of the coherence peaks, and they are plotted in the enlarged scale in Fig. S10. Figures 2(c) and 2(d) show the EDCs above $T_c$ (25 K) and below $T_c$ (2.5 K) at $k = k_F$ and $k \sim \Gamma$, respectively. We can see that the electron band just above $E_F$ at the $\Gamma$ point also shows a sharp superconducting coherence peak, although this band does not cross the $E_F$ in the normal state. The solid lines are fits to the BCS spectral function $A_\mathrm{BCS}(k,\omega)$\cite{Matsui2003PRL,Shimojima2011Science,Okazaki2012Science}, which can be expressed as
\[
A_\mathrm{BCS}(k,\omega) = \frac{1}{\pi}\left\{\frac{|u_k|^2\Gamma}{\left(\omega-E_k\right)^2+\Gamma^2}+\frac{|v_k|^2\Gamma}{\left(\omega+E_k\right)^2+\Gamma^2}\right\},
\]
where $E_k$ and $|u_k|^2$, $|v_k|^2$ are the quasiparticle energy and the coherence factors of Bogoliubov quasiparticles (BQPs), respectively. Using the normal-state dispersion $\epsilon_k$ with respect to the chemical potential $\mu$ and the SC gap $\Delta(k)$, $E_k$ can be expressed
\[
E_k = \sqrt{(\epsilon_k-\mu)^2 + |\Delta(k)|^2}
\]
and
\[
|u_k|^2 = 1 - |v_k|^2 = \frac{1}{2}\left(1+\frac{\epsilon_k}{E_k}\right),
\]
respectively. From the fits to the data, we estimate the Bogoliubov quasiparticle energy ($E_k = \sqrt{\epsilon_k^2 + |\Delta(k)|^2}$) of the hole band to be 2.3 meV (= $\Delta(k)$, because $\epsilon_k$ = 0 at $k$ = $k_F$ of this band) and of the electron band to be 1.3 meV, respectively. From the value of $\epsilon_k$ $\sim$ 0.7 meV, $\Delta(k)$ is estimated to be $\sim$ 1.1 meV for the electron band. This indicates different pairing strengths and reduced gap values 2$\Delta$/$k_BT_c$ for the electron and hole bands.
In addition, $\Delta$/$\epsilon_F$ for the hole band is estimated to be $\sim$ 0.3, corresponding to a relatively weak coupling. On the other hand,
since the energy position of the electron band is just 0.7 meV ($\sim$ 8 K) above $E_F$, it means that its occupancy in the normal state will strongly depend on temperature. Accordingly, the exact value of $\epsilon_F$ of the electron band cannot be described in the usual way.
If we regard $T_c$ as a measure of $\epsilon_F$, based on the fact that $T_c$ is the lowest temperature representing the normal state ($T_c$ = 14.5 K $\sim$ 1.2 meV), we obtain $\Delta$/$\epsilon_F$ $\geq$ 1,
indicative of the strong coupling limit. This estimation may seem to be fairly rough. However, if $\epsilon_F$ equals to $\Delta$, the bottom of the electron band should be located at $E$ = -$\Delta$ below $E_F$. Hence, we can say at least $\Delta$/$\epsilon_F$ $\geq$ 1. Thus, the electron band with a smaller $\Delta$ is actually in the strong-coupling regime. This represents the condition of an electron band with only a small number of carriers, but with a strong pairing interaction and a finite $\Delta$ exists for this band. On the otherhand, the hole band with a larger $\Delta$ lies in the relatively weaker-coupling regime.
It is suggestive of Cooper pairing for the hole band and Boson condensation for the electron band.
We note that even for the strong-coupling electron band, we have used a BCS spectral function to estimate the value of $\Delta$. This is not a problem as the obtained value of $\Delta$ represent the lower bound of $\Delta$, because a smaller value of $\epsilon_k$-$\mu$ for the BEC regime will give a larger value of $\Delta$\cite{Gaebler2010NP,Lubashevsky2012NP}.
Figures 2(e) and 2(f) show the intensity maps of the spectra above and below $T_c$, respectively, after dividing by the corresponding FD functions. The open circles are the same as in Fig. 2(a) and the solid line is a fitting result to the open circles using a polynomial function, representing the normal-state dispersion $\epsilon_k$~\cite{note_dispersion}. The solid lines in Fig. 2(f) are the BQP dispersions using $\epsilon_k$ in Fig. 2(e) and the $\Delta_k$ values obtained above for the electron and hole bands. Colors of the lines corresponds to the amplitude of the coherence factors $|u_k|^2$ above $E_F$ and $|v_k|^2$ below $E_F$. The red and blue regions correspond to the higher and lower values, respectively. It is noted that the BQP dispersion does not cross the brightest intensity around the $\Gamma$ point above $E_F$. This is attributed to the tail of the hole-band top, which is also positioned at the $\Gamma$-point. The normal-state and BQP dispersions have been plotted in the same panel of Fig. 2(g). The open circles plotted in Fig. 2(g) correspond to the dispersion of the coherence peaks at $T$ = 2.5 K shown in Fig. 2(b). The gray-scale density of the BQP dispersion corresponds to the amplitude of coherence factors.
It is interesting to note that the BQP dispersions merge for the electron and hole bands, indicative of a composite BCS-BEC superconductivity. The results indicate that irrespective of weak or strong coupling
, both the hole and electron bands in the superconducting state exhibit Bogoliubov quasiparticle dispersions due to particle-hole mixing~\cite{Campuzano1996PRB}. However, the superconductivity in the electron band can be expected to be very sensitive to the occupancy of the electron band with Se substitution for Te, as well as pressure/strain. This possibly explains the reported large variation in $T_c$ with pressure for FeSe ($T_c$ = 8.5-36.7 K)~\cite{Medvedev2009NM}.
\section*{Discussion}
Another difference can be recognized between the weak-coupling hole band and the strong-coupling electron band. Figures 3(a) and 3(b) show the temperature dependence of EDCs at $k_F$ for the hole band and $k \sim \Gamma$ for the electron band, respectively. The black solid lines indicate the fitting results using the BCS spectral function. The estimated SC-gap sizes are shown in Fig. 3(c). The existence of the pseudogap only for the hole band is clearer from the symmetrized EDCs (Fig. S11) or the FD-divided EDCs (Fig. S12).
The temperature dependence of the gap opening indicates another important difference for the weaker-coupling hole band compared to the strong-coupling electron band.
A pseudogap behavior can be recognized for the weaker coupling hole band in the spectra above $T_c$~\cite{note_pseudogap}. However, in strong contrast to the currently available BCS-BEC crossover theory~\cite{S'adeMelo1993PRL} which predict existence of a pseudogap in the BEC strong coupling regime, the electron band does not show a pseudogap above $T_c$~\cite{note_BQPcontinuation}. Thus, we find a coexistence of the weak coupling and strong coupling superconductivity in the same material but with attributes not fully consistent with our present understanding of weak and strong coupling superconductivity.
Our study identifies the required band structure for composite superconductivity, which is closely related to Dirac point dispersions, coexisting with a simple electron or hole band as schematically shown in Fig. 4.
\section*{Methods}
Single crystals of FeTe$_{0.6}$Se$_{0.4}$ were prepared by a melt-growth technique. Chemical composition of the grown crystals was determined by electron probe microanalysis (EPMA) and inductively coupled plasma (ICP) atomic emission spectrometry. Details have been described in Ref.~\onlinecite{Hanaguri2010Science}. ARPES data were collected using the laser ARPES apparatus developed at ISSP with the 6.994 eV, 6th harmonic of Nd:YVO$_4$ quasi continuous wave (q-CW, repetition rate = 120 MHz) laser and VG-Scienta HR8000 electron analyzer~\cite{Okazaki2012Science}. While this apparatus achieves the maximum energy resolution of 70 $\mu$eV, the overall energy resolution was set to $\sim$ 1.2 meV for the measurements of EDCs and MDCs near $E_F$ and 5 meV for $E$-$k$ map measurements, The angular resolution was 0.1 deg, corresponding to the momentum resolution of 0.0015 {\AA}$^{-1}$. Polarization of incident excitation laser was adjusted using a half-wave ($\lambda$/2) plate and a quarter-wave ($\lambda$/4) plate.
The $E_F$ positions were calibrated by measuring the Fermi edge of a gold film evaporated onto the sample substrate.
\subsection*{Band-structure calculations and dominant orbital characters}
Figure S1(a) shows band dispersions calculated by the Wien2k code for the pure FeTe. The lattice parameters were taken from those obtained by the powder neutron diffraction measurements~\cite{Li2009PRB} as in the previous report by Miyake {\it et al.}~\cite{Miyake2010JPSJ}. We confirmed that the obtained band dispersions are in accord with those by Miyake {\it et al}. We deduced the dominant orbital contribution of each band as indicated by different colors. Figure S1(b) shows the enlarged band dispersions in the vicinity of $E_F$ along the measured $X$-$\Gamma$-$X$ line. The 28th, 29th, and 30th bands show hole-like dispersions and cross the $E_F$, whereas the 31st band shows a electron-like dispersion just above $E_F$ around the $\Gamma$ point.
\subsection*{Orbital characters from polarization dependent measurements}
Figure S2(a) shows the experimental configuration of the laser ARPES measurements for this study. In this configuration, $xy$ and $yz$ orbitals can be measured only for the $p$ polarization, whereas the $x^2-y^2$, $z^2$, and $xz$ orbitals can be measured for both the polarizations from the parity selection rule~\cite{Damascelli2003RMP}. By taking account of the parity of each $d$ orbital and orbital characters obtained from the band-structure calculation, we assigned the dominant orbital characters of the observed three hole bands
as well as the electron band just above $E_F$,
as shown in Fig. S2(c).
\subsection*{Determination of the band dispersions above $E_F$ from three different methods}
We used three different methods to determine the band dispersions above $E_F$ at 25 K as described in the following. Each method has its own advantages and disadvantages. However, the three methods provide consistent band dispersions, and we can safely conclude that an electron band exists just above $E_F$ at the $\Gamma$ point. We could then determine the positions of the bottom of the electron band and the top of $x^2-y^2$ hole band.
\subsubsection*{Second derivative spectra with respect to energy}
Figure S3(a) shows a second derivative map with respect to energy obtained from the map shown in Fig. 1(a). The open circles indicate the peak positions. Figure S3(b) is obtained by first dividing the intensity map by the Fermi-Dirac (FD) function at $T$ = 25 K broadened with the experimental energy resolution, and then taking the second derivative.
\subsubsection*{Fitting to EDCs}
Figure S4 shows the results of fitting to several EDC cuts along the $X$-$\Gamma$-$X$ direction without dividing by the FD function. The solid lines indicate the fitting results. The fitting functions were obtained by first multiplying the FD function to the three Lorentzians, and then taking the convolution with the Gaussian corresponding to the experimental energy resolution.
\subsubsection*{Fitting to MDCs}
Figures S5(a) and S5(b) show the results of fitting to several cuts of MDCs at (a) 25 K and (b) 2.5 K, respectively, after dividing by the FD function convoluted with the Gaussian. We note that dividing by the FD function does not affect the lineshape of MDCs. The solid lines and vertical bars indicate the fitting functions and their peak positions. The fitting was performed in the region of $k\ge0$ with the symmetrized Lorentzians to avoid matrix element effects, i.e., the MDC fitting function $I(\omega)$ is given by $I(\omega) = \sum_{i}I_{i}(k) + \sum_{i}I_{i}(-k)$, where $I_{i}(k)$ is a component Lorentzian.
\subsection*{Band dispersions determined by various methods}
Figures S6(a) and S6(b) show the $E$-$k$ map measured at (a) 25 K and (b) 2.5 K, respectively. The open circles, rectangles, and triangles indicate the band dispersions determined by the second derivative spectra, fitting to the EDCs, and fitting to the MDCs. As mentioned above, each of these methods has advantages and disadvantages. Because the peak positions of the second derivative spectra are located where the gradient of the spectra shows large changes, it can detect the lower-energy side of the dispersion around the $\Gamma$ point. The fitting to the EDCs is the most appropriate to determine the band dispersion where the gradient of the dispersion is small, but it is difficult to determine the dispersions above $E_F$ without ambiguity. On the other hand, there is no ambiguity for the peak positions of MDCs even for those above $E_F$, but it is difficult to determine the dispersions around the top and bottom of the bands from MDC fits. However, the combination of three methods allows us to conclude that the top of the hole band is located at 6-7 meV above $E_F$ and an electron band exists just above $E_F$ for the dispersions at 25 K. At 2.5 K, we can recognize the superconducting coherence peaks separately at $k$ = $k_F$ and the $\Gamma$ point.
\subsection*{Fitting to EDCs at higher temperatures}
We performed measurements with another sample at higher temperatures of 35 K and 50 K in addition to 25 K. Figures S7(a)-(c) show the FD-divided EDCs at the $\Gamma$ point for these temperatures. The solid lines indicate the fitting functions which are the same as those used in Fig. 1(c), and the dashed lines are the component Lorentzians. The peak positions are consistent with data shown in Fig. 1, although the relative intensities of the two peaks are different. Figure S7(d) shows the FD-divided EDCs along the $\Gamma$-$X$ line at 50 K and the corresponding intensity plot is shown in Fig. S7(e). The rectangles indicate the band dispersions deduced from the fitting to the EDCs in the same way as Fig. 1(b). The solid lines are the same as those in Fig. 1(b) and extended up to 20 meV above $E_F$. It confirms that the band dispersions at 50 K are consistent with the band dispersions at 25 K.
\subsection*{Polarization dependent spectra below $T_c$ and above $T_c$}
Figures S8 and S9 show the polarization-dependent intensity map and second derivative map measured below $T_c$ (= 2.5 K) and above $T_c$ (= 25 K), respectively. They were measured with right circular, left circular, $s$-, and $p$-polarizations.
As described above in the section of {\bf ``Orbital characters from polarization dependent measurements''}, the orbitals with the odd parity can be
measured with the $p$-polarization. In the second derivative spectra below $T_c$, a clear feature at $\Gamma$ point can be seen only for the $p$-polarization just below $E_F$, while the structure for $k_F$ crossings away from $\Gamma$ point can be seen for the both polarizations. This indicates a difference in orbital characters between the $x^2-y^2$ hole band and the electron band just above $E_F$. The right and left circular polarizations were used to avoid the selection rules for bands of particular symmetry which arise in photoemission with linear polarization. This ensures we have measured all the band dispersions with minimal matrix element effects.
\subsection*{Dispersion of the superconducting coherence peak around the $\Gamma$ point}
The superconducting coherence peaks and BQP dispersions shown in Figs. 2(b) and 2(g) were enlarged in Fig. S10. These plots clearly shows that the coherence peaks at $k$ $\sim$ $\Gamma$ is originated from the electron band just above $E_F$. The BQP disperion originated from the $x^2-y^2$ hole band is almost flat around $k$ = $\pm$0.1 {\AA}$^{-1}$ and shows an indication of a bending-back behaviour, while the curvature of the dispersion around $k$ $\sim$ $\Gamma$ is very different from the almost flat dispersion of the BQP originated from the $x^2-y^2$ hole band and corresponds a reflection of the electronic dispersion just above $E_F$ in the normal state.
\subsection*{Temperature dependence of symmetrized EDCs and FD-divided EDCs}
The temperature dependent EDCs shown in Fig. 3 were symmetrized with respect to $E_F$ and the results are shown in Fig. S11, and they were divided by the FD functions of corresponding temperatures and the results are shown in Fig. S12. The existence of the pseudogap for the $x^2-y^2$ hole band can be clearly recognized also from both symmetrized EDCs and FD-divided EDCs as well as the temperature dependence of the SC-gap size estimated from fitting.
\clearpage
\begin{figure*} [htbp]
\begin{center}
\includegraphics[width=12cm]{FigS1}
\end{center}
\begin{flushleft}
{\bf Fig. S1.} Band-structure calculations for the parent FeTe based on the density functional theory. ({\bf a}), Band dispersion along the high symmetric line in the Brillouin zone calculated by Wien2k code. The dominant orbital character of each band is indicated by different colors. The bands near the $E_F$ are mainly composed of $x^2-y^2$ and $xz/yz$ orbitals. ({\bf b}), Band dispersion near the $E_F$ is along the $X$-$\Gamma$-$X$ line. The 28th, 29th, and 30th bands show hole-like dispersions and cross the $E_F$, whereas the 31st band shows a electron-like dispersion just above $E_F$ around the $\Gamma$ point. The dispersion of the 28th and 31st bands looks like a Dirac cone.
\label{FigS1}
\end{flushleft}
\end{figure*}
\begin{figure*} [htbp]
\begin{center}
\includegraphics[width=15cm]{FigS2}
\end{center}
\begin{flushleft}
{\bf Fig. S2.} Experimental configuration and linear-polarization dependence of ARPES intensity. ({\bf a}), Experimental configuration and parity of each $d$ orbital with respect to the mirror plane including the analyzer slit. ARPES intensity plotted as a function of momentum and energy measured at 25 K with ({\bf b}) $s$- and ({\bf c}) $p$-polarizations, respectively. Dominant $d$ orbital for each band is indicated in (c).
\label{FigS2}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=15cm]{FigS3}
\end{center}
\begin{flushleft}
{\bf Fig. S3.} Band dispersions from the second derivative spectra ({\bf a}), Second derivative map with respect to energy. ({\bf b}), Second derivative map after dividing by the FD function convoluted with a Gaussian of the experimental resolution. The open circles indicate band dispersions deduced from the peak positions of each map.
\label{FigS3}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=8cm]{FigS4}
\end{center}
\begin{flushleft}
{\bf Fig. S4.} Fitting to several cuts of EDCs along the $\Gamma$-$X$ line without dividing by the FD function. The solid lines indicate the fitting results.
\label{FigS4}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=15cm]{FigS5}
\end{center}
\begin{flushleft}
{\bf Fig. S5.} Fits to several cuts of MDCs along the $\Gamma$-$X$ line. {\bf a} MDCs at 25 K. {\bf b} MDCs at 2.5 K.
\label{FigS5}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=12cm]{FigS6}
\end{center}
\begin{flushleft}
{\bf Fig. S6.} Band dispersions determined by three methods. ({\bf a} and {\bf b}), $E$-$k$ map measured at (a) 25 K and (b) 2.5K. Black circles, red rectangles, blue triangles are determined by the second derivative spectra, fitting to the EDCs, and fitting to the MDCs, respectively.
\label{FigS6}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=15cm]{FigS7}
\end{center}
\begin{flushleft}
{\bf Fig. S7.} Fitting to FD-divided EDCs at higher temperatures. ({\bf a}-{\bf c}), FD-divided EDCs at the $\Gamma$ point measured at 25 K (a), 35 K (b), and 50 K (c) for another sample. The solid and dashed lines indicate the fitting functions and component Lorentzians. ({\bf d}) FD-divided EDCs along the $\Gamma$-$X$ line at 50 K. ({\bf e}) Intensity plot of the ARPES spectra at 50 K. The rectangles indicate the band dispersions deduced from the fitting to the EDCs in the same way as Fig. 1(b). The solid lines are the same as those in Fig. 1(b), thereby confirming that the band dispersions at 50 K are consistent with the band dispersions at 25 K.
\label{FigS7}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=15cm]{FigS8}
\end{center}
\begin{flushleft}
{\bf Fig. S8.} Polarization-dependent intensity map below $T_c$ (= 2.5 K) and above $T_c$ (= 25 K). ({\bf a}-{\bf d}) Intensity map at 2.5 K measured with (a) right circular, (b) left circular, (c) $s$-, and (d) $p$-polarizations, respectively. ({\bf e}-{\bf h}) Intensity map at 25 K measured with (e) right circular, (f) left circular, (g) $s$-, and (h) $p$-polarizations, respectively.
\label{FigS8}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=15cm]{FigS9}
\end{center}
\begin{flushleft}
{\bf Fig. S9.} Polarization-dependent second derivative map with respect to energy below $T_c$ (= 2.5 K) and above $T_c$ (= 25 K). ({\bf a}-{\bf d}) Intensity map at 2.5 K measured with (a) right circular, (b) left circular, (c) $s$-, and (d) $p$-polarizations, respectively. ({\bf e}-{\bf h}) Intensity map at 25 K measured with (e) right circular, (f) left circular, (g) $s$-, and (h) $p$-polarizations, respectively.
\label{FigS9}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=14cm]{FigS10}
\end{center}
\begin{flushleft}
{\bf Fig. S10.} The superconducting coherence peaks and BQP dispersions shown in Figs. 2(b) and 2(g) are plotted in an enlarged scale. These plots clearly show that the coherence peaks at $k$ $\sim$ $\Gamma$ is originated from the electron band just above $E_F$.\label{FigS10}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=14cm]{FigS11}
\end{center}
\begin{flushleft}
{\bf Fig. S11.} Temperature dependence of symmetrized EDCs at $k$ = $k_F$ and $k$ $\sim$ $\Gamma$. Temperature dependence of symmetrized EDCs at ({\bf a}) $k$ = $k_F$ of $x^2-y^2$ hole-like band and ({\bf b}) $k$ $\sim$ $\Gamma$ (bottom of the electron-like band), respectively.\label{FigS11}
\end{flushleft}
\end{figure*}
\begin{figure*} [h]
\begin{center}
\includegraphics[width=14cm]{FigS12}
\end{center}
\begin{flushleft}
{\bf Fig. S12.} Temperature dependence of FD-divided EDCs at $k$ = $k_F$ and $k$ $\sim$ $\Gamma$. Temperature dependence of FD-divided EDCs at ({\bf a}) $k$ = $k_F$ of $x^2-y^2$ hole-like band and ({\bf b}) $k$ $\sim$ $\Gamma$ (bottom of the electron-like band), respectively.\label{FigS12}
\end{flushleft}
\end{figure*}
|
2,869,038,155,672 | arxiv | \section{St\"uckelberg Qubits.}
The scaling of Bekenstein entropy \cite{Bekenstein} as the black hole area in Planck units, $S_{BH} = (R/L_P)^2$, suggests that the
information is stored in some boundary qubit degrees of freedom, one per each Planck-area-size pixel.
This approach is usually called the principle of black hole holography \cite{tHooft}.
Without committing to the literal validity of such a picture, it is nevertheless necessary to understand the fundamental reason behind the area law. One microscopic explanation \cite{us,critical} (see, \cite{gold, hair, portrait} for further discussions) is that the area law is a reflection of the scaling of the quantum gravitational coupling, $\alpha(R)$, of gravitons of wavelength $R$. Notice, that the most basic property of the inverse gravitational coupling is to ``run" with the length-scale $R$ as the {\it area},
\begin{equation}
{1 \over \alpha (R)} \, = \, {R^2 \over L_P^2} \,.
\label{gravitycoupling}
\end{equation}
Moreover note that the above relation between the quantum gravitational coupling and the area holds in an arbitrary number of dimensions.
Thus, in the picture of \cite{us,critical}, the black hole entropy is fundamentally defined by
the inverse graviton coupling
\begin{equation}
S_{BH} \, = \, {1 \over \alpha (R)} \,,
\label{entropy}
\end{equation}
and the area law is a direct consequence of (\ref{gravitycoupling}). The microscopic reason behind
(\ref{entropy}) according to \cite{critical, gold} is the quantum criticality of the black hole state, which
is characterized by the appearance of $N \, = \, \alpha^{-1}(R)$ collective Bogoliubov-Goldstone modes with energy gap $\Delta = \alpha(R) { \hbar \over R}$. The corresponding entropy is
obviously given by (\ref{entropy}).
Hence, in this picture holography is an emergent phenomenon that results from the
{\it quantum criticality} of the underlying gravitational system, for which both the number $N$ as well as the energy gap $\Delta$ of the qubits is determined by the gravitational coupling (\ref{gravitycoupling}). As we shall see, we shall discover a connection, strikingly similar to (\ref{entropy}), between the information capacity and the inverse coupling of the system, from a completely different approach.
This allows us to draw an interesting conclusion about the nature of the information qubits of holographic systems, in which gauge-redundancy and quantum criticality appear as two sides
of the same coin.
In order to formulate this alternative approach, in the present paper we would like to ask whether there exists a simple way for understanding the area law from a fundamental gauge principle, without the knowledge of the microscopic picture of the black hole interior.
We shall be interested in systems that contain sub-systems separated by surfaces that act
as boundaries from the information point of view.
Consider two regions of the space, $A$ and $B$, separated by a closed surface $S$. Let us assume that due to some dynamical reason the retrieval of information from the region $B$ into $A$ is highly suppressed or simply impossible. We shall say that for an observer in the region $A$ (we shall call her Alice) the surface $S$ acts as an {\it information boundary}.
The suppression of the information-retrieval probability can be, for example, due to an energy barrier that prevents some particle species crossing over from
$B$ to $A$ (see examples below). One natural example that fits within our definition of the information boundary is a black hole horizon. In this case the region $B$ is identified with the interior of the black hole and Alice is an external observer.
The general question is: How can Alice characterize the information-storage capacity of the hidden region $B$?
One known possibility is that Alice traces over the degrees of freedom in $B$ and ends up with a density
matrix. Naturally, in such a treatment an entanglement
entropy is attributed to the region $B$. The problem with this information measure is that it is based on a complete ignorance about the region
$B$. Consequently, the resulting entanglement entropy, instead of giving a real information measure, gives only an upper bound on the information-storage capacity of the region $B$. In reality, the true information-capacity may be much smaller. For black holes, the entanglement
entropy gives the same scaling with area as the Bekenstein entropy. However, this fact does not constitute a satisfactory microscopic explanation of the black hole entropy, but rather a reflection of a simple coincidence, namely that a black hole happens to saturate the Bekenstein entropy bound. Correspondingly, the two entropies agree in scaling.
Therefore, in our approach we shall not rely on entanglement entropy.
The question we would like to ask is whether there is a more efficient measure of information-storage capacity that Alice could derive without the detailed knowledge of the microscopic properties of the system $B$.
In the present paper we would like to point out that in systems in which the information carriers are some sort of gauge degrees of freedom, the information-storage capacity of the interior region can be understood in terms of the energy gap of the boundary
St\"uckelberg degrees of freedom that are necessary for maintaining gauge redundancy of Alice's description in the presence of the information boundary $S$.
In order to outline the general argument, let us assume that Alice deals with a bulk gauge theory of a gauge field
$h$ with a Lagrangian density $L(h)$, that under gauge transformation shifts by a total derivative $\delta L = d\Omega$.
In the presence of the information boundary $S$, this generates a surface term $\int_{S} \Omega$, where
the integral is taken over the world-volume of the boundary surface $S$.
By gauge redundancy, such a term must be cancelled by a shift of some St\"uckelberg field, $\eta$.
Thus, the effective theory
(from the information perspective) that Alice deals with is a bulk gauge theory plus the world-volume action of surface $S$ with the St\"uckelberg degrees of freedom,
\begin{equation}
{\mathcal S} \, = \, \int_{A} L(h) + \int_{S} \eta \,,
\label{action}
\end{equation}
such that the variation of $\eta$ is $\delta \eta = - \Omega$.
It is important to stress here what is the key point of the former argument. Alice needs to have in her region {\it manifest} gauge invariance. The necessity of imposing this condition on her physics forces her to decorate what for her is acting as an information boundary with new physical degrees of freedom. In other words, she is deriving part of the actual dynamics of the information boundary by imposing a manifest gauge invariance in the region that is, in information terms, accessible to her.
The new degree of freedom $\eta$ provides the natural qubits that store (part of the) information about the region $B$. The elementary qubits can be labelled either by the world-volume coordinates of the boundary, or equivalently by the world-volume momentum modes up to some maximal momentum. For example, for a spherical boundary these can be labelled by the usual spherical harmonics $l,m$.
Obviously, the total number of elementary qubits scales as the {\it area} of the boundary $S$. Thus, the quantum information can be stored in the occupation numbers of different world-volume momentum modes. For example, in the spherical case the quantum information can be included in a set of occupation numbers $n_{m,l}$ labeled by the
spherical harmonics $m,l$.
The capacity of information-storage is measured by the energy gap of the qubit $\eta$.
Let this gap be $\Delta$. Let us also assume that the size of the region $B$ is $R$.
Then, a useful measure of the information-storage capacity is the following quantity,
\begin{equation}
{\mathcal I} \equiv {\hbar \over R\Delta} \, .
\label{Inform}
\end{equation}
The physical meaning of the quantity $ {\mathcal I} $ is the following. In a generic system of size $R$, with weakly interacting degrees of freedom, we can store a single bit of information at the energy expense $\sim \hbar /R$. Indeed,
this energy roughly measures the spacing between the ground-state and the first excited energy-level in such systems.
Thus, the quantity $ {\mathcal I} $ gives us a measure of how costly is, relative to this energy,
the storage of a single bit of information in the St\"uckelberg degree of freedom $\eta$.
If $ {\mathcal I} \gg 1$ the system is an efficient storer of information. In the opposite case, $ {\mathcal I} \ll 1$, the information-storage capacity of the system is very poor \footnote{There exist a different natural interpretation of ${\mathcal I}$ that Alice may immediately think about. Since Alice knows the size $R$ of the region $B$, she can infer the quantum uncertainty in the energy stored in this region as $\frac{\hbar}{R}$. However, for her this uncertainty is {\it resolved} into ${\mathcal I}$ St\"uckelberg boundary states per each qubit given by $\eta$. Hence, she can already infer that ${\mathcal I}$ measures a part of the {\it degeneracy} of the ground state, whatever may be the underlying reason from the microscopic physics point of view, describing the region-$B$.}.
Notice that the quantity ${\mathcal I}$ measures the part of the entropy that is stored in the
St\"uckelberg modes. This is simple to understand.
In order to evaluate this part of the entropy, we have to compute the number of
St\"uckelberg states that populate the energy levels within the energy gap $\hbar /R$ and then take the log of this number. Restricting ourselves by considering
the first excited level for each qubit, the state with $N$ excited qubits has energy
$E = N \Delta$. Restricting this energy to be $ E < \hbar /R$, we easily get that the maximal
allowed number of the excited qubits is $N_{max}={\mathcal I}$. The corresponding number of states
obviously is $\sim 2^{\mathcal I}$ and the resulting contribution to the entropy is ${\mathcal I}$.
We can derive a more general expression relating the entropy to the number of St\"uckelberg modes $N$ and the maximal number of excited modes ${\cal I}$. Through simple combinatorics, we obtain
\begin{equation}
\label{eq:entropy}
e^S = \sum_n^{\cal I}{N + n - 1 \choose n} = \frac{\Gamma(1 + N + {\cal I})}{\Gamma(1 + {\cal I})\Gamma(1 + N)}\,,
\end{equation}
where the sum is to count all states with occupation $n \leq {\cal I}$.
For large $N$ and ${\cal I}$, the leading order expression for general $N$ and $\Delta$ is
\begin{equation}
\label{eq:entropy_large}
S \approx {\cal I}\left(1 + \log\frac{N}{{\cal I}}\right)
\end{equation}
How big is $\Delta$? On very general physical grounds, we can estimate it in the following way.
Let $m$ be the characteristic mass scale of the theory in the boundary region. For example,
if the mass gap is generated by fluctuations of some massive particles composing the boundary,
the scale $m$ will be set by the mass of these degrees of freedom. In case of no pre-existing mass gap in the theory, the natural scale will be given by the inverse size of the system,
$m= \hbar/R$. Let $\alpha$ be the quantum coupling of the fluctuating degrees of freedom.
Then the expected energy gap for the St\"uckelberg modes is
\begin{equation}
\Delta_\text{St\"uckelberg} = \alpha(m) \, m \, ,
\label{gap!}
\end{equation}
where $\alpha(m)$ is the coupling evaluated at the energy scale $m$ (or equivalently
at the length-scale $\hbar/m$).
Thus, the information capacity is,
\begin{equation}
{\mathcal I} \equiv {1 \over \alpha (m)} \, {\hbar \over Rm} \, .
\label{Im}
\end{equation}
On the other hand, if the gauge theory in question is massless, then we must take $m= \hbar/R$ and
the St\"uckelberg qubit energy gap becomes,
\begin{equation}
\Delta_\text{St\"uckelberg} \, = \, \alpha(R) {\hbar \over R} \, .
\label{gapmzero}
\end{equation}
The information measure is therefore given by,
\begin{equation}
{\mathcal I} \, = \, {1 \over \alpha (R)} \, .
\label{Imzero}
\end{equation}
Thus, for a gapless gauge theory that can dynamically produce the information boundary, the
information capacity is measured by the inverse quantum coupling, exactly in the same way
as this was happening, according to (\ref{entropy}), for Bekenstein entropy!
In other words, applying (\ref{Imzero}) to the black hole horizon, and taking into account that the gravitational coupling at the scale $R$ is given by (\ref{gravitycoupling}), yields
\begin{equation}
{\mathcal I}_{BH} \, = \, {R^2 \over L_P^2} \,
\end{equation}
For a black hole, the number of St\"uckelberg modes $N$ and the information measure ${\cal I}$ coincide. Hence we recover from Eq.\eqref{eq:entropy_large} the well-known area law,
\begin{equation}
S = {R^2 \over L_P^2} = S_{BH} \, .
\label{areaI}
\end{equation}
Let us summarize our main message. In any gauge theory, Alice can characterize the part of the quantum information hidden in the region $B$ in terms of the St\"uckelberg qubits.
For an arbitrary gapless gauge system, this information is measured by
(\ref{Imzero}). But, for a generic system, it accounts only for a part of the hidden information.
For black holes, due to the scaling of the gravitational quantum coupling $\alpha(R)$, the
amount of information that can be accounted by St\"uckelberg qubits saturates the bound and thus accounts for the entire information of the hidden region.
Finally, an interesting connection emerges between the St\"uckelberg formulation presented here and the previous picture of holography developed in \cite{us,critical}. As explained above, in the latter picture,
holography results from the many-body description of the black hole interior in terms of a
critical graviton condensate delivering Bogoliubov modes with tiny energy gaps. In the present paper
we understand holography in the gauge-redundancy language of an external observer in terms of a
very similar energy gap of the St\"uckelberg modes. This coincidence suggests that we are dealing with two languages describing the same physics.
The striking similarity between (\ref{Imzero}) and (\ref{entropy}) indicates a
possible fundamental connection between holography and quantum criticality. This connection maps the St\"uckelberg qubits of Alice's description onto the Bogoliubov modes of the many-body quantum critical
state of \cite{us,critical}.
In the rest of the paper we shall elaborate on these ideas.
\section{Spin-2 Case}
We shall now discuss our idea in more details.
The first part of our argument is generic for any bounded system with gauge redundancy.
Let us therefore illustrate it for linearized gravity. The non-linearities can be taken into account, but they do not add anything to the essence of the phenomenon.
Consider the action of linearized Einstein gravity,
\begin{equation}
S_E \, = \, \int d^4x L(h) \, = \, \int d^4x h^{\mu\nu} {\mathcal E} h_{\mu\nu} \,,
\label{actionE}
\end{equation}
where ${\mathcal E} h_{\mu\nu} \, \equiv \, \Box h_{\mu\nu} \, - \, \eta_{\mu\nu} \Box h \, - \, \partial_{\mu} \partial^{\alpha} h_{\alpha\nu} \, - \, \partial_{\nu} \partial^{\alpha} h_{\alpha\mu} \, + \, \partial_{\mu} \partial_{\nu} h \, + \,
\eta_{\mu\nu} \partial^{\alpha}\partial^{\beta} h_{\alpha\beta}$
is the linearized Einstein tensor and $h\equiv h_{\alpha}^{\alpha}$. This system exhibits a gauge redundancy under the following transformation,
\begin{equation}
h_{\mu\nu} \rightarrow h_{\mu\nu} + \partial_{\mu} \xi_{\nu}
+ \partial_{\nu} \xi_{\mu}\,,
\label{gauge}
\end{equation}
where $\xi_{\mu}(x)$ is a transformation parameter vector.
Under this transformation, the Lagrangian density shifts by a total derivative,
\begin{equation}
\delta L \, = \, \partial_{\mu} \Omega^{\mu}\, ~~ {\rm where}~~ \Omega_{\mu} \equiv 2\xi^{\nu} {\mathcal E} h_{\mu\nu} \, .
\label{changeL}
\end{equation}
Let us now introduce a boundary. The role of it can be played by an arbitrary closed two-surface described by the target space coordinates
$X^{\mu},~\mu=0,1,2,3$ and the world-volume coordinates $y^{a}, ~a=0,1,2$. At the moment, the precise origin of the
boundary is not important for us. For instance, it can represent a dynamical solution of the theory, such as, e.g., a black hole horizon or a brane bubble. Alternatively, it can be imposed by hand as a mathematical surface. In either case, what is important for us is that the boundary separates the system into two sub-systems. We now wish to describe the gauge redundancy from the point of view of the exterior sub-system.
Alice sees a gauge theory on a space with boundary. In this description,
naively, under the gauge shift (\ref{gauge}) the action changes by a boundary term
\begin{equation}
\delta S \, = \int dX^{\mu}\wedge dX^{\nu}\wedge dX^{\alpha} \epsilon_{\mu\nu\alpha\beta} \Omega^{\beta} \, .
\label{varaction}
\end{equation}
However, since we started by a manifestly gauge-redundant theory, the same redundancy must hold
in the correct description of the sub-system.
In particular, this is obvious in cases in which the boundary represents a solution of the full gauge-invariant theory.
The only way to accommodate gauge invariance is to admit for the boundary to host a new degree of freedom, $\eta_{\mu}$, that acts as a St\"uckelberg field for maintaining the original gauge invariance. The action of this new degree of freedom is fixed from the above condition and has the following form,
\begin{equation}
S_{\eta} \, = \int dX^{\mu}\wedge dX^{\nu}\wedge dX^{\alpha} \epsilon_{\mu\nu\alpha\beta} \eta^{\beta} \, .
\label{actionstuck}
\end{equation}
Obviously, the combined action,
\begin{equation}
S = S_E + S_{\eta} \,,
\label{totalaction}
\end{equation}
is invariant under (\ref{gauge}) provided $\eta_{\mu}$ shifts as
\begin{equation}
\eta_{\mu} \rightarrow \eta_{\mu} - \Omega_{\mu} \,,
\label{shifteta}
\end{equation}
with $\Omega_{\mu}$ given by (\ref{changeL}).
Notice that $\eta_{\mu}$ is a fully legitimate degree of freedom that transforms as a scalar from the point of view
of the boundary world-volume theory. The very existence of this St\"uckelberg degree of freedom follows solely
from the requirement of symmetry under small diffeomorphisms. However, what promotes them into the physical carriers
of information is the interaction.
Classically these modes are exactly gapless and can be labeled by world-volume coordinates of the boundary theory.
This means that the number of St\"uckelberg qubits scales as area of the boundary!
This certainly rings a bell, and makes it tempting to identify the St\"uckelberg fields as the holographic degrees of freedom
for a black hole. However, the same coincidence also raises the question: Why are not
all information boundaries holographic? We shall address this question in the rest of the paper.
\section{Quantum Generation of Energy Gap and Information Counting}
In our derivation of boundary St\"uckelberg degrees of freedom, we have never used black hole properties. Our argument solely relied on gauge invariance and is therefore generic for an arbitrary boundary. So, what is special about the black hole horizon?
As we shall explain now, the question is closely related to the question of generation of an energy gap in
boundary St\"uckelberg modes. As discussed above, this gap defines the information-capacity (\ref{Inform}) of the St\"uckelberg modes, which in general can account for a certain fraction of the
total information capacity of the bounded inner system. Black holes turn out to be the systems with the smallest St\"uckelberg energy gap, which is just enough to saturate the
bound on information. With such a small gap, black hole horizon St\"uckelbergs can account for the entire information of the system. This is why black holes are holographic.
The source of the energy gap are quantum fluctuations.
The crucial point is that in a quantum theory, we must allow the boundary to fluctuate.
For example, for a closed brane bubble, the fluctuations are due to degrees of freedom that compose it. Similarly, for any finite mass black hole there is at least one model-independent source of horizon fluctuation: The back-reaction from Hawking evaporation.
Once the boundary fluctuations are taken into account, they create
bilinear (and higher) terms in the effective St\"uckelberg action. Such terms generate the energy gap in
St\"uckelberg qubits. On very general grounds, the energy gap can be estimated as
(\ref{gap!}), where $\alpha(R)$ is the characteristic quantum coupling of the fluctuating degrees of freedom, evaluated at the scale $R$, and
$m$ is their characteristic energy scale. In a massless theory it is given by the inverse size of the system, $m=\hbar/R$. Or, if the theory has an intrinsic mass gap in the spectrum,
$m$ is set by the latter.
In order to see more formally why the gap is generated, let us rewrite the action (\ref{actionstuck}) in a four-dimensional bulk language
\begin{equation}
{\mathcal S}_{\eta} \, = \int d^4x \, \eta_{\mu}(x) \,J^{\mu}(x)\,,
\label{coupling}
\end{equation}
where, $J^{\mu}$ is the Hodge-dual,
$J^{\mu} \equiv \, \epsilon^{\mu\nu\alpha\beta} J_{\nu\alpha\beta}$ \,,
of the boundary current,
\begin{equation}
J^{\nu\alpha\beta}(x) \equiv \int dX^{\nu}\wedge dX^{\alpha}\wedge dX^{\beta} \, \delta^4(x^{\mu} - X^{\mu}) \, .
\label{current}
\end{equation}
Due to virtual quantum processes the current-current correlator is non-vanishing (once interactions are taken into account) and induces a bilinear term in the action,
\begin{equation}
\eta_{\mu}\eta_{\nu} \langle J^{\mu} J^{\nu} \rangle \,.
\label{term}
\end{equation}
Of course, the bilinear term will be generated for the full gauge-invariant combination,
\begin{equation}
\left( L(h) \, + \, \partial^{\mu} \eta_{\mu} \right)^2 \,.
\label{coupling}
\end{equation}
Thus, as it is usual, the St\"uckelberg field acquires an energy gap via a Higgs-type mechanism. The
coefficient of this term can only vanish in the limit in which the fluctuations of the boundary can be ignored.
The higher is the ability of the boundary to fluctuate, the bigger is the generated mass gap.
It is now clear what limits the capacity of information-storage (\ref{Inform}). It is the energy gap, $\Delta$, in the
St\"uckelberg qubit. If $\Delta$ is large, the storage of information in qubits is very costly, and the corresponding
system is far from being holographic. This is the reason why most of the systems with information
boundaries are not holographic.
We are now ready to understand, at least qualitatively, why the black hole horizon is special among all possible information boundaries. We shall show this using two different arguments.
The first argument comes from a straightforward application of (\ref{gap!}) to the black hole case.
If we assume that the quantum fluctuations are due to gravitons of wavelength $R$, we have to take for $\alpha(R)$ the expression (\ref{gravitycoupling}) and $m=\hbar/R$. Substituting both into (\ref{gap!}), we get
\begin{equation}
\Delta_{BH} \, = \, {\hbar L_P^2 \over R^3} \,.
\label{deltablack}
\end{equation}
Remarkably, as already discussed above, this gives an information measure (\ref{Inform}) which, as shown at the end of section I, reproduces
Bekenstein entropy.
An alternative argument that leads us to the same result is based on estimating the strength of quantum fluctuations using the measure of back-reaction from the Hawking radiation.
For this, first recall that the quantum fluctuations of the horizon vanish in the semi-classical limit,
in which $L_P \rightarrow 0$ and the black hole size $R$ is kept finite.
In this limit the black hole horizon becomes {\it infinitely rigid} and is insensitive to any type of quantum back-reaction. In the case of finite $L_P$, the back-reaction on the horizon from Hawking radiation can easily be estimated to be of order $L_P^2/R^2$, by taking into account the relative change of the black hole mass, $\delta M_{BH}$, or its temperature,
due to each Hawking emission,
\begin{equation}
{\delta M_{BH} \over M_{BH}} \, = \,- \, \alpha(R) \, .
\label{backreaction}
\end{equation}
As it is clear, this back-reaction parameter coincides with the coupling of gravitons of wavelength $R$ discussed above, so we have denoted both by the
same symbol $\alpha(R)$.
Thus, the parameter $\alpha(R)$ is the measure of the black hole quantumness. It is
natural to expect that the quantum-mechanically generated energy gap in St\"uckelberg qubits must be
suppressed by this additional factor relative to a gap expected for an ordinary quantum system of size $R$, which would be
$\hbar /R$. This gives a simple alternative estimate for the quantum-generated energy gap of the St\"uckelberg qubit,
\begin{equation}
\Delta_{BH} = \alpha(R) { \hbar \over R} \, ,
\label{gapalpha}
\end{equation}
which agrees with (\ref{deltablack}).
Hence, both estimates give us the same energy gap of St\"uckelberg modes, which leads to the expressions (\ref{Imzero}) and (\ref{areaI})
for the quantity ${\mathcal I} $ that matches the Bekenstein entropy of a black hole of radius $R$.
From (\ref{gap!}) it is easy to understand why there are no systems of size $R$ with
the energy gap in the St\"uckelberg fields smaller than the black hole horizon. Indeed, no matter how weakly the quantum constituents are interacting by other forces, they must interact gravitationally. Correspondingly, putting aside accidental cancellations, the gravitational coupling $\alpha$ sets the lowest bound on the quantum interaction strength and subsequently on the
generated energy gap \footnote{In this light, it would be interesting to see if the energy gap can be maintained smaller due to cancellations, e.g., by supersymmetry.}.
For a larger energy gap $\Delta$, the entropy of the bounded inner system would be much less than the black hole entropy. This allows us to understand the high capacity of black hole information-storage
in terms of the lowest mass gap in St\"uckelberg fields.
\section{Examples with Large Energy Gap for the Boundary St\"uckelberg Mode}
In order to get a broader view, let us now consider other gauge systems, with dynamically-generated information boundaries. As we shall see, the information measure ${\mathcal I}$ in these systems is very small, due to the energy gap which goes hand in hand with the generation of the boundary. We shall consider two examples, which in a certain sense are
dual to each other.
\subsection{Classical energy gap: Information boundary from Meissner effect}
The first example is given by a slightly-deformed Higgs model that in addition to the usual Higgs vacuum, with a massive
photon, admits a second vacuum in which the photon is massless and the $U(1)$-gauge theory is in the Coulomb
phase.
The simple model is given by the following Lagrangian,
\begin{equation}
L_{\Phi} \, = \, |D_{\mu} \Phi|^2 \, - \, F_{\mu\nu} F^{\mu\nu} \, - \, \lambda^2 {|\Phi|^2 \over v^2} \left (|\Phi|^2 - v^2 \right )^2 \,,
\label{Higgs}
\end{equation}
where $\Phi (x) = |\Phi (x)| e^{i\theta(x)}$ is a complex scalar field, $F_{\mu\nu} \equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ is the Maxwellian field strength and
$D_{\mu} \equiv \partial_{\mu} - ig A_{\mu}$ with $g$ being the gauge coupling.
The two physically inequivalent vacua are, $\langle |\Phi| \rangle \, = \, 0$ and $\langle |\Phi| \rangle \, = \, v$.
In the first vacuum, the propagating fields are the massless photon and a complex scalar of mass
$m_{\Phi} \, = \, \hbar \lambda v$, whereas in the second one we have a massive photon with the mass
$m_A = \hbar gv$ and a real Higgs scalar with the mass $m_{\Phi} = \hbar \lambda v$. In both vacua the total number of propagating degrees of freedom is equal to four.
This theory also admits configurations in which the two vacua coexist and are separated
by a domain wall across which the Higgs expectation value interpolates from $0$ to $v$. Without loss
of generality we can place the wall at $z=0$.
The thickness of the wall
is set by the Compton wavelength of the Higgs particle $ \delta = (\lambda v)^{-1}$. At energies $\ll m_{\Phi}$ we can integrate out the thickness of the wall and the effective theory of the photon field becomes,
\begin{equation}
L_{eff} = - F_{\mu\nu} F^{\mu\nu} + \hbar^{-2}m^2(z) \left(A_{\mu} - {1 \over g} \partial_{\mu} \theta\right )^2\, ,
\label{photoneff}
\end{equation}
where the function $m^2(z)$ can be approximated as a step function $m^2(z < 0 ) = 0,~~ m^2(z > 0 ) = m_A$.
At energies $\ll m_A$ the photon cannot penetrate the Higgs region and the effective theory of the photon
is a Coulomb domain bounded by the wall. The eaten-up Goldstone $\theta$ plays the role
of the St\"uckelberg field that is maintaining the gauge redundancy, $A_{\mu} \rightarrow A_{\mu} + {1 \over g} \partial_{\mu} \omega, ~~~ \theta \rightarrow \theta + \omega$, despite the presence of the boundary.
However, there is a mass gap present already at the classical level. In fact, this allows an observer to conclude that the other side of the wall is in the Higgs phase.
Let us take for a region $B$ the interior of a Higgs vacuum bubble of radius $R$, embedded into
the Coulomb vacuum (region $A$)\footnote{If the two vacua are exactly degenerate, as it is the case for the Lagrangian
(\ref{Higgs}), the finite size bubble cannot be static and will tend to collapse because of the tension of the bubble wall. This is not a big complication for out purposes, if we are interested in information-storage properties
over short time intervals. Moreover, we can make the bubble of a given radius static by balancing the tension force
by a pressure-difference created via a small difference between the energy densities of the two vacua. An alternative
way is to stabilize the bubble by introducing some charged particles that are localized within the bubble wall. }. Then the region $B$, which is in the Higgs phase, at low energies is bounded by the sphere. The St\"uckelberg field is the eaten-up Goldstone that maintains the gauge redundancy throughout the space. But there is a large energy gap. The information measure $I$ is always very small, unless $m_A < \hbar /R$. The latter choice however
would mean that effectively there is no information boundary for the photon. In such a regime, we need to
consider the information stored in the Goldstone mode $\theta$ separately.
The latter statement becomes obvious by taking an extreme case when we switch off the gauge coupling completely
$g =0$, but keep $v$ finite. Then there is a massless Goldstone mode localized within the bubble and the information can be stored in
its zero momentum mode. However, in this limit the bubble is transparent for the photon and no information boundary exists for the gauge field.
\subsection{Quantum gap: Boundary from dual Meissner effect}
Let us consider a situation in which the information boundary is dynamically formed by a magnetic monopole condensate. This can be achieved by employing the massless gauge field localization mechanism of \cite{giamisha}, which is based on interpolation between the confining and Coulomb phases.
We can achieve this situation by modification of the
model (\ref{Higgs}) by embedding the $U(1)$ into a gauge $SU(2)$-symmetry and promoting the Higgs field into a real triplet
representation $\Phi^a,~a=1,2,3$. The Lagrangian now becomes,
\begin{equation}
L_{SU(2)} \, = \, |D_{\mu} \Phi^a|^2 \, - \, F_{\mu\nu}^a F^{a\mu\nu} \, - \, \lambda^2 {\Phi^a\Phi^a \over v^2} \left (\Phi^b\Phi^b - v^2 \right )^2 \, ,
\label{Higgstripled}
\end{equation}
where $D_{\mu}$ is the usual $SU(2)$-covariant derivative and $F_{\mu\nu}^a$ is a non-abelian
field strength. We have chosen the Higgs potential as in \cite{monopole1}.
Now, the two degenerate vacua are $\langle \Phi^a \rangle =0$ and $\langle \Phi^a \rangle = \delta^a_3 v$. In the first one, the theory is in the
$SU(2)$-phase.
In the second one, the $SU(2)$-group is Higgsed down to a $U(1)$ subgroup, which is in the Coulomb phase.
Again, we can consider a configuration in which the two phases are separated by a domain wall placed
at $z=0$. Notice, unlike the previous example, here at the classical level the wall is transparent for the $U(1)$-photon.
However, in the quantum theory the story changes dramatically.
The $SU(2)$-phase becomes confining and develops a mass gap given by the QCD scale $\Lambda$.
The photon traveling across the wall can only penetrate the $SU(2)$-domain in form of a glueball
of mass $\Lambda$. Correspondingly, at energies below $\Lambda$ the effective theory of
the $U(1)$-photon is a gauge theory with a boundary. The gauge invariance is again maintained by a St\"uckelberg field, the role of which is played by the phase of the monopole condensate that gives mass to a magnetic photon (see \cite{monopole1,monopole} for detailed discussion of different aspects of such a picture).
But, the very same physics that generates the monopole condensate also generates the mass gap due to the confinement of the electric charges.
In order to make the example more transparent let us place Alice in the $SU(2)$-domain.
Imagine that Alice is observing the bubble of the $U(1)$-vacuum. Let us take the size of the bubble $R$ to be much larger than the QCD length, $R \gg \hbar/\Lambda$.
Inside this bubble information can be stored
in form of the low-energy photon quanta. If these quanta have energies $\ll \Lambda$, the information stored in them cannot penetrate into the $SU(2)$ -domain, where the energy gap is
$\Lambda$ \footnote{Recently, a coherent state picture of photons in a ball was studied in \cite{ball}.
It would be interesting to see if the $SU(2)$-confining region discussed here provides the boundary conditions imposed in this analysis and also to see if the critical transition can be modeled in the many body photon language, when the ball reaches the size of $SU(2)$ QCD-length, $R \sim \hbar/\Lambda$.}.
Thus, the domain wall separating the $U(1)$-Coulomb and $SU(2)$-confining
phases acts as an information boundary for Alice. However, the barrier in this case is
rather peculiar. Indeed, the $U(1)$ factor is nowhere Higgsed throughout the space and consequently
there is no potential energy barrier for the photon at the level of the fundamental Lagrangian of the form (\ref{photoneff}). What prevents the photon to escape out of the bubble is not
a charge condensate, but rather a condensate of magnetic monopoles.
However, the magnetic monopole condensate, unlike the charge condensate, is {\it not} generating any fundamental mass-term for a photon, but
rather is making it {\it non-propagating}.
The mass gap generated for the St\"uckelberg field satisfies the general relation of the type (\ref{Im}), with the only caveat that the role of $m$ is played by $\Lambda$ and
the role of $\alpha(m)$ is played by the QCD coupling $\alpha_{SU(2)}(\Lambda)$ evaluated
at the scale $\Lambda$, which is of order one.
Thus, in both considered examples the mass gap in the St\"uckelberg field is generated simultaneously with the
generation of the boundary. In both cases the information boundary is bounding the regions of condensed charges, which are
either electric or magnetic. In both cases, the St\"uckelberg field is the phase of the condensate and in both cases
the generation of the mass gap in this degree of freedom is inseparable from the generation of the information boundary
\footnote{In all these examples for information measure ${\mathcal I}$ we get (\ref{Im})
where $m$ is the mass gap of Alice's physics. Obviously this mass gap must be bigger than $\frac{\hbar}{R}$ to allow Alice to know the actual existence of a hidden region of size $R$, so
in these cases ${\mathcal I}$ is trivially bounded by (\ref{Imzero}).}.
The difference is that in the electric condensate case, this phenomenon can take place already at the classical level, whereas in the case of magnetic charges the entire effect is quantum.
\section{Relation between Holography and Quantum Criticality}
According to (\ref{gapmzero}), in massless gauge systems with an information boundary the energy
gap in St\"uckelberg qubits is suppressed by the coupling $\alpha$, relative to the typical
energy gap, which would be expected to be $\sim \hbar /R$. This expression suggest an interesting underlying relation with
quantum criticality. According to the studies of \cite{critical,gold}, systems at the quantum critical point are very efficient storers of information, due to the appearance of qubits with an energy
gap that exhibits a suppression very similar to (\ref{gapmzero}). In particular, this is a property
of a gas of bosons with attractive interaction strength $\alpha$. In this case, the criticality is reached when the occupation number is equal to $\alpha^{-1}$. This system has been studied in a series of papers \cite{critical,gold} from a quantum information perspective, as a prototype model for black holes. By now it is well established that the quantum critical point is populated by
nearly-gapless qubits.
These qubits can be described in different languages. In the many-body language they can be described as Bogoliubov modes of the critical Bose-gas. Alternatively, they can be characterized
as the Goldstone modes of a non-linearly realized symmetry of the condensate \cite{gold}.
Based on this connection, the black hole portrait of \cite{us, critical, gold} suggests that black hole information is carried by the Bogoliubov-Goldstone qubits of the critical graviton condensate, with the energy gap
given by,
\begin{equation}
\Delta_\text{Bogoliubov} \, = \, \alpha(R) {\hbar \over R} \, .
\label{gapBH}
\end{equation}
The approach developed in the present paper suggests yet another description of these modes,
which naturally leads us to a potential link between quantum criticality and holography. Namely, the striking similarity between the energy gaps (\ref{gapmzero}) and (\ref{gapBH}) for the St\"uckelberg modes in a gauge description and Bogoliubov
modes in a many-body description of black holes, suggests that we are dealing with the two sides of the same coin and that the holographic systems microscopically represent the systems at a quantum critical point. The modes that by Alice are seen as St\"uckelberg degrees of freedom, are, in the microscopic
many-body description of the black hole, represented as the Bogoliubov-Goldstone modes of the critical graviton condensate. It is tempting to generalize this correspondence to other types
of holographic systems. This relation can also work in the opposite direction suggesting that
critical systems must exhibit some sort of holography in the sense presented here.
In this respect would interesting to investigate the observation of equivalence, in large $N$-limit, of
the grounds states of one dimensional $N$-particle Bose-gas on a ring and a gauge theory on a two-sphere \cite{andredaniel}.
The connection between the St\"uckelberg formulation of holography and quantum criticality gives an exciting possibility of experimentally studying yet another black hole property in critical systems that can be manufactured in table-top labs,
in the spirit of \cite{critical, gold}.
\section{Conclusions}
In this note we have suggested that some key features of holography can be understood in terms of the basic principles of gauge redundancy. The maintenance of this redundancy requires the appearance of St\"uckelberg degrees of freedom at the information boundary. These modes
then act as qubits that store a fraction or the entire information of the inaccessible region.
The amount of information stored by St\"uckelberg qubits is determined by their energy gap
$\Delta$.
Even if absent classically, this gap is generated by quantum effects and is given by
(\ref{gap!}).
Correspondingly, we have introduced a measure of information capacity ${\mathcal I}$ given by (\ref{Im}), which quantifies how energetically cheap is the excitation of St\"uckelberg qubits relative to the usual energy gap of a system of the same size, typically expected to be $\hbar/R$. For gauge theories without an intrinsic
mass gap in the particle spectrum, the quantities $\Delta$ and ${\mathcal I}$ are given by
(\ref{gapmzero}) and (\ref{Imzero}) respectively,
\begin{equation}
\Delta_\text{St\"uckelberg} \, = \, \alpha(R) {\hbar \over R}\,, ~~~
{\mathcal I} = {1 \over \alpha (R)} \, ,
\label{Ianddelta}
\end{equation}
which we have copied here for the reader's convenience. For the gravitational coupling $\alpha(R)$,
which scales as (\ref{gravitycoupling}), the above expression for ${\mathcal I}$
coincides with the Bekenstein entropy.
The expression (\ref{Ianddelta}) makes it clear, why among all possible information boundaries
the black hole horizon can store a maximal amount of information. The reason is simply that gravity
sets the lower bound on how weakly the quantum modes can interact. Even if particles in the theory
interact via additional hypothetical forces much weaker than gravity, they must still couple gravitationally.
Correspondingly, black holes house St\"uckelberg modes with the smallest possible gap in nature.
In this respect, it would be interesting to study what happens for extremal black holes or
other systems (e.g., supersymmetric ones) in which gravitational attraction of the constituents can be compensated
by the repulsion due to other forces. One may expect that the St\"uckelbergs for such systems remain gapless.
We thus conclude that black holes carry St\"uckelberg hair, which is closely analogous to the
Bogoliubov-Goldstone hair of the quantum critical picture of \cite{us,critical,hair,gold}. Both types of hair become infinite but unresolvable in the classical limit, because in this limit $\alpha(R)$ vanishes and
${\mathcal I}$ diverges.
Finally, taking into account the striking similarities in suppressions of their energy gaps, we have suggested a connection between the St\"uckelbergs and Bogoliubov modes of the critical graviton condensate. We have proposed that holographic systems are secretly systems at a quantum critical point. Generalizing this hypothesis in the opposite direction, the usual critical systems that can be obtained in table top labs, such as critical Bose-Einstein condensates of cold atoms, must also exhibit some notion of holography, which can open up very interesting experimental prospects.
\section*{Acknowledgements}
The work of G.D. was supported in part by Humboldt Foundation under Humboldt Professorship,
ERC Advanced Grant 339169 "Selfcompletion'', by TRR 33 "The Dark
Universe" and by the DFG cluster of excellence "Origin and Structure of the Universe".
The work of C.G. was supported in part by Humboldt Foundation and by Grants: FPA 2009-07908, CPAN (CSD2007-00042) and by the ERC Advanced Grant 339169 "Selfcompletion''.
The work of N.W. was supported by
the Swedish Research Council (VR) through the Oskar Klein Centre.
|
2,869,038,155,673 | arxiv | \section{Introduction and main results}
\noindent
In this paper, we consider a maximizing problem associated with
Sobolev type embedding $BV(\Bbb R^N)\hookrightarrow L^r(\Bbb R^N)$ for $1\leq r\leq 1^*:=\frac{N}{N-1}$ with $N\geq 2$, where $BV$ denotes the space of bounded variation,
see \cite{G} and Section 2. The inequality associated with the embedding $BV\hookrightarrow L^{1^*}$ is Mazya's inequality with its best-constant $E$ given by
\begin{align}\label{mazya}
E:=\sup_{u\in BV\setminus\{0\}}\left(\frac{\|u\|_{1^*}}{\|u\|_{TV}}\right)^{1^*}
=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{\frac{1}{N-1}},
\end{align}
where $\omega_{N-1}$ denotes the surface area of the $N$-dimensional unit ball, see \cite{M}.
It is well-known that \eqref{mazya} is
equivalent to the isoperimetric inequality,
and maximizers of $E$ consist of functions of the form $\lambda\chi_{B}\in BV$ with $\lambda\in\Bbb R\setminus\{0\}$ and a ball $B\subset\Bbb R^N$. A variational problem investigated in this paper is formulated as follows : for given $\alpha>0$,
\begin{align}\label{basic-pro}
D_\alpha:=\sup_{u\in BV, \,\,\|u\|_{TV}^a+\|u\|_{1}^b=1}\left(
\|u\|_{1}+\alpha \|u\|_{q}^q
\right),
\end{align}
where $1<q\leq 1^*$ and $a, b>0$. Especially for the critical case $q=1^*$, the maximizing problem associated with $D_\alpha$ suffers from both of the non-compactness of $BV\hookrightarrow L^1$ and $BV\hookrightarrow L^{1^*}$ called vanishing and concentrating phenomena, respectively.
One of our goals is to clarify an effect of the exponents $a$ and $b$ in
the inhomogeneous constraints on the (non-)attainability of $D_\alpha$.
\medskip
The attainability of maximizing problems corresponding to
the Sobolev embedding $W^{1,p}\hookrightarrow L^r$, where $1<p<N$,
$p\leq r\leq p^*:=\frac{Np}{N-p}$, were studied in \cite{IW2, N2}.
The authors in \cite{IW2} treated the variational problem given by
\begin{align*}
\sup_{u\in W^{1,p}, \,\,\|\nabla u\|_p^a+\|u\|_p^b=1}\left(
\|u\|_p^p+\alpha \|u\|_{q}^q
\right),
\end{align*}
where $p<q<p^*$ and $a,b>0$. This problem contains a difficulty coming from the
non-compactness of $W^{1,p}\hookrightarrow L^p$
due to a vanishing phenomenon.
After that, the author in \cite{N2} considered the same problem for the critical case $q=p^*$.
In this case, the problem becomes more complicated since one needs to exclude
both of vanishing and concentrating behaviors of a maximizing sequence
due to the non-compact embeddings $W^{1,p}\hookrightarrow L^p$ and $W^{1,p}\hookrightarrow L^{p^*}$, respectively.
The usual way in attacking this problem will be to compute the thresholds with respect to vanishing and concentrating phenomena and to investigate behaviors of a maximizing sequence in order to recover the compactness of the functional,
which was a strategy used in \cite{IW2}. However, the author in \cite{N2} gave
an alternative way in discussing the problem without a use
of the variational method directly. A main key used in \cite{N2} is to give
another expression of the functional in terms of the corresponding $1$-dimensional function
by a scaling argument. Based on these known results,
we consider the remaining case $p=1$, which leads to the problem \eqref{basic-pro}.
In fact, we observe that the method used in \cite{N2} can work for the marginal case $p=1$
by replacing $W^{1,1}$ with $BV$. Also, as an advantage of the case $p=1$,
we know the exact forms of maximizers of $E$ through the isoperimetric inequality,
and as a result, we obtain a characterization of maximizers of $D_\alpha$,
see Theorem \ref{thm5}.
\medskip
In order to state our main results, we start from the problem \eqref{basic-pro} with the subcritical case $1<q<1^*$.
In this case, the embedding $BV_{rad}\hookrightarrow L^q$ is compact,
where $BV_{rad}$ denotes the set of radially symmetric functions in $BV$,
and hence, the term $\|u\|_{q}$ in the functional will make an aid
to admit a maximizer of $D_\alpha$, see \cite{FP}.
On the other hand, $D_\alpha$ suffers from the non-compactness
of $BV\hookrightarrow L^1$, which comes from
the scaling $u_n(x):=\frac{1}{n^N}u(\frac{x}{n})$ with a fixed
$u\in BV\setminus\{0\}$. In general, we call $\{u_n\}_n\subset BV$ ``\,a vanishing sequence\,'' if $\{u_n\}_n$ satisfies the conditions :
$$
\sup_n\|u_n\|_{BV}<\infty,\quad\inf_n\|u_n\|_{1}>0,\quad\lim_{n\to\infty}\|u_n\|_{TV}=0.
$$
We also introduce the value $\alpha_v=\alpha_v(a,b,q)\in[0,\infty)$ defined by
\begin{align*}
\alpha_v:=\displaystyle\inf_{u\in BV, \,\,\|u\|_{TV}^a+\|u\|_{1}^b=1}\frac{1-\|u\|_{1}}{\|u\|_{q}^q}.
\end{align*}
If there exists a maximizing sequence $\{u_n\}_n$ of $D_\alpha$
such that $\{u_n\}_n$ is also a vanishing sequence,
we easily see $D_\alpha\leq1$. On the other hand, since $\alpha>\alpha_v$ is equivalent to $D_\alpha>1$, the value $\alpha_v$ is expected to be the threshold of $\alpha$ on the attainability of $D_\alpha$.
Our first result is stated as follows :
\begin{thm}\label{thm1}
Let $1<q<1^*$, $a>0$ and $b>0$.
\medskip
\noindent
{\bf (Non-threshold case $\alpha\ne\alpha_v$)}
\medskip
\noindent
{\rm (i)} Let $a>N(q-1)$. Then there holds $\alpha_v=0$, and $D_\alpha$ is attained for $\alpha>0$.
\medskip
\noindent
{\rm(ii)} Let $a\leq N(q-1)$. Then there holds $\alpha_v>0$, and $D_\alpha$ is attained for $\alpha>\alpha_v$, while $D_\alpha$ is not attained for $\alpha<\alpha_v$.
\medskip
\noindent
{\bf(Threshold case $\alpha=\alpha_v$)}
\medskip
\noindent
{\rm (iii)} Let $a<N(q-1)$, or let $a=N(q-1)$, $\frac{2N-1}{2(N-1)}<q<1^*$ and $b<b_0:=(q-1)(N-1)-\left(N-(N-1)q\right)$.
Then $D_{\alpha_v}$ is attained.
\medskip
\noindent
{\rm (iv)} Let $a=N(q-1)$ and $\begin{cases}
&\hspace{-.2cm}\frac{2N-1}{2(N-1)}<q<1^*\text{ \,and \,}b\geq b_0,\\
&\hspace{-.2cm}\text{or \,}1<q\leq\frac{2N-1}{2(N-1)}.
\end{cases}$ Then $D_{\alpha_v}$ is not attained.
\end{thm}
Next, we estimate the value $\alpha_v$. To this end, we introduce
the best-constant of the Gagliardo-Nirenberg type inequality $E_q$ :
$$
E_q:=\sup_{u\in BV\setminus\{0\}}\frac{
\|u\|_{q}^q
}{
\|u\|_{1}^{q-(q-1)N}\|u\|_{TV}^{(q-1)N}.
}
$$
One can calculate $E_q=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}$
and remark that $E_{1^*}=E$ is Mazya's best-constant, see Proposition \ref{GN-attain} (i).
By means of $E_q$, the value $\alpha_v$ is estimated as follows :
\begin{thm}\label{thm2}
\noindent
Let $1<q<1^*$, $a>0$ and $b>0$.
\medskip
\noindent
{\rm(i)} There hold $
\alpha_v
\begin{cases}
=0\quad\text{when \,}a>N(q-1),\\
>0\quad\text{when \,}a\leq N(q-1).
\end{cases}
$
\medskip
\noindent
{\rm(ii)} Let $a=N(q-1)$. Then there hold
\begin{align*}
\begin{cases}
&\hspace{-.3cm}
\alpha_v=\frac{1}{b E_q}\text{ \,when }
\begin{cases}
&\hspace{-.2cm}\frac{2N-1}{2(N-1)}<q<1^*\text{ \,and \,}b\geq b_0,\\
&\hspace{-.2cm}\text{or \,}1<q\leq\frac{2N-1}{2(N-1)},
\end{cases}\vspace{.2cm}\\
&\hspace{-.3cm}0<\alpha_v<\frac{1}{b E_q}
\text{ \,when \,}
\frac{2N-1}{2(N-1)}<q<1^*\text{ \,and \,}b<b_0.
\end{cases}
\end{align*}
\noindent
{\rm(iii)} {\bf(Asymptotic behaviors of $\alpha_v$ on the parameters $a$ and $b$)}
\medskip
\noindent
{\rm (a)} There holds $\lim_{a\downarrow 0}\alpha_v=\infty$.
\medskip
\noindent
{\rm (b)} Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b\geq b_0$, or let $1<q\leq\frac{2N-1}{2(N-1)}$.
Then there holds $\lim_{a\uparrow N(q-1)}\alpha_v=\frac{1}{b E_q}$.
\medskip
\noindent
{\rm(c)} Let $a\leq N(q-1)$. Then there hold $\lim_{b\downarrow 0}\alpha_v=\infty$
and $\lim_{b\to\infty}\alpha_v=0$.
\medskip
\noindent
{\rm(d)} Let $a=N(q-1)$ and $\frac{2N-1}{2(N-1)}<q<1^*$. Then there holds
$\lim_{b\uparrow b_0}\alpha_v=\frac{1}{b_0 E_q}$.
\end{thm}
We proceed to the critical case $q=1^*$.
In this case, $D_\alpha(a,b,1^*)$ suffers from
the non-compactness of not only $BV\hookrightarrow L^1$ but also $BV\hookrightarrow L^{1^*}$. The latter non-compactness comes from the scaling
$u_n(x):=n^{N-1}u(nx)$ with a fixed $u\in BV\setminus\{0\}$.
In general, we call $\{u_n\}_n\subset BV$ ``\,a concentrating sequence\,'' if
$\{u_n\}_n$ satisfies the conditions :
$$
\sup_n\|u_n\|_{BV}<\infty,\quad \inf_n\|u_n\|_{1^*}>0,\quad\lim_{n\to\infty}\|u_n\|_{1}=0.
$$
We also introduce the value $\alpha_c=\alpha_c(a,b)\in(0,\infty]$ defined by
$$
\alpha_c:=\displaystyle\sup_{u\in BV, \,\,\|u\|_{TV}^a+\|u\|_{1}^b=1}\frac{
\|u\|_{1}
}{E-\|u\|_{1^*}^{1^*}},
$$
where note $E-\|u\|_{1^*}^{1^*}\geq E\left(1-\|u\|_{TV}^{1^*}\right)>0$ since $0<\|u\|_{TV}<1$.
If there exists a maximizing sequence $\{u_n\}_n$ of $D_\alpha$
such that $\{u_n\}_n$ is also a concentrating sequence,
it is easy to see $D_\alpha\leq\alpha E$. On the other hand, since $\alpha<\alpha_c$ is equivalent to $D_\alpha>\alpha E$, the value $\alpha_c$ is expected to be the threshold of $\alpha$ on the attainability of $D_\alpha$ regarding to the concentrating phenomenon.
In fact, we can show that $D_\alpha$ with $\alpha$ in the region $(\alpha_v,\alpha_c)$ admits a maximizer whenever $\alpha_v<\alpha_c$, see Lemma \ref{vani-conce-ge} (iii).
We now state the attainability result on $D_\alpha=D_\alpha(a,b,1^*)$ :
\begin{thm}\label{thm3}
Let $a>0$ and $b>0$.
\medskip
\noindent
{\bf(Non-threshold case $\alpha\ne\alpha_v$ and $\alpha\ne\alpha_c$)}
\medskip
\noindent
{\rm(i)} Let $a>1^*$ and $b>1$. Then there hold $\alpha_v=0$ and $\alpha_c=\infty$,
and $D_\alpha$ is attained for $\alpha>0$.
\medskip
\noindent
{\rm(ii)} Let $a>1^*$ and $b\leq 1$. Then there hold $\alpha_v=0$ and $\alpha_c<\infty$,
and $D_\alpha$ is attained for $0<\alpha<\alpha_c$,
while $D_\alpha$ is not attained for $\alpha>\alpha_c$.
\medskip
\noindent
{\rm(iii)} Let $a\leq 1^*$ and $b>1$. Then there hold $\alpha_v>0$ and $\alpha_c=\infty$,
and $D_\alpha$ is attained for $\alpha>\alpha_v$, while $D_\alpha$ is not attained for $\alpha<\alpha_v$.
\medskip
\noindent
{\rm(iv)} Let $a\leq 1^*$ and $b\leq 1$. Then there holds $0<\alpha_v=\alpha_c<\infty$,
and $D_\alpha$ is not attained for $\alpha\ne\alpha_v(=\alpha_c)$.
\medskip
\noindent
{\bf(Threshold case $\alpha=\alpha_v$ or $\alpha=\alpha_c$)}
\medskip
\noindent
{\rm(v)} Let $a>1^*$. Then $D_{\alpha_c}\begin{cases}
&\text{ is attained when $b<1$},\\
&\text{ is not attained when $b=1$}.
\end{cases}
$
\medskip
\noindent
{\rm(vi)} Let $a=1^*$. Then $D_{\alpha_v}\begin{cases}
&\text{ is attained when $b=1$},\\
&\text{ is not attained when $b\ne1$}.
\end{cases}$
\medskip
\noindent
{\rm(vii)} Let $a<1^*$. Then $D_{\alpha_v}\begin{cases}
&\text{ is attained when $b>1$},\\
&\text{ is not attained when $b\leq1$}.
\end{cases}$
\end{thm}
Next, we estimate $\alpha_v$ and $\alpha_c$ by means of $E$ as follows :
\begin{thm}\label{thm4}
\noindent
Let $a>0$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $a>1^*$. Then there hold $\alpha_v=0$
and $\begin{cases}
&\alpha_c=\infty\text{ \,when \,}b>1,\\
&\frac{1}{E}<\alpha_c<\infty\text{ \,when \,}b\leq 1.
\end{cases}
$
\noindent
In particular, there holds $\alpha_c=\frac{a}{1^*E}$ when $b=1$.
\medskip
\noindent
{\rm(ii)} Let $a\leq 1^*$. Then there hold $
\begin{cases}
&0<\alpha_v<\frac{1}{E}\text{ \,and \,}\alpha_c=\infty\text{ \,when \,}b>1,\\
&\alpha_v=\alpha_c=\frac{1}{E}\text{ \,when \,}b\leq1.
\end{cases}
$
\noindent
In particular, there holds $\alpha_v=\frac{1}{bE}$ when $a=1^*$ and $b>1$.
\medskip
\noindent
{\rm(iii)} {\bf(Asymptotic behaviors of $\alpha_v$ and $\alpha_c$ on the parameters $a$ and $b$)}
\medskip
\noindent
{\rm(a)} Let $b>1$. Then there hold $\lim_{a\downarrow 0}\alpha_v=\frac{1}{E}$ and $\lim_{a\uparrow 1^*}\alpha_v=\frac{1}{bE}$.
\medskip
\noindent
{\rm(b)} Let $b\leq1$. Then there hold $\lim_{a\downarrow 1^*}\alpha_c=\frac{1}{E}$ and $\lim_{a\to\infty}\alpha_c=\infty$.
\medskip
\noindent
{\rm(c)} Let $a>1^*$. Then there hold $\lim_{b\downarrow 0}\alpha_c=\frac{1}{E}$
and $\lim_{b\uparrow 1}\alpha_c=\frac{a}{1^*E}$.
\medskip
\noindent
{\rm(d)} Let $a\leq 1^*$. Then there hold $\lim_{b\downarrow 1}\alpha_v=\frac{1}{E}$
and $\lim_{b\to\infty}\alpha_v=0$.
\end{thm}
In the end, we characterize the set of all maximizers of $D_\alpha$ for $1<q\leq 1^*$
by means of the corresponding $1$-dimensional function :
\begin{thm}\label{thm5}
Let $1<q\leq 1^*$, $\alpha>0$, $a>0$ and $b>0$.
Introduce $1$-dimensional functions $r(t)$, $\mu(t)$ and $f_\alpha(t)$ by
\begin{align*}
\begin{cases}
&r(t):=\frac{N}{t^{\frac{1}{a}}(t+1)^{\frac{a-b}{ab}}},\vspace{.2cm}\\
&\mu(t):=\frac{t^{\frac{N}{a}}(t+1)^{\frac{(a-b)N}{ab}-\frac{1}{b}}}{\omega_{N-1}N^{N-1}},\vspace{.2cm}\\
&f_\alpha(t):=\displaystyle\frac{
(1+t)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}+\alpha E_q\,t^{\frac{(q-1)N}{a}}
}{
(1+t)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}
\end{cases}
\end{align*}
for $t>0$. Then it holds $D_\alpha=\sup_{t>0}f_\alpha(t)$.
Furthermore, letting $\Sigma$ and $\Pi$ be sets defined by
\begin{align*}
\begin{cases}
&\Sigma:=\{
u_0\in BV\,|\,u_0\text{ is a maximizer of }D_\alpha
\},\\
&\Pi:=\{
t_0>0\,|\,\text{$t_0$ is a maximal point of $\sup_{t>0}f_\alpha(t)$}
\},
\end{cases}
\end{align*}
we obtain
$
\Sigma=\{
\pm\mu(t_0)\chi_{B_{r(t_0)}(x_0)}\in BV\,|\, t_0\in\Pi\text{ and }x_0\in\Bbb R^N
\}$.
\end{thm}
\noindent
Theorem \ref{thm5} is essentially proved by a scaling argument in \cite{N2}
together with the fact that maximizers of $E_q$ consist of functions of the form
$u=\lambda\chi_B$ with $\lambda\in\Bbb R\setminus\{0\}$ and a ball $B\subset\Bbb R^N$, see Proposition \ref{GN-attain} (ii), namely the information on maximizers of $E_{1^*}$ (the isoperimetric inequality) is transmitted to $E_q$ for any $1<q\leq 1^*$. On the other hand, it seems to be difficult to obtain a similar characterization to the problem based on $W^{1,p}$ with $1<p<N$
since we do not know the relation between maximizers of the Sobolev inequality (called Talenti's function) and those of the corresponding Gagliardo-Nirenberg inequality.
\medskip
For the limiting case $p=N$, a maximizing problem on $W^{1,N}$ corresponding to \eqref{basic-pro} was considered in \cite{IW}.
As another characterization of Sobolev's embedding in this case,
we know the Moser-Trudinger type inequalities.
Attainability problems associated with those inequalities
also have been investigated in rich literature.
Among others, we refer to \cite{IIW, I, LR, OST, R} and related works \cite{L, L2, LLZ, Li, LY, N}, in which similar problems to $D_\alpha$ were studied.
\medskip
This paper is organized as follows. Section 2 is devoted to prepare preliminary facts
and to prove Theorem \ref{thm5}. We show Theorem \ref{thm1}-\ref{thm2} and Theorem \ref{thm3}-\ref{thm4} in Section 3 and Section 4, respectively. Throughout the paper, the notation $\|\cdot\|_{p}$ denotes the standard $L^p$-norm. We pass to subsequences freely.
\section{Preliminaries}
\noindent
In this section, we collect several lemmas needed for the proofs of main theorems.
First, we recall the definition of the space of bounded variation $BV$.
$BV$ is a Banach space endowed with the norm $\|u\|_{BV}:=\|u\|_{TV}+\|u\|_{1}$,
where the total variation $\|u\|_{TV}$ is given by
$$
\|u\|_{TV}:=\sup\left\{
\int_{\Bbb R^N}u\operatorname{div}\psi\,\bigg|\,\psi=\{\psi_1,\cdots,\psi_N\}\subset C^1_c, \,
\|\psi\|_{\infty}:=\left\|\left(\sum_{i=1}^N|\psi_i|^2\right)^{\frac{1}{2}}\right\|_{\infty}\leq 1
\right\}.
$$
The Sobolev type embedding on $BV$ states $W^{1,1}\hookrightarrow BV\hookrightarrow L^r$ for $1\leq r\leq 1^*$.
We introduce $E_q$ and $\tilde E_q$ by
\begin{align*}
\begin{cases}
&E_q=E_q(u):=\sup_{u\in BV\setminus\{0\}}\frac{
\|u\|_{q}^q
}{
\|u\|_{1}^{q-(q-1)N}\|u\|_{TV}^{(q-1)N}
},\\
&\tilde E_q=\tilde E_q(u):=\sup_{u\in W^{1,1}\setminus\{0\}}\frac{
\|u\|_{q}^q
}{
\|u\|_{1}^{q-(q-1)N}\|\nabla u\|_{1}^{(q-1)N}
}
\end{cases}
\end{align*}
and similarly $D_\alpha$ and $\tilde D_\alpha$ by
\begin{align*}
\begin{cases}
&D_\alpha:=\sup_{\|u\|_{TV}^a+\|u\|_{1}^b=1}\left(\|u\|_{1}+\alpha\|u\|_{q}^q\right),\\
&\tilde D_\alpha:=\sup_{\|\nabla u\|_{1}^a+\|u\|_{1}^b=1}\left(
\|u\|_{1}+\alpha\|u\|_{q}^q
\right)
\end{cases}
\end{align*}
for $1<q\leq 1^*$ and $a,b,\alpha>0$.
\begin{prop}\label{GN-attain}
Let $1<q\leq 1^*$.
\medskip
\noindent
{\rm(i)} There holds $E_q=\tilde E_q=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}$.
\medskip
\noindent
{\rm(ii)} $E_q$ is attained by functions of the form $u=\lambda\chi_B\in BV$ with $\lambda\in \Bbb R\setminus\{0\}$ and a ball $B\subset \Bbb R^N$.
Moreover, the maximizer of $E_q$ necessarily has this form.
\medskip
\noindent
{\rm(iii)} $\tilde E_q$ is not attained in $W^{1,1}\setminus\{0\}$.
\end{prop}
\begin{proof}
First, recall the facts that it holds
$E_{1^*}=\frac{1}{N\omega_{N-1}^{\frac{1}{N-1}}}$ and $E_{1^*}$ is attained only by functions of the form $u=\lambda\chi_B\in BV$ with $\lambda\in \Bbb R\setminus\{0\}$ and a ball $B\subset \Bbb R^N$.
\medskip
\noindent
(i) By H\"older's inequality and Mazya's inequality, we have for $u\in BV$
\begin{align*}
&\|u\|_{q}^q\leq\|u\|_{1}^{q-(q-1)N}\|u\|_{1^*}^{(q-1)N}\\
&\leq\|u\|_{1}^{q-(q-1)N}\left(\frac{1}{N^{\frac{N-1}{N}}\omega_{N-1}^{\frac{1}{N}}}\|u\|_{TV}\right)^{(q-1)N}
=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}\|u\|_{1}^{q-(q-1)N}\|u\|_{TV}^{(q-1)N},
\end{align*}
which implies $E_q\leq\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}$.
Let $u_0=\chi_{B_1(0)}\in BV$. Then we can compute
$\|u_0\|_{1}=\|u_0\|_{q}^q=\frac{\omega_{N-1}}{N}$ and $\|u_0\|_{TV}=\omega_{N-1}$,
and then we observe $E_q(u_0)=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}$.
Hence, $u_0$ is a maximizer of $E_q$ and it follows $E_q=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}$.
\medskip
Next, we prove $E_q=\tilde E_q$.
It is enough to show $E_q\leq \tilde E_q$ since the converse inequality is obtained
by the facts $W^{1,1}\subset BV$ and $\|\nabla u\|_{1}=\|u\|_{TV}$ for $u\in W^{1,1}$.
Let $u_0\in BV\setminus\{0\}$ be a maximizer of $E_q$,
where note that the existence of $u_0$ has been already established as above.
By an approximation argument, there exists a sequence $\{u_n\}_{n=1}^\infty\subset BV\cap C^\infty$
such that $u_n\to u_0$ in $L^1$ and $\|u_n\|_{TV}\to \|u_0\|_{TV}$,
and up to a subsequence, $u_n\to u_0$ a.e. on $\Bbb R^N$.
We observe that $u_n\in W^{1,1}$ with $\|u_n\|_{TV}=\|\nabla u_n\|_{1}$.
Indeed, by using the fact that there holds $\|v\|_{TV(\Omega)}=\int_\Omega|\nabla v|$ for any $v\in BV(\Omega)\cap C^\infty(\Omega)$ with a bounded domain having its sufficiently smooth boundary,
we see
\begin{align*}
&\|u_n\|_{TV}=\sup_{R>0}\|u_n\|_{TV(B_R(0))}
=\sup_{R>0}\int_{B_R(0)}|\nabla u_n|=\lim_{R\to\infty}\int_{B_R(0)}|\nabla u_n|=\|\nabla u_n\|_{1}<\infty,
\end{align*}
where the last equality is shown by Lebesgue's monotone convergence theorem.
Then it holds $u_n\ne 0$ in $W^{1,1}$ for large $n$ since
$\|\nabla u_n\|_{1}=\|u_n\|_{TV}\to \|u_0\|_{TV}>0$ as $n\to\infty$.
Now we see by the convergences of $u_n$ together with Fatou's lemma,
\begin{align*}
E_q=E_q(u_0)\leq\liminf_{n\to\infty}E_q(u_n)\leq\limsup_{n\to\infty}E_q(u_n)=\limsup_{n\to\infty}\tilde E_q(u_n)
\leq \tilde E_q.
\end{align*}
Thus the assertion (i) has been proved.
\medskip
\noindent
(ii) Let $u_0=\lambda\chi_B\in BV$ for $\lambda\in\Bbb R\setminus\{0\}$ and a ball $B=B_R(x_0)$
with a radius $R>0$ centered at $x_0\in\Bbb R^N$.
Then we can compute
\begin{align*}
\|u_0\|_{1}=|\lambda|R^N\frac{\omega_{N-1}}{N},\quad \|u_0\|_{q}^q=|\lambda|^q R^N\frac{\omega_{N-1}}{N}\quad\text{and}\quad
\|u_0\|_{TV}=|\lambda|R^{N-1}\omega_{N-1},
\end{align*}
and thus these relations together with the assertion (i) show
$E_q(u_0)=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}=E_q$.
Hence, $u_0$ is a maximizer of $E_q$.
\medskip
Next, assume that $E_q$ is attained by $u_0\in BV\setminus\{0\}$.
Then by H\"older's inequality and the assertion (i), we have
\begin{align*}
&\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1}=E_q=E_q(u_0)\leq E_{1^*}(u_0) ^{(q-1)(N-1)}\leq E_{1^*}^{(q-1)(N-1)}
=\left(\frac{1}{N^{N-1}\omega_{N-1}}\right)^{q-1},
\end{align*}
which shows that $u_0$ is a maximizer of $E_{1^*}$.
Hence, $u_0=\lambda \chi_{B}$ for some $\lambda\in\Bbb R\setminus\{0\}$ and a ball $B\subset\Bbb R^N$.
The assertion (ii) has been proved.
\medskip
\noindent
(iii) By contradiction, assume that $\tilde E_q$ is attained by $u_0\in W^{1,1}\setminus\{0\}$.
Then the assertion (i) and the facts $W^{1,1}\subset BV$ and $\|\nabla u\|_{1}=\|u\|_{TV}$
for $u\in W^{1,1}$ imply that $u_0\in BV\setminus\{0\}$ is a maximizer of $E_q$.
Then the assertion (ii) shows that $u_0=\lambda \chi_B$ for $\lambda\in\Bbb R\setminus\{0\}$ and a ball $B\subset\Bbb R^N$,
which is a contradiction to $u_0\in W^{1,1}$.
The assertion (iii) has been proved.
\end{proof}
\begin{lem}\label{D-f-g}
Let $1<q\leq 1^*$, $\alpha>0$, $a>0$ and $b>0$.
Then there hold $D_\alpha=\sup_{t>0}f_\alpha(t)$ and $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)$ where for $t>0$,
\begin{align*}
\begin{cases}
&f_\alpha(t):=\displaystyle\frac{
(1+t)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}+\alpha E_q\,t^{\frac{(q-1)N}{a}}
}{
(1+t)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
},\\
&g(t):=\displaystyle\frac{
\left((1+t)^{\frac{1}{b}}-1\right)(1+t)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
}{
t^{\frac{(q-1)N}{a}}
}.
\end{cases}
\end{align*}
\end{lem}
\begin{proof}
For $u\in BV$ with $\|u\|_{TV}^a+\|u\|_{1}^b=1$, we see
\begin{align*}
&\|u\|_{1}+\alpha\|u\|_{q}^q\leq\|u\|_{1}+\alpha E_q\|u\|_{1}^{q-(q-1)N}\|u\|_{TV}^{(q-1)N}\\
&=\frac{
\|u\|_{1}(\|u\|_{TV}^a+\|u\|_{1}^b)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\|u\|_{1}^{q-(q-1)N}\|u\|_{TV}^{(q-1)N}
}{
(\|u\|_{TV}^a+\|u\|_{1}^b)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}\\
&=\frac{
\left(1+\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\left(\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}{\left(1+\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}}\\
&=f_\alpha\left(\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)
\leq\sup_{t>0}f_\alpha(t),
\end{align*}
which yields $D_\alpha\leq\sup_{t>0}f_\alpha(t)$.
On the other hand, let $v\in BV\setminus\{0\}$ be a maximizer of $E_q$.
The existence of $v$ was obtained by Proposition \ref{GN-attain} (ii).
For $\lambda>0$, we define $v_\lambda(x):=K\lambda v(\lambda^{\frac{1}{N}}x)$,
where $K=K(\lambda)>0$ is determined uniquely by
\begin{align}\label{K-def1}
\|v_\lambda\|_{TV}^a+\|v_\lambda\|_{1}^b
=K^a\lambda^{\frac{a}{N}}\|v\|_{TV}^a+K^b\|v\|_{1}^b=1.
\end{align}
Then we observe for $\lambda>0$,
\begin{align*}
&D_\alpha\geq\|v_\lambda\|_{1}+\alpha\|v_\lambda\|_{q}^q\\
&=\frac{
\|v_\lambda\|_{1}(\|v_\lambda\|_{TV}^a+\|v_\lambda\|_{1}^b)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\|v_\lambda\|_{1}^{q-(q-1)N}\|v_\lambda\|_{TV}^{(q-1)N}
}{
(\|v_\lambda\|_{TV}^a+\|v_\lambda\|_{1}^b)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}\\
&=\frac{
\left(1+\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\left(\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}{
\left(1+\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}\\
&=f_\alpha\left(\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)
=f_\alpha\left(K^{a-b}\lambda^{\frac{a}{N}}\frac{\|v\|_{TV}^a}{\|v\|_{1}^b}\right)
=f_\alpha\left(\frac{1}{K^b\|v\|_{1}^b}-1\right).
\end{align*}
By the equation \eqref{K-def1}, we see that $K=K(\lambda)$ is a continuous
function on $(0,\infty)$ satisfying $K<\frac{1}{\|v\|_{1}}$ for $\lambda>0$, $\lim_{\lambda\downarrow 0}K=\frac{1}{\|v\|_{1}}$ and $\lim_{\lambda\to\infty}K=0$. Thus we obtain
$D_\alpha\geq\sup_{\lambda>0}f_\alpha\left(\frac{1}{K^b\|v\|_{1}^b}-1\right)
=\sup_{t>0}f_\alpha(t)$. Thus we have proved $D_\alpha=\sup_{t>0}f_\alpha(t)$.
\medskip
Next, for $u\in BV$ with $\|u\|_{TV}^a+\|u\|_{1}^b=1$, we see
\begin{align*}
&\frac{1-\|u\|_{1}}{\|u\|_{q}^q}\geq\frac{1-\|u\|_{1}}{E_q\|u\|_{1}^{q-(q-1)N}\|u\|_{TV}^{(q-1)N}}\\
&=\frac{
\left(\left(1+\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)^{\frac{1}{b}}-1\right)
\left(
1+\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}
\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
}{
E_q \left(\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}=\frac{1}{E_q}g\left(
\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}
\right)\geq\frac{1}{E_q}\inf_{t>0}g(t),
\end{align*}
and thus there holds $\alpha_v\geq\frac{1}{E_q}\inf_{t>0}g(t)$.
On the other hand, let $v\in BV\setminus\{0\}$ be a maximizer of $E_q$
and define $v_\lambda$ as above. Then we see for $\lambda>0$,
\begin{align*}
&\alpha_v\leq\frac{1-\|v_\lambda\|_{1}}{\|v_\lambda\|_{q}^q}
=\frac{1-\|v_\lambda\|_{1}}{E_q\|v_\lambda\|_{1}^{q-(q-1)N}\|v_\lambda\|_{TV}^{(q-1)N}}\\
&=\frac{
\left(\left(1+\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)^{\frac{1}{b}}-1\right)
\left(1+\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
}{
E_q\left(\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}\\
&=\frac{1}{E_q}g\left(\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)
=\frac{1}{E_q}g\left(K^{a-b}\lambda^{\frac{a}{N}}\frac{\|v\|_{TV}^a}{\|v\|_{1}^b}\right)
=\frac{1}{E_q}g\left(\frac{1}{K^b\|v\|_{1}^b}-1\right),
\end{align*}
and thus we get $\alpha_v\leq\frac{1}{E_q}\inf_{\lambda>0}g\left(\frac{1}{K^b\|v\|_{1}^b}-1\right)=\frac{1}{E_q}\inf_{t>0}g(t)$.
Thus we have proved $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)$.
\end{proof}
\begin{lem}\label{D-imply-f(t)}
Let $1<q\leq 1^*$, $\alpha>0$, $a>0$ and $b>0$. Assume that $D_\alpha$ is attained by $u_0\in BV$. Then there exist $R>0$, $x_0\in\Bbb R^N$ and $\lambda_0\in\Bbb R\setminus\{0\}$ such that
$u_0=\lambda_0\chi_{B_R(x_0)}$, where the coefficient $\lambda_0$ satisfies
\begin{align}\label{lambda0-condition}
(|\lambda_0|R^{N-1}\omega_{N-1})^a+\left(|\lambda_0|R^N\frac{\omega_{N-1}}{N}\right)^b=1.
\end{align}
In addition, $\sup_{t>0}f_\alpha(t)$ is attained at $t=(\frac{N}{R})^b(|\lambda_0|R^{N-1}\omega_{N-1})^{a-b}$.
\end{lem}
\begin{proof}
By Lemma \ref{D-f-g}, we see
\begin{align}
&\notag\sup_{t>0}f_\alpha(t)=D_\alpha=\|u_0\|_{1}+\alpha\|u_0\|_{q}^q
\leq\|u_0\|_{1}+\alpha E_q\|u_0\|_{1}^{q-(q-1)N}\|u_0\|_{TV}^{(q-1)N}\\
&\notag=\frac{
\|u_0\|_{1}\left(\|u_0\|_{TV}^a+\|u_0\|_{1}^b\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\|u_0\|_{1}^{q-(q-1)N}\|u_0\|_{TV}^{(q-1)N}
}{
\left(\|u_0\|_{TV}^a+\|u_0\|_{1}^b\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}\\
&\label{dfg-appli}=\frac{
\left(1+\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}+\alpha E_q\left(\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}{\left(1+\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}}
=f_\alpha\left(\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)\leq\sup_{t>0}f_\alpha(t),
\end{align}
which implies that $u_0$ is a maximizer of $E_q$. By applying Proposition \ref{GN-attain} (ii),
we can write $u_0=\lambda_0\chi_{B_R(x_0)}$ for some $\lambda_0\in\Bbb R\setminus\{0\}$, $R>0$ and $x_0\in\Bbb R^N$.
Moreover, since $\|u_0\|_{1}=|\lambda_0|R^N\frac{\omega_{N-1}}{N}$ and $\|u_0\|_{TV}=|\lambda_0|R^{N-1}\omega_{N-1}$,
the coefficient $\lambda_0$ satisfies
\begin{align*}
\|u_0\|_{TV}^a+\|u_0\|_{1}^b=
(|\lambda_0|R^{N-1}\omega_{N-1})^a+\left(|\lambda_0|R^N\frac{\omega_{N-1}}{N}\right)^b=1.
\end{align*}
In addition, the relation \eqref{dfg-appli} also implies that $\sup_{t>0}f_\alpha(t)$ is attained at
\begin{align*}
t=\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}=\left(\frac{N}{R}\right)^b(|\lambda_0|R^{N-1}\omega_{N-1})^{a-b}.
\end{align*}
The proof of Lemma \ref{D-imply-f(t)} is complete.
\end{proof}
\begin{cor}\label{non-attain-cor}
Let $1<q\leq 1^*$, $\alpha>0$, $a>0$ and $b>0$. Then $\tilde D_\alpha$ is not attained.
\end{cor}
\begin{proof}
On the contrary, assume that $\tilde D_\alpha$ is attained by $u_0\in W^{1,1}$.
Then recalling $\|\nabla u_0\|_{1}=\|u_0\|_{TV}$, we see that
$u_0$ also becomes a maximizer of $D_\alpha$. Then by Lemma \ref{D-imply-f(t)},
we have $u_0=\lambda_0\chi_B$ with some $\lambda_0\in\Bbb R\setminus\{0\}$
and some ball $B\subset\Bbb R^N$, which is a contradiction to $u_0\in W^{1,1}$.
\end{proof}
We are ready to prove Theorem \ref{thm5} :
\begin{proof}[{\rm \bf Proof of Theorem \ref{thm5}}]
First, we show $\Sigma\supset\{
\pm\mu(t_0)\chi_{B_{r(t_0)}(x_0)}\in BV\,|\, t_0\in\Pi\text{ and }x_0\in\Bbb R^N
\}$. To this end, let $t_0\in\Pi$, $x_0\in\Bbb R^N$ and $R>0$.
Recall that $v:=\pm\chi_{B_R(x_0')}$ is a maximizer of $E_q$,
where $x_0':=\frac{R t_0^{\frac{1}{a}}(t_0+1)^{\frac{a-b}{ab}}}{N}x_0$.
For $\lambda>0$, define $v_\lambda(x):=K\lambda v(\lambda^{\frac{1}{N}}x)$,
where $K=K(\lambda)>0$ is determined uniquely by
\begin{align}\label{K-def}
\|v_\lambda\|_{TV}^a+\|v_\lambda\|_{1}^b=K^a\lambda^{\frac{a}{N}}\|v\|_{TV}^a+K^b\|v\|_{1}^b=1.
\end{align}
Then we see
\begin{align}\label{K-trans}
\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}=K^{a-b}\lambda^{\frac{a}{N}}\frac{\|v\|_{TV}^a}{\|v\|_{1}^b}
=\frac{1}{K^b\|v\|_{1}^b}-1.
\end{align}
Note that the relation \eqref{K-def} shows $\lim_{\lambda\downarrow 0}K=\frac{1}{\|v\|_{1}}$, $\lim_{\lambda\to\infty}K=0$ and
\begin{align*}
K'=-\frac{
\frac{a}{N}K^a\lambda^{\frac{a}{N}-1}\|v\|_{TV}^a
}{
aK^{a-1}\lambda^{\frac{a}{N}}\|v\|_{TV}^a+bK^{b-1}\|v\|_{1}^b
}<0.
\end{align*}
Hence, the relation \eqref{K-trans} implies that there exists $\lambda_0>0$ uniquely such that
\begin{align}\label{lambda0t0}
\frac{\|v_{\lambda_0}\|_{TV}^a}{\|v_{\lambda_0}\|_{1}^b}=\frac{1}{K(\lambda_0)^b\|v\|_{1}^b}-1=t_0.
\end{align}
Combining \eqref{K-def} with \eqref{lambda0t0}, we can compute
\begin{align}\label{value-flam0}
&\lambda_0=\left(\frac{\|v\|_{1}}{\|v\|_{TV}}
t_0^{\frac{1}{a}}(t_0+1)^{\frac{a-b}{ab}}
\right)^N=\left(\frac{R}{N}t_0^{\frac{1}{a}}(t_0+1)^{\frac{a-b}{ab}}\right)^N,
\end{align}
where we have used $\|v\|_{1}=R^N\frac{\omega_{N-1}}{N}$ and $\|v\|_{TV}=R^{N-1}\omega_{N-1}$.
By Lemma \ref{D-f-g}, we see
\begin{align*}
&\sup_{t>0}f_\alpha(t)=D_\alpha\geq\|v_{\lambda_0}\|_{1}+\alpha\|v_{\lambda_0}\|_{q}^q\\
&=\frac{
\|v_{\lambda_0}\|_{1}\left(\|v_{\lambda_0}\|_{TV}^a+\|v_{\lambda_0}\|_{1}^b\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}+\alpha E_q\|v_{\lambda_0}\|_{1}^{q-(q-1)N}\|v_{\lambda_0}\|_{TV}^{(q-1)N}
}{
\left(\|v_{\lambda_0}\|_{TV}^a+\|v_{\lambda_0}\|_{1}^b\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}\\
&=\frac{
\left(1+\frac{\|v_{\lambda_0}\|_{TV}^a}{\|v_{\lambda_0}\|_{1}^b}\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}+\alpha E_q\left(\frac{\|v_{\lambda_0}\|_{TV}^a}{\|v_{\lambda_0}\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}{
\left(
1+\frac{\|v_{\lambda_0}\|_{TV}^a}{\|v_{\lambda_0}\|_{1}^b}
\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}=f_\alpha\left(\frac{\|v_{\lambda_0}\|_{TV}^a}{\|v_{\lambda_0}\|_{1}^b}\right)\\
&=f_\alpha\left(\frac{1}{K(\lambda_0)^b\|v\|_{1}^b}-1\right)=f_\alpha(t_0)=\sup_{t>0}f_\alpha(t),
\end{align*}
which implies that $v_{\lambda_0}$ is a maximizer of $D_\alpha$. Moreover, by \eqref{lambda0t0} and \eqref{value-flam0}, we can compute
$v_{\lambda_0}=\pm\mu(t_0)\chi_{B_{r(t_0)}(x_0)}$.
\medskip
Next, we show $\Sigma\subset\{
\pm\mu(t_0)\chi_{B_{r(t_0)}(x_0)}\in BV\,|\, t_0\in\Pi\text{ and }x_0\in\Bbb R^N
\}$. To this end, let $u_0\in BV$ be a maximizer of $D_\alpha$.
Then by Lemma \ref{D-imply-f(t)}, we can write $u_0=\lambda_0\chi_{B_R(x_0)}$ with some $R>0$, $x_0\in\Bbb R^N$ and $\lambda_0\in\Bbb R\setminus\{0\}$,
where $\lambda_0$ satisfies \eqref{lambda0-condition}.
We take $t_0>0$ uniquely determined by the equation $R=r(t_0)$
and put $\nu:=\mu(t_0)$.
Then we observe that $R$ and $\nu$ satisfy $(\nu R^{N-1}\omega_{N-1})^a+(\nu R^N\frac{\omega_{N-1}}{N})^b=1$,
which implies $\nu=|\lambda_0|$ since $|\lambda_0|$ satisfies the same equation by \eqref{lambda0-condition}.
Therefore, in order to complete the proof, it suffices to prove $t_0\in\Pi$.
Noting that $u_0$ is a maximizer both of $D_\alpha$ and $E_q$ together with
Lemma \ref{D-f-g}, we see
\begin{align*}
&\sup_{t>0}f_\alpha(t)=D_\alpha=\|u_0\|_{1}+\alpha\|u_0\|_{q}^q
=\|u_0\|_{1}+\alpha E_q\|u_0\|_{1}^{q-(q-1)N}\|u_0\|_{TV}^{(q-1)N}\\
&=\frac{
\|u_0\|_{1}\left(\|u_0\|_{TV}^a+\|u_0\|_{1}^b\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\|u_0\|_{1}^{q-(q-1)N}\|u_0\|_{TV}^{(q-1)N}
}{
\left(\|u_0\|_{TV}^a+\|u_0\|_{1}^b\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}\\
&=\frac{
\left(1+\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
+\alpha E_q\left(\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)^{\frac{(q-1)N}{a}}
}{
\left(1+\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)^{\frac{(q-1)N}{a}+\frac{N-q(N-1)}{b}}
}=f_\alpha\left(\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}\right)=f_\alpha(t_0),
\end{align*}
where we have used $\frac{\|u_0\|_{TV}^a}{\|u_0\|_{1}^b}=t_0$.
Hence, it follows $\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)$, which means $t_0\in\Pi$.
The proof of Theorem \ref{thm5} is complete.
\end{proof}
\section{Proof of Theorems \ref{thm1}-\ref{thm2}}
\noindent
In this section, we shall prove Theorems \ref{thm1}-\ref{thm2}.
We start from the following lemma :
\begin{lem}\label{routine-attain-lem}
Let $1<q<1^*$, $a>0$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $\alpha>\alpha_v$. Then $D_\alpha$ is attained.
\medskip
\noindent
{\rm(ii)} Assume $\alpha_v>0$ and let $0<\alpha<\alpha_v$. Then $D_\alpha$ is not attained.
\end{lem}
\begin{proof}
By Theorem \ref{thm5}, we see that $D_\alpha$ is attained if and only if $\sup_{t>0}f_\alpha(t)$
is attained.
\medskip
\noindent
(i) Let $\alpha>\alpha_v$. Note that the condition $q<1^*$ shows $\lim_{t\to\infty}f_\alpha(t)=0$. By the assumption $\alpha>\alpha_v$ and Lemma \ref{D-f-g},
there exists $t_0>0$ such that $\alpha>\frac{1}{E_q}g(t_0)$, which implies $f_\alpha(t_0)>1=\lim_{t\downarrow 0}f_\alpha(t)$. Hence, $\sup_{t>0}f_\alpha(t)$ is attained.
\medskip
\noindent
(ii) Assume $\alpha_v>0$ and let $0<\alpha<\alpha_v$.
By contradiction, assume that there exists $t_0>0$ such that
$\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)$. First, note $\sup_{t>0}f_\alpha(t)\geq\lim_{t\downarrow 0}f_\alpha(t)=1$. By the assumption $\alpha<\alpha_v$ and Lemma \ref{D-f-g}, we obtain
$\alpha<\alpha_v\leq\frac{1}{E_q}g(t_0)$, which implies $f_\alpha(t_0)<1$.
Then we see $1\leq\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)<1$, which is a contradiction.
Thus $\sup_{t>0}f_\alpha(t)$ is not attained.
\end{proof}
\begin{lem}\label{sub-pro1}
Let $1<q<1^*$, $a>0$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $a>N(q-1)$. Then there holds $\alpha_v=0$,
and $D_\alpha$ is attained for $\alpha>0$.
\medskip
\noindent
{\rm(ii)} Let $a<N(q-1)$. Then there holds $\alpha_v>0$,
and $D_\alpha$ is attained for $\alpha\geq\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha<\alpha_v$.
\end{lem}
\begin{proof}
(i) Let $a>N(q-1)$. In this case, since $\lim_{t\downarrow 0}g(t)=0$ and $g(t)>0$ for $t>0$,
by Lemma \ref{D-f-g}, we obtain $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)=\frac{1}{E_q}\lim_{t\downarrow 0}g(t)=0$,
and then Lemma \ref{routine-attain-lem} (i) implies that $D_\alpha$ is attained for $\alpha>0$.
\medskip
\noindent
(ii) Let $a<N(q-1)$. The conditions $a<N(q-1)$ and $q<1^*$ imply $\lim_{t\downarrow 0}g(t)=\lim_{t\to\infty}g(t)=\infty$.
Since $g(t)>0$ for $t>0$, there exists $t_0>0$ such that $\inf_{t>0}g(t)=g(t_0)>0$,
and then Lemma \ref{D-f-g} shows $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)=\frac{1}{E_q}g(t_0)>0$.
By Lemma \ref{routine-attain-lem}, it remains to prove $D_{\alpha_v}$ is attained,
which is equivalent to $\sup_{t>0}f_{\alpha_v}(t)$ is attained.
Note that $\alpha_v=\frac{1}{E_q}g(t_0)$ implies $f_{\alpha_v}(t_0)=1$. Recalling $\lim_{t\downarrow 0}f_{\alpha_v}(t)=1$
and $\lim_{t\to\infty}f_{\alpha_v}(t)=0$, we can conclude that $\sup_{t>0}f_{\alpha_v}(t)$ is attained.
\end{proof}
\begin{lem}\label{attain-n(q-1)}
Let $1<q<1^*$, $a=N(q-1)$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b\geq b_0:=(q-1)(N-1)-\left(N-(N-1)q\right)>0$.
Then there holds $\alpha_v=\frac{1}{b E_q}$, and $D_\alpha$ is attained for $\alpha>\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha\leq\alpha_v$.
\medskip
\noindent
{\rm(ii)} Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b<b_0$.
Then there holds $0<\alpha_v<\frac{1}{b E_q}$, and $D_\alpha$ is attained for $\alpha\geq\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha<\alpha_v$.
\medskip
\noindent
{\rm(iii)} Let $1<q\leq\frac{2N-1}{2(N-1)}$. Then there holds $\alpha_v=\frac{1}{bE_q}$,
and $D_\alpha$ is attained for $\alpha>\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha\leq\alpha_v$.
\end{lem}
\begin{proof}
First, note $\lim_{t\downarrow 0}g(t)=\frac{1}{b}$, and then $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)\leq\frac{1}{bE_q}$.
\medskip
\noindent
(i) Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b\geq b_0$. Define the function $\phi(t)$ for $t>0$ by
\begin{align*}
\phi(t):=(1+t)^{1+\frac{N-q(N-1)}{b}}-(1+t)^{1-\frac{(q-1)(N-1)}{b}}-\frac{t}{b}.
\end{align*}
We can compute
\begin{align*}
&\phi'(t)=\left(1+\frac{N-q(N-1)}{b}\right)(1+t)^{\frac{N-q(N-1)}{b}}
-\left(1-\frac{(q-1)(N-1)}{b}\right)(1+t)^{-\frac{(q-1)(N-1)}{b}}-\frac{1}{b}
\end{align*}
and $\phi''(t)=(1+t)^{-1-\frac{(q-1)(N-1)}{b}}\varphi(t)$, where
\begin{align*}
\varphi(t):=\frac{N-q(N-1)}{b}\left(1+\frac{N-q(N-1)}{b}\right)(1+t)^{\frac{1}{b}}+\frac{(q-1)(N-1)}{b}\left(1-\frac{(q-1)(N-1)}{b}\right).
\end{align*}
We see that the condition $b\geq b_0$ implies $\varphi(t)>\varphi(0)\geq 0$ for $t>0$.
Hence, $\phi'(t)>\phi'(0)=0$ for $t>0$, and then $\phi(t)>\phi(0)=0$ for $t>0$,
which is equivalent to $g(t)>\frac{1}{b}$ for $t>0$. Therefore, there holds $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)\geq\frac{1}{bE_q}$,
and it follows $\alpha_v=\frac{1}{bE_q}$. It remains to prove that $D_{\alpha_v}$ is not attained.
Indeed, since $g(t)>\frac{1}{b}$ for $t>0$ is equivalent to $f_{\alpha_v}(t)<1$ for $t>0$,
we know that $D_{\alpha_v}=\sup_{t>0}f_{\alpha_v}(t)=1$ is not attained.
\medskip
\noindent
(ii) Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b<b_0$. In this case, since the condition $b<b_0$ implies $\varphi(0)<0$,
there exists a unique $t_0>0$ such that $\phi''(t_0)=0$, $\phi''(t)<0$ for $t\in(0,t_0)$ and $\phi''(t)>0$ for $t\in(t_0,\infty)$.
Then by noting $\phi'(0)=0$ and $\lim_{t\to\infty}\phi'(t)=\infty$, we see that
there exists a unique $t_1\in(t_0,\infty)$ such that $\phi'(t_1)=0$,
$\phi'(t)<0$ for $t\in(0,t_1)$ and $\phi'(t)>0$ for $t\in(t_1,\infty)$.
Similarly, the facts $\phi(0)=0$ and $\lim_{t\to\infty}\phi(t)=\infty$ imply that
there exists a unique $t_2\in(t_1,\infty)$ such that $\phi(t_2)=0$,
$\phi(t)<0$ for $t\in(0,t_2)$ and $\phi(t)>0$ for $t\in(t_2,\infty)$.
Note that $\phi(t)<0$ for $t\in(0,t_2)$ is equivalent to $g(t)<\frac{1}{b}$ for $t\in(0,t_2)$,
which implies $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)<\frac{1}{bE_q}$.
Moreover, since $\lim_{t\downarrow 0}g(t)=\frac{1}{b}$, $\lim_{t\to\infty}g(t)=\infty$ and $g(t)>0$ for $t>0$,
we obtain $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)>0$. It remains to check that $D_{\alpha_v}$ is attained.
The signs of $\phi$ from the above observations give $g(t_2)=\frac{1}{b}$, $g(t)<\frac{1}{b}$ for $t\in(0,t_2)$ and $g(t)>\frac{1}{b}$ for $t\in(t_2,\infty)$.
These facts together with $\lim_{t\downarrow 0}g(t)=\frac{1}{b}$ yield that there exists $t_3\in(0,t_2)$ satisfying $\inf_{t>0}g(t)=g(t_3)$,
and hence, $\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)=\frac{g(t_3)}{E_q}$.
Therefore, we obtain $\alpha_vE_q=g(t_3)$ which is equivalent to $f_{\alpha_v}(t_3)=1=\lim_{t\downarrow 0}f_{\alpha_v}(t)$.
Recalling $\lim_{t\to\infty}f_{\alpha_v}(t)=0$, we see that $D_{\alpha_v}=\sup_{t>0}f_{\alpha_v}(t)$ is attained.
\medskip
\noindent
(iii) Let $1<q\leq\frac{2N-1}{2(N-1)}$. In this case, we see $\phi''(t)>0$ for $t>0$,
and hence, in the same way as in the case (i), we get the desired result.
\end{proof}
\begin{prop}\label{sub-pro2-est}
Let $1<q<1^*$, $a>0$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $a=N(q-1)$. Then there hold
$\lim_{b\downarrow 0}\alpha_v=\infty$ and $\lim_{b\uparrow b_0}\alpha_v=\frac{1}{b_0E_q}$.
\medskip
\noindent
{\rm(ii)} Let $a<N(q-1)$. Then there hold $\lim_{b\downarrow 0}\alpha_v=\infty$ and $\lim_{b\to\infty}\alpha_v=0$.
\medskip
\noindent
{\rm(iii)} There holds $\lim_{a\downarrow 0}\alpha_v=\infty$.
\medskip
\noindent
{\rm(iv)} Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b\geq b_0$, or let $1<q\leq\frac{2N-1}{2(N-1)}$.
Then there holds $\lim_{a\uparrow N(q-1)}\alpha_v=\frac{1}{bE_q}$.
\end{prop}
\begin{proof}
(i) Let $\frac{2N-1}{2(N-1)}<q<1^*$, $a=N(q-1)$ and $b<b_0$.
In the proof of Lemma \ref{attain-n(q-1)} (ii), we proved that $\inf_{t>0}g(t)$ is attained by some $t=t_b\in(0,\infty)$, and then
\begin{align*}
&\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)=\frac{g(t_b)}{E_q}
=\frac{\left((1+t_b)^{\frac{1}{b}}-1\right)(1+t_b)^{1-\frac{(q-1)(N-1)}{b}}}{E_qt_b}\\
&=\frac{(1+t_b)^{1+\frac{N-q(N-1)}{b}}-(1+t_b)^{1-\frac{(q-1)(N-1)}{b}}}{E_qt_b}.
\end{align*}
Noting $b<b_0<(q-1)(N-1)$ and thus $1-\frac{(q-1)(N-1)}{b}<0$, and using the inequality
$\frac{(1+t)^{1+\frac{N-q(N-1)}{b}}-1}{t}\geq 1+\frac{N-q(N-1)}{b}$ for $t>0$, we see
\begin{align*}
\alpha_v\geq\frac{(1+t_b)^{1+\frac{N-q(N-1)}{b}}-1}{E_qt_b}\geq\frac{1}{E_q}\left(1+\frac{N-q(N-1)}{b}\right),
\end{align*}
which implies $\lim_{b\downarrow 0}\alpha_v=\infty$.
Next, we prove $\lim_{b\uparrow b_0}\alpha_v=\frac{1}{b_0E_q}$.
To this end, we claim $\lim_{b\uparrow b_0}t_2=0$, which implies $\lim_{b\uparrow b_0}t_3=0$ since $0<t_3<t_2$,
where the numbers $t_2$ and $t_3$ are the ones introduced in the proof of Lemma \ref{attain-n(q-1)} (ii).
We write $t_2=t_2(b)$ and $t_3=t_3(b)$ for $b<b_0$. Take any positive sequence $\{b_i\}_{i=1}^\infty$ satisfying
$b_i\uparrow b_0$ as $i\to\infty$. Recall that for each $i$, $t_2(b_i)$ satisfies
\begin{align}\label{t2-equation}
\phi(t_2(b_i))
=\left(
1+t_2(b_i)
\right)^{1+\frac{N-q(N-1)}{b_i}}
-\left(1+t_2(b_i)\right)^{1-\frac{(q-1)(N-1)}{b_i}}-\frac{t_2(b_i)}{b_i}=0.
\end{align}
Since $1-\frac{(q-1)(N-1)}{b_i}\to 1-\frac{(q-1)(N-1)}{b_0}<0$ as $i\to\infty$,
the equation \eqref{t2-equation} shows that the sequence $\{t_2(b_i)\}_{i=1}^\infty$ is bounded,
and hence, we may assume that $\lim_{i\to\infty}t_2(b_i)=c_0$ for some $c_0\geq 0$.
Then letting $i\to\infty$ in \eqref{t2-equation} gives $\phi(c_0)=0$, which implies $c_0=0$
since we proved $\phi(t)>0$ for $t>0$ when $b=b_0$ in the proof of Lemma \ref{attain-n(q-1)} (i).
Therefore, the fact $\lim_{b\uparrow b_0}t_3(b)=0$ has been proved. Now we see
\begin{align*}
&\liminf_{b\uparrow b_0}\alpha_v
=\frac{1}{E_q}\liminf_{b\uparrow b_0}\left(\inf_{t>0}g(t)\right)
=\frac{1}{E_q}\liminf_{b\uparrow b_0}g(t_3(b))\\
&=\frac{1}{E_q}\liminf_{b\uparrow b_0}\frac{
\left((1+t_3(b))^{\frac{1}{b}}-1\right)(1+t_3(b))^{1-\frac{(q-1)(N-1)}{b}}
}{t_3(b)}
=\frac{1}{E_q}\liminf_{b\uparrow b_0}\frac{
(1+t_3(b))^{\frac{1}{b}}-1
}{t_3(b)}\\
&\geq\frac{1}{E_q}\lim_{b\uparrow b_0}\frac{
(1+t_3(b))^{\frac{1}{b_0}}-1
}{t_3(b)}
=\frac{1}{E_q}\lim_{t\downarrow 0}\frac{(1+t)^{\frac{1}{b_0}}-1}{t}
=\frac{1}{b_0E_q}.
\end{align*}
On the other hand, since $\inf_{t>0}g(t)\leq\lim_{t\downarrow 0}g(t)=\frac{1}{b}$ for $b<b_0$,
we obtain $\limsup_{b\uparrow b_0}\alpha_v=\frac{1}{E_q}\limsup_{b\uparrow b_0}(\inf_{t>0}g(t))\leq\frac{1}{b_0E_q}$.
As a result, the fact $\lim_{b\uparrow b_0}\alpha_v=\frac{1}{b_0E_q}$ has been proved.
\medskip
\noindent
(ii) Let $a<N(q-1)$. We first prove $\lim_{b\downarrow 0}\alpha_v=\infty$.
Since $\lim_{t\downarrow 0}g(t)=\lim_{t\to\infty}g(t)=\infty$, there exists $t_b\in(0,\infty)$ such that $\inf_{t>0}g(t)=g(t_b)>0$.
Take any positive sequence $\{b_j\}_{j=1}^\infty$ satisfying $b_j\downarrow 0$ as $j\to\infty$.
First, suppose $\liminf_{j\to\infty}t_{b_j}\in(0,\infty]$. In this case, we may assume $t_{b_j}\geq c$ for $j$ with some $c>0$.
Then we see for $j$,
\begin{align*}
&\alpha_v=\frac{g(t_{b_j})}{E_q}
=\frac{
(1+t_{b_j})^{\frac{N(q-1)}{a}+\frac{N-q(N-1)}{b_j}}\left(1-(1+t_{b_j})^{-\frac{1}{b_j}}\right)
}{E_q
t_{b_j}^{\frac{N(q-1)}{a}}
}\\
&\geq\frac{1}{E_q}(1+c)^{\frac{N-q(N-1)}{b_j}}\left(1-(1+c)^{-\frac{1}{b_j}}\right)\to\infty
\end{align*}
as $j\to\infty$. Hence, there holds $\lim_{b\downarrow 0}\alpha_v=\infty$.
Next, suppose $\liminf_{j\to\infty}t_{b_j}=0$, and we may assume $t_{b_j}\downarrow 0$ as $j\to\infty$.
Since $b_j<1$ for large $j\in\Bbb N$, there holds
$(1+t)^{\frac{1}{b_j}}-1\geq t(1+t)^{\frac{1}{b_j}-1}$ for $t>0$.
By using this inequality and the condition $a<N(q-1)$, we see
\begin{align*}
&\alpha_v=\frac{g(t_{b_j})}{E_q}
=\frac{
\left((1+t_{b_j})^{\frac{1}{b_j}}-1\right)(1+t_{b_j})^{(q-1)(\frac{N}{a}-\frac{N-1}{b_j})}
}{
E_qt_{b_j}^{\frac{N(q-1)}{a}}
}\\
&\geq\frac{
(1+t_{b_j})^{
\frac{N(q-1)}{a}-1+\frac{N-q(N-1)}{b_j}
}
}{E_qt_{b_j}^{\frac{N(q-1)}{a}-1}}
\geq\frac{1}{E_qt_{b_j}^{\frac{N(q-1)}{a}-1}}\to\infty
\end{align*}
as $j\to\infty$. Then there holds $\lim_{b\downarrow 0}\alpha_v=\infty$. Next, we see
\begin{align*}
\alpha_v=\frac{1}{E_q}\inf_{t>0}g(t)\leq\frac{g(1)}{E_q}
=\frac{1}{E_q}(2^{\frac{1}{b}}-1)2^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}\to0
\end{align*}
as $b\to\infty$, and thus it follows $\lim_{b\to\infty}\alpha_v=0$.
\medskip
\noindent
(iii) We prove $\lim_{a\downarrow 0}\alpha_v=\infty$.
Let $a<N(q-1)$. Then since $\lim_{t\downarrow 0}g(t)=\lim_{t\to\infty}g(t)=\infty$,
there exists $t_a\in(0,\infty)$ such that $\inf_{t>0}g(t)=g(t_a)>0$.
Take any positive sequence $\{a_j\}_{j=1}^\infty$ satisfying $N(q-1)>a_j\downarrow 0$ as $j\to\infty$.
First, suppose $\liminf_{j\to\infty}t_{a_j}\in(0,\infty]$.
In this case, we may assume $t_{a_j}\geq c$ for $j$ with some $c>0$.
Then we see for $j$,
\begin{align}
&\notag\alpha_v=\frac{g(t_{a_j})}{E_q}=
\frac{
(1+t_{a_j})^{\frac{N(q-1)}{a_j}+\frac{N-q(N-1)}{b}}\left(1-(1+t_{a_j})^{-\frac{1}{b}}\right)
}{E_qt_{a_j}^{\frac{N(q-1)}{a_j}}}\\
&\label{ato0-est}\geq\frac{1}{E_q}\left(1+\frac{1}{t_{a_j}}\right)^{\frac{N(q-1)}{a_j}}(1+t_{a_j})^{\frac{N-q(N-1)}{b}}\left(1-(1+c)^{-\frac{1}{b}}\right).
\end{align}
Suppose $\liminf_{j\to\infty}t_{a_j}=\infty$. Then by \eqref{ato0-est}, we have
\begin{align*}
\alpha_v\geq\frac{1}{E_q}(1+t_{a_j})^{\frac{N-q(N-1)}{b}}\left(1-(1+c)^{-\frac{1}{b}}\right)\to\infty
\end{align*}
as $j\to\infty$, and hence $\lim_{j\to\infty}\alpha_v=\infty$.
Suppose $\liminf_{j\to\infty}t_{a_j}\in(0,\infty)$. Then we may assume that $c\leq t_{a_j}\leq\tilde c$ for $j$ with some $\tilde c>c$.
Hence, by \eqref{ato0-est}, we see
\begin{align*}
\alpha_v\geq\frac{1}{E_q}\left(1+\frac{1}{\tilde c}\right)^{\frac{N(q-1)}{a_j}}(1+c)^{\frac{N-q(N-1)}{b}}\left(1-(1+c)^{-\frac{1}{b}}\right)\to\infty
\end{align*}
as $j\to\infty$, and hence $\lim_{j\to\infty}\alpha_v=\infty$.
Next, suppose $\liminf_{j\to\infty}t_{a_j}=0$, and we may assume $t_{a_j}\downarrow 0$ as $j\to\infty$.
Then we see
\begin{align*}
\alpha_v=\frac{g(t_{a_j})}{E_q}\geq\frac{1-(1+t_{a_j})^{-\frac{1}{b}}}{E_q t_{a_j}^{\frac{N(q-1)}{a_j}}}\to\infty
\end{align*}
as $j\to\infty$, and hence, $\lim_{j\to\infty}\alpha_v=\infty$.
As a conclusion, we have proved $\lim_{a\downarrow 0}\alpha_v=\infty$.
\medskip
\noindent
(iv) Let $\frac{2N-1}{2(N-1)}<q<1^*$ and $b\geq b_0$, or let $1<q\leq\frac{2N-1}{2(N-1)}$.
We prove $\lim_{a\uparrow N(q-1)}\alpha_v=\frac{1}{bE_q}$.
For $a>0$, we write $\alpha_v=\alpha_v(a)$ and $g=g_a$. Letting $a<N(q-1)$, we see
$\alpha_v(a)\leq\frac{g_a(t)}{E_q}$ for $t>0$, and then
$\limsup_{a\uparrow N(q-1)}\alpha_v(a)\leq\frac{g_{N(q-1)}(t)}{E_q}$ for $t>0$.
Taking the infimum for $t\in(0,\infty)$ in this relation yields $\limsup_{a\uparrow N(q-1)}\alpha_v(a)\leq\alpha_v(N(q-1))=\frac{1}{bE_q}$,
where we have used Lemma \ref{attain-n(q-1)} (i) and (iii).
Next, let $a<N(q-1)$, and let $t_a\in(0,\infty)$ be a point satisfying $\inf_{t>0}g_a(t)=g_a(t_a)$,
and hence, $\alpha_v(a)=\frac{g_a(t_a)}{E_q}$.
Take any positive sequence $\{a_j\}_{j=1}^\infty$ satisfying
$a_j\uparrow N(q-1)$ as $j\to\infty$. First, suppose $\overline\lim_{j\to\infty}t_{a_j}\in(0,\infty)$, and then we may assume
$t_{a_j}\to t_0\in(0,\infty)$ as $j\to\infty$. We see
\begin{align*}
\lim_{j\to\infty}\alpha_v(a_j)=\frac{1}{E_q}\lim_{j\to\infty}g_{a_j}(t_{a_j})=\frac{g_{N(q-1)}(t_0)}{E_q}
\geq\frac{1}{E_q}\inf_{t>0}g_{N(q-1)}g(t)=\alpha_v(N(q-1))=\frac{1}{bE_q},
\end{align*}
where we have used Lemma \ref{attain-n(q-1)} (i) and (iii).
On the other hand, since we have already proved $\lim_{j\to\infty}\alpha_v(a_j)\leq\frac{1}{bE_q}$,
we obtain $g_{N(q-1)}(t_0)=\frac{1}{b}$. However, this is impossible since we observed $g_{N(q-1)}(t)>\frac{1}{b}$ for $t>0$
in the proof of Lemma \ref{attain-n(q-1)} (i) and (iii). Next, suppose $\overline\lim_{j\to\infty}t_{a_j}=\infty$, and then we may assume
$t_{a_j}\to\infty$ as $j\to\infty$. We see for $j$,
\begin{align*}
&\alpha_v(a_j)=\frac{g_{a_j}(t_{a_j})}{E_q}
=\frac{(1+t_{a_j})^{\frac{N(q-1)}{a_j}+\frac{N-q(N-1)}{b}}\left(1-(1+t_{a_j})^{-\frac{1}{b}}\right)}{
E_qt_{a_j}^{\frac{N(q-1)}{a_j}}
}\\&\geq\frac{1}{E_q}(1+t_{a_j})^{\frac{N-q(N-1)}{b}}\left(1-(1+t_{a_j})^{-\frac{1}{b}}\right)\to\infty
\end{align*}
as $j\to\infty$, and hence, $\lim_{j\to\infty}\alpha_v(a_j)=\infty$,
which is a contradiction since we have already proved $\limsup_{j\to\infty}\alpha_v(a_j)\leq\frac{1}{bE_q}$.
As a result, it holds $\lim_{j\to\infty}t_{a_j}=0$, and hence, $\lim_{a\uparrow N(q-1)}t_a=0$.
Then we see
\begin{align*}
&\liminf_{a\uparrow N(q-1)}\alpha_v(a)=\frac{1}{E_q}\liminf_{a\uparrow N(q-1)}g_a(t_a)
=\frac{1}{E_q}\liminf_{a\uparrow N(q-1)}\frac{
\left((1+t_a)^{\frac{1}{b}}-1\right)(1+t_a)^{(q-1)(\frac{N}{a}-\frac{N-1}{b})}
}{t_a^{\frac{N(q-1)}{a}}}\\
&\geq\frac{1}{E_q}\liminf_{a\uparrow N(q-1)}\frac{(1+t_a)^{\frac{1}{b}}-1}{t_a^{\frac{N(q-1)}{a}}}
\geq\frac{1}{E_q}\lim_{a\uparrow N(q-1)}\frac{(1+t_a)^{\frac{1}{b}}-1}{t_a}=\frac{1}{bE_q}.
\end{align*}
As a conclusion, we have $\lim_{a\uparrow N(q-1)}\alpha_v(a)=\frac{1}{bE_q}$.
\end{proof}
\begin{proof}[{\rm \bf Proof of Theorems \ref{thm1}-\ref{thm2}}]
Gathering up Lemmas \ref{sub-pro1}-\ref{attain-n(q-1)} and Proposition \ref{sub-pro2-est}, we have the results stated in Theorems \ref{thm1}-\ref{thm2}.
\end{proof}
\section{Proof of Theorems \ref{thm3}-\ref{thm4}}
\noindent
In this section, we shall prove Theorems \ref{thm3}-\ref{thm4}.
We start from the following lemma :
\begin{lem}\label{alpha^*-h}
Let $\alpha>0$, $a>0$ and $b>0$. Then there hold $D_\alpha=\sup_{t>0}f_\alpha(t)$,
$\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)$ and $\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)$ where for $t>0$,
\begin{align*}
\begin{cases}
&f_\alpha(t):=\displaystyle\frac{(1+t)^{\frac{1^*}{a}-\frac{1}{b}}+\alpha E_{1^*}t^{\frac{1^*}{a}}}{(1+t)^{\frac{1^*}{a}}},\\
&g(t):=\displaystyle\frac{(1+t)^{\frac{1^*}{a}}-(1+t)^{\frac{1^*}{a}-\frac{1}{b}}}{t^{\frac{1^*}{a}}},\\
&h(t):=\displaystyle\frac{(1+t)^{\frac{1^*}{a}-\frac{1}{b}}}{(1+t)^{\frac{1^*}{a}}-t^{\frac{1^*}{a}}}.
\end{cases}
\end{align*}
\end{lem}
\begin{proof}
The former two equalities are obtained by putting $q=1^*$ in Lemma \ref{D-f-g}.
Hence, we consider $\alpha_c$. For $u\in BV$ with $\|u\|_{TV}^a++\|u\|_{1}^b=1$, we see
\begin{align*}
&\frac{\|u\|_{1}}{E_{1^*}-\|u\|_{1^*}^{1^*}}\leq\frac{\|u\|_{1}}{E_{1^*}(1-\|u\|_{TV}^{1^*})}
=\frac{
\left(
1+\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}
\right)^{\frac{1^*}{a}-\frac{1}{b}}
}{
E_{1^*}\left(
\left(
1+\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}
\right)^{\frac{1^*}{a}}-\left(\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)^{\frac{1^*}{a}}
\right)
}\\
&=\frac{1}{E_{1^*}}h\left(\frac{\|u\|_{TV}^a}{\|u\|_{1}^b}\right)
\leq\frac{1}{E_{1^*}}\sup_{t>0}h(t),
\end{align*}
which shows $\alpha_c\leq\frac{1}{E_{1^*}}\sup_{t>0}h(t)$.
On the other hand, let $v\in BV\setminus\{0\}$ be a maximizer of $E_{1^*}$.
For $\lambda>0$, define $v_\lambda(x):=K\lambda v(\lambda^{\frac{1}{N}}x)$,
where $K=K(\lambda)>0$ is uniquely determined by
\begin{align*}
\|v_\lambda\|_{TV}^a+\|v_\lambda\|_{1}^b=K^a\lambda^{\frac{a}{N}}\|v\|_{TV}^a+K^b\|v\|_{1}^b=1.
\end{align*}
Then for $\lambda>0$, we observe
\begin{align*}
&\alpha_c\geq\frac{\|v_\lambda\|_{1}}{E_{1^*}-\|v_\lambda\|_{1^*}^{1^*}}
=\frac{\|v_\lambda\|_{1}}{E_{1^*}(1-\|v_\lambda\|_{TV}^{1^*})}
=\frac{1}{E_{1^*}}h\left(\frac{\|v_\lambda\|_{TV}^a}{\|v_\lambda\|_{1}^b}\right)\\
&=\frac{1}{E_{1^*}}h\left(K^{a-b}\lambda^{\frac{a}{N}}\frac{\|v\|_{TV}^a}{\|v\|_{1}^b}\right)
=\frac{1}{E_{1^*}}h\left(\frac{1}{K^b\|v\|_{1}^b}-1\right).
\end{align*}
Since $K=K(\lambda)$ is a continuous function on $(0,\infty)$
satisfying $K<\frac{1}{\|v\|_{1}}$ for $\lambda>0$,
$\lim_{\lambda\downarrow 0}K=\frac{1}{\|v\|_{1}}$ and $\lim_{\lambda\to\infty}K=0$,
we obtain
\begin{align*}
&\alpha_c\geq\frac{1}{E_{1^*}}\sup_{\lambda>0}h\left(\frac{1}{K^b\|v\|_{1}^b}-1\right)
=\frac{1}{E_{1^*}}\sup_{t>0}h(t).
\end{align*}
Hence, the proof of Lemma \ref{alpha^*-h} is complete.
\end{proof}
\begin{lem}\label{vani-conce-ge}Let $a>0$ and $b>0$.
\medskip
\noindent
{\rm(i)} Assume $\alpha_c<\infty$ and let $\alpha>\alpha_c$.
Then $D_\alpha$ is not attained.
\medskip
\noindent
{\rm(ii)} Assume $\alpha_v>0$ and let $\alpha<\alpha_v$.
Then $D_\alpha$ is not attained.
\medskip
\noindent
{\rm(iii)} Assume $\alpha_v<\alpha_c$ and let $\alpha_v<\alpha<\alpha_c$.
Then $D_\alpha$ is attained.
\end{lem}
\begin{proof}
By Theorem \ref{thm5}, we see that $D_\alpha$ is attained if and only if $\sup_{t>0}f_\alpha(t)$ is attained.
\medskip
\noindent
(i) Assume $\alpha_c<\infty$ and let $\alpha>\alpha_c$.
By contradiction, assume that there exists $t_0>0$ such that
$\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)$. First, note $\sup_{t>0}f_\alpha(t)\geq\lim_{t\to\infty}f_\alpha(t)=\alpha E_{1^*}$. By Lemma \ref{alpha^*-h} and the assumption $\alpha>\alpha_c$,
we obtain $\alpha>\alpha_c\geq\frac{1}{E_{1^*}}h(t_0)$, which implies $f_\alpha(t_0)<\alpha E_{1^*}$.
Then we see $\alpha E_{1^*}\leq\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)<\alpha E_{1^*}$,
which is a contradiction. Thus $\sup_{t>0}f_\alpha(t)$ is not attained.
\medskip
\noindent
(ii) Assume $\alpha_v>0$ and let $\alpha<\alpha_v$.
By contradiction, assume that there exists $t_0>0$ such that $\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)$.
First, note $\sup_{t>0}f_\alpha(t)\geq\lim_{t\downarrow 0}f_\alpha(t)=1$.
By Lemma \ref{alpha^*-h} and the assumption $\alpha<\alpha_v$, we obtain
$\alpha<\alpha_v\leq\frac{1}{E_{1^*}}g(t_0)$, which implies $f_\alpha(t_0)<1$.
Then we see $1\leq\sup_{t>0}f_\alpha(t)=f_\alpha(t_0)<1$, which is a contradiction. Thus $\sup_{t>0}f_\alpha(t)$ is not attained.
\medskip
\noindent
(iii) Assume $\alpha_v<\alpha_c$ and let $\alpha_v<\alpha<\alpha_c$.
First, note that $\lim_{t\downarrow 0}f_\alpha(t)=1$ and $\lim_{t\to\infty}f_\alpha(t)=\alpha E_{1^*}$.
By the assumption $\alpha>\alpha_v$, there exists $t_0>0$ such that
$\alpha>\frac{1}{E_{1^*}}g(t_0)$, which implies $f_\alpha(t_0)>1$.
On the other hand, by the assumption $\alpha<\alpha_c$,
there exists $t_1>0$ such that $\alpha<\frac{1}{E_{1^*}}h(t_1)$, which implies $f_\alpha(t_1)>\alpha E_{1^*}$. As a result, $\sup_{t>0}f_\alpha(t)$ is attained.
\end{proof}
\begin{lem}\label{a>1^*-ests}
Let $a>1^*$ and $b>0$. Then there hold
\begin{align*}
\alpha_v=0\text{ \,and \,}
\alpha_c=\begin{cases}
&\infty\text{ \,when \,}b>1,\\
&\frac{1}{E_{1^*}}<\alpha_c<\infty\text{ \,when \,}b\leq1.
\end{cases}
\end{align*} In particular, there hold
$\lim_{b\downarrow 0}\alpha_c=\frac{1}{E_{1^*}}$,
$\lim_{b\uparrow 1}\alpha_c=\frac{a}{1^*E_{1^*}}$, $\alpha_c=\frac{a}{1^*E_{1^*}}$ when $b=1$,
$\lim_{a\downarrow 1^*}\alpha_c=\frac{1}{E_{1^*}}$ when $b\leq1$ and $\lim_{a\to\infty}\alpha_c=\infty$ when $b\leq1$.
Moreover,
\begin{align*}
D_\alpha\text{ is attained for }
\begin{cases}
&\alpha>0\text{ \,when \,}b>1,\\
&0<\alpha<\alpha_c\text{ \,when \,}b=1,\\
&0<\alpha\leq\alpha_c\text{ \,when \,}b<1,
\end{cases}
\end{align*}
while
\begin{align*}
D_\alpha\text{ is not attained for }
\begin{cases}
&\alpha\geq\alpha_c\text{ \,when \,}b=1,\\
&\alpha>\alpha_c\text{ \,when \,}b<1.
\end{cases}
\end{align*}
\end{lem}
\begin{proof}
Let $a>1^*$. First, we can compute
\begin{align*}
\lim_{t\downarrow 0}g(t)=\lim_{t\downarrow 0}\frac{(1+t)^{\frac{1}{b}}-1}{t^{\frac{1^*}{a}}}
=\lim_{t\downarrow 0}\frac{a}{b1^*}t^{1-\frac{1^*}{a}}(1+t)^{\frac{1}{b}-1}=0,
\end{align*}
and then noting $g(t)>0$ for $t>0$, we have
$\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)=\frac{1}{E_{1^*}}\lim_{t\downarrow 0}g(t)=0$.
Next, we see $\lim_{t\downarrow 0}h(t)=1$ and
\begin{align*}
\lim_{t\to\infty}h(t)
=\lim_{t\to\infty}\frac{(1+t)^{-\frac{1}{b}}}{1-(\frac{t}{1+t})^{\frac{1^*}{a}}}
=\lim_{t\to\infty}\frac{a}{b1^*}\frac{(1+t)^{-\frac{1}{b}+1}}{(\frac{t}{1+t})^{\frac{1^*}{a}-1}}
=\begin{cases}
&\infty\text{ \,when \,}b>1,\\
&\frac{a}{1^*}\text{ \,when \,}b=1,\\
&0\text{ \,when \,}b<1.
\end{cases}
\end{align*}
We distinguish between three cases.
First, let $b>1$. Then it holds $\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=
\frac{1}{E_{1^*}}\lim_{t\to\infty}h(t)=\infty$,
and then by Lemma \ref{vani-conce-ge} together with $\alpha_v=0$,
$D_\alpha$ is attained for $\alpha>0$.
Next, let $b=1$. In this case, it follows $\lim_{t\to\infty}h(t)=\frac{a}{1^*}$.
By a direct computation, we obtain for $t>0$,
\begin{align*}
h'(t)=\frac{t^{\frac{1^*}{a}}(1+t)^{\frac{1^*}{a}-2}}{\left((1+t)^{\frac{1^*}{a}}-t^{\frac{1^*}{a}}\right)^2}\tilde h(t)\text{ \,and \,}
\tilde h'(t)=\frac{1^*}{at^2}\left(\left(\frac{t}{1+t}\right)^{1-\frac{1^*}{a}}-1\right)<0,
\end{align*}
where $\tilde h(t):=1+\frac{1^*}{at}-(\frac{1+t}{t})^{\frac{1^*}{a}}$.
Since $\lim_{t\to\infty}\tilde h(t)=0$, we observe $\tilde h(t)>0$ for $t>0$,
which implies $h'(t)>0$ for $t>0$. Summing-up, we have $\lim_{t\downarrow 0}h(t)=1$,
$\lim_{t\to\infty}h(t)=\frac{a}{1^*}>1$ and $h'(t)>0$ for $t>0$, which show
$\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\frac{1}{E_{1^*}}\lim_{t\to\infty}h(t)=\frac{a}{1^*E_{1^*}}$ and $1<h(t)<\frac{a}{1^*}$ for $t>0$.
Thus by Lemma \ref{vani-conce-ge}, $D_\alpha$ is attained for $0<\alpha<\alpha_c(=\frac{a}{1^*E_{1^*}})$, while $D_\alpha$ is not attained for $\alpha>\alpha_c$.
Furthermore, the relation $h(t)<\frac{a}{1^*}=\alpha_c E_{1^*}$ for $t>0$
implies $f_{\alpha_c}(t)<\alpha_c E_{1^*}=\lim_{s\to\infty}f_{\alpha_c}(s)\leq\sup_{s>0}f_{\alpha_c}(s)$ for $t>0$. Hence, $\sup_{t>0}f_{\alpha_c}(t)$ is not attained,
which is equivalent to the non-attainability of $D_{\alpha_c}$ by Theorem \ref{thm5}. Next, let $b<1$. By a direct computation, we have for $t>0$,
\begin{align*}
h'(t)=\frac{t^{\frac{1^*}{a}}(1+t)^{\frac{1^*}{a}-\frac{1}{b}-1}}{b\left((1+t)^{\frac{1^*}{a}}-t^{\frac{1^*}{a}}\right)^2}\tilde h(t)\text{ \,and \,}\tilde h'(t)=\frac{1^*}{at^2}\left(
\left(\frac{t}{1+t}\right)^{1-\frac{1^*}{a}}-b
\right),
\end{align*}
where $\tilde h(t):=1+\frac{b1^*}{at}-(\frac{1+t}{t})^{\frac{1^*}{a}}$. Then we obtain
\begin{align*}
\tilde h'(t)\begin{cases}
&<0\text{ \,for \,}0<t<t_0,\\
&=0\text{ \,for \,}t=t_0,\\
&>0\text{ \,for \,}t>t_0,
\end{cases}
\end{align*}
where $t_0:=\frac{b^{\frac{a}{a-1^*}}}{1-b^{\frac{a}{a-1^*}}}>0$.
Since $\lim_{t\to\infty}\tilde h(t)=0$ and
$\tilde h(t)=\frac{1}{t}\left(\frac{b1^*}{a}+t-t^{1-\frac{1^*}{a}}(1+t)^{\frac{1^*}{a}}\right)\to\infty$ as $t\downarrow 0$, there exists $t_1\in(0,t_0)$ such that
\begin{align*}
\tilde h(t)
\begin{cases}
&>0\text{ \,for \,}0<t<t_1,\\
&=0\text{ \,for \,}t=t_1,\\
&<0\text{ \,for \,}t>t_1,
\end{cases}
\end{align*}
which implies
\begin{align*}
h'(t)
\begin{cases}
&>0\text{ \,for \,}0<t<t_1,\\
&=0\text{ \,for \,}t=t_1,\\
&<0\text{ \,for \,}t>t_1.
\end{cases}
\end{align*}
This fact together with $\lim_{t\downarrow 0}h(t)=1$ and $\lim_{t\to\infty}h(t)=0$ shows
$\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\frac{1}{E_{1^*}}h(t_1)$,
and then it follows $\frac{1}{E_{1^*}}<\alpha_c<\infty$.
Thus by Lemma \ref{vani-conce-ge}, $D_\alpha$ is attained for $0<\alpha<\alpha_c$, while $D_\alpha$ is not attained for $\alpha>\alpha_c$.
Furthermore, note that $\alpha_c=\frac{1}{E_{1^*}}h(t_1)$
is equivalent to $f_{\alpha_c}(t_1)=\alpha_c E_{1^*}$.
This fact together with $\lim_{t\downarrow 0}f_{\alpha_c}(t)=1$
and $\lim_{t\to\infty}f_{\alpha_c}(t)=\alpha_c E_{1^*}=h(t_1)>1$,
we can conclude that $\sup_{t>0}f_{\alpha_c}(t)$ is attained,
and hence, $D_{\alpha_c}$ is attained by Theorem \ref{thm5}.
It remains to show the asymptotic behaviors of $\alpha_c$ on $a$ and $b$.
First, we prove $\lim_{b\downarrow 0}\alpha_c=\frac{1}{E_{1^*}}$.
Since $0<t_1<t_0\to0$ as $b\downarrow 0$, we have $t_1\to 0$ as $b\downarrow 0$,
and then we see $\lim_{b\downarrow 0}h(t_1)=1$, which implies
$\lim_{b\downarrow 0}\alpha_c=\frac{1}{E_{1^*}}\lim_{b\downarrow 0}h(t_1)=\frac{1}{E_{1^*}}$.
Next, we prove $\lim_{b\uparrow 1}\alpha_c=\frac{a}{1^*E_{1^*}}$.
We write $t_1=t_1(b)$ for $0<b<1$. First, we claim $\lim_{b\uparrow 1}t_1(b)=\infty$.
On the contrary, assume $\underline\lim_{b\uparrow 1}t_1(b)<\infty$.
Then we can pick up a sequence $\{b_j\}_{j=1}^\infty\subset(0,1)$ satisfying
$b_j\uparrow 1$ as $j\to\infty$ and $\lim_{j\to\infty}t_1(b_j)=\overline t_1\in[0,\infty)$.
Recall that $t_1(b_j)$ satisfies $\tilde h(t_1(b_j))=0$, which implies
\begin{align}\label{b_j-eq}
\frac{b_j1^*}{a}+t_1(b_j)-t_1(b_j)^{1-\frac{1^*}{a}}\left(1+t_1(b_j)\right)^{\frac{1^*}{a}}=0
\end{align}
for each $j$. Letting $j\to\infty$ in \eqref{b_j-eq}, we obtain
\begin{align*}
\frac{1^*}{a}+\overline t_1-\overline t_1^{1-\frac{1^*}{a}}\left(1+\overline t_1\right)^{\frac{1^*}{a}}=0,
\end{align*}
which shows that $\overline t_1>0$ is a solution of $\tilde h(t)=0$ for $t>0$ with $b=1$.
On the other hand, in the same way as above, we see that $\tilde h(t)$
for $t>0$ with $b=1$ satisfies $\lim_{t\downarrow 0}\tilde h(t)=\infty$,
$\lim_{t\to\infty}\tilde h(t)=0$ and $\tilde h'(t)<0$ for $t>0$,
and hence, it holds $\tilde h(t)>0$ for $t>0$,
which is a contradiction to $\tilde h(\overline t_1)=0$.
As a result, we obtain $\underline\lim_{b\uparrow 1}t_1(b)=\infty$,
which is equivalent to $\lim_{b\uparrow 1}t_1(b)=\infty$.
Now we compute $\lim_{b\uparrow 1}h(t_1)$.
Since $t_1$ satisfies $\tilde h(t_1)=0$, we have
$(1+t_1)^{\frac{1^*}{a}}=t_1^{-\frac{a-1^*}{a}}\left(\frac{b1^*}{a}+t_1\right)$.
Plugging this relation to $h(t_1)$, we obtain
\begin{align*}
h(t_1)
=\frac{a}{b1^*}\left(\frac{t_1}{t_1+\frac{b1^*}{a}}\right)^{\frac{a-1^*}{1^*}}
\frac{
t_1^{\frac{(a-1^*)(1-b)}{b1^*}}
}{
\left(t_1+\frac{b1^*}{a}\right)^{\frac{a(1-b)}{b1^*}}
}.
\end{align*}
Since $t_1(b)\to\infty$ as $b\uparrow 1$, in order to prove $\lim_{b\uparrow 1}h(t_1)=\frac{a}{1^*}$, it is enough to show that $\lim_{b\uparrow 1}t_1^{1-b}=1$.
Recalling $0<t_1<t_0$, we have $(1-b)\log t_1\leq(1-b)\log t_0\to 0$ as $b\uparrow 1$,
which shows $\lim_{b\uparrow 1}(1-b)\log t_1(b)=0$,
and hence, it holds $\lim_{b\uparrow 1}t_1^{1-b}=1$.
As a conclusion, we obtain $\lim_{b\uparrow 1}h(t_1)=\frac{a}{1^*}$,
which gives $\lim_{b\uparrow 1}\alpha_c=\frac{a}{1^*E_{1^*}}$.
Next, we prove $\lim_{a\to\infty}\alpha_c=\infty$ when $b\leq 1$.
We may assume $b<1$ since we have already proved $\alpha_c=\frac{a}{1^*E_{1^*}}$ when $a>1^*$ and $b=1$.
Noting $t_0\to\frac{b}{1-b}$ as $a\to\infty$, we see $h(t_0)\to\infty$ as $a\to\infty$.
Then since $t_1(<t_0)$ is the maximum point of $h(t)$ for $t>0$, we see $h(t_1)>h(t_0)\to\infty$ as $a\to\infty$,
which shows $\lim_{a\to\infty}h(t_1)=\infty$, and hence, it holds $\lim_{a\to\infty}\alpha_c=\frac{1}{E_{1^*}}\lim_{a\to\infty}h(t_1)=\infty$.
Next we show $\lim_{a\downarrow 1^*}\alpha_c=\frac{1}{E_{1^*}}$ when $b\leq 1$.
In the same reason as above, we may assume $b<1$.
Since $b<1$, we see $0<t_1<t_0\to 0$ as $a\downarrow 1^*$, and hence, it holds $\lim_{a\downarrow 1^*}t_1=0$.
Then we have $\lim_{a\downarrow 1^*}h(t_1)=1$, which is equivalent to $\lim_{a\downarrow 1^*}\alpha_c=\frac{1}{E_{1^*}}$.
The proof of Lemma \ref{a>1^*-ests} is complete.
\end{proof}
\begin{lem}
Let $a=1^*$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $b>1$. Then there hold $\alpha_v=\frac{1}{bE_{1^*}}$ and $\alpha_c=\infty$,
and $D_\alpha$ is attained for $\alpha>\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha\leq\alpha_v$.
\medskip
\noindent
{\rm(ii)} Let $b=1$. Then there holds $\alpha_v=\alpha_c=\frac{1}{E_{1^*}}$,
and $D_\alpha$ is not attained for $\alpha\ne \alpha_v$,
while $D_{\alpha_v}$ is attained.
\medskip
\noindent
{\rm(iii)} Let $b<1$. Then there holds $\alpha_v=\alpha_c=\frac{1}{E_{1^*}}$,
and $D_\alpha$ is not attained for $\alpha>0$.
\end{lem}
\begin{proof}
(i) Let $a=1^*$ and $b>1$. Since $g(t)=\frac{(1+t)\left(1-(1+t)^{-\frac{1}{b}}\right)}{t}$
for $t>0$, we see $\lim_{t\downarrow 0}g(t)=\frac{1}{b}$ and $\lim_{t\to\infty}g(t)=1$.
We can compute for $t>0$, $g'(t)=t^{-2}(1+t)^{-\frac{1}{b}}\tilde g(t)$,
where $\tilde g(t):=1+\frac{t}{b}-(1+t)^{\frac{1}{b}}$,
and we obtain for $t>0$,
$\tilde g'(t)=\frac{1}{b}-\frac{1}{b}(1+t)^{\frac{1}{b}-1}>0$ since $b>1$.
Then noting $\lim_{t\downarrow 0}\tilde g(t)=0$, we have $\tilde g(t)>0$ for $t>0$,
which implies $g'(t)>0$ for $t>0$.
Here, recalling $\lim_{t\downarrow 0}g(t)=\frac{1}{b}<1$ and $\lim_{t\to\infty}g(t)=1$,
we obtain $\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)=\frac{1}{bE_{1^*}}$.
On the other hand, since $h(t)=(1+t)^{1-\frac{1}{b}}$ for $t>0$, we obtain
$\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\infty$ since $b>1$.
Thus by Lemma \ref{vani-conce-ge}, $D_\alpha$ is attained for $\alpha>\alpha_v$, while $D_\alpha$ is not attained for $0<\alpha<\alpha_v$.
Next, we consider the case $\alpha=\alpha_v$.
Note $\alpha_v E_{1^*}=\frac{1}{b}<g(t)$ for $t>0$, which implies
$f_{\alpha_v}(t)<1=\lim_{s\downarrow 0}f_{\alpha_v}(s)\leq\sup_{s>0}f_{\alpha_v}(s)$
for $t>0$. Hence, $\sup_{t>0}f_{\alpha_v}(t)$ is not attained,
which is equivalent to the non-attainability of $D_{\alpha_v}$ by Theorem \ref{thm5}.
\medskip
\noindent
(ii) Let $a=1^*$ and $b=1$.
In this case, since $g(t)=1$ for $t>0$, it follows
$\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)=\frac{1}{E_{1^*}}$.
On the other hand, since $h(t)=1$ for $t>0$, it follows
$\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\frac{1}{E_{1^*}}$.
Thus there holds $\alpha_v=\alpha_c=\frac{1}{E_{1^*}}$,
and by Lemma \ref{vani-conce-ge}, $D_\alpha$ is not attained for $\alpha\ne\alpha_v(=\alpha_c)$. Next, we consider the case $\alpha=\alpha_v$.
In this case, we see $f_{\alpha_v}(t)=1$ for $t>0$,
and hence, $\sup_{t>0}f_{\alpha_v}(t)$ is attained, which is equivalent
to the attainability of $D_{\alpha_v}$ by Theorem \ref{thm5}.
\medskip
\noindent
(iii) Let $a=1^*$ and $b<1$. First, recall $\lim_{t\downarrow 0}g(t)=\frac{1}{b}$
and $\lim_{t\to\infty}g(t)=1$. In the same way as in the case (i), we see
$g'(t)=t^{-2}(1+t)^{-\frac{1}{b}}\tilde g(t)$
with $\tilde g(t):=1+\frac{t}{b}-(1+t)^{\frac{1}{b}}$ for $t>0$.
Then we obtain $\tilde g'(t)=\frac{1}{b}-\frac{1}{b}(1+t)^{\frac{1}{b}-1}<0$
for $t>0$ since $b<1$. Thus noting $\lim_{t\downarrow 0}\tilde g(t)=0$, we have
$\tilde g(t)<0$ for $t>0$, which implies $g'(t)<0$ for $t>0$.
Since $\lim_{t\downarrow 0}g(t)=\frac{1}{b}>1$ and $\lim_{t\to\infty}g(t)=1$,
it follows $\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)=\frac{1}{E_{1^*}}$.
On the other hand, since $h(t)=(1+t)^{1-\frac{1}{b}}$ for $t>0$,
it follows $\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\frac{1}{E_{1^*}}$.
Hence, we obtain $\alpha_v=\alpha_c=\frac{1}{E_{1^*}}$,
and then by Lemma \ref{vani-conce-ge}, $D_\alpha$ is not attained for $\alpha\ne\alpha_v(=\alpha_c)$. Next, we consider the case $\alpha=\alpha_v$.
Note $\alpha_v E_{1^*}=1<g(t)$ for $t>0$, which implies $f_{\alpha_v}(t)<1=\lim_{s\downarrow 0}f_{\alpha_v}(s)\leq\sup_{s>0}f_{\alpha_v}(s)$ for $t>0$. Hence, $\sup_{t>0}f_{\alpha_v}(t)$
is not attained, which is equivalent to the non-attainability of $D_{\alpha_v}$ by Theorem \ref{thm5}.
\end{proof}
\begin{lem}\label{last-lem-cri}
Let $a<1^*$ and $b>0$.
\medskip
\noindent
{\rm(i)} Let $b>1$. Then there hold $0<\alpha_v<\frac{1}{E_{1^*}}$ and $\alpha_c=\infty$,
and $D_\alpha$ is attained for $\alpha\geq\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha<\alpha_v$.
Moreover, there hold $\lim_{b\downarrow 1}\alpha_v=\frac{1}{E_{1^*}}$, $\lim_{b\to\infty}\alpha_v=0$,
$\lim_{a\downarrow 0}\alpha_v=\frac{1}{E_{1^*}}$ and $\lim_{a\uparrow 1^*}\alpha_v=\frac{1}{b E_{1^*}}$.
\medskip
\noindent
{\rm(ii)} Let $b\leq1$. Then there holds $\alpha_v=\alpha_c=\frac{1}{E_{1^*}}$,
and $D_\alpha$ is not attained for $\alpha>0$.
\end{lem}
\begin{proof}
(i) Let $a<1^*$ and $b>1$. First, we see $\lim_{t\downarrow 0}g(t)=\infty$ and $\lim_{t\to\infty}g(t)=1$. By a direct computation, we have for $t>0$,
$g'(t)=t^{-\frac{1^*}{a}-1}(1+t)^{\frac{1^*}{a}-\frac{1}{b}-1}\tilde g(t)$,
where $\tilde g(t):=\frac{t}{b}+\frac{1^*}{a}\left(1-(1+t)^{\frac{1}{b}}\right)$,
and $\tilde g'(t)=\frac{1}{b}-\frac{1^*}{ab}(1+t)^{\frac{1}{b}-1}$.
Then we observe
\begin{align*}
\tilde g'(t)\begin{cases}
&<0\text{ \,for \,}0<t<t_0,\\
&=0\text{ \,for \,}t=t_0,\\
&>0\text{ \,for \,}t>t_0,
\end{cases}
\end{align*}
where $t_0:=(\frac{1^*}{a})^{\frac{b}{b-1}}-1>0$.
Note
\begin{align*}
\lim_{t\downarrow 0}\tilde g(t)=0\text{ \,and \,}
\lim_{t\to\infty}\tilde g(t)=\lim_{t\to\infty}t\left(\frac{1}{b}+\frac{1^*}{at}-\frac{1^*(1+t)^{\frac{1}{b}}}{at}\right)=\infty
\end{align*}
since $b>1$. Hence, there exists $t_1>t_0$ such that
\begin{align*}
\tilde g(t)\begin{cases}
&<0\text{ \,for \,}0<t<t_1,\\
&=0\text{ \,for \,}t=t_1,\\
&>0\text{ \,for \,}t>t_1,
\end{cases}
\end{align*}
which implies
\begin{align*}
g'(t)\begin{cases}
&<0\text{ \,for \,}0<t<t_1,\\
&=0\text{ \,for \,}t=t_1,\\
&>0\text{ \,for \,}t>t_1.
\end{cases}
\end{align*}
Then recalling $\lim_{t\downarrow 0}g(t)=\infty$ and $\lim_{t\to\infty}g(t)=1$,
we have $\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)=\frac{1}{E_{1^*}}g(t_1)>0$,
which gives $0<\alpha_v<\frac{1}{E_{1^*}}$.
On the other hand, we see
\begin{align*}
\lim_{t\downarrow 0}h(t)=1\text{ \,and \,}
\lim_{t\to\infty}h(t)=\lim_{t\to\infty}\frac{(1+t)^{-\frac{1}{b}}}{1-(\frac{t}{1+t})^{\frac{1^*}{a}}}
=\frac{a}{b1^*}\lim_{t\to\infty}(1+t)^{-\frac{1}{b}+1}=\infty
\end{align*}
since $b>1$. Hence, we obtain $\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\infty$.
As a result, $D_\alpha$ is attained for $\alpha>\alpha_v$,
while $D_\alpha$ is not attained for $0<\alpha<\alpha_v$.
Next, we consider the case $\alpha=\alpha_v$.
Note $\alpha_v E_{1^*}=g(t_1)$ implies $f_{\alpha_v}(t_1)=1$.
Combining this fact with $\lim_{t\downarrow 0}f_{\alpha_v}(t)=1$
and $\lim_{t\to\infty}f_{\alpha_v}(t)=\alpha_v E_{1^*}=g(t_1)<1$,
we can conclude that $\sup_{t>0}f_{\alpha_v}(t)$ is attained, which is equivalent
to the attainability of $D_{\alpha_v}$ by Theorem \ref{thm5}.
Next, we prove the asymptotic behaviors of $\alpha_v$ on $a$ and $b$.
First, we show $\lim_{b\to\infty}\alpha_v=0$.
By a direct computation, we have for $b>1$,
\begin{align*}
g(t_0)
=\left(
\frac{1}{1-(\frac{a}{1^*})^{\frac{b}{b-1}}}
\right)^{\frac{1^*}{a}}\left(1-\left(\frac{a}{1^*}\right)^{\frac{1}{b-1}}\right)\to 0
\end{align*}
as $b\to\infty$. Since $t_1$ is the minimum point of $g(t)$ for $t>0$,
we have $0<g(t_1)<g(t_0)\to 0$ as $b\to\infty$,
and thus it holds $\lim_{b\to\infty}g(t_1)=0$,
which shows $\alpha_v=\frac{1}{E_{1^*}}g(t_1)\to 0$ as $b\to\infty$.
Next, we show $\lim_{b\downarrow 1}\alpha_v=\frac{1}{E_{1^*}}$.
First, since $g(t_1)<1$ for $b>1$, we obtain $\overline\lim_{b\downarrow 1}g(t_1)\leq1$.
On the other hand, recall that $t_1$ satisfies $\tilde g(t_1)=0$, which implies
$1+t_1=\left(1+\frac{at_1}{b1^*}\right)^b$. Plugging this relation to $g(t_1)$, we see for $b>1$,
\begin{align}
&\notag g(t_1)=\left(\frac{a}{b1^*}+\frac{1}{t_1}\right)^{\frac{1^*}{a}}
\left(\frac{a}{b1^*}\right)^{\frac{(b-1)1^*}{a}}\left(\frac{1}{1+\frac{b1^*}{at_1}}\right)
\left(t_1+\frac{b1^*}{a}\right)^{\frac{(b-1)1^*}{a}}\\
&\label{bto1-beha2}\geq\left(\left(\frac{a}{1^*}\right)^{\frac{1^*}{a}}+o(1)\right)
\left(t_0+\frac{b1^*}{a}\right)^{\frac{(b-1)1^*}{a}}
\end{align}
as $b\downarrow 1$, where we used $t_1>t_0\to\infty$ as $b\downarrow 1$,
which gives $\lim_{b\downarrow 1}t_1=\infty$. Furthermore, we observe for $b>1$,
\begin{align*}
&\left(t_0+\frac{b1^*}{a}\right)^{\frac{(b-1)1^*}{a}}
=\left(\frac{1^*}{a}\right)^{\frac{b1^*}{a}}
\left(1+\frac{\frac{b1^*}{a}-1}{(\frac{1^*}{a})^{\frac{b}{b-1}}}\right)^{\frac{(b-1)1^*}{a}}
=\left(\frac{1^*}{a}\right)^{\frac{b1^*}{a}}\left(1+o(1)\right)^{\frac{(b-1)1^*}{a}}
\end{align*}
as $b\downarrow 1$, and hence, it holds
\begin{align}\label{bto1-beha1}
\lim_{b\downarrow 1}\left(t_0+\frac{b1^*}{a}\right)^{\frac{(b-1)1^*}{a}}=\left(\frac{1^*}{a}\right)^{\frac{1^*}{a}}
\end{align}
Combining \eqref{bto1-beha2} with \eqref{bto1-beha1}, we obtain
$\underline\lim_{b\downarrow 1}g(t_1)\geq\left(\frac{a}{1^*}\right)^{\frac{1^*}{a}}\left(\frac{1^*}{a}\right)^{\frac{1^*}{a}}=1$.
As a conclusion, we have $\lim_{b\downarrow 1}g(t_1)=1$,
which yields $\lim_{b\downarrow 1}\alpha_v=\frac{1}{E_{1^*}}\lim_{b\downarrow 1}g(t_1)=\frac{1}{E_{1^*}}$.
Next, we show $\lim_{a\downarrow 0}\alpha_v=\frac{1}{E_{1^*}}$.
Since $g(t_1)<1$, we have $\overline\lim_{a\downarrow 0}g(t_1)\leq 1$.
On the other hand, noting $t_1>t_0\to\infty$ as $a\downarrow 0$, we see
\begin{align*}
g(t_1)=\left(1+\frac{1}{t_1}\right)^{\frac{1^*}{a}}\left(1-(1+t_1)^{-\frac{1}{b}}\right)\geq
1-(1+t_1)^{-\frac{1}{b}}\to 1
\end{align*}
as $a\downarrow 0$, and thus it holds $\underline\lim_{a\downarrow 0}g(t_1)\geq1$.
As a conclusion, we obtain $\lim_{a\downarrow 0}g(t_1)=1$, which implies $\lim_{a\downarrow 0}\alpha_v=\frac{1}{E_{1^*}}$.
Next, we show $\lim_{a\uparrow 1^*}\alpha_v=\frac{1}{bE_{1^*}}$.
We write $t_1=t_1(a)$ for $a<1^*$.
First, we claim $\lim_{a\uparrow 1^*}t_1(a)=0$.
To this end, assume that $\overline\lim_{a\uparrow 1^*}t_1(a)=\overline t_1\in(0,\infty]$.
Then we can pick up a sequence $\{a_j\}_{j=1}^\infty\in(0,1^*)$ such that $a_j\uparrow 1^*$ as $j\to\infty$
and $\lim_{j\to\infty}t_1(a_j)=\overline t_1$. Recall that $t_1(a_j)$ satisfies $\tilde g(t_1(a_j))=0$, which implies
\begin{align}\label{a_j-rela}
\frac{1}{b}+\frac{1^*}{a_jt_1(a_j)}-\frac{1^*(1+t_1(a_j))^{\frac{1}{b}}}{a_jt_1(a_j)}=0
\end{align}
for each $j$. First, assume that $\overline t_1=\infty$. Then letting $j\to\infty$ in \eqref{a_j-rela}, we obtain $\frac{1}{b}=0$, which is a contradiction.
Hence, it holds $\overline t_1\in(0,\infty)$. Now letting $j\to\infty$ in \eqref{a_j-rela} again, we have
\begin{align*}
\frac{1}{b}+\frac{1}{\overline t_1}-\frac{(1+\overline t_1)^{\frac{1}{b}}}{\overline t_1}=0,
\end{align*}
which yields $1+\overline t_1-(1+\frac{\overline t_1}{b})^b=0$. However, this is a contradiction since we can check $1+t-(1+\frac{t}{b})^b<0$ for $t>0$.
As a result, we have $\overline\lim_{a\uparrow 1^*}t_1=0$, which is equivalent to $\lim_{a\uparrow 1^*}t_1=0$.
Now we compute $g(t_1)$. By a direct computation, we see
\begin{align*}
&g(t_1)=\frac{(1+t_1)^{\frac{1^*}{a}-\frac{1}{b}}}{t_1^{\frac{1^*}{a}-1}}\frac{(1+t_1)^{\frac{1}{b}}-1}{t_1}
=\frac{\frac{1}{b}+o(1)}{t_1^{\frac{1^*}{a}-1}}
\end{align*}
as $a\uparrow 1^*$, where we used $\lim_{a\uparrow 1^*}t_1=0$. Hence, in order to prove $\lim_{a\uparrow 1^*}g(t_1)=\frac{1}{b}$,
it is enough to show $\lim_{a\uparrow 1^*}t_1^{\frac{1^*}{a}-1}=1$.
Since $0<t_0<t_1\to 0$ as $a\uparrow 1^*$, we see for $a<1^*$ close enough to $1^*$,
\begin{align*}
\left|\left(\frac{1^*}{a}-1\right)\log t_1\right|=\left(\frac{1^*}{a}-1\right)\log\frac{1}{t_1}
<\left(\frac{1^*}{a}-1\right)\log\frac{1}{t_0}\to 0
\end{align*}
as $a\uparrow 1^*$, which gives $\lim_{a\uparrow 1^*}\left(\frac{1^*}{a}-1\right)\log t_1=0$, and hence, it holds
$\lim_{a\uparrow 1^*}t_1^{\frac{1^*}{a}-1}=1$. As a conclusion, we obtain $\lim_{a\uparrow 1^*}g(t_1)=\frac{1}{b}$,
which shows $\lim_{a\uparrow 1^*}\alpha_v=\frac{1}{E_{1^*}}\lim_{a\uparrow 1^*}g(t_1)=\frac{1}{bE_{1^*}}$.
\medskip
\noindent
(ii) Let $a<1^*$ and $b\leq1$.
In the same way as in the case (i), we have for $t>0$,
\begin{align*}
g'(t)=t^{-\frac{1^*}{a}-1}(1+t)^{\frac{1^*}{a}-\frac{1}{b}-1}\tilde g(t)
\text{ \,and \,}\tilde g'(t)=\frac{1}{b}-\frac{1^*}{ab}(1+t)^{\frac{1}{b}-1},
\end{align*}
where $\tilde g(t):=\frac{t}{b}+\frac{1^*}{a}\left(1-(1+t)^{\frac{1}{b}}\right)$.
Then we see $b\tilde g'(t)=1-\frac{1^*}{a}(1+t)^{\frac{1}{b}-1}\leq 1-\frac{1^*}{a}<0$
for $t>0$ since $a<1^*$ and $b\leq 1$, and thus it holds $\tilde g'(t)<0$ for $t>0$.
Then since $\lim_{t\downarrow 0}\tilde g(t)=0$, we obtain $\tilde g(t)<0$ for $t>0$,
which implies $g'(t)<0$ for $t>0$.
This fact together with $\lim_{t\to\infty}g(t)=1$, we have
$\alpha_v=\frac{1}{E_{1^*}}\inf_{t>0}g(t)=\frac{1}{E_{1^*}}$.
On the other hand, we see $\lim_{t\downarrow 0}h(t)=1$ and
\begin{align*}
\lim_{t\to\infty}h(t)=\lim_{t\to\infty}\frac{
(1+t)^{-\frac{1}{b}}
}{
1-(\frac{t}{1+t})^{\frac{1^*}{a}}
}=\frac{a}{b1^*}\lim_{t\to\infty}(1+t)^{-\frac{1}{b}+1}=
\begin{cases}
&\frac{a}{1^*}\text{ \,when \,}b=1,\\
&0\text{ \,when \,}b<1.
\end{cases}
\end{align*}
In the same way as in the case (i), we see for $t>0$,
\begin{align*}
h'(t)=\frac{
t^{\frac{1^*}{a}}(1+t)^{\frac{1^*}{a}-\frac{1}{b}-1}
}{
b\left((1+t)^{\frac{1^*}{a}}-t^{\frac{1^*}{a}}\right)^2
}\tilde h(t)\text{ \,and \,}
\tilde h'(t)=\frac{1^*}{at^2}\left(\left(\frac{1+t}{t}\right)^{\frac{1^*}{a}-1}-b\right)>\frac{1^*}{at^2}(1-b)\geq0
\end{align*}
since $a<1^*$ and $b\leq1$, where $\tilde h(t):=\frac{b1^*}{at}+1-(\frac{1+t}{t})^{\frac{1^*}{a}}$.
Hence, it follows $\tilde h'(t)>0$ for $t>0$. Since $\lim_{t\to\infty}\tilde h(t)=0$,
we obtain $\tilde h(t)<0$ for $t>0$, which shows $h'(t)<0$ for $t>0$.
This fact together with $\lim_{t\downarrow 0}h(t)=1$ and
$\lim_{t\to\infty}h(t)=\begin{cases}
&\frac{a}{1^*}<1\text{ \,when \,}b=1,\\
&0\text{ \,when \,}b<1
\end{cases}
$ gives $\alpha_c=\frac{1}{E_{1^*}}\sup_{t>0}h(t)=\frac{1}{E_{1^*}}$.
As a result, we have $\alpha_v=\alpha_c=\frac{1}{E_{1^*}}$,
and then $D_\alpha$ is not attained for $\alpha\ne\alpha_v(=\alpha_c)$.
Next, we consider the case $\alpha=\alpha_v$.
Note that $(1=)\alpha_v E_{1^*}<g(t)$ for $t>0$ implies
$f_{\alpha_v}(t)<1=\lim_{s\downarrow 0}f_{\alpha_v}(s)\leq\sup_{s>0}f_{\alpha_v}(s)$
for $t>0$. Hence, $\sup_{t>0}f_{\alpha_v}(t)$ is not attained,
which is equivalent to the non-attainability of $D_{\alpha_v}$ by Theorem \ref{thm5}.
The proof of Lemma \ref{last-lem-cri} is complete.
\end{proof}
\begin{proof}[{\rm \bf Proof of Theorems \ref{thm3}-\ref{thm4}}]
Gathering up Lemmas \ref{a>1^*-ests}-\ref{last-lem-cri} we have the results stated in Theorems \ref{thm3}-\ref{thm4}.
\end{proof}
|
2,869,038,155,674 | arxiv | \section{Side-gate controlled QPCs}
Our device consists of a monolayer graphene crystal encapsulated in hexagonal boron nitride~\cite{Dean2010}. The stack is deposited on a SiO$_2$/Si substrate, which serves as the back-gate (BG). We use conventional electron-beam lithography and reactive ion etching to fabricate three QPCs in the same graphene crystal (Fig.~\ref{fig1}(a), see details in Methods). The constriction widths of the three QPCs are $w=$ 550, 350 and 150 nm, respectively. Both the QPC constrictions and the side gates are formed in the same etching step, as shown in Fig.~\ref{fig1}(b). Here, the QPC channels are horizontal, and the triangular-shaped graphene regions that are etched away from the channel are used as the side gates. Both the side gates and the main region of graphene are contacted via thermally evaporated Cr/Au. The gap between the channel and the side gate at their closest separation is 50 nm and grows wider as one moves further away from the QPC. This design is intended to reduce the undesired gating away from the QPCs.
To characterize the efficiency of the QPCs as a function of constriction width, we measure their two-probe differential conductance, $G$ (Methods). All data presented in this paper are taken at 100 mK unless noted otherwise. In Fig.~\ref{fig1} (c), we show the conductance of all three QPCs as a function of $V_{QPC}$ at a magnetic field of $B=4$ T and a back-gate voltage $V_{BG}=4$ V (bulk filling factor $\nu=2$). When the bulk is n-doped, applying a negative side gate voltage gradually reduces $G$ to zero. This is due to the carrier depletion in the constriction, which causes a reduction of the transmission probability of the edge states. Applying a high positive side gate voltage can introduce additional Landau-levels inside the constrictions, which helps to equilibrate the chemical potential of the counter-propagating edge states~\cite{Zimmermann2017}. Further increasing $V_{QPC}$ results in a reduction of $G$, which saturates at $e^2/h$ when the equilibration process is complete. At this point the constriction effectively serves as a floating contact, thereby doubling the QPC resistance.
As expected, the tuning efficiency of the side gates is higher for the narrower QPCs. Indeed, the extent of the $G\approx 2e^2/h$ plateau in Fig.~\ref{fig1}(c) shrinks as the constriction becomes progressively narrower. While the resulting higher tunability of the QPC is an attractive feature, the degree of conductance quantization eventually degrades. Patches of reduced conductance are seen in the middle of the $\nu=2$ plateau of QPC3. Moreover, the counterpropagating channels rapidly equilibrate with increasing $V_{QPC}$, as clearly visible for QPC3 at $\nu=6$ around $V_{BG}=10$ V in Fig.~\ref{fig1}(d). The degradation is expected due to the smaller separation between the counter-propagating edge states which enables backscattering; the edge disorder introduced by reactive-ion etching~\cite{Bischoff2016} should also have the most effect on the narrowest constriction. The lifting of the quantization becomes even more noticeable for symmetry-broken states which have a smaller energy gap. (These states start to show up at $B= 6$ T in this device and are shown in Supporting Information Fig.~\ref{sup0}.) We therefore conclude that for our design the optimal compromise is obtained for QPC2, which has the medium constriction width of 350 nm.
Fig.~\ref{fig1}(d) shows the conductance maps as a function of the back and side gates in a wide range of bulk filling factors from $\nu= -6$ to $6$ at $B=4$ T. Ideally, QPC conductance should change monotonically between the plateaus~\cite{Zimmermann2017}. However, here multiple resonances appear as lines in the transition regions, corresponding to oscillations in the line cuts presented in Fig.~\ref{fig1} (c). Note that the slope of these resonances in the gate-gate map is negative, apparently tracing lines of constant electron density in the QPC regions. This is an indication that the resonances are caused by the charging of localized states within the constrictions~\cite{Bischoff2015,Bischoff2016}. From the slopes, we roughly estimate the side-gates efficiency in tuning of electron density as 1/10, 1/6 and 1/2 times the back-gate efficiency for QPC1, QPC2 and QPC3 respectively.
\begin{figure}
\includegraphics[width=0.9\textwidth]{Figs/fig2/fig2V2.pdf}
\caption{\textbf{Phase jumps of the Fabry-P\'erot interferometer.}
(a) Optical image of the Fabry-P\'erot interferometer. Scale bar is 1 $\mu$m. (b) Sketch showing the propagation of the $\nu=$ 2 edge states. Two distinct edge channels are obtained at $B=$ 6 T and $V_{BG}\approx$ 3.05 V. The QPC voltages are $V_{QPC1}=- 20$ V and $V_{QPC2}=- 8$ V, resulting in the outer channel being partially transmitted while the inner channel is fully reflected. (c,d) Two-probe conductance $G$ across the interferometer measured as a function of c) TPG and BPG and d) TPG and BG. The dashed lines mark the position of the phase jumps. (e) Simulated conductance map corresponding to panel (c) with parameters obtained from the fit in panel (g). (f) The capacitance network model we use to explain the phase jumps observed in (c,d). The unintentional quantum dot (QD) is a part of the top plunger gate (see text). The estimated capacitances are $C_{TPG}\approx77$ aF, $C_{BG}\approx800$ aF, $C_1\approx15$ aF and $C_2\approx130$ aF. (g) A vertical cut of the conductance map in panel (c) at $V_{BPG}=0$ (black dots) agrees well with the fit (red curve) corresponding to the model in panel (f). }
\label{fig2}
\end{figure}
\section{Fabry-P\'erot Interferometer and Graphene Plunger Gates}
To further explore the operation of the graphene gates, we designed a Fabry-P\'erot interferometer using QPC1 and QPC2, which both showed good quantization of the $\nu=1$ and 2 states at $B=6$ T (see Supporting Information Fig.~\ref{sup0}). To this end, we etched $\sim100$ nm wide trenches into the graphene region between QPC1 and QPC2, thereby detaching the two middle metal contacts that were used to measure Fig.~\ref{fig1}. The advantage of severing the contacts is two-fold. First, the edge states that used to be equilibrated by these contacts can now preserve their phase coherence in the region between the two QPCs, thereby forming the interferometer, see Fig.~\ref{fig2}(a). Second, the two graphene strips attached to the former contacts now serve as the top and bottom plunger gates (TPG and BPG). Voltages $V_{TPG}$ and $V_{BPG}$ applied to the plunger gates tune the positions of the quantum Hall edge states, which affect the effective area of the cavity between the two QPCs. The resulting interferometer cavity is about 8 $\mu m^2$ in area and 12 $\mu$m in perimeter.
We first study the $\nu=2$ case, when the interferometer has two distinct edge channels. The back-gate voltage is set at $V_{BG}$ = 3.05 V, corresponding to the beginning of the bulk $\nu=$ 2 plateau, which extends from 2.8 V to 6.8 V. The QPC voltages are fixed at $V_{QPC1}=-$ 20 V and $V_{QPC2}=-$ 8 V. Here, the inner channel is fully reflected while the outer channel is partially transmitted into the cavity and is the one causing the interference pattern, see schematics in Fig.~\ref{fig2}(b).
With these parameters, the transmission probabilities of the outer channel are estimated to be $\sim0.5$ (see Supporting Information).
We start by applying voltages to both plunger gates; increasing these voltages moves the interfering channel towards the graphene edge, thereby increasing the enclosed area $A$. If the magnetic field $B$ is held constant, a change $\delta A$ in the area modulates the Aharonov-Bohm (AB) phase of electrons traveling around the cavity by $\delta \phi=2 \pi e B \delta A /h$. This modulation can be observed by measuring the differential conductance across the interferometer while tuning both $V_{TPG}$ and $V_{BPG}$ (Fig.~\ref{fig2}(c)). We observe periodic interference fringes in a wide range of gate voltages, with similar periods of about 2 mV, indicating that the two gates have similar efficiency.
We next focus on the small phase jumps observed in Fig.~\ref{fig2}(c) (one of them is marked by the black dashed line). The features are periodic in TPG, indicating regular charging of some localized states. The features are not affected by the BPG, resulting in their appearance as horizontal lines. Apparently, these states are located in the TPG region, and the BPG is too far away to have a sizable effect. In Fig.~\ref{fig2}(d), we plot the same interference pattern as a function of TPG and BG. The features now acquire a slope vs. $V_{TPG}$ and $V_{BG}$ (see the black dashed line as an example), which indicates that the states are influenced by both gates. In comparison to the AB fringes associated with the interferometer, these phase jumps appear twice more frequently in $V_{TPG}$, but five times less frequently in $V_{BG}$ (see the separation between jumps in Fig.~\ref{fig2}(d)). Most importantly, their slope is positive, opposite to that of the AB fringes.
These observations suggest that the states responsible for the small phase jumps are localized not in the interferometer, but in the region of graphene attached to the TPG, which effectively forms a small quantum dot. The states in this dot are coupled to the gate electrode via tunneling, and applying positive TPG voltage depletes this dot of electrons, in contrast to the same voltage adding electrons in the interferometer. This is the reason why the resulting phase jumps follow lines of a positive slope in the $V_{TPG} - V_{BG}$ plane of Fig.~\ref{fig2}(d). The graphene region connected to TPG is about 4-6 times smaller than the interferometer, in agreement with the observed smaller capacitance of these states to the BG as compared to the states of the interferometer.
Once the origin of the phase jumps has been identified, it is straightforward to account for them by considering the capacitive network shown in Fig.~\ref{fig2}(f). Here, the localized states in the TPG region (QD) are tunnel coupled to $V_{TPG}$. Their capacitances to the interferometer (FP) and BG are $C_1$ and $C_2$ respectively. The capacitances of the interferometer to the TPG and BG are $C_{TPG}$ and $C_{BG}$. The amplitude of the phase jumps is given by $2\pi C_1/(C_1+C_2)$.
We approximate the oscillatory part of conductance by a simple $\delta G \propto \cos(\delta \phi)$ appropriate for our relatively high base temperature of 100 mK where higher-harmonics have been sufficiently suppressed by thermal smearing effect (see the next section).
In Fig.~\ref{fig2}(g), we plot the conductance vs. TPG (dots), corresponding to the vertical cross-section on Figs.~\ref{fig2}(c). The distortions of the oscillations introduced by the phase jumps are well captured by the fit (red line). The fit gives $C_{TPG}\approx77$ aF and $C_1\approx C_2/9$.
Using the parameters obtained in this fit, we can calculate the expected interference pattern. The resulting conductance pattern is shown in Fig.~\ref{fig2}(e), demonstrating a good qualitative agreement with the experiment (Fig.~\ref{fig2}(c)). From the $V_{BG}$ period of the interference pattern $\sim0.2$ mV and of the phase jumps $\sim1.1$ mV in Fig.~\ref{fig2}(d), we further estimate $C_{BG}\approx800$ aF (correponding to a parallel plate capacitor of $\sim$ 7 um$^2$) and $C_1+C_2\approx145$ aF. Therefore, we have $C_1\approx15$ aF and $C_2\approx 130$ aF ($\sim$ 1 um$^2$).
We note that the BPG does not produce similar features, indicating that the metal electrode is better coupled to the nearby graphene region.
\section{Energy dependence of conductance oscillations}
We next explore the energy dependence of the interference fringes by tuning the temperature and bias. We chose to set the bulk filling factor at $\nu=1$, when only a single spin-polarized edge channel is present. We plot the conductance pattern as a function of bias $V$ and $V_{BG}$ in Fig.~\ref{fig3}(a). The map shows contours of constant phase, which are sometimes interpreted as non-interacting AB oscillations, although it has been demonstrated that the charging effects are significant~\cite{ofek_role_2010,sivan_interaction-induced_2018} even at $\nu=1$~\cite{roosli_observation_2020}. We estimate (see Supporting Information) that in our case the charging energy exceeds the level spacing. In fact, the familiar Coulomb diamonds can be recovered by integrating the differential conductance to plot the current through the dot (Fig.~\ref{diamond}). The diamonds are strongly asymmetric because the capacitance to the gates dominates the capacitance to the source and drain contacts. The vertical boundaries of the diamonds are not visible in the differential conductance but are revealed in the current map.
The strong role of the Coulomb interactions is confirmed by Fig.~\ref{fig3}(b), which shows the map of conductance in the $V_{TPG} - \Delta B$ plane, where $\Delta B$ is the deviation of magnetic field from 6 T. The AB period of the interferometer should be $\sim 0.5$ mT, and on that scale the interference fringes show negligible slope, which indicates the dominant role of the charging energy~\cite{halperin_theory_2011}. Similar nearly vertical conductance fringes were observed in the other measurement at $\nu=1$ dominated by Coulomb interactions~\cite{roosli_observation_2020}.
Nonetheless, the description in terms of Fabry-P\'erot interference is still possible~\cite{Feldman2022}. To that end, we average the conductance over the gate voltage, $\langle G(V) \rangle_{BG}$, as shown in the right inset of Fig.~\ref{fig3}(a). We then subtract $\langle G(V) \rangle_{BG}$ from the data in Fig.~\ref{fig3}(a) and obtain a simple stripe pattern of Fig.~\ref{fig3}(c), which strongly resembles the simulated conductance map of resonant tunneling plotted in Fig.~\ref{simmap}. Note that the background subtraction does not affect the amplitude of conductance oscillations vs. $V_{BG}$.
\begin{figure}
\includegraphics[width=1\textwidth]{Figs/fig3/fig3.pdf}
\caption{\textbf{Energy dependence of AB interference.}
(a) The interferometer conductance measured as a function of bias $V$ and $V_{BG}$ at $\nu=$ 1 (schematic indicated in the inset). The QPC transmission probabilities are $\mathcal{T}_{QPC1}\sim0.7$ and $\mathcal{T}_{QPC2}\sim0.6$. The right inset shows the conductance averaged over the gate voltage, $\langle G(V) \rangle_{BG}$, plotted as a function of bias. (b) Interference pattern vs. $V_{BG}$ and the incremental magnetic field change $\Delta B$. (c) Conductance of panel (a) with gate-averaged background $\langle G(V) \rangle_{BG}$ (inset of panel a) subtracted to reveal the interference fringes. (d) The visibility of zero-bias oscillation pattern plotted vs. temperature. The blue solid curve is a fit in the form of Eq.~(\ref{eq2}) with $T_0=46$ mK and the red dashed line is obtained from numerically convolving the data in panel (a) with the derivative of the Fermi function.}
\label{fig3}
\end{figure}
The conductance fringes in Fig.~\ref{fig3}(c) have a positive slope, where increasing $V_{BG}$ increases the AB phase $\delta \phi$, as we have seen in Fig.~\ref{fig2}. On the other hand, increasing $V$ reduces the ``traveling phase'' $\phi_0=(-eV/\hbar v) L$, where $L$ is the perimeter of the interferometer and $v$ is the velocity of the edge states, which is renormalized by interactions. We use Fig.~\ref{fig3}(c) to obtain the bias oscillation period of $V_0\approx$ 80 $\mu$V. From here, we estimate the edge state velocity $v=e V_0 L/h\approx$ 2.3$\times$10$^5$ m/s, which agrees with the previously reported values~\cite{deprez_tunable_2020,ronen_aharonov-bohm_2020}.
The variation of the geometrical phase $\phi_0=\frac{2 \pi E}{eV_0}$ with energy results in smearing of the conductance pattern. We find that in our temperature range $T=100-250$~mK it is sufficient to approximate the conductance oscillations as $\delta G \propto \cos(\delta \phi+\phi_0)$. From here, we obtain the thermal dependence of the oscillations:
\begin{equation}
\delta G\propto \int_{-\infty}^{\infty} \cos(\delta\phi+\frac{2 \pi E}{eV_0})\frac{\partial f(E,T)}{\partial E}dE \propto \frac{T/T_0}{\sinh(T/T_0)} \cos(\delta \phi),
\label{eq2}
\end{equation}
where $f$ is the Fermi distribution function and $T_0=eV_0/2\pi^2k_B$. Therefore, the visibility of the oscillations, defined as $(G_{max}-G_{min})/(G_{max}+G_{min})$, is expected to be proportional to $T/\sinh(T/T_0)$. To extract the visibility from the experimental data, we measure the conductance maximum (minimum) as the average height of several peaks (valleys) in $G(T)$ measurement of Fig.~\ref{tem}. The resulting Fig.~\ref{fig3}(d) shows the measured zero-bias visibility vs. temperature (black dots) and the fit (blue solid line). Note that
the expression of Eq.~(\ref{eq2}) does not simply reduce to $\exp(-T/T_0)$. The fit yields $T_0=46$ mK, which closely matches the value $eV_0/2\pi^2k_B\approx 47$ mK with the value of $V_0$ obtained from Fig.~\ref{fig3}(c).
Another way to compare the energy and temperature scales is to numerically convolve the transmission extracted from the $G(V)$ pattern in Fig.~\ref{fig3}(a) with the Fermi-Dirac distribution at elevated temperatures. The result is shown as the red dashed line in Fig.~\ref{fig3}(d), also agreeing with the measured temperature dependence. The fact that both numerical convolution of $G(V)$ and fitting with Eq.~(\ref{eq2}) produce consistent results verifies that thermal broadening is the main origin of the visibility suppression at elevated temperatures.
In summary, we present a simple recipe to fabricate quantum point contacts with self-aligned side-gates in graphene, which operate in the quantum Hall regime. A reduction in the constriction widths of the QPCs improves their tunability at the expense of the degree of the conductance quantization, suggesting an optimal width in the $300-400$ nm range. We use this technique to define a quantum Hall Fabry-P\'erot interferometer and explore the charging patterns both for $\nu= 1$ and $2$, studying their dependence on gate voltages, bias and temperature. In the future, additional screening layers such as top and bottom graphite gates can be used to reduce the charging effects. Such designs, pioneered in GaAs could enable exploration of the non-Abelian statistics in the fractional quantum Hall regime~\cite{nakamura_direct_2020}. The quality of the side-gates and QPCs can also be substantially improved with emerging cryoetching~\cite{Cleric2019} and AFM lithography~\cite{Kun2020} techniques.
\section{Methods}
\subsection{Sample Fabrication}
The graphene and h-BN crystals are exfoliated onto SiO$_2$ substrates and then assembled together by the conventional dry transfer technique using PC/PDMS stamps. The contacts and trenches are patterned by electron beam lithography using a layer of PMMA resist, developed in cold IPA/DI water mixture (ration 3:1) and then further etched by CHF$_3$/O$_2$ plasma in a reactive ion etcher. The Cr/Au (1 nm/100 nm) metal leads are deposited at 10$^{-7}$ mbar base pressure in a thermal evaporator.
\subsection{Measurement}
The device is cooled down to the base temperature of $\sim 100$ mK in a Leiden dilution refrigerator wired with resistive coaxial lines and low pass filters. The two-terminal differential resistance of individual QPCs is measured with 1 nA, 15 Hz square-wave excitations with a digital acquisition board and a homemade voltage preamplifier. For a better signal-to-noise ratio, the two-terminal differential conductance of the interferometer is measured with a lock-in amplifier and homemade low noise current preamplifier with a current to voltage gain of 10$^7$. The amplitude of the voltage excitation applied from the lock-in amplifier is 7 $\mu$V and the frequency is 577 Hz. In this case, the device is measured in series with two 5 kOhm filters, whose resistance is then subtracted from the measured resistance. Similarly, the voltage drop across the filters is subtracted from the applied bias to obtain
the bias across the sample. By calibrating against the quantum Hall resistance of the sample, an overall gain correction of 1.0638 is obtained to account for the RC filters, voltage dividers and buffers used in the system. The DC bias and gate voltages are applied by a combination of NI USB-6363 and DAC8728 controlled by NI USB-6501.
\acknowledgement
We greatly appreciate stimulating discussion with H. U. Baranger.
Transport measurements conducted by L.Z., E.G.A., and T.F.Q.L. were supported by NSF award DMR-2004870.
Lithographic fabrication and characterization of the samples by L.Z. and A.S., as well as the project guidence by Z.I and G.F were supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy, under Award No. DE-SC0002765.
T.F. and F.A. acknowledge the ARO under Award W911NF-16-1-0132.
K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, (grant no. JPMXP0112101001), JSPS KAKENHI (grant no. JP20H00354) and CREST (no. JPMJCR15F3, JST).
The sample fabrication was performed in part at the Duke University Shared Materials Instrumentation Facility (SMIF), a member of the North Carolina Research Triangle Nanotechnology Network (RTNN), which is supported by the National Science Foundation (Grant ECCS-1542015) as part of the National Nanotechnology Coordinated Infrastructure (NNCI).
\newpage
\global\long\defS\arabic{equation}{S\arabic{equation}}
\global\long\defS\arabic{figure}{S\arabic{figure}}
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\section{Supporting Information}
\subsection{Characterization of QPC transmissions}
\begin{figure}
\includegraphics[width=1\textwidth]{Figs/sup/6TQPC.pdf}
\caption{Two-probe differential conductance of a) QPC1 and b) QPC2 before the device was etched into an interferometer. The data measured at 6T shows quantized $\nu=1$ regions. In comparison, $\nu=1$ has not formed yet for the narrowest QPC3, shown in (c).}
\label{sup0}
\end{figure}
Before etching the device to form an interferometer, QPC1 and QPC2 are characterized at 6 T showing good quantization for $\nu=1$ (Fig.~\ref{sup0}). Once the sample was etched to form the interferometer, the transmission coefficient of a particular QPC (1 or 2) can be estimated by measuring the two-terminal conductance of the interferometer, $G$, while keeping the other QPC (2 or 1) open.
\begin{figure}
\includegraphics[width=1\textwidth]{Figs/sup/v2T_QPC.pdf}
\caption{\textbf{QPC operation for Fig.~2.}
(a) The interferometer conductance $G$ as a function of QPC voltages at $V_{BG}=3$ V ($\nu=2$). The green diamond marks the QPC voltges used in Fig.~2 ($V_{QPC1}=-20$ V and $V_{QPC2}=-8$ V.).
(b) $G$ as a function of $V_{BG}$ at $V_{QPC1}=-20$ V, $V_{QPC2}=0$ (blue) V and $V_{QPC1}=0$ V, $V_{QPC2}=-8$ V (red). The vertical green line shows the position of the back-gate voltage in Fig.~2 ($V_{BG}=3.05$ V). From the intersections we determine $\mathcal{T}_{QPC1}\sim0.6$ and $\mathcal{T}_{QPC2}\sim0.4$.}
\label{sup1}
\end{figure}
As shown in Fig.~\ref{sup1}(a), we first measure $G$ as a function of both QPC voltages at $V_{BG}=3$ V ($\nu=2$). The green diamond marker labels the QPC operation point for Fig. 2 ($V_{QPC1}=-20$ V and $V_{QPC2}=-8$ V). At $V_{QPC1,2}=0$ V, both QPCs are open ($G=e^2/h$). From the conductance at $(V_{QPC1},V_{QPC2})=(-20,0)$ V and $(0,-8)$ V, we find the QPC transmissions to be around 0.4 for both QPC1 and QPC2. For the data shown in Fig. 2, we use a slightly higher $V_{BG}=3.05$ V and the QPC transmissions are found to be higher (see Fig.~\ref{sup1}(b) where the green line marks $V_{BG}=3.05$ V). Here, the estimation of QPC transmissions is $\mathcal{T}_{QPC1}\sim0.6$ and $\mathcal{T}_{QPC2}\sim0.4$.
\begin{figure}
\includegraphics[width=1\textwidth]{Figs/sup/v1T_QPC.pdf}
\caption{\textbf{QPC operation map for Fig.~3.}
(a) The interferometer conductance $G$ as a function of QPC voltages at $V_{BG}=2.15$ V ($\nu=1$). The green diamond marks the QPC voltges used in Fig.~3 ($V_{QPC1}=-13.5$ V and $V_{QPC2}=-6.63$ V.).
(b) A cut of (a) along the vertical white dashed line showing $G$ as a function of $V_{QPC1}$ while QPC2 is open. The green line labels QPC1 operation voltage. (c) A cut of (a) along the horizontal white dashed line showing $G$ as a function of $V_{QPC2}$ while QPC1 is open. The green line labels QPC2 operation voltage. From the intersections in (b, c) we determine $\mathcal{T}_{QPC1}\sim0.7$ and $\mathcal{T}_{QPC2}\sim0.6$.}
\label{sup2}
\end{figure}
To determine the QPC transmissions at the $\nu=1$ operating point, in Fig.~\ref{sup2} (a), we plot $G$ as a function of both QPC voltages at the same back-gate voltage used in Fig. 3, i.e. $V_{BG}=2.15$ V. The green diamond marker labels the QPC operation point for Fig. 3 ($V_{QPC1}=-13.5$ V and $V_{QPC2}=-6.63$ V). Compared to $\nu=2$, the QPC transmissions at $\nu=1$ are much more sensitive to the gate voltages, and the cross-talk between the two QPC gates becomes noticeable. In order to estimate the transmission of QPC1 more accurately in the presence of cross-talk, in Fig.~\ref{sup2}(b) we plot $G$ as a function of $V_{QPC1}$ at $V_{QPC2}=-4$ V, labeled by the vertical white dashed line in panel (a). This value of $V_{QPC2}$ is not too far from the operation point, yet QPC2 is already open. Similarly, for determining transmission of QPC2, in Fig.~\ref{sup2}(c) we plot a cut at $V_{QPC1}=-10.5$ V, corresponding to the horizontal white dashed line in panel (a).The QPC operation gate voltages labeled by green lines in panel (b) and (c), from which we estimate $\mathcal{T}_{QPC1}\sim0.7$ and $\mathcal{T}_{QPC2}\sim0.6$.
Note that between measuring in the $\nu=2$ and $\nu=1$ regimes, the sample condition changed due to a sudden power-off of the electronics controlling the gates. Although the QPCs were still operating properly, the charge neutrality point shifted towards the p-side by $\sim0.5$ V in $V_{BG}$.
\subsection{Map of current through the interferometer}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Figs/sup/Imap.pdf}
\caption{The integral of the differential conductance in Fig.~3(a), showing the current through the interferometer as a function of bias and back-gate voltage.}
\label{diamond}
\end{figure}
As demonstrated in Ref.~\cite{roosli_observation_2020}, quantum Hall Fabry-P\'erot interferometers have close relation to quantum dots in the quantum Hall regime. When the transmissions across the QPCs are high, the device is operated in the interferometer regime. Decreasing the transmissions gradually tune an interferometer smoothly into a closed quantum dot. Here, we operate the device in the intermediate regime. The close relation to a quantum dot can be seen in the map of the DC current through the interferometer as a function of the applied bias and the back-gate voltage, Fig.~\ref{diamond}. This map is obtained by integrating the striped interference pattern of the differential conductance shown in Fig.~\ref{fig3}a of the main text. ``Coulomb diamonds" can be clearly seen in the center of the map and the size of the diamonds agrees with the oscillation period $V_0$.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{Figs/sup/heatingmapcut.pdf}
\caption{(a) Zero-bias conductance ($\nu=1$) as a function of $V_{BG}$ and temperature, from which the visibility curve in Fig. 3(d) is extracted. (b) A cut of (a) along the dashed line showing the peak conductance as a function of temperature. The red line is a guide for the eye. (c) Zero-bias differential conductance averaged over back-gate voltage as a function of temperature. (d) Differential conductance averaged over back-gate voltage as a function of bias at $\nu=$ 1 and various QPC configurations. Here, $V_{BG}=$ 2.04 V, slightly closer to the Dirac point than the data shown in Fig.~3.}
\label{tem}
\end{figure}
\subsection{Interference pattern at higher energies and level spacing}
Fig.~\ref{tem}(a) shows the zero-bias conductance map vs. gate voltage and temperature, measured in the same range as Fig.~3 of the main text.
In Fig~\ref{tem}(b), we plot the peak conductance as a function of temperature following the dashed line in panel (a). Clearly, the conductance maximum, $G_{max}$, varies non-monotonically with increasing temperature. For $T<0.2$ K, $G_{max}$ decays with increasing temperature, as expected for transport through a single level.
$G_{max}$ flattens and starts to increase above $0.2$ K, suggesting that multiple energy levels are contributing to the transport in this regime. Similarly, the contribution of the excited states is visible in the conductance averaged over gate voltage, $\langle G \rangle_{BG}$, which grows with temperature, Fig.~\ref{tem}(c).
Finally, Fig.~\ref{tem}(d) shows $\langle G \rangle_{BG}$ measured vs. bias for several QPC configurations. In a Fabry-P\'erot interferometer with negligible charging energy, $\langle G \rangle_{BG}$ should not depend on bias (neglecting dephasing), because the same levels are contributing to transport. Both here, and in Fig.~3(a), $\langle G \rangle_{BG}$ universally demonstrates a pronounced dip around zero bias. This dip is consistent with the observed $\langle G \rangle_{BG}$ increase with temperature (Fig.~\ref{tem}(c)), thereby providing an additional indication that multiple excited levels are involved.
We conclude that in our sample, the charging energy should dominate over the level spacing. This conclusion is consistent with the near lack of $B$ dependence in Fig.~3b.
\subsection{Visibility decay at finite bias}
In Fig.~3(a), we observe a rapid decay of the visibility of the interference pattern with increasing bias and the visibility becomes negligible above $\sim 100 \mu$V, close to the oscillation period $V_0$. Similar rapid decay of visibility with bias has been widely observed in both Fabry-P\'erot and Mach-Zehnder interferometers, regardless of the hosting material. Unlike Mach-Zehnder interferometers, in Fabry-P\'erot interferometers, it is hard to distinguish the effects of phase averaging and true decoherence. While the effects of decoherence can be modeled by an exponential or Gaussian decay~\cite{deprez_tunable_2020}, in the following we show that the observed suppression of visibility can also be phenomenologically attributed to an effective electron heating, resulting in phase averaging.
When weakly coupled to the source and drain leads, the non-equilibrium distribution function of the edge state in the interferometer, $f_{R}$, is a weighted average of the distribution function of source and drain, $f_{S}$ and $f_{D}$~\cite{Davies1993_current}.
\begin{equation}
f_R=\frac{\Gamma_S f_S + \Gamma_D f_D}{\Gamma_S+\Gamma_D}=\gamma f_S + (1-\gamma) f_D,
\end{equation}
where $\gamma=\Gamma_S/(\Gamma_S+\Gamma_D)$ and $\Gamma_S$ ($\Gamma_D$) is the tunneling rate to the source (drain) proportional to the QPC transmissions. Strictly, this equation is only valid in the strong backscattering regime. But since it's a continuous change from strong backscattering to weak backscattering, we use it as a crude approximation for the intermediate scattering regime studied here.
For the $\nu=1$ case presented in Fig.~3, we use $\gamma=$ 1/2 since the estimated QPC transmissions are very close to each other ($\mathcal{T}_{QPC1}\sim0.7$ and $\mathcal{T}_{QPC2}\sim0.6$).
Without energy loss to the environment, $e$-$e$ scattering tends to relax this non-equilibrium distribution into thermal equilibrium at elevated temperatures due to the excess energy in this double step distribution function~\cite{PhysRevLett.105.056803}. The effective temperature at full equilibrium is found to be~\cite{Altimiras2009}
\begin{equation}
T_{max} (V)=\sqrt{T_{SD}^2+3\gamma(1-\gamma)\left(\frac{eV}{\pi k_B}\right)^2},
\label{tmax}
\end{equation}
where $T_{SD}$ is the electron temperature of source and drain, $V$ is the applied voltage bias and $k_B$ is the Boltzmann constant. The second term under the square root is an effective heating from the relaxation of excess energy provided by the chemical potential difference.
Since the edge channel likely does not achieve full equilibrium, we multiply this term by a parameter $\alpha<1$, which describes the amount of excess energy that has relaxed. Then, Eq.~(\ref{tmax}) becomes
\begin{equation}
T_{eff} (V,\alpha)=\sqrt{T_{SD}^2+3\gamma(1-\gamma)\left(\frac{eV}{\pi k_B}\right)^2\alpha}.
\label{teff}
\end{equation}
One can relate the relaxation parameter $\alpha$ to the electron-electron scattering rate via Boltzmann equation~\cite{Dubi2019}, which may depend on bias voltage.
As shown in the main text, the visibility, $v=(G_{max}-G_{min})/(G_{max}+G_{min})$, of AB oscillations in single-particle regime is proportional to $T/\sinh(T/T_0)$ to leading order. Therefore, at finite bias we have
\begin{equation}
v(V)\propto \frac{T_{eff}(V,\alpha)}{\sinh(T_{eff}(V,\alpha)/T_{0})},
\label{vV}
\end{equation}
suggesting a fast decay of visibility with increasing bias.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{Figs/sup/visTeff.pdf}
\caption{(a) The visibility as a function of bias. The symmetrized visibility is then further converted into an effective temperature $T_{eff}$. (b) $T_{eff}^2-T_0^2$ (black dots) plotted versus $V^2$. The red dashed line plots the expectation value $\frac{3}{4}(\frac{eV}{\pi k_B})^2$ from full thermalization of a double-step distribution function with equal contribution from source and drain and the blue line plots 0.4 times this value.}
\label{visTeff}
\end{figure}
To find the dependence of $\alpha$ on $V$, we use the relation between visibility and temperature obtained in Fig.~\ref{fig3}(d) to convert the visibility at finite bias (plotted in Fig.~\ref{visTeff} (a)) into an effective electronic temperature $T_{eff}$. Here, we symmetrize the visibility measured at positive and negative bias since the visibility decay is mostly independent from the small bias asymmetry observed in Fig.~\ref{visTeff}(a).
In Fig.~\ref{visTeff}(b), we plot $T_{eff}^2-T_{SD}^2$ inferred from the visibility as a function of $V^2$ (black dots), together with $T_{max}^2-T_{SD}^2$ from Eq.~(\ref{tmax}) (red dashed line). First, we see that $T_{eff}$ does not exceed $T_{max}$, confirming that attributing the visibility drop to effective electron equilibration is not unreasonable. Second, $T_{eff}^2-T_{SD}^2$ is close to about 0.4 times $T_{max}^2-T_{SD}^2$ (blue line), indicating that about 40$\%$ of the excess energy provided by the chemical potential difference is converted into thermal energy. Surprisingly, this fraction does not strongly depends on bias.
Therefore, we find a bias independent $\alpha=0.4$ that can well explain the finite bias visibility decay and from now on we treat it as a known constant for simulating the conductance map shown in Fig.~3(a).
Historically, the bias dependence of visibility of interference patterns has been fitted to either exponential decay~\cite{mcclure_edge-state_2009} for Fabry-P\'erot interferomters or Gaussian decay~\cite{Roulleau_finitebias_2007} for Mach-Zehnder interferometers. A comparison of both fits in Fabry-P\'erot interferomters has been presented in Ref.~\cite{deprez_tunable_2020}, where both fitting expressions can capture the fast decay but not the exact trend. Here, Eq.~(\ref{teff}\&\ref{vV}) produces a Gaussian decay at low bias ($eV<k_BT_{SD}$) and an exponential decay at high bias in agreement with the refined Gaussian phase randomization model presented in Ref.~\cite{Roulleau2008}.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{Figs/sup/9maps.pdf}
\caption{Calculated conductance maps as a function of bias and back-gate voltage for various $x$ and $\alpha$. From top to bottom, $x=$ 0, 0.1 and 0.5. From left to right, $\alpha=$ 0, 0.4 and 1. The unit of the colorbar is $e^2/h$.}
\label{9map}
\end{figure}
Next, instead of approximating the interference pattern as a sinusoidal function, we also want to take into account higher order terms and the bias symmetrization effect observed widely in prior works.
Neglecting the renormalization of QPC transmissions which can give a power law modulation~\cite{NgoDinh2012}, the total current through the interferometer at zero temperature is
\begin{equation}
I=\frac{-e}{h}\int_{-eV_D}^{-eV_S}\frac{\mathcal{T}_{QPC1}\mathcal{T}_{QPC2}}{1+\mathcal{R}_{QPC1}\mathcal{R}_{QPC2}-2\sqrt{\mathcal{R}_{QPC1}\mathcal{R}_{QPC2}}\cos(\delta\phi+\frac{2\pi E}{eV_0})}dE,
\end{equation}
where $\mathcal{R}_{QPC1}=1-\mathcal{T}_{QPC1}$ and $\mathcal{R}_{QPC2}=1-\mathcal{T}_{QPC2}$. The distribution of the voltage drop is determined by the capacitive coupling between the interferometer and the source, drain, and gate electrodes. To take this effect into account, we follow Ref.~\cite{deprez_tunable_2020} to set the source and drain bias drops as $V_{S}=(1-x)V$ and $V_{D}=-xV$.
Therefore, the zero-temperature differential conductance $G_0=dI/dV$ at finite bias $V$ is
\begin{equation}
G_0(V,x)=(1-x)g_0(-eV_S)+xg_0(-eV_D),
\end{equation}
where
\begin{equation}
g_0(E)=\frac{e^2}{h}\frac{\mathcal{T}_{QPC1}\mathcal{T}_{QPC2}}{1+\mathcal{R}_{QPC1}\mathcal{R}_{QPC2}-2\sqrt{\mathcal{R}_{QPC1}\mathcal{R}_{QPC2}}\cos(\delta\phi+\frac{2\pi E}{eV_0})}
\end{equation}
is the differential conductance contribution from electrons at energy $E$.
The differential conductance at $T_{eff} (V,f)$ is then
\begin{equation}
G(V,\alpha,x)=(1-x)g(-eV_S,V,\alpha)+xg(-eV_D,V,\alpha),
\end{equation}
where
\begin{equation}
g(E,V,\alpha)=\int_{-\infty}^{\infty} g_0 (E+\epsilon) \frac{\partial}{\partial \epsilon}\left(\frac{1}{e^{\epsilon/k_BT_{eff} (V,\alpha)}+1}\right) d\epsilon
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Figs/sup/simmap.pdf}
\caption{Calculated conductance maps as a function of bias and back-gate voltage for to simulate the experimental result in Fig.~3(c).}
\label{simmap}
\end{figure}
To demonstrate the bias symmetrization effect together with the electron heating effect, we plot the calculated conductance maps for $x=$ 0, 0.1, 0.5 and $\alpha=$ 0, 0.4, 1 in Fig.~\ref{9map}. As the voltage drop on drain increases, the pattern gradually evolves into a checkerboard. Meanwhile, increasing electron thermalization results in faster visibility decay at high bias. To simulate the background subtracted map shown in Fig.~3(c), in Fig.~\ref{simmap}, we use $\mathcal{T}_{QPC1}=\mathcal{T}_{QPC2}=0.6$, $T_0=$ 0.1 K, $\alpha=$ 0.4 and $x=$ 0. The map clearly reproduces the rapid decay of visibility above $\sim 100 \mu$V seen in Figs.~\ref{fig3}(a,c). Since there is no voltage drop on drain ($x=$ 0), the obtained interference pattern is stripe in contrast to the widely observed checkerboard pattern in GaAs. The checkerboard pattern corresponds to a symmetric voltage drop on source and drain due to the electrochemical potential of the cavity being tied to the source-drain voltages~\cite{NgoDinh2012}. Ref.~\cite{deprez_tunable_2020} also observed a small voltage drop at the drain in a graphene interferometer of similar size and the effect of symmetrization increases with the interferometer size.
|
2,869,038,155,675 | arxiv | \section{\label{}}
\section{Introduction}
The search for $\gamma$-rays from radio galaxies is important for
the understanding of the dynamics and structure of jets in active galactic nuclei (AGN).
Even though radio galaxies are AGN with jets, their jet is not oriented toward the observer
and therefore the radiation produced by the jet is not Doppler-boosted towards
higher energies and luminosities, making them more challenging to detect in the
very high energy (VHE: $E>100$~GeV) regime.
The discovery of VHE $\gamma$-rays from the radio galaxy M~87 by the HEGRA
collaboration~\citep{Aharonian2003}, detected later by VERITAS~\citep{Acciari2008},
and from NGC~5128 (Centaurus~A) by the HESS collaboration~\citep{Aharonian2009} has shown that
non-blazar AGN can produce very energetic photons from non-thermal processes.
Radio galaxies are classified into two main
families based on the morphology of their radio emission~\citep{FanaroffRiley},
whether it is core dominated (FR~I) or lobe dominated (FR~II),
with differences in the radio energetics and
in the discrete spectral properties~\citep{Zirbel1995}. The large number of features that
FR~I radio galaxies share with BL Lac type blazars suggests a possible unification between
the two sub-classes of AGN, in which FR I radio galaxies are BL Lac objects observed at larger jet viewing
angles~\citep{UrryPadovani}.
Evidence for synchrotron emission in radio to X-ray energies from both the extended structures and
the core is well explained by relativistic particles moving in a beamed
relativistic jet~\citep{Ghisellini1993}.
A commonly considered mechanism for HE-VHE (HE: high energy, 100~MeV$<E<$100~GeV) radiation is the synchrotron-self-Compton (SSC)
process~\citep{Jones1974}, where the optical and UV synchrotron photons are up-scattered by the same
relativistic electrons in the jet. Predictions concerning the inverse Compton (IC) component
have long been established for the $\gamma$-ray
emission~\citep{BloomMarscher1996} and frequency-dependent
variability~\citep{Ghisellini1989}. Besides leptonic scenarios, several models also consider a hadronic origin for
non-thermal emission in jets. Accelerated protons can initiate electromagnetic cascades or
photomeson processes~\citep{Mannheim1993}, or directly emit synchrotron radiation \citep{Aharonian2002, Reimer2004}
and produce $\gamma$-rays through collisions with ambient gas \cite{Beall1999, Pohl2000}.
Modelling the blazar jet emission with a homogeneous SSC mechanism may imply particularly
high Lorentz factors, $\Gamma \gtrsim 50$, with consequent high Doppler factors and small beaming angles $\theta \simeq 1^\circ$
\citep{Kraw2002}. Such a small beaming angle is in conflict with the unification scheme according to which FR~I radio galaxies
and BL~Lac objects are the same kind of object observed at different viewing angles. Moreover,
these high values for the Doppler factor are in disagreement with the small apparent velocities observed
in the sub-parsec region of the TeV BL Lac objects Mrk~421 and Mrk~501 \citep{Marscher1999}.
These considerations suggest a more complicated geometry, for example
a decelerating flow in the jet with a consequent gradient in the Lorentz
factor of the accelerated particles and a smaller average $\Gamma$ \citep{Georganopoulos2003}.
As a result of this gradient, the fast upstream particles interact with the downstream
seed photons with an amplified energy density, because of the Doppler boost due to the relative Lorentz factor
$\Gamma_\mathrm{rel}$. The IC process then requires less extreme values for the
Lorentz factor and allows larger values for the beaming angle.
In a similar way, a jet spine-sheath structure consisting of a faster internal spine
surrounded by a slower layer has been also suggested for the broadband non-thermal emission of VHE BL Lac
objects~\citep{Ghisellini2005}. An inhomogeneous jet with a slow component may explain the HE-VHE emission observed in radio galaxies at larger angles
($\theta_\mathrm{layer} = 1/\Gamma_\mathrm{layer} \sim 20^\circ$).
Observation of the VHE component from radio galaxies
is therefore significant for the AGN jet modeling. In this work an overview of the observations
of radio galaxies by VERITAS is presented.
\section{The VERITAS Instrument}
The VERITAS detector is an array of four 12-m diameter imaging
atmospheric Cherenkov telescopes located in southern Arizona~\cite{Weekes}.
Designed to detect emission from astrophysical
objects in the energy range from 100~GeV to greater than 30~TeV,
VERITAS has an energy resolution of $\sim$15\% and an angular
resolution (68\% containment) of $\sim$0.1$^\circ$ per event at 1~TeV.
A source with a flux of 1\% of the Crab Nebula flux is detected in $\sim$25~hours
of observations, while a \mbox{5\% Crab Nebula} flux source is detected in less than
2~hours. The field of view of the VERITAS telescopes is
3.5$^\circ$. For more details on the VERITAS instrument and the imaging atmospheric-Cherenkov
technique, see~\cite{Perkins2009}.
\section{Observations}
Most of the VERITAS observations of radio galaxies are on the radio galaxy M~87.
This AGN is located in the center of the Virgo cluster at a distance of $\sim$16~Mpc
and is currently the brightest detected VHE radio galaxy.
M~87 was originally detected with marginal significance by HEGRA at TeV
energies~\cite{Aharonian2003},
and later also by HESS~\cite{Aharonian2006}, VERITAS~\cite{Acciari2008} and MAGIC~\cite{Albert2008}.
This giant radio galaxy has always been of particular interest because
its jet lies at $\sim$20$^\circ$ respect to the line of sight and
its core and the structure of the jet
are spatially resolved in X-ray, optical and radio observations,
thus it is an ideal candidate for correlated MWL studies~\cite{Wilson2002}.
In 2008 VERITAS coordinated an observational campaign with two other major VHE observatories
(MAGIC, HESS), overlapping with VLBA radio observations~\cite{M87Science}.
Three Chandra X-ray pointed observations have also been performed during the first half of 2008.
Multiple flares at VHE have been detected. In X-rays, the inner-most knot in the jet (HST-1) was found in low
state, while the core region was in high state since 2000. Progressive brightening of the core region
in radio was also seen along the VHE flare development. This is an indication that
the $\gamma$-ray emission originates from a region close to the core rather than from more distant regions.
In April 2010, during the seasonal monitoring of M~87, VERITAS detected another flare with peak flux
of $\sim$20\% of the Crab Nebula flux. During the six-month observation period, M~87 was detected
at a level of 25.6$\sigma$ above the background, with an average flux above 350~GeV equivalent
to 5\% of the Crab Nebula flux. Dedicated analysis in 20-minute bins has been performed on the
April 2010 flaring episode. A spectral analysis has been done on three different phases of the
flaring episode: the rising phase, the peak and the falling phase. A power-law fit has been applied to each phase,
showing a hint of spectral variability: $\Gamma_\mathrm{rise}=2.60\pm0.31$,
$\Gamma_\mathrm{peak}=2.19\pm0.07$, $\Gamma_\mathrm{fall}=2.62\pm0.18$. Figure~\ref{fig_M87-2}
shows the 2010 seasonal lightcurve and the spectral analysis at different times for the flaring episode
that occurred in April 2010. Details on the analysis and results of the 2010 observational campaign on M~87
are presented in a publication currently in the process of peer-review. Results of the extensive multi-year
MWL observational campaign on M~87 will be presented soon too.
\begin{figure}[!t]
\vspace{5mm}
\centering
\includegraphics[width=2.8in]{icrc0781_fig02.epsi}
\includegraphics[width=2.8in]{icrc0781_fig03.epsi}
\caption{\emph{(upper plot)} VERITAS light curve of the 2010 seasonal monitoring campaign.
\emph{(lower plot)} Spectral analysis for three phases of the April 2010 flaring event: rising phase ( circles),
peak (squares) and decreasing phase (triangles).}
\label{fig_M87-2}
\end{figure}
\section{Conclusions}
The radio galaxy M~87 is a unique laboratory for studying the acceleration and emission processes
around the supermassive black hole of AGNs. Its relatively high brightness in VHE $\gamma$-rays
enables to perform cross-correlated MWL observational campaigns and to study variability and spectral evolution
features. VERITAS VHE observations have been crucial during past MWL observational campaigns in
identifying a close-region to the core as responsible for the $\gamma$-ray emission. During the 2009-2010 observational
season, VERITAS detected the strongest flare ever observed in $\gamma$-rays on M~87.
This observation enabled for the first time the study of flux and spectral temporal properties on a radio galaxy.
Further details on the long-term MWL observational campaign on M~87 are in the process of publication.
\bigskip
\begin{acknowledgments}
This research is supported by grants from the US Department of Energy, the US National Science Foundation,
and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland, and by STFC in the UK.
We acknowledge the excellent work of the technical support staff at the FLWO and at the collaborating
institutions in the construction and operation of the instrument.
\end{acknowledgments}
\bigskip
|
2,869,038,155,676 | arxiv | \section{Introduction}
\subsection{Motivation}
\subsubsection{Background}
Since Marstrand's seminal paper \cite{Marstrand1954}, several authors have sought to improve and generalize the so-called
Marstrand projection Theorem.
Let us recall the basic Euclidean setup in any dimension, as in \cite{Mattila1995} (Marstrand's paper dealt only with the plane).
Let $n \geq 2$ and $1 \leq k \leq n-1$. Fix a Borel subset $A \subset \mathbf{R}^n$ of Hausdorff dimension $s$.
Pick a vector subspace of dimension $k$ at random (with respect to the Lebesgue measure on the space of
$k$--dimensional vector subspaces of $\mathbf{R}^n$). Project $A$ along the vector subspace, \emph{e.g.}
pushing it down the quotient mapping. The projected set (sitting in a space isometric to $\mathbf{R}^{n-k}$) has,
almost surely, Hausdorff dimension $\inf\{s, n-k\}$.
A Marstrand-type result is a Theorem of this kind: an almost sure equality for the dimension of a given Borel set projected
through a random projection.
It could also be said that a Marstrand-type result deals with the almost sure dimension of a Borel set, transverse to
a random foliation (according to a fixed measure on some space of foliations). This is our point of view in this paper. For
a definition, see \ref{ss.transverse-dimension}.
There are (at least) two natural ways to generalize this result:
\begin{itemize}
\item To look at a restricted family of foliations, \emph{e.g.} to consider, in $\mathbf{R}^3$, vector lines spanned not
by any vector but by a one-parameter family of vectors, as in \cite{Orponen2017}.
\item To look at foliations defined in the same fashion in non-Euclidean spaces, \emph{e.g.} in Heisenberg group.
See, for example, \cite{Balogh2012}.
\end{itemize}
\begin{table}
\caption{Notations}
\vspace{1em}
\begin{tabular}{|c|c|c|}\hline
Symbol & Page & Definition \\ \hline
$\mathbf{G}(E)$ & \pageref{def.GB} & Grassmann algebra of $E$ \\
$\mathbf{G}^k (E)$ & \pageref{def.GBk} & $k$-vectors of $E$\\
$\vee$ & \pageref{def.vee} & Progressive (exterior) product \\
$\mathbf{D}^k(E)$ & \pageref{def.DBk} & Decomposable $k$-vectors of $E$ \\
$\mathrm{Span}(u_1 \vee \cdots \vee u_k)$ & \pageref{def.span} & $\mathbf{K} u_1 \oplus \cdots \oplus \mathbf{K} u_k$\\
$\wedge$ & \pageref{def.wedge} & Regressive product \\
$\mathbf{G}^k (\Phi)$ & \pageref{def.GBkPhi} & Canonical extension of $\Phi$ to $\mathbf{G}^k(E)$ \\
$\mathbf{P}_\mathbf{K}^n$ & \pageref{def.projective-space} & Projective space of dimension $n$ over $\mathbf{K}$ \\
$\mathbf{P}(E)$ & \pageref{def.PB} & Projective space associated to $E$\\
$\mathbf{P} \mathbf{D}^k(\mathbf{K}^{n+1})$ & \pageref{def.PBDBK} & Image of $\mathbf{D}^k (\mathbf{K}^{n+1})$ in the projective space $\mathbf{P} \mathbf{G}^k (\mathbf{K}^{n+1})$ \\
$d$ & \pageref{eq.angular-metric} & Angular metric on a projective space \\
$\perp$ & \pageref{def.perp} & Orthogonality w.r.t. inner product \\
$\mathrm{Proj}_U$ & \pageref{def.projU} & Generalized radial projection at $U$\\
$\tau$ & \pageref{def.tau} & See formula \eqref{def.tau} \\
$\phi_U$ & \pageref{def.phiU} & Lipschitz modulus; see formula \eqref{def.phiU} \\
$\mathrm{Leb}$ & \pageref{def.Leb} & Lebesgue measure \\
$I_\sigma (\mu)$ & \pageref{def.energy} & $\sigma$-energy of $\mu$ \\
$\mathbf{S},\mathbf{B}$ & \pageref{def.S-B} & Sphere and open ball in $\mathbf{P}_\mathbf{K}^n$ \\
$\mathcal L_\mathbf{K}^k$ & \pageref{def.Lk} & Space of small $(k-1)$--spheres if $\mathbf{K}=\mathbf{R}$, resp. $k$--chains if $\mathbf{K}=\mathbf{C}$ \\ \hline
\end{tabular}
\end{table}
\subsubsection{Description the of results}
In this paper, we look at a quite obvious generalization: namely,
we generalize Marstrand projection Theorem to the \emph{complex} setting. The word \emph{complex sphere}
in the title of this paper refers to the Euclidean spheres of odd dimension; these spheres sit naturally in
complex projective spaces, and the complex Marstrand projection Theorem we will obtain can be restricted to complex spheres
to yield interesting projection Theorems with respect to some special families of so-called ``small spheres''.
The most notable feature of our approach is that everything happens in the projective space; this allows us to do
coordinate-free computations, using extensively the Grassmann algebra. This adds some conceptual and notational difficulty.
The first half of our results (dealing with linear foliations of projective spaces) could be obtained using coordinates
and standard computations as in \cite{Mattila1995} (where the real setting is handled).
The other half of our results would be very awkward to formulate without the language of Grassmann algebra, especially in the complex
setting, which is of interest to us.
Informally, a reason for this is the fact that there is no good analogue, for Heisenberg groups,
of projective spaces associated to vector spaces. The projective space $\mathbf{P}_\mathbf{R}^n$ can be defined at the space of
``infinite circles of $\mathbf{R}^{n+1}$ passing through the origin''. This definition would also make sense in Heisenberg group,
replacing ``infinite circles'' with ``infinite chains'' (see \cite{Goldman1999} for the definition
of finite and infinite chains in Heisenberg group, or \ref{ss.general-setup} for the definition
of chains we will be using); the point is that there is only one infinite chain passing
through the origin. This is why, in this paper, we have to consider ``finite chains''; this also explains why it is
much more efficient to work without coordinates.
Throughout this paper, we will deal with the real and complex settings at the same time. In the real setting,
none of the results we obtain is new: they are all essentially equivalent to the basic Theorem
of Marstrand--Kaufman--Mattila. We state them nonetheless because they serve to provide some geometric
intuition to the reader, and to convince them that we are indeed generalizing the
classical, real Euclidean, Marstrand Theorem -- this may not be obvious at first.
\subsubsection{Plan of the paper}
In \ref{ss.transverse-dimension} we give a precise meaning to the notion of transverse dimension with respect to a foliation.
In \ref{sss.grassmann-algebra} to \ref{sss.grassmann-extensions} we introduce the needed algebraic device: the Hermitian Grassmann (bi)algebra
associated to a Hermitian space. In \ref{ss.basic-properties} we state and prove useful properties of the
inner product in the Hermitian Grassmann algebra. The distance formula in \ref{ss.first-distance-formula}
is a first hint of the geometric significance of the Hermitian structure on the Grassmann algebra.
Section \ref{s.linear-foliations} is the part of the paper that deals with linear foliations of projective
spaces and this is where we apply the algebraic tools described previously. In \ref{ss.radial-foliations}
we define the generalized radial projections we will use to parametrize our linear foliations. We then endow, in \ref{ss.codomain-metric}, the codomain
of these radial projections with a canonical metric. The analysis of transversality of generalized radial projections is made easy by the
product formula contained in \ref{ss.product-formula}. The result of \ref{ss.generalized-distance} is not needed in this paper; it is stated
because it answers a question that arises naturally in this context. In \ref{ss.first-transversality} we prove transversality of the basic
linear foliations of projective spaces, and this is applied to obtain a coordinate-free version of Marstrand's projection Theorem.
We improve on this in \ref{ss.transversality-pointed} by looking at a lower-dimensional family of foliations; in the real setting the
result we then obtain is equivalent to the classical Theorem of Marstrand--Kaufman--Mattila.
In Section \ref{s.spheres-foliations} we look at the results of the previous Sections in restriction to spheres.
The special case of $1$--chains has to
be dealt with separately in \ref{ss.foliations-one-chains}. In \ref{ss.concrete-look} we describe our foliations in coordinates
in order to help the reader get an idea of the geometry behind the algebra; a description of the geometry of
chains is out of the scope of this paper and we refer to \cite{Goldman1999} for details, and suggestive computer-generated pictures.
\subsection{Transverse dimension}\label{ss.transverse-dimension}
We give a precise meaning to the notion of \emph{dimension of a Borel set transverse to
a given foliation}.
Let $X$ be a locally compact metric space. A \emph{foliation} on $X$ is a partition
$\xi$, all atoms of which are closed subsets of $X$. We denote by $\xi(x)$ the atom
of $\xi$ a given point $x \in X$ belongs to. The quotient space $X/\xi$ has elements
the atoms of $\xi$ and is endowed with the final topology for the projection
mapping $X \to X/\xi$. In general the metric of $X$ does not pass to the quotient,
because two distinct atoms of $\xi$ may be at zero distance from one another.
On the other hand, for any compact subset $K$ of $X$, the trace $\xi|K$ of $\xi$
on $K$ has compact atoms; the metric of $X$, restricted to $K$, passes
to the quotient, and the projection mapping $K \to K/\xi$ is, by definition,
Lipschitz.
\begin{definition} \label{def.transverse-dimension}
In this situation, the \emph{transverse dimension of $K$ with respect
to $\xi$} is the Hausdorff dimension of the quotient metric space $K/\xi$.
If $A$ is a Borel subset of $X$, the \emph{transverse dimension of $A$
with respect to $\xi$} is the supremum of the transverse dimensions
of all compact subsets $K \subset A$, with respect to $\xi$.
\end{definition}
The transverse dimension of $A$ with respect to $\xi$ is at most equal to the Hausdorff dimension of $A$, $\dim A$;
indeed, $\dim A = \sup_K \dim K$ where $K$ goes through the family of compact subsets of $A$; and for any compact $K$,
the quotient mapping $K \to K/\xi$ is Lipschitz and cannot increase Hausdorff dimension.
This definition of transverse dimension highlights the fact that in general,
we do not have to be too concerned with the choice of the Lipschitz
mapping we use to parametrize the foliation. In the most classical situa-
tion, $X$ is the Euclidean plane and $\xi$ is the foliation of $X$ by affine lines
of some given angle $\theta$. The orthogonal projection onto the vector line
of angle $\theta + \pi/2$ is a suitable Lipschitz mapping, and so is the quotient
mapping with respect to the vector line of angle $\theta$.
In more complex situations, the adequate projection may have a less
elementary description, and it may also not be Lipschitz on the whole
space $X$, but only on every compact subspace of $X$. Our emphasis in this
paper will be on the geometry of foliations. We will introduce suitable
Lipschitz projections to work with but in our perspective the foliations
come first.
Let us provide a simple example to explain why it is useful to think in terms
of foliations rather than projections. Let $X$ be the Euclidean plane minus
the origin, and let $\xi$ be the foliation of $X$ by vector lines with the origin removed.
Now let $X'$ be the Euclidean plane minus two points $x$ and $y$, and
let $\xi'$ be the foliation of $X'$ by circles passing through $x$ and $y$ (with $x$
and $y$ removed from the circle). From the point of view of transverse dimension,
there is no difference between $(X, \xi)$ and $(X', \xi')$.
We may identify $X'$ with $X$ (we have to remove one more point from $X$)
via a M\"{o}bius transformation $f$; now $f$ maps $\xi'$ onto $\xi$ and it is locally
biLipschitz, so any local dimension property is preserved by $f$.
The fact that $\xi$ can be parametrized by the radial mapping at $0$ is
irrelevant and there is no need to find a corresponding projection mapping
for $\xi$.
Of course the transverse dimension of a given subset $A$ with
respect to $\xi$ depends of how $A$ is sitting with respect to $\xi$;
in general this is a very difficult problem. A \emph{Marstrand-type} Theorem deals not with a
fixed foliation but rather with almost every foliation in a given foliations space
endowed with some version of the Lebesgue measure.
We will usually make an abuse of language and speak of a foliation $\mathcal F$ of a space $X$
when in fact something has to be removed from both $X$ and $\mathcal F$ in order to get a genuine partition.
For example we may say ``look at the foliation of the plane by lines passing through the origin''. What we mean
in this case is ``look at the foliation of the plane minus the origin by lines passing through the origin with
the origin removed''. Likewise, if some line $L$ is fixed in $\mathbf{R}^3$, the family of all affine planes containing $L$,
with $L$ removed, is a foliation of $\mathbf{R}^3 \setminus L$, but we will actually write ``Let $\mathcal F$ be the
foliation of $\mathbf{R}^3$ by affine planes containing $L$''. It would be very unpleasant and quite pedantic to write down in
every case which subset should be removed from the space and the leaves, and we leave it to the reader to make the obvious
corrections.
\subsubsection{Transversality and Kaufman's argument}
Transversality of a family of foliations (or a family of projections) is the crucial property to look for
when one sets out to prove a Marstrand-type result. In the presence of transversality, a quite general argument,
due to Kaufman \cite{Kaufman1975}, allows to prove a version of Marstrand's projection Theorem, as well as some
improvements which we do not discuss in this paper in order to keep things short.
In this paper, we are not going to improve on Kaufman's argument. Our purpose is to introduce ``good foliations''
and transversality will follow quite naturally from the definition (and the product formula, see \ref{ss.product-formula}).
A general exposition of Kaufman's argument in an abstract setting (dealing with parametrized families of projections)
can be found in \cite{Peres2000}. We write down a detailed proof of Corollary \ref{cor.first-transversality} because
there are some issues, requiring us to work ``locally'' (cutting the measure
into small pieces), that do not appear in Kaufman's usual argument.
Our statements deal with dimension of sets; it would be equally sensible to concern ourselves with dimension of measures
and in the proofs this is what we actually do, implicitly using Frostman's Lemma, as in \cite{Mattila1995}.
\subsection{Hermitian forms on the Grassmann algebra}\label{ss.hermitian-forms}
\subsubsection{The Grassmann algebra}\label{sss.grassmann-algebra}
We refer to \cite{Barnabei1985} for a good elementary exposition of the Grassmann exterior (bi)algebra
associated to a vector space.
Another good reference is \cite{Bourbaki1998}, but beware of the conflicting notations.
In this paper we will use the same notations as in \cite{Barnabei1985}.
We now recall some basic definitions and fix appropriate notations.
Let $\mathbf{K}$ be $\mathbf{R}$ or $\mathbf{C}$ and fix a finite-dimensional $\mathbf{K}$-vector space $E$. The
elements of $E$ will usually be denoted by the letters $u, v, w$.
We denote by $\mathbf{G}_\mathbf{K}(E)$\label{def.GB} the Grassmann algebra (over $\mathbf{K}$) associated with
$E$. The Grassmann algebra is also called the exterior algebra of $E$. The
\emph{progressive product} (also called \emph{exterior product}) will be denoted by the
symbol $\vee$\label{def.vee}
The \emph{regressive product} (to be introduced later) will be denoted
by $\wedge$.
\begin{remark}
If $\mathbf{K}$ is $\mathbf{C}$, we may consider the Grassmann algebra over $\mathbf{R}$,
$\mathbf{G}_\mathbf{R}(E)$, as well as the Grassmann algebra over $\mathbf{C}$, $\mathbf{G}_\mathbf{C}(E)$.
In this paper we will always work with the Grassmann algebra over $\mathbf{C}$ (when $\mathbf{K} = \mathbf{C}$).
In a later paper, we will also consider the Grassmann algebra, over $\mathbf{R}$,
of a complex space, and this will allow us to define and study a family
of foliations (called \emph{real spheres} or \emph{Ptolemy circles}) different from the
\emph{chains} which are the main focus of this paper. See \emph{e.g.} \cite{Goldman1999}
for definitions.
Henceforth, we drop the subscript $\mathbf{K}$ in the notation for the Grassmann
algebra.
\end{remark}
The subspace of $k$--vectors will be denoted by $\mathbf{G}^k (E)$\label{def.GBk}. If $n$ is the
dimension of $E$ (over $\mathbf{K}$),
\begin{equation}
\mathbf{G}(E) = \bigoplus_{k=0}^n \mathbf{G}^k(E)
\end{equation}
where $\mathbf{G}^0(E)$ is \emph{canonically isomorphic} to $\mathbf{K}$, $\mathbf{G}^1(E)$ is
\emph{canonically isomorphic} to $E$, $\mathbf{G}^{n-1}(E)$ is \emph{non-canonically isomorphic} to the algebraic dual of
$E$, and $\mathbf{G}^n(E)$ is \emph{non-canonically isomorphic} to $\mathbf{K}$ (the choice of a basis of
$\mathbf{G}^n(E)$ is equivalent to the choice of a non-degenerate alternating $n$--linear form).
The above direct sum is \emph{graded}: for $U \in \mathbf{G}^k(E)$ and $V \in \mathbf{G}^\ell(E)$,
the progressive product $U \vee V$ belongs to $\mathbf{G}^{k+\ell}(E)$ (where by definition
$\mathbf{G}^i(E) = 0$ if $i > n$).
The set of \emph{pure} or \emph{decomposable} $k$--vectors, i.e. $k$--vectors of the form $u_1 \vee \cdots \vee u_k$ ($u_1,\ldots, u_k \in E$)
will be denoted by $\mathbf{D}^k(E)$\label{def.DBk}. We will often use capital letters $U, V, W$ to denote $k$--vectors, and most of the time we
will consider pure $k$--vectors only.
Bear in mind that unless $k = 1$ or $k = \dim(E) - 1$, $\mathbf{D}^k(E)$ is not a
vector space.
If $U = u_1 \vee \cdots \vee u_k$ is a non-zero element of $\mathbf{D}^k(E)$, we denote by
$\mathrm{Span}(U)$\label{def.span} the vector subspace $\mathbf{K} u_1\oplus \cdots \oplus \mathbf{K} u_k$ of $E$. This is the smallest
vector subspace $E'$ of $E$ such that $U$ belongs to $\mathbf{G}^k(E')$.
Thus, if $U$ and $V$ are, respectively, a pure $k$--vector and a pure $\ell$--vector,
such that $U \vee V \neq 0$, then $\mathrm{Span}(U \vee V ) = \mathrm{Span}(U) \oplus \mathrm{Span}(V)$.
The basic fact that $k$--dimensional vector subspaces of $E$ are in one-to-one
correspondance with projective classes of elements of $\mathbf{D}^{k+1}(E)$ will
be used at every moment throughout this paper.
\subsubsection{The regressive product}
We now recall briefly the definition of the \emph{regressive product}\label{def.wedge}. Let $n$ be
the dimension of $E$ (over $\mathbf{K}$). The choice of a non-degenerate alternating
$n$-linear form $\omega$ on $E$ yields a \emph{Hodge isomorphism}
\begin{equation}
* : \mathbf{G}(E) \to \mathbf{G}(E^*)
\end{equation}
that identifies $\mathbf{G}^k(E)$ with $\mathbf{G}^{n-k}(E^*)$ (were $E^*$ is the algebraic dual space
of E in the usual sense).
The pull-back, through this isomorphism, of the progressive product
in $\mathbf{G}(E^*)$ is, by definition, the regressive product in $\mathbf{G}(E)$, denoted by $\wedge$.
The regressive product depends on the choice of $\omega$; to put it differently,
it depends on the choice of a basis of the 1--dimensional space $\mathbf{G}^n(E)$.
By definition,
\begin{equation}
U \wedge V = (U^* \vee V^*)^*
\end{equation}
Let us also recall the definition of the Hodge isomorphism. The bilinear mapping
\begin{equation}
\mathbf{G}^k(E) \times \mathbf{G}^{n-k}(E) \to \mathbf{G}^n(E)
\end{equation}
defined by
\begin{equation}
(u_1 \vee \cdots \vee u_k, u_{k+1} \vee \cdots \vee u_n) \mapsto u_1 \vee \cdots \vee u_n
\end{equation}
(and extended by linearity) is composed with the isomorphism
$\mathbf{G}^n(E) \to \mathbf{K}$ associated to $\omega$,
\begin{equation}
u_1 \vee \cdots \vee u_n \mapsto \omega(u_1, \ldots, u_n)
\end{equation}
and identifies $\mathbf{G}^k(E)$ with the dual of $\mathbf{G}^{n-k}(E)$; this dual is also canonically
isomorphic to $\mathbf{G}^{n-k}(E^*)$. In this way, we obtain for every $k$ an
isomorphism $\mathbf{G}^k(E) \to \mathbf{G}^{n-k}(E^*)$ which is, by definition, the Hodge isomorphism restricted
to $\mathbf{G}^k(E)$.
For details, see \cite{Bourbaki1998} or \cite{Barnabei1985}.
The geometric significance of the regressive product should be clear:
if $U \wedge V \neq 0$, where $U$, $V$ are, respectively, a pure $k$--vector and a pure $\ell$--vector,
and $k+\ell \geq n$, then $U \wedge V$ is a pure $(k+\ell-n)$--vector such that
$\mathrm{Span}(U \wedge V) = \mathrm{Span}(U) \cap \mathrm{Span}(V)$; if $k+\ell <n$, $U \wedge V=0$.
Endowed with $\vee$ and $\wedge$, $\mathbf{G}(E)$ is the Grassmann bialgebra of $E$.
\subsubsection{Grassmann extensions of Hermitian forms}\label{sss.grassmann-extensions}
Let $E$ be, as before, a finite-dimensional $\mathbf{K}$-vector space ($\mathbf{K} = \mathbf{R}$ or
$\mathbf{C}$), now endowed with a sesquilinear form $\Phi$; by definition $\Phi(\alpha u, \beta v) =
\overline{\alpha} \beta \Phi(u, v)$ for $\alpha, \beta \in \mathbf{K}$ and $u, v \in E$.
If $\mathbf{K}$ is $\mathbf{R}$ this is bilinearity in the usual sense.
Denote by $\overline{E}$ the $\mathbf{K}$--vector space with the same underlying additive
group as $E$ and the $\mathbf{K}$--operation law defined by $\alpha \cdot u= \overline{\alpha} u$
(where the right-hand side denotes the operation of $\alpha$ on $u$ in $E$).
If $\mathbf{K}$ is $\mathbf{R}$, $\overline E$ is equal to $E$, whereas if $\mathbf{K} = \mathbf{C}$ the identity mapping is
an anti-isomorphism $\overline E \to E$.
We now recall, as in \cite{Bourbaki2007}, the canonical extension of the sesquilinear form $\Phi$
to the Grassmann algebra $\mathbf{G}(E)$.
Fix $k \geq 1$. The mapping $\overline{E}^k \times E^k \to \mathbf{K}$ defined by
\begin{equation}
(u_1, \ldots, u_k; v_1, \ldots, v_k) \mapsto \mathrm{det} (\Phi(u_i, v_j ))
\end{equation}
(where $\mathrm{det}$ is the usual determinant of a $k \times k$ matrix) is $\mathbf{K}$--multilinear.
Since, also, the right-hand side is zero as soon as $u_i = u_j$ or $v_i = v_j$ for some $i \neq j$,
this mapping yields, by the universal property of the Grassmann algebra
(see \cite{Bourbaki1998}, \S 7, Proposition 7), a $\mathbf{K}$--bilinear form $\mathbf{G}^k(\overline{E}) \times \mathbf{G}^k(E) \to \mathbf{K}$.
Using the canonical antilinear identication of $\mathbf{G}^k(\overline{E})$ with $\overline{\mathbf{G}^k(E)}$, we
obtain a sesquilinear form on $\mathbf{G}^k(E)$ which we denote by $\mathbf{G}^k(\Phi)$\label{def.GBkPhi}.
If $\Phi$ is Hermitian, meaning that $\Phi(v, u) = \overline{\Phi(u, v)}$, so is $\mathbf{G}^k(\Phi)$. Also if $\Phi$
is non-degenerate, $\mathbf{G}^k(\Phi)$ is non-degenerate as well; if $\Phi$ is definite, $\mathbf{G}^k(\Phi)$
is definite, and of the same sign.
The following result is basic.
\begin{lemma} \label{lemma.basic}
If $U_1,U_2 \in \mathbf{D}^k(E)$, and $V \in \mathbf{D}^\ell(E)$, are such that $\mathrm{Span}(U_1)$ and $\mathrm{Span}(U_2)$ are both $\Phi$--orthogonal
to $\mathrm{Span}(V)$,
\begin{equation}
\mathbf{G}^{k+\ell} (\Phi) (U_1 \vee V, U_2 \vee V) = \mathbf{G}^k (U_1,U_2) \times \mathbf{G}^\ell (V,V)
\end{equation}
\end{lemma}
\subsubsection{Basic properties of the Hermitian norm}\label{ss.basic-properties}
Fix $n \geq 1$ and let again $\mathbf{K}$ be $\mathbf{R}$ or $\mathbf{C}$. We denote the canonical basis
of $\mathbf{K}^{n+1}$ by $(e_0,\ldots,e_n)$. For any $u, v \in \mathbf{K}^{n+1}$, we denote
the usual Hermitian inner product by
\begin{equation}\label{def.inner-product}
\inprod{u}{v} = \sum_{i=0}^{n} \overline{x_i} y_i
\end{equation}
(where $u=(x_0,\ldots,x_n)$ and $v=(y_0,\ldots,y_n)$)
and we use the same symbol for the canonical Grassmann extension, i.e.
\begin{equation}\label{def.extended-inner-product}
\inprod{u_1 \vee \cdots \vee u_k}{v_1 \vee \cdots \vee v_k} = \mathrm{det} (\inprod{u_i}{v_j})
\end{equation}
(where the right-hand side is the determinant of the $k \times k$ matrix whose $(i, j)$--component
is $\inprod{u_i}{v_j}$. This Grassmann extension is still a Hermitian
inner product on $\mathbf{G}^k (\mathbf{K}^{n+1})$ and the associated Hermitian norm is denoted, as usual,
by $\| \cdot \|$.
The $n$--dimensional projective space over $\mathbf{K}$ is denoted by $\mathbf{P}_\mathbf{K}^n$\label{def.projective-space} ; this is
the space of $\mathbf{K}$--vector lines in $\mathbf{K}^{n+1}$. If $\mathbf{K} = \mathbf{R}$, respectively $\mathbf{K} = \mathbf{C}$, $\mathbf{P}^n_\mathbf{K}$ is a
Riemannian manifold of dimension $n$, respectively a Hermitian manifold
of complex dimension $n$ (and real dimension $2n$). In general, if $E$ is some $\mathbf{K}$--vector space,
the symbol $\mathbf{P} E$\label{def.PB} denotes the projective space associated to $E$ over $\mathbf{K}$.
We will also use the notation $\mathbf{P} \mathbf{D}^k (\mathbf{K}^{n+1})$\label{def.PBDBK} to denote the space of projective classes of
elements of $\mathbf{D}^k (\mathbf{K}^{n+1})$. (Note that $\mathbf{D}^k (\mathbf{K}^{n+1})$ is not a vector space in general.)
In this paper, the letter $d$ will always denote the \emph{angular metric} on $\mathbf{P}^n_\mathbf{K}$ defined by
\begin{equation}\label{eq.angular-metric}
d(u,v) = \Dist{u}{v}
\end{equation}
where $u, v$ are non-zero elements of $\mathbf{K}^{n+1}$. In the left-hand side we are abusing notations and denoting elements of
$\mathbf{P}^n_\mathbf{K}$ by corresponding elements of $\mathbf{K}^{n+1}$. It seems preferable to slightly abuse notations rather than
use the cumbersome notation $[u],[v]$ when we are dealing with projective classes.
From this definition, the following formula follows at once:
\begin{equation}\label{eq.basic-formula}
d(u,v)^2 = 1 - \frac{|\inprod{u}{v}|^2}{\|u\|^2 \cdot \|v\|^2} = \sin^2 (\theta)
\end{equation}
where $\theta$ is the (non-oriented) angle from $u$ to $v$.
The orthogonal complement with respect to the Hermitian inner product will be denoted by $\perp$\label{def.perp}.
For example, $v^\perp $ is the space of all vectors
$u \in \mathbf{K}^{n+1}$ such that $\inprod{u}{v} = 0$.
\begin{lemma}\label{lemma.basic-inequality}
For any $U \in \mathbf{D}^k(\mathbf{K}^{n+1}), V \in \mathbf{D}^\ell(\mathbf{K}^{n+1})$,
\begin{equation}
\| U \vee V \| \leq \| U \| \cdot \| V \|
\end{equation}
\end{lemma}
and this is an equality if and only if $\mathrm{Span}(U)$ and $\mathrm{Span}(V)$ are orthogonal.
This follows from the following Lemma which we state separately for
future reference.
\begin{lemma}\label{lemma.basic-orthogonality}
Let $V \in \mathbf{D}^\ell(\mathbf{K}^{n+1})$ and denote
\begin{itemize}
\item $\pi_V^\perp$ the orthogonal projection $\mathbf{K}^{n+1} \to \mathrm{Span}(V)^\perp$
\item $\Pi_V^\perp$ the orthogonal projection $\mathbf{G}^k (\mathbf{K}^{n+1}) \to \mathbf{G}^k(\mathrm{Span}(V)^\perp)$.
\end{itemize}
Then $\mathbf{G}^k (\pi_V^\perp)=\Pi_V^\perp$ and for any $U \in \mathbf{D}^k (\mathbf{K}^{n+1})$,
\begin{equation}
\| U \vee V \| = \| \Pi_V^\perp (U) \| \cdot \| V \|
\end{equation}
\end{lemma}
The notation $\mathbf{G}^k(f)$, where $f$ is a linear mapping with domain $E$,
stands for the extension of $f$ to $\mathbf{G}^k(E)$, which is characterized by the
relation $\mathbf{G}^k(f)(u_1 \vee \cdots \vee u_k) = f(u_1) \vee \cdots \vee f(u_k)$ for any $u_1, \ldots, u_k \in E$.
\begin{proof}
Recall the basic property that a linear projection $\pi$ is orthogonal if and only if
for any vector $x$ in the image of $\pi$ and any other vector $y$, $\inprod{x}{\pi(y)}$ is equal to $\inprod{x}{y}$.
To show that $\mathbf{G}^k (\pi_V^\perp)$ is the orthogonal projection onto $\mathbf{G}^k (\mathrm{Span}(V)^\perp)$, it is enough
to check that for any $U \in \mathbf{D}^k (\mathbf{K}^{n+1})$ and any $U' \in \mathbf{D}^k (\mathrm{Span}(V)^\perp)$,
\begin{equation}
\inprod{U}{U'} = \inprod{\mathbf{G}^k (\pi_V^\perp) (U)}{U'}
\end{equation}
Now by definition the right-hand side is equal to $\mathrm{det}(\inprod{\pi_V^\perp(u_i)}{u_j'})$ and $\inprod{\pi_V^\perp(u_i)}{u_j'}=\inprod{u_i}{u_j'}$
by the basic property of orthogonal projections; this determinant is thus equal to $\mathrm{det}(\inprod{u_i}{u_j'}) = \inprod{U}{U'}$.
The formula for norms then follows from the fact that $U \vee V = \mathbf{G}^k (\pi_V^\perp) (U) \vee V$ by definition of the progressive product.
\end{proof}
\subsubsection{First distance formula}\label{ss.first-distance-formula}
\begin{theorem}\label{th.first-distance-formula}
Let $U= u_0 \vee \cdots \vee u_k$ be a non-zero element of $\mathbf{D}^{k+1}(\mathbf{K}^{n+1})$ and let
$w \in \mathbf{K}^{n+1}$ be non-zero. The quantity
\begin{equation}
\tau(U,w)= \Dist{U}{w}
\end{equation}
is equal to the distance between $w$ and the $k$--dimensional projective subspace $\mathbf{P}_\mathbf{K} (\mathrm{Span}(U))$ in $\mathbf{P}_\mathbf{K}^{n+1}$.
\end{theorem}
We commit the usual abuse of language of denoting by $w$ both a non-zero vector and its image in $\mathbf{P}_\mathbf{K}^n$.
\begin{proof}
Let $\pi$ be the orthogonal projection from $\mathbf{K}^{n+1}$ onto $\mathrm{Span}(U)$. Then $w - \pi(w)$ is orthogonal to $\mathrm{Span}(U)$,
so $\tau(U,w)^2$ is equal to
\begin{equation}
\frac{\| w - \pi(w)\|^2}{\|w\|^2} = 1 - \frac{\| \pi(w) \|^2}{\|w\|^2} = 1 - \sup_{v \in \mathrm{Span}(U)} \frac{| \inprod{w}{v} |^2}{\|w\|^2 \cdot \|v\|^2}
\end{equation}
(where we used the convexity of orthogonal projections and ). By using formula \eqref{eq.basic-formula}, we see
that the right-hand side of the last equation is equal to
\begin{equation}
\inf_{v \in \mathrm{Span}(U)} d(w,v)^2 = d(w,\mathbf{P} (\mathrm{Span}(U)))^2
\end{equation}
\end{proof}
\section{Linear foliations in real and complex projective spaces}\label{s.linear-foliations}
In this section, we fix an integer $n \geq 2$ and we work in the $n$--dimensional
$\mathbf{K}$--projective space $\mathbf{P}_\mathbf{K}^n$.
If $U = u_0 \vee \cdots \vee u_k$ is a non-zero element of $\mathbf{D}^{k+1}(\mathbf{K}^{n+1})$ ($k \leq n$), we will
denote by $L_U$ the projective subspace $\mathbf{P}(\mathrm{Span}(U))$ of $\mathbf{P}_\mathbf{K}^n$. The mapping
$[U] \mapsto L_U$ is a bijection from $\mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$ to the space of $k$--dimensional
$\mathbf{K}$--projective subspaces of $\mathbf{P}^n_\mathbf{K}$. We will identify these spaces and say ``let
$[u_0 \vee \cdots \vee u_k]$ be a $k$--dimensional projective subspace of $\mathbf{P}^n_\mathbf{K}$.''
\subsection{Generalized radial foliations}\label{ss.radial-foliations}
Fix an integer $k$, $0 \leq k \leq n-2$. For any $k$--dimensional projective subspace
$L$ of $\mathbf{P}^n_\mathbf{K}$, and any $x \in \mathbf{P}^n_\mathbf{K} \setminus L$, there is one and only one $(k+1)$--dimensional
projective subspace of $\mathbf{P}^n_\mathbf{K}$ containing $L \cup \{x\}$.
To $L$ we may thus associate a foliation of $\mathbf{P}^n_\mathbf{K}$ by $(k+1)$--dimensional
projective subspaces. We exclude the case $k = n - 1$ because the foliation
is then trivial (it has only one leaf).
(Recall that when we say ``a foliation of $\mathbf{P}^n_\mathbf{K}$ by projective subspaces'' here we
actually mean ``a foliation of $\mathbf{P}^n_\mathbf{K} \setminus L_U$ by projective subspaces containing $L_U$,
with $L_U$ removed''.)
Let us describe this foliation algebraically, thanks to the Grassmann
algebra, in order to perform computations.
Fix a $k$--dimensional projective subspace $[U] = [u_0\vee\cdots\vee u_k] \in \mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$.
of $\mathbf{P}^n_\mathbf{K}$ and denote by $\mathrm{Proj}_U$\label{def.projU}
\begin{equation}
\begin{array}{lccc}
\mathrm{Proj}_U: & \mathbf{P}_\mathbf{K}^n \setminus L_U & \to & \mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1}) \\
& [w] & \mapsto & [u_0 \vee \cdots \vee u_k \vee w] \\
\end{array}
\end{equation}
By definition, the fibers of this mapping are exactly the $(k + 1)$--dimensional projective subspaces of $\mathbf{P}_\mathbf{K}^n$ containing $L_U$.
We are now going to endow $\mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$ with a natural metric, in order to be able
to prove the needed transversality properties for our linear foliations.
\subsection{The angular metric on the codomain}\label{ss.codomain-metric}
We endowed earlier $\mathbf{G}^\ell (\mathbf{K}^{n+1})$ with a Hermitian structure for $0 \leq \ell \leq n+1$. To this Hermitian space
we may associate its degree 2 Grassmann algebra, $\mathbf{G}^2(\mathbf{G}^\ell(\mathbf{K}^{n+1}))$ and this is in turn a Hermitian space in a natural
way. We are now looking at the Grassman algebra arising from the vector space underlying a Grassmann algebra;
its elements are of the form $U \vee V$ , where $U$ and $V$ belong
to $\mathbf{G}^\ell(\mathbf{K}^{n+1})$, and the reader should be careful not to believe that, somehow,
if $U = u_1 \vee \cdots \vee u_\ell$ and $V = v_1 \vee \cdots \vee v_\ell$, the element $U \vee V$ of
$\mathbf{G}^2(\mathbf{G}^\ell(\mathbf{K}^{n+1}))$ could be equal to the element $U \vee V$ of $\mathbf{G}^{2 \ell}(\mathbf{K}^{n+1})$. These
elements do not sit in the same space. They are the same thing if and
only if $\ell = 1$.
This construction allows to endow $\mathbf{P} \mathbf{G}^\ell(\mathbf{K}^{n+1})$ (the projective space associated to
$\mathbf{G}^\ell (\mathbf{K}^{n+1}))$ with the metric defined as in \eqref{eq.angular-metric}.
In turn, the restriction of this metric to $\mathbf{P} \mathbf{D}^\ell(\mathbf{K}^{n+1})$ endows the space
of $(\ell - 1)$-projective subspaces of $\mathbf{P}_\mathbf{K}^n$ with a natural metric.
Our aim in the paragraph to follow is to study the distance
between $\mathrm{Proj}_U(w_1)$ and $\mathrm{Proj}_U(w_2)$ for $w_1,w_2 \in \mathbf{P}_\mathbf{K}^n$.
\subsection{The product formula}\label{ss.product-formula}
\begin{theorem}\label{th.product-formula}
Let $p \geq 1$. For any element $V \in \mathbf{G}^{p-1} (\mathbf{K}^{n+1})$ and any $w_1,w_2 \in \mathbf{P}_\mathbf{K}^n \setminus L_V$,
\begin{equation}
\| (V \vee w_1) \vee (V \vee w_2) \| = \| V \| \cdot \| V \vee w_1 \vee w_2 \| \label{eq.product-formula}
\end{equation}
where $(V \vee w_1) \vee (V \vee w_2)$ belongs to $\mathbf{G}^2 (\mathbf{G}^p (\mathbf{K}^{n+1}))$ and $V \vee w_1 \vee w_2$ belongs to $\mathbf{G}^{p+1}(\mathbf{K}^{n+1})$.
\end{theorem}
\begin{proof}
Denote by $\pi$ the orthogonal projection $\mathbf{K}^{n+1} \to \mathrm{Span}(V)^\perp$. By the basic properties of progressive product
we have
\begin{equation}
V \vee w_1 = V \vee \pi(w_1)
\end{equation}
and similarly for $w_2$; we also have, for the same reasons,
\begin{equation}
V \vee w_1 \vee w_2 = V \vee \pi(w_1) \vee \pi(w_2)
\end{equation}
Without loss of generality, we can thus assume that $w_1$ and $w_2$ are orthogonal to $\mathrm{Span}(V)$.
Now, by definition, the square of the left-hand side in equation \eqref{eq.product-formula} is equal to the $2 \times 2$ determinant
\begin{equation}
\left| \begin{array}{cc}
\|V \vee w_1 \|^2 & \inprod{V \vee w_1}{V \vee w_2} \\
\inprod{V \vee w_2}{V \vee w_1} & \| V \vee w_2\|^2\end{array} \right|
\end{equation}
We can apply Lemma \ref{lemma.basic} and the previous determinant is equal to
\begin{equation}
\|V\|^4 \cdot \left| \begin{array}{cc} \|w_1\|^2 & \inprod{w_1}{w_2} \\
\inprod{w_2}{w_1} & \| w_2 \|^2\end{array} \right| = \|V\|^4 \cdot \| w_1 \vee w_2 \|^2
\end{equation}
On the other hand, using again orthogonality of $w_1$ and $w_2$ with respect to $\mathrm{Span}(V)$,
as well as Lemma \ref{lemma.basic} we see that
\begin{equation}
\| V \vee w_1 \vee w_2 \| = \| V \| \cdot \|w_1 \vee w_2 \|
\end{equation}
and the proof is over.
\end{proof}
For any elements $U \in \mathbf{D}^k (\mathbf{K}^{n+1})$, $V \in \mathbf{D}^\ell (\mathbf{K}^{n+1})$, we denote by $\tau(U,V)$ the number
\begin{equation} \label{def.tau}
\tau(U,V) = \Dist{U}{V}
\end{equation}
where $U \vee V \in \mathbf{D}^{k+\ell}(\mathbf{K}^{n+1})$. If $k$ (resp. $\ell$) is equal to $1$, this is the distance from
$U$ (resp. $V$) to $\mathbf{P} (\mathrm{Span}(V))$ (resp. $\mathbf{P} (\mathrm{Span}(U))$) (Theorem \ref{th.first-distance-formula}). Also, note that $\tau(U,V) \leq 1$.
With this notation, the previous Theorem has the following consequence:
\begin{equation}\label{eq.proj-dist-formula}
d (\mathrm{Proj}_U (w_1),\mathrm{Proj}_U(w_2)) = \frac{\tau(U,w_1 \vee w_2)}{\tau(U,w_1) \tau(U,w_2)} d(w_1,w_2)
\end{equation}
which will play a crucial role in our analysis.
\subsection{Generalized distance formula}\label{ss.generalized-distance}
In this paragraph, we elucidate the geometric significance of the number
$\tau(U,V)$ introduced above; this is not needed in the rest of the paper.
We start with some notations and a lemma. Fix $q \geq 2$ and $k,\ell \geq 1$.
such that $k + \ell \leq q$. If $V$ is some non-zero decomposable $\ell$-vector of $\mathbf{K}^q$,
i.e. $V \in \mathbf{D}^\ell (\mathbf{K}^q)$, we let
\begin{equation}
\mathbf{G}_0^k (V;\mathbf{K}^q) = \{ U \in \mathbf{G}^k (\mathbf{K}^q)\ ;\ U \vee V = 0 \}
\end{equation}
This is the \emph{annihilator of $V$ in $\mathbf{G}^k (\mathbf{K}^q)$}, a vector subspace of $\mathbf{G}^k (\mathbf{K}^q)$.
\begin{lemma}
The orthogonal complement of $\mathbf{G}^k (\mathrm{Span}(V)^\perp)$ in $\mathbf{G}^k (\mathbf{K}^q)$ is equal to $\mathbf{G}_0^k (V;\mathbf{K}^q)$.
\end{lemma}
\begin{proof}
Let $\pi$ be the orthogonal projection $\mathbf{K}^q \to \mathrm{Span}(V)^\perp$. Then $\mathbf{G}^k (\pi)$
is the orthogonal projection $\mathbf{G}^k (\mathbf{K}^q) \to \mathbf{G}^k(\mathrm{Span}(V)^\perp)$. Here, we denote by
$\mathbf{G}^k(\pi)$ the extension of $\pi$ to $\mathbf{G}^k (\mathbf{K}^q)$.
Also, it follows from the definition of progressive product that $U \vee V = \mathbf{G}^k (\pi) (U) \vee V$.
This $(k+\ell)$-vector is zero if and only if $\mathbf{G}^k(\pi) (U)=0$, which is equivalent to saying
that $U$ is orthogonal to $\mathbf{G}^k (\mathrm{Span}(V)^\perp)$.
\end{proof}
\begin{theorem} \label{th.generalized-distance}
For $U \in \mathbf{D}^k (\mathbf{K}^q)$ and $V \in \mathbf{D}^\ell (\mathbf{K}^q)$,
\begin{equation}
\Dist{U}{V} = d(U,\mathbf{P} \mathbf{G}_0^k(V;\mathbf{K}^q))
\end{equation}
where $U \vee V$ belongs to $\mathbf{D}^{k+\ell}(\mathbf{K}^q)$.
\end{theorem}
In other words, the number $\tau(U,V)$ is equal to the distance, in $\mathbf{P} \mathbf{G}^k (\mathbf{K}^q)$,
between $U$ and the projective subspace $\mathbf{P} \mathbf{G}_0^k (V;\mathbf{K}^q)$.
\begin{proof}
Let $\pi$ be the orthogonal projection from $\mathbf{K}^q$ onto $\mathrm{Span}(V)^\perp$.
We compute, taking into account the previous Lemma,
\begin{equation}
1 - \left( \Dist{U}{V} \right)^2 = 1 - \frac{\| \mathbf{G}^k (\pi) (U) \|^2}{\|U\|^2} =
\frac{\|U-\mathbf{G}^k (\pi)(U)\|^2}{\|U\|^2}
\end{equation}
and $U - \mathbf{G}^k (\pi)(U)$ is the image of $U$ through the orthogonal projection onto
$\mathbf{G}^k (\mathrm{Span}(V)^\perp)^\perp$. The convexity property of inner product then yields
\begin{equation}
\frac{\|U-\mathbf{G}^k (\pi)(U)\|^2}{\|U\|^2} = \sup_{W} \frac{|\inprod{U}{W}|^2}{\|U\|^2 \cdot \|W\|^2}
\end{equation}
where the supremum is relative to $W \in \mathbf{G}^k(\mathrm{Span}(V)^\perp)^\perp = \mathbf{G}_0^k (V;\mathbf{K}^q)$. The right-hand side is equal to
\begin{equation}
1 - \inf_W \left( \Dist{U}{W} \right)^2 = 1 - \inf_W d(U,W)^2
\end{equation}
(where still $W \in \mathbf{G}^k_0 (V;\mathbf{K}^q)$ and $U \vee W \in \mathbf{G}^2 (\mathbf{G}^k (\mathbf{K}^q))$) and the Lemma is proved.
\end{proof}
\subsection{Lipschitz property; transversality; Marstrand-type Theorem}\label{ss.first-transversality}
Recall the setting from the beginning of the section: $n \geq 2$ is fixed, $0 \leq k \leq n-2$
and $U=u_0 \vee \cdots \vee u_k \in \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$ is a (momentarily fixed)
decomposable $(k+1)$--vector of $\mathbf{K}^{n+1}$.
Introduce the Lipschitz modulus function
\begin{equation} \label{def.phiU}
\phi_U (w_1,w_2) = \frac{d(\mathrm{Proj}_U(w_1),\mathrm{Proj}_U(w_2))}{d(w_1,w_2)}
\end{equation}
for any pair of distinct $w_1,w_2 \in \mathbf{P}_\mathbf{K}^n \setminus L_U$.
Recall the formula \eqref{eq.proj-dist-formula}
\begin{equation} \label{eq.phiU-formula}
\phi_U(w_1,w_2) = \frac{\tau(U,w_1 \vee w_2)}{\tau(U,w_1) \tau(U,w_2)}
\end{equation}
A basic fact is that $\mathrm{Proj}_U$ is ``locally Lipschitz''.
\begin{proposition} \label{prop.lipschitz}
The restriction of $\mathrm{Proj}_U$ to any compact subspace of $\mathbf{P}_\mathbf{K}^n \setminus L_U$
enjoys the Lipschitz property.
\end{proposition}
\begin{proof}
By the above formula \eqref{eq.phiU-formula}, the Proposition follows from the fact
that the function $w \mapsto \tau(w,U)$ is continuous and non-zero
in $\mathbf{P}_\mathbf{K}^n \setminus L_U$.
\end{proof}
The space $\mathbf{P} \mathbf{D}^{k+1} (\mathbf{K}^{n+})$ is a Hermitian manifold and carries a natural Lebesgue
measure $\mathrm{Leb}$\label{def.Leb}. This measure can be defined in an elementary way: endow
$(\mathbf{P}_\mathbf{K}^n)^{k+1} = \mathbf{P}_\mathbf{K}^n \times \cdots \times \mathbf{P}_\mathbf{K}^n$ with the product ($k+1$ times)
of the Lebesgue measure of $\mathbf{P}_\mathbf{K}^n$, and push this product measure forward through the
almost-everywhere defined mapping
\begin{equation}
\begin{array}{rcl}
(\mathbf{P}_\mathbf{K}^n)^{k+1} = \mathbf{P}_\mathbf{K}^n \times \cdots \times \mathbf{P}_\mathbf{K}^n & \to &
\mathbf{P} \mathbf{D}^{k+1} (\mathbf{K}^{n+1}) \\
([u_0],\ldots,[u_k]) & \mapsto & [u_0 \vee \cdots \vee u_k] \\\end{array}
\end{equation}
We now state our first transversality result.
\begin{proposition} \label{prop.first-transversality}
For any distinct $w_1,w_2 \in \mathbf{P}_\mathbf{K}^n$ and any $r>0$,
\begin{equation}
\mathrm{Leb} \{ U \in \mathbf{P} \mathbf{D}^{k+1} (\mathbf{K}^{n+1})\ ;\ \phi_U(w_1,w_2) < r \} \lesssim r^{\delta_\mathbf{K} (n-k-1)}
\end{equation}
uniformly in $w_1,w_2$, where $\delta_\mathbf{K}$ is $1$ if $\mathbf{K}=\mathbf{R}$ and $2$ if $\mathbf{K}=\mathbf{C}$.
\end{proposition}
\begin{proof}
Since $\phi_U(w_1,w_2) \geq \tau(U,W)$ by \eqref{eq.phiU-formula}, where we let $W=w_1 \vee w_2$, it is enough
to show that
\begin{equation}
\mathrm{Leb} \{ U \in \mathbf{P} \mathbf{D}^{k+1} (\mathbf{K}^{n+1}) \ ;\ \tau(U,W) < r \} \lesssim r^{n-k-1}
\end{equation}
If $k=0$ this is a special case of Lemma \ref{lemma.first-transversality} below. If $k \geq 1$ we argue by induction
using the inequality
\begin{equation}
\tau(U,W) \geq \tau(u_k,U' \vee W) \tau(U',W)
\end{equation}
where $U=u_0 \vee \cdots \vee u_k$ and $U' = u_0 \vee \cdots \vee u_{k-1}$. Lemma \ref{lemma.first-transversality}
along with Fubini's Theorem yield
\begin{multline}
\mathrm{Leb} \{ U \in \mathbf{P} \mathbf{D}^{k+1} (\mathbf{K}^{n+1})\ ;\ \tau(U,W) < r \}\\ \lesssim
r^{\delta_\mathbf{K} (n-k-1)} \int \mathrm{d} \mathrm{Leb}(U') \ \tau(U',W)^{-\delta_\mathbf{K} (n-k-1)}
\end{multline}
The induction hypothesis implies that $\int \mathrm{d} \mathrm{Leb}(U') \ \tau(U',W)^{-\delta_\mathbf{K} (n-k-1)}$ is finite,
and the Proposition follows.
\end{proof}
\begin{lemma}\label{lemma.first-transversality}
For any $\ell$-dimensional projective subspace $L$ of $\mathbf{P}_\mathbf{K}^n$
\begin{equation}
\mathrm{Leb} \{ u \in \mathbf{P}_\mathbf{K}^n\ ;\ d(u,L) \leq r\} \lesssim r^{\delta_\mathbf{K} (n-\ell)}
\end{equation}
uniformly in $r$.
\end{lemma}
This Lemma and its proof are standard.
\begin{corollary} \label{cor.first-transversality}
Fix $k$, $0 \leq k \leq n-2$. Let $A$ be a Borel subset of $\mathbf{P}_\mathbf{K}^n$ of Hausdorff dimension $s$. For almost
every $k$--dimensional projective subspace $L$ of $\mathbf{P}_\mathbf{K}^n$, the transverse dimension of $A$, with respect to
the foliation of $\mathbf{P}_\mathbf{K}^n \setminus L$ by $(k+1)$-dimensional projective subspace containing $L$, is equal
to
\begin{equation}
\inf \{\delta_\mathbf{K}(n-k-1),s\}
\end{equation}
\end{corollary}
\begin{proof}
We apply Kaufman's classical argument using the transversality property stated in Proposition
\ref{prop.first-transversality}. Let us provide some details. We assume $s>0$.
First, note that if $v_1,v_2$ are two different points of $\mathbf{P}_\mathbf{K}^n$, the set of $k$--dimensional
projective subspaces passing through $v_1$ and $v_2$ has Lebesgue measure $0$ in $\mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$.
Taking this fact into account, pick $2$ disjoint closed balls
$B_1, B_2$ such that the Hausdorff dimension of $A \cap B_i$ is $s$ for each $i$,
and small enough that the set of $k$--dimensional projective subspace of $\mathbf{P}^n_\mathbf{K}$ meeting both $B_1$ and $B_2$
has very small Lebesgue measure. This is possible because the Hausdorff dimension of a finite union
$\cup X_i$ is the supremum of the Hausdorff dimensions of the $X_i$.
Let $O_1,O_2$ be open subsets of $\mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$ such that for any $U \in \overline{O_i}$, the projective subspace
$L_U$ does not meet $B_i$, and that the complement of $O_1 \cup O_2$ has very small Lebesgue measure in
$\mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$. Now fix $i=1$ or $2$.
Let $\sigma < \inf \{ s, \delta_\mathbf{K} (n-k-1)\}$ and $\mu$ be a Borel probability measure supported on $A \cap B_i$
such that the $\sigma$--energy of $\mu$ is finite:
\begin{equation} \label{def.energy}
I_{\sigma} (\mu) = \int \frac{\mathrm{d} \mu (w_1) \mathrm{d} \mu(w_2)}{d(w_1,w_2)^{\sigma}} < \infty
\end{equation}
(see \cite{Mattila1995} 8.8 and 8.9).
We apply Fubini's Theorem:
\begin{equation}
\int_{O_i} \mathrm{d} \mathrm{Leb} (U) I_{\sigma} (\mathrm{Proj}_U (\mu)) =
\int \frac{\mathrm{d} \mu (w_1) \mathrm{d} \mu(w_2)}{d(w_1,w_2)^{\sigma}} \int_{O_i} \mathrm{d} \mathrm{Leb} (U)
\phi_U (w_1,w_2)^{-\sigma}
\end{equation}
and we will show that the right-hand side is finite by checking that
\begin{equation}
\int_O \mathrm{d} \mathrm{Leb}(U) \phi_U(w_1,w_2)^{-\sigma}
\end{equation}
is bounded by a uniform constant for any distinct $w_1,w_2 \in B_i$.
A standard application of Fubini's Theorem followed by a change of variable yields
\begin{equation}
\int_{O_i} \mathrm{d} \mathrm{Leb}(U) \phi_U(w_1,w_2)^{-\sigma} \lesssim 1+\int_0^1 \mathrm{Leb}\{ U \in O_i\ ;\ \phi_U(w_1,w_2) < t\} t^{-(1+\sigma)} \ \mathrm{d}t
\end{equation}
(where the constant implied by the notation $\lesssim$ does not depend on $w_1,w_2$). Taking into account
Proposition \ref{prop.first-transversality}, we see that the right-hand side is bounded by a uniform constant as soon as
$\sigma < \delta_\mathbf{K} (n-k-1)$, which holds by assumption.
All in all, we get
\begin{equation}
\int_{O_i} \mathrm{d} \mathrm{Leb} (U) I_{\sigma} (\mathrm{Proj}_U (\mu)) \lesssim I_\sigma (\mu) < \infty
\end{equation}
showing that for Lebesgue--almost every $U \in O_i$, the transverse dimension of $A \cap B_i$, along the foliation
by $(k+1)$--dimensional projective subspaces containing $U$, is at least equal to $\sigma-\varepsilon$.
Thus for almost every $U \in O_1 \cup O_2$, the transverse dimension of $A$, along the foliation by
$(k+1)$--dimensional projective subspaces containing $U$, is at least equal to $\sigma-\varepsilon$.
Since $\varepsilon$ was arbitrary, we get the desired conclusion for almost every $U \in O_1 \cup O_2$.
The Theorem follows from this, because the complement of $O_1 \cup O_2$ has arbitrarily small Lebesgue measure.
\end{proof}
The previous results will now be improved by looking at a special subfamily of foliations.
\subsection{Transversality of pointed foliations}\label{ss.transversality-pointed}
As before, $n \geq 2$ is fixed and we work in $\mathbf{P}_\mathbf{K}^n$.
\begin{lemma}\label{lemma.intersection-transversality}
Fix $V \in \mathbf{D}^n (\mathbf{K}^{n+1})$. For any $U \in \mathbf{D}^k (\mathbf{K}^{n+1})$ and $W \in \mathbf{D}^2 (\mathbf{K}^{n+1})$ such that
\begin{itemize}
\item $\mathrm{Span}(U) \subset \mathrm{Span}(V)$;
\item $\mathrm{Span}(W) \not \subset \mathrm{Span}(V)$
\end{itemize}
we have
\begin{equation}
\Dist{U}{W} \geq \frac{\| U \vee (W \wedge V)\|}{\|U\| \cdot \|W \wedge V \|}
\end{equation}
\end{lemma}
\begin{proof}
Let $V=v_1 \vee \cdots \vee v_n$ and assume, as we may, that $(v_1,\ldots,v_n)$ is an orthonormal
basis of $\mathrm{Span}(V)$. Also, choose $w_2 \in \mathrm{Span}(V)$ and $w_1$ orthogonal to $\mathrm{Span}(V)$ such that
$W=w_1 \vee w_2$.
It follows that $W \wedge V$ is colinear to $w_2$; hence
\begin{equation}
\frac{\| U \vee (W \wedge V)\|}{\|U\| \cdot \|W \wedge V \|} = \Dist{U}{w_2}
\end{equation}
On the other hand,
\begin{equation}
\Dist{U}{W} = \Dist{U \vee w_2}{w_1} \times \Dist{U}{w_2} \times d(w_1,w_2)^{-1}
\end{equation}
where the first term of the right-hand side is equal to $1$ because $w_1$ was chosen to be orthogonal to $\mathrm{Span}(V)$
and both $\mathrm{Span}(U)$ and $w_2$ are inside $\mathrm{Span}(V)$; also, we know that $d(w_1,w_2) \leq 1$ (\eqref{eq.angular-metric} and Lemma
\ref{lemma.basic-inequality}).
Hence the Lemma.
\end{proof}
\begin{proposition} \label{prop.second-transversality}
Fix $V$ as in the Lemma, and let $K$ be some compact subset of $\mathbf{P}_\mathbf{K}^n \setminus L_V$. For any
distinct $w_1,w_2 \in K$
\begin{equation}
\mathrm{Leb} \{ U \in \mathbf{P} \mathbf{D}^{k+1} (\mathrm{Span}(V))\ ;\ \phi_U (w_1,w_2) < r \} \lesssim r^{\delta_\mathbf{K} (n-k-1)}
\end{equation}
where as previously $\delta_\mathbf{K}$ is $1$ or $2$ according as $\mathbf{K}$ is $\mathbf{R}$ or $\mathbf{C}$.
\end{proposition}
\begin{proof}
Follow the line of the proof of Proposition \ref{prop.first-transversality}, replacing
the $2$-vector $W$ with the (genuine) vector $V \wedge W$.
\end{proof}
\begin{corollary} \label{cor.second-transversality}
Fix $k$ and $V$ as before. Let $A$ be a Borel subset of $\mathbf{P}_\mathbf{K}^n \setminus L_V$ of Hausdorff
dimension $s$. For almost every $k$--dimensional projective subspace $L$ of
$\mathbf{P}_\mathbf{K} (\mathrm{Span}(V))$, the transverse dimension of $A$ with respect to the foliation of
$\mathbf{P}_\mathbf{K}^n \setminus L_V$ by $(k+1)$--dimensional projective subspaces containing $L$,
is equal to
\begin{equation}
\inf \{ \delta_\mathbf{K} (n-k-1),s\}
\end{equation}
\end{corollary}
This is very similar to the previous Corollary, but we are now looking at $k$--dimensional
projective subspaces of a fixed projective hyperplane $L_V$, effectively lowering the dimension
of the space of foliations.
More precisely, the space of $k$--dimensional projective subspaces of $\mathbf{P}_\mathbf{K}^n$ has
dimension $(k+1)(n-k)$ whereas the space of $k$--dimensional projective subspaces of $L_V$
has dimension $(k+1)(n-k-1)$.
In restricting our space of foliations, we did not lose anything dimension-wise. We will see later
that this is not really surprising, by showing how, when $\mathbf{K}=\mathbf{R}$, this Corollary is actually
equivalent to the classical Marstrand--Kaufman--Mattila projection Theorem.
\section{Linear foliations of spheres}\label{s.spheres-foliations}
\subsection{General setup} \label{ss.general-setup}
Let $n \geq 2$. We will deal at the same time with the $(n-1)$-sphere in $\mathbf{P}_\mathbf{R}^n$ and
the $(2n-1)$-sphere in $\mathbf{P}_\mathbf{C}^n$, so let us introduce suitable notations: denote by $\mathbf{S}, \mathbf{B} \subset \mathbf{P}_\mathbf{K}^n$
the sets\label{def.S-B}
\begin{equation}
\left. \begin{array}{lcc}
\mathbf{S}^{n-1} & = & \{ [1:x_1:\ldots:x_n] \in \mathbf{P}_\mathbf{R}^n\ ;\ x_1^2+\cdots+x_n^2 = 1\} \\
\mathbf{B}^n & = & \{ [1:x_1:\ldots:x_n] \in \mathbf{P}_\mathbf{R}^n\ ;\ x_1^2+\cdots+x_n^2 < 1\}
\end{array} \right\} \quad \mathrm{if} \quad \mathbf{K}=\mathbf{R}
\end{equation}
\begin{equation}
\left. \begin{array}{lcc}
\mathbf{S}^{2n-1} & = & \{ [1:z_1:\ldots:z_n]\in \mathbf{P}_\mathbf{C}^n\ ;\ |z_1|^2+\cdots+|z_n|^2 = 1\} \\
\mathbf{B}^{2n} & = & \{ [1:z_1:\ldots:z_n]\in \mathbf{P}_\mathbf{C}^n\ ;\ |z_1|^2+\cdots+|z_n|^2 < 1\}
\end{array} \right\} \quad \mathrm{if} \quad \mathbf{K}=\mathbf{C}
\end{equation}
If $L$ is some $k$--dimensional projective subspace of $\mathbf{P}_\mathbf{K}^n$, let $\mathcal F_L$ be the
foliation of $\mathbf{S}$ the leaves of which are the intersections of
$\mathbf{S}$ with $(k+1)$--dimensional projective subspaces of $\mathbf{P}_\mathbf{K}^n$
containing $L$.
(Remember that we are abusing the language and that what we really mean here is that $\mathcal F_L$
is the foliation of $\mathbf{S}\setminus (L \cap \mathbf{S})$ the leaves of which are the intersections of
$\mathbf{S} \setminus (L \cap \mathbf{S})$ with $(k+1)$--dimensional projective subspaces of $\mathbf{P}_\mathbf{K}^n$
containing $L$.)
If $\mathbf{K}=\mathbf{R}$, the case $k=0$ is essentially empty and should be removed from consideration.
The leaves of $\mathcal F_L$ are small $k$--spheres if $\mathbf{K}=\mathbf{R}$, respectively small $(2k+1)$-spheres if
$\mathbf{K}=\mathbf{C}$.
Our previous projection results in $\mathbf{P}_\mathbf{K}^n$ (Corollaries \ref{cor.first-transversality} and \ref{cor.second-transversality})
translate without any further work
to interesting projection results in $\mathbf{S}$. We need only remark that the restriction of $d$
to $\mathbf{S}$ is equal to the usual angular metric on $\mathbf{S}$.
\begin{theorem}
Fix $k$, $0\leq k \leq n-2$, and let $A$ be a Borel subset of $\mathbf{S}$ of Hausdorff dimension $s$.
For almost every $k$--dimensional projective subspace $L$ of $\mathbf{P}_\mathbf{K}^n$, the transverse
dimension of $A$ with respect to $\mathcal F_L$ is equal to $\inf \{ s, \delta_\mathbf{K} (n-k-1)\}$.
\end{theorem}
We are going to restrict this family of foliations in order to obtain foliations which can be
described purely in terms of spherical geometry.
For $k \geq 1$, let $\mathcal L_\mathbf{K}^k$\label{def.Lk} be the space of $k$--dimensional projective subspaces
$L$ of $\mathbf{P}_\mathbf{K}^n$ that meet $\mathbf{B}$. If $\mathbf{K}=\mathbf{R}$, this is the same thing as the space of small $(k-1)$--spheres
of $\mathbf{S}^{n-1}$.
\begin{definition}\label{def.kchain}
Assume $\mathbf{K}=\mathbf{C}$. For $k \geq 0$, a $k$--chain is the intersection of $\mathbf{S}^{2n-1}$ with
a $k$--dimensional projective subspace of $\mathbf{P}_\mathbf{C}^n$ that meets $\mathbf{B}^{2n}$.
\end{definition}
A $k$--chain is the complex analogue of a small $k$--sphere; it is also a special case of small $(2k-1)$--sphere.
For example, the space of all small $1$-spheres (i.e. all small circles) of $\mathbf{S}^3$ has dimension $6$, whereas the
space of all $1$-chains of $\mathbf{S}^3$ has dimension $4$.
\begin{lemma}
If $k \geq 1$, $\mathcal L_\mathbf{K}^k$ is an open subset of $\mathbf{P} \mathbf{D}^{k+1}(\mathbf{K}^{n+1})$, the space of all
$k$--dimensional projective subspaces of $\mathbf{P}_\mathbf{K}^n$.
\end{lemma}
In particular, the Lebesgue measure of this space is non-zero. Corollary \ref{cor.first-transversality}
thus implies the following
\begin{theorem}\label{th.sphere-first}
Fix $k$, $1 \leq k \leq n-2$ and let $A$ be a Borel subset of $\mathbf{S}$ of Hausdorff dimension $s$.
\begin{description}
\item[$\mathbf{K}=\mathbf{R}$ :] For almost every $(k+1)$--tuple $(u_0,\ldots,u_k)$ of points of
$\mathbf{S}=\mathbf{S}^{n-1}$, the transverse dimension of $A$, with respect to the foliation of
$\mathbf{S}^{n-1}$ by small $k$--spheres passing through each of these points, is equal to
\begin{equation}
\inf\{n-k-1,s\}
\end{equation}
\item[$\mathbf{K}=\mathbf{C}$ :] For almost every $(k+1)$--tuple $(u_0,\ldots,u_k)$ of points of
$\mathbf{S}=\mathbf{S}^{2n-1}$, the transverse dimension of $A$, with respect to the foliation of
$\mathbf{S}^{2n-1}$ by $(k+1)$--chains passing through each of these points, is equal to
\begin{equation}
\inf\{2(n-k-1),s\}
\end{equation}
\end{description}
\end{theorem}
This follows from Corollary \ref{cor.first-transversality} because the restriction of
the Lebesgue measure to $\mathcal L_\mathbf{K}^k$ is equivalent to the probability measure obtained
by picking $k+1$ points at random on $\mathbf{S}$ and looking at the only $k$--dimensional projective subspace
passing through each of these points.
In the same fashion, Corollary \ref{cor.second-transversality} implies the following
\begin{theorem}\label{th.sphere-second}
Fix $k$, $1 \leq k \leq n-2$ and let $L \subset \mathbf{S}$ be a small $(n-2)$--sphere if $\mathbf{K}=\mathbf{R}$,
respectively a $(n-2)$--chain if $\mathbf{K}=\mathbf{C}$. Let $A$ be a Borel subset of $\mathbf{S}$ of Hausdorff
dimension $s$.
\begin{description}
\item[$\mathbf{K}=\mathbf{R}$ :] For almost every $(k+1)$--tuple $(u_0,\ldots,u_k)$ of points of
$L$, the transverse dimension of $A$, with respect to the foliation of
$\mathbf{S}^{n-1}$ by small $k$--spheres passing through each of these points, is equal to
\begin{equation}
\inf\{n-k-1,s\}
\end{equation}
\item[$\mathbf{K}=\mathbf{C}$ :] For almost every $(k+1)$--tuple $(u_0,\ldots,u_k)$ of points of
$L$, the transverse dimension of $A$, with respect to the foliation of
$\mathbf{S}^{2n-1}$ by $(k+1)$--chains passing through each of these points, is equal to
\begin{equation}
\inf\{2(n-k-1),s\}
\end{equation}
\end{description}
\end{theorem}
When $\mathbf{K}=\mathbf{C}$, the case $k=0$ is missing (because $\mathbf{S}^{2n-1}$ is negligible for the
Lebesgue measure on $\mathbf{P}_\mathbf{C}^n$) and we have to handle it separately.
\subsection{Foliations by $1$-chains} \label{ss.foliations-one-chains}
We now fix $\mathbf{K}=\mathbf{C}$. For every $u \in \mathbf{S}^{2n-1}\subset\mathbf{P}^n_\mathbf{C}$, the foliation of $\mathbf{P}_\mathbf{C}^n$ by projective lines
passing through $u$ induces a foliation of $\mathbf{S}^{2n-1}$ by 1-chains passing through $u$.
\begin{theorem}\label{th.sphere-third}
Let $A$ be a Borel subset of $\mathbf{S}^{2n-1}$ of Hausdorff dimension
$s$. For almost every $u \in \mathbf{S}^{2n-1}$, the transverse dimension of $A$
with respect to the foliation of $\mathbf{S}^{2n-1}$ by $1$--chains passing through $u$ is equal to
\begin{equation}
\inf\{s, 2n - 2\}
\end{equation}
\end{theorem}
\begin{proof}
The Hausdorff dimension of a Borel set $A$ is the supremum of the
Hausdorff dimensions of its compact subsets. Using this fact, we can
assume, without loss of generality, that $A$ is a compact subset of $\mathbf{S}^{2n-1}$.
Let $O$ be an open subset of $\mathbf{S}^{2n-1}$ that is non-empty and such that the closure
$\overline{O}$ does not meet $A$. We first show that the conclusion holds for almost
every $u \in O$.
Introduce the set $C_\varepsilon(A, O)$ of all projective lines $[W] \in \mathbf{P} \mathbf{D}^2(\mathbf{C}^{n+1})$
that meet $A$ and $\overline{O_\varepsilon}$, where $O_\varepsilon$ is the $\varepsilon$--neighbourhood
of $O$ in $\mathbf{P}^n_\mathbf{C}$, i.e.
Fix $\varepsilon$ small enough that $A$ does not meet $\overline{O_\varepsilon}$. Let $G = \mathbf{PU}(1, n)$ and
fix as in Lemma \ref{lemma.sphere-third} a compact subset $\mathcal G$ of $G$ such that any 1-chain meeting
$A$ and $\overline{O_\varepsilon}$ is of the form $gL_0$, where $g \in \mathcal G$ and $L_0$ is the 1-chain passing
through, say, $[e_0 + e_1]$ and $[e_0 - e_1]$. (Recall that $(e_0,\ldots,e_n)$ is the canonical basis of $\mathbf{K}^{n+1}$.)
I claim that for any $r > 0$, and any distinct $w_1, w_2 \in A$
\begin{equation}
\mathrm{Leb}_{\mathbf{S}^{2n-1}} \{u \in O\ ;\ \tau (u, W ) \leq r\} \lesssim r^{2n-2}
\end{equation}
where $W = w_1 \vee w_2$ and the constant implied by the notation $\lesssim$ is uniform
in $r, W$ and $u$.
We can assume that $r$ is small enough (with respect to the $\varepsilon$ fixed
above) that the projective line $[W ]$ has to meet both $A$ and $\overline{O_\varepsilon}$ in order for the left-hand side to be non-zero.
Let $g$ be an element of $\mathcal G$ such that $[W ] = [gW_0]$
where $W_0 = e_0 \vee e_1$. This is possible because $(e_0 + e_1) \vee (e_0 - e_1)$ is a
scalar multiple of $e_0 \vee e_1$.
Now
\begin{multline}
\mathrm{Leb}_{\mathbf{S}^{2n-1}} \{u \in O\ ;\ \tau (u, gW_0) \leq r\} \lesssim \mathrm{Leb}_{\mathbf{S}^{2n-1}} \{u \in O\ ;\ \tau (u, W_0) \lesssim r\} \\
\lesssim \mathrm{Leb}_{\mathbf{S}^{2n-1}} \{u \in \mathbf{S}^{2n-1}\ ;\ \tau (u, W_0) \leq r\} \lesssim r^{2n-2}
\end{multline}
where the first inequality follows from the compacity of $\mathcal G$ (and the
subsequent fact that the singular values of $g$ belong to a compact subset of
$]0, +\infty[$) and the last inequality is an easy computation.
At this point we can apply Kaufman's argument; this yields that for Lebesgue-almost every $u \in O$, the transverse dimension of $A$ with respect
to the foliation of $\mathbf{S}^{2n-1}$ by 1-chains passing through $u$ is equal to
$\inf\{s, 2(n - 1)\}$.
Now let $x$ be any point of $\mathbf{S}^{2n-1}$. For any $\varepsilon > 0$ we can find $\delta > 0$
such that
\begin{equation}
\dim(\mathbf{K} \setminus B(x, \delta)) \geq s - \varepsilon
\end{equation}
(where $\dim$ is the Hausdorff dimension). Taking into account the previous statement,
it follows that for Lebesgue-almost every $u \in B(x, \delta/2)$, the transverse dimension of $K$, with respect
to the foliation by 1-chains passing through $u$, is at least equal to $\inf\{s - \varepsilon, 2(n - 1)\}$.
The compacity of $\mathbf{S}^{2n-1}$ then implies that for Lebesgue-almost every
$u \in \mathbf{S}^{2n-1}$, the transverse dimension of $K$, with respect to the foliation
by 1-chains passing through $u$, is at least $\inf\{s - \varepsilon, 2(n - 1)\}$.
The Theorem follows by letting $\varepsilon$ go to 0 along a countable sequence.
\end{proof}
\begin{lemma}\label{lemma.sphere-third}
Let $K^-$, $K^+$ be disjoint non-empty compact subsets of $\mathbf{S}^{2n-1}$
and let $L_0$ be a fixed $1$--chain. There is a compact subset $\mathcal G$ of G such that
any $1$--chain meeting both $K^-$ and $K^+$ is of the form $gL_0$ for some $ g \in \mathcal G$.
\end{lemma}
\begin{proof}
Fix $\xi^-$, $\xi^+$ two distinct elements of $L_0$ and let $KAN$ be an Iwasawa
decomposition of $G$ in which the Cartan subgroup $A$ fixes both $\xi^-$ and
$\xi^+$ (and, consequently, leaves $L_0$ globally invariant).
The mapping $g \mapsto (g\xi^-, g\xi^+)$ defines, by passing to the quotient, a proper and onto mapping
\begin{equation}
\omega : G/A \to \{(\eta^-, \eta^+) \in \mathbf{S}^{2n-1} \times \mathbf{S}^{2n-1}\ ;\ \eta^- \neq \eta^+ \}
\end{equation}
(Recall that a continuous mapping is proper if the inverse image of any
compact subset is a compact subset.)
Now, $K^- \times K^+$ being compact, so must be its inverse image $\omega^{-1}(K^-\times K^+)$.
To conclude, use the fact that any compact subset of $G/A$ is the image
(through the quotient mapping $G \to G/A$) of a compact subset of $G$.
\end{proof}
\begin{theorem}\label{th.sphere-chain}
Let $A$ be a Borel subset of $\mathbf{S}^{2n-1}$ of Hausdorff dimension $s$. Fix a $(n - 1)$--chain $L$.
For Lebesgue-almost every $u \in L$, the transverse dimension of A with respect to the foliation
of $\mathbf{S}^{2n-1}$ by 1-chains passing through $u$ is at least
\begin{equation}
\inf\{s, 2n - 3\}
\end{equation}
\end{theorem}
\begin{proof}
We argue as in the proof of the previous Theorem. Using transitivity of $G=\mathbf{PU}(1, n)$ in the same fashion as before, we can safely assume
that $L$ is the intersection of $\mathbf{S}^{2n-1}$ with $\mathbf{P}(\mathbf{C} e_0 \oplus \cdots \oplus \mathbf{C} e_{n-1})$. Let
$V = e_0 \vee \cdots \vee e_{n-1}$.
Now fix a compact subset $K \subset \mathbf{S}^{2n-1}$ that does not meet $L = L_V$.
For any distinct $w_1, w_2 \in K$, letting $W = w_1 \vee w_2$, we know by Lemma \ref{lemma.intersection-transversality}
that
\begin{equation}
\tau (u, W ) \geq \tau (u, W \vee V )
\end{equation}
for any $u \in \mathbf{S}^{2n-1}$. It follows that
\begin{equation}
\mathrm{Leb}_L \{ u \in L\ ;\ \tau (u, W ) \leq r\} \leq \mathrm{Leb}_L \{ u \in L\ ;\ d(u, W \vee V ) \leq r\}
\end{equation}
and $W \vee V$ is just a point of the $(n - 1)$--chain $L$.
The $(n - 1)$--chain $L$ is equal to the $(2n - 3)$--sphere $\mathbf{S}^{2n-3} \subset \mathbf{P}_\mathbf{C}^{n-1}$
where we identify $\mathbf{P}_\mathbf{C}(\mathrm{Span}(V))$ with $\mathbf{P}_\mathbf{C}^{n-1}$. The angular metric defined on $\mathbf{P}^{n-1}_\mathbf{C}$
as in formula \eqref{eq.angular-metric} is equal to the restriction of $d$ to $\mathbf{P}^{n-1}_\mathbf{C}$. Its restriction to
$L$ is just the spherical metric of $\mathbf{S}^{2n-3}$. The right-hand side in the previous equation
is thus $\leq r^{2n-3}$. The Theorem follows from this estimate, as above.
\end{proof}
\subsection{A concrete look at these foliations}\label{ss.concrete-look}
We now look at some concrete examples in order to get a better idea of
what is actually going on in the previous Theorems and how our results
are related to Marstrand's classical projection Theorem.
\subsubsection{The real case}
We fix $\mathbf{K} = \mathbf{R}$.
\paragraph{In affine coordinates.} Let $k = 0$ and $n \geq 2$. For any $u \in \mathbf{P}_\mathbf{R}^n$, we
consider the foliation of $\mathbf{P}_\mathbf{R}^n$ by projective lines passing through $u$
(with $u$ removed). If we pick affine coordinates $\mathbf{R}^n \subset \mathbf{P}_\mathbf{R}^n$ and send some
projective hyperspace $\mathbf{P}^{n-1}$ to infinity, we are looking, when $u$ belongs to
$\mathbf{R}^n$, at the usual radial projection in $\mathbf{R}^n$.
On the other hand, if $u$ belongs to $\mathbf{P}^{n-1}$, the resulting foliation of $\mathbf{R}^n$
has leaves affine lines parallel to the vector line associated to $u$.
We can now translate the conclusion of Corollary \ref{cor.second-transversality} in affine terms:
for Lebesgue-almost every $u \in \mathbf{P}^{n-1}$, the dimension of $A$ transverse to
the foliation of $\mathbf{R}^n$ by affine lines parallel to the vector line associated to $u$, is equal to $\inf\{s, n - 1\}$.
This is exactly the statement of the usual Marstrand projection Theorem along lines.
We see that this statement is equivalent to the following one: for
any affine hyperspace $L$ in $\mathbf{R}^n$, for Lebesgue-almost every $u \in L$, the
dimension of $A$ transverse to the foliation of $\mathbf{R}^n$ by affine lines
passing through $u$ is equal to $\inf\{s, n - 1\}$.
We leave it to the reader to inspect the case when $k \geq 1$ and come to
the conclusion that Corollary \ref{cor.second-transversality} is again essentially equivalent to
Marstrand's classical projection Theorem (which, in this case, is due to Kaufman and
Mattila).
\paragraph{Foliations of the sphere.} If $k = 0$, there is no interesting
foliation induced because a (real) projective line meets the sphere in at most two points.
Assume $1 \leq k \leq n - 2$. Let us fix already a $(n - 2)$--sphere $L \subset \mathbf{S}^{n-1}$
and let $A$ be a Borel subset of $\mathbf{S}^{n-1}$ of Hausdorff dimension $s$. According
to Theorem \ref{th.sphere-second} for almost every $u_0, \ldots, u_k$ in $L$, the transverse dimension
of $A$ with respect to the foliation of $\mathbf{S}^{n-1}$ by $k$--spheres passing through
$u_0, \ldots, u_k$ is equal to $\inf\{s, n - 1 - k\}$.
Let us look at this result in the Euclidean space: send $u_0$ at infinity via
the stereographic projection, in such a way that $L$ is the subspace $\mathbf{R}^{n-2}$ of $\mathbf{R}^{n-1}$.
For any $u_1, \ldots, u_k \in L$, the foliation of $\mathbf{R}^{n-1}$ we obtain has leaves the affine $k$--spaces containing
$u_1, \ldots, u_k.$ We thus see that Theorem \ref{th.sphere-second} is actually weaker than Corollary \ref{cor.second-transversality} when $\mathbf{K} = \mathbf{R}$.
\subsubsection{The complex case}
\paragraph{In affine coordinates.} Let us look already at the affine version of
Corollary \ref{cor.second-transversality}. First, fix $k = 0$. We are looking at foliations of $\mathbf{C}^n$ by
complex affine lines parallel to a given complex vector line (associated to
some point $u \in \mathbf{P}_\mathbf{C}^{n-1}$). We can recast this in real terms: we are looking
at foliations of $\mathbf{R}^{2n}$ with real affine 2-planes parallel to a given real vector 2-plane.
Now this real vector 2-plane cannot be just any 2-plane: it has
to be the real plane underlying some complex line.
It becomes apparent that we are effectively improving on Marstrand's
projection Theorem. This Theorem deals with the family of every real
vector plane of $\mathbf{R}^{2n}$, and we see that the conclusion still holds when we
restrict to this subspace of foliations, which is is Lebesgue-negligible.
It is perhaps enlightening to compare the dimension of the space of all
foliations of $\mathbf{R}^{2n}$ by parallel affine 2-planes, which is equal to $2(n-1)\times 2 = 4(n-1)$,
to the dimension of the subspace of those special foliations coming from complex lines,
which is equal to $2(n - 1)$ (this is of course the real dimension of $\mathbf{P}_\mathbf{C}^{n-1}$).
For an arbitrary $k \geq 0$ (and $k \leq n - 2$), we are looking, in the complex
case, at a family of foliations of $\mathbf{R}^{2n}$ by $2(k + 1)$--dimensional real spaces coming
from $(k + 1)$--dimensional complex spaces; the dimension of this
space of foliations is equal to $2(n - 1 - k)(k + 1)$, whereas the dimension of the space
of all $2(k + 1)$--dimensional real vector subspaces of $\mathbf{R}^{2n}$ is
equal to $4(n - 1 - k)(k + 1)$.
\paragraph{Foliations of the sphere.} A $k$--chain of $\mathbf{S}^{2n-1}$ is a special case of
a small $(2k - 1)$--sphere of this sphere. It is only natural to wonder what kind
of object that is. In fact, chains appear naturally in complex hyperbolic
geometry. The complex hyperbolic space of (complex) dimension $n$, $\mathbf{H}^n_\mathbf{C}$,
has $\mathbf{S}^{2n-1}$ as its boundary at infinity. A totally geodesic submanifold $S$
of $\mathbf{H}^n_\mathbf{C}$ is one of two types:
\begin{itemize}
\item $S$ is isometric to a complex hyperbolic space $\mathbf{H}_\mathbf{C}^k$, $1 \leq k \leq n$;
\item Or $S$ is isometric to a real hyperbolic space $\mathbf{H}_\mathbf{R}^k$, $1 \leq k \leq 2(n - 1)$.
\end{itemize}
In the first case, the boundary of $S$ is a $k$--chain. In the second
case, it is, in Cartan's terminology, a real $k$--sphere. This is quite an unfortunate term;
a complex chain is a sphere just as much as a real sphere is.
We will come back to so-called real spheres in a later paper.
\bibliographystyle{plain}
|
2,869,038,155,677 | arxiv | \section{Introduction} \label{sec:INT}
Large-scale spiral magnetic field (B-field) structures are frequently observed in spiral galaxies \citep[e.g.,][]{Beck2019,SALSAVI}.
These B-fields are thought to be generated via a mean-field dynamo driven by differential rotation of the galactic disk and turbulent helical motions \citep{ss21}. The three-dimensional structure of galactic B-fields can be decomposed into radial ($B_{\rm r}$), azimuthal ($B_{\phi}$), and vertical ($B_{\rm z}$) components, where the coordinate system is typically defined relative to the core of the galaxy. The structure of the disk magnetic field (e.g., at the midplane $z = 0$) is often summarized using the pitch angle $\Psi_{\rm B}=\arctan(B_{\rm r}/B_{\rm \phi})$ \citep{Krasheninnikova1989}. In this formalism, a perfectly toroidal B-field has $\Psi_{\rm B} =0^{\circ}$, and a perfectly radial B-field has $\Psi_{\rm B} = 90^{\circ}$.
The full three-dimensional structure of galactic B-fields is not directly measurable, but $\Psi_{\rm B}$ can be estimated from polarimetric measurements of a galactic disk. For any point in the galaxy, $\Psi_{\rm B}$ is the angle between the local magnetic field orientation and the tangent to a circle with origin at the galaxy's center that passes through that point. The latest compilation of $\Psi_{\rm B}$ from radio polarimetric observations of $19$ nearby galaxies shows that the $\Psi_{\rm B}$ is mostly constant within the central $5-10$ kpc, with values in the range of $20-35^{\circ}$ \citep{Beck2019}. The $\Psi_{\rm B}$ were found to be systematically offset by $5–10^{\circ}$ when compared with the molecular (CO) spiral arms \citep{VanEck2015,Frick2016}---i.e, the magnetic pitch angles are more open than the molecular gas arms.
Far-infrared (FIR) polarimetric observations have shown to reveal different components of the large-scale B-fields in the disk of galaxies \citep[e.g.,][]{SALSAI,SALSAII,SALSAVI}. The detailed study performed in the spiral galaxy M51 showed that the radio and FIR magnetic pitch angles are similar within the central $6$ kpc, but at larger radii the FIR $\Psi_{\rm B}$ wrapped tighter than at radio wavelengths \citep{SALSAI}. The reason for this difference may be caused by the interaction of M51 with M51b and/or the injection of kinetic energy driven by the strong star formation region in the outskirts of the spiral arms of M51. The difference in the observed B-field structure arises from the different nature of the interstellar medium (ISM) associated with the FIR and radio wavelengths. The FIR polarization arises from thermal emission of magnetically aligned dust grains tracing a weighted density medium along the line-of-sight (LOS) and within the beam (i.e., full-width-at-half-maximum, FWHM) of the observations of a dense ($\log_{10}(N_{\rm{HI+H2}} [\rm{cm}^{-2}]) = [19.96, 22.91]$) and cold ($T_{\rm d} = [19, 48]$ K) component of the ISM \citep[SALSA~IV, ][]{SALSAIV}.
The pitch angles have been characterized assuming a priori function of a spiral arm---i.e., logaritmic spirals. Specifically, the pitch angles have been derived using a logarithmic spiral function fitted to the spiral arms of the gas tracers and/or a multi-mode logarithmic spiral B-field fitted to the magnetic arms \citep{Fletcher2011,VanEck2015}. Then, a mean pitch angle value of the entire galaxy is estimated and compared between tracers. A wavelet-based approach was used in the spiral galaxy M83 \citep{Frick2016} with the shape and width of the kernel as user-defined parameters. \citet{SALSAI} performed the same wavelet analysis to the morphological spiral arms and estimated the mean pitch angle per annulus as a function of galactocentric radius for the magnetic pitch angles. A model-independent systematic study of the magnetic pitch angles as a function of galactocentric radii and tracers is required.
Our goal is to characterize the B-field morphology of spiral galaxies using a model-independent approach. We make use of the linear polarimetric composition approach \citep{Palumbo2020} applied to analyze the B-field structure around the supermassive black hole of M87 from the Event Horizon Telescope (EHT) \citep{EHT2021_VII,EHT2021_VIII}. This method is a specific case ($m=2$) of the general E/B decomposition widely used to analyze the polarization from the cosmic microwave background \citep[CMB; e.g.,][]{Zaldarriaga2001}. \citet[][SALSA~II]{SALSAII} applied this method to the B-fields in the starburst ring of NGC~1097 showing that the radio B-field is dominated by a spiral B-field (with an azimuthal mode $m=2$), while a constant ($m=0$) B-field dominates at FIR wavelengths. The $m=2$ B-field was attributed to a magnetohydrodynamic (MHD) dynamo, and the $m=0$ B-field was associated with galactic shocks between the bar and the starburst ring. These results showed the potential of this method to analyze the B-fields in the multiphase of galaxies. Here, we apply the linear polarimetric decomposition to analyze the B-field orientation in a sample of five spiral galaxies with resolved radio and FIR polarimetric observations. We describe in Section \ref{sec:MET} the methodology of the linear polarimetric decomposition. The results of the decomposition of the B-field morphology using FIR and radio observations are shown in Section \ref{sec:APPLICATION}. Our discussions are described in Section \ref{sec:DIS}, and our main conclusions are summarized in Section \ref{sec:CON}.
\section{Methods}\label{sec:MET}
We adapt the method \citep{Palumbo2020} proposed for the analysis of EHT measurements of the polarized emission around the supermassive black hole in M87 \citep{EHT2021_VII,EHT2021_VIII}. Here, we summarize the method and describe how this linear polarimetric decomposition can be used to estimate the underlying B-field structure of a spiral galaxy.
\begin{figure
\centering
\includegraphics[width=\columnwidth]{fig1}
\caption{Examples of ring-valued linear polarization fields. The morphology of the linear polarization field corresponding to $0 \leq m \leq 3$ periodic modes with different values of the $\beta_m$ coefficient are presented.}
\label{fig:fig1}
\end{figure}
\subsection{Decomposition into azimuthal B-Field modes}\label{subsec:ModelDefinition}
We start by describing the linear polarization via the complex polarized intensity
\begin{equation}\label{eq:complexP}
P_{\rm B}(\rho, \phi) \equiv -Q(\rho, \phi) - iU(\rho, \phi),
\end{equation}
\noindent
where $Q$ and $U$ are the Stokes parameters of linear polarization, and $\rho$ and $\phi$ are radial and azimuthal coordinates, respectively. The sign convention in the definition of $P_{\rm B}$ represents our interest in the B-field orientation, which is rotated by $90^{\circ}$ from the electric vector position angle measured in radio and FIR polarimetric observations. The measured polarization field is decomposed into azimuthal modes, $m$, with amplitudes of $\beta_m$ via the decomposition definition
\begin{equation}\label{1}
\beta_m = \frac{1}{I_{\rm ann}} \int_{\rho_{\rm min}}^{\rho_{\rm max}} \int_0^{2\pi} P(\rho, \phi) e^{im\phi} \rho d\phi d\rho
\end{equation}
$\beta_m$ is a dimensionless complex number. Its absolute value, $|\beta_{\rm m}|$, corresponds to the amount of coherent power in the $m^\mathrm{th}$ mode, and the argument, $\angle \beta_{\rm m}$, corresponds to the average pointwise rotation of the B-field orientation within an annulus of radius $[\rho_{\rm{min}},\rho_{\rm{max}}]$. We define $\phi = 0^{\circ}$ as the B-field orientation in the north direction with positive values increasing along the counterclockwise direction (East of North). This decomposition can be thought of as a radially averaged azimuthal Fourier transform of the complex polarization field, where the $\beta_m$ coefficients are Fourier coefficients corresponding to the internal Fourier modes. We provide a collection of examples of ring-valued linear polarization fields corresponding to $0 \leq m \leq 3$ periodic modes with different values of the $\beta_m$ coefficient in Figure \ref{fig:fig1} \citep[see also figure 1 of][]{Palumbo2020}. The B-field orientations along its respective ring-valued linear polarization field are offset by an angle $\theta_m$ given by half of the complex phase of $\beta_m$ $=(\Re{(\beta_m)} + \Im{(\beta_m}$)) within $[-90, 90]^{\circ}$
\begin{equation}\label{2}
\theta_m = \frac{1}{2}\arctan{\left( \frac{\Im{(\beta_m)}}{\Re{(\beta_m)}} \right)}
\end{equation}
The $m = 0$ mode corresponds to a constant B-field orientation, $m = 1$ mode corresponds to a half dipole field structure, and $m = 2$ corresponds to a radial and toroidal distribution in the real space and a spiral structure in the complex space. Note that the $m=2$ mode is analogous to the $E$ and $B$ mode decomposition commonly used in studies of CMB polarization \citep[e.g.,][]{Kamionkowski1997,SK1997,Zaldarriaga2001}, where the real part of $\beta_2$ is the $E$-mode and the imaginary part is the $B$-mode. We show the reconstruction of a non-trivial B-field orientation with a combination of $m=0$ and $m=2$ modes in Figure \ref{fig:fig2}.
\subsection{Implementation of the algorithm}\label{subsec:ModelSteps}
We estimate the B-field orientation over a spiral galaxy as follows.
\begin{enumerate}
\item We construct a two-dimensional map of azimuthal angles such that $\phi=0^{\circ}$ corresponds to North (up), and $\phi$ increases in the counterclockwise direction (East from North).
\item We define a set of radial masks centered at the peak of the galaxy's central emission at a given wavelength. The first step is to define the grid of radial distances, i.e., the projected distance of each pixel from the galaxy's center in the $x, y$ plane (with the line of sight along $z$). If every galaxy were perfectly face-on, the radial masks would be simple circular annuli of this grid. In practice, we rotate the radial distance grid by the galaxy's inclination, $i$, and tilt, $\theta$, angles. We calculate $r' = R_x[i]R_z[\theta]r$, where $r$ is the original radius, $r'$ is the new radial distance, and $R_x[i]$, $R_z[\theta]$ are the rotation matrices for the inclination and tilt, respectively. We can thus define the projected annulus for any given inner and outer radius.
\item We compute $P_{\rm B}(\rho, \phi) \equiv -Q(\rho, \phi) - iU(\rho, \phi)$, the complex-valued polarized intensity (Equation \ref{eq:complexP}), where $Q$ and $U$ are the measured Stokes parameters at the galactrocentric radius of $\rho = \sqrt{x^{2} + y^{2}}$ and azimuthal angle $\phi$.
\item Using Equation \ref{1}, we calculate the amplitude $|\beta_m|$ and angle $\angle{\beta_m}$ for every annulus.
\item For $m = 2$ only, we take the product of the basis function $e^{im\phi}$ and the complex-valued polarization field $P(\rho, \phi) = Q(\rho, \phi) + iU(\rho, \phi)$. Note that this definition of the polarized intensity is the opposite sign to Equation \ref{eq:complexP}. We define this for $m=2$ to facilitate comparison with other measurements of galaxy magnetic pitch angles: this quantity represents the tangent to the local circumference at a given distance from the galaxy’s center, which is equivalent to the pitch angle of the B-field, $\Psi_2$. The angles $\angle \beta_{2}$ and $\Psi_{2}$ are thus complementary. Figure \ref{fig:fig2} illustrates the definition of each.
\end{enumerate}
We estimate the uncertainty on our decomposition parameters via a Monte Carlo technique. We generate $5000$ realizations of the Stokes $I$, $Q$, and $U$ fields by randomly sampling each pixel from a Gaussian distribution centered on the measured value, and with a standard deviation equal to the measurement uncertainty. From each realization we compute $|\beta_m|$, $\angle{\beta_m}$, and $\Psi_{2}$ for each annulus defined by radial range $[\rho_{\rm{min}}, \rho_{\rm{max}}]$. We compute the mean and standard deviation of each quantity over the $5000$ samples. This method was implemented in \textsc{python}, and the code is available in the Appendix.
\begin{figure}
\includegraphics[width=\columnwidth]{fig2}
\caption{Example of the B-field decomposition of a galaxy. The composition of the $m = 2$ mode with ${\beta_2} = -i$ and the $m = 0$ mode with ${\beta_0} = -i$ is shown. Additionally, the relationship between the averaged pointwise rotation of the B-field orientation, $\angle{\beta_2}$, and the magnetic pitch angle, $\Psi_2$, is illustrated at the top of the $m = 2$ mode on the left.
\label{fig:fig2}}
\end{figure}
\begin{deluxetable*}{lcccccl}
\centering
\tablecaption{Galaxy Sample. \emph{Columns, from left to right:} (a) Galaxy name. (b) Galaxy distance in Mpc. (c) Physical scale in pc per arcsec. (d) Galaxy type. (e) Inclination of the galaxy in degrees. (f) Position angle of the long axis of the galaxy in the plane of the sky. (g) References for the distance, inclination, and tilt angles.
\label{tab:GalaxySample}
}
\tablecolumns{6}
\tablewidth{0pt}
\tablehead{\colhead{Galaxy} & \colhead{Distance$^{1}$} & \colhead{Scale} & \colhead{Type$^{\star}$} &
\colhead{Inclination (i)$^{2}$} & \colhead{Tilt (PA)$^{2}$} & \colhead{References} \\
& \colhead{(Mpc)} & \colhead{(pc/\arcsec)} &
\colhead{($^{\circ}$)} & \colhead{($^{\circ}$)} & \colhead{($^{\circ}$)} \\
\colhead{(a)} & \colhead{(b)} & \colhead{(c)} & \colhead{(d)} & \colhead{(e)} & \colhead{(f)} & \colhead{(g)}}
\startdata
M51 & $8.58$ & $41.21$ & Sa & $22.5\pm5$ & $-7\pm3$ &
$^{1}$\citet{McQuinn2017}; $^{2}$\citet{Colombo2014} \\
M83 & $4.66$ & $22.38$ & SAB(s)c & $25\pm5$ & $226\pm5$ &
$^{1}$\citet{Tully2013}; $^{2}$\citet{Crosthwaite2002} \\
NGC~3627 & $8.90$ & $42.75$ & SAB(s)b & $52\pm1$ & $176\pm1$ &
$^{1}$\citet{Kennicutt2003}; $^{2}$\citet{Kuno2007} \\
NGC~4736 & $5.30$ & $25.46$ & SA(r)ab & $36\pm7$ & $292\pm2$ &
$^{1}$\citet{Kennicutt2003}; $^{2}$\citet{Dicaire2008} \\
NGC~6946 & $6.80$ & $32.66$ & Sc & $38.4\pm3.0$ & $239\pm1$ &
$^{1}$\citet{Karachentsev2000}; $^{2}$\citet{Daigle2006} \\
\enddata
\tablenotetext{{\star}}{Galaxy type from NASA/IPAC Extragalactic Database (NED; \url{https://ned.ipac.caltech.edu/})}
\end{deluxetable*}
\section{Application}\label{sec:APPLICATION}
We describe the data used in this work and show the results of this method to quantify the B-field structure of spiral galaxies.
\subsection{Archival data}\label{subsec:ArchivalData}
We apply the method presented in Section \ref{sec:MET} to a sample of five spiral galaxies. These spiral galaxies are the only publicly available objects with combined FIR and radio polarimetric observations. The application to both wavelengths allows us to characterize the B-field morphology in two different phases of the ISM. Table \ref{tab:GalaxySample} lists the properties of the galaxy sample used in this work.
The FIR data were taken from the Survey of ExtragALactic magnetiSm with SOFIA (SALSA\footnote{Data from SALSA can be found at \url{http://galmagfields.com/}}) published by \citet[][SALSA IV]{SALSAIV}. All FIR polarimetric observations were performed using SOFIA/HAWC+ at $154$ $\mu$m~with a beam size (FWHM) of $13.6$\arcsec~and a pixel scale of $6.90$\arcsec~(i.e., Nyquist sampling). For a detailed description of the data reduction see \citet[][SALSA III]{SALSAIII} and for an analysis of the polarization fraction see \citet[][SALSA IV]{SALSAIV}.
The radio polarimetric observations were obtained with the Very Large Array (VLA) and the Effelsberg 100-m radio telescope at $6$ cm with a typical angular resolution of $8$\arcsec. For a detailed description of the data reduction of each galaxy we defer to \citet[][M51]{Fletcher2011}, \citet[][M83]{Frick2016}, \citet[][NGC~3627]{Soida2001}, \citet[][NGC~4736]{Chyzy2008}, and \citet[][NGC6946]{Beck1991,Beck2007}. The $6$ cm polarimetric observations were selected because they are the common radio wavelength of all galaxies and have higher signal-to-noise ratios than the $3$ cm. At longer radio wavelengths ($18$, $20$ cm), the observations can be strongly affected by Faraday rotation \citep{Beck2019}. For all radio observations, the Stokes $I$, $Q$, and $U$ were convolved with a 2D Gaussian kernel to match the beam size of the HAWC+ observations. Then each galaxy was reprojected to match the footprint and pixelization of the HAWC+ observations. The smoothed and reprojected radio polarimetric observations are publicly available on the SALSA website. Figure \ref{fig:fig3} shows the measured B-field orientation at $154$ $\mu$m~and $6$ cm for each of the spiral galaxies used for our study. For visualization purposes, we display one B-field orientation per beam and only polarization measurements with $PI/\sigma_{PI} \ge 3$, where $PI$ is the polarized intensity and $\sigma_{PI}$ is the associated uncertainty.
\subsection{Results of the B-field orientation decomposition}\label{subsec:results}
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{fig3.pdf}
\caption{B-field orientation of our sample of spiral galaxies. Polarization measurements are shown with constant
length to illustrate the inferred B-field orientations at $154$ $\mu$m~(black lines) and $6$ cm (white lines) overlaid on the $154$~$\mu$m~total intensity (color scale). All figures share the same colorscale as shown in the colorbar of NGC~6946. Polarization measurements per beam (red circle) with $PI/\sigma_{PI} \ge 3.0$ are shown, where $PI$ and $\sigma_{PI}$ are the polarized intensity and its uncertainty, respectively.
\label{fig:fig3}}
\end{figure*}
We apply the steps presented in Section \ref{subsec:ModelSteps} to the five galaxies shown in Figure \ref{fig:fig3}. For each galaxy, we select data with $I/\sigma_{I} \ge 10$, where $I$ and $\sigma_{I}$ are the Stokes $I$ and its uncertainty, respectively. Table \ref{tab:GalaxySample} shows the galaxy’s inclination, $i$, and tilt, $\theta$, angles used to define the projected annuli. We calculate the radial profiles selecting the width of each annulus to be equal to the beam size of the HAWC+ observations, i.e., $13\farcs6$ ($2$ pixels). The core ($2$ beams = $27\farcs2$ = $4$ pixels) of each galaxy is masked because of the limited number of independent measurements in that innermost region. Each decomposition is centered at the location of the peak total intensity of the radio emission. All of these galaxies have an unresolved core at radio wavelengths, while the FIR emission shows an extended core (e.g., M83) or a dearth of central emission (e.g., M51). We test the robustness of the central coordinate selection by varying the central coordinates by $\pm1$ pixel in all directions. Specifically, we moved the central coordinates by $\pm1$ pixel and estimated a mean error $<10^{\circ}$ at FIR wavelengths and $<4^{\circ}$ at radio wavelengths in the final pitch angles, $\Psi_{2}$, across the entire disk. We show the results (Figures \ref{fig:fig4} and \ref{fig:fig5}) out to the largest radius where we can measure an uncertainty on $\Psi_{2}$ $\le30^{\circ}$.
\begin{figure*
\includegraphics[width=\textwidth]{m51.png}\
\includegraphics[width=\textwidth]{m83.png}\
\caption{Linear decomposition results of the FIR and radio B-fields for M51 and M83. The titles of each galaxy show the intensity map at FIR wavelengths to guide the reader with the morphological structure of the galaxy. We show the FIR and radio magnetic pitch angles, $\Psi_2$, as a function of the galactocentric radius with the middle solid line corresponding to the mean values and the width of the line corresponding to $\pm 1\sigma$ (Left). Note that for M51, we have included $\angle \beta_0$ which is the angle associated with the $m = 0$ mode. The center panel shows the relative amplitude of each mode as a function of distance from the galactic center at FIR wavelengths with the width of each band corresponding to the value of the mode's relative amplitude. Additionally, on the top of each band for each mode are error bars, showing $\pm 1\sigma$. The right panel shows the same but at radio wavelengths.
\label{fig:fig4}}
\end{figure*}
\begin{figure*}[ht!]
\includegraphics[width=\textwidth]{NGC3627.png}\
\includegraphics[width=\textwidth]{NGC4736.png}\
\includegraphics[width=\textwidth]{NGC6946.png}
\caption{As in Figure \ref{fig:fig4} with NGC~3627, NGC~4736, and NGC~6946.
\label{fig:fig5}}
\end{figure*}
\subsubsection{Amplitudes}\label{subsubsec:amplitudes}
We compute $\beta_{m}$ for the $-3 \le m \le 3$ modes and show their relative amplitudes as a function of the radii. The fractional amplitude per annulus, per galaxy is
\begin{equation}
\tilde{|\beta_{m}|} = \frac{|\beta_{m}|}{\sum_{m=-3}^{m=3}|\beta_{m}|},
\end{equation}
where the uncertainties are estimated using the Monte Carlo approach described in Section \ref{subsec:ModelSteps}. We can further average over all annuli and all galaxies to estimate the mean relative mode amplitude of our spiral galaxies (Figure \ref{fig:fig6} and Table \ref{tab:beta_m}).
At radio ($6$ cm) wavelengths, we find that $m=2$ is the most dominant mode for our spiral galaxies. We estimate that the dominant $m=2$ mode at $6$ cm has a mean relative contribution of $\tilde{|\beta_{2}|} = 0.39\pm0.04$, averaged over annuli. The B-field modes with $m=0$, $m=3$, and $m=1$ have similar relative contributions to the radio polarization of $\tilde{|\beta_{0}|} = 0.14\pm0.03$, $ \tilde{|\beta_{3}|} = 0.12\pm0.02$, and $ \tilde{|\beta_{1}|}= 0.11\pm0.02$, respectively. All negative modes have relative amplitudes $<0.1$ in the radio, but together they account for $\tilde{|\beta_{m<0}|} = 0.23\pm0.04$ of the total average mode amplitude.
At FIR ($154~\mu$m) wavelengths, $m=2$ and $m=0$ have similar relative contributions of $\tilde{|\beta_{2}|} = 0.18\pm0.04$ and $\tilde{|\beta_{0}|} = 0.18\pm0.03$ averaged over the full disk. The modes $m=1$ and $m=3$ have the same relative amplitude of $0.13$. The negative modes have relative amplitudes in the range of $0.10-0.12$, and they sum to $|\tilde{\beta}_{m<0}| = 0.33\pm0.05$ of the total average FIR polarization mode amplitude. The FIR polarization data thus show significantly weaker mode preference compared to the radio polarization data in general. This is evident in the individual galaxy mode decompositions (Figures \ref{fig:fig4} and \ref{fig:fig5}), as well as the mean relative mode contribution for each data set (Figure \ref{fig:fig6}).
NGC~6946 displays the most extreme difference between the mode decomposition in the radio and the FIR data (Figure \ref{fig:fig5}, last row). In the radio, the $m=2$ mode is strongly dominant, with $\tilde{|\beta_{2}|}= 0.42\pm0.07$. By contrast, all modes contribute roughly equally to the FIR polarization ($\tilde{|\beta_{m}|} = [0.12-0.17]$). This is perhaps not surprising because of the galaxies in our sample, NGC~6946 has the most irregular B-field morphology in the FIR. We recalculate the mean relative mode amplitudes for the four galaxies excluding NGC~6946, and we find that the Table \ref{tab:beta_m} values do not change within the stated uncertainties.
\begin{figure}[ht!]
\includegraphics[width=\columnwidth]{fig6.png}
\caption{Mean relative amplitudes of the B-field modes of a composited spiral galaxy. FIR (red) and radio (blue) relative amplitudes for modes $-3\le m \le 3$ are shown. The B-field pattern associated with each mode is shown at the top.
\label{fig:fig6}}
\end{figure}
\begin{deluxetable}{lccccc}
\tablecaption{Mean relative amplitudes of a spiral galaxy. \emph{Columns, from left to right:}
(a) Mode.
(b) FIR relative amplitude.
(c) Radio relative amplitude. The errors represent the standard deviation of the pitch angle profile, not the uncertainties of the average value.
\label{tab:beta_m}
}
\tablecolumns{9}
\tablewidth{0pt}
\tablehead{\colhead{Mode} & \colhead{$\langle |\tilde{\beta_{m}^{\rm{FIR}}} |\rangle$} & \colhead{$\langle |\tilde{\beta_{m}^{\rm{Radio}}} |\rangle$}
}
\startdata
3 & $0.13\pm0.03$ & $0.12\pm0.02$ \\
2 & $0.18\pm0.04$ & $0.39\pm0.04$ \\
1 & $0.13\pm0.02$ & $0.11\pm0.02$ \\
0 & $0.18\pm0.03$ & $0.14\pm0.03$ \\
-1 & $0.11\pm0.02$ & $0.09\pm0.02$ \\
-2 & $0.12\pm0.02$ & $0.08\pm0.02$ \\
-3 & $0.10\pm0.03$ & $0.06\pm0.02$
\enddata
\end{deluxetable}
\subsubsection{Magnetic pitch angles}\label{subsubsec:pitchangles}
The magnetic pitch angle, $\Psi_{2}$, is the complementary angle to the $\angle \beta_{2}$ angle estimated by the B-field mode decomposition method (Figure \ref{fig:fig2}). We show radial profiles of the pitch angles for each galaxy in Figures \ref{fig:fig4} and \ref{fig:fig5}. We also estimate the mean pitch angles, $\langle \Psi_{2} \rangle$, per galaxy at FIR and radio wavelengths (Table \ref{tab:pB}).
We estimate that the mean pitch angle at FIR wavelengths is smaller than at radio wavelengths (Table \ref{tab:pB}), i.e., $\tilde{\langle |\Psi_{2}^{\rm{FIR}}|\rangle} < \tilde{\langle |\Psi_{2}^{\rm{Radio}}|\rangle}$. This result indicates that radio spiral B-fields are more open than FIR spiral B-fields in our sample. In addition, the FIR wavelengths have pitch angles with a larger angular dispersion, $\pm24^{\circ}$, than at radio wavelengths, $\pm8^{\circ}$. This result shows that radio spiral B-fields are more ordered than FIR spiral B-fields.
At radio ($6$ cm) wavelengths, we estimate that $\Psi_{2}^{\rm{Radio}}$ increases as the galaxy radius increases. This result is consistent with the literature \citep{Beck2019} and implies that $\Psi_{2}^{\rm{Radio}}$ opens up toward the outskirts of the disk. It is interesting to note the drastic change of the pitch angle from $-30^{\circ}$ to $-10^{\circ}$ at a projected radial distance of $\sim2.5$ kpc in M83 (Figure \ref{fig:fig4}). At $\sim2.5$ kpc, the pitch angle varies in the interface between the bar region and the spiral arms as determined from the velocity fields of gas tracers (i.e., HII, CO) and stellar dynamics \citep{Kenney1991}.
At FIR ($154$ $\mu$m) wavelengths, $\Psi_{2}^{\rm{FIR}}$ shows many variations as a function of the galactrocentric radius, although in all cases except NGC~6946 there is a large-scale spiral ordered B-field evident in the FIR polarization. We find that at certain radii (e.g. $2.5-3.5$ kpc) in NGC~6946 the FIR B-field has similar pitch angles to those measured at radio wavelengths (Fig. \ref{fig:fig5}). However, the angular dispersion of NGC~6946 is large, $\pm41^{\circ}$, across the disk when compared to the radio B-fields, $\pm5^{\circ}$ (Table \ref{tab:pB}).
\begin{deluxetable*}{lcccccl}
\tablecaption{Mean magnetic pitch angle per galaxy and wavelength from this work vs. literature. \emph{Columns, from left to right:}
(a) Galaxy name.
(b) FIR magnetic pitch angle.
(c) Radio magnetic pitch angle.
(d) Range of radio magnetic pitch angle.
(e) Radio magnetic pitch angles of the ordered B-fields from the literature.
(f) Range of radio magnetic pitch angles of the ordered B-fields from the literature.
(g) References of (d, e). In this table, the errors represent the standard deviation of the pitch angle profile, not the uncertainties of the average value.
\label{tab:pB}
}
\tablecolumns{9}
\tablewidth{0pt}
\tablehead{\colhead{Galaxy} &
\colhead{$\langle \Psi_{2}^{\rm{FIR}} \rangle$} &
\colhead{$\langle \Psi_{2}^{\rm{Radio}} \rangle$} &
\colhead{$\Psi_{2}^{\rm{Radio}}$} &
\colhead{$\langle p_{\rm{o}}^{\rm{Radio}} \rangle$} &
\colhead{ $p_{\rm{o}}^{\rm{Radio}}$} &
\colhead{References} \\
& \colhead{($^{\circ}$)} & \colhead{($^{\circ}$)} & \colhead{($^{\circ}$)} \\
\colhead{(a)} & \colhead{(b)} & \colhead{(c)} & \colhead{(d)} & \colhead{(e)} & \colhead{(f)} & \colhead{(g)} }
\startdata
M51 & $25\pm17$ & $28\pm5$ & $[16-34]$ & $22\pm2$ & $[19-27]$ & \citet{Fletcher2011}\\
M83 & $-41\pm17$ & $-29\pm8$ & $-[12-36]$ & $-30\pm3$ & $-[23-35]$ & \citet{Beck2019}\\
NGC~3627 & $4\pm17$ & $49\pm8$ & $[41-51]$ & $37\pm8$ & $[16-68]$ & \citet{Soida2001}\\
NGC~4736 & $-21\pm6$ & $-30\pm3$ & $-[24-32]$ & $-35\pm5$ & - & \citet{Chyzy2008} \\
NGC~6946 & $6\pm41$ & $-17\pm5$ & $-[8-25]$ & $-27\pm2$ & $[30-32]$ & \citet{Ehle1993,Beck2019}\\
$\langle\tilde{|\Psi_{2}|\rangle}$ & $21\pm24$ & $29\pm8$ & - & $30\pm7$ & -
\enddata
\end{deluxetable*}
\section{Discussion}\label{sec:DIS}
\subsection{Galactic dynamo}\label{subsec:GalDyn}
We quantified the large-scale ordered B-fields in the disk of galaxies as a linear combination of axisymmetric modes. We found that B-fields with $m=2$ (spiral pattern) modes dominate at radio wavelengths, while $m=2$ and $m=0$ (constant) modes have similar contributions at FIR wavelengths. The FIR data show a larger relative contribution from higher modes than the radio wavelengths. We discuss these measurements in the context of galactic dynamo theory.
Turbulent dynamo theory explains the measured B-fields as a combination of fluctuation (or small-scale) dynamos and mean-field (or large-scale) dynamos \citep{Subramanian1998,BS2005,ss21}. In this picture, the large-scale B-fields are generated by differential rotation of the galaxy disk and turbulent helical motions. The turbulent B-fields are generated by turbulent gas motions at scales $\lesssim 50-100$ pc scales of energy injection by supernova explosions and stellar feedback \citep{Ruzmaikin1988,BS2005,Haverkorn2008}. Present-day FIR and radio polarimetric observations \citep[][]{Beck2019,SALSAIV} with spatial resolutions of $\ge100$ pc cannot resolve the turbulent B-fields in galaxies. The measured B-fields are dominated by the large-scale B-fields, although angular fluctuations of polarimetric properties across the disk can be estimated and compared to expectations for sub-beam-scale physics like star formation, shear, and shocks.
The total B-field can be described as the sum of a regular (or coherent) component and a random component \citep[e.g.,][]{Haverkorn:2015,Beck2019}. A well-defined B-field direction within the beam size of the observations is described as a regular B-field. The random B-field component may have spatial reversals within the beam of the observations, which can be isotropic or anisotropic. The directions of the isotropic random B-fields have the same dispersion in all spatial dimensions. An anisotropic random B-field has a well-defined average orientation in addition to sub-beam-scale reversals. Observationally, the combination of anisotropic and regular B-fields is known as ordered B-fields. Polarized radio synchrotron emission traces the ordered (regular and anisotropic random) B-fields in the plane of the sky, which depends on the strength and geometry of the B-fields, and the cosmic ray electron density. Regular B-fields can only be traced using Faraday rotation measures, which are sensitive to the direction of the B-field along the LOS. Polarized FIR dust emission is sensitive to the density-weighted line-of-sight average of the plane-of-sky B-field orientation, in addition to dust properties like column density and temperature. In this work, we have measured and quantified the large-scale ordered B-fields in spiral galaxies traced by both FIR and radio polarimetry.
Other works have analyzed the regular B-field structure of galaxies using linear models of the mean-field galactic dynamo \citep[e.g.,][]{KW1991,Berkhuijsen1997,Fletcher2004,Fletcher2011}. These models assume an expanded B-field pattern in Fourier series in the azimuthal angle. Each mode is a logarithmic spiral with a constant magnetic pitch angle with the sum of all modes providing a non-axisymmetric B-field. The radio B-field orientations, corrected for Faraday rotation, are fitted using a linear superposition of logarithmic spiral B-fields in three dimensions. The azimuthal wave number $m_{\rm{d}} = 0$ is an axisymmetric B-field with constant B-field direction, $m_{\rm{d}} = 1$ is a bisymmetric B-field with two opposite spiral B-field directions, and $m_{\rm{d}} = 2$ is a quadrisymmetric B-field with alternating B-field directions. Most of the studied galaxies in radio polarimetric observations are dominated by $m_{\rm{d}}=0$ \citep{Beck2019}, although higher modes are sometimes required \citep[e.g.,][]{Ehle1993,Rohde1999}. $m_{\rm{d}}>2$ cannot be studied with the spatial resolutions provided by current radio polarimetric observations because they are not sensitive to small spatial variations of the B-field direction. For some galaxies, any combination of modes provides a good fit for the B-field orientations \citep[][table 6]{Beck2019}. For the galaxies and wavelengths in our sample, this approach finds that the disk of M51 is dominated by $m_{\rm{d}}=0$ and $m_{\rm{d}}=2$ with a relative amplitude of $0.72\pm0.06$ \citep{Fletcher2011}. The halo is dominated by $m_{\rm{d}} =1$ and also has $m_{\rm{d}}=2$ with a relative amplitude of $0.30\pm0.09$. M83 is dominated by $m_{\rm{d}}=1$ and has $m_{\rm{d}}=0$ with a relative amplitude of $0.43\pm0.3$ \citep{Beck2019}. NGC~6946 has similar contributions of $m_{\rm{d}}=0$ and $m_{\rm{d}}=2$ \citep{Ehle1993,Rohde1999}. The rotation measures of NGC~3627 did not show distinguishable patterns, which was attributed to Faraday rotation from an extended hot, low-density ionized magnetized halo \citep{Soida2001}.
Table \ref{tab:pB} shows a comparison of the magnetic pitch angles between our measurements, $\Psi_{2}$, and the literature, $p_{\rm{o}}$. We show the mean and ranges of the ordered B-field pitch angles estimated using the B-field orientations from radio polarimetric observations obtained from the literature. The range of $p_{\rm{o}}$ in some of the galaxies shows the minimum and maximum values of the literature magnetic pitch angles within the galactocentric radii used in our analysis. We see that $p_{\rm{o}}$ is similar to our measured $\langle \Psi_{2}^{\rm{Radio}} \rangle$. They are not equal because a) $p_{\rm{o}}$ is affected by several B-field modes, and b) we only used the LOS that have high-SNR measurements in both the FIR and the radio data. Despite that the LOS associated with the FIR measurements is not well-sampled in the inter-arm regions (Figure \ref{fig:fig3}), the mean magnetic pitch angles across the disk at radio wavelengths are similar using both methods, $\langle \tilde{\Psi_{2}^{\rm{Radio}}} \rangle = 29\pm8^{\circ}$ vs. $\langle \tilde{p_{\rm{o}}} \rangle = 30\pm7^{\circ}$. This result implies that the radio B-fields in the arms are very similar to the B-fields dominating in the interarms, although the arms have larger contributions from star formation activity.
Galactic dynamo models provide the azimuthal wave number, $m_{\rm{d}}$, and their associated pitch angles for the disk B-field and the helical B-field. Because the large-scale regular B-field is modeled as a linear combination of logarithmic spiral modes, all dynamo modes are equal to our $m=2$ (spiral B-field). Although $m=2$ is dominant, our method shows that the measured B-field is a combination of several B-field patterns (Figure \ref{fig:fig6}). These modes (and the combination of them) can be interpreted as non-axisymmetric ordered B-fields showing deviations from the large-scale spiral ordered B-fields, perhaps due to particular physics (e.g., star-forming regions, shearing, compression, and/or shocks) across the disk. We note that the rotation measure distribution across a galaxy can also be used to measure pitch angles and B-field direction, providing complementary information \citep{Beck2019}.
\subsection{Comparison with geometrical models}\label{subsec:Comp}
The B-field morphologies in galaxies have also been quantified using pure geometrical models. We describe these methods and discuss their advantages and caveats.
\textit{Axisymmetric B-fields}: This approach estimates the pixel-by-pixel pitch angles across the galaxy disk. The measured B-field orientations are reprojected and tilted to obtain a face-on view of the galaxy, and an azimuthal template is subtracted from the data. The radial pitch angles are estimated as the mean of the pitch angles at a given annulus. This method was developed by \citet{SALSAI} and applied to the same M51 observations presented here. The advantages of this method are that a) the pitch angles on a pixel-to-pixel basis can be estimated, and b) the pitch angles are estimated without prior assumptions about the morphology of the B-field pattern. The pixel-by-pixel map can be used to estimate the means of the magnetic pitch angles from specific areas of the disk, like spiral arms or inter-arm regions, by applying user-defined masks \citep{SALSAI}. The disadvantage is that the angular offsets between the measured B-field orientations and the azimuthal profiles are assumed to be the magnetic pitch angles. The B-field modes cannot be estimated.
\textit{Three-dimensional axisymmetric spiral B-fields}: This approach uses a three-dimensional regular B-field model with an axisymmetric spiral B-field configuration. This B-field model is a mode of a galactic dynamo with a symmetric spiral pattern in the galactic midplane with a helical component. This method has been used in the FIR polarimetric observations of Centaurus A \citep{ELR2021}, radio polarimetric observations of other galaxies \citep{Braun2010}, and in our Galaxy \citep{RG2010}. The advantages of this method are that the three-dimensional B-field component, and the pixel-by-pixel pitch angles across the disk can be obtained. The disadvantages are that a parametric B-field model has to be assumed, and the best-fit B-field is not unique due to the large number of free parameters and the ambiguity of the three-dimensional information of the B-fields from observations \citep[e.g.,][]{Braun2010,RG2010}. In addition, the B-field modes cannot be estimated.
\begin{figure}[ht!]
\includegraphics[width=\columnwidth]{fig7.png}
\caption{Comparison of magnetic pitch angles measurements, $\Psi_2$, between methods at FIR and radio wavelengths. Our method and the pitch angles by \citet{SALSAI} are shown.
\label{fig:fig7}}
\end{figure}
The aforementioned models provide the magnetic pitch angle from the measured B-field orientations. Note that our estimated $\Psi_{2}$ is the intrinsic magnetic pitch angle associated with a purely spiral axisymmetric B-field. Figure \ref{fig:fig7} shows a comparison of the measured magnetic pitch angles using \textsc{mohawc} by \citet{SALSAI} and our pitch angle, $\Psi_{2}$, for M51. \citet{SALSAI} showed that the radio and FIR polarimetric observations do not necessarily trace the same B-field component. This result is clearly detected at galactrocentric distances of $r\ge5$ kpc in \citet{SALSAI}, but only if the spiral arms are analyzed separately from the interarm regions. Those authors used a mask to separate the arm and interarm regions based on the total integrated emission (i.e., moment 0) of HI. The measured FIR and radio magnetic pitch angles are identical when the full disk was analyzed at once \citep[][Fig. 7]{SALSAI}. Our new method obtains the same result -- a difference in the FIR and radio pitch angles at large radius -- but without the need to mask the data to separate physical components of the disk. Figure \ref{fig:fig7} emphasizes the potential of our method to characterize B-field morphologies in the multi-phase ISM using a model-independent approach that does not require masking to separate different galactic components.
\subsection{Broader applications}\label{subsec:FutureApps}
The method presented here is adapted from \citet{Palumbo2020}. The polarimetric linear decomposition has been applied to the B-field orientation generated by magnetohydrodynamic accretion disk simulations and proposed as a model-independent approach to measuring the accretion state of the M87 black hole observed with the EHT \citep[see also][]{EHT2021_VII,EHT2021_VIII,Emami:2022}. While we adapted this decomposition and applied to the B-fields of spiral galaxies, our method can also be applied to any vector field where a circle or ellipse is a geometry of particular interest. Apart from galaxies, this method could also be adapted to other ISM morphologies, such as supernova remnants or wind-blown bubbles in star-forming regions \citep[e.g.,][]{Tahani2022}, or radio synchrotron loops \citep[e.g.,][]{Vidal:2015}.
One straightforward extension of the method presented here would be to quantify the morphology of galaxy structure observed via the total intensity distribution at different wavelengths. One very simple approach to perform this analysis would be to compute the spatial gradient of the emission in order to encode morphological information as a vector field. One could then apply this method directly and compare the results to the magnetic field structure. Similarly, one could quantify the intensity morphology using the Hessian, which measures local curvature in the image plane and thus has been widely used for measuring the orientations of filamentary structures in astrophysical observations \citep[e.g.,][]{PlanckCollaborationXXXII:2016}.
Using this approach to compare galaxy structure to galaxy magnetic field structure opens up intriguing possibilities for morphological insights beyond pitch angle comparisons. Here we can draw further upon analogies to the $E/B$ decomposition of the polarization field that is frequently used to characterize CMB polarization and has recently been widely applied to Galactic emission of diverse physical origins \citep[e.g.][]{Clark:2015, Krachmalnicoff:2018, PlanckCollaboration:2016, PlanckCollaboration:2020}. The correlation between the total intensity and $E$- or $B$-mode polarization field in Galactic dust emission is related to the degree of alignment or misalignment of filamentary density structures and the magnetic field \citep{Huffenberger:2020, Clark:2021, Cukierman:2022}. The quantification of these correlations in Galactic dust polarization is clearly extended to the formalism presented here, in order to compute the correlation between various modes of magnetic structure and various tracers of galactic emission structure.
\section{Conclusions}\label{sec:CON}
We have adapted and successfully applied a new model-independent B-field decomposition approach to measure the large-scale ordered B-field orientations associated with five spiral galaxies using FIR and radio polarimetric observations. With radio ($6$ cm) measurements, we found that the B-fields of spiral galaxies were mainly composed of $m = 2$ with additional but subdominant contributions from $m = 0, m = 3,$ and $m = 1$. With FIR ($154$ $\mu$m) measurements, the most dominant modes were $m = 2$ and $m = 0$ with smaller relative contributions from $m = 1$ and $m = 3$. At both radio and FIR wavelengths, the overall contribution of $\tilde{|\beta_{m < 0}|}$ was less than $\tilde{|\beta_{m \ge 0}|}$. NGC~6946 is the extreme case in our sample. In this galaxy, radio measurements still showed $m = 2$ to be dominant, with the rest of the modes contributing roughly the same to the overall B-field orientation. By contrast, in the FIR, no particular mode dominates the B-field structure.
We also found that the mean pitch angle of these galaxies is smaller in the FIR data than in the radio, i.e. $\langle |\Psi_{2}^{\rm{FIR}}| \rangle < \langle |\Psi_{2}^{\rm{Radio}}| \rangle$. If this trend holds, the implication is that radio spiral B-fields are more open than FIR spiral B-fields. Overall, we find that $\Psi_2$ increases with increasing radius at radio wavelengths, meaning that the magnetic field structure opens out toward the outskirts of the galaxy. With FIR wavelengths we found greater angular dispersion than with radio wavelengths, indicating that FIR spiral B-fields are less ordered than radio spiral B-fields.
\begin{acknowledgments}
Based on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 08\_0012 Program. SOFIA is jointly operated by the Universities Space Research Association, Inc. (USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK 0901 to the University of Stuttgart.
\end{acknowledgments}
\vspace{5mm}
\facilities{SOFIA (HAWC+), VLA}
\software{\textsc{aplpy} \citep{aplpy2012,aplpy2019},
\textsc{astropy} \citep{astropy:2022},
\textsc{pandas} \citep{pandas},
\textsc{matplotlib} \citep{matplotlib},
\textsc{scipy} \citep{scipy}}
|
2,869,038,155,678 | arxiv | \section{Introduction}
The subject of this paper is the rigorous derivation of quasistatic evolution models for nonlinearly elastic - finitely plastic plates. The problem of deriving lower dimensional models for thin structures has been intensively studied since the early 90's by means of a rigorous approach based on $\Gamma$-convergence \cite{A-B-P, L-R}. Starting from the seminal paper \cite{FJM}, this approach has led to establish a hierarchy of limit models for plates \cite{FJM, FJM2}, rods \cite{M-M3, M-M, S0, S1}, and shells \cite{F-J-M-M, L-M-P, L-M-P2}, in the stationary framework and in the context of nonlinear elasticity.
More recently, the $\Gamma$-convergence approach to dimension reduction has gained attention also in the evolutionary framework: in nonlinear elasticity \cite{AMM}, crack propagation \cite {B, FPZ}, linearized elastoplasticity \cite{DM, LM, LR}, and delamination problems \cite{MRT}.
In this paper we justify via $\Gamma$-convergence some linearized quasistatic evolution models for a thin plate, whose elastic behaviour is nonlinear and whose plastic response is governed by finite plasticity with hardening. {{We remark that different schools in finite plasticity are still competing and a generally accepted model is still lacking (see e.g. \cite{Ber}). We shall adopt here a mathematical model introduced in \cite{CHM, Men, M}. }}We assume that the reference configuration of the plate is the set
$$\Omega_{\varepsilon}:=\omega\times\big(-\tfrac{\varepsilon}{2},\tfrac{\varepsilon}{2}\big),$$
where $\omega$ is a domain in $\mathbb R^2$ and $\varepsilon>0$ represents the thickness of the plate. {{Following the lines of \cite{Lee} and \cite{Man},}} we consider deformations of the plate $\eta\in W^{1,2}(\Omega_{\varepsilon};\mathbb R^3)$ satisfying the multiplicative decomposition
$$\nabla \eta(x)=F_{el}(x)F_{pl}(x)\quad\text{for a.e. }x\in\Omega_\ep,$$
where $F_{el}\in L^2(\Omega;\mathbb{M}^{3\times 3})$ is the elastic strain, $F_{pl}\in L^2(\Omega_{\varepsilon};SL(3))$ is the plastic strain and $SL(3):=\{F\in\mathbb{M}^{3\times 3}:\, \det F=1 \}$. { To guarantee coercivity in the plastic strain variable, we suppose to be in a hardening regime. More precisely,} the stored energy associated to a deformation $\eta$ and to its elastic and plastic strains is expressed as follows:
\begin{eqnarray*}
\nonumber \cal{E}(\eta,F_{pl})&:=&\intome{W_{el}(\nabla \eta(x) F_{pl}^{-1}(x))}+\intome{W_{hard}(F_{pl}(x))}\\
&=&\intome{W_{el}(F_{el}(x))}+\intome{W_{hard}(F_{pl}(x))},
\end{eqnarray*}
where $W_{el}$ is a frame-indifferent elastic energy density satisfying the standard assumptions of nonlinear elasticity, and $W_{hard}$ describes hardening.
The plastic dissipation is given in terms of a dissipation distance $D:\mathbb{M}^{3\times 3}\times\mathbb{M}^{3\times 3}\to [0,+\infty]$, which is defined via a positively 1-homogeneous potential $H_D$ (see Section \ref{prel}).
We consider a subset $\gamma_d$ of $\partial\omega$ and for every $t\in [0,T]$ we prescribe on $\gamma_d\times \big(-\tfrac{\varepsilon}{2},\tfrac{\varepsilon}{2}\big)$ a boundary datum for the deformations, of the form
$$\pep(t,x):=\Big(\begin{array}{c}x'\\ x_3\end{array}\Big)+\varepsilon^{\alpha-1}\Big(\begin{array}{c}u^0(t,x')\\0\end{array}\Big)+\varepsilon^{\alpha-2}\Big(\begin{array}{c}-x_3\nabla' v^0(t,x')\\v^0(t,x')\end{array}\Big)\quad\text{for every }x=(x',\varepsilon x_3)\in\overline{\Omega}_{\varepsilon},$$
where $\alpha\geq 3$, $u^0\in C^1([0,T];C^1(\overline{\omega};\mathbb R^2))$, $v^0\in C^1([0,T];C^2(\overline{\omega}))$ and $\nabla'$ denotes the gradient with respect to $x'$.
As usual in dimension reduction, we perform a change of variable to state the problem on a fixed domain independent of $\varepsilon$. We consider the set $\Omega:=\omega\times \big(-\tfrac 12,\tfrac 12\big)$ and the map $\psi^{\varepsilon}:\overline{\Omega}\to\overline{\Omega}_{\varepsilon}$, given by
$$\psi^{\varepsilon}(x):=(x',\varepsilon x_3)\quad\text{for every }x=(x',x_3)\in\overline{\Omega}.$$
To deal with the nonlinear structure of the energy, we follow the approach of \cite{FM}: we assume $\pep(t)$ to be a $C^1$ diffeomorphism on $\mathbb R^3$ and we write deformations $\eta\in W^{1,2}(\Omega_{\varepsilon};\mathbb R^3)$ as
$$\eta \circ \psi^{\varepsilon}=\pep(t) \circ z,$$
where $z\in W^{1,2}(\Omega;\mathbb R^3)$ satisfies
$$z(x)=\psi^{\varepsilon}(x)=(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\gamma_d\times \big(-\tfrac 12,\tfrac 12\big).$$
To any plastic strain $F_{pl}\in L^2(\Omega_{\varepsilon};SL(3))$ we associate a scaled plastic strain $P\in L^2(\Omega;SL(3))$ defined as
$$P:=F_{pl}\circ \psi^{\varepsilon}$$
and we rewrite the stored energy as
$$\cal{F}_{\varepsilon}(t,z,P):=\intom{W_{el}(\nabla \pep(t,z(x))\nep z(x))}+\intom{W_{hard}(P(x))}=\frac{1}{\varepsilon}\cal{E}(\eta,F_{pl}),$$
where $\nep z:=(\nabla' z|\tfrac{1}{\varepsilon}\partial_3 z)$.
In this setting, according to the variational theory for rate-independent processes developed in \cite{MM}, a quasistatic evolution for the boundary datum $\pep$ is a function $t\mapsto(z(t),P(t))\in W^{1,2}(\Omega;\mathbb R^3)\times L^2(\Omega;SL(3))$ such that for every $t\in [0,T]$ the following two conditions are satisfied:
\begin{enumerate}
\item[(qs1)] \emph{global stability:} there holds
$$z(t)=\psi^{\varepsilon}\quad\cal{H}^2\text{ - a.e. on }\gamma_d\times \big(-\tfrac 12,\tfrac 12\big)$$ and $(z(t), P(t))$ minimizes
\begin{eqnarray}
\nonumber \cal{F}_{\varepsilon}(t,\tilde{z},\tilde{P})+{\varepsilon^{\alpha-1}}\intom{D(P(t),\tilde{P})},
\end{eqnarray}
among all $(\tilde{z},\tilde{P})\in W^{1,2}(\Omega;\mathbb R^3)\times L^2(\Omega;SL(3))$ such that $\tilde{z}=\psi^{\varepsilon}$ $\cal{H}^2$ - a.e. on $\gamma_d\times \big(-\tfrac 12,\tfrac 12\big)$;
\item[(qs2)] \emph{energy balance:}
\begin{eqnarray}
\nonumber &&\cal{F}_{\varepsilon}(t,z(t),P(t))+{\varepsilon^{\alpha-1}}\cal{D}(P;0,t)\\
\nonumber &&=\cal{F}_{\varepsilon}(0,z(0),P(0))+{\varepsilon^{\alpha-1}}\int_0^t{\intom{E^{\varepsilon}(s):\Big(\nabla \dot{\pep}(s,z(s))(\nabla \pep)^{-1}(s,z(s))\Big)}\,ds}.
\end{eqnarray}
\end{enumerate}
In the previous formula, $\cal{D}(P;0,t)$ is the plastic dissipation in the interval $[0,t]$ (see Section \ref{quas}), $E^{\varepsilon}(t)$ is the stress tensor, defined as
$$E^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-1}}DW_{el}\big(\nabla \pep(t,z(t))\nep z(t)(P)^{-1}(t)\big)\big(\nabla \pep(t,z(t))\nep z(t)(P)^{-1}(t)\big)^T,$$
and $\alpha\geq 3$ is the same exponent as in the expression of the boundary datum.
Our main result is the characterization of the asymptotic behaviour of quasistatic evolutions as $\varepsilon\to 0$. More precisely, in Theorem \ref{cvstress} and Corollaries \ref{sp} and \ref{su} we show that, given a sequence of initial data $(\zep_0,P^{\varepsilon}_0)$ which is compact in a suitable sense, if $t\mapsto (\zep(t),P^{\varepsilon}(t))$ is a quasistatic evolution for the boundary datum $\pep$ (according to (qs1)--(qs2)), satisfying $\zep(0)=\zep_0$ and $P^{\varepsilon}(0)=P^{\varepsilon}_0$, then defining the in-plane displacement
$$u^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-1}}\intt{\Big(\Big(\begin{array}{c}\pep_1(t,\zep(t))\\\pep_2(t,\zep(t))\end{array}\Big)-x'\Big)},$$
the out-of-plane displacement
$$v^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-2}}\intt{\pep_3(t,\zep(t))}$$
and the scaled linearized plastic strain
$$p^{\varepsilon}(t):=\frac{P^{\varepsilon}(t)-Id}{\varepsilon^{\alpha-1}},$$
for every $t\in [0,T]$ we have
\begin{equation}
\nonumber
p^{\varepsilon}(t)\to p(t)\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}),
\end{equation}
where $p(t)\in L^2(\Omega;\mathbb{M}^{3\times 3})$ with $\tr{p(t)}=0$ a.e. in $\Omega$. If $\alpha>3$ there hold
\begin{eqnarray}
&&\label{1ci}u^{\varepsilon}(t)\to u(t)\quad\text{strongly in }W^{1,2}(\omega;\mathbb R^2),\\
&&\label{2ci}v^{\varepsilon}(t)\to v(t)\quad\text{strongly in }W^{1,2}(\omega),
\end{eqnarray}
for every $t\in [0,T]$, where $u(t)\in W^{1,2}(\omega;\mathbb R^2)$ and $v(t)\in W^{2,2}(\omega)$. If $\alpha=3$, the convergence of the in-plane and the out-of-plane displacements holds only on a $t$-dependent subsequence. Moreover, $t\mapsto(u(t),v(t),p(t))$ is a solution of the following reduced quasistatic evolution problem: for every $t\in [0,T]$
\begin{enumerate}
\item[(qs1)$_{r\alpha}$]\emph{reduced global stability:}
$$u(t)=u^0(t),\quad v(t)=v^0(t),\quad \nabla' v(t)=\nabla' v^0(t)\quad\cal{H}^1\text { - a.e. on }\gamma_d$$
and $(u(t), v(t), p(t))$ minimizes
$$\intom{Q_2\big(\mathrm{sym} \nabla' \tilde{u}-x_3(\nabla')^2 \tilde{v}+\tfrac {L_{\alpha}}{2} \nabla' \tilde{v}\otimes \nabla' \tilde{v}-\tilde{p}'\big)}+\intom{\B{\tilde{p}}}+\intom{H_D(\tilde{p}-p(t))}$$
among all triples $(\tilde{u},\tilde{v},\tilde{p})\in W^{1,2}(\omega;\mathbb R^2)\times W^{2,2}(\omega)\times L^2(\Omega;\mathbb{M}^{3\times 3})$, such that $\tr{\tilde{p}}=0$ a.e. in $\Omega$, and
$$\tilde{u}=u^0(t),\quad \tilde{v}=v^0(t)\quad\text{and}\quad \nabla' \tilde{v}=\nabla' v^0(t)\quad\cal{H}^1\text{ - a.e. on }\gamma_d;$$
\item[(qs2)$_{r\alpha}$]\emph{reduced energy balance:}
\begin{eqnarray*}
&&\intom{Q_2(e_{\alpha}(t))}+\intom{\B{p(t)}}+\cal{D}_{H_D}(p;0,t)=\intom{Q_2(e_{\alpha}(0))}+\intom{\B{p(0)}}\\
&&+\int_0^t{\intom{\mathbb C_2e_{\alpha}(s):\Big(\begin{array}{cc}\nabla' \dot{u}^0(s)+L_{\alpha}\nabla' \dot{v}^0(s)\otimes\nabla' v(s)-x_3(\nabla')^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)}\,ds}
\end{eqnarray*}
\end{enumerate}
where
$$e_{\alpha}(t):=\mathrm{sym} \nabla' u(t)-x_3(\nabla')^2 v(t)+\tfrac{L_{\alpha}} {2} \nabla' v(t)\otimes \nabla' v(t)-p'(t)\quad\text{for every }t\in [0,T]$$
and $$L_{\alpha}:=\begin{cases}0&\text{if }\alpha>3,\\
1&\text{if }\alpha=3.\end{cases}$$
In the above formulas, $\tilde{p}'$ and $p'(t)$ are the $2\times 2$ minors of $\tilde{p}$ and $p(t)$ given by the first two rows and columns, $\nabla'$ denotes the gradient with respect to $x'$, $Q_2$ and $B$ are two symmetric, positive definite quadratic forms on $\mathbb M^{2\times 2}$ and $\mathbb{M}^{3\times 3}$, respectively, for which an explicit formula is provided (see Sections \ref{prel} and \ref{comp}), $\mathbb C_2$ is the tensor associated to $Q_2$ and $\cal{D}_{H_D}$ is the plastic dissipation in the interval $[0,t]$ for the reduced model (see \eqref{NUMpag13}). \\
We remark that Theorem \ref{cvstress} is only a convergence result. In fact, the issue of the existence of a quasistatic evolution in finite plasticity according to (qs1)--(qs2), is quite delicate, and it has only recently been solved in \cite{MM09} by adding to the stored-energy functional some further regularizing terms in the plastic component. We shall not add these further terms here, we rather show, in the last section, that our convergence result can be extended to sequences of approximate discrete-time quasistatic evolutions, whose existence is always guaranteed (see Theorem \ref{cvapp}). The limit quasistatic evolution problem identified in (qs1)$_{r\alpha}$--(qs2)$_{r\alpha}$, on the other hand, has always a solution (see Remark \ref{csd}).\\
The constant $L_{\alpha}$ in the limit problem encodes the main differences between the cases $\alpha>3$ and $\alpha=3$. Indeed, for $\alpha=3$, the limit energy contains the nonlinear term $\tfrac12 \nabla' v\otimes \nabla' v$, which is related to the stretching due to the out-of-plane displacement. For $\alpha>3$ the limit problem is completely linearized and, in the absence of hardening, coincides with that identified in \cite{DM} starting from three-dimensional linearized plasticity. However, we point out that the role of the hardening term in the present formulation is fundamental to deduce compactness of the three-dimensional evolutions (see Step 1, Proof of Theorem \ref{cvstress}).
The limit stored energy and the limit plastic dissipation potential have both been deduced in the static case by $\Gamma$-convergence arguments. Indeed, in the absence of plastic deformations $(p=0)$ the stored energy reduces to the Von K\'arm\'an functional for $\alpha=3$ and to the linear plate functional for $\alpha>3$, which have been rigorously justified via $\Gamma$-convergence in \cite{FJM2} as low energy limits of three-dimensional nonlinear elasticity. In the case where plastic deformation is allowed, the energy in (qs1)$_{r\alpha}$ has been obtained in \cite{D1} as $\Gamma$-limit of a suitable scaling of the three-dimensional energy in (qs1). Our particular choice of the boundary datum and the scaling of the displacements are motivated by these results.\\
The setting of the problem and some arguments in the proofs are close to those of \cite{MS}. In particular, the proof of Theorem \ref{cvstress} follows along the general lines of \cite{MRS}, where an abstract criterion for convergence of quasistatic evolutions is provided.
A major difficulty in the proof of the reduced energy balance is related to the compactness of the stress tensors $E^{\varepsilon}(t)$. In fact, due to the physical growth assumptions on $W_{el}$, weak $L^2$ compactness of $E^{\varepsilon}(t)$ is in general not guaranteed. However, the sequence of stress tensors satisfies the following properties: there exists a sequence of sets $O_{\varepsilon}(t)$, which converges in measure to $\Omega$, such that on $O_{\varepsilon}(t)$ the stresses $E^{\varepsilon}(t)$ are weakly compact in $L^2$, while in the complement of $O_{\varepsilon}(t)$ their contribution is negligible in the $L^1$ norm. This mixed-type convergence is enough to pass to the limit in the three-dimensional energy balance. This argument of proof is similar to the one used in \cite{M-S} by Mora and Scardia, to prove convergence of critical points for thin plates under physical growth conditions for the energy density.
A further difficulty arises because of the physical growth conditions on $W_{el}$: the global stability (qs1) does not secure that $\zep(t)$ fulfills the usual Euler-Lagrange equations. This is crucial to identify the limit stress tensor. This issue is overcome by proving that $\zep(t)$ satisfies the analogue of an alternative first order condition introduced by Ball in \cite[Theorem 2.4]{B} in the context of nonlinear elasticity, and by adapting some techniques in \cite{M-S}.
Finally, to obtain the reduced global stability condition, we need an approximation result for triples $(u,v,p)\in W^{1,2}(\omega;\mathbb R^2)\times W^{2,2}(\omega)\times L^2(\Omega;\mathbb{M}^{3\times 3})$ such that
\begin{equation}
\label{diri}u=0,\quad v=0,\quad\nabla' v=0\quad\cal{H}^1\text{ - a.e. on }\gamma_d
\end{equation}
in terms of smooth triples. {Arguing as in \cite[Section 3]{DM}, such a density result is proved under additional regularity assumptions on $\partial \omega$ and on $\gamma_d$ (see Lemma \ref{bdc}).\\}
The paper is organized as follows: in Section \ref{prel} we set the static problem and we describe the limit functional. In Section \ref{comp} {we recall the compactness results proved in \cite{D1} and we prove an approximation result for triples $(u,v,p)$ satisfying \eqref{diri}.} Section \ref{quas} concerns the formulation of the quasistatic evolution problems, the statement of the main results of the paper and the construction of the mutual recovery sequence, whereas Section \ref{pquas} is entirely devoted to the proofs of the convergence of quasistatic evolutions. Finally, in Section \ref{appr} we discuss convergence of approximate discrete-time quasistatic evolutions.
\smallskip
\noindent{\bf{Notation}}
We shall write any point $x\in\mathbb R^3$ as a pair $(x',x_3)$, where $x'\in \mathbb R^2$ and $x_3\in\mathbb R$.
We shall use the following notation: given $\varphi:\Omega\to \mathbb R^3$, we denote by $\varphi':\Omega\to \mathbb R^2$ the map
$$\varphi':=\Big(\begin{array}{c}\varphi_1\\\varphi_2\end{array}\Big)$$
and for every $\eta\in W^{1,2}(\Omega)$ we denote by $\nabla'\eta$ the vector $\Big(\begin{array}{c}\partial_1 \eta\\\partial_2\eta\end{array}\Big)$.
Analogously, given a matrix $M\in \mathbb{M}^{3\times 3}$, we use the notation $M'$ to represent the minor
$$M':=\Big(\begin{array}{cc}M_{11}&M_{12}\\mathbb M_{21}&M_{22}\end{array}\Big).$$
\section{Preliminaries and setting of the problem}
\label{prel}
Let $\omega\subset \mathbb R^2$ be a connected, bounded open set with { $C^2$} boundary. Let $\varepsilon>0$. We assume that the set $\Omega_{\varepsilon}:=\omega\times\big(-\tfrac \ep2,\tfrac \ep2\big)$ is the reference configuration of a finite-strain elastoplastic plate, and every deformation $\eta\in W^{1,2}(\Omega_\ep;\mathbb R^3)$ fulfills the multiplicative decomposition
$$\nabla \eta(x)=F_{el}(x)F_{pl}(x)\quad\text{for a.e. }x\in\Omega_\ep,$$
where $F_{el}\in L^2(\Omega_\ep;\mathbb{M}^{3\times 3})$ represents the elastic strain, $F_{pl}\in L^2(\Omega_\ep;SL(3))$ is the plastic strain and $SL(3):=\{F\in\mathbb{M}^{3\times 3}: \det F=1\}.$ The stored energy (per unit thickness) associated to a deformation $\eta$ and to its elastic and plastic strains can be expressed as follows:
\begin{eqnarray}
\nonumber \cal{E}(\eta,F_{pl})&:=&\intome{W_{el}(\nabla \eta(x) F_{pl}^{-1}(x))}+\intome{W_{hard}(F_{pl}(x))},\\
\label{nsenergy}&=&\intome{W_{el}(F_{el}(x))}+\intome{W_{hard}(F_{pl}(x))}
\end{eqnarray}
where $W_{el}$ is the elastic energy density and $W_{hard}$ describes hardening.\\
{\bf Properties of the elastic energy}\\
We assume that $W_{el}:\mathbb{M}^{3\times 3}\to [0,+\infty]$ satisfies
\begin{itemize}
\item[(H1)] $W_{el}\in C^1(\mathbb{M}^{3\times 3}_+),\quad W_{el}\equiv +\infty \text{ on }\mathbb{M}^{3\times 3}\setminus \mathbb{M}^{3\times 3}_+$,
\item [(H2)] $W_{el}(Id)=0,$
\item [(H3)] $W_{el}(RF)=W_{el}(F)\quad\text{for every }R\in SO(3),\, F\in \mathbb{M}^{3\times 3}_+,$
\item [(H4)] $W_{el}(F)\geq c_1 \mathrm{dist}^2(F;SO(3))\quad\text{for every }F\in \mathbb{M}^{3\times 3}_+,$
\item [(H5)] $|F^T DW_{el}(F)|\leq c_2 (W_{el}(F)+1) \quad\text{for every }F\in \mathbb{M}^{3\times 3}_+.$
\end{itemize}
Here $c_1,c_2$ are positive constants, $\mathbb{M}^{3\times 3}_+:=\{F\in\mathbb{M}^{3\times 3}:\, \det F>0\}$ and $SO(3):=\{F\in\mathbb{M}^{3\times 3}_+:\, F^T F=Id\}$.
We also assume that there exists a symmetric, positive semi-definite tensor $\mathbb C:\mathbb{M}^{3\times 3}\to \mathbb M^{3\times 3}_{sym}$ such that, setting
$$Q(F):=\frac{1}{2}\mathbb C F:F\quad\text{for every }F\in\mathbb{M}^{3\times 3},$$
the quadratic form $Q$ encodes the local behaviour of $W_{el}$ around the identity, namely
\begin{equation}
\label{quadrwel}
\forall \delta>0\,\, \exists c_{el}(\delta)>0 \text{ such that }\forall F\in B_{c_{el}(\delta)}(0)\text{ there holds }|W_{el}(Id+F)-Q(F)|\leq \delta |F|^2.
\end{equation}
\begin{oss}
By \cite[Proposition 1.5]{DL} and by (H3) and (H5), there holds
\begin{equation}
\label{mandel2}
|DW_{el}(F)F^T|\leq c_3(W_{el}(F)+1)\quad\text{for every }F\in\mathbb{M}^{3\times 3}_+,
\end{equation}
where $c_3$ is a positive constant.
Moreover, by (H1) and (H5), there exist $ c_4,\, c_5,\,\gamma > 0$ such that, for every $G_1,\, G_2\in B_{\gamma}(Id)$ and for every $F \in \mathbb{M}^{3\times 3}_+$ the following estimate holds true
\begin{equation}
\label{lemmams}
|W_{el}(G_1FG_2) - W_{el}(F)| \leq c_4(W_{el}(F) + c_5)(|G_1-Id| + |G_2-Id|)
\end{equation}
(see \cite[Lemma 4.1]{MS}).\\
\end{oss}
\begin{oss}
As remarked in \cite[Section 2]{MS}, the frame-indifference condition (H3) yields
$$\mathbb C_{ijkl}=\mathbb C_{jikl}=\mathbb C_{ijlk}\text{ for every }i,j,k,l\in\{1,2,3\}$$
and
$$\mathbb C F=\mathbb C\, (\mathrm{sym}\, F) \quad\text{for every }F\in\mathbb{M}^{3\times 3}.$$
Hence, the quadratic form $Q$ satisfies:
$$Q(F)=Q(\mathrm{sym}\,F)\quad\text{for every }F\in\mathbb{M}^{3\times 3}$$
and by (H4) it is positive definite on symmetric matrices. This, in turn, implies that there exist two constants $r_\C$ and $R_\C$ such that
\begin{equation}
\label{growthcondQ}
r_\C |F|^2\leq Q(F)\leq R_\C |F|^2\quad\text{for every }F\in\mathbb{M}^{3\times 3}_{\mathrm{sym}},
\end{equation}
and
\begin{equation}
\label{growthcondC}
|\mathbb C F|\leq 2R_\C |F| \quad\text{for every }F\in\mathbb{M}^{3\times 3}_{\mathrm{sym}}.
\end{equation}
\end{oss}
\begin{oss}
We note that \eqref{quadrwel} entails, in particular,
$$W_{el}(Id)=0,\quad DW_{el}(Id)=0$$
and
$$\mathbb C =D^2W_{el}(Id),\quad \mathbb C_{ijkl}=\frac{\partial^2 W}{\partial F_{ij}\partial F_{kl}}(Id)\text{ for every }i,j,k,l\in\{1,2,3\}.$$
By combining \eqref{quadrwel} with \eqref{growthcondC} we deduce also that there exists a constant $c_{el_2}$ such that
\begin{equation}
\label{locquad}
|DW_{el}(Id+F)|\leq (2R_\C+1)|F|
\end{equation}
for every $F\in\mathbb{M}^{3\times 3}$, $|F|<c_{el_2}$.
\end{oss}
\noindent{\bf Properties of the hardening functional}\\
We assume that the hardening map $W_{hard}:\mathbb{M}^{3\times 3}\to [0,+\infty]$ is of the form
\begin{equation}
\nonumber
W_{hard}(F):=\begin{cases}\twh(F)&\text{for every }F\in K,\\
+\infty&\text{otherwise}.
\end{cases}
\end{equation}
Here $K$ is a compact set in $SL(3)$ that contains the identity as a relative interior point, and the map $\twh:\mathbb{M}^{3\times 3}\to [0,+\infty)$ fulfills
\begin{eqnarray}
\nonumber &&\twh\text{ is locally Lipschitz continuous},\\
\label{prh3} && \twh(Id+F)\geq c_6 |F|^2\quad\text{for every }F\in\mathbb{M}^{3\times 3},
\end{eqnarray}
where $c_6$ is a positive constant.
We also assume that there exists a symmetric, positive definite tensor $\mathbb{B}:\mathbb{M}^{3\times 3}\to\mathbb{M}^{3\times 3}$ such that, setting
$$B(F):=\frac{1}{2}\mathbb{B}F:F\quad\text{ for every }F\in\mathbb{M}^{3\times 3},$$
the quadratic form $B$ satisfies
\begin{eqnarray}
\nonumber && \forall \delta>0\, \exists c_h(\delta)>0 \text{ such that }\forall F\in B_{c_h(\delta)}(0)\text{ there holds }|\twh(Id+F)-\B{F}|\leq \delta \B{F}.\\
\label{prh4}
\end{eqnarray}
In particular, by the hypotheses on $K$ there exists a constant $c_k$ such that
\begin{eqnarray}
\label{prk1}&& |F|+|F^{-1}|\leq c_k\quad\text{for every }F\in K,\\
\label{prk2}&& |F-Id|\geq \frac{1}{c_k}\quad\text{for every }F\in SL(3)\setminus K.
\end{eqnarray}
Combining \eqref{prh3} and \eqref{prh4} we deduce also
\begin{equation}
\label{grbelowh}
\frac{c_6}{2} |F|^2\leq \B{F}\quad\text{for every }F\in\mathbb{M}^{3\times 3}.
\end{equation}
{\bf Dissipation functional}\\
Denote by $\mathbb M^{3\times 3}_D$ the set of symmetric trace-free matrices, namely
$$\mathbb M^{3\times 3}_D:=\{F\in\mathbb{M}^{3\times 3}_{\mathrm{sym}}: \tr F=0\}.$$
Let $H_{D}:\mathbb M^{3\times 3}_D \to [0,+\infty)$ be a convex, positively one-homogeneous function such that
\begin{equation}
\label{growthh}
r_K |F|\leq H_{D}(F)\leq R_K |F|\quad\text{for every }F\in\mathbb M^{3\times 3}_D.
\end{equation}
We define the dissipation potential $H:\mathbb{M}^{3\times 3}\to [0,+\infty]$ as
\begin{equation}
\nonumber
H(F):=\begin{cases}H_{D}(F)&\text{if }F\in \mathbb M^{3\times 3}_D,\\
+\infty &\text{otherwise.}\end{cases}
\end{equation}
For every $F\in\mathbb{M}^{3\times 3}$ consider the quantity
\begin{equation}
\label{distid}
D(Id,F):=\inf \Big\{\int_0^1{H(\dot{c}(t)c^{-1}(t))\,dt}: c\in C^1([0,1];\mathbb{M}^{3\times 3}_+),\, c(0)=Id,\, c(1)=F \Big\}.
\end{equation}
Note that, by the Jacobi's formula for the derivative of the determinant of a differentiable matrix-valued map, if $D(Id, F)<+\infty$, then $F\in SL(3)$.
We define the dissipation distance as the map $D:\mathbb{M}^{3\times 3}\times \mathbb{M}^{3\times 3}\to [0,+\infty]$, given by
$$D(F_1,F_2):=\begin{cases}D(Id, {F_2}F_1^{-1})& \text{if }F_1\in\mathbb{M}^{3\times 3}_{+}, F_2\in\mathbb{M}^{3\times 3}\\ +\infty& \text{if }F_1\notin \mathbb{M}^{3\times 3}_{+}, F_2\in\mathbb{M}^{3\times 3}.
\end{cases}$$
We note that the map $D$ satisfies the triangle inequality
\begin{equation}
\label{triang}
D(F_1,F_2)\leq D(F_1,F_3)+D(F_3,F_2)
\end{equation}
for every $F_1,F_2, F_3\in\mathbb{M}^{3\times 3}$.
Given a preexistent plastic strain $F_{pl}^0\in L^2(\Omega_{\varepsilon};SL(3))$, we define the plastic dissipation potential associated to a plastic configuration $F\in L^2(\Omega_{\varepsilon};SL(3))$ as
\begin{equation}
\label{dissord}
\varepsilon^{\alpha-1}\intome{D(F_{pl}^0;F)},
\end{equation}
where $\alpha\geq 3$ is a given parameter.
\begin{oss}
We remark that there exists a positive constant $c_7$ such that
\begin{eqnarray}
\label{prd1} &&D(F_1,F_2)\leq c_7\quad\text{for every }F_1,F_2\in K,\\
\label{prd2} && D(Id,F)\leq c_7|F-Id|\quad\text{for every }F\in K.
\end{eqnarray}
Indeed, by the compactness of $K$ and the continuity of the map $D$ on $SL(3)\times SL(3)$ (see \cite{M}), there exists a constant $\tilde{c}_7$ such that
\begin{equation}
\label{quasiprd1}
D(F_1,F_2)\leq \tilde{c}_7\quad\text{for every }F_1,F_2\in K.
\end{equation}
By the previous estimate, \eqref{prd2} needs only to be proved in a neighbourhood of the identity. More precisely, let $\delta>0$ be such that $\log F$ is well defined for $F\in K$ and $|F-Id|<\delta$. If $F\in K$ is such that $|F-Id|\geq \delta$, by \eqref{quasiprd1} we have
$$D(Id,F)\leq \frac{\tilde{c}_7}{\delta}|F-Id|.$$
If $|F-Id|<\delta$, taking $c(t)=\exp({t\log F})$ in \eqref{distid}, inequality \eqref{growthh} yields
$$D(Id,F)\leq H_{D}(\log F)\leq R_K |\log F|\leq C|F-Id|$$
for every $F\in K$. Collecting the previous estimates we deduce \eqref{prd1} and \eqref{prd2}.
\end{oss}
\subsection{Change of variable and formulation of the problem}
{
We suppose that the boundary $\partial\omega$ is partitioned into two disjoint open subsets $\gamma_d$ and $\gamma_n$, and their common boundary $\partial\lfloor_{\partial\omega}\gamma_d = \partial\lfloor_{\partial\omega}\gamma_n$(topological notions refer here to the relative topology of $\partial\omega$). We assume that $\gamma_d$ is nonempty and that $\partial\lfloor_{\partial\omega}\gamma_d= \{P_1, P_2\}$, where $P_1, P_2$ are two points in $\partial\omega$. We denote by $\Gamma_{\varepsilon}$ the portion of the lateral surface of the plate given by $\Gamma_{\varepsilon}:=\gamma_d\times\big(-\tfrac \ep2,\tfrac \ep2\big)$. On $\Gamma_{\varepsilon}$ we prescribe a boundary datum of the form
\begin{equation}
\label{defbddat}
\phi^{\varepsilon}(x):=\Big(\begin{array}{c}x'\\x_3\end{array}\Big)+\Big(\begin{array}{c}\varepsilon^{\alpha-1}u^0(x')\\0\end{array}\Big)+\varepsilon^{\alpha-2}\Big(\begin{array}{c}-x_3\nabla' v^0(x')\\v^0(x')\end{array}\Big)\end{equation}
for every $x=(x',\varepsilon x_3)\in\Omega_{\varepsilon}$, where $u^0\in C^{1}(\overline{\omega};\mathbb R^2)$, $v^0\in C^{2}(\overline{\omega})$ and $\alpha\geq 3$ is the same parameter as in \eqref{dissord}.
We consider deformations $\eta\in W^{1,2}(\Omega_{\varepsilon};\mathbb R^3)$ satisfying \begin{equation}
\label{bddatunsc}
\eta=\pep\quad\cal{H}^2\text{ - a.e. on }\Gamma_{\varepsilon}.
\end{equation}}
As usual in dimension reduction, we perform a change of variable to formulate the problem on a domain independent of $\varepsilon$. We consider the set $\Omega:=\omega \times \big(-\tfrac 12, \tfrac 12\big)$ and the map $\psi^{\varepsilon}:\overline{\Omega}\to \overline{\Omega}_{\varepsilon}$ given by
\begin{equation}
\label{cov}
\psi^{\varepsilon}(x):=(x',\varepsilon x_3)\quad\text{for every }x\in\overline{\Omega}.\end{equation}
To every deformation $\eta \in W^{1,2}(\Omega_\ep;\mathbb R^3)$ satisfying
\eqref{bddatunsc}
and to every plastic strain $F_{pl}\in L^2(\Omega_\ep;SL(3))$, we associate the scaled deformation $y:=\eta\circ \psi^{\varepsilon}$ and the scaled plastic strain $P:=F_{pl}\circ \psi^{\varepsilon}$. Denoting by $\Gamma_d$ the set $\gamma_d\times \big(-\tfrac 12, \tfrac 12\big),$ the scaled deformation satisfies the boundary condition
{\begin{equation}
\label{bddatum}
y=\phi^{\varepsilon}\circ \psi^{\varepsilon}\quad\cal{H}^2\text{ - a.e. on }\Gamma_d.
\end{equation}}
Denote by $\cal{A}_{\varepsilon}(\phi^{\varepsilon})$ the class of pairs $(y^{\varepsilon},P^{\varepsilon})\in W^{1,2}(\Omega;\mathbb R^3)\times L^2(\Omega;SL(3))$ such that \eqref{bddatum} is satisfied.
Applying the change of variable \eqref{cov} to \eqref{nsenergy} and \eqref{dissord}, the energy functional is now given by
\begin{equation}
\label{encov}
\cal{I}(y,P):=\frac{1}{\varepsilon}\cal{E}(\eta,F_{pl})=\intom{W_{el}(\nep y(x) P^{-1}(x))}+\intom{W_{hard}(P(x))},
\end{equation}
where $\nep y(x):=\big(\partial_1 y(x)\big|\partial_2 y(x)\big|\frac{1}{\varepsilon} \partial_3 y(x)\big)$ for a.e. $x\in\Omega$. The plastic dissipation potential is given by
\begin{equation}
\label{disscov}
\varepsilon^{\alpha-1}\intom{D(P^{\varepsilon,0},P)}
\end{equation}
where $P^{\varepsilon,0}:=F_{pl}^0\circ \psi^{\varepsilon}$ is a preexistent plastic strain. We remark here that the asymptotic behaviour of sequences of pairs $(y^{\varepsilon}, P^{\varepsilon})\in\cal{A}_{\varepsilon}(\pep)$ such that
$$\cal{I}(y^{\varepsilon},P^{\varepsilon})+\varepsilon^{\alpha-1}\intom{D(P^{\varepsilon,0},P^{\varepsilon})}$$
is of order $\varepsilon^{2\alpha-2}$ has been studied in \cite{D1} under suitable assumptions on the maps $P^{\varepsilon,0}$.\\
\section{Compactness results}
\label{comp}
In this section we collect two compactness results that were {obtained} in \cite{D1} { and we state an approximation result which will be crucial in the proof of the reduced global stability condition}. In the first theorem, the rigidity estimate proved by Friesecke, James and M\"uller in \cite[Theorem 3.1]{FJM} allow us to
approximate sequences of deformations whose distance of the gradient from $SO(3)$ is uniformly bounded, by means of rotations (see \cite[Theorem 3.3]{D1}).
\begin{teo}
\label{compactbd1}
Assume that $\alpha\geq 3$. Let $(y^{\varepsilon})$ be a sequence of deformations in $W^{1,2}(\Omega;\mathbb R^3)$ satisfying \eqref{bddatum} and such that
\begin{equation}
\label{elasticbd}
\|\mathrm{dist}(\nep y^{\varepsilon}, SO(3))\|_{L^2(\Omega; \mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1}.
\end{equation}
Then, there exists a sequence $(R^{\varepsilon})\subset W^{1,\infty}(\omega; \mathbb{M}^{3\times 3})$ such that for every $\varepsilon>0$
\begin{eqnarray}
&&\label{rt1} R^{\varepsilon}(x')\in SO(3)\quad\text{for every }x'\in \omega,\\
&&\label{rt2} \|\nep y^{\varepsilon}-R^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1},\\
&&\label{rt3} \|\partial_i R^{\varepsilon}\|_{L^2(\omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-2},\,i=1,2\\
&&\label{rt4} \|R^{\varepsilon}-Id\|_{L^2(\omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-2}.
\end{eqnarray}
\end{teo}
Let $\A:\mathbb{M}^{2\times 2}\to \mathbb{M}^{3\times 3}_{\mathrm{sym}}$ be the operator given by
$$\A F:=\Bigg(\begin{array}{cc}\mathrm{sym}\,F&\hspace{-0.5 cm}\begin{array}{c}\lambda_1(F)\vspace{-0.1 cm}\\\lambda_2(F)\end{array}\vspace{-0.1 cm}\\\begin{array}{cc}\lambda_1(F)&\lambda_2(F)\end{array}&\hspace{-0.5 cm}\lambda_3(F)\end{array}\Bigg)\quad\text{for every }F\in\mathbb M^{2\times 2},$$
where for every $F\in\mathbb M^{2\times 2}$ the triple $(\lambda_1(F),\lambda_2(F),\lambda_3(F))$ is the unique solution to the minimum problem
$$\min_{\lambda_i\in\mathbb R}Q\Bigg(\begin{array}{cc}\mathrm{sym}\,F&\hspace{-0.5 cm}\begin{array}{c}\lambda_1\vspace{-0.1 cm}\\\lambda_2\end{array}\vspace{-0.1 cm}\\\begin{array}{cc}\lambda_1&\lambda_2\end{array}&\hspace{-0.5 cm}\lambda_3\end{array}\Bigg).$$
We remark that for every $F\in\mathbb M^{2\times 2}$, $\A(F)$ is given by the unique solution to the linear equation
\begin{equation}
\label{linearmin}
\mathbb C \A (F):\Bigg(\begin{array}{ccc}0&0&\lambda_1\\0&0&\lambda_2\\\lambda_1&\lambda_2&\lambda_3\end{array}\Bigg)=0\quad\text{for every }\lambda_1,\lambda_2,\lambda_3\in\mathbb R.
\end{equation}
This implies, in particular, that $\A$ is linear.
We define the quadratic form $Q_2:\mathbb{M}^{2\times 2}\to [0,+\infty)$ as
$$Q_2(F)=Q(\A (F))\quad\text{for every }F\in \mathbb{M}^{2\times 2}.$$
By properties of $Q$, we have that $Q_2$ is positive definite on symmetric matrices. We also define the tensor $\mathbb C_2:\mathbb{M}^{2\times 2}\to \mathbb{M}^{3\times 3}_{\mathrm{sym}}$, given by
\begin{equation}
\label{defc2}
\mathbb C_2 F:=\mathbb C\A (F)\quad\text{for every }F\in \mathbb{M}^{2\times 2}.
\end{equation}
We remark that by \eqref{linearmin} there holds
\begin{equation}
\label{nothird}
\mathbb C_2 F:G=\mathbb C_2 F:\Big(\begin{array}{cc}\mathrm{sym}\,G&0\\0&0\end{array}\Big)\quad\text{for every }F\in \mathbb{M}^{2\times 2},\,G\in\mathbb{M}^{3\times 3}
\end{equation}
and
$$Q_2(F)=\frac{1}{2}\mathbb C_2 F:\Big(\begin{array}{cc}\mathrm{sym}\,F&0\\0&0\end{array}\Big)\quad\text{for every }F\in \mathbb{M}^{2\times 2}.$$
Given a sequence of deformations $(y^{\varepsilon})\subset W^{1,2}(\Omega;\mathbb R^3)$, we consider some associated quantities: the in-plane displacements
\begin{equation}
\label{inplane}
u^{\varepsilon}(x'):=\frac{1}{\varepsilon^{\alpha-1}}\intt{\big(\m{{y}^{\varepsilon}}(x',x_3)-x'\big)}\quad\text{for a.e. }x'\in \omega,
\end{equation}
and the out-of-plane displacements
\begin{equation}
\label{outofplane}
v^{\varepsilon}(x'):=\frac{1}{\varepsilon^{\alpha-2}}\intt{y^{\varepsilon}_3(x',x_3)}\quad\text{for a.e. }x'\in \omega.
\end{equation}
For every sequence $(y^{\varepsilon})$ in $W^{1,2}(\Omega;\mathbb R^3)$ satisfying both \eqref{bddatum} and \eqref{elasticbd}, we introduce also the strains
\begin{equation}
\label{defgep}
G^{\varepsilon}(x):=\frac{(R^{\varepsilon}(x))^T\nep y^{\varepsilon}(x)-Id}{\varepsilon^{\alpha-1}}\quad\text{for a.e. }x\in\Omega,
\end{equation}
where the maps $R^{\varepsilon}$ are the pointwise rotations provided by Theorem \ref{compactbd1}.
Denote by $\cal{A}(u^0,v^0)$ the set of triples $(u,v,p)\in W^{1,2}(\Omega;\mathbb R^2)\times W^{2,2}(\Omega)\times L^2(\Omega;\mathbb M^{3\times 3}_D)$ such that
$u=u^0,\,v=v^0,\text{ and }\nabla'v=\nabla'v^0$ $\cal{H}^1\text{ - a.e. on }\gamma_d$.
The next theorem allows us to deduce some compactness properties for the displacements and strains and a liminf inequality for the scaled stored energy and plastic dissipation potential, introduced in \eqref{encov} and \eqref{disscov} (see \cite[Theorem 3.4]{D1}).
\begin{teo}
\label{liminfineq}
Assume that $\alpha\geq 3$. Let $(y^{\varepsilon},P^{\varepsilon})$ be a sequence of pairs in $\cal{A}_{\varepsilon}(\phi^{\varepsilon})$ satisfying
\begin{equation}
\label{engest2}
\cal{I}(y^{\varepsilon},P^{\varepsilon})\leq C\varepsilon^{2\alpha-2}
\end{equation}
for every $\varepsilon>0$.
Let $u^{\varepsilon}$, $v^{\varepsilon}$ and $G^{\varepsilon}$ be defined as in \eqref{inplane}, \eqref{outofplane}, and \eqref{defgep}, respectively. Then, there exist $(u,v,p)\in \cal{A}(u^0,v^0)$ such that, up to subsequences, there hold
\begin{eqnarray}
&&\label{cptyep}y^{\varepsilon}\to \Big(\begin{array}{c}x'\\0\end{array}\Big)\quad\text{strongly in }W^{1,2}(\Omega;\mathbb R^3),\\
&&\label{cptuep} u^{\varepsilon}\rightharpoonup u\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2),\\
&&\label{cptvep} v^{\varepsilon}\to v\quad\text{strongly in }W^{1,2}(\omega),\\
&&\label{cptnep3}\frac{\nabla' y^{\varepsilon}_3}{\varepsilon^{\alpha-2}}\to \nabla' v\quad\text{strongly in }L^2(\Omega;\mathbb R^2),
\end{eqnarray}
and the following estimate holds true
\begin{equation}
\label{3comphest}
\big\|\frac{y^{\varepsilon}_3}{\varepsilon}-x_3-\varepsilon^{\alpha-3}v^{\varepsilon}\big\|_{L^2(\Omega)}\leq C\varepsilon^{\alpha-2}.
\end{equation}
Moreover, there exists $G\in L^2(\Omega;\mathbb{M}^{3\times 3})$ such that
\begin{equation}
\label{cptGep}
G^{\varepsilon}\rightharpoonup G\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}),
\end{equation}
and the $2\times 2$ submatrix $G'$ satisfies
\begin{equation}
\label{gaff}
G'(x', x_3) = G_0(x') - x_3 (\nabla')^2 v(x')\quad\text{for a.e. }x\in\Omega,
\end{equation}
where
\begin{eqnarray}
&&\label{Ga3} \mathrm{sym}\, G_0 = \frac{(\nabla' u+(\nabla' u)^T +\nabla' v\otimes \nabla' v)}{2}\quad\text{if } \alpha=3,\\
&&\label{Ga>3} \mathrm{sym}\, G_0 = \mathrm{sym} \nabla' u\quad\text{if }\alpha> 3.
\end{eqnarray}
The sequence of plastic strains $(P^{\varepsilon})$ fulfills
\begin{equation}
\label{Pep1} P^{\varepsilon}(x)\in K\quad\text{for a.e. }x\in\Omega,
\end{equation}
and
\begin{equation}
\label{Pep2} \|P^{\varepsilon}-Id\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1}
\end{equation}
for every $\varepsilon$. Moreover, setting
\begin{equation}
\label{defpep}
p^{\varepsilon}:=\frac{P^{\varepsilon}-Id}{\varepsilon^{\alpha-1}},
\end{equation} up to subsequences
\begin{equation}
\label{wconvpep}
p^{\varepsilon}\rightharpoonup p\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Finally,
\begin{eqnarray}
\label{liminftot}\intom{Q_2(\mathrm{sym}\, G'-p')}+\intom{B(p)}\leq \liminf_{\varepsilon\to 0} \frac{1}{\varepsilon^{2\alpha-2}}\cal{I}(y^{\varepsilon},P^{\varepsilon}).
\end{eqnarray}
If in addition
\begin{equation}
\label{engest3}
\frac{1}{\varepsilon^{\alpha-1}}\intom{D({P}^{\varepsilon,0},{P}^{\varepsilon})}\leq C\quad\text{for every }\varepsilon>0
\end{equation}
and there exist a map $p^0\in L^2(\Omega;\mathbb{M}^{3\times 3}_D)$ and a sequence $({p}^{\varepsilon,0})\subset L^2(\Omega;\mathbb{M}^{3\times 3})$ such that $P^{\varepsilon,0}=Id+\varepsilon^{\alpha-1}p^{\varepsilon,0}$, with ${p}^{\varepsilon,0}\rightharpoonup p^0$ weakly in $L^2(\Omega;\mathbb{M}^{3\times 3})$, then
\begin{equation}
\label{liminfdiss}
\intom{H_{D}({p}-p^0)}\leq \liminf_{\varepsilon\to 0} \frac{1}{\varepsilon^{\alpha-1}}\intom{D({P}^{\varepsilon,0},{P}^{\varepsilon})}.
\end{equation}
\end{teo}
{\begin{proof}
The proof follows easily by adapting \cite[Proof of Theorem 3.4]{D1}.
\end{proof}
{We conclude this section by providing an approximation result for triples $(u,v,p)\in \cal{A}(0,0)$ by means of smooth triples.}
More precisely, denoting by $C^{\infty}_c(\omega\cup\gamma_n)$ the sets of smooth maps having compact support in $\omega\cup\gamma_n$, the following lemma holds true.
\begin{lem}
\label{bdc}
{{
Let $(u,v,p)\in \cal{A}(0,0)$. Then there exists a sequence of triples $(u^{k},v^{k},p^{k})\in C^{\infty}_c(\omega\cup\gamma_n;\mathbb R^2)\times C^{\infty}_c(\omega\cup\gamma_n)\times C^{\infty}_c(\Omega;\mathbb M^{3\times 3}_D)$ such that
\begin{eqnarray*}
&&u^{k}\to u\quad\text{strongly in }W^{1,2}(\omega;\mathbb R^2),\\
&&v^{k}\to v\quad\text{strongly in }W^{2,2}(\omega),\\
&&p^{k}\to p\quad\text{strongly in }L^2(\Omega;\mathbb M^{3\times 3}_D).
\end{eqnarray*}
}} \end{lem}
\begin{proof}
The approximation of the plastic strain $p$ is obtained by standard arguments. The approximation of the in-plane displacements and out-of-plane displacements follows by adapting the arguments in \cite[Theorem 3.9 and Lemma 6.10]{DM}.
\end{proof}
\section{The quasistatic evolution problem}
\label{quas}
In this section we set the quasistatic evolution problem.
For every $t\in [0,T]$ we prescribe a boundary datum $\pep(t)\in W^{1,\infty}(\Omega;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$, defined as
\begin{equation}
\nonumber
\pep(t,x):=\Big(\begin{array}{c}x'\\x_3\end{array}\Big)+\varepsilon^{\alpha-1}\Big(\begin{array}{c}u^0(t,x')\\0\end{array}\Big)+\varepsilon^{\alpha-2}\Big(\begin{array}{c}-x_3\nabla' v^0(t,x')\\v^0(t,x')\end{array}\Big),
\end{equation}
for every $x\in\mathbb R^3$, where the map $t\mapsto u^0(t)$ is assumed to be $C^{1}([0,T];C^1(\mathbb R^2;\mathbb R^2))$ and the map $t\mapsto v^0(t)$ is $C^{1}([0,T];C^2(\mathbb R^2))$. We consider deformations $t\mapsto y^{\varepsilon}(t)$ from $[0,T]$ into $W^{1,2}(\Omega;\mathbb R^3)$ that satisfy
\begin{equation}
\nonumber
y^{\varepsilon}(t,x)=\pep(t,(x',\varepsilon x_3))\quad\cal{H}^2\text{ -a.e. on }\Gamma_d,
\end{equation}
and plastic strains $t\mapsto P^{\varepsilon}(t)$ from $[0,T]$ into $L^2(\Omega;SL(3))$.
For technical reasons, it is convenient to modify the map $t\mapsto\pep(t)$ outside the set $\Omega$. We consider a truncation function $\tep\in W^{1,\infty}(\mathbb R)\cap {C^{1}}(\mathbb R)$ satisfying
\begin{eqnarray}
\label{treq1}&&\tep(s)=s\quad\text{ in }(-\ell_{\varepsilon}, \ell_{\varepsilon}),\\
\label{treq2}&&|\tep(s)|\leq |s|\text{ for every }s\in \mathbb R,\\
\label{treq3}&&\|\tep\|_{L^{\infty}(\mathbb{R})}\leq 2 \ell_{\varepsilon},\\
\label{treq4}&&\dot{\tep}(s)=0\quad\text{ if }|x_3|\geq \ell_{\varepsilon}+1,\\
\label{treq5}&&\|\dot{\tep}(s)\|_{L^{\infty}(\mathbb R)}\leq 2,
\end{eqnarray}
where $\ell_{\varepsilon}$ is such that
\begin{eqnarray}
\label{lp1}&&\varepsilon^{\alpha-1-\gamma}\ell_{\varepsilon}\to 0,\\
\label{lp2}&& \varepsilon \ell_{\varepsilon}\to +\infty,\\
\label{lp3}&&\varepsilon^{2\alpha-2}\ell_{\varepsilon}^3\to 0,
\end{eqnarray}
for some $0<\gamma<\alpha-2$.
For $\alpha>3$ we also require
\begin{equation}
\label{lp4}
\varepsilon^{\alpha-1}\ell_{\varepsilon}^2\to 0.
\end{equation}
\begin{oss}
A possible choice of $\ell_{\varepsilon}$ is $\ell_{\varepsilon}=\frac{1}{\varepsilon^{1+\lambda}}$, with $0<\lambda<\min\{\frac{\alpha-3}{2},\alpha-2-\gamma\}$ when $\alpha>3$ and $0<\lambda<\min\{\frac{1}{3},1-\gamma\}$ in the case $\alpha=3$.
\end{oss}
With a slight abuse of notation, for every $t\in [0,T]$ we still denote by $\pep(t)$ the map defined as
\begin{equation}
\label{defphiep}
\pep(t,x):=\Big(\begin{array}{c}x'\\x_3\end{array}\Big)+\varepsilon^{\alpha-1}\Big(\begin{array}{c}u^0(t,x')-\tep\big(\frac{x_3}{\varepsilon}\big)\nabla' v^0(t,x')\\0\end{array}\Big)+\varepsilon^{\alpha-2}\Big(\begin{array}{c}0\\v^0(t,x')\end{array}\Big)
\end{equation}
for every $x\in\mathbb R^3$.
\begin{oss}
\label{propinverse}
Conditions \eqref{treq1} and \eqref{lp2} guarantee that $\pep(t)$ is indeed an extension of the originally prescribed boundary datum, for $\varepsilon$ small enough. Conditions \eqref{treq3} and \eqref{treq5} provide a uniform bound with respect to $t$ on the $W^{1,\infty}(\mathbb R^3;\mathbb R^3)$ norm of $\pep(t)-id$. By \eqref{treq3}, \eqref{treq5} and \eqref{lp1}, there exists $\varepsilon_0>0$ such that, for every $t\in [0,T]$ and $\varepsilon<\varepsilon_0$, the map $\pep(t):\mathbb R^3\to \mathbb R^3$ is invertible with smooth inverse $\vep(t)$. Since $$\pep(t, \vep(t,x))=x\quad\text{for every }x\in\mathbb R^3,$$ by \eqref{defphiep} there holds
\begin{eqnarray}
\label{fcomp} && \bvep(t)-x'=-\varepsilon^{\alpha-1}u^0(t,\bvep(t))+\varepsilon^{\alpha-1}\tep\Big(\frac{\vep_3(t)}{\varepsilon}\Big)\nabla' v^0(t,\bvep(t)),\\
\label{3comp}&& \vep_3(t)-x_3=-\varepsilon^{\alpha-2}v^0(t,\bvep(t)),
\end{eqnarray}
for every $t\in [0,T]$. Hence, by the smoothness of $u^0$ and $v^0$ and by \eqref{treq3}, we deduce the estimates
\begin{equation}
\label{distinv1}
\|\bvep(t)-x'\|_{L^{\infty}(\mathbb R^3;\mathbb R^2)}\leq C\varepsilon^{\alpha-1}\ell_{\varepsilon},
\end{equation}
and
\begin{equation}
\label{distinv13}
\|\vep_3(t)-x_3\|_{L^{\infty}(\mathbb R^3)}\leq C\varepsilon^{\alpha-2},
\end{equation}
where both constants are independent of $t$. In particular, \eqref{fcomp} yields
\begin{eqnarray}
\nonumber \nabla \bvep(t)-\Big(\begin{array}{ccc}1&0&0\\0&1&0\end{array}\Big)&&=-\varepsilon^{\alpha-1}\nabla' u^0(t,\bvep(t))\nabla \bvep(t)\\
\nonumber &&+\varepsilon^{\alpha-1}\tep\Big(\frac{\vep_3(t)}{\varepsilon}\Big)(\nabla')^2 v^0(t,\bvep(t))\nabla \bvep(t)\\
\label{expgr}&&+\varepsilon^{\alpha-2}\dot{\tep}\Big(\frac{\vep_3(t)}{\varepsilon}\Big)\nabla' v^0(t,\bvep(t))\otimes\nabla \vep_3(t),
\end{eqnarray}
and \eqref{3comp} implies
\begin{equation}
\label{gradinv0}
\nabla \vep_3(t)-e_3=-\varepsilon^{\alpha-2} (\nabla \bvep(t))^T\nabla' v^0(t,\bvep(t)),
\end{equation}
for every $t\in [0,T]$.
A direct computation shows that
\begin{eqnarray}
\nonumber&&\nabla \pep(t,x)=Id+\varepsilon^{\alpha-1}\Big(\begin{array}{cc}\nabla' u^0(t,x')&0\\0&0\end{array}\Big)-\varepsilon^{\alpha-1}\Big(\begin{array}{cc}\tep\big(\frac{x_3}{\varepsilon}\big)(\nabla')^2 v^0(t,x')&0\\0&0\end{array}\Big)\\
\label{gradpep}&&+\varepsilon^{\alpha-2}\Big(\begin{array}{cc}0&-\dot{\tep}\big(\frac{x_3}{\varepsilon}\big)\nabla' v^0(t,x')\\(\nabla' v^0(t,x'))^T&0\end{array}\Big)\quad\text{for every }x\in\mathbb R^3.
\end{eqnarray}
Hence by \eqref{treq3}, \eqref{treq5} and \eqref{lp1} there holds
\begin{equation}
\label{bddinverse}
\|\nabla \vep (t)\|_{L^{\infty}(\mathbb R^3;\mathbb{M}^{3\times 3})}\leq \|(\nabla \pep (t))^{-1}\|_{L^{\infty}(\mathbb R^3;\mathbb{M}^{3\times 3})}\leq C,\end{equation}
for every $t\in[0,T]$ and for every $\varepsilon<\varepsilon_0$. Therefore, \eqref{treq3}, \eqref{treq5}, \eqref{lp2}, \eqref{expgr} and \eqref{gradinv0} yield
\begin{equation}
\label{gradinv}
\|\nabla \bvep(t)-(e_1|e_2|0)\|_{L^{\infty}(\mathbb R^3;\mathbb M^{3\times 2})}\leq C\varepsilon^{\alpha-1}\ell_{\varepsilon},
\end{equation}
and
\begin{equation}
\label{gradinv1}
\|\nabla \vep_3(t)-e_3\|_{L^{\infty}(\mathbb R^3;\mathbb R^3)}\leq C\varepsilon^{\alpha-2}.
\end{equation}
\end{oss}
By Remark \ref{propinverse} for $\varepsilon$ small enough the function $\pep(t)$ is a smooth diffeomorphism for every $t\in [0,T]$. This implies that we are allowed to define a map $t\mapsto \zep(t)$ from $[0,T]$ into $W^{1,2}(\Omega;\mathbb R^3)$ as the pointwise solution of
\begin{equation}
\nonumber
y^{\varepsilon}(t,x)=\pep(t,\zep(t,x))
\end{equation}
for every $t\in [0,T]$. {We note that
\begin{equation}
\label{rsze}
\zep(t)=(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d
\end{equation}
for every $t\in [0,T]$.} According to this change of variable, the elastic energy at time $t$ associated to the deformation $y^{\varepsilon}(t)$ can be written in terms of $\zep(t)$ as
$$\intom{W_{el}(\nep y^{\varepsilon}(t) (P^{\varepsilon})^{-1}(t))}=\intom{W_{el}\big(\nabla \pep (t,\zep(t))\nep \zep (t)(P^{\varepsilon})^{-1}(t)\big)}.$$
For every $t\in [0,T]$ we define the three-dimensional stress as
\begin{equation}
\nonumber
E^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-1}}DW_{el}\Big(\nabla \pep (t,\zep(t))\nep \zep (t)(P^{\varepsilon})^{-1}(t)\Big)\Big(\nabla \pep (t,\zep(t))\nep \zep (t)(P^{\varepsilon})^{-1}(t)\Big)^T.
\end{equation}
Let $s_1,s_2\in [0,T]$, with $s_1\leq s_2$. For every function $t\mapsto P(t)$ from $[0,T]$ into $L^2(\Omega;SL(3))$, we define its dissipation as
$$\cal{D}(P;s_1,s_2):=\sup\Big\{\sum_{i=1}^N\intom{D(P(t_{i-1}), P(t_i))}: s_1=t_0<t_1<\cdots<t_N=s_2\Big\}.$$
Analogously, for every function $t\mapsto p(t)$ from $[0,T]$ into $L^2(\Omega;\mathbb M^{3\times 3}_D)$, we define its $H_D$-dissipation as
\begin{equation}
\label{NUMpag13}
\cal{D}_{H_D}(p;s_1,s_2):=\sup\Big\{\sum_{i=1}^N\intom{H_D( p(t_i)-p(t_{i-1}))}: s_1=t_0<t_1<\cdots<t_N=s_2\Big\}.
\end{equation}
Finally, we denote by $\cal{F}_{\varepsilon}(t,z,P)$ the quantity
$$\cal{F}_{\varepsilon}(t,z,P):=\intom{W_{el}\big(\nabla \pep (t,z)\nep z P^{-1}\big)}+\intom{W_{hard}(P)}$$
for every $t\in [0,T]$, $z\in W^{1,2}(\Omega;\mathbb R^3)$ and $P\in L^2(\Omega;SL(3))$.
We are now in a position to give the definition of quasistatic evolution associated to the boundary datum $t\mapsto\pep(t)$.
\begin{defin}\label{epquasevol}
Let $\varepsilon>0$.
An {\em$\varepsilon$-quasistatic evolution} for the boundary datum $t\mapsto\pep(t)$ is a function $t\mapsto(\zep(t),P^{\varepsilon}(t))$ from $[0,T]$ into $W^{1,2}(\Omega;\mathbb R^3)\times L^2(\Omega;SL(3))$ that satisfies the following conditions:
\begin{enumerate}
\item[(qs1)] for every $t\in [0,T]$ we have {$\zep(t,x)=(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}, $P^{\varepsilon}(t,x)\in K$ for a.e. $x\in\Omega$ and
\begin{eqnarray}
\nonumber\cal{F}_{\varepsilon}(t,\zep(t),P^{\varepsilon}(t))\leq \cal{F}_{\varepsilon}(t,\tilde{z},\tilde{P})+{\varepsilon^{\alpha-1}}\intom{D(P^{\varepsilon}(t),\tilde{P})},
\end{eqnarray}
for every $(\tilde{z},\tilde{P})\in W^{1,2}(\Omega;\mathbb R^3)\times L^2(\Omega;SL(3))$ such that {$\tilde{z}(x)=(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$} and $\tilde{P}(x)\in K$ for a.e. $x\in\Omega$;
\item[(qs2)] the map
$$s\mapsto \intom{E^{\varepsilon}(s):\Big(\nabla \dot{\pep}(s,\zep(s))(\nabla \pep)^{-1}(s,\zep(s))\Big)}$$
is integrable in $[0,T]$ and for every $t\in[0,T]$
\begin{eqnarray}
\nonumber &&\cal{F}_{\varepsilon}(t,\zep(t),P^{\varepsilon}(t))+{\varepsilon^{\alpha-1}}\cal{D}(P^{\varepsilon};0,t)\\
\nonumber &&=\cal{F}_{\varepsilon}(0,\zep(0),P^{\varepsilon}(0))+{\varepsilon^{\alpha-1}}\int_0^t{\intom{E^{\varepsilon}(s):\Big(\nabla \dot{\pep}(s,\zep(s))(\nabla \pep)^{-1}(s,\zep(s))\Big)}\,ds}.
\end{eqnarray}
\end{enumerate}
\end{defin}
\begin{oss}
We remark that if the function $t\to (\zep(t),P^{\varepsilon}(t))$ satisfies condition (qs1), then $E^{\varepsilon}(t)\in L^1(\Omega;\mathbb{M}^{3\times 3})$ for every $t\in [0,T]$. Indeed, by (qs1), taking $\tilde{z}(x)=(x',\varepsilon x_3)$ for every $x\in\Omega$ and $\tilde{P}=P^{\varepsilon}(t)$, we deduce
\begin{equation}
\label{preles}
\intom{W_{el}\big(\nabla \pep(t,\zep(t))\nep \zep(t)(P^{\varepsilon})^{-1}(t)\big)}\leq \intom{W_{el}\big(\nabla \pep(t,(x',\varepsilon x_3))(P^{\varepsilon})^{-1}(t)\big)}.
\end{equation}
On the other hand, $P^{\varepsilon}(t)\in K$ a.e. in $\Omega$ and for $\varepsilon$ small enough there exists two constants $C_1$ and $C_2$ such that $\det (\nabla \pep(t,(x',\varepsilon x_3)))\geq C_1$ and $\|\nabla \pep (t,(x',\varepsilon x_3))\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C_2$. Therefore, by hypothesis (H1) the quantity in \eqref{preles} is finite and
\begin{equation}
\label{detpo}
\det \big(\nabla \pep(t,\zep(t))\nep \zep(t)(P^{\varepsilon})^{-1}(t)\big)>0\quad\text{a.e. in }\Omega
\end{equation}
for $\varepsilon$ small enough. Finally, by \eqref{mandel2} we obtain
$$\intom{|E^{\varepsilon}(t)|}\leq \frac{c_4}{\varepsilon^{\alpha-1}}\Big(\intom{W_{el}(\nabla \pep(t,\zep(t))\nep \zep(t)(P^{\varepsilon})^{-1}(t))}+1\Big)<+\infty.$$
\end{oss}
\begin{oss}
\label{Esym}
By the frame-indifference (H3) of $W_{el}$, there holds
\begin{equation}
\nonumber
DW_{el}(F)F^T=F(DW_{el}(F))^T\quad\text{for every }F\in\mathbb{M}^{3\times 3}_+.
\end{equation}
Therefore, by \eqref{detpo}, for $\varepsilon$ small enough $E^{\varepsilon}(t,x)\in\mathbb M^{3\times 3}_{sym}$ for every $t\in [0,T]$ and for a.e. $x\in\Omega$.
\end{oss}
Set
$$L_{\alpha}:=\begin{cases}0&\text{if }\alpha>3\\
1&\text{if }\alpha=3.
\end{cases}$$For every $\alpha\geq 3$ we define a reduced quasistatic evolution as follows.
\begin{defin}
\label{reda>3}
For $\alpha\geq 3$, a {\emph{reduced quasistatic evolution}} for the boundary data $t\mapsto u^0(t)$ and $t\mapsto v^0(t)$ is a map $t\mapsto (u(t),v(t),p(t))$ from $[0,T]$ into $W^{1,2}(\omega;\mathbb R^2)\times W^{2,2}(\omega)\times L^2(\Omega;\mathbb M^{3\times 3}_D)$, that satisfies the following conditions:
\begin{enumerate}
\item [(qs1$_{r\alpha}$)] for every $t\in [0,T]$ there holds $(u(t),v(t),p(t))\in\cal{A}(u^0(t),v^0(t))$, and setting
\begin{equation}
\label{dsigmat}
e_{\alpha}(t):=\mathrm{sym}\nabla' u(t)+\tfrac{L_{\alpha}}{2}\nabla' v(t)\otimes\nabla' v(t)-x_3(\nabla')^2 v(t)-p'(t),
\end{equation}
we have
\begin{eqnarray*}
&&\intom{Q_2(e_{\alpha}(t))}+\intom{\B{p(t)}}\leq\intom{Q_2\big(\mathrm{sym}\,\nabla' \hat{u}+\tfrac{L_{\alpha}}{2}\nabla' \hat{v}\otimes\nabla' \hat{v}-x_3(\nabla')^2 \hat{v}-\hat{p}'\big)}\\&&+\intom{\B{\hat{p}}}+\intom{H_D(\hat{p}-p(t))},
\end{eqnarray*}
for every $(\hat{u},\hat{v},\hat{p})\in\cal{A}(u^0(t),v^0(t))$;
\item[(qs2$_{r\alpha}$)] the map
$$s\to \intom{\mathbb C_2e_{\alpha}(s):\Big(\begin{array}{cc}\nabla' \dot{u}^0(s)+L_{\alpha}\nabla' \dot{v}^0(s)\otimes\nabla' v(s)-x_3(\nabla')^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)}$$ is integrable in $[0,T]$. Moreover
for every $t\in [0,T]$ there holds
\begin{eqnarray*}
&&\intom{Q_2(e_{\alpha}(t))}+\intom{\B{p(t)}}+\cal{D}_{H_D}(p;0,t)=\intom{Q_2(e_{\alpha}(0))}+\intom{\B{p(0)}}\\
&&+\int_0^t{\intom{\mathbb C_2e_{\alpha}(s):\Big(\begin{array}{cc}\nabla' \dot{u}^0(s)+L_{\alpha}\nabla' \dot{v}^0(s)\otimes\nabla' v(s)-x_3(\nabla')^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)}\,ds}.
\end{eqnarray*}
\end{enumerate}
\end{defin}
\begin{oss}
\label{csd}
An adaptation of \cite[Theorem 4.5]{DDM} guarantees that, if $\alpha>3$, for every triple $(\overline{u},\overline{v},\overline{p})\in\cal{A}(u^0(0),v^0(0))$ satisfying
\begin{eqnarray*}
&&\intom{Q_2\big(\mathrm{sym} \nabla' \overline{u}-x_3(\nabla')^2 \overline{v}+\tfrac{L_{\alpha}}{2}\nabla' \overline{v}\otimes \nabla' \overline{v}-\overline{p}')}+\intom{\B{\overline{p}}}\\
&&\leq\intom{Q_2\big(\mathrm{sym}\nabla' \hat{u}-x_3(\nabla')^2 \hat{v}+\tfrac{L_{\alpha}}{2}\nabla' \hat{v}\otimes \nabla' \hat{v}-\hat{p}'\big)}+\intom{\B{\hat{p}}}+\intom{H_D(\hat{p}-\overline{p})},
\end{eqnarray*}
for every $(\hat{u},\hat{v},\hat{p})\in\cal{A}(u^0(0),v^0(0))$, there exists a reduced quasistatic evolution $t\mapsto (u(t),v(t),p(t))$ (according to Definition \ref{reda>3}) such that $u(0)=\overline{u}$, $v(0)=\overline{v}$ and $p(0)=\overline{p}$. Moreover, by adapting \cite[Theorem 5.2 and Remark 5.4]{DDM} one can show that the maps $t\mapsto u(t)$, $t\mapsto v(t)$ and $t\mapsto p(t)$ are Lipschitz continuous from $[0,T]$ into $W^{1,2}(\omega;\mathbb R^2)$, $W^{2,2}(\omega)$ and $L^2(\Omega;\mathbb M^{3\times 3}_D)$, respectively.
In the case $\alpha=3$, the existence of a reduced quasistatic evolution $t\mapsto (u(t),v(t),p(t))$ such that $(u(0),v(0),p(0))=(\overline{u},\overline{v},\overline{p})$ can still be proved by adapting \cite[Theorem 4.5]{DDM}. We remark that the proof of this result is more subtle than its counterpart in the case $\alpha>3$, due to the presence of the nonlinear term $\frac{1}{2}\nabla'v\otimes \nabla'v$. {{In fact, for $\alpha=3$ one can not prove the analogous of \cite[Theorem 3.8]{DDM} and can not guarantee that the set of discontinuity points of the function $t\mapsto e_3(t)$ is at most countable. Hence, when trying to prove the analogous of \cite[Theorem 4.7]{DDM}, that is, to deduce the converse energy inequality by the minimality, some further difficulties arise to study convergence of the piecewise constant interpolants of $t\mapsto e_3(t)$. To cope with this problem}}, one can apply \cite[Lemma 4.12]{DFT}, which guarantees the existence of partitions of $[0,T]$ on which the Bochner integrals of some relevant quantities can be approximated by Riemann sums, and argue as in \cite[Lemma 5.7]{B}.
\end{oss}
\begin{oss}
\label{eul3}
By taking $\hat{p}=p(t)$ in (qs1$_{r\alpha}$), it follows that a reduced quasistatic evolution $t\mapsto (u(t),v(t),p(t))$ satisfies
\begin{eqnarray*}
\intom{Q_2(e_{\alpha}(t))}\leq \intom{Q_2(\mathrm{sym}\,\nabla' \hat{u}+\tfrac{L_{\alpha}}{2}\nabla' \hat{v}\otimes\nabla' \hat{v}-x_3(\nabla')^2 \hat{v}-{p}'(t))}
\end{eqnarray*}
for every $(\hat{u},\hat{v})\in W^{1,2}(\omega;\mathbb R^2)\times W^{2,2}(\omega)$ such that
{$$\hat{u}=u^0(t),\,\hat{v}=v^0(t)\text{ and }\nabla'\hat{v}=\nabla' v^0(t) \quad\cal{H}^1\text{ - a.e. on }\gamma_d.$$
Hence, in particular, there holds
$$\intom{\mathbb C_2 e(t):\nabla' \zeta}=0$$
for every $\zeta \in W^{1,2}(\omega;\mathbb R^2)$ such that $\zeta=0$ $\cal{H}^1$ - a.e. on $\gamma_d$.}
\end{oss}
With the previous definitions at hand we are in a position to state the main result of the paper.
\begin{teo}
\label{cvstress}
Let $\alpha\geq 3$. Assume that $t\mapsto u^0(t)$ belongs to $C^1([0,T];W^{1,\infty}(\mathbb R^2;\mathbb R^2)\cap C^{1}(\mathbb R^2;\mathbb R^2))$ and $t\mapsto v^0(t)$ belongs to $C^1([0,T];W^{2,\infty}(\mathbb R^2)\cap C^{2}(\mathbb R^2))$, respectively. For every $t\in [0,T]$, let $\pep(t)$ be defined as in \eqref{defphiep}.
Let $(\mathring{u},\mathring{v},\mathring{p})\in \cal{A}(u^0(0),v^0(0))$ be such that
\begin{eqnarray}
\nonumber &&\intom{Q_2(\mathrm{sym}\nabla' \mathring{u}-x_3(\nabla')^2 \mathring{v}+\tfrac{L_{\alpha}}{2}\nabla' \mathring{v}\otimes \nabla' \mathring{v}-\mathring{p}')}+\intom{\B{\mathring{p}}}\\
\nonumber&&
\leq \intomm{Q_2(\nabla' \hat{u}-x_3(\nabla')^2 \hat{v}+\tfrac{L_{\alpha}}{2}\nabla' \hat{v}\otimes \nabla' \hat{v}-\hat{p}')}+\intom{\B{\hat{p}}}+\intom{H(\hat{p}-\mathring{p})},\\
\label{servedopo1}
\end{eqnarray}
for every $(\hat{u},\hat{v},\hat{p})\in\cal{A}(u^0(0),v^0(0))$.
Assume there exists a sequence of pairs $(y_0^{\varepsilon},P_0^{\varepsilon})\in \cal{A}_{\varepsilon}(\pep(0))$ such that
\begin{equation}
\label{servedopo2}\cal{I}(y^{\varepsilon}_0,P^{\varepsilon}_0) \leq \cal{I}(\hat{y},\hat{P})+{\varepsilon^{\alpha-1}}\intom{D(P^{\varepsilon}_0,\hat{P})},
\end{equation}
for every $(\hat{y},\hat{P})\in\cal{A}_{\varepsilon}(\pep(0))$, and
\begin{eqnarray}
\label{convu0} && u^{\varepsilon}_0:=\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((y^{\varepsilon}_0)'-x'\big)}\to \mathring{u}\quad\text{strongly in }W^{1,2}(\omega;\mathbb R^2),\\
\label{convv0} && v^{\varepsilon}_0:=\frac{1}{\varepsilon^{\alpha-2}}\intt{(y^{\varepsilon}_0)_3}\to \mathring{v}\quad\text{strongly in }W^{1,2}(\omega),\\
\label{convP0} && p^{\varepsilon}_0:=\frac{P^{\varepsilon}_0-Id}{\varepsilon^{\alpha-1}}\to \mathring{p}\quad\text{strongly in }L^2(\Omega;\mathbb M^{3\times 3}_D),\\
\nonumber && \lim_{\varepsilon\to 0}\,\frac{1}{\varepsilon^{2\alpha-2}}\cal{I}(y^{\varepsilon}_0, P^{\varepsilon}_0) =\intom{Q_2(\mathrm{sym} \nabla' \mathring{u}-x_3(\nabla')^2 \mathring{v}+\tfrac{L_{\alpha}}{2}\nabla' \mathring{v}\otimes\nabla' \mathring{v}-{\mathring{p}}')}\\
\label{convE0} &&+\intom{\B{\mathring{p}}}.
\end{eqnarray}
Finally, for every $\varepsilon>0$, let $t\mapsto (\zep(t),P^{\varepsilon}(t))$ be an $\varepsilon$-quasistatic evolution for the boundary datum $\pep(t)$ such that
$$\zep(0)=\vep(0,y^{\varepsilon}_0)\quad\text{a.e. in }\Omega$$
and $$P^{\varepsilon}(0)=P^{\varepsilon}_0.$$
Then, there exists a reduced quasistatic evolution $t\mapsto (u(t),v(t),p(t))$ for the boundary data $(u^0(t),v^0(t))$ (according to Definition \ref{reda>3}), such that $u(0)=\mathring{u}$, $v(0)=\mathring{v}$, $p(0)=\mathring{p}$ and, up to subsequences,
\begin{equation}
\label{cvpt} p^{\varepsilon}(t):=\frac{P^{\varepsilon}(t)-Id}{\varepsilon^{\alpha-1}}\rightharpoonup p(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3})
\end{equation}
for every $t\in [0,T]$. Moreover, for $\alpha>3$ up to subsequences there holds
\begin{eqnarray}
&&\label{cvut}u^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-1}}\intt{\big(({\phi}^{\varepsilon})'(t,\zep(t))-x'\big)}\rightharpoonup u(t)\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2),\\
&&\label{cvvt}v^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-2}}\intt{\pep_3(t,\zep(t))}\to v(t)\quad\text{strongly in }W^{1,2}(\omega),
\end{eqnarray}
for every $t\in [0,T]$. For $\alpha=3$, for every $t\in [0,T]$ there exists a $t$-dependent subsequence $\varepsilon_j\to 0$ such that
\begin{eqnarray}
&&\label{cvut3}u^{\ep_{j}}(t):=\frac{1}{\ep_{j}^{\alpha-1}}\intt{\big(({\phi}^{\ep_{j}})'(t,\zepjt(t))-x'\big)}\rightharpoonup u(t)\,\text{ weakly in }W^{1,2}(\omega;\mathbb R^2),\\
&&\label{cvvt3}v^{\ep_{j}}(t):=\frac{1}{\ep_{j}^{\alpha-2}}\intt{\pepjt_3(t,\zepjt(t))}\to v(t)\quad\text{strongly in }W^{1,2}(\omega).
\end{eqnarray}
\end{teo}
In the case $\alpha>3$ the convergence result is stronger than the analogous result for $\alpha=3$ as the convergence of $u^{\varepsilon}(t)$ and $v^{\varepsilon}(t)$ holds on a subsequence independent of $t$. This is related to the fact that, for $\alpha>3$, once $t\mapsto p(t)$ is identified, both $t\mapsto u(t)$ and $t\mapsto v(t)$ are uniquely determined. In the case $\alpha=3$ this property is not true anymore because of the presence of the nonlinear term $\frac{1}{2}\nabla' v(t)\otimes \nabla' v(t)$.
We shall prove the previous theorem in the next section. To conclude this section, we prove a technical lemma concerning some properties of the truncation maps $\tep$ and we provide the construction of the so-called ``joint recovery sequence", that will be used in the proof of Theorem \ref{cvstress}.
\begin{lem}
\label{cvproptep}
Let $\tep\in W^{1,\infty}(\mathbb R)\cap {C^{1}(\mathbb R)}$ be such that \eqref{treq1}--\eqref{lp2} hold and let $(\zeta^{\varepsilon})$ be a sequence in $L^{2}(\Omega)$ such that
\begin{equation}
\label{bdz}
\|\zeta^{\varepsilon}\|_{L^2(\Omega)}\leq C\varepsilon.
\end{equation}
Then,
\begin{equation}
\label{tep1h}\Big\|1-\dot{\tep}\Big(\frac{\zeta^{\varepsilon}}{\varepsilon}\Big)\Big\|_{L^2(\Omega)}\leq \frac{3}{ \ell_{\varepsilon}}.
\end{equation}
Moreover, if $\zeta^{\varepsilon}$ satisfies
\begin{equation}
\label{3compus}
\big\|\frac{\zeta^{\varepsilon}}{\varepsilon}-x_3-\varepsilon^{\alpha-3}v\big\|_{L^2(\Omega)}\to 0,
\end{equation}
for some $v\in L^{2}(\omega)$, then
\begin{equation}
\label{claimvh} \tep\Big(\frac{\zeta^{\varepsilon}}{\varepsilon}\Big)\to\begin{cases}x_3&\text{if }\alpha>3\\
x_3+v&\text{if }\alpha=3\end{cases}\quad\text{strongly in }L^2(\Omega).\\
\end{equation}
\end{lem}
\begin{proof}
Denoting by $O_{\varepsilon}$ the set
$$O_{\varepsilon}:=\Big\{x\in \Omega: |\zeta^{\varepsilon}(x)|\geq \varepsilon\ell_{\varepsilon}\Big\},$$
by \eqref{bdz} and by Chebychev inequality, there holds
$$\cal{L}^3(O_{\varepsilon})\leq \frac{C}{\ell_{\varepsilon}^2}.$$
Hence, by \eqref{treq1} and \eqref{treq5},
$$\Big\|1-\dot{\tep}\Big(\frac{\zeta^{\varepsilon}}{\varepsilon}\Big)\Big\|_{L^2(\Omega)}=\Big\|1-\dot{\tep}\Big(\frac{\zeta^{\varepsilon}}{\varepsilon}\Big)\Big\|_{L^2(O_{\varepsilon})}\leq \frac{2}{ \ell_{\varepsilon}}.$$
To prove \eqref{claimvh}, we note that by \eqref{3compus} there holds
$$\tep\Big(\frac{\zeta^{\varepsilon}}{\varepsilon}\Big)\to\begin{cases}x_3&\text{if }\alpha>3\\
x_3+v&\text{if }\alpha=3\end{cases}\quad\text{a.e. in }\Omega.$$
On the other hand, \eqref{treq2} yields $\big|\tep\big(\frac{\zeta^{\varepsilon}}{\varepsilon}\big)\big|\leq \big|\frac{\zeta^{\varepsilon}}{\varepsilon}\big|$ for every $\varepsilon$ and for a.e. $x\in \Omega$.
Therefore \eqref{claimvh} follows by the dominated convergence theorem. \end{proof}
For the sake of simplicity, in the next theorem we omit the time dependence of $u^0$ and $v^0$. With a slight abuse of notation, we denote by $\pep$ the map
\begin{equation}
\nonumber\pep(x):=\Big(\begin{array}{c}x'\\x_3\end{array}\Big)+\varepsilon^{\alpha-1}\Big(\begin{array}{c}u^0(x')-\tep(\frac{x_3}{\varepsilon})\nabla' v^0(x')\\0\end{array}\Big)+\varepsilon^{\alpha-2}\Big(\begin{array}{c}0\\v^0(x')\end{array}\Big),
\end{equation}
for a.e. $x\in\Omega$, where $u^0\in W^{1,\infty}(\mathbb R^2;\mathbb R^2)\cap C^{1}(\mathbb R^2;\mathbb R^2)$ and $v^0\in W^{2,\infty}(\mathbb R^2)\cap C^{2}(\mathbb R^2)$.
We are now in a position to construct the joint recovery sequence.
\begin{teo}
\label{mutrecseq}
Let $(y^{\varepsilon}, P^{\varepsilon})\in \cal{A}_{\varepsilon}(\pep)$ satisfy \eqref{engest2}
for every $\varepsilon>0$. Let $u,v,G,p$ be defined as in Theorem \ref{liminfineq} and let
$\hat{u}:=u+\tilde{u}$, $\hat{v}:=v+\tilde{v}$, and $\hat{p}:=p+\tilde{p}$, where {$\tilde{u}\in C^{\infty}_c(\omega\cup\gamma_n;\mathbb R^2)$, $\tilde{v}\in C^{\infty}_c(\omega\cup\gamma_n)$} and $\tilde{p}\in C^{\infty}_c(\Omega;\mathbb M^{3\times 3}_D)$.
Then, there exists a sequence of pairs $(\hye,\hpe)\in \cal{A}_{\varepsilon}(\pep)$, such that
\begin{eqnarray}
&&\label{cvidh} \hye\to \Big(\begin{array}{c}x'\\0\end{array}\Big)\quad\text{strongly in }W^{1,2}(\Omega;\mathbb R^3),\\
&&\label{4cptuepmrsh} \hat{u}^{\varepsilon}:=\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((\hye)'-x'\big)}\rightharpoonup \hat{u}\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2)\text{ for }\alpha>3,\\
&&\label{4cptuepmrsh2} \hat{u}^{\varepsilon}\rightharpoonup \hat{u}-v\nabla \tilde{v}\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2)\text{ for }\alpha=3,\\
&&\label{cptvepmrsh} \hat{v}^{\varepsilon}:=\frac{1}{\varepsilon^{\alpha-2}}\intt{\hye_3}\to \hat{v}\quad\text{strongly in }W^{1,2}(\omega),\\
\label{Pep1mrsh} &&\hpe(x)\in K\quad\text{for a.e. }x\in\Omega,\\
\label{wconvpepmrsh} &&
\hppe:=\frac{\hpe-Id}{\varepsilon^{\alpha-1}}\rightharpoonup \hat{p}\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{eqnarray}
Moreover, the following inequalities hold true:
\begin{equation}
\label{limsuphardening}
\limsup_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\Big(\intom{W_{hard}(\hpe)}-\intom{W_{hard}(P^{\varepsilon})}\Big)\leq \intom{\B{\hat{p}}}-\intom{\B{{p}}},
\end{equation}
\begin{equation}
\label{limsupdiss}
\limsup_{\varepsilon\to 0}\frac{1}{\varepsilon^{\alpha-1}}\intom{D(P^{\varepsilon},\hpe)}\leq \intom{H_D(\hat{p}-p)},
\end{equation}
and
\begin{eqnarray}
&&\nonumber\limsup_{\varepsilon\to 0} \frac{1}{\varepsilon^{2\alpha-2}}\Big(\intom{W_{el}(\nep \hye (\hpe)^{-1})}-\intom{W_{el}(\nep y^{\varepsilon} (P^{\varepsilon})^{-1})}\Big)\\
&&\nonumber\leq \intom{Q_2(\mathrm{sym}\,\hat{G}'-\hat{p}')}-\intom{Q_2(\mathrm{sym}\,{G}'-{p}')},\\
\label{limsupeng}
\end{eqnarray}
where the submatrix $\hat{G}'$ satisfies
\begin{equation}
\nonumber
\hat{G}'(x', x_3) := \hat{G}_0(x') - x_3 (\nabla')^2 \hat{v}(x')\quad\text{for a.e. }x\in\Omega,
\end{equation}
and
\begin{eqnarray*}
&&\nonumber \mathrm{sym}\, \hat{G}_0 = \frac{(\nabla' \hat{u}+(\nabla' \hat{u})^T +\nabla' \hat{v}\otimes \nabla' \hat{v})}{2}\quad\text{for } \alpha=3,\\
&&\nonumber \mathrm{sym}\, \hat{G}_0 = \mathrm{sym}\, \nabla' \hat{u}\quad\text{for }\alpha> 3.
\end{eqnarray*}
\end{teo}
\begin{proof}
We divide the proof into four steps. In the first step we exhibit a sequence of deformations $(\hye)$ satisfying \eqref{cvidh}--\eqref{cptvepmrsh}. In the second step we construct a sequence $(\hpe)$ of plastic strains and we prove the limsup inequality for the hardening and the dissipation terms. In the third step we rewrite the elastic energy in terms of some auxiliary quantities and in the fourth step we prove the limsup inequality for the elastic energy.
We first remark that by \eqref{engest2} and the boundary condition
{\begin{equation}
\label{bddatum2}
y^{\varepsilon}(x)=\pep(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d,
\end{equation}}
the sequence $(y^{\varepsilon},P^{\varepsilon})$ fulfills the hypotheses of Theorems \ref{compactbd1} and \ref{liminfineq}. Hence, there exists a sequence $(R^{\varepsilon})\subset W^{1,\infty}(\omega;\mathbb{M}^{3\times 3})$ such that \eqref{rt1}--\eqref{rt4} hold true, and $(y^{\varepsilon})$ satisfies \eqref{cptyep}. Moreover, defining $u^{\varepsilon}, v^{\varepsilon},$ and $G^{\varepsilon}$ according to \eqref{inplane}, \eqref{outofplane} and \eqref{defgep}, properties \eqref{cptuep}--\eqref{Ga>3} hold true. The sequence of plastic strains $(P^{\varepsilon})$ satisfies
\begin{equation}
\label{Pep1mrs} P^{\varepsilon}(x)\in K\quad\text{for a.e. }x\in\Omega,
\end{equation}
and defining $p^{\varepsilon}$ as in \eqref{defpep}, there holds
\begin{equation}
\label{wconvpepmrs}
p^{\varepsilon}\rightharpoonup p\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Finally, by Theorem \ref{liminfineq}, $(u,v,p)\in\cal{A}(u^0,v^0)$ and, by \eqref{cptvep} and \eqref{3comphest}, the sequence $(y^{\varepsilon}_3)$ fulfills the hypothesis of Lemma \ref{cvproptep}, hence
\begin{equation}
\label{claimvhm} \tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\to\begin{cases}x_3&\text{if }\alpha>3\\
x_3+v&\text{if }\alpha=3\end{cases}\quad\text{strongly in }L^2(\Omega),
\end{equation}
and by \eqref{lp2} and \eqref{tep1h}
\begin{equation}
\label{tep1hm}
\frac{1}{\varepsilon}-\frac{1}{\varepsilon}\dot{\tep}\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\to 0\quad\text{strongly in }L^2(\Omega).
\end{equation}
{\em Step 1: Construction of the deformations}\\
Let $d\in C^{\infty}_c(\mathbb R^3;\mathbb R^3)$ with $\mathrm{supp}\, d\subset \Omega$. Consider the map
$$\etp(x):=\int_{-\frac{1}{2}}^{\frac{x_3}{\varepsilon}}{d(x',s)\,ds}\quad\text{for every }x\in \mathbb R^3.$$
Since $d$ has compact support in $\Omega$, there holds
$$|\etp(x)|\leq \int_{-\frac{1}{2}}^{\big|\frac{x_3}{\varepsilon}\big|}{|d(x',s)|\,ds}\leq \int_{-\frac{1}{2}}^{\frac{1}{2}}{|d(x',s)|\,ds}\quad\text{for every }x\in\mathbb R^3$$
and analogously
\begin{equation}
\label{etaep0}
\|{\nabla}'{\etp}\|_{L^{\infty}(\mathbb R^3;\mathbb M^{3\times 2})}\leq \|{\nabla}' d\|_{L^{\infty}(\mathbb R^3;\mathbb M^{3\times 2})}.
\end{equation}
A straightforward computation yields
\begin{equation}
\label{etaep12}
\partial_3 \etp(x)=\frac{1}{\varepsilon}d\Big(x',\frac{x_3}{\varepsilon}\Big)\quad\text{for every }x\in\mathbb R^3.
\end{equation}
Hence,
\begin{equation}
\label{etaep1}
\|\etp\|_{W^{1,\infty}(\mathbb R^3;\mathbb R^3)}\leq \frac{C}{\varepsilon}.
\end{equation}
In particular, the map $\etp\circ y^{\varepsilon}$ satisfies
\begin{eqnarray}
&&\label{etaep2} \|\etp\circ y^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb R^3)}\leq C,\\
&&\label{etaep3} \|{\nabla}'(\etp\circ y^{\varepsilon})\|_{L^2(\Omega;\mathbb M^{3\times 2})}\leq C \|\nabla'(y^{\varepsilon})'\|_{L^2(\Omega;\mathbb M^{2\times 2})}+\frac{C}{\varepsilon}\|\nabla'y^{\varepsilon}_3\|_{L^2(\Omega;\mathbb R^2)}.
\end{eqnarray}
We { extend $\tilde{u}$ and $\tilde{v}$ to zero outside their support, we} consider the functions
$$f^{\varepsilon}(x):=x+\Big(\begin{array}{c}\varepsilon^{\alpha-1}\tilde{u}(x')\\\varepsilon^{\alpha-2}\tilde{v}(x')\end{array}\Big)-\Big(\begin{array}{c}\varepsilon^{\alpha-1}\tep\big(\frac{x_3}{\varepsilon}\big)\nabla' \tilde{v}(x')\\0\end{array}\Big)+\varepsilon^{\alpha}\etp(x)$$
for every $x\in\mathbb R^3$, and we set
$$\hye:=f^{\varepsilon}\circ y^{\varepsilon}.$$
It is easy to see that $\hye\in W^{1,2}(\Omega;\mathbb R^3)$, we now check that
{\begin{equation}
\label{bddat}
\hye=\phi^{\varepsilon}(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d.
\end{equation}
To prove it, we first remark that by \eqref{bddatum2}
\begin{equation}
\label{almbddat}
\hye=f^{\varepsilon}(\pep(x',\varepsilon x_3))\quad\cal{H}^2\text{ - a.e. on }\Gamma_d.\end{equation}
Hence, it remains only to show that
\begin{equation}
\label{almbddat2}
f^{\varepsilon}(\pep(x',\varepsilon x_3))=\pep(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d.
\end{equation}
Let $A\subset \mathbb R^2$ be an open set such that $\overline{\gamma_d}\subset (A\cap\partial\omega)$ and $\tilde{u},\tilde{v},\nabla'\tilde{v}=0$ in $A$.
Since $d$ has compact support in $\Omega$, without loss of generality we may assume that $\etp(x)=0$ for all $x\in A\times \mathbb R$ and every $\varepsilon$. Therefore, we have $f^{\varepsilon}(x)=x$ in $A\times\big(-\tfrac 12,\tfrac 12\big)$. Let now $O\subset \mathbb R^2$ be an open set such that $\overline{\gamma_d}\subset (O\cap\partial\omega)$ and $\overline{O}\subset A$, and let $0<\delta_0<\mathrm{dist}(O,\partial A)$. By \eqref{treq2}, there holds
$$|(\pep)'(x',\varepsilon x_3)-x'|\leq \varepsilon^{\alpha-1}\|u^0\|_{L^{\infty}(\mathbb R^2;\mathbb R^2)}+\frac{1}{2}\varepsilon^{\alpha-2}\|\nabla' v^0\|_{L^{\infty}(\mathbb R^2;\mathbb M^{2\times 2})}< \frac{\delta_0}{2}$$
for every $x\in O\times\big(-\tfrac 12,\tfrac 12\big)$, for $\varepsilon$ small enough. Hence, $\pep(x',\varepsilon x_3)\in A$ for every $x\in O\times\big(-\tfrac 12,\tfrac 12\big)$, and $f^{\varepsilon}(\phi^{\varepsilon}(x',\varepsilon x_3))=\phi^{\varepsilon}(x',\varepsilon x_3)$ for every $x\in O\times\big(-\tfrac 12,\tfrac 12\big)$. This implies \eqref{almbddat2} and \eqref{bddat}.}
To prove \eqref{cvidh}, we remark that by the smoothness of $\tilde{u}$ and $\tilde{v}$, estimates \eqref{treq3}, \eqref{treq5}, \eqref{lp2} and \eqref{etaep1} imply
\begin{equation}
\label{unifbdvarphi}
\|f^{\varepsilon}-id\|_{W^{1,\infty}(\mathbb R^3;\mathbb R^3)}\leq C\varepsilon^{\alpha-1}\ell_{\varepsilon}.
\end{equation}
On the other hand, we have
\begin{eqnarray*}
&&\Big\|\hye-\Big(\begin{array}{c}x'\\0\end{array}\Big)\Big\|_{W^{1,2}(\Omega;\mathbb R^3)}\leq \|\hye-y^{\varepsilon}\|_{W^{1,2}(\Omega;\mathbb R^3)}+\Big\|y^{\varepsilon}-\Big(\begin{array}{c}x'\\0\end{array}\Big)\Big\|_{W^{1,2}(\Omega;\mathbb R^3)}\\
&&\leq C\|f^{\varepsilon}-id\|_{W^{1,\infty}(\mathbb R^3;\mathbb R^3)}\|\nabla y^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}+\Big\|y^{\varepsilon}-\Big(\begin{array}{c}x'\\0\end{array}\Big)\Big\|_{W^{1,2}(\Omega;\mathbb R^3)},
\end{eqnarray*}
so that \eqref{cvidh} follows by \eqref{cptyep}, \eqref{lp1} and \eqref{unifbdvarphi}.
We now prove convergence of the out-of-plane displacements associated to $(\hye)$.
To show \eqref{cptvepmrsh} we note that
\begin{equation}
\nonumber
\hat{v}^{\varepsilon}=\frac{1}{\varepsilon^{\alpha-2}}\intt{f^{\varepsilon}_3(y^{\varepsilon})}
=v^{\varepsilon}+\intt{\tilde{v}((y^{\varepsilon})')}+\varepsilon^2\intt{\etp_3(y^{\varepsilon})}.
\end{equation}
By \eqref{cptyep}, up to subsequences, we can assume
\begin{equation}
(y^{\varepsilon})'\to x'\quad\text{and }\nabla'(y^{\varepsilon})'\to Id\quad\text{a.e. in }\Omega.
\label{ptcvid}
\end{equation}
Hence, by the dominated convergence theorem and the smoothness of $\tilde{v}$ we obtain
$$\tilde{v}((y^{\varepsilon})')\to \tilde{v}\quad\text{strongly in }L^2(\Omega)$$
and
$$\nabla'\tilde{v}( (y^{\varepsilon})')\to \nabla'\tilde{v}\quad\text{strongly in }L^2(\Omega;\mathbb R^2).$$
By \eqref{cptyep}, \eqref{cptvep}, \eqref{etaep2} and \eqref{etaep3} we conclude
\begin{equation}
\nonumber
\hat{v}^{\varepsilon}\to v+\tilde{v}=\hat{v}\quad\text{strongly in }W^{1,2}(\omega).
\end{equation}
To prove \eqref{4cptuepmrsh} and \eqref{4cptuepmrsh2} we note that
\begin{eqnarray}
\nonumber &&
\hat{u}^{\varepsilon}=u^{\varepsilon}+\intt{\tilde{u}((y^{\varepsilon})')}-\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla' \tilde{v}((y^{\varepsilon})')}\\
\label{expuh} &&+\varepsilon\intt{(\etp)'(y^{\varepsilon})}.
\end{eqnarray}
By \eqref{claimvhm}, \eqref{ptcvid} and the dominated convergence theorem,
\begin{eqnarray}
&&\nonumber \tilde{u}((y^{\varepsilon})')\to \tilde{u}\quad\text{strongly in }L^2(\Omega;\mathbb R^2),\\
&&\nonumber
\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla' \tilde{v}((y^{\varepsilon})')}\to 0\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha>3,\\
&&\nonumber
\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla' \tilde{v}((y^{\varepsilon})')}\to v\nabla\tilde{v}\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha=3.
\end{eqnarray}
Hence, by \eqref{etaep2}, we have
$$\hat{u}^{\varepsilon}\to \hat{u}\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha>3,$$
and
$$\hat{u}^{\varepsilon}\to \hat{u}-v\nabla \tilde{v}\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha=3.$$
To complete the proof of \eqref{4cptuepmrsh} and \eqref{4cptuepmrsh2}, it remains to show that
\begin{equation}
\label{wcvgraduh}
\frac{1}{\varepsilon^{\alpha-1}}\nabla'\hat{u}^{\varepsilon}\text{ is bounded in }L^2(\Omega;\mathbb M^{2\times 2}).
\end{equation}
By \eqref{expuh} there holds
\begin{eqnarray*}
&&\frac{1}{\varepsilon^{\alpha-1}}\nabla'\hat{u}^{\varepsilon}=\nabla'u^{\varepsilon}+\intt{\nabla'\tilde{u}((y^{\varepsilon})') \nabla'(y^{\varepsilon})'}\\
&&-\intt{\tepp(\nabla')^2 \tilde{v}((y^{\varepsilon})') \nabla'(y^{\varepsilon})'}-\frac{1}{\varepsilon}\intt{\teppp\nabla' \tilde{v}((y^{\varepsilon})')\otimes \nabla'y^{\varepsilon}_3}\\
&&+\varepsilon\intt{\nabla'(\etp\circ y^{\varepsilon})}.
\end{eqnarray*}
By adding and subtracting the matrix $(R^{\varepsilon})'$ we obtain
\begin{eqnarray*}
&&\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\nabla' (y^{\varepsilon})'}=\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\big(\nabla' (y^{\varepsilon})'-(R^{\varepsilon})'\big)}\\
&&+\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')(R^{\varepsilon})'}.
\end{eqnarray*}
Combining \eqref{rt2} and \eqref{treq3}, we deduce
\begin{equation}
\nonumber
\Big\|\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\big(\nabla' (y^{\varepsilon})'-(R^{\varepsilon})'\big)\Big\|_{L^2(\Omega;\mathbb{M}^{2\times 2})}\leq C\varepsilon^{\alpha-1}\ell_{\varepsilon}.
\end{equation}
On the other hand, by \eqref{rt1} and \eqref{claimvhm}, the maps
$\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla^2 \tilde{v}((y^{\varepsilon})')(R^{\varepsilon})'\text{ are bounded in }L^2(\Omega;\mathbb M^{2\times 2}).$ The $L^2$-boundedness of the quantity in \eqref{wcvgraduh} follows now by combining \eqref{cptyep}, \eqref{cptuep}, \eqref{cptnep3}, \eqref{treq5} and \eqref{etaep3}.\\
\begin{comment}
To prove \eqref{cptuepmrsh} and \eqref{cptuepmrsh2} we note that
\begin{eqnarray}
\nonumber &&
\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((\hye)'-x'\big)}=u^{\varepsilon}+\intt{\tilde{u}((y^{\varepsilon})')}-\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla' \tilde{v}((y^{\varepsilon})')}\\
\label{expuh} &&+\varepsilon\intt{(\etp)'(y^{\varepsilon})}.
\end{eqnarray}
By \eqref{claimvhm}, \eqref{ptcvid} and the dominated convergence theorem,
\begin{eqnarray}
&&\nonumber \tilde{u}((y^{\varepsilon})')\to \tilde{u}\quad\text{strongly in }L^2(\Omega;\mathbb R^2),\\
&&\nonumber
\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla' \tilde{v}((y^{\varepsilon})')}\to 0\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha>3,\\
&&\nonumber
\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla' \tilde{v}((y^{\varepsilon})')}\to v\nabla\tilde{v}\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha=3.
\end{eqnarray}
Hence, by \eqref{etaep2}, we have
$$\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((\hye)'-x'\big)}\to \hat{u}\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha>3,$$
and
$$\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((\hye)'-x'\big)}\to \hat{u}-v\nabla \tilde{v}\quad\text{strongly in }L^2(\omega;\mathbb R^2)\quad\text{for }\alpha=3.$$
To complete the proof of \eqref{cptuepmrsh} and \eqref{cptuepmrsh2}, it remains to show that
\begin{equation}
\label{wcvgraduh}
\frac{1}{\varepsilon^{\alpha-1}}\nabla'\Big(\intt{\big((\hye)'-x'\big)}\Big)\text{ is bounded in }L^2(\Omega;\mathbb M^{2\times 2}).
\end{equation}
By \eqref{expuh} there holds
\begin{eqnarray*}
&&\frac{1}{\varepsilon^{\alpha-1}}\nabla'\Big(\intt{\big((\hye)'-x'\big)}\Big)=\nabla'u^{\varepsilon}+\intt{\nabla'\tilde{u}((y^{\varepsilon})') \nabla'(y^{\varepsilon})'}\\
&&-\intt{\tepp(\nabla')^2 \tilde{v}((y^{\varepsilon})') \nabla'(y^{\varepsilon})'}-\frac{1}{\varepsilon}\intt{\teppp\nabla' \tilde{v}((y^{\varepsilon})')\otimes \nabla'y^{\varepsilon}_3}\\
&&+\varepsilon\intt{\nabla'(\etp\circ y^{\varepsilon})}.
\end{eqnarray*}
By adding and subtracting the matrix $(R^{\varepsilon})'$ we obtain
\begin{eqnarray*}
&&\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\nabla' (y^{\varepsilon})'}=\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\big(\nabla' (y^{\varepsilon})'-(R^{\varepsilon})'\big)}\\
&&+\intt{\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')(R^{\varepsilon})'}.
\end{eqnarray*}
Combining \eqref{rt2} and \eqref{treq3}, we deduce
\begin{equation}
\nonumber
\Big\|\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\big(\nabla' (y^{\varepsilon})'-(R^{\varepsilon})'\big)\Big\|_{L^2(\Omega;\mathbb{M}^{2\times 2})}\leq C\varepsilon^{\alpha-1}\ell_{\varepsilon}.
\end{equation}
On the other hand, by \eqref{rt1} and \eqref{claimvhm}, the maps
$\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)\nabla^2 \tilde{v}((y^{\varepsilon})')(R^{\varepsilon})'\text{ are bounded in }L^2(\Omega;\mathbb M^{2\times 2}).$ The $L^2$-boundedness of the quantity in \eqref{wcvgraduh} follows now by combining \eqref{cptyep}, \eqref{cptuep}, \eqref{cptnep3}, \eqref{treq5} and \eqref{etaep3}.\\
\end{comment}
{\em Step 2: Construction of the plastic strains}\\
Arguing as in \cite[Proof of Lemma 3.6]{MS}, we introduce the sets
$$\sep:=\{x\in \Omega: \exp(\varepsilon^{\alpha-1}\tilde{p}(x))P^{\varepsilon}(x)\in K\},$$
we define
$$\hppe:=\begin{cases}\frac{1}{\varepsilon^{\alpha-1}}\big(\exp(\varepsilon^{\alpha-1}\tilde{p})P^{\varepsilon}-Id\big)&\text{in }\sep,\\
p^{\varepsilon}&\text{in }\Omega\setminus S_{\varepsilon},\end{cases}$$
and
\begin{equation}
\nonumber
\hpe:=Id+\varepsilon^{\alpha-1}\hppe,
\end{equation}
so that, by \eqref{Pep1mrs}, the sequence $(\hpe)$ satisfies \eqref{Pep1mrsh}. Since $\tr{\tilde{p}}=0$,
$$\det(\exp(\varepsilon^{\alpha-1}\tilde{p}))=\exp(\varepsilon^{\alpha-1}\tr{\tilde{p}})=1,$$
therefore
\begin{equation}
\label{sl3ae}
\exp(\varepsilon^{\alpha-1}\tilde{p}(x))P^{\varepsilon}(x)\in SL(3)\quad\text{for a.e. }x\in \Omega.
\end{equation}
By \eqref{sl3ae} we can estimate $\cal{L}^3(\Omega\setminus \sep)$. Indeed by \eqref{prk2} and \eqref{wconvpepmrs} there holds
\begin{eqnarray}
\nonumber &&
\cal{L}^3(\Omega\setminus \sep)\leq c_k^2\intom{|\big(\exp(\varepsilon^{\alpha-1}\tilde{p}(x))P^{\varepsilon}(x)-Id|^2} \\
\nonumber &&=c_k^2\intom{|\big(\exp(\varepsilon^{\alpha-1}\tilde{p}(x))+\varepsilon^{\alpha-1}\exp(\varepsilon^{\alpha-1}\tilde{p}(x))p^{\varepsilon}(x)-Id|^2}\\
\label{measnotk} &&
\leq C\varepsilon^{2(\alpha-1)}\intom{(1+|p^{\varepsilon}(x)|^2)}\leq C\varepsilon^{2(\alpha-1)}.
\end{eqnarray}
Now,
\begin{equation}
\label{diffhen}
\hppe-p^{\varepsilon}=\begin{cases} \frac{1}{\varepsilon^{\alpha-1}}\big(\exp(\varepsilon^{\alpha-1}\tilde{p})-Id\big)P^{\varepsilon}&\text{in }\sep,\\
0&\text{in }\Omega\setminus S_{\varepsilon}.\end{cases}
\end{equation}
By \eqref{Pep1mrs}, \eqref{wconvpepmrs} and \eqref{measnotk} we deduce the following convergence properties:
\begin{equation}
\label{sumdiffph}
\begin{cases}
\|\hppe-p^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C,&\\
\hppe-p^{\varepsilon}\to \tilde{p}\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}),&\\
\hppe+p^{\varepsilon}\rightharpoonup \hat{p}+p\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}),&
\end{cases}
\end{equation}
hence in particular \eqref{wconvpepmrsh} holds true.
Arguing exactly as in \cite[Proof of Lemma 3.6, Step 2 and Step 4]{MS}, we obtain \eqref{limsuphardening} and \eqref{limsupdiss}.\\
{\em Step 3: Convergence properties of the elastic energy}\\
To complete the proof of the theorem it remains to prove \eqref{limsupeng}.
To this purpose, let $w^{\varepsilon}$ be the map defined as
\begin{equation}
\label{defwep}
w^{\varepsilon}:=\frac{(P^{\varepsilon})^{-1}-Id+\varepsilon^{\alpha-1}p^{\varepsilon}}{\varepsilon^{\alpha-1}}=\varepsilon^{\alpha-1}(P^{\varepsilon})^{-1}(p^{\varepsilon})^2.
\end{equation}
By \eqref{prk1} and \eqref{Pep1mrs}, there exists a constant $C$ such that
\begin{equation}
\nonumber
\varepsilon^{\alpha-1}\|p^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
and
\begin{equation}
\label{linftywephm}
\varepsilon^{\alpha-1}\|w^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
for every $\varepsilon$. Furthermore, by \eqref{wconvpepmrs},
$$\|w^{\varepsilon}\|_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1}\quad\text{for every }\varepsilon.$$
By the two previous estimates it follows that $(w^{\varepsilon})$ is uniformly bounded in $L^2(\Omega;\mathbb{M}^{3\times 3})$ and
\begin{equation}
\label{l2wephm}
w^{\varepsilon}\rightharpoonup 0\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Now, by \eqref{defgep} and the frame-indifference property (H3) of $W_{el}$ there holds
\begin{eqnarray}
\nonumber W_{el}(\nep y^{\varepsilon}(P^{\varepsilon})^{-1})&=&W_{el}\big((Id+\varepsilon^{\alpha-1}G^{\varepsilon})(Id+\varepsilon^{\alpha-1}(w^{\varepsilon}-p^{\varepsilon}))\big)\\
\label{id+fep}&=&W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}),
\end{eqnarray}
for a.e. $x\in\Omega$, where
\begin{equation}
\label{deffep1}
F^{\varepsilon}:=G^{\varepsilon}+w^{\varepsilon}-p^{\varepsilon}+\varepsilon^{\alpha-1}G^{\varepsilon}(w^{\varepsilon}-p^{\varepsilon}).
\end{equation}
We note that $$\|G^{\varepsilon}(w^{\varepsilon}-p^{\varepsilon})\|_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C$$
by \eqref{cptGep}, \eqref{wconvpepmrs} and \eqref{l2wephm}. Moreover, by \eqref{cptGep}, \eqref{Pep1mrs} and \eqref{linftywephm},
$$\varepsilon^{\alpha-1}\|G^{\varepsilon}(w^{\varepsilon}-p^{\varepsilon})\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq\varepsilon^{\alpha-1}\|G^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\|(w^{\varepsilon}-p^{\varepsilon})\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C$$
for every $\varepsilon$. Hence
$$\varepsilon^{\alpha-1}G^{\varepsilon}(w^{\varepsilon}-p^{\varepsilon})\rightharpoonup 0\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}),$$
which in turn, by \eqref{cptGep}, \eqref{wconvpepmrs} and \eqref{l2wephm}, yields
\begin{equation}
\label{wconvFeh}
F^{\varepsilon}\rightharpoonup G-p\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Analogously, we define
\begin{equation}
\label{defhwep}
\hat{w}^{\varepsilon}:=\frac{(\hpe)^{-1}-Id+\varepsilon^{\alpha-1}\hppe}{\varepsilon^{\alpha-1}}=\varepsilon^{\alpha-1}(\hpe)^{-1}(\hppe)^2.
\end{equation}
Then,
$$(\hat{P}^{\varepsilon})^{-1}=Id+\varepsilon^{\alpha-1}(\hat{w}^{\varepsilon}-\hat{p}^{\varepsilon}),$$
by \eqref{prk1} and \eqref{Pep1mrsh} we deduce
\begin{equation}
\nonumber
\varepsilon^{\alpha-1}\|\hat{w}^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C,
\end{equation}
and by \eqref{wconvpepmrsh},
\begin{equation}
\nonumber
\hat{w}^{\varepsilon}\rightharpoonup 0\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
We define
\begin{equation}
\label{defhgep}
\hat{G}^{\varepsilon}:=G^{\varepsilon}+\hat{w}^{\varepsilon}-\hppe+\varepsilon^{\alpha-1}G^{\varepsilon}(\hat{w}^{\varepsilon}-\hppe).
\end{equation}
Arguing as before, we can prove that
\begin{equation}
\label{partconvfh}
\hat{G}^{\varepsilon}\rightharpoonup G-\hat{p}\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
We shall prove that there exists a sequence $(\hat{F}^{\varepsilon})\subset L^2(\Omega;\mathbb{M}^{3\times 3})$ satisfying $$W_{el}(\nep \hye(\hpe)^{-1})=W_{el}(Id+\varepsilon^{\alpha-1}\hat{F}^{\varepsilon})$$
and such that
\begin{equation}
\label{propfhep}
\hat{F}^{\varepsilon}-\hat{G}^{\varepsilon}\to N_{\alpha}\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}),
\end{equation}
where
\begin{equation}
\label{deffa}
{N}_{\alpha}:=\mathrm{sym}\Big(\begin{array}{c}\nabla' \tilde{u}-x_3(\nabla')^2 \tilde{v}\\0\end{array}\Big|d\Big)\quad\text{for }\alpha>3,
\end{equation}
and
\begin{eqnarray}
N_{3}&:=&\mathrm{sym}\Big(\begin{array}{c}\nabla \tilde{u}-(x_3+v)(\nabla')^2 \tilde{v}+\frac{\nabla' \tilde{v}\otimes\nabla' \tilde{v}}{2}\\0\end{array}\Big|\begin{array}{c}d'(x',x_3+v)\\d_3(x',x_3+v)+\frac{1}{2}|\nabla'\tilde{v}|^2 \end{array}\Big)
\label{deffa3}
\end{eqnarray}
a.e. in $\Omega$.
To this purpose, we first observe that by \eqref{defgep}, \eqref{defhgep} and the frame-indifference hypothesis (H3) there holds
\begin{eqnarray}
\nonumber&&W_{el}(\nep \hye (\hpe)^{-1})=W_{el}\big(\nabla f^{\varepsilon}(y^{\varepsilon})\nep y^{\varepsilon}(\hpe)^{-1}\big)\\
\label{eqwel}&&=W_{el}\Big((R^{\varepsilon})^T\sqrt{(\nabla f^{\varepsilon}(y^{\varepsilon}))^T\nabla f^{\varepsilon}(y^{\varepsilon})}R^{\varepsilon}(Id+\varepsilon^{\alpha-1}\hat{G}^{\varepsilon})\Big).
\end{eqnarray}
We set
$$M^{\varepsilon}(x):=\frac{\nabla f^{\varepsilon}(x)-Id}{\varepsilon^{\alpha-1}\ell_{\varepsilon}}.$$
By \eqref{unifbdvarphi} there holds
\begin{equation}
\label{unifbdmep}
\|M^{\varepsilon}(y^{\varepsilon})\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
for every $\varepsilon$.
We claim that, to prove \eqref{propfhep} it is enough to show that
\begin{equation}
\label{symmep}
\ell_{\varepsilon}(R^{\varepsilon})^T\mathrm{sym} \big(M^{\varepsilon}(y^{\varepsilon})\big)R^{\varepsilon}\to\begin{cases} \mathrm{sym}\Big(\begin{array}{c}\nabla' \tilde{u}-x_3(\nabla')^2 \tilde{v}\\0\end{array}\Big|d\Big)&\text{if }\alpha>3\\
\mathrm{sym}\Big(\begin{array}{c}\nabla' \tilde{u}-(x_3+v)(\nabla')^2 \tilde{v}\\0\end{array}\Big|d(x',x_3+v)\Big)&\text{if }\alpha=3\end{cases}
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$, and
\begin{equation}
\label{meptmep}
\varepsilon^2\ell_{\varepsilon}^2(R^{\varepsilon})^T(M^{\varepsilon}(y^{\varepsilon}))^TM^{\varepsilon}(y^{\varepsilon})R^{\varepsilon}\to\Big(\begin{array}{cc}\nabla' \tilde{v}\otimes\nabla' \tilde{v}&0\\0&|\nabla'\tilde{v}|^2\end{array}\Big)\quad\text{if }\alpha=3
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. Indeed, a Taylor expansion around the identity yields
$$\sqrt{(Id+F)^T(Id+F)}=Id+\mathrm{sym}\, F+\frac{F^TF}{2}-\frac{(\mathrm{sym}\, F)^2}{2}+O(|F|^3)$$
for every $F\in\mathbb{M}^{3\times 3}$.
Hence,
\begin{eqnarray*}
&&\sqrt{(\nabla f^{\varepsilon}(y^{\varepsilon}))^T\nabla f^{\varepsilon}(y^{\varepsilon})}=Id+\varepsilon^{\alpha-1}\ell_{\varepsilon}\,\mathrm{sym}\, M^{\varepsilon}(y^{\varepsilon})+\frac{\varepsilon^{2\alpha-2}\ell_{\varepsilon}^2}{2}(M^{\varepsilon}(y^{\varepsilon}))^TM^{\varepsilon}(y^{\varepsilon})\\
&&-\frac{\varepsilon^{2\alpha-2}\ell_{\varepsilon}^2}{2}\big(\mathrm{sym}\, M^{\varepsilon}(y^{\varepsilon})\big)^2+O(\varepsilon^{3\alpha-3}\ell_{\varepsilon}^3).
\end{eqnarray*}
Substituting the previous expression into \eqref{eqwel} we obtain
\begin{equation}
\label{id+hfep}
W_{el}(\nep \hye(\hpe)^{-1})=W_{el}(Id+\varepsilon^{\alpha-1}\hat{F}^{\varepsilon})
\end{equation}
where
\begin{eqnarray}
\nonumber\hat{F}^{\varepsilon}&=&\hat{G}^{\varepsilon}+\ell_{\varepsilon}(R^{\varepsilon})^T\mathrm{sym}\, M^{\varepsilon}(y^{\varepsilon})R^{\varepsilon}+\frac{\varepsilon^{\alpha-1}\ell_{\varepsilon}^2}{2}(R^{\varepsilon})^T(M^{\varepsilon}(y^{\varepsilon}))^TM^{\varepsilon}(y^{\varepsilon})R^{\varepsilon}\\
\nonumber &&-\frac{\varepsilon^{\alpha-1}\ell_{\varepsilon}^2}{2}(R^{\varepsilon})^T\big(\mathrm{sym}\, M^{\varepsilon}(y^{\varepsilon})\big)^2R^{\varepsilon}+\varepsilon^{\alpha-1}\ell_{\varepsilon}(R^{\varepsilon})^T\mathrm{sym}\, M^{\varepsilon}(y^{\varepsilon})R^{\varepsilon}\hat{G}^{\varepsilon}+O(\varepsilon^{2\alpha-2}\ell_{\varepsilon}^3)\\
\nonumber &&+O(\varepsilon^{2\alpha-2}\ell_{\varepsilon}^2)\hat{G}^{\varepsilon}
\end{eqnarray}
Now, if $\alpha>3$, by \eqref{rt1} and \eqref{unifbdmep} there holds
\begin{eqnarray*}
\|\hat{F}^{\varepsilon}-\hat{G}^{\varepsilon}-\ell_{\varepsilon}(R^{\varepsilon})^T\mathrm{sym} (M^{\varepsilon}(y^{\varepsilon}))R^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}&\leq& C\varepsilon^{\alpha-1}\ell_{\varepsilon}^2+C\varepsilon^{\alpha-1}\ell_{\varepsilon}\|\hat{G}^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
&+& C\varepsilon^{2\alpha-2}\ell_{\varepsilon}^3+C\varepsilon^{2\alpha-2}\ell_{\varepsilon}^2\|\hat{G}^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}.
\end{eqnarray*}
Hence, by combining \eqref{lp1}, \eqref{lp4}, \eqref{partconvfh} and \eqref{symmep} we deduce \eqref{propfhep}.
In the case $\alpha=3$, by \eqref{unifbdmep} and \eqref{symmep} there holds
$$\varepsilon^4\ell_{\varepsilon}^4\intom{|\mathrm{sym} (M^{\varepsilon}(y^{\varepsilon}))|^4}\leq C\varepsilon^4\ell_{\varepsilon}^4\intom{|\mathrm{sym} (M^{\varepsilon}(y^{\varepsilon}))|^2}\leq C\varepsilon^4\ell_{\varepsilon}^2.$$
Therefore, by \eqref{rt1} and \eqref{unifbdmep} we have
\begin{eqnarray*}
\|\hat{F}^{\varepsilon}-\hat{G}^{\varepsilon}-\ell_{\varepsilon}(R^{\varepsilon})^T\mathrm{sym} (M^{\varepsilon}(y^{\varepsilon}))R^{\varepsilon}-\frac{\varepsilon^2\ell_{\varepsilon}^2}{2}(R^{\varepsilon})^T (M^{\varepsilon}(y^{\varepsilon}))^TM^{\varepsilon}(y^{\varepsilon})R^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
\leq C\varepsilon^2\ell_{\varepsilon}+C\varepsilon^2\ell_{\varepsilon}\|\hat{G}^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}+C\varepsilon^4\ell_{\varepsilon}^3\\
+C\varepsilon^4\ell_{\varepsilon}^2\|\hat{G}^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}.
\end{eqnarray*}
Therefore, once \eqref{symmep} and \eqref{meptmep} are proved, \eqref{propfhep} follows by \eqref{lp1}, \eqref{lp3} and \eqref{partconvfh}.
We now prove \eqref{symmep} and \eqref{meptmep}. By straightforward computations we have
\begin{eqnarray*}
&&\ell_{\varepsilon}\mathrm{sym} \,(M^{\varepsilon}(y^{\varepsilon}))=\mathrm{sym}\Big(\begin{array}{cc}\nabla' \tilde{u}((y^{\varepsilon})')-\teppm(\nabla')^2 \tilde{v}((y^{\varepsilon})')&0\\0&0\end{array}\Big)\\
&&+\frac{1}{\varepsilon}\mathrm{sym}\Big(\begin{array}{cc}0&\big(1-\tepppm\big)\nabla' \tilde{v}((y^{\varepsilon})')\\0&0\end{array}\Big)+\varepsilon\,\mathrm{sym} (\nabla'\etp(y^{\varepsilon})|\partial_3 \etp(y^{\varepsilon})).
\end{eqnarray*}
Now, $\varepsilon \nabla'\etp(y^{\varepsilon})\to 0$ strongly in $L^2(\Omega;\mathbb M^{3\times 2})$ by \eqref{etaep0}. Moreover, \eqref{etaep12} yields
$$\varepsilon \partial_3 \etp(y^{\varepsilon}(x))=d\Big((y^{\varepsilon})'(x),\frac{y^{\varepsilon}_3(x)}{\varepsilon}\Big)\quad\text{for a.e. }x\in\Omega.$$
Hence, by \eqref{3comphest} and \eqref{ptcvid}, there holds
\begin{equation}
\label{star1}
\varepsilon\,(\nabla'\etp(y^{\varepsilon})|\partial_3 \etp(y^{\varepsilon}))\to\begin{cases} (0|d)&\text{if }\alpha>3\\(0|d(x',x_3+v))&\text{if }\alpha=3\end{cases}
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3}).$
On the other hand, by \eqref{claimvhm}, \eqref{ptcvid}, and the dominated convergence theorem
\begin{equation}
\label{star2}
\nabla' \tilde{u}((y^{\varepsilon})')-\tep\Big(\frac{y^{\varepsilon}_3}{\varepsilon}\Big)(\nabla')^2 \tilde{v}((y^{\varepsilon})')\to\begin{cases}\nabla' \tilde{u}-x_3(\nabla')^2 \tilde{v}&\text{if }\alpha>3\\
\nabla' \tilde{u}-(x_3+v)(\nabla')^2 \tilde{v}&\text{if }\alpha=3\end{cases}
\end{equation}
strongly in $L^2(\Omega;\mathbb M^{2\times 2})$.
Claim \eqref{symmep} follows now by combining \eqref{rt1}, \eqref{rt4}, \eqref{tep1hm}, \eqref{star1} and \eqref{star2}.
To prove \eqref{meptmep}, we observe that by \eqref{tep1hm}, \eqref{star1} and \eqref{star2}, if $\alpha=3$ there exists a constant $C$ such that
$$\Big\|\ell_{\varepsilon}M^{\varepsilon}(y^{\varepsilon})-\frac{1}{\varepsilon}\Big(\begin{array}{cc}0&-\nabla' \tilde{v}((y^{\varepsilon})')\\(\nabla' \tilde{v}((y^{\varepsilon})'))^T&0\end{array}\Big)\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C$$
for every $\varepsilon$. Hence, by \eqref{rt1} there holds
\begin{eqnarray}
\nonumber&&\Big\|\varepsilon^2\ell_{\varepsilon}^2(R^{\varepsilon})^T(M^{\varepsilon}(y^{\varepsilon}))^TM^{\varepsilon}(y^{\varepsilon})R^{\varepsilon}\\
\nonumber&&-(R^{\varepsilon})^T\Big(\begin{array}{cc}0&-\nabla' \tilde{v}((y^{\varepsilon})')\\(\nabla' \tilde{v}((y^{\varepsilon})'))^T&0\end{array}\Big)^T\Big(\begin{array}{cc}0&-\nabla' \tilde{v}((y^{\varepsilon})')\\(\nabla' \tilde{v}((y^{\varepsilon})'))^T&0\end{array}\Big)R^{\varepsilon}\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
\nonumber&&\leq C\varepsilon^2\ell_{\varepsilon}\|M^{\varepsilon}(y^{\varepsilon})\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}+C\varepsilon\Big\|\Big(\begin{array}{cc}0&-\nabla' \tilde{v}((y^{\varepsilon})')\\(\nabla' \tilde{v}((y^{\varepsilon})'))^T&0\end{array}\Big)\Big\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})},\\
\label{est1a3}
\end{eqnarray}
which converges to zero due to \eqref{lp1} and \eqref{unifbdmep}.
On the other hand,
\begin{eqnarray*}
&&\Big(\begin{array}{cc}0&-\nabla' \tilde{v}((y^{\varepsilon})')\\(\nabla' \tilde{v}((y^{\varepsilon})'))^T&0\end{array}\Big)^T\Big(\begin{array}{cc}0&-\nabla' \tilde{v}((y^{\varepsilon})')\\(\nabla' \tilde{v}((y^{\varepsilon})'))^T&0\end{array}\Big)\\
&&=\Big(\begin{array}{cc}\nabla' \tilde{v}((y^{\varepsilon})')\otimes \nabla' \tilde{v}((y^{\varepsilon})')&0\\0&|\nabla' \tilde{v}((y^{\varepsilon})')|^2\end{array}\Big).
\end{eqnarray*}
Moreover, by \eqref{ptcvid} and by the dominated convergence theorem there holds
\begin{equation}
\label{est2a3}\Big(\begin{array}{cc}\nabla' \tilde{v}((y^{\varepsilon})')\otimes \nabla' \tilde{v}((y^{\varepsilon})')&0\\0&|\nabla' \tilde{v}((y^{\varepsilon})')|^2\end{array}\Big)\to \Big(\begin{array}{cc}\nabla' \tilde{v}\otimes\nabla' \tilde{v}&0\\0&|\nabla' \tilde{v}|^2\end{array}\Big)
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. Combining \eqref{est1a3} and \eqref{est2a3}, we deduce \eqref{meptmep} and therefore \eqref{propfhep}.\\
{\emph{Step 4: Limsup inequality for the elastic energy}}\\
We are now in a position to prove \eqref{limsupeng}. We argue as in \cite[Lemma 3.6]{MS}. We fix $\delta>0$ and we introduce the sets
$$U_{\varepsilon}:=\{x\in \Omega: \varepsilon^{\alpha-1}(|F^{\varepsilon}|+|\hat{F}^{\varepsilon}|)\leq c_{el}(\delta)\},$$
where $c_{el}(\delta)$ is the constant in \eqref{quadrwel}.
By \eqref{partconvfh} and \eqref{propfhep} it follows that
\begin{equation}
\label{propfhep2}
\hat{F}^{\varepsilon}\rightharpoonup \hat{F}_{\alpha}:=N_{\alpha}+G-\hat{p}\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}),\text{ for }\alpha\geq 3.
\end{equation}
By \eqref{wconvFeh} and by Chebychev inequality we deduce
\begin{equation}
\label{measUeph}
\cal{L}^3(\Omega\setminus U_{\varepsilon})\leq C\varepsilon^{2\alpha-2}.
\end{equation}
Since $$\nep \hye (\hpe)^{-1}=\nabla f^{\varepsilon}(y^{\varepsilon})\big(\nep y^{\varepsilon}(P^{\varepsilon})^{-1}\big)P^{\varepsilon}(\hpe)^{-1},$$
property \eqref{lemmams} yields
\begin{eqnarray}
\nonumber &&|W_{el}(\nep \hye (\hpe)^{-1})-W_{el}(\nep y^{\varepsilon}(P^{\varepsilon})^{-1})|\\
\label{diffwel} &&
\leq C(1+W_{el}(\nep y^{\varepsilon}(P^{\varepsilon})^{-1}))(|\nabla f^{\varepsilon}(y^{\varepsilon})-Id|+|P^{\varepsilon}(\hpe)^{-1}-Id|)
\end{eqnarray}
a.e. in $\Omega$. By \eqref{prk1} and \eqref{Pep1mrsh} there holds
\begin{eqnarray*}
\|P^{\varepsilon}(\hpe)^{-1}-Id\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq c_k\|P^{\varepsilon}-\hpe\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\\
\leq c_k\|(Id-\exp(\varepsilon^{\alpha-1}\tilde{p}))P^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}
\leq C\varepsilon^{\alpha-1},
\end{eqnarray*}
hence, by combining \eqref{unifbdvarphi}, \eqref{measUeph} and \eqref{diffwel} we deduce
\begin{eqnarray}
\nonumber &&\frac{1}{\varepsilon^{2\alpha-2}}\Big|\int_{\Omega\setminus U_{\varepsilon}}{W_{el}(\nep \hye (\hpe)^{-1})}-\int_{\Omega\setminus U_{\varepsilon}}{W_{el}(\nep y^{\varepsilon}(P^{\varepsilon})^{-1})}\Big|\\
\label{badsetestimate}&&\leq C\varepsilon^{\alpha-1}(1+\ell_{\varepsilon})\Big(1+\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep y^{\varepsilon}(P^{\varepsilon})^{-1})}\Big),
\end{eqnarray}
which tends to zero owing to \eqref{engest2} and \eqref{lp1}.
On the other hand, on the sets $U_{\varepsilon}$ we can use the estimate \eqref{quadrwel}. Hence, by \eqref{id+fep}, \eqref{id+hfep} and the quadratic structure of $Q$ we obtain
\begin{eqnarray}
\nonumber &&\frac{1}{\varepsilon^{2\alpha-2}}\int_{U_{\varepsilon}}{W_{el}(\nep \hye (\hpe)^{-1})\,dx}-\frac{1}{\varepsilon^{2\alpha-2}}\int_{U^{\varepsilon}}{W_{el}(\nep y^{\varepsilon}(P^{\varepsilon})^{-1})\,dx}\\
\nonumber && \leq\delta \intom{(|F^{\varepsilon}|^2+|\hat{F}^{\varepsilon}|^2)}+\intom{(Q(\hat{F}^{\varepsilon})-Q(F^{\varepsilon}))}\\
\label{cit1}&&= \delta \intom{(|F^{\varepsilon}|^2+|\hat{F}^{\varepsilon}|^2)}+\frac{1}{2}\intom{\mathbb C(\hat{F}^{\varepsilon}-F^{\varepsilon}):(\hat{F}^{\varepsilon}+F^{\varepsilon})}.
\end{eqnarray}
Now, by \eqref{wconvFeh} and \eqref{propfhep2} there holds
\begin{equation}
\label{cit2}
\hat{F}^{\varepsilon}+F^{\varepsilon}\rightharpoonup \hat{F}_{\alpha}+G-p\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Moreover,
\begin{equation}
\label{diffefhe}
\hat{F}^{\varepsilon}-F^{\varepsilon}\to \hat{F}_{\alpha}-G+p\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Indeed, by \eqref{propfhep} and \eqref{propfhep2} it is enough to show that
$$\hat{G}^{\varepsilon}-F^{\varepsilon}\to p-\hat{p}\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}).$$
By \eqref{deffep1} and \eqref{defhgep} we have
$$\hat{G}^{\varepsilon}-F^{\varepsilon}=(Id+\varepsilon^{\alpha-1}G^{\varepsilon})(\hat{w}^{\varepsilon}-\hppe-w^{\varepsilon}+p^{\varepsilon}).$$
Now, by \eqref{diffhen}, \eqref{defwep} and \eqref{defhwep}, $\hat{w}^{\varepsilon}-w^{\varepsilon}=0$ in $\Omega\setminus S^{\varepsilon}$, whereas in the sets $S^{\varepsilon}$ we have
\begin{eqnarray*}
&&\hat{w}^{\varepsilon}-w^{\varepsilon}=\varepsilon^{\alpha-1}(\hpe)^{-1}(\hppe)^2-\varepsilon^{\alpha-1}(P^{\varepsilon})^{-1}(p^{\varepsilon})^2\\
&&=\varepsilon^{\alpha-1}(P^{\varepsilon})^{-1}\big(\exp(-\varepsilon^{\alpha-1}\tilde{p})(\hppe)^2-(p^{\varepsilon})^2\big)\\
&&=\varepsilon^{\alpha-1}(P^{\varepsilon})^{-1}(\exp(-\varepsilon^{\alpha-1}\tilde{p})-Id)(\hppe)^2+\varepsilon^{\alpha-1}(P^{\varepsilon})^{-1}((\hppe)^2-(p^{\varepsilon})^2).
\end{eqnarray*}
Therefore, by \eqref{Pep1mrsh}, \eqref{wconvpepmrsh}, \eqref{Pep1mrs} and \eqref{wconvpepmrs}, we deduce
\begin{eqnarray*}
&&\|\hat{w}^{\varepsilon}-w^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C(\varepsilon^{\alpha-1}+\|\hppe-p^{\varepsilon}\|_{L^2(\Omega;\mathbb{M}^{3\times 3})})\leq C,\\
&&\|\hat{w}^{\varepsilon}-w^{\varepsilon}\|_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1},\\
&&\|\hat{w}^{\varepsilon}-w^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C(1+\|\hppe-p^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}).
\end{eqnarray*}
Combining these estimates with \eqref{cptGep} and \eqref{sumdiffph} we obtain \eqref{diffefhe}.
Consider now the case $\alpha>3$. By \eqref{badsetestimate}--\eqref{diffefhe} we have
\begin{eqnarray*}
&&\nonumber\limsup_{\varepsilon\to 0} \Big\{\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep \hye (\hpe)^{-1})}-\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep y^{\varepsilon} (P^{\varepsilon})^{-1})}\Big\}\\
&&\leq\frac{1}{2}\intom{\mathbb C(\hat{F}_{\alpha}-G+p):(\hat{F}_{\alpha}+G-p)}+C\delta.
\end{eqnarray*}
Since $\delta$ is arbitrary, we deduce
\begin{eqnarray}
&&\nonumber\limsup_{\varepsilon\to 0} \Big\{\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep \hye (\hpe)^{-1})}-\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep y^{\varepsilon} (P^{\varepsilon})^{-1})}\Big\}\\
&&\nonumber \leq\frac{1}{2}\intom{\mathbb C(\hat{F}_{\alpha}-G+p):(\hat{F}_{\alpha}+G-p)}\\
&&\label{qtrick}=\intom{Q(\hat{F}_{\alpha})}-\intom{Q(G-p)}\leq \intom{Q(\hat{F}_{\alpha})}-\intom{Q_2(G'-p')}.
\end{eqnarray}
By \eqref{deffa} and \eqref{propfhep2}, up to an approximation argument, we may assume that $d$ is such that
$$Q(\hat{F}_{\alpha})=Q_2(\mathrm{sym}\,\nabla' \hat{u}-x_3(\nabla')^2\hat v-\hat{p}').$$
This, together with \eqref{qtrick}, implies \eqref{limsupeng}.
In the case $\alpha=3$ a preliminary approximation argument is needed. {Let $(\tilde{u}^k)$ be a sequence in $C^{\infty}_c(\omega\cup\gamma_n;\mathbb R^2)$, such that
$$\tilde{u}^k\to \tilde{u}+v\nabla' \tilde{v}\quad\text{strongly in }W^{1,2}(\omega;\mathbb R^2)$$
(such a sequence exists by Lemma \ref{bdc} because $\tilde{u}\in C^{\infty}_c(\omega\cup\gamma_n;\mathbb R^2)$ and $\tilde{v}\in C^{\infty}_c(\omega\cup\gamma_n)$).
Let also ${v}^k\in C^{\infty}_c(\omega)$ be such that
$${v}^k\to {v}\quad\text{strongly in }L^2(\omega)$$
and set}
$${d}^k(x):=d(x',x_3-{v}^k(x'))\quad\text{for a.e. }x\in\Omega.$$
Since $d\in C^{\infty}_c(\Omega;\mathbb R^3)$, there exists an open set $O\subset \mathbb R^2$ such that $\overline{O}\subset \omega$ and $d^k(x',x_3)=0$ for every $x\in (\omega\setminus \overline{O})\times \mathbb R$. Moreover, $d^k(x',x_3)=0$ for every $x\in\mathbb R^3$ such that $|x_3|>\frac{1}{2}+\|{v}^k\|_{L^{\infty}(\mathbb R^2)}$. Hence, $d^k\in C^{\infty}(\mathbb R^3;\mathbb R^3)$ and $$\mathrm{supp}\, {d^k}\subset \overline{O}\times \big(-\tfrac12-\|{v}^k\|_{L^{\infty}(\mathbb R^2)}, \tfrac 12+\|{v}^k\|_{L^{\infty}(\mathbb R^2)}\big).$$
It is easy to see that \eqref{deffa3}, \eqref{propfhep2} and \eqref{badsetestimate}--\eqref{diffefhe} can still be deduced, and for every $k$ we can construct a sequence $(\hye_k, \hpe_k)$ that satisfies \eqref{cvidh}--\eqref{wconvpepmrsh} with $\hat{u}$ replaced by $u+\tilde{u}^k$, and
\begin{eqnarray*}
&&\nonumber\limsup_{\varepsilon\to 0} \Big\{\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep \hye_k (\hpe_k)^{-1})}-\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep y^{\varepsilon} (P^{\varepsilon})^{-1})}\Big\}\\
&&\leq\frac{1}{2}\intom{\mathbb C(\hat{F}^k-G+p):(\hat{F}^k+G-p)},
\end{eqnarray*}
where
$$\hat{F}^k:=\mathrm{sym}\Big(\begin{array}{c}\nabla' \tilde{u}^k-(x_3+v)(\nabla')^2 \tilde{v}+\frac{\nabla' \tilde{v}\otimes\nabla' \tilde{v}}{2}\\0\end{array}\Big|\begin{array}{c}d'(x', x_3+{v}-{v}^k)\\d_3(x',x_3+{v}-{v}^k)+\frac{1}{2}|\nabla' \tilde{v}|^2\end{array}\Big)+G-\hat{p}.$$
On the other hand,
$$\hat{F}^k\to \mathrm{sym}\Big(\begin{array}{c}\nabla' \tilde{u}-x_3(\nabla')^2 \tilde{v}+\nabla' {v}\otimes\nabla' \tilde{v}+\frac{\nabla' \tilde{v}\otimes\nabla' \tilde{v}}{2}\\0\end{array}\Big|\begin{array}{c}d'\\d_3+\frac{1}{2}|\nabla' \tilde{v}|^2\end{array}\Big)+G-\hat{p}=:\hat{F}$$
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$, as $k\to +\infty$. A diagonal argument leads then to the estimate
\begin{eqnarray}
&&\nonumber\limsup_{\varepsilon\to 0} \frac{1}{\varepsilon^{2\alpha-2}}\Big(\intom{W_{el}(\nep \hye (\hpe)^{-1})}-\intom{W_{el}(\nep y^{\varepsilon} (P^{\varepsilon})^{-1})}\Big)\\
&&\label{aend}\leq \frac{1}{2}\intom{\mathbb C(\hat{F}-G+p):(\hat{F}+G-p)}.
\end{eqnarray}
Up to a further approximation, we may assume that $d$ is such that
$$Q(\hat{F})=Q_2\Big(\mathrm{sym}\,\nabla' \hat{u}-x_3(\nabla')^2 \hat{v}+\frac{1}{2}\nabla' \hat{v}\otimes\nabla' \hat{v}-\hat{p}'\Big),$$
hence \eqref{limsupeng} follows by \eqref{aend}.
\end{proof}
\section{Convergence of quasistatic evolutions}
\label{pquas}
The first part of this section is devoted to the proof of Theorem \ref{cvstress}. We first prove the theorem for $\alpha>3$ and then we show how the proof must be modified for $\alpha=3$.
\begin{proof}[Proof of Theorem \ref{cvstress} in the case $\alpha>3$]
The proof is divided into five steps.\\
{\em{Step 0: A priori estimates on the elasto-plastic energy}}\\
Set $y^{\varepsilon}(t):=\pep(t,\zep(t))$ for every $t\in [0,T]$. Arguing as in the proof of \eqref{rsze}, it is immediate to see that
{\begin{equation}
\label{bdtep}
y^{\varepsilon}(t,x)=\pep(t,(x',\varepsilon x_3))\quad\cal{H}^2\text{ - a.e. on }\Gamma_d.
\end{equation}}
In this step we shall show that there exists a constant $C$ such that for every $t\in [0,T]$ and every $\varepsilon$ there holds
\begin{equation}
\label{unifapest}
\frac{1}{\varepsilon^{\alpha-1}}\|\mathrm{dist}(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t),SO(3))\|_{L^2(\Omega)}+\|p^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}+\|\varepsilon^{\alpha-1}p^{\varepsilon}(t)\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C.
\end{equation}
To this purpose, we first remark that since $t\mapsto (\zep(t), P^{\varepsilon}(t))$ is an $\varepsilon$-quasistatic evolution, then
\begin{equation}
\label{isink}
P^{\varepsilon}(t,x)\in K\quad\text{for a.e. }x\in\Omega,\text{ for every }\varepsilon\text{ and }t,
\end{equation}
hence $\varepsilon^{\alpha-1}p^{\varepsilon}(t)\in K-Id$ for every $\varepsilon$ and $t$ and by \eqref{prk1} there exists a constant $C$ such that
\begin{equation}
\label{unifbdpep}
\|\varepsilon^{\alpha-1}p^{\varepsilon}(t)\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C\quad\text{for every }\varepsilon\text{ and }t.
\end{equation}
By the minimality condition (qs1), taking $\tilde{z}(x)=(x',\varepsilon x_3)$ and $\tilde{P}(x)= Id$ for every $x\in \Omega$, and observing that $W_{hard}(Id)=0$ a.e. in $\Omega$ by \eqref{prh4}, we deduce
\begin{eqnarray}
\label{estiniz} \frac{1}{\varepsilon^{2\alpha-2}}\cal{F}_{\varepsilon}(t,\zep(t),P^{\varepsilon}(t))\leq \frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}\big(\nabla \pep (t, (x',\varepsilon x_3))\big)}
+\frac{1}{\varepsilon^{\alpha-1}}\intom{D(P^{\varepsilon}(t),Id)},
\end{eqnarray}
for every $t\in [0,T]$ and for all $\varepsilon$. By \eqref{prd2} and \eqref{isink}, there holds
$$D(P^{\varepsilon}(t),Id)=D(Id, (P^{\varepsilon})^{-1}(t))\leq c_7|(P^{\varepsilon})^{-1}(t)-Id|\leq c_7c_K|Id-P^{\varepsilon}(t)|,$$
where the last inequality follows by \eqref{prk1}. Hence, Holder inequality yields
\begin{equation}
\label{estiniz2}
\frac{1}{\varepsilon^{\alpha-1}}\intom{D(P^{\varepsilon}(t),Id)}\leq\frac{C}{\varepsilon^{\alpha-1}}\|Id-P^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}.
\end{equation}
On the other hand, by frame indifference (H3) of $W_{el}$ we obtain
\begin{eqnarray*}
{W_{el}\big(\nabla \pep(t, (x',\varepsilon x_3))\big)}={W_{el}\Big(\sqrt{(\nabla \pep)^T (t, (x',\varepsilon x_3))\nabla \pep (t,(x',\varepsilon x_3)}\Big)}
\end{eqnarray*}
for every $x\in\Omega$ and for all $t\in [0,T]$. By \eqref{treq1}, \eqref{lp2} and \eqref{gradpep} there holds
\begin{eqnarray*}
\nonumber&&\nabla \pep(t, (x',\varepsilon x_3))=Id+\varepsilon^{\alpha-1}\Big(\begin{array}{cc}\nabla' {u}^0(t,x')-x_3(\nabla')^2 {v}^0(t,x')&0\\0&0\end{array}\Big)\\
&&+\varepsilon^{\alpha-2}\Big(\begin{array}{cc}0&-\nabla' {v}^0(t,x')\\(\nabla' {v}^0(t,x'))^T&0\end{array}\Big),
\end{eqnarray*}
for every $x\in\Omega$. Since $\alpha>3$, we deduce
\begin{eqnarray*}
&&(\nabla \pep)^T (t,(x',\varepsilon x_3))\nabla \pep (t, (x',\varepsilon x_3))\\
&&=Id+2\varepsilon^{\alpha-1}\mathrm{sym}\Big(\begin{array}{cc}\nabla' {u}^0(t,x')-x_3(\nabla')^2 {v}^0(t,x')&0\\0&0\end{array}\Big)+o(\varepsilon^{\alpha-1}),
\end{eqnarray*}
and
\begin{equation}
\label{sqrtdeco}
\sqrt{(\nabla \pep)^T (t,(x',\varepsilon x_3))\nabla \pep (t, (x',\varepsilon x_3))}=Id+\varepsilon^{\alpha-1}M(t,x)+o(\varepsilon^{\alpha-1}),
\end{equation}
where
$$M(t,x)=\mathrm{sym}\Big(\begin{array}{cc}\nabla' {u}^0(t,x')-x_3(\nabla')^2 {v}^0(t,x')&0\\0&0\end{array}\Big)\quad\text{for every }x\in\Omega.$$
Therefore,
$$\frac{1}{\varepsilon^{2\alpha-2}}W_{el}\big(\nabla \pep(t, (x',\varepsilon x_3))\big)=\frac{1}{\varepsilon^{2\alpha-2}}W_{el}\big(Id+\varepsilon^{\alpha-1}M(t,x)+o(\varepsilon^{\alpha-1})\big)$$
for every $x\in \Omega$. Now, by the smoothness of $u^0$ and $v^0$, there exists a constant $C$ such that
\begin{equation}
\label{unifquad}
\sup_{t\in [0,T]}\|M(t)\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation} and there exist $\overline{\varepsilon}$ such that, if $\varepsilon<\overline{\varepsilon}$, for every $t\in [0,T]$
$$|\varepsilon^{\alpha-1}M(t)+o(\varepsilon^{\alpha-1})|\leq c_{el}(1),$$
where $c_{el}$ is the constant in \eqref{quadrwel}.
Therefore, by \eqref{quadrwel}, \eqref{growthcondQ}, and \eqref{unifquad} we have
\begin{equation}
\label{ubnp}\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}\big(\nabla \pep(t, (x',\varepsilon x_3))\big)}\leq C\Big(\intom{|M(t)|^2}+1\Big)\leq C
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$.
By combining \eqref{estiniz}, \eqref{estiniz2} and \eqref{ubnp} we obtain
\begin{eqnarray}
\nonumber &&\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}\big(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t)\big)}+\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{hard}(P^{\varepsilon}(t))}\\
\nonumber &&\leq C\Big(1+\frac{1}{\varepsilon^{\alpha-1}}\|Id-P^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\Big).\\
\label{estiniz3}
\end{eqnarray}
Now, by \eqref{prh3} there holds
$$\frac{c_6}{\varepsilon^{2\alpha-2}}\intom{|Id-P^{\varepsilon}(t)|^2}\leq C\Big(1+\frac{1}{\varepsilon^{\alpha-1}}\|Id-P^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\Big),$$
which in turn, by Cauchy inequality implies
\begin{equation}
\label{part1ubd}
\|p^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}=\frac{1}{\varepsilon^{\alpha-1}}\|Id-P^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$. On the other hand, by \eqref{estiniz3} and \eqref{part1ubd}, we deduce
\begin{equation}
\label{unelt}
\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}\big(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t)\big)}\leq C,
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$. Estimate \eqref{unifapest} follows now by \eqref{unifbdpep}, \eqref{part1ubd}, \eqref{unelt} and the growth condition (H4).\\
{\em{Step 1: A priori estimate on the dissipation functional}}.\\
In this step we shall show that there exists a constant $C$, such that
\begin{equation}
\label{bunifapest}
\frac{1}{\varepsilon^{\alpha-1}}\cal{D}(P^{\varepsilon};0,t)\leq C\quad\text{for every }\varepsilon\text{ and for all }t\in [0,T].
\end{equation}
By (qs2), \eqref{convE0} and \eqref{estiniz3}--\eqref{unelt} it is enough to show that there exists a constant $C$ such that
\begin{equation}
\label{unifbdeep}
\Big|\frac{1}{\varepsilon^{\alpha-1}}{\intom{E^{\varepsilon}(t):\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))}}\Big|\leq C
\end{equation}
for every $\varepsilon$ and $t\in [0,T]$. To prove \eqref{unifbdeep}, we first deduce some properties of the map $t\mapsto E^{\varepsilon}(t)$.
Let $R\in SO(3)$. By \eqref{prk1} and \eqref{isink} there holds
\begin{eqnarray*}
&&|\nep y^{\varepsilon}(t)-R|^2=|\nep y^{\varepsilon}(t)-RP^{\varepsilon}(t)+\varepsilon^{\alpha-1}Rp^{\varepsilon}(t)|^2\\
&&\leq 2|\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t)-R|^2|P^{\varepsilon}(t)|^2+2\varepsilon^{2\alpha-2}|p^{\varepsilon}(t)|^2\\
&&\leq 2\,c_K^2|\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t)-R|^2+2\varepsilon^{2\alpha-2}|p^{\varepsilon}(t)|^2.
\end{eqnarray*}
Hence, the growth condition (H4) implies
$$\|\mathrm{dist}(\nep y^{\varepsilon}(t),SO(3))\|^2_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\Big(\intom{W_{el}(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t))}+\varepsilon^{2\alpha-2}\|p^{\varepsilon}(t)\|^2_{L^2(\Omega;\mathbb{M}^{3\times 3})}\Big),$$
which in turn yields
\begin{equation}
\label{unifdist}
\|\mathrm{dist}(\nep y^{\varepsilon}(t),SO(3))\|^2_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{2\alpha-2}
\end{equation}
by \eqref{unifapest} and \eqref{unelt}.
By \eqref{bdtep} and \eqref{unifdist}, the sequence $y^{\varepsilon}(t)$ fulfills the hypotheses of Theorem \ref{compactbd1}. Hence, for every $t\in [0,T]$ there exists a sequence of maps $(R^{\varepsilon}(t))\subset W^{1,\infty}(\omega;\mathbb{M}^{3\times 3})$ such that
\begin{eqnarray}
&&\label{rt1t} R^{\varepsilon}(t,x')\in SO(3)\quad\text{for every }x'\in \omega,\\
&&\label{rt2t} \|\nep y^{\varepsilon}(t)-R^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1},\\
&&\label{rt3t} \|\partial_i R^{\varepsilon}(t)\|_{L^2(\omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-2},\quad i=1,2,\\
&&\label{rt4t} \|R^{\varepsilon}(t)-Id\|_{L^2(\omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-2},
\end{eqnarray}
where the constant $C$ is independent of $\varepsilon$ and $t$.
We consider the auxiliary maps
$$w^{\varepsilon}(t):=\frac{(Id+\varepsilon^{\alpha-1}p^{\varepsilon}(t))^{-1}-Id+\varepsilon^{\alpha-1}p^{\varepsilon}(t)}{\varepsilon^{\alpha-1}},$$
the elastic strains
$$G^{\varepsilon}(t):=\frac{(R^{\varepsilon}(t))^T\nep y^{\varepsilon}(t)-Id}{\varepsilon^{\alpha-1}},$$
and the matrices
\begin{equation}
\label{deffept}
F^{\varepsilon}(t):=G^{\varepsilon}(t)+w^{\varepsilon}(t)-p^{\varepsilon}(t)+\varepsilon^{\alpha-1}G^{\varepsilon}(t)(w^{\varepsilon}(t)-p^{\varepsilon}(t)),
\end{equation}
for all $t\in [0,T]$. Clearly we have
\begin{equation}
\label{decompep}
(P^{\varepsilon})^{-1}(t)=Id+\varepsilon^{\alpha-1}(w^{\varepsilon}(t)-p^{\varepsilon}(t))\quad\text{and}\quad \nep y^{\varepsilon}(t)=R^{\varepsilon}(t)(Id+\varepsilon^{\alpha-1}G^{\varepsilon}(t)).
\end{equation}
Since
\begin{equation}
\label{numerarepag35}
w^{\varepsilon}(t)=\varepsilon^{\alpha-1}(Id+\varepsilon^{\alpha-1}p^{\varepsilon}(t))^{-1}(p^{\varepsilon}(t))^2
\end{equation}
for every $t\in [0,T]$, by \eqref{unifapest} and \eqref{isink} there holds
\begin{equation}
\label{linftybdw}
\|\varepsilon^{\alpha-1}w^{\varepsilon}(t)\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C\quad\text{for every }t\in [0,T],
\end{equation}
\begin{equation}
\label{numerarepag352}
\|w^{\varepsilon}(t)\|_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1}\quad\text{for every }t\in [0,T],
\end{equation}
and
\begin{equation}
\label{l2bdw}
\|w^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\quad\text{for every }t\in [0,T].
\end{equation}
Combining \eqref{numerarepag352} and \eqref{l2bdw} we deduce
\begin{equation}
\label{weakl2w}
w^{\varepsilon}(t)\rightharpoonup 0\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3})\text{ for every }t\in [0,T].
\end{equation}
On the other hand, \eqref{rt1t} and \eqref{rt2t} yield
\begin{equation}
\label{unifbdrt2}
\|G^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$.
Collecting \eqref{unifapest}, \eqref{linftybdw}, \eqref{l2bdw} and \eqref{unifbdrt2}, we obtain
\begin{eqnarray}
\nonumber&&\|F^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq \|G^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}+\|w^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}+\|p^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
\label{unifbdfep}&&+\|G^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\|\varepsilon^{\alpha-1}(w^{\varepsilon}(t)-p^{\varepsilon}(t))\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{eqnarray}
for every $\varepsilon$ and for all $t\in [0,T]$.
Now, by \eqref{deffept}, \eqref{decompep} and the frame-indifference (H3) of $W_{el}$ we deduce the decomposition
\begin{equation}
\label{stressrot}
\eep(t)=R^{\varepsilon}(t)\eepp(t)(R^{\varepsilon}(t))^T
\end{equation}
for every $t\in [0,T]$, where
\begin{equation}
\nonumber
\eepp(t):=\frac{1}{\varepsilon^{\alpha-1}}DW_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))^T.
\end{equation}
We argue as in \cite[Proof of Theorem 3.1, Steps 2--3]{M-S} and we first show that there exist two positive constants $k_1, k_2$, independent of $\varepsilon$, such that
\begin{equation}
\label{claimunifinteg}
|\eepp(t)|\leq k_1\Big(\frac{W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))}{\varepsilon^{\alpha-1}}+k_2|F^{\varepsilon}(t)|\Big)
\end{equation}
for every $t\in [0,T]$ and for a.e. $x\in \Omega$.
Indeed, let $c_{el_2}$ be the constant in \eqref{locquad}.
Suppose that $\varepsilon^{\alpha-1}|F^{\varepsilon}(t)|\geq c_{el_2}$. We remark that (H1), \eqref{isink} and \eqref{unelt} imply in particular that
$$\det (\nep y^{\varepsilon}(t))>0\quad\text{a.e. in }\Omega.$$
Therefore, by \eqref{mandel2} there holds
\begin{equation}
\label{toolclaimstress1}
|\eepp(t)|\leq \frac{c_3}{\varepsilon^{\alpha-1}}\Big(W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))+1\Big)\leq {c_3}\Big(\frac{W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))}{\varepsilon^{\alpha-1}}+\frac{1}{c_{el_2}}|F^{\varepsilon}(t)|\Big).
\end{equation}
Consider now the case where $\varepsilon^{\alpha-1}|F^{\varepsilon}(t)|< c_{el_2}$. Then, by \eqref{locquad} there holds
$$DW_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))\leq\varepsilon^{\alpha-1}(2R_\C+1)|F^{\varepsilon}(t)|,$$
which in turn implies
\begin{equation}
\label{toolclaimstress2}
|\eepp(t)|\leq C|F^{\varepsilon}(t)|(|Id|+|\varepsilon^{\alpha-1}F^{\varepsilon}(t)|)\leq C|F^{\varepsilon}(t)|.
\end{equation}
Collecting \eqref{toolclaimstress1} and \eqref{toolclaimstress2}, we obtain \eqref{claimunifinteg}.
By \eqref{unelt}, \eqref{unifbdfep} and \eqref{claimunifinteg}, for every measurable $\Lambda\subset \Omega$, the following estimate holds true:
\begin{equation}
\label{unifinteg}
\int_{\Lambda}{|\eepp(t)|\,dx}\leq k_1\int_{\Lambda}{\Big(\frac{W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))}{\varepsilon^{\alpha-1}}+k_2|F^{\varepsilon}(t)|\Big)}\leq C(|\Lambda|^{\frac{1}{2}}+\varepsilon^{\alpha-1}),
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$. By \eqref{rt1t} there holds also
\begin{equation}
\label{unifintegnt}
\int_{\Lambda}{|E^{\varepsilon}(t)|\,dx}\leq C(|\Lambda|^{\frac{1}{2}}+\varepsilon^{\alpha-1}),
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$.
Let now $\gamma\in (0,\alpha-2)$ be the positive constant in the definition of the maps $\tep$. Let $O_{\varepsilon}(t)$ be the set given by
$$O_{\varepsilon}(t):=\{x\in \Omega:\,{\varepsilon}^{\alpha-1-\gamma}|F^{\varepsilon}(t,x)|\leq c_{el_2}\},$$
and let $\chi_{\varepsilon}(t):\Omega\to \{0,1\}$ be the map
$$\chi_{\varepsilon}(t,x)=\begin{cases}1&\text{if }x\in O_{\varepsilon}(t),\\
0&\text{otherwise}.
\end{cases}$$ By Chebychev inequality and \eqref{unifbdfep} we deduce \begin{equation}
\label{measbep}
\cal{L}^3(\Omega \setminus O_{\varepsilon}(t))\leq C {\varepsilon}^{2(\alpha-1-\gamma)},
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$. By combining \eqref{unifinteg} and \eqref{measbep} we conclude that
\begin{equation}
\label{badsetstress}
\|(1-\chi_{\varepsilon}(t)){\tilde{E}}^{\varepsilon}(t)\|_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1-\gamma}\quad\text{for every }t\in [0,T].
\end{equation}
By \eqref{unifintegnt} the previous estimate implies also
\begin{equation}
\label{badsetstressnt}
\|(1-\chi_{\varepsilon}(t)){{E}}^{\varepsilon}(t)\|_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1-\gamma}\quad\text{for every }t\in [0,T].
\end{equation}
On the other hand \eqref{locquad} yields the following estimate on the sets $O_{\varepsilon}(t)$:
\begin{eqnarray*}
|\chi_{\varepsilon}(t){\tilde{E}}^{\varepsilon}(t)|\leq (2R_\C+1)|F^{\varepsilon}(t)||Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t)|\leq C(1+c_{el_2}\varepsilon^{\gamma})|F^{\varepsilon}(t)|,
\end{eqnarray*}
which in turn, by \eqref{unifbdfep}, implies
\begin{equation}
\label{goodsetsufb}
\|\chi_{\varepsilon}(t){\tilde{E}}^{\varepsilon}(t)\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$.
By \eqref{lp1}, \eqref{rt1t}, \eqref{badsetstressnt} and \eqref{goodsetsufb}, and since $E^{\varepsilon}(t)$ is symmetric by Remark \ref{Esym}, to prove \eqref{unifbdeep} it is enough to show that there exists a constant $C$ such that
\begin{equation}
\label{linftybdwk}
\Big\|\frac{1}{\varepsilon^{\alpha-1}}\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))\Big\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C\ell_{\varepsilon}
\end{equation}
and
\begin{equation}
\label{l2bdwk}
\Big\|\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))\Big)\Big\|_{L^{2}(\Omega;\mathbb{M}^{3\times 3})}\leq C
\end{equation}
for every $\varepsilon$ and for all $t\in [0,T]$. By \eqref{gradpep}, there holds
\begin{eqnarray}
\nonumber&&\frac{1}{\varepsilon^{\alpha-1}}\nabla \dot{\pep}(t,\zep(t))=\Big(\begin{array}{cc}\nabla' \dot{u}^0(t,\bzep(t))-\tep\big(\frac{\zep_3(t)}{\varepsilon}\big)(\nabla')^2 \dot{v}^0(t,\bzep(t))&0\\0&0\end{array}\Big)\\
\label{gradpept}&&+\frac{1}{\varepsilon}\Big(\begin{array}{cc}0&-\dot{\tep}\big(\frac{\zep_3(t)}{\varepsilon}\big)\nabla' \dot{v}^0(t,\bzep(t))\\(\nabla' \dot{v}^0(t,\bzep(t)))^T&0\end{array}\Big).
\end{eqnarray}
Estimate \eqref{linftybdwk} follows directly by \eqref{treq3}, \eqref{treq5}, \eqref{lp2}, and \eqref{bddinverse}. To prove \eqref{l2bdwk}, we first provide an estimate for the $L^2$ norm of the maps $\frac{1}{\varepsilon}\zep_3(t)$. To this purpose, let $v^{\varepsilon}(t)$ be defined as in \eqref{cvvt}. It is easy to see that
$$v^{\varepsilon}(t)=\frac{1}{\varepsilon^{\alpha-2}}\intt{y^{\varepsilon}_3(t)}\quad\text{and}\quad\nabla' v^{\varepsilon}(t)=\frac{1}{\varepsilon^{\alpha-2}}\intt{{\nabla'y^{\varepsilon}_3(t)}}$$
for every $\varepsilon$ and for all $t\in [0,T]$.
By \eqref{bdtep}, arguing as in the proof of Theorem \ref{compactbd1},
$$v^{\varepsilon}(t)=v^0(t)\quad\cal{H}^1\text{ - a.e. on }\gamma_d.$$
By \eqref{rt2t} and \eqref{rt4t}, we have
$$\|\nabla' v^{\varepsilon}(t)\|_{L^2(\omega;\mathbb R^2)}\leq C$$
for every $\varepsilon$ and $t\in [0,T]$. Hence, by Poincar\'e inequality we deduce
$$\|v^{\varepsilon}(t)-v^0(t)\|_{L^2(\omega)}\leq C\|\nabla' v^{\varepsilon}(t)-\nabla' v^0(t)\|_{L^2(\omega;\mathbb R^2)}\leq C,$$
which in turn, by the smoothness of $v^{0}$, yields
$$\|v^{\varepsilon}(t)\|_{L^2(\omega)}\leq C\quad\text{for every }\varepsilon\text{ and for all }t\in [0,T].$$
By \eqref{rt2t}, \eqref{rt4t} and Poincar\'e-Wirtinger inequality, we deduce
\begin{equation}
\label{estimate3}
\Big\|\frac{y^{\varepsilon}_3(t)}{\varepsilon}-x_3-\varepsilon^{\alpha-3}v^{\varepsilon}(t)\Big\|_{L^2(\Omega)}\leq C\Big\|\frac{\partial_3 y^{\varepsilon}_3(t)}{\varepsilon}-1\Big\|_{L^2(\Omega)}\leq C\varepsilon^{\alpha-2}
\end{equation}
for every $t\in [0,T]$, which implies
\begin{equation}
\label{uf3}
\Big\|\frac{y^{\varepsilon}_3(t)}{\varepsilon}\Big\|_{L^2(\Omega)}\leq C\quad\text{for every }\varepsilon\text{ and }t\in [0,T].
\end{equation}
On the other hand,
\begin{equation}
\label{relzep}
z^{\varepsilon}(t)=\vep(t,y^{\varepsilon}(t))\quad\text{a.e. in }\Omega,
\end{equation}
hence by \eqref{3comp},
\begin{equation}
\label{zep3}
\frac{\zep_3(t)}{\varepsilon}=\frac{y^{\varepsilon}_3(t)}{\varepsilon}-\varepsilon^{\alpha-3}v^0(t,({\varphi}^{\varepsilon})'(t,y^{\varepsilon}(t))).
\end{equation}
Therefore \eqref{treq2} and \eqref{uf3} yield
\begin{equation}
\label{l2usf}
\Big\|\tep\Big(\frac{\zep_3(t)}{\varepsilon}\Big)\Big\|_{L^2(\Omega)}\leq \Big\|\frac{\zep_3(t)}{\varepsilon}\Big\|_{L^2(\Omega)}\leq C \quad\text{for every }\varepsilon\text{ and }t\in [0,T].
\end{equation}
By Lemma \ref{cvproptep}, we deduce
\begin{equation}
\label{l2usf2}
\Big\|1-\dot{\tep}\Big(\frac{\zep_3(t)}{\varepsilon}\Big)\Big\|_{L^2(\Omega)}\leq \frac{2}{\ell_{\varepsilon}}\quad\text{for every }\varepsilon\text{ and }t\in [0,T].
\end{equation}
Collecting \eqref{lp2}, \eqref{gradpept}, \eqref{l2usf} and \eqref{l2usf2}, we deduce that there exists a constant $C$ such that
$$\Big\|\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym} \nabla \dot{\pep}(t,\zep(t))\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C$$
for every $\varepsilon$ and for all $t\in [0,T]$. Therefore, to prove \eqref{l2bdwk}, it remains only to study the quantity
$$\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))\big((\nabla \pep)^{-1}(t,\zep(t))-Id\big)\Big).$$
By \eqref{bddinverse},
$$\|(\nabla \pep(t))^{-1}-Id\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C\quad\text{for every }\varepsilon\text{ and }t\in [0,T].$$
By \eqref{l2usf}, the first term in the right hand side of \eqref{gradpept} is uniformly bounded in $L^2(\Omega;\mathbb{M}^{3\times 3})$. Therefore, it remains to show that
\begin{equation}
\label{newstar}
\frac{1}{\varepsilon}\Big(\begin{array}{cc}0&-\dot{\tep}\big(\frac{\zep_3(t)}{\varepsilon}\big)\nabla' \dot{v}^0(t,\bzep(t))\\(\nabla' \dot{v}^0(t,\bzep(t)))^T&0\end{array}\Big)\Big((\nabla \pep)^{-1}(t,\zep(t))-Id\Big)
\end{equation}
is uniformly bounded in $L^2(\Omega;\mathbb{M}^{3\times 3})$.
By \eqref{treq5} and by the smoothness of $v^0$, there holds
\begin{equation}
\label{estt1}
\Big\|\frac{1}{\varepsilon}\Big(\begin{array}{cc}0&-\dot{\tep}\big(\frac{\zep_3(t)}{\varepsilon}\big)\nabla' \dot{v}^0(t,\bzep(t))\\(\nabla' \dot{v}^0(t,\bzep(t)))^T&0\end{array}\Big)\Big\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq \frac{C}{\varepsilon}
\end{equation}
for every $t\in [0,T]$.
On the other hand,
\begin{equation}
\label{estt2bis}
(\nabla \pep)^{-1}(t,\zep(t))=\nabla \vep(t,y^{\varepsilon}(t))\quad\text{a.e. in }\Omega.
\end{equation}
Property \eqref{gradinv1} yields the estimate
\begin{equation}
\label{estt3}
\|\nabla \vep_3(t,y^{\varepsilon}(t))-e_3\|_{L^{\infty}(\Omega;\mathbb R^3)}\leq C\varepsilon^{\alpha-2}
\end{equation}
for every $t\in[0,T]$, whereas by \eqref{treq5}, \eqref{expgr} and \eqref{bddinverse}
$$\|\nabla \vep_i(t,y^{\varepsilon}(t))-e_i\|_{L^{2}(\Omega;\mathbb R^3)}\leq C\varepsilon^{\alpha-1}\Big\|\tep\Big(\frac{\vep_3(t,y^{\varepsilon}(t))}{\varepsilon}\Big)\Big\|_{L^2(\omega)}+C\varepsilon^{\alpha-2},$$
hence by \eqref{relzep} and \eqref{l2usf} we obtain
\begin{equation}
\label{estt4}
\|\nabla \vep_i(t,y^{\varepsilon}(t))-e_i\|_{L^{2}(\Omega;\mathbb R^3)}\leq C\varepsilon^{\alpha-2}.
\end{equation}
By combining \eqref{estt1}--\eqref{estt4}, we deduce
\begin{eqnarray}
\nonumber
&&\Big\|\frac{1}{\varepsilon}\Big(\begin{array}{cc}0&-\dot{\tep}\big(\frac{\zep_3(t)}{\varepsilon}\big)\nabla' \dot{v}^0(t,\bzep(t))\\(\nabla' \dot{v}^0(t,\bzep(t)))^T&0\end{array}\Big)\Big((\nabla \pep)^{-1}(t,\zep(t))-Id\Big)\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
\label{utiledopo}&&\leq C\varepsilon^{\alpha-3}
\end{eqnarray}
for every $\varepsilon$ and $t\in [0,T]$, therefore the quantity in \eqref{newstar} is uniformly bounded in $L^2(\Omega;\mathbb M^{3\times 3})$, and the proof of \eqref{l2bdwk} is complete.
By \eqref{badsetstress}--\eqref{l2bdwk}, since all estimates are uniform both in $\varepsilon$ and $t$, we deduce \eqref{unifbdeep}, which in turn yields \eqref{bunifapest}.\\
{\em{Step 2: Reduced Stability}}\\
Owing to the a priori bounds \eqref{unifapest} and \eqref{bunifapest}, we can apply the generalized version of Helly's Selection Principle in \cite[Theorem A.1]{MRS}. To show it, take $\cal{Z}:=L^2(\Omega;\mathbb{M}^{3\times 3})$ endowed with the weak topology of $L^2$, and set
$$\cal{D}_{\varepsilon}(z_1,z_2):=\frac{1}{\varepsilon^{\alpha-1}}\intom{D(Id+\varepsilon^{\alpha-1}z_1, Id+\varepsilon^{\alpha-1}z_2)}$$
and
$$\cal{D}_{\infty}(z_1,z_2):=\intom{H(z_2-z_1)}$$
for every $z_1,z_2\in L^2(\Omega;\mathbb{M}^{3\times 3})$. Hypotheses (A.1) and (A.2) of \cite[Theorem A.1]{MRS} are satisfied by \eqref{growthh}--\eqref{liminfdiss}. Hypothesis (A.3) follows by adapting \cite[Lemmas 3.4 and 3.5]{MS}, whereas condition (A.4) follows directly by \eqref{unifapest} and \eqref{bunifapest}.
Hence, by \cite[Theorem A.1]{MRS}, there holds
\begin{equation}
\label{convpept}
\begin{array}{c} p^{\varepsilon}(t)\rightharpoonup p(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3})\quad\text{for every }t\in [0,T],\vspace{0.2 cm}\\
\cal{D}_{H_D}(p;0,t)\leq \displaystyle{\liminf_{\varepsilon\to 0}}\frac{1}{\varepsilon^{\alpha-1}}\cal{D}(P^{\varepsilon};0,t)\quad\text{for every }t\in [0,T].
\end{array}
\end{equation}
Moreover, by $\eqref{convP0}$, $p(0)=\mathring{p}$.
Let now $t\in [0,T]$ be fixed. By \eqref{bdtep}, \eqref{rt2t}, \eqref{rt4t} and Poincar\'e inequality, up to subsequences there holds
\begin{equation}
\label{cvitsb}
y^{\varepsilon}(t)\to\Big(\begin{array}{c}x'\\0\end{array}\Big)\quad\text{strongly in }W^{1,2}(\Omega;\mathbb R^3).
\end{equation}
Arguing as in the proof of Theorem \ref{liminfineq} and owing to \eqref{unifapest}, we deduce the existence of a pair $(u^*(t),v^*(t))\in W^{1,2}(\omega;\mathbb R^2)\times W^{2,2}(\omega)$ such that $(u^*(t),v^*(t),p(t))\in\cal{A}(u^0(t),v^0(t))$ and a sequence $\varepsilon_j\to 0$ such that
\begin{eqnarray}
\label{cvu+}&&u^{\varepsilon_j}(t)\rightharpoonup u^*(t)\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2),\\
\label{cvv+}&&v^{\varepsilon_j}(t)\to v^*(t)\quad\text{strongly in }W^{1,2}(\omega).
\end{eqnarray}
In particular, by \eqref{convu0} and \eqref{convv0} we have $u^*(0)=\mathring{u}$ and $v^*(0)=\mathring{v}$.
By \eqref{unifbdrt2} up to extracting a further subsequence, there exists a map $G^*(t)\in L^2(\Omega;\mathbb{M}^{3\times 3})$ such that
\begin{equation}
\label{debsubsgep}
G^{\varepsilon_j}(t)\rightharpoonup G^*(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3})
\end{equation}
and the $2\times 2$ submatrix $(G^*)'(t)$ satisfies
\begin{equation}
\label{gaff+}
(G^*)'(t,x) = G^*_0(t,x') - x_3 (\nabla')^2 v^*(t,x')\quad\text{for a.e. }x\in\Omega,
\end{equation}
where
\begin{eqnarray}
&&\label{Ga3+} \mathrm{sym}\, G^*_0(t) = \mathrm{sym}\,\nabla' u^*(t).
\end{eqnarray}
We shall show that the triple $(u^*(t),v^*(t),p(t))$ satisfies the reduced stability condition (qs1$_{r\alpha}$). By {Lemma \ref{bdc}}, it is enough to prove the inequality for triples $(\hat{u},\hat{v},\hat{p})\in\cal{A}(u^0(t),v^0(t))$ such that
{\begin{eqnarray*}
&&\tilde{u}:=\hat{u}-u^*(t)\in C^{\infty}_c(\omega\cup\gamma_n;\mathbb R^2),\\
&&\tilde{v}:=\hat{v}-v^*(t)\in C^{\infty}_c(\omega\cup\gamma_n),\\
&&\tilde{p}:=\hat{p}-p^*(t)\in C^{\infty}_c(\Omega;\mathbb M^{3\times 3}_D).
\end{eqnarray*}}
By Theorem \ref{mutrecseq} there exists a sequence $(\hat{y}^{\varepsilon_j},\hat{P}^{\varepsilon_j})\in\cal{A}_{\varepsilon_j}(\pepjt(t))$ satisfying
\begin{eqnarray*}
&& \intom{Q_2(\mathrm{sym}\,\hat{G}'-\hat{p}')}+\intom{\B{\hat{p}}}\\
&&-\intom{Q_2(\mathrm{sym}\,({G}^*)'(t)-{p}'(t))}-\intom{\B{p(t)}}+\intom{H_D(\hat{p}-p(t))}\\
&&\geq \limsup_{\varepsilon_j\to 0} \Big\{\frac{1}{{\varepsilon_j}^{2\alpha-2}}\intom{W_{el}({\nep}_j {\hat{y}}^{\varepsilon_j} ({\hat{P}}^{\varepsilon_j})^{-1})}+\frac{1}{{\varepsilon_j}^{2\alpha-2}}\intom{W_{hard}({\hat{P}}^{\varepsilon_j})}\\
&&-\frac{1}{{\varepsilon_j}^{2\alpha-2}}\intom{W_{el}({\nep}_j y^{\varepsilon_j}(t) (P^{\varepsilon_j})^{-1}(t))}-\frac{1}{{\varepsilon_j}^{2\alpha-2}}\intom{W_{hard}(P^{\varepsilon_j}(t))}\\
&&+\frac{1}{{\varepsilon_j}^{\alpha-1}}\intom{D(P^{\varepsilon_j}(t),\hat{P}^{\varepsilon_j})}\Big\}\\
\end{eqnarray*}
where
\begin{equation}
\nonumber
\hat{G}'(x', x_3) := \hat{G}_0(x') - x_3 (\nabla')^2 \hat{v}(x')\quad\text{a.e. in }\Omega,
\end{equation}
and
\begin{eqnarray}
&&\nonumber \mathrm{sym}\, \hat{G}_0 = \mathrm{sym}\, \nabla' \hat{u}.
\end{eqnarray}
Inequality (qs1$_{r\alpha}$) follows now by the $\varepsilon$-stability (qs1) of $(y^{\varepsilon}(t),P^{\varepsilon}(t))$.
By strict convexity of the quadratic form $Q_2$, an adaptation of \cite[Theorem 3.8]{DDM} yields that, once $p(t)$ is identified, there exist unique $u(t)\in W^{1,2}(\omega;\mathbb R^2)$ and $v(t)\in W^{2,2}(\omega)$ such that (qs1$_{r\alpha}$) holds at time $t$. This implies that $u^*(t)=u(t)$, $v^*(t)=v(t)$ for every $t\in [0,T]$ and both \eqref{cvu+} and \eqref{cvv+} hold for the whole sequences $u^{\varepsilon}(t)$ and $v^{\varepsilon}(t)$ and for every $t\in [0,T]$. Moreover, by \eqref{debsubsgep}--\eqref{Ga3+} we have
\begin{equation}
\nonumber
\mathrm{sym}\,{(G^*)}'(t) = \mathrm{sym}\,\nabla'u(t) - x_3 (\nabla')^2 v(t)
\end{equation}
and
$$\mathrm{sym}\,(G^{\varepsilon})'(t)\rightharpoonup \mathrm{sym}\,\nabla'u(t) - x_3 (\nabla')^2 v(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3})\quad\text{for every }t\in [0,T].$$
{\em{Step 3: Convergence of the scaled stress}}\\
In this step we shall show that for every $t\in[0,T]$ there exists a subsequence $\varepsilon_j$, possibly depending on $t$, such that
\begin{equation}
\label{goodsetstress}
\chi_{\varepsilon_j}(t)E^{\varepsilon_j}(t)\rightharpoonup E^*(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}),
\end{equation}
where
\begin{equation}
\label{cestar}
E^*(t)=\mathbb C(G^*(t)-p(t)).
\end{equation}
To this purpose, for $t\in [0,T]$ fixed, let $\varepsilon_j\to 0$ be such that \eqref{debsubsgep} holds and let $F^{\varepsilon_j}(t)$ be the map defined in \eqref{deffept}.
By \eqref{unifapest}, \eqref{linftybdw} and \eqref{debsubsgep} we deduce
$$\|\varepsilon^{\alpha-1}G^{\varepsilon_j}(t)(w^{\varepsilon_j}(t)-p^{\varepsilon_j}(t))\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\leq C\quad\text{for every }\varepsilon_j.$$
On the other hand, by \eqref{unifapest}, \eqref{l2bdw}, and \eqref{debsubsgep}, there holds
\begin{equation}
\label{numerarepag39}
\varepsilon^{\alpha-1}G^{\varepsilon_j}(t)(w^{\varepsilon_j}(t)-p^{\varepsilon_j}(t))\to 0\quad\text{strongly in }L^1(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Hence, we conclude that
\begin{equation}
\label{toolconvfep}
\varepsilon^{\alpha-1}G^{\varepsilon_j}(t)(w^{\varepsilon_j}(t)-p^{\varepsilon_j}(t))\rightharpoonup 0\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Collecting \eqref{deffept}, \eqref{weakl2w}, \eqref{convpept}, \eqref{debsubsgep} and \eqref{toolconvfep} we obtain
\begin{equation}
\label{convfep}
F^{\varepsilon_j}(t)\rightharpoonup G^*(t)-p(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
By \eqref{measbep} we deduce that $\chi_{\varepsilon_j}(t)\to 1$ boundedly in measure, therefore by \eqref{convfep} there holds
\begin{equation}
\nonumber
\chi_{\varepsilon_j}(t)F^{\varepsilon_j}(t)\rightharpoonup G^*(t)-p(t)\quad \text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Now, estimate \eqref{unifinteg} implies that the sequence $({\tilde{E}}^{\varepsilon_j}(t))$ is uniformly bounded in $L^1(\Omega;\mathbb{M}^{3\times 3})$ and is equiintegrable, hence by Dunford-Pettis Theorem, up to extracting a further subsequence, there exists ${E}^*(t)\in L^1(\Omega;\mathbb M^{3\times 3}_{sym})$ such that
\begin{equation}
\nonumber
{\tilde{E}}^{\varepsilon_j}(t)\rightharpoonup {E}^*(t)\quad\text{weakly in }L^1(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Using a Taylor expansion argument in $O_{\varepsilon}(t)$, and arguing as in \cite[Proof of Theorem 3.1, Step 3]{M-S} we deduce
\begin{equation}
\nonumber
\chi_{\varepsilon_j}(t){\tilde{E}}^{\varepsilon_j}(t)\rightharpoonup \mathbb C (G^*(t)-p(t))\quad\text{weakly in }L^2(\Omega;\mathbb M^{3\times 3}_{sym}).
\end{equation}
By \eqref{rt1t} and \eqref{rt4t}, the sequence $(R^{\varepsilon}(t))$ converges boundedly in measure to the identity, hence the previous convergence implies in particular \eqref{goodsetstress} and \eqref{cestar}.\\
{\em{Step 4: Characterization of the limit stress}}\\
In this step we shall show that
\begin{equation}
\label{claimstress}
E^*(t)=\mathbb C_2 (\mathrm{sym}\,\nabla' u(t)-x_3(\nabla')^2v(t)-p'(t)):=E(t)\quad\text{for every }t\in [0,T].
\end{equation}
This, in turn, will imply that all convergence properties established in the previous step hold for the entire sequences and for every $t\in [0,T]$.
We first remark that, choosing $\tilde{P}=P^{\varepsilon}(t)$ in (qs1) there holds
\begin{equation}
\label{minimel}
\intom{W_{el}(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t))}\leq \intom{W_{el}(\nep \tilde{y} (P^{\varepsilon})^{-1}(t))},
\end{equation}
for every $\tilde{y}\in W^{1,2}(\Omega;\mathbb R^3)$ such that {$\tilde{y}=\phi^{\varepsilon}(t,(x',\varepsilon x_3))\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}.
{Let $\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ be such that $\eta\circ \phi^{\varepsilon}(t,(x',\varepsilon x_3))=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$.} Then, in particular, we can consider in \eqref{minimel} inner variations of the form $y^{\varepsilon}+\lambda \eta \circ y^{\varepsilon}$, where $\lambda \in \mathbb R$. By the growth hypothesis \eqref{mandel2} and by the minimality condition \eqref{minimel}, an adaptation of \cite[Theorem 2.4]{Ba} shows that $y^{\varepsilon}(t)$ satisfies the following Euler-Lagrange equation:
\begin{equation}
\label{euler}
\intom{DW_{el}(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t))(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t))^T:\nabla \eta (y^{\varepsilon}(t))}=0
\end{equation}
for every $t\in [0,T]$ and for every $\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ such that {$\eta \circ \phi^{\varepsilon}(t,(x',\varepsilon x_3))=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}.
Hence,
\begin{equation}
\label{eulerstress}
\intom{E^{\varepsilon}(t):\nabla \eta (y^{\varepsilon}(t))}=0
\end{equation}
for every $t\in [0,T]$ and for every $\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ such that {$\eta \circ \phi^{\varepsilon}(t,(x',\varepsilon x_3))=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}.
Now, fix $t\in [0,T]$ and let $\varepsilon_j$ be the sequence selected in the previous step.
Let {$\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ be such that $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}. We consider the maps $\eta^{\varepsilon_j}(t)\in W^{1,\infty}(\mathbb R^3,\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ defined as
$$\eta^{\varepsilon_j}(t):=\varepsilon_j \eta\big(\vepj_1(t), \vepj_2(t), \tfrac{1}{\varepsilon_j}\vepj_3(t)\big).$$
It is clear that {$\eta^{\varepsilon_j}(t)\circ \phi^{\varepsilon_j}(t,(x',\varepsilon_j x_3))=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}, hence we can use $\eta^{\varepsilon_j}(t)$ as a test function in \eqref{eulerstress} and we obtain
\begin{equation}
\label{testfctpart}
\intom{E^{\ep_{j}}(t):\nabla\eta^{\varepsilon_j} (y^{\varepsilon_j}(t))}=0
\end{equation}
for every $j$.
Now, set $\xi^{\varepsilon_j}(x)=\big(\vepj_1(t,x), \vepj_2(t,x), \tfrac{1}{\varepsilon_j}\vepj_3(t,x)\big)$ for every $x\in \mathbb R^3$. We can rewrite \eqref{testfctpart} as
\begin{eqnarray}
\nonumber &&\sum_{i=1,2,3}\varepsilon_j\intom{E^{\ep_{j}}(t)e_i\cdot\sum_{k=1,2}\partial_k\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))\partial_i \xi^{\varepsilon_j}_k(y^{\varepsilon_j}(t))}\\
\nonumber &&+\varepsilon_j\sum_{i=1,2}\intom{E^{\ep_{j}}(t)e_i\cdot\partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))\partial_i \xi^{\varepsilon_j}_3(y^{\varepsilon_j}(t))}\\
\label{eulerconti}&&+\varepsilon_j\intom{E^{\ep_{j}}(t)e_3\cdot \partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))\partial_3 \xi^{\varepsilon_j}_3(y^{\varepsilon_j}(t))}=0.
\end{eqnarray}
Since $\eta\in W^{1,\infty}(\mathbb R^3,\mathbb R^3)$ and $E^{\ep_{j}}(t)$ is uniformly bounded in $L^1(\Omega;\mathbb{M}^{3\times 3})$ by \eqref{unifintegnt}, estimate \eqref{bddinverse} yields that the term in the first row of \eqref{eulerconti} converges to zero. By \eqref{gradinv1}, the term in the second row of \eqref{eulerconti} can be bounded as follows:
\begin{eqnarray*}
&&\Big|\varepsilon_j\sum_{i=1,2}\intom{E^{\ep_{j}}(t)e_i\cdot\partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))\partial_i \xi^{\varepsilon_j}_3(y^{\varepsilon_j}(t))}\Big|\leq C\varepsilon_j^{\alpha-2}\|E^{\ep_{j}}(t)e_i\|_{L^1(\Omega;\mathbb R^3)}
\end{eqnarray*}
and hence converges to zero due to \eqref{unifintegnt}.
By \eqref{gradinv1}, there holds
\begin{eqnarray}
\nonumber &&\Big|\varepsilon_j\intom{E^{\ep_{j}}(t)e_3\cdot \partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))\partial_3 \xi^{\varepsilon_j}_3(y^{\varepsilon_j}(t))}-\intom{E^{\ep_{j}}(t)e_3\cdot \partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))}\Big|\\
\nonumber &&\leq C{\varepsilon_j}^{\alpha-2}\|E^{\ep_{j}}(t)e_3\|_{L^1(\Omega;\mathbb R^3)}.
\end{eqnarray}
which converges to zero, owing to \eqref{unifintegnt}. Therefore, \eqref{eulerconti} yields
\begin{equation}
\label{remaineuler}
\lim_{\varepsilon_j\to 0}\intom{E^{\ep_{j}}(t)e_3\cdot \partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))}= 0.
\end{equation}
By \eqref{lp1}, \eqref{distinv1} and \eqref{cvitsb} we deduce
\begin{equation}
\nonumber
\xi^{\varepsilon_j}_k(y^{\varepsilon_j}(t))\to x_k\quad\text{strongly in }L^2(\Omega)\quad\text{for }k=1,2.
\end{equation}
Since $\alpha>3$, by \eqref{distinv13} and \eqref{estimate3} we have $\xi^{\varepsilon_j}_3(y^{\varepsilon_j}(t))\to x_3$ strongly in $L^{2}(\Omega)$. Hence, by the regularity of $\eta$,
\begin{eqnarray}
\nonumber && \partial_3\eta (\xi^{\varepsilon_j}(y^{\varepsilon_j}(t)))\to \partial_3 \eta (t,x)\quad\text{a.e. in }\Omega\text{ as }\varepsilon_j\to 0.
\end{eqnarray}
By the dominated convergence theorem and by combining \eqref{lp1}, \eqref{badsetstressnt}, \eqref{goodsetstress} and \eqref{remaineuler}, we conclude that
$$\intom{E^*(t)e_3\cdot \partial_3 \eta (t)}=0,$$
for every {$\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ such that $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}. Hence,
\begin{equation}
\label{3colstress}
E^*(t)e_3=0\quad\text{a.e. in } \Omega.
\end{equation}
By combining \eqref{linearmin}, \eqref{cestar} and \eqref{3colstress} we deduce \eqref{claimstress}. Moreover, by \eqref{linearmin}
{\begin{equation}
\label{nnpserve}
\mathrm{sym}\,G^*(t)-p(t)=\A(\mathrm{sym}\, \nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t)),\text{ for every }t\in [0,T].
\end{equation}}
{\em{Step 5: Reduced energy balance}}\\
To complete the proof of the theorem it remains to show that the triple $(u(t),v(t),p(t))$ satisfies
\begin{eqnarray}
\nonumber &&\intom{Q_2\big(\mathrm{sym}\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t)\big)}+\intom{\B{p(t)}}+\cal{D}(p;0,t)\\
\nonumber &&\leq\intom{Q_2\big(\mathrm{sym}\nabla' u(0)-x_3(\nabla')^2 v(0)-p'(0)\big)}+\intom{\B{p(0)}}\\
\nonumber &&+\int_0^t{\intom{\mathbb C_2(\mathrm{sym}\,\nabla' u(s)-x_3(\nabla')^2 v(s)-p'(s)):\Big(\begin{array}{cc}\nabla \dot{u}^0(s)-x_3(\nabla')^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)}\,ds}.\\
\label{rin1}
\end{eqnarray}
Once \eqref{rin1} is proved, the opposite inequality in (qs2$_{r\alpha}$) follows by adapting of \cite[Theorem 4.7]{DDM}.
We claim that, to prove \eqref{rin1} it is enough to show that
\begin{equation}
\label{claimtool21}
\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))\Big)\to \mathrm{sym}\Big(\begin{array}{cc}\nabla' \dot{u^0}(t)-x_3(\nabla')^2 \dot{v}^0(t)&0\\0&0\end{array}\Big)
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$, for all $t\in [0,T]$.
\begin{comment}
\begin{eqnarray}
\nonumber &&\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))(\nabla \pep(t,\zep(t)))^{-1}\Big)\\
\nonumber &&\to \mathrm{sym}\Big(\begin{array}{cc}\nabla \dot{u^0}(t)-x_3\nabla^2 \dot{v}^0(t)+(v(t)-v^0(t))\nabla^2 \dot{v}^0(t)+\nabla \dot{v}^0(t)\otimes \nabla v^0(t)&0\\0&-\frac{d}{dt}\frac{|\nabla v^0(t)|^2}{2}\end{array}\Big)\\
\label{claimtool22}
\end{eqnarray}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$ for every $t\in [0,T]$.
\end{comment}
Indeed, if \eqref{claimtool21} holds, by \eqref{lp1}, \eqref{badsetstressnt}, \eqref{linftybdwk}, \eqref{goodsetstress} and \eqref{claimstress}, one has
$$\frac{1}{\varepsilon^{\alpha-1}}\intom{E^{\varepsilon}(s):\nabla \dot{\pep}(s,\zep(s))(\nabla \pep)^{-1}(s,\zep(s))}\to \intom{E(s): \mathrm{sym}\Big(\begin{array}{cc}\nabla' \dot{u^0}(s)-x_3(\nabla')^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)},
$$
for every $s\in [0,t]$. Hence, by \eqref{unifbdeep} and the dominated convergence theorem we deduce
\begin{eqnarray}
\nonumber &&
\frac{1}{\varepsilon^{\alpha-1}}\int_0^t{\intom{E^{\varepsilon}(s):\nabla \dot{\pep}(s,\zep(s))(\nabla \pep)^{-1}(s,\zep(s))}\,ds}\\
\label{claimwork1}&&\to\int_0^t{\intom{E(s): \mathrm{sym}\Big(\begin{array}{cc}\nabla \dot{u^0}(s)-x_3\nabla^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)}\,ds}.
\end{eqnarray}
On the other hand, by Theorem \ref{liminfineq} there holds
\begin{eqnarray}
\nonumber &&\intom{Q_2\big(\mathrm{sym}\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t)\big)}+\intom{\B{p(t)}}\\
\nonumber &&\leq\liminf_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\cal{F}_{\varepsilon}(t,\zep(t),P^{\varepsilon}(t)).
\end{eqnarray}
Therefore, once \eqref{claimtool21} is proved, by \eqref{convpept} and \eqref{claimwork1}, passing to the liminf in the $\varepsilon$ energy balance (qs2), inequality \eqref{rin1} follows by \eqref{convE0}.
To prove \eqref{claimtool21}, we first study some properties of the maps $\zep(t)$. By \eqref{fcomp} and \eqref{relzep} there holds
$$\zep_i(t)=y^{\varepsilon}_i(t)-\varepsilon^{\alpha-1}u^0_i(t,\bvep(t,y^{\varepsilon}(t)))+\varepsilon^{\alpha-1}\tep\Big(\frac{\vep_3(t,y^{\varepsilon}(t))}{\varepsilon}\Big)\partial_i v^0(t,\bvep(t,y^{\varepsilon}(t)))$$
for every $t\in [0,T]$, $i=1,2$. Hence, by \eqref{treq3}, \eqref{lp1} and \eqref{cvitsb} we deduce
\begin{equation}
\label{distzepi}
\zep_i(t)\to x_i\quad\text{strongly in }L^2(\Omega)\quad\text{for every }t\in[0,T],\,i=1,2.
\end{equation}
Moreover, by \eqref{zep3} we have
\begin{eqnarray}
\nonumber
&&\Big\|\frac{\zep_3(t)}{\varepsilon}-x_3-\varepsilon^{\alpha-3}v(t)+\varepsilon^{\alpha-3}v^0(t)\Big\|_{L^2(\Omega)}\leq \Big\|\frac{y^{\varepsilon}_3(t)}{\varepsilon}-x_3-\varepsilon^{\alpha-3}v^{\varepsilon}(t)\Big\|_{L^2(\Omega)}\\
\nonumber &&+\varepsilon^{\alpha-3}\|v^{\varepsilon}(t)-v(t)\|_{L^2(\Omega)}+\varepsilon^{\alpha-3}\|v^0(t)-v^0(t,\bvep(t,y^{\varepsilon}(t)))\|_{L^2(\Omega)}.
\end{eqnarray}
Hence, by \eqref{distinv1}, \eqref{cvvt}, \eqref{estimate3} and \eqref{cvitsb},
\begin{equation}
\label{distzep3}
\Big\|\frac{\zep_3(t)}{\varepsilon}-x_3-\varepsilon^{\alpha-3}v(t)+\varepsilon^{\alpha-3}v^0(t)\Big\|_{L^2(\Omega)}\to 0
\end{equation}
for every $t\in [0,T]$. In particular, by Lemma \ref{cvproptep},
\begin{equation}
\label{usfz}
\tep\Big(\frac{\zep_3(t)}{\varepsilon}\Big)\to x_3\quad\text{strongly in }L^2(\Omega).
\end{equation}
Arguing as in the proof of \eqref{l2bdwk}, we perform the decomposition
\begin{eqnarray}
\nonumber&& \frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))\Big)=\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\big(\nabla\dot{\pep}(t,\zep(t))\big)\\
\label{decompst}&&+\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla\dot{\pep}(t,\zep(t))\Big((\nabla \pep)^{-1}(t,\zep(t))-Id\Big)\Big).
\end{eqnarray}
By \eqref{lp2}, \eqref{gradpept}, \eqref{l2usf2}, \eqref{distzepi} and \eqref{usfz}, we obtain
\begin{equation}
\label{term11}
\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\big(\nabla\dot{\pep}(t,\zep(t))\big)\to \mathrm{sym}\Big(\begin{array}{cc}\nabla \dot{u}^0(t)-x_3\nabla^2 \dot{v}^0(t)&0\\0&0\end{array}\Big)
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. To study the second term in the right-hand side of \eqref{decompst}, we remark that by \eqref{gradpept} and \eqref{utiledopo}, there holds
\begin{eqnarray*}
&&\Big\|\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t)(\zep(t))-Id)\Big)\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
&&\leq C \Big(1+\Big\|\tep\Big(\frac{\zep_3(t)}{\varepsilon}\Big)\Big\|_{L^2(\Omega)}\Big)\|(\nabla \pep(t))^{-1}(\zep(t))-Id\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}+C\varepsilon^{\alpha-3}.
\end{eqnarray*}
On the other hand, \eqref{lp2}, \eqref{gradinv}, \eqref{gradinv1} and \eqref{estt2bis} yield
$$\|(\nabla \pep(t))^{-1}(\zep(t))-Id\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1}\ell_{\varepsilon}.$$
Hence, by \eqref{lp1} and \eqref{usfz} we have
\begin{eqnarray}
\frac{1}{\varepsilon^{\alpha-1}}\mathrm{sym}\Big(\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))-Id)\Big)\to 0
\label{term11bis}
\end{eqnarray}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. By combining \eqref{term11} and \eqref{term11bis} we obtain \eqref{claimtool21}. This completes the proof of the theorem.
\end{proof}
We give only a sketch of the proof of Theorem \ref{cvstress} in the case $\alpha=3$, as it follows closely that of Theorem \ref{cvstress} for $\alpha>3$.
\begin{proof}[Proof of Theorem \ref{cvstress} in the case $\alpha=3$]{\quad}\\
\emph{Steps 0--3}\\
Steps 0--3 follow as a straightforward adaptation of the corresponding steps in the case $\alpha>3$,
where now \eqref{sqrtdeco} holds with
\begin{eqnarray*}M(t,x)&:=&\mathrm{sym}\Big(\begin{array}{cc}\nabla' {u}^0(t,x')-x_3(\nabla')^2 {v}^0(t,x')&0\\0&0\end{array}\Big)\\
&+&\frac{1}{2}\Big(\begin{array}{cc}\nabla' v^0(t,x')\otimes \nabla'v^0(t,x')&0\\0&|\nabla' v^0(t,x')|^2 \end{array}\Big)
\end{eqnarray*}
for every $x\in\Omega$ and for all $t\in [0,T]$. The only relevant difference is that we can not conclude that $u(t)$ and $v(t)$ are uniquely determined once $p(t)$ is identified. Hence, now all convergence properties hold on $t$-dependent subsequences.\\
{\em{Step 4: Characterization of the limit stress}}\\
Arguing exactly as in Step 4 of the proof of Theorem \ref{cvstress} for $\alpha>3$, we obtain
\begin{equation}
\label{almeuler}
\intom{E(t,x)e_3\cdot \partial_3 \eta (t,(x',x_3+v(t,x')-v^0(t,x')))}=0
\end{equation}
for every {$\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ such that $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}. We consider a sequence $(w_k)\subset C^\infty_c(\omega)$ that converges to $v(t)-v^0(t)$ strongly in $L^2(\omega)$. We take as test functions in \eqref{almeuler} the maps $\eta_k(x):=\eta(x',x_3-w_k(x'))$, where {$\eta\in W^{1,\infty}(\mathbb R^3,\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ and $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}. We have
$$\intom{E(t,x)e_3\cdot \partial_3 \eta (t,(x',x_3+v(t,x')-v^0(t,x')-w_k(x')))}=0\quad\text{for every }k.$$
Passing to the limit as $k\to +\infty$ in the previous equation, by the dominated convergence theorem we deduce
$$\intom{E(t)e_3\cdot \partial_3 \eta}=0$$
for every {$\eta\in W^{1,\infty}(\mathbb R^3,\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ such that $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}, which implies $E(t)e_3=0$ a.e. in $\Omega$. Hence, \eqref{linearmin} yields
\begin{equation}
\nonumber
E(t)=\mathbb C_2(e(t)),
\end{equation}
and
{\begin{equation}
\label{strlima3}
\mathrm{sym}\,G(t)-p(t)=\A (\mathrm{sym}\nabla' u(t)+\tfrac{1}{2}\nabla' v(t)\otimes\nabla' v(t)-x_3(\nabla')^2 v(t)-p'(t)).
\end{equation}}
{\em{Step 5: Reduced energy balance}}\\
Arguing as in Step 5 of the case $\alpha>3$, to prove (qs2$_{r3}$) it is enough to show that
\begin{eqnarray}
\nonumber &&\intom{Q_2(e_3(t))}+\intom{\B{p(t)}}+\cal{D}_{H_D}(p;0,t)\\
\nonumber &&\leq\intom{Q_2(e_3(0))}+\intom{\B{p(0)}}\\
\nonumber &&+\int_0^t{\intom{\mathbb C_2(e_3(s)):\Big(\begin{array}{cc}\nabla \dot{u}^0(s)+\nabla' v(s)\otimes \nabla' \dot{v}^0(s)-x_3(\nabla')^2 \dot{v}^0(s)&0\\0&0\end{array}\Big)}\,ds},\\
\label{rin13}
\end{eqnarray}
where $t\mapsto e_3(t)$ is the map defined in \eqref{dsigmat}.
Indeed, once \eqref{rin13} is proved, (qs2$_{r3}$) follows by adapting \cite[Theorem 4.7]{DDM} according to Remark \ref{csd}.
To prove \eqref{rin13}, we argue as in \cite[Lemma 5.1]{B} and we set
$$\Theta^{\varepsilon}(t):=\frac{1}{\varepsilon^2}\intom{E^{\varepsilon}(t):\nabla \dot{\pep}(t,\zep(t))(\nabla \pep)^{-1}(t,\zep(t))},$$
$$\Theta(t):=\limsup_{\varepsilon\to 0}\Theta^{\varepsilon}(t)$$
for every $t\in [0,T]$. By \eqref{unifbdeep} (which is still true for $\alpha=3$), $\Theta(t)\in L^1([0,T])$ and by Fatou's lemma there holds
\begin{equation}
\label{limsuptau}
\limsup_{\varepsilon\to 0}\int_0^t{\Theta^{\varepsilon}(s)\,ds}\leq \int_0^t{\Theta(s)\,ds}.
\end{equation}
Now, by Theorem \ref{liminfineq} we know that
\begin{eqnarray*}
\nonumber &&\intom{Q_2(e(t))}+\intom{\B{p(t)}}\leq \liminf_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\cal{F}^{\varepsilon}(t,\zep(t), P^{\varepsilon}(t)).
\end{eqnarray*}
By (qs2), \eqref{convE0}, \eqref{convpept} and \eqref{limsuptau} we deduce
\begin{eqnarray*}
\nonumber &&\intom{Q_2(e(t))}+\intom{\B{p(t)}}+\cal{D}_{H_D}(p;0,t)\leq\intom{Q_2(e(0))}+\intom{\B{p(0)}}+\int_0^t{\Theta(s)\,ds}.
\end{eqnarray*}
Hence, to prove \eqref{rin13} it is enough to show that
\begin{equation}
\label{identtaue}
\Theta(t)=\intom{E(t):\Big(\begin{array}{cc}\nabla \dot{u}^0(t)+\nabla' v(t)\otimes \nabla' \dot{v}^0(t)-x_3(\nabla')^2 \dot{v}^0(t)&0\\0&0\end{array}\Big)}
\end{equation}
for a.e. $t\in [0,T]$.
To this purpose, fix $t\in [0,T]$ and let $\ep_{j}\to 0$ be such that
$$\Theta(t)=\lim_{\ep_{j}\to 0}\Theta^{\ep_{j}}(t).$$
Up to extracting a further subsequence, we may assume that $\ep_{j}$ is the same subsequence we selected in the previous steps. We claim that
\begin{eqnarray}
\nonumber
&&\frac{1}{\ep_{j}^{2}}\mathrm{sym}\Big(\nabla \dot{\pepjt}(t,\zepjt(t))(\nabla \pepjt)^{-1}(t,\zepjt(t))\Big)\\
\nonumber&&\to \mathrm{sym}\Big(\begin{array}{cc}\nabla' \dot{u}^0(t)+\nabla' \dot{v}^0(t)\otimes \nabla' {v}^0(t)-(x_3+v(t)-v^0(t))(\nabla')^2 \dot{v}^0(t)&0\\0&\frac{d}{dt}\frac{|\nabla' v^{0}(t)|^2}{2}\end{array}\Big)\\
\label{lhtg}
\end{eqnarray}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. To prove the claim, we perform the decomposition \eqref{decompst}. Now, arguing as in the proof of \eqref{term11}, and using \eqref{distzep3} and Lemma \ref{cvproptep} we obtain
\begin{equation}
\label{term12}
\frac{1}{\ep_{j}^{2}}\mathrm{sym}(\nabla \dot{\pepjt}(t,\zepjt(t)))\to \mathrm{sym}\Big(\begin{array}{cc}\nabla' \dot{u}^0(t)-(x_3+v(t)-v^0(t))(\nabla')^2 \dot{v}^0(t)&0\\0&0\end{array}\Big)
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. To study the second term in the right-hand side of \eqref{decompst}, we remark that by
\eqref{lp2}, \eqref{gradinv}, \eqref{gradinv1} and \eqref{estt2bis}, one has
\begin{equation}
\nonumber
\Big\|(\nabla \pepjt)^{-1}(t,\zepjt(t))-Id\Big\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C\ep_{j}^{2}\ell_{\ep_{j}}.
\end{equation}
By \eqref{l2usf}, there holds
\begin{eqnarray}
\nonumber
&&\Big\|\Big(\begin{array}{cc}\nabla' \dot{u}^0(t,\bzepjt(t))-\tepjt\big(\frac{\zepjt_3(t)}{\ep_{j}}\big)(\nabla')^2 \dot{v}^0(t,\bzepjt(t))&0\\0&0\end{array}\Big)\Big((\nabla \pepjt)^{-1}(t,\zepjt(t))-Id\Big)\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\\
\label{term13}
&&\leq C\ep_{j}^{2}\ell_{\ep_{j}},
\end{eqnarray}
which tends to zero due to \eqref{lp1}.
By \eqref{gradpept}, it remains only to study the asymptotic behaviour of
$$\frac{1}{\ep_{j}}\Big(\begin{array}{cc}0&-\dot{\theta}^{\ep_{j}}\big(\frac{\zepjt_3(t)}{\ep_{j}}\big)\nabla' \dot{v}^0(t,\bzepjt(t))\\(\nabla' \dot{v}^0(t,\bzepjt(t)))^T&0\end{array}\Big)\Big((\nabla \pepjt)^{-1}(t,\zepjt(t))-Id\Big).$$
By \eqref{estt2bis}, this is the same as studying the quantity
$$\frac{1}{\ep_{j}}\Big(\begin{array}{cc}0&-\dot{\theta}^{\ep_{j}}\big(\frac{\zepjt_3(t)}{\ep_{j}}\big)\nabla' \dot{v}^0(t,\bzepjt(t))\\(\nabla' \dot{v}^0(t,\bzepjt(t)))^T&0\end{array}\Big)\Big(\nabla \vepjt(t,y^{\ep_{j}}(t))-Id\Big).$$
We claim that
\begin{equation}
\label{fclhp}\frac{1}{\ep_{j}}\Big(\nabla \vepjt(t,y^{\ep_{j}}(t))-Id\Big)\to \Big(\begin{array}{cc}0&\nabla' v^0(t)\\-(\nabla' v^0(t))^T&0\end{array}\Big)
\end{equation}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. Indeed, by \eqref{expgr} and \eqref{bddinverse} and the smoothness of $u^0$ and $v^0$,
\begin{eqnarray*}
&&\Big\|\frac{1}{{\ep_{j}}}\Big(\nabla (\vepjt)'(t,y^{\ep_{j}}(t))-\Big(\begin{array}{ccc}1&0&0\\0&1&0\end{array}\Big)\Big)-(0|\nabla' v^0(t))\Big\|_{L^2(\Omega;\mathbb M^{2\times 3})}\leq C\ep_{j}\Big\|\tepjt\Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)\Big\|_{L^2(\Omega)}\\
&&+\Big\|\dot{\theta}^{\ep_{j}}\Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)\nabla' v^0(t, (\vep)'(t, y^{\ep_{j}}(t)))\otimes (\nabla \vepjt_3(t, y^{\ep_{j}}(t))-e_3)\Big\|_{L^2(\Omega;\mathbb M^{2\times 3})}\\
&&+\Big\|\dot{\theta}^{\ep_{j}}\Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)\nabla' v^0(t, (\vep)'(t, y^{\ep_{j}}(t)))-\nabla' v^0(t)\Big\|_{L^2(\Omega;\mathbb R^2)}+C\varepsilon_j.
\end{eqnarray*}
By \eqref{treq2}, \eqref{3comp}, and \eqref{uf3}(which can be proved arguing exactly as in Step 1 of the case $\alpha>3$), we deduce
\begin{equation}
\label{nnpp}
\Big\|\tepjt \Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)\Big\|_{L^2(\Omega)}\leq\Big\|\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big\|_{L^2(\Omega)}\leq C\Big(\Big\|\frac{y^{\ep_{j}}_3(t)}{\ep_{j}}\Big\|_{L^2(\Omega)}+\|v^0\|_{L^{\infty}(\omega;\mathbb R^2)}\Big)\leq C.
\end{equation}
On the other hand, by \eqref{treq5} and \eqref{gradinv1}
\begin{eqnarray*}
&&\Big\|\dot{\theta}^{\ep_{j}}\Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)\nabla' v^0(t, (\vep)'(t, y^{\ep_{j}}(t)))\otimes (\nabla \vepjt_3(t, y^{\ep_{j}}(t))-e_3)\Big\|_{L^2(\Omega;\mathbb M^{2\times 3})}\\
&&\leq C\|\nabla \vepjt_3(t,y^{\ep_{j}}(t))-e_3\|_{L^{\infty}(\Omega;\mathbb R^3)}\leq C\ep_{j}.
\end{eqnarray*}
Finally, by \eqref{nnpp} and Lemma \ref{cvproptep}
\begin{eqnarray*}
&&\Big\|\dot{\theta}^{\ep_{j}}\Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)\nabla' v^0(t, (\vep)'(t, y^{\ep_{j}}(t)))-\nabla' v^0(t)\Big\|_{L^2(\Omega;\mathbb R^2)}\\
&&\leq C \Big\|\dot{\theta}^{\ep_{j}}\Big(\frac{\vepjt_3(t,y^{\ep_{j}}(t))}{\ep_{j}}\Big)-1\Big\|_{L^2(\Omega)}+\|\nabla' v^0(t, (\vep)'(t, y^{\ep_{j}}(t)))-\nabla' v^0(t)\|_{L^2(\omega;\mathbb R^2)}\\
&&\leq \frac{C}{\ell_{\ep_{j}}}+\|\nabla' v^0(t, (\vep)'(t, y^{\ep_{j}}(t)))-\nabla' v^0(t)\|_{L^2(\omega;\mathbb R^2)}
\end{eqnarray*}
which converges to zero owing to \eqref{lp1}, \eqref{distinv1}, \eqref{distinv13}, \eqref{cvitsb} (which can be proved arguing exactly as in Step 2 of the case $\alpha>3$) and the dominated convergence theorem. By collecting the previous remarks, we obtain
$$\Big\|\frac{1}{{\ep_{j}}}\Big(\nabla (\vepjt)'(t,y^{\ep_{j}}(t))-\Big(\begin{array}{ccc}1&0&0\\0&1&0\end{array}\Big)\Big)-(0|\nabla' v^0(t))\Big\|_{L^2(\Omega;\mathbb{M}^{3\times 3})}\to 0.$$
On the other hand, by \eqref{gradinv0} there holds
\begin{eqnarray*}
&&\Big\|\frac{\nabla \vepjt_3(t,y^{\ep_{j}}(t))-e_3}{\ep_{j}}+\Big(\begin{array}{c}\nabla'v^0\\0\end{array}\Big)\Big\|_{L^2(\Omega;\mathbb R^3)}\leq C\Big\|\nabla (\vepjt)'(t)-\Big(\begin{array}{ccc}1&0&0\\0&1&0\end{array}\Big)\Big\|_{L^{\infty}(\Omega;\mathbb M^{2\times 3})}\\
&&+\|\nabla' v^0(t, (\vepjt)'(t, y^{\ep_{j}}(t)))-\nabla' v^0(t)\|_{L^2(\Omega;\mathbb R^2)}
\end{eqnarray*}
which tends to zero owing to \eqref{lp1}, \eqref{distinv1}, \eqref{distinv13}, \eqref{gradinv}, \eqref{cvitsb} and the dominated convergence theorem. Therefore, the proof of claim \eqref{fclhp} is completed.
Now, by \eqref{cvitsb}, \eqref{fclhp} and the dominated convergence theorem we conclude that
\begin{eqnarray}
\nonumber &&\frac{1}{\ep_{j}}\Big(\begin{array}{cc}0&-\dot{\theta}^{\ep_{j}}\big(\frac{\zepjt_3(t)}{\ep_{j}}\big)\nabla' \dot{v}^0(t,\bzepjt(t))\\(\nabla' \dot{v}^0(t,\bzepjt(t)))^T&0\end{array}\Big)\Big(\nabla \vepjt(t,y^{\ep_{j}}(t))-Id\Big)\\
&&\label{term15} \to \Big(\begin{array}{cc}\nabla' \dot{v}^0(t)\otimes \nabla' {v}^0(t)&0\\0&\frac{d}{dt}\frac{|\nabla' v^0(t)|^2}{2}\end{array}\Big)
\end{eqnarray}
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. By combining \eqref{term12}, \eqref{term13} and \eqref{term15} we deduce \eqref{lhtg}. Now, by \eqref{nothird}, \eqref{lp1}, \eqref{badsetstressnt}, \eqref{linftybdwk}, \eqref{goodsetstress} (which still hold true for $\alpha=3$), \eqref{strlima3} and \eqref{lhtg} we obtain
\begin{equation}
\label{A1}
\Theta(t)=\intom{E(t):\Big(\begin{array}{cc}\nabla' \dot{u}^0(t)+\nabla' \dot{v}^0(t)\otimes \nabla' {v}^0(t)-(x_3+v(t)-v^0(t))(\nabla')^2 \dot{v}^0(t)&0\\0&0\end{array}\Big)}.
\end{equation}
On the other hand,
\begin{eqnarray*}
&&\mathrm{sym}(\nabla' \dot{v}^0(t)\otimes \nabla' {v}^0(t)-(v(t)-v^0(t))(\nabla')^2 \dot{v}^0(t))\\
&&=-\mathrm{sym}\,\nabla' \big((v(t)-v^0(t))\nabla' \dot{v}^0(t)\big)+\mathrm{sym} \big(\nabla' v(t)\otimes \nabla'\dot{v}^0(t)\big)
\end{eqnarray*}
and
\begin{equation}
\label{A2}
\intom{\mathbb C_2 E(t):\nabla' \big((v(t)-v^0(t))\nabla' \dot{v}^0(t)\big)}=0
\end{equation}
by Remark \ref{eul3}. By combining \eqref{A1} and \eqref{A2}, the proof of \eqref{identtaue} and of the theorem is complete.
\end{proof}
To conclude this section we show some corollaries of Theorem \ref{cvstress}. We first prove that under the hypotheses of the theorem we can deduce convergence of the elastic energy and of the hardening functional. More precisely, the following result holds true.
\begin{cor}
Under the assumptions of Theorem \ref{cvstress}, for $\alpha>3$ for every $t\in [0,T]$, setting $y^{\varepsilon}(t):=\pep(t,\zep(t))$ there holds
\begin{equation}
\nonumber
\lim_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{el}(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t))}= \intom{Q_2(\mathrm{sym}\,\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))},
\end{equation}
and
\begin{equation}
\label{cvs2}
\lim_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{hard}(P^{\varepsilon}(t))}= \intom{\B{p(t)}}.
\end{equation}
The analogous result holds true for $\alpha=3$ on the $t$-dependent subsequence $\ep_{j}\to 0$ selected in Theorem \ref{cvstress}.
\end{cor}
\begin{proof}
The result follows by combining the liminf inequalities in Theorem \ref{liminfineq}, the $\varepsilon$-energy balance (qs2) and the reduced energy balance (qs1$_{r\alpha}$).
\end{proof}
In particular, we can deduce strong convergence of the sequence of scaled plastic strains by the convergence of the energies.
\begin{cor}
\label{sp}
Under the hypotheses of Theorem \ref{cvstress}, {for $\alpha>3$} there holds
\begin{equation}
\label{scpt}
p^{\varepsilon}(t)\to p(t)\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3})
\end{equation}
for every $t\in [0,T]$. { The analogous result holds true for $\alpha=3$ on the $t$-dependent subsequence $\ep_{j}\to 0$ selected in Theorem \ref{cvstress}.}
\end{cor}
\begin{proof}
Fix $\delta>0$ and let $c_h(\delta)$ be the constant in \eqref{prh4}. By \eqref{prh4} there holds
\begin{equation}
\label{tay}
W_{hard}(Id+F)\geq \B{F}-C\delta|F|^2\quad\text{for every }F\in\mathbb{M}^{3\times 3},\,|F|<c_h(\delta).
\end{equation}
Fix $t\in [0,T]$ and for every $\varepsilon$ consider the set
$$S_{\varepsilon}(t):=\Big\{x\in\Omega: |p^{\varepsilon}(t,x)|<\frac{c_h(\delta)}{\varepsilon}\Big\}.$$
Denoting by $\mu_{\varepsilon}(t)$ the characteristic function of the set $S_{\varepsilon}(t)$, by \eqref{cvpt} and Chebychev inequality,
\begin{equation}
\label{iceq0}
{\mu_{\varepsilon}(t)}\to 1\quad\text{boundedly in measure as }\varepsilon\to 0.
\end{equation}
and thus
\begin{equation}
\label{wcrset}
\mu_{\varepsilon}(t)p^{\varepsilon}(t)\rightharpoonup p(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
We remark that in the set $S_{\varepsilon}(t)$ we have $\varepsilon^{\alpha-1}|p^{\varepsilon}(t)|<\varepsilon^{\alpha-2}c_h(\delta).$
Hence, by \eqref{tay} for $\varepsilon$ small enough there holds
\begin{eqnarray*}
\frac{1}{\varepsilon^{2\alpha-2}}W_{hard}(P^{\varepsilon}(t))\geq \frac{1}{\varepsilon^{2\alpha-2}}\mu_{\varepsilon}(t)W_{hard}(P^{\varepsilon}(t))
\geq \mu_{\varepsilon}(t)\big(\B{p^{\varepsilon}(t)}-C\delta|p^{\varepsilon}(t)|^2\big).
\end{eqnarray*}
In particular, by \eqref{cvpt}, \eqref{cvs2} and the lower semicontinuity of $B$ with respect to weak $L^2$ convergence, we have
\begin{eqnarray*}
&&\intom{\B{p(t)}}= \lim_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\intom{W_{hard}(P^{\varepsilon}(t))}\geq\limsup_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\intom{\mu_{\varepsilon}(t)W_{hard}(P^{\varepsilon}(t))}\\
&&\geq \limsup_{\varepsilon\to 0}\intom{\mu_{\varepsilon}(t)\B{p^{\varepsilon}(t)}}-C\delta
\geq \liminf_{\varepsilon\to 0}\intom{\mu_{\varepsilon}(t)\B{p^{\varepsilon}(t)}}-C\delta\geq \intom{\B{p(t)}}-C\delta.
\end{eqnarray*}
Since $\delta$ is arbitrary, we obtain
\begin{eqnarray}
&& \label{iceq1}\lim_{\varepsilon\to 0}\intom{\mu_{\varepsilon}(t)\B{p^{\varepsilon}(t)}}= \intom{\B{p(t)}}
\end{eqnarray}
and by \eqref{cvs2}
\begin{equation}
\label{iceq1bis}
\lim_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\intom{(1-\mu_{\varepsilon}(t))W_{hard}(P^{\varepsilon}(t))}=0.
\end{equation}
By \eqref{prh3} and \eqref{iceq1bis} we deduce
\begin{equation}
\label{iceq2}
\lim_{\varepsilon\to 0}\intom{(1-\mu_{\varepsilon}(t))|p^{\varepsilon}(t)|^2}\leq \frac{2}{c_6}\lim_{\varepsilon\to 0}\frac{1}{\varepsilon^{2\alpha-2}}\intom{(1-\mu_{\varepsilon}(t))W_{hard}(P^{\varepsilon}(t))}=0.
\end{equation}
Hence, by \eqref{grbelowh} there holds
\begin{eqnarray}
\nonumber \intom{|p^{\varepsilon}(t)-p(t)|^2}&=&\intom{\mu_{\varepsilon}(t)|p^{\varepsilon}(t)-p(t)|^2}+\intom{(1-\mu_{\varepsilon}(t))|p^{\varepsilon}(t)-p(t)|^2}\\
\nonumber&\leq& \frac{2}{c_6}\intom{\mu_{\varepsilon}(t)\B{p^{\varepsilon}(t)-p(t)}}+ 2\intom{(1-\mu_{\varepsilon}(t))(|p^{\varepsilon}(t)|^2+|p(t)|^2)}.\\
\label{iceq3}
\end{eqnarray}
Recalling the quadratic structure of $B$, the first term in the second row of \eqref{iceq3} can be decomposed as
\begin{eqnarray*}
\frac{2}{c_6}\intom{\mu_{\varepsilon}(t)\B{p^{\varepsilon}(t)-p(t)}}&=&\frac{2}{c_6}\intom{\mu_{\varepsilon}(t)\B{p^{\varepsilon}(t)}}+\frac{2}{c_6}\intom{\mu_{\varepsilon}(t)\B{p(t)}}\\
&-&\frac{4}{c_6}\intom{\mu_{\varepsilon}(t)\mathbb{B}p^{\varepsilon}(t):p(t)}
\end{eqnarray*}
and tends to zero due to \eqref{iceq0}--\eqref{iceq1}.
On the other hand, by \eqref{iceq0} and \eqref{iceq2}
$$\intom{(1-\mu_{\varepsilon}(t))(|p^{\varepsilon}(t)|^2+|p(t)|^2)}\to 0.$$
By combining the previous results, we deduce \eqref{scpt}.
\end{proof}
Convergence of the energy implies also strong convergence of the in-plane displacements. More precisely, the following result holds true.
\begin{cor}
\label{su}
Under the assumptions of Theorem \ref{cvstress}, for $\alpha>3$, for every $t\in [0,T]$ there holds
\begin{equation}
\label{scut}
u^{\varepsilon}(t)\to u(t)\quad\text{strongly in }W^{1,2}(\omega;\mathbb R^2).
\end{equation}
The same result holds true for $\alpha=3$, on the $t$-dependent subsequence $\ep_{j}\to 0$ selected in Theorem \ref{cvstress}.
\end{cor}
\begin{proof}
We prove the corollary for $\alpha>3$. The case where $\alpha=3$ follows by simple adaptations. Fix $t\in [0,T]$ and let $F^{\varepsilon}(t)$ be the map defined in \eqref{deffept}. Fix $\delta>0$ and consider the set
$$U_{\varepsilon}(t):=\Big\{x\in\Omega: |F^{\varepsilon}(t,x)|<\frac{c_{el}(\delta)}{\varepsilon}\Big\},$$
where $c_{el}(\delta)$ is the constant in \eqref{quadrwel}.
In particular, in the set $U_{\varepsilon}(t)$ there holds $\varepsilon^{\alpha-1}|F^{\varepsilon}(t)|\leq \varepsilon^{\alpha-2}c_{el}(\delta)$. Hence, denoting by $\mu_{\varepsilon}(t)$ the characteristic function of $U_{\varepsilon}(t)$, by (H3), \eqref{quadrwel} and \eqref{decompep}, we have
\begin{eqnarray*}
\frac{1}{\varepsilon^{2\alpha-2}}W_{el}(\nep y^{\varepsilon}(t)(P^{\varepsilon})^{-1}(t))=\frac{1}{\varepsilon^{2\alpha-2}}W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))\geq \mu_{\varepsilon}(t){Q(F^{\varepsilon}(t))}-\mu_{\varepsilon}(t)C\delta|F^{\varepsilon}(t)|^2.
\end{eqnarray*}
By Chebychev inequality and \eqref{unifbdfep},
\begin{equation}
\label{nnpmeas}
\mu_{\varepsilon}(t)\to 1\quad\text{boundedly in measure,}
\end{equation}
whereas by \eqref{convfep} and \eqref{nnpserve},
\begin{equation}
\label{nnpfep}
\mu_{\varepsilon}(t)\mathrm{sym}\,F^{\varepsilon}(t)\rightharpoonup \A(\mathrm{sym} \,\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Arguing as in the proof of \eqref{iceq1} we obtain
\begin{eqnarray}
\label{iceq11}&& \lim_{\varepsilon\to 0}\intom{\mu_{\varepsilon}(t)Q(F^{\varepsilon}(t))}=\intom{Q_2(\mathrm{sym}\,\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))}
\end{eqnarray}
and
\begin{eqnarray}
\nonumber\lim_{\varepsilon \to 0} \frac{1}{\varepsilon^{2\alpha-2}}\intom{(1-\mu_{\varepsilon}(t))W_{el}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))}= 0.
\end{eqnarray}
By (H4), this implies that
\begin{equation}
\label{iceq12bis}
\lim_{\varepsilon \to 0} \frac{1}{\varepsilon^{2\alpha-2}}\intom{(1-\mu_{\varepsilon}(t))\mathrm{dist}^2(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t), SO(3))}\to 0.
\end{equation}
On the other hand, \eqref{growthcondQ} and \eqref{nothird} yield
\begin{eqnarray*}
&&\intom{\big|\mu_{\varepsilon}(t)\mathrm{sym}\, F^{\varepsilon}(t)- \A\big(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t)\big)\big|^2}\\
&&\leq \frac{1}{r_{\mathbb C}}\intom{Q\big(\mu_{\varepsilon}(t)\mathrm{sym}\, F^{\varepsilon}(t)- \A\big(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t)\big)\big)}\\
&&=\frac{1}{r_{\mathbb C}}\intom{Q(\mu_{\varepsilon}(t) F^{\varepsilon}(t))}+\frac{1}{r_{\mathbb C}}\intom{Q_2(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))}\\
&&-\frac{2}{r_{\mathbb C}}\intom{\mu_{\varepsilon}(t)\mathbb C_2F^{\varepsilon}(t):(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))}.
\end{eqnarray*}
Hence, by \eqref{nnpfep} and \eqref{iceq11}
\begin{equation}
\label{iceq14}
\mu_{\varepsilon}(t)\mathrm{sym}\, F^{\varepsilon}(t)\to \A(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Moreover,
\begin{eqnarray}
&&\nonumber\frac{1}{\varepsilon^{\alpha-1}}\mu_{\varepsilon}(t) \mathrm{dist}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t), SO(3))\\
&&\nonumber=\mu_{\varepsilon}(t)|\mathrm{sym}\, F^{\varepsilon}(t)|+\mu_{\varepsilon}(t)O(\varepsilon^{\alpha-1}|F^{\varepsilon}(t)|^2)\to | \A(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))|\\
\label{iceq13}
\end{eqnarray}
strongly in $L^2(\Omega)$. By combining \eqref{iceq12bis} and \eqref{iceq13} we deduce
$$\frac{1}{\varepsilon^{\alpha-1}}\mathrm{dist}(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t), SO(3))\to | \A(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))|$$
strongly in $L^2(\Omega)$. In particular, the sequence $\frac{1}{\varepsilon^{2\alpha-2}}\mathrm{dist}^2(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t), SO(3))$ is equi-integrable.
Now, recalling that by \eqref{deffept} there holds
$$Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t)=(Id+\varepsilon^{\alpha-1}G^{\varepsilon}(t))(Id+\varepsilon^{\alpha-1}p^{\varepsilon}(t))^{-1},$$
by \eqref{prk1} and \eqref{isink} for every $R\in SO(3)$ we deduce
\begin{eqnarray*}
&&\frac{1}{\varepsilon^{2\alpha-2}}|Id+\varepsilon^{\alpha-1}G^{\varepsilon}(t)-R|^2=\frac{1}{\varepsilon^{2\alpha-2}}|(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t))(Id+\varepsilon^{\alpha-1}p^{\varepsilon}(t))-R|^2\\
&&\leq \frac{c_k^2}{\varepsilon^{2\alpha-2}}|Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t)-R|^2+|p^{\varepsilon}(t)|^2,
\end{eqnarray*}
which in turn implies
\begin{eqnarray*}
\frac{1}{\varepsilon^{2\alpha-2}}\mathrm{dist}^2(Id+\varepsilon^{\alpha-1}G^{\varepsilon}(t), SO(3))\leq \frac{c_k^2}{\varepsilon^{2\alpha-2}}\mathrm{dist}^2(Id+\varepsilon^{\alpha-1}F^{\varepsilon}(t), SO(3))+|p^{\varepsilon}(t)|^2.
\end{eqnarray*}
Hence, by \eqref{scpt} $\frac{1}{\varepsilon^{2\alpha-2}}\mathrm{dist}^2(Id+\varepsilon^{\alpha-1}G^{\varepsilon}(t), SO(3))$ is equi-integrable. Arguing as in \cite[Section 7.2, Proof of Theorem 2]{FJM2} we obtain the equi-integrability of $|G^{\varepsilon}(t)|^2$.
We claim that also $|F^{\varepsilon}(t)|^2$ is equi-integrable. Indeed, by \eqref{deffept}, there holds
$$|F^{\varepsilon}(t)|^2\leq C(|G^{\varepsilon}(t)|^2+|w^{\varepsilon}(t)|^2+|p^{\varepsilon}(t)|^2+\varepsilon^{2\alpha-2}|G^{\varepsilon}(t)w^{\varepsilon}(t)|^2+\varepsilon^{2\alpha-2}|G^{\varepsilon}(t)p^{\varepsilon}(t)|^2).$$
Now, by \eqref{unifapest}, \eqref{isink} and \eqref{numerarepag35}, we have
$$|w^{\varepsilon}(t)|^2\leq c_K^2\varepsilon^{2\alpha-2}|p^{\varepsilon}(t)|^4\leq C|p^{\varepsilon}(t)|^2.$$
Hence, by \eqref{scpt} the maps $|w^{\varepsilon}(t)|^2$ are equi-integrable. Moreover, by \eqref{unifapest} there holds
$$\varepsilon^{2\alpha-2}|G^{\varepsilon}(t)p^{\varepsilon}(t)|^2\leq C|G^{\varepsilon}(t)|^2$$
and by \eqref{linftybdw}
$$\varepsilon^{2\alpha-2}|G^{\varepsilon}(t)w^{\varepsilon}(t)|^2\leq C|G^{\varepsilon}(t)|^2.$$
Therefore, the equi-integrability of $|F^{\varepsilon}(t)|^2$ follows from the equi-integrability of $|G^{\varepsilon}(t)|^2$.
By \eqref{iceq14}, this implies that
$$\mathrm{sym}\, F^{\varepsilon}(t)\to \A(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))$$
strongly in $L^2(\Omega;\mathbb{M}^{3\times 3})$. On the other hand, by \eqref{numerarepag352} and \eqref{numerarepag39},
$$w^{\varepsilon}(t)-\varepsilon^{\alpha-1}G^{\varepsilon}(t)(p^{\varepsilon}(t)-w^{\varepsilon}(t))\to 0$$
strongly in $L^1(\Omega;\mathbb{M}^{3\times 3})$. Therefore, by \eqref{deffept} and \eqref{scpt} we obtain
{\begin{eqnarray*}
\mathrm{sym}\, G^{\varepsilon}(t)\to \A(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))+p(t)\quad\text{strongly in }L^1(\Omega;\mathbb{M}^{3\times 3}).
\end{eqnarray*}}
By the equi-integrability of $|G^{\varepsilon}(t)|^2$, it follows that {$$\mathrm{sym}\, G^{\varepsilon}(t)\to \A(\nabla' u(t)-x_3(\nabla')^2 v(t)-p'(t))+p(t)\quad\text{strongly in }L^2(\Omega;\mathbb{M}^{3\times 3}).$$}
The conclusion follows then arguing as in \cite[Section 7.2, Proof of Theorem 2]{FJM2}.
\end{proof}
\section{Convergence of approximate minimizers}
\label{appr}
Theorems \ref{cvstress} is actually only a convergence result. Indeed, under our assumptions the existence of an $\varepsilon$-quasistatic evolution according to Definition \ref{epquasevol} is not guaranteed. Howewer, following the same approach as in \cite[Theorem 2.3]{MS}, we can extend our convergence result to sequences of approximate discrete-time ${\varepsilon}$-quasistatic evolutions. More precisely, setting
{\begin{eqnarray*}
\cal{A}_{\varepsilon}&:=&\{(z,P)\in W^{1,2}(\Omega;\mathbb R^3)\times L^2(\Omega;SL(3)):\\
&&z=(x',\varepsilon x_3)\quad\cal{H}^2\text{ - a.e. on }\Gamma_d\quad\text{ and }P(x)\in K\quad\text{a.e. in }\Omega\},
\end{eqnarray*}}
we give the following definition.
\begin{defin}
\label{apmin}
Given a sequence of time-partitions
$$\{0=t^0_{\varepsilon}<t^1_{\varepsilon}<\cdots t^{N^{\varepsilon}}_{\varepsilon}=T\},$$
with time-steps
\begin{equation}
\label{ss49}\tau_{\varepsilon}:=\max_{i=1,\cdots N^{\varepsilon}}(t^{i}_{\varepsilon}-t^{i-1}_{\varepsilon})\to 0\quad\text{as }\varepsilon\to 0,
\end{equation}
and a sequence of positive parameters $\delta_{\varepsilon}\to 0$, we call $\{(z^i_{\varepsilon}, P^i_{\varepsilon})\}$ a sequence of \emph{approximate minimizers} if, for every $\varepsilon>0$, $(z^0_{\varepsilon}, P^0_{\varepsilon})\in\cal{A}_{\varepsilon}$, and $(z^i_{\varepsilon},P^i_{\varepsilon})\in \cal{A}_{\varepsilon}$ satisfies
\begin{eqnarray}
\nonumber&&\cal{F}_{\varepsilon}(t^i_{\varepsilon},z^i_{\varepsilon},P^i_{\varepsilon})+{\varepsilon^{\alpha-1}}\intom{D(P^{i-1}_{\varepsilon},P^i_{\varepsilon})}\\
\label{mind}
&&\leq \varepsilon^{2\alpha-2}\delta_{\varepsilon}(t^i_{\varepsilon}-t^{i-1}_{\varepsilon})+\inf_{(z,P)\in\cal{A}_{\varepsilon}}\Big\{\cal{F}_{\varepsilon}(t^i_{\varepsilon},z,P)+{\varepsilon^{\alpha-1}}\intom{D(P^{i-1}_{\varepsilon},P)}\Big\}
\end{eqnarray}
for every $i=1,\cdots, N^{\varepsilon}$.
\end{defin}
Our final result is to show that every sequence of approximate minimizers converges, as $\varepsilon\to 0$, to a reduced quasistatic evolution.
{
\begin{teo}
\label{cvapp}
Let $\alpha\geq 3$. Assume that $t\mapsto u^0(t)$ belongs to $C^1([0,T];W^{1,\infty}(\mathbb R^2;\mathbb R^2)\cap C^{1}(\mathbb R^2;\mathbb R^2))$ and $t\mapsto v^0(t)$ belongs to $C^1([0,T];W^{2,\infty}(\mathbb R^2)\cap C^{2}(\mathbb R^2))$, respectively. For every $t\in [0,T]$, let $\pep(t)$ be defined as in \eqref{defphiep} and let $(\mathring{u},\mathring{v},\mathring{p})\in \cal{A}(u^0(0),v^0(0))$ be such that
\begin{eqnarray}
\nonumber &&\intom{Q_2(\mathrm{sym}\nabla' \mathring{u}-x_3(\nabla')^2 \mathring{v}+\tfrac{L_{\alpha}}{2}\nabla' \mathring{v}\otimes \nabla' \mathring{v}-\mathring{p}')}+\intom{\B{\mathring{p}}}\\
\nonumber&&
\leq \intomm{Q_2(\nabla' \hat{u}-x_3(\nabla')^2 \hat{v}+\tfrac{L_{\alpha}}{2}\nabla' \hat{v}\otimes \nabla' \hat{v}-\hat{p}')}+\intom{\B{\hat{p}}}+\intom{H_D(\hat{p}-\mathring{p})},\\
\label{servedopo1}
\end{eqnarray}
for every $(\hat{u},\hat{v},\hat{p})\in\cal{A}(u^0(0),v^0(0))$.
Given a sequence of time-partitions
$$\{0=t^0_{\varepsilon}<t^1_{\varepsilon}<\cdots t^{N^{\varepsilon}}_{\varepsilon}=T\},$$
with time-steps
\begin{equation}
\nonumber \tau_{\varepsilon}:=\max_{i=1,\cdots N^{\varepsilon}}(t^{i}_{\varepsilon}-t^{i-1}_{\varepsilon})\to 0\quad\text{as }\varepsilon\to 0,
\end{equation}
and a sequence of positive parameters $\delta_{\varepsilon}\to 0$, assume there exists a sequence of pairs $(y_0^{\varepsilon},P_0^{\varepsilon})\in \cal{A}_{\varepsilon}(\pep(0))$ such that
\begin{equation}
\label{appminimg}\cal{I}(y^{\varepsilon}_0,P^{\varepsilon}_0) \leq \cal{I}(\hat{y},\hat{P})+{\varepsilon^{\alpha-1}}\intom{D(P^{\varepsilon}_0,\hat{P})}+\delta_{\varepsilon}\tau_{\varepsilon}\varepsilon^{2\alpha-2},
\end{equation}
for every $(\hat{y},\hat{P})\in\cal{A}_{\varepsilon}(\pep(0))$, and
\begin{eqnarray}
\label{convu0d} && u^{\varepsilon}_0:=\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((y^{\varepsilon}_0)'-x'\big)}\to \mathring{u}\quad\text{strongly in }W^{1,2}(\omega;\mathbb R^2),\\
\label{convv0d} && v^{\varepsilon}_0:=\frac{1}{\varepsilon^{\alpha-2}}\intt{(y^{\varepsilon}_0)_3}\to \mathring{v}\quad\text{strongly in }W^{1,2}(\omega),\\
\label{convP0d} && p^{\varepsilon}_0:=\frac{P^{\varepsilon}_0-Id}{\varepsilon^{\alpha-1}}\to \mathring{p}\quad\text{strongly in }L^2(\Omega;\mathbb M^{3\times 3}_D),\\
\nonumber && \lim_{\varepsilon\to 0}\,\frac{1}{\varepsilon^{2\alpha-2}}\cal{I}(y^{\varepsilon}_0, P^{\varepsilon}_0) =\intom{Q_2(\mathrm{sym} \nabla' \mathring{u}-x_3(\nabla')^2 \mathring{v}+\tfrac{L_{\alpha}}{2}\nabla' \mathring{v}\otimes\nabla' \mathring{v}-{\mathring{p}}')}\\
\label{convE0d} &&+\intom{\B{\mathring{p}}}.
\end{eqnarray}
Let $(z^i_{\varepsilon}, P^{i}_{\varepsilon})$ be a sequence of approximate minimizers and let $(\overline{z}^{\varepsilon}(t), \overline{P}^{\varepsilon}(t))$ be the corresponding right-continuous, piecewise constant interpolants on the time partitions. Let $\overline{\phi}^{\varepsilon}(t)$ be the associated interpolant of $t\mapsto\pep(t)$. Then, for every $t\in [0,T]$
\begin{equation}
\nonumber
\overline{p}^{\varepsilon}(t):=\frac{\overline{P}^{\varepsilon}(t)-Id}{\varepsilon^{\alpha-1}}\rightharpoonup p(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).
\end{equation}
Moreover, for $\alpha>3$, for every $t\in [0,T]$ the following convergence properties hold true:
\begin{eqnarray*}
&&\overline{u}^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-1}}\intt{\big((\overline{\phi}^{\varepsilon})'(t,\overline{z}^{\varepsilon}(t))-x'\big)}\rightharpoonup u(t)\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2),\\
&&\overline{v}^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-2}}\intt{\overline{\phi}^{\varepsilon}_3(t, \overline{z}^{\varepsilon}(t))}\to v(t)\quad\text{strongly in }W^{1,2}(\omega),
\end{eqnarray*}
where $t\mapsto (u(t),v(t),p(t))$ is a reduced quasistatic evolution.
For $\alpha=3$, up to extracting a $t$-dependent subsequence $\ep_{j}\to 0$, there holds
\begin{eqnarray*}
&&\overline{u}^{\ep_{j}}(t):=\frac{1}{\ep_{j}^{\alpha-1}}\intt{\big((\overline{\phi}^{\ep_{j}})'(t,\overline{z}^{\ep_{j}}(t))-x'\big)}\rightharpoonup u(t)\quad\text{weakly in }W^{1,2}(\omega;\mathbb R^2),\\
&&\overline{v}^{\ep_{j}}(t):=\frac{1}{\ep_{j}^{\alpha-2}}\intt{\overline{\phi}^{\ep_{j}}_3(t, \overline{z}^{\ep_{j}}(t))}\to v(t)\quad\text{strongly in }W^{1,2}(\omega),
\end{eqnarray*}
where $t\mapsto (u(t),v(t),p(t))$ is a reduced quasistatic evolution.
\end{teo}
\begin{oss}
The set of admissible data $(\mathring{u},\mathring{v},\mathring{p})$ for Theorem \ref{cvapp} is nonempty.
Indeed, for every $\varepsilon>0$ let $(y^{\varepsilon}_0,P^{\varepsilon}_0)\in\cal{A}_{\varepsilon}(\pep(0))$ be such that
$$\cal{I}(y^{\varepsilon}_0,P^{\varepsilon}_0)+\varepsilon^{\alpha-1}\intom{D(Id,P^{\varepsilon}_0)}\leq \inf_{(\hat{y},\hat{P})\in\cal{A}_{\varepsilon}(\pep(0))}\Big\{\cal{I}(\hat{y},\hat{P})+\varepsilon^{\alpha-1}\intom{D(Id,\hat{P})}\Big\}+\delta_{\varepsilon}\tau_{\varepsilon}\varepsilon^{2\alpha-2}.$$
Since by \eqref{triang}
$$D(Id,\hat{P})\leq D(Id,P^{\varepsilon}_0)+D(P^{\varepsilon}_0,\hat{P}),$$
we deduce that $(y^{\varepsilon}_0,P^{\varepsilon}_0)$ fulfills \eqref{appminimg}. By the regularity of $\partial\omega$, the set $\gamma_d$ coincides $\cal{H}^1$ - a.e. with its closure in the relative topology of $\partial\omega$, which in turn is a closed (nontrivial) interval in $\partial\omega$.
Hence, by \cite[Theorem 5.1]{D1}, choosing $p^{\varepsilon,0}=p^0=0$ for every $\varepsilon>0$, and $s_{\varepsilon}=\delta_{\varepsilon}\tau_{\varepsilon}\varepsilon^{2\alpha-2}$, we infer the existence of a triple $(\mathring{u},\mathring{v},\mathring{p})\in\cal{A}(u^0(0),v^0(0))$ such that \eqref{servedopo1} is satisfied and \eqref{convu0d}--\eqref{convE0d} hold true.
\end{oss}}
\begin{proof}[Proof of Theorem \ref{cvapp}]
The proof follows along the general lines of the proof of Theorems \ref{cvstress}. We sketch the main steps in the case $\alpha>3$. The case $\alpha=3$ follows by straightforward adaptations.\\
\emph{Quasi-stability condition}\\
By \eqref{triang} the piecewise constant interpolants fullfill
\begin{equation}
\label{quasst}
\cal{F}_{\varepsilon}(t, \overline{z}^{\varepsilon}(t),\overline{P}^{\varepsilon}(t))\leq \cal{F}_{\varepsilon}(t,\hat{z},\hat{P})+{\varepsilon^{\alpha-1}}\intom{D(\overline{P}^{\varepsilon}(t),\hat{P})}+\delta_{\varepsilon}\tau_{\varepsilon}\varepsilon^{2\alpha-2}
\end{equation}
for every $(\hat{z},\hat{P})\in\cal{A}_{\varepsilon}$. The previous inequality will play the role of the $\varepsilon$-stability condition (qs1). \\
\emph{Discrete energy inequality}\\
To adapt the proof of Theorem \ref{cvstress} we shall need an analogous of condition (qs2). To this purpose, we notice that, by \eqref{mind} the following discrete energy inequality holds true
\begin{eqnarray*}
&&\cal{F}_{\varepsilon}(t^i_{\varepsilon},z^i_{\varepsilon},P^i_{\varepsilon})+{\varepsilon^{\alpha-1}}\intom{D(P^{i-1}_{\varepsilon},P^i_{\varepsilon})}\leq \varepsilon^{2\alpha-2}\delta_{\varepsilon}(t^i_{\varepsilon}-t^{i-1}_\varepsilon)+\cal{F}_{\varepsilon}(t^i_{\varepsilon}, z^{i-1}_{\varepsilon},P^{i-1}_{\varepsilon})\\
&&=\varepsilon^{2\alpha-2}\delta_{\varepsilon}(t^i_{\varepsilon}-t^{i-1}_\varepsilon)+\cal{F}_{\varepsilon}(t^{i-1}_{\varepsilon}, z^{i-1}_{\varepsilon},P^{i-1}_{\varepsilon})+\int_{t^{i-1}_{\varepsilon}}^{t^i_{\varepsilon}}{\partial_s \cal{F}_{\varepsilon}(s, z^{i-1}_{\varepsilon}, P^{i-1}_{\varepsilon})\,ds}\\
&&=\varepsilon^{2\alpha-2}\delta_{\varepsilon}(t^i_{\varepsilon}-t^{i-1}_\varepsilon)+\cal{F}_{\varepsilon}(t^{i-1}_{\varepsilon}, z^{i-1}_{\varepsilon},P^{i-1}_{\varepsilon})\\
&&+{\varepsilon^{2\alpha-2}}\int_{t^{i-1}_{\varepsilon}}^{t^i_{\varepsilon}}{\intom{DW_{el}\big(\nabla \pep(s,z^{i-1}_{\varepsilon})\nep z^{i-1}_{\varepsilon}(P^{i-1}_{\varepsilon})^{-1}\big):\nabla \dot{\pep}(s,z^{i-1}_{\varepsilon})\nep z^{i-1}_{\varepsilon}(P^{i-1}_{\varepsilon})^{-1}}\,ds}\\
&&=\varepsilon^{2\alpha-2}\delta_{\varepsilon}(t^i_{\varepsilon}-t^{i-1}_{\varepsilon})+\cal{F}_{\varepsilon}(t^{i-1}_{\varepsilon}, z^{i-1}_{\varepsilon},P^{i-1}_{\varepsilon})\\
&&+{\varepsilon^{\alpha-1}}\int_{t^{i-1}_{\varepsilon}}^{t^i_{\varepsilon}}{\intom{E_{\varepsilon}^{i-1}(s):\nabla \dot{\pep}(s,z^{i-1}_{\varepsilon})(\nabla \pep)^{-1}(s,z^{i-1}_{\varepsilon})}\,ds},
\end{eqnarray*}
where
$$E_{\varepsilon}^{i-1}(s):=\frac{1}{\varepsilon^{\alpha-1}}DW_{el}\big(\nabla \pep(s,z^{i-1}_{\varepsilon})\nep z^{i-1}_{\varepsilon}(P^{i-1}_{\varepsilon})^{-1}\big)\big(\nabla \pep(s,z^{i-1}_{\varepsilon})\nep z^{i-1}_{\varepsilon}(P^{i-1}_{\varepsilon})^{-1}\big)^T$$
for every $s\in [t^{i-1}_{\varepsilon}, t^i_{\varepsilon}]$.
By iterating the discrete energy inequality, recalling that $\overline{P}^{\varepsilon}(t)$ is locally constant, we obtain
\begin{eqnarray}
&&\nonumber\cal{F}_{\varepsilon}(t,\overline{z}^{\varepsilon}(t),\overline{P}^{\varepsilon}(t))+{\varepsilon^{\alpha-1}}\cal{D}(\overline{P}^{\varepsilon};0,t)\\
&&\nonumber\leq \varepsilon^{2\alpha-2}\delta_{\varepsilon}T +\cal{F}_{\varepsilon}(0, \zep_0, P^{\varepsilon}_0)+{\varepsilon^{\alpha-1}}\int_0^t{\intom{\overline{E}^{\varepsilon}(s):\nabla \dot{\pep}(s,\overline{z}^{\varepsilon}(s))(\nabla \pep)^{-1}(s,\overline{z}^{\varepsilon}(s))}\,ds},\\
\label{notenough}
\end{eqnarray}
where $\zep_0:=\vep(0,y^{\varepsilon}_0)$ and
$$\overline{E}^{\varepsilon}(s):=\frac{1}{\varepsilon^{\alpha-1}}DW_{el}\big(\nabla \pep(s,\overline{z}^{\varepsilon}(s))\nep\overline{z}^{\varepsilon}(s)(\overline{P}^{\varepsilon})^{-1}(s)\big)\big(\nabla \pep(s,\overline{z}^{\varepsilon}(s))\nep \overline{z}^{\varepsilon}(s)(\overline{P}^{\varepsilon})^{-1}(s)\big)^T$$
for every $s\in [0,t]$.\\
\emph{Proof of the reduced stability condition and energy balance}\\
The reduced stability condition can be deduced as in Step 2 of the proof of Theorem \ref{cvstress}. Moreover, arguing as in the proof of Theorem \ref{cvstress} one can show that $\overline{E}^{\varepsilon}(t)$ converges in the sense of \eqref{badsetstressnt} and \eqref{goodsetstress} to a limit stress $E(t)$ such that
$$E(t)=\mathbb C(G(t)-p(t)).$$
The crucial step to deduce the reduced energy balance is to show that $E(t)e_3=0$ a.e. in $\Omega$, that is,
\begin{equation}
\label{minimality}
E(t)=\mathbb C_2(G'(t)-p'(t)).
\end{equation}
The main difference with respect to Theorem \ref{cvstress} is that in this case we can not deduce this condition starting from the three-dimensional Euler-Lagrange equations because \eqref{notenough} does not imply \eqref{euler}.
To cope with this problem, set $\overline{y}^{\varepsilon}(t)=\overline{\phi}^{\varepsilon}(t,\overline{z}^{\varepsilon}(t))$ for every $t\in [0,T]$. Let {$\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ be such that $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}. We consider variations of the form
$$\hat{y}=\overline{y}^{\varepsilon}(t)+\tau_{\varepsilon}\varepsilon^{\alpha-1}\eta^{\varepsilon}\circ \overline{y^{\varepsilon}},$$
where $\eta^{\varepsilon}$ is the test function considered in Step 4 of the proof of Theorem \ref{cvstress}. By \eqref{quasst}, taking $\hat{P}=\overline{P}^{\varepsilon}(t)$, we deduce
\begin{eqnarray*}
-\delta_{\varepsilon}&&\leq\frac{1}{\varepsilon^{\alpha-1}}\intom{\frac{W_{el}\Big(\Big(Id+\tau_{\varepsilon}\varepsilon^{\alpha-1}\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t))\Big)\nep \overline{y}^{\varepsilon}(t)(\overline{P}^{\varepsilon})^{-1}(t)\Big)-W_{el}(\nep \overline{y}^{\varepsilon}(t)(\overline{P}^{\varepsilon})^{-1}(t))}{\tau_{\varepsilon}\varepsilon^{\alpha-1}}}\\
&&=\frac{1}{\varepsilon^{\alpha-1}}\intom{\int_0^1{\frac{d}{ds}\frac{W_{el}\Big(\Big(Id+s\tau_{\varepsilon}\varepsilon^{\alpha-1}\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t))\Big)\nep\overline{y}^{\varepsilon}(t)(\overline{P}^{\varepsilon})^{-1}(t)\Big)}{ \tau_{\varepsilon}\varepsilon^{\alpha-1}}}\,ds}\\
&&=\intom{\Phi^{\varepsilon}(t):\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t))},
\end{eqnarray*}
where
$$\Phi^{\varepsilon}(t):=\frac{1}{\varepsilon^{\alpha-1}}\int_0^1{DW_{el}\Big(\Big(Id+s\tau_{\varepsilon}\varepsilon^{\alpha-1}\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t))\Big)\nep\overline{y}^{\varepsilon}(t)(\overline{P}^{\varepsilon})^{-1}(t)\Big)(\nep\overline{y}^{\varepsilon}(t)(\overline{P}^{\varepsilon})^{-1}(t))^T\,ds}.$$
Since $\overline{P}^{\varepsilon}(t)\in L^2(\Omega;SL(3))$, $\det\,\overline{P}^{\varepsilon}(t)=1$ a.e. in $\Omega$. Moreover, by (H1) and \eqref{quasst} we deduce that $\det\, \nep\overline{y}^{\varepsilon}(t)>0$ a.e. in $\Omega$. On the other hand, since $\|\nabla \eta^{\varepsilon}\|_{L^{\infty}(\Omega;\mathbb{M}^{3\times 3})}\leq C$ for every $\varepsilon$ (see Step 4 of the proof of Theorem \ref{cvstress} and \eqref{bddinverse}), by \eqref{ss49},
$$\det\,(Id+s\tau_{\varepsilon}\varepsilon^{\alpha-1}\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t)))>0\quad\text{ for every }s\in [0,1],$$
for $\varepsilon$ small enough. Hence, by combining \eqref{mandel2} and \eqref{lemmams} we deduce that $\Phi^{\varepsilon}(t)$ is well defined for $\varepsilon$ small enough.
Now, there holds
\begin{equation}
\label{star5}
\liminf_{\varepsilon\to 0}\Big\{\intom{\Phi^{\varepsilon}(t):\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t))}\Big\}\geq 0.
\end{equation}
We claim that
\begin{equation}
\label{claimdisc}
\lim_{\varepsilon\to 0}\intom{\Phi^{\varepsilon}(t):\nabla \eta^{\varepsilon}(\overline{y}^{\varepsilon}(t))}=\intom{E(t)e_3:\partial_3\eta}.
\end{equation}
We note that, once \eqref{claimdisc} is proved, from \eqref{star5} it follows that
$$\intom{E(t)e_3:\partial_3\eta}\geq 0$$
for every {$\eta\in W^{1,\infty}(\mathbb R^3;\mathbb R^3)\cap C^{\infty}(\mathbb R^3;\mathbb R^3)$ such that $\eta=0\quad\cal{H}^2\text{ - a.e. on }\Gamma_d$}, hence the proof of \eqref{minimality} is complete.
To prove \eqref{claimdisc}, it is enough to consider the sets
$$O_{\varepsilon}(t):=\{x:\varepsilon^{\alpha-1-\gamma}|\overline{F}^{\varepsilon}(t)|<1\},$$
where the maps $\overline{F}^{\varepsilon}(t)$ are the piecewise constant interpolants of the maps $F^{\varepsilon}(t)$ defined in \eqref{deffept}. Arguing as in the proof of \eqref{badsetstress} and \eqref{goodsetstress}, one can show that, denoting by $\chi_{\varepsilon}(t)$ the characteristic function of the set $O_{\varepsilon}(t)$, there holds
$$||(1-\chi_{\varepsilon}(t))\Phi^{\varepsilon}(t)||_{L^1(\Omega;\mathbb{M}^{3\times 3})}\leq C\varepsilon^{\alpha-1}$$
and
$$\chi_{\varepsilon}(t)\Phi^{\varepsilon}(t)\rightharpoonup E(t)\quad\text{weakly in }L^2(\Omega;\mathbb{M}^{3\times 3}).$$
Claim \eqref{claimdisc} follows now arguing as in Step 4 of the proof of Theorem \ref{cvstress}.
\end{proof}
\noindent
\textbf{Acknowledgements.}
I warmly thank Maria Giovanna Mora for having proposed to me the study of this problem and for many helpful and stimulating discussions and suggestions.\\
This work was partially supported by MIUR under PRIN 2008.
\bigskip
|
2,869,038,155,679 | arxiv | \section{Introduction}\label{sec:intro}
EXO~0748--676\ is an intensively studied low-mass X-ray binary that was initially discovered with the European X-ray Observatory SATellite (\textit{EXOSAT}) in 1985 February \citep{parmar1985}. However, in retrospect the source already appeared active in \textit{EXOSAT}\ slew survey observations several times beginning 1984 July \citep{reynolds1999}, whereas the earliest detection dates back to 1980 May, when EXO~0748--676\ was serendipitously observed with the \textit{EINSTEIN}\ satellite \citep[][]{parmar1986}. The system exhibits irregular X-ray dips and displays eclipses that last for $\sim 8.3$~min and recur every 3.82~hr, which allow the unambiguous determination of the orbital period of the binary \citep[][]{parmar1986,wolff2008c}.
The detection of type-I X-ray bursts \citep[e.g.,][]{gottwald1986} conclusively identify the compact primary as a neutron star. A few X-ray bursts have been observed that exhibited photospheric radius expansion (PRE), which indicates that the Eddington luminosity is reached near the burst peak and allows for a distance estimate towards the source \citep[][]{wolff2005,galloway06}. For a Helium-dominated photosphere, a distance of $D=7.4\pm0.9$~kpc can be derived, while assuming solar composition results in a distance estimate of $D=5.9\pm0.9$~kpc \citep[][]{galloway06}. The rise time and duration of the PRE bursts observed from EXO~0748--676\ suggest pure Helium ignition, rendering 7.4~kpc as the best distance estimate \citep[][]{galloway06}, although this value is subject to several uncertainties \citep[][]{wolff2005,galloway2008}.
At the time of its discovery, EXO~0748--676\ was detected at 2--10 keV luminosities of $\sim(1-7)\times10^{36}~\mathrm{(D/7.4~kpc)^2}~\mathrm{erg~s}^{-1}$ \citep[][]{parmar1986}. However, during the \textit{EINSTEIN}\ observation of 1980, several years prior to the \textit{EXOSAT}\ detections, it displayed a 0.5--10 keV luminosity of $\sim 5 \times 10^{33}~\mathrm{(D/7.4~kpc)^2}~\mathrm{erg~s}^{-1}$ \citep[][]{parmar1986,garcia1999}. The source can therefore be classified as a transient X-ray binary. Nevertheless, such systems typically exhibit accretion outbursts that last only weeks to months \citep[e.g.,][]{chen97}, whereas EXO~0748--676\ was persistently detected at luminosities of $\sim10^{36-37}~\mathrm{(D/7.4~kpc)^2}~\mathrm{erg~s}^{-1}$ by various satellites for over 24 years. Similar prolonged accretion episodes continuing for years to decades have been observed for a few other systems, which are termed quasi-persistent X-ray binaries \citep[e.g.,][]{wijnands04_quasip}.
In 2008 August--September, observations with the Proportional Counter Array (PCA) onboard the \textit{Rossi X-ray Timing Explorer} (\textit{RXTE}) and \textit{Swift}'s X-ray Telescope (XRT) indicated that the X-ray flux of EXO~0748--676\ was declining \citep[][]{wolff2008,wolff2008b}. Optical and near-IR observations of the optical counterpart, UY~Vol, performed in 2008 October showed that the optical emission had also faded compared to the brighter X-ray state \citep[][]{hynes2008,hynes09,torres2008}. These events indicated that the accretion was ceasing and that the system was transitioning from outburst to quiescence. This is also illustrated by Fig.~\ref{fig:asm}, which displays the X-ray lightcurve of EXO~0748--676\ as observed with the All-Sky Monitor (ASM) onboard \textit{RXTE}\ since 1996. The decrease in source activity is clearly seen around $\sim4600$ days.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{asm_lc_chan_extended.eps}
\end{center}
\caption[]{{\textit{} \textit{RXTE}/ASM 20-day averaged lightcurve (1.5--12 keV) of EXO~0748--676, illustrating the cessation of the outburst in 2008 August/September.
For reference: the dashed vertical line corresponds to 2008 December 31.
The arrows indicate the times of our four sequences of \textit{Chandra}\ observations, which were performed when the source dropped below the detection limit of \textit{RXTE}\ (both of the ASM and the PCA).}}
\label{fig:asm}
\end{figure}
\textit{Chandra}\ observations carried out in 2008 mid-October (i.e., after the transition to quiescence started) revealed an X-ray spectrum composed of a soft, thermal component joined by a hard powerlaw tail that dominates the spectrum above $\sim2-3$~keV \citep[][see also Section~\ref{subsec:spectraldata}]{degenaar09_exo1}. This is frequently seen for neutron star X-ray binaries in quiescence \citep[e.g.,][]{rutledge1999,zand2001_qNS,tomsick2004}. The non-thermal component is usually well-fitted by a simple powerlaw with index 1--2 \citep[e.g.,][]{asai1996}. The fractional contribution of the hard powerlaw tail to the 0.5--10 keV X-ray flux widely varies amongst sources and possibly also with changing luminosity \citep[][]{jonker2004,jonker07_eos}. The physical process that is responsible for the powerlaw spectral component remains elusive \citep[see e.g.,][]{campana1998, campana2003}.
Although the soft spectral component has been ascribed to low-level accretion \citep[][]{zampieri1995}, it is most often interpreted as thermal surface radiation from the cooling neutron star \citep[][]{brown1998}. According to this model, the accretion of matter compresses the neutron star crust, which induces a series of electron captures, neutron emissions and pycnonuclear fusion reactions \citep[e.g.,][]{haensel1990a,haensel2003,haensel2008,gupta07}. The heat energy released in these processes is spread over the neutron star via thermal conduction.
The neutron star cools primarily via neutrino emissions from the stellar core, as well as photon radiation from the surface. The former depends on the equation of state of cold nuclear matter and the central density of the neutron star \citep[e.g.,][]{yakovlev2004,page2006}.
The neutron star core reaches a thermal steady state in $\sim10^{4}$~years, yielding an incandescent emission from the neutron star surface set by the time-averaged accretion rate of the system, as well as the rate of neutrino emissions from the stellar core \citep[e.g.,][]{brown1998,colpi2001}. When combined with estimates of the outburst history, observations of quiescent neutron stars can constrain the rate of neutrino emissions, thereby providing insight into the interior properties of the neutron star \citep[e.g.,][]{heinke2009}.
Once the steady state is reached, the neutron star core temperature will not change appreciably during a single outburst, but the temperature of the crust can be dramatically altered. In regular transients that have a typical outburst duration of weeks to months, the crustal heating processes will only cause a slight increase in the crust temperature \citep[][]{brown1998}. However, in quasi-persistent X-ray binaries the prolonged accretion episodes can cause a significant temperature gradient between the neutron star crust and core. Once the accretion ceases, the crust is expected to thermally relax on a time scale of years, until equilibrium with the core is re-established \citep[][]{rutledge2002}. During the initial stages of the quiescent phase the thermal emission will therefore be dominated by the cooling crust, whereas eventually a quiescent base level is reached that is set by the thermal state of the core \citep[][]{wijnands2001,rutledge2002}. This provides the special opportunity to separately probe the properties of the neutron star crust \citep[][]{haensel2008,brown08}.
In 2001, the neutron star X-ray binaries KS~1731--260\ and MXB~1659--29\ both made the transition to quiescence, following accretion episodes of 12.5 and 2.5 years, respectively \citep[][]{wijnands2001,wijnands2002,wijnands2003,wijnands2004,cackett2006,cackett2008}. More recently, in 2007, the $\sim1.6$-year long outburst of XTE~J1701--462\ came to a halt \citep[][]{altamirano2007,homan2007,fridriksson2010}. All three systems were subsequently monitored with \textit{Chandra}\ and \textit{XMM-Newton}, which revealed that thermal flux and neutron star temperature were gradually decreasing over the course of years (see also Section~\ref{sec:discussion}). This can be interpreted as cooling of the neutron star crust that has been heated during the prolonged accretion outburst. Successful modelling of the observed quiescent X-ray lightcurves with neutron star thermal evolution models supports this hypothesis and provides important constraints on the crust properties, such as the thermal conductivity \citep[][]{shternin07,brown08}.
Along these lines we have pursued an observational campaign of EXO~0748--676\ to study the time evolution of the quiescent X-ray emission following its long accretion outburst. In \citet[][]{degenaar09_exo1}, we discussed \textit{Chandra}\ and \textit{Swift}\ observations obtained between 2008 September 28 and 2009 January 30. We found a relatively hot and luminous quiescent system with a temperature of $kT^{\infty}_{\mathrm{eff}}\sim0.11-0.13$~keV and a thermal 0.01--100 keV luminosity of $\sim(8-16)\times10^{33}~(\mathrm{D/7.4~kpc})^2~\mathrm{erg~s}^{-1}$. No clear decrease in effective temperature and thermal bolometric flux was found over the five-month time span.
In this paper we report on continued \textit{Swift}\ and \textit{Chandra}\ observations of EXO~0748--676\ during its quiescent state. In addition, we include an archival \textit{XMM-Newton}\ observation performed $\sim 2$ months after the cessation of the outburst. Previous \textit{Chandra}\ and \textit{Swift}\ observations discussed by \citet{degenaar09_exo1} were re-analysed in this work in order to obtain a homogeneous quiescent lightcurve.
\section{Observations and data analysis}
Table~\ref{tab:obs} gives an overview of all new observations of EXO~0748--676\ discussed in this paper. A list of earlier \textit{Chandra}\ and \textit{Swift}\ observations obtained during the quiescent phase can be found in \citet{degenaar09_exo1}.
\begin{table}
\caption{Observation log.}
\begin{threeparttable}
\begin{tabular}{l l l l}
\hline \hline
Satellite & Obs ID & Date & Exp. time \\
& & & (ks) \\
\hline
\textit{XMM} & 0560180701* & 2008-11-06 & 29.0 (MOS) \\
& & & 22.9 (PN) \\
\textit{Swift}\ & 31272016 & 2009-02-13 & 3.5 \\
\textit{Swift}\ & 31272017 & 2009-02-20 & 4.1 \\
\textit{Swift} & 31272018* & 2009-02-23 & 5.1 \\
\textit{Chandra}\ & 9071* & 2009-02-23/24 & 15.8 \\
& 10871* & 2009-02-25 & 9.6 \\
\textit{Swift}\ & 31272019* & 2009-03-01 & 3.2 \\
\textit{Swift}\ & 31272020 & 2009-03-10 & 5.1 \\
\textit{Swift}\ & 31272021 & 2009-03-16 & 4.6 \\
\textit{Swift}\ & 31272022 & 2009-04-09 & 3.5 \\
\textit{Swift}\ & 31272023 & 2009-04-16 & 2.8 \\
\textit{Swift}\ & 31272024 & 2009-04-23 & 4.8 \\
\textit{Swift}\ & 31272025 & 2009-05-07 & 4.5 \\
\textit{Swift}\ & 31272026 & 2009-05-14 & 3.6 \\
\textit{Swift}\ & 31272027 & 2009-05-28 & 3.4 \\
\textit{Swift}\ & 31272028 & 2009-06-05 & 4.1 \\
\textit{Chandra}\ & 9072* & 2009-06-10 & 27.2 \\
\textit{Swift}\ & 31272029* & 2009-06-11 & 4.3 \\
\textit{Swift}\ & 31272030 & 2009-06-18 & 3.9 \\
\textit{Swift}\ & 31272031* & 2009-06-26 & 5.5 \\
\textit{Swift}\ & 31272032* & 2009-07-03 & 4.8 \\
\textit{Swift}\ & 31272033 & 2009-07-18 & 5.5 \\
\textit{Swift}\ & 31272034* & 2009-07-25 & 5.8 \\
\textit{Swift}\ & 31272035* & 2009-07-31 & 10.3 \\
\textit{Swift}\ & 31272036* & 2009-08-15 & 9.4 \\
\textit{Swift}\ & 31272037 & 2009-08-25 & 1.1 \\
\textit{Swift}\ & 31272038 & 2009-08-26 & 7.4 \\
\textit{Swift}\ & 31272039* & 2009-09-08 & 4.7 \\
\textit{Swift}\ & 31272040* & 2009-09-09 & 4.3 \\
\textit{Swift}\ & 31272041 & 2009-10-01 & 1.9 \\
\textit{Swift}\ & 31272042 & 2009-10-02 & 1.8 \\
\textit{Swift}\ & 31272043 & 2009-10-07 & 2.0 \\
\textit{Swift}\ & 31272044 & 2009-10-08 & 2.4 \\
\textit{Swift}\ & 31272045 & 2009-10-09 & 2.3 \\
\textit{Swift}\ & 31272046* & 2009-11-05 & 4.2 \\
\textit{Swift}\ & 31272047* & 2009-12-21 & 9.4 \\
\textit{Swift}\ & 31272048* & 2010-10-01 & 9.6 \\
\textit{Swift}\ & 31272049 & 2010-02-12/13 & 11.3 \\
\textit{Swift}\ & 31272050* & 2010-03-12/13 & 9.5 \\
\textit{Chandra}\ & 11059* & 2010-04-20 & 27.4 \\
\hline
\end{tabular}
\label{tab:obs}
\begin{tablenotes}
\item[]Note. -- The observations marked with an asterisk contain (part of) eclipses. The listed exposure times represent the duration of the observations uncorrected for eclipses.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{\textit{XMM-Newton}}\label{subsec:xmm}
EXO~0748--676\ was observed with the European Photon Imaging Camera (EPIC) onboard \textit{XMM-Newton}\ on 2008 November 6 from 08:30--16:42 \textsc{ut} \citep[see also][]{bassa09}. The EPIC instrument consists of two MOS detectors \citep[][]{turner2001_mos} and one PN camera \citep[][]{struder2001_pn}, which are sensitive in the 0.1--15 keV energy range and have effective areas of 922~cm$^2$ and 1227~cm$^2$ (at 1 keV), respectively. Both the PN and the two MOS instruments were operated in full window mode and using the medium optical blocking filter.
Data reduction and analysis was carried out with the Science Analysis Software (\textsc{SAS}; v. 9.0.0). We reprocessed the Original Data Files (ODF) using the tasks \textsc{emproc} and \textsc{epproc}. To identify possible periods of high particle background, we extracted high-energy lightcurves ($\geq 10$~keV for the MOS and between 10--12~keV for the PN). No strong background flares occurred during the observation. The net exposure times are 29.0 and 22.9~ks for the MOS and PN, respectively. EXO~0748--676\ is detected at count rates of $0.16\pm0.01~\mathrm{counts~s}^{-1}$ (MOS) and $0.55\pm0.01~\mathrm{counts~s}^{-1}$ (PN).
Source spectra and lightcurves were obtained with the software task \textsc{evselect}, using a 35~arcsec circular region and applying pattern selections 0--12 and 0--4 for the MOS and PN data, respectively. Corresponding background events were extracted from a circular region with a radius of 70 arcsec. For the MOS cameras, the background was positioned on a source-free region on the same CCD as the source. For the PN instrument, the background events were extracted from an adjacent CCD, at the same distance from the readout node to ensure similar low-energy noise. The ancillary response files (arf) and redistribution matrices (rmf) were generated for each of two MOS and the PN cameras with the tasks \textsc{arfgen} and \textsc{rmfgen}.
The EPIC lightcurves show two full eclipses \citep[see also][]{bassa09}, corresponding to eclipse cycles 54384 and 54385 in the numbering system of \citet{parmar1986}. To calculate the correct non-eclipse time-averaged fluxes, we reduce the exposure times for each instrument by 500~s per eclipse, which is the approximate length of the eclipses of EXO~0748--676\ \citep[][]{wolff2008c}.\footnote{As shown by \citet{wolff2008c}, the duration of the eclipses of EXO~0748--676\ varied between $\sim484$ and 512 s over the years 1996--2008. These small uncertainties in the eclipse duration do not affect our results.} Using the tool \textsc{grppha}, the spectra were grouped to contain a minimum of 20 photons per bin.
\subsection{\textit{Chandra}}\label{subsec:chan}
We obtained three new \textit{Chandra}\ observations of EXO~0748--676\ using the S3 chip of the Advanced CCD Imaging Spectrometer \citep[ACIS;][]{garmire2003_acis}. The ACIS detector is sensitive in the 0.1--10 keV passband and has an effective area of 340~cm$^2$ at 1 keV. The first observation consists of two separate exposures obtained on 2009 February 23--24 22:07--03:15 \textsc{ut} (obs ID 9071) and 2009 February 25 12:32--15:59 \textsc{ut} (obs ID 10871), lasting for $\sim15.8$ and $\sim9.6$~ks, respectively. In both data sets, EXO~0748--676\ is clearly detected at a count rate of $0.17\pm0.01~\mathrm{counts~s}^{-1}$. This is a factor $\sim1.5$ lower than observed in 2008 October, when the source was detected with \textit{Chandra}/ACIS-S at a rate of $0.24\pm0.01~\mathrm{counts~s}^{-1}$. Two full eclipses are seen in the lightcurve of observation 9071, while one eclipse is present in that of 10871 (eclipse cycle numbers 55071, 55072 and 55080, respectively).
A second \textit{Chandra}\ observation was carried out on 2009 June 10, from 12:36--21:16 \textsc{ut}, with an exposure time of 27.2~ks (obs ID 9072). In this observation EXO~0748--676\ is detected at a count rate of $0.16\pm0.01~\mathrm{counts~s}^{-1}$ and the lightcurve shows two full eclipses (cycles 55740 and 55741). Furthermore, a 27.4 ks exposure was taken on 2010 April 20 from 02:37--11:28 \textsc{ut} (obs ID 11059), which captured three full eclipses (see Fig.~\ref{fig:eclipse}, these correspond to eclipse cycle numbers 57708, 57709 and 57710), and detected the source at a count rate of $0.14\pm0.01~\mathrm{counts~s}^{-1}$. Similar to our treatment of the \textit{XMM-Newton}\ data, we reduce the exposure times of all \textit{Chandra}\ observations by 500~s per eclipse. There are no indications of background flares, so the full data set was used in further analysis.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{chan_acis_lc_11059.eps}
\end{center}
\caption[]{{\textit{Chandra}/ACIS-S 0.5--8 keV lightcurve of EXO~0748--676\ obtained on 2010 April 20 (obs ID 11059). Each point represents 200~s of data. Three eclipses are visible.}}
\label{fig:eclipse}
\end{figure}
We reduced the data employing the \textsc{ciao} tools (v. 4.2) and standard \textit{Chandra}\ analysis threads. For all three observation sequences, the ACIS-S3 CCD was operated in a 1/8 sub-array, resulting in a frame-time of 0.4~s. For the observed count rates, the pile-up fraction is $<2$ percent, so no further corrections were made. Spectra were extracted with the tool \textsc{psextract}, while the rmf and arf files were created using \textsc{mkacisrmf} and \textsc{mkarf}, respectively. We employ a circular region of 3 arcsec to obtain source events and a 10--25 arcsec annulus for the background. We also reprocessed the \textit{Chandra}\ observation obtained in 2008 October \citep[see][]{degenaar09_exo1} to benefit from the calibration update that was released in 2009 December. Prior to spectral fitting, the spectra were grouped to contain at least 20 photons per bin.
\subsection{\textit{Swift}}\label{subsec:swift}
In addition to the \textit{Chandra}\ and \textit{XMM-Newton}\ observations, we have been monitoring EXO~0748--676\ on a regular basis with the XRT \citep[][]{burrows05} aboard the \textit{Swift}\ satellite. The instrument has an effective area of 110~cm$^2$ at 1.5 keV and is operated in the energy range of 0.2--10 keV. Starting in 2008 late-September, approximately $2-3$ pointings were performed each month with a typical duration of $\sim3-5$~ks per observation, and a separation of $\sim1-2$ weeks. From 2009 November onwards, the cadence was lowered to one observation per month with a longer exposure time when possible (see Table~\ref{tab:obs}). EXO~0748--676\ is detected in the XRT observations at count rates of $\sim(1-5)\times10^{-2}~\mathrm{counts~s}^{-1}$.
All \textit{Swift}/XRT observations were obtained in the photon-counting (pc) mode and were processed using the \textsc{xrtpipeline} with standard quality cuts (event grade 0--12). Using \textsc{xselect} (v. 2.4), we extracted source spectra from a circular region with a radius of 35 arcsec ($\sim15$~pixels), which optimises the signal to noise ratio at the observed count rates \citep[][]{evans2007}. Corresponding background events were averaged over three source-free regions of similar shape and size. Employing the tool \textsc{xrtexpomap}, we created exposure maps to account for the effective area of the CDD, while arfs generated with \textsc{xrtmkarf} account for vignetting and point-spread-function corrections. The latest rmf (v. 11) was obtained from the \textsc{caldb} database.
Due to low statistics, it is not possible to identify eclipses in the \textit{Swift}\ lightcurves. Therefore, we used the ephemeris of \citet{wolff2008c} to determine during which observations eclipses were occurring (see Table~\ref{tab:obs}). To calculate the correct non-eclipse time-averaged fluxes, the exposure times of these observations were reduced with the duration of the eclipses contained in the data (500 s if a full eclipse was present, but less if only part of an eclipse was expected). Furthermore, \textit{Swift}\ observations obtained within a 2-day time span were summed to improve the data statistics.\footnote{This is the case for obs IDs 31272037/38, 31272039/40 and 31272043/44/45; see Table~\ref{tab:obs}.} This seems justified, since the \textit{Chandra}\ data do not reveal any spectral changes on such time scales (see Section~\ref{subsec:spectraldata}).
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{chan1_chan4_spec_red.eps}
\end{center}
\caption[]{{Spectra of the \textit{Chandra}\ observations of 2008 October (black) and 2010 April (red), along with the model fits (solid lines). The separate contributions of the \textsc{nsatmos} and \textsc{powerlaw} components are represented by the dotted (2008 data) and dashed (2010 data) lines. }}
\label{fig:spec}
\end{figure}
\subsection{Spectral models}\label{subsec:spectraldata}
We fitted the spectral data in the 0.5--10 keV energy range using \textsc{xspec} \citep[v. 12.0;][]{xspec}.
This software package facilitates fitting a spectral model simultaneously to multiple data files, which each have their own response and background files. As is common practise, we fit the \textit{XMM-Newton}\ data with all spectral parameters tied between the different detectors (i.e., the model parameters are not allowed to vary independently between the PN and two MOS detectors). For all fits throughout this paper, we included the effect of neutral hydrogen absorption, $N_{\mathrm{H}}$, along the line of sight using the \textsc{phabs} model with the default \textsc{xspec} abundances \citep[][]{anders1989_phabs_abun} and cross-sections \citep[][]{balucinska1992_phabs_cross}.
We first investigate the shape of the quiescent spectrum of EXO~0748--676\ by considering the \textit{XMM-Newton}\ observation, which provides the highest statistics. A single absorbed powerlaw (\textsc{powerlaw} in \textsc{xspec}) provides an acceptable fit to the data ($\chi^2_{\nu}=1.3$ for 466 d.o.f.). However, the spectral index is unusually large for an X-ray binary ($\Gamma=4.7\pm0.1$) and suggests that the spectrum has a thermal shape. Using a simple absorbed blackbody model, \textsc{bbodyrad}, results in an adequate fit ($\chi^2_{\nu}=1.2$ for 466 d.o.f.), although the inferred emitting region has a much smaller radius than expected for a neutron star ($\sim2-4$~km for distances of $5-10$~kpc). Nevertheless, it is thought that radiative transfer effects in the neutron star atmosphere cause the emergent spectrum to deviate from a blackbody \citep[e.g.,][]{zavlin1996,rutledge1999}. There are several neutron star atmosphere models available within \textsc{xspec}, which yield equivalent results \citep[see e.g.,][]{heinke2006,webb2007}. In the remainder of this work, we concentrate on fitting the data with a neutron star atmosphere model \textsc{nsatmos} \citep[][]{heinke2006}.
The \textsc{nsatmos} model consists of five parameters, which are the neutron star mass and radius ($M_{\mathrm{NS}}$ and $R_{\mathrm{NS}}$), the effective temperature in the neutron star frame (i.e., non-redshifted; $kT_{\mathrm{eff}}$), the source distance ($D$) and a normalisation factor, which parametrizes the fraction of the surface that is radiating. We keep the latter fixed at 1 throughout this work, which corresponds to the entire neutron star surface emitting. The effective temperature as seen by an observer at infinity is given by $kT^{\infty}_{\mathrm{eff}}= kT_{\mathrm{eff}}/(1+z)$, where $1+z = (1-R_{\mathrm{s}}/R_{\mathrm{NS}})^{-1/2}$ is the gravitational redshift factor, with $R_{\mathrm{s}}=2GM_{\mathrm{NS}}/c^2$ being the Schwarzschild radius, $G$ the gravitational constant and $c$ the speed of light.
The \textit{XMM-Newton}\ data is well-fitted by an absorbed \textsc{nsatmos} model ($\chi^2_{\nu}=1.1$ for 466 d.o.f.), although significant residuals above the model fit are present for energies $\gtrsim2-3$~keV. We model this non-thermal emission by adding a powerlaw component, which significantly improves the fit ($\chi^2_{\nu}=1.0$ for 464 d.o.f.; an F-test suggests a $\sim1\times10^{-14}$ probability of achieving this level of improvement by chance). \textit{Chandra}\ observations carried out in 2008 mid-October, three weeks prior to this \textit{XMM-Newton}\ observation, also indicated the presence of a non-thermal component in the quiescent spectrum of EXO~0748--676\ \citep[][]{degenaar09_exo1}. Whereas the \textit{Chandra}\ data could not constrain the powerlaw index, the larger collective area of \textit{XMM-Newton}\ provides better constraints for the fluxes under consideration.
By using a combined \textsc{nsatmos} and \textsc{powerlaw} model to fit the \textit{XMM-Newton}\ data, we obtain a powerlaw index of $\Gamma=1.7\pm0.5$, i.e., in between the values of $\Gamma=1$ and $\Gamma=2$ considered by \citet[][]{degenaar09_exo1}. This fit furthermore yields $N_{\mathrm{H}}=(7\pm2)\times10^{20}~\mathrm{cm}^{-2}$ and $R_{\mathrm{NS}}=17.8\pm1$~km, when fixing the neutron star mass to a canonical value of $M_{\mathrm{NS}}=1.4~\mathrm{M}_{\odot}$ and the distance to $D=7.4$~kpc \citep[the best estimate from type-I X-ray burst analysis;][]{galloway06}. The resulting powerlaw component contributes $\sim10$ percent to the total unabsorbed 0.5--10 keV flux. This is lower than the $\sim15-20$ percent inferred from the \textit{Chandra}\ observations performed in 2008 mid-October \citep[][]{degenaar09_exo1}. The obtained hydrogen column density is consistent with values found for EXO~0748--676\ during its outburst \citep[$N_{\mathrm{H}}\sim 7\times10^{20}-1.2\times10^{21}~\mathrm{cm}^{-2}$; e.g.,][]{sidoli05}.
The \textit{Chandra}\ observations obtained in 2009 February and June are well-fitted by an absorbed \textsc{nsatmos} model and do not require an additional powerlaw component. However, the 2010 April data shows evidence for such a hard tail, as significant residuals are present above the \textsc{nsatmos} model fit for energies $\gtrsim2-3$~keV. If we include a powerlaw with photon index $\Gamma=1.7$, as was found from fitting the \textit{XMM-Newton}\ data (see above), this model component contributes $\sim10$, $\sim5$ and $\sim15$ percent to the total unabsorbed 0.5--10 keV flux for the data taken in 2009 February, June and 2010 April, respectively. Fig.~\ref{fig:spec} compares the \textit{Chandra}\ spectral data obtained on 2008 October and 2010 April, showing that both spectral components decreased over the 18-month time span that separates the two observations. We found no spectral differences between the two separate exposures performed in 2009 February and therefore we tied all spectral parameters between these two spectra in the fits.
The \textit{Swift}\ data do not provide sufficient statistics to constrain the presence of a hard spectral component. We do include a powerlaw in the fits, but fix both the index and the normalisation of this component (see Section~\ref{subsec:evolution}). Since it is unclear how the powerlaw exactly evolves over time, we adjust the powerlaw normalisation for the \textit{Swift}\ observations such that it always contributes $10$ percent of the total unabsorbed 0.5--10 keV flux. After treating each \textit{Swift}\ observation separately, we found that the thermal flux and neutron star temperature did not evolve significantly between consecutive observations. To improve the statistics, we therefore sum the \textit{Swift}\ data into groups spanning $\sim1-4$ weeks of observations, resulting in exposure times of $\sim10-20$~ks (see Table~\ref{tab:spec}). The summed spectra were grouped to contain a minimum of 20 photons per bin.
\begin{table*}
\caption{Results from fitting the spectral data.}
\begin{threeparttable}
\begin{tabular}{l l l l l l l l l}
\hline \hline
Satellite & Date & $\Delta t$ & Pow. frac. & $kT^{\infty}_{\mathrm{eff}}$ & $F_{\mathrm{X}}$ & $F_{\mathrm{bol}}^{\mathrm{th}}$ & $L_{\mathrm{bol}}$ & $\chi^2_{\nu} $ \\
& & (days) & (\%)& (eV) & & & & (d.o.f.)\\
\hline
\textit{Swift}$\dagger$ & 2008-09-28 -- 2008-10-07 & 4.9 & 10 fix & $123.7\pm5.4$ & $1.31\pm0.22$ & $1.53\pm0.26$ & $10.0\pm1.7$ & 0.93 (8) \\%2.5
\textit{Chandra}$\dagger$ & 2008-10-12/13/15 & 1.4 & $20\pm3$ & $118.8\pm0.9$ & $1.23\pm0.02$ & $1.31\pm0.04$ & $8.6\pm0.3$ & 1.03 (175) \\
\textit{Swift}$\dagger$ & 2008-10-29 -- 2008-11-02 & 2.2 & 10 fix & $118.3\pm2.6$ & $1.10\pm0.09$ & $1.28\pm0.11$ & $8.4\pm0.7$ & 0.67 (14) \\%10.4
\textit{XMM} & 2008-11-06 & 0.2 & $7\pm2$ & $120.7\pm0.4$ & $1.14\pm0.01$ & $1.39\pm0.02$ & $9.1\pm0.1$ & 1.08 (467) \\
\textit{Swift}$\dagger$ & 2008-11-28 -- 2008-12-20 & 11.0 & 10 fix & $118.7\pm2.6$ & $1.11\pm0.10$ & $1.30\pm0.12$ & $8.5\pm0.8$ & 1.09 (14) \\
\textit{Swift}$\dagger$ & 2009-01-10 -- 2009-01-30 & 9.8 & 10 fix & $116.2\pm2.2$ & $0.99\pm0.07$ & $1.19\pm0.09$ & $7.8\pm0.6$ & 1.09 (18) \\
\textit{Swift}\ & 2009-02-13 -- 2009-02-23 & 5.1 & 10 fix & $117.2\pm2.4$ & $1.02\pm0.08$ & $1.23\pm0.10$ & $8.1\pm0.7$ & 0.50 (15) \\
\textit{Chandra}\ & 2009-02-23/25 & 0.9 & $12\pm4$ & $113.5\pm1.3$ & $0.91\pm0.03$ & $1.09\pm0.05$ & $7.1\pm0.3$ & 0.90 (139) \\
\textit{Swift}\ & 2009-03-01 -- 2009-03-16 & 7.2 & 10 fix & $115.6\pm2.3$ & $0.97\pm0.08$ & $1.17\pm0.09$ & $7.7\pm0.6$ & 0.79 (16) \\
\textit{Swift}\ & 2009-04-09 -- 2009-04-23 & 7.1 & 10 fix & $112.2\pm2.8$ & $0.86\pm0.09$ & $1.03\pm0.11$ & $6.8\pm0.7$ & 1.16 (10) \\
\textit{Swift}\ & 2009-05-07 -- 2009-06-05 & 14.7 & 10 fix & $114.2\pm2.3$ & $0.92\pm0.07$ & $1.11\pm0.09$ & $7.3\pm0.6$ & 0.88 (16) \\
\textit{Chandra}\ & 2009-06-10 & 0.2 & $4\pm3$ & $111.0\pm0.7$ & $0.75\pm0.01$ & $0.99\pm0.03$ & $6.5\pm0.2$ & 1.19 (93) \\
\textit{Swift}\ & 2009-06-11 -- 2009-07-03 & 11.5 & 10 fix & $111.9\pm2.2$ & $0.75\pm0.07$ & $1.03\pm0.08$ & $6.7\pm0.5$ & 0.99 (17) \\
\textit{Swift}\ & 2009-07-18 -- 2009-07-31 & 6.9 & 10 fix & $110.5\pm2.1$ & $0.79\pm0.06$ & $0.98\pm0.07$ & $6.4\pm0.5$ & 0.42 (18) \\
\textit{Swift}\ & 2009-08-15 -- 2009-09-09 & 12.7 & 10 fix & $110.0\pm1.8$ & $0.78\pm0.05$ & $0.96\pm0.06$ & $6.3\pm0.4$ & 1.30 (26) \\
\textit{Swift}\ & 2009-10-01 -- 2009-11-05 & 17.4 & 10 fix & $108.0\pm2.6$ & $0.69\pm0.07$ & $0.88\pm0.09$ & $5.8\pm0.6$ & 1.49 (11) \\
\textit{Swift}\ & 2009-12-21 -- 2010-10-01 & 10.2 & 10 fix & $109.4\pm2.0$ & $0.74\pm0.06$ & $0.94\pm0.07$ & $6.1\pm0.5$ & 1.51 (19) \\
\textit{Swift}\ & 2010-02-12 -- 2010-03-13 & 15.0 & 10 fix & $109.4\pm2.0$ & $0.76\pm0.06$ & $0.94\pm0.07$ & $6.1\pm0.5$ & 1.25 (20) \\
\textit{Chandra}\ & 2010-04-20 & 0.2 & $15\pm4$ & $108.6\pm1.1$ & $0.77\pm0.02$ & $0.91\pm0.04$ & $6.0\pm0.2$ & 0.79 (91) \\
\hline
\end{tabular}
\label{tab:spec}
\begin{tablenotes}
\item[]Note. -- The observations marked by a dagger were already discussed in \citet{degenaar09_exo1}, but re-fitted in this work. These results were obtained by using a combined absorbed \textsc{nsatmos} and \textsc{powerlaw} model, where $N_{\mathrm{H}}=7\times10^{20}~\mathrm{cm}^{-2}$, $M_{\mathrm{NS}}=1.4~\mathrm{M}_{\odot}$, $R_{\mathrm{NS}}=15.6$~km, $D=7.4$~kpc and $\Gamma=1.7$ were kept fixed. The quoted errors represent 90 percent confidence levels. $F_{\mathrm{X}}$ represents the 0.5--10~keV total model flux and $F_{\mathrm{bol}}^{\mathrm{th}}$ gives the 0.01--100 keV \textsc{nsatmos} flux; both are unabsorbed and in units of $10^{-12}~\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$. $L_{\mathrm{bol}}$ gives the 0.01--100 keV luminosity of the \textsc{nsatmos} model component in units of $10^{33}~\mathrm{erg~s}^{-1}$ and assuming a source distance of D=7.4~kpc. $\Delta t$ represents the time interval of the observations in days and the fractional powerlaw contribution is given in a percentage of the total unabsorbed 0.5--10 keV flux.
\end{tablenotes}
\end{threeparttable}
\end{table*}
\section{Results}
\subsection{Spectral fits}\label{subsec:evolution}
As discussed in Section~\ref{subsec:spectraldata}, the quiescent spectrum of EXO~0748--676\ can be described by a combination of a neutron star atmosphere model and a non-thermal powerlaw tail. We fitted the \textit{Chandra}\ and \textit{XMM-Newton}\ data simultaneously within \textsc{xspec} to a combined \textsc{nsatmos} and \textsc{powerlaw} model subject to interstellar absorption, to explore the best-fit values for the neutron star mass and radius, source distance and hydrogen column density. We include the first set of \textit{Chandra}\ observations obtained in 2008 October \citep[discussed in][]{degenaar09_exo1} in the analysis. As before, we use the \textsc{phabs} model with the default \textsc{xspec} abundances and cross-sections to take into account the neutral hydrogen absorption along the line of sight. The powerlaw index is fixed to $\Gamma=1.7$ (the best fit-value obtained from \textit{XMM-Newton}\ observations; see Section~\ref{subsec:spectraldata}), because there are not sufficient counts at higher energies in the \textit{Chandra}\ spectra to allow this component to vary. The powerlaw normalisation is left as a free parameter.
If the neutron star mass and radius are fixed to canonical values of $M_{\mathrm{NS}}=1.4~\mathrm{M}_{\odot}$ and $R_{\mathrm{NS}}=10$~km, and in addition the source distance is fixed to $D=7.4$~kpc, the hydrogen column density pegs at its lower limit ($N_{\mathrm{H}}=0$). When the distance is left to vary freely, the best-fit value is $4.6\pm0.3$~kpc, which is just outside the range obtained from X-ray burst analysis \citep[5--8.3~kpc;][]{galloway06}. Therefore, we choose to keep the distance fixed at 7.4~kpc, and instead allow the neutron star radius to vary. This way, we obtain best-fit values of $N_{\mathrm{H}}=(7\pm1)\times10^{20}~\mathrm{cm}^{-2}$ and $R=15.6\pm0.8$~km. If additionally the neutron star mass is left free to vary in the fit, this parameter is not strongly constrained ($M_{\mathrm{NS}}\sim 1.6\pm 0.6~\mathrm{M}_{\odot}$). In the final fits we choose to fix the neutron star mass to $M_{\mathrm{NS}}= 1.4~\mathrm{M}_{\odot}$, because otherwise the uncertainty in this quantity will dominate the errors of the other parameters.
For the final spectral analysis, we fit all \textit{XMM-Newton}, \textit{Chandra}\ and \textit{Swift}\ data with an absorbed \textsc{nsatmos} plus \textsc{powerlaw} model, where $N_{\mathrm{H}}=7\times10^{20}~\mathrm{cm}^{-2}$, $M_{\mathrm{NS}}=1.4~\mathrm{M}_{\odot}$, $R_{\mathrm{NS}}=15.6$~km, $D=7.4$~kpc and $\Gamma=1.7$ are fixed, while the neutron star effective temperature is left as a free parameter. The powerlaw normalisation is left to vary freely for the \textit{Chandra}\ and \textit{XMM-Newton}\ observations, but fixed for the \textit{Swift}\ data (so that this component contributes 10 percent to the total unabsorbed 0.5--10 keV flux).
We fit all data in the 0.5--10~keV energy range and deduce the absorbed and unabsorbed fluxes in this band. The thermal model fit is extrapolated to the energy range of 0.01--100 keV to estimate the thermal bolometric flux.
The results from fitting the X-ray spectra in this way are presented in Table~\ref{tab:spec}. The effective temperatures and thermal bolometric fluxes derived from \textit{Chandra}, \textit{Swift}\ and \textit{XMM-Newton}\ data are displayed in Fig.~\ref{fig:temp}. Examination of Fig.~\ref{fig:temp} suggests that there is a small but discernible offset in the thermal flux and neutron star temperature as deduced from the different satellites. This is briefly discussed in Section~\ref{subsec:crosscal}.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{PAPER_II_flux_temp_combi_sum_May2010_POW10_v2.eps}
\end{center}
\caption[]{Evolution of the bolometric flux (top) and effective temperature (bottom) of EXO~0748--676, deduced from \textit{Chandra}/ACIS-S (black squares), \textit{Swift}/XRT (grey triangles) and \textit{XMM-Newton}/EPIC (black star) data. Multiple \textit{Swift}\ observations were summed to improve the data statistics (see Section~\ref{subsec:spectraldata}).
}
\label{fig:temp}
\end{figure}
\subsection{Lightcurve fits}\label{subsec:decay}
Fig.~\ref{fig:temp} clearly reveals a decaying trend in thermal flux and temperature. To investigate the decay shape, we fit the temperature curve with an exponential decay function of the form $y(t)=a~e^{-(t-t_0)/\tau}$, where $a$ is a normalisation constant, $t_0$ is the start time of the cooling curve and $\tau$ the e-folding time. Given the apparent offset between the different instruments (see Section~\ref{subsec:crosscal}), we perform different fits to the \textit{Chandra}\ and \textit{Swift}\ data. We fix $t_0$ to 2009 September 5 (MJD 54714), which is in between the first non-detection by \textit{RXTE}/PCA and the first \textit{Swift}/XRT observation of the source \citep[][]{degenaar09_exo1}.
The simple exponential decay, represented by the dotted lines in Fig.~\ref{fig:temp_chanfit}, yields an e-folding time of $6121.7\pm2004.0$~days for the \textit{Chandra}\ data, but does not provide a good fit ($\chi^2_{\nu}=6.0$ for 2 d.o.f.).
For the \textit{Swift}\ lightcurve we find $\tau=5328.1\pm674.7$~days ($\chi^2_{\nu}=0.5$ for 12 d.o.f.). If we include a constant offset (i.e., $y(t)=a~e^{-(t-t_0)/\tau}+b$; solid lines in Fig.~\ref{fig:temp_chanfit}), we obtain a better fit for the \textit{Chandra}\ data, yielding a normalisation of $a=13.4\pm0.2$~eV, an e-folding decay time of $\tau=191.6\pm9.7$~days and a constant offset of $b=107.9\pm0.2$~eV ($\chi^2_{\nu}=0.02$ for 1 d.o.f.). For the \textit{Swift}\ data we find $a=17.2\pm1.8$~eV, $\tau=265.6 \pm 100.0$~days and $b=106.2\pm2.5$~eV ($\chi^2_{\nu}=0.34$ for 11 d.o.f.), which is consistent with the \textit{Chandra}\ fit.
Although an exponential decay provides an adequate description of the data of EXO~0748--676, as has been found for other sources \citep[e.g.,][]{cackett2006,fridriksson2010}, mathematically a neutron star crust is expected to cool via a (broken) powerlaw \citep[][]{eichler1989,brown08}. If we fit a single powerlaw of the form $y(t)=A(t-t_0)^{B}$ to the \textit{Chandra}\ data, we find an index of $B=-0.03\pm0.01$ and a normalisation of $A=134.4\pm1.0$~eV ($\chi^2_{\nu}=0.13$ for 2 d.o.f.). For the \textit{Swift}\ observations we find $B=-0.05\pm0.01$ and $A=144.7\pm3.8$~eV ($\chi^2_{\nu}=0.4$ for 12 d.o.f.). These powerlaw fits are indicated by the dashed lines in Fig.~\ref{fig:temp_chanfit}.
A broken powerlaw also yields an acceptable fit to the \textit{Swift}\ data ($\chi^2_{\nu}=0.3$ for 10 d.o.f.). We find a normalisation of $A=135.0\pm17.8$~eV, a break at $166.0\pm99.2$~days and decay indices of $-0.03\pm0.03$ and $-0.06\pm0.02$ before and after the break, respectively. This fit is indicated by the dashed-dotted curve in Fig.~\ref{fig:temp_chanfit}. There are not sufficient \textit{Chandra}\ observations to fit a broken powerlaw decay. We note that the shape of the decay curve of EXO~0748--676\ is not strongly affected by our choice of spectral parameters ($N_{\mathrm{H}}$, $M_{\mathrm{NS}}$, $R_{\mathrm{NS}}$, and $\Gamma$) or assumed distance \citep[see also previous studies by e.g.,][]{wijnands2004,cackett2008}.
\subsection{Instrument cross-calibration}\label{subsec:crosscal}
The quiescent lightcurve presented in Fig.~\ref{fig:temp} shows indications that the thermal flux and temperature inferred from the \textit{Chandra}\ observations lie below the trend of the \textit{Swift}\ data points. This possible shift ($\sim 6$ percent for the flux lightcurve) may be due to cross-calibration issues between the two satellites. A study of the Crab nebula indeed revealed an offset between \textit{Chandra}\ and \textit{Swift}, whereas such a discrepancy was not found between \textit{Swift}\ and \textit{XMM-Newton}\ \citep[][]{kirsch2005}. This might be reflected in our results as well, since the \textit{XMM-Newton}\ data point appears to line up with the trend indicated by the \textit{Swift}\ data. However, our \textit{Chandra}\ and \textit{Swift}\ data points may also be (partly) offset due to the fact that we cannot constrain the powerlaw component in the \textit{Swift}\ data, which we therefore fixed to contribute $10$ percent of the total 0.5--10 keV unabsorbed flux (see Section~\ref{subsec:spectraldata}).
\begin{figure*}
\begin{center}
\includegraphics[width=8.0cm]{PAPER_II_chan_temp_fits_May2010_LOG_extended_v2.eps}
\hspace{0.4cm}
\includegraphics[width=8.0cm]{PAPER_II_swift_temp_fits_May2010_LOG_extended.eps}
\end{center}
\caption[]{Evolution of the effective temperature of EXO~0748--676\ fitted to different decay functions (see Section~\ref{subsec:decay}). The left image displays \textit{Chandra}\ data and exponential decay fits both with and without a constant offset (solid and dotted line, respectively), as well as a decaying powerlaw (dashed curve). The right image shows \textit{Swift}\ observations, where the dashed line is again a powerlaw fit, while the solid and dotted curves are exponential decays. In addition, this plot includes a fit to a broken powerlaw, which is represented by the dashed-dotted line.}
\label{fig:temp_chanfit}
\end{figure*}
\section{Discussion}\label{sec:discussion}
We discuss \textit{Chandra}, \textit{Swift}\ and \textit{XMM-Newton}\ observations obtained after the cessation of the very long ($\sim$24 year) active period of EXO~0748--676. Fitting the spectral data with a neutron star atmosphere model \textsc{nsatmos}, did not reveal clear indications of a changing thermal spectrum during the first five months of the quiescent phase \citep[][]{degenaar09_exo1}. However, now that the quiescent monitoring has extended to 19 months (1.6 years), we find a significant decrease in neutron star effective temperature from $kT^{\infty}_{\mathrm{eff}}\sim124$ to $109$~eV. The thermal bolometric flux was observed to decay from $F_{\mathrm{bol}}^{\mathrm{th}}\sim1.5\times10^{-12}$ to $0.9\times10^{-12}~\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$.
In addition to a soft, thermal component, the \textit{Chandra}\ and \textit{XMM-Newton}\ observations show evidence for a hard powerlaw tail with index $\Gamma=1.7$. The fractional contribution of the hard spectral component to the total unabsorbed 0.5--10 keV flux initially decreased from $\sim20$ percent in 2008 October to $\sim4$ percent in 2009 June. However, observations carried out in 2010 April suggest that the powerlaw fraction increased again to $\sim15$ percent. Similar behaviour has been observed for several other quiescent neutron star systems \citep{jonker2004,jonker07_eos}, although others show more irregular behaviour \citep[][]{fridriksson2010}. In Cen X-4, the powerlaw tail in the quiescent spectrum shows variations that appear to be linked to changes in the thermal component, possibly caused by low-level accretion \citep[][]{cackett2010_cenx4}.
The gradual decrease in thermal flux and neutron star temperature observed for EXO~0748--676\ can be interpreted as the neutron star crust cooling down in quiescence after it has been heated during its long accretion outburst. Fig.~\ref{fig:sources} compares our data of EXO~0748--676\ with the crust cooling curves observed for the neutron star X-ray binaries KS~1731--260, MXB~1659--29\ and XTE~J1701--462. This plot shows that the amount of cooling following the end of the outburst is markedly smaller for EXO~0748--676\ than for the other three sources. We have observed our target over the first 19~months after the cessation of the outburst and during this time the thermal bolometric flux has decreased by a factor of $\sim1.7$. In a similar time span, the thermal bolometric fluxes of KS~1731--260, MXB~1659--29\ and XTE~J1701--462\ had decreased by a factor of $\sim3.5$, 6 and 2.5, respectively \citep[see][]{cackett2006,fridriksson2010}. The effective neutron star temperature of EXO~0748--676\ has decreased by about 10 percent, compared to $\sim30$, 40 and 20 percent for KS~1731--260, MXB~1659--29\ and XTE~J1701--462.
Although the observed fractional changes in neutron star temperature and thermal bolometric flux are smaller for EXO~0748--676\ than for the other three sources, the decay itself may not be markedly different. The quiescent lightcurves of KS~1731--260, MXB~1659--29\ and XTE~J1701--462\ can be fit with an exponential decay function levelling off to a constant value, yielding e-folding times of $\sim305\pm50$, $\sim465\pm25$ and $\sim120\pm25$~days, respectively \citep[][]{cackett2008,fridriksson2010}. For the \textit{Chandra}\ data of EXO~0748--676, we find an e-folding time of $\sim192\pm10$~days (see Section~\ref{subsec:decay}). These decay times provide a measure of the thermal relaxation time of the neutron star crust, which depends on the composition and structure of the lattice, the distribution of heating sources and the thickness of the crust \citep[e.g.,][]{lattimer1994,rutledge2002,shternin07,brown08}.
\citet{rutledge2002} and \citet{shternin07} calculate theoretical cooling curves for KS~1731--260, assuming different physics for the crust and core. These authors present simulations for both an amorphous crust and an ordered crystalline lattice. For the latter, the spread of nuclide charge numbers ($Z$) in the crust matter is small, which is referred to as a low level of impurities and results in a highly conductive crust. A large number of impurities gives an amorphous structure, which affects the thermal properties of the crust and results in a low conductivity. In addition, \citet{rutledge2002} explore standard (i.e., slow) and enhanced neutrino cooling mechanisms, yielding different core temperatures. Comparing our results on EXO~0748--676\ with the decay shapes resulting from those calculations suggests that the neutron star has a highly conductive crust, similar to what has been inferred for the other three sources \citep[][]{wijnands2002,wijnands2004,cackett2006,shternin07,brown08,fridriksson2010}. The fact that the decay curve of EXO~0748--676\ is rather shallow may be explained in terms of a relatively small temperature gradient and thus lower thermal flux across the core-crust boundary \citep[cf. the model curves for a highly conductive crust and different core temperatures presented by][]{rutledge2002}. This can be due to a combination of a warm neutron star core and a relatively low mass-accretion rate during outburst.
The exponential decay fit to the \textit{Chandra}\ data of EXO~0748--676\ indicates that the neutron star crust might already be close to restoring equilibrium with the core. The fit results in a quiescent base level of $107.9\pm0.2$~eV, while we found a temperature of $108.6\pm1.1$~eV for the observation performed in 2010 April. Prior to its last outburst, EXO~0748--676\ was observed in quiescence with the \textit{EINSTEIN}\ observatory, displaying a 0.5--10 keV unabsorbed flux of $8.4^{+4.2}_{-1.7} \times 10^{-13}~\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$ \citep[][]{garcia1999}. Our \textit{Chandra}\ observations of 2010 April detected EXO~0748--676\ at a 0.5--10 keV unabsorbed flux of $(7.7\pm0.2) \times 10^{-13}~\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$ (see Table~\ref{tab:spec}). Assuming that the \textit{EINSTEIN}\ detection caught EXO~0748--676\ at its quiescent base level, this supports the idea that the crust has nearly cooled down. This would imply that the neutron star core in EXO~0748--676\ is relatively hot \citep[cf.][]{heinke2009}, suggesting that either standard cooling mechanisms are operating and that the neutron star is not very massive, or that the time-averaged mass-accretion rate of the system is very high due to a short recurrence time (see below).
The energy deposited during outburst is given by $L_{\mathrm{nuc}} \sim \langle \dot{M} \rangle Q_{\mathrm{nuc}}/m_{\mathrm{u}}$ \citep[e.g.,][]{brown1998,colpi2001}. Here, $Q_{\mathrm{nuc}}\sim2$~MeV is the nuclear energy deposited per accreted baryon \citep[][]{gupta07,haensel2008}, $m_{\mathrm{u}}$ is the atomic mass unit and $\langle \dot{M} \rangle$ is the time-averaged accretion rate of the system. The latter can be expressed as $\langle \dot{M} \rangle = \langle \dot{M}_{\mathrm{ob}} \rangle \times t_{\mathrm{ob}} / t_{\mathrm{rec}}$, where $\langle \dot{M}_{\mathrm{ob}} \rangle$ is the average accretion rate during outburst episodes, $t_{\mathrm{ob}}$ is the outburst duration and $t_{\mathrm{rec}}$ is the system's recurrence time. The factor $t_{\mathrm{ob}} / t_{\mathrm{rec}}$ represents the duty cycle of the system. The neutron star core is expected to be in a steady state, in which the energy radiated during quiescence balances the heat deposited during outburst. We can thus obtain an estimate of the duty cycle of EXO~0748--676\ by equating the heating and cooling rates.
A neutron star cools primarily via photon radiation from the surface and neutrino emissions from the stellar core. If the lightcurve of EXO~0748--676\ has indeed (nearly) levelled off, the bolometric luminosity emitted as photons is thus $L_{\gamma} \sim6\times10^{33}~\mathrm{(D/7.4~kpc)^2}~\mathrm{erg~s}^{-1}$ (as measured during the \textit{Chandra}\ observation of 2010 April). The rate of neutrino emissions depends on the temperature of the neutron star core, which can be estimated from the effective surface temperature once the crust has thermally relaxed. A quiescent base level of $kT^{\infty}_{\mathrm{eff}}\sim108$~eV (as suggested by exponential decay fits to the \textit{Chandra}\ data), implies an effective surface temperature in the neutron star frame of $kT_{\mathrm{eff}}\sim140$~eV ($\sim1.6\times10^{6}$~K), for a canonical values of $M_{\mathrm{NS}}=1.4~\mathrm{M}_{\odot}$ and $R_{\mathrm{NS}}=10$~km (i.e., $1+z=1.3$). Using the relation between the effective surface temperature and the interior temperature calculated by \citet{brown08}, yields $T_{\mathrm{core}}\sim1.3\times10^{8}$~K. For such a core temperature, the minimum energy escaping the neutron star as neutrino's (i.e., assuming standard core cooling) is $L_{\nu}\sim10^{34-35}~\mathrm{erg~s}^{-1}$ \citep[][]{page2006}.
Equating the energy losses via photon radiation from the neutron star surface ($L_{\gamma}$) and neutrino emissions from the stellar core ($L_{\nu}$) with the energy gained via crustal reactions during outburst ($L_{\mathrm{nuc}}$), suggests that EXO~0748--676\ must have a time-averaged mass-accretion rate of $\langle \dot{M} \rangle \gtrsim 8 \times 10^{15}~\mathrm{g~s}^{-1}$. During outburst, EXO~0748--676\ displayed an average bolometric luminosity of $\sim 6 \times10^{36}~\mathrm{(D/7.4~kpc)^2}~\mathrm{erg~s}^{-1}$ \citep[][]{sidoli05,boirin2007}. Assuming that the accretion luminosity is given by $L_{\mathrm{acc}}=(GM_{\mathrm{NS}}/R_{\mathrm{NS}}) \langle \dot{M}_{\mathrm{ob}} \rangle$, this translates into a mass-accretion rate during outburst of $\langle \dot{M}_{\mathrm{ob}} \rangle \sim 3 \times 10^{16}~\mathrm{g~s}^{-1}$ for a canonical neutron star with $M=1.4~\mathrm{M}_{\odot}$ and $R=10$~km.\footnote{We note that EXO~0748--676\ is an eclipsing system and therefore part of the central X-ray flux may be intercepted from our line of sight. However, the X-ray burst behaviour of the source is consistent with the mass-accretion rate inferred from the observed X-ray luminosity \citep[][]{boirin2007}.}
If the crust has indeed thermally relaxed, the above estimates show that EXO~0748--676\ must have a duty cycle of $\gtrsim30$ percent to explain the observed quiescent bolometric luminosity of $\sim6\times10^{33}~\mathrm{(D/7.4~kpc)^2}~\mathrm{erg~s}^{-1}$ in terms of thermal emission from the cooling neutron star (i.e., opposed to continued accretion). The outburst of EXO~0748--676\ started between 1980 May and 1984 July and the system returned to quiescence in 2008 September, i.e., $t_{\mathrm{ob}}=24-28$ years. If the observed outburst is typical for the long-term behaviour of this source, the expected recurrence time is thus $\lesssim100$~years. In case the neutron star cools via more efficient core neutrino emission processes, the recurrence time required to explain the observed quiescent luminosity is shorter (i.e., the duty cycle is higher). Although the above calculation is only a crude approximation \citep[e.g., there is a significant uncertainty in the relation between the surface- and interior temperature of the neutron star, depending on the atmospheric composition and the depth of the light element layer;][]{brown08}, it illustrates that EXO~0748--676\ must have a high duty cycle if the cooling curve has indeed reached its quiescent base level.
\citet{brown1998}, \citet{rutledge2000} and \citet{colpi2001} have suggested that EXO~0748--676\ continues to accrete in quiescence, because the quiescent luminosity inferred from the 1980 \textit{EINSTEIN}\ observation is higher than predicted by standard cooling models. However, these conclusions are based on an assumed duty cycle of $\sim1$ percent, but we have no a priori knowledge about this. Although we cannot exclude that the system is indeed accreting in quiescence, the above estimates show that a duty cycle of $\gtrsim30$ percent can explain the observed quiescent level of EXO~0748--676\ as being due to thermal emission from the cooling neutron star. A duty cycle of $\gtrsim30$ percent is high, although not unprecedented for neutron star transients \citep[e.g.,][]{chen97,degenaar09_gc}
Recently, \citet[][]{brown08} demonstrated that the cooling of a neutron star crust is expected to follow a broken powerlaw decay. A break is predicted to occur due to a transition in the crystal structure of the crust matter, and the slope before the break reflects the heat flux from the outer crustal layers. Therefore, we also fitted the neutron star temperatures obtained for EXO~0748--676\ to a powerlaw and found decay indices of $-0.03\pm0.01$ and $-0.05\pm0.01$ for the \textit{Chandra}\ and \textit{Swift}\ data sets, respectively. The \textit{Swift}\ observations indicate that a possible break in the quiescent lightcurve may have occurred $\sim67-265$~days after the cessation of the outburst (see Section~\ref{subsec:decay}). By fitting a broken powerlaw function, we obtain a decay index of $-0.03\pm0.03$ before the break, which steepens to $-0.06\pm0.02$ thereafter. However, since these slopes are consistent with being equal, further observations are required to confirm whether a break has indeed occurred.
The decay parameters that we find for EXO~0748--676\ are comparable to that obtained by \citet{fridriksson2010} for XTE~J1701--462. These authors found that the quiescent lightcurve breaks $\sim20-150$ days post-outburst and report decay indices of $\sim-0.03$ and $\sim-0.07$ before and after the break, respectively. \citet{fridriksson2010} note that possible cross-calibration effects between \textit{Chandra}\ and \textit{XMM-Newton}\ might introduce small shifts that also allow a single powerlaw decay with slope $\sim-0.05$. The cooling curves of KS~1731--260\ and MXB~1659--29\ appear to have steeper decays with indices of $\sim-0.12$ and $\sim-0.33$, respectively \citep{cackett2008}. Due to the scarcity of data points it is unclear whether a break occurred in the quiescent lightcurves of the latter two sources \citep{cackett2008,brown08}.
The powerlaw fits show no indications that the quiescent lightcurve of EXO~0748--676\ is levelling off. Thus, it is also possible that the neutron star temperature continues to decay further and that the core is cooler than suggested by the exponential decay fits and the 1980 \textit{EINSTEIN}\ detection. The relatively slow decrease of EXO~0748--676\ might then reflect that the crust has a high conductivity, albeit lower than that of the neutron stars in KS~1731--260\ and MXB~1659--29. Further observations are thus required to determine whether the neutron star crust in EXO~0748--676\ has nearly cooled down and to be able to draw firm conclusions on the crust and core properties.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{PAPER_II_all_sources_May2010_POW10_v2.eps}
\end{center}
\caption[]{The effective temperatures of KS~1731--260\ \citep[green diamonds; from][]{cackett2006}, MXB~1659--29\ \citep[red bullets; from][]{cackett2006,cackett2008}, XTE~J1701--462\ \citep[grey crosses; from][]{fridriksson2010} and EXO~0748--676\ (black squares, triangles and star). Exponential decay fits to the data of KS~1731--260, MXB~1659--29\ and XTE~J1701--462\ are shown to guide the eye (green dashed, red dashed-dotted and grey dotted line, respectively). The two data points of XTE~J1701--462\ that lie above the decay fit are likely due to a temporary increase in the accretion rate causing reheating of the neutron star \citep[][]{fridriksson2010}.
}
\label{fig:sources}
\end{figure}
\section*{Acknowledgements}
This work was supported by the Netherlands Organisation for Scientific Research (NWO) and made use of the \textit{Swift}\ public data archive. We acknowledge \textit{Swift}\ PI N. Gehrels and the \textit{Swift}\ planning team for their help in carrying out the ToO campaign. EMC was supported by NASA through the Chandra Fellowship Program. MTW, PSR and KSW acknowledge the United States Office of Naval Research. JH\ and WHGL\ acknowledge support from Chandra grant GO8-9045X.
\bibliographystyle{mn2e}
|
2,869,038,155,680 | arxiv | \section{Introduction}
The observational indications for the dark matter and dark energy have stipulated the active development of variety of models, including those of modifications of Newton's gravity and of General Relativity (GR).
Various principles are taken as bases in those models, such as scalar-tensor, f(R) theories, MOND (see \cite{Sand,Clif,Bert,Bah} and refs therein) with different motivations and with the natural aim to satisfy the observational data or to suggest verification tests for future observations.
Below we reconsider a principle which is in the very roots of Newton's gravity, i.e. the theorem that a sphere is acting gravitationally as a point mass situated in its center. That theorem had enabled Newton to attribute the gravitational law $r^{-2}$ to the motion of planets which are definitely extended spheres and are not point masses. Later the Newton's gravity became a key element GR, acting as its weak field limit and enabling to correspond the predicted effects with the observed ones. That concerns, however, the Einstein equations without the cosmological term and Einstein's motivation \cite{E} for introducing the cosmological constant was the static Universe. In the approach below we show that the cosmological term appears in Einstein's equations from the above mentioned theorem proved in \cite{N}.
As shown in \cite{G}, the most general function satisfying that theorem, besides the usual $r^{-2}$ term, contains also another term with a cosmological constant.
Taking that modified Newton's law with a cosmological constant as the weak field limit of GR one arrives to a modification of GR containing the cosmological term naturally! We then show that this approach both to Newton's gravity theory and to GR enables one to link two observational facts, i.e. the dark matter in the galaxies and the cosmological constant. The dark matter then appears as an observational contribution of repulsive gravity at large scales, i.e. in galactic halos and clusters of galaxies.
\section{Newton's theorem and General Relativity}
As shown in \cite{G} the most general function for the force satisfying Newton's theorem i.e. the condition for the sphere to attract as a point of the sphere's mass and situated in its center has the form
\begin{equation}
f(r)= Ar^{-2} + \Lambda r,
\end{equation}
as the solution of equation
\begin{equation}
\frac{r^2}{2}f''(r) + r f'(r)-f(r)=0,
\end{equation}
where $A$ and $\Lambda$ are constants.
The second term corresponds the cosmological constant term if one turns to the Newtonian form of the Friedmann cosmological equation \cite{G}.
Eq. (1), however, within the context of the shell theorem defines a force-free field only in the center of a shell, but preserves the O(3) symmetry.
Turning to the GR, instead of the usual Newtonian limit for its weak field approximation \cite{W}
now metric tensor components $g_{00}$ and $g_{rr}$ will be modified and one will have the metric for the point mass as
\begin{equation}
ds^2 = (1 - 2A r^{-1} - \Lambda r^2/3)c^2 dt^2 + (1 - 2A r^{-1} - \Lambda r^2/3)^{-1}dr^2 + r^2 d\Omega^2.
\end{equation}
The principal fact here is in the following. The Einstein equations without cosmological term are considered to have the usual Newtonian limit in weak-field approximation, while the Einstein equations with cosmological term will formally violate that limit, e.g. according to Weinberg "{\it ..$\Lambda$ must be very small so as not to interfere with the successes of Newton's theory of gravitation.}" (Chapter 7.1, \cite{W}). We now see that via Eq.(1) that violation is removed.
Eq.(3) is the covariant metric having its weak-field limit Eq.(1) \cite{No}.
Instead, we see that Eq.(1) ensures the weak-field limit for Einstein equations with cosmological constant
\begin{equation}
G_{\mu\nu} + \Lambda g_{\mu\nu}= \kappa T_{\mu\nu}.
\end{equation}
The adoption of the Newton's "sphere = point mass" theorem and hence of the Eq. (1) will readily lead to renormalization in various predictions of GR. Although such modification of GR coincides with its original form of Einstein's equations, there is drastic difference in the motivations.
Here the starting point is the Newton's gravity and his sphere-point theorem and $\Lambda$ is appearing in Einstein's equations readily and not as an extra term added by hand to fulfill a static universe concept. Namely, if Newton might have found the general function Eq. (1), then Einstein initially would have written GR equations with $\Lambda$ i.e. with the link of O(3) and the Lorentz group.
Of course, the $\Lambda$-term in Eq.(3) has been considered previously (Schwarzschild -- de Sitter metric), however, the Newton theorem's approach described here, as we will see below leads to insights on the common nature of the dark matter and dark energy (cosmological constant).
\section{The Eq.(1), the cosmological constant and the dark matter}
If the Einstein equations with the cosmological constant have the Newtonian limit Eq.(1), then one will have a link e.g. to the two currently adopted principal observational facts on the dark energy and the dark matter. Indeed, while the cosmological constant in the Einstein equations is considered to describe the acceleration of the Universe, the Newtonian potential and its modifications are attributed e.g. to the observational indications for the dark matter in the galaxies.
The value of the cosmological constant is deduced in several ways, the one indicated by the Planck data \cite{P} is
\begin{equation}
\Lambda \simeq 1.1\,\, 10^{-52} m^{-2}.
\end{equation}
Regarding the dark matter in the galaxies, it is shown that the parameters of the halos determine the late disk and early spheroidal structures of the galaxies \cite{Kr}. Then, for virialized (for "oscillator" term $\propto R^k$ with $k=2$, see Eq. (10.7) in \cite{LL}) structures we have
\begin{equation}
\Lambda=\frac{3\sigma^2}{2 c^2 R^2}\simeq 3\,\, 10^{-52} (\frac{\sigma}{50\, km s^{-1}})^2(\frac{R}{300 \,kpc})^{-2} m^{-2},
\end{equation}
normalized to the velocity dispersion at a given radius of halo.
Numerically $\Lambda$s in (5) and (6) are close and this fact can be interpreted as follows. The positive cosmological constant corresponds to the accelerating Universe and hence to negative pressure and to repulsion as evidence of vacuum energy \cite{Z}. The crucial point regarding the dark matter is that Eq.(1) defines non-force-free field within the sphere except its center, increasing from center, thus mimicking increase of the central mass. Thus, within this interpretation the dark matter is a gravitating mass with force (1) and revealing its repulsion at large scales e.g. in galactic halos. The effective increase of the central attracting mass will support the effect of "flat rotation curves", although, obviously, the description of given observational rotation curves will need extensive modeling and numerical simulations for the input parameters of the disk and halo. For the analysis of the dark matter problem within $f(R)$ theories see \cite{Cap}.
\section{Galaxy groups and Eq.(6)}
We now test this approach on $\Lambda$-nature of dark matter using the data by Karachentsev et al \cite{Kar} for a sample of 17 galaxy groups of Hercules–--Bootes region which (the data) include the $\it rms$ galactic velocities $\sigma$ and the harmonic average radii $R_h$ of the groups. Then, using Eq.(6) we obtain the $\Lambda$ as presented in the Table 1; the galaxy groups are denoted by the name of the brightest galaxy (first column).
\begin{center}
{\bf Table 1.} $\Lambda$ obtained using Eq.(6) for galaxy groups of the Hercules---Bootes region.
{
\renewcommand{\baselinestretch}{1.2}
\renewcommand{\tabcolsep}{3.5mm}
\small
\begin{tabular}{ | l | r | r | r | }
\hline
Galaxy group & $\sigma (km/s^{-1})$ & $R_h(kpc)$ & $\Lambda(m^{-2})$\\ \hline
\hline
NGC4736 & 50 & 338 & 3.84E-52 \\ \hline
NGC4866 & 58 & 168 & 2.09E-51 \\ \hline
NGC5005 & 114 & 224 & 4.55E-51 \\ \hline
NGC5117 & 27 & 424 & 7.12E-53 \\ \hline
NGC5353 & 195 & 455 & 3.23E-51 \\ \hline
NGC5375 & 47 & 66 & 8.91E-51 \\ \hline
NGC5582 & 106 & 93 & 2.28E-50 \\ \hline
NGC5600 & 81 & 275 & 1.52E-51 \\ \hline
UGC9389 & 45 & 204 & 8.55E-52 \\ \hline
PGC55227 & 14 & 17 & 1.19E-50 \\ \hline
NGC5961 & 63 & 86 & 9.43E-51 \\ \hline
NGC5962 & 97 & 60 & 4.59E-50 \\ \hline
NGC5970 & 92 & 141 & 7.48E-51 \\ \hline
UGC10043 & 67 & 65 & 1.87E-50 \\ \hline
NGC6181 & 53 & 196 & 1.28E-51 \\ \hline
UGC10445 & 23 & 230 & 1.76E-52 \\ \hline
NGC6574 & 15 & 70 & 8.07E-52 \\ \hline
\hline
Average & & & 8.24E-51 \\ \hline
St.deviation & & & 1.15E-50 \\ \hline
\hline
\end{tabular}
}
\end{center}
The correspondence of values of $\Lambda$ in Table 1 to those in Eqs.(5) and (6) is visible.
\section{Conclusions}
We demonstrated a natural way for the appearance of the cosmological constant in Einstein equations. This follows while adopting for the weak field approximation of General Relativity the Eq.(1) as the general function satisfying the Newton's theorem of 1687, that the gravitating sphere acts as a point mass located in its center. This drastically differs from Einstein's original motivation for the introduction of the cosmological constant.
This approach enables to draw the following conclusions:
(a) the nature of the dark matter in galaxies is in the gravity;
(b) the dark energy (cosmological constant) and the dark matter are of the common nature;
(c) the dark matter is the signature of repulsive gravity of the ordinary matter determined by $\Lambda$ constant and dominating at large scales, i.e. larger than of galactic halos.
Thus, the key constituents of the dark universe are naturally linked here. The repulsive gravity nature of the dark matter is responsible for the observed increase of the mass-to-luminosity M/L ratio while moving from the scales of galaxies to those of galaxy clusters. The non-force-free shell (galactic halo) determines the internal structures of galaxies (disks), as indicate the observations.
The observational data of a sample of galaxy groups \cite{Kar}, i.e. systems containing 3 or more galaxies, are shown to confirm Eq.(6) on the $\Lambda$-nature of the dark matter.
Obviously, far more consequences of this modified GR, i.e. with weak-field of Eq.(1), are of interest regarding the experimental and theoretical aspects. The current experimental tests for GR, including the recent 5\% accuracy for the Lense-Thirring effect obtained via the LARES satellite \cite{Lares}, are obviously far from detecting the potential contribution of the $\Lambda$-term in GR. However, the observations of black holes and pulsars including the detections of gravitational waves via LIGO \cite{LIGO}, on the one hand, and of galactic halos (e.g. \cite{G2}), dynamics of galaxy groups and clusters for weak-field gravity, on the other hand, can provide efficient tests. Then, the cosmological constant determines not only expansion of the Universe but also the weak-field gravity and even possibly is linked to the arrow of time \cite{AG}. The modified GR reveals also the obvious link to the AdS/CFT correspondence, since instead of Poincare group now one has a SO(2,3) group. Further consequences of this approach are in \cite{GS1,GS2}.
|
2,869,038,155,681 | arxiv | \section{Introduction}
The anomalous magnetic moment of the muon, $a_{\mu} = (g-2)_{\mu}/2$,
stands as an enduring test of the Standard Model (SM), where the
$\sim3.5\sigma$ (or higher) discrepancy between the experimental
measurement $a_{\mu}^{\rm exp} = 11\ 659 \ 209.1 \ (5.4) \ (3.3) \times 10^{-10}$~\cite{Bennett:2002jb,PDG2016} and the SM prediction
$a_{\mu}^{\rm SM}$ could be an indication of the existence of new
physics beyond the SM. Efforts to improve the experimental estimate at Fermilab
(FNAL)~\cite{Grange:2015fou} and at J-PARC~\cite{Mibe:2010zz,Abe:2019thb} aim to
reduce the experimental uncertainty by a factor of four compared to
the BNL measurement. It is therefore imperative that the SM prediction
is also improved to determine whether the $g-2$ discrepancy is well
established.
The uncertainty of $a_{\mu}^{\rm SM}$ is entirely dominated by the
hadronic contributions, where the hadronic vacuum polarisation contributions can be
separated into the leading-order (LO) and higher-order contributions. These are calculated utilising dispersion integrals
and the experimentally measured cross section $\sigma^0_{{\rm had},\gamma} (s) \equiv \sigma^0(e^+e^-\rightarrow
\gamma^* \rightarrow {\rm hadrons} + \gamma)$, where the superscript 0 denotes the bare cross section (undressed of
all vacuum polarisation (VP) effects) and the subscript $\gamma$ indicates
the inclusion of effects from final state photon radiation (FSR). At
LO, the dispersion relation reads
\begin{equation} \label{eq:amu}
a_{\mu}^{\rm had,\,LO\,VP} =
\frac{\alpha^2}{3\pi^2}\int^{\infty}_{m_{\pi}^2} \frac{{\rm d}s}{s}
R(s)K(s) \ \ \ \ ; \ \ \ \ R(s) = \frac{\sigma^0_{{\rm had},\gamma} (s)}{\sigma_{\rm pt}(s)}
\equiv \frac{\sigma^0_{{\rm had},\gamma} (s)}{4\pi\alpha^2/3s} \, ,
\end{equation}
where $R(s)$ denotes the hadronic $R$-ratio and $K(s)$ is a well known kernel function.
Below $\sim2$GeV, the estimates of $a_{\mu}^{\rm had,\, VP}$ and the corresponding uncertainties are determined from the experimentally measured cross sections of individual hadronic final states, where the hadronic $R$-ratio in this region is predominantly constructed from the sum of the determined cross sections. Above $\sim2$GeV, data for the measured total hadronic $R$-ratio (all hadronic final states) are combined. For nearly all these channels, the available data from numerous different experiments must be analysed, combined and then integrated over according to equation~\eqref{eq:amu} to give a corresponding estimate of the contribution to $a_{\mu}^{\rm had,\, LO\, VP}$. Therefore, the dependence of this calculation on the quality and precision of these measured cross sections is substantial and many experiments have dedicated programmes focused on the accurate measurement of these processes. This document focuses on the effect of these measurements on the recent KNT18 analysis of $a_{\mu}^{\rm had,\, VP}$~\cite{Keshavarzi:2018mgv}. Details of other similar analyses can be found in~\cite{Davier:2017zfy,Jegerlehner:2017gek,Jegerlehner:2017lbd,Benayoun:2015gxa,Colangelo:2018mtw}.
\section{Experimental measurements of $e^+e^-\rightarrow {\rm hadrons}$}
Experimental measurements of the cross sections of exclusive hadronic final states are obtained via two approaches. The first is the standard direct energy scan approach, where data is collected at fixed centre of mass (C.M.) energy intervals. The second is achieved through radiative return, where the differential cross section is measured as a function of the invariant mass of the hadronic final state, $\sqrt{s} = M_{\rm had}$. The cross section $\sigma_{\rm had} \equiv \sigma(e^+e^-\rightarrow \ {\rm hadrons})$ is then determined, for example, according to~\cite{Binner:1999bt} using the relation
\begin{equation} \label{pipidiffxSec}
s\frac{{\rm d}\sigma\big({\rm had}+\gamma\big)}{dM_{\rm had}^2} = \sigma_{\rm had}(M_{\rm had}^2)H(M_{\rm had}^2,s) \ ,
\end{equation}
where $H$ is the radiator function describing the emission of photons in the initial state~\cite{Rodrigo:2001kf,Czyz:2002np,Czyz:2003ue,Czyz:2004rj}.
\subsection{Direct energy scan experiments}
\subsubsection{CMD-3}
The CMD-3 detector~\cite{Khazin:2008zz} is the first of two direct energy scan experiments at the $e^+e^-$ collider VEPP-2000~\cite{Khazin:2010zz}. The VEPP-2000 machine has a C.M. energy range of $0.3 \leq \sqrt{s} \leq 2$ GeV, with a design luminosity of $L = 10^{32}cm^{-2}s^{-1}$ at $\sqrt{s} = 2$ GeV. The CMD-3 experimental programme has already published cross section measurements for many final states (see e.g.~\cite{Akhmetshin:2018mqd,Solodov:2017pyu,CMD-3_PhiPsi19}). Of these, major improvements have been seen in the measurements of the $K\bar{K}$ cross sections, with both the $K^+K^-$~\cite{Kozyrev:2017agm} and $K^0_S K^0_L$~\cite{Kozyrev:2016raz} analyses yielding very precise results of the narrow $\phi$ resonance that dominates in both these channels. In the $K^+K^-$ channel in particular, these new data replace those previously measured by CMD-2~\cite{Akhmetshin:2008gz}, which are currently awaiting reanalysis as they suffer from an overestimation of the trigger efficiency for slow kaons~\cite{Kozyrev:2017agm,CMD-2trigger}. The cross section values of these new CMD-3 data are higher than all other existing data in this channel~\cite{Kozyrev:2017agm}, leading to significant new data tensions in the overall combination of all available $K^+K^-$ data (see Section~\ref{sec:DataTensions}). Notably, the CMD-3 experiment has also recently released data for the $3\pi^+3\pi^-\pi^0$ final state, which had not previously been measured~\cite{CMD-3:2019ufp}.
Of particular importance to future determinations of $a_{\mu}^{\rm had,\, VP}$ is the announced new measurement of the $\pi^+\pi^-$ cross section by CMD-3, which is currently undergoing an extensive analysis~\cite{CMD-3_PhiPsi19}.\footnote{The $\pi^+\pi^-$ channel accounts for over 70\% of the total value of $a_{\mu}^{\rm had, \, LO \, VP}$, due to the large $\rho$ resonance structure in the low energy region below 1 GeV that is highly weighted by $K(s)$ in equation~\eqref{eq:amu}. Consequently, it also dominates the overall uncertainty of the hadronic vacuum polarisation contributions, resulting in CMD-3 (and other experiments) re-measuring this final state in an attempt to more precisely determine $a_\mu^{\rm had, \, VP}$.} With high-precision in mind, this measurement aims to be the most precise in terms of statistical precision of all the current data sets being combined in the $\pi^+\pi^-$ channel and to also achieve a systematic uncertainty budget of $\sim0.4-0.5\%$, compared to the $\sim0.6-0.8\%$ achieved by CMD-2~\cite{Akhmetshin:2006wh,Akhmetshin:2006bx,Aulchenko:2006na}.
\subsubsection{SND}
The SND experiment~\cite{Achasov:2009zza} is the second general purpose detector at VEPP-2000~\cite{Khazin:2010zz}. It also collected data at the VEPP-2M collider~\cite{Aulchenko:1975dc} that predated this, where data for exclusive hadronic final states were collected between 1996-2000 in the energy range $0.4\leq\sqrt{s}\leq1.4$ GeV. This was then extended to $0.3\leq\sqrt{s}\leq2.0$ GeV as part of the upgrade to the VEPP-2000 machine. In recent years, SND have released new data for several hadronic modes~\cite{Kupich:2017luv,Logashenko:2016xhy}, notably the $\pi^0\gamma$~\cite{Achasov:2016bfr,Achasov:2018ujw} cross section and the $K^+K^-$ cross section above the $\phi$ resonance~\cite{Achasov:2016lbc}. The SND experiment is also currently analysing a new measurement of the $\pi^+\pi^-$ cross section, having collected an integrated luminosity of $5{\rm pb}^{-1}$ of data for this important final state~\cite{SND_PhiPsi19}. The systematic uncertainties of this measurement are predicted to be in the range of $0.8-0.9\%$.
\subsubsection{KEDR}
The KEDR detector~\cite{KEDR} at the VEPP-4M $e^+e^-$ collider~\cite{VEPP-4M} is an experimental facility dedicated to the measurement of the full multi-hadron cross section, or total hadronic $R$-ratio. It has already published measurements of $R(s)$ at 22 C.M. energies between $1.84\leq\sqrt{s}\leq3.72$ GeV, with total uncertainties ranging from $3.9\%$ (2.4\% systematic uncertainties) at lower energies to 2.6\% ($\sim1.9\%$ systematic uncertainties) above the $J/\psi$ resonance~\cite{KEDR_PhiPsi19}. The agreement between these data and pQCD in this energy range is much improved compared to the previous measurements of the $R$-ratio by BES/BES-II in this region\cite{Bai:1999pk,Bai:2001ct,Ablikim:2006aj,Ablikim:2006mb,Ablikim:2009ad}. The KEDR experiment also plans to complete two scans of $R(s)$ from $4.56\leq\sqrt{s}\leq6.96$ GeV by the end of 2019~\cite{KEDR_PhiPsi19}.
\subsection{Radiative return experiments}
\subsubsection{BABAR}
The BABAR detector~\cite{Aubert:2001tu} resides at the PEP-II $e^+e^-$ storage ring at SLAC~\cite{PEP-II}, which operates predominantly at the C.M. energy $\sqrt{s} = 10.6$ GeV. The experiment detects large-angle photons with energies $E^*_{\gamma} > 3$ GeV, which defines the C.M. energy $\sqrt{s'}$ of the leptonic or hadronic final state determined via radiative return. This allows for precise cross sections measurements from production threshold up to 3-5 GeV~\cite{Michel_BABAR}.
The experimental programme at BABAR dedicated to low-energy hadronic cross sections has measured an almost complete set of exclusive hadronic channels below 2 GeV, missing only the $\pi^+\pi^-\pi^0$, $\pi^+\pi^-4\pi^0$ and $\geq7\pi$ modes. Arguably its most impressive measurement is that of the $\pi^+\pi^-(\gamma)$ cross section from threshold to $\sqrt{s'} \leq 3$ GeV, which is statistically the most precise of all measured $\pi^+\pi^-$ cross sections and has a systematic uncertainty of only 0.5\% in the region of the all-important $\rho$ resonance~\cite{Aubert:2009ad,Lees:2012cj}. BABAR have also announced a forthcoming release of a new measurement of the $\pi^+\pi^-(\gamma)$ cross section which should have twice the number of the statistics of the data published in~\cite{Lees:2012cj} and have even better control of the systematic uncertainties~\cite{Michel_BABAR}.
With their broad experimental programme, BABAR measurements have also greatly improved the determination of many other channels. A new measurement of the $\pi^+\pi^-\pi^0\pi^0$ channel~\cite{TheBABAR:2017vzo} has provided the only new data in this channel since 2003. The $K^+K^-$ channel now includes a precise and finely-binned measurement, supplemented with full statistical and systematic covariance matrices~\cite{Lees:2013gzt}. The neutral final state $K^0_S K^0_L\pi^0$ has also been measured, completing all modes that contribute to the $KK\pi$ final state. In addition, BABAR have also completed all modes that contribute to the $KK\pi\pi$ channel~\cite{TheBABAR:2017vgl}. Finally, a very recent measurement of the $\pi^+\pi^-3\pi^0$ cross section is the first published data of this final state to be included in the overall compilation~\cite{Lees:2018dnv}.
\subsubsection{KLOE/KLOE-2}
DA$\Phi$NE~\cite{DAFNE} is a high luminosity $e^+e^-$ collider that operates predominantly at the centre of mass energy equal to the $\phi$ meson mass, $\sqrt{s} = m_{\phi} = 1.0194 \text{ GeV} $~\cite{PDG2016}. The KLOE detector has been used to obtain three precise measurements of the cross section $\allowbreak\sigma\big(e^+e^-\allowbreak\rightarrow\pi^+\pi^-\gamma(\gamma)\big)$ in 2008~\cite{Ambrosino:2008aa,KLOE08-KLOEnote}, 2010~\cite{Ambrosino:2010bv,KLOE10-KLOEnote} and 2012~\cite{Babusci:2012rp,KLOE12-KLOEnote}.\footnote{The KLOE collaboration also made a measurement of $\sigma\big(e^+e^-\rightarrow\pi^+\pi^-\gamma(\gamma)\big)$ in 2005~\cite{Aloisio:2004bu}. However, this is now considered to be superseded by the 2008 measurement, as discussed in~\cite{Ambrosino:2008aa}.} Each of these measurements results in two-pion contribution to the anomalous magnetic moment of the muon of~\cite{Anastasi:2017eio}
align}{\begin{eqnarray}{align}
a_{\mu}^{\pi^+\pi^-}({\rm KLOE08}, \ 0.35< s <0.95 \text{ GeV}^2) & = (386.6 \pm 0.4_{\rm stat} \pm 3.3_{\rm sys}) \times 10^{-10} , \nonumber
\\
\
a_{\mu}^{\pi^+\pi^-}({\rm KLOE10}, \ 0.10 < s <0.85 \text{ GeV}^2) & = (477.9 \pm 2.0_{\rm stat} \pm 6.7_{\rm sys}) \times 10^{-10} , \nonumber
\\
\
a_{\mu}^{\pi^+\pi^-}({\rm KLOE12}, \ 0.35< s <0.95 \text{ GeV}^2) & = (385.1 \pm 1.2_{\rm stat} \pm 2.3_{\rm sys}) \times 10^{-10} .
align}{\end{eqnarray}{align}
The simultaneous use of the KLOE measurements required a detailed analysis to attain the correct combination of the three, which have a non-trivial influence on $a_{\mu}^{\pi^+\pi^-}$ and provide an important comparison with other experimental measurements of $\sigma_{\pi\pi}$. The KLOE measurements of $\sigma_{\pi\pi(\gamma)}$ are, in part, highly correlated, necessitating the construction of full statistical and systematic covariance matrices to be used in any combination of these data. The construction of these matrices and the combination of the three measurements was achieved in~\cite{Anastasi:2017eio}, which resulted in a single vector for the two-pion cross section $\sigma_{\pi\pi(\gamma)}$, along with a corresponding covariance matrix. This combination of the KLOE cross section data resulted in an estimate of the two-pion contribution to the anomalous magnetic moment of the muon of
\begin{equation} \label{KLOEcombination}
a_{\mu}^{\pi^+\pi^-}({\rm KLOE \ combination}, \ 0.10< s <0.95 \text{ GeV}^2) = (489.8 \pm 1.7_{\rm stat} \pm 4.8_{\rm sys} ) \times 10^{-10} .
\end{equation}
\subsubsection{BESIII}
The BESIII detector~\cite{Ablikim:2009aa} is a general purpose detector at the BEPCII $e^+e^-$ collider~\cite{BEPCII}, which operates at C.M. energies between $2.0\leq\sqrt{s}\leq4.6$ GeV and has a design luminosity of $L = 10.0^{33}cm^{-2}s^{-1}$ at the $\psi(3770)$ resonance~\cite{BESIII_Redmer}. The BESIII experiment have published a measurement of the $e^+e^-\rightarrow\pi^+\pi^-$ cross section focused solely on the $\rho$ resonance contribution between $0.6\leq\sqrt{s}\leq0.9$ GeV~\cite{Ablikim:2015orh}. With a total uncertainty of $\sim0.9\%$ and the evident disagreement between the BABAR and KLOE cross sections, this measurement provides an interesting comparison to the existing data. BESIII have also announced future releases of the $\pi^+\pi^-\pi^0$, $\pi^+\pi^-\pi^0\pi^0$, $\omega\pi^0$ and $\pi^+\pi^-3\pi^0$ cross sections~\cite{BESIII_TGm2}, along with measurements of the total hadronic $R$-ratio~\cite{BESIII_PhiPsi19}.
\section{Results for $a_{\mu}^{\rm had,\, VP}$ from KNT18}
align}{\begin{eqnarray}{table}[!t]
\centering
\scalebox{0.9}{
{\renewcommand{\arraystretch}{0.9}
align}{\begin{eqnarray}{tabular}{|l|c|c|}
\hline
{\bf Channel} & {\bf Energy range (GeV)} & $a_{\mu}^{\rm had, \, LO \, VP} \times 10^{10}$ \\
\hline
$\pi^+\pi^-$ & $ 0.305 \leq \sqrt{s} \leq 1.937 $ & $ 502.97 \pm 1.97\hphantom{0} $ \\ $\pi^+\pi^-\pi^0$ & $ 0.660 \leq \sqrt{s} \leq 1.937 $ & $ 47.79 \pm 0.89 $ \\
$\pi^+\pi^-\pi^+\pi^-$ & $ 0.613 \leq \sqrt{s} \leq 1.937 $ & $ 14.87 \pm 0.20 $ \\
$\pi^+\pi^-\pi^0\pi^0$ & $ 0.850 \leq \sqrt{s} \leq 1.937 $ & $ 19.39 \pm 0.78 $ \\
$K^+K^-$ & $ 0.988 \leq \sqrt{s} \leq 1.937 $ & $ 23.03 \pm 0.22 $\\
$K^0_S K^0_L$ & $ 1.004 \leq \sqrt{s} \leq 1.937 $ & $ 13.04 \pm 0.19 $ \\
$KK\pi$ & $ 1.260 \leq \sqrt{s} \leq 1.937 $ & $ \hphantom{0}2.71 \pm 0.12 $ \\
$KK2\pi$ & $ 1.350 \leq \sqrt{s} \leq 1.937 $ & $ \hphantom{0}1.93 \pm 0.08 $ \\
Inclusive channel & $ 1.937 \leq \sqrt{s} \leq 11.200 $ & $ 43.67 \pm 0.67 $ \\
\hline align}{\end{eqnarray}{tabular}
}
}\caption{Contributions to $a_{\mu}^{\rm had, \, LO \, VP}$~\cite{Keshavarzi:2018mgv}.}\label{tab:amuhadexc}
align}{\end{eqnarray}{table}
align}{\begin{eqnarray}{figure}[!t]
\centering
\subfloat[$\sigma^{0}(e^+e^-\rightarrow\pi^+\pi^-)$]{%
\includegraphics[width= 0.33\textwidth]{ch-10_rho-omega-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow\pi^+\pi^-\pi^0)$]{%
\includegraphics[width= 0.33\textwidth]{ch-16_full-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow\pi^+\pi^-\pi^+\pi^-)$]{%
\includegraphics[width= 0.33\textwidth]{ch-12-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow\pi^+\pi^-\pi^0\pi^0)$]{%
\includegraphics[width= 0.33\textwidth]{ch-17-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow K^+K^-)$]{%
\includegraphics[width= 0.33\textwidth]{ch-02_full_range-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow K^0_S K^0_L)$]{%
\includegraphics[width= 0.33\textwidth]{ch-04_full_range-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow KK\pi)$]{%
\includegraphics[width= 0.33\textwidth]{KKpi_Data-vs-Isospin-eps-converted-to.pdf}}\hfill
\subfloat[$\sigma^{0}(e^+e^-\rightarrow KK\pi\pi)$]{%
\includegraphics[width= 0.33\textwidth]{KKpipi_Data-vs-Isospin-eps-converted-to.pdf}}\hfill
\subfloat[Inclusive data]{%
\includegraphics[width= 0.33\textwidth]{Inclusive-High_2-4GeV-eps-converted-to.pdf}}\hfill
\caption{The resulting cross sections of the leading and major sub-leading hadronic final states~\cite{Keshavarzi:2018mgv}.}\label{fig:excxSec}
align}{\end{eqnarray}{figure}
The KNT18 analysis~\cite{Keshavarzi:2018mgv} is a complete re-evaluation of the hadronic vacuum polarisation contributions, $a_{\mu}^{\rm had, \ VP}$. The results from this work for contributions to $a_{\mu}^{\rm had, \, LO \, VP}$ and cross sections from the major channels are given in Table~\ref{tab:amuhadexc} and Figure~\ref{fig:excxSec}, respectively. In the $\pi^+\pi^-$ channel, the radiative return measurements from BABAR, KLOE and BESIII in the $\rho$ region have greatly improved the estimate of this final state. The cross section in the $\rho$ region is displayed in plot (a) of Figure~\ref{fig:excxSec}. For all displayed channels, the data combinations include new measurements which, coupled with updates in the KNT data combination routine~\cite{Keshavarzi:2018mgv}, have improved the estimates of $a_{\mu}^{\rm had, \, LO \, VP}$ from these final states. The uncertainty contribution from $\pi^+\pi^-\pi^0\pi^0$ is still relatively large in comparison with its contribution to $a_{\mu}^{\rm had, \, LO \, VP}$ and requires better new data. Plot (g) of Figure~\ref{fig:excxSec} demonstrates good agreement between the previously used estimate from isospin relations and the data-based approach in the $KK\pi$ final state. Examining plot (h) of Figure~\ref{fig:excxSec}, it is evident that the isospin relations provided a poor estimate of the $KK\pi\pi$ final state. The inclusive hadronic $R$-ratio compilation is shown in plot (i) of Figure~\ref{fig:excxSec}, which demonstrates that the inclusive data combination is much improved. With the new KEDR data, the differences between the inclusive data and pQCD are not as large as previously and, hence, the contributions in the entire inclusive data region are now estimated using the inclusive data alone.
\subsection{Data tensions in the KNT18 analysis}\label{sec:DataTensions in the KNT}
align}{\begin{eqnarray}{figure}[!t]
\centering
{\includegraphics[width=0.5\textwidth]{Ch10-KNT2017_RhoCompare-eps-converted-to.pdf}
\caption{\small The comparison of the integration of the individual radiative return measurements and the combination of direct scan $\pi^+\pi^-$ measurements between $0.6 \leq \sqrt{s} \leq 0.9$ GeV~\cite{Keshavarzi:2018mgv}.} \label{fig:RadRetCompare}}
align}{\end{eqnarray}{figure}
align}{\begin{eqnarray}{figure}[!t]
\centering
\subfloat[$\pi^+\pi^-$]{%
\includegraphics[width= 0.535\textwidth]{Ch10-KNT2017_Diff-RhoInt-eps-converted-to.pdf}\label{fig:RadRetFit}}\hfill
\subfloat[$K^+K^-$]{%
\includegraphics[width= 0.465\textwidth]{Ch02_Diff_phi-eps-converted-to.pdf}\label{fig:K+K-diff}}\hfill
\caption{Relative difference plots of data in the $\pi^+\pi^-$ channel on the $\rho$ resonance and in the $K^+K^-$ channel on the $\phi$ resonance, against the fit all of all data for the respective channel. The width of the coloured bands represent the propagation of the statistical and systematic uncertainties added in quadrature~\cite{Keshavarzi:2018mgv}.}
align}{\end{eqnarray}{figure}
In the $\pi^+\pi^-$ channel, the BABAR data are noticeably higher than the average, causing an increase to the two-pion contribution to $a_{\mu}^{\rm SM}$. This is evident from Figure~\ref{fig:RadRetCompare}, which compares the estimates of $a_{\mu}^{\pi^+\pi^-}$ from the full data combination, the radiative return measurements and all other measurements (direct energy scan) in the dominant $\rho$ region. Notably, the deviation between the estimates from KLOE combination and the BABAR data in this range is $\sim 2.8\sigma$. With the highly correlated nature of the KLOE combination now having a dominating influence overall in the KNT18 analysis, a large disagreement is also noted between the full $\pi^+\pi^-$ data combination and the integral of the BABAR data alone.\footnote{This effect is more prominent when considering the evaluation of $a_{\mu}^{\pi^+\pi^-}$ from the BABAR data alone over the full available energy range. This results in an estimate of $a_{\mu}^{\pi^+\pi^-}(\text{BaBar data only}) = 513.2 \pm 3.8$ compared to the result given in Table~\ref{tab:amuhadexc}. It should be noted that similar differences are observed between the integral of the BABAR data alone and full evaluations of $a_{\mu}^{\pi^+\pi^-}$ from other recent analyses~\cite{Davier:2017zfy,Jegerlehner:2017gek,Jegerlehner:2017lbd,Benayoun:2015gxa,Colangelo:2018mtw}.} This is made more apparent when considering Figure~\ref{fig:RadRetFit}. From Figure~\ref{fig:RadRetCompare}, is clear that the full $\pi^+\pi^-$ data combination agrees well with the new BESIII data, the KLOE data and the combination of the remaining direct scan data. Interestingly, however, the BESIII data is in better agreement with the BABAR data at the peak of the resonance where the cross section is largest, slightly alleviating the disagreement between the full $\pi^+\pi^-$ data combination and the BABAR data. The tension between data sets is, however, reflected and accounted for in the local $\chi^2$ error inflation, which results in an $\sim15\%$ increase in the uncertainty of $a_{\mu}^{\pi^+\pi^-}$~\cite{Keshavarzi:2018mgv}.
This estimate of $a_{\mu}^{K^+K^-}$ exhibits an increase of the mean value of more than 1$\sigma$ attributed to the inclusion of the new BABAR and CMD-3 data. This can be seen in Figure~\ref{fig:K+K-diff}. Previously, the data combination in the $\phi$ resonance region for this channel was dominated by the SND scans~\cite{Achasov:2000am} visible in Figure~\ref{fig:K+K-diff} and the now omitted CMD-2 scans~\cite{Akhmetshin:2008gz}, which were in good agreement. The BABAR data~\cite{Lees:2013gzt}, which due to their precision and correlated uncertainties now dominate the $K^+K^-$ data combination, are higher in this region than both the SND and CMD-2 data. The most recent CMD-3 data are higher still~\cite{Kozyrev:2017agm}. The reanalysis of the CMD-2 data will prove crucial in resolving the current differences in this channel and, should they agree further with the BABAR and CMD-3 data, would result in a further increase of the estimate from this channel. Overall, the uncertainty has drastically improved, with much of the change being due to a finer clustering over the $\phi$ resonance after the inclusion of the new high statistics BABAR data. However, the disagreement between the data seen in Figure~\ref{fig:K+K-diff} is accounted for in the local error inflation which provides an increase to the uncertainty of $a_{\mu}^{K^+K^-}$ of $\sim20\%$~\cite{Keshavarzi:2018mgv}.
\subsection{Total results for $a_{\mu}^{\rm had,\,LO\,VP}$ and $a_{\mu}^{\rm had,\,NLO\,VP}$}
align}{\begin{eqnarray}{figure}[!t]
\centering
\subfloat[The hadronic $R$ ratio]{%
\includegraphics[width= 0.49\textwidth]{Combination_full_range-eps-converted-to.pdf}}\hfill
\vspace{0.5cm}
\subfloat[The uncertainty of the hadronic $R$ ratio]{%
\includegraphics[width= 0.49\textwidth]{dR_full_range-eps-converted-to.pdf}}\hfill
\vspace{-0.5cm}
\caption{Contributions to the total hadronic $R$ ratio from the different final states (left panel) and their uncertainties (right panel) below 1.937 GeV. The full $R$ ratio and its uncertainty is shown in light blue in each plot, respectively. Each final state is included as a new layer on top in decreasing order of the size of its contribution to $a_{\mu}^{\rm had, \, LO \, VP}$~\cite{Keshavarzi:2018mgv}.} \label{hadxSec}
align}{\end{eqnarray}{figure}
From the sum of all hadronic contributions shown in Figure~\ref{hadxSec}, the estimate for $a_{\mu}^{\rm had, \, LO \, VP}$ from this analysis is~\cite{Keshavarzi:2018mgv}
\begin{equation}\label{LOHVP_KNT18}
a_{\mu}^{\rm had, \, LO \, VP} = (693.26 \pm 1.19_{\rm stat} \pm 2.01_{\rm sys} \pm 0.22_{\rm vp} \pm 0.71_{\rm fsr}) \times 10^{-10} = (693.26 \pm 2.46_{\rm tot}) \times 10^{-10} \ ,
\end{equation}
where the uncertainties include all available correlations and local $\chi^2_{\rm min}/{\rm d.o.f.}$ inflation.
Using the same data compilation as for the calculation of $a_{\mu}^{\rm had, \, LO \, VP}$, the next-to-leading order (NLO) contribution to $a_{\mu}^{\rm had, VP}$ is determined to be $a_{\mu}^{\rm had, NLOVP} = (-9.82 \pm 0.04) \times 10^{-10}$.
\subsection{SM prediction of $g-2$ of the muon} \label{g-2muon}
From these results for $a_{\mu}^{\rm had, \, LO \, VP}$ and $a_{\mu}^{\rm had, \, NLO \, VP}$, the SM prediction of the anomalous magnetic moment of the muon is found to be~\cite{Keshavarzi:2018mgv}
\begin{equation} \label{amuSMfinal}
a_{\mu}^{\rm SM} = (11\ 659 \ 182.04 \pm 3.56) \times 10^{-10} \, .
\end{equation}
Comparing this with the current experimental measurement results in a deviation of $\Delta a_{\mu} = (27.06 \pm 7.26)\times 10^{-10}$, corresponding to a $3.7\sigma$ discrepancy.
\section{Conclusions}
The uncertainty of $a_{\mu}^{\rm SM}$ is entirely dominated by the hadronic contributions, where below $\sim2$GeV, the estimates of $a_{\mu}^{\rm had,\, VP}$ and the corresponding uncertainties are determined from the experimentally measured cross sections of individual hadronic final states. These measurements are achieved experimentally via direct energy scan or through radiative return. Many experiments have dedicated programmes to precisely measure these final states, meaning that a vast amount of data is now available and that, in some cases, overall precision has reached the sub-percent level. However, data tensions are evident between measurements of the same hadronic channels from different experiments, which reduces the overall quality of the data combinations used to determine $a_{\mu}^{\rm had, \, VP}$.
The KNT18 analysis has completed a full re-evaluation of the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the muon, $a_{\mu}^{\rm had, \, VP}$. Combining all available $e^+e^- \rightarrow {\rm hadrons}$ cross section data, this analysis found $a_{\mu}^{\rm had, \, LO \, VP} = (693.26 \pm 2.46)\times 10^{-10}$ and $a_{\mu}^{\rm had, \, NLO \, VP} = (-9.82 \pm 0.04)\times 10^{-10}$. This has resulted in a new estimate for the Standard Model prediction of $a_{\mu}^{\rm SM} = (11\ 659 \ 182.04 \pm 3.56) \times 10^{-10}$, which deviates from the current experimental measurement by $3.7\sigma$.
|
2,869,038,155,682 | arxiv | |
2,869,038,155,683 | arxiv | \section{Introduction}
\section{Introduction}
Turbulence in ordinary fluids such as air or water consists of rotational eddies of different sizes
which we term vortices\index{subject}{vortices}. Vortices\index{subject}{vortices!quantum vortices} therefore
are a hallmark signature of a turbulent flow~\citep{Barenghi01}.\index{authors}{Barenghi, C. F.}\index{Donnelly,
R. J.}
In superfluids, quantum
vortices\index{subject}{vortices!quantum vortices} differ from their classical counterparts because of the
quantization of circulation. This means that
the rotational motion of a superfluid is constrained to discrete
vortices\index{subject}{vortices!quantum vortices} which all have the same core structure. Turbulence in
superfluid helium has been the subject of many recent
experimental and theoretical investigations~\citep{Skrbek12}\index {authors}{Sreenivasan, K. R.}.
Recently, experimentalists have been
able to visualise individual vortex lines and reconnection events
using tracer particles~\citep{Fonda12}\index{authors}{Fonda, E.}\index{authors}{Meichle,
D.P.}\index{authors}{Ouellette, N. T.}\index{authors}{Hormoz, S.}\index{authors}{Sreenivasan, K. R.}. Weakly
interacting, dilute atomic Bose-Einstein
condensates (henceforth referred to as BECs)\index{subject}{Bose Einstein condensates} present a distinct platform to view and probe
quantum turbulence\index{subject}{quantum turbulence!turbulence}. A key feature here is the ability to
directly resolve the structure of individual vortices\index{subject}{vortices!quantum vortices} and in
turn the dynamics of a turbulent vortex tangle~\citep{Henn09}.\index{authors}{Henn, E. A. L.}
\index{authors}{Seman, J. A.}\index{authors}{Roati, G.}\index{authors}{Magalh\~aes, K. M. F.}
\index{authors}{Bagnato, V. S.}\index{subject}{vortices!vortex tangle}
As a result of the quantized nature of vorticity,
quantum turbulence in superfluid helium
and in BECs\index{subject}{Bose-Einstein condensates} can be viewed as a
simpler, idealized analog of turbulence in ordinary fluids,
and opens the possibility of studying problems
which may be relevant to our general understanding of turbulence.
\section{Why atomic Bose-Einstein condensates?}
Since their first generation
in 1995~\citep{Davis95,Anderson95},\index{authors}{Davis, K. B.}\index{authors}{Mewes, M. O.}\index{authors}{Andrews, M. R.}\index{authors}{van Druten, N. J.}\index{authors}{Durfee, D. S.}
\index{authors}{Kurn, D. M.}\index{authors}{Ketterle, W.}\index{authors}{Anderson, M. H.}
\index{authors}{Ensher, J.}\index{authors}{Matthews, M. R.}\index{authors}{Wieman, C. E.} atomic BECs have been used to study a wide variety of
nonlinear dynamics, for example, solitons, vortices\index{subject}{vortices!quantum vortices} and four-wave
mixing \citep{Kevrekidis08}\index{authors}{Kevrekidis, P. G.}\index{authors}{Carretero-Gonzalez, R.}.
A merit of exploiting BECs as a testbed of nonlinear physics lies with
the immense control and flexibility they offer. For example:
\begin{itemize}
\item \emph{Trapping geometry, shape and dimensionality}\\
Optical and magnetic fields can be employed to precisely create a potential landscape for the atoms in the BEC,
which in turn enables control of the shape and effective dimensionality of the system
\citep{Gorlitz01}.\index{authors}{G\"orlitz, A.}\index{authors}{Vogels, J. M.}\index{authors}{Leanhardt, A. E.}
\index{authors}{Raman, C.}\index{authors}{Gustavson, T. L.}\index{authors}{Abo-Shaeer, J. R.}
\index{authors}{Chikkatur, A. P.}\index{authors}{Gupta, S.}\index{authors}{Inouye, S.}\index{authors}{Rosenband, T.}
\index{authors}{Ketterle, W.} A
basic requirement of these gases is confinement in space to prevent contact with hot surfaces. This is
typically provided by magnetic traps which are harmonic in shape and have the form
\citep{Fortagh07}\index{authors}{Fort\'agh, J.}\index{authors}{Zimmermann, C.}
\begin{eqnarray}
V_{\rm{ext}}({\bf r}) = \frac{1}{2}m\omega (x^2 + y^2 + z^2),
\end{eqnarray}
where $\omega$ is the trapping frequency and $m$ the mass of the atom.
This type of trap results in an atomic cloud with a radial density profile which resembles an inverted parabola.
If a harmonic trap is used which is very strongly confining in one direction, for example
\begin{eqnarray}
V_{\rm{ext}}({\bf r}) = \frac{1}{2}m\omega (x^2 + y^2 + \epsilon z^2),
\end{eqnarray}
where $\epsilon \gg 1$, the dynamics in that direction
is effectively inhibited and the system becomes effectively two-dimensional (2D). By changing
$\epsilon$, one can easily change the effective dimensionality, which is particularly important
in turbulence (2D turbulence is very different from 3D turbulence).
In the same way, if the trap is
very tightly confining in two directions, the dynamics
is mainly in the third direction, and the system is effectively
one-dimensional (1D).
More complicated trapping geometries can be realised, for example
a toroidal ring or a periodic optical lattice. Traps can also
be made time-dependent by rotating or shaking the trapping potential.
Furthermore, one can create localized potentials using optical fields, which can mimic an obstacle and be moved through the system on demand.
\item \emph{Interaction strength}\\
Typically, the dominant atomic interaction in a BEC\index{subject}{Bose-Einstein condensates} is the short-range
and isotropic {\it s}-wave interaction. Experimentalists can employ magnetic Feshbach
resonances~\citep{Inouye98}\index{authors}{Inouye, S.}\index{authors}{Andrews, M. R.}\index{authors}{Stenger,
J.}\index{authors}{Miesner, H. J.}\index{authors}{Stamper-Kurn, D. M.}\index{authors}{Ketterle, W.} to change the strength of
these interactions and even their nature, i.e. whether they are attractive or repulsive
\citep{Roberts98}\index{authors}{Roberts, J. L.}\index{authors}{Claussen, N. R.}\index{authors}{Burke, J.
P.}\index{authors}{Greene, C. H.}\index{authors}{Cornell, E. A.}. Furthermore, by using atoms with relatively large
magnetic dipole moments, e.g. $^{52}$Cr, it is possible to create a BEC\index{subject}{Bose-Einstein condensates}
where the atoms also experience significant dipole-dipole interactions, which are long-range and anisotropic, and
greatly modify the static and dynamical properties of the system \citep{Lahaye09}\index{authors}{Lahaye, T.}\index{authors}{Menotti, C.}\index{authors}{Santos, L.}\index{authors}{Lewenstein, M.}\index{authors}{Pfau, T}.
\item \emph{Vortex core optical imaging} \\
The healing length which characterizes the vortex core size is typically around $10^{-7}$m in a BEC\index{subject}{Bose-Einstein condensates} (c.f.
$10^{-10}$m in superfluid Helium). By expanding the BEC\index{subject}{Bose-Einstein condensates} (following
release from the trapping potential), the vortex can be directly imaged and resolved via optical
absorption~\citep{Madison00,Raman01}\index{authors}{Madison, K. W.}\index{authors}{Chevy, F.}\index{authors}{Wohlleben, W.}\index{authors}{Dalibard, J.}\index{authors}{Raman, C.}\index{authors}{Abo-Shaeer, J. R.}\index{authors}{Vogels, J. M.}\index{authors}{Xu, K.}\index{authors}{Ketterle, W.}. Advanced real time imaging of condensates
containing vortices\index{subject}{vortices!quantum vortices} has also recently been developed~\citep{Freilich10}\index{authors}{Freilich, D. V.}\index{authors}{Bianchi, D. M.}\index{authors}{Kaufman, A. M.}\index{authors}{Langin, T. K.}\index{authors}{Hall, D. S.}, allowing the precession of vortices\index{subject}{vortices!quantum vortices} to be observed.
\end{itemize}
In the limit of zero temperature and weak interactions, the evolution equation for the macroscopic
condensate wavefunction, $\phi ({\bf{r}},t)$, is a form of nonlinear Schr{\"o}dinger equation, commonly known
as the Gross-Pitaevskii Equation (GPE)\index{subject}{Gross-Pitaevskii Equation}:
\begin{eqnarray}
i\hbar \frac{ \partial \phi(\mathbf{r},t)}{ \partial t} = \left[ - \frac{\hbar ^2}{2m} \nabla^2 +
V_{\rm ext}(\mathbf {r},t) + g|\phi(\mathbf{r},t)|^2 - \mu \right] \phi(\mathbf{r},t),
\label{eqn_gpe}
\end{eqnarray}
where $\bf{r}$ is the position in space, $t$ is time and $\hbar$ is Planck's constant divided by $2\pi$.
The GPE\index{subject}{Gross-Pitaevskii Equation} provides a good quantitative description of the dynamics of a BEC\index{subject}{Bose-Einstein condensates} over all lengthscales
available, from the vortex core to the system size, up to temperatures of approximately
$T \simeq 0.5T_c$ (where the critical temperature $T_c$ is of the order of mK in typical
experiments). At the right-hand side we recognise the kinetic energy term, $(-\hbar^2/2m) \nabla^2$,
the trapping potential $V_{\rm ext}(\mathbf {r},t)$ (which in general may be time-dependent), the interaction term, $g|\phi(\mathbf{r},t)|^2$ where
$g = 4\pi \hbar^2 a_s/m$ and $a_s$ is the 3D $s-$wave scattering length, and the chemical potential $\mu$.
The GPE\index{subject}{Gross-Pitaevskii Equation} can be almost exactly mapped to the classical Euler equation;
the small difference, namely the
quantum stress, regularizes
the solutions, preventing singularities which may arise in an Euler
fluid~\citep{Barenghi08}\index{authors}{Barenghi, C. F.}.
The GPE\index{subject}{Gross-Pitaevskii Equation} is a practically exact model in the limit of zero temperature, where
essentially all of the atoms exist in the Bose-Einstein condensate phase\index{subject}{Bose-Einstein condensates}. In many experiments the condensate exists at well below the BEC\index{subject}{Bose-Einstein condensates} transition temperature such that this approximation is justified.
Extensions of the GPE\index{subject}{Gross-Pitaevskii Equation} to include the
effect of thermal atoms provide a more complete (albeit not exact) physical model of a real
BEC~\citep{Jackson09}\index{authors}{Jackson, B.}\index{authors}{Proukakis, N. P.}
\index{authors}{Barenghi, C. F.}\index{authors}{Zaremba, E.} (see
e.g.~\cite{Proukakis08}\index{authors}{Proukakis, N. P.}\index{authors}{Jackson, B.} for an in depth review of finite
temperature models).
However BECs\index{subject}{Bose-Einstein condensates} suffer an
important limitation. The systems which can be currently created in the
laboratory contain a small number of atoms,
typically $10^3$ to $10^9$, hence do not sustain the number of
quantum vortices\index{subject}{vortices!quantum vortices} present in helium
experiments. For example, up to a few hundred vortices\index{subject}{vortices!quantum vortices} have been achieved in the largest 2D BECs \citep{Abo-shaeer02}\index{authors}{Abo-Shaeer, J. R.}\index{authors}{Raman, C.}\index{authors}{Ketterle, W.}.
This brings to light the issue of length scales. A defining
feature of classical turbulence, besides nonlinearity, is the huge number of length scales which are excited. The range of length scales available in
turbulent superfluid helium at very small temperatures is perhaps
even larger, since short wavelength helical waves
along the vortices\index{subject}{vortices!quantum vortices} can be
generated by nonlinear interactions, producing a turbulent cascade
called the Kelvin wave cascade~\citep{Vinen06}\index{authors}{Vinen, W.}. A simple question arises:
\emph{can a BEC, containing a limited number of
vortices\index{subject}{vortices!quantum vortices}, really become turbulent?}
Our tentative answer is yes.
Numerical results thus far \citep{Nore97,Berloff02,Kobayashi05b,Yepez09}\index{authors}{Nore, C.}\index{authors}{Abid, M.}\index{authors}{Brachet, M. E.}\index{authors}{Berloff, N. G.}\index{authors}{Svistunov, B. V.}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}\index{authors}{Yepez, J.}\index{authors}{Vahala, G.}\index{authors}{Vahala, L.}\index{authors}{Soe, M.}
suggest that kinetic energy is distributed over the length scales in agreement
with the $k^{-5/3}$ Kolmogorov scaling which is observed in ordinary turbulence
(where $k$ is the wavenumber) even over this small range of length scales.
Therefore, the study of turbulence in a BEC\index{subject}{Bose-Einstein condensates} represents an exciting
opportunity to probe a new regime residing somewhere between chaos and turbulence.
In the remainder of this paper, we aim to identify some important open questions about
turbulence in BECs; where appropriate we will
review some of the work which has been carried out to date.
\section{Quantum Turbulence in atomic BECs, where are we?}
\label{sec:qt_BEC}
The following is an extensive, but by no means exhaustive,
list of aspects yet to be understood regarding turbulence
in atomic BECs\index{subject}{Bose-Einstein condensates}.
To structure our discussion, we distinguish
the evolution of turbulent flow into three stages, namely;
\begin{itemize}
\item[(i)] \underline{The generation of the turbulence.} \emph{What are the most effective and efficient ways to generate turbulence?} \emph{Does the way in which the turbulence is generated affect the `type' of turbulence created?}
\item[(ii)] \underline{The statistical steady state.} \emph{Are there universal features of turbulence, for instance, is the Kolmogorov energy spectrum present? What are the statistics of the turbulence velocity field?}
\item[(iii)] \underline{The decay of the turbulence.} \emph{How does the turbulence decay?} \emph{What is the best way to measure the decay?}
\end{itemize}~\\
\noindent{\bf{(i) The generation of the turbulence}}\\
To understand the generation of a vortex
tangle\index{subject}{vortices!vortex tangle} in a quantum gas, we must
first understand how individual
vortices\index{subject}{vortices!quantum vortices} are nucleated.
The very first creation of such a vortex took place in
a two-component condensate and was driven by the rotation of one component around
the other. The subsequent removal of the inner component resulted in the formation of a hollow core of a
singly quantized vortex~\citep{Anderson00}\index{authors}{Anderson, B. P.}\index{authors}{Haljan, P. C.}\index{authors}{Wieman, C. E.}\index{authors}{Cornell, E. A.}. Further techniques
for generating vortex structures soon followed, including the creation of vortex rings following the ``snake instability" decay of a dark soliton ~\citep{Anderson01}\index{authors}{Anderson, B. P.}\index{authors}{Haljan, P. C.}\index{authors}{Regal, C. A.}\index{authors}{Feder, D. L.}\index{authors}{Collins, L. A.}\index{authors}{Clark, C. W.}\index{authors}{Cornell, E. A.},
phase imprinting~\citep{Leanhardt02}\index{authors}{Leanhardt, A. E.}\index{authors}{G\"orlitz, A.}\index{authors}{Chikkatur, A. P.}\index{authors}{Kielpinski, D.}\index{authors}{Shin, Y.}\index{authors}{Pritchard, D. E.}\index{authors}{Ketterle, W.} and by
a rapid quench through the transition temperature for the onset of
Bose-Einstein condensation\index{subject}{Bose-Einstein condensates}, i.e. the Kibble-Zurek mechanism~\citep{Weiler08,Freilich10}\index{authors}{Weiler, C.}\index{authors}{Neely, T. W.}\index{authors}{Scherer, D. R.}\index{authors}{Bradley, A. S.}\index{authors}{Davis, M. J.}\index{authors}{Anderson, B. P.}.
However, to generate a large number of vortices\index{subject}{vortices!quantum vortices} in the system at any one time,
two other techniques have proved to be more effective:
\begin{itemize}
\item[(i)] \emph{Rotation} of an anisotropic BEC\index{subject}{Bose-Einstein condensates} excites surface modes
leading to the nucleation of vortices at the edge which then drift into the bulk of the BEC. If the rotation is performed about only one axis, a vortex lattice\index{subject}{vortices!lattice} is created
~\citep{Hodby01,Abo-shaeer02,Abo-shaeer01,Madison00,Madison01}\index{authors}{Hodby, E.}\index{authors}{Hechenblaikner,
G.}\index{authors}{Hopkins, S. A.}\index{authors}{Marag\`o, O. M.}\index{authors}{Foot, C. J.}\index{authors}{Abo-Shaeer, J. R.}\index{authors}{Raman, C.}\index{authors}{Ketterle, W.}\index{authors}{Abo-Shaeer, J. R.}\index{authors}{Raman, C.}\index{authors}{Vogels, J. M.}\index{authors}{Ketterle, W.}\index{authors}{Madison, K.
W.}\index{authors}{Chevy, F.}\index{authors}{Wohlleben, W.}\index{authors}{Dalibard, J.}\index{authors}{Madison, K.
W.}\index{authors}{Chevy, F.}\index{authors}{Bretin, F.}\index{authors}{Dalibard, J.}.
In 3D, if the rotation is performed about more than one axis,
a vortex tangle\index{subject}{vortices!vortex tangle} has
been predicted to form~\citep{Kobayashi07}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}.
\item[(ii)] \emph{A moving (cylindrical) obstacle}, such as that created by the potential from a blue-detuned laser beam moving
through a quantum fluid, generates
pairs of vortices\index{subject}{vortices!quantum vortices}
in its wake when its speed exceeds a critical value~\citep{Raman01}\index{authors}{Raman, C.}\index{authors}{Abo-Shaeer, J. R.}\index{authors}{Vogels, J. M.}\index{authors}{Xu, K.}\index{authors}{Ketterle, W.}. Recently, this method has been used to generate and study a collection of vortex dipoles in a 2D BEC~\citep{Neely10}\index{authors}{Neely, T. W.}\index{authors}{Samson, E. C.}\index{authors}{Bradley, A. S.}\index{authors}{Davis, M. J.}\index{authors}{Anderson, B. P.}.
\end{itemize}
Both methods generate a large number of
vortices\index{subject}{vortices!quantum vortices}, in 2D as well as in 3D. However, one can
bypass the initial transient period and begin with a nonequilibrium state of vortices\index{subject}{vortices!quantum vortices}. Experimentally, this can be achieved via
imprinting a phase profile onto the condensate via laser beams, as performed by~\cite{Leanhardt02} for
generating single vortices\index{subject}{vortices!quantum vortices} of arbitrary charge. The use of such imprinting to generate a vortex tangle has
been implemented theoretically~\citep{White10}, with the resulting tangle similar to that depicted in Fig~\ref{white_tangle}.
\\
Our preliminary results with method (ii) (laser stirring), suggest that it is possible to
generate a large number of vortices\index{subject}{vortices!quantum vortices}; we have found qualitative evidence
that, by moving the obstacle along different paths,
we can change the isotropy of the resulting tangle of vortices.
\begin{figure}[h!]
\centering{
a)
\includegraphics[scale = 0.3]{ajallen_fig1a.png}
\hspace{0.1cm}
b)
\includegraphics[scale = 0.3]{ajallen_fig1b.png}
\hspace{0.1cm}
c)
\includegraphics[scale = 0.3]{ajallen_fig1c.png}
}\\
\includegraphics[clip,scale = 0.4,angle = 270]{ajallen_fig1d.png}
\caption{Top row: Density isosurfaces of a 3D spherical BEC\index{subject}{Bose-Einstein condensates} at times
$t\omega = $ a) 60, b) 100 and c) 240 after stirring
the condensate along the $z-$direction in a circular path. We see here that the surface plot picks up the vortex
cores as well as some of the condensate edge. The resulting vortex length in each direction is shown over time
in the bottom part of this figure where it has been normalised by the peak vortex length.}
\label{fig:iso_cir}
\end{figure}
Fig.~\ref{fig:iso_cir} (top row) shows the density isosurface of a 3D spherical condensate at three
instants in time, after the condensate has been stirred for a time $t_{\rm{stir}}$, along a
circular path with a Gaussian-shaped laser stirrer oriented in the $z-$direction.
For a simple measure of the isotropy of the
tangle\index{subject}{vortices!vortex tangle}, we plot the projected vortex lengths
$L_x$, $L_y$ and $L_z$ in each Cartesian direction
(bottom part of Fig.~\ref{fig:iso_cir}).
All projected lengths rapidly increase during the stirring period
($t< t_{\rm{stir}}$); after the laser has been removed ($t> t_{\rm{stirr}}$),
$L_x$, $L_y$ and $L_z$ all decrease for a short period of time.
Later, only the vortex lengths $L_x$ and $L_y$ in the transverse
($x$ and $y$) directions further decrease, whereas $L_z$ remains
approximately constant because the vortex tangle decays into a
regular lattice, as it is apparent from the final density isosurface
(Top row, c)). This is as expected: in stirring the condensate circularly we impart angular
momentum about the $z$-axis, and it is well known that the ground state of a superfluid with
sufficient angular momentum features a lattice of regularly-spaced, vortex lines aligned along the
$z$-axis.
\begin{figure}[h!]
\centering{
\includegraphics[scale = 0.4,angle = 270]{ajallen_fig2.png}
}
\caption{Vortex length, normalised by the peak length, in the $x,y$ and $z$ directions for a stirrer
along the $z-$direction, moving in a Figure-eight path.}
\label{fig:infin}
\end{figure}
In Fig.~\ref{fig:infin} the vortex length is shown when the stirring
takes place, for the same amount of time, along a Figure-eight path.
Again, the vortex length increases over the duration of the laser stirring
($t< t_{\rm{stir}}$); however, after the laser is removed ($t> t_{\rm{stir}}$),
the tangle decays isotropically, i.e. all projected lengths $L_x$,
$L_y$ and $L_z$ decay together.
For the Figure-eight path, the laser also moves through the centre
of the condensate, where the density is higher, the vortices which
are generated are longer than those generated
at the edge of the condensate; therefore, this laser stirring path
is more efficient at creating a dense, random vortex
tangle\index{subject}{vortices!vortex tangle} than simply stirring the
condensate in a circular fashion. \\~\\
\noindent{\bf{(ii) Statistical state}}\\
Once the turbulence state has been created (by whichever means),
its steady state properties can be investigated.
BECs offer the possibility to study 2D and 3D turbulence and the cross-over
region between the two \citep{Parker05}\index{authors}{Parker, N. G.}\index{authors}{Adams, C. S.}. We now review our current understanding of the properties
of a turbulent tangle of vortices\index{subject}{vortices!quantum vortices}\index{subject}{vortices!vortex tangle} in a BEC, first in 2D and then in 3D. \\
\noindent\emph{Quantum turbulence in 2D}\\
In 2D classical turbulence, the conservation of enstrophy dictates that energy must flow
from small scales of energy injection to large scales forming, for example, large
clusters of vortices~\citep{Kraichnan67}\index{authors}{Kraichnan, R. H.}\index{subject}{vortices!quantum vortices}. This inverse cascade is
thought to underly Jupiter's giant
Red Spot and has been experimentally examined in classical
fluids~\citep{Sommeria88,Marcus88}\index{authors}{Sommeria, J.}\index{authors}{Meyers, S. D.}\index{authors}{Swinney, H. L.}\index{authors}{Marcus, P.} (for a review see~\cite{Kellay02}\index{authors}{Kellay, H.}\index{authors}{Goldburg, W. I.}). Attempts to observe the inverse cascade
effect in quantum gases have lead to
modelling vortex generation~\citep{Parker05,White12,Fujimoto11,Reeves12},\index{authors}{Parker, N. G.}
\index{authors}{Adams, C. S.}\index{authors}{White, A. C.}\index{authors}{Barenghi, C. F.}
\index{authors}{Proukakis, N. P.}\index{authors}{Fujimoto, K.}\index{authors}{Tsubota, M.}\index{authors}{Reeves, M.}
\index{authors}{Billam, T.}\index{authors}{Anderson, B. P.}\index{authors}{Bradley, A. S.} and to experiments on
the dynamics of vortex dipoles created by a moving
potential~\citep{Neely10,Neely12}.\index{authors}{Neely, T. W.}\index{authors}{Samson, E.
C.}\index{authors}{Bradley, A. S.}\index{authors}{Davis, M. J.}\index{authors}{Anderson, B.
P.}\index{authors}{Neely, T. W.}\index{authors}{Bradley, A. S.}\index{authors}{Samson, E. C.}
\index{authors}{Rooney, S.}\index{authors}{Wright, E.}\index{authors}{Law,
K.}\index{authors}{Carretero-Gonzalez, R.}\index{authors}{Kevrekidis, P. G.}\index{authors}{Davis,
M. J.}\index{authors}{Anderson, B. P.}
~\cite{Numasato10}\index{authors}{Numasato, R.}\index{authors}{Tsubota, M.}\index{authors}{L'vov, V. S.}, evolved the 2D GPE\index{subject}{Gross-Pitaevskii Equation} to a turbulent state by initially
imposing a random phase on the wavefunction. They did not observe a reverse
cascade but rather a direct cascade. They argued that since the
total number of vortices\index{subject}{vortices!quantum vortices}, and therefore the enstrophy, is not conserved in simulations of the GPE\index{subject}{Gross-Pitaevskii Equation} because
of vortex pair annhilation, the inverse energy cascade is irrelevant for 2D quantum
turbulence\index{subject}{quantum turbulence!turbulence}.
Conversely, ~\cite{Reeves12}\index{authors}{Reeves, M.}\index{authors}{Billam, T.}\index{authors}{Anderson, B. P.}\index{authors}{Bradley, A.
S.} reported the numerical observation of the inverse cascade.
They solved the 2D damped GPE\index{subject}{Gross-Pitaevskii Equation} (DGPE) and generated a turbulent state
by imposing the fluid to flow past 4 stationary potential barriers. The speed of the flow was
sufficiently high ($v \simeq 0.822c$, where $c$ is the sound speed of the quantum fluid) so as to create
many vortices\index{subject}{vortices!quantum vortices} and thereby a turbulent flow.
Using a statistical algorithm they measured how prone `like-winding' vortices\index{subject}{vortices!quantum vortices}, i.e.
vortices\index{subject}{vortices!quantum vortices} of the same charge, were to cluster together depending on the amount of
dissipation imposed. They found that an intermediate level of damping lead
to small clusters of like-winding vortices\index{subject}{vortices!quantum vortices} being formed and inferred an inverse energy
cascade from analysis of the incompressible energy spectrum (energy associated with vortices\index{subject}{vortices!quantum vortices}).
However, this analysis was carried out in the wake of the obstacle where like-winding vortices\index{subject}{vortices!quantum vortices} are naturally in the vicinity of each other. To
make this result more convincing, a larger box for analysis would be desirable.
Similar work by~\cite{White12}\index{authors}{White, A. C.}\index{authors}{Barenghi, C. F.}\index{authors}{Proukakis, N. P.} implemented a rotating elliptical paddle
to generate large numbers of
vortices\index{subject}{vortices!quantum vortices}.
Again, clustering of like-winding
vortices\index{subject}{vortices!quantum vortices} was observed, but no
inverse cascade was reported.
A possible reason for the lack of definitive evidence for or against
the inverse cascade in quantum gases is their relatively small size,
i.e. the systems which were studied lacked enough length scales.
However, this limitation has not prevented the observation of the direct
Kolmogorov-3D cascade in systems of similar
size~\citep{Nore97,Kobayashi05a,Kobayashi05b,Yepez09,Kobayashi07}\index{authors}{Nore, C.}\index{authors}{Abid,
M.}\index{authors}{Brachet, M. E.}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}\index{authors}{Kobayashi,
M.}\index{authors}{Tsubota, M.}\index{authors}{Yepez, J.}\index{authors}{Vahala, G.}\index{authors}{Vahala,
L.}\index{authors}{Soe, M.}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}.\\
\noindent\emph{Quantum turbulence in 3D}\\
In their seminal work, ~\cite{Nore97}\index{authors}{Abid, M.}\index{authors}{Brachet, M. E.} used the GPE\index{subject}{Gross-Pitaevskii Equation} to investigate 3D
quantum turbulence\index{subject}{quantum turbulence!turbulence}
in a homogeneous box by evolving an initial, large scale Taylor Green vortex.
By decomposing the velocity field into divergence-free and
curl-free parts, they obtained incompressible
(associated with vortex motion) and compressible
(associated with acoustic excitations) kinetic energy spectra respectively.
They showed that the incompressible kinetic
energy spectrum is similar to
the classical $k^{-5/3}$ Kolmogorov energy spectrum at
scales down to the intervortex spacing.
Similarly,~\cite{Berloff02}\index{authors}{Berloff, N. G.}\index{authors}{Svistunov, B. V.} began with a non-equilibrium state, generated
by imprinting a random phase on the equilibrium wavefunction of a homogeneous box.
They then observed the evolution of the quantum turbulence\index{subject}{quantum turbulence!turbulence} by solving the GPE\index{subject}{Gross-Pitaevskii Equation} and further
allowing the system to evolve to phase coherence.
Similarly,~\cite{Kobayashi05a,Kobayashi05b}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.} imprinted a random phase on
the equilibrium wavefunction of a homogeneous box and evolved according to the GPE\index{subject}{Gross-Pitaevskii Equation}.
They introduced a dissipative term
which only acted on scales smaller than the healing length
to represent thermal dissipation in the system. They also obtained a decaying incompressible energy
spectrum which has the Kolmogorov power law over the inertial range.
In order to clarify the extent of this range, statistical steady turbulence
was created by a moving random potential which continuously injected
energy into the system at large scales; a damping term removed energy
at small length scales. They found that
the inertial range was slightly narrower for the continuously forced
turbulence because the moving potential sets the
energy containing range.
Further to these methods, \cite{White10}\index{authors}{White, A. C.}\index{authors}{Barenghi, C. F.}\index{authors}{Proukakis, N. P.}\index{authors}{Youd, A. J.}\index{authors}{Wacks, D. H.} imprinted a
staggered vortex array onto a
harmonically trapped BEC\index{subject}{Bose-Einstein condensates} and evolved the system according to
the GPE\index{subject}{Gross-Pitaevskii Equation} (see Fig.~\ref{white_tangle}).
In this work, they calculated the probability density function of the velocity
components and found that (in both 2D and 3D) it is not a Gaussian like
in ordinary turbulence, in agreement with experimental results obtained
in superfluid helium~\citep{Paoletti08}\index{authors}{Paoletti, M. S.}\index{authors}{Fisher, M. E.}\index{authors}{Sreenivasan, K. R.}\index{authors}{Lathrop, D. P.}.
This is an important observation which distinguishes quantum from classical turbulence.
\begin{figure}[h!]
\centering{
\includegraphics[scale = 0.5]{ajallen_fig3.png}
}
\caption{3D turbulent state of a harmonically trapped BEC. Condensate edge shown by blue shading, turbulent
vortex tangle\index{subject}{vortices!vortex tangle} shown in purple \citep{White10}\index{authors}{White, A.
C.}\index{authors}{Barenghi, C. F.}\index{authors}{Proukakis, N. P.}\index{authors}{Youd, A.
J.}\index{authors}{Wacks, D. H.}. Figure courtesy of Angela C. White.
}
\label{white_tangle}
\end{figure}
Using a similar method of imprinting,~\citep{Yepez09}\index{authors}{Yepez, J.}\index{authors}{Vahala, G.}\index{authors}{Vahala, L.}\index{authors}{Soe, M.} performed
impressively large simulations in a homogeneous box using a
quantum lattice gas algorithm (up to $5760^3$ grid points) and resolved
scales smaller than the vortex core radius.
Most of the methods of generating quantum
turbulence\index{subject}{quantum turbulence!turbulence} discussed so far
in this section have a common aim: the incompressible
energy spectrum of quantum turbulence\index{subject}{quantum turbulence!turbulence} after an initial, turbulent state has been set up
(with the exception of~\citep{Kobayashi05a,Kobayashi05b}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}\index{authors}{Kobayashi, M.}\index{authors}{Tsubota, M.}).
However, the group of Tsubota have also carried out
simulations where they dynamically create a vortex tangle\index{subject}{vortices!vortex tangle} by solving the DGPE\index{subject}{Gross-Pitaevskii Equation} with combined rotation
along two axis of a harmonically trapped BEC. They found that by changing the ratio between the rotation
frequencies in both directions, they could generate a
vortex lattice\index{subject}{vortices!lattice} or a more disordered array of
vortices\index{subject}{vortices!quantum vortices} which formed a vortex tangle\index{subject}{vortices!vortex tangle} in
which individual
vortices\index{subject}{vortices!quantum vortices} appear to be nucleated with no preferred direction.
They measured the incompressible energy spectrum and found it to be
consistent with Kolmogorov law.
Experimentally, a small vortex
tangle\index{subject}{vortices!vortex tangle} has been created
in a harmonically trapped BEC\index{subject}{Bose-Einstein condensates}
through the combination of rotation and an external oscillating
perturbation
by Bagnato's group~\citep{Henn09,Henn10,Seman11,Shiozaki11}.\index{authors}{Henn, E. A.
L.}\index{authors}{Seman, J. A.}\index{authors}{Roati, G.}\index{authors}{Magalh\~aes, K. M.
F.}\index{authors}{Bagnato, V. S.}\index{authors}{Henn, E. A. L.}\index{authors}{Seman, J. A.}
\index{authors}{Roati, G.}\index{authors}{Magalh\~aes, K. M. F.}\index{authors}{Bagnato, V. S.}
They noticed that upon expansion of the condensate
the usual inversion of aspect ratio of the gas~\citep{Mewes96,Castin96} did not happen.
\index{authors}{Mewes, M. O.}\index{authors}{Andrews, M. R.}\index{authors}{van Druten, N. J.}
\index{authors}{Kurn, D. M.}\index{authors}{Durfee, D. S.}\index{authors}{Ketterle, W.}
\index{authors}{Castin, Y.}\index{authors}{Dum, R.} This effect
could be a possible signature of the creation of a tangle of
vortices\index{subject}{vortices!vortex tangle}.
This brings us to the final stage of the evolution of turbulence, the decay.\\~\\
\noindent{\bf{(iii) The decay of the turbulence}}\\
In classical turbulence, the cascade of kinetic energy over the length scales
terminates at some very short scale where viscosity dissipates kinetic energy
into heat. The absence of viscosity in quantum fluids
means there must exist other mechanisms of energy dissipation. The most
likely is acoustic emission.
When two vortices\index{subject}{vortices!quantum vortices} reconnect,
some energy is lost in the form of sound. Reconnections
are also thought to create high frequency Kelvin waves on
vortices\index{subject}{vortices!quantum vortices}. It is thought that,
in superfluid helium,
Kelvin waves interact nonlinearly and create shorter and shorter waves,
until sound waves are emitted at high frequency~\citep{Vinen06}\index{authors}{Vinen, W.}. This
energy transfer is called the Kelvin wave cascade.
Experiments in superfluid helium show that, depending on the scale at which
energy is injected, the decay of the turbulence can be one of two
forms~\citep{Baggaley12}\index{authors}{Baggaley, A. W.}\index{authors}{Barenghi, C. F.}\index{authors}{Sergeev, Y. A.}:
\begin{itemize}
\item[(i)] \emph{`Semiclassical' or `Kolmogorov' turbulence:} The vortex tangle\index{subject}{vortices!vortex tangle}
seems polarised and
structured over many length scales. This type of turbulence is
generated when the forcing is at length scales larger than the average
intervortex spacing. In this regime, the vortex length $L$ decays
as $L \simeq t^{-3/2}$, which is consistent with the decay of a Kolmogorov
spectrum.
\item[(ii)] \emph{`Ultraquantum' or `Vinen' turbulence:} The scale of
the forcing is less than the intervortex spacing,
the vortex tangle\index{subject}{vortices!vortex tangle} seems random and possesses a single length scale, and the vortex
length decays as $L \simeq t^{-1}$~\citep{Walmsley08}\index{authors}{Walmsley, P. M.}\index{authors}{Golov, A. I.}.
\end{itemize}
How to measure the decay in a BEC is an open question.
\cite{White10}\index{authors}{White, A. C.}\index{authors}{Barenghi, C. F.}\index{authors}{Proukakis, N. P.}
\index{authors}{Youd, A. J.}\index{authors}{Wacks, D. H.} studied the decay of the turbulent
tangle\index{subject}{vortices!vortex tangle} by numerically monitoring the vortex length, $L$, over time.
They showed that the line length increases initially
as reconnections take place before decaying
over time. By further solving the dissipative GPE\index{subject}{Gross-Pitaevskii Equation}, they confirmed that
thermal dissipation leads to a faster decay of line length but could not clearly distinguish between
$L \simeq t^{-1}$ or $L \simeq e ^{-\alpha t}$ behaviour ($\alpha$ being some decay parameter).
However, is vortex linelength the best measure for this decay? Or can we again look to the
incompressible energy spectrum to visualise the decay of vortices\index{subject}{vortices!quantum vortices} and draw some conclusions
from both of these quantities? Furthermore, there is the question of how this is best achieved
experimentally. The images taken of condensate density are typically column-integrated over
the imaging direction which means that depth information becomes lost and an extraction of the
true 3D vortex line length is not possible. Just as the attenuation of second sound is used to
measure vortex length in helium, what surrogate measures of vortex line length are accessible
experimentally in BECs\index{subject}{Bose-Einstein condensates}? Such questions, both
fundamental and practical in nature, will provide a rich source of research in these systems in the future.
\section{Summary}
We have discussed weakly interacting, dilute atomic Bose-Einstein condensates\index{subject}{Bose-Einstein condensates} as tools for understanding the nature of
quantum turbulence\index{subject}{quantum turbulence!turbulence}, motivated largely by the huge degree of control they offer.
Even though the range of lengthscales excited in these systems is much less than in superfluid helium, the direct
Kolmogorov energy cascade has been predicted to exist and the regimes of turbulence accessible is vast and
interesting its own right. We have distinguished three distinct phases of quantum turbulence - its generation of
turbulence, its steady state and its decay - and briefly
reviewed work done in understanding these so far, whilst highlighting fundamental questions about each phase.
\ack
We acknowledge A. C. White for useful discussions and for providing us with an image for use in this article.
This work was supported by the grant EP/I019413/1 from the EPSRC.
\bibliographystyle{jfm2}
|
2,869,038,155,684 | arxiv | \section{Introduction}
LiDAR, radar, and camera are the three main sensory modalities employed by the perception system of an autonomous driving vehicle. Though LiDAR-based 3D object detection is very popular in high-level autonomy, its wide adoption is still limited by some unsolved issues. First, LiDAR is prone to adversarial conditions (e.g. rainy weather); second, current LiDAR systems still exhibit prohibitively high maintenance need and cost; third, the mass-production of LiDAR is not ready to meet the growing demand.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{imgs/front_page_showcase.png}
\end{center}
\caption[]{An illustration of the associations between radar detections (radar pins) and camera detections (2D bounding boxes). The context of the scene is illustrated in the top picture, with the image captured by the camera along with the detected bounding boxes and the projected radar pins (shown as numbered blue circles). The bottom picture highlights the association relationships (shown in red lines
) between radar pins and bounding boxes. The orange line in the middle denotes \textit{uncertain} association relationship, which will be explained later.}
\label{fig:showcase}
\end{figure}
An automotive millimeter-wave radar can also provide a certain level of geometrical information with relatively precise range and speed estimates. Moreover, as a widely-adopted sensor in automobiles for decades, radar is relatively robust, low-cost, and low-maintenance. The fusion between radar and camera combines radar's geometrical information and camera's appearance and semantic information, which is still the mainstream perception solution in many practical autonomous driving and assisted driving systems.
Traditionally, the radar-camera fusion is achieved by the combination of rule-based association algorithms and kinematic model based tracking. The key is data association between radar and camera detections. The noisy and sparse nature of radar detection and the depth ambiguity from a mono camera makes such association problem very challenging. Traditionally, the association process is hand-crafted based on minimizing certain distance metrics along with some heuristic rules. It not only requires a large amount of engineering and tuning but is also hard to adapt to ever-growing data.
An emerging solution is to use learning-based methods to replace the rule-based radar-camera fusion. The latest advances focus on direct 3D object detection with the combined radar and camera data as the input \cite{john2019rvnet, nabati2019rrpn, nobis2019deep}. These approaches all rely on LiDAR-based ground-truth to build the link between radar and camera. This is feasible on most public datasets such as nuScenes~\cite{nuscenes2019}, Waymo~\cite{sun2019scalability} etc. However, it cannot be applied to a large fleet of commercial autonomous vehicles, often equipped with only radars and cameras. In this study, we propose a scalable learning-based framework to associate radar and camera information without the costly LiDAR-based ground-truth.
Our goal is to find representations of radar and camera detection results, such that matched pairs are close and unmatched ones are far. We convert the detection results into image channels and combine them with the original image to feed into a convolutional neural networks (CNN), namely, \textit{AssociationNet}. Training is performed based on imperfect labels obtained from a traditional rule-based association method. A loss sampling mechanism is introduced to mitigate false labels. To further boost the performance, we guide the reasoning logic of AssociationNet by adding a novel ordinal loss. The proposed AssociationNet significantly outperforms the rule-based method through global reasoning about the actual scene.
Our main contributions are summarized as follows:
\begin{itemize}
\item We proposed a scalable learning-based radar-camera fusion framework without using ground-truth labels from LiDAR, which is suitable for building a low-cost, production-ready perception system in real autonomous driving applications.
\item We designed a loss sampling mechanism to alleviate the impact of the label noise, and also invented an ordinal loss to enforce critical association logic into the model for performance enhancement.
\item We developed a robust model via representation learning, which is capable of handling various challenging scenarios, and also outperforms the traditional rule-based algorithm by 11.6\% regarding the F1 score.
\end{itemize}
\section{Related Work}
\subsection{Sensor Fusion}
Traditionally, different sensory modules process their data separately. A downstream sensor fusion module augments the sensory outputs (typically detected objects) to form a more comprehensive understanding of the surroundings. Such an object-level fusion method is the mainstream approach~\cite{cho2014multi, kawasaki2004standard, langer1996fusing, garcia2012data, zhong2018camera} and is still widely used on many Advanced Driver Assistance Systems (ADAS). In object-level fusion, object detection is independently performed on each sensor, and the fusion algorithm combines such object detection results to create so-called global tracks for kinematic tracking~\cite{aeberhard2012track}.
Data association is the most critical and challenging task in object-level fusion. Precise association can easily lead to 3D object detection and multiple objects tracking solutions~\cite{aeberhard2012track, cesic2016radar}. Traditional approaches tend to manually craft various distance metrics to represent the similarities between different sensory outputs. Distance minimization~\cite{cho2014multi} and other heuristic rules are applied to find the associations. To handle the complexity and uncertainty, probabilistic models are also sometimes adopted in the association process~\cite{bar2009probabilistic}.
\subsection{Learning-Based Radar-Camera Fusion}
The learning-based radar-camera fusion algorithms can be primarily categorized into three groups, data-level fusion, feature-level fusion, and object-level fusion. The data-level fusion and feature-level fusion combine the radar and camera information at the early stage \cite{nobis2019deep, guo2018pedestrian} and the middle stage \cite{john2019rvnet, chang2020spatial, nabati2021centerfusion}, respectively, but both directly perform 3D object detection. Hence, they rely on LiDAR to provide ground-truth labels during training, which prohibits their usage to autonomous vehicles without LiDAR.
The learning-based object-level fusion remains under-explored due to the lesser amount of information contained in the detected results. In this study, our proposed method belongs to this category in that we focus on associating radar and camera detection results. Thus, our method is more compatible with traditional sensor fusion pipeline. On the other hand, our method also directly takes the raw camera and radar data for further performance enhancement.
\subsection{CNN for Heterogeneous Data}
The tremendous success of CNN on structured image data inspires its application to many other types of heterogeneous data, such as sensor parameters, point clouds, and association relationships between two groups of data \cite{newell2017pixels}. In order to get compatible with CNN, a popular approach is to adapt the heterogeneous data into a form of pseudo-images. Examples include encoding camera intrinsic into images with normalized coordinates and field of view maps \cite{Facil_2019_CVPR}, projecting radar data into the image plane to form new image channels \cite{chadwick2019distant, nobis2019deep}, and the various forms of projection-based LiDAR point-cloud representations \cite{wang2020pillar, Meyer_2019_CVPR}. We adopted a similar approach in this study to handle the heterogeneous radar and camera outputs.
\subsection{Representation Learning}
Representation learning has been considered as the key to understanding complex environments and problems~\cite{bengio2013representation, lecun2015deep, kolesnikov2019revisiting}. Representation learning has been widely used in many natural language processing tasks such as word embedding \cite{mikolov2013efficient}, and many computer vision tasks, such as image classification \cite{chen2020simple}, object detection \cite{Girshick_2014_CVPR}, and keypoint matching \cite{detone2018superpoint}. In this study, we aim at learning a vector in the high-dimensional feature space as the representation for each object in the scene, in order to establish the interactions between objects as well as enable global reasoning about the scene.
\section{Problem Formulation}
We use a front-facing camera and a millimeter-wave mid-range front radar for the proposed radar-camera fusion, while the approach can be easily generalized to 360 perception with proper hardware setups. The camera intrinsic and the extrinsics of both sensors are obtained through offline calibration. The radar and camera operate asynchronously at 20Hz and 10Hz, respectively. Both the radar and the camera are front-facing and have the field-of-views (FOVs) of 120 degrees and 52 degrees, respectively. The output of the camera sensor at each frame is an RGB image with the size of 1828 pixels (width) by 948 pixels (height), whereas the output of the radar sensor at each frame is a list of processed points with many attributes (conventionally referred to as radar pins). Since the radar used here performs internal clustering, each output radar pin is on the object level.\footnote{The proposed fusion technique also applies to lower level detection, e.g., radar locations.} There are several tens of radar pins per-frame depending on the actual scene and traffic. The attributes of each radar pin are listed in Table \ref{tab:radarpin_features}. There are two noteworthy characteristics of the radar pins. First, we only consume the 2D position information in the Bird's-Eye View (BEV) without the elevation angle, due to poor resolution and large measurement noise in the elevation dimension. Second, each radar pin either corresponds to a movable object (cars, cyclists, pedestrians etc.) or an interfering static structure such as a traffic sign, a street light, or a bridge.
In this study, we focus on associating 2D bounding boxes detected from a camera image to radar pins detected in the corresponding radar frame. With precise associations, many subsequent tasks like 3D object detection and tracking become much easier if not trivial.
\begin{table}[h]
\caption{The Features of Each Radar Pin}
\label{tab:radarpin_features}
\begin{center}
\renewcommand{\arraystretch}{2}
\begin{tabular}{llp{10cm}}
\hline\hline
\textbf{Feature} & \textbf{Explanation}\\[1pt]
\hline
object id & the id of the radar pin \\[2pt]
obstacle prob & \parbox{6cm}{the probability of the existence of an obstacle being detected by the radar pin} \\[2pt]
position x & \parbox{6cm}{the x coordinate of the position of the detected obstacle in radar frame} \\[2pt]
position y & \parbox{6cm}{the y coordinate of the position of the detected obstacle in radar frame} \\[2pt]
velocity x & \parbox{6cm}{the velocity of the detected obstacle along the x coordinate in radar frame} \\[2pt]
velocity y & \parbox{6cm}{the velocity of the detected obstacle along the y coordinate in radar frame} \\[2pt]
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{The Features of Each 2D Bounding Box}
\label{tab:bbox_features}
\begin{center}
\renewcommand{\arraystretch}{2.5}
\begin{tabular}{llp{10cm}}
\hline\hline
\textbf{Feature} & \textbf{Explanation}\\[1pt]
\hline
center x & \parbox{6cm}{the x coordinate in the image plane of the center of the bounding box} \\[2pt]
center y & \parbox{6cm}{the y coordinate in the image plane of the center of the bounding box} \\[2pt]
height & \parbox{6cm}{the height of the bounding box in the image plane} \\[2pt]
width & \parbox{6cm}{the width of the bounding box in the image plane} \\[2pt]
category & \parbox{6cm}{the category of the detected moving object, including \textit{sedan}, \textit{suv}, \textit{truck}, \textit{bus}, \textit{bicycle}, \textit{tricycle}, \textit{motorcycle}, \textit{person}, and \textit{unknown}} \\[2pt]
\hline \hline
\end{tabular}
\end{center}
\end{table}
\section{Methods}
Our proposed method mainly consists of a preprocessing step to align radar and camera data, a CNN-based deep representation learning network, AssociationNet, and a postprocessing step to extract representations and make associations. An overview of the method is shown in Fig. \ref{fig:overview} and details are explained the following sections.
\begin{figure*}
\begin{center}
\includegraphics[width=1.0\linewidth]{imgs/method_overview3.png}
\end{center}
\caption{An overview of AssociationNet. \textit{Process \textbf{a}} illustrates how the radar pins and 2D bounding boxes are first projected into the camera image plane and then produce a pseudo-image. \textit{Process \textbf{b}} illustrates how the final pseudo-image is composed by concatenating all the features of radar pins, bounding boxes, and the original RGB camera image. The pseudo-image is then fed into a neural network to learn high-level semantic representations. \textit{Process \textbf{c}} illustrates how the learned representation vectors for objects are finally extracted from the feature-map generated in the last layer of the neural network.}
\label{fig:overview}
\end{figure*}
\subsection{Radar and Camera Data Preprocessing}
Temporal and spatial alignment is performed in the preprocessing stage. For each camera frame, we look for the nearest radar frame to perform data alignment. We align the nearest radar frame to the time instant of the camera frame, by moving the radar pin locations forward/backward along the time axis under a constant velocity assumption. After the temporal alignment, the radar pins are further transformed from the radar coordinate to the camera coordinate using the known extrinsics. All the attributes of the aligned radar pins will be used in AssociationNet.
Each camera frame is first fed into a 2D object detection network to produce a list of 2D bounding boxes corresponding to the movable objects in the scene. The output attributes for each detected 2D bounding box are displayed in Table \ref{tab:bbox_features}. Though the network used in this study is an anchor-based RetinaNet ~\cite{lin2017focal} network, any 2D object detector will serve the purpose. After preprocessing, a list of temporally and spatially aligned radar pins and bounding boxes will be ready for association.
\subsection{Deep Association by Representation Learning}
We employ AssociationNet to learn a semantic representation (or a descriptor) of each radar pin and each bounding box. Under such representation, a pair of matched radar pin and bounding box will \say{look} similar, in the sense that the distance between the learned representations are small. An overview of the general process is shown in Fig.~\ref{fig:overview}.
To leverage the powerful CNN architecture, we project each radar pin and 2D bounding box into the image plane to generate a pseudo-image, with each attribute occupying an independent channel. Specifically, each bounding box is assigned to the pixel location of its center. Each radar pin is assigned to the pixel location which is obtained through projecting its 3D location into the image plane using the camera intrinsic. The process is illustrated in \textit{Process \textbf{a}} of the Fig. \ref{fig:overview}. Next, we concatenate the raw RGB camera image with the corresponding pseudo-image to incorporate the rich pixel-level information. AssociationNet is then applied to perform representation learning.
As shown in Fig. \ref{fig:network}, the network consists of a ResNet-50 \cite{he2016deep} as the backbone, a Feature Pyramid Network \cite{lin2017feature} for feature-map decoding, along with two extra layers to restore the size of the output feature-map to the original input size. The output feature-map contains the high-level semantic representations of radar pins and bounding boxes. As each radar pin or bounding box has a unique pixel location in the feature-map, we extract the representation vector of each on the output feature-map at its corresponding pixel location. The process is illustrated in \textit{Process \textbf{c}} of the Fig. \ref{fig:overview}.
The input pseudo-image contains seven radar pin channels, four bounding box channels, and three raw camera image \textit{RGB} channels. The radar pin channels include \textit{object-id}, \textit{obstacle-prob}, \textit{position-x}, \textit{position-y}, \textit{velocity-x}, \textit{velocity-y}\footnote{Positions and velocities used here are under camera coordinate, as it is after the spatial alignment step in the preprocessing.}, and a \textit{heatmap} to indicate the projected pixel location. The bounding box channels include \textit{height}, \textit{width}, \textit{category}, and also a \textit{heatmap} to indicate the pixel location. The output feature-map contains 128 channels, resulting in the dimension of the representation vector to be 64 for each radar pin and bounding box.
\begin{figure}
\begin{center}
\includegraphics[width=1.00\linewidth]{imgs/network_architecture3.png}
\end{center}
\caption{The architecture of the neural network. It consists of a ResNet-50 as the backbone, a feature pyramid for feature decoding, and two extra layers to restore the feature-map size. The feature-map in the last layer will be used to extract the representation vectors as shown in the \textit{Process \textbf{c}} of the Fig. \ref{fig:overview}}
\label{fig:network}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.00\linewidth]{imgs/inference_diagram3.png}
\end{center}
\caption{An overview of the process of obtaining final associations from the learned representation vectors.}
\label{fig:infer}
\end{figure}
The obtained representation vector captures the semantic meaning of each radar pin and each bounding box in a high dimension space. If a radar pin and a bounding box come from the same object in real world, we treat the pair of radar pin and bounding box as a positive sample, otherwise it is considered as a negative sample. We try to minimize the distance between the representation vectors of any positive sample and maximize the distance between the representation vectors of any negative sample. Based on such logic, we design loss functions according to the association ground-truth labels. We pull together the representation vectors of positive samples with the following pull loss:
\begin{equation}
\label{eq:pull}
L_{pull} = \frac{1}{n_{pos}} \sum_{(i_1, i_2) \in \mathbb{POS}} \max(0, \lVert h_{i_1} - h_{i_2} \rVert - m_1)
\end{equation}
And we push apart the representation vectors of negative samples with the following push loss:
\begin{equation}
\label{eq:push}
L_{push} = \frac{1}{n_{neg}} \sum_{(i_1, i_2) \in \mathbb{NEG}} \max(0, m_2 - \lVert h_{i_1} - h_{i_2} \rVert)
\end{equation}
Here $\mathbb{POS}$ and $\mathbb{NGE}$ are the set of positive samples and the set of negative samples, respectively in each frame; $n_{pos}$ and $n_{neg}$ are the total number of associations in $\mathbb{POS}$ and $\mathbb{NGE}$ respectively; $(i_1, i_2)$ denotes the $i^{th}$ association pair consisting of radar pin $i_1$ and bounding box $i_2$; $h_{i_1}$ and $h_{i_2}$ denotes the learned representation vectors; and $m_1$ and $m_2$ are the thresholds for the desired distances of representations among positive associations and negative associations, which were preset to be 2.0 and 8.0 in our experiments.
During inference, we calculate the Euclidean distance between the representation vectors of all possible radar-pin-bounding-box pairs. If the distance falls below a certain threshold, the radar pin and the bounding box will be considered as a successful association. More details of the inference process will be explained later.
\subsubsection{Loss Sampling}
The association labels used for supervising the learning process is ultimately from the traditional rule-based method, which are far from 100\% accurate and contain some noise. To mitigate the impact of the inaccurate labels, we first purify the labels by applying some simple filters to remove low-confidence associations, which increases the precision in the remaining association labels at the cost of the undermined recall. During the push loss calculation in the training of AssociationNet, instead of exhausting all negative pairs (a pair of a radar pin and a bounding box that is not present in the association labels), we only sample a fraction of those to be used for push loss calculation to alleviate pushing apart positive pairs by mistake. The number of sampled negative pairs is set to be equal to the number of positive ones at each frame.
\subsubsection{Ordinal Loss}
One particular kind of error made by AssociationNet is that it could violate the simple ordinal rule, i.e., a farther radar pin associates to a closer bounding box and a closer radar pin associates to a farther bounding box. To solve this issue, an ordinal loss is introduced.
Denote the $y$ coordinate of bounding box $i$'s bottom edge as $y_{\text{max}}^i$ and the corresponding depth in 3D world as $d^i$. For any two bounding boxes on the same image we have the property:
\begin{equation}
\label{eq:y2d}
y_{\text{max}}^i > y_{\text{max}}^j \iff d^i > d^j
\end{equation}
The ordering of the objects in the 3D world can be interpreted as the relative vertical ordering of the bottom edges of the corresponding bounding boxes. Hence, we design an additional ordinal loss to enforce self-consistency according to the ordinal rule, which is written as:
\begin{dmath}
\label{eq:order}
L_{ord} = \frac{2}{\widehat{n_{pos}} \cdot (\widehat{n_{pos}} - 1)} \cdot \\
\sum_{\substack{
(i_1, i_2) \in \widehat{\mathbb{POS}} \\
(j_1, j_2) \in \widehat{\mathbb{POS}}
}}
\sigma(- (d^{i_1} - d^{j_1}) \cdot (y_{\text{max}}^{i_2} - y_{\text{max}}^{j_2}) ),
\end{dmath}
where $\widehat{\mathbb{POS}}$ denotes the set of predicted positive associations and $\widehat{n_{pos}}$ is the size of the set; $i_1$ and $i_2$ represent the associated radar pin and bounding box of the $i^{th}$ predicted association in $\widehat{\mathbb{POS}}$; similarly, $j_1$ and $j_2$ represent the associated radar pin and bounding box of the $j^{th}$ predicted association in $\widehat{\mathbb{POS}}$; $d^*$ represents the depth of a radar pin in camera coordinate, and $y_{\text{max}}^*$ represents the $y$ coordinate of a bounding box's bottom edge; $\sigma$ is the sigmoid function to smooth the loss values.
Finally, the total loss is calculated as:
\begin{equation}
\label{eq:tot}
L_{tot} = L_{pull} + L_{push} + w_{ord} \cdot L_{ord},
\end{equation}
where the $w_{ord}$ is the adjustable weight to balance losses.
\subsection{Training and Inference}
The AssociationNet was trained with a batch size of 48 frames at four NVIDIA GeForce RTX 2080 Ti GPUs. The SGD optimizer was used for training at a total of 10K iterations. The learning rate was set to be $10^{-4} $ initially, and then was decreased by a factor of 10 at the end of 8K iterations and 9K iterations, respectively.
At the inference time, the representation vectors for all radar pins and bounding boxes are first predicted using the trained model. An affinity matrix is then calculated, where each matrix element corresponds to the distance between the representations of a radar pin and a bounding box. In reality, each bounding box may be associated with multiple radar pin\footnote{This is usually the case where the corresponding vehicles are of large sizes, such as trailer trucks and buses.}, while each radar pin can only match to at most one bounding box. As a result, we devise each radar pin to get associated to the bounding box with the smallest distance in the affinity matrix. Lastly, the improbable associations with a distance larger than a threshold are filtered out, which usually consists of radar pins from interfering static objects. The whole inference process is summarized in Fig.~\ref{fig:infer}.
\subsection{Evaluation}
The predicted associations are compared against human-annotated ground-truth associations in the test dataset. We use precision, recall, and F1 score as the metrics for evaluating the performance.
In some very complicated scenes, correctly associating all radar pins and bounding boxes is very challenging even for human annotators. Therefore, we mark those plausible but dubious associations as \say{\textit{uncertain}} in the evaluation process. An example is shown in Fig. \ref{fig:bev_demo}. For those \say{\textit{uncertain}} associations, they are counted as neither positive nor negative associations, which will be excluded from both true and false positive predictions.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=0.9\linewidth]{imgs/example_1582824111473599054_bev_wi_legend5.png}
\end{center}
\caption[]{An illustration of radar pins, bounding boxes, and their association relationships under BEV perspective. This BEV image corresponds to the same scene as displayed in Fig. \ref{fig:showcase}. Each grid in the image represents a 10-meter-by-10-meter square in physical space. The bounding boxes are represented as solid cycles in this image. The location of a bounding box is estimated by the Inverse Projective Mapping (IPM) method from the bounding box's center, to provide a rough reference for its real 3D location. A truncated frustum accompanying each bounding box is also plotted, for better assisting human curators to determine the association relationships\footnotemark.}
\label{fig:bev_demo}
\end{figure*}
\begin{figure*}[h]
\centering
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[width=1\linewidth]{imgs/vis_for_paper_1582827956419278932.png}
\end{subfigure}
\begin{subfigure}[b]{1\textwidth}
\centering
\includegraphics[width=1\linewidth]{imgs/vis_for_paper_1582822828792179750.png}
\end{subfigure}
\caption{Examples of AssociationNet predictions. Here, the red solid lines represent the true-positive associations; and the pink solid lines represent predicted positive associations but labeled as \textit{uncertain} in the ground-truth. In the second example, the added green lines represent the false-positive predictions; and the added black lines represent the false-negative predictions.
Also note that each bounding box on the left corresponds to a solid circle with the same color on the right.
}
\label{fig:result_demo}
\end{figure*}
\section{Experiments and Discussion}
\subsection{Dataset}
The AssociationNet was trained and evaluated on an in-house dataset with 12 driving sequences collected by a testing fleet, which consists of 14.8 hours of driving in various driving scenarios, including highway, urban, and city roads. The radar and camera were synchronized at 10 Hz initially and further downsampled to 2 Hz, in order to reduce the temporal correlation among adjacent frames. Eleven sequences out of the twelve were used for training with the other one left for the test. Therefore, there are 104,314 synchronized radar and camera frames in the training dataset, and 2,714 in the test dataset. For the training data, the association labels were generated by a traditional rule-based algorithm with additional filtering to increase the precision. For the test data, we manually curated the labels with human annotators to obtain high-quality ground-truth.
\subsection{Effect of Loss Sampling}
We studied the effect of loss sampling on the AssociationNet's performance. Experiments were conducted with \textit{no sampling} (meaning that all the negative pairs present in the label are used for push loss calculation), and loss sampling with different sampling ratios. The sampling ratio is defined as the ratio between the number of positive pairs and the number of negative pairs at each frame. The result is shown in Table \ref{tab:effect_sample}. We can see that the best sampling ratio is 1:1 with the loss sampling mechanism, which boosts the performance by 1.1\% in terms of the F1 score.
\begin{table}[]
\caption{The Effect of Loss Sampling}
\label{tab:effect_sample}
\begin{center}
\begin{tabular}{c|c}
\hline\hline
\textbf{Sample Ratio} & \textbf{Performance} \\[1pt]
\textbf{ } & Precision / Recall / F1 \\[1pt]
\hline
\textit{no sampling} & 0.896 / 0.925 / 0.911 \\[1pt]
1:2 & 0.901 / 0.931 / 0.916 \\[1pt]
1:1 & 0.906 / 0.939 / 0.922 \\[1pt]
2:1 & 0.899 / 0.933 / 0.915 \\[1pt]
3:1 & 0.899 / 0.929 / 0.914 \\[1pt]
\hline\hline
\end{tabular}
\end{center}
\end{table}
\footnotetext{The frustum is calculated also by the IPM method, from the two side-edges of each bounding box. According to projective geometry, the real object detected by a bounding box has to be within the bounding box's frustum, and hence the possibly matched radar pins as well. We truncated the frustums for the ease of visualizing. The widths of each frustum at the truncated positions are set to be one meter and five meters, respectively. As the physical width of a vehicle is most likely to be within the range, the possibly matched radar pins also tend to lie within the truncated frustum.}
\subsection{Effect of Ordinal Loss}
The effect of the ordinal loss is shown in Table \ref{tab:effect_ord}. The ordinal loss can facilitate both precision and recall to some degree. With the optimal loss weight, the performance is boosted by 1.8\% in terms of the F1 score.
\begin{table}[]
\caption{The Effect of Ordinal Loss}
\label{tab:effect_ord}
\begin{center}
\begin{tabular}{c|c}
\hline\hline
\textbf{Loss Weight} & \textbf{Performance} \\[1pt]
$w_{ord}$ & Precision / Recall / F1 \\[1pt]
\hline
0.0 & 0.897 / 0.912 / 0.904 \\[1pt]
0.5 & 0.897 / 0.923 / 0.910 \\[1pt]
1.0 & 0.899 / 0.931 / 0.915 \\[1pt]
2.0 & 0.906 / 0.939 / 0.922 \\[1pt]
5.0 & 0.889 / 0.918 / 0.903 \\[1pt]
\hline\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Comparison with Rule-Based Algorithm}
We compared the performance of AssociationNet with the traditional rule-based algorithm, as shown in Table \ref{tab:compare}. Notably, though the traditional rule-based algorithm was used to generate association labels to supervise the training of AssociationNet, AssociationNet significantly outperforms the rule-based alternative. This demonstrates the inherent robustness of learning-based algorithms in handling complex scenarios.
\begin{table}[]
\caption{Comparison with Rule-based Algorithm}
\label{tab:compare}
\begin{center}
\begin{tabular}{c|c}
\hline\hline
\textbf{Algorithm} & \textbf{Performance} \\[1pt]
\textbf{ } & Precision / Recall / F1 \\[1pt]
\hline
Rule-based & 0.890 / 0.736 / 0.806 \\[1pt]
Learning-based & 0.906 / 0.939 / 0.922 \\[1pt]
\hline\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Visualization}
Examples of the predicted associations are shown in Fig. \ref{fig:result_demo}. Despite multiple big trucks present in both examples, AssociationNet correctly predicted their associations, which demonstrates the robustness of the algorithm. On the other hand, in the second example, there are two bounding boxes incorrectly associated, with one bounding box having no predicted associations and the other associated to a wrong radar pin. The two bounding boxes correspond to vehicles at the very far range. The mistakes are largely due to the small sizes of the objects in the camera image and also the heavy occlusions.
\section{Conclusion}
In this work, we developed a scalable learning-based radar-camera fusion algorithm, without using LiDAR for ground-truth labels generation. Such a solution has many practical merits at the current technological stage, including low cost, low maintenance, high reliability, and more importantly, readiness for mass-production. We employed deep representation learning to tackle the challenging association problem, with the benefits of enabled feature-level interaction and global reasoning. We also designed a loss sampling mechanism and a novel ordinal loss to mitigate the impact of label noise and enforce critical human logic into the learning process. Although imperfect labels generated by a traditional rule-based algorithm were used to train the network, our proposed algorithm outperforms the rule-based teacher by 11.6\% in terms of the F1 score.
{\small
\bibliographystyle{ieee_fullname}
|
2,869,038,155,685 | arxiv | \section{Introduction}
The study of field theory in Anti-de Sitter (AdS) spaces, which
topologically are hyperbolic maximally symmetric spaces, has been
revived over the past two years following the so-called Maldacena
conjecture relating type IIB supergravity on AdS$_5$ $\times$ $S^5$
with $\mathcal{N}=4$, $U(N)$ Super Yang Mills theory in four
dimensions.
More than a decade ago, the calculation of correlation functions in
maximally symmetric spaces using only intrinsic geometric objects was
presented in a series of papers starting with
\cite{Burgess85,Allen86a,Allen86b}. In one
of them \cite{Allen86b}, Green's functions for two-component spinors
in maximally symmetric four-spaces were considered using the $SL(2,R)$
formulation. To our knowledge, this analysis has not been extended
since to Dirac spinors in other space-time dimensions. However, it
should be mentioned
that spinor Green's functions in AdS spaces have been considered and
calculated by other means in the context of the AdS/CFT correspondence
\cite{Kawano99,Rashkov99}.
In the present paper, we present an intrinsically geometric approach
to spinor Green's functions in maximally symmetric spaces. In
section~\ref{pprop}, we introduce the spinor parallel propagator for
maximally symmetric spaces of dimension $n$ and find its covariant
derivatives. Then, in section~\ref{sprop}, we calculate the spinor
Green's functions for the spaces $\mathbb{R}^n$, $S^n$ and
$H^n$. Finally, section~\ref{conc} contains conclusions.
In the remainder of this section, we would like to review the
elementary maximally symmetric bi-tensors, which
have been discussed in detail by Allen and Jacobsen
\cite{Allen86a}.
Consider a maximally symmetric space of dimension $n$ with constant
scalar curvature $n(n-1)/R^2$. For the space $S^n$, the radius $R$ is
real and positive, whereas for the hyperbolic space $H^n$, $R=il$ with
$l$ positive, and in the flat case, $\mathbb{R}^n$, $R=\infty$.
Consider further two points $x$ and $x'$, which can be connected
uniquely by a shortest geodesic. Let $\mu$ be the proper geodesic
distance along this shortest geodesic between $x$ and $x'$.
Then, the vectors
\begin{equation}
\label{ndef}
n_\nu(x,x') = D_\nu \mu(x,x') \quad \text{and} \quad
n_{\nu'}(x,x') = D_{\nu'} \mu(x,x')
\end{equation}
are tangent to the geodesic and have unit length. Furthermore, denote
by $g^\mu_{\;\nu'}(x,x')$ the vector parallel propagator along the
geodesic. Notice the relation $n^{\nu'} = -g^{\nu'}_{\;\mu} n^\mu$.
These elementary maximally bi-tensors $n^\mu$, $n^{\mu'}$ and
$g^\mu_{\;\nu'}$ satisfy the following properties:
\begin{subequations}
\begin{align}
\label{dn}
D_\mu n_\nu &= A(g_{\mu\nu} -n_\mu n_\nu)\\
\label{dnprime}
D_{\mu'} n_\nu &= C(g_{\mu'\nu} +n_{\mu'} n_\nu)\\
\label{dg}
D_\mu g_{\nu\lambda'} &= -(A+C) (g_{\mu\nu} n_{\lambda'} +
g_{\mu\lambda'} n_\nu),
\end{align}
\end{subequations}
where $A$ and $C$ are functions of the geodesic distance $\mu$ and are
given by
\begin{equation}
\label{AC}
A = \frac1R \cot \frac{\mu}R \quad \text{and} \quad
C = -\frac1{R\sin(\mu/R)}.
\end{equation}
Therefore, they satisfy the relations
\begin{equation}
\label{ACrel}
dA/d\mu =-C^2, \quad dC/d\mu =-AC \quad \text{and} \quad C^2-A^2
=1/R^2.
\end{equation}
Finally, our convention for covariant gamma matrices is
$\{\Gamma^\mu,\Gamma^\nu\} =2 g^{\mu\nu}$.
\section{Spinor Parallel Propagator}
\label{pprop}
To start, consider a bi-spinor $\Lambda(x',x)^{\alpha'}_{\;\beta}$,
which acts as parallel propagator for Dirac spinors in a maximally
symmetric space-time, i.e.\ it performs the parallel transport
\[{\Psi'}(x')^{\alpha'} = \Lambda(x',x)^{\alpha'}_{\;\beta} \Psi(x)^\beta.\]
The spinor parallel propagator $\Lambda(x',x)$ can be uniquely defined
for any space-time dimension by the following properties:
\begin{subequations}
\begin{align}
\label{lambdadef1}
\Lambda(x',x) &= [\Lambda(x,x')]^{-1},\\
\label{lambdadef2}
\Gamma^{\nu'}(x') &= \Lambda(x',x) \Gamma^\mu(x) \Lambda(x,x')
g^{\nu'}_\mu(x',x),\\
\label{lambdadef3}
n^\mu D_\mu \Lambda(x,x') &=0.
\end{align}
\end{subequations}
Eqn.\ \eqref{lambdadef1} implies that $\Lambda(x,x)^{\alpha'}_\beta =
\delta^{\alpha'}_\beta$, whereas eqn.\ \eqref{lambdadef2} conveniently
formulates the parallel transport of the covariant gamma
matrices. Finally, eqn.\ \eqref{lambdadef3} says that $\Lambda(x,x')$
is covariantly constant along the geodesic of parallel transport.
We would like to evaluate now a particular property of
$\Lambda(x,x')$, namely its covariant derivative. Therefore, combine
eqns.\ \eqref{lambdadef1} and \eqref{lambdadef2} to
\begin{equation}
\label{gammaprop}
\Gamma^\nu \Lambda(x,x') = \Lambda(x,x') \Gamma^{\mu'} g^\nu_{\mu'}
\end{equation}
and differentiate covariantly with respect to $x$ to obtain
\begin{equation}
\label{defdiff}
\Gamma^\nu D_\lambda \Lambda(x,x') = D_\lambda \Lambda(x,x') \Gamma^{\mu'}
g^\nu_{\mu'} - (A+C) \Lambda(x,x') \Gamma^{\mu'} (\delta^\nu_\lambda
n_{\mu'} + g_{\lambda\mu'} n^\nu),
\end{equation}
where we have used the property \eqref{dg} of the vector parallel
propagator. Now, use eqn.\ \eqref{gammaprop} for the second term on
the right hand side of eqn.\ \eqref{defdiff} and multiply with
$\Gamma^\lambda$, which yields
\begin{equation}
\label{defdiff2}
2 D^\nu \Lambda(x,x') - \Gamma^\nu \slashD \Lambda(x,x') = \slashD
\Lambda(x,x') \Gamma^{\mu'}g^\nu_{\mu'} + (A+C) (\Gamma^\nu
\Gamma^\rho n_\rho - n n^\nu) \Lambda(x,x').
\end{equation}
Thus, a multiplication with $\Gamma_\nu$ leads to
\[ (2-n) \slashD \Lambda(x,x') = \Gamma_\nu \slashD \Lambda(x,x')
\Gamma^{\mu'} g^\nu_{\mu'},\]
the solution of which is
\begin{equation}
\label{slashDlambda}
\slashD \Lambda(x,x') = B n_\mu \Gamma^\mu \Lambda(x,x'),
\end{equation}
where $B$ is some function of the geodesic distance $\mu$.
Then, substituting eqn.\ \eqref{slashDlambda} into eqn.\
\eqref{defdiff2} yields
\[ 2 D^\nu \Lambda(x,x') = 2 B n^\nu \Lambda(x,x')
+ (A+C) (\Gamma^\nu \Gamma^\rho n_\rho - n n^\nu) \Lambda(x,x').\]
Moreover, multiplying this with $n_\nu$ and using eqn.\
\eqref{lambdadef3} one determines $B$ to be
\[ B= \frac12 (n-1)(A+C).\]
Therefore, one finally obtains
\begin{equation}
\label{dlambda}
D_\mu \Lambda(x,x') = \frac12 (A+C) \left(\Gamma_\mu \Gamma^\nu n_\nu
-n_\mu\right) \Lambda(x,x').
\end{equation}
For completeness, we also give the expression for $D_{\mu'}
\Lambda(x,x')$. It is easily obtained from eqn.\ \eqref{dlambda} using
eqn.\ \eqref{lambdadef1} and is given by
\begin{equation}
\label{dlambda2}
D_{\mu'} \Lambda(x,x') = -\frac12 (A+C) \Lambda(x,x') \left(\Gamma_{\mu'}
\Gamma^{\nu'} n_{\nu'} - n_{\mu'}\right).
\end{equation}
\section{Spinor Green's Function}
\label{sprop}
Using the spinor parallel propagator $\Lambda(x,x')$ calculated in
section~\ref{pprop}, we would now like to find the spinor Green's
function $S(x,x')$ satisfying
\begin{equation}
\label{greendef}
\left[(\slashD -m) S(x,x')\right]^\alpha_{\;\beta'} =
\frac{\delta(x-x')}{\sqrt{g(x)}} \delta^\alpha_{\beta'}.
\end{equation}
Here, we have written the indices explicitly in order to emphasize
that this is a bi-spinor equation. We shall henceforth omit the
indices.
Now, we make the general ansatz
\begin{equation}
\label{greenans}
S(x,x') = \left[ \alpha(\mu)+\beta(\mu) n_\nu \Gamma^\nu \right]
\Lambda(x,x'),
\end{equation}
where $\alpha$ and $\beta$ are functions of the geodesic distance
$\mu$ still to be determined. We substitute the ansatz
\eqref{greenans} into eqn.\ \eqref{greendef} and, after using eqn.\
\eqref{dlambda}, obtain the two coupled differential equations
\begin{align}
\label{sys1}
\beta' +\frac12(n-1)(A-C) \beta -m\alpha &=
\frac{\delta(x-x')}{\sqrt{g(x)}},\\
\label{sys2}
\alpha' +\frac12(n-1)(A+C)\alpha - m\beta &=0,
\end{align}
where the prime denotes differentiation with respect to $\mu$.
In order to proceed, multiply eqn.\ \eqref{sys1}
with $m$ and substitute $m\beta$ from eqn.\ \eqref{sys2}. One finds
\begin{equation}
\label{alphaeq}
\alpha''+(n-1) A\alpha' -\frac12(n-1)C(A+C) \alpha -\left[
\frac{(n-1)^2}{4R^2} +m^2 \right] \alpha
= m \frac{\delta(x-x')}{\sqrt{g(x)}},
\end{equation}
where eqn.\ \eqref{ACrel} has been used. We shall solve equation
\eqref{alphaeq} separately for the spaces $\mathbb{R}^n$, $S^n$ and
$H^n$.
\subsection{Green's Function for \boldmath$\mathbb{R}^n$}
For $\mathbb{R}^n$, we have $A=-C=1/\mu$, $R=\infty$ and
$\mu=|x-x'|$. Thus, eqn.\ \eqref{alphaeq} becomes
\begin{equation}
\label{alphaeqR}
\alpha''+\frac{n-1}\mu \alpha' -m^2\alpha
= m\, \delta(x-x').
\end{equation}
The solution to eqn.\ \eqref{alphaeqR} is
\begin{equation}
\label{alphaR}
\alpha(\mu) = - \left(\frac{m}{2\pi}\right)^\frac{n}2
\mu^{1-\frac{n}2}\, \mathrm{K}_{\frac{n}2-1}(m\mu),
\end{equation}
where the functional form was obtained by solving eqn.\
\eqref{alphaeqR} for $\mu\ne0$, and the constant was found by
matching the singularity. Furthermore, one finds from eqn.\
\eqref{sys2} $m\beta =\alpha'$, i.e.\ $n_\nu \beta = \partial_\nu
\alpha/m$, so that the final result for the spinor Green's function
in $\mathbb{R}^n$ is
\begin{equation}
\label{greenR}
S(x,x') = - \frac1m \left(\frac{m}{2\pi}\right)^\frac{n}2
(\slashdel + m) \mu^{1-\frac{n}2}\,
\mathrm{K}_{\frac{n}2-1}(m\mu).
\end{equation}
Upon Fourier transforming it, one obtains the more familiar expression
\begin{equation}
\label{greenRfour}
S(x,x') = - (\slashdel + m)
\int\frac{d^n\!k}{(2\pi)^n}\; \mathrm{e}^{-i
k\cdot(x-x')} \frac1{k^2+m^2}.
\end{equation}
\subsection{Green's Function for \boldmath$S^n$}
In order to solve eqn.\ \eqref{alphaeq}, we consider first $x\ne x'$
and make the substitution
\begin{equation}
\label{zsub}
z = \cos^2 \frac{\mu}{2R}.
\end{equation}
This yields the differential equation
\begin{equation}
\label{alphaeqS}
\left[ z(1-z) \frac{d^2}{dz^2} + \frac{n}2 (1-2z)\frac{d}{dz} -
\frac{(n-1)^2}4 -m^2 R^2 - \frac{n-1}{4z} \right] \alpha(z) =0.
\end{equation}
Then, writing $\alpha(z)=\sqrt{z}\gamma(z)$, one obtains a
hypergeometric equation for $\gamma$,
\begin{subequations}
\label{gammaeq}
\begin{gather}
\label{gammaeq1}
H(a,b;c;z) \gamma(z) = 0,\\
\intertext{where}
\label{H}
H(a,b;c;z) = z(1-z) \frac{d^2}{dz^2} + [c-(a+b+1)z] \frac{d}{dz}
-ab\\
\intertext{is the hypergeometric operator, and its parameters are}
\label{abc}
a = \frac{n}2 -i|m|R, \quad b = \frac{n}2+i|m|R, \quad c =
\frac{n}2+1.
\end{gather}
\end{subequations}
The solution of eqn.\ \eqref{gammaeq} which is singular at $z=1$
is \cite{Gradshteyn}
\begin{equation}
\label{gammaS}
\gamma(z) = \lambda \,\mathrm{F}(a,b;c;z) = \lambda
\,\mathrm{F}(n/2-i|m|R,n/2+i|m|R;n/2+1;z),
\end{equation}
where $\lambda$ is a proportionality constant. Therefore, $\alpha(z)$
is
\begin{equation}
\label{alphaS}
\alpha(z) = \lambda \sqrt{z}\,\mathrm{F}(n/2-i|m|R,n/2+i|m|R;n/2+1;z).
\end{equation}
We can now determine the constants $\lambda$ by matching
the singularity in eqn.\ \eqref{alphaeq}. This is equivalent to
demanding the singularity of $\alpha$ at $\mu=0$ to have the same
strength as in the case of $\mathbb{R}^n$. One finds from eqn.\
\eqref{alphaS}
\[ \alpha \to \lambda
\frac{\Gamma(n/2+1)\Gamma(n/2-1)}{\Gamma(n/2-i|m|R)\Gamma(n/2+i|m|R)}
\left(\frac{\mu}{2R}\right)^{2-n},\]
whereas in $\mathbb{R}^n$ we have, from eqn.\ \eqref{alphaR},
\begin{equation}
\label{Rasym}
\alpha \to - \frac{m}4 \Gamma(n/2-1) \pi^{-n/2} \mu^{2-n}.
\end{equation}
Comparing these two expressions we find
\begin{equation}
\lambda = - m
\frac{\Gamma(n/2-i|m|R)\Gamma(n/2+i|m|R)}{\Gamma(n/2+1) \pi^{n/2}
2^n} R^{2-n}.
\end{equation}
Finally, one can calculate $\beta$ from eqn.\ \eqref{sys2}, which yields
\begin{align}
\notag
\beta(z) &= -\frac1m \left[ \frac1R \sqrt{z(1-z)} \frac{d}{dz} +
\frac{n-1}{2R} \sqrt{\frac{1-z}z} \right] \alpha(z)\\
\label{betaS}
&= -\frac{\lambda}{mR} \sqrt{1-z} \left[ z \,
\mathrm{F}(n/2+1-i|m|R,n/2+1+i|m|R;n/2+2;z) \phantom{\frac{n}2}
\right. \\ \notag &\quad
+ \left.\frac{n}2 \, \mathrm{F}(n/2-i|m|R,n/2+i|m|R;n/2+1;z) \right].
\end{align}
It should be noticed that $\beta$ has a finite $m\to0$ limit, whereas
$\alpha$ vanishes.
\subsection{Green's Function for \boldmath$H^n$}
For $H^n$, we can start with eqn.\ \eqref{gammaeq} and set $R=il$,
i.e.\ we have to solve
\begin{subequations}
\label{gammaeqH}
\begin{gather}
\label{gammaeqH1}
H(a,b;c;z) \gamma(z) = 0\\
\intertext{with}
a = \frac{n}2 + |m|l, \quad b = \frac{n}2-|m|l, \quad c =
\frac{n}2+1.
\end{gather}
\end{subequations}
There are two solutions to eqn.\ \eqref{gammaeqH} which behave
asymptotically like a power of $z$ for $z\to\infty$. These are
\begin{equation}
\label{gammaH}
\gamma_\pm(z) = \lambda_\pm z^{-\left(\frac{n}2 \pm |m|l\right)}
\,\mathrm{F} \left(\frac{n}2\pm |m|l, \pm |m|l; 1\pm 2|m|l; \frac1z
\right),
\end{equation}
where $\lambda_\pm$ are constants. The choice of the minus sign is not
always possible. In fact, for $1-2|m|l=0,-1,-2,\ldots$ the
hypergeometric series is indeterminate. Thus, we shall include the
solution with the minus sign only, if $|m|l<1/2$. Hence, we have
two solutions for $\alpha$,
\begin{equation}
\label{alphaH}
\alpha_\pm(z) = \lambda_\pm z^{-\left(\frac{n-1}2 \pm |m|l\right)}
\,\mathrm{F} \left(\frac{n}2\pm |m|l, \pm |m|l; 1\pm 2|m|l; \frac1z
\right),
\end{equation}
and we can now proceed to determine the constants $\lambda_\pm$ in a
similar fashion as in the $S^n$ case.
From eqn.\ \eqref{alphaH} we find for $\mu\to0$
\[ \alpha \to \lambda_\pm \left(\frac\mu{2l}\right)^{2-n}
\frac{\Gamma(1\pm2|m|l) \Gamma(n/2-1)}{\Gamma(n/2\pm|m|l)
\Gamma(\pm|m|l)}.\]
Comparing this expression to the $\mathbb{R}^n$ case, eqn.\
\eqref{Rasym}, we find
\begin{equation}
\label{lpm}
\lambda_\pm = \mp \sgn m\, 2^{-(n\pm2|m|l)}
l^{1-n}\frac{\Gamma(n/2\pm|m|l)}{\pi^{(n-1)/2} \Gamma(1/2\pm|m|l)},
\end{equation}
where the doubling formula for Gamma functions has been used.
Finally, let us calculate $\beta$ from eqn.\ \eqref{sys2}. Using a
recursion formula for hypergeometric functions we find
\begin{align}
\notag
\beta_\pm(z) &= \frac1m \left[ \frac1l \sqrt{z(z-1)} \frac{d}{dz} +
\frac{n-1}{2l} \sqrt{\frac{z-1}z} \right] \alpha_\pm (z) \\
\label{betaH}
&= \mp \sgn m\, \lambda_\pm \sqrt{z-1}\, z^{-\left(\frac{n}2 \pm
|m|l\right)} \,\mathrm{F} \left(\frac{n}2\pm |m|l, 1\pm |m|l; 1\pm
2|m|l; \frac1z \right).
\end{align}
It is interesting to note that in the limit $m\to0$ the functions
$\beta_+$ and $\beta_-$ become identical, whereas $\alpha_+$ and
$\alpha_-$ do not, but differ in their signs. The reason is, of
course, that, for $m=0$, eqns.\ \eqref{sys1} and \eqref{sys2}
decouple, and $\alpha$ can be a solution of eqn.\ \eqref{sys2} with
arbitrary proportionality constant.
Moreover, for $m=0$, the common value of $\beta_\pm$ is a rational
function of $z$,
\[ \beta_\pm (z)= \frac{\Gamma(n/2)}{(2\pi)^n} l^{1-n}
(z-1)^{-(n-1)/2}.\]
\section{Conclusions}
\label{conc}
We have introduced the spinor parallel propagator for maximally
symmetric spaces in any dimension. This enabled us to
find expressions for the Dirac spinor Green's functions in the
maximally symmetric spaces $\mathbb{R}^n$, $S^n$ and $H^n$ in terms
of intrinsic geometric objects. Although there are obstructions to the
quantization of spinors in odd dimensional manifolds with boundary
\cite{Carey99}, our results should be applicable to the AdS/CFT
correspondence, because of the classical dynamics in AdS space.
\section*{Acknowledgments}
I would like to thank K.~S.~Viswanathan and R.~C.~Rashkov for
stimulating discussions. Moreover, financial support from Simon Fraser
University is gratefully acknowledged.
|
2,869,038,155,686 | arxiv | \section{Introduction}
In 1997 we put forward the idea that the remarkable sharpness of the knee at 3-4 PeV in
the cosmic ray (CR) energy spectrum is due to the dominant contribution of a single
source. This sharpness could be the consequence of the sharp cutoff of the maximum
accelerated energy in the source. Another argument in favour of the single source model
was the fine structure of the spectrum in the vicinity of the knee ie, not just a
single smooth transition, albeit a sharp one. Specifically, we found evidence for two
'knees' \cite{Erl1}. Initially we thought that the dominant primary nucleus in the knee
was oxygen. In this case the second structure, observed at an energy about 3.5-4 times
higher than the energy of the first knee (~usually just referred to as 'the knee'~),
is due to primary iron. 4 years later we updated the analysis using the 40
size spectra of extensive air showers (EAS) available at that time. We confirmed
the conclusion that the observed sharpness of the knee is higher than expected in the
classic diffusion model. We extended the analysis on the spectra of Cherenkov light
from EAS and found the same fine structure as observed with the detectors of EAS
charged particles \cite{Erl2}. Later on, we came to the conclusion that the most likely
nucleus, dominant in the knee, was helium, not oxygen. In this case the second observed
structure in the energy spectrum at 12-16 PeV is not due to iron, but to oxygen
\cite{Erl3}. If this is true, at even higher energies, about 40-50 PeV, one can expect
another structure due to iron. \\
The application of the theoretical model of the supernova explosion to the
observations yielded an estimate of the most likely distance (230-350 pc) and age
(84-350 kyears) of the source \cite{Erl4}. American astronomers found that the
pulsar B0656+14, located inside the Monogem Ring supernova remnant (SNR), fits well
this distance and age. Therefore this complex can be the candidate for the single
source. Unfortunately, it is difficult to observe this source by means of the
anisotropy of charged CR, since the particles with PV-rigidity have a giroradius less
than 1 pc, which is two orders of magnitude less than the estimated distance to the
source. Similarly, it is difficult to confirm the location by detecting the Monogem
Ring with TeV gamma-rays since it is not a discrete, but a very extended, SNR with an
angular diameter about 18$^\circ$. \\
The aim of this paper is to review the present situation around the knee and update the
status of the single source model. \\
\section{New data}
\begin{figure}[!h]
\centering
\includegraphics[height=3.4in,angle=-90]{icrc0301_fig01.eps}
\caption{\footnotesize Energy spectra of primary CR, measured by Tibet-III (a), KASCADE
(b), GAMMA (c), Yakutsk (d), Maket-Ani (e) and Tunka (f) arrays. Symbols {\em e},
$\mu$ and {\em C} in brackets indicate the measured EAS component: electromagnetic,
muon or Cherenkov light respectively. Notations QGSJET+HD ({\em heavy dominant}) in (a)
and QGSJET-01 in (b) indicate the interaction model used for the transition from the
measured parameters to the primary energy. Full lines are fits by the expression (2)
with best fit parameters shown in the last five columns of Table 1. Dashed lines are
extrapolations of these fits to the energy above the fitted range.}
\label{fig1}
\end{figure}
Since the beginning of this decade several new measurements of the CR energy spectrum
have been published \cite{Tibet,Kasc,Gamma,Yakut,Maket,Tunka}. They are shown in Figure
1. We do not present here the EAS-TOP+MACRO measurements \cite{EASTOP} since their
authors showed spectra of light and heavy elements separately. We omitted also the
results of KASCADE-Grande \cite{Kasc-Gr} since they are still preliminary.
\subsection{Sharpness}
The position of the knee (~$logE^k_1$~) has been determined as the point with the
maximum sharpness of the spectrum, the sharpness $S_1$ being defined as \cite{Erl1}
\begin{equation}
S_1=-\frac{d^2(logI)}{d(logE)^2}
\end{equation}
The results are shown in columns 2 and 3 of Table 1. \\
It is interesting to note that as a rule measurements of muons and Cherenkov light give
higher sharpness than the electromagnetic component (~except for Tunka~). It
is understandable since both muons and Cherenkov light are proxies of the energy lost
by the cascade in the atmosphere which is quite close to the primary energy.
\begin{table*}[!th]
\label{table1}
\centering
\begin{tabular}{||c||c|c||c|c|c|c|c||} \hline
Array & $logE^k_1$ & $S_1$ & $logE^k_2$ & $\gamma$ & $\Delta \gamma$ & $\delta$ & $S_2$ \\
\hline
Tibet-III & 6.60 & 1.34$\pm$0.21 & 6.59 & 2.64 & 0.48 & 8.85 & 2.44$\pm$0.77 \\
KASCADE & 6.60 & 3.07$\pm$0.77 & 6.55 & 2.60 & 0.49 & 8.43 & 2.34$\pm$4.86 \\
GAMMA & 6.61 & 1.19$\pm$0.30 & 6.76 & 2.76 & 0.32 & 10.0 & 1.84$\pm$0.43 \\
Yakutsk & 6.69 & 3.56$\pm$0.65 & 6.57 & 2.64 & 0.46 & 15.7 & 4.17$\pm$4.31 \\
Maket-ANI & 6.70 & 1.53$\pm$0.53 & 6.78 & 2.75 & 0.44 & 12.9 & 3.30$\pm$3.30 \\
Tunka & 6.50 & 1.44$\pm$0.60 & 6.64 & 2.59 & 0.75 & 1.64 & 0.71$\pm$0.13 \\ \hline
\end{tabular}
\caption{\footnotesize Best fit parameters of the energy spectra presented in
Fig.1. Columns 2 and 3 show parameters determined by expression (1), columns 4-8 -
by expressions (2) and (3)}
\end{table*}
The result is that the primary energy spectrum is indeed rather sharp. In any case,
Table 1
shows that in all new measurements the sharpness of the knee is substantially
higher than 0.3 which is the characteristic value for the Galactic Diffusion
Model. \\
At this stage it is important to make some more remarks about what might be expected
for the value of $S$ in the conventional case of many sources contributing. As pointed
out in \cite{Erl5}, the value $S=0.3$ is, in fact, an upper limit for a 'normal
composition'. Further analysis, involving the most likely generation of CR particles
with a variety of exponents for their energy spectra - conditioned by different SNR
expanding in regions of different density and magnetic field (~some of which will be
increased by the CR shocks themselves~) - leads to $S$ value even lower than 0.3. The
case for the Single Source Model, in which the sharpness is due to the vicinity of the
single source ( SNR or pulsar ) with a sharp cutoff of the emission spectrum at the
maximum acceleration energy is thus strengthened.
\subsection{Fine structure}
We define 'fine structure' of the spectrum as the existence of reliable deviations from
the power law fits both below and above the knee. To find these deviations we fitted
the spectra in Fig.1 by the expression proposed in \cite{Samvel}:
\begin{equation}
I(E)=AE^{-\gamma}(1+(\frac{E}{E^k})^\delta)^{-\frac{\Delta \gamma}{\delta}}
\end{equation}
Here $\gamma$ is the power law index of the spectrum below the knee, which changed by
$\Delta \gamma$ above the knee. Between these regions there is a transition range
which is described by the sharpness parameter $\delta$. The true sharpness $S_2$ is
connected with $\delta$ as
\begin{equation}
S_2 = \delta \Delta \gamma \frac{ln10}{4}
\end{equation}
We have found best fit values for these parameters using the least squares method with
MINUIT code \cite{MINUIT}. They are shown in the 5 last columns of Table 1. The energy
range, in which we fitted spectra, and the fit itself, are shown by the full line in
Fig.1. The linear extrapolation of the fit to higher energies is indicated by the
dashed line. \\
Deviations of the actual intensities from this fit and its extrapolation for all 6
spectra are shown in Fig.2a. In order to remove the difference between energy scale of
all spectra and
reveal the peculiarities in their {\em shape} we referred the deviations from the fit
(2) in the individual spectra to the individual energy of the knee $logE^k_2$. Mean
values of the deviation are shown in Fig.2b. \\
\begin{figure}[thb]
\centering
\includegraphics[height=3.4in,angle=-90]{icrc0301_fig02.eps}
\caption{\footnotesize Fine structure of the PCR energy spectrum. The irregularity at
the position of the knee, $log(E/E^k) = 0$, is not seen since the expression (2) gives
a good fit of the spectrum in the knee region.}
\label{fig:fig2}
\end{figure}
The irregularity at $log(E/E^k)=0.5-0.6$ found in \cite{Erl1,Erl2} is confirmed in
the new spectra too. The progress in these new measurements lets us proceed to
higher energies. Here, a new feature can be seen at $log(E/E^k)=1-1.2$. It was
first noticed in \cite{Gamma} and now confirmed by 5 other spectra. \\
If indeed $He$ nuclei dominate in the knee then the first irregularity at
$log(E/E^k)=0.5-0.6$ can be referred to the $CNO$-group of nuclei and the second one at
$log(E/E^k)=1-1.2$ - to the $Fe$-group. If true, the existence of these groups is in
favour of the single source model. The peaks marked {\em 'CNO(?)'} and {\em 'Fe(?)'}
are at the correct places for the nuclei which are thought to form the bulk of the CR
after hydrogen and helium.
\subsection{Other evidence}
During the last year the PAMELA and ATIC collaborations claimed that they observed an
excess of positrons \cite{Adrian1} and electrons ($e^-+e^+$) \cite{Chang} in the
primary CR. The evidence was in the form of a sudden upturn in the positron spectrum at
$\sim3-5$ GeV leading to a bump in the $e^-+e^+$ spectrum at $\sim500$ GeV. The peak
is some 3-4 times the 'background level'formed by a smooth steepening of the spectrum
from energies below the bump. The other measurements although showing somewhat smaller
intensity in the bump confirmed an irregular behavior of the spectrum in this region
\cite{Abdo} and its sharp steepening at TeV energies above the bump \cite{HESS1,HESS2},
which creates the feature similar or even sharper than the knee at PeV energies. \\
These publications caused great interest and inspired many attempts at
their explanation (~see references in \cite{Barger}~). The bulk of the
proposed models suggested mechanisms in which the additional
electrons and positrons were created by the interaction, annihilation or decay of dark
matter particles. Other models proposed astrophysical scenarios with extra electrons
and positrons created, accelerated and emitted by various astrophysical sources. We
consider these latter scenarios as more likely not only because dark matter particles
are still elusive, but such models face the difficulty of an absence of extra
antiprotons in the PAMELA data \cite{Adrian2}, which should inevitably be produced in
processes
including dark matter particles. To suppress the production of antiprotons the models
with dark matter include additional assumptions which make them more complicated and
speculative.
\section{Discussion of the knee and of the 'electron bump'}
We discuss astrophysical models here because they seem to us more realistic and
according to our view they give support to our single source model. Here we present
arguments in favor of such a view.
\begin{itemize}
\item The essence of the single source model is a concept that CR sources are
non-uniformly distributed in space and time. As a consequence the CR energy spectrum
observed at the Earth can carry traces of this non-uniformity, i.e.
irregularities of some kind or a fine structure. In particular the substantial
contribution of just the nearby and recent single source (~SNR or pulsar~) to the flux
of CR protons and nuclei can be the cause of the knee at PeV energies.
\item The observed sharpness of the knee is due to several reasons: (i) the source is
relatively close to the solar system and the energy spectrum of its CR is not distorted
by the propagation effects. Its shape is close to the slope of the production spectrum.
Below the knee it is rather flat (~$\gamma \approx 2.1$~) compared with the bulk of CR
(~$\gamma \approx 2.7$~) and its contribution is more pronounced at high energies.
(ii) The CR energy spectrum has a sharp cutoff at the maximum acceleration energy.
(iii) If the {\em He} component is dominant at the knee it gives an additional
sharpness since we must expect a gap between the {\em He} and {\em CNO} group of
nuclei. (iv) Since the source is 'single' the smoothing effect on the knee sharpness
due to the spread of characteristics inevitable in the case of multiple sources is at
a minimum.
\item The electron component of CR is even more sensitive to the presence of nearby and
recent sources than protons and nuclei, since electrons from remote and old sources
suffer not only from diffusive losses, but also from rising energy losses.
Indeed, in our paper devoted to SNR and the electron component \cite{Erl6} we have
shown how different electron spectra could be for different samples of the time-space
distribution of SNR in our Galaxy. We show this collection of electron spectra in
Figure 3. The simulations were made assuming that the electrons are accelerated in
the SNR in a similar manner to that for protons. It is seen that the fine structure of
the spectrum appears already at energies above 100 Gev and clearly seen as bumps in
the TeV region, which is due to the presence of recent and nearby SNR.
\begin{figure}[thb]
\centering
\includegraphics[height=3.4in,angle=-90]{icrc0301_fig03.eps}
\caption{\footnotesize Examples of predicted electron spectra for different time-space
distributions of SNR. Experimental data: crosses - ATIC2 \cite{Chang}, open circles -
Fermi LAT \cite{Abdo}, full squares and triangles - HESS \cite{HESS1,HESS2}.
No normalization has been applied. The difference by the factor of 2-3 in absolute
intensities between most simulated samples and the experimental data is not important
and can be easily reduced by reasonable adjusting the model parameters. The bumps in
the simulated samples are generally at an energy higher than the observed
electron-positron bump, which is at $\sim$500 GeV. This is interpreted by us as
indicating that these electrons+positrons are secondaries to 'our' SNR-accelerated
protons and nuclei. The thick solid line in the TeV region shows the contribution to
electrons expected from the Monogem Ring SNR \cite{Erl6}.}
\label{fig:fig3}
\end{figure}
\item In the absence of likely sources of positrons within the SNR, the observations
dictate that the positrons come from another process. SNR and pulsars are usually
immersed in an the envelopes of gas:
remnants of the SN explosion (~SNR~), and pulsar wind nebulae (~PWN~). The accelerated
particles interact with this gas and create electron-positron pairs. These secondary
positrons and electrons will have energies less than electrons directly accelerated and
their ensuing bump will be at lower energy: tens of GeV is not unreasonable.
\item the energy spectrum of electrons from nearby and recent sources
would be as flat as the production spectrum of the accelerated particles and create a
feature similar to the knee in the spectrum of primary CR nuclei. The sharpness of this
'electron knee' in this scenario is supported by HESS measurements \cite{HESS1,HESS2}
and is due to the sharpness of the knee for primary nucleons, rising energy losses of
electrons and positrons at TeV energies and a small pile up of electrons which
initially had an energy higher than the knee, but lost it during the propagation.
The calculations made in some works support such a possibility
\cite{Barger,Profumo,Hong,Shaviv,Malysh,Blasi,Fujita,Piran}. There are, however,
attempts to give a methodical explanation of the PAMELA and ATIC excess by the rising
contamination of positrons and electrons from primary protons \cite{Fazely,Schub}.
\item the magnitude of the bump from the single source with respect to the background
in electrons can be greater than that in nuclei and this can be understood. In view of
the rapidly increasing energy losses for electrons in the general interstellar medium,
compared with only a diffusive loss for nuclei, the electron background is relatively
lower. We estimate that the bump (ATIC) contains an energy of $\sim 10^{-5}$eVcm$^{-3}$
; for reference, our single source contributes about $2\cdot 10^{-4}$eVcm$^{-3}$ in the
CR spectrum.
\end{itemize}
\section{Conclusions}
We have analysed the six new energy spectra which have appeared in the papers published
since the last update of our single source model. All the previous findings
(~sharpness and the fine structure of the knee~) are confirmed by this new data. The
advance to a higher
energy of about 10$^8$GeV lead us to confirm the existence of a new feature - another
irregularity in the spectrum at energies of 50-80 PeV, claimed first in \cite{Gamma}.
If the dominant contribution to the knee is due to primary {\em He}-nuclei, this new
irregularity is just where primary iron nuclei should appear. \\
We consider that the latest findings of the irregularities in the electron and
positron spectra (~'electron bump or knee'~) can have the same origin as the 'hadron
knee' in the spectra of the primary nuclei, i.e. they are due to the non-uniformity of
the time-space distribution of CR sources, and if true, it is an additional support of
the single source model. \\
\newpage
It is interesting to postulate that the last HESS point at $\sim$4500 GeV could
represent part of a signature of SNR-accelerated electrons from a source such as
Monogem Ring; certainly it is in the energy region where we expect a bump. \\
{\em Acknowledgments} \\
The authors are thankful to the Ralph Kohn Foundation for their interest and
financial support of this work. \\
|
2,869,038,155,687 | arxiv |
\section{Introduction}
\label{sec:int}
The production of charm quarks at HERA has been studied both in deep inelastic
scattering
(DIS)~\cite{pl:b407:402,np:b545:21,epj:c12:35,pl:b528:199,pr:d69:012004} and
photoproduction~\cite{np:b472:32,epj:c6:67,np:b729:492,epj:c47:597,hep-ex-0608042}.
In general, reasonable agreement
is seen with next-to-leading-order (NLO) QCD predictions.
This paper presents measurements of the \dstar{} cross section in the range
$0.05 < \ensuremath{{Q^{2}}} < 0.7{\,\text{Ge}\eVdist\text{V\/}}^{2}$.
The beampipe calorimeter of
ZEUS~\cite{pl:b407:432,pl:b487:53} was used for the measurement of the
scattered lepton, which allows the first measurements of the transition region
between photoproduction (photon virtuality, $\ensuremath{{Q^{2}}} \sim 0{\,\text{Ge}\eVdist\text{V\/}}^2$) and DIS
($\ensuremath{{Q^{2}}} > 1{\,\text{Ge}\eVdist\text{V\/}}^{2}$).
The cross sections are compared to the predictions of two different NLO QCD
calculations, one designed for DIS, the other for the photoproduction region.
This paper investigates whether the calculations remain valid in this
transition region.
\section{Experimental set-up}
\label{sec:exp}
This analysis was performed with data taken from 1998 to 2000, when HERA
collided electrons or positrons\footnote{Hereafter, both electrons and
positrons are referred to as electrons.} with energy $E_e=27.5{\,\text{Ge}\eVdist\text{V\/}}$ with
protons of energy $E_p=920{\,\text{Ge}\eVdist\text{V\/}}$. The combined data sample has an integrated
luminosity of $\mathcal{L}=81.9 \pm 1.8$\,pb$^{-1}$.
\Zdetdesc
\Zctddesc\footnote{\ZcoosysB}
\Zcaldesc
The scattered electron was detected in the beampipe
calorimeter (BPC). The BPC
allowed the detection of low-\ensuremath{{Q^{2}}}{} events, where the electron is scattered
through a small angle. The BPC was used in previous measurements of the proton
structure function, $\ftwo$, at low \ensuremath{{Q^{2}}}~\cite{pl:b407:432,pl:b487:53}. It
originally consisted of two tungsten--scintillator sampling calorimeters with
the front faces located at $Z=-293.7\,\text{cm}$, the centre at $Y=0.0\,\text{cm}$, and the
inner edge of the active area at $X=\pm 4.4\,\text{cm}$, as close as possible to the
electron-beam trajectory.
At the end of 1997 one of the two BPC calorimeters was removed; hence, for the
analysis in this paper, only the calorimeter located on the $+X$ side of the
beampipe was utilised.
It had an active area of $12.0 \times
12.8\,\text{cm}^2$ in $X\times Y$ and a depth of $24$ radiation lengths. The relative
energy resolution as determined in test-beam measurements with 1\hbox{$\,\text{--}\,$}{}6{\,\text{Ge}\eVdist\text{V\/}}{}
electrons was ${\Delta E}/{E}={17\%}/{\sqrt{E\,({\,\text{Ge}\eVdist\text{V\/}})}}$.
The luminosity was measured from the rate of the bremsstrahlung process $ep
\rightarrow e\gamma p$, where the photon was measured in a lead--scintillator
calorimeter\cite{desy-92-066,*zfp:c63:391,*acpp:b32:2025} placed in the HERA
tunnel at Z=-107m.
A three-level trigger system was used to select events
online\cite{zeus:1993:bluebook,proc:chep:1992:222}. At all three
levels, the event was required to contain a scattered electron candidate in
the BPC. Additionally, at the third level, a reconstructed \dstar{} candidate
was required for the event to be kept for further analysis. The efficiency of
the online \dstar{} reconstruction, determined relative to an inclusive
DIS trigger, was above 95$\%$\cite{pr:d69:012004}.
\section{Kinematic reconstruction and event selection}
\label{sec:kinvar}
Deep inelastic electron-proton scattering, $ep\to eX$, can be described in
terms of two kinematic variables, chosen here to be $y$ and \ensuremath{{Q^{2}}}, where $y$ is
the inelasticity.
They are defined as $\ensuremath{{Q^{2}}}=-q^2=-(k-k')^2$ and $y=\ensuremath{{Q^{2}}}/(2P\cdot q)$,
where $k$ and $P$ are the four-momenta of the incoming electron and proton,
respectively, and $k'$ is the four-momentum of the scattered electron. The
inelasticity, which is the fractional energy transferred to the proton in its
rest frame, is related to the Bjorken scaling variable $x$ and \ensuremath{{Q^{2}}}{} by
$\ensuremath{{Q^{2}}}=sxy$, where $s=4E_{e}E_{p}$ is the square of the electron-proton
centre-of-mass energy of 318{\,\text{Ge}\eVdist\text{V\/}}.
The values of $y$ and \ensuremath{{Q^{2}}}{} were calculated using the measured electron
scattering angle and the energy deposited in the BPC as detailed in a previous
analysis\cite{pl:b407:432}, which also describes the method used for the
energy calibration of the BPC. A time dependent re-calibration of the energy
response was necessary~\cite{thesis:tandler:2003},
as radiation damage of the
scintillator resulted in a degradation of about $10\%$ by the end of the 2000
running period.
A series of cuts was applied
to reject background. The events were required to have a primary vertex
within 50\,\text{cm}{} in $Z$ of the nominal interaction point. The electron
candidates in the BPC were required to have $E_{\mathrm{BPC}}>4{\,\text{Ge}\eVdist\text{V\/}}$, as the
trigger efficiency is low below this energy. The electron impact point on the
face of the BPC was required to be more than 0.7\,\text{cm}{} from the inner edge to
ensure good shower containment.
Photoproduction events were efficiently rejected by requiring the events to
have $35 < E-P_{Z} < 65 {\,\text{Ge}\eVdist\text{V\/}}$, where $E-P_{Z} = \sum_{i} (E-P_{Z})_{i}$ is
summed over all CAL deposits, including the scattered electron candidate in
the BPC. Finally, events with an additional well-reconstructed electron
candidate in the CAL with energy greater than 5{\,\text{Ge}\eVdist\text{V\/}}{} were rejected to reduce
background from DIS events with $\ensuremath{{Q^{2}}} > 1{\,\text{Ge}\eVdist\text{V\/}}^{2}$.
The measured kinematic region in $y$ and $\ensuremath{{Q^{2}}}$ was restricted to the range of
high acceptance, $0.02 < y < 0.85$, $0.05 < \ensuremath{{Q^{2}}}{} < 0.7{\,\text{Ge}\eVdist\text{V\/}}^2$. With these
cuts, the reconstructed invariant mass of the hadronic system, $W$, lies
between 50 and 300{\,\text{Ge}\eVdist\text{V\/}}, with a
mean of 190{\,\text{Ge}\eVdist\text{V\/}}.
\section{\boldmath Selection of \dstar{} candidates}
\label{sub:dssel}
The \dstar{} mesons were identified using the decay channel $\dstarp \to \dzero
\pi^{+}_s$ with the subsequent decay $\dzero \to K^-\pi^+$ and the
corresponding antiparticle decay chain, where $\pi^+_s$ refers to a
low-momentum (``slow'') pion accompanying the \dzero.
Charged tracks measured by the CTD and assigned to the primary event
vertex\footnote{The resolution of such tracks is not good enough to separate
primary and secondary vertices from $c$ and $b$ hadron decays.}
were selected. The transverse momentum was required to be greater than
0.12{\,\text{Ge}\eVdist\text{V\/}}. The \ensuremath{p_{T}}{} cut was raised to 0.25{\,\text{Ge}\eVdist\text{V\/}}{} for a data subsample
corresponding to $(16.9 \pm 0.4)\,\text{pb}^{-1}$, for which the
low-momentum track-reconstruction efficiency was lower due to the operating
conditions of the CTD\cite{nim:a515:37}. Each track was required
to reach at least the third superlayer of the CTD. These restrictions ensured
that the track acceptance was high and the momentum resolution was good.
Tracks in the CTD with opposite charges and transverse momenta $p_T >
0.45{\,\text{Ge}\eVdist\text{V\/}}$ were combined in pairs to form $D^0$ candidates. The tracks were
alternately assigned the kaon and the pion mass and the invariant mass of
the pair, $M_{K\pi}$, was determined. Each additional track, with charge
opposite to that of the kaon track, was assigned the pion mass and combined
with the $D^0$-meson candidate to form a \dstar{} candidate.
A mass window for the signal region of the \dzero{} varying from $1.82<
M_{K\pi} < 1.91{\,\text{Ge}\eVdist\text{V\/}}$ to $1.79< M_{K\pi} < 1.94{\,\text{Ge}\eVdist\text{V\/}}$ was used,
reflecting the dependence of the CTD resolution on $\ensuremath{p_T(D^{\ast})}$. The signal
region for the reconstructed mass difference $\Delta M=(M_{K\pi\pi_s} -
M_{K\pi})$ was \mbox{$0.1435 < \Delta M < 0.1475{\,\text{Ge}\eVdist\text{V\/}}$}. The requirement of
$\ensuremath{p_T(D^{\ast})}/\ensuremath{E_{T}^{\theta>10^{\circ}}} > 0.1$ was also applied, where \ensuremath{E_{T}^{\theta>10^{\circ}}}{} is the transverse
energy outside a cone of $\theta = 10^{\circ}$ defined with respect to the
proton direction. This cut rejects background without significantly
affecting the signal.
The \dstar{} mesons were selected in the kinematic region $1.5<\ensuremath{p_T(D^{\ast})}<9{\,\text{Ge}\eVdist\text{V\/}}$
and $|\ensuremath{\eta(D^{\ast})}|<1.5$. The $\Delta M$ distribution for events with an
electron reconstructed in the BPC is shown in Fig.~\ref{fig1}.
To extract
the number of \dstar{} mesons, the $\Delta M$ distribution was fit using an
unbinned likelihood method, with a Gaussian to describe the signal and a
threshold function to describe the combinatorial background.
A first estimate of the background was given by \dstar{} candidates with
wrong-sign combinations, in which both tracks forming the \dzero{} candidates
have the same charge and the third track has the opposite charge.
These are
shown as the shaded region in Fig.~\ref{fig1}.
The number of \dstar{} mesons obtained from the fit was $N(\dstar)=253 \pm 25$.
\section{Acceptance corrections and systematic uncertainties}
\label{sec:evsim}
The acceptances were calculated using the {\sc
Herwig~6.1}~\cite{hep-ph-9912396,*cpc:67:465} and {\sc
Rapgap~2.08}~\cite{cpc:86:147} Monte Carlo (MC) models.
Both models simulate charm and beauty production and include contributions
from both direct and resolved photoproduction.
In direct photoproduction the photon participates as a point-like particle in
the hard scattering process,
while in resolved photoproduction a parton in the photon
scatters on a parton in the proton.
The generated events were passed
through a full simulation of the detector,
using {\sc Geant 3.13}~\cite{tech:cern-dd-ee-84-1} and then processed and
selected with the same programs as used for the data. The
CTEQ5L~\cite{epj:c12:375} parton density function (PDF) was used for the
proton and GRV-LO~\cite{pr:d46:1973} was used for the photon. The charm-quark
mass was set to 1.5{\,\text{Ge}\eVdist\text{V\/}}.
The {\sc Herwig} predictions are in good agreement with the data distributions
for both the scattered lepton and hadronic variables and so this Monte Carlo
was used to correct the data for detector effects. For the kinematic region
of the measurement
$0.05<\ensuremath{{Q^{2}}}<0.7{\,\text{Ge}\eVdist\text{V\/}}^2$, $0.02<y<0.85$, $1.5<\ensuremath{p_T(D^{\ast})}<9{\,\text{Ge}\eVdist\text{V\/}}$, and
$|\ensuremath{\eta(D^{\ast})}|<1.5$ the acceptance was $(1.11\pm 0.03)\%$. This includes
the geometrical acceptance of the BPC, which was about 9\%, and the
reconstruction efficiency for the \dstar{} decay chain.
The {\sc Rapgap} MC gives a similarly good representation of the
data and was used to estimate part of the systematic
uncertainties, as described below.
The differential cross section for a given observable $Y$ was determined
using
\begin{eqnarray*}
\frac {d\sigma}{dY} & = &
\frac {N} {A \cdot \mathcal {L} \cdot B \cdot \Delta Y},
\end{eqnarray*}
where $N$ is the number of \dstar{} events in a bin of size $\Delta Y$, $A$ is
the acceptance (which takes into account migrations and efficiencies for that
bin) and $\mathcal {L}$ is the integrated luminosity. The product, $B$, of the
appropriate branching ratios for the \dstar{} and \dzero{} decays was set to
$(2.57\pm 0.05)\%$~\cite{jphys:g33:1}.
The systematic uncertainties of the measured cross sections were determined by
changing in turn the selection cuts or the analysis procedure within their
uncertainties and repeating the extraction of the cross
sections~\cite{thesis:irrgang:2004}. The major experimental sources of
systematic uncertainty were (the variation of the total cross section is given
in parentheses): the BPC alignment ($^{+2.5}_{-3.1} \%$) and energy scale
($^{+0.4}_{-1.2} \%$); the uncertainty in the CTD momentum scale
($^{+0.2}_{-1.5} \%$) and the CAL energy scale ($\pm 1 \%$); the
$\ensuremath{p_T(D^{\ast})}/\ensuremath{E_{T}^{\theta>10^{\circ}}}$ cut ($^{+3.0}_{-1.7} \%$) and the \dstar{} signal extraction
($^{+0.1}_{-1.5} \%$). The uncertainty due to the MC model ($^{+9.5}_{-4.8}
\%$) was determined by using {\sc Rapgap} to evaluate the acceptance
correction rather than {\sc Herwig}, as well as by varying the fraction of
resolved and direct photoproduction processes in the simulation.
All the above errors were added in quadrature separately for the positive and
negative variations to determine the overall systematic uncertainty. The
overall normalisation has additional uncertainties of 2.2\% due to the
luminosity measurement and 2.0\%
due to knowledge of branching ratios. These are
included in the error quoted for the total cross section but not in
the systematic uncertainties of the differential cross sections.
\section{Theoretical predictions}
\label{sec:theory}
Two different calculations were used to
evaluate the theoretical expectation for charm production.
The HVQDIS program~\cite{pr:d57:2806} implements an NLO calculation of charm
production in DIS. At low \ensuremath{{Q^{2}}}{}, the hadron-like structure of the photon, not
included in HVQDIS, is needed to regularise the NLO calculation. Therefore
predictions from this program are expected to lose accuracy in the
limit $\ensuremath{{Q^{2}}}{} \rightarrow 0$. The ZEUS measurements of \dstar{} production in
DIS for
$\ensuremath{{Q^{2}}} > 1.5{\,\text{Ge}\eVdist\text{V\/}}^{2}$ are in good agreement with the HVQDIS
prediction~\cite{pr:d69:012004}.
The FMNR program~\cite{np:b454:3} implements an NLO calculation of charm
photoproduction which includes the hadron-like component of the photon.
Electroproduction cross sections can be obtained with FMNR using the
Weizs\"acker-Williams approximation~\cite{pl:b319:339} and are therefore
expected to be reliable only at low \ensuremath{{Q^{2}}}, where this approximation is valid.
The FMNR predictions are in reasonable agreement with ZEUS
measurements of \dstar{} photoproduction~\cite{epj:c6:67}, considering the
theoretical uncertainties.
It is therefore interesting to see whether these calculations are able to
reproduce the data in the transition region between photoproduction and DIS.
The following parameters were used in the calculations for both programs.
They were chosen to be the same as in a previous
publication~\cite{pr:d69:012004}. A variant of the ZEUS-S NLO QCD global
fit~\cite{pr:d67:012007} to structure-function data was used as the
parameterisation of the proton PDFs. This fit was repeated in the
fixed-flavour-number scheme, FFNS, in which the PDF has three active quark
flavours in the proton, and $\Lambda^{(3)}_{\rm QCD}$ is set to 0.363{\,\text{Ge}\eVdist\text{V\/}}. The
mass of the charm quark was set to 1.35{\,\text{Ge}\eVdist\text{V\/}}. The renormalisation and
factorisation scales were set to $\mu_{R} = \mu_{F} = \sqrt{\ensuremath{{Q^{2}}}+4m_c^2}$ in
HVQDIS, while for FMNR they were set to the usual choice of $\mu_{R} = \mu_{F}
= \sqrt{\ensuremath{p_{T}}^2+m_c^2}$, where $\ensuremath{p_{T}}^2$ is the average transverse momentum
squared of the charm quarks. The charm fragmentation to a \dstar{} is carried
out using the Peterson function~\cite{pr:d27:105}. The hadronisation fraction,
$f(c \to \dstar)$, was taken to be $0.238$~\cite{hep-ex-9912064,*epj:c44:351}
and the Peterson parameter, $\epsilon$, was set to 0.035\cite{np:b565:245}.
The parameters used here for the FMNR calculation are different from those
used in a previous photoproduction analysis~\cite{epj:c6:67} (which used $m_c
= 1.5{\,\text{Ge}\eVdist\text{V\/}}$) leading to a 20\% larger predicted photoproduction cross section.
For the FMNR calculation the electroproduction cross section, $\sigma_{ep}$, was
obtained from the photoproduction cross section, $\sigma_{\gamma p}(W)$, using
\begin{eqnarray*}
\sigma_{ep} & = & \int_{y_{\mathrm{min}}}^{y_\mathrm{max}} dy\,
\Phi(y,Q^{2}_{\mathrm{min}},Q^{2}_{\mathrm{max}})
\sigma_{\gamma p}(\sqrt{y s}),
\end{eqnarray*}
where
\begin{eqnarray}
\label{eq:flux}
\Phi(y,Q^{2}_{\mathrm{min}},Q^{2}_{\mathrm{max}}) & = &
\frac{\alpha_{\mathrm{em}}}{2\pi} \left[
\frac{(1 + (1-y)^{2})}{y}
\ln{\frac{Q^{2}_{\mathrm{max}}}{Q^{2}_{\mathrm{min}}}} -
2 m_{e} y \left(
\frac{1}{Q^{2}_{\mathrm{min}}} -
\frac{1}{Q^{2}_{\mathrm{max}}}
\right)
\right]
\end{eqnarray}
is the photon flux and $y_{\mathrm{min}}$, $y_{\mathrm{max}}$,
$Q^{2}_{\mathrm{min}}$, $Q^{2}_{\mathrm{max}}$ define the measurement range in
$y$ and $Q^{2}$.
The NLO QCD predictions for \dstar{} production are affected by systematic
uncertainties, which were also evaluated as in a previous ZEUS
paper~\cite{pr:d69:012004}\footnote{For the HVQDIS case, following
\cite{pr:d69:012004}, the minimum value for the scales was
set to $2 m_c$.}.
The sources of systematic uncertainties on the total cross section are:
charm quark mass ($^{+15}_{-13}\%$ for HVQDIS,
$^{+16}_{-14}\%$ for FMNR);
renormalisation and factorisation scale
($^{+1}_{-13}\%$ for HVQDIS,
$^{+23}_{-10}\%$ for FMNR);
ZEUS PDF ($\pm 5\%$);
fragmentation ($^{+10}_{-6}\%$).
For both programs, the systematic uncertainties were added in quadrature and
are displayed as a band in the figures.
Theoretical calculations of the total charm cross section in this
\ensuremath{{Q^{2}}}{} range can not be compared to the present data since
\dstar{} are only measured in a limited \ensuremath{p_{T}}{} and $\eta$ range.
\section{Cross section measurements}
\label{sec:res}
The total cross section for $0.05<\ensuremath{{Q^{2}}}<0.7{\,\text{Ge}\eVdist\text{V\/}}^2$, $0.02<y<0.85$,
\mbox{$1.5<\ensuremath{p_T(D^{\ast})}<9{\,\text{Ge}\eVdist\text{V\/}}$} and $|\ensuremath{\eta(D^{\ast})}|<1.5$ is:
$$
\sigma(ep \rightarrow e \dstar{} X) =
10.1 \pm 1.0(\mbox{stat.})^{+1.1}_{-0.8}(\mbox{syst.}) \pm 0.20(\mbox{BR}) \,\text{nb},
$$
where the first uncertainty is statistical, the second from systematic effects
(including the luminosity uncertainty) and the third from the uncertainties in
the branching ratios.
The prediction from the HVQDIS program is $8.6^{+1.9}_{-1.8}\,\text{nb}$, in
agreement with the data,
while the prediction from FMNR is
$8.9^{+2.4}_{-1.4}\,\text{nb}$\footnote{The contribution from the hadron-like
component of the photon is 9\%.}, also in good agreement.
The measured differential \dstar{} cross sections as a function of \ensuremath{{Q^{2}}}, $y$,
\ensuremath{p_T(D^{\ast})}{} and \ensuremath{\eta(D^{\ast})}{} for the data are shown in
Fig.~\ref{fig:hvqq2yetapt} and given in Table~\ref{tab:dxs}.
The predictions of the NLO calculations, including their uncertainties, are
shown as bands.
The measured differential cross sections
are well described over the full measured kinematic region by both
calculations.
This analysis was also compared to previous ZEUS measurements of \dstar{}
production in DIS~\cite{pr:d69:012004} made in the kinematic region
$1.5<\ensuremath{{Q^{2}}}<1000{\,\text{Ge}\eVdist\text{V\/}}^2$, $0.02<y<0.7$, $1.5<\ensuremath{p_T(D^{\ast})}<15{\,\text{Ge}\eVdist\text{V\/}}$ and $|\ensuremath{\eta(D^{\ast})}|<1.5$.
In order to directly compare with the results presented there, the cross
sections were recalculated in the modified kinematic region $0.02<y<0.7$. No
correction was made for the different upper cut on \ensuremath{p_T(D^{\ast})}{}, as the size of the
effect is $\approx 1\%$.
For this modified kinematic region, the differential cross section as a
function of \ensuremath{{Q^{2}}}{} is presented in Fig.~\ref{fig:dis2003ext} and given in
Table~\ref{tab:dxsr}. The systematic errors were assumed to be the same as
those in the full $y$ range. Figure~\ref{fig:dis2003ext} also shows the
previous ZEUS measurement and the HVQDIS prediction. The combination of both
measurements shows that the slope of $d\sigma/d\ensuremath{{Q^{2}}}$ changes with \ensuremath{{Q^{2}}}; at
high \ensuremath{{Q^{2}}}{} the slope is steeper than at low \ensuremath{{Q^{2}}}.
The NLO calculation describes the measured data well over the
full \ensuremath{{Q^{2}}}{} range.
The \dstar{} electroproduction cross sections were converted to
$\gamma p$ cross sections, \ensuremath{\sigma_{\gamma{} p}}, in the range $1.5 < \ensuremath{p_T(D^{\ast})} < 9{\,\text{Ge}\eVdist\text{V\/}}$ and
$|\ensuremath{\eta(D^{\ast})}| < 1.5$ (measured in the laboratory frame) using the photon flux from
Eq.~\ref{eq:flux}. The cross sections are given for $W = 160{\,\text{Ge}\eVdist\text{V\/}}$, which
corresponds to $y=0.25$, close to the mean $y$ of the measured cross sections.
The $W$ dependence of \ensuremath{\sigma_{\gamma{} p}}{} was evaluated from the data. The uncertainty of
this procedure was estimated to be 10\%. A comparison of the charm
photoproduction cross section~\cite{epj:c6:67}, this measurement and the DIS
cross sections~\cite{pr:d69:012004} is shown in Fig.~\ref{fig:gammapxsect}.
The numbers are tabulated in Table~\ref{tab:gammapxsect}. The photoproduction
point was corrected for the different kinematic range and centre-of-mass
energy used here using the FMNR
program.
As can be seen,
the present measurements are consistent with the photoproduction cross
section. A fit using a function of the form
$\sigma(\ensuremath{{Q^{2}}}) = S M^{2} / (\ensuremath{{Q^{2}}} + M^2)$,
where $S$ is the photoproduction cross section at $\ensuremath{{Q^{2}}} = 0$ and $M^2$
is the scale at which the $\gamma p$ cross section changes from the
photoproduction value to the DIS $1/\ensuremath{{Q^{2}}}$ behaviour, gives a good description
of the data over the whole \ensuremath{{Q^{2}}}{} range with $S = 823 \pm 63\,\text{nb}$ and $M^{2} =
13 \pm 2{\,\text{Ge}\eVdist\text{V\/}}^{2}$. The value of $M^{2}$ found here for charm production is
close to $4 m_{c}^{2}$~\cite{prep:15:181} and significantly larger than that
found for inclusive data $M_{0}^{2} = 0.52 \pm 0.05{\,\text{Ge}\eVdist\text{V\/}}^{2}$~\cite{pl:b487:53}.
\section{Conclusions}
\label{sec:concl}
Charm production has been measured as a function of \ensuremath{{Q^{2}}}, $y$, $\ensuremath{p_T(D^{\ast})}$
and $\ensuremath{\eta(D^{\ast})}$ in the kinematic region $0.05 < \ensuremath{{Q^{2}}} < 0.7{\,\text{Ge}\eVdist\text{V\/}}^2$, $0.02 < y <
0.85$, $1.5 < \ensuremath{p_T(D^{\ast})} < 9.0{\,\text{Ge}\eVdist\text{V\/}}$ and $|\ensuremath{\eta(D^{\ast})} | < 1.5$. These measurements
extend the previous ZEUS measurements in DIS to lower \ensuremath{{Q^{2}}}. The measured
differential cross sections are well described by two different NLO QCD
calculations: one (FMNR) is designed for the photoproduction region;
while the other (HVQDIS) is designed for DIS. Both calculations
predict similar cross sections in the intermediate \ensuremath{{Q^{2}}}{} region measured
here, which agree well with the measurements.
The measurements, converted to $\gamma p$ cross sections, also agree well
with the \dstar{} photoproduction data.
\section*{Acknowledgements}
We would like to thank B.~Harris, E.~Laenen and S.~Frixione for helpful
discussions on the application of QCD calculations in this intermediate
regime. We thank the DESY Directorate for their strong support and
encouragement. The remarkable achievements of the HERA machine group were
essential for the successful completion of this work. The design,
construction and installation of the ZEUS detector have been made possible by
the effort of many people who are not listed as authors.
|
2,869,038,155,688 | arxiv | \section{Introduction}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.98\linewidth]{IR_vit_curves.png}
\end{center}
\caption{An estimation of the intrinsic dimension of the hidden activations of $5,000$ ImageNet images using MLE and TwoNN within a ViT model. We include the stable rank for comparison. Input and output layers are omitted, so $0$ corresponds to the first hidden representation. Shaded regions indicate $95\%$ confidence intervals over $40$ randomly selected ImageNet images for stable rank and three random samplings of $5,000$ ImageNet images for MLE and TwoNN. \label{fig-ID-vs-SR-vit}}
\end{figure}
It is a common assumption within deep learning that models can achieve strong performance on high-dimensional data because this data actually lives on much lower-dimensional submanifolds that are embedded in the high-dimensional ambient Euclidean space. This idea is often termed the {\emph{manifold hypothesis}}, and it underlies much of the conceptual framework for modern deep learning. Despite the centrality of this idea, such hypothetical data manifolds remain mostly mysterious even for routinely used datasets such as ImageNet \cite{russakovsky2015imagenet}.
Even the simplest statistics, such as intrinsic dimension, remain challenging to compute with a high degree of certainty. Most intrinsic dimension estimators are only theoretically guaranteed to recover the true dimension of a data manifold as the number of samples, and hence the density of samples, goes to infinity (see for example the analysis in \cite[\S 3]{mle}). This suggests that when attempting to estimate the dimension of a data manifold, one should use as many samples as possible, i.e. the entire dataset. However, even for a large image dataset, the closest neighbors of any given point may be quite distant in pixel space and quite distinct semantically. Thus intrinsic dimensionality estimation of image datasets is data-hungry, and even in relatively data-rich situations, it is unclear whether the hypotheses guaranteeing accurate dimension estimation are met.
In this work, we describe a new tool to illuminate how models process local neighborhoods of a data manifold. Instead of tracing an outline of the manifold via a collection of nearby points from the dataset, we instead perturb a single base point in specific ways that keep it on the manifold. To do this we leverage the notion of a frame from differential geometry. A $k$-frame is an assignment of $k$ linearly independent vectors to the tangent space of each point $x$ in a $m$-dimensional manifold $M$. The span of these vectors defines a subbundle of the tangent bundle of $M$. When we apply the first $\ell$-layers of a deep learning model to this $k$-frame, we call the resulting set of hidden activations a {\emph{neural $k$-frame}} (since it will generally not be a true frame). We can then analyze how a deep learning model transforms the data manifold $M$ locally by analyzing how it transforms neural $k$-frames around various datapoints. Neural $k$-frames detect the compression and distortion of the data manifold’s tangent spaces.
Neural frames have several advantages over related methods. The first is that each neural frame calculation requires only a single datapoint, thus avoiding one of the primary challenges of intrinsic dimensionality estimators. Perhaps more importantly though, we can choose particular directions of interest in the tangent space of a data manifold at a given point and then construct neural frames that capture these. For example, we describe a number of interesting $k$-frames including an {\emph{augmentation frame}} constructed via small continuous augmentations (e.g., subtle changes in hue, small rotations of the image, see Figure \ref{fig-example-augmentations}) and a $k$-frame built from local sampling performed using a diffusion model \cite{Rombach_2022_CVPR} and the recently proposed Boomerang method \cite{luzi2022boomerang}. Thus, neural frames are a more targeted analytic than intrinsic dimensionality estimators. Note that in both the frames mentioned above, we avoid the problem of needing densely sampled data by either leveraging our knowledge of naturalistic image augmentations or the availability of robust generative models (both of which approximately produce points which remain on the image manifold).
We apply neural $k$-frames to a range of different models with distinct architectures and training routines and draw a range of conclusions about how different computer vision models process image manifolds. For example, we show that well-trained CNNs and vision transformers exhibit starkly different behavior when processing the augmentation frames, diffusion frames, and random collections of linearly independent (off-manifold) frames, suggesting that these models are at some level “aware” of the tangent space of the data manifold. We also find qualitative similarities between measurements of augmentation neural frames and intrinsic dimension of hidden features, suggesting that in data poor settings, neural frames may be used in place of intrinsic dimension estimators to analyze some questions. Finally, we find that certain aspects of training leave strong imprints on the way that models process the local neighborhood around a data point. In particular, adversarially robust models, models trained with heavy augmentation, and models trained using more vanilla approaches all have a distinct signature in terms of their neural frames.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.7\linewidth]{diagram.png}\\
\end{center}
\caption{A cartoon visualizing an augmentation frame with three tangent vectors (derived from hue shift, brightness shift, and a small rotation and crop). These are linearly dependent in the picture because we are restricted to showing a 2-dimensional tangent space. In reality, they will all be linearly independent. Augmentations are exaggerated to make them more easily visible. \label{fig-diagram}}
\end{figure*}
In summary, our contributions in this work include the following:
\begin{itemize}\itemsep0em
\item We describe {\emph{neural frames}}, a flexible tool inspired from the notion of a $k$-frame from differential geometry which can probe how deep learning models process data manifolds at the local level (even when limited data is available).
\item We investigate the geometry behind neural frames as well as outline a few concrete examples of specific frames which give especially interesting results in experiments.
\item We use neural frames to reach several conclusions about the impact of training method and model architecture on the way that a deep learning model processes local patches of an image manifold.
\end{itemize}
\section{Related work}
Data manifolds and their relationship to deep learning have been studied through the lens of various tools. One approach has been to study local properties of manifolds (that is, those properties that can be measure by only considering a small neighborhood around a datapoint). The most notable such property is intrinsic dimension. There is a rich collection of methods for estimating the intrinsic dimension of a dataset, though most of these were adapted for lower-dimensional data than what we consider in this work (see for example \cite{grassberger2004measuring,fukunaga1971algorithm,farahmand2007manifold,albergante2019estimating,kvinge2018monitoring} and \cite{campadelli2015intrinsic} for a helpful survey).
In this paper we compare to maximum likelihood estimation (MLE) \cite{mle} and TwoNN \cite{facco2017estimating} since these are popular methods that have already been used to understand how deep learning models learn. TwoNN was used in \cite{ansuini2019intrinsic} to show that the intrinsic dimension of data decreases as it progressively passes deeper in a network and that the intrinsic dimension of data in the final layers of the model is negatively correlated with model performance. In a similar vein, MLE was used by \cite{pope2021intrinsic} to conduct experiments suggesting that the intrinsic dimension of a data manifold may determine how much data is required to train a successful model. Local manifold properties like intrinsic dimensionality have been especially useful when studying detection of adversarial examples and other anomalous data \cite{ma2018characterizing,weerasinghe2021local,houle2017local,kvinge2019rare}. More generally, \cite{amsaleg2017vulnerability} linked high intrinsic dimensionality of a dataset to the adversarial vulnerability of models trained on that data. Manifolds can also be characterized by their global properties and tools to extract these properties have also been used to understand deep learning (e.g., \cite{naitzat} used synthetic datasets to investigate how deep learning models unravel and simplify topologically complex datasets as data passes through the model).
Like many of the works summarized above, neural frames are a tool to analyze local neighborhoods of a data manifold. We show that when neural frames are applied to the hidden representations of deep learning models, we pick up some of the same phenomena detected by previous tools like intrinsic dimension. This is significant since neural frames can be applied even for datasets that are very small or sparsely sampled. On the other hand, neural frames can be viewed as complementary to other approaches. For example, while they can lower bound the intrinsic dimension, in most cases neural frames do not capture it explicitly. Neural frames can probe specific subspaces of the tangent space of a data manifold (e.g., perturbations corresponding to specific types of continuous augmentations) whereas intrinsic dimension and related statistics are blind to these differences. Finally, since they do not depend on additional datapoints to infer manifold structure, neural frames allow us to probe a much smaller neighborhood of the image manifold than other methods. For example, the average distance between an ImageNet image and its closest neighbor is approximately $82 \pm 16$ in terms of $\ell_2$-distance in pixel space, whereas our neural frames use perturbations that push a point a distance of approximately $10$ on average.
\section{Natural images, manifolds, tangent spaces, and vector bundles}
Since we will utilize the term `natural image' frequently in this work, we here give an explicit example of what we mean.
\begin{definition}
A \emph{natural image} is an image that was or could have been captured with a handheld camera (e.g., smartphone, point-and-shoot, or DSLR).
\end{definition}
Note that our definition is generous in that we allow it to include images that have been altered via transformations that many digital cameras are already performing behind the scenes (e.g., shifts in hue, etc). This is important since one of our primary examples of a neural frame, the augmentation frame, is built from small perturbations in the direction of these types of transformations. In this paper we take the word `manifold' to mean a smooth manifold in the formal sense.
A comprehensive reference on manifolds is \cite{lee2013smooth} --- here for convenience we briefly introduce key geometric objects of interest: tangent bundles, their sub-bundles, and frames.
Let $M \subset \mathbb{R}^n$ be an $m$-dimensional smooth manifold, $x$ a point on $M$, and suppose $\gamma: (-1, 1) \rightarrow M$ is a smooth path on $M$ such that $\gamma(0) = x$. The {\emph{tangent vector}} associated with $\gamma$ at $x$ is the derivative of $\gamma$ at $0$, $\gamma'(0)$. The tangent space $T_xM$ of $M$ at $x$ is the vector space of all such tangent vectors \(\gamma'(0)\) at $x$ (here \(\gamma\) varies over all possible smooth paths in $M$ that pass through $x$). The {\emph{tangent bundle}} $TM$ of $M$ is the union of all tangent spaces for each $x \in M$, $\coprod_{x \in M} T_xM$ --- it is a manifold in its own right of dimension $2m$. For more details see \cite[\S 3]{lee2013smooth}.
The tangent bundle can be thought of as the assignment of a vector space to each point in $M$; the concept of a vector bundle generalizes the tangent bundle of a manifold. Informally, a vector bundle is a smooth map $\pi: E \rightarrow M$ such that the fibers \(E_x := \pi^{-1}(x) \subseteq E\) are finite-dimensional real vector spaces isomorphic to $\mathbb{R}^l$ and they vary smoothly with respect to \(x\) in the sense that for any $x \in M$ there is a neighborhood $U$ of $x$ with a diffeomorphism $\varphi_x: \pi^{-1}(U) \to U \times \mathbb{R}^l$. A sub-vector bundle \(F \subseteq E \) is an embedded submanifold which is itself a vector bundle over \(M\) (with respect to the induced smooth map \(F \subseteq E \xrightarrow{\pi} M\)) such that for each point \(x\in M\), \(F_x \subseteq E_x\) is a linear subspace. For formal definitions we refer to \cite[\S 10]{lee2013smooth}. Our interest is in sub-vector bundles of tangent bundles $TM$.
A common way to obtain such sub-vector bundles in practice is from vector fields. Recall that in the case of the tangent bundle, the map $\pi: TM \rightarrow M$ is the map such that if $z \in T_xM \subset TM$, then $\pi(z) = x$. Then a smooth vector field is formally a smooth function \(v: M \to TM \) with the property that \(\pi(v(x)) = x \) for all \(x \in M\) (informally this corresponds to the usual notion that a vector field consists of a choice of tangent vector for each $x \in M$). We can use parametrized functions from our manifold to itself to construct vector fields.
\begin{lemma}[{cf. \cite[Prop. 9.7]{lee2013smooth}}]
\label{lem:flows}
If $f: (-1,1) \times M \rightarrow M$ is a smooth function on smooth manifold $M$ with the property that \(f(0, x) = x\) for all \(x \in M\), then the function \(v(x) = \frac{\partial}{\partial t}f(t,x)|_{t=0}\) is a smooth vector field.
\end{lemma}
A {\emph{$k$-frame}} of a finite vector space $V$ of dimension $m \geq k$ is a set of $k$ linearly independent vectors. A {\emph{$k$-frame on $m$-dimensional manifold $M$}} is a choice of $k$-frame for each tangent space $T_xM$ which varies smoothly with respect to the structure of $M$. Given \(k\) vector fields, there is always an open set where they form a \(k\)-frame, whose span is a subbundle of the tangent bundle.
\begin{lemma}
\label{lem:non-vanishing-determinants}
Let \(v_1, \dots, v_k : M \to TM \) be smooth vector fields on an $m$-dimensional manifold $M$.
\begin{enumerate}[(i), nosep]
\item The set $U$ of all $x$ in $M$ such that $v_1(x), \dots, v_k(x)$ are linearly independent is open.\footnote{although possibly empty}
\item The (sub)spaces \(\mathrm{span}(v_1(x), \dots, v_k(x)) \subseteq T_x M\) form a sub-vector bundle of the tangent bundle of \(U\).
\end{enumerate}
\end{lemma}
We put the two lemmas above together to give the statement that will form the basis for one type of neural frames that we introduce in the next section.
\begin{corollary}
\label{cor-bundle-from-functions}
Suppose that $\mathcal{F} = \{f_i: (-1,1)\times M \to M \, | \, i = 1,\dots, k\}$ is a collection of smooth maps such that $f_i(0,x)=x$ for all \(x \in M\), and let $v_i(x) = \frac{\partial}{\partial t}f(t,x)|_{t=0}$. If $v_1(x), \dots, v_k(x)$ are linearly independent in $T_xM$ for all $x \in M$, then $f_1,
\dots, f_k$ define a $k$-dimensional vector bundle on $M$ which we denote by $V_{\mathcal{F}}$ and $v_1(x), \dots, v_k(x)$ is a $k$-frame of this vector bundle.
\end{corollary}
The geometric machinery we have introduced in this section will provide the framework for neural frames, which we introduce below. We note however that this framework is supposed to act as a guide, not a guarantee. Indeed, by necessity, we will have to violate certain assumptions when running experiments. For example, many popular deep learning architectures are not actually smooth everywhere and some of our augmentations will not be smooth either (largely due to the discrete nature of digital images). However, we have tried to choose functions that are at least moderately well-behaved.
\section{Neural frames}
\label{sect-neural-frames}
Suppose that we have a data manifold $M$ embedded in ambient space $\mathbb{R}^n$ along with smooth functions $\mathcal{F} = \{f_1, \dots, f_k\}$ with $f_i:(-1,1) \times M \rightarrow M$. If $f_1, \dots, f_k$ satisfy the conditions of \cref{cor-bundle-from-functions}, then we obtain a sub-vector bundle of the tangent bundle of $M$, $V_{\mathcal{F}}$, along with a frame $v_1(x), \dots, v_k(x)$.
Suppose that
\begin{equation*}
\begin{tikzcd}
\mathbb{R}^n \arrow[r, "F_1"] & \mathbb{R}^{n_2} \arrow[r, "F_2"] & \mathbb{R}^{n_3} \arrow[r, "F_3"] & \cdots \arrow[r, "F_l"] & \mathbb{R}^k
\end{tikzcd}
\end{equation*}
is a neural network that decomposes into $\ell$ layers, i.e. $\ell$ is the depth of the network.
We write $F_{\leq i} := F_i \circ \dots \circ F_1: \mathbb{R}^n \rightarrow \mathbb{R}^{n_{i+1}}$ to denote the function that consists of the first $i$ layers of $F$. While there is in general no way to use $F_{\leq i}$ to push forward a vector bundle $V_{\mathcal{F}}$ and hence a $k$-frame, if $F_{\leq i}$ is smooth, then its differential is a linear map $dF_{\leq i}: T_xM \to T_{F_{\leq i}(x)}\mathbb{R}^{n_{i+1}}$.
This setup allows us to define neural frames.
\begin{definition}
Let $v_1, \dots, v_k$ be a $k$-frame on $m$-dimensional manifold $M$. Let $F$ be a neural network as above. Then $dF_{\leq i}(v_j(x))$ is a tangent vector in $T_{F_{\leq i}(x)}\mathbb{R}^{n_{i+1}}$ and the {\emph{neural $k$-frame at layer $i$ of $F$ and point $x$}} is the set of vectors $dF_{\leq i}(v_1(x)), \dots, dF_{\leq i}(v_k(x))$.
\end{definition}
Informally, a neural $k$-frame is simply the object we get when we push a true $k$-frame through the first $i$-layers of a model.
Even when we have $F_{\leq i}$ and $v_j(x)$, how do we actually compute $dF_{\leq i}(v_j(x))$? This is fortunately simpler than it perhaps looks. Assuming that $v_j(x)$ is derived from a function $f_j: (-1,1)\times M \to M$ as above, fixing \(x \in M \) and letting \(t \in (-1,1)\) vary, the composition \(F_{\leq i} (f(t, x))\) is a smooth path in $\mathbb{R}^{n_{i+1}}$ such that \(F_{\leq i} (f(0, x)) = F_{\leq i}(x)\), and
\begin{equation}
\label{eq:chain}
dF_{\leq i}(v_j(x)) = \frac{\partial F_{\leq i} (f(t, x))}{\partial t}\Big|_{t=0}.
\end{equation}
In practice, we compute $v_i(x) = \frac{\partial}{\partial t}f_i(t,x)|_{t=0}$ by numerically approximating the partial derivative and we compute $dF_{\leq i}(v_j(x))$ by approximating the derivative on the right hand side of \cref{eq:chain}.
Once we have a neural frame, $dF_{\leq i}(v_1(x)), \dots, dF_{\leq i}(v_k(x))$, we can extract a range of statistics that help diagnose what $F_{\leq j}$ is doing to the data manifold at point $x$. Since we assume that $v_1(x), \dots, v_k(x)$ belong to a sufficiently small neighborhood $U$ of $M$ such that $U$ is approximately linear, an initial idea may be to look for changes in rank of the matrices $A_{x,i}$ with columns $dF_{\leq i}(v_1(x)), \dots, dF_{\leq i}(v_k(x))$, as $i$ varies.
Unfortunately, rank is sensitive to noise and is hence unsuitable for this application. An appealing alternative is \emph{stable rank} \cite{rudelson2007sampling}, which is the ratio between squared Frobenius norm and the squared spectral norm of a matrix. For $A_{x,i}$ this is
\begin{equation*}
r(A) := \frac{||A_{x,i}||_{\mathrm{frob}}^2}{||A_{x,i}||_{\mathrm{spec}}^2}.
\end{equation*}
It can easily be calculated by computing (in terms of the sorted singular values $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_k$ of $A_{x,i}$) \(r(A) = \frac{1}{\sigma_1^2} \sum_i \sigma_i^2\).
Since $k$ is small in practice ($< 50$), computing stable rank via a singular value decomposition of $A_{x,i}$ is quick. We note that stable rank has found a number of useful applications to deep learning \cite{sanyal2019stable}.
Stable rank is a lower bound on rank, and we can interpret a decrease in stable rank when moving from $v_1(x),\dots,v_k(x)$ to $dF_{\leq i}(v_1(x)), \dots, dF_{\leq i}(v_1(x))$ to mean that $F_{\leq j}$ compresses the data manifold $M$ in the directions captured by $v_1(x),\dots,v_k(x)$. This of course does not mean that $F_{\leq j}$ compresses all of $M$ at $x$ since in general $v_1(x),\dots,v_k(x)$ will only span a subspace of the tangent space of $M$ at $x$. Nevertheless we will see that some of results we obtain using stable rank reflect patterns seen in past intrinsic dimension experiments. It is worth mentioning that while true linear algebraic rank has the property that the rank of a product \(AB\) is at most the minimum of the ranks of \(A\) and \(B\), this fails for stable rank. However, the extent of this failure is controlled by the behavior of the top singular values of \(A, B\) and \(AB\) --- precisely:
\begin{lemma}
\label{lem:stable-rank-noninc}
If \(A \) and \(B\) are \(l \times m \) and \(m \times n\) matrices respectively, then
\begin{equation}
\label{eq:stable-rank-noninc}
r(AB) \leq \Big(\frac{\lVert A \rVert_{\mathrm{spec}} \lVert B \rVert_{\mathrm{spec}}}{\lVert AB \rVert_{\mathrm{spec}}}\Big)^2 \min \{r(A), r(B) \}.
\end{equation}
\end{lemma}
We include \cref{lem:stable-rank-noninc} since when taking \(A = dF_{i+1} \) and \(B = dF_{\leq i} \), so that \(AB = dF_{\leq i+1}\), \cref{lem:stable-rank-noninc} seems to explain to some extent the general decreasing trend in the curves of \cref{fig-stable-rank-robust-resnet50-aug,fig-stable-rank-augmentation,fig-stable-rank-different-frames-resnet50,fig-ID-vs-SR-vit}. Note that by the multiplicative property of the spectral norm, the first factor on the right hand side of \cref{eq:stable-rank-noninc} is always \(\geq 1 \). Of course, \cref{lem:stable-rank-noninc} is only an approximation when applied to deep learning models since it ignores the non-linearities.
We end this section by discussing two different types of frames which are particularly interesting to explore in computer vision.\\
\vspace{1mm}
\noindent\textbf{Augmentation frames:} This neural frame is generated by image augmentations $f_i: (a,c) \times M \rightarrow M$ with the following properties: (1) as implied by the domain and range of $f_i$ above, $f_i$ transforms one natural image (on $M$) to another natural image (on $M$). While this latter image was not actually captured by a camera (instead being produced in software) it should be plausible that it could have been. (2) Aside from the input image, $f_i$ is also controlled by a parameter $t \in (a,c)$ for $a < c \in \mathbb{R}$ such that for some $b \in (a,c)$, $f_i(b,x) = x$ for all $x \in M$. For example, if $f_{\text{rot}}: (-180,180) \times M \rightarrow M$ is image rotation, with the first parameter measuring the number of degrees that an image will be rotated, then it is always the case that $f_{\text{rot}}(0,x) = x$. Note that such functions can always be reparametrized to fit the form in \cref{lem:flows} and \cref{cor-bundle-from-functions}.
The frame $v_1,\dots,v_k$ derived from $f_1, \dots, f_k$ describes a number of pseudo-naturalistic directions in which an image can vary without leaving the image manifold. The neural frame associated with this frame tells us how a model handles change in these directions locally. In Table \ref{tab:augmentation-transformations} in the Appendix, we list the image augmentations that we used, the library we used to implement them, and the augmentation parameters that were used in our experiments.
Some image augmentations come with more than a single real parameter that a user can choose from. For example, when rotating an image, one can often pick the pixel coordinates of the point which will be the folcrum of the rotation (for example, in \cite{marcel2010torchvision}). How many versions of the augmentation should one add to the augmentation frame in such cases? In a $224 \times 224$ image we have $50176$ pixels that we could rotate around. How many can be added before the corresponding tangent vectors become linearly independent? In cases where the underlying augmentation corresponds to the action of a Lie group (including this case, where the Lie group is the special Euclidean group $SE(2)$), Lie theory can provide an answer. We begin by recalling that the action of a Lie group $G$ on a manifold $M$ induces a linear map from the Lie algebra $\mathfrak{g}$ to $T_xM$ for any $x \in M$.
\begin{proposition} \cite[Theorem 20.15]{lee2013smooth}
\label{prop-lie-algebra}
Let \(\mathfrak{g} = T_0 G\) be the Lie algebra of \(G\), suppose \(x \in M \), and define \(\mathrm{ev}_x: G \to M \) as \(\mathrm{ev}_x(g) = \rho(g, x) \). Then $\mathrm{ev}_x$ is a smooth map and the differential of \(\mathrm{ev}_x \) at the identity element \(e \in G\) is a linear map \(d \mathrm{ev}_x: \mathfrak{g} \to T_x M\).
\end{proposition}
Given Proposition \ref{prop-lie-algebra}, our problem is equivalent to identifying the dimension of the image of \(d \mathrm{ev}_x\)
This will tell us the maximum number of linearly independent tangent vectors that can be generated by the action of $G$.
To state the solution, we require a piece of terminology: the \emph{stabilizer} of a point \(x \in M\) is the subgroup \( G_x = \{g \in G \, | \, gx =x \}\).
\begin{proposition}
\label{prop:orbit-stab}
The natural map \(\mathfrak{g} \to T_x M\) is injective if and only if the stabilizer \(G_x \) is discrete.
\end{proposition}
In our rotation example
the stabilizer $SE(2)_x$ of a natural image \(x\) almost always consists of the identity alone, hence is in particular discrete.
Since the dimension of Lie group $SE(2)$ is 3 and a Lie algebra's dimension (as a vector space) is equal to the manifold dimension of its corresponding Lie group, the subspace of $T_xM$ spanned by tangent vectors generated by all possible rotations at different points in a image is 3. Thus we conclude that for most images we only need to include tangent vector approximations for rotations at 3 points in an image. In our experiments we choose to rotate at pixels $(0,0), (50,50),$ and $(-50,50)$.
\\
\vspace{.1mm}
\noindent\textbf{Diffusion frames:} In the recent work \cite{luzi2022boomerang}, Luzi et. al. describe how diffusion models can be used to sample locally around an image. In essence, the method they describe, called Boomerang, adds a user chosen amount of noise to an image (driving it toward the latent space of the model) and then uses the diffusion model to bring it back to the space of natural images. In this process the image will be subtly altered in a naturalistic way. This method fits nicely within our scheme of neural frames: we use Boomerang to produce $k$ distinct perturbations of an image, then we define a $k$-frame with these. We assume that the perturbations generated by the diffusion model are small enough so that the linear path from a perturbed image to the real image lies on the image manifold.
Note that since we generate random diffusion-based perturbations around each image independently, our diffusion frames are only frames on a small neighborhood around an image. This is in contrast to augmentation frames which, being systematically and nearly-smoothly defined for all images, are an approximation of a frame on the entire manifold.
\section{Experiments}
We use neural frames to probe the local behavior of deep learning models.
We give full experimental details in Section \ref{appendix-experimental-details} in the supplementary material.
Unless noted otherwise, we use publicly available weights from torchvision \cite{marcel2010torchvision} or timm \cite{rw2019timm}.
To simplify diagrams, we omit layer names providing their numerical correspondence in Tables \ref{table-vit-layers}-\ref{table-alexnet-layers} in the supplementary material. \\
\noindent\textbf{The semantic content of a frame impacts the way that a model processes it:} It is reasonable to ask whether a CNN or transformer actually ``sees'' different frames differently. One might worry that on the small scale that we work, the semantic differences in different frames are not registered by the model. For example, might a model process tangent vectors pointed in the direction of hue change the same way it processes a random vector?
To test this, we look at the stable rank of neural frames at different layers of a ResNet50 \cite{he2016deep} with ImageNet trained weights \cite{marcel2010torchvision}. The frames we use include:
(1) \emph{Gaussian noise:} We perturb an image with random Gaussian noise with mean and variance which we normalize to match the statistics of vectors in our augmentation frame. Note this is not a frame on the image manifold itself. (2) \emph{Augmentation frame:} We use the augmentations listed in Table \ref{tab:augmentation-transformations} to generate an augmentation frame (example images of this frame are found in Figure \ref{fig-example-augmentations} in the supplementary material).
(3) \emph{Random rotation of augmentation frame:} We randomly rotate the augmentation frame above so it retains its geometric structure but loses its semantic meaning.
(4) \emph{Perturbations via stable diffusion:} We use the Boomerang method \cite{luzi2022boomerang} to sample around an ImageNet image and take these samples as perturbations to build a frame (example images of this frame are found in Figure \ref{fig-example-sd}).
The results are shown in Figure \ref{fig-stable-rank-different-frames-resnet50} (Figure \ref{fig-stable-rank-different-frames-vit} in the supplementary material shows the same experiment performed on a ViT). We can see that even at the coarse level of stable rank, these neural frames each exhibit quite distinct behavior when processed by the models. The neural frame generated from Gaussian noise predictably has the highest stable rank in the ambient space (layer 0), but this drops quickly in both models as the frame is processed. This process occurs over a single block in the ResNet50 but more gradually in the ViT. Do these results suggest that vision transformers retain certain types of information to a greater depth than CNNs? As we will show in the rest of our experiments below, sensitivity to specific directions tends to be highly impacted by training that includes perturbations in those directions.
As can be seen in layer 0, the noise frame has significantly higher stable rank in ambient space than the augmentation frame (pairs of vectors in the noise frame tend to be more orthogonal when compared to pairs of vectors in the augmentation frame).
To test whether the differences that we see between the augmentation frame and noise frame in Figure \ref{fig-stable-rank-different-frames-resnet50} actually arise from their semantic meaning and not this structural property, we take the augmentation frames and randomly rotate them with an orthogonal matrix at each point in the input space.
This does not change their intrinsic structure (such as angles between pairs of vectors), but it does erase the semantic information associated with the augmentations. Indeed, we see that after rotation, the stable rank of the augmentation frame and its randomly rotated counterpart are the same in the ambient space. On the other hand, the stable rank of the rotated neural frame changes significantly compared to the augmentation neural frame as these pass through the networks. For both the ViT and ResNet50, the rotated augmentation neural frame tends to have lower stable rank, indicating that the semantic content of the augmentation perturbations impacts the way that a model processes these frames. Finally, the diffusion frame has low stable rank in the ambient space and remains relatively constant as it travels through the model.\\
\noindent\textbf{Summary:} The semantic content of a frame impacts the way that a model processes it. Stable rank detects these differences.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.98\linewidth]{stable_rank_resnet50_frames_compare.png}
\end{center}
\caption{The stable rank of different types of frames measured at various layers of a ResNet50 model trained on ImageNet. Layer zero corresponds to model input and Layer 6 corresponds to model output. Shaded regions indicate $95\%$ confidence intervals over $40$ randomly selected ImageNet images. \label{fig-stable-rank-different-frames-resnet50}}
\end{figure}
\subsection{A comparison to intrinsic dimensionality estimation techniques}
As mentioned in the introduction, our tool shares some similarities to intrinsic dimensionality estimators. To explore this connection, we measured the intrinsic dimensionality of the hidden activations of an ImageNet trained ViT \cite{marcel2010torchvision} over $5,000$ images from the ImageNet training set. We used MLE \cite{mle} and TwoNN \cite{ facco2017estimating} to measure intrinsic dimensionality. In Figure \ref{fig-ID-vs-SR-vit} we plot these (left axis) alongside the stable rank of these two models with respect to the augmentation frame. We see that both plots are surprisingly similar up to scaling, suggesting that many of the patterns that are picked up by these intrinsic dimensionality estimators can also be detected locally using our method. We stress that while each intrinsic dimensionality calculation required running $5,000$ images through the model, each calculation of stable rank required just a single image that was then augmented in a small number of ways. Furthermore, as we saw in the previous section, one can pick many different types of neural frames that provide different information about the behavior of a model. Indeed, in the next section we see that by exploring different choices of frames we can make interesting observations about adversarial robust models vs. models trained with strong forms of augmentation.
While the trends in Figure \ref{fig-ID-vs-SR-vit} are broadly similar, they do contain some differences. Further differences can be found in the supplementary material in Figure \ref{fig-ID-vs-SR-resnet50}, which shows the result of the same experiment for an ImageNet trained ResNet50 \cite{marcel2010torchvision}. \\
\noindent\textbf{Summary:} While neural frames do not capture intrinsic dimension explicitly, they do appear to capture many of the same qualitative patterns as this statistic, but with far fewer datapoints.
\subsection{Adversarial training and stable rank}
It has been empirically confirmed via a range of different methods that adversarial training has effects on the way that computer vision models process data at the local level \cite{engstrom2019adversarial}. It thus makes sense to ask whether such models process frames differently than models trained with standard methods. To investigate this question, we calculated the stable rank for 5 layers of several different ResNet50 models, each trained with a different $l_2$-robust $\epsilon$ bound of adversarial training. These weights were obtained from \cite{salman2020adversarially}.
On the left in Figure \ref{fig-stable-rank-robust-resnet50-aug} we show the stable rank over $40$ random ImageNet images with respect to the augmentation frame and models with various strengths of adversarial training (here $\epsilon$ gives the $\ell_2$ bound on adversarial examples shown to the model during training). We observe that the stable rank of our augmentation frames generally decreases slightly as the strength of adversarial training increases. Furthermore, these differences are most pronounced at earlier layers of the model. On the other hand, we can see that when we substitute the augmentation neural frame for the off-manifold noise neural frame (right, Figure \ref{fig-stable-rank-robust-resnet50-aug}) that stable rank generally increases as the strength of adversarial training increases. We conjecture that this different behavior for different frames may be related to the well-known trade-off between adversarial robustness and clean accuracy \cite{tsipras2018robustness,pmlr-v97-zhang19p}. An adversarially trained model’s effort to learn noise directions may come at the expense of its ability to preserve on-manifold directions. Finally, in the supplementary material (Figure \ref{fig-stable-rank-sd}, left), we show that when evaluated on the diffusion frame, increased adversarial robustness results in growth of stable rank in early layers of the model. This may be related to the fact that noise (which adversarially robust models encounter during training) is a fundamental component in the generation of the diffusion frame.\\
\noindent\textbf{Summary:} Adversarial training tends to preserve off-manifold (noise) frames in early layers, but behaves like standard training for on-manifold frames.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.48\linewidth]{stable_rank_resnet50_robust.png}
\includegraphics[width=.48\linewidth]{stable_rank_resnet50_robust_noise.png}
\end{center}
\caption{{\textbf{(Left)}} The stable rank (by layer) of adversarially trained $l_2$-robust ResNet50 models with varying $\epsilon$ values evaluated with respect to augmentation frames on ImageNet images. Layer zero corresponds to model input and the last layer corresponds to model output. {\textbf{(Right)}} The same models evaluated on noise frames. Shaded regions indicate $95\%$ confidence intervals over $40$ randomly selected ImageNet images. \label{fig-stable-rank-robust-resnet50-aug}}
\end{figure*}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.48\linewidth]{stable_rank_resnet50_robust_SD_frame.png}
\includegraphics[width=.48\linewidth]{stable_rank_resnet50_augmentation_SD-frame.png}
\end{center}
\caption{\textbf{(Left)} The stable rank (by layer) for ResNet50s with various levels of adversarial training evaluated on diffusion frames. The larger $\epsilon$ is, the more adversarially robust we consider the model to be. {\textbf{(Right)} The same experiments run with models using different types of augmentation (and one adversarially trained model for comparison). Shaded regions indicate $95\%$ confidence intervals over $40$ random ImageNet images.} \label{fig-stable-rank-sd}}
\end{figure*}
\subsection{Stable rank and augmentation}
We perform an experiment analogous to the one above, except that we substitute adversarially robust ResNet50 models for ResNet50 models trained with different types of augmentation. We consider ResNet50 models trained with the augmentation methods PRIME \cite{modas2021prime},
Deep Augmentation \cite{hendrycks2021many},
and Stylized ImageNet \cite{geirhos2018imagenet}. Our results are shown in Figure \ref{fig-stable-rank-augmentation}. We see that the behavior of these models is significantly different than what we saw with adversarially robust models. Those models trained with Deep Augmentation preserve the augmentation neural frames substantially better than an adversarially robust model (with $\epsilon = 5$) does, and somewhat better than models with standard training or augmentation training using style transfer (red and pink). This may reflect the fact that Deep Augmentation is more similar to the kinds of shifts present in the augmentation frame, again pointing to the fact that the variation a model sees during training has a strong impact on the way that it processes a manifold in the neighborhood of a datapoint.
Interestingly, we find that while models trained with Deep Augmentation do not preserve noise neural frames as well as adversarially robust models in earlier layers. In deep layers, the opposite is true. Figure \ref{fig-stable-rank-aug-resnet50-noise} in the supplementary material shows the result of such an experiment.\\
\noindent\textbf{Summary:} Training with certain augmentations tends to preserve on-manifold augmentation frames in deeper layers of a model, and it preserves off-manifold (noise) frames if noise was an augmentation used.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.98\linewidth]{stable_rank_resnet50_augmentations_aug_frame.png}
\end{center}
\caption{The stable rank (by layer) of ResNet50 models evaluated with respect to augmentation frames on ImageNet images where each model was trained with a different augmentation method. Layer zero corresponds to model input and the last layer corresponds to model output. Shaded regions indicate $95\%$ confidence intervals over $40$ randomly selected ImageNet images. \label{fig-stable-rank-augmentation}}
\end{figure}
\subsection{Stable rank over the course of training}
To better understand how the stable rank of a frame changes over the course of training, we saved the weights of a ResNet18 \cite{he2016deep} trained from scratch on ImageNet every $10$ iterations (for $1000$ iterations) and then every $100$ iterations for the approximately $16$ remaining epochs. The training hyperparameters that we used can be found in Table \ref{table-hyperparameters}.
In Figure \ref{fig-stable-training} we show the stable rank (by layer) for this ResNet18 with respect to an augmentation frame at different stages of training. We see that at a large scale the general trend is for stable rank to increase as training increases, but that these changes are most significant in the later layers of the model. For example, the latent space layer (layer 5), increases from an initial stable rank around $1.5$ to a stable rank of $3.5$, an increase of $2$, while the stable rank of layer 1 (in one of the first blocks of the model), only increases from $3.5$ to $4$, an increase of only $.5$. We conjecture that one effect of the later stages of training is that a model gains the tendency to preserve those frames related to natural changes of an image. This guess is supported by Figure \ref{fig-training-early} (right) which shows that while stable rank increases throughout training for frames of naturalistic directions, it decreases for noise frames whose directions lack any connection to the content of the image.
Interestingly, we find stable rank also peaks (though not as high as later) once in the early iterations of training. In Figure \ref{fig-training-early} (left) we see that stable rank increases for approximately the first 50 iterations of training and then decreases again until around iteration 200. It then slowly increases again for the rest of training. It would be interesting to understand what drives these dynamics.\\
\noindent\textbf{Summary:} Stable rank tends to increase (especially in later layers) over the course of training for augmentation frames. It decreases for noise frames. There is a small spike in stable rank for augmentation frames in the first $100$ iterations of training.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.48\linewidth]{stable_rank_resnet18_training_low-res.png}
\includegraphics[width=.48\linewidth]{stable_rank_resnet18_training_layer_curves.png}
\end{center}
\caption{\textbf{(Left)} The stable rank of an augmentation frame (by layer) for a ResNet18 trained from scratch. Different colored curves correspond to the number of iterations of training that the model has undergone. {\textbf{(Right)}} The stable rank of different layers of the model as a function of the number of training iterations. Shaded regions in both plots indicate $95\%$ confidence intervals over $40$ random ImageNet images. \label{fig-stable-training}}
\end{figure*}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.48\linewidth]{general_CNNs.png}
\includegraphics[width=.48\linewidth]{general_transformers.png}
\end{center}
\caption{The stable rank, by layer, for a range of different deep learning architectures. {\textbf{(Left)}} Different CNN architectures, {\textbf{(right)}} transformer architectures. Shaded regions indicate $95\%$ confidence intervals over $40$ random ImageNet images. \label{fig-stable-rank-architectures}}
\end{figure*}
\subsection{Stable rank and architecture}
In Figure \ref{fig-stable-rank-architectures} we plot the stable rank (as a function of layer) for an augmentation frame and two different families of architectures: CNNs (left) and vision transformers (right). Shaded regions depict $95\%$ confidence intervals calculated over $40$ random ImageNet images. The CNN architectures that we plot (left) are DenseNet121 \cite{huang2017densely}, InceptionV3 \cite{szegedy2016rethinking}, ResNet50 \cite{he2016deep}, and ResNeXT50 \cite{xie2017aggregated}. On the right we plot hidden layers from transformers: ViT \cite{dosovitskiy2020image} and Swin \cite{liu2021swin}. All use the default ImageNet torchvision \cite{marcel2010torchvision} weights.
We note two trends in these plots:
\begin{enumerate}[(i)]
\item \label{item:plateau} All curves consist of a plateau spanning most layers of the model followed by a dramatic dropoff in stable rank at the last layers.
\item \label{item:osc} The transfomer models exhibit significantly more \emph{fluctuation} in stable rank than the CNNs.
\end{enumerate}
\Cref{item:plateau} could be partially explained by the fact that \emph{all} models, both CNNs and transformers, include residual connections. Note that a toy residual network with \(n\)-dimensional feature spaces and identity activations consists of a composition of layers of the form \( I_n + W \), where \(I_n\) is an identity matrix and \(W\) a \(n\times n\) weight matrix. These have singular values of the form \(1 + \sigma_i\), where \(\{\sigma_i\}\) are the singular values of \(W\), and thus stable rank
\begin{equation}
\label{eq:res-stable-rank}
\frac{1}{(1+ \sigma_1)^2} \sum_i (1+ \sigma_i)^2.
\end{equation}
Suppose \(W\) is a random matrix with IID entries sampled from \(\mathcal{N}(0, \frac{2}{n})\) (this is true before training with He normal initialization \cite{he2016deep}). Then a calculation using \cite{mpdist} shows that for large \(n\) the expected stable rank of \(I_n + W \) is approximately
\begin{equation}
\frac{n}{(1+ 2 \sqrt{2})^2} \int_0^{2\sqrt{2}} (1+y^2)\sqrt{8-y^2} \frac{dy}{2 \pi} \approx 0.37 n.
\end{equation}
Provided this is larger than the stable rank of the neural frame in the preceding layer, \cref{lem:stable-rank-noninc} might suggest that \(I_n + W \) preserves the stable rank of the neural frame.\footnote{The expected stable rank of \(W \) itself is \(\frac{n}{4} \).}
As for \cref{item:osc}, while these experiments alone are insufficient to identify a reason for this apparent difference, one could make a number of different conjectures. It could be, for instance, that the priors hardcoded into CNNs dampen the extent to which the models stretch and compress an image manifold. Alternatively, it might be that the nonlinearity of attention layers leads to more geometric changes layer-to-layer. This nonlinearity is in contrast to the convolutions in CNNs, which are linear in isolation (that is, not considering the nonlinear layers that often follow them). These questions would be interesting to address in follow-up work.
As for \cref{item:osc}, while these experiments alone are insufficient to identify a reason for this apparent difference, one could make a number of different conjectures. It could be, for instance, that the priors hardcoded into CNNs dampen the extent to which the models stretch and compress an image manifold. Alternatively, it might be that the nonlinearity of attention layers leads to more geometric changes layer-to-layer. This nonlinearity is in contrast to the convolutions in CNNs, which are linear in isolation (that is, not considering the nonlinear layers that often follow them). These questions would be interesting to address in follow-up work.\\
\noindent\textbf{Summary:} The stable rank of most architectures tend to exhibit plateau behavior that may relate to residual connections. CNNs tend to have less volatility than transformers.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.48\linewidth]{stable_rank_resnet18_training_high-res.png}
\includegraphics[width=.48\linewidth]{stable_rank_resnet18_training_noise.png}
\end{center}
\caption{\textbf{(Left)} The stable rank of an augmentation frame (by layer) during early iterations of a ResNet18 trained from scratch. {\textbf{(Right)}} The stable rank of the ResNet18 model (by layer) evaluated on a noise frame. In both plots, different colored curves correspond to the number of iterations of training that the model has undergone.\label{fig-training-early}}
\end{figure*}
\section{Limitations}
While neural frames provide unique and valuable information about how a network processes the tangent bundle of a data manifold, this information is never the full story. For example, in most cases the frames we use span proper subspaces of the tangent space. Thus, there may be changes to the tangent space that we miss because they are orthogonal or nearly orthogonal to all vectors in the frame that we use. Augmentation frames may be challenging to use in specialized scenarios where augmentations that preserve the particular image manifold under consideration are not a priori known. Finally, as noted above, augmentations are not actually smooth because of the discrete nature of images. This means that the mathematical framework that we outline in Section \ref{sect-neural-frames}, while providing useful structure, will not hold ``on the nose.''
\section{Conclusion}
While data manifolds play a central role in our understanding of how and why deep learning works, extracting any tangible information about them is challenging. In this paper we provide a new tool, neural frames, to help probe the ways that deep learning models interact with data manifolds. We show that neural frames capture some of the information that can be obtained by other more data-hungry approaches, but also provide their own set of unique insights.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.19\linewidth]{brightness.png}
\includegraphics[width=.19\linewidth]{contrast_0.png}
\includegraphics[width=.19\linewidth]{crop_bilinear.png}
\includegraphics[width=.19\linewidth]{hue.png}
\includegraphics[width=.19\linewidth]{sharpness.png}\\
\includegraphics[width=.19\linewidth]{diff_brightness.png}
\includegraphics[width=.19\linewidth]{diff_contrast_0.png}
\includegraphics[width=.19\linewidth]{diff_crop_bilinear.png}
\includegraphics[width=.19\linewidth]{diff_hue.png}
\includegraphics[width=.19\linewidth]{diff_sharpness.png}\\
\includegraphics[width=.19\linewidth]{crop_linear.png}
\includegraphics[width=.19\linewidth]{rotation0.png}
\includegraphics[width=.19\linewidth]{saturation.png}
\includegraphics[width=.19\linewidth]{jpeg.png}\\
\includegraphics[width=.19\linewidth]{diff_crop_linear.png}
\includegraphics[width=.19\linewidth]{diff_rotation.png}
\includegraphics[width=.19\linewidth]{diff_saturation.png}
\includegraphics[width=.19\linewidth]{diff_jpeg.png}
\end{center}
\caption{\textbf{(First and third rows)} A subset of the sample perturbations, from left to right and top to bottom: brightness, contrast, crop with bilinear interpolation, hue, sharpness, crop with linear interpolation, rotation, saturation, jpeg compression. \textbf{(Second and fourth rows)} The difference between the original and perturbed images (above). \label{fig-example-augmentations}}
\end{figure*}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=.19\linewidth]{sd_original.png}
\includegraphics[width=.19\linewidth]{sd_0-0.png}
\includegraphics[width=.19\linewidth]{sd_0-1.png}
\includegraphics[width=.19\linewidth]{sd_0-2.png}\\
\hspace{33mm}\includegraphics[width=.19\linewidth]{sd_diff_0-0.png}
\includegraphics[width=.19\linewidth]{sd_diff_0-1.png}
\includegraphics[width=.19\linewidth]{sd_diff_0-2.png}
\end{center}
\caption{\textbf{(First row)} The original ImageNet image and perturbations of this image sampled using the Boomerang method and stable diffusion. \textbf{(Second row)} The difference between the original image and each perturbation. \label{fig-example-sd}}
\end{figure*}
{\small
\bibliographystyle{ieee_fullname}
|
2,869,038,155,689 | arxiv | \section{Introduction}
\label{intro}
The number and devastating impacts of natural disasters have grown significantly worldwide. The latest report released by Munich Re (2020) states that natural disasters such as hurricanes, floods, and other disasters caused more than \$210 billion estimated damage worldwide, while \$95 billion of the damage occurred in the US. According to the same report, summer floods in China were the costliest natural disaster worldwide in 2020, and the number of fatalities in floods was higher than other natural disasters \cite{munichre}. Recent studies show that the frequency and impact of flooding increases in certain regions as a result of climate change \cite{davenport2021contribution, ncei, tabari2020climate} due to an increase in sea level \cite{strauss2016unnatural} and frequency of extreme precipitation \cite{diffenbaugh2017quantifying}, or intensifying hurricane rainfall \cite{trenberth2018hurricane}. Thus, it is crucial to predict streamflow and, consequently, flooding to mitigate its devastating effects in terms of damage and fatalities.
Many physical and data-driven methods have been proposed to achieve accurate streamflow predictions, and recent studies show that deep learning models often provide more accurate results than physical-based models \cite{gauch2019data, xiang2020distributed, xiang2021regional}. Recurrent neural networks based approaches are mainly used for the task as a result of the success of these architectures on time series problems \cite{sit2019decentralized}. \cite{hu2018deep} proposed a Long short-term memory (LSTM) \cite{hochreiter1997long} model that predicts the hourly streamflow from 1 to 6 hours lead time. \cite{kratzert2018rainfall} developed another LSTM model that predicts daily runoff for the first time. \cite{xiang2020distributed} developed a model that uses multiple GRUs \cite{cho2014learning} and Time-distributed layers in order to predict up to 120 hours of streamflow. Many studies aim to predict streamflow based on various data such as evapotranspiration, current streamflow, or weather data. More detailed information about deep learning studies on streamflow prediction can be found in \cite{sit2020comprehensive}.
Graph Neural Networks (GNNs) and variants including Graph Convolutional Networks (GCNs), Graph Attention Networks (GATs), or Graph Recurrent Networks (GRNs) have gained much attention as a result of their performance on many deep learning tasks comprising sequenced data points which can be expressed as a graph \cite{wu2020comprehensive, zhou2020graph}. For instance, \cite{seo2018structured} proposes a convolutional recurrent neural network architecture that captures spatial dependencies with CNNs and identifies dynamic patterns with GRUs in the structured data sequences by showing two use cases; predicting moving MNIST data and modeling natural language with the Penn Treebank dataset. \cite{bai2020adaptive} uses a graph convolutional recurrent network with two new modules that allow to learn node-specific patterns and discover spatial correlations from data separately for a traffic prediction task. Based on successful implementations of GNNs on structured sequences, in this paper, we present a model based on Graph Convolutional GRUs for streamflow prediction that we will refer to as StreamGConvGRU. Furthermore, we show our preliminary results for StreamGConvGRU using real-world data for a small network of streamflow sensors. To the best of our knowledge, this is the first work that presents at least some preliminary results for streamflow prediction using Graph Neural Networks.
\section{Methodology}
GNNs expect the input data as a graph and successively outputs a graph. Since most rivers are connected to each other and form a network of rivers, locations along rivers, more specifically stream gauge locations along a river network, could be converted into a graph and fed to a GNN.
\begin{figure}[!htb]
\vskip 0.2in
\begin{center}
\centerline{\frame{\includegraphics[width=\columnwidth]{map_marked.png}}}
\caption{Visualization of study area and USGS sensor locations on Google Maps with their \textit{id\_in\_graph: usgs\_id}}
\label{sensors}
\end{center}
\vskip -0.2in
\end{figure}
\subsection{Dataset}
The United States Geological Survey (USGS) maintains a network of stream gauges while actively deploying new ones all around the United States. The historical measurements for each of these sensors with a temporal resolution of 15 minutes are publicly available. Even though there are hundreds of USGS stream gauges in each watershed, in order to provide a proof of concept model setup, we chose to use eight sensors in Johnson County, Iowa, that form a small subnetwork. Seven of these stream gauges are within the watershed of the eighth one, and they feed the water to the eighth stream gauge located at the outlet of the watershed. In this study, we aim to predict the next 36 hours of measurements for the eighth stream gauge at the outlet by predominantly using 36 hours of historical streamflow data for all eight stream gauges. USGS sensor ids of these stream gauges are as follows, 05453520, 05454000, 05454090, 05454300, 05455100, 05454500, 05455500, 05455700, the last one being the sensor for that we aim to predict the future streamflow values (Figure \ref{sensors}).
Along with historic stream measurements, the StreamGConvGRU is also fed with precipitation measurements for the past 36 hours and the next 36 hours for all the stream gauge locations. The resource we employed for the rainfall data is the Stage IV hourly multi-sensor precipitation analysis. Since all the precipitation that falls into a river’s watershed affects the streamflow, the rainfall data within every sensor’s watershed were aggregated by summing up. Also, in order to match stream gauge temporal resolution with Stage IV, the USGS sensor measurements were aggregated to hourly by averaging them.
\begin{figure}[!htb]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{data_arch_gnn.png}}
\caption{Summary of data sources and how each snapshot was formed.}
\label{data}
\end{center}
\vskip -0.2in
\end{figure}
\begin{table}[!htb]
\caption{Dataset splits and, start and end dates for each split.}
\label{dataset-split}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccr}
\toprule
Split & Start Date & End Date & \# of Snapshots \\
\midrule
Train & 10.01.2011 & 10.01.2016 & 32789\\
Validation & 10.01.2016 & 10.01.2017 & 6896\\
Test & 10.01.2017 & 10.01.2018 & 8253 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
\begin{figure*}[!ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[height=5cm,keepaspectratio]{gnn.png}}
\caption{Architecture proposed for StreamGConvGRU.}
\label{arch}
\end{center}
\vskip -0.2in
\end{figure*}
Within the selected date range (10/1/2011 - 10/01/2018), a snapshot was created for each hour except for the times when the USGS sensor 05455700 is down. In order to ensure a continuous dataset and introduce noise to the training process, we simply used zeros for when any of the seven upstream sensors did not have any measurements. The data acquisition step ends with 47,938 total snapshots. Train, validation, test split (Table \ref{dataset-split}) then was made by snapshot timestamps, and a normalization to map values to $[0-4]$ was applied to all subsets using minimum and maximum values of streamflow and aggregated precipitation observations for the USGS sensor id 05455700 in the training subset. In the end, each snapshot had an input with the size of [8 x 3 x 36] (number of stream gauges x number of time-series sequences used x length of the sequence). The output size similarly was [8 x 36] (Figure \ref{data}). Since a GNN expects graph properties, the graph for sensors was built by considering the hydrological connectivity of the nodes and the distance between USGS sensors as weights.
\begin{figure*}[!ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[height=4cm,keepaspectratio]{gru.png}}
\caption{Convolutional Bidirectional GRU network’s architecture.}
\label{gru}
\end{center}
\vskip -0.2in
\end{figure*}
\subsection{Network}
StreamGConvGRU is a GNN model based on Graph Convolutional Gated Recurrent Unit Networks (GConvGRUs) \cite{seo2018structured}. While GConvGRUs have the ability to understand spatial structures with convolutional layers, they also incorporate Gated Recurrent Unit Networks (GRUs) to understand dynamic temporal patterns. For details about GConvGRUs and Graph Convolutional Recurrent Neural Networks, please refer to \cite{seo2018structured}.
The StreamGConvGRU uses three separate GConvGRU subnetworks (Figure \ref{arch}) for each of the three sequences explained in the previous subsection, that is, previous 36 hours of discharge measurements, previous 36 hours of precipitation observations, and next 36 hours of precipitation measurements. The outputs of each of these subnetworks then are summed up and fed to a linear layer that outputs 36 hours of predictions for each of the eight sensors. It should be noted that since a GNN outputs a graph, the product of the network is also a set of sequences, but since the goal is to predict the next discharge levels for the USGS sensor 05455700, the training is done to optimize one sequence while other seven sequences were simply ignored.
We also built various GRU networks for comparison purposes. The best performing GRU network for our dataset was a Convolutional Bidirectional GRU model, which we will refer to as ConvBiGRU (Figure \ref{gru}). ConvBiGRU was trained by feeding a matrix with the shape of [24 x 36] for each snapshot instead of the graph. The matrix was created for each snapshot by combining all sequences for all USGS sensor locations within the river network. The output of the GRU network is only a sequence of subsequent discharge measurements rather than measurements for each of the sensors in the sensor network.
Networks described here were trained using RMSprop \cite{hinton2012neural} optimizer with L1 loss as the cost function on NVIDIA Titan V GPUs. During the training, the best scoring networks’ weights over the validation set were saved, and the results that will be presented in the next section were generated by predicting streamflow for the test dataset. While the ConvBiGRU was implemented using PyTorch numeric computing library \cite{paszke2017automatic}, the StreamGConvGRU was implemented using pytorch-geometric-temporal library \cite{rozemberczki2021pytorch} that is built on top of Pytorch, and pytorch-geometric libraries \cite{fey2019fast}.
\section{Results and Discussions}
Besides the neural network architecture described in the previous section, we define a naive baseline model here. Namely, persistence, which is inherently data-driven, is proposed to be used as a baseline model in streamflow prediction \cite{ghimire2020exploring}. Persistence is built relying on the idea that “tomorrow will be like today.” It has a straightforward implementation, when prediction is being done at timestep $t$, every future measurement of sensor $s$ for timestep $t’$ is predicted as $s[t-t’]$. In other words, persistence assumes all predictions will be the same as the latest measurement at time $t$.
\begin{figure}
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{results.png}}
\caption{Hourly NSE scores for StreamGConvGRU, persistence and ConvBiGRU.}
\label{results}
\end{center}
\vskip -0.2in
\end{figure}
To compare persistence, ConvBiGRU, and StreamGConvGRU, we used Nash-Sutcliffe Efficiency (NSE) score \cite{nash1970river}. NSE score is a widely used efficiency metric for streamflow prediction models, both physically based and data-driven applications, and defined as
\[NSE = 1 - \frac{\sum_{t=1}^{T} (Q_m^t - Q_o^t)^2 }{\sum_{t=1}^{T} (Q_o^t - \overline{Q_o})^2}\]
where $Q_m^t$, $Q_o^t$ and $\overline{Q_o}$ mean modeled discharge at time $t$, observed discharge at time $t$ and mean of observed discharges, respectively. Please note that the best NSE score a streamflow prediction can get is $1.0$. Thus, out of two streamflow prediction models, the model with the NSE score closer to $1.0$ is the better one.
The preliminary results for hourly NSE scores are presented in Figure \ref{results}. As it can be seen in Figure \ref{results}, even though it was the best GRU-based model we built, ConvBiGRU does not get closer to persistence. The StreamGConvGRU model we present in this paper, on the other hand, outperforms both persistence and the ConvBiGRU model by a significant margin after the fifth hour. The performance of the persistence baseline for the first few hours could be explained with the approach it uses. Streamflow rates typically do not change drastically in the first few hours. However, when it starts to change, the StreamGConvGRU model takes the lead as it takes advantage of upstream nodes' data and connectivity. A similar relationship between the proposed model and persistence can be seen in \cite{xiang2020distributed}. Also, we want to stress that various others RNNs we built to compare with StreamGConvGRU did not produce comparable NSE scores; consequently, we decided not to include them here for the sake of simplicity.
\section{Conclusions}
The importance of streamflow and, consequently, flood prediction increases as a result of the devastating effects of climate change. This paper presented an approach where we employed a Graph Convolutional GRU Networks based model, StreamGConvGRU, to predict 36 hours of streamflow for a sensor location using the upstream river network. As shown in the preliminary results, the StreamGConvGRU provides better performance than the persistence baseline and a Convolutional Bidirectional GRU model in our study region for short-term hourly streamflow prediction. For the future work, we aim to focus on the following three points: (1) exploring GConvGRUs’ abilities with a more extensive river network; (2) prediction of greater lead time on streamflow; (3) prediction of streamflow on each node in the river network instead of focusing only on one node at the outlet.
|
2,869,038,155,690 | arxiv | \section{Introduction}
\setcounter{equation}0
The statistical-mechanical origin of the Bekenstein-Hawking entropy
\cite{Hawk:75}, \cite{Beke:72} is one of the most intriguing problems
of the black hole physics. There exist several promising approaches to
this problem: string theory approach (see for a review Ref.
\cite{Peet:97}), calculations of the entropy of some 3D black holes
\cite{Carlip:95}, \cite{Strominger:97}, an explanation in the framework
of the loop quantum gravity \cite{ABCK:97}, a mechanism suggested in
Sakharov's induced gravity \cite{Sakharov} and others.
In the induced
gravity approach \cite{Jacobson}--\cite{FF:97} the Bekenstein-Hawking
entropy is related to the statistical-mechanical entropy of heavy
constituent fields which induce the Einstein theory in the low-energy
limit. Gravitons in the induced gravity are analogous to
phonon excitations in condensed matter systems \cite{Volovik:98}.
A special class of induced gravity models was investigated in Refs.
\cite{FFZ}, \cite{FF:97}. These models contain heavy spinors and
scalar constituents propagating in an external
gravitational field. The dynamics of the gravitational field arises as
the result of quantum effects. The one-loop effective action for
quantum constituents gives the low energy classical action for
the Einstein gravity. The constructed models of induced gravity are
free from the leading ultraviolet divergences. The induced Newton
constant $G$ is completely determined by the parameters of the
constituents, and it is finite only if non-minimally coupled scalar
fields are present. It was demonstrated that the Bekenstein-Hawking
entropy $S^{BH}$ in the induced gravity can be written as
\begin{equation}\label{i1}
S^{BH}={{\cal A}\over 4G}=S^{SM}-Q\, .
\end{equation}
Here ${\cal A}$ is the surface area of the horizon, and
$S^{SM}=-\mbox{Tr}(\hat{\rho}\ln\hat{\rho})$ is the
statistical-mechanical (or entanglement) entropy of the thermally
excited (with thermal density matrix $\hat{\rho}$) constituent fields
propagating near the horizon. The quantity
\begin{equation}\label{i1a}
Q=\sum_s\xi_s \int_{\Sigma}\langle \hat{\phi_s}^2\rangle
\end{equation}
is the sum of contributions of the nonminimally coupled scalar fields
$\hat{\phi_s}$. In this relation $\xi_s$ are parameters of nonminimal
coupling and $\langle \hat{\phi_s}^2\rangle$ is the
quantum average of the squares of the scalar operators on the
bifurcation surface $\Sigma$. In these particular models the origin of
$Q$ is related to the nonminmimal couplings of the scalar fields. It
was shown in \cite{FF:97}, $Q$ can be interpreted as a Noether charge
\cite{Wald:93}--\cite{JKM:94} and it determines
the difference between the energy of
the fields and their canonical Hamiltonian.
The subtraction in (\ref{i1}) is unavoidable for the following reasons.
The contribution of each (Bose and Fermi) constituent field into
$S^{SM}$ is positive and divergent. Thus, the entropy $S^{SM}$ is
divergent, while black hole entropy $S^{BH}$ is finite. In formula
(\ref{i1}) the divergence of $S^{SM}$ is exactly compensated by the
divergence of the quantity $Q$.
There is a more profound reason why the Noether charge $Q$ appears in
(\ref{i1}). The Bekenstein-Hawking entropy $S^{BH}$ determines the
degeneracy of states of a black hole. It was argued in \cite{FF:97}
that this degeneracy can be calculated by counting states of
constituents with fixed total {\em energy}. On the other hand, the
entropy $S^{SM}$ is directly related to the distribution over the
levels of the {\em Hamiltonian} of constituent fields. The
additional term $Q$ is required to relate it to the distribution over
the energy levels.
\bigskip
In the present work we consider a wider class of induced gravity models
which besides scalar and spinor constituents contains also massive
vector fields. For briefness we call such models {\em vector models}.
We demonstrate that the parameters of vector models can be chosen
so that to
exclude the leading ultraviolet divergences even if all scalar fields
are minimally coupled. The remarkable fact is that the relation
(\ref{i1}) is still valid. The Noether charge $Q$ in
equation (\ref{i1}) is related now to the ``natural'' coupling
of vector fields with the curvature. The universality of the form
of Eq. (\ref{i1}) seams to be quite general property of the induced
gravity theories.
The important property of a vector model is that its only free
parameters are the masses of the fields, while the "nonminimal
couplings" are fixed by the form of the action of the vector fields. As
we will see, this property makes it possible a new interesting
interpretation of the Bekenstein-Hawking entropy in induced gravity in
terms of a two-dimensional quantum theory on $\Sigma$. Thus, the
induced gravity models provide a simple realization of the holographic
principle: the black hole entropy is encoded in "surface" degrees of
freedom, i.e., in the degrees of freedom of the theory which propagate
very close to the black hole horizon. The holographic principle
was formulated in
\cite{Hooft:93}, \cite{Susskind} (see also recent paper
\cite{Hooft:98}) and at the present moment it is actively
discussed in the framework of the string theory
\cite{Maldacena:97}--\cite{SuWi:98}.
The paper is organized as follows. In Section 2 we describe
the models of induced gravity with vector fields.
Section 3 is devoted to the derivation of Eq. (\ref{i1}) for these
models.
Special attention here is paid to the calculation of
statistical-mechanical entropy of vector fields in the presence of
the Killing horizon
and to the properties of the Noether charge which is connected with
nonminimal vector couplings. These results
enable us to adopt the statistical-mechanical
explanation of the Bekenstein-Hawking entropy given in Ref. \cite{FF:97}
to more general class of induced gravity models.
In Section 4 we establish the relation between
the Bekenstein-Hawking entropy
and the effective action of a 2D free massive quantum fields ``living''
on the bifurcation surface $\Sigma$ of the horizons.
As we show this relation is satisfied for induced gravity
obtained from a theory with partly broken supersymmetry.
Concluding remarks and a brief discussion of the holographic
property of the black hole entropy in induced
gravity theories are presented in Section 5.
The relation between energy, Hamiltonian, and the Noether charge
for massive vector fields is derived in Appendix.
We use sign conventions of book \cite{MTW:73} and, thus,
we work with the signature $(-+++)$ for the Lorentzian metric.
\section{Induced gravity models with vector fields}
\setcounter{equation}0
The vector model\footnote{A similar model of induced gravity was discussed in
\cite{Zelnikov:97}.}
consists of $N_s$
minimally coupled
scalar fields $\phi_i$ with masses $m_{s,i}$, $N_d$ spinors $\psi_{j}$
with masses
$m_{d,j}$, and $N_v$ vector fields $V_k$ with masses
$m_{v,k}$. The classical actions of the fields are standard
\begin{equation}\label{1.1a}
I_{s}[\phi_i]=-\frac 12\int dV\left[(\nabla \phi_i)^2+
m_{s,i}^2\phi^2_i\right]~~~,
\end{equation}
\begin{equation}\label{1.2a}
I_{d}[\psi_j]= \int dV \bar{\psi}_{j}
(\gamma^\mu\nabla_\mu+m_{d,j})\psi_j~~~,
\end{equation}
\begin{equation}\label{1.3a}
I_{v}[V_k]=-\int dV\left[\frac 14 F_k^{\mu\nu}F_{k\mu\nu}+
\frac 12 m_{v,k}^2V_k^\mu V_{k\mu}\right]~~~,
\end{equation}
where $dV=\sqrt{-g}d^4x$ is the volume element of 4D space-time ${\cal
M}$ and $F_{k\mu\nu}=\nabla_\mu V_{k\nu}-\nabla_\nu V_{k\mu}$. The
corresponding quantum effective action of the model is
\begin{equation}\label{1.4a}
\Gamma=\sum_{i=1}^{N_s}\Gamma_s(m_{s,i})+\sum_{j=1}^{N_d} \Gamma(m_{d,j})+
\sum_{k=1}^{N_v} \Gamma(m_{v,k})~~~.
\end{equation}
$\Gamma$ is a functional of the metric $g_{\mu\nu}$ of the background
space-time.
The scalar and spinor actions follow immediately from
Eqs. (\ref{1.1a}) and (\ref{1.2a}),
\begin{equation}\label{1.5a}
\Gamma_s(m_{s,i})=\frac 12 \log\det(-\nabla^2+m_{s,i}^2)~~~,
\end{equation}
\begin{equation}\label{1.6a}
\Gamma_d(m_{d,j})=-\log\det(\gamma^\mu\nabla_\mu+m_{d,j})~~~.
\end{equation}
As a result of equation of motion, a massive vector field $V_\mu$ obeys
the condition $\nabla^\mu V_\mu=0$, which leaves only
three independent components. Under quantization this condition
can be realized as a constraint so that the effective action
for vector fields takes the form
\begin{equation}\label{1.7a}
\Gamma_v(m_{v,k})=\tilde{\Gamma}_v(m_{v,k})- \Gamma_s(m_{v,k})~~~,
\end{equation}
\begin{equation}\label{1.8a}
\tilde{\Gamma}_v(m_{v,k})=
\frac 12 \log\det(-\nabla^2\delta^\mu_\nu+R^\mu_\nu+
m_{v,k}^2\delta^\mu_\nu)~~~,
\end{equation}
where $R^\mu_\nu$ is the Ricci tensor. The functional
$\tilde{\Gamma}_v(m_{v,k})$
represents the effective action for a massive vector field
which we will denote as $A_{k,\mu}$. The classical action
for $A_{k,\mu}$ which results in (\ref{1.8a}) is
\begin{equation}\label{1.9a}
\tilde{I}_{v}[A_k]=-\frac 12\int dV\left[\nabla^{\mu}A_k^\nu
\nabla_\mu A_{k\nu}+R_{\mu\nu}A_k^\mu A_k^\nu+
m_{v,k}^2A_k^\mu A_{k\mu}\right]~~~.
\end{equation}
The field $A_k^\mu$ obeys no constraints and carries an
extra degree of freedom.
The contribution of this unphysical degree of freedom
in (\ref{1.8a}) is compensated by subtracting the
action $\Gamma_s(m_{v,k})$ of a scalar field with the mass $m_{v,k}$,
see Eq. (\ref{1.7a}).
\bigskip
In general, the effective action (\ref{1.4a}) is ultraviolet
divergent quantity. Let us discuss now the constraints which have
to be imposed on the masses of the constituents to eliminate
the leading divergences in $\Gamma$.
The divergences related to
each particular field follow from the Schwinger-DeWitt
representation
\begin{equation}\label{1.10a}
\Gamma_i=-{\eta_i \over 2}\int^{\infty}_{\delta}{ds \over s} e^{-m_i^2s}
\mbox{Tr} e^{-s\Delta_i}~~~,
\end{equation}
where $\eta_i=+1$ for Bose fields and $-1$ for Fermi fields, and
$\delta$ is an ultraviolet cutoff. The divergences come from the
lower integration limit where one can use the asymptotic expansion
of the trace of the heat kernel operator of $\Delta_i$
\begin{equation}\label{1.11a}
\mbox{Tr} e^{-s\Delta_i}\simeq {1 \over (4\pi s)^2}\int dV
(a_{i,0}+sa_{i,1}+...)~~~.
\end{equation}
For the fields under consideration we have
\begin{equation}\label{1.12a}
\Delta_s=-\nabla^\mu\nabla_\mu~~,~~a_{s,0}=1~~,~~a_{s,1}=\frac 16 R~~~,
\end{equation}
\begin{equation}\label{1.13a}
\Delta_d=-(\gamma^\mu\nabla_\mu)^2~~,~~a_{d,0}=4~~,~~a_{d,1}=-\frac 13 R~~~,
\end{equation}
\begin{equation}\label{1.14a}
(\Delta_v)^\mu_\nu=-\nabla^\rho\nabla_\rho\delta^\mu_\nu
+R^\mu_\nu~~,~~a_{v,0}=4~~,~~a_{v,1}=-\frac 13 R~~.
\end{equation}
As in the case of the model considered in Ref. \cite{FFZ}, we require
vanishing of the cosmological
constant and cancelation of the divergences of the
induced Newton constant.
These conditions can be written down with the help of the
following two functions
\begin{equation}\label{1.5}
p(z)=\sum_{i=1}^{N_s} m_{s,i}^{2z}-4\sum_{j=1}^{N_d} m_{d,j}^{2z}+
3\sum_{k=1}^{N_v} m_{v,k}^{2z}~~~,
~~~
q(z)=\sum_{i=1}^{N_s} m_{s,i}^{2z}+2\sum_{j=1}^{N_d} m_{d,j}^{2z}-3
\sum_{k=1}^{N_v} m_{v,k}^{2z}~~~.
\end{equation}
As can be shown by using Eqs. (\ref{1.4a}), (\ref{1.10a})--(\ref{1.14a}),
the induced cosmological constant vanishes when
\begin{equation}\label{1.6}
p(0)=p(1)=p(2)=p'(2)=0~~~.
\end{equation}
The induced Newton constant $G$ is finite
if
\begin{equation}\label{1.7}
q(0)=q(1)=0~~~.
\end{equation}
The constraints
result in simple relations
\begin{equation}\label{1.6aa}
N_s=N_d=N_v~~~,~~~\sum_{i=1}^{N_s} m_{s,i}^2=\sum_{j=1}^{N_d} m_{d,j}^2=
\sum_{k=1}^{N_v} m_{v,k}^2~~~.
\end{equation}
They show that one cannot construct the theory
with finite cosmological and Newton constants
from vector and spinor fields only.
\bigskip
The low-energy limit of the theory corresponds to the regime when the
curvature radius $L$ of the spacetime ${\cal M}$ is much greater than
the Planck length $m_{Pl}^{-1}$. In this limit the effective action
$\Gamma$ of the theory can be expanded in the curvature. The terms in
this series are local and the leading terms can be calculated
explicitly. In the linear in curvature approximation $\Gamma$ coincides
with the Einstein action \footnote{To induce the correct boundary term
in (\ref{1.15a}) one has to add to $\Gamma$ an integral of
averages of field operators on the spatial boundary ${\partial {\cal
M}}$, see \cite{BaSo:96}. These terms are not relevant for our
analysis. Let us emphasize that we are interested in the
statistical-mechanical computation of the black hole entropy for which
only the region near the horizon is important.}
\begin{equation}\label{1.15a}
\Gamma[g]\simeq{1 \over 16\pi G}\left(\int_{\cal M}dV R
+2\int_{\partial{\cal M}}dv K\right)~~~.
\end{equation}
Here $dv$ is the volume element of $\partial {\cal M}$. The Newton
constant is determined by the following expression
\begin{equation}\label{1.8}
{1 \over G}={1 \over 12\pi}q'(1)=
{1 \over 12\pi}\sum_{i=1}^{N}\left( m_{s,i}^2\ln m_{s,i}^2+
2m_{d,i}^2\ln m_{d,i}^2
-3m_{v,i}^2\ln m_{v,i}^2
\right)~~~.
\end{equation}
Here, according to (\ref{1.6aa}), we put $N=N_s=N_d=N_v$. From this
expression it is easy to conclude that at least some of the
constituents must be heavy and have mass comparable with the Planck
mass $m_{Pl}$. For simplicity in what follows we assume that all the
constituents are heavy.
\bigskip
Let us analyze models where conditions (\ref{1.5}) and (\ref{1.6})
are satisfied. Equations (\ref{1.6aa}) are trivially satisfied when all
fields are in supersymmetric multiplets. However in such
supersymmetric models $p(z)=q(z)\equiv 0$ (because masses of the fields
in the same supermultiplet coincide) and the induced gravitational
constant vanishes. A nontrivial induced gravity theory can be obtained
if the supersymmetry is partly broken by splitting the masses of the
fields in the supermultiplets.
Let us demonstrate this by an example. Consider the model with $N$
massive supermultiplets. Each multiplet consists of one scalar, one
Dirac spinor and one vector field, so that the numbers of Bose and
Fermi degrees of freedom coincide\footnote{Supersymmetric models with
free massive scalar, spinor and vector fields are discussed, for
instance, in Ref. \cite{LoWo:81}}. We suggest that masses of vector and
spinor fields are equal, $m_{v,i}=m_{d,i}\equiv m_i$ (here $i$ is the
number of the multiplet). The masses of the scalar partners are assumed
to be $m_{s,i}=(1+x_i)m_i$, where $x_i$ is a dimensionless coefficient.
The case when $|x_i|\ll 1$ corresponds to slightly broken
supersymmetry. For this case
\begin{equation}\label{1.16a}
p(z)=q(z)=\sum_{i=1}^N m_i^{2z}\left[(1+x_i)^{2z}-1\right]\simeq
2z\sum_{i=1}^N x_i m_i^{2z}~~~.
\end{equation}
Now equations (\ref{1.6}), (\ref{1.7}), and (\ref{1.8}) take the simple
form
\begin{equation}\label{1.17a}
\sum_{i=1}^N x_i m_i^2=0~~~,~~~\sum_{i=1}^N x_i m_i^4=0~~~,
\end{equation}
\begin{equation}\label{1.18a}
\sum_{i=1}^N x_i m_i^4\ln m_i^2=0~~~,~~~{1 \over G}\simeq{1 \over 6\pi}
\sum_{i=1}^N x_i m_i^2\ln m_i^2~~~.
\end{equation}
This is a system of linear equations for $x_i$
which for $N\ge 4$ has nontrivial
solutions.
The induced gravity constraints provide cancelation of the leading
ultraviolet divergences. However, some logarithmical divergences are
still present on general backgrounds. On the Schwarzschild background
the logarithmic divergences are pure topological and can be neglected.
That is why in what follows we restrict the analysis to black holes of
this type\footnote{ At least some of the logarithmic divergences can
be eliminated in more complicated models, for instance in models which
contain both vector and non-minimally coupled scalar fields. These
models allow one to generalize the analysis of the black hole entropy
problem in induced gravity to charged black holes.}.
\section{Statistical calculation of the
black hole entropy}
\setcounter{equation}0
Let us now calculate the statistical-mechanical entropy $S^{SM}$ in
the vector models of induced gravity and compare it with the
Bekenstein-Hawking entropy of a black hole. As a result of this
comparison, we prove the validity of equation (\ref{i1}) for these
models.
The canonical ensemble of constituent fields on a static,
asymptotically flat background can be described by standard methods.
The statistical-mechanical entropy of the fields
is determined from the free energy
\begin{equation}\label{2.1a}
F(\beta)=-\beta^{-1}\ln \mbox{Tr} \exp(-\beta :\!\hat{H}\!:)
=\eta \beta^{-1}\int_0^{\infty}d\omega {dn \over d\omega}
\ln\left(1-\eta e^{-\beta\omega}\right)~~~.
\end{equation}
Here $\beta$ is the inverse temperature measured at infinity and
$:\!\hat{H}\!:$ is the Hamiltonian of the system which is defined as
the generator of canonical transformations along the Killing time. The
factor $\eta= 1$ for bosons and $\eta= -1$ for fermions, $\omega$ are
the frequencies of single-particle excitations, $dn/ d\omega$ is the
density of levels $\omega$.
When the background space-time is the exterior region of a black hole
the single particle-spectra have a number of specific properties
because of the presence of the Killing horizons \cite{FF:98}. In
particular, the density of states $dn/ d\omega$ infinitely grows near
the horizon. Although this divergence has an infrared origin,
regularizations of the ultraviolet type can be applied to make $dn
/d\omega$ finite. For scalar and spinor fields on general static
backgrounds the divergences of $dn /d\omega$ were computed in
\cite{Fursaev:97}. In the Pauli-Villars regularization the leading
divergences for scalar and Dirac spinor fields of the mass $m$ are
\begin{equation}\label{2.2a}
{dn_s(m) \over d\omega}= {b(m) \over 8\pi^2\kappa}\, {\cal A}~~~,
~~~
{dn_d(m) \over d\omega}= {b(m) \over 2\pi^2\kappa} \, {\cal A}~~~,
\end{equation}
\begin{equation}\label{2.3a}
b(m)=c\mu^2-m^2\ln{\mu^2 \over m^2}~~~.
\end{equation}
Here $\kappa=(4M)^{-1}$ is the surface gravity of the black hole,
$\mu$ is the Pauli-Villars cutoff, and $c=\ln{729 \over 256} >0$.
Modes propagating in the vicinity of the horizon give the main
contribution to the densities of levels. That is why the quantity $dn
/d\omega$ scales as the surface area ${\cal A}$ of the horizon. This
also means that to get (\ref{2.2a}) it is sufficient to restrict
oneself by the Rindler approximation of the black hole metric
\begin{equation}\label{1.9aa}
ds^2=-\kappa^2\rho^2dt^2+d\rho^2+dz_1^2+dz_2^2~~~.
\end{equation}
Here $\rho>0$, and $t$ is the Rindler time coordinate. In this
approximation the densities of levels for high spin fields can be
computed by using expression (\ref{2.2a}) for scalars and spinors.
Let us consider a massive vector field in the Minkowski spacetime.
We denote by $X^{m}$ ($m=0,\ldots, 4$) the Cartesian coordinates in
this space and by $V_m=(V_0,V_a)$, $a=1,2,3$,
the components of the vector field with respect to the Cartesian
frame. Then the equations of motion which extremize vector
field action (\ref{1.3a}) are simply a set of Klein-Gordon equations
for four ``scalars'' $V_m$ plus the additional constraint $\partial_m
V^m=0$. The constraint serves to express the time-component $V_0$ in
terms of other components $V_a$. The contribution of this component to
the energy is negative and $V_0$ cannot be considered as an independent
physical degree of freedom. The density of levels of the vector field
$dn_v /d\omega$ multiplied by $d\omega$ is the number of independent
solutions $V_m(t,\rho,z)=e^{-i\omega t}V_m(\rho,z)$ of the field
equations with the frequencies in the interval $(\omega,
\omega+d\omega)$. The solutions are determined by three independent
functions $V_\alpha$. Therefore, in the Rindler approximation $dn_v /
d\omega$ is greater by the factor 3 than the density of levels of a
scalar field of the same mass $m$. From (\ref{2.2a}) we find
\begin{equation}\label{2.4a}
{dn_v(m) \over d\omega}=3{dn_s(m) \over d\omega}=
{3b(m) \over 8\pi^2\kappa} {\cal A}~~~.
\end{equation}
The curvature corrections may change this relation
but they are not important for further analysis.
The statistical-mechanical entropy
\begin{equation}\label{2.5ab}
S=\beta^2 {\partial F \over \partial \beta}~~,~~
\end{equation}
of scalar, spinor and vector fields follows from Eqs. (\ref{2.1a}),
(\ref{2.2a}) and (\ref{2.4a})
\begin{equation}\label{2.5a}
S_s(m_{s,i})={b(m_{s,i}) \over 48\pi}{\cal A}~~,~~
S_d(m_{d,i})={2b(m_{d,i}) \over 48\pi}{\cal A}~~,~~
S_v(m_{v,i})={3b(m_{v,i}) \over 48\pi}{\cal A}~~.
\end{equation}
Expressions (\ref{2.5a}) are obtained from formula (\ref{2.5ab})
at the Hawking temperature, i.e., at $\beta=2\pi/\kappa=8\pi M$.
The statistical-mechanical entropy of the constituents in
the induced gravity model is
\begin{equation}\label{2.6a}
S^{SM}=\sum_{i=1}^N\left[S_s(m_{s,i})+S_d(m_{d,i})+S_v(m_{v,i})\right]~~~.
\end{equation}
By substituting (\ref{2.5a}) into (\ref{2.6a})
and taking into account (\ref{2.3a}), (\ref{1.6aa})
we get
$$
S^{SM}={1 \over 48\pi}\sum_{i=1}^N\left[m_{s,i}^2\ln m_{s,i}^2+
2m_{d,i}^2\ln m_{d,i}^2+3m_{v,i}^2\ln m_{v,i}^2\right]{\cal A}
$$
\begin{equation}\label{2.7a}
+{1 \over 8\pi}\left[c N\mu^2-\ln \mu^2
\sum_{i=1}^N m_{v,i}^2\right]{\cal A}~~~.
\end{equation}
Let us now calculate the Noether charge $Q$ for our model.
It is instructive to
discuss first the entropy of a black hole in a classical theory. According
to Wald et al. \cite{Wald:93}--\cite{JKM:94}, the black hole entropy
can be interpreted as a Noether charge and obtained from the
Lagrangian $L$ of the theory. For theories which do not include
the derivatives of the metric higher than second order the entropy
can be written in the form
\begin{equation}\label{2.8a}
S=-8\pi\int_{\Sigma} t_\mu n_\nu t_\lambda n_\rho
{\partial L \over
\partial R_{\mu\nu\lambda\rho}}d\sigma~~~,
\end{equation}
where $R_{\mu\nu\lambda\rho}$ is the Riemann tensor. The integration
in (\ref{2.8a})
goes over the bifurcation surface $\Sigma$ of the horizon, $d\sigma$
is the volume element of $\Sigma$ ($\int_{\Sigma} d\sigma={\cal A}$).
Vectors $t_\mu$ and $n_\mu$ are two mutually orthogonal
vectors normal to $\Sigma$ such that $t^2=-1$ and $n^2=1$.
For the Einstein theory equation (\ref{2.8a}) reproduces the
Bekenstein-Hawking formula for the black hole entropy. The important
consequence of (\ref{2.8a}) is that coupling of the
matter fields with the curvature gives a nonzero contribution
$\Delta S$ to the Bekenstein-Hawking entropy. In quantum theory $\Delta
S$ becomes an average of the corresponding field operator on $\Sigma$.
Let us now consider the vector model of induced gravity. According to
(\ref{1.4a}) the effective action of the theory can be written as a
path integral
\begin{equation}\label{2.9a}
\exp{i\Gamma[g]}=\int [D\Phi]\exp (iI[g,\Phi])~~~,
\end{equation}
\begin{equation}\label{2.10a}
I[g,\Phi]\equiv\sum_{i=1}^N\left(I_s[\phi_i]+I_d[\psi_i]+\tilde{I}_v[A_i]
+I_s[\varphi_i]\right)~~~,
\end{equation}
where $\Phi=\{\phi_i,\psi_i, A_i,\varphi_i\}$. The
functionals $I_s$, $I_d$, and $\tilde{I}_v$ are defined by
Eqs. (\ref{1.1a}), (\ref{1.2a}), and (\ref{1.9a}), respectively.
The origin of the scalar fields $\varphi_i$ in (\ref{2.9a})
is related to the quantization of the
massive vector fields $A_i$. It is assumed that $\varphi_i$
obey the ``wrong'' (Fermi) statistics in order to reproduce Eq. (\ref{1.7a}).
As follows from
(\ref{1.9a}), the total ``classical action'' $I[g,\Phi]$ includes
the nonminimal couplings of the vector fields $A_i$. By using
formula (\ref{2.8a}) in the theory with the action $\tilde{I}_v[A]$
one obtains the nonzero term
\begin{equation}\label{2.11a}
\Delta S=-8\pi\int_{\Sigma} t_\mu n_\nu t_\lambda n_\rho
{\partial \tilde{L}_v \over
\partial R_{\mu\nu\lambda\rho}}d\sigma=\pi
\int_{\Sigma}(t^\mu t^\nu-n^\mu n^\nu) A_\mu A_\nu d\sigma~~~,
\end{equation}
where we put $\tilde{I}_v[A]=\int \tilde{L}_v dV$.
In the induced gravity such terms result in a correction
to the entropy of a black hole. In the first order in the Planck constant
this correction simply is
\begin{equation}\label{2.12a}
\Delta S=\pi
\int_{\Sigma}(t^\mu t^\nu-n^\mu n^\nu) \sum_{i=1}^N
\langle \hat{A}_{i\mu} \hat{A}_{i\nu} \rangle d\sigma\equiv
-Q~~~.
\end{equation}
Here the average $\langle \hat{A}_{i\mu} \hat{A}_{i\nu} \rangle$ is
understood as a regularized quantity. The quantity $Q$ has a meaning of
Wald's Noether charge associated with nonminimal interaction terms of
the vector field. The sign minus in the r.h.s. of (\ref{2.12a}) is chosen
so that $Q$ be positive.
By using the Pauli-Villars regularization one finds that in the Rindler
approximation
\begin{equation}\label{2.13a}
\langle \hat{A}_{i\mu} \hat{A}_{i\nu} \rangle=\eta_{\mu\nu}
{b(m_{v,i}) \over 16\pi^2}~~~,
\end{equation}
where $\eta_{\mu\nu}$ is the Minkowski metric and function $b(m_{v,i})$
is defined by (\ref{2.3a}). Equation (\ref{2.13a}) gives
\begin{equation}\label{2.14a}
Q={1 \over 8\pi}\sum_{i=1}^N b(m_i) {\cal A}={1 \over 8\pi}
\left(cN\mu^2-\ln \mu^2\sum_{i=1}^N m_{v,i}^2+
\sum_{i=1}^N m_{v,i}^2\ln m_{v,i}^2
\right){\cal A}~~~.
\end{equation}
This result allows one to show that the total Bekenstein-Hawking
entropy $S^{BH}$ in induced gravity is the difference of
statistical-mechanical entropy $S^{SM}$, see Eq. (\ref{2.7a}), and the
Noether charge $Q$. As can be easily seen, the divergences of $S^{SM}$
are exactly canceled by the divergences of the charge $Q$, so that one
gets the finite expression
\begin{equation}\label{2.15a}
S^{SM}-Q={1 \over 48\pi}\sum_{i=1}^N\left[m_{s,i}^2\ln m_{s,i}^2+
2m_{d,i}^2\ln m_{d,i}^2-3m_{v,i}^2\ln m_{v,i}^2\right]{\cal A}=
{ {\cal A}\over 4G}=S^{BH}~~~.
\end{equation}
This expression coincides exactly with the Bekenstein-Hawking entropy
in induced gravity where the induced Newton constant is determined by
formula (\ref{1.8}).
As was argued in Ref. \cite{FF:97}, the statistical-mechanical reason
why the Noether charge appears in (\ref{2.15a}) is related to the fact
that the canonical Hamiltonian $H$ and the energy $E$ of the system are
different. $H$ defines free energy (\ref{2.1a}) and entropy $S^{SM}$
while the energy $E$ is connected with the spectrum of the mass of the
black hole.
In Appendix we show that for the vector model
\begin{equation}\label{2.21a}
H-E={\kappa \over 2\pi} Q~~~.
\end{equation}
This is the same relation which was found in \cite{FF:97} for the
induced gravity model with non-minimally coupled scalar fields.
Relation (\ref{2.21a}) can be used to provide the
statistical-mechanical interpretation of the subtraction of the charge
$Q$ in the black hole entropy formula (\ref{2.15a}). This
interpretation repeats the one already given in \cite{FF:97}: the
subtraction of $Q$ is needed in order to pass from the distribution over
the energy in canonical ensemble of constituent fields to the
distribution over the black hole mass in the black hole canonical
ensemble which determines $S^{BH}$.
\section{Black hole entropy and 2D quantum theory on $\Sigma$}
\setcounter{equation}0
Our aim now is to relate the Bekenstein-Hawking entropy $S^{BH}$,
Eq. (\ref{2.15a}), to a 2D quantum theory of free massive fields
``living'' on the bifurcation surface $\Sigma$ of the horizon.
To this aim it is instructive to represent expression (\ref{2.15a})
in another equivalent form. First, let us note that in the
Rindler approximation the regularized correlators of the
scalar, spinor and vector fields of the mass $m$ have the simple form
\begin{equation}\label{1.9}
\langle \hat{\phi}^2 \rangle={b(m) \over 16\pi^2}~~,~~
\langle \hat{\bar{\psi}}\hat{\psi}\rangle=4m\langle \hat{\phi}^2 \rangle
~~,~~
\langle \hat{V}_\mu\hat{V}^\mu \rangle=
3\langle \hat{\phi}^2 \rangle~~~.
\end{equation}
In the Pauli-Villars regularization the function $b$ is defined by
(\ref{2.3a}). From Eqs. (\ref{2.15a}) and (\ref{1.9})
we easily find that
\begin{equation}\label{1.14}
S^{BH}={\pi\over 6}\sum_{i=1}^{N}\int_{\Sigma}d\sigma \left[
2\langle \hat{\phi}_{i}^2 \rangle
+{1 \over m_{d,i}}
\langle \hat{\bar{\psi}}_{i}\hat{\psi}_{i}\rangle
-2\langle \hat{V}^2_{i} \rangle
\right]~~~.
\end{equation}
One can check that
the divergences in correlators in (\ref{1.14})
are canceled because of induced gravity
constraint $q(1)=0$, see Eqs. (\ref{1.5}) and (\ref{1.7}).
Since the surface $\Sigma$ of bifurcation of horizons is a set of fixed
points of the Killing vector, only zero-frequency (``soft'') modes
contribute to the correlators on $\Sigma$ (for a detailed discussion of
this point, see \cite{FF:97}).
As was shown in \cite{FF:97}, the correlator of scalar
fields taken on the bifurcation surface of the Killing horizons
behaves effectively as a two-dimensional operator.
Namely, if $z$ and $z'$ are the coordinates
of the points $x$ and $x'$ on $\Sigma$, see (\ref{1.9aa}), then
\begin{equation}\label{4.1}
\langle\hat{\phi}(x(z))\hat{\phi}(x(z'))
\rangle=-
{1 \over 4\pi}
\langle z|\ln O_{\Sigma}|z'\rangle~~~,
\end{equation}
\begin{equation}\label{4.3}
O_{\Sigma}=-\nabla^2_{\Sigma}+m^2~~~,
\end{equation}
where $-\nabla^2_{\Sigma}$ is the Laplacian on $\Sigma$. The left and
right parts of (\ref{4.1}) should be calculated in the same
regularization. It should be emphasized that (\ref{4.1}) is exact
relation for the Rindler space\footnote{It can be generalized to curved
backgrounds with a Killing horizon. In general case the operator
$O_\Sigma$ for very massive fields can be found by comparing
Schwinger-DeWitt asymptotics of four and two dimensional operators, see
for the details \cite{FF:97}. The key property which allows two
dimensional interpretation of 4D correlators on $\Sigma$ is that
$\Sigma$ is a geodesic surface. That is, any 4D geodesic which begins
and ends on $\Sigma$ coincides with 2D geodesic on $\Sigma$.}. For r.h.s.
of (\ref{4.1}) we find that
\begin{equation}\label{4.7}
-{1 \over 4\pi} \int_{\Sigma}\langle z|\ln O_{\Sigma}|z \rangle ~
d\sigma=-{1 \over 4\pi} \ln \det(
-\nabla^2_{\Sigma}+m^2)
\equiv -{1 \over 2\pi} W_s(m)~~~.
\end{equation}
The functional $W_s(m)$ has the meaning of the effective action of a 2D
quantum field $\chi$ given on $\Sigma$. It can be expressed in terms of
the Euclidean path integral as
\begin{equation}\label{a4.10}
e^{-W_s(m)}=\int D[\chi]
\exp\left[-\frac 12\int_{\Sigma}((\nabla_\Sigma\chi)^2+m^2\chi^2)
d\sigma\right]
~~~,
\end{equation}
where $D[\chi]$ is a covariant measure. In Ref. \cite{FF:97} a
two-dimensional auxiliary field $\chi$ ``living'' on the bifurcation
surface $\Sigma$ was called a {\it flucton} field to distinguish it from
the 4D fields in the black hole exterior. From (\ref{4.1}) and
(\ref{4.7}) one obtains
\begin{equation}\label{a4.9a}
\int_{\Sigma}d\sigma \langle \hat{\phi}^2(x(z))\rangle
=-{1 \over 2\pi}W_s(m)~~~.
\end{equation}
It follows from (\ref{1.14}) and (\ref{a4.9a}) that the contribution
of scalar fields to the black hole entropy $S^{BH}$ can be interpreted in
terms of 2D quantum theory of fluctons on $\Sigma$.
\bigskip
We now find an analogous representation for the
contribution to $S^{BH}$ from spinor and vector
fields. The correlators of these fields in the coinciding points
are tensors in the certain representations of the Lorentz group.
Different parts of these tensors have different two-dimensional
interpretation.
We begin with the correlator (\ref{4.11}) of vector fields
restricted on $\Sigma$
\begin{equation}\label{4.11}
\langle\hat{V}_\mu(x(z))\hat{V}_\nu(x(z'))
\rangle=\langle\hat{A}_\mu(x(z))\hat{A}_\nu(x(z'))\rangle+
m^{-2}\nabla_\mu\nabla'_\nu\langle\hat{\varphi}(x(z))\hat{\varphi}(x(z'))
\rangle~~~,
\end{equation}
where $\hat{\varphi}$ is a scalar field of the same mass as
$\hat{V}_\mu$.
Let us consider the components of tensor quantities in
the Minkowski coordinates $X^m$. With respect to the coordinate
transformations on $\Sigma$, components of a tensor with indices $0$
and $3$ behave as scalars while components with indices $1$ and $2$
transform as vectors on $\Sigma$. By using arguments of Ref.
\cite{FF:97} one can express the correlator
$\langle\hat{A}_\mu\hat{A}_\nu\rangle$ with $\mu,\nu=1,2$ in terms of
the 2D vector Laplacian on $\Sigma$. Analogously, components of the
correlator with $\mu,\nu=0,3$ can be represented in terms of the 2D scalar
Laplacian. Thus, we find that
\begin{equation}\label{4.12}
\int_{\Sigma}d\sigma\langle
\hat{A}^1(x(z))\hat{A}_1(x(z))+\hat{A}^2(x(z))\hat{A}_2(x(z))
\rangle=-{1 \over 2\pi}\tilde{W}_v(m)~~~,
\end{equation}
\begin{equation}\label{4.13}
\int_{\Sigma}d\sigma\langle
\hat{A}^0(x(z))\hat{A}_0(x(z))+\hat{A}^3(x(z))\hat{A}_3(x(z))
\rangle=-2{1 \over 2\pi}W_s(m)~~~.
\end{equation}
$W_s$ is the effective action of a scalar field on $\Sigma$ and
$\tilde{W}_v(m)$ is the effective action of a vector field on $\Sigma$
with the same mass $m$ as that of 4D field
\begin{equation}\label{4.12ab}
\tilde{W}_v(m)=\frac 12 \log\det((-\nabla_{\Sigma}^2+\frac 12 R_\Sigma
+m^2)\delta^A_B)~~~.
\end{equation}
Here $A,B=1,2$ and $R_\Sigma$ is the curvature of $\Sigma$
which can be neglected in the Rindler approximation.
It follows from (\ref{4.11})--(\ref{4.13})
that
$$
\int_{\Sigma}d\sigma\langle \hat{V}^\mu(x(z))\hat{V}_\mu(x(z))
\rangle
$$
\begin{equation}\label{4.13a}
=\int_{\Sigma}d\sigma\langle \hat{A}^\mu(x(z))\hat{A}_\mu(x(z))
-\hat{\varphi}(x(z))\hat{\varphi}(x(z))\rangle
=-{1 \over 2\pi}(W_v(m)+2W_s(m))~~~,
\end{equation}
\begin{equation}\label{4.13ab}
W_v(m)=\tilde{W}_v(m)-W_s(m)~~~.
\end{equation}
The functional $W_v(m)$ corresponds
to the quantization of the massive 2D vector field described by
the classical action analogous to 4D action (\ref{1.3a}).
Similar relations can be obtained for spinor fields. One 4D
Dirac spinor corresponds to two 2D Dirac spinors on $\Sigma$. One
easily finds that
\begin{equation}\label{4.14}
\int_{\Sigma}d\sigma\langle \hat{\bar{\psi}}(x(z))\hat{\psi}(x(z))
\rangle={m \over \pi}W_d(m)~~~,
\end{equation}
where $W_d(m)$ is the effective action of 2D spinors
on $\Sigma$ with mass $m$.
\bigskip
By using equations (\ref{a4.9a}), (\ref{4.13a}) and (\ref{4.14})
in expression (\ref{1.14}) for the Bekenstein-Hawking entropy in
induced gravity we find
\begin{equation}\label{2.10}
S^{BH}=\frac 16\sum_{i=1}^{N}\left[-
W_{s}(m_{s,i})+W_{d}(m_{d,i})+W_{v}(m_{v,i})+2W_{s}(m_{v,i})
\right]~~~.
\end{equation}
This form of the entropy looks similar to the effective action of a two
dimensional quantum field model on the surface $\Sigma$. To make this
similarity more evident let us consider the concrete induced gravity
model with partially broken supersymmetry which was discussed in
Section 2. In this model the masses of vector and spinor fields
coincide, $m_{v,i}=m_{d,i}=m_i$. As a result, $W_{v}(m)=W_{s}(m)=-\frac
12 W_d(m)$ and Eq. (\ref{2.10}) takes the form
\begin{equation}\label{2.10aa}
S^{BH}=-{1 \over 12} \sum_{i=1}^{N}
\left[2W_s(m_{s,i})+W_{d}(m_{i})
\right]\equiv -{1 \over 12}\Gamma^{(2)}~~~.
\end{equation}
The quantity $\Gamma^{(2)}$ is the effective action of a 2D
model which consists of $N$ spinor fields with masses $m_{i}$
and $2N$ scalar fields with
masses $m_{s,i}$.
In fact, we have 2D induced gravity on $\Sigma$.
The condition that the 4D curvature is small compared to the masses
of the fields guarantees that the two-dimensional curvature of
$\Sigma$ is small as well. So the
2D effective action $\Gamma^{(2)}$ can be computed as an expansion
in curvature. The leading term in this expansion is
the cosmological constant term
\begin{equation}\label{2.3}
\Gamma^{(2)}[\gamma]\simeq\int_{\Sigma} \sqrt{\gamma} \, \, d^2x\, \,
\lambda~~~.
\end{equation}
Here $\lambda$ is the ``induced'' 2D cosmological constant which
is expressed in terms of induced 4D Newton constant (\ref{1.8})
as $\lambda=-3/G$.
The constraints which provide the ultraviolet finiteness
of 4D Newton constant, see Eqs. (\ref{1.7}), automatically
guarantee the finiteness of 2D cosmological constant.
The 2D model described by functional $\Gamma^{(2)}$
can be obtained from
the supersymmetric model with $N$ multiplets
consisting of a spinor and 2 scalar fields. The split of the masses of
spinor and scalar fields breaks the supersymmetry and yields nonvanishing
2D cosmological constant.
Of course, the suggested connection between 4D and 2D theories is not
unique, and one may expect that in general the coefficient in r.h.s. of
(\ref{2.10aa}) can be another rational number. Let us emphasize that
the considered models of induced gravity are phenomenological and
admit a large arbitrariness in the choice of masses of the constituent
fields. One may hope that if the induced gravity is obtained from an
underlying fundamental theory the masses of the fields will be fixed by
some principle which will determine the coefficient in (\ref{2.10aa}).
A remark is also in order about two-dimensional interpretation
of the Noether charge $Q$. By taking into account Eq. (\ref{2.14a}) it
is easy to show that in the Rindler approximation
\begin{equation}\label{xx}
Q=-\sum_{i}W_v(m_{v,i})~~~.
\end{equation}
This relation holds in any induced gravity model with vector fields and
does not require additional conditions on the masses of the constituent
fields. It enables one to relate $Q$ to a quantum theory of 2D vector
fields on $\Sigma$.
\section{Discussion}
\setcounter{equation}0
To summarize, we consider a class of induced gravity models
where the low
energy gravitational field is generated by quantum
one-loop effects in a system of heavy constituents. The vector
models presented
here consist of massive scalar, spinor, and vector constituent fields,
and do not require non-minimal couplings of the scalar constituents.
We demonstrated that
the general mechanism of the entropy generation in the induced gravity
proposed in Ref. \cite{FF:97} does work, and the Bekenstein-Hawking
entropy can be derived by statistical-mechanical counting the energy
states of heavy constituents.
It was further demonstrated that the expression for the
Bekenstein-Hawking entropy in the induced gravity can be identically
rewritten in terms of fluctuations of the constituent fields at the
event horizon. The latter are determined only by zero-frequency
(``soft'') solutions of the corresponding field equations. These
``soft'' modes are uniquely defined by their asymptotics at the
bifurcation sphere of horizons $\Sigma$. Using this property, it was
explicitly demonstrated that the fluctuations of the constituent
fields at the horizon coincide with the effective action of
two-dimensional (flucton) fields on $\Sigma$. This mechanism is
somewhat similar to the idea of the holography
\cite{Hooft:93}--\cite{SuWi:98}.
We hope to discuss this relation in more details somewhere
else.
As a result of the two-dimensional reduction, the Bekenstein-Hawking
entropy appears to be equal to $|\lambda|{\cal A}/12$, where $\lambda$
is the 2D cosmological constant induced on the surface of the horizon
by 2D flucton fields. This implies that the degrees of freedom
responsible for the black hole entropy in the induced gravity can be
related to surface degrees of freedom of the black hole horizon. Such
conclusion is supported by the observation that since the masses of the
constituents are very high (of order of Planckian mass) the
fluctuations of the constituent fields near the horizon can be directly
connected with the fluctuations of the 2D geometry of the horizon.
This might bring connection with the well-known results of statistical
computations of black hole entropy of 3D black holes \cite{Carlip:95},
\cite{Strominger:97}.
It should be emphasized once again, that the induced gravity approach
does not pretend to explain the black hole entropy from the first
principles of the fundamental theory of quantum gravity (such as the
string theory) but it allows one to demonstrate the universality of the
entropy and its independence of the concrete details of such theory. It
gives us a hint that only a few quite general properties of the
fundamental theory (such as the low energy gravity as induced
phenomenon, finiteness of the low energy coupling constants, holography,
and so on) are really required for the statistical-mechanical
explanation of the black hole entropy.
\vspace{12pt}
{\bf Acknowledgements}:\ \ This work was partially supported by the
Natural Sciences and Engineering Research Council of Canada. One of the
authors (V.F.) is grateful to the Killam Trust for its financial
support.
\newpage
|
2,869,038,155,691 | arxiv | \section*{Abstract}{\small
The boundary between professional and amateur astronomers gets narrower and narrower.
We present several real examples, most of them published in refereed journals, of works resulting from fruitful collaborations between key amateur astronomers in Spain and professional colleagues.
The common denominator of these works is the search for binaries, mostly nearby, wide, common proper-motion pairs with low-mass stellar components, including some of the most fragile systems ever found.
\normalsize}
\end{minipage}
\section{Today: {\em in media res} \label{section.1}}
In the 17th century some good practices for scientific writing were already established, and since then scientists have given more importance to plain and accurate description than rhetorical flourishes.
However, authors in that epoch also emphasised the importance of not boring the reader.
Here we present the results of several professional-amateur collaborations on low-mass stars in fragile multiple systems using {\em in media res} (``into the midst of things''), the literary and artistic narrative technique wherein the relation of a story begins either at the midpoint or at the conclusion, rather than at the beginning.
We wish the reader to feel like enjoying a bloodless Quentin Tarantino's film script: into the middle of a sequence of events that led a group of amateur astronomers make great discoveries with only sporadic supervision from professionals.
\section{\label{section.200807} July 2008: proper-motion companions to Luyten stars}
\begin{figure}
\center
\includegraphics[width=0.75\textwidth]{caballerojaF1.pdf}
\caption{\label{F1} Inner part of the proper-motion diagram ($\mu_\alpha \cos{\delta}$ vs. $\mu_\delta$) of the Luyten stars in Salim \& Gould (2003).
The [red] corona indicates the interval of proper motions studied by Caballero et~al. (2010a).}
\end{figure}
This contribution corresponds to a talk given by the first author during the outreach session of the X meeting of the Spanish Astronomical Society (Sociedad Espa\~nola de Astronom\'{\i}a, SEA) in Valencia in summer 2012.
Four years before, at the VIII SEA meeting in Santander, he gave two talks: one on formation, evolution and multiplicity of brown dwarfs and giant exoplanets (Caballero 2010a), the other on a Virtual Observatory search for companions to Luyten stars in close collaboration with some of the authors of this proceeding.
There, Caballero et~al. (2010a) showed preliminary results of an Aladin-based survey for companions to high-proper-motion Luyten stars in the Salim \& Gould (2003) catalogue.
Among the 1947 studied stars (and white dwarfs) with proper motions in the interval 0.5 $< \mu <$ 1.0\,arcsec\,a$^{-1}$ (see Fig.~\ref{F1}), they recovered 76 double, triple and quadruple systems, many of which are still poorly known.
Measured angular separations ranged from about 5 to over 500\,arcsec.
\begin{table}[]
\caption{Some investigated systems with Luyten primaries wider than $\rho$ = 100\,arcsec}
\center
\begin{minipage}{1.0\textwidth}
\center
\begin{tabular}{l llc cc}
\hline\hline
WDS\footnote{Washington Double Star catalogue.} & Primary & Secondary & Sp. & $\rho$ & $\theta$ \\
code & & & types & [arcsec] & [deg] \\ [0.5ex]
\hline
STT 547 AF & HD 38 AB & BD+44 4548 & (K6\,V+M0\,V) + M1\,Ve & 327.9 & 253.8 \\
LDS 71 & L 512--14 & L 512--15 & M3\,Ve + M: & 105.4 & 312.1 \\
...\footnote{The system AB--C, discovered by Caldwell et al. (1984), has no WDS entry yet.} & HD 15468 AB & HD 15468 C & (K4.5\,V+K:\,V) + M2.5\,V & 473 & 242.1 \\
LDS 3508 & CD--31 1454 & LP 888--25 & K2:\,V + K:\,V & 223.0 & 263.5 \\
RUB 4 & BD--17 3088 & WT 1759 & K7\,V + DZ\,7 & 399.0 & 192.8 \\
LDS 4172 & LP 794--17 & LP 794--16 & K: + M: & 323.7 & 293.7 \\ [0.5ex]
\hline
\end{tabular}
\end{minipage}
\label{T1}
\end{table}
Six of the widest systems identified by Caballero et~al. (2010a), with angular separations greater than 100\,arcsec, are listed in Table~\ref{T1}.
Only three secondaries, two early-M dwarfs and one white dwarf, have been spectroscopically studied in detail.
The other three secondaries have only rough spectral-type estimations and deserve a further astrometric, photometric, and spectroscopic analysis.
Caballero et~al. (2010a) also serendipitously re-discovered half a dozen poorly-known physical pairs, one of which outstood because of its supposed youth, late spectral type, nearness and wide projected physical separation: GIC~158.
\section{\label{section.201007} July 2010: G~125--15\,AB + G~125--14 = GIC 158}
Caballero et~al. (2010b) investigated in detail the system GIC~158, whose primary (G~125--15\,AB) is an active M4.5\,Ve star previously inferred to be young ($\tau \sim$ 300--500\,Ma) based on its high X-ray luminosity.
Actually, it is an inflated, double-lined, spectroscopic binary with a short period of photometric variability of 1.6 days.
The observed X-ray and H$\alpha$ emissions, photometric variability, and abnormal radius and effective temperature of the primary are indicative of strong magnetic activity, possibly because of orbital synchronisation and rapid rotation.
The secondary (G~125--14) has the same spectral type but is more than one magnitude fainter than the primary.
At $d \sim$ 26\,pc, the estimated projected physical separation between the two components of about 1.2\,kAU ensures that GIC~158 is one of the widest systems with intermediate M-type primaries yet found in the solar neighbourhood.
\section{\label{section.200700} From November 2006 to March 2007: Koenigstuhl 1,~2,~3}
Years earlier, while preparing a near-infrared colour-colour diagram of young brown dwarfs, Caballero (2007a) serendipitously discovered the widest ``ultracool'' binary at that time.
Koenigstuhl~1 is a common proper-motion pair of M6.0:\,V and M9.5\,V stars separated by 78\,arcsec.
At $d \sim$ 23\,pc, the projected physical separation becomes 1.80$\pm$0.17\,kAU, which was three orders of magnitude greater than the widest separation between ``normal'' ultracool binaries then known.
Because of its low total mass, of only about 0.18\,$M_\odot$, its binding energy makes the pair be still one of the most fragile systems known to date.
Caballero (2007b) went on looking for common proper-motion companions to almost 200 stars and brown dwarfs with spectral types later than M5.0\,V in the southern hemisphere, and discovered Koenigstuhl~2 and 3.
The latter is a multiple system formed by a bright, slightly metal-poor, F8\,V star, an M8.0\,V+L3\,V close pair at the astonishing projected physical separation (for its mass) of 11.9$\pm$0.3\,kAU, and another L1\,V close companion to the primary recently discovered by Gauza et~al. (2012).
\section{From May 2009 to December 2009: the widest double stars}
Our Aladin-based proper-motion survey of Luyten stars (Section~\ref{section.200807}), which actually was a heritage of the Koenigstuhl survey (Section~\ref{section.200700}), resulted not only in the characterisation of the magnetically-active, low-mass, triple system GIC~158 (Section~\ref{section.201007}), but also in the recovery of fragile pairs with angular separations greater than 100\,arcsec and total masses less than 1\,M$_\odot$.
But are they actually bound?
Before going ahead with our study, we had to make sure of that some of our discoveries had a physical meaning.
There were already very wide binaries and candidates in the literature with projected physical separations $s >$ 50\,kAU (e.g., Gliese 1969; Allen et~al. 2000; Zapatero Osorio \& Mart\'{\i}n 2004; L\'epine \& Bongiorno 2007; Makarov et~al. 2008; Poveda et~al. 2009; see also Scholz et~al. 2008).
However, numerous works had concluded that dynamical evolution dictates a sharp cutoff in the number of very wide binaries with physical separations greater than about 20\,kAU (one tenth of a parsec -- e.g., Tolbert 1964; Bahcall \& Soneira 1981; Abt 1988; Wasserman \& Weinberg 1991; Poveda \& Allen 2004).
In his first issue of the series ``Reaching the boundary between stellar kinematic groups and very wide binaries'' at Astronomy \& Astrophysics, Caballero (2009) concluded that the key parameter for ascertaining the physical meaning of an ultrawide pair ($s >$ 20\,kAU) is not the projected physical separation itself, but the gravitational binding energy, defined by $U_g^* = -G M_1 M_2 / s$.
Besides, he found that the most fragile systems, regardless of their total mass, have binding energy moduli of the order of 10$^{33}$--10$^{34}$\,J.
Caballero (2010b) took the statement ``size {\em and mass} matter'' to the limit with the identification of the star KU~Lib as the distant fifth member of the Zubenelgenubi ($\alpha$~Lib) multiple system based on common proper motion, parallax, radial velocity and age.
The stars, which are part of the young Castor moving group ($\tau \sim$ 200\,Ma), may be in the process of disruption.
With a total mass of 6.7\,M$_\odot$, in spite of the gargantuan projected physical separation of over 200\,kAU (about 2.6\,deg projected in the sky), the binding energy modulus is not lower than, for example, the nearby system $\alpha$~Cen~AB + Proxima Centauri.
The confirmation that this kind of ultrawide systems does exist was afterwards brought by Shaya \& Olling (2011) and Makarov (2012).
However, Zubenelgenubi + KU~Lib will for ever have the honour of being the first to break the one-parsec-separation barrier.
\section{From February to October 2012: more cool wide binaries}
\begin{table}[]
\caption{Wide multiple systems with cool dwarfs found in The Observatory series}
\center
\begin{minipage}{1.0\textwidth}
\center
\begin{tabular}{l ll ccc}
\hline\hline
WDS & Primary & Secondary & ${\mathcal M}_{\rm total}$ & $s$ & $-U_g^*$ \\
code & & & [M$_\odot$] & [kAU] & [10$^{33}$\,J] \\ [0.5ex]
\hline
KO 4 & NLTT 6496 & NLTT 6491 & 0.34$^{+0.4}_{-0.5}$ & 5.7$^{+1.8}_{-1.2}$ & 7.7--8.2 \\ %
KO 5 & HD 212168 (A) & DE2226--75 (C) & 1.20$\pm$0.10 & 6.09$\pm$0.08 & 30$\pm$3 \\ %
KO 6 & LP 209--28 & LP 209--27 & 0.88:--1.04: & 133--167 & 2.4--2.8 \\ %
FMR 83 & LSPM J0651+1843 & LSPM J0651+1845 & 0.50$\pm$0.10 & 10$^{+6}_{-4}$ & 11$\pm$1 \\ [0.5ex]
\hline
\end{tabular}
\end{minipage}
\label{T2}
\end{table}
Koenigstuhl~1 and Zubenelgenubi + KU~Lib, with total masses and projected physical separations of 0.2\,M$_\odot$ and 1--2\,kAU, and 6--7\,M$_\odot$ and 200\,kAU, respectively, are the archetypes of the most fragile systems in their total-mass intervals.
Because of the large abundance of M dwarfs, it is natural to consider that there can exist a large number of systems with intermediate total masses and projected physical separations, i.e., 0.5--2.0\,M$_\odot$ and 10--100\,kAU.
Some of them were found by Gliese, Giclas or Luyten and have next gone unnoticed; others simply await discovery.
The main aim of the series ``Cool dwarfs in wide multiple systems'' at The Observatory is to find and characterise such intermediate-mass ultrafragile systems.
The four systems studied in The Observatory series are summarised in Table~\ref{T2}.
Koenig\-stuhl~4 (Caballero 2012) is a loosely-bound common-proper-motion pair of two bright mid-M dwarfs at about 19\,pc.
Koenigstuhl~5 (Caballero \& Montes 2012) is a triple system formed by a Sun-like {\em Hipparcos} star, a poorly investigated K dwarf, and an M8.5\,V-type star at 6.09\,kAU to the primary.
While these two systems were discovered serendipitously, Koenigstuhl~6 (Caballero et~al. 2012), an extremely fragile system of two late-K/early-M dwarfs, was found during the proper-motion survey for companions to Luyten stars (Section~\ref{section.200807}).
It was not only discovered by amateurs, but part of its photometric and astrometric characterisation was performed with an ``amateur'' telescope of 40\,cm.
Something similar happened with FMR~83, a common proper-motion pair of two identical mid-M dwarfs separated by about 10\,kAU, which was discovered by Rica (2012) and characterised by Rica \& Caballero (2012).
\section{October 2012: back to the future}
At the time of writing these lines, our near-future project is the ``Observatori Astron\`omic del Garraf Wide Pairs Survey''.
It involves over 20 Spanish double-star amateur astronomers supported by the Spanish Virtual Observatory.
It is a continuation of the Koenigstuhl and Luyten surveys, but in which the team uses the Aladin sky atlas for blinking two images of the Digital Sky Survey taken at very separated epochs (several decades) and for looking for common proper-motion pair candidates {\em by eye}.
The difference with the visual blinking method that was widely used in the middle of the 20th century is that we take advantage of virtual observatory tools and powerful computers that allow loading images and catalogues in seconds.
After surveying only 17\,\% of the sky ($\alpha$ = 0 to 12\,h, $\delta$ = --20 to +20\,deg), the team has discovered 1725 new pairs, which have been already tabulated by the Washington Double Star catalogue (T.~Tobal, priv.~comm. to the WDS) and will be presented in forthcoming publications (J.\,A.~Caballero, D.~Valls-Gabaud, E.~Solano et~al., in prep.).
The strengths of the wide pair survey are illustrated by GWP~1519, which is formed by G~45--26 (M1:\,V, $\sim$0.5\,M$_\odot$) and LSPM~J1101+0059 (M7:\,V, $\sim$0.1\,M$_\odot$).
At the most probable distance to the double, the angular separation of 41.3\,arcsec translates into a projected physical separation of $s \sim$ 2.4\,kAU, from where one deduces a low binding energy modulus of $|U_g^*| \sim$ --50\,10$^{33}$\,J.
Curiously enough, GWP~1519 was discovered during the very first days of the survey by N\'uria Miret, who at that time was a... High school student!
\small
\section*{Acknowledgments}
JAC is an {\em investigador Ram\'on y Cajal} of the CSIC at CAB.
Financial support was provided by the Spanish Ministerio de Ciencia e Innovaci\'on under grants
AyA2011-30147-C03-02 and~-03.
|
2,869,038,155,692 | arxiv | \section{Introduction}
\label{sec:introduction}
In the Solar System, the orbits of all the planets are nearly coplanar (within 4 degrees, except for Mercury). The ecliptic (the plane of the Earth's orbit) is also close to the equatorial plane of the Sun: the spin-orbit misalignment is only $\beta_\oplus=7.5^\circ$. The low inclination of the massive planets with respect to the ecliptic is normally taken as an indication that planets form within a flattened protoplanetary disc, itself closely aligned with the stellar equator. The newly discovered Kepler-30 system \citep{2012Natur.487..449S} is even flatter, and confirms this view. However, exo-planets with strong spin-orbit misalignment have been detected (e.g. $\beta>50^\circ$ \citet{2011A&A...533A.113M, 2011epsc.conf..596M, 2011A&A...527L..11H, 2011MNRAS.414.3023S}). Considering that the plane of the past protoplanetary disc should be identical to the present stellar equator\footnote{This is generally accepted, but is actually the subject of debate (see e.g. \citet{2011EPJWC..1103003C, 2012Natur.491..418B}).}, the orbital plane of these planets must have been changed by some mechanism.
One process generally invoked to explain inclined orbits is scattering by multiple planets in the system after the protoplanetary disc has dissipated (e.g. \citet{2002Icar..156..570M,2008ApJ...686..580C,2008ApJ...686..603J}). These works assume that unstable crowded systems are formed, and undergo planet-planet scattering after a relatively short time when the gas nebula dissipates. However, recent work suggests that unstable systems reach instability while still embedded in the gas disc \citep{Legaetal2013}. A second process is planet-planet interactions during migration in the protoplanetary disc \citep{2003ApJ...597..566T, 2009MNRAS.400.1373L, 2011MNRAS.412.2353L, 2011CeMDA.111..201L}. During the gas-driven migration, the system can enter an inclination-type resonance or the resonant configuration becomes unstable as the resonance excites the eccentricities of the planets and planet-planet scattering sets in. All this affirms the need for a better understanding of the interactions between giant planets and a gaseous protoplanetary disc when the orbit of the former is highly inclined with respect to the midplane of the later. Here, we study this phenomenon, in detail.
\citet{2004ApJ...602..388T} have shown in linear studies that the inclination of a low-mass planet embedded in a disc is exponentially damped by planet-disc interactions for any non-vanishing inclination. Such results are formally valid only for $i \ll H/r$. However, numerical simulations of more highly inclined planets have shown that the exponential damping might be valid up to $i \approx 2 H/r$. If the planet has an even greater inclination, the damping rates deviate from being exponential and it can be fitted by a $di/dt \propto i^{-2}$ function \citep{2007A&A...473..329C, 2011A&A...530A..41B}. However, for high-mass planets, the linear regime is no longer valid. \citet{0004-637X-705-2-1575} considered Jovian-type planets on inclined and eccentric orbits. They find highly inclined and eccentric planets with Jovian masses lose their inclination and eccentricity very quickly (on a time-scale of the order of $10^3$ years) when entering the disc again (when $i<H/r$). Since a highly inclined planet is only slightly disturbed by the accretion disc (and vice versa), this kind of planet is only able to open a gap in the disc when the inclination drops to $i<10.0^\circ$.
Planet-disc interactions also influence the eccentricity of embedded planets, as has been shown by \cite{1980ApJ...241..425G}. It has been suggested, by performing linear analysis, that the planetary eccentricity can be increased through planet-disc interaction under some conditions \citep{2003ApJ...585.1024G, 2004ApJ...606L..77S, 2008Icar..193..475M}. They estimate that eccentric Lindblad resonances can cause eccentricity growth for gap-forming planets. However, numerical simulations show that eccentricity in the disc is damped for a variety of masses \citep{2007A&A...473..329C, 2009arXiv0904.3336M, 2010A&A.523...A30}.
For very-high-mass planets, on the other hand an eccentric instability in the disc can arise \citep{2006A&A...447..369K}. In turn, this eccentric disc can possibly increase the planetary eccentricity \citep{2001A&A...366..263P,2006ApJ...652.1698D}. However, this process can only explain the eccentricity of very massive ($\approx 5-10 M_{Jup}$) planets. \citet{2013MNRAS.431.1320X} have recently studied the interactions between Jupiter-mass planets and circumstellar discs as well. However, they did not consider planets on eccentric orbits and they were using SPH simulations, while we use a grid-based code.
In this paper, we investigate planet-disc interactions for planets above $1M_{\rm Jup}$, considering different inclination and eccentricity values. Our analysis also aims at deriving a formula for the change of eccentricity and inclination due to planet-disc interactions, in order to study the long-term evolution of systems with massive planets. Indeed, long-term evolution studies of planetary systems cannot be done with hydrodynamical simulations, as the computation time is too long, and N-Body codes that consider the gravitational effects only are used. A correct damping rate of eccentricity and inclination is needed in order to simulate the evolution correctly. This study will be the topic of our paper~II.
We use isothermal three-dimensional (3D) simulations to determine the change of inclination and eccentricity due to planet disc interactions. In Sect.~\ref{sec:methods} we describe the numerical methods used, as well as the procedure to calculate the forces acting on the embedded planets to determine $di/dt$ and $de/dt$. In Sect.~\ref{sec:change} we show $di/dt$ and $de/dt$ as a function of inclination $i$ and eccentricity $e$, and provide fitting formulae. Additionally an observed oscillatory behaviour is discussed in this section. The implications for single-planet systems are shown in Sect.~\ref{sec:application}.
\section{Physical modelling}
\label{sec:methods}
The protoplanetary disc is modelled as a 3D, non-self-gravitating gas whose motion is described by the Navier-Stokes equations. We use the code {\tt Nirvana} \citep{1997ZiegYork,2001ApJ...547..457K}, which uses the {\tt FARGO}-algorithm \citep{2000A&AS..141..165M} and was described in our earlier work on planets on inclined orbits \citep{2011A&A...530A..41B}. We note that the use of the {\tt FARGO}-algorithm may not be straight forward in the case of highly inclined planets ($i=75.0^\circ$). Our test simulations, however, show that this algorithm can also be used in highly inclined planets, see Appendix.~\ref{ap:numerics}. Here we treat the disc as a viscous medium in the locally isothermal regime. We do not use radiation transport, as we focus here on high-mass planets that open a gap inside a disc, where the effects of heating and cooling of the disc are much less important than for low-mass planets \citep{2009A&A...506..971K}. A more detailed description of the used code can be found in \citet{2009A&A...506..971K}.
\subsection{Smoothing of the planetary potential}
\label{subsec:pot}
An important issue in modelling planetary dynamics in discs is the gravitational potential of the planet since this has to be artificially smoothed to avoid singularities. While in two dimensions a potential smoothing takes care of the otherwise neglected vertical extension of the disc, in three dimensional simulations the most accurate potential should be used. As the planetary radius is much smaller than a typical grid cell, and the planet is treated as a point mass, a smoothing of the potential is required to ensure numerical stability.
In \citet{2009A&A...506..971K} two different kinds of planetary potentials for 3D discs have been discussed. The first is the classic $\epsilon_{sm}$-potential
\begin{equation}
\label{eq:epsilon}
\Phi_p^{\epsilon_{sm}} = - \frac{G M_p}{\sqrt{d^2 + \epsilon_{sm}^2}} \, .
\end{equation}
Here $M_P$ is the planetary mass, and $d=| \mathbf{r} - \mathbf{r_P}|$ denotes the distance of the disc element to the planet. This potential has the advantage that it leads to very stable evolutions when the parameter $\epsilon_{sm}$ is a significant fraction of the Roche radius. The disadvantage is that for smaller $\epsilon_{sm}$, which would yield a higher accuracy at larger $d$, the potential becomes very deep at the planetary position. Additionally, the potential differs from the exact $1/r$ potential even for medium to larger distances $d$ from the planet.
To resolve these problems at small and large $d$ simultaneously, the following {\it cubic}-potential has been suggested \citep{2006A&A...445..747K,2009A&A...506..971K}
\begin{equation}
\label{eq:cubic}
\Phi_p^{cub} = \left\{
\begin{array}{cc}
- \frac{G M_p}{d} \, \left[ \left(\frac{d}{r_\mathrm{sm}}\right)^4
- 2 \left(\frac{d}{r_\mathrm{sm}}\right)^3
+ 2 \frac{d}{r_\mathrm{sm}} \right]
\quad & \mbox{for} \quad d \leq r_\mathrm{sm} \ \textcolor{white}{.} \\
- \frac{G M_p}{d} \quad & \mbox{for} \quad d > r_\mathrm{sm} \ .
\end{array}
\right.
\end{equation}
The construction of the planetary potential is such that for distances larger than $r_{sm}$ the potential matches the correct $1/r$ potential. Inside this radius ($d < r_{sm}$) it is smoothed by a cubic polynomial. This potential has the advantage of exactness outside the specified distance $r_{sm}$, while being finite inside.
For $1M_{\rm Jup}$ and $5M_{Jup}$ we use the cubic potential with $r_{sm}=0.8 R_H$. For the $10M_{\rm Jup}$ planet, we use the $\epsilon_{sm}$-potential with $r_{sm}=0.8 R_H$, with the Hill radius $R_H$ given by
\begin{equation}
R_{H} = a_p \left(\frac{M_p}{3 M_\star}\right)^{1/3} \ ,
\end{equation}
where $a_p$ is the semi major axis of the planet, and $M_\star$ is the mass of the central star.
As the planetary mass increases, so does the amount of material accumulated near the planet. In order to resolve the gradients of density in that region correctly, a much higher resolution is required. Therefore, we change the cubic potential to the $\epsilon_{sm}$-potential for the $10M_{\rm Jup}$ planet. For the torque acting on the planets, the consequences are minimal, as we use a torque cut-off function in the Hill sphere of the planet, as described below. Additional information regarding the smoothing length can be found in Appendix~\ref{ap:numerics}.
\subsection{Initial setup}
The three-dimensional ($r, \theta, \phi$) computational domain consists of a complete annulus of the protoplanetary disc centred on the star, extending from $r_{min}=0.2$ to $r_{max}=4.2$ in units of $r_0=a_{Jup}=5.2 AU$, where we put the planet. The planet is held on a fixed orbit during the evolution. The eccentricity of the planet can be $e_0=0.0$, $e_0=0.2$, or $e_0=0.4$. We use $390 \times 48 \times 576$ active cells for the simulations with $1M_{Jup}$ and $260 \times 32 \times 384$ active cells for $5M_{Jup}$ and $10M_{Jup}$. This resolution is sufficient, as we still resolve the horseshoe width with a few grid cells for all planetary masses. The horseshoe width is defined for large planets as $x_s = \sqrt{12} a_P (q/3)^{1/3}$ \citep{2006ApJ...652..730M}, where $q$ is the planet-star mass ratio. Tests regarding the numerical resolution can be found in Appendix~\ref{ap:numerics}.
In the vertical direction, the annulus extends $7^\circ$ below and above the disc's midplane, meaning $83^\circ < \theta < 97^\circ$. Here $\theta$ denotes the polar angle of our spherical polar coordinate system measured from the polar axis, therefore the midplane of the disc is at $\theta=90.0^\circ$. We use closed boundary conditions in the radial and vertical directions. In the azimuthal direction, periodic boundary conditions are used. The central star has one solar mass $M_\ast = M_\odot$, and the total disc mass inside [$r_{min}, r_{max}$] is $M_{disc} = 0.01 M_\odot$. The aspect ratio of the disc is $H/r=0.05$. We use an $\alpha$ prescription of the viscosity, where $\nu = \alpha c_s^2 / \Omega_K$ \citep{1973A&A....24..337S} with $\alpha=0.005$\,; $\Omega_K$ is the Kepler frequency\,; $c_s = \sqrt{P/\rho}$ denotes the isothermal sound speed, $P$ the pressure, $\rho$ the volume density of the gas, and $H=c_s/\Omega$.
The models are initialised with constant temperatures on cylinders with a profile $T(s) \propto s^{-1}$ with $s=r \sin \theta$. This yields a constant ratio of the disc's vertical height $H$ to the radius $s$. The initial vertical density stratification is given approximately by a Gaussian
\begin{equation}
\rho(r,\theta)= \rho_0 (r) \, \exp \left[ - \frac{(\pi/2 - \theta)^2 \, r^2}{2 H^2} \right] \ .
\end{equation}
Here, the density in the midplane is $\rho_0 (r) \propto r^{-1.5}$ which leads to a $\Sigma(r) \propto \, r^{-1/2}$ profile of the vertically integrated surface density. In the radial and $\theta$-direction we set the initial velocities to zero, while for the azimuthal component the initial velocity $u_\phi$ is given by the equilibrium of gravity, centrifugal acceleration and the radial pressure gradient. This corresponds to the equilibrium configuration for a purely isothermal disc with constant viscosity. However, as the massive planets in the disc start to open gaps, the density and surface density profile get distorted.
\subsection{Calculation of forces}
\label{subsec:forces}
To determine the change of orbital elements for planets on fixed inclined orbits, we follow \citet{burns:944} and compute the forces as described in \citet{2011A&A...530A..41B}. The gravitational torques and forces acting on the planet are calculated by integrating over the whole disc, where we apply a tapering function to exclude the inner parts of the Hill sphere of the planet. Specifically, we use the smooth (Fermi-type) function
\begin{equation}
\label{eq:fermi}
f_b (d)=\left[\exp\left(-\frac{d/R_H-b}{b/10}\right)+1\right]^{-1}
\end{equation}
which increases from 0 at the planet location ($d=0$) to 1 outside $d \geq R_{H}$ with a midpoint $f_b = 1/2$ at $d = b R_{H}$, i.e. the quantity $b$ denotes the torque-cutoff radius in units of the Hill radius. This torque-cutoff is necessary to avoid large, probably noisy contributions from the inner parts of the Roche lobe and to disregard material that is possibly gravitationally bound to the planet \citep{2009A&A...502..679C}. Here we assume $b= 0.8$, as a change in $b$ did not influence the results significantly \citep{2009A&A...506..971K}.
If a small disturbing force $\mathbf{dF}$ (given per unit mass) due to the disc is acting on the planet, the planet changes its orbit. This small disturbing force $\mathbf{dF}$ may change the planetary orbit in size (semi-major axis $a$), eccentricity $e$, and inclination $i$. The inclination $i$ gives the angle between the orbital plane and an arbitrary fixed plane, which is in our case the equatorial plane ($\theta = 90^\circ$), which corresponds to the midplane of the disc. Only forces lying in the orbit plane can change the orbit's size and shape, while these forces cannot change the orientation of the orbital plane. In \citet{burns:944} the specific disturbing force is written as
\begin{equation}
\label{eq:distforce}
\mathbf{dF} = \mathbf{R} + \mathbf{T} + \mathbf{N} = R \mathbf{e}_{R} + T \mathbf{e}_T + N \mathbf{e}_N \ ,
\end{equation}
where each $\mathbf{e}$ represents the relevant orthogonal component of the unit vector. The perturbing force can be split into these components: $\mathbf{R}$ is radially outwards along $\mathbf{r}$; $\mathbf{T}$ is transverse to the radial vector in the orbit plane (positive in the direction of motion of the planet); and $\mathbf{N}$ is normal to the orbit planet in the direction $\mathbf{R} \times \mathbf{T}$.
\citet{burns:944} finds for the change of inclination
\begin{equation}
\label{eq:ichange}
\frac{di}{dt} = \frac{a N \cos \xi}{H} \ ,
\end{equation}
where the numerator is the component of the torque which rotates the specific angular momentum $\mathbf{H} = \mathbf{r} \times \mathbf{\dot{r}}$ about the line of nodes (and which thereby changes the inclination of the orbital plane). The specific angular momentum $H$ is defined as
\begin{equation}
H = \sqrt{ G M_\star a_p (1 - e^2)} \ .
\end{equation}
The angle $\xi$ is related to the true anomaly $f$ by $f = \xi - \omega$, with $\omega$ being the argument of periapsis and $\xi$ describes the angle between the line of nodes and the planet on its orbit around the star. For the case of circular orbits, the argument of periapsis $\omega$ is zero.
The change of eccentricity is given by \citet{burns:944} as
\begin{equation}
\label{eq:eccchange}
\frac{de}{dt} = \left[ \frac{a (1-e^2)}{GM_\star} \right]^{1/2} \left[R \sin f + T (\cos f + \cos \epsilon) \right] \ ,
\end{equation}
where $\epsilon$ is the eccentric anomaly, which is given by
\begin{equation}
\cos \epsilon = \frac{e + \cos f}{1 + e \cos f} \ .
\end{equation}
With this set of equations, we are able to calculate the forces acting on planets on fixed orbits and determine $di/dt$ and $de/dt$.
\section{Planets on inclined and eccentric orbits}
\label{sec:change}
In this section we investigate the changes of the planetary orbit due to planet-disc interactions. The planets are put in fixed orbits with inclinations ranging from $i_0=1.0^\circ$ to $i_0=75^\circ$, with a total of ten different inclinations. For each inclination we also adopt three different eccentricities, which are $e_0=0.0$, $e_0=0.2$ and $e_0=0.4$.
We note that the orbit of highly inclined planets is not embedded completely in the hydrodynamical grid, since the grid is only extended up to $7^\circ$ above and below midplane. However, the density distribution in the vertical direction follows a Gaussian profile and for an aspect ratio of $0.05$ we are at about $2.5\sigma$ at $7^\circ$ so that the contribution of the gas can be neglected at larger $\theta$.
\subsection{Gaps in discs}
The criterion for gap opening depends on the viscosity, the pressure, and the planetary mass \citep{2006Icar..181..587C}. Giant planets ($M \lower 3pt\hbox{${\buildrel > \over \sim}$}\ 0.5 M_{\rm Jup}$) are generally massive enough to split the disc. However, the inclination of a giant planet plays a very important role in opening a gap as well, as can be seen in Fig.~\ref{fig:Sig10MJup}, where we display the surface density profile of discs with embedded $10M_{\rm Jup}$ planets on different inclinations.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{Sig10MJup.eps}}
\caption{Surface density for disc simulations with $10M_{Jup}$ planets in circular and eccentric orbits with different inclinations. The surface density is plotted after $400$ planetary orbits. The evolution has reached an equilibrium state, meaning that the surface density does not change in time any more.
\label{fig:Sig10MJup}
}
\end{figure}
Clearly, a lower inclination produces a much wider and deeper gap inside the disc. For larger inclinations, the gap opening is reduced, as the planet spends less and less time inside the disc to push material away from its orbit. Additionally, eccentric planets open up gaps that are less deep than their circular counter parts. This effect is very important for the damping of inclination and eccentricity, as an open gap inside the disc prolongs the damping time-scale of inclination \citep{2011A&A...530A..41B} and of eccentricity \citep{2010A&A.523...A30}. Gap opening also indicates that linear analysis of the situation is no longer applicable.
In Fig.~\ref{fig:10MJupxyrho} we present slices in the $x-z$-plane for the disc's density for $10M_{Jup}$ planets on inclinations of $1^\circ$, $20^\circ$, and $75^\circ$ degrees. The inclinations correspond to those shown in the surface density plot (Fig.~\ref{fig:Sig10MJup}). Clearly the depth of the gap shown in the surface density is reflected in the 2D plots. Additionally, the density structures show no effects at the upper and lower boundaries because of boundary conditions, indicating that an opening angle of $7^\circ$ is sufficient for our simulations.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=1.0\linwx]{10MJupxydeg1.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=1.0\linwx]{10MJupxydeg20.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=1.0\linwx]{10MJupxydeg75.eps}}
\caption{Density (in $g/cm^3$) of a $r-\theta$-slice through the disc at the azimuth of an embedded $10 M_{Jup}$ planet on a fixed circular inclined orbit with $i_0=1.0^\circ$ (top), $i_0=20^\circ$ (middle), and $i_0=75.0^\circ$ (bottom). The planet is at its lowest point in orbit (lower culmination) at the time of the snapshot, which was taken after $400$ planetary orbits. We note the slightly different colour scale for each plot. The black line indicates the midplane of the grid to which the inclination of the disc is measured (see Sect.~\ref{subsec:eichange}).
\label{fig:10MJupxyrho}
}
\end{figure}
\subsection{Change of the disc structure}
\label{subsec:eichange}
It has been known since several years that massive planets are able not only to open up a gap in the disc, but are also able to change the shape of the whole disc by turning it eccentric \citep{2001A&A...366..263P, 2006A&A...447..369K}. Additionally, the inclination of the disc will change due to the interactions with the inclined planet. In this section, we discuss the impact of a massive planet on the eccentricity and inclination of the disc.
In Fig.~\ref{fig:eidisc10MJup} we display the eccentricity (top) and inclination (bottom) of the disc interacting with a $10M_{Jup}$ planet with different inclinations ($1^\circ$ and $75^\circ$) and eccentricities. The calculations for deriving the eccentricity and inclination of the disc can be found in Appendix~\ref{ap:eidisc}.
For low planetary inclinations, the influence of the planet on the eccentricity of the protoplanetary disc is greater than on high planetary inclinations, simply, because the planet is closer to midplane and can therefore influence the eccentricity of the disc more strongly by pushing the material away. The eccentricity increase of the disc is stronger for planets in circular orbits than for planets that are already in an eccentric orbit. For highly inclined planets, the situation is reversed. The disc is most eccentric for planets that are already in an eccentric orbit and the disc is less eccentric for planets in circular orbits. Additionally, the eccentricity of the disc is highest close to the planet and drops with distance from the planet, independent of the inclination of the planet.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=1.0\linwx]{edisc10MJup.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=1.0\linwx]{idisc10MJup.eps}}
\caption{Eccentricity (top) and inclination (bottom) of the disc with a $10M_{Jup}$ planet influencing the disc structure after $400$ planetary orbits.
\label{fig:eidisc10MJup}
}
\end{figure}
The inclination of the disc for the $i_0=1^\circ$ planets is greater mostly around the planet's location (at $r=1.0a_{Jup}$) because the influence of the planet is strongest there. The inclination of the disc can be larger than the inclination of the planet. This is possible because the planet opens a gap inside the disc and pushes the material away from the planet (Fig.~\ref{fig:Sig10MJup}, top), which can also be seen in the 2D density configuration (top of Fig.~\ref{fig:10MJupxyrho}). In the outer parts of the disc the disc remains non-inclined.
For planets with high inclinations the situation is slightly different than for planets with low inclinations. The maximum of inclination is lower and there is no distinct maximum of inclination visible inside the planetary orbit ($r<1.0a_{Jup}$) compared to the case of low inclinations. However, the outer parts of the disc show a non-zero inclination (which has a tendency to be larger for larger planetary eccentricities), which was not visible for the low-inclination planets. Additionally, they show a small peak of inclination at $r \approx 1.25a_{Jup}$.
\subsection{Change of orbital parameters}
\subsubsection{Eccentricity}
\label{subsubsec:Eccentricity}
As stated in Sect.~\ref{subsec:forces}, the forces acting on a planet on a fixed orbit can be calculated and then used to determine a rate of change for the inclination and eccentricity. The damping rates are taken when the planet-disc interactions are in an equilibrium state and do not change on average any more. The damping given by Eq.~(\ref{eq:ichange}) varies strongly within the time of an orbit and slightly from one orbit to an other. Thus, we averaged it over $40$ planetary orbits.
In Fig.~\ref{fig:eccentricity} we present the change of eccentricity $de/dt$ for planets of $1M_{Jup}$, $5M_{Jup}$, and $10M_{Jup}$ on orbits with different inclinations and eccentricities. The change of $de/dt$ has been studied in the past for coplanar low-mass planets \citep{2007A&A...473..329C, 2010A&A.523...A30} and for high-mass planets \citep{2001A&A...366..263P, 2006A&A...447..369K}.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{eccentricity.eps}}
\caption{Change of eccentricity $de/dt$ for planets with $1M_{Jup}$, $5M_{Jup}$, and $10M_{Jup}$ with different eccentricities. Points are results from numerical simulations, while lines indicate the fitting of the data. The $1M_{Jup}$ planets have been evolved for $200$ planetary orbits, the $5M_{Jup}$ and $10M_{Jup}$ planets have been evolved for 400 orbits. The forces used to calculate the data points have been averaged over $40$ planetary orbits for all simulations.
\label{fig:eccentricity}
}
\end{figure}
For low inclinations ($i_0<10^\circ$) the damping of eccentricity is stronger than for larger inclinations in the case of $1M_{Jup}$. The maximum damping rate is also dependent on the initial eccentricity $e_0$, where a larger $e_0$ provides a faster damping. The damping of eccentricity is reduced significantly for larger inclinations $i_0>20^\circ$. As soon as the planet is no longer embedded in the disc, the damping reduces, as it is most efficient when the planet is inside the disc and not high above or below the disc for most of its orbit.
For low inclinations ($i_0<10^\circ$) and low eccentricities ($e_0<0.2$), the $5M_{Jup}$ planet opens up a large gap inside the disc. As the planet opens a gap inside the disc the damping is reduced because there is less material close to the planet to damp its orbit. For large initial eccentricities ($e_0=0.4$), an increase of eccentricity is observable for low planetary inclinations. But for higher inclinations, the damping of eccentricity increases as well, because the planet does not open up such a deep gap (Fig.~\ref{fig:Sig10MJup}). However, for $i_0>40^\circ$ the damping of eccentricity becomes smaller again, because the planet spends less and less time in the midplane of the disc where most of the disc material is, which is responsible for damping.
For even higher masses ($10M_{Jup}$), we observe an eccentricity increase for low planetary inclinations for all non-zero eccentricities. But for larger inclinations ($i_0>15^\circ$), the eccentricity is damped again. The largest value of damping is at $i_0\approx 30^\circ-50^\circ$, depending on the planet's eccentricity and is then reduced for higher inclinations again, following the trend described for the $5M_{Jup}$ planet.
For large planets with low inclinations, the eccentricity of the planet can rise, which has been observed in \citet{2001A&A...366..263P} and \citet{2006A&A...447..369K}. \citet{2001A&A...366..263P} stated that if the planet opens up a large gap, the $m=2$ spiral wave at the $1:3$ outer eccentric Lindblad resonance becomes dominant (because the order $1$ resonances lie inside the gap) and induces eccentricity growth. However, they found an eccentricity increase only for $M_P>20M_{Jup}$, while our simulations indicate it clearly already for $M_P>5M_{Jup}$ (Fig.~\ref{fig:eccentricity}, bottom). The differences between their 2D simulations and our 3D simulations can be the cause of the change in the required planetary mass for eccentricity growth.
Additionally, by embedding a high-mass planet inside a disc, the disc can become eccentric, as shown in Sect.~\ref{subsec:eichange}. The disc's eccentricity is dependent on the planet's inclination and slightly dependent on its eccentricity as well (see Fig.~\ref{fig:eidisc10MJup}).
It seems that the coupling between a large disc eccentricity at $r \approx 1-1.5 a_P$ and a large planetary eccentricity (the $i_0=1^\circ$ with $e=0.4$ case) results in a large force on the planet. This effect is increased as the planet in an eccentric orbit opens a small gap leaving more material at that location. This leads then to a greater increase of eccentricity for highly eccentric planets, compared to those with small eccentricity.
\subsubsection{Inclination}
\label{subsubsec:Inclination}
In Fig.~\ref{fig:inclination} we present the rate of change of inclination $di/dt$, presented in degrees per orbit, for planets with different masses and different eccentricities. For $1M_{Jup}$ the inclination is damped for all initial inclinations. For increasing inclinations with $i_0<15^\circ$ (smaller for increasing eccentricity), the damping of inclination increases. This increase is nearly linear, as has been shown for low-mass planets in theory \citep{2004ApJ...602..388T} and in numerical simulations \citep{2007A&A...473..329C, 2011A&A...530A..41B}. The rates of inclination damping for zero-eccentricity planets are comparable to those stated in \citet{2013MNRAS.431.1320X}.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{inclination.eps}}
\caption{Change of inclination $di/dt$ for planets with $1M_{Jup}$, $5M_{Jup}$, and $10M_{Jup}$ with different eccentricities. $di/dt$ is in degrees per orbit at the planet's location $r_P=1.0a_{Jup}$. Points are results from numerical simulations, while lines indicate the fitting of the data. The $1M_{Jup}$ planets have been evolved for $200$ planetary orbits, the $5M_{Jup}$ and $10M_{Jup}$ planets have been evolved for $400$ orbits. The forces used to calculate the data points have been averaged over $40$ planetary orbits for all simulations.
\label{fig:inclination}
}
\end{figure}
For $i_0>15^\circ$, the damping rate of the inclination is a decreasing function of inclination\,; this is consistent with the planet-disc interaction being weaker when the planet spends more time farther from the midplane.
For $5M_{Jup}$ the damping of inclination is almost the same as for the $1M_{Jup}$ planet, but with a maximum at $i_0 \approx 20^\circ$. However, there is a significant difference for high inclined and low eccentric planets\,: the inclination is not damped if $i_0>50^\circ$, but it increases for $e_0<0.1$. This behaviour will be discussed in Sect.~\ref{subsec:moving}.
The $10M_{Jup}$ planet shows the same general behaviour as the $5M_{Jup}$ planet, but the inclination increase already sets in at $i_0\geqslant 45^\circ$, depending on $e_0$. Still, no inclination increase is observed in the high eccentricity simulations ($e_0=0.4$). We also want to stress here that the damping rate significantly increases with increasing planetary eccentricity for all planetary masses.
The increase of inclination for high-mass planets due to interactions with the disc has been studied in \citet{2001ApJ...560..997L}. They state that the $1:3$ mean-motion resonance also acts to increase inclination. This resonance is at $r_r=2.08r_P$, which clearly is not inside an open gap in the case of $i_0=75^\circ$ (see Fig.~\ref{fig:Sig10MJup}). However, the resonances closer to the planet ($1:2$ and $2:3$) are also not completely inside the gap, so that there should be some damping effects, but the damping of inclination through these resonances is weaker than the increase from the $1:3$ resonance because in total the inclination increases for high inclined planets (Fig.~\ref{fig:inclination}). \citet{2001ApJ...560..997L} also used small $i_0$ for their calculations in order to apply linear theory, which does not apply for large inclinations. The situation for our high inclination planets might therefore be completely different from their calculations.
\subsection{Moving planets in discs}
\label{subsec:moving}
\subsubsection{Short-term evolution}
In order to verify the results of inclination and eccentricity change, we present in this section simulations of planets evolving freely in the disc. The planets are moving because of the influences of the discs forces. We present here several interesting cases for planets with high inclinations. The first case is for $5M_{Jup}$ and $10M_{Jup}$ with an inclination of $i_0=40^\circ$ and $i_0=75^\circ$ in circular orbits with an evolution time of $80$ planetary orbits. In Fig.~\ref{fig:Inc40ev} the evolution of inclination with time is presented for the two different planets and inclinations. The evolution is nearly identical, as was predicted by the measured forces for the planets on fixed orbits, which is shown by the solid lines (rates from Fig.~\ref{fig:inclination}).
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{Inc40evo.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{Inc75evo.eps}}
\caption{Evolution of inclination of $5M_{Jup}$ and $10M_{Jup}$ planets with an initial inclination of $i_0=40.0^\circ$ (top) and $i_0=75.0^\circ$ (bottom) in circular orbits. The simulations have been restarted with a moving planet after the disc was evolved for a fixed planet for $400$ planetary orbits. The time index has been reset to zero and the lines correspond to the expected damping rates from Fig.~\ref{fig:inclination}.
\label{fig:Inc40ev}
}
\end{figure}
One should be aware, however, that by keeping the planet in a fixed orbit, angular momentum in the system is not conserved because, for example, the inclination of the disc is rising (see Fig.~\ref{fig:eidisc10MJup}) while the planet remains in a fixed orbit. The effect of conserving angular momentum is not a problem for low-mass planets, where the measured forces match perfectly with the inclination damping rates for moving planets \citep{2011A&A...530A..41B}, but for big planets of several Jupiter masses this can lead to small differences because the angular momentum transfer from disc to planet and vice-versa is much larger.
\subsubsection{Long-term evolution}
The long-term evolution of planets with different inclinations and eccentricities is displayed in Fig.~\ref{fig:kozai}. At the beginning of the evolution, the change of inclination and eccentricity matches those presented in Figs.~\ref{fig:eccentricity} and \ref{fig:inclination} for planets in fixed orbits. However, the evolution after the initial orbits is quite different from what was expected by the previous simulations.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{Inckozai.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{Ecckozai.eps}}
\caption{Long-term evolution of planets with different inclinations, eccentricities, and masses in discs. The simulations are started from the equilibrium structures with fixed planets, where the planets are then released and allowed to move freely inside the disc. The top plot features the inclination of the planets, while the bottom plot shows the eccentricity of the planets. In the beginning the changes of eccentricity and inclination match the ones displayed in Figs.~\ref{fig:eccentricity} and \ref{fig:inclination}.
\label{fig:kozai}
}
\end{figure}
In the $10M_{Jup}$, $e_0=0.0$, $i_0=75^\circ$ case, the inclination initially increases slightly with a rate that corresponds to the predicted rate (see also Fig.~\ref{fig:Inc40ev}). At the same time, the eccentricity of the planet increases and after about $250$ orbits it reaches $e\approx 0.25$. This eccentricity corresponds to inclination damping (Fig.~\ref{fig:inclination}), which is what happens in the evolution of the planet: the inclination drops. However, the eccentricity still increases at the same time, which was not predicted by the analysis of planets in fixed orbits (Fig.~\ref{fig:eccentricity}). The eccentricity then rises until a peak of $e\approx 0.9$, where it starts to drop again. At the same time, the inclination decrease stops and the inclination starts to rise again. As soon as the eccentricity has dropped to $e\approx 0.4$, the inclination starts to decrease again.
This exchange of inclination and eccentricity is representative of
the Kozai mechanism, introduced initially to describe the evolution of
a highly inclined asteroid perturbed by Jupiter \citep{1962P&SS....9..719L, 1962AJ.....67..591K}. A similar Kozai
mechanism affects the orbits of highly inclined planets with respect
to a disc \citep{2010MNRAS.404..409T, 2013MNRAS.428..658T}. For inclinations above a critical value, the gravitational force
exerted by the disc on the planet produces Kozai cycles where the
eccentricity of the planet can be pumped to high values, in antiphase
with its inclination. We note that the Kozai mechanism is visible
in the given computation time because of the high mass values
considered in our study ($5$ and $10 M_{Jup}$), comparable to the total mass
of the disc ($0.01 M_\odot$). Indeed high masses induce faster dynamical
evolution.
When the planet starts at a larger initial eccentricity ($e_0=0.2$ or
$e_0=0.4$), the general behaviour is similar as can be seen in
Fig.~\ref{fig:kozai}, but the first Kozai cycle occurs earlier. Circular orbits at high inclination constitute an unstable
equilibrium of the secular dynamics, so the evolution at zero initial
eccentricity remains for a while close to the separatrix associated with
the equilibrium \citep{2007Icar..191..469L}. The Kozai effect does not act for initial inclinations smaller
than a critical value ($i_0<~40^\circ$ in the restricted problem of
\citet{1962AJ.....67..591K}). We therefore also display a planet with $i_0=40^\circ$,
and show that the eccentricity and inclination oscillations are
significantly reduced.
Even if eccentricity can be pumped to high values, the Kozai mechanism
only postpones the alignment with the disc and the circularization of
the orbit induced by damping forces of the disc on the planet (given
in Figs.~\ref{fig:eccentricity} and \ref{fig:inclination}). As clearly shown by the evolution of the planet with
$e_0=0.2$, Kozai cycles repeat with reduced intensity. The drop of
inclination is much larger than the raise of inclination, after the
eccentricity increase/decrease cycle. These results are in agreement
with \citet{2013MNRAS.428..658T}, showing that low-mass planets would
remain on inclined and eccentric orbits over the disc lifetime, while
higher mass planets would align and circularize. We also illustrate in
Fig.~\ref{fig:kozai} the influence of the planet mass by considering a planet of $5
M_{Jup} (e=0.0)$: the eccentricity value reached during the second cycle
of inclination increase (at $450$ orbits) is higher for the $5 M_{Jup}$ planet, as expected.
The effect of Kozai oscillations between a disc and planet was also stated in \citet{2013MNRAS.431.1320X} however, \citet{2013MNRAS.431.1320X} were not able to resolve a full Kozai cycle, probably because their mass-ratio between planet and disc is smaller than in our case. This shows that for $i_0>40^\circ$, the measure of the forces on a planet on a fixed orbit is not relevant. In this case, damped Kozai oscillations will govern the long-term evolution of the orbital parameters. This phenomenon can be of crucial importance for the study of the fate of planets scattered on high-inclination orbits.
\subsection{Fitting for e and i}
\label{subsec:fit}
In Figs.~\ref{fig:inclination} and \ref{fig:eccentricity} we provided the change of $di/dt$ and $de/dt$ for different planetary masses. In these plots, lines indicate a fit for these data points. We now present the fitting formulae, which depend on the planet mass $M_P$, the eccentricity $e_P$, and inclination $i_P$. The inclination $i_P$ used in the presented formulae is given in degrees, as is the resulting $di/dt$. As discussed in the previous section, these formulae are only relevant for $i_0<40^\circ$ where no complex cycles are observed. Therefore, in fitting our parameters we have ignored the data points corresponding to high inclinations, in particular the ones showing inclination increase. This applies to the fitting of inclination and eccentricity.
As can be seen in the figures, the results of the numerical simulations are all but smooth. Therefore, one should not expect the fit to be very accurate with simple functions. However, our goal is to catch the big picture, and to provide an acceptable order of magnitude of the effect of the disc on the inclination and eccentricity. In log scale, the data appear relatively close to an increasing power law of $i_P$ for small $i_P$, and a decreasing power law of $i_P$ for large $i_P$. Therefore, we base our fits on the general form for the damping rates
\begin{equation}
\mathcal{F}(i_P)=-\frac{M_{disc}}{0.01\,M_\star}\left(ai_P^{-2b}+ci_P^{-2d}\right)^{-1/2} \ ,
\label{eq:general}
\end{equation}
where $b$ is positive and $d$ is negative. This way, for small $i_P$, $\mathcal{F}(i_P)\approx i_P^{b}\left(M_{disc}/0.01\,M_\star\sqrt{a}\right)$, and for large $i_P$, $\mathcal{F}(i_P)\approx i_P^{d}\left(M_{disc}/0.01\,M_\star\sqrt{c}\right)$. The coefficients $a$, $b$, $c$, and $d$ depend on the planet mass and eccentricity, and are fitted to the data as follows. The damping rate also has to be linear dependent on the disc mass $M_{disc}/M_\star$, as our simulations linearly scale with the gas density.
\subsubsection{Eccentricity}
We do not want ($de/dt$) to tend to zero when $i_P$ tends to zero. A pure increasing power law of $i_P$ is inappropriate here. The damping function will be given by
\begin{equation}
\mathcal{F}_e(i_P)=-\frac{M_{disc}}{0.01\,M_\star}\left(a(i_P+i_D)^{-2b}+ci_P^{-2d}\right)^{-1/2}\ ,
\label{eq:FF_e}
\end{equation}
where $i_D$ is a small inclination so that for $i_P\approx 0$, $de/dt\approx -\frac{M_{disc}}{0.01\,M_\star}\frac{i_D^b}{\sqrt{a}}$. We are using $i_D=\tilde{M}_p/3$ degrees in Eq.~\ref{eq:FF_e}, where $\tilde{M}_p=1000\,M_p/M_\star$ is the planet mass in Jupiter masses. For small eccentricities, it is expected that $e_P/(de/dt)=\tau_e$ is constant. This makes the coefficient $a$ proportional to $e_P^{-2}$. We find that $de/dt$ is well fitted by the above general form using the coefficients
\begin{eqnarray}
a_e(M_P,e_P) &= &80\,e_P^{-2}\, \exp \left(-e_P^2 \tilde{M}_p / 0.26 \right) 15^{\tilde{M}_p} \, \left(20 + 11\tilde{M}_p-\tilde{M}_p^2 \right)
\nonumber \\
b_e(M_P) &= &0.3\tilde{M}_p \nonumber \\
c_e(M_P) &= &450+2^{\tilde{M}_p} \nonumber \\
d_e(M_P) &= &-1.4+\sqrt{\tilde{M}_p}/6 \ .
\label{eq:F_e}
\end{eqnarray}
The second degree polynomial function of $\tilde{M}_p$ in the expression of $a$ is just a refinement, its value being between $30$ and $50$ for $1<\tilde{M_p}<10$. We note, however, that it is negative for $\tilde{M_p}>12$ so this expression only applies for $\tilde{M_p}<11$, but this covers the range of giant planets. To describe the change of eccentricity we add a second function ${\cal G}_e$, which describes the eccentricity increase for high-mass planets. The damping and excitation of $e_P$ are two different mechanisms that add on the planet, and one of them finally dominates, setting the sign of $de/dt$. The fits in Fig.~\ref{fig:eccentricity} are the added functions.
For ${\cal G}_e$ we use the result of \citet{2001A&A...366..263P} who calculated the eccentricity excitation for $i_0=0^\circ$ high mass-planets. Our calculation is presented in Appendix~\ref{ap:eccentricity} and gives
\begin{equation}
{\cal G}_e|_{i=0} = 12.65\, \frac{M_P M_{disc}}{M_{\star}^2}\, e_P\ .
\end{equation}
Then, we find that this excitation decreases with $i$ as a Gaussian function, finally making
\begin{eqnarray}
\label{eq:gaue}
{\cal G}_e (i_P,M_P,e_P) = 12.65 \, \frac{M_P M_{disc}}{M_{\star}^2}\, e_P \, \exp\left(-\left(\frac{(i_P/1^\circ)}{\tilde{M}_p}\right)^2\right) \ .
\label{eq:G_e}
\end{eqnarray}
In principle planets with $M_P < 5 M_{Jup}$ and $e<0.3$ do not show any signs of eccentricity increase and the Gaussian function should not be added in that case. However, the function is designed to scale with the planetary mass, so that lower mass planets are not affected by it. The change of eccentricity is then given by the sum of ${\cal F}_e$ and ${\cal G}_e$.
\subsubsection{Inclination}
In the case of inclination damping data, we notice that the decreasing power law dominates actually before the intersection with the increasing power law\,; thus, we multiply the term $ai_P^{-2b}$ in our general formula by a Gaussian function of $i_P$ centred on $0^\circ$, so that this term is not affected for small $i_P$ but vanishes more quickly than is natural. It allows our fitting formula to catch the peak of damping in inclination observed around $5$ to $20$ degrees in Figure~\ref{fig:inclination}. For small $i_P$, $di/dt$ should be close to linear in $i_P$, so the coefficient $b$ should be close to $1$. The damping function for inclination ${\cal F}_i$ is then given, in degrees per orbit, by
\begin{eqnarray}
a_i(M_P,e_P) &= &1.5 \cdot 10^4 (2-3e_P){\tilde{M}_p}^3\nonumber \\
b_i(M_P,e_P) &= &1+\tilde{M}_p e_P^2 /10 \nonumber \\
c_i(M_P,e_P) &= &1.2 \cdot 10^{6}/\big[ (2-3e_P) (5+e_P^2 (\tilde{M}_p+2)^3) \big] \nonumber \\
d_i(e_P) &= &-3+2e_P \nonumber \\
g_i(M_P,e_P) &= &\sqrt{3\tilde{M}_p / (e_P+0.001)}\times 1^\circ \nonumber \\
\mathcal{F}_i(M_P,e_P,i_P) &= & -\frac{M_{disc}}{0.01\,M_\star}\left[ a_i\, \left(\frac{i_P}{1^\circ}\right)^{-2b_i}\exp(-(i_P/g_i)^2/2) \right.\\
& & \hspace{3.5cm} + \left.c_i\,\left(\frac{i_P}{40^\circ}\right)^{-2d_i}\ \right]^{-1/2} \ . \nonumber
\label{eq:F_i}
\end{eqnarray}
We note that the expression for coefficient $c_i$ is clearly not valid for $e>2/3$.
From our formulae for $de/dt$ and $di/dt$ we can now estimate how the eccentricity and inclination of a planet will evolve for all $e_P$ and $i_P$. In Fig.~\ref{fig:2Ddidt} the $di/dt$ for different inclinations and eccentricities for $5M_{Jup}$ and $10M_{Jup}$ according to the formulae is presented. In Fig.~\ref{fig:2Ddedt} the $de/dt$ for the same two planetary masses is plotted.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{5MJupi.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{10MJupi.eps}}
\caption{{\bf top:} $di/dt$ for a $5M_{Jup}$ planet with different eccentricities and inclinations. The values of $di/dt$ have been determined with the formula given in Sect.~\ref{subsec:fit}. {\bf bottom:} same, but for $10M_{Jup}$. We note the different colour coding as the change is dependent on the planetary mass.
\label{fig:2Ddidt}
}
\end{figure}
Figure~\ref{fig:2Ddidt} clearly indicates that the damping rate of inclination is highest for planets with a large eccentricity that are moderately inclined above the midplane ($i_P\approx 15^\circ$). The inclination damping rate indicates that planets that are scattered during the gas disc phase in orbits with moderate inclination ($i_P<40^\circ$), would lose their inclination well within the gas dispersal of the disc.
As already indicated in Fig.~\ref{fig:eccentricity}, the eccentricity is always damped for high inclinations. For high planetary masses, the eccentricity of the planet can increase for low planetary inclinations because of interactions with disc. We find an eccentricity increase for both high-mass cases, but the increase of eccentricity declines with increasing eccentricity and inclination. Additionally, the threshold of $e_P$ and $i_P$ for which eccentricity can increase is larger for higher mass planets, which is indicated by the white line in Fig.~\ref{fig:2Ddedt} that represents the transition from eccentricity increase to decrease. Below the line eccentricity increases, above the line eccentricity decreases.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{5MJupe.eps}}
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{10MJupe.eps}}
\caption{{\bf top:} $de/dt$ for a $5M_{Jup}$ planet with different eccentricities and inclinations. The values of $de/dt$ have been determined with the formula given in Sect.~\ref{subsec:fit}. {\bf bottom:} same, but for $10M_{Jup}$. The white line in the figure indicates the transition between eccentricity increase and eccentricity damping. Below the white line, the eccentricity increases, above the line eccentricity decreases. We note the different colour coding as the change is dependent on the planetary mass.
\label{fig:2Ddedt}
}
\end{figure}
\section{Application to single-planet systems}
\label{sec:application}
The movement of a single planet in the disc can only be predicted if $i<40^\circ$ and $e<0.65$ as the planet would undergo a Kozai-oscillation for larger $i$. Additionally, the fitting formula might not be totally accurate for $e>0.5$, since our simulations only cover an eccentricity space of up to $e=0.4$. In Fig.~\ref{fig:Kozaimove} the trajectory of the $10 M_{Jup}$ planet with $i_0=75^\circ$ and $e_0=0.4$, which was shown in Fig.~\ref{fig:kozai} is displayed. This illustrates that the movement of the planet is a complex process as long as the Kozai-oscillations are still operational, but as soon as $i<40^\circ$, the planet loses inclination, which is then not converted back into eccentricity. The planet is damped towards midplane on a non-zero eccentricity. This non-zero eccentricity will actually hold in this case (see Section.~\ref{subsubsec:Eccentricity}).
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[width=0.9\linwx]{10MJupemove.eps}}
\caption{Evolution of $e$ and $i$ of the $10 M_{Jup}$ planet (shown in Fig.~\ref{fig:kozai}) with $i_0=75^\circ$ and $e_0=0.4$ in the $e$-$i$ plane (black line). The background is the extended formula of the fit for $de/dt$ and the white line marks the transition between eccentricity increase and damping as in Fig.~\ref{fig:2Ddidt}. The blue lines indicate calculated trajectories of $10 M_{Jup}$ planets from the fitting formulae.
\label{fig:Kozaimove}
}
\end{figure}
A typical damping rate of $de/dt=0.001 /orbit$ would suggest that the planet will lose $\approx 0.085$ in eccentricity in the period of $10^4$ years. A typical damping rate of $di/dt = 0.01 deg/orbit$ indicates that the planet will lose $\approx 8.5^\circ$ of inclination in $10^4$ years.
The important parameters for the evolution of the orbit of a planet are the damping timescales $\tau_e=e/(de/dt)$ and $\tau_i=i/(di/dt)$. We find that $e/\mathcal{F}_e$ is much smaller than $i/\mathcal{F}_i$ for $i>10-20^\circ$, depending on the planet mass and eccentricity. Thus, planets scattered on highly inclined orbits will follow a certain pattern. While the inclination is damped slowly and still high, the eccentricity is damped to zero. After the inclination is damped further, the eccentricity of the planet can rise because of interactions with the disc (if $e$ is below the white line in Fig.~\ref{fig:Kozaimove}). Finally the inclination is damped to zero and the planet remains with a non-zero eccentricity. This is illustrated by the blue lines in Fig.~\ref{fig:Kozaimove} that correspond to calculated trajectories of $10M_{Jup}$ planets.
Nevertheless, this suggests that at the time of the disc dispersal, the favoured endstate for the planet's evolution is an eccentric orbit in midplane of the disc. This implies that the scattering process of inclined planets must have taken place after the gas is depleted or gone.
\section{Summary}
\label{sec:summary}
We have presented the evolution of eccentricity $e$ and inclination $i$ of high-mass planets ($M_P \geq 1M_{Jup}$) in isothermal protoplanetary discs. The planets have been kept on fixed orbits around the host star, and the forces from the disc acting onto the planet have been calculated. By using these forces, a change of $de/dt$ and $di/dt$ has been determined.
Inclination and eccentricity are in general damped by the interactions with the disc. For $1M_{Jup}$ the damping rate of $e$ and $i$ is highest for only very small inclinations ($i_0\approx 3^\circ$), while the maximal damping rate is shifted to larger inclinations for more massive planets. As the more massive planets carve deeper gaps inside the disc, the damping interactions with the disc are reduced. But for larger inclinations, the planet can feel the full damping potential of the disc and is therefore damped in $e$ and $i$ at a faster rate.
There are two exceptions. The first is for low-inclination planets with a sufficient mass ($M_P > 4-5 M_{Jup}$). In this case, the interactions of the planet with the disc result in an increase of eccentricity of the planet, which has already been observed and studied \citep{2001A&A...366..263P, 2006A&A...447..369K}. However, our 3D results predict an increase of eccentricity for lower planetary masses than the previous studies.
The second exception arises for massive planets ($M_P \approx M_{disc}$, in our case for $M_P>5M_{Jup}$) on high initial inclinations ($i_0>40^\circ$). In the long-term evolution of the planet, eccentricity can increase, while inclination is damped and vice-versa. The planet undergoes a Kozai-cycle with the disc, but in time the oscillations of the planet in $e$ and $i$ diminish, as $e$ and $i$ get damped by the disc at the same time. The planet will end up in midplane through the interactions with the disc.
In Sect.~\ref{subsec:fit} we provided formulae for $di/dt$ and $de/dt$ for high-mass planets, which we fitted to the numerical hydrodynamical simulations. The formulae can now be used to calculate the long-term evolution of planetary systems during the gas phase of the disc with N-Body codes. However, we recommend not using the fitting formula, if the planetary eccentricity is $e>0.65$ and if $i>40^\circ$ (because of the Kozai interactions, a fit that can be used for the long-term evolution of planets is hard to predict).
In the end, the planet's inclination will be damped to zero. Low-mass planets ($M_P< 4-5 M_{Jup}$) will end up in circular orbits in the midplane of the disc, while higher mass planets ($M_P > 5M_{Jup}$) will pump their eccentricity to larger values because of interactions with the disc. This implies that the scattering process of inclined planets must have taken place after the gas is well depleted.
The influence of the gaseous protoplanetary disc on the inclination is also of crucial importance, if multiple planets are present in the disc that excite each other's inclination during their migration \citep{2009MNRAS.400.1373L, 2011MNRAS.412.2353L}. The influence of the disc on the long-term evolution of multi-body systems will be studied in a future paper.
\begin{acknowledgements}
B. Bitsch has been sponsored through the Helmholtz Alliance {\it Planetary Evolution and Life}. The work of A.-S. Libert is supported by an FNRS Postdoctoral Research Fellowship. The calculations were performed on systems of the Computer centre (ZDV) of the University of T\"ubingen and systems operated by the ZDV on behalf of bwGRiD, the grid of the Baden W\"urttemberg state. We thank the referee Willy Kley for his useful and helpful remarks that improve the paper.
\end{acknowledgements}
|
2,869,038,155,693 | arxiv | \section{Preliminary results}
Throughout $L$ will denote a finite-dimensional Lie algebra over a field $F$. We denote algebra direct sums by `$\oplus$', whereas vector space direct sums will be denoted by `$\dot{+}$'. If $B$ is a subalgebra of $L$ we define $B_L$, the {\em core} (with respect to $L$) of $B$ to be the largest ideal of $L$ contained in $B$. In \cite{cideal} we defined a subalgebra $B$ of $L$ to be a {\em c-ideal} of $L$ if there is an ideal $C$ of $L$ such that $L=B+C$ and $B \cap C \subseteq B_L$.
\par
Let $M$ be a maximal subalgebra of $L$. We say that a chief factor $C/D$ of $L$ {\em supplements} $M$ in $L$ if $L=C+M$ and $D \subseteq C \cap M$; if $D=C \cap M$ we say that $C/D$ {\em complements} $M$ in $L$. In \cite{idealindex} we defined the {\em ideal index} of a maximal subalgebra $M$ of $L$, denoted by $\eta(L:M)$, to be the well-defined dimension of a chief factor $C/D$ where $C$ is an ideal minimal with respect to supplementing $M$ in $L$. Here we introduce a further concept which is related to the previous two.
\par
Let $M$ be a maximal subalgebra of $L$ and let $C/D$ be a chief factor of $L$ with $D \subseteq M$ and $L=M+C$. Then $(M \cap C)/D$ is called a {\em c-section} of $M$ in $L$. The analogous concept for groups was introduced by Wang and Shirong in \cite{w-s} and studied further by Li and Shi in \cite{l-s}.
\par
We say that $L$ is {\em primitive} if it has a maximal subalgebra $M$ with $M_L=0$. First we show that all c-sections of $M$ are isomorphic.
\begin{lemma}\label{l:unique} For every maximal subalgebra $M$ of $L$ there is a unique c-section up to isomorphism.
\end{lemma}
\begin{proof} Clearly c-sections exist. Let $(M \cap C)/D$ be a c-section of $M$ in $L$, where $C/D$ is a chief factor of $L$, $D \subseteq M$ and $L=M+C$. First we show that this c-section is isomorphic to one in which $D=M_L$. Clearly $D \subseteq M_L \cap C \subseteq C$, so either $M_L \cap C = C$ or $M_L \cap C=D$. If the former holds, then $C \subseteq M_L$, giving $L=M$, a contradiction. In the latter case put $E=C+M_L$. Then $E/M_L \cong C/D$ is a chief factor and $(M \cap E)/M_L$ is a c-section. Moreover,
\[ \frac{M \cap E}{M_L} = \frac{M_L +M \cap C}{M_L} \cong \frac{M \cap C}{M_L \cap C} = \frac{M \cap C}{D}.
\]
So suppose that $(M \cap C_1)/M_L$ and $(M \cap C_2)/M_L$ are two c-sections, where $C_1/M_L$, $C_2/M_L$ are chief factors and $L=M+C_1=M+C_2$. Then $L/M_L$ is primitive and so either $C_1=C_2$ or else $C_1/M_L \cong C_2/M_L$ and $C_1 \cap M = M_L = C_2 \cap M$, by \cite[Theorem 1.1]{primitive}. In the latter case both c-sections are trivial.
\end{proof}
\bigskip
Given a Lie algebra $L$ with a maximal subalgebra $M$ we define $Sec(M)$ to be the Lie algebra which is isomorphic to any c-section of $M$; we call the natural number $\eta^*(L:M) = \dim Sec(M)$ the {\em c-index} of $M$ in $L$.
\par
The relationship between c-ideals and c-sections, and between ideal index and c-index, for a maximal subalgebra $M$ of $L$ is given by the following lemma.
\begin{lemma} Let $M$ be a maximal subalgebra of a Lie algebra $L$. Then
\begin{itemize}
\item[(i)] $M$ is a c-ideal of $L$ if and only if $Sec(M)=0$; and
\item[(ii)] $\eta^*(L:M) = \eta(L:M)- \dim(L/M)$.
\end{itemize}
\end{lemma}
\begin{proof}
\begin{itemize}
\item[(i)] Suppose first that $M$ is a c-ideal of $L$. Then there is an ideal $C$ of $L$ such that $L=M+C$ and $M \cap C \subseteq M_L$. Then $M \cap C = M_L \cap C$ is an ideal of $L$. Let $K$ be an ideal of $L$ with $M \cap C \subset K \subseteq C$. Then $K \not \subseteq M$, so $L=M+K$ and $M \cap C = M \cap K$. This yields that
$\dim L = \dim M + \dim K - \dim (M \cap K) = \dim M + \dim C - \dim (M \cap C),
$ so $K=C$ and $C/(M \cap C)$ is a chief factor of $L$. It follows that $Sec(M)=0$.
\par
The converse is clear.
\item[(ii)] Let $C/D$ be a chief factor such that $L=M+C$ and $C$ is minimal in the set of ideals supplementing $M$ in $L$. Then $\eta(L:M) = \dim(C/D)$, by the definition of ideal index. Thus,
\begin{align} \eta(L:M) & = \dim(C/D) = \dim C - \dim D \nonumber \\
& = \dim C - \dim C \cap M + \dim C \cap M - \dim D \nonumber \\
& = \dim L - \dim M + \dim(C \cap M/D) \nonumber \\
& = \dim(L/M) + \eta^*(L:M). \nonumber
\end{align}
\end{itemize}
\end{proof}
\begin{lemma}\label{l:supp} Let $A/B$ be an abelian chief factor of $L$. Then any maximal subalgebra of $L$ that supplements $A/B$ must complement $A/B$.
\end{lemma}
\begin{proof} Let $M$ supplement $A/B$, so $L=A+M$ and $B \subseteq M$. Then $[L,M \cap A]=[A+M,M \cap A] \subseteq B+M \cap A = M \cap A$. So $M \cap A$ is an ideal of $L$ and $M \cap A=B$.
\end{proof}
\bigskip
The following lemma will also be useful.
\begin{lemma}\label{l:factor} Let $B \subseteq M \subseteq L$, where $M$ is maximal in $L$ and $B$ is an ideal of $L$. Then $Sec(M) \cong Sec(M/B)$.
\end{lemma}
\begin{proof} Clearly $M/B$ is a maximal subalgebra of $L/B$. Let $(C/B)/(D/B)$ be a chief factor of $L/B$ such that $D/B \subseteq M/B$ and $C/B + M/B = L/B$. Then $C/D$ is a chief factor of $L$ such that $L=C+M$ and $D \subseteq M$. Hence $Sec(M) \cong C \cap M/D \cong Sec(M/B)$.
\end{proof}
\bigskip
In \cite{primitive} it was shown that a primitive Lie algebra can be one of three types: it is said to be
\begin{itemize}
\item[1.] {\em primitive of type $1$} if it has a unique minimal ideal that is abelian;
\item[2.] {\em primitive of type $2$} if it has a unique minimal ideal that is non-abelian; and
\item[3.] {\em primitive of type $3$} if it has precisely two distinct minimal ideals each of which is non-abelian.
\end{itemize}
If $M$ is a maximal subalgebra of $L$, then $L/M_L$ is clearly primitive; we say that $M$ is of type $i$ if $L/M_L$ is primitive of type $i$ for $i=1,2,3$. Then we have the following result.
\begin{lemma}\label{l:prim} Let $L$ be a Lie algebra over a field $F$ and let $M$ be a maximal subalgebra of $L$.
\begin{itemize}
\item[(i)] If $M$ is of type $1$ or $3$ then Sec$(M)=0$.
\item[(ii)] If $F$ has characteristic zero and $M$ is of type $2$ then Sec$(M) \cong M/M_L$.
\end{itemize}
\end{lemma}
\begin{proof} \begin{itemize}
\item[(i)] This follows from \cite[Theorem 1.1 3(a),(c)]{primitive}.
\item[(ii)] Let $A/B$ be a nonabelian chief factor that is supplemented by $M$, so $L=A+M$ and $B=A \cap M_L$. Then $L/M_L$ is simple, by \cite[Theorem 1.7 2]{primitive}, which implies that $L=A+M_L$. Hence
\[ \frac{M}{M_L} = \frac{M \cap (A+M_L)}{M_L} = \frac{M \cap A +M_L}{M_L} \cong \frac{M \cap A}{M_L \cap A} = \frac{M \cap A}{B} = Sec(M).
\]
\end{itemize}
\end{proof}
\section{Main results}
First we can state Theorems 3.1, 3.2 and 3.3 of \cite{cideal} in terms of c-sections as follows.
\begin{theor}\label{t:trivial} Let $L$ be a Lie algebra over a field $F$. Then
\begin{itemize}
\item[(i)] every maximal subalgebra $M$ of $L$ has trivial c-section if and only if $L$ is solvable; and
\item[(ii)] if $F$ has characteristic zero, or is algebraically closed of characteristic greater than 5, then $L$ has a maximal subalgebra with trivial c-section if and only if $L$ is solvable.
\end{itemize}
\end{theor}
\begin{theor}\label{t:char0} Let $L$ be a Lie algebra over a field $F$ of characteristic zero. Then $Sec(M)$ is solvable for all maximal subalgebras $M$ of $L$ if and only if $L=R \dot{+} S$, where $R$ is the (solvable) radical of $L$ and $S$ is a direct sum of simple algebras which are minimal non-abelian or isomorphic to $sl_2(F)$.
\end{theor}
\begin{proof} Suppose first that $Sec(M)$ is solvable for all maximal subalgebras $M$ of $L$, and let $L=R \dot{+} S$ be the Levi decomposition of $L$. Then $Sec(M)$ is solvable for all maximal subalgebras $M$ of $S$, by Lemma \ref{l:factor}. Let $S = S_1 \oplus \ldots \oplus S_n$, where $S_i$ is simple for each $1 \leq i \leq n$. If $M$ contains all $S_i$ apart from $S_j$, then $Sec(M) \cong M \cap S_j$, so every subalgebra of $S_j$ is solvable. It follows from \cite[Theorem 2.2 and the remarks following it]{supsolv} that $S_j$ is minimal non-abelian or isomorphic to $sl_2(F)$ for each $1 \leq j \leq n$.
\par
Suppose conversely that $L$ has the claimed form and let $M$ be a maximal subalgebra of $L$. Every chief factor of $L$ is either abelian or simple, and so every c-section of $M$ is either abelian or isomorphic to a proper subalgebra of one of the simple components of $S$. In either case $Sec(M)$ is solvable.
\end{proof}
\begin{coro}\label{c:char0} Let $L$ be a Lie algebra over a field $F$ and suppose that every maximal subalgebra has $c$-index $k$. Then
\begin{itemize}
\item[(i)] if $k >0$, $L$ must be semisimple.
\medskip
Suppose further that $F$ has characteristic zero. Then
\item[(ii)] every simple ideal of its Levi factor must have all of its maximal subalgebras of dimension $k$;
\item[(iii)] $k=0$ if and only if $L$ is solvable;
\item[(iv)] $k=1$ if and only if $\sqrt{F} \not \subseteq F$ and $L$ is a direct sum of non-isomorphic three-dimensional non-split simple ideals; and
\item[(v)] $k=2$ if and only if $L$ is a direct sum of non-isomorphic ideals and either (a) each of these ideals is a minimal non-abelian simple Lie algebra with all maximal subalgebras of dimension $2$, or (b) $\sqrt{F} \subseteq F$ and one of the ideals is isomorphic to $sl_2(F)$, whilst any others are minimal non-abelian simple Lie algebras with all maximal subalgebras of dimension $2$.
\end{itemize}
\end{coro}
\begin{proof} \begin{itemize}
\item[(i)] If $L$ has non-trivial radical, it has an abelian chief factor which is supplemented, and hence complemented, by Lemma \ref{l:supp}, so $k=0$.
\item[(ii)] This is clear.
\item[(iii)] This is Theorem \ref{t:trivial} (i).
\item[(iv)] Suppose that $k=1$. Then $L$ is semisimple and each simple component has all of its maximal subalgebras one dimensional, by (i) and (ii). It follows that they are three-dimensional simple and $\sqrt{F} \not \subseteq F$, by \cite[Theorem 3.4]{chief}. Moreover, they must be non-split. If there are two that are isomorphic, say $S$ and $\theta(S)$, where $\theta$ is an isomorphism, then the diagonal subalgebra $\{s+ \theta(s): s \in S\}$ is maximal in $S \oplus \theta(S)$. But this together with the simple components other than $S$ and $\theta(S)$ gives a maximal subalgebra $M$ of $L$ with $c$-index $0$ in $L$.
\par
Conversely, suppose that $L$ is a direct sum of non-isomorphic three-dimensional simple ideals, $S_1 \oplus \ldots \oplus S_n$, and $\sqrt{F} \not \subseteq F$. Let $M$ be a maximal subalgebra of $L$ with $S_i \not \subseteq M$ and $S_j \not \subseteq M$ for some $1 \leq i, j \leq n$ with $i \neq j$. Then $L=M+S_i=M+S_j$ which yields that $M \cap S_i$ and $M \cap S_j$ are ideals of $L$ and hence are trivial. But then $S_i \cong L/M \cong S_j$, a contradiction. It follows that every maximal subalgebra contains all but one of the simple components and hence that $k=1$.
\item[(v)] This is similar to (iv), using Theorem \ref{t:char0} and noting that $sl_2(F)$ has a one-dimensional maximal subalgebra if and only if $\sqrt{F} \not \subseteq F$, by \cite[Theorem 3.4]{chief}.
\end{itemize}
\end{proof}
\bigskip
Note that algebras as described in Corollary \ref{c:char0} do exist as the following example shows. This example was constructed by Gejn (see \cite[Example 3.5]{gejn}).
\begin{ex} Let $L$ be the Lie algebra generated by the matrices
\[ f_1 = \left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -E \\
0 & E & 0 \end{array} \right),
f_2 = \left( \begin{array}{ccc}
0 & 0 & A \\
0 & 0 & 0 \\
-E & 0 & 0 \end{array} \right),
f_3 = \left( \begin{array}{ccc}
0 & -A & 0 \\
E & 0 & 0 \\
0 & 0 & 0 \end{array} \right)
\]
\[
g_1 = \left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -A \\
0 & A & 0 \end{array} \right),
g_2 = \left( \begin{array}{ccc}
0 & 0 & 2E \\
0 & 0 & 0 \\
-A & 0 & 0 \end{array} \right),
g_3 = \left( \begin{array}{ccc}
0 & -2E & 0 \\
A & 0 & 0 \\
0 & 0 & 0 \end{array} \right)
\]
where $A= \left( \begin{array}{cc} 0 & 2 \\ 1 & 0 \end{array} \right)$, $E= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ and $0= \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)$, with repect to the operation $[,]$, over the rational numbers $\Q$. Then $L$ is simple nonabelian (see \cite[Example 3.5]{gejn}), and the maximal subalgebras are $\Q f_i + \Q g_i$ for $i=1,2,3$.
\end{ex}
\begin{ex}
Gejn also goes on to construct simple minimal nonabelian Lie algebras over $\Q$ of dimension $3k$ for $k \geq 1$ by putting
\[ A= \left( \begin{array}{cccccc}
0 & 0 & 0 & \ldots & 0 & 2 \\
1 & 0 & 0 & \ldots & 0 & 0 \\
0 & 1 & 0 & \ldots & 0 & 0 \\
. & . & . & \ldots & . & . \\
0 & 0 & 0 & \dots & 1 & 0 \end{array} \right)
\]
$E$ as the $k \times k$ identity matrix and $0$ as the $k \times k$ zero matrix (see \cite[Example 3.6]{gejn}). It is straightforward to check that in these every maximal subalgebra has $c$-index $k$.
\end{ex}
The following corollary is straightforward.
\begin{coro}\label{c:onemax} Let $L=R\dot{+}S$ be a Lie algebra over a field $F$ of characteristic zero, where $R$ is the radical and $S$ is a Levi factor, and suppose that $L$ has a maximal subalgebra with $c$-index $k$. Then
\begin{itemize}
\item[(i)] if $k>0$ then $S \neq0$;
\item[(ii)] $k=1$ if and only if $\sqrt{F} \not \subseteq F$ and $S$ has a minimal ideal $A$ which is three-dimensional simple;
\item[(iii)] $k>1$ if and only if $S$ has a minimal ideal with a maximal subalgebra of dimension $k$.
\end{itemize}
\end{coro}
\bigskip
Recall that a triple $(G,[p],\iota)$ consisting of a restricted Lie algebra $(G,[p])$ and a homomorphism $\iota : L \rightarrow G$ is called a {\em $p$-envelope} of $L$ if (a) $\iota$ is injective and (b) the $p$-algebra generated by $\iota(L)$ equals $G$. If $L$ is finite-dimensional then it has a finite-dimensional $p$-envelope (see, for example, \cite[Section 2.5]{s-f}). Let $(L_p,[p],\iota)$ be a $p$-envelope of $L$. If $S$ is a subalgebra of $L$ we denote by $S_p$ the restricted subalgebra of $L_p$ generated by $\iota(S)$. Then the {\em (absolute) toral rank} of $S$ in $L$, $TR(S,L)$, is defined by
\[
TR(S,L) = \hbox{max} \{\hbox{dim}(T) : T \hbox{ is a torus of } (S_p + Z(L_p))/Z(L_p)\}.
\]
This definition is independent of the $p$-envelope chosen (see \cite{strade1}). We write $TR(L,L) = TR(L)$. A Lie algebra $L$ is {\em monolithic} if it has a unique minimal ideal (the {\em monolith} of $L$). The {\em Frattini ideal}, $\phi(L)$, is the largest ideal contained in every maximal subalgebra of $L$. We put $L^{(0)}=L$, $L^{(n)}=[L^{(n-1)},L^{(n-1)}]$ for $n \in \N$ and $L^{(\infty)}= \cap_{n=0}^{\infty} L^{(n)}$.
\begin{theor}\label{t:nilp} Let $L$ be a Lie algebra over an algebraically closed field $F$ of characteristic $p>0$. Then $Sec(M)$ is nilpotent for every maximal subalgebra $M$ of $L$ if and only $L$ is solvable.
\end{theor}
\begin{proof} Let $L$ be a minimal non-solvable Lie algebra such that $Sec(M)$ is nilpotent for every maximal subalgebra $M$ of $L$, and let $R$ be the (solvable) radical of $L$. If $L$ is simple then every maximal subalgebra of $L$ is nilpotent, and no such Lie algebra exists over an algebraically closed field. So $L$ has a minimal ideal $A$, and $L/A$ is solvable. If there are two distinct minimal ideals $A_1$ and $A_2$, then $L/A_1$ and $L/A_2$ are solvable, whence $L \cong L/(A_1 \cap A_2)$ is solvable, a contradiction. Hence $L$ is monolithic with monolith $A$. If $A \subseteq R$ then again $L$ would be solvable, so $L$ is semisimple and $\phi(L)=0$. Thus, there is a maximal subalgebra $M$ of $L$ such that $L=M+A$.
\par
Put $C=M \cap A$ which is an ideal of $M$. If ad\,$a$ is nilpotent for all $a \in A$ then $L$ is solvable, a contradiction. Hence there exists $a \in A$ such that ad\,$a$ is not nilpotent. Let $L=L_0 \dot{+} L_1$ be the Fitting decomposition of $L$ relative to ad\,$a$. Then $L_0 \neq L$ and $L_1 \subseteq A$, so that if $P$ is a maximal subalgebra containing $L_0$, we have $L=A+P$ and $a \in A \cap P$. We can, therefore, assume that $C \neq 0$.
\par
Then $C$ is nilpotent and $L/A \cong M/C$ is solvable, whence $M$ is solvable. Now $[M,N_A(C)] \subseteq N_A(C)$, so $M+N_A(C)$ is a subalgebra of $L$. But $L=M+N_A(C)$ implies that $C$ is an ideal of $L$, from which $C=A$ and $L$ is solvable, a contradiction. It follows that $M=M+N_A(C)$, and so $N_A(C)=M \cap A = C$, and $C$ is a Cartan subalgebra of $A$. Now $C_p$ is a Cartan subalgebra of $A_p$, by \cite[Lemma]{wilson}, and so there is a maximal torus $T \subseteq A_p$ such that $C_p = C_{L_p}(T)$ (see \cite{seligman}).
\par
Let $A_0(T)+ \sum_{i \in \Z_p} A_{i \alpha}$ be a $1$-section with respect to $T$. Then every element of $C$ acts nilpotently on $L_0$, the Fitting null-component relative to $T$, and thus so does every element of $C_p$. It follows that $L= L_0+ \sum_{i \in \Z_p} A_{i \alpha}$ so $L^{(\infty)}=A$ is simple with $TR(A)=1$. We therefore have that
\[p \neq 2, \hspace{.3cm} A \in \{sl_2(F), W(1:\underline{1}), H(2:\underline{1})^{(1)}\} \hbox{ if } p>3
\]
\[ \hbox{and } \hspace{.3cm} A \in \{sl_2(F), psl_3(F)\} \hbox{ if } p=3,
\]
by \cite{premet} and \cite{sk}. But now, $\dim A_{\alpha} = 1$ (by \cite[Corollary 3.8]{bo} for all but $psl_3(F)$, and this is straightforward to check) and $M=L_0 \subset L_0+A_{\alpha} \subset L$, a contradiction. It follows that $L$ is solvable.
\par
The converse is clear.
\end{proof}
\bigskip
A subalgebra $U$ of $L$ is {\em nil} if ad\,$u$ acts nilpotently on $L$ for all $u \in U$. Notice that we cannot replace `nilpotent' in Theorem \ref{t:nilp} by `solvable' or `supersolvable' and draw the same conclusion, as $sl_2(F)$ is a counter-example. However, we can prove the same result with `nilpotent' replaced by the stronger condition `nil' without any restrictions on the field $F$.
\begin{theor}\label{t:res} Let $L$ be a Lie algebra over any field $F$. Then $Sec(M)$ is nil for every maximal subalgebra $M$ of $L$ if and only if $L$ is solvable.
\end{theor}
\begin{proof} Let $L$ be a minimal non-solvable Lie algebra such that $Sec(M)$ is nil for every maximal subalgebra $M$ of $L$. If $L$ is simple then every maximal subalgebra of $L$ is nil. It follows that every element of $L$ is nil and $L$ is nilpotent, by Engel's Theorem. Hence no such Lie algebra exists. So, arguing as in paragraphs $1$ and $2$ of Theorem \ref{t:nilp} above, $L$ is monolithic with monolith $A$, $L/A$ is solvable, and there is a maximal subalgebra $M$ of $L$ such that $L=M+A$ with an element $a \in M \cap A$ such that ad\,$a$ is not nilpotent. But this is a contradiction, since $A \cap M=Sec(M)$ is nil.
\par
Once again, the converse is clear.
\end{proof}
\bigskip
Let $(L,[p])$ be a restricted Lie algebra. Recall that an element $x \in L$ is called {\em $p$-nilpotent} if there exists an $n \in \N$ such that $x^{[p]^n}=0$. Then we have the following immediate corollary.
\begin{coro}\label{c:res} Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Then $Sec(M)$ is $p$-nilpotent for every maximal subalgebra $M$ of $L$ if and only if $L$ is solvable.
\end{coro}
\begin{proof} Simply note that that a $p$-nilpotent subalgebra is nil.
\end{proof}
|
2,869,038,155,694 | arxiv | \section{Introduction}
Emphasis Selection recently proposed by \cite{shirani-etal-2019-learning} aims to select candidate words for emphasis in short sentences. By emphasizing words, people's intent can be better conveyed, which is useful in a variety of applications. For example, it can be used in spoken language processing to generate more expressive sentences and be used to enable automated design assistance in authoring, i.e., labeling important parts in a paragraph or in a poster title.
Although it seems that this task is highly similar to the task of keyword extraction ~\cite{Beliga20141KE}, these two tasks are fundamentally different. The first difference is that keyword extraction focuses on a paragraph which is composed of multiple sentences while
emphasis selection aims to choose words from a short sentence. This difference implies that modeling sentence structure is more effective in emphasis selection.
The second difference is that many global word statistics methods employed in keyword extraction such as TF-IDF and word co-occurence frequency will not work in this task, because for short sentences, it is meaningless to count word frequency and whether the word should be emphasized has nothing to do with the frequency of the word. In addition, keyword extraction requires that the collected keywords are diverse, which means that if two words have similar meaning, only one should be kept. However, in emphasis selection, similar words tend to be emphasized together. Emphasis selection also shares some resemblances with entity recognition~\cite{yadav-bethard-2018-survey}. But one major difference is that
the parts of speech of emphasized words are more diverse and the relation of adjacent words is weaker in the emphasis selection task.
\begin{figure}[!t]
\includegraphics[width=\linewidth]{figure/example.png}
\caption{Two examples of the emphasis selection. Words with darker background indicate that more people agree to emphasize.}
\label{fig:example}
\vspace{-0.2cm}
\end{figure}
Generally speaking, emphasis selection can be modeled as a sequence classification task where the input is a sentence and the output is each word's probability to be emphasized. Shirani et al. ~\cite{shirani-etal-2019-learning} propose a model which is based on the Recurrent Neural Network~\cite{conf/interspeech/MikolovKBCK10} and KL-Divergence loss function.
Despite the fact that it looks like a straightforward task, there still exist some challenges.
The first challenge is about how to incorporate sentence structure information into the model. Sentence structure information includes what role (subject, predicate, object, etc.) the word plays as well as the position of the word in a sentence. Obviously, this kind of information is very useful. Existing works ~\cite{shirani-etal-2019-learning} fail to model the global structure of a sentence.
The second challenge is that there is no given context except a short sentence, so it requires the model to be able to capture some common patterns or regularities of most people.
More concretely, if two words are similar, they are more likely to be emphasized together. For example, in Figure~\ref{fig:example}, \textbf{persistence} and \textbf{victory} are more likely to be emphasized together. This observation can also be found in the second example: \textbf{Never} and \textbf{impossible}. Moreover, we analyze the training dataset and get a more concrete understanding of this phenomenon through the following procedures: For each training sentence, we consider the most popular emphasized word called word A. Then, we identify the most similar word called word B to the word A based on GloVe embedding ~\cite{pennington2014glove}. We find that the word B is also emphasized with a higher probability than other words in this sentence and this phenomenon occurs in about 26\% of the training dataset.
Therefore, modeling this kind of relationship between words definitely can help improve the performance of models.
In this paper, we propose a sentence structure graph to handle the sentence structure issue.
Specifically, the sentence structure graph is derived from the parse tree of a sentence which contains useful information for this task. For example, as illustrated in Figure ~\ref{fig:Tag Graph}, when the path is S$\rightarrow$NP$\rightarrow$PRP, the word \textbf{I} is not inclined to be emphasized since this path indicates that this word is a subject. However, when the path is S$\rightarrow$VP$\rightarrow$S$\rightarrow$VP$\rightarrow$NP$\rightarrow$NN, the word \textbf{basketball} is likely to be emphasized since this word is a noun in a verb phrase. Generally, such sentence structure graph can reveal the role of words in a sentence which is beneficial for emphasis selection. Another important information - word relationship information is captured by a
word similarity graph.
Through the word similarity graph, words can share information with their neighbours, resulting in similar emphasized probabilities of similar words. Next, graph neural networks~\cite{DBLP:journals/corr/VaswaniSPUJGKP17,cai-lam-2020-graph,DBLP:journals/corr/KipfW16,unknown,yun2019graph,vel2018graph} which has been demonstrated effective in modeling graph structure data are employed to learn the representation of each node of these two graphs.
We conduct extensive experiments based on different word embeddings, i.e., GloVe~\cite{pennington2014glove}, ELMo~\cite{peters2018contextualized}, RoBERTa~\cite{roberta} and the experimental results show that our model can achieve superior performance.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figure/tagGraph2.png}
\caption{The part above the curve is the sentence structure graph constructed from the sentence: I love playing basketball. After the whole parse tree is encoded, the embeddings of the green nodes are used as the structure information for further classification.}
\label{fig:Tag Graph}
\vspace{-3pt}
\end{figure}
\section{Related Work}
Emphasis selection is a new task proposed by ~\cite{shirani-etal-2019-learning} which aims to choose a subset of words to emphasize in a sentence.
Shirani et al. ~\cite{shirani-etal-2019-learning} propose a model which is based on the Recurrent Neural Network~\cite{conf/interspeech/MikolovKBCK10}.
KL-Divergence loss function is adopted to conduct the label distribution learning (LDL)~\cite{DBLP:journals/corr/GengZ14}. This method achieves competitive performance over the sequence labeling model: CRF~\cite{Lafferty:2001:CRF:645530.655813}.
In Recent years, graph neural networks ~\cite{unknown,DBLP:journals/corr/KipfW16,yun2019graph,vel2018graph,cai-lam-2020-graph} have demonstrated superiority in modeling the structure of graphs. Kipf et al.~\cite{DBLP:journals/corr/KipfW16} propose a graph convolutional network which is based on the fourier theory. One drawback of this model is that the edge weight of the graph needs to be known in advance. To overcome this shortcoming, Petar et al.~\cite{vel2018graph} use a masked self-attention layer to calculate the weight of node's neighbours dynamically and then aggregate information by conducting a weighted addition operation. Currently, graph neural networks are applied to various tasks. Feria et al. ~\cite{DBLP:journals/corr/abs-1807-03012} construct a word graph by calculating the word embedding similarity and apply the community detection algorithm to find different communities. Through the graph, they can find named entities for a bilingual language base in an unsupervised manner. Sun et al.~\cite{DBLP:journals/corr/abs-1905-07689} put forward a diverse graph pointer network for keyword extraction. They first construct a word graph based on the distance of two words and then use the graph convolutional network as an encoder to obtain each node's representation, finally a pointer network decoder and the diverse mechanism are employed to generate diverse keywords. The graph encoder can capture document-level word salience and overcome the long-range dependency problem of RNN.
\section{Methodology}
We follow the same problem setting given by ~\cite{shirani-etal-2019-learning}. Suppose a sentence is composed of $n$ words $C=(x_1,x_2,...,x_n)$.
Our goal is to obtain a subset $S$ of words in $C$ as selected words for emphasis where $1\leq \left|S\right| \leq n$.
We model this task as a prediction problem:
\begin{equation}
(p_1,p_2,..,p_n) = model(x_1,x_2,...,x_n)
\end{equation}
where $p_i$ is $i$-th word's probability to be emphasized. Then $S$ contains the top-$\left|S\right|$ words with high probability.
Figure~\ref{fig:model} depicts the architecture of our proposed model which is composed of three parts: (i) the middle part - sequence encoder (ii) the left part - word similarity graph encoder (iii) the right part - sentence structure graph encoder. Next, we will provide a detailed description of each part.
\subsection{Sequence Encoder}
The sequence encoder is composed of an embedding layer and a bidirectional GRU. It is mainly used to model the sequence information, i.e., word sequence and tag sequence. Formally,
given a sentence $C=(x_1,x_2,...,x_n)$ with $n$ words, the embedding layer is responsible for converting each word into a $d_1$-dimensional vector and converting the corresponding POS tag into a $d_2$-dimensional vector:
\begin{equation}
(w_1,...,w_n) = WordEmbed(x_1,...,x_n)
\end{equation}
\begin{equation}
(e_1,..,e_n) = TagEmbed(t_1,...,t_n)
\end{equation}
where $(t_1,...,t_n)$ is the POS tag sequence and $w_i \in \mathbb{R}^{d_1}, e_i \in \mathbb{R}^{d_2}$. Then the word embedding and the tag embedding are concatenated and fed into a encoder $E$ to encode the sequence information. We can obtain the outputted hidden state of the encoder:
\begin{equation}
(h_1,...,h_n) = E([w_1,e_1],...[w_n,e_n])
\end{equation}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure/model6.png}
\caption{An overview of our model, the left part is the word similarity graph encoder, the middle part is the sequence encoder and the right part is the sentence structure graph encoder. Lx representes that there are L such blocks.}
\label{fig:model}
\end{figure}
\subsection{Word Relationship Modeling}
Given a sentence, we take each word as a node and the weight of the edge is calculated by the word embedding similarity. The weight matrix is denoted by $A \in \mathbb{R}^{n\times n}$.
After the graph is constructed, a $L$-layer graph convolutional network (GCN) ~\cite{DBLP:journals/corr/KipfW16} is employed to encode the word similarity graph:
\begin{equation}
H^{l+1} = ReLU(D^{-\frac{1}{2}}AD^{-\frac{1}{2}}H^lW^l \label{GCN})
\end{equation}
where $W^l$ is a parameter and $H^l$ denotes the nodes' representation in the $l$-th layer. $D \in \mathbb{R}^{n\times n}$ is a diagonal matrix and $D_{ii}=\sum_j A_{ij}$.
Recall that the WSG is a complete graph since each two words are connected by a weighted edge. There exists a serious problem: Useful information may be overwhelmed by useless information, because a majority number of words do not need to be emphasized, causing the information in words that are not emphasized dominates the words that should be emphasized. To alleviate this problem, we adopt two strategies: residual module ~\cite{DBLP:journals/corr/HeZRS15} and gate mechanism~\cite{DBLP:journals/corr/GehringAGYD17,DBLP:journals/corr/DauphinFAG16}. The residual module makes the current node's representation as the addition between the former representation and the aggregated information from its neighbours. The gate mechanism controls the magnitude of the aggregated information. Through this way, the current node's representation will not be significantly affected by its neighbours.
Therefore Equation \eqref{GCN} can be rewritten as:
\begin{align}
& \qquad \qquad M^{l+1} = H^lW^l \\
& \qquad \qquad C = s(H^lW^g) \\
&H^{l+1} = M^{l+1} +D^{-\frac{1}{2}}AD^{-\frac{1}{2}}M^{l+1}\otimes C
\end{align}
where $s(\cdot)$ is the sigmoid function and $\otimes$ is the point-wise multiplication.
We obtain $H^0$ from the word embedding matrix and obtain the $L$-th layer output $H^L=(w^L_1,...,w^L_n)$ as each node's features of the word similarity graph.
\subsection{Sentence Structure Modeling}
SSG is constructed by parsing the sentence using NLTK\footnote{\url{https://www.nltk.org/}} and StandfordNLP~\footnote{\url{https://stanfordnlp.github.io/CoreNLP/}}. Then, we remove the leaf nodes (which are the words) and the remaining part is the SSG. Each node of the graph is a kind of POS tag and the path from the root to a specific word can reveal what role the word plays in the sentence.
Apparently, the weight of edges is important. For example, in Figure~\ref{fig:Tag Graph}, the root node S has two children nodes NP and VP. The edge (S, NP) should have a smaller weight than the edge (S, VP) since people tend not to emphasize the subject in most circumstances. Different from WSG where the weight can be calculated by the word embedding similarity explicitly, it is not appropriate to calculate the weight in the SSG by the node similarity.
Hence, we integrate the idea of Transformer~\cite{DBLP:journals/corr/VaswaniSPUJGKP17,cai-lam-2020-graph} and masked self-attention ~\cite{vel2018graph} to the SSG modeling.
Firstly, we generate three vectors: key, query, value, according to the current node's representation:
\begin{align}
k^{l+1}_{i} = \mathbf{W}^l_k(v^l_{i}) \\
q^{l+1}_{i} = \mathbf{W}^l_q(v^l_{i}) \\
v^{l+1}_{i} = \mathbf{W}^l_v(v^l_{i})
\end{align}
where $\mathbf{W}^l_k, \mathbf{W}^l_q, \mathbf{W}^l_v$ are parameters. $k^{l}_{i},q^{l}_{i},v^{l}_{i}$ correspond to the $l$-th layer key, query, value vector respectively. $v^0_i$ is initialized from the tag embedding matrix. Then, a masked self-attention is employed to allow nodes aggregating information only from their neighbours.
\begin{equation}
v^{l+1}_i = \sum_{j \in \mathcal{N}(i)} a_{ij}v^{l+1}_j
\end{equation}
\begin{equation}
a_{ij} = \frac{exp(q^{l+1}_i k^{l+1}_j)}{\sum_{z \in \mathcal{N}(i)} exp(q^{l+1}_i k^{l+1}_z)}
\end{equation}
where $ \mathcal{N}(i)$ is the neighbour set of the node $i$. After the graph is encoded with a $L$-layer network, we obtain the leaf nodes (the green nodes shown in Figure ~\ref{fig:Tag Graph}) representation $V=(v^L_1,v^L_2,...,v^L_n)$.
\subsection{Loss Function}
After obtaining these three modules' output, we conduct a concatenation operation and calculate the probability:
\begin{equation}
p_i = softmax ( f([h_i,v^L_i,w^L_i]) )
\end{equation}
where $p_i \in \mathbb{R}^3$ is $i$-th word's probability distribution. $f$ represents a fully connected neural network.
We adopt negative log likelihood as the loss function:
\begin{equation}
L = -\sum_{C \in D_{train}}\sum_{i=1}^{\left|C\right|} \log p_{iy_i}
\end{equation}
\section{Experiment and Results}
\begin{table}[h]
\small
\centering
\begin{tabular}{lllllll}
\hline
\toprule
DIY &ideas&for&leafing&up&your&home\\
\hline
\midrule
B& O & O & O & O & B & I \\
B& I & O & O & O & O& O \\
B& O & O & O & O & O & O\\
O& B & O & O & O & O & O\\
O& O & O & B & O & O & B\\
O& O & O & B & I & I & I\\
O& O & O & B & O & O & O\\
B& O & O & B & O & O & B\\
B& I & O & O & O & O & O\\
\bottomrule
\hline
\end{tabular}
\caption{An example of the labeled dataset}
\label{tab:one_example}
\end{table}
\begin{table*}[t]
\centering
\begin{tabular}{llllll}
\hline
\toprule
Methods &Match-1&Match-2&Match-3&Match-4&Average\\
\hline
GloVe \\
\hline
CNN & 0.541 & 0.678 & 0.754 & 0.805 & 0.695 \\
RNN~\cite{shirani-etal-2019-learning} & 0.536 & \textbf{0.712} & 0.777 & 0.811 & 0.709 \\
Ours &\textbf{0.569} & 0.703 & 0.772 & \textbf{0.813} & 0.714 \\
Ours w/o WSG & 0.563 & 0.710 & \textbf{0.778} & 0.810 & \textbf{0.715} \\
Ours w/o SSG & 0.561 & 0.710 & 0.769 & 0.811 & 0.713\\
\hline
\midrule
ELMo \\
\hline
CNN & 0.574 & 0.729 & 0.795 & 0.832 & 0.733\\
RNN-based~\cite{shirani-etal-2019-learning} & 0.592 & 0.752 & 0.804 & 0.822 & 0.743 \\
Ours & \textbf{0.610} & \textbf{0.768} & \textbf{0.813} & \textbf{0.836} & \textbf{0.757} \\
Ours w/o WSG & 0.604 & 0.742 & 0.804 & 0.827 & 0.744 \\
Ours w/o SSG & 0.597 & 0.753 & 0.801 & 0.836 & 0.747 \\
\bottomrule
\hline
\end{tabular}
\caption{Results of our model and baselines on GloVe and ELMo. The best performance is boldfaced.}
\label{tab:results}
\end{table*}
\begin{table*}[t]
\centering
\begin{tabular}{c|cccccc}
\hline
\toprule
&Stay &foolish&to&stay&sane&.\\
\hline
\midrule
Annotator &0.333(4) &0.889(1/2)&0.222(5/6)&0.444(3)&0.889(1/2)&0.222(5/6)\\
RNN-based &0.502(3)&0.565(2)&0.227(6)&0.460(4)&0.798(1)&0.357(5) \\
Ours & 0.502(4) &0.784(2) &0.210(6)&0.595(3)&0.805(1)&0.288(5)\\
\bottomrule
\hline
\end{tabular}
\caption{A sample case.
Numbers outside the brackets indicate the word's probability of being emphasized. Numbers in the brackets are the ranking of the corresponding word. (a/b) means that two words have the same ranking.}
\label{tab:case}
\end{table*}
\subsection{Dataset}
We use the dataset\footnote{\url{https://github.com/RiTUAL-UH/SemEval2020_Task10_Emphasis_Selection}} provided by ~\cite{shirani-etal-2019-learning}. The dataset contains $2742$ training sentences and 392 test sentences. Each sentence is labeled by nine annotators. Table ~\ref{tab:one_example} gives a sample record of one sentence. B, I, O represent the beginning word to be emphasized, the interior word to be emphasized, and the word not to be emphasized respectively. Since there exists different opinions about whether the word should be emphasized, the labels given by nine annotators are slightly different.
\subsection{Experimental Setup}
We regard each annotator's labeling as a sample in the dataset. In other words, each sentence is associate with nine samples.
In order to verify the robustness of our model, we conduct experiments on two pre-trained word embeddings: 300-$d$ GloVe~\cite{pennington2014glove} and 2048-$d$ ELMo~\cite{peters2018contextualized}. For the above two kinds of embeddings, we adopt GRU as the encoder $E$. The GRU hidden state size is $512$ and $1024$ respectively. The word similarity graph's node embedding size is $300$ and $2048$ respectively. The sentence structure graph's node embedding size is $300$ and $512$ respectively. Moreover, we initialize the sentence structure graph's node embedding by training a classifier which only uses the sentence structure graph encoder. We adopt a two-layer bidirectional GRU. The sentence structure graph and the word similarity graph are encoded by a two-layer graph neural network. The batch size is set to 16. The negative slope of the ReLU function is set to $0.2$. We use the Adam optimizer and the learning rate is $0.0001$. The number of epoch is $100$. We also add a dropout layer and the dropout rate is $0.5$.
Since generalized pretrained language models such as BERT~\cite{bert}, RoBERTa~\cite{roberta} are demonstrated effective in a large bunch of downstream tasks, we also report results obtained by fine-tuning the RoBERTa on the emphasis selection dataset. There are two different experimental settings. The first setting is that only the RoBERTa model is used as the encoder $E$. The second setting is that a GRU layer is added on the top of the RoBERTa model, i.e., RoBERTa+GRU is the encoder $E$. The sentence structure model and the word relationship model remain unchanged. Adam optimizer is adopted and the learning rate is set to 1e-5.
\subsection{Evaluation Metric}
We adopt \textbf{Match-m} ~\cite{shirani-etal-2019-learning} as the evaluation metric which is defined as below:
\begin{table*}[t]
\centering
\begin{tabular}{llllll}
\hline
\toprule
Methods &Match-1&Match-2&Match-3&Match-4&Average\\
\hline
RoBERTa \\
\hline
Ours w/o both &0.635 & 0.756 & 0.803 & 0.832 & 0.757 \\
Ours w/o WSG & \textbf{0.640} & 0.775 & 0.793 & 0.827 & 0.759 \\
Ours w/o SSG & 0.633 & 0.760 & \textbf{0.804} & \textbf{0.839} & 0.759\\
Ours &0.633 & \textbf{0.779} & 0.803 & 0.833 & \textbf{0.762}\\
\hline
\midrule
RoBERTa+GRU \\
\hline
Ours w/o both & 0.607 & 0.755 & 0.795 & 0.822 & 0.745 \\
Ours w/o WSG & 0.602 & \textbf{0.766} & 0.798 & 0.825 & 0.748 \\
Ours w/o SSG & \textbf{0.607} & 0.758 & 0.801 & 0.837 & 0.747 \\
Ours & 0.600 & 0.761 & \textbf{0.806} & \textbf{0.838} & \textbf{0.751}\\
\bottomrule
\hline
\end{tabular}
\caption{Results of our model and baselines based on two different architectures, RoBERTa and RoBERTa+GRU. The best performance is boldfaced.}
\label{tab:results_lm}
\end{table*}
\begin{table*}[t]
\centering
\scalebox{0.9}{\begin{tabular}{c|ccccccccc}
\hline
\toprule
&Thanks &for &showing &me &all &the & best &dance &moves\\
\hline
\midrule
Annotator &0.444 &0.111&0.111&0.111&0.111&0&0.888&0.555&0.444\\
Ours & 0.412 &0.057 &0.332&0.092&0.063&0.024&0.406&0.599&0.387\\
\bottomrule
\hline
\end{tabular}}
\caption{A failed case.
Numbers are the word's probability of being emphasized.}
\label{tab:case_failed}
\end{table*}
\textbf{Match-m}: For a sentence $C$,
we choose $m$ words (denoted by $S^g_m(C)$) with the top-$m$ probability (probability of the label B + probability of the label I) in the ground truth and $m$ words (denoted by $S^p_m(C)$) based on the predicted probability. The formula is defined as:
\begin{equation}
\text{Match-m} = \frac{\sum_{C \in D_{test}} \frac{\left| S^g_m(C) \cap S^p_m(C)\right|}{min(\left|C\right|,m)}}{\left|D_{test} \right|}
\end{equation}
\subsection{Results and Analysis}
\subsubsection{Experimental Results}
We compare our model with the existing model based on RNN proposed by ~\citet{shirani-etal-2019-learning} and the convolutional neural network (CNN). We report results evaluated by the metrics Match-1, Match-2, Match-3, Match-4 and the average of these four metrics.
From Table \ref{tab:results}, we can see that CNN lags behind other models on the whole.
When the word embedding is GloVe, models with at least one graph surpass RNN on almost all the metrics except Match-2. In particular, our model can achieve an improvement on Match-1 and Match-4. Our model without WSG (word similarity graph) achieves an excellent performance on Match-3 and Average. When the word embedding is ELMo,
ours is superior to RNN-based on all the evaluation metrics. Compared to these two ablated models, Ours can also achieve better performance. Ours w/o WSG is better than RNN-based on all the evaluation metrics except Match-2 and Ours w/o SSG is better than RNN-based except Match-3.
On the whole, models with graphs can obtain better results on most metrics compared to the baseline models, which shows the advantage of these two components.
Experimental results based on RoBERTa are listed in Table~\ref{tab:results_lm}. Compared with the results based on GloVe and ELMo, RoBERTa and its variants achieve higher average match score which shows that a better initialized word embedding is helpful for a better performance. For the same RoBERTa encoder, Ours can obtain the highest score on Average and Match-2. For RoBERTa+GRU encoder, Ours can obtain the highest score on Average, Match-3 and Match-4. However, one interesting finding is that RoBERTa encoder performs much better than RoBERTa+GRU encoder. Two possible reasons may interpret this phenomenon. The first reason is the overfitting problem and the second reason is that the larger network is harder to train due to some optimization issues, e.g., gradient vanishing.
\subsubsection{Case Study}
To gain some insights of our proposed model, we present a sample case generated by the ELMo-based model as shown in Table~\ref{tab:case}. We can see that Ours not only predicts the ranking accurately, but also obtains very close probability to the ground truth probability derived by annotators.
Besides that, the probabilities of foolish and sane predicted by our model are very close than that predicted by RNN-based, which shows that the word similarity graph can impel similar words to have similar probabilities.
We also provide a failed case in Table~\ref{tab:case_failed}. It is intrinsically harder to rank the words in this sentence even for human beings. Our model does not rank them correctly on these cases where multiple words may be emphasized.
\subsubsection{Some Useful Tips}
We conclude some tips on the experiment that leads to better performance. (1) We can firstly train a classifier only using the SSG, then use the pre-trained embeddings as an initialization of the sentence graph nodes embeddings. It can obtain higher score and faster convergence of the model.
(2) We also consider another method to model the relationships between words using a self-attention operation proposed by ~\citet{article} above the hidden vectors of RNN. However, the performance is slightly degraded compared to removing this operation. So we think it is much better to model words relationships and sequence information separately.
\section{Conclusions}
The sentence structure graph and the word similarity graph are proposed to solve two issues found in emphasis selection.
The sentence structure graph helps to model the structure information of the sentence and the word similarity graph is useful in
modeling relationships between words.
With the development of graph neural network, the two graphs can be properly encoded and integrated into existing models. Experimental results demonstrate that our framework can achieve superior performance.
|
2,869,038,155,695 | arxiv | \section{INTRODUCTION}
The Langvin model is a very important model to describe the
diffusion behavior in non-equilibrium systems, and
it has been widely applied to various phenomena in
physics, chemistry and biology.
The Langevin model is usually transformed to the Fokker-Planck
equation (FPE) which deals with the probability distribution
function (PDF) of a state variable \cite{Risken96}.
It is generally not possible to obtain analytic solutions
of the second-order partial equations.
Indeed, exact analytical solutions of the FPE are
known for only a few cases.
In most cases, approximate solutions are obtained by
using analytic or numerical methods.
Typical analytic methods are an appropriate change of variables,
eigenfunction expansion, perturbation expansion,
path integral, Green's function, moment method, and the
continued-fraction method \cite{Risken96}.
When no analytic solutions are available, numerical methods
such as finite-difference and finite-element methods
have been employed.
For some Langevin models subjected to additive noise only,
exact solutions have been obtained. For the linear Langevin model,
the exact dynamical solution is expressed by the Gaussian distribution
with time-dependent mean and variance of a state variable.
For the FPE including a nonlinear diffusion term,
some authors have obtained exact dynamical solutions
\cite{Borland99,Plastino00,Malacarne02}.
The generalized FPEs in which time dependences
are introduced in drift and diffusion terms have been
investigated \cite{Malacarne02}-\cite{Heinsale07}.
When multiplicative noise is incorporated to the Langevin model,
the problem becomes much difficult \cite{Munoz04}.
For the linear Langevin model subjected to
additive and multiplicative noise,
the exact stationary solution is available, and it has been
considerably discussed in connection with the non-Gaussian
PDF in the nonextensive statistics
\cite{Tsallis88}-\cite{Hasegawa07b}.
An exact dynamical solution for the linear Langevin model
subjected to multiplicative noise only
is obtained in Ref. \cite{Fa03}
although it does not represent the stationary solution.
Approximate dynamical solutions of
the linear and nonlinear FPEs subjected to multiplicative noise
have been discussed with some sophisticated methods
such as the polynomial expansion of the logarithmic PDF
\cite{Paola02}, the linearizing transformation \cite{Unal08} and
the direct quadrature method for moment solution \cite{Attar08}.
Numerical methods are powerful approaches when exact dynamical
solutions are not available. Analytical solutions are, however,
indispensable in some subjects.
A typical example is a calculation of the time-dependent
Fisher information which is expressed by the derivatives of the
dynamical PDF with respect to its parameters.
In a recent paper \cite{Hasegawa08a},
we calculated the Fisher information
in a typical nonextensive system described by the linear
Langevin model subjected to additive and multiplicative noise.
We developed an analytic dynamical approach to the FPE combined with
the $q$-moment method in which moments are
evaluated over the escort probability distribution \cite{Tsallis98}.
The dynamical PDFs calculated by our moment method
are shown to be in good agreement with those obtained by the
partial difference equation method (PDEM) \cite{Hasegawa08a}.
By using the calculated time-dependent PDF, we discussed
the dynamical properties of the Fisher information
\cite{Hasegawa08a}.
It is the purpose of the present study to extend such an analytical
approach so as to be applied to a wide class of Langevin model
with the use of the conventional (normal) moment method instead of the
$q$-moment method.
The paper is organized as follows. In Sec. 2, we discuss the adopted
Langevin model and moment method to obtain the dynamical PDF.
The developed method has been applied to the three Langevin models.
We present some numerical calculations of the time-dependent
PDF in response to an applied signal and force.
Section 3 is devoted to conclusion and discussion on the dynamics of
Fisher information of the inverse-gamma distribution.
\section{METHOD AND RESULT}
\subsection{Fokker-Planck equation}
We have adopted the Langevin model subjected to
cross-correlated additive ($\xi$)
and multiplicative noise ($\eta$) given by
\begin{eqnarray}
\frac{dx}{dt}\!\!&=&\!\! F(x) + G(x) \eta(t)
+ \xi(t)+I(t).
\label{eq:A1}
\end{eqnarray}
Here
$F(x)$ and $G(x)$ are arbitrary functions of $x$,
$I(t)$ stands for an external input, and $\eta(t)$ and $\xi(t)$
express zero-mean Gaussian white noises with correlations given by
\begin{eqnarray}
\langle \eta(t)\:\eta(t') \rangle
&=& \alpha^2 \:\delta(t-t'),\\
\langle \xi(t)\:\xi(t') \rangle
&=& \beta^2 \: \delta(t-t'),\\
\langle \eta(t)\:\xi(t') \rangle &=&
\epsilon \alpha \beta \: \delta(t-t'),
\label{eq:A2}
\end{eqnarray}
where $\alpha$ and $\beta$ denote the strengths of multiplicative
and additive noise, respectively, and $\epsilon$
the degree of the cross-correlation between the two noise.
The FPE is expressed by \cite{Tessone98,Liang04,Jin05}
\begin{eqnarray}
\frac{\partial}{\partial t}\: p( x,t)
&=&- \frac{\partial}{\partial x}\left( \left[F(x) +I
+\left( \frac{\phi}{2} \right)
[\alpha^2 G(x)G'(x)+ \epsilon \alpha \beta\: G'(x)]
\right] \:p( x,t)\right)
\nonumber \\
&+& \left(\frac{1}{2} \right) \frac{\partial^2}{\partial x^2}
\{[\alpha^2 G(x)^2+ 2 \epsilon \alpha\beta G(x)+\beta^2]\:p(x,t) \},
\label{eq:A3}
\end{eqnarray}
where $G'(x)=dG(x)/dx$,
and $\phi=0$ and 1 in the Ito and Stratonovich
representations, respectively.
Although we have adopted the Langevin model for a single
variable in this study, it is straightforward to extend it to the
coupled Langevin model with the use of the mean-field
approximation \cite{Hasegawa08a}.
For $I(t)=I$, the stationary PDF of $p(x)$
is expressed by \cite{Hasegawa07c}
\begin{eqnarray}
\ln p(x) &=& X(x)+Y(x)
-\left(1- \frac{\phi}{2} \right)
\ln \left(\frac{1}{2}
[\alpha^2 G(x)^2 +2\epsilon \alpha \beta G(x)+ \beta^2] \right),
\label{eq:A4}
\end{eqnarray}
with
\begin{eqnarray}
X(x) &=& 2 \int \:dx \:
\left[ \frac{F(x)}{\alpha^2 G(x)^2+2\epsilon \alpha \beta G(x)
+\beta^2} \right], \\
Y(x) &=& 2 \int \:dx \:
\left[ \frac{I}{\alpha^2 G(x)^2+2\epsilon \alpha \beta G(x)
+\beta^2} \right].
\label{eq:A5}
\end{eqnarray}
\subsection{Equations of motion for the moments}
An equation of motion for the $n$th moment is given by
\begin{eqnarray}
\frac{\partial \langle x^n \rangle}{\partial t}
&=& \int \frac{\partial p(x,t)}{\partial t}\:x^n \:dx, \\
&=& n \left( \left< x^{n-1} F(x) \right>+ \left< x^{n-1} I(t) \right>
+\frac{\phi}{2}\left[\alpha^2\left< x^{n-1} G(x)G'(x) \right>
+ \epsilon \alpha \beta \left< x^{n-1} G'(x) \right>\right] \right)\nonumber \\
&+& \frac{n(n-1)}{2}\left [\alpha^2 \left< x^{n-2}G(x)^2 \right>
+ 2 \epsilon \alpha \beta \left< x^{n-2}G(x) \right>
+ \beta^2 \left< x^{n-2} \right> \right],
\label{eq:A10}
\end{eqnarray}
where suitable boundary conditions are adopted.
For $n=1,\:2$, we obtain
\begin{eqnarray}
\frac{\partial \langle x \rangle}{\partial t}
&=& \langle F(x) \rangle + \langle I(t) \rangle
+ \frac{\phi}{2} [\alpha^2 \langle G(x) G'(x) \rangle
+ \epsilon \alpha \beta \langle G'(x) \rangle],
\label{eq:A6} \\
\frac{\partial \langle x^2 \rangle}{\partial t}
&=& 2 \langle x F(x) \rangle+ 2 \langle x I(t) \rangle
+ \phi [\alpha^2 \langle x G(x) G'(x) \rangle
+ \epsilon \alpha \beta \langle x G'(x) \rangle] \nonumber \\
&+& \alpha^2 \langle G(x)^2 \rangle
+ 2 \epsilon \alpha \beta \langle G(x) \rangle +\beta^2.
\label{eq:A7}
\end{eqnarray}
Expanding $x$ as $x=\mu+\delta x$ and retaining up to
$O(\langle (\delta x)^2 \rangle)$, we obtain
equations of motion for the
average $\mu$ [$=\langle x \rangle $] and
variance $\sigma^2$ [$=\langle x^2 \rangle -\langle x \rangle^2$]
given by \cite{Hasegawa07b}
\begin{eqnarray}
\frac{d \mu}{dt}&=& f_0+f_2\sigma^2
+\frac{\phi }{2}
\left( \alpha^2[g_0g_1+3(g_1g_2+g_0g_3)\sigma^2]
+ \epsilon \alpha \beta(g_1+3 g_3 \sigma^2) \right)+I(t),
\label{eq:A8}\\
\frac{d \sigma^2}{dt} &=& 2f_1 \sigma^2
+ (\phi+1) (g_1^2+2 g_0g_2)\alpha^2 \sigma^2
+2 \epsilon \alpha \beta (\phi+1) g_2 \sigma^2 \nonumber \\
&+& \alpha^2 g_0^2+ 2 \epsilon \alpha \beta g_0 +\beta^2,
\label{eq:A9}
\end{eqnarray}
where $f_{\ell}=(1/\ell !)
\partial^{\ell} F(\mu)/\partial x^{\ell}$ and
$g_{\ell}=(1/\ell !)
\partial^{\ell} G(\mu)/\partial x^{\ell}$.
\subsection{Model A}
\subsubsection{Stationary distribution}
Our dynamical moment approach
will be applied to the three Langevin models
A, B and C, which will be separately discussed
in Secs. 2,3, 2.4 and 2.5, respectively.
First we consider the model A in which $F(x)$ and $G(x)$ are given by
\begin{eqnarray}
F(x) &=& -\lambda x,
\label{eq:B0} \\
G(x) &=& x,
\label{eq:B1}
\end{eqnarray}
with $\epsilon=0.0$ ({\it i.e.,} without the cross-correlation),
where $\lambda $ expresses the relaxation rate.
The model A has been adopted as a microscopic model
for nonextensive systems \cite{Sakaguchi01}-\cite{Hasegawa07b}.
From Eq. (\ref{eq:A3}), the FPE in the Stratonovich representation
is given by
\begin{eqnarray}
\frac{\partial}{\partial t}\: p(x,t)
&=& \frac{\partial}{\partial x} \left[\lambda x -I(t)\right] p(x,t)
+ \left( \frac{\beta^2}{2} \right)
\frac{\partial^2}{\partial x^2}p(x,t) \nonumber
\\
&+& \left( \frac{\alpha^2}{2} \right)
\frac{\partial}{\partial x}
\left[ x \frac{\partial}{\partial x} \{ x p(x,t) \} \right].
\label{eq:B2}
\end{eqnarray}
By using Eqs. (\ref{eq:A4})-(\ref{eq:A5}),
we obtain the stationary PDF given by \cite{Hasegawa08a}
\begin{eqnarray}
p(x) &=& \left( \frac{1}{Z} \right)
\frac{\exp [2c \tan^{-1}(ax)]}{(1+a^2x^2)^b},
\label{eq:B3}
\end{eqnarray}
with
\begin{eqnarray}
a &=& \frac{\alpha}{\beta},
\label{eq:B4} \\
b &=& \frac{(2\lambda+\alpha^2)}{2 \alpha^2},
\label{eq:B5} \\
c &=& \frac{I}{\alpha \beta},
\label{eq:B6} \\
Z &=& \frac{\sqrt{\pi}\:\Gamma(b)\Gamma(b-\frac{1}{2})}
{a\:\mid \Gamma(b+ i c) \mid^2}.
\label{eq:B7}
\end{eqnarray}
By using Eq. (\ref{eq:B3}), we obtain the mean and variance in
the stationary state given by
\begin{eqnarray}
\mu &=& \frac{c}{a (b-1)}=\frac{2I}{(2 \lambda-\alpha^2)},
\label{eq:B8} \\
\sigma^2 &=& \frac{[(b-1)^2+c^2]}{a^2(b-1)^2(2b-3)}
=\frac{(\alpha^2 \mu^2+\beta^2)}{2(\lambda-\alpha^2)}.
\label{eq:B9}
\end{eqnarray}
Depending on the model parameters, the stationary PDF
given by Eq. (\ref{eq:B3}) may reproduce various
PDFs such as the Gaussian, $q$-Gaussian,
Cauchy and inverse-gamma PDFs \cite{Hasegawa08a}.
\subsubsection{Dynamical distribution}
It is worthwhile to remind the dynamical solution of the FPE
given by Eq. (\ref{eq:B2}) in the limit of $\alpha = 0.0$
({\it i.e.,} additive noise only), for which
the time-dependent solution is given by
\begin{equation}
p(x,t)= \frac{1}{\sqrt{2 \pi \:\sigma(t)^2}}
\;e^{-[x-\mu(t)]^2/2 \sigma(t)^2},
\label{eq:C1}
\end{equation}
with $\mu(t)$ and $\sigma(t)^2$ satisfying equations of motion given by
\begin{eqnarray}
\frac{d \mu(t)}{dt} &=& -\lambda \mu(t)+ I(t),
\label{eq:C2}
\\
\frac{d \sigma(t)^2}{dt} &=& -2 \lambda \sigma(t)^2 + \beta^2.
\label{eq:C3}
\end{eqnarray}
In order to derive the dynamical solution of the FPE
for $\alpha \neq 0.0$ given by Eq. (\ref{eq:B2}),
we adopt the moment approach with the following steps:
\noindent
(1) We assume that dynamical PDF has the
same structure as the stationary one, as given by
\begin{eqnarray}
p(x,t) &=& \left( \frac{1}{Z(t)} \right)
\frac{ \exp [2 c(t) \:\tan^{-1} \{a(t) x\} ] }
{ [1+ a(t)^2 x^2 ]^{b(t)}},
\label{eq:C4}
\end{eqnarray}
with
\begin{eqnarray}
Z(t) &=& \frac{\sqrt{\pi}\:\Gamma[b(t)]\:\Gamma[b(t)-\frac{1}{2}]}
{a(t)\:\mid \Gamma[b(t)+ i c(t)] \mid^2}.
\label{eq:C5}
\end{eqnarray}
\noindent
(2) With the assumption (1), we first tried to derive equations of motion
for the parameters of $a(t)$, $b(t)$ and $c(t)$, by using the FPE after
Refs. \cite{Borland99,Plastino00,Malacarne02}. Unfortunately, it did
not work because functional forms
in the left and right sides of the FPE become different.
Then we tried to express the parameters
in terms of importance quantities of $\mu(t)$ and $\sigma(t)^2$
such as to be consistent with the relations for
the stationary state given by
Eqs. (\ref{eq:B8}) and (\ref{eq:B9}).
Because the number of parameters (three)
is larger than two for $\mu(t)$ and $\sigma(t)^2$,
the parameters of $a(t)$, $b(t)$ and $c(t)$ cannot be uniquely
expressed in terms of $\mu(t)$
and $\sigma(t)^2$ from Eqs. (\ref{eq:B8}) and (\ref{eq:B9}).
If the first three moments in the stationary state are available,
it is possible to uniquely express $a(t)$, $b(t)$ and $c(t)$
in terms of them, though such a calculation is laborious.
In order to overcome the above problem,
we have imposed an additional condition that
expressions for the parameters
yield the consistent result in the two limiting cases of
$\alpha \rightarrow 0$ and $\beta \rightarrow 0$.
After several tries, we have decided that $b(t)$ and $c(t)$
in Eqs. (\ref{eq:C4}) and (\ref{eq:C5}) are expressed as
\begin{eqnarray}
b(t) &=& \frac{[1+a^2\{\mu(t)^2+3 \sigma(t)^2\}]}
{2 a^2 \sigma(t)^2},
\label{eq:C6}\\
c(t) &=& \frac{[1+a^2\{\mu(t)^2+\sigma(t)^2\}] \mu(t)}
{2 a \:\sigma(t)^2},
\label{eq:C7}
\end{eqnarray}
with the time-independent $a$
($=\alpha/\beta$) given by Eq. (\ref{eq:B4}).
The relations given by Eqs. (\ref{eq:C6}) and (\ref{eq:C7})
are consistent with Eqs. (\ref{eq:B8}) and (\ref{eq:B9})
for the stationary state and they satisfy the above-mentioned
limiting conditions, as will be shown shortly.
\noindent
(3) Equations of motion for $\mu(t)$ and $\sigma(t)^2$
in Eqs. (\ref{eq:C6}) and (\ref{eq:C7}) are obtained from
Eqs. (\ref{eq:A8})-(\ref{eq:B1}),
as given by
\begin{eqnarray}
\frac{d \mu(t)}{dt}&=&-\lambda \mu(t) + I(t)
+ \frac{\alpha^2 \mu(t)}{2},
\label{eq:C8}\\
\frac{d \sigma(t)^2}{dt} &=& -2 \lambda \sigma(t)^2
+ 2 \alpha^2 \sigma(t)^2 + \alpha^2 \mu(t)^2 + \beta^2.
\label{eq:C9}
\end{eqnarray}
Thus the dynamical solution of the FPE given by Eq. (\ref{eq:B2})
is expressed by Eqs. (\ref{eq:C4})-(\ref{eq:C9}).
In the following, we will show that
the relations given by Eqs. (\ref{eq:C6}) and (\ref{eq:C7})
lead to results consistent in the two limiting cases of
$\alpha \rightarrow 0.0$ and $\beta \rightarrow 0.0$.
\vspace{0.5cm}
\noindent
{\bf (a) $\alpha \rightarrow 0$ case}
In the limit of $\alpha \rightarrow 0.0$
({\it i.e.,} additive noise only),
$p(x,t)$ given by Eq. (\ref{eq:C4}) reduces to
\begin{eqnarray}
p(x,t) &\propto& e^{-a(t)^2b(t) x^2+2a(t)c(t) x} \\
&\rightarrow & e^{-[x-\mu(t)]^2/2\sigma(t)^2},
\label{eq:D3}
\end{eqnarray}
because Eqs. (\ref{eq:C6}) and (\ref{eq:C7})
with $a \rightarrow 0.0$ yield
\begin{eqnarray}
a(t)^2 b(t) &=& \frac{[1+a^2\{\mu(t)^2+3 \sigma(t)^2\}]}{2 \sigma(t)^2}
\rightarrow \frac{1}{2 \sigma(t)^2}, \\
2a(t)c(t) & = &
\frac{[1+a^2\{\mu(t)^2+\sigma(t)^2\}]\:\mu(t)}{\sigma(t)^2}
\rightarrow \frac{\mu(t)}{\sigma(t)^2}.
\end{eqnarray}
Equation (\ref{eq:D3}) agrees with the Gaussian distribution
given by Eq. (\ref{eq:C1})
\vspace{0.5cm}
\noindent
{\bf (b) $\beta \rightarrow 0$ case}
In the opposite limit of $\beta=0.0$
({\it i.e.,} multiplicative noise only), the stationary PDF
given by Eqs. (\ref{eq:B3}) and (\ref{eq:B7}) with $I > 0$
leads to the inverse-gamma distribution expressed by
\begin{eqnarray}
p(x) &=& \frac{\kappa^{\delta-1}}{\Gamma[\delta-1]}
\:x^{-\delta} e^{-\kappa/x}\: \Theta(x),
\label{eq:E1}
\end{eqnarray}
where
\begin{eqnarray}
\delta &=& 2b,
\label{eq:E2} \\
\kappa &=& \frac{2c}{a}.
\label{eq:E3}
\end{eqnarray}
Here $\Gamma(x)$ denotes the gamma function and
$\Theta(t)$ the Heaviside
function: $\Theta(t)=1$ for $t > 0$ and zero otherwise.
From Eq. (\ref{eq:E1}), we obtain
the average and variance in the stationary state given by
\begin{eqnarray}
\mu &=& \frac{\kappa}{(\delta-2)},
\label{eq:E4}\\
\sigma^2 &=& \frac{\kappa^2}{(\delta-2)^2 (\delta-3)},
\label{eq:E5}
\end{eqnarray}
from which $\delta$ and $\kappa$ are expressed
in terms of $\mu$ and $\sigma^2$ as
\begin{eqnarray}
\delta &=& \frac{\mu^2+3 \sigma^2}{\sigma^2},
\label{eq:E6}\\
\kappa &=& \frac{\mu^2+\sigma^2}{\sigma^2}\:\mu.
\label{eq:E7}
\end{eqnarray}
On the contrary, the dynamical PDF given by
Eq. (\ref{eq:C4}) in the limit of $\beta \rightarrow 0.0$
(and $I > 0 $) reduces to
\begin{eqnarray}
p(x, t) \propto \frac{e^{-2c(t)/a(t)x}}{x^{2b(t)}}\:\Theta(x)
\rightarrow \frac{e^{-\kappa(t)/x}}{x^{\delta(t)}}\:\Theta(x),
\label{eq:E8}
\end{eqnarray}
because Eqs. (\ref{eq:C6}) and (\ref{eq:C7}) with
$\beta \rightarrow 0.0$ ($a \rightarrow \infty$) lead to
\begin{eqnarray}
\delta(t) &=& 2b(t)
= \frac{[1+a^2\{\mu(t)^2+3 \sigma(t)^2\}]}{a^2\sigma(t)^2}
\rightarrow \frac{\mu(t)^2+3\sigma(t)^2}{\sigma(t)^2},
\label{eq:E9} \\
\kappa(t) &=& \frac{2c(t)}{a(t)}
=\frac{[1+a^2\{\mu(t)^2+\sigma(t)^2\}]\mu(t)}{a^2\sigma(t)^2}
\rightarrow \frac{\mu(t)^2+\sigma(t)^2}{\sigma(t)^2}\:\mu(t).
\label{eq:E10}
\end{eqnarray}
Equations (\ref{eq:E8}), (\ref{eq:E9}) and (\ref{eq:E10}) agree
with Eqs. (\ref{eq:E1}), (\ref{eq:E6}) and (\ref{eq:E7}), respectively.
Thus the expressions given by Eqs. (\ref{eq:C6}) and (\ref{eq:C7})
yield the consistent result
covering the two limits of $\alpha \rightarrow 0.0$ and
$\beta \rightarrow 0.0$.
\subsection{Model B}
\subsubsection{Stationary distribution}
Next we consider the model B in which $F(x)$ and $G(x)$ are given
by Eqs. (\ref{eq:B0}) and (\ref{eq:B1}) with $\epsilon \neq 0$.
From Eqs. (\ref{eq:A4})-(\ref{eq:A5}), the stationary PDF
is given by
\begin{eqnarray}
p(x)&=& \left( \frac{1}{Z} \right)
\frac{ \exp[2c \tan^{-1} \{a(x+f)\} ] }
{[1+a^2(x+f)^2]^b},
\label{eq:F1}
\end{eqnarray}
with
\begin{eqnarray}
a &=& \frac{\alpha}{\beta \sqrt{1-\epsilon^2}},
\label{eq:F2} \\
b &=& \frac{2\lambda+\alpha^2}{2 \alpha^2},
\label{eq:F3}\\
c &=& \frac{(I+\lambda f)}
{\alpha \beta \sqrt{1-\epsilon^2}},
\label{eq:F4}\\
f &=& \frac{\epsilon \beta}{\alpha},
\label{eq:F5}\\
Z &=& \frac{\sqrt{\pi}\:\Gamma(b)\Gamma(b-\frac{1}{2})}
{a\:\mid \Gamma(b+ i c) \mid^2}.
\label{eq:F6}
\end{eqnarray}
Equations (\ref{eq:F1}) and (\ref{eq:F6}) yield the average
and variance in the stationary state given by
\begin{eqnarray}
\mu &=& \frac{c}{a(b-1)}-f
= \frac{(2 I +\epsilon \alpha \beta)}{(2 \lambda-\alpha^2)},
\label{eq:F7} \\
\sigma^2 &=& \frac{[(b-1)^2+c^2]}{a^2(b-1)^2(2b-3)}
= \frac{(\alpha^2 \mu^2+2 \epsilon \alpha\beta \mu+\beta^2)}
{2(\lambda-\alpha^2)}.
\label{eq:F8}
\end{eqnarray}
\subsubsection{Dynamical distribution}
With the use of the procedure mentioned for the model A in Sec. 2.3,
the dynamical PDF $p(x,t)$
of the model B is assumed to be
given by Eq. (\ref{eq:F1}) but with $b$ and $c$ replaced by
\begin{eqnarray}
b(t) &=& \frac{[1+a^2\{[\mu(t)+f]^2+3 \sigma(t)^2\}]}
{2 a^2 \sigma(t)^2},
\label{eq:G1} \\
c(t) &=& \frac{[1+a^2\{[\mu(t)+f]^2+\sigma(t)^2\}] [\mu(t)+f]}
{2 a \sigma(t)^2},
\label{eq:G2}
\end{eqnarray}
which agree with Eqs. (\ref{eq:F7}) and (\ref{eq:F8})
in the stationary state.
Equations of motion for $\mu(t)$ and $\sigma(t)^2$
in Eqs. (\ref{eq:G1}) and (\ref{eq:G2}) are given by
\begin{eqnarray}
\frac{d \mu(t)}{dt}&=&-\lambda \mu(t) + I(t)
+ \frac{\alpha^2 \mu(t)}{2}+ \frac{\epsilon \alpha \beta}{2},
\label{eq:G3}\\
\frac{d \sigma(t)^2}{dt} &=& -2 \lambda \sigma(t)^2
+ 2 \alpha^2 \sigma(t)^2
+ \alpha^2 \mu(t)^2 + 2 \epsilon \alpha \beta \mu(t) + \beta^2,
\label{eq:G4}
\end{eqnarray}
which are derived from Eqs. (\ref{eq:A8})-(\ref{eq:B1}).
It is noted that in the limits of $\alpha \rightarrow 0.0$
and $\beta \rightarrow 0.0$, the cross-correlation
between additive and multiplicative noise
does not work, and the result for the model B reduces to that
for the model A.
Thus the moment method
with the use of Eqs. (\ref{eq:G1}) and (\ref{eq:G2})
leads to the results consistent in the limits of $\alpha \rightarrow 0.0$
and $\beta \rightarrow 0.0$, where $p(x,t)$ becomes
the Gaussian and inverse-gamma distributions, respectively.
\subsection{Model C}
\subsubsection{Stationary distribution}
Now we consider the model C in which $F(x)$ and $G(x)$ are given by
\begin{eqnarray}
F(x) &=& -\lambda (x+s),
\label{eq:H1} \\
G(x) &=& \sqrt{x^2+ 2 sx+r^2 },
\label{eq:H2}
\end{eqnarray}
with $\epsilon=0.0$ where $\lambda $ expresses the relaxation rate,
and $r$ and $s$ are parameters.
We assume that $\epsilon=0.0$ because we cannot obtain
the analytic stationary PDF for $\epsilon \neq 0.0$.
The model C with $r=s=0.0$ is nothing but the model A.
From Eqs. (\ref{eq:A4})-(\ref{eq:A5}),
the stationary PDF for the model C is given by
\begin{eqnarray}
p(x) & \propto &
\left[1-\frac{\alpha^2}{D}(x+s)^2 \right]^{-b}\:e^{Y(x)},
\label{eq:H3}
\end{eqnarray}
where
\begin{eqnarray}
Y(x) &=& \left( \frac{I}{\alpha \sqrt{D}} \right)
\ln \left| \frac{x+s-\sqrt{D}}{x+s+\sqrt{D}} \right|,
\hspace{1cm}\mbox{for $D > 0$}
\label{eq:H4} \\
&=& \left( \frac{2I}{\alpha \sqrt{-D}} \right)
\tan^{-1} \left( \frac{\alpha(x+s)}{\sqrt{-D}} \right),
\hspace{1cm}\mbox{for $D < 0$}
\label{eq:H5}\\
&=& - \frac{2I}{\alpha^2 (x+s)},
\label{eq:H6}
\hspace{3cm}\mbox{for $D = 0$}
\end{eqnarray}
with
\begin{eqnarray}
D &=& \alpha^2(s^2-r^2)-\beta^2,
\label{eq:H7} \\
b &=& \frac{(2\lambda+\alpha^2)}{2 \alpha^2}.
\label{eq:H8}
\end{eqnarray}
When we consider the case of $D < 0$,
the stationary PDF is rewritten as
\begin{eqnarray}
p(x) &=& \left( \frac{1}{Z} \right)
\frac{ \exp[2c \tan^{-1} \{a(x+s)\} ] }
{[1+a^2(x+s)^2]^b},
\label{eq:H9}
\end{eqnarray}
with
\begin{eqnarray}
a &=& \frac{\alpha}{\sqrt{\beta^2+\alpha^2(r^2-s^2)}},
\label{eq:H10} \\
c &=& \frac{I}{\alpha \sqrt{\beta^2+\alpha^2(r^2-s^2)} }, \\
\label{eq:H11}
Z &=& \frac{\sqrt{\pi}\:\Gamma(b)\Gamma(b-\frac{1}{2})}
{a\:\mid \Gamma(b+ i c) \mid^2}.
\label{eq:H12}
\end{eqnarray}
From Eq. (\ref{eq:H9}), we obtain $\mu$ and $\sigma^2$ in
the stationary state expressed by
\begin{eqnarray}
\mu &=& \frac{c}{a(b-1)}-s
=\frac{2I}{(2\lambda-\alpha^2)}-s,
\label{eq:H13} \\
\sigma^2 &=& \frac{[(b-1)^2+c^2]}{a^2 (b-1)^2(2b-3)}
=\frac{[\alpha^2 (\mu^2+2s\mu+r^2) + \beta^2]}
{2(\lambda-\alpha^2)}.
\label{eq:H14}
\end{eqnarray}
\subsubsection{Dynamical distribution}
In order to obtain the dynamical PDF for the model C,
we adopt the same procedure
as those for the models A and B.
We assume that the dynamical solution $p(x,t)$ of the model C
is given by Eq. (\ref{eq:H9}) but with $b$ and $c$ replaced by
\begin{eqnarray}
b(t) &=& \frac{[1+a^2\{[\mu(t)+s]^2+3 \sigma(t)^2\}]}
{2 a^2 \sigma(t)^2},
\label{eq:J1} \\
c(t) &=& \frac{[1+a^2\{[\mu(t)+s]^2+\sigma(t)^2\}] [\mu(t)+s]}
{2 a \sigma(t)^2},
\label{eq:J2}
\end{eqnarray}
which agree with Eqs. (\ref{eq:H13}) and (\ref{eq:H14})
in the stationary state.
Equations of motion for
$\mu(t)$ and $\sigma(t)^2$ in Eqs. (\ref{eq:J1}) and (\ref{eq:J2})
are given by
\begin{eqnarray}
\frac{d \mu(t)}{dt}&=& -\left(\lambda -\frac{\alpha^2}{2} \right)
[\mu(t)+s] + I(t),
\label{eq:J3} \\
\frac{d \sigma(t)^2}{dt} &=& -2 (\lambda-\alpha^2) \sigma(t)^2
+ \alpha^2 [\mu(t)^2+2s\mu(t)+r^2]+\beta^2,
\label{eq:J4}
\end{eqnarray}
which are derived from Eqs. (\ref{eq:A8}), (\ref{eq:A9}),
(\ref{eq:H1}) and (\ref{eq:H2}).
It is easy to see that the moment method
with the use of Eqs. (\ref{eq:J1}) and (\ref{eq:J2})
lead to the results consistent in the limits of $\alpha \rightarrow 0.0$
and $\beta \rightarrow 0.0$, where $p(x,t)$ becomes
the Gaussian and inverse-gamma distributions, respectively.
\subsection{Model calculations}
We will present some numerical calculations in this subsection.
In order to examine the validity of the moment
approach, we have employed the partial difference
equation derived from Eq. (\ref{eq:A3})
with $\phi=1$, as given by
\begin{eqnarray}
p(x,t+v) &=& p(x,t)
+\left(-F'
+ \frac{\alpha^2}{2}[(G')^2+GG^{(2)}]
+ \frac{\epsilon \alpha \beta}{2} G^{(2)} \right) v \:p(x,t) \nonumber\\
&+&\left[ -F-I(t)
+ \frac{3 \alpha^2}{2} G G'
+ \frac{3 \epsilon \alpha \beta}{2}G' \right]
\left(\frac{v}{2 u}\right)[p(x+u)-p(x-u)] \nonumber \\
&+& \left(\frac{\alpha^2}{2} G^2 + \epsilon \alpha \beta G
+ \frac{\beta^2}{2} \right)
\left(\frac{v}{u^2}\right)[p(x+u,t)+p(x-u,t)-2p(x,t)],
\label{eq:L1}
\end{eqnarray}
where $u$ and $v$ denote incremental steps of $x$ and $t$, respectively.
We impose the boundary condition:
\begin{eqnarray}
p(x,t)=0, \hspace{1cm}\mbox{for $ \mid x \mid \ge x_m$}
\label{eq:L2}
\end{eqnarray}
with $x_m=5$, and the initial condition of $p(x,0)=p_0(x)$ where $p_0(x)$
is the stationary PDF.
We have chosen parameters of $u=0.05$ and $v=0.0001$ such as to satisfy
the condition: $(\alpha^2 x_m^2 v/2 u^2) < 1/2$, which is required for
stable, convergent solutions of the PDEM.
First we apply a pulse input signal given by
\begin{equation}
I(t) = \Delta I \:\Theta(t-2)\Theta(6-t)+I_b,
\label{eq:M1}
\end{equation}
with $\Delta I=0.5$ and $I_b=0.0$
to the model B with $\lambda=1.0$, $\alpha=0.5$,
$\beta=0.5$ and $\epsilon=0.5$.
Figure \ref{figK} shows the time-dependence of the
PDF at various $t$ in response to an applied input.
Solid curves express the results of the moment method
calculated with the use of
Eqs. (\ref{eq:F1}), (\ref{eq:F6}), (\ref{eq:G1})-(\ref{eq:G4}),
and dashed curves denote those of the PDEM
with Eq. (\ref{eq:L1}).
Figure \ref{figJ} shows the time-dependence
of $\mu(t)$ and $\sigma(t)^2$ calculated by the moment method
with Eqs. (\ref{eq:G3}) and (\ref{eq:G4}) (solid curves)
and by the PDEM with Eq. (\ref{eq:L1}) (dashed curves).
At $0 \leq t < 2.0$ where no input signal is applied,
we obtain $\mu(t)=0.071$ and $\sigma(t)^2=0.179$.
The PDF at $ t < 2.0$ is not symmetric with respect to its
center because of the introduced correlation of $\epsilon=0.5$.
By an applied pulse at $2.0 \leq t < 6.0$,
$\mu(t)$ and $\sigma(t)^2$ are increased, and
the position of $p(x,t)$ moves rightward
with slightly distorted shapes.
After an input pulse diminishes at $t \geq 6.0$, the PDF
gradually restores to its original stationary shape.
Next we apply the pulse input given by Eq. (\ref{eq:M1})
with $\Delta I=0.5$ and $I_b=0.0$
to the model C with $\lambda=1.0$, $\alpha=0.5$,
$\beta=0.5$, $r=0.5$ and $s=0.2$.
Figure \ref{figB} shows the time-dependent
PDF at various $t$ in response to an applied input.
Results of the moment method
calculated with the use of
Eqs. (\ref{eq:H9}), (\ref{eq:H12}), (\ref{eq:J1})-(\ref{eq:J4})
are shown by solid curves while
those of the PDEM with Eq. (\ref{eq:L1})
are expressed by dashed curves.
The time-dependent $\mu(t)$ and $\sigma(t)^2$ are shown in
Fig. \ref{figA}: solid curves denote the results calculated
by the moment method with Eqs. (\ref{eq:J3}) and (\ref{eq:J4}):
dashed curves express those by the PDEM with Eq. (\ref{eq:L1}).
In the stationary state at $t < 2.0$,
we obtain $\mu(t)=-0.2$ and $\sigma(t)^2=0.2$.
By an applied pulse at $2.0 \leq t < 6.0$,
$\mu(t)$ and $\sigma(t)^2$ are increased and
the position of $p(x,t)$ moves rightward
with slightly changed shapes.
It is possible to calculate the response to temporal changes
in the model parameters such as $\lambda$, $\alpha$ and $\beta$.
As an example, we introduce the time-dependent
relaxation rate $\lambda(t)$ given by
\begin{equation}
\lambda(t) = \Delta\lambda \:\Theta(t-2)\Theta(6-t)+\lambda_b,
\label{eq:M2}
\end{equation}
with $\Delta \lambda=1.0$ and $\lambda_b=1.0$
to the model C with $\alpha=0.5$, $\beta=0.0$,
$r=0.5$, $s=0.2$ and $I=0.0$.
Equation (\ref{eq:M2}) stands for an application of
an external force of $-\Delta \lambda \:x$
at $2.0 \leq t < 6.0$.
The time dependent $p(x,t)$ is plotted in Fig. \ref{figF},
where solid and dashed curves
denote the results of the moment method and PDEM, respectively.
Figure \ref{figG} shows the time dependences of
$\mu(t)$ and $\sigma(t)^2$.
When an external force is applied at $2.0 \leq t < 6.0$,
the width of the PDF is reduced and
$\sigma(t)^2$ is decreased while $\mu(t)$ has no changes.
It is noted in Figs. \ref{figK}-\ref{figG} that
results of $p(x, t)$, $\mu(t)$ and $\sigma(t)^2$
calculated by the moment method are in good
agreement with those obtained by the PDEM.
\section{CONCLUSION AND DISCUSSION}
The moment approach to the FPE
discussed in preceding Sec. 2
may be applied to various Langevin models provided
analytic expressions for the PDF and for
the first- and second-order moments in the
stationary state are available.
For example, when $F(x)$ and $G(x)$ are given by
\begin{eqnarray}
F(x) &=& -\lambda x \vert x \vert^{r-1},
\label{eq:P1} \\
G(x) &=& x \vert x \vert^{s-1},
\label{eq:P2}
\end{eqnarray}
for $r \geq 0$, $s \geq 0$,
the stationary PDF with $\epsilon=0.0$
is given by \cite{Anten02,Hasegawa07c}
\begin{eqnarray}
p(x) \propto (\alpha^2 \vert x \vert^{2s}+\beta^2)^{-1/2}
\exp[X(x)+Y(x)],
\label{eq:P3}
\end{eqnarray}
with
\begin{eqnarray}
X(x) &=& -\left( \frac{2\lambda \vert x \vert^{r+1}}{\beta^2 (r+1)}\right)
F\left(1, \frac{r+1}{2s}, \frac{r+1}{2s}+1;
-\frac{\alpha^2 \vert x \vert^{2s}}{\beta^2} \right), \\
Y(x) &=& \left( \frac{2 I \vert x \vert }{\beta^2}\right)
F\left(1, \frac{1}{2s}, \frac{1}{2s}+1;
-\frac{\alpha^2 \vert x \vert^{2s}}{\beta^2} \right),
\label{eq:P4}
\end{eqnarray}
where $F(a,b,c;z)$ denotes the hypergeometric function.
Equations of motion for $\mu$ and $\sigma^2$ are given by
\cite{Hasegawa07b}
\begin{eqnarray}
\frac{d\mu}{dt} &=& -\lambda \mu \vert \mu\vert^{r-1}+I
-\left( \frac{\lambda}{2} \right) r(r-1)
\mu \vert \mu\vert^{r-3}\sigma^2 \nonumber \\
&+& \left( \frac{\alpha^2}{2}\right)
[s \mu \vert \mu \vert^{2s-2}
+s(s-1)(2s-1)\mu \vert \mu\vert ^{2s-4}\sigma^2],
\label{eq:P5} \\
\frac{d \sigma^2}{dt} &=& -2 \lambda r \vert \mu \vert^{r-1} \sigma^2
+2s(2s-1)\alpha^2 \vert \mu \vert^{2s-2} \sigma^2
+\alpha^2 \vert \mu \vert^{2s}+\beta^2.
\label{eq:P6}
\end{eqnarray}
If analytic expressions for stationary values
of $\mu$ and $\sigma^2$ are obtainable
from Eqs. (\ref{eq:P5}) and (\ref{eq:P6}),
we may apply our moment method to the FPE given by
Eqs. (\ref{eq:A3}), (\ref{eq:P1}) and (\ref{eq:P2})
with the following steps:
(1) adopting the stationary PDF
given by Eqs. (\ref{eq:P3})-(\ref{eq:P4}), and
(2) expressing its parameters in terms of
the time-dependent $\mu(t)$ and $\sigma(t)^2$
in an appropriate way, as mentioned for models A, B and C.
As an application of our method, we have calculated
the Fisher information for the dynamical inverse-gamma distribution,
which is realized for $\beta=0.0$ in the model A [Eq. (\ref{eq:E8})],
\begin{eqnarray}
p(x, t) &=& \frac{\kappa^{\delta(t)-1}}{\Gamma[\delta(t)-1]}
\:x^{-\delta(t)} e^{-\kappa(t)/x } \: \Theta(x),
\label{eq:N1}
\end{eqnarray}
with the time-dependent $\delta(t)$ and $\kappa(t)$
given by Eqs. (\ref{eq:E9}) and (\ref{eq:E10}).
With the use of Eq. (\ref{eq:N1}),
the Fisher information matrix given by
\begin{eqnarray}
g_{ij} &=& \left< \left( \frac{\partial \ln p(x)}{\partial \theta_i}\right)
\left( \frac{\partial \ln p(x)}{\partial \theta_j} \right) \right>,
\label{eq:N2}
\end{eqnarray}
is expressed by
\begin{eqnarray}
g_{\delta\delta} &=& \psi'[\delta(t)-1],
\label{eq:N3}\\
g_{\kappa \kappa} &=& \frac{[\delta(t)-1]^2}{\kappa(t)^2},
\label{eq:N4} \\
g_{\kappa \delta} &=& \frac{[\delta(t)-1]}{\kappa(t)}
\{ \psi[\delta(t)-1]-\psi[\delta(t)] \},
\label{eq:N5}
\end{eqnarray}
where $\psi(x)$ and $\psi'(x)$ are
di- and tri-gamma functions, respectively.
Figure \ref{figD} shows
the time-dependent inverse-gamma distribution $p(x,t)$
when an input pulse given by Eq. (\ref{eq:M1})
with $\Delta I=0.3$ and $I_b=0.2$ is applied
to the model A with $\lambda=1.0$, $\alpha=0.5$ and $\beta=0.0$.
Solid and dashed curves show the results of the moment method
and the PDEM, respectively.
The time dependences of $\mu(t)$ and $\sigma(t)^2$
and of Fisher information are plotted
in Fig. \ref{figE}(a) and (b), respectively,
where solid (dashed) curves express
the result of the moment method (PDEM).
By an applied pulse at $2.0 \leq t < 6.0$,
$\mu(t)$ and $\sigma(t)^2$ are increased and
the position of the PDF moves
rightward with an increased width.
An applied pulse increases $g_{\kappa \kappa}$
while it decreases $g_{\delta \kappa}$.
An interesting behavior is observed in $g_{\delta \delta}$
which is decreased at $t=2.0$ when a pulse is applied,
but afterward it seems to gradually reduce to the stationary value.
A similar behavior is realized in $g_{\delta \delta}$ also at $t \geq 6.0$
when the applied pulse is off.
In our previous paper \cite{Hasegawa08a},
we applied the $q$-moment approach to the model A,
deriving equations similar to Eqs. (\ref{eq:C6})-(\ref{eq:C9})
in the (normal) moment approach.
There are some differences between the $q$- and normal-moment approaches.
The stationary variance $\sigma_q^2$ in the $q$-moment approach
evaluated over the escort distribution is stable
for $0 \leq \alpha^2/\lambda < \infty $
whereas $\sigma^2$ in the normal-moment approach
is stable for $0 \leq \alpha^2/\lambda < 1.0 $
[Eqs. (\ref{eq:B9}), (\ref{eq:F8}) or (\ref{eq:H14})].
Although the time dependences of $\mu_q(t)$ and $\sigma(t)_q^2$
calculated by the $q$-moment approach are
similar to those of $\mu(t)$ and $\sigma(t)^2$ for a small
$ \alpha $, the difference between them becomes significant
for a large $ \alpha $ (see Fig. 13 in Ref. \cite{Hasegawa08a}).
These differences yield the quantitative difference in $p(x,t)$
calculated by the normal- and $q$-moment methods, although both
the methods lead to qualitatively similar results.
We note that in the limit of $\alpha=0.0$ ({\it i.e.,} additive noise only),
the dynamical solution given by the $q$- or normal-moment method reduces to
the Gaussian solution given by Eqs. (\ref{eq:C1})-(\ref{eq:C3}).
Thus the $q$- or normal-moment approach is a
generalization of the Gaussian solution to the FPE
with $\alpha \neq 0.0$ given by Eq. (\ref{eq:A3}).
In summary, by using the second-order moment method,
we have discussed the analytic time-dependent solution of
the FPE which includes additive and multiplicative noise
as well as external perturbations. It has been demonstrated that
dynamical PDFs calculated by the moment approach are in good agreement
with those obtained by the PDEM. Our moment method has some disadvantages.
The variance $\sigma^2$ diverges at $\alpha^2/\lambda \geq 1.0$,
for which our method cannot be applied.
If an applied perturbation induces a large $\mu(t)$ and/or $\sigma(t)^2$,
our method leads to poor results which are
not in good agreement to those calculated by the PDEM.
These are inherent in the moment approximation in which each moment
is required to be small.
Despite these disadvantages, however,
our moment method has following advantages:
(i) obtained dynamical solutions are compatible with
the exact stationary solutions in the Langevin models A, B and C,
(ii) it is useful for various subjects
in which analytical dynamical PDFs are indispensable
({\it e.g.,} Ref. \cite{Hasegawa08a}), and
(iii) the second-order moment approach is more tractable
than sophisticated methods \cite{Paola02}-\cite{Attar08}
for the FPE subjected to multiplicative noise.
As for the item (iii), it is possible to take account of contributions
from higher-order moments than second-order ones
with the use of Eq. (\ref{eq:A10}),
though actual calculations become tedious.
\section*{Acknowledgments}
This work is partly supported by
a Grant-in-Aid for Scientific Research from the Japanese
Ministry of Education, Culture, Sports, Science and Technology.
\newpage
|
2,869,038,155,696 | arxiv |
\section{Implementation Details}
\label{sec:implementation-details}
We train all models on a RTX8000 (48GB) GPU. The hyperparameters for pretraining and fine-tuning are given in Table ~\ref{tab:hyperparameters}. The only hyperparameter we sweeped over is learning rate (1e-5, 5e-5, 1e-4, 5e-4, 1e-3). The number of epochs were set to a large number with early stopping based on validation score. We've used Adafactor optimizer for all our experiments~\cite{adafactor}.
\begin{table}[ht]
\centering
\setlength{\tabcolsep}{2.5pt}
\begin{tabular}{cccccc}\toprule
Model & Dataset & LR & Epochs & BS \\
\midrule
T5 & \textsc{TeaBReaC}\xspace & $10^{-4}$ & 20 & 32 \\
Bart & \textsc{TeaBReaC}\xspace & $10^{-5}$ & 20 & 32 \\
NT5 & \textsc{TeaBReaC}\xspace & $10^{-3}$ & 20 & 32 \\
Preasm & \textsc{TeaBReaC}\xspace & $5\times10^{-5}$ & 20 & 32 \\
\midrule
T5 & DROP & $10^{-4}$ & 20 & 32 \\
Bart & DROP & $10^{-5}$ & 20 & 32 \\
NT5 & DROP & $10^{-3}$ & 40 & 32 \\
Preasm & DROP & $5\times10^{-5}$ & 20 & 32 \\
T5 & TAT-QA & $10^{-4}$ & 20 & 32 \\
Bart & TAT-QA & $10^{-5}$ & 20 & 32 \\
NT5 & TAT-QA & $10^{-3}$ & 40 & 32 \\
Preasm & TAT-QA & $5\times10^{-5}$& 20 & 32 \\
T5 & IIRC & $10^{-4}$ & 20 & 32 \\
Bart & IIRC & $10^{-5}$ & 20 & 32 \\
NT5 & IIRC & $10^{-3}$ & 40 & 32 \\
Preasm & IIRC & $5\times10^{-5}$ & 20 & 32 \\
\bottomrule
\end{tabular}
\caption{
(Top) Hyperparameters for pretraining language models on \textsc{TeaBReaC}\xspace. For large sized models (T5, Bart, Preasm), each epoch constitutes 100000/32=3125 steps. For small sized model (NT5), each epoch constitutes 1000000/32=31250 steps. For each step, we uniformly randomly sample a batch of \textsc{TeaBReaC}\xspace compositional (multihop) instances or primitive instance. (Bottom) Hyperparameters for training language models on target datasets from scratch or fine-tuning language models pretrained on \textsc{TeaBReaC}\xspace on target datasets. The hyperparameters for IIRC-gold and IIRC-retrieved experiments are the same. NT5 is a small-sized model, all others are large-sized. LR refers to learning rate and BS refers to batch size.}
\label{tab:hyperparameters}
\end{table}
\section{Examples of Multihop QA Instances}
Example multihop QA instances involving \texttt{project} and \texttt{boolean} primitives are given in Fig. ~\ref{fig:synthetic-instance-examples-apdx}.
\section{List of Primitives (Python Functions)}
List of primitives (python functions) and a corresponding example is given in Table ~\ref{tab:primitives}.
\section{Examples of Instances for Individual Primitives}
Examples of template based QA instances for teaching individual primitives are given in Table ~\ref{tab:primitives-examples}.
\begin{figure*}[ht]
\centering
\includegraphics[width=1\textwidth]{images/synthetic-instance-examples-apdx.pdf}
\caption{Synthetic reading comprehension QA instances involving \texttt{project} (top) and \texttt{boolean} (bottom) primitives.}
\label{fig:synthetic-instance-examples-apdx}
\end{figure*}
\fontsize{10}{10}\selectfont
\clearpage
\onecolumn
\renewcommand*{\arraystretch}{1.9}
\begin{longtable}{p{5.2cm}p{9.5cm}}
\caption{List of primitives (python functions) and a corresponding example.}\\
\toprule
\textbf{Primitive} & \textbf{Example} \\
\midrule
\endfirsthead
\multicolumn{2}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\toprule
\textbf{Primitive} & \textbf{Example} \\
\midrule
\endhead
\hline \multicolumn{2}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
compare\_numbers &
\begin{minipage}{8cm}
compare\_numbers(\#1, \#2, ``>") $\Rightarrow$ False \\ \\
\textbf{State}: \\
\#1: 25 \\
\#2: 28
\end{minipage} \\
\midrule
compare\_dates &
\begin{minipage}{8cm}
compare\_dates(\#1, \#2, ``>") $\Rightarrow$ False \\ \\
\textbf{State}: \\
\#1: 25 Jan 2012 \\
\#2: 28 Jan 2012
\end{minipage} \\
\midrule
maximum\_date &
\begin{minipage}{8cm}
maximum\_date([\#1, \#2]) $\Rightarrow$ 28 Jan 2012 \\ \\
\textbf{State}: \\
\#1: 25 Jan 2012 \\
\#2: 28 Jan 2012
\end{minipage} \\
\midrule
minimum\_date &
\begin{minipage}{8cm}
minimum\_date([\#1, \#2]) $\Rightarrow$ 25 Jan 2012 \\ \\
\textbf{State}: \\
\#1: 25 Jan 2012 \\
\#2: 28 Jan 2012
\end{minipage} \\
\midrule
date\_subtraction &
\begin{minipage}{8cm}
date\_subtraction(\#1, \#2, ``days") $\Rightarrow$ 3 \\ \\
\textbf{State}: \\
\#1: 25 Jan 2012 \\
\#2: 28 Jan 2012
\end{minipage} \\
\midrule
arg\_maximum\_date &
\begin{minipage}{8cm}
arg\_maximum\_date([\#1, \#2]) $\Rightarrow$ \#2 \\ \\
\textbf{State}: \\
\#1: 25 Jan 2012 \\
\#2: 28 Jan 2012
\end{minipage} \\
\midrule
arg\_minimum\_date &
\begin{minipage}{8cm}
arg\_minimum\_date([\#1, \#2]) $\Rightarrow$ \#1 \\ \\
\textbf{State}: \\
\#1: 25 Jan 2012 \\
\#2: 28 Jan 2012
\end{minipage} \\
\midrule
arg\_bool &
\begin{minipage}{8cm}
arg\_bool([\#1, \#2], ``true") $\Rightarrow$ \#1 \\ \\
\textbf{State}: \\
\#1: True \\
\#2: False
\end{minipage} \\
\midrule
count &
\begin{minipage}{8cm}
count(\#1) $\Rightarrow$ 3 \\ \\
\textbf{State}: \\
\#1: [ABC, XZE, PQR]
\end{minipage} \\
\midrule
addition &
\begin{minipage}{8cm}
addition(\#1) $\Rightarrow$ 2657.3 \\ \\
\textbf{State}: \\
\#1: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
subtraction &
\begin{minipage}{8cm}
subtraction(100, \#1): 75 \\ \\
\textbf{State}: \\
\#1: 25
\end{minipage} \\
\midrule
multiplication &
\begin{minipage}{8cm}
multiplication(\#1, 5): 125 \\ \\
\textbf{State}: \\
\#1: 25
\end{minipage} \\
\midrule
division &
\begin{minipage}{8cm}
division(\#1, 100): 254.2 \\ \\
\textbf{State}: \\
\#1: 25420
\end{minipage} \\
\midrule
mean &
\begin{minipage}{8cm}
mean(\#1) $\Rightarrow$ 885.8 \\ \\
\textbf{State}: \\
\#1: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
maximum\_number &
\begin{minipage}{8cm}
maximum\_number(\#1) $\Rightarrow$ 2564.2 \\ \\
\textbf{State}: \\
\#1: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
minimum\_number &
\begin{minipage}{8cm}
minimum\_number(\#1) $\Rightarrow$ 3 \\ \\
\textbf{State}: \\
\#1: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
arg\_maximum\_number &
\begin{minipage}{8cm}
arg\_maximum\_number([\#1, \#2, \#3]) $\Rightarrow$ \#2 \\ \\
\textbf{State}: \\
\#1: 3 \\
\#2: 2564.2 \\
\#3: 90.1
\end{minipage} \\
\midrule
arg\_minimum\_number &
\begin{minipage}{8cm}
arg\_minimum\_number([\#1, \#2, \#3]) $\Rightarrow$ \#1 \\ \\
\textbf{State}: \\
\#1: 3 \\
\#2: 2564.2 \\
\#3: 90.1
\end{minipage} \\
\midrule
kth\_highest &
\begin{minipage}{8cm}
kth\_highest(\#1, 2) $\Rightarrow$ 90.1 \\ \\
\textbf{State}: \\
\#1: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
kth\_lowest &
\begin{minipage}{8cm}
kth\_lowest(\#1, 2) $\Rightarrow$ 90.1 \\ \\
\textbf{State}: \\
\#1: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
are\_items\_same &
\begin{minipage}{8cm}
are\_items\_same(\#1, \#2) $\Rightarrow$ False \\ \\
\textbf{State}: \\
\#1: ABC \\
\#2: EDX
\end{minipage} \\
\midrule
are\_items\_different &
\begin{minipage}{8cm}
are\_items\_different(\#1, \#2) $\Rightarrow$ True \\ \\
\textbf{State}: \\
\#1: ABC \\
\#2: EDX
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_max\_num &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_max\_num(\#1, \#2) $\Rightarrow$ PQR \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_min\_num &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_min\_num(\#1, \#2) $\Rightarrow$ ABC \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_given\_value &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_given\_value(\#1, \#2, MNO) $\Rightarrow$ ABC \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [MNO, XER, OIY]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_compared\_to &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_compared\_to(\#1, \#2, 80, >) $\Rightarrow$ [PQR, MNZ] \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_in\_range &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_in\_range\_num(\#1, \#2, 80, 100) $\Rightarrow$ [MNZ] \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [3, 2564.2, 90.1]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_compared\_to\_date &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_compared\_to\_date(\#1, \#2, 25 Feb 2012, >) $\Rightarrow$ [PQR, MNZ] \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [25 Jan 2012, 18 March 2012, 13 Oct 2019]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_in\_range\_date &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_in\_range\_date(\#1, \#2, 25 Feb 2012, 1 Nov 2021, 100) $\Rightarrow$ [PQR, MNZ] \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [25 Jan 2012, 18 March 2012, 13 Oct 2019]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_max\_date &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_max\_date(\#1, \#2) $\Rightarrow$ MNZ \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [25 Jan 2012, 18 March 2012, 13 Oct 2019]
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_min\_date &
\begin{minipage}{8cm}
filter\_a\_where\_b\_is\_min\_date(\#1, \#2) $\Rightarrow$ ABC \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNZ] \\
\#2: [25 Jan 2012, 18 March 2012, 13 Oct 2019]
\end{minipage} \\
\midrule
grouped\_count &
\begin{minipage}{8cm}
grouped\_count(\#1, \#2) $\Rightarrow$ {ABC: 2, XYI: 2, PQR: 1} \\ \\
\textbf{State}: \\
\#1: [ABC, XYI, ABC, PQR, XYI] \\
\#2: [UIQ, QWA, OUE, UHI, RVC]
\end{minipage} \\
\midrule
grouped\_sum &
\begin{minipage}{8cm}
grouped\_sum(\#1, \#2) $\Rightarrow$ {ABC: 4, XYI: 7, PQR: 4} \\ \\
\textbf{State}: \\
\#1: [ABC, XYI, ABC, PQR, XYI] \\
\#2: [1, 2, 3, 4, 5]
\end{minipage} \\
\midrule
grouped\_mean &
\begin{minipage}{8cm}
grouped\_mean(\#1, \#2) $\Rightarrow$ {ABC: 2, XYI: 3.5, PQR: 4} \\ \\
\textbf{State}: \\
\#1: [ABC, XYI, ABC, PQR, XYI] \\
\#2: [1, 2, 3, 4, 5]
\end{minipage} \\
\midrule
union &
\begin{minipage}{8cm}
union(\#1, \#2, \#3) $\Rightarrow$ [ABC, PQR, MNO, JHI, KMR] \\ \\
\textbf{State}: \\
\#1: [ABC, PQR] \\
\#2: [MNO] \\
\#3: [JHI, KMR]
\end{minipage} \\
\midrule
intersection &
\begin{minipage}{8cm}
intersection(\#1, \#2) $\Rightarrow$ [PQR] \\ \\
\textbf{State}: \\
\#1: [ABC, PQR, MNO] \\
\#2: [PQR]
\end{minipage} \\
\midrule
arg\_intersection &
\begin{minipage}{8cm}
arg\_intersection(\#1, \#2, \#3) $\Rightarrow$ [WEC] \\ \\
\textbf{State}: \\
\#1: [XYI, ORE, WEC] \\
\#2: [ABC, PQR, MNO] \\
\#3: [null, null, MNO]
\end{minipage} \\
\midrule
list\_subtraction &
\begin{minipage}{8cm}
list\_subtraction(\#1, \#2) $\Rightarrow$ [XYI, WEC] \\ \\
\textbf{State}: \\
\#1: [XYI, ORE, WEC] ;
\#2: [ORE]
\end{minipage} \\
\midrule
logical\_and &
\begin{minipage}{8cm}
logical\_and(\#1, \#2) $\Rightarrow$ False \\ \\
\textbf{State}: \\
\#1: False ;
\#2: True
\end{minipage} \\
\midrule
logical\_or &
\begin{minipage}{8cm}
logical\_or(\#1, \#2) $\Rightarrow$ True \\ \\
\textbf{State}: \\
\#1: False ;
\#2: True
\end{minipage} \\
\midrule
select &
\begin{minipage}{8cm}
select("touchdowns by Edwards") $\Rightarrow$ [ABC, DXE, FGH] \\ \\
\textbf{Facts in context}: \\
touchdowns by Edwards $\Rightarrow$ ABC \\
touchdowns by Edwards $\Rightarrow$ DXE \\
touchdowns by Edwards $\Rightarrow$ FGH
\end{minipage} \\
\midrule
filter &
\begin{minipage}{8cm}
filter(``\#1 from 1st quarter") $\Rightarrow$ [ABC, DXE] \\ \\
\textbf{State}: \\
\#1: [ABC, DXE] \\ \\
\textbf{Facts in context}: \\
what is from 1st quarter? $\Rightarrow$ ABC \\
what is from 1st quarter? $\Rightarrow$ DXE \\
what is from 1st quarter? $\Rightarrow$ MNF \\
what is from 1st quarter? $\Rightarrow$ IOU
\end{minipage} \\
\midrule
project &
\begin{minipage}{8cm}
project(``when \#1 died") $\Rightarrow$ [March 22, 1958] \\ \\
\textbf{State}: \\
\#1: [MNS] \\ \\
\textbf{Facts in context}: \\
when PYS died $\Rightarrow$ March 22, 1958 \\
when MNS died $\Rightarrow$ March 22, 1958 \\
when QFY died $\Rightarrow$ August 16, 1533
\end{minipage} \\
\midrule
boolean &
\begin{minipage}{8cm}
boolean(``if Aikmen started the game at quarterback for the cowboys") $\Rightarrow$ True \\ \\
\textbf{Facts in context}: \\
Aikmen started the game at quarterback for the cowboys
\end{minipage}
\label{tab:primitives}
\end{longtable}
\clearpage
\twocolumn
\fontsize{10}{10}\selectfont
\clearpage
\onecolumn
\renewcommand*{\arraystretch}{1.9}
\begin{longtable}{p{5.2cm}p{9.5cm}}
\caption{Examples QA instances for individual primitives (python functions)}\\
\toprule
\textbf{Primitive} & \textbf{Example} \\
\midrule
\endfirsthead
\multicolumn{2}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\toprule
\textbf{Primitive} & \textbf{Example} \\
\midrule
\endhead
\hline \multicolumn{2}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
compare\_numbers &
\begin{minipage}{8cm}
\textbf{Quesion:} Is 984,486.24 greater than 594147.75? \\
\textbf{Context:} \\
\textbf{Answer :} ['yes']
\end{minipage} \\
\midrule
compare\_dates &
\begin{minipage}{8cm}
\textbf{Quesion:} Is 1934-9-4 greater than 27 May 1899? \\
\textbf{Context:} \\
\textbf{Answer :} ['yes']
\end{minipage} \\
\midrule
maximum\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} Which of the following dates come later? \\
\textbf{Context:} 11/30/1690 , 1690-05-17 \\
\textbf{Answer :} ['November 30, 1690']
\end{minipage} \\
\midrule
minimum\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} Which of the following dates come before the other? \\
\textbf{Context:} 1925-4-12 , 18 Apr 1696 \\
\textbf{Answer :} ['April 18, 1696']
\end{minipage} \\
\midrule
date\_subtraction &
\begin{minipage}{8cm}
\textbf{Quesion:} How many days passed between 1567-6-29 and May 28, 1567? \\
\textbf{Context:} \\
\textbf{Answer :} ['32']
\end{minipage} \\
\midrule
arg\_maximum\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} Which event has highest date: OUM or NKE? \\
\textbf{Context:} Event OUM has date 1977-3-13. Event NKE has date November, 5 2011. \\
\textbf{Answer :} ['NKE']
\end{minipage} \\
\midrule
arg\_minimum\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} Which event happened earliest: KSX or KBO or JJT? \\
\textbf{Context:} Event KSX has date 11/9/1705. Event KBO has date 04 Jul, 1786. Event JJT has date 04/11/1729. \\
\textbf{Answer :} ['KSX']
\end{minipage} \\
\midrule
count &
\begin{minipage}{8cm}
\textbf{Quesion:} How many total entities the following list has? \\
\textbf{Context:} DMX NQX LFD RJN AMG \\
\textbf{Answer :} ['5']
\end{minipage} \\
\midrule
addition &
\begin{minipage}{8cm}
\textbf{Quesion:} Given the list of numbers, give their total sum. \\
\textbf{Context:} 977.98 ; 710 ; seven ; 4.72 \\
\textbf{Answer :} ['1699.7']
\end{minipage} \\
\midrule
subtraction &
\begin{minipage}{8cm}
\textbf{Quesion:} What is 721,251 - 32561? \\
\textbf{Context:} \\
\textbf{Answer :} ['688690']
\end{minipage} \\
\midrule
multiplication &
\begin{minipage}{8cm}
\textbf{Quesion:} If you multiply forty-eight with 41, what do you get? \\
\textbf{Context:} \\
\textbf{Answer :} ['1968']
\end{minipage} \\
\midrule
division &
\begin{minipage}{8cm}
\textbf{Quesion:} What is 47 divided by 6 in nearest integer? \\
\textbf{Context:} \\
\textbf{Answer :} ['7']
\end{minipage} \\
\midrule
mean &
\begin{minipage}{8cm}
\textbf{Quesion:} What is the average of the following numbers in nearest integer? \\
\textbf{Context:} 172 ; 691 \\
\textbf{Answer :} ['431']
\end{minipage} \\
\midrule
maximum\_number &
\begin{minipage}{8cm}
\textbf{Quesion:} Given the following list, what is the largest number? \\
\textbf{Context:} 6603 ; 3.76 ; 636,337.65 ; 91.72 \\
\textbf{Answer :} ['636337.65']
\end{minipage} \\
\midrule
minimum\_number &
\begin{minipage}{8cm}
\textbf{Quesion:} What is the smallest of the following numbers? \\
\textbf{Context:} 60,810.74 ; 2.24 ; 48.8 \\
\textbf{Answer :} ['2.24']
\end{minipage} \\
\midrule
arg\_maximum\_number &
\begin{minipage}{8cm}
\textbf{Quesion:} Which entity has biggest value: ROJ or ZZH or KFI? \\
\textbf{Context:} Entity ROJ has value 91,889. Entity ZZH has value 0.93. Entity KFI has value 9,223.7. \\
\textbf{Answer :} ['ROJ']
\end{minipage} \\
\midrule
arg\_minimum\_number &
\begin{minipage}{8cm}
\textbf{Quesion:} Which entity has lowest value: TXM or KPG or JLD? \\
\textbf{Context:} Entity TXM has value 195.35. Entity KPG has value 861878. Entity JLD has value 41. \\
\textbf{Answer :} ['JLD']
\end{minipage} \\
\midrule
kth\_highest &
\begin{minipage}{8cm}
\textbf{Quesion:} Give the 2nd maximum value of \#17? \\
\textbf{Context:} \#17 has values 20787.56, 8265.18. \#9 has values January 25, 1787, January 27, 1787, January 08, 1787, January 18, 1787. \#3 has values February 14, 1994. \#18 has values 3.47, 4692.13, 735.31. \\
\textbf{Answer :} ['8265.18']
\end{minipage} \\
\midrule
kth\_lowest &
\begin{minipage}{8cm}
\textbf{Quesion:} Which is the 3rd lowest value of \#1? \\
\textbf{Context:} \#7 has values July 24, 1506, July 04, 1506, July 02, 1506, July 15, 1506. \#1 has values 2, 9, 23866. \#11 has values KFI, DXK, TFM. \\
\textbf{Answer :} ['23866']
\end{minipage} \\
\midrule
are\_items\_same &
\begin{minipage}{8cm}
\textbf{Quesion:} Are the following entities the same? \\
\textbf{Context:} Jan 07, 1696 and 01-7-1696. \\
\textbf{Answer :} ['yes']
\end{minipage} \\
\midrule
are\_items\_different &
\begin{minipage}{8cm}
\textbf{Quesion:} Are the following entities different? \\
\textbf{Context:} HUU and 09-29-1771. \\
\textbf{Answer :} ['yes']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_max\_num &
\begin{minipage}{8cm}
\textbf{Quesion:} What entity has biggest value? \\
\textbf{Context:} Entity OGQ has value 59. Entity HDU has value 94. Entity KLM has value 28,742. Entity LGV has value 713. Entity KGH has value 701. Entity DXK has value 373. \\
\textbf{Answer :} ['KLM']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_min\_num &
\begin{minipage}{8cm}
\textbf{Quesion:} Which entity has the minimum value? \\
\textbf{Context:} Entity FYO has value 266. Entity XHY has value 199052. Entity EQO has value 534. \\
\textbf{Answer :} ['FYO']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_given\_value &
\begin{minipage}{8cm}
\textbf{Quesion:} Which entities with value equal to 6.45? \\
\textbf{Context:} Entity KSX has value 6.45. Entity NLV has value 887.41. Entity OJP has value 603145.31. \\
\textbf{Answer :} ['KSX']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_compared\_to &
\begin{minipage}{8cm}
\textbf{Quesion:} Entities that have value larger than 948768.92? \\
\textbf{Context:} Entity AFE has value 871781. Entity RQX has value 989,517.24. \\
\textbf{Answer :} ['RQX']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_compared\_to\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} List the entities with date below Jul 20 1646? \\
\textbf{Context:} Entity ZBK has date 9-12-1560. Entity AGU has date July 17 1953. \\
\textbf{Answer :} ['ZBK']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_max\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} Which entity has latest date? \\
\textbf{Context:} Entity SML has value 11-28-1882. Entity PYS has value Nov 19 1882. \\
\textbf{Answer :} ['SML']
\end{minipage} \\
\midrule
filter\_a\_where\_b\_is\_min\_date &
\begin{minipage}{8cm}
\textbf{Quesion:} What entity has least recent date? \\
\textbf{Context:} Entity SDA has value 5 March, 1523. Entity HXJ has value 14 March 1523. Entity RZO has value 1-26-1523. Entity ZMH has value 23 Jul, 1523. \\
\textbf{Answer :} ['RZO']
\end{minipage} \\
\midrule
grouped\_count &
\begin{minipage}{8cm}
\textbf{Quesion:} How many times do each of EBC, HNQ occur in \#14? \\
\textbf{Context:} \#14 has HNQ, EBC, HNQ. \#3 has OZB, LNW, LYP, AGU, HVP, SDA. \#17 has ULN, ZZH, RZO \\
\textbf{Answer :} ['1', '2']
\end{minipage} \\
\midrule
grouped\_sum &
\begin{minipage}{8cm}
\textbf{Quesion:} What are the addition of values for each of QWU, JLD? \\
\textbf{Context:} QWU has value 179541.17. JLD has value 6,641.78. JLD has value 3.15. QWU has value 6,053.93. QWU has value 44,251.33. JLD has value 411.83. \\
\textbf{Answer :} ['229846.43', '7056.76']
\end{minipage} \\
\midrule
grouped\_mean &
\begin{minipage}{8cm}
\textbf{Quesion:} For each of TKR, NLV, what are the mean of values in integers? \\
\textbf{Context:} TKR has value 929. TKR has value 737. TKR has value ninety-five. NLV has value 928. \\
\textbf{Answer :} ['587', '928']
\end{minipage} \\
\midrule
union &
\begin{minipage}{8cm}
\textbf{Quesion:} Give answer union of \#20, \#12, \#13? \\
\textbf{Context:} \#20 has answer 29.77. \#12 has answer KBE. \#11 has answer June 10, 1701. \#13 has answer January 23, 1503. \\
\textbf{Answer :} ['29.77', 'KBE', 'January 23, 1503']
\end{minipage} \\
\midrule
intersection &
\begin{minipage}{8cm}
\textbf{Quesion:} List the entities that occur in both \#10 and \#7? \\
\textbf{Context:} \#1 has entities ICU, WAT. \#10 has entities WAT, ICU. \#7 has entities WAT, ICU. \\
\textbf{Answer :} ['ICU', 'WAT']
\end{minipage} \\
\midrule
arg\_intersection &
\begin{minipage}{8cm}
\textbf{Quesion:} List the entities contain values common in both \#9 and \#20? \\
\textbf{Context:} Entity KBE has value UJI for \#20. Entity KLM has value ARU for \#20. Entity KBE has no value for \#9. Entity KLM has value ARU for \#9. \\
\textbf{Answer :} ['KLM']
\end{minipage} \\
\midrule
logical\_and &
\begin{minipage}{8cm}
\textbf{Quesion:} What is logical AND of the given booleans? \\
\textbf{Context:} True False \\
\textbf{Answer :} ['no']
\end{minipage} \\
\midrule
logical\_or &
\begin{minipage}{8cm}
\textbf{Quesion:} What is logical OR of the given booleans? \\
\textbf{Context:} False False \\
\textbf{Answer :} ['no']
\end{minipage} \\
\label{tab:primitives-examples}
\end{longtable}
\clearpage
\twocolumn
\section{Conclusions}
Large language models demonstrate impressive reading comprehension abilities and a wide variety of reasoning skills. Despite these abilities and the availability of large scale multihop QA datasets, large LM-based QA models do not reliably learn to use such reasoning skills for answering complex questions.
In this work, we show that the greater control that synthetic contexts offer can be leveraged to create a teaching dataset where models can learn a broad range of reasoning skills in a reliable manner, especially for more complex questions. Our transfer results on actual QA datasets also add to the line of work that shows synthetic datasets can be used to inject useful skills that transfer over to real natural language tasks. Given the artifact issues in real datasets (specifically, in their contexts) and the difficulty in controlling for them via perturbations,
leveraging existing multihop questions for their broad reasoning patterns but using synthetic contexts appears to be a viable alternative for carefully constructing teaching datasets, where models can learn the \textit{right way} to reason.
\section{\textsc{TeaBReaC}\xspace Dataset Construction}
\label{sec:construction}
The overview of \textsc{TeaBReaC}\xspace construction pipeline is shown in Fig.~\ref{fig:pipeline}. We discuss the QA instance generator in \S~\ref{subsec:instance-generator}, and discuss the dataset generator used to create QA dataset in \S~\ref{subsec:dataset-generator}.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{images/teabreac-generator-overview.pdf}
\caption{High-level schematic of \textsc{TeaBReaC}\xspace construction process.
}
\label{fig:pipeline}
\end{figure}
\subsection{Instance Generator}
\label{subsec:instance-generator}
The Instance Generator takes a question $Q$ and its QDMR decomposition $D$ as input, and generates a synthetic context $C$ and the corresponding answer $A$ as its output. The tuple ($Q$, $C$, $A$) is the generated reading comprehension QA instance. This conversion happens in two steps: (i) QDMR to Typed Program, (ii) Typed Program to Context and Answer.
\subsubsection*{QDMR to Typed Program:}
Our goal is to generate a synthetic context $C$ that can be used to answer the question $Q$ (based on the QDMR $D$), and to also provide the answer $A$. To be able to generate $C$ and $A$, we must be able to create facts corresponding to each step in the QDMR reasoning graph (i.e., ground the QDMR predicates\footnote{E.g., the step \texttt{``return players who kicked \#1"} has the predicate \texttt{``return players who kicked \_\_"}.}) and compute the final answer by stepping through the QDMR program.
To achieve this, we need a formal representation (a \textbf{Program}) that captures the precise reasoning implied by the QDMR decomposition $D$, and that can be executed step-by-step (e.g., in a programming language such as Python). This isn't possible directly via QDMRs as (i) although structured, they are written in natural language and have variation inherent in natural language;
(ii) they don't have input and output type information, e.g., it is unclear whether the \texttt{project} operator should generate a dictionary, a list, or a scalar, making it difficult to have the full program be executable.
To convert a QDMR $D$ into a Program $P$, we define a set of functions (in Python) such as \texttt{select}, \texttt{filter}, \texttt{grouped\_count}, etc, and then parse QDMRs into these functions using rules and heuristics. An example conversion is shown in Fig.~\ref{fig:qdmr-to-program}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{images/qdmr-to-program.pdf}
\caption{Example conversion of a QDMR decomposition (top) to a Typed Program (bottom).}
\label{fig:qdmr-to-program}
\end{figure}
In our representation, we have 44 Python functions (primitives) operating over various types (number, date, named entity) and structures (scalar, list, dictionary) of inputs and outputs. The full list is given in Appendix~\ref{tab:primitives}.
Note that these primitives don't always have a clearly defined output type. While in most cases the output type is obvious (e.g., \texttt{arithmetic\_sum} returns a number), for some of these primitives (\texttt{select}, \texttt{project}, \texttt{filter}), it's under-defined.
E.g., \texttt{select("number of soldiers in USA")} should output a single number, \texttt{select("when did India get independence")} should output a single date, and \texttt{select("countries surrounding India")} should output a list of named entities. For such operations, we again use heuristic rules and type propagation on the global structure of $P$ to infer expected types and structures of output. We call the program with type information inferred for each step a \textbf{Typed Program} $\tilde{P}$, an example of which is shown in Fig.~\ref{fig:qdmr-to-program}.
\subsubsection*{Synthetic Context + Answer:}
Next, we generate $C$ and $A$ from the typed program $\tilde{P}$. We first describe the construction at a high level, then discuss an example generation in the context of the desirable properties we want our QA instances to have, and finally describe the general construction algorithm.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{images/synthetic-instance-example.pdf}
\caption{A simplified example of a QA instance in \textsc{TeaBReaC}\xspace, with a (simplified) real question from the DROP dataset and the synthetic context we construct for it using the question's 3-step decomposition. The instance satisfies desirable properties \textbf{P1}, \textbf{P2}, and \textbf{P3}, and thus helps robustly teach multihop reasoning skills.}
\label{fig:synthetic-instance-example}
\end{figure*}
We generate $C$ by grounding the \textbf{predicates} derived from the QDMR $D$ with random \textbf{entities}. Fig.~\ref{fig:synthetic-instance-example} shows an example of $C$ for a simple program with three steps. \textbf{Predicates}: The predicates that need to be grounded can belong to 4 of the 44 operators (Table~\ref{tab:primitives}), which we refer to as grounding operators: \texttt{select}, \texttt{project}, \texttt{filter}, \texttt{boolean}. E.g., Fig.~\ref{fig:synthetic-instance-example} uses \texttt{select} and \texttt{filter}. Examples involving \texttt{project} and \texttt{boolean} are shown in Appendix~\ref{fig:synthetic-instance-examples-apdx}. \textbf{Entities:} The grounded entities can be of 3 types: number, date, or named entity.
A random number can be anywhere from 0 to 1 million, a random date can be anywhere from year 1100 to 2022, and a random named entity can be any sequence of 3 letters (e.g., \texttt{ADC}).
Since our programs are typed, we know which predicate should be grounded with which entity type. E.g., \texttt{select("number of soldiers in USA")} should be grounded using a number, whereas \texttt{select("countries surrounding India)"} should be grounded with a list of named entities.
\paragraph{Minimizing reasoning shortcuts.} Naively creating $C$ using QDMR $D$ can introduce simple shortcuts that models can exploit, thus circumventing the necessary reasoning. Note that QDMR is a sequence of steps where each step $s_i$ can use answers from zero or more previous steps; e.g., \texttt{``return number \#2 where \#3 is least goals"} in Fig~\ref{fig:qdmr-to-program} (top) uses the answer from step \#2 and \#3. However, if there is only one player who scored field goals, all reasoning steps can be ignored. To ensure models do learn the intended reasoning, our goal is to create $C$ such that one cannot bypass the intended reasoning (or program) steps and still arrive at the correct answer $A$. To this end, we ground the predicates with entities such that the following three properties hold:
\paragraph{P1: Answers to dependent steps can't be ignored.}
If step $s_j$ is dependent on step $s_i$, then the answer to $s_j$ can't be identified without knowing the answer to $s_i$. E.g., in Fig~\ref{fig:synthetic-instance-example}, step \#2 is asking about \textit{``1st quarter"}, in particular it's asking \textit{``which of the touchdowns by Edward are from the first quarter"}. Since there are many touchdowns
\textit{``from the 1st quarter"}, and only some of them are \textit{``touchdowns by Edward"} (indicated in \textcolor{blue}{blue}), one cannot narrow down the answer to step \#2 without knowing step \#1's answer. We ensure this property holds for different operators in different ways. E.g., for \texttt{filter} operator, we ensure the answer to the step is always a \textit{proper} subset of all the entities grounded with that predicate (\{ABC, DXE\} $\subset$ \{ABC, DXE, MNF, IOU\} in Fig.~\ref{fig:synthetic-instance-example}).
\paragraph{P2: Steps can't be no-op.} The input and output of any step cannot be the same, as otherwise the reasoning in that step can be bypassed. E.g., in Fig~\ref{fig:synthetic-instance-example}, step \#2 is asking about \textit{``1st quarter"}, in particular it's asking \textit{``which of the touchdowns by Edwards are from the 1st quarter"}. Again, there are many \textit{``touchdowns by Edwards"}, and only some of them are \textit{``from the 1st quarter"} (indicated in \textcolor{blue}{blue}). Therefore, ignoring step \#2 (i.e., treating it as a no-op) would result in an incorrect answer being used for subsequent steps.
Similar to the first property, we again ensure this property holds for different operators in different ways. E.g., for \texttt{filter} operator, we ensure the answer to the step is always a \textit{proper} subset of the answer to the dependent step (\{ABC, DXE\} $\subset$ \{ABC, DXE, FGH, PQR\} in Fig.~\ref{fig:synthetic-instance-example}).
\paragraph{}
Properties \textbf{P1} and \textbf{P2} describe the upper half of the synthetic context in Fig.~\ref{fig:synthetic-instance-example}, that is, facts pertaining to the gold reasoning chain.
Although this ensures step-by-step execution will result in the gold answer, there is only one possible complete execution that leads to an answer. As a result, the question can be completely ignored. To fix this, we introduce a third property:
\paragraph{P3: Context also supports a different answer to a contrastive question.} Just the way we generate facts corresponding to the gold chain of reasoning (upper half in Fig.~\ref{fig:synthetic-instance-example}), we also generate facts corresponding to a distractor chain (lower half of Fig.~\ref{fig:synthetic-instance-example}), potentially using perturbed predicates (e.g., Edwards $\Rightarrow$ Tom, 1st $\Rightarrow$ 2nd). This ensures there is always one minimally different (contrastive~\cite{contrastsets}) question that results in a different answer in the same context. E.g., \textit{``How many touchdowns did Tom throw in the 2nd quarter"} results in the answer \textcolor{red}{1}, which is different from the gold answer \textcolor{teal}{2} in Fig.~\ref{fig:synthetic-instance-example}). To generate predicate perturbations, we swap numbers, dates, and named entities (PERSON, ORG, etc) with a similar entity of the same type. The cases where predicate doesn't have any entity of these types, we use a similar but different and type-consistent predicate from a different question as a perturbed predicate. E.g., \texttt{yards of rushing touchdowns} could be perturbed to \texttt{yards of passing touchdowns}. To do this, we retrieve the top 30 type-consistent predicates with the highest word-overlap not exceeding 75\%, and sample one at random.
\paragraph{General Algorithm.}
Algorithm \ref{alg:instance_gen} shows the pseudo-code for generating QA instances satisfying the properties mentioned above.
The \texttt{GenQAInstance} function takes question Q, QDMR D and expected answer cardinality \texttt{N} of the answer, and attempts to generate a QA instance with desirable properties for 200 maximum tries. For a given question, QDMR pair, we vary \texttt{N} $\in \{1,2,3,4\}$.
The \texttt{facts} represent list of grounded predicates that form the context, \texttt{state.ans} represents stepwise answers for gold reasoning chain (e.g., green boxes in Fig.~\ref{fig:synthetic-instance-example}), and \texttt{state.dis} represents stepwise answers for distractor reasoning chain (e.g., red boxes in Fig.~\ref{fig:synthetic-instance-example}). These are initialized to $\emptyset$(\texttt{L3}) and updated during the instance generation.
To construct a QA instance, we iterate through the program (or QDMR) steps. For each step, we create facts for the gold reasoning chain by grounding the predicate in the QDMR and update the facts and answer state accordingly using the \texttt{execute} function. e.g., In step \#2 in Fig.~\ref{fig:synthetic-instance-example}, the facts in the top-half are added and $\{$ABC, DXE$\}$ is marked as the current answer state.
The \texttt{execute} function will generate these facts and answers such that P1 and P2 are satisfied or return False if it can't. We similarly generate facts and update the state for the distractor reasoning chain (\texttt{L7)} by using a perturbed (\texttt{L6}) QDMR predicate (e.g., Edward $\Rightarrow$ Tom, 1st $\Rightarrow$ 2nd in Fig.~\ref{fig:synthetic-instance-example}). This generates the facts and reasoning chain shown in the lower half of Fig.~\ref{fig:synthetic-instance-example} ensuring P3 is satisfied.
The implementation of
\texttt{execute} function is dependent on the program primitives (Table~\ref{tab:primitives}) and will be provided in the released code~\url{https://github.com/stonybrooknlp/teabreac}. But broadly speaking there are two classes of primitives: (1) primitives like \texttt{select} and \texttt{filter} that need to first add facts by grounding the predicate, and then update the answer state for that step (e.g., step \#1 and \#2 in Fig.~\ref{fig:synthetic-instance-example}) (2) primitives like count with no additional grounding of facts and only need to update the state based on the underlying computation (e.g., step \#3 in Fig.~\ref{fig:synthetic-instance-example}).
\begin{algorithm}[t!]
\caption{Pseudo-code for generating QA instances from question Q, QDMR D, and answer cardinality N}
\label{alg:instance_gen}
\begin{footnotesize}
\begin{algorithmic}[1]
\renewcommand{\algorithmicindent}{1em}
\renewcommand{\algorithmiccomment}[1]{\hfill$ \triangleright$ \textit{#1}}
\Function{GenQAInstance}{
Q, D, N}
\For{$1 \leq i \leq 200$} \Comment{Max retries}
\State state.ans, state.dis, facts $\gets$ $\emptyset$
\For{step $\in$ qdmr.steps}
\State ans\_succ $\gets$ execute(step, state.ans, facts) \Comment{Update for gold reasoning chain}
\State maybe\_perturb(step) \Comment{Perturb predicate for distractor chain}
\State dis\_succ $\gets$ execute(step, state.dis, facts)
\Comment{Update for distractor reasoning chain}
\If{\textbf{not} ans\_succ \textbf{or} \textbf{not} dis\_succ}
\State failed $\gets$ True
\State break
\EndIf
\EndFor
\If{(\textbf{not} failed \textbf{and} \\
\hspace{3em} accept(state, facts, ans\_num))}
\State return QA(Q, \Comment{question} \\
\hspace{7em} facts, \Comment{context}\\
\hspace{7em} state.ans[-1]) \Comment{gold answer}
\EndIf
\EndFor
\EndFunction
\end{algorithmic}
\end{footnotesize}
\end{algorithm}
If all the steps finish with success, we check if the generation is \texttt{acceptable} (\texttt{L14}) before creating a QA instance. For it to be acceptable, the generated answer cardinality must match the expected value, the number of facts must be within 25, and the final answer for gold and distractor reasoning chains must be different. We create a reading comprehension QA instance with the input question Q as question, facts as the context (concatenated after shuffling), and the answer at the final step as the gold answer.
\subsection{Dataset Generator}
\label{subsec:dataset-generator}
Now that we have a way to generate QA instance from a (question, QDMR) pair, we can generate a dataset by just using questions from datasets with annotated QDMRs. However, we find that the natural distribution of the \emph{reasoning patterns} in these datasets is extremely long-tailed. We define \textbf{reasoning pattern} as a unique sequence of primitives present in the program. e.g., program in Fig.~\ref{fig:synthetic-instance-example} has 3 steps having \texttt{select}, \texttt{filter} and \texttt{count} primitives, in that order, so the reasoning pattern is \texttt{``select filter count"}
If we invoke instance generator uniformly over the available QDMRs, we get a QA dataset that is extremely skewed towards popular patterns. Pre-training models on such a dataset overfits it only on few reasoning patterns and prevents it from learning a broad range of reasoning skills. To fix this, we employ the following strategy in the dataset generator: (i) sample a reasoning pattern (ii) sample a question-QDMR pair from that reasoning pattern (iii) possibly perturb the entities (named entities, dates, numbers, ordinals) in the question (and accordingly in the QDMR) with a closely similar entity of the same type\footnote{
e.g., Edwards $\Rightarrow$ Tom to create a new question: "How many touchdowns did Tom throw in the 1st quarter?". Since this perturbation is similar to the one used to create distractor chains, it makes distinguishing these distractor chains in the unperturbed questions from the gold chains in the perturbed questions much harder and better enforces property P3 in \S \ref{subsec:instance-generator}.}
(iv) invoke the instance generator for $N\in \{1, 2, 3, 4\}$. The resulting training dataset has about 900 reasoning patterns with the top 10 common patterns having only 4\% of examples (compared to 50\% in the source typed programs).
\subsection{Additional QA Instances for Primitives}
\label{subsec:primitive-data-generator}
Lastly, in addition to these synthetic multihop instances, we also have instances to teach 44 individual primitives, similar to \citet{genbert}. These instances are created based on simple templates. E.g., for primitive \texttt{filter\_a\_where\_b\_is\_compared\_to}, a question could be ``Entities that have value larger than 948768.92?" and context could be ``Entity AFE has value 871781. Entity RQX has value 989,517.24." resulting in answer [`RQX']. Example QA instances for various other primitives are given in the appendix (Table~\ref{tab:primitives-examples}). In all, we've 30K training and 1K development instances for each primitive.
\subsection{Final Dataset}
Final \textsc{TeaBReaC}\xspace dataset has 525K and 15K train and dev multihop QA instances respectively, and has about 900 reasoning patterns.
To create it we used QDMRs from QA and semantic parsing datasets, DROP~\cite{drop}, ComplexWebQuestions~\cite{complexwebqns}, HotpotQA~\cite{hotpotqa}, SPIDER~\cite{spider}, ComQA~\cite{comqa}, ATIS~\cite{atis}.\footnote{We did not use visual QA datasets CLEVR-humans~\cite{clevr} and NLVR2~\cite{nlvr2} as the questions in it are vastly different from ones we see in reading comprehension-based datasets.} We use both \texttt{low} and \texttt{high} level decompositions from QDMR limited to two to six reasoning steps.
\section{Teaching Broad-Coverage Reasoning Skills in a Robust Fashion}
Multihop questions come in a wide variety. Some involve numeric operations~\cite{drop}, some involve assessing whether complete information is present or not~\cite{iirc}, some involve tables and text~\cite{tatqa}, and so on. One way to surface the reasoning needed for answering these questions is to look at their \emph{decomposition} into smaller reasoning steps that can be composed together in order to arrive at the correct answer. For example, consider the question in Fig.~\ref{fig:introduction}, \emph{From what yard-line did Shayne kick two field goals?}. This can be decomposed as follows: list the field goals by Shayne Graham, identify the yard-lines corresponding to each of them, map each yard-line with the field goal and count them, and select the yard-line with two field goals.
While questions in multihop datasets are authored with the intent that such multi-step reasoning will be used to answer them, the context associated with the questions often allows models to cheat by taking shortcuts~\cite{min2019compositional,dire}. E.g., if the context mentions field goals only by Shayne Graham and no one else, models can ignore the player name and still succeed.
Our key observation is that the decomposition of a question can be leveraged to carefully design a synthetic context for this question that avoids cheating, thereby allowing us to teach models a broad range of reasoning skills in a robust fashion. To achieve this, we procedurally create a large pretraining RC QA dataset, \textsc{TeaBReaC}\xspace, by using real multihop questions (from existing datasets) and their decomposition annotations (already available in the form of QDMRs), and carefully constructing synthetic contexts.
QDMR or Question Decomposition Meaning Representation~\cite{break} is a common way to represent the reasoning in many types of multihop questions as a structured decomposition graph. QDMR has standardized operators (represented as nodes) such as \texttt{select}, \texttt{project}, \texttt{group}, \texttt{comparative}, etc., that transform their input. These are connected together to a final node which produces the answer. Figure~\ref{fig:introduction} shows the above example question paired with its QDMR graph. Importantly, QDMRs are already available for several multihop datasets. This allows us to take on our challenge of building contexts without shortcuts. It also provides evidence that QDMR is a general representation covering a broad spectrum of multihop reasoning phenomena.
Briefly, our method involves the following main steps; these are described in more detail in \S\ref{sec:construction}.
\paragraph{Making QDMRs more precise.}
To create QA instances that teach the precise reasoning captured in QDMR, we need a precise and formal representation of reasoning captured in QDMRs. QDMRs, although structured, don't quite do so, as they are written in natural language and don't specify the datatypes of their input and output. Since this will be crucial for our approach, we will convert QDMRs into formal programs with over 44 executable primitive operations along with their input/output types (\S ~\ref{subsec:instance-generator}).
\paragraph{Teaching robust compositional skills.}
Past work has shown that compositional questions don't necessitate multihop reasoning as datasets often have reasoning shortcuts~\cite{min2019compositional,chen2019understanding,dire}. To teach the reasoning reflected our formal programs robustly, our QA instances must be such that models cannot bypass the reasoning steps and still arrive at the correct answer. To achieve this goal, we create a synthetic QA instance from a question-program pair, where the question is the same as the original question, but the context is procedurally constructed by grounding the predicates in QDMR in a careful way such that models can't cheat their way to the correct answer.
\paragraph{Teaching a broad range of reasoning patterns}
Although QDMRs cover a broad range of reasoning patterns, we find that the natural distribution of reasoning patterns in QDMRs is extremely skewed towards popular reasoning patterns (\S~\ref{subsec:dataset-generator}). Training on QA instances generated from such a distribution leads models to overfit to only a few most representative reasoning patterns, and not learn broad-range reasoning skills. To ensure this doesn't happen, we make sure our synthetic dataset is more balanced in terms of reasoning patterns (\S~\ref{subsec:dataset-generator}).
\paragraph{Teaching a broad range of reasoning primitives.}
In addition to our process of constructing a pretraining dataset to teach compositional skills described thus far, we observe that it also helps if we teach models the constituent primitive reasoning skills. To achieve this, similar to prior work~\cite{genbert}, we procedurally generate QA instances based on fixed templates for each of the 44 primitives present in our formal programs (\S~\ref{subsec:primitive-data-generator}).
\section{Experiments}
\subsection{Experimental Setup}
To test the effectiveness of \textsc{TeaBReaC}\xspace pretraining, we compare models directly fine-tuned on target datasets with models first pretrained on \textsc{TeaBReaC}\xspace and then fine-tuned on target datasets. We report the exact match metric (EM) for all evaluations.
\subsubsection{Datasets}
We evaluate \textbf{in-domain performance} using DROP, TAT-QA and IIRC. The reported numbers are on their dev sets.\footnote{We selected training hyper-parameter (learning rate) for each baseline model and dataset, based on the dev set performance. Our experiments with \textsc{TeaBReaC}\xspace use this identical learning rate. Thus, the reported results slightly favor the baseline models that \emph{don't} use \textsc{TeaBReaC}\xspace. A proper evaluation on unseen data will be included in the next revision of this paper.} For IIRC, we consider two settings: gold-setting (IIRC-G) which uses only gold supporting sentences as reading comprehension context, and retrieved-setting (IIRC-R) which retrieves paragraphs using a retrieval marginalization method~\cite{mitigating-retrieval-negs}. We evaluate \textbf{robustness} using DROP contrast set ~\cite{contrastsets} and DROP BPB contrast set ~\cite{bpb}\footnote{We use the human validated set.}. For robustness evaluation, we only fine-tune on DROP dataset and evaluate on the contrast sets directly.
\subsubsection{Models}
We evaluate \textsc{TeaBReaC}\xspace pretraining on two kinds of (language) models. \textbf{Plain language models}: T5-Large ~\cite{t5} and Bart-Large ~\cite{bart}. \textbf{Numeracy-aware language models}: These are language models pretrained on synthetic dataset from two past works. NT5 ~\cite{nt5}, which is a T5-small\footnote{NT5 has only been released in small size.} model pretrained on a dataset released by ~\citet{genbert}, and PReasM-Large, which is a T5-Large model pretrained on dataset released by ~\citet{preasm}. Our models are implemented using PyTorch~\cite{pytorch}, Huggingface Transformers~\cite{transformers} and AllenNLP~\cite{allennlp}. The implementation details and training hyperparameters are given in the appendix, \S ~\ref{sec:implementation-details}.
\paragraph{Tokenization.}
We find that character tokenization for numbers (a trick adopted from NT5~\cite{nt5}) significantly improves model performance. For instance, as shown in Table~\ref{table:digit-tokenization}, PReasM-large fine-tuned on DROP without any such tokenization gets 69.4 (as reported in ~\cite{preasm}), but with such tokenization (our implementation) gets 76.6. We see similar trends on T5 and Bart as well. So we use this tokenization as a default for all models across all our experiments.
\begin{table}[ht]
\centering
\small
\setlength{\tabcolsep}{1.5pt}
\begin{tabular}{p{5.4cm}cc}\toprule
Method & Dig. Tok. & DROP \\
\midrule
T5 (reported in ~\cite{preasm}) & \faTimes & 61.8 \\
T5 (our implementation) & \faCheck & \textbf{70.3} \\
\midrule
PReasM (reported in ~\cite{preasm}) & \faTimes & 69.4 \\
PReasM (our implementation) & \faCheck & \textbf{76.6} \\
\bottomrule
\end{tabular}
\caption{Digit tokenization (character tokenization for numbers) siginifantly helps downstream perforamnce, so we use it everywhere.}
\label{table:digit-tokenization}
\end{table}
\subsection{Experimental Results}
\subsubsection*{Learnability of \textsc{TeaBReaC}\xspace}
Since our goal is to teach models the reasoning skills in \textsc{TeaBReaC}\xspace, we first assess how well models do on the \textsc{TeaBReaC}\xspace dataset. As shown in Table ~\ref{table:teabreac-results}, models are able to learn both primitive and multihop QA skills required in \textsc{TeaBReaC}\xspace. On primitives instances models get 92-99 accuracy, and on multihop instances, models get 82-86 accuracy. We'll later show that these scores are good enough to make progress on real datasets. At the same time, these aren't perfect scores, demonstrating limitations of vanilla LM-based neural models. Thus, \textsc{TeaBReaC}\xspace can also serve as a benchmark to help design better multihop models, especially for questions requiring 4 or more steps of reasoning.
\begin{table}[t]
\centering
\setlength{\tabcolsep}{2.5pt}
\begin{tabular}{p{1.8cm}p{2.2cm}p{2.2cm}}\toprule
Model & \img{images/teabreac.png} \xspace Primitive & \img{images/teabreac.png} \xspace Multihop \\
\midrule
T5 \img{images/teabreac.png} & \quad \quad 94.6 & \quad 83.7 \\
Bart \img{images/teabreac.png} & \quad \quad 92.1 & \quad 83.0 \\
PReasM \img{images/teabreac.png} & \quad \quad 99.2 & \quad 81.6 \\
NT5 \img{images/teabreac.png} & \quad \quad 95.0 & \quad 86.2 \\
\bottomrule
\end{tabular}
\caption{Models learn the skills required in \textsc{TeaBReaC}\xspace during pretraining well, but achieving perfect score is challenging for vanilla LM-based neural models.}
\label{table:teabreac-results}
\end{table}
\eat{
}
\begin{table}[ht]
\centering
\setlength{\tabcolsep}{2.0pt}
\begin{tabular}{cp{1.85cm}cccc}\toprule
& Model & DROP & TAT-QA & IIRC-G & IIRC-R \\
\midrule
\multirow{2}{*}{
\rotatebox[origin=c]{90}{\parbox[c]{2.0cm}{
\centering Plain LMs}
}
}
& T5 & 70.3 & 38.6 & 64.2 & 42.3 \\
& T5 \img{images/teabreac.png} & \textbf{78.3} & \textbf{50.9} & \textbf{69.2} & \textbf{43.1} \\
\cmidrule{2-6}
& Bart & 66.5 & 35.5 & 62.1 & 42.0 \\
& Bart \img{images/teabreac.png} & \textbf{78.0} & \textbf{44.4} & \textbf{72.0} & \textbf{45.4} \\
\midrule
\multirow{2}{*}{
\rotatebox[origin=c]{90}{\parbox[c]{2.0cm}{
\centering Num. LMs}
}
}
& NT5 & 69.2 & 44.3 & \textbf{66.6} & \textbf{41.8} \\
& NT5 \img{images/teabreac.png} & \textbf{71.6} & \textbf{47.0} & 66.0 & 40.7 \\
\cmidrule{2-6}
& PReasM & 76.9 & 40.0 & 70.0 & 42.0 \\
& PReasM \img{images/teabreac.png} & \textbf{80.1} & \textbf{54.2} & \textbf{73.3} & \textbf{47.2} \\
\midrule
\multirow{2}{*}{
\rotatebox[origin=c]{90}{\parbox[c]{0.90cm}{
\centering Others}
}
}
& GenBERT & 68.8 & -- & -- & -- \\
& POET\textsubscript{BART} & 77.7 & 41.5 & -- & -- \\
\bottomrule
\end{tabular}
\caption{Model Performance: EM scores with and without \img{images/teabreac.png} \xspace \textsc{TeaBReaC}\xspace pretraining (on dev sets). Pretraining language models (LMs) on \textsc{TeaBReaC}\xspace improves their performance across multiple QA datasets, for both plain and numeracy-aware (Num.) LMs.
NT5 is the only small-sized LM considered and exhibits a somewhat different trend.
}
\label{table:performance-results}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{images/improvements-plot-by-complexity.png}
\caption{
(Left) EM scores with and without \img{images/teabreac.png} \xspace \textsc{TeaBReaC}\xspace pretraining on DROP across varying numbers of hops, as determined by QDMR decompositions. (Middle) Same as left, but with PReasM model as the starting point. (Right) Number of questions for each number of hops in DROP dev set. \textsc{TeaBReaC}\xspace pretraining helps more on more complex questions (Left, Middle). The average performance metric doesn't show such large improvements because more complex questions are less frequent (Right).}
\label{fig:improvements-plot-by-complexity}
\end{figure*}
\subsubsection*{\textsc{TeaBReaC}\xspace improves model performance}
Table~\ref{table:performance-results} compares performance on DROP, TAT-QA, IIRC-G and IIRC-R. For both plain language models, T5 and Bart, \textsc{TeaBReaC}\xspace pretraining results in substantial improvements across all datasets --- 8-11 points gain on DROP, 9-13 points on TAT-QA, 6-10 points on IIRC-G and 1-3 points on IIRC-R. For numeracy-aware language models, NT5 and PReasM, \textsc{TeaBReaC}\xspace pretraining results in 2-3 points of improvement on DROP and 3-14 points of improvement on TAT-QA. \textsc{TeaBReaC}\xspace pretraining doesn't improve NT5 performance on IIRC-G and IIRC-R, but it improves PReasM performance on both datasets by 3-5 points.
We also compare with GenBERT~\cite{genbert} and POET\textsubscript{BART}~\cite{poet}\footnote{Training data and model checkpoints of POET aren't publicly available at the time of the submission.} numbers as reported in the respective papers. GenBERT is a BERT-large model specialized for DROP dataset trained with synthetic data proposed by~\citet{genbert}. Our models of comparable size (T5 and Bart) when pretrained with \textsc{TeaBReaC}\xspace perform substantial better than GenBERT (9-10 points). POET is a pretraining method on arithmetic, logic-based, and SQL-based synthetic data. Our Bart-large model of comparable size, performs 0.3 and 3 points better than POET\textsubscript{BART}.
\subsubsection*{\textsc{TeaBReaC}\xspace improves model robustness}
Table~\ref{table:robustness-results} compares robustness via performance on the DROP contrast set and the DROP BPB set.
\begin{table}[ht]
\centering
\setlength{\tabcolsep}{3pt}
\begin{tabular}{cp{2.7cm}cc}\toprule
& Model & DROP-CS & DROP-BPB \\
\midrule
\multirow{2}{*}{
\rotatebox[origin=c]{90}{\parbox[c]{2.0cm}{
\centering Plain LMs}
}
}
& T5 & 44.6 & 51.1 \\
& T5 \img{images/teabreac.png} & \textbf{52.6} & \textbf{57.0} \\
\cmidrule{2-4}
& Bart & 43.2 & 45.3 \\
& Bart \img{images/teabreac.png} & \textbf{53.6} & \textbf{56.3} \\
\midrule
\multirow{2}{*}{
\rotatebox[origin=c]{90}{\parbox[c]{2.0cm}{
\centering Num. LMs}
}
}
& NT5 & 38.6 & 46.2 \\
& NT5 \img{images/teabreac.png} & \textbf{43.4} & \textbf{47.6} \\
\cmidrule{2-4}
& PReasM & 49.8 & 50.3 \\
& PReasM \img{images/teabreac.png} & \textbf{53.5} & \textbf{58.8} \\
\bottomrule
\end{tabular}
\caption{Robustness Evaluation: EM scores with and without \img{images/teabreac.png} \xspace \textsc{TeaBReaC}\xspace pretraining on two DROP contrast sets. Pretraining LMs on \textsc{TeaBReaC}\xspace improves models robustness, for both plain and numeracy aware (num.) LMs. NT5 is the only small-sized LM considered.}
\label{table:robustness-results}
\end{table}
For both plain language models, T5 and Bart, \textsc{TeaBReaC}\xspace pretraining shows substantial improvements in robustness --- 8-10 points improvements on DROP contrast set, and 6-11 points on DROP BPB Set. For numeracy-aware language models, NT5 and PReasM, \textsc{TeaBReaC}\xspace pretraining results in 4-5 points of improvement on DROP contrast set and 1-8 points of improvement on DROP BPB set. The fact that \textsc{TeaBReaC}\xspace improves models pretrained on previous synthetic datasets demonstrates its complementary nature.
\subsubsection*{\img{images/teabreac.png}\xspace improves more on more complex questions}
We further investigate how the improvements provided by \textsc{TeaBReaC}\xspace vary based on the complexity or number of reasoning steps of the question. We can obtain the number of reasoning steps from QDMRs, but the QDMRs are not available for all questions in DROP dev set. Therefore, we train a T5-Large based QDMR parser, run it on the entire DROP dev set, and use the number of reasoning steps of the \textit{predicted} QDMR as a proxy.
Figure~\ref{fig:improvements-plot-by-complexity} compares the performance of \textsc{TeaBReaC}\xspace pretraining on questions with increasing (estimated) hop lengths. For plain language model, T5, the baseline model significantly drops from 75 to 45 as the number of hops or complexity of the question increases. In contrast, model with \textsc{TeaBReaC}\xspace pretraining stays the same or improves with increasing number of hops. In other words, there is a significantly larger improvement for more complex questions, where the original T5 model struggles (e.g., 30 points gain on 4+ hops vs 8 points gain on average). Similarly, for numeracy-aware language model, PReasM-Large, we see more improvement on more complex questions (e.g., 4.1-6.5 points on 3+ hops, 3.5 points on average).
We also observe that more complex questions are significantly less frequent in the DROP dev set as shown in the rightmost plot of Fig.~\ref{fig:improvements-plot-by-complexity}. This makes our large gains on more complex questions (achieved in part via the balancing of reasoning patterns in \textsc{TeaBReaC}\xspace) not quite visible in the aggregate dataset metric reported earlier in Table~\ref{table:performance-results}.
\eat{
}
\section{Introduction}
Multihop Question Answering (QA) is a complex problem that requires a wide variety of reasoning skills. In addition to basic reading comprehension (RC), models must connect multiple pieces of information, sometimes employ numerical and other forms of discrete reasoning, and compose these skills as required by the question. However, even though questions in multihop datasets often cover a broad range of interesting reasoning patterns, the datasets are dominated by only a few patterns, which is what trained models naturally focus on. Moreover, the contexts occurring in existing RC datasets often contain artifacts and reasoning shortcuts~\cite{min2019compositional,chen2019understanding,dire}. Such contexts allow models to find the answer while bypassing some reasoning steps, in turn preventing models from learning the intended reasoning skills.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{images/introduction.pdf}
\caption{\textsc{TeaBReaC}\xspace \img{images/teabreac.png}\ Dataset Construction: We leverage widely available question decomposition annotations (QDMRs) for real questions from a broad range of datasets to carefully construct synthetic contexts such that answering the resulting \img{images/teabreac.png}\ question requires proper multihop reasoning. These questions are further re-balanced to help teach a broad set of reasoning skills.}
\label{fig:introduction}
\end{figure}
How, then, can we teach models broad multihop reasoning skills? One way is to have greater control over the types of input contexts models see during training---contexts that cover a wide variety of reasoning patterns while not allowing models to easily succeed via shortcuts.
We observe that questions in existing datasets (henceforth referred to as ``real questions'') already cover a wide variety of reasoning patterns. The challenge, then, is to teach these reasoning patterns \emph{robustly}, even when they are relatively rare in multihop datasets (e.g., 4-6 hops reasoning).
As a means to this end, we turn to \emph{synthetic context generation for real questions}.
Specifically, we propose to construct contexts for real questions synthetically from scratch (instead of perturbing existing contexts), resulting in much greater control over reasoning shortcuts.
Further, context generation also enables us to balance out the distribution of reasoning patterns in our dataset, e.g., by synthesizing additional contexts, and thereby examples, for questions from the long-tail of underrepresented reasoning patterns.
Our use of synthetic contexts to reliably teach broad skills is inspired by three strands of recent RC QA research. One strand has shown that skills learnt over synthetic data can indeed transfer to real datasets~\cite{genbert,nt5,preasm,poet}. A second strand has shown that perturbing the existing (natural) contexts of RC instances in a targeted fashion can reduce artifact-based reasoning~\cite{jia2017adversarial,dire}. A third strand has shown that carefully constructing contexts (for synthetic questions) to have sufficient distractors can reduce artifacts exploitable by current models~\cite{musique,commaqa}.
Building upon these three strands, we introduce \textsc{TeaBReaC}\xspace,\footnote{\textsc{TeaBReaC}\xspace stands for ``\textbf{T}eaching \textbf{B}road \textbf{Rea}soning skills via decomposition-guided \textbf{C}ontexts'', and is pronounced as ``Tea Break".} a teaching dataset that includes carefully constructed synthetic contexts for a broad set of real multihop questions sourced from six existing datasets. \textsc{TeaBReaC}\xspace was designed with the goals of strong control over cheatability and balanced coverage of reasoning patterns. To identify the intended reasoning, we leverage question decomposition annotations, specifically Question Decomposition Meaning Representation or QDMR annotations which are widely available for a broad set of datasets~\cite{break}.
Figure~\ref{fig:introduction} shows the overview of our construction process for \textsc{TeaBReaC}\xspace. We turn the basic structured representation of the reasoning steps in QDMRs into a precise \textit{program} that can be executed against a synthetic context to arrive at an answer. We then construct a synthetic context by asserting a set facts that relate to the parts of the multihop question. We do this by grounding the predicates of QDMR (e.g., \textit{field goals of Shayne Graham} in Fig.~\ref{fig:introduction}) with randomly generated entities. We also add distractor statements to the context to ensure that bypassing reasoning steps results in an incorrect answer. This forces models to learn the intended reasoning. We then add an outer loop around this process that ensures that the reasoning patterns---as measured by the program signatures of the questions---remain balanced in the final dataset. This forces models to learn a broad range of reasoning patterns instead of focusing on the few dominant ones. Finally, similar to prior work~\cite{genbert}, we also add simpler single-step questions to teach individual primitive skills underlying our formal programs.
Our experiments demonstrate that pretraining large language models (LMs) on \textsc{TeaBReaC}\xspace before fine-tuning on target multihop QA datasets results in significant improvements on multiple in-distribution evaluation sets (DROP~\cite{drop}, TAT-QA~\cite{tatqa}, IIRC~\cite{iirc}) by up to 13 points (exact match), as well as on two contrastive evaluation sets of DROP by upto 11 points. Furthermore, even if we start with numeracy-aware LMs already pretrained on similar past work~\cite{genbert,nt5,preasm}, \textsc{TeaBReaC}\xspace provides further improvement by upto 14 EM points. Interestingly, \textsc{TeaBReaC}\xspace is substantially more beneficial for more complex questions (those with more reasoning steps), improving the T5 model by about 30 EM points on questions with 5 or more steps.
In summary, we make three contributions:
(1) A novel methodology to create a teaching dataset (a) with broad reasoning skills covering a wide range of multihop reasoning patterns and (b) leveraging existing QDMR annotations to carefully construct contexts that require true multi-hop reasoning.
(2) The \textsc{TeaBReaC}\xspace teaching dataset with over 525K questions covering about 900 reasoning patterns or program signatures. (3) An empirical demonstration that pretraining on \textsc{TeaBReaC}\xspace before fine-tuning makes both regular and numeracy-aware LMs is much more effective and robust at multihop reasoning, especially for more complex questions.
\section{Related Work}
Many strands of research have pursued the goals of building robust models with broad reasoning skills. These include teaching these skills to models using data augmentation (natural or synthetic), ensuring robust reasoning using datasets with minimal reasoning shortcuts and adversarial perturbation, or directly building multi-step reasoning models using question decompositions. Our work builds on these ideas by creating a synthetic dataset with minimal reasoning shortcuts to teach robust reasoning skills to existing models. Additionally, we develop a novel method to leverage existing decompositions to capture broad reasoning skills in our dataset.
\paragraph{Question Decomposition.}
This aims to represent complex questions in terms of their component reasoning steps. These decompositions can be useful for building more interpretable and modular systems or just provide a shared question meaning representation across multiple datasets.
To enable development of better systems, several recent multihop QA datasets come with question decompostion annotations~\cite{qasc,complexwebqns,strategyqa,musique,commaqa}. These works have enabled the development of \textit{explicit} multistep reasoning systems that first decomposes a multihop question into sub-questions, and answers the sub-questions step-by-step to arrive at the answer~\cite{decomprc,tmn,musique}. Our goal in this work is to use decompositions to instead teach black-box language to perform multi-step reasoning implicitly (within the model).
Since each dataset uses its own decomposition format, they have also led to narrow dataset-specific solutions. The BREAK dataset~\cite{break} on the other hand, defined a standardized meaning representation format (inspired by semantic parsing) for several QA datasets. This shared representation has allowed the development of contrastive datasets~\cite{bpb}. In this work, we leverage these annotations to build a dataset to teach broad reasoning skills to models.
\paragraph{Robust Multihop Reasoning.}
Past work has shown how to perturb existing multihop QA instances to prevent shortcuts and incentivize robust reasoning. \citet{jiang2019avoiding,ding2021reasoning} created adversarial multihop question by perturbing the reasoning chains in HotpotQA~\cite{hotpotqa}. Other datasets~\cite{dire,musique,multihop-robustifying} ensure robust reasoning via minimally perturbed unanswerable questions. Our approach targets a broader set of questions and eliminates multiple reasoning shortcuts.
The closest work in this line is the Break-Perturb-Build dataset~\cite{bpb}. BPB dataset also used QDMR but with the goal of creating contrastive questions via minor question perturbation~\cite{learningdifference,contrastsets}. Importantly, they use the existing context with reasoning shortcuts that can be hard to eliminate with only question perturbation (e.g., no distractors). Additionally this dataset is mainly used as an evaluation set (as we also do) and has not been shown to result in better models by training on it.
\paragraph{Data Augmentation for QA.}
Several past works have used data augmentation via synthetic datasets to improve QA performance. Following recent works are most relevant to our approach. ~\citet{genbert} created a synthetic dataset using a few hand-crafted templates for injecting numerical reasoning skills (along with a specialized architecture). This dataset was also later used to build a numeracy-aware T5~\cite{t5} model: NT5~\citet{nt5}. ~\citet{preasm} created a synthetic dataset using 13 handcrafted multihop QA reasoning patterns applied on wikipedia tables. Lastly, ~\citet{poet} showed that pretraining language models on synthetic dataset derived from input and output of program executors (arithmetic, logic-based and SQL-based) can also improve downstream QA performance. In contrast to these works, we use actual questions from a wide range of real datasets to teach a broad range of multi-hop reasoning skills.
Past work has also explored synthetic QA dataset generation leveraging generation capabilities of large language models(LM). In addition to many works on generating single-hop QA datasets~\cite{bartolo-etal-2021-improving,alberti-etal-2019-synthetic,puri-etal-2020-training}, ~\citet{pan-etal-2021-unsupervised} have used LMs to create a multihop QA dataset. However, they focus on generating questions for only two types of reasoning (composition and comparison) given real paragraphs. We leave the prospect of combining our synthetic context generation method with the generation capabilities of LMs as a future work.
Multi-task learning over several QA datasets~\cite{2020unifiedqa,multiqa} can also help models learn a broad range of QA skills. However, these works have only targeted simple QA datasets. We view such multi-task learning as orthogonal to our method; it can be used to train models on \textsc{TeaBReaC}\xspace along with other QA datasets.
|
2,869,038,155,697 | arxiv | \section{Introduction}
Let $\varphi:(M^m,g)\longrightarrow (N^n,h)$ be a smooth map between Riemannian manifolds.
The $p$-energy functional of $\varphi$ is defined by
\begin{equation}\label{eq1.1}
E_{p}(\varphi;D)=\frac{1}{p}\int_{D}|d\varphi|^pv_{g},
\end{equation}
where $D$ is a compact domain in $M$, $|d\varphi|$ the Hilbert-Schmidt norm of the differential $d\varphi$, $v^g$ the volume element on $(M^m,g)$, and $p\geq2$.\\
A smooth map is called $p$-harmonic if it is a critical point of the $p$-energy functional (\ref{eq1.1}). We have
\begin{equation}\label{eq1.2}
\frac{d}{dt}E_{p}(\varphi_{t};D)\Big|_{t=0}=-\int_{D}h(\tau_{p}(\varphi),v)v_{g},
\end{equation}
where $\{\varphi_{t}\}_{t\in (-\epsilon,\epsilon)}$ is a smooth variation of $\varphi$ supported in $D$,
$\displaystyle v=\frac{\partial \varphi_{t}}{\partial t}\Big|_{t=0}$ the variation vector field of $\varphi$,
and $\tau_{p}(\varphi)=\operatorname{div}^M(|d\varphi|^{p-2}d\varphi)$ the $p$-tension field of $\varphi$.\\
Let $\nabla^{M}$ the Levi-Civita connection of $(M^m,g)$, and $\nabla^{\varphi}$ the pull-back connection on $\varphi^{-1}TN$, the map $\varphi$ is $p$-harmonic if and only if (see \cite{BG,BI,ali})
\begin{equation}\label{eq1.3}
|d\varphi|^{p-2}\tau(\varphi)+(p-2)|d\varphi|^{p-3} d\varphi(\operatorname{grad}^M|d\varphi|)=0,
\end{equation}
where $\tau(\varphi)=\operatorname{trace}_g\nabla d\varphi$ is the tension field of $\varphi$ (see \cite{BW,ES}). The $p$-bienergy functional of $\varphi$ is defined by
\begin{equation}\label{eq1.4}
E_{2,p}(\varphi;D)=\frac{1}{2}\int_D|\tau_p(\varphi)|^2 v^g.
\end{equation}
We say that $\varphi$ is a $p$-biharmonic map if it is a critical point of the $p$-bienergy functional \eqref{eq1.4}, the Euler-Lagrange equation of the $p$-bienergy functional is given by (see \cite{cherif2})
\begin{eqnarray}\label{eq1.7}
\tau_{2,p}(\varphi)
&=&\nonumber -|d\varphi|^{p-2}\operatorname{trace}_gR^{N}(\tau_{p}(\varphi),d\varphi)d\varphi
-\operatorname{trace}_g\nabla^\varphi |d\varphi|^{p-2} \nabla^\varphi \tau_{p}(\varphi)\\
&&-(p-2)\operatorname{trace}_g\nabla <\nabla^\varphi\tau_{p}(\varphi),d\varphi>|d\varphi|^{p-4}d\varphi=0,
\end{eqnarray}
where $R^N$ is the curvature tensor of $(N^n,h)$ defined by
\begin{equation*}
R^N(X,Y)Z=\nabla^N_X \nabla^N_Y Z-\nabla^N_Y \nabla^N_X Z-\nabla^N_{[X,Y]}Z,\quad\forall X,Y,Z\in\Gamma(TN),
\end{equation*}
and $\nabla^N$ the Levi-Civita connection of $(N^n,h)$. The $p$-energy functional (resp. $p$-bienergy functional) includes as a special case $(p = 2)$ the energy functional (resp. bienergy functional), whose critical points are the usual harmonic maps (resp. biharmonic maps \cite{Jiang}).\\
A submanifold in a Riemannian manifold is called a $p$-harmonic submanifold (resp. $p$-biharmonic submanifold) if the isometric immersion defining the submanifold is a $p$-harmonic map (resp. $p$-biharmonic map). Will call proper $p$-biharmonic submanifolds a $p$-biharmonic submanifols which is non $p$-harmonic.
\section{Main Results}
Let $(M^m,g)$ be a hypersurface of $(N^{m+1},\langle ,\rangle )$, and $\mathbf{i} : (M^m,g) \hookrightarrow (N^{m+1},\langle ,\rangle ) $ the canonical inclusion.
We denote by $\nabla^M$ (resp. $\nabla^N$) the Levi-Civita connection of $(M^m,g)$ (resp. of $(N^{m+1},\langle ,\rangle )$), $\operatorname{grad}^M$ (resp. $\operatorname{grad}^N$) the gradient operator in $(M^m,g)$ (resp. in $(N^{m+1},\langle ,\rangle )$, $B$ the second fundamental form of the hypersurface $(M^m,g)$, $A$ the shape operator with respect to the unit normal vector field $\eta$, $H$ the mean curvature of $(M^m,g)$, $\nabla^\perp$ the normal connection of $(M^m,g)$, and by $\Delta$ (resp. $\Delta^{\perp}$) the Laplacian on $(M^m,g)$ (resp. on the normal bundle of $(M^m,g)$ in $(N^{m+1},\langle ,\rangle )$ (see \cite{BW,ON,YX}).
Under the notation above we have the following results.
\begin{theorem}\label{th1}
The hypersurface $(M^m,g)$ with the mean curvature vector $H= f \eta $ is $p$-bihamronic if and only if
\begin{equation}\label{sys1}
\left\{
\begin{array}{lll}
-\Delta^M(f) + f |A|^2 -f \operatorname{Ric}^N(\eta , \eta ) + m(p-2) f^3 &=& 0; \\\\
2A(\operatorname{grad}^M f) -2 f (\operatorname{Ricci}^N \eta)^\top + ( p-2 + \dfrac{m}{2} ) \operatorname{grad}^M f^2 &=& 0,
\end{array}
\right.
\end{equation}
where $\operatorname{Ric}^N $ (resp. $\operatorname{Ricci}^N$) is the Ricci curvature (resp. Ricci tensor) of $(N^{m+1},\langle ,\rangle )$.
\end{theorem}
\begin{proof}
Choose a normal orthonormal frame $\{ e_i \}_{i=1,... , m }$ on $(M^m,g)$ at $x$, so that $\{ e_i , \eta \}_{i=1,...,m} $ is an orthonormal frame on the ambient space $(N^{m+1},\langle ,\rangle )$.
Note that, $d\mathbf{i}(X)=X$, $\nabla^{\mathbf{i}}_X Y = \nabla^N_X Y $, and the $p$-tension field of $\mathbf{i}$ is given by $ \tau_p(\mathbf{i}) = m^{\frac{p}{2} } f \eta $.
We compute the $p$-bitension field of $\mathbf{i}$
\begin{eqnarray}\label{eq2.2}
\nonumber \tau_{2,p}(\mathbf{i})& =& -|d\mathbf{i}|^{p-2} \operatorname{trace}_g R^N(\tau_p(\mathbf{i}) , d\mathbf{i} ) d\mathbf{i} \\
\nonumber & & -(p-2) \operatorname{trace}_g \nabla \langle \nabla^\mathbf{i} \tau_p(\mathbf{i}) , d\mathbf{i} \rangle |d\mathbf{i}|^{p-4} d\mathbf{i} \\
& &- \operatorname{trace}_g \nabla^\mathbf{i}|d\mathbf{i}|^{p-2} \nabla^\mathbf{i}\tau_p(\mathbf{i}).
\end{eqnarray}
The first term of (\ref{eq2.2}) is given by
\begin{eqnarray}\label{eq2.3}
\nonumber -|d\mathbf{i}|^{p-2} \operatorname{trace}_g R^N (\tau_p(\mathbf{i}) , d\mathbf{i} )d\mathbf{i}
&=& -|d\mathbf{i}|^{p-2} \sum_{i=1}^mR^N(\tau_p(\mathbf{i}) , d\mathbf{i}(e_i))d\mathbf{i}(e_i)\\
\nonumber &=& - m^{p-1} f \sum_{i=1}^m R^N(\eta , e_i)e_i \\
\nonumber &=& -m^{p-1} f \operatorname{Ricci}^N \eta \\
\nonumber &=& -m^{p-1} f \left[ (\operatorname{Ricci}^N \eta )^\perp +(\operatorname{Ricci}^N \eta)^\top \right].\\
\end{eqnarray}
We compute the second term of (\ref{eq2.2})
\begin{eqnarray*}
-(p-2)\operatorname{trace}_g \nabla\langle \nabla^\mathbf{i} \tau_p(\mathbf{i}) , d\mathbf{i} \rangle |d\mathbf{i}|^{p-4}d\mathbf{i}
&=& -(p-2)m^{p-2} \sum_{i,j=1}^m\nabla^N_{e_j} \langle \nabla^N_{e_i} f \eta , e_i \rangle e_j,
\end{eqnarray*}
\begin{eqnarray*}
\sum_{i=1}^m\langle \nabla^N_{e_i} f \eta , e_i \rangle
&=& \sum_{i=1}^m\left[\langle e_i(f) \eta , e_i \rangle + f\langle \nabla^N_{e_i} \eta , e_i \rangle \right]\\
&=& -f \sum_{i=1}^m \langle \eta , B(e_i , e_i) \rangle \\
&=& -m f^2.
\end{eqnarray*}
By the last two equations, we have the following
\begin{equation}\label{eq2.4}
-(p-2)\operatorname{trace}_g \nabla\langle \nabla^\mathbf{i} \tau_p(\mathbf{i}) , d\mathbf{i} \rangle |d\mathbf{i}|^{p-4}d\mathbf{i}
= m^{p-1}(p-2) \left( \operatorname{grad}^M f^2 + m f^3\eta\right).
\end{equation}
The third term of (\ref{eq2.2}) is given by
\begin{eqnarray}\label{eq2.5}
\nonumber - \operatorname{trace}_g \nabla^{\mathbf{i}}|d\mathbf{i}|^{p-2} \nabla^{\mathbf{i}}\tau_p(\mathbf{i})
&=& -m^{p-1} \sum_{i=1}^m\nabla^N_{e_i} \nabla^N_{e_i} f \eta \\
\nonumber &=& -m^{p-1} \sum_{i=1}^m\nabla^N_{e_i}[e_i(f) \eta +f \nabla^N_{e_i} \eta ] \\
\nonumber &=& -m^{p-1}\left[ \Delta^M (f) \eta + 2 \nabla^N_{\operatorname{grad}^M f } \eta + f \sum_{i=1}^m\nabla^N_{e_i}\nabla^N_{e_i} \eta \right].
\nonumber \\
\end{eqnarray}
Thus, at $x$, we obtain
\begin{eqnarray} \label{eq2.6}
\sum_{i=1}^m\nabla_{e_i}^N \nabla_{e_i}^N \eta
&=& \nonumber \sum_{i=1}^m\nabla_{e_i}^N\left[(\nabla_{e_i}^N \eta)^\perp +(\nabla_{e_i}^N \eta)^\top \right] \\
&=&\nonumber -\sum_{i=1}^m\nabla_{e_i}^NA(e_i) \\
&=& - \sum_{i=1}^m\nabla_{e_i}^M A(e_i)-\sum_{i=1}^mB(e_i , A(e_i)).
\end{eqnarray}
Since $\langle A(X),Y\rangle = \langle B(X,Y) , \eta\rangle $ for all $X,Y \in \Gamma(TM) $, we get
\begin{eqnarray}\label{eq2.7}
\nonumber \sum_{i=1}^m\nabla_{e_i}^M A(e_i)
&=&\sum_{i,j=1}^m \langle \nabla_{e_i}^M A(e_i), e_j\rangle e_j \\
\nonumber &=&\sum_{i,j=1}^m\left[ e_i \langle A(e_i) , e_j \rangle e_j - \langle A(e_i) , \nabla^M_{e_i} e_j \rangle e_j \right]\\
\nonumber &=&\sum_{i,j=1}^m e_i \langle B(e_i,e_j) , \eta \rangle e_j \\
\nonumber &=&\sum_{i,j=1}^m e_i \langle \nabla_{e_j}^Ne_i , \eta \rangle e_j\\
&=&\sum_{i,j=1}^m \langle \nabla_{e_i}^N\nabla_{e_j}^Ne_i , \eta \rangle e_j.
\end{eqnarray}
By using the definition of curvature tensor of $(N^{m+1},\langle ,\rangle)$, we conclude
\begin{eqnarray} \label{eq2.8}
\nonumber \sum_{i=1}^m\nabla_{e_i}^M A(e_i)
&=&\sum_{i,j=1}^m \left[\langle R^N (e_i,e_j)e_i , \eta\rangle e_j + \langle \nabla_{e_j}^N\nabla_{e_i}^Ne_i , \eta\rangle e_j\right] \\
\nonumber
&=&\sum_{i,j=1}^m \left[ -\langle R^N (\eta , e_i) e_i ,e_j \rangle e_j + \langle \nabla_{e_j}^N\nabla_{e_i}^Ne_i , \eta\rangle e_j \right]\\
\nonumber
&=& - \sum_{j=1}^m\langle \operatorname{Ricci}^N \eta , e_j \rangle e_j + \sum_{i,j=1}^m e_j\langle \nabla_{e_i}^Ne_i , \eta\rangle e_j-\sum_{i,j=1}^m \langle \nabla_{e_i}^N{e_i}, \nabla_{e_i}^N \eta \rangle e_j \\
&=& -( \operatorname{Ricci}^N \eta )^\top + m \operatorname{grad}^M f.
\end{eqnarray}
On the other hand, we have
\begin{eqnarray} \label{eq2.9}
\nonumber \sum_{i=1}^mB(e_i,A(e_i )) &=& \sum_{i=1}^m\langle B(e_i , A(e_i)) , \eta\rangle \eta \\
\nonumber &=& \sum_{i=1}^m\langle A(e_i) , A(e_i )\rangle \eta \\
&=& |A|^2 \eta.
\end{eqnarray}
Substituting (\ref{eq2.6}), (\ref{eq2.8}) and (\ref{eq2.9}) in (\ref{eq2.5}), we obtain
\begin{eqnarray}\label{eq2.10}
- \operatorname{trace}_g \nabla^{\mathbf{i}}|d\mathbf{i}|^{p-2} \nabla^{\mathbf{i}}\tau_p(\mathbf{i})
&=&\nonumber - m^{p-1} \big[ \Delta^M(f)\eta-2 A (\operatorname{grad}^M f) + f (\operatorname{Ricci}^N \eta )^\top \\
&&- \dfrac{m}{2} \operatorname{grad}^M f^2 - f |A|^2 \eta \big].
\end{eqnarray}
The Theorem \ref{th1} follows by (\ref{eq2.2})-(\ref{eq2.4}), and (\ref{eq2.10}).
\end{proof}
As an immediate consequence of Theorem \ref{th1} we have.
\begin{corollary}\label{corollary1}
A hypersurface $(M^m,g) $ in an Einstein space $(N^{m+1},\langle ,\rangle )$ is $p$-biharmonic if and only if it's mean curvature function $f$ is a solution of the following PDEs
\begin{equation}\label{sys2}
\left\{
\begin{array}{lll}
-\Delta^M(f) + f |A|^2 + m(p-2) f^3 - \frac{S}{m+1}f &=& 0; \\\\
2A(\operatorname{grad}^M f) + ( p-2 + \dfrac{m}{2} ) \operatorname{grad}^M f^2 &=& 0,
\end{array}
\right.
\end{equation}
where $S$ is the scalar curvature of the ambient space.
\end{corollary}
\begin{proof}
It is well known that if $(N^{m+1} ,\langle , \rangle ) $ is an Einstein manifold then $\operatorname{Ric}^N(X,Y) = \lambda \langle X,Y\rangle $
for some constant $\lambda$, for any $X,Y \in \Gamma(TN)$. So that
\begin{eqnarray*}
S&=& \operatorname{trace}_{\langle , \rangle} \operatorname{Ric}^N\\
&=& \sum_{i=1}^m\operatorname{Ric}^N(e_i,e_i) + \operatorname{Ric}^N(\eta, \eta) \\
&=& \lambda (m+1),
\end{eqnarray*}
where $\{ e_i \}_{i=1,... , m }$ is a normal orthonormal frame on $(M^m,g)$ at $x$.
Since $\operatorname{Ric}^N(\eta, \eta) = \lambda $, on conclude that
$$\operatorname{Ric}^N(\eta, \eta) = \frac{S}{m+1}. $$
On the other hand, we have
\begin{eqnarray*}
(\operatorname{Ricci}^N\eta)^{\top}&=& \sum_{i=1}^m\langle \operatorname{Ricci}^N\eta , e_i\rangle e_i \\
&=& \sum_{i=1}^m\operatorname{Ric}^N(\eta , e_i)e_i\\
&=& \sum_{i=1}^m\lambda\langle \eta , e_i\rangle e_i \\
&=& 0.
\end{eqnarray*}
The Corollary \ref{corollary1} follows by Theorem \ref{th1}.
\end{proof}
\begin{theorem} \label{th2}
A totally umbilical hypersurface $(M^m,g) $ in an Einstein space $(N^{m+1},\langle ,\rangle )$ with non-positive scalar curvature is $p$-biharmonic if and only if it is minimal.
\end{theorem}
\begin{proof}
Take an orthonormal frame $\{ e_i, \eta \}_{i=1,...,m}$ on the ambient space $(N^{m+1} , \langle ,\rangle )$ such that
$\{ e_i \}_{i=1,...,m}$ is an orthonormal frame on $(M^m,g)$. We have
\begin{eqnarray*}
f &=& \langle H,\eta\rangle \\
&=& \frac{1}{m} \sum_{i=1}^m\langle B(e_i, e_i), \eta \rangle \\
&=& \frac{1}{m} \sum_{i=1}^m\langle g(e_i, e_i) \beta \eta , \eta \rangle \\
&=& \beta,
\end{eqnarray*}
where $\beta\in C^\infty(M)$. The $p$-biharmonic hypersurface equation $(\ref{sys2})$ becomes
\begin{equation}\label{sys3}
\left\{
\begin{array}{lll}
-\Delta^M(\beta) + m (p-1)\beta^3 - \frac{S}{m+1} \beta &=& 0; \\\\
(p-1 + \frac{m}{2} ) \beta \operatorname{grad}^M \beta &=& 0,
\end{array}
\right.
\end{equation}
Solving the last system, we have $\beta= 0$ and hence $f=0$, or
$$\beta= \pm\sqrt{\frac{S}{m(m+1)(p-1)}},$$
it's constant and this happens only if $S\geq0$. The proof is complete.
\end{proof}
\section{$p$-biharmonic hypersurface in conformally flat space }
Let $ \mathbf{i} : M^m \hookrightarrow \mathbb{R}^{m+1} $ be a minimal hypersurface with the unit normal vector field $\eta$,
$ \widetilde{ \mathbf{i} } : (M^m , \widetilde{g}) \hookrightarrow (\mathbb{R}^{m+1} , \widetilde{h} = e^{2\gamma} h ) $, $x\longmapsto
\widetilde{ \mathbf{i} }(x)=\mathbf{i}(x)=x$, where $\gamma \in C^{\infty}(\mathbb{R}^{m+1} )$, $h= \langle ,\rangle _{\mathbb{R}^{m+1}} $, and $\widetilde{g} $ is the induced metric by $\widetilde{h}$, that is
$$ \widetilde{g}(X,Y ) = e^{2\gamma} g(X,Y)= e^{2\gamma} \langle X,Y\rangle _{\mathbb{R}^{m+1}} ,$$ where $g$ is the induced metric by $h$.
Let $\{e_i, \eta \}_{i=1,...,m} $ be an orthonormal frame adapted to the $p$-harmonic hypersurface on $(\mathbb{R}^{m+1} , h )$, thus $ \{\widetilde{e}_i, \widetilde{\eta} \}_{i=1,...,m} $ becomes an orthonormal frame on $ (\mathbb{R}^{m+1} , \widetilde{h} ) $, where $\widetilde{e}_i=e^{-\gamma}e_i$ for all $i=1,..., m $, and $ \widetilde{\eta}=e^{-\gamma} \eta $.
\begin{theorem}\label{th3}
The hypersurface $(M^m , \widetilde{g})$ in the conformally flat space $(\mathbb{R}^{m+1} , \widetilde{h}) $ is $p$-biharmonic if and only if
\begin{equation}\label{sys4}
\left\{
\begin{array}{lll}
\eta(\gamma) e^{-\gamma} \big[-\Delta^M(\gamma) -m \operatorname{Hess}^{\mathbb{R}^{m+1}}_{\gamma} ( \eta , \eta ) + (1-m)|\operatorname{grad}^M \gamma |^2
\\\\- |A|^2+m (1-p) \eta(\gamma)^2\big]
+ \Delta^M(\eta(\gamma) e^{-\gamma}) + (m-2) ( \operatorname{grad}^M \gamma ) (\eta(\gamma) e^{-\gamma} )= 0; \\\\
-2 A(\operatorname{grad}^M( \eta(\gamma) e^{-\gamma})) +2 (1-m) \eta(\gamma) e^{-\gamma} A(\operatorname{grad}^M \gamma) \\\\+ (2p-m)\eta(\gamma)\operatorname{grad}^M (\eta(\gamma) e^{-\gamma}) = 0,
\end{array}
\right.
\end{equation}
where $\operatorname{Hess}^{\mathbb{R}^{m+1}}_{\gamma}$ is the Hessian of the smooth function $\gamma$ in $(\mathbb{R}^{m+1} , h) $.
\end{theorem}
\begin{proof}
By using the Kozul's formula, we have
$$\left\{
\begin{array}{ll}
\widetilde{\nabla}_X^M Y = \nabla_X^M Y + X(\gamma) Y + Y(\gamma)X -g(X,Y) \operatorname{grad}^M \gamma; \\\\
\widetilde{\nabla}_U^{\mathbb{R}^{m+1}} V = \nabla_U^{\mathbb{R}^{m+1}} V + U(\gamma) V + V(\gamma)U -h(U,V) \operatorname{grad}^{\mathbb{R}^{m+1}} \gamma,
\end{array}
\right.$$
for all $X,Y \in\Gamma(TM )$, and $ U,V \in \Gamma(T\mathbb{R}^{m+1} )$.
Consequently
\begin{eqnarray}\label{eq3.2}
\nabla_X^{\widetilde{\mathbf{i}}} d\widetilde{\mathbf{i}}(Y) &=&\nonumber\nabla_{X}^{\widetilde{\mathbf{i}}} Y \\
&=&\nonumber \widetilde{\nabla}_{d\mathbf{i} (X)}^{\mathbb{R}^{m+1}} Y\\
&=&\nonumber \widetilde{\nabla}_X^{\mathbb{R}^{m+1}} Y \\
&=& \nabla_X^{\mathbb{R}^{m+1}} Y + X(\gamma) Y+ Y(\gamma)X -h(X,Y) \operatorname{grad}^{\mathbb{R}^{m+1}} \gamma,\qquad
\end{eqnarray}
and the following
\begin{eqnarray}\label{eq3.3}
d\widetilde{\mathbf{i}} ( \widetilde{\nabla}_X^M Y )
&=&\nonumber d\mathbf{i}(\nabla_X^M Y) + X(\gamma) d\mathbf{i} (Y) + Y(\gamma) d\mathbf{i}(X) - g(X,Y) d\mathbf{i}( \operatorname{grad}^M \gamma )\\
&=&\nabla_X^M Y + X(\gamma) Y + Y(\gamma) X - g(X,Y) \operatorname{grad}^M \gamma.
\end{eqnarray}
From equations (\ref{eq3.2}) and (\ref{eq3.3}), we get
\begin{eqnarray}\label{eq3.4}
(\nabla d\widetilde{\mathbf{i}})(X,Y)
&=&\nonumber \nabla_X^{\widetilde {\mathbf{i}} } d \widetilde {\mathbf{i}} (Y) -d \widetilde {\mathbf{i}}(\widetilde{\nabla}_X^M Y)\\
&=&\nonumber (\nabla d\mathbf{i})(X,Y) + g(X,Y)[\operatorname{grad}^M\gamma -\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma]\\
&=& B(X,Y) -g(X,Y) \eta(\gamma)\eta.
\end{eqnarray}
So that, the mean curvature function $\widetilde{f}$ of $(M^m,\widetilde{g})$ in $ (\mathbb{R}^{m+1} , \widetilde{h} ) $ is given by $ \widetilde{f}=-\eta(\gamma)e^{-\gamma} $. Indeed, by taking traces in (\ref{eq3.4}), we obtain $$e^{2\gamma}\widetilde{H} = H - \eta(\gamma)\eta.$$
Since $(M^m,g)$ is minimal in $ (\mathbb{R}^{m+1} , h ) $, we find that $\widetilde{H} = - e^{-2\gamma} \eta(\gamma)\eta$, that is
$\widetilde{H} = - e^{-\gamma} \eta(\gamma)\widetilde{\eta}$.\\
With the new notations the equation $(\ref{sys1})$ for $p$-biharmonic hypersurface in the conformally flat space becomes
\begin{equation}\label{sys5}
\left\{
\begin{array}{lll}
-\widetilde{\Delta}(\widetilde{f}) + \widetilde{f} |\widetilde{A}|_{\widetilde{g}}^2 -\widetilde{f} \, \widetilde{\operatorname{Ric}}^{\mathbb{R}^{m+1} }(\widetilde{\eta} , \widetilde{\eta} ) + m(p-2) \widetilde{f}^3 &=& 0; \\\\
2\widetilde{A}(\widetilde{\operatorname{grad}}^M \widetilde{f}) -2 \widetilde{f} (\widetilde{\operatorname{Ricci}}^{\mathbb{R}^{m+1} } \widetilde{\eta})^\top + ( p-2 + \dfrac{m}{2} ) \widetilde{\operatorname{grad}}^M \widetilde{f}^2 &=& 0,
\end{array}
\right.
\end{equation}
A straightforward computation yields
\begin{eqnarray*}
\nonumber \widetilde{\operatorname{Ricci}}^{\mathbb{R}^{m+1}} \eta
&=& e^{-2\gamma} \big[\operatorname{Ricci}^{\mathbb{R}^{m+1}}\eta -\Delta^{\mathbb{R}^{m+1}}(\gamma)\eta+(1-m)\nabla_{\eta}^{\mathbb{R}^{m+1}} \operatorname{grad}^{\mathbb{R}^{m+1}} \gamma
\\
\nonumber &&+(1-m) |\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma|^2 \eta
- (1-m) \eta(\gamma)\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma \big];
\end{eqnarray*}
\begin{eqnarray}\label{eq5}
\nonumber \widetilde{\operatorname{Ric}}^{\mathbb{R}^{m+1} }(\widetilde{\eta} , \widetilde{\eta} )
&=& \widetilde{h}(\widetilde{\operatorname{Ricci}}^{\mathbb{R}^{m+1}} \widetilde{\eta},\widetilde{\eta}) \\
\nonumber &=&h(\widetilde{\operatorname{Ricci}}^{\mathbb{R}^{m+1}}\eta,\eta)\\
\nonumber&=& e^{-2\gamma} h\big(\operatorname{Ricci}^{\mathbb{R}^{m+1}}\eta - \Delta^{\mathbb{R}^{m+1}}(\gamma) \eta+(1-m)\nabla_{\eta}^{\mathbb{R}^{m+1}} \operatorname{grad}^{\mathbb{R}^{m+1}} \gamma\\
\nonumber &&+ (1-m) |\operatorname{grad}^{\mathbb{R}^{m+1}}\gamma|^2 \eta - (1-m) \eta(\gamma)\operatorname{grad}^{\mathbb{R}^{m+1}}\gamma,\eta\big) \\
\nonumber &=& e^{-2\gamma} \big[-\Delta^{\mathbb{R}^{m+1}}(\gamma) +(1-m) \operatorname{Hess}_{\gamma}^{\mathbb{R}^{m+1}}(\eta,\eta) +(1-m) |\operatorname{grad}^{\mathbb{R}^{m+1}}\gamma|^2\\
&& -(1-m)\eta(\gamma)^2\big];
\end{eqnarray}
\begin{eqnarray}\label{eq8}
(\widetilde{\operatorname{Ricci}}^{\mathbb{R}^{m+1}} \widetilde{\eta})^\top
\nonumber&=&\sum_{i=1}^mh(\widetilde{\operatorname{Ricci}}^{\mathbb{R}^{m+1} } \widetilde{\eta} ,e_i)e_i \\
\nonumber &=& (1-m)e^{-3\gamma}\sum_{i=1}^m\left[ h ( \nabla_{\eta}^{\mathbb{R}^{m+1}}\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma , e_i)e_i -\eta(\gamma)h(\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma , e_i)e_i \right]\\
\nonumber &=& (1-m) e^{-3\gamma}\Big[\sum_{i=1}^m h ( \nabla_{e_i}^{\mathbb{R}^{m+1}}\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma , \eta)e_i
-\eta(\gamma) \operatorname{grad}^M \gamma\Big] \\
\nonumber &=& (1-m) e^{-3\gamma}\Big[\sum_{i=1}^m e_i h(\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma , \eta)e_i
-\sum_{i=1}^m h(\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma , \nabla_{e_i}^{\mathbb{R}^{m+1}} \eta ) e_i\\
\nonumber &&- \eta(\gamma) \operatorname{grad}^{M} \gamma\Big] \\
\nonumber &=& (1-m) e^{-3\gamma} \big[\operatorname{grad}^M \eta(\gamma)
+\sum_{i=1}^m h(\operatorname{grad}^{\mathbb{R}^{m+1}} \gamma , Ae_i)e_i
-\eta(\gamma) \operatorname{grad}^M \gamma\big]\\
&=& (1-m) e^{-3\gamma} \big[\operatorname{grad}^M \eta(\gamma)
+ A(\operatorname{grad}^{M} \gamma) - \eta(\gamma) \operatorname{grad}^M \gamma\big];
\end{eqnarray}
\begin{eqnarray}\label{eq6}
\nonumber \widetilde{\Delta}(\widetilde{f})
&=& e^{-2\gamma}[\Delta(\widetilde{f})+(m-2)d\widetilde{f}(\operatorname{grad}^M \gamma )] \\
&=& e^{-2\gamma}[-\Delta(\eta(\gamma)e^{-\gamma})-(m-2)(\operatorname{grad}^M \gamma)(\eta(\gamma)e^{-\gamma})];
\end{eqnarray}
\begin{eqnarray}\label{eq3.8}
|\widetilde{A}|_{\widetilde{g}}^2
&=&\nonumber\sum_{i=1}^m\widetilde{g}(\widetilde{A}\widetilde{e}_i,\widetilde{A}\widetilde{e}_i)\\
&=&\nonumber\sum_{i=1}^m g(\widetilde{A}e_i,\widetilde{A}e_i)\\
&=&\nonumber\sum_{i=1}^m h(\widetilde{\nabla}_{e_i}^{\mathbb{R}^{m+1}} \widetilde{\eta},\widetilde{\nabla}_{e_i}^{\mathbb{R}^{m+1}} \widetilde{\eta}) \\
&=&\nonumber\sum_{i=1}^m h(\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta}+e_i(\gamma) \widetilde{\eta} +\widetilde{\eta}(\gamma)e_i,\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta}+e_i(\gamma) \widetilde{\eta} +\widetilde{\eta}(\gamma)e_i)\\
&=&\nonumber\sum_{i=1}^m \big[h(\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta},\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta}) +2 \widetilde{\eta}(\gamma)h(\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta},e_i)+e_i(\gamma)^2e^{-2\gamma}\\
&&+2e_i(\gamma)h(\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta},\widetilde{\eta})\big]+m\widetilde{\eta}(\gamma)^2.
\end{eqnarray}
The first term of (\ref{eq3.8}) is given by
\begin{eqnarray*}
\sum_{i=1}^mh(\nabla_{e_i}^{\mathbb{R}^{m+1}}e^{-\gamma}\eta,\nabla_{e_i}^{\mathbb{R}^{m+1}}e^{-\gamma}\eta)
&=&\sum_{i=1}^m h(-e^{-\gamma}e_i(\gamma) \eta+e^{-\gamma}\nabla_{e_i}^{\mathbb{R}^{m+1}}\eta , -e^{-\gamma}e_i(\gamma) \eta+e^{-\gamma}\nabla_{e_i}^{\mathbb{R}^{m+1}}\eta )\\
&=&\sum_{i=1}^m[ e^{-2\gamma}e_i(\gamma)^2+e^{-2\gamma}h(\nabla_{e_i}^{\mathbb{R}^{m+1}}\eta,\nabla_{e_i}^{\mathbb{R}^{m+1}}\eta)]\\
&=& e^{-2\gamma} |\operatorname{grad}^M \gamma|^2 + e^{-2\gamma}|A|^2.
\end{eqnarray*}
The second term of (\ref{eq3.8}) is given by
\begin{eqnarray*}
2\widetilde{\eta}(\gamma)\sum_{i=1}^m h(\nabla_{e_i}^{\mathbb{R}^{m+1}}\widetilde{\eta},e_i)
&=& -2e^{-\gamma} \eta(\gamma)\sum_{i=1}^m h(e^{-\gamma}\eta,\nabla_{e_i}^{\mathbb{R}^{m+1}} e_i ) \\
&=& -2me^{-2\gamma} \eta(\gamma)h(\eta, H) \\
&=&0.
\end{eqnarray*}
Here $H=0$. We have also
\begin{eqnarray*}
2\sum_{i=1}^me_i(\gamma)h(\nabla_{e_i}^{\mathbb{R}^{m+1}} \widetilde{\eta},\widetilde{\eta})
& =&\sum_{i=1}^m e_i(\gamma) e_i h(\widetilde{\eta} , \widetilde{\eta}) \\
&=& \sum_{i=1}^me_i(\gamma) e_i(e^{-2\gamma}) \\
&=& -2e^{-2\gamma}\sum_{i=1}^m e_i(\gamma)^2 \\
&=& -2e^{-2\gamma} |\operatorname{grad}^M \gamma|^2.
\end{eqnarray*}
Thus
\begin{equation}\label{eq7}
|\widetilde{A}|_{\widetilde{h}}^2= e^{-2\gamma} |A|^2 + m e^{-2\gamma}\eta(\gamma)^2.
\end{equation}
We compute
\begin{eqnarray}\label{eq9}
\nonumber \widetilde{\operatorname{grad}}^M \widetilde{f}&=& e^{-2\gamma} \sum_{i=1}^me_i(\widetilde{f})e_i\\
&=& -e^{-2\gamma}\operatorname{grad}^M (\eta(\gamma)e^{-\gamma});
\end{eqnarray}
and the following
\begin{eqnarray}\label{eq10}
\nonumber \widetilde{A}(\widetilde{\operatorname{grad}}^M \widetilde{f})
&=& -\widetilde{\nabla}_{\widetilde{\operatorname{grad}}^M \widetilde{f}}^{\mathbb{R}^{m+1}} \widetilde{\eta} \\
\nonumber &=&-\widetilde{\nabla}_{\widetilde{\operatorname{grad}}^M \widetilde{f}}^{\mathbb{R}^{m+1}} e^{-\gamma} \eta \\
\nonumber &=& e^{-\gamma}(\widetilde{\operatorname{grad}}^M \widetilde{f}) (\gamma) \eta - e^{-\gamma}\widetilde{\nabla}_{\widetilde{\operatorname{grad}}^M \widetilde{f}}^{\mathbb{R}^{m+1}}\eta \\
\nonumber &=& -e^{-3\gamma} \operatorname{grad}^M (\eta(\gamma)e^{-\gamma} ) (\gamma) \eta + e^{-3\gamma} \widetilde{\nabla}_{\operatorname{grad}^M (\eta(\gamma)e^{-\gamma})}^{\mathbb{R}^{m+1}}\eta\\
\nonumber &=& -e^{-3\gamma} \operatorname{grad}^M(\eta(\gamma)e^{-\gamma})(\gamma) \eta + e^{-3\gamma} \eta(\gamma) \operatorname{grad}^M(\eta(\gamma)e^{-\gamma})\\
\nonumber && +e^{-3\gamma} \operatorname{grad}^M(\eta(\gamma)e^{-\gamma})(\gamma) \eta + e^{-3\gamma}\nabla_{\operatorname{grad}^M(\eta(\gamma)e^{-\gamma})}^{\mathbb{R}^{m+1}} \eta \\
&=& e^{-3\gamma} \eta(\gamma) \operatorname{grad}^M(\eta(\gamma)e^{-\gamma}) - e^{-3\gamma } A(\operatorname{grad}^M \eta(\gamma) e^{-\gamma}).\quad\qquad
\end{eqnarray}
Substituting $(\ref{eq5})-(\ref{eq10})$ in $(\ref{sys5})$, and by simplifying the resulting equation we obtain the system $(\ref{sys4})$.
\end{proof}
\begin{remark}\label{remark}\quad
\begin{enumerate}
\item Using Theorem \ref{th3}, we can construct many examples for proper $p$-biharmonic hypersurfaces in the conformally flat space.
\item If the functions $\gamma$ and $\eta(\gamma)$ are non-zero constants on $M$, then according to Theorem \ref{th3}, the hypersurface $(M^m , \widetilde{g})$ is $p$-biharmonic in $(\mathbb{R}^{m+1} , \widetilde{h}) $ if and only if
$$|A|^2=m (1-p) \eta(\gamma)^2-m\eta(\eta(\gamma)).$$
\end{enumerate}
\end{remark}
\begin{example}
The hyperplane $\mathbf{i} :\mathbb{R}^m \hookrightarrow (\mathbb{R}^{m+1} , e^{2\gamma(z)} h ) $, $x \longmapsto(x , c) $, where $ \gamma \in C^{\infty}(\mathbb{R}) $, $h= \sum_{i=1}^{m}dx_i^2 + dz^2 $, and $c\in \mathbb{R}$, is proper $p$-biharmonic if and only if
$(1-p)\gamma'(c)^2-\gamma''(c)=0$. Note that, the smooth function
$$\gamma(z)=\frac{\ln\left(c_1(p-1)z+c_2(p-1)\right)}{p-1},\quad c_1,c_2\in\mathbb{R},$$
is a solution of the previous differential equation (for all $c$).
\end{example}
\begin{example}
Let $M$ be a surface of revolution in $\{(x,y,z)\in\mathbb{R}^3\,|\,z>0\}$.
If $M$ is part of a plane orthogonal to the axis of revolution, so that $M$ is parametrized by $$(x_1,x_2)\longmapsto(f(x_2)\cos(x_1),f(x_2)\sin(x_1),c),$$ for some constant $c>0$. Here $f(x_2)>0$. Then, $M$ is minimal, and according to Theorem \ref{th3}, the surface $M$ is proper $p$-biharmonic in
$3$-dimensional hyperbolic space $(\mathbb{H}^3,z^{\frac{2}{p-1}}h)$, where $h=dx^2+dy^2+dz^2$.
\end{example}
\textbf{Open Problems.}
\begin{enumerate}
\item If $M$ is a minimal surface of revolution contained in a catenoid, that is $M$ is parametrized by $$(x_1,x_2)\longmapsto\left(a\cosh\left(\frac{x_2}{a}+b\right)\cos(x_1),a\cosh\left(\frac{x_2}{a}+b\right)\sin(x_1),x_2\right),$$ where $a\neq0$ and $b$ are constants.
Is there $p\geq2$ and $\gamma\in C^\infty(\mathbb{R}^3)$ such that $M$ is proper $p$-biharmonic in $\left(\mathbb{R}^3,e^{2\gamma}(dx^2+dy^2+dz^2)\right)$?
\item Is there a proper $p$-biharmonic submanifolds in Euclidean space $(\mathbb{R}^n,dx_1^2+...+dx_n^2)$?
\end{enumerate}
\footnotesize{
|
2,869,038,155,698 | arxiv | \section{Introduction}
Electron transport through nanoscale structures is a stochastic process due to the randomness of the individual tunneling events. Quantum correlations and electron-electron interactions can strongly influence the transport process and thus the statistics of transferred charges. Full counting statistics\cite{Levitov1993,Levitov1996,Nazarov2003} concerns the distribution of the number of transferred charge, or equivalently, all corresponding cumulants (irreducible moments) of the distribution. Conventional transport measurements have focused on the first cumulant, the mean current, and in some cases also the second cumulant, the noise.\cite{Blanter2000} Higher order cumulants, however, reveal additional information concerning a variety of physical phenomena, including quantum coherence, entanglement, disorder, and dissipation.\cite{Nazarov2003} For example, non-zero higher-order cumulants reflect non-Gaussian behavior. Counting statistics in mesoscopic physics has been a subject of intensive theoretical interest for almost two decades, but recently it has also gained considerable experimental interest: in a series of experiments,\cite{Reulet2003,Bomze2005,Bylander2005,Fujisawa2006,Gustavsson2006,Fricke2007,Timofeev2007,Gershon2007,Gabelli2009,Flindt2009,Gustavsson2009,Fricke2010,Fricke2010b} high order cumulants
and even the entire distribution function of transferred charge have been measured, clearly demonstrating that counting statistics now has become an important concept also in experimental physics.
The theory of counting statistics was first formulated by Levitov and Lesovik for non-interacting electrons using a scattering formalism.\cite{Levitov1993,Levitov1996} Subsequent works have focused on the inclusion of interaction effects in the theory.\cite{Nazarov1999,Kindermann2003} In one approach, Coulomb interactions are incorporated via Markovian (generalized) master equations as originally developed by Bagrets and Nazarov.\cite{Bagrets2003} This approach is often convenient when considering systems with strong interactions, e.g., Coulomb-blockade structures. More recent developments include theories for finite-frequency counting statistics,\cite{Emary2007} conditional counting statistics,\cite{Sukhorukov2007} connections to entanglement entropy\cite{Klich2009} and to fluctuation theorems,\cite{Foerster2008,Esposito2009} and extensions to systems with non-Markovian dynamics.\cite{Braggio2006,Flindt2008,Schaller2009} The last topic forms the central theme of this paper.
We have recently published a series of papers on counting statistics.\cite{Flindt2005,Braggio2006,Flindt2008} Previous methods for evaluating the counting statistics of systems described by master equations had in practice been limited to systems with only a few states, and in Ref.\ \onlinecite{Flindt2005} we thus developed techniques for calculating the first few cumulants of Markovian systems with many states, for example nano-electromechanical systems.\cite{Flindt2004} In Ref.\ \onlinecite{Braggio2006}, Braggio and co-workers generalized the approach by Bagrets and Nazarov by including non-Markovian effects that may arise for example when the coupling to the electronic leads is not weak. The methods presented in these papers were subsequently unified and extended in Ref.\ \onlinecite{Flindt2008}, where we presented a general approach to calculations of cumulants of arbitrary order for systems with many states as well as with non-Markovian dynamics. The aim of the present paper is to provide a detailed derivation and description of this method, which recently has been used in a number of works,\cite{Zedler2009,Emary2009a,Emary2009b,Urban2009,Lindebaum2009,Zhong2009,Dominguez2010} as well as to illustrate its use with three examples of current experimental relevance.
The paper is organized as follows: In Sec.\ \ref{sec:non-markov} we
introduce the generic non-Markovian generalized master equation
(GME) which is the starting point of this work. The GME describes the
evolution of the reduced density matrix of the system, which has
been resolved with respect to the number of transferred particles.
Memory effects due to the coupling to the
environment as well as initial system-environment correlations are included in the GME.
Within this framework it is possible to calculate the finite-frequency current noise for non-Markovian GMEs\cite{Flindt2008} as we will discuss in future works. Section \ref{sec:non-markov} concludes with details of the superoperator notation used throughout the paper.
In Sec.\ \ref{sec:zerofreqcumu} we develop a theory for the
zero-frequency cumulants of the current. The cumulant generating
function (CGF) is determined by a single dominating pole of the
resolvent of the memory kernel, and its derivatives with respect to
the counting field evaluated at zero counting field yield the
cumulants of the current. Even in the Markovian case it is difficult
to determine analytically the dominating pole and in many cases one
would have to find it numerically. Numerical differentiation, however, is notoriously unstable,
and often one can only obtain accurate results for the first few
derivatives with respect to the counting field, i.\ e.\ the cumulants. In
order to circumvent this problem, we develop a numerically stable
recursive scheme based on a perturbation expansion in the counting
field. The scheme enables calculations of zero-frequency current
cumulants of very high orders, also for non-Markovian systems. Some
notes on the evaluation of the cumulants are presented, with the
more technical numerical details deferred to App.\ \ref{app:QR}.
Section~\ref{sec:univosc} gives a discussion of the generic behavior of high-order cumulants. As some of us have recently shown,\cite{Flindt2009} the high-order cumulants for basically any system (with or without memory effects) are expected to grow factorially in magnitude with the cumulant order and oscillate as functions of essentially any parameter as well as of the cumulant order. We describe the theory behind this prediction which is subsequently illustrated with
examples in Sec.\ \ref{sec:Markexamples}.
Section~\ref{sec:Markexamples} is devoted to two Markovian
transport models of current research interest, which we use to
illustrate our recursive scheme and the generic behavior of high-order cumulants discussed in Sec.\ \ref{sec:univosc}. We start with a model of
transport through a two-level quantum dot developed by
Belzig.\cite{Belzig2005} Due to the relatively simple analytic
structure of the model, it is possible to write down a closed-form
expression for the CGF, allowing us to develop a thorough
understanding of the behavior of high-order cumulants obtained
using our recursive scheme. We study the large deviation function of the system,\cite{Touchette2009}
which describes the tails of the distribution of
measurable currents, and discuss how it is related to the cumulants.
The second example concerns charge transport coupled to quantized
mechanical vibrations as considered in a recent series of papers on
transport through single
molecules\cite{Boese2001,Braig2003,McCarthy2003,Millis2004,Koch2005,Koch2005b,Pistolesi2008}
and other nano-electromechanical
systems.\cite{Gorelik1997,Armour2002,Fedorets2002,Novotny2003,Novotny2004,Flindt2004,Fedorets2004,Haupt2006,Rodrigues2006,Huebener2007,Harvey2008,Koerting2009,Huebener2009,Cavaliere2009prep,Harvey2009}
Due to the many oscillator states participating in transport the matrix
representations of the involved operators are of large dimensions
and it is necessary to resort to numerics. We demonstrate the
numerical stability of our recursive algorithm up to very high
cumulant orders ($\sim 100$) and show how
oscillations of the cumulants can be used to extract information
about the analytic structure of the cumulant generating function.
We calculate the large deviation function and show that it is highly sensitive to the
damping of the vibrational mode.
Section~\ref{sec:nonMarkov} concerns the counting statistics of non-Markovian systems. We consider a model of non-Markovian electron
transport through a Coulomb-blockade double quantum dot embedded in
a dissipative heat bath and coupled to electronic leads. The dynamics of the charge populations of the double dot
can be described using a non-Markovian GME whose detailed derivation
is presented in App.\ \ref{app:doubledot}. We study the behavior
of the first three cumulants thus extending previous studies that
have been restricted to the noise.\cite{Aguado2004a,Aguado2004b} We
focus in particular on the influence of
decoherence\cite{Kiesslich2007} on the charge transport statistics. Finally, we discuss possible subtleties associated with non-Markovian dynamics and we provide the
reader with a unifying point of view on a number of results reported
in previous studies as well as in the examples discussed in this
paper.
Our conclusions are stated in Sec.~\ref{sec:conclusions}.
Appendix \ref{app:QR} describes the numerical algorithms
used to solve the recursive equations for high-order cumulants,
while Apps.\ \ref{app:vibmol} and \ref{app:doubledot} give detailed derivations of
the Markovian GME for the vibrating molecule and the non-Markovian GME for the double-dot system, respectively.
\section{Generalized Master equation}
\label{sec:non-markov}
The generic transport setup under consideration in this work is
depicted in Fig.\ \ref{fig:gensetup}: A nanoscopic quantum system is
connected by tunneling barriers to two electronic leads, allowing
for charge and energy exchange with the leads. Typically, the
quantum system consists of a discrete set of (many-body) quantum
states. Moreover, the system is coupled to an external heat bath to
and from which energy can flow. We will be considering a transport
configuration, where a bias difference between the leads drives
electrons through the system.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{gensetup.eps}
\caption{Generic transport setup. A quantum system is connected to
electronic leads and a heat bath. A bias difference between the
leads drives electrons through the system, which can exchange energy
with the surrounding heat bath. The system is described by the
$n$-resolved density matrix $\hatrho(n,t)$ (see text), where $n$ is
the number of electrons that have been collected in the right lead
during the time span $[0,t]$. The probability distribution of $n$ is
denoted as $P(n,t)$.} \label{fig:gensetup}
\end{center}
\end{figure}
The quantum system is completely described by its (reduced) density
matrix $\hatrho(t)$, obtained by tracing out the environmental degrees of freedoms, i.\ e.\ the electronic leads and the heat bath. It is, however, advantageous to resolve
$\hatrho(t)$ into the components $\hatrho(n,t)$, corresponding to
the number of electrons $n$ that have tunneled through the system
during the time span $[0,t]$.\cite{Makhlin2001,Shelankov2003,Wabnig2005} The $n$-resolved density matrix
allows us to study the statistics of the number of transferred
charges, similarly to well-known techniques from quantum optics.\cite{Cook1981,Lenstra1982,Plenio1998} We note that the (un-resolved) density matrix can always be recovered by summing over $n$, $\hatrho(t)=\sum_n\hatrho(n,t)$. For bi-directional processes, the number of tunneled electrons $n$ can be both positive and negative.
The major focus in the literature has been on systems obeying
Markovian dynamics;\cite{Blum1996,Gardiner2008,Alicki2007} however
recent years have witnessed an increased interest in non-Markovian
processes as well.\cite{Wilkie2000,Aissani2003,Breuer2006,Bellomo2007,Breuer2007,Budini2008,Timm2008,Breuer2009} In this
spirit we consider a generic non-Markovian generalized master
equation (GME) of the form
\begin{equation}
\frac{d}{dt}\hatrho(n,t)=\sum_{n'}\int_{0}^{t}dt'\W(n-n',t-t')\hatrho(n',t')+\hatgamma(n,t),
\label{eq:GME}
\end{equation}
obtained by tracing out the electronic leads and the heat bath. An
equation of this type arises for example in the partitioning scheme
devised by Nakajima and Zwanzig,\cite{Zwanzig2001} and in the
real-time diagrammatic technique for the dynamics of the reduced
density matrix on the Keldysh contour as described in Refs.\
\onlinecite{Schoeller1994,Konig1996,Konig1996b,Makhlin2001,Braggio2006}. The
memory kernel $\W$ accounts for the dynamics of the system
taking into account the influence of the degrees of freedom that
have been projected out, e.\ g.\ the electronic leads and the heat
bath. Here we have assumed that the
system is not explicitly driven by any time-varying fields, such
that the kernel $\W$ only depends on the time difference $t-t'$.
Additionally, we assume that the number of electrons $n$ that have
been collected in the right lead does not affect the system
dynamics, and the kernel consequently only depends on the difference
$n-n'$. The generic non-Markovian GME also contains the
inhomogeneity $\hatgamma(n,t)$ which accounts for initial
correlations between system and
environment.\cite{Zwanzig2001,Flindt2008} Typically, $\W(n,t)$ and
$\hatgamma(n,t)$ decay on comparable time scales, and $\hatgamma(n,t)$ thus vanishes in the long-time limit of Eq.\ (\ref{eq:GME}).
At this point, we note that while our method for extracting cumulants works
for {\it any} GME which satisfies certain, rather general conditions
specified in detail in Sec.~\ref{sec:zerofreqcumu},
the physical meaningfulness of the results nevertheless depends crucially on a
consistent derivation of the $n$-resolved memory kernel $\W(n,t)$. In Sec.\ \ref{subsec:nonMarkcorr} we discuss various subtleties associated with a proper derivation of the memory kernel for non-Markovian systems. In this work we use the notion of the Markovian limit of a general
non-Markovian GME in a somewhat loose manner, namely by referring to
the ``Markovian" limit of Eq.\ (\ref{eq:GME}) as the case, where
$\W(n,t)=\W(n)\delta(t)$ and $\hatgamma(n,t)=0$. We use this
terminology for the ease of notation, although we are aware that the
proper Markovian limit under certain circumstances may actually be
different. For an example of this, we refer the reader to
Ref.~\onlinecite{Spohn1979}, where it is demonstrated that the
correct Markovian limit for weak coupling theories should be
performed in the interaction picture. Since this procedure only
influences the off-diagonal elements in the weak coupling regime, we
ignore this subtlety in the rest of the paper as we will not be
considering such cases. In relevant situations this difference
should be taken into account --- it would, however, only lead to a
reinterpretation of the non-Markovian corrections studied in
Sec.~\ref{subsec:nonMarkcorr}. The issue of non-Markovian behavior,
its nature and distinction from Markovian approximations, is a
nontrivial and timely topic\cite{Wolf2008,Breuer2009,Rivas2009} which we only touch
upon briefly in this work, but our formalism paves the way for systematic studies of
such problems in the context of electronic noise and counting
statistics, for example as in Ref.~\onlinecite{Zedler2009}.
\subsection{Counting statistics}
In the following we introduce the
notion of cumulants of the charge transfer probability distribution,
and derive a formal expression for the cumulant generating function
CGF from the GME (\ref{eq:GME}). The probability
distribution for the number of transferred particles is obtained
from the $n$-resolved density by tracing over the system degrees of
freedom,
\begin{equation}
P(n,t)=\mathrm{Tr}\{\hatrho(n,t)\}.
\end{equation}
Obviously, probability must be conserved, such that $\sum_n P(n,t)=1$. In order to study the cumulants of
$P(n,t)$ it is convenient to introduce a cumulant generating function
(CGF) $S(\chi,t)$ via the definition
\begin{equation}
e^{S(\chi,t)}\equiv\sum_{n}P(n,t)e^{in\chi},
\end{equation}
from which the cumulants $\llangle n^m\rrangle$ follow as
derivatives with respect to the counting field $\chi$ at $\chi=0$,
\begin{equation}
\llangle n^m\rrangle(t)\equiv\left.\frac{\partial^mS(\chi,t)}{\partial(i\chi)^m}\right|_{\chi\rightarrow0}.
\end{equation}
Alternatively, one can write
\begin{equation}
e^{S(\chi,t)}=\mathrm{Tr}\{\hatrho(\chi,t)\},
\end{equation}
which defines the $\chi$-dependent density matrix
\begin{equation}
\hatrho(\chi,t)\equiv\sum_{n}\hatrho(n,t)e^{in\chi}.
\end{equation}
By going to Laplace space via the transformation
\begin{equation}
\hatrho(\chi,z)\equiv\int_0^{\infty}dt\hatrho(\chi,t)e^{-zt},
\end{equation}
Equation (\ref{eq:GME}) transforms to an algebraic equation reading
\begin{equation}
z\hatrho(\chi,z)-\hatrho(\chi,t=0)=\W(\chi,z)\hatrho(\chi,z)+\hatgamma(\chi,z).
\label{eq:EOMinLaplace}
\end{equation}
This equation can be solved formally by introducing the resolvent
\begin{equation}
\mathcal{G}(\chi,z)\equiv[z-\W(\chi,z)]^{-1}.
\end{equation}
and writing
\begin{equation}
\hatrho(\chi,z)=\mathcal{G}(\chi,z)[\hatrho(\chi,t=0)+\hatgamma(\chi,z)].
\end{equation}
Finally, inverting the Laplace transform using the Bromwich integral
we obtain for the CGF\cite{Flindt2008}
\begin{equation}
e^{S(\chi,t)}=\frac{1}{2\pi
i}\int_{a-i\infty}^{a+i\infty}\!\!\!\!\!\!\!\!
dz\,\mathrm{Tr}\{\mathcal{G}(\chi,z)
[\hatrho(\chi,t=0)+\hatgamma(\chi,z)]\} e^{zt}, \label{eq:CGF2}
\end{equation}
where $a$ is larger than the real parts of all singularities of the
integrand.
Equation (\ref{eq:CGF2}) is a powerful formal result for the CGF, and, as we shall see,
it also leads to practical schemes for calculating current fluctuations.
In this work, we concentrate on the zero-frequency cumulants,
determined by the long-time limit of the CGF. The case of
finite-frequency noise,\cite{Flindt2008} where the inhomogeneity
$\hatgamma(\chi,z)$ plays an important role, will be considered
in future works.
\subsection{Notational details}
Throughout this paper we will use the superoperator notation
previously described in Ref.\ \onlinecite{Flindt2004}
and also used in a number of other
works.\cite{Flindt2005,Flindt2005b,Huebener2007,Flindt2008,Brandes2008,Harvey2008,Koerting2009,Huebener2009,Emary2009a,Emary2009b,Schaller2009,Harvey2009,Urban2009,Lindebaum2009,Zhong2009,Wu2010,Dominguez2010}
Using this notation, standard linear algebra operations
can conveniently be performed, analytically and numerically. Within
the formalism, the memory kernel $\W$, the resolvent $\G$, and other
operators that act linearly on density matrices, are referred to as
superoperators and denoted by calligraphic characters. Conventional
quantum mechanical operators, like the density matrix $\hatrho$,
acting in the conventional quantum mechanical Hilbert space, can be
considered themselves to span a Hilbert space, referred to as the
superspace. The superoperators act in the superspace, while
conventional quantum mechanical operators are considered as vectors
using a bra(c)ket notation, i.e.,
$\hat{V}\leftrightarrow|v\rrangle$, where $\hat{V}$ is a
conventional quantum mechanical operator, and $|v\rrangle$ is the
corresponding ket in the superspace. Double angle brackets are used
here in order to avoid confusion with conventional kets. In
numerical calculations, bras and kets are represented by vectors,
while superoperators are represented by matrices. The inner product
between bras and kets is defined as $\llangle
v|u\rrangle\equiv\mathrm{Tr}\{\hat{V}^{\dagger}\hat{U}\}$. Since the
involved superoperators, like $\W$ and $\G$, are not hermitian,
their eigenvalues are generally complex. In such cases, left and
right eigenvectors corresponding to a particular eigenvalue are not
related by hermitian conjugation. The left eigenvector, or bra,
corresponding to an eigenvalue $\lambda_k$ is therefore denoted with
a tilde, e.g.\ $\llangle \tilde{\lambda}_k|$, to avoid confusion
with the hermitian conjugate $|\lambda_k\rrangle^{\dagger}$ of the corresponding right eigenvector,
or ket, $|\lambda_k\rrangle$.
\section{Zero-frequency current cumulants}
\label{sec:zerofreqcumu}
In this section we derive the recursive method for evaluating
the zero-frequency current cumulants. We first define the zero-frequency
cumulants of the current as
\begin{equation} \llangle
I^m\rrangle\equiv\left.\frac{d}{dt}\llangle
n^m\rrangle(t)\right|_{t\rightarrow\infty}=\left.\frac{d}{dt}\frac{\partial^m
S(\chi,t)}{\partial(i\chi)^m}\right|_{\chi\rightarrow0,t\rightarrow\infty},
\label{eq:zerofreqcum}
\end{equation}
where $m=1,2,\ldots$. As we shall show below, the
cumulants of the passed charge become linear in $t$ at long times
such that $\llangle n^m\rrangle(t)\rightarrow \llangle I^m\rrangle
t$, and the zero-frequency current cumulants are thus intensive
quantities (with respect to time). Thus, in the long-time limit
$\llangle I^m\rrangle/\llangle I\rrangle=\llangle
n^m\rrangle/\llangle n\rrangle$, and we use these two normalized
quantities interchangeably throughout the paper.
In order to find the long-time limit of the CGF, we consider the
formal solution Eq.\ (\ref{eq:CGF2}). The memory kernel
$\W(\chi,z)$ is assumed to have a single isolated eigenvalue
$\lambda_0(\chi,z)$, which for $\chi,z=0$ is zero, corresponding to
the stationary limit of $\hatrho(t)$, i.e.,
$\hatrho(t)\rightarrow\hatrho^{\mathrm{stat}}$ for large $t$. Here,
$\hatrho^{\mathrm{stat}}$ is the normalized solution to
$\W(\chi=0,z=0)\hatrho^{\mathrm{stat}}=0$. We exclude cases, where
the zero-eigenvalue is degenerate due to two or more uncoupled
sub-systems.\cite{vankampen2007} In the bracket notation
$\hatrho^{\mathrm{stat}}$ is denoted as $|0\rrangle$. The
corresponding left eigenvector can be found be noting that the
memory kernel with $\chi=0$ conserves probability for any $z$.
This can be inferred from the GME in Laplace space: For normalized
density matrices with $\mathrm{Tr}\{\hatrho(\chi=0,t)\}=1$, we have
$\mathrm{Tr}\{\hatrho(0,z)\}=1/z$, and Eq.\ (\ref{eq:EOMinLaplace})
yields
\begin{equation}
\mathrm{Tr}\{\W(0,z)\hatrho(0,z)\}+\mathrm{Tr}\{\hat{\gamma}(0,z)\}=0.
\label{eq:probconserv}
\end{equation}
It is generally possible to choose an initial state such that $\mathrm{Tr}\{\hat{\gamma}(0,z)\}=0$. The kernel does not depend on the choice of initial state and since Eq.\ (\ref{eq:probconserv}) holds for any normalized density matrix $\hatrho(0,z)$ we
deduce that $\mathrm{Tr}\{\W(0,z)\,\bullet\}=0$. In the bracket notation this equality can be expressed as $\llangle\tilde{0}|\W(0,z)=0$
with the left eigenvector $\llangle\tilde{0}|$ in the superspace
corresponding to the identity operator $\hat{1}$ in the conventional
Hilbert space. This moreover implies that\cite{Braggio2006}
\begin{equation}
\lambda_0(0,z)=0 \,\,\mathrm{for\,\, all}\,\, z.
\label{eq:lambda0}
\end{equation}
We next examine the
eigenvalue $\lambda_0(\chi,z)$ which we assume evolves adiabatically
from $\lambda_0(0,0)=0$ with small $\chi$ and $z$. It is convenient
to introduce the mutually orthogonal projectors
\begin{equation}
\PP(\chi,z)=\PP^2(\chi,z)=|0(\chi,z)\rrllangle\tilde{0}(\chi,z)|
\end{equation}
and
\begin{equation}
\QQ(\chi,z)=\QQ^2(\chi,z)\equiv 1-\PP(\chi,z)
\end{equation}
with $\PP(\chi,z)$ developing adiabatically from
$\PP(0,0)\equiv|0\rrllangle\tilde{0}|$ for small $\chi$ and $z$. Here, $\llangle\tilde{0}(\chi,z)|$ and $|0(\chi,z)\rrangle$ are the
left and right eigenvectors corresponding to $\lambda_0(\chi,z)$,
which develop adiabatically from $\llangle\tilde{0}|$ and
$|0\rrangle$, respectively. In
terms of $\PP(\chi,z)$ and $\QQ(\chi,z)$ the memory kernel can be
partitioned as
\begin{equation}
\W(\chi,z)=\lambda_0(\chi,z)\PP(\chi,z)+\QQ(\chi,z)\W(\chi,z)\QQ(\chi,z).
\label{eq:parti1}
\end{equation}
In deriving this expression we used
\begin{equation}
\PP(\chi,z)\W(\chi,z)\PP(\chi,z)=\lambda_0(\chi,z)\PP(\chi,z).
\end{equation}
Using the partitioning, Eq.\ (\ref{eq:parti1}), the resolvent becomes
\begin{equation}
\G(\chi,z)=\frac{\PP(\chi,z)}{z-\lambda_0(\chi,z)}+\QQ(\chi,z)\frac{1}{z-\W(\chi,z)}\QQ(\chi,z).
\label{eq:partioning}
\end{equation}
For $\chi=0$ the first term of the resolvent has a simple pole at
$z=0$, which determines the long-time limit, i.e. it corresponds to
the stationary state $\hatrho^{\mathrm{stat}}$. We denote the pole at
$z=0$ by $z_0$. All singularities of the second term have negative real
parts and do not contribute in the long-time limit. Again, we assume
adiabatic evolution of the pole $z_0(\chi)$ from $z_0(0)=0$ with
small $\chi$, such that $z_0(\chi)$ is the particular pole that
solves\cite{Braggio2006,Flindt2008}
\begin{equation}
z_0-\lambda_0(\chi,z_0)=0 \label{eq:pole}.
\end{equation}
With small $\chi$, the other
singularities still have more negative real parts and the pole
$z_0(\chi)$ again determines the long-time behavior. From Eq.\
(\ref{eq:CGF2}) we then find for large $t$
\begin{equation}
e^{S(\chi,t)}\rightarrow D(\chi,z_0)e^{z_0(\chi)t}, \label{eq:CGF3}
\end{equation}
where
\begin{equation}
D(\chi,z_0)=\mathrm{Tr}\{\PP(\chi,z_0)[\hatrho(\chi,t=0)+\hat{\gamma}(\chi,z_0)]\}.
\end{equation}
From the definition of the zero-frequency current cumulants in Eq.\
(\ref{eq:zerofreqcum}) we then establish that
\begin{equation}
z_0(\chi)=\sum_{n=1}^{\infty}\frac{(i\chi)^{n}}{n
!}\llangle I^{n} \rrangle.
\label{eq:poleCGF}
\end{equation}
We note that the CGF in the long-time limit and thus the
zero-frequency cumulants do not depend on the initial state
$\hatrho(\chi,t=0)$ and the inhomogeneity $\hat{\gamma}(\chi,z)$. In contrast, both $\hatrho(\chi,t=0)$ and $\hat{\gamma}(\chi,z)$ must be appropriately incorporated in order to calculate the finite-frequency noise.\cite{Flindt2008} Equations (\ref{eq:pole}) and (\ref{eq:poleCGF}) form the main
theoretical result of this section, generalizing earlier results for
Markovian systems.\cite{Bagrets2003,Flindt2005} In the Markovian
limit, the memory kernel and the corresponding eigenvalue close to 0
have no $z$-dependence, and Eq.\ (\ref{eq:pole}) immediately yields
$z_0(\chi)=\lambda_0(\chi)$,\cite{Bagrets2003,Flindt2005} where $\lambda_0(\chi)$ is the
eigenvalue of the $z$-independent kernel, which goes to zero with
$\chi$ going to zero, i.e. $\lambda_0(0)=0$.
Although, we have formally derived an expression for the CGF, it may
in practice, given a specific memory kernel $\W(\chi,z)$, be
difficult to determine the eigenvalue $\lambda_0(\chi,z)$ including
its dependence on $\chi$ and $z$. Moreover, the solution of Eq.\
(\ref{eq:pole}) itself poses an additional problem, which needs to
be addressed in the non-Markovian case. In the Markovian limit, only
derivatives of the eigenvalue $\lambda_0(\chi)$ with respect to the
counting field $\chi$ need to be determined. However, with the superoperator $\W(\chi)$ being represented by a matrix of size $N\times N$, there is no closed-form expression for the eigenvalue $\lambda_0(\chi)$ already with $N>4$. The immediate alternative
strategy would then be to calculate numerically the eigenvalue and the derivatives with respect to $\chi$.
Typically, however, this is a numerically unstable
procedure, which is limited to the first few
derivatives.\cite{Press2007} Consequently, we devote the rest of
this section to the development of a numerically stable, recursive
scheme that solves Eqs.\ (\ref{eq:pole}) and (\ref{eq:poleCGF}) for
high orders of cumulants, including in the non-Markovian case.
\subsection{Recursive scheme}
\subsubsection{The Markovian case}
We consider first the Markovian case,\cite{Flindt2005,Baiesi2009}
before proceeding with the general non-Markovian case. In the
Markovian case, the memory kernel $\W$ has no $z$-dependence, and
the current cumulants are determined by the eigenvalue
$\lambda_0(\chi)$ which solves the eigenvalue problem
\begin{equation}
\W(\chi)|0(\chi)\rrangle=\lambda_0(\chi)|0(\chi)\rrangle,
\label{eq:eigprob1}
\end{equation}
where $\lambda_0(0)=0$. We find the eigenvalue using perturbation
theory in the counting field $\chi$ in a spirit similar to that of
standard Rayleigh-Schr\"{o}dinger perturbation theory. To this end
we introduce the unperturbed operator $\W\equiv\W(0)$ and the
perturbation $\Delta\W(\chi)$ such that
\begin{equation}
\W(\chi)=\W+\Delta\W(\chi).
\end{equation}
We can then write
\begin{equation}
\lambda_0(\chi)=\llangle\tilde{0}|\Delta\W(\chi)|0(\chi)\rrangle,
\label{eq:eigenvalue}
\end{equation}
where we have used $\llangle\tilde{0}|\W=0$ and chosen the
conventional normalization $\llangle\tilde{0}|0(\chi)\rrangle=1$. We
moreover employ the shorthand notation
$\PP=\PP^2\equiv\PP(0,0)=|0\rrllangle\tilde{0}|$ and
$\QQ=\QQ^2\equiv 1-\PP$ for the projectors introduced in the
previous section, and write
\begin{equation}
|0(\chi)\rrangle=|0\rrangle+\QQ|0(\chi)\rrangle, \label{eq:vector}
\end{equation}
consistently with the choice of normalization. Using that
$\W=\QQ\W\QQ$, Eq.\ (\ref{eq:eigprob1}) can be written
\begin{equation}
\QQ\W\QQ|0(\chi)\rrangle=[\lambda_0(\chi)-\Delta\W(\chi)]|0(\chi)\rrangle.
\label{eq:QWQ}
\end{equation}
Next, we introduce the pseudo-inverse\cite{Flindt2004,Flindt2005}
defined as\footnote{We note that our definition of the
pseudo-inverse differs from the Moore-Penrose pseudo-inverse $\RR_{M\!P}$, which
is implemented in many numerical software packages, e.g., in Matlab.
However, when projected on the regular subspace by
$\QQ$ it reduces to our pseudo-inverse, i.e. $\QQ\RR_{M\!P}\QQ=\RR$.}
\begin{equation}
\RR=\QQ\W^{-1}\QQ.
\end{equation}
The pseudo-inverse is a well-defined object, since the inversion is
performed in the subspace corresponding to $\QQ$, where $\W$ is
regular. By applying $\RR$ on both sides of Eq.\ (\ref{eq:QWQ}) we
find
\begin{equation}
\QQ|0(\chi)\rrangle=\RR[\lambda_0(\chi)-\Delta\W(\chi)]|0(\chi)\rrangle,
\end{equation}
which combined with Eq.\ (\ref{eq:vector}) yields
\begin{equation}
|0(\chi)\rrangle=|0\rrangle+\RR[\lambda_0(\chi)-\Delta\W(\chi)]|0(\chi)\rrangle.
\label{eq:eigenvector}
\end{equation}
Equations (\ref{eq:eigenvalue}) and (\ref{eq:eigenvector}) form the basis of
the recursive scheme developed below.
We first Taylor expand the eigenvalue $\lambda_0(\chi)$, the
eigenvector $|0(\chi)\rrangle$, and the perturbation
$\Delta\W(\chi)$, around $\chi=0$ as
\begin{equation}
\begin{split}
\lambda_0(\chi)=&\sum_{n=1}^{\infty}\frac{(i\chi)^n}{n!}\llangle I^n\rrangle,\\
|0(\chi)\rrangle=&\sum_{n=0}^{\infty}\frac{(i\chi)^n}{n!}|0^{(n)}\rrangle,\\
\Delta\W(\chi)=&\sum_{n=1}^{\infty}\frac{(i\chi)^n}{n!}\W^{(n)},
\end{split}
\end{equation}
where we have used that $\lambda_0(0)=0$ and $\Delta\W(0)=0$. Inserting these expansions into
Eqs.\ (\ref{eq:eigenvalue}) and (\ref{eq:eigenvector}), and
collecting terms to same order in $\chi$, we arrive at a recursive
scheme reading
\begin{equation}
\begin{split}
\llangle I^n\rrangle_=&\sum_{m=1}^n{n\choose m}\llangle\tilde{0}|\W^{(m)}|0^{(n-m)}\rrangle,\\
|0^{(n)}\rrangle = &\mathcal{R}\sum_{m=1}^n{n \choose m}
\left[\llangle I^{m}\rrangle-\W^{(m)}\right]|0^{(n-m)}\rrangle,
\end{split}
\label{eq:recursivescheme}
\end{equation}
for $n=1,2,\ldots$. The recursive scheme allows for systematic calculations of
cumulants of high orders.
As illustrative examples we evaluate the first three current
cumulants using the recursive scheme,
\begin{equation}
\begin{split}
\llangle
I^1\rrangle_M=&\llangle\tilde{0}|\W^{(1)}|0\rrangle,\\
\llangle I^2\rrangle_M=&\llangle\tilde{0}|\left(\W^{(2)}-2\W^{(1)}\RR\W^{(1)}\right)|0\rrangle,\\
\llangle I^3\rrangle_M=&\llangle\tilde{0}|\left(\W^{(3)}+6\W^{(1)}\RR\W^{(1)}\RR\W^{(1)}\right.\\
&-3\{\W^{(2)}\RR\W^{(1)}+\W^{(1)}\RR\W^{(2)}\}\\
&\left.-6\llangle I^1\rrangle_M\W^{(1)}\RR^2\W^{(1)}\right) |0\rrangle,\\
\end{split}\label{eq:firstthree}
\end{equation}
having used
$|0\rrangle\equiv|0^{(0)}\rrangle$ and $\RR|0\rrangle=0$, since
$\QQ|0\rrangle=0$. The subscript $M$ reminds us that these results hold
for the Markovian case. The expressions (\ref{eq:firstthree}) for
the first three cumulants are equivalent to the ones derived in
Ref.\ \onlinecite{Flindt2005}, albeit using a slightly different
notation. Importantly, the recursive scheme presented here allows
for an easy generation of higher order cumulants, either
analytically or numerically.
\subsubsection{The non-Markovian case}
We now proceed with the non-Markovian case, where we first need to
consider the eigenvalue problem
\begin{equation}
\W(\chi,z)|0(\chi,z)\rrangle=\lambda_0(\chi,z)|0(\chi,z)\rrangle
\label{eq:eigprob2}
\end{equation}
where $\lambda_0(\chi,z)$ is the particular eigenvalue for which
$\lambda_0(0,z)=0$. The basic equations, Eqs.\ (\ref{eq:eigenvalue})
and (\ref{eq:eigenvector}), are still valid, provided that
$\lambda_0(\chi)$, $|0(\chi)\rrangle$, $\W$, and $\Delta\W(\chi)$
are replaced by $\lambda_0(\chi,z)$, $|0(\chi,z)\rrangle$,
$\W=\W(0,0)$, and $\Delta\W(\chi,z)=\W(\chi,z)-\W(0,0)$,
respectively, i.e.,
\begin{equation}
\lambda_0(\chi,z)=\llangle\tilde{0}|\Delta\W(\chi,z)|0(\chi,z)\rrangle,
\label{eq:eigenvalue2}
\end{equation}
and
\begin{equation}
|0(\chi,z)\rrangle=|0\rrangle+\RR[\lambda_0(\chi,z)-\Delta\W(\chi,z)]|0(\chi,z)\rrangle.
\label{eq:eigenvector2}
\end{equation}
Again, we Taylor expand all objects around $\chi=0$, but in this
case also around $z=0$,
\begin{equation}
\begin{split}
\lambda_0(\chi,z)=&\sum_{n,l=0}^{\infty}\frac{(i\chi)^n}{n!}\frac{z^l}{l!}c^{(n,l)},\\
|0(\chi,z)\rrangle=&\sum_{n,l=0}^{\infty}\frac{(i\chi)^n}{n!}\frac{z^l}{l!}|0^{(n,l)}\rrangle,\\
\Delta\W(\chi,z)=&\sum_{n,l=0}^{\infty}\frac{(i\chi)^n}{n!}\frac{z^l}{l!}\W^{(n,l)},
\end{split}
\label{eq:expansions}
\end{equation}
with $\W^{(0,0)}=0$ by definition and $c^{(0,l)}=0$, since
$\lambda_0(0,z)=0$. Inserting these expansions into Eqs.\
(\ref{eq:eigenvalue2}) and (\ref{eq:eigenvector2}) and collecting
terms to same orders in $\chi$ and $z$, we find a recursive scheme
reading
\begin{equation}
\begin{split}
c^{(n,l)}=&\sum_{m=1}^n{n\choose m}\sum_{k=0}^l{l\choose k}\llangle\tilde{0}|\W^{(m,k)}|0^{(n-m,l-k)}\rrangle,\\
|0^{(n,l)}\rrangle = &\mathcal{R}\sum_{m=0}^n\!{n \choose
m}\!\sum_{k=0}^l\!{l \choose
k}\!\left[c^{(m,k)}\!-\!\W^{(m,k)}\right]\!|0^{(n-m,l-k)}\rrangle.
\end{split}
\label{eq:recursivescheme1}
\end{equation}
In case the memory kernel has no $z$-dependence, corresponding to
the Markovian case, only terms with $l=0$ are non-zero, and the
recursive scheme reduces to the one given in Eq.\
(\ref{eq:recursivescheme}). In particular, the coefficients
$c^{(n,0)}$ equal the current cumulants $\llangle I^n\rrangle_M$ in
the Markovian limit of the kernel, $z\rightarrow 0$.
In the non-Markovian case, we need to proceed with the solution of
Eq.\ (\ref{eq:pole}) for $z_0$ and extract the current cumulants
$\llangle I^{n}\rrangle$. Inserting the expression for $z_0$ in Eq.\
(\ref{eq:poleCGF}) into Eq.\ (\ref{eq:pole}) and using the expansion
of $\lambda_0(\chi,z)$ given in Eq.\ (\ref{eq:expansions}), we find
\begin{equation}
\sum_{n=1}^{\infty}\frac{(i\chi)^{n}}{n !}\llangle
I^{n}\rrangle=\sum_{k,l=0}^{\infty}\frac{(i\chi)^k}{k!}\frac{1}{l!}\left\{\sum_{n=1}^{\infty}\frac{(i\chi)^{n}
}{n !}\llangle I^{n}\rrangle\right\}^lc^{(k,l)}.
\end{equation}
Collecting terms to same order in $\chi$, we find
\begin{equation}
\llangle
I^n\rrangle=n!\sum_{k,l=0}^n\frac{1}{k!}\frac{1}{l!}P^{(n-k,l)}c^{(k,l)},
\label{eq:recursivescheme2}
\end{equation}
in terms of the auxiliary quantity
\begin{equation}
P^{(k,l)}\equiv\sum_{\substack{n_1,\ldots,n_l=1\\
n_1+\ldots+n_l=k}}^{k}\frac{\llangle
I^{n_1}\rrangle}{n_1!}\cdots\frac{\llangle
I^{n_l}\rrangle}{n_l!},\,\, l\geq 1,
\end{equation}
where only terms in the sums for which $n_1+\ldots+n_l=k$ should be
included. For $l=0$, we have $P^{(k,0)}\equiv\delta_{k,0}$. The
auxiliary quantity can also be evaluated recursively by noting that
\begin{equation}
P^{(k,l)}=\sum_{n=1}^{k}\frac{\llangle
I^{n}\rrangle}{n!}P^{(k-n,l-1)}, \label{eq:recursivescheme3}
\end{equation}
with the boundary conditions $P^{(k,0)}=\delta_{k,0}$,
$P^{(0,l)}=\delta_{0,l}$, and $P^{(k,-1)}\equiv 0$.
When combined, Eqs.\ (\ref{eq:recursivescheme1},
\ref{eq:recursivescheme2}, \ref{eq:recursivescheme3}) constitute a
recursive scheme which allows for numerical or analytic calculations
of cumulants of high orders in the general non-Markovian case. As
simple examples, we show the first three cumulants\cite{Flindt2008} obtained from
Eqs.\ (\ref{eq:recursivescheme2}), (\ref{eq:recursivescheme3}), in
terms of the coefficients $c^{(n,l)}$
\begin{equation}
\begin{split}
\llangle
I^1\rrangle =& c^{(1,0)},\\
\llangle I^2\rrangle =& c^{(2,0)}+2c^{(1,0)}c^{(1,1)},\\
\llangle I^3\rrangle =& c^{(3,0)}+3 c^{(2,0)}c^{(1,1)}\\
&+3c^{(1, 0)}\left[c^{(1, 0)} c^{(1, 2)}+2(c^{(1,1)})^2+ c^{(2,1)}\right].\\
\end{split}
\label{eq:nonMarkovcumulants}
\end{equation}
In general, the $n$'th current cumulant $\llangle I^n\rrangle$
contains the coefficients
\begin{equation}
c^{(k,l)}=\partial_{(i\chi)}^k\partial_{z}^l\lambda_0(\chi,z)|_{\chi,z\rightarrow
0}
\end{equation}
with $1\leq k+l\leq n$. However, coefficients of the form
$c^{(0,l)}$ are zero since $\lambda_0(0,z)\equiv 0$ as discussed
below Eq.~\eqref{eq:probconserv} and it thus suffices to consider
$l\leq n-1$. From Eq.~\eqref{eq:recursivescheme1} it follows that
$c^{(k,l)}$ depend only on $\W^{(m,n)}$ with $m\leq k$ and $n\leq l$
so that we can conclude that the $n$'th cumulant of the current
depends at maximum on the $(n-1)$'th time-moment of the memory
kernel $\int_0^{\infty}dt \,t^{n-1} \W(\chi,t)$. In particular, this
implies that the mean current is a purely Markovian quantity
depending only on the time-integrated memory kernel while the second
and higher order cumulants deviate from the results in the Markovian
case.\cite{Braggio2006}
The coefficients $c^{(n,l)}$ can be found from Eq.\
(\ref{eq:recursivescheme1}). Coefficients of the form $c^{(n,0)}$ only contain zeroth order terms
in $z$ and are, as already mentioned, equal to the current cumulants
$\llangle I^n\rrangle_M$ in the Markovian limit, i.e.,
\begin{equation}
c^{(n,0)}=\llangle I^n\rrangle_M,\,\, n=1,2,3,\ldots.
\end{equation}
For the other coefficients entering the expressions in Eq.\ (\ref{eq:nonMarkovcumulants}) for the first three non-Markovian current cumulants, we find
\begin{equation}
\begin{split}
c^{(1,1)}=&\llangle\tilde{0}|\left(\W^{(1,1)}-\W^{(1,0)}\RR\W^{(0,1)}\right)|0\rrangle,\\
c^{(1,2)}=&\llangle\tilde{0}|\left(\W^{(1,2)}-2\W^{(1,1)}\RR\W^{(0,1)}-\W^{(1,0)}\RR\W^{(0,2)}\right.\\
&\left.+2\W^{(1,0)}\RR\W^{(0,1)}\RR\W^{(0,1)}\right)|0\rrangle,\\
c^{(2,1)}=&\llangle\tilde{0}|\left(\W^{(2,1)}+2\W^{(1,0)}\RR\W^{(0,1)}\RR\W^{(1,0)}\right.\\
&+2\W^{(1,0)}\RR\W^{(1,0)}\RR\W^{(0,1)}-2\W^{(1,1)}\RR\W^{(1,0)}\\
&\left.-2\W^{(1,0)}\RR\W^{(1,1)}-\W^{(2,0)}\RR\W^{(0,1)}\right)|0\rrangle.\\
\end{split}
\label{eq:nonMarkovcoeff}
\end{equation}
Again, as in the Markovian case, higher order cumulants including the coefficients $c^{(k,l)}$ are readily
generated, analytically or numerically. The results presented here can be generalized to the
statistics of several different counted quantities as in Ref.\ \onlinecite{Sanchez2007,Sanchez2008b}, and cross-correlations can be evaluated using the same compact notation developed in this work.\cite{Braggio2009b}
\subsection{Notes on evaluation}
\label{subsec:evaluation}
As previously mentioned, the size of the memory kernel $\W(\chi,z)$
could in practice hinder the calculation of $\lambda_0(\chi,z)$ and
the solution of Eq.\ (\ref{eq:pole}), and thus the evaluation of the
current cumulants. The recursive scheme described above,
however, only relies on the ability to solve matrix equations and
perform matrix multiplications. Both of these operations are
numerically feasible and stable, even when the involved matrices are of large
dimensions. In general, the recursive scheme requires the following
steps: The stationary state must be found by solving
\begin{equation}
\W|0\rrangle=0, \label{eq:hom_eq}
\end{equation}
with the normalization requirement
$\llangle\tilde{0}|0\rrangle=\mathrm{Tr}\{\hat{1}^{\dagger}\hatrho^{\mathrm{stat}}\}=1$.
Secondly, the $\chi$ and $z$ derivatives of the memory kernel must
be found
\begin{equation}
\W^{(n,l)}=\left.\partial^n_{(i\chi)}\partial^l_{z}\W(\chi,z)\right|_{\chi,z\rightarrow
0}
\end{equation}
for $(n,l)\neq (0,0)$. Typically, the dependence on the counting
field $\chi$ enters matrix elements in an exponential function (see
e.\ g.\ Refs.\ \onlinecite{Nazarov2003}, \onlinecite{Bagrets2003}, \onlinecite{Braggio2006} and
examples in Secs. \ref{sec:Markexamples} and \ref{sec:nonMarkov}), e.\ g.\
as a factor of $e^{i\chi}$, for which the derivatives with respect
to $\chi$ are easily found analytically. The $z$-dependence of the
matrix element $[\W(\chi,z)]_{kj}$ can be written
\begin{equation}
[\W(\chi,z)]_{kj}=\int_0^{\infty}dt [\W(\chi,t)]_{kj} e^{-zt},
\end{equation}
such that
\begin{equation}
[\W^{(n,l)}]_{kj}=\int_0^{\infty}dt (-t)^l
\left[\partial^n_{(i\chi)}\W(\chi,t)|_{\chi\rightarrow
0}\right]_{kj}.
\end{equation}
The integration over time can be performed in a numerically stable
manner for arbitrary $n$,\cite{Press2007} thereby avoiding taking
numerical derivatives with respect to $z$.
Finally, matrix multiplications have to be performed. Here, special
attention has to be paid to terms involving the pseudo-inverse
$\RR$, i.e.\ $\RR|x\rrangle$, where $|x\rrangle$ for example has the
form $\W^{(0,1)}|0\rrangle$ in the expression for the coefficient $c^{(1,1)}$ in Eq.\ (\ref{eq:nonMarkovcoeff}). In order to evaluate such expressions we introduce
$|y\rrangle$ as the solution(s) to
\begin{equation}
\W|y\rrangle=\QQ|x\rrangle, \label{eq:Req}
\end{equation}
such that
\begin{equation}
\QQ|y\rrangle=\RR|x\rrangle, \label{eq:RRx}
\end{equation}
which can be verified by applying $\RR$ on both sides of Eq.\
(\ref{eq:Req}) and using that $\RR\W=\QQ\W^{-1}\QQ\W=\QQ$ and
$\RR\QQ=\RR$. The projector $\QQ$ in Eq.\ (\ref{eq:Req}) ensures
that the right hand side lies in the range of $\W$, and since $\W$
is singular, the equation has infinitely many solutions. The
solutions can be written
\begin{equation}
|y\rrangle=|y_0\rrangle+c|0\rrangle,\,\, c\in\mathbb{C},
\end{equation}
where $|y_0\rrangle$ is a particular solution to Eq.\
(\ref{eq:Req}), which can be found numerically. We then obtain
$\RR|x\rrangle$ by applying $\QQ$ to $|y\rrangle$ according to Eq.\
(\ref{eq:RRx}) and find
\begin{equation}
\RR|x\rrangle=\QQ\left(|y_0\rrangle+c|0\rrangle\right)=\QQ|y_0\rrangle,
\end{equation}
since $\QQ|0\rrangle=0$.
In App.\ \ref{app:QR} we describe a simple numerical
algorithm for solving Eqs.\ (\ref{eq:hom_eq}) and (\ref{eq:Req}).
For very large dimensions of the involved matrices, it may
be necessary to invoke more advanced numerical methods to solve
these equations.\cite{Flindt2004} Numerically, the recursive scheme
is stable for very high orders of cumulants (up to order $\sim100$), which we have
tested on simple models. The results presented in this work have all
been obtained using standard numerical methods as the one described
in App.\ \ref{app:QR}.
\section{Asymptotics of high-order cumulants}
\label{sec:univosc}
Before illustrating our methods in terms of specific examples,
we discuss the asymptotic behavior of
high-order cumulants. As some of us have recently shown certain
ubiquitous features are expected for the high-order
cumulants.\cite{Flindt2009} In particular, the absolute values
of the high-order cumulants are expected to grow factorially with the cumulant
order. Moreover, the high-order cumulants are predicted to oscillate as functions of
basically any parameter, as well as of the cumulant order. This
behavior was confirmed experimentally by measurements of the
high-order transient cumulants of electron transport through a
quantum dot.\cite{Flindt2009} In the experiment, the transient
cumulants indeed grew factorially with the cumulant order and
oscillated as functions of time (before reaching the long-time
limit), in agreement with the general prediction. For completeness,
we repeat here the essentials of the theory underlying these
asymptotic properties of high-order cumulants.
The asymptotic behavior of high-order cumulants follows from
straightforward considerations. In the following we denote the CGF
by $S(\chi,\{\lambda\})$, where $\{\lambda\}$ represents the set of
all parameters needed to specify the system; whether the dynamics
is Markovian or non-Markovian is irrelevant. In general, we can
assume that the CGF has a number of singularities in the
complex-$i\chi$ plane at $i\chi=i\chi_j$, $j=1,2,3\ldots$, which can
be either poles or branch-points. Typically, the positions of the
singularities depend on $\{\lambda\}$. Exceptions, where the CGF
has no singularities, do exist, e.\ g.\ the Poisson process, whose CGF
is given by an exponential function, but we exclude such cases in
the following.
Close to a singularity $i\chi\simeq i\chi_j$, we can write the CGF as
\begin{equation}\label{eq:singularity}
S(\chi,\{\lambda\})\simeq \frac{A_j}{(i\chi_j-i\chi)^{\mu_j}}
\end{equation}
for some $A_j$ and $\mu_j$, determined by the nature of the
singularity. For example, for a finite-order pole $\mu_j$ denotes the order of the pole, while $\mu_j=-1/2$ would correspond to the branch point of a square-root function. Logarithmic singularities can be treated on a similar footing with only slight modifications.\cite{Flindt2009} The derivatives with respect to the counting field are now
\begin{equation}
\frac{\partial^m S(\chi,\lambda)}{\partial(i\chi)^m}\simeq
\frac{A_jB_{m,\mu_j}}{(i\chi_j-i\chi)^{m+\mu_j}}
\end{equation}
with
\begin{equation}
B_{m,\mu_j}\equiv\mu_j(\mu_j+1)\cdots (\mu_j+m-1) \label{eq:Bmu}
\end{equation}
for $m\geq1$. As the order $m$ is increased this approximation
becomes better away from the singularity at
$\chi=\chi_j$ according to the Darboux theorem.\cite{Dingle1973,Berry2005,Flindt2009} For sufficiently high $m$,
the cumulants of the passed charge can thus be written
\begin{equation}
\begin{split}
\llangle
n^m\rrangle&=\left.\frac{\partial^mS(\chi,\lambda)}{\partial(i\chi)^m}\right|_{\chi\rightarrow0}\\
&\simeq\sum_j\frac{A_jB_{m,\mu_j}}{|i\chi_j|^{m+\mu_j}}e^{-i(m+\mu_j)\arg(i\chi_j)},
\end{split}
\label{eq:uniosc}
\end{equation}
where the sum runs over all singularities of the CGF. Here, we have written the singularities as
\begin{equation}
i\chi_j=|i\chi_j|e^{i\arg(i\chi_j)},
\end{equation}
where $|i\chi_j|$ is the modulus of the singularity $i\chi_j$ and
$\arg(i\chi_j)$ is the corresponding complex argument. In general,
the singularities $i\chi_j$ together with the factors $A_j$ come in complex conjugate pairs, ensuring that
the expression in Eq.\ (\ref{eq:uniosc}) is real.
From Eq.\ (\ref{eq:uniosc}) we deduce that the cumulants grow factorially
in magnitude with the order $m$ due to the factors $B_{m,\mu_j}$ given in Eq.\ (\ref{eq:Bmu}). We
also see that the high-order cumulants are determined primarily by
the singularities closest to zero. Contributions from other
singularities are suppressed with the relative
distance from zero and the order $m$, and can thus be neglected for
large $m$. Importantly, we observe
that the high-order cumulants become oscillatory functions of
\emph{any} parameter among $\{\lambda\}$ that changes
$\arg(i\chi_j)$ as well as of the cumulant order $m$ [see also Eq.\ (\ref{eq:twosingexp}) below]. We refer to these ubiquitous features, which should occur in a large class of
transport processes, as universal oscillations. For example, we expect oscillations of high-order cumulants for basically any transport process described by a GME, since the CGF for these systems typically have logarithmic singularities at finite times\cite{Flindt2009} or square-root branch points in the long-time limit.\cite{Bender1978} Factorial growth and oscillations as functions of various parameters can be found in several independent studies of high-order cumulants,\cite{Pilgram2003,Foerster2005,Foerster2007,Flindt2008,Urban2008,Khoruzhenko2009,Prolhac2009,Golubev2010,Hassler2010} as well as in the recent experiment described in Ref.\ \onlinecite{Flindt2009}, demonstrating the generality of the phenomenon. Similar observations and discussions can also be found in quantum optics\cite{Dodonov1994} and high-energy physics,\cite{Dremin1994,Bhalerao2003,Bhalerao2004} further confirming the prediction. We note that in the long-time limit, the positions of the dominating singularities are
no longer time dependent,\cite{Flindt2009} and the cumulants cease
to oscillate as functions of time. Instead, the cumulants of the
passed charge become linear in time, as previously discussed in Sec.\
\ref{sec:zerofreqcumu}.
A simple (and common) situation arises if only two complex conjugate singularities, $|i\chi_0|e^{i\arg{i\chi_0}}$ and $|i\chi_0|e^{-i\arg{i\chi_0}}$, are closest to zero.
In that case, Equation (\ref{eq:uniosc}) immediately yields
\begin{equation}\label{eq:twosingexp}
\llangle n^m\rrangle\simeq \!\frac{2|A_0|
B_{m,\mu_0}}{|i\chi_0|^{m+\mu_0}}\cos\!\left[(m\!+\!\mu_0)\arg i\chi_0\!-\!\arg
A_0\right].
\end{equation}
Using this expression we can determine the positions of the dominating singularities from numerical calculations of the high-order cumulants as we shall demonstrate in the second example considered in Sec.\ \ref{sec:Markexamples}. We note that while the factorial growth and the oscillations are system independent, other features, for example the frequency of the oscillations, are determined by the particular details of the system under consideration.
Finally, we mention the Perron-Frobenius theorem regarding stochastic matrices\cite{Baiesi2009,Demboo1998} which implies that the CGFs considered in Sec.\ \ref{sec:Markexamples} must be analytical functions at least in a strip along the real axis in the
complex-$i\chi$ plane. This has important consequences especially for the nature of the high-order
cumulants which rests heavily on the analytical properties of the CGF. We illustrate this statement in both examples in Sec.\ \ref{sec:Markexamples}.
\section{Markovian Systems}
\label{sec:Markexamples}
\subsection{Electron bunching in a two-level quantum dot}
\label{sec:Belzig}
In our first example we study electron bunching in transport
through a two-level quantum dot as described by Belzig in
Ref.~\onlinecite{Belzig2005}. Due to the relatively simple
analytical structure of the model, it is possible to
illustrate the concepts of universal oscillations introduced
above. The model allows us to test the accuracy of our numerical
calculations of high-order cumulants against analytic
expressions.
We start by summarizing the setup in Belzig's model. Consider a
single quantum dot with two single-particle levels coupled to
voltage-biased source and drain electrodes. The two levels serve as
parallel transport channels. Due to strong Coulomb interactions on
the quantum dot only one of the levels can be occupied at a time.
The system exhibits super-Poissonian bunching transport in cases
where both levels are coupled by the same rate $\Gamma_L$ to the,
say, left lead, whose Fermi level is kept well above both levels,
while the couplings to the other lead are markedly different, such
that one level is coupled to the right lead by the rate $\Gamma_R\ll
\Gamma_L$ and the other by $x\Gamma_R$ with $x\ll 1$. This situation
can arise for example, if the two levels are situated above and
slightly below, respectively, the Fermi level of the right lead at a
finite electron temperature.
This particular configuration leads to bunching of electrons in
the transport due to the existence of the blocking state: if the
dot is empty there is equal probability for either of the two
levels to be filled. Current runs easily through the first level,
while the other level effectively is blocked, or more precisely,
the transport through the level is limited by the very small right
rate $x\Gamma_R$, constituting a bottleneck. The transport thus proceeds in
bunches of electrons passing intermittently through the first
level separated by quiet periods of blocked transport when
the other level is occupied. This bunching effects leads to super-Poissonian noise with a
Fano factor above unity. For more detailed discussions of the model
as well as its generalizations to many levels, the reader is
referred to Ref.~\onlinecite{Belzig2005}.
The counting statistics of the system can be obtained from a
Markovian rate equation for the probability vector
$\hat{p}=(p_0,p_+,p_-)^T$, containing the ($n$-resolved)
probabilities $p_{0,-,+}$ for the quantum dot to be empty, or the
first ($+$, non-blocking) or second ($-$, blocking) level being
occupied, respectively. The corresponding $\chi$-dependent rate
matrix reads
\begin{equation}
\label{eq:Wolfgangmodel}
\W (\chi)
=\begin{pmatrix}
-2-\Gamma(1-x) & \Gamma e^{i\chi} & x\Gamma e^{i\chi}\\
1 & -\Gamma & 0\\
1+e^{-i\chi}\Gamma(1-x) & 0 & -x\Gamma
\end{pmatrix}
\end{equation}
Here, we have rescaled the time and set $\Gamma_L\equiv 1$ while
renaming $\Gamma_R\equiv\Gamma$ in order to simplify the analytic
results in the following. We have also made a minor modification
of the model in Ref.\ \onlinecite{Belzig2005} by including the
back-flow into the blocking level from the right lead. This
modification, however, changes only slightly the detailed
quantitative results, while leaving the main qualitative features
identical in the limit of interest $x,\Gamma\ll 1$.
Since the model involves only three states, the CGF can be found
analytically in the long-time limit. The full expression is too
lengthy to be presented here, but in the limit $x, \Gamma \ll 1$,
it reduces to the result by Belzig\cite{Belzig2005} (also for our
slightly modified model; note, however, the opposite sign
convention for the CGF in Ref.~\onlinecite{Belzig2005})
\begin{equation}
S(\chi,t)\rightarrow 2\Gamma xt\frac{e^{i\chi}-1}{2-e^{i\chi}}.
\label{eq:Belzigsinglepole}
\end{equation}
Clearly, the CGF has simple poles at
\begin{equation}
i\chi_j=\ln 2+j2\pi i,\,\,\, j=\ldots,-1,0,1,\ldots
\end{equation}
with the pole $i\chi_0=\ln 2$ being closest to 0. However, according to the Perron-Frobenius theorem mentioned in Sec.\ \ref{sec:univosc} the CGF cannot have singularities on the real $i\chi$-axis.
In order to illustrate this point, we consider the expected behavior
of the high-order cumulants based on the CGF above. Close to the
singularity $i\chi_0$, we approximate the CGF by the first non-zero term of the Laurent series
\begin{equation}
S(\chi,t)\simeq \frac{\Gamma x t}{i\chi_0-i\chi}.
\end{equation}
This corresponds to Eq.\ (\ref{eq:singularity}) with $A_0=\Gamma
x t$ and $\mu_0=1$. From Eq.\ (\ref{eq:uniosc}) we then obtain a
simple asymptotic expression for the high-order cumulants reading
\begin{equation}
\llangle I^m\rrangle_{\!1s}/\Gamma x=\llangle n^m\rrangle_{\!1s}/\Gamma x t \simeq m!/(\ln 2)^{m+1}.
\label{eq:Belzig1pApprox}
\end{equation}
Here, the subscript $_{1s}$ indicates that the expression has been
obtained using the approximate CGF in Eq.\ (\ref{eq:Belzigsinglepole}) with
only a single singularity closest to zero. In Table
\ref{table:comparison} we compare the asymptotic expression with
results for the first six cumulants obtained by direct
differentiation of the CGF in Eq.\ (\ref{eq:Belzigsinglepole}).
The asymptotic results are very close to the exact derivatives of
the approximate CGF. Despite the good agreement with the
approximate results, the asymptotic expression in Eq.\
(\ref{eq:Belzig1pApprox}) does not reproduce our numerically exact
results, also shown in the table, obtained using our recursive scheme. In particular for high orders,
the asymptotic expression starts to deviate significantly from the
numerically exact results.
\begin{table}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\hline
$\llangle I^m\rrangle/\Gamma x$ & $m=1$ & 2 & 3 & 4 & 5 & 6 \\
\hline
Single-pole approx. & 2.000 & 6.000 & 26.00 & 150.0 & 1082 & 9366\\
\hline
Single-pole asympt. & 2.081 & 6.006 & 25.99 & 150.0 & 1082 & 9366\\
\hline
Numerics & 1.978 & 5.880 & 25.18 & 143.3 & 1017 & 8644\\
\hline
\hline
\end{tabular}
\caption{Normalized zero-frequency current cumulants for transport
through a two-level quantum dot. Single-pole approximation results
have been obtained by direct differentiation of the CGF in Eq.\
(\ref{eq:Belzigsinglepole}) or its asymptotic expression Eq.\
(\ref{eq:Belzig1pApprox}), respectively. The numerically exact
results have been obtained using our recursive scheme and the rate
matrix in Eq.\ (\ref{eq:Wolfgangmodel}) with $x=0.001$ and
$\Gamma=0.01$.} \label{table:comparison}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=0.90\textwidth, trim = 0 0 0 0, clip]{bunchingorder.eps}
\caption{High-order cumulants and large deviation function for bunching
transport through a two-level quantum dot. Left and central panels show comparisons between exact
numerics and the single
pole approximation stemming from Eq.\ (\ref{eq:Belzigsinglepole}) for two different
values of $x=0.001,\,0.01$ and $\Gamma=0.01$. The asymptotic expression in Eq.\ (\ref{eq:twosingexp}) based on a pair of complex conjugate singularities is shown with full lines. Notice that $B_{m,-1/2}<0$. The right panel shows a comparison of the large deviation function (LDF) obtained from exact numerics and the single pole approximation in Eq.\ (\ref{eq:Belzig1pLDF}), respectively.} \label{fig:bunchingorder}
\end{center}
\end{figure*}
As anticipated above, these deviations can be traced back to the
expression in Eq.\ (\ref{eq:Belzigsinglepole}), that we obtained
in the limit $x, \Gamma \ll 1$. In order to proceed from here, we return to the full expression
for the CGF in the long-time limit (not shown). A careful analysis
reveals that, in fact, there is a pair of complex conjugate
singularities closest to zero, and not just a single pole. The two
singularities, denoted as $i\tilde{\chi}_0$ and
$(i\tilde{\chi}_0)^*$, correspond to branch points of a
square-root, and for small $x\ll1$ the position of the branch
point $i\tilde{\chi}_0$ is
\begin{equation}\label{eq:Belzigpoleposition}
i\tilde{\chi}_0 = \ln(2+\Gamma)-2 x\frac{4 + \Gamma (6 + \Gamma)}{(2 +
\Gamma)^2}+ 4i \sqrt{x}\frac{1 + \Gamma}{2 + \Gamma}.
\end{equation}
Clearly, for small $x,\Gamma\ll 1$ the branch points are close to
the position of the single pole $i\chi_0=\ln 2$. However, for any
finite $x$, the two branch points have small, but finite,
imaginary parts thus complying with the Perron-Frobenius theorem. The singularity structure around the branch point $i\tilde{\chi}_0$
is characterized by Eq.~\eqref{eq:singularity} with $\mu_0=-1/2$ and
$A_0\approx \Gamma t\sqrt[4]{x}e^{i\pi/4}$, and we can then use the asymptotic expressions in Eq.\ (\ref{eq:twosingexp}) for the high-order cumulants.
In the left and central panels of Fig.~\ref{fig:bunchingorder} we compare this expression, and the single-pole
approximation in Eq.\ (\ref{eq:Belzig1pApprox}), with numerically exact results obtained using our
recursive scheme for $\Gamma=0.01$ and two different values of $x=0.001,\,0.01$.
Figure~\ref{fig:bunchingorder} shows several important features.
Firstly, the (scaled) high-order cumulants indeed behave in an
oscillatory manner as function of the cumulant order $m$, which
coincides with the cosine part of Eq.\
(\ref{eq:twosingexp}). Obviously, for smaller $x=0.001$ the period
of the oscillations, determined by $\arg i\tilde{\chi}_0$, is longer
in accordance with Eq.\ (\ref{eq:twosingexp}). Furthermore,
for the small value of $x=0.001$, the asymptotic form of the
high-order cumulants is reached around $m\simeq 30$, while the
single-pole approximation agrees well for lower orders, $m\lesssim
10$. For the higher value of $x=0.01$, significant deviations from
the single-pole behavior begin already for the fourth cumulant,
while the asymptotic oscillatory form holds from around $m=12$.
Notice the importance of the exact analytical knowledge of the
singularities
--- even though $x=0.01\ll 1$ (together with $\Gamma=0.01\ll 1$) may
seem a very small number justifying the usage of the single pole
approximation, we see from Eq.\ (\ref{eq:Belzigpoleposition}) that
the imaginary part of the pole and its argument scale like
$\sqrt{x}=0.1$, thus invalidating the single-pole approximation
far earlier than expected from a linear-in-$x$ scaling assumption.
A complementary view on the charge transport statistics is provided by the large deviation function (LDF),\cite{Touchette2009} which quantifies deviations of measurable
currents from the average value. The LDF is obtained from the probability
distribution
\begin{equation}
P(n,t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}d\chi e^{S(\chi,t)-in\chi}
\end{equation}
and is defined as the long-time limit of $\ln[P(I,t)]/t$, where $I\equiv n/t$ is the current. For long times, we have $S(\chi,t)\rightarrow \lambda_0(\chi) t$ and the integral can be evaluated in the saddle-point approximation with the saddle-point $\chi=\chi_0$ given by the solution to the saddle-point equation
\begin{equation}
\lambda_0'(\chi_0)=iI, \label{eq:saddlepointeq}
\end{equation}
The saddle-point equation implies a parametric dependence of the saddle-point $\chi_0=\chi_0(I)$ on the current $I$. Using the saddle-point approximation, the LDF becomes
\begin{equation}
\frac{\ln [P(I,t)]}{t}\rightarrow \lambda_0(\chi_0)-iI\chi_0. \label{eq:largedeviation}
\end{equation}
We first solve the saddle-point equation for the approximate CGF in Eq.\ (\ref{eq:Belzigsinglepole}) and find
\begin{equation}
\frac{\ln [P_{1s}(I,t)]}{\langle I\rangle t}\rightarrow\frac{\sqrt{1+8 \kappa}-3}{4}
-\kappa\log\left[\frac{16\kappa}{(1+\sqrt{1+8\kappa})^2}\right], \label{eq:Belzig1pLDF}
\end{equation}
where $\kappa\equiv I/\langle I\rangle$ and the subscript $_{1s}$ again reminds us that the expression has been
obtained using the approximate CGF with only a single singularity closest to zero. Obviously, the current must be positive ($\kappa>0$), since transport is unidirectional.
Also for the LDF, we can compare the analytic approximation with numerical exact results. To this end, we need to solve
the saddle-point equation taking as starting point the kernel in Eq.\ (\ref{eq:Wolfgangmodel}).
The derivative of the eigenvalue $\lambda_0(\chi)$ is now calculated using the
Hellman-Feynman theorem, writing
\begin{equation}
\begin{split}
\lambda_0'(\chi)&=\frac{\partial}{\partial\chi}\llangle \tilde{0}(\chi)|\W(\chi)|0(\chi)\rrangle \\
&=\llangle \tilde{0}(\chi)|\W'(\chi)|0(\chi)\rrangle,
\end{split}
\label{eq:Hellman-Feynman}
\end{equation}
where $\llangle\tilde{0}(\chi)|$ and $|0(\chi)\rrangle$ are left and
right eigenvectors of $\W(\chi)$, respectively, corresponding to the
eigenvalue $\lambda_0(\chi)$, and
$\llangle\tilde{0}(\chi)|0(\chi)\rrangle=1$. For a given value of $\chi$ we calculate numerically the left and right eigenvectors $\llangle \tilde{0}(\chi)|$ and $|0(\chi)\rrangle$ and find $\lambda_0'(\chi)$ using the expression for the derivative in Eq.\ (\ref{eq:Hellman-Feynman}). With this procedure we search numerically for the value of $\chi=\chi_0$ that solves Eq.\ (\ref{eq:saddlepointeq}) for a given value of $I$, and with the solution $\chi_0$ we evaluate the LDF using Eq.\ (\ref{eq:largedeviation}). We find that $\chi_0$ is purely imaginary.\cite{Bagrets2003} We note that, in principle, the existence of a saddle point solution is not guaranteed in the whole range of currents, and there are examples,\cite{Bagrets2006,Visco2006} where the behavior of the LDF changes abruptly at finite values of $I$ due to singularities of the CGF on the real $i\chi$-axis. In our case, however, the Perron-Frobenius theorem ensures that the CGF is analytic on the real $i\chi$-axis and the LDF is smooth as function of $I$. In the right panel of Fig.\ \ref{fig:bunchingorder} we show a comparison between exact numerics and the analytic result (\ref{eq:Belzig1pLDF}) for the large deviation function in the single pole approximation. Around the mean value $I\simeq \langle I\rangle$ the analytic result agrees well with numerics. However, in the tails of the distribution a clear disagreement between the analytic approximation and numerics is visible. The disagreement reflects the deviations for the cumulants seen in the central panel of Fig.\ \ref{fig:bunchingorder}. We remark that measurements of the LDF recently have become accessible in experiments on real-time electron counting.\cite{Fricke2010}
The discussion in this subsection illustrates the need for careful considerations when
manipulating CGFs analytically. Concerning cumulants, we deal with
two opposite and non-commutative orders of limits: for a fixed order
of cumulants, a limiting procedure with changing parameters
converges to the approximate form given by the appropriate limit of
the CGF, such as the single-pole approximation in Eq.\
(\ref{eq:Belzigsinglepole}) in our case. However, the convergence of
the CGF is not uniform in $\chi$ due to potential singularities and
thus for fixed parameters, high order cumulants generically take on
the universal oscillatory form discussed above. One should thus be careful when using limiting forms of a
CGF to extract cumulants of arbitrary orders. In general, the
low-order cumulants follow the predicted pattern reasonably well,
but at some point significant deviations appear and the universal
oscillatory behavior should emerge. The order at which this crossover
occurs depends on details of the analytical structure of the CGF and
may be hard to predict. As we have shown explicitly, deviations of the cumulants from exact results are also clearly visible in the large deviation function.
\subsection{Transport through a vibrating molecule}
\label{subsec:vibration}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{vibmole.eps}
\caption{(color online). Transport through vibrating molecule. The
molecule is coupled to the left (right) lead with coupling
$\Gamma_L$ ($\Gamma_R$). The bias difference $eV=\mu_L-\mu_R$ drives
single electrons through the molecule. The system is operated in the
Coulomb blockade regime, where only $m=0$ or $m=1$ additional
electrons are allowed on the molecule. As an electron tunnels onto
the molecule, the equilibrium position of the molecule is shifted
due to the electric field $E$. The two harmonic
potentials corresponding to $m=0,1$ are shown. The damping rate of the vibrating molecule is denoted as $K$.} \label{fig:vibmole}
\end{center}
\end{figure}
In our next example, we consider a model of charge transport through a
molecule coupled to quantized vibrations.\cite{McCarthy2003,Millis2004,Boese2001,Flindt2005,Koch2005,Koch2005b,Haupt2006}
In the regime of weak coupling to the electronic leads, electron tunneling can be described using Fermi's golden rule rates for transitions between different vibrational and charge occupation states. For strong electron-phonon coupling, the large shift of the oscillator equilibrium position due to an electron tunneling onto the molecule suppresses the (Franck-Condon) overlap between the initial and final vibrational state for low-lying oscillator states. This leads to surpressed tunnel rates at low
bias-voltages, so-called Franck-Condon blockade. For larger bias-voltages, higher-excited oscillator
states become available, and the system can escape the blockade
regime. For weak oscillator dampings, several electrons can be
transferred through the molecule, once the blockade is lifted, until
a charge transfer event eventually leaves the oscillator in the
ground state and the current is suppressed again. Such dynamical
Franck-Condon blockade processes have been predicted to lead to very
large enhancements of the zero-frequency noise.\cite{Koch2005} Recently, Frank-Condon blockade was observed in experiments on suspended carbon nanotube quantum dots.\cite{Leturcq2009}
The system considered in the following is depicted in Fig.\ \ref{fig:vibmole}. Here we follow to
a large extent the description of the model given in Refs.\ \onlinecite{McCarthy2003}, \onlinecite{Flindt2005}. The Hamiltonian of the system and the detailed derivation of the resulting Markovian GME are given in App.\ \ref{app:vibmol}, where the various parameters of the model are also defined.
Due to the large number of
oscillator states, there is little hope for obtaining a closed-form
expression for the CGF that would allow for any analytic
manipulations. Instead, as we shall see, the numerically
evaluated high-order cumulants can be used to extract the precise
location of the dominating singularities of the CGF. We concentrate in the following on the unequilibrated oscillator
regime, where the damping rate of the oscillator is much smaller than the electron tunneling rates, $K\ll\Gamma_{L/R}$. As explained above, the combination of
strong electron-phonon coupling and weak oscillator damping leads to
dynamical Franck-Condon blockade, resulting in a large enhancement
of the current noise as demonstrated in Ref.\ \onlinecite{Koch2005}
using Monte-Carlo simulations. In Ref.\ \onlinecite{Koch2005b} the
analysis was extended to the full distribution of the transferred
charge and an analytic approximation for the CGF was presented based
on an avalanche-type of transport, where ``quiet'' periods of
transport are interrupted by a sequel of self-similar charge
avalanches. The analytic result for the CGF was shown to agree very
well with Monte-Carlo simulations of the probability distribution
$P(n,t)$. However, similarly to the previous
example, the approximate CGF has a single, simple pole on the
real-$i\chi$ axis, violating the required properties of the CGF,
mentioned at the end of Sec.\ \ref{sec:univosc}, thus making it unsuited for
predictions of the high-order cumulants. In particular, within this
approximation, the high-order cumulants would not oscillate, which contradicts our numerical findings.
Oscillations of the high-order cumulants with system parameters must be due to singularities located away from the real-$i\chi$. In the following, we assume that the CGF has a pair of complex-conjugate
singularities, $i\chi_0=|i\chi_0|e^{i\arg i\chi_0}$ and $|i\chi_0|e^{-i\arg i\chi_0}$, closest
to zero. As we will now show, the positions of these singularities can be found from our numerical calculations of high-order cumulants. To this end, we define
\begin{equation}
a_0=A_0/(i\chi_0)^{\mu_0}
\label{eq:a0}
\end{equation}
and rewrite Eq.\ (\ref{eq:twosingexp}) as
\begin{equation}
\llangle n^m\rrangle\simeq \!\frac{2|a_0|
B_{m,\mu_0}}{|i\chi_0|^{m}}\cos\!\left[m\arg i\chi_0\!-\!\arg
a_0\right].
\label{eq:twosingexpre}
\end{equation}
Following the ideas of Ref.\ \onlinecite{Zamastil2005}, Sec.\ 4, we find for the ratios of two successive cumulants
\begin{equation}
\begin{split}
\frac{\llangle n^{m-1}\rrangle}{\llangle n^m\rrangle}\frac{m+\mu_0-1}{|i\chi_0|}&=\cos\!\left[\arg i\chi_0\right]\\
&+\sin\!\left[\arg i\chi_0\right]\tan\!\left[m\arg i\chi_0-\!\arg a_0\right]
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\frac{\llangle n^{m+1}\rrangle}{\llangle n^m\rrangle}\frac{|i\chi_0|}{m+\mu_0}&=\cos\!\left[\arg i\chi_0\right]\\
&-\sin\!\left[\arg i\chi_0\right]\tan\!\left[m\arg i\chi_0-\!\arg a_0\right].
\end{split}
\end{equation}
Adding the two left and right hand sides, respectively, and rearranging, we obtain the equation
\begin{equation}
\begin{split}
2(m+\mu_0)\llangle n^{m}\rrangle |i\chi_0|\cos\!\left[\arg i\chi_0\right]-\llangle n^{m+1}\rrangle|i\chi_0|^2=\\
\llangle n^{m-1}\rrangle(m+\mu_0-1)(m+\mu_0).
\end{split}
\end{equation}
Using the substitution $m\rightarrow m+1$, we obtain an additional equation and thus arrive at a linear system of two equations which we solve for
$|i\chi_0|\cos\!\left[\arg i\chi_0\right]$ and $|i\chi_0|^2$ and thereby find $i\chi_0$. The method takes as input $\llangle n^{m-1}\rrangle$, $\llangle n^{m}\rrangle$, $\llangle n^{m+1}\rrangle$, and $\llangle n^{m+2}\rrangle$, and the accuracy is expected to improve with increasing cumulant order $m$.\cite{Zamastil2005}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, trim = 0 0 0 0, clip]{C60results2.eps}
\caption{(color online). High-order (normalized) cumulants for
unequilibrated molecule. Numerically exact results are shown
together with the asymptotics described by Eq.\
(\ref{eq:twosingexp}). Parameters entering Eq.\ (\ref{eq:twosingexp}) are
$\mu_0=-1/2$, $A_0=1.4810\times 10^{-7}e^{-i0.7378}$, and
$i\chi_0=0.0113 e^{i0.6262}$. System parameters (defined in App.\ \ref{app:vibmol}) are given in units of the natural oscillator frequency (with
$e,\hbar,k_B=1$) $V=3\omega_0$, $\Gamma=\Gamma_L=\Gamma_R=0.001\omega_0$,
$T=0.05\omega_0$, $K=10^{-10}\omega_0$, $\varepsilon=16\omega_0$, $c_1=4$, $c_2=0$. In the numerical calculations we have used $N=15$ oscillator states.} \label{fig:C60res2}
\end{center}
\end{figure}
Having determined $i\chi_0$, we find $a_0$ in a similar spirit by rewriting Eq.\ (\ref{eq:twosingexpre}) as
\begin{equation}
\begin{split}
\llangle n^m\rrangle\simeq &2B_{m,\mu_0}\left[\mathrm{Re}\{(i\chi_0)^{-m}\}\mathrm{Re}\{a_0\}\right.\\
&\left.
-\mathrm{Im}\{(i\chi_0)^{-m}\}\mathrm{Im}\{a_0\}\right].
\end{split}
\end{equation}
Again, we obtain via the substitution $m\rightarrow m+1$ a linear system of two equations that we solve for $\mathrm{Re}\{a_0\}$ and $\mathrm{Im}\{a_0\}$ and thus find $a_0=\mathrm{Re}\{a_0\}+i\mathrm{Im}\{a_0\}$. Finally, we determine $A_0$ from Eq.\ (\ref{eq:a0}). More advance methods for extracting the positions of singularities are available,\cite{Zamastil2005} but they require solutions of non-linear equations and will not be considered here.
In order to extract $i\chi_0$ and $A_0$ from the high-order cumulants, we need to know the nature of the singularities and hence $\mu_0$. Typically, the singularities are square-root branch points (see Ref. \onlinecite{Bender1978}, Sec.\ 7.5) and we thus take $\mu_0=-1/2$. In Fig.\ \ref{fig:C60res2} we show numerical results for the
(normalized) cumulants as function of the order $m$ together with
the asymptotic expression Eq.\ (\ref{eq:twosingexp}) for the high-order cumulants with $i\chi_0$ and $A_0$ found using the method described above. The asymptotic expression shows excellent agreement with the numerically exact results. For $m\gtrsim 5$, we see trigonometric oscillations whose frequency is determined by $\arg{i\chi_0}$. We note that a good agreement between our numerical results and the asymptotic expression could only be obtained with $\mu_0=-1/2$, thus confirming that the singularities stem from square-root branch points.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, trim = 0 0 0 0, clip]{C60results3.eps}
\caption{(color online). Large deviation function for the vibrating
molecule. Results are shown for different values of the damping
$K$, going from the unequilibrated regime, at low $K\ll\Gamma$, to
$K\sim\Gamma$, where the molecule equilibrates between each
tunneling event. For the unequilibrated case, a much larger range
of currents are probable, compared to transport through the
equilibrated molecule. System parameters (defined in App.\ \ref{app:vibmol}) are given in units of the natural oscillator frequency (with
$e,\hbar,k_B=1$) $V=3\omega_0$, $\Gamma_L=\Gamma_R=0.001\omega_0$,
$T=0.05\omega_0$, $\varepsilon=16\omega_0$, $c_1=4$, $c_2=0$.} \label{fig:C60res3}
\end{center}
\end{figure}
The large deviation function can also be evaluated numerically using the method described in the previous subsection.
In Fig.\ \ref{fig:C60res3} we show numerical results for the large
deviation function with different values of the damping $K$. For
large dampings, the oscillator is essentially equilibrated and the
measurable currents are closely centered around the mean current
$\llangle I\rrangle$. As the damping is lowered, we approach the
unequilibrated regime, where the transport statistics is dominated
by avalanche transport with a corresponding large zero-frequency
noise. Accordingly, the large deviation function is considerably
broadened and a much wider range of currents
become measurable.
\section{Non-Markovian systems}
\label{sec:nonMarkov}
\subsection{Dissipative double quantum dot}
\label{subsec:DQD}
In the previous two examples, we focused on the asymptotic behavior of
the high-order cumulants for two Markovian systems. We now turn
our attention to a model for which a weak coupling prescription
does not suffice and non-Markovian effects become significant. We
focus here on the influence of memory effects on the first few
cumulants, while referring the reader to Ref.\
\onlinecite{Flindt2008} for a discussion of the high-order
cumulants for the non-Markovian system presented in this example.
We consider a model of charge transport through a double
quantum dot (DQD) coupled to a heat bath which causes dephasing and
relaxation. Such systems were studied experimentally in Refs.\ \onlinecite{Fujisawa1998,Barthold2006}. The counting statistics in the transition between
coherent and sequential tunneling through DQDs has
been studied theoretically by Kie{\ss}lich and
co-workers.\cite{Kiesslich2006} In their work, decoherence was
described using either a charge detector model or via
phenomenological voltage probes.\cite{Blanter2000} More elaborate descriptions of
decoherence caused by a weakly coupled heat bath were given in Refs.\
\onlinecite{Kiesslich2007,Sanchez2008} and shown to agree well with experiments.
Here, we take these ideas further and go beyond the perturbative
treatment of the heat bath. This situation has previously been investigated by
Aguado and Brandes using a polaron transformation, assuming weak
coupling to the electronic leads in the high-bias
limit.\cite{Brandes1999,Aguado2004a,Aguado2004b} In the following, we apply an
alternative non-perturbative scheme for the coupling to the heat
bath, enabling us to fully include broadening due to the electronic leads.
Within this approach, we can study the cross-over between weak and
strong couplings to the heat bath and evaluate the effects of strong
decoherence on the charge transport statistics. In particular, we
show that only in the limit of weak coupling and high temperatures,
the dephasing caused by the heat bath can be accounted for by a
charge detector model with a single effective dephasing rate.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{DQDsetup.eps}
\caption{Dissipative double quantum dot. The Coulomb blockaded
double quantum dot consists of the left and right levels $|L\rangle$
and $|R\rangle$, coherently coupled with tunnel coupling $T_c$ and
dealigned by $\varepsilon$. A large bias across the system drives
electrons through the double quantum dot from the left lead with
rate $\Gamma_L$ to the right lead with rate $\Gamma_R$. The system
is coupled with dissipation strength $\alpha$ to a heat bath
at temperature $T$ and with Ohmic spectral function $J_{\Omega}(\omega)$. The
probability distribution of the number of transferred charges $n$ is
denoted $P(n,t)$.} \label{fig:DQDsetup}
\end{center}
\end{figure}
The model of charge transport through a Coulomb blockaded
DQD\cite{Aguado2004a,Aguado2004b} is
illustrated in Fig.\ \ref{fig:DQDsetup}. The DQD is coupled
to source and drain electrodes, while dissipation is provided by an
external heat bath. The DQD is operated in the Coulomb blockade regime close to a charge
degeneracy point, where only a single additional electron is allowed
on the double dot. Again, we consider for simplicity spinless electrons. The Hamiltonian of the double dot
can be written
\begin{equation}
\hatH_S=\epsilon_0|0\rlangle
0|+\frac{\varepsilon}{2}\hat{s}_z+T_c\hat{s}_x,
\end{equation}
where the pseudo-spin operators are
\begin{equation}
\hat{s}_z\equiv |L\rlangle L|-|R\rlangle R|
\end{equation}
and
\begin{equation}
\hat{s}_x\equiv |L\rlangle R|+|R\rlangle L|,
\end{equation}
respectively. Here, the two quantum dot levels $|L\rangle$ and
$|R\rangle$ are dealigned by $\varepsilon$ and their tunnel coupling
is $T_c$. The energy of the `empty' state $|0\rangle$ is
$\epsilon_0$. The pseudo-spin interacts with an external heat bath
consisting of harmonic oscillators,
\begin{equation}
\hatH_B=\sum_j\hbar\omega_j\hat{a}_j^{\dagger}\hat{a}_j,
\end{equation}
whose positions are coupled to the $z$-component of the pseudo-spin,
adding the term $\hat{V}_B\hat{s}_z$ to the full Hamiltonian with
\begin{equation}
\hat{V}_B= \sum_j\frac{g_j}{2}(\hat{a}_j^{\dagger}+\hat{a}_{j}).
\end{equation}
Finally, the spin-boson system is tunnel-coupled to left ($L$) and
right ($R$) leads via the tunnel-Hamiltonian
\begin{equation}
\hatH_T=\sum_{k_{\alpha},\alpha=L,R}(t_{k_{\alpha}}\hat{c}^{\dagger}_{k_\alpha}|0\rlangle
\alpha|+\mathrm{h.c.}),
\end{equation}
with both leads described as non-interacting fermions, i.e.,
\begin{equation}
\hatH_{\alpha}=\sum_{k_\alpha}\varepsilon_{k_\alpha}\hat{c}^{\dagger}_{k_\alpha}\hat{c}_{k_\alpha},\,\,\,
\alpha=L,R,
\end{equation}
kept at chemical potentials $\mu_{\alpha}$, $\alpha=L,R$, and temperature $T$. The full
Hamiltonian then reads
\begin{equation}
\hatH=\hatH_S+\hatH_T+\hatH_L+\hatH_R+\hatH_B+\hat{V}_B\hat{s}_z.
\label{eq:fullHamiltonian}
\end{equation}
As previously pointed out,\cite{Aguado2004a,Aguado2004b} the model
can be mapped onto that of transport through a superconducting single
electron transistor, when the charging energy is much larger than
the Josephson energy. Throughout this example we take $\hbar=k_B=e=1$.
As explained in App.\ \ref{app:doubledot}, transport through the
double dot can be described using a non-Markovian equation of motion
of the form in Eq.\ (\ref{eq:GME}) for the three electronic
occupations of the double dot collected in the vector
$\hatrho=(\rho_{0},\rho_{L},\rho_{R})^T$. The occupation
probabilities of the empty, left, and right states, are denoted
$\rho_{0}$, $\rho_{L}$, and $\rho_{R}$, respectively. The
corresponding memory kernel in Laplace space reads
\begin{equation}
\W(\chi,z)=
\begin{pmatrix}
-\Gamma_L & 0 & \Gamma_Re^{i\chi}\\
\Gamma_L & -\Gamma_{B}^{(+)}(z) & \Gamma_{B}^{(-)}(z) \\
0 & \Gamma_{B}^{(+)}(z) & -\Gamma_{B}^{(-)}(z)-\Gamma_R \\
\end{pmatrix}.
\label{eq:kernel}
\end{equation}
We note that the kernel with $\chi=0$ has a single zero-eigenvalue $\lambda_0(0,z)=0$ for all $z$, in agreement with Eq.\ (\ref{eq:lambda0}). The kernel has been derived under the assumption that
the symmetrically applied
bias $eV=|\mu_L-\mu_R|$ between the electronic leads is much larger than
the tunneling rates to the leads and the temperature $T$. The tunneling rates are defined as
\begin{equation}
\Gamma_{\alpha}(\epsilon)=2\pi\sum_{k}|t_{k_{\alpha}}|^2\delta(\epsilon-\varepsilon_{k_\alpha}),\,\,\,
\alpha=L,R,
\end{equation}
and are assumed energy-independent, such that
$\Gamma_{\alpha}(\epsilon)\equiv\Gamma_{\alpha}$, $\alpha=L,R$. We count the
number of electrons that have been collected in the right lead, and
consistently with this choice, the counting field $\chi$ has been
introduced in the off-diagonal element of the memory kernel that
contains the rate $\Gamma_R$.
The expressions for the bath-assisted hopping rates
are derived in App.\ \ref{app:doubledot} and for real $z$ they
read
\begin{equation}
\Gamma_B^{(\pm)}(z)=T_c^2[g^{(+)}(z_\pm)+g^{(-)}(z_\mp )],
\end{equation}
where $z_\pm\equiv z\pm i\varepsilon+\Gamma_R/2$. These expression are valid to the lowest order in the tunnel coupling $T_c$. The
bath-correlation functions in Laplace space are
\begin{equation}
g^{(\pm)}(z)=\int_0^{\infty}dt e^{-W(\mp t)-zt}
\end{equation}
with\cite{Weiss2001}
\begin{equation}
W(t)=\int_0^{\infty}d\omega
\frac{J(\omega)}{\omega^2}\{[1-\cos(\omega
t)]\coth{(\beta\omega/2)}+i\sin(\omega t)\},
\end{equation}
and
\begin{equation}
J(\omega)\equiv\sum_j|g_j|^2\delta(\omega-\omega_j)
\end{equation}
being the spectral function of the heat bath. In this work we
consider Ohmic dissipation characterized by a coupling strength
$\alpha$ such that the spectral
function reads
\begin{equation}
J_\Omega(\omega)=2\alpha \omega e^{-\omega/\omega_c}.
\end{equation}
where $\omega_c$ is the frequency cut-off, assumed to be the highest energy scale of the system.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, trim = 0 0 0 0, clip]{DQDresults1.eps}
\caption{(color online). Cumulants for weakly coupled heat bath.
The first three cumulants are shown as functions of the
dealignment $\varepsilon$. We show results for a low ($T=0.8\Gamma_R$)
and a high temperature ($T=8\Gamma_R$) as well as for the uncoupled
case ($\alpha=0$). For the high temperature case, we compare with
results obtained using a charge detector dephasing
model (short-dashed black line) and results without inclusion of memory
effects (long-dashed black line), see text. Parameters are
$\Gamma_L=0.1\Gamma_R$, $T_c=0.1\Gamma_R$, $\alpha=0.01$, and
$\omega_c=5\times10^4\Gamma_R$, and thus $\Gamma_d\simeq 0.8\Gamma_R$ according to Eq.\ (\ref{eq:dephaserate}).} \label{fig:DQDres1}
\end{center}
\end{figure}
In Fig.\ \ref{fig:DQDres1} we show results for weak couplings to
the heat bath, $\alpha\ll 1$. As the two quantum dot levels are
tuned into resonance ($\varepsilon=0$), the current reaches a
maximum with a width mainly determined by $\Gamma_R$. The
corresponding values for the second and third cumulant, normalized
with respect to the current, are suppressed below unity. The
suppression is stronger for the third cumulant. Away from
resonance, the mean current falls off, and the second and third
cumulants approach unity, corresponding to a Poisson process.
Without coupling to the heat bath, $\alpha=0$ (dotted blue line), the only broadening
mechanism is the escape of electrons through the right barrier at
rate $\Gamma_R$. This rate also defines the relevant energy scale to which we compare
the temperature of the heat bath. Away from resonance, the uncoupled case captures
well the results obtained at low temperatures, $T< \Gamma_R$. (dashed-dotted green line). At higher temperatures, $T\gg\Gamma_R$ (full red line), the peak in the current and the dips in the second and third cumulants are considerably
broadened due to the strong temperature induced dephasing.
Further results for the weak coupling limit are presented in Ref.~\onlinecite{Braggio2008}.
In order to understand the behavior at high temperatures (full red line), we
imagine replacing the heat bath by a charge detector which
measures the position of electrons on the DQD, thereby causing
dephasing.\cite{Gurvitz1997,Kiesslich2006,Braggio2009} The effects of the
charge detector can be described by a single dephasing rate
$\Gamma_d$, entering as an additional exponential decay of the
off-diagonal elements between the left and right quantum dot
states. As we show in App.\ \ref{app:doubledot}, this picture
follows from the high-temperature limit of the kernel in Eq.\
(\ref{eq:kernel}), and the corresponding dephasing rate is
\begin{equation}
\Gamma_d=2\alpha \pi T.
\label{eq:dephaserate}
\end{equation}
The dynamics of the system effectively becomes Markovian at high temperatures $T\gg\Gamma_R$, where the characteristic memory time $\sim(\Gamma_R/2+\Gamma_d)^{-1}$ of the kernel is shorter than the timescale $\sim\Gamma_R^{-1}$ over which the populations of the DQD evolve.
In Fig.\ \ref{fig:DQDres1} we see that the counting statistics at
high temperatures (full red line) are well approximated by the charge detector
model (short-dashed black line), which captures the broadening of the peak in the current
and the dips in the second and third cumulants. For high
temperatures (full red line), the large value of the dephasing rate indicates that
the system is strongly dephased. The charge detector model,
however, cannot account for the weak asymmetry between the phonon
emission ($\varepsilon>0$) and absorption ($\varepsilon<0$) sides at low temperatures (dashed-dotted green line).
In our description of the DQD system we have traced out the
electronic off-diagonal elements, the coherencies, together with
the electronic leads and the heat bath. Our derivation allows us to combine
strong coupling to the heat bath with broadening of the electronic levels due to the electrodes.
However, even without coupling to the heat bath, the kernel must be time-dependent in order to account for the coherent oscillations between the left
and right quantum dot states. These coherent effects are suppressed, when the dephasing is
strong, and in that limit we thus expect that a Markovian description would suffice. We check this assumption by
plotting in Fig.\ \ref{fig:DQDres1} only the Markovian parts of
the cumulants at high temperatures (long-dashed black line). Away from resonance, the
Markovian parts agree well for the second and third cumulants
showing that the system at high temperatures effectively is
Markovian. The mean current, as previously mentioned, is already a
Markovian quantity and it coincides with the Markovian
contribution as expected.\cite{Braggio2006} Closer to resonance,
some deviations for the second and third cumulants are seen as the system is not completely dephased.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth, trim = 0 0 0 0, clip]{DQDresults2.eps}
\caption{(color online). Cumulants in the intermediate regime
between weak and strong coupling to the heat bath. The first three
cumulants are shown as functions of the dealignment $\varepsilon$.
The values of the coupling to the heat bath are $\alpha=0$, 0.01,
0.05, and 0.1. For large couplings ($\alpha=0.5,1$), we compare
with results obtained without inclusion of memory
effects (dot-dashed lines), see text. Parameters are
$\Gamma_L=\Gamma_R$, $T_c=0.1\Gamma_R$, $T=0.5\Gamma_R$, and
$\omega_c=5\times 10^4\Gamma_R$.}
\label{fig:DQDres2}
\end{center}
\end{figure}
In Fig.\ \ref{fig:DQDres2} we show results for larger values of
the coupling to the heat bath. Also in this case, the peak in the
current and the dips in the second and third cumulants are
suppressed as the coupling is increased and dephasing becomes
stronger. Contrary to the weak coupling regime, however, no
broadening of these features are observed. This is not consistent
with the charge detector model, which would predict an increased
width together with suppression of the height of the current peak
and the depth of the dips in the second and third cumulants.
Additionally, as the coupling $\alpha$ is increased, the
emission/absorption asymmetry becomes stronger as exchange of
energy quanta with the heat bath becomes increasingly important.
As noted before, neither the absence of broadening nor the
asymmetry of the peaks and dips can be accounted for by the
charge detector model. For large values of the coupling to the
heat bath, the DQD system completely dephases and the Markovian
contribution describes well the behavior of the counting
statistics, in particular away from resonance, where coherent
effects are less relevant. At even larger couplings, the heat bath
tends to localize electrons to one of the two quantum dots, and
the effective tunnel rate between the two quantum dots becomes
highly suppressed. In that case, the current through the system is
very low and the statistics is Poissonian.
\subsection{Discussion of non-Markovian systems}
\label{subsec:nonMarkcorr}
We close this section by pointing out various possible subtleties associated with
Markovian\cite{Spohn1979,Kohen1997} and, in particular,
non-Markovian\cite{Stenholm2001} GMEs. As we shall argue,
special attention should be paid to the
sometimes paradoxical nature of heuristically derived non-Markovian
GMEs (see e.\ g.\ Ref.~\onlinecite{Stenholm2001}) in order to ensure physically meaningful results.
We moreover discuss the interpretation of the ``mean memory time" for non-Markovian systems and we show that it in certain cases can turn negative. As an illustrative example, we consider unidirectional transport through a single electronic level. This generic model provides us with a unifying explanation of previous results obtained for several different systems and, in addition, it displays a few possible peculiarities of non-Markovian transport.
The Markovian master equation for the simple two-state model of unidirectional transport through a single
electronic level is determined by the rate matrix\cite{Blanter2000,Bagrets2003,Gustavsson2006,Flindt2009}
\begin{equation}\W(\chi)=
\begin{pmatrix}
-\Gamma_L & \Gamma_R e^{i\chi} \\
\Gamma_L & -\Gamma_R
\end{pmatrix}
\end{equation}
with the counting field $\chi$ corresponding to tunneling across the
right barrier. An intuitive way to generalize the rate matrix to
the non-Markovian case would be simply to replace the rates $\Gamma_{L/R}$
by time-dependent rates, so that the rate matrix in Laplace
space instead reads
\begin{equation}
\label{eq:twostatekernel}
\W(\chi,z)=
\begin{pmatrix}
-\Gamma_L(z) & \Gamma_R(z) e^{i\chi} \\
\Gamma_L(z) & -\Gamma_R(z)
\end{pmatrix}.
\end{equation}
The non-Markovian character could be caused by
external degrees of freedom that have
been traced out, for example a
harmonic oscillator mode coupled to the occupation of the
electronic level, as studied in Sec.~\ref{subsec:vibration}.
Alternatively, it could be due to an energy-dependent tunneling density of states as in
Ref.~\onlinecite{Zedler2009}, or many-body-induced effects
as in the Fermi-edge-singularity problem in
transport.\cite{Matveev1992,Hapke-Wurst2000,Frahm2006,Maire2007,Ruth2008}
The microscopic origin of the memory, however, is not important in the following and its
effect on the current noise is qualitatively captured by the ``mean
memory times"
\begin{equation}
\tau_{L,R}\equiv-\left.\tfrac{d}{dz}\log\Gamma_{L,R}(z)\right|_{z=0}=\frac{\int_0^{\infty}t\Gamma_{L,R}(t)dt}{\int_0^{\infty}\Gamma_{L,R}(t)dt}.
\label{eq:memtime}
\end{equation}
If $\Gamma(t)\propto e^{-t/\tau}$ the mean memory time has a direct physical interpretation as the characteristic memory time $\tau$. However, the following statements are valid for any memory kernel as long as it has a finite mean memory time $\tau$. Using Eqs.\ (\ref{eq:nonMarkovcumulants},\ref{eq:nonMarkovcoeff}) we find the following expression for the
Fano factor [see also Eq.\ (13) of Ref.\ \onlinecite{Flindt2007}]
\begin{equation}
\label{eq:twostate}
F=\frac{\llangle I^2\rrangle}{\llangle
I\rrangle}=\frac{\Gamma_L^2+\Gamma_R^2}{(\Gamma_L+\Gamma_R)^2}-2\frac{\Gamma_L^2\Gamma_R\tau_R+\Gamma_R^2\Gamma_L\tau_L}{(\Gamma_L+\Gamma_R)^2}
\end{equation}
where $\Gamma_{L,R}\equiv\Gamma_{L,R}(z=0)>0$ is the Markovian limit
of the rates. The second term is a non-Markovian correction to the
well-known expression for the Fano factor of transport through a
single electronic level.\cite{Blanter2000} For certain parameter
values, however, this non-Markovian correction can make the Fano factor turn
negative --- a clearly unphysical result (the zero-frequency noise
must be positive, see e.g. Refs.~\onlinecite{Blanter2000,Flindt2004}). We discuss
this issue in further detail below.
Obviously, different baths producing the same mean
memory times give rise to the same electronic noise. Intuitively, one would also expect the mean
memory times to be positive. Since the Markovian limit of the $\Gamma$'s must be
positive (being rates) the non-Markovian correction appears
negative; thus the general effect of memory on transport through a single level
is a decrease of the noise compared to the corresponding
Markovian limit. This statement is in line with a number of previous findings: It explains the anomalous suppression of the Fano factor below $1/2$ in
transport through a single electronic level coupled to a mechanical
resonator reported in Ref.~\onlinecite{Haupt2006}. Additionally, the
non-Markovian correction due to strong spectral features in the
Fermi edge singularity
problem\cite{Matveev1992,Hapke-Wurst2000,Frahm2006,Ruth2008} is
responsible for the observed discrepancy\cite{Maire2007} between the
measured Fano factor and the expected result based on the Markovian
part only. A more
detailed account of this problem will be presented
elsewhere.\cite{Roszakinprep} Finally, the suppression of the Fano
factor compared to the Markovian approximation is also confirmed by
the study of an exactly solvable case\cite{Zedler2009} in the regime
where the non-Markovian GME provides a good approximation to the
exact dynamics.
Although the above expression (\ref{eq:twostate}) provides a
unifying explanation of these three examples, it obviously cannot be
correct in general as mentioned above. For weak non-Markovian behavior, where the $\tau$'s are small, the Fano factor stays positive, and the non-Markovian corrections lead to a reduction of noise. In general, however, there is no guaranty that the
Fano factor in Eq.\ (\ref{eq:twostate}) is always non-negative. This can be traced back to the heuristic
inclusion of the non-Markovian kernel in Eq.\
(\ref{eq:twostatekernel}). While the non-Markovian kernel for
unidirectional transport through a single level in general may be
written in the form \eqref{eq:twostatekernel} {\em without} counting
fields, the inclusion of the counting field must be carried out
carefully, for example by using well-controlled
systematic derivation procedures, such as those based on
perturbation theories.\cite{Braggio2006}
Non-Markovian GMEs may, however, still lead to unphysical results, when employed outside their regime of validity, as recently discussed by Zedler and co-workers.\cite{Zedler2009} In
the Markovian case, the heuristic addition of the counting field in $\W$ usually leads to correct results for the counting statistics, although exceptions do exist.\cite{Prachar2009} In the non-Markovian double dot system studied in Sec.\ \ref{subsec:DQD}, the counting field enters the Markovian ($z$-independent) part of kernel in Eq.\ (\ref{eq:kernel}), and the inclusion of the counting field does not lead to any of the issues discussed above.
Finally, we discuss another subtlety associated with strongly non-Markovian
systems. Under certain circumstances the mean memory time $\tau$,
defined in Eq.\ (\ref{eq:memtime}) may in fact become negative. This
happens for example for the dissipative double quantum dot studied in Sec.~\ref{subsec:DQD}.
For a sufficiently small dissipation rate, we find for the bath-assisted rates $\partial_z\Gamma_B^{(\pm)}(z=0)>0$, resulting in negative mean memory times. Formally, there is no problem associated with this phenomenon (the GME still describes a
positivity-preserving evolution), but the physical interpretation of the non-Markovian corrections is less clear due to this counterintuitive behavior. The problem is purely
interpretation-related and concerns the issue of a proper Markovian
limit.
In cases with large memory effects, the formal Markovian limit, corresponding to $\W(z\rightarrow 0)$, does
not give a reasonable description of the system {\em dynamics}, although it yields correct
{\em stationary} quantities, like the mean current.
Since the noise (and also higher-order cumulants) is a time
integral of a transient quantity, namely a current-current
correlation function, the formal Markovian limit of the noise in these cases is a somewhat
unphysical quantity. The problems with the interpretation of a Markovian
limit also influence the interpretation of the non-Markovian
corrections (via, e.\ g.\ negative memory times). Bluntly, a
physically meaningful result is
arbitrarily split into two additive parts, Markovian and non-Markovian, that each do not necessarily have a reasonable physical interpretation. The full result, however, is correct and
physically plausible. These effects are well
illustrated and can be understood by studying exactly solvable cases such
as the one in Ref.~\onlinecite{Zedler2009}.
In this section, we have only briefly touched upon various open questions and subtleties associated with interpretations of non-Markovian dynamics. However, the exact method developed in this paper paves the way for future systematic studies of memory effects in connection with electronic noise and counting statistics.
\section{Conclusions}
\label{sec:conclusions}
We have presented a detailed derivation of a recursive scheme for evaluating high-order cumulants of transport through Coulomb-blockade nanostructures with many states and non-Markovian dynamics. In order to illustrate the use of our method for Markovian systems we considered the counting statistics of transport through a two-level quantum dot and a vibrating molecule. In both cases, we have shown how the behavior of high-order cumulants is determined by dominating singularities of the cumulant generating functions. Oscillations of the high-order cumulants as function of the cumulant order can be used to locate the positions of singularities as we have demonstrated. We have also calculated the distribution of measurable currents, the so-called large deviation function, and shown how the tails of the distributions reflect the high-order cumulants. In order to illustrate the use of our method for a non-Markovian system, we considered transport through a dissipative double quantum dot. For this system, we have studied how bath-induced dephasing affects the first three cumulants and found that effects of the heat bath cannot be accounted for by an effective detector model, when the coupling becomes strong. Finally, we have discussed the nature and significance of non-Markovian dynamics in relation to counting statistics.
The research presented in this work points to several interesting directions to follow. While we have focused on the zero-frequency current cumulants of non-Markovian processes, it would be interesting to see, if the methods presented here could be extended to finite frequencies, as it was recently done for Markovian processes.\cite{Emary2007} It has now been firmly established that high-order cumulants of the counting statistics generally grow factorially with the cumulant order and oscillate as functions of basically any system parameters as well as of the cumulant order. It would be interesting to study in further detail how microscopic details of a system are reflected, for example, in the frequency of these oscillations. Such a study would shed new light on the information contained in high-order cumulants.
Finally, we believe that the methods presented here will pave the way for future systematic studies of counting statistics in connection with non-Markovian dynamics.
\acknowledgements
We thank R.~Aguado, T.~Brandes, C.\ Emary, D.\ Kambly, S.~Kohler, D.~Marcos, K.~Neto\v{c}n\'{y},
M.~Sassetti, P.~Talkner, J.\ Zamastil, and P.\ Zedler for fruitful discussions and
suggestions. We thank the group of R.\ J.\ Haug for enlightening discussions about
experimental aspects of counting statistics. The work was supported by the Villum Kann Rasmussen
Foundation, INFM--CNR Seed Project, European Science Foundation
(`Arrays of Quantum Dots and Josephson Junctions'), Czech Science
Foundation (grant 202/07/J051), and FiDiPro of the Finnish Academy.
The work of T.~N.\ is a part of the research plan MSM 0021620834
financed by the Ministry of Education of the Czech Republic.
|
2,869,038,155,699 | arxiv | \section{Introduction}
The recent discovery of QAH effect in a magnetic insulator has attracted considerable interest in this new state of quantum matter~\cite{qi2006,qi2008,liu2008,li2010,wang2011,yu2010,onoda2003,xiao2011,ruegg2011,chang2013b,wang2013a,wang2013b}.
In a QAH insulator, theoretically predicted in magnetic topological insulators (TIs)~\cite{qi2006,qi2008,liu2008,li2010,wang2011,yu2010}, the strong spin-orbit coupling and ferromagnetic (FM) ordering combine to give rise to an insulating state with a topologically nontrivial band structure characterized by a finite Chern number~\cite{thouless1982,haldane1988}. In a beautiful experiment, the QAH effect has been discovered in Cr-doped (Bi,Sb)$_2$Te$_3$ magnetic TI~\cite{chang2013b}, where at zero magnetic field, the gate-tuned Hall resistance $\rho_{xy}$ exhibits quantized plateau at values $\pm h/e^2$ while the longitudinal resistance $\rho_{xx}\rightarrow 0$. The plateau transition is of particular interest, in which $\rho_{xy}$ changes from one quantized value to another over a narrow interval of external magnetic field at the coercivity, and $\rho_{xx}$ exhibits peaks~\cite{chang2013b}. In this paper, we address the critical properties of the quantum phase transition between adjacent QAH phases, and some of the theoretical predictions are already confirmed in the QAH experiment~\cite{chang2013b}.
This issue is closely related to the integer quantum Hall effect (QHE) plateau transition~\cite{prange1990}. In a strong magnetic field $B$, a two-dimensional (2D) electron gas exhibits the QHE over a wide range of sample disorder. The plateau transition between different quantized value for $\rho_{xy}$ reflects delocalization transition in each Landau level (LL). This delocalization has shown to be a critical phenomena~\cite{kivelson1992,huckestein1995,sondhi1997,kramer2005}, where the localization length $\xi$ diverges as a power law $\xi\sim\left(B-B_c\right)^{-\nu}$ with a universal critical exponent $\nu$~\cite{pruisken1988,huckestein1990,huo1992}. Scaling behavior in transport coefficients has been observed as the zero-temperature critical point is approached, as a function of temperature $T$, sample size, and frequency, which yield the value $\nu\approx2.38$~\cite{li2009,koch1991b,engel1993}. Chalker and Coddington proposed a network model to describe the quantum percolation of 2D electrons in a strong magnetic field and a smooth random potential~\cite{chalker1988}. The semiclassical cyclotron orbits propagate along the equipotential lines of the disorder potential, and the tunneling processes occur whenever two orbits approach each other on a distance less than the cyclotron radius. Extensive numerical simulations~\cite{chalker1988,lee1993,slevin2009} show that the network model has a plateau transition with $\nu=2.4\pm0.2$, in excellent agreement with the experimental results.
\begin{figure}[b]
\begin{center}
\includegraphics[width=3.2in]{fig1.pdf}
\end{center}
\caption{Chiral edge states along domain walls at the coercivity in a magnetic TI. $+$ (grey region) and $-$ (white region) denotes the upward and downward magnetic domains with $|\Delta|>|m_0|$, respectively. The shadow region denotes $|\Delta|<|m_0|$. The arrowed lines are chiral states and correspond to the links in network model. The circles enclose the tunneling point between chiral states which correspond to the saddle points (nodes).}
\label{fig1}
\end{figure}
The magnetic TI studied in the QAH experiment~\cite{chang2013b} develop robust ferromagnetism at low temperature, possibly mediated by van Vleck mechanism~\cite{yu2010}. In the magnetized states, the magnetic domains of the material can be viewed as a single domain with up or down magnetization, and the system is in a QAH state with quantized $\rho_{xy}$ being $+h/e^2$ or $-h/e^2$. The magnetization reversal in this system leads to a quantum phase transition between two QAH states. At the coercive field, the magnetic domains are being switched from up to down \emph{randomly}, so many upward and downward domains coexist [marked as $+$ and $-$ in Fig.~\ref{fig1}]. At the boundary of each domain, there exists a chiral edge state~\cite{qi2008} with spatial decay length $\lambda$. Each edge state is characterized by a random phase change along the domain boundary. Tunneling between two edge states will occur whenever they are separated less than $\lambda$. Therefore, the QAH plateau transition at the coercivity in a magnetic TI is very much like the network model of the integer QHE plateau transition in the lowest LL. Although these two cases belong to quite different limits, the symmetries of the systems are common, i.e., the \emph{unitary} class without time-reversal nor spin-rotational symmetry~\cite{kramer2005}. One purpose of the present work is to propose a microscopic model for the QAH plateau transition, and establish its relation to the network model, so that the critical exponent obtained for the latter can be used for the former.
The organization of this paper is as follows. After this
introductory section, Sec. II describes the microscopic model
for the QAH plateau transition. Section III describes the mapping
from the model for QAH plateau transition to the network model for
the integer QHE transition. Section IV presents the
results and discussion on coercivity transition and experimental
proposal in a magnetic TI. Section V concludes this paper. Some auxiliary materials are relegated into an Appendix.
\section{Model}
Now, we turn to the QAH state in 2D thin film of a magnetic TI with spontaneous FM order. The low-energy bands of this system consist of Dirac-type surface states only~\cite{qi2008,yu2010,wang2013a}, for the bulk states are always gapped. The generic form of the effective Hamiltonian is
\begin{eqnarray}\label{model0}
\widetilde{\mathcal{H}}_0(k_x,k_y)
&=&v_F k_y\widetilde{\sigma}_1\otimes\widetilde{\tau}_3-v_F k_x\widetilde{\sigma}_2\otimes\widetilde{\tau}_3+\Delta\widetilde{\sigma}_3\otimes1
\nonumber
\\
&&+m(k)1\otimes\widetilde{\tau}_2,
\end{eqnarray}
with the basis of $|t\uparrow\rangle$, $|t\downarrow\rangle$, $|b\uparrow\rangle$ and $|b\downarrow\rangle$, where $t$, $b$ denote the top and bottom surface states and $\uparrow$, $\downarrow$ represent the spin up and down states, respectively. $\widetilde{\sigma}_i$ and $\widetilde{\tau}_i$ ($i=1,2,3$) are Pauli matrices acting on spin and layer, respectively. $v_F$ is the Fermi velocity and we set $v_F\equiv1$. $\Delta$ is the exchange field along the $z$ axis introduced by the FM ordering. Here, $\Delta\propto \langle S\rangle$ with $\langle S\rangle$ the mean field expectation value of the local spin~\cite{yu2010}. The magnetization $M\propto\langle S\rangle_{\text{ave}}$ where $\langle S\rangle_{\text{ave}}$ is the spatial average of $\langle S\rangle$. $m(k)$ describes the tunneling effect between the top and bottom surface states. To the lowest order in $k$, $m(k)=m_0+m_1(k_x^2+k_y^2)$, and $\left|m_0\right|<\left|\Delta\right|$ guarantees the system is in the QAH state. For simplicity, the spatial inversion symmetry is assumed, which requires that $v_F$, $\Delta$ and effective $g$-factor take the same values for top and bottom surfaces.
In terms of the new basis $|+\uparrow\rangle$, $|-\downarrow\rangle$, $|+\downarrow\rangle$, $|-\uparrow\rangle$ with $|\pm\uparrow\rangle=(|t\uparrow\rangle\pm|b\uparrow\rangle)/\sqrt{2}$
and $|\pm\downarrow\rangle=(|t\downarrow\rangle\pm|b\downarrow\rangle)/\sqrt{2}$, the system is decoupled into two models with opposite chirality~\cite{wang2013a}
\begin{eqnarray}\label{model1}
\mathcal{H}_0(k_x,k_y) &=&
\begin{pmatrix}
\mathcal{H}_+(k) & 0\\
0 & \mathcal{H}_-(k)
\end{pmatrix},\\
\mathcal{H}_{\pm}(k) &=& k_y\tau_1\mp k_x\tau_2+\left(m(k)\pm\Delta\right)\tau_3
\end{eqnarray}
where $\tau_i$ are Pauli matrices. At half filling, $\mathcal{H}_{\pm}(k)$ have Chern number $\mp 1$ or $0$ depending on whether the Dirac mass is inverted ($m(k)\pm\Delta<0$) or not ($m(k)\pm\Delta>0$) at $\Gamma$ point. Thus the total Chern number of the system is
\begin{equation}\label{chern}
C = \begin{cases}
\Delta/|\Delta|, & \text{for} \ \ \left|\Delta\right|>\left|m_0\right|\\
0, & \text{for} \ \ \left|\Delta\right|<\left|m_0\right|
\end{cases}
\end{equation}
The Chern number changes by 1 at $\Delta=\pm m_0$. In the QAH state, the Hall conductance $\sigma_{xy}=Ce^2/h$ is in a quantized plateau and depends only on the sign of $\Delta$.
Magnetization reversal will change the sign of $M$, leading to the QAH plateau transition at $\Delta=\pm m_0$. Here we consider how the random magnetic domains at the coercivity will effect the QAH phase transition at $\Delta^*_1=m_0$ and $\Delta^*_2=-m_0$. In general, the disorder will generate spatially random perturbations to the pure Hamiltonian $\mathcal{H}_0$ in Eq.~(\ref{model1}). Specifically, at the coercivity, the system mainly has three types of randomness,
\begin{eqnarray}\label{randompotential}
\mathcal{H}_A &=& A_x(x,y)\tau_2\otimes\sigma_3-A_y(x,y)\tau_1\otimes1,
\nonumber
\\
\mathcal{H}_{\Delta} &=& \Delta(x,y)\tau_3\otimes\sigma_3,
\nonumber
\\
\mathcal{H}_{V} &=& V(x,y),
\end{eqnarray}
where $\sigma_3$ is Pauli matrix. $\vec{A}\equiv(A_x,A_y)$, $\Delta$, and $V$ are nonuniform and random in space, but constant in time. Thus they mix up the momenta but not the frequencies. $\mathcal{H}_A$ corresponds to a random vector potential, which comes from the gauge coupling ($\vec{k}\rightarrow\vec{k}-\vec{A}$) with the random magnetic field in the system. $\mathcal{H}_\Delta$ is the random exchange field along $z$ axis induced by the local spin in magnetic domains. $\mathcal{H}_V$ is the random scalar potential induced by impurities in the materials. Here the random exchange field within the $x$-$y$ plane is ignored, for effectively it only contributes to a negligible small random exchange field along $z$ axis at the transition point [see Appendix~\ref{randomperturbation}]. Obviously, $\mathcal{H}_A$ and $\mathcal{H}_\Delta$ break time-reversal symmetry, while $\mathcal{H}_V$ preserves time-reversal symmetry. To be concrete, at $\Delta=\pm m_0$, we will assume that all three random potentials are symmetrically distributed about zero mean. We also assume the interaction between the electrons can be neglected.
Here we mention that the model introduced above is very similar to the random Dirac model for the description of the integer QHE transition~\cite{fisher1985,ludwig1994}. The fixed point of the random Dirac model with all three different kinds of disorder is in a strong coupling regime, and is conjectured to be a generic integer QHE fixed point~\cite{ludwig1994}. This suggests the QAH plateau transition should have a similar critical behavior. However, the critical properties of the random Dirac model have not yet been accessible analytically. In order to get the critical exponents for QAH plateau transition, we construct a general mapping from the model for QAH transition to the network model.
\section{Mapping to network model}
Now, we consider $\mathcal{H}_+(k)$ in presence of disorders $\mathcal{H}_A$, $\mathcal{H}_\Delta$ and $\mathcal{H}_V$, which describes the phase transition from $C=+1$ to $C=0$ at $\Delta=-m_0$. In real space, the Hamiltonian has the form
\begin{equation}
\mathcal{H}_+=(-i\partial_y-A_y)\tau_1-(-i\partial_x-A_x)\tau_2+\delta\tau_3+V\ ,
\end{equation}
where $\delta(x,y)\equiv m_0+\Delta(x,y)$ is the Dirac mass. The $m_1$ term has been neglected, for it does not affect the plateau transition. For convenience, we make a unitary transformation $\widetilde{\mathcal{H}}_+\equiv G\mathcal{H}_+G^\dag$, and obtain
\begin{equation}
\widetilde{\mathcal{H}}_+ = (-i\partial_x-A_x)\tau_3-(-i\partial_y-A_y)\tau_1-\delta\tau_2+V,
\end{equation}
with $G=(\tau_2-\tau_3)/\sqrt{2}$. In the low-energy limit, the unitary evolution operator in a unit time for $\widetilde{\mathcal{H}}_+$ is
\begin{equation}\label{evolution}
\mathcal{U}=e^{-i\widetilde{\mathcal{H}}_+}\approx 1-i\widetilde{\mathcal{H}}_+-\frac{\widetilde{\mathcal{H}}_+^2}{2}
\approx e^{-iV}
\begin{pmatrix}
\gamma &\alpha \\
-\alpha^* & \gamma^*
\end{pmatrix},
\end{equation}
where
\begin{eqnarray}
\gamma(x,y) &=& \cos\delta\cos\left(-i\partial_y-A_y\right)e^{-i\left(-i\partial_x-A_x\right)},
\nonumber
\\
\alpha(x,y) &=& e^{i\left(-i\partial_y-A_y\right)}\left[\sin\delta+i\sin\left(-i\partial_y-A_y\right)\right].
\nonumber
\end{eqnarray}
Here, $\gamma^*$, $\alpha^*$ are the corresponding complex conjugates.
\begin{figure}[b]
\begin{center}
\includegraphics[width=3.3in]{fig2.pdf}
\end{center}
\caption{The network model. (a) shows the coordinate system for plaquettes and the labeling of the four links.
(b) Amplitudes associated with possible scattering paths at nodes.}
\label{fig2}
\end{figure}
Then, we turn to the network model as shown in Fig.~\ref{fig2}. Such model is defined using the language of scattering theory~\cite{chalker1988}.
It consists of a square lattice of plaquettes. At the boundary of each plaquette, there is an edge state at the Fermi energy representing equipotentials, in which an electron drifts along the direction indicated by the arrow. Plaquettes are labeled by integer coordinates $(x,y)$, and we denote the four links $i$ making up a plaquette by $i=1, 2, 3, 4$, so that a link is specified by the combination $(x,y,i)$ where $x+y$ is even. The wave function for the electron on the link $(x,y,i)$ is represented by the current amplitude $Z_i(x,y)$, which is characterized by the phase change $\phi_i$ along the link ($0\leq\phi_i\leq2\pi$). The tunneling process at the nodes [denoted as $\mathbf{S}$ and $\mathbf{S}'$ in Fig.~\ref{fig2}(b)] may be related by a scattering matrix with a parameter $\vartheta$ ($0\leq\vartheta\leq\pi/2$) as
\begin{equation}
\begin{pmatrix}
Z_2\\
Z_4
\end{pmatrix}
=\begin{pmatrix}
\cos\vartheta & \sin\vartheta\\
-\sin\vartheta & \cos\vartheta
\end{pmatrix}
\begin{pmatrix}
Z_1\\
Z_3
\end{pmatrix}.
\end{equation}
Now, we associate a unitary scattering matrix with the model~\cite{ho1996}, which is roughly a time evolution operator. In the basis of $\left(Z_1(x,y), Z_3(x,y); Z_2(x,y), Z_4(x,y)\right)$, the one-step scattering matrix between the nearest-neighbor links is
\begin{equation}
\mathcal{S} = \begin{pmatrix}
0 & \mathcal{N}_1 \\
\mathcal{N}_2 & 0
\end{pmatrix},
\end{equation}
where
\begin{equation}
\mathcal{N}_1 = \begin{pmatrix}
\sin\vartheta e^{i\phi_1}\tau^x_-\tau^y_+ & \cos\vartheta e^{i\phi_1} \\
\cos\vartheta e^{i\phi_3} & -\sin\vartheta e^{i\phi_3}t^x_+t^y_-
\end{pmatrix},
\nonumber
\end{equation}
and
\begin{equation}
\mathcal{N}_2 = \begin{pmatrix}
\cos\vartheta e^{i\phi_2} & \sin\vartheta e^{i\phi_2}\tau^x_+\tau^y_+ \\
\sin\vartheta e^{i\phi_4}\tau^x_-\tau^y_- & -\cos\vartheta e^{i\phi_4}
\end{pmatrix}.
\nonumber
\end{equation}
Here, $\tau^x_\pm$ and $\tau^y_\pm$ are the translation operators defined as $\tau^x_{\pm}Z_i(x,y)=Z_i(x\pm1,y)$ and $\tau^y_{\pm}Z_i(x,y)=Z_i(x,y\pm1)$. The two-step scattering matrix then decouples as
\begin{equation}
\mathcal{S}^2 = \begin{pmatrix}
\mathcal{N}_1\mathcal{N}_2 & 0 \\
0 & \mathcal{N}_2\mathcal{N}_1
\end{pmatrix}.
\end{equation}
To extract the localization length, it is sufficient to just deal with the upper-left block $\mathcal{N}_1\mathcal{N}_2$~\cite{ho1996}.
If the phases $\phi_i$ are uniformly distributed between $0$ and $2\pi$, the network model is critical at $\vartheta=\vartheta_c=\pi/4$ where $\xi$ diverges, and in the localized phase otherwise~\cite{chalker1988}.
In the continuum limit, the translation operators can be written as $\tau^{x}_\pm=e^{\pm\partial_{x}}$ and $\tau^{y}_\pm=e^{\pm\partial_{y}}$. By identifying $A_x=(\phi_1-\phi_3)/2$, $A_y=(\phi_4-\phi_2)/2$, $V=-\sum_{i=1}^{4}\phi_i/2$ and $\vartheta=\vartheta_c+\delta/2$, we find that the unitary matrix $\mathcal{N}_1\mathcal{N}_2$ is exactly the same as the evolution operator $\mathcal{U}$ defined in Eq.~(\ref{evolution}). Specifically, the randomness in the individual link phases arise from fluctuation in the vector potential $\vec{A}$, variations in the total Aharonov-Bohm phase associated with each plaquette come from fluctuations in the scalar potential $V$, and the random tunneling parameter is not constant everywhere if the fluctuations in the mass $\Delta$ are present. Similar procedure can be done for $\mathcal{H}_-(k)$ for $C=-1$ to $C=0$ transition. Therefore, by using of the time evolution operator, we have established in detail a mapping from the QAH plateau transition to the network model.
\section{Results and Discussion}
\subsection{Coercivity transition}
The QAH plateau transition at the coercivity should have the same critical behavior as the network model. More specifically, the localization length $\xi$ of the levels near the Fermi energy diverges like a universal power law in $\Delta$ as $\xi = \xi_{\Delta}\left|\Delta-\Delta^*\right|^{-\nu}$. For $\Delta\propto M$, and at the coercivity $M\propto H$, therefore,
\begin{equation}
\xi(H)=\xi_0\left|H-H^*\right|^{-\nu},
\end{equation}
with the critical exponent $\nu\approx2.4$ and $H^*$ is the critical external field of the plateau transition.
As there exist two critical points at $\Delta_1^*$ and $\Delta_2^*$, we predict there should be \emph{four} critical magnetic field $\pm H_1^*$ and $\pm H_2^*$ at which $\xi$ diverges as shown in Fig.~\ref{fig3}.
In the finite-size scaling theory, the conductance tensor depends on the parameter $H$ only through a single variable with the ansatz~\cite{pruisken1988},
\begin{equation}\label{scaling}
\sigma_{\alpha\beta}(H) = f_{\alpha\beta}\big[L^{1/\nu}_{\mathrm{eff}}\left(H-H^*\right)\big],
\end{equation}
where $\alpha,\beta=x,y$. $\sigma_{xx}$ is the longitudinal conductance. $L_{\mathrm{eff}}$ is the effective system size.
$f_{\alpha\beta}$ is a regular function (power series) of its argument except near the QAH plateaus.
Such power-law behavior of the transport coefficients reflects the two-parameter scaling of the conductance tensor~\cite{huckestein1995,pruisken1988}. When $L_{\text{eff}}\gg\xi$, one expect $f_{xx}\propto\exp(-L_{\text{eff}}/\xi)$.
At $T=0$~K, $L_{\mathrm{eff}}$ is equal to the system size $L$. At finite $T$, $L_{\mathrm{eff}}$ is given by the phase coherence length $L_{\mathrm{in}}$~\cite{thouless1977}, which behaves as $L_{\mathrm{in}}(T)\propto T^{-p/2}$ as $T\rightarrow0$~\cite{abrahams1981}. Then $L_{\text{eff}}^{1/\nu}\propto T^{-\kappa}$ with $\kappa=p/2\nu$. The $n$th derivative of the conductance tensor at the critical point is
\begin{equation}\label{Tscaling}
\frac{\partial^n\sigma_{\alpha\beta}(H^*)}{\partial H^n} \propto L_{\text{eff}}^{n/\nu} \propto T^{-n\kappa}.
\end{equation}
This is the $T$-dependent scaling of QAH plateau transition. More specifically, as shown in Fig.~\ref{fig3}(a), the maximum slope in the $\sigma_{xy}$ curve diverges as a power law in temperature $T$ as
\begin{equation}
(\partial\sigma_{xy}/\partial H)_{\text{max}} \propto T^{-\kappa}.
\end{equation}
In addition, the half-width of $\sigma_{xx}$ peak vanishes like
\begin{equation}
\Delta_{1/2}H \propto T^{\kappa}.
\end{equation}
The statement of Eq.~(\ref{Tscaling}) can be directly translated into resistance [see Appendix~\ref{res_con_tensor}].
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.4in]{fig3.pdf}
\end{center}
\caption{(color online) Magnetic field dependence of $\sigma_{xy}$ and $\sigma_{xx}$. (a) Sketch of $\sigma_{xy}$ and $\sigma_{xx}$ as a function of applied magnetic field $H$. An intermediate plateau with $\sigma_{xy}=0$ appears at the hysteresis loop, while $\sigma_{xx}$ shows two peaks around the coercive field. (b) $\sigma_{xy}$ vs. $H$ at three different $T$ with $T_1<T_2<T_3$. (c) The corresponding $\sigma_{xx}$ vs. $H$.}
\label{fig3}
\end{figure}
The exponent $\nu$ can be measured directly by studying same Hall-bar geometries but different sizes. For sufficiently small samples, $(\partial\sigma_{xy}/\partial H)_{\text{max}}$ and $\Delta_{1/2}H$ should saturate at low $T$, and the saturation temperature would decrease with increasing system size. This is because that as the temperature when $L_{\text{in}}\sim L$, the $T$-dependent scaling at higher $T$ crosses over to size-dependent scaling. The saturation value of $\Delta_{1/2}H$ at low $T$ is then given by the condition $L/\xi\approx 1$, i.e.,
\begin{equation}
\Delta_{1/2}H \propto L^{-1/\nu}.
\end{equation}
The universal power-law behavior in temperature shows the characteristics of a second-order phase transition. And the magnetization $M$ is used as a continuous parameter for the phase transition between adjacent QAH phases. One may be concerned with this assumption, since in a FM material, $M$ is usually thought to reverse abruptly (known as the ``infinite avalanche'') at the coercivity, marking the occurrence of a first-order transition~\cite{vladimir1983}. Such discontinuity will completely conceal the above second-order phase transition. However, as studied extensively by materials scientists, the hysteresis curve of FM materials are often smooth. This is due to inevitable dissipations (such as the presence of disorders) in the process of magnetization~\cite{sethna1993}. The existence of dissipations make the magnetization process no longer a first-order transition, but a smooth crossover. Therefore, one could observe the critical behavior of QAH plateau transition on the hysteresis loop in a magnetic TI.
\subsection{Experimental proposal}
For the recent QAH experiment in a Cr$_x$(Bi,Sb)$_{2-x}$Te$_3$ thin film, at low enough $T$, one would observe the zero Hall plateau with $\rho_{xy}=0$ and $\sigma_{xy}=0$. The corresponding $\sigma_{xx}$ would have two peaks at the coercivity as shown in Fig.~\ref{fig3}(a), while $\rho_{xx}$ only has one peak. This remarkable theoretical prediction is already borne out in experiment~\cite{chang2013b}, by inverting the experimental data of $\rho_{xx}$ into $\sigma_{xx}$, $\sigma_{xx}$ shows double peaks at the coercivity while $\rho_{xx}$ only has single peak~\cite{ke_note}. However, the $\rho_{xy}=0$ and $\sigma_{xy}=0$ plateau are not yet observed, possibly because $T$ is still not low enough or the transitions in $\mathcal{H}_+(k)$ and $\mathcal{H}_-(k)$ are nearly degenerate~\cite{reason_note}.
As shown in Fig.~\ref{fig3}(b), the $\sigma_{xy}=0$ plateau disappears as $T$ increases.
Even without the signature of zero Hall plateau in $\rho_{xy}$, one can still measure the critical behavior by studying the $T$-dependent and size-dependent scaling predicted above. For a definite system size, the maximum slope in $\rho_{xy}$ should diverge in $T$ as
\begin{equation}
(\partial\rho_{xy}/\partial H)_{\text{max}} \propto T^{-\kappa}.
\end{equation}
However, the temperature dependence of the Fermi-Dirac distribution leads to a temperature dependence of the resistance,
$\rho_{\alpha\beta}(T)=\int dE (-\partial f(T)/\partial E)\rho_{\alpha\beta}(T=0)$. In order to observe the universal scaling
behavior, the temperature must be low enough that the influences of the finite width of Fermi-Dirac distribution can be neglected.
While for a definite low temperature, the maximum slope in $\rho_{xy}$ scales in $L$ as
\begin{equation}
(\partial\rho_{xy}/\partial H)_{\text{max}} \propto L^{-1/\nu}.
\end{equation}
Moreover, $\rho_{xx}\propto\exp\left(-L_{\text{eff}}\left|H-H^*\right|^\nu/\xi_0\right)$ when $\rho_{xy}$ is close to the quantized value with $L_{\text{eff}}\gg\xi$. The critical exponent $\nu\approx2.4$, independent of the transition is degenerate or not~\cite{li2009,koch1991b,engel1993,chalker1988,lee1993,slevin2009}.
\section{Conclusion}
In summary, starting from the microscopic model for QAH plateau transition, we construct a mapping to the network model for integer QHE transition. We predict that $\sigma_{xx}$ would show two peaks at the coercivity while $\rho_{xx}$ only has single peak. Remarkably, this theoretical prediction is already borne out in experiment~\cite{chang2013b}. The scaling theory of Hall plateau transition in QAH effect is proposed. To observe the universal scaling behavior, $T$ must be low enough. However, the absolute scale in $T$ is very much dependent on the microscopic details of the randomness in magnetic domains. Only the value of the exponent $\nu$ is universal~\cite{kappa_note}. Moreover, without LLs, QAH plateau transition at the coercivity in a magnetic TI provides an experimental platform to test the random Dirac model~\cite{ludwig1994}, which was originally proposed for the description of integer QHE plateau transition. A field theory description of the QAH transition including the renormalization group flow of $\sigma_{xx}$ and $\sigma_{xy}$ will be studied in future work.
\begin{acknowledgments}
We are deeply indebted to Steven Kivelson for many valuable discussions. This work is supported
by the Defense Advanced Research Projects Agency Microsystems
Technology Office, MesoDynamic Architecture Program (MESO) through the Contract No.~N66001-11-1-4105; the DARPA Program on
``Topological Insulators -- Solid State Chemistry, New Materials and Properties,'' under the Grant No.~N66001-12-1-4034; and by the US Department of
Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No.~DE-AC02-76SF00515.
\end{acknowledgments}
\begin{appendix}
\section{Plateau transition point}
\label{plateautransition}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=6.7in]{fig4}
\end{center}
\caption{The bulk and edge state spectrum of the QAH model described by Eq.~(1) in the presence of external magnetic field. (a), (b) \& (c) shows the bulk LLs, where in (a) $m_0+m_1/\ell_c^2<\Delta<-m_0-m_1/\ell_c^2$, in (b) $\Delta=-m_0-m_1/\ell_c^2$, and in (c) $\Delta>-m_0-m_1/\ell_c^2$. The Chern number (a) $C=0$, (b) transition point, (c) $C=+1$. The coercivity $B_c\approx0.097$~T, in (a)-(c) it clearly shows $|m_1/\ell_c^2|\ll |m_0|$. (d)-(h) shows the low-lying bulk and edge state energies as a function of the centers of the Landau orbitals when varying $\Delta$. $\Delta$ and corrosponding Chern number, (d) $\Delta<m_0+m_1/\ell_c^2$ and $C=-1$, (e) transition point $\Delta=m_0+m_1/\ell_c^2$, (f) $m_0+m_1\ell_c^2<\Delta<-m_0-m_1/\ell_c^2$ and $C=0$, (g) transition point $\Delta=-m_0-m_1/\ell_c^2$, (h) $\Delta>-m_0-m_1/\ell_c^2$ and $C=+1$. In (f),
the Fermi energy lies in-between the two bulk inverted LLs. The Fermi energy crosses the LLs, giving rise to the pair of counterpropagating edge states. It is the case for (a). (g) corresponds to (b). (h) corresponds to (c), where the Fermi energy only cross one LL, give rise to $C=1$.}
\label{fig4}
\end{figure*}
An external magnetic field is required to induce the coercivity transition, and in experiment the coercive field is small ($B_c<0.1$~T)~\cite{chang2013b}. There is no Landau levels (LLs) in this system as the cyclotron energy at the coercivity is much smaller than the potential fluctuation. Such small coercivity will shift the plateau transition point from $\Delta=\pm m_0$ to $\Delta=\pm (m_0+m_1/\ell_c^2)$, where $\ell_c=\sqrt{\hbar/eB_c}$ is the magnetic length. With $B_c<0.1$~T, $m_1/\ell_c^2<0.1$~meV. Since for the magnetic TI thin film studied in experiment $m_0\gg 1$~meV~\cite{chang2013b,wang2013b}, the shift of plateau transition point due to the coercivity is \emph{negligible}. This can be obtained by including the magnetic field in the Hamiltonian $\mathcal{H}_0$, and study the Chern number change as $\Delta$ varies.
At the coercivity, the external magnetic field enters into Eq.~(\ref{model1}) via minimal coupling: $\vec{k}\rightarrow \vec{k}+e\vec{A}$, where in the symmetric gauge the vector potential
\begin{equation}
\vec{A}=\frac{B}{2}\left(-y,x\right).
\end{equation}
We define the new operators
\begin{eqnarray}
\pi_{+} &=& \hbar\left(k_{+}+\frac{ieB}{2\hbar}z\right),\\
\pi_{-} &=& \hbar\left(k_{-}+\frac{ieB}{2\hbar}z^*\right),
\end{eqnarray}
where $k_\pm=k_x\pm ik_y$ and $z=x\pm iy$. These operators obey the commutation relations
\begin{equation}
\left[\pi_+,\pi_-\right]=-\frac{2\hbar^2}{\ell_c^2}.
\end{equation}
with the magnetic length $\ell_c=\sqrt{\hbar/eB}$. Using these commutation relation we define rasing and lowering operators
\begin{eqnarray}
&&a = \frac{\ell_c}{\sqrt{2}}\pi_-, \ \ a^{\dag} = \frac{\ell_c}{\sqrt{2}}\pi_+,
\\
&&\left[a, a^{\dag}\right] = 1.
\end{eqnarray}
The Hamiltonian can be rewritten as
\begin{eqnarray}
\mathcal{H}_0 &=& \begin{pmatrix}
\mathcal{H}_+(a,a^{\dag}) & 0\\
0 & \mathcal{H}_-(a,a^{\dag})
\end{pmatrix},
\\
\mathcal{H}_\pm(a,a^{\dag}) &=& \left[m_0\pm\Delta+\frac{2m_1}{\ell_c^2}\left(a^{\dag}a+\frac{1}{2}\right)\right]\tau_3
\nonumber
\\
&&+\frac{\sqrt{2}v_F}{\ell_c}\left(ia\tau_\pm-ia^{\dag}\tau_\mp\right).
\end{eqnarray}
where $\tau_j$ ($j=1,2,3$) are Pauli matrices, $\tau_\pm=(\tau_1\pm i\tau_2)/2$.
The spectrum of this Hamiltonian can be solved since only a finite number of harmonic oscillator Landau levels are coupled. The energy spectrum is
\begin{equation}
E_s = -s\frac{m_1}{\ell_c^2}\pm\sqrt{\frac{2v_F^2}{\ell_c^2}\mathcal{N}+\left(m_0+s\Delta+\frac{2m_1}{\ell_c^2}\mathcal{N}\right)^2}
\end{equation}
with $s=\pm$, and $\mathcal{N}=0,1,2,3,...$. This spectrum has ``zero mode'' given by
\begin{eqnarray}
E^0_+ &=& -m_0-\Delta-\frac{m_1}{\ell_c^2},\\
E^0_- &=& m_0-\Delta+\frac{m_1}{\ell_c^2}.
\end{eqnarray}
At the coercivity $B_c$, $E^0_{\pm}=0$ gives the transition point. Thus at half filling, the total Chern number of the system with the magnetic field becomes
\begin{equation}\label{chern}
C = \begin{cases}
\Delta/|\Delta|, & \text{for} \ \ \left|\Delta\right|>\left|m_0+m_1/\ell^2\right|\\
0, & \text{for} \ \ \left|\Delta\right|<\left|m_0+m_1/\ell^2\right|
\end{cases}
\end{equation}
Now the plateau transition point becomes $\Delta=\pm(m_0+m_1/\ell_c^2)$ with $B=B_c$.
The bulk LL and edge state spectrum for 5-quintuple layers (QLs) of Cr$_x$(Bi,Sb)$_{2-x}$Te$_3$ magnetic TI with different $\Delta$ is shown in Fig.~\ref{fig4}. The parameters are taken from Ref.~\onlinecite{wang2013b}, where $m_0<0$ and $m_1>0$. Fig.~\ref{fig4}(a) shows bulk LLs with $m_0+m_1/\ell_c^2<\Delta<-m_0-m_1/\ell_c^2$, in Fig.~\ref{fig4}(f) it shows the corresponding edge states, and there should be counter-propagating edge states that carry opposite Hall current. In Fig.~\ref{fig4}(c) and (h) with $\Delta>-m_0-m_1/\ell_c^2$, the LL spectrum changes where the Fermi energy is slightly above the two zero modes, and only one of them will provide edge state, which gives $C=1$.
\section{Random perturbations}
\label{randomperturbation}
Now we consider the random perturbations to the pure Hamiltonian of Eq.~(\ref{model0}). First, at the coercivity, the magnetic domains are being switched from up to down \emph{randomly}. The exchange field induced by local spin $\langle S\rangle$ in such random magnetic domains will give rise to
\begin{equation}
\widetilde{H}_{\Delta} = \Delta_z\widetilde{\sigma}_3\otimes1+\Delta_x\widetilde{\sigma}_1\otimes1+\Delta_y\widetilde{\sigma}_2\otimes1,
\end{equation}
where $\Delta_z$ is the exchange field along $z$ axis, $\Delta_{x,y}$ are the exchange field in the $x$-$y$ plane.
Second, top and bottom surface state will feel different random scalar potentials $V_1$ and $V_2$, respectively,
\begin{equation}
\widetilde{H}_{V} = \overline{V} 1\otimes1 +\delta V1\otimes\widetilde{\tau}_3,
\end{equation}
with $\overline{V}=(V_1+V_2)/2$, and $\delta V=(V_1-V_2)/2$.
Third, a small external magnetizing field $H$ is required to induce the coercivity transition. At the coercivity, the magnetization of the system $M$ is spatially random. So the magnetic field $B=\mu_0(H+M)$ in this system is also random in space, which couples to the system through a random vector potential $\vec{A}=(A_x,A_y)$, with the minimal coupling $\vec{k}\rightarrow\vec{k}-\vec{A}$, we have
\begin{equation}
\widetilde{H}_{A} = -A_y\widetilde{\sigma}_1\otimes\widetilde{\tau}_3+A_x\widetilde{\sigma}_2\otimes\widetilde{\tau}_3.
\end{equation}
All three types of randomness have been taken into account.
Then we make a unitary transformation to the basis of $|+\uparrow\rangle$, $|-\downarrow\rangle$, $|+\downarrow\rangle$, $|-\uparrow\rangle$ with $|\pm\uparrow\rangle=(|t\uparrow\rangle\pm|b\uparrow\rangle)/\sqrt{2}$
and $|\pm\downarrow\rangle=(|t\downarrow\rangle\pm|b\downarrow\rangle)/\sqrt{2}$. The pure Hamiltonian decouples as
\begin{eqnarray}
\mathcal{H}_0(k_x,k_y) &=&
\begin{pmatrix}
\mathcal{H}_+(k) & 0\\
0 & \mathcal{H}_-(k)
\end{pmatrix},
\\
\mathcal{H}_{\pm}(k) &=& k_y\tau_1\mp k_x\tau_2+\left(m(k)\pm\Delta\right)\tau_3.
\end{eqnarray}
$\tau_i$ are Pauli matrices. The random perturbations in the new basis are
\begin{equation}
\mathcal{H}_{\Delta} = \begin{pmatrix}
\Delta_z\tau_3 & \Delta_x1_{2\times2}-i\Delta_y\tau_3\\
\Delta_x1_{2\times2}+i\Delta_y\tau_3 & -\Delta_z\tau_3
\end{pmatrix},
\end{equation}
\begin{equation}
\mathcal{H}_{V} = \begin{pmatrix}
\overline{V} & \delta V\tau_1\\
\delta V\tau_1 & \overline{V}
\end{pmatrix},
\end{equation}
\begin{equation}
\mathcal{H}_{A} = \begin{pmatrix}
-A_y\tau_1+A_x\tau_2 & 0\\
0 & -A_y\tau_1-A_x\tau_2
\end{pmatrix}.
\end{equation}
The $\Delta_{x,y}$ will in general mix $\mathcal{H}_+(k)$ and $\mathcal{H}_-(k)$. However, we only consider the plateau transition, and the transition point of $\mathcal{H}_+(k)$ and $\mathcal{H}_-(k)$ are different, $\Delta=-m_0$ for $\mathcal{H}_+(k)$ and $\Delta=m_0$ for $\mathcal{H}_-(k)$. For plateau transition at $\Delta=-m_0$, $\mathcal{H}_-(k)$ is gapped, and the low energy physics is only determined by $\mathcal{H}_+(k)$. The $\Delta_{x,y}$ term can be perturbatively added into $\mathcal{H}_+(k)$ as
\begin{equation}
\mathcal{H}_{\Delta}^{x,y} \approx \frac{\Delta_x^2+\Delta_y^2}{2\Delta}\tau_3,
\end{equation}
which gives a random exchange field along $z$ axis. In general, in the system the fluctuation $\Delta_{x,y}\ll\Delta$, thus $(\Delta_x^2+\Delta_y^2)/2\Delta\ll\Delta_z$. Therefore, to the first order, this term can be neglected. Similarly for the transition at $\Delta=m_0$.
$\delta V$ term will also mix $\mathcal{H}_+(k)$ and $\mathcal{H}_-(k)$. Following the same discussion above for $\Delta_{x,y}$. At $\Delta=-m_0$, $\delta V$ contributes a random exchange field term along $z$ axis in $\mathcal{H}_+(k)$ as
\begin{equation}
\mathcal{H}_{\delta V} \approx \frac{(\delta V)^2}{2m_0}\tau_3,
\end{equation}
$\delta V\ll |m_0|$, so this term is negligibly small compared to $\Delta_z$. Besides, the 2D film of magnetic topological insulator is very thin (less than 5~nm), the random scalar potential feeled by the top and bottom surface states are almost the same $V_1\approx V_2$. Therefore, $\delta V$ can be ignored. Similarly for the transition at $\Delta=m_0$.
Finally, we have build up the model for the QAH plateau transition,
\begin{eqnarray}
\mathcal{H} =\mathcal{H}_0 + \mathcal{H}_{\Delta} +\mathcal{H}_{V}+\mathcal{H}_A,
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{H}_{\Delta} &=& \begin{pmatrix}
\Delta_z\tau_3 & 0\\
0 & -\Delta_z\tau_3
\end{pmatrix},
\\
\mathcal{H}_{V} &=& \begin{pmatrix}
\overline{V} & 0\\
0 & \overline{V}
\end{pmatrix},
\\
\mathcal{H}_{A} &=& \begin{pmatrix}
-A_y\tau_1+A_x\tau_2 & 0\\
0 & -A_y\tau_1-A_x\tau_2
\end{pmatrix}.
\end{eqnarray}
Redefine $\Delta_z(x,y)=\Delta(x,y)$ and $\overline{V}(x,y)=V(x,y)$, this gives Eq.~(\ref{randompotential}) in the paper.
\section{Resistivity and conductivity tensor}
\label{res_con_tensor}
The resistivity tensor is
\begin{equation}
\boldsymbol\rho = \begin{pmatrix}
\rho_{xx} & \rho_{xy}\\
-\rho_{xy} & \rho_{yy}
\end{pmatrix},
\end{equation}
and the conductivity tensor is
\begin{equation}
\boldsymbol\sigma = \begin{pmatrix}
\sigma_{xx} & \sigma_{xy}\\
-\sigma_{xy} & \sigma_{yy}
\end{pmatrix},
\end{equation}
with $\boldsymbol{\sigma}=\boldsymbol{\rho}^{-1}$, we have
\begin{equation}
\rho_{xx} = \frac{\sigma_{xx}}{\sigma_{xx}\sigma_{yy}+\sigma_{xy}^2}=\frac{\sigma_{xx}}{\sigma_{xx}^2+\sigma_{xy}^2},
\end{equation}
and
\begin{equation}
\rho_{xy} = \frac{-\sigma_{xy}}{\sigma_{xx}\sigma_{yy}+\sigma_{xy}^2}=\frac{-\sigma_{xy}}{\sigma_{xx}^2+\sigma_{xy}^2}.
\end{equation}
This transforms $\sigma_{xx}$ and $\sigma_{xy}$ into $\rho_{xx}$ and $\rho_{xy}$.
When $\Delta<-|m_0|$, the system is insulating with Chern number $C=-1$, thus we have
\begin{equation}
\boldsymbol\sigma = \frac{e^2}{h}\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix},
\end{equation}
and corresponding resistivity tensor is
\begin{equation}
\boldsymbol\rho = \frac{h}{e^2}\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}.
\end{equation}
Similar case for $\Delta>|m_0|$. When $-|m_0|<\Delta<|m_0|$, the system is insulating with Chern number $C=0$, thus we expect the conductivity tensor
\begin{equation}
\boldsymbol\sigma = \begin{pmatrix}
\eta & 0\\
0 & \eta
\end{pmatrix},
\end{equation}
where in large sample at zero-temperature ($T=0$), $\eta\rightarrow 0^+$; for finite sample with finite $T$, $\eta$ is very small (possibly due to variable range hopping). Thus the corresponding resistivity tensor is
\begin{equation}
\boldsymbol\rho = \begin{pmatrix}
1/\eta & 0\\
0 & 1/\eta
\end{pmatrix}.
\end{equation}
For the QAH effect in magnetic TI, at low $T$, there should exist zero Hall plateau with $\sigma_{xy}=0$ and $\rho_{xy}=0$. From the scaling theory, we predict that $\sigma_{xx}$ generally become nonzero between the plateau transition from $\sigma_{xy}=-e^2/h$ to $\sigma_{xy}=0$ and $\sigma_{xy}=0$ to $\sigma_{xy}=e^2/h$. At $\sigma_{xy}=0$ plateau, $\sigma_{xx}\rightarrow0$. Therefore, $\sigma_{xx}$ shows two peaks at the coercivity. However, $\rho_{xx}$ only shows one peak at the coercivity. Because at $\rho_{xy}=0$ plateau, $\rho_{xx}=1/\eta\rightarrow\infty$. In fact, this remarkable theoretical prediction is already borne out in experiment, by inverting the experimental data of $\rho_{xx}$ into $\sigma_{xx}$, at the coercivity, $\sigma_{xx}$ shows double peaks with two critical fields while $\rho_{xx}$ only has single peak~\cite{chang2013b}.
The critical field $H_{1}^*$ and $H_2^*$ is not universal. For example, a slightly macroscopic inhomogeneity in the electron density across the sample will in general result a slightly different $H_{1}^*$ and $H_2^*$. Such inhomogeneities do not affect the power-law behaviors in $\rho_{xx}$ and $\rho_{xy}$.
\end{appendix}
|
2,869,038,155,700 | arxiv | \section{Introduction}
Coherence, the most fundamental quantum feature of a nonclassical system,
stems from quantum superposition principle which reveals the wave particle
duality of matter. It has been shown that coherence plays the key roles in
the physical dynamics in biology \cite{Engel,Plenio,Coll,loyd,licm,Huel}, transport theory \cite{reb,wit} , and thermodynamics \cite{berg,Nar,Horo,Los1,Los2}. In
particular, some typical approaches such as phase space distributions and
higher order correlation functions have been developed in quantum optics to
reveal quantum coherence even as an irrigorous quantification \cite{Glauber,Sudarshan,Scully}. Quite
recently, quantum coherence has been attracting increasing
interest in various aspects \cite%
{Pleniom, Giro, Napoli, Lewen,Rast,Piani,Winter,Du,Chi,Chi2,Chi3,Marvian,Marvian2,Yao,Sing,Radha} including the quantification of coherence \cite{Pleniom, Giro, Napoli, Lewen,Rast,Piani}, the operational resource theory \cite{Winter, Du, Chi,Chi2,Chi3}, the distribution \cite{Radha}, the different
understandings \cite{Yu09,Stre,Ma,Tan} and so on.
Quantification of coherence is the most essential ingredient not only in the
quantum theory but also in the practical application. Various quantities
have been proposed to serve as a coherence quantifier, however the available candidates are still quite limited. Up to now, only two
alternatives, i.e., the coherence measures based on $l_{1}$ norm and the relative entropy, have turned
out to be a satisfactory coherence measure \cite{Pleniom}. In contrast, the usual $l_{p}$ ($p\neq 1$) norm can not directly induce a good
measure \cite{Lewen}. In addition, the coherence quantifier based on the Fidelity is easily shown to
satisfy the monotonicity that the coherence of the post-incoherent-operation state doesn't increase,
but it violates the strong monotonicity that average coherence doesn't increase under the
sub-selective incoherent operations \cite{Pleniom,Fan}. Similarly, even though the coherence based on the trace norm
also satisfies the monotonicity but lacks a strict proof for the strong monotonicity \cite{Lewen,Fan}. However, we know
that the strong monotonicity is much more important than the monotonicity
not only because the sub-selection of the measurement outcomes required by
the strong monotonicity can be well controlled in experiment as is stated in
Ref. \cite{Pleniom,Lewen}, but also because the realizable sub-selection would lead to the real
increment of the coherence from the point of resource theory of view if the
strong monotonicity was violated. In this sense, the quantitative
characterization of coherence still needs to be paid more attention.
Recently, Ref. \cite{Rast} has also proposed a coherence quantifier in terms of the
Tsallis relative $\alpha $ entropy which lays the foundation to the
non-extensive thermo-statistics and plays the same role as the standard
logarithmic entropy does in the information theory \cite{lisa,Tsallis}. However, it is
unfortunate that the Tsallis relative $\alpha $ entropy isn't an ideal
coherence measure either because Ref. \cite{Rast} showed that it only satisfies the
monotonicity and a variational monotonicity rather than the strong
monotonicity. Is it possible to bridge the Tsallis relative $%
\alpha $ entropy with the strong monotonicity by some particular and
elaborate design? In this paper, we build such a bridge between the Tsallis relative $\alpha $ entropy with
the strong monotonicity, hence present a family of good coherence
quantifiers. By considering the special case in this family, one can find that the $l_{2}$ norm
can be validly employed to quantify the coherence. The remaining of this
paper is organized as follows. In Sec. II, we introduce the coherence measure and the Tsallis
relative $\alpha $ entropy. In Sec. III, we present the
family of coherence quantifier and mainly prove them to be strongly
monotonic. In Sec. IV, we study the maximal coherence and several particular coherence measure. Finally, we finish the paper by
the conclusion and some discussions.
\section{The coherence and the Tsallis relative $\protect%
\alpha $ entropy}
The resource theory includes three ingredients: the free states, the
resource states and the free operations \cite{Winter, Gour}. For coherence, the free states are
referred as to the incoherent states which are defined in a given fixed
basis $\left\{ \left\vert i\right\rangle \right\} $ by the states with the
density matrices in the diagonal form, i.e., $\delta =\sum\limits_{i}\delta
_{i}\left\vert i\right\rangle \left\langle i\right\vert $ with $%
\sum\limits_{i}\delta _{i}=1$ for the positive $\delta _{i}$. All the states
without the above diagonal form are the coherent states, i.e., the resource
states. The quantum operations described by the Kraus operators $\left\{
K_{n}\right\} $ with $K_{n}^{\dag }K_{n}=\mathbf{I}$ are called as the
incoherent operations and serve as the free operations for coherence, if $%
K_{n}\delta K_{n}^{\dag }\in \mathcal{I}$ for any incoherent $\delta $. In
this sense, the standard criteria of a good coherence quantifier $C(\rho )$
for the state $\rho $ can be rigorously rewritten as \cite{Pleniom} (i) (Null) $C(\delta
)=0 $ for $\delta \in \mathcal{I}$; (ii) (Strong monotonicity) for any state
$\rho $ and incoherent operations $\left\{ K_{n}\right\} $, $C(\rho
)\geqslant \sum\limits_{n}p_{n}C(\rho _{n})$ with $p_{n}=$Tr$K_{n}\rho
K_{n}^{\dag }$ and $\rho _{n}=K_{n}\rho K_{n}^{\dag }/p_{n}$; (iii)
(Convexity) For any ensemble $\left\{ q_{i},\sigma _{i}\right\} $, $%
C(\sum\limits_{i}q_{i}\sigma _{i})\leq \sum\limits_{i}q_{i}C(\sigma _{i})$.
In addition, the monotonicity requires $C(\rho )\geqslant
C(\sum\limits_{n}p_{n}\rho _{n})$, however, it alone isn't laid in an
important position because the measurement outcomes of $\left\{
K_{n}\right\} $ can be well controlled (subselected) in practical
experiments, or in other words, the violation of the strong monotonicity
means that the ultimate coherence is actually increased by the incoherent
operations $\left\{ K_{n}\right\} $ even though it can be automatically
implied by (ii) and (iii). With these criteria, any measure of
distinguishability such as the (pseudo-) distance norm could induce a
potential candidate for a coherence quantifier. But it has been shown that
some candidates only satisfy the monotonicity rather than the strong
monotonicity, so they are not ideal and could be only used in the limited
cases. Ref. \cite{Rast} found that the coherence based on the Tsallis relative $%
\alpha $ entropy is also such a coherence quantifier without the strong
monotonicity.
The Tsallis relative $\alpha $ entropy is a special case of the quantum $f$%
-divergences \cite{Rast,Hiai}. For two density matrices $\rho $ and $\sigma $, it is defined
as%
\begin{equation}
D_{\alpha }\left( \rho ||\sigma \right) =\frac{1}{\alpha -1}\left( \text{Tr}%
\rho ^{\alpha }\sigma ^{1-\alpha }-1\right)
\end{equation}%
for $\alpha \in (0,2]$. It is shown that for $\alpha \longrightarrow 1,$ $%
D_{\alpha }\left( \rho ||\sigma \right) $ will reduce to the relative
entropy $S\left( \rho ||\sigma \right) =Tr\rho \log _{2}\rho -\rho \log
_{2}\sigma $. The Tsallis relative $\alpha $ entropy $D_{\alpha }\left( \rho
||\sigma \right) $ inherits many important properties of the quantum $f$%
-divergences, for example, (Positivity) $D_{\alpha }\left( \rho ||\sigma
\right) \geq 0$ with equality if and only if $\rho =\sigma $, (Isometry) $%
D_{\alpha }\left( U\rho U^{\dag }||U\sigma U^{\dag }\right) =D_{\alpha
}\left( \rho ||\sigma \right) $ for any unitary operations, (Contractibility) $%
D_{\alpha }\left( \$\left( \rho \right) ||\$(\sigma )\right) \leq D_{\alpha
}\left( \rho ||\sigma \right) $ under any trace-preserving and completely
positive (TPCP) map $\$$ and (Joint convexity) $D_{\alpha }\left(
\sum_{n}p_{n}\rho _{n}||\sum_{n}p_{n}\sigma _{n}\right) \leq
\sum_{n}p_{n}D_{\alpha }\left( \rho _{n}||\sigma _{n}\right) $ for the
density matrices $\rho _{n}$ and $\sigma _{n}$ and the corresponding
probability distribution $p_{n}$.
Based on the Tsallis relative $\alpha $ entropy $D_{\alpha }\left( \rho
||\sigma \right) $, the coherence in the fixed reference basis $\left\{
\left\vert j\right\rangle \right\} $ can be characterized by \cite{Rast}
\begin{eqnarray}
\tilde{C}_{\alpha }(\rho ) &=&\min_{\delta \in \mathcal{I}}D_{\alpha }\left(
\rho ||\delta \right) \notag \\
&=&\frac{1}{\alpha -1}\left[ \left( \sum_{j}\left\langle j\right\vert \rho
^{\alpha }\left\vert j\right\rangle ^{1/\alpha }\right) ^{\alpha }-1\right] .
\label{Tsc}
\end{eqnarray}%
However, it is shown that $\tilde{C}_{\alpha }(\rho )$ satisfies all the
criteria for a good coherence measure but the strong monotonicity. Since $%
D_{\alpha \rightarrow 1}\left( \rho ||\sigma \right) $ reduces to the
relative entropy $S\left( \rho ||\sigma \right) $ which has induced the good
coherence measure, throughout the paper we are mainly interested in $\alpha
\in (0,1)\cup (1,2]$.
In addition, the Tsallis relative $\alpha $ entropy $D_{\alpha }\left( \rho
||\sigma \right) $ can also be reformulated by a very useful function as
\begin{equation}
D_{\alpha }\left( \rho ||\sigma \right) =\frac{1}{\alpha -1}\left( f_{\alpha
}\left( \rho ,\sigma \right) -1\right)
\end{equation}
with
\begin{equation}
f_{\alpha }\left( \rho
,\sigma \right) =\mathrm{Tr}\rho ^{\alpha }\sigma ^{1-\alpha }.
\end{equation}
Accordingly, the
coherence $\tilde{C}_{\alpha }(\rho )$ can also be rewritten as
\begin{equation}
\tilde{C}%
_{\alpha }(\rho )=\frac{1}{\alpha -1}\left[ \mathrm{sgn}_{1}(\alpha
)\min_{\delta \in \mathcal{I}}\mathrm{sgn}_{1}(\alpha )f_{\alpha }\left( \rho
,\delta \right) -1\right]
\end{equation} which, based on Eq. (\ref{Tsc}), leads to the
conclusion%
\begin{equation}
\min_{\delta \in \mathcal{I}}\text{sgn}_{1}(\alpha )f_{\alpha }\left( \rho
,\delta \right) =\left( \sum_{j}\left\langle j\right\vert \rho ^{\alpha
}\left\vert j\right\rangle ^{1/\alpha }\right) ^{\alpha }. \label{concl}
\end{equation}%
Based on Eq. (\ref{concl}) and the properties of $D_{\alpha }\left( \rho ||\sigma \right) $ mentioned above, one can have the following observations for the function $f_{\alpha }\left( \rho ,\sigma \right) $ \cite{Rast, Hiai}.
\textbf{Observations}: $f_{\alpha }\left( \rho ,\sigma \right) $ satisfies the following properties:
(I) $f_{\alpha }\left( \rho ,\sigma \right) \geq 1$ for $\alpha \in (1,2]$
and $f_{\alpha }\left( \rho ,\sigma \right) \leq 1$ for $\alpha \in (0,1)$
with equality if and only if $\rho =\sigma $;
(II) For a unitary operation $U$, $f_{\alpha }\left( U\rho U^{\dagger
},U\sigma U^{\dagger }\right) =f_{\alpha }\left( \rho ,\sigma \right) $;
(III) For any TPCP map $\$$, $f_{\alpha }\left( \rho ,\sigma \right) $
doesn't decrease for $\alpha \in (0,1)$, and doesn't increased for $\alpha
\in (1,2]$, namely,
\begin{equation}
\text{sgn}_{1}(\alpha )f_{\alpha }\left( \$\left[ \rho \right] ,\$\left[
\sigma \right] \right) \leq \text{sgn}_{1}(\alpha )f_{\alpha }\left( \rho
,\sigma \right) ,
\end{equation}%
where the function is defined by sgn$_{1}(\alpha )=\left\{
\begin{array}{cc}
-1, & \alpha \in (0,1) \\
1, & \alpha \in (1,2]%
\end{array}%
\right. $;
(IV) The function sgn$_{1}(\alpha )f_{\alpha }\left( \rho ,\sigma \right) $
is jointly convex;
(V) For a state $\delta $, $f_{\alpha }\left( \rho \otimes \delta ,\sigma
\otimes \delta \right) =f_{\alpha }\left( \rho ||\sigma \right) $, which can
be easily found from the function itself.
\section{The coherence measures based on the Tsallis relative $\protect%
\alpha $ entropy}
To proceed, we would like to present a very important lemma for the function
$f_{\alpha }\left( \rho ,\sigma \right) $, which is the key to show our main
result.
\textbf{Lemma 1}.-Suppose both $\rho $ and $\sigma $ simultaneously undergo
a TPCP map $\$:=\left\{ M_{n}:\sum\limits_{n}{M}_{n}^{\dagger }{M}%
_{n}=\mathbb{I}_{S}\right\} $ which transforms the states $\rho $ and $%
\sigma $ into the ensemble $\left\{ p_{n},\rho _{n}\right\} $ and $\left\{
q_{n},\sigma _{n}\right\} $, respectively, then we have
\begin{equation}
\text{sgn}_{1}(\alpha )f_{\alpha }\left( \rho _{S},\delta _{S}\right) \geq
\text{sgn}_{1}(\alpha )\sum\limits_{n}p_{n}^{\alpha }q_{n}^{1-\alpha
}f_{\alpha }\left( \rho _{n},\sigma _{n}\right) . \label{lem1}
\end{equation}%
\textbf{Proof. }Any TPCP map can be realized by a unitary operation on a
composite system followed by a local projective measurement \cite{Nielsen}. Suppose system
S is of our interest and A is an auxiliary system. For a TPCP map $%
\$:=\left\{ M_{n}:\sum\limits_{n}{M}_{n}^{\dagger }{M}_{n}=\mathbb{%
I}_{S}\right\} $, one can always find a unitary operation $U_{SA}$ and a
group of projectors $\left\{ \Pi _{n}^{A}=\left\vert n\right\rangle
_{A}\left\langle n\right\vert \right\} $ such that
\begin{eqnarray}
&&M_{n}\rho _{S}M_{n}^{\dagger }\otimes \Pi _{n}^{A} \notag \\
&=&\left( \mathbb{I}_{S}\otimes \Pi _{n}^{A}\right) U_{SA}\left( \rho
_{S}\otimes \Pi _{0}^{A}\right) U_{SA}^{\dag }\left( \mathbb{I}_{S}\otimes
\Pi _{n}^{A}\right) . \label{eq}
\end{eqnarray}%
Using Properties (I) and (II), we have
\begin{eqnarray}
&&f_{\alpha }\left( \rho _{S},\delta _{S}\right) \notag \\
&=&f_{\alpha }\left( U_{SA}\left( \rho _{S}\otimes \Pi _{0}^{A}\right)
U_{SA}^{\dag },U_{SA}\left( \sigma _{S}\otimes \Pi _{0}^{A}\right)
U_{SA}^{\dag }\right)
\end{eqnarray}%
holds for any two states $\rho _{S}$ and $\sigma _{S}$. Let $\rho _{Sf}=\$_{SA}%
\left[ U_{SA}\left( \rho _{S}\otimes \Pi _{0}^{A}\right) U_{SA}^{\dag }%
\right] $ and $\sigma _{Sf}=\$_{SA}\left[ U_{SA}\left( \sigma _{S}\otimes
\Pi _{0}^{A}\right) U_{SA}^{\dag }\right] $ which describe the states $%
U_{SA}\left( \rho _{S}\otimes \Pi _{0}^{A}\right) U_{SA}^{\dag }$ and $%
U_{SA}\left( \sigma _{S}\otimes \Pi _{0}^{A}\right) U_{SA}^{\dag }$ undergo
an arbitrary TPCP map $\$_{SA}$ performed on the composite system S plus A.
Based on Property (III), one can easily find
\begin{equation}
\text{sgn}_{1}(\alpha )f_{\alpha }\left( \rho _{S},\delta _{S}\right) \geq
\text{sgn}_{1}(\alpha )f_{\alpha }\left( \rho _{Sf},\sigma _{Sf}\right) .
\label{eq1r}
\end{equation}%
Suppose the TPCP map $\$_{SA}:=\left\{ \mathbb{I}_{S}\otimes \Pi
_{n}^{A}\right\} $, according to Eq. (\ref{eq}), one can replace $\rho _{Sf}$
and $\sigma _{Sf}$ in Eq. (\ref{eq1r}), respectively, by
\begin{equation}
\rho _{Sf}\rightarrow \tilde{\rho%
}_{Sf}=\sum\limits_{n}M_{n}\rho _{S}M_{n}^{\dagger }\otimes \Pi _{n}^{A}
\end{equation}
and \begin{equation}
\sigma _{Sf}\rightarrow \tilde{\sigma}_{Sf}=\sum\limits_{n}M_{n}\sigma
_{S}M_{n}^{\dagger }\otimes \Pi _{n}^{A}.\end{equation} Therefore, we get
\begin{gather}
\text{sgn}_{1}(\alpha )f_{\alpha }\left( \rho _{S},\delta _{S}\right) \geq
\text{sgn}_{1}(\alpha )f_{\alpha }\left( \tilde{\rho}_{Sf},\tilde{\sigma}%
_{Sf}\right) \notag \\
=\text{sgn}_{1}(\alpha )\sum\limits_{n}f_{\alpha }\left( M_{n}\rho
_{S}M_{n}^{\dagger }\otimes \Pi _{n}^{A},M_{n}\sigma _{S}M_{n}^{\dagger
}\otimes \Pi _{n}^{A}\right) \notag \\
=\text{sgn}_{1}(\alpha )\sum\limits_{n}f_{\alpha }\left( M_{n}\rho
_{S}M_{n}^{\dag },M_{n}\sigma _{S}M_{n}^{\dagger }\right) \notag \\
=\text{sgn}_{1}(\alpha )\sum\limits_{n}p_{n}^{\alpha }q_{n}^{1-\alpha
}f_{\alpha }\left( \rho _{n},\sigma _{n}\right) , \label{main1}
\end{gather}%
which completes the proof.\hfill{}$\blacksquare$
Based on Lemma 1 and the preliminaries given in the previous section, we can present our main theorem as follows.
\textbf{Theorem 1.}-The coherence of a quantum state $\rho $
can be measured by
\begin{eqnarray}
C_{\alpha }\left( \rho \right) &=&\min_{\delta \in \mathcal{I}}\frac{1}{%
\alpha -1}\left( f_{\alpha }^{1/\alpha }\left( \rho ,\delta \right)
-1\right) \label{t1} \\
&=&\frac{1}{\alpha -1}\left( \sum_{j}\left\langle j\right\vert \rho ^{\alpha
}\left\vert j\right\rangle ^{1/\alpha }-1\right) , \label{firstd}
\end{eqnarray}%
where $\alpha \in (0,2]$, $\left\{ \left\vert j\right\rangle \right\} $ is
the reference basis and $f_{\alpha }\left( \rho ,\delta \right) =\left(
\alpha -1\right) D_{\alpha }\left( \rho ||\sigma \right) +1$ with $D_{\alpha
}\left( \rho ||\sigma \right) $ representing the Tsallis relative $\alpha $
entropy.
\textbf{Proof.}-At first, one can note that the function $x^{\alpha }$ is a
monotonically increasing function on $x$, so Eq. (\ref{firstd}) obviously
holds for positive $x$ due to Eq. (\ref{concl}).
\textit{Null.- } Since the original Tsallis entropy defined by Eq. (\ref{Tsc}) can unambiguously distinguish a coherent state from the incoherent one. Eq. (\ref{Tsc}) implies that $\sum_{j}\left\langle
j\right\vert \rho ^{\alpha }\left\vert j\right\rangle ^{1/\alpha }=1$ is
sufficient and necessary condition for incoherent states. Thus the zero $C_{\alpha }\left( \rho \right)$ is also a sufficient and necessary condition for incoherent state $\rho$.
\textit{Convexity.-} From Ref. \cite{Liebp}, one can learn that the function $%
g(A)=$Tr$\left( XA^{p}X^{\dag }\right) ^{s}$ is convex in positive matrix $A$
for $p\in \lbrack 1,2]$ and $s\geq \frac{1}{p}$, and concave in $A$ for $p\in
(0,1]$ and $1\leq s\leq \frac{1}{p}$. Now let's assume $A=\rho $ , $%
X=\left\vert j\right\rangle \left\langle j\right\vert $ and $p=\alpha $ and $%
s=\frac{1}{\alpha }$, thus one has
\begin{eqnarray}
g^j_\alpha(\rho)&=&\mathrm{Tr}\left( \left\vert j\right\rangle \left\langle j\right\vert\rho^{\alpha}\left\vert j\right\rangle \left\langle j\right\vert\right) ^{1/\alpha}= \left\langle j\right\vert\rho^{\alpha}\left\vert j\right\rangle ^{1/\alpha},\end{eqnarray}
which implies $g^j_\alpha(\rho)$ is convex in density matrix $\rho$
for $\alpha \in \lbrack 1,2]$ and $s= \frac{1}{\alpha}$, and concave in $\rho$ for $\alpha\in
(0,1]$ and $s= \frac{1}{\alpha}$. Here the subscript $\alpha$ and the superscript $j$ in $g^j_\alpha$ specifies the particular choice. So it is easy to find that
$\frac{1}{\alpha-1}\sum_j g^j_\alpha(\rho)$ is convex for $\alpha\in (0,2]$. Considering Eq. (\ref{firstd}), one can easily show $C_{\alpha }\left( \rho \right) $
is convex in $\rho $.
\textit{Strong monotonicity.-} Now let $\{M_{n}\}$ denote the incoherent
operation, so the ensemble after the incoherent operation on the state $\rho
$ can be given by $\left\{ p_{n},\rho _{n}\right\} $ with $p_{n}=$Tr$%
M_{n}\rho M_{n}^{\dag }$ and $\rho _{n}=M_{n}\rho M_{n}^{\dag }/p_{n}$. Thus
the average coherence $\bar{C}_{\alpha }$ is
\begin{eqnarray}
\bar{C}_{\alpha } &=&\sum_{n}p_{n}C_{\alpha }\left( \rho _{n}\right) \notag
\\
&=&\min_{\delta _{n}\in \mathcal{I}}\frac{1}{\alpha -1}\left(
\sum_{n}p_{n}f_{\alpha }^{1/\alpha }\left( \rho _{n},\delta _{n}\right)
-1\right) . \label{diyi}
\end{eqnarray}%
Let $\delta ^{o}$ denote the optimal incoherent state such that
\begin{equation}
C_{\alpha}\left( \rho \right) =\frac{1}{\alpha -1}\left( f_{\alpha }^{1/\alpha
}\left( \rho ,\delta ^{o}\right) -1\right) ,
\end{equation} i.e.,
\begin{equation}
f_{\alpha }(\rho
,\delta ^{o})=\min_{\delta \in \mathcal{I}}\mathrm{sgn}_{1}(\alpha )f_{\alpha
}(\rho ,\delta ).
\end{equation} Considering the incoherent operation $\{M_{n}\},$ we have $%
\sigma _{n}^{o}=M_{n}\delta ^{o}M_{n}^{\dagger }/q_{n}\in \mathcal{I}$ with $%
q_{n}=$Tr$M_{n}\delta ^{o}M_{n}^{\dagger }$. Therefore, one can immediately
find that%
\begin{equation}
\min_{\delta \in \mathcal{I}}\text{sgn}_{1}(\alpha )f_{\alpha }^{1/\alpha
}(\rho ,\delta )\leq \text{sgn}_{1}(\alpha )f_{\alpha }^{1/\alpha }\left(
\rho _{n},\sigma _{n}^{o}\right) ,\label{dier}
\end{equation}%
where we use the function $x^{1/\alpha }$ is monotonically increasing on $x$%
. According to Eqs. (\ref{diyi}) and (\ref{dier}), we obtain
\begin{equation}
\bar{C}_{\alpha }\leq \frac{1}{\alpha -1}\left( \sum_{n}p_{n}f_{\alpha
}^{1/\alpha }\left( \rho _{n},\sigma _{n}^{o}\right) -1\right) . \label{tit}
\end{equation}%
In addition, the H\"{o}lder inequality \cite{Holder} implies that for $\alpha \in (0,1),$%
\begin{equation}
\left[ \sum_{n}q_{n}\right] ^{1-\alpha }\left[ \sum_{n}p_{n}f_{\alpha
}^{1/\alpha }\left( \rho _{n},\sigma _{n}^{o}\right) \right] ^{\alpha }\geq
\sum_{n}p_{n}^{\alpha }q_{n}^{1-\alpha }f_{\alpha }\left( \rho _{n},\sigma
_{n}^{o}\right) ,
\end{equation}%
and the inequality sign is reverse for $\alpha \in (1,2],$ so Eq. (\ref{tit}%
) becomes
\begin{eqnarray}
\bar{C}_{\alpha } &\leq &\frac{1}{\alpha -1}\left( \left[ \sum_{n}p_{n}^{%
\alpha }q_{n}^{1-\alpha }f_{\alpha }\left( \rho _{n},\sigma _{n}^{o}\right) %
\right] ^{1/\alpha }-1\right) \notag \\
&\leq &\frac{1}{\alpha -1}\left( f_{\alpha }^{1/\alpha }\left( \rho ,\delta
^{o}\right) -1\right) =C_{\alpha }, \label{comp1}
\end{eqnarray}%
which is due to Lemma 1. Eq. (\ref{comp1}) shows the strong monotonicity of $%
C_{\alpha }.$\hfill{}$\blacksquare$
\section{Maximal coherence and several typical quantifiers}
Next, we will show that the maximal coherence can be achieved by the maximally coherent states.
At first, we assume $\alpha\in(0,1)$. Based on the
eigen-decomposition of a $d$-dimensional state $\rho :\rho =\sum\limits_{k}\lambda _{k}\left\vert
\psi _{k}\right\rangle \left\langle \psi _{k}\right\vert $ with $\lambda
_{k} $ and $\left\vert \psi _{k}\right\rangle $ representing the eigenvalue
and eigenvectors, we have
\begin{eqnarray}
\sum_{j}\left\langle j\right\vert \rho ^{\alpha }\left\vert j\right\rangle
^{1/\alpha } &=&\sum_{j}\left( \sum_{k}\lambda _{k}^{\alpha }\left\vert
\left\langle \psi _{k}\right. \left\vert j\right\rangle \right\vert
^{2}\right) ^{1/\alpha } \notag \\
&\geq &d\left( \sum_{jk}\frac{\lambda _{k}^{\alpha }}{d}\left\vert
\left\langle \psi _{k}\right. \left\vert j\right\rangle \right\vert
^{2}\right) ^{1/\alpha } \notag \\
&\geq &d\left( \sum_{k}\frac{\lambda _{k}^{\alpha }}{d}\right) ^{1/\alpha
}\geq d^{\frac{\alpha -1}{\alpha }}. \label{lb}
\end{eqnarray}%
One can easily find that the lower bound Eq. (\ref{lb}) can be attained by the maximally coherent states $\rho_m=\left\vert \Psi\right\rangle\left\langle\Psi\right\vert$ with $\left\vert\Psi\right\rangle=\frac{1}{\sqrt{d}}\sum_je^{i\phi_j}\left\vert j\right\rangle$. Correspondingly, the coherence is given by \begin{equation}C_{0<\alpha<1}(\rho_m)=\frac{1}{1-\alpha}(1-d^{\frac{\alpha -1}{\alpha }}).\end{equation} Similarly, for $\alpha \in (1,2]$, the function $x^{1/\alpha }$ is
concave, which leads to that Eq. (\ref{lb}) with the
inverse inequality sign holds. The inequality can also saturate for $\rho_m$. The corresponding coherence is given by \begin{equation}C_{1<\alpha\leq2}(\rho_m)=\frac{1}{\alpha-1}(d^{\frac{\alpha -1}{\alpha }}-1).\end{equation}
$C_{\alpha }\left( \rho \right) $ actually defines a family of
coherence measures related to the Tsallis relative $\alpha $ entropy. This
family includes several typical coherence measures. As mentioned above, the
most prominent coherence measure belonging to this family is the coherence
in terms of relative entropy, i.e., $C_{1}\left( \rho \right) =S(\rho )$.
One can also find that
\begin{eqnarray}
C_{1/2}\left( \rho \right) &=&\min_{\delta \in \mathcal{I}}2\left( 1-\left[
Tr\sqrt{\rho }\sqrt{\delta }\right] ^{2}\right) \notag \\
&=&\min_{\delta \in \mathcal{I}}\left\Vert \sqrt{\rho }-\sqrt{\delta }%
\right\Vert _{2}^{2} \notag \\
&=&1-\sum_{i}\left\langle i\right\vert \sqrt{\rho }\left\vert i\right\rangle
^{2}
\end{eqnarray}%
with $\left\Vert \cdot \right\Vert _{2}$ denoting $l_{2}$ norm. So the $%
l_{2}$ norm has been revived for coherence measure by considering the square
root of the density matrices. This is much like the quantification of
quantum correlation proposed in Ref. \cite{luoq}. In addition, $C_{1/2}(\rho)$ can also
be rewritten as
\begin{equation}
C_{1/2}\left( \rho \right) =-\frac{1}{2}\sum_i\mathrm{Tr}\left\{\left [\sqrt{\rho},\left\vert i\right\rangle\left\langle i\right\vert\right]^2\right\}
\end{equation}
which is just the coherence measure based on the skew information \cite{skew1,skew2}.
Finally, one can also see
that
\begin{eqnarray}
C_{2}\left( \rho \right) &=&\min_{\delta \in \mathcal{I}}\left( \sqrt{%
Tr\rho ^{2}\delta ^{-1}}-1\right) \notag \\
&=&\sum_{i}\left\langle i\right\vert \rho ^{2}\left\vert i\right\rangle
^{1/2}-1
\end{eqnarray}%
which is a simple function of the density matrix.
\section{Discussions and conclusion}
We establish a family of coherence measures that are closely related to the Tsallis relative $\alpha$ entropy.
We prove that these coherence measures satisfy all the required criteria for a satisfactory coherence measure especially including
the strong monotonicity.
We also
show this family of coherence measures includes several typical coherence measures such as the coherences measure based on von Neumann entropy, skew information and so on. Additionally, we show how to validate the $l_2$ norm as a coherence measure. Finally, we would like to emphasize that the convexity and the strong
monotonicity could be two key points which couldn't easily be compatible with each other to some extent. Fortunately, Ref. \cite{Liebp} provides the important knowledge to harmonize both points in this paper. This work builds the bridge between the Tsallis relative $\alpha$ entropy and the strong monotonicity and provides the important alternative quantifiers for the coherence quantification. This could shed new light on the strong monotonicity of other candidates for coherence measure.
\section{Acknowledgements}
This work was supported by the National Natural Science
Foundation of China, under Grant No.11375036, the Xinghai Scholar
Cultivation Plan and the Fundamental Research Funds for the Central
Universities under Grant No. DUT15LK35 and No. DUT15TD47.
|
2,869,038,155,701 | arxiv | \section{Introduction}
In this paper we study locally finite CW complexes $X_{m,n}$ which
are combinatorial models for the Baumslag--Solitar groups
$BS(m,n)$. The group $BS(m,n)$ acts freely and cocompactly on
$X_{m,n}$ with quotient space $Z_{m,n}$ homeomorphic to the standard
presentation $2$--complex of $BS(m,n)$. We wish to
understand lattices in the locally compact group $\Aut(X_{m,n})$ of
combinatorial automorphisms. $BS(m,n)$ is one such
lattice but we are interested in others.
If $T$ is a locally finite tree then Bass has shown that any two
uniform lattices in $\Aut(T)$ are commensurable up to conjugacy
\cite{bass}, so in a sense there is only ``one'' lattice. On the other
hand, if $T_1$ and $T_2$ are non-isomorphic locally finite trees then
$\Aut(T_1 \times T_2) \cong \Aut(T_1) \times \Aut(T_2)$ has a very
rich lattice theory, including the celebrated examples of Burger and
Mozes \cite{burgermozes} and Wise \cite{wisethesis,wisecsc}.
The group $\Aut(X_{m,n})$ has some similarities with both $\Aut(T)$
and $\Aut(T_1) \times \Aut(T_2)$, and can be viewed as lying
intermediate between the two. Let $T_{m,n}$ denote the regular
directed tree in which every vertex has $m$ incoming and $n$ outgoing
edges. This is the usual Bass-Serre tree for $BS(m,n)$. The complex
$X_{m,n}$ is homeomorphic to $T_{m,n} \times {\mathbb R}$, with projection
inducing a homomorphism $\pi_*\colon \thinspace \Aut(X_{m,n}) \to
\Aut(T_{m,n})$. This homomorphism is continuous and \emph{injective}
when $m\not= n$ (though not an embedding; see Remark
\ref{notembedding}). Thus, while an element of $\Aut(T_1 \times T_2)$
is an arbitrary pair of tree automorphisms, an element of
$\Aut(X_{m,n})$ is just a single one.
Nevertheless, our main result shows that for many values of $m$ and
$n$, $\Aut(X_{m,n})$ is large enough to accommodate
incommensurable uniform lattices, just as $\Aut(T_1 \times T_2)$
does. Our main construction appears in Theorem \ref{mainthm}, restated
as follows.
\begin{theorem}\label{main1}
If\/ $\gcd(k,n) \not= 1$ then $\Aut(X_{k,kn})$ contains uniform lattices
$G_1, G_2$ that are not abstractly commensurable.
\end{theorem}
A similar statement holds for other cases of $\Aut(X_{k,kn})$, by a
different construction. Combining Theorems \ref{g3thm} and \ref{g4thm}
we have:
\begin{theorem}\label{main2}
If either
\begin{enumerate}
\item $n$ has a non-trivial divisor $p\not= n$ such that $p < k$, or
\item $n < k$ and $k \equiv 1 \mod n$
\end{enumerate}
then\/ $\Aut(X_{k,kn})$ contains uniform lattices that are not abstractly
commensurable.
\end{theorem}
The lattices in the previous results are all torsion-free. A
combinatorial characterization of the torsion-free lattices in
$\Aut(X_{m,n})$ is given in Theorem \ref{gbslatticechar}.
In the other direction, there is just one situation in which we can
say for sure that incommensurable lattices do not exist, namely, when
$\Aut(X_{m,n})$ is discrete. This occurs if and only if
$\gcd(m,n)=1$, by Theorem \ref{discrete}.
It is perhaps not surprising that the incommensurable lattices of
Theorems \ref{main1} and \ref{main2} lie in $\Aut(X_{m,n})$ for which $m$
divides $n$. This behavior illustrates the large degree of symmetry
enjoyed by these complexes. By the same token, the groups $BS(k,kn)$
are unusually symmetric. When $k,\abs{n}>1$, the groups
$\Aut(BS(k,kn))$ and $\Out(BS(k,kn))$ fail to be finitely generated,
by \cite{collinslevin} (see also \cite{clay} for a newer, more
geometric proof).
\medskip
Every torsion-free lattice in $\Aut(X_{m,n})$ is a generalized
Baumslag--Solitar group (GBS group). These are the fundamental
groups of graphs of groups in which all edge and vertex groups are
${\mathbb Z}$. We make use of many of the standard tools for studying these
groups, such as deformations and covering theory for
finite-index subgroups. The main new tool developed here is a
commensurability invariant called the \emph{depth profile}. It is a
subset of ${\mathbb N}$, modulo an equivalence relation on subsets of ${\mathbb N}$
based on ``truncated'' division; see Section
\ref{invariantsec}. Computing the depth profile of a GBS group is not
completely straightforward and we do it only under certain
assumptions. It becomes helpful to pass to finite index subgroups in
order to meet these assumptions; this is where the covering theory
comes in.
One case where the depth profile can be computed easily is the case of
Baumslag--Solitar groups. Using the depth profile we obtain a new and
much simpler proof of one of the main results of \cite{CRKZ}, in which
it is shown that various Baumslag--Solitar groups are not abstractly
commensurable. See Example \ref{bsexample} and Corollary \ref{crkz}.
In Section \ref{cayleysec} we show that the incommensurable lattices
of Theorems \ref{main1} and \ref{main2} all admit isomorphic Cayley
graphs; see Corollary \ref{cayleycor}. This result is not completely
obvious since their actions on $X_{m,n}$ are not transitive on
the vertices. However, it is enough that they act on the vertices with
the same orbits, which is what happens here.
There are many basic questions remaining regarding lattices in
$\Aut(X_{m,n})$, such as classifying the pairs $m,n$ for which
incommensurable lattices exist. We have said nothing about lattices
with torsion, either. Also, there is a phenomenon (see Figure
\ref{easycase} and Remark \ref{envelope}) of different groups
$\Aut(X_{m,n})$ and $\Aut(X_{m',n'})$ admitting isomorphic lattices in
an unexpected way. These questions are gathered in Section
\ref{questions}.
The motivation for this work came in part from the following
question of Stark and Woodhouse \cite[Question 1.9]{starkwoodhouse}:
If $H$ and $H'$ are one-ended residually finite hyperbolic groups that
act geometrically on the same simplicial complex, are $H$ and $H'$
abstractly commensurable? The groups considered here are certainly not
hyperbolic or residually finite, but they provide another setting,
in addition to lattices in products of trees, where their question has
an interesting answer. See also \cite{dergachevaklyachko} for yet
another example of incommensurable groups with a common combinatorial
model.
\begin{ack}
The author was partially supported by Simons Foundation award
\#638026. He thanks Alex Margolis for helpful comments and for
suggesting Theorem \ref{discrete}.
\end{ack}
\section{Preliminaries}\label{sec:prelim}
\subsection{Graphs}
A \emph{graph} $A$ is a pair of sets $(V(A),E(A))$ with maps
$\partial_0, \partial_1 \colon \thinspace E(A) \to V(A)$ and a free involution $e
\mapsto \overline{e}$ on $E(A)$, such that $\partial_i(\overline{e})
= \partial_{1-i}(e)$ for all $e$. An element of $E(A)$ is an oriented
edge with initial vertex $\partial_0(e)$ and terminal vertex
$\partial_1(e)$; then $\overline{e}$ is the ``same'' edge with the
opposite orientation. An edge is a \emph{loop} if $\partial_0(e)
= \partial_1(e)$. For each $v\in V(A)$ we define $E_0(v) = \{ e \in
E(A) \mid \partial_0(e) = v \}$.
A \emph{directed graph} is a graph $A$ together with a partition $E(A)
= E^+(A) \sqcup E^-(A)$ that separates every pair $\{e,
\overline{e}\}$. The edges in $E^+(A)$ are called \emph{directed}
edges. The ``direction'' is from $\partial_0(e)$ to
$\partial_1(e)$. For each $v \in V(A)$ we define $E^+_0(v) = E^+(A)
\cap E_0(v)$ and $E^-_0(v) = E^-(A) \cap E_0(v)$.
\subsection{$G$--trees}
By a \emph{$G$--tree} we mean a simplicial tree $X$ with an action of
$G$ by simplicial automorphisms, without inversions. Given $X$, if
$g\in G$ fixes a vertex we call $g$ \emph{elliptic}. If it is not
elliptic, then there is a $g$--invariant line in $X$, called the
\emph{axis} of $g$, on which $g$ acts by a non-trivial
translation. In this case, we call $g$ \emph{hyperbolic}.
An \emph{elliptic subgroup} is a subgroup $H < G$ that
fixes a vertex.
\subsection{Automorphisms of CW complexes}
Let $X$ be a CW complex. A \emph{topological automorphism} of $X$ is a
homeomorphism which preserves the cell structure, and in particular,
the partition of $X$ into open cells. The group of such automorphisms
will be denoted by $\Aut_{top}(X)$.
We are interested in a more combinatorial
notion, however. The \emph{combinatorial automorphism group},
denoted
by $\Aut(X)$, is the quotient of $\Aut_{top}(X)$
in which two automorphisms are considered the same if they induce the
same permutation on the set of cells of $X$.
In many cases one can lift $\Aut(X)$ to a subgroup of
$\Aut_{top}(X)$ by imposing a metric constraint on topological
automorphisms. For the complexes $X_{m,n}$ considered in this paper,
there are piecewise hyperbolic or Euclidean metrics available that can
be used in this way; see Section \ref{metricsect}.
\subsection{Locally finite complexes}
If $X$ is connected and locally finite then $\Aut(X)$ is a locally
compact group, as we explain here. For any CW complex $X$ the
topology on $\Aut(X)$ has a subbasis given by the sets
\[ U^X_{\sigma \to \tau} \ = \ \{ \, f \in \Aut(X) \mid f(\sigma) = \tau
\, \}\]
for all pairs of cells $\sigma, \tau$ of $X$. A basis is given by
finite intersections of these sets. That is, two automorphisms are
close if they agree on a large finite collection of cells of $X$.
Define two cells to be \emph{adjacent} if their closures
intersect nontrivially. For any cell $\sigma$ let $B_{\sigma}(r)$
denote the combinatorial ball of radius $r$ about $\sigma$. This is
defined recursively as follows: $B_{\sigma}(0) = \{\sigma\}$, and
$\tau \in B_{\sigma}(r)$ if $\tau$ is equal or adjacent to a cell in
$B_{\sigma}(r-1)$. In a locally finite CW complex, every cell is
adjacent to only finitely many cells, and each $B_{\sigma}(r)$ is
finite. If $X$ is connected then $X = \bigcup_r B_{\sigma}(r)$ for any
$\sigma$.
Now let $G = \Aut(X)$ with $X$ connected and locally finite. If
$\sigma$ is a cell, its stabilizer $G_{\sigma}$
is open (it equals $U^X_{\sigma \to \sigma}$) and has the structure
\[ G_{\sigma} \ = \ \underset{r}{\lim_{\longleftarrow}} \,
\image\bigl(G_{\sigma} \overset{\text{res}}{\longrightarrow}
\Aut(B_{\sigma}(r))\bigr). \]
As an inverse limit of finite groups, $G_{\sigma}$ is profinite and
compact. Since compact open subgroups are always commensurable, all
cell stabilizers are commensurable. Finally, the existence of compact
open subgroups implies that $\Aut(X)$ is locally compact.
\subsection{Lattices}
In a locally compact group $G$, a discrete subgroup $\Gamma < G$ is
called a \emph{lattice} if $G/\Gamma$ carries a finite positive
$G$-invariant measure, and a \emph{uniform lattice} if $G/\Gamma$ is
compact.
Now let $X$ be a connected, locally finite CW
complex. Let $G = \Aut(X)$. A subgroup $\Gamma < G$ is discrete
if and only if every cell stabilizer $\Gamma_{\sigma}$ is finite. In
this case, define the \emph{covolume} of $\Gamma$ to be
\[ \Vol(X/\Gamma) \ = \ \sum_{[\sigma] \in
\cells(X/\Gamma)} 1/\abs{\Gamma_{\sigma}}.\]
The sum is taken over a set of representatives of the
$\Gamma$--orbits of cells of $X$.
The next result follows directly from \cite[1.5--1.6]{basslubotzky}.
\nocite{basslubotzky}
\begin{proposition}\label{blprop}
Suppose that $G = \Aut(X)$ acts cocompactly on $X$ and let $\Gamma <
G$ be a discrete subgroup. Then
\begin{enumerate}
\item\label{bl1} $\Gamma$ is a lattice if and only if\/
$\Vol(X/\Gamma)< \infty$
\item\label{bl2} $\Gamma$ is a uniform lattice if and only if\/
$X/\Gamma$ is compact.
\end{enumerate}
\end{proposition}
Note that it follows from \ref{blprop} that every torsion-free lattice
is uniform. (Indeed, every lattice with bounded torsion is uniform.)
The lattices considered in this paper will generally be torsion-free.
\section{Generalized Baumslag--Solitar groups}
\subsection{Definitions}
A \emph{generalized Baumslag--Solitar group} (or \emph{GBS group}) is
a group that admits a
graph of groups decomposition in which all vertex and edge groups are
${\mathbb Z}$. Equivalently, it is a group that acts without inversions on a
simplicial tree such that all vertex and edge stabilzers are
infinite cyclic. We refer to \cite{forester3,forester2,levitt-auto} for
general background on these groups.
If $G$ is a GBS group with corresponding Bass--Serre tree $X$, we call
$X$ a \emph{GBS tree} for $G$. Its quotient graph of groups has every
edge and vertex group isomorphic to ${\mathbb Z}$, with inclusion maps given by
multiplication by various non-zero integers. Thus this graph of groups
is specified by a connected graph $A$ ($=X/G$) and a \emph{label
function} $\lambda \colon \thinspace E(A) \to ({\mathbb Z}-\{0\})$. We denote the
corresponding graph of groups by $(A,\lambda)_{{\mathbb Z}}$. Letting $\lambda
\colon \thinspace E(X) \to ({\mathbb Z} - \{0\})$ denote the induced label function on $X$,
we have
\begin{equation}\label{labelindex}
\abs{\lambda(e)} \ = \ [G_{\partial_0(e)} : G_e]
\end{equation}
for all edges $e\in E(X)$.
\subsection{Fibered $2$--complexes}
It is often useful to take a topological viewpoint as in
\cite{scottwall}. A GBS group is then the fundamental group of the
total space of a graph of spaces in which each edge and vertex space
is a circle. Given a labeled graph $(A,\lambda)$ there are oriented
circles $C_v$ for each vertex $v$ and $C_e = C_{\overline{e}}$ for
each $e \in E(A)$. For each $e$ let $M_e$ be the mapping cylinder of
the degree $\lambda(e)$ covering map $C_e \to C_{\partial_0(e)}$. Note
that $M_e$ contains embedded copies of both $C_e$ and
$C_{\partial_0(e)}$. The total space of the graph of spaces is now
defined as the quotient \[Z_{(A,\lambda)} \ = \ \bigsqcup_{e\in E(A)}
M_e \Big\slash \!\sim\] where all copies of $C_v$ (in $M_e$ for $e
\in E_0(v)$) are identified by the identity map, and $C_e$ and
$C_{\overline{e}}$ (in $M_e$ and $M_{\overline{e}}$) are identified by
the identity, for all $v\in V(A)$ and $e\in E(A)$. We will call
$Z_{(A,\lambda)}$ the \emph{fibered $2$--complex} associated to
$(A,\lambda)$. It is naturally foliated by circles, and may be thought
of as a $2$--dimensional analogue of a Seifert fibered space, with
singular fibers the vertex circles $C_v$. The leaf space of
$Z_{(A,\lambda)}$ is $A$ (realized topologically as a
$1$--complex). Each $M_e$ is embedded in $Z_{(A,\lambda)}$, and under
the map to the leaf space, $M_e$ is the pre-image of a closed
half-edge in $A$.
\begin{example}\label{seifert}
Let $M$ be a Seifert fibered $3$--manifold that is not closed. Let
$\Sigma$ be the quotient $2$--orbifold, which has only isolated cone
singularities. The underlying surface (also non-closed) contains a
\emph{spine}, i.e., a $1$--complex embedded as a deformation
retract. This spine $S$ may be chosen so that the singularities of
$\Sigma$ are all vertices of $S$. The union of the fibers over $S$ is
then a deformation retract of $M$, and it has the structure of a
fibered $2$--complex. Hence $\pi_1(M)$ is a GBS group.
\end{example}
Other well known examples of GBS groups include free-by-cyclic groups
$F_n \rtimes_{\phi} {\mathbb Z}$ with periodic monodromy $\phi$ \cite{levitt1}
and one-relator groups with non-trivial center
\cite{pietrowski}. Every finite index subgroup of a GBS group is
a GBS group.
\begin{example}\label{bsname}
Let $(A,\lambda)$ be the labeled graph having one vertex and one loop
$e$ with labels $\lambda(e) = m$, $\lambda(\overline{e}) = n$. The
corresponding GBS group is the \emph{Baumslag--Solitar group}
\begin{equation}\label{presentation}
BS(m,n) \ = \ \langle \, a, t \mid ta^m
t^{-1} = a^n\, \rangle.
\end{equation}
This is the \emph{standard}
GBS structure for $BS(m,n)$. In this case we give the space
$Z_{(A,\lambda)}$ the additional name $Z_{m,n}$.
\end{example}
\begin{notation}
We denote by $\bigvee_{i=1}^k BS(m_i,n_i)$ the GBS group defined by
the labeled graph having one vertex and $k$ loops, each with labels
$m_i$ and $n_i$.
\end{notation}
\subsection{Non-elementary GBS groups}
A $G$--tree is \emph{elementary} if there is an invariant point or
line, and \emph{non-elementary} otherwise. In the case of a GBS tree,
it is elementary if and only if the group is isomorphic to ${\mathbb Z}$,
${\mathbb Z}\times {\mathbb Z}$, or the Klein bottle group \cite[Lemma
2.6]{forester2}. Thus, this is a property of the GBS group itself.
A fundmental property of non-elementary GBS groups is that any two GBS
trees for the same group $G$ define the same partition of $G$ into
elliptic and hyperbolic elements \cite[Corollary 6.10]{forester1}, and
moreover have the same elliptic subgroups.
\subsection{Segments and index}
Let $X$ be a locally finite $G$--tree. A \emph{segment} is an embedded
edge path $\sigma = (e_1, \dotsc, e_k)$. Its \emph{initial} and
\emph{terminal vertices} are $\partial_0(\sigma)= \partial_0 (e_1)$
and $\partial_1(\sigma) = \partial_1(e_k)$ respectively, and its
\emph{reverse} is $\overline{\sigma} = (\overline{e}_k, \dots,
\overline{e}_1)$. The pointwise stabilizer of $\sigma$ is $G_{\sigma}
= G_{\partial_0 \sigma} \cap G_{\partial_1\sigma}$. If $x,y$ are
vertices of $X$ then $[x,y]$ denotes the segment with initial vertex
$x$ and terminal vertex $y$. The \emph{index} of $\sigma$ is the
number $i(\sigma) \ = \ [G_{\partial_0 \sigma}: G_{\sigma}]$. This
index is also defined for edges, since an edge is a segment of length
one.
If $X$ is a GBS tree with label function $\lambda$ induced by the
quotient graph of groups, then $i(e) = \abs{\lambda(e)}$ for all $e
\in E(X)$, as noted in \eqref{labelindex}. The main difference between
\emph{labels} and \emph{indices} is that labels may be negative.
The following Lemma is Corollary 3.6 of \cite{forester3}. The latter
result has edges and labels instead of segments and indices, but the
proof is the same.
\begin{lemma}\label{fixpathlemma}
Let $X$ be a GBS tree and suppose the segment $\sigma$ is a
concatenation of segments $\sigma = \sigma_1 \dotsm \sigma_k$. Let
$n_i = i(\sigma_i)$ and $m_i = i(\overline{\sigma}_i)$ for each
$i$. Then $G_{\sigma_1}$ fixes $\sigma$ if and only if
\begin{equation}\label{fixpath}
\prod_{i=2}^{\ell} n_i \ \text{ divides } \ \prod_{i=1}^{\ell - 1} m_i
\end{equation}
for all $\ell = 2, \dotsc, k$. \qed
\end{lemma}
The index of a segment can now be determined as follows (for instance,
by taking each $\sigma_i$ to be an edge):
\begin{lemma}\label{segmentindex}
Let $X$ be a GBS tree and suppose the segment $\sigma$ is a
concatenation of segments $\sigma = \sigma_1 \dotsm \sigma_k$. Let
$n_i = i(\sigma_i)$ and $m_i = i(\overline{\sigma}_i)$ for each
$i$. Then $i(\sigma)$ is the smallest positive integer $r$ such that
\begin{equation}\label{pathindex}
n_1 \mid r \ \text{ and } \ \prod_{i=1}^{\ell} n_i \ \text{
divides } \ r \prod_{i=1}^{\ell - 1} m_i \ \text{ for } \ \ell = 2,
\dotsc, k.
\end{equation}
\end{lemma}
\begin{proof}
Write $G_{\partial_0 \sigma} = \langle \, x \, \rangle$. We want $r$
such that $G_{\sigma} = \langle \, x^r \, \rangle$, which occurs if
and only if $r$ is smallest such that $\langle \, x^r \, \rangle$
fixes $\sigma$. Consider a longer segment $\sigma_0 \cdot \sigma$
where $i(\overline{\sigma}_0) = r$ (perhaps in a larger GBS
tree). Then $G_{\sigma_0} = \langle \, x^r \, \rangle$, and so
$\langle \, x^r \, \rangle$ fixes $\sigma$ if and only if condition
\eqref{fixpath} holds for the concatenation $\sigma_0 \dotsm \sigma_k$
(for $\ell = 1, \dotsc, k$). These conditions are exactly statement
\eqref{pathindex}.
\end{proof}
\begin{remark}\label{shortindex}
When $k=2$ there is a closed formula $i(\sigma) = n_1 n_2 /
\gcd(m_1, n_2))$.
\end{remark}
More important than \eqref{pathindex}, perhaps, is
that $i(\sigma)$ and $i(\overline{\sigma})$ are determined completely
by the indices seen along $\sigma$, in any expression as a
concatenation.
\subsection{The modular homomorphism}
Let $G$ be a GBS group with GBS tree $X$ and quotient
labeled graph $(A,\lambda)$. Fix a non-trivial elliptic element $a \in
G$. Since elliptic elements are all commensurable, every element $g$
satisfies a relation $g^{-1}a^mg = a^n $ for
some non-zero integers $m,n$. The function $q(g) = m/n$ defines a
homomorphism $G \to {\mathbb Q}^{\times}$ called the \emph{modular
homomorphism} (cf. \cite{kropholler-centrality,basskulkarni}). It does
not depend on the choice of $a$.
The modular homomorphism takes the value $1$ on every elliptic
element, so it factors through $\pi_1(A)$, indeed through $H_1(A)$
since ${\mathbb Q}^{\times}$ is abelian. It can be computed from $(A,\lambda)$
as follows. If $g\in G$ maps to $\alpha \in H_1(A)$ represented by
the $1$--cycle $(e_1, \dotsc, e_n)$, then
\begin{equation}\label{mh}
q(g) \ = \ \prod_{i=1}^n \lambda(e_i)/\lambda(\overline{e}_i).
\end{equation}
Recall that when $G$ is non-elementary, all GBS trees for $G$ have the
same elliptic elements. Thus, in this case, $q$ is well defined in
terms of $G$ alone.
If $V$ is any non-trivial elliptic subgroup of $G$, then there is a
formula
\begin{equation}\label{modulus}
\abs{q(g)} \ = \ [V: V \cap V^g] / [V^g : V \cap V^g]
\end{equation}
(see \cite[Section 6]{forester3}). Note that for $V = G_x$, the right
hand side is the ratio of indices $i(\sigma) / i(\overline{\sigma})$
for the segment $\sigma = [x, gx]$.
A GBS group $G$ is called \emph{unimodular} if $q(G) \subset \{\pm
1\}$.
\subsection{The orientation character}
Let $G$ be a GBS group with GBS tree $X$. The \emph{orientation
character} of $G$ (which a priori depends on $X$) is a homomorphism
$\chi\colon \thinspace G \to \{\pm 1\}$ defined by
\[\chi(g) \ = \ q(g) / \abs{q(g)}.\]
When $G$ is non-elementary $\chi$ is well defined in terms of $G$
alone, since this is true of $q$. When $G$ is ${\mathbb Z}$ or ${\mathbb Z}\times {\mathbb Z}$,
every orientation character is trivial. In the case of the Klein
bottle group, there are two deformation spaces of GBS trees, one with
trivial and one with non-trivial orientation character. Note that
every GBS group has a subgroup of index at most $2$ with trivial
orientation character.
\begin{remark}\label{signchange}
Let $G$ be a GBS group with GBS tree $X$ and quotient labeled graph
$(A,\lambda)$. If the orientation character is trivial then the
label function $\lambda$ can be made positive by \emph{admissible sign
changes}; see \cite[Lemma 2.7]{clayforester}.
An admissible sign change is the change in $\lambda$ that results from
changing the choices of generators in the graph of groups defined by
$(A,\lambda)$. The graph of groups itself does not change, only its
description in terms of labels. If one changes the generator of a
vertex group $G_v$, then the labels of edges in $E_0(v)$ all change
sign. If one changes the generator of an edge group $G_e$, then
$\lambda(e)$ and $\lambda(\overline{e})$ both change sign.
\end{remark}
\subsection{Labeled graphs and deformations}
It often happens that different labeled graphs define isomorphic
GBS groups. A given GBS group generally has no preferred GBS tree (or
labeled graph). In fact, it is still an open problem to find an
algorithm which determines whether two labeled graphs define the same
group. (This problem has been solved in some special cases; see
especially \cite{forester3, levitt-auto, clayforester, dudkin, CRKZ}.)
It is true, however, that any two GBS trees for a non-elementary $G$
are related by an \emph{elementary deformation}
\cite{forester1}. This means that they are related by a finite
sequence of elementary moves, called elementary collapses and
expansions. There is also a \emph{slide move}, which can be expressed
as an expansion followed by a colllapse. Viewed topologically, the
moves are homotopy equivalences between fibered $2$--complexes in
which annuli are collapsed or expanded or slid over one another. The
next proposition gives an explicit description of these moves.
\begin{proposition}[Prop. 2.4 of \cite{forester3}] \label{gbsmoves}
If an elementary move is performed on a generalized Baumslag--Solitar
tree, then the quotient graph of groups changes locally as follows:
\begin{pict}{90}{12}
\thicklines
\put(10,5){\circle*{1}}
\put(25,5){\circle*{1}}
\put(10,5){\line(1,0){15}}
\thinlines
\put(45,6.5){\vector(1,0){15}}
\put(60,2.2){\vector(-1,0){15}}
\zindex{52.5}{8.2}{\mbox{collapse}}
\zindex{52.5}{4}{\mbox{expansion}}
\put(25,5){\line(3,5){3}}
\put(25,5){\line(3,-5){3}}
\put(10,5){\line(-5,3){5}}
\put(10,5){\line(-5,-3){5}}
\put(80,5){\circle*{1}}
\put(80,5){\line(-5,3){5}}
\put(80,5){\line(-5,-3){5}}
\put(80,5){\line(3,5){3}}
\put(80,5){\line(3,-5){3}}
\scriptsize
\zindex{8.5}{8}{a}
\zindex{8.5}{2}{b}
\zindex{12}{6.8}{n}
\zindex{22}{7}{\pm 1}
\zindex{28.5}{7.5}{c}
\zindex{28.5}{2.8}{d}
\zindex{78.5}{8}{a}
\zindex{78.5}{2}{b}
\zindex{85.5}{7.5}{\pm nc}
\zindex{85.9}{2.8}{\pm nd}
\end{pict}
A slide move has the following description:
\begin{pict}{100}{13}
\thicklines
\put(75,3){\circle*{1}}
\put(90,3){\circle*{1}}
\put(75,3){\line(1,0){15}}
\put(90,3){\line(-1,2){4}}
\thinlines
\put(47.5,3){\vector(1,0){10}}
\zindex{52.5}{5}{\mbox{slide}}
\put(90,3){\line(5,3){5}}
\put(90,3){\line(5,-3){5}}
\put(75,3){\line(-5,3){5}}
\put(69,3){\line(1,0){6}}
\put(75,3){\line(-5,-3){5}}
\scriptsize
\zindex{77}{1.5}{m}
\zindex{88}{1.5}{n}
\zindex{86}{6}{ln}
\thicklines
\put(10,3){\circle*{1}}
\put(25,3){\circle*{1}}
\put(10,3){\line(1,0){15}}
\put(10,3){\line(1,2){4}}
\thinlines
\put(25,3){\line(5,3){5}}
\put(25,3){\line(5,-3){5}}
\put(10,3){\line(-5,3){5}}
\put(4,3){\line(1,0){6}}
\put(10,3){\line(-5,-3){5}}
\scriptsize
\zindex{12}{1.5}{m}
\zindex{23}{1.5}{n}
\zindex{9}{7}{lm}
\end{pict}
or
\begin{pict}{100}{10}
\thicklines
\put(82.5,5){\circle*{1}}
\put(87.5,5){\circle{10}}
\put(72.5,5){\line(1,0){10}}
\thinlines
\put(47.5,5){\vector(1,0){10}}
\zindex{52.5}{7}{\mbox{slide}}
\put(82.5,5){\line(-5,-3){4.5}}
\put(82.5,5){\line(-1,-4){1.2}}
\put(82.5,5){\line(-1,4){1.2}}
\scriptsize
\zindex{85.1}{3.5}{m}
\zindex{84.7}{6.5}{n}
\zindex{79.4}{7.3}{ln}
\thicklines
\put(17.5,5){\circle*{1}}
\put(22.5,5){\circle{10}}
\put(7.5,5){\line(1,0){10}}
\thinlines
\put(17.5,5){\line(-5,-3){4.5}}
\put(17.5,5){\line(-1,-4){1.2}}
\put(17.5,5){\line(-1,4){1.2}}
\scriptsize
\zindex{20.1}{3.5}{m}
\zindex{19.7}{6.5}{n}
\zindex{14}{7.3}{lm}
\end{pict}
\end{proposition}
\subsection{Subgroups of GBS groups}
Every subgroup of a GBS group is either a GBS group or a free
group. In the former case the inclusion $H \hookrightarrow G$ is
induced by a covering map from one fibered $2$--complex to
another. Such covering maps are encoded by \emph{admissible branched
coverings} of labeled graphs, defined as follows (cf. \cite[Lemma
5.3]{levitt1}).
\begin{definition}\label{admissibledef}
An \emph{admissible branched covering} of labeled graphs, from
$(A,\lambda)$ to $(B,\mu)$, consists of a surjective graph morphism $
p \colon \thinspace A \to B$ between connected graphs together with a \emph{degree}
function
\[d \colon \thinspace E(A) \sqcup V(A) \to {\mathbb N}\]
satisfying $d(e) = d(\overline{e})$ for all $e\in E(A)$ such that the
following holds. Given an edge $e \in E(B)$ with $\partial_0(e) = v$
and a vertex $u\in p^{-1}(v)$, let $k_{u,e} = \gcd(d(u),\mu(e))$. Then
\begin{enumerate}
\item $\abs{p^{-1}(e) \cap E_0(u)} = k_{u,e}$
\item $\lambda(e') = \mu(e)/k_{u,e}$ for each edge $e' \in p^{-1}(e)
\cap E_0(u)$
\item $d(e') = d(u)/k_{u,e}$ for each edge $e' \in p^{-1}(e) \cap
E_0(u)$.
\end{enumerate}
See Figure \ref{admissiblefig}.
\end{definition}
\begin{figure}[!ht]
\hspace*{3.5cm}
\begin{tikzpicture}
\small
\draw[very thick] (1,4) -- (4,5);
\draw[very thick] (1,4) -- (4,3);
\draw[rotate around={-10:(1,4)},very thick] (1,4) -- (4,5);
\filldraw[fill=white,thick] (1,4) circle (.7mm);
\draw (0.8,4) node[anchor=east] {$u$};
\draw[rotate around={-5:(1,4)}] (4.05,4.112) node[transform
shape,anchor=center] {$\vdots$};
\draw (4,4) node[anchor=west] {$\begin{rcases} & \\ & \\ &
\\ & \end{rcases} \ k_{u,e} \ = \
\gcd(\textcolor{mygreen}{d(u)},\textcolor{violet}{\mu(e)})$};
\draw[rotate around={18.435:(1,4)}] (1.15,4) node[transform
shape,anchor=south west,violet] {$\mu(e)\big\slash k_{u,e}$};
\draw[thick,->,color=mygreen] (1,3.3) -- (1,1.7);
\draw[color=mygreen] (1,2.5) node[anchor=west] {$d(u)$};
\draw[thick,->,color=mygreen] (2.6,2.83) -- (2.6,1.7);
\draw[color=mygreen] (2.6,2.2) node[anchor=west] {$d(u)\big\slash
k_{u,e}$};
\draw[very thick] (1,1) -- (4,1);
\filldraw[fill=white,thick] (1,1) circle (.7mm);
\draw (0.8,1) node[anchor=east] {$v$};
\draw (2.6,0.9) node[anchor=north] {$e$};
\draw (1.15,1) node[anchor=south west,violet] {$\mu(e)$};
\end{tikzpicture}
\caption{The admissibility condition. Each edge of $p^{-1}(e) \cap
E_0(u)$ has label $\mu(e)/k_{u,e}$ and degree $d(u)/k_{u,e}$. There
are $k_{u,e}$ such edges.}\label{admissiblefig}
\end{figure}
By lifting the graph of spaces structure, every covering space of a
fibered $2$--complex has a compatible graph of spaces structure, where
either every vertex and edge space is a line, or every vertex and
edge space is a circle. The two cases correspond to whether the
subgroup acts freely on the Bass--Serre tree
or is a GBS group (acting elliptically if it is cyclic); see \cite[Lemmas
2.6--2.7]{forester2}.
Note that there is an induced surjective morphism $p \colon \thinspace A \to B$ of
underlying graphs, and the covering map is fiber-preserving, with $p$
the induced map on leaf spaces.
\begin{proposition}\label{admissibleprop}
Let $G$ be a GBS group with labeled graph $(B,\mu)$. There is a
one-to-one correspondence between conjugacy classes of GBS
subgroups of $G$ (excluding hyperbolic cyclic subgroups) and
admissible branched coverings $(A,\lambda) \to (B,\mu)$.
\end{proposition}
This result is essentially the same as \cite[Lemma 6.3]{levitt1}, but
we explain it here somewhat differently.
\begin{proof}
It suffices to classify the fibered $2$--complex covering spaces of
$Z_{(B,\mu)}$. Recall that $Z_{(B,\mu)}$ is a union of mapping
cylinders $M_e$. The admissibility condition is simply a description
of the finite-sheeted covers of these subspaces $M_e$. As a mapping
cylinder, $M_e$ deformation retracts onto $C_{\partial_0(e)}$, and has
infinite cyclic fundamental group. However it is also a fiber bundle
over this circle, giving it the structure of a mapping
\emph{torus}. The fiber is a cone on $n = \abs{\mu(e)}$ points,
denoted by $C(n)$, and the monodromy $\phi\colon \thinspace C(n) \to C(n)$ is the
automorphism which permutes the $n$ points by an $n$--cycle
$\sigma$. Let $z_1, \dotsc, z_n$ be the $n$ points and $c$ the cone
point. Writing $M_e$ as
\[C(n) \times [0,1] \big\slash (x,0) \sim (\phi(x),1),\]
the image of $c \times [0,1]$ is the singular circle
$C_{\partial_0(e)}$ and the images of $z_i \times [0,1]$ join up to
form the circle $C_e$. The map $C_e \to C_{\partial_0(e)}$ in which
each $z_i \times \{t\}$ maps to $c\times \{t\}$ is the degree
$\mu(e)$ covering map defining $M_e$.
For any $d\in {\mathbb N}$, the $d$--sheeted covering space $N$ of $M_e$ is the
mapping torus of
\[\phi^d \colon \thinspace C(n) \to C(n).\]
The permutation $\sigma^d$ decomposes into $k = \gcd(d,n)$ disjoint
$(n/k)$--cycles. Thus the pre-image of $C_e$ in $N$ is $k$ disjoint
circles and $N$ is the mapping cylinder of the map $\bigsqcup_{i=1}^k
S^1 \to S^1$ in which each component maps by degree $\mu(e)/k$. Each
of the circles above $C_e$ covers with degree $d/k$, and the circle
above $C_{\partial_0(e)}$ covers with degree $d$. Now $N$ is exactly
described by the admissibility condition as shown in Figure
\ref{admissiblefig}, with $d = d(u)$. Thus for any covering map of
fibered $2$--complexes, the induced morphism of indexed graphs is an
admissible branched covering. The degree function records the degrees
of each fiber in the cover mapping to its image. (Orientations of the
fibers in the cover are chosen so that these degrees are all
positive.)
Conversely, let $p \colon \thinspace (A,\lambda) \to (B,\mu)$ be an admissible
branched covering. Recall that $Z_{(A,\lambda)}$ and $Z_{(B,\mu)}$
are, respectively, unions of the subspaces $M_{e'}$ ($e' \in E(A)$)
and $M_e$ ($e \in E(B)$). For each $e \in E(B)$ with $\partial_0(e) =
v$ and each $u \in p^{-1}(v)$ the subspace
\[N_{u,e} \ = \ \bigcup_{e' \in p^{-1}(e) \cap E_0(u)} M_{e'} \]
of $Z_{(A,\lambda)}$ has a degree $d(u)$ covering map $p_{u,e} \colon \thinspace
N_{u,e} \to M_e$, by the admissibility condition. These subspaces
$N_{u,e}$ intersect each other only along the circles $C_u$ and
$C_{e'}$. The restrictions $p_{u,e} \vert_{C_u}$ and $p_{u,e}
\vert_{C_{e'}}$ are coverings of degrees $d(u)$ and $d(e')$
respectively. By adjusting the covering maps on $N_{u,e}$ by
fiber-preserving isotopies, the restrictions of the coverings to the
circles $C_u$ and $C_{e'}$ (for $u \in V(A)$, $e' \in E(A)$) can be
made to all agree on each circle. Then the covering maps $p_{u,e}$
join to give a covering of fibered $2$--complexes.
\end{proof}
\begin{remark}
Every \emph{finite index} subgroup of $G$ is a GBS subgroup, and these
correspond to the branched coverings $(A,\lambda) \to (B,\mu)$ for
which the morphism $A \to B$ has finite fibers. More generally, if
$H<G$ corresponds to the branched covering $p\colon \thinspace (A,\lambda) \to
(B,\mu)$, then $[G:H] \ = \ \sum_{u \in p^{-1}(v)} d(u)$, for any
vertex $v$ of $B$.
\end{remark}
\begin{remark}
Every GBS group is \emph{coherent}, meaning that every finitely
generated subgroup is finitely presented. Since every subgroup is free
or GBS, it suffices to note that a GBS group $G$ with minimal GBS tree
$X$ is finitely generated if and only if $X$ is cocompact. In that
case, $G$ is the fundamental group of a compact fibered $2$--complex,
and is finitely presented.
\end{remark}
\section{The $2$--complexes $X_{m,n}$ and $X_{(A,\lambda)}$}
\label{complexes-sec}
Fix positive integers $m,n$ and let $k = \gcd(m,n)$. Let $T_{m,n}$
denote the directed simplicial tree in which every vertex has $m$
incoming edges and $n$ outgoing edges. This tree is the Bass--Serre
tree of $BS(m,n)$ with its standard labeled graph. The orientations
are such that the stable letter $t$ in the presentation
\eqref{presentation} is directed forward, and has axis in $T_{m,n}$
that is a directed line which $t$ shifts forward one unit.
Recall from Example \ref{bsname} the fibered $2$--complex $Z_{m,n}$.
It is homeomorphic to the presentation $2$--complex for the
presentation of $BS(m,n)$ given in \eqref{presentation}. That CW
complex has one vertex, two edges labeled $a$ and $t$, and one
$2$--cell attached by the boundary word $t a^m t^{-1} a^{-n}$. The
cell structure we want to put on $Z_{m,n}$ is obtained from this by
subdivision and is illustrated in Figure \ref{Zcells}. The initial
$2$--cell has been split into $k$ $2$--cells, with new $1$--cells
labeled $t_1, \dots, t_{k-1}$. The new $2$--cells are attached by the
words $t_i a^{m/k} t^{-1}_{i+1} a^{-n/k}$ (indices modulo $k$), with
$t_0$ understood to mean $t$. There is still only one vertex.
\begin{figure}[!ht]
\begin{tikzpicture}[scale=0.9]
\small
\filldraw[fill=gray!15,thick] (1,1) rectangle (4,2);
\filldraw[fill=gray!15,thick] (5,1) rectangle (8,2);
\filldraw[fill=gray!15,thick] (11,1) rectangle (14,2);
\filldraw[fill=black,thick] (1,1) circle (.5mm);
\filldraw[fill=black,thick] (2.5,1) circle (.5mm);
\filldraw[fill=black,thick] (4,1) circle (.5mm);
\filldraw[fill=black,thick] (5,1) circle (.5mm);
\filldraw[fill=black,thick] (6.5,1) circle (.5mm);
\filldraw[fill=black,thick] (8,1) circle (.5mm);
\filldraw[fill=black,thick] (11,1) circle (.5mm);
\filldraw[fill=black,thick] (12.5,1) circle (.5mm);
\filldraw[fill=black,thick] (14,1) circle (.5mm);
\filldraw[fill=black,thick] (1,2) circle (.5mm);
\filldraw[fill=black,thick] (2,2) circle (.5mm);
\filldraw[fill=black,thick] (3,2) circle (.5mm);
\filldraw[fill=black,thick] (4,2) circle (.5mm);
\filldraw[fill=black,thick] (5,2) circle (.5mm);
\filldraw[fill=black,thick] (6,2) circle (.5mm);
\filldraw[fill=black,thick] (7,2) circle (.5mm);
\filldraw[fill=black,thick] (8,2) circle (.5mm);
\filldraw[fill=black,thick] (11,2) circle (.5mm);
\filldraw[fill=black,thick] (12,2) circle (.5mm);
\filldraw[fill=black,thick] (13,2) circle (.5mm);
\filldraw[fill=black,thick] (14,2) circle (.5mm);
\draw[thick,->] (1,1.54) -- (1,1.55);
\draw[thick,->] (4,1.54) -- (4,1.55);
\draw[thick,->] (5,1.54) -- (5,1.55);
\draw[thick,->] (8,1.54) -- (8,1.55);
\draw[thick,->] (11,1.54) -- (11,1.55);
\draw[thick,->] (14,1.54) -- (14,1.55);
\draw[thick,->] (1.84,1) -- (1.85,1);
\draw[thick,->] (3.34,1) -- (3.35,1);
\draw[thick,->] (5.84,1) -- (5.85,1);
\draw[thick,->] (7.34,1) -- (7.35,1);
\draw[thick,->] (11.84,1) -- (11.85,1);
\draw[thick,->] (13.34,1) -- (13.35,1);
\draw[thick,->] (1.5,2) -- (1.6,2);
\draw[thick,->] (2.5,2) -- (2.6,2);
\draw[thick,->] (3.5,2) -- (3.6,2);
\draw[thick,->] (5.5,2) -- (5.6,2);
\draw[thick,->] (6.5,2) -- (6.6,2);
\draw[thick,->] (7.5,2) -- (7.6,2);
\draw[thick,->] (11.5,2) -- (11.6,2);
\draw[thick,->] (12.5,2) -- (12.6,2);
\draw[thick,->] (13.5,2) -- (13.6,2);
\draw (9.52,1.46) node {$\dotsm$};
\draw (1.75,0.9) node[anchor=north] {$a$};
\draw (3.25,0.9) node[anchor=north] {$a$};
\draw (5.75,0.9) node[anchor=north] {$a$};
\draw (7.25,0.9) node[anchor=north] {$a$};
\draw (11.75,0.9) node[anchor=north] {$a$};
\draw (13.25,0.9) node[anchor=north] {$a$};
\draw (1.5,2.1) node[anchor=south] {$a$};
\draw (2.5,2.1) node[anchor=south] {$a$};
\draw (3.5,2.1) node[anchor=south] {$a$};
\draw (5.5,2.1) node[anchor=south] {$a$};
\draw (6.5,2.1) node[anchor=south] {$a$};
\draw (7.5,2.1) node[anchor=south] {$a$};
\draw (11.5,2.1) node[anchor=south] {$a$};
\draw (12.5,2.1) node[anchor=south] {$a$};
\draw (13.5,2.1) node[anchor=south] {$a$};
\draw (0.9,1.5) node[anchor=east] {$t$};
\draw (4,1.5) node[anchor=east] {$t_1$};
\draw (5.05,1.5) node[anchor=west] {$t_1$};
\draw (8.1,1.5) node[anchor=west] {$t_2$};
\draw (11,1.5) node[anchor=east] {$t_{k-1}$};
\draw (14.1,1.5) node[anchor=west] {$t$};
\end{tikzpicture}
\caption{The cell structure for $Z_{m,n}$ when $m=3k$,
$n=2k$.
}\label{Zcells}
\end{figure}
The $2$--complex $X_{m,n}$ is defined to the the universal cover of
$Z_{m,n}$ with the induced cell structure. It is tiled entirely by
quadrilateral $2$--cells of the kind shown in Figure \ref{Zcells},
with sides of length $1, m/k, 1, n/k$. The reason for subdividing the
initial cell structure of $Z_{m,n}$ is to increase the homogeneity of
$X_{m,n}$ and make it as symmetric as possible.
We extend the definition to allow $m,n \in {\mathbb Z}-\{0\}$ by declaring
$X_{m,n} = X_{\abs{m},\abs{n}}$. Then $BS(m,n)$ is a lattice in
$\Aut(X_{m,n})$ for any $m,n$ (since $X_{m,n}$ is the universal cover
of $Z_{\pm m, \pm n}$). However, when discussing an unspecified
$X_{m,n}$, the default assumption will be that $m,n > 0$.
\subsection{Combinatorial description}\label{combinatorialsect}
The space $X_{m,n}$ is homeomorphic to
$T_{m,n} \times {\mathbb R}$, but combinatorially and geometrically it is very
far from being a product (unless $m=n$). Let $\pi \colon \thinspace X_{m,n} \to
T_{m,n}$ be the projection map. The $1$--cells of $X_{m,n}$ will be
called \emph{horizontal} if they map to a vertex of $T_{m,n}$ and
\emph{vertical} if they map to an edge. The vertical edges inherit
orientations from the edges of $T_{m,n}$, consistent with the
orientations labeled $t$ or $t_i$ in Figure \ref{Zcells}. We view
this direction as the ``upward'' direction in $X_{m,n}$ and in
$T_{m,n}$. (The horizontal $1$--cells are \emph{not} oriented; the
orientations labeled $a$ in Figure \ref{Zcells} should be ignored when
considered as cells in $X_{m,n}$.)
For any $v \in V(T_{m,n})$ the pre-image $\pi^{-1}(v)$ will be called
a \emph{branching line}. The pre-image of a closed edge of $T_{m,n}$
will be called a \emph{strip}. Note that the branching lines are tiled
by horizontal edges. We define an \emph{$(i,j)$--cell} to be a
$2$--cell attached by a combinatorial path consisting of one upward
vertical edge, $i$ horizontal edges, one downward vertical edge, and
$j$ horizontal edges. An \emph{$(i,j)$--strip} is a bi-infinite
sequence of $(i,j)$--cells, joined along their vertical edges. Every
strip in $X_{m,n}$ is a $(m/k,n/k)$--strip.
We may regard $X_{m,n}$ as being assembled from branching lines and
$(m/k,n/k)$--strips just as $T_{m,n}$ is made of vertices and
edges. Each branching line has $n$ strips above it and $m$ strips
below it. When attaching a strip \emph{above} a branching line, there
are $n/k$ ways to do this; if we identify the vertices of the
branching line with ${\mathbb Z}$, the vertical edges of the strip will meet a
coset $i + (n/k){\mathbb Z}$ for some $i$. In $X_{m,n}$, the $n$ strips are
joined along the cosets $i + (n/k){\mathbb Z}$ for $i = 1, \dotsc, n$. Thus
every vertex on the line has $k$ outgoing vertical edges.
The strips below the branching line are attached in a similar manner;
there are $m$ of them, attached along the cosets $i + (m/k){\mathbb Z}$ for $i
= 1, \dotsc m$. Each vertex in the line has $k$ incoming vertical
edges. This description of the neighborhood of every branching line,
together with the projection to $T_{m,n}$, completely determines
$X_{m,n}$ as a CW complex.
\subsection{Metric structure}\label{metricsect}
The complex $X_{m,n}$ admits a piecewise-Riemannian metric on which
$\Aut(X_{m,n})$ acts by isometries. If $m = n$ then each $2$--cell is
isometric to a Euclidean $m/k \times 1$ rectangle, and all $1$--cells
have length $1$. If $m \not= n$ then each $2$--cell is isometric to a
right-angled quadrilateral region in the hyperbolic plane whose
vertical sides are geodesics of length $\abs{\log(m/n)}$ and whose
horizontal sides are concentric horocyclic arcs of lengths $m/k$ and
$n/k$.
When $m \not= n$, each strip is isometric to the region in ${\mathbb H}^2$
between two concentric horocycles of distance $\abs{\log(m/n)}$
apart. Since the branching lines are horocycles, $X_{m,n}$ has
concentrated positive curvature along these lines, and is definitely
not hyperbolic in any global sense. On the other hand, each strip in
$X_{m,m}$ is isometric to a Euclidean strip of width $1$, and
$X_{m,m}$ is isometric to $T_{m,m} \times {\mathbb R}$.
\subsection{Projection}
The complex $X_{1,1}$ is the Euclidean plane tiled by unit squares. It
admits rotations and the branching lines are not invariant. In all
other cases, branching lines and strips in $X_{m,n}$ are preserved by
$\Aut(X_{m,n})$. Hence the projection $\pi \colon \thinspace X_{m,n} \to T_{m,n}$
induces a homomorphism $\pi_*\colon \thinspace \Aut(X_{m,n}) \to \Aut(T_{m,n})$. Now
choose consistent orientations for all of the branching lines. Every
$g \in \Aut(X_{m,n})$ either preserves the orientations of all
branching lines, or reverses all of them. We will call $g$
\emph{orientation preserving} or \emph{orientation reversing}
accordingly.
\begin{lemma}\label{fixedray}
Suppose $m < n$. Let $\alpha$ be a directed ray in $T_{m,n}$ and
suppose $\pi_*(g)$ fixes $\alpha$ pointwise, for some orientation
preserving $g \in \Aut(X_{m,n})$. Then $g$ fixes $\pi^{-1}(\alpha)$
pointwise.
\end{lemma}
A similar statement holds for ``anti-directed'' rays if $m > n$.
\begin{proof}
The metric on $\pi^{-1}(\alpha)$ is isometric to a closed horoball in
${\mathbb H}^2$. Thus $g$ acts by a hyperbolic isometry preserving this
horoball. In the upper half plane model with the center of the
horoball at infinity, the only such isometries are reflections about
vertical lines and horizontal translations. Since $g$ is orientation
preserving, it is a translation. However, in this model, the
$2$--cells are rectangular regions of the form $[a,b] \times [c, d]$
with width $\abs{b-a}$ getting arbitrarily large as one moves upward
in the plane. No horizontal translation can preserve such a cell
structure, except for the identity.
\end{proof}
\begin{corollary}\label{injective}
If $m \not= n$ then $\pi_*\colon \thinspace \Aut(X_{m,n}) \to \Aut(T_{m,n})$ is
continuous and injective.
\end{corollary}
\begin{proof}
For continuity note that the pre-image of $U^{T_{m,n}}_{\sigma
\to \tau}$ is the union of the sets $U^{X_{m,n}}_{\tilde{\sigma} \to
\tilde{\tau}}$ for $\tilde{\sigma} \in \pi^{-1}(\sigma)$,
$\tilde{\tau}\in \pi^{-1}(\tau)$. For injectivity, suppose first that
$g \in \ker(\pi_*)$ is orientation preserving. Then $g = \id$ by Lemma
\ref{fixedray}, since $T_{m,n}$ is a union of rays of the relevant
type (directed or anti-directed).
If $g$ is orientation reversing, then it acts on every branching line
as a reflection with a unique fixed point, and similarly on every
strip as a reflection. In the latter case, the line segment of fixed
points is either a vertical edge or it passes through the center of a
$2$--cell. If, say, $m < n$ and $n/k > 2$, consider two strips whose
lower sides are the same branching line, with vertical edges joined
along cosets one unit apart. Then the fixed point sets of the strips
cannot agree on the branching line and we have a contradiction. If
$n/k = 2, m/k=1$ then one finds a similar contradiction by considering
an arrangement of four strips whose projection to $T_{m,n}$ is a
segment of two downward edges followed by two upward edges.
\end{proof}
\begin{remark}\label{notembedding}
The map $\pi_*$ is not an embedding. The topology on $\Aut(X_{m,n})$
is strictly finer than the subspace topology on
$\pi_*(\Aut(X_{m,n}))$. One can show that $\pi_*(U^{X_{m,n}}_{\sigma
\to \tau})$ is not open in $\pi_*(\Aut(X_{m,n}))$, for any cells
$\sigma, \tau$ with $U^{X_{m,n}}_{\sigma \to \tau} \not=\emptyset$.
\begin{proof}[Proof in the case $\sigma = \tau$]
Note that $\pi_*(U^X_{\sigma \to \sigma})$ contains the identity
element of $\Aut(T) \cap \pi_*(\Aut(X))$. Every basic neighborhood of
the identity has the form $U^T_{\sigma_1 \to \sigma_1} \cap \dotsm
\cap U^T_{\sigma_{\ell} \to \sigma_{\ell}} \cap \pi_*(\Aut(X))$. Let
$S \subset T$ be a finite subtree containing $\pi(\sigma), \sigma_1,
\dotsc, \sigma_{\ell}$. The subcomplex $\pi^{-1}(S)$ admits a
non-trivial shift which projects to the identity on $S$. This shift
extends to an automorphism $g \in \Aut(X_{m,n})$. Now $\pi_*(g) \in
U^T_{\sigma_1 \to \sigma_1} \cap \dotsm \cap U^T_{\sigma_{\ell} \to
\sigma_{\ell}} \cap \pi_*(\Aut(X))$ while $g \not\in U^X_{\sigma \to
\sigma}$. Since $\pi_*$ is injective, $\pi_*(g) \not\in
\pi_*(U^X_{\sigma \to \sigma})$. Hence no basic neighborhood of the
identity is contained in $\pi_*(U^X_{\sigma \to \sigma})$.
\end{proof}
\end{remark}
\subsection{General labeled graphs}
For any labeled graph $(A,\lambda)$ let $T_{(A,\lambda)}$ denote the
corresponding Bass--Serre tree. We will define $X_{(A,\lambda)}$ to be
the universal cover of $Z_{(A,\lambda)}$ with a suitable cell
structure.
For each $v$ put a cell structure on the circle $C_v$ with one
vertex and one edge. Given $e \in E(A)$, let $n_e = \abs{\lambda(e)}$,
$m_e = \abs{\lambda(\overline{e})}$, and $k_e = \gcd(m_e,n_e)$. The
annulus $M_e \cup M_{\overline{e}}$ has boundary curves of lengths
$n_e$ and $m_e$ attached to $C_{\partial_0(e)}$ and
$C_{\partial_1(e)}$ respectively. Tile this annulus with $k_e$
$(m_e/k_e, n_e/k_e)$--cells. Doing this for every edge we obtain a
cell structure for $Z_{(A,\lambda)}$, which then induces one on
$X_{(A,\lambda)}$.
Every strip above the annulus $M_e \cup M_{\overline{e}}$ is an
$(m_e/k_e, n_e/k_e)$--strip, and $X_{(A,\lambda)}$ is
homeomorphic to $T_{(A,\lambda)} \times {\mathbb R}$. For each $e$, every
branching line covering $C_{\partial_0(e)}$ has $n_e$ strips
above it covering $M_e \cup M_{\overline{e}}$, attached along the
cosets $i + (n_e/k_e){\mathbb Z}$ for $i = 1, \dotsc, n_e$. Every branching
line covering $\partial_1(e)$ has $m_e$ strips below it covering $M_e
\cup M_{\overline{e}}$, attached along the cosets $i + (m_e/k_e){\mathbb Z}$
for $i = 1, \dotsc, m_e$.
\begin{remark}
The notions of \emph{orientation preserving} and \emph{reversing}
automorphisms of $X_{m,n}$ apply equally well to $X_{(A,\lambda)}$
whenever $X_{(A,\lambda)} \not\cong X_{1,1}$. If $G$ is the GBS group
defined by $(A,\lambda)$, then these notions agree with the
orientation character on $G$. That is, $g\in G <
\Aut(X_{(A,\lambda)})$ is orientation preserving if and only if
$\chi(g) = 1$.
In this way, the orientation character extends to a homomorphism $\chi
\colon \thinspace \Aut(X_{(A,\lambda)}) \to \{\pm 1\}$, even if the modular
homomorphism $q$ does not.
\end{remark}
\subsection{Torsion-free lattices in $X_{m,n}$}\label{latticesec}
One simple way for a GBS group to be a lattice in $\Aut(X_{m,n})$ is
if its labeled graph $(A, \lambda)$ satisfies $X_{(A,\lambda)} \cong
X_{m,n}$. The next result gives a criterion for this.
\begin{proposition}\label{latticeprop}
Let G be the GBS group defined by $(A,\lambda)$ and suppose there is a
directed graph structure $E(A) = E^+(A) \sqcup E^-(A)$ on $A$ such
that
\begin{enumerate}
\item\label{i1} for every $v \in V(A)$,
\[ \sum_{e \in E^+_0(v)} \abs{\lambda(e)} \ = \ n \ \text{
and } \ \sum_{e \in E^-_0(v)} \abs{\lambda(e)} \ = \ m \]
\item\label{i2} for every $e \in E^+(A)$, let $n_e =
\abs{\lambda(e)}$, $m_e = \abs{\lambda(\overline{e})}$, $k_e =
\gcd(m_e, n_e)$, and $k = \gcd(m,n)$; then
\[ n_e / k_e \ = \ n/k
\ \text{ and } \ m_e / k_e \ = \ m/k.\]
\end{enumerate}
Then $X_{(A,\lambda)} \cong X_{m,n}$, and hence $G$ is a lattice in\/
$\Aut(X_{m,n})$.
\end{proposition}
\begin{proof}
Condition \eqref{i1} says that the tree $T_{(A,\lambda)}$, with
directed graph structure induced from $A$, is isomorphic to
$T_{m,n}$. This is so because the two sums count the numbers of strips
entering (resp. exiting) each branching line in
$X_{(A,\lambda)}$. Condition \eqref{i2} says that every strip in
$X_{(A,\lambda)}$ is a $(m/k,n/k)$--strip. It remains to examine how
these strips join the branching lines.
Fix a vertex $v\in V(A)$ and a branching line $L$ covering $C_v$. For
each $e \in E^+(A) \cap E_0(v)$ there are $\abs{\lambda(e)}$ strips
above $L$ mapping to $M_e \cup M_{\overline{e}}$. These are
attached along the cosets $i + (n/k){\mathbb Z}$ for $i = 1, \dotsc,
\abs{\lambda(e)}$. Each coset has the same number of such strips
attached to it (namely, $k_e$). Hence, overall, the $n$ strips above
$L$ are distributed evenly among the cosets of $(n/k){\mathbb Z}$, with $k$ of
them attached along each one. The strips attached below $L$ are
also equidistributed among the cosets of $(m/k){\mathbb Z}$. Now
$X_{(A,\lambda)}$ matches the description of $X_{m,n}$ from Section
\ref{combinatorialsect}.
\end{proof}
Now suppose $m\not= n$ and consider a general torsion-free uniform
lattice $G$ in $\Aut(X_{m,n})$. It acts freely and cocompactly on
$X_{m,n}$ by Proposition \ref{blprop}. Note that every
automorphism preserves the directed structure on $T_{m,n}$ because
$m\not= n$. In particular, no strip has its sides exchanged, so every
strip covers an annulus in $X_{m,n}/G$. Each branching line covers a
circle. The quotient is then a compact fibered $2$--complex,
homeomorphic to some $Z_{(A,\lambda)}$. Moreover the graph $A$ is a
directed graph, with directed structure induced by that of $T_{m,n}$.
We put a cell structure on $Z_{(A,\lambda)}$ by identifying it with
$X_{m,n}/G$. There is a \emph{length function} $\ell \colon \thinspace V(A) \sqcup
E(A) \to {\mathbb N}$ defined as follows. For $v \in V(A)$, $\ell(v)$ is the
combinatorial length of the circle $C_v$. For $e \in E(A)$, $\ell(e)$
is the number of $2$--cells tiling the annulus $M_e \cup
M_{\overline{e}}$ (that is, its combinatorial girth). Note that
$\ell(e) = \ell(\overline{e})$ for all $e\in E(A)$. The edges in
$M_e\cup M_{\overline{e}}$ crossing from one boundary component to the
other are called \emph{vertical edges}, since they are the images of
vertical edges of $X_{m,n}$. They are directed, consistently with
$e$.
\begin{proposition}\label{fiveconditions}
Suppose $m\not= n$ and let $G$ be a torsion-free uniform lattice in
$\Aut(X_{m,n})$. Let $k = \gcd(m,n)$, $m' = m/k$, and
$n'=n/k$. Let $(A,\lambda)$ be a directed labeled graph such that
$X_{m,n}/G$ is homeomorphic to $Z_{(A,\lambda)}$, with
associated length function $\ell \colon \thinspace V(A) \sqcup E(A) \to {\mathbb N}$ and
directed structure induced from $T_{m,n}$. Then
\begin{enumerate}
\item\label{j1}
for every $v \in V(A)$,
\[ \sum_{e \in E^+_0(v)} \abs{\lambda(e)} \ = \ n \ \text{
and } \ \sum_{e \in E^-_0(v)} \abs{\lambda(e)} \ = \ m \]
\item\label{j2}
for every $e \in E^+(A)$,
\begin{align*}
\ell(\partial_0(e)) \abs{\lambda(e)} \ &= \ n' \ell(e), \\
\ell(\partial_1(e)) \abs{\lambda(\overline{e})} \ &= \ m' \ell(e)
\end{align*}
\item\label{j5}
for every $v \in V(A)$, let $k_0(v) = \gcd(\ell(v),
n')$ and $k_1(v) = \gcd(\ell(v),m')$; then there exist partitions
\[ E^+_0(v) \ = \ E^+_1 \sqcup \dotsm \sqcup E^+_{k_0(v)}, \quad
E^-_0(v) \ = \ E^-_1 \sqcup \dotsm \sqcup E^-_{k_1(v)}\]
such that the sums
$\sum_{e \in E^+_i} \abs{\lambda(e)}$ are equal for all $i$,
and the sums $\sum_{e \in E^-_j} \abs{\lambda(e)}$ are equal for all
$j$.
\item\label{j3} for every $v \in V(A)$,
\[\sum_{e \in E^+_0(v)} \ell(e) \ = \ k \ell(v) \ = \ \sum_{e \in
E^-_0(v)} \ell(e)\]
\item\label{j4} the directed graph $A$ is strongly connected.
\end{enumerate}
\end{proposition}
\begin{proof}
Conclusion \eqref{j1} follows exactly as in Proposition
\ref{latticeprop}. Conclusion \eqref{j2} is evident from the cell
structure on the annulus $M_e \cup M_{\overline{e}}$, which is tiled
by $(m',n')$--cells. Its boundary curves have lengths $n'\ell(e)$ and
$m'\ell(e)$, and they wrap $\abs{\lambda(e)}$ times and
$\abs{\lambda(\overline{e})}$ times respectively around
$C_{\partial_0(e)}$ and $C_{\partial_1(e)}$.
For \eqref{j5}, identify the vertices of $C_v$ with the cyclic group
${\mathbb Z}/\ell(v){\mathbb Z}$ (in their natural cyclic ordering). The element $n' +
\ell(v){\mathbb Z}$ generates the subgroup $k_0(v){\mathbb Z} / \ell(v){\mathbb Z}$. Given $e \in
E^+_0(v)$ the annulus $M_e \cup M_{\overline{e}}$ has $\ell(e)$
outgoing vertical edges incident to vertices of $C_v$, spaced $n'$ units
apart. They meet the vertices along a coset of $k_0(v){\mathbb Z} / \ell(v){\mathbb Z}$
in ${\mathbb Z}/\ell(v){\mathbb Z}$, with $\ell(e) k_0(v)/\ell(v)$ outgoing edges
incident to each such vertex.
For each $i = 1, \dotsc, k_0(v)$ let $E_i^+$ be the set of edges $e
\in E^+_0(v)$ such that the vertical edges of $M_e \cup
M_{\overline{e}}$ are joined to $C_v$ along the coset $i + (k_0(v){\mathbb Z} /
\ell(v){\mathbb Z})$ in ${\mathbb Z}/\ell(v){\mathbb Z}$. Now every vertex in the coset has
\[ \sum_{e\in E^+_i} \ell(e) k_0(v) / \ell(v) \
\overset{\eqref{j2}}{=} \ \sum_{e\in
E^+_i} \abs{\lambda(e)} k_0(v) / n' \ = \ (k_0(v)/n') \sum_{e\in
E^+_i} \abs{\lambda(e)}\]
outgoing vertical edges incident to it. In the universal cover
$X_{m,n}$, every vertex has the same number of outgoing vertical
edges; hence the same is true of $X_{m,n}/G$ and the sums $\sum_{e \in
E^+_i} \abs{\lambda(e)}$ are the same for all $i$. The statement for
$E^-_0(v)$ is proved similarly.
Conclusion \eqref{j3} follows from \eqref{j1} and \eqref{j2}:
\[
\sum_{e \in E^+_0(v)} \ell(e) \ = \ \sum_{e \in E^+_0(v)}
\ell(v)\abs{\lambda(e)} / n'
\ = \ \ell(v) n / n' \ = \ k \ell(v)
\]
and similarly for the second sum.
For \eqref{j4}, define a new directed graph $A'$ from $A$ by
replacing each directed edge $e$ with $\ell(e)$ directed edges. By
\eqref{j3}, each vertex of $A'$ has equal numbers of incoming and
outgoing edges. It follows that $E^+(A')$ admits a partition into
directed cycles. Hence every directed edge is part of a directed
cycle. The same is then true of $A$. Now consider the decomposition of
$A$ into strongly connected components. If this decomposition is
non-trivial then there is a directed edge between two such components
that cannot be part of a directed cycle, which is a
contradiction. Hence $A$ is strongly connected.
\end{proof}
\begin{theorem}\label{gbslatticechar}
Suppose $m\not= n$ and let $G$ be a torsion-free group. Then $G$ is
isomorphic to a uniform lattice in $\Aut(X_{m,n})$ if and only if
there exist a compact GBS structure $(A,\lambda)$ for $G$, a directed
graph structure $E(A) = E^+(A) \sqcup E^-(A)$, and a function $\ell
\colon \thinspace V(A) \cup E(A) \to {\mathbb N}$ satisfying $\ell(e) = \ell(\overline{e})$
for all $e \in E(A)$ such that conclusions \eqref{j1}, \eqref{j2}, and
\eqref{j5} of Proposition \ref{fiveconditions} hold.
\end{theorem}
\begin{proof}
The ``only if'' direction holds by Proposition
\ref{fiveconditions}. For the converse let $(A,\lambda)$, the directed
graph structure, and $\ell$ be given. Let $k = \gcd(m,n)$, $m' = m/k$,
and $n' = n/k$. We put a cell stucture on $Z_{(A,\lambda)}$ as
follows. For each $v\in V(A)$ let $C_v$ have $\ell(v)$ vertices and
$\ell(v)$ edges. Choose an identification of the cyclically ordered
vertices with ${\mathbb Z}/\ell(v){\mathbb Z}$. For each $e \in E^+(A)$ let $A_e$ be an
annulus tiled by $\ell(e)$ $(m',n')$-cells, so that the initial end is
a circle of length $n'\ell(e)$ and the opposite end has length
$m'\ell(e)$. The $\ell(e)$ edges joining the two ends are called
vertical edges, and are directed from the initial end to the other
end.
For each $v$ and $e \in E^+_0(v)$ there are $k_0(v)$ ways,
combinatorially, to attach the initial end of $A_e$ to $C_v$ by a
degree $\lambda(e)$ map, corresponding to the coset of
$k_0(v){\mathbb Z}/\ell(v){\mathbb Z}$ in ${\mathbb Z}/\ell(v){\mathbb Z}$ met by the vertical edges of
$A_e$. (The existence of these combinatorial attaching maps follows
from \eqref{j2}.) Using the partition $E^+_0(v) = E^+_1 \sqcup \dotsm
\sqcup E^+_{k_0(v)}$, attach each annulus $A_e$ along the coset $i +
(k_0(v){\mathbb Z}/\ell(v){\mathbb Z})$ where $e \in E^+_i$. Attach the other ends of
each annulus similarly along the cosets of $k_1(v){\mathbb Z}/\ell(v){\mathbb Z}$. The
resulting space is homeomorphic to $Z_{(A,\lambda)}$.
By \eqref{j1} we have $T_{(A,\lambda)} \cong T_{m,n}$ as directed
graphs. By construction, each strip in the universal cover of
$Z_{(A,\lambda)}$ is an $(m',n')$--strip. Finally, by \eqref{j5}, each
vertex of $Z_{(A,\lambda)}$ has equal numbers of outgoing (respectively,
incoming) vertical edges. It follows that the strips in the universal
cover are joined to the branching lines in the manner described in
Section \ref{combinatorialsect}. Hence the universal cover of
$Z_{(A,\lambda)}$ is isomorphic to $X_{m,n}$ as a CW complex, and $G$
acts freely and cocompactly on $X_{m,n}$.
\end{proof}
\subsection{Discreteness}
One clear situation in which $\Aut(X_{m,n})$ cannot contain
incommensurable lattices is when it is discrete. When this occurs,
every lattice has finite index in $\Aut(X_{m,n})$, by
\cite[1.7]{basslubotzky}.
\begin{theorem}\label{discrete}
For any $m,n\geq 1$ the group $\Aut(X_{m,n})$ is discrete if and only
if\/ $\gcd(m,n) = 1$. When this occurs, there is a short exact
sequence
\[ 1 \to BS(m,n) \to \Aut(X_{m,n}) \overset{\chi}{\to} \{\pm 1\} \to 1.\]
\end{theorem}
\begin{proof}
Suppose $\gcd(m,n) = 1$ and let $K$ be the kernel of $\chi \colon \thinspace
\Aut(X_{m,n}) \to \{\pm 1\}$. We claim that $K$ acts freely on
the vertices of $X_{m,n}$. If $g\in K$ fixes a vertex $x \in X_{m,n}$
then it also fixes pointwise the branching line $L$ containing
$x$. Now note from Section \ref{combinatorialsect} that the strips
above this branching line are all attached along different cosets $i +
n{\mathbb Z}$, and the same is true of the strips below $L$. Hence $g$ cannot
permute these strips and it acts trivially on them. By similar
reasoning, $g$ acts trivially on the radius $r$ neighborhood of $L$,
for every $r$, so $g = 1$.
The lattice $BS(m,n)$ is a subgroup of $K$ acting transitively on
the vertices of $X_{m,n}$. Since $K$ acts freely on these vertices, we
must have $K = BS(m,n)$.
Now suppose that $\gcd(m,n) = k \not= 1$. Let $K = \ker(\chi)$ and
consider a vertex stabilizer $K_x$. Choose any directed ray
$\alpha$ in $T_{m,n}$ with initial vertex $\pi(x)$. Let $L_x$ be the
branching line through $x$ and for any vertex $v \in \alpha$ let $L_v$
be the branching line $\pi^{-1}(v)$. There are $k$ strips attached
above $L_v$ along each coset $i + (n/k){\mathbb Z}$. In particular there are
elements $g \in K_x$ which fix $L_x$ and $L_v$ pointwise, but permute
the strips above $L_v$ nontrivially. Since $v$ is arbitrarily far
from $x$, $K_x$ is infinite and $K$ is non-discrete. (With a little
more care one may express $K_x$ explicitly as an inverse limit of
products of permutation groups.)
\end{proof}
\section{A commensurability invariant}\label{invariantsec}
Consider a non-elementary GBS group $G$ and a non-trivial elliptic
subgroup $V < G$. Recall that because $G$ is non-elementary, the
elliptic subgroups of $G$ are well defined, as is the modular
homomorphism $q \colon \thinspace G \to {\mathbb Q}^{\times}$.
We define the \emph{$V$\!-depth} of an element $g\in G$ to be $D_V(g)
= [V: V \cap V^g]$ . Next we define the \emph{depth profile}:
\[\mathcal{D}(G,V) \ = \ \{ D_V(g) \mid g \in G \text{ and } q(g) =
\pm 1\} \ \subset \ {\mathbb N}. \]
The depth profile is a commensurability invariant in the following
sense.
\begin{theorem}\label{invariant}
Let $G$ be a non-elementary GBS group and $V<G$ a non-trivial elliptic
subgroup.
\begin{enumerate}
\item\label{t1} If\/ $G'<G$ is a subgroup of finite index
and $V \subset G'$ then \[\mathcal{D}(G,V) \ = \ \mathcal{D}(G',V).\]
\item\label{t2} If\/ $V' < V$ with $[V:V'] = r$ then
\[\mathcal{D}(G,V') \ = \ \{ \, n/\gcd(r,n) \mid n \in
\mathcal{D}(G,V)\, \}.\]
\end{enumerate}
\end{theorem}
Once $\mathcal{D}(G,V)$ is known, using \eqref{t2} one can compute
depth profiles for every finite index subgroup of $G$, and
hence for every GBS group commensurable with $G$, by \eqref{t1}.
Alternatively, one may define an equivalence relation on the set of
subsets of ${\mathbb N}$, by declaring $S \subset {\mathbb N}$ equivalent to $\{
\, n/\gcd(r,n) \mid n \in S \, \}$ for each $r \in {\mathbb N}$ and taking the
symmetric and transitive closure. Then the equivalence class of
$\mathcal{D}(G,V)$ is a true commensurability invariant of $G$.
Given $S \subset {\mathbb N}$ and $r\in {\mathbb N}$ let $S/r$ denote the set $\{
\, n/\gcd(r,n) \mid n \in S \, \} \subset {\mathbb N}$.
\begin{proposition}
Two subsets $S, S' \subset {\mathbb N}$ are equivalent if and only if there
exist $r, r' \in {\mathbb N}$ such that $S/r = S'/r'$.
\end{proposition}
\begin{proof}
Let $\divideby_r \colon \thinspace {\mathbb N} \to {\mathbb N}$ denote the function $n \mapsto
n/\gcd(r,n)$. One easily verifies that $\divideby_s \circ
\divideby_r = \divideby_{rs}$, and therefore $(S/r)/s$ = $S/(rs)$
for all $S \subset {\mathbb N}$ and $r,s \in {\mathbb N}$.
Let $\sim$ denote the smallest equivalence relation on
$\mathcal{P}({\mathbb N})$ such that $S \sim S/r$ for all $r\in {\mathbb N}$. Define $S
\approx S'$ if there exist $r, r'\in {\mathbb N}$ such that $S/r =
S'/r'$. Clearly, $S \approx S'$ implies $S \sim S'$. For the converse
it suffices to show that $\approx$ is an equivalence relation,
i.e. that it is transitive. If $S \approx S'$ and $S' \approx S''$
then there exist $r, r', s', s'' \in {\mathbb N}$ such that $S/r = S'/r'$ and
$S'/s' = S''/s''$. Then
\[S/(rs') = (S/r)/s' = (S'/r')/s' = S' / (r's') = (S' /
s')/r' = (S'' / s'')/r' = S'' / (s'' r'),\]
so indeed $S \approx S''$.
\end{proof}
\begin{example}\label{bsexample}
Let $G = BS(k, kn)$ with $k,{n} > 1$. There is an index $k$ normal
subgroup $G' < G$ isomorphic to $\bigvee_{i=1}^k BS(1,n)$, with vertex
group $V$. Since every edge in this decomposition of $G'$ has indices
$1$ and $n$, every element of modulus $\pm 1$ has $V$--depth a power
of ${n}$. All powers are realized, and
\[ \mathcal{D}(G,V) \ = \ \mathcal{D}(G',V) \ = \ \{ \, {n}^i \mid
i \in {\mathbb N} \cup \{0\}\,\}.\]
See Proposition \ref{wedgecomputation} ahead for a more detailed proof
(since by slide moves we can write $G' \cong BS(1,n) \vee
\bigvee_{i=1}^{k-1} BS(1,1)$). It is important that $k > 1$; otherwise
there are no hyperbolic elements of modulus $\pm 1$, and all elliptic
elements have $V$--depth $1$. In that case, none of the powers of
${n}$ are realized, and the depth profile of $G'$ is $\{ 1\}$.
Changing the elliptic subgroup to $V' < V$ with $[V,V']=r$, Theorem
\ref{invariant}\eqref{t2} gives
\[ \mathcal{D}(G, V') \ = \ \mathcal{D}(G,V)/r\ = \ \{ \,
{n}^i/\gcd(r, {n}^i) \mid i
\in {\mathbb N} \cup \{0\}\,\}.\]
This set (for any $r$) has the property that, with finitely many
exceptions, any two successive elements have ratio ${n}$. Hence the
modulus ${n}$ is an invariant of the equivalence class of depth
profiles of $BS(k,kn)$. This yields a new proof of the most difficult
case of \cite[Theorem 1.1]{CRKZ}:
\end{example}
\begin{corollary}[Lemma 7.2 of \cite{CRKZ}]\label{crkz}
The groups $BS(k_1, k_1 n_1)$ and $BS(k_2, k_2n_2)$ with $k_i,
{n_i}>1$ are commensurable only if\/ ${n_1} = {n_2}$.
\end{corollary}
Theorem \ref{invariant} will follow quickly from the next lemma.
\begin{lemma}\label{invariantlemma}
Let $G$ be a non-elementary GBS group and $V < G$ a non-trivial
elliptic subgroup. Suppose $q(g) = \pm 1$.
\begin{enumerate}
\item\label{l1} $D_V(g) = D_V(g^k)$ for all $k \geq 1$.
\item\label{l2} If\/ $V' < V$ with $[V:V']=r$ then $D_{V'}(g) = D_V(g) /
\gcd(r,D_V(g))$.
\end{enumerate}
\end{lemma}
\begin{proof}
First we prove \eqref{l2}. Let $d =
D_V(g)$ and let $V = \langle x \rangle$, $V^g = \langle y
\rangle$. Then $V \cap V^g$ is the subgroup $\langle
x^d\rangle$. Since $q(g) = \pm 1$ we have $[V^g: V \cap V^g] = d$ and
so $V \cap V^g = \langle y^d\rangle$ also. We wish to identify the
subgroup $V' \cap V'^g$, given that $V' = \langle x^r\rangle$ and
$V'^g = \langle y^r \rangle$.
Since $\langle x^r\rangle \cap \langle y^r\rangle \subset \langle x
\rangle \cap \langle y \rangle = \langle x^d \rangle = \langle y^d
\rangle$, it follows that
\begin{align}
\langle x^r \rangle \cap \langle y^r\rangle \
&= \ \langle x^r \rangle \cap \langle x^d \rangle \cap \langle y^r
\rangle \cap \langle y^d \rangle \notag\\
&= \ \langle x^{rd/\gcd(r,d)} \rangle \cap \langle
y^{rd/\gcd(r,d)}\rangle \notag\\
&= \ \langle x^{rd/\gcd(r,d)} \rangle \ = \ \langle
y^{rd/\gcd(r,d)}\rangle. \label{line3}
\end{align}
The equalities of line \eqref{line3} hold because both $\langle
x^{rd/\gcd(r,d)} \rangle$ and $\langle y^{rd/\gcd(r,d)}\rangle$ are
the unique subgroup of $\langle x^d \rangle = \langle y^d \rangle$ of
index $r/\gcd(r,d)$. Since $V' \cap V'^g = \langle x^{rd/\gcd(r,d)}
\rangle$ and $V' = \langle x^r\rangle$, we conclude that $D_{V'}(g) =
d/\gcd(r,d)$.
For \eqref{l1}, let $X$ be a GBS tree for $G$. Then $V< G_x$ for some
vertex $x\in X$ and $[G_x:V] < \infty$. From \eqref{l2} we can say
that $D_{G_x}(g) = D_{G_x}(g^k)$ implies $D_V(g) = D_V(g^k)$, and thus
it suffices to establish \eqref{l1} when $V$ is a vertex stabilizer.
Let $V = G_x$ for some vertex $x \in X$. Suppose first that $g$ is
hyperbolic with axis $X_g \subset X$. Let $y\in X_g$ be the vertex
closest to $x$. Define the segments $\tau = [x,y]$, $\sigma_1 =
[y,gy]$, $\sigma_i = g^{i-1}\sigma_1$ (for $1 < i \leq k$) and $\sigma
= \sigma_1 \dotsm \sigma_k = [y, g^k y]$. Then $[x,gx] = \tau \cdot
\sigma_1 \cdot g \overline{\tau}$ and $[x, g^k x] = \tau \cdot \sigma
\cdot g^k \overline{\tau}$.
Let $d = i(\sigma_1)$. Since $q(g) = \pm 1$ we have
$i(\overline{\sigma}_1) = d$ as well, and so $i(\sigma_i)=
i(\overline{\sigma}_i) = d$ for all $i$. By Lemma \ref{segmentindex}
it follows that $i(\sigma) = i(\overline{\sigma}) = d$. Now both
$[x,gx]$ and $[x,g^kx]$ are expressed as concatenations of three
segments, with matching indices along each. Hence (by Lemma
\ref{segmentindex}) $i([x,gx]) = i([x,g^kx])$. Since $D_V(g) =
i([x,gx])$ and $D_V(g^k) = [x, g^k x])$, we are done in this case.
If $g$ is elliptic let $y$ be a vertex closest
to $x$ that is fixed by $g$. Let $\tau = [x,y]$. Then $[x,gx] = \tau
\cdot g \overline{\tau}$ and $[x,g^kx] = \tau \cdot g^k
\overline{\tau}$. These two concatenations have matching indices along
them, so $D_V(g) = i([x,gx]) = i([x,g^kx]) = D_V(g^k)$ as in the
previous case.
\end{proof}
\begin{proof}[Proof of Theorem \ref{invariant}.] For \eqref{t1}, it is
immediate that $\mathcal{D}(G',V) \subset \mathcal{D}(G,V)$. For the
reverse inclusion, given $D_V(g) \in \mathcal{D}(G,V)$, choose $k$
such that $g^k \in G'$. Then $D_V(g) \in \mathcal{D}(G',V)$ by
Lemma \ref{invariantlemma}\eqref{l1}. Conclusion \eqref{t2} follows
directly from Lemma \ref{invariantlemma}\eqref{l2}.
\end{proof}
\section{The main examples}
Our main examples will be lattices in $\Aut(X_{k, k n})$ for $k, n >
1$. We fix some notation: define $a,b,c$ such that $b = \gcd(k,n)$, $k
= ab$, and $n = bc$.
\subsection{The lattice $G_1$}
This group is the index $2$ subgroup
of $BS(k, kn)$ generated by $a$, $tat^{-1}$, and $t^2$. It has a
labeled graph description as shown below, with two vertices and two
edges.
\[G_1: \quad
\vcenter{\hbox{
\begin{tikzpicture}
\scriptsize
\draw[very thick] (2,2) ellipse (1 and 1);
\filldraw[fill=white,thick] (2,1) circle (.7mm);
\filldraw[fill=black,thick] (2,3) circle (.7mm);
\draw[very thick,->] (3,1.96) -- (3,1.95);
\draw[very thick,->] (1,2.04) -- (1,2.05);
\draw[violet] (1.9,2.97) node[anchor=south east] {$kn$};
\draw[violet] (2.1,2.97) node[anchor=south west] {$k$};
\draw[violet] (2.4,.75) node[anchor=base] {$kn$};
\draw[violet] (1.7,.75) node[anchor=base] {$k$};
\end{tikzpicture}
\hspace*{1.02cm}
}}\]
Depth profiles of $BS(k,kn)$ were computed in Example
\ref{bsexample} (see also Proposition \ref{wedgecomputation}
below). Hence, for a suitable choice of elliptic subgroup
$V_1 < G_1$, the depth profile is
\[ \mathcal{D}(G_1, V_1) \ = \ \{\, n^i \mid i \in {\mathbb N} \cup
\{0\}\,\}.\]
\subsection{The lattice $G_2$}
This group is defined by the directed labeled graph $(B,\mu)$
below. It is bipartite with two vertices $v$ (white) and $u$ (black),
and $k+1$ directed edges. The edges $e_1, \dotsc, e_k$ are directed
from $u$ to $v$ and the edge $e_0$ is directed from $v$ to $u$. We
have $\mu(e_0) = k$, $\mu(\overline{e}_0) = kn$ and $\mu(e_i) = 1$,
$\mu(\overline{e}_i) = n$ for $i \not= 0$.
\[G_2: \quad
\vcenter{\hbox{
\begin{tikzpicture}
\small
\draw[very thick] (2,2) ellipse (1.2 and 1);
\draw[very thick] (2,1) arc(-45:45:1.4142126);
\filldraw[fill=white,thick] (2,1) circle (.7mm);
\filldraw[fill=black,thick] (2,3) circle (.7mm);
\draw[very thick,->] (3.2,1.96) -- (3.2,1.95);
\draw[very thick,->] (2.4142126,1.96) -- (2.4142126,1.95);
\draw[very thick,->] (0.8,2.04) -- (0.8,2.05);
\draw[color=black] (2.83,1.97) node {$\dotsm$};
\draw (2.1,2) node {$e_1$};
\draw (3.25,2) node[anchor=west] {$e_k$};
\draw (0.75,2) node[anchor=east] {$e_0$};
\scriptsize
\draw[violet] (1.9,2.97) node[anchor=south east] {$kn$};
\draw[violet] (2.85,2.97) node {$1$};
\draw[violet] (2.4,2.75) node {$1$};
\draw[violet] (2.4,1.25) node {$n$};
\draw[violet] (2.85,1.03) node {$n$};
\draw[violet] (1.7,.75) node[anchor=base] {$k$};
\end{tikzpicture}
\hspace*{1.02cm}
}}\]
Both $G_1$ and $G_2$ are lattices in $\Aut(X_{k,kn})$ by Proposition
\ref{latticeprop}.
\subsection{The subgroup $H_2$}
We will define a finite index subgroup $H_2 < G_2$
by constructing an admissible branched covering $(A, \lambda)$ of
$(B,\mu)$. This subgroup will aid us in computing a depth profile
for $G_2$.
The graph $A$ has one vertex $v_1$ above $v$ and $b$ vertices $u_1,
\dotsc, u_b$ above $u$. For each $i$ there are $k$ directed edges from
$u_i$ to $v_1$, mapping to $e_1, \dotsc, e_k$ respectively, and $a$
directed edges from $v_1$ to
$u_i$, mapping to $e_0$. If $e$ is a directed edge above $e_0$ then
its labels are $\lambda(e) = 1$ and $\lambda(\overline{e}) = b^2
c$. If $e$ is a directed edge above $e_i$ ($i \geq 1$) then its labels
are $\lambda(e) = 1$ and $\lambda(\overline{e}) = c$.
Finally, the
degree function of the branched covering is given by $d(v_1) = k$,
$d(u_i) = a$, $d(e) = 1$ for every edge $e$ above $e_0$, and $d(e) =
a$ for every edge $e$ above $e_i$ ($i \geq 1$). See Figure
\ref{h2graph}. One may verify that the conditions of Definition
\ref{admissibledef} are met.
\begin{figure}[!ht]
$\vcenter{\hbox{
\begin{tikzpicture} [scale=1.2]
\begin{scope}[xshift=-0.3cm]
\small
\begin{scope}[rotate around={-30:(2,3)}]
\draw[very thick] (0,3) -- (2,3);
\filldraw[fill=black,thick] (0,3) circle (.5mm);
\draw (0,3) node[anchor=east] {$u_b \ $};
\draw[very thick,->] (1.04,3) -- (1.05,3);
\end{scope}
\begin{scope}[rotate around={30:(2,3)}]
\draw[very thick] (0,3) -- (2,3);
\filldraw[fill=black,thick] (0,3) circle (.5mm);
\draw (0,3) node[anchor=east] {$u_1 \ $};
\draw[very thick,->] (1.04,3) -- (1.05,3);
\end{scope}
\filldraw[fill=white,thick] (2,3) circle (.5mm);
\draw (2,3) node[anchor=west] {$v_1$};
\draw[color=black] (0.268,3.09) node {$\vdots$};
\draw[thick,color=mygreen,->] (0.268,1.6) -- (0.268,1);
\draw[thick,color=mygreen,->] (1.174,1.9) -- (1.174,1.1);
\draw[thick,color=mygreen,->] (2,2.2) -- (2,1.2);
\draw[very thick] (0.268,.5) -- (2,.5);
\draw[very thick,->] (1.174,.5) -- (1.184,.5);
\filldraw[fill=black,thick] (0.268,.5) circle (.5mm);
\draw (.268,.5) node[anchor=east] {$u \ $};
\filldraw[fill=white,thick] (2,.5) circle (.5mm);
\draw (2,.5) node[anchor=west] {$\, v$};
\draw (1.18,.4) node[anchor=north] {$e_i$};
\scriptsize
\draw[violet] (0.5,4.09) node {$1$};
\draw[violet] (1.85,3.3) node {$c$};
\draw[violet] (0.5,1.91) node {$1$};
\draw[violet] (1.85,2.7) node {$c$};
\draw[color=mygreen] (0.268,1.3) node[anchor=east] {$a$};
\draw[color=mygreen] (1.174,1.5) node[anchor=east] {$a$};
\draw[color=mygreen] (2,1.72) node[anchor=east] {$k$};
\draw[violet] (0.5,.53) node[anchor=south] {$1$};
\draw[violet] (1.81,.53) node[anchor=south] {$n$};
\end{scope}
\small
\begin{scope}[rotate around={30:(3,3)}]
\draw[very thick] (3,3) arc(120:60:2);
\draw[very thick] (3,3) arc(240:300:2);
\filldraw[fill=black,thick] (5,3) circle (.5mm);
\draw[color=black] (4,3.08) node[transform shape,scale=.7] {$\vdots$};
\draw (5,3) node[anchor=west] {$ \ u_b$};
\draw[very thick,->] (4.04,3.2679) -- (4.05,3.2679);
\draw[very thick,->] (4.04,2.7321) -- (4.05,2.7321);
\end{scope}
\begin{scope}[rotate around={-30:(3,3)}]
\draw[very thick] (3,3) arc(120:60:2);
\draw[very thick] (3,3) arc(240:300:2);
\filldraw[fill=black,thick] (5,3) circle (.5mm);
\draw[color=black] (4,3.08) node[transform shape,scale=.7] {$\vdots$};
\draw (5,3) node[anchor=west] {$ \ u_1$};
\draw[very thick,->] (4.04,3.2679) -- (4.05,3.2679);
\draw[very thick,->] (4.04,2.7321) -- (4.05,2.7321);
\end{scope}
\filldraw[fill=white,thick] (3,3) circle (.5mm);
\draw (3,3) node[anchor=east] {$v_1$};
\draw[color=black] (4.732,3.09) node {$\vdots$};
\draw[thick,color=mygreen,->] (4.732,1.6) -- (4.732,1);
\draw[thick,color=mygreen,->] (3.906,1.8) -- (3.906,1);
\draw[thick,color=mygreen,->] (3,2.2) -- (3,1.2);
\draw[very thick] (3,.5) -- (4.732,.5);
\draw[very thick,->] (3.906,.5) -- (3.916,.5);
\filldraw[fill=white,thick] (3,.5) circle (.5mm);
\draw (3,.5) node[anchor=east] {$v \,$};
\filldraw[fill=black,thick] (4.732,.5) circle (.5mm);
\draw (4.732,.5) node[anchor=west] {$ \ u$};
\draw (3.92,.4) node[anchor=north] {$e_0$};
\scriptsize
\draw[color=black] (4.05,3.6) node {$a$};
\draw[color=black] (4.05,2.4) node {$a$};
\draw[violet] (3.01,3.35) node {$1$};
\draw[violet] (3.41,3.23) node {$1$};
\draw[violet] (4.38,4.2) node {$b^2 c$};
\draw[violet] (4.83,3.67) node {$b^2 c$};
\draw[violet] (4.83,2.4) node {$b^2 c$};
\draw[violet] (4.38,1.83) node {$b^2 c$};
\draw[violet] (3.02,2.65) node {$1$};
\draw[violet] (3.41,2.77) node {$1$};
\draw[color=mygreen] (3,1.72) node[anchor=east] {$k$};
\draw[color=mygreen] (3.906,1.42) node[anchor=east] {$1$};
\draw[color=mygreen] (4.732,1.3) node[anchor=west] {$a$};
\draw[violet] (3.19,.53) node[anchor=south] {$k$};
\draw[violet] (4.45,.53) node[anchor=south] {$kn$};
\end{tikzpicture}
}} \quad
\quad H_2\colon \ \ {\displaystyle \bigvee_{i=1}^b} \ \
\vcenter{\hbox{
{
\begin{tikzpicture}[scale=1.2]
\small
\draw[very thick] (2,2) ellipse (1.2 and 1);
\draw[very thick] (2,1) arc(-45:45:1.4142126);
\draw[very thick] (2,1) arc(225:135:1.4142126);
\filldraw[fill=white,thick] (2,1) circle (.5mm);
\filldraw[fill=black,thick] (2,3) circle (.5mm);
\draw[very thick,->] (3.2,1.96) -- (3.2,1.95);
\draw[very thick,->] (2.4142126,1.96) -- (2.4142126,1.95);
\draw[very thick,->] (1.5857874,2.04) -- (1.5857874,2.05);
\draw[very thick,->] (0.8,2.04) -- (0.8,2.05);
\draw[color=black] (2.83,1.97) node {$\dotsm$};
\draw[color=black] (1.21,1.97) node {$\dotsm$};
\draw (2,0.9) node[anchor=north] {$v_1$};
\draw (2,3.1) node[anchor=south] {$u_i$};
\scriptsize
\draw[color=black] (1.1928937,2.03) node[anchor=south] {$a$};
\draw[color=black] (2.8071063,2.03) node[anchor=south] {$k$};
\draw[violet] (0.96,2.9) node {$b^2 c$};
\draw[violet] (2.89,2.9) node {$1$};
\draw[violet] (2.43,2.68) node {$1$};
\draw[violet] (1.44,2.68) node {$b^2 c$};
\draw[violet] (2.38,1.25) node {$c$};
\draw[violet] (1.63,1.25) node {$1$};
\draw[violet] (2.83,1.03) node[anchor=base] {$c$};
\draw[violet] (1.16,1.0) node[anchor=base] {$1$};
\end{tikzpicture}
}
}}$
\caption{The admissible branched covering defining $H_2$. Above each
$e_i$ ($i \geq 1$) there are $b$ edges as shown. Above $e_0$ there
are $k = ab$ edges as shown. Overall, $(A,\lambda)$ is a wedge
product of $b$ copies of a graph, joined at the vertex $v_1$.
}\label{h2graph}
\end{figure}
By collapsing each of the edges above $e_k$, and then performing $k-1$
slide moves, we find that
\begin{align}
H_2 \ &\cong \ \bigvee_k BS(1,n^2) \ \vee \bigvee_{b(k-1)} BS(c,c)\notag \\
&\cong \ BS(1,n^2) \ \vee \ \bigvee_{k-1} BS(1,1) \ \vee
\bigvee_{b(k-1)} BS(c,c). \label{h2-final}
\end{align}
A depth profile of $H_2$ can now be computed using the next result.
\begin{proposition}\label{wedgecomputation}
Suppose $G = BS(1,N) \vee \bigvee_{i=1}^r BS(n_i,n_i)$ for some $r
\geq 1$ and suppose that $N>1$, each $n_i$ divides $N$, and the set
$\{ n_1, \dotsc, n_r, N \, \}$ is closed under taking $\lcm$ and
contains $1$. Let $V$ be
the vertex group. Then
\begin{equation}\label{formula}
\mathcal{D}(G,V) \ = \
\{ \, N^in_j \mid i\in {\mathbb N} \cup\{0\}, j = 1, \dotsc, r \, \}.
\end{equation}
\end{proposition}
\begin{proof}
Let $X$ be the Bass--Serre tree for the given GBS structure of
$G$. Note that $G$ acts transitively on $V(X)$. If $\sigma$ is any
non-trivial segment in $X$ then there is an element $g$ taking
$\partial_0(\sigma)$ to $\partial_1(\sigma)$, and from \eqref{modulus}
we have that $\abs{q(g)} = i(\sigma)/i(\overline{\sigma})$. Thus we
will call the segment \emph{unimodular} if $i(\sigma) =
i(\overline{\sigma})$.
The subgroup $V$ is the stabilizer of a vertex $v$. Now
$\mathcal{D}(G,V)$ is the set of indices $i(\sigma)$ of
unimodular segments starting at $v$. But every segment can be
translated to start at $v$, with no change to its index. Hence
$\mathcal{D}(G,V)$ is simply the set of indices of all unimodular
segments in $X$. These observations apply to any GBS tree having one
vertex orbit. (In a general GBS tree with $V = G_x$,
$\mathcal{D}(G,V)$ is the set of indices of unimodular segments having
both endpoints in $Gx$.)
Call an edge $e \in E(X)$ \emph{ascending} if $i(e) = 1$ and
$i(\overline{e}) = N$, and \emph{descending} if $\overline{e}$ is
ascending. Note that every edge in $X$ is either ascending,
descending, or unimodular. Now every unimodular segment $\sigma$ of
length $> 1$ has one of the following forms:
\begin{enumerate}
\item\label{c1} $\sigma_1 \sigma_2$ with $\sigma_1, \sigma_2$ unimodular
\item\label{c2} $e_1 \tau e_2$ with $\tau$ unimodular and $e_1$
ascending, $e_2$ descending
\item\label{c3} $e_1 \tau e_2$ with $\tau$ unimodular and $e_1$
descending, $e_2$ ascending.
\end{enumerate}
Let $D$ denote the right hand side of \eqref{formula}. It is easily
verified that $D$ is closed under taking $\lcm$. We can now show that
every unimodular $\sigma$ has index in $D$ by induction on
length. Unimodular edges have index $n_j$ for some $j$, which are in
$D$. If $\sigma$ is of type \eqref{c1} then by Remark \ref{shortindex}
we have $i(\sigma) = \lcm(i(\sigma_1),i(\sigma_2)) \in D$. If $\sigma$
is of type \eqref{c2} with $i(\tau) = N^i n_j$, using Remark
\ref{shortindex} one finds that $i(\sigma) = N^{i-1}n_j$ if $i \geq 1$
and $i(\sigma) = 1$ if $i=0$. Hence $i(\sigma)\in D$. If $\sigma$ is
of type \eqref{c3} then $i(\sigma) = N i(\tau) \in D$.
Finally, consider a segment $\sigma = e_1 \dotsm e_i \tau e'_1
\dotsm e'_i$ where each $e_k$ is descending, $\tau$ is a unimodular
edge with index $n_j$, and each $e'_k$ is ascending. Then $i(\sigma) =
N^in_j$.
\end{proof}
\begin{theorem}\label{mainthm}
If\/ $\gcd(k,n) \not= 1$ then the lattices $G_1, G_2 < \Aut(X_{k,kn})$
are not abstractly commensurable.
\end{theorem}
\begin{proof}
Consider the GBS structure \eqref{h2-final} for $H_2$ and let $V_2$ be
its vertex group. By Theorem \ref{invariant} and Proposition
\ref{wedgecomputation} we have
\[ \mathcal{D}(G_2, V_2) \ = \ \mathcal{D}(H_2,V_2) \ = \ \{\, n^{2i}
\mid i \in {\mathbb N}\cup \{0\}\,\} \ \cup \ \{\, n^{2i}c \mid i \in {\mathbb N}\cup
\{0\}\,\}. \]
We also have
\[ \mathcal{D}(G_1, V_1) \ = \ \{\, n^i \mid i \in {\mathbb N} \cup \{0\}\,\}\]
as mentioned earlier.
Enumerating the elements of $\mathcal{D}(G_i,V_i)$ in order, notice
that each element divides the next one. Taking the ratios of
successive elements one obtains the sequences $(n, n, n, \dotsc )$ for
$i=1$ and $(c, n^2/c, c, n^2/c, \dotsc )$ for $i=2$. The ``tails'' of
these ratio sequences are unchanged when passing from
$\mathcal{D}(G_i, V_i)$ to $\mathcal{D}(G_i,V_i)/r$ for any $r \in
{\mathbb N}$, because the values of $\gcd(r,n^i)$, $\gcd(r,n^{2i})$, and
$\gcd(r,n^{2i}c)$ stabilize as $i \to \infty$, all to the same
number. (This number is $r$ with all of its prime factors not
dividing $n$ removed.) Since $c\not= n$, the
two tails will never agree, and so the two depth profiles are
inequivalent.
\end{proof}
\begin{remark}\label{mainthm-remark}
If $\gcd(k,n) =1$ then $c = n$ and the depth profiles of $G_1$ and
$G_2$ coincide. However, the depth profile is not failing us as an
invariant, as it turns out the groups are commensurable in this
case. The labeled graph on the right hand side of Figure \ref{h2graph}
is an admissible branched covering of the labeled graph for $G_1$,
since $b=1$ and $k=a$.
\end{remark}
\begin{example}
Figure \ref{easycase} illustrates $G_1$ and $G_2$ in the simplest
case, when $X_{k,kn} = X_{2,4}$. In this case the lattices are
commensurable to $BS(2,4)$ and $BS(4,16)$ respectively, which are
incommensurable by \cite{CRKZ}. The vertical maps are admissible
branched coverings and the horizontal arrows are elementary
deformations. A similar phenomenon occurs whenever $n=k>1$: in
$\Aut(X_{k,k^2})$ there are incommensurable lattices commensurable to
$BS(k,k^2)$ and $BS(k^2,k^4)$ respectively.
\begin{figure}[!ht]
\begin{tikzpicture}[scale=1.2]
\small
\draw[very thick] (2,1.5) circle (0.5);
\filldraw[fill=black,thick] (2,1) circle (.7mm);
\filldraw[fill=white,thick] (2,2) circle (.7mm);
\draw (1.4,1.5) node[anchor=east] {$G_1\colon$};
\draw[very thick] (5,1.5) ellipse (0.6 and 0.5);
\draw[very thick] (5,1) arc(-45:45:0.706063);
\filldraw[fill=black,thick] (5,1) circle (.7mm);
\filldraw[fill=white,thick] (5,2) circle (.7mm);
\draw (4.3,1.5) node[anchor=east] {$G_2\colon$};
\draw[very thick,->] (5,3.4) -- (5,2.6);
\draw[very thick] (5,4.5) ellipse (0.6 and 0.5);
\draw[very thick] (5,4) arc(-45:45:0.706063);
\draw[very thick] (5,5.5) ellipse (0.6 and 0.5);
\draw[very thick] (5,5) arc(-45:45:0.706063);
\filldraw[fill=black,thick] (5,4) circle (.7mm);
\filldraw[fill=white,thick] (5,5) circle (.7mm);
\filldraw[fill=black,thick] (5,6) circle (.7mm);
\draw[very thick,->] (6.1,5) -- (6.9,5);
\draw (6.5,5) node[anchor=south] {$\cong$};
\begin{scope}[rotate around={7:(8,5)}]
\draw[very thick] (8,5) .. controls (8.675,6.35) and (9.35,5.675) .. (8,5);
\draw[very thick] (8,5) .. controls (7.325,3.65) and (6.65,4.325) .. (8,5);
\end{scope}
\begin{scope}[rotate around={-7:(8,5)}]
\draw[very thick] (8,5) .. controls (9.35,4.325) and (8.675,3.65) .. (8,5);
\draw[very thick] (8,5) .. controls (6.65,5.675) and (7.325,6.35) .. (8,5);
\end{scope}
\filldraw[fill=white,thick] (8,5) circle (.7mm);
\draw[very thick,->] (9.1,5) -- (9.9,5);
\draw (9.5,5) node[anchor=south] {$\cong$};
\begin{scope}[rotate around={7:(11,5)}]
\draw[very thick] (11,5) .. controls (11.675,6.35) and (12.35,5.675) .. (11,5);
\draw[very thick] (11,5) .. controls (10.325,3.65) and (9.65,4.325) .. (11,5);
\end{scope}
\begin{scope}[rotate around={-7:(11,5)}]
\draw[very thick] (11,5) .. controls (12.35,4.325) and (11.675,3.65) .. (11,5);
\draw[very thick] (11,5) .. controls (9.65,5.675) and (10.325,6.35) .. (11,5);
\end{scope}
\filldraw[fill=white,thick] (11,5) circle (.7mm);
\draw[very thick,->] (11,3.4) -- (11,2.6);
\draw[very thick] (11,1.5) circle (0.5);
\filldraw[fill=white,thick] (10.5,1.5) circle (.7mm);
\draw (5.1,3) node[anchor=west] {$2:1$};
\draw (11.1,3) node[anchor=west] {$4:1$};
\scriptsize
\draw (6.5,4.9) node[anchor=north] {$\text{collapses}$};
\draw (9.5,4.9) node[anchor=north] {$\text{slides}$};
\draw[violet] (1.75,2.14) node {$2$};
\draw[violet] (2.25,2.14) node {$4$};
\draw[violet] (1.75,0.87) node {$4$};
\draw[violet] (2.25,0.87) node {$2$};
\draw[violet] (4.75,2.14) node {$2$};
\draw[violet] (4.75,0.87) node {$4$};
\draw[violet] (5.32,1.75) node {$2$};
\draw[violet] (5.33,1.3) node {$1$};
\draw[violet] (5.32,2.14) node {$2$};
\draw[violet] (5.33,0.87) node {$1$};
\draw[violet] (4.75,3.87) node {$4$};
\draw[violet] (5.33,3.87) node {$1$};
\draw[violet] (5.33,4.3) node {$1$};
\draw[violet] (5.33,4.71) node {$1$};
\draw[violet] (5.33,5.3) node {$1$};
\draw[violet] (5.33,5.71) node {$1$};
\draw[violet] (4.75,6.14) node {$4$};
\draw[violet] (5.32,6.14) node {$1$};
\draw[violet] (5.65,4.85) node {$1$};
\draw[violet] (5.65,5.16) node {$1$};
\draw[violet] (4.76,4.75) node {$1$};
\draw[violet] (4.76,5.25) node {$1$};
\draw[violet] (7.5,4.82) node {$1$};
\draw[violet] (7.5,5.19) node {$1$};
\draw[violet] (8.5,4.82) node {$1$};
\draw[violet] (8.5,5.19) node {$1$};
\draw[violet] (7.87,5.65) node {$4$};
\draw[violet] (8.14,5.65) node {$1$};
\draw[violet] (7.87,4.37) node {$4$};
\draw[violet] (8.14,4.37) node {$1$};
\draw[violet] (10.5,4.82) node {$4$};
\draw[violet] (10.5,5.19) node {$1$};
\draw[violet] (11.5,4.82) node {$1$};
\draw[violet] (11.5,5.19) node {$4$};
\draw[violet] (10.87,5.65) node {$4$};
\draw[violet] (11.14,5.65) node {$1$};
\draw[violet] (10.87,4.37) node {$1$};
\draw[violet] (11.14,4.37) node {$4$};
\draw[violet] (10.36,1.75) node {$4$};
\draw[violet] (10.3,1.25) node {$16$};
\end{tikzpicture}
\caption{Lattices $G_1, G_2$ in $\Aut(X_{2,4})$. $G_1$ is an index 2
subgroup of $BS(2,4)$. $G_2$ contains an index 2 subgroup isomorphic
to an index 4 subgroup of $BS(4,16)$ ($ \not\sim BS(2,4)$). For
appropriate choices of elliptic subgroups $V_i < G_i$ the depth
profiles are $\mathcal{D}(G_1,V_1) = \{\, 2^i \,\}$ and
$\mathcal{D}(G_2,V_2) = \{\, 4^i \,\}$.
}\label{easycase}
\end{figure}
\end{example}
\begin{remark}\label{envelope}
Regarding Figure \ref{easycase}, notice that the finite-index subgroup
of $G_2$ is a lattice both in $\Aut(X_{2,4})$ and
$\Aut(X_{4,16})$, even though their ``standard'' lattices $BS(2,4)$
and $BS(4,16)$ are not commensurable.
\end{remark}
\section{Further cases of $\Aut(X_{k,kn})$}
We return to the situation of Remark \ref{mainthm-remark},
when $\gcd(k,n) = 1$.
\subsection{The lattice $G_3$}
Suppose that $p$ is a non-trivial divisor of $n$ (not necessarily
prime) such that $p < k$. Let $l = k-p$. The labeled graph below
defines a lattice $G_3 < \Aut(X_{k,kn})$ by Proposition
\ref{latticeprop}. It has vertices $v$ (white) and $u$ (black),
directed edges $e_1, \dotsc, e_k$ from $u$ to $v$, directed edges
$f_1, \dotsc, f_l$ from $v$
to $u$, and a directed edge $f_0$ from $v$ to $u$. The labels are
given by $\lambda(e_i) = 1$, $\lambda(\overline{e}_i) = n$,
$\lambda(f_0) = p$, $\lambda(\overline{f}_0) = pn$, and $\lambda(f_i)
= 1$, $\lambda(\overline{f}_i) = n$ ($i > 0$).
\[G_3\colon \ \ \vcenter{\hbox{
\begin{tikzpicture}
\small
\draw[very thick] (2,2) ellipse (1.5 and 1);
\draw[very thick] (2,1) arc(-45:45:2.5 and 1.4142126);
\draw[very thick] (2,1) arc(225:135:2.5 and 1.4142126);
\draw[very thick] (2,1) arc(210:150:2);
\filldraw[fill=white,thick] (2,1) circle (.7mm);
\filldraw[fill=black,thick] (2,3) circle (.7mm);
\draw[very thick,->] (3.5,1.96) -- (3.5,1.95);
\draw[very thick,->] (2.735,1.96) -- (2.735,1.95);
\draw[very thick,->] (0.5,2.04) -- (0.5,2.05);
\draw[very thick,->] (1.265,2.04) -- (1.265,2.05);
\draw[very thick,->] (1.732,2.04) -- (1.732,2.05);
\draw[color=black] (0.9,1.97) node {$\dotsm$};
\draw[color=black] (3.15,1.97) node {$\dotsm$};
\scriptsize
\draw[color=black] (0.9, 2.01) node[anchor=south] {$l$};
\draw[color=black] (3.1, 2.01) node[anchor=south] {$k$};
\draw[violet] (0.82,2.9) node {$n$};
\draw[violet] (1.32,2.68) node {$n$};
\draw[violet] (2.1,2.58) node {$pn$};
\draw[violet] (2.68,2.68) node {$1$};
\draw[violet] (3.18,2.9) node {$1$};
\draw[violet] (2.06,1.37) node {$p$};
\draw[violet] (2.66,1.32) node {$n$};
\draw[violet] (3.16,1.1) node {$n$};
\draw[violet] (1.32,1.35) node {$1$};
\draw[violet] (0.82,1.13) node {$1$};
\end{tikzpicture}
\hspace*{.85cm}
}}\]
\begin{theorem}\label{g3thm}
Suppose $n$ has a non-trivial divisor $p\not= n$ such that $p <
k$. Then the lattices $G_1, G_3 < \Aut(X_{k,kn})$ are not abstractly
commensurable.
\end{theorem}
\begin{proof}
Collapsing $f_1$ and performing $k-1$ slide moves, we obtain
\begin{align*}
G_3 \ &\cong \ \bigvee_k BS(1,n^2) \ \vee \!
\bigvee_{k-p-1} BS(n,n) \vee BS(pn,pn) \\
&\cong \ BS(1,n^2) \ \vee \ \bigvee_{k-1} BS(1,1) \ \vee \!
\bigvee_{k-p-1} BS(n,n) \ \vee \ BS(pn, pn).
\end{align*}
Let $V_3$ be the vertex group. Proposition \ref{wedgecomputation}
provides the depth profile:
\[ \mathcal{D}(G_3, V_3) \ = \
\begin{cases}
\{\, n^{2i}\, \} \cup \{\, n^{2i+1}\, \} \cup \{\, n^{2i+1} p\, \} &
\text{ if } p<k-1 \\
\{\, n^{2i}\, \} \cup \{\, n^{2i+1} p\, \} & \text{ if }
p=k-1. \\
\end{cases} \]
In both cases, enumerating the elements in order, each element divides
the next. Hence there is a sequence of successive ratios that can be
compared to the sequence $(n, n, n, \dotsc)$ arising from
$\mathcal{D}(G_1,V_1)$. Exactly as in the proof of Theorem
\ref{mainthm}, the tails of the sequences, modulo shifting, are
invariants of the equivalence classes of depth profiles.
The sequences of ratios are
\[\begin{array}{ll}
\text{the $3$--periodic sequence } n, p, n/p, \ \dotsc & \text{ if }
p<k-1 \\
\text{the $2$--periodic sequence } np, n/p, \ \dotsc & \text{ if }
p=k-1. \\
\end{array}\]
Neither of these sequences eventually agree with $(n, n, n, \dotsc)$,
so the two depth profiles are inequivalent in both cases.
\end{proof}
\subsection{The lattice $G_4$} Suppose $n<k$ and $k \equiv 1 \mod
n$. Let $l = (k-1)/n$. The labeled graph below defines a lattice $G_4
< \Aut(X_{k,kn})$. It is bipartite with vertices $v$ (white) and $u$
(black), has directed edges $e_1, \dotsc, e_k$ from $u$ to $v$ with
labels $\lambda(e_i) = 1$, $\lambda(\overline{e}_i) = n$, and directed
edges $f_0, \dotsc, f_l$ from $v$ to $u$ with labels $\lambda(f_0) =
1$, $\lambda(\overline{f}_0) = n$, $\lambda(f_i) = n$,
$\lambda(\overline{f}_i) = n^2$ ($i \geq 1$).
\[ G_4\colon \ \
\vcenter{\hbox{
\begin{tikzpicture}
\small
\draw[very thick] (2,2) ellipse (1.5 and 1);
\draw[very thick] (2,1) arc(-45:45:2.5 and 1.4142126);
\draw[very thick] (2,1) arc(225:135:2.5 and 1.4142126);
\draw[very thick] (2,1) arc(210:150:2);
\filldraw[fill=white,thick] (2,1) circle (.7mm);
\filldraw[fill=black,thick] (2,3) circle (.7mm);
\draw[very thick,->] (3.5,1.96) -- (3.5,1.95);
\draw[very thick,->] (2.735,1.96) -- (2.735,1.95);
\draw[very thick,->] (0.5,2.04) -- (0.5,2.05);
\draw[very thick,->] (1.265,2.04) -- (1.265,2.05);
\draw[very thick,->] (1.732,2.04) -- (1.732,2.05);
\draw[color=black] (0.9,1.97) node {$\dotsm$};
\draw[color=black] (3.15,1.97) node {$\dotsm$};
\scriptsize
\draw[color=black] (0.9, 2.01) node[anchor=south] {$l$};
\draw[color=black] (3.1, 2.01) node[anchor=south] {$k$};
\draw[violet] (0.845,2.933) node {$n^2$};
\draw[violet] (1.345,2.713) node {$n^2$};
\draw[violet] (2.02,2.58) node {$n$};
\draw[violet] (2.68,2.68) node {$1$};
\draw[violet] (3.18,2.9) node {$1$};
\draw[violet] (2.02,1.4) node {$1$};
\draw[violet] (2.66,1.32) node {$n$};
\draw[violet] (3.16,1.1) node {$n$};
\draw[violet] (1.32,1.32) node {$n$};
\draw[violet] (0.82,1.1) node {$n$};
\end{tikzpicture}
\hspace*{.85cm}
}}\]
\begin{theorem}\label{g4thm}
Suppose $n < k$ and $k \equiv 1 \mod n$. Then the lattices $G_1, G_4 <
\Aut(X_{k,kn})$ are not abstractly commensurable.
\end{theorem}
\begin{proof}
Collapsing $f_0$ and then performing $2l+k-1$ slide moves yields
\begin{align*}
G_4 \ &\cong \ \bigvee_l BS(n^2,n^2) \ \vee \
\bigvee_{k} BS(1,n^2) \\
&\cong \ BS(1,n^2) \ \vee \bigvee_{l+ k-1} BS(1,1).
\end{align*}
The depth profile is $\{ \, n^{2i} \mid i \in {\mathbb N} \cup \{0\} \, \}$,
which is inequivalent to the depth profile of $G_1$.
\end{proof}
\subsection{The remaining cases}\label{lastcase}
There are two remaining cases of $X_{k,kn}$ not covered by Theorems
\ref{mainthm}, \ref{g3thm}, and \ref{g4thm}. The first is when every
prime factor of $n$ is greater than $k$. The second is when $n$ is
prime, $n <k$, and $k \not\equiv 0,1 \mod n$ (for instance,
$X_{5,15}$). The constructions seen thus far all result in lattices
having equivalent depth profiles. Ad hoc arguments seem to indicate
that some of these examples (in the second case) are still
incommensurable, but a complete presentable proof is elusive so far.
\section{Cayley graphs}\label{cayleysec}
Given a finitely generated group $G$ let $S$ be a symmetric finite
generating set that does not contain $1$. The \emph{Cayley graph}
$\Cay(G,S)$ is a connected graph with edges labeled by elements of
$S$, defined as follows. The vertex set is $G$. For every $g\in G$ and
$s \in S$ there is a unique edge $e$ with $\partial_0(e) = g$ and
$\partial_1(e) = gs$. This edge is assigned the label $s$. Thus,
$\overline{e}$ has label $s^{-1}$.
We say that $G_1$ and $G_2$ \emph{admit isomorphic Cayley graphs} if
there exist generating sets $S_1, S_2$ as above such that $\Cay(G_1,S_1)$
and $\Cay(G_2,S_2)$ are isomorphic as unlabeled graphs. Note that this
is not a transitive relation (see \cite{hainkescheele}).
There are several interesting examples of groups that are quite
different from one another admitting isomorphic Cayley graphs. Most
involve torsion in an essential way, since finite groups of the same
cardinality always admit isomorphic Cayley graphs. If $A$ and $B$ are
two such finite groups, then ${\mathbb Z} \wr A$ and ${\mathbb Z} \wr B$ admit
isomorphic Cayley graphs, by \cite{dyubina}. (Such groups are never
finitely presented, however.) Similarly, if $G_A$ and $G_B$ are
extensions of $A$ and $B$ respectively by a group $Q$, then
$G_A$ and $G_B$ admit isomorphic Cayley graphs
(see \cite[proof of Corollary 1.13]{MMV} for instance).
If one seeks incommensurable \emph{torsion-free} groups with
isomorphic Cayley graphs, then some well known examples can be found
among lattices in products of locally finite trees. This phenomenon
was first observed and discussed by Wise in \cite{wisethesis}; the
lattices of Burger and Mozes \cite{burgermozes} also provide
examples. In \cite{dergachevaklyachko} Dergacheva and Klyachko
constructed a pair of incommensurable torison-free groups with
isomorphic Cayley graphs, via amalgams of Baumslag--Solitar
groups. Our lattices here provide new examples, which moreover are
also coherent. It is an open question whether there exist coherent
lattices in products of trees \cite[Problem 10.10]{wisecsc}.
If a group acts cocompactly on a connected CW complex $X$, freely and
transitively on the vertices, then the $1$--skeleton $X^{(1)}$ is a
Cayley graph for $G$. This is the situation for the torsion-free
examples just mentioned. Unfortunately, our lattices $G_i$ don't act
in this way on $X_{k,kn}$. Instead we have the following result.
\begin{proposition}\label{cayleyprop}
Suppose $G_1$ and $G_2$ act cocompactly on a connected graph $\Gamma$,
freely on the vertices, with a common vertex orbit. Then $G_1$ and
$G_2$ admit isomorphic Cayley graphs.
\end{proposition}
Having a vertex orbit in common is important. For instance, two groups
may act on the same graph, with the same number of vertex orbits, and
that is not enough. In the group $G = {\mathbb Z} \times {\mathbb Z}/2$ with any Cayley
graph $\Gamma$, the subgroups ${\mathbb Z} \times \{0\}$ and $2{\mathbb Z} \times {\mathbb Z}/2$
both act on $\Gamma$ with two vertex orbits, but they do not admit
isomorphic Cayley graphs, by \cite{hainkescheele} (see also
\cite{loh}).
\begin{proof}
The assumptions imply that $\Gamma$ is locally finite with bounded
valence. Also, there is a vertex $v_0 \in V(\Gamma)$ such that $G_1 v_0
= G_2 v_0$. Let $V_0$ denote this vertex orbit. Because the action of
$G_1$ is cocompact, there is a number $C$ such that every vertex of
$\Gamma$ has distance at most $C$ from $V_0$.
Now let $P$ be the set of paths in $\Gamma$ of the form $\alpha \cdot
e \cdot \beta$ where
\begin{itemize}
\item $e \in E(\Gamma)$
\item $\alpha$ is a shortest path in $\Gamma$ from $V_0$
to $\partial_0 e$
\item $\beta$ is a shortest path in $\Gamma$ from $\partial_1 e$ to
$V_0$.
\end{itemize}
We build a new graph $\Delta'$ as follows: $V(\Delta')$ is the set
$V_0$ and $E(\Delta')$ is $P$. That is, each $p\in P$ has initial and
terminal endpoints $\partial_0 p$ and $\partial_1 p$ in $V_0$, which
defines an edge in $\Delta'$. (Note that $P$ is closed under the
involution $p \mapsto \overline{p}$.) Finally, define $\Delta$ from
$\Delta'$ by eliminating all loops and duplicate edges, if any exist.
We claim that $\Delta$ is connected. Suppose $\gamma = (e_1, \dotsc,
e_n)$ is any path in $\Gamma$ with endpoints in $V_0$. For each $i$
let $\alpha_i$ be a shortest path from $V_0$ to $\partial_0 e_i$. Then
$(e_1 \cdot \overline{\alpha}_2) \cdot (\alpha_2 \cdot e_2 \cdot
\overline{\alpha}_3) \dotsm (\alpha_n \cdot e_n)$ is a concatenation
of paths in $P$ from $\partial_0 \gamma$ to $\partial_1 \gamma$.
It is immediate that both $G_1$ and $G_2$ act on $\Delta$, freely and
transitively on the vertices. Next let
\[S_i \ = \ \{\, g \in G_i \mid d_{\Delta}(v_0, gv_0) = 1\,\}\]
for $i = 1,2$. These sets are finite because $d_{\Delta}(v_0, gv_0) =
1$ implies $d_{\Gamma}(v_0, gv_0) \leq 2C+1$ and $\Gamma$ is locally
finite. Finally, $\Cay(G_i, S_i) \cong \Delta$ since every path in $P$
has a unique translate under the action of $G_i$ with initial vertex
$v_0$. That is, every edge of $\Delta $ has a unique $G_i$--translate
which is an edge from $v_0$ to $s v_0$ for some $s \in S_i$.
\end{proof}
\begin{corollary}\label{cayleycor}
The lattices $G_1, G_2, G_3, G_4 < \Aut(X_{k,kn})$ admit isomorphic Cayley
graphs.
\end{corollary}
\begin{proof}
Recall that the labeled graphs defining $G_i$ have two vertices, black
and white. Lifting this coloring to the common Bass--Serre tree
$T_{k,kn}$ we get a bipartite vertex coloring. Now lift this vertex
coloring to the vertices of $X_{k,kn}$ using $\pi$. Each branching
line has only vertices of one color, and every strip has opposite
vertex colors on its two sides. The key observation now is that the white
vertices and the black vertices are exactly the vertex orbits under
any of the group actions. Hence Proposition \ref{cayleyprop} applies,
with $\Gamma$ the $1$--skeleton of $X_{k,kn}$.
\end{proof}
\section{Questions}\label{questions}
The main question is the following:
\begin{question}
For which pairs $(m,n)$ does $\Aut(X_{m,n})$ contain incommensurable
lattices?
\end{question}
The main open cases are the two cases of $\Aut(X_{k,kn})$ from Section
\ref{lastcase} and the case when $\gcd(m,n) \not= 1$, $m \nmid n$, and
$n \nmid m$.
\begin{question}
What are the uniform lattices in $\Aut(X_{m,n})$ with torsion?
Are there uniform lattices that are not virtually torsion-free?
\end{question}
\iffalse
For any $(A,\lambda)$ there is a straightforward construction of a
lattice in $\Aut(X_{(A,\lambda)})$ with $2$--torsion, as a
graph of infinite dihedral groups. These examples contain torsion-free
subgroups of index $2$. If $\lambda$ is positive then this latter
statement is easy to see; every torsion element is detected by
the orientation character, so its kernel is torsion-free. It would be
much more interesting to find lattices with higher-order torsion.
\fi
\begin{question}
Does $\Aut(X_{m,n})$ contain non-uniform lattices?
\end{question}
\begin{question}
When do $\Aut(X_{m,n})$ and $\Aut(X_{m',n'})$ contain isomorphic
lattices?
\end{question}
We have seen that $\Aut(X_{k,k^2})$ and $\Aut(X_{k^2,k^4})$ contain
isomorphic lattices (namely, the finite-index subgroup from Figure
\ref{easycase}), even though their ``standard'' lattices $BS(k,k^2)$
and $BS(k^2,k^4)$ are not commensurable. This lack of rigidity for
lattices in $\Aut(X_{m,n})$ is intriguing.
Finally, all of the above can be studied for the more general groups
$\Aut(X_{(A,\lambda)})$.
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2,869,038,155,702 | arxiv | \section{Introduction}
\label{secintroduction}
Current traffic laws represent relatively stable driving regulations followed by the majority of drivers, which is essential for ensuring driving safety. However, monitoring a vehicle behaviour’s law compliance can provide substantial evidence for the traceability of traffic accidents. Due to the rapid development of autonomous vehicles (AVs), in the foreseeable future, there is going to be a period when AVs and human drivers will drive on the road together~\cite{nair2021sharing}. This requires AVs to follow the traffic laws strictly in the same way as human drivers follow them~\cite{cascetta2022autonomous,quante2021human}; otherwise, differences between humans and AVs' driving behaviours will lead to misunderstanding and distrust between humans and AVs, leading to chaos in the traffic flow and severely reducing driving safety. However, how to make AVs follow the laws has been a challenge because the systematic solutions to traffic law compliance definitions and decision-making have still been under slow progress~\cite{li2020deep,shetty2021safety,liu2022road}. As the first step of traffic law compliance driving, defining which behaviours follow the law has still not been realised because of the fuzziness of human-driver-oriented natural language traffic laws. Under current technical conditions, it has been challenging for AVs to understand natural language, particularly its part based on human knowledge in severe safety-related laws. Namely, the digital system of an AV can interact only with digital information that has the exact meanings, which brings the issue of how to express the current fuzzy natural language traffic laws accurately in a digital way.
In complex traffic scenarios, AVs not only need to avoid pedestrians, motor vehicles, and other traffic participants but also must follow the traffic law constraints, such as traffic signs, traffic markings, and the right-way law. However, the current natural language traffic laws cannot be directly transformed into executable automatic driving commands. The compliance understanding of fuzzy natural language in the research on traffic laws has varied from person to person. Thus, it is challenging to map the traffic law’s natural language sentences into static, fixed compliance logic judgement expressions. Therefore, constructing a standardised mathematical description language of traffic laws covering individual understanding differences and traffic law complex constraints has been crucial to breaking through the bottleneck of intelligent vehicle-compliant driving technology.
\begin{figure*}[thbp]
\centering
\includegraphics[width = 1\linewidth]{fig1.pdf}
\caption{Online traffic law violation monitoring of autonomous vehicle.This monitor can be deployed on the ego vehicle and monitoring the traffic law violation behaviors of the ego vehicle in both highway and intersection scenarios. When deployed on every vehicle in the dataset, the statistical chart of monitoring results are shown as the two figure at the left lower corner.}
\label{violation monitor}
\end{figure*}
\cite{buchanan1970some} conducted the systematical law standardisation research for the first time using expert systems and a rule-based standardised approach. Sergot et al. and Bench Capon et al. published their research findings in \emph{British Nationality Act} and \emph{Supplementary Benefit Act}, and their works represent two critical milestones in the standardised description of regulations using the first-order logic statements to describe the traffic rules~\cite{1986The,1987Logic}. However, their research has been limited in expressing the original concepts. In addition to the first-order logic, there has been a deontic logic-based method for standardising the rules, which can explicitly express the notions of permission and obligation~\cite{1983Permissions,governatori2018practical}. ~\cite{bhuiyan2019methodology, villata2020traffic} proposed a defeasible deontic logic to handle rule exceptions and resolve conflicts in rule norms. For road traffic regulations, ~\cite{Royakkers0Extending} extended the obligation logic to solve the problem of speed limit conflict in Dutch traffic regulations. In addition to these methods, most studies have identified temporal logic as a suitable specification formalism. Linear temporal logic (LTL) \cite{pnueli1977temporal} is based on propositional logic and temporal operators, which are used for specifying motions and formalising traffic laws~\cite{esterle2020formalizing,esterle2019specifications}. Further, metric temporal logic (MTL) represents an extension of propositional linear temporal logic with discrete-time-bounded temporal operators~\cite{thati2005monitoring}. Most commonly-occurring real-time properties, such as invariance and bounded response, can be expressed in MTL fragments~\cite{ouaknine2008some}. The MTL has been used for traffic laws monitoring because it can specify an interval over which a particular property must be fulfilled~\cite{maierhofer2020formalization}. Unlike LTL and MTL that provide Boolean results, the result of signal temporal logic (STL)~\cite{maler2004monitoring} with quantitative semantics \cite{fainekos2009robustness} represents a degree of satisfaction or violation of a property \cite{hekmatnejad2019encoding, arechiga2019specifying,fremont2022scenic}.
Based on the above-mentioned research on the logic expressions of traffic laws, ~\cite{rizaldi2015formalising,rizaldi2016formally,rizaldi2017formalising} decomposed the traffic laws into a set of multiple atomic propositions and employed LTL to express the logic combinations using atomic propositions, which based on the Isabelle theorem could achieve a standardised description of traffic laws. The effectiveness of the proposed overtaking and safe distance monitoring was verified on a dataset. However, this method can perform only offline monitoring based on data recorded by a vehicle but cannot be applied to the actual vehicle driving process. To achieve real-time online monitoring, ~\cite{esterle2020formalizing} used LTL to describe traffic laws and transformed these laws into a deterministic finite automaton to monitor the compliance of vehicles' behaviours and combined them with datasets to eliminate misjudgements iteratively. However, this method is applicable only to laws of interactive behaviours of road participants and does not consider the rules for the representation of static participants, such as traffic signs and markings. ~\cite{9488998} proposed an online monitoring method but from a third-party view. They used machine RCNN to distinguish moving objects first and then estimated their parameters; next, binary thresholds were used to judge whether the object broke the laws. This method is faster and more efficient than humans but can be applied only to detecting traffic marking law violations and can be used only in roadside equipment. In \cite{wang2013video,csentacs2019real,bachtiar2020vehicle}, roadside videos were used to perform real-time online monitoring of simple law violation behaviours. This type of method can be used for government supervision and as a reference for developing law-compliance decision-making systems, but it has fewer direct contributions to autonomous driving. In terms of vehicle side monitoring and simultaneous monitoring of static participant laws and interactive behaviours, ~\cite{esterle2019specifications} defined vehicle behaviour through discrete spatial sequence combinations, extended monitoring to decision trajectories, and judged the compliance of the action sequence of each trajectory. However, this method requires road space division based on other vehicles and environmental information, which is time-consuming in complex scenes and can reduce accuracy. Further, ~\cite{ozkul2018police} proposed a vehicle side but not Ego vehicle online monitor system, which uses communication with nearby vehicles to monitor other vehicles' law-violation behaviours and reports the detected behaviours to the transportation authority. This system preserves privacy and has no false positives, but the original study paid more attention to introducing the proposed framework while fewer law violation monitor algorithms were discussed. For right-of-way monitoring, ~\cite{karimi2020formalizing} set the corresponding laws for each scene to determine whether a vehicle needed to give way in the scene. However, the judgement was very conservative and required other vehicles to use the turn signals correctly.
The above-mentioned research has provided a useful theoretical basis for the digitization of traffic laws and law violation monitoring of the high-level AVs, but a more standardized, systematic mathematical description of traffic laws and a more comprehensive violation monitoring incorporating different types of law constraints are still serious challenges. It can be found that current standardized or formalized descriptions of traffic laws are only in the stage of trying different logical expressions to digitize some law articles. However, there is no complete scheme for online traffic law violation monitoring of ego vehicle behavior at the vehicle end. This paper proposed a systematic method to monitor the self traffic law violation behaviors from the ego vehicle side as shown in Fig.\ref{violation monitor}. With this method applying to each car in highway and intersection dataset, all the violation behaviors can be found and counted as the two figures at the lower left quarter of Fig.\ref{violation monitor}.
The contributions of this paper include the following.
1) An ego vehicle side self traffic law violation online monitor system that only use perception and decision-making information, whose result can be used for better decision-making. A traffic law constraints classifications and online monitor-oriented trigger domain-based layered architecture for traffic law digitization.
2) The detailed computer-friendly atomic proposition list involved traffic law digitization and the example MTL expressions for primary highway and intersection laws.
3) The essential thresholds selection method and the dataset verification method with its results of the example online monitor.
\section{Traffic Regulations and Constraints}
\subsection{Chinese traffic laws}
The analysis presented in this paper is based on \textit{Road traffic Safety law of the People's Republic of China}, which is a law that was defined and adopted by the Standing Committee of the National People's Congress and issued by the president in the form of a presidential order. However, the traffic law has been too abstract and general for specific actions to maintain road traffic order, improve traffic efficiency and protect property safety. There, there has been an implementing regulation called \textit{The Regulation on the Implementation of the Law of the People's Republic of China on Road Traffic Safety}, which makes the traffic law much more specific. In this implementation regulation, concrete actions and restrictions are given in eight chapters and 115 articles, including road traffic conditions, traffic regulations, and traffic accident handling. Although the regulation belongs to administrative regulation and has lower potency than traffic law, this implementing regulation still plays a vital role in daily traffic accident liability judgement. This paper is primarily based on chapter 4: Provisions on Road Passage in this regulation. Moreover, to contain severe violations and reduce road traffic accidents, the Chinese traffic law stipulated a scoring system in 2004. The traffic violation score was adjusted according to the impact on road traffic safety and smoothness in 2021. The \textit{Administrative Measures for Scoring Violations of Road Traffic Safety} went into effect on April 1, 2022, which will surely be an essential reference for the law and safety balance of the AV decision-making system.
It should be noted that different countries have different traffic conditions. However, Chinese traffic laws have various driving behaviour restrictions from other countries, particularly in terms of highway regulations and right-of-way regulations. In China, the highest highway driving speed is 120 km/h if signs or lanes do not specify it. In contrast, there has been no upper-speed limit on highways in Germany. In England, the upper-speed limit is slightly lower than that in China, which is 70 mph (112.6 km/h). Meanwhile, in Japan, the speed is limited to as slow as 100 km/h. When driving on a Chinese highway, the required distance between vehicles is 100 m when going faster than 100 km/h and 50 m when driving slower than 100 km/h. This differs from regulations used in other countries, including Germany, Japan and America, where the distance is defined by the time interval between adjacent vehicles. Moreover, under particular weather conditions or other conditions’ limitations, China's upper speed is up to 30 km/h. In comparison, this speed is 50 km/h in Germany, whereas there has been no specified restriction in England, Japan and America. For overtaking, all countries specify that vehicles should stop overtaking if it is possible to conflict with a car coming from the opposite direction or when the vehicle to be overtaken is overtaking. All countries have defined numerous restrictions to make the overtaking progress safer. However, Chinese restriction laws are more specific and detailed compared to those used in other countries; for instance, the laws prevent vehicles from overtaking when the front vehicle is doing particular actions or is a special vehicle for emergency tasks or during given road situations. As for the right of way, American law requires vehicles to stop 3 s before stop signs or stop lines, while in China, there has been no such law, vehicles stop according to traffic lights. Like in all other countries, In China, pedestrians have the highest priority on the sidewalks, but at crossroads without traffic lights, a vehicle that goes straight has the highest priority, while vehicles turning left or right have the lowest priority. However, in America and Germany, vehicles that turn right have higher priority than those that turn left.
Except for the above-mentioned laws, Chinese traffic laws have more detailed restrictions than other countries when facing different road conditions, weather conditions, traffic signs, and lines and situations. This will be discussed in detail in the next section.
\subsection{Traffic law constraints and their classification}
Although traffic laws differ among countries, most of them restrict driving behaviour from four aspects: vehicle speed, distance between vehicles, driving actions, and right of way. According to \textit{The Regulation on the Implementation of the Law of the People's Republic of China on Road Traffic Safety}, Chapter 4: Road traffic regulations, there are 49 traffic regulations, but only 25 articles are related to motor vehicle driving behaviours. Due to the overlapping, there are 11 articles restricting the vehicle speed; three articles restrict the distance between vehicles, 12 articles regulate driving behaviour and four articles concern right of way. A detailed classification is shown in Fig.\ref{classification}.
\begin{figure}[htbp]
\centering
\includegraphics[width = 0.8\linewidth]{fig2.pdf}
\caption{The classifications of Chinese traffic law restrictions. Each circle means a different class of traffic law constraint. The size of the circle represents the proportion of this class, and the overlap means both constraint classes appear in the same article. The solid edge linetype of the circle meanings there is no ambiguous expression involved, while dash dot line means in certain articles fuzzy parameter is involved and the dot line means the corresponding expressions in each article need to clarify.}
\label{classification}
\end{figure}
Among all vehicle speed restrictions, the traffic law usually restricts driving behaviour through three types of descriptions: 1) defining upper or lower speed limits; 2) giving the driving behaviour suggestions; 3) giving driving state suggestions. As for the first type, a law restrains the vehicle speed in a certain range during a given scenario. The second type of description stipulates drivers slow down when performing certain actions or meeting certain situations without defining a clear speed reduction claim. The third type of description only suggests drivers driving at a low speed in specific situations but does not define the speed. In distance-related restrictions, the distance between vehicles and other traffic participants is usually constrained by descriptions such as: 1) maintaining a distance of at least a specific distance in certain scenes; however, in most cases, there will be no given distance value; 2) the law only gives the recommendation to maintain a safe distance. As for the restriction actions, eight types of actions are regulated. The law specifies whether a vehicle shall pass, not hinder, overtake, change lanes, give way, turn around, reverse or stop under certain working conditions. Among all eight types of restrictions, only the “not hinder” is non-direct driving action and without any further definitions. Thus, how to avoid hindering other traffic participants is different from person to person. The “stop” and “turn around” actions clearly define what to do in certain situations or when facing traffic signs or lines. Other restrictions are defined clearly only in certain situations when a vehicle is required to do or not to do a particular action, but not all restrictions are clearly defined since they are sometimes combined with other fuzzy sentences like "passing slowly". As for the restriction regarding the right of way, in a specific situation, it is stipulated when a vehicle has the right of way priority: for instance, the opposite-driving vehicles go first, straight-forward or right-forward vehicles go first, left-turning vehicles go first, or driving in proper order.
Therefore, the Chinese traffic law and its regulations restrict vehicles’ driving behaviours under certain situations by direct or fuzzy natural language-based descriptions. Due to the fuzzy part of the descriptions, certain driving actions vary with drivers’ experiences, ages, risks and tolerances. However, the technical level of automatic driving can be considered the same at any development stage, and it is hard for machines to understand so many regulations, especially fuzzy ones. Therefore, it is necessary to digitize traffic laws using the same standard for all AVs based on the exact and executable logic descriptions.
\section{Traffic Law digitization}
\subsection{Different purposes of law digitization}
\begin{table*}[!t]
\footnotesize
\caption{The comparisons of different law codification purposes\label{table1}}
\centering
\begin{tabular}{p{2.2cm} p{3cm}<{\raggedright} p{3.4cm}<{\raggedright} p{3.4cm}<{\raggedright} p{3.4cm}<{\raggedright}}
\toprule
\multirow{3}{2.4cm}{\textbf{Comparison}} & \multirow{3}{3cm}{\textbf{Offline monitoring}} & \multicolumn{3}{c}{\textbf{Online monitoring}}\\
\cmidrule{3-5}
& & Fact-based law violation monitoring & Decision-based law violation monitoring & Prediction-based law violation monitoring\\
\midrule
\multirow{8}{2.4cm}{\textbf{Application}} & {1). Used for offline evaluation of AV tasks of a vehicle; 2).Used in the roadside equipment to provide third-party monitoring of vehicle behaviours} & {1). Used at an AV for online monitor of AV's behaviours and division of accident responsibility; 2). Used in the roadside equipment for online third-party monitor of vehicle behaviour violations} & Used in a vehicle for online management of autonomous driving behaviours and online interference response for illegal decision- making & Used in a vehicle for online management of autonomous driving behaviours and online interference response for possible violations\\
\midrule
\multirow{8}{2.4cm}{\textbf{Information}} & \textbf{Ego vehicle}: all time periods of data & \textbf{Ego vehicle}: current and past periods of data & \textbf{Ego vehicle}: current and past periods of data and division- making data & \textbf{Ego vehicle}: current and past periods of data and division- making data \\
& \textbf{Other vehicle}: all time periods of data & \textbf{Other vehicle}: current and past periods of collected data& \textbf{Other vehicle}: current and past periods of collected data& \textbf{Other vehicle}: current and past periods of collected data and the corresponding future prediction data\\
\midrule
{\textbf{Overall judgment difficulty}} & \multirow{2}{3cm}{Easy} & \multirow{2}{3.4cm}{Hard} & \multirow{2}{3.4cm}{Ordinary} & \multirow{2}{3.4cm}{Ordinary}\\
\midrule
\multirow{3}{2.4cm}{\textbf{Results}} & \multirow{3}{3cm}{Unchangeable} & \multirow{3}{3.4cm}{Unchangeable} & \multirow{3}{3.4cm}{Varying with decisions} & Varying with decisions, control outputs and others’ behaviours\\
\midrule
{\textbf{Decision friendly}} & None & Bad & Good & Best\\
\bottomrule
\end{tabular}
\end{table*}
When digitizing the traffic laws, not only the fuzzy parts of the laws will result in different digital logic judgements and progress, but also the digitization's purpose will greatly influence the digitization progress and results. The purposes of traffic law digitization can be roughly divided into two main categories, offline and online law violation monitoring types. In online monitoring, three types of purposes have been defined: 1) reality-based law violation monitoring; 2) decision-based law violation monitoring; 3) prediction-based violation monitoring. Offline monitoring is a law violation monitoring that judges the driving behaviour law-compliance of one or more vehicles in the whole scenario by obtaining the whole-period vehicle behaviour information. In contrast, online monitoring is a law violation monitoring that judges the driving behaviour law-compliance of the ego vehicle or all vehicles from the start of the monitoring process to the current time using the observed or collected data. It should be noted that different types of online monitoring show certain differences. The monitoring types are compared in detail in Table~\ref{table1}. In Table~\ref{table1}, online monitoring is divided into three types according to the information used for monitoring. The fact-based monitoring uses historical and current vehicle behaviour data. In this type of monitoring, once a law violation is found, the violation result is an established fact that cannot be changed. Therefore, this type of monitoring can be used on both the vehicle side and the road equipment side for recording violation behaviours of vehicles, and the result can be used as a reference for accident responsibility division. Except for historical and current vehicle behaviour data, when the ego vehicle’s decision data are involved, online monitoring is given the ability to foresee the future actions of the ego vehicle. This type of monitoring is regarded as decision-based law violation monitoring, and it can be used only for ego vehicles. However, this monitoring type can tell whether the ego vehicle will break the traffic law if following the current decision. Also, the decision-making system can read the monitor’s output to adjust its decision to comply with the traffic law. Therefore, the monitoring result changes with the decision of the ego vehicle. Furthermore, the traffic law restrains the relationships between traffic participants. Thus, if it is required that the monitoring result has the best law compliance guiding significance, the prediction behaviours of other participants should also be considered, and this represents the prediction-based law violation monitoring. This monitoring type combines historical and current vehicle behaviour data with the ego vehicle’s decision and perception data to judge the law compliance of the decision in the prediction range. However, the monitoring result is unstable and varies with the decisions, other participants’ behaviours and their predictions. However, this monitoring type is the most decision-friendly monitoring, and its result gives the best advance quantity to adjust the decision.
The used information, difficulty and resulting stability for each monitoring type when monitoring the four types of laws are presented in Table~\ref{table1}, where difficulties are evaluated based on the authors' best knowledge. Speed and distance violations are easy to distinguish as their values can be obtained easily from the current data. The only difficult point in distance monitoring is finding vehicles in the same lane ahead. However, the behaviour and right-of-way monitoring types are difficult to perform. If the monitored vehicle is a white box for offline monitoring or only in certain given scenarios, it will be easy to select law monitoring algorithms to determine behaviour violations for vehicle decisions or scenario types. Still, when facing a black box vehicle and in a free-run situation, performing offline monitoring is relatively challenging because it is necessary to estimate a vehicle' next action on the whole trajectory and which law is convenient for a particular case. Therefore, without the whole-trajectory data and using only past and current data that ego vehicle collected, it is difficult to monitor behaviour violations for the fact-based monitor because it is needed to set more judgement conditions to determine which law is suitable for the current scenario. Furthermore, right-of-way monitoring is even more challenging because other traffic participants are involved. These participants’ behaviours can lead to a situation where much more judgement conditions need to be discussed, and more thresholds should be considered. By using the whole-trajectory data of all participants or the ego vehicle’s decision and prediction data, the monitor task will become easier to perform because the future data can reduce the condition classification discussions. This is the main reason the fact-based monitor is the most challenging to achieve.
\begin{figure*}[htbp]
\centering
\includegraphics[width = 0.8\linewidth]{fig3.pdf}
\caption{The architecture of the proposed trigger domain-based layered traffic law digitization method}
\label{architecture}
\end{figure*}
It should be noted that an online monitor, which is installed on a vehicle, is more meaningful than that installed on the roadside equipment as the roadside equipment cannot cover all the road net. Moreover, in addition to the onboard law violation records, the ability to distinguish law violation behaviours can provide suggestions for every decision an autonomous system makes. Therefore, according to the aforementioned analyses and reasons, this study selects a decision-based online law violation monitoring as a research objective, which is a balance between the difficulty and result meanings of the decision-making system.
\subsection{Trigger domain-based layered architecture of traffic law digitization}
Although traffic laws usually define different constraints under certain conditions, much research on law digitization and monitoring has been conducted for improving the offline methods. In the related research, conditions have usually been simplified for digitized laws because the scenario classifications or certain behaviour monitoring groups imply the conditions that the digitized violation judgement part can confront with. Different from offline monitoring, in online monitoring, scenarios appear randomly. Thus, it is necessary to classify the current situation, select the right law article, and monitor a vehicle’s behaviour. Therefore, condition classification plays an essential role in online monitoring. In view of that, this paper proposes a trigger domain-based layered architecture for law digitization. The proposed architecture is described in the following.
As shown in Fig.~\ref{architecture}, an online decision-based law violation monitor is deployed on the ego vehicle. The monitor reads data obtained from the perception system and decision-making system and outputs the results to the decision-making system to help it make compliance decisions. Since some law articles in traffic laws stipulate a series of vehicle actions, those law articles can be broken down into simple ones. That is why the laws can be grouped up and digitized into layers. The upper the layer is, the more the lower layer’s articles parts will be involved. The proposed architecture can avoid monitoring the same action in different places at the same time, as the upper layer’s law articles can borrow the lower layer’s digital expressions; the lowest article code will run only once during the entire monitoring process. Each article’s digital expression constitutes a domain, where the trigger is the entrance into the domain. It should be noted that only when a certain trigger is satisfied, the data flow can enter the domain, and then the violation judgements will start. These triggers play the roles of condition classifications and are logical results of a series of logical judgements, which run from the start of the system operation and can activate the right law article to monitor under the right conditions. Because the monitor is an online monitor, only current and past data can be obtained. The violation judgements inside the domain can be roughly divided into two types: the current state judgements and the continuous state judgements. The current state judgements can directly determine whether a vehicle’s behaviour is violating a certain law or not; meanwhile, the continuous state judgement determines whether a whole process of vehicle behaviour or a cumulative quantity is satisfied. Using the proposed architecture, the traffic law can be digitized logically, and the correctness of the monitoring progress can be ensured.
\subsection{Atomic proposition for application}
In most studies on traffic laws digitization and monitoring, temporal logic, including metric temporal logic (MTL) and linear temporal logic (LTL), has been widely used. As a monitor contains continuous state judgement, the MTL is used in this study to interpret the definition of the trigger domain and logic proposition. The MTL introduces the more abundant temporal operators into the Boolean operators, using their good performance to describe the relationship between the behavioural logic within a limited time and spatial coordinates.
The research on using MTL to digitize the laws usually uses only the MTL expressions, which can show the logical relationships of each atomic proposition clearly. The logic relationships are essential, and researchers have decomposed and recombined the law’s judgements with atomic propositions. However, using only logical relationships cannot provide the expression’s results. The atomic propositions between logic operation symbols still have a non-negligible contribution to the MTL expressions. Implementing the MTL expressions into executable programs requires using both logic relationships and calculable atomic propositions. However, most research has rarely decomposed laws into the smallest calculable atomic propositions. This makes the MTL expressions look like the natural language laws but in a logical combination way. To avoid this problem and to improve the practicability of the traffic law monitor, this study splits each law article into fundamental atomic propositions, as shown in Fig.~\ref{overallthinking}.
\begin{figure*}[htbp]
\centering
\includegraphics[width = 0.8\linewidth]{fig4.pdf}
\caption{The overall process of traffic law digitization}
\label{overallthinking}
\end{figure*}
According to the proposed architecture, each law article is composed of triggers, judgements, and thresholds when digitized. Every formulation in traffic law is expressed using calculable formulas or logical comparisons, and the logical comparisons in triggers are defined. Then, for each comparison, a suitable threshold is set. Using the MTL, the law article is first transferred into MTL expressions, and then, using all the
definitions, the atomic propositions in MTL are further converted into triggers and logical judgements with thresholds. When digitizing traffic laws, it is necessary to express each law article’s trigger domain ranges using the MTL expressions and clarify the atomic propositions and their logical relationships used in the digitization process. Therefore, in a traffic scenario, the more scenes a specific law article involves, the more parameters or fuzzy parameters need to be determined based on the actual data in a digitized atomic proposition, the more complex the logical relationship between atomic propositions is, and the more difficult it is to digitize law articles. Based on different types of restrictions in regulations, this study subdivides 25 driving-related articles in \textit{The Regulation on the Implementation of the Law of the People's Republic of China on Road Traffic Safety} into more than 90 detailed clauses. The digitization difficulty of each clause is measured according to the scene fitness, the atomic proposition complexity, and the number of fuzzy thresholds. Among them, scene fitness is the scene level that describes how narrow a scene is when a particular clause takes effect. The scene fitness is divided into four levels denoted by I--IV. The higher the level is, the more restrictive conditions of the scene will be, and the more difficult it will be to determine the trigger of the scene, which will increase the difficulty in digitization at the triggering condition level. The atomic proposition complexity is the number of atomic propositions required for the logical judgement of clauses after the triggering scene is determined. The more atomic propositions are required, the more complex the logical judgement will be. At the same time, the more ambiguous the statutes describe the behaviour, the more complex the logical judgement is required. As shown in Fig.~\ref{complexity}, all the driving relevant traffic law articles are evaluated, and the number in each circle represents the entry of the article, which can be found in our website in detail. The atomic propositions with a complexity of five or higher are basically motion and right-of-way restrictions because the speed and other clauses limitations, such as light usage, are indicated by a scene and thresholds. Furthermore, they do not involve interaction with other traffic participants. The higher the number of interactions involved in a clause is, the higher the atomic proposition complexity will be. Action and right-of-way restrictions mainly need to consider the interactions with other traffic participants and include behaviours that a single atomic proposition cannot simply express, such as “keep the necessary safety distance”, and “slow down and drive to the right”. Fuzzy thresholds are necessary for logical judgement in the digitization process, but these thresholds are not clearly defined by regulations.
Therefore, how to define the fuzzy thresholds has been a challenge.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\linewidth]{fig5.pdf}
\caption{Digital complexity of regulations}
\label{complexity}
\end{figure}
Therefore, it is necessary to analyse reasonable thresholds from a large amount of data and explore their rationality. Thus, the number of fuzzy thresholds, the scence fitness and the atomic proposition complexity can affect digitization complexity to a different extent. As shown in Fig.~\ref{complexity}, the larger the index value is, the more complex the regulation’s digitization is from each aspect.
This paper takes the principal laws of highways and intersections as examples. All the logical definitions that the examples involved are listed in APPENDIX \ref{appendixc}, which are all calculable with given information at our best known.
\subsection{Highway and intersection traffic law digitization examples}
\subsubsection{Highway traffic law violation monitor}
This study selects the five most important and common traffic law articles related to the driving process on highways under non-adverse weather conditions from \textit{The Regulation on the Implementation of the Law of the People's Republic of China on Road Traffic Safety}. These five articles define the speed, distance and behaviour restrictions, and the original texts are as follows:
\noindent 1) Article-44 The motor vehicles changing lanes shall not affect the motor vehicles driving normally along the corresponding lanes.
\noindent 2) Article-47 After confirming there is a sufficient safe distance, the latter shall overtake the vehicle mentioned first from its left side, and after there is a second necessary safe distance between them, the overtaking vehicle shall turn on right turn light and return to the original lane.
\noindent 3) Article-78 Driving speeds for different vehicle lanes of an expressway shall be indicated, the maximum speed shall not exceed 120 kilometers per hour while the minimum speed shall not be lower than 60 kilometers per hour. Where there are two vehicle lanes in the same direction, the minimum speed for the left lane is 100 kilometers per hour; and where there are three vehicle lanes or more in the same direction, the minimum speed for the farthest left lane is 110 kilometers per hour, and 90 kilometers per hour for the middle lane. Where there is any discrepancy between the speed indicated by a speed limit sign put up on a road and the driving speeds mentioned above, a motor vehicle shall be driven at the speed indicated by the speed limit sign on the road.
\noindent 4) Article-80 Where a motor vehicle is running on expressway at a speed which exceeds 100 kilometers per hour, a distance of 100 meters or more shall be maintained from the vehicle in front in the same vehicle lane; and when the speed is lower than 100 kilometers per hour, the distance from the vehicle in front may be narrowed appropriately, but the minimum distance may not be less than 50 meters.
\noindent 5) Article-82.3 When driving a motor vehicle on expressway, the driver shall not drive over or on the dividing line of vehicle lanes or on the shoulder.
\begin{table}[htbp]
\footnotesize
\centering
\caption{{\normalsize Trigger domain and logic judgements of typical Articles of highway}}
\renewcommand{\arraystretch}{1.0}
\label{table2}
\begin{tabular}{p{1.2cm} p{2cm} p{4.2cm}}
\toprule
Article & Trigger domain & Logical requirement \\
\midrule
\multirow{1}{2cm} {Article44} & $\textit{T}_{\emph{ChangeLeftlane}}$ & \textit{ChangeLeftLaneCompliance} = $\bm T$\\
& & $\Leftrightarrow$ \textit{\quad FrontViolation}=$\bm F$ \\
& & \textit{$\qquad$ $\wedge$ RearLeftViolation}=$\bm F$ \\
& & $\qquad$ $\wedge$ \textit{FrontLeftViolation} = $\bm F$\\
& & $\qquad$ $\wedge$ {LngTmOnLine} = $\bm F$\\
& $\textit{T}_{ChangeRightlane}$ &\textit{ChangeRightLaneCompliance} = $\bm T$\\
& & $\Leftrightarrow$ \textit{\quad FrontViolation}=$\bm F$ \\
& & \textit{\qquad $\wedge$ RearRightViolation}=$\bm F$ \\
& & $\qquad \wedge$ \textit{FrontRightViolation} = $\bm F$\\
& & $\qquad \wedge$ {LngTmOnLine} = $\bm F$\\
\multirow{1}{2cm} {Article47} & $\textit{T}_{Overtake1}$ & \textit{FrontnotOvertake} = $\bm T$\\
& & $\wedge$ \textit{ChangeLeftLaneCompliance} = $\bm T$\\
& $\textit{T}_{Overtake2}$ & \textit{RecommendedSpeed} = $\bm T$\\
& $\textit{T}_{Overtake3}$ & \textit{ChangeRightLaneCompliance} = $\bm T$ \\
& $\textit{T}_{OvertakeR}$ & \textit{OvertakeonRight} = $\bm F$\\
{Article78} & $\textit{T}_{SpeedLimit}$ & \textit{SpeedCompliance=} $\bm T$\\
Article80 & $\textit{T}_{\emph{KeepFollowingDistance}}$ & \textit{FollowingCompliance}=$\bm T$\\
{Article82.3} & $\textit{T}_{DriveonLaneline}$ & {LngTmOnLine}= $\bm F$ \\
\bottomrule
\end{tabular}
\end{table}
Article 82.3 ensures normal traffic flow, suggesting that vehicles shall not drive on the lane line for a long time. Article 44 is the law related to lane-changing behaviour. Namely, when a vehicle changes driving lanes, the vehicle will drive on the line for a certain period. Therefore, lane-changing monitoring must include the monitoring of article 82.3 to judge whether a vehicle changes lanes too slowly. Article 47 is the law defining overtaking behaviour. A complete overtaking behaviour can be divided into three stages: the lane-changing stage, the target vehicle passing stage and the original lane-returning stage. The first and third stages include lane-changing behaviour, so they also include monitoring Article~44. The traffic law monitor can be layered and modular based on those relationships above. The upper-layer modules can call the lower-layer modules to reduce the digitization work and computational power consumption.
The traffic law violation monitoring of each law article represents the combination and invocation of related modules, and each module includes the trigger domain and logical judgements. Table~\ref{table2} presents the trigger domain and logical judgements of laws considered in this study.
In order to make a better logical judgement, this study uses the combination of the global coordinate system and the ego vehicle coordinate system to facilitate the usage of information obtained from the HD map and a vehicle sensor. The lane to which \textit{obj} belongs is defined as Lane$({obj})$. The innermost lane of the road in the same direction is defined as Lane 1, The IDs of other lanes and lane lines are increasing outward. The lane line with ID $i$ is expressed as LanLine$(i)$, which is represented by a cubic fitting curve. In this study, the area around the ego vehicle is divided into six areas: front left area, rear left area, front area, rear area, front right area and rear right area. For vehicles in each area, only the vehicle nearest to the ego vehicle is considered. Use subscripts to express the position relationship of \textit{Tgt} relative to \textit{Ego}, for instance, $\emph{Tgt}_{f}$ represent \textit{Tgt} in the front area of \textit{Ego}. The subscripts of the other five areas are similar.
It should be noted that Article 78 should always be monitored when driving on the main way of the highway. Thus, the triggering condition that the ego vehicle is located on the main way of the highway is expressed as follows:
\begin{equation}
\begin{aligned}
T_{Speed Limit}\Leftrightarrow \text{RoadType}(Ego)=highway_{main}\\
\end{aligned}
\end{equation}
where RoadType is the road type code (see APPENDIX \ref{appendixc}), and highwaymain represents the main way of the highway.
The vehicle’s speed must follow traffic sign regulations, and if there are no traffic signs, the vehicle's speed should follow the requirement specified for a particular lane. When the triggering condition is satisfied, the speed limit monitoring module starts monitoring.
\begin{equation}
\begin{aligned}
&\emph{SpeedCompliance} \Leftrightarrow\\
& \left(\text{SpdSignArea}(Ego)\wedge vx(Ego)\in \left[V_{sign\_min}, V_{sign\_max}\right]\right)\\
&\vee \left(\begin{array}{l}
\neg \text{SpdSignArea}(Ego) \wedge vx(Ego)\in[60,120]\text{km/h}\\
\wedge \left(\begin{array}{l}
\left(\begin{array}
{l}N_{mw}\geq3 \wedge \text{Lane}(Ego) \in \left[2,N_{mw}\right)\\
\wedge vx(Ego)\geq90\text{km/h}
\end{array}\right)\\
\vee \left(\begin{array}{l}
N_{mw}\geq 3 \wedge \text{Lane}(Ego)=1 \\
\wedge vx(Ego)\geq110\text{km/h}
\end{array}\right)\\
\vee \left(\begin{array}{l}
N_{mw}=2 \wedge \text{Lane}(Ego)=1 \\
\wedge vx(Ego)\geq100\text{km/h}
\end{array}\right)\\
\end{array}\right)
\end{array}\right)
\end{aligned}
\end{equation}
where $V_{sign\_max}$ and $V_{sign\_min}$ represent the upper and lower limits indicated by the speed limit sign, respectively; $N_{mw}$ is the number of main ways in the same direction.
For Article 80, the triggering condition is similar to that of Article 78, but there should be the in-front vehicle:
\begin{equation}
\begin{aligned}
T_{\emph{KeepFollowingDistance}}\Leftrightarrow &(\text{RoadType}(Ego)=highway_{main})\\
&\wedge \exists \emph{Tgt}_{f}
\end{aligned}
\end{equation}
When the triggering condition is satisfied, the following distance monitoring module starts monitoring. The corresponding mathematical expression is as follows:
\begin{equation}
\begin{aligned}
&\emph{FollowingCompliance} \Leftrightarrow \\
&(vx(Ego)>100\text{km/h} \wedge \textit{distance}(Ego,\emph{Tgt}_{f})>100\text{m})\\
&\vee (vx(Ego)\leq 100\text{km/h} \wedge \textit{distance}(\emph{Ego},\emph{Tgt}_{f})>50\text{m})
\end{aligned}
\end{equation}
For Article 82.3, the triggering condition is that the ego vehicle overlaps with the lane line, which can be expressed as follows:
\begin{equation}
\begin{aligned}
&T_{DriveonLaneline}\Leftrightarrow \\
& \text{overlap}({Area}(Ego), \text{LaneLine}(\text{Lane}({Ego})))\\
& \vee \text{overlap}({Area}(Ego), \text{LaneLine}(\text{Lane}({Ego})+1))\\
\end{aligned}
\end{equation}
When the triggering condition is satisfied, the driving-on-lane-line monitoring module starts monitoring. The moment when the triggering condition is satisfied is denoted by $t_{in}$, and the moment when the triggering condition is not satisfied is denoted by $t_{out}$. “Vehicle drives on the lane line for a long time”, is defined as follows:
\begin{equation}
\begin{aligned}
&\text{LngTmOnLine}(Ego) \\
&\Leftrightarrow (t-t_{in }>t_{max\_cl}) \wedge (t_{out}=\varnothing)
\end{aligned}
\end{equation}
where $t_{max\_cl}$ is the maximum allowable time for driving on the lane line.
As described in Section 3.1, this study uses the decision-based online law violation monitor. For Article 44, the process of lane changing is divided into changing lanes to the left and changing lanes to the right. For instance, for changing lanes to the left, the triggering condition is that the ego vehicle makes a decision to change lanes to the left and overlaps with the left lane line. During this task, the ego vehicle shall keep a compliant distance from relevant vehicles. The relevant vehicles include the front vehicle, front left vehicle and left rear vehicle. As mentioned above, the lane-changing process includes a certain period when a vehicle is driving on the lane line. Therefore, the lane-changing monitoring module includes the monitoring module of driving on a lane line. The corresponding mathematical expression is as follows:
\begin{equation}
\begin{aligned}
&\textit{T}_{\emph{ChangeLeftlane}} \Leftrightarrow \\
& \text{Decision}({Ego}) = \emph{ChangeLeftlane}\\
& \wedge \text{overlap}({Area}(Ego), \text{LaneLine}(\text{Lane}({Ego})))\\
& \wedge vy(Ego)>0 \\
\end{aligned}
\end{equation}
When the triggering condition is satisfied, the changing-left-lane monitoring module starts monitoring.
As defined by Article 44 and presented in Table~\ref{table2},
longtime\_on\_laneline belongs to the driving-on-lane-line monitoring module, and it will be called directly when the changing-left-lane monitoring module starts to monitor. The specific definitions of the other three sub-propositions are as follows:
\begin{equation}
\begin{aligned}
&\emph{FrontViolation} \Leftrightarrow \\
& \exists \emph{Tgt}_{f}\wedge\left(\begin{array}{l}
{TTCX}(Ego, \emph{Tgt}_{f})\leq TTCx\\
\vee \ \textit{distance}(Ego, \emph{Tgt}_{f})\leq d_{clmin}
\end{array}\right)\\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&\emph{RearLeftViolation} \Leftrightarrow\\
& \exists \emph{Tgt}_{rl}\wedge\left(\begin{array}{l}
{TTCX}(Ego, \emph{Tgt}_{rl})\leq TTCx\\
\vee \ \textit{distance}(Ego, \emph{Tgt}_{rl})\leq d_{clmin}
\end{array}\right)\\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&\emph{FrontLeftViolation} \Leftrightarrow\\
& \exists \emph{Tgt}_{fl}\wedge\left(\begin{array}{l}
{TTCX}(Ego, \emph{Tgt}_{fl})\leq TTCx\\
\vee \ \textit{distance}(Ego, \emph{Tgt}_{fl})\leq d_{clmin}
\end{array}\right)\\
\end{aligned}
\end{equation}
In this study, the minimum actual distance and time to the collision are used to define the judgement threshold of compliance distance; $TTCx$ is the minimum time to longitudinal collision; $d_{clmin}$ is the minimum actual distance of a vehicle from relevant vehicles. In addition, it is needed to monitor $\text{LngTmOnLine}$.
Further, according to Article 47, the overtaking process is divided into three consecutive stages and each of them is monitored. Namely, overtake stage 1 is the lane-changing stage, where the ego vehicle can change lanes to the left when overtaking. Thus, the monitoring module of overtake stage 1 should include the monitoring module of changing the left lane. Overtake stage 2 is the target vehicle passing stage. In this stage, it is recommended to maintain a proper speed difference between the ego vehicle and the target vehicle. Overtake stage 3 is the original lane returning stage, where the ego vehicle changes lanes to the right to return to its original lane. Therefore, the monitoring module of overtake stage 3 should also include the monitoring module of changing the right lane. It should be noted that in this stage, overtaking from the right is not allowed. The detailed violation monitor of Article 47 is given in APPENDIX \ref{appendixa}.
\subsubsection{Intersection traffic law violation monitor}
At intersections, traffic laws are mainly defined by traffic lights and the right of way. According to the implementation, Articles 38--41 define how a vehicle can cross an intersection. In this study, the intersection traffic laws are divided into three groups: 1) traffic-light rules; 2) virtual-lane-follow rules; 3) right-of-way rules. Accordingly, Article 38 is divided into six sub-articles, which are shown in Table~\ref{table3}.
\begin{table}[htbp]
\footnotesize
\centering
\caption{The sub-articles of Article 38}
\renewcommand{\arraystretch}{1.0}
\begin{tabular}{p{1.6cm} p{6.4cm} }
\toprule
\multicolumn{1}{c}{Article 38} & \multicolumn{1}{c}{Content of regulation} \\
\midrule
\multirow{3}{2cm} {Traffic lights rules} & Green light means that vehicles are allowed to proceed; \\
& Yellow light means that vehicles across the stop line may keep on driving; \\
& Red light means that vehicles are prohibited from passing \\ \hline
\multirow{1}{2cm} {Virtual lane follow rules} & Vehicles shall try to follow the best theoretical route when passing through intersections \\ \hline
\multirow{4}{2cm} {Right of way rules} & The making-a-turn vehicles shall not interfere with the straight-moving vehicles and pedestrians that are allowed to pass \\
& At the red light, the right-turn vehicles may proceed without interfering with other vehicles and pedestrians that are allowed to pass \\
\bottomrule
\end{tabular}
\label{table3}
\end{table}
Among the traffic lights rules, only red and yellow lights restrict vehicle moving. Under the red light, a vehicle should neither touch the stop line nor pass the stop line from the outside of an intersection to the inside of the intersection. Under the yellow light, a vehicle should not pass the stop line from the outside of an intersection to the inside of the intersection. According to the virtual-lane-follow rules, vehicles should drive inside the virtual lane, which is calculated by an algorithm based on the map information. Virtual lane defines which areas a vehicle can drive in and which areas are not recommended for driving. If a vehicle does not drive within these areas, the vehicle will break the law. The right-of-way rules are the most complex rules among all rules related to intersections. They define that a vehicle should avoid high right-of-way vehicles, pedestrians, and non-motor vehicles. To monitor the compliance of the right-of-way rules, virtual stop lines and check lines are defined to determine if a target vehicle is a high right-of-way vehicle and where the ego vehicle should stop. The map information required for compliance monitoring is presented in Fig.~\ref{cross}.
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{fig6.pdf}
\caption{Map information of intersection}
\label{cross}
\end{figure}
In Fig.~\ref{cross}, the intersection area is enclosed by the dotted rectangular area formed by the stop lines, and the usual virtual lane for passing vehicles is represented by dot-dash lines; meanwhile, the legal but unusual virtual lane is recognised as a lane created by the dotted line. For every entrance lane, there is the corresponding direction-judgement line that is used to determine the intention of moving direction for each passing vehicle. The direction-judgement line is denoted by the solid line in Fig.~\ref{cross}. When the ego vehicle makes a moving direction decision, the virtual lane is then determined, and the other virtual lane with a higher right-of-way that intersects with this virtual lane will be selected, the corresponding virtual stop line and a check area will be generated. If there exists a target vehicle with a higher right-of-way passing through the check area, the ego vehicle should not pass the virtual stop line to ensure unobstructed passing of the target vehicle. The virtual stop line is presented by the solid line in Fig.~\ref{cross}, and the check area is indicated by the rectangular area without boundary lines. When it is judged based on the direction-judgement line that another vehicle intends to drive along this lane, and the midpoint of the front end of this vehicle is within this check area, it is considered that there is a high right-of-way vehicle in the check area, and the ego vehicle shall not pass the virtual stop line. The crosswalk sub-area is part of the crosswalk corresponding to the lane width. It aims to judge whether the passing vehicles interfere with the nearby pedestrians and non-motor vehicles. The trigger domain and logical judgements of relevant rules considered in this section are presented in Table~\ref{table4}.
The detailed violation monitor of Article 38 is listed in the APPENDIX \ref{appendixb}.
\begin{table}[!htbp]
\footnotesize
\centering
\caption{{\normalsize Trigger domain and logic judgements of Article 38 for crossroad}}
\renewcommand{\arraystretch}{1.0}
\label{table4}
\begin{tabular}{p{1.5cm} p{2cm} p{4cm}}
\toprule
Article38 & Trigger domain & Logical requirement \\
\midrule
{Traffic light rules} & $\textit{T}_{\textit{TrafficLightRules}}$ & \textit{IllegalPass}=$\bm F$ \\
{Virtual lane follow rules} & $\textit{T}_{\textit{VirtualLane}}$ & \textit{FollowUsualVirtualLane}=$\bm T$ $\vee \textit{FollowUnusualVirtualLane}$=$\bm T$\\
\multirow{2}{1.5cm}{Right of way rules} & $\textit{T}_{\textit{RightofWay}}$ & \textit{ViolationRightofWay}=$\bm F$\\
& $\textit{T}_{\textit{AvoidPedestrian}}$ & \textit{ImpedePedestrian}=$\bm F$\\
\bottomrule
\end{tabular}
\end{table}
\section{Monitor validation and data analysis}
\subsection{Vehicle trajectory datasets and high precision map}
All aforementioned MTL expressions of highway and intersection traffic laws were programmed in MATLAB/Simulink using different enable subsystems as part of the proposed online monitor. The proposed traffic law violation monitor was verified on Chinese vehicle trajectory datasets.
The highway monitor was verified on the AD4CHE dataset~\cite{zhang2022aerial}. Like the HighD dataset, the AD4CHE dataset is also an overhead viewing angle dataset, which was acquired with drones around Chinese highways and annotated by DJI. Most data in the AD4CHE dataset are arranged in the same way as in the HighD dataset. The difference between the HighD and AD4CHE datasets is that the road in the AD4CHE dataset is not straight but curved with a number of inward and outward sections. Thus, a lane map for the AD4CHE dataset was also provided for fast determination of a lane an object vehicle belonged to. Furthermore, it is worth mentioning that the AD4CHE dataset includes congested sections of highways, so in this dataset, vehicles drive at a relatively slow speed, which results in high traffic law violation rates of vehicle speed and distance.
The intersection monitor was verified on the SIND dataset~\cite{xu2022drone}. The SIND dataset was collected at a signalised intersection in China, and it contains information on traffic participant trajectories, traffic light status and high-definition maps. This dataset has been the only Chinese intersection dataset with traffic light information.
The proposed monitor is an online monitor, so in the highway monitor test, the data were transferred into the ego vehicle’s view first. Once the ego vehicle was selected, the vehicle coordinate system was established according to the ego vehicle’s states, and all the other vehicles’ coordinates were transferred into this coordinate system. Considering that not all vehicles could provide complex information about themselves and the surrounding environment, this study aimed to use a minimal amount of data to make the monitor run to improve the applicability of the proposed monitor. The ego vehicle’s parameters (e.g., width and length) and states (e.g., velocities and accelerations in the \textit{X} and \textit{Y} directions and head angle) were combined as the ego vehicle's data bus. The other vehicles’ data (e.g., relative coordinates, velocities, accelerations, and dimensions) and pedestrians’ states (e.g., relative coordinates and velocities) that could be easily obtained by mature algorithms were combined and used as an object data bus. In addition, each point of the road lines was transferred and used in a map data bus. Using the three data buses as inputs, the highway monitor could monitor the law violation behaviours of any vehicle from the dataset.
Meanwhile, in the intersection monitor test, the data maintained the global coordinate system as the high-definition map was used. The data were combined into different data buses as highway data, and an extra intersection map data bus was constructed for the intersection monitor test. The intersection bus included roads, signs, stop lines, pedestrian crossings and crosshatches. The road data provided information on the road type, lane ID and lane direction (i.e., enter or leave the intersection). For each lane, there were right and left line types, points, and line ending point coordinates used to describe lanes in detail. For each sign, there was a type and a position, which were included in the data. There were also types, start and ending points coordinates for the stop lines, and lane IDs indicated which lane the stop lines belonged to. And for the pedestrian crossing and crosshatch areas, there were vertex coordinates. The monitors used the data buses to monitor the traffic law violation behaviours of the ego vehicle.
It should be mentioned that there was no decision information on any vehicle in the dataset. Therefore, for each vehicle selected as an ego vehicle, the behaviour decision was approximated according to the states and global trajectory of the vehicle. All decisions of selected vehicles were generated using the following methods.
For the highway traffic law violation monitor, the lane-changing and overtaking decision information was obtained as both lane-changing and vehicle-overtaking behaviours had an overlap with the lane lines period. Namely, it was challenging for the online monitor to distinguish whether a vehicle would perform a lane change or an overtake based on vehicle states only. In both the lane change and the overtake, there was a lane-changing behaviour, and it was considered that this behaviour started with a lateral speed higher than 0.25~m/s\cite{konigshof2022parameter}.
\begin{equation}
\begin{aligned}
&{vy(Ego)}>0.25m/s\Rightarrow \\
&\text{Decision}({Ego}) = \emph{ChangeLeftlane}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&{vy(Ego)}<-0.25m/s\Rightarrow \\
&\text{Decision}({Ego}) = ChangeRightlane
\end{aligned}
\end{equation}
Similarly, if the lateral speed of the ego vehicle was higher than 0.25 m/s, and if there was a vehicle in front of the ego vehicle having a speed that was lower than the ego vehicle's speed, their TTC was less more 20 s. Thus, it was considered that the ego vehicle would make a decision to overtake.
\begin{equation}
\begin{aligned}
&\left(\begin{array}{l}
\exists \emph{Tgt} \in \text{front}(Ego) \wedge vx(\emph{Tgt})<vx(Ego) \\
\wedge {TTCX}(Ego, \emph{Tgt})<20s \\
\wedge|vy(Ego)|>0.25m/s
\end{array}\right)\\
&\Rightarrow \text{Decision}({Ego}) = Overtake\\
\end{aligned}
\end{equation}
For the intersection traffic law violation monitor, the decision information that needed to be obtained was the driving direction decision. In SIND dataset, each vehicle passed through the intersection. The ego vehicle’s road ID in the first and last frames showed how the vehicle went. For the ego vehicle, the road ID of the first frame was denoted by $\text{RoadID}_{in}(Ego)$, and the road ID of the last frame was denoted by $\text{RoadID}_{out}(Ego)$. According to the relationship between these two IDs, the direction decision of a vehicle was determined as follows:
\noindent 1. Go-straight decision:
\begin{equation}
\begin{aligned}
&\text{RoadID}_{out}(Ego)=(\text{RoadID}_{in}(Ego)+2)\|\\
&(\text{RoadID}_{in}(Ego)-2)\Rightarrow \text{Decision}({Ego}) = \emph{GoStraight}\\
\end{aligned}
\end{equation}
\noindent 2. Turn-left decision:
\begin{equation}
\begin{aligned}
&\text{RoadID}_{out}(Ego)=(\text{RoadID}_{in}(Ego)+3)\|\\
&(\text{RoadID}_{in}(Ego)-1)\Rightarrow \text{Decision}({Ego}) = \emph{TurnLeft}\\
\end{aligned}
\end{equation}
\noindent 3. Turn-right decision:
\begin{equation}
\begin{aligned}
&\text{RoadID}_{out}(Ego)=(\text{RoadID}_{in}(Ego)+1)\|\\
&(\text{RoadID}_{in}(Ego)-3)\Rightarrow \text{Decision}({Ego}) = \emph{TurnRight}\\
\end{aligned}
\end{equation}
\subsection{Compliance thresholds and statistical analysis of results}
\subsubsection{Compliance thresholds}
In the traffic law digitization process, setting a proper threshold represents an essential part. A suitable threshold will make the monitor more reasonable and accurate. However, some thresholds are not defined by traffic laws, so how to select appropriate values has been the main challenge. In this study, the driving habits of human drivers in the dataset were analysed in the law article relative driving scenarios to obtain the compliance behaviour threshold with a certain degree of confidence.
In Section 3.4, for highway traffic law violation monitoring, three thresholds were used. However, there have been no clear definitions for thresholds in the law articles, such as the minimum distance ($d_{clmin}$) to other vehicles when changing the lane, the maximum allowed time of driving on the line ($t_{max\_cl}$) and the minimum time to longitudinal collision ($TTCx$) with other vehicles when changing the lane. Using the AD4CHE dataset, the traffic law concerning vehicle driving behaviours was analysed, and the concerned thresholds were calculated.
\begin{figure}[htb]
\centering
\subfigure[distance-deceleration scatter diagram]{
\label{dmin}
\includegraphics[scale=0.48]{fig7-1.pdf}}
\subfigure[beta fitting histogram of $T_{cl}$]{
\label{Vycl}
\includegraphics[scale=0.48]{fig7-2.pdf}}
\subfigure[TTC-$T_{cl}$/TTC scatter diagram]{
\label{ttcx}
\includegraphics[scale=0.48]{fig7-3.pdf}}
\caption{Statistical results of threshold analysis}
\label{threshold}
\end{figure}
Considering that an acceptable lane change progress should not affect other vehicles’ driving behaviours, the distances from behind vehicles and their deceleration behaviours can show whether they are affected by the front vehicle. To calculate the $d_{clmin}$ value, all effective lane-changing behaviours in the dataset were used. When the lane change began, the data on a particular scenario were extracted if a vehicle in the target lane appeared within 30 m behind. A scatter plot figure is presented in Fig.~\ref{dmin}, where the horizontal axis indicates the distance from the vehicle behind at the beginning of the lane-changing process, and the vertical axis represented the average deceleration of vehicles behind the target lane during the ego vehicle's lane changing. As shown in Fig.~\ref{dmin}, the distances between the ego vehicle and the vehicles behind were between 14~m and 20~m when the ego vehicle started to change lanes and overlapped with the lane line. Namely, most drivers decided to enter the target lane when they were 14~m--20~m away from the vehicle behind. The traffic law required that when another vehicle cut in, the ego vehicle needed to slow down properly and keep to the right to avoid the collision if conditions permit. Thus, the driver would decelerate properly for careful driving when a vehicle was cut in. As shown in Fig.~\ref{dmin}, at the speed of 0.35 m/s$^2$, there was an obvious deceleration dividing line. Namely, the deceleration values of most vehicles behind the ego vehicle were below 0.35 m/s$^2$ when the other vehicle cut in. The statistical graphic of vehicles' distances is presented at the bottom of Fig.~\ref{dmin}, where it can be seen that within the distance range of~14 m--20~m, the number of behind vehicles that decelerated less than 0.35~m/s$^2$ was larger than the number of behind vehicles that decelerate more than 0.35~m/s$^2$. Therefore, it was considered that when the dclmin was 14 m, and the other thresholds were met, the behind vehicles would not be obstructed by the lane-changing vehicles but only decelerate for prudent driving.
Furthermore, the time of vehicle overlapped with the lane line during the complete lane-changing behaviour was calculated, and the statistical results are shown in Fig.~\ref{Vycl}. The vehicle-line overlapping time during the lane change varied from 0.8~s to 8.8~s, but most of them were concentrated within the range of 1.5~s--3~s, which obeyed the Beta distribution. A threshold of 6~s was selected as $t_{max\_cl}$ because 99.92\% of all vehicles’ lane changings could be completed within that time. Moreover, considering that the lane-changing time should be longer than the TTC to the front vehicle when changing the line, the initial TTC to the front vehicle and the ratio between the lane change time and the initial TTC were calculated, as shown in Fig.~\ref{ttcx}. Considering that the TTC to the front vehicle shoule be at least longer than the lane changing time, a normal ratio threshold of one was selected, and the fitting curve of the ratio was drawn. The intersection point indicated that the $TTCx$ should be 2.3~s.
\subsubsection{Statistical results}
\begin{figure}[htbp]
\centering
\subfigure[Statistical results of the illegal cases]{
\label{highwaynumber}
\includegraphics[scale=0.5]{fig8-1.pdf}}
\subfigure[Statistical results of the proportion of violation types]{
\label{highwaypercentage}
\includegraphics[scale=0.5]{fig8-2.pdf}}
\caption{Statistical results on the AD4CHE dataset fragment.}
\label{results_hw}
\end{figure}
According to the set thresholds, traffic law violation monitoring was performed on each vehicle in the AD4CHE dataset. All vehicles in the dataset were marked by an ID. The IDs were assigned in a sequence; the selected vehicle was treated as the ego vehicle, while other vehicles in the scenario were treated as sounding vehicles. The traffic law violation monitor monitored the ego vehicle’s behaviour. After the last vehicle’s behaviour was monitored, the statistical results of different traffic law violation types on the dataset were obtained. The violation statistical results of the 25th fragment in the AD4CHE dataset are presented in Fig.~\ref{results_hw}. This fragment lasted for 290 s; since the right part of the dataset was severely congested while the left part was better, only vehicles in the left part were monitored. The statistics of various types of highway traffic law violations on every 5 s are presented in Fig.~\ref{highwaynumber}. The proportion of each violation in the dataset is displayed in Fig.~\ref{highwaypercentage}. Due to the impact of traffic congestion, it was difficult for vehicles to reach the minimum speed limit specified on the highway. Maintaining compliance following distance was also challenging. Therefore, there were many violations of the speed limit and following distance. In contrast, there were fewer violations of overtaking and lane changing.
\begin{figure}[htbp]
\centering
\subfigure[Statistical results of the illegal cases]{
\label{crossnumber}
\includegraphics[scale=0.5]{fig9-1.pdf}
}
\subfigure[Statistical results of the proportions of different violation types]{
\label{crosspercentage}
\includegraphics[scale=0.5]{fig9-2.pdf}}
\caption{Statistical results on the SIND dataset fragment.}
\label{results_c}
\end{figure}
No ambiguous threshold was introduced in the digitization process of intersection-related traffic laws. Similar to the monitoring test on the AD4CHE dataset, every vehicle in the SIND dataset was monitored for traffic law violations. The statistical results of the 8\_2\_1 fragment in the SIND dataset are presented in Fig.~\ref{results_c}; this fragment lasted for approximately 1,200~s. The statistics of intersection violations for each type on every 20~s are presented in Fig.~\ref{crossnumber}. Fig~\ref{crosspercentage} shows the proportions of different violations in the entire dataset. Since the virtual lanes were set as a circular arc without any relaxation value, the violation of this law was relatively high, and the follow-up research could provide a better virtual lane range. In addition, China has imposed severe punishment on the red light running while the yellow light running is treated more leniently. However, all traffic violations in the statistical result were yellow light running. The statistical results also showed that Chinese drivers' compliance with traffic laws was directly proportional to safety and punishment for violations.
\subsubsection{Illegal examples}
The scenario of the each dataset was reproduced by the visualisation program. Through the corresponding violation fragments in the visualisation program, it was more intuitive to verify the monitoring program's accuracy. Scenarios of vehicles violating the traffic law in the 44th fragment of the AD4CHE dataset are presented in Fig.~\ref{example}. Among them, the ID 15117 vehicle violated the lane-changing law due to its low TTCx value and distance from the vehicle behind. The ID 15228 vehicle ran at a speed of 70.24~km/h, and its following distance was only 8.12~m, which was much less than required. The ID 15306 vehicle ran at the innermost lane at a speed of 58~km/h, which was much lower than the required speed of 110~km/h.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\linewidth]{fig10.pdf}
\caption{Illegal examples of highway}
\label{example}
\end{figure}
Further, Fig.~\ref{example2} displays a number of scenarios where vehicles violated the traffic law in the fragment 8\_2\_1 in the SIND dataset. Among them, the ID 44 vehicle violated the right of way; it obstructed the straight ahead of the ID 42 vehicle. The ID 218 vehicle violated the virtual lane rule. Namely, when passing the intersection, this vehicle did not follow the best virtual lane. Even in some periods of time, it did not follow the not-recommended virtual lane. The ID 551 vehicle violated the traffic light rules. It crossed the stop line at the red light.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\linewidth]{fig11.pdf}
\caption{Illegal examples of intersection}
\label{example2}
\end{figure}
\section{Conclusion and future work}
This paper proposes an online traffic law violation monitor for monitoring an AV’s behaviours. The proposed online monitor runs with the autonomous system and provides the traffic law violation results in real-time based on the information from the vehicle’s perception and decision systems. The obtained results can be further used for making better law compliance decisions. Unlike offline monitors, the proposed online monitor can operate in different scenarios lacking overall information. To adapt to all scenarios and monitor appropriate law articles using little information, this paper proposes the trigger domain-based architecture, where triggers are used to classify environment situations and distinguish the behaviours of a vehicle with the decision information. In addition, the atomic propositions that are necessary for the trigger and judgement MTL expressions are shared in detail for better understanding. The characteristics and challenges in digitizing traffic laws for online monitoring purposes are discussed. The main traffic laws of highways and intersections are taken as digitization examples. A method for calculating fuzzy thresholds in digitized laws based on datasets is introduced, and all the digitized traffic laws are verified on datasets.
Future work directions could include: (1) a traffic law violation state cancel module behind the judgement part in the trigger domain that considers the around driving conditions to cancel the non-mandatory law violation states for achieving more flexible violation judgement and avoiding subsequent decision-making system having an unsolvable problem; (2) a mapping system based on ontology mapping the atomic propositions to the traffic law nature language, which can ensure that in situations that temporary traffic law changes or driving area changes vehicles can direct access to the violation monitoring MTL expressions without any manual intervention; (3) a traffic law compliance decision-making system that can transfer the MTL expressions into decisions, ensuring full traffic law compliance when safe.
\section*{Acknowledgement}
The research presented in this paper is financially supported by the National Natural Science Foundation of China Project under Grant No.~52072215~and~U1964203 and the National key R\&D Program of China under Grant No. 2020YFB1600303.
\bibliographystyle{IEEEtran}
|
2,869,038,155,703 | arxiv | \section{Note to Practitioners}
\section{Introduction}
Sampled optimisation problems can comprise a great number of sampled constraints. They constitute an important category of problems that are widely encountered in the scenario approach~\citep{calafiore_uncertain_2004,calafiore2012mixed,CampiBook2018}, and also with learning paradigms at large, including statistical learning theory~\citep{vidyasagar_randomized_2001,vidyasagar_learning_2002,alamo_randomized_2009}. In these fields of study, semi-infinite optimisation problems---i.e., robust optimisation problems with an infinite number of constraints---are usually approximated and reformulated as a sampled optimisation problem associated with a finite set of random constraints. Effectively, the number of random samples is established so as to achieve a required level of probabilistic robustness when solving this approximate sampled optimisation problem.
If a too stringent robustness criterion is sought, it is expected that the sample complexity---i.e., the size of the set of samples necessary to achieve this robustness criterion in a probabilistic senses---becomes high, thus yielding a computationally intensive sampled optimisation.
Dealing with the computational complexity of this particular class of optimisation problems has not attracted much attention despite its critical practical importance. In~\cite{chamanbaz_sequential_2013,chamanbaz_sequential_TAC_2016}, a sequential strategy has been considered in the frame of the scenario optimisation approach. This strategy consists in reducing at each iteration step $k$ the sample complexity, denoted as $N(k)$, as compared to the scenario bound. In a sequential fashion, this is followed by solving the sampled optimisation problem based on $N(k)$, and ultimately by checking the robustness of the obtained solution by means of a validation test. The termination criterion is simply based on having the identified solution passing this test. Should the solution fail the validation test, a larger sample complexity $N(k+1)>N(k)$ is considered, and the previous steps are iterated until the algorithm ultimately satisfies the termination criterion. In \cite{chamanbaz_statistical_2014}, concepts from statistical learning theory are applied to solve robust linear and bilinear matrix inequality problems. Specifically, in this approach, the sample complexity is first computed and then a sequential randomized method is considered for the solution of the sampled optimisation problem. It is worth noting that the approaches reported in \cite{chamanbaz_sequential_2013,chamanbaz_statistical_2014,chamanbaz_sequential_TAC_2016} may end up demanding a large number of validation samples so as to satisfy the robustness condition of the candidate solution at each iteration. Furthermore, there are applications for which the set of samples is obtained from actual experiments, thereby making it a limited resource, and somehow restricting their availability. In \cite{calafiore2016repetitive}, a solution to the scenario problem is proposed based on concepts similar to those in~\cite{chamanbaz_sequential_TAC_2016}. Specifically, the algorithm introduced in~\cite{calafiore2016repetitive} does not rest on iterative increases of the cardinality of the set of design samples. Instead, it hinges on a probabilistic characterisation of the length of the iterative process required to arrive at a solution. Still using a sequential process, a `wait-and-judge scenario' optimisation \citep{Campi2018} has been developed without the need to test the validity of the candidate solution~\cite{GarattiIncremental2019}. In this algorithm, a sampled optimisation along with an estimation of the the number of support constraints are iteratively carried out. At each iteration step, the robustness---in a probabilistic sense and based on set accuracy and confidence levels---controls the selection of the sample complexity. Practically, the method proposed in \cite{GarattiIncremental2019} primarily aims at minimising the number of scenarios without necessarily keeping the computational complexity low. Indeed, the repetitive evaluation of the number of support constraints may becoming computationally prohibitive when the size of the set of sample constraints grows.
Optimisation has long been at the core of Machine Learning (ML) and it continues to support the development of novel ML strategies. Interestingly, ML techniques have recently been considered to tackle demanding optimisation problems, such as in solving continuous and mixed-integer optimisation problems, see \cite{bengio2020machine} for an extensive review. Authors in \cite{lopez2016irace} used machine learning to automatically tune parameters of the optimisation algorithm. Reinforcement learning \cite{sutton2018reinforcement} has been used in \cite{dai2017learning} to solve a diverse range of combinatorial problems defined over graphs where a neural network is trained to learn a heuristic algorithm which suggests the next node to visit contributing towards the optimal solution. In \cite{misra2018learning}, a statistical approach is developed to uncover the set of optimal active constraints---i.e., constraints that hold with equality in the optimal solution---for parametric optimisation problems. The algorithm needs to be fine-tuned for every problem and is limited to continuous optimisation problems---the paper claims that it can capture mixed-integer and non-convex problem but, no numerical evidence has been reported supporting that claim. The approach presented in \cite{bertsimas2019online,bertsimas2021voice} does not have the limitations of \cite{misra2018learning}. the authors of \cite{bertsimas2019online,bertsimas2021voice} consider parametric optimisation problems---optimisation problems in which several parameters vary within a range each time the problem is formulated---and present a multiclass classifier to identify an \textit{optimal strategy} based on which one can recover the optimal solution without explicitly solving the problem. The optimal strategy is defined as the set of basis---also referred to as support constraints, see e.g. \cite[Definition 2.1]{calafiore2010random}, which constitutes the minimal set of constraints underpinning the optimal solution---for continuous problems and the set of tight constraints together with the integer part of the optimal solution for mixed-integer problems. It is worth adding that none of the aforementioned approaches are designed to solve robust optimisation problems.
In this paper---which is an extended version of \cite{CHAMANBAZ20206749}, we propose two algorithms for solving sampled optimisation problems. First, a sequential deterministic algorithm---Algorithm \ref{alg:sequential algorithm}---solves the sampled optimisation problem using concepts borrowed from the classical Las Vegas algorithm for linear and integer programming (LIP) \citep{Clarkson:1995:LVA:201019.201036}. Algorithm \ref{alg:sequential algorithm} as introduced in~\cite{CHAMANBAZ20206749} is intrinsically deterministic and does resort to any probabilistic validation at any stage. These features make it distinctive from other approaches primarily seeking to maintain a low level of sampled constraints---as a way of keeping the computational burden for the optimisation task as reduced as possible---while maintaining probabilistic guarantees at the same level as those for the original problem. Here, we instead consider a computationally lean approach towards the solution of sampled optimisation problems for cases where the number of sampled constraints is large.
The impetus for Algorithm \ref{alg:sequential algorithm} originates from a \textit{distributed} randomized constraints consensus approach introduced in \cite{chamanbaz_randomized_MILP_2017,chamanbaz2019randomized}, and that is effective at solving robust distributed mixed-integer problems. This specific approach is probabilistic---whereas both algorithms presented in the present paper are deterministic---with agents carrying out local computation and communication with the goal of achieving consensus on a candidate solution~\cite{chamanbaz_randomized_MILP_2017,chamanbaz2019randomized}. Note that a probabilistic validation step is included in \cite{chamanbaz_randomized_MILP_2017,chamanbaz2019randomized} which has the same nature as the one used in~\cite{chamanbaz_sequential_TAC_2016}.
In Algorithm \ref{alg:learning sequential algorithm}, we borrow ideas from the classification technique used in \cite{bertsimas2021voice,bertsimas2019online} to further reduce the convergence time compared to Algorithm \ref{alg:sequential algorithm}. In particular, a neural network classifier is developed to predict the basis corresponding to the optimisation problem formulated at an uncertain point. The classifier is used in reducing the computational complexity corresponding to the optimisation step of the algorithm.
Given a sampled optimisation problem, the proposed two algorithms achieve a finite-time convergence towards the optimal solution. Both algorithms are sequential by design. Moreover, each iteration can be subdivided into the following steps: \emph{(i)} a validation step, and \emph{(ii)} an optimisation one. In that first step of this sampled optimisation, each and every constraint is verified for the candidate solution, and violating ones are singled out. In itself, stage \emph{(i)} is not computationally demanding since it is limited to validating the candidate solution without any optimisation \emph{per se}. In step \emph{(ii)}, the optimisation takes place with a set of constraints that comprises: \emph{(a)} the limited subset of constraints that have been identified in step \emph{(i)} as being violated, and \emph{(b)} the current \emph{basis}, which is constitutes the smallest set of constraints fully characterizing the test solution. These two steps---validation and optimisation---are iteratively repeated until not a single constraint is violated by the test solution, which is guaranteed to occur within a finite number of iterations.
It is worth highlighting that the deterministic character of both algorithms enable their use for any optimisation problem exhibiting the Helly-type property; see \cite{amenta_helly-type_1994,amenta_hellys_2015}.
As a matter of fact, this particular class of optimisation problems constitutes a sizeable share of non-convex problems. Lastly, it is worth stressing that a key novelty of this work---beyond the concepts reported in~\cite{CHAMANBAZ20206749} and the existing literature for this class of problems---is the use of machine learning (ML), and specifically a neural network multiclass classifier to solve robust optimisation problems. Compared to~\cite{CHAMANBAZ20206749}, the present paper contains an entirely new
ML-based algorithm, which seeks to identify participating constraints and thereby, exhibit a much reduced computational complexity.
\noindent
{\bf Notations\\}
In what follows, uppercase and italic letters, e.g., $F$, refer to constraints, whereas calligraphic uppercase letters, e.g., $\mathcal{F}$, refer to the set induced by the specific constraint $F$. Using these notations, if $A=B\cup C$ with $B$ and $C$ being collections of
constraints, then $\mathcal{A}=\mathcal{B}\cap \mathcal{C}$, that is, the set induced by the union of constraints $B$ and $C$ is the intersection of $\mathcal{B}$ and $\mathcal{C}$. Still with these notations, an optimisation problem
\begin{align*}
\min\,\,& c^T{\mathbf{x}}\\
\text{subject to }& {\mathbf{x}}\in \FF,
\end{align*}
is fully characterized by the pair $(F,c)$. Lastly, $J(F)$ is the smallest value of
$c^T{\mathbf{x}}$ while ${\mathbf{x}}\in\mathcal{F}$. Moreover, \texttt{randint(a,b)} generates a random integer within the interval $[a,b]$ and $\lceil a\rceil$ returns the smallest integer number greater than $a$.
\section{Preliminaries and Problem Statement}
Let us consider the following robust optimisation problem
\begin{align}\nonumber
\min\,\,& c^T{\mathbf{x}}\\ \nonumber
\text{subject to }& {\mathbf{x}}\in \FF(q),\,\,\, \forall q\in{\mathbb{Q}} \\ \label{eq:MICP}
& {\mathbf{x}}\in {\mathbb{R}}^{d_R}\times{\mathbb{Z}}^{d_Z},
\end{align}
where ${\mathbf{x}}\in {\mathbb{R}}^{d_R}\times{\mathbb{Z}}^{d_Z}$ constitutes the vector of decision variables, the vector $q$ contains the entire set of uncertain parameters acting on the system, such that $\FF(q)=\{{\mathbf{x}}\in {\mathbb{R}}^d:f({\mathbf{x}},q)\leq0\}$, with
$d=d_R+d_Z$ and, $f({\mathbf{x}},q): \mathbb{R}^d\times{\mathbb{Q}}\rightarrow{\mathbb{R}}$,
is the constraint function of Problem \eqref{eq:MICP}. Note that when all decision variables ${\mathbf{x}}$ are assumed to be continuous, then for any given value of $q$, this constraint $f({\mathbf{x}},q)$ is a convex function. Without any loss of generality, a linear objective function may be assumed. Indeed, should a nonlinear convex objective function be considered, it could readily be transformed into the epigraph form by introducing an extra decision variable.
Furthermore, if $d_Z=0$ then Problem \eqref{eq:MICP} reduces to a classical continuous convex optimisation task; if $d_R =0$, Problem \eqref{eq:MICP} constitutes an integer optimisation problem, while in the general case associated with $d_R\neq0,d_Z\neq 0$, we are in the presence of a mixed-integer optimisation.
An efficient method to identify an approximate solution to \eqref{eq:MICP} with specific robustness guarantees---in the probabilistic sense---is recast it as a sampled optimisation problem by means of the scenario approach \citep{calafiore_uncertain_2004,calafiore_scenario_2006,calafiore2010random,calafiore2012mixed,CampiBook2018}.
With this approach, the semi-infinite optimisation problem \eqref{eq:MICP} is recast as a constrained optimisation with a finite number of constraints. These constraints are constructed based on some specifically samples, which are extracted randomly from uncertainty set ${\mathbb{Q}}$. Formally, $N$ independent and identically distributed (i.i.d) samples are extracted from the set ${\mathbb{Q}}$:
\[
{{\rm\bf q}}=\{q^{(1)},\ldots,q^{(N)}\}\in{\mathbb{Q}}^N,
\]
and thus, one can formulate the following sampled optimisation problem
\begin{align}\nonumber
{\mathbf{x}}^*_N=\arg\min\,\,& c^T{\mathbf{x}}\\ \nonumber
\text{subject to }& {\mathbf{x}}\in \bigcap_{i=1}^N\FF(q^{(i)})\\ \label{eq:sampled optimisation problem}
& {\mathbf{x}}\in {\mathbb{R}}^{d_R}\times{\mathbb{Z}}^{d_Z}.
\end{align}
It is important noting that only a few constraints are required to solve Problem~\eqref{eq:sampled optimisation problem}. Indeed, the concept of \emph{basis}---the minimal set of constraints defining the solution---is key. This concept is closely connected to the Helly-type theorems originally proposed by E. Helly in \cite{helly1923mengen}, see \cite{amenta_helly-type_1994,amenta_hellys_2015} for more details. The central aim of this work is the identification of that basis for Problem \eqref{eq:sampled optimisation problem}, which offers a means to determine its solution ${\mathbf{x}}^*_N$ in a computationally efficient fashion.
\begin{definition}[Basis]
Given a collection of constraints $F$, a subset of minimal cardinality $B\subseteq F$ is a basis of $F$ if the
optimal cost of the problem defined by $(F,c)$ is identical to the one
defined by $(B,c)$, and the optimal cost decreases if any constraint is removed.
\end{definition}
The \emph{combinatorial dimension} of any problem $(F,c)$ is defined by the size of its largest basis. Based on the following theorem adopted from \cite[Corollary 1]{calafiore2012mixed}, and \cite[Theorem 3.11]{amenta_hellys_2015}, it can be explicitly expressed in terms of $d_R$ and $d_Z$ for the mixed-integer problem \eqref{eq:sampled optimisation problem}.
\begin{theorem}\label{thm:combinatorial dim}
The combinatorial dimension of Problem \eqref{eq:sampled optimisation problem} is $(d_R+1)2^{d_Z}-1$.
\end{theorem}
The following two assumptions are considered when seeking a solution to any subproblem of~\eqref{eq:sampled optimisation problem}.
\begin{assumption}[Uniqueness]~\label{assum: uniqueness} For every $N\in\mathbb{N}$ and every multisample ${{\rm\bf q}}\in\mathbb{Q}^N$, there is a unique solution to Problem~\eqref{eq:sampled optimisation problem}.
\end{assumption}
Assumption~\ref{assum: uniqueness} is not \emph{per se} too restrictive. Indeed, to guarantee the uniqueness of the optimal point, one can resort to a number of strategies, such as considering a strictly-convex objective function, a lexicographic ordering, or a universal tie-breaking rule, see \cite[Observation 8.1]{amenta_helly-type_1994} for more details.
\begin{assumption}[Non-degeneracy]~\label{assum: degeneracy} The solution of any subproblem formed by every $N\in\mathbb{N}$ and every multisample ${{\rm\bf q}}\in\mathbb{Q}^N$ coincides with the solution of a problem which involves only basis constraints of the subproblem.
\end{assumption}
Based on~\cite[Theorem 3 and Corollary 2]{calafiore2012mixed}, the robustness property of ${\mathbf{x}}^*_N$ is fully characterized by the following theorem.
\begin{theorem}\label{thm:Scenario}
Suppose Assumption \ref{assum: uniqueness} holds. Given probabilistic accuracy $\varepsilon\in(0,1)$ and confidence levels $\delta\in(0,1)$, let $N$ be the smallest integer satisfying
\begin{equation}\label{eq:scenario bound}
\delta\geq \sum_{\ell=0}^{(d_R+1)2^{d_Z}-2} {N \choose
\ell} \varepsilon^\ell(1-\varepsilon)^{N-\ell}.
\end{equation}
Then the solution of \eqref{eq:sampled optimisation problem} ${\mathbf{x}}^*_N$ satisfies
\begin{align*}
\Pr^{N}\bigg\{{{\rm\bf q}}\in\mathbb{Q}^{N}:
\Pr\bigg\{q\in\mathbb{Q}:{\mathbf{x}}^*_N\notin \FF(q)\bigg\}
\leq\varepsilon \bigg\}\geq1-\delta,
\end{align*}
where $\Pr^N$ is the product probability measure on ${\mathbb{Q}}^N$.
\end{theorem}
Note that~\eqref{eq:scenario bound} defines a binomial tail relating the number of samples $N$, with the accuracy level $\varepsilon$, confidence $\delta$, and the dimension of decision variables $d_R$ and $d_Z$. The sample complexity $N$ can be computed by numerically solving \eqref{eq:scenario bound} for any problems given desired accuracy and confidence levels.
Given that the number of scenario samples $N$ is inversely proportional to $\varepsilon$, and that there is logarithmic relationship with $1/\delta$, so if $\varepsilon,\delta$ must somehow be made small, then $N$ in \eqref{eq:scenario bound} can become excessively large, thereby possibly yielding a challenging sampled optimisation. This possibility highlights the critical need for a computationally efficient algorithm towards the solution of the sampled optimisation problem \eqref{eq:sampled optimisation problem}.
\section{Sequential Algorithm}
In this section, we detail the actual steps involved in our proposed sequential algorithm for the solution of sampled mixed-integer problems that can be formulated as in \eqref{eq:sampled optimisation problem}. As a first step, we introduce two fundamental primitives. The first, $[q^\texttt{Viol},{\texttt{feasible}}]={\texttt{Verification}}(F(q^{(1)}), \ldots,F(q^{(N)}),{\mathbf{x}},r) $, checks the feasibility of a test solution ${\mathbf{x}}$ for all the sampled constraints involved in \eqref{eq:sampled optimisation problem}, i.e. $F(q^{(1)}), \ldots,F(q^{(N)})$ and---if there exist---finds $r$ violating samples. In the event of a violation, the flag ${\texttt{feasible}}$ is set to $0$ otherwise, ${\texttt{feasible}}=1$. The second primitive is $[{\mathbf{x}},B]={\texttt{Solve\textsubscript{MIP}}}(F,c)$. This primitive deals with the actual solution of the optimisation problem constructed from $(F,c)$, and outputs the optimal point ${\mathbf{x}}$ along with the associated basis $B$. The primitive needs to first solve the problem to find the optimal solution ${\mathbf{x}}$, and then depending on the nature of the problem---being continuous, integer, or mixed-integer---it uses different methods to find the basis. See Remark \ref{rem:finding basis} for a detailed explanation on how to find the basis. These primitives form the backbone of the algorithm. First, a test solution ${\mathbf{x}}$ is examined---using the ${\texttt{Verification}}$ primitive---to check if it is compatible with the full set of $N$ constraints of \eqref{eq:sampled optimisation problem} and---whenever possible---$r$ violating samples are identified; we note that there might be only $r'<r$ samples violating the candidate solution. Subsequently, the algorithm enters the optimisation phase---using the primitive ${\texttt{Solve\textsubscript{MIP}}}$---whose constraint set consists of $(i)$ the constraints formed at $r$ violating samples and $(ii)$ the current set of basis. The algorithm then iterates over these two steps until no more violating samples can be found. The full process is formally detailed in Algorithm~\ref{alg:sequential algorithm}.
\begin{algorithm}[t]
\begin{algorithmic}[1]
\caption{Sequential Algorithm}
\label{alg:sequential algorithm}
\STATE\algorithmicrequire{ $c,\,r,\,d_R,\,d_Z,\,F(q^{(i)}), i = 1, \ldots N,$}
\STATE\algorithmicensure{ ${\mathbf{x}}_{\texttt{seq}},B_{\texttt{seq}}$}\\
{\bf Initialisation:}
\STATE Set $m=(d_R+1)2^{d_Z}$, ${\texttt{feasible}}=0$, $t=0$
\STATE $[{\mathbf{x}}(0),B(0)]={\texttt{Solve\textsubscript{MIP}}}(F(q^{(1)})\cup\ldots\cup F(q^{(m)}),c)$\\
{\bf Evolution:}
\WHILE{${\texttt{feasible}}==0$}{
\STATE $[q^\texttt{Viol},{\texttt{feasible}}]={\texttt{Verification}}(F(q^{(1)}), \ldots,F(q^{(N)}), {\mathbf{x}}(t),r)$
\STATE $[{\mathbf{x}}(t+1),B(t+1)]={\texttt{Solve\textsubscript{MIP}}}(F(q^\texttt{Viol})\cup B(t),c)$
}
\STATE $t = t + 1$
\ENDWHILE
\STATE Set ${\mathbf{x}}_{\texttt{seq}} = {\mathbf{x}}(t+1) $ and $B_{\texttt{seq}}=B(t+1)$
\RETURN ${\mathbf{x}}_{\texttt{seq}},B_{\texttt{seq}}$
\end{algorithmic}
\end{algorithm}
Here are some important remarks related to Algorithm~\ref{alg:sequential algorithm}.
\begin{remark}[Complexity of an iteration step]
A key attribute of Algorithm \ref{alg:sequential algorithm} is that the complexity of the optimisation task under consideration at each iteration step would not increase with the iteration count. Indeed, the \textit{maximum} possible number of constraints involved at the optimisation stage is $r+(d_R+1)2^{d_Z}-1$. For instance, for a mixed-integer optimisation problem with $d_R = 5,\, d_Z = 3$ if we set $r = 10$, the number of constraints can be at most $57$; or in a continuous optimisation problem in which the dimension of the solution space is $d_R=5$, the number of constraints can be at most $15$. Therefore, at each iteration step, an optimisation of fixed---and small---complexity is solved.
\end{remark}
\begin{remark}[Complexity of verification step]
The verification step of Algorithm \ref{alg:sequential algorithm} is computationally inexpensive since it only requires checking the feasibility of a candidate solution for the $N$ sampled uncertainties present in Problem \eqref{eq:sampled optimisation problem}.
\end{remark}
\begin{remark}[Identification of the basis]\label{rem:finding basis}
In a continuous optimisation problem, the basis is fully characterized by the smallest set of active constraints and it coincides with the set of active constraints if Assumption~\ref{assum: degeneracy} holds. Hence, in a continues optimisation problem it is straightforward to identify the basis. With a mixed-integer problem, however, identifying that basis has the potential of becoming computationally demanding. Alternatively, a more tractable way of computing this basis---without necessarily seeking it to be of minimal cardinality---is to individually test each constraint to confirm whether or not they can belong to the basis.
In practice, one may consider dropping the $i$-th constraint for the optimisation sought. Should the objective value returned by this modified optimisation problem be smaller than the objective value of the original one, than the discarded constraint can be integrated into the basis. As previously highlighted, however, the number of constraints of the original problem---see line $7$ of Algorithm \ref{alg:sequential algorithm}---is at most $r+(d_R+1)2^{d_Z}-1$. Thus, it is computationally inexpensive to identify the basis for problems with a limited number of constraints. We further note that a comparable greedy approach has been considered in the frame of algorithms aimed at solving scenario with discarded constraints \citep{campi2011sampling,calafiore2010random}, and more recently in the wait-and-judge scenario optimisation case \citep{Campi2018,GarattiIncremental2019}. In Section \ref{sec:learning}, we present a learning-based strategy to further reduce the overall computational complexity, while focusing on this particular step.
\end{remark}
\begin{remark}[Choice of the number of violating samples $r$]\label{rem:basis computation}
The number of violating samples $r$ in Algorithm \ref{alg:sequential algorithm} is the byproduct of a trade-off between, on the one hand the complexity of the optimisation task at each step, and on the other hand the number of steps required for convergence. Specifically, a larger value for $r$ would yield a more complex optimisation at line $7$ of the algorithm, but with a smaller iteration count. Depending on the computational capacity of the platform running Algorithm~\ref{alg:sequential algorithm}, one can tune $r$ to achieve the best performance in terms of total computational time.
\end{remark}
The main features of Algorithm \ref{alg:sequential algorithm} are encapsulated by the following theorem.
\begin{theorem}\label{thm:property of algorithm}
Let Assumptions \ref{assum: uniqueness} and \ref{assum: degeneracy} hold. Then, the following statements hold.
\begin{enumerate}
\item The objective value of the candidate solution $c^T{\mathbf{x}}(t)=J(B(t))$ is monotonically increasing while Algorithm \ref{alg:sequential algorithm} is progressing.
\item
Algorithm \ref{alg:sequential algorithm} terminates in finite time.
\item
The solution returned by Algorithm \ref{alg:sequential algorithm}, ${\mathbf{x}}_{\texttt{seq}}$ is identical to ${\mathbf{x}}^*_N$.
\end{enumerate}
\end{theorem}
\vskip 1ex
{\bf Proof: } Note that when forming the basis at time $t+1$, i.e. $B(t+1)$, we use $B(t)$ from the previous time step $t$. Thus, $J(B(t+1))\geq J(B(t))$. Moreover, there has to be at least one violating constraint $F^\texttt{viol}$ in all the iterations of Algorithm \ref{alg:sequential algorithm}, except of course at the last iteration. Indeed, if there had not been a violating constraint, the condition at line $5$ would have been satisfied and the algorithm would have terminated. This means that at line $7$ of Algorithm \ref{alg:sequential algorithm}, we solve an optimisation problem whose constraints set involves the current basis $B(t)$ and at least one violating constraint $F^\texttt{viol}$. Therefore, owing to the presence of the violating constraint(s) $F^\texttt{viol}$, and due to Assumption \ref{assum: uniqueness}, the cost has to increase, i.e. $J(B(t+1))> J(B(t))$. This completes the proof of statement (i) of Theorem \ref{thm:property of algorithm}.
Since the number of constraints involved in \eqref{eq:sampled optimisation problem} is finite, then the number of candidate bases leading to a finite number of candidate costs $J(B(t))$ is also finite. Furthermore, the cost $J(B(t))$ is strictly increasing with the iteration counter $t$---as proved in the first statement. Since, the sequence $\{J(B(t))\}_{t>0}$ is strictly increasing and has a finite number of elements, it will converge in a finite number of iterations leading to the finite-time termination of the algorithm. This completes the proof of statement (ii) of the theorem.
We first note that since at any iteration $t$ of the algorithm, a subproblem of mixed-integer problem \eqref{eq:sampled optimisation problem} is being solved, $J(B(t))$ cannot be greater than $J(F)=c^T{\mathbf{x}}^*_N$, where $F \doteq \bigcup_{i=1}^N F(q^{(i)})$; then, $J(B(t))\leq J(F),\,\forall t>0$ and as a result $J(B_{\texttt{seq}})\leq J(F)$. We now show that $J(B_{\texttt{seq}})$ cannot be smaller than $J(F)$. Assume by contradiction that $J(B_{\texttt{seq}})<J(F)$ or equivalently $J(B_{\texttt{seq}})<J(B_{\texttt{seq}}\cup F)$ as $B_{\texttt{seq}}\subseteq F$. By construction, ${\mathbf{x}}_{\texttt{seq}}$ must satisfy all the constraints in \eqref{eq:sampled optimisation problem} as it has passed the verification step of Algorithm \ref{alg:sequential algorithm}; then, ${\mathbf{x}}_{\texttt{seq}}\in\FF$ with $\FF\doteq \bigcap_{i=1}^N \FF(q^{(i)})$. Moreover, by definition $\FF\subseteq \BB_{\texttt{seq}}$, which implies that ${\mathbf{x}}_{\texttt{seq}}\in\FF\cap\BB_{\texttt{seq}}$. Now, taking into account the fact that $\BB_{\texttt{seq}}$ is the set generated by the basis $B_{\texttt{seq}}$ associated with ${\mathbf{x}}_{\texttt{seq}}$, we can state that $J(B_{\texttt{seq}})\geq J(F\cup B_{\texttt{seq}})$, which is in direct contradiction with our earlier assumption that $J(B_{\texttt{seq}})<J(B_{\texttt{seq}}\cup F)$. Thus, $J(B_{\texttt{seq}})$ can neither be greater nor smaller than $J(F)=c^T{\mathbf{x}}^*_N$. Therefore, $J(B_{\texttt{seq}})=c^T{\mathbf{x}}^*_N$, which can be recast in the following equivalent form $c^T{\mathbf{x}}_{\texttt{seq}}=c^T{\mathbf{x}}^*_N$. The latter combined with Assumptions \ref{assum: uniqueness} and \ref{assum: degeneracy} concludes the proof of Theorem \ref{thm:property of algorithm}.
\section{Learning-based Sequential Algorithm}\label{sec:learning}
We first briefly discuss the approach first introduced in \cite{bertsimas2021voice}. Given a parametric optimisation problem with parameter $q$
\begin{align}\nonumber
\min\,\,& c^T{\mathbf{x}}\\ \nonumber
\text{subject to }& {\mathbf{x}}\in \FF(q),\,\,\, \\ \label{eq:parametric_MILP}
& {\mathbf{x}}\in {\mathbb{R}}^{d_R}\times{\mathbb{Z}}^{d_Z},
\end{align}
an optimal strategy $S(q)$ is defined using which one can solve a reduced problem returning a solution identical to the optimal solution of \eqref{eq:parametric_MILP}. We remark that Problem \eqref{eq:parametric_MILP} is identical to \eqref{eq:MICP} when the set of uncertainty $\mathbb{Q}$ reduces to a singleton. The optimal strategy is defined as the set of basis for continuous problems and the group of active constraints jointly with the integer part of decision variable at the optimal point for mixed-integer problems. For continuous problems, one can solve the problem only subject to the basis constraints and for mixed-integer problems, the optimal value of the continuous part of the decision variable can be obtained by fixing the integer part to the value provided by the optimal strategy and solving a reduced problem whose constraint set involves only the set of active constraints. In \cite{bertsimas2021voice}, a multiclass classification problem is formulated to learn the mapping from parameter $q$ to the optimal strategy $S(q)$. A set of parameters $q_i\in\mathbb{Q}, i=1,\ldots,M$ is generated randomly and for each parameter $q_i$ the optimal strategy $s_i$ is computed. The training data $(q_i,s_i), i=1,\ldots,M$, with $q_i$ as parameters and $s_i$ as the corresponding labels identifying the optimal strategy is used in training a multiclass classifier $\widehat{S}$. Given an unseen parameter $q_i$, the goal of the classifier $\widehat{S}$ is to identify a strategy as close as possible to the optimal strategy. The classifier can assist in reducing the computational complexity of solving the parametric optimisation problem. The approach is very useful for parametric online optimisation problems where we repeatedly want to solve Problem \eqref{eq:parametric_MILP} for slightly different parameters $q$. In the online phase, where the goal is to solve \eqref{eq:parametric_MILP} for a particular parameter $q$, the parameter is first given to the trained classifier to estimate the optimal strategy, $\widehat{s} = \widehat{S}(q)$. Subsequently, the optimal strategy is used for finding the optimal solution. Performing the mentioned two steps is much less computationally complex than directly solving \eqref{eq:parametric_MILP}.
The most computationally demanding part of Algorithm~\ref{alg:sequential algorithm} is to solve the mixed-integer problem and identify the corresponding basis at line 7. This step involves solving and finding the basis of an optimisation problem for which the set of constraints includes the current basis $B(t)$ along with the constraints formed by the violating samples $F(q^{\texttt{Viol}})$. The computational complexity of solving the problem and finding its basis has a direct relationship with the actual number of constraints involved in the problem. Hence, if we reduce the number of constraints without changing its solution or the set of basis, it is very likely that the complexity of the step presented at line 7 of Algorithm~\ref{alg:sequential algorithm} is reduced. This observation was the main motivation for using a learning-based strategy to reduce the computational complexity of Algorithm~\ref{alg:sequential algorithm}.
The computational burden can in fact be reduced using the approach presented in \cite{bertsimas2021voice}. To this end, we take a similar approach as \cite{bertsimas2021voice} to train a multiclass classifier $\widehat{S}$ which---having the violating sample $q^{\texttt{Viol}}$---can identify the basis constraint of an optimisation problem of the form \eqref{eq:parametric_MILP}. We use the classifier as an intermediate step to compute the basis of the problem $(F(q^\texttt{Viol})\cup B(t),c)$ (see line 7 of Algorithm \ref{alg:sequential algorithm}). The violating sample is first fed to the classifier $\widehat{S}$ to estimate basis of the problem $B_{q^\texttt{Viol}} = \widehat{S}(F(q^\texttt{Viol}),c)$ and next, the estimated basis is used in $[{\mathbf{x}}(t+1), B(t+1)]={\texttt{Solve\textsubscript{MIP}}}(B_{q^\texttt{Viol}}\cup B(t),c)$ to find basis of the problem $(F(q^\texttt{Viol})\cup B(t),c)$. It is worth noting that since the number of constraints in $B_{q^\texttt{Viol}}$ is much smaller than the ones in $F(q^\texttt{Viol})$, the primitive ${\texttt{Solve\textsubscript{MIP}}}(B_{q^\texttt{Viol}}\cup B(t),c)$ would be computationally much cheaper than ${\texttt{Solve\textsubscript{MIP}}}(F(q^\texttt{Viol})\cup B(t),c)$, see Tables \ref{tab:OPF results}, \ref{tab:unit commitment results}, and \ref{tab: simulation results MILP} which support this claim.
We remark that the optimal strategy used in \cite{bertsimas2021voice} for mixed-integer problems is the set of active constraints together with the integer part of decision variables at the optimal point. However, this strategy would not be useful in our case. For this reason, we modify the optimal strategy for the mixed-integer problem to be the basis, see Remark \ref{rem:basis computation} on how to compute the basis. To train the classifier, we first randomly generate a number of samples from the set of uncertainty $\mathbb{Q}$ and for each sample compute the basis. After generating all the training samples, we use a one-hot encoding on the collection of the basis generated for all the training samples to define labels suitable for the multiclass classifier. There are a number of multiclass classifiers in the literature, however we used a deep neural network to model multiclass classification \cite{bengio2007learning,lecun2015deep} as it shows superior performance compared to similar approaches such as support vector machine \cite{vapnik_statistical_1998} or random forest \cite{breiman2001random}. More details on deep neural network architectures are provided in Section \ref{sec:simulation}.
Any multiclass classifier has certain probability of misclassification. In order to handle misclassification, we modify Algorithm~\ref{alg:sequential algorithm} so to minimise the effect or incorrect basis estimation. From Theorem \ref{thm:property of algorithm}, we know that the objective value of the candidate solution monotonically increases while Algorithm \ref{alg:sequential algorithm} progresses. A problematic misclassification is when a basis is incorrectly estimated, which results in the objective value of the candidate solution to stop increasing or even decrease. Such a problematic misclassification can easily be recognised by comparing the objective value of the current candidate solution with the objective value of the candidate solution at the previous iteration. If the objective value is non-increasing, we use $[{\mathbf{x}}(t+1),B(t+1)]={\texttt{Solve\textsubscript{MIP}}}(F(q^\texttt{Viol})\cup B(t),c)$ to find the candidate basis and update the candidate solution. However, given that the classifier has usually very low probability of misclassification, the algorithm would rarely need to find the basis of the full problem. The full modified algorithm is detailed in Algorithm \ref{alg:learning sequential algorithm}.
We limit the value of $r$---the number of violating samples returned by the verification primitive---to $1$ to simplify the classifier training procedure.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\caption{Learning-based Sequential Algorithm}
\label{alg:learning sequential algorithm}
\STATE\algorithmicrequire{ $c,\,\widehat{S},\,d_R,\,d_Z,\,F(q^{(i)}), i = 1, \ldots N,$}
\STATE\algorithmicensure{ ${\mathbf{x}}_{\texttt{seq}},B_{\texttt{seq}}$}\\
{\bf Initialization:}
\STATE Set $m=(d_R+1)2^{d_Z}$, ${\texttt{feasible}}=0$, $t=0$
\STATE $[{\mathbf{x}}(0),B(0)]={\texttt{Solve\textsubscript{MIP}}}(F(q^{(1)})\cup\ldots\cup F(q^{(m)}),c)$\\
{\bf Evolution:}
\WHILE{${\texttt{feasible}}==0$}{
\STATE $[q^\texttt{Viol},{\texttt{feasible}}]={\texttt{Verification}}(F(q^{(1)}), \ldots,F(q^{(N)}), {\mathbf{x}}(t),1)$
\STATE $B_{q^\texttt{Viol}} = \widehat{S}(q^\texttt{Viol})$
\STATE $[{\mathbf{x}}(t+1),B(t+1)]={\texttt{Solve\textsubscript{MIP}}}(B_{q^\texttt{Viol}}\cup B(t),c)$
\IF{$J(B(t+1))\leq J(B(t))$}{
\STATE $[{\mathbf{x}}(t+1),B(t+1)]={\texttt{Solve\textsubscript{MIP}}}(F(q^\texttt{Viol})\cup B(t),c)$
}
\ENDIF
}
\STATE $t = t + 1$
\ENDWHILE
\STATE Set ${\mathbf{x}}_{\texttt{seq}} = {\mathbf{x}}(t+1) $ and $B_{\texttt{seq}}=B(t+1)$
\RETURN ${\mathbf{x}}_{\texttt{seq}},B_{\texttt{seq}}$
\end{algorithmic}
\end{algorithm}
\section{Numerical Examples}\label{sec:simulation}
To thoroughly test the performance of Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm}, we have considered a comprehensive series of numerical simulations. Specifically, we consider a wide range of different problems including robust optimal power flow, robust unit commitment, and robust mixed-integer linear programming to quantitatively assess the effectiveness of the proposed algorithms. To this aim, the performance of these algorithms are compared in terms of time required to complete the optimisation task. Furthermore, the performance of the presented algorithms is benchmarked against a direct solution of sampled optimisation problem obtained with widely used commercial solvers such as Gurobi \cite{gurobi} and Mosek \cite{andersen2000mosek}---we use Mosek for the robust optimal power flow problem which includes semidefinite constraints since Gurobi is unable to handle such constraints.
All simulations are performed on a Linux computing cluster in Rio Tinto Centre for Mine Automation. For all simulations, we allocated $12$ CPUs and $64$ GB of RAM.
\subsection{Classifier Training}
Algorithm~\ref{alg:learning sequential algorithm} requires a classifier to estimate the basis at line 7. In this subsection we discuss training the classifier and tuning its hyperparameters for all the numerical examples presented in the subsequent subsections. In order to train the classifier we first need to generate several training samples $(q_i,s_i), i=1,\ldots, M$. The training sample is a tuple which includes uncertainty instance $q_i$ and its corresponding label $s_i$ which defines the optimal strategy. In order to generate training data, we start by extracting $M$ samples $q_i, i=1,\ldots,M$ from the uncertainty set $\mathbb{Q}$, and solve the parametric optimisation problem of the form \eqref{eq:parametric_MILP} formed at the extracted samples. Next, the basis is identified using the procedure mentioned in Remark \ref{rem:finding basis} and is encoded to form the optimal strategy. There are several ways to encode the optimal strategy; we use a one-hot encoding approach. In this method a vector of all zeroes is created whose dimension is equal to the number of unique strategies found in the training data. If a data point belongs to the $i$th unique strategy, the $i$th component of the vector is set to $1$.
We need to train a multiclass classifier to estimate the optimal strategy. We use a classical Neural Network (NN) approach to design the classifier. The neural network classifier has an input layer with dimension equal the number of uncertain parameters, a number of inner (hidden) layers each with a depth that needs to be tuned, and an output layer whose dimension is equal to the number of unique strategies. The activation function is selected to be the rectified linear unit (ReLU) for all the layers except the last (output) layer which has a softmax activation due to the multiclass nature of the classification problem.
We use the Keras library \cite{chollet2015keras} from TensorFlow \cite{tensorflow2015-whitepaper} to implement the NN model. There are several hyperparameters---such as the number of hidden layers, depth of each hidden layer, batch size, number or epochs, and optimiser---that should be tuned to design a classifier with the smallest misclassification error. Classically, $80\%$ of the data is used for training and the rest is used for testing the performance of the NN. We use a grid search method and $K$-fold cross-validation from the scikit-learn library~\cite{scikit-learn} to tune the hyperparameters. Moreover, we design and tune three classifiers for the three problems discussed in the subsequent sections. The configuration and hyperparameters used for training the three NN classifiers are shown in Table~\ref{tab:classifier parameters}. We also report accuracy observed over the training and test sets. All classifiers exhibit a very high accuracy on both training and test sets.
We use Matlab to model, generate the training samples, and solve the sampled optimisation problem for the optimal power flow and unit commitment problems while mixed-integer linear programming problem is modelled and solved in Python. For the optimal power flow and unit commitment problems the trained classifiers---which are trained using TensorFlow library---are exported to Matlab to predict the optimal strategy when using Algorithm~\ref{alg:learning sequential algorithm}.
\subsection{Robust Optimal Power Flow}
Optimal Power Flow (OPF) is an optimisation problem solved at regular intervals to define the operating point of controllable generators in power grids. Given the predicted demand, and network configuration, resources and limitations, OPF defines active power of controllable generators and their magnitude of complex bus voltage so that the generation cost is minimised and the network constraints---such as line loading, min/max power rating of generators, and bus voltage---are respected. The increasing penetration of renewable energy resources introduces a large amount of uncertainty in the OPF. When uncontrollable resources fluctuate, the classical OPF solution can be very inefficient and may result in line overloads and potentially cascading outages. This calls for a robust strategy that generates policies which minimise the generation cost, and at the same time ensure that the network constraints are not frequently violated.
One of the successful approaches in designing a robust strategy is to use stochastic methods based on the scenario approach \cite{chamanbaz2019probabilistically,ETH-AC}, see \cite{chamanbaz2019Encyclopedia} for a full survey on available techniques. However, due to the complexity of the OPF problem and the fact that the number of decision variables is large, sampled optimisation Problem \eqref{eq:sampled optimisation problem} becomes very complex. For instance, for New England 39-bus systems case, choosing $\varepsilon=0.1, \delta = 1 \times10^{-10}$ the approach presented in \cite{ETH-AC} requires $11,831$ scenario samples and the sampled optimisation problem takes an impractical amount of time to get solved, see \cite[Section V]{chamanbaz2019probabilistically}.
\begin{table*}[t]
\caption{Configuration and hyperparameters used for training neural network classifiers.}
\begin{center}
\scalebox{1}{
\resizebox{1\textwidth}{!}{
\begin{tabular}{c||c|c|c|c|c|c|c||c|c}
\toprule
Problem & \# Training & \# Unique& \# Hidden & \# Epochs & Batch & Optimiser & Width of & Training & Test \tabularnewline
& Samples & Strategies & Layers & & Size & & Hidden Layers & Accuracy & Accuracy \tabularnewline
\midrule
\midrule
Optimal Power Flow & $28,000$ & $35$ & $3$ & $200$ & $256$ & Adam &$512$ & $98.27\%$ & $98.25\%$ \tabularnewline \midrule
Unit Commitment & $28,000$ & $138$ & $3$ & $300$ & $128$ & Adam & $512$ & $96.58\%$ & $96.41\%$ \tabularnewline \midrule
Mixed-integer Linear Program & $51,000$ & $300$ & $2$ & $200$ & $1024$ & Adam &$512$ & $96.47\%$ & $96.42\%$ \tabularnewline
\bottomrule
\end{tabular}}
}
\end{center}
\label{tab:classifier parameters}
\end{table*}
We modified the New England 39-bus system case to include $4$ wind generators connected to buses $5,\,6,\,14$ and $17$ and used the scenario based stochastic method presented in \cite{ETH-AC} to formulate the sampled optimisation problem. The number of uncertain parameters in the problem is 4---corresponding to uncertain active power generated by renewable generators. The penetration level of renewable generator is $30\%$ meaning that renewable generators can provide up to $30\%$ of the total demand. The uncertainty distribution is chosen by Pearson system with a standard deviation equal to $0.2\times$ (predicted generation power) and kurtosis of $3.5$ leading to a leptokurtic distribution with a heavier tail than that of a Gaussian.
The configuration and parameters for training the neural network classifier are tabulated in Table \ref{tab:classifier parameters}.
In Table \ref{tab:OPF results}, we report the time that it takes to directly solve the sampled problem using Mosek \cite{andersen2000mosek}, Algorithm \ref{alg:sequential algorithm}, and Algorithm \ref{alg:learning sequential algorithm} for different number of scenario samples. For small number of scenario samples, e.g. $N =100$, Algorithm \ref{alg:sequential algorithm} is slower than directly solving the sampled problem using Mosek, however, Algorithm\ref{alg:learning sequential algorithm} is still faster than Mosek. For large number of samples, Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} both outperform Mosek. For instance, when the number of scenario samples is $10^4$, Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} are respectively $21$, and $32$ times faster than Mosek.
\begin{table}[t]
\caption{CPU time taken to directly solve the sampled optimal power flow problem using Mosek, and the CPU time it takes for Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} to solve the problem for different values of the scenario samples.}
\begin{center}
\begin{tabular}{c||c|c|c}
\toprule
\# Scenario & CPU Time & CPU Time & CPU Time \tabularnewline
Samples & Mosek & Algorithm \ref{alg:sequential algorithm} & Algorithm \ref{alg:learning sequential algorithm} \tabularnewline
\midrule
\midrule
$10^2$ & $121.4$ & $127.9$ & $56.3$ \tabularnewline \midrule
$10^3$ & $1598.8$ & $363.9$ & $165.7$ \tabularnewline \midrule
$5\times 10^3$ & $3.06\times 10^{4}$ & $2594.4$ & $1286.7$ \tabularnewline \midrule
$10^4$ & $1.18\times 10^5$ & $5657.9$ & $3686.5$ \tabularnewline
\bottomrule
\end{tabular}
\end{center}
\label{tab:OPF results}
\end{table}
\subsection{Robust Unit Commitment}
Unit commitment is a mathematical optimisation problem solved in power grids to determine the commitment of each generator. It considers a time horizon and given a predicted demand over the considered horizon, and generators' minimum and maximum power ratings, its solution defines which generators should be on-line and which ones should be off-line so that the total generation cost is minimised. There are many models developed for the unit commitment problem in the literature, see \cite{haaberg2019fundamentals} and references therein for a full review on the topic. In this subsection, we consider a simplistic version of this problem. Sets, indices, and variables used in defining the model are introduced first.
\begin{tabular}{p{0.2\columnwidth} p{0.75 \columnwidth}}
$\mathcal{G}$& set of generators in the grid with cardinality $n_g,\, |\mathcal{G}|=n_g$\\
$T\in\mathbb{N}$ & time horizon over which the problem is solved \\
$t\in\{1,\ldots,T\}$ & time periods\\
$P_{i,t}\in\mathbb{R}$ & active power generated by generator $i$ at time period $t$\\
$U_{i,t}\in\{0,1\}$ & on-off status of generator $i$ at time $t$\\
$D_t\in \mathbb{R}$ & demand at time $t$\\
$P^{\min}_{i,t}\in\mathbb{R}$ & minimum active power generator $i$ should provide at time $t$ \\
$P^{\max}_{i,t}\in\mathbb{R}$ & maximum active power generator $i$ can provide at time $t$ \\
$\tau^\text{OFF}_i$ & unit $i$ must be off-line for $\tau^\text{OFF}_i$ before it can be on-line\\
$\tau^\text{ON}_i$ & unit $i$ must be on-line for $\tau^\text{ON}_i$ before it can be off-line\\
$\Delta P^{\max}_{i,t}$ & maximum allowed difference between power generated by generator $i$ at time $t$ and $t-1$
\end{tabular}
\textbf{Objective}\\
The objective is to minimise the total operating cost of all generators across the grid
\[
f(P) = \sum_{i}^{n_g}\sum_{t=1}^{T} Q_{ii}P^2_{i,t}+C_iP_{i,t},
\]
where $Q\in\mathbb{R}^{n_g\times n_g}$ and $C\in\mathbb{R}^{n_g}$ are respectively diagonal matrix, and vector defining the running cost of generators.
\textbf{Constraints}\\
The amount of power each generator can provide is constrained by the following constraint
\begin{equation*}
U_{i,t}P^{\min}_{i,t}\leq P_{i,t}\leq U_{i,t}P^{\max}_{i,t}.
\end{equation*}
The total power generated by active generators should meet demand at all time
\begin{equation*}
\sum_{i=1}^{n_g}P_{i,t}\geq D_t, \,\, \forall t=1,\ldots,T.
\end{equation*}
Minimum up-time and down-time of each generator is defined using the following constraint
\begin{equation*}
U_{i,\tau}\geq U_{i,t}-U_{i,t-1},\, \tau = t,t+1,\ldots,\min(T,t+\tau^\text{ON}_i-1),\,\forall i\in\mathcal{G},\,t = 2,\ldots,T.
\end{equation*}
\begin{equation*}
U_{i,\tau}\leq 1- U_{i,t-1}-U_{i,t},\, \tau = t,t+1,\ldots,\min(T,t+\tau^\text{OFF}_i-1),\,\forall i\in\mathcal{G},\,t = 2,\ldots,T.
\end{equation*}
The above two constraints require generator $i$ to remain on-line (resp. off-line) for $\tau^\text{ON}_i$ (resp. $\tau^\text{OFF}_i$) time periods before they can go off-line (resp. on-line).
The following ramp constraint limits the rate of change of active power generated by each generator at each sampling time
\begin{equation*}
P_{i,t}-P_{i,t-1}\leq\Delta P_{i,t}^{\max}, \, \forall t = 1,\ldots,T,\, \forall i\in\mathcal{G}.
\end{equation*}
The demand at time $t$, denoted as $D_t$, is not fully known. To capture the uncertainty associated with demand, we assume that $D_t$ is constructed by a nominal predicted demand $D^0_t$ and an uncertain demand $D^q_t$:
\[
D_t = D^0_t + D^q_t, \,\,t = 1,\ldots,T.
\]
\begin{table}[t]
\caption{CPU time it takes to directly solve the sampled unit commitment problem using Gurobi, and the CPU time it takes for Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} to solve the problem for different values of the scenario samples.}
\begin{center}
\begin{tabular}{c||c|c|c}
\toprule
\# Scenario & CPU Time & CPU Time & CPU Time \tabularnewline
Samples & Gurobi & Algorithm \ref{alg:sequential algorithm} & Algorithm \ref{alg:learning sequential algorithm} \tabularnewline
\midrule
\midrule
$10^2$ & $17.2$ & $27.1$ & $11.6$ \tabularnewline \midrule
$10^3$ & $288.2$ & $36.1$ & $14.5$ \tabularnewline \midrule
$5\times 10^3$ & $3904.1$ & $68.7$ & $42.2$ \tabularnewline \midrule
$10^4$ & $2.3\times 10^4$ & $106.9$ & $72.9$ \tabularnewline
\bottomrule
\end{tabular}
\end{center}
\label{tab:unit commitment results}
\end{table}
For computational purposes, we select $n_g=4,\,T=12,\,P^{\max}_{i,t} = \texttt{randint(1,115)},\,P^{\min}_{i,t}=\lceil P^{\max}_{i,t}/2\rceil, \, D_t^0=150\sin(2\pi/24\,t),t=1,\ldots,T,\,Q=\text{diag}(\texttt{randint(1,50)}),\,C = \texttt{randint(1,50)},\, \tau^{\text{ON}}_i=\texttt{randint(1,T)},\,\tau^{\text{OFF}}_i=\texttt{randint(1,T)},\forall i \in \mathcal{G}$. The uncertain component of the demand is bounded in $[-1,1]$, i.e. $D_t^q\in[-1,1], \forall t = 1,\ldots,T$. The configuration and parameters used for training the neural network classifier used in Algorithm \ref{alg:learning sequential algorithm} is shown in Table \ref{tab:classifier parameters}. We used Gurobi version $9.1.2$ to solve sampled optimisation problem for different number of scenario samples and compared its performance in terms of the time it takes to solve the problem with Algorithms \ref{alg:sequential algorithm}, and \ref{alg:learning sequential algorithm}. The result of this simulation is shown in Table \ref{tab:unit commitment results}. Similar to Table \ref{tab:OPF results}, for a small number of samples directly solving the sampled optimisation problem using Gurobi results in a smaller solution time than Algorithm \ref{alg:sequential algorithm}. However, for a large number of scenarios, both Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} notably outperform Gurobi. For $N=10,000$ (last row of Table \ref{tab:unit commitment results}), Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} are respectively $217$ and $319$ times faster than the direct solution obtained using Gurobi. This shows the significant computational improvement one can achieve by using Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm}.
\subsection{Robust Mixed-integer Linear Programs}\label{sec:lp}
Classical robust Mixed-Integer Linear Programming (MILP) problems admit the following formulation
\begin{align}\label{eq:milp}
&\text{minimize} \qquad c^T{\mathbf{x}}\\ \nonumber
&\text{subject to: } A{\mathbf{x}}\leq b+b_q, \\ \nonumber
& \qquad\qquad{\mathbf{x}}\in{\mathbb{R}}^{25}\times {\mathbb{Z}}^{5},
\end{align}
where the objective definition is defined by the vector $c\in{\mathbb{R}}^{30}$, while $A\in{\mathbb{R}}^{500\times 30},\, b\in{\mathbb{R}}^{500}$ constitute the (fixed) matrix and vector used to define the set of nominal constraints of Problem \eqref{eq:milp}, and $b_q\in{\mathbb{R}}^{500}$ is a so-called interval vector---i.e., a vector whose entries vary in given intervals---characterizing the uncertainty in the optimisation problem \eqref{eq:milp}. The vectors $b,c$ and nominal matrix $A$ are generated such that problem \eqref{eq:milp} is feasible.
To this end, we follow the methodology presented in
\citep{dunham_experimental_1977}.
The distribution of uncertain vector $b_q$ is uniform and its entries are bounded in $b\times[-0.01, 0.01]$.
The sampled version of problem \eqref{eq:milp} is constructed by extracting random samples $\{b_q^{(i)}\}_{i=1}^N$ from the set of uncertainty
\begin{align}\label{eq:sampled milp}
&\text{minimize} \qquad c^T{\mathbf{x}}\\ \nonumber
&\text{subject to: } A{\mathbf{x}}\leq b+b_q^{(i)}, \,\, i=1,\ldots,N\\ \nonumber
& \qquad\qquad{\mathbf{x}}\in{\mathbb{R}}^{25}\times {\mathbb{Z}}^{5}.
\end{align}
The hyper-parameters used in training the deep neural network classifier are listed in Table \ref{tab:classifier parameters}.
In Table \ref{tab: simulation results MILP}, we vary the number of scenario samples $N$ and solve problem \eqref{eq:sampled milp} using Algorithms \ref{alg:sequential algorithm} and, \ref{alg:learning sequential algorithm} and compare their performance against directly solving Problem \eqref{eq:sampled milp} using Gurobi \cite{gurobi}.
For a small number of scenario samples, Algorithm \ref{alg:sequential algorithm} is slower than directly solving the sampled optimisation problem \eqref{eq:sampled milp} using Gurobi. However, for all the scenario samples, Algorithm \ref{alg:learning sequential algorithm} outperforms Gurobi and Algorithm \ref{alg:sequential algorithm}. It is worth noting that for some entries in Table \ref{tab: simulation results MILP}, the Gurobi solver has been found to require more than $64$ GB of RAM to complete the solution process, thus preventing it from completing this task on the cluster. This highlights yet another major advantage of Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} in the fact that they achieve significant memory savings compared to classical algorithms meant to solve sampled optimisation problems associated with large number of constraints.
\begin{table}[t]
\caption{CPU time it takes to directly solve the sampled MILP problem using Gurobi, and the CPU time it takes for Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm} to solve the problem for different values of the scenario samples. NA refers to the case that Gurobi requires more resources to solve the problem.}
\begin{center}
\begin{tabular}{c||c|c|c}
\toprule
\# Scenario & CPU Time & CPU Time & CPU Time \tabularnewline
Samples & Gurobi & Algorithm \ref{alg:sequential algorithm} & Algorithm \ref{alg:learning sequential algorithm} \tabularnewline
\midrule
\midrule
$10^4$ & $89.3$ & $232$ & $78.8$ \tabularnewline \midrule
$5\times 10^4$ & $466.5$ & $292.2$ & $109.9$ \tabularnewline \midrule
$10^5$ & $1220$ & $295.2$ & $161.1$ \tabularnewline \midrule
$5\times 10^5$ & NA & $615.5$ & $452.1$ \tabularnewline
\bottomrule
\end{tabular}
\end{center}
\label{tab: simulation results MILP}
\end{table}
\section{Conclusion}
In this paper, we presented two algorithms for solving sampled optimisation problems. Both algorithms exhibit a significant saving in time and memory required for solving this class of optimisation problems.
Both algorithms involve two main steps: verification and optimisation which are performed sequentially to converge toward the optimal solution. At each step of these algorithms, we need to compute the basis---minimal set of constraints defining the current solution. Algorithm \ref{alg:learning sequential algorithm} features a neural network multiclass classifier to reduce the complexity associated with finding the basis at each iteration of the algorithm.
The convergence properties of both algorithms are analysed and extensive numerical simulations are performed to compare their performance---in solving various non-trivial sampled optimisation problems---against widely used commercial solvers.
The two proposed algorithms significantly reduce the computational time of solving problems for which the number of constraints is much larger than the number of decision variables. If the number of decision variables is large, the combinatorial dimension of the problem might grow---see Theorem \ref{thm:combinatorial dim} for the exact upper bound---leading to a possible increase in the complexity of Algorithms \ref{alg:sequential algorithm} and \ref{alg:learning sequential algorithm}. A possible future direction is to combine the sequential nature of the two proposed algorithms with column generation methods \cite{desaulniers2006column,ford1958suggested} to reduce the computational complexity for the case that the number of decision variables is large.
|
2,869,038,155,704 | arxiv | \section{Introduction}
Owing to the high precision and long duration time series provided during the last decade by space missions such as {\it Kepler} \citep{Borucki2010,Koch2010}, K2 \citep{2014PASP..126..398H}, and CoRoT \citep{Auvergne2009}, the field of asteroseismology has led a revolution in stellar astrophysics. The power of the method relies in accessing the stellar interiors through the study of the surface manifestation of internal resonant oscillations. In addition to its contribution to stellar physics \citep[e.g.,][]{chaplin2013,Hekker2017,Bowman2017,Garcia2019}, asteroseismology has also helped advance the field of exoplanets \citep{VanEylen2014,Lundkvist2016}.
In April, 2018, the Transiting Exoplanet Survey Satellite (TESS) joined the short list of space-based telescopes dedicated to finding planets by means of the transit method \citep{Ricker2015}. TESS hosts four charge coupled device (CCD) cameras aligned with the ecliptical poles, that stare at the same fraction of the sky for two of TESS orbits (2$\times$13.7 days, approximately). The observations collected by the four CCDs during two consecutive orbits are defined as a sector. Due to the large field of view of the CCDs (24$\times$24 degrees each), the Ecliptic hemispheres are divided in 13 sectors, specifically 13 in the southern hemisphere and 13 in the northern hemisphere during the primary mission. Different from {\it Kepler}, TESS is designed to detect transiting planets around very bright stars, which permits us to easily carry out ground-based radial velocity follow-ups to determine planetary masses \citep{Trifonov2019,Rodriguez2019}. However, using TESS for asteroseismology introduces strong timing requirements \citep{Lund2017}. Although the internal clock of TESS might be very accurate in its own time, it can have a constant drift or offset or variation in the length of a second, caused by hardware limitations, software errors, lags in electronics after safe-modes/downlinks, missed leap seconds, and wrong reference frames, among others. In consequence, time stamps need verification and possibly calibration.
The TESS Asteroseismic Science Consortium (TASC) hosts the group ``TESS Data for Asteroseismology'' (T'DA) which is in charge of delivering light curves for all of TASC, hence encompassing many different types of stars, including all targets found in full frame images. Requested by the TESS Science Processing Operations Center (SPOC), T'DA was also asked to carry out independent verification of TESS timestamps. This exercise is required to ensure the highest level of asteroseismic inference from TESS data, and works as a mechanism to prevent and diagnose any timing malfunction, as it happened to {\it Kepler} timestamps\footnote{\url{https://archive.stsci.edu/kepler/release_notes/release_notes19/DataRelease_19_20130204.pdf}}. To carry out this work, TESS has been continuously observing a modest list of eclipsing binary systems (EBSs) with relatively short periods, most of them between 0.7 and 4.5 days with the exception of TV Nor, which has an orbital period of 8.5 days. In order to achieve accurate timing measurements, the EBSs are mostly of Algol type presenting deep, V-shaped, and relatively short eclipses. They cover a range of latitudinal and longitudinal ecliptic coordinates, to ensure observability throughout TESS's first year.
In this work we present the timing requirements to be able to carry out asteroseismology using TESS data in Section~\ref{sec:req}, and we show the photometric data collected from two ground-based telescopes located in Argentina and gathered by TESS in Section~\ref{sec:Data}. We detail our strategy for determining the mid-eclipse times and the model functions used in Section~\ref{sec:Mid-Eclipses}, and we present the timing verification computed from contemporaneous ground and ground, and ground and space-based data in Sections~\ref{sec:time_test} and~\ref{sec:time_offset}, along with the timing verification carried out solely using TESS data in Section~\ref{sec:time_drift}. We close this work with our final remarks in Section~\ref{sec:DyC}.
\section{Timing requirements for asteroseismology}
\label{sec:req}
The requirements for timing for asteroseismology are mainly of importance for high amplitude coherent oscillators (such as $\delta$ Sct and RR-lyr stars), while the requirements for stochastic oscillators are less strict. The formal requirements are specified in the internal document SAC\_TESS\_0002\_5\footnote{\url{https://tasoc.dk/docs/SAC_TESS_0002_5.pdf}}, which discuss three main categories: (1) accurate values for the exposure length, required to reach the photon noise limit (requirement RS-TASC-01); (2) accurate knowledge of differential times within one month of observations, required to reach the theoretical accuracy on oscillation mode frequencies and amplitudes (especially important for coherent oscillators). Important here is also the conversion of spacecraft times to barycentric julian date (BJD), which should be as accurate as the determination of differential times (requirements RS-TASC-02 and RS-TASC-03)\footnote{Measured to 10 ms (rms) (R.~K.~Vanderspek, private communication)}; (3) to compare observations from TESS with ground-based facilities the absolute time in BJD needed (requirement RS-TASC-04).
The requirements are strongest for bright high-amplitude coherent oscillators. Considering a $m_{V}=4$ star with an amplitude of $10\%$ relative variability and a period of a few hours, target values have been set to 5 msec over the course of an observing sector for points (1) and (2), while the target value is 0.5 sec for point (3). For a solar-like oscillator the times should be accurate over a period of 10 days to better than 1 sec (3 sec for a red giant oscillator).
In this analysis we consider points (2) and (3) of the above, and refer to SAC\_TESS\_0002\_5 for more details \citep[see also][]{Montgomery1999}.
\section{Observations and data analysis}
\label{sec:Data}
\subsection{Ground-based photometry}
The ground-based observations presented in this work were collected using mainly the 2.15 meter telescope, {\it Jorge Sahade} (henceforth, CASLEO-2.15, programs JS-2018B-14, JS-2019A-02) and to a lesser extent the 0.6 meter telescope {\it Helen Sawyer Hogg} (henceforth, CASLEO-0.60, Director's Discretionary Time). Both telescopes are located at the Argentinian Complejo Astron\'omico El Leoncito (CASLEO). For CASLEO-2.15 we used a Roper Scientific model VersArray 2048B camera with a charge coupled device (CCD) detector (manufactured by Princeton Instruments) to collect the photometry. The imaging area is 2048$\times$2048 pixels, where each pixel is 13.5$\times$13.5 $\mu$m. The CCD is sensitive to wavelengths between 300 and \mbox{1000 nm}. To reduce dark current, the camera is cooled with liquid nitrogen and kept at approximately $-120$ degrees Celsius. With the mounted focal reducer, the circular, unvignietted field-of-view has a diameter of ${\sim}$9 arcminutes. CASLEO-0.60 has a SBIG STL-1001E CCD, which is exclusively used for photometry. The imaging area is 1024$\times$1024 pixels, with a pixel size of 24$\times$24 $\mu$m. The CCD is sensitive to wavelengths between 400 and \mbox{1000 nm}, and is cooled down with a Peltier system. The telescope doesn't suffer vignetting, so the total fiel of view is 9.26$\times$9.26 arcminutes. All our observations were performed using an \textit{R} filter, with an effective central wavelength, $\lambda_o$, of 635 nm and a full-width at half maximum (FWHM) of 107 nm. The main reason for this choice was to use a filter with a transmission response as similar as possible to the transmission response of TESS ($\lambda_o$ = 785 nm, FWHM = 400 nm), minimizing differences in the light curves associated with the wavelength-dependent stellar limb darkening. For a better overall photometric quality, this filter also circumvents the large telluric contamination around the \textit{I}-band. Contrary to TESS's constant $120$-sec cadence, the exposure time of the ground-based light curves depends mainly on the brightness of the star of interest, the altitude of the star during observations, and the photometric quality of the night during observations. In consequence, during an observing run we adjusted the exposure time so that the peak of the target point spread function was kept at around half the dynamic range of the CCD. This choice allows for an adequate compromise between linearity and good signal.
To achieve high precision photometry from the ground, we observed with the telescopes slightly defocused \citep{Kjeldsen1992,Southworth2009}. The achieved photometric precision per observing run is listed in column 4 of Table~\ref{tab:ObsCond}, along with other quantities derived from our observations.
\begin{table*}[ht!]
\caption{\label{tab:ObsCond} Parameters derived from our ground-based observations. From left to right: the TESS Input Catalogue (TIC) and name of the observed target; the magnitude of the target in the TESS bandpass, m$_\mathrm{TESS}$; the date corresponding to the beginning of the local night; the telescope performing the observations; the standard deviation of the residual light curves in parts-per-thousand (ppt), $\sigma_{\rm res}$; the number of data points per light curve, $N$; the average cadence in seconds, CAD; the total observing time, $\Delta T_{\rm tot}$, in hours; the airmass range, $\chi_{\rm min,max}$, showing minimum and maximum values, respectively; the eclipse coverage, EC; a parameter to account for correlated noise, $\beta$ (see Section~\ref{sec:corrnoise}); and the derived time shifts, $\Delta t$, in seconds as compared to TESS data. The letter code specifying the eclipse coverage during each observation is as follows: O: out of eclipse, before ingress. I: ingress. B: bottom. E: egress. O: out of eclipse, after egress. Following the name the letter code (C) corresponds to the eclipses that have ground observations contemporaneous with TESS; (CG) corresponds to the eclipses that have contemporaneous observations between two ground-based stations; (NC) correspond to those that do not have contemporaneous observations, but a primary or secondary eclipse is clearly observed.}
\centering
\scalebox{0.92}{\hspace{-2cm}
\begin{tabular}{l c c c c c c c c c c c}
\hline \hline
TIC/Name & m$_\mathrm{TESS}$ & Date & Telescope & $\sigma_{\rm res}$ &$N$& CAD & $\Delta T_{\rm tot}$ & $\chi_{\rm min,max}$ & EC & $\beta$ & $\Delta t$ \\
& & yyyy.mm.dd & & (ppt) & & (sec) & (hours) & & & & (seconds) \\
\hline
69819180/KX Aqr (CG) & 7.66 & 2018.06.07 & CASLEO-2.15 & 4.2 & 297 & 51.8 & 4.27 & 1.01,1.96 & -IBE- & 1.05 & 17 $\pm$ 138 \\
69819180/KX Aqr (CG) & 7.66 & 2018.06.07 & CASLEO-0.60 & 8.3 & 460 & 35.6 & 4.55 & 1.01,1.69 & -IBE- & 1.05 & 17 $\pm$ 138 \\
349797905/NV Tel (NC) & 9.66 & 2018.07.08 & CASLEO-2.15 & 1.3 & 343 & 34.9 & 3.32 & 1.14,1.53 & -IBE- & 1.03 & 302 $\pm$ 276 \\
54018297/WY Cet (C) & 8.85 & 2018.09.27 & CASLEO-2.15 & 7.9 & 1719 & 14.6 & 6.98 & 1.09,1.87 & OI--- & 1.98 & 52 $\pm$ 77 \\
9945183/VV Eri (C) & 11.29 & 2018.10.30 & CASLEO-2.15 & 10.1 & 430 & 41.6 & 4.97 & 1.07,1.35 & -IBEO & 1.09 & -2.5 $\pm$ 10 \\
220402294/BD Dor (C) & 11.22 & 2018.11.10 & CASLEO-2.15 & 2.5 & 851 & 28.9 & 6.84 & 1.10,2.10 & OIBEO & 1.98 & -29 $\pm$ 26 \\
260161144/AO Pic (C) & 9.11 & 2018.12.10 & CASLEO-2.15 & 6.2 & 1462 & 18.4 & 7.45 & 1.11,1.58 & OI--- & 1.46 & 7 $\pm$ 12 \\
220402294/BD Dor (NC) & 11.22 & 2018.12.11 & CASLEO-2.15 & 2.8 & 955 & 28.4 & 7.53 & 1.10,1.50 & OIBEO & 2.54 & 37 $\pm$ 15 \\
80659292/AW Vel (C) & 10.32 & 2019.01.25 & CASLEO-2.15 & 6.7 & 1280 & 21.7 & 7.73 & 1.02,1.55 & OIBEO & 1.86 & 30 $\pm$ 4 \\
260161144/AO Pic (C) & 9.11 & 2019.01.26 & CASLEO-2.15 & 5.4 & 351 & 58.7 & 5.72 & 1.35,1.89 & OIBE- & 1.39 & -9 $\pm$ 11 \\
80659292/AW Vel (C) & 10.32 & 2019.01.27 & CASLEO-2.15 & 1.7 & 1109 & 24.1 & 7.41 & 1.08,1.45 & OIBEO & 2.88 & -1 $\pm$ 8 \\
220402294/BD Dor (C) & 11.22 & 2019.03.24 & CASLEO-2.15 & 2.2 & 445 & 22.8 & 2.75 & 1.15,1.25 & -IBE- & 1.01 & 8 $\pm$ 14 \\
219373406/X Pic (C) & 10.51 & 2019.04.19 & CASLEO-2.15 & 5.3 & 288 & 21.7 & 1.73 & 1.33,1.85 & -IBE- & 1.03 & 2 $\pm$ 8 \\
331183881/V636 Cen (NC)& 8.06 & 2019.05.20 & CASLEO-0.60 & 3.9 & 313 & 66.6 & 5.79 & 1.05,1.43 & OIBEO & 1.01 & -34 $\pm$ 15 \\
41561453/RR Nor (C) & 10.19 & 2019.05.23 & CASLEO-0.60 & 3.8 & 349 & 52.1 & 5.05 & 1.09,1.23 & -IBE- & 1.83 & -32 $\pm$ 7 \\
214716930/TV Nor (C) & 8.78 & 2019.06.07 & CASLEO-2.15 & 3.1 & 2188 & 12.4 & 7.58 & 1.06,1.53 & OIBE- & 1.01 & -30 $\pm$ 10 \\
41561453/RR Nor (C) & 10.19 & 2019.06.09 & CASLEO-2.15 & 11.6 & 811 & 17.0 & 3.83 & 1.12,1.70 & OI--- & 1.23 & 86 $\pm$ 60 \\
349797905/NV Tel (NC) & 9.66 & 2019.07.17 & CASLEO-2.15 & 5.9 & 484 & 34.8 & 4.69 & 1.04,1.77 & -IBE- & 3.12 & -224 $\pm$ 216 \\
\hline
\end{tabular}}
\end{table*}
The ground-based data are reduced and the light curves are constructed by means of the {\it Differential Photometry Pipelines for Optimum Lightcurves}, DIP$^2$OL. A full description of DIP$^2$OL can be found in \citet{vonEssen2018}. In brief, the first component of the pipeline is based on \texttt{IRAF}'s command language \citep{iraf1993}, and it does aperture photometry. First, normal calibration sequences take place, depending on the availability of bias, darks and flatfield frames. The reduction continues with cosmic ray rejection and posterior alignment of the science frames. Afterwards, reference stars within the field are automatically chosen, usually of similar brightness to the target star to minimize the noise in the differential light curves \citep{Howell2006}. Photometric fluxes and errors are measured for all stars with different apertures, usually dividing the range from 0.5 to 3 times the nightly averaged FWHM in ten, and for each of these we use three different background rings. In this work we do not detrend the data, as the eclipses are deep (usually $\Delta$Flux$\sim$50-80\%). Instead, we treat their noise as explained in Section~\ref{sec:corrnoise}. The second part of DIP$^2$OL is written in \texttt{Python}. The routine produces several light curves using different combinations of reference stars. The final differential light curve is the unweighted sum of the flux of the target star divided by the sum of the unweighted fluxes of the reference stars that produced the light curve with the smallest point-to-point scatter. The pipeline repeats this process per aperture and sky ring. The code outputs the time in Julian dates shifted to the center of each exposure, the differential fluxes, photometric error bars and the detrending quantities that are ignored in this work. In particular, a high degree and unphysical time-dependent polynomial is fitted to the light curve through least-squares minimization. A residual light curve is constructed producing the difference between the final light curve and the best-fit polynomial. From this residual light curve we compute the standard deviation, and we use this value to enlarge the photometric error bars, so that their average is at the same level of this standard deviation. With the light curves fully constructed, we convert the time stamps from Julian dates to Barycentric Julian dates, BJD$_\mathrm{TDB}$, using the web tool provided by \cite{Eastman2010}. The stars listed in Table~\ref{tab:ObsCond} are used in three different ways. Those having a (C) correspond to the eclipses that have contemporaneous observations with TESS, while those with a (NC) do not have contemporaneous observations, but an eclipse is clearly observed. Both were used to compute potential time offsets (Section~\ref{sec:time_offset}). Those with an (CG) correspond to eclipses that have contemporaneous observations from the ground, and were used to test the timings between CASLEO-2.15 and CASLEO-0.60 (Section~\ref{sec:time_test}).
\subsection{Correlated noise for the ground-based light curves}
\label{sec:corrnoise}
Several problems arise when observing stars from Earth as compared to space. Fluctuations in the atmosphere causing an abrupt dimming of the star, clouds suddenly appearing and poor tracking of the observed stellar fields are only some of the many nuisances that have to be overcome in order to obtain accurate, reliable data. Thus, different techniques have been developed in order to eliminate these nuisances. As the eclipses are deep and the target stars are relatively bright, we do not detrend the photometry, but rather increase the individual photometric uncertainties by the so-called $\beta$ factor to account for correlated noise \citep{Pont2006,Carter2009}.
In order to compute the $\beta$ factor, as described in \cite{vonEssen2013}, we first compute residuals by fitting a high order, non-physical, polynomial to the ground-based light curves. Then, we divide the residuals into $M$ bins of $N$ averaged data points. This average accounts for changes in exposure time that might be needed to compensate for changes in airmass or transparency during the observing runs. Due to the usual length of our ground-based data sets, we consider bins of four different lengths, namely 10, 15, 20, and 25 minutes.
In general, if the data have no correlated noise then the noise in the residuals should follow the expectation of independent random numbers:
\begin{equation}
\hat{\sigma}_N = \sigma N^{-1/2}[M/(M-1)]^{1/2}\ ,
\end{equation}
\noindent where $\sigma$ is the standard deviation of the unbinned residual light curve, and $\sigma_N$ corresponds to the standard deviation of the data binned with $N$ averaged data points per bin:
\begin{equation}
\sigma_N = \sqrt{\frac{1}{M}\sum_{i = 1}^{M}\left(\langle\hat{\mu}_i\rangle - \hat{\mu}_i\right)^2}\ .
\end{equation}
\noindent In the equation above, $\hat{\mu_i}$ corresponds to the mean value of the residuals per bin ($i$) and $\langle\hat{\mu_i}\rangle$ is the mean value of the means. $\beta$ is computed averaging $\beta_N$ = $\hat{\sigma}_N$/$\sigma_N$, computed in the time bins mentioned before. When we found $\beta$ to be larger than 1, we enlarged the individual photometric errors of the ground-based light curves by this factor, and only then we carried out the determination of the individual mid-eclipse times.
\subsection{TESS data}
During the first thirteen sectors, the eclipsing binary systems comprising our timing verification list were observed with a cadence of 120 seconds. For the 120-sec cadence data we adopted the PDCSAP light curves provided by the Science Processing Operations Center \citep[SPOC;][]{Jenkins2016} pipeline in the Target Pixel Files (TPFs)\footnote{https://archive.stsci.edu/missions/tess/doc/EXP-TESS-ARC-ICD-TM-0014.pdf}, which were downloaded from the TASOC database\footnote{tasoc.dk}. Only for \mbox{BD Dor} was it necessary for us to create custom light curves for Sectors 2-5, as during these sectors \mbox{BD Dor} was incorrectly associated with the target \mbox{TIC 220402290}. As this target lies only ${\sim}3$ pixels away from the correct target, \mbox{TIC 220402294}, both stars were included in the photometric aperture. The eclipses of \mbox{BD Dor} were in consequence observable, but were highly diluted by the contribution of \mbox{TIC 220402290} to the total flux. This missidentification, that was also found in other catalogues, was reported by our group to the Centre de Données astronomiques de Strasbourg and corrected. As previously mentioned, given the proximity of the two stars they are both contained in the TPFs for \mbox{TIC 220402290}, so we simply defined a new aperture around the correct target. In later Sectors, \mbox{BD Dor} is correctly associated with \mbox{TIC 220402294}.
\section{Determination of the mid-eclipse times}
\label{sec:Mid-Eclipses}
Depending on the specific binary system observed, and thus the spectral type of the stars, their relative sizes and their mutual distances, the overall shape of the eclipses will significantly change from one system to the other. One model to determine mid-eclipse timings would not accommodate a wide range of difference eclipse shapes. To overcome this we have developed three different ways to extract the eclipse timings of ground and space-based data.
The first involves the use of a time-dependent second order polynomial, the second an inverted Gaussian function, and the third is similar to a cross-correlation between two contemporaneous light curves. The first two techniques are specified in Section~\ref{sec:functions}, while the cross-correlation method is detailed in Section~\ref{sec:crosscorr}. Regardless of the model used, timing offsets between TESS and ground-based data are computed in three ways. From the three results, we always report the one with the smallest difference.
\subsection{Model functions for the mid-eclipse times}
\label{sec:functions}
A method for computing accurately the mid-eclipse times was first given by \cite{Kwee1956}. Following their approach, our first model corresponds to a time-dependent, second order polynomial,
\begin{equation}
f(t) = a t^2 + b t + c,\,
\end{equation}
\noindent where $a$, $b$ and $c$ are the fitting parameters. Here, the mid-eclipse time is computed as \mbox{$T_o = -b/2a$}, and its associated error is computed from standard error propagation.
The second model is an inverted Gaussian function,
\begin{equation}
g(t) = \beta - \alpha e^{-\frac{(t - \mu)^2}{2\sigma^2}},\,
\end{equation}
\noindent where $\alpha$, $\beta$, $\mu$ and $\sigma$ are the fitting parameters, and the mid-eclipse time is computed as \mbox{$T_o = \mu$}.
\subsection{Optimum window around mid-eclipse to derive accurate timings}
\label{sec:crosscorr}
While TESS data are largely continuous within a sector, ground-based observations face other challenges, mainly imposed by the diurnal rotation of the Earth and the cloud coverage. In consequence, the coverage from CASLEO does not resemble that from TESS. In some cases, the eclipse coverage is asymmetric, in some eclipses the instant of minimum flux is missing, and in some others there are gaps without data. This inconvenient coverage will have an impact in the precision of the derived timings. To overcome this, before calculating the mid-eclipse times we sort the data to find an optimum number of data points (and thus, eclipse coverage) that best match our models. The sorting function will gradually remove data points with a flux larger than a specified value. After each round of trimming, the remaining data points are fitted with our models (Section~\ref{sec:functions}). The gradual chopping starts at the maximal observed flux, $f_\mathrm{max}$, and ends at \mbox{$f_\mathrm{min}$ + $0.1 (f_\mathrm{max} - f_\mathrm{min})$}, where $f_\mathrm{min}$ corresponds to the minimum observed flux value, and 0.1 is user specified. The reason why the sorting does not reach $f_\mathrm{min}$ is to ensure that the amount of fitting parameters does not exceed the number of data points. An example of the use of this sorting strategy can be seen in Figure~\ref{fig:sorting}, where three different fits are shown for three different chopping values. As we are only determining the optimum window that best match our models, the fits are carried out by means of a simple least-squares minimization. After performing each fit, we compute the reduced chi-squared, $\chi^2_\mathrm{red}$, considering at each step the changing number of data points. The final eclipse coverage of the ground-based light curves used to determine mid-eclipse times is the one corresponding to a $\chi^2_\mathrm{red}$ value equal to (or close to) one. In the figure, the eclipse coverage that best matches our model lies between the blue and cyan lines.
\begin{figure}[ht!]
\centering
\includegraphics[width=.5\textwidth]{chopping.pdf}
\caption{\label{fig:sorting} Flux in arbitrary units as a function of hours from the time of mid-eclipse. The black points show one eclipse of \mbox{BD Dor} from TESS observations. Overplotted are three second-order polynomial fits to all the visible flux in violet, flux values lower than 0.87 in red, and flux values lower than 0.72 in cyan. Dashed horizontal lines indicate these levels and are placed to guide the eye.}
\end{figure}
\subsection{Cross-correlation}
\label{sec:shift_squeeze}
Our third method resembles a cross-correlation between TESS and CASLEO data. Here, the proper time lag between data sets is determined by minimizing the sum of the squared residuals. This sum should approach zero as the correlation between the two data sets become larger. It is generally straightforward to visualize the method in the following way. As one data set is kept fixed through the entire process, the other set is shifted along the abscissa with a given time lag and a scaling is applied to the ordinate values to scale one light curve to the other. Due to the continuity of the TESS data the full eclipse is typically covered, unlike the ground-based light curves (see column 9 of Table~\ref{tab:ObsCond} for the eclipse coverage). We therefore always considered the TESS data as the data being shifted, because here one can better scale the light curve. The scaling is applied because the light curves may appear different due to limb darkening or due to errors while constructing them, such as aperture losses or intrapixel variations. The first step in the program is to center the eclipses around zero in the ordinate so that the applied scale will squeeze or stretch the light curve from TESS, and not simply multiply the flux by a given factor. This is done by calculating and subtracting the mean of each data set separately. With the data sets varying in size, the mean value is computed considering data points where both TESS and CASLEO observations are defined. A time lag is then applied to the TESS data and the flux is linearly interpolated and evaluated at the times of the ground-based data. The mean is then recalculated and subtracted once more from the TESS data, which is necessary to account for the potential change after interpolating to CASLEO's timings. We then proceed in calculating the sum of the squared residuals.
Both time lags and scaling factors are obtained from grids with sensible ranges: $\pm$60 seconds with a step of 1 second for the time lag, and $\pm$10\% variability with a step of 0.5\% for the scaling. For each combination of parameters the sum of squared residuals is computed. The final time lag is the one that minimizes the sum of squared residuals.
\subsection{Errors on the mid-eclipse times}
\label{sec:errors_Tos}
To compute reliable uncertainties for the mid-eclipse times determined from TESS and CASLEO data using the three approaches described in Section~\ref{sec:Mid-Eclipses}, we determine the timing uncertainties by fitting the data and models using a Markov-Chain Monte Carlo (MCMC) approach, as implemented in \texttt{PyAstronomy}\footnote{\label{note1}\url{https://github.com/sczesla/PyAstronomy}}, a collection of \texttt{Python} routines implemented in the \texttt{PyMC} \citep{Patil2010} and \texttt{SciPy} \citep{Jones2001} packages. The best-fit mid-eclipse times and their uncertainties are derived from the mean and standard deviation \mbox{(1-$\sigma$)} of the posterior distributions of the fitted parameters, which are drawn from $10^5$ iterations after carrying out a conservative burn-in of 20\% of the initial samples. This burn-in was determined from prior visual inspection of the chains.
\section{Results}
\label{sec:results}
\subsection{Testing timings between CASLEO-2.15 and CASLEO-0.60}
\label{sec:time_test}
Verifying TESS timestamps from ground-based observations means that the TESS timestamps will be limited by the accuracy of the ground-based observatios. In consequence, the success of this technique relies on how accurately CASLEO's telescopes can report their own timestamps. Both telescopes collect the Universal time from two identical global positioning systems (GPS). The sidereal time is based on a micro-controller synchronized with the GPS that sends its timing to the Programmable Logical Controllers, which in turn are in charge of collecting the data. Despite the professional setup, we carried out an independent check of their timing resemblance. To do so, on the night of June 7, 2018, we observed the eclipsing binary \mbox{KX Aqr} contemporaneously with CASLEO-2.15 and CASLEO-0.60. The target was not observed by TESS. Figure~\ref{fig:KX_Aqr} shows CASLEO-2.15 data in red, and CASLEO-0.60 data in blue. The timing difference between the two data sets was obtained using the cross-correlation method described in Section~\ref{sec:shift_squeeze}. Its value, of \mbox{19 $\pm$ 85 seconds}, is consistent with zero at 1-$\sigma$ level. The large uncertainty, in this case, reflects the high noise in the CASLEO-0.60 light curve. A detailed description of the observations can be found in the first two lines of Table~\ref{tab:ObsCond}.
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{KX_Aqr.pdf}
\caption{\label{fig:KX_Aqr} Ground-based eclipse observation of \mbox{KX Aqr} collected with CASLEO-2.15 (red filled circles) and with CASLEO-0.60 (blue empty circles). The figure displays arbitrary flux units as a function of hours from mid-eclipse time.}
\end{figure}
\subsection{Testing for a time offset}
\label{sec:time_offset}
Our work is based on the determination of mid-eclipse times of selected binary systems observed from TESS and from CASLEO's telescopes. Thus, it is expected that the times of minimum flux will occur simultaneously. This will not necessarily be observed if there is a time offset or a time drift in the clock on-board TESS. To determine a potential time offset of the TESS timestamps, we observed eclipses from several binary systems during TESS's first thirteen sectors. Several aspects reduced the number of good contemporaneous data sets. Some examples are outdated ephemerides, which produced inaccurate windows at which to observe from the ground, and poor weather conditions during observations leading to poor photometric quality in the derived light curves. As a consequence, not all the eclipses listed in Table~\ref{tab:ObsCond} have contemporaneous ground-space observations. Only the twelve specified with a (C) next to their names do. For each one of these eclipse observations, we computed the mid-eclipse times as detailed in Sections~\ref{sec:functions} and \ref{sec:shift_squeeze}, and the timing differences between TESS and ground (Table~\ref{tab:ObsCond}). Figure~\ref{fig:eclipses} shows the corresponding light curves. The errors in the Observed-minus-Calculated (O-C) points are computed from simple error propagation, taking into account the individual timing uncertainties. Averaging the timing differences computed from the twelve contemporaneous eclipses, the derived mean timing offset is \mbox{5.8 $\pm$ 2.5 seconds}. As some of our points in the O-C diagram have a larger offset and a corresponding large uncertainty, in order to properly take them into account our reported offset was obtained computing the weighted mean, and its uncertainty was derived from the standard error of the weighted mean. The timing differences are shown as black squares in Figure~\ref{fig:OC_TESS}. If no time offset exist the O-C points should be normally distributed with zero mean. Only recently, the TESS team discovered a time offset of 2 seconds\footnote{\url{https://archive.stsci.edu/missions/tess/doc/tess_drn/tess_sector_22_drn31_v01.pdf}}. By taking this offset into consideration, our results improve to \mbox{3.8 $\pm$ 2.5 seconds}, only 1.5-$\sigma$ away from zero. It is worth to mention that as of sector 20, the data products on the Mikulski Archive for Space Telescopes (MAST) are corrected by the 2 second offset.
If our derived mid-eclipse times are properly computed and don't show any systematic effect that arises purely from our procedures, they should follow a normal distribution. To assess this, we performed a Kolmogorov-Smirnov test \citep{KolmogorovS} in which we compared our TESS-ground timing differences against a normal distribution. The derived p-value of \mbox{p = 0.913} does not allow us to reject the null hypothesis that both distributions are the same. In addition, we used the best-fit orbital periods and mid-eclipse times of reference listed in Table~\ref{tab:ephemeris} to determine the timing differences between the observed mid-eclipse times corresponding to the four (NC) data sets and the corresponding ones computed from the ephemeris. Two of the O-C points are shown in Figure~\ref{fig:OC_TESS} in blue triangles, as the other two are off-range to allow for proper visual inspection. The derived timing offset is \mbox{2 $\pm$ 11 seconds}, consistent with zero at 1-$\sigma$ level.
\begin{figure*}[ht!]
\centering
\includegraphics[width=.33\textwidth]{WY_Cet_2018-09-27.pdf}
\includegraphics[width=.33\textwidth]{VV_Eri_2018-10-30.pdf}
\includegraphics[width=.33\textwidth]{BD_Dor_2018-11-10.pdf}
\includegraphics[width=.33\textwidth]{AO_Pic_2018-12-10.pdf}
\includegraphics[width=.33\textwidth]{AW_Vel_2019-01-25.pdf}
\includegraphics[width=.33\textwidth]{AO_Pic_2019-01-26.pdf}
\includegraphics[width=.33\textwidth]{AW_Vel_2019-01-27.pdf}
\includegraphics[width=.33\textwidth]{BD_Dor_2019-03-24.pdf}
\includegraphics[width=.33\textwidth]{X_Pic_2019-04-19.pdf}
\includegraphics[width=.33\textwidth]{RR_Nor_2019-05-23.pdf}
\includegraphics[width=.33\textwidth]{TV_Nor_2019-06-07.pdf}
\includegraphics[width=.33\textwidth]{RR_Nor_2019-06-09.pdf}
%
\caption{\label{fig:eclipses} Ground versus space-based photometry. The figures are arranged chronologically from left to right and top to bottom. All figures show arbitrary flux units as a function of the hours from mid-eclipse time. Overplotted are CASLEO-2.15 and CASLEO-0.60 data in red open circles, and TESS data in black filled circles. The data sets are shifted to maximize their overlap. The applied time shift is specified in the last column of Table~\ref{tab:ObsCond}. In each sub-figure we specify the name of the target, the telescope that performed the observations and the date corresponding to the beginning of CASLEO's observing night.}
\end{figure*}
\subsection{Testing for a time drift}
\label{sec:time_drift}
The time drift method relies solely on space-based data, exploiting the power of the continuous observations of all the short-period binary systems followed by TESS for the time verification. The advantage of this method is that it can be run without having ground-based data. The disadvantage is that care must be taken when trying to interpret the derived O-C diagrams. Even though a trend may occur, this does not necessarily stem from TESS timing-drifts. It could instead be a result of the physics in the system itself. Thereby, the same trend must occur in the O-C diagram for several binary systems. It will also not be possible to infer anything about the absolute timing with this method, as the TESS time is not compared to an outside source, so the only possible result from this method is an assessment of a potential drift in the times.
To determine the potential time drift in TESS timings we proceed as follows. For each system we determined the individual mid-eclipse times by carrying out the mid-eclipse timing strategy presented in Section~\ref{sec:Mid-Eclipses}. From the individual mid-eclipse times we determined the orbital period and mid-eclipse time of reference per system. To compute the timing deviation compared to a constant period, we fitted the observed mid-eclipse times, T$_{o,i}$, to the expression:
\begin{equation}
T_{o,i} = P \times E_i + T_0\,.
\label{eq:nosigma}
\end{equation}
\noindent Here, the orbital period, P, and the mid-eclipse time of reference, $T_0$, are the previously mentioned fitting parameters. $E_i$ denotes the epochs with respect to the mid-eclipse time of reference. Both orbital period and mid-eclipse time of reference determined in this work are listed in Table~\ref{tab:ephemeris} for the 26 eclipsing binary systems that were followed by TESS during the first year of observations to fulfill its timing verification. For the fitted parameters, errors are obtained from the 68.27\% confidence level of the marginalized posterior distribution. While the individual mid-eclipse times and their uncertainties are computed from the posterior distributions obtained from 10$^5$ MCMC steps, the ephemerides refinement are created by 10$^6$ MCMC steps. In both cases, we apply a conservative 25\% burn-in of the initial chains. For each of the binary systems we visually inspect the posterior distributions for normality. We check for convergence of the chains by sub-dividing the remaining 75\% in three, computing the usual statistics in each case, and checking for 1-$\sigma$ consistency in the periods and mid-eclipse times of reference. We carried out this procedure to reject the stars showing either a large spread in their O-C diagrams or intrinsic timing variability. Figure~\ref{fig:OC_TESS} shows, in red points, the O-C values of the binary systems that did not show a large spread. As usual, the O-C points were constructed subtracting to each mid-eclipse time (O) the mid-eclipse time assuming a constant period (C). As timing requirement, we considered a standard deviation of the O-C points smaller than 30 seconds. This limit rejected a few binary systems which O-C points were clearly showing intrinsic variability. The figure includes 405 O-C points, and has been made from primary eclipses only, as the secondary eclipses in most cases were shallow ($\Delta$Flux $\sim$ 0.1\%) and thus not providing timings as precise as their primary counterparts.
\begin{figure*}[ht!]
\centering
\includegraphics[width=.95\textwidth]{OC_TESS.pdf}
\caption{\label{fig:OC_TESS} O-C diagram of the 120-sec TESS data in red; timing differences between the contemporaneous TESS and CASLEO data in black squares, and the non-contemporaneous in blue triangles. Uncertainties are in all cases given at $1-\sigma$ level. The large uncertainty and offset of the last black square corresponds to RR Nor, and it is the product of a partial observation.}
\end{figure*}
If a time drift is taking place in TESS photometry, this should be manifested equally in all the O-C points. In consequence, rather than fitting to the individual mid-eclipse times Eq.~\ref{eq:nosigma}, we considered the following expression, which was fitted to the 405 eclipse times jointly:
\begin{equation}
T_{{\rm obs},i,j} = P_j E_{i,j} + T_{0,j} + \sigma_{\rm drift} \times (t-t_{\rm ref})\,,
\label{eq:sigma}
\end{equation}
\noindent where $j$ runs over the different eclipsing binary systems, and $i$ over the number of eclipses for a given system. The shared constant drift amongst all systems is given by $\sigma_\mathrm{drift}$ in seconds per day, and $t_{\rm ref}$ is a common reference time for the drift. For a given system $P_j$, $T_{0,j}$ and $E_{i,j}$ correspond to the orbital period, time of reference, and eclipse ephemeris. In this work and for simplicity, we have always considered the drift to vary linearly with time, and we tested $\sigma_\mathrm{drift}$ against both a monotonic growth and decay. As a simple example to stress the power of the method, if we consider a time drift of \mbox{$\sigma_\mathrm{drift}$ = +1 second/day}, after a year of observations the last eclipse of a star located in the continuous viewing zone (CVZ) would be shifted about 6 minutes with respect to its non-shifted counterpart. This difference can easily be detected by eye when comparing contemporaneous observations from space and from the ground. However, a shift like this could pass unnoticed if only a single space-based data set is analyzed, rather than the whole sample. If, for instance, the individual mid-eclipse times of this time-drifted binary system are fitted only, the orbital period would slightly change when compared to that of a non-shifted data set, compensating for the drift. In this case, the individual O-C diagram would appear most likely flat, as only drifts of several seconds per day would create a curvature in an O-C diagram. In consequence, it is fundamental to carry out this exercise analyzing all the stars in the sample at once, because in doing so a small time drift could resurface from the noise.
Assuming then that there is a time drift growing (or decaying) linearly in time, we re-fitted the individual mid-eclipse times of all the stars together, but this time using Equation~\ref{eq:sigma} as model and, as previously mentioned, considering $\sigma_\mathrm{drift}$ to be equal for all the stars. As starting values for the period and the mid-eclipse times of reference we used the ones obtained before, assuming no shift, along with uniform priors covering $\pm50\%$ the starting values. If a time drift exists in TESS data, this would reflect into a $\sigma_\mathrm{drift}$ inconsistent with zero. After performing $10^6$ MCMC iterations along with a conservative burn-in of the first $20\%$ of the samples, the derived drift computed from the posterior distribution of the parameter was determined to be $\rm \sigma_\mathrm{drift} = 0.009 \pm 0.015$ seconds/day, fully consistent with zero at 1-$\sigma$ level. The corresponding posterior distribution and evolution of the traces can be seen in Figure~\ref{fig:tracesigma}. To allow for visual inspection, the triangle plots of twelve randomly selected best-fit individual periods and corresponding mid-eclipse times of reference can be seen in the Appendix, under Figures~\ref{fig:TP_per} and \ref{fig:TP_T0}, respectively, showcasing the rather low correlations between the parameters.
\begin{figure}[ht!]
\includegraphics[width=.5\textwidth]{trace_sigma.pdf}
\caption{\label{fig:tracesigma} The resulting Markov chain for $\sigma_{\rm drift}$. Left: posterior distribution. Right: evolution of the traces.}
\end{figure}
In order to test the robustness of our method, we injected into the TESS photometry drifts corresponding to 0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, and 5 seconds per day. If the technique works, we should be able to recover the injected signal with a similar precision as before. After shifting all TESS timestamps we re-computed the individual mid-eclipse times, we fitted the individual periods and mid-eclipse times of reference as if these shifts would not exist, and then we attempted at recovering the injected drift by fitting all the individual mid-eclipse times following Eq.~\ref{eq:sigma}. \fref{fig:sigma_in_out} shows the difference between the recovered and the injected time drifts, in seconds per day. As the figure shows, within uncertainties all the recovered values are fully consistent with the injected ones.
\begin{figure}[ht!]
\centering
\includegraphics[width=.5\textwidth]{sigma_in_out.pdf}
\caption{\label{fig:sigma_in_out}Injected versus recovered time drift. The figure shows the difference between the recovered and the injected drift, as a function of the injected drift. Uncertainties are given at 1-$\sigma$ level.}
\end{figure}
\section{Conclusion}
\label{sec:DyC}
\begin{figure*}[ht!]
\centering
\includegraphics[width=\textwidth]{timing_limits2.pdf}
\caption{\label{fig:timelim} Reachable amplitudes (in relative variability in percent) as a function of oscillation period in minutes and TESS magnitude (colour). Markers indicate stars listed in the document SAC\_TESS\_0002\_5, for which a TESS magnitude could be found -- in cases where the measured timing values are of sufficient quality for asteroseismic analysis the marker is filled, with a colour corresponding to the TESS magnitude (i.e., where the stellar amplitude lie below the upper uncertainty limit of the computed relations for the given stellar magnitude and period), otherwise it is left open. Left: amplitudes corresponding to the measured time drift over a 27 day observing sector. The TESS magnitudes of the five relations shows are indicated on the colourbar. In each case the thick line gives the median value for the amplitude from a Monte Carlo sampling of the time drift (or offset) and the shaded region gives the corresponding uncertainty from the 16th and 84th percentiles. The vertical lines indicate for each star the median amplitude of the corresponding relation at the period and TESS magnitude of the star. Right: amplitudes corresponding to the measured absolute time offset.}
\end{figure*}
To reliably carry out asteroseismology studies from TESS data, potential drifts and the absolute time for TESS observations must be known to a high accuracy. Even though the internal clock on board TESS is very accurate in its own time, as it happened to the {\it Kepler} space telescope drifts and offsets could take place. In consequence, we have carried out a photometric follow-up of several eclipsing binary systems from TESS and from the ground, using two telescopes located at the Complejo Astron\'omico El Leoncito, in Argentina. Comparing the timings of twelve primary eclipses of binary systems of Algol type from the ground to those observed by TESS we find a time offset of 5.8 $\pm$ 2.5 seconds (in the sense that the barycentric time measured by TESS is ahead of real time), indicative of a small offset but still consistent with zero at the $2.3-\sigma$ level. It is worth to mention that the TESS team has recently discovered a time offset of 2 seconds that accounts for some portion of our detected time offset. As of sector 20, the data products on MAST are corrected. Taking this offset into consideration improves our results to a total time offset of 3.8 $\pm$ 2.5 seconds, consistent with zero at the $1.5-\sigma$ level. Carrying out a joint analysis of 405 individual mid-eclipse times collected from 26 eclipsing binary systems, we find TESS to have a time drift consistent with zero, and equal to $\sigma_{\rm drift} = 0.009 \pm 0.015$ seconds/day. For this, we assumed a monotonic, linearly growing --and decaying-- time-dependent drift. To the precision that our joined data can achieve, we can confirm that the TESS clock does not present neither a clear time offset nor a time drift.
It is clear that we cannot reach a precision on the estimation of the time drift or offset that satisfy the requirements given in \sref{sec:req}. It is, however, worth remembering that these were defined based on the very brightest, highest amplitude, and shortest period pulsators. So, while our current analysis cannot guarantee TESS observations with timing specifications that ensure an optimum asteroseismic analysis for these, there will still be many fainter, lower amplitude, longer period pulsators whose requirements are fulfilled. In \fref{fig:timelim} we show the amplitudes that can be reached for a given pulsation period and TESS magnitude given the estimated drift and absolute offset. Given the relatively large uncertainties on our estimates the amplitude values were obtained from a Monte Carlo sampling rather than using standard error propagation. To compute the noise per measurement that enters in the calculations we used the prescription by \citet{Sullivan2015}, even though we are aware that the mission will do better than the estimates here. We combined this with measured values from the TASOC pipeline for mean flux and number of pixels in an aperture as a function of TESS magnitude \citep{TASOC}. We adopt a systematic noise of $5 \rm \, ppm\, hr^{-1}$, which mainly affect the noise at the very bright end ($T_{\rm mag} \lesssim 4$).
As seen from \fref{fig:timelim} it will be possible to compare stellar oscillations observed by TESS with ground-based observations for several of the stars listed in SAC\_TESS\_0002\_5 based on the measured absolute time offset. We find that the measured offset is of a size that will not become an issue for comparing ground-based and space data for coherent oscillations for most of the targets observed with TESS. Specifically we find that for all TESS stars fainter than $T_{\rm mag} =4$ oscillations with periods longer than one hour and amplitudes below ${\sim}5$ mmag ($0.5 \%$) are unaffected. For stars fainter than $T_{\rm mag} =9$ oscillations with periods longer than on hour and amplitudes below ${\sim}50$ mmag ($5 \%$) are unaffected.
Only for one of the stars in SAC\_TESS\_0002\_5 does the measured time drift allow for the theoretical accuracy to be reached. In the case of solar-like oscillators, with amplitudes of a few ppm and periods of the order of a few minutes on the main-sequence, to a few hundred ppm and periods of the order a day on the red-giant branch, the current timing measurements are sufficient to reach the theoretical accuracy on the determination of frequencies and comparison with ground-based facilities.
We note that the pulsators listed in SAC\_TESS\_0002\_5 represent some of the stars with the very strongest timing requirements within their respective variability class, and the requirements for most stars observed by TESS will therefore be less strict. Also, the model used for the photometric noise represents the lower envelope, so for many stars the photometry will be noisier and as a consequence the timing requirement will be reduced.
\acknowledgments
{
Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant DNRF106). The work is based on data obtained at Complejo Astron\'omico El Leoncito, operated under agreement between the Consejo Nacional de Investigaciones Cient\'ificas y T\'ecnicas de la Rep\'ublica Argentina and the National Universities of La Plata, C\'ordoba and San Juan. CvE and HK acknowledge support from the European Social Fund via the Lithuanian Science Council (LMTLT) grant No. 09.3.3-LMT-K-712-01-0103. MNL and RH acknowledge support from the ESA PRODEX programme.}
|
2,869,038,155,705 | arxiv | \section{Introduction} \label{Intro}
Red giants are evolved low mass stars which left the main sequence after burning all the hydrogen in their core.
In the Hertzsprung--Russell (HR) diagram, they are located in a narrow temperature range: $3500~\leqslant~T_{\mathrm{eff}}~\leqslant~5600$ K
and in a broader luminosity range: $0.5~\leqslant~\log(L/\mathrm{L_\odot})~\leqslant~3.5$ \citep[see e.g.][]{2009A&A...503L..21M}.
Among red giants, one distinguishes stars ascending the red giant branch (RGB) and those in the red clump.
The first ones have not yet started helium burning in their core.
Consequently, while their core is contracting, their radius increases and they become brighter and brighter.
The second ones have started helium burning in their core.
Therefore, they are in a relatively stable state with a nearly constant radius and luminosity.
\cite{2009Natur.459..398D} presented the first study of several hundreds of red giants observed with CoRoT,
showing that they exhibit non-radial oscillations with common patterns.
Indeed, red giants are solar-type pulsators,
exhibiting oscillation modes intrinsically stable and stochastically excited by turbulent convection in the upper parts of their convective envelope.
Stellar oscillations appear in the power spectrum of the light curve as an excess power, resulting from a balance between mode driving and damping.
\cite{2011A&A...525L...9M, 2013A&A...559A.137M} showed that the frequency distribution of modes follow a universal pattern (hereafter UP),
valid for a large range of evolutionary stages, from main-sequence to AGB stars.
Thus, as for all solar-like pulsators, the pressure modes (p-modes) in red giants can be characterised to first order by three seismic indices
\citep[e.g.][]{2013ARA&A..51..353C}:
the frequency of the maximum power $\nu_{\mathrm{max}}$, its height $H_{\mathrm{env}}$ and an equidistant frequency spacing $\Delta\nu$, called the mean large separation.
Both seismic indices, $\Delta\nu$ and $\nu_{\mathrm{max}}$ are directly related to stellar physical properties
such as the mean density $ <\rho>$, the surface gravity $g$ and the effective temperature $T_{\mathrm{eff}}$
\citep[see e.g.][]{1986ApJ...306L..37U, 1991ApJ...368..599B, 2011A&A...530A.142B, 1995A&A...293...87K}.
Therefore, it is possible to deduce the \textit{seismic} mass and radius \citep{2010A&A...522A...1K} to a very good precision and therefore,
study stellar populations \citep[see e.g.][]{2009A&A...503L..21M}.
Turbulent convection at the stellar photosphere induces an other signal observable in the light curve: granulation.
At the stellar surface, granulation appears under the form of irregular cellular patterns evolving with time.
In the power spectrum of the light curve, the signature of stellar granulation is located at low frequency.
It can be characterised by two parameters \citep[e.g.][]{2014A&A...570A..41K}:
the characteristic amplitude $\sigma_\mathrm{g}^2$, which is the RMS (Root Mean Square) brightness fluctuation ($\sigma_{\rm g}^2$, thus corresponds to the total integrated energy of the granulation),
and the effective timescale $\tau_{\mathrm{eff}}$, or \textit{e-folding time}, which measures the temporal coherence of the granulation in the time domain \citep[e.g.][]{2011ApJ...741..119M,2014A&A...570A..41K}.
Thus, the granulation parameters carry information about stellar convection \citep[see e.g.][and references therein]{2013A26A...559A..39S}.
Various methods to automatically extract the stellar seismic indices and/or granulation parameters from light-curves exist.
Seismic indices extraction methods have been reported in \cite{2011MNRAS.415.3539V, 2011A&A...525A.131H}
and granulation extraction methods in \cite{2011ApJ...741..119M}.
Regarding seismic indices extraction, pipelines usually deduce $\nu_{\mathrm{max}}$ from the centroid of a Gaussian fit to the smoothed power spectrum,
except in the automated Bayesian method of \cite[][hereafter CAN]{2010A&A...522A...1K, 2014A&A...570A..41K}.
For $\Delta\nu$, there are three main approaches sharing same mathematical basis: the autocorrelation of the time series, the autocorrelation of the power spectrum
and the power spectrum of the power spectrum.
In addition, the autocorrelation method of \cite[][hereafter COR]{2011A&A...525L...9M}
refines the $\Delta\nu$ estimate in a second step by maximizing the correlation between the raw spectrum and the UP.
The granulation parameters are extracted by fitting one or two Harvey-like functions \citep{1985ESASP.235..199H} on the smoothed spectrum.
For most methods, the least-squares algorithm is used,
except for \cite{2010A&A...511A..46M}'s method (hereafter A2Z), which uses the MLE,
coupled with the Levenberg-Marquardt (LM) algorithm \citep{numerical_recipes} for the optimization.
The CAN method fits the raw spectrum using a Bayesian Monte Carlo Markov Chain (MCMC) algorithm.
These studies show that it is difficult to simultaneously and consistently extract all these parameters at the same time.
The problem comes mainly from the fact that smoothing the power spectrum is generally used to measure $\nu_{\mathrm{max}}$ and the granulation parameters.
However, smoothing alters both the width and the height of the pulsation pics and the granulation profile.
On the other hand, the unsmoothed power spectrum follows a $\chi^2$ statistics with two degrees of freedom
\citep{1984PhDT........34W} which requires for the optimisation the use of the Maximum Likelihood Estimator \citep[][hereafter MLE]{1994A&A...289..649T}
or the Bayesian approach \citep[e.g.][]{2010A&A...522A...1K}.
By observing several tens of thousands of red giants,
CoRoT \citep{2009IAUS..253...71B, 2006ESASP1306...33B} and $\textit{Kepler}$ \citep{2010AAS...21510101B, 2010PASP..122..131G}
have enabled a breakthrough in our understanding of and way of studying these stars.
In this context, the Stellar Seismic Indices (SSI\footnote{\label{SSI_footnote}SSI database website: \url{http://ssi.lesia.obspm.fr/}})
database intends to provide the scientific community with a homogeneous set of parameters characterizing solar-type pulsators observed by CoRoT and $\textit{Kepler}$.
For this purpose, we analyse this large set of stars,
extracting simultaneously and consistently the seismic indices ($\Delta\nu$, $\nu_{\mathrm{max}}$ and $H_{\mathrm{env}}$)
together with the granulation parameters ($\tau_{\mathrm{eff}}$ and $\sigma_\mathrm{g}^2$).
Furthermore, we want to characterise the error bars on the measurements provided.
Hence, we developed a new automated method, called MLEUP \citep{2017EPJWC.16001012D}.
The final estimates of seismic indices and granulation parameters (as well as their respective uncertainties)
are obtained via the MLE adjustment of the unsmoothed Fourier
spectrum with a parametric model including components for the pulsations, the granulation, the activity and the
intrumental white noise.
In Sect. \ref{Desciption_pipeline}, we describe our method.
In Sect. \ref{simulations}, we assess its performances and limitations using simulated light-curves.
Accordingly, we have developed a light curve simulator designed to be as representative of red giants as possible.
We use simulated light-curves to quantify the bias on the MLEUP results for seismic indices and granulation parameter
values. We also quantify the biases between the real dispersion of the results and the formal errors provided by the MLE adjustment.
In Sect. \ref{application},
we apply our method to almost all stars observed by $\textit{Kepler}$ and CoRoT and discuss the results obtained,
while taking into account correction of the biases and the uncertainty estimates.
Finally, Sect. \ref{conclusion} is devoted to the discussion and conclusion.
\section{Description of the MLEUP method} \label{Desciption_pipeline}
MLEUP is an automated method based on the UP and designed to extract simultaneously the seismic indices ($\Delta\nu$, $\nu_{\mathrm{max}}$ and $H_{\mathrm{env}}$)
and the granulation parameters ($\tau_{\mathrm{eff}}$ and $\sigma_\mathrm{g}^2$) of red-giant stars (see Sect. \ref{Intro}).
In this section, we first describe the global model used in this method.
Then, we describe the different steps used to adjust this model to the spectra.
\subsection{The stellar background and oscillations model} \label{BG+osc_model}
The global theoretical model we consider (Eq. (\ref{equation_BG+UP})) is composed of two main contributions:
the stellar background model and the oscillation model.
\subsubsection{The background model} \label{background_model}
\cite{1985ESASP.235..199H} approximated the solar background signal as the sum of pseudo-Lorentzian functions:
\begin{equation}
P(\nu) = \sum_{i}^{N} \frac{4 \sigma_{\mathrm{g},i}^2 \tau_i}{1+(2\pi\nu\tau_i)^{\alpha_i}} \, ,
\end{equation}
with $P(\nu)$, the total power of the signal at frequency $\nu$; $N$, the number of background components;
$\sigma_\mathrm{g,i}$, the characteristic amplitude of a given component; $\tau$, the characteristic timescale; and $\alpha_i$,
the slope characterizing the frequency decay of pseudo-Lorentzian functions.
For a Lorentzian function, as used by \cite{1985ESASP.235..199H}, $\alpha_i=2$, which corresponds to an exponential decay function in the temporal domain.
Each Lorentzian is associated with a different background component, such as the granulation or the activity.
Later, it was noticed that to better model solar granulation,
it was necessary not only to change the value of the exponent $\alpha_{\rm gran}=2$ to $\alpha_{\rm gran}=4$ \citep[e.g.][]{Andersen1998}
or higher \citep[$\alpha_i=6.2$,][]{2012MNRAS.421.3170K}, but also to use two components instead of one \citep[e.g.][]{2005A&A...443L..11V}.
Then, with the arrival of the quasi-continuous long time series of CoRoT,
granulation has also been observed in other solar type oscillators \citep{2008Sci...322..558M}.
In a few cases with the $\textit{Kepler}$ data, it has even been possible to observe the double component of the granulation in the sub-giant phase \citep{2013ApJ...767...34K}
as well as in the red giant phase \citep{2010A&A...522A...1K}.
However, the origin of this double component characterizing granulation is not well understood. \\
In the framework of the SSI database, we want to be able to analyse, in a homogeneous way, as many of the stars observed by CoRoT and $\textit{Kepler}$ as possible.
However, these data are disparate and for most of them,
the signal-to-noise level and/or the resolution does not allow us to reliably fit two granulation components.
Hence, we only use one granulation component, modelled by a Lorentzian-like function (with $\alpha$ as a free parameter).
Nonetheless, this induces a bias in the measurements that we will quantify and correct a posteriori using simulated light-curves
(see Sect. \ref{simulations}).
In addition to the granulation, we consider a second Lorentzian function ($\alpha=2$),
in order to take into account the signal located at very low frequencies,
usually accepted to be the signature of the stellar activity
\citep[][]{1985ESASP.235..199H, 2010Sci...329.1032G, 2014A&A...562A.124M} and a possible residual of instrumental effects.
Finally, the background model is completed with a constant component to fit the white noise,
corresponding to the photon and instrumental noise.
We obtain the following background model
\begin{equation}
BG(\nu) = W + \sum_{i=1}^{N=2} \frac {P_i} {1+(2 \pi \tau_i \nu)^{\alpha_i}} \, ,
\label{equation_BG}
\end{equation}
with $N$, the number of background components; $W$, a constant for the white noise;
$P_i$, the power of the Lorentzian-like profiles ($P_{\mathrm{act}}$, $P_{\mathrm{gran}}$) at a frequency of zero;
$\tau_i$, the characteristic timescales ($\tau_{\mathrm{act}}$, $\tau_{\mathrm{gran}}$) and $\alpha_i$, the slope ($\alpha_{\mathrm{act}}=2$; $\alpha_{\mathrm{gran}}$).\\
We note that $P_{\mathrm{gran}}$, $\tau_{\mathrm{gran}}$ and $\alpha_{\mathrm{gran}}$ are highly dependent on the type or number of the functions used.
However, they can be related in a simple way to the intrinsic parameters of the granulation, $\sigma_{\rm g}$ and $\tau_{\mathrm{eff}}$. Indeed, $\sigma_{\rm g}$ is given by the following relation \citep{2013ApJ...767...34K}
\begin{equation}
\sigma_\mathrm{g}^2 = \sum_{i}^{N_\mathrm{g}} \frac {1} {2} \frac {P_i} {\tau_i \alpha_i \sin(\frac {\pi} {\alpha_i})} \, ,
\label{sigma}
\end{equation}
with $N_\mathrm{g}$, the number of granulation components.
The ``e-folding time'', $\tau_{\mathrm{eff}}$, is measured using the autocorrelation function (hereafter ACF) of the granulation component \footnote{We compute $\tau_{\mathrm{eff}}$ numerically.
First, we computed the inverse Fourier transform ($\mathrm{FFT}^{-1}$) of the granulation component (one pseudo-Lorentzian) to get the corresponding ACF:
$ACF=\mathrm{FFT}^{-1} [P_{\mathrm{gran}}/(1 + (2 \pi \tau_{\mathrm{gran}} \nu)^{\alpha_{\mathrm{gran}}})]$.
Then, we normalise the ACF by the first bin. $\tau_{\mathrm{eff}}$ corresponds to where the normalised ACF is equal to $1/e$.}.
Thus, unlike $\tau_{\mathrm{gran}}$, $\tau_{\mathrm{eff}}$ does not explicitly dependent on the number of granulation components, nor the function used.
If one uses a Lorentzian function to fit the granulation ($\alpha = 2$), $\tau_{\mathrm{eff}} = \tau_{\mathrm{gran}}$,
while for a pseudo-Lorentzian, $\tau_{\mathrm{eff}} \neq \tau_{\mathrm{gran}}$.
\subsubsection{The oscillation model} \label{oscillations_model}
In previous models \citep{2011ApJ...741..119M}, the oscillation spectrum is modeled using a Gaussian component and $\nu_{\mathrm{max}}$ is determined by fitting this envelope on the raw or smoothed spectrum.
However, we know that the oscillations spectrum follows a discrete pattern which can be parametrized by the so-called {\it Universal Pattern} (UP) \citep{2011A&A...525L...9M}.
Thus, in order to improve the seismic indices estimates, we replace the Gaussian profile by the UP. \\
To develop a synthetic and parametric pattern, we first generate a frequency comb following the asymptotic relation \citep{1980ApJS...43..469T, 2011A&A...525L...9M}.
\begin{equation}
\nu_{n,\ell} = n + \frac{\ell}{2} + \varepsilon(\Delta\nu) - d_{0\ell}(\Delta\nu) + \frac{\gamma}{2} \left(n - \frac{\nu_{\mathrm{max}}}{\Delta\nu}\right)^{2} \Delta\nu \, ,
\label{equation_freq_comb}
\end{equation}
where $n$ and $\ell$ are respectively the radial order and the degree of a given mode.
$\varepsilon$ is an offset; $d_ {0\ell}$, the small separation; and $\gamma$, the curvature.
For each $n$, we use the first four angular degrees ($\ell=0,1,2,3$).
$\Delta\nu$ and $\nu_{\mathrm{max}}$ are input parameters while $\varepsilon$, $d_{0\ell}$ and $\gamma$ are deduced from scaling relations depending on $\Delta\nu$.
$\varepsilon$ and $d_{0\ell}$ follow a relation taken from \cite{2011A&A...525L...9M}, and $\gamma$ from \cite{2013A&A...550A.126M}.
Then, we describe each individual mode (Eq. (\ref{equation_freq_comb})) with a Lorentzian shape modulated by its individual visibility:
\begin{equation}
L_{n,\ell} (\nu) = \frac {V_{\ell}^2} {1 + \lbrack 2 (\nu-\nu_{n,\ell}) / \Gamma \rbrack ^2 } \, ,
\label{equation_Lorentzian}
\end{equation}
where the linewidth $\Gamma$ is taken from \cite{2012sf2a.conf..173B} and the visibility $V_{\ell}$,
the height of the individual modes of degree $\ell$, from \cite{2012A&A...537A..30M}.
Last of all, we multiply the Lorentzian profiles $L_{n,\ell}$ by a Gaussian envelope $G_\mathrm{env}(\nu)$, centred at $\nu_{\mathrm{max}}$:
\begin{equation}
G_\mathrm{env} (\nu) = H_{\mathrm{env}} \exp \left[\frac {-(\nu-\nu_{\mathrm{max}})^2} {\delta \nu_{\mathrm{env}}^2 / 4 \ln 2 } \right] \, ,
\label{equation_Gaussian}
\end{equation}
where the full width at half maximum ($\delta \nu_{\mathrm{env}}$) is taken as a function of $\nu_{\mathrm{max}}$ following the scaling relation proposed by \cite{2012A&A...537A..30M}. The height of the Gaussian envelope ($H_{\mathrm{env}}$) is a fitted parameter.\\
Thus we finally get the UP, an oscillation pattern parametrized by three input parameters $\nu_{\mathrm{max}}$, $\Delta\nu$ and $H_{\mathrm{env}}$ (see Fig. \ref{UP}):
\begin{equation}
UP (\nu) = G_\mathrm{env}(\nu) \times \sum_{n=1}^{n_{\mathrm{env}}} \sum_{\ell=0}^{3} L_{n,\ell} (\nu) \, ,
\label{equation_UP}
\end{equation}
with $n_{\mathrm{env}}$, the total number of radial orders considered. It is deduced from the scaling relation taken in \cite{2012A&A...537A..30M}.
\begin{figure}
\centering
\includegraphics[scale=0.35]{Figures/test_gen_up_1.pdf}
\caption{The \textit{Universal Pattern} (UP) is the red giant parametric oscillations pattern, used as model to fit the observed one.}
\label{UP}
\end{figure}
\subsubsection{Global model} \label{global_model}
The global model has to consider a damping factor $\eta(\nu)$ which takes into account the distortion of the spectrum due to the integration time
\citep{1993DSSN....6...19M, 2014A&A...570A..41K, 2014MNRAS.445..946C}.
This factor is particularly important for $\textit{Kepler}$ long cadence data because of a low Nyquist frequency ($\nu_\mathrm{Nyq} \sim 287~\mu$Hz).
When the integration time is equal to the sampling time, the damping factor is expressed as:
\begin{equation}
\eta(\nu) = \mathrm{sinc} \left( \frac{\pi \nu}{2\nu_{\mathrm{Nyq}}} \right) \, .
\label{damping_factor}
\end{equation}
This factor does not affect the white noise component.\\
The global model used to fit spectra is:
\begin{equation}
P (\nu) = W + \eta^2(\nu) \left[ \sum_{i=1}^{N} \frac {P_i} {1+(2 \pi \tau_i \nu)^{\alpha_i}} + UP(\nu) \right] \, ,
\label{equation_BG+UP}
\end{equation}
with $N=2$ if the activity is taken into account in the fit, and $N=1$ otherwise.
\subsection{Detailed explanations of the algorithm} \label{Modelling}
\begin{figure*}
\includegraphics[width=160mm]{Figures/methode_globale_mle_up.pdf}
\caption{Diagram describing the different steps of the MLEUP method. In red, the outputs of MLEUP. \label{method}}
\end{figure*}
To adjust our model to the power spectrum, we use the Maximum-Likelihood Estimator (MLE) algorithm \citep{1994A&A...289..649T}
coupled with the Levenberg-Marquardt (LM) algorithm \citep{numerical_recipes} for the optimization.
Since we do not smooth the spectrum in order to preserve all the information (see Sect. \ref{Intro}),
we could not use the least-squares method which is suited for a Gaussian statistics only,
whereas the raw spectrum follows a $\chi^2$ statistics with two degrees of freedom \citep{1984PhDT........34W}.
An alternative approach to the MLE would have been to use the Bayesian/Monte-Carlo-Markov-Chain (MCMC) method
to adjust our model to the spectrum (as for the CAN method).
However, it has been considered too time consuming and therefore not acceptable for analyzing all CoRoT and $\textit{Kepler}$ data.
Nonetheless, since the MLE/LM method is sensitive to guesses,
we perform several steps before fitting all parameters together in order to improve the robustness of the final fit.
Figure \ref{method} summarizes the method and in the following section we describe each step.
\subsubsection{1\textsuperscript{st} step: Determining initial guesses} \label{step_1}
The first step consists in determining guesses for $\Delta\nu$ and $\nu_{\mathrm{max}}$.
For this purpose, we follow the COR method \citep{2009A&A...508..877M} which is based on the ACF method.
It deduces the large separation by calculating the autocorrelation of the time series, more precisely,
the inverse Fourier transform of the filtered PSD.
The Power Spectral Density (PSD) is computed from the light curve with the Fast Lomb-Scargle periodogram algorithm
developed by Leroy (2012)\footnote{The code with the Python interface developed by R\'eza Samadi can be found at: \url{https://pypi.python.org/pypi/pynfftls/1.2}}.
In the case of CoRoT, due to signatures of the low-Earth orbit \citep{2009A&A...506..411A}, data are contaminated by the 1 c.d$^{-1}$ frequency ($11.57~\mu$Hz) and its harmonics,
as well as by the orbital frequency ($161.71~\mu$Hz). Thus, since the ACF is sensitive to regularities,
we replace \textbf{(only for this step)} these frequencies by noise following the statistics of the PSD ($\chi^2$ with two degrees of freedom).
As was done in \cite{2009A&A...508..877M}, we filter the PSD using a cosine function centred at the frequency $\nu_\mathrm{c}$ with a width $\delta\nu_\mathrm{c}$. Here,
$\nu_\mathrm{c}$ takes values from $3~\mu$Hz to $110~\mu$Hz for CoRoT (to avoid orbital frequencies above) and from $1~\mu$Hz to the Nyquist frequency for $\textit{Kepler}$,
following a geometric progression with a ratio $g=2^{0.1}$, which allows a filter overlap.
$\delta\nu_c$ is equal to $3\Delta\nu_\mathrm{c}$, with $\Delta\nu_\mathrm{c}$ proportional to $\nu_\mathrm{c}$ following the scaling law established in \cite{2012A&A...537A..30M}.
Afterwards, we compute the inverse Fourier transform of the PSD multiplied by the filter.
For each filter, we obtain the envelope autocorrelation function (EACF) which reaches a maximum equal to $\mathcal{A}_{\mathrm{max}}$ at time $\tau_{\Delta\nu}$ = 2/$\Delta\nu$.
The filter with the highest envelope $\mathcal{A}_{\mathrm{max}}$ gives an estimate of $\Delta\nu^{\mathrm{ACF}}$ as well as $\nu_{\mathrm{max}}^{\mathrm{ACF}}$, which is
equal to the frequency position $\nu_{\mathrm{c}}$ of the corresponding filter.
In order to improve these results, the process is repeated around $\nu_{\mathrm{max}}^{\mathrm{ACF}}$ at $\pm \nu_{\mathrm{max}}^{\mathrm{ACF}}/3 $
following an arithmetic progression with a ratio equal to the resolution of the spectrum.
We get our final estimate of $\Delta\nu^{\mathrm{ACF}}$ and $\nu_{\mathrm{max}}^{\mathrm{ACF}}$.
With $\nu_{\mathrm{max}}^{\mathrm{ACF}}$, we deduce guesses for the following parameters:
$H_{\mathrm{env}}$ from the scaling law in \cite{2012A&A...537A..30M} and $\tau_{\mathrm{gran}}$ \citep{2011ApJ...741..119M}.
For $P_{\mathrm{gran}}$, we compute the median of the spectrum between $\nu_{\mathrm{max}}^{\mathrm{ACF}}/25$ and $\nu_{\mathrm{max}}^{\mathrm{ACF}}/10$.
We estimate the white noise component $W$ by computing the median of the last points of the spectrum over a width of $3 \Delta\nu^{\mathrm{ACF}}$.
$\alpha_{\mathrm{gran}}$ is initialized to 2.
Finally, as initial guesses, we set $\tau_{\mathrm{act}} = 10 \tau_{\mathrm{gran}}$ and $P_{\mathrm{act}}$ equal to the first bin of the power of the spectrum. Meanwhile $\alpha_{\mathrm{act}}$ is fixed to 2.
\subsubsection{2\textsuperscript{sd} step: Background fit} \label{step_2}
The aim of this step is to obtain a better estimate of the background parameters. Thus, only its parameters will be free during the fit.
Since the spectrum is not smoothed in order to preserve all its informations, the Maximum-Likelihood Estimator (MLE) algorithm is used.
The MLE consists in maximising the likelihood function $L$, which is the probability to obtain the observed power spectrum $S(\nu)$
with a spectrum model $M(\nu,\lambda)$ given by Eq.~\ref{equation_BG+UP} and a set of free parameters $\lambda$ \citep[e.g.][]{1990ApJ...364..699A, 1994A&A...289..649T}.
Knowing that the raw power spectrum of solar oscillations follows a $\chi^2$ probability distribution with two degrees of freedom (Woodard 1984),
the likelihood function $L$ is defined as:
\begin{equation}
L = \prod_{i=1}^{N} \frac{1}{M(\nu_i, \lambda)} e^{- \frac{S(\nu_i)}{M(\nu_i, \lambda)} } \, ,
\label{densite_proba}
\end{equation}
where $N$ is the number of independent frequencies $\nu_i$ of the power spectrum $S(\nu)$.\\
In practice, one will minimize the logarithmic likelihood function $\mathcal{L}$ instead of maximising $L$. $\mathcal{L}$ is calculated as:
\begin{equation}
\mathcal{L} = - \ln L = - \sum_{i=1}^{N} \frac{S(\nu_i)}{M(\nu_i, \lambda)} + \ln M(\nu_i, \lambda) \, .
\label{densite_proba}
\end{equation}
Consequently, the position of the minimum of $\mathcal{L}$ in the $\lambda$-space gives the most likely value of $\lambda$ \citep{1998A&AS..132..107A},
i.e. the set of optimal parameters.
The formal error bars are then deduced by taking the diagonal elements of the inverse of the Hessian matrix $h$ \citep{numerical_recipes}:
\begin{equation}
h_{ij} = \frac{\partial^2 \mathcal{L}}{\partial \lambda_i \partial \lambda_j} \, .
\label{matrice_hessienne}
\end{equation}
The minimization of $\mathcal{L}$ is done using a modified version of Powell's method \citep{Powell1964}.\\
The signal at very low frequency depends on the intensity of the stellar activity, instrumental effects and on the resolution of the spectrum.
Also, in some cases, it is important to take into account this component in order to improve the robustness and the quality of the fit.
Thus, we perform two fits with two different models.
One {\it with} the activity component ($N=2$ in Eq. (\ref{equation_BG+UP})), using six free parameters ($W$, $P_{\mathrm{gran}}$, $\tau_{\mathrm{gran}}$, $\alpha_{\mathrm{gran}}$, $P_{\mathrm{act}}$ and $\tau_{\mathrm{act}}$)
and one {\it without} the activity component ($N=1$ in Eq. (\ref{equation_BG+UP})), using only four parameters ($W$, $P_{\mathrm{gran}}$, $\tau_{\mathrm{gran}}$, $\alpha_{\mathrm{gran}}$).
In both cases, we consider all the spectrum.
Then, to distinguish the best fit, we proceed to a statistical test computing the logarithmic likelihood ratio $\Lambda$
following \cite{wilks1938} \citep[see also][]{1998A&AS..132..107A,2012MNRAS.421.3170K}:
\begin{equation}
\ln \Lambda = \mathcal{L}(\lambda_{p+q}) - \mathcal{L}(\lambda_p) \, ,
\label{log_ratio}
\end{equation}
with $\mathcal{L}$, the logarithmic likelihood function given by the set of parameters $\lambda$;
the index represents the number of free parameters considered for a given fit:
$p$, in the case without the activity component and $p+q$, in the case with the activity component (here $p=4$ and $q=2$).
Next, we compare the value of $-2 \ln \Lambda$ to the confidence level (CL) which follows a $\chi^2$ distribution with $q$ degrees of freedom, fixed at a given probability $P$.
Here, $q=2$ and we adopt $P=99\%$. Consequently, the confidence level is equal to $\mathrm{CL}=9.21$.\\
Thus, we can get three possible scenarios:
\begin{enumerate}
\item[a.] $-2 \ln \Lambda \geq +\mathrm{CL}$: The fit with the activity is more significant than without, given the probability $P$.
So, we keep the activity component in the model.
\item[b.] $-2 \ln \Lambda \leq -\mathrm{CL}$: The activity component is not significant. So, we remove the activity from the model.
\item[c.] $-\mathrm{CL} < -2 \ln \Lambda < +\mathrm{CL}$: This third case is less clear-cut.
So, we remove the activity and we ignore frequencies below $\nu_{\mathrm{max}}/20$.
\end{enumerate}
From new granulation parameters, we deduce new guesses for the seismic indices: $\nu_{\mathrm{max}}^{\mathrm{gran}}$ \citep{2011ApJ...741..119M} and $\Delta\nu^{\mathrm{gran}}$ \citep{2012A&A...537A..30M}.
We compare these new guesses with the ones given by the ACF ($\nu_{\mathrm{max}}^{\mathrm{ACF}}$ and $\Delta\nu^{\mathrm{ACF}}$),
according to their logarithmic likelihood function (hereafter LLF).
The pair giving the lowest LLF (denoted $\mathcal{L}_1$) is kept as seismic guesses ($\nu_{\mathrm{max,1}}$ and $\Delta\nu_{1}$) for the next step. $H_{\mathrm{env,1}}$ is deduced from the scaling relation given in \cite{2012A&A...537A..30M}.
\subsubsection{3\textsuperscript{rd} step: mapping of $\nu_{\mathrm{max}}$ and $\Delta\nu$ } \label{step_3}
\begin{figure}
\centering
\includegraphics[scale=0.37, trim=0cm 0cm 1.5cm 1cm, clip=True]{Figures/mapping_Dnu_KIC5527304.png}
\caption{Topology of the logarithmic likelihood function $\mathcal{L}$ as a function of $\Delta\nu$ over the interval $\Delta\nu \pm 20\%$ for KIC 5527304.
The red cross indicates the lowest value of $\mathcal{L}$ corresponding to the best match between the UP and the observed oscillations spectrum.}
\label{Dnu_mapping}
\end{figure}
In order to improve $\nu_{\mathrm{max}}$ and $\Delta\nu$ estimates, we perform the mapping of the parameters.
The mapping consists in computing the LLF for several values of a given parameter around a guess.
We do not use in this step a simple minimization algorithm in order to avoid as much as possible local minima
due to the complexity of the likelihood topology,
especially for $\Delta\nu$, which exhibits several valleys (cf. Fig. \ref{Dnu_mapping}).
Indeed, when $\Delta\nu$ changes, three others parameters change proportionally $\varepsilon$, $d_{0\ell}$ and $\gamma$ (cf. Sect. \ref{oscillations_model}), thereby
modifying the oscillation component.
The value giving the lowest likelihood corresponds to the combination where the synthetic modes best match the observations.
Since $\nu_{\mathrm{max}}$ and $\Delta\nu$ are strongly dependent on each other, the best would be to do the mapping of both indices simultaneously.
However, this is very time consuming. Therefore, we obtain two separate mappings.
The quality of the mapping, and therefore of the deduced results, depend on previous guesses of the background and oscillations.
So, it is more efficient to start finding the mapping in some cases by one parameter rather than by the other.
Thus, we proceed with two optimization strategies:
\begin{enumerate}
\item[(1)] We obtain the mapping of $\nu_{\mathrm{max}}$ around $\nu_{\mathrm{max,1}} \pm 25\%$.
The lowest LLF gives us $\nu_{\mathrm{max}}^{(1)}$ with which we deduce new guesses:
$\Delta\nu^{\mathrm{map}}$ and $H_{\mathrm{env}}^{(1)}$ following the scaling law from \cite{2012A&A...537A..30M}.
Then, we obtain the mapping of $\Delta\nu$ around $\Delta\nu^{\mathrm{map}} \pm 20\%$.
The value of $\Delta\nu$ with the lowest LLF gives $\Delta\nu^{(1)}$.
\item[(2)] The strategy is reversed: First, the mapping of $\Delta\nu$ is performed,
giving $\Delta\nu^{(2)}$ from which $\nu_{\mathrm{max}}^{\mathrm{map}}$ and $H_{\mathrm{env}}^{\mathrm{map}}$ are deduced following the scaling laws from \cite{2012A&A...537A..30M}.
Then, we obtain the mapping of $\nu_{\mathrm{max}}$ around $\nu_{\mathrm{max}}^{\mathrm{map}}$ to obtain $\nu_{\mathrm{max}}^{(2)}$. We then deduce $H_{\mathrm{env}}^{(2)}$ from the scaling relation taken from \cite{2012A&A...537A..30M}.
\end{enumerate}
At last, we compare final LLF values given by the strategy (1), (2), as well as the one obtained in the second step ($\mathcal{L}_1$).
Results with the lowest LLF are kept, giving new seismic indices estimates: $\nu_{\mathrm{max,2}}$, $\Delta\nu_2$ and $H_\mathrm{env,2}$.
\subsubsection{4\textsuperscript{th} step: Oscillations fit} \label{step_4}
The aim of this step is to optimize the three seismic indices determined in the previous step using the minimization algorithm
within $\nu_{\mathrm{max,2}} \pm 6 \Delta\nu_2$.
At the end, we get new seismic guesses ($\nu_\mathrm{max,3}$, $\Delta\nu_{3}$ and $H_\mathrm{env,3}$).
\subsubsection{5\textsuperscript{th} step: Background and oscillations global fit} \label{step_5}
For this last fit, the background and oscillations are fitted simultaneously (cf. Fig. \ref{fit_final}).
Thus, all parameters of the model (Eq. (\ref{equation_BG+UP})) are free, except $\alpha_{\mathrm{act}}$ which is fixed to 2 (if we have kept the activity component).
We obtain final estimates of seismic indices: $\nu_{\mathrm{max}}^\mathrm{f}$, $\Delta\nu^\mathrm{f}$ and $H_{\mathrm{env}}^\mathrm{f}$ and background parameters,
including the granulation: $P_{\mathrm{gran}}^\mathrm{f}$, $\tau_{\mathrm{gran}}^\mathrm{f}$, $\alpha_{\mathrm{gran}}^\mathrm{f}$;
the white noise $W^\mathrm{f}$ and, depending of the case, the activity: $P_{\mathrm{act}}^\mathrm{f}$, $\tau_{\mathrm{act}}^\mathrm{f}$.
Then, we compute internal errors from each parameter by inverting the Hessian matrix \citep{numerical_recipes}.
If the fit did not converge while the activity was kept, the process returns to the second step,
removing the activity component and skipping low frequencies ($\nu < \nu_{\mathrm{max}}^{\mathrm{ACF}}/20$).
\begin{figure}
\centering
\includegraphics[scale=0.38, trim=0cm 0cm 1.5cm 1cm, clip=True]{Figures/PSD_fit_KIC2850913.png}
\caption{Results of the simultaneous adjustment of the background and oscillations at the 5\textsuperscript{th} step of MLEUP (black line).
In grey, the raw PSD of KIC 2850913.
The dash-dot green line and the dashed blue one correspond respectively to the activity and granulation component.
The dotted magenta line represents the white noise component and the solid red one is the Universal Pattern.}
\label{fit_final}
\end{figure}
\subsubsection{6\textsuperscript{th} step: null hypothesis $H_0$ test} \label{step_7}
In this additional step, we use the ACF to produce an independent rejection criterion.
We recompute the ACF around $\nu_{\mathrm{max}}^\mathrm{f}$ at $\pm \nu_{\mathrm{max}}^\mathrm{f}/3$,
following an arithmetic progression with a common difference equal to the resolution of the spectrum and we keep the highest $\mathcal{A}_{\mathrm{max}}$ value.
Following \cite{2009A&A...508..877M}, we apply to this $\mathcal{A}_{\mathrm{max}}$ value the null hypothesis $ H_0$ test.
We keep the results when $\mathcal{A}_{\mathrm{max}}$ is above the threshold $\mathcal{A}_{\mathrm{lim}} = 8$ which corresponds to a probability of $P= 1 \%$.
\section{Biases and error calibration using synthetic light-curves} \label{simulations}
In this section, our goal is to test the MLEUP algorithm on synthetic light-curves in order to calibrate
(and be able to correct a posteriori) possible biases in the estimates of seismic indices and
granulation parameters as well as in the associated error estimates.
This approach is motivated by two main reasons.
The first one is that we anticipate possible biases, since the model used in the MLEUP
algorithm (see Sect. \ref{Desciption_pipeline}) is different, simpler in fact,
than the one suggested by
our present understanding of solar-like pulsations and granulation. For the reasons developed in Sect. \ref{Desciption_pipeline}, our MLEUP-model
only includes pressure dominated modes and only one component for granulation.
Synthetic light-curves including mixed modes and a two components description of the granulation will allow us to investigate the possible
impact of such differences on the estimates of the various seismic indices and granulation parameters.
The second reason is that we want to test the strategy adopted for the MLEUP algorithm
(see Sect. \ref{Modelling}) and in particular to which extent it allows us to reach the best solution,
avoiding as much as possible solutions associated with secondary minima in the optimization process.
This approach relies on the assumption that the present understanding of solar-like pulsation
and granulation patterns is advanced enough to allow us to produce synthetic light-curves which are realistic
and representative enough of the data obtained with CoRoT and $\textit{Kepler}$.
We thus developed a solar-like light-curve simulator (SLS) to produce such synthetic light-curves
representative of CoRoT and $\textit{Kepler}$ data\footnote{The code is available at: \url{https://psls.lesia.obspm.fr}}.
We generated sets of light-curves,
for several representative durations, stellar magnitudes and evolutionary stages.
The detailed description of these synthetic light-curves and of the
statistical study of their MLEUP analysis are given in \ref{full_simulations}.
Here we stress the main specificities of the input model considered for the SLS and we describe
the corrections applied to the results obtained with
the MLEUP algorithm when analysing CoRoT and $\textit{Kepler}$ data in Sect. \ref{application}.
\subsection{The input model for synthetic light-curves} \label{simulations_parameters}
\begin{figure}
\centering
\includegraphics[scale=0.35]{Figures/test_gen_up_2.pdf}
\caption{The red curve corresponds to the theoretical spectrum pattern and the blue one illustrates
the simulated power spectrum for a given realisation generated with the SLS for a CoRoT-type star.}
\label{fig_simus}
\end{figure}
The SLS generates light-curves following an input model which features the oscillations, the granulation and the white noise
to the best of our present knowledge as based on the CoRoT and $\textit{Kepler}$ experiments.
The activity signal is not considered because, contrarily to pulsations or granulation, we lack any prescription
about how it behaves with stellar mass or evolution stage so far.
Besides these inputs, the SLS mimics the stochastic nature of these phenomena thus producing a spectrum (and the associated light-curve) representative of a
given observational realization.
This is achieved by applying an artificial random dispersion, following a $\chi^2$ statistics with two degrees of freedom statistics,
around the theoretical mean model (see Fig. \ref{fig_simus}).
The $\ell=0,2,3$ oscillations components are simulated following the UP \citep{2011A&A...525L...9M}, just as in our MLEUP-model
(see Sect. \ref{oscillations_model}).
The dipole $\ell=1$ mixed modes component is considered following \cite{2012A&A...540A.143M} and as detailed in \ref{full_simulations}.
Finally, the granulation is modeled with two components, following the model F
of \cite{2014A&A...570A..41K}, as described in Sect. \ref{background_model}.
For both satellites, simulations take into account the sampling time ($dt$),
the typical white noise level (see \ref{full_simulations}), and the attenuation factor as described in Sect. \ref{global_model}.
\subsection{Biases and internal error corrections} \label{biases_correction}
The study of synthetic light-curves (\ref{full_simulations}) allowed us to estimate biases and to characterize them quantitatively (Tab. \ref{tableau_coefficient_simus}).
They are corrected for in the analysis of real data in Sect. \ref{application}.
This study also reveals that internal errors sometimes significantly underestimate real errors.
This can be understood since formal errors derived form the Hessian matrix are at best lower limits for the error estimates, according to Kramer-Rao theorem \citep[see][and references therein]{Kendall1967}.
We therefore established conservative correction functions summarized in Tab. \ref{tableau_correction_erreur_interne} that are applied to the analysis of real data in Sect. \ref{application}.
\section{Application to large sets of stars} \label{application}
We applied the MLEUP analysis on a large set of stars observed with CoRoT and $\textit{Kepler}$.
Here we present and comment the results of this analysis.
\subsection{Observations} \label{observations}
\subsubsection{Target selection} \label{target_selection}
The CoRoT space mission was dedicated to seismology and the detection of exoplanets \citep{2006ESASP1306...33B}.
In 18 runs, from 2006 to 2013, CoRoT observed about 150,000 stars located in two opposite directions in our galaxy.
We analysed all the ready-to-use CoRoT legacy
data\footnote{CoRoT legacy data archive: \url{http://idoc-corot.ias.u-psud.fr}} \citep{2016cole.book.....C}
for which the observations lasted longer than 50 days in order to get a sufficient signal-to-noise ratio
and frequency resolution to detect the oscillations.
This represents a total number of 113,677 CoRoT stars.
The CoRoT legacy data were processed as described in \cite{2016cole.book...41O}.
When stars were observed several times,
we concatenated their light-curves (assuming that there is no temporal coherence) by joining them together and adjusting their average level of intensity.
The second space mission, $\textit{Kepler}$, a NASA spacecraft \citep{2010AAS...21510101B},
was launched in 2009 and observed the same galactic region during more than four years, divided in 17 quarters.
We analysed all long cadence ($dt = 29.42$ min) $\textit{Kepler}$ data\footnote{$\textit{Kepler}$ data archive: \url{http://archive.stsci.edu/kepler/}}
for which the observations lasted longer than 10 days.
These data were corrected as documented in
the $\textit{Kepler}$ Data Processing Handbook\footnote{The $\textit{Kepler}$ Data Processing Handbook website:
\url{https://archive.stsci.edu/kepler/manuals/KSCI-19081-001_Data_Processing_Handbook.pdf}}.
Then, from the effective temperature $T_{\mathrm{eff}}$ extracted from different catalogues (see Sect. \ref{scaling_relations} for more detail),
we selected the stars in the temperature range $3800$ K $\leq~T_{\mathrm{eff}}~\leq~5700$ K.
Outside this range, we assume they are not red giants.
As was done for CoRoT data, we concatenated light-curves of stars observed several times.
Gaps smaller than $10^4$ s were filled using a linear interpolation.
This dataset represents 207,610 stars, most of which were observed during all the duration of the mission.
\subsubsection{Rejection of outliers} \label{Rejection_outliers}
After analysing data from both satellites separately, we kept the results which satisfy the following criteria:
\begin{itemize}
\item The fit of the three seismic indices must be properly converged.
\item $\mathcal{A}_{\mathrm{lim}} = 8.0 \leq \mathcal{A}_{\mathrm{max}} \leq \mathcal{A}_{\mathrm{lim}} = 700$ (see Sect. \ref{simulations_parameters} for more details).
The upper limit was defined in order to avoid false detections caused by possible artefacts in the spectrum.
\item The signal-to-noise ratio (SNR) is defined as: $SNR = H_{\mathrm{env}}/B_\mathrm{max}$, with $B_\mathrm{max}$ the height of the background at the frequency $\nu_{\mathrm{max}}$.
We keep results with a \textit{SNR} between: $1.0 \leq SNR \leq 70$. The upper limit is used for the same reasons as for $\mathcal{A}_{\mathrm{max}}$.
\item We restrict the frequency interval of $\nu_{\mathrm{max}}$ to $2~<~\nu_{\mathrm{max}}~<~100~\mu$Hz for CoRoT and to $2~<~\nu_{\mathrm{max}}~<~250~\mu$Hz for $\textit{Kepler}$.
Indeed, as shown by the simulations (see Sect.~\ref{simulations} and \ref{full_simulations}), for higher values of $\nu_{\mathrm{max}}$,
the dispersion of parameters becomes too high, especially for $\nu_{\mathrm{max}}$ and $\Delta\nu$, and is poorly represented by the internal errors.
The lower limit is due to the presence of some peaks below $2~\mu$Hz that we suspect to be artefacts for $\textit{Kepler}$
and due to the limited resolution for CoRoT.
\item Values of the granulation slope $\alpha_{\mathrm{gran}}$ are limited to $\alpha_{\mathrm{gran}} < 5.0$,
because higher slopes generally correspond to a bad fit, typically due to an artefact.
\item For each parameter, the ratio between the internal errors and the associated values must be less than 50\%.
\item Outliers are removed using the following combination of seismic indices:
\begin{equation}
\left( \frac{\Delta\nu - \Delta\nu^\mathrm{SR}}{\alpha_\pm \Delta\nu^\mathrm{SR}}\right )^{2} +
\left( \frac{H_{\mathrm{env}} - H_{\mathrm{env}}^\mathrm{SR}}{\beta_\pm H_{\mathrm{env}}^\mathrm{SR}}\right)^{2} \leqslant 1 \, ,
\end{equation}
with $\alpha_+ = \alpha_- = 0.30$, $\beta_+ = 3$ and $\beta_- = 1$.
$\beta_+ \neq \beta_-$ because the distribution of results for the parameters $H_{\mathrm{env}}$ is asymmetric.
$X^\mathrm{SR}$ is the value of the parameter $X$ corresponding to the scaling relation.
The scaling laws are initially taken from the literature (see Tab. \ref{relation_echelle_ref})
and iteratively based on the considered data sample.
\end{itemize}
For the sub-sample, for which we have both the seismic indices and granulation parameters, we used these additional criteria:
\begin{itemize}
\item The fit of both granulation parameters must be properly converged.
\item Values of the granulation slope $\alpha_{\mathrm{gran}}$ are also limited to $\alpha_{\mathrm{gran}} \geq 1.0$ since $\alpha_{\mathrm{gran}}~<1.0$ does not give physical results.
\item Outliers are removed using the following combination of seismic indices (see above) and granulation parameters:
\begin{equation}
\left( \frac{P_{\mathrm{gran}} - P_{\mathrm{gran}}^\mathrm{SR}}{\beta_\pm P_{\mathrm{gran}}^\mathrm{SR}}\right )^{2} +
\left( \frac{\tau_{\mathrm{gran}} - \tau_{\mathrm{gran}}^\mathrm{SR}}{\beta_\pm \tau_{\mathrm{gran}}^\mathrm{SR}}\right )^{2} \leqslant 1 \, ,
\end{equation}
with $\beta_+ = 3$ and $\beta_- = 1$ (the distribution of $P_{\mathrm{gran}}$ and $\tau_{\mathrm{gran}}$ are also asymmetric).\\
\end{itemize}
Finally, we yield 20,122 stars (2943 CoRoT stars and 17,179 $\textit{Kepler}$ stars) for which we extracted the seismic indices. They form the $S_\mathrm{s}$ dataset. Besides, for 17,109 stars (806 CoRoT stars, and 16,303 $\textit{Kepler}$ stars) we obtained both the seismic indices and the granulation parameters ($S_\mathrm{s+g}$ dataset).
As noted recently \citep[see e.g.][]{2016ApJ...827...50M}, some stars in KIC have been unclassified or misclassified.
Here among the 15,626 $\textit{Kepler}$ stars identified as red giants in the $\textit{Kepler}$ Input Catalogue \citep[KIC,][]{2011AJ....142..112B},
we detect oscillations for 13,277 stars.
In addition, we found 3902 new oscillating red-giant stars not identified in KIC as red giants.
\subsection{Results for the various parameters} \label{scaling_relations}
We present in this subsection the results obtained with MLEUP for the selected CoRoT and $\textit{Kepler}$ datasets, and for the various parameters.
For some $\textit{Kepler}$ targets, we have additional informations allowing us to enrich and improve our understanding of the results:
\begin{itemize}
\item Evolutionary stages: \cite{2016A&A...588A..87V}
have determined the evolutionary stages of more than five thousands $\textit{Kepler}$ stars classified as red giants
using an automatic measurement of the gravity period spacing of dipole modes $\Delta\Pi_1$.
Thanks to their results, we are able to discern between stars belonging to the RGB from those in the clump for about 25\% of our selected $\textit{Kepler}$ dataset.
However, this method is limited to $\nu_{\mathrm{max}} \geq 35~\mu$Hz for RGB stars and to $\nu_{\mathrm{max}} \geq 25~\mu$Hz for clump stars.
Indeed, it is difficult to automatically measure the $\Delta\Pi_1$ at low frequency (i.e. for evolved stars)
since the mixed modes are less visible and therefore more difficult to detect \citep[also see][]{2014A&A...572A..11G}. \\
\item
Mass and radius estimates:
Combining $T_{\mathrm{eff}}$, $\Delta\nu$ and $\nu_{\mathrm{max}}$ allows us to calculate the seismic mass,
radius and luminosity from the following scaling relations \citep[e.g.][]{2010A&A...509A..77K}:
\begin{equation}
\frac{M}{\mathrm{M_\odot}} \propto \left( \frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{ref}}} \right)^3 \left( \frac{\Delta\nu}{\Delta\nu_{\mathrm{ref}}} \right)^{-4} \left( \frac{T_{\mathrm{eff}}}{T_\odot} \right)^{3/2} \, ,
\label{eq_masse}
\end{equation}
\begin{equation}
\frac{R}{\mathrm{R_\odot}} \propto \left( \frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{ref}}} \right) \left( \frac{\Delta\nu}{\Delta\nu_{\mathrm{ref}}} \right)^{-2} \left( \frac{T_{\mathrm{eff}}}{T_\odot} \right)^{1/2} \, ,
\label{eq_rayon}
\end{equation}
and
\begin{equation}
\frac{L}{\mathrm{L_\odot}} \propto \left( \frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{ref}}} \right)^2 \left( \frac{\Delta\nu}{\Delta\nu_{\mathrm{ref}}} \right)^{-4} \left( \frac{T_{\mathrm{eff}}}{T_\odot} \right)^{5} \, .
\label{eq_luminosite}
\end{equation}
These relations are normalised with the reference values given in \cite{2013A&A...550A.126M}:
$\Delta\nu_{\mathrm{ref}} = 138.8~\mu$Hz, $\nu_{\mathrm{ref}} = 3104~\mu$Hz and $T_\odot=5777$ K.\\
The effective temperatures $T_{\mathrm{eff}}$ are extracted from several catalogues.
First, we took $T_{\mathrm{eff}}$ from both spectroscopic surveys APOGEE DR12 \citep{2016arXiv160802013S} and LAMOST DR2 \citep{2016yCat.5149....0L}.
We got 7205 and 1809 effective temperatures respectively.
Then, from the photometric Str\"{o}mgren survey for Asteroseismology and Galactic Archaeology (SAGA) \citep{2014ApJ...787..110C}, we got 377 effective temperatures .
Finally, from \cite{2017ApJS..229...30M}, which is an updated of the \cite{2014ApJS..211....2H} catalogue reporting $T_{\mathrm{eff}}$ for stars
observed by $\textit{Kepler}$ for quarters Q1 to Q17 (DR25), we obtained 7788 more effectives temperatures.
Altogether, these catalogues provide us with effective temperatures for the entire set of selected $\textit{Kepler}$ targets.
\end{itemize}
The analysis of the seismic indices and the fundamental parameters is done using the dataset $S_\mathrm{s}$, and for the granulation parameters, we use the $S_\mathrm{s+g}$ dataset (see Sect. \ref{Rejection_outliers}).
\subsubsection{Height of the Gaussian envelope $H_{\mathrm{env}}$} \label{Henv}
\begin{figure*}
\centering
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Henv_numax_corot.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Henv_numax_kepler.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Henv_numax_kepler_evo.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Henv_numax_kepler_M.png}
\caption {Height of the Gaussian envelope $H_{\mathrm{env}}$ obtained with the CoRoT (\textit{a}) and $\textit{Kepler}$ (\textit{b}) datasets $S_\mathrm{s}$.
Black crosses represent the values obtained, with their error bars in grey.
The red line is the deduced scaling relations. \newline
\textit{c:} Same as figure \textit{b} with the 1333 RGB stars in blue and the 3152 clump stars in red.
The green dashed line and the dotted one represent respectively the scaling relation obtained with the RGB and clump stars. \newline
\textit{d:} Same as figure \textit{b} with the stellar mass indicated via the colour code.
For better visibility of the mass variation, only stars with a mass in the range $0.5 \leqslant M/\mathrm{M_\odot} \leqslant 3.0$ are plotted.
\label{fig_resultat_Henv}}
\end{figure*}
Overall, the values of $H_{\mathrm{env}}$ obtained with CoRoT and $\textit{Kepler}$ data are comparable (cf. Fig. \ref{fig_resultat_Henv}a and b).
The dispersion is larger in the case of CoRoT as expected and as it had been previously suggested by the simulations (cf. Sect.~\ref{simulations} and \ref{full_simulations}).
This is mainly due to shorter observation time $T$ in the case of CoRoT.
Concerning $\textit{Kepler}$ results, the $H_{\mathrm{env}}$ distribution shows a broadening below the deduced scaling relation for $\nu_{\mathrm{max}} \gtrsim 30~\mu$Hz.
Figure \ref{fig_resultat_Henv}c, with the information on the evolutionary stage for some stars,
reveals that this broadening is mainly due to clump stars (red crosses) which exhibit an envelope height which decreases significantly
faster with $\nu_{\mathrm{max}}$ than in RGB stars (blue crosses).
The RGB component is close to the reference because the majority of stars without information on the evolutionary stage belongs to the RGB,
especially below $\nu_{\mathrm{max}} = 30~\mu$Hz.
This dependency of $H_{\mathrm{env}}$ on the evolutionary stage has already been observed by \cite{2011A&A...532A..86M,2012A&A...537A..30M} on a much smaller sample.
The information on the mass (cf. Fig. \ref{fig_resultat_Henv}d) confirms the distribution of the clump component since,
according to the theory of the stellar evolution,
higher-mass stars belong essentially to the clump and its extension, the secondary clump \citep[e.g.][]{1994sse..book.....K}.
\subsubsection{Mean large separation $\Delta\nu$} \label{Dnu}
\begin{figure*}
\centering
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Dnu_numax_corot.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Dnu_numax_kepler.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Dnu_numax_kepler_evo.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/Dnu_numax_kepler_M.png}
\caption {Same as Fig. \ref{fig_resultat_Henv} but for the mean large separation $\Delta\nu$. \label{fig_resultat_Dnu}}
\end{figure*}
For CoRoT and $\textit{Kepler}$, the dispersion for the parameter $\Delta\nu$ is quite small (see Fig. \ref{fig_resultat_Dnu}a et b),
in agreement with the simulations (cf. Sect.~\ref{simulations} and \ref{full_simulations}).
Interestingly, as for $H_{\mathrm{env}}$, the information on the evolutionary stage reveals a clear difference between
the clump (in red) and the RGB (in blue) (cf. Fig. \ref{fig_resultat_Dnu}c).
This may be explained for example by the difference in mass range covered by the RGB and clump stars as we discuss in Sect.~\ref{param_mass}.
Consequently, the $\Delta\nu$ scaling laws depend slightly on the evolutionary stage.
To our knowledge, this result has never been observed before.
Figure \ref{fig_resultat_Dnu}d illustrates the distribution in mass of our sample. We must stress that the apparent gradient in mass is a direct consequence of the scaling law used to estimate masses (Eq.~\ref{eq_masse}).
\subsubsection{Granulation effective timescale $\tau_{\mathrm{eff}}$} \label{taueff}
\begin{figure*}
\centering
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/taueff_numax_corot.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/taueff_numax_kepler.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/taueff_numax_kepler_evo.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/taueff_numax_kepler_M.png}
\caption {Granulation effective timescale $\tau_{\mathrm{eff}}$ obtained with the CoRoT (\textit{a}) and $\textit{Kepler}$ (\textit{b}) datasets $S_\mathrm{s+g}$.
Black crosses represent the obtained values, with their error bars in grey.
The red line is the deduced scaling relations.
In the case of $\textit{Kepler}$, the power law representations were deduced from results with $\nu_{\mathrm{max}}<100~\mu$Hz. \newline
\textit{c:} Same as figure \textit{b} with the 1256 RGB stars (in blue) and the 3142 clump stars (in red).
The green dashed line and the dotted one represent respectively the scaling relation obtained with the RGB and clump stars with $\nu_{\mathrm{max}}<100~\mu$Hz. \newline
\textit{d:} Same as figure \textit{b} with the stellar mass indicated via the colour code.
For better visibility of the mass variation, only stars with a mass in the range $0.5 \leqslant M/\mathrm{M_\odot} \leqslant 3.0$ are plotted.}
\label{fig_resultat_taueff}
\end{figure*}
Results obtained for the parameter $\tau_{\mathrm{eff}}$
have a larger dispersion in the case of CoRoT (cf. Fig. \ref{fig_resultat_taueff}a) than in the case of $\textit{Kepler}$.
As suggested by the simulations (cf. Sect.~\ref{simulations} and \ref{full_simulations}), this is mainly due to shorter observation time $T$ in the case of CoRoT.
Considering the $\textit{Kepler}$ results, one can see in Fig. \ref{fig_resultat_taueff}b two characteristic structures.
The first one is in the interval $20<\nu_{\mathrm{max}}<100~\mu$Hz, where one can see a greater dispersion for $\tau_{\mathrm{eff}}$.
Figure \ref{fig_resultat_taueff}c reveals that this broadening is due to the clump component
{ even if the two components almost fully overlap each other,
thereby emphasizing that $\tau_{\mathrm{eff}}$ depends poorly on the evolutionary stage.}
This is not surprising since it depends essentially on the surface parameters of the star.
The second structure is in the interval $100<\nu_{\mathrm{max}}<200~\mu$Hz, where values of $\tau_{\mathrm{eff}}$ are significantly below the scaling relation.
This feature is explained by the values taken by the slope $\alpha_{\mathrm{gran}}$ of the Lorentzian function which fits the granulation
(see Sect. \ref{resultats_agran} for more details).
\subsubsection{Granulation characteristic amplitude $\sigma_\mathrm{g}^2$} \label{sigma2}
\begin{figure*}
\centering
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/sigma2_numax_corot.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/sigma2_numax_kepler.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/sigma2_numax_kepler_evo.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/sigma2_numax_kepler_M.png}
\caption{Same as Fig. \ref{fig_resultat_taueff}, but for the granulation characteristic amplitude $\sigma_\mathrm{g}^2$. }
\label{fig_resultat_sigma2}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale=0.38, trim=0cm 0cm 1.5cm 1cm, clip=True]{Figures/resultats/sigmaR_numax_kepler_evo.png}
\caption {Granulation characteristic amplitude $\sigma_\mathrm{g}^2$ multiplied by the seismic radius $R^2$ as a function of $\nu_{\mathrm{max}}$
obtained with the $\textit{Kepler}$ dataset $S_\mathrm{s+g}$.
Red crosses represent clump stars and the blue ones, RGB stars.
The red line is the scaling relation determined with a least-square fit considering the internal errors on both axes.}
\label{fig_sigmaR_numax}
\end{figure}
Overall, results for $\sigma_\mathrm{g}^2$ are comparable for the CoRoT and $\textit{Kepler}$ datasets (cf. figure \ref{fig_resultat_sigma2}a and b),
with a stronger dispersion for CoRoT as expected.
In the case of $\textit{Kepler}$ results, structures can once again be distinguished.
The structure located in the interval $100<\nu_{\mathrm{max}}<200~\mu$Hz is associated with the behaviour of the slope $\alpha_{\mathrm{gran}}$ (see Sect. \ref{resultats_agran}) as was the case for $\tau_{\mathrm{eff}}$.
The bulge from $\nu_{\mathrm{max}} \simeq 30$ to $\simeq100~\mu$Hz is associated with the clump component as revealed in figure \ref{fig_resultat_sigma2}c
by the evolutionary stage as well as in figure \ref{fig_resultat_sigma2}d with the mass distribution.
As for the parameters $H_{\mathrm{env}}$ and $\Delta\nu$, we find that $\sigma_\mathrm{g}^2$ depends significantly on the evolutionary stage
{(cf. Fig. \ref{fig_resultat_sigma2}c and d).}
This dependency of $\sigma_\mathrm{g}^2$ with the evolutionary stage has already been observed by \cite{2012A&A...544A..90H} on a much smaller sample.
\subsubsection{Granulation slope $\alpha_{\mathrm{gran}}$} \label{resultats_agran}
\begin{figure*}
\centering
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/agran_numax_kepler.png}
\includegraphics[scale=0.5, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/simus_kepler/MLEUP_agran_corrected.png}
\caption{\textit{a:} Granulation slope $\alpha_{\mathrm{gran}}$ obtained with the $\textit{Kepler}$ dataset $S_\mathrm{s+g}$.
The colour code represents the stellar flux normalised by the flux of a 12.5 magnitude of star and the error bars are in grey.
Black dots are the median of $\alpha_{\mathrm{gran}}$ results in a box of $30~\mu$Hz around the corresponding $\nu_{\mathrm{max}}$ of the simulations. \newline
\textit{b:} Results of the $\textit{Kepler}$-type simulations for RGB stars.
Triangles represent the median of $\alpha_{\mathrm{gran}}$, error bars are the $\pm 1 \sigma$ dispersion and colours indicate different simulation parameters.
Red corresponds to the simulations described in Section \ref{simulations_parameters},
cyan represents simulations with a magnitude 8.0 (instead of 12.0) and black corresponds to the simulations with only one granulation component. }
\label{fig_resultat_agran}
\end{figure*}
The distribution of the results obtained with the $\textit{Kepler}$ dataset (Fig. \ref{fig_resultat_agran}a)
is qualitatively consistent with the simulations (red line in Fig. \ref{fig_resultat_agran}b).
$\alpha_{\mathrm{gran}}$ is about 4 at $\nu_{\mathrm{max}}=10~\mu$Hz, then it decreases until $\nu_{\mathrm{max}} \simeq 150-200~\mu$Hz after which it increases again.
This pattern, observed in both observations and simulations, is an artefact of our method which uses one component to fit the granulation.
As shown by the simulations, the decrease of $\alpha_{\mathrm{gran}}$ values can be explained by the difference of lifetime of both granulation components with $\nu_{\mathrm{max}}$.
This is corroborated by the almost flat curve of the simulations with only one component (black line in Fig. \ref{fig_resultat_agran}b).
Indeed, the more $\nu_{\mathrm{max}}$ increases, the more the two components move away from each other.
Therefore, since we adjust only one granulation component, $\alpha_{\mathrm{gran}}$ needs to take increasingly low values
in order to correctly take into account the whole contribution of both components.
Consequently, $\alpha_{\mathrm{gran}}$ can take very low values in the interval $100<\nu_{\mathrm{max}}<200~\mu$Hz (Fig. \ref{fig_resultat_agran}a),
corresponding physically to an artificially too rapid decrease of the granulation coherence time,
and thus, induces the low values of $\tau_{\mathrm{eff}}$ and the high values of $\sigma_\mathrm{g}^2$ observed in sections \ref{taueff} and \ref{sigma2}.
There is also a clear correlation between $\alpha_{\mathrm{gran}}$ and the stellar flux
showing that brighter stars systematically have lower $\alpha_{\mathrm{gran}}$ values in the interval $10\leqslant\nu_{\mathrm{max}}\leqslant150-200~\mu$Hz,
as it can be seen in Figure \ref{fig_resultat_agran}a for the observations and in Figure \ref{fig_resultat_agran}b for the simulations (cyan line).
This is due to the fact that the fainter the star is, the higher the noise will be and consequently, the flatter the Lorentzian function giving low $\alpha_{\mathrm{gran}}$ will be.
Finally, from $\nu_{\mathrm{max}} \simeq 150-200~\mu$Hz, $\alpha_{\mathrm{gran}}$ increases (Fig. \ref{fig_resultat_agran}a and b) because of the Nyquist frequency which causes a degeneracy of the fit, thus causing an overestimate
of the white noise and an underestimate of the power of the granulation at high frequencies.
For the observations, $\alpha_{\mathrm{gran}}$ reaches higher values than for simulations because of a broader stellar magnitude range.
\subsection{Comparison with others published studies} \label{comparaison_methods}
In this section, we first compare the results of our analysis with those available for the same stars in the literature.
Then, we use power law representations to compare the general trends associated with the whole set of stars we analysed.
This comparison is made with the power laws obtained with various methods on different sets of stars published in the litterature. Additionaly, we compare results obtained on CoRoT and $\textit{Kepler}$ data sets.
\subsubsection{Star by star comparison} \label{starbystar_comparaison}
We compared our $\textit{Kepler}$ results with those available in the APOKASC catalogue \citep{2014ApJS..215...19P}.
These seismic indices were obtained with the COR method \citep{2010MNRAS.402.2049H}.
For the 1850 stars in common, we made a one to one comparison between our $\Delta\nu$ and $\nu_{\mathrm{max}}$ results and APOKASC values.
In case of $\Delta\nu$, our results are within the internal errors.
However, it can be noticed that uncertainties given by \cite{2014ApJS..215...19P} have been estimated by adding
in quadrature the formal uncertainty returned by the \cite{2010MNRAS.402.2049H}'s method to the standard deviation of the values returned by all methods.
They are thus already representative of the dispersion between various methods.
In the case of $\nu_{\mathrm{max}}$, we found a small but significant bias.
It is negligible at low frequencies ($\nu_{\mathrm{max}} \sim 10~\mu$Hz) but can reach about 10\% at high frequencies ($\nu_{\mathrm{max}} > 200~\mu$Hz).
We attribute this difference to the fact that in our approach we do not smooth the power spectrum in order to preserve the oscillation modes heights
and the stellar background shape.\\
We also compared our CoRoT results with those derived by \cite{2011A&A...525L...9M}. There are 352 common stars.
The comparison shows that our results are on the whole consistent with \cite{2011A&A...525L...9M} values.
However, the measurements show a larger dispersion than with the APOKASC data, with higher outliers which cannot be explained by the internal errors.
This larger dispersion can probably be explained by the fact that the analysis performed by \cite{2011A&A...525L...9M}
were based on the first CoRoT data processing pipeline which has been substantially improved since then \citep{2016cole.book...41O}.
For $\Delta\nu$, this comparison shows that there is no significant bias between our results and those derived by \cite{2011A&A...525L...9M}.
For $\nu_{\mathrm{max}}$, we found that the dispersion can be explained by the internal errors.
However, a small bias increasing with the frequency is observed and is of the same order as the one observed with the APOKASC data.
\subsubsection{Power law representations} \label{scaling_relation_discution}
Another way to compare our results with results found in the literature is to consider power law representations of our results as a function of $\nu_{\mathrm{max}}$.
We adjusted each stellar index dataset by a power law of the form $y = \alpha (\nu_{\mathrm{max}}) ^\beta$ using a least-square fit
while taking into account error bars on the axes ($y$ and $\nu_{\mathrm{max}}$). We obtained the fits for both CoRoT and $\textit{Kepler}$ datasets.
Values of the adjusted coefficients $\alpha$ and $\beta$ are reported in Table\ \ref{relation_echelle_resultats}
together with the reduced $\chi^2$ values obtained for each fit.
The reduced $\chi^2$ values are systematically significantly larger than three.
This means that the difference between the observations and the power law are statistically
significant (with a probability higher than $99\%$) and hence cannot be explained by the noise.
Indeed, the measured indices exhibit complex structures that explain the important departure from those power laws.
This highlights the fact that such representations do not fully represent the diversity of the red giant sample. Accordingly, one must pay attention that the comparison made in terms of power laws should in principle be made with the same sample of stars, which is in practice difficult or often not possible. However, in some cases, such comparisons do reveal large differences which must be attributed to differences in the analysis methods as will be shown hereafter.
The power laws derived from the CoRoT data are all found to significantly depart from those derived from the $\textit{Kepler}$ data.
These departures may be explained by differences in the instrumental response function of both instruments (e.g. bandwidths) but more likely from the fact that the observed population of star are not exactly the same \citep[also see][]{2010ApJ...723.1607H}.
This latter hypothesis is supported by the fact that the clump and the RGB stars exhibit very different structures.
We now turn to the power law representations published in the literature (cf. Tab. \ref{relation_echelle_ref} for some references).
Several studies have focussed on the $\Delta\nu-\nu_{\mathrm{max}}$ relation
on $\textit{Kepler}$ data \citep[e.g.][]{2010ApJ...723.1607H,2012A&A...537A..30M}
and some for CoRoT data \citep[e.g.][]{2010A&A...517A..22M, 2009A&A...506..465H, 2009MNRAS.400L..80S}.
The $\Delta\nu-\nu_{\mathrm{max}}$ power law derived from $\textit{Kepler}$ data set is consistent with the one by \cite{2012A&A...537A..30M},
which was determined by an average of power law obtained by various methods.
Concerning the $\sigma_\mathrm{g}^2-\nu_{\mathrm{max}}$ and the $\tau_{\mathrm{eff}}-\nu_{\mathrm{max}}$ power law representations,
our $\textit{Kepler}$ power law and those derived by \cite{2014A&A...570A..41K} have similar slopes $\beta$. However, considering error bars, the difference remains statistically significant.
Furthermore, our power laws for $H_{\mathrm{env}}$ strongly (and significantly) differ from the one deduced by \cite{2012A&A...537A..30M}.
Nonetheless, \cite{2012A&A...537A..30M} do not measure exactly the same quantity as we do.
Indeed, unlike our method, these authors smooth the spectrum before fitting it by a Gaussian function.
Except for $\tau_{\mathrm{eff}}$, all the stellar indices result in power law representation significantly different between the clump and the RGB stars.
The values of the $\alpha$ and $\beta$ coefficients are given in table \ref{relation_echelle_resultats} for both evolutionary stages.
This shows again how sensitive are the coefficients of the power law w.r.t. stellar samples with different proportions of RGB and clump stars. Thus part of the differences with the literature found here are necessarily due to difference in the stellar samples.
For the $\Delta\nu-\nu_{\mathrm{max}}$ power law, the difference between the clump and the RGB sets can either be due to difference in the mass distribution between the two populations or to big differences in the core structures between clump and RGB stars (cf. Sect. \ref{param_mass}).
For $\sigma_\mathrm{g}^2-\nu_{\mathrm{max}}$ relation, the difference between the clump and RGB stars is essentially due to a radius dependence of $\sigma_\mathrm{g}^2$.
Indeed, \cite{2006A&A...445..661L} showed that $\sigma_\mathrm{g}^2 \propto 1/R^2$.
Our results confirm this power law as shown in figure \ref{fig_sigmaR_numax}, where values of $\sigma_\mathrm{g}^2 R^2$
for RGB and clump stars follow very similar power laws.
($\beta_\mathrm{clump}=-2.187 \pm 0.002$ and $\beta_\mathrm{RGB}=-1.929 \pm 0.004$).
Based on all the stars with $\nu_{\mathrm{max}}<100~\mu$Hz of the dataset $S_\mathrm{s+g}$,
we deduced the following power law while taking into account the internal errors on both axes
\begin{equation} \label{eq_sigmaR_rayon}
\sigma_\mathrm{g}^2 R^2 = (8.59 \pm 0.02)10^{10}~\nu_{\mathrm{max}}^{-2.1574 \pm 0.0006} \; .
\end{equation}
Finally, concerning the $H_{\mathrm{env}}-\nu_{\mathrm{max}}$ relation, the dependence on evolutionary stage is also found to be statically significant but has not yet been explained.
\subsection{Stellar parameters inferred from seismic indices} \label{param_fondamentaux}
In order to illustrate the results obtained with this large sample of stars, we estimated the stellar seismic masses, radii and luminosities via equations (\ref{eq_masse}), (\ref{eq_rayon}) and (\ref{eq_luminosite}) respectively, using the seismic indices from the $\textit{Kepler}$ dataset $S_\mathrm{s}$ and the effective temperatures from different catalogs
(see Sect. \ref{scaling_relations} for more details).
In the following sub-sections, we comment on these results in the light of stellar evolution theory.
\begin{figure*}
\centering
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/M_numax.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/R_numax.png}
\caption {Mass (\textit{a}) and radius (\textit{b}) distribution as a function of $\nu_{\mathrm{max}}$ obtained with the $\textit{Kepler}$ dataset $S_\mathrm{s}$.
Red crosses represent clump stars and the blue ones, RGB stars.
The red line (right panel) is the scaling relation determined with a least-square fit considering the internal errors on both axes.}
\label{fig_resultat_M_R}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/HRdiagram_kepler_evo.png}
\includegraphics[scale=0.39, trim=0.37cm 0cm 1.cm 0.7cm, clip=True]{Figures/resultats/HRdiagram_kepler_M.png}
\caption{Hertzsprung--Russell diagram with the information on the evolutionary stage (\textit{a}) and the mass (\textit{b})
for the $\textit{Kepler}$ dataset $S_\mathrm{s}$.\newline
\textit{a:} Red crosses represent clump stars and the blue ones, RGB stars. \newline
\textit{b:} The colour code represents the stellar mass.
For better visibility of the mass variation, only stars with a mass included in the range $0.5 \leqslant M/\mathrm{M_\odot} \leqslant 3.0$ are represented.
The black cross indicate the median 1-$\sigma$ error bars in both axes. In order to better reveal the fine structures, we only plotted stars for which the effective temperature was determined by the spectroscopy surveys APOGEE and LAMOST.}
\label{fig_HR_diagram}
\end{figure*}
\subsubsection{Mass distribution} \label{param_mass}
Figure \ref{fig_resultat_M_R}a presents the mass distribution as a function of $\nu_{\mathrm{max}}$.
We can clearly see that RGB stars (in blue) mainly have masses between $\sim1$ and $2~\mathrm{M_\odot}$.
Indeed, less massive stars are non-existent on the RGB because their lifetimes on the main sequence exceed the age of the Universe, while more-massive stars evolve faster and follow different evolutionary tracks.
Clump stars (in red) cover a wider mass range than RGB stars, from $\sim0.5$ to $3.0~\mathrm{M_\odot}$.
Clump stars with low masses ($<1~\mathrm{M_\odot}$) are stars which have travelled all along the RGB until the tip where they suffer strong mass loss.
Masses higher than $\sim 2~\mathrm{M_\odot}$ belong to the so-called secondary clump \citep{1999MNRAS.308..818G}, close to the primary clump in the HR diagram.
Typically, these stars have switched to the helium reactions in a non-degenerate core.\\
In this context, the different slopes found for $\Delta\nu$-$\nu_{\mathrm{max}}$ relation between clump and RGB stars in section \ref{Dnu} can have several causes. On one hand, as already commented, this can be due to difference in mass distribution between the two samples.
However, to demonstrate this argument, one would require independent mass measurements.
The different slopes could also be explained by the very different structures of clump and RGB stars.
Indeed, scaling relations assume homologous structures \citep[see e.g.][]{2013ASPC..479...61B}. Due to big differences in the core structures between clump and RGB stars, we may expect a difference in scaling relations, as suggested by \citet{2012MNRAS.419.2077M}.
\subsubsection{Radius distribution} \label{param_rayon}
In the radius-$\nu_{\mathrm{max}}$ diagram (Fig. \ref{fig_resultat_M_R}b), clump stars exhibit a small radius range.
This is consistent with the fact that they have a rather constant radius during this evolutionary stage while RGB stars experience a strong increase
of their radius during their ascent of the RGB up to the tip. \\
There is a strong correlation between the radius and $\nu_{\mathrm{max}}$.
The scaling relation obtained from the $\textit{Kepler}$ dataset $S_\mathrm{s}$ (cf. Tab. \ref{relation_echelle_resultats})
is consistent with the value measured by \cite{2010A&A...517A..22M} and the theoretical value \citep[][c.f. Tab.~\ref{relation_echelle_ref}]{2010A&A...517A..22M}.
\subsubsection{Hertzsprung--Russell diagram} \label{param_HR}
Figure \ref{fig_HR_diagram} shows the location of our sample in the Hertzsprung--Russell (HR) diagram.
On the right and left panels, one can recognize the red giant branch from $\log(T_{\mathrm{eff}})~\sim~3.6$ to $\sim~3.75$ K.
From classical observing techniques, it is not possible to distinguish the stars belonging to the red clump
from those ascending the red giant branch
because of their overlap in luminosity and temperature.
However, this is possible thanks to the seismic constraints related to the physical conditions in the stellar cores \citep[e.g.][cf. Sect. \ref{scaling_relations}]{2016A&A...588A..87V}.
In Fig. \ref{fig_HR_diagram}a, all stars identified as RGB stars (in blue) are at the bottom of the RGB, corresponding to relatively young red-giant stars.
Regarding stars identified as clump stars (in red), their position is consistent with the red clump in the HR diagram.
However, some of them are above ($\log(L/\mathrm{L_\odot}) > 2.1$). Those stars are probably leaving the red clump on their way to the asymptotic giant branch (AGB).
Another feature is highlighted by the figure \ref{fig_HR_diagram}a: the presence of the bump as predicted by models of stellar evolution \citep[e.g.][]{2012A&A...543A.108L}
can be seen as an over-density of stars on the RGB below the clump around $\log L \sim 1.5$.
Note also that a similar signature of the presence of the bump and secondary clump has recently been revealed by \cite{2018A&A...609A.116R} using the Gaia Data Release 1.
In the figure \ref{fig_HR_diagram}b,
one notes different extension of the RGB stars in the HR diagram depending on mass. Indeed, low stellar masses extend toward lower temperature and to a less extent lower luminosity than the higher ones.
This is again in qualitative agreement with what is obtained with models which show
that the higher mass stars start the RGB at higher luminosity and follow hotter tracks \citep[e.g.][]{2012A&A...543A.108L}.
The quantitative analysis of these observations is out of the scope of the present study.
Nevertheless, they confim that seismic indices coupled with classical observational constraints open new perspective for quantitative comparison with theoretical stellar evolution models.
\section{Conclusion} \label{conclusion}
The method MLEUP developed and described here allows analysing automatically and homogeneously large datasets of light-curves. It extracts simultaneously
the fundamental seismic indices of the oscillations ($\Delta\nu$, $\nu_{\mathrm{max}}$ and $H_{\mathrm{env}}$)
and the parameters characterizing the granulation ($\tau_{\mathrm{eff}}$ and $\sigma_\mathrm{g}$) of red-giant solar type pulsators.
The performances of MLEUP were first evaluated
using sets of simulated light-curves representative for both evolutionary stages, RGB and clump, and both CoRoT and $\textit{Kepler}$ observation conditions.
These tests were used to characterize biases on the values and associated uncertainties obtained with MLEUP (cf. Sect. \ref{simulations} and \ref{full_simulations}).
These biases were then used to correct the measurements obtained with real data (cf. Tab. \ref{tableau_coefficient_simus}).
We applied MLEUP to all CoRoT data with a duration of observation larger than 50 days and to all long cadence $\textit{Kepler}$ data.
We successfully extracted the seismic indices for 2943 CoRoT stars and 17,179 $\textit{Kepler}$ stars,
increasing significantly the number of $\textit{Kepler}$ stars known as oscillating red giants.
We were able to extract simultaneously the seismic indices and the granulation parameters for 806 CoRoT stars and 16,303 $\textit{Kepler}$ stars.
To our knowledge, the number of seismic indices and granulation parameters derived by MLEUP is significantly higher than any previously published analyses.
Those indices and parameters are available in the \textit{Stellar Seismic Indices} (SSI) database.
For some $\textit{Kepler}$ targets, we have additional informations.
The asymptotic period spacing $\Delta\Pi_1$ is available for $\sim 25\%$ of our $\textit{Kepler}$ datasets \citep{2016A&A...588A..87V},
allowing us to distinguish the RGB stars from those of the red-clump.
The effective temperature, taken from a combination of different catalogs, is available for the whole $\textit{Kepler}$ dataset (see Sect. \ref{scaling_relations} for more details).
Thanks to those additional constraints we were able to deduce power law representations for both evolutionary stages individually (cf. Tab. \ref{relation_echelle_resultats}) and
we estimated the mass, radius and luminosity of numerous stars via scaling relations combining the effective temperature, $\nu_{\mathrm{max}}$ and $\Delta\nu$.
Our results firmly establish trends previously observed with less objects,
such as the dependency of $H_{\mathrm{env}}$ and $\sigma_\mathrm{g}^2$ with the evolutionary stage \citep{2011A&A...532A..86M, 2012A&A...537A..30M, 2012A&A...544A..90H}.
We also revealed an other dependency which has never been observed to our knowledge:
the faster variation of $\Delta\nu$ with $\nu_{\mathrm{max}}$ for stars belonging to the secondary clump than for those of the RGB.
Based on theoretical scaling relations, we showed that the dependency with the evolutionary stage in the case of $\sigma_\mathrm{g}^2$ is essentially due to a radius dependence.
Concerning the $H_{\mathrm{env}}$ trend, it is not well understood and call for dedicated theoretical studies. \\
By now, the MLEUP method has been optimized for red-giant stars because CoRoT and $\textit{Kepler}$ have detected solar type oscillations for several tens of thousands of such object
in comparison to the few hundred main-sequence solar type pulsators.
However, it should be possible to adapt MLEUP to analyse sub-giant and main-sequence stars.
Indeed, the oscillations spectra of those solar type pulsators contain less mixed modes than those of red giants.
Consequently, it should be easier to fit the Universal Pattern for these kind of stars.
Likewise, MLEUP could be adapted to evolutionary stages later than red giants, such as asymptotic giant branch (AGB), with $\nu_{\mathrm{max}}$ below $1~\mu$Hz.
In this respect, data from OGLE \citep[e.g.][]{2013A&A...559A.137M} offer a great perspective, but the methods will have to be adapted to handle these very long time series with low duty cycles compared to CoRoT or $\textit{Kepler}$ data.
In the future, TESS \citep[\textit{Transiting Exoplanet Survey Satellite},][]{2015JATIS...1a4003R}
and PLATO \citep[\textit{PLAnetary Transits and Oscillation of stars}][]{2014ExA....38..249R}
will provide data for a large number of bright main-sequence and sub-giant objects
for which the extraction of seismic indices and granulation parameters will be possible.
Thereby, a method such as MLEUP will be valuable to analyse automatically all these data.
\section*{Acknowledgements} \label{Acknowledgements}
This paper is based on data from the CoRoT Archive.
The CoRoT space mission has been developed and operated by the CNES,
with contributions from Austria, Belgium, Brazil, ESA (RSSD and Science Program), Germany, and Spain.
This paper also includes data collected by the $\textit{Kepler}$ mission.
Funding for the $\textit{Kepler}$ mission was provided by the NASA Science Mission directorate.
The authors acknowledge the entire $\textit{Kepler}$ and CoRoT team, whose efforts made these results possible.
We acknowledge financial support from the SPACEInn FP7 project (SPACEInn.eu) and from
the \textquotedblleft Programme National de Physique Stellaire\textquotedblright\ (PNPS, INSU, France) of CNRS/INSU.
We thank Carine Babusiaux and Laura Ruiz-Dern for having provided us spectroscopic measurements
of effective temperatures together than their expertise in this domain.
We thank Daniel Reese for improving the text in many places. Finally, we thank the referee for her/his valuable comments"
|
2,869,038,155,706 | arxiv |
\section{Introduction}\label{section:introduction}
Optimization techniques have been widely applied in many scientific and engineering applications.
For instance, in the field of artificial intelligence, researchers often attempt to optimize various machine learning models, e.g. tuning hyper-parameters of support vector machines (SVMs)~\cite{RN1} and optimizing deep neural network architecture~\cite{RN2, RN3}, to obtain a better performance. In the areas of industrial design and manufacturing, engineers always encounter numerous optimization problems for various products and scenarios, such as the optimization of aerodynamic shapes for aircraft, cars, bridges, etc.~\cite{RN5} and the optimization of supply chain management~\cite{RN6}.
In finance, investors usually pursue an optimal portfolio aiming to maximize the return while minimizing the risk~\cite{RN10, RN9}. There are many optimization problems in our daily lives like finding the shortest vehicle route to a destination~\cite{RN7}, resource allocation to satisfy performance goals~\cite{RN8}, and so on.
Since many real-world optimization problems are too complex to be solved with a good solution
by conventional optimization approaches in a reasonable time,
meta-heuristic optimization algorithms have recently captured much attention and achieved some success~\cite{RN11}.
In the past decades, researchers have invented several nature-inspired meta-heuristic optimization algorithms
to imitate some phenomena or behaviors of the nature.
Such algorithms can be classified into five categories: evolution-based, swarm-intelligence-based, physics-based, chemistry-based and human-based algorithms.
Evolutionary algorithms (EAs) are inspired by the biological evolutionary process. Genetic algorithm (GA)~\cite{RN12}, evolution strategies (ES)~\cite{RN13} and differential evolution (DE)~\cite{RN14} can be regarded as representative algorithms in EAs.
For the second category, swarm intelligence algorithms (SIs) imitate the intelligent behaviors of creatures in nature. Particle swarm optimization (PSO) is the most pioneering work of SIs~\cite{RN16}. Up to now, the research of SIs has been very active such that new algorithms are being proposed from time to time. Some well-known examples of SIs include: Ant colony optimization (ACO)~\cite{RN15}, artificial bee colony (ABC)~\cite{RN17}, social spider algorithm (SSA)~\cite{RN23}, whale optimization algorithm (WOA)~\cite{RN18}, grey wolf optimizer (GWO)~\cite{RN19}, etc.
For both physics-based and chemistry-based optimization algorithms, that are motivated by physical phenomena and chemical reactions, examples include simulated annealing (SA)~\cite{RN20}, chemical reaction optimization (CRO)~\cite{RN21}, nuclear reaction optimization (NRO)~\cite{RN22} and so on. Lastly, collective decision optimization algorithm (CDOA)~\cite{RN25} and queuing search algorithm (QSA)~\cite{RN24} are examples of the last category.
According to the no-free-lunch theorem (NFL), there is no single meta-heuristic algorithm that can optimally tackle all optimization problems~\cite{RN26}. Undoubtedly, this motivates researchers to continuously develop new algorithms for various applications. In particular, the proposed algorithm should be very competitive with the few existing successful optimization approaches such as PSO for solving the well-known benchmark functions as well as various real-world problems in terms of the solution quality, rate of convergence, scalability, reliability and flexibility, etc.
In this paper, we propose a novel, powerful and nature-inspired meta-heuristic algorithm namely the virus spread optimization (VSO) for tackling continuous optimization problems. VSO mimics the mighty spread of viruses among hosts.
Here, we devise a new representation scheme and operations that are radically different from all the previously proposed virus-based optimization algorithms.
First, the viral ribonucleic acid (RNA) of each host in VSO denotes a potential solution to the problem at hand for which different viral infection, mutation and recovery operations will help to diversify the searching strategies
in order to largely enhance the solution quality.
In addition, an imported infection mechanism, inheriting the searched optima from another colony,
is introduced to possibly avoid the prematuration of any potential solution in solving complex problems.
The VSO algorithm has an excellent capability to conduct adaptive neighborhood searches
around the discovered local and global optima for achieving better solutions.
Furthermore, with a flexible infection mechanism, VSO can quickly escape from local optima in order to look for other globally (sub-)optimal solution(s).
To evaluate the performance of the proposed optimization algorithm, experiments are conducted
on a series of well-known benchmark functions including $16$ classical examples listed in~\cite{RN31}~\cite{RN66}~\cite{RN77} and $30$ problems specially designed by the IEEE CEC 2014 for competition~\cite{RN29}. In addition, VSO is applied to two real-world applications such as the financial portfolio optimization and optimization of hyper-parameters of SVMs for classification problems. To investigate the scalability, the algorithm was well-tested on the classical benchmark functions and portfolio optimization problems with different ranges of dimensions including: low ($30$ \& $100$ dimensions), medium ($300$ \& $500$ dimensions) and high ($1,000$ dimensions) for the benchmark functions, and different numbers as $30$, $100$ and $250$ of stocks for portfolio optimization. A standardized running environment and settings are used for a fair comparison of the performance of the VSO algorithm with those of the conventional meta-heuristic algorithms including GA~\cite{RN12}, DE~\cite{RN14}, PSO~\cite{RN16}, ABC~\cite{RN17}, as well as state-of-the-art ones, i.e. SSA~\cite{RN23}, WOA~\cite{RN18} and covariance matrix adaptation evolution strategy (CMA-ES)~\cite{RN32} with their outstanding performance reported in literature.
The experimental results verify that VSO achieves impressive performances
in terms of solution quality, convergence rate, scalability, reliability and flexibility when compared to those of
the above conventional and state-of-the-art meta-heuristic algorithms.
To the best of our knowledge, two virus-based algorithms namely the virus colony search (VCS)~\cite{RN28} and virus optimization algorithm (VOA)~\cite{RN27} have been proposed to tackle various optimization problems.
However, VSO is radically different from these two existing algorithms in their analogies, motivation, implementations and search behaviors. We will further reveal the details in~\Autoref{section:vso}.
In summary, the major contributions of this paper are as follows.
\begin{itemize}
\item A new meta-heuristic algorithm as a very competitive and potential approach is proposed to solve challenging and continuous optimization problems;
\item The proposed optimization algorithm, combining the strengths of EAs and SIs, can achieve an excellent trade-off between exploitation and exploration by the unique design of the diversification of the search strategies. This makes the algorithm applicable to a wider range of problems in practice;
\item The imported infection mechanism, as a novel search strategy cooperating with other meta-heuristic algorithms, helps to significantly enhance the overall optimization algorithm for tackling more complex problems;
\item The outstanding performance of the proposed algorithm is demonstrated not only on the solution quality but also the rate of convergence, scalability and reliability through performing a series of experiments on $46$ well-recognized benchmark functions and two real-world optimization problems.
\end{itemize}
The rest of this paper is organized as follows.
~\Autoref{section:vso} describes the analogies, operations, implementations and work flow of the VSO algorithm in details.
The experimental results and related discussion on the benchmark functions are presented in~\Autoref{section:benchmark_function_evaluations}.
The performances for two real-world applications including financial portfolio optimization and optimization of hyper-parameters of SVMs for classification problems are shown and discussed in \Autoref{section:portfolio_optimization} and~\Autoref{section:SVM} respectively. We conclude this work and shed lights on various potential future directions in~\Autoref{section:conclusion}.
\section{The Virus Spread Optimization Algorithm}\label{section:vso}
\subsection{Analogies and Definitions}
Considering the powerful spread of viruses with a great diversity of viral behaviors, VSO is proposed to simulate such process loosely. The analogies of VSO are listed in ~\autoref{tab:analogy}. The host and virus are essential components of the algorithm.
\input{tables/analogy}
In VSO, the population is composed of hosts. There are four types of hosts imitating the spread of viruses and the immunological differences in nature: {\em healthy, mild, severe} and {\em critical\/}. Each host including the healthy one carries a virus.
In fact, many animals including humans may carry all kinds of non-infectious viruses in nature~\cite{RN35}. For instance, a healthy human may carry a few viruses like endogenous retrovirus (ERV) that are in fact beneficial to our immune system~\cite{RN34}. Besides, bats carry a lot of unknown viruses yet may not get sick from those viruses~\cite{RN33}.
The main difference between $healthy$ hosts and other hosts is that $healthy$ hosts act as healthy carriers with non-infectious viruses while the infected (also called infectious) ones, i.e. {\em mild, severe, and critical\/} hosts, can infect the $healthy$ hosts.
More importantly, there are different viral infection and mutation operations for each type of hosts in VSO to diversify the searching strategies so that the optimizing capability and flexibility can be largely enhanced. More definitions are provided as follows.
\begin{itemize}
\item Definition 1: A Viral RNA\\
Each host has a viral RNA that represents a possible solution as shown in~\eqref{eq:eq1}.
\begin{equation}\label{eq:eq1}
X_i=[x_i^1,x_i^2,\ldots,x_i^D]
\end{equation}
where $ X_i$ (vector) is the RNA of the virus denoting a possible solution to the problem at hand, $i$ is the iteration number, and $D$ is the dimensionality, i.e. the number of decision variables, of the problem.
\item Definition 2: A Healthy Host\\
A $healthy$ host is a host carrying a non-infectious virus whose RNA is generated randomly in every iteration. The host conducts a random search in the solution domain as listed in~\eqref{eq:eq2}.
\begin{equation}\label{eq:eq2}
X_i=\ U(S)
\end{equation}
where $S$ is the whole search space while $U$ is a random number generator function based on the uniform distribution of $S$.
\item Definition 3: A Mild Host\\
A $mild$ host is carrying an infectious virus. As shown in~\eqref{eq:eq3}-\eqref{eq:eq4}, the virus of this host can mutate with a mutation intensity $intensity_i^M$ and also infect other $healthy$ hosts with a rate $R^M$ that is relatively low when compared to other infectious hosts.
\begin{equation}\label{eq:eq3} {intensity}_i^M=\alpha\ast{intensity}_{i-1}^M+\gamma\ast\ rand\left(0,1\right)\ast\ ({gbest}_{i-1}-X_i)
\end{equation}
\begin{equation}\label{eq:eq4}
X_{i+1}=\ X_i+\ {intensity}_i^M
\end{equation}
where $intensity_i^M$ (vector) is the mutation intensity of the $mild$ host at the iteration $i$, $\alpha\in [0,1]$ and $\gamma\in[1,2]$ are the scaling factors, ${gbest}_{i-1}$ is the best solution obtained by the population at the iteration $i-1$, and $rand(0,1)$ is a random number between $0$ and $1$.
\item Definition 4: A Severe Host\\
As shown in~\eqref{eq:eq5}-\eqref{eq:eq6}, a $severe$ host carries an infectious virus that can mutate with a mutation intensity $intensity_i^S$ and also infect other $healthy$ hosts with its own rate $R^S$. Overall speaking, its infectious ability is medium as compared to that of the $critical$ host.
\begin{equation}\label{eq:eq5}
{intensity}_i^S=\delta_s\ast{intensity}_{i-1}^S\\
\end{equation}
\begin{equation}\label{eq:eq6}
X_{i+1}=\ X_i+\ Gaussian(0,{intensity}_i^S) * \ X_i\
\end{equation}
where ${intensity}_i^S$ (scalar) is the mutation intensity of the $severe$ host at the iteration $i$, $\delta_s \in (0,1]$ is the decay rate, and $Gaussian(0,{intensity}_i^S)$ is the Gaussian function with the mean as $0$ and the standard deviation as ${intensity}_i^S$.
\item Definition 5: A Critical Host\\
In VSO, there is only one $critical$ host which represents the currently most optimal solution obtained so far. As shown in~\eqref{eq:eq7}, its viral mutation is paused yet with the highest infection rate $R^C$ to carry its relatively good solution quality to other $healthy$ hosts.
\begin{equation}\label{eq:eq7}
X_{i+1}=\ X_i
\end{equation}
\end{itemize}
\subsection{Operations}\label{section:operations}
In VSO, the {\em initialization, selection, mutation, infection\/} and {\em recovery\/} are five essential operations while the {\em imported infection\/} serves as an additional operation to enhance the optimizing performance.
\subsubsection{Initialization}\label{section:initialization}
At the starting point with the number of iterations as $0$, the whole population is initialized as $healthy$ hosts. The viral RNA of each host is randomly generated in the search space according to~\eqref{eq:eq8}.
\begin{equation}\label{eq:eq8}
X_{i=0}=bound_{l} + rand(0,1) * (bound_{u} - bound_{l})
\end{equation}
where $i$ is the iteration number, $bound_{l}$ and $bound_{u}$ denote the lower and upper bounds of the corresponding domain of the variable being considered.
The mutation intensities ${intensity}_i^S$ and ${ intensity}_i^M$ of the mild and $severe$ hosts are initialized in~\eqref{eq:eq9} and~\eqref{eq:eq10} as below.
\begin{equation}\label{eq:eq9}
intensity^M_{i=0} = \frac{U(bound_{l},bound_{u})}{10}
\end{equation}
\begin{equation}\label{eq:eq10}
intensity^S_{i=0} = \frac{1}{rand(0,1)}
\end{equation}
Algorithm~\ref{algo:initialization} shows the detailed initialization process.
\input{algorithms/initialization}
\subsubsection{Selection}\label{section:selection}
In VSO, the host with the best solution will be selected as the $critical$ host after calculating fitness for all hosts at each iteration. As presented in Algorithm \ref{algo:selection}, the host that has achieved the best solution up to current iteration will be designated as the $critical$ host while the previous $critical$ one will be downgraded to the $severe$ host.
In nature, due to the complicated viral mutation, immune response and outside environment, some viruses infecting a $healthy$ host may develop into deadly viruses shortly. Analogously, a $healthy$ host conducting a random search will possibly become the $critical$ one directly as well in VSO.
\input{algorithms/selection}
\subsubsection{Mutation}\label{section:mutation}
The mutation behavior of the searching strategy is one of the key factors to the success of VSO. Depending on the type of hosts, the mutation operation will work according to~\eqref{eq:eq2}-\eqref{eq:eq7}. Algorithm \ref{algo:mutation} clearly shows the pseudo-code of the mutation operation. The viral RNAs of all hosts will be updated accordingly by the mutation operation at each iteration.
\input{algorithms/mutation}
\subsubsection{Infection}\label{section:infection}
The main objective of the infection mechanism is to spread the viral information among all the hosts so as to empower the search effectiveness of the VSO algorithm. In the real world, the transmission route, such as direct contact, is necessary for the spread of many viral diseases~\cite{killingley2013routes}. We hereby design a three-step mechanism for the infection operation in VSO.
At first, every infectious host has one or more chances to contact $healthy$ hosts at each iteration.
Secondly, we have to decide whether that contacted $healthy$ host will be infected or not. Therefore, different infection rates are assigned to the hosts according to their types as shown in~\eqref{eq:eq11}.
\begin{equation}\label{eq:eq11}
R_{infect} = [R^M,R^S,R^C]
\end{equation}
where $ 0<R^M \leq R^S < R^C <1$. They are the infection rates for $mild$, $severe$, and $critical$ host, respectively. More specifically, the infection rate is the probability of an infectious host infecting a $healthy$ host when they contact.
Lastly, in case of a $healthy$ host infected by an infectious host successfully, it will become a $severe$ or $mild$ host at different probabilities. We hereby design a transformation matrix as illustrated in~\eqref{eq:eq12}.
\begin{equation}\label{eq:eq12}
P_{trans} =
\left[
\begin{matrix}
P^C_{H->M} & P^C_{H->S} \\ \\
P^S_{H->M} & P^S_{H->S} \\ \\
P^M_{H->M} & P^M_{H->S}
\end{matrix}
\right]
\end{equation}
where $P_{trans}$ is the matrix of transformation probabilities. For instance, $P^S_{H->M}$ is the conditional probability of a $healthy$ host becoming the $mild$ host
given by being infected by a $severe$ host. As there are only two events here that are mutually exclusive, i.e. becoming a mild or severe host, the summation of each row of probabilities is equal to $1$.
\vbox{}
At each iteration, the specific procedure of the infection is summarized as follows.
\begin{itemize}
\item As indicated in~\eqref{eq:eq11}, the $healthy$ host contacting with an infectious host will be infected with probabilities $R^C$, $R^S$and $R^M$ respectively as dependent on the type of the infectious host;
\item During the infection, the $healthy$ host may be infected as the severe or $mild$ host according to the transformation probabilities as described in~\eqref{eq:eq12}. Specially, the host infected by the $mild$ host can become the $mild$ host only so that the transformation probability $P^M_{H->M}$ is always 1 and $P^M_{H->S}$ is equal to $0$;
\item In addition, two solution sharing mechanisms may be performed during the infection process. When a $healthy$ host (destination) infected by an infectious host (source) to become a $severe$ host, the viral RNA of the source will be copied to the destination directly as shown in ~\autoref{fig:infected_as_severe_host}. In addition, when a $healthy$ host is infected by a $mild$ host, each assigned value of the viral RNA of the destination will be randomly replaced by the source with a fixed probability of $0.5$ as shown in ~\autoref{fig:infected_as_mild_host}.
\end{itemize}
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{images/infected-as-severe-host.png}
\caption{The host to be infected as the $severe$ host}
\label{fig:infected_as_severe_host}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{images/infected-as-mild-host.png}
\caption{The host to be infected as the $mild$ host}
\label{fig:infected_as_mild_host}
\end{figure}
The implementation of the above viral infection is described in Algorithm~\ref{algo:infection}. The infectious and $healthy$ hosts are firstly sorted according to the ascending and descending order of their fitness values respectively. Since the VSO algorithm is designated for solving minimization problems, the smaller the fitness value, the better the solution. Thus, the infectious host with a better solution quality will be more likely selected to infect a $healthy$ host. Conversely, a $healthy$ host with a worse fitness value will be more likely to be infected. Moreover, an integer parameter $H$ as mentioned above is used to limit the maximum number of $healthy$ hosts to be contacted by each infectious host. This can help to avoid any premature convergence of the whole population to any local minimum.
\input{algorithms/infection}
\subsubsection{Recovery}\label{section:recovery}
The recovery operation is another key mechanism of the VSO algorithm. Due to the powerful viral spread in the infection, all hosts may be infected very soon so that the searching capacity of the algorithm may still be quickly converged into a local minima even though with the aforementioned parameter $H$ to restrain the maximum number of contacted hosts.
Thus, in case all the hosts are infected, the recovery operation will be performed to carefully reset some of the infected hosts to continue with the exploration process. We have not adopted the simple random or scheduled restart approaches used by many algorithms such as~\cite{RN44,RN45,RN46}. Instead, an interesting mechanism to gradually downgrade the infected hosts is devised as inspired by the nature in which an infected host has to recover gradually. Likewise, each infected host will be downgraded to the less severe host type of the VSO framework. For example, a $severe$ host will be recovered to the $mild$ host while a $mild$ host will become the $healthy$ host. As the searching restrictions will be relaxed for the ``recovered'' host types, the searching capacity of the algorithm will be enhanced gradually as well so as to explore the other parts of the search space.
Furthermore, a parameter $recPercent$ called the recovery rate is used to specify the percentage of the infected hosts with the worst solution quality to be recovered. This can help to avoid losing all the search information accumulated so far during the search process. The detailed implementation is given in Algorithm~\ref{algo:recovery}.
\input{algorithms/recovery}
\subsubsection{Imported Infection}\label{section:imported_infection}
As inspired by the possible migration of hosts from one place to another that may increase the spread of a viral disease in the real world, the concept of ``imported infection'' is introduced as an additional operation of the VSO framework to enhance its search performance for solving complex optimization problems.
Accordingly, a new colony is developed through the DE algorithm to construct some potentially better solution to the whole population of VSO. However, this simple heuristic operation may break the searching patterns of the concerned VSO algorithm, thus possibly leading to a poorer performance. Therefore, an adaptive probability is predefined to export the DE colony to the whole population of VSO in a probabilistic manner as illustrated in Algorithm \ref{algo:imported_infection}.
\input{algorithms/imported_infection}
As an additional operation, the imported infection may help to improve the search performance in some complex cases yet it will also increase the overall computational complexity of the VSO algorithm. Hence, we may flexibly skip this additional operation in some cases. More importantly, this novel design provides a useful interface for researchers or users to integrate their own algorithms for some specific problems.
\subsection{The Algorithmic Flow of VSO}\label{section:algorithmic_flow}
~\autoref{fig:flow} manifests the algorithmic flow of VSO. Firstly, the concerned parameters of the algorithm and the features of the problem are provided as the input to start the execution of VSO. Then, the involved operations as described in~\Autoref{section:operations} are performed successively.
\begin{figure}[!htb]
\centerfloat
\includegraphics[scale=0.7]{images/flow.png}
\caption{The Algorithmic Flow of VSO}
\label{fig:flow}
\end{figure}
\subsection{A Detailed Analysis on the Search Behavior}\label{section:searching_behavior}
Exploitation and exploration are the two cornerstones of search techniques in solving optimization algorithms. If the exploitation ability is too strong, the algorithm may easily fall into local optima. On the other hand, the algorithm may not be able to converge to any possible solution of a relatively high fitness value in case it solely relies on a very powerful exploration mechanism.
With the novel design of VSO as clearly explained in the previous subsections, it is obvious that VSO combines both advantages of SIs and EAs in order to achieve an excellent balance between exploitation and exploration. The search behavior of the VSO algorithm is summarized as follows.
\subsubsection{Exploitation}
\begin{itemize}
\item As the $critical$ host representing the best solution obtained so far, it has the highest infection rate. Thus, it is more likely to infect $healthy$ hosts to become the $severe$ hosts in the next iteration. To perform such infection operation, the viral RNA of the $critical$ host will be directly replicated to the newly infected host according to~\Autoref{section:infection}. This implies that an increasing number of $healthy$ hosts may acquire the valuable search information of the currently best solution to become $severe$ hosts. On the other hand, since the mutation intensity of the $severe$ hosts will be decreased rapidly as described in~\Autoref{section:mutation}, the $severe$ hosts will conduct neighborhood searches around the locally optimal solution. Overall speaking, this will surely help to enhance the exploitation ability of VSO to improve its solution quality;
\item Each time when a better solution is found, the previous $critical$ host will be automatically downgraded to a $severe$ host to continue its neighborhood search around the previous best solution for a certain duration as seen in~\Autoref{section:infection} before any possible transformation to another host type. Meanwhile, the downgraded host is able to infect other healthy hosts to search this area together.
\end{itemize}
\subsubsection{Exploration}
\begin{itemize}
\item All $healthy$ hosts of the VSO algorithm perform random exploration to try to find a better solution of the whole search space;
\item The main role of $mild$ hosts is to improve the exploration capacity of the VSO algorithm. When a $healthy$ host is infected to become a $mild$ host, the viral RNA of the infectious one will not be replicated directly to the $healthy$ host. Instead, a uni-directional infection mechanism as presented in~\Autoref{section:infection} is performed, that is different from the two-sided crossover operation used in EAs. Moreover, a $mild$ host can always mutate with a higher degree of freedom as guided by the computed intensity. This infection scheme empowers the VSO algorithm with an outstanding exploration ability;
\item Due to the recovery mechanism, the infected hosts will be recovered and re-initialized from time to time. The recovery mechanism helps to escape from any local minimum for a better exploration;
\item The imported infection mechanism hybridizes the whole population of the VSO algorithm with another new colony using a totally different searching approach. This may possibly enlarge the search scope of the VSO algorithm for tackling more complex optimization problems.
\end{itemize}
As illustrated in ~\autoref{fig:searching_pattern}, the red cross denotes the globally optimal solution of the specific function while the only red dot represents the $critical$ host as the best solution obtained so far. Clearly, this $critical$ host infect several $severe$ hosts as denoted by gray dots around the central circle to look better solutions whereas the $mild$ hosts as represented by the orange dots will continue to search toward the red dot that is very likely to achieve a near optimal solution.
\begin{figure}[!htb]
\centerfloat
\includegraphics[scale=0.42]{images/searching_pattern.png}
\caption{The Searching Pattern of VSO}
\label{fig:searching_pattern}
\end{figure}
\subsubsection{Parameters Setting}
Below is a few basic rules for the parameters setting of the VSO algorithm.
\begin{itemize}
\item The higher the value of $R^M$, $P^C_{H->M}$ or $\gamma$, the better the global search ability of the VSO algorithm, and vice versa;
\item On the other hand, the larger the value of $R^C$, $R^S$, $P^C_{H->S}$, $P^S_{H->S}$, or $\alpha$, or the smaller the value of $\delta_s$, the better the local search capability of the VSO algorithm, and vice versa;
\item $R$ and $revPercent$ are the conflicting parameters to balance the convergence of the algorithm. $R$ should not be very large, and generally depends on the population size $N_{pop}$ of the VSO algorithm. For instance, $R$ can be set to $1$ for a specific problem with the population size as $50$ to be discussed in the subsequent section;
\item A larger value of $P_{im}$ may sometimes help to get some quick improvement in solving specific complex optimization problems. Yet for a relatively large value of $P_{im}$, it may also break the good searching patterns. From the empirical observations, $P_{im} \in (0,0.5]$ is typically a good choice for most benchmark problem sets carefully examined in this work.
\end{itemize}
Because of the diverse searching strategies utilized in the VSO algorithm, the number of parameters is relatively larger than other popular meta-heurisitc algorithms such as GA, PSO, etc.
Yet from the preliminary observations, the performance of the VSO algorithm is relatively robust when only a few of the aforementioned parameters are changed at the same time. Moreover, it is found that the VSO algorithm can flexibly tackle a variety of optimization problems using the same parameter settings without much tuning. As revealed in~\Autoref{section:benchmark_function_evaluations}, the same parameter settings of the VSO algorithm are consistently used in all the following experiments.
\section{Evaluations on Benchmark Functions}\label{section:benchmark_function_evaluations}
To validate both the efficiency and effectiveness of the proposed algorithm, the VSO algorithm is utilized to solve two benchmark function groups including the classical and IEEE CEC 2014 benchmark functions.
For the classical benchmark functions, a total of $16$ well-known functions given in \autoref{tab:classical_benchmarking_functions} are used. These functions have been well-tested in all kinds of studies of meta-heuristic algorithms in previous research. Among the functions, $F1-F8$ are uni-modal functions while $F9-F16$ are multi-modal functions. Besides, all functions can be scalable from $2$ to $1,000$ dimensions so that the scalability of the concerned algorithms can be investigated. The motivation for testing these classical functions is outlined as follows:
\begin{itemize}
\item To quickly evaluate the searching capability of VSO when compared to those of other popular meta-heuristic algorithms, especially in terms of the solution quality;
\item To evaluate the rate of convergence;
\item To test the reliability of the algorithm;
\item To investigate the scalability of the algorithm.
\end{itemize}
On the other hand, the IEEE CEC 2014 benchmark functions, as shown in \autoref{tab:CEC_benchmark_functions}, are specially designed for evaluating the performance of meta-heuristic algorithms in the competition of single objective real-parameter numerical optimization problems. The functions ($CEC1-CEC30$) contain various novel characteristics such as shiftings and rotations, it is much more difficult to solve them than the classical set. Up to our understanding, no algorithm has solved all functions optimally. More details about these functions can be found in~\cite{RN29}. Despite the difficulty, we evaluate the effectiveness and robustness of the VSO algorithm on this set of challenging functions.
In the following experiments, all results are collected on the same computer with the Intel Core i9-7900X CPU running at $3.3\sim4.5$ GHz and $64$ GB of RAM. All algorithms were implemented in $Python3$. ~\autoref{tab:parameters-setting} lists the parameter settings of each concerned algorithm according to the recommended values reported in the literature. Except for the population size, there are totally $11$ unique parameters in VSO as listed in \autoref{tab:parameters-setting}. In fact, for other parameters that are not listed, they can be derived according to the relationships mentioned in~\Autoref{section:operations}. For instance, since $P^C_{H->S}$ is set to $0.8$, $P^C_{H->M} $is $0.2$. Furthermore, with the imported infection operation, the population size of the main process of the VSO algorithm is consistently set as $30$ while that of the imported infection is $20$. It is worth noting that the parameters of each algorithm remain unchanged in all experiments in order to evaluate the adaptability of the underlying algorithm with the same parameter settings on various problem sets for a fair comparison.
\input{tables/parameters}
\subsection{Classical Benchmark Functions}\label{section: classical_benchmark_functions_test}
The classical benchmark functions with $30$ dimensions have been widely used for evaluating many meta-heurisitic algorithms like PSO, GA, etc., in many previous studies. In the following evaluation, each function is tested over $31$ runs for each algorithm. The maximum number of iterations in each run is $10^4$. \autoref{tab:classical-func-30} shows the relevant results with the mean as the average value of the fitness values obtained over all runs. The standard deviation of the fitness values is calculated to examine the robustness of the algorithms. Furthermore, the best and worst results are carefully considered. To investigate the computational complexity, the average computational time in CPU seconds is recorded. Finally, two rankings in terms of the averaged fitness values and computational times are listed in order to make more precise and objective comparisons on the different performance measures of the underlying algorithms.
\begin{itemize}
\item In respect of the uni-modal functions $F1$-$F8$, VSO consistently beats other algorithms in all the rankings. For multi-modal functions $F9$-$F16$, the VSO algorithm gets the first places for $6$ functions as well. More importantly, VSO achieves the exact global optima for all the $12$ functions, i.e. $F1$-$F9$, $F11$, $F13$, and $F15$. The standard deviations are $0$ for all these cases, thus showing the excellent robustness of VSO;
\item As for other algorithms, we can observe that the performance of CMA-ES and WOA are not bad for the uni-modal functions. Regarding multi-modal ones ($F9$-$F16$), it is clear that CMA-ES goes worse but WOA still works well;
\item For both GA and ABC, their performances are not satisfied for multi-modal functions because they may not be good at solving these relatively high dimensional and complex problems;
\item Regarding the DE and SSA algorithms, although they acquire very small errors in some functions, they cannot find the exact global optima;
\item Due to the simple and efficient searching strategies, PSO is very fast. It ranks as the first place in computational time in $9$ cases. Unfortunately, the performance of fitness is worst among all the algorithms.
\end{itemize}
~\autoref{tab:classical-30-summary} shows the summary of classical function evaluations where the average of the rankings in all functions for each algorithm is computed. VSO ranks as the first place with respect to the fitness values whereas it is ranked as the fourth place in terms of the computational time.
\begingroup
\setlength{\LTleft}{-20cm plus -1fill}
\setlength{\LTright}{\LTleft}
\input{tables/classical_funs_30_results}
\endgroup
\input{tables/classical_funs_30_summary}
\subsection{CEC benchmark Functions Test}
Following the CEC 2014 recommendation~\cite{RN29}, the dimension of problems is selected as $30$ as well. Function $CEC1$-$CEC3$ are uni-modal functions with rotations. $CEC4$-$CEC16$ are simple multi-modal functions but with various shiftings and rotations. $CEC17$-$CEC22$ are hybrid functions while $CEC23$-$CEC30$ are the composition functions.
As the global optimum of each function is different (from $100$ to $3000$), the fitness result is converted to the error as calculated in~\eqref{eq:eq13} to make the comparison more straightforward. In other words, when the result gets closer to $0$, it implies that the best solution obtained by the algorithm is closer to the global optimum of the corresponding function.
\begin{equation}\label{eq:eq13}
fitness = f(x)-f(x^*)
\end{equation}
where $x$ is the best solution obtained by the algorithm while $f(x^*)$ is the real global optimum of the function.
~\autoref{tab:cec-results} reports the results over $31$ independent runs on each function for each algorithm. A few observations are specified as follows.
\begin{itemize}
\item For the uni-modal functions $CEC1$-$CEC3$, no algorithm is dominated. Because of the complicated rotation, the errors for $CEC1$ are huge for all algorithms. Only GA and VSO perform relatively better. Both errors are on the same order of magnitude, i.e. $10^6$. In the case of $CEC2$, it is similar to $CEC1$, only SSA and VSO are on the smallest order ($10^2$) of magnitude. Meanwhile, the best metric of all runs for VSO is $0$, which demonstrates that only VSO has once achieved the exact global optima. Interestingly, WOA performs very well in the classical uni-modal functions but does not work in these complicated uni-modal cases. It is remarkable that only VSO is able to achieve the global optimum exactly (with fitness error $0$) in $CEC3$;
\item For multi-modal functions $CEC4$-$CEC16$, it is obvious that the VSO algorithm attains a better performance than those of other algorithms. In terms of the mean of fitness, VSO ranks the best for over half of the functions, including the $CEC6$, $CEC9$, $CEC10$, $CEC13$, $CEC14$, $CEC15$ and $CEC16$. SSA followed by VSO acquires the best performance of fitness in $4$ functions;
\item For $6$ hybrid functions $CEC17$-$CEC22$, the VSO algorithm achieves the best performance in all functions except for $CEC18$ where it ranks the second place;
\item For the composition functions of $CEC23$-$CEC30$, VSO outperforms all compared algorithms in the first $6$ functions, i.e. $CEC23$-$CEC28$. It can also be observed that WOA obtains the best performance in $CEC23$ and $CEC25$. This verifies the outstanding optimizing capacity of VSO on such complex functions.
\end{itemize}
~\autoref{tab:cec-summary} indicates that VSO generally outperforms all other algorithms in terms of the fitness values.In addition, VSO with the imported infection operation powered by DE works well here yet the performance of the standalone DE is the worst of all.
\clearpage
\begingroup
\setlength{\LTleft}{-20cm plus -1fill}
\setlength{\LTright}{\LTleft}
\input{tables/cec_results}
\endgroup
\clearpage
\input{tables/cec_summary}
\subsection{Convergence Test}
In addition to the solution quality, we are also interested in the rate of convergence. Therefore, the convergence test on those classical benchmark functions is conducted.
~\autoref{fig: convergence_plot} displays the convergence results based on the median fitness of all trials. The results are given as below.
\input{plots/convergence_plot}
\begin{itemize}
\item VSO generally converges faster than other algorithms and hence possesses superior convergence capability for such optimization problems;
\item For the uni-modal functions, it seems that almost all algorithms can quickly converge. This is because most algorithms can achieve small errors as stated in \autoref{tab:classical-func-30}. However, only VSO achieves the exact global optima for these functions which has been discussed in~\Autoref{section: classical_benchmark_functions_test};
\item With the more complicated multi-modal functions $F9$-$F16$, it is obvious that the VSO performs very well with respect to the rate of convergence. The convergence rates of some other algorithms decrease, such as GA, DE and SSA on $F9$; DE, ABC and SSA on F10; and CMA-ES and DE on $F16$. WOA has a fast rate of convergence as well.
\end{itemize}
\subsection{Reliability test}
~\autoref{fig: box_plot} plots a series of box plots through all runs for the classical benchmark functions for each algorithm. From the obtained results, the following observations can be drawn.
\begin{itemize}
\item For the uni-modal functions, the reliability of VSO is impressive over other algorithms. For example, the performances of both ABC and PSO are quite unstable;
\item For the multi-modal functions, VSO can constantly generate stable results. The only exception is $F12$ where DE achieves the best reliability but it fails to acquire a good solution. The reliability of WOA is followed by VSO. But it becomes much worse in $F16$.
\end{itemize}
\input{plots/box_plot}
\subsection{Scalability Test}\label{section:scalability_test}
In addition to the above low dimensional benchmark functions, a series of evaluations are performed on the medium and high dimensional classical benchmark problems, including $100$, $300$, $500$ and $1000$ dimensions to test the scalability of VSO. To make a thorough comparison, we also employ other algorithms in this evaluation. As aforementioned, the parameter settings of each algorithm are the same as the test on the $30$-$D$ problems.
From the results listed in \autoref{tab:scalability-test-results} we have the following observations:
\begin{itemize}
\item VSO achieves the best performance in almost all functions with all dimensions except for $F12$ and $F14$ with $100$ dimensions. In other words, the VSO algorithm ranks first on $59$ out of the total $64$ cases ($\approx$ 92.19\%). More importantly, VSO attains the exact globally optimal solutions for most of the cases. Take $F4$ with $500$ dimensions and $F8$ with $1000$ dimensions as examples. Only VSO obtains the globally optima in both cases. For the latter one, other algorithms except for WOA not only fail to get the exact globally optimal solution but also their values are very bad. Similar cases include $F4$ with $100$ to $1000$ dimensions, $F7$ with $100$ to $1000$ dimensions, $F8$ with $100$ to $1000$ dimensions, etc., in which VSO is the only algorithm with the fitness error as $0$. The findings demonstrate the excellent scalability of our proposed algorithm;
\item More specifically, VSO shows the advantage of the computational time for some $1000$-$D$ high dimensional problems. For instance, the ranking of VSO for the computational time goes up to the second or even the first place in $F2$, $F3$, $F5$, $F6$, $F12$, $F16$;
\item As for other algorithms, the solution qualities drop down with increasing dimensions. For instance, for CMA-ES, the mean of fitness of $30$-$D$ $F3$ is $5.25E-284$ as reported in \autoref{tab:classical-func-30}. Nevertheless, the values are $1.74E+01$, $2.24E+04$, $1.14E+05$, and $6.38E+05$ for $100$, $300$, $500$, and $1000$-$D$ problems, respectively. Likewise, the mean value of SSA for $F9$ with $30$-$D$ is $2.25E-13$ while the values become $1.77E+02$, $1.42E+03$, $2.93E+03$ and $6.98E+03$ for $100$, $300$, $500$, and $1000$-$D$ problems, accordingly;
\item The only competitor is WOA probably due to its sophisticated design of the searching strategies as inspired by searching for prey and attacking the prey of the whales~\cite{RN18}.
\end{itemize}
\autoref{tab:summary-scalability-test} summaries the results of the scalability test.
\clearpage
\begingroup
\setlength{\LTleft}{-20cm plus -1fill}
\setlength{\LTright}{\LTleft}
\input{tables/scalability_test_results}
\endgroup
\input{tables/scalability_test_summary}
\subsection{Evaluation on VSO without Imported Infection}\label{section:vso_no_im}
To investigate the performance of VSO without the imported infection operation, which is an additional function, another evaluation was conducted on the same set of classical and CEC benchmark functions. From \autoref{tab:vso-without-im-results} \& \ref{tab:vso-without-im-scalability}, we can observe that:
\input{tables/vso_without_im_results}
\begin{itemize}
\item Through a comparison of \autoref{tab:classical-func-30} and \ref{tab:vso-without-im-results}, the mean values of fitness by VSO without any imported infection operation are same as VSO with DE except for $F12$ and $F16$;
\item As for the complicated CEC benchmark functions, the performances of two approaches are same for $5$ cases, i.e. $CEC23$, $CEC24$, $CEC25$, $CEC27$, and $CEC28$. VSO without DE even achieves better in the cases of $CEC1$, $CEC17$, $CEC29$, and $CEC30$. For the remaining $21$ functions, the VSO algorithm with the imported infection powered by DE is readily better;
\item In terms of the computational time, the average time of running all $46$ classical and CEC $30$-$D$ functions for VSO with and without DE are $50.84s$ and $26.69s$, respectively. This means the introduction of such imported infection operation almost doubles the computational time;
\item Taking $100$-$D$ classical benchmark functions as examples, the scalability of VSO without the imported infection mechanism is also impressive.
\end{itemize}
On the other hand, a more thorough investigation should be conducted in the future work on what specific condition(s) and how this additional operator can actually help to enhance the search performance of VSO in handling various complex real-world applications. Furthermore, other meta-heuristic algorithms can be studied as the algorithm in the imported infection operation.
\input{tables/vso_without_im_scalability}
\section{Real-world Application I: Financial Portfolio Optimization}\label{section:portfolio_optimization}
\subsection{Problem Description}
Portfolio optimization is one of the most important problems in finance. Investors usually want to maximize returns and minimize risks through allocating a fixed amount of capital into a collection of assets.
According to the mean-variance model, which is a well-known and widely-used portfolio optimization theory formulated by Markowitz~\cite{RN51}, the variance is a risk measure. The optimization problem is presented in~\eqref{eq:eq14}-\eqref{eq:eq15} as below.
\begin{equation}
max\ E(R(x))=\sum_{i=1}^nx_i u_i
\label{eq:eq14}
\end{equation}
\begin{equation}
min\ V(R(x))=\sum_{i=1}^n\sum_{j=1}^nx_i x_j\sigma_{ij}
\label{eq:eq15}
\end{equation}
\begin{equation}\nonumber
subject \; to:\, x \in X = \{x \in R \,| \sum_{i=1}^nx_i = 1, x_i \geq 0\}
\end{equation}
where $x_i$ is the proportion weight of the initial capital that will be allocated in the $i^{th}$ asset, $u_i$ is the return of the $i^{th}$ asset, $\sigma_{ij}$ stand for the covariance of returns of the $i^{th}$ and $j^{th}$ assets, $E(R(x))$ and $V(R(x))$ are the expected return and variance of the whole portfolio, respectively.
To optimize above two objectives simultaneously, we combine them into one single objective function as shown in~\eqref{eq:eq16}.
\begin{equation}
max\ SR = \frac{E(R(x)) - R_f}{V(R(x))}
\label{eq:eq16}
\end{equation}
where SR is called the sharpe ratio that represents the return and risk simultaneously of the portfolio, $R_f$ is a risk-free rate.
Also, the sharpe ratio has been one of the most important measurement tools to evaluate the performance of investment portfolio in the real-world financial industry.
Since the VSO and other comparative algorithms are designated for solving minimization problems, the problem should be changed to the minimization problem as given in~\eqref{eq:eq17}.
\begin{equation}
min\ fitness = \frac{1}{SR}
\label{eq:eq17}
\end{equation}
\begin{equation}\nonumber
subject \; to:\, SR=10^{-10} \; if \; SR \leq 0
\end{equation}
As it is very possible that the return is zero or even negative in the financial market, a very small number $10^{-10}$ is assigned to the SR for this case.
In order to avoid handling the equality constraint, the solution can be converted to the unconstrained form as shown in \ref{eq:eq18}.
\begin{equation}
x_i' = \frac{x_i}{\sum_{i=1}^n|x_i|}
\label{eq:eq18}
\end{equation}
\subsection{Experimental Setting}
Considering that the U.S. stock market is the biggest developed market and the Chinese stock market is the biggest emerging market all over the world, we select these two markets as our experimental targets. For the U.S. market, $S\&P500$ represents $500$ large companies listed on stock exchanges in the U.S. Likewise, $CSI300$ constituent stocks are the top $300$ stocks traded on the Shanghai Stock Exchange and the Shenzhen Stock Exchange. As the lists of both $S\&P500$ and $CSI300$ were adjusted from time to time, we selected the maximum number up to $250$ stocks in each group according to the order of their stock symbols to make a fair comparison. The full stocks list is illustrated in \autoref{tab:stocks_list}.
Following the previous practice, the information of mean and covariance is acquired from the historical data. In the experiment, we calculated such values through the $5$-year historical daily data of the candidate stocks, i.e. from 1 Jan 2015 to 31 Dec 2019 excluding non-transaction days. More specifically, the average daily return on the historical data is computed as the expected return of each stock.
We also tried different number of stocks, i.e. $10$, $30$, $100$ and $250$, to further investigate the scalability of the algorithm on this practical application. More specially, in addition to longing the stocks, Additionally, we studied a real-world scenario in which short-selling is allowed, i.e. $x_i$ can be negative, which enlarges the searching space of the problem. The US $5$-$year$ treasury yield of $2.57\%$ and China $3$-$year$ fixed deposit interest rate of $4.22\%$ are performed as the risk-free rates for $S\&P500$ and $CSI300$, accordingly.
We utilized the same parameters set for each algorithm as the benchmark functions test. But we set the maximum iteration as $10^3$ due to a large amount of the data.
\subsection{Results and Discussion}
Different from the benchmark functions tests, the portfolio optimization is a maximization problem here. From \autoref{tab:portfolio-results}-\ref{tab:porfolio-summary}, we can see that:
\begin{itemize}
\item For the group of $CSI300$, VSO achieves best in $6$ cases. In particular, VSO gets the impressive sharpe ratio of $2.802$ in the case of $CSI300$ Long/Short $250$ stocks, which totally beat other algorithms (the second best $1.7184$ is generated by ABC). In this group, the performance of SSA is also good. Regarding the case of Long $250$ stocks, SSA obtains $1.8787$ v.s. $1.6666$ by VSO;
\item For the group of $S\&P500$, VSO performs best in $5$ cases. Similarly, VSO acquires the result, i.e. $3.6203$, that is much better than others for the case of Long/Short $250$ stocks;
\item Above phenomenon may imply that VSO is good at optimizing high-dimensional problems with large searching spaces;
\end{itemize}
\begingroup
\setlength{\LTleft}{-20cm plus -1fill}
\setlength{\LTright}{\LTleft}
\input{tables/portfolio_results}
\endgroup
\input{tables/portfolio_summary}
\section{Real-world Application II: Optimization of Hyper-parameters of Support Vector Machines}\label{section:SVM}
\subsection{Problem Description}
SVMs are widely adopted machine learning algorithms particularly useful for some limited sample datasets within the framework of the statistical learning theory. According to the literature, SVMs have achieved impressive success in various applications, such as image classification~\cite{RN52}, natural language processing~\cite{RN53}, and financial prediction~\cite{RN54}, etc.
In practice, the performance of SVMs usually depends on its hyper-parameters. There are two major types of algorithms in SVMs: classification and regression. In this experiment, we apply SVMs to classify some real-world practical datasets.
The mathematical expressions of SVM is shown as in~\eqref{eq:eq19}.
\begin{eqnarray}
\begin{split}
\max_\alpha \sum_j \alpha_j - \frac{1}{2} \sum_{j,k} \alpha_j, \alpha_k y_j y_k k(x_j, x_k)\\
subject \; to: 0 \leq \alpha_j \leq C \; and \; \sum_{j} \alpha_j y_j = 0
\end {split}
\label{eq:eq19}
\end{eqnarray}
where $C$ is the tunable penalty factor and $K$ is the kernal function.
Due to the outstanding performance of RBF kernel function, it is used in this test as stated in~\eqref{eq:eq20}.
\begin{eqnarray}
k(x_j, x_k) = exp\Bigg( - \frac{ \parallel x_j - x_k \parallel ^2 }{2 \sigma^2} \Bigg)
\label{eq:eq20}
\end{eqnarray}
where $\sigma$ is another tunable parameter.
Using this kernel in the SVM classifier, we can get the decision function as shown in~\eqref{eq:eq21}.
\begin{eqnarray}
f(x) = \textrm{sign}\Bigg[ \sum_i \alpha_i y_i
exp\Bigg( - \frac{ \parallel x - x_i \parallel ^2 }{2 \sigma^2} \Bigg)
+ b \Bigg]
\label{eq:eq21}
\end{eqnarray}
In this test, we have to optimize two hyper-parameters: the penalty factor $C$ and $\sigma$ for classification problems.
\subsection{Experimental Setting}
There are five datasets involved:
\begin{itemize}
\item Australian Credit Approval: A well-known dataset that concerns credit card applications approval in Australia~\cite{RN49};
\item HCC Survival: HCC dataset was obtained at a University Hospital in Portugal and contais several demographic, risk factors, laboratory and overall survival features of 165 real patients diagnosed with HCC~\cite{RN47};
\item Iris:This is perhaps the best known dataset to be found in machine learning. It is to classify type of iris plant~\cite{RN49};
\item Somerville Happiness Survey: A dataset about life survey~\cite{RN48}.
\item Wine: This dataset is the results of a chemical analysis of wines grown in the same region in Italy but derived from three different cultivars~\cite{RN49};
\end{itemize}
All datasets as listed above are publicly available at~\cite{RN49}. As for the searching space, we set $C, \sigma \in [10^{-5},10^5]$.
The accuracy of $10$-$fold$ cross validation is computed as the fitness in the evaluation. The maximum iteration is set as $500$.
\subsection{Results and Discussion}
The detailed results are illustrated into \autoref{tab:svm-results} where the mean fitness represents the average classification accuracy of 30 runs. The findings are stated as follows.
\begin{itemize}
\item VSO outperforms over all other algorithms on $4$ out of $5$ datasets. For example, VSO achieves $83.83\%$ of mean accuracy for the first dataset;
\item The performance of CMA-ES is very bad in this test. On the other hand, ABC that performs unwell in previous benchmark functions tests becomes not bad here;
\item Although VSO can get an enhancement of accuracy, it does not show a big advantage compared with other candidates in this low-dimensional problem optimization.
\end{itemize}
\begingroup
\setlength{\LTleft}{-20cm plus -1fill}
\setlength{\LTright}{\LTleft}
\input{tables/svm_results}
\endgroup
\input{tables/svm_summary}
\section{Conclusion}\label{section:conclusion}
In summary, a novel and powerful meta-heuristic optimization algorithm
called VSO is proposed
for tackling challenging continuous optimization problems in many real-life applications.
Inspired by the spread and behavior of viruses, the algorithm is carefully devised with
different viral operations to diversify the searching strategies in order
to highly improve its optimizing capacity.
In this paper, VSO is firstly evaluated on a total of $46$ well-known benchmark functions
covering many different types of optimization problems.
The rate of convergence, scalability, and reliability of the algorithm are well-validated
on all these benchmark functions.
Moreover, VSO is used to solve two real-world applications
including the financial portfolio optimization and optimization of hyper-parameters of SVMs for classification problems.
All the obtained results are carefully compared and analyzed with those of classical algorithms
such as GA, PSO, and DE as well as the state-of-the-art optimization approaches including CMA-ES, WOA, and SSA.
The results demonstrate the outstanding performance of our proposed algorithm in terms of
solution fitness, convergence rate, scalability, reliability, and flexibility. Especially, VSO shows a unique potential for high-dimensional continuous optimization problems. Additionally, the algorithmic framework is much flexible to provide an interface to hybridization with other algorithms.
The drawbacks of VSO are summarized as follows.
First, the number of algorithmic parameters is larger than those of the existing popular optimization approaches like GA.
Second, the implementation is a bit complicated.
Last but not least,
the computational speed is not mostly in the dominant position due to its multiple searching strategies.
Concerning the future work, how to make the parameters of VSO to be self-adaptive is worth exploring. A more thorough investigation should be conducted on the imported infection operation.
In addition, the applicability of VSO can be further investigated in various real-world applications.
Lastly, VSO has a great potential to be extended for solving mixed continuous-discrete as well as
multi-objective optimization problems.
|
2,869,038,155,707 | arxiv | \section{Introduction}
This paper concerns the energy distribution in the final state of
deeply inelastic lepton scattering. Using a naive parton model, one
would predict that the scattered parton appears as a single narrow
jet at a certain angle $(\theta_*,\phi_*)$ in the detector. Taking
hard QCD interactions into account, one predicts a much richer
structure for the final state energy distribution. In a previous
paper\citex{mos} (henceforth referred to as~I), we investigated this
structure, using an energy distribution function defined in analogy to the
energy-energy correlation function in $e^+ e^-$
annihilation.\citex{energycor,ellis}
We studied
this energy distribution as a function of angle $(\theta_B,\phi_B)$
in the detector in the region not too near to the direction
$(\theta_*,\phi_*)$. In this region, simple QCD perturbation theory is
applicable, and we presented calculations at order $\alpha_s$. In this
paper, we extend the analysis to the region of $(\theta_B,\phi_B)$
near to $(\theta_*,\phi_*)$ . Here, multiple soft gluon radiation is
important. Thus we use a summation of perturbation theory.
\subsection{The energy distribution function}
There is extensive literature on semi-inclusive deeply inelastic
scattering;\citex{early,late,todd,mirkes,graudenz,peccei,morfin}
a brief history and complete set of references can be found
in paper~I. We begin here with a concise review of how the energy
distribution function is defined, and then discuss how we sum the
contributions that are important in the region
$(\theta_B,\phi_B) \simeq (\theta_*,\phi_*)$ to obtain a result which
is valid for all values of $(\theta_B,\phi_B)$.
The reaction that we study is $e+ {\rm A} \to e +{\rm B} + {\rm X}$ at
the HERA electron-proton collider.\citex{wsmith}
Let us describe the particles by
their energies and angles in the HERA laboratory frame, with the
positive z-axis chosen in the direction of the proton beam and the
negative z-axis in the direction of the electron beam. In completely
inclusive deeply inelastic scattering, one measures only $E'$ and
$\theta'$, the energy and angle of the scattered electron. In the
semi-inclusive case studied in this paper, one also measures some basic
features of the hadronic final state. In principle, one can measure the
energy $E_B$ and the angles $(\theta_B,\phi_B)$ of the outgoing hadron
B. However, it is much simpler to perform a purely calorimetric
measurement, in which only the total energy coming into a calorimeter
cell at angles $(\theta_B,\phi_B)$ is measured. This calorimetric
measurement gives the energy distribution
\begin{equation}
{d \Sigma \over
dE'\ d\cos\theta'\ d\cos\theta_B\ d\phi_B}
=
\sum_B \int dE_B\ (1 - \cos\theta_B) E_B\
{d \sigma (e + A \to B + X)
\over
dE'\ d\cos\theta'\ dE_B\ d\cos\theta_B\ d\phi_B} \ .
\label{eq:edist}
\end{equation}
The sum runs over all species of produced hadrons B. We have included
a factor $(1 - \cos\theta_B)$ in the definition because this factor is
part of the Lorentz invariant dot product $P_{A,\mu} P_B^\mu = E_A E_B
(1 - \cos\theta_B)$.
Notice that $d\Sigma$ measures the distribution of energy in the final
state as a function of angle without asking how that energy is split
into individual hadrons moving in the same direction.\citex{feynman}
For this reason,
the theoretical expression for $d\Sigma$ will not involve parton decay
functions that describe how partons decay into hadrons.
\subsection{Partonic variables and their relation to HERA lab frame}
At the Born level, the hard scattering process for the reaction is
$electron + quark \to electron + quark$ by means of virtual photon or
${\rm Z}_0$ exchange. At order $\alpha_s$, one can have virtual
corrections to the Born graph. In addition, one can have processes in
which there are two scattered partons in the final state. Then the
initial parton can be either a quark (or antiquark) or a gluon, while
the observed hadron can come from the decay of either of the final
state partons. Some of these possibilities are illustrated in \xfig{diai},
\xfig{diaii}, and \xfig{diaiii}.
\figdiagi
\figdiagii
\figdiagiii
Let us consider the effect of the emission of the additional,
unobserved, ``bremsstrahlung'' parton. We can define the part of the
vector boson momentum $q^\mu$ that is transverse to the momentum of the
incoming hadron $P_A^\mu$ and to the momentum of the outgoing hadron
$P_B^\mu$. One merely subtracts from $q^\mu$ its projections along
$P_A^\mu$ and $P_B^\mu$ (taking $P_A^2 = P_B^2 = 0$),
\begin{eqnarray}
q_T^\mu &=&
q^\mu
- \dfrac{q \cdot P_B}{P_A \cdot P_B} \, P_A^\mu
- \dfrac{q \cdot P_A}{P_A \cdot P_B} \, P_B^\mu
\quad .
\end{eqnarray}
We let ${q_T} = [- q_T^\mu \cdot {{q_T}}_{\mu}]^{1/2}$ represent the
magnitude of the transverse momentum. It is ${q_T}$ that is analogous to
the transverse momentum of produced W's and Z's or lepton pairs in the
Drell Yan process. In the naive parton model, there are no
bremsstrahlung partons and all parton momenta are exactly collinear
with the corresponding hadron momenta, so one has ${q_T} = 0$. At
order $\alpha_s$, unobserved parton emission allows ${q_T}$ to be
nonzero.
In order to properly describe the parton kinematics we need four more
variables besides ${q_T}$. Two are the standard variables for deeply
inelastic scattering, $Q^2 = -q^\mu q_\mu$ and $x = Q^2/(2q\cdot
P_A)$. The third is a momentum fraction for the outgoing hadron $B$,
\begin{equation}
z = { P_B\cdot P_A \over q\cdot P_A} \qquad.
\end{equation}
(Thus the integration over the energy of hadron $B$ in
definition~(\ref{eq:edist}) of the energy distribution is equivalent to an
integration over $z$.) The fourth variable is an azimuthal angle
$\phi$. To define $\phi$, we choose a frame, called the hadron frame,
\xfig{hadframe}, in which the incoming hadron $A$ has its three-momentum
$P_A$ along the positive $z$-axis and the virtual photon four-momentum
$q^{\mu}$
lies along the negative $z$-axis. Then as long as ${q_T}\not=0$, hadron $B$
has some transverse momentum, and we align the $x$- and $y$-axes so that
$P_B^x>0$ and $P_B^y = 0$. We now
define $\phi$ as the azimuthal angle of the incoming lepton in the
hadron frame. These variables are described more fully in paper~I, and
relevant formulas are given in the Appendix of this paper.
\fighadronframe
\figkini
\figqtphi
The variables ${q_T}$ and $\phi$ can be translated to the observables
of the HERA lab frame, \xfig{figkin1}. In the naive parton model, the
outgoing hadron
$B$ (along with all the other hadrons arising from the decay of the
struck quark) emerges in the plane defined by the incoming and outgoing
electrons at a precisely defined angle $(\theta_*,\phi_*)$, which
can be computed from the incoming particle momenta and the momentum of
the scattered electron. The point ${q_T} = 0$ corresponds to
$(\theta_B,\phi_B) = (\theta_*,\phi_*)$.
We choose our $x$-axis such that $\phi_*=0$.
Lines of constant positive
${q_T}$ are curves in the $(\theta_B,\phi_B)$ plane that encircle the
point $(\theta_*,\phi_*)$. Lines of constant $\phi$ radiate out of the
point $(\theta_*,\phi_*)$, crossing the lines of constant ${q_T}$.
This is illustrated in \xfig{phietaii}. The precise formulas for the map
relating
$(\theta_B,\phi_B)$ and $({q_T},\phi)$ are given in the Appendix.
In paper~I and in this paper, we find it convenient to convert from the
laboratory frame variables $\{E^{\prime}, \theta^{\prime}\}$ of the
scattered lepton and $\{\theta_B,\phi_B\}$ of the observed hadron to
$\{x,Q^2\}$ for the lepton and $\{{q_T},\phi\}$ for the observed
hadron. We also convert from $E_B$ to $z$. With this change of
variables, \eq{edist} becomes
\begin{equation}
{ d \Sigma \over dx \, dQ^2 \, d{q_T^2}\, d\phi}
=
\sum_B\int \, dz \, z \ \left(\dfrac{Q^2}{2 x \, E_A}\right) \
{ d \sigma \over dx \, dQ^2 \, d{q_T^2}\, d\phi \, dz}
\ .
\end{equation}
\subsection{The Sudakov summation of logarithms of ${q_T}$ }
The main object of study in this paper is the distribution of energy
as a function of ${q_T}$ for ${q_T^2} \ll Q^2$. In paper~I, we applied
straightforward perturbation theory to analyze the energy
distribution in the region ${q_T^2} \sim Q^2$, and $\alpha_s(Q^2) \ll 1$.
Here there is a rich structure as a function of the angles that relate
the hadron momenta to the lepton momenta. In fact, a complete
description requires nine structure functions.
When one examines the region ${q_T^2} \ll Q^2$, one finds that the
angular structure simplifies greatly. However, the dependence on ${q_T}$
becomes richer than the dependence on ${q_T}$ of the lowest order graphs.
By summing the most important parts of graphs at arbitrarily high
order, one finds a structure that is sensitive to the fact that QCD is
a gauge theory.
Briefly, the physical picture\citex{parisi} is as follows. At the
Born level of deeply inelastic scattering, a quark in the incoming
proton enters the scattering with momentum $\xi P_A^\mu$ that is
precisely along the beam axis. This quark is scattered by a virtual
photon,
$Z$- or
$W$-boson. Its momentum $\xi P_A^\mu + q^\mu$ is in a direction
$(\theta_*,\phi_*)$ that can be reconstructed by knowing the lepton
momenta. However at higher orders of perturbation theory, the
momentum of the final state parton is
\begin{equation}
(\xi P_A^\mu + q^\mu) - (k_1^\mu + k_2^\mu +\cdots + k_N^\mu)
\qquad ,
\end{equation}
where the $k_i^\mu$ are momenta of gluons that emitted in the
process. In a renormalizable field theory, it is very easy to emit
gluons that are nearly collinear to either the initial or final parton
directions. In addition, in a gauge theory such as QCD, it is very
easy to emit gluons that are soft ($k^\mu \ll Q$). Each gluon
emission displaces ${q_T}$ by a small amount, so that one may think of
the parton as undergoing a random walk in the space of transverse
momenta. With one gluon emission, one finds a cross section that is
singular as ${q_T} \to 0$:
\begin{equation}
{d\sigma \over d\, q_T^2}\
\propto \
\alpha_s\,{a + b \log(q_T^2/Q^2) \over q_T^2}\ .
\end{equation}
At order $\alpha_s^N$ the $1/q_T^2$ singularity is multiplied by a
polynomial in $\log(q_T^2/Q^2)$ of order $2N-1$. This series sums to
a function of $q_T$ that is peaked at $q_T = 0$ but is not singular
there. The width of this distribution is much bigger than the $300\
{\rm MeV}$ that one would guess based on experience with soft hadronic
physics. On the other hand the width is quite small compared to the
hard momentum scale $Q$.
Essentially this same physics has been studied in the two crossed
versions of the process $e + A \to e + B + X$ that can be studied at
HERA. In electron-positron annihilation, $e + \bar e \to A + B +
X$, one looks at the energy-energy correlation function for hadrons
$A$ and $B$ nearly back-to-back.\citex{CSS,eedata,LEPSLC}
In $A+B\to \ell + \bar{\ell} + X$, one studies the distribution of
the lepton pair as a function of its transverse momentum $q_T$
with respect to the beam
axis.\citex{CSS,DWS,drellyan,qiu,fnalexp,naexp,MirkesOhnemus}
The same analysis applies also to the distribution of the transverse
momentum of $W$ or $Z$ bosons produced in hadron
colliders.\citex{altarelliwzpt,wzpt,ArnoldKauffman,yuanladinsky}
{}From these studies, the following picture emerges. First,
the leading logs ($n=2N-1$) can be summed to all orders,
and dominate the perturbation theory in the region $\alpha_s(Q^2) \ll
1$ and $\alpha_s(Q^2) \, \ln^2(Q^2/{q_T^2})
{\lower0.5ex\hbox{$\stackrel{<}{\sim}$}} 1$. Unfortunately,
most of the interesting physics, and most of the data, lie outside this
region of validity of the leading logarithm approximation. Fortunately,
one can go beyond the leading logarithm summation to obtain a result
that is valid even when $\alpha_s(Q^2)
\ln^2(Q^2/{q_T^2})$ is large.\citex{CSS,ArnoldKauffman}
The plan for the remainder of the paper is as follows.
In Sec.~2, we
use our $\alpha_s$ calculation in paper~I to calculate the
asymptotic form of the energy distribution functions in the
${q_T}\to 0$ limit.
In Sec.~3, we
introduce the Sudakov form factor which sums the
soft gluon radiation in the limit ${q_T}\to 0$.
In Sec.~4, we
compare the asymptotic form of the energy distribution functions to
extract the order $\alpha_s$ contributions to the perturbative
coefficients $A$, $B$,
$C^{\rm IN}$ and $C^{\rm OUT}$.
In Sec.~5, we
address the issue of matching the small ${q_T}$ region to the large ${q_T}$
region.
In Sec.~6 we investigate the form of the non-perturbative corrections
in the small ${q_T}$ region, and relate these to the Drell-Yan and $e^+e^-$
processes.
In Sec.~7, we review the principle steps in the calculation.
In Sec.~8, we
present results for the energy distribution functions throughout the
full ${q_T}$ range.
Conclusions are presented in Sec.~9, and the Appendix contains a
set of relevant formulas.
\section{The Energy Distribution Functions \label{ECF}}
In this section we review the order $\alpha_s$ perturbative results of
paper~I in order to extract the terms in
$d\Sigma/ dx\, dQ^2\, dq_T^2\, d\phi$ that behave like $1/q_T^2$ times
logs as $q_T \to 0$. In Sec.~3, we display the structure of $d\Sigma/
dx\, dQ^2\, dq_T^2\, d\phi$ with the Sudakov summation of logarithms.
Then in Sec.~4, by comparing the summed form with the order $\alpha_s$
form of $d\Sigma/ dx\, dQ^2\, dq_T^2\, d\phi$, we will be able to extract
the coefficients that appear in the summed form.
\subsection{Energy Distribution Formulas}
The process we consider is
$e^- + A \to e^- + B + X$, and
the fundamental formula for the energy distribution is:
\begin{eqnarray}
{ d \Sigma \over dx \, dQ^2 \, d{q_T^2}\, d\phi}
&=&
\sum_{k=1}^{9} \
{\cal A}_k(\psi,\phi)
\sum_{V_1,V_2} \
\sum_{j, j'} \
\Sigma_0(Q^2;V_1,V_2,j, j',k) \
\Gamma_k(x,Q^2,q_T^2;j,j')
\qquad .
\label{eq:master}
\end{eqnarray}
The hyperbolic boost angle, $\psi$, that connects the natural hadron
and lepton frame is given by\citex{angulartheory}
\begin{eqnarray}
{{\rm cosh}} \psi
&=&
\frac{2 x s}{Q^2} - 1
\qquad ,
\end{eqnarray}
and $\phi$ is the azimuthal angle in the hadron frame.
The nine angular functions ${\cal A}_k(\psi,\phi)$
arise from hyperbolic $D^1(\psi,\phi)$ rotation matrices.
The complete set of ${\cal A}_k(\psi,\phi)$ are listed in the Appendix,
but the two we shall focus on are
\begin{eqnarray}
{\cal A}_1(\psi,\phi) &=& 1 + {{\rm cosh}}^2(\psi) \nonumber\\
{\cal A}_6(\psi,\phi) &=& 2 \ {{\rm cosh}}(\psi)
\qquad .
\end{eqnarray}
We sum over the intermediate vector bosons $\{V_1, V_2\}=
\{\gamma,Z^0\}$ or $\{W^\pm\}$, as appropriate, and we also sum over
the initial and final partons, $\{j, j'\}$.
The factor $\Sigma_0(Q^2;V_1,V_2,j, j',k)$ contains the leptonic and
partonic couplings, the boson propagators, and numerical factors; it is
defined in the Appendix, \eq{sigzero}.
It is the hadronic energy distribution functions,
$\Gamma_k(x,Q^2,q_T^2;j,j')$,
that we shall calculate.
If we expand the $\Gamma_k$ in the form of
perturbative coefficients convoluted with parton distribution
functions, then two of the functions $\Gamma_k$, namely $\Gamma_1$
and $\Gamma_6$, behave like $\log^n({q_T^2}/Q^2)/{q_T^2}$
with $n \ge 0$ for ${q_T}\to 0$. The others behave like $1/{q_T}$ or $1$
times possible logarithms. In this paper we are interested in small
${q_T}$ behavior, so we concentrate our attention on $\Gamma_1$ and
$\Gamma_6$.
What of the less singular structure functions $\Gamma_2$,
$\Gamma_3$, $\Gamma_4$, $\Gamma_5$, $\Gamma_7$, $\Gamma_8$ and
$\Gamma_9$? Fixed order perturbation theory is not applicable for
the calculation of these $\Gamma_k$ for small $q_T$. We note, on the
grounds of analyticity, that these $\Gamma_k$ must be finite or, for
certain $k$, vanish as $q_T \to 0$, even though they have weak
singularities in finite order perturbation theory. Our perturbative
results in the region of moderate $q_T$ indicate that the fraction
of $d \Sigma/dx\,dQ^2\,dq_T^2\,d\phi$ contributed by these
$\Gamma_k$ is small and dropping as $q_T$ decreases. We thus
conclude that these contributions would be hard to detect
experimentally for small $q_T$. For this reason, we do not address
the problem of summing perturbation theory for $\Gamma_2$,
$\Gamma_3$, $\Gamma_4$, $\Gamma_5$, $\Gamma_7$, $\Gamma_8$ and
$\Gamma_9$.
Applying the methods of Refs.~\cite{CS,CSS} to deeply inelastic
scattering, we write $\Gamma_1$ in the form
\begin{eqnarray}
\Gamma_1(x,Q^2,{q_T^2};j,j') &=&
\Gamma_1^{Pert}(x,Q^2,{q_T^2};j,j')
- \Gamma_1^{Asym}(x,Q^2,{q_T^2};j,j')
+ W(x,Q^2,q_T^2;j,j')
\quad .
\nonumber \\
\label{eq:smallqTdecompose}
\end{eqnarray}
Here $W(x,Q^2,q_T^2;j,j')$ sums the singular terms to all orders, and
contains the leading behavior of $\Gamma_1$ as ${q_T} \to 0$.
$\Gamma_1^{Pert}(x,Q^2,{q_T^2};j,j')$ is simply $\Gamma_1(x,Q^2,{q_T^2};j,j')$
evaluated at a finite order ($\alpha_s^1$ for our purpose) in perturbation
theory.
$\Gamma_1^{Asym}(x,Q^2,{q_T^2};j,j')$ equals $W(x,Q^2,q_T^2;j,j')$ truncated
at a finite order of $\alpha_s$ in perturbation theory.
Specifically, if we expand $W(x,Q^2,q_T^2;j,j')$ in the form
of perturbative coefficients convoluted with parton distribution functions,
then the coefficients have the form of $\log^n({q_T^2}/Q^2)/{q_T^2}$ with $n
\ge
0$. There are, by definition, no terms that behave like
$({q_T^2}/Q^2)^p$ times possible logarithms for $p > -1$. Such terms
exist in $\Gamma_1$, but they are associated with
$(\Gamma_1^{Pert} - \Gamma_1^{Asym})$ in \eq{smallqTdecompose}.
The angular function ${\cal A}_1(\psi,\phi)= 1 + {{\rm cosh}}^2(\psi)$
that multiplies $W$ in
the small ${q_T}$ limit arises from the numerator factor
\begin{equation}
{\rm Tr} \{ {{\ell \kern -4.6 pt \raise 2 pt \hbox{/}}} \gamma_\mu {{\ell \kern
-4.6 pt \raise 2 pt \hbox{/}}}^{\prime} \gamma_\nu\}\
{\rm Tr} \{ {{P \kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\!A} \gamma^\mu {{P
\kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\! B} \gamma^\nu\}
\qquad .
\end{equation}
Here
${\rm Tr} \{{{\ell \kern -4.6 pt \raise 2 pt \hbox{/}}}
\gamma_\mu {{\ell \kern -4.6 pt \raise 2
pt \hbox{/}}}^{\prime} \gamma_\nu\}$ is associated with the lepton scattering,
and the factor
${{P \kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\!A}
\cdots {{P \kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\! B}$
gives the Dirac structure of
the hadronic part of the cut diagram in
\xfig{nonpertiii}(b) in the limit ${q_T} \to 0$.
We will discuss \xfig{nonpertiii}
further in Sec.~\ref{NONPERT}.
\fignonpertiii
The weak currents also contain $\gamma_5\gamma^\mu$ terms. This gives
the possibility of another angular function in the small ${q_T}$
limit. With the same limiting hadronic structure,
${{P \kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\!A}\cdots {{P \kern -6.5 pt
\raise 2 pt \hbox{/}}}_{\!\! B}$ we can have
\begin{equation}
{\rm Tr} \{ {{\ell \kern -4.6 pt \raise 2 pt \hbox{/}}} \gamma_5\gamma_\mu
{{\ell \kern -4.6 pt \raise 2 pt \hbox{/}}}^{\prime}
\gamma_\nu\}\
{\rm Tr} \{ {{P \kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\!A} \gamma_5\gamma^\mu
{{P \kern -6.5 pt \raise 2 pt \hbox{/}}}_{\!\! B}
\gamma^\nu\}
,
\nonumber \\
\end{equation}
which is proportional to the angular function
${\cal A}_6(\psi,\phi) = 2 \, {{\rm cosh}}(\psi)$
at ${q_T} = 0$.
(Note that both
${\cal A}_1(\psi,\phi)$ and ${\cal A}_6(\psi,\phi)$
are independent of the azimuthal angle $\phi$.)
Thus $\Gamma_6$ has the structure
\begin{eqnarray}
\Gamma_6(x,Q^2,{q_T^2};j,j') &=&
\Gamma_6^{Pert}(x,Q^2,{q_T^2};j,j')
- \Gamma_6^{Asym}(x,Q^2,{q_T^2};j,j')
+ (-1) W^{}(x,Q^2,q_T^2;j,j')
,
\nonumber \\
\label{eq:smallqTdecomposeii}
\end{eqnarray}
with the same function\footnote{
The minus sign in front of $W(x,Q^2,{q_T^2};j,j')$ in
\eq{smallqTdecomposeii} arises from our convention
for the functions
${\cal A}_k(\psi,\phi)$ and couplings
$\Sigma_0(Q^2;V_1,V_2,j, j',k)$ that multiply
$\Gamma_1$
and $\Gamma_6$.
}
$W$ as in \eq{smallqTdecompose}.
Again
$W$ contains the terms that behave like $\log^n({q_T^2}/Q^2)/{q_T^2}$ in
perturbation theory, while $(\Gamma_6^{Pert} - \Gamma_6^{Asym})$ contain
the less singular terms. Our object now will be to study the small ${q_T}$
function $W$.
\subsection{Parton Level Distributions}
The above hadronic process takes place via the partonic sub-process
$V(q) + a(k_a) \to b(k_b) + X $
where $V$ is an intermediate vector boson, and
$a$ and $b$ denote parton species.
The hadron structure function $W(x,Q^2,{q_T^2};j,j')$ is
related to a perturbatively
calculable parton level structure function
$w_a({\widehat x },Q^2,{q_T^2};j,j')$ via
\begin{eqnarray}
W(x,Q^2,{q_T^2};j,j') &=&
f_{a/A} \otimes w_a
=
\int_x^1 { d\xi \over \xi} \
\sum_a f_{a/A}(\xi,\mu) \
w_a({\widehat x },Q^2,{q_T^2};j,j')
\qquad ,
\label{eq:partonhadron}
\end{eqnarray}
with $\xi_a = k_a^+/P_A^+$ and ${\widehat x }=x/\xi_a$.
Here $f_{a/A}$ is the
{\vbox{\hrule\kern 1pt\hbox{\rm MS}}}\ parton distribution function.
Note that the decay distribution function $d_{B/b}(\xi_b,\mu)$ is
absent since we have used the extra $\int z \, dz$
and the sum over hadrons
from the definition of
the
energy distribution to integrate out the $d_{B/b}(\xi_b,\mu)$
via the momentum sum rule,
\begin{eqnarray}
\sum_{B} \ \int \ d\xi_b \ \xi_b \ d_{B/b} (\xi_b,\mu) &=& 1
\qquad .
\label{}
\end{eqnarray}
The partonic structure
function, $w_a({\widehat x },Q^2,{q_T^2};j,j')$,
is obtained by first computing the partonic tensor
\begin{eqnarray}
w^{\mu\nu}(k_a,k_b,q) &=&
\dfrac{1}{2} \,
\sum_{X,s,s^\prime} \
\int \
d^4 x \ e^{-i q\cdot x} \
\langle k_a,s | j^\nu(0) | k_b, s^\prime; X \rangle \
\langle k_b, s^\prime; X | j^\mu(0) | k_a,s \rangle
,
\end{eqnarray}
which is a matrix element of current operators.
We then project out the appropriate angular component
({\it cf.}, paper~I), and extract the leading term in the
${q_T}\to 0$ limit.
Explicit calculation will show that these limits
(up to overall factors) are
identical for the projection of the 1 and 6 tensors.
In the small ${q_T}$ limit, the energy distribution function
is then given by:
\begin{eqnarray}
{d \Sigma \over
dx\ dQ^2\ d{q_T^2}\ d\phi}
&\simeq&
{\cal A}_1(\psi,\phi) \
\sum_{V_1,V_2} \
\sum_{j, j'} \
\Sigma_0(Q^2;V_1,V_2,j, j',1) \
\sum_a \
f_{a/A}(\xi,\mu) \otimes
w_a({\widehat x },Q^2,{q_T^2};j,j')
\nonumber \\ &-&
{\cal A}_6(\psi,\phi) \
\sum_{V_1,V_2} \
\sum_{j, j'} \
\Sigma_0(Q^2;V_1,V_2,j, j',6) \
\sum_a \
f_{a/A}(\xi,\mu) \otimes
w_a({\widehat x },Q^2,{q_T^2};j,j')
\nonumber \\ &+&
{\rm \ plus\ terms\ less\ singular\ than\ 1/{q_T^2}}
\qquad .
\end{eqnarray}
Again, the relative minus sign is simply due to the definition of
${\cal A}_k(\psi,\phi)$ and $\Sigma_0(Q^2;V_1,V_2,j, j',k)$.
\subsection{The Asymptotic Energy Distribution Functions}
We observe (from the
results of paper~I) that the perturbative
$\Gamma_1^{Pert}(x,Q^2,q_T^2;j,j')$ and
$\Gamma_6^{Pert}(x,Q^2,q_T^2;j,j')$ diverge as $1/{q_T^2}$ for ${q_T} \to 0$.
To identify the singular terms, we can
expand the on-shell delta function for small ${q_T}$ using
\begin{eqnarray}
\ 2\pi\ \delta\!\left[ (q^\mu + k_a^\mu - k_b^\mu)^2\right]
&=&
{ 2 \pi {\widehat x } \over Q^2} \
\Biggl\{
\ln \left( {Q^2 \over {q_T^2}} \right) \delta(1-{\widehat x }) \
\delta(1-{\widehat z })
+
{ \delta(1-{\widehat z }) \over (1-{\widehat x })_+ }
+
{ \delta(1-{\widehat x }) \over (1-{\widehat z })_+ }
\Biggr\}
,
\nonumber \\
\label{}
\end{eqnarray}
where the ``$+$"-prescriptions is defined as usual by:
\begin{eqnarray}
\int_z^1 dy \ \dfrac{G(y)}{(1-y)_+}
&=&
G(1) \ln(1-z) +
\int_z^1 dy \ \dfrac{[G(y)-G(1)]}{(1-y)}
\qquad .
\label{}
\end{eqnarray}
Taking the ${q_T} \to 0$ limit for the results of paper~I,
we find the partonic energy distribution to be
\begin{eqnarray}
w_a^{Asym}({\widehat x },Q^2,{q_T^2};j,j')
&=&
\left[
\dfrac{16 \pi^2 \, \alpha_s}{{q_T^2}}
\right] \
\Biggl\{
\delta_{a,j} \, \delta(1-{\widehat x }) \ C_F \
\left[ 2 \ln \left(\dfrac{Q^2}{{q_T^2}} \right) - 3 \right]
\nonumber \\[10pt]
&+& \delta_{a,j} \ C_F \ \left[ \dfrac{1+{\widehat x }^2}{1-{\widehat x }}
\right]_+ \
+ \delta_{a,g} \ \left[ \dfrac{{\widehat x }^2+ (1-{\widehat x })^2}{2}
\right]
\Biggl\}
\qquad ,
\label{eq:eqbmu}
\end{eqnarray}
where we use $\delta_{a,j}$ and $\delta_{a,g}$ for the quark and gluon
contributions, respectively.
For convenience, we denote the asymptotic limit ${q_T}\to 0$ of
$w_a$ by $w_a^{Asym}$.
In this limit, we can greatly simplify this expression by
identifying the QCD splitting functions. We present the result for the
hadronic structure function convoluted with the parton distributions,
({\it cf.}, \eq{partonhadron}):
\begin{eqnarray}
\Gamma^{Asym}(x,Q^2,{q_T^2};j,j')
&=&
\left[
\dfrac{16 \pi^2 \, \alpha_s}{{q_T^2}}
\right] \
\Bigl\{
f_{j/A}(x) \ C_F \
\left[
2 \ \ln \left(\dfrac{Q^2}{{q_T^2}} \right) - 3
\right]
\nonumber \\
&+&
f_{j/A} \otimes P_{q/q}
+ f_{g/A} \otimes P_{q/g}
\Bigr\}
\qquad ,
\label{eq:sudexp}
\end{eqnarray}
where $\otimes$ represents a convolution in ${\widehat x }$.
In the simple form above, it is easy to identify the separate
contributions.
The last two terms arise from the collinear singularities, and are
proportional to
the appropriate first order splitting kernel, $P_{q/q}$ and $P_{q/g}$.
It is the remaining term in which we are interested as these arise from
the
soft gluon processes.
We note that
$\Gamma^{Asym}$ is defined such that the combination
$\Gamma^{Pert}-\Gamma^{Asym}$
has only logarithmic singularities
as ${q_T}\to 0$.
\section{Sudakov Form Factor}
In this section, we display the structure of $d\Sigma/ dx\, dQ^2\,
dq_T^2\, d\phi$ with the Sudakov summation of logarithms. This provides
the basis a formula that includes nonperturbative effects, developed in
Sec.~6. In addition, in Sec.~4 we compare the summed form of this
section with the order $\alpha_s$ form of $d\Sigma/ dx\, dQ^2\,
dq_T^2\, d\phi$ from Sec.~2, in order to extract the coefficients that
appear in the summed form.
\subsection{Bessel Transform
of $w_a({\widehat x },Q^2,{q_T^2};j,j')$}
It proves
convenient to introduce a Fourier transform between transverse momentum
space (${q_T}$) and impact parameter space ($b$),
\begin{eqnarray}
w_a({\widehat x },Q^2,{q_T^2};j,j') &=&
\int {d^2 b \over (2\pi)^2}
\ e^{i {q_T} \cdot b } \
\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')
\nonumber \\
&=&
\int_0^{\infty} {d b \over 2\pi} \ b \ J_0(b \, {q_T}) \
\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')
\qquad ,
\label{eq:btransform}
\end{eqnarray}
as
$\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')$ will have a simple
structure.\citex{CSS}
Effectively, we make use of the renormalization group equation
to sum the logs of $Q^2$, and gauge invariance to sum the logs of ${q_T}
\sim 1/b$.
The Fourier transform also maps the ${q_T}$ singularities at the
origin to the large $b$ behavior of
$\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')$; we will take advantage of
this when we consider non-perturbative contributions.
\subsection{Sudakov Form Factor}
The structure function in impact parameter space
$\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')$ has the factorized form:
\begin{eqnarray}
\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')
&=&
C^{IN }_{ja} \big({\widehat x } ,b\mu \big) \
\sum_{a'}
\int \, d{\widehat z } \, {\widehat z } \
C^{OUT}_{a^{\prime}\, j^{\prime}} \big({\widehat z } ,b\mu \big) \
e^{-S(b)}
\qquad .
\label{eq:suddefi}
\end{eqnarray}
This form is from references \cite{CS} and \cite{CSS} applied to the DIS
process, and generalized to include vector bosons other than the photon.
The last
exponential factor is the Sudakov form factor:
\begin{eqnarray}
S(b)
&=&
\int_{C_1^2/b^2}^{C_2^2 Q^2} \
{d\mu^2 \over \mu^2}
\left\{\ln\left[{C_2^2 Q^2\over \mu^2}\right]
A( \alpha_s(\mu) ) +
B( \alpha_s(\mu) )\right\}
\qquad .
\label{eq:suddefii}
\end{eqnarray}
The logarithm in the exponential is characteristic of the
gauge theory. It arises from the soft gluon summation in QCD at the low
transverse momentum ${q_T^2} \ll Q^2$.
The arbitrary constants $\{ C_1, C_2 \}$ reflect the freedom in the choice of
renormalization scale.
We choose $\{ C_1, C_2 \}$ to be
\begin{eqnarray}
C_1 &=& 2 e^{-\gamma_E} \\
C_2 &=& 1 \qquad .
\label{eq:c1c2}
\end{eqnarray}
The functions $A$, $B$ and the
hard scattering functions
$C$'s are simple power series in the strong coupling constant
$\alpha_s$ with numerical
coefficients:\footnote{
Collins and Soper\citex{CSS} (CS)
expand in powers of ${\alpha_s}/\pi$, and
Davies, Webber, and Stirling\citex{DWS} (DWS)
expand in powers of ${\alpha_s}/(2\pi)$.
We carry the extra factor of (2)
explicitly to facilitate comparison between these references.}
\begin{eqnarray}
A( \alpha_s(\mu) ) &=&
\sum_{N=1}^\infty\ \left\{{\alpha_s(\mu)\over (2) \pi}\right\}^N \ A_N
\\
B( \alpha_s(\mu) ) &=&
\sum_{N=1}^\infty\ \left\{{\alpha_s(\mu)\over (2) \pi}\right\}^N \ B_N
\label{eq:abterms}
\end{eqnarray}
\begin{eqnarray}
C_{ja}^{IN}( {\widehat x },b\mu ) &=&
\delta(1-{\widehat x }) \, \delta_{ja} + \sum_{N=1}^\infty \
C_{ja}^{IN(N)} ({\widehat x },b\mu ) \
\left\{{\alpha_s(\mu)\over (2) \pi}\right\}^N
\label{eq:ctermsi}
\\
C_{a^{\prime}j^{\prime}}^{OUT}( {\widehat z },b\mu ) &=&
\delta(1-{\widehat z }) \, \delta_{a^{\prime}j^{\prime}} + \sum_{N=1}^\infty \
C_{a^{\prime}j^{\prime}}^{OUT(N)}({\widehat z },b\mu ) \
\left\{{\alpha_s(\mu)\over (2) \pi}\right\}^N
\qquad .
\label{eq:ctermsii}
\end{eqnarray}
The normalization has been chosen such that each hard scattering function
$C$ equals a
\hbox{$\delta$-function} at leading order.
As noted in reference \cite{CSS}, in the limit $Q\to \infty$, all
logarithms may be counted as being equally large. Therefore, to evaluate
the cross section at ${q_T}\simeq0$ to an approximation of ``degree $N$,"
one must evaluate
$A$ to order ${\alpha_s}^{N+2}$,
$B$ to order ${\alpha_s}^{N+1}$,
$C^{IN}$ and $C^{OUT}$ to order ${\alpha_s}^{N}$,
and the $\beta$ function order ${\alpha_s}^{N+2}$.
In particular, an extra order in $A$ is necessary due to the extra
logarithmic factor in \eq{suddefii}.
For the present calculation, we evaluate
$A$ to order ${\alpha_s}^{2}$,
$B$ to order ${\alpha_s}^{1}$,
$C^{IN}$ and $C^{OUT}$ to order ${\alpha_s}^{1}$,
and the $\beta$ function to order ${\alpha_s}^{2}$.
This yields the cross section
to order ${\alpha_s}^{1}$ for large ${q_T}$,
to order ${\alpha_s}^{0}$ for small ${q_T}$,
and the cross section integrated over ${q_T}$ to
${\alpha_s}^{1}$.
\subsection{Perturbative Expansion of the Sudakov Form Factor}
We can extract the $A_i$ and $B_i$ coefficients of the Sudakov factor by
expanding $\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')$
of \eq{suddefi} in ${\alpha_s}$, and comparing with the perturbative
calculation of paper~I.
Here, we take a fixed momentum scale $\mu_0$ in $\alpha_s(\mu_0)$ as the
running of
$\alpha_s(\mu)$
contributes only to higher orders. We can now compute the integral over
$\mu^2$ analytically to obtain:
\begin{eqnarray}
S(b)
&=&
\int_{C_1^2/b^2}^{C_2^2 Q^2} \
{d\mu^2 \over \mu^2} \
\left[\ln\left[{C_2^2 Q^2\over \mu^2}\right] A( \alpha_s(\mu) ) +
B( \alpha_s(\mu) )\right]
\nonumber\\[5pt]
&\simeq&
{\alpha_s(\mu_0) \over (2) \, \pi }
\left[
A_1 \, {L^2 \over 2}
+ B_1 L \right]
\qquad ,
\end{eqnarray}
where
\begin{eqnarray}
L&=& \ln\left[ {C_2^2 \over C_1^2 } \ b^2 \, Q^2\right]
\qquad .
\end{eqnarray}
We expand the Sudakov exponential out to order ${\alpha_s}^1$,
\begin{eqnarray}
e^{-S(b)} &\simeq&
1 - S(b) + {\cal O}({\alpha_s}^2)
\qquad ,
\end{eqnarray}
and perform the Bessel transform of
$\widetilde{w}_a({\widehat x },Q^2,b^2;j,j')$
({\it cf.}, \eq{suddefi})
to obtain the partonic
structure function in momentum space:
\begin{eqnarray}
w_{a}^{}({\widehat x },Q^2,{q_T^2};j,j')
&=&
\int_0^{\infty} {d b \over 2\pi} \ b \ J_0(b \, {q_T}) \
\left[
\delta_{a,j} \, \delta(1-{\widehat x }) +
\dfrac{{\alpha_s}(\mu)}{(2) \pi} \ C_{a,j}^{IN\, (1)}({\widehat x },b\mu)
\right]
\nonumber \\ &\times&
\sum_{a'} \
\left[
\delta_{a^\prime,j^\prime} +
\int_0^1 \, {\widehat z } \, d{\widehat z } \ \dfrac{{\alpha_s}(\mu)}{(2)
\pi} \
C_{a^\prime,j^\prime}^{OUT\, (1)}({\widehat z },b\mu)
\right]
\nonumber \\[10pt] &\times&
\left[
1 - S(b) + {\cal O}({\alpha_s}^2)
\right]
\qquad ,
\end{eqnarray}
where we have used the first order expressions for
$C^{IN \, (N)}_{jk} ({\widehat x },\mu b)$
and
$C^{OUT \, (N)}_{jk} ({\widehat z },\mu b)$.
Finally, we integrate to obtain the ${\cal
O}({\alpha_s}^1)$ terms for finite ${q_T}$:
\begin{eqnarray}
w_a^{}({\widehat x },Q^2,{q_T^2};j,j')
&\simeq&
\left[
\dfrac{16 \pi^2 \, \alpha_s}{{q_T^2}}
\right] \
\Biggl\{
\delta_{a,j} \,
\delta(1-{\widehat x }) \
\left[
\dfrac{2 A_1}{(2)} \,
\left\{
\ln \left(\dfrac{Q^2}{{q_T^2}} \right)
-2\ln \left(\dfrac{e^{\gamma_E} C_1}{2 C_2} \right)
\right\}
+
\dfrac{2 B_1}{(2)}
\right]
\,
\nonumber\\[10pt] && \qquad \qquad \qquad
+ \ \delta_{a,j} \ P_{q/q}(x)
+ \ \delta_{a,g} \ P_{q/g}(x)
\nonumber\\ && \qquad \qquad \qquad
+ {\rm \ terms\ proportional\ to\ } \delta({q_T^2})
\Biggl\}
\qquad .
\label{eq:sudexpi}
\end{eqnarray}
Here, we have use the fact that the renormalization group equation tells us
the
form of $C^{IN(1)}({\widehat x },\mu b)$ and $C^{OUT(1)}({\widehat z },\mu b)$
must be
a splitting kernel times $\log[\mu b]$, plus a function independent of $\mu$
and $b$.
Equivalently, for the hadronic structure function, we find:
\begin{eqnarray}
W^{}(x,Q^2,{q_T^2};j,j')
&\simeq&
\left[
\dfrac{16 \pi^2 \, \alpha_s}{{q_T^2}}
\right] \
\Biggl\{
f_{j/A}(x) \
\left[
\dfrac{2 A_1}{(2)} \,
\left\{
\ln \left(\dfrac{Q^2}{{q_T^2}} \right)
-2\ln \left(\dfrac{e^{\gamma_E} C_1}{2 C_2} \right)
\right\}
+
\dfrac{2 B_1}{(2)}
\right] \,
\nonumber\\[10pt] && \qquad \qquad \qquad
+ \ f_{j/A} \otimes \ P_{q/q}
+ \ f_{g/A} \otimes \ P_{q/g}
\nonumber\\ && \qquad \qquad \qquad
+ {\rm \ terms\ proportional\ to\ } \delta({q_T^2})
\Biggl\}
\qquad .
\label{eq:sudexpii}
\end{eqnarray}
We will compare the first-order expansions in \eq{sudexpi} and
\eq{sudexpii} with the asymptotic limit of the perturbative calculations
of Sec.~\ref{ECF} to extract the desired $A_1$ and $B_1$ coefficients.
\section{Comparing Asymptotic and Sudakov Contributions}
In this section, we compare the summed form
of $d\Sigma/ dx\, dQ^2\, dq_T^2\, d\phi$
with the order $\alpha_s$
form, and thus extract the
coefficients that appear in the summed form.
\subsection{Extraction of $A$ and $B$}
Comparing the expansion of the Sudakov expression
[\eq{sudexpii}]
with the asymptotic results [\eq{sudexp}], we
obtain the order $\alpha_s^1$ coefficients $A_1$ and $B_1$,
\begin{eqnarray}
A_1 &=& (2) \ C_F \\
B_1 &=& (2) \ 2\, C_F \,
\ln \left[ {C_1\over 2 C_2} \, e^{\gamma_E -(3/4)} \right]
\qquad .
\end{eqnarray}
With our particular choice of the arbitrary constants $\{ C_1, C_2 \}$
in \eq{c1c2}, we have:
\begin{eqnarray}
A_1 &=& (2) \ C_F \\
B_1 &=& (2) \ \left[ \dfrac{-3}{\phantom{-} 2} \right] \ C_F
\qquad .
\end{eqnarray}
We find that the results for $A_1$ and $B_1$ obtained above are
identical to those found in reference~\cite{CS} for Drell-Yan
production, as well as those found in reference~\cite{eedata} for $e^+\,
e^-$ annihilation. This apparent crossing symmetry has been demonstrated at
order
$\alpha_s^2$ by Trentadue.\citex{trentadue} In light of this result,
we shall make use of the $A_2$ coefficient\citex{trentadue}
\begin{eqnarray}
A_2 &=& (4) \left\{
{67\over9} - {\pi^2\over 3} - {10\over 27} \, N_f
+ {2\over 9} (33-2N_f) \ln\left({C_1 \over 2 \, e^{-\gamma_E}} \right)
\right\}
\qquad .
\end{eqnarray}
The extra order in the $A_i$ expansion will compensate extra logarithm
$L$ which is not present for the $B_i$ terms.
\subsection{Expansion of $C^{IN}$ and $C^{OUT}$}
$C^{IN}$ and $C^{OUT}$ terms are obtained by comparing the terms in the
perturbative expansion proportional to $\delta({q_T})$
with the expanded summed form.
Since the virtual graphs yield contributions only proportional to
$\delta({q_T})$, they will only enter $C^{IN}$ and $C^{OUT}$.
The real graphs yield {\it both} zero and finite ${q_T}$ terms;
therefore, they will contribute to both $A_i$, $B_i$, and
the $C^{IN}$ and $C^{OUT}$ coefficients.
The calculation of the virtual graphs has been performed by
Meng\rlap,\citex{meng} and we make use of those results.
We have defined the $C ({\widehat x },\mu b)$ coefficients
such that at leading order, they are
\begin{eqnarray}
C^{IN \, (0)}_{jk} ({\widehat x },\mu b)
&=&
\delta_{jk} \, \delta(1-{\widehat x } )
\nonumber \\[10pt]
C^{OUT \, (0)}_{jk} ({\widehat z },\mu b)
&=&
\delta_{jk} \, \delta(1-{\widehat z } )
\nonumber \\[10pt]
C^{IN \, (0)}_{jg} ({\widehat x },\mu b)
&=&
C^{OUT \, (0)}_{gk} ({\widehat z },\mu b)
= 0
\qquad .
\end{eqnarray}
(Here, $j$ and $k$ denote quarks and anti-quarks, and $g$ denotes
gluons.)
At next to leading order,
we find that $C^{IN \, (1)} ({\widehat x },\mu b)$
match those calculated by CS for the Drell-Yan process:\citex{CS}
\begin{eqnarray}
C^{IN \, (1)}_{jk} ({\widehat x },\mu b)
&=&
\delta_{jk} \Biggl\{
{2\over 3} (1-{\widehat x })
+P_{q/q}({\widehat x }) \
\ln \left( { \lambda \over \mu b} \right)
\nonumber \\[10pt]
&+& \delta(1-{\widehat x }) \ \Biggl[
-C_F \, \ln^2 \left( {C_1 \ e^{-3/4} \over C_2 \, \lambda } \right)
+ {\pi^2 \over 3} - {23 \over 12}
\Biggr]
\Biggr\}
\end{eqnarray}
\begin{eqnarray}
C^{IN \, (1)}_{jg} ({\widehat x },\mu b)
&=&
{1\over 2} {\widehat x } (1-{\widehat x })
+P_{j/g}({\widehat x }) \
\ln \left( { \lambda \over \mu b} \right)
\qquad .
\end{eqnarray}
The $C^{OUT \, (1)} ({\widehat z },\mu b)$ are
simply those for $e^+ e^-$ as given in reference~\cite{eedata}:
\begin{eqnarray}
C^{OUT \, (1)}_{jk} ({\widehat z },\mu b)
&=&
\delta_{jk} \Biggl\{
{2\over 3} (1-{\widehat z })
+P_{q/q}({\widehat z }) \
\ln \left( { \lambda \over \mu b} \right)
\nonumber \\[10pt]
&+& \delta(1-{\widehat z }) \ \Biggl[
-C_F \, \ln^2 \left( {C_1 \ e^{-3/4} \over C_2 \, \lambda } \right)
+ {\pi^2 \over 3} - {29 \over 12}
\Biggr]
\Biggr\}
\end{eqnarray}
\begin{eqnarray}
C^{OUT \, (1)}_{gk} ({\widehat z },\mu b)
&=&
{2\over 3} \, {\widehat z }
+P_{g/k}({\widehat z }) \
\ln \left( { \lambda \over \mu b} \right)
\qquad ,
\end{eqnarray}
where we define $\lambda=2e^{-\gamma_E}$ to simplify the notation. Note that
$C^{IN}$ and
$C^{OUT}$ are only a function of the ratio $C_1/C_2$.
\subsection{Complete Expression}
Now that we have obtained $A_1$, $A_2$, and $B_1$, we can substitute into
equation~\eq{suddefii} to obtain the complete Sudakov contribution
(including the full
$\alpha_s(\mu)$
dependence). We choose to perform the $\mu$ integral analytically, as the
Bessel transform would be prohibitively CPU intensive if we did not.
To facilitate this computation we provide an integral table in
Appendix~\ref{APPA} including all the necessary terms
We are now ready to combine the separate parts of the calculation.
\section{Matching}
We now have computed the contributions to the energy distribution
functions for the
perturbative $\Gamma_k^{Pert}$ in paper~I [\eq{partonhadron}],
the summed (or Sudakov) $W^{}(x,Q^2,q_T^2;j,j')$ in \eq{sudexpii}, and
the asymptotic $\Gamma_k^{Asym}$ in \eq{sudexp}.
We can simply assemble these pieces
to form the total structure functions via:
\begin{eqnarray}
\Gamma_1(x,Q^2,{q_T^2};j,j') &=&
\Gamma_1^{Pert}(x,Q^2,{q_T^2};j,j')
+ \phantom{(-1)} W^{}(x,Q^2,q_T^2;j,j')
- \Gamma_1^{Asym}(x,Q^2,{q_T^2};j,j')
\nonumber\\
\Gamma_6(x,Q^2,{q_T^2};j,j') &=&
\Gamma_6^{Pert}(x,Q^2,{q_T^2};j,j')
+ (-1) W^{}(x,Q^2,q_T^2;j,j')
- \Gamma_6^{Asym}(x,Q^2,{q_T^2};j,j')
.
\nonumber \\
\label{eq:}
\end{eqnarray}
Here, $\Gamma_k^{Pert}$ and $\Gamma_k^{Asym}$ are evaluated
at order ${\alpha_s}^1$, while
$W$ contains a summation of perturbation theory.
In the limit ${q_T}\to0$,
$\Gamma_k^{Pert}$ and $\Gamma_k^{Asym}$ will cancel
each other leaving $W$ as we desire.
In the limit ${q_T}\simeq Q$, $W$ and
$\Gamma_k^{Asym}$ will cancel to leading order in $\alpha_s$;
however, the finite difference may not be
negligible. To ensure that we recover the proper
result ($\Gamma_k^{Pert}$) for large ${q_T}$, we
define the total energy distribution
function ($\Gamma_k$) to be:
\begin{eqnarray}
\Gamma_1(x,Q^2,{q_T^2};j,j') &=&
\Gamma_1^{Pert}(x,Q^2,{q_T^2};j,j')
\nonumber \\
&+&
{\cal T}\left(\dfrac{{q_T}}{Q}\right)
\left\{ \phantom{(-1)} W^{}(x,Q^2,q_T^2;j,j') -
\Gamma_1^{Asym}(x,Q^2,{q_T^2};j,j') \right\}
\nonumber\\
\Gamma_6(x,Q^2,{q_T^2};j,j') &=&
\Gamma_6^{Pert}(x,Q^2,{q_T^2};j,j')
\nonumber \\
&+&
{\cal T}\left(\dfrac{{q_T}}{Q}\right)
\left\{(-1) W^{}(x,Q^2,q_T^2;j,j') - \Gamma_6^{Asym}(x,Q^2,{q_T^2};j,j')
\right\}
\qquad ,
\nonumber \\
\label{eq:}
\end{eqnarray}
where we introduce the arbitrary function
\begin{eqnarray}
{\cal T}\left( {{q_T} \over Q} \right)
&=&
{1 \over 1+
\left(\rho \, \dfrac{{q_T}}{Q}
\right)^4 }
\qquad .
\label{eq:rhoeq}
\end{eqnarray}
The transition function
${\cal T}({q_T}/Q)$ serves to switch smoothly
from the matched formulas to the perturbative
formula, and
$\rho$ is an arbitrary parameter which determines
the details of the matching.
\xfig{tempvii} displays ${\cal T}({q_T}/Q)$ for a range of $\rho$ values.
We will choose $\rho=5$ which ensures that
$\Gamma_k \simeq \Gamma_k^{Pert}$ for
${q_T}/Q \, {\lower0.5ex\hbox{$\stackrel{>}{\sim}$}} \, 0.4$, a conservative
value.
\figtempvii
\section{Non-Perturbative Contributions \label{NONPERT}}
In analogy with \eq{partonhadron} and \eq{suddefi},
the Bessel transform of the hadronic structure function is defined
as:
\begin{eqnarray}
W^{}(x,Q^2,q_T^2;j,j')
&=&
\int \, {d^2 b \over (2\pi)^2 } \
e^{i {q_T} \cdot b } \
\widetilde{W}^{}(x,Q^2,b^2;j,j')
\qquad .
\label{eq:wdefii}
\end{eqnarray}
When $b$ is small, we have:
\begin{eqnarray}
\widetilde W(x,Q^2,b^2;j,j')
&=&
\int_{x}^{1}
\dfrac{d\xi}{\xi} \
\sum_a \
f_{a/A}(\xi,\mu) \
C^{IN }_{ja} \big({\widehat x } ,b\mu \big) \
\sum_{a'}
\int \, d{\widehat z } \, {\widehat z } \
C^{OUT}_{a^{\prime}\, j^{\prime}} \big({\widehat z } ,b\mu \big) \
e^{-S(b)}
.
\label{eq:wtildedef}
\end{eqnarray}
The perturbative calculation of $\widetilde W(x,Q^2,b^2;j,j')$ is not
reliable for $b {\lower0.5ex\hbox{$\stackrel{>}{\sim}$}} 1/\Lambda$. However,
the integration over
$b$ in \eq{wdefii} runs to infinitely large $b^2$, and the
region $b {\lower0.5ex\hbox{$\stackrel{>}{\sim}$}} 1/\Lambda$ is important for
values of $Q^2$ and
${q_T^2}$ of practical interest. In order to deal with the large
$b^2$ region, we follow the method introduced in
Refs.~\cite{CS,CSS}. We define a value $b_{\rm max}$ such that we can
consider perturbation theory to be reliable for $b < b_{\rm max}$.
(In our numerical examples, we take $1/b_{\rm max} =2 \, {{\rm GeV}}$.)
Then we define a function $b_*$ of $b$ such that $b_* \approx b$ for
small $b$ and $b_* < b_{\rm max}$ for all $b$:
\begin{equation}
b_* =
{ b \over \sqrt{1 + b^2/b_{\rm max}^2}}
\qquad .
\end{equation}
We define a version of $\widetilde W(x,Q^2,b^2;j,j')$ for which
perturbation theory is always reliable by $\widetilde
W(x,Q^2,b_*^2;j,j')$. Note that for small $b$, the difference between
$\widetilde W(b_*)$ and $\widetilde W(b)$ is negligible because $b_* \approx
b$. Conversely, perturbation theory is always applicable for
the calculation of $\widetilde W(b_*)$ because $b_*$ is small even when
$b$ is large.
Next, we define a nonperturbative function $\exp(- S_{\rm NP}(b))$ as
the ratio of $\widetilde W(b)$ and $\widetilde W(b_*)$:
\begin{equation}
\widetilde W(x,Q^2,b^2;j,j')
=
\widetilde W(x,Q^2,b_*^2;j,j')\
e^{- S_{\rm NP}(x,Q^2,b^2;j,j') }
\qquad .
\label{eq:SNPdef}
\end{equation}
Ultimately, we will have to use nonperturbative information to
determine $S_{\rm NP}(b)$. However, some important information is
available to us. From \eq{wtildedef}, we see that
\begin{equation}
\dfrac{\partial \log[\widetilde W(x,Q^2,b^2;j,j')]}{\partial \log Q^2}
\end{equation}
is independent of $x,j,j'$ and $Q^2$. This result is derived in
perturbation theory, but at arbitrary order, so we presume that it
holds even beyond perturbation theory. Then
\begin{equation}
\dfrac{\partial S_{\rm NP}(x,Q^2,b^2;j,j')}{\partial \log Q^2 }
\end{equation}
is also independent of $x,j,j',k$ and $Q^2$. That is, $S_{\rm NP}$
has the form
\begin{equation}
S_{\rm NP}(x,Q^2,b^2;j,j') =
\log\left(Q^2/Q_0^2\right)g_1(b)
+ \Delta S_{\rm NP}(x,b^2;j,j')
\qquad .
\label{eq:sudakovi}
\end{equation}
(Here $Q_0$ is an arbitrary constant with dimensions of mass, inserted
to keep the argument of the logarithm dimensionless.) Furthermore, in
$\widetilde W$, the $x$ and $j$ dependence occurs in a separate factor from
the $j'$ dependence. Thus the second term in \eq{sudakovi} above
can be simplified to
\begin{equation}
S_{\rm NP}(x,Q^2,b^2;j,j') =
\log\left(Q^2/Q_0^2\right)g_1(b)
+ g_A(x,b^2;j)
+ g_B(b^2;j')
\qquad .
\end{equation}
(Recall, we have integrated over ${\widehat z }$.)
Perturbation theory is not applicable for the calculation of the
functions $g_1(b)$, $g_A(x,b^2;j)$ and $g_B(b^2;j')$ for large $b$.
For small $b$, perturbation theory tells us only that these functions
approach 0 as $b \to 0$. This follows from \eq{SNPdef}, and
the fact that $b_*/b \to 0$ when $b \to 0$. (See Ref.~\cite{CSS} for
further discussion.) Since we learn little from perturbation theory,
we turn to non-perturbative sources of information. Fortunately, the
analogous functions in $e^+e^-$ annihilation and in the Drell-Yan
process have been fit using experimental
results.\citex{CS,DWS,yuanladinsky}
\figtempv
We therefore ask whether the functions $g_1(b)$, $g_A(x,b^2;j)$ and
$g_B(b^2;j')$ in deeply inelastic scattering are related to the
analogous functions in the other two processes. Consider first
$g_1(b)$, the coefficient of $\log(Q^2/Q_0^2)$. According to the
analysis of Ref.~\cite{CS}, this function receives contributions from the
two jet subdiagrams in \xfig{nonpertiii}(b). (In this
figure, we use a space-like axial gauge.) The soft gluon connections
in \xfig{nonpertiii}(b) affect $g_A(x,b^2;j)$ and
$g_B(b^2;j')$, but do not contribute ``double logarithms,'' and thus
do not affect $g_1(b)$. Thus
\begin{equation}
g_1(b) \equiv g_1^{DIS}(b) = g_1^{\rm IN}(b) + g_1^{\rm OUT}(b)
\qquad ,
\end{equation}
where $g_1^{\rm IN}(b)$ is associated with the incoming beam jet
(the lower subdiagram in \xfig{nonpertiii}(b)) while
$g_1^{\rm OUT}(b)$ is associated with the outgoing struck-quark jet
(the upper subdiagram in \xfig{nonpertiii}(b)). In the
Drell-Yan process, depicted in \xfig{nonpertiii}(a),
there are two incoming beam jets and one has
\begin{equation}
g_1^{\rm DY}(b) = 2 g_1^{\rm IN}(b)
\qquad .
\end{equation}
In $e^+ e^-$ annihilation, depicted in
\xfig{nonpertiii}(c), there are two outgoing quark
jets and one has
\begin{equation}
g_1^{e\bar e}(b) = 2 g_1^{\rm OUT}(b)
\qquad .
\end{equation}
Thus
\begin{equation}
g_1(b) \equiv g_1^{DIS}(b) =
(1/2)\,g_1^{\rm DY}(b) +(1/2)\, g_1^{e\bar e}(b)
\qquad .
\end{equation}
In the following section, we show numerical results using
Ref.~\cite{CS} for $g_1^{e\bar e}(b)$ and
Ref.~\cite{DWS} for $g_1^{\rm DY}(b)$.
The situation for $g_A(x,b^2;j)$ and $g_B(b^2;j')$ is not so simple.
Let us write
\begin{equation}
S_{\rm NP}^{\rm DY}(x,Q^2,b^2;j,j') =
\log\left(Q^2/Q_0^2\right) g_1^{\rm DY}(b)
+ g_2^{\rm DY}(x_A,b^2;j)
+ g_2^{\rm DY}(x_B,b^2;j')
\qquad .
\end{equation}
for the Drell-Yan process and
\begin{equation}
S_{\rm NP}^{\bar e e}(x,Q^2,b^2;j,j') =
\log\left(Q^2/Q_0^2\right) g_1^{\bar e e}(b)
+ g_2^{\bar e e}(b^2;j)
+ g_2^{\bar e e}(b^2;j')
\qquad .
\end{equation}
for the energy-energy correlation function in $e^+e^-$ annihilation.
({\it Cf.}, \xfig{tempv}.)
One might like to assume that $g_A(x,b^2;j)$ is the same function as
$g_2^{\rm DY}(x,b^2;j)$ while $g_B(b^2;j')$ is the same function as
$g_2^{\bar e e}(b^2;j')$. However, this may not be true because all
of these functions get contributions from the soft gluon exchanges
that link the two jets in \xfig{nonpertiii},
(represented by the function $U(b)$ in Ref.~\cite{CS}).
Furthermore, the
dependence of the functions $g_2^{\rm DY}(x,b^2;j)$ and
$g_2^{\bar e e}(b^2;j')$ on the flavors $j$ and $j'$ has not been
determined from experimental data. What we know are flavor averaged
functions $g_2^{\rm DY}(x,b^2)$ and $g_2^{\bar e e}(b^2)$. Thus the
best we can do is propose a model for the functions we need:
\begin{equation}
g_A(x,b^2;j) + g_B(b^2;j')
= t\, g_2^{\rm DY}(x,b^2)
+ (1-t)\,g_2^{\bar e e}(b^2)
\qquad ,
\label{eq:sudparm}
\end{equation}
where $0<t<1$, with $g_2^{\bar e e}(b^2)$ taken from
Ref.~\cite{CS} and $g_2^{\rm DY}(x,b^2)$ taken from
Ref.~\cite{DWS}. We vary the parameter $t$
between 0 and 1 to get an estimate of the uncertainty involved.
({\it Cf.}, \xfig{tempvi}.)
\figtempvi
\figtempvia
For comparison, we present the above parameterizations for the
non-perturbative contributions with the recent fit by Ladinsky and
Yuan\citex{yuanladinsky} for W-production in \xfig{tempvia}. Ladinsky and
Yuan introduce an extra degree of freedom by allowing for a $\tau=x_A\,
x_B$ dependence. We present the comparison for a range of $\tau$; this
allows one to gauge the effects of different non-perturbative
estimates, and correlate the Ladinsky and
Yuan parameterization with that presented in \eq{sudakovi} and
\eq{sudparm}.
\section{Reprise}
For the benefit of the reader, we review the principal steps
in the calculation of the energy distribution. The energy
distribution is given by:
\begin{eqnarray}
{ d \Sigma \over dx \, dQ^2 \, d{q_T^2}\, d\phi}
&=&
\sum_{k=1}^{9} \
{ d \Sigma_k \over dx \, dQ^2 \, d{q_T^2}\, d\phi}
\nonumber \\[10pt]
&=&
\sum_{k=1}^{9} \
{\cal A}_k(\psi,\phi)
\sum_{V_1,V_2} \
\sum_{j, j'} \
\Sigma_0(Q^2;V_1,V_2,j, j',k) \
\Gamma_k(x,Q^2,q_T^2;j,j')
\qquad ,
\label{eq:}
\end{eqnarray}
where ${\cal A}_k(\psi,\phi)$ are the nine angular
functions arising from hyperbolic $D^1(\psi,\phi)$
rotation matrices. The sum on $V_1$
and $V_2$ runs over vector boson types, $\{\gamma,Z\}$ or
$\{W^\pm\}$ as appropriate.
The sums over $j$ and $j^\prime$ include all quark
flavors, $\{u,\bar u,d,\bar d,\dots\}$; for neutral
currents, this sum is diagonal
$(j=j^\prime)$.
The function
$\Sigma_0(Q^2;V_1,V_2,j, j',k)$ includes factors for the
coupling of the electron to the vector bosons as well as
factors for the propagation of the vector bosons.
The energy distribution
function that we have computed is
$\Gamma_k(x,Q^2,q_T^2;j,j')$.
In the limit ${q_T}\to 0$, the $\Gamma_1$ and $\Gamma_6$
will contain the dominant singularities as their angular
structure is proportional to the Born process.
We define:
\begin{eqnarray}
\Gamma_1(x,Q^2,{q_T^2};j,j') &=&
\Gamma_1^{Pert}(x,Q^2,{q_T^2};j,j')
\nonumber \\
&+&
{\cal T}\left(\dfrac{{q_T}}{Q}\right)
\left\{ \phantom{(-1)} W^{}(x,Q^2,q_T^2;j,j') -
\Gamma_1^{Asym}(x,Q^2,{q_T^2};j,j') \right\}
\nonumber\\
\Gamma_6(x,Q^2,{q_T^2};j,j') &=&
\Gamma_6^{Pert}(x,Q^2,{q_T^2};j,j')
\nonumber \\
&+&
{\cal T}\left(\dfrac{{q_T}}{Q}\right)
\left\{(-1) W^{}(x,Q^2,q_T^2;j,j') - \Gamma_6^{Asym}(x,Q^2,{q_T^2};j,j')
\right\}
\label{eq:}
\end{eqnarray}
where the matching function ${\cal T}({q_T}/Q)$ [\eq{rhoeq}] is provided
to ensure proper behavior as ${q_T}\to Q$.
$\Gamma_k^{Pert}$ represents the perturbative results of
paper~I [\eq{partonhadron}] calculated at order ${\alpha_s}^1$,
$\Gamma_k^{Asym}$ represents the asymptotic limit
(${q_T}\to 0$) of
$\Gamma_k^{Pert}$ [\eq{sudexp}], and
$W^{}(x,Q^2,q_T^2;j,j')$ represents the summed
(Sudakov) term [\eq{suddefi}] which is finite as ${q_T}\to 0$.
Note the function $W^{}(x,Q^2,q_T^2;j,j')$ the same for
both $\Gamma_1$ and $\Gamma_6$.
The form of the Sudakov structure function
is particularly simple in impact parameter space:
\begin{eqnarray}
W^{}(x,Q^2,q_T^2;j,j')
&=&
\int \, {d^2 b \over (2\pi)^2 } \
e^{i {q_T} \cdot b } \
\widetilde{W}^{}(x,Q^2,b^2;j,j')
\qquad .
\label{eq:}
\end{eqnarray}
To ensure that the calculation is reliable for
large $b$ (small ${q_T}$), we introduce:
\begin{equation}
\widetilde W(x,Q^2,b^2;j,j')
=
\widetilde W(x,Q^2,b_*^2;j,j')\
e^{- S_{\rm NP}(x,Q^2,b^2;j,j') }
\qquad ,
\label{eq:}
\end{equation}
where $b_* \in [0,b_{max}]$ for $b \in [0,\infty]$.
The perturbative function $\widetilde W(x,Q^2,b_*^2;j,j')$
is given by:
\begin{eqnarray}
\widetilde W(x,Q^2,b_*^2;j,j')
&=&
\int_{x}^{1}
\dfrac{d\xi}{\xi} \
\sum_a \
f_{a/A}(\xi,\mu) \
C^{IN }_{ja} \big({\widehat x } ,b_*\mu \big) \
\int \, d{\widehat z } \, {\widehat z } \ \sum_{a'} \
C^{OUT}_{a^{\prime}\, j^{\prime}} \big({\widehat z } ,b_*\mu \big) \
e^{-S(b_*)}
,
\nonumber \\
\label{eq:}
\end{eqnarray}
where ${\widehat x }=x/\xi$.
For the incoming particles, there is an integration over a parton momentum
fraction
$\xi$, a sum over parton types
$a = g, u, \bar u, d, \bar d, \dots$, a parton distribution
function $f_{a/A}$ and a set of perturbative coefficients
$C^{\rm IN }$.
For the outgoing partons, there is an integration over
parton momentum fraction $\hat z$, weighted by $\hat z$,
a sum over parton types
$a' = g, u, \bar u, d, \bar d, \dots$, and there are
perturbative coefficients $C^{\rm OUT}$ associated with
the outgoing states.
The heart of the formula is the Sudakov factor
$\exp[-S(b_*)]$, defined as:
\begin{eqnarray}
S(b_*)
&=&
\int_{C_1^2/b_*^2}^{C_2^2 Q^2} \
{d\mu^2 \over \mu^2}
\left\{\ln\left[{C_2^2 Q^2\over \mu^2}\right]
A( \alpha_s(\mu) ) +
B( \alpha_s(\mu) )\right\}
\qquad .
\label{eq:}
\end{eqnarray}
The functions $A$, $B$, as well as $C^{\rm IN}$ and
$C^{\rm OUT}$, have perturbative expansions in
powers of $\alpha_s$.
We choose the arbitrary constants $\{ C_1, C_2\}$ as in \eq{c1c2}.
The non-perturbative
contribution is parameterized in terms of the fits to
$e^+ e^-$ and Drell-Yan data\rlap.\citex{eedata,CS,DWS}
\begin{equation}
S_{\rm NP} (x,Q^2,b^2;j,j') =
\log\left[\dfrac{Q^2}{Q_0^2}\right]
\left\{
\dfrac{g_1^{\rm DY}(b) + g_1^{e\bar e}(b) }{2}
\right\}
+ t \ g_2^{\rm DY}(x,b^2)
+ (1-t) \ g_2^{\bar e e}(b^2)
\qquad .
\end{equation}
The arbitrary parameter $t \in [0,1]$
interpolates between the
$e^+ e^-$ and Drell-Yan form.
\section{Results}
We present numerical results of the energy distribution function
for representative values of $\{x,Q^2\}$ using the CTEQ3 parton
distributions\rlap.\citex{cteq3}
We present results only for the $\Gamma_1$ set of structure functions,
as the $\Gamma_6$ set have the identical ${q_T}\to 0$ structure (up to a
sign).
Recall that the structure functions are given by:
\begin{eqnarray}
\Gamma_1(x,Q^2,{q_T^2};j,j') &=&
\Gamma_1^{Pert}(x,Q^2,{q_T^2};j,j')
\nonumber \\
&+&
{\cal T}\left(\dfrac{{q_T}}{Q}\right)
\left\{ W^{}(x,Q^2,q_T^2;j,j') - \Gamma_1^{Asym}(x,Q^2,{q_T^2};j,j') \right\}
\qquad .
\label{eq:}
\end{eqnarray}
Making use of \eq{master}, we have a parallel relation for the
energy distribution function:
\begin{eqnarray}
\dfrac{d\Sigma_1(x,Q^2,{q_T^2};j,j')}{dx\, dQ^2 \, d{q_T^2}\, d\phi} &=&
\dfrac{d\Sigma_1^{Pert}(x,Q^2,{q_T^2};j,j')}{dx\, dQ^2 \, d{q_T^2}\, d\phi}
\nonumber \\
&+&
{\cal T}\left(\dfrac{{q_T}}{Q}\right)
\left\{
\dfrac{d\Sigma_1^{Sum}(x,Q^2,{q_T^2};j,j')}{dx\, dQ^2 \, d{q_T^2}\, d\phi}
- \dfrac{d\Sigma_1^{Asym}(x,Q^2,{q_T^2};j,j')}{dx\, dQ^2 \, d{q_T^2}\, d\phi}
\right\}
,
\label{eq:}
\end{eqnarray}
where we use the ``Sum" superscript to denote the summed Sudakov contribution
derived from $W$.
We will examine both the individual terms as well as the total in the
following.
We will use the shorthand
$
d\Sigma_1 \equiv
d\Sigma_1(x,Q^2,{q_T^2};j,j')/(dx\, dQ^2 \, d{q_T^2}\, d\phi)
$
\subsection{${q_T}$ Distributions}
\figtempi
\figtempiii
In \xfig{tempi} and \xfig{tempiii}, we show the separate contributions
to
$d\Sigma_1$ as a function of ${q_T}$
for
two choices of $\{x,Q^2\}$.\footnote{
In the small ${q_T}$ region, $d\Sigma_1$ and $d\Sigma_6$ are independent of
$\phi$; therefore we need not specify it.
}
We have included an extra factor of ${q_T^2}$ to make the features of the
plot more legible.
As anticipated, we see that
$d\Sigma_1^{Pert}\simeq d\Sigma_1^{Asym}$ as ${q_T}\to 0$ leaving
$d\Sigma_1 \simeq d\Sigma_1^{Sum}$.
For large ${q_T}$, we find $d\Sigma_1^{Sum}- d\Sigma_1^{Asym} \simeq 0$,
but this cancellation is not as precise as the above because the relation
$\Gamma^{Sum} - \Gamma^{Asym} \simeq 0$ holds only to first-order.
Therefore, in the following figures
we shall include the ${\cal T}({q_T^2}/Q^2)$ factor to ensure that
$d\Sigma_1^{Sum}- d\Sigma_1^{Asym}$ is smoothly turned off at large
${q_T}$.
The fact that $d\Sigma_1^{Sum}$ and $d\Sigma_1^{Asym}$
become negative for large ${q_T}$ reminds us that these expressions
were approximations valid only for ${q_T} \ll Q$.
\figtempii
\figtempiv
Having examined the separate terms, we now turn our attention to
the energy distribution function,
$d\Sigma_1$.
Again, we have included an extra factor of ${q_T^2}$
in \xfig{tempii}(a) and \xfig{tempiv}(a) to make the features of the
plot more legible.
In \xfig{tempii}(b) and
\xfig{tempiv}(b), we plot $d\Sigma_1$ in
the small ${q_T}$ region (without an extra ${q_T^2}$ factor) to demonstrate
that the summed results approach a finite limit as ${q_T}\to 0$.
We present the results for three choices of the non-perturbative
function $S_{\rm NP}(x,Q^2,b^2;j,j')$ as parameterized in \eq{sudparm}.
The choice $t=0$ corresponds to the $e^+e^-$ limit\rlap,\citex{CS}
while $t=1$ corresponds to the Drell-Yan limit\rlap,\citex{DWS}
and $t=1/2$ corresponds to an even mix of the above.
The difference due to the non-perturbative contribution is quite
significant for low ${q_T}$.
The $t=0$ ($e^+e^-$)
non-perturbative function, which is much narrower in $b$-space, yields a
broader energy distribution; this is clearly evident in the
figures as we see the peak move to lower ${q_T}$ values as we shift from
the $t=0$ ($e^+e^-$)
to $t=1$ (Drell-Yan).
At large ${q_T}$, $d\Sigma_1$ is independent
of the non-perturbative contributions, since it is dominated by
$d\Sigma_1^{Pert}$.
Clearly, the HERA data should be able to distinguish between this range
of distributions, particularly in the small ${q_T}$ regime where the
span of the non-perturbative contributions are
significant.\citex{wsmith,olnesstung}
\section{Conclusions}
Measurement of the distribution of hadronic energy in the final state in
deeply inelastic electron scattering at HERA can provide a good test of
our understanding of perturbative QCD.
Furthermore, we can probe non-perturbative physics because the
the energy distribution functions are sensitive to the non-perturbative
Sudakov form factor $S_{{{\scriptscriptstyle N\kern-0.16667em P}}}(b)$ in the
small ${q_T}$ region.
We have evaluated the energy distribution function
for finite transverse momentum ${q_T}$ at order
$\alpha_s$ in paper~I.
Because the distribution is weighted by the final state hadron energy,
this physical observable is infrared safe, and independent of the
decay distribution functions.
In this paper, we sum the soft gluon radiation into a Sudakov form
factor to evaluate the energy distribution function in the small
${q_T}$ limit.
By matching the small and large ${q_T}$ regions, we obtain a complete
description throughout the kinematic range.
This result is significant phenomenologically as a the bulk of the events
occur at small ${q_T}$ values, where perturbation theory by itself is
divergent. This technique can provide an incisive tool for the
study of deeply inelastic scattering.
Additionally, crossing relations allow us to relate the
non-perturbative contribution in
deeply inelastic
scattering energy distributions to analogous quantities in the Drell-Yan
and $e^+e^-$ annihilation processes.
\acknowledgments
We would like to thank
E. Berger,
S. Ellis,
K. Meier,
W. Tung,
for valuable discussions.
We also thank R. Mertig for assistance with FeynCalc, and
S. Riemersma for carefully reading the manuscript.
R.M and F.O. would also like to acknowledge the support and gracious
hospitality of Dr. A. Ali and the
Deutsches Elektronen Synchrotron.
This work is supported in part by
the U.S. Department of Energy, Division of High Energy Physics.
|
2,869,038,155,708 | arxiv | \section{Introduction}
Reliable maps of the coronal density are important for linking various solar wind structures to the low solar atmosphere, for studies of the coronal response to the solar cycle, and for space weather applications: either as an inner boundary conditions for solar wind models, or for direct ballistic extrapolation into interplanetary space. Estimates of the coronal electron density can be made through inversion of coronal visible light observations. This has been achieved using several methods of varying complexity during eclipses, or by coronagraphs, for several decades. The introduction of \citet{morgan2015} gives a summary of the field, including discussion of the difficulties involved and examples of applications. A comprehensive review is given by \citet{aschwanden2011b}.
This paper presents a new inversion method based on spherical harmonics for the extended inner solar corona, valid for regions where the large-scale structure is close to radial. Spherical harmonics as a basis for 3D reconstruction is used in some branches of medicine and geophysics \citep[e.g.][and references within]{merrill1996,arridge2009,levis2015}. The method is described in section \ref{method}, and is tested on a simple set of synthetic data in section \ref{simple}. A more complicated set of synthetic data is discussed in section \ref{complex}. An approach to regularizing the higher-order spherical harmonics is presented in section \ref{regular}. A discussion of datagaps, noise and temporal changes is given in section \ref{noise}. Application to observations are demonstrated in section \ref{observations}. Conclusions are in section \ref{conclusions}. The appendix presents an alternative method to calculate the spherical harmonic coefficients based on iteration rather than least-squares.
\section{Inversion using spherical harmonics}
\label{method}
\subsection{Outline}
\label{outline}
For a spherical surface at a constant height $r=r_0$, the coronal density $\rho$ at Carrington longitude $\phi$ and latitude $\theta$ may be approximated by a spherical harmonic basis,
\begin{equation}
\label{eq1}
\rho (\phi,\theta,r_0) = \sum_{i=0}^{n_{sph}-1} c_i S_i (\phi,\theta)
\end{equation}
where the $c_i$ are coefficients and $S_i$ are the real-valued spherical harmonics, with the $i$ index related to latitudinal order $l$ ($l \leq L$, where $L$ is the highest order) and longitudinal order $m$ ($-l \leq m \leq l$) by:
\begin{table}[h]
\centering
\begin{tabular}{ccc}
$i$ & $l$ & $m$ \\
\tableline
\\
0 & 0 & 0\\
1 & 1 & -1 \\
2 & 1 & 0 \\
3 & 1 & 1 \\
. & . & . \\
$n_{sph}-1$ & $L$ & $L$ \\
\end{tabular}
\end{table}
Note that $S_0$ is the mean density component (a constant at all $\phi$ and $\theta$) and $n_{sph}=(L+1)^2$. By increasing the order $L$ to large values, any sufficiently continuous density structure can be well approximated by equation \ref{eq1}.
If a radial coronal density structure is assumed above the height of interest, the profile $f(r \geq r_0)$ of density with height can be described by a simple function. For example, considering mass flux conservation for a spherically-expanding corona under acceleration for heights at around 5$ R_{\odot}$,
\begin{equation}
\label{eq2}
f(r)=\left( \frac{r_0}{r} \right)^\alpha, \hspace{3em} r \geq r_0
\end{equation}
with $\alpha=2.2$. Thus the coronal density can be described by
\begin{equation}
\label{eq3}
\rho (\phi,\theta,r) = \rho (\phi,\theta,r_0) f(r), \hspace{3em} r \geq r_0
\end{equation}
For a volume segmented into discrete voxels, the observed K-coronal (electron) brightness $B_{k}$\ is the line-of-sight summation of the product of density and a factor $g$ which contains known constants, Thomson scattering coefficients and the length of each line-of-sight segment through each voxel (see for example section 2.1 of \citet{quemerais2002}, and references within):
\begin{equation}
\label{eq5}
B_k = \sum_{j=1}^{n_{los}}g_j \rho_j = \sum_{j=1}^{n_{los}}g_j f(r_j)\sum_{i=0}^{n_{sph}} c_i S_{ij},
\end{equation}
where the $j$ index labels voxels lying along the line of sight, thus $S_{ij}$ is the value of the spherical harmonic at order level $i$ and voxel $j$.
Each spherical harmonic $S_{ij}$ may be summed independently of the other harmonics along the line of sight to give the brightness contribution resulting from each harmonic. Defining $A_i$:
\begin{equation}
\label{eq6}
A_i = \sum_{j=1}^{n_{los}}g_j f(r_j) S_{ij},
\end{equation}
the total brightness is given by
\begin{equation}
\label{eq7}
B_k = \sum_{i=0}^{n_{sph}} c_i A_i.
\end{equation}
This describes a linear relationship between the contribution from each spherical harmonic density distribution and the observed brightness. For the purpose of finding an unknown density distribution from observed brightness, a reconstruction space with prescribed $S_{ij}$, $f(r_{j})$ and $g_j$ is created. The line of sight summations of equation \ref{eq6} are calculated, and the problem is reduced to finding the coefficients $c_i$ - thus the line-of-sight integrations are made only once, leading to high efficiency. Given a large number of observations ($n_{obs} \gg n_{sph}$), the system is overdetermined and can be solved using least squares. The ability to perform the line of sight summations independently for each spherical harmonic is based on the assumption of a radially-structured corona at heights above the height of interest, and a uniform profile to the decrease in density with height (e.g. equation \ref{eq2}). The assumption of a radial corona is reasonable at $r=$5$ R_{\odot}$, and the approximation of an assumed radial density profile is discussed later.
\subsection{Application}
Consider a set of observed coronal images recording brightness $B_k$, taken over an extended time period (e.g. half a solar rotation, $\sim$2 weeks). Circular samples of data at constant distance from Sun center, at a height at which the coronal structure is deemed close to radial (e.g. 5$ R_{\odot}$), are extracted over the time period, giving $b$, which records $B_k$ as a function of position angle and time. For each member of $b$, a geometrical line-of-sight is defined through the corona, extending to large heights behind and in front of the point of closest approach to the Sun (similar to the description in the following section for the creation of synthetic observations). A set of $S_{ij}$, $g_j$, and $f(r)$ are prepared (with the unknown $f(r)$ set according to equation \ref{eq2}). The line-of-sight summation of equation \ref{eq6} is then implemented. This gives a set $A_{i}$, one for each spherical harmonic, each of size $n_{obs}$. Assuming a normal distribution to observational errors, the problem is reduced to solving
\begin{equation}
\label{eq8}
\min_c \; |\matr{b-Ac}|^2,
\end{equation}
with matrix $\matr{A}$ of size $n_{sph} \times n_{obs}$, $\matr{b}$ of size $n_{obs}$ and $\matr{c}$ the coefficients of size $n_{sph}$. The least-squares solution to equation \ref{eq8} is
\begin{equation}
\label{eq10}
\matr{c=(A^\intercal A)^{-1}A^\intercal b}.
\end{equation}
For numerical stability, before applying equation \ref{eq10}, $\matr A$ and $\matr b$ are divided by the mean of the absolute values of $\matr A$ (both contain very small numbers).
\section{A simple test}
\label{simple}
Synthetic observations are made from a known density distribution. For this example, a spherical distribution of density at height 5$ R_{\odot}$\ is created using equation \ref{eq1}, with $L=11$ ($n_{sph}=144$). The $c_i$ are created from a set of random numbers in the range $-1$ to 1, divided by weight $l+m+1$, so that higher-order components are reduced in amplitude. The distribution is then scaled between a minimum at a typical value for electron density in a coronal hole \citep{doyle1999}, and a maximum within a streamer \citep{gibson2003}. This distribution is shown in figure \ref{density0}a. This will be the target density distribution against which the method is tested. The distribution is simple in the sense that it is based directly on spherical harmonics - it is not similar to a true coronal density distribution, yet it serves as an initial test of the method.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{f01.jpg}
\end{center}
\caption{(a) The density distribution created using spherical harmonics of order $L=11$ with weighted random coefficients (see text) for a spherical shell at height 5$ R_{\odot}$. (b) The reconstructed density. (c) The percentage difference between target and reconstructed densities. The longitude and colatitudes are Carrington coordinates.}
\label{density0}
\end{figure}
Synthetic observations are made by specifying an uniform vector of pixels describing a circle centered on the solar disk as observed from the perspective of LASCO C2 during 2007/03/15-30. One observation per hour is synthesized throughout this period, for 360 pixels distributed at each degree around the circle (or position angle, measured counter clockwise from north). Thus $n_{obs}=\sim1.2 \times 10^5$ pixels are defined. A line of sight (LOS) is created for each pixel, with 200 points along each LOS extending to $\pm 10$$ R_{\odot}$\ from the point of closest approach to the Sun. Appropriate diverging LOS are used (extending in a narrow cone from the position of the coronagraph through the corona). Spherical Carrington coordinates are calculated for each point, and the density set by equation \ref{eq1} and the random coefficients. For this test case, $f(r)$ is not set according to equation \ref{eq2}, since we can directly use the radial description of density decrease with height in a coronal hole given by \citet{doyle1999} to fix the minimum density at each height. Similarly, the formulation of \citet{gibson2003} can be used to set the maximum density at each height. The $g_i$ are then calculated, and the resulting emission summed along each line of sight. The `observed' K-coronal brightness $b$, as a function of position angle and time, is shown in figure \ref{b0}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{f02.jpg}
\end{center}
\caption{$B_k$ values created from the line-of-sight integration of the density distribution of figure \ref{density0}a. The brightness is given for an `observational' height of 5$ R_{\odot}$, giving a synoptic-type map as a function of position angle and time.}
\label{b0}
\end{figure}
An important choice in reconstructing the density is the choice of $L$, or the maximum number of orders. For the sake of this first simple test, this is set at $L=11$, to match the order of the input distribution. Solving equation \ref{eq8} takes a few seconds on a 2.8GHz Intel Core i7 desktop computer with 16Gb memory. The reconstructed density map is shown in figure \ref{density0}b. The percentage difference between target ($\rho_t$) and reconstructed ($\rho_r$) density is shown in figure \ref{density0}c. The mean absolute percentage deviation is 3.8\%, whilst the distribution correlation $C$ over the sphere, given by
\begin{equation}
C= \frac{\sum (\rho_r-\tilde{\rho}_r)(\rho_t-\tilde{\rho}_t)}{[(\sum (\rho_r-\tilde{\rho}_r)^2)(\sum (\rho_t-\tilde{\rho}_t)^2)]^{0.5}},
\end{equation}
\noindent
is 99.8\% (the $\tilde{\rho}$ are means). Figure \ref{dens0slice} compares latitudinal slices of the observed and reconstructed density for several different longitudes. The residual, or the difference between the reconstructed and observed brightness, is close to zero as shown in figure \ref{b0slice}, which directly compares slices of the observed and reconstructed brightness as a function of position angle for several different times over the `observation' period. The mean absolute fractional deviation of the observed and reconstructed brightness is 0.5\%.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{f03.jpg}
\end{center}
\caption{Slices of the target density (solid line) and reconstructed density (dashed) as a function of latitude, for various longitudes at a height of 5$ R_{\odot}$.}
\label{dens0slice}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{f04.jpg}
\end{center}
\caption{Slices of the `observed' (crosses) and reconstructed (line) $B_k$ as a function of position angle, for various `dates' during the test period. They are almost identical.}
\label{b0slice}
\end{figure}
The algorithm is close to giving a perfect reconstruction for this simple test case. This is perhaps not surprising given that the test density is based directly on spherical harmonics, and that the information on the number of orders ($L=11$) has been used for the solution. Note that the original density distribution used to create the synthetic observations has a density decrease with height based on the formulation of \citet{doyle1999} and \citet{gibson2003}. This gives a decrease with height which is proportional to the relative density of each point, but which does not follow the spherically uniform decrease of equation \ref{eq2}. For the reconstruction, the true decrease is assumed unknown, and equation \ref{eq2} is used. It is obvious from the success of the reconstruction that this leads to only a minor error.
The Appendix describes an alternative method for finding the coefficients $\matr{c}$, based on the properties of the spherical harmonics and iteration. The alternative method performs well in the case where the target density is directly based on spherical harmonics. In general, and for the rest of this work, it is not used since its performance degrades (in both accuracy and efficiency) in comparison to the least-squares method on more complicated density distributions. It is included in the Appendix since it is an interesting approach and may prove useful in other contexts.
\section{A more realistic test}
\label{complex}
In this section, a complicated, narrowly-peaked, density distribution is used to test the reconstruction method. In contrast to the previous simple test, the density distribution is not based directly on a spherical harmonic basis, and therefore the distribution cannot be exactly fitted by a limited order of spherical harmonics, and the number of orders required in the reconstruction cannot be determined beforehand. This distribution is
\begin{equation}
\label{eq14}
\rho (\phi,\theta)= (\rho_1(\phi,\theta) + 1) \left[ \exp \left( -\frac{\rho_2(\phi,\theta)^2}{\omega} \right)+0.2 \right],
\end{equation}
where $\rho_1$, $\rho_2$ are summed spherical harmonic series with weighted random coefficients (as in the simple case of the preceding section), with $L=11$ and $M=9$, and with $\rho_1$ scaled between 0 and 1. The exponential term forms ridges centered on where the $\rho_2$ function passes through zero, and these ridges can be made narrow by setting $\omega$ to a small value. The $\rho_1$ term introduces variability to the value of both the ridges and the background. This initial density distribution is scaled to appropriate coronal values of density in a similar way to the simple case above. The resulting density distribution is shown in figure \ref{density1}a. Through the exponential function, this distribution has extended, narrow and intricate structures, and is more similar to the expected form of the true coronal density distribution, being distributed along narrow sheets along polarity inversion regions and pseudostreamers \citep[e.g.][]{morgan2010structure}. The brightness resulting from LOS integration of the density is shown in figure \ref{b1}a, again for an `observation' period of half a solar rotation towards the end of March 2007, from the perspective of LASCO C2.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f05.jpg}
\end{center}
\caption{As figure \ref{density0}, but for the complicated, narrowly-peaked density distribution of equation \ref{eq14}.}
\label{density1}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f06.jpg}
\end{center}
\caption{(a) $B_k$ values created from the line-of-sight integration of the density distribution of figure \ref{density1}a. The brightness is given for an `observational' height of 5$ R_{\odot}$, giving a synoptic-type map as a function of position angle and time. (b) The model brightness as created from the reconstructed density of figure \ref{density1}b.}
\label{b1}
\end{figure}
A high order spherical harmonic basis is required to reconstruct the target density, and for this test we set $L=25$ ($n=676$). The calculation of the LOS integrations of the $A_i$ takes around five minutes on the desktop computer, and the least-squares estimation takes also around five minutes - the calculation of the covariance matrix $\matr{A^\intercal A}$ accounts for most of this time. The reconstructed density has a mean absolute fractional deviation of 13.7\% from the target, with a structural correlation of 94.0\%. The comparison is shown in figure \ref{density1}. The reconstructed brightness, shown in figure \ref{b1}b is almost identical to the `observed', with a mean absolute fractional deviation of 1.2\%.
Despite the decent structural correlation in density distribution, and the almost identical match between model and observed brightness, the reconstruction suffers from high-frequency longitudinal oscillations, leading to large inaccuracy near the equator and regions of low density (including a small negative region). These oscillations are caused by large spherical harmonic coefficient values at high frequencies as the data is overfitted. Figure \ref{gibbs} shows the optimal density that can be achieved using a $25^{th}$ order spherical harmonic basis. The coefficients are calculated directly from integrating the product of each spherical harmonic basis with the true input density over the spherical shell by
\begin{equation}
\label{eqint}
c_i = \int_\phi \int_\theta \rho(\theta,\phi) S_i(\theta,\phi) \sin\phi \; d_\theta d_\phi .
\end{equation}
Steep jumps in density cause high-frequency oscillations (Gibbs oscillations), which can be seen in figure \ref{gibbs}b. These are minor compared to the large-amplitude errors in the least-squares tomographical reconstruction.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{f07.jpg}
\end{center}
\caption{(a) The density arising from a direct (non-tomographical) calculation of harmonic coefficients (see text). (b) A slice along the equator comparing true density (black) and spherical harmonic density (red). }
\label{gibbs}
\end{figure}
The tendency of the reconstruction to contain negative densities near high-density regions is a problem which plagues coronal tomography. This test shows that it is a problem which arises not solely due to rapid temporal changes in the streamer belt or due to contamination by coronal mass ejections (this test data has zero noise and no temporal changes). It is a problem intrinsic to the observations - of convolution of linear lines of sight through an extended spherical structure, and related to missing information at heights below the height of interest $r_0$ for any single observation. Even for tomography at heights below 5$ R_{\odot}$, this problem is unavoidable at the limit of the instrument field of view. The problem of extreme oscillations in reconstructed density is worse near the equator: for a given observation, the LOS integrations at the equator pass through only a limited range of longitude and through only a very small range of latitude. At the poles, the LOS observations pass through the whole polar corona, near to the axis of rotation, giving a more stable reconstruction.
The results of this section show that some form of regularization is required to impose smoothness on the reconstruction and to avoid negative densities.
\section{Regularisation of the higher-order harmonics}
\label{regular}
Other coronal tomography methods impose a condition on the spatial smoothness of the reconstruction \citep[e.g.][]{frazin2000} to avoid unphysical high-frequency components. A similar and necessary extension of the spherical harmonic approach is given here. It is desirable to increase the highest order of the spherical harmonics in order to reconstruct the density structure at the finest possible resolution, yet this leads to greater instability of the highest orders. Coronal tomography methods achieve stability by imposing a weighted penalty term for lack of spatial smoothness in the reconstructed density - thus the optimal reconstruction is given by a compromise between the best fit to the data and the spatial smoothness of the reconstruction (regularization).
The noise $\sigma$ at each position angle and time bin is estimated from the original pre-binned data by isolating the highest-frequency spatial and temporal component. To achieve this, a datacube is created of dimensions position angle, height, and time. The height range is a narrow strip ($\pm 0.2$$ R_{\odot}$) centered on the height of interest. The datacube is convolved with a narrow Gaussian kernel over position angle and time, and this smoothed data subtracted from the original. This leaves the high-frequency residual containing noise, rapid temporal changes, and some residual from very sharp gradients. The narrow height range serves to increase the number of pixels at each point, giving an improved estimate of noise.
Defining $\matr{A_\sigma}=[\sigma]^{-1}.\matr{A}$ and $\matr{b_\sigma}=[\sigma]^{-1}.\matr{b}$, a regularized solution weighted by the noise reciprocal is given by
\begin{equation}
\label{eq11}
\matr{c=(A_\sigma^\intercal A_\sigma + \lambda w)^{-1}A_\sigma^\intercal b_\sigma},
\end{equation}
where $\lambda$ is a regularisation factor that sets the balance between fitting the data and imposing \emph{a priori} constraints on the solution. $\matr w$ is a square matrix, with diagonal elements $i=0,1,...,n_{sph}-1$ given by
\begin{equation}
\label{eq12}
\matr w_i = \frac{l_i + |m_i|}{\sum_{i}^{n_{sph}-1} l_i + |m_i|},
\end{equation}
and non-diagonal elements are zero (the $l$ and $m$ are the spherical harmonic longitudinal and latitudinal order). $\matr w$ takes the place of the more commonly-used identity matrix so that the regularisation has a larger direct impact on higher frequency harmonics.
In previous work on regularization in coronal tomography, the commonly-used positivity constraint on the density selects values of $\lambda$ where density is everywhere zero or positive. From our own tests on this approach, this gives an overly-smooth solution - that is, for all small values of $\lambda$ the positivity constraint is not satisfied, and only at large values does the density become everywhere positive. A different approach is taken here. Our fitting routine finds an optimal solution using two parameters. One is $\lambda$ (the smoothing parameter), and the other is a minimum density threshold $\rho^\prime$. The main steps in this approach are:
\begin{enumerate}
\item Values $\lambda_k$, with index $k=0,1,...,n_k-1$ are set by a logarithmic increment between the minimum entry of the diagonal of the co-variance matrix $\matr{A_\sigma^\intercal A_\sigma}$ divided by 10, and the maximum entry multiplied by 2. Typically we set $n_k=25$.
\item A minimum density is estimated from the observed brightness values through a spherically-symmetric inversion of the $2^{nd}$ percentile minimum of brightness. Values of $\rho_{j}^\prime$, with index $j=0,1,...,n_j-1$ are set between the minimum density divided by 5, and the minimum density multiplied by 2. Typically we set $n_j=20$.
\item For each value of $\lambda_k$, an initial solution is given by equation \ref{eq11}. This solution gives an initial density distribution on a longitude-latitude map at the coronal height of interest (e.g. 5$ R_{\odot}$).
\item For each value of $\rho^\prime_j$ the initial reconstruction solution at the current $\lambda_k$ is thresholded to a minimum value of $\rho^\prime_j$. A new set of spherical harmonic coefficients are calculated directly from this thresholded density map via equation \ref{eqint}. These adjusted coefficients $\matr{c}_{k,j}$ are used to give a measure of goodness-of-fit to data for the current value of $\lambda$ and $\rho^\prime$ by:
\begin{equation}
\label{eq13}
\chi_{k,j} = \frac{1}{n_{obs}} \sum \frac{\sqrt{\left(\matr{b_\sigma-A_\sigma c}_{k,j}\right)^2}}{\matr{\sigma}}.
\end{equation}
\end{enumerate}
Thus a 2D array $\chi_{k,j}$ is gained that maps the goodness of fit as a function of $\lambda$ and $\rho^\prime$. The final task is to define an optimal point within this array. Figure \ref{regularfig} shows $\chi_{k,j}$ for the complicated density distribution, calculated over a grid of 25 $\lambda$ and 20 $\rho^\prime$ points. As expected, $\chi$ increases with increasing $\lambda$ - a smoother density reconstruction gives a poorer fit to data. $\chi$ also increases with increasing $\rho^\prime$, since the reconstructed density is thresholded to a higher minimum value, taking it further from the initial least-squares solution. There is a broad region within this array that contains the lowest values of $\chi$ and has very low gradients of $\chi$ (i.e. low variability): $\chi$ increases only slowly in this region as a function of both $\lambda$ and $\rho^\prime$. This region is identified by the 15\%\ percentile minimum value of $\chi$, shown by the white contour. Through tests using several different density distributions, addition of various noise levels (and datagaps), and tests on real data, we define the optimum point within this region as halfway between the region centroid and the point on the region boundary furthest from the origin, shown as the triangle symbol. This point defines our final solution for density. The solution, for this example, has a minimum density threshold of $\rho^\prime=10.4 \times 10^3$cm$^{-3}$, and $\lambda = 1.74 \times 10^3$ (for interpolated grid position $k=7.65$ and $k=2.98$). The true minimum density of the synthetic density distribution is $1.19 \times 10^3$cm$^{-3}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f08.jpg}
\end{center}
\caption{The goodness-of-fit to data $\chi_{k,j}$, as defined by equation \ref{eq13} as a function of the regularization parameter $\lambda$ and minimum density threshold $\rho^\prime$. The white contour shows the 15\%\ minimum percentile. The triangle symbol shows the optimal point as described in the text.}
\label{regularfig}
\end{figure}
Application of this fitting routine results in a considerable improvement in reconstructed density, as shown in figure \ref{densweights}. The high-frequency oscillations near the equator are greatly reduced. The density has a mean absolute fractional deviation of 12.3\% from the target, with a structural correlation of 95\%. The brightness values are fitted with a mean absolute deviation of 1.1\%. As inherent to the fitting method, there are no regions of negative density. The fitting routine adds around 5 minutes to the computational time: the $\matr{A^\intercal A}$ covariance matrix is pre-computed, and calculations of modeled brightness and density for equations \ref{eq13} and \ref{eq14} are efficient due to the spherical harmonic basis. Note that for this test case, there is no noise, so an arbitrary constant value of noise is set for each data point (i.e. no weighting in equation \ref{eq11}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f09.jpg}
\end{center}
\caption{The reconstructed density as gained from the regularized fitting method.}
\label{densweights}
\end{figure}
\section{Missing data, noise and rapid temporal changes}
\label{noise}
Figure \ref{bnoise}a shows the brightness test data degraded through the addition of random normally-distributed noise at 5\%\ of the mean signal level. Regularized tomography applied to this noisy dataset gives the density of figure \ref{densnoise}a. The reconstructed density has a mean absolute fractional deviation of 12.1\% from the target, with a structural correlation of 95\%. The brightness values are fitted with a mean absolute deviation of 4.3\%. The solution has a minimum density of $\rho^\prime=9.96 \times 10^3$cm$^{-3}$, and $\lambda = 1.75 \times 10^3$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f10.jpg}
\includegraphics[width=8.5cm]{./f11.jpg}
\end{center}
\caption{(a) The synthetic brightness data degraded by 5\%\ normally-distributed random noise. (b) A set of synthetic observations with three periods of missing data (rectangular black blocks) centered on 2007/03/18, 22 and 25. The first period lasts for two days, the two other periods for a day each. Noise with amplitude 5\%\ of the mean signal is also present in this data.}
\label{bnoise}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f12.jpg}
\includegraphics[width=8.5cm]{./f13.jpg}
\end{center}
\caption{(a) The reconstructed density for the input data degraded by noise. (b) As (a), but for the noisy input data including datagaps.}
\label{densnoise}
\end{figure}
The largest reconstruction errors are near the equator, where high-density regions are underestimated, and low-density regions overestimated - that is, the reconstruction gives density which is too smooth over longitude compared to the sharply-defined structures and large gradients of the true density. This is an important point to remember when interpreting tomography results applied to real data - the equatorial regions are the most important regions in the context of space weather studies, yet is where the reconstruction errors are greatest.
All coronagraphs suffer from occasional datagaps, with the potential to seriously degrade tomographical reconstructions. Figure \ref{bnoise}b shows a half-solar-rotation set of noisy synthetic observations with 4 missing days of data (around one-third missing) split into 3 gaps of 2 days, 1 day and 1 day. The reconstructed density for this data is shown in figure \ref{densnoise}b. It deviates from the target density by 14.1\%, with a spatial correlation of 94\%. The reconstructed and observed brightness deviate by 4.3\%. The solution has a minimum density of $\rho^\prime=9.9 \times 10^3$cm$^{-3}$, and $\lambda = 2.19 \times 10^3$. Thus the spherical harmonic basis provides stability in the presence of even quite substantial datagaps.
The most detrimental noise in coronagraph data is perhaps not a normal distribution, but rather isolated pixels or groups of pixels of spurious high/low values caused by, for example, sporadic bursts of energetic particles which can seriously deteriorate some images, or the passage of bright planets. The weighted fitting can help reduce the impact of these on the results. More importantly, rapid changes in brightness and structure caused by CMEs have a large detrimental effect on reconstruction. Paper \Rmnum{1}\ introduces several processing steps to reduce these problems. In particular, the dynamic separation technique (DST) reduces the effect of CMEs, and also results in a smoother signal with reduced salt-and-pepper noise. Observations which are seriously degraded (possibly due to bursts of energetic particles), can be identified and discarded, as described in Paper \Rmnum{1}. Occasionally, telemetry or read errors can lead to missing blocks of data within an image. Discarding bad images, or missing data blocks, will result in short datagaps, which seems acceptable for the spherical harmonic method as shown above.
Lastly, coronal structure must change, either slowly, or rapidly, and may reconfigure very rapidly during the passage of large CMEs. Time-dependent coronal tomography (based on regularisation methods) has been successfully applied by \citet{vibert2016}. In principle, the spherical harmonic approach can be extended to include time-dependency, with the coefficients becoming functions of time. Initial experiments with a time-dependent density model shows that this is a very challenging task - particularly if a step-change in density is needed to account for rapid changes. Further development is necessary, reserved for a future publication.
\section{Application to observations}
\label{observations}
This section applies the tomography to observations made by the LASCO C2 and the STEREO SECCHI COR2 A coronagraphs for a half-Carrington rotation period centered on 2009/03/20 12:00. At this time, the STEREO A spacecraft is separated by $60^\circ$ from SOHO. The data are processed and calibrated according to the method of Paper \Rmnum{1}. The height of interest is set at 5.5$ R_{\odot}$, and the data rebinned into a position-angle and time array with 180 position angle bins, 200 time steps. The data array for LASCO C2 is shown in figure \ref{realdata}a, and for COR2 A in figure \ref{realdata}c. The data binning can be set at higher resolution, at the expense of computational time. The binning here allows reconstructions to be made in approximately 10 minutes.
The choice of period, and height, is to allow convenient comparison with figure 5 of \citet{frazin2010}. The density reconstruction for LASCO C2 is shown in figure \ref{realdens}a, and for COR2 A in figure \ref{realdens}b. The LASCO C2 data is fitted with a mean absolute deviation of 10.6\%, with a smoothing parameter of $\lambda=6.2 \times 10^4$ and minimum density $\rho_{min}=1.4 \times 10^3$cm$^{-3}$. For COR2 A the values are 7.0\%, $\lambda=5.1 \times 10^4$ and $\rho_{min}=6.5 \times 10^3$cm$^{-3}$. The mean absolute fractional difference between the two reconstructed densities is 38\%, with a spatial correlation of 81\%. Comparing with figure 5 of \citet{frazin2010}, these density maps are smoother, and have maximum densities at around half the values of \citet{frazin2010}. Currently there is no other empirical verification for density maps such as these. From figure \ref{realdens}, COR2 A seems to give a better reconstruction, in that the streamer belt is narrow, and is fitting the data more closely. Comparison with future \emph{in situ} measurements of the coronal density by the Parker Solar Probe will be invaluable for coronal tomography.
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f14.jpg}
\includegraphics[width=8.5cm]{./f15.jpg}
\end{center}
\caption{(a) The brightness of the corona observed at 5.5$ R_{\odot}$\ by LASCO C2 for a two-week period centered on 2009/03/20. (b) Model brightness gained from reconstructed density for LASCO C2. (c) As (a), but observed by the COR2 A instrument. (d) Model brightness for the COR2 A reconstructed density.}
\label{realdata}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.5cm]{./f16.jpg}
\includegraphics[width=8.5cm]{./f17.jpg}
\end{center}
\caption{(a) Reconstructed density at a height of 5.5$ R_{\odot}$\ gained from the LASCO C2 observations shown in figure \ref{realdata}a. (b) As (a), but for the COR2 A observations shown in figure \ref{realdata}c.}
\label{realdens}
\end{figure}
\section{Conclusions and future work}
\label{conclusions}
For heights where the coronal structure can be well-approximated as radial with an uniform density decrease with increasing height (i.e. the extended inner corona), a model of the density based on spherical harmonics leads to a very efficient and stable method for reconstruction. This is demonstrated for a simple and complex model coronal density distribution. The method is robust to large datagaps of several days. Without regularisation, the smoothness of the reconstructed density is dictated by the highest order of the spherical harmonic basis. However, the true coronal density is likely to have steep gradients between regions of low and high density, or very narrow regions of high density, and a high order is required to approximate these. To counteract this problem, we provide a method for regularised solutions where the smoothness of the reconstructed density, and a minimum density threshold, is taken into consideration.
The application of this method to a large dataset will be presented in the third paper of this series. Other future work involves finding a robust time-dependent extension to the spherical harmonic approach, where the harmonic coefficients can change as a function of time. We also aim to experiment with other approaches similar to spherical harmonics that have proved useful in geophysics, including wavelet-based spherical functions \citep{chambodut2005}. We anticipate these may prove useful for the non-radial corona, in particular for extreme ultraviolet (EUV) observations of the low corona.
Spherical harmonics are a simple yet powerful basis for inversion of coronal density, and should be a consideration for other coronal applications such as EUV diagnostics in the low corona, or 3D reconstructions of the coronal magnetic field with future spectropolarimetric instruments.
\hskip 20pt
|
2,869,038,155,709 | arxiv | \section{Introduction}
It is the characteristic feature of soft matter systems that a macromolecular
component of nano- to micrometer size is dispersed in a solvent of much
smaller molecules. The mesoscopic length scale of the dispersed
component implies that crystalline phases have a very small shear modulus --
which roughly scales
like the inverse of the third power of the structural length scale --
and that both crystalline and fluid phases are characterized by long
structural relaxation times. Soft matter systems have
therefore interesting dynamical properties, because the time scale of
an external perturbation can easily become comparable with the
intrinsic relaxation time of the dispersed macromolecules.
One of the unique properties of soft matter is its viscoelastic
behavior\cite{lars99b}. Due to the long structural relaxation time,
the internal
degrees of freedom cannot relax sufficiently fast in an oscillatory
shear flow, so that there is some elastic restoring force which
pushes the system back to its previous state. A very well studied
example of viscoelastic fluids are polymer solutions and polymer
melts\cite{ferr80b,doi86,lars99b}. In the case of polymer melts,
the characteristic time scale
is given by the reptation time, {\em i.e.} by the time
it takes a chain to slide by its contour length along the tube formed
by other polymer chains\cite{doi86}.
In order to bridge the length- and time-scale gap between the solvent
and macromolecular or colloidal scales, several mesoscopic simulation
techniques -- such as the lattice-Boltzmann method, dissipative-particle
dynamics (DPD), and multi-particle collision dynamics (MPC) -- have
been suggested in recent
years, and are in the process of being developed further. The idea of
all these methods is to strongly simplify the microscopic dynamics in
order to gain computational efficiency, but at the same time to exactly
satisfy the conservation laws of mass, momentum and energy, so that
hydrodynamic behavior emerges naturally on larger length scales.
We will focus here on the multi-particle collision dynamics (MPC)
technique\cite{male99,male00a,lamura2001}, also called stochastic rotation
dynamics\cite{ihle01} (SRD), originally developed for Newtonian fluids.
This particle-based hydrodynamics method
consists of alternating streaming and collision steps. In the streaming
step, point particles move ballistically. In the collision step, particles are
sorted into the cells of a simple cubic (or square) lattice. All particles
in a cell collide by a rotation of their velocities relative to the
center-of-mass velocity around a random axis\cite{male99}. A random
shift of the cell lattice is performed before each collision step in
order to restore Galilean invariance\cite{ihle01}.
This method has been applied very successfully to study the hydrodynamic
behavior of many complex fluids, such as polymer
solutions in equilibrium\cite{gg:gomp05h,lee06} and
flow\cite{webster2005,ryde06,gg:gomp06d},
colloidal dispersions\cite{padd04,hech05},
vesicle suspensions\cite{gg:gomp04h,gg:gomp05c},
and reactive fluids\cite{tucc04,eche07}.
The viscoelastic behavior of polymer solutions leads to many unusual
flow phenomena, such as shear-induced phase
separation\cite{helf89,onuk89,miln93},
viscoelastic phase separation\cite{tana00b},
and elastic turbulence\cite{groi00}.
A coarse-grained description of viscoelastic fluids is necessary
in order to obtain a detailed understanding of the role of elastic forces
in such flow instabilities.
However, there is a second level of complexity in soft matter system,
in which a colloidal component is dispersed in a solvent, which is
itself a complex fluid. Examples are spherical or rod-like colloids
dispersed in polymer solutions or melts, which are exposed to a shear
flow\cite{lyon01,suen02,hwan04,scir04,verm05}. Shear flow can induce
particle aggregation and alignment in these systems. This is important,
for example, in the processing of nanocomposites\cite{verm05}.
The aim of this paper is therefore the development of a MPC algorithm,
which is able to describe viscoelastic phenomena, but at the same time
retains the computational simplicity of standard MPC for Newtonian
fluids, and thereby allows to take advantage of this mesoscale simulation
for the investigation of flow instabilities as well as suspensions with
viscoelastic solvents. We show that this goal can be achieved by
replacing the point particles of standard MPC by harmonic dumbbells.
In order to obtain a strong elastic contribution, we consider a fluid,
which consists of dumbbells only. However, it is of course straightforward
to mix dumbbells with a point-particle solvent. A similar idea has been
suggested recently for DPD fluids\cite{somf06}.
\section{The Model}
\label{sec:model}
\subsection{Algorithm}
In our MPC model, we consider $N_\mathrm{p}$ point particles of mass $\mathrm{m}$,
which are pairwise connected by a harmonic potential
$\mathbf{V}(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{1}{2} \mathrm{K} (\mathbf{r}_{1} - \mathbf{r}_{2} )^2$
to form dumbbells, where $\mathrm{K}$ is the spring constant.
The center-of-mass position $\mathbf{r}^{\mathrm{c}}_{i}$ and velocity
$\mathbf{v}^{\mathrm{c}}_{i}$
for each dumbbell $i$, with $i=1,2,...,N_\mathrm{p}/2$, are represented by
\begin{equation}
\mathbf{r}^{\mathrm{c}}_{i} = \frac{1}{2} (\mathbf{r}_{i1} + \mathbf{r}_{i2} ) \;; \; \; \;
\mathbf{v}^{\mathrm{c}}_{i} = \frac{1}{2} (\mathbf{v}_{i1} + \mathbf{v}_{i2} ) \;.
\end{equation}
Here $\mathbf{r}_{i1}$, $\mathbf{r}_{i2}$ and $\mathbf{v}_{i1}$,
$\mathbf{v}_{i2}$ denote the position and velocity of the two point
particles composing a dumbbell $i$, respectively.
The MPC algorithm consists of two steps, streaming and
collisions\cite{male99,male00a,malevanets2004}.
In the streaming step, within a time interval $h$,
the motion of all dumbbells is governed by Newton's equations of motion,
\begin{equation}
\mathrm{m}^\mathrm{c} \frac{d\mathbf{v}^{\mathrm{c}}_{i}}{dt} = \mathbf{f}^{\mathrm{c}}_{i} \;; \; \;
\frac{d\mathbf{r}^{\mathrm{c}}_{i}}{dt} = \mathbf{v}^{\mathrm{c}}_{i} \;,
\end{equation}
where $\mathrm{m}^\mathrm{c}=2\mathrm{m}$ is the mass of a dumbbell,
and $\mathbf{f}^{\mathrm{c}}_{i}$ is the total external force on dumbbell $i$.
We consider only constant force fields.
The center-of-mass positions and velocities of dumbbells are then given
by a simple ballistic motion.
The evolution of the relative coordinates of each dumbbell
are determined by the harmonic interaction potential, so that
\begin{eqnarray}
\label{rr}
\mathbf{r}_{i1}(t+h) - \mathbf{r}_{i2} (t+h)
& = & \mathbf{A}_{i}(t) \cos(\omega_0 h) \nonumber \\ && + \mathbf{B}_{i}(t) \sin(\omega_0 h)\;; \\
\label{rv}
\mathbf{v}_{i1}(t+h) - \mathbf{v}_{i2} (t+h)
& = & - \omega_0 \mathbf{A}_{i}(t) \sin(\omega_0 h) \nonumber \\
& & + \omega_0 \mathbf{B}_{i}(t) \cos(\omega_0 h) \;,
\end{eqnarray}
with angular frequency $\omega_0 = \sqrt{2\mathrm{K} / \mathrm{m}}$.
The vectors $\mathbf{A}_{i}$ and $\mathbf{B}_{i}$ are
different for each time step, and are
calculated from the relative positions and velocities
of the point particles of dumbbell $i$ before the streaming step,
\begin{equation}
\mathbf{A}_{i}(t) = \mathbf{r}_{i1}(t) - \mathbf{r}_{i2} (t)\;; \; \;
\mathbf{B}_{i}(t) = \frac{1}{\omega_0}\left(\mathbf{v}_{i1}(t) - \mathbf{v}_{i2} (t)\right) \;.
\end{equation}
In the MPC algorithm described here, $\mathbf{r}^{\mathrm{c}}$,
$\mathbf{v}^{\mathrm{c}}$, $\mathbf{A}$ and $\mathbf{B}$ are continuous variables,
evolving in discrete increments of time.
In the absence of shear flow, the average length of the dumbbell is
$\mathrm{r_0}^{(d)} \equiv \sqrt{\langle \mathrm{r}^2 \rangle_{\rm eq}}=
\sqrt{d~k_{\mathrm{B}}T / \mathrm{K}}$ for a $d$-dimensional system.
In the collision step, the point particles are sorted into the cells of
a cubic lattice with lattice constant $a_0$.
Multi-particle collisions are performed for all particles in a cell $J$,
by the same SRD algorithm\cite{male99} as for point particle fluids. The
velocity of each particle relative
to the center-of-mass velocity $\mathbf{v}_{\mathrm{cm},J}$ of the cell
is rotated around a randomly chosen axis by a fixed angle $\alpha$,
\begin{equation}
\label{vcm}
\mathbf{v}'_j(t+h) = \mathbf{v}_{\mathrm{cm},J} + \mbox{$\hat{\mathcal{R}}$}(\alpha) \left[\mathbf{v}_j(t+h)-
\mathbf{v}_{\mathrm{cm},J} \right]\;,
\end{equation}
where $\mbox{$\hat{\mathcal{R}}$}(\alpha)$ is a stochastic rotation matrix, and
\begin{equation}
\label{vcm2}
\mathbf{v}_{\mathrm{cm},J} = \sum^{N_J}_{j=1}\mathbf{v}_j/N_J\;,
\end{equation}
with $N_J$ the number of particles within cell $J$.
This step guarantees that each particle changes the direction as well as
the magnitude of its velocity during the multi-particle collisions, while
the local momentum and the kinetic energy are conserved.
Random shifts are applied in each direction, so that the Galilean
invariance is ensured
even in case of small mean free path\cite{ihle01,ihle2003}.
In order to describe Couette or oscillatory shear flow, the system is
confined within two parallel hard walls in the $y$ direction,
which are moving oppositely along the $x$ direction.
Here, $L_x$, $L_y$ and $L_z$ are used to denote the dimension of the
simulation box along the corresponding directions.
For a steady shear flow, the shear rate is given by $\dot{\gamma} =2 v_{\mathrm{wall},x} / L_y$,
with $v_{\mathrm{wall},x}$ the $x$ component of
the velocity of the wall moving along the positive direction.
Periodic boundary conditions are applied in the $x$ and $z$ direction,
bounce-back boundary condition in the $y$ direction.
The system is therefore divided into $L_x/a_0$ and $L_z/a_0$ cells
in the $x$ and $z$ directions (parallel to the walls),
but $L_y/a_0 +1$ cells in the $y$ direction because of the random shifts.
At the walls, for collision cells which are not completely filled by particles,
extra virtual point particles are added to conserve the monomer number
density, $\rho$, defined by the average number of monomers per
cell\cite{lamura2001}.
In principle, the velocities of the virtual particles can be drawn from
a Maxwell-Boltzmann distribution of average velocity equal to the wall
velocity and variance $\sqrt{k_{\mathrm{B}}T / \mathrm{m}}$, where $k_{\mathrm{B}}T$ the bulk
temperature. In the simulation code, it is not necessary to sample the
velocity of virtual wall particles individually.
A random vector from Maxwell-Boltzmann distribution with wall velocity
and variance $\sqrt{(\rho - n)k_{\mathrm{B}}T / \mathrm{m}}$ is then used instead
of the contribution of the entire virtual particles in the cell,
where $n$ is the number of real particles in that cell.
For point particles, the combination of bounce-back boundary condition and
virtual wall particles has been shown to guarantee no-slip boundary
condition to a very good approximation\cite{lamura2001}.
\subsection{Thermostats}
In order to keep the system temperature constant, various thermostats
can be employed. In the first case, the MPC method with collisions by
stochastic rotations (MPC-SRD)
of relative velocities is augmented by velocity rescaling. The simulation
box is subdivided into $L_y/a_0$ layers parallel to the walls.
In each layer, the new velocity $\mathbf{v}'_j$ of each particle
$j$ in cell $J$ is obtained by rescaling the velocity relative to the
center-of-mass velocity of that cell,
\begin{equation}
\label{th1}
\mathbf{v}'_j = \mathbf{v}_{\mathrm{cm},J} +
( \mathbf{v}_j - \mathbf{v}_{\mathrm{cm},J} ) \sqrt{\frac{k_{\mathrm{B}}T}{k_{\mathrm{B}}T'}}\;.
\end{equation}
Here $k_{\mathrm{B}}T'$ is calculated from the actual velocity distribution
\begin{equation}
\sum_{J\in {\rm layer}} \sum^{N_J}_{j=1} \frac{1}{2} \mathrm{m} ( \mathbf{v}_j - \mathbf{v}_{\mathrm{cm},J} )^2 =
( \sum_{J\in {\rm layer}} N_J - \tilde{N}_{\rm layer} ) k_{\mathrm{B}}T' \;,
\end{equation}
where $N_J$ denotes the number of particles in cell $J$ and
$\tilde{N}_{\rm layer}$ the number of cells which contains particles
within a layer.
In the second case, the Anderson's thermostat version of MPC, denoted MPC-AT,
is applied\cite{noguchi2007,gg:gomp02c}.
This thermostat employs a different collision rule instead of Eq.~(\ref{vcm}).
In the MPC-AT$-a$ version of the algorithm (without angular momentum
conservation, compare Sec.~\ref{sec:angular} below),
the new velocities of point particles in the collision step are assigned
as\cite{noguchi2007}
\begin{equation}
\mathbf{v}'_j = \mathbf{v}_{\mathrm{cm},J} + \mathbf{v}^{\mathrm{ran}}_j
- \sum^{N_K}_{k=1}\frac{\mathbf{v}^{\mathrm{ran}}_k}{N_K} \;.
\end{equation}
Here $\mathbf{v}^{\mathrm{ran}}_j$ is a velocity chosen from the
Maxwell-Boltzmann distribution and $N_K$ the number of particles within
cell $\mathrm{K}$. Instead of energy conservation in MPC, the temperature is kept
constant in MPC-AT.
\subsection{Angular Momentum Conservation}
\label{sec:angular}
The standard MPC algorithm as well as the Anderson thermostat version
do not conserve angular momentum.
It has been shown recently\cite{gg:gomp07h} that this lack of
angular-momentum conservation may lead to quantitative or even
qualitative incorrect results, like non-physical torques in circular
Couette flows.
We therefore also consider the angular-momentum conserving modification of
MPC-AT\cite{noguchi2007,gg:gomp07h}, denoted MPC-AT$+a$.
Here, the velocities in the collision step are calculated by
\begin{eqnarray}
\mathbf{v}'_j & = & \mathbf{v}_{\mathrm{cm},J} + \mathbf{v}^{\mathrm{ran}}_j
- \sum^{N_K}_{k=1}\frac{\mathbf{v}^{\mathrm{ran}}_k}{N_K} \\
& + & \left\{ \mathrm{m}~\mathbf{\Pi}^{-1}~ \sum^{N_K}_{k=1}
\left(\mathbf{r}_k - \mathbf{r}_{\mathrm{cm},K}\right)
\times \left( \mathbf{v}_k - \mathbf{v}^{\mathrm{ran}}_k \right) \right\}
\nonumber \\
& \times & \left(\mathbf{r}_j - \mathbf{r}_{\mathrm{cm},J}\right) \;, \nonumber
\end{eqnarray}
where $\mathbf{\Pi}$ and $\mathbf{r}_{\mathrm{cm},J}$ denote the
moment-of-inertia
tensor and the center of mass of particles in the cell, respectively.
\subsection{Wall Potential}
In the absence of shear flow, the monomer density profile $\rho(y)$ can be
calculated from the interaction potentials $V$ of the dumbbells,
\begin{eqnarray}
\label{wall}
\rho(y) & = & \frac{1}{Z} \int^{L_y}_{0} d y'~
\mathrm{e}^{-\frac{1}{2} \frac{\mathrm{K}}{k_{\mathrm{B}}T} (y-y')^2} \;,\\
\label{wall0}
\rho(y) / \rho_\mathrm{b} & = & \frac{1}{2}
\left[ \mathrm{erf}\left(\sqrt{\mathrm{K} /2 k_{\mathrm{B}}T}~y\right) \right. \nonumber \\
& + & \left. \mathrm{erf}\left(\sqrt{\mathrm{K} /2 k_{\mathrm{B}}T}~(L_y-y)\right)\right] \;,
\end{eqnarray}
where $\rho_\mathrm{b}$ is the bulk monomer density, $Z$ the partition function,
and \textit{erf} the error function.
\Fig{den} shows excellent agreement of the theoretical prediction
\eq{wall0} with simulation data.
The particles are not equally distributed along the wall direction;
instead, at both walls, the density is only half of the bulk density.
In order to reduce possible slip effects,
it seems desirable to make the particle distribution as uniform as possible.
An attractive potential is therefore applied
when the center-of-mass position of the dumbbells approaches one of the walls,
\begin{eqnarray}
\label{wall1}
V_{\mathrm{wall}}(y_{i1},y_{i2}) & = & -2c_2~k_{\mathrm{B}}T~
\left(1 - \frac{y_{i1}+y_{i2}}{2c_1\mathrm{r}_0^{(1)}}~\right) \nonumber \\
&& \mathrm{for}~~\frac{y_{i1}+y_{i2}}{2} \leq c_1\mathrm{r}_0^{(1)} \;; \nonumber \\
V_{\mathrm{wall}}(y_{i1},y_{i2}) & = & -2c_2~k_{\mathrm{B}}T~
\left(1 - \frac{2L_y - y_{i1} - y_{i2}}{2c_1\mathrm{r}_0^{(1)}}~\right) \nonumber \\
&& \mathrm{for}~~\frac{y_{i1}+y_{i2}}{2} \geq L_y- c_1\mathrm{r}_0^{(1)} \;,
\end{eqnarray}
where $\mathrm{r}_0^{(1)}=\sqrt{k_{\mathrm{B}}T / \mathrm{K}}$ is the one-dimensional average
extension of a dumbbell. The density profile is now given by
\begin{equation}
\label{wall2}
\rho(y) = \frac{1}{Z} \int^{L_y}_{0} d y'~
\mathrm{e}^{-\frac{1}{2} \frac{\mathrm{K}}{k_{\mathrm{B}}T} (y-y')^2}
\mathrm{e}^{- V_{\mathrm{wall}}(y,y') } \;.
\end{equation}
The advantages of the piecewise linear form \eq{wall1} of the wall
potential are twofold. Firstly and most importantly, it allows for
an analytical integration of the equations of motion during the
streaming step. Secondly, the density profile in the absence of
flow can again be calculated analytically (see Appendix for details).
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP1.eps}
\end{center}
\caption{
\label{den}
(Color online)
Monomer density profiles with (squares) and without (circles)
attractive wall potentials applied along the wall direction
when particles approach close to walls.
The dashed and dotted lines are the theoretical prediction
described in \Eq{wall} and \Eq{wall2}, respectively.
The spring constant of dumbbells and the collision time are
$\mathrm{K}=0.2$ and $h=0.02$, respectively.
Both simulations are with absence of shear flow.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP2.eps}
\end{center}
\caption{
\label{rho}
(Color online)
Monomer density profiles at various dimensionless shear rates
$\dot{\gamma} / \omega_0$, ranging from $\dot{\gamma} / \omega_0=0.0$ to $0.447$,
when attractive wall potentials are applied.
The spring constant of dumbbells and the collision time are
as same as those in \Fig{den}. The small ripples in the profile at
large $\dot\gamma/\omega_0$ are due to inhomogeneities in the
temperature profile, since velocity rescaling is not sufficiently
efficient at high shear rates. The ripples to not appear for MPC-AT.
}
\end{figure}
The simulated density profile shows excellent agreement with
the analytical solution of \Eqs{wall1} and \eq{wall2} (see Appendix).
The factors $c_1$ and $c_2$ are chosen to obtain a nearly uniform density
distribution. This is achieved for $c_1=1.3$ and $c_2=0.4$.
As shown in \Fig{den}, the densities of point particles at both wall
boundaries deviate by less than $10\%$ lower from the bulk value, when
the attractive wall potentials are applied.
Simulations are also performed on systems of dumbbells with various
spring constants, ranging from $\mathrm{K} a_0^2/k_{\mathrm{B}}T = 0.1$ to
$\mathrm{K} a_0^2/k_{\mathrm{B}}T = 5.0$, in the absence of shear flow.
It is found that for the given values of $c_1$ and $c_2$, the density
profiles are essentially independent of the spring constant of the
dumbbell in the range $0.1 < \mathrm{K} a_0^2/k_{\mathrm{B}}T < 1.0$.
In \Fig{rho}, we plot density profiles in shear flow.
At lower shear rates, {\em i.e.~} $\dot{\gamma} /\omega_0 \leq 0.1$, nearly identical
profiles are obtained as without flow.
For higher shear rates, deviations of the density profile from
the equilibrium profile become significant.
Nevertheless, these profiles are still more uniform than those without an
attractive wall potential.
Our investigations are mainly
focusing on relatively low shear rates, where
the non-uniformity of the density profile is not significant.
\subsection{Stress Tensor and Shear Viscosity}
\label{sec:stress}
In the MPC model, the viscosity $\eta$ consists of a
kinetic and collisional contribution\cite{kikuchi2003,tuez03}.
At steady shear rates,
with flow along the $x$ direction and gradient along the $y$ direction,
$\eta$ is calculated by measuring the $xy$ component of the stress tensor,
$\sigma_{xy} = \sigma^{\mathrm{kin}}_{xy} + \sigma^{\mathrm{col}}_{xy}$,
so that $\eta=\sigma_{xy} / \dot{\gamma}$.
In the streaming step, $\sigma^{\mathrm{kin}}_{xy}$ is proportional to
the flux of the $x$ momentum crossing a plane normal to the $y$ direction.
Since the stress tensor is independent of the position of the plane,
we choose $y=0$ or $y=L_y$ to measure the momentum transfer.
In two-dimensional simulations,
\begin{equation}
\label{stresskin}
\sigma^{\mathrm{kin}}_{xy} = \frac{\mathrm{m}}{L_x h}
\sum^{N_1}_{i=1} \left[ v'_{x,i}(t_w) - v_{x,i}(t_w) \right]\;,
\end{equation}
where $t_w \in [t,t+h]$ is the time at which particle $i$ bounces back
from the wall, $v_{x,i}(t_w)$ and $v'_{x,i}(t_w)$ are the velocities just
before and after the collision with the wall, and $N_1$ denotes the
number of particles which hit one of the walls in the time interval
$[t,t+h]$. In the collision step,
particles close to the wall will change their velocities
due to the multi-particle collisions with virtual wall particles
with average velocity $v_x=\pm \frac{1}{2}\dot{\gamma} L_y$,
so that
\begin{equation}
\label{stresscol}
\sigma^{\mathrm{col}}_{xy} = \frac{\mathrm{m}}{L_x h}
\sum^{N_2}_{i=1} \left[ v'_{x,i}(t+h) - v_{x,i}(t+h) \right]\;.
\end{equation}
Here $N_2$ denotes the number of particles
which have multi-particle collisions with virtual particles,
while $v_{x,i}(t+h)$ and $v'_{x,i}(t+h)$ are the velocities of particle $i$
before and after the collision step, respectively.
In our simulations, $N_2$ is found to be much larger than $N_1$
for small collision times $h$,
indicating that the collisional part dominates the shear viscosity.
Simulations are first performed on a system of pure point-like fluid
particles to verify the measurement of the zero-shear viscosity
from \Eqs{stresskin} and \eq{stresscol}. We get perfect agreement with
the theoretical predictions\cite{kikuchi2003,tuez03,ripoll2005} for $\eta$.
The shear viscosity can also be measured from system
under Poiseuille flow\cite{lamura2001,ripoll2006} by
\begin{equation}
\label{Poiseuille}
\eta = \frac{\rho g L_y^2}{8v_\mathrm{max}}\;,
\end{equation}
where $g$ is the gravitation field, and $v_\mathrm{max}$ the maximum
flow velocity.
\subsection{Storage and Loss Moduli}
In an oscillatory shear flow, the shear rate $\dot{\gamma}(t)$ is time-dependent,
\begin{equation}
\label{dgt}
\dot{\gamma}(t) = \gamma_0~\omega~\cos{(\omega t)}\;,
\end{equation}
where $\gamma_0$ and $\omega$ are
the strain amplitude and the oscillation frequency, respectively.
Note that the frequency $\omega$ in \Eq{dgt} is independent of
the angular frequency $\omega_0$ of harmonic dumbbells in \Sec{sec:model}.
In our simulations, we choose $\gamma_0 \ll 1$ in order to investigate
the linear viscoelastic regime.
The stress tensor is divided into two contributions,
the viscous part $\sigma'$ and the elastic part $\sigma''$, so
that\cite{lars99b,macosko_book}
\begin{eqnarray}
\label{ss}
\sigma_{xy}(t) & = & \sigma' \sin{(\omega t)} + \sigma'' \cos{(\omega t)} \nonumber \\
& = & \gamma_0 \left[G'(\omega) \sin{(\omega t)} + G''(\omega) \cos{(\omega t)}\right]\;,
\end{eqnarray}
where $G'$ is the storage modulus, which measures the in-phase storage
of the elastic energy, and $G''$ is the loss modulus, which measures
the out-of-phase energy dissipation.
For a simple Maxwell fluid, $G'$ and $G''$ are given by\cite{macosko_book}
\begin{eqnarray}
\label{mg1}
G' & = & G^*
\frac{(\omega/\omega^*)^2}{1+(\omega/\omega^*)^2} \;; \\
\label{mg2}
G'' & = & G^*
\frac{\omega/\omega^*}{1+(\omega/\omega^*)^2} \; \;,
\end{eqnarray}
where $\omega^*$ is a characteristic relaxation frequency, and $G^*$
is a characteristic shear modulus.
In the limit of $\omega \ll \omega^*$, the loss modulus is
$G''=\eta~\omega$, where $\eta$ is the zero-shear viscosity.
\subsection{Kinetic Theory of Dumbbells in Solution}
In order to estimate the rheological properties of our model fluid, we
modify the kinetic theory for dilute solutions of elastic
dumbbells \cite{bird87}.
For Hookean dumbbells in a solvent, the viscosity $\eta_0$, the storage
modulus $G_0^\prime$ and the loss modulus $G_0^{\prime\prime}$ are given
by\cite{bird87}
\begin{equation}
\eta_0 = \eta_s + \frac{\rho}{2} \, \frac{k_BT}{\omega_s},
\label{visc_theo0}
\end{equation}
\begin{equation}
G_0^\prime=\frac{\rho k_BT}{2} \, \frac{(\omega / \omega_s)^2}{1+(\omega / \omega_s)^2},
\end{equation}
\begin{equation}
G_0^{\prime\prime}=\eta_s\omega+\frac{\rho k_BT}{2} \,
\frac{\omega / \omega_s}{1+(\omega / \omega_s)^2},
\end{equation}
where
\begin{equation}
\omega_s=\frac{4\mathrm{K}}{\zeta_s}
\end{equation}
with solvent viscosity $\eta_s$ and friction coefficient $\zeta_s$ of a
monomer.
Moreover, the expectation value for the square of the monomer separation,
divided by its equilibrium value, is given by\cite{bird87}
\begin{equation}
\frac{\langle r^2 \rangle}{\langle r^2 \rangle_{\rm eq}}=1+\frac{2}{3}(\dot{\gamma} / \omega_s)^2.
\end{equation}
In Ref.~\onlinecite{bird87}, the friction coefficient is obtained from
Stokes' law for a bead of radius $r$ in the solvent, {\em i.e.~}
$\zeta_s=6 \pi \eta_s r$.
However, in the MPC dumbbell fluid, there exists no explicit solvent and
the monomers are point particles instead of spheres.
Nevertheless, the motion of the monomers is governed by the friction
caused by the surrounding monomers which can be considered to take the
role of the solvent.
Using $\zeta=k_BT/D$, which follows from the Stokes-Einstein relation,
we can thus relate the friction to the diffusion constant $D$ of a
MPC fluid of point particles with the same monomer density.
Similarly, we substitute the viscosity of the solvent, $\eta_s$, by the
corresponding viscosity $\eta_{\rm MPC}$ of a MPC fluid of point
particles.
Theoretical expression for $\eta_{\rm MPC}$ and $D$ for the different
collision methods can be found in
Refs.~\onlinecite{kikuchi2003,tuez03,gg:gomp07h,gg:gomp07xxe} and
Refs.~\onlinecite{ripoll2005,tuez06,gg:gomp07xxe}, respectively.
The zero-shear viscosity then reads
\begin{equation}
\eta = \eta_{\rm MPC} + \frac{\rho}{2} \, \frac{k_BT}{\omega_H},
\label{eq:visc_theo_MPC}
\end{equation}
where we have introduced
\begin{equation}
\label{eq:omega_H}
\omega_H=\frac{4\mathrm{K}}{\zeta}=\frac{4D\mathrm{K}}{k_BT}.
\end{equation}
Note that the limit $\mathrm{K} \rightarrow \infty$ corresponds to a MPC fluid of
$N_p/2$ point particles of mass $\mathrm{m}^\mathrm{c}$. Here, the second term in
Eq.~(\ref{eq:visc_theo_MPC}) vanishes, and since
$\eta_{\rm MPC}(\rho/2,2\mathrm{m}) \approx \eta_{\rm MPC}(\rho,\mathrm{m})$ for not too
small $\rho$ and sufficiently small $h$ (so that the collisional part of
the viscosity dominates),
the viscosity resulting from this simple theory approaches the correct
value in this limit.
Consequently, we use the same substitutions for the storage and loss
modulus, and for the average dumbbell extension, and obtain
\begin{equation}
G^\prime=\frac{\rho k_BT}{2} \, \frac{(\omega / \omega_H)^2}{1+(\omega / \omega_H)^2}
\end{equation}
\begin{equation}
G^{\prime\prime}=\eta_{\rm MPC}\omega
+ \frac{\rho k_BT}{2} \, \frac{\omega / \omega_H}{1+(\omega / \omega_H)^2},
\end{equation}
and
\begin{equation}
\frac{\langle r^2 \rangle}{\langle r^2 \rangle_{\rm eq}}=1+\frac{2}{3}(\dot{\gamma} / \omega_H)^2.
\label{eq:R2}
\end{equation}
We emphasize that the above expressions only serve as a semi-quantitative
description of the MPC dumbbell fluid. For example, the employed expressions
for the diffusion constant neglect hydrodynamic interactions, which
become important for small time steps $h$.
\section{Results}
\label{sec:results}
\subsection{Dimensionless Variables and Parameters}
In the remainder of this article, we introduce dimensionless
quantities by measuring
length in unit of the lattice constant $a_0$,
mass in unit of the dumbbell mass $\mathrm{m}^\mathrm{c}$,
time in units of $a_0\sqrt{\mathrm{m}^\mathrm{c}/k_{\mathrm{B}}T}$,
velocity in units of $\sqrt{k_{\mathrm{B}}T/\mathrm{m}^\mathrm{c}}$,
monomer number density $\rho$ in units of $a_0^{-d}$, where $d$ is the
spatial dimension, and the spring constant $\mathrm{K}$ in units of $k_{\mathrm{B}}T/a^2_0$.
The shear rate $\dot{\gamma}$ and all kinds of frequencies
are measured in units of $\sqrt{k_{\mathrm{B}}T/\mathrm{m}^\mathrm{c} a^2_0}$.
Finally the viscosity $\eta$ is in units of $\sqrt{\mathrm{m}^\mathrm{c} k_{\mathrm{B}}T/a^2_0}$,
and the storage modulus $G'$ and the loss modulus $G''$ are in units of
$k_{\mathrm{B}}T/a^3_0$.
In these dimensionless units, the mean free path $\lambda$ (in units
of the lattice constant) becomes equivalent to the time step $h$.
In our simulations, harmonic dumbbells with $N_\mathrm{p}$ point particles
are initially placed in a two- or three-dimensional rectangular box at random.
We choose the average number density of point particles $\rho=20$
and $L_x=50$ for all two-dimensional simulations
which results in $N_\mathrm{p}=1000L_y$.
The collision time ranges from $h=0.01$ to $h=0.2$,
while the spring constant ranges from $\mathrm{K}=0.1$ to $\mathrm{K}=5.0$.
The rotational angle is chosen $\alpha=90^\mathrm{o}$
and $\alpha=130^\mathrm{o}$ for two- or three-dimensional simulations,
respectively.
We use small $h$ and large $\alpha$ to obtain large Schmidt numbers required
for fluid-like behavior\cite{ripoll2004,ripoll2005}.
Most of the results shown are obtained from two-dimensional systems,
except in a few cases where this is explicitly mentioned.
In Tab.~\ref{tab:omegaH}, the theoretical values for the diffusion
constant $D$ are given for $h=0.1$ for the different collision methods
and various monomer densities \cite{ripoll2005,gg:gomp07xxe}.
The corresponding results for other time steps $h$ can be obtained
by employing the linear relationship between $D$ and $h$.
\begin{table}[h]
\begin{tabular}[t]{c|c|c|c}
$\rho$ \ & $D^{\rm (SRD)}$ & $D^{{\rm (AT}-a{\rm)}}$ & \ $D^{{\rm (AT}+a{\rm)}}$ \\
\hline
10 \ \ & \ \ 0.1222 \ \ & \ \ 0.1222 \ \ & \ \ 0.1353\\
20 \ \ & \ \ 0.1105 \ \ & \ \ 0.1105 \ \ & \ \ 0.1162\\
40 \ \ & \ \ 0.1051 \ \ & \ \ 0.1051 \ \ & \ \ 0.1078\\
\end{tabular}
\caption{Diffusion constants $D$ of point-particle fluids
for the standard MPC-SRD algorithm, as well as for
MPC-AT$-a$ and MPC-AT$+a$ simulations for various monomer densities,
in two dimensions. All data are calculated for collision time $h=0.1$.
Diffusion constants for other time steps $h$ can be obtained
by employing the linear relationship between $D$ and $h$.
Note that the values for MPC-AT$-a$ are identical
with those for MPC-SRD with collision angle $\alpha=90^\mathrm{o}$.
}
\label{tab:omegaH}
\end{table}
\subsection{Steady Shear Flow}
\label{sec:steadyflow}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm,clip]{029809JCP3.eps}
\end{center}
\caption{
\label{snap}
(Color online)
Snapshots of dumbbell configurations in steady shear flow.
The system size is $L_x=L_y=50$.
Half of each dumbbell is colored red, the other half yellow for reason
of visualization. In each frame only 2500 dumbbells are shown,
so that the density is $10$ times as high as appears from the pictures.
The spring constant and the collision time are $\mathrm{K}=0.2$ and
$h=0.02$, respectively.
From (A) to (D), the applied shear rates are
$\dot{\gamma} / \omega_H=0.0565$, $0.565$, $2.83$ and $5.65$.
}
\end{figure}
In \Fig{snap}, we present snapshots for steady shear flow
with a simulation box containing 25000 dumbbells.
At lower shear rates, {\em i.e.~} $\dot{\gamma} / \omega_H \leq 0.6$,
see \Fig{snap}A and \ref{snap}B, the average extension of the dumbbells
is hardly distinguishable from the equilibrium value.
In these two cases, the shear flow is not strong enough
to align the dumbbells along the flow direction,
so that both systems are still isotropic.
With increasing $\dot{\gamma}$, shear forces overwhelm entropic forces.
As a result, dumbbells are largely stretched,
at the same time reorientated along the flow direction,
as presented in \Fig{snap}C and \ref{snap}D.
Note that near both the walls, the average size
$\langle \mathrm{r^2} \rangle ^{1/2}$
of the dumbbells in flow is larger than in the bulk.
Also, an alignment of the dumbbells is found near
the walls, both with and without shear flow, with peaks at $y=0$ and $y=L_y$.
This is an effect of the geometrical constraints imposed on anisotropic
particles by a hard wall.
Furthermore, a maximum of the extension occurs at a {\em finite} distance
from the wall, which we attribute to the combined effect of the wall
and the flow conditions; dumbbells very close to the wall
are sterically oriented parallel to the wall and thus experience only
a very small shear force, while those a little further away are close to
the average inclination angle (see \Fig{e2e} below), which corresponds
to the largest stretching. The distance of the position of the maximum
from the wall decreases with increasing shear rate, and seems to
approach the size of the collision cells for large
$\dot\gamma$.
The relative peak height increases with increasing shear rate.
For example, we find that the maximum extension
$\langle \mathrm{r^2} \rangle ^{1/2}$ near the wall is about $11\%$ larger
than the bulk extension for $\dot{\gamma}/\omega_H=1.13$, while it is about
$28\%$ larger than in the bulk for $\dot{\gamma}/\omega_H=2.83$.
\begin{figure}[ht]
\label{e2e}
\begin{center}
\includegraphics*[width=7cm,clip]{029809JCP4.eps}\\
\end{center}
\caption{
(Color online)
Distribution of dumbbell configurations for the system shown
in Fig.~\ref{snap}. Each dot indicates the end-to-end vector
of a dumbbell.
}
\end{figure}
\Fig{e2e} presents the extensional and orientational distribution of the
dumbbells for various shear rates.
At lower shear rates, $\dot{\gamma} / \omega_H \leq 1$, the end-to-end vector
of the dumbbells is distributed on a circle, see \Fig{e2e}A and B,
indicating an isotropic orientation.
At a higher shear rate, $\dot{\gamma} / \omega_H = 2.83$,
the orientational distribution becomes an elongated ellipse, see \Fig{e2e}C.
With increasing shear rate, the distribution elongates further.
Simultaneously, dumbbells become more aligned with the flow direction,
as can be seen quantitatively from the inclination angle $\theta$ shown in
\Fig{angle}. Here, the inclination angle is defined as the angle between
the average orientation of the end-to-end vector of a dumbbell and the
the flow direction.
At lower shear rates, $\dot{\gamma} / \omega_H \leq 1$, the inclination angle
approaches $\theta=45^\mathrm{o}$,
while it decays to zero for large shear rates with a
power law $\dot\gamma^{-1}$.
\begin{figure}[ht]
\begin{center}
\includegraphics*[clip]{029809JCP5.eps}
\end{center}
\caption{
\label{angle}
(Color online)
The inclination angle $\theta$ as a function of dimensionless shear rate
$\dot{\gamma} / \omega_H$ for the system of \Fig{snap}, with spring constant
$\mathrm{K}=0.2$, collision time $h=0.02$, and system size $L_x=L_y=50$.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics*[clip]{029809JCP6.eps}
\end{center}
\caption{
\label{eta-r-gamma-y}
(Color online)
Shear viscosity $\eta$ and scaled average dumbbell length
$\langle \mathrm{r}^2 \rangle^{1/2} / \mathrm{r}_0^{(2)}$
as a function of dimensionless shear rate $\dot{\gamma} / \omega_H$.
Systems with the wall separation
$L_y=10,~20,~30$, and $50$ are investigated.
The spring constant is $\mathrm{K}=0.2$ and the collision time $h=0.02$.
}
\end{figure}
In \Fig{eta-r-gamma-y}, we plot the shear viscosity $\eta$
as a function of dimensionless shear rate $\dot{\gamma} / \omega_H$ for various
wall separations $L_y$ ranging from $10$ to $50$.
In each system, $\eta$ remains constant until the applied shear rate
reaches a critical value $\dot{\gamma}_c / \omega_H \approx 5$.
The shear viscosity then decays rapidly as $\dot{\gamma}$ further increases,
showing a typical ``shear-thinning'' behavior.
\Fig{eta-r-gamma-y} also shows the average extension of dumbbells
$\langle \mathrm{r}^2 \rangle^{1/2} / \mathrm{r}_0 ^{(2)}$
as a function of the shear rate.
Two comments are required here. Firstly, in our MPC model, an entanglement
between dumbbells is not taken into account, so that they can
freely cross each other.
Also, the absence of an excluded-volume interaction implies that there
is no benefit of a para-nematic ordering in terms of an increased
sliding of parallel dumbbells along each other as in solutions of
rod-like colloids; instead,
parallel dumbbells interact very similarly to isotropically
oriented dumbbells, since in both cases the monomers colloide
with other monomers in exactly the same fashion. Thus, our system
is very similar to a solution of non-interacting harmonic dumbbells,
for which -- in the absence of a finite extensibility --
neither ``shear-thinning" nor ``shear-thickening"
is expected\cite{bird87,doi86}, compare Eq.~(\ref{visc_theo0}).
Secondly, the size of the simulation box should have no influence
on the bulk viscosity at a given shear rate.
However, the plateau value of the viscosity
increases strongly with the wall separation $L_y$.
This indicates that boundary effects could be responsible for the observed
``shear-thinning'' behavior.
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP7.eps}
\end{center}
\caption{
\label{vp-y}
(Color online)
Velocity profiles for wall separations
$L_y=10$, $20$, $30$, and $50$. Data are obtained for
spring constant $\mathrm{K}=0.2$, collision time $h=0.02$, and shear
rate $\dot{\gamma} / \omega_H=0.565$.
The dashed line corresponds to the applied shear rate.
Solid lines represent fits to the bulk part of the velocity profiles,
their slopes yield the effective shear rates $\dot{\gamma}_{\mathrm{eff}}$.
}
\end{figure}
We therefore examine the velocity profiles for systems with various wall
separations $L_y$. In \Fig{vp-y}, the average velocities $v_x$ of the
monomers along the
flow direction are plotted as function of $L_y$ for a fixed shear rate
of $\dot{\gamma} / \omega_H = 0.565$. The velocities at the boundaries deviate
only very little from the wall velocities, {\em i.e.~} there is very little
slip at the walls, as expected.
However, the velocity decays rapidly in a boundary layer of thickness $\Delta$,
then decays linearly to zero at the middle plane.
Obviously the applied shear rate $\dot{\gamma}$ is not appropriate to calculate
the shear viscosities
from the stress tensor $\sigma_{xy}$ by $\eta = \sigma_{xy} / \dot{\gamma}$.
An effective shear rate $\dot{\gamma}_{\mathrm{eff}}$ is therefore introduced instead,
which characterizes the linear bulk part of the velocity profile.
At a given shear rate, the larger the wall separation,
the less the effective shear rate deviates from $\dot{\gamma}$,
since the finite-size effect is much stronger in smaller systems.
The ratio $\dot{\gamma} / \dot{\gamma}_{\mathrm{eff}}$ between the applied
and the effective shear rates is plotted in \Fig{gamma-y},
as a function of $\dot{\gamma} /\omega_H$.
At lower shear rates, {\em i.e.~} $\dot{\gamma} < \dot{\gamma}_\mathrm{c}$,
where $\dot{\gamma}_\mathrm{c}$ is the critical shear rate,
the ratio $\dot{\gamma} / \dot{\gamma}_{\mathrm{eff}}$ is independent of the shear rate.
When the applied shear rate becomes larger than this critical value,
the effective shear rate $\dot{\gamma}_{\mathrm{eff}}$ increases more slowly
than $\dot{\gamma}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP8.eps}
\end{center}
\caption{
\label{gamma-y}
(Color online)
Ratios between the applied shear rates $\dot{\gamma}$ and the effective
shear rates $\dot{\gamma}_{\mathrm{eff}}$ as a function of $\dot{\gamma}/\omega_H$
for the same systems as in \Fig{eta-r-gamma-y}.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP9.eps}
\end{center}
\caption{
\label{eta-eff-y}
(Color online)
Master curve of the viscosity $\eta_\mathrm{eff}$ as a function of the
effective shear rate $\dot{\gamma}_\mathrm{eff}$ on a semi-logarithmic scale.
Symbols are the same as in \Fig{eta-r-gamma-y} and \Fig{gamma-y}.
For strong shear flow, {\em i.e.~} $\dot{\gamma}_\mathrm{eff} / \omega_H \gtrsim 2$,
the viscosity increases (``shear thickening'').
The dashed line is fitted to the data with wall separations $L_y \geq 20$.
In the inset, $\eta_\mathrm{eff}$ is plotted as a function of
$\dot{\gamma}_\mathrm{eff} / \omega_H$ for three-dimensional systems.
A $20 \times 20 \times 10$ simulation box is chosen,
while the spring constant $\mathrm{K}$, the collision time $h$
and the average number density $\rho$ are
the same as in the two-dimensional systems.
}
\end{figure}
Consequently, the effective shear viscosity can be calculated by
\begin{equation}
\label{etaeff}
\eta_{\mathrm{eff}}=\frac{\sigma_{xy}}{\dot{\gamma}_\mathrm{eff}} \;.
\end{equation}
In \Fig{eta-eff-y}, $\eta_{\mathrm{eff}}$ is shown against
$\dot{\gamma}_{\mathrm{eff}} / \omega_H$ for various wall separations $L_y$.
The data for different system sizes now all fall onto a single
master curve, which describes the bulk shear viscosity.
Now, instead of ``shear thinning'' shown in \Fig{eta-r-gamma-y}, a
very weak ``shear thickening'' behavior is observed when
$\dot{\gamma}_{\mathrm{eff}} / \omega_H > 1$.
Three-dimensional simulations are also carried out for systems of
$20 \times 20 \times 10$ boxes along the $x$, $y$, and $z$ directions.
For the same parameters $\mathrm{K}=0.2$ and $h=0.02$,
weak ``shear thickening'' behavior is also observed, as shown in the inset
of \Fig{eta-eff-y}, when $\dot{\gamma}_\mathrm{eff} / \omega_H$ reaches the critical
value, $\dot{\gamma}_{\mathrm{c},\mathrm{eff}} / \omega_H \approx 2$.
The value of the critical
shear rates are found to be very similar in two and three dimensions.
\Fig{eta-eff-y} shows that the effective shear viscosity
$\eta_\mathrm{eff}$ is nearly independent of the shear rate for
$\dot{\gamma}_\mathrm{eff} /\omega_H \leq \dot{\gamma}_\mathrm{c,eff} /\omega_H \approx 2$.
This critical shear rate corresponds to the onset of the apparent
``shear thinning'' observed in \Fig{eta-r-gamma-y}, as well as the
deviation of
$\dot{\gamma} / \dot{\gamma}_\mathrm{eff}$ from its low-shear-rate value in \Fig{gamma-y}.
It should be noticed that the value of
$\dot{\gamma}_{\mathrm{c},\mathrm{eff}} / \omega_H \approx 2$
implies $\dot{\gamma}_{\mathrm{c}} / \omega_H$ is in the range $[3,6.4]$ for
system sizes $L_y \in [10,50]$, compare \Fig{gamma-y}.
However, it is important to note that there is already a pronounced
alignment and stretching of the dumbbells for smaller shear rates;
\Fig{angle} shows that the inclination angle $\theta$ has decreased from
$\theta=45^\mathrm{o}$ in the absence of shear flow to
$\theta \approx 15^\mathrm{o}$ at $\dot{\gamma} /\omega_H=3$, while
\Fig{eta-r-gamma-y} indicates that
$\langle \mathrm{r}^2 \rangle^{1/2} / \mathrm{r}_0 ^{(2)} \approx 2 $ at
$\dot{\gamma} /\omega_H=3$.
The spring constant $\mathrm{K}$ of the dumbbells is of great importance,
since it controls the elasticity of the fluid. We have therefore
examined velocity profiles of systems of dumbbells with various
spring constants. In \Fig{vp-k}, the simulation results are plotted for
a fixed applied shear rate $\dot{\gamma}=0.01$.
The effect of the boundary layer becomes more pronounced with decreasing
spring constant. By fitting the linear parts of the velocity profiles,
we find that, for the same shear rate $\dot{\gamma}=0.01$,
the effective shear rate $\dot{\gamma}_\mathrm{eff}$ for dumbbells with $\mathrm{K}=0.1$
is about 10 times lower than that with the highest spring constant studied
here, $\mathrm{K}=4.0$. The thickness of the boundary layer is
proportional to the equilibrium average extension
$\mathrm{r}_0^{(2)} = \sqrt{2k_{\mathrm{B}}T/\mathrm{K}}$, as shown in the inset of \Fig{vp-k}.
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP10.eps}
\end{center}
\caption{
\label{vp-k}
(Color online)
Velocity profiles for various spring constants,
ranging from $\mathrm{K}=0.1$ to $\mathrm{K}=4.0$.
The wall separation in each case is $L_y=10$.
The dashed line corresponds to the applied shear rate $\dot{\gamma}=0.01$,
while solid lines are the fitted effective velocity profiles.
The inset shows the thickness of the boundary layer $\Delta$ as a
function of equilibrium extension
$\mathrm{r}_0^{(2)} = \sqrt{2k_{\mathrm{B}}T/\mathrm{K}}$. The dashed line is a
linear fit.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP11.eps}
\end{center}
\caption{
\label{3d-k}
(Color online)
The zero-shear viscosity $\eta_\mathrm{eff}$
as a function of spring constant $\mathrm{K}$
in both two-dimensional (circles) and three-dimensional
(squares) systems.
The solid lines are linear fits, the dashed lines indicate
the theoretical predictions (\ref{eq:visc_theo_MPC}).
In all simulations, the collision time is $h=0.02$.
Two-dimensional simulations are performed in systems of
$50 \times 10$ boxes, while three-dimensional simulations are in
$30 \times 30 \times 20$ boxes
along the $x$, $y$ and $z$ directions, respectively.
The average number density in three-dimensional systems is $\rho=10$,
which is half of value in two-dimensional systems.
}
\end{figure}
The zero-shear viscosity $\eta_\mathrm{eff}$ is found to depend linearly
on $1/\mathrm{K}$, see \Fig{3d-k}. As $\mathrm{K}$ increases,
the effective viscosity $\eta_\mathrm{eff}$ approaches the expected
value of system of point particles with mass of $\mathrm{m}^{\mathrm{c}}$ and
density $\rho/2$.
The same linear relationship between $\eta_\mathrm{eff}$ and $1/\mathrm{K}$ is
also obtained in three-dimensional systems, as shown in \Fig{3d-k}.
Not only the linear dependence of $\eta_\mathrm{eff}$ on $1/\mathrm{K}$ but
also the prefactors are in very good agreement with the theoretical
predictions (\ref{eq:visc_theo_MPC}).
The scaled mean free path $\lambda$,
which determines how far a point particle travels between collisions,
is another important parameter which affects the shear viscosity.
We always employ small mean free paths\cite{ripoll2004,ripoll2005},
so that the collisional viscosity is dominant compared to the kinetic
viscosity. The data of \Fig{dt-y}(a) demonstrate that the zero-shear
viscosity increases linearly with $1/\lambda$,
for all spring constants $\mathrm{K}$ studied here,
as it does for a system of point particles\cite{kikuchi2003,tuez03,ripoll2005}.
However, the slope decreases with increasing $\mathrm{K}$, in good agreement
with the analytical results obtained from Eq.~(\ref{eq:visc_theo_MPC}),
as shown in the inset of \Fig{dt-y}(a).
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP12.eps}
\end{center}
\caption{
\label{dt-y}
(Color online)
The effective shear viscosity $\eta_\mathrm{eff}$ as a function of
the dimensionless effective shear rate $\dot{\gamma}_\mathrm{eff} / \omega_0$,
on a double-logarithmic scale.
(a) For fixed spring constant $\mathrm{K}=0.2$ and various mean free paths
$\lambda=0.01$, $0.02$, $0.05$, $0.1$ and $0.2$.
(b) For fixed mean free path $\lambda=0.02$ and various spring constants
$\mathrm{K}=0.2$, $0.5$, $1.0$ and $5.0$.
In both cases, the wall separation is $L_y=10$.
The inset in (a) shows the zero-shear viscosity $\eta_\mathrm{eff}$
as a function of $1/\lambda$.
The solid line indicates the theoretical result for
$\mathrm{K}\rightarrow\infty$, while the other lines show the predictions
(\ref{eq:visc_theo_MPC}) for $\mathrm{K}=0.2$, $0.5$, and $5.0$.
}
\end{figure}
The weak ``shear-thickening'' behavior is observed for all mean free
paths investigated here, see \Fig{dt-y}. Thus, this weak
``shear-thickening'' behavior is intrinsic to the MPC algorithm,
and cannot be avoided by a variation of the collision time.
\Fig{dt-y}(a) indicates that the critical shear rate
$\dot{\gamma}_\mathrm{c,eff} /\omega_0$ depends only very weakly on the
mean free path $\lambda$. Therefore, we present the simulation
data in \Fig{dt-y} as a function of $\dot{\gamma}_\mathrm{eff} /\omega_0$,
since $\omega_0=(2\mathrm{K}/m)^{1/2}$ is independent of $\lambda$, while
$\omega_H$ decreases linearly with $\lambda$.
The ``shear-thickening'' behavior becomes more pronounced and
slowly shifts to smaller values of $\dot{\gamma}_\mathrm{eff} /\omega_0$
for system of dumbbells with smaller spring constants, see \Fig{dt-y}(b).
In the range of investigated spring constants and mean free paths,
the shear thickening occurs roughly at
$\dot{\gamma}_\mathrm{c,eff} /\omega_0 \simeq 0.1$.
It is important to note that
the viscosity of the standard point-particle MPC fluid is also not
independent of the shear rate, but shows a weak {\em shear-thinning} behavior
at high shear rates \cite{kikuchi2003}. For our model parameters and
in two dimensions, this shear-thinning behavior sets in at a shear
rate $\dot{\gamma}_\mathrm{c} \simeq 1$. Thus, with increasing $\mathrm{K}$, shear
thickening occurs at a slowly increasing
$\dot{\gamma}_\mathrm{c,eff} /\omega_0$ for $\mathrm{K}\le 1$; for larger spring
constants $\mathrm{K} \ge 5$, shear thinning is observed instead, and
$\dot{\gamma}_\mathrm{c,eff} /\omega_0$ decreases again (since
$\dot{\gamma}_\mathrm{c,eff}\to 1$ and $\omega_0 \to \infty$ for $\mathrm{K}\to \infty$).
\subsection{Small-amplitude Oscillatory Shear Flow}
Another way to explore the viscoelastic properties of a fluid is
to apply a small-amplitude oscillatory shear flow.
We use here the strain amplitudes $\gamma_0 = \dot{\gamma} / \omega$ in the range $0.1$ to $0.5$ to mimic a
small amplitude shearing.
The frequencies $ \omega$ ranges from $10^{-4}$ to $10^{-1}$ in our simulations, which
provides a wide range of shear rate from $10^{-5}$ to $5 \times 10^{-2}$.
The storage and loss moduli as a function of oscillation frequency are plotted in \Fig{y-g}.
Similarly to the simulations of steady shear flow,
effective shear rates are measured from
the bulk velocity profiles at times when $\cos(\omega t)=\pm1$.
By doing so, all the simulation data fall onto master curves
at various wall separations from $L_y=10$ to $50$.
As can been seen from \Fig{y-g}(a), the storage modulus $G'$ is well
fitted by \Eq{mg1}, indicating that the dumbbell system exhibits a
typical behavior of a Maxwell fluid.
The relaxation frequency $\omega^*$ obtained from the fit of the storage
modulus $G'$ against $\omega$ is then used in \Eq{mg2} to fit
the loss modulus $G''$.
In \Fig{y-g}(b), at low frequencies, $\omega \leq 0.02$,
the simulation data follow the expected linear $\omega$-dependence very well.
In this linear regime, the shear viscosity is then calculated by
$\eta = G''(\omega) / \omega$, which yields $\eta = 565$,
in excellent agreement with the result in steady shear flow,
compare \Fig{eta-eff-y}.
Note that the fitted values for the amplitude $G^*$ in \Eqs{mg1} and \eq{mg2}
differ by about a factor 2.
This indicates that the system investigated here does not behave exactly like a simple Maxwell fluid.
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP13.eps}
\end{center}
\caption{
\label{y-g}
(Color online)
(a) Storage $G'$ and (b) loss modulus $G''$,
as a function of oscillation
frequency $\omega$ on a double-logarithmic scale, for systems with
various wall separations ranging from $L_y=10$ to $50$.
The spring constant and the collision time are $\mathrm{K}=0.2$ and
$h=0.02$, respectively.
The dashed line in (a) is fitted by the Maxwell model, \Eq{mg1}, on
the basis of all simulation data,
while the one shown in (b) is based on data
for oscillation frequencies $\omega < 0.02$.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP14.eps}
\end{center}
\caption{
\label{k-g}
(Color online)
(a) The storage $G'$ and (b) the loss moduli $G''$,
as function of oscillation
frequency $\omega$ on a double-logarithmic scale, for systems of
dumbbells with various spring constants ranging from
$\mathrm{K}=0.2$ to $\mathrm{K}=1.0$.
The wall separation and the collision time are $L_y=10$ and $h=0.02$,
respectively. The inset shows the fitted relaxation frequencies
$\omega^*$ as a function of the frequency $\omega_H$ predicted by
Eq.~(\ref{eq:omega_H}). The dashed line shows the identity
$\omega^*=\omega_H$.
}
\end{figure}
In \Fig{k-g}, we examine the storage and loss moduli of system of dumbbells
with various spring constants.
As in \Fig{y-g}, simulation results are all well fitted by
the Maxwell equations \eq{mg1} and \eq{mg2}, except for somewhat different
amplitudes $G^*$. The relaxation frequency $\omega^*$ is found
to agree very well with $\omega_H$, as shown in the inset of \Fig{k-g}.
At lower oscillation frequency in \Fig{k-g}(b), the viscosities calculated
from $G''(\omega)/ \omega$ are $\eta=565$, $253$ and $144$
for systems with $\mathrm{K}=0.2$, $0.5$ and $1.0$, respectively.
These values are again in excellent agreement with those calculated from
\Eq{etaeff} in steady shear flow. For all spring constants $\mathrm{K}$,
the fitted amplitudes $G^*$ for the storage moduli $G'$ are about half of
those calculated for the loss moduli $G''$.
This indicates that even for a system of dumbbell with high spring constant,
a simple Maxwell model is not appropriate for a quantitative description.
\subsection{Angular Momentum Conservation}
The viscosity of a simple MPC-AT$+a$ fluid (with angular-momentum
conservation) is about a factor
$1/2$ smaller than of a MPC-AT$-a$ fluid \cite{gg:gomp07h,gg:gomp07xxe}.
We thus expect the viscosity of the dumbbell fluid to be affected
by angular-momentum conservation as well.
The simulation results for both MPC-AT$-a$ and MPC-AT$+a$ methods are
compared in Fig.~\ref{fig:eta_rho}. We find that the effective zero-shear
viscosity $\eta_{\rm eff}$ increases linearly with the monomer density
$\rho$ for $\rho \gtrsim 5$.
The corresponding theoretical results (\ref{eq:visc_theo_MPC}) are in
good agreement with the simulation results for both investigated spring
constants.
Minor deviations from the linear relationship of $\eta_{\rm eff}$ with
$\rho$ originate from the variation of the diffusion constant at low
densities, which approaches a constant value for high $\rho$.
The viscosity of MPC-AT$+a$ is lower than for MPC-AT$-a$, although
this effect is less pronounced than for pure point-particle MPC fluids,
since the main contribution to the viscosity originates from the spring
tension.
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP15.eps}
\end{center}
\caption{
\label{fig:eta_rho}
(Color online)
Effective shear viscosities $\eta_{\rm eff}$ as a function of the
density $\rho$ for MPC-AT$-a$ and MPC-AT$+a$, each for
spring constants $\mathrm{K}=0.2$ and $\mathrm{K}=0.4$.
The lines represent the theoretical results obtained from
Eq.~(\ref{eq:visc_theo_MPC}).
The wall separation and the collision time are $L_y=20$ and
$h=0.014$, respectively.
}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[clip]{029809JCP16.eps}
\end{center}
\caption{
\label{fig:R2}
(Color online)
Scaled average of the dumbbell extension,
$\langle r^2 \rangle / \langle r^2 \rangle_{\rm eq} -1$,
as a function of the effective
shear rate $\dot{\gamma}_\mathrm{eff}/\omega_H$ for angular-momentum conserving
and non-conserving methods.
The spring constant and collision time are $\mathrm{K}=0.2$ and
time step $h=0.014$, respectively, the density is $\rho=10$ and
the wall separation is $L_y=20$.
The dashed line represents the theoretical result (\ref{eq:R2}).
}
\end{figure}
In Fig.~\ref{fig:R2}, we present the average squared dumbbell extension,
determined in the bulk as a function of the effective shear rate, along
with the theoretical results (\ref{eq:R2}), for both the MPC-AT$-a$ and
MPC-AT$+a$ methods.
Note that the diffusion constant $D$ in Eq.~(\ref{eq:R2}) is different
for angular-momentum conserving and non-conserving methods.
The angular-momentum conservation has only little effect on the spring
extension; for a given effective shear rate, the extension is slightly
lower for the angular-momentum conserving method.
The agreement of the simulation data with the theoretical
result (\ref{eq:R2}) is again remarkably good.
\section{Discussion}
As a further test for the correct calculation of the effective viscosity
by the procedure described in Secs.~\ref{sec:stress} and
\ref{sec:steadyflow}, we have also determined the
viscosity from Poiseuille flow. As in Ref.~\onlinecite{gg:gomp02c}, we apply
a gravitational force of strength $g$ parallel to the walls, with $g$ in
the range from $g=0.0001$ to $g=0.01$ (in units of $k_{\mathrm{B}}T/a_0$).
We fit the central part of the
velocity profile to a parabolic flow curve. The value of this curve at
the wall positions determines the effective wall slip. When this
slip velocity is subtracted, Eq.~(\ref{Poiseuille}) in
Sec.~\ref{sec:stress} is employed to determine the viscosity\cite{zhang2007}.
We have used this method for a system of dumbbells with $\mathrm{K}=0.2$
in a $30 \times 30$ box. Excellent agreement between the two methods
to calculate the zero-shear viscosity is obtained.
Our results for the dependence of the inclination angle $\theta$ on the
shear rate can be compared with the decay
of the inclination angle of flexible and semi-flexible
polymers. For dilute polymer solutions in the asymptotic regime of
high shear rates (where the finite extensibility is important), $\theta$
has been predicted from Brownian dynamics simulations \cite{schr05} and
theory \cite{wink06} to decay with a power law $\dot\gamma^{-0.3}$
and $\dot\gamma^{-1/3}$, respectively. For extensible dumbbells, the theory of
Ref.~\onlinecite{wink06} predicts\cite{wink07} $\theta\sim\dot\gamma^{-1}$, in
excellent agreement with our simulation results.
The wall slip in polymer melts has been studied extensively. In this case,
molecular dynamics simulations of polymer fluids with Lennard-Jones interactions
between monomers give a wall slip with a boundary layer thickness, which is on
the order of the monomer diameter $\sigma$ or less \cite{prie04,zhang2007}. Our
model could be compared more easily with results for polymer solutions,
because our model does not include excluded-volume interactions. However,
there is little knowledge about semi-dilute polymer solutions near a wall
under flow conditions. Nevertheless, some comparisons with polymer melts with
moderate chain lengths are possible, where entanglement effects are absent.
For example, the molecular dynamics simulations of Zhang et al.\cite{zhang2007}
show a maximum of mean squared radius of gyration at a finite distance
$\Delta_m$ from the wall, which shifts from $\Delta_m \simeq 1.5 \sigma$
for chains with $4$ monomers to $\Delta_m \simeq 2.2 \sigma$ for
$10$ monomers.
\section{Summary and Conclusions}
A multi-particle collision dynamics (MPC) algorithm has been developed
to investigate the viscoelastic properties of harmonic-dumbbell fluid
in shear flow. The method is based on alternating streaming and
collision steps, just as the original MPC method for Newtonian fluids.
The only modification is to replace the ballistic motion of fluid
point particles by harmonic oscillations during the streaming step.
In this model, the entanglement between dumbbells is neglected.
Moreover, the storage and loss moduli are calculated
by introducing a small amplitude oscillatory shear flow.
Our results can be summarized as follows:
First,
under steady shear flow, the dumbbells keep their isotropic distribution
at low shear rates, but get highly stretched and orientated along the
flow direction at high shear rates.
The velocity profile is not uniform along the gradient direction, but
boundary layers of high shear develop near the walls. The thickness of
these boundary layers is found to scale with the size of the
dumbbells in the absence of flow.
The effective shear viscosity, calculated from the ratio
between the off-diagonal component of the stress tenor, $\sigma_{xy}$,
and the effective shear rate $\dot{\gamma}_{\mathrm{eff}}$,
expresses a very weak ``shear thickening'' behavior at high shear rates.
Second,
the dependence of the viscosity on two parameters, the spring constant
$\mathrm{K}$ of the dumbbells and the collision time $h$, has been investigated.
These two parameters are of central importance, since the former
controls the elastic energy of the system, while the latter determines
the mean free path $\lambda$, which measures the fraction a the cell size
that a fluid particle travels on average between collisions.
We find that the shear viscosity of the dumbbell fluid increases
linearly with $1/\mathrm{K}$ and with $h$.
Third,
the storage and loss moduli of our viscoelastic solvent are studied
by imposing an oscillatory velocity on the two solid walls.
The storage modulus $G'$ is found to be proportional to $\omega^2$
at low frequencies, and to level off at $\omega^*=\omega_H$.
Its behavior over the whole frequency range studied here is well
described by a Maxwell fluid.
The loss modulus $G''$ increases linearly with $\omega$ for low
frequencies. The shear viscosities obtained from the ratio
$G''/\omega$ at low shear rates agree very well with
those obtained from simulations with steady shear.
On the other hand, for $\omega > \omega_H$, we find that the data
approach a plateau value, while for a Maxwell fluid $G''$ would
decrease again for higher frequencies.
Our numerical results are quantitatively in good agreement with a
simple theory,
based on the kinetic theory of dilute solutions of dumbbells, where
the transport coefficients of the standard MPC point-particle fluid
are employed for the viscosity and the diffusion constant of the
solvent.
In our MPC algorithm of harmonic dumbbells, both elastic and viscous
behaviors of solvent particles can be modeled properly,
while hydrodynamic interactions are efficiently taken into account.
These are valuable assets to guide future simulations
on investigating rheological properties of suspensions of
spherical, rod-like or polymeric solute molecules in viscoelastic fluids.
\section*{Acknowledgment}
We thank R.G.~Winkler, M.~Ripoll and H.~Noguchi for
many stimulating and helpful discussions. Partial support of this work
by the DFG through the Sonderforschungsbereich TR6 ``Physics of
Colloidal Dispersion in External Fields'' is gratefully acknowledged.
\section*{Appendix: Analytical solution of the density profile
with attractive wall potentials\label{App}}
Combining Equations \eq{wall1} and \eq{wall2}, the density profile,
when attractive wall potentials are introduced, can be solved analytically.
Considering the symmetry of the density profile, $\rho(y)=\rho(L_y-y)$,
only the initial half part need to be taken into account.
For $0 < y < 2c_1\mathrm{r}_0^{(1)}$,
we then arrive at
\begin{widetext}
\begin{eqnarray}
\rho(y) & = & \frac{1}{Z} \left\{ \int^{L_y}_{0} d y'~
\mathrm{e}^{- \mathrm{K} (y-y')^2 / 2k_{\mathrm{B}}T} \right.
\nonumber \\
& + &
\left. \int^{2c_1\mathrm{r}_0^{(1)}-y}_{0} d y'~
\mathrm{e}^{-\- \mathrm{K} (y-y')^2 / 2k_{\mathrm{B}}T} ~
\left[~\mathrm{e}^{ 2c_2 k_{\mathrm{B}}T (1 - (y+y')/2c_1\mathrm{r}_0^{(1)} ~)} -1~\right] \right\} \;.
\nonumber
\end{eqnarray}
which implies
\begin{eqnarray}
\rho(y) & = & \frac{1}{Z} \left\{ \mathrm{erf}\left(\sqrt{\mathrm{K} /2k_{\mathrm{B}}T}~(L_y-y)\right)
+ \mathrm{erf}\left(\sqrt{\mathrm{K} /2 k_{\mathrm{B}}T}~(2y-2c_1\mathrm{r}_0^{(1)})\right) \right.
\nonumber \\
&+& \left. \exp\left[\frac{(c_2k_{\mathrm{B}}T\mathrm{r}_0^{(1)})^2 +
4c_2k_{\mathrm{B}}T(c_1\mathrm{r}_0^{(1)}-y)}{2c_1\mathrm{r}_0^{(1)}}\right]
\left[\mathrm{erf}\left( \frac{-c_2k_{\mathrm{B}}T + c_1\mathrm{r}_0^{(1)}y \mathrm{K} /k_{\mathrm{B}}T}
{c_1\mathrm{r}_0^{(1)} \sqrt{2\mathrm{K} / k_{\mathrm{B}}T}} \right) \right. \right.
\nonumber \\
&+& \left. \left. \mathrm{erf}\left(\frac{ c_2k_{\mathrm{B}}T + 2(c_1\mathrm{r}_0^{(1)}-y) c_1\mathrm{r}_0^{(1)} \mathrm{K}/ k_{\mathrm{B}}T}
{c_1\mathrm{r}_0^{(1)} \sqrt{2\mathrm{K} / k_{\mathrm{B}}T}} \right) \right]
\right\} \;,
\nonumber
\end{eqnarray}
\end{widetext}
while for $2c_1\mathrm{r}_0^{(1)} < y < L_y/2$, the density profile is given by \Eq{wall0}.
|
2,869,038,155,710 | arxiv | \subsection{Homogeneous flat spaces} \label {homogeneous flat spaces}\
\smallskip
In this section we will prove the well known result that any connected transitive Lie subgroup of the isometry group of $\mathbb{R}^n$ must contain a pure translation. As it will become clear in the following sections, the constructions and techniques presented here can be easily adapted to homogenous space with non-trivial nullity.
\smallskip
Let $G$ be a connected Lie subgroup of the full isometry group $I(\mathbb R^n)= \mathrm {O}(n)\ltimes \mathbb R^n$. Assume that $G$ acts transitively on $\mathbb R^n$. We identify any element $X$ of the Lie algebra $\mathfrak {so}(n)\ltimes \mathbb R^n$ as a Killing field of the Euclidean space. Namely, if $X = B +v$, $B\in \mathfrak {so}(n)$ and $v\in \mathbb R^n$, then
$X_q = Bq +v$. Observe that for any $q\in \mathbb R^n$ and any $w\in T_q\mathbb R^n$, $$\nabla_wX = Bw,$$ and so $\nabla X$ is a parallel skew-symmetric tensor. Therefore, $X$ is a transvection if and only if $B=0$, i.e., $X$ is a pure translation.
We shall see that there always exists a non trivial transvection in the Lie algebra $\mathfrak g$ of $G$.
In fact, let $X = B +v \in \mathfrak g$, $B\neq 0$, and let $w\in \mathbb R^n$ such that $Bw\neq 0$.
For each $t \in \mathbb{R}$ put $q_t = tw$. Then $\Vert X_{ q_t}\Vert \to \infty$, if $t\to +\infty$. Since $G$ acts transitively on $\mathbb R^n$, for each $t\in \mathbb{R}$ we can choose an element $g_t \in G$ such that $g_t(q_t) = 0$. Set $X^t = (g_t)_* (X)\in \mathfrak{g}$.
Then $X^t_0 = t\mathrm {d}g_t (Bw) +\mathrm {d}g_t (v)$ and $\nabla X^t =
\mathrm {d}g_tB \mathrm {d}(g_t)^{-1}$. Let now $t_k\to +\infty$ be such that the sequence (of constant norm)
$\mathrm {d}g_{t_k} (Bw)$ converges to some $0\neq u\in \mathbb R^n$. Then
$\frac { 1\, }{t_k}X^{t_k}$ converges to the transvection associated to $u$.
Let now $\mathfrak a\neq \{0\}$ be the ideal of $\mathfrak g$ that consists of all the transvections in $\mathfrak g$. Let $\mathbb V = \{X_0:X\in \mathfrak a\}$. One has, from the fact that
$\mathfrak a$ is an ideal of $\mathfrak g$, that
$B \mathbb V \subset \mathbb V$, where $X= B +v \in \mathfrak g$ is arbitrary. Therefore, one also has that $B\mathbb{V}^{\perp}\subset \mathbb{V}^{\perp}$.
Consider the $G$-invariant distribution $\mathcal{D} ^\perp$ on $\mathbb R ^ n$ defined by $\mathbb V^\perp$ such that $\mathcal D^\perp _0 = \mathbb V^\perp $. Let us identify $\mathbb V^\perp$ with the integral manifold of $\mathcal D^\perp$ by $0$.
Since $G$ acts transitively on $\mathbb R^n$, for any $w\in \mathbb V^\perp$ there exists $X \in \mathfrak g$ of the form $X = B + w$, where $B\neq 0$. Then $X_0\in \mathbb{V}^{\perp}$ and so $X_{|\mathbb{V}^\perp}$ is always tangent to $\mathbb{V}^\perp$. Therefore, the Lie subgroup $G'$ of $G$ that leaves $\mathbb V^\perp$ invariant is transitive on this subspace.
Let $X=B+v\in \mathfrak{g}$ be arbitrary. Let us see that the restriction $B_{\vert \mathbb V^\perp} =0$. In fact, if $B_{\vert \mathbb V^\perp} \neq 0$, the same limit argument used before with
$g_t\in G'$, would lead to the construction of a non-trivial transvection of $\mathbb R^n$ in a direction perpendicular to $\mathbb{V}$. This is a contradiction that proves that $B_{\vert \mathbb V^\perp} =0$.
Observe that all the integral curves of $X\in \mathfrak a$ are geodesics (i.e., lines) that lie in the $G$-invariant distribution $\mathcal D$ defined by $\mathbb V$. If $X\in \mathfrak g$ is such that $X_0\in \mathbb V^\perp$, then the integral line of $X$ with initial condition $0$ is also a geodesic.
Let $g\in I(\mathbb R^n)$ be such that $g_* (X) = X$ for all $X =B + z \in \mathfrak g$.
Let us write $g(x) = Ux + u$, where $U\in \mathrm O(n)$, $u\in \mathbb R^n$. Then, taking $X \in \mathfrak a$, one has that $U$ leaves $\mathbb V$ invariant and $U_{\vert \mathbb V}$ is the identity. Then, for an arbitrary $X = B + z$, one has that $UBU^{-1} = B$, since
$B_{\vert \mathbb V^\perp} = 0$. Let us assume that $z\in \mathbb V ^\perp$. Then
\begin {eqnarray} \label {EU} g_* (X)_q &=& \mathrm {d}g (X_{g^{-1}q}) =
U (B ( U^{-1}(q-u))+ z) \\
& =& B(q-u) + Uz = X_q = Bq +z\nonumber
\end {eqnarray}
and so $$(U-Id)z = Bu.$$
But the right hand side belongs to $\mathbb V ^\perp$ and the left one to $\mathbb V$. Then $Uz = z$, for all $z\in \mathbb V^\perp$, since there is a Killing field induced by $G$ in any direction. Then $U= Id$, since $U$ acts trivially on $\mathbb V$, and $g$ is a translation.
Let $\Gamma \subset I(\mathbb R^n)$ be a discrete subgroup that acts properly discontinuously and commutes with $G$. Then $\Gamma$ consists of translations. So the quotient space of $\mathbb R^n$ by $\Gamma$ is a $G$-homogeneous spaces which is the Riemannian product of a flat torus by a Euclidean space.
This implies that any flat homogeneous Riemannian manifold is such a product and so it is isometric to an abelian Lie group with an invariant metric (the presentation group $G$ may be non-abelian). So,
any homogeneous flat Riemannian manifold is a product of a torus and a Euclidean space (cf. \cite{AK}).
Before concluding this section, let us recall the well-known fact that a Lie group of isometries that acts simply transitively on $\mathbb R^n$ must be ($2$-step) solvable (see e.g. \cite{Al}). In fact, it is not hard to obtain it form the general description given in the first part of this remark (and using the fact that there is no isotropy).
\section {The nullity of homogeneous spaces} \label{secnul}
\subsection {The osculating distributions of the nullity}\label {der}\
\smallskip
Let $M=G/H$ be a Riemannian homogeneous manifold and assume that its nullity distribution $\nu$ is non-trivial. We also assume that $\nu$ is not parallel. Otherwise, $M$ would split off, locally, a flat factor.
Let us consider the {\it osculating distribution} $\nu^{(1)}$ associated to the nullity distribution. Namely, if $C^\infty (\nu)$ are the tangent fields of $M$ that lie in $\nu$,
$$\nu ^{ (1)}_q = \nu _q + \mathrm{span}\, \{ \nabla _w X : X \in C^\infty (\nu), w\in T_qM\}$$
Then $\nu ^{ (1)}$ is a $G$-invariant distribution that properly contains $\nu$, since
$\nu$ is non-parallel. It is not hard to see that one only needs to consider $\nabla _w X$, for $X$ in some family of fields that lie in $\nu$ and such that $X_q$ span $\nu _q$.
The osculating distribution can be defined for any $G$-invariant distribution $\mathcal H$ and $\mathcal H ^{(1)}$ is also a $G$-invariant distribution. The
osculating distribution of order $k$ is defined as
$$\mathcal H ^{k} = (\mathcal H ^{k-1})^{(1)}, $$
where $\mathcal H ^{(0)}:= \mathcal H$.
\begin {lema}\label {derived} Let $\mathcal H$ be a $G$-invariant distribution
of $M$. Then
$$ \mathcal H ^{ (1)}_q = \mathcal H _q + \mathrm {span}\, \{\nabla _v Z: Z\in \mathcal K ^G (M) , v\in \mathcal H_q\}$$
\end {lema}
\begin {proof} Since $\mathcal H$ is $G$-invariant, the flow of any Killing field $Z$, induced by $G$, leaves $\mathcal H$ invariant. Then, differentiating the flow, one obtains that
$[Z,X]$ lies in $\mathcal H$, if $X$ lies in $\mathcal H$. So, if $X$ lies in $\mathcal H$,
$$\nabla _{Z_q}X = \nabla _{X_q}Z + [Z, X]_q \sim \nabla _{X_p}Z \ (\text {mod }
\mathcal H _q).$$
Since $M$ is homogeneous, there are Killing fields in any arbitrary direction. This proves the lemma.
\end {proof}
We will show later that
$\nu ^{(1)}$ is an autoparallel and flat distribution (that properly contains $\nu$). Moreover, we will show that $\nu ^{(2)}$ is contained in a (natural) proper $G$-invariant integrable distribution.
\subsection {Homogeneous geodesics tangent to the nullity}\label {homogeo}
\
\smallskip
Let $M =G/H$ be a presentation of a homogeneous Riemannian manifold, where $G$ is a connected Lie group which acts on $M$ by isometries. Fix $p\in M$ and let $N(p)$ be the leaf of nullity by $p$.
If $X$ is a Killing field that is tangent to $N(p)$ at $p$, then $X_{\vert N(p)}$ must be always tangent to $N(p)$. This follows from the fact that $X$ is projectable to the quotient of
$M$ by the integral manifolds of $\nu$ (see Section \ref{LTA}).
Let, for $p\in M$,
$$E^p = \{g\in G: gN(p)=N(p)\}.$$
Then $E^p$ is a Lie subgroup of $G$ which acts smoothly and transitively on $N(p)$ (this action may be non effective).
The Killing fields of $M$ induced by $E^p$ are
those Killing fields induced by $G$ that are tangent to $N(p)$ at $p$ (or equivalently, are always tangent to $N(p)$). They form a Lie subalgebra of $\mathcal K(M)$ which we denote by
$\mathfrak{e}^p \simeq \text Lie (E ^p)$.
Observe that the totally geodesic submanifold $N(p)$ of $M$ is extrinsically homogeneous and flat. Moreover, from Section \ref
{homogeneous flat spaces}, it is globally flat. Recall that for any Killing field $Z$ of $M$,
$\nabla Z$ is parallel along $N(p)$ (see Lemma \ref {ABC}). So, any transvection
$X\in \mathcal K ^G(M)$ at $p$ must be also a transvection at all $q\in N(p)$. If in addition it is tangent to $N(p)$, it is called an {\it extrinsic transvection} of $N(p)$.
\begin {lema}\label {SC} Assume that $M=G/H$ has non-trivial nullity distribution $\nu$ and does not split off, locally, a Euclidean factor. Then any leaf of nullity $N(p)$ is simply connected and so isometric to a Euclidean space. Moreover, the pullback $i^*(TM)$ over the inclusion $i: N(p)\to M$ is globally flat.
\end {lema}
\begin {proof}
Assume that $N(p)$ is not simply connected. We have, from Section \ref {homogeneous flat spaces}, that $N(p)$ has a non trivial closed geodesic
$\gamma _v$ (in fact, this is a general fact about homogeneous spaces since any geodesic loop must be a closed geodesic). From Lemma \ref {ABC} any Killing field $X$ must be parallel along $\gamma _v$ and thus $\nabla _vX = 0$. Then, by Corollary \ref {KostantdeRham}, $\mathbb R v$ extends locally to a parallel distribution and so $M$ locally splits off a line. A contradiction which proves the first assertion. The second assertion follows from the fact that $R_{u,v} =0$, if $u,v \in \nu _q$, and the first part.
\end {proof}
\
With the same arguments as in Section \ref {homogeneous flat spaces}, by considering isometries
$g_t \in E ^p$ we have the following results
\begin {lema}\label {transvectionN(p)1} If there exists $X\in \mathfrak{e} ^p$ such that
$X_{\vert N(p)}$ is not an intrinsic transvection of $N(p)$, then there exists
$0\neq Y \in \mathfrak{e} ^p$ such that it is an extrinsic transvection at any point of $N(p)$ (i.e. $\mathfrak p ^p \cap \mathfrak{e} ^p \neq \{0\}$).
\end {lema}
\begin {lema}\label {transvectionN(p)2} Let $\mathcal D$ the parallel distribution of
$N(p)$ which is given by the directions of the extrinsic transvections and let
$\mathcal D^\perp$ be its complementary perpendicular distribution on $N(p)$. Then, for any $q\in N(p)$,
{\it a}) If $v\in \mathcal D_q$, then the geodesic $\gamma _v(t)$ is the integral curve of an extrinsic transvection $X \in \mathfrak{e} ^p$ of $N(p)$.
{\it b}) If $v\in \mathcal D^\perp _q$, then the geodesic $\gamma _v (t)$ is the integral curve of some $X\in \mathfrak{e} ^p$.
\end {lema}
\
Let $\hat \nu$ be the $G$-invariant distribution of $M$ defined by
\begin {equation} \label {hat}
\hat {\nu} _q := \mathrm {span} \, \{ \nabla _wZ: Z\in \mathcal K ^G (M), w\in \nu _q\}.
\end {equation}
Then, from Lemma \ref {derived},
\begin {equation} \label {01nu} \nu ^{(1)} = \nu + \hat {\nu},
\end {equation}
where $\nu ^{(1)}$ is the osculating distribution.
Note that $\nu \subsetneq \nu ^{(1)}$, since $\nu$ is not a parallel distribution.
\begin {defi}\label {defiadapted} The distribution $\hat \nu$ will be called the {\it adapted distribution} of $\nu$.
\end {defi}
\begin {prop} [Existence of transvections] \label {existstransv}
Let $M=G/H$ be a homogeneous Riemannian manifold, which does not split off, (locally) a flat factor and with a non-trivial nullity distribution $\nu$.
Then, for any $y\in \hat \nu _p$, there exists a transvection $Y\in \mathcal K ^G(M)$ such that $Y_p = y$.
\end {prop}
\begin {proof}
Let $Z\in \mathcal K ^G$ and let $v\in \nu _p$ that either belongs to $\mathcal D_p$ or, to $\mathcal D_p^\perp$ (we keep the notation of Lemma \ref {transvectionN(p)2}). Then, by the above mentioned lemma, there exits $X\in
\mathfrak{e} ^p$ such that $\gamma _v (t) = \phi _t (p)$, where $\phi _t$ is the flow associated to
$X$.
\color {black}
Let $Z\in \mathcal K^G(M)$ with $\nabla _vZ\neq 0$. Then, from Lemma \ref {ABC},
$$ Z_{\gamma_v (t)} =
\tau_t (Z_p + t\nabla _v Z),$$
where $\tau _t$ is the parallel transport along $\gamma (t)$, and
$$ \nabla _{\gamma_v'(t)}(\nabla Z )=0,$$
or equivalently
$$ (\nabla Z)_{\gamma_v (t)} = \tau _t ((\nabla Z)_p) = \tau _t \circ (\nabla Z)_p\circ \tau ^{-1}_t .$$
Let us consider the family
$$Z^t: = (\phi _{-t})_*(Z)\ \ \ \ \ \ \ \ (t\in \mathbb R)$$of Killing fields induced by $G$.
Let us compute their initial conditions at $p$. First recall that from (\ref {etnabla}),
$\tau _t ^{-1} \circ \mathrm d _p \phi _t = \mathrm e ^{t(\nabla X)_p}$
and so, since $(d _p \phi _t )^{-1}= \mathrm d _{\gamma _v (t)}\phi _{-t}$,
$$
\mathrm d _{\gamma _v (t)}\phi _{-t}\circ \tau _t = \mathrm e ^{-t(\nabla Z)_p}$$
\begin{eqnarray*}Z^t_p &=& \mathrm {d}_{\gamma _v(t)}\phi _{-t} (Z_{\gamma (t)})
= \mathrm {d}_{\gamma _v(t)}\phi _{-t} (\tau_t (Z_p + t\nabla _v Z)) \\
&=&\mathrm e ^{-t(\nabla X)_p}Z_p + t \mathrm e ^{-t(\nabla X)_p}\nabla _vZ.
\end {eqnarray*}
\begin{eqnarray*}(\nabla Z^t)_p &=&
\mathrm {d}_{\gamma_v (t)}\phi _{-t} ((\nabla Z)_{\gamma (t)}) =
\mathrm {d}_{\gamma_v (t)}\phi _{-t} (\tau _t ((\nabla Z)_p)) \\
&=& \mathrm e ^{-t(\nabla X)_p}((\nabla Z)_p) =
\mathrm e ^{-t(\nabla X)_p}\circ (\nabla Z)_p \circ
\mathrm e ^{t(\nabla X)_p}
\end {eqnarray*}
Consider now the family $\frac 1 t Z^t$, $t\neq 0$. They are also Killing fields induced by $G$ and, by Remark \ref {accumulation}, we can choose a sequence of real numbers
$\{t_n\} \to +\infty$ such that $\mathrm e ^{-t_n(\nabla X)_p}$ tends to the identity transformation of $T_pM$.
Then $\frac {1\, } {t_n} Z^{t_n}$ converges
to a Killing field $Y$ with initial conditions (see Remark \ref{limitK})
$$(Y)^p = (\nabla _vZ, 0)$$
Since $\nu _p = \mathcal D_p \oplus \mathcal D^\perp _p $, and the sum of two transvection at $p$ is a transvection at $p$, we finish the proof.
\end {proof}
\begin {nota}\label {corpropo} Since $\nu \subsetneq \nu ^{(1)} =
\nu + \hat {\nu}$, there must exist $Z\in \mathcal K ^G(M)$ and $v\in \nu _p$ such that $v$ either belongs to $\mathcal D_p$ or to
$\mathcal D^\perp _p$ and $\nabla _vZ\notin \nu _p$.
Then, from Proposition \ref {existstransv}, there exists a non-trivial transvection
$Y$ at $p$ such that $Y_p = \nabla _vZ \notin \nu _p$.
\end {nota}
\
\begin {defi}\label {adapted} A transvection $Y$ at $p$ in the direction of
$\nabla_v Z\in \hat \nu _p$, $Z \in \mathcal K ^G (M)$, $v\in \nu _p$,
will be called an {\it adapted transvection} to the vector $v$.
(see Proposition \ref {existstransv}).
\end {defi}
\
\subsection{Curvature of adapted transvections} \label{secmainteo}
\begin {prop}\label {R=0} Let $M=G/H$ be a homogeneous Riemannian manifold which does not split off a local flat factor.
Assume that $M$ has a non-trivial nullity distribution and let $Y$ be an adapted transvection to the vector $v\in \nu _p$. Then, for any $U\in \mathcal K ^G(M)$ that is bounded along $\gamma _v (t)$, one has that
$$R_{U_p, Y_p}= 0.$$
\end {prop}
\begin{proof}
From Lemma \ref{SC} the integral manifold $N(p)$ of the nullity $\nu$ is complete and simply connected, and hence isometric to $\mathbb R ^m$. Moreover, the pull-back
$i_* (TM)$ is globally flat.
Let $\gamma_v (t)$ be a geodesic in $N(p)$, where $v\in T_pN(p) = \nu _p$. Write $v = v_1 + v_2$, where $v_1\in \mathcal D_p$,
$v_2\in \mathcal D^\perp _p$ (see Lemma \ref {transvectionN(p)2}). Then the parallel transport $\tau _t$ along $\gamma _v$, from $0$ to $t$, can be achieved as the composition of the parallel transport $ \bar \tau _t^t \circ \bar \tau _t$ where
$\bar \tau _t $ is the parallel transport along the geodesics $\gamma _{v_2}(t)$, from $0$ to $t$, and
$\bar \tau _t^t$ is the parallel transport along the geodesic $\gamma ^t (s) = \gamma _{v_2}(t) + \gamma _{v_1}(s)$, from $s=0$ to $s=t$ (we identify $N(p)\simeq \mathbb R^m$).
Observe, that for any $t$, there is a transvection $H^t$ such that its associated flow
$s\mapsto \psi^t_s$, satisfies
$\gamma ^t(s) = \psi^t_s(\gamma _{v_2} (t))$.
The geodesic $\gamma _{v_2}$ is homogeneous, i.e $\gamma _{v_2} (t) = \phi _t (p)$, where $\phi _t$ is the flow associated to a Killing field $X\in \mathfrak{e} ^p$, with $X_p= v_2$.
Then, from formula (\ref {etnabla}),
$$\bar \tau _t =\mathrm d _{p}\phi _t \circ \mathrm {e}^{-t(\nabla X)_p}, $$
and
$$\bar\tau ^t_t = \mathrm {d}_{\gamma _{v_2}(t)}\psi^t_t =
\mathrm {d}_{\phi _t (p)}\psi^t_t $$
Then
$$ \tau _t = \mathrm {d}_{\phi _t (p)}\psi^t_t
\circ \mathrm d _{p}\phi _t \circ \mathrm {e}^{-t(\nabla X)_p}. $$
If $g^t= \psi _t^t\circ \phi _t$, then $g^t$ is an isometry and
\begin {equation}\label {FFF}
\tau _t = d_pg^t \circ \mathrm {e}^{-t(\nabla X)_p}
\end {equation}
Let $Z\in \mathcal K ^G(M)\simeq \mathfrak g $ be such that
$Z$ is not bounded along $\gamma _v (t)$, or equivalently, by Lemma \ref {ABC},
$w = \nabla _vZ \neq 0$. Let $Y\in \mathcal K ^G(M)$ be a transvection adapted to $v$ whose initial conditions at $p$ are
$$(Y)^p = (w, 0).$$
Let $U\in \mathcal K ^G (M)$ be bounded along $\gamma _v(t)$, or equivalently, $\nabla _v U =0$. We will determine the initial conditions of the bracket $[U,Z]$ at a point $\gamma _v (t)$, for an arbitrary $t$. More precisely, we are interested in the
second component, $(\nabla [U,Z])_{\gamma _v (t)}$.
Recall, from Lemma \ref {ABC} (ii), that for any $\hat Z\in \mathcal K ^G(M)$, $\nabla \hat Z$ is parallel along $\gamma _v $.
From (\ref {initialbracket}), one has that
\begin {equation} \label {UZ}
(\nabla [U,Z])_{\gamma _v (t)} = R_{U_{\gamma _v (t)} , Z _{\gamma _v (t)}} - [(\nabla U)_{\gamma _v (t)}, (\nabla Z)_{\gamma _v (t)}]
\end {equation}
Since $\nabla U$ and $\nabla Z$ are parallel along $\gamma _v $, so is the bracket $[\nabla U, \nabla Z]$.
But $\nabla [U,Z]$ is parallel as well, and hence, from (\ref {UZ}),
\begin {equation} \label {RUZ}
R_{U_{\gamma _v (t)} , Z _{\gamma _v (t)}}
\end {equation}
must be parallel along $\gamma _v$.
One has, from Lemma \ref{ABC}, that $U_{\gamma _v (t)} =
\tau _t (U_p)$ and that $Z_{\gamma _v (t)} =
\tau _t (Z_p) + t \tau _t w$, where $w = \nabla _vZ$. So, replacing in (\ref {RUZ}), one obtains that
\begin {equation}
R_{\tau _t (U_p), \tau _t (Z_p)} + t R_{\tau _t (U_p), \tau _t (w)}
\end {equation}must be parallel along $\gamma _v (t)$.
In particular, this expression must be bounded along $\gamma _v(t)$.
Since $M$ is homogeneous, both curvature operators $R_{\tau _t (U_p), \tau _t (Z_p)}$ and
$R_{\tau _t (U_p), \tau _t (w)}$ are bounded by
the supremum $$\mathrm{sup}\{\,\Vert R_{x,y}\Vert\,:\,x,y\in T_pM,\, \Vert x\Vert,\Vert y\Vert\leq C\}<\infty$$ for a suitable constant $C$.
This implies that $\Vert R_{\tau _t (U_p), \tau _t (w)}\Vert $ should tend to $0$ as $t$ tends to infinity.
Then, recalling (\ref {FFF}),
\begin {eqnarray} \label {RUZ4}
\Vert R_{\tau _t (U_p), \tau _t (w)}\Vert &=&
\Vert R_ { \mathrm d _{p}g^t (a(t)),
\mathrm d _{p}g^t (b(t))} \Vert
\\ & =& \Vert d _{p}g^t\circ R_ { a(t),
b(t)}\circ \mathrm (d _{p}g^t)^{-1} \Vert \\
&=& \Vert R_{a(t), b(t)}\Vert,
\end {eqnarray}
where
$a (t) = \mathrm {e}^{-t(\nabla X)_p}(U_p)$,
$b(t) = \mathrm {e}^{-t(\nabla X)_p}(w)$.
By Remark \ref {accumulation} one can take a sequence $\{t_n\}$ tending to infinity such that $\mathrm {e}^{-t_n(\nabla X)_p}$
tends to the identity of $T_pM$. Then one concludes that
$\Vert R_{a(t_n), b(t_n)}\Vert $ tends to $ \Vert R_{U_p, w}\Vert$.
Since $\Vert R_{\tau _{t_n} (U_p), \tau _{t_n} (w)}\Vert =
\Vert R_{a(t_n), b(t_n)}\Vert $ must tend to $0$ as $t\to \infty$,
we conclude that $R_{U_p, w}=0$ \end{proof}
\begin {nota}\label {generalizedR=0} In the proof of Proposition \ref {R=0} we only used that the projection of $U$ to $\nu ^\perp$ is bounded. Then: {\it $ R_{U_p, Y_p}= 0$ if $Y$ is an adapted transvection at $p$ and $U$ belongs to the bounded algebra $\mathfrak u ^p$}
\noindent (see Definition \ref {BA} and its preceding paragraph).
\end {nota}
\begin {lema}\label {mainL}
Let $M=G/H$ be a homogeneous Riemannian manifold which does not split off a local flat factor and with a non-trivial nullity distribution. Let $0\neq Y$ be a
transvection adapted to $v\in \nu _p$.
\color {black} Then
\begin {enumerate}
\item If $Z$ is any Killing field of $M$, then $[Y,Z]$ is bounded along $\gamma _v$ (or equivalently, $\nabla _v[Y,Z] = 0$).
\item If $U$ is any bounded Killing field of $M$, then
$[Y, U]$ is a transvection at $p$ (i.e., $(\nabla[Y, U])_p= 0$).
\item If $\bar Y$ is any transvection at $p$, then
$[Y, \bar Y] =0$.
\item
$[Y,[Y,[Y, \mathcal K (M)]]]= 0$, or equivalently,
identifying Killings fields with elements of the isometry algebra,
$\mathrm {ad}^3_Y = 0$, in the Lie algebra of the full isometry group of $M$.
\end{enumerate}
\end {lema}
\begin {proof}
Let $Z\in \mathcal K (M)$. Then, from
(\ref{initialbracket}), $(\nabla [Y,Z])_p = R_{Y_p, Z_p}$.
So, $$\nabla _v [Y,Z] = R_{Y_p, Z_p}v=0 ,$$ since $v$ is in the nullity.
Then $[Y,Z]$ is bounded along $\gamma _v$. This proves (i).\\
To see (ii) observe that $(\nabla [Y,U])_p = R_{Y_p, U_p} = 0$ by Proposition \ref {R=0}. Thus, $[Y,U]$ is a transvection at $p$.\\
Let now $\bar Y$ be a transvection at $p$. In particular $\nabla _v\bar Y =0$ and so $\bar Y$ is bounded along $\gamma _v$. Then part (ii) applies and $(\nabla [Y, \bar Y])_p = 0$. Since $Y, \bar Y$ are both transvections at $p$, by (\ref {initialbracket}), $ [Y, \bar Y]_p =0$.
Then $([Y, \bar Y])^p =(0,0)$ and so $[Y, \bar Y]$ vanishes identically on $M$. This proves (iii).\\
Finally if $Z\in \mathcal K (M)$ is arbitrary, then $[Y,Z]$ is bounded
by part (i). Then applying (ii) $[Y,[Y,Z]]$ is a transvection at $p$.
Then, by part (iii), $$[Y,[Y,[Y,Z]] = 0.$$
\end {proof}
We improve part (iv) of Lemma \ref {mainL}. Namely,
\begin {teo}\label {ST}
Let $M=G/H$ be a homogeneous Riemannian manifold which does not split off a local flat factor and with a non-trivial nullity distribution. Let $0\neq Y$ be a transvection adapted to $v\in \nu _p$. Then
$[Y,[Y, \mathcal K (M)]]=0$, or equivalently, identifying Killings fields with elements of the isometry algebra,
$\mathrm {ad}^2_Y=0$, in the Lie algebra of the full isometry group of $M$. Moreover, $[Y, \mathcal K ^G(M)]\neq 0$.
\end{teo}
\begin {proof}
We will not regard, as before, Killing fields along $\gamma _v$, but along the geodesic $\beta (t) = \phi _t (p)$, where $\phi _t$ is the flow associated to $Y$.
Since $Y$ is a transvection at $p$, $d_p\phi _t$ coincides with the parallel transport along $\beta (t)$. Then if $\psi$ is any field in $M$,
$[Y,\psi]_{\beta (t)}$ is the covariant derivative $\frac {\, \mathrm D}
{\mathrm d t} \psi _{\beta (t)}$. Let us apply this for $\psi = Z\in \mathcal
K (M)$. Keep in mind that $Z_{\beta (t)}$ is a Jacobi field along $\beta$. So, from
Lemma \ref {mainL}, (iv)
\begin {equation}\label {D3}
\frac {\, \mathrm D^3}
{\mathrm d t^3} Z _{\beta (t)} =0
\end {equation}
Now in general the curvature tensor $R$ is invariant under isometries and
$d_p\phi _t$ coincides with the parallel transport along $\beta (t)$.
This implies that the Jacobi operator $R_{\, \cdot , \beta' (t)} \beta' (t)$
diagonalizes in a parallel basis with constant distinct eigenvalues $\lambda _0 =0, \lambda _1, \cdots , \lambda _r$ (as in symmetric spaces (see \cite {BOR})).
Let $V^0 (t), V^1 (t), \cdots , V^r(t)$ be the eigenspaces of the Jacobi operator $R_{\, \cdot , \beta' (t)} \beta' (t)$ associated to
$0, \lambda _1, \cdots , \lambda _r$, respectively. Any of such subspaces must be parallel along $\beta (t)$.
Then the orthogonal projection $Z^i(t)$ of $Z_{\beta (t)}$ to
$V^i(t)$ is of one of the following types, according with the sign of $\lambda _i$.
(a) $Z^0(t) = a(t) + t b(t)$,
where $a(t), b(t)\in V^0(t)$ are parallel fields along $\beta (t)$.
(b) If $\lambda _i >0$,
$Z^i(t) = \cos (\sqrt {\lambda_i} t) a(t) + \sin (\sqrt {\lambda _ i }t) b(t)$, where $a(t), b(t)\in V^i(t)$ are parallel fields along $\beta (t)$.
(c) If $\lambda _i <0$, $Z^i(t) = \cosh (\sqrt {-\lambda_i} t) a(t) + \sinh (\sqrt {-\lambda_i} t) b(t)$, where $a(t), b(t)\in V^i(t)$ are parallel fields along $\beta (t)$.\\
But (\ref {D3}) implies that $Z^i(t) = 0$, for $i= 1, \dots , r$ hence $Z _{\beta (t)} = Z^0(t)$ and so
\begin{equation}\label {D4}
\frac {\, \mathrm D^2}
{\mathrm d t^2} Z _{\beta (t)} =0
\end {equation}
Then $[Y,[Y, Z]]_{\beta (t)}\equiv 0$. In particular,
$[Y,[Y, Z]]_p = 0$. But, by Lemma \ref {mainL} (i) and (ii),
$(\nabla[Y, [Y,Z]])_p = 0$. Then $[Y,[Y, Z]] =0$.
This proves that $$[Y,[Y, \mathcal K (M)]]=0 .$$
It only remains to show that $[Y, \mathcal K ^G(M)]\neq 0$.
Assume, on the contrary, that $[Y, Z]= 0$, for all $Z\in \mathcal K^G(M)$.
Since $Y$ is a transvection at $\beta (t)$, for all $t$,
the covariant derivative along $\beta (t)$ of
$Z_{\beta (t)}$, as we have seen, coincides with $[Y, Z]_{\beta (t)} =0$ as assumed.
Then $Z_{\beta (t)}$ is parallel along $\beta (t)$ hence
$\nabla _{\beta'(0)}Z = \nabla _{Y_p}Z = 0$. Since $Z$ is arbitrary in $\mathcal K ^G(M)$ we conclude, from Corollary \ref {KostantdeRham}, that $M$ splits locally the direction of $Y_p$. A contradiction.
\end {proof}
In the proof of the above theorem it was shown that the adapted transvection $Y$ has null Jacobi operator along $\beta (t)$ or equivalently at $p$. Indeed being $M$ homogeneous, there is a Killing field in any direction, and we conclude that the Jacobi operator has only one eigenvalue $\lambda _0 = 0$. From Remark \ref {corpropo} we may assume that
$Y_p \notin \nu _p$. Then we have the following result that will be very useful for finding irreducible homogeneous Riemannian manifolds with non-trivial nullity distribution.
\begin {cor}\label {STcor} Let $M=G/H$ be a homogeneous Riemannian manifold which does not split off a local flat factor and with a non-trivial nullity distribution $\nu$. Then any adapted transvection $Y$ to $v\in \nu _p$ has a null Jacobi operator $R_{\, \cdot , Y_p}Y_p$.
Moreover, there exists such an adapted transvection $Y$ with $Y_p\notin \nu _p$.
\end {cor}
Any transvection at $p$ belongs, by definition, to the Cartan
subspace $\mathfrak p ^p$ at $p$ (see (\ref {Cs})).
Those with trivial Jacobi operator must lie in the abelian part $\mathfrak
p ^p_0$ of the Cartan subspace. Namely,
\begin {cor}\label {ABE} Let $M=G/H$ be a homogeneous Riemannian manifold which does not split off a local flat factor and with a non-trivial nullity distribution. Then any transvection $Y$, adapted to $v\in \nu _p$, belongs to the abelian part
$\mathfrak p^p_0$ of the Cartan subspace at $p$. In particular, the distribution of symmetry $\mathfrak s$ of $M$ is non-trivial and so the index of symmetry of $M$ is positive.
\end {cor}
\subsection {Transvections with null Jacobi operator}\label {NuJaOp} \
\smallskip
We have the following result for homogeneous spaces that have transvections with null Jacobi operator (not depending on the existence of a non-trivial nullity).
\begin {prop} \label {JacNul} Let $M=G/H$ be a homogeneous Riemannian manifold where $G = I(M)^o$ and such that $M$ does not split off, locally, a flat factor. Let $\mathfrak a_0^p$ be the set of transvections $Y$ at $p\in M$ with null Jacobi operator $R_{\cdot , Y_p}Y_p$. Then
\begin {itemize}
\item [(i)] If $0\neq Y\in \mathfrak a_0^p$, then $[Y,[Y, \mathcal K (M)]] = 0$ and
$[Y, \mathcal K^G(M)]\neq 0$.
\item [(ii)] The set $\mathfrak a_0^p$ is a vector space. Moreover, it is an abelian Lie algebra and the distribution $q \to \mathfrak a_0^q.q$ is autoparallel and flat.
\end {itemize}
\end {prop}
\begin {proof}
Let $Y\in \mathfrak a_0^p$. Let $\phi _t$ the flow associated to $Y$. Then
$\gamma (t) =\phi _t(p)$ is the geodesic with initial condition $v=Y_p$. Let $Z\in \mathcal K (M)$. Then, from the Jacobi equation,
$Z_{\gamma (t)} = u(t) + t w(t)$, where $u(t), w(t)$ are the parallel transports along $\gamma (t)$ of $Z_p$ and $\nabla _vZ$, respectively. Since
$[Y,Z] = \frac {\mathrm \, d}{\mathrm d t}_{\vert 0}
(\phi _{-t})_*(Z)$, by formula
(\ref {etnabla}), we obtain that
$$[Y, Z]_{\gamma (t)} =\nabla _{\gamma ^{\prime}(t)}Z_{\gamma (t)}= w(t), $$
i.e., $[Y, Z]_{\gamma (t)}$ is parallel along $\gamma (t)$.
So,
$$ [Y,[Y, Z]]_{\gamma (t)}=0. $$
Then $U =[Y,[Y, Z]]$ belongs to the double isotropy algebra at $v$. In particular
$U$ belongs to the isotropy algebra $\mathcal{K}(M)_p \subset \mathfrak {so}(T_pM)$
(via the isotropy representation at $p$). Moreover, if $\psi _s$ is the flow associated to $U$, $\mathrm d _p \psi _s (v)= v$, for all $s$. Or , equivalently,
$$\psi _s (\gamma (t)) = \gamma (t).$$
Since isometries map transvections into transvections, from the above equalities we obtain that $(\psi _s)_* (Y) = Y$ and therefore, differentiating the flow, $-[U,Y]= [Y,U]=0$.
Let $B$ be the Killing form of $\mathcal K(M)$. Then $B_{\vert \mathcal{K}(M)_p\times \mathcal{K}(M)_p} $
is negative definite (see the beginning of the proof of Lemma \ref {EFF}).
$$B(U,U) = B([Y,[Y, Z]],U) = - B([Y, Z],[Y,U]) = 0.$$
Then $U=0$, since $U\in \mathcal{K}(M)_p$. This proves the first assertion of (i).
If $[Y, \mathcal K^G(M)] = 0$, then $g_*(Y) = Y$, for all $g\in G$. Then, since $G$ acts transitively on $M$, $Y$ is a transvection at any point and so a parallel field.
Then $M$ locally splits off a line. A contradiction. This finishes the proof of (i).
Since $\mathfrak a_0^p$ is invariant by scalar multiplications, one has to show that if $X, Y\in \mathfrak a_0^p$ then $Z= X+ Y \in \mathfrak a_0^p$
(observe that the Jacobi null condition is not a linear condition). In the notation of
Section \ref {index}, we have that $\mathfrak a_0^p \subset \mathfrak p ^p_0$, the abelian part of the Cartan subspace at $p$. Observe that $[X, Y]=0$, since this field is zero when restricted to the symmetry leaf $L(p)$ (see Lemma \ref {EFF}).
Then $[X, [Y, \cdot]] = [Y, [X, \cdot]]$. Taking this and the fact that
$[X, [X, \cdot]] = 0 = [Y, [Y, \cdot]] $ into account, we obtain that
$$[Z, [Z, [Z, \cdot]] = 0.$$
The same argument used in the proof of Theorem \ref {ST} shows that the Jacobi operator $R_{\cdot , Z_p}Z_p$ is zero (see also the paragraph just after this theorem). Then $Z\in \mathfrak a^p_0$. So $\mathfrak a_0^p$ is a vector subspace of the abelian Lie algebra $\mathfrak p _0 ^p$. If $L(p)$ is the leaf of symmetry by $p$, then its Euclidean factor is given by $A.p$, where $A$ is the abelian Lie group of isometries with Lie algebra $\mathfrak p _0 ^p$. If $A'$ is the Lie group associated to $\mathfrak a^p_0$, then $A'.p$ is totally geodesic in $A.p$. Since $A.p$ is totally geodesic in $M$, then $A'.p$ is totally geodesic in $M$ and it is an integral manifold of the distribution $q \to \mathfrak a_0^q.q$. This concludes the proof.
\end {proof}
Observe, from Proposition \ref {JacNul} and Theorem \ref {ST}, that in any direction of $\hat \nu_p$ there exists a transvection
$X$, with $\mathrm {ad}^2_X = 0$ and $\mathrm {ad}_X \neq 0$.
\vspace{.2cm}
\color {black}
\subsection{Applications to semisimple and nilpotent homogeneous spaces}$\ $
\begin{cor}\label{corsemisimple}
Let $M=G/H$ be a simply connected homogeneous Riemannian manifold without a Euclidean de Rham factor.
Assume that Lie algebra $\mathfrak g$ of $G$ is reductive.
Then $M$ has a trivial nullity.
\end{cor}
\begin{proof}
Assume that the nullity $\nu$ is non-trivial and let $p= [e] \in M$.
We will regard $\mathfrak g$ as the Lie algebra
$\mathcal K^G$ of Killing fields induced by $G$.
Then, by Proposition \ref {existstransv} and Theorem \ref {ST},
there exists an adapted transvection
$0\neq Y\in \mathfrak g$ of $M$ at $p$, such that
$\mathrm {ad}^2 _Y= 0$ and $\mathrm {ad} _Y\neq 0$. Let us decompose
$\mathfrak g = \mathfrak a \oplus \mathfrak k \oplus \mathfrak i$
into the direct sum the ideals which are abelian, of the compact type, and of the noncompact type, respectively.
Let us write $Y = Y^0+
Y^1 + Y^2$, where $Y^0\in \mathfrak a$, $Y^1\in \mathfrak k$, and
$Y^2\in \mathfrak i$. Since $\mathrm {ad}^2 _Y= 0$, then $\mathrm {ad}^2 _{Y^1}=0$. Hence $Y^1=0$, since $\mathfrak k$ is of the compact type.
On the one hand, since $\langle \nabla _{\mathfrak g}Y,\mathfrak a\rangle_p = 0$, we obtain, from equation (\ref{fundamentalequation}), that
\begin {equation}\label {perAb}
\langle [Y, \mathfrak g], \mathfrak a\rangle _p =0
\end {equation}
On the other hand, since $\mathrm {ad} _Y\neq 0$, $Y^2\neq 0$. Hence $Y^2$ is a 2-step nilpotent element of $\mathfrak i$.
Then, by the Jacobson-Morozov theorem \cite{OVG}, $Y^2$ belongs to a $\mathfrak {sl}(2)$-triple, in $\mathfrak i$, $\{Y^2, Z,W\}$ such that
$[W,Y^2]=2Y^2$, $[W,Z]=-2Z$ and $[Y^2,Z]=W$.
Since $Y$ is a transvection at $p$, by equation
(\ref{fundamentalequation}), have that
\begin {eqnarray*} 0&=&\langle \nabla_Y Y,W\rangle_p=\langle[Y,W],Y\rangle_p
= \langle[Y,W],Y ^2\rangle_p \text {\small \ \ by (\ref {perAb})} \\
&=& \langle[Y^2,W],Y^2\rangle_p=
-2\Vert Y^2_p\Vert^2
\end {eqnarray*}
Then $Y^2_p=0$ and so $Y^2$ belongs to the isotropy algebra
$\mathfrak h = \mathrm {Lie} (H)$. Let $(\, , \, )$ be an $\mathrm {Ad}(H)$-invariant inner product in $\mathfrak g$. Then $\mathrm {ad} _{Y^2}$ is skew-symmetric and so
$\mathrm {ad} _{Y^2}=0$, since $\mathrm {ad}^2 _{Y^2}=0$.
Then $Y^2\in \mathfrak a$. A contradiction.
\end {proof}
\bigskip
The same argument of the above corollary shows that there are no transvections of order $2$ in a homogeneous Riemannian manifold $M=G/H$ with $\mathfrak g$ reductive.
\
\color {black}
\begin{cor}\label{cornilpotent}
Let $M = G$ be a 2-step nilpotent Lie group with a left invariant metric
which does not split off a local flat factor.
Then the nullity distribution of $M$ is trivial.
\end{cor}
\begin{proof} \label {2-step}
We proceed by contradiction assuming that the nullity distribution $\nu$ is non trivial.
Let $Y$ be the transvection at $p \in M$ given by Theorem \ref{ST} such that $[Y, \mathfrak g]\neq {0}$ i.e. $Y$ does not belong to the center $\mathfrak c$ of $\mathfrak g$. From (\ref
{fundamentalequation}),
$$0 =2\langle \nabla _{\mathfrak c} Y , \mathfrak g \rangle_p =
\langle [Y, \mathfrak g], \mathfrak c\rangle _p \, \, .$$
Since $[Y, \mathfrak g] \subset \mathfrak c$ due that $G$ is 2-step nilpotent we get that $[Y,\mathfrak g]_p = 0$
hence $[Y, \mathfrak g] = {0}$. A contradiction.
\end{proof}
\begin{nota} The above corollary also follows from well known facts about the Ricci tensor and the de Rham factor of a 2-step nilpotent Lie group with a left invariant metric \cite[Proposition (2.5) and Proposition (2.7)]{Eb94} .
\end{nota}
\subsection {The nullity is not a homogeneous foliation} \
We finish this section by showing that the nullity foliation is far from being a homogeneous foliation (i.e. given by the orbits of an isometric group action).
Observe, from the affine Killing equation (\ref {affK}), that a Killing field $X$ lies in the nullity
distribution if and only if the Nomizu tensor $\nabla X$ is parallel. Below we show that a Killing field $X\neq 0$ that lies in the nullity must be tangent to the (local) Euclidean de Rham factor. We were unable to find this result in the literature.
\begin {prop}\label {KillTanNu} Let $M$ be a homogeneous Riemannian manifold without Euclidean (local) de Rham factor. Assume that the nullity distribution $\nu $ of $M$ is non-trivial. Then there exists no Killing field $ X\neq 0$ of $M$ such that $X$ lies in
$\nu$.
\end {prop}
\begin {proof}
We may assume that $M$ is simply connected. Let
$M = M_1 \times \cdots \times M_r$ be the de Rham decomposition of $M$, where
$M_1, \cdots , M_r$ are irreducible Riemannian manifolds. The nullity $\nu$ of $M$ is the direct sum of the nullities $\nu _i$ of $M_i$, $i=1, \cdots , r$. Let $X\in \mathcal K(M)$ be always tangent to $\nu$. The projection $X_i$ of $X$ to any given factor $M_i$ belongs to
$\mathcal K (M_i)$. If $M_i$ has a trivial nullity, then $X_i=0$. So we may assume that
$\nu _i \neq 0$. Assume that $X_i\neq 0$. From formula (\ref {affK}) we have that $\nabla X_i$ is a parallel
skew-symmetric $(1,1)$-tensor of $M_i$. If $\ker (\nabla X_i) =TM$, then $X_i$ is a parallel field and so $M_i$ splits off a line. A contradiction. Let $p\in M$. Then $(\nabla X_i)_p$ has a complex eigenvalue $\lambda\notin \mathbb R$. Let $\{0\}\neq \mathbb V\subset T_pM$ be the $(\nabla X_i)_p$-invariant subspace
associated to $\lambda$ and $\bar {\lambda}$. Since $\nabla X_i$ is a parallel tensor, then $\mathbb V$ extends to a parallel distribution of $M_i$ and since $M_i$ is irreducible, then $\mathbb V = T_pM$. In this case, eventually by rescaling $X$, we have that $J=\nabla X_i$ is a K\"ahler structure on $M_i$ and so $M_i$ is a K\"ahler (homogeneous) manifold. Then the field $\xi = JX_i$ lies in the nullity distribution $\nu _i$.
Moreover, as it is standard to check, $\nabla ^2\xi = 0$. So
$\xi$ satisfies the affine Killing equation
$\nabla_{u,v} ^2 \xi = R_{u,\xi}v$ (see \ref {affK}). If $\phi _t$ is the flow associated to $\xi$, for any given $t$, $\phi _t$ is an homothetic transformation of $M$ associated to the constant $\mathrm {e}^{ta} $, where $A = a Id$ is the symmetric part of
$\nabla \xi$ (cf. \cite {KN}, Lemma 1, pg. 242). In fact, this symmetric part $A$, from the affine Killing equation, must be parallel. Moreover, since $M_i$ is irreducible, $A$ has only one (constant) eigenvalue.
In our particular case $\nabla \xi = J \nabla X_i= -Id $, and so $a=-1$.
But, in a homogeneous non-flat irreducible space, any homothetic transformation is an isometry. This is a general fact for a complete Riemannian manifolds (see \cite {KN}, Theorem 3.6, pg. 242). For the sake of self-completeness, we will show this in our homogeneous context. In fact,
$\mathrm {d}_p \phi _t : T_pM \to T_{\phi _t(p)}$ is a homothetic map. Namely,
$$\langle \mathrm {d}_p \phi _t (v), \mathrm {d}_p \phi _t (v)\rangle =
\mathrm {e}^{-2t}\langle v, w\rangle$$
One has, since $\phi _t$ preserves the Levi-Civita connection,
that $\mathrm {d}_p \phi _t$ maps the curvature tensor $R^p$ of $M_i$ at $p$ into the curvature tensor $R^{\phi _t(p)}$ at
$\phi _t(p)$. Then, by a standard calculation,
$$\mathrm {e}^{-2t}\Vert R^{\phi _t(p)}\Vert = \Vert R^{p}\Vert$$
This is a contradiction, since $M_i$ is homogeneous and non-flat. Then $X_i =0$, for any $i=1, \cdots , r$. Then $X=0$.
\end {proof}
The same proof works assuming $M$ to be complete not necessarily homogeneous.
\vspace {.2cm}
\section {Symmetry and nullity}\label{symnul}
Let $M=G/H$ be a homogeneous locally irreducible Riemannian manifold with a non-trivial distribution of symmetry $\mathfrak s$. Recall that $\mathfrak s$ is not contained in the nullity distribution $\nu$, see Remark \ref {corpropo}. Since both distributions $\nu$ and $\mathfrak s$ are $G$-invariant their sum
\begin {equation} \label {proper2}
\tilde \nu = \nu + \mathfrak s
\end {equation} has constant rank, and hence $\tilde \nu$ is a distribution on $M$.
Observe that the above sum could be non direct.
\begin{lema}\label {proper}
The distribution $\tilde \nu$ is autoparallel.
Moreover, if $\tilde N (p)$ is an integral manifold of $\tilde \nu$ then
the restrictions $\mathfrak s_{\vert \tilde N (p)}$, $\nu_{\vert \tilde N (p)}$ are parallel distributions of
$\tilde N (p)$.
\end{lema}
\begin{proof}
Let $Y\in \mathfrak p^p$, the Cartan subspace at $p$ (see (\ref {Cs})), and let $c(t)$ be a curve contained in the leaf of nullity $N(p)$ joining $p$ and an arbitrary point $q \in N(p)$.
From the affine Killing equation (\ref {affK}) one has that $\nabla Y$ is parallel along $c(t)$. This implies that
$(\nabla Y)_q = 0 $ for all $q\in N(p)$ hence $Y_q\in \mathfrak s _q$.
Since $p$ is arbitrary, we get
$$\nabla _{\nu}\mathfrak s \subset \mathfrak s \, \, .$$
Let $\phi _t$ be the flow associated to $Y$.
Since $\nu$ is $G$-invariant and, by equation (\ref {etnabla}), $d_p\phi _t$ gives the parallel transport along (the geodesic) $\phi _t (p)$, we must have that
$\nu$ is parallel along the leaf of symmetry $L(p)$ at $p$.
Since $p$ is arbitrary we conclude that
$$\nabla _{\mathfrak s} \nu \subset \nu \, .$$
Then, since $\nu $ and $\mathfrak s$ are both autoparallel, we conclude that $\tilde \nu$ is autoparallel.
\end{proof}
Let now $\mathfrak s ^0 $ be the flat part of the distribution of symmetry (see equation (\ref{flatfactor})) and consider the distribution
\begin {equation}\label {proper0}
\tilde \nu ^0 = \mathfrak s ^0 + \nu
\end {equation}
which is not in general a direct sum.
\
We have the following lemma.
\begin{lema} The nullity distribution is properly contained
in the $I(M)$-invariant distribution $ \tilde \nu ^0 $
which is autoparallel and flat.
\end{lema}
\begin{proof}
First observe that, from Corollary \ref{ABE}, there is a transvection $Y\in\mathfrak{s}^0$ which does not lie in $\nu$. So $\nu$ is properly contained in $\tilde{\nu}^0$. Since both $\mathfrak{s}^0$ and $\nu$ are $I(M)$-invariant, so is $\tilde{\nu}^0$.
By the above Lemma we have that locally $\tilde N (p) = L(p) \times W$ as Riemannian product, where $W$ is a Riemannian manifold.
Now $\mathfrak s ^0 $ is the flat parallel distribution tangent to the whole flat de Rham factor of any leaf of symmetry $L(q)\subset \tilde N (q)$. So we conclude that the restriction $\mathfrak s ^0 _{\vert \tilde N (p)}$ is parallel.
Then
$$\tilde \nu ^0_{\vert \tilde N (p)} = \mathfrak s ^0 _{\vert \tilde N (p)}+ \nu_{\vert \tilde N (p)}$$
is a parallel distribution of $\tilde N(p)$. This implies that $\tilde \nu ^0 $ is an autoparallel distribution of $M$. Moreover, it must be flat, since
$ \mathfrak s ^0 $ and $\nu$ are parallel and flat distributions of
$ \tilde N (p)$.
\end{proof}
\section {The osculating distributions and the isotropy }\label{section5}
Here we give the details of the proofs of part (1) and (3) of Theorem A.
We already showed that
\begin {equation}\label {123} \nu ^{(1)} = \nu + \hat {\nu}
\end {equation}
and that $ \nu \subsetneq \nu ^{(1)} = \nu + \hat {\nu}$, see equation (\ref{01nu}).
Moreover, $\hat {\nu}\subset \mathfrak s^0$ by Corollary \ref{ABE}. By using that $\hat {\nu}$ is $G$-invariant and making the same arguments as in Section \ref {symnul} one has that $\nu ^{(1)}$
is an autoparallel and flat, and so a proper, distribution of $M$.
One can also prove this fact by using that $R_{X,Y} = 0$, if $X, Y$ are transvections that belong to $\mathfrak p _0^p$ (and $\hat {\nu}_p \subset \mathfrak p _0^p.\, p = \mathfrak s _0^p$).
\vspace{.25cm}
The inclusion $\nu ^{(1)} \subsetneq \nu ^{(2)}$ is proper since $\nu ^{(1)}$
is not a parallel distribution.
The osculating distribution $\nu ^{(2)}$ is $G$-invariant and by (\ref {123}) one has that
$$\nu ^{(2)} = \nu ^{(1)} + \hat {\nu}^{(1)}.$$
Then, from Lemma \ref{derived}, and taking into account that $\hat {\nu}\subset
\nu ^{(1)}$, it follows that
\begin {equation} \label {nu2span}
\nu ^{(2)}_p = \nu ^{(1)}_p + \mathrm {span} \, \{\nabla _v Z: Z\in \mathcal K ^G (M) , v\in \hat {\nu} _p\}
\end {equation}
But, if $X$ is a transvection with $X_p =v\in \hat {\nu}_p$, $\nabla_vZ = [X,Z]_p+\nabla _{Z_p}X=[X,Z]_p$. Then
\begin {equation} \label {biss}
\nu ^{(2)}_p = \nu ^{(1)} +\mathrm {span} \, \{[X,Z]_p: Z\in \mathcal K ^G (M), X \text { is a transvection at } p \text { with } X_p \in
\hat {\nu} _p\}
\end {equation}
Let $X$ be a transvection at $p$, with $X_p\in \hat {\nu}_p$, and let
$Z\in \mathcal K ^G (M)$ be arbitrary. Then, from (\ref {initialbracket}), the initial conditions at $p$ of $[X,Z]$ are
\begin {equation} \label {br}
([X,Z])^p = (([X,Z]_p , (\nabla [X,Z])_p) = (\nabla _{X_p}Z , R_{X_p,Z_p})
\end {equation}
Observe that
\begin {equation}\label {uuuu}
\nabla_{\nu _p} [X,Z] = R_{X_p,Z_p}\nu _p = \{0\}
\end {equation}
\
For $p\in M$, define the subspace $\mathfrak u^p$ of $\mathcal{K}^G(M)$ as $$\mathfrak u^p=\{ U\in \mathcal{K}^G(M)\,:\, \nabla_{\nu_p}U\subset \nu_p \}.$$
From Lemma \ref{ABC} it follows that for any $q\in N(p)$, $\mathfrak u^p=\mathfrak u^q$. Moreover, from formula (\ref {curvatureformula}) it follows that $\mathfrak u^p $ is a Lie subalgebra of $\mathcal{K}^G(M)$. Observe that $U\in \mathfrak u$ if and only if the normal component to $\nu$ of $U_{\vert N(p)}$ is parallel (since $TN$ is a parallel sub-bundle of the pullback $i_*(TM)$, cf. Lemma \ref{SC}).
If $g$ is an isometry, then $\mathfrak u ^{g(p)}=
g_*(\mathfrak u ^p)$.
\
\begin {defi}\label {BA} The Lie algebra $\mathfrak u ^p$ is called the {\it bounded algebra at $p$.} The $G$-invariant distribution $\mathcal U$, defined by
$\mathcal U_p = \mathfrak u^p .\, p$ is called the {\it bounded distribution}.
\end {defi}
\
\noindent
{\bf Observation.} The Lie algebra $\mathfrak u ^p$ contains:
\begin {itemize}
\item Any transvection at $p$.
\item Any element in the linear span of
$$\{[X,Z]: Z\in \mathcal K ^G (M), X \text { a transvection at } p \text { with } X_p \in \hat {\nu} _p\}.$$
\item Any Killing field which is tangent to $N(p)$ at $p$ (and so always tangent to $N(p)$). In particular, any Killing field in the isotropy algebra at $p$.
\item The bounded distribution does not depend on the presentation group $G$ of $M$. This follows from the fact the bounded algebra contains the isotropy algebra.
\end {itemize}
\rm
\
From the above properties we have that
\begin {equation} \label {332}
\nu ^{(2)}_p \subset \mathfrak u ^p.\, p
\end {equation}
Let $\bar G^p \subset G$ be the Lie subgroup associated to the Lie subalgebra
$\mathfrak u ^p \subset \mathfrak g =
\mathrm {Lie}(G) \simeq \mathcal K^G(M)$.
Then, from the previous observation, $\bar G^p$ contains the isotropy subgroup $G_p$ (since $M$ is simply connected and so $G_p$ is connected).
Note that $\bar G^{gp} = g\bar G ^p g^{-1}$.
We have the following result, whose proof is standard, from the fact that
$G_p \subset \bar G^p$.
\begin {lema} \label {aaab} $\mathcal U$ is an integrable distribution with integral
manifolds $\bar G^q\cdot q$, $q\in M$.
\end {lema}
\begin {lema} \label {aaabb} $\mathcal U$ is a proper distribution of $M$ (or equivalently, since the bounded algebra $\mathfrak u ^p$ contains the isotropy algebra at $p$, any bounded algebra is a proper subalgebra of $\mathfrak g$).
\end {lema}
\begin {proof} Suppose on the contrary that $\mathcal{U}=TM$. Then for any $p\in M$, $\bar{ G}^p\cdot p = M$. Since
$\mathfrak u ^p = \mathrm {Lie}(\bar G^p)$, we would have that
$$\nabla _{\nu _p}X \subset \nu _p\, ,$$
for any Killing field induced by $\bar G^p$. Then, from Corollary
\ref {KostantdeRham}, $\nu $ is a parallel distribution. A contradiction.
\end {proof}
Summarizing, one has the following $G$-invariant distributions:
\begin {equation}\label {summarizing}
\{0\} \neq \nu \subsetneq \nu ^{(1)} \subsetneq
\nu ^{(2)}\subset \mathcal U \subsetneq TM\, ,
\end {equation}
where $\nu$ and $\nu ^{(1)}$ are autoparallel and flat, and $\mathcal U$ is integrable and $G$-invariant.
\
In the examples of last section we have that $\nu ^{(2)} = \mathcal U$ and this distribution is not autoparallel, see Remark \ref{distriExamples}.
\begin {nota}\label {1321} Observe that (\ref {summarizing}) implies that the codimension $k$ of the nullity must be at least $3$. Moreover, if $k=3$ then
$\mathrm {codim}\, \nu ^{(1)} = 2$,
$\nu ^{(2)}= \mathcal U $, and $\mathrm {codim}\, \mathcal U =1$.
\end {nota}
\
\begin {teo} \label {isodim} Let $M^n=G/H$ be a simply connected irreducible homogeneous manifold with nullity distribution $\nu$ of codimension
$k$ ($G$ is not assumed to be connected). Then
\begin {enumerate}
\item [({\it i})]The representation $\rho$ of $H$ on $\nu _p^\perp$ is faithful ($p=[e]$).
\item [({\it ii})] $\dim H \leq \frac 12 (k-2)(k-3)$. In particular, if $k= 3$, the isotropy is trivial (if $G$ is connected).
\end {enumerate}
\end {teo}
\begin {proof} Assume that $\ker (\rho)$ is a non-trivial normal subgroup of $H$.
Let $\mathbb V$ be the set of fixed vectors of $\ker (\rho)$.
Then $\nu _p ^\perp \subset \mathbb V \subsetneq T_pM$. Then, by \cite {BCO}, Theorem 9.1.2, $\mathbb V$ extends to an autoparallel $G$-invariant distribution $\mathcal D$ which contains the foliation $\nu ^\perp$. Observe that for any $q \in M$, $v\in \mathcal D_q,\, w \in \nu _q$ $R_{v,w} = 0$.
Then by Proposition 3.3 in \cite{DS}, $M$ splits. A contradiction. This proves (i).
From (\ref {summarizing}), $H$ leaves any of the following subspaces invariant:
$\nu _p \subsetneq \nu ^{(1)}_p\subsetneq \mathcal U _p\subsetneq T_pM$. Let us consider the representation $\rho$ of $H$ on the orthogonal complement $\nu_p^\perp$. Then there exist three non trivial $H$-invariant subspaces $\mathbb V_1, \mathbb V _2, \mathbb V_3$, mutually perpendicular and of dimensions $d_1$, $d_2$ and $d_3$ respectively such that
$$\nu_p^\perp = \mathbb V_1 \oplus \mathbb V _2 \oplus \mathbb V_3, $$
where $d_1\leq d_2\leq d_3$, and $d_1+ d_2+ d_3 =k$.
Then, by making use of part (i),
\begin {eqnarray*}\dim H &=& \dim \rho (H) \leq \dim \mathrm {SO}(d_1) + \dim \mathrm {SO}(d_2) +
\dim \mathrm {SO}(d_3) \\ &\leq& \dim \mathrm {SO}(k-2) = \frac 12 (k-2)(k-3),
\end{eqnarray*}
where the last inequality is standard to show. This proves (ii).
\end {proof}
We finish this section by proving that $\nu ^{(1)}$ is a parallel distribution when restricted to any leaf of the bounded distribution.
\begin {lema}\label{parnu1} Let $X,\, U$ be vector fields of $M$ such that $X$ lies in $\nu ^{(1)}$ and $U$ lies in $\mathcal U$. Then $\nabla _UX$ lies in $\nu ^{(1)}$ (and so $\nu ^{(1)}$, restricted to any leaf of $\mathcal U$, is a parallel and flat distribution).
\end{lema}
\begin {proof} Recall that $\nu ^{(1)} = \nu + \hat \nu$.
We may assume that either $X$ belongs to $\nu$ or $X$ belongs to $\hat \nu$
({\it a}) Let $p\in M$ be arbitrary, let $X$ belong to $\nu$ and let $Z$ be a Killing field that belongs to the bounded algebra $\mathfrak u ^p$, with $Z_p= U_p$. Since $\nu $ is $G$-invariant $[Z, X]$ lies in $\nu$.
Then $\nabla _{Z_p} X\in \nu _p$ if and only if
$\nabla _{X_p} Z \in \nu _p$, which follows from the definition of the bounded algebra.
({\it b}) Let $X$ belong to $\hat \nu$. From the definition of $\hat \nu$, the fields of the form $\nabla _W Z$ span $\hat \nu$, where $Z\in \mathcal K ^G(M)$ and $W$ is a vector field of $M$ that lies in $\nu $. So we may assume that
$X= \nabla _W Z$. Let $U$ be a vector field of $M$ that lies in $\mathcal U$.
From (\ref {affK})
$\nabla ^2_{U,W} Z = R_{U,Z}W = 0$. So,
$$0= \nabla ^2_{U,W} Z = \nabla _U\nabla _W Z -
\nabla _{\nabla _UW} Z.$$
Observe, from part (a), that ${\nabla _UW}$ lies in $\nu$. Then
$\nabla _{\nabla _UW} Z$ lies in $\hat \nu$ and hence
$\nabla _U\nabla _W Z$ lies in $\hat \nu$.
\end {proof}
\section {Homogeneous spaces with co-nullity $3$}
Let $M^n=G/H$ be a simply connected Riemannian manifold with non trivial nullity
distribution $\nu$ of codimension $3$. Then, by Theorem \ref {isodim},
$H= \{e\}$ and so $M= G$, with a left invariant metric. Then the autoparallel and
$G$-invariant distribution
$ \nu ^{(1)}$ has codimension $2$, and the integrable $G$-invariant distribution
$\mathcal U\supset \nu ^{(1)}$ has codimension $1$.
Let $p\in M$ be fixed. Then there exist Lie subgroups $H_1\subset H_2$ of $G$ such that $H_1\cdot p \subset H_2\cdot p$ are the integral manifolds by $p$ of
$ \nu ^{(1)}$ and $\mathcal U$, respectively (or, equivalently $\mathrm {Lie}
(H_2) = \mathfrak u ^p$).
\begin {lema} \label {setermina} \
\begin {enumerate}
\item $H_1\cdot p$ is isometric to $\mathbb R^{n-2}$.
\item $H_2\cdot p$ is intrinsically flat.
\
\end {enumerate}
\end {lema}
\begin {proof}
That $H_1.p$ is flat was proved at the beginning of Section \ref{section5}.
By Corollary \ref {STcor}, and part $(ii)$ of Proposition \ref{JacNul}, the Jacobi operator in any vector tangent to $\nu ^{(1)}$ is null.
So the proof of Lemma \ref {SC} also shows that the integral manifolds of $\nu ^{(1)}$ are simply connected. This proves $(i)$.
Observe, since $k=3$, that is $\nu ^{(1)}$ an autoparallel and flat sub-distribution of codimension $1$
of $\mathcal U$.
From Lemma \ref {parnu1} it follows that $\nu ^{(1)}$, restricted to any leaf $S= H_2 \cdot p$ of $\mathcal U$ is a parallel and flat distribution of codimension $1$. Then $S$ is flat, which proves (ii).
\end {proof}
\vspace{.3cm}
\subsection{Excluding the Levi factors if $\mathbf {k=3}$ }
$\ $
Let $M$ be a homogeneous simply connected Riemannian manifold without Euclidean de Rham factor.
Assume that $M$ has a non-trivial nullity distribution of codimension 3. By Theorem \ref {isodim}, $M$ has no isotropy and so $M= G$, where $G=I(M)^o$ is endowed with a left invariant metric.
Let $\mathcal U$ be the bounded distribution that has codimension $1$ (see Remark \ref{1321}). The integral manifold by $p$ of $\mathcal U$ is given by
$H_2\cdot p$, where the Lie algebra of $H_2$ is the bounded algebra
$\mathfrak u ^p$.
Since $G$ acts freely $\mathfrak u ^p$ has codimension $1$ in the Lie algebra
$\mathfrak g \simeq \mathcal K (M)$ of $G$.
Recall, from Lemma \ref {setermina} (ii), that $H_2\cdot p$ is intrinsically flat. Then, since there is no isotropy,
$\mathfrak u^p$ is solvable (see Section \ref {homogeneous flat spaces}).
Assume that the Levi decomposition of $\mathfrak g$ has a non-trivial Levi factor. Namely,
$$\mathfrak g =\mathfrak h \ltimes s, $$
where $\mathfrak h$ is a semisimple Lie algebra and $s$ is the (solvable) radical of $\mathfrak g$.
Observe that the intersection $\mathfrak h\cap \mathfrak u^p$ has codimension $1$ in $\mathfrak h$.
Moreover, since $\mathfrak h$ is semisimple, the projection of $\mathfrak u ^p$ to $\mathfrak h$ cannot be onto. This implies, since the codimension of $\mathfrak u ^p$ in $\mathfrak g$ is $1$, that
\begin {equation}\label {Levi9}
\mathfrak u ^p = \left( \mathfrak h\cap \mathfrak u^p \right) \ltimes \mathfrak s
\end {equation}
Assume that $\mathfrak h = \mathfrak h _1 \oplus \cdots \oplus \mathfrak h _r$
is a direct sum of simple ideals. With the same arguments as before the projection $\mathfrak h'_i$ of
$\mathfrak u ^p$ to $\mathfrak h _i$ has codimension $1$ in $\mathfrak h _i$ and coincides with $\mathfrak h_i\cap \mathfrak u^p $, $i=1, \cdots , r$.
We must have
$\mathfrak u ^p = (\mathfrak h'_1 \oplus \cdots \oplus \mathfrak h'_r) \ltimes \mathfrak s$, which implies,
$r=1$, since $\mathfrak u ^p$ has codimension $1$.
If $\mathfrak h$ is of the compact type, then $\mathfrak h\cap \mathfrak u^p$ is solvable and so abelian. Since $\mathfrak h$ is simple, this intersection must be properly contained in $\mathfrak h$ and so of codimension $1$. Then $\mathfrak h\cap \mathfrak u^p$ is in the center of $\mathfrak h$ since each $\mathrm{ad}(x)$ is skew-symmetric w.r.t. the Killing form. A contradiction that shows that $\mathfrak h$ is of the non-compact type.
Let $\mathfrak h = \mathfrak p \oplus \mathfrak k$ be the Cartan decomposition of
$\mathfrak h$. Then $\mathfrak k$ has an $\mathrm {ad}_{\mathfrak k}$-invariant positive definite
inner product $\langle \, , \,\rangle$. The intersection $ \mathfrak k\cap \mathfrak u ^p $ is solvable hence abelian and has codimension at most $1$. Since each $\mathrm{ad}(x)$, $x \in {\mathfrak k}$ is skew-symmetric w.r.t. $\langle \, , \,\rangle$ we conclude that $\mathfrak k$ is abelian. So, in any case, $\mathfrak h = \mathfrak {sl}_2$.
We have shown, if $\mathfrak g$ is not solvable, that
$$\mathfrak g = \mathfrak {sl}_2 \ltimes \mathfrak s, $$
where $\mathfrak s$ is the radical of $\mathfrak g$.
Let
$\mathfrak b := \mathfrak {sl}_2\cap \mathfrak u ^p$, which is solvable since $\mathfrak u ^p$ is so.
As previously observed,
$$\mathfrak u ^p= \mathfrak b \ltimes \mathfrak s. $$
Since $\mathfrak b $ is solvable, there exist, as it is well-known, a basis $A, B, C$ of
$\mathfrak {sl} _2$, such that $A, B$ span $\mathfrak b$ and
\begin {equation} \label {sl2tri}
[A, B] = 2B, \ \ \ [A, C] = -2C, \ \ \ [B, C] = A
\end {equation}
i.e. $A, B, C$ is a so-called $\mathfrak {sl}_2$-triple.
We will identify any element $v\in \mathfrak g$ with the Killing field $q\mapsto v.q$ of $M$. This identification is a Lie algebra anti isomorphism. With this identification, after replacing $A$ by $-A$, we have the same relations of (\ref {sl2tri}) for $A, B, C$.
\
\begin {lema}\label {nablaAB}
\
\begin {enumerate}
\item $ \nabla _{\nu _p} B = \{0\}$
\item $\nabla _{\nu _p} A = \{0\}$
\item $\nabla _{\nu _p} C = \{0\}$
\end {enumerate}
\end {lema}
\begin {proof}
From equation \ref{curvatureformula} we have that
$$ 2\nabla B = \nabla [A, B] = R_{A, B} - [\nabla A , \nabla B] . $$
Then
\begin {eqnarray}\label {12453}
2(\nabla B)_{\vert \nu _p} = (\nabla [A, B])_{\vert \nu _p}&=& - [\nabla A , \nabla B]_{\vert \nu _p}\\
&=& -[(\nabla A)_{\vert \nu _p} , (\nabla B)_{\vert \nu _p}].
\end {eqnarray}
where last equality follows from the fact that $A, B \in \mathfrak u ^p$ and so $\nabla _{\nu _p}A , \nabla _{\nu _p}B \subset \nu _p$.
But the skew-symmetric endomorphism $ -[(\nabla A)_{\vert \nu _p} , (\nabla B)_{\vert \nu _p}]$ of
$\nu _p$ is perpendicular to $(\nabla B)_{\vert \nu _p}$ (with the usual inner product).
Then $\nabla _{\nu _p}B = \{0\}$ which proves (i).
\vspace {.2cm}
From (2.5) we have that
\begin {eqnarray}\label {aa994}
\nabla _{\nu _p}A &=& \nabla_ {\nu _p} [B, C]
= - (\nabla _{\nabla _{\nu _p} C} B - \nabla _{\nabla _{\nu _p}B} C) \\
&=& - \nabla _{\nabla _{\nu _p}C} B,
\end {eqnarray}
where the last equality is due to (i).
On the one hand, $ - \nabla _{\nabla C _{\nu _p} }B$ must be perpendicular to
$\ker (\nabla B)_p \supset \nu _p$. On the other hand, $\nabla _{\nu _p}A \subset \nu _p $. Then, from \ref {aa994}, we obtain that $\nabla _{\nu _p}A = \{0\}$, and so (ii).
\vspace {.2cm}
Recall that $[A, C] = -2C$, and let $v\in \nu _p$ be arbitrary. Then, by equation \ref {curvatureformula}
\begin {eqnarray}
-2\nabla _{v}C &=& \nabla _{v}[A, C]= - (\nabla _{\nabla _{v} C} A - \nabla _{\nabla _{v}A} C) \\
& = &
- \nabla _{\nabla _{v} C} A,
\end {eqnarray}
where the last equality is due to (ii).
Since $(\nabla A)_p$ is skew-symmetric the last term of the above equality is perpendicular to
$\nabla _{v} C$. But the first term of this equality is proportional to $\nabla _{v} C$.
Then $\nabla _{v} C= 0$, which proves (iii).
\end {proof}
\
Lemma \ref {nablaAB} implies that $C$ belongs to the bounded algebra $\mathfrak u ^p$.
But $\mathfrak g$ is linearly spanned by $C$ and $\mathfrak u ^p$. Then
$$\mathfrak g ^p =\mathfrak u ^p.$$
This contradicts Lemma \ref {aaabb}.
Then $\mathfrak g$ has no Levi factor and so we obtain the following result:
\begin {teo}\label {Gsolvable} Let $M=G/H$ be a simply connected homogeneous Riemannian manifold without Euclidean de Rham factor. Assume that the nullity distribution of $M$ is non-trivial and of codimension $k=3$. Then $H= \{e\}$ and $G$ is solvable.
\end {teo}
\medskip
\section {The leaves of $\nu$ are closed and $\nu ^\perp$ is completely non-integrable} \label {LTA}
\smallskip
\begin {lema}\label {EDF} Let $M= G/H$ be a (non-simply connected) homogeneous Riemannian manifold and let $\mathcal {D}^0$ be the parallel distribution of $M$, associated to its local Euclidean de Rham factor of $M$ ($G$ connected). Let $\bar F (p)$ be the closure of a (maximal) integral manifold $F(p)$ by $p$ of $\mathcal {D}^0$. Then there is a closed abelian normal subgroup $A$ of $ I(M)$ (not depending on $p$ and non-necessarily contained in $G$) such that
$\bar F (p) = A\cdot p$, for all $p\in M$. In particular, $\bar F (p)$ is a
flat (embedded) homogeneous submanifold $M$.
\end {lema}
\begin {proof}
Let $\tilde M$ be the universal cover of $M$ and write it as
$\tilde M = \mathbb R^k\times M_1$, where $\mathbb R^k$ is the Euclidean
de Rham factor. Let $\tilde G \subset I(\tilde {M})$ be the (connected) lift of $G$. Let
$\Gamma $ be the deck transformations of $\tilde {M}$. Then $\Gamma$ commutes with
$\tilde G$. Let $\Gamma ^0$ be the image of the projection of $\Gamma$ to $I(\mathbb R^k)$. Let $\tilde G^0$ and $\tilde G^1$ be the images of the projections of $\tilde G$ into
$I(\mathbb R^k)$ and $I(M_1)$, respectively. Then $\Gamma ^0$ commutes with the transitive group $\tilde G^0$ of isometries. Then the elements of $\Gamma ^0$ are translations (see Section \ref {homogeneous flat spaces}).
Let $\tilde {T} \simeq \mathbb R ^k$ be group of translations of $\mathbb R ^k$.
Then $\tilde {T} \times \tilde G^1$ acts transitively on $\tilde M$ and commutes with $\Gamma$. Then $T \times \tilde G^1$ projects to a transitive group of isometries of $M$. Let $T \subset I(M)$ be the projection of $\tilde T$. Observe, since $\tilde {T}$ is a normal subgroup of $I(\tilde M)$, that $T$ is a normal subgroup of $I(M)$.
Let $A$ be the closure of $T$ in $I(M)$. Then $A$ is a normal subgroup of $I(M)$ and
$\bar {F}(p) = A\cdot p $.
\end {proof}
\begin {nota}\label {REDF} We are in the assumptions and notation of Lemma \ref {EDF}.
Observe, from the above Proposition, that the family of closures of the integral manifolds of
$\mathcal D^0$ are a $I(M)$-invariant foliation of $M$.
\end {nota}
\medskip
Let $M= G/H$ be a simply connected irreducible homogeneous Riemannian manifold with a non-trivial nullity distribution $\nu$, where $G=I(M)$.
Let, for $p\in M$,
$E^p$ be the Lie subgroup of $G$ that leaves invariant the integral manifold
$N(p)$ of $\nu$ by $p$ (see Section \ref{homogeo}). Observe that since $\nu$ is $G$-invariant, $G_p= (E^p)_p$. We may assume that $p=[e]$ so that $G_p=H$.
Then, since $N(p)$ and $H$ are connected, $E^p$ is connected.
Let $\bar E^p$ be the closure of $E^p$ in $G=I(M)$ and let
$\bar N (p)$ be the closure of $N(p)$ in $M$.
Then
$$ \bar N (p) = \bar E^p\cdot p $$
Since $\nu _p\subset T_p\bar N (p)$, then $\bar \nu = \nu _{\vert \bar N (p)}$ is a distribution
of $\bar N(p)$. Since
$E^p$ is a normal subgroup of $\bar E^p$, then the integral manifolds of the autoparallel distribution $\bar \nu$ are given by
\begin {equation}\label {766}
E^p\cdot x \ \ \ \ \ (x\in \bar N(p)).
\end {equation}
\begin {lema}\label {nubarN} $\bar \nu $ is contained in the nullity distribution of
$\bar N(p)$.
\end {lema}
\begin {proof} Let us first show that $\bar \nu$ is in the nullity of the second fundamental form $\alpha$ of $\bar N(p)$.
Let $X$ be a Killing field of $M$ induced by $\bar E^p$ and let $\gamma _v (t) =
\mathrm {Exp} (tu)p$ be a homogeneous geodesic in $N(p)$, $u\in \mathrm {Lie}
(E^p)$ with $u.p=v$.
The proof of Proposition \ref {existstransv} shows that there is transvection on the direction of $\nabla _vX$. Moreover, from the construction of such a transvection,
one has that $\nabla _vX\in T_p\bar N(p)$. Since the Killing field $X$, induced by
$\bar E^p$, is arbitrary, we conclude that $\nu_p$ belongs to the nullity of
$\alpha$ at $p$ (and the same is true for any $q\in \bar N(p)$.
Then, from the Gauss equation, one obtains that $\bar \nu$ is contained in the nullity distribution of $\bar N(p)$.
\end {proof}
\begin {nota} The proof of Lemma \ref {nubarN} shows the following: let $G'$ be a Lie subgroup of $G$ that contains $E^p$ and let $S = G'\cdot p$.
Then $\nu _{\vert S}$ is contained in the nullity of $S$.
\end {nota}
\begin {teo}\label {MMLPQTP} Let $M= G/H$ be a simply connected irreducible homogeneous Riemannian manifold with non trivial nullity distribution $\nu$, where $G =I(M)^o$.Then any (maximal) integral manifold of $\nu$ is a closed (embedded) submanifold of $M$.
\end {teo}
\begin {proof}
Applying Proposition \ref {KillTanNu} (and the comment below it) any Killing field, induced by
$\bar E^p$, that lies in the nullity of
$\bar N (p)$ must be tangent to $\mathcal D ^0$, the distribution associated to the
local Euclidean de Rham factor of $\bar N(p)$. Since the Killing fields induced by $E^p$ lie in $\nu$, which is included in the nullity of $\bar N (p)$, then
$N(p)$ is included in the integral manifold of $F(p)$ of $\mathcal D ^0$.
By Lemma \ref {EDF} the closure $\bar F (p)$ is a flat embedded submanifold of
$M$. Observe that $\bar F (p) = T^r\times \mathbb R ^s$, since it is flat and homogeneous (see Section \ref {homogeneous flat spaces}).
By Remark \ref {REDF} there is a maximal (connected) Lie subgroup $S$ of $\bar E^p$, that contains $E^p$, such that
$S\cdot p = \bar F (p)$. Then $E^p$ is a normal subgroup of $S$.
Then the orbits $N(y) = E^p\cdot y$ are parallel totally geodesic submanifolds of
$\bar F (p)$, $y\in \bar F (p)$
(see Lemma \ref {wwqq}).
Let us consider the intersection $T_pN(p)\cap T_pT^r$. If this intersection is different from $\{0\}$, then $N(p)$ has a closed geodesic. A contradiction, since
$N(p)$ is isometric to a Euclidean spaces (see Lemma \ref {SC}).
Then $T_pN(p)\cap T_pT^r =\{0\}$. Then, as it is not hard to see,
$N(p)$ is a closed submanifold of $\bar F (p)$. Then $N(p)$ is closed or, equivalently, $E^p$ is a closed subgroup of $I(M)$.
\end {proof}
\begin {lema}\label {wwqq}
Let $G$ be a connected subgroup of $I(\mathbb R ^n)$ which acts transitively on
$\mathbb R ^n$. Let $G'$ be connected normal subgroup of $G$.
Then the orbits of $G'$ are parallel affine subspaces of $I(\mathbb R ^n)$.
\end {lema}
\begin {proof} It is well known that any (connected) Lie subgroup of $I(\mathbb R ^n)$ has a totally geodesic orbit (see e.g. Theorem 3.5, pg. 100 in \cite{AVS}). Then, since $G'$ is a normal subgroup of $G$, all orbits of $G'$ are affine subspaces of
$\mathbb R ^n$. Let $d$ be the distance between
the affine subspaces $G'\cdot x$ and $G'\cdot y$. We may assume that $d(x,y) = d$
So for any $g'x\in G'\cdot x$, $d(g'x, g'y) = d$. Then any point of $G'\cdot x$ is at a distance $d$ to some point in $G'\cdot x$. Then the affine subspace $G'\cdot x$ must be parallel to the affine subspace $G'\cdot y$.
\end {proof}
\smallskip
\subsection {The affine bundle and the connection given by $\mathbf {\nu ^\perp}$}\
\smallskip
We keep the assumptions and notation of this section. From Theorem \ref{MMLPQTP}
we have that $N(p)$ is closed, or equivalently, $E^p$ is a closed subgroup
of $G=I(M)$ ($p=[e]$). Then $M=G/H$ is the total space of a fiber bundle over
$B= G/E^p$, with standard fiber $E^p/H= N(p)\simeq \mathbb R^k$ (with a Euclidean affine structure, see Lemma \ref {SC}). The projection of $M$ onto $B$ will be denoted by $\pi$.
Observe that $B$ is the quotient space $M/\mathcal N$ of $M$
by the leaves of the nullity foliation $\mathcal N = \{N(q): q\in M\}$.
Since the elements of $\mathcal N$ are, in a natural way, Euclidean affine spaces, one has that $M$ is an Euclidean affine bundle (and so an affine combination of local sections of $M$ is a local section). Observe that $G$ leaves $\mathcal N$ invariant and, for any
$g\in G$, $g$ is an isometry between $\pi ^{-1}(\pi (q)) = N(q)$ and $\pi ^{-1}(\pi (gq))=N(gq) $. Moreover, Lemma \ref {KillTanNu} implies that $G$ acts almost effectively on $M/\mathcal{N}=B$.
Let us consider the natural affine connection on $M\overset {\pi}{\to}B$ given by the distribution $\nu ^\perp$. In fact, a perpendicular variation of totally geodesic manifolds, is by isometries. So, the local horizontal lift of curves in $B$ gives rise to local isometries between the involved fibers. From this particular situation, it is well known, and standard to show, that any piece-wise differentiable curve
$c:[0,1]\to B$ can be lifted to a (unique) horizontal curve $\tilde c_u:[0,1]\to M$
with $\tilde c_u (0) = u$, for any
$u\in \ \pi ^{-1}(c(0))$. Then there is a well defined parallel transport
$\tau _c: \pi ^{-1}(c(0)) \to \pi ^{-1}(c(1))$, which is an isometry, given by
$\tau _c (u) = \tilde c _u (1)$. Then $\nu ^\perp$ is an affine connection.
For each $b\in B$, let $\Phi (b)\subset I(\pi ^{-1}(b))$ denote the holonomy group of
$\nu ^\perp$ at $p$ (holonomy groups are conjugated by parallel transport). Note that $B$ is simply connected, since $M$ is simply connected and the fibers are connected. Then the holonomy groups $\Phi (b)$ are connected.
Let us consider, for $q\in M$, the holonomy subbundle
$\mathrm {Hol}(q)$.
Namely, $\mathrm {Hol}(q)$ consists of all the elements of $M$ that can be reached
from $q$ by a horizontal curve. The holonomy subbundles foliate $M$
. Moreover, any holonomy subbundle intersects any given fiber $\pi ^{-1}(b)$ in an orbit of the holonomy group $\Phi (b)$.
The holonomy subbundles, despite what happens in a principal bundle,
may have different dimensions depending on the dimensions of the orbits of the holonomy group. But in our case the holonomy subbundles have all the same dimension (and so their tangent spaces define a smooth distribution), since, for any $g\in G$, $u\in M$,
\begin{equation} \label {holhol}
\mathrm {Hol}(gq) = g\mathrm {Hol}(q) \text { \ \ \ and \ \ \ }
\Phi (\pi (g q)) =g (\Phi (\pi (q))g^{-1}.
\end{equation}
In particular, if $g\in G^q$ then $\Phi (\pi (q)) =g (\Phi (\pi (q))g^{-1}$.
This implies that $\bar G^q = \{g_{\vert N(q)} : g\in G^q\}$ is included in the normalizer of $\Phi (\pi (q))$ in $I(N(q))$.
Then $L = \Phi (\pi (q)) . \bar G^q$ is a Lie group of isometries, which is transitive
on $N(q)\simeq \mathbb R ^k$, and $\Phi (\pi (q))$ is a normal subgroup of $L$.
Then, by Lemma \ref {wwqq}, we have that:
\centerline { \ ($*$) \ {\it The orbits of $\Phi (\pi (q))$ are parallel affine subspaces of $N(q)$.}}
\
Observe that the above property implies that $\Phi (\pi (q))$ {\it acts polarly} on
$N(q)$.
\medskip
Let $\mathcal {Y}$ be the distribution of $M$ defined by the normal spaces of
the holonomy subbundles. Namely,
\begin {equation}\label {abcrrr} \mathcal {Y}_q = (T_q\mathrm {Hol}(q))^\perp \subset \nu _q
\end {equation}
Then $\mathcal {Y}\subset \nu$ and, from ($*$),
$\mathcal {Y}_{\vert N(q)}$ is a parallel (i.e. constant) foliation of $N(q)$, for all $q\in M$ (which is perpendicular to the holonomy orbits). Moreover, since all the orbits of $\Phi (\pi (q))$ are principal orbits:
\smallskip
\centerline { ($**$) \ \ {\it any $w\in \mathcal {Y}_q$ is a fixed vector of the isotropy $\Phi (\pi (q))_q$.}}
Let $p\in M$ be fixed and let $\xi \in \mathcal {Y}_p$. Then $\xi$ induces a normal vector field of $\mathrm {Hol}(p)$ in the following way: if $q\in \mathrm {Hol}(p)$, choose $\tilde c_p : [0,1] \to M$ be a horizontal piece-wise differentiable curve
with $\tilde c_p(0)=p$, $\tilde c_p(1)=q$. Let $c=\pi \circ \tilde c _p$, then
define
$$\tilde \xi (q)= \mathrm {d}\tau _c (\xi)$$
From ($**$) one obtains that $\tilde \xi$ is well defined (and it is standard to show that it is smooth). Let us show that $\tilde \xi$ is a parallel normal vector field.
Observe that
$$T_p\mathrm {Hol}(p) = \nu ^\perp_p
\oplus T_p(\Phi (\pi (p))\cdot p)$$
From the construction of $\tilde \xi$, taking into account that $\Phi (\pi (p))$ acts polarly on $N(p)$, one obtains that
$$\nabla ^\perp _v\tilde \xi= 0, \text {\ \ \ \
for all } v\in {T_p(\Phi (\pi (p))\cdot p))}$$
Let now $u\in \nu ^\perp _p$ and let $\tilde c _p (s)$ be a horizontal curve with
$\tilde c^\prime _p (0)= u$ and let $c(t) =\pi (\tilde c_p(t))$. Let $\gamma _\xi (t)$ be a geodesic (i.e. a line) in $N(p)$ with $\gamma ^\prime_\xi (0) =\xi$.
Let $\tau ^s$ be the parallel transport along $c(s)$, form $0$ to $s$.
Let us consider $$f(s,t) = \tau ^s (\gamma _\xi (t)), $$
which is variation of geodesics that are tangent to $\nu$.
Then
$$J_ \xi (t):=\tfrac {\partial f }{\partial s}_ {\vert (0, t)}$$
is a Jacobi field along the geodesic
$\gamma _{\xi}(t)$.
Note that $s\mapsto f(s,t_0)$ is a horizontal curve,
i.e. tangent to $\nu ^\perp$, and so $J^\xi (t)$ is a horizontal field along
$\gamma _\xi (t)$.
Since $N(p)$ is totally geodesic in $M$, $\nu ^\perp _{\vert N(p)}$ is a parallel subbundle of the pull-back of $TM$, via the inclusion of
$N(p)$ in $M$.
Then
\begin {equation} \label {4544}
J_\xi ^\prime(t) := \tfrac {\mathrm {D}\, }{ \mathrm {d}t}J_\xi (t),
\end {equation}
as well as $J(t)$, are horizontal fields along $\gamma _\xi (t)$.
Observe that
\begin {eqnarray*}\label {2233}
J_\xi ^\prime(0) &=& \tfrac {\mathrm {D}\, }{ \partial t}_{\vert (0,0)}
\tfrac {\partial \,}{\partial s}f (s,t)=
\tfrac {\mathrm {D}\, }{ \partial s}_{\vert (0,0)}
\tfrac {\partial \,}{\partial t}f (s,t) \\
&=& \tfrac {\mathrm {D}\, }{ \mathrm {d} s}_{\vert 0}\mathrm {d}\tau ^s (\xi)
= \tfrac {\mathrm {D}\, }{ \mathrm {d} s}_{\vert 0}\tilde \xi (\tilde c_p(s)) \\
&=& \nabla _u\tilde \xi \in \nu ^\perp _p
\end{eqnarray*}
Since $u\in \nu ^\perp_p$ is arbitrary, then $\tilde \xi$ is also $\nabla ^\perp$-parallel in the horizontal directions and so
$$\nabla ^\perp \tilde \xi =0. $$
Moreover, from (\ref {2233}) one also obtains that
any shape operator of $\mathrm {Hol}(p)$ leaves $\nu ^\perp _p$ invariant.
Observe, since $\xi\in (T_p(\mathrm {Hol}(p)))^\perp$ is arbitrary, that:
\smallskip
\centerline {\ ($***$) \ \ {\it The normal bundle of
$\mathrm {Hol}(p)$ is globally flat. }}
\smallskip
Let $A$ be the shape operator of $\mathrm {Hol}(p)$. It is standard to show, since $\nabla ^\perp\tilde \xi =0$, that $J_\xi ^\prime(0) = -A_\xi (u)$. So, the Jacobi field $J_\xi (t)$ has the following initial conditions:
\begin {eqnarray*}\label {7545}
\ J_\xi (0)&=& u, \\
J_\xi ^\prime (0)&=& -A_\xi (u)
\end {eqnarray*}
Note that $J_\xi (t) = \tilde c _{\gamma _\xi (t)}^\prime (0)$, where
$\tilde c _{\gamma _\xi} (t)$ is the horizontal lift of $c(t)=\pi (\tilde c _p(t))$.
Then
\begin {equation}\label {3976} \mathrm {d}\pi (J_\xi (t) ) = c^\prime (0)= \mathrm {d}\pi (u)
\end {equation}
Assume that $\lambda \neq 0$ is an eigenvalue of $A_\xi$ and let $u\neq 0$ be an eigenvector of $A_\xi$ associated to $\lambda$. Since $\gamma _\xi (t)$ lies in
$N(p)$, then
$$J_\xi (t) = (1-t\lambda)\hat u (t),$$
where $\hat u (t)$ is the parallel transport of $u$ along $\gamma _\xi (t)$.
Then $J(1/\lambda) =0$, which contradicts (\ref {3976}). This shows that
$$A_{\xi \vert \nu ^\perp_p}=0= A_{\tilde \xi (p) \vert \nu ^\perp_p}$$
From its construction, $\tilde \xi$ is constant along the holonomy orbit
$\Phi (\pi (p))\cdot p$ (see ($*$) and ($**$)).
Then
$$A_{\xi \vert T_p(\Phi (\pi (p))\cdot p)}=0$$
and so
$$A_\xi =0.$$
Since $p$ and $\xi \in (T_p(\mathrm {Hol}(p)))^\perp =\mathcal Y _p$ are arbitrary, we conclude that any holonomy subbundle $\mathrm {Hol}(q)$ is a totally geodesic submanifold of $M$.
Observe that the distribution $\mathcal Y$ is autoparallel, since $\mathcal Y\subset \nu$ and is parallel inside the leaves of the nullity
(see the paragraph below (\ref
{abcrrr})).
Since the holonomy subbundles are totally geodesic and its perpendicular distribution $\mathcal Y$ is autoparallel, we conclude that $\mathcal Y$ is a parallel distribution. This is a contradiction, since $M$ is irreducible, unless $\mathcal Y =0$. Then $\mathrm {Hol}(p) =M$ and so $\Phi (p)$ is transitive on $N(p)$.
By summarizing our main results in the section we obtain:
\begin {teo}\label{transihol} Let $M=G/H$ be a simply connected irreducible homogeneous Riemannian manifold with a non-trivial nullity distribution $\nu$.
Then the quotient space $B$ of $M$ by the leaves of the nullity is a manifold and
$M$ is a Euclidean affine fiber bundle over $B$, with standard fiber isometric to $\mathbb R^k$. Moreover, $\nu ^\perp$ defines a metric affine connection on $M$ with a transitive holonomy group (and so $\nu ^\perp$ is completely non-integrable).
\end {teo}
For Euclidean or spherical submanifolds, the complete non-integrability of the distribution perpendicular to the relative nullity (i.e. the nullity of the second fundamental form) was proved in
\cite {Vi} (this is not true in hyperbolic space).
\section{The proof of the main results}\label{demostraciones}
The main theorems stated in the introduction were proved throughout the paper or are direct consequences of previous results. We sum them up here.
\medskip
\noindent
{\it Proof of Theorem A.} The first part of (1) was proved throughout Section \ref{section5} and concluding with equation (\ref {summarizing}).
The fact that the integral manifolds of $\nu$ are simply connected was proved in Lemma \ref {SC}.
By Corollary \ref {STcor}, and part $(ii)$ of Proposition \ref{JacNul}, the Jacobi operator in any vector tangent to $\nu ^{(1)}$ is null.
So the proof of Lemma \ref {SC} also shows that the integral manifolds of $\nu ^{(1)}$ are simply connected.
Part (2). The existence of an adapted transvection $Y$, see Definition \ref{adapted}, with $Y_p=v \notin \nu_p$ was proved in Proposition \ref{existstransv}. The fact that the Jacobi operator $R_{,v}v$ is null was proved in Theorem \ref {ST} and stated in Corollary \ref{STcor}. Theorem \ref {ST} also states that $[Y,[Y, \mathcal K(M)]] = 0$ and that $Y$ does not belong to the center of $\mathcal K ^G(M)$.
For an arbitrary $v\in \hat {\nu} _p$ the existence of a transvection $Y$ with the stated properties follows from Proposition \ref {JacNul}, since the adapted transvections at $p$ span $\hat \nu _p$.
Part (3) is Theorem \ref {isodim}.
The first part of (4) is Theorem \ref {Gsolvable}.
In Section \ref {EX33} we construct non trivial examples, in any dimension, with $k=3$ and $G$ not unimodular. For the non-unimodularity see Remark \ref{Milnor}. \qed
\vspace {.2cm}
\noindent
{\it Proof of Theorem B.} It follows from Theorem \ref {MMLPQTP} and Theorem
\ref {transihol}. \qed
\vspace {.2cm}
\noindent
{\it Proof of Proposition C.} Part (a) is Corollary \ref{corsemisimple}. If $M$ is compact see Proposition \ref{compact2} for a direct proof. Part (b) is Corollary \ref {cornilpotent}. \qed
\color {black}
\section{Examples with nullity of codimension $3$ and co-index of symmetry $2$}
\label{EX33}
In this section we construct examples of irreducible Riemannian homogeneous spaces with nullity of codimension 3 and co-index of symmetry 2 in any dimension greater or equal to $4$. As explained in the introduction such examples are optimal since neither the nullity can be greater nor the co-index of symmetry can be smaller due to Reggiani's Theorem \cite{R}.
Let $G = \mathbb{R}^{d} \rtimes \mathbb{R} $ be the semidirect product of the abelian groups where $\mathbb{R}$ acts on $\mathbb{R}^d$ as
$\exp(t A)$, $t \in \mathbb{R}$ and $A$ is defined as
\[ A = \begin{bmatrix}a M_{d-1} & a e_1 \\
-a e_1^t & a
\end{bmatrix} \]
where $M_{d-1} = (m_{ij})$ is $(d-1) \times (d-1)$ skew-symmetric with $m_{ij} = 1$ for $i < j$, $e_1$ is the first canonical column of $\mathbb{R}^{d-1}$,$e_1^T$ its transpose and the constant $a$ is chosen so $1 = trace(A A^T)$ , i.e. $a^2 = \frac{1}{3 + (n-2)(n-3)}$. \\
We can regard $G$ as Lie subgroup of $\mathrm{GL}(d+1,\mathbb{R})$ whose Lie algebra is generated by the following $d+1$ matrices:
\[ E_i := \begin{bmatrix} 0 & e_i \\
0 & 0
\end{bmatrix} , \hspace{1cm} \mathbf{A} := \begin{bmatrix} A & 0\\
0 & 0
\end{bmatrix} \]
where $i=1,\dots,d$ and $e_1,\cdots,e_d$ are the canonical columns of $\mathbb{R}^d$.
Let $g$ be the left invariant metric on $G \subset \mathrm{GL}(d+1,\mathbb{R})$ given (on $T_{e}G$ the tangent space at the identity $e \in G$) by $g(X,Y) = \mathrm{trace}(X Y^t)$, where $Y^t$ indicates the transpose matrix.
Observe that $E_1,\cdots,E_d,\mathbf{A}$ is a orthonormal base of $T_{e}G$.
\begin{lema}\label{nablas} Let $\nabla$ be the Levi-Civita connection of $(G,g)$ and $R$ its curvature tensor.
Then at $e \in G$ the following holds:
\begin{itemize}
\item[i)] $\nabla E_1 = \nabla E_2 = \cdots = \nabla E_{d-1} = 0$, i.e. $E_1,\cdots,E_{d-1}$ are transvections at $e \in G$,
\item[ii)] $\nabla E_d =
\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & a \\
0 &-a & 0 \\
\end{bmatrix} $,
\item[iii)] $\nabla \mathbf{A} =
\begin{bmatrix}
-a M_{d-1} & -a e_1 & 0 \\
a e_1^T & 0 & 0\\
0 & 0 & 0
\end{bmatrix}$
\end{itemize}
\end{lema}
\begin{proof} The proof is a computation by using equation (\ref{fundamentalequation}).
To show $i)$ notice that equation (\ref{fundamentalequation}) gives $\langle \nabla_{E_i} E_j, E_k \rangle = 0 $ for any $i,j,k \in {1,\cdots,d}$. Equation (\ref{fundamentalequation}) gives
\[ 2 \langle \nabla_{\mathbf{A}} E_j, E_k \rangle = \langle [\mathbf{A} , E_j],E_k \rangle + \langle [\mathbf{A} , E_k],E_j \rangle \]
so for $ 1 \leq j \leq d-1$ we get
\[ 2 \langle \nabla_{\mathbf{A}} E_j, E_k \rangle = \langle A.e_j ,e_k \rangle + \langle A.e_k ,e_j \rangle = a m_{jk} + a m_{kj} = 0 \]
this shows $i)$.\\
To show $ii)$ observe that from the definition of $A$, if $1<k<d$:
\[ 2 \langle \nabla_{\mathbf{A}} E_d, E_k \rangle = \langle A.e_d ,e_k \rangle + \langle A.e_k ,e_d \rangle = \langle a e_1 + a e_d ,e_k \rangle + \langle A.e_k ,e_d \rangle = 0 \, , \]
\[ 2 \langle \nabla_{\mathbf{A}} E_d, E_d \rangle = \langle A.e_d ,e_d \rangle + \langle A.e_d ,e_d \rangle = 2a \, . \]
this show $ii)$.
Finally, \[ 2 \langle \nabla_{E_i} {\mathbf{A}} , E_j \rangle = \langle [E_i,\mathbf{A}],E_j \rangle + \langle [\mathbf{A},E_j],E_i \rangle \]
then \[ 2 \langle \nabla_{E_i} {\mathbf{A}} , E_j \rangle = -\langle A e_i,e_j \rangle + \langle A e_j,e_i \rangle \]
son if $i,j \in \{1,\cdots,d-1\}$ we have $2\langle \nabla_{E_i} {\mathbf{A}} , E_j \rangle = -a m_{ij}+a m_{ji} = -2a m_{ij} $.\\
Now for $1<j<d$ we get
$2\langle \nabla_{E_d} {\mathbf{A}} , E_j \rangle = -\langle A e_d,e_j \rangle + \langle A e_j,e_d \rangle = -\langle a e_1 + a e_d,e_j \rangle + \langle A e_j,e_d \rangle = 0$.
We need
$2\langle \nabla_{E_d} {\mathbf{A}} , E_1 \rangle = -\langle A e_d,e_1 \rangle + \langle A e_1,e_d \rangle = -2a$.
and for any $j$ we get
$2\langle \nabla_{\mathbf{A}} {\mathbf{A}} , E_j \rangle = 0$. By using that $\nabla \mathbf{A}$ is skew-symmetric this shows $iii)$.
\end{proof}
\begin{lema} The nullity $\nu_{e}$ of $R$ at $e \in G$ is generated by $E_2,\cdots,E_{d-1}$.
\end{lema}
\begin{proof} We are going to compute $R_{X E_i}$ for $i=1,\cdots,d-1$ by using formula (\ref{curvatureformula}).
We have $R_{X E_i} = \nabla [X,E_i]$ since $\nabla E_i = 0$ by $i)$ in Lemma \ref{nablas}. So
\[R_{X E_i} = \nabla [\sum_{j=1}^{j=d} \langle X,E_j \rangle E_j + \langle X,\mathbf{A} \rangle \mathbf{A} ,E_i] = \langle X,\mathbf{A} \rangle \nabla [\mathbf{A} ,E_i] = \langle X,\mathbf{A} \rangle R_{\mathbf{A} E_i} \, .\]
By formula (\ref{curvatureformula}) $R_{\mathbf{A} E_i}=\nabla [\mathbf{A} , E_i]$ then
\[ R_{\mathbf{A} E_i} = \nabla \left( \sum_{k=1}^{d-1} a m_{ki} E_k + -a \delta_{1i} E_d \right) \]
and by Lemma \ref{nablas} we get $R_{\mathbf{A} E_i} = -a \delta_{1i} \nabla E_d$. Then $E_2,\cdots,E_{d-1}$ belongs to the nullity $\nu_{e}$ of $R$.
Since $R_{\mathbf{A} E_1} = -a \nabla E_d$ $ii)$ in Lemma $ii)$ \ref{nablas} implies that \[ \nu_{e} = span\{E_2,\cdots,E_d\} \]
because $R_{\mathbf{A} E_1} \neq 0$ and $ker(\nabla E_d) = span\{E_1,\cdots,E_{d-1}\}$.
\end{proof}
\begin{nota}\label{distriExamples} Observe that in these examples the Killing vector field $Y$ defined as
\[ Y:= E_1 - \sum_{k=3}^{d-1} E_k \] is a transvection as in $ii)$ of Theorem {\bf A}. Indeed, by $iii)$ of Lemma \ref{nablas} $Y = \nabla_{E_2} \mathbf{A}$ hence $Y \in {\hat \nu}_1$ and $Y \notin \nu_1$ and the Jacobi operator $R_{\cdot, Y}Y$ is null:
\[ R_{X, Y}Y = R_{X , E_1}E_1 = \langle X,\mathbf{A} \rangle R_{\mathbf{A} E_1}E_1 = \langle X,\mathbf{A} \rangle (-a) \nabla_{E_1} E_d = 0 \, .\]
Actually, in our examples $\nu^{(1)}_{e}=span\{E_1, E_2,\cdots,E_{d-1}\}$ and
$$\nu^{(2)}_{e}={\mathcal{U}}_{e} = span\{E_1, E_2,\cdots,E_d\} \, \, .$$
\end{nota}
\begin{lema}\label{nonablainv} There is non (non-trivial) $\nabla \mathbf{A}$-invariant subspace in $\nu_{e}$.
\end{lema}
\begin{proof} According to $iii)$ of Lemma \ref{nablas} and the previous lemma
a non-trivial subspace of $\nabla \mathbf{A}$ in $\nu_{e}$ produces a non trivial subspace of the matrix $M_{d-1}$ in the subspace of $\mathbb{R}^{d-1}$ generated by the vectors $e_2,\cdots,e_{d-1}$. But this is not possible by the lemma in Appendix.
\end{proof}
To show that $(G,g)$ is an irreducible Riemannian manifold we use the following Lemma
\begin{lema} Let $M=G/H$ be a homogeneous Riemannian manifold whose nullity distribution has codimension 3.
If $M$ is not locally irreducible then the flat factor of the local de Rham decomposition is non-trivial.
\end{lema}
\begin{proof} If there is not local flat factor then there is an irreducible local factor whose nullity has codimension 1 hence this factor is flat. Contradiction.
\end{proof}
\begin{teo}\label{examples} For each $n \geq 4$ the $n$-dimensional simply connected homogeneous Riemannian manifold is irreducible. Its nullity distribution has codimension 3 and its co-index of symmetry is $2$. Moreover its Ricci tensor has four eigenvalues: zero with multiplicity $n-3$, $-a^2$ and $a^2 \left( \frac{-1 \pm \sqrt{5}}{2}\right)$. So $(G,g)$ has sectional curvatures of both signs and its scalar curvature is $-2a^2 = \frac{-2}{3 + (n-2)(n-3)}$.
\end{teo}
\begin{proof}
That the index of symmetry $i_\mathfrak{s}(G)$ is $n-2$ follows from $i)$ in Lemma \ref{nablas}.
To show that our examples are irreducible Riemannian manifolds assume, by contradiction, that for some $(G,d)$ is reducible. Then by the above Lemma there is a non-trivial flat factor $\mathbf{E}$ of $(G,g)$.
The tangent bundle $T\mathbf{E}$ is contained in the nullity distribution $\nu$.
Moreover $[\mathbf{A} , T \mathbf{E}] \subset T\mathbf{E}$ since $T\mathbf{E}$ is $I(G,g)$-invariant.
Let $Z$ be a parallel vector field of $(G,g)$ tangent to $\mathbf{E}$.
Then $([\mathbf{A} , Z])_{e} = (\nabla_Z \mathbf{A})_{e} - (\nabla_{\mathbf{A}} Z)_{e} = (\nabla_Z \mathbf{A})_{e} \in (T\mathbf{E})_{e} \subset \nu_{e}$.
Since the parallel vector fields $Z$ generate $T\mathbf{E}$, it follows that $\nabla \mathbf{A}$ leaves invariant the non-trivial subspace $(T\mathbf{E})_{e}$ of $\nu_{e}$. This contradicts Lemma \ref{nonablainv}.
A direct computation using Lemma \ref{nablas} and formula (\ref{curvatureformula}) shows that the Ricci tensor restricted to $\nu_{e}^{\perp} = \mathrm{span}\{E_1,E_2,\mathbf{A} \}$ is given by the matrix $\begin{bmatrix} 0 & a^2 & 0 \\ a^2 & -a^2 & 0 \\ 0 & 0 & -a^2 \end{bmatrix}$.
\end{proof}
\begin{nota} For $d=3$, by changing the matrix $A$, it is possible to show the existence
of 1-parameter family $(G_{\lambda},g_{\lambda})$ of 4-dimensional non homothetic irreducible homogeneous
metrics with nullity of dimension $1$.
\end{nota}
\begin{nota} With the same ideas and modifying the matrix $A$ it is possible to construct examples with $k>3$.
\end{nota}
\begin{nota}\label{Milnor} Observe that $\mathrm{trace}(\mathrm{ad}(\mathbf{A})) \neq 0$ i.e. our solvable groups are not unimodular hence they do not admit finite volume quotients \cite[Remark, Lemma 6.2.]{Mi76}.
\end{nota}
\section{Appendix: invariant subspaces of the skew-symmetric matrix $M$}\label{skewM}
Let $M = (m_{ij})$ be the real $d \times d$, $(d>2)$, skew-symmetric matrix with $m_{ij} = 1$ if $i<j$.
Let $\mathbb{W} \subset \mathbb{R}^d$ be the subspace generated by canonical vectors $e_2,\cdots,e_d$ i.e. the orthogonal complement of the first canonical vector $e_1$.
The goal of this appendix is to prove the following:
\begin{lema} There are no $M$-invariant (non-trivial) subspace contained in $\mathbb{W}$.
\end{lema}
\begin{proof}
By contradiction assume that there is a non-trivial $M$-invariant subspace $\mathbb{U} \subset \mathbb{W}$.
We have to consider two cases $dim(\mathbb{U})=1$ or $dim(\mathbb{U})=2$.
Case $dim(\mathbb{U})=1$. Since $M$ is skew-symmetric we have that $\mathbb{U} \subset \ker(M)$. Let $(a_1,\cdots,a_d) \neq 0 \in \mathbb{U} \subset \ker(M)$.
Then
\begin{equation} \label{defM} M.a = (\sum_{k=2}^{d}a_k , \cdots , -\sum_{k=1}^{i-1}a_k + \sum_{k=i+1}^d a_k , \cdots, -\sum_{k=1}^{d-1}a_k) \, .
\end{equation}
So subtracting two consecutive components we obtain:
\begin{center}
$\begin{cases} 0 = a_2 - a_1 , \\ 0 = a_3 - a_2 \\ \cdots \\ 0 = a_d - a_{d-1} .
\end{cases}$
\end{center}
Since $a \in \mathbb{U} \subset \mathbb{W}$ we have $a_1 = 0$ hence $a=0$. Contradiction.
Case $dim(\mathbb{U})=2$. In this case $\mathbb{U}$ is spanned by two vectors $a, b:=M.a$. Since $M$ is skew-symmetric there is $r \neq 0 \in \mathbb{R}$ such that $M.b = r.a$. We can assume $r \neq 1$ since otherwise the vector $a+b$ is $M$-invariant which was excluded in Case 1.
By using equation \ref{defM} we get \begin{center}
$\begin{cases} b_1 - b_2 = a_1 + a_2 \\ ra_1 - ra_2 = b_1 + b_2
\end{cases}$
\end{center}
By using $a,b \in \mathbb{U} \subset \mathbb{W}$ we get that $a_1=b_1=0$ and so $\begin{cases} - b_2 = a_2 \\ - ra_2 = b_2
\end{cases}$ hence $a_2=b_2=0$ due to $r\neq1$.
Now equation \ref{defM} gives us for $i=1,\cdots,d-1$
\begin{center}
$\begin{cases} b_i - b_{i+1} = a_i + a_{i+1} \\ ra_i - ra_{i+1} = b_i + b_{i+1}
\end{cases}$ \,.
\end{center}
So if we assume, as inductive hypothesis, that $a_1=a_2=\cdots=a_i = b_1=b_2=\cdots=b_i = 0 $ we obtain for $a_{i+1},b_{i+1}$:
\begin{center}
$\begin{cases} - b_{i+1} = a_{i+1} \\ - ra_{i+1} = b_{i+1}
\end{cases}$ \,.
\end{center}
hence $a_{i+1},b_{i+1} = 0$ due to $r \neq 1$. Then $a=b=0$ a contradiction.
\end{proof}
|
2,869,038,155,711 | arxiv | \section{INTRODUCTION}
The HIE-LINAC strives to increase the energy and quality of post-accelerated radioactive ion beams (RIBs) delivered by the ISOLDE nuclear facility at CERN~\cite{tech_opts}. The linac will comprise of two dedicated sections for low and high energy, containing cavities designed with geometric reduced velocities, $\beta_{0}$, of 6.3\% and 10.3\% respectively. The high energy section will be built first and boost the energy of the existing facility from 3~MeV/u to 10~MeV/u. The low energy section will replace the NC 7-gap and 9-gap resonators and provide full energy variability of the RIB. We present studies of the high-$\beta$ cavity and full beam dynamics simulations of stage 2a of the HIE-LINAC, which is outlined schematically in Fig.~\ref{stage2a}.
\begin{figure}[htb]
\centering
\includegraphics*[width=82mm]{TH6PFP026f1.eps}
\caption{The stage 2a layout of the HIE-LINAC.}
\label{stage2a}
\end{figure}
The design of the HIE-LINAC lattice has been discussed in~\cite{bd_LINAC08} and the high energy cryomodule lattice is presented schematically in Fig.~\ref{high_cryo}.
\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{TH6PFP026f2.eps}
\caption{The lattice design of the high-energy cryomodule. Dimensions in~mm.}
\label{high_cryo}
\end{figure}
The design of the solenoid is being developed at CERN~\cite{private_remo}, with the capability of delivering a longitudinal axial peak field of 120~kG and a stray field at the adjacent cavities of only 0.02~G. A three dimensional particle tracking routine has been written in order to study the single particle dynamics within the high-$\beta$ cavity. The electromagnetic cavity fields implemented in the tracking routine were simulated using the \texttt{MWS}~\cite{MWS} code. The multi-particle beam dynamics simulations were performed by implementing the field maps in the program \texttt{TRACK}~\cite{TRACK}.
\section{SINGLE PARTICLE BEAM DYNAMICS IN THE HIGH-$\beta$ CAVITY}
The SC QWRs operate at a frequency of 101.28~MHz and are designed to deliver a gradient of 6~MV/m. They posses two gaps and operate in $\pi$-mode, providing efficient acceleration over a wide velocity range. The cavities are independently phased and able to deliver beams of optimal energy for the specified range of mass-to-charge states. More details about the cavity design can be found in~\cite{cav}. All stated results consider the QWRs accelerating the beam at a synchronous phase of -20 degrees with respect to the phase of maximum energy gain.
The energy gain on axis was calculated numerically and compared to an approximation in which the accelerating field is constant within the gaps and the ion transit velocity through the cavity is constant. This first-order approximation is good to better that 1\% for velocities above $\beta_{0}$, however, a discrepancy of more than 5\% is observed for the lightest ions at injection energy. In this case, the velocity change in the cavity can reach almost 10\% and in order to quickly and accurately predict the energy gain we are forced to use a second-order analytic approximation, as derived by Delayen in~\cite{delayen}. The analytic expression is shown in Fig.~\ref{delayen}, where all the symbols have the same meaning as in the quoted reference: $T$ is the first-order transit time factor (TTF) and $T^{(2)}$ and $T^{(2)}_{s}$ are the second-order TTFs. The expression allows for a quick computation of the energy gain to better than 0.5\% for ions entering the cavity with any phase and reduced velocities as low as 7.8\% of the speed of light.
\begin{figure}[htb]
\centering
\includegraphics*[width=80mm]{TH6PFP026f3.eps}
\caption{The energy gain as a function of phase for ions with A/q = 2.5, injected into the high-$\beta$ cavity at 8\% the speed of light.}
\label{delayen}
\end{figure}
As a result of the change in velocity, lighter ions at injection energy don't receive maximum acceleration by crossing the cavity centre when the electromagnetic fields are zero and reversing. In the extreme case, with a beam of A/q = 2.5 at 3 MeV/u, we calculate that the maximum energy gain occurs for ions that cross the cavity centre 7.2 degrees in phase later than the switch of sign of the fields. In order to maintain the correct synchronous phase and design energy the bunches must be shifted accordingly in phase.
\section{COMPENSATION OF BEAM STEERING}
The presence of a non-negligible horizontal magnetic field component on the axis of the QWR necessitates a compensation scheme in order to reduce the kick on the beam within the cavity. The beam steering effect is strong and has a magnitude of several tenths of milliradians at the cavity axis. The phase dependency of the steering force can couple the longitudinal and transverse motions causing emittance growth and it is for this reason that we consider a compensation scheme within the cavity, in addition to correctors placed outside of the cryomodules. We employ the compensation scheme described by Ostroumov in~\cite{beam_steer} and deliberately offset the beam axis vertically by 2.6 mm in the cavity. This introduces a compensating electric field component which creates a force in opposition to the magnetic steering force acting on the beam. The amount of offset was chosen by minimising the integrated beam steering effect along the linac and selecting the best compromise for the design specification of mass-to-charge states. After compensation, the beam steering effect is significantly reduced over a broad velocity range as shown in Fig.~\ref{comp}.
\begin{figure}[htb]
\centering
\includegraphics*[width=80mm]{TH6PFP026f4.eps}
\caption{The compensation of a beam with A/q = 4.5.}
\label{comp}
\end{figure}
The difference in the velocity dependence of the electric and magnetic forces causes a deterioration in the compensation quality at low velocity. The RF defocusing forces in the horizontal, \emph{x}, and vertical, \emph{y}, directions at 1 mm either side of the offset beam axis are shown in the Fig.~\ref{RF_defocus}. The asymmetry of the RF defocusing force has consequences for the transverse beam emittance when the beam is rotated in the solenoid focusing channel. The emittance growth of the beam in the solenoid focusing channel is studied in the realistic field beam dynamic simulations.
\begin{figure}[htb]
\centering
\includegraphics*[width=80mm]{TH6PFP026f5.eps}
\caption{The defocusing kick on ions of A/q = 2.5 offset 1~mm from the axis in the horizontal and vertical directions.}
\label{RF_defocus}
\end{figure}
\section{REALISTIC FIELD BEAM DYNAMICS SIMULATIONS}
The latest electromagnetic field maps were implemented into the \texttt{TRACK} code in order to run multi-particle simulations and study the evolution of the beam emittance along stage 2a of the HIE-LINAC. Space charge forces are neglected because of the low beam intensity. In Figs.~\ref{stage2a_bd_2_5} and~\ref{stage2a_bd_4_5}, we present the simulation results of 50000 macro-particles generated with a 6D waterbag distribution and with A/q = 2.5 and 4.5, entering an aligned linac operating at 90 degrees transverse phase advance per period. The transverse emittance growth of the beam is minimal and its source is dominated by the phase spread of the beam at injection in the QWRs of the first cryomodule. The emittance growth arising from the asymmetric transverse distortion of the beam coupled with rotation in the solenoids contributes less than 1\% to the RMS emittance growth. The beam parameters are summarised in Table~\ref{bp}.
\begin{figure}[tb]
\centering
\includegraphics[width=80mm]{TH6PFP026f6.eps}
\caption{The beam dynamics with a RIB of A/q = 2.5.}
\label{stage2a_bd_2_5}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=80mm]{TH6PFP026f7.eps}
\caption{The beam dynamics with a RIB of A/q = 4.5.}
\label{stage2a_bd_4_5}
\end{figure}
\begin{table}[hbt]
\centering
\small
\caption{Beam Parameters for Stage 2a of the HIE-LINAC}
\begin{tabular}{lcc}
\toprule
\textbf{Parameter (A/q)} & \textbf{Input (2.5/4.5)} & \textbf{Output (2.5/4.5)} \\
\midrule
$\alpha_{t}$ & -0.05/-0.15 & -0.1/-0.21 \\
$\beta_{t}$ (mm/mrad) & 1.0/1.0 & 1.0/1.0 \\
$\epsilon^{99.5\%}_{t,norm}$ ($\pi$ mm mrad) & 0.25/0.25 & 0.28/0.26 \\
$\alpha_{l}$ & 1.37/1.37 & -1.05/0.34 \\
$\beta_{l}$ (deg/keV) & 0.043/0.026 &0.015/0.012 \\
$\epsilon^{99.5\%}_{l,norm}$ ($\pi$ ns keV/u) & 1.66/1.66 & 2.05/1.74 \\
Energy (MeV/u) & 3.07/2.80 & 14.49/9.31 \\
\bottomrule
\end{tabular}
\label{bp}
\end{table}
\section{MISALIGNMENT STUDY}
A misalignment study is being carried out in order to specify the tolerances on the alignment of the elements based on the demands of the beam dynamics. The elements will be aligned to an internal reference within each cryomodule and each cryomodule will be aligned separately with the beam. For this reason we first consider a misalignment study of the elements within an individual cryomodule. We use the \texttt{TRACK} code to randomly misalign elements within the first cryomodule in the transverse plane. The procedure simulates the beam centroid as a paraxial single particle at injection and no correction routine is applied. The code is looped over 1000 randomly misaligned linacs within a given tolerance and the output distribution of centroids in phase-space is statistically analysed. The misalignment tolerances are symmetric in the horizontal and vertical transverse directions. The distribution of centroids at output from the first cryomodule is Gaussian and closely independent of the mass-to-charge state of the ion. We present the results of separately misaligning the solenoid and cavities in Fig.~\ref{error_study}. The tolerance on the alignment of the solenoid in the transverse plane must be on average 5.7 times more stringent than the cavities in order to attain the same RMS spread in the centroid distribution after the first cryomodule. The solenoid will require a separate and specialised alignment system within the cryomodule in order to cope with a greater demand for alignment.
\begin{figure}[tb]
\centering
\includegraphics[width=80mm]{TH6PFP026f8.eps}
\caption{The transverse spread of the beam centroid distribution resulting from random misalignments of either the solenoid or the cavities in the HIE-LINAC.}
\label{error_study}
\end{figure}
|
2,869,038,155,712 | arxiv | \section{Introduction}
The relatively long wavelengths and large oscillator strengths of the Mg\,II absorbers make these systems
accessible to ground-based spectrographs at $z\gtrsim0.11$, a redshift range where large galaxy samples can
be effectively collected using 4-m class telescopes.
Early studies of host candidates of Mg\,II absorbers have confirmed the circumgalactic nature of these
systems (e.g.,\ \citealt{bergeron1986a,lanzetta1990a,bergeron1991a,bechtold1992a,steidel1994a,bowen1995a}).
At $z\mathrel{\hbox{\rlap{\hbox{\lower3pt\hbox{$\sim$}}}\hbox{\raise2pt\hbox{$<$}}}} 1$, Mg\,II absorbers are found at projected physical separations $\rho \mathrel{\hbox{\rlap{\hbox{\lower3pt\hbox{$\sim$}}}\hbox{\raise2pt\hbox{$<$}}}} 100$ kpc of normal galaxies
characterized by a wide range of colors and luminosities \citep{steidel1994a,chen2008a}. These absorbers
originate in cool, photo-ionized gas at temperature $T\sim 10^4$ K and trace high
column densities of neutral hydrogen N(H\,I)$ \gtrsim 10^{18}$ cm$^{-2}$ \citep{bergeron1986a,churchill2003a,rao2006a}.
Because of the relatively large oscillator strength, most of the Mg\,II absorbers are likely saturated, especially in SDSS
spectra. In such cases, the absorption equivalent width reflects the underlying gas kinematics rather than the total gas
column density. On average, stronger absorbers are found at smaller
projected distances from the host galaxy (e.g.,\ \citealt{lanzetta1990a,churchill2005a,tripp2005a,kacprzak2008a,barton2009a,chen2010b,nielsen2013a}).
At the same time, Mg\,II absorbers have also been found to be tracers of galactic-scale outflows in star-forming galaxies
(e.g.,\ \citealt{weiner2009a,rubin2010a}).
While several studies have been able to characterize the properties of the CGM based on observations of Mg\,II
absorption (see \citealt{nielsen2013a} for a recent compilation), there is still a lack of a general prescription that
relates these absorbers to the overall galaxy population. Yet it is possible to gain insights into the connection
between the absorbing gas and luminous matter by measuring their two-point correlation function.
The spatial two-point correlation function of astrophysical objects is a powerful tool to
characterize the dark matter halos in which baryons inhabit (e.g.,\ \citealt{davis1983a,zehavi2002a}).
In recent years, many studies have employed the large-scale clustering signal of various astrophysical objects to
constrain the mass of the underlying dark matter halos. This technique relies on
the understanding that the large-scale bias of dark matter halos is monotonically increasing with mass leading to a
direct relationship between the amplitude of the two-point correlation function and the associated mean halo mass.
The two-point correlation function of a wide range of astrophysical objects has been studied.
These objects include, but are not limited to, QSO metal line absorption systems (e.g., \citealt{adelberger2003a,chen2009a}),
damped-Ly$\alpha$ systems (e.g., \citealt{bouche2004b,cooke2006a}), low- and intermediate-redshift SDSS galaxies
(e.g., \citealt{zehavi2002a,tinker2005a}), Quasars (e.g.\ \citealt{ross2009a}), Lyman-break selected
(e.g.,\ \citealt{bullock2002a,trainor2012a}) and red galaxies at high-$z$ (e.g.,\ \citealt{daddi2003a,hartley2013a}).
The two-point correlation function of intervening Mg\,II $\lambda\lambda 2796,2803$ absorbers found in QSO spectra was among the early analyses aimed at characterizing the masses of absorber hosts at intermediate redshifts $z\sim0.5$ (e.g.,\ \citealt{bouche2004a}). In Gauthier et al. (2009, hereafter G09), the authors calculated the large-scale two-point cross-correlation function
between a sample of $\approx 500$ Mg\,II absorbers of strength $W_r^{\lambda2796}>1$\AA\ and a
\emph{volume-limited} sample of $\approx 200$k luminous red galaxies (LRGs) at $z=0.45-0.60$.
These authors found a mild decline in the mean halo bias of Mg\,II absorbers with increasing rest-frame absorption
equivalent width, $W_r(2796)$, similar to the findings of \citet{bouche2006a} and \citet{lundgren2009a} based on
a flux-limited sample.
In addition, G09 found a strong clustering signal of Mg\,II absorbers on scales
$r_p \mathrel{\hbox{\rlap{\hbox{\lower3pt\hbox{$\sim$}}}\hbox{\raise2pt\hbox{$<$}}}} 0.3$ \mbox{$h^{-1}~{\rm Mpc}~$}, indicating the presence of cool gas inside the virial radii of the dark matter halos hosting the passive
LRGs. This result was later confirmed by the spectroscopic follow-up surveys of \citet{gauthier2010a} and \citet{gauthier2011a}.
The observed $b$--$W_r^{\lambda2796}$ relation has profound implications
for the physical origin of the Mg\,II absorbing gas. For instance, if more massive halos contain, on average more Mg\,II gas
along a given sightine, one would expect a monotonically increasing bias with increasing $W_r^{\lambda2796}$. \citet{tinker2008a}
developed a halo occupation distribution model and showed that the observed mildly decreasing trend in the $b$--$W_r^{\lambda2796}$ relation is consistent with a transition in the halo gas properties, from primarily cool
in low mass halos to predominantly hot but with a small fraction of cool gas surviving in high mass halos.
Here we expand upon the G09 analysis, utilizing the largest Mg\,II absorber catalog available from SDSS DR7.
We extend the
two-point correlation function to weaker absorbers with $W_r^{\lambda2796}<1$\AA\ while improving
the precision of the bias measurements of the stronger systems. Specifically, we use a combined Mg\,II catalog
from \citet{zhu2013a} and \citet{seyffert2013a}. As described in the following section, the combined absorber catalog
is five times larger than what was used in G09, increasing the statistical significance of the halo bias
measurements and allowing a clustering analysis based on absorber subsamples from smaller $W_r(2796)$ intervals.
This paper is organized as follows. In section 2, we present the samples of LRGs and Mg\,II absorbers
along with the methodology adopted to compute the two-point correlation functions. The
two-point correlation functions and the derived bias and halo masses of absorber
hosts are presented in section 3. We discuss the implications of our results for the nature of these absorbers
in section 4. We adopt a $\Lambda$ cosmology with $\Omega_M = 0.25$ and $\Omega_{\Lambda} = 0.75$
throughout the paper. All projected distances are in co-moving units unless otherwise stated and all
magnitudes are in the AB system. Stellar and halo masses are in units of solar masses.
\section{Observations and data analysis}
\subsection{LRG catalog}
The clustering signal of Mg\,II absorbers is computed with respect to a reference population of astrophysical
objects acting as tracers of the underlying dark
matter distribution. As discussed in G09, this tracer should be distributed over the
same imaging footprint as the QSO sighlines that were searched for Mg\,II absorbers. In addition, this reference population
must have a redshift distribution similar to that of the absorbers. Marked differences in survey mask definition or redshift distributions
would alter both the shape and the amplitude of the correlation signal in an undesirable fashion.
Since Mg\,II absorbers can be detected at $z\gtrsim0.35$ in SDSS spectroscopic data, the SDSS LRG sample
offers the largest reference population yielding a sufficient
number of galaxy--Mg\,II pairs necessary for a precise calculation of the two-point correlation function.
LRGs are massive, super-$L_*$ galaxies which are routinely observed from the ground
with 4-m class telescopes out to $z\sim 0.7$. The clustering signal of LRGs has confirmed
that these galaxies reside in bias environments characterized by halo masses $\sim 10^{13}$ \mbox{$h^{-1}\,{\rm M}_\odot$}\
(e.g., \citealt{padmanabhan2008a,blake2008a}). They are excellent tracers of the large-scale structures in the universe.
We used the \citet{thomas2011a} photometrically-selected
LRG catalog (hereafter MegaZDR7) which is based on the SDSS DR7 imaging footprint.
The catalog comprises 1.4~M entries distributed over 7746 deg$^2$
encompassing the spring fields of SDSS located at $\alpha=7^h-19^h$. MegaZDR7 covers
the photometric redshift range $z_{\rm ph}= 0.38-0.75$. Note that the SEGUE stripes were excluded
from this catalog. In addition, the three SDSS fall stripes
(76, 82, 86) were excluded from the survey window function of \citet{thomas2011a} mainly to
simplify their survey window function. Given the relatively small contribution of these
stripes, we did not modify the \citet{thomas2011a} galaxy catalog to include these stripes.
\begin{figure}
\centerline{
\includegraphics[angle=0,scale=0.55]{distributions_new2.pdf}
}
\caption{\emph{Top:} Rest-frame equivalent width distribution of the 2211 Mg\,II
absorbers included in the two-point correlation function calculation. \emph{Bottom:} Redshift distribution
of the volume-limited sample of LRGs (\emph{thin line}) at $z=0.45-0.60$ and Mg\,II absorbers
(\emph{bold line}).}
\label{distributions}
\end{figure}
The LRGs were first selected via a series of cuts in a multidimensional color diagram \citep{collister2007a,blake2008a}
while the photometric redshifts were constructed by an artificial neural network code. In all respects, the
LRGs were selected in an identical fashion to the galaxy sample employed in G09. As shown in Figure 2 of
G09, galaxies with $i'>20$ mag have uncertain photo-$z$'s. We thus restricted ourselves to objects
with $i'_{\rm deV}<20$.\footnote{The $i_{\rm dev}$ magnitudes were corrected for Galactic
extinction according to the \citet{schlegel1998a} maps.} Typical errors in the photometric redshifts of bright LRGs
with $i'<20$ are $\sigma_z/(1+z) \approx 0.03 $. Moreover, we applied further cuts by requiring that
$\delta_{\rm sg}$, the star-galaxy separation parameter, to be $\delta_{\rm sg} > 0.2$. According to
\citet{collister2007a}, selecting objects with $\delta_{\rm sg}>0.2$ limits the contamination fraction of M
dwarf stars to $\approx 1.5$\%. These cuts yielded a flux-limited catalog of 1.1M objects.
As discussed in G09, a flux-limited selection criterion creates an inhomogeneous sample of
LRGs, excluding intrinsically fainter and thus less massive objects at higher redshifts. These authors showed
that absorber bias may have been overestimated by as much as $\approx 20$\% in previous studies based on
flux-limited samples of galaxies (e.g.\, \citealt{bouche2006a,lundgren2009a}). Consequently, we adopted
a volume-limited sample of limiting magnitude
\begin{equation}
M_{i'} - 5 \log h < -22
\end{equation}
over the redshift range $z=0.45-0.60$. The limiting magnitude corresponds to the absolute magnitude of our
faintest galaxies while the redshift range was selected to maximize both the number of LRGs and
absorbers included in the calculation. Increasing the upper-limit of the redshift range would select intrinsically brighter
galaxies resulting in a much
smaller sample size. After excluding all galaxies falling outside our survey mask (see section 2.4.2), our
final LRG catalog comprises 333k entries, an increase of $\approx 70$\% compared to G09. The photometric
redshift distribution of the LRGs is presented in the bottom panel of Figure 1.
\begin{figure}
\centerline{
\includegraphics[angle=0,scale=0.40]{ewr_comp2.pdf}
}
\caption{Comparison between the rest-frame equivalent width $W_r^{\lambda2796}$ measured by
both ZM13 and S13 for a subsample of $\sim$ 1500 Mg\,II absorbers listed in both catalogs.
The solid line corresponds to the best-fit linear regression of the data
$W_{r,\rm S13}(2796) = 1.07W_{r,\rm{ZM13}}(2796)-0.07$ (\AA).}
\label{ewr_comp}
\end{figure}
\subsection{Mg\,II catalog}
\citet{zhu2013a} and Seyffert et al. (2013) carried out independent searches of Mg\,II
absorption features in the spectra of distant QSOs found in the SDSS DR7 QSO spectroscopic
catalog, producing the two largest Mg\,II absorber catalogs that are available in the literature.
These separate efforts allow us to evaluate the completeness and confirm the accuracy of each
absorber catalog. By comparing these catalogues, our goal is to establish the most
\emph{complete} Mg\,II catalog while eliminating as many false positives as possible.
Although both catalogs are derived from essentially the same QSO spectra, we found
that they significantly differ in the number of detected absorbers, completeness, and the rate of likely false detections.
Even when a given absorber is identified by both groups, we found systematic differences in the
measurements of $W_r^{\lambda2796}$ (see Figure 2). In this section, we describe how we compared and combined these two
catalogs to yield the final absorber sample adopted for the two-point correlation calculation.
We first describe each catalog separately along with their respective detection techniques.
We then discuss the methods we employed to combine the two catalogues
and produce the final Mg\,II absorber sample.
\subsubsection{Seyffert et al. (2013) catalog}
The Seyffert et al (2013, hereafter S13) Mg\,II absorber catalog was constructed using
a subset of the SDSS DR7 quasar catalog and
following an equivalent methodology to the C\,IV survey described
in detail in \citet{cooksey2013a}. Here we briefly outline the procedure.
Of the 105,783 quasar spectra in Schneider et al. (2010), 79,595 were
searched for Mg\,II systems if they satisfy the following criteria: (a) The QSOs were not
broad-absorption-line QSOs (i.e. were not listed in \citealt{shen2011a}); (b) The median
signal-to-noise ratio exceeds $\langle {S/N} \rangle = 4$ per pixel in the region
where intervening absorbers could be detected, outside of the Ly$\alpha$ forest.
Every quasar spectrum was normalized with a ``hybrid continuum," a fit
combining principle-component analysis, $b$-spline correction, and
pixel/absorption-line rejection. Absorption line candidates were
automatically detected by convolving the normalized flux and error
arrays with a Gaussian kernel with FWHM = 1 pixel, roughly an SDSS
resolution element (resel). The candidate lines with convolved $({
S/N})_{\rm conv} \ge 3.5$ per resolution element in the $\lambda$2796 line and
$2.5\,{\rm resel}^{-1}$ in $\lambda$2803 were paired into candidate Mg\,II
doublets if the separations of the two lines fell within $\pm 150$ km/s of the
expected doublet separation $\Delta\,v=767$ km/s. Any
automatically detected absorption feature with $({S/N})_{\rm conv}
\ge 3.5$ per resolution element and broad enough to enclose a Mg\,II
doublet was included in the candidate list. S13 excluded absorbers
blueshifted by less than 3000 km/s from the QSO redshift. For the purposes of this
constraint, the quasar redshifts were taken from Schneider et al. (2010),
not from \citet{hewett2010a}, as was used in \citet{zhu2013a} and the
rest of the current work.
All candidates were visually inspected by at least one author
of S13 and most by two. They were rated on a four-point scale from
0 (definitely false) to 3 (definitely true). The systems were judged
largely on the basis of the expected properties of the Mg\,II doublet (e.g., centroid alignment,
correlated profiles) but also including possible, associated ions for verification. Any
system with rating of 2 or 3 were included in subsequent analyses.
The wavelength bounds of the absorption lines were automatically
defined by where the convolved $S/N$ array began increasing when
stepping away from the automatically detected line centroid. The new
centroid was then set to be the flux-weighted mean wavelength within
the bounds, and the Mg\,II doublet redshift was defined by the
new centroid of the $\lambda$2796 transition. The sum of the absorbed flux within the bounds sets
the equivalent width. In summary, the S13 catalog contains 35,629 absorbers
over the redshift range $z=0.4-2.3$.
\subsubsection{Zhu \& Menard (2013) catalog}
We also examined the recently published Zhu \& M\'enard (2013, hereafter ZM13) Mg\,II catalog. The ZM13
catalog is based on a sample of 85,533 QSO sightlines distributed over the SDSS DR7
Legacy and SEGUE spectroscopic footprints. In addition, ZM13 included 1411 QSOs from
the \citet{hewett2010a} sample that were not identified in Schneider et al. (2010). Their catalog consists of 35,752 intervening
Mg\,II absorbers over the redshift range $z=0.4-2.3$.
In brief, ZM13 used a non-negative principle component analysis (PCA)
and a set of eigenspectra to estimate the continuum level of each QSO spectrum. Further improvements
on the continuum estimation was done by applying two median filters of 141 and 71 pixels in width. This process
was repeated three times until convergence was achieved.
Once the continuum is determined, candidate Mg\,II
absorbers were selected via a matching filter search involving the Mg\,II doublet and four Fe\,II lines. Absorbers
were identified if their $S/N$ was above a minimum threshold of 4 for the $\lambda2796$ line and 2 for the $\lambda2803$ line.
Each candidate doublet was then fitted with double-Gaussian profiles and candidates were rejected if the measured doublet separation
exceeded 1\AA. To further eliminate false positives, ZM13 made use of the Fe\,II $\lambda2586$ and $\lambda2600$ lines to measure the $S/N$
of the four lines and applied a cut on the $S/N$ to reject false positives. Finally, $W_r^{\lambda2796}$ and $z_{\rm Mg\,II}$ were
determined by fitting a Gaussian (or double Gaussian) to the candidate profile. In summary, the ZM13 method is fully automated and
involves little human intervention.
\begin{figure}
\centerline{
\includegraphics[angle=0,scale=0.44]{wr_snr_s13_2.pdf}
}
\caption{The rest-frame equivalent width $W_r^{\lambda2796}$ of Mg\,II absorbers as measured by S13 with respect to the median
$S/N$ per pixel, $\langle S/N \rangle$, of the QSO spectra over the redshift range $z_{\rm Mg\,II}=0.45-0.60$.
The solid line is a 3rd-order polynomial fit to the bottom 5th percentile of the $W_r^{\lambda2796}$ distribution. We interpret
this fit as the typical lower-limit on $W_r^{\lambda2796}$ that can
be measured along a QSO sightline of given $\langle S/N \rangle$. The determination of this lower limit on $W_{r}(S/N)$
is necessary to generate the catalog of random absorbers (see section 2.3.4).
}
\label{snr_g_wr}
\end{figure}
\subsubsection{Comparing and combining the Mg\,II catalogs}
We first compared the two catalogs and identified common absorbers by selecting those with the same
RA and DEC coordinates. In addition, we made sure that for each matched absorber, the observed wavelength
listed in one catalog's was falling within the wavelength bounds of the other and vice versa.
In the redshift range $z_{\rm Mg\,II}=0.45-0.60$, at velocity separation $\delta v>10000$ \kms\
below the QSO redshift and within our survey mask (see section 2.3.2) we found 1491 matched
absorbers in the ZM13 and S13 catalogs. The redshifts of these absorbers, as measured by S13
is consistent with the published values of ZM13. We adopted the redshifts
of S13.
However, we found that $W_r^{\lambda2796}$ measured by ZM13 is systematically
lower than S13. In Figure \ref{ewr_comp}, we show a comparison of $W_r^{\lambda2796}$ measured
for $\sim 1500$ matched absorbers randomly selected from both catalogs. In this Figure, the solid line corresponds to the best-fit linear regression
of $W_r^{\lambda2796}$ S13 with respect to $W_r^{\lambda2796}$ ZM13. We found
\begin{equation}
W_{r,\rm{S13}}(2796) = (1.07\pm 0.01) \times W_{r,\rm{ZM13}}(2796) - (0.07\pm0.01)
\end{equation}
where $W_{r,\rm{S13}}(2796)$ is measured by S13 and $W_{r,\rm{ZM13}}(2796)$ by ZM13.
To determine which measurement of $W_r^{\lambda2796}$ should be adopted in our final absorber catalog, we examined
$\approx$ 100 randomly selected absorbers in ZM13 and S13. We first fitted the QSO spectra with a series
of third-order $b-$splines to set the continuum level in the spectral regions of the absorbers and we visually
established the boundaries of the absorbing region. We determined $W_r^{\lambda2796}$ by integrating the
flux decrement between the absorption boundaries. Our measurements are in agreement with S13. We found
that ZM13 tends to underestimate the continuum level, although the differences are typically $\lesssim0.1$\AA. We thus adopted
the $W_r^{\lambda2796}$ estimates of S13 and we applied the above $W_r^{\lambda2796}$ correction for absorbers found
only by ZM13.
To establish our final absorber catalogs, we applied further selection criteria. In addition to the absorber
redshift range, QSO--absorber velocity separation, and survey footprint cuts discussed above, we also applied
cuts to eliminate false detections. We first applied a series of $W_r$ ratios to eliminate systems with either
very large ($W_r^{\lambda2796}/W_r^{\lambda2803}>2.2$) or very low ($W_r^{\lambda2796}/W_r^{\lambda2803}<0.59$)
column density ratios that are inconsistent with Mg\,II absorbers being either optically thin or completely saturated.
In addition, we applied a combined 2.9$\sigma$ detection threshold on the doublet components decomposed into a 2.5$\sigma$ ($1.5\sigma$)
threshold on the $\lambda2796$ ($\lambda2803$) transition. Furthermore, one of us (KLC) visually inspected
all the remaining systems and further eliminated likely false positives. Finally, we eliminated four absorbers
found by ZM13 that occur in the Hewett \& Wild (2010) sample of 1411 visually-inspected QSO sightlines because there is no
$S/N$ estimate readily available for these QSO sightlines (see section 2.3.4).
These selection criteria yielded a sample 2323 absorbers. We further restricted the sample to absorbers
with $W_r^{\lambda 2796}>0.4$\AA\ to ensure a large enough sample ($\approx 10000$) of QSO sightlines with
sufficient $S/N$ to detect the \emph{weakest} absorbers. This selection criterion is particularly
important when generating the random absorber catalog for the weakest systems (see section 2.3.4). This final
cut reduces the number of absorbers to 2211.
Of these 2211 absorbers, 1260 were found by ZM13 and S13,
793 were only found by S13, and 158 only by ZM13. A number of reasons explain why 793 absorbers were found
by S13 and not by ZM13. Among them, 265 absorbers were found within $|\delta z| = 0.02$ redward of the
QSO C\,IV emission with the QSO emission redshift defined by Hewett \& Wild (2010). In addition, 40 absorbers
were found within $|\delta z| = 0.04$ blueward of the QSO Mg\,II emission, and 8 were found in near Galactic
Ca\,II H\&K absorbers. A combination of factors, including differences in the automatic line detection
algorithm and user biases could potentially explain why the remaining 480 systems were missed by ZM13
although estimating the relative importance of each factor is beyond the scope of this paper (see S13 for further
details).
As for the 158 absorbers detected by ZM13 and not by S13, 6 occur in sightlines with $\langle S/N \rangle< 4$ pix$^{-1}$ in
the region searched for Mg\,II absorbers, 53 are found in sightlines identified as BALs by \citet{shen2011a}, and 99 have too low $\langle S/N \rangle_{\rm conv}$
in either the $\lambda 2796$ or $\lambda2803$ spectral regions to be automatically detected by S13.
In summary, our final catalog consists of $2211$ absorbers with
$W_r^{\lambda2796}=0.40-5.6$\AA\ and distributed over the redshift range $z=0.45-0.60$. Note
that the typical redshift error of these absorbers is very small and of order $10-20$km/s.
In Figure \ref{distributions}, we show the redshift distributions of LRGs and Mg\,II absorbers
as well as the $W_r^{\lambda2796}$ distribution. The distribution of $z_{\rm Mg\,II}$ is flat while the photo-$z$ distribution increases
sightly toward higher $z$. We divided this Mg\,II sample according to $W_r^{\lambda2796}$ into four bins of approximately equal
number of absorbers. Each bin contains roughly the same number of absorbers as the entire Mg\,II sample considered
by G09. The bins considered have $W_r=[0.4-0.78],[0.78-1.08],[1.08-1.59]$, and $[1.59-5.6]$ \AA.
\subsection{Two-point correlation statistics}
\subsubsection{Method}
A detailed description of the method to measure the two-point correlation function is presented
in G09. In summary, we adopted the Landy \& Szalay (1993, hereafter LS93) minimum variance
estimator to calculate the projected two-point auto- and cross-correlation functions ($w_p$)
between Mg\,II absorbers and LRGs. The LS93 estimator is
\begin{equation}
w_p(r_p) = \frac{D_aD_g-D_aR_g - D_gR_a +R_aR_g}{R_aR_g}
\label{ls93}
\end{equation}
where $D$ and $R$ are data and random points; $a$ and $g$ refer to absorbers
and galaxies; and $r_p$ is the projected co-moving separation on the sky between two objects.
We adopted the same binning as in G09 and divided the pairs into eight $r_p$ bins equally
spaced in logarithmic space and covering the range $0.2-35$ \mbox{$h^{-1}~{\rm Mpc}~$}. The upper limit of 35
\mbox{$h^{-1}~{\rm Mpc}~$} is a few times smaller than the size of our jackknife cells (see section 2.4.3). In the
following subsections, we discuss the selection of a survey mask and the methodology
adopted to generate random galaxies $R_g$ and absorbers $R_a$.
\subsubsection{Survey mask}
When computing the $w_p(r_p)$ statistics, both data and randoms should be distributed on
the same survey mask. If galaxies and absorbers occupy survey windows that are not
completely overlapping on the sky, the shape and amplitude of the correlation signal
would be altered in an undesirable fashion. Hence it is crucial to identify a survey mask that is
common to both LRGs and Mg\,II absorbers and large enough to minimize the number of
objects falling outside the survey mask. In addition,
the same survey mask shall be used to distribute random LRGs and Mg\,II absorbers.
We adopted the SDSS DR7 spectroscopic angular selection function mask\footnote{The file used was $\rm{sdssdr72safe0res6d.pol}$
available
at http://space.mit.edu/~molly/mangle/download/data.html.}
provided by the NYU Value-Added Galaxy Catalog team \citep{blanton2005a} and assembled
with the Mangle 2.1 software \citep{hamilton2004a,swanson2008a}. This mask
represents the completeness of the SDSS spectroscopic survey as a function of the
angular position on the sky. Since LRGs were not identified by \citet{thomas2011a} in
stripes 76,82 and 86, we also excluded these stripes from the survey window.
In addition, our adopted survey mask does not include the SEGUE spectroscopic mask.
\begin{figure*}
\centerline{
\includegraphics[angle=0,scale=0.77]{XX_all5_new3.pdf}
}
\caption{Two-point cross- and auto-correlation functions for the volume-limited sample of LRGs
at $z=0.45-0.60$. In the \emph{top left} panel, the LRG auto-correlation signal is represented by
\emph{open circles} while the Mg\,II--LRG cross-correlation is shown in \emph{solid grey circles}.
For comparison, we show the LRG--LRG auto-correlation function from G09 in \emph{crosses}.
To guide the eye, we included a thick solid line corresponding to the best-fit power-law model $f(x) = ax^b$ of
the LRG auto-correlation signal at $r_p \gtrsim 2$ \mbox{$h^{-1}~{\rm Mpc}~$}. The vertical \emph{dashed} lines demarcate the small-scale (one-halo)
$w_p$ from the large-scale (two-halo) $w_p$ on which the bias calculation is based. In this case, $b=-0.787$ and $a=0.335$. To facilitate the
comparison between the clustering amplitude of absorbers and LRGs, we repeated this solid curve in all five remaining panels.
The \emph{thin black} curve corresponds to the best-fit power-law
model of the cross-correlation function in which we fixed the power-law slope to the value derived
for the LRG auto-correlation signal ($b=-0.787$). The dotted lines and and shaded areas correspond to the 1$-\sigma$
error bars on the amplitude $a$.
The Mg\,II absorbers are divided in five bins according to the rest-frame equivalent
width of the $\lambda2796$ transition $W_r^{\lambda2796}$. Each panel corresponds
to a different $W_r^{\lambda2796}$ bin labelled in the upper right corner. No correction for photometric
redshifts is applied in this figure. The correction will be applied when estimating
the relative bias of Mg\,II absorber hosts (see section 3.2). G09 showed that uncertainties in photometric redshifts of the LRGs
decreases the clustering amplitude ratio between Mg\,II absorbers and the LRGs. }
\label{xx}
\end{figure*}
\subsubsection{The errors on $w_p$}
We consider two independent sources contributing to the errors in the measured $w_p$:
cosmic variance and photometric redshift uncertainties. The contribution of cosmic variance can be estimated
by using the jackknife resampling technique applied to the survey mask. The
sky was separated into $N=192$ cells of similar survey area ($\approx 40$ deg$^2$). The choice of the
number of cells was obtained after running convergence
tests on a varying number of cells (see G09 for more details). The cosmic variance estimate for each $r_{p,i}$
corresponds to the $i$th-diagonal element of the covariance matrix calculated with
the jackknife method,
\begin{equation}
{\rm COV}(w_i,w_j)=\frac{N-1}{N}\sum_{k=1}^{N}(w_i^k-\overline{w_i})(w_j^k-\overline{w_j})
\label{covariance}
\end{equation}
where $k$ represents the iteration in which box $k$ was removed from the calculation.
The mean value $\bar{w_i}$ was calculated for bin $i$ over all $w_p^k$'s. Although $r_p$
bins are correlated on large scales we consider only the diagonal elements of the covariance
matrix when estimating the contribution of cosmic variance to the error on $w_p$.
The second source of errors comes from large uncertainties in the photometric redshifts
of the galaxies. As discussed in G09, photometric redshift uncertainties affect $w_p$ in two distinct ways.
The first effect is to \emph{lower} the overall clustering amplitude by introducing uncorrelated
pairs in the calculation (see section 3.2). The second effect is to add random noise in the
calculation. This is particularly important for the inner $r_p$ bins which particularly suffer from
an uncertainty on $z_{\rm ph}$ which translates into a large fractional uncertainty on $r_p$.
To estimate the random noise in $w_p$ as a result of photo-$z$ uncertainties, we followed
the procedure described in G09. In brief, for each cross- and auto-correlation calculation,
we generated 100 independent realizations of the LRG catalog by sampling the photo-$z$
distribution of each galaxy. We considered that each photo-$z$ distribution can be characterized
by a normal distribution with $\sigma_z=0.03(1+z_{\rm ph})$ and a mean value $z_{\rm ph}$ \citep{collister2007a}.
We then calculated $w_p$ for each realization and determined the photo-$z$ contribution
to the error on $w_p$ by calculating the dispersion among these 100 realizations for each
$r_p$ bin.
We found that while photometric redshift uncertainties can increase the random error by
$\mathrel{\hbox{\rlap{\hbox{\lower3pt\hbox{$\sim$}}}\hbox{\raise2pt\hbox{$<$}}}} 20$\% at $r_p\mathrel{\hbox{\rlap{\hbox{\lower3pt\hbox{$\sim$}}}\hbox{\raise2pt\hbox{$<$}}}} 1$\mbox{$h^{-1}~{\rm Mpc}~$}, the effect is negligible for larger values of $r_p$. Typically
at these larger $r_p$ values, the increase is $<3$\% for the LRG auto-correlation signal. Not surprisingly, the effect is smaller
for the cross-correlation calculation since only one of the two pair members has photometric redshift.
These findings are consistent with G09. Since the contribution of
photo-$z$'s uncertainties is negligible on large scale where the clustering amplitude is calculated,
we thus only considered cosmic variance in the error budget of $w_p$. Note however that photo-$z$'s
also lowers the amplitude of the clustering signal and this effect is significant and is discussed
in section 3.2.
\subsubsection{Generating random LRGs and Mg\,II absorbers}
The RA and DEC positions of each random LRG was generated using the
\emph{ransack} routine available through the Mangle software package.
The redshifts of the random galaxies were determined by sampling the
redshift distribution of the LRGs (see Figure 1). The number of random galaxies was
determined after running convergence tests. As discussed in G09,
having $\sim$ 10 times more random galaxies than the actual number of
LRGs is sufficient to achieve convergence. Consequently, we generated
a catalog of 4M random galaxies.
We randomly assigned the the RA and DEC positions of the random absorbers
among the QSO sightlines that have been surveyed by ZM13 and S13 and fell within our
survey mask. Of these sightlines, 63,294 were
found within our spectroscopic survey mask. The redshift of each random
Mg\,II was determined by sampling the redshift distribution of the absorber
data while $W_r^{\lambda2796}$ was assigned by sampling the distribution
$dN/W_r \propto \exp(-W_r/W_*)$ where $W_*$ is the typical absorber strength
given by equation (5) in ZM13.
To allocate a given random absorber to a QSO sightline, one has to consider
two additional limitations. First, the absorber should be found at velocity
separation $|\delta_v|>10,000$\kms\ from the QSO redshift and outside of the Ly$\alpha$
forest. This effectively
eliminates all QSO sightlines with $z_{\rm QSO}<0.45$ and $z_{\rm QSO}>2.67$.
Second, the QSO spectra should have high enough $S/N$
to detect a random absorber of a given strength.
This constraint becomes particularly important when generating
random absorbers for the \rm{weak} absorber bin ($W_r^{\lambda2796}=0.40-0.78$\AA).
To establish whether or not a given sightline
could harbor a random absorber of strength $W_r^{\lambda2796}$, we empirically determined
the minimum $W_r^{\lambda2796}$ ($W_{r,\rm min}$) that could be detected at a given $S/N$
by adopting the following procedure. First, we determined the median $S/N$ per pixel,
$\langle S/N \rangle$, over the redshift range $z_{\rm Mg\,II}=0.45-0.60$ for all QSO
spectra in the Schneider et al. (2010) catalog. Next, we compared the strength of the
Mg\,II absorbers listed in S13 with $\langle S/N \rangle$. The results are shown in Figure
\ref{snr_g_wr}. We computed a 3rd-order polynomial fit to the bottom 5th-percentile
of the $W_r^{\lambda2796}$ distribution and interpret this fit as the minimum absorber strength, $W_{r,\rm min}(S/N)$,
that could be detected given $\langle S/N \rangle$. The fit is
shown by the solid line in Figure \ref{snr_g_wr}. Note that the value of $W_{r,\rm min}(S/N)$ corresponds
approximately to a 2.5-$\sigma$ detection across the whole range of $\langle S/N \rangle$ considered.
A random absorber of strength $W_r^{\lambda2796}>W_{r,\rm min}$
could be detected and is included in our random absorber catalog.
We repeated the procedures described above until we collected a sample
of 100K random absorbers. We adopted this number of random absorbers
after performing convergence tests, as described in G09.
\section{Results}
In this section, we present the projected two-point
cross-correlation functions LRG--Mg\,II along with the LRG
auto-correlation function for a volume-limited sample of LRGs at $z=0.45-0.60$.
We explored the dependence of the clustering signal on $W_r^{\lambda2796}$
by computing the LRG--Mg\,II cross-correlation for five $W_r^{\lambda2796}$ bins
with each bin having a similar number of absorbers. A summary of each
calculation, including the adopted $W_r^{\lambda2796}$ binning, the number of
absorbers and LRGs along with the number of $D_aD_g$ pairs found in the
first $r_p$ bin is presented in Table 1.
In Figure \ref{xx}, we show the auto- and cross-correlation functions. In the top left panel, we show
the LRG--LRG auto-correlation function in \emph{open circles}. The error bars on the auto-correlation
function are typically smaller than the size of the symbol. We calculated
the best-fit power law model $f(x)=ax^b$ of the auto-correlation signal at $r_p\geq $ 2 \mbox{$h^{-1}~{\rm Mpc}~$}\
and displayed the result as the thick solid line. We limited the fit to data points in the two-halo regime where
the bias is determined. As discussed in Zehavi et al. (2004), correlation functions of galaxies display departures
from power law models, especially at large separations. In the LRG auto-correlation function, this departure
is more pronounced at $r_p>10$ \mbox{$h^{-1}~{\rm Mpc}~$}\ and demonstrates the limitation of using such power law fits to
determine the bias of galaxies and absorber hosts. In \emph{grey circles}, we show the LRG--Mg\,II cross-correlation
signal. We also adopted a power-law model for the cross-correlation function, but we fixed the slope to the
value obtained for the LRGs ($b=-0.787$). The thin black line and the shaded areas correspond to the
best-fit and 1-$\sigma$ errors on the amplitude $a$. Note that the best-fit power-law models are only shown
to guide the eye and do not enter in the calculation of the relative bias of absorber hosts.
The other five panels display the cross-correlation signals for the five $W_r^{\lambda2796}$ bins
considered. The range of $W_r^{\lambda2796}$ for each bin is labeled in the upper-right corner
of each panel. In all panels, the error bars on $w_p$ correspond to cosmic variance, estimated
using the jackknife resampling technique (see section 2.4.3). Note that the data
points shown in Figure \ref{xx} have \emph{not} been corrected for the systematic effects of
photometric redshifts on the clustering amplitude. These considerations
will enter in the calculation of the relative bias of absorber hosts and are discussed
in section 3.2.
\begin{figure*}
\centerline{
\includegraphics[angle=0,scale=0.70]{rel_bias_DR_new_3.pdf}
}
\caption{(\emph{a}) Relative bias of Mg\,II hosts derived from the direct ratio method for points at $r_p\geq 2$\mbox{$h^{-1}~{\rm Mpc}~$}.
We show the results for the four $W_r^{\lambda2796}$ bins considered
in this analysis. (\emph{b}) Absolute bias of Mg\,II absorber hosts. To compute the bias, we adopted the bias of
LRGs $b_g=2.023 \pm 0.006$, as measured in G09.
(\emph{c}) In \emph{open} circles, we show halo masses derived by inverting the $b(M)$ relationship. Note
that the lower error-bar on four
of the five $W_r^{\lambda2796}$ bins are arbitrary. This occurs when the lower error bar reaches
$b \approx 0.7$ giving us no constraint on the
halo mass. The points represent the median of the $W_r^{\lambda2796}$ distribution
in each bin and the error bars on $W_r^{\lambda 2796}$ represent the extent of each bin.
}
\label{rel_bias}
\end{figure*}
At $r_p\gtrsim$ 2 \mbox{$h^{-1}~{\rm Mpc}~$}, where LRG--LRG and Mg\,II--LRG pairs are probing distinct dark
matter halos (i.e.,\ in the two-halo regime), the weakest absorbers
($W_r^{\lambda2796} < 1.08$\AA) have the largest cross-correlation amplitude
similar to the LRG auto-correlation signal.
In contrast, absorbers in the bin $W_r^{\lambda2796}=1.08-1.59$\AA\ show, on average, the lowest
clustering amplitude on large scales.
\subsection{Theoretical framework on bias and halo mass determination}
Here we provide a brief discussion of the theoretical framework behind the determination of the
Mg\,II absorber host bias and halo mass from the projected two-point correlation function. A more detailed
overview can be found in \citet{berlind2002a}, \citet{zheng2004a}, and \citet{tinker2005a}.
The bias of dark matter halos of mass $M$ is the ratio of the real-space two-point correlation function of these
dark matter halos, $\xi_h(M,r)$, and the dark matter auto-correlation function $\xi_m( r)$
\begin{equation}
b_h^2(M,r)=\frac{\xi_h(M,r)}{\xi_m(r)} \; .
\end{equation}
In practice, we compute the projected two-point correlation statistics by integrating the real-space
correlation function along the line-of-sight ($\pi$) direction
\begin{equation}
w_p(r_p) = \int_l \xi(r_p,\pi)d\pi \; .
\end{equation}
The LRG--LRG auto-correlation signal $\xi_{gg}$ can be decomposed into a mass dependent $b_g^2(M_g)$
and scale dependent $f_g^2( r)$ term
\begin{equation}
\xi_{gg}(M_g, r) = b_g^2(M_g) f_g^2( r) \xi_m( r)
\end{equation}
where $M_g$ is the bias-weighted mean halo mass. Similarly, the cross-correlation absorber--galaxy term can be written as
\begin{equation}
\xi_{ga}(M_g,M_a,r) = b_g(M_g) b_a(M_a) f_g( r) f_a( r)\xi_m( r) ,
\end{equation}
where the terms with subscript $a$ refer to the absorbers.
On large scales and for the halo masses considered in this paper, the scale dependence of $b_h^2(M,r)$, $f( r)$, is
almost independent of halo mass and is divided out when computing the relative bias
of absorber hosts, $\hat{b}$. Consequently, one can write
\begin{equation}
\hat{b} \equiv \frac{b_a(M_a)}{b_g(M_g)} = \frac{\xi_{ga}( r)}{\xi_{gg}( r)} = \frac{w_{ag}(r_p)}{w_{gg}(r_p)}
\end{equation}
where $\hat{b}$ is the scale-independent relative bias of absorber hosts $a$ with respect to a population of galaxies $g$.
In other words, $\hat{b}$ can be calculated from the ratio of the projected two-point correlation functions $w$ on
large scales. From $\hat{b}$, the bias (and mass) of the absorber hosts can be derived if the absolute bias of
the tracer galaxy population ($b_g$) is known.
For this purpose, we adopted the same galaxy bias as in G09. Since the LRG selection criteria and
redshift range are the same as G09, the auto-correlation signal of LRGs should, in principle
be the same. We verified that this was the case. In the upper-left panel of Figure \ref{xx}, we show the LRG
auto-correlation signal of G09 in \emph{crosses}. As expected, the differences from
G09 are negligible. We thus adopted the bias value of $b_{g}=2.023\pm0.006$ as found by G09.
\subsection{Calculating the relative bias }
Before calculating the relative bias of Mg\,II absorber hosts, it is important to consider the effects of
photometric redshifts on the clustering amplitude and bias measurements. LRGs typically have
photo-$z$ accuracy of $\sigma_z\approx 0.03(1+z_{\rm ph})$. Large uncertainties on the galaxy
redshifts affect the clustering signal in two different ways. Since the redshifts are uncertain, the
angular diameter distance is uncertain too, introducing uncertainties in the projected co-moving
separations between LRGs and Mg\,II absorbers and effectively ``smoothing" out sharp features present in the
intrinsic two-point correlation signal. As discussed in section 2.4.3, this source of uncertainty mostly affects
the inner bins of the correlation function and is negligible at $r_p \gtrsim$ 2 \mbox{$h^{-1}~{\rm Mpc}~$} where the clustering
amplitude is measured.
Furthermore, photometric redshift errors effectively \emph{broadens} the redshift range included in the
calculation and adds uncorrelated galaxy--Mg\,II pairs in the calculation. This effect results in a
systematic lowering of the \emph{amplitude} of the clustering signal. G09 addressed this issue by calculating
the projected clustering signal on mock LRG distributions. Their mock catalog was produced by
populating dark matter halos of an $N-$body simulation with a halo occupation distribution function
determined from a spectroscopic LRG sample. Then the redshift of each galaxy was perturbed to mimic
the effects of photometric redshifts and the authors re-calculated the two-point correlation statistics.
In a nutshell, G09 found that the MgII--LRG cross-correlation
amplitude on large scales ($>1$ \mbox{$h^{-1}~{\rm Mpc}~$}) is reduced by a factor 0.79$\pm$0.02 compared to what is found for
a spectroscopic sample of LRGs while the LRG--LRG auto-correlation amplitude is reduced by 0.71$\pm$0.01.
Consequently, we determined the relative bias, $\hat{b}$, by multiplying the uncorrected $\hat{b}$
values by the correction factor $\mathcal{C}=0.90\pm0.02$. Hereafter, all $\hat{b}$ and absolute bias measurements $b$
are corrected for this systematic effect introduced by photometric redshifts.
In G09, they discussed the direct ratio (DR) method to calculate the relative bias of absorber hosts. This technique
employs the weighted mean ratio of all data points at $r_p \gtrsim 2$ \mbox{$h^{-1}~{\rm Mpc}~$}\ to obtain $\hat{b}$
\begin{equation}
\hat{b}=\sum_{i=4}^{8} \omega_i \frac{w_{ag,i}}{w_{gg,i}} \times \mathcal{C}
\end{equation}
where the weights, $\omega_i$ are given by
\begin{equation}
\omega_i = \frac{w_{ag,i}}{\sigma_i^2 w_{gg,i}} \; .
\end{equation}
The index $i$ denotes the $r_p$ bin and $\sigma_i$ is the error on $w_{ag,i}/w_{gg,i}$ using
error propagation technique as is the error on $\hat{b}$ (see G09). In Table 2, we provide estimates for the relative
bias $\hat{b}$, bias $b$, and halo masses for each one of the five $W_r^{\lambda2796}$ bins
considered and the entire Mg\,II sample.
\subsection{Absolute bias and halo mass}
As discussed in section 3.1, the halo mass of absorber hosts could be obtained simply by inverting
the $b(M)$ relationship. We refer to this method as \emph{bias-inverted} halo mass.
Moreover, one can estimate halo masses by integrating the bias-weighted
halo mass function down in mass until the bias value reaches $b$. The minimum
halo mass corresponding to $b$ can thus be used as a lower-limit in the integral of the
halo mass function. This \emph{bias-weighted} halo mass estimate is simply derived by integrating the
mass-weighted halo mass function using this lower-limit.
These two techniques were discussed in details in G09. The authors show that the methods give halo mass estimates
within 0.1 dex of each other for both LRGs and Mg\,II absorber hosts. Consequently, we followed the G09 methodology
and used the bias-inverted masses as our Mg\,II hosts mass estimates.
In Figure \ref{rel_bias}, we show the relative bias, bias, and halo masses of Mg\,II absorber hosts for the
five $W_r^{\lambda2796}$ bins considered. Except for the weakest absorbers,
the lower error bars we quoted on the halo masses are arbitrary. This lack of constraints arises because when the lower error
bar on the bias reaches 0.7, there is no constraint on the halo mass. In fact, $b(M)$ reaches a minimum value
of $\approx0.7$ and becomes nearly independent of halo mass at log $M_h \mathrel{\hbox{\rlap{\hbox{\lower3pt\hbox{$\sim$}}}\hbox{\raise2pt\hbox{$<$}}}} 9$ (e.g.,\ \citealt{tinker2008b}).
\begin{figure}
\centerline{
\includegraphics[angle=0,scale=0.70]{rel_bias_models3_gray3_2.pdf}
}
\caption{ \emph{Top:} Comparison between
the relative bias estimates of Mg\,II absorber hosts and
the predictions from a simple model in which the gas distribution
follows an isothermal profile (see section 4 for further details). In \emph{open circles} are the $\hat{b}$
values found in this paper. In \emph{grey triangles} we show the results
from G09.
The \emph{solid} line corresponds to our best-fit model. Interestingly, this model predicts a
monotonically increasing $b$--$W_r$ relation. We also present a direct comparison between the predictions of the
``transition" model from Tinker \& Chen (2008) (TC08) in \emph{dashed} curve
(see section 4 for further details). This transition models predicts an anti-correlation
between $b$ and $W_r$ and is in better agreement with our data, especially for the
weakest $W_r$ bins.
\emph{Bottom:} Comparison between the best-fit model and the
frequency distribution function of absorbers, $f(W_r)$, measured by S13 (\emph{open circles}) in the redshift range $z_{\rm Mg\,II}=0.45-0.60$.
We also plotted the ZM13 $f(W_r)$ values in \emph{grey} circles over the range $z_{\rm Mg\,II}=0.43-0.55$ for comparison.
Because of the small $f(W_r)$ error bars for points at $W_r\lesssim2$\AA, the $\chi^2$ is driven by the weak
$W_r$ bins. The model tends to underestimate $f(W_r)$ for strong absorbers although a larger covering fraction for
massive halos or a higher $A_{W_0}$ would over predict the number of weaker systems while flattening the overall $f(W_r)$ slope.
For 32 degrees of freedom, this model has a reduced $\chi^2$, $\chi_r^2 = 1.81$ which has an associated $P$-value $P=0.003$.
Accordingly, this simple model can be rejected at a $\approx 2.8\sigma$ (one-sided) confidence level.
}
\label{rel_bias_models}
\end{figure}
\begin{table}
\centering
\begin{minipage}{140mm}
\caption{Description of the cross-correlation calculations}
\begin{tabular}{@{}cccc@{}}
\hline
$W_r^{\lambda2796}$ [\AA] & N$_{\rm Mg\,II}$ & N$_{\rm LRGs}$ & DD pairs (1st bin) \\
\hline
All & 2211 & 333334 & 130 \\
$0.40-0.78$ & 559 & 333334 & 36 \\
$0.78-1.08$ & 531 & 333334 & 26\\
$1.08-1.59$ & 564 & 333334 & 25 \\
$>1.59$ & 557 & 333334 & 24 \\
\hline
\hline
\end{tabular}
\end{minipage}
\end{table}
\section{Discussion}
We presented the cross-correlation function of Mg\,II absorbers with respect to a volume-limited sample
of LRGs at $z=0.45-0.60$ using the SDSS DR7 galaxy and absorber catalogs. We benefited from
an absorber catalog four times larger and an increase of 70\% in the number of LRGs compared
to the samples used in G09. We extended the clustering analysis to weaker absorbers with $W_r^{\lambda2796}<1$\AA\
that were excluded from the G09 analysis.
The clustering signal of Mg\,II absorbers was calculated in four $W_r^{\lambda2796}$ bins of roughly equal
number of absorbers spanning the range $W_r^{\lambda2796}=0.4-5.6$\AA. On average, stronger absorbers
with $W_r^{\lambda2796}>1$\AA\ reside in less massive halos than weaker ones.
\begin{table}
\begin{minipage}{80mm}
\caption{Relative bias, absolute bias, and halo mass estimates of Mg\,II absorber hosts}
\begin{tabular}{@{}ccccccccccc@{}}
\hline
$W_r^{\lambda2796}$ [\AA] & $\hat{b}$\footnote{We applied a correction factor of 0.90$\pm$0.02
that takes into account the broader redshift interval introduced by photo-$z$, as discussed in G09} & $b$\footnote{We adopted the bias of LRGs $b_g=2.023\pm0.006$ as
measured in G09.} & $\langle \log M_h \rangle $\footnote{The halo mass corresponds to the \emph{bias-inverted} halo mass, i.e. we invert the $b$ vs $M_h$ relationship
to find the halo mass corresponding to the measured bias.} \\
\hline
All & 0.56 $\pm$ 0.12 & 1.14 $\pm$ 0.23 & $12.1^{+0.4}_{-0.7}$ \\
$0.4-0.78$ & 0.65 $\pm$ 0.24 & 1.32 $\pm$ 0.48 & 12.5$^{+0.6}_{-1.4}$\\
$0.78-1.08$ & 0.68 $\pm$ 0.24 & 1.38 $\pm$ 0.48 & 12.6$^{+0.5}_{-1.2}$\\
$1.08-1.59$ & 0.42 $\pm$ 0.19 & 0.86 $\pm$ 0.39 & 11.2$^{+1.1}$\\
$>1.59$ & 0.54 $\pm$ 0.21 & 1.10 $\pm$ 0.42 & 12.0$^{+0.7}$ \\
\hline
\hline
\end{tabular}
\end{minipage}
\end{table}
The observed $b$--$W_r^{\lambda2796}$ relation offers a statistical characterization of the origin of the Mg\,II absorbers as
a whole, but the interpretation requires a comprehensive model of halo gas content as a function of halo mass. Similarly,
the frequency distribution function $f(W_r)$ of absorbers, namely the number of absorbers per unit $W_r$ per unit comoving length,
also provides important constraints on halo gas models. We thus compared these two datasets with the predictions of a simple analytical
model in which the Mg\,II gas is distributed in dark matter halos according to an isothermal profile. In this model, the gas clumps are distributed according to an isothermal
profile of finite radial extent. This radius is denoted by $R_{\rm gas}$, the gaseous radius of the halo. In our model, the value of
$R_{\rm gas}$ is set to $R_{\rm gas} =1/3 \times R_{\rm vir}$ where $R_{\rm vir}$ is the virial radius of the dark matter
halo following \citet{chen2010a}. For example, a typical $L_*$ galaxy with halo mass $\sim 10^{12}$ \mbox{$h^{-1}\,{\rm M}_\odot$}\ at $z\sim 0.25$ has $R_{\rm gas}=75$ \mbox{$h^{-1}~{\rm kpc}~$} \citep{chen2010a}.
The total absorption equivalent width is the sum over all clumps encountered along the sightline
and thus corresponds to the integral of the isothermal profile along a given sightline located at impact parameter $s$
\begin{equation}
W_r(s,M) = A_W\frac{2\mathcal{G}_0}{\sqrt{s^2+a_h^2}} \arctan \sqrt{\frac{R_{\rm gas}^2-s^2}{s^2+a_h^2}}
\label{wr_model}
\end{equation}
where $a_h$ is the core radius of the isothermal profile, $A_W$ is the mean absorption equivalent width per
unit surface mass density of the cold gas and $\mathcal{G}_0$ is
\begin{equation}
\mathcal{G}_0 = \frac{M(<R_{\rm gas})/4\pi}{R_{\rm gas}-a_h \arctan (R_{\rm gas}/a_h)} \; .
\end{equation}
Following Tinker \& Chen (2008, hereafter TC08), we set $a_h=0.2R_{\rm gas}$ and adopted a power-law for $A_W$
\begin{equation}
A_W(M)= A_{W_0} \bigg( \frac{M}{10^{12} h^{-1} M_{\odot} }\bigg)^{-0.2}~[h^{-1} \rm{\AA} ~\rm{cm}^{2}~\rm{g}^{-1} ]
\end{equation}
where $A_{W_0}$ is a free parameter independent of halo mass. The slope of this power-law corresponds to the
best-fit value of TC08. The frequency distribution function can be written as
\begin{equation}
f(W_r) \equiv \frac{d^2N}{dW_r dl} = \int dM \frac{dn}{dM} \sigma_g(M) P(W_r|M)
\end{equation}
where $dn/dM$ is the halo mass function (Warren et al. 2006), $\sigma_g(M)=\pi R_{\rm gas}^2$ is the gas cross section and
$P(W_r|M)$ is the probability that a halo of mass $M$ hosts an absorber of strength $W_r$. In turn, $P(W_r|M)$ can
be written as :
\begin{equation}
P(W_r|M) = \kappa_g(M) \frac{2s(W_r|M)}{R_{\rm gas}^2} \frac{ds}{dW_r}
\label{prob}
\end{equation}
(see equation 10 of TC08). In this expression, the impact parameter $s(W_r|M)$ is found by inverting equation
\ref{wr_model} which is performed numerically. The derivative of $s$ with respect to
$dW_r$, $ds/dW_r$, is calculated following equation 11 in TC08. In equation \ref{prob}, $\kappa_g(M)$ is the gas
covering fraction. We adopted a double power law to account for the mass dependence of $\kappa$
\begin{equation}
\kappa(M) = \begin{cases} \kappa_{12}(M/10^{12})^{\gamma}, & \mbox{if } M \leq 10^{12} \mbox{$h^{-1}\,{\rm M}_\odot$} \\ \kappa_{12}(M/10^{12})^{\alpha}, & \mbox{if } M>10^{12} \mbox{$h^{-1}\,{\rm M}_\odot$} \end{cases}
\end{equation}
Following the empirical findings
of \citet{chen2010a}, we assigned $\kappa_{12}=0.7$ which corresponds to the covering fraction of $W_r^{\lambda 2796}>0.3$ \AA\
absorbers in $L_*$-galaxy halos. Both slopes $\alpha$ and $\gamma$ are free parameters.
In addition to $f(W_r)$, the model described above allows us to predict the $b$--$W_r$ relation.
The bias corresponds to the mean
halo bias weighted by the probability of finding an absorber $W_r$ in a halo of mass $M$
\begin{equation}
b=\frac{1}{f(W_r)} \int dM \frac{dn}{dM} \sigma_g(M) b_h(M) P(W_r|M)
\end{equation}
where $b_h$ is the halo bias taken from \citet{tinker2008b}.
In summary, this model has three free parameters: $A_{W_0}$, $\alpha$, and $\gamma$. We generated a grid of
models by varying these three parameters independently and we compared our model predictions for $f(W_r)$ and $b$ with the
empirical data of S13 in the range $z_{\rm Mg\,II}=0.45-0.60$ and the bias data points of Figure 5a. In addition, we plotted in \emph{grey}
the ZM13 $f(W_r)$ data over the redshift range $z_{\rm Mg\,II}=0.43-0.55$ for comparison.
For each model, we computed a $\chi^2$ goodness-of-fit test \begin{equation}
\chi^2 = \chi_b^2 + \chi_f^2 = \sum_{i=0}^4 \frac{(b_i-\bar{b}_i)^2}{\sigma_{b_i}^2} + \sum_{j=0}^{31} \frac{(f_j - \bar{f}_j)^2}{\sigma_{f_j}^2}
\end{equation}
where $\chi_b^2$ and $\chi_f^2$ correspond to the values from the bias and frequency distribution function. The parameters
$\bar{b}$ and $\bar{f}$ are the model predictions for the bias and frequency distribution function respectively.
Because of the larger number of datapoints of the frequency data (31 vs 4), $\chi_f^2$ will dominate over $\chi_b^2$ for
most values of $(A_{W_0},\alpha,\gamma)$. The best-fit model has $\chi^2=57.9$ with $\chi_b^2=0.96$ and $\chi_f^2=56.97$
The corresponding best-fit parameters are $(126,-1.40,2.18)$. For a $10^{13} \mbox{$h^{-1}\,{\rm M}_\odot$}$ halo, this model predicts $\kappa=0.03$,
which is lower than the value obtained by \citet{gauthier2011a} for LRG halos ($\kappa_{\rm LRG}=0.22\pm0.13$). We show the best-fit model in Figure 6. Because
of the small $f(W_r)$ error bars for points at $W_r\lesssim2$\AA, $\chi_f^2$ is driven by the weak $W_r$ bins. The model tends to
underestimate $f(W_r)$ for strong absorbers although a larger $\kappa$ for massive halo or a larger $A_{W_0}$ would over predict
the number of weaker systems while flattening the overall $f(W_r)$ slope. Interestingly, this model
predicts a monotonically increasing $b$--$W_r$ relation. For 32 degrees of freedom, the reduced
$\chi^2$ is $\chi_r^2 = 1.81$ which has an associated $P$-value of $P\approx0.003$. Accordingly, this simple model can be rejected at a
$\approx 2.8\sigma$ (one-sided) confidence level.
In contrast with the simple model presented above, TC08
also developed a halo occupation distribution model (HOD) in which they introduced a transition mass
scale of $\sim 3\times10^{11}$ \mbox{$h^{-1}\,{\rm M}_\odot$}\ above which a shock develops and reduce the amount of Mg\,II absorbing
gas within the shock radius. This transition mass scale was motivated by recent hydrodynamical simulations
of galaxy formation showing two distinct channels of gas accretion for halos above and below the
transition scale (e.g. \citealt{keres2009a}), although the conclusions drawn from these early studies
are now being challenged by recent hydrodynamical simulations (e.g.,\ \citealt{nelson2013a}).
TC08 showed that their model predicts near unity covering fraction of Mg\,II absorbing gas over a wide
range of halo masses, suggesting that Mg\,II absorbers are probing an unbiased sample of galaxies,
not preferentially selected for their recent star formation activity.
Although this ``transition" model provides a better fit to $f(W_r)$, the most important differences occur
in the $b$--$W_r$ relationship. Because of the shock radius developing in massive halos, the transition model
predicts a suppressed contribution from massive halos yielding a lower $b$ for stronger absorbers. This
results in an overall $b$--$W_r$ anti-correlation. In Figure \ref{rel_bias_models} we show a direct comparison
between the relative bias and the model predictions taken from TC08.
The best-fit TC08 ``transition" model is shown in \emph{dashed} and provides a better fit to the bias data although
most of the discriminative power lies in the weakest $W_r$ bin.
The bias data shown in Figure 5 suggest a possible flattening or upturn in the $b$--$W_r$ relation for absorbers
with $W_r^{\lambda2796}\gtrsim 1.59$\AA . A similar, trend was also seen in \citet{bouche2006a} and \citet{lundgren2009a}.
These results are consistent with \citet{gauthier2013a} who argue that ultra-strong Mg\,II absorbers with
$W_r^{\lambda2796}\gtrsim 3$\AA\ trace gas dynamics of the intragroup medium. Gauthier (2013)
estimated halo masses of $\log M_h=12-13.3$ for the galaxy groups presented in their paper and in \citet{nestor2011a}.
Similar group environments have also been found around $W_r^{\lambda2796}\approx 2$\AA\
absorbers at intermediate redshifts \citep{whiting2006a,kacprzak2010b}.
Nevertheless, the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) is collecting a large sample of spectroscopically
identified LRGs with a redshift precision of $dz<0.0005$ instead of $dz\sim0.05$. The availablity of a spectroscopic
LRG sample offers an exciting opportunity to study the real-space clustering signal of Mg\,II absorbers and to examine the
two-point function on small scales ($<$1Mpc). These new measurements will provide further insights into large-scale
motion of cool gas uncovered by Mg\,II and allow a detailed investigation of the cool halo gas content of massive halos hosting the LRGs.
\section*{Acknowledgments}
It is a pleasure to thank Michael Rauch and Jeremy Tinker for helpful comments and discussions.
JRG gratefully acknowledges the financial support of a Millikan Fellowship provided by Caltech.
The SDSS MgII catalog (Seyffert et al. 2013) was funded largely by the National Science
Foundation Astronomy \& Astrophysics Postdoctoral Fellowship (AST-1003139) and in part by
the MIT Undergraduate Research Opportunity Program (UROP) Direct Funding, from the
Office of Undergraduate Advising and Academic Programming and the John Reed UROP Fund.
We are grateful to the SDSS collaboration for producing and maintaining the SDSS public data archive.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration,
the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.
The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating
Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel,
University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab,
the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute
for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group,
the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA),
the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University,
University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory,
and the University of Washington.
\footnotesize{
\bibliographystyle{mn2e}
|
2,869,038,155,713 | arxiv | \section{Introduction}
For a variety of reasons, including unequal access to primary education, family support, and enrichment activities, different demographic groups can vary widely in their level of preparation by the time they reach their senior year of high school, when they apply for college. In an attempt to correct for this unfortunate reality, many colleges in the United States follow some sort of affirmative action policy in their admissions, which is to say, their admissions decisions explicitly take demographics into account. What is often unstated (and perhaps not even explicitly considered by the colleges) is what exactly the long term goals of these policies are, beyond the short term goal of having a diverse freshman class. In this paper, we consider two explicit goals, and study the extent to which they can be met in a simple two stage model:
\begin{enumerate}
\item \textbf{Equal opportunity}: The probability that an individual is accepted to college \emph{and then} ultimately hired by an employer may depend on an individual's type, but conditioned on their type, should not depend on their demographic group.
\item \textbf{Elimination of Downstream Bias}: Rational employers selecting employees from the college population should not make hiring decisions based on group membership.
\end{enumerate}
Neither of these desiderata will necessarily be achieved by admissions rules that ignore demographic information. For example, suppose college admissions is set by a uniform admissions threshold on entrance exam scores. Assuming these scores are equally informative about all groups, this will guarantee that conditioned on a student's type, whether or not she is \emph{admitted to college} will be independent of her group membership, but it does not imply that whether or not she is ultimately hired is independent of her group! This is because exam scores are only a noisy signal about student type. Therefore, if two groups have different prior distributions on type, they will have different posterior distributions on type when conditioned on being admitted to college according to a group-blind admissions rule. The result will be that a Bayesian employer will insist that students from a group with lower mean or higher variance will have to cross a higher threshold on their college grades in order to be hired. In addition to incentivizing explicit group-based discrimination by the employer, this also results in a failure of equal opportunity for the students, because once admitted to college, two individuals of the same type might have to cross different grade thresholds in order to be hired. Thus, a simple ``group blind'' admissions rule fails to achieve either goal 1 or 2 as laid out above. In this paper, we study the extent to which these goals can be achieved via other means available to the college: in particular, how it admits and grades students.
\subsection{Limitations of our Model}
When interpreting our results, it is important to understand the scope and limitations of our model. First, this paper considers fairness goals that are limited to preventing inequity from being further propagated --- treating opportunities at the high school level and earlier as fixed --- and that do not attempt to correct for past inequity. This manifests itself in that our ``equal opportunity'' goal takes as given that the prospects for employment may ``fairly'' vary as a function of an individual's type \emph{at the time at which they apply for college}, and does not attempt to address or correct the historical forces that might have resulted in different groups having different type distributions to begin with. Attempting to correct for this kind of historical inequity would require a ``value-added model'' of education, in which colleges can \emph{change} the type distributions of their student population either through the direct effect of education, or through a second order effect on student behavior before they apply. In our model, colleges do not change student types, they only serve as signaling mechanisms. Similarly, our ``equal opportunity'' goal aims to equalize the probability that students are hired conditioned on their types --- but one might reasonably instead ask for a corrective notion of fairness, in which the probability of passing through the pipeline is \emph{higher} for the historically disadvantaged group conditioned on type. We do not consider this.
Our model also ignores the possibility that exam scores and grades are themselves \emph{biased}. We explicitly assume the opposite --- that exam scores and grades are unbiased estimators of student types, for both groups. If instead exam scores were systematically biased downwards for one group, then the response of a rational employer to an admissions policy would be very different --- because students who made it through the college pipeline \emph{despite} negative bias would have a higher relative posterior probability of having a high type. There is evidence that effects of this sort are real \cite{bohren2017dynamics}.
The two kinds of fairness goals that we study do not speak to the size of the student of employee population coming from each group. For example, in principle, one could satisfy both the equal opportunity and elimination-of-downstream-bias goals that we propose, but at a cost of employing very few individuals from one of the groups. However, we show that even without an additional goal of having large representation from both groups, the fairness goals we set out cannot generally be achieved.
Finally, we assume that employers are single-minded expectation maximizers, with no explicit desire for fairness or diversity. Of course this is often not the case.
Despite these limitations and simplifying assumptions, we find that in the model we study, many natural fairness goals are already impossible. We think that these negative results are likely to persist in more complex models that attempt to capture additional realism.
\subsection{Our Model and Results}
We consider a simple model of admissions, grading and hiring that views the role of colleges only as a means of signaling quality and performing a gatekeeping function, rather than as providing explicit value added\footnote{This is consistent with the signaling view of the role of colleges in the economics literature, beginning with \cite{spence}}. We consider two \emph{groups} representing pre-defined subsets of the population, divided according to socio-economic or other demographic lines. Each student from group $i$ is endowed with a \emph{type} $t$, which is drawn independently from a Gaussian type distribution $P_i$ that is dependent on the students' group membership. A student's type ultimately measures her value to an employer. We model employers as having a fixed cost $C$ for hiring an individual, and a gain that is proportional to their type. If the employer hires an individual who has type $t$, they obtain utility $t-C$. A college can choose an admissions rule and a grading policy. Although students types are unobservable, each student has an admissions exam score that is an observable unbiased estimator of their type. We model exam scores as being distributed as a unit variance Gaussian, centered at the student's type. An admissions policy for the school is a mapping between exam scores and admissions probabilities. We allow schools to set different admissions policies for different groups, but for most of our results, we require the natural condition that admissions probabilities within a group be monotonically non-decreasing in exam scores\footnote{A non-monotone admissions rule would have the property that sometimes a student with a lower exam score would have a higher probability of admission that a student with a higher exam score. Non-monotonicity within a group is highly undesirable, because it would give some students a perverse incentive to intentionally try and lower their exam scores. If such incentives were present, it would no longer be reasonable to model exam scores as unbiased estimators of student types.}. Deterministic monotone admissions policies simply correspond to setting admissions thresholds based on exam scores. For simplicity, in the body of the paper, we restrict attention to deterministic admissions rules, but in the Appendix, we extend our results to cover probabilistic admissions rules as well.
Schools may also set a grading policy. A grade is also modeled as a Gaussian centered at a student's true type, but the school may choose the variance of the distribution. We assume that a student's grade is conditionally independent of her entrance exam score, conditioned on her type. One limiting extreme (infinite variance) corresponds to committing not to report grades at all. This limiting case is actually achievable because schools can simply opt not to share grades --- in fact, this practice has been adopted at several top business schools \cite{nogrades}. At the other limiting extreme, types are perfectly observable. This extreme is generally not achievable, and we do not consider it in this paper. In between, the school can modulate the strength of the signal that employers get about student type, beyond the simple indicator that they were admitted to college.
Employers know the prior distributions $P_i$ on student types, as well as the admissions and grading policy of the school. They are rational expectation maximizers. When deciding whether or not to hire a student, they will condition on all information available to them --- a student'a group membership, the fact that she was admitted to college under the college's admissions policy, and the grade that she received under the college's grading policy --- to form a posterior distribution about the student's type. They will hire exactly those students for whom they have positive expected utility under this posterior distribution.
In order to incentivize a particular employer to use a hiring rule that is independent of group membership, it is necessary to set admissions and grading policies such that for every student admitted to the school, and for every grade $g$ that she may receive, the indicator that the conditional expectation of her type $t$ is above the employer's hiring cost $C$ is independent of the student's group membership. If there is uncertainty about what the employer's hiring cost $C$ is, or if there are multiple employers, then it is necessary to guarantee this property for an interval of hiring costs $C \in [C^-, C^+]$ rather than for just a fixed cost. We distinguish these two cases. We call this property Irrelevance of Group Membership (IGM), in the single threshold and multiple threshold case respectively. A seemingly stronger property that we might desire is that the posterior distribution on student types conditional on admission to college is \emph{identical} for both groups. We call this property \emph{strong} Irrelevance of Group Membership (sIGM). Because it symmetrizes the two groups, it in particular guarantees that members of both groups will be treated identically by rational decision makers at any further stage down the decision making pipeline. We show that in the presence of finite, nonzero variance in both exam scores and grades, IGM in the multiple threshold case implies sIGM. Finally, we say that an admissions rule and grading policy satisfy the \emph{equal opportunity} condition, if a student's probability of making it all the way through the pipeline --- i.e. being admitted to college \emph{and then} being hired by the employer, is independent of her group conditioned on her type. Trivially, any group-symmetric admissions policy will satisfy both conditions if the two group type distributions are identical, so for the results that follow, we always assume that the group type distributions are distinct --- differing in their mean, their variance, or both.
First, to emphasize that our impossibility results will crucially depend on the fact that exam scores are only a noisy signal of student ability, we consider the noiseless case, in which college admissions can be decided \emph{directly} as a function of student type (this corresponds to the case in which exam scores have no noise). In this case, we can ``have it all'': there is a simple monotone admissions rule that guarantees both the equal opportunity condition, and satisfies IGM for multiple thresholds --- for any grading policy that the school might choose. After establishing this simple result, in the rest of the paper we move on to the more realistic case in which exam scores are only a noisy signal of student type.
Next, we study what is possible if the college chooses to not report grades at all. In this case, we can also ``have it all'' --- simply by setting a sufficiently high, group independent admissions threshold, a school can achieve both equal opportunity and IGM for multiple thresholds. This gives another view of the effects of practicing grade non-disclosure at highly selective schools \cite{nogrades}.
Finally, in the bulk of the paper, we study the common case in which the college uses informative grades --- i.e. sets the variance of its grade distribution to be some finite value. In this case, we show that it \emph{is} possible to obtain IGM in the single threshold case, but that no monotone admissions rule can obtain sIGM. Because of the equivalence between sIGM and IGM for the multiple threshold case, this implies that no monotone admissions rule can obtain IGM in the multiple threshold case, even in isolation. Next, we consider the equal opportunity condition. One trivial way to obtain it is to simply admit nobody to college. We show that this is in general the only way in the multiple thresholds case: no non-zero monotone admissions rule can satisfy the equal opportunity condition, even in isolation.
\subsection{Related Work}
Our work fits into two streams of research. Within the recent line of work on algorithmic fairness, the most closely related work is that of Chouldechova \cite{Chou16} and Kleinberg, Mullainathan, and Raghavan \cite{KMR16}. Both of these papers prove the impossibility of simultaneously satisfying certain fairness desiderata in batch classification and regression settings. Broadly speaking, both papers show the impossibility of simultaneously equalizing false positive and false negative rates (related to our equal opportunity goal --- see also \cite{HPS16}) and positive predictive value or calibration (related to our IGM goals). Our work is quite different, however: the goals that we study are not direct properties of the classification rule in question (in our case, the college admissions rule), but instead properties of its downstream effects. And while the work of \cite{Chou16,KMR16} shows the impossibility of simultaneously satisfying these fairness criteria, in our setting, we show that they are often impossible to satisfy even in isolation.
Our paper also fits into an older line of work studying economic models of discrimination and affirmative action, which has its modern roots in \cite{arrow} and \cite{phelps}. For example, Coate and Loury \cite{CL93} and Foster and Vohra \cite{FV92} study two stage models in which students from two different groups (who are a-priori identical) can in the first stage choose whether or not to make a costly investment in themselves, which will increase their value to employers. In the 2nd stage, employers may set a hiring rule that acts on a noisy signal about student quality. These works show the existence of a self-confirming equilibrium, in which only one group makes investments in themselves and are subsequently given employment opportunities, and consider interventions which can escape these discriminatory equilibria. These works can be viewed as studying the ``upstream effects'' of affirmative action policies, and explaining the mechanics by which different student populations may end up with different type distributions. The effect of the interventions proposed in these models is very slow, because it requires a new generation of students to recognize the opportunities made available to them via affirmative action policies and make costly investments in their education in response, well before they enter the job market. In contrast, our work can be viewed as studying the ``downstream effects'' of these policies and examining shorter term effects which can be realized in a time frame that need not be long enough for type distributions to change.
More recently, the computer science community has begun studying fairness desiderata in dynamic models. Jabbari et al study the costs (measured as their effect on the rate of learning) of imposing fairness constraints on learners in general Markov decision processes \cite{JJKMR17}. Hu and Chen \cite{HC18} study a dynamic model of the labor market similar to that of \cite{CL93,FV92} in which two populations are symmetric, but can choose to exert costly effort in order to improve their value to an employer. They study a two stage model of a labor market in which interventions in a ``temporary'' labor market can lead to high welfare symmetric equilibrium in the long run. Liu et al. \cite{delayed} study a two round model of lending in which lending decisions in the first round can change the type distribution of applicants in the 2nd round, according to a known, exogenously specified function. They study how statistical constraints on the lending rule can improve or harm outcomes as compared to a myopic (i.e. ignoring dynamic effects) profit maximizing rule, and find that for two kinds of interventions, both improvement and harm are possible, depending on the details of how lending effects the type distribution. Finally, \cite{incentives} studied the regulator's problem of providing financial incentives for a lender to satisfy fairness constraints in an online classification setting.
\section{Model}
We consider two populations of students, $1$ and $2$. In population $i \in \{1,2\}$, each student has a type drawn from a Gaussian distribution $P_i = \mathcal{N}\left(\mu_i,\sigma_i^2\right)$ with mean $\mu_i$ and variance $\sigma_i^2$. Since our problem is trivial if $P_1 = P_2$, in this paper we assume always that $P_1 \neq P_2$, i.e. the type distributions differ either in their mean, or their variance, or both. We denote by $T_i$ the random variable that represents the type of a student from population $i$. Throughout the paper, $\phi$ denotes the probability density function and $\Phi$ the cumulative density function of a standard normal random variable with mean $0$ and variance $1$.
Each student takes a standardized test (SAT, etc.) and obtains a score given by
\[
S_i = T_i + X
\]
where $X$ follows a normal distribution with mean $0$ and variance $1$, that does not depend on the population $i$, i.e., the student's score is a noisy but unbiased estimate of his type.
Additionally, we consider a university that admits students from both populations. The university designs an admission rule $A_i: \mathbb{R} \to [0,1]$ for each population $i$, such that a student from population $i$ with score $s$ is accepted with probability $A_i(s)$. We also abuse notation and let $A_i$ denote the binary random variable whose value is $1$ if a student is accepted, and $0$ otherwise. This admission rule is required to be monotone non-decreasing; i.e. an increase in exam score cannot lead to a \emph{decrease} in admissions probability. We say that an admissions rule is deterministic if $A_i(s) \in \{0,1\}$. A deterministic monotone admissions rule is characterized by a threshold $\beta_i$ such that a student is accepted if and only if $S_i \geq \beta_i$. We call such rules``thresholding admissions rules''. We focus on thresholding admissions rules in the body of this paper, but extend our results to probabilistic admissions rules to the Appendix. For simplicity of notation, we will often write $x_i(t)=\mathbb{P} \left[A_i = 1 |T_i = t \right]$ (Note that $x_i(t) = \mathbb{P} \left[S_i \geq \beta_i |T_i = t \right]$ in the deterministic case).
Every student who is admitted to the university receives a grade, given by:
\[
G_i = T_i + Y
\]
where $Y$ follows a normal distribution with mean $0$ and variance $\gamma^2$ that does not depend on the population $i$. $\gamma$ can be set by the university, and represents the strength of the signal provided by a grading policy\footnote{In actuality, of course, students receive many grades, not just one. But note that when one averages two normally distributed random variables, the result is also normally distributed, but with lower variance. Hence, one way to modulate the variance of a grade signal is to modulate the \emph{number} of grades computed. The more assignments and exams that are graded, the lower the variance of the signal. The fewer that are graded, the higher the variance.}. In our model, the University must commit to a single grading policy to use across groups.
Finally, an employer makes a hiring decision for each student that graduates from the university. The employer knows the priors $P_i$, the admission rules $A_1,~A_2$ used by the school, the grading policy $\gamma$, and observes the grades of the students (as well as the fact that they were admitted to the school). The employer's expected utility for accepting a university graduate from population $i$ with grade $g$ is then given by
\[
\mathbb{E} \left[ T_i |G_i = g, A_i = 1\right] - C
\]
where $C$ is the cost for the employer to hire a student.
The employer hires a university graduate from population $i$ with grade $g$ if and only if
\[
\mathbb{E} \left[ T_i | G_i = g, A_i = 1\right] \geq C
\]
Throughout the paper, we study the feasibility of achieving the following fairness goals:
\begin{definition}[Equal opportunity]
Equal opportunity holds if and only if the probability of a student being hired by the employer conditional on his type is independent of the student's group. I.e. if for all types $t \in \mathbb{R}$,
\begin{align*}
&\int_{g} \mathbb{P} \left[G_1 = g, A_1 = 1 | T_1 = t \right] \mathbbm{1} \{ \mathbb{E} \left[T_1 | G_1 = g, A_1 = 1 \right] \geq C\} dg
\\& = \int_{g} \mathbb{P} \left[G_2 = g, A_2 = 1 | T_2 = t\right] \mathbbm{1} \{ \mathbb{E} \left[T_2 | G_2 = g, A_2 = 1 \right] \geq C\} dg
\end{align*}
\end{definition}
\begin{definition}[Irrelevance of Group Membership]
\textit{Irrelevance of Group Membership (IGM)} holds if and only if, conditional on admission by the school and on grade $g$, the employer's decision on whether to hire a student is independent of the student's group. I.e. if for all grades $g \in \mathbb{R}$,
\begin{align*}
\mathbb{E} \left[T_1 | G_1 = g, A_1 = 1 \right] \geq C \Leftrightarrow \mathbb{E} \left[T_2 | G_2 = g, A_2 = 1 \right] \geq C
\end{align*}
\end{definition}
We further introduce a robust version of IGM, called \textit{strong} Irrelevance of Group Membership, that symmetrizes the two populations and guarantees that members of both populations will be treated identically by rational decision makers at any further stage of the decision making pipeline.
\begin{definition}[strong Irrelevance of Group Membership]
\textit{Strong Irrelevance of Group Membership (sIGM)} holds if and only if, conditional on admission by the school and on grade $g$, the employer's posterior on a student's type is independent of the student's population. I.e., for all $g \in \mathbb{R}$, for all $t \in \mathbb{R}$,
\begin{align*}
\mathbb{P} \left[T_1 = t | G_1 = g, A_1 = 1 \right] = \mathbb{P} \left[T_2 = t | G_2 = g, A_2 = 1 \right]
\end{align*}
\end{definition}
We note that sIGM holds if and only if the posterior on students' types conditional on admission by the school are identical:
\begin{claim}\label{clm: sIGM_equivalence}
sIGM holds if and only if for all $t \in \mathbb{R}$:
\begin{align*}
\mathbb{P} \left[T_1 = t | A_1 = 1 \right] = \mathbb{P} \left[T_2 = t | A_2 = 1 \right]
\end{align*}
\end{claim}
\begin{proof}
See Appendix~\ref{app: sIGM_equivalence}
\end{proof}
\section{Inference Preliminaries}
In this section, we derive some basic properties of the joint distributions on student types, exam scores, admissions rules, and grades that are relevant for reasoning about the employer's Bayesian inference task. We will draw upon these basic results in the coming sections.
\subsection{Preliminaries on Gaussians and Multivariate Gaussians
First, we observe that together, student types, exam scores, and grades are distributed according to a multi-variate Gaussian.
\begin{claim}
$(T_i,S_i,G_i)$ follows a multivariate normal distribution.
\end{claim}
\begin{proof}
A set of random variables is distributed according to a multivariate normal distribution if every linear combination of the variables is distributed as a univariate normal distribution.
For all $a,b,c \in \mathbb{R}$, $a T_i + b S_i + c G_i = (a+b+c) T_i + b X_i + c Y_i$ follows a normal distribution as the sum of independent normal random variables.
\end{proof}
We now quote a basic fact about the conditional distribution that results when one starts with a multi-variate normal distribution, and conditions on the realization of a subset of its coordinates.
\begin{claim}\label{clm: conditional_MVN}
Let $n \geq 2$ be an integer. Let $Z \in \mathbb{R}^n$ be a random variable following a multi-variate normal distribution. Let $Z = (Z_1,Z_2)$ where $Z_i \in \mathbb{R}^{n_i}$ with $n_1 + n_2 = n$. Suppose $Z$ has mean $m =(m_1,m_2)$ where $m_i \in \mathbb{R}^{n_i}$, and covariance matrix
\[
\Sigma =
\left[
\begin{array}{c|c}
\Sigma_{11} & \Sigma_{12} \\
\hline
\Sigma_{21} & \Sigma_{22}
\end{array}
\right]
\]
where $\Sigma_{ij} \in \mathbb{R}^{n_i \times n_j}$. Then $\mathbb{E} \left[ Z_1 | Z_2 = z_2 \right] = m_1 + \Sigma_{12} \Sigma_{22}^{-1} (z_2 - m_2)$ and $\mathrm{Var} \left[ Z_1 | Z_2 = z_2 \right] = \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}$.
\end{claim}
\begin{proof}
See lecture notes \cite{MVN2008}.
\end{proof}
The following technical lemma will also be useful for us.
\begin{claim}\label{clm: hazard_rate}
The hazard rate $H(x) = \frac{\phi(x)}{1-\Phi(x)}$ of a standard normal random variable is increasing, and satisfies
\[
\lim_{x \to -\infty} H(x) = 0,~H(x) = x + o_{x \to +\infty}(1)
\]
\end{claim}
This is a commonly known result in the literature on probability theory and statistics. For completeness, we provide a proof in Appendix \ref{app: hazard_rate}.
\subsection{Employer's First Moment Inference}
The main lemma of this section characterizes the employer's Bayesian inference task when the college is using a threshold admissions rule: the posterior expectation of a student's type, conditioned on their exam score being sufficiently high to cross the admissions threshold, and on their observed grade. In the appendix, we give the corresponding inference rule for the employer when the college can use an arbitrary monotone admissions rule.
\begin{lemma}\label{lem: closed_form_bayes_posterior}
\begin{align*}
&\mathbb{E} \left[ T_i | S_i \geq \beta_i, G_i = g \right]
\\&= \frac{\gamma^2}{\sigma_i^2 + \gamma^2} \mu_i
+ \frac{\sigma_i^2}{\sigma^2 + \gamma^2} g
\\&+ \frac{\gamma^2 \sigma_i^2}{\sqrt{(\sigma_i^2+\gamma^2) (\sigma_i^2 + \gamma^2 + \gamma^2 \sigma_i^2)}}
\cdot
H \left( \frac{(\sigma_i^2 + \gamma^2) \cdot \beta_i - \gamma^2 \mu_i - \sigma_i^2 g}{ \sqrt{(\sigma_i^2+ \gamma^2) (\sigma_i^2+ \gamma^2 + \gamma^2 \sigma_i^2)}} \right)
\end{align*}
where $H(x) = \frac{\phi(x)}{1-\Phi(x)}$ is the Hazard function of a standard normal random variable.
\end{lemma}
\begin{proof}
The proof is given in Appendix~\ref{app: closed_form_bayes_posterior}.
\end{proof}
A corollary of the previous lemma is that the posterior expectation computed by the employer will satisfy a number of nice regularity conditions which will be useful in proving our impossibility results:
\begin{corollary}\label{cor: increasing_limits}
$e_i(\mu_i,\sigma_i,\beta_i,g) = \mathbb{E} \left[ T_i | S_i \geq \beta_i, G_i = g \right]$ is continuous, differentiable, and strictly increasing in each of $\mu_i,~g$ and $\beta_i$. Further,
\begin{align*}
&\lim_{g \to -\infty} e(\mu_i,\sigma_i,\beta_i,g) = -\infty,
\\& \lim_{g \to +\infty} e_i(\mu_i,\sigma_i,\beta_i,g) = +\infty,
\end{align*}
and
\begin{align*}
&\lim_{\beta_i \to -\infty} e(\mu_i,\sigma_i,\beta_i,g) = \frac{\gamma^2}{\sigma_i^2 + \gamma^2} \mu_i
+ \frac{\sigma_i^2}{\sigma^2 + \gamma^2} g,
\\
&\lim_{\beta_i \to +\infty} e(\mu_i,\sigma_i,\beta_i,g) = + \infty.
\end{align*}
\end{corollary}
\begin{proof}
See Appendix \ref{app: increasing_limits}
\end{proof}
Finally, we define a quantity that will be useful to make reference to in a number of our forthcoming arguments: the minimum grade that results in a student from group $i$ being hired by the employer, given a fixed admissions rule.
\begin{definition}[Hiring threshold on grades]\label{def: g*}
We define $g_i^*(C) = \min \{g:~\mathbb{E} \left[ T_i | S_i \geq \beta_i, G_i = g \right] \geq C\}$ the inverse function of $g \to \mathbb{E} \left[ T_i | S_i \geq \beta_i, G_i = g \right]$.\
\end{definition}
By Corollary~\ref{cor: increasing_limits}, $g^*_i(.)$ is a well-defined function on domain $\mathbb{R}$, and is continuous, differentiable, and strictly increasing.
\subsection{Moments of the posterior distribution for monotone admission rules}
The following lemma holds for the general case of monotone, randomized admission rules, and is useful in characterizing the moments of the distribution of types conditional on $A_i = 1$ and $G=g$ in population $i$:
\begin{lemma}\label{lem: conditional_derivatives}
Let $A_i(.)$ be a non-decreasing, non-zero, possibly randomized admission rule. For all $g \in \mathbb{R}$, $\mathbb{E} \left[T_i^k| G_i=g, A_i = 1 \right]$ is finite and differentiable in $g$, and its derivative satisfies the following equation:
\begin{align*}
&\frac{\partial}{\partial g} \mathbb{E}_i \left[T_i^k \middle\vert G_i=g, A_i = 1 \right]
\\&= \frac{1}{\gamma^2} \mathbb{E}_i \left[T_i^{k+1}\middle\vert G_i = g, A_i=1\right]
\\&- \frac{1}{\gamma^2} \mathbb{E}_i \left[T_i^k \middle\vert G_i =g, A_i=1 \right] \cdot \mathbb{E}_i \left[T_i |G_i=g,A_i=1 \right].
\end{align*}
\end{lemma}
\begin{proof}
The proof is given in Appendix~\ref{app: conditional_derivatives}.
\end{proof}
\section{When Both Conditions are Satisfiable}
In this section, we observe that there are two settings in which it is possible to ``have it all'' --- satisfying both IGM and equal opportunity even in the multiple threshold case. The first setting is that of noiseless exam scores: when student types are perfectly observable by the school. The second setting is when the school opts not to report grades. We view the first setting as generally unrealisable, since any student evaluation will involve some degree of stochasticity. However the 2nd case --- in which a school opts not to report grades --- can be realized.
\subsection{Noiseless Exam Scores (Observable Types)}
First, we observe that if schools can perfectly observe student types (we have noiseless exam scores with $S_i = T_i$), then there is a simple threshold admissions rule that simultaneously achieves IGM and equal opportunity, even in the multiple threshold case. The ideas is simple: Given a range of employer costs $[C^-, C^+]$, the college simply sets an admissions threshold of $C^+$ or higher, using the same threshold for members of both groups. Because the threshold is the same for both groups, the probability of being admitted to college is a function only of type, and independent of group membership conditioned on type. Because scores were noiseless, admissions to college deterministically certifies that a student's type $t_i \geq C^+$, and so the employer chooses to hire everyone, independently of the grade they receive (and independently of their group membership). Hence, the probability of being hired is the same as the probability of being accepted to college, and is independent of group membership conditioned on type, and the employer's hiring rule is independent of group membership.
\begin{claim}\label{clm: fairness_nonoise}
Suppose $S_i = T_i$, i.e. a student's score perfectly reveals his type. Then for any hiring interval of hiring costs $[C^-, C^+] \in \mathbb{R}$, the non-zero admissions rule:
$$A_i(s) = 1 \Leftrightarrow s \geq C^+$$
for both groups $i \in \{1,2\}$ satisfies IGM and equal opportunity when paired with \emph{any} grading policy.
\end{claim}
\begin{proof}
See Appendix \ref{app: fairness_nonoise}.
\end{proof}
\begin{claim}
Suppose the school does not assign grades to students. Then for any hiring interval of hiring costs $[C^-, C^+] \in \mathbb{R}$, the non-zero thresholding admissions rule:
$$A_i(s) = 1 \Leftrightarrow s \geq \beta$$
for both groups $i \in \{1,2\}$ satisfies IGM and equal opportunity
when $\beta$ is large enough.
\end{claim}
\begin{proof}
For $\beta$ big enough, $\mathbb{E}\left[T_i | S_i \geq \beta \right] \geq C^+$ as
\\$\lim_{\beta \to +\infty} \mathbb{E} \left[T_i | S_i \geq \beta\right]= +\infty$; this can be seen either by following the same steps as in the proof of Lemma \ref{lem: closed_form_bayes_posterior} to obtain that
\[
\mathbb{E}\left[T_i | S_i \geq \beta \right] = \mu_i + \frac{\sigma_i^2}{\sqrt{1+\sigma_i^2}} H \left(\frac{\beta_i-\mu_i}{\sqrt{1+\sigma_i^2}} \right)
\]
which tends to $+\infty$ when $\beta_i \to +\infty$ by Claim \ref{clm: hazard_rate}. Another way of deriving this expression is by noting that not having a grade is equivalent to having an uninformative grade, i.e. to having $\gamma \to +\infty$.
Now, let $\beta$ be large enough such that in both populations, such that $\mathbb{E}\left[T_i | S_i \geq \beta \right] \geq C^+$. IGM immediately holds as every student that is accepted by the school is hired by the employer. Equal opportunity holds because the probability of a student with type $t$ being hired by the employer is exactly the probability that he is admitted by the school (every student admitted by the school is hired by the employer), hence is given by
\[
\mathbb{P} \left[ S_i \geq \beta | T_i = t\right] = \int_{s \geq \beta} \phi(s-t) dt,
\]
and is independent of the student's population.
\end{proof}
Note that this result is achieved by having the school set a very high admissions threshold (uniformly for both groups), and declining to give grades. Hence, declining to give grades may be a reasonable strategy for promoting our fairness goals in a highly selective school, but does not work when admissions thresholds must be lower. We note that the practice of grade witholding in MBA programs seems to be limited to the very top programs \cite{nogrades}.
In the remainder of the paper we consider the case in which exam scores have positive finite variance, and in which the college uses a grading policy with positive finite variance. What will be possible will depend on whether we are in the single or multiple threshold case.
\section{The Single Threshold Case}
In this section, we consider what is possible when there is only a single employer with a hiring cost $C$ that is known to the college. We show that in this case, IGM can always be achieved, but that it is impossible to achieve sIGM.
\subsection{IGM can always be achieved}
The main idea is as follows: For any grading scheme, and with a single threshold $C$ in mind, the college can separately set different admissions thresholds $\beta_1^*$ and $\beta_2^*$ for the two groups respectively such that the posterior expectation for a student type from each group crosses the threshold of $C$ at a grade $g^*$, which can be made to be the same for both populations. Since the only thing that matters in the employer's hiring decision is whether or not the student's expected type is above or below $C$, this is enough to cause the employer's hiring decision to be independent of group membership. The next lemma establishes that it is always possible to find such thresholds:
\begin{lemma}\label{lem: weak_calibration}
For any $C$ in $\mathbb{R}$, there exists thresholds $\beta_1^*$ and $\beta_2^*$ and a grade $g^*$ such that
\[
\mathbb{E} \left[T_1 | G_1 = g^*, S_1 \geq \beta_1^* \right] = \mathbb{E} \left[T_2 | G_2 = g^*, S_2 \geq \beta_2^* \right] = C
\]
\end{lemma}
\begin{proof}
It follows by Corollary~\ref{cor: increasing_limits} that
\[
\mathbb{E} \left[T_i | G_i = g, S_i \geq \beta_i \right]
\]
is continuous in $\beta_i$ and must reach any value between $\frac{\gamma^2}{\sigma_i^2 + \gamma^2} \mu_i + \frac{\sigma_i^2}{\sigma_i^2 + \gamma^2} g$ and $+\infty$. For $g^*$ small enough, it must be the case that
\[
\frac{\gamma^2}{\sigma_i^2 + \gamma^2} \mu_i + \frac{\sigma_i^2}{\sigma_i^2 + \gamma^2} g^* \leq C < +\infty,
\]
hence there exists $\beta_i^*$ such that
\[
\mathbb{E} \left[T_i | G_i = g^*, S_i \geq \beta_i^* \right] = C.
\]
\end{proof}
\begin{corollary}
Fix any $C$ in $\mathbb{R}$. When the school uses thresholding admission rules with thresholds $\beta_1^*$ and $\beta_2^*$, IGM holds for that $C$.
\end{corollary}
\begin{proof}
$\mathbb{E} \left[T_i | G_i = g, S_i \geq \beta_i^* \right]$ is a strictly increasing function of $g$ by Corollary~\ref{cor: increasing_limits} , therefore the employer accepts students from any population if and only if $g \geq g^*$ where $g^*$ is population-independent, which proves the results.
\end{proof}
\subsection{sIGM is impossible}
We now show that strong IGM --- making the posterior distributions for both groups identical --- is impossible. In addition to its intrinsic interest, this result will be a key ingredient in our impossibility results for the multiple threshold setting.
\begin{lemma}\label{lem: strong_calib_gaussian}
Suppose the priors are distinct. For any two thresholds $\beta_1$ and $\beta_2$, there must exists $t \in \mathbb{R}$ such that
\[
\mathbb{P} \left[T_1 = t | S_1 \geq \beta_1 \right] \neq \mathbb{P} \left[T_2 = t | S_2 \geq \beta_2 \right]
\]
I.e., sIGM cannot hold.
\end{lemma}
\begin{proof}
Let $x_i(t) = \mathbb{P} \left[ S_i \geq \beta_i | T_i = t\right]$. Suppose for all $t \in \mathbb{R}$, sIGM holds, i.e.
\[
\mathbb{P} \left[T_1 = t | S_1 \geq \beta_1 \right] \neq \mathbb{P} \left[T_2 = t | S_2 \geq \beta_2 \right]
\]
by Claim~\ref{clm: sIGM_equivalence}. Then
\[
\frac{x_1(t) \phi\left(\frac{t-\mu_1}{\sigma_1}\right)}{\mathbb{P} \left[S_1 \geq \beta_1 \right]} = \frac{x_2(t) \phi \left(\frac{t-\mu_2}{\sigma_2}\right)}{\mathbb{P} \left[S_2 \geq \beta_2 \right]}
\]
hence
\[
\frac{x_1(t)}{x_2(t)} = \frac{\sigma_1 \mathbb{P} \left[S_1 \geq \beta_1 \right]}{\sigma_2 \mathbb{P} \left[S_2 \geq \beta_2 \right]} \cdot \exp\left( \frac{(t-\mu_2)^2}{2\sigma_2} - \frac{(t-\mu_1)^2}{2\sigma_1^2} \right)
\]
$x_1(.)$ and $x_2(.)$ are non-decreasing functions with values in $[0,1]$, and $x_i(t) = \int_{s \geq \beta_i} \phi(s-t) ds$ is non-zero; therefore, $\lim_{t = +\infty} x_i(t)$ exists and is strictly positive. It must then be the case that $\frac{x_1(t)}{x_2(t)}$ has a finite and strictly positive limit in $+\infty$. On the other hand,
\[
\exp\left( \frac{(t-\mu_2)^2}{2\sigma_2} - \frac{(t-\mu_1)^2}{2\sigma_1^2} \right) = K \exp \left(\frac{t^2}{2} \left(\frac{1}{\sigma_2^2}-\frac{1}{\sigma_1^2}\right) + \left( \frac{\mu_1}{\sigma_1^2} - \frac{\mu_2}{\sigma_2^2}\right) t \right)
\]
for some constant $K$. It is easy to see that the above quantity tends to either $+\infty$ or $0$ as $t \to +\infty$ as long as either $\sigma_1 \neq \sigma_2$ or $\mu_1 \neq \mu_2$ (one of $\frac{1}{\sigma_2^2}-\frac{1}{\sigma_1^2}$ and $\frac{\mu_1}{\sigma_1^2} - \frac{\mu_2}{\sigma_2^2}$ must be non-zero). This leads to a contradiction.
\end{proof}
\subsection{Equal opportunity}
We defer the technical results of this section to Appendix~\ref{app: equalodd_partial}.
Lemma~\ref{lem: equalodd_partial_1} shows that for thresholding admission rules, IGM and equal opportunity cannot simultaneously hold for Gaussian priors with the same variance but different mean. This shows that obtaining fairness in the general case is significantly more difficult than in the simple cases in which the types are observable and the school does not assign grades.
Lemma~\ref{lem: equalodd_partial_2} shows that arguably stringent conditions on the grade accuracy and the thresholds set by the school must hold for equal opportunity to be possible. We conjecture that these conditions are, in general, impossible to satisfy, making equal opportunity impossible to satisfy even in isolation, in the single threshold case. As we will see in the next section, it is impossible to satisfy in the multiple-threshold case.
\section{The Multiple Threshold Case}
In this section, we turn to the multiple threshold case, which we view as the main setting of interest. In this case, we ask whether we can achieve IGM and equal opportunity not just with respect to a single known hiring cost $C$, but with respect to an entire interval of hiring costs $C \in [C^-, C^+]$. This will be the case when there are multiple employers, or simply when there is some uncertainty about the hiring threshold used by a single employer.
\subsection{IGM is Impossible}
In this section, we show that IGM is impossible to achieve even in isolation. The proof proceeds by showing that in the multiple threshold case, IGM must imply sIGM --- i.e. that the posterior distributions conditional on admission to college are identical for both groups. Impossibility then follows from the impossibility of achieving sIGM (even for a single threshold), which we proved in the last section.
We first state a technical lemma, showing that if we satisfy IGM for every employer cost $C$ in a continuous interval, we must actually be equalizing the posterior expected type across groups for every grade $g$ in some other continuous interval.
\begin{claim}\label{clm: IGM_threshold_to_grade}
Let $\ubar{C} < \bar{C}$. Suppose that for all $C \in (\ubar{C},\bar{C})$,
\[
\mathbb{E}\left[T_1|G_1=g, S_1 \geq \beta_1\right] \geq C \Leftrightarrow \mathbb{E}\left[T_2|G_2=g, S_2 \geq \beta_2\right] \geq C,
\]
then it must be the case that for $g$ in some interval $(a,b)$,
\[
\mathbb{E}\left[T_1|G_1=g, S_1 \geq \beta_1\right] = \mathbb{E}\left[T_2|G_2=g, S_2 \geq \beta_2\right]
\]
\end{claim}
\begin{proof}
Let $a = g_1^*(\ubar{C})$ and $b = g_1^*(\bar{C})$ where $g_1^*(.)$ is the strictly increasing inverse of $g \to \mathbb{E} \left[T_1 | G_1 = g, A_1 = 1 \right]$ as per Claim \ref{cor: increasing_limits} and Definition \ref{def: g*}. Suppose there exists $g \in (a,b)$ such that
\[
\mathbb{E}\left[T_1|A_1=1,G_1=g\right] > \mathbb{E}\left[T_2|A_2=1,G_2=g\right],
\]
then for $C = \mathbb{E}\left[T_1|A_1=1,G_1=g\right] \in (\ubar{C},\bar{C})$, it must be the case that
\[
\mathbb{E}\left[T_1|A_1=1,G_1=g\right] \geq C > \mathbb{E}\left[T_2|A_2=1,G_2=g\right]
\]
which contradicts the assumption of the claim. Now, suppose there exists $g \in (a,b)$ such that
\[
\mathbb{E}\left[T_1|A_1=1,G_1=g\right] < \mathbb{E}\left[T_2|A_2=1,G_2=g\right],
\]
then for $\epsilon > 0$ small enough, $C = \mathbb{E}\left[T_1|A_1=1,G_1=g\right]+\epsilon \in (\ubar{C},\bar{C})$ and
\[
\mathbb{E}\left[T_1|A_1=1,G_1=g\right] < C \leq \mathbb{E}\left[T_2|A_2=1,G_2=g\right]
\]
which also contradicts the assumption of the claim. Therefore, it must be the case that for all $g \in (a,b)$,
\[
\mathbb{E}\left[T_1|A_1=1,G_1=g\right] = \mathbb{E}\left[T_2|A_2=1,G_2=g\right]
\]
\end{proof}
We can now go on to prove the main theorem in this section:
\begin{theorem}\label{thm: robust_IGM}
Suppose the priors are distinct,
then IGM cannot for all hiring costs $C \in (\ubar{C},\bar{C})$.
\end{theorem}
\begin{proof}
By Claim \ref{clm: IGM_threshold_to_grade}, it must be the case that for all $g \in (a,b)$ for some interval $(a,b)$, $\mathbb{E}\left[T_1|G_1=g, S_1 \geq \beta_1\right] = \mathbb{E}\left[T_2|G_2=g, S_2 \geq \beta_2 \right]$. For all $g$ in $(a,b)$, by Lemma~\ref{lem: conditional_derivatives}
\[
\frac{\partial}{\partial g} \mathbb{E} \left[T_1 | G_1=g, S_1 \geq \beta_1 \right] = \frac{\partial}{\partial g} \mathbb{E} \left[T_2 | G_2=g, S_2 \geq \beta_2 \right]
\]
and hence $E\left[T_1^2|G_1=g, S_1 \geq \beta_1\right] = \mathbb{E}\left[T_2^2|G_2=g, S_2 \geq \beta_2\right]$. It is easy to see that using the same argument by induction yields that for all integers $k$,
\[
\mathbb{E}\left[T_1^k|G_1=g, S_1 \geq \beta_1\right] = \mathbb{E}\left[T_2^k|G_2=g,S_1 \geq \beta_1\right]
\]
Since the distributions of types for the two populations conditional on $G_i=g, S_i \geq \beta_i$ admit a moment generating function (this follows immediately from the fact that $P_i$ admits a moment generating function) and have identical moments, it must be that the distributions are the same for all $g \in (a,b)$. I.e., for all $g \in (a,b)$, we have
\[
\mathbb{P} \left[T_1 = t |G_1=g, S_1 \geq \beta_1\right] = \mathbb{P} \left[T_2 = t |G_2=g, S_1 \geq \beta_1\right]
\]
We have that in population $i$,
\[
\mathbb{P} \left[T_i = t | G_i=g, S_i \geq \beta_i \right] = \frac{\mathbb{P} \left[T_i = t | S_i \geq \beta_i \right] \phi \left(\frac{g-t}{\gamma} \right)}{\int_t \mathbb{P} \left[T_i = t | S_i \geq \beta_i \right] \phi \left(\frac{g-t}{\gamma} \right) dt}
\]
Note that $\int_t \mathbb{P} \left[T_i = t | S_i \geq \beta_i \right] \phi \left(\frac{g-t}{\gamma} \right) dt$ is a function of $g$ only, that we will denote $p_i(g)$ from now on.
\[
\mathbb{P} \left[T_1 = t | G_1=g, S_1 \geq \beta_i \right] = \mathbb{P} \left[T_2 = t | G_2=g, S_2 \geq \beta_2 \right]
\]
implies
\[
\frac{\mathbb{P} \left[T_1 = t | S_1 \geq \beta_1 \right]}{\mathbb{P} \left[T_2 = t | S_2 \geq \beta_2 \right]} = \frac{p_1(g)}{p_2(g)}
\]
for all $g \in (a,b)$ and $t \in \mathbb{R}$. $\mathbb{P} \left[T_1 = t | S_1 \geq \beta_1 \right]$ and $\mathbb{P} \left[T_2 = t | S_2 \geq \beta_2 \right]$ are both probability density functions that integrate to $1$, so it must be the case that $\frac{p_1(g)}{p_2(g)} = 1$ and $\mathbb{P} \left[T_1 = t | S_1 \geq \beta_1 \right] = \mathbb{P} \left[T_2 = t | S_2 \geq \beta_2 \right]$. Therefore, sIGM must hold, which we have shown is impossible in Lemma \ref{lem: strong_calib_gaussian}.
\end{proof}
\subsection{Equal opportunity cannot hold}
Finally, we show that in the multiple threshold case, it is also impossible to satisfy the equal opportunity condition.
\begin{theorem}\label{thm: impossibility_equalodds}
Suppose the priors are distinct. There exist no thresholding admission rules such that equal opportunity is guaranteed for all $C \in (\ubar{C},\bar{C})$, for any $\ubar{C} < \bar{C}$.
\end{theorem}
\begin{proof}
It is easy to see $x_i(t) = \int_s A_i(s) \phi(s-t) ds = \int_u A_i(u+t) \phi(u) du$ is monotone non-decreasing in $t$ and non-zero. Remember
\[
e_i(g) = \mathbb{E} \left[T_i | G_i = g, S_i \geq \beta_i \right]
\]
has a strictly increasing and differentiable inverse $g^*_i(.)$ on $(-\infty,+\infty)$ by Corollary~\ref{cor: increasing_limits}, and a student is hired by the employer if and only if $g \geq g^*_i(C)$. A student with type $t$ in population $i$ gets therefore hired with probability
\[
\int_{g \geq g^*(C)} x_i(t) \phi \left(\frac{g-t}{\gamma} \right) dt = x_i(t) \left( 1 - \Phi \left(\frac{g^*_i(C)-t}{\gamma} \right) \right)
\]
equal opportunity then imply that $\forall t \in \mathbb{R}, C \in (\ubar{C},\bar{C})$,
\[
\frac{x_1(t)}{x_2(t)} \cdot \left( 1 - \Phi \left(\frac{g^*_1(C)-t}{\gamma} \right) \right)
= \left( 1 - \Phi \left(\frac{g^*_2(C)-t}{\gamma} \right) \right)
\]
Taking the first order derivative in $C$ of both sides of the above equation, we have that for all $C \in (\ubar{C},\bar{C})$, for all $t \in \mathbb{R}$,
\[
\frac{x_1(t)}{x_2(t)} \cdot \frac{\frac{\partial g_1^*}{\partial C}(C)}{\frac{\partial g_2^*}{\partial C}(C)} = \frac{\phi \left(\frac{g^*_2(C)-t}{\gamma}\right)}{\phi \left( \frac{g^*_1(C)-t}{\gamma}\right)}
\]
Suppose for some $C \in (\ubar{C},\bar{C})$, $g^*_1(C) \neq g^*_2(C)$. Without loss of generality, renumber the populations such that $g^*_2(C) > g^*_1(C)$. We have that
\[
\frac{\phi \left(\frac{g^*_2(C)-t}{\gamma}\right)}{\phi \left( \frac{g^*_1(C)-t}{\gamma}\right)} = \exp \left(\frac{2 \left( g_2^*(C) - g_1^*(C)\right) t + g_1^*(C)^2-g_2^*(C)^2}{2 \gamma^2} \right)
\]
and we know that $g_i^*(.)$ is a strictly increasing function so $\frac{\partial g_2^*}{\partial C}(C) > 0$ so it must be the case that
\[
\lim_{t \to +\infty} \frac{x_1(t)}{x_2(t)} = +\infty.
\]
Since $x_1(t)$ is upper-bounded by $1$, this implies in particular that $x_2(t) \to 0$ as $t \to +\infty$, which contradicts $x_2(.)$ being a non-zero, non-decreasing function. Hence, it must be the case that for all $C \in (\ubar{C},\bar{C})$, $g_1^*(C) = g_2^*(C)$, i.e. IGM holds. By Lemma~\ref{thm: robust_IGM}, this is impossible.
\end{proof}
\section{Conclusion}
We consider two natural fairness goals that a college might have for its affirmative action policies: granting equal opportunity to individuals with the same type when graduating from high school, independent of their group membership, and incentivizing downstream employers to make hiring decisions that are independent of group membership. We show that these goals can be simultaneously achieved by highly selective colleges (i.e. those with very high admissions thresholds) --- \emph{but only if they do not report grades to employers}. This provides another view on this practice, which is followed by several highly selective MBA programs. On the other hand, we find that these goals are generally unachievable even in isolation if schools report informative grades. These impossibility results crucially hinge on the fact that exam scores and grades provide only \emph{noisy} signals about student types, and hence require rational expectation maximizers to reason about prior type distributions, which can vary by group.
Our paper leaves open a natural technical question: can a college set admissions and informative grading policies to realize the equal opportunity condition, in the \emph{single threshold} case? We conjecture that the answer to this question is \emph{no}, and in the Appendix, we give a theorem supporting this conjecture --- ruling out the possibility for deterministic admissions rules in every case except when the grading variance is exactly $1$.
\subsection*{Acknowledgements}
We thank Mallesh Pai and Jonathan Ullman for helpful discussions at an early stage of this work.
\bibliographystyle{plainnat}
|
2,869,038,155,714 | arxiv | \section{Introduction}
The following observation, now referred to as the Unruh effect, was made by W.
Unruh in 1976 \cite{u}. When a detector, coupled to a
relativistic quantum field in its vacuum
state, is uniformly accelerated through Minkowski
spacetime, with proper acceleration $a$, it registers a {\em thermal} black body
radiation at temperature
$T=\frac{\hbar a}{2\pi c k_{\mathrm B}}$. This is the so-called Unruh
temperature. In more anthropomorphic terms \cite{uw}, ``for a free quantum
field in its vacuum state in Minkowski spacetime $M$ an observer with uniform
acceleration $a$ will feel that he is bathed by a thermal distribution of
quanta of the field at temperature $T$.'' This result has attracted a fair amount of
attention, and generated considerable surprise and even some scepticism. For a review of
various aspects of the subject, nice physical discussions of the phenomenon,
and further references, we refer to \cite{ta}, \cite{wa} \cite{fuu}. The
reason for the surprise is that, if you think of the vacuum as ``empty
space'', then you will find it puzzling that a detector, accelerated or not,
which may itself initially be in its ground state, will ``see particles'',
since, after all, in the vacuum, there aren't any. In order not to be
surprised, one has to remember that, of course, the vacuum is not ``empty
space'', but
the ground state of the field, and one should {\em expect} the detector
to react to the presence of the field when it is accelerated through space.
For example, if you were to drag a detector along a non-relativistic chain of
oscillators in its ground state, you would certainly expect the coupling
between the detector and the oscillators to excite both. The energy for this
process is, in final analysis, furnished by the agent that drags the detector
along the oscillator chain.
What is nevertheless still surprising in connection with the Unruh effect is
the claim that the detector ``perceives'' a {\em thermal} distribution of
radiation at some particular temperature that only depends on the
acceleration. To see what is precisely meant by these statements, it is
helpful to get rid of the anthropomorphic terminology used above and in much
of the literature as well as of all reference to particles or quanta, which
turn out to be irrelevant to the discussion. This is what we will do below. It
is worth pointing out in this connection that already in \cite{u},
``detection of a particle'' is defined by ``excitation of the detector'', and
does therefore not presuppose the actual definition of what a particle
precisely is, which is a tricky thing to do, as is well known \cite{fu}. In
fact, the computations in the physics literature of the excitation probability
of the detector can be seen to be perturbative computations of the asymptotic
state of the detector (see \cite{uw} for example). We therefore adopt the
following simple formulation of the Unruh effect. Consider the coupled
detector-field system. Suppose that initially it is in a product
state with the field in the vacuum state. Now let the coupled system evolve.
At some later (detector proper) time, the state of the system will no longer
be a product state. Now trace out the field variables, to obtain the reduced
state of the detector (which will be a mixed state, even if the initial state
was pure). The Unruh effect states that the latter converges,
asymptotically in the observer's proper time, to the Gibbs state at
the aforementioned temperature $T$. Note that this is not by any means
obvious: after all, a priori, it is not clear why the detector state should,
asymptotically in time, converge at all, and even if it does, it is not
obvious it should tend to a positive temperature state: a priori, it could
have been any other mixed state.
It is our goal in this paper to give a complete and rigorous proof of the
above statement. The way we have formulated it makes it clear already that we
think of it as a problem in the theory of open quantum systems in which a
small system, here the detector, is coupled to a reservoir, here the field.
Let us formulate our result somewhat more precisely. For a completely rigorous
statement, we refer to Section \ref{s:model}. The model we consider is the one
proposed in \cite{uw}, which is itself a simplification of the model
considered in \cite{u}. The detector is modeled by a two-level system and the
field is taken to be a massive or massless Klein-Gordon field. The observable
algebra of the detector is therefore generated by ``fermionic''
creation/annihilation operators $A$, $A^\dagger(\sigma)$. The free Heisenberg
evolution of the detector is
$\dot A(\sigma)=-i E A(\sigma)$, where $\sigma$ is the detector's proper
time. In other words, the free detector Hamiltonian is
$$
H_{\mathrm D}=E A^\dagger A.
$$
The coupling between the field and the detector
is realized via a monopole, and is ultraviolet regularized; it is sometimes referred to as a de
Witt monopole detector (see \cite{ta}). Suppose initially the detector-field
system is in a product state $\omega_0$ with the detector in a state described
by some density matrix $\rho$ and the field in the Minkowski vacuum state. Let
$B$ be a detector observable and $\alpha_\sigma^\lambda(B)$ its Heisenberg
evolution under the coupled dynamics, with coupling constant $\lambda$. Then
we prove that
\begin{equation}\label{eq:unruh1}
\lim_{\sigma\to\infty}\omega_0(\alpha_\sigma^\lambda(B))
=\frac{1}{Z_{\beta,\mathrm D}} \mathrm{Tr}\ \mathrm{e}^{-\beta
H_{\mathrm D}}B+ O(\lambda^2).
\end{equation}
Here $\beta=(k_{\mathrm B}T)^{-1}$ with $T$ the Unruh temperature and $Z_{\beta, \mathrm
D}=\mathrm{Tr}\mathrm{e}^{-\beta H_{\mathrm D}}$. The approach to the equilibrium
state is exponentially fast.
Our proof of this result is based on techniques developed in the last decade to
prove ``return to equilibrium'' in open quantum systems \cite {JP1,JP2,BFS,M,DJ,DJP}. We combine these with the
Bisognano-Wichman theorem \cite{biwi}, which states that the vacuum is a KMS
state for the Lorentz boosts on the Rindler wedge. The relevance of this last
result to the Unruh effect (and a generalization to more general spacetimes)
was explained a long time ago by Sewell in \cite{se}. Let us point out that
the work of Sewell, together with known stability results of KMS states
(see e.g. \cite{Da,KFGV}) imply a result somewhat similar to but considerably weaker than
(\ref{eq:unruh1}), namely
\begin{equation}\label{eq:unruh2}
\lim_{\sigma\to\infty,
\lambda\to0,\lambda\sigma^2=1}\omega_0(\alpha_\sigma^\lambda(B))
=\frac{1}{Z_{\beta,\mathrm D}} \mathrm{Tr}\ \mathrm{e}^{-\beta
H_{\mathrm D}}B.
\end{equation}
This is the so-called van Hove weak coupling limit.
In our result, the
limit $\sigma\to\infty$ is shown to exist for all sufficiently small $\lambda$,
and to coincide with the right hand side of (\ref{eq:unruh1}).
The paper is organized as follows. In Section \ref{s:model}, we describe the
model in detail and state our main result. We will also comment on the precise
role played by the choice of the form factor determining the ultraviolet cutoff
in the interaction term. Section \ref{s:proof} is devoted to
its proof. The latter uses Araki's perturbation theory for KMS states and its
recent extensions, together with the spectral approach to the problem of
return to equilibrium developed in the cited references. Since this material
is rather technical, we have made an effort to state the result in Section
\ref{s:model} with as little reference to it as possible.\\
\noindent{\bf Acknowledgements:} SDB gratefully acknowledges the hospitality of McGill University,
Universit\'e de Montr\'eal and Concordia University, where part of this
work was performed. MM is grateful to Universit\'e des Sciences et
Technologies de Lille 1 for support and hospitality.
\section{The model and the result} \label{s:model}
We need to give a precise description of the model and in particular of its
dynamics. This requires some preliminaries.
\subsection{The free field}
Let us start by describing in detail the field to which the detector will be
coupled. The field operators are represented on the symmetric Fock space
${\cal F}$ over $L^2({\mathbb R}^d, \mathrm{d} \underline x)$. Here $d\geq 1$ is the dimension of
space and $x=(x^0,{\underline x})$ be is a point in Minkowski space--time ${\mathbb R}\times{\mathbb R}^d$
(with metric signature $(+,-,\dots,-)$). So ${\cal F}:=\oplus_{n\in{\mathbb N}} {\cal F}^{(n)}$,
where ${\cal F}^{(n)}$ is the $n$-fold symmetric tensor product of the
one-particle space $L^2({\mathbb R}^d, \mathrm{d} \underline x)$.
Let ${\cal S}({\mathbb R}^{d+1};{\mathbb R})$ and ${\cal S}({\mathbb R}^{d+1};{\mathbb C})$ denote the real and
the complex valued Schwartz functions on ${\mathbb R}^{d+1}$, respectively. For
$f\in{\cal S}({\mathbb R}^{d+1};{\mathbb C})$, one defines the field operators in the usual way:
$$
\Omega=(-\Delta+m^2)^{1/2}, S^{\pm}f=\int_{\mathbb R} \mathrm{d} t \frac1{\sqrt\Omega}\mathrm{e}^{\pm{\mathrm i}\Omega t}f_t, \
Q[f]=\frac1{\sqrt2}(a^\dagger(S^+ f) + a(\overline{S^-f})).
$$
Here $\Delta$ is the Laplacian, $m\geq0$ the mass, and $a, a^\dagger$ are the
usual creation and annihilation operators on ${\cal F}$ (we follow the convention
that $f\mapsto a^\dagger(f)$ is linear while $f\mapsto a(f)$ is antilinear),
and the bar denotes complex conjugation. When $m=0$, we will
suppose $d>1$. Writing formally
$$
Q[f]=\int_{{\mathbb R}^{d+1}}\mathrm{d} x f(x) Q(x), \quad x=(x^0, {\underline x}),
$$
this leads to the familiar
\begin{equation}
Q(x)=\int_{{\mathbb R}^d}\frac{\mathrm{d}{\underline k}}{\sqrt{2\omega({\underline k})}}\big[ \mathrm{e}^{\i {\underline k}\,{\underline x}
-\i\omega({\underline k}) x^0}a({\underline k}) +\mathrm{e}^{-\i {\underline k}\,{\underline x} +\i\omega({\underline k}) x^0}a^*({\underline k})\big],
\label{b2}
\end{equation}
where $\omega({\underline k})=\sqrt{\underline k^2 + m^2}$. The field satsifies the
Klein-Gordon equation $\square Q(x) + m^2 Q(x)=0$, where
$\square=\partial_{x^0}^2-\Delta$. We use units in which $\hbar=1=c$.
As can be learned in any book on special relativity (such as \cite{ri}), in an
adapted choice of inertial coordinate frame, a uniformly accelerated
worldline of proper acceleration $a>0$, parametrized by its proper time
$\sigma$, has the form
$$
x^0(\sigma)=\frac1a\sinh a\sigma,\quad x^1(\sigma)=\frac1a\cosh a\sigma,\quad
x^2(\sigma)=0= x^3(\sigma).
$$
Associated to this worldline is the right wedge (or Rindler wedge) $
W_{\mathrm R}:=\{x\in{\mathbb R}^4 |\ x^1>|x^0|\}. $ It is the intersection of the
causal future and past of the worldline, or the collection of spacetime points
to which the observer on the worldline can send signals and from which he can
also receive signals. Note for later reference that the left wedge
$W_{\mathrm L}:=-W_{\mathrm R}$ is the causal complement of $\overline
W_{\mathrm R}$.
There exists a global coordinate system on $W_{\mathrm R}$ that is
particularly well adapted to the description of the problem at hand. It is
given by the so-called Rindler coordinates $(\tau,u,\underline
x_\perp)\in{\mathbb R}\times {\mathbb R}_+^*\times {\mathbb R}^{d-1}$, defined by
\begin{equation}\label{eq:coordchange}
x^0= u\sinh \tau,\ x^1=u\cosh \tau,\ \underline x_\perp=(x_2, \dots x_d).
\end{equation}
Here $\tau$ is a global time coordinate on the right wedge. Note that, given
$a\in{\mathbb R}^+, (\alpha_2,\dots \alpha_d)\in{\mathbb R}^{d-1}$, the curve $u=1/a$,
$\underline x_\perp = (\alpha_2, \dots, \alpha_d)$ is the worldline of a
uniformly accelerated observer with proper acceleration $a$ and proper time
$\sigma = a^{-1} \tau$. In addition, two points in the right wedge with the
same value for the $\tau$-coordinate are considered as simultaneous in the
instantaneous rest frame of any such observer (see \cite{ri}). Among the
Lorentz boosts, only the boosts in the $x_1$-direction leave the right wedge
invariant. In inertial coordinates they are given by the linear
transformations
$$
B_{\tau'}= \left[
\begin{array}{ccccc}
\cosh\tau' &\sinh\tau'&0&\dots &0\\
\sinh\tau'&\cosh\tau'&0&\dots &0\\
0&0&1&\dots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\dots&1
\end{array}
\right].
$$
In the Rindler coordinates, this becomes $B_{\tau'}(\tau, u, \underline
x_\perp)= (\tau +\tau', u, \underline x_\perp)$. In this sense, the boosts in the
$x_1$-direction act as time translations on the Rindler wedge.
Since the field satisfies the Klein-Gordon equation, one has, in Rindler
coordinates on $W_{\mathrm R}$:
\begin{equation}\label{eq:kleingordonwedge}
\left(u^{-2}\partial_\tau^2 - u^{-1}\partial_{u} u
\partial_{u} +(-\Delta_{\perp}+ m^2)\right) Q(\tau, u,\underline
x_\perp)=0.
\end{equation}
Moreover, the covariance of the free field under the Poincar\'e group yields,
for all $\tau\in{\mathbb R}$,
\begin{equation}\label{eq:boostfield}
Q[f\circ B_{-\tau}]=\mathrm{e}^{{\mathrm i} L_{\mathrm F} \tau}Q[f]\mathrm{e}^{-{\mathrm i}
L_{\mathrm F} \tau},
\end{equation}
with
\begin{equation}\label{eq:liouvfreefield}
L_{\mathrm F}=\mathrm{d} \Gamma (K), \ K=\Omega^{1/2} X^1\Omega^{1/2},
\end{equation}
where $X^1$ is the operator of multiplication by $x^1$.
In particular, for $x=(\tau, u, \underline x_\perp)\in W_{\mathrm R}$,
$$
Q(\tau, u, \underline x_\perp) = \mathrm{e}^{{\mathrm i} L_{\mathrm F}
\tau}Q(0,u,\underline x_\perp)\mathrm{e}^{-{\mathrm i} L_{\mathrm F} \tau}.
$$
In other words, $L_{\mathrm F}$ generates the free Heisenberg dynamics of the
field operators associated to the right wedge. Let us furthermore introduce,
for later purposes, the conjugate field
\begin{equation}\label{eq:conjfield}
P[f]:=\frac{\mathrm{d} }{\mathrm{d} \tau} Q[f\circ B_{-\tau}]\mid_{\tau=0}\,={\mathrm i}\left[L_{\mathrm
F}, Q[f]\right].
\end{equation}
It then follows from the basic properties of the free field that the equal
time commutation relations of the field and the conjugate field are, at
$\tau=0$,
\begin{equation}\label{eq:equaltime}
\left[ Q(0, u, \underline x_\perp), P(0, u', \underline x_\perp')\right] ={\mathrm i} u
\delta_u(u')\ \delta_{\underline x_\perp}(\underline x_{\perp}').
\end{equation}
The following useful identity follows from (\ref{eq:kleingordonwedge}) and
(\ref{eq:conjfield}):
\begin{equation}\label{eq:useful}
{\mathrm i}\left[ L_{\mathrm F}, P(0,u, \underline x_\perp)\right] = -\left(-u\partial_u
u\partial_u + u^2(-\Delta_{\perp} + m^2)\right)Q(0,u,\underline x_\perp).
\end{equation}
For an algebraic formulation of the dynamics, indispensable in what follows,
we need to identify the observable algebra of the theory. The observable
algebra of the field is ${\mathcal A}_{\mathrm F}:=\{W(f) | f\in {\cal S}({\mathbb R}^{d+1}, {\mathbb R})\}''$, with
$W(f)=\mathrm{e}^{-{\mathrm i} Q[f]}$ the usual Weyl operators. One should think of the
observable algebra as containing all bounded functions of the (smeared) field
operators $Q[f]$ or, more pictorially, all observables that can be constructed
from the $Q(x)$, $x\in {\mathbb R}^{d+1}$. Associated to the right and left wedges are
local algebras of observables ${\mathcal
A}_{\mathrm F;\mathrm{R, L}}:=\{W(f) | f\in
{\cal S}(W_{\mathrm{R,L}},{\mathbb R})\}''$. Again, those should be thought of as
containing all observables that can be constructed with the field operators
$Q(x)$, for $x$ belonging to the wedge considered. As pointed out above, one
can define on ${\mathcal A}_{\mathrm F}$ an automorphism group $\alpha_\tau^0$ by
\begin{equation}\label{eq:freedynamfield}
\alpha_{{\mathrm F},\tau}^0(A)=\mathrm{e}^{{\mathrm i} L_{\mathrm F}\tau} A\ \mathrm{e}^{-{\mathrm i} L_{\mathrm
F}\tau},\qquad A\in{\mathcal A_{\mathrm F}}.
\end{equation}
We note that $\alpha^0_{{\mathrm F},\tau}$ leaves ${\mathcal A}_{\mathrm F;\mathrm{R}}$ invariant.
\subsection{The free detector}\label{s:freedetector}
As pointed out in the introduction, we think of the detector as a two-level
system. Our results extend without problem to an $N$-level system, at the cost
of irrelevant notational complications. So we follow the physics literature on
the subject and limit ourselves to a highly idealized two-level detector. Its
observable algebra is simply the algebra of two by two matrices ${\mathcal
B}({\mathbb C}^2)$. It will be convenient to use a representation of this algebra in
which both the ground state and the Gibbs state at inverse temperature $\beta$
are represented by vectors. This representation, well known in the
mathematical physics literature on quantum statistical mechanics, is of course
different from the usual one in the standard physics literature in which the
latter is represented by a density matrix.
It is defined as follows. One represents the observable algebra ${\mathcal
B}({\mathbb C}^2)$ as ${\mathcal A}_{\mathrm D}:={\mathcal
B}({\mathbb C}^2)\otimes\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2$ on ${\mathcal H}_{\mathrm
D}={\mathbb C}^2\otimes{\mathbb C}^2$, with in particular $A^\dagger:=\left[
\begin{array}{cc} 0&1\\0&0\end{array}\right]\otimes \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2$. The algebra ${\mathcal A}_{\mathrm D}$ is generated by the identity
operator, $A^\dagger$, $A$ and $A^\dagger A$ and one has $A A^\dagger +
A^\dagger A=\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l$.
In this representation, the free Heisenberg evolution of the detector with
respect to its proper time $\sigma$ is generated by the self-adjoint operator
$L_{\mathrm D}:=H_{\mathrm D}\otimes \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2-\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2\otimes H_{\mathrm
D}$, with
\begin{equation}
H_{\mathrm D}=
\left[\begin{array}{cc}
E & 0\\ 0 & 0\end{array}\right],
\label{hd}
\end{equation}
for some $E>0$, where $E$ represents the excitation energy of the detector; $L_{\mathrm D}$ is referred to as the
free Liouvillean of the detector. To see this it is enough to remark that
$$
A^\dagger(\sigma):= \alpha_{{\mathrm D},\sigma}^0(A):=\mathrm{e}^{{\mathrm i} L_{\mathrm
D}\sigma} A^\dagger \mathrm{e}^{-{\mathrm i}
L_{\mathrm D}\sigma}
$$
satisfies the correct Heisenberg equation of motion
\begin{equation}\label{eq:heisenbergdetector}
\dot A^\dagger(\sigma)=i E A^\dagger(\sigma)
\end{equation}
of an unperturbed two-level system. Note that the energy levels of the
detector are thought of in this model as pertaining to internal degrees of
freedom (\cite{u,uw, ta}). One should think of the two-level system as being
dragged through spacetime by an external agent that ensures it has constant
acceleration $a$. So the translational degrees of freedom of the detector are
not dynamical variables in this kind of model. In the representation above,
the ground state of the detector can be represented by the vector $|--\rangle$
and the Gibbs state at inverse temperature $\beta$ by the vector
$$
|\beta, \mathrm D\rangle:=(1+\mathrm{e}^{-\beta E})^{-1/2}(|--\rangle + \mathrm{e}^{-\beta
E/2}|++\rangle)\in {\mathcal H}_{\mathrm D}.
$$
Indeed, one easily checks that, for any $B\in{\mathcal B}({\mathbb C}^2)$,
$$
\langle \beta, \mathrm D| B\otimes \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2|\beta, \mathrm D\rangle
=\frac1{Z_{\beta, \mathrm D}} \mathrm{Tr}\ \mathrm{e}^{-\beta H_{\mathrm D}}B,\quad
Z_{\beta, \mathrm D}=\mathrm{Tr}\ \mathrm{e}^{-\beta H_{\mathrm D}}.
$$
It is the fact that both the ground state and positive temperature states of
the detector can be represented by vectors that makes this representation
particularly suitable for the problem at hand.
\subsection{The uncoupled field-detector system}\label{s:uncoupldsystem}
It is now easy to describe the observable algebra of the joint detector-field
system, as well as its uncoupled dynamics. On the Hilbert space
${\mathcal H}:={\mathcal H}_{\mathrm D}\otimes {\cal F}$ we consider the observable algebra ${\mathcal
A}:={\mathcal A}_{\mathrm D}\otimes{\mathcal A}_{\mathrm F, R}$ and the self-adjoint operator $L_0=L_{\mathrm D}\otimes \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{\mathrm {\cal F}}+
\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{{\mathcal H}_{\mathrm D}}\otimes a L_{\mathrm F}$. The latter determines an automorphism
group
$$
\alpha^0_{\sigma}=\alpha^0_{{\mathrm D}, \sigma}\otimes \alpha^0_{{\mathrm
F},a\sigma}
$$
of $\mathcal A$ in the usual way: $\alpha^0_{\sigma}(B)=\mathrm{e}^{{\mathrm i} L_0\sigma}
B\mathrm{e}^{-{\mathrm i} L_0\sigma}$, $B\in\mathcal A$. Setting
$B(\sigma):=\alpha_\sigma^0(B)$ this yields a solution of the Heisenberg
equations of motion of the uncoupled detector-field system on the Rindler
wedge $W_{\mathrm R}$, which are given by (\ref{eq:kleingordonwedge}) and
(\ref{eq:heisenbergdetector}), with $\tau=a\sigma$.
We will be mostly interested in the state of the system where, initially, the
detector is in its ground state, and the field in its Minkowski vacuum. This
state is represented by the vector $|\mathrm g\rangle:=|--\rangle\otimes
|0\rangle\in {\mathcal H}$. We will write, for any $B\in\mathcal A$:
\begin{equation}\label{eq:groundstate}
\langle B\rangle_{\mathrm g} := \langle \mathrm g| B|\mathrm g\rangle
\end{equation}
\subsection{The coupled field-detector system}
For the coupled system we will use the same representation of the observable
algebra, but change the dynamics. We will give a precise and mathematically
rigorous definition of the dynamics below but to link it with the physics
literature on the subject, we start with a formal computation. Let
$C(\sigma)=[A(\sigma), A^\dagger(\sigma)]$. According to
\cite{uw}, the Heisenberg equations of motion of the observables of the
coupled system are
\begin{eqnarray}\label{eq:heisenbergcoupled}
(\Box + m^2)Q(x)&=&-\lambda\rho( x_* )(A+A^\dagger)(\frac{\tau(x)}{a})\\
\dot A(\sigma)&=&-{\mathrm i} E A(\sigma) +{\mathrm i}\lambda C(\sigma)\int\mathrm{d} u\mathrm{d} \underline
x_\perp a u\ \rho(x_* ) Q(a\sigma, u, \underline x_\perp),\nonumber
\end{eqnarray}
The function $\rho$ tunes the coupling between the detector and the field. It
is evaluated at
$$
x_*:=x-x(\tau(x)/a),
$$
the spacelike vector linking
$x$ in the right wedge to the instantaneous position $x(\sigma)$ of the
detector, at proper time $\sigma=\tau(x)/a$, where
$\tau(x)$ is the Rindler time coordinate defined in (\ref{eq:coordchange}).
Let $(\tau,u,\underline x_\perp)$ be the Rindler coordinates of the point $x$
then the ones of $x(\sigma)$ are $(\tau,1/a,\underline 0_\perp)$ and hence we
may identify $x_*$, whose coordinates are $(0,u-1/a,\underline x_\perp)$, with an element of
$(-1/a, +\infty)\times {\mathbb R}^{d-1}$. We take the coupling function $\rho$ to be
in $C_0^\infty( (-1/a,
+\infty)\times {\mathbb R}^{d-1} )$, normalized as $\int \rho(\underline x) \mathrm{d}
\underline x^d=1$. Typically we imagine $\rho$ to be peaked at the origin, so that the field is
coupled strongest at the position of the detector. Only for such couplings
does it
make sense to interpret $\sigma$ as the proper time of the detector. Indeed,
if the detector is coupled to the field over a large spatial region,
different parts of the detector undergo a different acceleration and have
a different proper time. The mathematical result we obtain then still
holds, but does no longer have the same physical interpretation. A coupling
{\it strictly} localized at the position of the detector is
formally given by $\rho(\underline x)=\delta(\underline x)$, a situation
which does not fit the rigorous mathematical setup presented in this
work. We will comment further on the role played by the choice of coupling in
Section \ref{sectfgrc}.
Using (\ref{eq:equaltime}) and (\ref{eq:useful}), it is easy to
show through a formal computation that these equations are satisfied by the
operators $Q^{(\lambda)}(\tau, u, \underline x_\perp)$ and
$A^{(\lambda)}(\sigma)$ defined as follows:
$$
Q^{(\lambda)}(\tau, u, \underline x_\perp):=\mathrm{e}^{\i \tilde
L_\lambda\frac{\tau}{a}}\ Q(0,u, \underline x_\perp)\ \mathrm{e}^{-\i \tilde
L_\lambda\frac{\tau}{a}},\quad A^{(\lambda)}(\sigma):=\mathrm{e}^{\i \tilde
L_\lambda\sigma}\ A\ \mathrm{e}^{-\i \tilde L_\lambda\sigma},
$$
where
\begin{equation}\label{eq:coupledliouvillean}
\tilde L_\lambda:=L_0 +\lambda I, I:=(A+A^\dagger) \int \mathrm{d} u\mathrm{d} \underline
x_\perp au\ \rho(x_*|_{\tau=0})Q(0,u,\underline x_\perp)
\end{equation}
and $x_*|_{\tau=0}$ is given in Rindler coordinates by $(0,u-1/a,\underline x_\perp)$. In
other words, the Liouvillean $\tilde L_\lambda$ generates the correct
Heisenberg dynamics of the observables in the representation at hand.
{\it Remark. } The analysis we carry out in this paper works for general interactions of the form $I =
G\cdot Q(g) + G^*\cdot Q(\overline g)$, and for sums of such terms, where $G$
are matrices acting on the detector space, and $g\in L^2({\mathbb R}^3,
\mathrm{d}\underline x)$ are ``form factors''.
The following result is proved in Section \ref{dynpropproofsect}.
\begin{proposition}
\label{dynproposition}
The operator $\tilde L_\lambda$ in (\ref{eq:coupledliouvillean}) is for all
$\lambda$ essentially
self-adjoint on $D(L_0)\cap D(I)$ and the maps $\alpha^\lambda_\sigma(B):= \mathrm{e}^{{\mathrm i} \tilde L_\lambda
\sigma}B \mathrm{e}^{-{\mathrm i} \tilde L_\lambda\sigma}$ with $\sigma\in{\mathbb R}$ and $B\in\mathcal
A$ define a weakly continuous one-parameter group of automorphisms of the
observable algebra $\mathcal A$.
\end{proposition}
\subsection{The result}\label{s:result}
We are now in a position to give a precise statement of our result. Define
\begin{equation}
g(\varkappa,\underline k_\perp)
=
\widehat{\Big(x_1\rho(x_*|_{\tau=0})\Big)}\big((|\underline k_\perp|^2+m^2)^{1/2} \sinh\varkappa,\underline k_\perp\big),
\label{m7}
\end{equation}
where $\widehat{\ }$ denotes the Fourier transform.
\begin{theorem}
\label{thm:partialresult}
Let $d\geq 1$ if $m>0$ and $d\geq 2$ if $m=0$, and suppose the following ``Fermi Golden Rule Condition'' holds,
\begin{equation}
\label{fgrc}
\int_{{\mathbb R}} \mathrm{d} \varkappa \ \mathrm{e}^{-{\mathrm i} \frac Ea \varkappa}\
g(\varkappa,\underline k_\perp)\neq 0
\mbox{\ \ \ \ for some $\underline k_\perp\in {\mathbb R}^{d-1}$}.
\end{equation}
Then there is a constant $\lambda_0>0$ s.t. if $0<|\lambda|<\lambda_0$ then
\begin{equation}
\lim_{\sigma\to\infty}\langle\alpha_\sigma^\lambda(B)\rangle_{\mathrm g}
=\frac{1}{Z_{\beta,\mathrm D}} \mathrm{Tr}\ \mathrm{e}^{-\beta
H_{\mathrm D}}B+ O(\lambda^2),
\label{mm1}
\end{equation}
for all $B\in \mathcal B({\mathbb C}^2)$, and where $\beta= \frac{2\pi}{a}$.
More generally, if $\varrho$ is any density matrix on ${\mathcal H}$ then
\begin{equation}
\lim_{\sigma\to\infty}{\mathrm Tr}\, \varrho\alpha_\sigma^\lambda(B F)
=\Big(\frac{1}{Z_{\beta,\mathrm D}} \mathrm{Tr}\ \mathrm{e}^{-\beta
H_{\mathrm D}}B\Big)\, \langle 0| F|0\rangle+ O(\lambda^2),
\label{mm0}
\end{equation}
for any detector observable $B\in \mathcal B({\mathbb C}^2)$ and any field observable
$F\in {\cal A}_{\mathrm F}$.
\end{theorem}
Result (\ref{mm1}) shows that if at $\sigma=0$ the detector-field system is in a state which
is a local perturbation of its ground state, then the reduced density matrix
of the detector converges asymptotically in time to the detector's Gibbs state
at inverse temperature $\beta= \frac{2\pi}{a}$. This is a (slightly) stronger
statement than the formulations usually found in the literature, since it
allows both the field and the detector to be initially in an excited state.
{\it Remarks.\ }
1) Theorem \ref{thm:partialresult} follows from a more complete result, stated
as Theorem \ref{thm:fullresult} below, where the r.h.s. of (\ref{mm1}) is identified as the equilibrium state
$|\lambda\rangle\in{\cal H}$ of the coupled system, see also (\ref{eq:kmscoupled}) below. An expansion of
$\langle\lambda|\cdot|\lambda\rangle$ for small $\lambda$ yields the uncoupled equilibrium state
plus an error of second order in $\lambda$ (the absence of a first order error
term is due to the fact that the expectation of the interaction $I$ in the
uncoupled equilibrium state vanishes).
2) The approach to the limit state in (\ref{mm1}) is exponentially fast,
$$
\left|
{\mathrm Tr}\, \varrho\alpha_\sigma^\lambda(B)
- \langle \lambda | B|\lambda\rangle\right| < C\|B\| \mathrm{e}^{-\lambda^2\eta
\sigma},
$$
where $C$ is a constant (depending on the interaction, but not on the initial
density matrix $\varrho$ nor on $B$) and $\eta = (1+\mathrm{e}^{-2\pi E/a})\xi$, with
\begin{equation}
\xi \equiv\xi(E) = \frac{1}{2a}\int_{{\mathbb R}^{d-1}}\mathrm{d}\underline k_\perp \left|\int_{\mathbb
R}\mathrm{d}\varkappa \ \mathrm{e}^{-{\mathrm i} \frac Ea \varkappa}g(\varkappa,\underline
k_\perp)\right|^2\geq 0.
\label{m19}
\end{equation}
The quantity $\tau_{\mathrm{relax}}=1/\lambda^2\eta$ is called the {\it
relaxation time} of the process.
The purpose of condition (\ref{fgrc}) is to ensure that $\xi>0$, i.e., that
$\tau_{\mathrm{relax}}<\infty$. We will show in the following subsection that
this is typically the case.
We finally remark that, whereas the leading term of the right hand side of (\ref{mm1}) does not
depend on the choice of form factor $\rho$ in the interaction term, the
relaxation time $\tau_{\mathrm{relax}}$ does, via (\ref{m7}) and (\ref{m19}). Nevertheless, we show in
the next subsection that $\tau_{\mathrm{relax}}$ is independent of the form
factor for interactions sharply localized at the position of the detector.
\subsection{The Fermi Golden Rule Condition}
\label{sectfgrc}
The goal of this section is to show that (\ref{fgrc}) is satisfied for
``generic'' interactions.
\begin{proposition}
\label{fgrcprop}
Take the coupling function $\rho$ in (\ref{eq:heisenbergcoupled}),
(\ref{eq:coupledliouvillean}) to be of the form
$\rho(\underline x)
= \rho_1(x_1)\rho_\perp(\underline x_\perp)$, (``square detector'') with
$\rho_1\geq 0$. Then condition
(\ref{fgrc}) is satisfied for all $E$ except for $E\in{\cal E}$,
where $\cal E$ is a discrete (possibly empty) subset of $\mathbb R$. In
particular, $\xi(E)>0$ for all $E\not\in\cal E$.
\end{proposition}
Values of $E$ satisfying $\xi(E)=0$ (which form necessarily a subset of $\cal E$ in the proposition) correspond to energy gaps of the detector
Hamiltonian for which thermalization of the detector occurs (if at all) with a
larger relaxation time at least of the order $\lambda^{-4}$ (as opposed to
$\lambda^{-2}$ for $E$ s.t. $\xi(E)>0$), see \cite{MLSO}.
For a particular choice of the coupling function $\rho$ one may
resort to a numerical study of the condition (\ref{fgrc}). On the analytic
side we can calculate $\xi$, (\ref{m19}), in the limit of a strictly localized
interaction. More precisely, we choose $\rho_1$, $\rho_\perp$ as in
Proposition \ref{fgrcprop}, and consider the family $\rho_\epsilon(\underline x)
= \epsilon^{-d}\rho_1(x_1/\epsilon)\rho_\perp(\underline x_\perp/\epsilon)
\rightarrow \delta(x_1-1/a)\delta(\underline x_\perp)$
which represents an interaction localized exactly at the position of
the detector in the limit $\epsilon\rightarrow 0$. Each $\epsilon$ defines
thus a $\xi_\epsilon(E)$ by (\ref{m19}), and we obtain, for $d=3$ and $m>0$,
\begin{eqnarray*}
\lefteqn{
\lim_{\epsilon\rightarrow 0}\xi_\epsilon(E)}\\
&=&
\frac a2 \int_{{\mathbb R}^2}\frac{\mathrm{d}\underline k_\perp}{\omega_\perp^4}\left| \int_{\mathbb R}\mathrm{d}\varkappa
\frac{2\sinh^2\varkappa-\frac{E}{\omega_\perp}\cosh\varkappa-1}{(\frac{E}{\omega_\perp} +\cosh\varkappa)^4} \mathrm{e}^{-\i\big[\frac Ea \varkappa +\frac{\omega_\perp}{a}
\sinh\varkappa\big]}\right|^2,
\end{eqnarray*}
where $\omega_\perp=\sqrt{|\underline k_\perp|^2+m^2}$. This limit does not
depend on the form of $\rho$ and the leading term of $\tau_{\mathrm{relax}}$
(as $\epsilon\rightarrow 0$) is thus independent of the detector form factor.
\medskip
{\it Proof of Proposition \ref{fgrcprop}.\ }
We denote the integral in (\ref{fgrc}) by ${\mathrm i}
\widehat{\rho_\perp}(\underline k_\perp) J(E,\omega_\perp)$,
where $\omega_\perp=\sqrt{|\underline k_\perp|^2+m^2}$, see also
(\ref{m7}). For $\omega_\perp\neq0$ we can make the change of variable
$y=\omega_\perp \sinh\varkappa$ to obtain the representation
\begin{equation}
J(E,\omega_\perp) =
\int_{\mathbb R}\mathrm{d}
y \frac{\mathrm{e}^{-{\mathrm i} \frac Ea \mathrm{argsinh}(y/\omega_\perp)}}{\sqrt{\omega_\perp^2 + y^2}} f(y),\ \ \ f(y):=
\mathrm{e}^{-{\mathrm i} y/a}\Big(\frac{-{\mathrm i}}{a}\widehat{\rho_1}(y) + \widehat{\rho_1}'(y)\Big).
\label{u1}
\end{equation}
We view $\omega_\perp^2=\mu$ in the integral as a parameter, $\mu>0$. We first
show that given any $\mu_0>0$, the integral in (\ref{u1}), for $E=0$, does
not vanish identically in any neighbourhood of $\mu_0$.
Let us consider $\mu_0=1$; a simple modification of the following argument
yields the general case. Assume {\it ad absurdum} that $J(0,\mu)=0$
for all $\mu$ in a neighbourhood of $1$. Then, by taking derivaties
of $J(0,\mu)$ with respect to $\mu$, at $\mu_0=1$,
we see that
\begin{equation}
\int_{\mathbb R}\mathrm{d} y \ (1+y^2)^{-n}\, (1 + y^2)^{-1/2} f(y)=0,
\label{u2}
\end{equation}
for all $n=0,1,\ldots$ Now, it is not difficult to verify that the linear span
of all functions $(1+y^2)^{-n}$, $n=1,2,\ldots$ is dense in the space of even
functions in $L^2({\mathbb R},dy)$. (One may prove this with little effort
via the Fourier transform, for example.) It thus follows from (\ref{u2}) that
the even part of $f$ must vanish,
$f(y)+f(-y)=0$ for all $y\geq 0$. In particular, $f(0)=0$, which means that
\begin{equation}
a^{-1} =-{\mathrm i} \widehat{\rho_1}'(0)
\label{u3}
\end{equation}
(we assume without loss of generality that $\rho_1$ is normalized as
$\int_{\mathbb R}\mathrm{d} x\, \rho_1(x)=1$). On the other hand, we have $-{\mathrm i}
\widehat{\rho_1}'(0) = -\int_{\mathbb R}\mathrm{d} x \, x\rho_1(x)<a^{-1}$, since in
the integral, $x>-a^{-1}$ due to the fact that $\rho_1$ is supported in
$(-1/a,\infty)$. Therefore condition (\ref{u3}) is not verified.
This shows that given any $\mu_0>0$ we can find a $\mu_1>0$ (arbitrarily close
to $\mu_0$) with the property that $J(0,\mu_1)\neq 0$.
Pick a nonzero $K_0\in {\mathbb R}^{d-1}$ satisfying
$\widehat{\rho_\perp}(K_0)\neq 0$ and set $\mu_0:=\sqrt{|K_0|^2+m^2}$. Then,
by the above argument and by the continuity of $\widehat{\rho_\perp}$ there is
a $\mu_1=:\sqrt{|K_1|^2+m^2}$ (which is close to $\mu_0$ and defines a $K_1$
close to $K_0$) s.t. $J(0,\mu_1)\neq 0$ {\it and}
$\widehat{\rho_\perp}(K_1)\neq 0$. Hence we have shown that there exists a
nonzero $K_1$ satisfying ${\mathrm i} \widehat{\rho_\perp}(K_1) J(0,\omega_1)\neq 0$,
where $\omega_1=\sqrt{|K_1|^2+m^2}$. Condition (\ref{fgrc}) is thus satisfied
for $E=0$.
Finally we pass to the other values of $E$ by an analyticity argument. Indeed,
one easily sees (best by using the form of $J$ in which one integrates over
$\varkappa$ rather than $y$, c.f. (\ref{m7}), (\ref{fgrc})) that the map $E\mapsto J(E,\omega_1)$ is analytic and by
the previous argument it does not vanish at $E=0$. Thus the zeroes of this
map are contained in a discrete set ${\cal E}(\omega_1)\subset{\mathbb
C}$. Any $E$ avoiding this set thus satisfies (\ref{fgrc}).
\hfill $\blacksquare$
\section{Proof of Theorem \ref{thm:partialresult}}\label{s:proof}
\subsection{Strategy}
As mentioned in the introduction, the first ingredient of the proof is the
observation that the Minkowski vacuum is (a realization of) the GNS
representation of a KMS state on the right wedge algebra for the Lorentz
boosts at the inverse temperature $\beta=2\pi$. This is the content of
Theorem \ref{thm:kmsfield} below. To give a precise statement, we need the
so-called modular conjugation operator, defined as follows:
\begin{equation}
J_{\mathrm F}=\Gamma(j_{_{\mathrm F}}),\ {\mathrm {where}}\ \forall \ \psi \in L^2({\mathbb R}^d, \mathrm{d}
\underline x), \ j_{_{\mathrm F}}\psi(\underline x)=\overline \psi(-x_1, \underline x_\perp);
\label{m10}
\end{equation}
here $\Gamma(j)$ stands for the second quantization of $j$.
\begin{theorem}[\cite{biwi}]
\label{thm:kmsfield} The Fock vacuum in
${\cal F}$ induces on $\mathcal A_{\mathrm{F,R}}$ a state which is KMS at inverse
temperature $\beta=2\pi$ for $\alpha_{\mathrm F,\tau}^0$. In particular, one
has
$$
{\mathcal A}_{\mathrm F, R}'={\mathcal A}_{\mathrm F, L},\quad J_{\mathrm F}{\mathcal A}_{\mathrm F, R} J_{\mathrm F}={\mathcal A}_{\mathrm F, L},\quad
\overline{{\mathcal A}_{\mathrm F, R}|0,\mathrm{F}\rangle}={\cal F}=\overline{{\mathcal A}_{\mathrm F, L}|0,\mathrm F\rangle},
$$
and for all $f\in {\cal S}(W_{\mathrm R},{\mathbb C})$: ${ L_{\mathrm F}|0,
\mathrm F\rangle=0}$ and $\mathrm{e}^{-\pi L_{\mathrm
F}}Q[f]|0,\mathrm F\rangle=J_{\mathrm F}Q[\overline f]|0,\mathrm F\rangle
$.
\end{theorem}
This result was proven in considerable generality in \cite{biwi}, for
relativistic fields satisfying the Wightman axioms. The result above for the free
scalar field can be obtained from essentially direct computations, and we shall not detail
it.
Similarly, the states $|\beta, \mathrm D\rangle$ introduced in Section
\ref{s:freedetector} are GNS representatives of the KMS states at inverse
temperature $\beta$ for the free detector dynamics $\alpha_{\mathrm D,
\sigma}$ on the detector observable algebra ${\mathcal B}({\mathbb C}^2)$. This well
known observation is for convenience summarized in the following lemma. The
appropriate conjugate operator is given by
$$
J_{\mathrm D}=E(C\otimes C),
$$
where $C$ is the antilinear operator of complex conjugation on ${\mathbb C}^2$ and $E$
is the exchange operator on ${\mathbb C}^2\otimes{\mathbb C}^2$, $E\varphi\otimes\chi=\chi\otimes\varphi$.
\begin{lemma} For any $\beta>0$, the vector $|\beta, \mathrm D\rangle$ induces on ${\mathcal B}({\mathbb C}^2)$ a state that is KMS at inverse temperature $\beta$ for $\alpha_{\mathrm D, \sigma}^0$. In particular, one has
$$
{\mathcal A}_{\mathrm D}'=\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2\otimes \mathcal B({\mathbb C}^2),\quad {\mathcal A}_{\mathrm D}'=J_{\mathrm D}{\mathcal A}_{\mathrm D} J_{\mathrm
D},\quad
{\mathcal A}_{\mathrm D}|\beta, D\rangle ={\mathcal H}={\mathcal A}_{\mathrm D}'|\beta, D\rangle,
$$
and
$$
\mathrm{e}^{-\beta L_{\mathrm D}/2}(B\otimes \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2)|\beta, \mathrm D\rangle
=J_{\mathrm D} (B^*\otimes \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2)|\beta, \mathrm D\rangle.
$$
\end{lemma}
Defining, on ${\mathcal H}={\mathcal H}_{\mathrm D}\otimes{\cal F}$, $J:=J_{\mathrm D}\otimes J_{\mathrm F}$,
it follows that the vector
\begin{equation}
\label{eq:kmsuncoupled}
|\mathrm 0\rangle:= |\beta =2\pi/a\rangle \otimes |0,\mathrm F\rangle
\end{equation}
is a GNS representative of the KMS state at inverse temperature $\beta=\frac{2\pi}{a}$
for the free dynamics $\alpha_\sigma^0$ on ${\mathcal A}={\mathcal A}_{\mathrm D}\otimes{\mathcal A}_{\mathrm F, R}$. This suggests
to treat the problem at hand as one of return to
equilibrium.
The rest of the argument then proceeds in three steps:
(a) One proves the existence of a GNS representative $|\lambda\rangle\in{\mathcal H}$,
defined below,
of the KMS state for the perturbed dynamics at the
same temperature (Section \ref{s:perturbation});
(b) One reduces the proof of Theorem \ref{thm:fullresult} and hence of
Theorem \ref{thm:partialresult} to showing that
the generator of the perturbed dynamics has a
simple eigenvalue at $0$ and otherwise absolutely continuous spectrum only;
(c) One finally uses spectral deformation theory to prove these two statements.
The strategy in (a)-(b)-(c) has been applied successfully to radiative problems
in atomic physics, the spin-boson model, and similar systems in \cite{JP1,JP2,BFS,M,DJP}, where we refer for further
references. A concise introduction to the field can be found in \cite{pi}. The
implementation of this strategy in the present context is reasonably
straightforward. We will detail those points that are specific to the
current situation.
\subsection{Perturbation theory}\label{s:perturbation}
We define on ${\mathcal H}={\mathcal H}_{\mathrm D}\otimes {\cal F}$,
in addition to $\tilde L_\lambda$ (see \ref{eq:coupledliouvillean}), the
so-called {\it standard Liouvillean}
\begin{equation}\label{eq:standardliouvillean}
L_\lambda =\tilde L_\lambda -\lambda JIJ.
\end{equation}
We outline the proof of the following result in Section
\ref{dynpropproofsect}.
\begin{lemma}
\label{lem:standardliouvillean} $L_\lambda$ is essentially
self-adjoint on $D(L_0)\cap D(I)\cap D(JIJ)$ and, for all $B\in{\mathcal A}$,
$$
\alpha_\sigma^\lambda(B) =\mathrm{e}^{\i L_\lambda \sigma}B\mathrm{e}^{-\i L_\lambda \sigma}.
$$
\end{lemma}
A useful feature of the standard Liouvillean (in fact, the motivation for its
definition!) is that the
unitary it generates leaves the equilibrium state of the coupled system
invariant, see (\ref{eq:kernel}) below.
\begin{proposition}
\label{prop:perturbedkms}
The vector $|0\rangle$ representing the uncoupled equilibrium state,
(\ref{eq:kmsuncoupled}), is in the domain of the unbounded operator
$\mathrm{e}^{-\frac{\pi}{a}\tilde L_\lambda}$, and the vector
\begin{equation}
\label{eq:kmscoupled}
|\lambda\rangle := \frac{\mathrm{e}^{-\frac{\pi}{a}\tilde L_\lambda}|0
\rangle}{\|\mathrm{e}^{-\frac{\pi}{a}\tilde L_\lambda}|0\rangle\|}\in{\mathcal H}
\end{equation}
defines a $(\frac{2\pi}{a}, \alpha_\sigma^\lambda)$-KMS state on ${\mathcal A}=
{\mathcal A}_{\mathrm D}\otimes {\mathcal A}_{\mathrm F, R}$ and it satisfies
\begin{equation}\label{eq:kernel}
L_\lambda |\lambda\rangle =0.
\end{equation}
\end{proposition}
{\it Proof. } To show that $|0\rangle \in{\rm Dom}(\mathrm{e}^{-\frac{\pi}{a}\tilde
L_\lambda})$ we check that the Dyson series
\begin{equation}
\sum_{n\geq 0}(-\lambda)^n \int_0^{\pi/a}\mathrm{d} t_1\cdots \int_0^{t_{n-1}}\mathrm{d} t_n\
\alpha_{{\mathrm i} t_n}^0(I)\cdots \alpha_{{\mathrm i} t_1}^0(I) |0\rangle
\label{m1}
\end{equation}
converges. We write the interaction operator conveniently as $I = G Q[g]$,
where $G=A+A^\dagger$ and $g(x)=a\delta(x^0)x^1\rho(x_*(\underline
x))$ has support in $W_{\mathrm R}$,
c.f. (\ref{eq:coupledliouvillean}). In (\ref{m1}) we have set, for real $s$,
$$
\alpha_{{\mathrm i} s}^0(I) = \mathrm{e}^{-s L_{\mathrm D}}G\mathrm{e}^{s L_{\mathrm D}} \
\mathrm{e}^{-s L_{\mathrm F}} Q[g] \mathrm{e}^{s L_{\mathrm F}}.
$$
To see that $\mathrm{e}^{-s L_{\mathrm F}} Q[g] \mathrm{e}^{s L_{\mathrm F}}$ is well defined
for $0\leq s\leq\pi/a$ one shows that since $g$ is supported in the right
wedge, the map $t\mapsto \mathrm{e}^{{\mathrm i} t L_{\mathrm F}} Q[g] \mathrm{e}^{-{\mathrm i} t L_{\mathrm F}}
= Q[g\circ B_{-t}]$ has an analytic continuation into the strip $0<{\rm Im}\,
t<\pi/a$, and it is continuous at the boundary of the strip ($t\in{\mathbb
R}$, $t\in {\mathrm i}\frac{\pi}{a}{\mathbb R}$). This argument is actually part of
the proof of the Bisognano--Wichmann theorem, \cite{biwi}. It follows in
particular that the integrals in (\ref{m1}) are well defined and that
furthermore
$$
\sup_{0\leq {\rm Im}\, s \leq \pi/a} \left\| \alpha_{{\mathrm i} s}^0(I)
(N+1)^{-1/2}\right \| = C<\infty,
$$
where $N$ is the number operator on Fock space. Since $|0\rangle$ is the
vacuum on the field part, and each interaction term
$\alpha_{{\mathrm i} s}^0(I)$ can increase the particle number by at most one we have
the bound $\|\alpha_{{\mathrm i} t_n}^0(I)\cdots \alpha_{{\mathrm i} t_1}^0(I) |0\rangle\|\leq
C^n\sqrt{n!}$. It follows that the series (\ref{m1}) converges (for all values
of $\lambda$) and hence $|0\rangle \in{\rm Dom}(\mathrm{e}^{-\frac{\pi}{a}\tilde
L_\lambda})$.
The facts that $|\lambda\rangle$ defines a
$(\frac{2\pi}{a},\alpha_\sigma^\lambda)$-KMS state and that $L_\lambda
|0\rangle =0$ follow from Araki's perturbation theory of KMS states, and from
perturbation theory of standard Liouville operators, see \cite{DJP}. \hfill
$\blacksquare$
We are now in a position to state the full result, of which Theorem \ref{thm:partialresult}
is an immediate consequence:
\begin{theorem}
\label{thm:fullresult}
Assume that the Fermi Golden Rule Condition (\ref{fgrc}) is satisfied. There exists $\lambda_0$ so that for all
$0<|\lambda|< \lambda_0$, for all density matrices $\varrho$ on ${\mathcal H}$ and for
all $B\in{\mathcal A}$
$$
\lim_{\sigma\to\infty} {\mathrm{Tr}}\ \varrho \ \alpha_\sigma^\lambda(B) =
\langle \lambda|B|\lambda\rangle.
$$
\end{theorem}
{\it Proof.} We show in Section \ref{redsection} that the result follows if
the spectrum of $L_\lambda$ is purely absolutely continuous
with the exception of a single simple eigenvalue at zero. These spectral
characteristics are shown in Theorem \ref{specthm}. \hfill $\blacksquare$
\subsection{Reduction to a spectral problem}
\label{redsection}
We reduce the proof of Theorem \ref{thm:fullresult} to a spectral problem
via the following simple lemma, which is a variant of the
Riemann-Lebesgue lemma:
\begin{lemma}
\label{stephansfamouslemma}Let ${\mathcal H}$ be a Hilbert space, $\phi\in{\mathcal H}$, $\mathcal A$
a subalgebra of $\mathcal B({\mathcal H})$ whose commutant we denote by ${\cal A}'$, and
let $L$ be a self-adjoint operator on ${\mathcal H}$. Suppose that ${\mathcal A'\phi}$
is dense in ${\mathcal H}$, that $\mathrm{e}^{{\mathrm i} L \tau} \mathcal A \mathrm{e}^{-{\mathrm i} L \tau}\subset
\mathcal A,\ \forall \tau$, that $L\phi=0$, and that on the orthogonal
complement of $\phi$, $L$ has purely absolutely continuous spectrum.
Then we have
\begin{equation}
\lim_{\tau \to \infty}\mathrm{Tr}\ \varrho\ \mathrm{e}^{{\mathrm i} L \tau} B \mathrm{e}^{-{\mathrm i} L
\tau}=\langle \phi, B\phi\rangle,
\label{m2}
\end{equation}
for all $A\in{\cal A}$ and for all density matrices $0\leq \varrho \in \mathcal
L^1({\mathcal H})$, ${\mathrm Tr}\varrho=1$.
\end{lemma}
{\it Proof.} We may diagonalize $\varrho=\sum_{n=1}^\infty p_n
|\psi_n\rangle\langle\psi_n|$, where $\psi_n\in{\cal H}$ and the probabilities
$0\leq p_n\leq 1$ sum up to one. So it suffices to show (\ref{m2}) for a
rank-one density matrix $\varrho=|\psi\rangle\langle\psi|$. Given any
$\epsilon>0$ there is a $B'\in {\cal A}'$
s.t. $\|\psi-B'\phi\|<\epsilon$. Thus by replacing $\psi$ by $B'\phi$,
commuting $B'$ and $\mathrm{e}^{{\mathrm i} L \tau} A \mathrm{e}^{-{\mathrm i} L \tau}$, and by using the
invariance of $\phi$ under $\mathrm{e}^{-{\mathrm i} L \tau}$ we obtain
\begin{equation}
{\mathrm Tr\,}\varrho\,\mathrm{e}^{{\mathrm i} L \tau} B \mathrm{e}^{-{\mathrm i} L \tau}= \langle\psi,\mathrm{e}^{{\mathrm i} L \tau} B \mathrm{e}^{-{\mathrm i} L \tau}\psi\rangle=\langle \psi, B'
\mathrm{e}^{{\mathrm i} L \tau}B\phi\rangle +O(\epsilon),
\label{m3}
\end{equation}
where the remainder is estimated uniformly in $\tau$. Since the spectrum of
$L$ is absolutely continuous except for a simple eigenvalue at zero with
eigenvector $\phi$, the propagator $\mathrm{e}^{{\mathrm i} L\tau}$ converges in the weak sense
to the rank-one projection $|\phi\rangle\langle\phi|$, as
$\tau\rightarrow\infty$. Using this in (\ref{m3}), together with the facts that
$\langle\psi,B'\phi\rangle = 1+O(\epsilon)$, and that $\epsilon$ can be chosen
arbitrarily small yields relation (\ref{m2}). \hfill $\blacksquare$
We apply Lemma \ref{stephansfamouslemma} with $L=L_\lambda$ and
$\phi=|\lambda\rangle$. The density of ${\cal A}'|\lambda\rangle$ follows
from the KMS property of $|\lambda\rangle$, the invariance of ${\cal A}$ under
$\mathrm{e}^{{\mathrm i} L_\lambda\tau}\cdot \mathrm{e}^{-{\mathrm i} L_\lambda\tau}$ follows from Lemma
\ref{lem:standardliouvillean} and the relation $L_\lambda|\lambda\rangle=0$ is
shown in Proposition \ref{prop:perturbedkms}. It remains to prove that on the orthogonal complement of $|\lambda\rangle$, $L_\lambda$ has purely
absolutely continuous spectrum.
\section{Spectral analysis of $L_\lambda$}
\label{specanal}
The spectrum of the operator $L_{\mathrm D}$ consists of two simple eigenvalues
$\pm E$ (eigenvectors $|\pm,\mp\rangle$) and a doubly degenerate eigenvalue at
$0$ (eigenvectors $|\pm,\pm\rangle $). $L_{\mathrm F}$ has absolutely
continuous spectrum covering the entire real axis, and a single embedded
eigenvalue at the origin. This eigenvalue is simple and has eigenvector
$|0,\mathrm F\rangle$. It follows that $L_0$ has absolutely continuous
spectrum covering the axis and three embedded eigenvalues at $0,\pm E$, the
one at $0$ being doubly degenerate.
Our goal is to show that the nonzero eigenvalues are unstable under the
perturbation $\lambda(I-JIJ)$, and that the degeneracy of the eigenvalue zero
is lifted. We do this via {\it spectral deformation theory}, showing that the
unstable (parts of the) eigenvalues turn into {\it resonances} located in the
lower complex plane.
\subsection{Spectral deformation}
For the spectral analysis it is useful to consider the unitarily transformed
Hilbert space $L^2({\mathbb R}^d,\mathrm{d}\varkappa\,\mathrm{d}^{d-1}\underline k_\perp)$ of
one-particle wave functions of the field, determined by $L^2({\mathbb
R^d,\mathrm{d}^d\underline x})\ni f\mapsto Wf$ with
\begin{equation}
(Wf)(\varkappa,\underline k_\perp):= \sqrt{\omega_\perp\cosh\varkappa}\
\widehat{f}(\omega_\perp\sinh\varkappa,\underline k_\perp),
\label{m4}
\end{equation}
where $\omega_\perp := \sqrt{|\underline k_\perp|^2 +m^2}$ and where
$\widehat{f}$ is the Fourier transform of $f$. The advantage of this
representation of the Hilbert space is that the operator $K$, defined in
(\ref{eq:liouvfreefield}), takes the particularly simple form $K = {\mathrm i}\partial_\varkappa$.
The transformation $W$ lifts to Fock space in the usual way. We do not
introduce new names for spaces and operators in the transformed system. The
Liouville operator (\ref{eq:coupledliouvillean}) is
\begin{eqnarray}
L_\lambda &=& L_{\mathrm D}+ L_{\mathrm F}+\lambda V,\nonumber\\
L_0 &=& L_{\mathrm D}+ a L_{\mathrm F},\ \ L_{\mathrm D}= H_{\mathrm
D}\otimes\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2 - \mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2\otimes H_{\mathrm D}, \ \ L_{\mathrm F}=\mathrm{d}\Gamma({\mathrm i}\partial_\varkappa),
\label{m5}\\
V &=& I- JIJ,\ \ \ I=G \otimes\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2\otimes\frac{a}{\sqrt 2}\left\{ a^\dagger(g) +
a(g)\right \}
\label{m6}
\end{eqnarray}
acting on the Hilbert space ${\cal H}= {\mathbb C}^2\otimes {\mathbb
C}^2\otimes{\cal F}$, where ${\cal F}$ is the bosonic Fock space over
$L^2({\mathbb R}^d,\mathrm{d}\varkappa\,\mathrm{d}^{d-1}\underline k_\perp)$. In (\ref{m6})
$G$ is the $2 \times 2$ matrix with $0$ on the diagonal and $1$ on the
off-diagonals, and $g(\varkappa,\underline k_\perp)
= \big(W \Omega^{-1/2}x_1\rho(x_*|_{\tau=0})\big)(\varkappa,\underline
k_\perp)$ is given in (\ref{m7}).
The action of $j_{_{\mathrm F}}$, (\ref{m10}), is given by $\big(j_{_{\mathrm
F}}f\big)(\varkappa,\underline k_\perp)= \overline{f}(-\varkappa,\underline k_\perp)$.
We describe now the complex deformation. Let $\theta\in {\mathbb R}$. The map
$$
\psi_\theta(\varkappa_1,\ldots,\varkappa_n):=\big(U_\theta\psi\big)(\varkappa_1,\ldots,\varkappa_n)
:=
\mathrm{e}^{{\mathrm i}\theta(\varkappa_1+\cdots \varkappa_n)}\psi(\varkappa_1,\ldots,\varkappa_n)
$$
defines a unitary group on $\cal F$ (we are not displaying the variables
$\underline k_\perp$ in the argument of $\psi$ since $U_\theta$ does not act on
them). An easy calculation shows that
\begin{equation}
L_\lambda(\theta):=U_\theta L_\lambda U_\theta^* = L_0(\theta) +\lambda V(\theta),\ \
L_0(\theta) = L_0 -a \theta N, \ \ V(\theta)= I(\theta) - J I(\theta) J,
\label{m8}
\end{equation}
where $N=\mathrm{d}\Gamma(\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l)$ is the number operator on $\cal F$ and
\begin{equation}
I(\theta) = G \otimes\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_2\otimes\frac{a}{\sqrt 2}\left\{ a^\dagger(\mathrm{e}^{{\mathrm i}\theta\varkappa}g) +
a(\mathrm{e}^{{\mathrm i}\overline\theta\varkappa}g)\right \},
\label{m11}
\end{equation}
where we have put the complex conjugate $\overline \theta$ in the argument of
the annihilation operator in (\ref{m11}) in view of the complexification of
$\theta$.
\begin{lemma}
\label{analyticityinteraction}
Let
\begin{equation}
\theta_0(m,d):=
\left\{
\begin{array}{cl}
\infty & \mbox{if $m\neq 0$ and $ d\geq 1$}\\
\frac{d-1}{2} & \mbox{if $m=0$ and $ d\geq 2$}
\end{array}
\right.
\label{m12}
\end{equation}
where $m\geq 0$ is the mass of the field and $d$ is the spatial
dimension. We have $\mathrm{e}^{
{\mathrm i}\theta\varkappa} W\Omega^{-1/2}h \in L^2({\mathbb R}^d,d\varkappa d\underline
k_\perp)$ for all $\theta\in {\mathbb C}$ satisfying $|\theta|<\theta_0$ and for all
$h\in{\cal S}({\mathbb R}^d,{\mathbb C})$. Moreover, for $|\theta|<\theta_0$, $L_\lambda(\theta)$ is a closed operator on the dense domain ${\cal D}={\rm Dom}(L_0)\cap{\rm Dom}(N)$.
\end{lemma}
{\it Proof.} $L_0(\theta)$ is a normal operator, so it is
closed. Assume we know that $\mathrm{e}^{{\mathrm i}\theta\varkappa} W \Omega^{-1/2}{\cal S}({\mathbb R}^d,{\mathbb C})\subset L^2({\mathbb R}^d,d\varkappa d\underline
k_\perp)$, and recall that $x_1\rho(x_*|_{\tau=0})\in {\cal S}({\mathbb R}^d,{\mathbb C})$. Then, for ${\rm Im}\theta\neq 0$ the perturbation $V(\theta)$ is
infinitesimally small w.r.t. $L_0(\theta)$, so $L_\lambda(\theta)$ is closed
by stability of closedness. For ${\rm Im}\theta= 0$ the operator
$L_\lambda(\theta)$ is even selfadjoint.
Let $h\in{\cal S}({\mathbb R}^d,{\mathbb C})$. According to (\ref{m4}) we have
$$
(W \Omega^{-1/2}h)(\varkappa,\underline k_\perp) =
\widehat{h}(\omega_\perp\sinh\varkappa,\underline k_\perp).
$$
Since $\widehat{h}\in \cal S$ we have that for any
integer $n$ there is a constant $C_n$ s.t.
$$
\left|\widehat{h}(\omega_\perp\sinh\varkappa,\underline k_\perp)\right| <
\frac{C_n}{1+[m^2\sinh^2\varkappa +|\underline
k_\perp|^2\cosh^2\varkappa]^n}.
$$
For $m=0$ we thus obtain (using an obvious change of variables) the estimate
\begin{equation}
\int_{\mathbb R} \mathrm{d}\varkappa\ \mathrm{e}^{2\theta'|\varkappa|}\int_{{\mathbb
R}^{d-1}}\mathrm{d}\underline k_\perp \
\left|\widehat{h}(\omega_\perp\sinh\varkappa,\underline k_\perp)\right|^2 <
\widetilde{C}_n\int_{\mathbb R} \mathrm{d}\varkappa\ \frac{\mathrm{e}^{2\theta'|\varkappa|}}{[\cosh\varkappa]^{d-1}}
\label{m13}
\end{equation}
which is finite provided $\theta'=|{\mathrm Im}\theta|<(d-1)/2$. If $m\neq 0$
then the l.h.s. of (\ref{m13}) is bounded from above by
$$
\int_{\mathbb R} \mathrm{d}\varkappa\int_{{\mathbb R}^{d-1}}\mathrm{d}\underline k_\perp
\ \mathrm{e}^{2\theta'|\varkappa|} \frac{C_n^2}{[1/2+m^2\sinh^2\varkappa]^n [1/2+|\underline
k_\perp|^2]^n}
$$
which is finite if $\theta'=|{\mathrm Im}\theta|<n$, and $n$ can be
chosen arbitrarily large. \hfill $\blacksquare$
\subsection{Spectra of $L_\lambda(\theta)$ and of $L_\lambda$}
The goal of this section is to prove the following result.
\begin{theorem}
\label{specthm}
Suppose the Fermi Golden Rule Condition (\ref{fgrc}) holds. There is a
$\lambda_0>0$ s.t. if $0<|\lambda|<\lambda_0$ then the spectrum of $L_\lambda$
consists of a simple eigenvalue at zero and is purely absolutely continuous on
the real axis otherwise.
\end{theorem}
Remark that the spectrum of $L_0(\theta)=L_0 -a \theta N$ consists of the {\it isolated
eigenvalues} $\pm E$ (simple) and $0$ (doubly degenerate), and of the lines
of continuous spectrum $\{{\mathbb R}-{\mathrm i} \, an\, {\mathrm
Im}\theta\}_{n=1,2,\ldots}$. We now analyze the behaviour of the
eigenvalues of $L_0(\theta)$ under the perturbation $\lambda V(\theta)$. The
strategy is to show that the eigenvalues $\pm E$ are unstable under the
perturbation, and that the degeneracy of the eigenvalue zero is lifted. Note
that the kernel of $L_\lambda$ is non-empty by construction, see
(\ref{eq:kernel}).
{\it Proof of Theorem \ref{specthm}.}
The central part of the proof is the control of the resonances bifurcating out of the
eigenvalues $\pm E$ and $0$, see Lemma \ref{lemmam1}. A standard
analyticity argument then implies Theorem \ref{specthm}. The latter can be
summarized as follows: one checks that for all complex $z$ with
${\mathrm{Im}z>0}$, $\theta\mapsto \langle
\psi_{\overline\theta},(L_\lambda(\theta)-z)^{-1}\phi_\theta\rangle$ is
analytic in $0<|\theta|<\theta_0$, $\mathrm{Im}\theta>0$, and continuous as
$\mathrm{Im}\theta\downarrow 0$, for a dense set of deformation analytic
vectors $\psi$, $\phi$ (take e.g. finite-particle vectors of Fock space built
from test functions $f(\varkappa)$ with compact support). As is well known, the real
eigenvalues of $L_\lambda(\theta)$ coincide with those of $L_\lambda$, and
away from eigenvalues the spectrum of $L_\lambda$ is purely absolutely continuous.
We now present in more detail the resonance theory.
\begin{lemma}
\label{lemmam1}
Let $\theta'={\mathrm Im}\theta>0$. There is a
$\lambda_1$ (independent of $\theta$) s.t. if
$|\lambda|<\lambda_1\min(1,\theta')$ then in the half-plane $\{{\mathrm
Im}z\geq -\theta'/2\}$ the spectrum of $L_\lambda(\theta)$ consists of four
eigenvalues $\varepsilon_\pm(\lambda)$ and $\varepsilon_0(\lambda)$ and $0$ only
(which do not depend on $\theta$).
Moerover, we have $\varepsilon_j(\lambda) =e_j-\lambda^2\varepsilon^{(2)}_j+
o(\lambda^2)$, where $j=+,-,0$, and
\begin{equation}
\varepsilon^{(2)}_0 ={\mathrm i} \, \mathrm{Im}\,\varepsilon^{(2)}_{\pm}= {\mathrm i} \,
(1+\mathrm{e}^{-2\pi \frac Ea})\xi,
\label{m18}
\end{equation}
whith $\xi$ given in (\ref{m19}).
\end{lemma}
We remark that the Fermi Golden Rule Condition (\ref{fgrc}) asserts that $\xi>0$.
{\it Proof of Lemma \ref{lemmam1}.}
By an argument of
stability of the spectrum it is not difficult to show that the spectrum in the
indicated half plane consists of four eigenvalues only.
The position (at second order in $\lambda$) is governed by so-called {\it
level shift operators}, see e.g. \cite{MLSO} and references therein. We
explain this with the help of the {\it Feshbach map} \cite{BFS}.
Let $e$ be an eigenvalue of $L_0(\theta)$ and denote the corresponding
(orthogonal) eigenprojection by $Q_e=P_e P_\mathrm{vac}$, where $P_e$ is the
spectral projection of $L_{\mathrm D}$ onto $e$ and $P_\mathrm{vac}$ projects
onto the vacuum in $\cal F$. Set $\overline Q_e:=\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l- Q_e$ and denote by
$\overline X^e=\overline Q_e X \overline
Q_e\upharpoonright_{\mathrm{Ran}\overline Q}$ the restriction of an operator
$X$ to $\mathrm{Ran}\overline Q$. A standard estimate using Neumann series
shows the following fact.
\begin{lemma}
\label{lemmam3}
There is a constant $\lambda_2$ (independent of $\theta$) s.t. if $|\lambda|<
\lambda_2 \min(E,\theta')$ then, for each eigenvalue $e$ of $L_0$, the open
ball of radius $\theta'/2$ around $e$, $B(e,\theta'/2)$, belongs to the
resolvent set of $\overline L_\lambda^e(\theta)$.
\end{lemma}
It follows from Lemma \ref{lemmam3} that the {\it Feshbach map}
\begin{equation}
F_{e,z}(L_\lambda(\theta)) := Q_e\left( e -\lambda^2 V(\theta)\overline Q_e
(\overline L_\lambda^e(\theta)-z)^{-1}\overline Q_e V(\theta)\right) Q_e
\label{m14}
\end{equation}
is well defined for all $z\in B(e,\theta'/2)$. This map has the following
remarkable {\it isospectrality property} \cite{BFS}: for all $z\in
B(e,\theta'/2)$,
\begin{equation}
z\in\mathrm{spec}(L_\lambda(\theta)) \Longleftrightarrow
z\in\mathrm{spec}\big(F_{e,z}(L_\lambda(\theta))\big).
\label{m15}
\end{equation}
Thus it suffices to examine the spectrum of the operator
$F_{e,z}(L_\lambda(\theta))$ which acts on the finite dimensional space
$\mathrm{Ran}Q_e$. We expand the resolvent in (\ref{m14}) around $\lambda=0$
and consider spectral parameters $z=e+O(\lambda)$ to obtain
$$
F_{e,z}(L_\lambda(\theta)) = Q_e\left( e -\lambda^2 V(\theta) \overline Q_e
(\overline L_0(\theta)-e)^{-1}\overline Q_e V(\theta)\right) Q_e +o(\lambda^2),
$$
where $\lim_{\lambda\rightarrow 0}o(\lambda^2)/\lambda^2=0$. We now use
analyticity in $\theta$ to conclude that
\begin{equation}
F_{e,z}(L_\lambda(\theta)) = Q_e\left( e -\lambda^2 V \overline Q_e
(\overline L_0-e -{\mathrm i} 0_+)^{-1}\overline Q_e V\right) Q_e +o(\lambda^2),
\label{m16}
\end{equation}
where ${\mathrm i} 0_+$ stands for the limit of ${\mathrm i} \varepsilon$ as
$\varepsilon\downarrow 0$. The operators
$$
\Lambda_e:= Q_e V \overline Q_e
(\overline L_0-e -{\mathrm i} 0_+)^{-1}\overline Q_e VQ_e
$$
are called level shift operators. For $e=\pm
E$ they reduce in the present case simply to numbers
($\dim\mathrm{Ran}Q_e=1$), while $\Lambda_0$
corresponds here to a $2\times 2$ matrix. Using the expression
(\ref{m6}) for $V$ one can calculate explicitly the level shift operators (see
also \cite{BFS,M,MLSO} for more detail on explicit calculations in related models).
\begin{lemma}
\label{lemmam4}
In the basis $\{|-,-\rangle, |+,+\rangle\}$ of $\mathrm{Ran}Q_0$ we have
\begin{equation*}
\Lambda_0={\mathrm i}\xi \mathrm{e}^{-\pi \frac Ea}
\left[
\begin{array}{cc}
\mathrm{e}^{-\pi \frac Ea} & -1\\
-1 & \mathrm{e}^{\pi \frac Ea}
\end{array}
\right],\ \ \mbox{and}\ \ \mathrm{Im}\Lambda_{\pm E} = (1+\mathrm{e}^{-2\pi \frac
Ea})\xi\, ,
\end{equation*}
where $\xi$ is given in (\ref{m19})
\end{lemma}
{\it Remark.} The Gibbs state of the detector at inverse temperature
$\beta=2\pi/a$ (represented by a vector $\propto [1, \mathrm{e}^{-\pi E/a}]$) spans
the kernel of $\Lambda_0$.
This lemma together with (\ref{m16}) and the isospectrality (\ref{m15}) shows
the expansions (\ref{m18}) and (\ref{m19}). This proves Lemma
\ref{lemmam1}, and at the same token, concludes the proof of Theorem \ref{specthm}. \hfill $\blacksquare$
\section{Proofs of Proposition \ref{dynproposition} and of Lemma \ref{lem:standardliouvillean} }
\label{dynpropproofsect}
{\it Proof of Proposition \ref{dynproposition}.\ }The coupled Liouville opertor (\ref{eq:coupledliouvillean}) has the form
$\widetilde{L}_\lambda= L_0+\lambda I$, where $I=\tilde G Q[\tilde g]$ with $\tilde G=A+A^\dagger$ and
$\tilde g(x)=a\delta(x^0)x^1\rho(x_*( x))$ has support in
$W_{\mathrm R}$. Essential selfadjointness of $\widetilde{L}_\lambda$ can
easily be shown using the Glimm--Jaffe--Nelson commutator theorem, see
e.g. \cite{FM}, Section 3.
The Araki-Dyson series expansion gives (weakly on a dense set)
\begin{eqnarray}
\mathrm{e}^{{\mathrm i} t \widetilde{L}_\lambda} M \mathrm{e}^{-{\mathrm i} t \widetilde{L}_\lambda}
&=&
\sum_{n=0}^\infty\lambda^n \int_0^t\mathrm{d} t_1\int_{t_1}^t\mathrm{d}
t_2\cdots\int_{t_{n-1}}^t \mathrm{d} t_n
\Big[ \tilde G(t_1)Q[\tilde g\circ B_{-at_1}], \Big[\cdots\nonumber \\
&& \cdots \Big[\tilde G(t_n)Q[\tilde g\circ B_{-at_n}], \mathrm{e}^{{\mathrm i} tL_0} M \mathrm{e}^{-{\mathrm i}
tL_0}\Big]\cdots\Big]\Big],
\label{arakidyson}
\end{eqnarray}
where we set $\tilde G(t)=\mathrm{e}^{{\mathrm i} t L_{\mathrm D}}G \mathrm{e}^{-{\mathrm i} t L_{\mathrm D}}$. For $M\in {\cal A}$ any element $M'\in {\cal A}'$ commutes
termwise with the series (\ref{arakidyson}), hence $M'\mathrm{e}^{{\mathrm i} t
\widetilde{L}_\lambda} M \mathrm{e}^{-{\mathrm i} t \widetilde{L}_\lambda}=\mathrm{e}^{{\mathrm i} t
\widetilde{L}_\lambda} M \mathrm{e}^{-{\mathrm i} t \widetilde{L}_\lambda} M'$. Therefore we have $\mathrm{e}^{{\mathrm i} t
\widetilde{L}_\lambda} {\cal A} \mathrm{e}^{-{\mathrm i} t \widetilde{L}_\lambda}\in {\cal
A}$.
\hfill $\blacksquare$
{\it Proof of Lemma \ref{lem:standardliouvillean}.\ }
Essential selfadjointness is shown using the Glimm--Jaffe--Nelson
commutator theorem, see e.g. \cite{FM} Section 3.1. The fact that $\tilde
L_\lambda$ and $L_\lambda$ define the same dynamics on $\mathcal A$ is easily
derived by using that $L_\lambda-\tilde L_\lambda$ belongs to the commutant
of $\mathcal A$ (and e.g. applying the Trotter product formula), see also
\cite{FM}. \hfill $\blacksquare$
|
2,869,038,155,715 | arxiv | \section{Introduction}
The emission from BL Lacertae objects is dominated by the intense
relativistic boosted non--thermal continuum produced within a
relativistic jet closely aligned with the line of sight, making
these objects (together with Flat Spectrum Radio Quasars, the other
subgroup of the {\it blazars} family) the best laboratories to study the
physics of relativistic jets.
The overall emission, from radio to $\gamma$--rays (extended in some
cases to the multi--TeV band), shows the presence of two well--defined
broad components, the first one peaking in the optical--soft-X--ray
bands, the second one in the GeV--TeV region. The low energy peak is
attributed to synchrotron emission by relativistic
electrons in the jet, while the second component is commonly believed
to be Inverse Compton emission (hereafter IC) from the same electron
population (although different scenarios have been proposed, see e.g.
Mannheim \cite{mannheim93}, Aharonian \cite{aharonian00}, Pohl \&
Schlickeiser \cite{pohl00}, Mucke et al. \cite{mucke03}).
BL Lacs are further subdivided in Low--energy Peak BL Lacs (LBL),
exhibiting the synchrotron peak in the IR-optical region of the
spectrum, and High--energy Peak BL Lacs (HBL), in which the
synchrotron peak can lie in the UV--X--ray band. The almost
featureless optical spectra of HBL clearly indicate that the
environment external to the jet is quite poor in soft photons,
suggesting that the high--energy emission is mainly produced through
the Comptonization of the synchrotron radiation (Synchrotron
Self--Compton emission, SSC). The few extragalactic TeV sources firmly
detected so far (Mrk~421, Mrk 501, PKS 2344+514, PKS 2155--304,
1ES1959+65, 1ES 1426+428, e.g. Krawczynski \cite{krawczynski04} for
a recent review) belong to this class.
Since the discovery of the first BL Lac object emitting TeV radiation,
Mrk~421, (Punch et al. \cite{punch92}) TeV blazars have been the target of
very intense observational and theoretical investigations. Indeed the
possibility of observing the emission produced by very high energy
electrons (up to Lorentz factors of the order of $10^7$) coupled with
observations in the X--ray band, where the synchrotron peak of these
sources is usually located, offers a unique tool to probe the
processes responsible for the cooling and the acceleration of
relativistic particles.
Studies conducted simultaneously in the X--ray and in the TeV bands
are of particular importance, since in the simple SSC framework one
expects that variations in X--rays and TeV should be closely
correlated, being produced by electrons with similar energies.
Assuming a typical magnetic field intensity $B\sim 0.1$ G and Doppler
factor $\delta\sim 10$, photons with energy E $\sim 1$ keV are emitted
by electrons with Lorentz factor $\gamma \sim 10^6$. The same
electrons will upscatter photons at energy $E\sim \gamma mc^2\sim 1$
TeV (since the Lorentz factor is extremely large the scattering will
occur in the Klein--Nishina regime even with optical target photons).
In fact observations at X--ray and TeV energies (Buckley et al. \cite{buckley96},
Catanese et al. \cite{catanese97}) yielded significant evidence of correlated and
simultaneous variability of the TeV and X--ray fluxes. During the
X/TeV 1998 campaign on Mrk~421 a flare was detected
simultaneously both at X--ray and TeV energies and the maxima were
simultaneous within 1 hour, confirming that variations in these two
bands are closely related (Maraschi et al. \cite{maraschi99}). Subsequent
more extensive analyses confirmed these first results also in other
sources. Note however that the correlation seems to be violated in
some cases, as indicated by the observation of an ``orphan'' (i.e. not
accompanied by the corresponding X--ray flare) TeV event in the BL Lac
1ES 1959+650 (Krawczynski et al. \cite{krawczynski04a}).
Although the study of TeV BL Lacs
has been the subject of great effort in the past (e.g.
B\"ottcher \cite{bottcher04}, Georganopoulos \& Kazanas \cite{georg03},
Moderski, Sikora \& Blazejowski \cite{moderski03}, B\"ottcher \&
Dermer \cite{bottcher02b}, Tavecchio et al. \cite{tavecchio01},
Takahashi et al. \cite{takahashi00}, Kirk, Rieger \& Mastichiadis \cite{kirk98}),
a detailed analysis of the expected correlations of the fluxes in different
bands is lacking. In this paper we present for the first time a
comprehensive study of the time--dependent emission expected in the
case of the homogeneous SSC model, following the evolution of
electrons and taking into account both radiative and adiabatic
losses. In particular we focus our attention on the correlation between
the variations observed in the X--rays and in the TeV band, which can
be quite informative of the processes responsible for the observed
variations. The present work has been stimulated (as described in
Sect. 2) by the observation in several cases of flares for which a
more than linear behavior was observed
(namely, $F_{\rm TeV} \propto F^x_{\rm X-ray}$, with $x>1$).
At first sight this does not pose particular
problems (an increase of the density of the emitting electrons
will make the self--Compton flux increase more than the synchrotron
flux), but we will show instead that these more than linear
correlations are quite difficult to reproduce,
with standard assumptions, when we consider that the
same type of correlation has been observed not only
during the rising phases of a flare, but also during the decay phase
(Fossati at al. \cite{fossati04}).
In Section 2 we discuss some of the clearest observational evidence for
the correlation between X--rays and $\gamma$--rays. In Section 3 and 4 we
describe the model used to calculate the expected correlations in the SSC
framework (assuming spherical and cylindrical geometries) and we present
the results, stressing the difficulty of obtaining more than linear relations
between TeV and X-ray variations. In Section 5 we briefly discuss the effects
that can be induced by considering the finite light crossing time in the
study of the variability. In Section 6 we conclude.
\section{Observed correlations}
\label{sec_obs-corr}
The correlation between the evolution of the X--ray and the $\gamma$--ray
emission of high--energy peaked BL Lac (HBL) objects may provide important
information on the emission mechanism of such sources and the physical
mechanisms producing the variability. However, to investigate such a
correlation it is necessary to compare two different light curves,
obtained in the same period of time by two different instruments. The
sampling rate for both light curves must be similar to provide a
reliable result. Since the variability timescale in HBL objects is very
short, the sampling of the light curves should also be dense, with
several data points per day. Moreover, additional simultaneous spectral
observations are required to provide basic information about the
evolution of the spectrum. The last condition is very important for the
modeling of the source activity. For example it is important to know if
the X--ray synchrotron emission ($F_s$) is generated below or above the
$\nu F_{s}(\nu)$ peak. It is difficult to meet all these conditions.
Therefore, there are just a few cases where the correlation between the
X--ray and the $\gamma$--ray activity can be investigated in detail.
In this section we present three correlations obtained for two very well
known HBL objects Mrk~421 and Mrk~501, and we also discuss the
correlations obtained by other investigators.
\phantom{nothing}
\begin{figure}[p]
\resizebox{\hsize}{!}{\includegraphics{1556fig1_c.ps}}
\caption{The activity of Mrk~501 observed in 1997 April. The upper panels show
the $\gamma$--ray and the X--ray light curves obtained by the Whipple (a), CGRO--OSSE (b)
and RXTE--ASM (c) experiments (Catanese et al. \cite{catanese97}). The shaded areas in
these figures indicate the observing periods of the $Beppo$SAX and CAT instruments.
In panel (d) we show the spectra obtained by the above mentioned instruments (Pian
et al. \cite{pian98}, Djannati--Atai et al. \cite{djannati99}). In this panel we
show also the energy bands used to obtain the light curves. The lower panel (e)
shows the two correlations made between the X--ray and the TeV gamma--ray fluxes.
For clarity, we multiply the OSSE data in this panel by the factor 0.5.
The dashed lines in this panel show a template for the linear and the quadratic
correlation.
}
\label{fig_501corr}
\end{figure}
\begin{figure}[!t]
\resizebox{\hsize}{!}{\includegraphics{1556fig2_c.ps}}
\caption{The activity of Mrk~421 observed on 2000 February. The upper panels show
the $\gamma$--ray and the X--ray light curves obtained by the HEGRA (a), RXTE--PCA (b)
and RXTE--ASM (c) experiments (Krawczynski et al. \cite{krawczynski01}). The shaded
areas in these panels indicate the observing periods for the spectra obtained by the
RXTE and HEGRA instruments presented in panel (d). The lower panel (e) shows the
correlation between the X--ray and the TeV $\gamma$--ray fluxes. The dashed lines
in this panel show a template for the linear and the quadratic correlation.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
}
\label{fig_421corr}
\end{figure}
In Figure \ref{fig_501corr} we show the activity of Mrk~501 observed in 1997 April. We show
in this figure the light curves obtained by the Whipple (E$>$350 GeV), CGRO--OSSE (50--150 keV),
and RXTE--ASM (3--20 keV) experiments (Figs \ref{fig_501corr}--a,b,c respectively, Catanese et al.
\cite{catanese97}). We show also the evolution of the spectral energy distribution (Fig.
\ref{fig_501corr}--d) observed by the $Beppo$SAX (0.1--200 keV) and CAT (E$>$250 GeV) instruments
during this activity (Pian et al. \cite{pian98}, Djannati--Atai et al. \cite{djannati99}). For
this data we calculate two correlations. The first correlation is calculated between the
fluxes obtained by the OSSE and the Whipple experiments. This correlation gives an almost
linear result (Fig. \ref{fig_501corr}--e). The precise computation gives in this case
$F_{\rm{Whipple}} \propto F_{\rm{OSSE}}^{1.71\pm0.50}$ with a reduced $\chi^2 = 0.67$.
The second correlation is calculated between the data from the ASM experiment and the fluxes
obtained by the Whipple telescope. This correlation gives a quadratic or even more than quadratic
result. The precise computation gives $F_{\rm{Whipple}} \propto
F_{\rm{ASM}}^{2.69\pm0.56}$ and the reduced $\chi^2 = 0.65$.
Note that to compare two light curves, where the observational points are randomly
distributed in time, it is necessary to select points gathered almost exactly at the same
time. This may practically mean that we are able to correlate only a few points in a light
curves where we have even dozens of points. In the alternate approach we may try to
interpolate one of the light curves. However, this method requires similar sampling of
the light curves. Moreover, the sampling should be comparable or less than the characteristic
variability time scale of a source. In the case of the OSSE-Whipple correlation we have
interpolated the Whipple light curve. The interpolation method in this particular case
has a minimal influence on the final result because the OSSE and Whipple observational
points were gathered almost at the same time. There is only one point in the
OSSE light curve (50549-50550 MJD) which has no counterpart in the TeV observations
and we have excluded this point from our analysis. In the
ASM-Whipple correlation we have interpolated the ASM light curve. The time shifts between
the observational points in these light curves are relatively large in comparison to the
previous correlation. However, each point in the ASM light
curve has been calculated as an average of many basic measurements made during a period
of about one day. Therefore, the interpolation in this case also provides meaningful
results. Moreover, our results are in agreement with the
previous correlation made by Djannati--Atai et al. (\cite{djannati99}).
They obtained a more than linear but less than quadratic correlation between the
$Beppo$SAX and the CAT telescope fluxes.
Note that the energy range of the $Beppo$SAX observations (0.1--200 keV)
covers the spectral bands used by the ASM and the OSSE experiments
(Fig. \ref{fig_501corr}--d).
Therefore, the correlation mentioned above gives an average of the two correlations presented in
this work.
The source was observed at that time also by other instruments. Krawczynski et al.
(\cite{krawczynski00}) correlated the evolution of the X--ray emission at 3 keV with the
evolution of the $\gamma$--ray radiation at 2 TeV. The correlation they obtained seems to
be more than quadratic (see Figure 6--a in their paper).
This is very similar to the
correlation presented here
between the ASM (2--10 keV) and the Whipple observations. The second correlation
performed by the team mentioned above concerns the evolution of the X--ray emission at 25 keV
and the evolution of the $\gamma$--rays at 2 TeV.
This correlation gives an almost quadratic result
(see Figure 6--b in their article).
The correlations discussed above show that there was significant
difference between the evolution of the synchrotron emission before
the $\nu F_s(\nu)$ peak and the evolution around the peak. It means
that the correlation may depend on the position of the spectral bands
(below, around or above the $\nu F_{s/c}(\nu)$ peak) used for the
calculations. The comparison between our correlations and the correlation
done by Djannati--Atai et al. (\cite{djannati99}) indicates that the
slope of the correlation may also depend on the width of the spectral band.
In Figure \ref{fig_421corr} we present the activity of Mrk~421 observed
in 2000 February. We show the light curves gathered by the HEGRA (E$>$1TeV),
RXTE--PCA (3--20 keV) and the RXTE--ASM (2--10 keV) experiments
(Figs \ref{fig_501corr}--a,b,c respectively, Krawczynski et al. \cite{krawczynski01}).
We show also the spectral energy distributions observed at that time
(Fig. \ref{fig_501corr}--d). We calculate for this source the correlation
between the PCA and the HEGRA fluxes. To obtain the correlation we have
interpolated the HEGRA light curve. However, the interpolation as
in the case of the OSSE-Whipple correlation for Mrk 501, has minimal
influence on the result because the observations in both light
curves were made almost simultaneously. In this case we have obtain
unclear results; the detailed fitting gives $F_{\rm{HEGRA}} \propto
F_{\rm{PCA}}^{0.47\pm0.15}$ with a poor reduced $\chi^2$
of 1.7. Note that also the correlation between the ASM and the HEGRA data,
which we do not present here, does not provide an informative result.
This example shows that the relatively long timescale observations with
one data point per day may not be sufficient to obtain meaningful
information on the TeV/X--ray correlation.
However, Fossati et al. (\cite{fossati04}) have recently reported very precise observations
of this source made in 2001 March. The object was observed during a period of one
week by the RXTE--PCA, Whipple and the HEGRA instruments. The rate of the measurements
was relatively high and several flaring events were well observed. The most significant
seems to be a flare observed on March 19 by the PCA and Whipple instruments.
The TeV/X--ray correlation for this flare is almost quadratic ($F_{\rm TeV} \propto
F_{\rm X}^{2.3\pm0.3}$) for the rising and decaying phase. However, the correlation
for data obtained by the Whipple and RXTE--PCA experiments during this campaign
gives $F_{\rm TeV} \propto F_{\rm X}^{1.3\pm0.1}$. This indicates that the longterm
correlation may give results significantly different from the correlation performed
for one single flaring event. The reason for the difference may be related to the fact
that for a long term light curve (e.g. a few days or even longer) we may correlate at
least a few flaring events. If for example the activity is generated in the framework
of the internal shock mechanism (e.g. Rees \cite{rees78}, Guetta et al. \cite{guetta04})
then the flares may overlap each other in time. If we observe a decay phase of a flare
and during this decay a new flare will start to rise then the observed correlation
would be completely different even if the base correlation for each flare was the same.
Recently Tanihata et al. (\cite{tanihata04}) published a spectral analysis of
the observations of Mrk~421 obtained during a 7 day campaign in 1998. They found
a correlation between the TeV flux and the synchrotron peak flux $F_{\rm TeV} \propto
F_{\rm s,peak}^{1.7\pm0.3}$.
In the following we will take particular care in looking for robust
solutions yielding linear or quadratic correlations between X-ray and TeV
bands. Although so far there are only few cases for which these
correlations (especially the quadratic one) have been clearly
established, our attention is motivated by the fact that, as we will
discuss below, the observations of such a correlations could impose
strong constraints on the scenario usually considered for the variability
observed in these sources. In particular the observation of a quadratic
relation during the {\it decaying phase} is particularly intriguing. In
fact, if the quadratic increase of the synchrotron and IC emission could
be easily reproduced, increasing the electron density within the source,
the decrease is more problematic. Escape of electrons from the
sources appears unrealistic in the small timescale shown by
variability. On the other hand radiative cooling can affect only the
high-energy particles. Indeed, the simple quadratic relation predicted
for the IC emission is strictly valid for IC in the Thomson regime.
However, the TeV emission of the HBL objects is probably generated in the
Klein-Nishina regime. In Appendix \ref{app_thomson} we estimate the physical
parameters of a source that could generate TeV emission in
the Thomson limit. The estimation gives unacceptably large values of the
Doppler factor ($\delta \gtrsim 1000$, Begelman et al. \cite{begelman94}).
Therefore, in all models presented in this work we assume that the TeV radiation
comes from the IC scattering in the Klein-Nishina regime.
The decline of the cross-section in the Klein-Nishina regime (valid for
sources with a plausible value of the physical quantities) has the effect
of decoupling the (high-energy) electrons producing the high-energy photons
and those (at low-energy) producing the optical--IR synchrotron seed
photons (e.g. Tavecchio, Maraschi \& Ghisellini 1998). In these
conditions, since the radiative cooling only affects the high-energy
particles, a {\it linear} relation between X-rays and TeV should be
expected, since the seed photons can be considered almost constant during
the decay. A plausible alternative, discussed in detail below, is to
admit adiabatic cooling, affecting all the electrons, irrespective of
their energy.
\section{Spherical homogeneous source}
\label{sec_spher-src}
A homogeneous synchrotron-SSC model is frequently proposed as a possible
explanation for the X--ray and the $\gamma$--ray emission of HBL
objects. Such model provide a very good and simple explanation for
the Spectral Energy Distributions (SED) observed in X--rays and the
TeV gamma rays (e.g. Dermer $\&$ Schlickeiser \cite{dermer93},
Bednarek $\&$ Protheroe \cite{bednarek97}, Mastichiadis
\& Kirk \cite{mastichiadis97}, Pian et al. \cite{pian98}, Katarzy\'nski et al.
\cite{katarzynski01}). Therefore, we decided to check if such a model is
also able to explain quite specific properties of the observed
variability. In the approach presented in this section we do not
consider Light Crossing Time Effects (hereafter LCTE) in the synchrotron
radiation field inside the source nor for the total observed emission.
This means that the results presented in this section are strictly
valid only if the physical processes, which modify the source brightness,
are slower than the source light crossing time. This condition may
not always be correct in the case of blazars. However, it allows us to
investigate in detail the most important physical processes which may
modify the emission level of the source. Detailed understanding of these
processes is very important for any more complex modeling.
\subsection{Description of the model}
\label{sub_sph-geom}
We assume for this modeling a spherical homogeneous source, which may
undergo expansion or compression. As a first step, to simplify the model and
to provide analytical formulae which describe the evolution of such a
source, we decided to consider only four physical processes which may
occur during the evolution. We calculate increase or decrease of the
source volume, increase or decay of the magnetic field intensity,
variations of the particle density and adiabatic heating or cooling of
the particles. In the model we separate as much as possible the
mathematical description for each process. This allows us to
investigate separately the influence of each process on the source
emission. In this first approach we do not consider the radiative cooling
of the particles. The impact of this process will be discussed in the next
part of this work (Subsection \ref{sub_rad-cool}).
The time evolution of the source radius ($R$) and the magnetic field
intensity ($B$) are assumed to be a power law functions
\begin{equation}
\label{equ_evol_magn}
R(t) = R_0 \left(t_0/t \right)^{-r_e},~~~~~
B(t) = B_0 \left(t_0/t \right)^{m},
\end{equation}
where $t_0$, $R_0$ and $B_0$ are initial time, magnetic field intensity and
radius respectively. The indices $r_e$ and $m$ are a free parameters in
our modeling.
The initial ($t= t_0$) distribution of the electron energy inside the source is defined
by a broken power law
\begin{equation}
\label{equ_ini_elec_spec}
N_0(\gamma) =
\left\{
\begin{array}{ll}
k_1 \gamma^{-n_1}, & \mbox{$ \gamma_{\rm min} \leq \gamma \leq \gamma^0_{\rm brk} $}\\
k_2 \gamma^{-n_2}, & \mbox{$ \gamma^0_{\rm brk} ~< \gamma \leq \gamma_{\rm max} $}\\
\end{array}
\right.,
\end{equation}
where $k_2 = k_1 (\gamma^0_{\rm brk})^{n_2-n_1}$, $\gamma$ is the Lorentz factor which is
equivalent to the electron energy, $\gamma^0_{\rm brk}$ describes initial position of the
break, $n_1$ and $n_2$ are spectral indices before and above the break respectively.
In the model we assume that the dominant part of the X-ray and TeV emission is produced by
the electrons with the Lorentz factors around $\gamma_{\rm brk}$. Therefore, the values of
$\gamma_{\rm min}$ and $\gamma_{\rm max}$ parameters are not important for the results of the
modeling. For all calculations presented in this section we use $\gamma_{\rm min} = 1$ and
$\gamma_{\rm max} = 10^8$.
The evolution of the electron energy spectrum is defined by a minimum
\begin{equation}
\label{equ_min_elec_spec}
N(\gamma, t) = \min\left\{N_1(\gamma, t), N_2(\gamma, t)\right\}
\end{equation}
of two power law functions
\begin{equation}
\label{equ_sph_pow_elec_spec}
\begin{array}{ll}
\vspace*{0.2cm}
N_1(\gamma, t) = K_1(t) \gamma^{-n_1}; &~~~~~K_1(t) = k_1 \left(\frac{t_0}{t} \right)^{3r_{d}}
\left(\frac{t_0}{t} \right)^{r_{a}(n_1-1)},\\
N_2(\gamma, t) = K_2(t) \gamma^{-n_2}; &~~~~~K_2(t) = k_2 \left(\frac{t_0}{t} \right)^{3r_{d}}
\left(\frac{t_0}{t} \right)^{r_{a}(n_2-1)},\\
\end{array}
\end{equation}
where $K_1$ and $K_2$ describe the evolution of the particle density before and after the break
($\gamma_{brk}$). The exponent $3r_d$ describes the decrease or increase of the particle density.
The adiabatic heating or cooling of the particles is described by the indices $r_{a}(n_{1|2}-1)$.
Note that if we assume for example adiabatic expansion or compression of the source with a constant
number of the particles, then the parameters $r_e$, $r_d$ and $r_a$ should be equal (e.g. Kardashev
\cite{kardashev62}, Longair \cite{longair92}). This is in fact the most realistic case. However, to
investigate separately the influence of the above mentioned processes, we decided to produce
such a detailed parameterization.
Assuming that the electron spectrum is defined by the minimum of two evolving power law functions we
can easily find the evolution of the break $\gamma_{\rm brk}(t) = \gamma^0_{\rm brk}
\left(t_0/t \right)^{r_{a}}.$
\subsubsection{Synchrotron emission}
The synchrotron emission coefficient which describes the radiation of the electrons is given by
the well-known formula $j_s \propto K B^{(\alpha+1)}$ (e.g. Rybicki $\&$ Lightman
\cite{rybicki79}) where $\alpha = (n-1)/2$. In our particular case we have to define
the evolution of two emission coefficients
\begin{equation}
j^{1|2}_s(t) \propto K_{1|2}(t) B(t)^{(\alpha_{1|2}+1)}
\propto k_{1|2} B_{1|2} \left(\frac{t_0}{t}\right)^{3r_d+r_a(n_{1|2}-1)+m(\alpha_{1|2}+1)},
\end{equation}
where $j^1_s$ describes emission of the low energy electrons ($\gamma < \gamma_{brk}$),
$j^2_s$ describes the radiation of the high energy electrons ($\gamma > \gamma_{brk}$) and
$B_{1|2} = B_0^{\alpha_{1|2}+1}$.\footnote{Note that in order to reduce the number of equations
we use in this work the following simplification
$f^{1|2}_x \propto a_{1|2}$ means $f^1_x \propto a_1$
or $f^2_x \propto a_2$.}
In our modeling we neglect the electron self--absorption process which is important only for the
emission at the radio frequencies which we do not analyze in this work. Therefore, we can
approximate the evolution of the intensity of the synchrotron radiation from the spherical
source by
\begin{equation}
\label{equ_sph_isyn12}
I^{1|2}_s(t) \propto R(t) j^{1|2}_s(t)
\propto R_0 k_{1|2} B_{1|2}
\left(\frac{t_0}{t}\right)^{-r_e + 3r_d+r_a(n_{1|2}-1)+m(\alpha_{1|2}+1)}.
\end{equation}
Finally we can write the evolution of the synchrotron flux multiplying the intensity by the
source surface
\begin{eqnarray}
\vspace*{0.1cm}
\label{equ_sph_fsyn1}
F^1_s(t) & \propto & R^2(t) I^1_s(t) ~\propto~ R_0^3 k_1 B_1 \left(\frac{t_0}{t}\right)^{-s_1},\\
s_1 & = & 3r_e - 3r_d - r_a(n_1-1) - m(\alpha_1+1),
\end{eqnarray}
\begin{eqnarray}
\vspace*{0.1cm}
\label{equ_sph_fsyn2}
F^2_s(t) & \propto & R^2(t) I^2_s(t) ~\propto~ R_0^3 k_2 B_2 \left(\frac{t_0}{t}\right)^{-s_2},\\
s_2 & = & 3r_e - 3r_d - r_a(n_2-1) - m(\alpha_2+1),
\end{eqnarray}
where the indices $s_1$ and $s_2$ describe the evolution of the synchrotron emission generated
by the low and the high energy electrons respectively. Note that there is a difference
between the evolution of the $F^1_s$ flux and the evolution of the $F^2_s$ radiation due to the
adiabatic processes ($r_a(n_{1|2}-1)$) and the evolution of the magnetic field intensity
($m(\alpha_{1|2}+1)$).
\subsubsection{Self Compton emission}
Tavecchio et al. (\cite{tavecchio98}) presented detailed studies of the SSC emission generated
by the electrons with an energy spectrum approximated by a broken power law. The IC spectrum in
such an approach can be divided into four basic components ($F^{1...4}_c$). However, only two of them are dominant
in the IC emission generated by HBL objects. The first dominant component ($F^1_c$) is generated
by the low energy electrons ($K_1, N_1$) and the first part of the synchrotron spectrum ($I^1_s,
F^1_s$). The second important component ($F^2_c$) is produced by the high energy electrons ($K_2,
N_2$) and the first part of the synchrotron emission ($I^1_s, F^1_s$). With these
results in mind, we can derive the formulae describing the evolution of
emission coefficients for the dominant components
\begin{equation}
j^{1}_c(t)
\propto K_1(t) I^1_s(t)
\propto R_0 k_1^2 B_1
\left(\frac{t_0}{t}\right)^{-r_e+6r_d+2r_a(n_1-1)+m(\alpha_1+1)}.
\end{equation}
\begin{eqnarray}
j^{2}_c(t)
& \propto & K_2(t) I^1_s(t)\nonumber\\
& \propto & R_0 k_1 k_2 B_1
\left(\frac{t_0}{t}\right)^{-r_e+6r_d+r_a(n_2-1)+r_a(n_1-1)+m(\alpha_1+1)}.
\end{eqnarray}
By analogy to the synchrotron emission we can write the evolution of the intensity of the
IC emission for both the discussed cases $I^{1|2}_c(t) \propto R(t) j^{1|2}_c(t).$
The evolution of the IC flux is given then by
\begin{eqnarray}
\vspace*{0.1cm}
F^{1}_c(t) & \propto & R^2(t) I^{1}_c(t)~~\propto~~
R_0^4 k^2_1 B_1 \left(\frac{t_0}{t}\right)^{-c_1},\\
c_1 & = & 4r_e-6r_d-2r_a(n_1-1) - m(\alpha_1+1),
\end{eqnarray}
\begin{eqnarray}
\vspace*{0.1cm}
F^{2}_c(t) & \propto & R^2(t) I^{2}_s(t)~~\propto~~R_0^4 k_1 k_2 B_1
\left(\frac{t_0}{t}\right)^{-c_2},\\
c_2 & = & 4r_e-6r_d-r_a(n_1-1)-r_a(n_2-1)-m(\alpha_1+1),
\end{eqnarray}
where $c_1$ describes the evolution of the IC radiation in the Thomson limit and
$c_2$ describes the evolution of the IC emission in the Klein--Nishina regime.
\subsubsection{Components of the synchrotron self Compton spectrum: an example}
To show that the approximations used in the derivation of the equations
describing the SSC emission are correct, in Figure \ref{fig_sph-geom} we
report an example of the evolution of the synchrotron and the IC emission
generated by the expanding spherical homogeneous source, calculated using
our numerical code. The results presented in this subsection are
correct for a wide range of values of the physical parameters usually
assumed in order to explain the X-ray and TeV emission of the HBL objects.
However, the results are correct only for case where the dominant part of the
synchrotron and the IC emission (the peaks) is generated by electrons
with the Lorentz factors close to $\gamma_{\rm brk}$ which practically
means $n_2 > 3$. The spectra presented in Figure \ref{fig_sph-geom} are divided
into different components. This test shows that indeed only two components are
important for the high energy IC emission for a large amplitude of the
variation. Note that the components are calculated numerically,
therefore the spectra shown are slightly different from the curves
obtained from the analytical approximations by Tavecchio et
al. (\cite{tavecchio98}). For the numerical calculations we use the SSC
emission mechanism described by Katarzy\'nski et
al. (\cite{katarzynski01}). In the test we use following parameters
$\delta = 50$, $R_0 = 2 \times 10^{16}$ [cm], $B_0 = 0.004$ [G], $k_1 =
10^3$ [cm$^{-3}]$, $\gamma_{\rm min} = 1$, $\gamma_{\rm brk} = 1.2 \times 10^6$,
$\gamma_{\rm max} = 10^8$, $r_e = r_d = r_a = m = 1$. Note that
in an ideal case where the source is perfectly spherical and homogeneous
and the magnetic field inside the source is well organized the parameter
$m$ which describes the evolution of the magnetic field should be equal to
$2 r_e$ (e.g. $m=2$ for a linear expansion of the source) in order to
keep the magnetic flux constant.
\begin{figure}[p]
\resizebox{\hsize}{!}{\includegraphics{1556fig3_c.ps}}
\caption{The evolution of the synchrotron and the IC emission of an expanding
spherical homogeneous source. Panel (a) shows the evolution of the
electron energy spectrum. The spectrum gives the synchrotron radiation
shown in panel (b). The evolution of the IC spectrum is shown
in panels (c) and (d). To simplify the figures we show only the initial
and the final spectra. Note that the initial spectra in all
figures are shown by bold--solid lines while the spectra
generated at the end of the simulation are shown by bold--dashed
lines. The basic components of the emission are shown by various thin
lines. A detailed description of the components and the parameters
used for the test are given in subsection \ref{sub_sph-geom}.
}
\label{fig_sph-geom}
\end{figure}
However, practically the occurrence of such ideal
sources is very unlikely. Usually we would expect a more complex geometry,
inhomogeneity and a turbulent magnetic field inside the source which may
give a more complex evolution of the magnetic field. For example a slower
than expected decrease of the magnetic field during the expansion may
be caused by the turbulent dynamo effect (e.g. Atoyan \& Aharonian
\cite{atoyan99} for discussion of this problem). Therefore, we assume the
parameter $m$ to be a free parameter in our modeling. For the linear expansion
we assume that the value of this parameter may vary from 1 to 2. The value
of this parameter has no influence on the results presented in this subsection.
However, in other tests presented in the next part of this work,
we had to use $m=1$ in order to explain the quadratic correlation observed
in the decaying phase of the flare.
Therefore, to be consistent and to provide the possibility of comparison of
the results, in all numerical calculations presented in this paper we have
selected $m=1$. Moreover, in this test and in all other simulations presented
in this work we assume that the $t_0$ parameter is equal to $10~R_0/c$. This gives
the radius expansion velocity equal to $0.1 c$ for $r_e = 1$. To change
the source emission level at least a few times we simulate the evolution for
a period of time equal to $3~R_0/c$.
This set of parameters is quite specific because we use a relatively small
value of the initial magnetic field and a relatively large value of the Doppler
factor (see estimations made by Tavecchio et al. \cite{tavecchio98} or
Katarzy\'nski et al. \cite{katarzynski01}). We assume a small value of the magnetic
field in order to neglect radiative cooling, which cannot be described easily
by the analytical formulae. On the other hand, to reach the observed level of
emission, we have to assume a large value of the Doppler factor.
The Doppler factor value $\ge 50$ has been
used by some authors (e.g. Krawczynski et al. \cite{krawczynski02},
Konopelko et. al. \cite{konopelko03}) to explain the emission of TeV blazars.
We will prove in the next part of this work (Subsection \ref{sub_rad-cool}) that
the radiative cooling is indeed negligible for this set of parameters.
In this test the radiative cooling is
negligible for electrons with the energy described by the Lorentz factor
around $\gamma_{\rm brk}$ which are producing the dominant part of the emission
observed as the synchrotron and IC peaks. On the other hand the radiative
cooling is still important for electrons with a Lorentz factor around
$\gamma_{\rm max}$. However, the radiation generated by these particles is
almost completely negligible due to the steepness of their spectrum.
Moreover, for this test and for other calculations presented in this work
we use the Hubble constant equal to 65 [km s$^{-1}$ Mpc$^{-1}$] and the
redshift equal to 0.03. In the above calculations and in all other
calculations of the IC scattering presented in this work, we neglect the
possible absorption of the $\gamma$--ray emission due to pair production
inside the source. This process can be neglected if the Doppler factor of
the source is of the order of ten or more (e.g. Bednarek \& Protheroe
\cite{bednarek97}). Moreover, to calculate the observed TeV flux, we should
also consider the absorption of this radiation by interaction with the Infrared
Intergalactic Background. However, in this particular case, we are mainly
interested in investigating the evolution of the TeV radiation with respect
to the evolution of the synchrotron X--ray emission. Therefore, absorption
of the TeV emission by the IR background, which is constant in time, does
not modify the TeV/X--ray flux correlation.
\subsection{Basic cases}
Correlating the X--ray and TeV emission of the HBL objects we can
distinguish two general cases. In the first case we compare the X--ray
emission before the synchrotron peak ($F^1_s$) with the $\gamma$--ray
radiation above the IC peak ($F^{2}_c$, e.g. Mrk~501, Figure~\ref{fig_501corr}).
In the second case we correlate the X--ray emission above the synchrotron
peak ($F^2_s$) with the $\gamma$--ray radiation above the IC peak
($F^{2}_c$, e.g. Mrk~421, Figure~\ref{fig_421corr}). To simplify the
discussion presented in this work we hereafter call these correlations
$c_2/s_1$ and $c_2/s_2$ respectively. We also present in this section
the estimates of other possible correlations ($F^1_s$ vs $F^1_c$ and
$F^2_s$ vs $F^1_c$). However, we will not discuss them in detail.
The more complex cases of the TeV/X--ray correlation, where the considered
bands correspond to the emission around the peaks, discussed
in subsection \ref{sub_peak-corr}.
For the tests presented in this section we use the same values of the
physical parameters as in the calculations presented in the previous
subsection. To simulate different evolution scenarios we change only
four model parameters $r_e, r_d, r_a, m$ setting 0 or 1 as the value for
these parameters. This means that in principle we can study only expansion of
the source. However, as long as we neglect the radiative cooling, the
results of such tests are opposite to the results which we could obtain
from the compression of the source. Having four parameters and two possible
values for these parameters (0 or 1) we have in principle sixteen different
combinations~($2^4$). However, one combination is not interesting
($r_e=r_d=r_a=m=0$) and four other combinations are not realistic
(all cases where we should calculate the adiabatic cooling
without a decrease of the particle density). In Table~\ref{tab_corr_spher}
we show eleven other possible combinations plus one test where we
use $m = 2$.
We discuss here in detail only a few tests which we consider to be the most
realistic:
\begin{itemize}
\item[$\bullet$]{
In our first case (Table~\ref{tab_corr_spher}--$a$) we assume the expansion
of the source volume ($r_e = 1$) with constant particle density and
magnetic field intensity ($r_d=m= 0$). This scenario could correspond to a constant
injection of particles (e.g. from a shock region) which extends the
dimension of the source instead of increasing the particle density. The
synchrotron radiation in this case is proportional to the source volume.
Therefore, the synchrotron emission produced by the low and the high energy
electrons increases with time as $R^3 \propto t^{3r_e}$ ($s_{1|2}= 3$.
Eqs~\ref{equ_sph_fsyn1}, ~\ref{equ_sph_fsyn2}). The IC radiation in this
test is also proportional to the source volume. However, the IC emission is
also proportional to the intensity of the synchrotron emission
(Eq.~\ref{equ_sph_isyn12}) which grows linearly with the radius of the
source ($I^{1|2}_s \propto R j^{1|2}_s \propto t^{r_e}$). Therefore, the IC
radiation increases in time as $t^{3r_e + r_e}$ ($c_{1|2}=4$). Finally we
obtain $c_{1|2}/s_{1|2} = 4/3$ for all possible correlations.
}
\item[$\bullet$]{
The second case ($b$) is opposite to the scenario just described. We increase
the particle density ($r_d = 1$) keeping the volume and the magnetic
field intensity constant ($r_e = m = 0$). A linear increase of the density provides
a linear increase of the synchrotron radiation. The IC emission is proportional
to the density of the particles and to the intensity of the synchrotron
radiation which is also proportional to the density. Therefore, the IC emission
is proportional to the square of the particle density. This is well-known
relation. In this particular case we obtain a quadratic correlation in all
possible cases ($c_{1|2}/s_{1|2} = 2/1$).
}
\begin{table}[!t]
\caption{Estimated correlation between evolution of the synchrotron emission and the
evolution of the IC emission for different evolutionary scenarios.
Columns 2--5 show the parameters used for the tests.
Columns 6--9 show the estimated values of the correlation in four possible
cases ($c_{1|2}/s_{1|2}$).
}
\label{tab_corr_spher}
\begin{center}
\begin{tabular}{ccccc|llll}
\hline
\hline
case &$r_e$ & $r_d$ & $r_a$ & $m$ & $c_1/s_1$ & $c_1/s_2$ & $c_2/s_1$ & $c_2/s_2$\\
\hline
$a$& 1 & 0 & 0 & 0 & 1.333 & 1.333 & 1.333 & 1.333 \\
$b$& 0 & 1 & 0 & 0 & 2 & 2 & 2 & 2 \\
$c$& 1 & 1 & 0 & 0 & inf & inf & inf & inf \\
$d$& 1 & 1 & 1 & 0 & 4 & 1 & 7 & 1.75 \\
$e$& 1 & 1 & 1 & 1 & 2.2 & 0.786 & 3.4 & 1.214 \\
$f$& 0 & 0 & 0 & 1 & 1 & 0.5 & 1 & 0.5 \\
$g$& 0 & 1 & 1 & 0 & 2 & 1.143 & 2.75 & 1.571 \\
$h$& 0 & 1 & 1 & 1 & 1.727 & 0.950 & 2.273 & 1.250 \\
$i$& 1 & 1 & 0 & 1 & 2.332 & 1.167 & 2.333 & 1.167 \\
$j$& 1 & 0 & 0 & 1 & 1.667 & inf & 1.667 & inf \\
$k$& 0 & 1 & 0 & 1 & 1.667 & 1.250 & 1.667 & 1.250 \\
$l$& 1 & 1 & 1 & 2 & 1.75 & 0.7 & 2.5 & 1 \\
\hline
\end{tabular}
\end{center}
\end{table}
\item[$\bullet$]{
In the next case ($d$) we assume an increase of the volume, decrease
of the particle density and with adiabatic cooling ($r_e=r_d=r_a=1$) with a
constant value of the magnetic field intensity ($m=0$) during the evolution.
Even if it is rather unlikely that the magnetic field can remain constant,
we describe this case to provide a sort of introduction to our last, more
complex test. The synchrotron emission decreases in this test due to
the adiabatic losses because the decrease of the density is fully
compensated for the increase in the volume if $r_e=r_d$. This decrease
depends on the slope ($n_{1|2}$) of the particle spectrum (Eq.~\ref{equ_sph_pow_elec_spec}).
The synchrotron emission produced by the low energy electrons decreases
more slowly ($t^{-r_a(n_1-1)}$, $s_1 = -1$) than the emission generated by the
high energy electrons ($t^{-r_a(n_2-1)}$, $s_2 = -4$). The IC emission
also decreases in this test.
In the Thomson limit, the IC flux is proportional to the increase
in the volume ($t^{3r_e=3}$) and the radius ($t^{r_e=1}$, which comes from
the intensity of the synchrotron emission $I^{1}_s \propto R j^{1}_s \propto
t^{r_e}$) and also to the square of the density of the low energy electrons
including the adiabatic losses ($K_1^2 \propto t^{-6r_d-2r_a(n_1-1)
= -8}$). This finally gives $c_1 = -4$. In the Klein--Nishina regime the IC
radiation is proportional to the evolution of the volume and radius as in the
previous case ($t^{3r_e+r_e=4}$). However, this time the scattering is
proportional to the density of the low energy electrons including the
adiabatic losses ($K_1 \propto t^{-3r_d-r_a(n_1-1)=-4}$, which comes from
the intensity of synchrotron radiation $I^1_s$) and is proportional to
the density of the high energy electrons including the adiabatic losses
($K_2 \propto t^{-3r_d-r_a(n_2-1)=-7}$). As a result, the IC flux in the
Klein--Nishina regime decreases as $t^{-7}$ ($c_2 = -7$). The influence
of the adiabatic losses provides in this test four different correlations
$c_1/s_1 = 4$, $c_1/s_2 = 1$, $c_2/s_1 = 7$, $c_2/s_2 = 1.75$.
}
\item[$\bullet$]{
In our last test ($e$) we assume an increase of the source radius, the decrease
of the particle density with adiabatic cooling and also decay of the magnetic
field ($r_e=r_d=r_a=m=1$). The decrease of the magnetic field causes a much
faster decrease of the synchrotron emission ($t^{-r_a(n_{1|2}-1)-m(\alpha_{1|2}+1)}$,
$s_1 = -2.5$, $s_2 = -7$) in comparison to the previous case. Moreover, this
decrease increases the difference between the synchrotron emission produced by low
and high energy electrons. The IC emission decreases faster as well. However,
the factor ($t^{-m(\alpha_{1}+1) = -1.5}$) which marks the difference to the
evolution discussed in our previous test is the same for the emission in the
Thomson and in the Klein--Nishina regime. This is a consequence of the fact
that the $\gamma$--rays in the Thomson and Klein--Nishina regime are produced
by the scattering of the synchrotron radiation field produced by the low
energy electrons ($I^1_s$). Therefore, the decay of the magnetic field does not
modify the difference between the two scattering regimes.
In conclusion, we have to add the factor --1.5 (with respect to the previous case)
to the indices describing the evolution of the IC emission,
to obtain $c_1 = -5.5$ and $c_2 = -8.5$.
Also in this case we obtain four different values for the correlations $c_1/s_1 =
2.2$, $c_1/s_2 = 0.786$, $c_2/s_1 = 3.4$, $c_2/s_2 = 1.214$.
}
\end{itemize}
For all estimations presented in this subsection we have
assumed constant values for the parameters $n_1$ and $n_2$. However, we have checked also
that for the most realistic scenarios ($e$ \& $l$) and for all combinations of the $n_1$ value
in the range from 1.5 to 2.5 and the $n_2$ value in the range from 3 to 8, the correlation $c_2/s_1$
is always more than quadratic while the correlation $c_2/s_2$ is always less than quadratic.
We provide also a simple mathematical formulae for these realistic evolutions
\begin{displaymath}
\begin{array}{lcccl}
\vspace{0.4cm}
c_2/s_1= & {\frac{\displaystyle 1+3n_1+2n_2}{\displaystyle 3n_1-1}}
& \mathrm{or} &
\frac{\displaystyle 1+2n_1+n_2}{\displaystyle 2n_1} & \textrm{TeV vs soft X--rays} \\
c_2/s_2= & \underbrace{\frac{1+3n_1+2n_2}{3n_2-1}}_{m=1}
& \mathrm{or} &
\underbrace{\frac{1+2n_1+n_2}{2n_2}}_{m=2} & \textrm{TeV vs hard X--rays} \\
\end{array}
\end{displaymath}
\subsection{Problem with ``the quadratic decay''}
\label{sub_quadr-prob}
This concerns the quadratic results of the $c_2/s_2$
correlations during the decay phase of the flare. Within the scenarios proposed in the
previous subsection there is only one solution that may explain exactly the quadratic
correlation (see column marked $c_2/s_2$ in Table~\ref{tab_corr_spher}). However, this well-known
test assumes only variations of the particle density. Therefore, this scenario can be
easily used to explain the rising phase of the flare, where the activity is generated for instance
by the injection of the particles into a constant volume of the source. We cannot easily explain the
decrease of the density during the decay of the flare. In principle such a decrease could
be related to the escape of the particles into a region where the magnetic field intensity
is significantly less than inside the source. However, this process does not guarantee that the
efficiency of the total IC emission will decrease two times faster than the efficiency of the
synchrotron radiation. The IC scattering may also occur efficiently in the region where the
magnetic field is significantly less, especially close to the source surface (distance $\lesssim
1.5R$), where the radiation field energy density is on average only two times less than on
the source surface (Gould \cite{gould79}).
The decrease of the density
could be also related to the expansion of the source. However, assuming the expansion we have to
consider also the influence of some additional physical process. We have to consider
the adiabatic cooling of the particles and probably also the decay of the magnetic field. All
these processes
can destroy the quadratic correlation which is related to the variation of the density.
Each additional physical process which can modify the evolution
of the synchrotron and the IC emission ($F_{s|c} \propto t^{s|c}$) in the same way ($t^{s|c+x}$)
can destroy this correlation.
This is a consequence of the simple fact that if $c/s = 2$ then $(c+x)/(s+x) \ne 2$.
If the influence of the additional process is different for the evolution
of the synchrotron emission ($t^{s+x}$) and for the evolution of the IC radiation ($t^{c+y}$)
then we have situations where $(c+y)/(s+x)$ should give 2 if we want to keep the quadratic
correlation.
Therefore, assuming $s=-1$ and $c=-2$, which is a simple consequence of the decrease
of the density, we obtain a simple relation $y=2x$ which should be fulfilled by the additional
physical process to keep the quadratic correlation.
For example the adiabatic cooling gives
$x=-r_a(n_2-1)$ for the evolution of $F^2_s$ and $y=-r_a(n_1-1)-r_a(n_2-1)$ for the evolution
of $F^{2}_c$. Therefore, the relation $y=2x$ can be fulfilled only if $n_1=n_2$ which does not
provide the correct solution. The efficiency of the synchrotron emission depends among other
things on the slope of the particle spectrum and the intensity of the magnetic field.
Therefore,
the decay of the magnetic field gives $x=-m(\alpha_2+1)$ for the evolution of $F^2_s$ and
$y=-m(\alpha_1+1)$ for the evolution of $F^{2}_c$. Therefore, the relation can be fulfilled
if $\alpha_1 = 2\alpha_2$ which also does not give the correct solution.
Moreover, we have
to stress that the decay of the magnetic field always results in a much faster decay of
the second part of the synchrotron radiation than the decay of the IC emission in the
Klein--Nishina regime and the Thomson limit as well. In principle this unwanted effect can be
compensated by the adiabatic cooling which increases the decrease speed of the $F^{2}_c$
flux in comparison to the decrease of the $F^2_s$ flux. However, detailed
computations show that the adiabatic cooling cannot fully compensate the influence of the decay
of the magnetic field; compare the correlations for the cases $d$ ($r_e=r_d=r_a=1, m=0$),
$e$ ($r_e=r_d=r_a=m=1$ and $l$ ($r_e=r_d=r_a=1, m=2$) in Table \ref{tab_corr_spher}.
\subsection{Change of the Doppler factor}
It is possible that in TeV sources the Doppler factor $\delta$ changes during
a single flare. Evidence for this come from the requirement of a large $\delta$--value
for the inner parts of the jet, generating most of the emission ($\delta \ge 10$, e.g.
Dondi \& Ghisellini \cite{dondi95}, Tavecchio et al. \cite{tavecchio98},
Kino et al. \cite{kino02}, Ghisellini et al. \cite{ghisellini02}, Katarzy\'nski et al.
\cite{katarzynski03}, or even $\delta\sim 50$, Krawczynski \cite{krawczynski02},
Konopelko et al. \cite{konopelko03}), while instead at the VLBI scale ($\sim$sub--pc,
including satellite VSOP data) there is no evidence of superluminal motion of the jet's
components (e.g. Piner et al. \cite{piner99}, Edwards \& Piner \cite{edwards02},
Piner \& Edwards \cite{piner04}, Giroletti et al., \cite{giroletti04}).
This suggests a strong deceleration of the source.
To analyze the impact of the change of the Doppler factor for the observed TeV/X--ray correlation
we assume the following evolution for $\delta(t)$
\begin{equation}
\delta(t) = \delta_0 \left( \frac{t_0}{t} \right)^d,
\end{equation}
where $d$ is a free parameter. The Doppler boosting effect may change the level of the emission
[$F(\nu) \propto \delta^3 F'(\nu')$] and the observed frequency ($\nu \propto \delta \nu'$).
For a power law spectrum ($F' \propto \nu'^{-\alpha}$) we then have
\begin{equation}
F(\nu, t) = \delta(t)^{3+\alpha} F'(\nu,t) \propto \delta_0 t^{-d(3+\alpha)} F'(\nu,t)
\end{equation}
Note that the
evolution of the flux at given frequency depends not only on the value of the Doppler factor,
but also on the spectral index $\alpha$. Therefore,
the evolution of the synchrotron emission can be different below and above the peak
($F^{1|2}_{\rm s} \propto \nu^{-\alpha_{1|2}}$). Also the evolution of the IC radiation
depends on the spectral index ($F^{1|2}_{\rm c} \propto \nu^{-\alpha_{1|{\rm KN}}}$, where
$\alpha_{\rm KN} \sim 2\alpha_2 - \alpha_1$, see Tavecchio et al. \cite{tavecchio98}).
We can describe the IC and synchrotron flux evolution due to the
change of the Doppler factor by the $y$ and $x$ coefficients respectively.
In this way the correlation which includes the change of $\delta$ becomes
$c'/s' = (c+y)/(s+x)$.
The change of $\delta$ will not modify the original ($c/s$) correlation
if $y/x = c/s$.
Values $y/x > c/s$ makes the new correlation $c'/s'$ steeper ($c'/s' > c/s$),
and values of $y/x < c/s$ makes the new correlation $c'/s'$ flatter.
This simple estimate shows that the decrease of $\delta$
can help to solve the problem of the quadratic decay,
but only slightly, since the change of the slope of the correlation
(for actual values of the synchrotron and IC spectral indices)
is only mild.
The coefficient $y$ for the evolution of the IC emission in the Klein--Nishina regime is
given by $y_2 = -3d-2d\alpha_{2}+d\alpha_1$. The parameter $x$ for the evolution of the
synchrotron emission below ($x_1$) and above ($x_2$) the peak is defined by
$ x_{1|2} = -3d-d\alpha_{1|2}$.
Setting $\alpha_1 = 0.5$, $\alpha_2 = 2$, and $\alpha_{\rm KN} = 3.5$,
as in the previous modeling, and assuming $d=1$, we obtain
$y_2/x_1 = 1.857$ and $y_2/x_2 = 1.3$.
For the most realistic evolution scenarios, cases $e$ and $l$ in the Table \ref{tab_corr_spher},
we obtain a decrease of the $c_2/s_1$ correlation slope
($[c_2/s_1 = 3.4 \to c'_2/s'_1 = 2.5]_e$ and
$[c_2/s_1 = 2.5 \to c'_2/s'_1 = 2.2]_l$) and a small increase of the $c_2/s_2$
correlation slope
($[c_2/s_2 = 1.214 \to c'_2/s'_2 = 1.25]_e$ and
$ [c_2/s_2 = 1 \to c'_2/s'_2 = 1.1]_l$).
This simple estimate shows that the impact of the change of the Doppler factor is much
stronger for the $c_2/s_1$ correlation than for the $c_2/s_2$ relation. This process does
not modify significantly the slope of the $c_2/s_2$ correlation, therefore it cannot solve the problem
of the quadratic decay.
\subsection{Emission around the peaks}
\label{sub_peak-corr}
Up to now, we have analyzed the evolution of the synchrotron and the IC
emission in a case were the observed radiation belongs to the spectrum
that can be well approximated by a power law function. In this case we
can well describe the evolution by the simple analytical formulae
derived in the previous subsections. Now we discuss the case where the
emission is observed at the peaks of the $\nu F_s(\nu)$ and $\nu F_c(\nu)$
spectra. In this more complex case, we use the numerical
code to calculate the correlation. Figure \ref{fig_peak-corr} shows an
\begin{figure}[!t]
\resizebox{\hsize}{!}{\includegraphics{1556fig4_c.ps}}
\caption{The upper panel shows the evolution of the SSC emission of an
expanding spherical homogeneous source. The areas indicated by the capital letters
show the spectral bands selected for the calculation of the correlation. The lower
panel shows the two main correlations (bold solid lines with arrows which indicate
the direction of the evolution) calculated for the evolutions presented in the
upper panel. The bold dashed line in the lower panel (A'-Y) shows the correlation
obtained after a small shift of the A--band. Thin dashed lines show a template for the
correlations.
}
\label{fig_peak-corr}
\end{figure}
example of an evolving SSC spectrum during the expansion of homogeneous
source. We use for this test the same set of physical parameters
used for the numerical modeling presented in subsection
\ref{sub_sph-geom}. For each emission process we selected two spectral
bands around the $\nu F(\nu)$ peak, named A ($10^{17}$--$10^{18}$ Hz),
B ($10^{18}$--$10^{19}$ Hz) and X ($10^{26}$--$10^{27}$ Hz),
Y ($10^{27}$--$10^{28}$ Hz). Note that most of the TeV emission is usually
observed in our X--band. Some observations are still possible above the
frequency $10^{27}$ Hz (see Figures \ref{fig_501corr}-d and
\ref{fig_421corr}-d) in our Y--band but currently we are not able to obtain
observations in the whole Y--band. However, the spectral index in the Y--band
is relatively steep. Therefore, radiation observed in this band is dominated
by the emission which comes from the low frequency part of this band which
can be observed. In our test we calculate only two correlations for
above described bands.
The first correlation (A--Y) is calculated between the evolution of the synchrotron emission (A)
in a transition phase (from $F^1_s$ to $F^2_s$) and the evolution of the IC radiation (Y) in
the Klein--Nishina regime ($F^{2}_c$). This gives an almost quadratic correlation.
This particular correlation is a kind of transition between the correlation $c_2/s_1$ (3.4 for this
particular test) and the correlation $c_2/s_2$ (which gives 1.214 for this particular case).
The second correlation (B--X) is calculated between the evolution of the second
part of the synchrotron radiation (B, $F^2_s$) and the evolution of the IC radiation (X) in a
transition phase from the emission in the Thomson limit to the radiation in the Klein--Nishina
regime. The correlation in this particular case is almost linear. This is the result of the
transition from the correlation described by $c_1/s_2$ (equal to 0.786 in this particular case)
and the correlation $c_2/s_2$ (equal 1.214 in this particular case).
The first test presented above shows that a quadratic correlation can be obtained in
some specific cases when we observe the radiation emitted close to the $\nu F(\nu)$
peaks. However, we stress that this result depends strongly on the spectral bands
used. This effect is clearly visible in Figure \ref{fig_peak-corr} where we show the
correlation (A'--Y) after a small shift of the A--band (A' $ \to 2 \times 10^{17}$--$
2 \times 10^{18}$ Hz). The almost quadratic correlation after this shift appears
significantly less than quadratic. Moreover, to obtain the
quadratic correlation we have assumed $m=1$. This means that the decrease of the
magnetic field intensity was slower than we could expect from the rule which
describes conservation of the magnetic flux ($m=2$).
\subsection{Evolution of the $n_2$ parameter}
\label{sub_n2}
In this subsection we analyse the effects related to the change of the
index of the high energy electrons spectrum ($n_2$). The main reason for
this test is to simulate the activity of Mrk~501 observed on April 1997.
The observations obtained at that time by the $Beppo$SAX satellite (Fig.
\ref{fig_501corr}, Pian et. al. \cite{pian98}) indicate that the X--ray
emission was almost stable around the energy 0.1 keV while the emission
around 100 keV was very variable.
To simulate such evolution we change the value of the $n_2$ parameter in
ten steps, assuming $n_2=2.5$ at the beginning of the simulation and
increasing this value by a factor of 0.25 after each step. Note that
the peak of the synchrotron spectrum, at the beginning of the test,
is generated by the highest energy electrons ($\gamma_{\rm max} = 10^7$,
for this test) because $n_2<3$. The other model parameters are the
same as in the previous modeling, except $\gamma_{\rm brk}$ which we
changed to $3 \times 10^5$ in order to obtain the constant emission
around 0.1 keV energy. The radiative cooling is negligible for
the physical parameters used in this simulation. In the next subsection
we discuss in detail this phenomenon and we show that indeed this
process is negligible.
\begin{figure}[!t]
\resizebox{\hsize}{!}{\includegraphics{1556fig5_c.ps}}
\caption{The upper panel shows the evolution of the SSC emission of the source where only
the slope $n_2$ of the high energy part of the electron spectrum was modified
during the simulation. To calculate the correlations presented in the lower panel
we selected the same spectral bands as in the previous modeling. Thin lines in the
lower panel show a template for the correlations.
}
\label{fig_n2}
\end{figure}
In the test we selected the same spectral bands for the calculation
of TeV/X--ray correlation as in the previous modeling and we also
discus three different correlations. The first correlation (A-Y) is
calculated selecting the TeV band decaying relatively fast and the
part of the X--ray spectrum decreasing relatively slowly. At the
beginning of the simulation the slope of the correlation is almost
quadratic and becomes 1.7 towards the end of the simulation.
In the second correlation (B-Y) we selected the parts of the TeV and
X--ray spectra decreasing relatively fast. At the beginning of the
simulation the two fluxes vary almost linearly. However, at the end
of the simulation the correlation slope is close to 0.5. The third
correlation (B-X) is opposite to the first relation (a fast decay of
the X-rays with relatively slow decay of the TeV emission) and gives
the slope close to 0.5 at the beginning of the simulation, changing
rapidly and reaching the value of 0.13 at the end of the simulation.
This simulation can reproduce the activity of Mrk~501 observed in April 1997. The
correlation between the Whipple and OSSE fluxes (Fig. \ref{fig_501corr}) can be explained
as the correlation between the spectral bands where the TeV and X--ray emission evolve
relatively fast (e.g. the beginning of our correlation B-Y). The quadratic correlation between
the Whipple and ASM fluxes can be explained as the correlation between the
part of the TeV band which evolves relatively fast and the part of the X--ray emission
which evolves relatively slowly (e.g. our correlation A-Y). However, the
presented examples depend strongly on the position of the peaks and on the
selected spectral bands.
\subsection{Impact of the radiative cooling}
\label{sub_rad-cool}
In all the scenarios discussed so far we have assumed that the
radiative cooling of the electrons is negligible.
Now we assume that radiative cooling is important and we
discuss its impact for the TeV/X--ray flux correlation.
To calculate the evolution of the electron energy spectrum including the
radiative cooling we use the kinetic equation
\begin{equation}
\frac{\partial N^*(\gamma,t)}{\partial t} = \frac{\partial}{\partial \gamma}
\left\{ \left[ C_{\rm cool}(t) \gamma^2 + C_{\rm adia}(t) \gamma \right] N^*(\gamma,t) \right\},
\label{equ_pde}
\end{equation}
where
\begin{displaymath}
C_{\rm cool}(t) = \frac{4}{3} \frac{\sigma_T}{m_e c} \left[U_B(t) + U_{\rm rad}(t) \right],~~~ U_B(t) = \frac{B(t)^2}{8\pi}
\end{displaymath}
describes the radiative cooling and $C_{\rm adia} = r_a/t$ describes the adiabatic cooling.
For simplicity, we assume that the radiation field energy density ($U_{\rm rad}$)
is equal to the magnetic field energy density ($U_B$).
The initial distribution of the electron energy spectrum is given by a
continuous broken power law
\begin{equation}
N_0(\gamma, t=t_0) = k_1 \gamma^{-n1} \left( 1 + \frac{\gamma}{\gamma^0_{\rm brk}} \right)^{n_1-n_2}.
\label{equ_ini_elec_spec_rad_cool}
\end{equation}
The solution of this kinetic equation is given in Appendix~\ref{app_solution}.
However, this solution must be converted to a unit volume $N(\gamma,t) = N^*(\gamma,t),
\left( t_0/t \right)^{3r_e}$ to be useful for the calculation of the emission coefficient
(e.g.~Kardashev \cite{kardashev62}). Since the evolution of the electron spectrum is
quite complex, we use the numerical code to check the influence of the radiative
cooling on the correlation.
\begin{figure}[t!]
\resizebox{\hsize}{!}{\includegraphics{1556fig6_c.ps}}
\caption{The impact of the radiative cooling for the correlation. Panel (a) shows the evolution
of the high energy part of the electron energy spectrum for the physical parameters
which we have selected for the modeling in the subsection \ref{sub_sph-geom}.
Panel (b) shows the
evolution of the electrons spectrum for the parameters used in the subsection
\ref{sub_rad-cool}. The evolution of the synchrotron and the IC emission is shown in
panel (c). We also show the spectral bands selected for the calculation of
the correlations. Panel (d) shows three different correlations calculated for
the tested evolution.
}
\label{fig_rad-cool}
\end{figure}
The radiative cooling may cause a fast decay of the high energy electrons.
This appears as a cut--off in the high energy part of the electron spectrum.
In Figure \ref{fig_rad-cool} we show two examples for the evolution of the spectrum.
For the first assumed evolution (Fig.~\ref{fig_rad-cool}--a)
the cooling is almost negligible while in the second example (Fig.~\ref{fig_rad-cool}--b)
the cooling significantly modifies the high energy part of the electron spectrum. In the
first test we use the same values of the physical parameters which we used for
numerical calculations presented in the subsections \ref{sub_sph-geom}, \ref{sub_peak-corr}
and \ref{sub_n2}. In the second test we use significantly different values.
We assume a larger initial intensity of the magnetic field ($B_0 = 0.004 \to 0.02$ [G])
to increase the radiative cooling rate. We assume a smaller value of the Doppler factor
($\delta=50 \to 30)$ and smaller initial source volume ($R_0 = 2 \times 10^{16} \to 1.2
\times 10^{16}$ [cm]) to keep the same level of the emission. Moreover, we also modify
the initial density of the particles ($k_1 = 10^{3} \to 3 \times 10^3$ [cm$^{-3}$]).
Figure~\ref{fig_rad-cool}--c shows the evolution of the synchrotron and the IC spectrum
produced by the electron spectrum given in Figure~\ref{fig_rad-cool}--b. The cut--off
in the electron spectrum produces an exponential decay of the synchrotron and the
IC emission. Figure \ref{fig_rad-cool}--c also shows the special bands selected for
the calculation of the correlations. In this test we have introduced one more X--ray
band (C, $10^{19}$ -- $10^{20}$ [Hz]) to check the correlation when using the
exponentially decaying part of the synchrotron emission.
The first correlation (A--Y), calculated for this test (Fig. \ref{fig_rad-cool}--d),
is analogous to the A--Y correlation computed neglecting radiative cooling.
This is the correlation between the X--ray emission in the transition phase
($F^1_s \to F^2_s$) and the IC radiation in the Klein--Nishina regime.
It gives, as in the previous test, an almost quadratic result.
The second correlation B--X was calculated between the second part of the
synchrotron emission ($F^2_s$) and the IC radiation in the Klein--Nishina
regime ($F^{2}_c$, $c_2/s_2$). In principle this correlation should give a
value of the correlation around 1.2 (see Table \ref{tab_corr_spher}). However,
the radiative cooling significantly modifies the evolution of the X--ray spectrum
in the B--band. Thus, the X--ray emission decreases faster than in the evolution
of the power law spectrum discussed in the previous subsection (see Figure
\ref{fig_peak-corr}). Therefore, we obtain an almost square root correlation
in this case. The last correlation (C--X) is the most extreme case where we
compare the evolution of the X--ray emission in the exponential decay part (C) with the evolution
of the IC radiation in the Klein--Nishina regime ($F^{2}_c$).
The exponential decay causes a very fast decrease of the synchrotron emission.
Since this decrease is more than two times faster than the decrease of the IC
radiation, the slope of the correlation is less than a square root.
The correlations presented in this subsection depend strongly on the position
of the spectral bands. Moreover, the radiative cooling which causes the exponential
decay of the spectra can destroy possible quadratic or linear correlations. This is
related to the fact that the synchrotron emission responds immediately to the
change of the high energy part of the electron spectrum while the relevant IC
radiation is generated in the Klein--Nishina regime, mainly by the electrons with
$\gamma$ less than $\gamma_{\rm brk}$ (see Figure \ref{fig_sph-geom}-c-d).
Moreover, the radiation field used in the scattering is not affected by the
radiative cooling. To obtain almost quadratic result for the A-Y correlation
we had to assume $m=1$.
\section{Cylindrical source - pizza or spaghetti}
\label{sec_cyl-src}
So far we investigated the evolution of a spherical source. The geometry of the source is
not important for the synchrotron emission as long as the electron self--absorption is neglected.
On the other hand the geometry of the source may be very important for the IC emission
which depends strongly on the radiation field available for the scattering. If the source is
strongly extended in one dimension (e.g. $x$) then only the local radiation field, proportional
to the other dimensions of the source ($y, z$), is important for the scattering. This effect
is important for the observed correlation especially if the source expands in only one or two
dimensions. To investigate this effect we approximate the emitting region by a
cylindrical geometry, described by the radius ($R$) and the length ($L$) of the cylinder.
We use this specific geometry to investigate two opposite cases. In the first case
we assume $L \ll R$ a ``pizza like'' geometry (hereafter we call this the {\it pizza case}).
In the second case we use the opposite assumption $L \gg R$ a ``spaghetti like'' geometry
(hereafter we call this the {\it spaghetti case}).
The main reason for such assumptions is that we are looking for robust solutions which could
explain a more than linear (or even quadratic) correlation during the decay of a flare.
We also investigate when such solutions are independent of the specific spectral bands
selected for the correlation.
Using the cylindrical geometry we can simulate the expansion of the source in one dimension.
This reduces the influence of the adiabatic cooling and the decay of the
magnetic field. Moreover, with this
geometry we can use a lower value of the Doppler factor (e.g. $\delta \sim 25$) and
we are still able to avoid the impact of the radiative cooling ($B_0 \sim 0.004$ [G]).
In addition, for some specific viewing angles, we can neglect the light crossing time effect
(e.g. a pizza--like source viewed face on, or a spaghetti--like source viewed at
$1/\Gamma$, corresponding to a viewing angle of $90^\circ$ in the comoving frame).
We assume that the radius and the length of the cylindrical source evolve in time as
power law functions
\begin{equation}
R(t) = R_0 \left(\frac{t_0}{t} \right)^{-r_e},~~~~~
L(t) = L_0 \left(\frac{t_0}{t} \right)^{-l_e},
\end{equation}
where $R_0$ and $L_0$ are initial radius and length respectively. This parameterization
gives a volume which evolves in time as $V(t) = \pi R_0^2 L_0 \left(t_0/t\right)^{-E}$,
where $E = (2r_e+l_e)$.
The initial distribution of the electron energy spectrum is assumed to be a broken power
law function as in the previous model (see Eq.~\ref{equ_ini_elec_spec}). The
evolution of the spectrum is given by a minimum of two power law functions as in the
previous modeling (see Eq.~\ref{equ_min_elec_spec}). However, for this modeling the
definition of the power law functions substituted for the minimum is different
according to the different parameterization of the cylindrical geometry. The functions
are given by the formulae
\begin{displaymath}
\begin{array}{ll}
\vspace*{0.1cm}
N_{1|2}(\gamma, t) = K_{1|2}(t) \gamma^{-n_{1|2}}; &~~~~~K_{1|2}(t) = k_{1|2} \left(\frac{t_0}{t} \right)^{D}
\left(\frac{t_0}{t} \right)^{A(n_{1|2}-1)},\\
\end{array}
\end{displaymath}
where $D = (2r_d + l_d)$ describes the evolution of the density and $A = (2r_a + l_a)/3$
describes the adiabatic heating or cooling of the particles.
The evolution of the magnetic field intensity for this model is defined in the same way as in
the previous modeling (see Eq.~\ref{equ_evol_magn}).
The evolution of the synchrotron emission coefficient is calculated for the low ($j^1_s$) and
high energy electrons ($j^2_s$) separately
\begin{equation}
j^{1|2}_s(t) \propto K_{1|2}(t) B(t)^{\alpha_{1|2}+1}
\propto k_{1|2} B_{1|2} \left(\frac{t_0}{t}\right)^{D+A(n_{1|2}-1)+m(\alpha_{1|2}+1)}.
\end{equation}
The evolution of the intensity of the synchrotron emission is calculated along the source length
\begin{equation}
I^{1|2}_s(t) \propto L(t) j^{1|2}_s(t) \propto L_0 k_{1|2} B_{1|2} \left(\frac{t_0}{t}\right)^{-l_e + D+A(n_{1|2}-1)+m(\alpha_{1|2}+1)},
\end{equation}
If we assume that the observer is located outside the source at some distance on the symmetry
axis of the cylinder then the observed flux is given by
\begin{eqnarray}
\vspace*{0.1cm}
F^{1|2}_s(t) & \propto & R^2(t) I^{1|2}_s(t) ~\propto~ R_0^2 L_0 k_{1|2} B_{1|2} \left(\frac{t_0}{t}\right)^{-s_{1|2}},\\
s_{1|2} & = & E - D - A(n_{1|2}-1) - m(\alpha_{1|2}+1).
\end{eqnarray}
The main difference between the previous modeling and the current calculations is
in the IC emission.
The difference comes from the energy density of the synchrotron
emission ($U_{rad}$) which is available for the scattering.
First we describe the radiation in the pizza case where the evolution of the emission
coefficient in the Thomson limit is defined by $j^{1}_{c_p}(t) \propto K_1(t)~I^1_s(t)$.
The evolution of the IC emission coefficient for the Klein--Nishina regime is given by
$j^{2}_{c_p}(t) \propto K_2(t)~I^1_s(t)$.
In this particular pizza case we use for the calculation of the scattering only local
radiation field that is proportional to the length of the source ($U_{rad} \propto I^1_s
\propto j^1_s L$). We assume that the contribution of the synchrotron emission from the
other parts of the source is negligibly small because the radiation field energy density
decreases with the square of the distance. According to this assumption the radiation
field used for the calculation of the IC scattering is isotropic. To obtain the evolution
of the observed flux we have to multiply the evolution of the emission coefficient defined
above by the source volume, which gives
\begin{eqnarray}
\vspace*{0.1cm}
F^{1}_{c_p}(t) & \propto & R_0^2 L_0^2 k^2_1 B_1 \left(\frac{t_0}{t}\right)^{-c_{1P}},\\
c_{1P} & = & 2r_e + 2l_e - 2D - 2A(n_1-1)-m(\alpha_1+1),
\end{eqnarray}
for the Thomson limit and
\begin{eqnarray}
\vspace*{0.1cm}
F^{2}_{c_p}(t) & \propto & R_0^2 L_0^2 k_1 k_2 B_1 \left(\frac{t_0}{t}\right)^{-c_{2P}},\\
\vspace*{0.1cm}
c_{2P} & = & 2r_e + 2l_e - 2D -A(n_1-1)-A(n_2-1)\nonumber\\
& - & m(\alpha_1+1),
\end{eqnarray}
for the Klein--Nishina regime.
In the spaghetti scenario we use only the local radiation field for the calculation of
the scattering. However, now the local radiation field is proportional to the source
radius ($U_{rad} \propto j^1_s R$).
Therefore, the evolution of the emission coefficient in the Thomson limit is given by
$j^{1}_{c_s}(t) \propto K_1(t)~R(t)~j^1_s(t)$.
The evolution of this coefficient for the Klein--Nishina regime is defined by
$j^{2}_{c_s}(t) \propto K_2(t)~R(t)~j^1_s(t)$. Finally, the evolution of the
observed fluxes are calculated in the same way as in the pizza scenario. For
the Thomson limit we have
\begin{eqnarray}
\vspace*{0.1cm}
F^{1}_{c_s}(t) & \propto & R_0^3 L_0 k^2_1 B_1 \left(\frac{t_0}{t}\right)^{-c_{1S}},\\
c_{1S} & = & 3r_e + l_e - 2D - 2A(n_1-1)-m(\alpha_1+1),
\end{eqnarray}
and for the Klein--Nishina regime we have
\begin{eqnarray}
\vspace*{0.1cm}
F^{2}_{c_s}(t) & \propto & R_0^3 L_0 k_1 k_2 B_1 \left(\frac{t_0}{t}\right)^{-c_{2S}},\\
\vspace*{0.1cm}
c_{2S} & = & 3r_e + l_e - 2D -A(n_1-1)-A(n_2-1)\nonumber\\
& - & m(\alpha_1+1).
\end{eqnarray}
\subsection{Specific solutions}
With respect to the spherical geometry, the cylinder is parametrized by two
more free parameters ($L_0$, $l_e$) which give more possible scenarios
for the time evolution of the flux. In this subsection we discuss
only a few possible cases that give directly a quadratic correlation between
the second part of the synchrotron emission ($F^2_s, s_2$) and the IC
radiation in the Klein--Nishina regime ($F^{2}_c, c_2$) during the decay of
a flare.
\begin{itemize}
\item[$\bullet$]{
Consider a pizza--like geometry and only linear increase of the source length
($l_e=1$, $r_e=r_d=r_a=l_d=l_a=m=0$). The increase causes a linear growth
of the synchrotron emission ($F^{1|2}_s \propto t^{l_e}, s_{1|2} = 1$) and
a quadratic ($c_{(1|2)P} = 2$) increase of IC radiation which is proportional
to the increase of the volume and the radiation field as well ($U_{rad}
\propto j_s L \propto t^{l_e}$). Thus in all possible cases we can
obtain the quadratic correlation ($c_{(1|2)P}/s_{1|2} = 2$). The same test
for the spaghetti case gives $c_{(1|2)P}/s_{1|2} = 1$ because the local
radiation field is proportional to the radius and is constant. Note that the
similar test for the spherical geometry ($r_e=1, r_d=r_a=m=0$) gives
$c_{1|2}/s_{1|2} = 1.333$.
}
\item[$\bullet$]{
In the next test we assume a pizza--like geometry with expansion of
the radius, decrease of the density and the adiabatic losses
that correspond to the change of the radius ($r_e=r_d=r_a=1$).
The other parameters are assumed to be constant ($l_e=l_d=l_a=m=0$).
For $r_e=r_d$ the expansion of the volume is compensated entirely by
the decrease of the density. Therefore, the synchrotron emission decreases
only due to the adiabatic cooling ($F^2_s \propto t^{-2r_a(n_2-1)/3},
s_2 = 2.67$). The IC emission in the Klein--Nishina regime depends on the
expanding volume ($t^{2r_e=2}$), the square of the decreasing density
($t^{-4r_d=-4}$) and the adiabatic losses of the low and high energy
electrons ($t^{-2r_a(n_{1|2}-1)/3 = -(0.67|2.67)}$). The combination of
the above processes gives a decrease $F^{2}_c \propto t^{-5.33}$,
which finally provides a quadratic correlation ($c_{2P}/s_2 = 2$).
However, the correlation depends on the value of the $n_1$ and $n_2$
parameter ($n_{1|2} = 2|5$, in the above test). We have checked that
for a given value of the parameter $n_1$ in the range from 1.5 to 2.5
there is always only one value of the index $n_2$ which provides the
quadratic result (e.g. $n_1 = 2.5$ and $n_2=5.5$). Other correlations
for this case, which we do not discuss in detail, are described by
$c_{1P}/s_1=5$, $c_{1P}/s_2=1.25$, $c_{2P}/s_1=8$.
}
\item[$\bullet$]{
The quadratic correlation can be obtained also in a three dimensional
expansion of a pizza--like source ($r_e=r_d=r_a=l_e=l_d=l_a=1$). However,
to get the quadratic result it is necessary to assume a constant
magnetic field ($m=0$) during the expansion and specific values of the
parameters $n_1$ and $n_2$ (e.g. $n_1=2$ and $n_2=4$). Note that similar
three dimensional expansion of the spherical source ($r_e=1=r_d=r_a=1,
m=0$) gives $c_2/s_2 = 1.75$.
}
\item[$\bullet$]{
The spaghetti scenario also provides a quadratic correlation if we assume the expansion of
the radius and the length of the source ($r_e=r_d=r_a=l_e=l_d=l_a=1$). However, also in this
case we have to assume a constant magnetic field ($m=0$) and specific slopes of the electron
spectrum (e.g. $n_1=2$ and $n_2=4$) to get the right correlation.
}
\end{itemize}
We conclude that the cylindrical geometry can provide
some quadratic solutions if we can assume a constant or almost constant value of the magnetic
field during the evolution of the source. Moreover, we have also to assume that the impact
of the radiative cooling is negligible. Otherwise with this particular
geometry the explanation of the quadratic result of the correlation $c_2/s_2$ is rather
problematic.
\section{Light crossing time effects (LCTE)}
\label{sec_lcte}
We distinguish two different types of LCTE. The first effect,
which we call the internal LCTE, is important for the
calculation of the evolution of the radiation field inside the source. This effect is
especially important for precise calculation of the IC emission. The second effect,
which we call the external LCTE, is important for the observer.
We investigate in detail in this subsection this second effect, which seems to
have a stronger impact for the correlation between the TeV and X--ray emission.
Note that recently Sokolov, Marscher \& McHardy (\cite{sokolov04})
proposed a new model to explain the rapid multifrequency variability
of blazars. They investigated in detail the influence of LCTE for the observed
emission. This model provide especially precise description of the IC scattering
in a case where the internal LCTE is important.
To check the influence of the external LCTE for the correlation we have used the model proposed
by Chiaberge $\&$ Ghisellini (\cite{chiaberge99}).
They assumed that the source is created by a shock wave which accelerates the electrons.
The electrons are injected from the shock region into a volume of the jet where
they generate most of the synchrotron and the IC radiation.
The geometry of the emitting region is approximated to be cubic and the region has been
divided into small homogeneous cells.
In figure \ref{fig_lcte} we sketch
the evolution of such a source.
On the right we show the intrinsic evolution.
On the left we show the evolution of the part of the source that can
be observed in the comoving frame at $90^{\circ}$ with respect to the shock velocity.
The number in each cell indicates the age of the particles inside the cell.
If the external LCTE is not important then the observer can
see immediately the changes of the total source structure.
This condition is true for this model if the shock velocity ($\beta_s$)
is significantly less than the speed of light.
Otherwise only part of the source volume can be observed by the observer at given time.
We examine two theoretical scenarios of the source evolution.
In the first we assume that the emission of a single cell is constant in time.
For the second scenario, opposite to the previous one,
we assume a very fast flux decay within a single cell.
\begin{figure}[!t]
\resizebox{\hsize}{!}{\includegraphics{1556fig7_c.ps}}
\caption{The evolution of the source according to the model proposed by Chiaberge $\&$
Ghisellini (\cite{chiaberge99}). The right column shows evolution of the
total volume of the source. The left column shows the cells that
contribute to the source emission observed at given time by the observer located at
$90^{\circ}$ (bottom of the figure) with respect to the shock velocity ($\beta_s$) in the
comoving frame. The number in each cell indicates the age of the particles. Note that
at time $4\Delta t$ the injection process was stopped.
}
\label{fig_lcte}
\end{figure}
\subsection{Constant emission of a single cell}
The assumption of constant emission requires the same age of the particles
in all cells.
This means that in cell presented in Figure \ref{fig_lcte} we should
have the same number--1.
To check the influence of the process for the evolution of the emission
we compare the intrinsic evolution of the source with the observed evolution.
For the intrinsic evolution of the source the constant injection of the cells
gives a linear increase of the synchrotron emission, which is proportional
to the increase of the source volume.
If we neglect the internal LCTE, the IC radiation also will increase proportionally to the
increase of the volume. However, this assumption is not precise and we will discuss the problem
of the internal LCTE at the end of this section. Immediately after the the injection stops
(for example due to destruction of the shock wave) we should observe a constant emission of the
source where the absolute level of the emission is proportional to the created volume.
If we consider the external LCTE then observer at the beginning of the evolution ($t=\Delta t $)
will receive only the emission from the cell which is nearest to him (bottom row of the cells).
At a later time the observer will start to receive successively emission produced by the cells
located at larger distances. If the emission of a single cell is constant in time
(number 1 in all observed cells) then the observed emission increases as a sum
\begin{equation}
\sum_{k=1}^n k = \frac{n(n+1)}{2},
\end{equation}
where $n$ is the number of cells in the bottom row.
Note that this formula is correct only for the source which at the end of the
injection is a square ($n \times n$ cells) and is correct only for the injection phase.
This simple estimation indicates that the observed flux will increase in time
as $t^{\sim 2}$ if $n \gg 10$.
This result is significantly different from the intrinsic evolution where
the increase of the emission in the injection phase was linear.
Therefore, this simple test shows that the external LCTE may
significantly change the evolution of the observed emission.
Note that in this test the level of the emission increases also after the
end of the injection.
However, in this phase the increase is slower
than $t^{\sim 2}$ and after some time ($2 \times l/c$ where $l$ is the
dimension of the square source) the flux level becomes constant,
and equal to the level of the intrinsic emission after the end of the injection.
\subsection{Very fast decay of a single cell}
We now assume that the flux, within a single cell, decays very fast.
The ``life time" of a single cell is $\Delta t$.
If the external LCTE is not important we observe constant emission of the source as long
as the shock is injecting particles.
The total volume of the source during the whole
injection process is represented only by the cells next to the shock front (e.g.
Figure \ref{fig_lcte}-d). The source disappears abruptly after the end of the injection.
When the external LCTE is important, at the beginning of the injection the
observer will see only the radiation produced by the bottom row, reduced in
this particular case to a single cell. At a later time the radiation produced
by the cells at larger distances will arrive at the observer.
This means that the observer will see a linear increase of the emission.
The duration of this increase depends on the number
of cells ($k$) in the column next to the shock front.
After the time $k \times \Delta t$ the
observed emission should be constant.
After the end of the injection we should observe a
linear decrease of the emission with a duration equal to the light crossing
time from the back to the near part of the source.
This is the opposite behavior to the linear increase observed at the beginning
of the simulation.
There are two important conclusions from this test.
The first is that even if
the intrinsic emission of the source is constant for some time at the beginning of the
evolution, the observer will see a linear increase of the radiation.
The second conclusion concerns the decay of the source.
Even if the source ``dies" abruptly in a very short
time, the observer will see a linear decay of the emission for a time corresponding to
the dimension of the source.
The conclusions obtained in this and in the previous subsection
are strictly valid for the specific adopted geometry.
In the next subsection we discuss more possible implications of LCTE.
\subsection{LCTE - general conclusions}
The model used to explore the influence of LCTE for the evolution of the source emission
gives the possibility to investigate only the external LCTE. However, the conclusion
obtained from the simple test which we have performed with this model seems to be quite
strong. If the emission produced by the total volume of the source evolves in time as $t^x$
then the observed emission evolves as $t^{x+1}$.
Besides being confirmed by our simple analytical tests,
this very simple rule is also confirmed by numerical calculations assuming several
different decay rates of a single cell emission.
However, this result is strictly valid only for the proposed geometry, source evolution and
specific position of the observer (at $90^{\circ}$).
The model used
cannot precisely describe the evolution of the radiation field inside the source
(internal LCTE) and cannot describe the emission of the expanding source.
This would require a more sophisticated numerical modeling.
If we assume that the external LCTE indeed increases by one the index which describes the
evolution of the emission, then we can analyze the impact of this process for the observed
correlation.
If for example the correlation between the synchrotron and the IC emission is
intrinsically quadratic then the observed correlation will not be quadratic.
This is a simple
consequence of the rule already discussed in subsection \ref{sub_quadr-prob}
(if $c/s = 2$ then $(c+x)/(s+x) \neq 2$).
On the other hand if we observe a quadratic
correlation it means that the intrinsic change of the IC emission must be much larger
that the change of the synchrotron radiation.
For example we need $F_s \propto t^{s= 1}$
and $F_c \propto t^{s= 3}$ intrinsically to obtain a quadratic correlation after the
basic transformation due to the external LCTE ($(c= 3+1)/(s= 1+1) = 2$). The transformation
indicates that only some specific intrinsic evolutions of the synchrotron and the IC emission
(e.g. $c=1~\&~s=0$, $c=3~\&~s=1$, $c=5~\&~s=2$) may provide a quadratic
correlation.
Similar arguments apply to any observed correlation slope.
The significantly faster evolution of the intrinsic IC flux with respect to the
evolution of the intrinsic synchrotron flux seems difficult to explain.
If we neglect
the internal LCTE, then for the calculation of the IC scattering inside a single cell we
use only the synchrotron radiation field produced within this cell.
With this assumption the evolution of the IC and synchrotron radiation will be exactly
the same ($c=s$).
However this approach is of course not always exact, since electrons
inside a given cell may scatter the seed photons produced also by the surrounding cells.
The cell at the center of the source is the one most affected by this effect.
For such a cell we could have a linear increase of the
radiation field during the injection phase.
The cells affected the least will be the ones at the corners of the source
(for them the effect should be four times less than for the cell in the center).
Therefore, the evolution of the
intrinsic IC emission of the whole source should be more than linear but less than
quadratic ($F_c \propto t^{1~<~c~<~2}$) which after the transformation gives $2<c<3$.
This estimation requires constant intrinsic synchrotron emission to explain a
quadratic correlation in the observer frame.
To conclude:
i) if the intrinsic evolution of the source produces a given
correlation between the synchrotron and IC emission, then the external LCTE yields
a correlation with a different slope;
ii) if we observe a given correlation and the external LCTE is important,
then the intrinsic relative IC/synchrotron change must be much stronger than
what is observed.
\section{Summary and conclusions}
We have presented a detailed study of the expected correlation between
the variations observed in the X-ray and TeV bands in HBL, in the
context of the widely-used homogeneous SSC model. This work has been
stimulated by the observations of both linear and quadratic
correlations in the few cases for which the available data are
suitable for a detailed study of the correlation in single flaring
events (Section \ref{sec_obs-corr}).
First we addressed the problem in the context of the widely used
spherical geometry (Section \ref{sec_spher-src}), presenting analytical
relations valid when the radiative cooling can be neglected and numerical
results valid in general. While a quadratic correlation during the
increasing phase of a flare is easy to reproduce through an increase
in the density (due, for instance, to a continuous injection of new
particles in the emitting volume), we found that the same correlation
observed during the decreasing phase poses a difficult problem. A case
close to a quadratic relation can be obtained only in the rather
physically implausible case of adiabatic expansion and a constant
magnetic field. When the observational bands comprise the peaks of
the SEDs the situation is more complex even if a close-to-quadratic
relation can be obtained, the solution strongly depends on the exact
position of the bands and a small change in the observational limits
inevitably would change the correlation.
The main conclusion that we can derive from the first part of our study
is that in order to to get a quadratic (or, even more general,
more-than-linear) correlation between X-rays and TeV we need rather
special conditions and/or fine-tuning in the temporal evolution of the
physical parameters. For special choices of the spectral bands under
study and of the parameters describing the evolution we can get even
more-than-quadratic relations. In all the cases these solutions are
quite ``delicate'' a small change in the observational band and/or
in the parameters inevitably changes the correlation.
Next (Section \ref{sec_cyl-src}) we investigated the changes related to
a source geometry, assuming that the emitting region is a cylinder.
Synchrotron emission is not affected by the actual shape of the emission
region (as long as absorption effects within the source are negligible).
On the other hand the value of the radiation energy density strongly
depends on the geometry, affecting the SSC emission. We distinguish
between two cases (called pizza or spaghetti) depending on the ratio
between the length and radius of the cylinder ($R/L>1$ or $R/L<1$
respectively). We found that, similar to the spherical case, quadratic
solution can be found only assuming that the magnetic field is not
affected by the expansion. Moreover, radiative cooling can destroy the
correlation.
Although light crossing time effects have not been directly taken into
account in the calculations, we have briefly discussed possible effects
of this phenomena (Section \ref{sec_lcte}). We show that the external
LCTE can ``weaken'' the intrinsic (within the source) correlation.
On the other hand if we observe for example a quadratic correlation and
the external LCTE is important then the intrinsic IC/synchrotron change
must be stronger than quadratic (e.g. $c/s = 3/1$).
As a last step we discussed the possibility that the SSC occurs in the
Thomson regime. This would easily produce a quadratic correlation
between X-rays and TeV, since the seed photons for the IC scattering
are produced by the same electrons. However this would require rather
implausible conditions for the emission region (extreme Doppler
factors, $\delta\sim 1000$, and an extremely compact source, see Appendix
\ref{app_thomson}).
The general conclusion is that if the quadratic correlation during
single flares will be found to be common, the simple homogeneous SSC
model will face a severe problem. As stressed many times,
the main difficulty is to explain the decaying phase, therefore
acceleration processes and/or particular injection mechanisms cannot
contribute to solving the problem.
\begin{acknowledgements}
We are grateful to E. Pian, A. Djannati-Atai, M. Catanese and H.
Krawczynski for the data obtained by the $Beppo$SAX, CAT, Whipple, OSSE,
RXTE-PCA and HEGRA experiments. This article made use of observations
gathered by RXTE/ASM experiment, provided by HEASARC, a service of
NASA/Goddard Space Fight Center. We acknowledge the EC funding under
contract HPRCN-CT-2002-00321 (ENIGMA network).
\end{acknowledgements}
|
2,869,038,155,716 | arxiv | \section{Introduction}
Spaces of genuine Bianchi modular forms have been used Berger, Demb\'el\'e, Pacetti and \c{S}eng\"un
to construct evidence for the Brumer--Kramer paramodularity conjecture and the Eichler--Shimura conjecture~\cite{BergerDembelePacettiSengun}.
This evidence however consists so far in just one Abelian surface
(that is paramodular in the Brumer--Kramer setting;
and they deduce from it an Abelian surface satisfying the Eichler--Shimura conjecture).
No other non-trivial example for the Eichler--Shimura conjecture in dimension $2$
is present in the literature.
This is due to the limitations of the database of genuine weight $2$ Bianchi modular forms that was available to the four above-named authors.
That database~\cite{RahmSengun} did treat only level One Bianchi modular forms,
yielding extremely few genuine forms.
So in order to systematically build up more evidence for the Brumer--Kramer paramodularity conjecture and the Eichler--Shimura conjecture in dimension $2$,
it will not be enough to simply continue on a larger database of level One genuine Bianchi modular forms,
but rather one should use a database of higher level genuine Bianchi modular forms.
The latter type of database is provided with the present paper~\cite{database}.
Again, as for level One, the higher weight spaces of genuine forms are very rare,
but the weight $2$ spaces in our database are more abundant, and give hope for the construction of an algorithm that could systematically produce the desired Abelian surfaces.
In order to find the spaces of genuine Bianchi modular forms in the present database,
a major task was to establish dimension formulas for the non-genuine forms:
For level One, formulas were already established by Finis and Grunewald~\cite{FGT},
yielding the dimensions of Langlands Base-Change forms and Complex Multiplication (CM) forms outside the Base-Change space.
These dimensions of the non-genuine subspace did, in~\cite{RahmSengun},
only need to be subtracted from the full dimension of the Bianchi modular forms space,
which was computed on the machine from the geometry of the Bianchi modular group.
The latter machine computations extend to higher level via the Eckmann--Shapiro lemma,
which is implemented for calculating the cohomology that corresponds to the Bianchi modular forms space through the Eichler--Shimura(--Harder) isomorphism.
The Base-Change dimension formulas however did require a greater effort for their extension to higher level.
What added a particular additional challenge here, is that at higher levels, also twists of Base-Change appear, and have to be taken care of as part of the non-genuine space.
\begin{result}
Let $K$ be an imaginary quadratic field of discriminant $D_K$, odd class number $h_K$
and ring of integers $\mathcal{O}_K$.
Let $N\geq1$ be a square-free integer, coprime to $D_K$.
Then we provide an explicit formula for the dimension of the space $S_k^{\textrm{nG}}(N\mathcal{O}_K)$
of non-genuine modular newforms of level $N\mathcal{O}_K$ and arbitrary weight $k$ in Theorem~\ref{thm:trivial}.
This formula comes with instructions (Section~\ref{Dimension formulas}) on how to evaluate it on a computer.
\end{result}
We also provide such formulas for class number $1$,
$K=\mathbb Q(\sqrt{-p})$ for some prime $p\equiv 3\mod 4$ and level
$\mathfrak{p}$ or $\mathfrak{p}^2$ where $\mathfrak{p}^2=(p)$
(see Section~\ref{level a power of the discriminant}).
But our dimension formulas need to be explicit enough to be evaluable on the machine,
so they involve a lot of case distinctions,
and consequently are provided only for a part of the levels.
The authors hope that it should however become clear for experts how to obtain formulas for the missing levels.
For instance, for some theorems we assume that the class number is odd,
which is equivalent to the discriminant being $-4$ or $-8$
or $-p$, where $p$ is a prime congruent to $3$ modulo $4$.
However, for even discriminants and class numbers, one should proceed analogously (see the remarks in Section~\ref{Final Remarks}).
In conclusion, we can say that the formulas and databases established with the present paper constitute a significant progress on the question raised in section 9.1 of~\cite{BergeronSengunVenkatesh}
on the exhaustion of spaces of newforms by non-genuine forms.
\subsection*{Organization of the paper}
In Section \ref{sec:setting}, we fix the notation and assumptions that we will use throughout the paper.
We also derive a first quantitative expression for the dimension of the Base-Change space.
In Section~\ref{twists}, we take the twists of Base-Change into account.
In Section~\ref{sec:CM}, we establish formulas for the dimension of the space of CM-forms that are not Base-Change.
In Section~\ref{sec:total}, we provide a formula for the total dimension of the space generated by newforms that are non-genuine,
i.e. of any of the three types mentioned above.
This formula is made explicit in Section~\ref{Dimension formulas}, in a way that it can be evaluated on the machine.
We make some concluding remarks on our formulas in Section~\ref{Final Remarks}.
In Section~\ref{sec:computational}, we present our machine results, in which the
non-genuine dimension is subtracted from the full dimension of the newforms space,
the latter being computed from the geometry of the Bianchi modular group.
Due to the sparsity of the genuine forms, those results can be considered as a treasure map to the conditions under which they exist.
\subsection*{Acknowledgements}
This research project was funded by Gabor Wiese's FNR Luxembourg grant INTER/DFG/FNR/12/10/COMFGREP.
We would like to thank John Cremona, Lassina Demb\'el\'e, Aurel Page and Gabor Wiese for helpful discussions.
Our very special thanks go to {M. Haluk \c{S}eng\"un},
for having initiated this research project,
contributed helpful advice and
having designed the MAGMA source code for computing the relevant cohomology of congruence subgroups via a version of the Eckmann-Shapiro lemma.
\newpage
\section{Setting}\label{sec:setting}
Let $D>1$ be a square-free integer and let $K=\mathbb Q(\sqrt{-D})$ and denote its ring of integers by $\mathcal{O}_K$. For any ideal $\mathfrak{n}$ of $\mathcal{O}_K$ and integer $k\geq2$ we let $S_{k,k}^K(\Gamma_0(\mathfrak{n}))$ denote the space of Cuspidal Bianchi modular forms over $K$ of weight $k,k$, level $\mathfrak{n}$ and trivial Nebentypus. This space admits an action of a Hecke algebra generated by operators indexed by the prime ideals of $\mathcal{O}_K$.
Each eigenvector $F$ of this algebra corresponds to an automorphic representation
$\Pi_F$ in $\mathcal{A}_2(K)$, the set of all cuspidal automorphic representations of GL$(2,\mathbb{A}_K)$.
We are interested in the ones appearing in the image of the Base-Change operator
defined by Langlands \cite{Langlands80},
$$\operatorname{\textrm{BC}}_\mathbb Q^K:\mathcal{A}_2(\mathbb Q)\to \mathcal{A}_2(K),$$
along with their twists. Here we generalise the formula of \cite{FGT} to arbitrary level $\mathfrak{n}$ coprime to the discriminant $D_K$ of $K$. More precisely, we provide an algorithm computing the dimension of this space without really computing the spaces in hand. In some simple cases we also provide explicit formulas. We also provide an algorithm for level $(p)=\mathfrak{p}^2$, where $p$ is a rational prime that ramifies in $K$. We hope that our proposed strategy outlines a complete plan for the interested reader to fill out the missing level cases without much effort.
It is clear that without loss of generality it is enough to study the Base-Change image problem on the new subspace of $S_{k,k}^K(\Gamma_0(\mathfrak{n}))$, which we will denote by $S_k(\mathfrak{n})$ in the rest of this paper.
A necessary condition for Base-Change to have non-trivial image is that the level is Galois stable, i.e. an ideal generated by a rational number. Let $\mathfrak{n}=N\mathcal{O}_K$ for some $N\geq 1$ coprime to~$D_K$.
The first crucial observation is that one can consider the Base-Change operator locally. Let $\Pi=\operatorname{\textrm{BC}}_\mathbb Q^K(\pi)$ be an automorphic representation in the Base-Change image.
Then for any prime ideal $\mathfrak{p}$ coprime to~$D_K$,
the conductor of $\Pi_\mathfrak{p}$ is the extension to $\mathcal{O}_K$ of the conductor of $\pi_p$,
where $p$ is the prime below $\mathfrak{p}$.
We therefore get that if $\pi$ contributes to the Base-Change subspace of
$S_k(\mathfrak{n})$, then its conductor away from $D_K$ is exactly $N$.
This leaves us with the primes dividing the discriminant. Let $\mathfrak{p}$ divide $D_K$. Then $\Pi_\mathfrak{p}$ is a principal unramified series, since we required that the level is coprime to the discriminant.
Let $\omega_p$ be the quadratic character of conductor $p$.
A complete list of all the possible local components $\pi_p$ of $\pi$ at $p$ is provided in \cite{FGT} which we also provide here for convenience:
\begin{itemize}
\item unramified Principal Series.
\item $1\oplus\omega_p$.
\item A certain supercuspidal representation ( of conductor $p^2$ when $p>2$).
\end{itemize}
We now have a complete description of the inverse image of the Base-Change operator in terms of its inertial type at the primes $p|ND_K$: If $p|N$, then the type can be anything of conductor equal to the power of $p$ dividing $N$.
And if $p|D_K$, then it is one of the types mentioned above.
In order to compute the dimension of the Base-Change subspace we would have to keep in mind that the Base-Change map is one-to-one when the component at a ramified prime is of the first kind and two-to-one if it is of the other two.
Moreover, if a newform in the preimage spaces has CM by $K$, then its image is Eisenstein, so it does not contribute to the dimension.
The above discussion yields the following formula.
\begin{prop} \label{prop:formula}
Let $S(d)$ be the set of prime divisors of $d$.
Let $\omega_d$ be the quadratic character of conductor $d$, under the convention that $\omega_d$ is trivial at a prime $p|d$ if and only if $p^2||d$.
\\Then we obtain
$$\dim S_k^{\textrm{BC}}(N\mathcal{O}_K) =
\sum_{d|D^2_K} \frac{1}{2^{|S(d)|}}\bigg( \dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d) - \dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)^{\textrm{CM}_K}\bigg),$$
where for each $d$, the space $S_k^{d-\textrm{sc,new}}(\Gamma_0(Nd), \omega_d)$ is the subspace of $S_k^{\textrm{new}}(\Gamma_0(Nd), \omega_d)$
spanned by newforms whose local type at every $p^2||d$ is the supercuspidal representation mentioned above.
And $S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)^{\textrm{CM}_K}$ is the $CM$ subspace.
\end{prop}
Of course many of the spaces appearing on the right hand side are trivial since the parity of their nebentypus does not match that of the weight $k$.
The final key observation is that we have formulas for all of the spaces appearing on the right hand side of the above formula (see \cite{CO} or \cite{FGT}).
\section{Twists of Base-Change}\label{twists}
It is fairly easy to see that it is possible to obtain non Base-Change forms just by twisting a Base-Change one by a suitable non Base-Change character. We would like to count these forms out of our genuine forms subspace and therefore we would like to find the dimension of the subspace they span as well.
Let us denote by $S_k^{\textrm{tBC}}(\mathfrak{n})$ the subspace of $S_k(\mathfrak{n})$ generated by those newforms that are twists of Base-Change.
For this purpose, we say that $f \in S_k(\mathfrak{n})$ is \textit{twist of Base-Change},
if there exists a classical weight $k$ modular form $g$ of level $M \in \mathbb N$ with $M$ dividing $\mathfrak{n}$,
such that $f = \operatorname{\textrm{BC}}_\mathbb Q^K(g) \otimes c$ for some non-trivial character $c$.
We first prove the following:
\begin{thm}\label{thm:twist-2}
Let $K$ be an imaginary quadratic field of discriminant $D_K$ and class number $h_K$.
Let $N\geq1$ be a square-free integer, coprime to $D_K$. Let $\mathfrak{n} = N \mathcal{O}_K$.
Then for all $k\geq 2$, $$\dim S_k^{\textrm{tBC}}(\mathfrak{n})=(h_K -1) \dim S_k^{\textrm{BC}}(\mathfrak{n}).$$
\end{thm}
\begin{proof}
Since we are interested in newforms of square-free level, we get that the local type at any prime $\mathfrak p$ is Steinberg or unramified principal series, depending on whether $\mathfrak p|\mathfrak{n}$ or not. Let $\epsilon$ be a non Base-Change character of $\mathcal{O}_K^*$ and assume it has non-trivial conductor $c(\epsilon)$. Let $\mathfrak p|c(\epsilon)$ be a prime ideal of $K$. Then $f\otimes\epsilon$ has level divisible by $\mathfrak p^2$ and its type at $\mathfrak p$ is $\epsilon\otimes \operatorname{\textrm{St}}$ or $\epsilon\oplus \epsilon$. In both cases, the type comes from Base-Change if and only if $\epsilon$ does. This shows that such an $\epsilon$ cannot be used to twist a non Base-Change form to a Base-Change one and vice versa. The only option left for $\epsilon$ is to be non Base-Change and to have trivial conductor. There precisely $h_k-1$ many such characters. Indeed each such character has the desired property which gives the desired result.
\end{proof}
\section{CM-Forms} \label{sec:CM}
Another subspace we would like to exclude as non genuine is the one generated by CM-newforms. They are the ones whose corresponding automorphic representation occurs as the automorphic induction of a suitable Hecke character of a quadratic extension $M/K$. Let us denote this subspace by $S_k^{\textrm{CM}}(\mathfrak{n})$. It is quite often the case that $S_k^{\textrm{CM}}(\mathfrak{n})\subseteq S_k^{\textrm{BC}}(\mathfrak{n})$, but not always. Nevertheless one can easily prove the following:
\begin{thm}\label{thm:CM}
Let $K$ and $\mathfrak{n}$ be as in Theorem \ref{thm:twist-2}. Moreover assume that $h_K$ is odd. Then for all $k\geq 2$,
$$\dim S_k^{\textrm{CM}}(\mathfrak{n}) = 0.$$
\end{thm}
\begin{proof}
Let $f$ be a newform that supposedly lies in $S_k^{\textrm{CM}}(\mathfrak{n})$. Let $\psi$ be the Hecke character of some extension $M/K$ whose automorphic induction is $f$. We then have that $L(f,s)=L(\psi,s)$. Equating local components at any prime $\mathfrak p$ dividing the discriminant $\mathfrak{d}_{M/K}$ we see that $\mathfrak p^2$ should divide~$\mathfrak{n}$.
Since $\mathfrak{n}$ is assumed to be square-free no such $\mathfrak p$ should exist and thus the extension $M/K$ is unramified.
This provides the desired contradiction since the class number of $K$ is assumed to be odd.
\end{proof}
\section{Total non-genuine subspace} \label{sec:total}
In this section we provide a formula for the total dimension of the space generated by newforms that are "non-genuine" in any of the three notions discussed above, i.e. a formula for
$$\dim \big(S_k^{\textrm{BC}}(\mathfrak{n}) + S_k^{\textrm{tBC}}(\mathfrak{n})+S_k^{\textrm{CM}}(\mathfrak{n})\big).$$
Let us denote this space by $S_k^{\textrm{nG}}(\mathfrak{n})$. Then combining Theorem~\ref{thm:twist-2}, Theorem~\ref{thm:CM} and Proposition~\ref{prop:formula} yields the following:
\begin{thm}\label{thm:trivial}
Let $K$ be an imaginary quadratic field of discriminant $D_K$ and odd class number $h_K$, and $N\geq1$ be a square-free integer, coprime to $D_K$.
Let $\mathfrak{n} = N \mathcal{O}_K$. Then
\begin{equation}\label{eq:main}
\begin{split}
\dim S_k^{\textrm{nG}}(\mathfrak{n}) &= h_K \dim S_k^{\textrm{BC}}(\mathfrak{n})\\
&= h_K\sum_{d|D^2_K} \frac{1}{2^{|S(d)|}}\bigg(\dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d) - \dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)^{\textrm{CM}_K}\bigg).
\end{split}
\end{equation}
\end{thm}
\section{Dimension formulas} \label{Dimension formulas}
In this section we provide explicit formulas/algorithm for the right hand side of the formula in Theorem \ref{thm:trivial}. We split the computation into two parts, each involving one of the two main ingredients of each summand in the right hand side of (\ref{eq:main}).
\subsection{Computing $\dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)$}
Our starting point is Proposition 4.18 in \cite{FGT} which we provide here for convenience.
Here, we denote by $S_k(\Gamma(N))$ the space of weight $k$ elliptic modular forms
for the principal congruence subgroup $\Gamma(N) \subseteq$ SL$(2,\mathbb Z)$ of level $N$.
\begin{prop}[Finis, Grunewald, Tirao]\label{prop:key}
Let $N \geq 1$ and $k \geq 2$ be integers and $\sigma$ a representation of $G_N = {\rm SL}(2,\mathbb Z/N\mathbb Z)$
such that $\sigma(-I_2)$ is the scalar $(-1)^k$. Let $U_N \subseteq G_N$ be the subgroup of all upper triangular unipotent elements and $S_3$ and $S_4$ the images in $G_N$ of elements of
${\rm SL}(2,\mathbb Z)$ of order $3$ and $4$, respectively. Then,
$$\dim \Hom_{G_N}(S_k(\Gamma(N)), \sigma) = \frac{k-1}{12}\dim\sigma - \frac{1}{2}\dim\sigma^{U_N} + \epsilon_k\Tr\sigma(S_3) + \mu_k \Tr\sigma(S_4) + \delta_{k,2}\dim \sigma^{G_N}.$$
\end{prop}
The constants $\epsilon_k$ and $\mu_k$ are explicit functions of $k$ and $\delta_{k,2}$ is the usual Kronecker delta notation.
In what follows, given a subspace of newforms $B\subseteq S_k(\Gamma(N))$, we will say that $\sigma$ \emph{defines} $B$ if $\Hom_{G_N}(S_k(\Gamma(N)), \sigma) \cong B$. We thus need to identify a suitable $\sigma$ for each subspace involved in the last line of (\ref{eq:main}). In fact we only need to compute the following five invariants associated to such
a sigma:
$$\dim \sigma,\ \dim\sigma^{U_N},\ \Tr\sigma(S_3),\ \Tr\sigma(S_4),\ \dim \sigma^{G_N}.$$
We will denote them by $I_i(\sigma)$, $i\in\{1,2,3,4,5\}$ respectively. It is important to notice that $\sigma$, and therefore the five invariants associated with it, depends only on the level structure of the
subspace that is of interest to us and not the weight. It is also clear that the following properties hold:
$$I_i(\sigma\oplus\sigma')=I_i(\sigma) + I_i(\sigma')$$
and
$$I_i(\otimes_p\sigma_p)=\prod_pI_i(\sigma_p)$$
for all $\sigma=\otimes_p\sigma_p$, $\sigma'$ and for all $i\in\{1,2,3,4,5\}$.
Let us fix one of the spaces in the right hand side of the formula in Theorem \ref{thm:trivial}, $S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)$ say, and let $\sigma$ be the representation defining it. In view of the properties just mentioned, we will compute the $I_i(\sigma)$ by determining the $N$-part and the $d$-part separately.
In order to compute the $N$-part, it is enough to notice that it corresponds to the one defining $S_k^{\textrm{new}}(\Gamma_0(N))$. Given an integer $N\geq1$, let $\sigma^{N}=\otimes_p \sigma^{N}_p$ and $\sigma^{N,\textrm{new}}=\otimes_p \sigma^{N,\textrm{new}}_p$ be the representation of $G_N$ defining $S_k(\Gamma_0(N))$ and $S_k^{\textrm{new}}(\Gamma_0(N))$ respectively. It is immediate then that $\sigma^{N}_p=\sigma^{p^e}$ and $\sigma^{N,\textrm{new}}_p=\sigma^{p^e,\textrm{new}}$, where $p^e||N$. Moreover it is easy to see that
$$\dim S_k^{\textrm{new}}(\Gamma_0(p^e)) = \dim S_k(\Gamma_0(p^e)) - 2\dim S_k(\Gamma_0(p^{e-1})) + \dim S_k(\Gamma_0(p^{e-2})),$$
for any $e\geq 0$, with the understanding that the dimensions mentioned are $0$ if the exponent of $p$ becomes negative. This in turn implies the following formula:
\begin{equation}\label{eq:newsubdim}
I_i(\sigma^{p^e,\textrm{new}}) = I_i(\sigma^{p^e}) - 2I_i(\sigma^{p^{e-1}}) + I_i(\sigma^{p^{e-2}}).
\end{equation}
As before, the $I_i$'s involved are $0$ if the corresponding exponent of $p$ is negative. Formulas for the right hand side are provided in \cite{CO} and we also state them here for convenience:
\begin{align*}
I_1(\sigma^{p^e}) & = \begin{cases}
1 & e=0\\
p^{e-1}(p+1) & e\geq 1\\
\end{cases}\\
I_2(\sigma^{p^e}) & = \lambda(e,0,p) = \begin{cases}
1 &e=0\\
2p^n &e=2n+1\\
p^n+p^{n-1} &e=2n\geq2\\
\end{cases}\\
I_3(\sigma^{p^e}) & = \#\{x \mod p^e| x^2+x+1=0\}=\begin{cases}
1 &e=0\textrm{ or }p^e=3\\
1 + \Big(\frac{-3}{p}\Big) &e\geq 1\textrm{ and }p\neq3\\
0 &e\geq2\textrm{ and }p=3\\
\end{cases}\\
I_4(\sigma^{p^e}) & = \#\{x \mod p^e| x^2+1=0\}=\begin{cases}
1 & e=0\textrm{ or }p^e=2\\
1 + \Big(\frac{-1}{p}\Big) &e\geq1\textrm{ and } p\neq 2\\
0 &e>1\textrm{ and } p=2\\
\end{cases}\\
I_5(\sigma^{p^e}) & = 1
\end{align*}
where $\lambda(sr_p, s_p, p)$ is the one defined in \cite{CO}. Using (\ref{eq:newsubdim}) and the above one easily gets:
\begin{align}
I_1(\sigma^{p^e,\textrm{new}}) &= \begin{cases}
1 & e= 0\\
p-1 & e = 1\\
p^2-p-1 & e = 2\\
p^{e-3}(p-1)^2(p+1) & e \geq 3\\
\end{cases}\label{eq:I1new}\\
I_2(\sigma^{p^e,\textrm{new}}) & = \begin{cases}
1 & e= 0\\
0 & e=2n+1\\
p-2 & e= 2\\
p^{n-2}(p-1)^2 & e= 2n\geq 4\\
\end{cases}\\
I_3(\sigma^{p^e,\textrm{new}}) &= \begin{cases}
1 & e= 0\textrm{ or }p^e=3^3\\
\Big(\frac{-3}{p}\Big)-1 & e=1\textrm{ and }p\neq3\\
-\Big(\frac{-3}{p}\Big) & e=2\textrm{ and }p\neq3\\
-1 &p^e=3\textrm{ or }3^2\\
0 &\textrm{otherwise}\\
\end{cases}
\end{align}
\begin{align}
I_4(\sigma^{p^e,\textrm{new}}) &= \begin{cases}
1 & e= 0\textrm{ or }p^e=2^3\\
\Big(\frac{-1}{p}\Big)-1 & e=1\textrm{ and }p\neq2\\
-\Big(\frac{-1}{p}\Big) & e=2\textrm{ and }p\neq2\\
-1 &p^e=2\textrm{ or }2^2\\
0 &\textrm{otherwise}\\
\end{cases}\\
I_5(\sigma^{p^e,\textrm{new}}) &= \begin{cases}
1 & e= 0\\
-1 & e=1\\
0 & e\geq 2\\
\end{cases}\label{eq:I5new}
\end{align}
Let $\tau^d$, where $d|D_K^2$, be the $d$-part of the representation $D_K$. The $I_i$'s for each $\tau^d_\ell$, with $\ell|d$, have already been determined in \cite{FGT}. We then have that $\sigma = \sigma^N\otimes \tau^d$ defines $S_k(\Gamma_0(Nd), \omega_d)$. Moreover $I_i(\sigma^N\otimes \tau^d) = I_i(\sigma^N)I_i(\tau^d)$ and we have explicit formulas for both terms in the right hand side.
$$\dim S_k(\Gamma_0(Nd), \omega_d) = \frac{k-1}{12}I_1(\sigma^N)I_1(\tau^d)+\cdots.$$
If one lets $I_i(\sigma_p)=1$ for all $i$ and all $p\nmid D_K$ then one gets the formulas derived in \cite{FGT}.
\subsection{Computing $\dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)^{\textrm{CM}_K}$}
We follow the method used in \cite{Tsaknias2012a} to count CM newforms of a given level, weight and Nebentypus. As explained there, every newform has associated to it a pair of characters $(\psi_f, \psi_\infty)$ that satisfies a certain compatibility condition. Moreover, for every such compatible pair, there are $h_K$ many newforms associated with it. The problem is thus reduced to counting compatible pairs that match the prescribed level, weight and Nebentypus. The infinity component is always uniquely determined by the weight, so we only have to count all the $\psi_f$'s that correspond to the given level and Nebentypus and are compatible with the given weight. Finally every such $\psi_f$ is determined by its local components at every prime $\mathfrak p$ of $\mathcal{O}_K$, so we count the possible choices for each of those and multiply them to get the final answer.
Recall that $Nd=\Nm(\mathfrak{m}) |D_K|$, where $\mathfrak{m}$ is the conductor of $\psi_f$. Since $N$ and $D_K$ are coprime, every prime dividing $N$ should divide $\Nm(\mathfrak{m})$ too. Moreover, a prime $\ell|d$ divides $\Nm(\mathfrak{m})$ if and only if $\ell^2|D_K$. We will treat primes $\mathfrak p$ separately, depending on their ramification and residue class degree, $e_\mathfrak p$ and $f_\mathfrak p$ respectively. We start with primes $p$ whose residue characteristic is greater than $2$.
\subsubsection{p inert} In this case $p=\mathfrak p$ and $\Nm(\mathfrak p)=p^2$, so the exponent of $p$ in $N$ should be even. Assume that this is the case: $p^{2n}||N$, for some $n\geq1$. We then get that the conductor of $\psi_f$ has to be divisible exactly by $\mathfrak p^n$. As we can see in \cite{Ranum1910}, $(\mathcal{O}/\mathfrak p^n)^*$ is generated by three independent generators: $\xi$, $1+p$ and $1+p\omega$ of order $p^2-1$, $p^{n-1}$ and $p^{n-1}$ respectively. The residues of rational integers form a subgroup isomorphic to $(\mathbb Z/p^n\mathbb Z)^*$ which is generated by $\xi^{p+1}$ and $1+p$.Since the restriction of $\psi_f$ on $\mathbb Z^*$ is predetermined by the nebentypus
we have unique choices for $\xi^{p+1}$ and $1+p$. In our case the nebentypus is trivial and therefore $\psi_f(\xi^{p+1})=1$ and $\psi_f(1+p)=1$. If $n=1$, then the only generator to consider is $\xi$ and since we need the conductor at $\mathfrak p$ to be $\mathfrak p$ we have to exclude $1$ from the possible values of $\psi_f(\xi)$, which leaves $p$ choices in total. If however $n>1$ then in order for the conductor at $\mathfrak p$ to be $\mathfrak p^n$ we need either $\psi_f(1+p)$ or $\psi_f(1+p\omega)$ to be a primitive $p^{n-1}$-th root of unity. Since $\psi_f(1+p)=1$ we get that $\psi_f(1+p\omega)$ must satisfy this condition which leaves $\varphi(p^{n-1})=p^{n-2}(p-1)$ choices. In this case all $p+1$ choices for $\psi_f(\xi)$
are permitted, so we get in total $p^{n-2}(p^2-1)$ many choices.
\subsubsection{$p$ split} In this case $p=\mathfrak p\bar\mathfrak p$ and $\Nm(\mathfrak p)=\Nm(\bar\mathfrak p)=p$. Let $p^t||N$ for some $t\geq1$. Then the $p$-part of the conductor of $\psi_f$ is apriori of the form $\mathfrak p^\alpha\bar\mathfrak p^\beta$ for any $0\leq\alpha,\beta\leq t$ such that $\alpha+\beta=t$. We will first show that in our case $\alpha=\beta=n\geq1$ and therefore $t$ must be even. We have the following group homomorphisms:
$$(\mathbb Z/p^{\max\{a,b\}}\mathbb Z)^*\hookrightarrow (\mathcal{O}_K/\mathfrak p^\alpha\bar\mathfrak p^\beta)^*$$
and
$$(\mathcal{O}_K/\mathfrak p^\alpha\bar\mathfrak p^\beta)^*\cong (\mathcal{O}_K/\mathfrak p^\alpha)^* \times (\mathcal{O}_K/\bar\mathfrak p^\beta)^*.$$
Notice that $1+p$ will map to $(1+p, 1+p)$ after composing the two maps above. Since we assume that the $p$-part of the conductor of $\psi_f$ is $\mathfrak p^\alpha\bar\mathfrak p^\beta$, we get that $\psi_f(1+p)$ must be a primitive $p^{\alpha-1}$-th root of unity, as well as a $p^{\beta-1}$-th one. This can of course only happen if $\alpha=\beta=n$.
Let's go back to counting all possible characters of $O_K^*$ of conductor $\mathfrak p^n\bar\mathfrak p^n$. Having in mind the isomorphism above, any such character is completely determined by the images of the generators of $ (\mathcal{O}_K/\mathfrak p^\alpha)^*$ and $(\mathcal{O}_K/\bar\mathfrak p^\beta)^*$. Wlog we can pick $1+p$ and $\delta$, of order $p^{n-1}$ and $p-1$ respectively, to generate both groups. Like before, the restriction of $\psi_f$ to $\mathbb Z$ is determined by the nebentypus. For the $p$-part we have that the two are in fact equal and since the nebentypus has trivial $p$-part we get the same for the $p$-part of $\psi_f|\mathbb Z$. This means that $(1+p, 1+p)$ and $(\delta, \delta)$ should map to $1$ and we therefore have that
$$\psi_f((1+p,1))=\psi_f((1,1+p))^{-1}$$
and
$$\psi_f((\delta,1))=\psi_f((1,\delta))^{-1}.$$
Apriori, $\psi_f((\delta,1))$ has $p-1$ choices. If however $n=1$, then this $\delta$ is the only generator and if it has trivial image, the conductor then becomes lower, which leaves $p-2$ choices. If $n>1$, the conductor condition is satisfied by restricting $\psi_f((1+p,1))$ to be a primitive $p^{n-1}$-th root of unity. This gives $p-1$ choices for the image of $\delta$ and $\varphi(p^{n-1})=p^{n-2}(p-1)$ many choices for that of $1+p$.
\subsubsection{p ramified} In this case $p=\mathfrak p^2$ and $\Nm(\mathfrak p)=p$. Assume that $p^t||d$ (remember that $N$ and $D_K$ are coprime). Assuming that $p^u||D_K$ (and since $p$ is ramified we also have that $u\geq1$), we get that $p^{t-u}||\Nm(\mathfrak{m})$ so the $p$-part of the conductor of $\psi_f$ should be $\mathfrak p^{t-u}$. If $t=1$, we clearly have a unique choice for the $p$-part of $\psi_f$ which happens to match the Nebentypus too. If $t=2$, we are looking for non-trivial characters of $(\mathcal{O}_K/\mathfrak p)^*$, which are apriori $p-2$ many. The nebentypus condition determines these characters uniquely on $(\mathbb Z/p\mathbb Z)^*$, which happens to be isomorphic to $(\mathcal{O}_K/\mathfrak p)^*$. The unique character that is left is actually non-trivial so we have a unique choice.
Summarizing all of the above:
$$CM(p^t)=\begin{cases}
1&t=0\\
1&t=1\textrm{ and $p$ ramified}\\
0&t=1\textrm{ and $p$ unramified}\\
1&t=2\textrm{ and $p$ ramified}\\
p-2&t=2\textrm{ and $p$ split}\\
p&t=2\textrm{ and $p$ inert}\\
0&t=2n+1\geq3\\
p^n(p-1)^2&t=2n\geq4\textrm{ and $p$ split}\\
p^n(p^2-1)&t=2n\geq4\textrm{ and $p$ inert}
\end{cases}$$
Recall that it is easy to determine whether a prime is split, inert or ramified in $K$ simply computing $\Bigg(\frac{D_K}{p}\Bigg)$. Putting everything together we get:
\begin{equation}\label{eq:CMEisdim}
\dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)^{\textrm{CM}_K} = \prod_{p^t||Nd}CM(p^t).
\end{equation}
Notice that the formula above works for any $N$ coprime to $D_K$, not just square-free. Specializing to square-free $N$ we get:
$$\dim S_k^{d\textrm{-sc,new}}(\Gamma_0(Nd), \omega_d)^{\textrm{CM}_K}=\begin{cases}
1 & N = 1 \textrm{ and } \textrm{rad}(D_K)|d,\\
0 & \textrm{otherwise}.
\end{cases}$$
where $\textrm{rad}(D_K)$ is the product of all primes dividing $D_K$.
Hopefully it should be obvious by now that we have sketched an algorithm that computes $\dim S_k^{\textrm{nG}}(\mathfrak{n})$ in an elementary way.
\subsection{The case $K=\mathbb Q(\sqrt{-p})$, $p\equiv3\mod 4$ and $\mathfrak{n}=(p)$.} \label{level a power of the discriminant}
Let $p\equiv3\mod 4$ be a prime number, and $(p) = \mathfrak{p}^2$ in $\mathcal{O}_K$
for $K=\mathbb Q(\sqrt{-p})$.
In this section, we essentially describe all the ingredients for a dimension formula in the cases where the level $\mathfrak{n}$ is $\mathfrak{p}$ or $\mathfrak{p}^2$.
For this purpose, we determine the parameters $I_i(\sigma_p)$ for all the suitable $\sigma_p$ in these two cases,
further the twists of Base-Change, and show that there is no CM outside Base-Change.
For simplicity, we restrict ourselves to the case where the class number $h_K$ of $K$ is 1.
Let us begin with $\mathfrak{n}=\mathfrak{p}$. In this case, the only possible type over $K$ of level $\mathfrak{p}$ is unramified Steinberg.
It is fairly easy to see that the only type over $\mathbb Q$ that can base-change to this is unramified Steinberg at $p$. This space over $\mathbb Q$ coincides with the new $\Gamma_0(p)$ space and the $\sigma$-parameters can be computed using (\ref{eq:I1new}) - (\ref{eq:I5new}):
$$I_1 = p-1, I_2 = 0, I_3= \Big(\frac{-3}{p}\Big) -1, I_4= -2, I_5 = -1.$$
Notice that the new $\Gamma_0(p)$ space over $\mathbb Q$ contains no CM forms.
Using the same arguments as in Theorem \ref{thm:twist-2} we see that there are no twists of Base-Change in $\Gamma_0(\mathfrak{p})$. We also claim that there are no CM forms: Indeed, assume there exists one of level $\Gamma_0(\mathfrak{p})$, $f$ say.
Then $f$ is automorphic induction of $\psi$ from $L$ to $K$,
where $L/K$ is a quadratic extension and $\psi$ is a Hecke character over~$L$.
Since the L-series for $f$ and $\psi$ must be the same, we get that $L$ must be ramified at $\mathfrak{p}$ only. This is absurd since class field theory for $K$ tells us that it does not have any even degree abelian extensions ramified only at $\mathfrak{p}$. Summing everything up:
$$\dim S_k^{\textrm{nG}}(\mathfrak{p}) = \dim S_k^{\textrm{new}}(\Gamma_0(p)).$$
One can then use the $I_i$ parameters given above or use classical dimension formulas to compute the right hand side.
We move to the $\mathfrak{n}=\mathfrak{p}^2=(p)$ case. Let $\omega_\mathfrak{p}$ be the quadratic character of conductor $\mathfrak{p}$ and recall that $\omega_p$ is the quadratic character of conductor $p$. The following list summarizes the possible types over $K$ of level $\mathfrak{p}^2$ and for each one gives the possible types over $\mathbb Q$
that base-change to them:
\begin{itemize}
\item All the spaces listed in the trivial level case: After twisting their Base-Change by a quadratic character of conductor $\mathfrak{p}$ they become of level $\mathfrak{p}^2$.
The type at $\mathfrak{p}$ after Base-Change is again Principal Series, $I(\omega_\mathfrak{p}, \omega_\mathfrak{p})$.
\item Principal Series $I(\chi, \chi^{-1})$, where $\chi$ is of conductor $p$, non quadratic. Their Base-Change (or their twist by a suitable character of conductor $\mathfrak{p}$) becomes a newform of level $\mathfrak{p}^2$ and trivial nebentypus. The type at $\mathfrak{p}$
after Base-Change is again Principal Series, $I(\eta, \eta^{-1})$ with $\eta\neq \omega_\mathfrak{p}$.
\item Unramified Steinberg Series $\operatorname{\textrm{St}}(p)$. As mentioned in the level $\mathfrak{n}=\mathfrak{p}$ case, their Base-Change image are the Unramified Steinberg Series $\operatorname{\textrm{St}}(\mathfrak{p})$ over $K$. After twisting by a quadratic character of conductor $\mathfrak{p}$ they become newforms of level $\mathfrak{p}^2$ and trivial nebentypus. The type at $\mathfrak{p}$ after Base-Change is ramified Steinberg, $\omega_\mathfrak{p}\otimes\operatorname{\textrm{St}}(\mathfrak{p})$.
\item Finally all Supercuspidal series of level $p^2$ that are not part of the third case described in the trivial level situation. Their Base-Change image (and their twists by $\omega_\mathfrak{p}$) are Supercuspidal Series of level $\mathfrak{p}^2$ and trivial nebentypus. The type at $\mathfrak{p}$ after Base-Change is again Supercuspidal Series.
\end{itemize}
These four components comprise all the possible Base-Change as well as twists of it that can occur for level $\mathfrak{p}^2$ and trivial nebentypus. Table \ref{tbl:psquaredtypes} provides the formulas to compute the parameters $I_i(\sigma)$ in each case, which one needs in order to use Proposition \ref{prop:key} to compute the corresponding dimensions. Here $SC_3(p)$ and $SC_4(p)$ are the functions $\Tr S_3(p)$ and $\Tr S_4(p)$ respectively that are defined in the statement of Lemma 4.19 of \cite{FGT}, and $\textrm{CPS}(p)$ is defines as follows:
$$\textrm{CPS}(p)=\begin{cases}
-2 &p\equiv 1 \mod 3\\
0 & \textrm{otherwise}\\
\end{cases}
$$
\begin{table}
\small
\begin{tabular}{|l||r|r|r|r|r|}
\hline
Type &dim $\sigma$ &dim $\sigma^{U_N}$ &tr$S_3$ &tr$S_4$ &dim $\sigma^{G_N}$ \\
\hline
$I(\omega_{\mathfrak{p}}, \omega_{\mathfrak{p}})$ &$\frac{p+1}{2}$ &$p-3$ &$1+\frac{1}{2}\textrm{SC}_3(p)$ &$0$ &$0$ \\
$I(\eta, \eta^{-1}), \eta \neq \omega_{\mathfrak{p}}$ &$\frac{(p-3)(p+1)}{2}$ &$1+h_K$ &$\textrm{CPS}(p)$ &$1+\frac{1}{2}\textrm{SC}_4(p)$ &$0$ \\
$\omega_{\mathfrak{p}}\otimes \operatorname{\textrm{St}}(\mathfrak{p})$ &$p-1$ &$0$ &$\Bigg(\frac{-3}{p}\Bigg)-1$ &$\Bigg(\frac{-1}{p}\Bigg)-1$ &$-1$ \\
Supercuspidal &$\frac{(p-3)(p-1)}{2}$ &$p-2+h_K$ &$-2\Bigg(\frac{-3}{p}\Bigg)-\textrm{CPS}(p)-\textrm{SC}_3(p)$ &$-1-\frac{1}{2}\Bigg(\frac{-1}{p}\Bigg)$ &$0$ \\
\hline
\end{tabular}
\normalsize
\caption{Parameter values for the types contributing to Base-Change of level $\mathfrak{p}^2$}\label{tbl:psquaredtypes}
\end{table}
Finally we need to account for any CM forms not already present in the Base-Change subspace. In fact one can easily see that there are none. The argument is almost identical to the one given for the level $\mathfrak{p}$ case: The existence of any such CM form would imply the existence of a quadratic extension of $K$ ramified only at $\mathfrak{p}$ but no such extension exists.
At this point, anyone wanting to compute $\dim S_k^{\textrm{nG}}(\mathfrak{p}^2)$ has everything needed to do so.
\section{Final Remarks} \label{Final Remarks}
We would like to sum up here the cases for which we provide a complete answer:
\begin{itemize}
\item Odd Class Number, square-free level, coprime to the discriminant, trivial nebentypus.
\item Class Number $1$, $K=\mathbb Q(\sqrt{-p})$ for some prime $p\equiv 3\mod 4$ and level $\mathfrak{p}$ or $\mathfrak{p}^2$ where $\mathfrak{p}^2=(p)$.
\end{itemize}
Many of our arguments however provide partial answers to a broader range of cases:
\begin{itemize}
\item We have a complete description in (\ref{eq:I1new}) - (\ref{eq:I5new}) of the $\sigma$ parameters away from the discriminant as long as the nebentypus is trivial for any level, not just square-free ones. The generalization to non-trivial nebentypus should be quite straightforward but even more cumbersome to write down as single formulas.
\item For primes $p$ dividing the discriminant, the $0$ exponent case is the one treated in \cite{FGT} and we provide an answer for exponents $1$ and $2$ and trivial nebentypus in the case $p\equiv3\mod 4$.
\item Theorem \ref{thm:twist-2} applies to imaginary quadratic fields of even class number too, not only to the ones with an odd class number.
\item Finally the formula in (\ref{eq:CMEisdim}) allows one to compute the dimension of the classical newforms that base-change to Eisenstein forms for any $N$ coprime to the discriminant and trivial nebentypus, not just for the square-free levels.
\end{itemize}
\section{Computational results for the spaces of genuine forms} \label{sec:computational}
The authors have used the software \textit{Bianchi.gp} (\cite{Rahm11}, \cite{Rahm13})
to compute the necessary geometric-topological information about the whole Bianchi group,
and then applied a MAGMA \cite{magma} implementation by \c{S}eng\"un
(for which we provide the algorithm in \cite{database})
to deduce the dimension of the relevant cohomology space for the congruence subgroup at level $\mathfrak{n}$,
passing by the Eckmann--Shapiro lemma.
They have then substracted the Eisenstein series space to get the cuspidal cohomology space,
which by the Eichler--Shimura(--Harder) isomorphism yields the cuspidal forms space.
Then they did substract the oldforms using a well-known recursive formula,
to get the dimensions of the newforms spaces $S_k(\mathfrak{n})$ defined in Section~\ref{sec:setting}.
\subsection{Level One}
We recall here a dimension computation of 4986 different spaces of cuspidal newforms at level $1$,
at varying discriminant $D$ (over 186 different imaginary quadratic fields) and varying weight $k+2$.
The precise scope of our computations is given in~\cite{RahmSengun}.
In only 22 of these spaces were we able to observe genuine forms.
The precise data about these exceptional cases is provided in Table~\ref{exceptional}.
We note that in \cite{RahmSengun},
some further subspaces are tabulated, which are in fact populated by CM-forms (arising through automorphic induction).
We need this table in order to spot the twists of genuine level One forms to deeper levels.
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
$|D|$ &{\bf 7} &{\bf 11} &{\bf 71} &{\bf 87} &{\bf 91} &{\bf 155} &{\bf 199} &{\bf 223} &{\bf 231} &{\bf 339} &{\bf 344}\\ \hline
$k+2$ &14 &12 &3 &4 &8 &6 &3 &2 &6 &3 &3 \\ \hline
$\dim$ &2 &2 &2 &2 &2 &2 &4 &2 &2 &2 &2 \\ \hline
\multicolumn{11}{c}{}\\ \hline
$|D|$ &{\bf 407} &{\bf 415} &{\bf 455} &{\bf 483} &{\bf 571} &{\bf 571} &{\bf 643} &{\bf 760} &{\bf 1003} &{\bf 1003 } &{\bf 1051} \\ \hline
$k+2$ &2 &2 &2 &3 &2 &3 &2 &4 &2 &3 &2 \\ \hline
$\dim$ &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 \\ \hline
\end{tabular}
\caption{Level One cases where there are genuine classes}\label{exceptional}
\end{table}
\subsection{Level: the square of minus the discriminant}
We have compared the evaluation of our above formulas against our numerical results for the full space of cuspidal newforms.
At discriminant $-m$, at level $(m)$ (of norm $m^2$ and Hermite Normal Form (HNF) $[m^2,0,m]$)
and its divisors, we have carried out this computation from (automorphic form) weight~$2$ up to the following upper limits for the weight.
$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Discriminant} & -7 & -11 & -19 & -43 & -67 \\
\hline
\text{weight up to} & 25 & 21 & 11 & 4 & 2 \\
\hline
\end{array}$$
The result is that of these 83 cuspidal newforms spaces, all are completely exhausted by (twists of) Base-Change, except for the following.
$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Discriminant} & -7 & -7 & -11 & -11 & -11 & -43 \\
\hline
\text{weight} & 6 & 14 & 12 & 3 & 5 & 2 \\
\hline
\text{potentially genuine space dimension}& \bf{2} & 2 & 2 & \bf{4} & \bf{4} & \bf{2} \\
\hline
\end{array}$$
Out of these, the forms at discriminant $-7$, weight $14$ and at discriminant $-11$, weight $12$
are twists of the genuine level One Bianchi modular forms already found by Grunewald~\cite{FGT}.
In the other cases, there are no level One forms that could be twisted~\cite{RahmSengun}.
That there are no CM-forms in the cases $m$ congruent to $3 \mod 4$,
is guaranteed for arbitrary weight by Theorem~\ref{prop:formula}.
So the remaining spaces must be genuine, and we print them in boldface.
\subsection{Square-free levels} \label{sec:Square-free levels}
\begin{table}
\centering
\begin{tabular}{|c|cc|cc|}
\hline & Weight 2 & & Higher weight (at least 3)& \\
\hline Discriminant & \# spaces & \# genuine spaces & \# spaces & \# genuine spaces \\
\hline
-7 & 1174 & 355 & 556 & 17 \\
-11 & 1307 & 353 & 683 & 14 \\
-19 & 504 & 151 & 531 & 6 \\
-43 & 318 & 61 & 103 & 1 \\
-67 & 123 & 17 & 33 & 1 \\
-163 & 24 & 4 & 3 & 0 \\
\hline
\end{tabular}
\caption{Overview of the sample presented in the Appendix.
Under ``\# spaces'', we count the newforms spaces that have been computed at the specified discriminant and varying level,
and under ``\# genuine spaces'', we count those of them which admit a non-trivial genuine subspace.}
\label{overview}
\end{table}
Let $n \in \mathbb Z$ be square-free and coprime to the discriminant of the imaginary quadratic field in question.
Consider the level $(n)$ of HNF $[n^2, 0, n]$.
Then we have the formulas of Section~\ref{Dimension formulas} for the dimension of the space of base-changed forms of level $(n)$.
We have compared them against the machine computed dimension of the space of cuspidal newforms.
{At discriminant -19,} the range of this machine computation was as follows.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Levels }n \text{ at discriminant -19}& 2,3 & 6 & 11 & 13 & 5 & 7, 15 & 14, 17 & 23 & 10 & 30 & 22, 31, 33\\
\hline
\text{weight up to} & 22 & 21 & 15 & 13 & 12 & 11 & 10 & 8 & 5 & 4 & 2\\
\hline
\end{array}$$
Out of these 154 spaces, only the following six spaces can admit genuine forms:
$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Level} & 6 & 6 & 11 & 15 & 17 & 30 \\
\hline
\text{weight} & 3 & 4 & 2 & 2 & 2 & 2 \\
\hline
\text{genuine space dimension} & 2 & 2 & 2 & 2 & 2 & 4 \\
\hline
\end{array}$$
As the level is square-free and the class group of the imaginary quadratic field is trivial, there are no twists of base-change forms.
By Theorem~\ref{thm:twist-2}, there are no CM forms.
So the above six spaces must be constituted of genuine forms.
The above sample of 154 spaces allows us to guess that genuine forms are more likely to occur at low weights than at high weights (supported by Table~\ref{overview});
and that levels which admit genuine forms at some low weights are rather unlikely to admit more of them at higher weights
(supported by the Appendix).
\section{Appendix: Detailed results in the square-free level case}
In addition to the sample of Section~\ref{sec:Square-free levels},
we include in the following tables a range of square-free ideals which are not Galois-stable (in our setting of imaginary quadratic fields, not totally real).
At each discriminant, we first specify the range of the pertinent machine computation,
and then in a separate table the spaces with a non-trivial genuine subspace.
The computation usually was run also at the Galois-conjugated level, but in the range tables,
we print only one HNF per pair of Galois-conjugate levels.
We omit the details for weight $2$, and outsource them to~\cite{database}.
A statistical overview is given in Table~\ref{overview}.
The bottleneck for the computations were the memory requirements; processor time aspects were completely eclipsed by them.
With the quadratic (in the weight) growth of the coefficient modules, the MAGMA program did grow its memory requirements quadratically.
\begin{center}
\footnotesize \begin{tabular}{|c|c|} \hline {\bf Range at discriminant -19.} Level HNFs, up to Galois conjugacy & {weights} \\
\hline
164 levels of norm up to 1145
and their Galois conjugates & only 2 \\
\hline \begin{tabular}{c} $
[ 140, 10, 2 ]
,
[ 140, 30, 2 ]
,
[ 153, 39, 3 ]
,
[ 161, 12, 1 ]
,
[ 161, 127, 1 ]
,
[ 172, 28, 2 ]
,
[ 175, 25, 5 ]
,
[ 187, 13, 1 ]
,
$ \\
$
[ 187, 156, 1 ]
,
[ 188, 12, 2 ]
,
[ 191, 154, 1 ]
,
[ 197, 150, 1 ]
,
[ 199, 128, 1 ]
,
[ 207, 30, 3 ]
,
[ 215, 100, 1 ]
,
[ 215, 14, 1 ]
,
$ \\
$
[ 220, 38, 2 ]
,
[ 220, 48, 2 ]
,
[ 229, 139, 1 ]
,
[ 233, 166, 1 ]
,
[ 235, 100, 1 ]
,
[ 235, 194, 1 ]
,
[ 239, 204, 1 ]
,
[ 244, 106, 2 ]
,
$ \\
$
[ 245, 0, 7 ]
,
[ 251, 198, 1 ]
,
[ 252, 30, 6 ]
,
[ 253, 173, 1 ]
,
[ 253, 217, 1 ]
,
[ 263, 113, 1 ]
,
[ 271, 227, 1 ]
,
[ 275, 10, 5 ]
,
$ \\
$
[ 277, 16, 1 ]
,
[ 283, 183, 1 ]
,
[ 289, 207, 1 ]
,
[ 292, 100, 2 ]
,
[ 301, 229, 1 ]
,
[ 301, 243, 1 ]
,
[ 305, 114, 1 ]
,
[ 305, 129, 1 ]
,
$ \\
$
[ 311, 17, 1 ]
,
[ 313, 273, 1 ]
,
[ 347, 18, 1 ]
,
[ 349, 110, 1 ]
,
[ 353, 131, 1 ]
,
[ 359, 140, 1 ]
,
[ 367, 303, 1 ]
,
[ 389, 168, 1 ]
,
$ \\
$
$ \end{tabular} & up to 3 \\
\hline \begin{tabular}{c} $
[ 101, 26, 1 ]
,
[ 115, 10, 1 ]
,
[ 115, 35, 1 ]
,
[ 119, 47, 1 ]
,
[ 119, 54, 1 ]
,
[ 121, 24, 1 ]
,
[ 131, 105, 1 ]
,
[ 137, 11, 1 ]
,
$ \\
$
[ 139, 56, 1 ]
,
[ 149, 58, 1 ]
,
[ 157, 115, 1 ]
,
[ 163, 75, 1 ]
,
[ 68, 26, 2 ]
,
[ 77, 19, 1 ]
,
[ 77, 68, 1 ]
,
[ 83, 37, 1 ]
,
$ \\
$
[ 85, 20, 1 ]
,
[ 85, 30, 1 ]
,
[ 900, 0, 30 ]
,
[ 92, 20, 2 ]
,
[ 99, 24, 3 ]
,
$ \end{tabular} & up to 4 \\
\hline \begin{tabular}{c} $
[ 100, 0, 10 ]
,
[ 180, 0, 6 ]
,
[ 35, 15, 1 ]
,
[ 35, 29, 1 ]
,
[ 43, 14, 1 ]
,
[ 44, 16, 2 ]
,
[ 47, 40, 1 ]
,
[ 49, 15, 1 ]
,
$ \\
$
[ 55, 19, 1 ]
,
[ 55, 24, 1 ]
,
[ 61, 53, 1 ]
,
[ 63, 15, 3 ]
,
[ 73, 22, 1 ]
,
$ \end{tabular} & up to 5 \\
\hline \begin{tabular}{c} $
[ 20, 0, 2 ]
,
[ 25, 20, 1 ]
,
$ \end{tabular} & up to 6 \\
\hline \begin{tabular}{c} $
[ 23, 10, 1 ]
,
[ 529, 0, 23 ]
,
$ \end{tabular} & up to 8 \\
\hline \begin{tabular}{c} $
[ 196, 0, 14 ]
,
[ 289, 0, 17 ]
,
$ \end{tabular} & up to 10 \\
\hline \begin{tabular}{c} $
[ 17, 13, 1 ]
,
[ 225, 0, 15 ]
,
[ 28, 10, 2 ]
,
[ 49, 0, 7 ]
,
[ 7, 1, 1 ]
,
$ \end{tabular} & up to 11 \\
\hline \begin{tabular}{c} $
[ 25, 0, 5 ]
,
[ 45, 0, 3 ]
,
[ 5, 0, 1 ]
,
$ \end{tabular} & up to 12 \\
\hline \begin{tabular}{c} $
[ 169, 0, 13 ]
,
$ \end{tabular} & up to 13 \\
\hline \begin{tabular}{c} $
[ 121, 0, 11 ]
,
$ \end{tabular} & up to 15 \\
\hline \begin{tabular}{c} $
[ 11, 2, 1 ]
,
$ \end{tabular} & up to 16 \\
\hline \begin{tabular}{c} $
[ 36, 0, 6 ]
,
$ \end{tabular} & up to 21 \\
\hline \begin{tabular}{c} $
[ 4, 0, 2 ]
,
[ 9, 0, 3 ].
$ \end{tabular} & up to 22 \\
\hline \end{tabular} \normalsize
\end{center}
Out of these, we get the following genuine spaces:
\begin{center}
\footnotesize \begin{tabular}{|c|c|c|} \hline {Weight} & $d$ & Level HNFs at discriminant -19 with
genuine space of dim. $d$ \\
\hline 2 & 1 &
73 levels of norm up to 1145
\\
\hline 2 & 2 &
31 levels of norm up to 1099
\\
\hline 2 & 3 &
16 levels of norm up to 935
\\
\hline 2 & 4 &
17 levels of norm up to 1081
\\
\hline 2 & 5 &
6 levels of norm up to 932
\\
\hline 2 & 6 &
6 levels of norm up to 940
\\
\hline 2 & 7 &
2 levels of norm up to 955
\\
\hline 3 & 2 &
\begin{tabular}{c} $
[ 289, 207, 1 ]
,
[ 289, 81, 1 ]
,
[ 36, 0, 6 ]
,
[ 49, 15, 1 ]
,
[ 49, 33, 1 ]
,
$ \end{tabular} \\
\hline 4 & 2 &
\begin{tabular}{c} $
[ 36, 0, 6 ]
.
$ \end{tabular} \\
\hline \end{tabular} \normalsize
\end{center}
That is, the genuine spaces at level $(6) = [36, 0, 6]$ already observed in Section~\ref{sec:Square-free levels},
as well as two further two-dimensional weight $3$ spaces and their Galois conjugates.
\begin{center}
\footnotesize \begin{tabular}{|c|c|} \hline {\bf Range at discriminant -43.} Level HNFs, up to Galois conjugacy & weights \\
\hline
133 levels of norm up to 787
and their Galois conjugates & only 2 \\
\hline \begin{tabular}{c} $
[ 52, 22, 2 ]
,
[ 59, 31, 1 ]
,
[ 67, 7, 1 ]
,
[ 79, 42, 1 ]
,
[ 68, 28, 2 ]
,
[ 83, 74, 1 ]
,
[ 103, 69, 1 ]
,
[ 109, 71, 1 ]
,
$ \\
$
[ 97, 64, 1 ]
,
[ 107, 49, 1 ]
,
[ 121, 110, 1 ]
,
[ 100, 0, 10 ]
,
[ 101, 9, 1 ]
,
[ 117, 33, 3 ]
,
[ 92, 38, 2 ]
,
[ 127, 40, 1 ]
,
$ \\
$
[ 99, 30, 3 ]
,
[ 121, 0, 11 ]
,
[ 139, 48, 1 ]
,
[ 167, 154, 1 ]
,
$ \end{tabular} & up to 3 \\
\hline \begin{tabular}{c} $
[ 53, 46, 1 ]
,
[ 31, 4, 1 ]
,
[ 41, 5, 1 ]
,
[ 49, 0, 7 ]
,
[ 44, 0, 2 ]
,
[ 47, 22, 1 ]
,
$ \end{tabular} & up to 4 \\
\hline \begin{tabular}{c} $
[ 17, 2, 1 ]
,
[ 23, 3, 1 ]
,
[ 25, 0, 5 ]
,
$ \end{tabular} & up to 5 \\
\hline \begin{tabular}{c} $
[ 13, 11, 1 ]
,
[ 11, 0, 1 ]
,
$ \end{tabular} & up to 6 \\
\hline \begin{tabular}{c} $
[ 9, 0, 3 ]
,
$ \end{tabular} & up to 7 \\
\hline \begin{tabular}{c} $
[ 4, 0, 2 ]
.
$ \end{tabular} & up to 8 \\
\hline \end{tabular} \normalsize
\end{center}
Out of these, we get the following genuine spaces:
\begin{center}
\footnotesize \begin{tabular}{|c|c|c|} \hline
{Weight} & $d$ & Level HNFs at discriminant -43 with
genuine space of dim. $d$ \\
\hline 2 & 1 &
32 levels of norm up to 737
\\
\hline 2 & 2 &
18 levels of norm up to 713
\\
\hline 2 & 3 &
4 levels of norm up to 719
\\
\hline 2 & 4 &
7 levels of norm up to 572
\\
\hline 6 & 2 &
\begin{tabular}{c} $
[ 9, 0, 3 ]
.
$ \end{tabular} \\
\hline \end{tabular} \normalsize
\end{center}
\begin{center}
\footnotesize \begin{tabular}{|c|c|} \hline {\bf Range at discriminant -67}. Level HNFs, up to Galois conjugacy & weights \\
\hline
54 levels of norm up to 361
and their Galois conjugates & only 2 \\
\hline \begin{tabular}{c} $
[ 29, 3, 1 ]
,
[ 37, 4, 1 ]
,
[ 36, 0, 6 ]
,
[ 49, 0, 7 ]
,
[ 47, 41, 1 ]
,
[ 59, 6, 1 ]
,
[ 71, 36, 1 ]
,
$ \end{tabular} & up to 3 \\
\hline \begin{tabular}{c} $
[ 17, 0, 1 ]
,
[ 19, 1, 1 ]
,
[ 23, 2, 1 ]
,
[ 25, 0, 5 ]
,
$ \end{tabular} & up to 4 \\
\hline \begin{tabular}{c} $
[ 9, 0, 3 ]
,
$ \end{tabular} & up to 5 \\
\hline \begin{tabular}{c} $
[ 4, 0, 2 ]
.
$ \end{tabular} & up to 6 \\
\hline \end{tabular} \normalsize
\end{center}
Out of these, we get the following genuine spaces:
\begin{center}
\footnotesize \begin{tabular}{|c|c|c|} \hline
{Weight} & $d$ & Level HNFs at discriminant -67 with
genuine space of dim. $d$ \\
\hline 2 & 1 &
6 levels of norm up to 323
\\
\hline 2 & 2 &
3 levels of norm up to 289
\\
\hline 2 & 3 &
6 levels of norm up to 289
\\
\hline 2 & 4 &
one level of norm 121
\\
\hline 2 & 8 &
one level of norm 196
\\
\hline 3 & 2 &
\begin{tabular}{c} $
[ 36, 0, 6 ]
.
$ \end{tabular} \\
\hline \end{tabular} \normalsize
\end{center}
\begin{center}
\footnotesize \begin{tabular}{|c|c|} \hline {\bf Range at discriminant -163.}
Level HNFs, up to Galois conjugacy & {weights} \\
\hline \begin{tabular}{c} $
[ 25, 0, 5 ]
,
[ 36, 0, 6 ]
,
[ 41, 40, 1 ]
,
[ 43, 41, 1 ]
,
[ 49, 0, 7 ]
,
[ 47, 2, 1 ]
,
[ 53, 49, 1 ]
,
[ 61, 4, 1 ]
,
$ \\
$
[ 71, 5, 1 ]
,
[ 83, 6, 1 ]
,
[ 97, 7, 1 ]
,
[ 121, 0, 11 ]
,
[ 113, 8, 1 ]
,
$ \end{tabular} & 2 \\
\hline \begin{tabular}{c} $
[ 9, 0, 3 ]
,
$ \end{tabular} & up to 3 \\
\hline \begin{tabular}{c} $
[ 4, 0, 2 ]
.
$ \end{tabular} & up to 4 \\
\hline \end{tabular} \normalsize
\end{center}
Out of these, we get the following genuine spaces:
\begin{center}
\footnotesize \begin{tabular}{|c|c|c|} \hline
{Weight} & $d$ & Level HNFs at discriminant -163 with
genuine space of dim. $d$ \\
\hline 2 & 2 &
3 levels of norm up to 47
\\
\hline 2 & 4 &
[49, 0, 7]
\\
\hline \end{tabular} \normalsize
\end{center}
\begin{center}
\scriptsize \begin{tabular}{|c|c|} \hline {\bf Range at discriminant -7.} Level HNFs, up to Galois conjugacy & {weights} \\
\hline
460 levels of norm up to 2767
and their Galois conjugates & only 2 \\
\hline \begin{tabular}{c} $
[ 214, 155, 1 ]
,
[ 218, 188, 1 ]
,
[ 218, 79, 1 ]
,
[ 214, 165, 1 ]
,
[ 212, 28, 2 ]
,
[ 172, 36, 2 ]
,
[ 253, 59, 1 ]
,
[ 198, 18, 3 ]
,
$ \\
$
[ 253, 147, 1 ]
,
[ 254, 104, 1 ]
,
[ 254, 231, 1 ]
,
[ 198, 12, 3 ]
,
[ 226, 42, 1 ]
,
[ 226, 70, 1 ]
,
[ 242, 0, 11 ]
,
[ 275, 20, 5 ]
,
$ \\
$
[ 298, 34, 1 ]
,
[ 261, 21, 3 ]
,
[ 274, 16, 1 ]
,
[ 274, 153, 1 ]
,
[ 277, 253, 1 ]
,
[ 281, 33, 1 ]
,
[ 268, 22, 2 ]
,
[ 289, 0, 17 ]
,
$ \\
$
[ 298, 183, 1 ]
,
[ 284, 78, 2 ]
,
[ 326, 25, 1 ]
,
[ 394, 341, 1 ]
,
[ 347, 272, 1 ]
,
[ 317, 233, 1 ]
,
[ 319, 94, 1 ]
,
[ 319, 268, 1 ]
,
$ \\
$
[ 302, 220, 1 ]
,
[ 302, 232, 1 ]
,
[ 326, 137, 1 ]
,
[ 359, 128, 1 ]
,
[ 361, 0, 19 ]
,
[ 373, 154, 1 ]
,
[ 379, 27, 1 ]
,
[ 148, 16, 2 ]
,
$ \\
$
[ 358, 125, 1 ]
,
[ 358, 304, 1 ]
,
[ 401, 248, 1 ]
,
[ 331, 174, 1 ]
,
[ 382, 19, 1 ]
,
[ 407, 378, 1 ]
,
[ 389, 296, 1 ]
,
[ 407, 193, 1 ]
,
$ \\
$
[ 387, 72, 3 ]
,
[ 382, 210, 1 ]
,
[ 333, 24, 3 ]
,
[ 431, 389, 1 ]
,
[ 386, 73, 1 ]
,
[ 386, 119, 1 ]
,
[ 394, 144, 1 ]
,
[ 422, 401, 1 ]
,
$ \\
$
[ 337, 212, 1 ]
,
[ 421, 244, 1 ]
,
[ 338, 0, 13 ]
,
[ 422, 190, 1 ]
,
[ 457, 85, 1 ]
,
[ 443, 285, 1 ]
,
[ 449, 196, 1 ]
,
[ 487, 103, 1 ]
,
$ \\
$
[ 541, 46, 1 ]
,
[ 547, 459, 1 ]
,
[ 463, 80, 1 ]
,
[ 473, 61, 1 ]
,
[ 473, 147, 1 ]
,
[ 477, 114, 3 ]
,
[ 529, 0, 23 ]
,
[ 529, 32, 1 ]
,
$ \\
$
[ 575, 45, 5 ]
,
[ 571, 158, 1 ]
,
[ 491, 234, 1 ]
,
[ 569, 178, 1 ]
,
[ 499, 151, 1 ]
,
[ 557, 133, 1 ]
,
[ 613, 563, 1 ]
,
[ 599, 129, 1 ]
,
$ \\
$
$ \end{tabular} & up to 3 \\
\hline \begin{tabular}{c} $
[ 86, 18, 1 ]
,
[ 106, 38, 1 ]
,
[ 106, 14, 1 ]
,
[ 116, 14, 2 ]
,
[ 86, 61, 1 ]
,
[ 92, 26, 2 ]
,
[ 100, 0, 10 ]
,
[ 211, 190, 1 ]
,
$ \\
$
[ 207, 27, 3 ]
,
[ 163, 25, 1 ]
,
[ 197, 144, 1 ]
,
[ 191, 19, 1 ]
,
[ 193, 119, 1 ]
,
[ 239, 72, 1 ]
,
[ 169, 0, 13 ]
,
[ 179, 53, 1 ]
,
$ \\
$
[ 225, 0, 15 ]
,
[ 233, 30, 1 ]
,
[ 263, 123, 1 ]
,
[ 121, 105, 1 ]
,
[ 121, 0, 11 ]
,
[ 134, 78, 1 ]
,
[ 134, 122, 1 ]
,
[ 127, 104, 1 ]
,
$ \\
$
[ 137, 16, 1 ]
,
[ 142, 39, 1 ]
,
[ 142, 31, 1 ]
,
[ 151, 81, 1 ]
,
[ 149, 114, 1 ]
,
[ 158, 91, 1 ]
,
[ 158, 145, 1 ]
,
$ \end{tabular} & up to 4 \\
\hline \begin{tabular}{c} $
[ 36, 0, 6 ]
,
[ 67, 55, 1 ]
,
[ 71, 31, 1 ]
,
[ 74, 45, 1 ]
,
[ 74, 65, 1 ]
,
[ 79, 66, 1 ]
,
[ 46, 32, 1 ]
,
[ 46, 36, 1 ]
,
$ \\
$
[ 44, 8, 2 ]
,
[ 50, 5, 5 ]
,
[ 58, 36, 1 ]
,
[ 58, 7, 1 ]
,
[ 107, 48, 1 ]
,
[ 109, 29, 1 ]
,
[ 99, 12, 3 ]
,
[ 113, 70, 1 ]
,
$ \\
$
$ \end{tabular} & up to 5 \\
\hline \begin{tabular}{c} $
[ 43, 18, 1 ]
,
[ 53, 14, 1 ]
,
$ \end{tabular} & up to 6 \\
\hline \begin{tabular}{c} $
[ 29, 7, 1 ]
,
[ 22, 17, 1 ]
,
[ 22, 15, 1 ]
,
[ 18, 3, 3 ]
,
[ 37, 28, 1 ]
,
$ \end{tabular} & up to 7 \\
\hline \begin{tabular}{c} $
[ 25, 0, 5 ]
,
[ 23, 13, 1 ]
,
$ \end{tabular} & up to 8 \\
\hline \begin{tabular}{c} $
[ 11, 4, 1 ]
,
$ \end{tabular} & up to 11 \\
\hline \begin{tabular}{c} $
[ 4, 0, 2 ]
,
$ \end{tabular} & up to 12 \\
\hline \begin{tabular}{c} $
[ 4, 2, 1 ]
,
$ \end{tabular} & up to 15 \\
\hline \begin{tabular}{c} $
[ 2, 0, 1 ]
,
$ \end{tabular} & up to 19 \\
\hline \begin{tabular}{c} $
[ 9, 0, 3 ]
.
$ \end{tabular} & up to 20 \\
\hline \end{tabular} \normalsize
\end{center}
Out of these, we get the following genuine spaces:
\begin{center}
\footnotesize \begin{tabular}{|c|c|c|} \hline {Weight} & $d$ & Level HNFs at discriminant -7 with genuine space of dim. $d$ \\
\hline 2 & 1 &
200 levels of norm up to 2657
\\
\hline 2 & 2 &
100 levels of norm up to 1913
\\
\hline 2 & 3 &
30 levels of norm up to 1814
\\
\hline 2 & 4 &
15 levels of norm up to 2563
\\
\hline 2 & 5 &
6 levels of norm up to 1439
\\
\hline 2 & 6 &
4 levels of norm up to 1702
\\
\hline 3 & 2 &
\begin{tabular}{c} $
[ 225, 0, 15 ]
,
$ \end{tabular} \\
\hline 4 & 1 &
\begin{tabular}{c} $
[ 11, 6, 1 ]
,
[ 11, 4, 1 ]
,
[ 22, 17, 1 ]
,
[ 22, 4, 1 ]
,
[ 46, 36, 1 ]
,
[ 46, 9, 1 ]
,
[ 92, 26, 2 ]
,
$ \\
$
[ 92, 18, 2 ]
,
[ 121, 105, 1 ]
,
[ 121, 15, 1 ]
,
[ 116, 14, 2 ]
,
[ 116, 42, 2 ]
,
$ \end{tabular} \\
\hline 4 & 2 &
\begin{tabular}{c} $
[ 22, 15, 1 ]
,
[ 22, 6, 1 ]
,
[ 58, 50, 1 ]
,
[ 58, 7, 1 ]
.
$ \end{tabular} \\
\hline \end{tabular} \normalsize
\end{center}
\begin{center}
\footnotesize \begin{tabular}{|c|c|} \hline {\bf Range at discriminant -11.}
Level HNFs, up to Galois conjugacy & {weights} \\
\hline
507 levels of norm up to 2803
and their Galois conjugates & only 2 \\
\hline \begin{tabular}{c} $
[ 180, 18, 6 ]
,
[ 207, 12, 3 ]
,
[ 213, 14, 1 ]
,
[ 213, 156, 1 ]
,
[ 225, 0, 15 ]
,
[ 235, 161, 1 ]
,
[ 235, 208, 1 ]
,
[ 236, 102, 2 ]
,
$ \\
$
[ 245, 21, 7 ]
,
[ 265, 118, 1 ]
,
[ 265, 171, 1 ]
,
[ 267, 125, 1 ]
,
[ 267, 230, 1 ]
,
[ 268, 48, 2 ]
,
[ 276, 100, 2 ]
,
[ 276, 54, 2 ]
,
$ \\
$
[ 279, 27, 3 ]
,
[ 284, 112, 2 ]
,
[ 289, 0, 17 ]
,
[ 291, 126, 1 ]
,
[ 291, 261, 1 ]
,
[ 295, 228, 1 ]
,
[ 295, 243, 1 ]
,
[ 309, 120, 1 ]
,
$ \\
$
[ 309, 17, 1 ]
,
[ 311, 280, 1 ]
,
[ 313, 244, 1 ]
,
[ 317, 224, 1 ]
,
[ 331, 104, 1 ]
,
[ 333, 39, 3 ]
,
[ 335, 158, 1 ]
,
[ 335, 243, 1 ]
,
$ \\
$
[ 339, 102, 1 ]
,
[ 339, 123, 1 ]
,
[ 345, 156, 1 ]
,
[ 345, 18, 1 ]
,
[ 345, 248, 1 ]
,
[ 345, 303, 1 ]
,
[ 353, 320, 1 ]
,
[ 355, 156, 1 ]
,
$ \\
$
[ 355, 298, 1 ]
,
[ 356, 104, 2 ]
,
[ 361, 0, 19 ]
,
[ 367, 128, 1 ]
,
[ 372, 142, 2 ]
,
[ 372, 166, 2 ]
,
[ 379, 335, 1 ]
,
[ 383, 19, 1 ]
,
$ \\
$
[ 388, 134, 2 ]
,
[ 389, 177, 1 ]
,
[ 397, 165, 1 ]
,
[ 401, 148, 1 ]
,
[ 411, 195, 1 ]
,
[ 411, 332, 1 ]
,
[ 412, 170, 2 ]
,
[ 419, 124, 1 ]
,
$ \\
$
[ 421, 35, 1 ]
,
[ 433, 386, 1 ]
,
[ 443, 361, 1 ]
,
[ 445, 141, 1 ]
,
[ 445, 36, 1 ]
,
[ 449, 183, 1 ]
,
[ 452, 20, 2 ]
,
[ 463, 207, 1 ]
,
$ \\
$
[ 467, 179, 1 ]
,
[ 471, 422, 1 ]
,
[ 485, 126, 1 ]
,
[ 485, 223, 1 ]
,
[ 487, 169, 1 ]
,
[ 489, 134, 1 ]
,
[ 489, 191, 1 ]
,
[ 499, 412, 1 ]
,
$ \\
$
[ 509, 22, 1 ]
,
[ 515, 188, 1 ]
,
[ 515, 223, 1 ]
,
[ 521, 39, 1 ]
,
[ 529, 0, 23 ]
,
[ 529, 119, 1 ]
,
[ 577, 184, 1 ]
,
[ 587, 281, 1 ]
,
$ \\
$
[ 599, 181, 1 ]
,
[ 619, 536, 1 ]
,
[ 631, 168, 1 ]
,
[ 643, 283, 1 ]
,
$ \end{tabular} & up to 3 \\
\hline \begin{tabular}{c} $
[ 100, 0, 10 ]
,
[ 111, 23, 1 ]
,
[ 111, 50, 1 ]
,
[ 115, 18, 1 ]
,
[ 115, 41, 1 ]
,
[ 124, 18, 2 ]
,
[ 137, 58, 1 ]
,
[ 141, 114, 1 ]
,
$ \\
$
[ 141, 120, 1 ]
,
[ 147, 0, 7 ]
,
[ 148, 26, 2 ]
,
[ 155, 133, 1 ]
,
[ 155, 71, 1 ]
,
[ 157, 108, 1 ]
,
[ 159, 12, 1 ]
,
[ 159, 65, 1 ]
,
$ \\
$
[ 163, 134, 1 ]
,
[ 177, 110, 1 ]
,
[ 177, 125, 1 ]
,
[ 179, 109, 1 ]
,
[ 181, 116, 1 ]
,
[ 185, 13, 1 ]
,
[ 185, 161, 1 ]
,
[ 188, 40, 2 ]
,
$ \\
$
[ 191, 137, 1 ]
,
[ 196, 0, 14 ]
,
[ 199, 167, 1 ]
,
[ 201, 158, 1 ]
,
[ 201, 176, 1 ]
,
[ 212, 24, 2 ]
,
[ 223, 173, 1 ]
,
[ 229, 101, 1 ]
,
$ \\
$
[ 251, 203, 1 ]
,
[ 257, 159, 1 ]
,
[ 269, 205, 1 ]
,
[ 75, 0, 5 ]
,
[ 93, 21, 1 ]
,
[ 93, 83, 1 ]
,
$ \end{tabular} & up to 4 \\
\hline \begin{tabular}{c} $
[ 15, 8, 1 ]
,
[ 169, 0, 13 ]
,
[ 507, 0, 13 ]
,
[ 1521, 0, 39 ]
,
[ 103, 17, 1 ]
,
[ 113, 10, 1 ]
,
[ 36, 0, 6 ]
,
[ 45, 3, 3 ]
,
$ \\
$
[ 60, 12, 2 ]
,
[ 60, 22, 2 ]
,
[ 67, 24, 1 ]
,
[ 69, 18, 1 ]
,
[ 69, 27, 1 ]
,
[ 71, 14, 1 ]
,
[ 89, 36, 1 ]
,
[ 92, 36, 2 ]
,
$ \\
$
[ 97, 29, 1 ]
,
$ \end{tabular} & up to 5 \\
\hline \begin{tabular}{c} $
[ 47, 20, 1 ]
,
[ 49, 0, 7 ]
,
[ 53, 12, 1 ]
,
[ 59, 51, 1 ]
,
$ \end{tabular} & up to 6 \\
\hline \begin{tabular}{c} $
[ 2209, 0, 47 ]
,
[ 20, 2, 2 ]
,
[ 31, 21, 1 ]
,
[ 37, 13, 1 ]
,
$ \end{tabular} & up to 7 \\
\hline \begin{tabular}{c} $
[ 15, 11, 1 ]
,
[ 23, 18, 1 ]
,
$ \end{tabular} & up to 8 \\
\hline \begin{tabular}{c} $
[ 12, 0, 2 ]
,
$ \end{tabular} & up to 9 \\
\hline \begin{tabular}{c} $
[ 25, 0, 5 ]
,
$ \end{tabular} & up to 10 \\
\hline \begin{tabular}{c} $
[ 9, 2, 1 ]
,
$ \end{tabular} & up to 11 \\
\hline \begin{tabular}{c} $
[ 9, 0, 3 ]
,
$ \end{tabular} & up to 12 \\
\hline \begin{tabular}{c} $
[ 4, 0, 2 ]
,
$ \end{tabular} & up to 15 \\
\hline \begin{tabular}{c} $
[ 3, 0, 1 ]
,
$ \end{tabular} & up to 17 \\
\hline \begin{tabular}{c} $
[ 25, 16, 1 ]
,
[ 5, 1, 1 ]
.
$ \end{tabular} & up to 20 \\
\hline \end{tabular} \normalsize
\end{center}
Out of these, we get the following genuine spaces:
\begin{center}
\footnotesize \begin{tabular}{|c|c|c|} \hline
{Weight} & $d$ & Level HNFs at discriminant -11 with
genuine space of dim. $d$ \\
\hline 2 & 1 &
178 levels of norm up to 2491
\\
\hline 2 & 2 &
107 levels of norm up to 2621
\\
\hline 2 & 3 &
32 levels of norm up to 2689
\\
\hline 2 & 4 &
18 levels of norm up to 2201
\\
\hline 2 & 5 &
4 levels of norm up to 1335
\\
\hline 2 & 6 &
6 levels of norm up to 2597
\\
\hline 2 & 7 &
4 levels of norm up to 1035
\\
\hline 2 & 12 &
4 levels of norm up to 2209
\\
\hline 4 & 1 &
\begin{tabular}{c} $
[ 15, 6, 1 ]
,
[ 15, 8, 1 ]
,
[ 185, 161, 1 ]
,
[ 185, 23, 1 ]
,
[ 20, 2, 2 ]
,
[ 20, 6, 2 ]
,
[ 45, 3, 3 ]
,
$ \\
$
[ 45, 9, 3 ]
,
$ \end{tabular} \\
\hline 4 & 2 &
\begin{tabular}{c} $
[ 100, 0, 10 ]
,
[ 92, 36, 2 ]
,
[ 92, 8, 2 ]
,
$ \end{tabular} \\
\hline 4 & 5 &
\begin{tabular}{c} $
[ 60, 12, 2 ]
,
[ 60, 16, 2 ]
,
$ \end{tabular} \\
\hline 6 & 2 &
\begin{tabular}{c} $
[ 25, 0, 5 ]
.
$ \end{tabular} \\
\hline \end{tabular} \normalsize
\end{center}
|
2,869,038,155,717 | arxiv | \section*{Compliance with ethical standards}
\begin{sloppypar}
\noindent \textbf{Funding} This study was funded by the Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo - FAPESP (grant no. 2016/13474-9), Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico - CNPq (grant no. 400284/2016-2) and Fundo de Apoio ao Ensino, Pesquisa e Extens\~ao da Unicamp - FAEPEX/UNICAMP (grant nos. 2210/18 and 2112/19).\\
\noindent \textbf{Conflict of interest} The authors declare that they have no conflict of interest.
\end{sloppypar}
\section{Introduction}
The interaction between granular matter and fluids is frequent in both nature and industry. One example commonly observed is the transport of sand by a fluid flow, such as happens in rivers, deserts, oceans, and pipelines. If sand is entrained in the horizontal (or nearly horizontal) direction as bed load, a moving layer that keeps contact with the static part of the bed, sand dunes usually grow \cite{Bagnold_1,Raudkivi_1,Yalin_1}. Their growth is the result of local erosion and deposition rates, where the perturbation of the fluid flow is known to be the unstable mechanism \cite{Andreotti_1,Andreotti_2,Elbelrhiti,Charru_3,Claudin_Andreotti}. Under one-directional flow and limited sand supply, sand dunes evolve to crescentic shape dunes, known as barchan dunes \cite{Herrmann_Sauermann,Sauermann_1}, or simply barchans, and their length depends on the length scale \cite{Hersen_1}:
\begin{equation}
L_{drag} = d \frac{\rho_s}{\rho},
\label{ldrag_equation}
\end{equation}
\noindent where $d$ is the diameter of grains, and $\rho_s$ and $\rho$ are the densities of the granular material and fluid, respectively. Therefore, the length and time scales of barchans go from 100 m and 1 year on deserts down to 10 cm and 1 minute for subaqueous barchans \cite{Claudin_Andreotti}.
Because barchan dunes are formed when grains are entrained as bed load, with the moving layer exchanging mass with the fixed part of the bed, the distribution of moving grains and their trajectories are important to understand the morphology and dynamics of barchans. For aeolian dunes, the moving grains effectuate ballistic flights over distances much larger than the grain diameter \cite{Bagnold_1}, while for subaqueous dunes the grains move mainly by rolling or sliding over each other, over distances comparable to the grain diameter \cite{Franklin_8}. If, on the one hand, there are models for the displacements of grains over aeolian and subaqueous dunes \cite{Sauermann_2,Sauermann_4,Kroy_C,Hersen_3,Pahtz_1}, on the other hand there are not, at present, precise measurements of the displacements of individual grains over barchans.
Few previous papers reported experiments on the displacements of individual grains within a bed-load layer under water flows. For the case of plane granular beds, Seizilles et al. \cite{Seizilles} investigated the displacements of individual grains under laminar flows, while Lajeunesse et al. \cite{Lajeunesse} and Penteado and Franklin \cite{Penteado} investigated them under turbulent flows. Lajeunesse et al. \cite{Lajeunesse} performed experiments on a water flume, where steady-state free-surface turbulent flows were imposed over different granular beds. The Reynolds numbers based on the water depth were within $1500$ and $6000$, and the beds consisted of quartz particles with median diameters of $1.15\,mm$, $2.24\,mm$, or $5.5\,mm$, corresponding to $12\,\leq\, Re_* \,\leq\,500$, where
\begin{equation}
Re_* = \frac{u_* d}{\nu}
\label{eq:Refric}
\end{equation}
\noindent is the Reynolds number at the grain scale. In Eq. \ref{eq:Refric}, $u_*$ is the shear velocity and $\nu$ the kinematic viscosity. Grains velocities, displacement lengths, and durations of flights were computed from the displacements of individual grains, which were obtained from images acquired with a high-speed camera. Concerning the grain velocities, Lajeunesse et al. \cite{Lajeunesse} found that the distributions of their transverse and longitudinal components follow Gaussian and decreasing exponential laws, respectively. In addition, they showed that the flight duration scales with the settling velocity of a single grain, the surface density of moving grains scales with $\theta - \theta_{c}$, and the grains velocity and flight length scale with $\theta^{1/2}-\theta_{c}^{1/2}$, where $\theta$ and $\theta_{c}$ are the Shields and the threshold Shields numbers. The Shields number is the ratio between the fluid drag and the particle friction (given by Coulomb's law), and typical values for bed load are within $0.01 < \theta < 1$. For turbulent flows, the Shields number is given by:
\begin{equation}
\theta = \frac{u_*^2}{\left( \rho_s / \rho -1 \right) g d},
\end{equation}
\noindent where $g$ is the magnitude of gravity. The threshold Shields number is the value of the Shields number at the inception of bed load.
Penteado and Franklin \cite{Penteado} performed their experiments in a closed-conduit channel of rectangular cross section. In the experiments, fully-developed turbulent flows were imposed over plane beds of known granulometry, and the granular bed was filmed with a high-speed camera. The granular beds consisted of glass spheres with density $\rho_s\,=\,2500\,kg/m^3$ and diameter ranging from $d\,=\,400\,\mu$m to $d\,=\,600\,\mu$m. The Reynolds numbers based on the hydraulic diameter (twice the channel height) were within $1.6 \times 10^4$ and $2.7 \times 10^4$, and $5\,\leq\, Re_* \,\leq\,11$. From the acquired images, Penteado and Franklin \cite{Penteado} computed the velocity fields and trajectories of individual grains. The authors found that the mean longitudinal displacement normalized by the grain diameter is of the order of 10, the mean displacement velocity normalized by the shear rate and grain diameter is of the order of 0.1, and the mean displacement time normalized by the grain diameter and settling velocity is of the order of 10.
Numerical studies have been conducted since the 2000s to investigate the formation of single barchans as well as of barchan fields. The first studies made use of continuum equations adapted for granular media, such as, for instance, Sauermann et al. \cite{Sauermann_4}, Kroy et al. \cite{Kroy_A,Kroy_C}, Hersen \cite{Hersen_3} and Schw{\"a}mmle and Herrmann \cite{Schwammle}. In those models, information such as the characteristic lengths and times are essential to fit adjustable constants. More recently, Lagrangian methods, such as the 3D cellular automaton model \cite{Zhang_2}, and Eulerian-Lagrangian methods, such as CFD-DEM (Computational Fluid Dynamics - Discrete Element Method) \cite{Khosronejad}, are being used in the study of barchan dunes. One of their advantages is that information on the grain scale can be obtained even inside a barchan dune, which is considerably difficult to be obtained from experiments. However, the coupling between the fluid flow and bed load rests to be validated in those simulations, and, therefore, experimental measurements at the grain scale are still important.
Zhang et al. \cite{Zhang_2} investigated numerically the mean residence time of individual grains in barchans in order to understand the contribution of each grain for the existence of a barchan dune. Using a 3D cellular automaton model, the authors tracked individual particles from the time they are incorporated to the barchan dune to the time they leave it by one of its horns. For different size and flow conditions, they showed the existence of two antagonistic lateral fluxes of grains, which are outward on the windward side (upstream the dune crest) and inward on the lee face (downstream the dune crest). Because of that antagonistic movement, grains in the central region tend to remain in this region (and in the dune), while grains close to lateral flanks tend to move to the horns and leave the dune. As a result, Zhang et al. \cite{Zhang_2} found that the mean residence time does not depend strongly on the dune size. Instead, it is given by the surface of the longitudinal central slice of the dune divided by the input sand flux.
\begin{sloppypar}
Recent studies investigated the formation of subaqueous barchans from initially conical heaps by considering the growth of horns \cite{Alvarez,Alvarez3}. Alvarez and Franklin \cite{Alvarez} measured experimentally the growth of horns on conical heaps under turbulent water flows. For the length of horns as a function of time, Alvarez and Franklin \cite{Alvarez} showed the existence of an initially positive slope, corresponding to its development, and a final plateau, corresponding to an equilibrium length for horns. Based on the evolution of horns, they proposed the characteristic times $0.5 t_c$ for the growth and $2.5 t_c$ for equilibrium of barchans, where $t_c$ is a characteristic time for the displacement of barchans computed as the length of the bedform divided by its celerity. Alvarez and Franklin \cite{Alvarez3} investigated the trajectories of individual grains during the formation of horns. The authors showed that most of grains forming the horns during their growth migrate from upstream regions on the periphery of the initial heap. In addition, they showed that individual grains have transverse displacements by rolling and sliding that are not negligible, contrasting with the general assertions for aeolian dunes that transverse displacements are due mainly to the diffusive effect of reptons and that horns grow mostly with grains originally in the flanks of the initial pile.
\end{sloppypar}
\begin{sloppypar}
The velocity fields and trajectories of grains over barchan dunes are important to understand not only the growth of these forms, but also their morphology and migration. In addition, this information is essential to fit adjustable constants in continuum models and to validate Lagrangian and Eulerian-Lagrangian methods. However, there are very few experimental studies on displacements of individual grains over subaqueous forms \cite{Alvarez,Alvarez3}. Because the water flow varies over the dune surface, the results obtained for bed load over plane granular beds are not valid for bed load over barchans. To the authors' knowledge, no previous study reported the velocity fields of grains over developed barchan dunes.
\end{sloppypar}
This paper presents an experimental investigation on the displacements of individual grains over subaqueous barchans. The experiments were performed in a closed conduit of transparent material and the dunes were filmed with a high-speed camera. The velocity fields and the trajectories of individual grains were computed from the acquired images with numerical scripts written in the course of this work. Our results show that the fields of grain velocities present local values within 1 and 10\% of the cross-sectional mean velocity of the fluid, and that, considering the average trajectories of grains moving over a given dune, grains have mean displacements within 30 and 60 grain diameters, with characteristic velocities within 10 and 20 \% of the cross-sectional mean velocity of the fluid. The displacement time varies between 30 and 90 \% of the settling time, and it seems to have two asymptotic behaviors: one close to bed load inception and other far from it. When compared with bed load over a plane bed, we observe that grains have the same mean velocity, but they travel distances up to 5 times larger, with higher densities of moving grains. Those findings contribute to increase our understanding of the dynamics of barchan dunes and can represent significant advancements on future numerical simulations involving barchans.
The next sections present the experimental setup and experimental results. The following section presents the conclusions.
\section{Experimental setup}
The experimental setup consisted of, in addition to a high speed camera and LED (Light-Emitting Diode) lamps, a water reservoir, centrifugal pumps, an electromagnetic flow meter, a flow straightener, a 5 m-long channel, and a settling tank, the water flowing in closed loop in the order just described. The channel was a closed conduit of rectangular cross section, 160 mm wide by 50 mm high, made of transparent material (plexiglass). Its downstream part contained the test section, which started 3 m downstream of the channel inlet and was 1 m long, the remaining 1 m section connecting the test section to the settling tank. The bottom wall of the test section was made of black plexiglass in order to minimize undesired reflection from that wall. The flow rates were adjusted with a set of valves, and the water flow was homogenized upstream of the channel entrance by the flow straightener, which consisted of a divergent-convergent nozzle filled with glass spheres. Fig. \ref{fig:1} shows a layout of the experimental setup and Fig. \ref{fig:test_section} presents a photograph of the test section.
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\columnwidth]{esquema_disp.eps}
\caption{Layout of the experimental setup.}
\label{fig:1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\columnwidth]{CrossSection1.eps}
\caption{Photograph of the test section.}
\label{fig:test_section}
\end{figure}
Prior to each test, the grains were poured in the test section, which was previously filled with water, and they formed a single conical heap at the bottom wall of the channel. Afterward, for each test run, a turbulent water flow was imposed in the channel, deforming the initial pile into a barchan dune. With that procedure, each experiment concerned one single barchan that loosed grains by its horns, and, therefore, decreased slowly in size while migrating. The employed fluid was tap water at temperatures within 24 and 26 $^o$C, and the employed grains were round glass beads with $\rho_s = 2500$ kg/m$^3$ and bulk density of 1500 kg/m$^3$. The grains were separated in two different populations according to their diameter: $0.40$ mm $\leq\,d\,\leq$ $0.60$ mm and $0.15$ mm $\leq\,d\,\leq$ $0.25$ mm, called here types 1 and 2, respectively; therefore, $d_1$ = 0.50 mm and $d_2$ = 0.20 mm are the mid-range mean diameters of types 1 and 2, respectively. The cross-sectional mean velocities $U$ were 0.243, 0.295 and 0.365 m/s, which correspond to Reynolds numbers based on the hydraulic diameter $Re=\rho U D_h /\mu$ of $1.9 \times 10^4$, $2.2 \times 10^4 $ and $2.8 \times 10^4 $, respectively, where $\mu$ is the dynamic viscosity of the fluid and $D_h$ is the hydraulic diameter of the channel. The initial heaps were formed with either 6.2 g or 10.3 g of glass beads, corresponding to initial volumes of 4.1 and 6.9 cm$^3$, respectively. We employed white and black glass beads, where 98\% of grains were white and 2\% black, the latter being used as tracers. Tracers were used because they are easier to track along images and, as they had the same diameter and surface characteristics as the other grains, we assumed that their displacements characterize the whole moving bed.
A high-speed camera of CMOS (Complementary Metal Oxide Semiconductor) type was placed above the channel in order to record the bed evolution (Fig. \ref{fig:test_section}). We used either a camera with a spatial resolution of 1600 px $\times$ 2560 px at frequencies up to 1400 Hz or one with a resolution of 1280 px $\times$ 1024 px at frequencies up to 1000 Hz, both controlled by a computer. In our tests, we set the frequency to values within 200 Hz and 300 Hz depending on the average velocity of grains, and we used a Nikkor lens of $60\,mm$ focal distance and F2.8 maximum aperture. LED lamps were branched to a continuous current source in order to supply the required light and, at the same time, avoid beating between the camera and light frequencies. Prior to the beginning of tests, a calibration procedure which consisted of taking one picture from a scale placed in the water channel was performed, allowing the conversion from pixels to a physical system of units. For the lower resolution camera, we set the ROI (region of interest) to 800 px $\times$ 1024 px to better fit a field of view of 79.4 mm $\times$ 101.7 mm, whereas for the higher resolution one we set the ROI to 1208 px $\times$ 1648 px in order to fit a field of 84.3 mm $\times$ 115.0 mm; therefore, in the acquired images the areas covered by particles of type 1 are of the order of 25 and 40 px for the lower and higher resolution cameras, respectively, and of the order of 5 and 10 px in the case of type 2 particles. The images were recorded in RGB (red, blue and green) format. An example of movie, showing the motion of grains over a barchan dune, is available as Supplementary Material \cite{Supplemental}.
During the tests we did not impose an influx of grains. In this way, the barchan dune decreased in size while migrating along the channel. However, changes in both size and shape were negligible within the duration of each test (between 4 and 65 s).
\subsection{Image processing}
\label{section_image_processing}
The acquired images were processed by numerical scripts written in the course of this work, which use a PTV (Particle Tracking Velocimetry) approach that compares pairs of images, finds the tracers, and follow them along images. Once the tracers are identified in each movie frame, an Eulerian or a Lagrangian framework can be adopted to find the velocity fields or the trajectories of tracers, respectively.
The first step in the image processing code is to remove the image background. For that, the scripts convert the images to grayscale in order to better differentiate the black beads from the background, which is also black but with whiter shades than the black beads. With the background removed, the images are binarized, the main properties of the dune such as its centroid and area are obtained, and the tracers are identified. Given the presence of noise with size of 1 px, caused by small changes in light or small displacements (due to vibration) of the camera, the next step is the filtering with an area filter to distinguish the tracers from noise. This is necessary because the tracers are relatively small objects (covering from 5 to 40 px in the images). The area filter consists in the removal of objects with areas equal or smaller than 2 to 10 px, depending on the tested field. Once applied the filter, all the tracers are identified in each image.
Since the proportion of tracers is small (2\% of the dune), they can be tracked along images. For that, the code follows a sequence of rules to match the tracers in consecutive images. Those rules take advantage of previous knowledge concerning bed load under water flows, namely that grains move mainly in the longitudinal direction with velocities of the order of 10\% of the cross-sectional mean velocity of water. Because the acquiring frequencies were sufficiently high, only four rules are necessary to match the tracers in image pairs: (i) to limit the maximum displacement of grains to within 30 and 50\% of the mean fluid displacement; (ii) to limit the variation of the area of a given tracer to 40 \%, i.e., a tracer in a given image must have an area within 60\% and 140\% of a tracer in the preceding image to allow matching; (iii) with the exception of tracers in the recirculation region, to assure that the longitudinal displacement is downstream; (iv) in case more than one tracer in one image can match a certain tracer in another image after the three previous rules were applied, to chose the tracer with the smallest transverse displacement between the considered images. Tracers that do not comply with those four rules are assumed to be no longer visible in the succeeding image, having left the dune by its horns or been buried under the bed, for instance. Once the tracers are identified and matched along images, they are labeled and their centroid positions stored together with the dune properties. Figs. \ref{fig:tracers}(a) and \ref{fig:tracers}(b) show, respectively, raw and processed images of the top view of a barchan dune. The black tracers over the surface of the barchan dune are visible in the raw image, and they are identified in the processed image by red circles with origin in the tracer centroids.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.90\columnwidth]{raw_developed_dune.eps}\\
(a)\\
\includegraphics[width=0.90\columnwidth]{pros_developed_dune.eps}\\
(b)
\end{tabular}
\end{center}
\caption{Top views of a barchan dune showing the visible black tracers: (a) raw image showing the tracers over the dune; (b) processed image with all the tracers identified by red circles. This dune was formed with $m$ = 10.3 g of type 1 grains ($d_1$ = 0.50 mm), under $U$ = 0.365 m/s. This corresponds to $Re_*$ = 10 and $\theta$ = 0.055. In the images, the flow is from top to bottom.}
\label{fig:tracers}
\end{figure}
The last steps are the computations in Eulerian and Lagrangian frameworks. For the Eulerian framework, the centroid positions of all matched tracers in consecutive images are subtracted and multiplied by the acquiring frequency. Next, Cartesian meshes with origin at the dune centroid are built, and the average velocity of grains in a cell is computed for each cell of the mesh. Finally, a temporal average is computed for each cell by considering all images of a test run (999 or 1299 images for tests with $Re$ = $2.8 \times 10^4 $ and 5997 for the remaining tests). With this procedure, fields of mean velocity of tracers are obtained for the considered mesh size. Sensitivity tests were performed in the meshes to determine the smallest size allowing the spatio-temporal averages. Based on that, we've chosen cell sizes of 25 px $\times$ 25 px.
For the Lagrangian framework, the evolution of centroid positions along time is determined for matched tracers, and the typical distances and velocities are computed.
\section{Results}
\subsection{Eulerian Framework}
Based on the centroid positions of tracers, Cartesian meshes were built, the mean velocities in each cell computed, and, afterward, a temporal average considering all images was computed for each cell, $<\overline{v}>$, as described in Subsection \ref{section_image_processing}. With this procedure we computed the mean velocity fields of tracers over a barchan dune. Because the only difference between tracers and the remaining grains was their color (they were both of colored glass), we assume in the following that the their behavior is representative of all grains forming the dune.
Figure \ref{fig:quiver1} presents the mean velocity field of grains over a barchan formed from 10.3 g of type 2 grains under a water flow with $U$ = 0.295 m/s. This corresponds to $Re$ = $2.2 \times 10^4$, $Re_*$ = 8 and $\theta$ = 0.038. In the figure, the abscissa and ordinate correspond, respectively, to the transverse ($x$) and longitudinal ($y$) coordinates, the magnitude of arrows is proportional to that of velocities, and the direction of arrows is the same as that of velocities.
\begin{sloppypar}
From Fig. \ref{fig:quiver1} we note that velocities increase from upstream position toward the dune centroid (and the dune crest), with a stronger longitudinal component than the transverse component, but with considerable transverse components close to the lateral flanks of the dune and in the region downstream of the crest. In this region, we observe small vectors pointing to the symmetry line of the barchan, corresponding to the mean velocities of grains falling by avalanches in the lee side. Just downstream of these vectors, we observe even smaller vectors with relatively strong transverse components pointing toward the symmetry line of the barchan, and which correspond to the mean velocities of grains in the recirculation region. Those regions and the motion of grains can be identified in the movie available as Supplementary Material \cite{Supplemental}. All the other test conditions showed similar behaviors, and their respective mean fields are also available as Supplementary Material \cite{Supplemental}.
\end{sloppypar}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.90\columnwidth]{TN14_quiver.eps}\\
\end{center}
\caption{Mean velocity field of grains over a dune formed with $m$ = 10.3 g of type 2 grains under $U$ = 0.295 m/s, corresponding to $Re$ = $2.2 \times 10^4$, $Re_*$ = 8 and $\theta$ = 0.038. The magnitude of arrows is proportional to that of velocities.}
\label{fig:quiver1}
\end{figure}
An easier way to localize the mean grain velocities is by identifying the mean values of velocities by using polar coordinates centered at the dune centroid. The radial and angular plots were used by Alvarez and Franklin \cite{Alvarez3} to localize the origin of grains migrating to horns, and are used in the following to localize the magnitude of mean velocities.
Figures \ref{fig:hist1}(a) and \ref{fig:hist1}(b) present the mean velocity $<\overline{v}>$ as functions of the radial position $r$ (with origin at the dune centroid) and the angle with respect to the transverse direction, respectively. In Fig. \ref{fig:hist1}(a) the abscissa corresponds to the radial position, the ordinate to the magnitude of the velocity vector, and the width of bars to the interval between the considered radial positions. In Fig. \ref{fig:hist1}(b), the numbers along the perimeter correspond to angles with respect to the transverse direction (the water flow direction is 270$^{\circ}$) and the height of bars to the magnitude of velocity, which can be measured using the radial scale along the 80$^{\circ}$ line.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.90\columnwidth]{TN14_hist.eps}\\
(a)\\
\includegraphics[width=0.90\columnwidth]{TN14_polarHist.eps}\\
(b)
\end{tabular}
\end{center}
\caption{Mean velocity $<\overline{v}>$ as a function of (a) the radial position $r$ and (b) the angle with respect to the transverse direction. Velocities obtained for a dune formed with $m$ = 10.3 g of type 2 grains under $U$ = 0.295 m/s, corresponding to $Re$ = $2.2 \times 10^4$, $Re_*$ = 8 and $\theta$ = 0.038.}
\label{fig:hist1}
\end{figure}
Figures \ref{fig:hist1}(a) and \ref{fig:hist1}(b) corroborate the observation made from Fig. \ref{fig:quiver1} about the existence of higher velocities in the centroid region, but now we can obtain readily the magnitude of velocities with respect to the dune centroid. The radial and angular charts for all the other test runs are available as Supplementary Material \cite{Supplemental}. We note from those charts that the mean velocities close to the centroid are around 8-10\% of the water velocity, and around 1-5\% of the water velocity far from the centroid, i.e., at the dune periphery.
It is important to remark here that the mean Eulerian velocities were computed by considering all displacements of tracers, even those which were accelerating or stopping. Such velocity criteria is different from the Langangian mean velocity presented in Subsection \ref{subsection_lagrangian}, where we did not considered strongly accelerating and decelerating tracers.
\subsection{Lagrangian Framework}
\label{subsection_lagrangian}
We obtained the centroid positions of tracers as functions of time with the procedure described in Subsection \ref{section_image_processing}, and from them we computed the distances and instantaneous velocities for the tracers. Figs. \ref{fig:lagrangian} and \ref{fig:lagrangian2} present examples of trajectories of tracers over a dune formed with $m$ = 10.3 g of type 1 grains under $U$ = 0.365 m/s, which corresponds to $Re_*$ = 10 and $\theta$ = 0.055. Figs. \ref{fig:lagrangian}a and \ref{fig:lagrangian}b show, respectively, the longitudinal and transverse positions, $y$ and $x$, and the longitudinal and transverse velocities, $v_y$ and $v_x$, of a tracer as functions of time. Fig. \ref{fig:lagrangian} corresponds to a tracer that was on the right side (with respect to the flow direction) of the barchan symmetry line. Figs. \ref{fig:lagrangian2}a and \ref{fig:lagrangian2}b show the same thing for a tracer that was on the left side of the barchan symmetry line. In Figs. \ref{fig:lagrangian} and \ref{fig:lagrangian2}, the red continuous and blue dashed curves correspond to the $y$ and $x$ components, respectively.
The behavior depicted in Figs. \ref{fig:lagrangian} and \ref{fig:lagrangian2} is typical of the ensemble of tracers tracked in our tests: they move mainly in the flow direction, with small transverse components, and present an intermittent motion, having periods of acceleration, deceleration and at rest. The intermittent motion is similar to that observed over plane beds for water velocities and grain diameters of the same order of magnitude as in the present study \cite{Penteado}. Transverse movements can be significant close to the dune flanks and downstream the lee side because the water flow direction changes significantly in these regions, as shown by Alvarez and Franklin \cite{Alvarez3} in the case of growing barchans, and as can be remarked in the movie available as Supplementary Material \cite{Supplemental}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.60\columnwidth]{PxT_right.eps}\\
(a)\\
\\
\\
\includegraphics[width=0.60\columnwidth]{VELxT_right.eps}\\
(b)
\end{tabular}
\end{center}
\caption{Example of tracking of a single grain over a dune formed with $m$ = 10.3 g of grains of type 1 under $U$ = 0.365 m/s, corresponding to $Re_*$ = 10 and $\theta$ = 0.055. (a) Longitudinal and transverse positions, $y$ and $x$, and (b) longitudinal and transverse velocities, $v_y$ and $v_x$, of a tracer as functions of time. The red continuous and blue dashed curves correspond to the $y$ and $x$ components, respectively. The origin of the $y(t),x(t)$ graphic is at the original position of the tracked particle, which in this case was to the right of the barchan symmetry line.}
\label{fig:lagrangian}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.60\columnwidth]{PxT_left.eps}\\
(a)\\
\\
\\
\includegraphics[width=0.60\columnwidth]{VELxT_left.eps}\\
(b)
\end{tabular}
\end{center}
\caption{Example of tracking of a single grain over a dune formed with $m$ = 10.3 g of grains of type 1 under $U$ = 0.365 m/s, corresponding to $Re_*$ = 10 and $\theta$ = 0.055. (a) Longitudinal and transverse positions, $y$ and $x$, and (b) longitudinal and transverse velocities, $v_y$ and $v_x$, of a tracer as functions of time. The red continuous and blue dashed curves correspond to the $y$ and $x$ components, respectively. The origin of the $y(t),x(t)$ graphic is at the original position of the tracked particle, which in this case was to the left of the barchan symmetry line.}
\label{fig:lagrangian2}
\end{figure}
For the moving grains, as the ones of Figs. \ref{fig:lagrangian} and \ref{fig:lagrangian2}, we computed displacements and velocities by considering only the moving periods. In Figs. \ref{fig:lagrangian}a and \ref{fig:lagrangian2}a, the moving periods correspond to positive slopes of the continuous line; therefore, our scripts identify those positive slopes for a given tracer, and then compute the displacements as the differences between the initial and final positions and the velocities as the slope values. This procedure gives the displacements between resting times and the corresponding velocities experienced by a single tracer. The scripts repeat the same computations for all the moving tracers of each test run, and afterward compute single averages and standard deviations for the entire test run. Because the mean lagrangian velocity is computed as the average of measured slopes, it does not take into account the accelerating and decelerating stages, being higher than the corresponding Eulerian averages.
\begin{sloppypar}
Table \ref{tab_dim} shows the mean values and standard deviations of longitudinal and transverse displacements and velocities, for each test run. The table presents the initial mass $m$, grain diameter $d$, mean cross-sectional velocity of the fluid $U$, shear velocity $u_*$, mean transverse displacement $\Delta x$, mean longitudinal displacement $\Delta y$, standard deviation of the mean transverse displacement $\sigma_{\Delta x}$, standard deviation of the mean longitudinal displacement $\sigma_{\Delta y}$, mean transverse velocity $<v_{x,lag}>$, mean longitudinal velocity $<v_{y,lag}>$, standard deviation of the mean transverse velocity $\sigma_{v x}$, standard deviation of the mean longitudinal velocity $\sigma_{v y}$, displacement time $t_d$, number of images $N_{im}$, total number of moving tracers $N_{mv}$, and movement density $\rho_{mv}$. The displacement time was computed as $\Delta y / <v_{y,lag}>$, and the shear velocity $u_*$ corresponds to the flow over the channel wall (of acrylic), the latter computed from velocity profiles measured with PIV (Particle Image Velocimetry).
\end{sloppypar}
\begin{table*}[t
\begin{center}
\caption{Initial mass $m$, grain diameter $d$, mean cross-sectional fluid velocity $U$, shear velocity $u_*$, mean transverse displacement $\Delta x$, mean longitudinal displacement $\Delta y$, standard deviation of the mean transverse displacement $\sigma_{\Delta x}$, standard deviation of the mean longitudinal displacement $\sigma_{\Delta y}$, mean transverse velocity $<v_{x,lag}>$ , mean longitudinal velocity $<v_{y,lag}>$, standard deviation of the mean transverse velocity $\sigma_{v x}$, standard deviation of the mean longitudinal velocity $\sigma_{v y}$, displacement time $t_d$, number of images $N_{im}$, total number of moving tracers $N_{mv}$, and movement density $\rho_{mv}$.}
\begin{tabular}{c c c c c c c c c c c c c c c c}
\hline\hline
$m$ & $d$ & $U$ & $u_*$& $\Delta_x$ & $\Delta_y$ & $\sigma_{\Delta x}$ & $\sigma_{\Delta y}$ & $<v_{x,lag}>$ & $<v_{y,lag}>$ & $\sigma_{v x}$ & $\sigma_{v y}$ & $t_d$ & $N_{im}$ & $N_{mv}$ & $\rho_{mv}$\\
g & mm & m/s & m/s & mm & mm & mm & mm & mm/s & mm/s & mm/s & mm/s & s & $\cdots$ & $\cdots$ & m$^{-2}$\\
\hline
10.3 & 0.5 & 0.365 & 0.0202 & 0.1 & 17.6 & 4.2 & 15.3 & 0.4 & 71.1 & 18.9 & 37.3 & 0.25 & 999 & 5994 & 217391\\
10.3 & 0.5 & 0.365 & 0.0202 & 0.3 & 18.0 & 4.5 & 16.4 & 0.9 & 69.7 & 18.7 & 38.2 & 0.26 & 999 & 5373 & 194869\\
10.3 & 0.5 & 0.365 & 0.0202 & 0.2 & 16.5 & 4.1 & 15.3 & 1.1 & 64.6 & 17.7 & 35.9 & 0.26 & 999 & 6208 & 225153\\
6.2 & 0.5 & 0.365 & 0.0202 & 0.1 & 15.7 & 3.9 & 12.9 & 0.1 & 69.8 & 19.4 & 36.2 & 0.22 & 1299 & 7400 & 347359\\
6.2 & 0.5 & 0.365 & 0.0202 & 0.0 & 15.7 & 4.1 & 12.6 & 0.3 & 72.2 & 20.0 & 36.9 & 0.22 & 999 & 7047 & 430125\\
6.2 & 0.5 & 0.365 & 0.0202 & 0.0 & 17.2 & 4.1 & 15.2 & 0.1 & 70.3 & 18.5 & 37.0 & 0.24 & 1299 & 3410 & 160067\\
10.3 & 0.5 & 0.295 & 0.0168 & -0.1 & 19.8 & 5.4 & 25.4 & -0.1 & 42.7 & 13.7 & 27.9 & 0.46 & 5997 & 8393 & 50708\\
10.3 & 0.5 & 0.295 & 0.0168 & 0.3 & 22.2 & 5.8 & 29.1 & 0.6 & 44.4 & 13.8 & 28.5 & 0.50 & 5997 & 8317 & 50249\\
10.3 & 0.5 & 0.295 & 0.0168 & 0.4 & 21.3 & 5.5 & 26.0 & 0.6 & 44.8 & 13.6 & 38.8 & 0.48 & 5997 & 11480 & 69358\\
6.2 & 0.5 & 0.295 & 0.0168 & 0.4 & 23.3 & 5.6 & 29.4 & 0.9 & 46.3 & 13.4 & 29.7 & 0.50 & 5997 & 8149 & 82856\\
6.2 & 0.5 & 0.295 & 0.0168 & 0.3 & 21.9 & 5.5 & 27.0 & 0.4 & 46.5 & 14.2 & 29.8 & 0.47 & 5997 & 7967 & 81006\\
6.2 & 0.5 & 0.295 & 0.0168 & 0.1 & 21.6 & 5.7 & 28.0 & 0.1 & 45.3 & 13.9 & 29.4 & 0.48 & 5997 & 7447 & 75719\\
10.3 & 0.5 & 0.243 & 0.0141 & -0.1 & 21.1 & 6.3 & 30.1 & -1.0 & 34.1 & 12.8 & 22.1 & 0.62 & 5997 & 2516 & 15201\\
10.3 & 0.5 & 0.243 & 0.0141 & 0.4 & 20.0 & 5.6 & 27.9 & 0.8 & 33.5 & 11.9 & 22.7 & 0.60 & 5997 & 2060 & 12446\\
10.3 & 0.5 & 0.243 & 0.0141 & 0.3 & 20.6 & 5.5 & 28.8 & 0.7 & 34.2 & 12.6 & 23.1 & 0.60 & 5997 & 2495 & 15074\\
6.2 & 0.5 & 0.243 & 0.0141 & -0.1 & 17.0 & 5.0 & 25.2 & -0.4 & 29.9 & 12.2 & 20.8 & 0.57 & 5997 & 1485 & 15099\\
6.2 & 0.5 & 0.243 & 0.0141 & -0.2 & 21.5 & 5.8 & 31.9 & -0.2 & 37.0 & 13.2 & 23.4 & 0.58 & 5997 & 1490 & 15150\\
6.2 & 0.5 & 0.243 & 0.0141 & -0.2 & 21.8 & 5.8 & 34.0 & -0.1 & 31.5 & 13.4 & 21.9 & 0.69 & 5997 & 1205 & 12252\\
10.3 & 0.2 & 0.295 & 0.0168 & 0.1 & 11.4 & 2.8 & 11.3 & 0.3 & 34.0 & 9.0 & 18.3 & 0.34 & 5997 & 70513 & 170406\\
10.3 & 0.2 & 0.295 & 0.0168 & 0.1 & 11.1 & 2.7 & 10.5 & 0.2 & 34.6 & 9.3 & 18.3 & 0.32 & 5997 & 78275 & 189165\\
10.3 & 0.2 & 0.295 & 0.0168 & 0.0 & 10.7 & 2.7 & 10.2 & 0.1 & 33.7 & 9.1 & 17.9 & 0.32 & 5997 & 77985 & 188464\\
6.2 & 0.2 & 0.295 & 0.0168 & -0.1 & 10.4 & 2.5 & 11.4 & -0.6 & 29.5 & 7.6 & 15.7 & 0.35 & 5997 & 53786 & 218752\\
6.2 & 0.2 & 0.295 & 0.0168 & -0.1 & 10.9 & 2.7 & 11.5 & -0.2 & 32.0 & 8.5 & 17.4 & 0.34 & 5997 & 55156 & 224324\\
\end{tabular}
\label{tab_dim}
\end{center}
\end{table*}
From Tab. \ref{tab_dim}, we observe that, by averaging over the whole dune, grains move mainly in the longitudinal direction, with both $\Delta_y / \Delta_x$ and $<v_{y,lag}> / <v_{x,lag}>$ having an order of magnitude equal to 100. It is important to note that, if we consider individual grains migrating from and to certain parts of the dune, the transverse components can be significant. As an example, that happens to the grains that move along the periphery of the dune, i.e., that go around the dune from its leading edge to one of its horns, as shown by Alvarez and Franklin \cite{Alvarez3}. In the present study, we investigate displacements averaged over the whole barchan. For these averages, as the fluid shearing increases, the longitudinal displacement decreases slightly while the longitudinal velocity increases and the displacement time decreases. This result is intriguing and was not \textit{a priori} expected. However, by analyzing the movies, we can observe that grains indeed travel shorter distances faster for higher shear velocities, but the quantity of moving grains is much higher, which indicates higher transport rates. The total number of moving tracers identified in all images $N_{mv}$, irrespective if they were the same from previous images (i.e., they could be already moving in previous images), are also shown in Tab. \ref{tab_dim}. A measure of the density of moving grains can be computed by considering the total field of each test, the total number of images of each test run $N_{im}$, and that the behavior of tracers is characteristic of the ensemble of grains. With these assumptions, the following quantity is proportional to the density of moving grains,
\begin{equation}
\rho_{mv} \,=\, \frac{50N_{mv}}{N_{im}A_{d}} \,,
\label{eq_density}
\end{equation}
\noindent where $A_{d}$ is the characteristic dune area (its volume divided by 10$d$) and $\rho_{mv}$ is called here movement density because it is not exactly the density of moving grains, but it is a reasonable approximation of it. Table \ref{tab_dim} shows that, indeed, the density of moving grains increases with the fluid shearing. Comparisons with Penteado and Franklin \cite{Penteado} show densities of the same order of magnitude for lower Shields numbers, and higher for higher Shields numbers (please see Tab. \ref{tab.1} for the Shields numbers). The computations employed for obtaining $\rho_{mv}$ in Penteado and Franklin \cite{Penteado} were different from ours; however, even considering uncertainties and errors, the results point to an increase in $\rho_{mv}$ over a barchan dune when compared to a flat bed, at least for Shields numbers equal or greater than 0.055. The higher density of moving grains is in accordance with the conclusions of Lajeunesse et al. \cite{Lajeunesse}, who found that the erosion rate depends locally on the shear stress caused by the liquid. Concerning variations with the grain diameter, both the longitudinal displacement and velocity increase with its augmentation.
Two important length scales for bed load are the grain diameter $d$ and the saturation length $L_{sat}\,\sim\,L_{drag}$, the latter being a length scale for the stabilization of sand flux after or downstream changes in fluid or sediment conditions \cite{Andreotti_1,Andreotti_2,Hersen_1}. One way to investigate if they are pertinent scales is by normalizing the longitudinal displacements by both parameters; therefore, we computed the dimensionless displacements $\Delta_{y,d}^{ad}\,=\,\Delta_y / d$ and $\Delta_{y,Ldrag}^{ad}\,=\,\Delta_y / L_{drag}$. Concerning the velocities, two important scales are the mean and shear velocities, $U$ and $u_*$, respectively. However, instead of using directly the shear velocity $u_*$ as a scale, the use the mean shear at the grain scale, $\partial_y u\,d$ = $u_*^2 d / \nu$, where $\partial_y u$ is the vertical gradient of the fluid velocity, may be more convenient. This scale represents the mean velocity at the grain scale. Therefore, we computed the dimensionless velocities $V_{y,U}^{ad} \,=\, <v_{y,lag}> / U$ and $V_{y,*}^{ad} \,=\, <v_{y,lag}> / ( u_*^2 d / \nu )$. Finally, a natural time scale for bed load is the settling time, $d / U_s$, where $U_s$ is the settling velocity of one single grain. This time scale represents the time that a grain takes to fall a distance equivalent to its diameter. Therefore, we computed the dimensionless displacement times $t_d^{ad} \,=\, t_d U_s/d$.
\begin{sloppypar}
Table \ref{tab.1} shows the average values of equivalent test runs. It presents the initial mass $m$, grain diameter $d$, Reynolds number based on the mean velocity and hydraulic diameter $Re$, Reynolds number based on the shear velocity and grain diameter $Re_{*}$, Shields number $\theta$, mean longitudinal displacement normalized by the grain diameter $\Delta_{y,d}^{ad}$, mean longitudinal displacement normalized by the drag length $\Delta_{y,Ldrag}^{ad}$, longitudinal velocity normalized by the cross-sectional mean velocity $V_{y,U}^{ad}$, longitudinal velocity normalized by the mean shear and grain diameter $V_{y,*}^{ad}$, and displacement time normalized by the settling velocity and grain diameter $t_d^{ad}$. In Tab. \ref{tab.1}, the dimensional values were averaged over the test runs under the same experimental conditions; therefore, it corresponds to a summary of all experiments.
\end{sloppypar}
\begin{table*}[t
\begin{center}
\caption{Initial mass $m$, grain diameter $d$, Reynolds number based on the mean velocity and hydraulic diameter $Re$, Reynolds number based on the shear velocity and grain diameter $Re_{*}$, Shields number $\theta$, mean longitudinal displacement normalized by the grain diameter $\Delta_{y,d}^{ad}$, mean longitudinal displacement normalized by the drag length $\Delta_{y,Ldrag}^{ad}$, longitudinal velocity normalized by the cross-sectional mean velocity $V_{y,U}^{ad}$, longitudinal velocity normalized by the mean shear and grain diameter $V_{y,*}^{ad}$, and displacement time normalized by the settling velocity and grain diameter $t_d^{ad}$. Averaged values for test runs corresponding to the same experimental conditions.}
\begin{tabular}{c c c c c c c c c c}
\hline\hline
$m$ & $d$ & $Re$ & $Re_*$ & $\theta$ & $\Delta_{y,d}^{ad}$ & $\Delta_{y,Ldrag}^{ad}$ & $V_{y,U}^{ad}$ & $V_{y,*}^{ad}$ & $t_d^{ad}$\\
g & mm & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ &$ \cdots$\\
\hline
10.3 & 0.5 & 2.8 $\times$ 10$^4$ & 10 & 0.055 & 34.7 & 13.9 & 0.19 & 0.34 & 37.1\\
6.2 & 0.5 & 2.8 $\times$ 10$^4$ & 10 & 0.055 & 32.4 & 13.0 & 0.19 & 0.35 & 33.6\\
10.3 & 0.5 & 2.2 $\times$ 10$^4$ & 8 & 0.038 & 42.2 & 16.9 & 0.15 & 0.31 & 70.4\\
6.2 & 0.5 & 2.2 $\times$ 10$^4$ & 8 & 0.038 & 44.5 & 17.8 & 0.16 & 0.33 & 71.0\\
10.3 & 0.5 & 1.9 $\times$ 10$^4$ & 7 & 0.027 & 41.1 & 16.4 & 0.14 & 0.34 & 90.0\\
6.2 & 0.5 & 1.9 $\times$ 10$^4$ & 7 & 0.027 & 40.2 & 16.1 & 0.13 & 0.33 & 90.1\\
10.3 & 0.2 & 2.2 $\times$ 10$^4$ & 3 & 0.095 & 55.3 & 22.1 & 0.12 & 0.61 & 37.2\\
6.2 & 0.2 & 2.2 $\times$ 10$^4$ & 3 & 0.095 & 53.3 & 21.3 & 0.10 & 0.55 & 39.7\\
\end{tabular}
\label{tab.1}
\end{center}
\end{table*}
For the ensemble of our tests, the mean displacement decreases slightly with the Shields number, indicating a decrease in traveled distances with the fluid shear. By considering that the shear velocities over a barchan dune reach, close to its crest, 1.4 of the value over the flat bed \cite{Andreotti_1,Andreotti_2}, Shields numbers over barchans reach values of approximately twice those over flat beds. The values of $u_*$ reported in the present paper were computed from profiles measured over the channel wall; however, $u_*$ values measured over plane beds with similar grains showed that the flow is close to hydraulic smooth regimes and the feedback effect is relatively weak \cite{Franklin_9}. Therefore, by considering the values of $2 \theta$ as representative of the normalized shear over the barchan, and by observing that displacement lengths over the barchan dune are between 13 and 18 $L_{drag}$ and 30 and 40 $d$ for type 1 grains and around 21-22 $L_{drag}$ and 55 $d$ for type 2, the values of $\Delta_{y,d}^{ad}$ can be directly compared with values reported by Lajeunesse et al. \cite{Lajeunesse} and Penteado and Franklin \cite{Penteado} for plane beds with the same range of Shields values. The comparison shows that mean values over barchan dunes are three to five times higher than those obtained over liquid-sheared plane beds. Considering that the threshold Shields $\theta_c$ is higher over the dune positive slope and that the comparison is made for the same range of Shields values, the longer displacements over the barchan dune indicates that details of the structure of the water flow over the dune affect substantially bed load characteristics. It has been proposed that the curvature of streamlines induce higher turbulent stresses at low regions of the boundary layer close to the dune leading edge and smaller ones at low regions close to the dune crest (first proposed by Wiggs et al. \cite{Wiggs} and afterward measured experimentally, for example, for 2D dunes by C\'u\~nez et al. \cite{Franklin_10}). The higher turbulent stresses over the upwind face of barchans would then be responsible for the longer distances traveled by grains, but this rests to be investigated further.
\begin{sloppypar}
The displacement velocities increase with the cross-sectional mean velocity of the fluid, varying roughly between 10 and 20 \% of $U$ for type 1 grains. When considering the grain diameter at the same $U$, the value of $V_{y,U}^{ad}$ decreases from approximately 0.15 for type 1 grains to approximately 0.10 for type 2. In terms of the mean shear at the grain scale, the displacement velocity remains at approximately 30 \% of $u_*^2 d / \nu$ for type 1 grains, at all flow conditions, and reaches approximately 60 \% of $u_*^2 d / \nu$ for type 2 grains. The values for type 1 grains are equal to those found by Penteado and Franklin \cite{Penteado} for grains of same mean diameter. This indicates that, although the fluid structure changes over the dune \cite{Wiggs}, causing displacements up to five times longer, the velocity of grains remains roughly the same as that over the flat bed. From the dimensional values, type 2 grains have displacement velocities that are smaller than that of larger grains; therefore, comparisons using $V_{y,*}^{ad}$ show that smaller grains move slower over the dune because they are exposed to lower mean velocities.
\end{sloppypar}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.85\columnwidth]{td_theta.eps}\\
(a)\\
\includegraphics[width=0.85\columnwidth]{td_thetadiff.eps}\\
(b)
\end{tabular}
\end{center}
\caption{Normalized displacement time $t_d^{ad}$ as functions of (a) the Shields number over the dune $\theta_d$ and (b) the excess of Shields number over the dune, $\theta_d \, - \, \theta_c$}
\label{fig:td_theta}
\end{figure}
Finally, we observe that the displacement time decreases with the shear stresses caused by the fluid and increases with the grain diameter, varying within roughly 30 and 90 \% of the settling time for type 1 grains and being approximately 40 \% of the settling time for type 2 grains. At first sight, the displacement time seems to not scale with the Shields number. However, by considering that most of dunes were close to incipient bed-load conditions, the type 1 grains may be too close to incipient conditions at the lower fluid velocities. Figures \ref{fig:td_theta}a and \ref{fig:td_theta}b present $t_d^{ad}$ as functions of $\theta_d$ and of $\theta_d - \theta_c$, respectively, where $\theta_d \,=\, 2\theta$ and $\theta_c$ was considered here as 0.04 \cite{Yalin_2,Buffington_1}. The figure shows a negative variation for $\theta_d < 0.1$ and a constant value for $\theta_d \geq 0.1$, indicating the presence two distinct asymptotic behaviors: one close to incipient conditions and other farther from it. However, given the relatively narrow variations of the pertinent parameters in the present study, the existence of different behaviors for $t_d^{ad}$ close and far from bed-load inception needs to be investigated further.
\section{Conclusions}
This paper presented an experimental investigation of the displacements of individual grains over subaqueous barchan dunes. Granular piles were placed in a closed conduit of transparent material where a turbulent water flow was imposed afterward. With this procedure, a single barchan dune was formed, which was filmed with a high-speed camera as it migrated along the channel. The velocity fields and the trajectories of individual grains were computed from the acquired images with numerical scripts written in the course of this work.
For the Eulerian framework, the instantaneous fields of bed-load velocity were obtained for each image by subtracting the positions of moving tracers (grains with different color) from the previous image. Time-averaged fields were then computed by generating a Cartesian mesh with origin at the dune centroid, computing the mean value in each cell, and computing a time average for each cell. These fields are available for all the test runs as Supplementary Material \cite{Supplemental}. The use of graphics of the magnitude of mean velocity as functions of the radial position $r$ (with origin at the dune centroid) and the angle with respect to the transverse direction allows the easy identification of the velocities magnitude and their respective positions over the dune. Depending on the region over the barchan, we found that the mean velocity of grains varies roughly within 1 and 10\% of the cross-sectional mean velocity of the fluid, with the highest velocities in the centroid region, and with important transverse components close to the lateral flanks of the dune and in avalanche and recirculation regions.
For the Lagrangian framework, we observed that individual grains move mainly in the flow direction, with small transverse components, and presenting an intermittent motion, with periods of acceleration, deceleration and at rest. For the range of diameters employed in the present study, our results show that, with the increase of the fluid shearing, the dimensional form of the longitudinal displacement decreases while the longitudinal velocity increases, which was not necessarily expected \textit{a priori}. By computing an average over the whole barchan, we found that the average displacement of grains varies within 30 and 60 grain diameters, which is three to five times higher than displacements obtained over liquid-sheared plane beds by Lajeunesse et al. \cite{Lajeunesse} and Penteado and Franklin \cite{Penteado}. One possible explanation for higher values over the barchan dune is the presence higher turbulent stresses over the upwind face of barchans, as proposed by Wiggs et al. \cite{Wiggs} and measured by C\'u\~nez et al. \cite{Franklin_10}, that would be responsible for the longer distances traveled by grains, but this rests to be investigated further. We found that the average velocity varies within 10 and 20 \% of the cross-sectional mean velocity of the fluid and 30 and 60 \% of the velocity based on the mean shear at the grain scale. The values for type 1 grains are equal to those found by Penteado and Franklin \cite{Penteado} for a flat bed, pointing out that changes in the fluid structure over the dune \cite{Wiggs} do not affect significantly the velocity of grains.
Our results showed that the displacement time varies between 30 and 90 \% of the settling time, decreasing with the Shields number for $\theta_d < 0.1$ and remaining constant for $\theta_d \geq 0.1$. This indicates the existence of two distinct asymptotic behaviors, one close to incipient bed load and other farther from it. However, this needs to be investigated further.
Those findings are a contribution to our knowledge on the dynamics of barchan dunes, and can bring significant advancements on numerical simulations involving barchans.
|
2,869,038,155,718 | arxiv | \section{Introduction}
Organizations are now using algorithmic models\footnote{Throughout this paper, we refer to a large class of algorithmic models using the now-common term ``algorithms'', especially those created through statistical modeling and machine learning.} (``algorithms'') in a variety of important domains, including healthcare~\cite{obermeyer-2019-dissecting}, credit~\cite{fourcade-2013-classification}, employment~\cite{hannak-2017-bias,chen-2018-chi}, and content distribution~\cite{ali-2019-discrimination}.
Sometimes, these algorithms can be beneficial.
But too often, they can lead to discriminatory effects.
For example, in the context of criminal justice, ProPublica showed that the COMPAS risk-assessment tool~\cite{COMPAS} used by judges to help make bail decisions was particularly likely to mislabel Black defendants as future criminals~\cite{courtland-2018-bias}.
Similarly, facial recognition algorithms have been shown to perform significantly worse for Black women~\cite{buolamwini-2018-gender}.
Facebook, in its quest to show relevant ads to users, diverts the delivery of employment and housing ads away from some demographic groups, even when an advertiser is trying to reach a broad, diverse audience~\cite{ali-2019-discrimination}.
These kinds of discriminatory effects can be challenging to detect, measure, and articulate.
Some have proposed mitigating discriminatory effects by removing demographic features from an algorithm's inputs.
For example, as we discuss in more detail in Section~\ref{subsec:hud}, the U.S. Department of Housing and Urban Development (HUD) recently proposed a rule that would apply this approach to housing discrimination~\cite{HUDProposedDisparateImpact}.
This approach is flawed, however, because algorithms can effectively use omitted demographic features by combining other inputs that are each {\em correlated} with the those features, potentially nullifying any protection from discriminatory effects.
This is particularly true in large-scale machine learning (ML) systems, which take can as input hundreds or thousands of features~\cite{blass-2019-algorithmic}.
In this paper, we leverage a unique opportunity created by a recent lawsuit settlement involving Facebook's advertising platform to explore the limits of this approach.
Specifically, we examine Facebook's {\em Lookalike Audiences}~\cite{FacebookLookalikeAudience} targeting tool, which takes a list of Facebook users provided by an advertiser (called the {\em source audience}) and creates a new audience of users who share ``common qualities'' with those in the source audience.
In March 2018, the National Fair Housing Alliance (NFHA) and others sued~\cite{FacebookCivilLawsuit} Facebook over violations of the Fair Housing Act (FHA)~\cite{FairHousingActDiscrimination}.
When the case was settled in March 2019, Facebook agreed to modify the functionality of Lookalike Audiences when used to target housing, credit, and employment ads.
In brief, Facebook created {\em Special Ad Audiences}~\cite{FacebookSpecialAdAudience}, which works like Lookalike Audiences, except its algorithm does {\em not} consider user profile fields of ``age, gender, relationship status, religious views, school, political views, interested in, or zip code'' when detecting common qualities~\cite{FacebookSpecialAdCategory}.
We seek to learn whether the Special Ad Audience algorithm actually produces significantly less {\em biased} audiences than the Lookalike Audience algorithm.
In other words, when provided with a source audience that skews heavily toward one demographic group over another, to what extent do each of these tools reproduce that skew?
We focus on skews along demographic features named in the settlement, enabling us to examine whether simply removing the protected features as input to an algorithm is sufficient eliminate bias along those features.
To do so, we develop a methodology to examine the delivery of the same ads when using the two types of audiences, measuring the skew along the lines of gender, age, race, and political views.
Our results show that:
\begin{itemize}
\item{} For gender, our Special Ad audiences\footnote{Throughout the paper, we use ``Lookalike Audience'' or ``Special Ad Audience'' to refer to the general tools provided by Facebook, and ``Lookalike audience'' or ``Special Ad audience'' to refer to a particular audience created using the tool.} are biased to almost the same degree as Lookalike audiences, with many of the results being statistically indistinguishable.
For example, when using a source audience that is all women, our Lookalike audience-targeted ad delivered to 96.1\% women, while the same ad targeted using Special Ad audiences delivered to 91.2\% women.
\item{} For age, our Special Ad audiences are almost as biased as Lookalike audiences when using source audiences that are from a single age range or a controlled mix of two age ranges.
\item{} For race, we use a different methodology to estimate the racial makeup of the Lookalike and Special Ad Audiences, as Facebook does not report the delivery along racial lines.
Our results suggest that Special Ad Audiences can skew along racial lines, as is true for Lookalike Audiences.
\item{} For political views, we use a similar methodology and find a similar skew for both Lookalike and Special Ad Audiences.
However, we observed less overall bias along political views than race in both audience types.
\item{} To underscore the real-world impact of these results, we place ads as an employer who is seeking to find candidates ``similar to'' to their current workforce using Special Ad Audiences.
Using a source audience consisting of Facebook employees---identified by {\tt @fb.com} email addresses---we find that the resulting Special Ad audience skews heavily towards 25--34-year-old men.
\item{} We confirm that previous findings on how Facebook's delivery mechanisms can cause further skews in who is shown ads hold for Special Ad Audiences.
We show that an ad for artificial intelligence jobs delivers mostly to young men, while an ad for supermarket jobs delivers mostly to middle-aged women, despite targeting the same gender- and age-balanced Special Ad audience.
\end{itemize}
\noindent Taken together, our results show that simply removing demographic features from the inputs of a large-scale, real-world algorithm will not always suffice to meaningfully change its outputs with respect to those features.
This work also demonstrates a methodology by which other algorithms could be studied.
To be clear, we are not claiming---and do not believe---that Facebook has {\em incorrectly} implemented Special Ad Audiences, or is in violation of its settlement agreement.
Rather, the findings in this paper are a natural result of how complex algorithmic systems work in practice.
The remainder of this paper is organized as follows:
Section~\ref{sec:background} provides background on Facebook's ad targeting tools and related work.
Section~\ref{sec:methodology} introduces our methodology and Section~\ref{sec:results} presents our results.
Section~\ref{sec:discussion} provides a concluding discussion.
\para{Ethics}
We took careful consideration of ethics when conducting the research in this paper.
{\em First}, we minimized harm to Facebook users by only running ``real'' ads, i.e.,\ if a user happened to click on one of our ads, they were sent by Facebook to a real-world site relevant to content the ad.
We therefore did not have any direct interaction with the users who were shown our ad, and did not collect any of their personally identifying information.
{\em Second}, we minimized harm to Facebook by running and paying for our ads just like any other advertiser.
In cases where we were running employment ads, we flagged them as such using Facebook's tools.
\begin{figure*}
\vskip-0.15in
\includegraphics[width=0.485\textwidth]{screenshot-lookalike}
\includegraphics[width=0.485\textwidth]{screenshot-special}
\caption{Screenshots of creation process for both Lookalike Audiences (left) and Special Ad Audiences (right). Both of them are the same from the advertiser's perspective: The advertiser first selects a source Custom audience, then selects a target country, and finally selects the fraction of that country's users to include in the new audience.}\label{fig:creation}
\end{figure*}
\section{Background}\label{sec:background}
In this section, we provide background on Facebook's ad targeting tools, including Special Ad Audiences, and related work.
\subsection{Facebook's ad targeting tools}
Facebook provides a range of {\em targeting} tools to help advertisers select an {\em audience} of users who will be eligible to see their ads.
For example, advertisers can select users through combinations of {\em targeting attributes}~\cite{FacebookAdAttributes}, including over 1,000 demographic, behavioral, and interest-based features.
More germane to this paper and its methods, Facebook also offers a number of other, more advanced targeting tools.
One such tool is {\em Custom Audiences}~\cite{FacebookCustomAudienceDef}, which allows advertisers indicate individual users that they wish to include in an audience.
To use Custom Audiences, an advertiser uploads a list of personally identifiable information (PII), potentially including names, email addresses, phone numbers, dates of birth, and mobile identifiers~\cite{FacebookCustomAudiencePII}.
Facebook then compares those identifiers against its database of active users, and lets the advertiser include matched users in their target audience.
Another tool is {\em Lookalike Audiences}~\cite{FacebookLookalikeAudience}, which creates an audience of users who share ``common qualities'' with users in a Custom audience provided by the advertiser (called the {\em source audience}).
A screenshot of Facebook's advertiser interface is shown in Figure~\ref{fig:creation} (left); the advertiser must select the country where they wish Facebook to select users from (``Audience Location'') and then must select the fraction of that country's population to include in the new Lookalike Audience (ranging from 1\% to 10\%).
Our prior work has demonstrated that Lookalike Audiences can reproduce demographic skews present in source audiences~\cite{speicher-2018-targeted}.
\subsection{Special Ad Audiences}
In March 2018, the NFHA and others sued Facebook for allowing landlords and real estate brokers to exclude members of protected groups from receiving housing ads~\cite{FacebookCivilLawsuit}.
The lawsuit was settled in March 2019, and Facebook agreed to make a number of changes to its ad targeting tools.
Relevant here, Facebook agreed to change how Lookalike Audiences (LAL) works when used with housing, credit, and employment (HEC) ads~\cite{FacebookNFHASettlement}:
\begin{displayquote}
5. Lookalike Audience (``LAL''): In the HEC Flow, LAL tool and marketing will be modified as follows:
\smallskip
(a) LAL tool may consider the following user profile fields: country, region, profession and field of study. LAL tool will not consider the following user profile fields: age, gender, relationship status, religious views, school, political views, interested in, or zip code.
\end{displayquote}
Facebook now refers to this modified Lookalike Audiences tool as {\em Special Ad Audiences}~\cite{FacebookSpecialAdAudience}.
Facebook says~\cite{FacebookSpecialAdCategory}:
\begin{displayquote}
[Special Ad Audiences] will create an audience based on similarities in online behavior and activity but that does not use certain categories, including age, gender, ZIP code or other similar categories.
\end{displayquote}
From an advertiser's perspective, Special Ad Audiences are created just like Lookalike Audiences (i.e., based on a source Custom audience).
A screenshot of Facebook's interface for creating Lookalike Audiences is shown in Figure~\ref{fig:creation} (left), and for Special Ad Audiences in Figure~\ref{fig:creation} (right).
\subsection{Related work}
We briefly overview related work on studying and mitigating algorithmic bias, as well as Facebook's advertising platform.
\para{Algorithmic bias}
Concerns over bias in algorithms have galvanized a growing research community.
This community has developed a number of approaches to {\em algorithmic auditing}~\cite{sandvig-2014-auditing}, a process of seeking to understand an algorithm's inputs, outputs, and potential for discriminatory effects.
Researchers have successfully studied a variety of widely deployed algorithmic systems including face-recognition systems~\cite{buolamwini-2018-gender}, e-commerce sites~\cite{hannak-2014-ecommerce}, search engines~\cite{hannak-2013-filterbubbles,kulshrestha-2017-quantifying,diakopoulos-2018-vote,robertson-2018-auditing,kay-2015-unequal}, job seeking sites~\cite{hannak-2017-bias,chen-2018-chi}, online translation services~\cite{bolukbasi-2016-man}, or health-management~\cite{obermeyer-2019-dissecting}.
A number of proposals have been put forward to mitigate the potential algorithmic biases; we refer the reader to a survey of both sources of bias and mitigation approaches~\cite{mehrabi-2019-survey} for a more in-depth treatment.
We highlight a few works most closely related to our topic of measurement.
Greenberg distinguishes two kinds of fairness concerns, {\em distributive} and {\em procedural}~\cite{greenberg-1987-taxonomy}.
The former aims to assure balanced outcomes, whereas the latter focuses on the process itself.
Elimination of features from an algorithm's input (as with Special Ad Audiences) falls into the {\em procedural} category.
Grgi\'c-Hla\v{c}a~et~al.\xspace~\cite{grgic-2018-beyond} propose a framework which relies on human moral judgments to determine which features are fair to use.
They point out that while people can accurately judge relevance and privacy aspects of a feature in decision making, they tend to fail at predicting the impact that feature might have on the decision outcomes.
Specifically, certain features might appear fair to human judges even though they are correlated with sensitive features.
In such cases process fairness does not lead to outcome fairness, and additional constraints must be enforced.
Further, there are cases in which none of the features is a strong proxy for a sensitive attribute but features can form a proxy when combined~\cite{yeom-2018-hunting,CfpbProxy}.
Finally, even if none of the features or their combinations are unfair, their predictive performance might be different across sub-populations.
Then, in an effort to to minimize the total error, the classifier will fit the majority group better than the minority~\cite{sapiezynski-2017-imbalance,chen-2018-my}.
Taken together, these prior works paint a clear picture of process fairness as insufficient to ensure fair outcomes.
More recently, there has been a growing agreement among scholars that focusing on particular algorithms is too narrow of a problem definition.
Real-world algorithmic decision systems are often composed of multiple algorithmic subsystems and can be discriminatory as a whole, even if built from a series of fair algorithms~\cite{dwork-2018-fairness}
Algorithms need to be modeled along with the other components of the {\em sociotechnical} systems they are embedded in~\cite{selbst-2019-fairness}.
The burden of these investigations lies on independent researchers and auditors since the companies who operate these algorhtms might not be incentivized to measure and address the externalities they cause~\cite{overdorf-2018-questioning}.
\para{Facebook's advertising platform}
Facebook runs one of the world's most powerful advertising platforms, and has been the object of study for a number of research projects.
Prior work has demonstrated that Facebook was using PII provided for security features (e.g., two-factor authentication) was used to allow advertisers to target users with ads~\cite{venkatadri-2019-pii}, that Custom Audiences can be used to leak user's PII~\cite{venkatadri-2018-targeting}, that Facebook's ad targeting options offer a variety of mechanisms to create discriminatory audiences~\cite{speicher-2018-targeted}, that political advertisers on Facebook with higher budgets target people using more privacy sensitive features~\cite{ghosh-2019-facebook} and that Facebook's ad delivery system {\em itself} may introduce unwanted biases when deciding which users should be presented with life opportunity~\cite{lambrecht-2018-algorithmic,ali-2019-discrimination} and political ads~\cite{ali-2019-arbiters}.
\section{Methodology}\label{sec:methodology}
We now describe the methodology we use to study Lookalike and Special Ad Audiences.
Recall that our goal is to measure whether Special Ad Audiences produce significantly less biased audiences than Lookalike Audiences.
We therefore need to be able to generate source audiences with controlled and known skew, from which we can create a Lookalike and a Special Ad audience.
To do so, we re-use an approach from prior work~\cite{ali-2019-discrimination}, relying on voter records from New York and North Carolina.
These records are available to the public, and include voters' gender, age, location (address), and (only in North Carolina) race.
Thus, for each demographic feature we wish to study, we first create a Custom audience based on the voter records (which we treat as ground truth).
For example, when studying gender, we select a subset of the voters who are listed as female and use that list to create a Custom audience.
We use each biased Custom audience to create both a Lookalike audience and a Special Ad audience.
For both types, we select users in the U.S. and choose the smallest size option (1\% of the population).
For some of our experiments, to measure the makeup of a target audience, we run actual ads and record how they are delivered.
For these experiments, we need to provide an {\em ad creative} (consisting of the ad text, headline, image, and destination URL).
Unless otherwise noted, we create a generic ad for Google Web Search, which has basic text (``Search the web for information'') and a link to Google Search.
We found that Facebook does not verify that an ad that is self-reported by an advertiser as a housing, credit, or employment ad is, in fact, such an ad.
Thus, we are able to run the same, generic ad creative using both Lookalike and Special Ad audiences.
\begin{figure}[t!]
\includegraphics[width=0.5\textwidth]{gender_mixes.pdf}
\caption{Gender breakdown of ad delivery to Lookalike and Special Ad audiences created from the same source audience with varying fraction of male users, using the same ad creative. We can observe that both Lookalike and Special Ad audiences reflect the gender distribution of the source audience, despite the lack of gender being provided as an input to Special Ad Audiences.}\label{fig:gender}
\end{figure}
\section{Results}\label{sec:results}
We now present our experiments and analyze the results.
We first examine whether Lookalike and Special Ad Audiences can be biased along the lines of gender and age, which are straightforward to measure as Facebook provides delivery statistics in the advertiser interface.
Next, we focus on race and political views, using a different methodology.
Finally, we show the real-world implications of these experiments using a series of employment and credit ads.
\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
\subsection{Gender}
We begin by focusing on gender. We create seven Custom audiences based on New York voter records.
Each audience contains 10,000 individuals, with varying fractions of men: 0\%, 20\%, 30\%, 40\%, 50\%, 60\%, 80\%, 100\%.
We then run ads to the resulting Lookalike and Special Ad audiences, and compare the results in ad delivery as reported by Facebook's advertiser interface.
Figure~\ref{fig:gender} presents a summary of the results of this experiment, and we make a number of observations.
{\em First}, we can see that each Lookalike audience clearly mirrors its source audience along gender lines: the Lookalike audience derived from a male-only source audience delivers to over 99\% men, and the the Lookalike audience derived from a female-only source audience delivers to over 97\% women.
{\em Second}, we observe a slight male bias in our delivery, relative to the source audience: for example, the Lookalike audience derived from a source audience of 50\% men actually delivered to approximately 70\% men.
This male skew has been observed by prior work~\cite{lambrecht-2018-algorithmic,ali-2019-discrimination} and may be due to market effects or ad delivery effects (which affect both Lookalike and Special Ad audiences equally).
{\em Third}, and most importantly, when we compare the delivery of each Special Ad audience to its corresponding Lookalike audience, we observe that a similar level of bias (that in some cases is statistically indistinguishable).
For example, the Special Ad audience derived from a male-only source audiences delivers to over 95\% men, despite being created without having access to users' genders.
Overall, the Special Ad audiences show a bit less bias when compared to the Lookalike audiences, but the trend is clear.
\begin{figure}[t!]
\includegraphics[width=0.5\textwidth]{age_full.pdf}
\caption{Age breakdown of ad delivery to Lookalike and Special Ad audiences created from the same source audience, using the same ad creative. We can observe extremely similar levels of bias, despite the lack of age as an input to Special Ad audiences. Panel A shows the results for source audiences consisting only of users in one age bracket. Panel B shows the results of mixing the youngest and the oldest users in different proportions.}\label{fig:age}
\end{figure}
\subsection{Age}
Next, we turn to age.
Facebook reports delivery based on users' age in terms of fixed ranges: 18--24, 25--34, 35--44, 45--54, 55--64, 65+.
We therefore design an experiment based off of these ranges. We create biased Custom audiences by selecting voter records that consist {\em only} of 10,000 users within a single age range.
As before, we then run two ads for each Custom audience: one to a Lookalike audience and another to a Special Ad audience.
Figure~\ref{fig:age}A presents the results, and the top six rows of Table~\ref{table:age} provide a more detailed breakdown (and includes the average age of the delivery audience in the final column).
{\em First}, we can immediately observe that that average age\footnote{We estimated the average age using weighted midpoints of each age range; it should be viewed simply as a summary of the aggregate distribution and not the precise average age.} for both the Lookalike audience and the Special Ad audience increases in the same manner, with the average age being 21.9 for the audiences created from the 18--24 age range and between 57 and 63 for the audiences created from the 65+ age range.
{\em Second}, examining the distribution of delivery to each age range shows very similar trends in each pair of audiences, with the 18--24 source audience delivering almost exclusively to young Facebook users in both cases, and the 65+ source audience delivering primarily to older Facebook users in both cases as well.
{\em Third}, as with gender, we may be observing that Special Ad audiences are slightly less biased---the average age of Special Ad audiences is closer to the median Facebook user age in all of the older user groups compared to that of the Lookalike audiences---but the overall effect is very strong.
Additionally, we look at what happens if we ``mix'' together users from different age ranges, as opposed to running experiments with a source audience from a single age range.
Specifically, we mix together both old (65+) and young (18--24) users in different proportions and examine the ultimate delivery audience for the Lookalike and Special Ad audiences.
Figure~\ref{fig:age}B and the bottom five rows of Table~\ref{table:age} present these results in the same format before.
As before, we can see that the average age follows a very similar trend for both types of audiences, and that the breakdown in delivery is quite similar as well.
Thus, the effect we are observing is not limited to homogeneous source audiences.
\subsection{Race}
Next, we turn to examine the extent to which Special Ad Audiences can be biased along racial lines, in the same manner we have observed Lookalike Audiences to be in past work~\cite{speicher-2018-targeted}.
We are unable to re-use the same methodology for age and gender, which relied on Facebook's ad delivery statistics.
Instead, we develop an alternative methodology that relies on {\em estimated daily results}~\cite{FacebookReachDiff}, which is an estimate provided by Facebook of the number of users that match the advertiser's targeting criteria and can be reached within the specified budget.
We set the daily budget to the maximum allowed value (\$1M) to best approximate the number of users that match the targeting criteria.
Facebook returns these values as a range (e.g., ``12,100 -- 20,400 users''); throughout this procedure, we always use the lower value.\footnote{We repeated the procedure using both the midpoint as well as the upper value and found similar results.}
The procedure has two steps: audience creation and targeting.
\paragraph{Audience Creation}
As before, we start with voter records from North Carolina (which provide race information).
We focus on two racial groups: Black (defined as users who self-report as Non-Hispanic Black) and white (defined as users who self-report as Non-Hispanic white).
For each race, we create two independent Custom audiences: one list of 10,000 randomly selected users with that race, and one list of 900,000 randomly selected users with that race.
We refer to these audiences as \texttt{w\_10k} and \texttt{w\_900k} (for the white audiences) and \texttt{b\_10k} and \texttt{b\_900k} (for the Black audiences).
Next, we use the Custom audiences with 10,000 users to create corresponding Lookalike and Special Ad audiences.
We refer to these audiences as $L_\texttt{w\_10k}$ (for the Lookalike audience based on \texttt{w\_10k}), $S_\texttt{w\_10k}$ (for the Special Ad audience), $L_\texttt{b\_10k}$, and $S_\texttt{b\_10k}$.
Our goal is then to estimate the racial bias of these Lookalike and Special Ad audiences.
\paragraph{Targeting}
We now use the ad targeting interface to obtain such estimates.
To do so, we begin the ad creation process and set our budget is the maximal value allowed (\$1M/day).
We also specify that we only target users in North Carolina.
Suppose we wish to obtain an estimate of the fraction of white users in $L_\texttt{w\_10k}$.
To do so, we first target the large white audience \texttt{w\_900k} audience and record the potential daily reach (e.g., 81,000).
We then target $L_\texttt{w\_10k}$ and record the potential daily reach (e.g., 397,000).
Finally, we target $L_\texttt{w\_10k}$ and {\em exclude} the \texttt{w\_900k} audience, and record the potential daily reach (e.g., 360,000).
Now, we can observe that excluding \texttt{w\_900k} from $L_\texttt{w\_10k}$ caused the potential daily reach to drop by 37,000, indicating that approximately 46\% (37,000/81,000) of \texttt{w\_900k} were present in $L_\texttt{w\_10k}$.
We can then repeat the process with excluding \texttt{b\_900k}, and measure the fraction of the large Black audience that is present in $L_\texttt{w\_10k}$.
By comparing the fraction of \texttt{w\_900k} and \texttt{b\_900k} that are present in $L_\texttt{w\_10k}$, we obtain an estimate of the racial bias of $L_\texttt{w\_10k}$.
\paragraph{Limitations}
It is important to note that, unlike in our experiments with gender and age, here we do not know the race of a vast majority of the audience.
This is because the Lookalike and Special Ad audiences that Facebook creates consist mostly of people who appear not to be in our voter records.
Thus, the results we present in this section only refer to the fraction of voters with known race who are included in each Lookalike and Special Ad audience, not the racial composition of these audiences overall, as further emphasized in Figure~\ref{fig:race}.
However, these estimates do give us a small window into the makeup of these Lookalike and Special Ad audiences.
\begin{figure}[t]
\includegraphics[width=0.92\linewidth]{total_venn.pdf}
\caption{Venn diagrams of overlap between Lookalike and Special audiences created from samples of 10,000 Black and white voters. While we do not know the race of the vast majority of the created audiences, the part that we do know shows racial bias.}\label{fig:race}
\end{figure}
\begin{table}[b!]
\begin{tabularx}{0.485\textwidth}{Xc|r|r}
\small
& & \multicolumn{2}{c}{\bf Percent overlap} \\
& & {\bf Black~~~\null } & {\bf White~~~\null } \\
{\bf Source} & {\bf Type} & (\texttt{b\_900k}) & (\texttt{w\_900k}) \\
\hline
\multirow{2}{*}{100\% Black} & Lookalike ($L_\texttt{b\_10k}$) & 61.0 & 16.0 \\
& Special ($S_\texttt{b\_10k}$) & 62.3 & 12.3 \\
\hline
\multirow{2}{*}{100\% white} & Lookalike ($L_\texttt{w\_10k}$) & 16.9 & 42.0 \\
& Special ($S_\texttt{w\_10k}$) & 10.4 & 35.8 \\
\end{tabularx}
\caption{Breakdown of overlap between audiences with known racial makeup and Lookalike and Special Ad audiences. While we do not know the race of the vast majority of the created audiences, we see large discrepancies in the race distribution among the known users.}\label{table:race}
\vskip-0.25in
\end{table}
\begin{figure}[t]
\includegraphics[width=1\linewidth]{overlaps.pdf}
\caption{Both Lookalike and Special Ad audiences created from source audiences of white users containing a higher fraction of white users than Black users. Conversely, audiences created from source audiences of Black users contain a higher fraction of Black users than white users.}\label{fig:overlaps}
\end{figure}
\paragraph{Results}
We begin by presenting Venn diagrams in Figure~\ref{fig:race} that capture the overlap between all of the audiences.
We summarize the overlap between the Lookalike and Special Ad audiences and the large white and Black audiences in Table~\ref{table:race}.
Focusing on the table, we can immediately observe that both the Lookalike audiences show significantly more overlap with the race of the source audience, suggesting that the makeup of the Lookalike audiences are racially biased.
For example, the Lookalike audience created from \texttt{b\_10k} contains 61\% of the \texttt{b\_900k} but only 16\% of \texttt{w\_900k}.
More importantly, the Special Ad audiences show a similar behavior (though as before, perhaps with slightly less of a bias).
Again, it is important to keep in mind that we can only make estimates of the fraction of \texttt{w\_900k} and \texttt{b\_900k} that overlap with the Lookalike and Special Ad audiences, and cannot comment on the majority of these audiences (as they likely fall outside of North Carolina).
Thus, our results are not conclusive---but only suggestive---that the overall audiences are similarly biased.
\paragraph{Robustness}
Here, we confirm that the presented results are robust to the random selection of seed from which Lookalike and Special Ad audiences are created.
To this end, we repeat the described process with 20 non-overlapping \texttt{b\_10k} and 20 non-overlapping \texttt{w\_10k} audiences.
We create a Lookalike and a Special Ad audience for each and compute the overlap with large \texttt{b\_700k} and \texttt{b\_700k} audiences and report the resulting fractions in Figure~\ref{fig:overlaps}.
We note that the racial skew observable in Lookalike audiences persists in Special Ad audiences but the effect is slightly smaller.
\begin{figure*}
\includegraphics[width=1\textwidth]{aud_ads.png}
\caption{Ad creatives used throughout the paper. All of our ads linked directly to the domains shown in the ad.}\label{fig:ads}
\end{figure*}
\subsection{Political views}
We next turn to measure the extent to which Lookalike and Special Ad Audiences can be biased along the lines of political views.
As with race, Facebook does not provide a breakdown of ad delivery by users' political views.
Thus, we repeat the methodology we used for race, using voter records from North Carolina and focusing on the differences in delivery to users registered as Republicans and Democrats.
Specifically, we create source audiences of Republicans and Democrats (\texttt{r\_10k} and \texttt{d\_10k}), as well as large Republican and Democrat audiences (\texttt{r\_900k} and \texttt{d\_900k}).
We then use the source audiences to create both Lookalike audiences ($L_\texttt{r\_10k}$ and $L_\texttt{d\_10k}$) and Special Ad audiences ($S_\texttt{r\_10k}$ and $S_\texttt{d\_10k}$).
As with race, we run the same generic ad to all audiences, and examine the fraction of the large audiences that are present in the Lookalike and Special Ad audiences.
\begin{table}[b!]
\begin{tabularx}{0.485\textwidth}{Xc|r|r}
\small
& & \multicolumn{2}{c}{\bf Percent overlap} \\
& & {\bf Democrat\null } & {\bf Republican\null } \\
{\bf Source} & {\bf Type} & (\texttt{d\_900k}) & (\texttt{r\_900k}) \\
\hline
\multirow{2}{*}{100\% Democrat} & Lookalike $L_\texttt{d\_10k}$& 51.6 & 31.8 \\
& Special $S_\texttt{d\_10k}$& 42.2 & 25.8 \\
\hline
\multirow{2}{*}{100\% Republican} & Lookalike $L_\texttt{r\_10k}$ & 28.1 & 50.0 \\
& Special $S_\texttt{r\_10k}$ & 25.0 & 47.0 \\
\end{tabularx}
\caption{Breakdown of overlap between source audiences with known political leaning and resulting Lookalike and Special Ad audiences. While we do not know the political leaning of the vast majority of the audiences, we see discrepancies in the distribution among the known users.}\label{table:politics}
\vskip-0.25in
\end{table}
We report the results in Table~\ref{table:politics}.
We can observe a skew along political views for Lookalike audiences (for example, the Lookalike audience created from users registered as Democrats contains 51\% of \texttt{d\_900k} but only 32\% of \texttt{r\_900k}).
We can also observe that the Special Ad audiences show a skew as well, though to a somewhat lesser degree than the Lookalike audiences.
As with the race experiments, we remind the reader that we can only observe the overlap between the created audiences and the large Democrat/Republican audiences; we are unable to measure the majority of the created audiences.
However, the demonstrated skew suggests that there is a bias in the overall makeup of the created audiences.
\subsection{Real-world use cases}\label{subsec:realads}
Next, we test a ``real-world'' use case of Special Ad Audiences.
We imagine an employer wants to use Facebook to advertise open positions to people who are similar to those already working for them.
The employer might assume that since the Special Ad Audiences algorithm is not provided with protected features as inputs, it will allow them to reach users who are similar to their current employees without dramatic gender, age, or racial biases.
The employer would therefore upload a list of their current employees to create a Custom audience, ask Facebook to create a Special Ad audience from that source audience, and then target job ads to the resulting Special Ad audience.
We play the role of this hypothetical employer (Facebook itself in this example, which provides employees with an \texttt{@fb.com} email address).
We then run the following experiment:
\begin{enumerate}
\item We create a Custom audience consisting of randomly generated American phone numbers, 11,000 of which Facebook matched to existing users. This is our baseline audience that we use to measure the bias in the created audience.
\item We create another Custom audience consisting of 12,356,604 generated email addresses: all 2--5 letter combinations + \texttt{@fb.com}, 11,000 of which Facebook matched to existing users. This is our audience of Facebook employees.
\item We create Special Ad audiences based on each of these two Custom audiences.
\item We run two generic job search ads (see Figure~\ref{fig:ads}A), each to one of these Special Ad audiences, at the same time, from the same account, with the same budget. This way we eliminate the side effects of optimization based on the ad content or budget.
\item We collect the delivery statistics with age/gender breakdown.
\end{enumerate}
\begin{figure}[b!]
\includegraphics[width=1\linewidth]{facebook_dist.pdf}
\caption{Gender and age breakdown of a generic job ad delivery to a Special Ad audience based on random American users (in orange) and a Special Ad audience based on Facebook employees (in blue). The based on Facebook employees is predominantly male and 25-34.}\label{fig:facebook_dist}
\vskip-0.15in
\end{figure}
Figure~\ref{fig:facebook_dist} presents the results of the experiment.
The Special Ad audience based on Facebook employees delivers to 88\% men, compared to 54\% in the baseline case.
Further, the Special Ad audience based on Facebook employees delivers to 48\% to men aged between 25-34, compared to 15\% for the baseline audience.
Finally, 47\% of all deliveries to the Facebook Special Ad audience are to users in California, compared to 2\% in the baseline audience.\footnote{While matching based on state is not prohibited in the settlement, these numbers indicate that our method of selecting Facebook employees based on random email addresses \texttt{@fb.com} is correct.}
Overall, our results show that our hypothetical employer's reliance on Special Ad audiences to avoid discrimination along protected classes was misplaced: their ad was ultimately delivered to an audience that was significantly biased along age and gender lines (and presumably reflective of Facebook's employee population).
We confirm these findings by running an additional experiment in the ``credit'' category, advertising tips for building credit (see ad copy in Figure~\ref{fig:ads}B) to the same two Special Ad audiences.
The results of this experiment are presented in a similar format in Figure~\ref{fig:real_credit}.
We observe that the ad targeting the Facebook-based audience still delivers predominantly towards male and 25--34-year-old users.
However, we also observe a shift towards delivering to male users in the random audience, even though it is the same audience as in the generic job ad.
This is likely an effect of the ad creative on how Facebook algorithms estimate relevance to different groups of people~\cite{ali-2019-discrimination}.
We further explore these effects in the next section.
\begin{figure}[t!]
\includegraphics[width=1\linewidth]{real_credit.pdf}
\caption{Gender and age breakdown of delivery of credit building ads to a Special Ad audience based on random American users (in orange) and a Special Ad audience based on Facebook employees (in blue). The Facebook-based audience still delivers predominantly males aged 25-34, but the random audience skews more male too.}\label{fig:real_credit}
\end{figure}
\begin{figure}[t!]
\includegraphics[width=1\linewidth]{real_jobs.pdf}
\caption{Gender and age breakdown of delivery of job ads to a Special Ad audience based on random American users. We observe that even if the source audience is held constant and is approximately gender-balanced, the content of the ad can still lead to large skews: ads for supermarket jobs deliver to 72\% female audience, whereas ads for jobs in AI delivery to 66\% male audience. Note also the skews in age: supermarket jobs deliver mostly to people aged 35 and older, whereas the AI jobs deliver nearly exclusively to people younger than 35. }\label{fig:real_jobs}
\end{figure}
\subsection{Content-based skew in delivery}\label{subsec:skew}
In previous work~\cite{ali-2019-discrimination}, we demonstrated that the skew in delivery can be driven by Facebook's estimated relevance of a particular ad copy to a particular group of people.
Specifically, even when we held the target audience constant, Facebook would deliver our ads to different subpopulations: ads for supermarket jobs were shown primarily to women, while ads for jobs in lumber industry were presented mostly to men.
Here, we show that these effects persist also when using Special Ad Audiences.
We re-use the Special Ad audience created from the random 11,000 users which we expect to be approximately gender- and age-balanced.
We then run generic job ads along with ads for supermarket and artificial intelligence pointing to search for either keyword on \texttt{indeed.com}, see Figure~\ref{fig:ads}A, C, and D.
We report the fraction of men in the reached audience in Figure~\ref{fig:real_jobs}; we can immediately observe that the different ads skew towards middle-aged women (in the case of supermarket jobs) or towards younger men (in the case of artificial intelligence jobs).
This skew when delivering ads to a gender-balanced audience underlines a crucial point: when designing fairness/anti-discrimination controls, one cannot just focus on one part of the {\em algorithmic} system.
Instead one must look at the whole {\em socio-technical} system, including how an algorithm is used by real people, how people adjust their behaviors in response to the algorithm, and how the algorithm adapts to people's behaviors.
\section{Discussion}\label{sec:discussion}
We have demonstrated that both Lookalike and Special Ad Audiences can create similarly biased target audiences from the same source audiences.
To reiterate, we are not claiming that Facebook has incorrectly implemented Special Ad Audiences, nor are we suggesting that the company has violated its settlement agreement.
Rather, our findings are a result of a complex algorithmic system at work.
Our findings have broad and narrow implications.
Broadly, we demonstrate that simply removing demographic features from a complex algorithmic system can be insufficient to remove bias from its outputs, which is an important lesson for government and corporate policymakers.
More specifically, we show that relative to Lookalike Audiences, Facebook's Special Ad Audiences do little to reduce demographic biases in target audiences. As a result, we believe Special Ad Audiences will do little to mitigate discriminatory outcomes.
\subsection{Policy implications}\label{subsec:hud}
In the U.S., President Trump's administration has has directed a range of civil rights officials to reexamine how `disparate impact' regulations might be changed or removed~\cite{TrumpAdministration}.
Disparate impact is a legal doctrine that says facially neutral practices in employment, housing, credit, and other areas can still lead to a finding of discrimination if they adversely affect a protected group. To date, at least one regulator is considering new rules that would insulate companies from disparate impact liability if their algorithms do not rely on protected demographic factors.
Our results strongly indicate that this is a flawed approach.
In August 2019, HUD published a proposed rule~\cite{HUDProposedDisparateImpact} to amend its interpretation of the FHA in light of the U.S.~Supreme Court's ruling in {\em Texas Department of Housing and Community Affairs v. Inclusive Communities Project, Inc}.
In particular, the proposed rule states in {\S}100.500(c)(2)(i) and (iii) that a defendant can claim a plaintiff has failed to establish a prima facie case for disparate impact when:
\begin{displayquote}
... a plaintiff alleges that the cause of a discriminatory effect is a model used by the defendant, such as a risk assessment algorithm, and the defendant:
(i) Provides the material factors that make up the inputs used in the challenged model and shows that these factors do not rely in any material part on factors that are substitutes or close proxies for protected classes under the Fair Housing Act and that the model is predictive of credit risk or other similar valid objective;
\end{displayquote}
This describes, in essence, the approach that the settlement proscribed and led to the creation of Special Ad Audiences.
Our results reinforce what dozens of companies, advocates, and researchers told HUD when its proposed rule was open for public comment: Removing protected features from an algorithm's inputs is not enough to prevent discriminatory effects.
\subsection{Legal implications}
At a high level, U.S. federal law prohibits discrimination in the marketing of housing, employment and credit opportunities.
Our findings might have near-term legal consequences for advertisers and even Facebook itself.
A creditor, employer, or housing provider who used biased Special Ad audiences in their marketing could run afoul of anti-discrimination laws.
This could be exceptionally frustrating for an advertiser who believed that Special Ad Audiences was an appropriate, legally-compliant way to target their ads.
Facebook itself could also face legal scrutiny. In the U.S., Section 230 of the Communications Act of 1934 (as amended by the Communications Decency Act)~\cite{CommunicationsDecencyAct} provides broad legal immunity to Internet platforms acting as publishers of third-party content.
This immunity was a central issue in the litigation resulting in the settlement analyzed above.
Although Facebook argued in court that advertisers are ``wholly responsible for deciding where, how, and when to publish their ads''~\cite{FacebookOnuohaMotionToDismiss}, this paper makes clear that Facebook can play a significant, opaque role by creating biased Lookalike and Special Ad audiences.
If a court found that the operation of these tools constituted a "material contribution" to illegal conduct, Facebook's ad platform could lose its immunity~\cite{UpturnFacebookAmicusBrief}.
\section*{Acknowledgements}
\small{
The authors thank Ava Kofman and Ariana Tobin for suggesting the experiments presented in Sections~\ref{subsec:realads}~\&~\ref{subsec:skew} as well as for going an extra mile (or two) for their ProPublica story around this work.
We also thank NaLette Brodnax for her feedback on the experimental design and Aleksandra Korolova for her comments on the manuscript.
This work was funded in part by a grant from the Data Transparency Lab, NSF grants CNS-1616234 and CNS-1916153, and Mozilla Research Grant 2019H1.
}
\setcounter{table}{0}
\renewcommand{\thetable}{A\arabic{table}}
\small{
\begin{table*}[h]
\begin{tabularx}{\textwidth}{Xc|r|r|r|r|r|r|r}
\small
& & \multicolumn{6}{c|}{\bf Delivery audience age breakdown (\%)} & ~~~{\bf Average} \\
{\bf Source} & {\bf Type} & 18--24 & 25--34 & 35--44 & 45--54 & 55--64 & ~~65+~ & {\bf Age} \\
\hline
\multirow{2}{*}{100\% 18--24} & Lookalike & 88.7 & 10.9 & 0.2 & 0.0 & 0.0 & 0.2 & 22.0 \\
& Special & 88.3 & 11.5 & 0.2 & 0.0 & 0.0 & 0.0 & 22.0 \\
\hline
\multirow{2}{*}{100\% 25--34} & Lookalike & 30.0 & 66.9 & 2.9 & 0.1 & 0.1 & 0.1 & 27.0 \\
& Special & 33.0 & 62.9 & 3.9 & 0.1 & 0.1 & 0.0 & 26.8 \\
\hline
\multirow{2}{*}{100\% 35--44} & Lookalike & 3.2 & 34.0 & 54.8 & 7.0 & 0.6 & 0.5 & 36.0 \\
& Special & 6.8 & 49.4 & 36.1 & 6.2 & 1.0 & 0.6 & 33.8 \\
\hline
\multirow{2}{*}{100\% 45--54} & Lookalike & 0.7 & 5.1 & 32.7 & 49.7 & 9.0 & 3.0 & 46.1 \\
& Special & 3.9 & 14.1 & 32.4 & 32.6 & 11.3 & 5.8 & 44.3 \\
\hline
\multirow{2}{*}{100\% 55--64} & Lookalike & 1.2 & 2.5 & 7.3 & 20.6 & 39.6 & 28.9 & 57.8 \\
& Special & 2.4 & 6.2 & 11.7 & 21.0 & 31.3 & 27.5 & 55.1 \\
\hline
\multirow{2}{*}{100\% 65+} & Lookalike & 3.3 & 4.2 & 3.2 & 3.3 & 19.6 & 66.4 & 63.5 \\
& Special & 6.7 & 9.6 & 5.2 & 6.9 & 23.2 & 48.5 & 57.7 \\
\hline
\hline
\multirow{2}{*}{20\% 18--24, 80\% 65+} & Lookalike & 70.1 & 14.5 & 1.1 & 0.9 & 3.3 & 10.0 & 28.9 \\
& Special & 65.1 & 18.1 & 2.3 & 2.4 & 4.4 & 7.7 & 29.1 \\
\hline
\multirow{2}{*}{40\% 18--24, 60\% 65+} & Lookalike & 78.6 & 14.8 & 0.7 & 0.5 & 1.5 & 3.9 & 25.0 \\
& Special & 80.3 & 13.9 & 1.2 & 0.5 & 1.2 & 2.9 & 24.4 \\
\hline
\multirow{2}{*}{50\% 18--24, 50\% 65+} & Lookalike & 84.1 & 12.5 & 0.6 & 0.1 & 0.6 & 2.0 & 23.4 \\
& Special & 81.3 & 16.0 & 0.7 & 0.6 & 0.5 & 0.9 & 23.2 \\
\hline
\multirow{2}{*}{60\% 18--24, 40\% 65+} & Lookalike & 83.2 & 14.1 & 0.4 & 0.1 & 0.9 & 1.2 & 23.2 \\
& Special & 83.3 & 14.9 & 0.8 & 0.4 & 0.3 & 0.7 & 22.9 \\
\hline
\multirow{2}{*}{80\% 18--24, 20\% 65+} & Lookalike & 89.1 & 10.1 & 0.4 & 0.1 & 0.1 & 0.3 & 22.1 \\
& Special & 85.3 & 13.8 & 0.6 & 0.1 & 0.2 & 0.0 & 22.3 \\
\end{tabularx}
\caption{Breakdown in age of delivery audience of ads to Lookalike and Special Ad Audiences created from the same source audience, using the same ad creative. The top six rows represent source audiences with a single age group; the bottom five rows represent source audiences with a mix of young and old users.
}\label{table:age}
\end{table*}
}
\balance
\newcommand{\etalchar}[1]{$^{#1}$}
|
2,869,038,155,719 | arxiv | \section{Introduction}
\label{sec:intro}
Thanks to the advance of modern X-ray telescopes such as \textit{Chandra} and \textit{XMM-Newton}, and the synergy with radio observations, we now know that isolated neutron stars (NSs) can manifest themselves as pulsars (PSRs) with a surface dipole magnetic field spanning more than five orders of magnitude, in the $\sim$10$^{10}$--10$^{15}$~G range\protect{\footnote{For the remainder of this work we use the equatorial surface dipole field, $B=3.2 \times 10^{19}\sqrt{ P \dot{P} }$~(G), inferred from the observed period, $P$(s), and period derivative, $\dot{P}$ (s s$^{-1}$).}}. Observationally, this has led to their organization into different classes, including (1) the rotationally powered radio and X-ray bright objects, like the Vela pulsar with $B$$\sim$$10^{11}$--$10^{13}$~G, (2) the magnetically powered pulsars (or magnetars) with $B$$\sim$$10^{14}$--$10^{15}$~G, exceeding the quantum electrodynamics (QED) limit of $4.4 \times 10^{13}$ G and observed primarily at high energies, (3) the highly magnetized pulsars (HBPs) with magnetic fields intermediate between the classical pulsars and magnetars, but still exceeding the QED limit, and (4) the central compact objects (CCOs) observed only in X-rays (so far), near the centres of supernova remnants (SNRs) and with inferred low magnetic fields, $B$$\sim$10$^{10}$--10$^{11}$~G. This diversity led several authors to attempt a unification through evolutionary models of NSs with their properties dictated primarily by a continuum of magnetic field strengths \citep[see e.g.][and references therein]{kaspi2010, dallOsso, mereghetti13, perna2013, vigano2013, safiharb15}.
The magnetic field is estimated using a standard model of NS evolution which assumes energy loss due to the emission of radiation from a point-like rotating magnetic dipole in vacuum, providing a spin-down torque with a braking index $n=3$ \citep{dipole}. This picture assumes rapid rotation of the NS after birth, so the observed period ($P$) differs from the initial period ($P_0$) by a large amount (i.e. $P_0 \ll P$) due to the constant torque acting to slow the NS spin \citep{spin}. However, the braking index has been measured for a small sample of young pulsars, and all so far differ from the prediction of the standard model with $n<3$ \citep{youngn}. A lower braking index can come about from a variety of mechanisms including a change in the moment of inertia of the star over time \citep{moi}, alignment of the magnetic field and rotation axis \citep{alignment1, alignment2}, the emission of a particle wind \citep{wind1, wind2}, magnetospheric effects \citep{magS1, magS2} and environmental interactions \citep{fallback}. Another serious problem with the standard picture concerns the NSs that are associated with SNRs. Generally, pulsar ages found from their `characteristic age' ($\tau_\text{PSR}=P/2 \dot{P}$) by assuming dipole radiation and the independently measured SNR ages are in disagreement, sometimes by orders of magnitude (in particular for the CCOs)\footnote{See http://www.physics.umanitoba.ca/snr/SNRcat for the high-energy catalogue of SNRs which compiles all known ages of SNRs and associated PSRs \citep{SNRCatRef}.}. The observed braking index and age discrepancy arise from the standard, and commonly adopted, assumption of a constant torque acting to brake the NS over its life span.
The growing evidence for NSs with X-ray luminosity in excess of their spin-down energy presents another difficulty for the standard scenario. Some of these objects are thought to be powered by the dissipation of magnetic energy rather than spin-down losses, examples of which include the anomalous X-ray pulsars (AXPs) and some soft gamma-ray repeaters (SGRs), unified under the class of `magnetars' \citep{magnetar1, magnetar3, magnetar2}. These are X-ray bright objects that are slowly rotating pulsars with exceptionally high magnetic fields and normally discovered through their bursting activity. However, a neat classification scheme for these objects proves elusive in the light of the discovery of `low-B magnetars' \citep[e.g.][]{rea2010}, and an HBP having behaved like a magnetar, yet thought to be a rotation-powered pulsar powering a bright pulsar wind nebula \citep{HBP2, HBP1}. The situation is further complicated by the CCOs with extremely low fields, dubbed as `anti-magnetars' \citep{CCO1}, yet still show an X-ray luminosity in excess of their spin-down energy. One recent interpretation for these objects is the suppression of their external field through magnetic field burial \citep{burial0, burial1, burial2, burial3}, also implied by spectroscopic models of these objects \citep{CCO2}. In this scenario, the accretion of supernova fall-back material occurs following the birth of the NS. This period of vigorous accretion has the effect of burying the dipole magnetic field component within the NS crust, reducing the spin-down energy loss and making the NS appear significantly older than its associated SNR. In this alternative model of NS evolution, field growth is needed to explain the initially small braking index and low surface fields, while a decaying toroidal component is invoked to explain the excess X-ray luminosity \citep{HoCCOReview}. The field burial scenario has been most recently described in significant detail by \citet{ho15}, who performed detailed calculations of the fall-back accretion process, including the inner structure of the NS, conductivity of the NS crust and a realistic equation of state. Besides an internal decaying toroidal component, the dipole field component also decays on large time-scales \citep{dallOsso, vigano2013}. The decaying external field is described by a parametrized model given by \citet{colpi}, expanded on by \citet{dallOsso} and used to describe the evolution of the AXP 1E~2259+586 by \citet{nakano15}.
In this paper we present a phenomenological parametrized family of models for magnetic field evolution in the NS population. Our model unifies the description of magnetic field growth and decay by making use of variations on the parametric forms from \citet{dallOsso} and \citet{colpi}, which we derive in Section \ref{sec:theory}. This model also reproduces the results of \citet{NB15} for exponential field growth and replicates the findings of \citet{dallOsso} for decaying fields. We fit our model to the observations of various NSs in Section \ref{sec:modelfitting}, testing our model against the detailed physical predictions found by \citet{ho15}. We discuss the results of our fits in Section \ref{sec:discussion}. Finally, our conclusions are summarized in Section \ref{sec:conclusions}.
\section{Theory and Parameter Space Exploration}
\label{sec:theory}
The standard model for NS spin-down from energy loss due to the emission of dipole radiation assumes a constant magnetic field $B \propto \sqrt{P \dot{P}}$, where $P$ and $\dot{P}$ are the period and period derivative, respectively. However, we are interested in the dynamical evolution of the magnetic field
\begin{equation}
B(t)=B_{\text{j}} f_{\text{j}}(t),
\label{Bt}
\end{equation}
where the time-dependence has been gathered into the function $f_{\text{j}}(t)$ \citep[see also ][]{IP_2} and $B_{\text{j}}$ is a constant reference field value, either the initial field strength in decay models (denoted by subscript $D$) or final field strength in growth models (labelled as $G$). We use the differential equation
\begin{equation}
P \dot{P} = b B^2,
\label{funDip}
\end{equation}
with $b=\text{constant}$, and do not consider the effect of spin-axis field alignment. Integrating this equation from the NSs birth at $t=0$ to an arbitrary later time $t$, we find
\begin{equation}
P^2=P_0^2+2bB_{\text{j}}^2 F_{\text{j}}^2
\label{P}
\end{equation}
where we have denoted the integral
\begin{equation}
F_{\text{j}}^2=\int_0^t f_{\text{j}}^2(t') dt'.
\label{F_j^2}
\end{equation}
From here on, we will generally suppress the time-dependence of the function $f$ for notational simplicity. The period derivative is
\begin{equation}
\dot{P}=\frac{bB_{\text{j}}^2f_{\text{j}}^2}{P},
\label{dotP}
\end{equation}
and we express the characteristic age, $\tau$, as
\begin{equation}
\tau=\frac{P}{2\dot{P}}=\frac{P^2}{2bB_{\text{j}}^2f_{\text{j}}^2},
\label{ttau1}
\end{equation}
where we have used equation (\ref{dotP}). Inserting equation (\ref{P}) in this expression gives us
\begin{equation}
\tau=\frac{P_0^2}{2bB_{\text{j}}^2f_{\text{j}}^2}+\frac{F_{\text{j}}^2}{f_{\text{j}}^2}.
\label{tau}
\end{equation}
Including a time-dependent field introduces an important distinction between the characteristic age $\tau$ and the model time $t$. The model time represents the amount of time elapsed from the birth of the NS to the present, and thus represents the ``true'' age of the NS. The characteristic age is the age that is determined from the period and period derivative of the NS (equation \ref{ttau1}), which differs from the true age due to the time-dependence of $P$ and $\dot{P}$. These dynamical quantities also introduce a time-dependence of the braking index $n=2-P\dot{P}/\ddot{P}$. Using equation (\ref{P}), the braking index is given by
\begin{equation}
n= 3 - 4 \tau \frac{\dot{f_{\text{j}}}}{f_{\text{j}}}.
\label{n}
\end{equation}
If the field decays, $\dot{f_{\text{j}}}$ is negative and $n>3$, while field growth has positive $\dot{f_{\text{j}}}$ and leads to a braking index $n<3$ as observed in many young NSs. In the case of $f_{\text{j}}=1$, $F_{\text{j}}^2=t$, and these formulae reduce to the standard spin-down from dipole radiation with a constant field.
\citet{dallOsso} use a parametrization to study field decay that was introduced by \citet{colpi}:
\begin{equation}
\frac{dB}{dt}=-aB(t)^{1+\alpha}=-\frac{B(t)}{\tau_{\text{D}}}
\end{equation}
where $\alpha$ is the decay index that describes how rapidly the decay proceeds and $a$ is a normalization parameter related to the specific physical mechanisms involved in the decay (e.g. Hall drift, ambipolar diffusion). The quantity $\tau_{\text{D}}=\left[ aB(t)^{\alpha} \right]^{-1}$ is the decay time-scale, which itself is time-dependent. The magnetic field described by these equations can be conveniently written as
\begin{equation}
B(t) = B_{\text{D}} \left\{
\begin{array}{ll}
\left( 1+ \alpha \frac{t} {\tau_{\text{m}}} \right)^{-\frac{1}{\alpha}} , & \alpha\neq0,2 \\
\exp\left( -\frac{t}{\tau_{\text{m}}} \right), & \alpha=0 \\
\end{array}\right.
\label{decayB}
\end{equation}
where $B_{\text{D}}$ is the initial field which decays over time, and $\tau_{\text{m}}=\tau_{\text{D}}(0)$ is the initial field decay time-scale. Note that the time-dependence of the decaying field given in equation (\ref{decayB}), $B(t)=B_{\text{D}} f_{\text{D}}(t)$, is contained entirely within the dimensionless function $f_\text{D}(t)$. This well-known parametrization is extremely useful as it can also be used to construct a model of field growth.
There are a number of properties the dimensionless function $f_\text{G}(t)$ must have in order to describe NS field growth. The growth must be bounded in time, in that the field begins at some minimum value and attains an asymptotic maximum value as time increases. This requires that the derivative of the field growth function is always positive and decreases to approach zero as $t$ becomes large. Furthermore, $f_\text{G}(t)$ should be parametrized in terms of a small number of quantities whose meaning has a clear physical interpretation. In fact, the field decay function $f_\text{D}(t)$ has attractive features that make it useful to also describe the derivative of a growing field $\text{d}{f_\text{G}}/\text{dt}$. First, the function decreases from a maximum $f_\text{D}(0)=1$ and becomes vanishingly small for large $t$. This bounded behaviour fulfills the exact criteria that is required to describe the derivative of a field $\text{d}{f_\text{G}}/\text{dt}$ that begins at a minimum value and grows to approach a constant strength as $t$ increases. Secondly, $f_\text{D}(t)$ is stated in terms of two parameters that have a well-understood interpretation. The index $\alpha$ controls the rate at which $f_\text{D}(t)$ changes with respect to $t$, with lower values giving the field evolution an exponential behaviour, and larger values slow the evolution providing a softer decay. The parameter $\tau_\text{m}$ controls the time-scale over which the magnetic field evolves. Therefore, let us consider the following basic form based on the field decay $f_\text{D}(t)$ from \citet{dallOsso}, with an appropriate normalization, as
\begin{equation}
\frac{\text{d}f_{\text{G}}}{\text{d}t}=\frac{(1-\alpha)}{\tau_{\text{m}}} f_{\text{D}}
\label{Bderiv}
\end{equation}
where $f_{\text{G}}$ is the time-dependent part of the growing field and $f_{\text{D}}$ contains the time-dependence of the decaying field model, normalized by the factor $(1-\alpha)/\tau_\text{m}$. Due to this normalization, growing fields require the field index to be in the range $0 \leq \alpha <1$. Equation \ref{Bderiv} results in a field that evolves in time as $B(t)=B_{\text{G}}f_{\text{G}}(t)$, where
\begin{equation}
f_{\text{G}}(t) = \epsilon + \left\{
\begin{array}{ll}
1-\left( 1+\alpha \frac{t}{\tau_{\text{m}}} \right)^{\frac{\alpha-1}{\alpha}}, & 0 < \alpha < 1 \\
1-\exp\left(-\frac{t}{\tau_{\text{m}}}\right), & \alpha=0
\end{array}\right. .
\label{fG}
\end{equation}
In the above expression the integration constant $\epsilon$ controls the initial field
\begin{equation}
B_0=B_{\text{G}} \epsilon
\label{BInit}
\end{equation}
in terms of the asymptotic value at large $t$, $B_{\text{G}}$. The growth model uses the boundary condition $B_{\text{j}}=B_{\text{G}}$ as the asymptotic field strength, and the decay model uses $B_{\text{j}}=B_{\text{D}}$ as the initial field. In terms of fields buried by fall-back accretion, the smaller $\epsilon$ is, the deeper the field has been buried within the NS, and the larger the difference between the initial and asymptotic field strength. The time-scale $\tau_{\text{m}}$ determines how long the field takes to emerge from the compact object. The field given by equation (\ref{fG}) describes a family of solutions in terms of the field index $\alpha$. When $\alpha=0$ the field evolves exponentially, which is particularly significant as this form was proposed by \citet{NB15} to describe growing NS fields. When $0<\alpha<1$, field growth occurs more slowly. Since the parameters of our growth model are given by the widely studied decaying field parametrization of \citet{dallOsso} and \citet{colpi}, we find this representation to be particularly intuitive.
The period $P$ and characteristic age $\tau$ given by equations (\ref{P}) and (\ref{tau}) depend on the integral of $f_{\text{G}}^2$ (equation \ref{F_j^2}). With the time-dependence from equation (\ref{fG}), we write
\begin{equation}
\begin{array}{ll}
F_{\text{G}}^2 =\int_0^t f_{\text{G}}^2(t')dt' = & (1+\epsilon)^2t \\
& -\frac{2(1+\epsilon)\tau_{\text{m}}}{2 \alpha-1}\left[ \left( 1 + \alpha \frac{t}{\tau_{\text{m}}}\right)^{\frac{2 \alpha-1}{\alpha}}-1\right] \\
& +\frac{\tau_{\text{m}}}{3 \alpha-2} \left[ \left( 1+\alpha \frac{ t}{\tau_{\text{m}}}\right)^{\frac{3\alpha-2}{\alpha}}-1\right].
\end{array}
\label{FG2}
\end{equation}
We give the special cases of this equation in the limits $\alpha \rightarrow 1/2$ and $\alpha \rightarrow 2/3$, and summarize the connection between the field decay and growth models in Table \ref{table1}.
\setcounter{table}{0}
\begin{table*}
\begin{center}
\begin{tabular}{|c|c|c|}
\multicolumn{3}{c}{Decay functions} \\ \hline
$\frac{\text{d}f_{\text{D}}}{\text{d}t}$ & $-\frac{1}{\tau_{\text{m}}}\left( 1+\alpha \frac{t}{\tau_{\text{m}}} \right)^{-\frac{(\alpha+1)}{\alpha}}$ & $0 \leq \alpha \leq 2$ \\
$$ & $-\frac{1}{\tau_{\text{m}}} \exp\left( -\frac{t}{\tau_{\text{m}}} \right)$ & $\alpha=0$ \\ \hline
$f_{\text{D}}$ & $\left(1+\alpha \frac{t}{\tau_{\text{m}}} \right)^{-\frac{1}{\alpha}} $ & $$ \\
$$ & $\exp\left( -\frac{t}{\tau_{\text{m}}} \right) $ & $\alpha = 0$ \\ \hline
$F_{\text{D}}^2=\int_0^t f_{\text{D}}^2(t')dt'$ & $\frac{\tau_{\text{m}}}{2-\alpha}\left[1-\left(1+\alpha \frac{t}{\tau_{\text{m}}} \right)^{\frac{\alpha-2}{\alpha}}\right]$ & $$ \\
$$ & $\frac{\tau_{\text{m}}}{2}\left[1-\exp\left( -2\frac{t}{\tau_{\text{m}}} \right)\right]$ & $\alpha=0$ \\
$$ & $\frac{\tau_{\text{m}}}{2}\ln \left( 1+2\frac{t}{\tau_{\text{m}}} \right)$ & $\alpha=2$ \\ \hline
\multicolumn{3}{c}{Growth functions} \\ \hline
$\frac{\text{d}f_{\text{G}}}{\text{d}t}$ & $\frac{(1-\alpha)}{\tau_{\text{m}}}\left( 1+ \alpha\frac{t}{\tau_{\text{m}}}\right)^{-\frac{1}{\alpha}}$ & $0 \leq \alpha < 1$\\
$$ & $\frac{1}{\tau_{\text{m}}}\exp\left( -\frac{t}{\tau_{\text{m}}} \right) $ & $\alpha = 0$ \\ \hline
$f_{\text{G}}$ & $1+\epsilon -\left( 1+ \alpha \frac{t}{\tau_{\text{m}}} \right)^{\frac{\alpha-1}{\alpha}}$ & $0<\alpha<1$ \\
$$ & $1+\epsilon -\exp\left( -\frac{t}{\tau_{\text{m}}} \right)$ & $\alpha = 0$ \\ \hline
$F_{\text{G}}^2=\int_0^t f_{\text{G}}^2(t')dt'$ & $(1+\epsilon)^2t-\frac{2(1+\epsilon)\tau_{\text{m}}}{2 \alpha-1}\left[ \left( 1 + \alpha \frac{t}{\tau_{\text{m}}}\right)^{\frac{2 \alpha-1}{\alpha}}-1\right] + \frac{\tau_{\text{m}}}{3 \alpha-2} \left[ \left( 1+\alpha \frac{ t}{\tau_{\text{m}}}\right)^{\frac{3\alpha-2}{\alpha}}-1\right]$ & $$ \\
$ \lim_{\alpha \rightarrow 1/2} F_{\text{G}}^2$ & $(1+\epsilon)^2t - 4(1+\epsilon)\tau_{\text{m}} \ln\left( 1+\frac{1}{2}\frac{t}{\tau_{\text{m}}}\right)+t\left( 1+\frac{1}{2} \frac{t}{\tau_{\text{m}}}\right)^{-1}$ & $\alpha = \frac{1}{2}$ \\
$ \lim_{\alpha \rightarrow 2/3} F_{\text{G}}^2$ & $(1+\epsilon)^2t -6(1+\epsilon)\tau_{\text{m}}\left[ \left(1+\frac{2}{3}\frac{t}{\tau_{\text{m}}}\right)^{\frac{1}{2}} -1 \right]+ \frac{3}{2} \tau_{\text{m}} \ln\left( 1+\frac{2}{3}\frac{t}{\tau_{\text{m}}} \right)$ & $\alpha = \frac{2}{3}$ \\ \hline
\end{tabular}
\caption{A summary of the time-dependent functions for describing magnetic field decay and growth.}
\label{table1}
\end{center}
\end{table*}
The model as stated has six parameters: the field index $\alpha$, growth factor $\epsilon$, time-scale $\tau_{\text{m}}$, asymptotic field $B_\text{G}$, the initial period $P_\text{0}$ and the model time $t$. The model time can be treated as a free parameter to vary between the lower and upper SNR age limits, $\tau_\text{SNR--}$ and $\tau_\text{SNR+}$, respectively, or can be fixed before beginning the optimization. The model then outputs the quantities we want to fit to the observed values: the period, period derivative and braking index (at the present time). The standard fitting problem is to vary the input parameters to produce a match with the output and the observed values. Thus, the problem is under constrained, in that there are fewer fit quantities than parameters, leading to a family of solutions. However, a closer inspection shows that the previously stated input parameters are not truly independent. In fact, a simplification can be achieved by changing our modelling approach.
Instead of fitting for $P$ and $\dot{P}$, let us assume their observed values a priori. We then know the magnetic field at the present time, which we call $B_\text{t}$, by equation (\ref{funDip}), and also by definition, the characteristic age $\tau=\tau_\text{PSR}$. With the definition of the growing field in equation \ref{Bt}, we can solve for the model time $t$ as a function of the present-day field $B_\text{t}$ and the four parameters ($\alpha$, $\epsilon$, $\tau_\text{m}$, $B_\text{G}$):
\begin{equation}
t = \left\{
\begin{array}{ll}
\frac{ \tau_\text{m} }{ \alpha }\left[ \left( 1+\epsilon -\frac{B_\text{t}}{B_\text{G}} \right)^\frac{\alpha}{\alpha-1} - 1 \right], & 0 < \alpha < 1 \\
-\tau_\text{m} \ln \left( 1+\epsilon -\frac{B_\text{t}}{B_\text{G}} \right), & \alpha=0
\end{array}\right. .
\label{tModel}
\end{equation}
Once we have calculated $t$, we find $f$ and $\dot{f}$ using equation (\ref{fG}), and obtain the braking index $n$ at the current time through equation (\ref{n}). The parameters are then varied to match $n$ and $t$ to the observed values. Knowledge of the characteristic age allows us to state the initial period as a function of the parameters, given by solving equation (\ref{tau}):
\begin{equation}
P_\text{0} = \sqrt{ 2 b B_\text{G}^2 \left( \tau_\text{PSR} f_\text{G}^2 - F_{G}^2 \right) }.
\label{P0eq}
\end{equation}
Restating the problem in this way is advantageous because it allows us to eliminate what was previously considered a free parameter, and treats the SNR age and braking index as quantities to fit. The simplification we introduce comes from reducing the number of parameters to a small enough set that a quantitative description of the model parameter space can be given, as described furthermore below. The introduction of additional physics beyond the phenomenological field growth also helps further simplify the situation.
We describe the model parameter space by fixing the values of the field index $\alpha$ and the growth factor $\epsilon$. For a given calculation we hold these values constant. Next, we form a grid of $\tau_\text{m}$ and $B_\text{G}$ values and calculate $t$ and $n$ for each ($\tau_\text{m}$, $B_\text{G}$) pair using equations (\ref{tModel}) and (\ref{n}). The regions of parameter space containing solutions with $t$ and $n$ within the observed limits are found by contouring these $2D$ functions to find the level curves corresponding to $\tau_\text{SNR+}$ and $\tau_\text{SNR--}$, and the measured limits on $n$. Solutions that satisfy the constraints live in the regions between these level curves. Solutions in the region where the sets intersect satisfy both of the constraints simultaneously. Changing the values of $\alpha$ and $\epsilon$ affects the morphology of the intersection regions. Using this approach, we study the regions of the parameter space that give physically realistic solutions without the need for an external optimization routine. For the remainder of this study, we will use this contouring approach for a variety of field index in the range $0.1 \leq \alpha \leq 0.9$ and growth factor $0.001 \leq \epsilon \leq 0.1$. In practice, we find that solutions with $\epsilon<10^{-3}$ do not significantly vary from one another for a given $\alpha$, so we do not consider any $\epsilon$ lower than this. Moreover, the largest asymptotic fields also typically correspond to small $\epsilon$ for a given $\alpha$, so we choose $\epsilon=0.1$ as an upper limit that still allows a significant field growth, though in general all $\epsilon < 1$ can be used. As a final point, we note that equation (\ref{P0eq}) can produce unphysical complex valued $P_\text{0}$. Thus, we impose a further constraint from the initial period:
\begin{equation}
y=\tau_\text{PSR} f_\text{G}^2 -F_{G}^2 \geq 0,
\end{equation}
with the equality corresponding to the limit $P_\text{0}=0$. This provides a boundary between the physical and unphysical solutions in the parameter space. Therefore along with $t$ and $n$, we also produce the corresponding $y$ on the ($\tau_\text{m}$, $B_\text{G}$) grid, and find the level curve $y=0$ using a contouring method. The unphysical region can then also be excluded by intersection.
\begin{center}
\begin{figure*}
\centerline{\includegraphics[scale=0.85, bb= 92 292 509 502, clip=true]{contour_plots.eps}}
\caption{Parameter constraint plots for the Vela pulsar, PSR B0833--45. The left-hand panel shows the parameter space region that satisfies the age constraint (red). The lower SNR age limit is the blue dashed line, and the solid blue line is the upper SNR age limit. On the right the same region of parameter space is shown but we include the braking index constraints (green curves). The red region shows the intersecting set, which satisfies both the age constraint and the braking index constraint. The lower group in both panels has $\alpha=0.1$, $\epsilon=0.1$, and the upper group $\alpha=0.9$, $\epsilon=0.001$. The horizontal black line in both panels marks the observed dipole field.}
\label{figContours}
\end{figure*}
\end{center}
As an example, in Fig. \ref{figContours} we show the parameter space for the Vela pulsar, PSR~B0833--45, which has a measured braking index and associated SNR age. On the left, we show the ($\tau_\text{m}$, $B_\text{G}$) parameter space using only the age constraint from the associated SNR. The lower limit $\tau_{\text{SNR}-}$ is the dashed blue line, and the upper limit $\tau_{\text{SNR}+}$ is the solid blue line. The region between the curves is the parameter space area that obeys the age constraint, and is coloured red. On the right we show the same region of parameter space, but include the braking index contours, denoted as green lines. Since the braking index is known to high accuracy, the red coloured regions satisfying both constraints simultaneously is significantly reduced. The lower group in both panels has $\alpha=0.1$, $\epsilon=0.1$, and the upper group $\alpha=0.9$, $\epsilon=0.001$. Note that there is degeneracy between the groups of solutions for a given $\tau_\text{m}$ depending on the chosen field index $\alpha$. Thus, while this method does not provide a unique solution, it allows us to quantify the regions of parameter space that contain physical solutions given a constant $\alpha$ and $\epsilon$ pair as input. We investigate the differences between the low- and high-$\alpha$ cases to study the limiting behaviour of the field growth model. Generally, the solutions that have observed fields close to the asymptotic field $B_\text{G}$ are near the end of their evolution. The $\alpha=0.1$, $\epsilon=0.1$ solutions with the lowest asymptotic fields have nearly finished their evolution and will grow by only $\approx 1 \%$ over the next few kyr. The braking index of these solutions will rapidly grow to the dipole value $n=3$ over this span of time. The solutions with $\alpha=0.9$ and $\epsilon=0.001$ have significantly larger asymptotic fields that are more than an order of magnitude higher than the low-$\alpha$ solutions. These NSs will take a significantly longer span of time for their fields to evolve to the final state and reach braking index $n=3$. For these solutions, the field growth significantly outlasts the observable life of the SNR and will appear as a highly magnetized, isolated NS with no apparent SNR association.
\section{Model Fitting and NS Evolution}
\label{sec:modelfitting}
Let us investigate the consequences of field growth in NSs using two initial approaches. First, we vary the field index $\alpha$ to demonstrate how this parameter affects the time evolution. Secondly, since this field growth model is phenomenological in nature, we investigate how well it can reproduce the results of numerical simulations, such as the detailed modelling of the burial and emergence of the magnetic fields in young accreting NSs explored recently by \citet{ho15}. That study focused on the young NSs with braking indices $n<2$, in particular the rotation-powered pulsars PSR J0537--6910 associated with the LMC SNR~N157B, the Vela pulsar B0833--45, and the HBP~J1734--3333 which has a proposed association with G354.8--0.8 \citep{manchester}. We note that the relationship between this SNR and HBP~J1734--3333 is tenuous and may be the result of a coincidental alignment. We list these systems in Table \ref{table2}, along with a carefully selected list of other NSs that are (1) securely associated with SNRs (thus providing an independent estimate of the true age) and (2) with `extremal' fields, namely from the class of magnetars, HBPs and CCOs.
\begin{table*}
\small
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\multicolumn{8}{c}{Observed properties of NSs} \\
\hline
PSR & $P$ & $\dot{P}$ & $n$ & $\tau_\text{PSR}$ & SNR & $\tau_\text{SNR--}$ & $\tau_\text{SNR+}$ \\
& (s) & ($10^{-11}s~s^{-1}$) & & (kyr) & & (kyr) & (kyr) \\ \hline
\hline
AXP~1E~1841--045 & $11.783$ & $3.930$ & $$ & 4.750 & G27.4+0.0 (Kes~73) & $0.750$ & $2.100 $ [1] \\
AXP~1E~2259+586 & $6.979$ & $4.843e-2$ & $$ & 228.317 & G109.1-01.0 (CTB~109)& $10.000$ & $16.000$ [2] \\
CXOU~J171405.7--381031 & $3.825$ & $6.400$ & $$ & 0.947 & G348.7+00.3 & $0.350$ & $3.150$ [3] \\
\hline
SGR~0526--66 & $8.054$ & $3.800$ & $$ & $3.358$ & N49 & $-$ & $4.800$ [4] \\
SGR~1627--41 & $2.595$ & $1.900$ & $$ & $2.164$ & G337.3--0.1 & $-$ & $5.000$ [5] \\
\hline
HBP~J1119--6127 & $0.408$ & $0.400$ & $2.684\pm 0.002$ {[14]}& 1.616 & G292.2--0.5 & $4.200$ & $7.100$ [6] \\
HBP~J1734--3333 & $1.170$ & $0.228$ & $ 0.9 \pm 0.2$ {[15]} & 8.131 & G354.8--0.8 & $1.300$ & $-$ [7] \\
HBP~J1846--0258 A & $0.325$ & $0.709$ & $2.64 \pm 0.01$ {[16]} & $0.726$ & G029.7--0.3 (Kes~75)& $0.900$ & $4.300$ [8] \\
HBP~J1846--0258 B & $0.327$ & $0.711$ & $2.16 \pm 0.13$ {[17]} & $0.728$ & & & \\
\hline
PSR~J0537--6910 & $0.016$ & $0.518$ & $-1.5 \pm 0.1$ {[18]} & $4.925$ & N157B & $1.000$ & $5.000$ [9] \\
PSR~B0833--45 & $0.089$ & $1.250$ & $1.4 \pm 0.2$ {[19]}& $11.319$ & G263.9--03.3 (Vela) & $5.400$ & $16.000$ [10] \\ \hline
\hline
RX~J0822.0--4300 & $0.112$ & $8.300e-4$ & $$ & $213.799$ & G260.4--3.4 (Puppis A) & $3.700$ & $5.200$ [11] \\
1E~1207.4--5209 & $0.424$ & $6.600e-6$ & $$ & $1.018e5$ & G296.5 +10.0 (PKS 1209--51/52) & $2.000$ & $20.000$ [12] \\
CXOU~J185238.6+004020 & $0.105$ & $8.680e-7$ & $$ & $1.917e5$ & G033.6+00.1 (Kes 79) & $5.400$ & $7.500$ [13] \\ \hline
\end{tabular}
\caption{For a given PSR, $P$ is the period, $\dot{P}$ the period derivative. The characteristic age is $\tau_\text{PSR}$ and the lower and upper SNR age limits are $\tau_\text{SNR--}$ and $\tau_\text{SNR+}$, respectively, from the McGill magnetar catalogue \citep[\protect\url{http://www.physics.mcgill.ca/~pulsar/magnetar/main.html}, ][]{mcGill}. The SNR ages have been compiled in the U. of Manitoba's High-Energy SNR Catalogue (SNRcat, \protect\url{http://www.physics.umanitoba.ca/snr/SNRcat/}). References to SNR ages in this table are [1]: \citet{1841Age}, [2]: \citet{nakano15}, [3]: \citet{1714Age2009}, [4]: \citet{0526Age}, [5]: \citet{1627Age}, [6]: \citet{g292age}, [7]: \citet{moi}, [8]: \citet{1846Age}, [9]: \citet{0537Age}, [10]: \citet{0833Age}, [11]: \citet{0822Age} , [12]: \citet{1207Age}, [13]: \citet{1852Age}. References to the braking indices included here are [14]: \citet{welt11}, [15]: \citet{youngn}, [16]: \citet{1846-3}, [17]: \citet{1846-4}, [18]: \citet{n0537}, [19]: \citet{n0833}.}
\label{table2}
\end{center}
\end{table*}
For the purpose of illustrating the effect that changing $\alpha$ has on the field evolution, we consider the HBP~J1734--3333 as an example and use the age derived in \citet{ho15}, $t=2.07$ kyr. We arbitrarily set the initial field to a typical NS field strength, $B_\text{0}=10^{11}$ G, which fixes $\epsilon$ for a given $B_\text{G}$ by equation (\ref{BInit}). We generate a family of curves using a constant field index that spans the full range $0 \leq \alpha<1$. For each value of $\alpha$, we follow the standard model fitting approach, treating the time-scale $\tau_\text{m}$, asymptotic field $B_\text{G}$ and initial period $P_\text{0}$ as fit parameters, which are varied numerically. The results are shown in Fig. \ref{fig:1734}, where we plot the period, period derivative, characteristic age, magnetic field, braking index and luminosity as functions of time for each of the $\alpha$ values, matching to the observed $P$, $\text{d}P/\text{dt}$ and $n$. The horizontal dashed lines represent the observed values and the vertical dashed line is the adopted current age $t$. In the luminosity panel, we show the spin-down luminosity ($\dot{E}$) as a horizontal dotted line and the 2--10 keV X-ray luminosity ($L_\text{x}$) as a dashed line. We call attention to two important features of this figure. First, the characteristic age decays rapidly from an initially high value regardless of $\alpha$. This general behaviour provides an explanation for young NSs that have a characteristic age larger than the corresponding SNR age. Secondly, the large characteristic age at early times gives a negative braking index at early times through equation (\ref{n}), which allows the field growth scenario to explain the observations of objects with $n<0$, such as PSR J0537--6910 with $n=-1.5$ \citep{n0537}. The field index $\alpha$ smoothly controls how quickly the braking index reaches the asymptotic value $n=3$.
\begin{center}
\begin{figure*}
\centerline{\includegraphics[scale=1, bb= 61 217 542 580, clip=true]{properties_1734_color.eps}}
\caption{Fits to HBP~J1734--3333 for a variety of field index $\alpha$. The vertical dashed line marks the adopted age $t=2.07$ kyr, chosen to facilitate comparison with the result from \citet{ho15}. The horizontal dashed lines mark the observed quantities. In the luminosity plot (lower right-hand panel) the horizontal dotted line marks the spin-down energy loss rate and the horizontal dashed line marks the observed X-ray luminosity.}
\label{fig:1734}
\end{figure*}
\end{center}
Next, we use our field growth model and contour approach to recover the quantitative behaviour of the detailed simulation performed by \citet{ho15}. We assume the value of the asymptotic field and age derived in that work a priori, and then we treat the time-scale $0.1 \leq \tau_\text{m} \leq 10$ kyr and field index $0 \leq \alpha \leq 0.99$ as free parameters. We construct a grid of parameter values ($\tau_\text{m}$, $\alpha$) and calculate $t$ and $n$ for each. Then we find the level sets of these functions at the derived age and mean braking index. The intersection of the two curves gives a unique $\tau_\text{m}$ and $\alpha$ pair. With $\epsilon=0.0021$ we find $P_\text{0}=1.0597$ s, compared to $1.06$ s given by \citet{ho15}. The initial period does not change significantly for lower $\epsilon$. The discrepancy grows slowly as $\epsilon$ is increased. Using the contour approach we find the parameters necessary for our model to reproduce the evolutionary trajectories of PSR~J1734--3333, PSR~B0833--45 and PSR~J0537--6910 from \citet{ho15}. The details of the fits are given in Table \ref{table:fitPar} and marked with an asterisk. It is a testament to the flexibility and usefulness of the parametric form that we were able to recover the behaviour of a simulation involving detailed and complex physical processes.
\begin{center}
\begin{table*}
\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{Fit parameters} \\
\hline
\hline
PSR & $\alpha$ & $\epsilon$ & $\tau_{\text{m}}$ & $B_{\text{G}}$ & $P_0$ & $t$ \\
& & & (kyr) & ($10^{13}$ G) & (s) & (kyr) \\ \hline
\hline
SGR~0526--66 & $0.100$ & $0.100$ & $1.642$ & $56.035$ & $3.246$ & $4.786$ \\
& $0.900$ & $0.001$ & $0.438$ & $240.000$ & $3.689$ & $4.779$ \\ \hline
SGR~1627--41 & $0.100$ & $0.100$ & $1.261$ & $22.488$ & $0.090$ & $3.677$ \\
& $0.900$ & $0.001$ & $3.201$ & $240.000$ & $0.477$ & $4.981$ \\ \hline
\hline
HBP~J1734--3333 & $0.100$ & $0.100$ & $1.193$ & $5.231$ & $1.012$ & $3.477$ \\
* & $0.633$ & $0.002$ & $0.085$ & $6.500$ & $1.060$ & $2.070$ \\
& $0.900$ & $0.001$ & $10.000$ & $75.236$ & $0.838$ & $9.939$ \\ \hline
HBP~J1846--0258 A & $0.100$ & $0.001$ & $0.074$ & $4.932$ & $0.247$ & $0.433$ \\
& $0.900$ & $0.100$ & $6.718$ & $43.273$ & $0.007$ & $0.810$ \\ \hline
HBP~J1846--0258 B & $0.100$ & $0.100$ & $0.286$ & $4.880$ & $0.187$ & $0.833$ \\
& $0.900$ & $0.100$ & $3.102$ & $43.187$ & $0.225$ & $0.430$ \\ \hline
\hline
PSR~J0537--6910 & $0.100$ & $0.100$ & $0.372$ & $0.094$ & $0.015$ & $1.007$ \\
* & $0.525$ & $0.053$ & $0.921$ & $0.170$ & $0.015$ & $1.950$ \\
& $0.900$ & $0.001$ & $10.000$ & $2.942$ & $0.014$ & $3.564$ \\ \hline
PSR~B0833--45 & $0.100$ & $0.100$ & $2.254$ & $0.338$ & $0.073$ & $6.570$ \\
* & $0.541$ & $0.005$ & $2.709$ & $0.550$ & $0.065$ & $10.200$ \\
& $0.900$ & $0.001$ & $10.000$ & $3.586$ & $0.058$ & $15.738$ \\ \hline
\hline
RX~J0822.0--4300 & $0.100$ & $0.100$ & $1.783$ & $0.098$ & $0.111$ & $5.200$ \\
& $0.900$ & $0.001$ & $10.000$ & $3.010$ & $0.112$ & $3.702$ \\ \hline
1E~1207.4--5200 & $0.100$ & $0.001$ & $6.860$ & $0.017$ & $0.420$ & $20.000$ \\
& $0.900$ & $0.001$ & $10.000$ & $0.880$ & $0.424$ & $2.004$ \\ \hline
CXOU~J185238.6+004020 & $0.100$ & $0.100$ & $2.572$ & $0.003$ & $0.105$ & $7.500$ \\
& $0.900$ & $0.001$ & $10.000$ & $0.069$ & $0.105$ & $5.401$ \\ \hline
\hline
\end{tabular}
\caption{Fit parameters of the NSs plotted in Figs. \ref{fig:phaseSpace} and \ref{fig:age}. The solutions with low and high asymptotic fields are listed, and solutions that recover the parameters of \citet{ho15} are marked with an asterisk.}
\label{table:fitPar}
\end{table*}
\end{center}
Finally, we apply the contour method to the remaining HBPs and PSRs listed in Table \ref{table:fitPar}, including the braking index when possible. We follow the prescription outlined for contouring in Section \ref{sec:theory}, holding $\alpha$ and $\epsilon$ constant and finding the level sets of $t$ and $n$ as functions of the time-scale $\tau_\text{m}$ and asymptotic field $B_\text{G}$. We do not consider any asymptotic field strength greater than the maximum observed magnetar field, $B_{\text{G}}=2.4\times 10^{15}$ G, of SGR 1806--20 \citep{maxField}. For each system shown in Table \ref{table1}, we provide example solutions with large and small asymptotic fields in Table \ref{table2}, and plot the trajectories of these example solutions in the $P$--$\dot{P}$ phase space in Fig. \ref{fig:phaseSpace}. In this plot, the evolutionary trajectories of the SGRs are given as green lines, the HBPs as red lines, the PSRs yellow lines and the CCOs as blue lines. The HBP~J1846--0258 is marked by a light grey diamond, HBP~J1119--6127 is a dark grey diamond and HBP~J1734--3333 is a white diamond. The PSRs J0537--6910 and B0833--45 are marked by grey and white stars, respectively. Markers that are black represent objects with X-ray luminosity in excess of spin-down luminosity. The parameters that describe the trajectories shown in this figure are likewise given in Table \ref{table:fitPar}. Note that we do not provide an example trajectory for HBP~J1119--6127, which will be discussed in the next section.
\begin{center}
\begin{figure*}
\centerline{\includegraphics[scale=0.90, bb= 123 221 476 572, clip=true]{phase_P-Pdot_color.eps}}
\caption{Phase space plot of evolutionary trajectories $P$--$\dot{P}$. The thin black dashed diagonal lines denote constant characteristic age from $100$ yr (upper left) to $1$ Gyr (lower right). The dotted black diagonal lines represent an increasing magnetic dipole field, from $10^{11}$ G (lower left) to $10^{15}$ G (upper right). Black symbols represent sources that have X-ray luminosity in excess of spin down luminosity. The low asymptotic field trajectories are marked as solid lines, and the high field solutions from Table \ref{table:fitPar} are dashed lines. The evolutionary tracks for HBPs are red, SGRs are green, PSRs are yellow and the CCOs denoted with blue. HBP~J1846--0258 is marked by a light grey diamond and the post-outburst trajectory is shown. HBP~J1734--3333 is a white diamond and HBP~J1119--6127 is a dark grey diamond (note that this object is not accompanied with a trajectory). The PSRs J0537--6910 and B0833--45 are marked by grey and white stars.}
\label{fig:phaseSpace}
\end{figure*}
\end{center}
In Fig. \ref{fig:age}, we plot the characteristic age against the model time, following the same conventions as Fig. \ref{fig:phaseSpace}. Despite the apparent similarity of many of the trajectories in the $P$--$\dot{P}$ phase space, the $\tau$--$t$ plot clearly shows the difference between these objects as a function of time. It is worth noting that the time evolution of the CCOs characteristic age explains the apparent large discrepancy between the pulsars adopted ages (appearing very old) and their associated young SNRs. In particular, for the three systems shown, the PSR and SNR ages match at times equal to or exceeding $\sim 10^{4.5}$ yr, by which time the SNR would have mostly dissipated. Therefore, the characteristic age for these objects considered will not reflect their true age as long as they are within their SNRs. This feature, along with the low asymptotic field strength (see Table \ref{table:fitPar}), also leads to the suggestion that CCOs could be ancestors of `old' isolated radio pulsars as long as they overcome the accretion or field-growth phase (which would explain their X-ray dominant emission) and their surface field grows to the critical limit required for radio emission. The late time evolution of the CCOs may also link them to the class of objects known as X-ray dim isolated NSs \citep[XDINS; ][]{xdin1}. These are radio-quiet X-ray pulsars with long periods ($3.45$--$11.37$ s) and no apparent SNR associations. Some of these objects are believed to have high magnetic fields in excess of $10^{13}$ G.
\begin{center}
\begin{figure*}
\centerline{\includegraphics[scale=0.90, bb= 130 220 476 575, clip=true]{ages_c_SNR_color.eps}}
\caption{Evolutionary trajectories plotted by NS characteristic age against model time for the systems shown in Table \ref{table:fitPar}. The SNR ages are represented as horizontal lines. The markers are placed at the mean value, or at the extreme when no upper or lower limit exists. The low asymptotic field trajectories are marked as solid lines, and the high field solutions from Table \ref{table:fitPar} are dashed lines. Colours are the same as in Fig. \ref{fig:phaseSpace}. The thick black line is $\tau_\text{PSR}=t$.}
\label{fig:age}
\end{figure*}
\end{center}
\section{Discussion}
\label{sec:discussion}
The $P$-$\dot{P}$ phase space trajectories shown in Fig. \ref{fig:phaseSpace} demonstrate possible evolutionary links between the apparently diverse set of NSs shown in Table \ref{table:fitPar}. As the NS fields grow, they evolve from the bottom of the figure upward, passing through the region of the phase space inhabited by the CCOs. Thus the PSRs J0537--6910 and B0833--45, and the HBPs J1846--0258 and J1734--3333 may have undergone a similar CCO stage during their evolution. The trajectories of these HBPs carry them towards the current position of AXP 1E2259+586. If HBPs and AXPs are related through evolution then field decay must begin once the buried field has emerged, raising the braking index to values $n>3$. We have also fit the CCOs to explore their potential future behaviour, and note that RX~J0822.0--4300 has asymptotic behaviour for both large and small fields which is very close to PSR~B0833--45. However, the time evolution of these objects is dramatically different as seen in Fig. \ref{fig:age}. Thus, objects like the HBPs may pass through the CCO stage relatively quickly, whereas objects such as CCO~1E~1207.4--5209 spend a more significant portion of their lives in this state. Finally, we note that CCO~CXOU~J185238.6+004020 requires an extremely low asymptotic field, with $B_{\text{G}}<6.9 \times 10^{11}$ G. Thus, even after the field emerges from this NS, it remains relatively low.
Since the SGRs 0527--66 and 1627--41 have characteristic ages less than the upper SNR age limit, we have also examined these objects using the growth model. However, the field growth mechanism is not generally expected to play a role in the evolution of the SGRs, since their characteristic ages are only smaller than the upper limit of the associated SNR age, and no lower limits are known. Moreover, these systems do not have a measured braking index, which is crucial in making the case for field growth ($n<3$) or field decay ($n>3$). Additionally, field decay has been proposed to explain the SGRs energetics as it has for AXPs, although their X-ray luminosity is not consistently larger than their spin-down energy. Due to the lack of a lower age limit, we consider solutions that produce $\tau_\text{PSR} < t \leq \tau_{\text{SNR}+}$. Generally solutions that satisfy this condition with large $B_\text{G}$ require a longer growth time-scale for a given $\alpha$ and $\epsilon$ pair, so we can find large fields $B_\text{G} > 2.0\times 10^{15}$ G provided we consider sufficiently large $\tau_\text{m}$. We plot the evolution of two example solutions in Figs \ref{fig:phaseSpace} and \ref{fig:age}. Interestingly, both SGR~1627--41 and SGR~0527--66 reach similar states in the limit of large asymptotic fields shown in Fig. \ref{fig:phaseSpace}, and the trajectories imply the SGR fields are still evolving. A lower limit for the SNR age would significantly constrain these results, provided that $\tau_{\text{SNR}-} > \tau_\text{PSR}$. For completion, we attempted fits to the AXPs as well, but these required unrealistically high initial spin periods. We stress that despite these interesting fits, field decay is necessary to reconcile the characteristic and SNR ages of the AXPs. This conclusion is supported by results from the literature \citep[e.g.][]{nakano15}, and is implied for the SGRs evolution as well \citep{dallOsso}.
HBP~J1846--0258 presents an interesting case since the braking index has been observed to decrease from $n=2.64$ to $2.16$ \citep{1846-3, 1846-4} following an outburst and spectral changes in 2008 \citep{HBP2, HBP1}. This braking index change was not accompanied by a change in luminosity or pulse profile which is difficult to explain on such short time-scales, but may represent a re-organization of the magnetosphere \citep{1846-5}. We fit the pre- and post-outburst configurations of the system which we label as A ($n=2.64$) and B($n=2.16$). However, we were not able to find any region of parameter space through our contour methods that could simultaneously satisfy both pre- and post-outburst configurations. This may not be the case if the field index were allowed to vary in time, but with constant $\alpha$, field growth cannot neatly explain the behaviour of this NS. The HBP~J1846--0258 is a complicated case, particularly because of the presence of a bright pulsar wind nebula powered by this object. This nebula implies wind-braking likely plays an important role in the evolution of this NS. For $\alpha=0.99$ and minimum $\epsilon=0.019$, we find a maximum field $B_\text{G}=6.5 \times 10^{14}$ G on a growth time-scale $9.2$ kyr. Changing $\alpha$ and $\epsilon$ results in a lower field on shorter time-scales. For a given pair of $\alpha$ and $\epsilon$, the system can be well constrained by the period condition and the SNR age, though low remnant ages are generally favoured.
Finally, there was a problem fitting to HBP~J1119--6127. For this system we were not able to simultaneously fit both the age of the associated SNR G292.2--0.5 \citep{g292age}, and the observed braking index $n=2.684$ \citep{welt11}. With the SNR age constraint the derived braking index is $n \approx 1.8$ assuming $t=4.2$ kyr. Alternatively, with the braking index constraint in place, the derived age was found to be close to the characteristic age, and cannot be reconciled with the observed SNR age. Intriguingly, a low value for the braking index of J1119--6127 was also proposed by \citet{g292age}, who suggested that the braking index may have recently changed from a lower value $n < 2$. Since neither of these scenarios satisfies the constraints, we have not included an evolutionary track for HBP~J1119--6127 in Figs \ref{fig:phaseSpace} and \ref{fig:age}, and also exclude it from our summary of solutions in Table \ref{table:fitPar}. We plan to investigate HBP~J1119--6127 with other emission mechanisms in future work.
It is also relevant that many of the systems included in Table \ref{table:fitPar} have large initial spin periods (i.e. $P_0 \approx 1$ s, and approaching $P$ for many systems), which are higher than expected for the traditional magnetar model \citep{magnetar1, IP_1}. This is notable because a problem with the magnetar model is the lack of super energetic SNRs which would be expected from an SNR hosting a rapidly spinning proto-NS \citep[for example,][]{vink, magnetarSNRs}.
\section{Conclusions}
\label{sec:conclusions}
We have devised a flexible and conveniently parametrized model for a growing magnetic field, which is based on the parametric forms used by \citet{colpi} and refined by \citet{dallOsso}. This parametrization can accommodate a variety of field time-dependence in addition to the exponential model suggested by \citet{NB15}, and the interpretation of the parameters is straightforward. By including the observationally measured period and period derivative and assuming the field index and growth parameter $\epsilon$ constant, we are able to study the portions of the parameter space containing solutions which reproduce the observables, without the need for an external optimization routine. We have shown that this phenomenological model is able to reproduce the detailed simulations of field growth by \citet{ho15} to high accuracy. By fitting the HBPs securely associated with SNRs with known ages and measured braking indices, we found interesting evolutionary trajectories for the systems in phase space. We conclude that if field growth is significant in the life cycle of HBPs, then they may be closely related to the CCOs early in their evolutionary histories. The end result of the field growth in CCOs may connect these objects to the HBPs and XDINSs. We also investigate the possibility of field growth in SGRs, however, the behaviour of these systems are largely unconstrained due to the absence of a lower SNR age limit and lack of measured braking index.
Field growth is not applicable to the AXPs, which require field decay to explain the observed difference in PSR and SNR ages, and a growing field is not necessary to explain the SGRs, provided that the characteristic age is larger than the true SNR age. Thus, in the context of magnetic field evolution, we conclude that both field growth and decay processes are required to explain the diverse population of NSs. Once the field has reached its asymptotic value, field decay may begin, increasing the braking index to values $n>3$ later in life. Thus, the time dependence of magnetic fields provides an interesting avenue to unify the population of NSs, and in particular, explain the apparently large characteristic ages for systems associated with relatively young SNRs.
While this evolutionary picture is simple and based on a phenomenological model, there are many emission mechanisms which have been proposed in the literature to solve the braking index and PSR--SNR age discrepancy problems. The field growth model was shown to be ineffective in explaining the constraints present in the HBP~J1119--6127 and the time evolution of HBP~J1846--0258, both of which are associated with pulsar wind nebulae. In a follow-up paper, we will thoroughly investigate alternatives to the physical emission mechanisms at work in these and various other classes of objects and the subsequent implications for the PSR--SNR association and evolution.
\section{Acknowledgements}
This work was primarily supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Canada Research Chairs Programme. SSH is also supported by an NSERC Discovery grant and the Canadian Space Agency. This research made use of NASA's Astrophysics Data System, McGill's magnetars catalogue and the U. of Manitoba's high-energy SNR catalogue (SNRcat). We also thank the referee for the thoughtful comments that helped significantly improve the clarity of this work.
|
2,869,038,155,720 | arxiv | \section{Introduction}
\vspace{0.5cm}
The problem of quantum quench, i.e. the response of a system to a time
dependent coupling,
has recently attracted a lot of attention in several areas of
many-body physics, particularly because of progress in cold atom
experiments \cite{sengupta},\cite{CCa},\cite{CCc}. This problem is
interesting for at least two reasons. The first relates to the
question of thermalization. Suppose we start with the ground state and
then turn on a time dependent coupling which again approaches a
constant at late times. Does the system evolve into some kind of
steady state ? If so, is the state "thermal" in any sense ?
The second question deals with the situation where the quench takes
place across a value of the parameter where there is an equilibriium
critical point. In this case there is some evidence that the time
evolution carries some universal features of the critical point.
Assuming that there is only one scale which governs the behavior of
the system in the critical region, one can, e.g. derive scaling
properties of one point functions following adaptations of the early
work of Kibble and Zurek \cite{kibble,zurek}. Suppose the coupling
approaches the critical coupling linearly, i.e.
\ben
(g-g_c) \sim vt
\een
These arguments then show that the one point function of an
operator with conformal dimension $x$ at the critical point has a
scaling behavior \cite{sengupta}, \ben <\cO (t)> \sim
(v)^{\frac{x\nu}{z\nu+1}} F(tv^{\frac{z\nu}{z\nu+1}}) \een where $z$
is the dynamical critical exponent and $\nu$ is the correlation length
exponent. Another manifestation of such universal behavior appears in
$1+1$ dimensional field theories which are quenched {\em suddenly} to
a critical point. In this case, powerful methods of boundary conformal
field theory can be used to obtain the time dependence of correlation
functions \cite{CCa,CCc}. For example the one point function of a
generic operator with conformal dimension $x$ behaves as \ben <\cO(t)>
\sim e^{-\frac{\pi x t}{2\tau_0}} \een where $\tau_0$ denotes a length
scale which characterizes the initial theory away from
criticality. $\tau_0$ is of course not universal, so neither is the
relaxation time $\tau = 2\tau_0/(\pi x)$. However, the ratio of the
relaxation times for two {\em different} operators $\cO_1$ and $\cO_2$
with conformal dimensions $x_1$ and $x_2$ is universal, \ben
\frac{\tau_1}{\tau_2} = \frac{x_2}{x_1} \een Unlike equilibrium
critical phenomena there is no general theoretical framework to
understand such scaling relations. In particular, there are very few
theoretical tools available to study such systems when they are
strongly coupled. It is, therefore, natural to explore if the AdS/CFT
correspondence \cite{Maldacena} - \cite{AdSR} is useful in this
problem.
In this contribution we will summarize results obtained in this
approach. The work on thermalization is in collaboration with Tatsuma
Nishioka and Tadashi Takayanagi \cite{dnt}. The work on quench across
critical points is in in collaboration with Pallab Basu \cite{bdas}.
In the AdS/CFT correspondence, couplings of the boundary field theory
are boundary values of a bulk field. In the regime where supergravity
is valid, the problem of quantum quench then reduces to a classical
problem with given initial and boundary conditions. This problem has
been studied when the time dependent coupling is the boundary metric
or the gauge theory coupling (i.e. the boundary value of the
dilaton). Suppose this coupling is a constant in the far past and
future, and has a smooth time dependent profile at intermediate
times. In the bulk description this corresponds to a disturbance
created on the boundary which propagates in the bulk. Under suitable
conditions, this leads to black hole formation in the bulk
\cite{janik,otherthermalization,holoentanglement}. The correlators at
future time would then be thermal with a temperature characterized by
the Hawking temperature. The time scale after which this happens
depends on the nature of the correlators, but turns out to be always
smaller than what one would expect from a conformally invariant system
evolving to a thermal state. Thus, in this case thermalization of the
field theory is signalled by black hole formation. For homogeneous
planar collapse (i.e. a space independent coupling in Poincare patch),
a black hole is always formed. In the case of homogeneous collapse in
global AdS, a black hole is formed when the rate of change is fast
enough compared to the scale set by the radius of the sphere on which
the boundary theory lives.
In other situations, e.g. a slow variation of the coupling for global
AdS, a black hole is not formed so long as the supergravity
approximation is valid. Rather, if the coupling becomes {\em weak} at
some time, the bulk string frame curvature grows large, leading to a
breakdown of the supergravity approximation - thus mimicking a
space-like singularity \cite{Awad}. For the case of a slow variation
of the coupling it turns out that the gauge theory remains well
defined and may be used to show that a smooth passage through this
region of small coupling is possible without formation of a large
black hole. Related scenarios appear in \cite{hertog} and \cite{eva}.
One of our main aims is to study quantum quench across critical
points. Many such critical points can be studied in setups where a
subset of the bulk fields can be treated in a probe approximation. In
this approximation, these probe fields provide the essential physics
and their backreaction to the background gravity can be ignored,
typically suppressed by $1/N$. This motivates us to study the problem
of quantum quench in situations where such a probe approximation is
valid.
We will first consider defect field theories which arise as dual
descriptions of a set of probe branes in the $AdS \times S$ bulk
\cite{KR}. This approach has been used extensively to study flavor
physics, as well as models with possible applications to condensed
matter systems. The nice feature of this approach is that the boundary
field theory is known, though they typically have supersymmetry. In
this case a quantum quench of couplings in this subsector becomes a
classical motion of these probe branes, with specified time dependent
boundary conditions at the $AdS$ boundary \cite{dnt}.
We investigate the
question of thermalization in this context. Since the background
geometry is unchanged in this approximation any black hole which is
formed due to the quench is not visible. We will find that
thermalization is nevetheless visible - this manifests itself as the
formation of an apparent horizon on the brane worldvolume.
In the second setup we consider a "bottom-up" bulk theory of gravity
with a neutral scalar field \cite{liu1}, where one writes down a bulk
theory and {\em assume} that there is some dual field theory on the
boundary. In this specific instance,
the background is a $AdS_4$ charged black
brane and the coupling of the scalar is large, so that its
backreaction to the geometry is small. When the mass of the scalar
lies in the range $ -\frac{9}{4} < m^2 < -\frac{3}{2}$ there is always
a critical temperature below which the trivial solution is
unstable. Equivalently, for a given temperature, whenever the mass is
below a certain value, the trivial solution is unstable. In this
regime there is a new nontrivial static, stable solution whose
"non-normalizable" part vanishes. This means that in
the dual theory there is a new phase where the expectation value of
the operator dual to this scalar is non-zero, even in the absence of
any source. For any nonzero temperature there is a continuous phase
transition with mean field exponents at the critical mass. At zero
temperature (i.e. when the background is an extremal brane) the
transition persists, but is of the Berezinskii-Kosterlitz-Thouless
type. This setup is similar to that of holographic superconductors
\cite{holosuper1, holosuper2,holosuper3} and
has been proposed as models for antiferromagentic transitions.
We consider quench across this critical point by working at the
critical mass, but turning on a time dependent source which crosses
zero (i.e. the critical point) at some time \cite{bdas}. We show that the dynamics
of the bulk scalar is dominated by a zero mode of the radial operator
in the critical region when the rate of change of the source is small,
This leads to a Landau-Ginsburg type dynamics with dynamical critical
exponent $z = 2$, and a resulting scaling behavior of the order
parameter.
\section{Probe Branes and Thermalization}
\vspace{0.5cm}
Probe branes in the bulk of $AdS$ have been used to introduce flavor
in the standard AdS/CFT correspondence. Consider for concreteness
$AdS_5 \times S^5$ whose dual is $\cN=4$ super-Yang-Mills in $3+1$
dimensions with gauge group $SU(N_c)$. Let us introduce $N_f$ Dp
branes which wrap a $AdS_m \times S^{p+1-m}$. Possible supersymmetric
wrappings are summarized in Table (\ref{tabl1}).
\begin{center}
\begin{table}[h]
\caption{\label{tabl1}Probe Branes in $AdS_5 \times S^5$.}
\centering
\begin{tabular}{|l|c|r|}
\hline
Brane&Wrapping&Dual Theory\\ \hline
D1& $AdS_2$ & $ 0+1 $ dim\\ \hline
D3& $AdS_3 \times S^1$ & $1+1$ dim\\ \hline
D5& $ AdS_4 \times S^2$ & $2+1$ dim\\ \hline
D7 & $AdS_5 \times S^3$ & $3+1$ dim \\ \hline
\end{tabular}
\end{table}
\end{center}
The Dp branes give rise to new hypermultiplet fields. These live on
the intersection of the Dp branes with the D3 branes which gave rise to
the $AdS_5 \times S^5$ geometry. From the point of view of the $N=4$
theory the hypermultiplets live on a lower dimensional defect. In the
strong coupling regime, the bulk theory is the original supergravity
together with the action of branes coupled to it.
In the limit of $N_f \ll N_c$ the backreaction of the probe branes on
the background $AdS_5 \times S^5$ geometry can be ignored and the
entire bulk theory is given by the action of these branes moving in
the fixed background geometry. We will take the brane action to be of
the DBI type. In the dual theory this means we can consider the defect
field theory by itself, and ignore the effect of hypermultiplet loops.
As is standard in the AdS/CFT correspondence, the boundary values of
the DBI fields are identified with sources for the dual operators in
the dual field theory. Consider for example the case of a D5
brane. Let us write the $AdS_5 \times S^5$ metric in the form
\ben
ds^2 =
(y^2+r^2)[-dt^2+dx_1^2+dx_2^2+dx_3^2]+
\frac{1}{y^2+r^2}[dr^2+r^2d\Omega_2^2+dy^2+y^2d(\Omega^\prime_2)^2]
\label{one}\een
The D5 brane is wrapped along $\xi^\alpha =
(t,r,\Omega_2,x_1,x_2)$. The fields in the DBI action are $y(\xi),
\Omega_2^\prime (\xi),x_3 (\xi)$, which are the transverse coordinates
to the brane. The value of $y(r=\infty)$ is then the mass of the
hypermultiplet fields coming from $(3,5)$ open strings joining the
D5 brane with the stack of $N_c$ three branes which produce the
background geometry. Thus a time dependent boundary value of $y$ is a
time dependent mass for the hypermultiplets.
Therefore a quantum quench in this dual theory may be implemented
simply by providing a time dependent boundary condition for the DBI
field. However the DBI fields are the transverse coordinates of the
branes - so this corresponds to a motion of the edge of the
brane. This disturbance sets up a wave along the brane and therefore
correponds to an excited state of the defect field theory. Our aim is
to figure out the nature of this state at late times.
In the full theory, such a disturbance would lead to a deformation of
the background geometry and possibly lead to black hole formation,
which would appear as thermalization in the boundary field theory. We
want to explore if any signature of thermalization remains in the
probe approximation. The following sections summarize some salient
points of work with Tatsuma Nishioka and Tadashi Takayanagi \cite{dnt}.
\subsection{Rotating D1 branes}
\vspace{0.5cm}
The essential physics is in fact apparent in the simplest example - D1
branes in $AdS_5 \times S^5$. For this purpose it is convenient to
write the $AdS_5 \times S^5$ metric as \ben ds^2 = 2drdv-f(r)dv^2+r^2
d\ts^2+(d\theta^2+\sin^2\theta d\vp^2+\cos^2\theta d\Omega^2_{3}) \een
If we are using the Poincare patch, $f(r) = r^2$ and $d\ts^2$ is the
flat metric on $R^3$, while in the global patch $f(r) = 1 +r^2$ and
$d\ts^2$ is the round metric on $S^3$. We have used
Eddington-Finkelstein coordinates in the $AdS_5$ part. The D1 brane is
along $(r,v)$ and its action is given by the standard DBI action
obtained from the induced metric. The dynamical fields on the brane
are $\theta (r,v), \vp (r,v), \Omega_3 (r,v)$ and the coordinates
contained in $d\ts^2$. It is clear from the symmetries that one can
have a class of solutions of the form \ben \vp (r,v),~~~~~~~~\theta =
\frac{\pi}{2} \een with all the other coordinates held constant. The
equations of motion which follow from the DBI action is best written
by using the advanced EF coordinate $u = v-2\int \frac{dr}{f(r)}$ as
well as $v$ \ben \partial_u\partial_v \vp + \frac{2}{L} \partial_v \vp
\partial_u \left( \frac{\partial_u \vp \partial_v \vp}{f(r)} \right) +
\frac{2}{L} \partial_u \vp \partial_v \left( \frac{\partial_u \vp
\partial_v \vp}{f(r)} \right) = 0 \ ,
\label{uveqnmotion}
\een where \ben L = 1-\frac{4}{f(r)}\partial_u \vp \partial_v \vp \ .
\een It is easy to see that any $\vp$ which satisfies either
$\partial_u \vp = 0$ or $\partial_v \vp = 0$ is a solution of
(\ref{uveqnmotion}). In particular, a solution which is a function of
$v$ alone represents the retarded effect of a boundary value of
$\vp$.
The induced metric produced by such a retarded solution is given by
\be
\label{indf} ds_{ind}^2 = -f(r) dudv +(\partial_v \vp)^2 dv^2 =
2drdv - [ f(r)-(\partial_v \vp)^2]dv^2 \ . \ee
This is a
two-dimensional AdS Vaidya metric which has an
apparent horizon at $f(r) = (\partial_v \vp)^2$, provided this
equation has a solution for real $r$. In the Poincare metric there is
always a solution for real $r$, while in global coordinates, this is
not guaranteed.
Depending on the profile of $\vp(v)$, the apparent horizon may or may
not develop into an event horizon. An example where it does is given
by the profile
\ben
\vp(v) = \vp_0 ( v + \frac{1}{k}\log \cosh (kv) )
\label{prof1}
\een
which leads to the following equation for the location of the
apparent horizon for the Poincare patch
\ben r = \vp_0 ( 1 + \tanh
(kv))
\label{ahprof1}
\een
This asymptotes to an event horizon at $r = 2\vp_0$. This
profile represents a D1 brane which starts from rest and spins with an
increasing spin, asymptoting to a constant rotation rate. The
function (\ref{prof1}) and the location of the apparent horizon is
shown in Figure (\ref{fig:prof1}) and (\ref{fig:ahprof1})
\begin{figure}[h]
\begin{minipage}{14pc}
\includegraphics[width=14pc]{prof1.eps}
\caption{\label{fig:prof1}The profile (\ref{prof1}) as a function of $v$}
\end{minipage}\hspace{2pc
\begin{minipage}{14pc}
\includegraphics[width=14pc]{ahprof1.eps}
\caption{\label{fig:ahprof1}Location of the apparent horizon. $v$
as a function of $r$ from equation (\ref{ahprof1}). At late time this becomes an event horizon}
\end{minipage}
\end{figure}
For strings
which eventually stop spinning, the apparent horizon does not develop
into an event horizon, but recedes back to $r=0$. For example if
\ben
\vp (v) = \vp_0(1 + \tanh(kv))
\label{prof2}
\een
the apparent horizon is located at
\ben
r = \frac{k \vp_0}{\cosh^2 (kv)}
\label{ahprof2}
\een
The
function (\ref{prof2}) and the location of the apparent horizon is
shown in Figure (\ref{fig:prof2}) and (\ref{fig:ahprof2})
\begin{figure}[h]
\begin{minipage}{14pc}
\includegraphics[width=14pc]{prof2.eps}
\caption{\label{fig:prof2}The profile (\ref{prof2}) as a function of $v$}
\end{minipage}\hspace{2pc
\begin{minipage}{14pc}
\includegraphics[width=14pc]{ahprof2.eps}
\caption{\label{fig:ahprof2}Location of the apparent horizon. $v$
as a function of $r$ from equation (\ref{ahprof2}). The apparent horizon now recedes back to $r=0$}
\end{minipage}
\end{figure}
In global $AdS$ the equation which determines the apparent horizon is
given by $1+r^2 = (\partial_v \vp)^2$ which does not always have a
solution for real $r$. For example, a brane whose end point is uniformly rotating
has $\vp (v) = \omega v$ - this would lead to an event horizon only
when $\omega > 1$.
The Poincare patch solutions represent injection of energy from the
boundary, which flows into the Poincare horizon. In the global
solutions, the energy flows from a point of the boundary to the
antipodal point of the $S^3$.
Fluctuations of the brane around this classical solution will feel the
effect of an apparent horizon on the worldsheet. Let us choose a static gauge
where the worldsheet coordinates are identified with two of the
space-time coordinates $\xi^a =
x^a,~~a=0,1$. The transverse coordinates are $x^I, I = 2 \cdots
9$. The $AdS \times S$ metric can be then written as \be ds^2 = g_{ab}
(x^a, x^I) dx^adx^b + G_{IJ}(x^a,x^K) dx^I dx^J \ .
\label{fluc1}
\ee
Expanding around a classical solution $x_0^I(x^a)$,
\be
x^I(x^a) = x_0^I (x^a) + y^I(x^a) \ ,
\label{fluc3}
\ee
This leads to the following action for quadratic fluctuations
\be
S_2 = \frac{T_{D1}}{2}\int d^{2}\xi \sqrt{-\gamma_0} \gamma_0^{ab} G_{IJ}
(\xi^a,x_0^I) \partial_a y^I \partial_b y^J \ .
\label{fluc8}
\ee
where $\gamma_0^{ab}$ denotes the induced metric due to the background
solution $x_o^I$.
In particular, the fluctuations of $\varphi,\theta$ i.e. all fluctuations in $S^q$ directions
are minimally coupled
massless scalars on the worldsheet, while the fluctuations of the
boundary gauge theory spatial directions $x^i, i
= 1\cdots 3$ have an additional factor of $r^2$ coming from the fact
that $G_{ij} = r^2 \delta_{ij}$ along these directions.
It is well known that fields which live on a space-time with an
apparent horizon behave approximately thermally if the apparent
horizon lasts long enough \cite{visser}. While the standard derivation of
Hawking radiation assumes the presence of an event horizon, the
essential physics is the large redshift, which is present near an
apparent horizon as well. In our case, profiles like (\ref{prof1})
lead to exact thermality at late times since the apparent horizon
evolves into an event horizon. On the other hand profiles like
(\ref{prof2}) lead to an effective ``time dependent temperature'' in
the dual theory, which of course makes sense when the time variation
is slow enough.
The thermal nature of the state produced by time dependence becomes
clear from a calculation of the fluctuation
of the end-point of the string.
In \cite{Bra} it has been shown that the fluctuations
of a string suspended from the horizon of a AdS black brane ended at
a flavor D-brane near the boundary of AdS are dual to Brownian
motion of the corresponding quark in the hot $\CN = 4$ gauge
theory. In this case the bulk black brane metric induces a worldsheet
metric which has a horizon. The fluctuations then reflect Hawking
radiation from the worldsheet horizon.
In the D-brane solutions considered above, the bulk metric has no
horizon. However due to the motion of the D-brane, the induced metric
on the worldvolume can develop a horizon. Since the fluctuations of
\cite{Bra} comes purely from properties of the induced metric it
is natural to expect that a similar phenomenon appears in our case.
The result of this calculation for fluctuations in the $\vp$
direction is
\ba \langle (\Delta y^\varphi (t-t^\prime))^2\rangle &
\sim & \frac{\pi (t-t^\prime)^2}{12 \beta^2} \ ,~~~~~~~~~~~~~~~~~~~\pi
(t-t^\prime) \ll \beta \ ,\no \langle (\Delta y^\varphi
(t-t^\prime))^2\rangle & \sim & \frac{(t-t^\prime)}{2 \beta}
-\frac{1}{2\pi} \log [2\pi (t-t^\prime)/\beta ] \ ,~~~~~\pi
(t-t^\prime) \gg \beta \ , \ea while for fluctuations in the $\vx$
direction are \ba \langle [\Delta y^i (t-t^\prime)]^2 \rangle & \sim &
\frac{(t-t^\prime)^2}{m\beta} \ ,~~~~t \ll m\beta^2 \ ,\no & \sim &
\beta |t-t^\prime| \ ,~~~~t \gg m\beta^2 \ . \ea
The rotating $D1$ brane solution corresponds to a time dependent
coupling in the $N=4$ theory coupled to hypermultiplets living on the
zero dimensional defect. The D1-D3 system is 1/4-BPS and the D1-D3
open strings lead to the two complex scalars $(Q,\ti{Q})$ of
hypermultiplets which belong to the fundamental and anti-fundamental
representations of the color $SU(N)$ gauge group. Let us
express the three complex adjoint scalar fields
in the $\CN=4$ super Yang-Mills by $(\Phi_1,\Phi_2,\Phi_3)$. These
correspond to cartesian coordinates in the transverse $C^3$ composed of
$(r,\Omega_5)$ where $\Omega_5$ represents the 5-sphere. We choose
$\Phi_3$ such that its phase rotation describes the one in the $\vp$
direction and that $\theta=\pi/2$ is equivalent to
$\Phi_1=\Phi_2=0$. The time dependent coupling term
which corresponds to a uniformly rotation D1-brane is given by
\be \int dt
\left[\ov{Q}~ \left[\mbox{Im}(\Phi_3 e^{-i\omega t})\right]^2~Q
+\ti{Q}~\left[ \mbox{Im}(\Phi_3 e^{-i\omega
t})\right]^2~\ov{\ti{Q}}\right] .
\label{intti}
\ee
The justification for this is given in \cite{dnt}.
For non-uniform rotation with a profile $\vp (v)$ the exponential
factors are simply replaced by $e^{\pm i \vp(t)}$.
Thus, from the boundary theory point of view we have a time dependent
coupling - this leads to thermalization. Note that this is
thermalization of only the hypermultiplet sector - the vector
multiplet sector is unchanged in the lowest order of this
approximation.
\subsection{Higher dimensional probes}
\vspace{0.5cm}
Thermalization in higher dimensional field theories can be also
investigated by considering higher dimensional branes in the bulk of
$AdS \times S$. It is possible to construct uniformly rotating D7 and
D5 branes using a combination of analytic and numerical methods.
While these D5 and D7 solutions have been obtained earlier in
\cite{evans},
the implications to thermalization was not realized.
Quench-like solutions (i.e. solutions where the rotation vanishes in the
asymptotic past and the asymptotic future) can be obtained numerically
as well \cite{dnt}.
Rotating D5 branes are dual to a time dependent mass of the
hypermultiplets in the dual 2+1 dimensional defect field
theory. Rotating D7 branes are dual to a time dependent phase of the
mass of fermions in the hypermultiplet, as well as a time dependent
bosonic potential. The essential physics is similar to the D1 brane
descibed in the previous subsection, viz. an apparent horizon is
formed and the fluctuations respond in a thermal fashion.
An additional signature of dissipation appears when we consider probe
D3 branes obtained by performing T-duality on the D1 brane solution
described above along $x_1$ and $x_2$ directions. In this case the
worldvolume is a $3+1$ dimensional theory and the dual defect field
theory is $2+1$ dimensions. It turns out that in this case one can
turn on a background electric field on the worldvolume of a uniformly
rotating D3 brane. This means we are turning on a chemical potential
and a charge density for the corresponding global charge in the dual
field theory, in addition to a time dependent coupling. The
fluctuations of the gauge field around this background now react to
the apparent horizon of the induced metric. This leads to an
electrical conductivity $\sigma(\nu)$ , whose behavior as a function
of the frequency $\nu$ is quite similar to Drude theory at low
frequencies. However the real part of $\sigma (\nu)$ approaches a
constant at large frequency, as is typical in a 2+1 dimensional
critical theories. This is shown in Figures (\ref{fig:one}) and
(\ref{fig:two})
\begin{figure}[h]
\begin{minipage}{14pc}
\includegraphics[width=14pc]{ReSigma.eps}
\caption{\label{fig:one}Real part of $\sigma(\nu)$.}
\end{minipage}\hspace{2pc
\begin{minipage}{14pc}
\includegraphics[width=14pc]{ImSigma.eps}
\caption{\label{fig:two}Imaginary part of $\sigma(\nu)$.}
\end{minipage}
\end{figure}
\subsection{Time Dependent Chemical Potential}
\vspace{0.5cm}
An interesting example of the process of thermalization due to
formation of an apparent horizon on the worldvolume concerns the
thermalization of the meson sector of $N=4$ Yang-Mills theory due to a
time dependent chemical potential \cite{hashimoto}. As usual, quarks
are introduced by placing a set of D7 branes and a chemical potential
corresponds to a worldvolume electric field. This is made time
dependent by coupling to a time dependent external current, i.e. by
injecting quarks from outside. As in the previous examples this
results in the formation of an apparent horizon with a characteristic
temperature. For other effects of this phenomenon see
\cite{otherapparent}.
Finally the formation of apparent horizons in these examples is similar to acceleration horizons on worldvolumes discussed in \cite{wa} and related phenomena have been studied in \cite{Gursoy}.
\section{Quench across a Holographic Critical Point}
\vspace{0.5cm}
As we discussed above, quantum quench is particularly interesting when
the time dependent coupling crosses a critical point.
The probe
approximation has turned out to be quite useful for studying
holographic critical points. Such phase transitions are known to occur
for various probe branes \cite{phase,karch,bktpapers}.
Another class of probe fields appear in discussions of holographic
superconductors \cite{holosuper1,holosuper2,holosuper3}.
This setup consists of a charged scalar field in the
presence of a charged black brane. When the gauge coupling is large,
the backreaction of the scalar and the gauge field to the background
geometry can be ignored. In this case, for a given mass of the scalar
field there is a critical temperature below which the scalar condenses
- this is interpreted as superfluidity in the boundary theory. The
phase transition between the ordered and disordered phases is a
critical point.
An even simpler setting consists of a neutral scalar field with a
quartic coupling which is large enough to ensure that a probe
approximation is reliable. As shown in \cite{liu1} for suitable values
of the parameters, this model displays a critical point of the type
encountered in antiferromagentic phase transitions. In this section I
will describe a recent study of quantum quench in this model in
collaboration with Pallab Basu
\cite{bdas}.
\subsection{The equlibrium phase transition}
\vspace{0.5cm}
The model of \cite{liu1} has a neutral scalar field $\phi(t,r,\vx)$ in
the background of a charged $AdS_4$ black brane. The lagrangian is
given by
\ben
\cL =
\frac{1}{2\kappa^2\lambda}\sqrt{-g}[-\frac{1}{2}(\partial \phi)^2
-\frac{1}{4}(\phi^2+m^2)^2-\frac{m^4}{4}]
\label{1-1}
\een
The background metric is given by (in $R_{AdS}=1$ units)
\ben
ds^2 = [-r^2 f(r)dt^2+r^2 d\vx^2]+\frac{dr^2}{r^2 f(r)}
\label{1-2}
\een
where
\ben
f(r) = [1+\frac{3\eta r_0^4}{r^4}-\frac{(1+3\eta) r_0^3}{r^3}]~~~~~~~~0 \leq \eta \leq 1
\label{1-3}
\een
The associated Hawking temperature is then given by
\ben
T = \frac{3}{4\pi r_0}(1-\eta)
\label{1-4}
\een
In the following we will replace $r \rightarrow r r_0$. This means all dimensional quantities are expressed in units of $r_0$.
In the limit of large $\lambda$ the
field $\phi$ can be regarded as a probe field.
In \cite{liu1} it was shown that when the mass lies in the range
\ben
-\frac{9}{4} < m^2 < -\frac{3}{2}
\label{1-5}
\een
there is a critical phase transition at
some value of $T = T_c(m)$ when the source to the dual operator
vanishes.
Conversely, for a given $T$ there is a
value of $m^2 = m_c^2$ where the theory is critical.
The upper limit in (\ref{1-5}) is the BF
bound for the near-horizon $AdS_2$ geometry which appears in the
extremal ($\eta = 0$) metric. (Note that the AdS scale for this
infrared $AdS_2$ is given by $1/\sqrt{6}$ in our units). The lower
bound is the BF bound for the asymptotic $AdS_4$. Field
configurations which are translationally invariant in the $\vx$
directions satisfy the equations of motion
\ben
\frac{1}{r^2}[-\frac{1}{f(r)}\partial_t^2+\partial_r(r^4
f(r)\partial_r)]\phi -m^2 \phi -\phi^3 = 0
\label{1-6}
\een
Near the $AdS_4$
boundary the asymptotic behavior of the solution to the linearized
equation is of the form \footnote{When we turn on $J(t)$, it is a valid concern whether we will be able to neglect the non-linear term near the boundary. This can be done as long as $\Delta>0$ or $m^2<0$.}
\ben
\phi (r) = J(t)r^{-\Delta_-}[1+O(1/r^2)]+<{\cal O}>(t) r^{-\Delta_+}[1+O(1/ r^2)]
\label{1-7}
\een
where $\Delta$ is given by
\ben
\Delta_\pm = \frac{3}{2} \pm \sqrt{m^2 + \frac{9}{4}}
\label{1-8}
\een
In the range of masses of interest, both the solutions are
normalizable, so that there is a choice of quantization. The standard
quantization considers the coefficient $J(t)$ as the source in the
dual field theory and $B(t)$ then gives the expectation value of the
dual operator. In the alternative quantization the expectation and source change the role.
Consider first the linearized problem, ignoring the cubic term. By a standard change of coordinates to tortoise coordinates $\rho$ and a field redefinition to
$\chi$,
\ben
d\rho =- \frac{dr}{r^2 f(r)}~~~~~~~~~~~~~~\phi (r,t) = \frac{\chi (\rho,t)}{r}
\label{1-9}
\een
The horizon is then at $\rho = \infty$ and the boundary is at $\rho = 0$.
At the linearized level, the equation (\ref{1-6}) becomes
\ben
-\partial_t^2 \chi = -\partial_\rho^2 \chi + V_0(\rho) \chi \equiv \cP_\rho \chi
\label{1-10}
\een
with
\ben
V_0(\rho) = r^2 f(r)[(m^2+2) - \frac{6\eta}{r^4}+\frac{1+3\eta}{r^3}]
\label{1-11}
\een
where in $V_0(\rho)$ we need to express $r$ in terms of $\rho$ using
(\ref{1-9}).
For solutions of the type $\chi \sim e^{-i\omega t}$, equation
(\ref{1-11}) is a Schrodinger problem in a potential $V_0(\rho)$. The
potential goes to zero at the horizon $\rho = \infty$ and behaves as
$\frac{(m^2+2)}{\rho^2}$ near the boundary $\rho = 0$. Note that for
a brane background at any finite temperature, $f(r) \sim (r-1)$ near
the horizon, while $\rho \sim -\log (r-1)$ so that $V_0 \sim
e^{-\rho}$ as we approach the horizon. A typical $V_0(\rho)$ for $m^2
> -2$ is shown in Figure(\ref{fig:potential1}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.75]{potential1.eps}
\end{center}
\caption{\label{fig:potential1}
The potential $V_0(\rho)$}
\end{figure}
In contrast, for the
extremal background $f(r) \sim (r-1)^2$ while $\rho \sim 1/(r-1)$ so
that $V_0 \sim 1/\rho^2$. This makes the analysis for the extremal
background rather subtle. We will work with the
non-extremal case.
In \cite{liu1} it was shown that when $m^2$ is below a critical value,
$m_c^2$ there are bound states of this Schrodinger problem, showing
that the trivial $\phi=0$ is unstable. At $m^2 = m_c^2$ a zero energy
bound state appears, which vanishes in an appropriate fashion at the
boundary and is in addition regular at the horizon. In the
complex frequency plane some quasinormal mode(s) hit the origin at
$m^2 = m_c^2$. This critical mass is $m_c^2 = -\frac{3}{2}$ when $\eta
= 1$ and decreases with decreasing $\eta$ or increasing temperature.
For $m^2 < m_c^2$ there is a stable nontrivial static solution $\phi_0
(r)$ of the full nonlinear equation of motion with the condition that
$J = 0$. This means that in the conventional quantization, the
expectation value of the dual operator is nonzero even in the absence
of a source, i.e. the field
condenses. The the critical point is at $m^2 =m_c^2$ and $J = 0$.
Similarly there is a nontrivial solution with $B = 0$ which means that
there is a condensate in the alternative quantization as well.
The critical point has mean field exponents at any finite
$T$. If the operator dual to the field $\phi$ is $\cO$ and the source
is $J$ then an analysis identical to that presented in \cite{liu1}
leads to (for $m^2-m_c^2 \rightarrow 0^+$) \ben <\cO>_{J=0}\sim
(m^2-m_c^2)^{1/2}~~~~~\frac{d<\cO>}{dJ}|_{J=0}\sim
(m^2-m_c^2)^{-1}~~~~~~<\cO>_{m=m_c}\sim J^{1/3}
\label{1-12}
\een
Exactly at zero temperature the phase transition is of BKT type
and the order parameter depends exponentially
\ben <\cO>_{J=0} \sim
\exp \left[-\frac{\pi\sqrt{6}}{2\sqrt{m_c^2-m^2}}\right]
\label{1-13}
\een
\subsection{Quenching across the Critical Point}
\vspace{0.5cm}
In the following, we will study quench across this critical point by considering a time dependent source $J(t)$ which asymptotes to constant values at early and late times and crosses zero at some time, e.g.
\ben
J(t) = J_0 \tanh (vt)
\een
In the conventional quantization this means that in the dual boundary field theory, we have a source coupling to the operator dual to $\phi$. For any static $J \neq 0$ we of course have a nontrivial $\phi(r)$ and hence a nonzero $<\cO>$. Our first aim is to get some insight into the time dependence of $<\cO (t)>$ when we have a nontrivial $J(t)$.
\subsection{Breakdown of Adiabaticity}
\vspace{0.5cm}
If the time dependence is slow enough one would expect that far away from the critical point the dynamics is adiabatic, while near the critical point adiabaticity should break down. It is instructive to examine the way this happens.
It is well known that to study low frequency modes in the background
of a black brane it is convenient to use ingoing Edddington-Finkelstein
coordinates,
\ben
u=t- \rho,~~~~~~\rho
\een
where $\rho$ is defined in (\ref{1-9}).
In terms of these coordinates the equation of motion \ref{1-6}) becomes
\ben
-2\partial_u\partial_\rho \chi = -\partial_\rho^2 \chi + V(\rho,\chi).
\label{4-1}
\een
where
\ben
V(\rho,\chi)= V_0(\rho)\chi+ f(r) \chi^3.
\een
This equation has to be solved with the boundary condition that the
field is {\em regular at the horizon}, which at the linearized level
is equivalent to requiring that the waves are purely ingoing at the
horizon \cite{bhmr,hubeny2}.
We need to solve (\ref{4-1}) with the condition
\ben
\chi(u,\rho) \rightarrow \rho^{-1+\Delta_-} J(u) ~~~~~~{\rm as}~~ \rho
\rightarrow 0
\label{4-1-1}
\een
where $\Delta_\pm$ are defined in (\ref{1-8}).
To perform the adiabatic expansion, let us decompose the field $\chi
(\rho, u)$ as
\ben
\chi(\rho,u)=\chi_l(\rho,u)+\chi_s(\rho,u)
\label{4-1-2}
\een
Where $\chi_l(\rho,u)=J(u) \rho^{-1+\Delta_-}$ and $\chi_s(\rho,u)
\sim \rho^{-1+\Delta_+}$ as $\rho \rightarrow 0$.
For a constant $J$, $\chi_l(\rho,u)=\chi_l(\rho)$ is time independent.
In this case there is a static
solution $\chi_s(\rho,u)=\chi_0(\rho)$, which is the equlibrium configuration. In the presence of a source which is {\em slowly varying} in units of the horizon radius, one can therefore expand the field $\chi_s(\rho,u)$ in an adiabatic expansion of the form
\ben \chi_s(\rho,u) = \chi_0(\rho, J(u)) +
\epsilon ~\chi_1(\rho,u) + \cdots.
\label{expan}
\een
Here $\epsilon \sim \partial_u$ (recall that we are using $r_0=1$ units) is an adiabaticity parameter which keeps track of the adiabatic
expansion. If we scale $u \rightarrow u/\epsilon$, each $u$ derivative
is of order $O(\epsilon)$. The idea then is to insert (\ref{expan})
into the equations of motion and obtain equations for $\chi_1,
\chi_2,\cdots$ order by order in $\epsilon$. To the lowest order one
gets
\ben
\cD_\rho^{(1)} \chi_1 = \{ [-\partial_\rho^2+ V_0(\rho)] +f(r)
(3\chi^2_0+6 \chi_l\chi_0+3\chi_l^2) \} \chi_1= - 2
\partial_u\partial_\rho \chi_l - 2 \partial_u\partial_\rho \chi_0
\label{1-16}
\een The solution to this equation is
\ben
\chi_1=\int d\rho'
G(\rho,\rho')\partial_{u^\prime}\partial_{\rho^\prime}
(\chi_0+\chi_l)(\rho^\prime).
\label{1-17a}
\een
where $G(\rho,\rho^\prime)$ is the Green's function of the
operator $\cD_\rho^{(1)}$ with the boundary conditions $G(0,\rho) =0$
and $G(\infty,\rho)$ is regular :
\bea G^{(1)}(\rho,\rho') & = &
\frac{1}{W(\ttxi_1,\ttxi_2)} \, \ttxi_1(\rho') \ttxi_2(\rho) , \quad
\rho < \rho' \nn \\ & = & \frac{1}{W(\ttxi_1,\ttxi_2)} \,
\ttxi_2(\rho') \ttxi_1(\rho), \quad \rho > \rho',
\eea where $\ttxi_1$
and $\ttxi_2$ are solutions of homogeneous part of eqn (\ref{1-17a})
satisfying appropriate boundary condition at the horizon $\rho
=\infty$ and the boundary $\rho = 0$ respectively, and
$W(\ttxi_1,\ttxi_2)$ is the Wronskian which is independent of $\rho$
in this case. We have normalized $\ttxi_1(\rho)$ and $\ttxi_2(\rho)$
in such a fashion that $\ttxi_1=1$ at the horizon and $\ttxi_2
\rightarrow \rho^{-1+\Delta_-}$ near the boundary. Regularity of the
functions $\chi_l$ and $\chi_0$ mean that $\partial_r \chi_l,
\partial_r \chi_0$ are finite at the horizon. Since $(r-1) \sim
e^{-\rho}$, this implies that $\partial_\rho(\chi_l+\chi_0) \sim
\exp(-\rho)$. This ensures that the integral in (\ref{1-17a}) is
finite even though the Green's function approaches a constant in the
region near the horizon $\rho^\prime \rightarrow \infty$. Furthermore, near the
horizon $\ttxi_2$ can be expressed as a linear combination of a
regular and irregular solution, i.e $\ttxi_2(\rho \rightarrow
\infty)=a \rho + b$. This implies that $W(\ttxi_1,\ttxi_2)
=a$. Thus $\chi_1(u,\rho)$ is finite so long as $a$ is finite.
At the critical point, $J$ becomes small. Then $\chi_0$ and $\chi_l$ in the left hand
side of (\ref{1-16}) vanish, and the operator is identical to the
operator acting on the linearized small fluctuations at $m^2 = m_c^2$
around the trivial solution $\chi_0 = 0$, i.e. the operator $\cP_\rho$
which
appears on the right hand side of (\ref{1-10}). We know that this
operator has a zero mode which is regular at the horizon {\em and}
vanishes as $\rho^{-1+\Delta_-}$ at the boundary $\rho = 0$. This
means that at this point $a = 0$. Therefore, the first adiabatic
correction diverges.
For small
$J(u)$, the leading departure from the critical operator comes from
the term which is proportional to $\chi_0^2 \sim J^{2/3}$. Thus we
can use perturbation theory in $J$ to estimate $a \propto J^{2/3}$. As
argued before, $\chi_0 \sim (-J)^\frac{1}{3}$, while $\chi_l \sim J$.
Hence the leading divergence in $\chi_1$ can be estimated as
$\chi_1(\rho,u) \sim \ J^{-4/3}\dot{J}$. Adiabaticity breaks down when
\ben
\chi_1(\rho,u) \sim \chi_0 \Rightarrow \dot{J} \sim J^{5/3}
\een
In particular for profiles of $J(u)$ where $J(u) \sim vu$ near the critical point at $J=0$, adiabaticity breakdown occurs at
\ben
u \sim v^{-2/5}
\een
\subsection{Quench in a Landau Ginsburg Model}
\vspace{0.5cm}
The scaling behavior found above is identical to that in a
Landau-Ginsburg dynamics with dynamical critical exponent $z=2$. The
dynamics of an spatially homogeneous order parameter $\vp$ is given by
\ben \frac{d\vp}{dt} +m^2 \vp + \vp^3+J(t) = 0
\label{three}
\een
The equilibrium critical point is at $m = J = 0$. For $m^2=0$ the
equilibrium value of the order parameter is
\ben
\vp_0(J) = [-J]^{1/3}
\een
As usual an adiabatic expansion is of the form
\ben
\vp (t) = \vp_0 (J(t)) + \epsilon \vp_1 (t) + \cdots
\een
and to lowest order
\ben
\vp_1 =\frac{1}{2\vp_0^2} {\dot{J}}\frac{\partial \vp_0}{\partial J}
\een
and adiabaticity breaks down when $\vp_1 \sim \vp_0$ which becomes the condition
\ben
{\dot{J}}\frac{\partial \vp_0}{\partial J} \sim J^{1/3} \Rightarrow {\dot{J}} \sim J^{5/3}
\een
exactly as in our system.
When adiabaticity breaks down our system enters a scaling
region. Suppose the function $J(t)$ behaves linearly with time in the
critical region. Then it is straightforward to see from (\ref{three})
that in this region the solution is of the form \ben \vp (t,v) =
v^{1/5} \vp(tv^{2/5},1) \een This means, in particular, that the time
at which the order parameter hits zero scales as $v^{-2/5}$ while the
value of the order parameter at $t=0$ scales as $v^{1/5}$. A numerical
solution of the equation (\ref{three}) with adiabatic initial
conditions is consistent with this scaling, as shown in Figure (\ref{fig:LG})
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.75]{LGsource2.ps}
\end{center}
\caption{\label{fig:LG}
The scaled order parameter as a function of scaled time for a $J(t) =
\tanh(vt)$ at $m^2=0$ with $v=10^{-0.5},10^{-1},10^{-1.5},10^{-2}$
(from the bottom on the left). The adiabatic solution (dashed) is
also shown as a comparison.}
\end{figure}
Note that the order parameter hits zero {\em later} than the location
of the equilibrium critical point. This is a manifestation of the
phenomenon of raising the critical temperature when the temperature is
time dependent \cite{bao1} which has been holographically realized in
\cite{bao2}.
We will now argue that the behavior of our holographic system in the
critical region is fairly well described by such a LG dynamics
\subsection{Small $v$ dynamics in the Holographic Model}
\vspace{0.5cm}
Consider the dynamics of the bulk field in the critical region in
the presence of a linear quench $J(u) = vu$ for small $v$. To do this,
first substitute (\ref{4-1-2}) in the equation (\ref{4-1}) and rescale
\ben
\chi_s \rightarrow v^\frac{1}{5} \tilde \chi_s,u \rightarrow
v^{-\frac{2}{5}}\tilde u
\label{rescale}
\een
The equation (\ref{4-1}) then becomes
\ben [-\partial_\rho^2+ V_0(\rho)] \tch_s+v^{\frac{2}{5}} [f(r)
(\tch_s)^3 + \tilde u [-\partial_\rho^2+ V_0(\rho)] \tch_l+2
\partial_{\tilde u}\partial_{\rho} \tch_s ] +\cdots=0
\label{veq}
\een
The ellipsis denote terms which contains higher powers of $v$.
Let us expand the sub-leading part of the scalar field in terms of eigenfunctions of the operator $\cP_\rho$ (defined in equation (\ref{1-10}) at the critical point,
\ben
\tch_s(\rho,u) = \int \tilde a_k(u) \ch_k(\rho) dk
\een
The $\ch_k$ satisfy
\ben
\cP_\rho^c \chi_k = [-\partial_\rho^2+ V_0^c(\rho)]\chi_k = k^2 \chi_k
\label{eigeneqn}
\een where $V_0^c$ denotes the potential in (\ref{1-11}) at
$m^2=m_c^2$. The eigenfunctions $\chi_k(\rho)$ are delta function
normalized and obey the condition
\ben
{\rm Lim}_{\rho \rightarrow 0}
[ \rho^{1-\Delta_-} \chi_k (\rho) ] = 0 \een In terms of the
eigen-coefficients $a_k(u)$ the equation (\ref{veq}) becomes,
\ben
k^2
\tilde a_k+ v^{\frac{2}{5}}\left( \tilde u {\cal J}_{k} + \int b_{kk'}
\partial_{\tilde u} \tilde a_{k'} dk' + \int \tilde a_{k'} \tilde
a_{k''} \tilde a_{k'''} C_{k,k',k'',k'''} d k' dk'' dk''' \right) +\cdots =0
\label{modeeqn}
\een where
\bea {\cal J}_{k}& = &\int \ch_k (\rho) [-\partial_\rho^2+
V_0(\rho)] \ch_l \, d\rho \nn \\
b_{kk'} & = & \int d \rho \, \ch_k
\partial_\rho \ch_k' \nn \\
C_{k,k',k'',k'''}& = &\int d\rho \, \ch_k
\ch_{k'} \ch_{k''} \ch_{k'''} f(r) .
\label{moments}
\eea
The equation
(\ref{modeeqn}) suggests that there is a solution in a perturbation
expansion of powers of $v^\frac{2}{5}$,
\ben \tilde
a_k(\tilde u) =\delta(k) \tilde \xi_0(\tilde u) + v^{\frac{2}{5}}
\tilde \eta_k(\tilde u)+\cdots,
\label{vexpn}
\een
If this expansion makes sense, the dominant behavior of the solution for $\chi_s$ is given by the zero mode $\tilde\xi_0$.
Substituting (\ref{vexpn}) in the equation (\ref{modeeqn}) we get to
the lowest order in the small $v$ expansion,
\bea &
& \tilde u \, {\cal J}_{0} + b_{00} \frac{d}{d\tilde u} \tilde
\xi_0(\tilde u) + C_{0000} \tilde \xi_0(\tilde u)^3 = 0 \nn \\ & &
\tilde \eta_k(\tilde u) = -\frac{1}{k^2}\left (\tilde u {\cal J}_{k}-
b_{k0} \frac{d}{d\tilde u} \tilde \xi_0(\tilde u)-C_{k000} \tilde
\xi_0(\tilde u)^3\right)
\label{scalingeqn}
\eea
The first equation in (\ref{scalingeqn}) determines the time
dependence of the zero mode $\tilde\xi_0$, while the second equation
determines the leading correction from nonzero modes in terms of the
solution for $\tilde\xi_0$.
It is useful to rewrite the second equation by subtracting the first from it,
\ben
\tilde \eta_k(\tilde u) = -\frac{1}{k^2}\left ( \tilde u ({\cal J}_{k}-{\cal J}_{0}) -
(b_{k0}-b_{00} )\frac{d}{d\tilde u} \tilde \xi_0(\tilde
u)-(C_{k000}-C_{0000}) \tilde \xi_0(\tilde u)^3\right)
\label{scalingeqn2}
\een
The expansion in powers of $v^{2/5}$ would be valid if $\eta_k(\tilde u)$ remains finite.
However, $k$ is a continuous parameter starting from zero. This means that there is a potential divergence in the $k \rightarrow 0$ limit.
Indeed, as will be argued in the next subsection, for generic
potential $V_0$ the numerator on the right hand side of
(\ref{scalingeqn2}) behaves as $k$ for small $k$, so that $\eta_k$
indeed diverges at $k=0$. However exactly at the critical point, the
small $k$ behavior changes to $k^2$ so that $\eta_k$ remains finite
and the expansion in $v^{2/5}$ remains valid.
\subsection{Validity of the small $v$ expansion}
\vspace{0.5cm}
To examine this issue we need to consider the eigenvalue
problem \ben [-\partial_\rho^2+ V_0(\rho)]\chi_k = k^2 \chi_k
\label{egen}
\een
As discussed above the potential $V_0(\rho) \rightarrow
-e^{-\rho}$ as $\rho \rightarrow \infty$. This potential is shown in Figure (\ref{fig:potential1}).
The basic features of the eigenfunctions can in fact be gleaned from a
simpler problem in which we replace the potential by the following
potential which has the same qualitative features. \bea & = &
\infty~~~~~~~~~~\rho = 0 \nn \\ U(\rho) & = & -U_0~~~~~~~~~~~~0 \leq
\rho \leq 1 \nn \\ & = & 0~~~~~~~~~~~~~~~1 \leq \rho \leq \infty
\label{squarewell}
\eea
This problem is of course solvable. The eigenfunctions of the
Schrodinger operator with eigenvalue $k^2 > 0$ are \bea \psi_k(\rho) &
= & \frac{A(k)}{\sqrt{\pi}} \sin (\sqrt{k^2+U_0}~\rho) ~~~~~~~~~~0
\leq \rho \leq 1 \nn \\ \psi_k(\rho) & = &
\frac{1}{\sqrt{\pi}}\sin(k\rho + \theta(\rho)) ~~~~~~~~~~\leq \rho
\leq \infty
\label{12-1}
\eea
where the constants $A(k)$ and $\theta(k)$ are determined by matching at $\rho = 1$,
\bea
A(k) &= & \frac{k}{\sqrt{k^2
\sin^2(\sqrt{k^2+U_0})+\sqrt{k^2+U_0}\cos^2(\sqrt{k^2+U_0}})}\nn \\ \theta(k)&=&
\tan ^{-1}\left(\frac{k \tan
\left(\sqrt{k^2+U_0}\right)}{\sqrt{k^2+U_0}}\right)-k.
\label{12-2}
\eea
The solution for $k=0$ is
\bea
\psi_0(\rho) & = & \frac{B}{\sqrt{\pi}} \sin (\sqrt{U_0}~\rho) ~~~~~~~~~~0 \leq \rho \leq 1 \nn \\
\psi_0(\rho) & = & a\rho + b ~~~~~~~~~~~~~~\leq \rho \leq \infty
\label{12-3}
\eea
The matching conditions at $\rho = 1$ now yield
\bea
\frac{B}{\sqrt{\pi}} \sin (\sqrt{U_0}~\rho) & = & a+b \nn \\
\frac{B\sqrt{U_0}}{\sqrt{\pi}} \cos (\sqrt{U_0}~\rho) & = & a
\label{12-4}
\eea
For any $a \neq 0$ the solution blows up at $\rho = \infty$. Thus regular solutions require $a = 0$. However the second equation in (\ref{12-4}) then imply that
\ben
\sqrt{U_0} = (n +\frac{1}{2}) \pi
\label{12-5}
\een
These are the zero modes. As we increase the depth of the potential, the first zero mode appears at $\sqrt{U_0} = \pi/2$. In the context of our model this is the potential where we have a critical point.
The small $k$ behavior of $A(k)$ and $\theta (k)$ can be read off from the expressions (\ref{12-2}). For a generic $U_0$ these are
\bea
A(k) & \sim & \frac{k}{\sqrt{U_0} \cos \sqrt{U_0}} + O(k^2) \nn \\
\theta (k) & \sim & k [ \frac{\tan \sqrt{U_0}}{\sqrt{U_0}} - 1] +O(k^3)
\label{12-6}
\eea
whereas for critical potentials we have
\bea
A(k) & \sim & 1 - \frac{k^2}{8} + O(k^4) \nn \\
\theta (k) & \sim & -\frac{\pi}{2} - \frac{k}{2}
\label{12-7}
\eea
Thus the small-$k$ behavior of the eigenfunctions are drastically different for the critical potentials. This has important implications for the coefficients like
$({\cal J}_{k}-{\cal J}_{0}), (b_{k0}-b_{00} )$ and $(C_{k000}-C_{0000})$ in (\ref{scalingeqn2}). Consider for example the quantity ${\cal J}_k$. This is an integral of the form
\ben
\int_0^\infty d\rho~J(\rho)~\chi_k (\rho)
\een
where $J(\rho)$ is a smooth function (which is $[-\partial_\rho^2+
V_0(\rho)] \ch_l $). If we replace the true eigenfunctions by those of our simplified problem, we get
\ben
{\cal J}_k = A(k) \int_0^1 d\rho \sin (\sqrt{k^2+U_0}\rho) J(\rho) + \int_1^\infty d\rho\sin(k\rho + \theta(k))
\een
Using (\ref{12-6}) and (\ref{12-7}) we therefore see that
\ben
{\cal J}_k -{\cal J}_0 \sim k~~~~~~~k \rightarrow 0
\een
for generic potentials, whereas
\ben
{\cal J}_k -{\cal J}_0 \sim k^2~~~~~~~k \rightarrow 0
\een
for critical potentials. It is straightforward to see that the behavior of the other coefficients $(b_{k0}-b_{00} )$ and $(C_{k000}-C_{0000})$ are similar.
The small-$k$ behavior of the eigenfunctions for the potential which
is relevant for us. $V_0^c(\rho)$
is quite similar. This has been discussed in the Appendix C of
\cite{bdas}. Going back to (\ref{scalingeqn2}) we therefore see that
the small $v$ expansion is generically {\em not valid} since the
corrections diverge at small $k$. However for the critical potential,
$\tilde\eta_k$ remain finite as $k \rightarrow 0$ and the expansion in
powers of $v^{2/5}$ makes sense.
\subsection{Scaling relation}
\vspace{0.5cm}
The validity of this expansion means that
the solution for $\chi_s$ is of the form
\ben
\chi_s(\rho,u) \approx v^{\frac{1}{5}}\tilde \xi_0 (v^{\frac{2}{5}}u)
\chi_0(\rho)+v^{\frac{3}{5}} \int \tilde \eta_k(v^{\frac{2}{5}}u)
\chi_k(\rho) dk + \cdots
\een
Thus the dynamics for small $v$ is dominated by the zero mode in the
critical region. The equation for $\tilde
\xi_0$ is, however, exactly the same as the equation for the order
parameter $\vp$ in the LG model in the previous subsection, after
going back to the original variables prior to the rescaling in (\ref{rescale}).
Thus in this region the system behaves as one with dynamical critical
exponent $z=2$.
Therefore
we conclude that to leading order in small $v$, we have
\ben
<\cO>(u,v) \sim {\rm Lim}_{\rho \rightarrow 0} [\rho^{1-\Delta_+}
\chi_s (\rho,u)] \sim v^{1/5} <\cO>(tv^{2/5},1)
\een
and the time scale behaves as $v^{-2/5}$.
The fact that the dynamics in the critical region is governed by an
equation with a first order time derivative is made manifest in our
treatment using Eddington-Finkelstein coordinates. The whole analysis
can be of course performed in principle in the $(t,r)$
coordinates. However, we suspect that in this case one has to exercise
extreme care, just as one had to extract out a leading horizon
behavior in the linearized problem of fields in a black brane
background \cite{starinets}.
Note that beyond the critical region, $J(u)$ departs from the form
$J(u) \sim vu$, and the expansion in powers of
$v^{2/5}$ is no longer valid. Now all the modes are important, and
integrating out higher modes can give rise to higher time derivatives
in the effective equation for $<\cO>$.
\subsection{Mass quench}
\vspace{0.5cm}
An analysis similar to the above can be carried out when the quenching
is performed by making the mass parameter of the bulk theory a
function of the retarded time $u$, keeping $J(u) = 0$. This does not have a direct
interpretation in the boundary field theory. However, as explained in
\cite{liu1}, the field $\phi$ can acquire a mass because of a
coupling to some other field $\phi^\prime$. A time dependent boundary
value of the field $\phi^\prime$ can then lead to a time dependent
mass. When $m^2(u) \sim m_c^2 + vu$ near the critical point, we now
get a behavior $<\cO> \sim v^{1/4}$.
\subsection{Numerical Results}
\vspace{0.5cm}
We have performed some preliminary numerical work for the case of a
mass quench. Our results clearly display the propagation of
disturbances towards the horizon along a light cone
, and a decay of
the order parameter in a fashion similar to the $z=2$ LG
dynamics, which is shown in Figure(\ref{fig:ord2}).
However our results are not accurate enough to verify the
scaling behavior found above. The late time decay should be governed
by the quasinormal modes \cite{hubeny2,quasi}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.75]{ord2.ps}
\end{center}
\caption{\label{fig:ord2}The order parameter $<\cO>(t)$ in boundary
theory.}
\end{figure}
\section{Outlook}
\vspace{0.5cm}
We have shown that holographic methods are useful in providing insight
into various questions related to quantum quench in strongly coupled
field theories which have a gravity dual. Interestingly, this is
possible in the probe approximation, which is typically much easier
to study.
So far we have been able to study in some detail quench
dynamics near holographic critical points which are described by mean
field exponents. Not surprisingly, we found scaling behavior
characteristic of Landau-Ginsburg models with $z=2$. This value of
the dynamical critical exponent is consistent with the results of
\cite{holosupdynamical}.
It is important to perform our quench analysis for zero temperature,
where the equilibrium transition is of the BKT type \cite{liu1},
similar to brane models of zero temperature chiral symmetry breaking
transition in \cite{bktpapers}. This case is rather subtle, but can be
studied using similar methods. We expect that in this case we will
have a $z=1$ theory dominating the critical region.
This would provide results for quench
dynamics which are not easily obtainable by other methods. Finally, a
more extensive numerical investigation should throw light on the
question of thermalization at late times, after the system has crossed
the critical region.
\section{Acknowledgements}
\vspace{0.5cm}
I would like to thank my collaborators Pallab Basu, Tatsuma Nishioka
and Tadashi Takayanagi for very enjoyable collaborations and many
insightful discussions. I would also like to thank Karl Landsteiner,
Satya Majumdar, Gautam Mandal, Shiraz Minwalla, Takeshi Morita,
Ganpathy Murthy, Omid Saremi, Alfred Shapere, Sandip Trivedi and
especially Kristan Jensen and Krishnendu Sengupta for
disucssions. S.R.D. would like to thank Institut de Fisica Teorica at
Madrid, Tata Institute of Fundamental Research at Mumbai and Indian
Association for the Cultivation of Science at Kolkata for hospitality
during the final stages of this work. Finally I thank the the organizers of 11th Workshop on Non-perturbative QCD in Paris, Sixth Crete Regional Meeting on String Theory in Milos and Quantum Theory and Symmetries 7 in Prague for organizing stimulating conferences.
This work is partially
supported by National Science Foundation grants PHY-0970069 and
PHY-0855614.
\section*{References}
|
2,869,038,155,721 | arxiv | \section{Introduction}
\label{sec:intro}
The electromagnetic current-current correlator at finite temperature
is one of the very important quantity to characterize the medium,
produced in high energy heavy ion collisions. The explicit dynamical
structure of this quantity for hadronic matter (HM) is directly linked with the in-medium
spectral function of neutral vector mesons and also with the thermal
dilepton and photon yields from HM sources, whereas its static limit provide the
estimation of an important transport coefficients like electrical
conductivity ($\sigma$) of the HM. According to
recent reviews~\cite{Rapp_rev,G_rev}, the effective field theoretical calculations
of hadrons at finite temperature are
very successful to describe the low mass dimuon enhancement measured by the NA60
collaboration~\cite{SPS}. This low mass enhancement also get boost from the quark
matter (QM) sources, which has been calculated by using Hard Thermal Loop (HTL)
technique in
Ref.~\cite{HTL} (see also Ref.~\cite{Sarbani} for effective QCD model calculation).
Therefore, it will be very interesting and
phenomenologically important to know
the static limit estimation of the dynamical structure of current-current correlator
by calculating $\sigma$ of hadronic medium in the frame work
of effective hadronic model, which is basically attempted by this present work.
The event by event analysis~\cite{Sokov} in relativistic heavy ion collisions
indicates about the possibility of generation of a high strength electric ($E$) and
magnetic ($B$) fields in the medium. For example, in the
relativistic heavy ion collider (RHIC) experiment, their approximate
values are $eB\approx m_\pi^2\approx 10^{18} G$ and
$eE\approx m_\pi^2\approx 10^{21}V/cm$~\cite{Tuchin}. Although a particular
magnetic field component becomes only non-zero in the average
scenario~\cite{Sokov,Tuchin}.
The time evolution of this average magnetic field~\cite{Tuchin} depends on
the $\sigma$ of the expanding medium, produced in
heavy ion collisions, which demands that we should have some good idea
on numerical values of this $\sigma$.
In Ref.~\cite{Akamatsu}, the electrical conductivity or electric
charge diffusion coefficient of evolving medium is used as input
to explain the low mass dilepton enhancement, observed experimentally
by PHENIX collaboration at RHIC. Whereas, Yin~\cite{Yin} have shown
that the electrical conductivity of quark-gluon-plasma (QGP) plays
important role to regulate the soft photon production via
realistic hydrodynamics simulation. Besides these indirect estimation
of electrical conductivity of QGP, it can directly be extracted from
charge dependent direct flow parameters in asymmetric heavy-ion
(Au+Cu) collisions~\cite{Hirano}. Along with these phenomenological
searching, different microscopic calculations for $\sigma$ of quark
~\cite{Cassing,Marty,Puglisi,Greif,PKS,Finazzo} and
hadronic phase~\cite{Lee,Nicola_PRD,Nicola,Greif2} have been done, although the results
of Cassing et al~\cite{Cassing} in the model of PHSD (parton hadron string dynamics)
and the NJL (Nambu-Jona-Lasino) model results of Marty et al.~\cite{Marty} have
covered $\sigma$ estimation for the temperature domain of both quark and hadronic matter.
On this problems, a large number of
Lattice QCD calculations have been done~\cite{LQCD_Ding,LQCD_Arts_2007,LQCD_Buividovich,
LQCD_Burnier,LQCD_Gupta,LQCD_Barndt,LQCD_Amato}, where their estimations
cover a large numerical band (see table~\ref{tab}, addressed in result section).
Now, from the calculations~\cite{Lee,Nicola_PRD,Nicola,Greif2} in the
hadronic temperature domain, we see that the results
of Ref.~\cite{Lee} and Refs.~\cite{Nicola_PRD,Nicola,Greif2} show
completely opposite nature of temperature ($T$) dependence of $\sigma$.
If we considered the results of Lee et al.~\cite{Lee} as an exceptional,
almost all of the earlier works~\cite{Cassing,Marty,Puglisi,Greif,PKS,Finazzo,
Nicola_PRD,Nicola,Greif2,LQCD_Amato} indicates that $\sigma/T$ decreases in
hadronic temperature domain~\cite{Cassing,Marty,Nicola_PRD,Nicola,Greif2}
and increases in the temperature domain of
quark phase~\cite{Cassing,Marty,Puglisi,Greif,PKS,Finazzo,LQCD_Amato}.
Their numerical values are located within the order - $\sigma/T\approx 10^{-3}$ to $10^{-2}$
for hadronic phase and $\sigma/T\approx 10^{-3}$ to $10^{-1}$ for quark phase.
These information from earlier studies indicate that the numerical strength as well as
the nature of $\sigma(T)$ both are not very settle issue till now.
In this context, the present investigation is similar kind of microscopic calculations
for $\sigma$, which is expected
to converge and update our understanding of $\sigma(T)$. Considering pion and nucleon as abundant
constituents of hadronic matter, we have calculated their electromagnetic current-current
correlators at finite temperature, whose static limit give the estimation of $\sigma$
for the respective components. As an interaction part, the effective hadronic
Lagrangian densities have been used to calculate
the in-medium scattering probabilities of pion and nucleon with other mesonic
and baryonic resonances, present in the hadronic medium.
Extending our investigations for
finite nucleon or baryon chemical potential $\mu_N$, the present results
provide the
estimation of
$\sigma$ in $T$-$\mu_N$ domain of hadronic matter.
The basic formalism of $\sigma$ is addressed in
the Sec.~\ref{sec:form}, where we will see that
the non-divergent values of current-current correlator
are mainly regulated by the thermal widths of medium components, which are
calculated and briefly described in Sec.~\ref{sec:width}.
Calculations of different loop diagrams are classified in three subsections.
After it, the numerical discussions have been
addressed in the result section (Sec.~\ref{sec:num}), which is followed by a summary in
Sec.~\ref{sec:concl}.
\section{Formalism of electrical conductivity}
\label{sec:form}
Owing to the famous Kubo formula~\cite{Zubarev,Kubo},
the electrical conductivity in momentum
space can be expressed in terms of spectral density
of current current correlator as~\cite{Nicola}
\begin{equation}
\sigma=\frac{1}{6}\lim_{q_0,\vq \rightarrow 0}\frac{A_\sigma(q_0,\vq)}{q_0}
\label{sigma_Nicola}
\end{equation}
where $A_\sigma(q_0,\vq)=\int d^4x e^{iq\cdot x}\langle[J^{\rm EM}_{i}(x),J^{i}_{EM}(0)]\rangle_\beta$
with
$\langle ..\rangle_\beta$
denotes the thermodynamical ensemble average.
In real-time thermal field-theory (RTF), any two point function at finite temperature
always gives a $2\times 2$ matrix structure. Hence, the thermal correlator
of electromagnetic current ($J^{\rm EM}_{\mu}(x)$) will be
\begin{equation}
\Pi^{ab}(q)=i\int d^4x e^{iqx}\langle T_c J^{\rm EM}_{\mu}(x) J_{\rm EM}^{\mu}(0)\rangle^{ab}_\beta~,
\label{pi_ab}
\end{equation}
where $T_c$ denotes the time ordering with respect to a
symmetric contour in the complex time plane. Because of the contour,
we get four possible set of two points and Therefore we get $2\times 2$
matrix structure of two point function. The superscripts
$a, b (=1,2)$ in Eq.~(\ref{pi_ab}) represent the (thermal) indices
of the matrix.
Retarded part of correlator $\Pi^R(q)$ and its corresponding
spectral density $A_\sigma(q)$ can be extracted from
11-component $\Pi^{11}(q)$ by using the relation
\begin{equation}
A_\sigma(q)=2{\rm Im}\Pi^R(q)=2{\rm tanh}(\frac{\beta q_0}{2})
{\rm Im}\Pi_{11}(q)~.
\label{A_piR}
\end{equation}
Using this relation (\ref{A_piR}),
the Eq.~(\ref{sigma_Nicola}) can alternatively be expressed as
\begin{eqnarray}
\sigma&=&\frac{1}{3}\lim_{q_0,\vq \rightarrow 0}
\frac{{\rm Im}\Pi^R(q_0,\vq)}{q_0}
\nonumber\\
&=&\frac{1}{3}\lim_{q_0,\vq \rightarrow 0}
\frac{{\rm tanh}(\frac{\beta q_0}{2}){\rm Im}\Pi^{11}(q_0,\vq)}{q_0}~.
\label{el_Im_R}
\end{eqnarray}
Since pion and nucleon constituents are our
matter of interest, so we should focus
on their electromagnetic currents:
\begin{eqnarray}
J^\mu_{\pi}&=&e\phi_\pi(\partial^\mu\phi_\pi)
\nonumber\\
{\rm and}~J^\mu_{N}&=&e{\overline\psi}_N\gamma^\mu\psi_N~,
\label{current}
\end{eqnarray}
which are electromagnetically coupled with photon via
interaction (QED) Lagrangian density
\begin{equation}
{\cal L}=-(J^\mu_\pi + J^\mu_N)A_\mu~.
\end{equation}
Since ($\phi_{\pi^+}$, $\phi_{\pi^-}$) from pion triplet
($\phi_{\pi^+}$, $\phi_{\pi^-}$, $\phi_{\pi^0}$) and proton
($\psi_p$) from nucleon doublet ($\psi_p$, $\psi_n$)
have non-zero electric charges, so
we have to keep in mind about relevant isospin factors $I^e_\pi=2$
and $I^e_N=1$, which should be multiplied during our calculations.
To calculate electrical conductivity of pionic ($\sigma_\pi$)
and nucleonic ($\sigma_N$) medium from their corresponding
spectral density or retarded part of correlator via Eq.~(\ref{el_Im_R}),
let us start from
11-component of the $\Pi_{ab}$ matrix.
The Wick contraction (see Appendix~\ref{cal_Nqk} ) of the
pion ($\phi_\pi$) and nucleon ($\psi_N$) fields give
one-loop diagrams of photon self-energy, which are shown
in Fig.~\ref{El_pi_N}(a) and ~\ref{El_N_piB}(a) respectively.
A general mathematical expression of these diagrams is
\begin{equation}
\Pi^{11}(q)=i e^2\int \frac{d^4k}{(2\pi)^4}N~ D^{11}(k)
D^{11}(p)~,
\label{self_eta}
\end{equation}
where $D^{11}(k)$ and $D^{11}(p)$ are the scalar parts of
propagators, appeared in RTF for 11-component; $p=q-k$ for
$\pi\pi$ loop in Fig.~\ref{El_pi_N}(a), $p=q+k$ for
$NN$ loop in Fig.~\ref{El_N_piB}(a).
Multiplication of vertex part and numerator part of two propagators
build the term $N$.
In RTF, a general form of $D^{11}(k)$ for boson or fermion is
\begin{eqnarray}
D^{11}(k)&=&\frac{-1}{k_0^2-\omega_k^2+i\epsilon}+2\pi i\epsilon_k
F_k(k_0)\delta(k_0^2-\omega_k^2)~,
\nonumber\\
&&~{\rm with}~ F_k(k_0)=n^+_k\theta(k_0)+
n^-_k\theta(-k_0)~,
\label{de11}
\end{eqnarray}
where $n^{\pm}_k(\omega_k)=\frac{1}{e^{\beta(\omega_k \mp \mu)} - \epsilon_k}$
are the thermal distribution functions and $\pm$ sign in
the superscript of $n_k$ stand for particle and
anti-particle respectively. Now when we proceed for special
cases - pion (boson) or nucleon (fermion) field, we have to
put
\begin{eqnarray}
&&\epsilon_k=+1,~\mu=\mu_\pi=0~{\rm i.e.}~ n^+_k=n^-_k~,
\nonumber\\
&&\omega_k=\omega^{\pi}_k=(\vec k^2+m_\pi^2)^{1/2}~{\rm for~pion}~,
\label{pi_case}
\end{eqnarray}
\begin{eqnarray}
&&\epsilon_k=-1,~\mu=\mu_N~({\rm nucleon~chemical~potential})~,
\nonumber\\
&&\omega_k=\omega^{N}_k=(\vec k^2+m_N^2)^{1/2}~{\rm for~nucleon}~.
\label{N_case}
\end{eqnarray}
However, for time being we will will continue our calculation
with the general form of $D^{11}$ from Eq.~(\ref{de11}) and
at latter stage, we will put these conditions (\ref{pi_case})
and (\ref{N_case}) in the general expression.
\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{El_pi_N.eps}
\caption{The diagram (a) is a schematic one-loop representation of
electromagnetic current-current correlator
for the medium with pionic constituents. The external photon lines
are coupled with double dashed internal lines of pions, which have some finite thermal
width. Thermal width of pion can be derived from its self-energy diagrams (b), (c)
and (d), where (b) represents pion self-energy for mesonic ($\pi M$) loops,
whereas diagrams (c) and (d) are direct and cross diagrams of pion self-energy
for $NB$ loops.}
\label{El_pi_N}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{El_N_piB.eps}
\caption{The diagram (a) is a schematic one-loop representation of
electromagnetic current-current correlator
for the medium with nucleonic constituents. Similar to double dashed lines
of pions in Fig.~(\ref{El_pi_N}), here double solid lines of nucleon
indicates that they have finite thermal width, which can be obtained from
the nucleon self-energy diagram (b) for $\pi B$ loops.}
\label{El_N_piB}
\end{center}
\end{figure}
After using (\ref{de11}) in Eq.~(\ref{self_eta}), if we do its $k_0$ integration
and put it in Eq.~(\ref{el_Im_R}), then we will get spectral
density of electromagnetic current-current correlator~\cite{G_IJMPA}:
\begin{eqnarray}
A_\sigma(q)&=&e^2\int\frac{d^3k}{(2\pi)^3}
\frac{(-\pi)N}{4\omega_k\omega_p}[C_1\delta(q_0 -\omega_k-\omega_p)
\nonumber\\
&&+C_2\delta(q_0-\omega_k+\omega_p)+C_3\delta(q_0 +\omega_k-\omega_p)
\nonumber\\
&&+C_4\delta(q_0 +\omega_k+\omega_p)]~,
\label{Pi_LU}
\end{eqnarray}
where $\omega_p=\omega_p^{\pi}=\{(\vq -\vec k)^2+m^2_{\pi}\}^{1/2}$ for pion field,
$\omega_p=\omega_p^{N}=\{(\vq +\vec k)^2+m^2_{N}\}^{1/2}$ for nucleon field.
Here $N$ are space component of $N(q,k_0=\pm \omega_k,\vec k)$ (see Appendix ~\ref{cal_Nqk}):
\begin{equation}
N= (-4)\{-\vec k\cdot\vq + \vec k^2\}~{\rm for}~\pi\pi~{\rm loop}~,
\label{N_qvkv_BB}
\end{equation}
and
\begin{eqnarray}
N&=&(-8)\{\vec k\cdot\vq +\vec k^2\} +4\vec k\cdot\vq ~{\rm for}~NN~{\rm loop}~.
\label{N_qvkv_FF}
\end{eqnarray}
The statistical probabilities,
attached with four different delta functions, are
\begin{eqnarray}
C_1&=& 1+n^{+}_k(\omega_k)+n^+_p(q_0 - \omega_k)~,
\nonumber\\
C_2&=&-n^{+}_k(\omega_k)+n^-_p(-q_0+\omega_k)~,
\nonumber\\
C_3&=&n^-_k(\omega_k)-n^+_p(q_0+\omega_k)~,
\nonumber\\
C_4&=&-1-n^-_k(\omega_k)-n^-_p(-q_0-\omega_k)~,~{\rm for}~\pi\pi~{\rm loop}~;
\nonumber\\
\end{eqnarray}
and
\begin{eqnarray}
C_1&=& -1+n^{-}_k(\omega_k)+n^+_p(q_0 + \omega_k)~,
\nonumber\\
C_2&=&-n^{-}_k(\omega_k)+n^-_p(-q_0+\omega_k)~,
\nonumber\\
C_3&=&n^+_k(\omega_k)-n^+_p(q_0+\omega_k)~,
\nonumber\\
C_4&=&1-n^+_k(\omega_k)-n^-_p(-q_0-\omega_k)~,~{\rm for}~NN~{\rm loop}~.
\nonumber\\
\end{eqnarray}
Four different delta functions are responsible for creating four
different regions of branch cuts in $q_0$-axis, where $A_\sigma(q_0,\vq)$
or Im$\Pi^R(q_0,\vq)$ becomes non-zero.
These regions are
\begin{eqnarray}
q_0&=&-\infty ~~~{\rm to}~~~ -\{\vq^2+4m_{\pi,N}^2\}^{1/2}~:~ {\rm unitary~ cut}~,
\nonumber\\
&=&\left.
\begin{array}{c}
-|\vq| ~~~~~{\rm to}~~~~ ~0 \\
~~~0~~~~~{\rm to}~~~~ ~|\vq|
\end{array}
\right\} ~:~ {\rm Landau~ cut}~,
\nonumber\\
&=&\{\vq^2+4m_{\pi,N}^2\}^{1/2} ~~~{\rm to}~~~\infty ~:~ {\rm unitary~ cut}~.
\end{eqnarray}
Since electrical conductivity $\sigma$ is the limiting value
of $A_\sigma(q_0,\vq)$ or Im$\Pi^R(q_0,\vq)$ at $q_0, \vq\rightarrow 0$,
therefore we should focus on Landau cuts only.
Hence, using the Landau part of Eq.~(\ref{Pi_LU}) in Eq.~(\ref{sigma_Nicola}), we have
\begin{eqnarray}
\sigma&=&\frac{e^2}{3}\lim_{q_0,\vq \rightarrow 0}\frac{1}{q_0}
\int\frac{d^3k}{(2\pi)^3}\frac{(-\pi)N}{4\omega_k\omega_p}
\{C_2\delta(q_0-\omega_k+\omega_p)
\nonumber\\
&&+C_3\delta(q_0+\omega_k-\omega_p)\}
\nonumber\\
&=&\frac{e^2}{3}\lim_{q_0,\vq \rightarrow 0}{\rm Im}
\left[\int\frac{d^3k}{(2\pi)^3}\frac{N}{4\omega_k\omega_p}\lim_{\Gamma \rightarrow 0}
\right.\nonumber\\
&&\left.
\left\{\frac{C_2/q_0}{(q_0-\omega_k+\omega_p)+i\Gamma}
+\frac{C_3/q_0}{(q_0+\omega_k-\omega_p)+i\Gamma}\right\}\right]~.
\nonumber\\
\label{el_G}
\end{eqnarray}
We will take finite value of $\Gamma$ in our further
calculations to get a non-divergent values of $\sigma$.
In Kubo approach, this traditional technique is widely
used to calculate different transport
coefficients like shear viscosity~\cite{Nicola,G_IJMPA},
electrical conductivity~\cite{Nicola_PRD}.
In this respect, this formalism
is very much close to quasi particle approximation.
The $\Gamma$ of medium constituents is basically their thermal width,
which is physically related with the probabilities of different
in-medium scattering. Inverse of $\Gamma$ measures the relaxation time
$\tau$, which is the average time of medium constituents
to reach their equilibrium conditions.
Next, applying the L'Hospital's rule in the Eq.~(\ref{el_G})
(see Appendix ~\ref{LHos}),
we get a generalized expression
of electrical conductivity for bosonic ($\phi_\pi$) or fermionic ($\psi_N$) field:
\begin{equation}
\sigma=\frac{\beta e^2}{3}\int\frac{d^3k}{(2\pi)^3}\frac{(-N^0)}{4\omega_k^2\Gamma}
[n^-_k(1+\epsilon_kn^-_k)+n^+_k(1+\epsilon_kn^+_k)]~,
\label{el_last}
\end{equation}
where
\begin{equation}
N^0=\lim_{q_0, \vq\rightarrow 0}N(k_0=\pm\omega_k,\vec k,q)~.
\label{N_0}
\end{equation}
Depending upon the sign of $\epsilon_k$, the statistical probability
becomes Bose enhanced ($\epsilon_k=+1$ for bosonic field) or Pauli blocked
($\epsilon_k=-1$ for fermionic field) probability.
Following the definition of $N^0$ in Eq.~(\ref{N_0}),
Eqs.~(\ref{N_qvkv_BB}) and (\ref{N_qvkv_FF}) can be simplified as
\begin{equation}
N^0=-I^e_\pi (4\vec k^2)~~~ {\rm for~}\pi\pi~{\rm loop}~,
\label{N0_pi}
\end{equation}
and
\begin{equation}
N^0=-I^e_N (8\vec k^2)~~~ {\rm for~}NN~{\rm loop}~.
\label{N0_N}
\end{equation}
Using the above Eqs.~(\ref{N0_pi}) and (\ref{N0_N}) in Eq.~(\ref{el_last})
as well as their relevant parameters from Eq.~(\ref{pi_case}) and (\ref{N_case}),
we get the electrical conductivity of the pionic
and nucleonic medium:
\begin{equation}
\sigma_\pi=\frac{\beta e^2}{3}\int^{\infty}_{0}
\frac{d^3\vec k}{(2\pi)^3}\frac{\vec k^2}{{\omega^\pi_k}^2\Gamma_\pi}n_k(\omega^\pi_k)
\{1+n_k(\omega_k^\pi)\}
\label{el_pi}
\end{equation}
and
\begin{eqnarray}
\sigma_N&=&\frac{2\beta e^2}{3}\int^{\infty}_{0}
\frac{d^3\vec k}{(2\pi)^3}\frac{\vec k^2}{{\omega^N_k}^2\Gamma_N}[n^+_k(\omega_k^N)\{1-n^+_k(\omega^N_k)\}
\nonumber\\
&&+n^-_k(\omega^N_k)\{1-n^-_k(\omega^N_k)\}]~.
\label{el_N}
\end{eqnarray}
Hence, adding the pionic and nucleonic components, we get the total electrical conductivity
\begin{equation}
\sigma_{\rm T}=\sigma_{\pi}+\sigma_N~.
\end{equation}
\section{Thermal width}
\label{sec:width}
Let us come to the thermal widths of pion ($\Gamma_\pi$) and nucleon
($\Gamma_N$). Pion thermal width can be obtained from the imaginary part
of pion self-energy for different mesonic and baryonic fluctuations.
Fig.~\ref{El_pi_N}(b) represents pion self-energy diagram for
$\pi M$ (mesonic) loops - ${\Pi}^R_{\pi(\pi M)}$, where $M=\sigma,~ \rho$.
Here subscript in ${\Pi}^R_{\pi(\pi M)}$ stands for the
external (outside the bracket) and internal (inside the bracket)
particles for the diagram~\ref{El_pi_N}(b). This notation will be followed
by latter diagrams also. Now, pion self-energy for different
baryonic loops (${\Pi}^R_{\pi(NB)}$) can have two possible diagrams
as shown in Fig.~\ref{El_pi_N}(c) and (d). Here internal lines $NB$
stand for nucleon ($N$) and baryon ($B$) respectively, where different
4-star spin $1/2$ and $3/2$ baryons are taken in our calculations. Adding
all those mesonic and baryonic loops, we get total thermal width of pion
$\Gamma_\pi$, which can be expressed as
\begin{eqnarray}
\Gamma_\pi&=& \sum_M\Gamma_{\pi(\pi M)} + \sum_B\Gamma_{\pi(NB)}
\nonumber\\
&=&-\sum_M{\rm Im}{\Pi}^R_{\pi(\pi M)}(k_0=\omega^\pi_k,\vec k)/m_\pi
\nonumber\\
&&~~~-\sum_B{\rm Im}{\Pi}^R_{\pi(NB)}(k_0=\omega^\pi_k,\vec k)/m_\pi~.
\label{Gam_pi}
\end{eqnarray}
Similarly, nucleon self-energy is shown in Fig.~\ref{El_N_piB}(b)
and it has been denoted as $\Sigma^R_{N(\pi B)}$, where in internal
lines, we have taken all those spin $1/2$ and $3/2$ baryons ($B$) as
taken in pion self-energy for baryonic loops. Hence, summing these all
$\pi B$ loops, we can express our nucleon thermal width as
\begin{equation}
\Gamma_N=\sum_B\Gamma_{N(\pi B)}=-\sum_B{\rm Im}\Sigma^R_{N(\pi B)}(k_0=\omega^N_k,\vec k)~.
\label{Gam_N}
\end{equation}
Next we discuss briefly the calculations of thermal widths from different one-loop self-energy
graphs as shown in Fig.~(\ref{El_pi_N}) and (\ref{El_N_piB}).
\subsection{Pion thermal width for different mesonic loops}
\label{subsec:pi_mes}
To calculate the mesonic loop contribution of pionic thermal width
$\Gamma_{\pi(\pi M)}$, the pion self-energy for $\pi M$ loops, where $M$ stands for $\sigma$
and $\rho$ mesons, have been evaluated and it is expressed as~\cite{GKS}
\begin{eqnarray}
\Gamma_{\pi(\pi M)}&=&{\rm Im}{\Pi}^R_{\pi(\pi M)}(k_0=\omega^\pi_k,\vec k)/m_\pi
\nonumber\\
&=&\frac{1}{m_\pi} \int \frac{d^3 \vec l}{32\pi^2 \omega_l^\pi\omega_u^M}
\nonumber\\
&& L(l_0=-\omega^\pi_l, \vec l, k_0=\omega^\pi_k, \vec k) \{n(\omega^\pi_l)
\nonumber\\
&& -n(\omega^M_u)\}\delta(\omega^\pi_k +\omega^\pi_l - \omega^M_u)~,
\label{G_pi_piM}
\end{eqnarray}
where $n(\omega^\pi_l)$, $n(\omega^M_u)$ are BE distribution functions of $\pi$, $M$
mesons with energies $\omega^\pi_l=(\vec l^2+m_\pi^2)^{1/2}$ and
$\omega^M_u=(|\vec k-\vec l|^2+m_M^2)^{1/2}$ respectively.
The vertex factors $L(k,l)$~\cite{GKS}
have been obtained from the effective Lagrangian density,
\begin{equation}
{\cal L} = g_\rho \, {\vec \rho}_\mu \cdot {\vec \pi} \times \partial^\mu {\vec \pi}
+ \frac{g_\sigma}{2} m_\sigma {\vec \pi}\cdot {\vec\pi}\,\sigma~.
\label{Lag_pipiM}
\end{equation}
\subsection{Pion thermal width for different baryonic loops}
Along with the mesonic fluctuations, different baryon fluctuations
may provide some contributions in pion thermal width. This component
can be derived from pion self-energy for different $NB$ loops,
where $B =N(940)$, $\Delta(1232)$, $N^*(1440)$, $N^*(1520)$,
$N^*(1535)$, $\Delta^*(1600)$, $\Delta^*(1620)$, $N^*(1650)$,
$\Delta^*(1700)$, $N^*(1700)$, $N^*(1710)$, $N^*(1720)$ are
taken~\cite{G_pi_JPG,G_eta_BJP}. The masses of all the 4-star
baryon resonances (in MeV) are presented inside the brackets.
The direct and cross diagrams of pion self-energy
for $NB$ loops are shown in Fig.~\ref{El_pi_N}(c) and (d).
Adding the relevant Landau cut contributions of both diagrams (c)
and (d), the total thermal width of pion for any $NB$ loop is
given by~\cite{G_pi_JPG,G_eta_BJP}
\begin{eqnarray}
\Gamma_{\pi(NB)}&=& {\rm Im}{\Pi}^R_{\pi(NB)}(k_0=\omega^\pi_k,\vec k)/m_\pi
\nonumber\\
&=&\frac{1}{m_\pi} \int \frac{d^3 \vec l}{32\pi^2 \omega_l^N\omega_u^B}
\nonumber\\
&& [L(l_0=\omega^N_l,\vec l, k_0=\omega_k^\pi,\vec k)\{n^+_l(\omega^N_l)
\nonumber\\
&& - n^+_u(\omega^B_u)\}\delta(\omega^\pi_k -\omega^N_l + \omega^B_u)
\nonumber\\
&&+L(l_0=-\omega^N_l,\vec l, k_0=\omega_k^\pi,\vec k)\{-n^-_l(\omega^N_l)
\nonumber\\
&&+ n^-_u(\omega^B_u)\}\delta(\omega^\pi_k +\omega^N_l - \omega^B_u)]~,
\label{G_pi_NB}
\end{eqnarray}
where $n^{\pm}(\omega^N_l)$, $n^{\pm}(\omega^B_u)$ are FD distribution functions of
$N$, $B$ ($\pm$ for particle and anti-particle)
with energies $\omega^N_l=(\vec l^2+m_N^2)^{1/2}$ and
$\omega^B_u=(|\pm\vec k +\vec l|^2+m_B^2)^{1/2}$ ($\pm$ for two different diagrams)
respectively.
With the help of the effective Lagrangian densities~\cite{Leopold},
\begin{eqnarray}
{\cal L}&=&\frac{f}{m_\pi}{\overline \psi}_B\gamma^\mu
\left\{
\begin{array}{c}
i\gamma^5 \\
1\!\!1
\end{array}
\right\}
\psi_N\partial_\mu\pi + {\rm h.c.}~{\rm for}~J_B^P=\frac{1}{2}^{\pm},
\nonumber\\
{\cal L}&=&\frac{f}{m_\pi}{\overline \psi}^\mu_B
\left\{
\begin{array}{c}
1\!\!1 \\
i\gamma^5
\end{array}
\right\}
\psi_N\partial_\mu\pi + {\rm h.c.}~{\rm for}~J_B^P=\frac{3}{2}^{\pm},
\label{Lag_BNpi}
\end{eqnarray}
the vertex factors $L(k,l)$~\cite{G_pi_JPG,G_eta_BJP}
can be found.
\subsection{Nucleon thermal width}
\label{subsec:N}
The nucleonic thermal width has been calculated from
nucleon self-energy for different possible $\pi B$ loops,
where $B$ stands for all the baryons as taken in pion self-energy
for baryonic loops. Evaluating the loop diagram, shown in
Fig.~\ref{El_N_piB}(b), we get~\cite{G_NNst_BJP,G_N}
\begin{eqnarray}
\Gamma_{N(\pi B)} &=&-\sum_B{\rm Im}\Sigma^R_{N(\pi B)}(k_0=\omega^N_k,\vec k)
\nonumber\\
&=&\int \frac{d^3 \vec l}{32\pi^2 \omega_l^\pi\omega_u^B}
\nonumber\\
&& L(l_0=-\omega^\pi_l,\vec l, k_0=\omega_k^N,\vec k)\{n(\omega^\pi_l)
\nonumber\\
&&+ n^+(\omega^B_u)\} \delta(\omega^N_k +\omega^\pi_l - \omega^B_u)
\label{G_N_piB}
\end{eqnarray}
where $n(\omega^\pi_l)$ and $n^+(\omega^B_u)$ are BE and
FD distribution functions for $\pi$ and $B$ with energies
$\omega_l^\pi=(\vec l^2+m_\pi^2)^{1/2}$ and $\omega^B_u=(|\vec k - \vec l|^2+m_B^2)^{1/2}$
respectively.
The vertex factors $L(k,l)$~\cite{G_NNst_BJP,G_N}
can be deduced by using the $\pi NB$ interaction Lagrangian
densities from Eq.~(\ref{Lag_BNpi}).
\section{Results and Discussion}
\label{sec:num}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{el_ppM_T.eps}
\caption{Temperature dependence of electrical conductivity
pionic medium due to its different mesonic fluctuations - $\pi\sigma$
(dotted line), $\pi\rho$ (dashed line) loops and their total (solid line).
With and without folding effect of resonances $M=\sigma, \rho$ are taken
in upper and lower panels respectively.}
\label{el_ppM_T}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{el_pNB_T.eps}
\caption{Effect of baryonic fluctuations ($N\Delta$ loop : dashed line,
$NB$ loops : solid line) after adding with mesonic fluctuations
($\pi M$ loops : dotted line) of pion on $\sigma_\pi(T)$ at
$\mu_N=0$ (a) and $\mu_N=0.300$ GeV (b).}
\label{el_pNB_T}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{el_pNB_mu.eps}
\caption{Same as Fig.~(\ref{el_pNB_T}) for $\sigma_\pi(\mu_N)$ at
$T=0.120$ GeV (a) and $T=0.150$ GeV (b).}
\label{el_pNB_mu}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{el_pNt_T.eps}
\caption{Temperature dependence of electrical conductivity for
pion (dotted line), nucleon (dashed line) components and their
total (solid line) at $\mu_N=0.500$ GeV (a) and $\mu_N=0.300$ GeV.}
\label{el_pNt_T}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{el_pNt_mu.eps}
\caption{$\mu_N$ dependence of electrical conductivity for
pion (dotted line), nucleon (dashed line) components and their
total (solid line) at $T=0.120$ GeV (a) and $T=0.150$ GeV.}
\label{el_pNt_mu}
\end{center}
\end{figure}
Using the $\Gamma_{\pi(\pi\sigma)}(\vec k,T)$, $\Gamma_{\pi(\pi\rho)}(\vec k,T)$ and
their total in the integrand of Eq.~(\ref{el_pi}), the dotted, dashed
and solid lines of Fig.~(\ref{el_ppM_T}) are generated, where folding~\cite{GKS}
by vacuum spectral functions of resonances $\sigma$ and $\rho$ are considered
in panel (a) but not in panel (b). Like the results of shear viscosity in
the earlier work~\cite{GKS}, $\sigma$ and $\rho$ resonances play dominant role in
the electrical conductivity at low ($T<0.100$ GeV) and high ($T>0.100$ GeV)
temperature domain respectively. We get $\sigma_\pi(T)$ as a decreasing function
in low and high temperature both, although a mild increasing function of shear
viscosity $\eta_\pi(T)$ has been observed in Ref.~\cite{GKS} at high temperature
domain of hadronic matter ($0.100$ GeV $<T<0.175$ GeV). The mathematical origin
for this differences in the nature of $\sigma_\pi(T)$ and $\eta_\pi(T)$ is
because of different power of momentum ($\vec k^4$ for $\sigma_\pi$ but $\vec k^6$
for $\eta_\pi$) in the numerator of their respective integrand.
Adding baryonic loop contributions with the mesonic loops of pion self-energy,
we get total thermal width of pion as described explicitly in Eq.~(\ref{Gam_pi}).
Fig.~\ref{el_pNB_T}(a) and (b) for $\mu_N=0$ and $0.300$ GeV reveal that $\sigma_\pi(T)$
reduces after adding baryonic loop contribution in pion self-energy and its reduction
strength becomes larger for larger values of $\mu_N$ as baryonic loop contribution,
$\Gamma_{\pi(NB)}(\vec k,T,\mu_N)$ depends sensitively on $\mu_N$. To display the
dominant contribution of $N\Delta$ loop ($\Gamma_{\pi(N\Delta)}$),
Fig.~(\ref{el_pNB_T}) shows individual contributions of meson loops,
meson loops + $N\Delta$ loop and meson + baryon loops by dotted,
dashed and solid lines respectively.
Next, Fig.~\ref{el_pNB_T}(a) and (b) for $T=0.120$ GeV and $0.150$ GeV
show $\mu_N$ dependence of electrical conductivity of pionic component
for meson loops (dotted line), meson loops + $N\Delta$ loop (dashed line)
and meson + baryon loops (solid line). As $\Gamma_{\pi(\pi M)}(\vec k,T)$
is independent of $\mu_N$, therefore corresponding $\sigma_\pi$
(dotted line) remain constant with the variation of $\mu_N$. After
adding $N\Delta$ loop (dashed line) and other baryon loops (solid line),
a decreasing nature of $\sigma_\pi(\mu_N)$ are clearly noticed. A sensitive
dependence of $\mu_N$ in $\Gamma_{\pi(NB)}$ for $N\Delta$ loop (dominant)
and other baryon loops are the main reason behind the decreasing nature
of $\sigma_\pi(\mu_N)$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{El_muT2.eps}
\caption{(Color online) Total electrical conductivity $\sigma_T$ in $T$-$\mu_N$ plane.}
\label{El_muT}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{comp.eps}
\caption{(Color online) Our results of $\sigma(T,\mu_N=0)/T$
are compared with the results of Refs.~\cite{Cassing,Marty,Lee,Nicola_PRD} (a).
Valley structure of $\sigma(T)$ at $\mu_N=0.400$ GeV, $0.500$ GeV, $0.600$ GeV
are shown by solid, dotted and dashed lines respectively in panel (b).}
\label{comp}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.35]{el_t_mu2.eps}
\caption{(a): Valley structure of $\sigma(\mu_N)$ at three different
$T$. (b): The points of minima (solid circles) and freeze
out line~\cite{freezeout} (solid line) are shown in $T$-$\mu_N$ plane.}
\label{el_t_mu2}
\end{center}
\end{figure}
In Fig.~\ref{el_pNt_T}(a) and (b) for $\mu_N=0.500$ GeV and $0.300$ GeV,
the $T$ dependence of pionic ($\sigma_\pi$), nucleonic ($\sigma_N$) components of
electrical conductivities and their total ($\sigma_T$) are shown by
dotted, dashed and solid lines respectively.
Corresponding results in $\mu_N$ axis are shown in
Fig.~\ref{el_pNt_mu}(a) and (b) for $T=0.120$ GeV and $0.150$ GeV.
Unlike to $\sigma_\pi$, the $\sigma_N$ increases with both $T$ and $\mu_N$.
The nucleon phase space factors or statistical weight
factors of FD distributions in $\sigma_N$ are playing a dominant over
the $\Gamma_N(T, \mu_N)$, whereas for pionic case, $\Gamma_\pi(T,\mu_N)$ becomes more
influential than pionic phase factors or statistical Bose enhanced
weight factors in $\sigma_\pi$. This is the mathematical reason for opposite
nature of $\sigma_\pi(T,\mu_N)$ and $\sigma_N(T,\mu_N)$. From a simultaneous
observation of Fig.~(\ref{el_pNt_T}) and (\ref{el_pNt_mu}), we can conclude
that the decreasing nature of $\sigma_T(T,\mu_N)$ becomes inverse beyond
a certain points of $T$ and $\mu_N$, where $\sigma_T$ exposes the points of minima.
This behavior can be visualized well from Fig.~(\ref{El_muT}), which
exhibits 3-dimensional plot of $\sigma_T(T,\mu_N)$.
Up to now, our results are presented as normalized values of $e^2$
(in other word we have taken $e^2=1$) but exact values of $\sigma_T$
(after multiplying by $e^2=4\pi/137$) have been shown in the last
two figures (\ref{comp}) and (\ref{el_t_mu2}).
Fig.~\ref{comp}(a) displays a comparison of present results
with the earlier results, obtained by Fraile et al.~\cite{Nicola}
(stars and triangles), Lee et al.~\cite{Lee} (open circles), Marty
et al.~\cite{Marty} (squares), Cassing et al.~\cite{Cassing}
(solid circles) at hadronic temperature domain for $\mu_N=0$.
Within $0.110$ GeV $<T<0.175$ GeV, present results more or less
agrees with the results of Ref.~\cite{Nicola,Lee} but quite smaller
than the results of Ref.~\cite{Marty,Cassing}. Fig.~\ref{comp}(b)
shows $\sigma_T$ vs $T$ at three different values of $\mu_N$, where
we notice the shifting of minimum values of $\sigma_T$ towards
lower $T$ as one increases $\mu_N$.
Alternatively, these minimum
values of $\sigma_T$ will also be shifted towards lower $\mu_N$ as $T$
will increase, which is explicitly shown in Fig.~\ref{el_t_mu2}(a).
Next, Fig.~\ref{el_t_mu2}(b) represents the points of minima for $\sigma_T$
in $T$-$\mu_N$ plane. An approximated freeze out line (solid line),
taken from Ref.~\cite{freezeout}, is also pasted in Fig.~\ref{el_t_mu2}(b).
The points of minima, which are located outside the freeze out line, can only
be covered by the expanding fireball, produced in different beam energies
of heavy ion collisions. Therefore, the minima or valley structure can be observed
from ($T_f\approx 0.166$ GeV, $\mu_f\approx 0$) to ($T_f\approx 0.140$ GeV,
$\mu_f\approx 0.420$ GeV), where subscript $f$ stands for freeze out.
In other word, from high beam energy $\sqrt{s}$
like RHIC experiment ($\sqrt{s}=200$ GeV), this valley structure can be
observed up to $\sqrt{s}\approx 8$ GeV. However, within this long range of beam
energy or freeze out line, some points of minima may cross the quark-hadron
transition line and therefore, they may not be observed in the experiment.
One should keep in mind that these minima or valley structure is completely
appeared due to phase space effect of hadronic medium and has nothing relation
with the quark-hadron transition. Therefore, one can observe only those points
of minima of hadronic medium, which will be in between freeze out and quark-hadron
transition lines. Although, there is some possibility for not observing any
points of minima, if they all are located in quark phase domain of $T$-$\mu_N$ plane.
In this regards, we can say at least that $\sigma(T,\mu_N)$ of hadronic medium
decreases as one goes towards quark-hadron transition line.
We have presented the numerical values of $\sigma(T,\mu_N=0)/T$,
estimated by earlier works in Table~\ref{tab}, where most of the works
are displaying the decreasing $\sigma(T)/T$ in hadronic
temperature~~\cite{Cassing,Marty,Nicola_PRD,Nicola}
and increasing $\sigma(T)/T$ in temperature domain of
quark phase~\cite{Cassing,Marty,Puglisi,Greif,PKS,Finazzo,LQCD_Amato}.
Among them, Refs.~\cite{Cassing,Marty}, covering the both temperature domain,
have exhibited the minimum value of $\sigma/T$ near transition temperature.
On the basis of these earlier results at $\mu_N=0$ and our estimations
at finite $\mu_N$ within the $T$-$\mu_N$ domain of hadronic matter, a valley
structure along quark-hadron transition line in $T$-$\mu_N$ plane may be expected
and this issue may be confirmed after further research on $\sigma$-calculations at finite
baryon density, based on different effective QCD model.
\begin{table}
\begin{center}
\label{tab}
\begin{tabular}{|c|c|c|}
\hline
& $\sigma/T$ at & $\sigma/T$ at \\
& $T=(0.120$ & $T=(0.175$ \\
& -$~0.175)$ GeV & -$~350)$ GeV \\
\hline
\hline
\underline{LQCD Results:} & & \\
Gupta~\cite{LQCD_Gupta} & - & $\approx0.375$ \\
Ding et al.~\cite{LQCD_Ding} & - & $\approx0.033(+0.018,$ \\
& & $~~~~~~-0.016)$ \\
Arts et al.~\cite{LQCD_Arts_2007} & - & $\approx0.020(\pm0.005)$ \\
Brandt et al.~\cite{LQCD_Barndt} & - & $\approx0.020(\pm 0.006)$ \\
Burnier et al.~\cite{LQCD_Burnier} & - & $\approx0.0064$ \\
Amato et al.~\cite{LQCD_Amato} & - & $\approx0.003(\pm 0.001)$ \\
& & -$~0.015(\pm 0.003)$ \\
Buividovich et al.~\cite{LQCD_Buividovich} & - & $\approx0.0021(\pm 0.0003)$ \\
\hline
Yin~\cite{Yin} & - & $\approx0.06(+0.04,$\\
& & $-0.02)$ \\
\hline
Puglisi et al.~\cite{Puglisi} & - & $\approx0.09~$-$~0.13$ \\
(PQCD in RTA) & & \\
Puglisi et al.~\cite{Puglisi} & - & $\approx0.01~$-$~0.07$ \\
(QP in RTA) & & \\
\hline
Greif et al.~\cite{Greif} & - & $\approx0.04~$-$~0.06$ \\
(BAMPS) & & \\
\hline
Marty et al.~\cite{Marty} & - & $\approx0.06~$-$~0.16$ \\
(DQPM) & & \\
Marty et al.~\cite{Marty} & $\approx0.06~$-$~0.05$ & $\approx0.05~$-$~0.5$ \\
(NJL) & & \\
\hline
Cassing et al.~\cite{Cassing} & $\approx0.088~$-$~0.025$ & $\approx0.025~$-$~0.2$ \\
(PHSD) & & \\
\hline
Finazzo et al.~\cite{Finazzo} & $\approx0.004~$-$~0.010$ & $\approx0.010~$-$~0.015$ \\
\hline
Lee et al.~\cite{Lee} & $\approx0.001~$-$~0.011$ & $\approx0.36~$-$~0.015$ \\
\hline
Fraile et al.~\cite{Nicola} & $\approx0.013~$-$~0.010$ & - \\
(Unitarization) & & \\
Fraile et al.~\cite{Nicola} & $\approx0.008~$-$~0.002$ & - \\
(ChPT) & & \\
\hline
Present Results & $\approx0.004~$-$~0.001$ & - \\
\hline
\end{tabular}
\caption{At $\mu_N=0$, the $\sigma(T)/T$ in approximated temperature
domain of hadronic ($T\approx 0.120$ GeV to $0.175$ GeV) and quark
($T\approx 0.175$ GeV to $0.350$ GeV) phases are presented in
2nd and 3rd columns, whereas in 1st column, the references
(with their methodologies) are addressed.}
\end{center}
\end{table}
\section{Summary and Conclusion}
\label{sec:concl}
The present work provide an estimation of electrical conductivity
of hadronic medium at finite temperature and baryon density. Assuming
pion and nucleon as most abundant medium constituents, we have first
deduced thermal correlators of their electromagnetic currents and then,
taking the static limit of these correlators, the expressions
of electrical conductivities for pionic and nucleonic components are derived.
For getting the non divergent values of these correlators in the static limit,
one has to include the finite thermal widths of the medium constituents - pion
and nucleon. This is a traditional quasi-particle technique of Kubo frame work,
used during the calculations of transport coefficients from the relevant correlators
in their static limits. Following the field theoretical version of optical theorem,
the thermal widths of pion and nucleon are obtained from the imaginary part
of their one-loop self-energy diagrams, which accommodate different mesonic
and baryonic resonances in the intermediate states. As a dynamical part,
the interaction of pion and nucleon with other mesonic and baryonic resonances
are guided by the effective hadronic Lagrangian densities, where their couplings
are tuned by the decay width of resonances, based on the experimental data from PDG.
The momentum distribution of
these thermal widths are integrated out during evaluation of electrical conductivities
of respective components.
The electrical conductivity for pionic component is obtained as a decreasing function
$T$ and $\mu_N$, where mesonic loops are dominant to fix its numerical strength.
The $\pi\sigma$ and $\pi\rho$ loops of pion self-energy control the strength
of electrical conductivity at low and high $T$ regions respectively. While a further
reduction of numerical values in conductivity at high $T$ domain is noticed after
addition of different baryonic loops in pion self-energy. Electrical conductivity
of pionic component due to mesonic loops remain constant with $\mu_N$ but it is
transformed to a decreasing function when the baryonic loops are added in the pion
self-energy. The nucleonic component give the increasing values of electrical
conductivity with the variation of $T$ and $\mu_N$. After adding these pionic
and nucleonic components, the total electrical conductivity first decreases
at pion dominating $T$-$\mu_N$ domain and then increases at nucleonic dominating
domain. Therefore, the numerical results show a set of $T$-$\mu_N$ points,
where total electrical conductivity becomes minimum and this valley structure
in $T$-$\mu_N$ plane can only be observed if the points of minima are located
between freeze out line and quark-hadronic transition line.
Comparing with earlier
estimations of electrical conductivity at $\mu_N=0$,
present work more or less agrees with Refs.~\cite{Nicola,Lee} quantitatively
and quantitatively it is similar with most of the earlier
works~\cite{Cassing,Marty,Nicola_PRD,Nicola,Greif2}, which show that electrical
conductivity at $\mu_N=0$ decreases with $T$. On the basis of these
earlier studies at $\mu_N=0$ and present investigation at finite $\mu_N$,
a general decreasing nature in the numerical values of electrical conductivity
for hadronic matter is observed when one goes from freeze out to quark-hadron
transition line in $T$-$\mu_N$ plane. Further research in different model
calculations at finite $\mu_N$ may confirm this conclusion.
{\bf Acknowledgment :}
The work is financially supported from UGC Dr. D. S. Kothari Post Doctoral Fellowship under
grant No. F.4-2/2006 (BSR)/PH/15-16/0060.
\section{Appendices}
\subsection{Calculation $N(\vq,\vec k)$}
\label{cal_Nqk}
Let us write the 11-component of two point function of current-current
correlator in terms of field operators. For $\phi_\pi$ field
it is given by
\begin{eqnarray}
\Pi_{11}(q)&=&i\int d^4x e^{iqx}\langle TJ^{\rm EM}_{\mu}(x) J_{\rm EM}^{\mu}(0) \rangle_\beta
\nonumber\\
&=&ie^2\int d^4x e^{iqx}\langle T \phi_\pi(x)\partial_\mu\phi_\pi(x)
\phi_\pi(0)\partial^\mu\phi_\pi(0)\rangle_\beta~.
\nonumber\\
\label{pi_ab_Ap}
\end{eqnarray}
With the help of the Wick's contraction technique, we have
\begin{eqnarray}
\Pi_{11}(q)&=&ie^2\int d^4x e^{iqx}
[\langle T \phi_\pi\underbrace{(x)\partial_\mu\phi_\pi\overbrace{(x)
\phi_\pi}(0)\partial^\mu\phi_\pi}(0)\rangle_\beta
\nonumber\\
&=&ie^2\int \frac{d^4k}{(2\pi)^4}N(q,k)
D_{11}(k)D_{11}(p=q-k)~,
\nonumber\\
\end{eqnarray}
where
\begin{equation}
N(q,k)= (-4)k^\mu(q-k)_\mu
\end{equation}
and its space component part is
\begin{equation}
N(\vq,\vec k)= (-4)\{-\vec k\cdot\vq + \vec k^2\}~.
\end{equation}
Similarly for $\psi_N$ field,
\begin{eqnarray}
\Pi_{11}(q)&=&ie^2\int d^4x e^{iqx}
\langle T{\overline \psi}_N\underbrace{(x)\gamma_\mu\psi_N\overbrace{(x)
{\overline \psi}_N}(0)\gamma^\mu\psi_N}(0)\rangle_\beta
\nonumber\\
&=&ie^2\int \frac{d^4k}{(2\pi)^4}N(q,k)
D_{11}(k)D_{11}(p=q+k)~,
\nonumber\\
\end{eqnarray}
where
\begin{eqnarray}
N(q,k)&=&{\rm Tr}[\gamma^\mu(q \!\!\! /+k \!\!\! /+m_\psi)
\gamma_\mu(k \!\!\! /+m_\psi)]
\nonumber\\
&=&8k^\mu(q+k)_\mu -4[k\cdot(q+k)-m_\psi^2]g^\mu_\mu
\end{eqnarray}
and the space component part of
\begin{equation}
N(q,k_0=\pm \omega_k,\vec k)=8k^\mu(q+k)_\mu -4[k\cdot q]g^\mu_\mu
\end{equation}
is
\begin{equation}
N(\vq,\vec k)=-8\vec k\cdot(\vq +\vec k) +4[\vec k\cdot \vq]g^i_i~.
\end{equation}
\subsection{Application of L'Hospital rule}
\label{LHos}
For finite value of $\Gamma$, the Eq.~(\ref{el_G}) becomes
\begin{equation}
\sigma=\frac{e^2}{3}\int\frac{d^3k}{(2\pi)^3}\frac{N^0}{4\omega_k^2\Gamma}
\lim_{q_0,\vq \rightarrow 0}\left[\frac{C_2}{q_0}
+\frac{C_3}{q_0}\right]~,
\label{el_G_finite}
\end{equation}
as
\begin{equation}
\lim_{\vq \rightarrow 0}\omega_p=\omega_k~.
\end{equation}
Applying L'Hospital's rule, we can write
\begin{eqnarray}
\lim_{q_0 \rightarrow 0}\frac{C_{2,3}(q_0)}{q_0}&=&\lim_{q_0\rightarrow 0}
\frac{\frac{d}{dq_0}\{C_{2,3}(q_0)\}}{\frac{d}{dq_0}\{q_0\}}
\nonumber\\
&=&\frac{d}{dq_0}\{ \pm n^{\mp}_p(\omega_p=\mp q_0+\omega_k) \}
\nonumber\\
&=&\beta [n^{\mp}_k(1 +\epsilon_k n^{\mp}_k)]~,
\end{eqnarray}
since
\begin{eqnarray}
\left(\pm \right)\frac{d}{dq_0}n^{\mp}_p(\omega_p=\mp q_0+\omega_k)&=&\left(\pm\right)
\frac{-\beta \frac{d\omega_p}{dq_0}
e^{\beta(\omega_p \pm \mu)}}{\{e^{\beta(\omega_p \pm \mu)}+\epsilon_k\}^2}
\nonumber\\
\lim_{q_0\rightarrow 0}\left(\pm \right)\frac{d}{dq_0}n^{\mp}_p(\omega_p=\mp q_0+\omega_k)
&=&\left(\pm\right)\frac{-\beta \left(\mp \right)
e^{\beta(\omega_k \pm \mu)}}{\{e^{\beta(\omega_k \pm \mu)}+\epsilon_k\}^2}
\nonumber\\
&=&\beta [n^{\mp}_k(1 +\epsilon_k n^{\mp}_k)]~.
\nonumber\\
\end{eqnarray}
|
2,869,038,155,722 | arxiv | \section{Introduction}\label{sec:Intro}
Most real life signals are non--stationary and non--linear. Standard techniques like Fourier or wavelet transform prove to be unable to capture properly their hidden features \cite{cicone2017dummies}. For this reason Huang et al. proposed in 1998 a new kind of algorithm, called Empirical Mode Decomposition (EMD) \cite{huang1998empirical}, which allows to unravel the hidden features of a non--stationary signal $s(x)$, $x\in\R$, by iteratively decomposing it into a finite sequence of simple components, called Intrinsic Mode Functions (IMFs). Such IMFs fulfill two properties:
the number of extrema and the number of zero crossings must either equal or differ at most by one;
considering upper and lower envelopes connecting respectively all the local maxima and minima of the function, their mean has to be zero at any point.
The wide variety of applications of this technique, see for instance \cite{cicone2017spectral,cicone2016hyperspectral,cicone2017Geophysics,sfarra2019thermal} and references therein, testified also by the high number of citations\footnote{The original work by Huang et al. \cite{huang1998empirical} as received so far, by itself, more than 10000 unique citations, according to Scopus} of the original paper by Huang et al. \cite{huang1998empirical}, together with the difficulty in analyzing it mathematically has attracted many researchers over the last two decades. Many alternative methods have been proposed, see \cite{cicone2017multidimensional} and references therein. All of these newly proposed methods are based on optimization with the only exception of the Iterative Filtering (IF) method, proposed by Lin et al. in \cite{lin2009iterative}, which is based instead on iterations.
The mathematical analysis of IF has been tackled by several authors in the last few years \cite{cicone2016adaptive,huang2009convergence,wang2013convergence,wang2012iterative} even for 2D or higher dimensional signals \cite{cicone2017multidimensional}. However several problems regarding this technique are still unsolved. In particular it is not yet clear how the stopping criterion used to discontinue the calculations of the IF algorithm influences the decomposition. Furthermore all the aforementioned analyses focused on the convergence of IF when applied to the extraction of a single IMF from a given signal, the so called inner loop. Regarding the decomposition of all the IMFs contained in a signal, which is related to the outer loop convergence and potential finiteness of the decomposition itself, nothing has been said so far. In this work we further analyze the IF technique addressing these and other questions.
The rest of this work is organized as follows: in Section \ref{sec:ContinuousIF} we review the details and properties of the method in the continuous setting and we provide new results regarding its inner loop convergence in presence of a stopping criterion as well as the outer loop convergence and finiteness. In Section \ref{sec:Discrete} we address the convergence analysis in the discrete setting for both the inner and outer loop of the algorithm. Based on these results in Section \ref{sec:speedup} we propose new ideas to increase the efficiency of the Iterative Filtering algorithm.
\section{IF algorithm in the continuous setting}\label{sec:ContinuousIF}
The key idea behind this decomposition technique is separating simple oscillatory components contained in a signal $s(x)$, $x\in\R$, the so called IMFs, by approximating the moving average of $s$ and subtracting it from $s$ itself. The approximated moving average is computed by convolution of $s$ with a window/filter function $w$
\begin{definition}\label{def:window}
A filter/window $w$ is a nonnegative and even function in $C^0\left([-L,\ L]\right)$, $L>0$, and such that $\int_\R w(z)\d z=\int_{-L}^{L} w(z)\d z=1$.
\end{definition}
We point out that the idea of iteratively subtracting the moving average comes from the Empirical Mode Decomposition (EMD) method \cite{huang1998empirical} where the moving average was computed as a local average between an envelope connecting the maxima and one connecting the minima of the signal under study. The use of envelopes in an iterative way is the reason why the EMD algorithm is still lacking a rigorous mathematical framework.
The pseudocode of IF is given in Algorithm \ref{algo:IF}
\begin{algorithm}
\caption{\textbf{Iterative Filtering} IMF = IF$(s)$}\label{algo:IF}
\begin{algorithmic}
\STATE IMF = $\left\{\right\}$
\WHILE{the number of extrema of $s$ $\geq 2$}
\STATE $s_1 = s$
\WHILE{the stopping criterion is not satisfied}
\STATE compute the filter length $l_m$ for $s_{m}(x)$
\STATE $s_{m+1}(x) = s_{m}(x) -\int_{-l_m}^{l_m} s_m(x+t)w_m(t)\dt$
\STATE $m = m+1$
\ENDWHILE
\STATE IMF = IMF$\,\cup\, \{ s_{m}\}$
\STATE $s=s-s_{m}$
\ENDWHILE
\STATE IMF = IMF$\,\cup\, \{ s\}$
\end{algorithmic}
\end{algorithm}
where $w_m(t)$ is a nonnegative and compactly supported window/filter with area equal to one and support in $[-l_m,\ l_m]$, where $l_m$ is called filter length and represents the half support length.
The IF algorithm contains two loops: the inner and the outer loop, the second and first while loop in the pseudocode respectively. The former captures a single IMF, while the latter produces all the IMFs embedded in a signal.
Assuming $s_1=s$, the key step of the algorithm consists in computing the moving average of $s_m$ as
\begin{equation}\label{eq:Mov_Average}
\mathcal{L}_m(s_m)(x)=\int_{-l_m}^{l_m} s_m(x+t)w_m(t)\dt,
\end{equation}
which represents the convolution of the signal itself with the window/filter $w_m(t)$.
The moving average is then subtracted from $s_m$ to capture the fluctuation part as
\begin{equation}\label{eq:flactuations}
\mathcal{M}_{m}(s_m)= s_m-\mathcal{L}_m(s_m)=s_{m+1}
\end{equation}
The first IMF, $\textrm{IMF}_1$, is computed repeating iteratively this procedure on the signal $s_m$, $m\in\N$, until a stopping criterion is satisfied, as described in the following section.
To produce the $2$-nd IMF we apply the same procedure to the remainder signal $r=s-\textrm{IMF}_1$.
Subsequent IMFs are produced iterating the previous steps.
The algorithm stops when $r$ becomes a trend signal, meaning it has at most one local extremum.
We observe that, even thought the algorithm allows potentially to recompute the filter length $l_m$ at every step of each inner loop, in practice we always compute the filter length only at the first step of an inner loop and then we keep it constant throughout the subsequent iterations. Hence $l_m=l_1=l$ for every $m\geq 1$.
Following \cite{lin2009iterative}, one possible way of computing the filter length $l$ is given by the formula
\begin{equation}\label{eq:Unif_Mask_length}
l:=2\left\lfloor\nu \frac{N}{k}\right\rfloor
\end{equation}
where $N$ is the total number of sample points of a signal $s(x)$, $k$ is the number of its extreme points, $\nu$ is a tuning parameter usually fixed around 1.6, and $\left\lfloor \cdot \right\rfloor$ rounds a positive number to the nearest integer closer to zero. In doing so we are computing some sort of average highest frequency contained in $s$.
Another possible way could be the calculation of the Fourier spectrum of $s$ and the identification of its highest frequency peak. The filter length $l$ can be chosen to be proportional to the reciprocal of this value.
The computation of the filter length $l$ is an important step of the IF technique. Clearly, $l$ is strictly positive and, more importantly, it is based solely on the signal itself. This last property makes the method nonlinear.
In fact, if we consider two signals $p$ and $q$ where $p\neq q$, assuming $\textrm{IMFs}(\bullet)$ represent the decomposition of a signal into IMFs by IF, the fact that we choose the half support length based on the signal itself implies that in general
$$\textrm{IMFs}(p+q)\neq \textrm{IMFs}(p)+\textrm{IMFs}(q)$$
Regarding the convergence analysis of the Iterative Filtering inner loop we recall here the following theorem
\begin{theorem}[Convergence of the Iterative Filtering method \cite{cicone2016adaptive,huang2009convergence}]\label{thm:theo_1}
Given the filter function $w(t), t\in[-l,l]$ be $L^2$, symmetric, nonnegative, $\int_{-l}^l w(t)\dt = 1$ and let $s(x)\in L^2(\mathbb{R})$. \newline
If $|1- \widehat{w}(\xi)| < 1 $ or $\widehat{w}(\xi)=0$, where $\widehat{w}(\xi)$ is the Fourier transform of $w$ computed at the frequency $\xi$,
Then
$\{\mathcal{M}^m(s)\}$ converges and
\begin{equation}\label{eq:IMF_cont}
\textrm{IMF}_1 = \lim\limits_{m\rightarrow \infty}{\mathcal{M}^m(s)(x)}= \int_{-\infty}^{\infty} \widehat{s}(\xi) \chi_{\{\widehat{w}(\xi)=0 \}}
e^{2\pi i \xi x} \textrm{d}\xi
\end{equation}
\end{theorem}
We observe here that given $h:[-\frac{l}{4},\frac{l}{4}]\rightarrow\R$, $z\mapsto h(z)$, nonnegative, symmetric, with $\int_\R h(z)\d z=\int_{-\frac{l}{4}}^{\frac{l}{4}} h(z)\d z=1$, if we construct the window $w_1$ as the convolution of $h$ with itself and we fix $w_m=w_1$ throughout all the steps $m$ of an inner loop, then the method converges for sure to the limit function \eqref{eq:IMF_cont} which depends only on the shape of the filter function chosen and the support length selected by the method \cite{cicone2016adaptive,cicone2017spectral}.
In general we can assume that the filter functions $w_m(u)$ are defined as some scaling of an a priori fixed filter shape $w:[-1,1]\rightarrow\R$. In particular we define the scaling function
\begin{equation}\label{eq:scalingFunc}
g_m:[-1,1]\rightarrow [-l_m,l_m],\qquad t\mapsto u=g_m(t),
\end{equation}
where $g_m$ is assumed to be invertible and monotone, such that $w_m(u)=C_m w(g_m^{-1}(u))=C_m w(t)$, where $t=g_m^{-1}(u)$, $u=g_m(t)$ and $C_m$ is a scaling coefficient which is required to ensure that $\int_\R w_m(u)\d u=\int_{-l_m}^{l_m} w_m(u)\d u=1$.
Regarding the computation of the scaling coefficient $C_m$, from the observation that $\d u = g_m'(t) \d t$, it follows that
\begin{equation}\label{eq:C_m}
\int_{-l_m}^{l_m} w_m(u)\d u=\int_{-l_m}^{l_m} C_m w(g_m^{-1}(u))\d u=C_m \int_{-1}^{1} w(t) |g_m'(t)| \d t
\end{equation}
hence
\begin{equation}\label{eq:C_x2}
C_m= \frac{1}{\int_{-1}^{1} w(t) |g_m'(t)| \d t}
\end{equation}
and
\begin{equation}\label{eq:w_m}
w_m(u)=C_m w(g_m^{-1}(u))=\frac{w(g_m^{-1}(u))}{\int_{-1}^{1} w(t) |g_m'(t)| \d t}
\end{equation}
As an example of a scaling function we can consider, for instance, linear or quadratic scalings: $g_m(t)=l_m t$ and $g_m(t)=l_m t^2$ respectively.
In the case of linear scaling we have that $g_m^{-1}(u)=\frac{u}{l_m}$, $g'_m(t)=l_m\geq 0$, for every $t\in\R$, and $C_m= \frac{1}{l_m}$. Hence
\begin{equation}\label{eq:w_m_linear}
w_m(u)=\frac{w\left(\frac{u}{l_m}\right)}{l_m}
\end{equation}
\subsection{IF inner loop convergence in presence of a stopping criterion}\label{sec:Stopping}
In Algorithm \ref{algo:IF} the inner loop has to be iterated infinitely many times. In numerical computations, however, some stopping criterion has to be introduced. One possible stopping criterion follows from the solution of
\begin{problem}\label{pb:IF_num_conv}
For a given $\delta > 0$ we want to find the value $N_0\in\N$ such that
$$\|\mathcal{M}^N(s)(x)-\mathcal{M}^{N+1}(s)(x)\|_{L^2}<\delta \qquad \forall N\geq N_0$$
\end{problem}
Applying the aforementioned stopping criterion, the inner loop of Algorithm \ref{algo:IF} converges in finite steps to an IMF whose explicit form is given in the following theorem where $\widehat{s}(\xi)$ represents the Fourier transform of $s$ at frequency $\xi$.
\begin{theorem}\label{thm:IF_inner_conv_stopping}
Given $s\in L^2(\R)$ and $w$ obtained as the convolution $\widetilde{w}\ast \widetilde{w}$, where $\widetilde{w}$ is a filter/window, Definition \ref{def:window}, and fixed $\delta>0$.
Then, for the minimum $N_0\in\N$ such that the following inequality holds true
\begin{equation}\label{eq:N0}
\frac{N_0^{N_0}}{(N_0+1)^{N_0+1}}<\frac{\delta}{\left\|\widehat{s}(\xi)\right\|_{L^2}} \quad \forall \xi\in\R
\end{equation}
we have that $\left\| \mathcal{M}^N(s)(x)-\mathcal{M}^{N+1}(s)(x)\right\|_{L^2}<\delta \quad \forall N\geq N_0$ and the first IMF is given by
\begin{equation}\label{eq:IMF_IF_stop}
\textrm{IMF}_1^\textrm{SC}=\mathcal{M}^N(s)(x)=\int_{\R} (1-\widehat{w}(\xi))^N \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \quad \forall N\geq N_0
\end{equation}
\end{theorem}
\begin{proof}
From the hypotheses on the filter $w$ it follows that its Fourier transform is in the interval $[0,\ 1]$, see \cite{cicone2016adaptive}. Furthermore from the linearity of the Fourier transform it follows that
$${\widehat{\mathcal{M}^N(s)(x)}(\xi)}= (1- \widehat{w}(\xi))^N \widehat{s}(\xi)=\left\{
\begin{array}{cc} \widehat{s}(\xi) & \textrm{ if } \widehat{w}(\xi)=0\\
(1- \widehat{w}(\xi))^N \widehat{s}(\xi) & \textrm{ if } |1- \widehat{w}(\xi)| < 1\\
\end{array}
\right.$$
since the Fourier Transform is a unitary operator, by the Parseval's Theorem, it follows that
$$\left\| \mathcal{M}^N(s)(x)-\mathcal{M}^{N+1}(s)(x)\right\|_{L^2} =\left\| \widehat{\mathcal{M}^N(s)(x)}(\xi)-\widehat{\mathcal{M}^{N+1}(s)(x)}(\xi)\right\|_{L^2}$$
$$=
\left\|(1-\widehat{w}(\xi))^N \left[1 -(1-\widehat{w}(\xi))\right]\widehat{s}(\xi)\right\|_{L^2}=
\left\|(1-\widehat{w}(\xi))^N \widehat{w}(\xi)\widehat{s}(\xi)\right\|_{L^2}
$$
We point out that this formula can also be interpreted as the $L^2$--norm of the moving average of $\mathcal{M}^N$ which is given by the convolution $\mathcal{M}^N\ast w$.
For a fixed $N$ we can compute the maximum of the function $(1-\widehat{w}(\xi))^N \widehat{w}$, for $\widehat{w}\in[0,\ 1]$, that is attained for $\widehat{w}(\xi)=\frac{1}{N+1}$. Therefore
$$\left\|(1-\widehat{w}(\xi))^N \widehat{w}(\xi)\widehat{s}(\xi)\right\|_{L^2}\leq \left\|\left(1-\frac{1}{N+1}\right)^N\frac{1}{N+1}\widehat{s}(\xi)\right\|_{L^2}$$
$$=\left\|\frac{N^N}{(N+1)^{N+1}}\widehat{s}(\xi)\right\|_{L^2}<\delta$$
Hence we consider the smallest $N_0\in\N$ such that
$$\frac{N_0^{N_0}}{(N_0+1)^{N_0+1}}<\frac{\delta}{\left\|\widehat{s}(\xi)\right\|_{L^2}}$$
\end{proof}
Equation \eqref{eq:IMF_IF_stop} provides a valuable insight on how the implemented algorithm is actually decomposing a signal into IMFs.
We recall that without any stopping criterion each IMF of a signal $s$ is given by the inverse Fourier transform of $\widehat{s}$ computed at the frequencies corresponding to zeros of $\widehat{w}$, as stated in \eqref{eq:IMF_cont}.
Therefore, from the observation that $\widehat{w}$ is a function not compactly supported and with isolated zeros, the IMFs produced with IF are given by the summation of pure and well separated tones.
Whereas, when we enforce a stopping criterion, we end up producing IMFs containing a much richer spectrum. In fact from \eqref{eq:IMF_IF_stop} we discover that an IMF is now given by the inverse Fourier transform of $\widehat{s}$ computed at every possible frequency in $\R$, each multiplied by the coefficient $(1-\widehat{w}(\xi))^N$. Since, by construction, $0 \leq \widehat{w}(\xi)\leq 1$, $\forall \xi\in\R$, then $(1-\widehat{w}(\xi))^N$ is equal to 1 if and only if $\widehat{w}(\xi)=0$, whereas for all the other frequencies it is smaller than 1 and it tends to zero as $N$ grows. The $(1-\widehat{w}(\xi))^N$ quantity represents in practice the percentage with which each frequency is contained in the reconstruction of an IMF from the Fourier transform of the original signal. The higher is the number of iterations $N$ the narrower are the intervals of frequencies that are almost completely captured in each IMF. And as $N\rightarrow\infty$ such intervals coalesce into isolated points corresponding to the zeros of $\widehat{w}$.
\subsubsection{Convergence with a threshold}
We start recalling a few properties regarding the filter functions $w$. Assuming $w(x)$, $x\in\R$, is a filter function supported on $(-1,\ 1)$, if we use the linear scaling described in \eqref{eq:w_m_linear}, then we can construct
\begin{equation}\label{eq:w^a}
w^a(x)=\frac{1}{a} w\left(\frac{x}{a}\right)
\end{equation}
where $w^a(x)$ is supported on $(-a,\ a)$.
If we define $\widehat{w}(\xi)=\int_{-\infty}^{+\infty} w(x) e^{-i\xi x 2 \pi} \d x$, then
\begin{equation}\label{eq:fft_w^a}
\widehat{w^a}(\xi)=\int_{-\infty}^{+\infty} \frac{1}{a}w\left(\frac{x}{a}\right) e^{-i\xi \frac{x}{a} a 2 \pi} \d x=\widehat{w}(a\xi)
\end{equation}
Therefore, if $\xi_0$ is a root of $\widehat{w}(\xi)=0$, then $\frac{\xi_0}{a}$ is a root of $\widehat{w^a}(\xi)=0$ because $\widehat{w^a}\left(\frac{\xi_0}{a}\right)=\widehat{w}\left(a\frac{\xi_0}{a}\right)=\widehat{w}(\xi_0)=0$.
We remind that, since $w$ are compactly supported functions, their Fourier transform are defined on $\R$ and they have zeros which are isolated points.
Given $0 < \gamma < 1$, we identify the set
\begin{equation}\label{eq:I_gamma}
I_{w,\gamma, N}=\left\{\xi\in\R\ :\ \widehat{w}(\xi) \leq 1 - \sqrt[N]{1-\gamma}\right\}.
\end{equation}
As $N\rightarrow\infty$ the quantity $1 - \sqrt[N]{1-\gamma}\rightarrow 0$, therefore $I_{w,\gamma, N}$ coalesces into isolated points corresponding to the zeros of $\widehat{w}$.
If we consider filters like the Fokker-Planck filters \cite{cicone2016adaptive} or any filter with smooth finite support properties we must have that, for a fixed $N\in\N$ and $\gamma>0$, there exists $\Xi_0>0$ such that
\begin{equation}\label{eq:Xi_0}
\widehat{w}(\xi) \leq 1 - \sqrt[N]{1-\gamma} < 1 \qquad \textrm{ for all }\ |\xi|\geq \Xi_0
\end{equation}
In fact, since $\int|w(x)|^2\d x < + \infty$ with $w(x)$ smooth function, then $\int|\widehat{w}(\xi)|^2\d \xi < + \infty$ which implies that $\widehat{w}(\xi)$ decays as $|\xi|\rightarrow \infty$.
So for a filter $w$ with smooth finite support properties the set $I_{w,\gamma, N}$ is made up of a finite number of disjoint compact intervals, containing zeros of $\widehat{w}$, together with the intervals $(-\infty,\ -\Xi_0]$ and $[\Xi_0,\ \infty)$.
Furthermore if we scale these filters using a linear scaling with coefficient $a>1$ it follows from the previous observations that $\Xi_0\rightarrow 0$ and, as a consequence, $I_{w,\gamma, N}$ converges to $\R\backslash \{0\}$.
As an example of a compactly supported filter we can consider the triangular filter function
\begin{equation}\label{eq:triangular}
w(x) = \left\{
\begin{array}{cc}
\frac{1}{L}-\frac{1}{L^2}|x| & \textrm{for} \; |x|\leq L\\
0 & \textrm{otherwise} \\
\end{array}
\right.
\end{equation}
whose Fourier transform is
\begin{equation}\label{eq:triangular_fourier}
\widehat{w}(\xi) = \frac{1}{L}\frac{\sin^2\left(L\pi \xi\right)}{\left(\pi \xi\right)^2}.
\end{equation}
The triangular filter and its Fourier transform are depicted in Fig. \ref{fig:Triangle_filter}
\begin{figure
\centering
\subfloat{{\includegraphics[width=0.48\textwidth]{./TriangularFilter.eps} }
~
\subfloat{{\includegraphics[width=0.48\textwidth]{./TriangularFilter_Fourier_transf.eps} }
\caption{ Left panel, triangular filter \eqref{eq:triangular} with $L=1$. Right panel, in black the Fourier transform \eqref{eq:triangular_fourier} and in red the threshold value $1 - \sqrt[N]{1-\gamma}$}\label{fig:Triangle_filter}
\end{figure}
Given the threshold value $1 - \sqrt[N]{1-\gamma}$ depicted in the right panel of Fig. \ref{fig:Triangle_filter} and the triangular filter \eqref{eq:triangular} with $L=1$, the set $I_{w,\gamma, N}$ is made up of four intervals: two compactly supported and centered around $1/2$ and $-1/2$, and other two starting around $0.8$ and $-0.8$ and ending at infinity and minus infinity, respectively.
We can use the threshold value $1 - \sqrt[N]{1-\gamma}$ in the computation of an IMF as follows: given \eqref{eq:IMF_IF_stop}, whenever $(1-\widehat{w}(\xi))^N \geq 1-\gamma$, we substitute $\widehat{w}(\xi)$ with zero. This is equivalent to setting $\widehat{w}(\xi)=0$ whenever $\xi\in I_{w,\gamma, N}$.
Therefore, using the previously described thresholding and based on Theorem \ref{thm:IF_inner_conv_stopping}, Algorithm \ref{algo:IF} converges to
\begin{equation}\label{eq:IMF_cont_stop_threshold}
\textrm{IMF}_1^\textrm{TH} =
\int_{\R\backslash I_{w,\gamma, N}} (1-\widehat{w}(\xi))^N \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi + \int_{I_{w,\gamma, N}} \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \quad \forall N\geq N_0
\end{equation}
where $I_{w,\gamma, N}$ is defined in \eqref{eq:I_gamma}.
We are now ready to prove the following
\begin{proposition}
Assuming that all the hypotheses of Theorem \ref{thm:IF_inner_conv_stopping} are fulfilled, then for every $\epsilon>0$ there exist a stopping criterion value $\delta>0$ and a threshold $0<\gamma<1$ such that
\begin{equation}\label{eq:NormInequalities_TH}
\left\|\textrm{IMF}_1-\textrm{IMF}_1^\textrm{TH}\right\|\leq \frac{\epsilon}{2}, \qquad \left\|\textrm{IMF}_1^\textrm{TH}-\textrm{IMF}_1^\textrm{SC}\right\|\leq \frac{\epsilon}{2}
\end{equation}
and
\begin{equation}\label{eq:NormInequality_SC}
\left\|\textrm{IMF}_1-\textrm{IMF}_1^\textrm{SC}\right\|\leq \epsilon
\end{equation}
where $\textrm{IMF}_1$, $\textrm{IMF}_1^\textrm{SC}$, and $\textrm{IMF}_1^\textrm{TH}$ are defined in \eqref{eq:IMF_cont}, \eqref{eq:IMF_IF_stop}, and \eqref{eq:IMF_cont_stop_threshold} respectively.
\end{proposition}
\begin{proof}
First of all we have that
$$\left\|\textrm{IMF}_1-\textrm{IMF}_1^\textrm{SC}\right\| \leq \left\|\textrm{IMF}_1-\textrm{IMF}_1^\textrm{TH}\right\| + \left\|\textrm{IMF}_1^\textrm{TH}-\textrm{IMF}_1^\textrm{SC}\right\|$$
where
\begin{eqnarray}
\nonumber \left\|\textrm{IMF}_1-\textrm{IMF}_1^\textrm{TH}\right\| & \leq & \left\|\int_\R \widehat{s}(\xi) \chi_{\left\{\xi\in\R\ |\ \widehat{w}(\xi)=0 \right\}} e^{2\pi i \xi x} \d \xi - \int_{\R\backslash I_{w,\gamma, N}} \!\!\!\!\!\!\!\!(1-\widehat{w}(\xi))^N\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi - \int_{I_{w,\gamma, N}} \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \right\| \leq \\
& & \left\|\int_{\R\backslash I_{w,\gamma, N}} \!\!\!\!\!\!\!\!(1-\widehat{w}(\xi))^N\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi\right\| + \left\|\int_{I_{w,\gamma, N}\backslash \left\{\xi\in\R\ |\ \widehat{w}(\xi)=0 \right\}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \right\|
\end{eqnarray}
and
\begin{eqnarray}
\nonumber \left\|\textrm{IMF}_1^\textrm{TH}-\textrm{IMF}_1^\textrm{SC}\right\| & \leq & \left\|\int_{\R\backslash I_{w,\gamma, N}} \!\!\!\!\!\!\!\!(1-\widehat{w}(\xi))^N\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi + \int_{I_{w,\gamma, N}} \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi - \int_\R (1-\widehat{w}(\xi))^N\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi\right\| \leq \\
& & \left\|\int_{I_{w,\gamma, N}} \left[1-(1-\widehat{w}(\xi))^N\right]\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \right\|
\end{eqnarray}
From \eqref{eq:I_gamma} and the fact that $\int_{I_{w,\gamma, N}} \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \rightarrow \int_{ \left\{\xi\in\R\ |\ \widehat{w}(\xi)=0 \right\}} \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi$ as $\gamma\rightarrow 0$ or $N\rightarrow\infty$, it follows that there exist $N_1\in\N$ big enough and $0<\gamma_1<1$ small enough such that
$$\left\|\int_{I_{w,\gamma_1, N_1}\backslash \left\{\xi\in\R\ |\ \widehat{w}(\xi)=0 \right\}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \right\| \leq \frac{\epsilon}{4}$$
Furthermore there exist $0<\gamma_2<1$ small enough and a $N_2\in\N$ so that
$$\left\|\int_{I_{w,\gamma_2, N_2}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left[1-(1-\widehat{w}(\xi))^{N_2}\right]\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \right\|\leq \frac{\epsilon}{2}$$
in fact as $\gamma_2\rightarrow 0$ the interval $I_{w,\gamma_2, N_2}$ tends to the set of frequencies corresponding to the zeros of $\widehat{w}(\xi)$.
Given $\gamma=\min\left\{\gamma_1,\ \gamma_2\right\}$, then there exists $N_3\in\N$ big enough such that $(1-\widehat{w}(\xi))^{N_3}$ is small enough in order to have
$$\left\|\int_{\R\backslash I_{w,\gamma, N}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!(1-\widehat{w}(\xi))^{N_3}\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi\right\|\leq \frac{\epsilon}{4}$$
If we consider $N_0=\max\left\{N_1,\ N_2,\ N_3\right\}$ there exists $\delta>0$ such that \eqref{eq:N0} holds true for every $N\geq N_0$.
\end{proof}
This proposition implies that $\textrm{IMF}_1^\textrm{TH}$ can be as close as we like to both $\textrm{IMF}_1^\textrm{SC}$ and $\textrm{IMF}_1$ if we choose wisely the stopping criterion value $\delta$ and the threshold $\gamma$.
\subsection{IF outer loop convergence}\label{subsec:outerLoopIF}
We do have now all the tools needed to study the Iterative Filtering outer loop convergence.
\begin{definition}[Significant IMFs with respect to $\eta>0$]
Fixed $\eta>0$ and given a signal $s$ and its decomposition in IMFs obtained using Algorithm \ref{algo:IF}, then we define \emph{significant IMFs} with respect to $\eta$ all the IMFs whose $L^\infty$-norm is bigger than $\eta$.
\end{definition}
\begin{theorem}\label{thm:IF_outer_conv}
Given a signal $s\in L^\infty(\R)$, whose continuous frequency spectrum is compactly supported with upper limit $B>0$ and lower limit $b>0$, and such that $\|\widehat{s}\|_{\infty} = c<\infty$, chosen a filter $w$ produced as convolution of a filter with itself, fixed $\delta>0$ and $\eta>0$.
\noindent Then the inner loop of Algorithm \ref{algo:IF} converges to \eqref{eq:IMF_IF_stop} and the outer loop produces only a finite number $M\in\N$ of significant IMFs whose norm is bigger than $\eta$.
\end{theorem}
\begin{proof}
Let us consider the Fourier transform of the signal $s$. From the hypotheses it follows that $|\widehat s(\xi)| = 0$ for every $\xi \geq B$.
We can assume that Algorithm \ref{algo:IF} in the first step of its outer loop starts selecting a filter $w_1$ such that the zero of $\widehat{w_1}$ with smallest frequency is at $B$. We recall in fact that one of the possible way to choose the filter length is based on the Fourier transform of $s$, as explained in Section 2. Given $\delta>0$ we can identify $N_1\in\N$ such that \eqref{eq:N0} is fulfilled for every $N\geq N_1$.
Now, from the hypothesis that $\|\widehat{s}\|_{\infty} = c<\infty$ it follows there exists the upper bound $c$ on $\widehat s(\xi)$ uniformly on $\xi \in \R$. From the hypotheses on the filter function it follows that $0<\widehat{w_1}<1$, ref. end of Section 2 in \cite{cicone2016adaptive}. Furthermore, from the assumption on the lower bound $b$ and upper bound $B$ of the continuous frequency spectrum of $s$, the fact that $\left\| e^{2\pi i \xi x} \right\|_{\infty}\leq 1$ for every $x,\xi\in\R$, by definition of the interval $I_{w_1,\gamma, \widetilde{N}_1}$, and for every $\widetilde{N}_1\geq N_1$ and $0<\gamma< \frac{\eta}{2c(B-b)}$, it follows that
\begin{eqnarray}
\nonumber \left\|\textrm{IMF}_1^\textrm{SC}-\textrm{IMF}_1^\textrm{TH}\right\|_{\infty} & \leq & \left\|\int_{[b,\ B]\cap I_{w_1,\gamma, \widetilde{N}_1}} \left[1-(1-\widehat{w_1}(\xi))^{\widetilde{N}_1}\right]\widehat{s}(\xi) e^{2\pi i \xi x}\d \xi \right\|_{\infty} \leq \\
\nonumber & \leq & \int_{[b,\ B] \cap I_{w_1,\gamma, \widetilde{N}_1}} \left\| \left[1-(1-\widehat{w_1}(\xi))^{\widetilde{N}_1}\right] \right\|_{\infty} \left\| \widehat{s}(\xi) \right\|_{\infty} \left\| e^{2\pi i \xi x} \right\|_{\infty} \d \xi \leq \\
& \leq& c \int_{[b,\ B]\cap I_{w_1,\gamma, \widetilde{N}_1}} \left\| \left[1-(1-\widehat{w_1}(\xi))^{\widetilde{N}_1}\right] \right\|_{\infty} \d \xi \leq c\gamma (B-b) < \frac{\eta}{2}
\end{eqnarray}
In particular we point out that $I_{w_1,\gamma, \widetilde{N}_1}$, defined as in \eqref{eq:I_gamma}, covers the interval of frequencies $[B-r_1,\ B+r_1]$, for some $r_1>\varepsilon>0$.
This last inequality follows from the fact that if we scale linearly the filter function $w$ to enlarge its support, as in \eqref{eq:w^a} for $a>1$, its Fourier transform is proportionally shrunk \eqref{eq:fft_w^a}. However the signal $s$ does have a lower bound $b$ in the continuous frequency spectrum which implies that the filter function $w_1$ cannot have a too wide support and as a consequence its Fourier transform cannot be too much squeezed. Therefore it does exist $\varepsilon>0$ which lower bounds the radius $r_1$.
If $\left\|\textrm{IMF}_1^\textrm{TH}\right\|_{\infty}< \frac{\eta}{2}$ then we can for sure regard this component as not significant because $\left\|\textrm{IMF}_1^\textrm{SC}\right\|_{\infty}\leq \left\|\textrm{IMF}_1^\textrm{SC}-\textrm{IMF}_1^\textrm{TH}\right\|_{\infty} + \left\|\textrm{IMF}_1^\textrm{TH}\right\|_{\infty}< \eta$. Otherwise, assuming $\left\|\textrm{IMF}_1^\textrm{TH}\right\|_{\infty} \geq \frac{\eta}{2}$, if $\left\|\textrm{IMF}_1^\textrm{SC}\right\|_{\infty}\geq \eta$, then $\textrm{IMF}_1^\textrm{SC}$ represents the first significant IMF in the decomposition. This conclude the first step of the outer loop in Algorithm \ref{algo:IF}.
In the second step of the outer loop Algorithm \ref{algo:IF} iterates the previous passages using now the remainder signal $s_2=s-\textrm{IMF}_1^{SC}$ and selecting a filter $w_2$ such that the zero of $\widehat{w_2}$ with smallest frequency is at $B-r_1$.
Also in this case, given $\delta>0$, we can identify $N_2\in\N$ such that \eqref{eq:N0} is fulfilled for every $N\geq N_2$. Furthermore $\widehat{s}_2(\xi)=\widehat{s}(\xi)-\widehat{\textrm{IMF}}_1^{SC}(\xi)=\left[1-\left(1-\widehat{w_2}(\xi)\right)^{\widetilde{N}_1}\right]\widehat{s}(\xi),\ \forall \xi \in \R$ which implies that
\begin{equation}
\left\|\widehat{s}_2\right\|_{\infty}\leq \left\|\left[1-\left(1-\widehat{w_2}(\xi)\right)^{\widetilde{N}_1}\right]\right\|_{\infty}\left\|\widehat{s}(\xi)\right\|_{\infty}\leq \left\|\widehat{s}(\xi)\right\|_{\infty}
\end{equation}
since $\widehat{w_2}(\xi)\in [0,\ 1],\ \forall \xi \in \R$ \cite{cicone2016adaptive}. Hence $\widehat{s}_2$ has the same uniform upper bound $c$ over all $\xi \in \R^+$ as $\widehat s(\xi)$.
Therefore
\begin{equation}\label{eq:IMF2}
\left\|\textrm{IMF}_2^\textrm{SC}-\textrm{IMF}_2^\textrm{TH}\right\|_{\infty}\leq \left\|\int_{I_{w_2,\gamma, \widetilde{N}_2}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \left[1-(1-\widehat{w_2}(\xi))^{\widetilde{N}_2}\right]\widehat{s_2}(\xi) e^{2\pi i \xi x}\d \xi \right\|_{\infty} \leq c\gamma (B-b) < \frac{\eta}{2}
\end{equation}
for every $\widetilde{N}_2\geq N_2$ and $0<\gamma< \frac{\eta}{2c(B-b)}$.
Furthermore $I_{w_2,\gamma, \widetilde{N}_2}$ covers the interval of frequencies $[B-r_2,\ B+r_2]$, for some $r_2>\varepsilon>0$. This last inequality follows from the same reasoning as before and the fact that the lower bound on the continuous frequency spectrum of $s_2$ is again $b$, by construction of $s_2$, the fact that $\gamma$ is fixed for every IMF and the Fourier transform of the scaled filter $w_2$ is a squeezed version of $\widehat w$, ref. equation \eqref{eq:fft_w^a}.
If $\left\|\textrm{IMF}_2^\textrm{TH}\right\|_{\infty}< \frac{\eta}{2}$ then we can regard this component as not significant. If instead $\left\|\textrm{IMF}_2^\textrm{TH}\right\|_{\infty}\geq \frac{\eta}{2}$ and $\left\|\textrm{IMF}_2^\textrm{SC}\right\|_{\infty}\geq \eta$, then $\textrm{IMF}_2^\textrm{SC}$ represents another significant IMF in the decomposition.
The subsequent outer loop steps follow similarly. The existence of the lower limit $\varepsilon$ for all $r_k>0$, $k\geq 1$, ensures that we can have a finite coverage of the interval of frequencies $[b,\ B]$.
In particular the algorithm generates a set $\left\{r_k\right\}_{k=1}^{R}$ such that $\sum_{k=1}^R r_k = B-b$ and there exists a natural number $0\leq M\leq R$ which represents the number of significant IMFs with respect to $\eta$.
\end{proof}
We point out that this theorem holds true also if we consider the $L^2$-norm instead of the $L^\infty$-norm thanks to the inclusion of $L^p$ spaces on a finite measure space.
From this Theorem it follows that IF with a stopping criterion allows to decompose a signal into a finite number of components given by \eqref{eq:IMF_IF_stop} each of which contains frequencies of the original signal filtered in a smart way.
We observe also that this theorem, together with Theorems \ref{thm:theo_1} and \ref{thm:IF_inner_conv_stopping}, allow to conclude that the IF method can not produce fake oscillations. Each IMF is in fact containing part of the oscillatory content of the original signal, as described in \eqref{eq:IMF_cont} and \eqref{eq:IMF_IF_stop}.
\section{IF algorithm in the discrete setting}\label{sec:Discrete}
Real life signals are discrete and compactly supported, therefore we want to analyze the IF algorithm discretization and study its properties.
Consider a signal $s(x)$, $x\in\R$, we assume for simplicity it is supported on $[0,\ 1]$, sampled at $n$ points $x_j= \frac{j}{n-1}$, with $j= 0,\ldots, n-1$, with a sampling rate which allows to capture all its fine details, so that aliasing will not play any role. The goal is to decompose the vector $\left[s(x_j)\right]_{j=0}^{n-1}$ into vectorial IMFs. Without loosing generality we can assume that $\|\left[s(x_j)\right]\|_2=1$.
From now on, to simplify the formulas, we use the notation $s=\left[s(x_j)\right]_{j=0}^{n-1}$. Furthermore, if not specified differently, we consider as matrix norm the so called Frobenius norm $\|A\|_2=\sqrt{\sum_{i,\ j=0}^{n-1}\left|a_{ij}\right|^2}$ which is unitarily invariant.
\begin{definition}\label{def:Discrete_filter}
A vector $w\in\R^n$, $n$ odd number, is called a \textbf{filter} if its values are symmetric with respect to the middle, nonnegative, and $\sum_{p=1}^n w_p = 1$.
\end{definition}
We assume that a filter shape has been selected a priori, like one of the Fokker-Planck filters described in \cite{cicone2016adaptive}, and that some invertible and monotone scaling function $g_m$ has been chosen so that $w_m(\xi)$ can be computed as described in \eqref{eq:w_m}.
Therefore, assuming $s_1=s$, the main step of the IF method becomes
\begin{equation}\label{eq:s_m+1}
s_{m+1}(x_i) = s_{m}(x_i)-\int_{x_i-l_m}^{x_i+l_m} \!\!\!\!\!\!\! s_m(y)w_m(x_i-y)\dy\approx s_{m}(x_i)-\!\!\!\!\! \sum_{x_j=x_i-l_m}^{x_i+l_m}\!\!\!\!\! s_m(x_j)w_m(x_i-x_j)\frac{1}{n}, \quad j=0,\ldots,n-1
\end{equation}
In matrix form we have
\begin{equation}\label{eq:MatrixForm}
s_{m+1}=(I-W_m)s_m
\end{equation}
where
\begin{equation}\label{eq:K}
W_m=\left[w_m(x_i-x_j)\cdot \frac{1}{n}\right]_{i,\ j=0}^{n-1}=\left[\frac{w(g_{m}^{-1}(x_i-x_j))}{\sum_{z_r=-1}^{1} w(z_r) |g'_{m}(z_r)| \Delta z_r}\cdot \frac{1}{n}\right]_{i,\ j=0}^{n-1}
\end{equation}
Algorithm \ref{algo:IF_discrete} provides the discrete version of Algorithm \ref{algo:IF}
\begin{algorithm}
\caption{\textbf{Discrete Iterative Filtering} IMF = DIF$(s)$}\label{algo:IF_discrete}
\begin{algorithmic}
\STATE IMF = $\left\{\right\}$
\WHILE{the number of extrema of $s$ $\geq 2$}
\STATE $s_1 = s$
\WHILE{the stopping criterion is not satisfied}
\STATE compute the function $w_m(\xi)$, whose half support length $l_m$ is based on the signal $\left[s_m(x_i)\right]_{i=0}^{n-1}$
\STATE $s_{m+1}(x_i) = s_{m}(x_i) - \sum_{j=0}^{n-1} s_m(x_j)w_m(|x_i-x_j|) \frac{1}{n},\qquad i= 0,\ldots, n-1$
\STATE $m = m+1$
\ENDWHILE
\STATE IMF = IMF$\,\cup\, \{ s_{m}\}$
\STATE $s=s-s_{m}$
\ENDWHILE
\STATE IMF = IMF$\,\cup\, \{ s\}$
\end{algorithmic}
\end{algorithm}
We remind that the first while loop is called outer loop, whereas the second one inner loop.
The first IMF is given by $\textrm{IMF}_1=\lim_{m\rightarrow\infty} (I-W_m)s_m$, where we point out that the matrix $W_m=[w_m(x_i-x_j)]_{i,\ j=0}^{n-1}$ depends on the half support length $l_m$ at every step $m$.
However in the implemented code the value $l_m$ is usually computed only in the first iteration of each inner loop and then kept constant in the subsequent steps, so that the matrix $W_m$ is equal to $W$ for every $m\in\N$. So the first IMF is given by
\begin{equation}\label{eq:First_IMF_fixed_length}
\textrm{IMF}_1=\lim_{m\rightarrow\infty} (I-W)^{m} s
\end{equation}
Furthermore in the implemented algorithm we do not let $m$ to go to infinity, instead we use a stopping criterion as described in section \ref{sec:Stopping}. For instance, we can define the following quantity
\begin{equation}\label{eq:SD}
SD:=\frac{\|s_{m+1}-s_{m}\|_2}{\|s_{m}\|_2}
\end{equation}
and we can stop the process when the value $SD$ reaches a certain threshold. Another possible option is to introduce a limit on the maximal number of iterations for all the inner loops. It is always possible to adopt different stopping criteria for different inner loops.
If we consider the case of linear scaling, making use of \eqref{eq:w_m_linear}, the matrix $W_m$ becomes
\begin{equation}\label{eq:W_linear}
W_m=\left[\frac{w\left(\frac{x_i-x_j}{l_m}\right)}{l_m}\cdot \frac{1}{n}\right]_{i,\ j=0}^{n-1}=\left[\frac{w\left(\frac{i-j}{(n-1)l_m}\right)}{l_m}\cdot \frac{1}{n}\right]_{i,\ j=0}^{n-1}
\end{equation}
We point out that the previous formula represent an ideal $W_m$, however we need to take into account the quadrature formula we use to compute the numerical convolution in order to build the appropriate $W_m$ to be used in the DIF algorithm.
For instance, if we use the rectangle rule, we need to substitute the exact value of $w(y)$ at $y$ with its average value in the interval of length $\frac{1}{n}$ centered in $y$ and multiply this value for the length of interval itself. Furthermore we should handle appropriately the boundaries of the support of $w(y)$, in fact the half length of the support is, in general, a non integer value. This can be done by handling separately the first and last interval in the quadrature formula. In fact we can scale the value of the integral on these two intervals proportionally to the actual length of the intervals themselves.
If we take into account all the aforementioned details we can reproduce a matrix $W_m$ which is row stochastic.
We observe that in the implemented code we simply scale each row of $W_m$ by its sum so that the matrix becomes row stochastic.
\subsection{Spectrum of $W_m$}
Since $W_m\in\R^{n\times n}$ represents the discrete convolution operator, it can be a circulant matrix, Toeplitz matrix or it can have a more complex structure. Its structure depends on the way we extend the signal outside its boundaries.
From now on we assume for simplicity that $n$ is an odd natural number, and that we have periodical extension of signals outside the boundaries, therefore $W_m$ is a circulant matrix given by
\begin{equation}\label{eq:W_m}
W_m=\left[
\begin{array}{cccc}
c_0 & c_{n-1} & \ldots & c_1 \\
c_{1} & c_0 & \ldots & c_2 \\
\vdots & \vdots & \ddots & \vdots \\
c_{n-1} & c_{n-2} & \ldots & c_0 \\
\end{array}
\right]
\end{equation}
where $c_j\geq 0$, for every $j=0,\ldots, \ n-1$, and $\sum_{j=0}^{n-1}c_j=1$.
Each row contains a circular shift of the entries of a chosen vector filter $w_m$. For the non periodical extension case we refer the reader to \cite{cicone2017BC}.
Denoting by $\sigma(W_m)$ the spectrum of the matrix, in the case of a circulant matrix it is well known that the eigenvalues $\lambda_j\in\sigma(W_m)$, $j=0,\ldots, \ n-1$ are given by the formula
\begin{equation}\label{eq:eig}
\lambda_j=c_0+c_{n-1}\omega_j+\ldots+c_1 \omega_j^{n-1},\quad \textrm{ for } \qquad j=0,\ldots, \ n-1
\end{equation}
where $i=\sqrt{-1}$, and $\omega_j=e^\frac{2\pi i j}{n}$ $j$--th power of the $n$--th root of unity, for $j=0,\ldots, \ n-1$.
Since we construct the matrices $W_m$ using symmetric filters $w_m$, we have that $c_{n-j}=c_j$ for every $j=1,\ldots,\frac{n-1}{2}$. Hence $W_m$ is circulant, symmetric and
\begin{equation*}
\lambda_j=c_0+c_{1}\left(\omega_j+\omega_j^{n-1}\right)+c_{2}\left(\omega_j^2+\omega_j^{n-2}\right)\ldots+
c_{\frac{n-1}{2}}\left(\omega_j^\frac{n-1}{2}+\omega_j^{\frac{n+1}{2}}\right)=
\end{equation*}
\begin{equation*}
c_0+\sum_{k=1}^{\frac{n-1}{2}}c_{k}\left(\omega_j^k+\omega_j^{n-k}\right)=c_0+\sum_{k=1}^{\frac{n-1}{2}}c_{k}\left(e^{\frac{2\pi i j}{n}k}+e^{\frac{2\pi i j}{n}(n-k)}\right)=
\end{equation*}
\begin{equation}\label{eq:eig1}
c_0+\sum_{k=1}^{\frac{n-1}{2}}c_{k}\left(e^{\frac{2\pi i j}{n}k}-e^{\frac{2\pi i j}{n}k}e^{2\pi i j}\right)
\end{equation}
Therefore
\begin{equation}\label{eq:eig2}
\lambda_j = c_0+2\sum_{k=1}^{\frac{n-1}{2}}c_{k} \cos\left(\frac{2\pi j k}{n}\right),\quad \textrm{ for } \qquad j=0,\ldots, \ n-1
\end{equation}
It is evident that, for any $j=0,\ldots, \ n-1$, $\lambda_j$ is real and $\sigma(W_m)\subseteq [-1,\ 1]$ since $W_m$ is a stochastic matrix.
Furthermore, if we make the assumption that the filter half supports length is always $l_m\leq \frac{n-1}{2}$, then the entries $c_j$ of the matrix $W_m$ are going to be zero at least for any $j\in[\frac{n-1}{4},\frac{3}{4}(n-1)]$.
We observe that the previous assumption is reasonable since it implies that we can study oscillations with periods at most equal to half of the length of a signal.
\begin{theorem}\label{thm:Spectra}
Considering the circulant matrix $W_m$ given in \eqref{eq:W_m}, assuming that $n>1$, $\sum_{j=0}^{n-1}c_j=1$, $c_j\geq 0$, and $c_{n-j}=c_j$, for every $j=1,\ldots,n-1$.
Then $W_m$ is non--defective, diagonalizable and has real eigenvalues.
Furthermore, if the filter half supports length $l_m$ is small enough so that $c_0=1$ and $c_j=0$, for every $j=1,\ldots, \ n-1$, then we have $n$ eigenvalues $\lambda_j$ all equal $1$.
Otherwise, if the filter half supports length $l_m$ is big enough so that $c_0<1$ and the values $c_k$ correspond to the discretization of a function with compact and connected support, then there is one and only one eigenvalue equal to $1$, which is $\lambda_0$, all the other eigenvalues $\lambda_j$ are real and strictly less than one in absolute value. So they belong to the interval $(-1,1)$.
\end{theorem}
\begin{proof}
First of all we recall that symmetric matrices are always non--defective, diagonalizable and with a real spectrum.
In the case of $c_0=1$ the conclusion follows immediately from the observation that $W_m$ reduces to an identity matrix.
When $c_0<1$ from \eqref{eq:eig2} it follows that $\lambda_0=1$ and all the other eigenvalues belong to the interval $[-1,1]$. Let us assume, by contradiction, that there exists another eigenvalue $\lambda_d=1$ for some $d\in\left\{1,\ 2,\ldots,\ n-1\right\}$. We assume for simplicity that $n$ is odd. The proof in the even case works in a similar way.
From \eqref{eq:eig2} and the fact that $c_{n-j}=c_j$, for every $j=1,\ldots,\frac{n-1}{2}$, it follows that
\begin{equation}\label{eq:lambda_d}
\lambda_d = c_0+2\sum_{k=1}^{\frac{n-1}{2}}c_{k} \cos\left(\frac{2\pi d k}{n}\right),\quad \textrm{ for } \qquad d\in\left\{1,\ 2,\ldots,\ n-1\right\}
\end{equation}
In the right hand side we have among the terms $c_{k}$, which by themselves would add up to 1, at least $c_1>0$ which is multiplied by $\cos\left(\frac{2\pi d}{n}\right)<1$ for any $d\in\left\{1,\ 2,\ldots,\ n-1\right\}$. Therefore the right hand side will never add up to 1. Hence we have a contradiction.
From \eqref{eq:eig2} it follows also that $\lambda_d\neq -1$ for any $d\in\left\{1,\ 2,\ldots,\ n-1\right\}$ because $\lambda_d$ is given by a convex combination of cosines and $+1$.
So all the eigenvalues of $W_m$ except $\lambda_0$ are real and strictly less than one in modulus.
\end{proof}
We observe that in the discrete iterative filtering algorithm the entries $c_k$ derive from the discretization of a filter function which is by Definition \ref{def:window} compactly supported. Furthermore, since the filter function is used to compute the moving average of a signal, it is reasonable to require its support to be connected.
Form this theorem it follows that
\begin{corollary}\label{cor:SpectraDoubleConvolution}
Considering the matrix $W_m$ given in the previous theorem, assuming $c_0<1$ and that $W_m$ is constructed using a filter $w_m$ that is produced as convolution of a symmetric filter $h_m$ with itself, then there is one and only one eigenvalue equal to $1$, all the other eigenvalues belong to the interval $[0,1)$.
\end{corollary}
\begin{proof}
The proof follows directly from the previous theorem and the fact that the matrix $W_m=\widetilde{W}_m^T*\widetilde{W}_m=\widetilde{W}_m^2$, where $\widetilde{W}_m$ is a circulant symmetric convolution matrix associated with the filter $\widetilde{w}_m$.
\end{proof}
\begin{corollary}\label{cor:Kernel_I-W}
Assuming $c_0<1$, the eigenvector of $W_m$ corresponding to $\lambda_0=1$ is a basis for the kernel of the matrix $(I-W_m)$, which has dimension one.
\end{corollary}
Before presenting the main proposition we recall that, given a circulant matrix $C=\left[c_{pq}\right]_{p,\ q=0,\ldots,n-1}$, its eigenvalues are
\begin{equation}\label{eq:Lambdas}
\lambda_p\ =\ \sum_{q=0}^{n-1} c_{1q}e^{-2\pi i p \frac{q}{n}} \qquad \qquad p\ =\ 0,\ldots,\ n-1
\end{equation}
and the corresponding eigenvectors are
\begin{equation}\label{eq:Eigenvectors}
u_p\ =\ \frac{1}{\sqrt{n}} \left[1,\ e^{-2\pi i p\frac{1}{n}},\ldots,\ e^{-2\pi i p\frac{n-1}{n}}\right]^T \qquad \qquad p\ =\ 0,\ldots,\ n-1
\end{equation}
which form an orthonormal set.
We recall that an eigenvalue of a matrix is called semisimple whenever its algebraic multiplicity coincides with its geometric multiplicity.
\begin{proposition}\label{pro:LimitValue}
Given a matrix $W_m$, assuming that all the assumptions of Theorem \ref{thm:Spectra} and Corollary \ref{cor:SpectraDoubleConvolution} hold true,
and assuming that $W_m=W$ for any $m\geq 1$. Given $\left\{\lambda_p\right\}_{p=0,\ldots,n-1}$, semisimple eigenvalues of $W$, and the corresponding eigenvectors $\left\{u_p\right\}_{p=0,\ldots,n-1}$, we define the matrix $U$ having as columns the eigenvectors $u_p$. Assuming that $W$ has $k$ zero eigenvalues, where $k$ is a number in the set $\in\{0,\ 1,\ldots,\ n-1\}$,
Then
\begin{equation}\label{eq:discreteLimit}
\lim_{m\rightarrow \infty}(I-W)^m=U Z U^T
\end{equation}
where $U$ is unitary and $Z$ is a diagonal matrix with entries all zero except $k$ elements in the diagonal which are equal to one.
\end{proposition}
\begin{proof}
From Theorem \ref{thm:Spectra} we know that $W$ is diagonalizable, therefore the matrix $U$ is orthogonal and all the eigenvalues of $W$ are semisimple. Furthermore, since the eigenvectors of $W$ are orthonormal, it follows that $U$ is a unitary matrix. Hence $W=U D U^T$, where $D$ is a diagonal matrix containing in its diagonal the eigenvalues of $W$.
From the assumption that $W$ is associated with a double convolved filter it follows that the spectrum of $W$ is contained in $[0,1]$, ref. Corollary \ref{cor:SpectraDoubleConvolution}. Therefore also the spectrum of $(I-W)$ is contained in $[0,1]$. Furthermore
\begin{equation*}
(I-W)=U(I-D)U^T
\end{equation*}
and $I-D$ is a diagonal matrix whose diagonal entries are in the interval $(0,1)$ except the first one which equals 0, ref. Corollary \ref{cor:Kernel_I-W}, and $k$ entries that are equal to 1.
Hence
\begin{equation*}
\lim_{m\rightarrow \infty}(I-W)^m=\lim_{m\rightarrow \infty}U(I-D)^m U^T= U Z U^T
\end{equation*}
where $Z$ is a diagonal matrix with entries all zero except $k$ elements in the diagonal which are equal to one.
\end{proof}
From the previous proposition it follows
\begin{corollary}\label{cor:ExplicitFormulaDiscreteIMF}
Given a signal $s\in\R^n$, assuming that we are considering a doubly convolved filter, and the half filter support length is constant throughout all the steps of an inner loop,
Then the first outer loop step of the DIF method converges to
\begin{equation}\label{eq:discreteIMF}
\textrm{IMF}_1=\lim_{m\rightarrow \infty}(I-W)^m s=U Z U^Ts
\end{equation}
\end{corollary}
So the DIF method in the limit produces IMFs that are projections of the given signal $s$ onto the eigenspace of $W$ corresponding to the zero eigenvalue which has algebraic and geometric multiplicity $k\in\{0,\ 1,\ldots,\ n-1\}$. Clearly, if $W$ has only a trivial kernel then the method converges to the zero vector. We point out that since (\ref{eq:Lambdas}) is also the Discrete Fourier Transform (DFT) formula of the sequence $\{c_{1q}\}_{q=0,\ldots,n-1}$, where $C=[c_{pq}]$ is a circulant matrix, it follows that the eigenvalues of $W$, can be computed directly as the DFT of the sequence $\{w_{1q}\}_{q=0,\ldots,n-1}$, by means of the Fast Fourier Transform (FFT). If we regard the DFT as a discretization of the Fourier Transform of the filter function $w$ it becomes clear that, since the latter has only isolated zeros, in many cases we will not have eigenvalues exactly equal to zero. So in general $W$ has only a trivial kernel and \eqref{eq:discreteIMF} converges to the zero vector. In order to ensure that the method produces a non zero vector we need to discontinue the calculation introducing some stopping criterion.
\subsection{DIF inner and outer loop convergence in presence of a stopping criterion}
If we assume that the half support length $l_m$ is computed only in the beginning of each inner loop, then the first IMF is given by \eqref{eq:First_IMF_fixed_length} and \eqref{eq:discreteIMF}.
In order to have a finite time method we may introduce a stopping criterion in the DIF algorithm, like the condition
\begin{equation}\label{eq:Discrete_Abs_StopCond}
\|s_{m+1}-s_m\|_2<\delta \qquad \forall m\geq N_0
\end{equation}
for some fixed $\delta>0$
Then, based on Corollary \ref{cor:ExplicitFormulaDiscreteIMF}, we produce an approximated first IMF given by
\begin{equation}\label{eq:approx_first_IMF}
\overline{\textrm{IMF}}_1=(I-W)^{N_0} s = U(I-D)^{N_0} U^T s
\end{equation}
\begin{theorem}\label{thm:DIF_conv_stop}
Given $s\in\R^n$, we consider the convolution matrix $W$ defined in \eqref{eq:W_m}, associated with a filter vector $w$ given as a symmetric filter $h$ convolved with itself. Assuming that $W$ has $k$ zero eigenvalues, where $k$ is a number in the set $\in\{0,\ 1,\ldots,\ n-1\}$, and fixed $\delta>0$,
Then, calling $\widetilde{s}=U^T s$, for the minimum $N_0\in\N$ such that it holds true the inequality
\begin{equation}\label{eq:N0_discrete}
\frac{N_0^{N_0}}{\left(N_0+1\right)^{N_0+1}}<\frac{\delta}{\|\widetilde{s}\|_\infty{\sqrt{n-1-k}}}
\end{equation}
we have that $\left\| s_{m+1}-s_m\right\|_{2}<\delta \quad \forall m\geq N_0$ and the first IMF is given by
\begin{equation}\label{eq:IMF1_direct}
\overline{\textrm{IMF}}_1=U(I-D)^{N_0} U^T s= U P \left[
\begin{array}{ccccccc}
0 & & & & & & \\
& (1-\lambda_1 )^{N_0} & & & & & \\
& & \ddots & & & & \\
& & & (1-\lambda_{n-1-k} )^{N_0} & & & \\
& & & & 1 & & \\
& & & & & \ddots & \\
& & & & & & 1 \\
\end{array}
\right] P^T U^T s
\end{equation}
where $P$ is a permutation matrix which allows to reorder the columns of $U$, which correspond to eigenvectors of $W$, so that the corresponding eigenvalues $\{\lambda_p\}_{p=1,\ldots,\ n-1}$ are in decreasing order.
\end{theorem}
\begin{proof}
\begin{eqnarray}
\nonumber \|s_{m+1}-s_m\|_2 &=& \|(I-W)^{m+1}-(I-W)^{m}\|_2=\|U(I-D)^{m}(I-D-I)U^Ts\|_2= \\
\|(I-D)^{m}(I-D-I)U^Ts\|_2 &=& \|(I-D)^{m}(I-D-I)\widetilde{s}\|_2
\end{eqnarray}
since $U$ is a unitary matrix and where $\widetilde{s}=U^T s$.
Given a permutation matrix $P$ such that the entries of the diagonal $PDP^T$ are the eigenvalues of $W$ in decreasing order of magnitude, starting from $\lambda_0=1$, and assuming that $W$ has $k$ zero eigenvalues, where $k$ is a number in the set $\in\{0,\ 1,\ldots,\ n-1\}$, then
\begin{eqnarray}
\nonumber \|(I-D)^{m}(I-D-I)\widetilde{s}\|_2 & \leq & \left\|P\left[
\begin{array}{ccccccc}
0 & & & & & & \\
& (1-\lambda_1 )^m \lambda_1 & & & & & \\
& & \ddots & & & & \\
& & & (1-\lambda_{n-1-k} )^m \lambda_{n-1-k} & & & \\
& & & & 0 & & \\
& & & & & \ddots & \\
& & & & & & 0 \\
\end{array}
\right]P^T\left[
\begin{array}{c}
\|\widetilde{s}\|_\infty \\
\vdots \\
\|\widetilde{s}\|_\infty \\
\end{array}
\right]
\right\|_2 \\
&\leq & {\sqrt{n-1-k}} \left(1-\frac{1}{m+1} \right)^m \frac{1}{m+1} \|\widetilde{s}\|_\infty = {\sqrt{n-1-k}} \frac{m^m}{(m+1)^{m+1}} \|\widetilde{s}\|_\infty
\end{eqnarray}
because the function $(1-\lambda)^m \lambda$ achieves its maximum at $\lambda=\frac{1}{m+1}$ for $\lambda\in[0,\ 1]$.
Hence the stopping criterion \eqref{eq:Discrete_Abs_StopCond} is fulfilled for $N_0$ minimum natural number such that \ref{eq:N0_discrete} holds true.
\end{proof}
We observe that, as we mentioned earlier, since (\ref{eq:Lambdas}) is also the Discrete Fourier Transform (DFT) formula of the sequence $\{c_{1q}\}_{q=0,\ldots,n-1}$, it follows that the eigenvalues of $W=\left[w_{pq}\right]_{p,\ q=0,\ldots,n-1}$, can be computed directly as the DFT of the sequence $\{w_{1q}\}_{q=0,\ldots,n-1}$, by means of the Fast Fourier Transform (FFT). This calculation can be done ``off line'', in fact, once the filter shape $w$ has been fixed, we can compute and store its FFT for different values of the size of its support. This fact, together with other previous results, can be used to improve the efficiency of the method as explained in the following section.
It is interesting to notice that each IMF is generated as a linear combination of elements in an orthonormal basis. Therefore we can regard the IMFs as elements of a frame which allows to decompose a given signal into a few significant components. From this prospective the IF algorithm can be viewed as a method that automatically produces elements of a frame associated with a signal. The possible connections between IF and the frame theory are fascinating, but out of the scope of the present work. We plan to follow this direction of research in a future work.
Regarding the DIF outer loop convergence they hold true the same results described in Section \ref{subsec:outerLoopIF} for the continuous setting. In fact, while the inner loop of the IF algorithm requires a discretization to deal with discrete signals, the outer loop does not require any form of discretization and it works the very same as in the continuous setting.
\section{Efficient implementation of the DIF algorithm}\label{sec:speedup}
In this section we want to review some ideas for an efficient implementation of the DIF algorithm applied to the decomposition of a signal $s$ of length $n$.
We underline that the following ideas apply only for periodical extension of the signal at the boundaries.
We start from Theorem \ref{thm:DIF_conv_stop} which allows to compute each IMF as fast as the FFT of a signal of length $n$.
The first idea is to precompute the number of iterations needed to achieve the required accuracy $\delta$ in the computation of a certain IMF. This number of iterations can be approximated by the minimum $N_0\in\N$ satisfying the inequality \eqref{eq:N0_discrete}. Then we can compute the IMF using \eqref{eq:IMF1_direct} where the eigenvalues $\{\lambda_k\}_{k=1,\ 2,\ldots,\ n}$ can be evaluated using \eqref{eq:Lambdas}, or by means of the Fast Fourier Transform since \eqref{eq:Lambdas} is equivalent to the Discrete Fourier Transform of the sequence $\{w_{1q}\}_{q=0,\ldots,n-1}$. Furthermore we recall that $U^Ts$ is the DFT of $s$ that can be computed using the FFT algorithm, whose computational complexity is $n\log(n)$, and that multiplying on the left by the matrix $U$ is equivalent to computing the Inverse DFT (IDFT) which can be done using the inverse FFT. Hence the IMF can be computed in one step as
\begin{equation}\label{eq:One step_algo}
\textrm{IMF} = \sum_{k=0}^{n-1} u_k (1-\lambda_k)^{N_0}\sigma_k = \textrm{IDFT}\left((I-D)^{N_0}\textrm{DFT}(s)\right)
\end{equation}
where $\sigma_k$ represents the $k$-th element of the DFT of the signal $s$.
The proposed a priori calculation of $N_0\in\N$ as the minimum value satisfying the inequality \eqref{eq:N0_discrete} is fast and easy, but provides only with an overestimation of the real number of iterations required. In order to compute the actual number of iterations required we can compute \eqref{eq:One step_algo} for subsequently bigger values of $N_0\in\N$ and stop whenever the quantity $SD$ defined in \eqref{eq:SD} is less or equal to $\delta$.
This is done in the so called Fast Iterative Filtering (FIF) method implemented for Matlab and available online\footnote{\url{www.cicone.com}}. By exploiting the FFT we speed up the calculations significantly. For a vector of tenths of millions of points the computational time passes from roughly two days for the standard IF to less than an hour on a personal computer with the FIF algorithm.
We point out that we can also precompute the eigenvalues $\lambda_k$ corresponding to any possible scaling of a filter $w$. In doing so we can reduce even further the computational time of the algorithm.
\section{Conclusions and Outlook}\label{sec:Conclusions}
In this work we tackle the problem of a complete analysis of the IF algorithm both in the continuous and discrete setting. In particular in the continuous setting we show how IF can decompose a signal into a finite number of so called IMFs and that each IMF contains frequencies of the original signal filtered in a ``smart way''.
In the discrete setting we prove that the DIF method is also convergente and, in the case of periodical extension at the boundaries of the given signal, we provide an explicit formula for the a priori calculation of each IMF. From this equation it follows that each IMF is a smart summation of eigenvectors of a circulant matrix.
We show that no fake oscillations can be produced neither in the continuous nor in the discrete setting.
From the properties of the DIF algorithm and the explicit formula for the IMFs produced by this method and derived in this work, we propose new ideas that has been directly incorporated in the implemented algorithm in order to increase its efficiency and reduce its computational complexity. The result is the so called FIF method\footnote{\url{www.cicone.com}} which allows to quickly decompose a signal by means of the FFT. This is an important result in this area of research which opens the doors to an almost instantaneous analysis of non stationary signals.
There are several open problems that remain unsolved. First of all from the proposed analysis it is clear that different filter functions have different Fourier transform and hence the decomposition produced by IF and DIF algorithms is directly influenced by this choice. In a future work we plan to study more in details the connections between the shape of the filters and the quality of the decomposition produced by these methods.
In the current work we analyzed the DIF assuming a periodical extension of the signals at the boundaries. We plan to study in a future work the behavior of the DIF method in the case of reflective, antireflective and other boundaries extensions of a signal.
Based on the numerical evidence \cite{cicone2016adaptive,cicone2017multidimensional} we claim that the Iterative Filtering method is stable under perturbations of the signal. We plan to study rigorously such stability in a future work.
The results about the DIF algorithm convergence suggest that the method allows, in general, to automatically generate a frame associated with a given signal. We plan to further analyze this connection in a future work.
Finally we recall that it is still an open problem how to extend all the results obtained for the Iterative Filtering technique to the case of the Adaptive Local Iterative Filtering method, whose convergence and stability analysis is still under investigation \cite{cicone2017spectral,cicone2016adaptive,cicone2017Geophysics}.
\section*{Acknowledgments}
This work was supported by NSF Awards DMS--1620345, DMS--1830225, ONR Award N00014--18--1--2852, the Istituto Nazionale di Alta Matematica (INdAM) ``INdAM Fellowships in Mathematics and/or Applications cofunded by Marie Curie Actions'', FP7--PEOPLE--2012--COFUND, Grant agreement n. PCOFUND--GA--2012--600198.
|
2,869,038,155,723 | arxiv | \section{Introduction}
\label{sec:intro}
The understanding of the structure of nucleons is one of the most important and interesting research subject in modern nuclear physics. The ultimate goal would be to have a complete description of quarks/gluons position and momenta inside a hadron, which is not easy because of the entanglement of initial/final states in all hadronic processes. In order to properly define all hadron constituent contributions, the cross sections should be factorized in some region of the phase space into properly defined hadronic matrix elements. Here we will consider the transverse momentum dependent distributions (TMD), which appear in the factorization of several processes like Drell-Yan, semi-inclusive deep-inelastic-scattering (SIDIS) and $e^+e^-$ hadron production \cite{Collins:2011zzd,GarciaEchevarria:2011rb,Echevarria:2014rua,Gaunt:2014ska,Vladimirov:2017ksc}. Drell-Yan processes directly test the TMD parton distribution functions (TMDPDF), while in SIDIS cross sections the TMDPDFs are coupled to a TMD fragmentation function in the final state. Finally in $e^+e^-$ hadron production only the TMD fragmentation is present. Because of the factorization theorem, the TMDs have several universal features like rapidity and renormalization scale evolution, which should be also tested including their (universal) non-perturbative part. Recently some of us have considered the possibility to define a jet-TMD, replacing a final state hadron with a jet \cite{Neill:2016vbi,Gutierrez-Reyes:2018qez,Gutierrez-Reyes:2019vbx} in SIDIS and $e^+e^-$ processes. The check of this possibility has revealed that standard jet definitions are compatible with a factorization theorem only in the case of small enough radii, which is a not obvious experimental condition in the planned electron-hadron collider like EIC or LHeC. Instead large jet-radii need a specific definition of jet, which allows soft radiation to be independent of radius. In \cite{Gutierrez-Reyes:2018qez,Gutierrez-Reyes:2019vbx} this was achieved using the winner-take-all (WTA) axis \cite{Bertolini:2013iqa}, and the perturbative calculations were done with a precision similar to the case of fragmenting hadrons.
In this work we consider the possibility of groomed jets in SIDIS or $e^+e^-\to \text{2\;jets}$. Developments in jet substructure have shown that applying a grooming algorithm to a jet, using for example the so called ``soft-drop'' procedure, robustly removes the contamination from both underlying event and non-global correlations. Since this process essentially removes wide angle soft radiation, retaining only a collinear core, it also dramatically reduces hadronization effects (see fig.~\ref{fig:hadronization}), thus allowing an easier access to the TMD non-perturbative physics which we want to probe. Groomed jets with an identified light/heavy hadron in the jet were also proposed as probes of TMD evolution and distribution in \cite{Makris:2017arq,Makris:2018npl}. The residual non-perturbative effects contain pieces that depend on the soft-drop grooming procedure and require careful analysis as was pointed out in \cite{Hoang:2019ceu}. In addition, with the use of soft-drop we can derive factorization theorems for large jet radius ($R\sim 1$), which we consider to be the relevant case for low energy experiments, such as EIC. In order to focus on collimated jet configurations, we also impose an upper cutoff in the groomed jet invariant mass.\footnote{Note that the small transverse momentum constraint does not necessarily ensure collimated configurations since topologies with two or more widely separated sub-jets are also permitted.} This constraint allows us to derive a factorization theorem involving the same universal soft function that appears in traditional hadronic TMD, and which is independent of the jet radius for $R\sim 1$. This is a key feature for groomed jets and it is necessary for the universality of TMDs and for this reason, in this paper, we only consider $R\sim1$. The cutoff is imposed using groomed jet-thrust, $e \equiv (m/Q)^2$, where $m$ is the groomed invariant mass and $Q$ is the center of mass energy. This allows us to introduce a single cutoff parameter, $e_{\text{cut}}$, independent of the jet energy or transverse momentum.
\begin{figure}[t!]
\centerline{\includegraphics[width = \textwidth]{ratios}}
\caption{Hadronization effects in a typical $e^+e^-\to \text{2\;jets}$ from \textsc{Pythia} 8~\cite{Sjostrand:2006za,Sjostrand:2007gs}.
at center of mass values, \textsc{Left}: $Q=100$ GeV, \textsc{Right}: $Q=50$ GeV.}
\label{fig:hadronization}
\end{figure}
The paper is organized as follows. In sec.~\ref{sec:main-1}, we give a review of soft-drop and discuss the factorization of the $e^+e^-\to \text{2\;jets}$, transverse momentum decorrelation within SCET and give detailed comparisons of our NNLL accurate prediction with simulations for this observable. In sec.~\ref{DIS-jet}, we consider the factorization for the corresponding observable in DIS. We carefully enumerate all the non-perturbative corrections and discuss their universality in sec.~\ref{hadronization}. We conclude in sec.~\ref{conclusion}. The details of the one loop calculations, resummation, and evolution are provided in the appendix.
\section{Di-jet events in electron-positron colliders}
\label{sec:main-1}
In this section we discuss the measurement of momentum de-correlation in electron-positron colliders. We identify events with two final state jets and we consider the transverse momentum of one jet w.r.t. the other. The measurement that we are considering in this work is a generalization of the di-hadron momentum de-correlation,
\begin{equation}
\bmat{q}_T = \frac{\bmat{p}_{T h_1}}{z_1} + \frac{\bmat{p}_{T h_2}}{z_2}
\label{observable}
\end{equation}
where one or both of the identified hadrons is replaced by a jet, defined through an infrared-safe jet algorithm. Here $\bmat{p}_{T h_i}$ and $z_{i}$ are the transverse momentum and energy fraction of the hadron $i$ respectively. The factorization theorem is usually written for this normalized vector sum of the transverse momenta rather than just the sum of the transverse momenta. It can be verified by momentum conservation and simple geometry that the quantity $\bmat{p}_{T h_1}/z_1$ represents the transverse momentum of the radiation recoiling against the hadron w.r.t the axis defined by the hadron itself. This makes it convenient to write a factorization theorem which matches onto the standard hadron fragmentation function as explained in \cite{Makris:2017arq}.
We consider three possible scenarios as illustrated in fig.~\ref{fig:measurment} and we refer to them as di-hadron, hadron-jet, and di-jet momentum de-correlation. To simplify the discussion we focus on the case of di-jets (fig.~\ref{fig:measurment}c) and we briefly comment how our results are generalized for the case of hadron-jet de-correlation. For the case of groomed jets the observable $\bmat{q}_T$ is defined with the groomed quantities, i.e., $p_{\text{jet}}^{\mu}$ is the \emph{groomed} jet four-momentum and $z_{\text{jet}} = 2 p^{0}_{\text{jet}}/Q$. The transverse component $\bmat{p}_{T\;\text{jet}}$ is measured with respect to an axis close to the full or groomed jet axes. The exact choice of the axis only differs by power corrections. For concreteness in the results that follow we make the choice of the axis to lie along one of the groomed jets.
\begin{figure}[t!]
\centerline{\includegraphics[width = \textwidth]{measurment}}
\caption{Three possible transverse momentum de-correlation measurements in $e^+ e^-$ annihilation: (a) Identify two hadrons $h_1$ and $h_2$ with momenta $p_{h_1}, p_{h_2}$ and energy fractions $z_{h_1},z_{h_2}$ respectively, (b) Identify a jet and a hadron with momenta $p_{\text{jet}}$, $p_{h}$ with energy fractions $z_{\text{jet}}, z_h$, (c) Identify two jets with momenta $p_{\text{jet}_1},p_{\text{jet}_2}$ and energy fractions $z_{\text{jet}_1}, z_{\text{jet}_2}$.}
\label{fig:measurment}
\end{figure}
Since we want to probe the non-perturbative physics, we wish to work in the small transverse momentum regime ($q_T \ll \sqrt{s}$ where $q_T \equiv \vert \pmb{q}_T \vert$). There are various ways one can define the jet axis and the choice of definition will impact the form of factorization. It was discussed in ref.~\cite{Gutierrez-Reyes:2018qez} that the standard jet axis choice suffers from factorization breakdown for large jet radius. This breakdown is due to energetic emissions at relatively wide angles. Such configurations will contribute to the small transverse momentum region when the energetic subjets are clustered in a single large radius jet. To avoid this problem in refs.~\cite{Gutierrez-Reyes:2018qez, Gutierrez-Reyes:2019vbx} the winner-take-all (WTA) axis was used instead. This way ensures that wide angle energetic emissions induce large transverse momentum ($q_T \sim \sqrt{s}$) pushing the $q_T$ measurement away from the observable region.
In this paper we propose, alternatively, the use of groomed jet-substructure to isolate the collimated configurations and choose the jet axis to be the groomed jet axis which is insensitive to jet boundary effects. Particularly we consider the normalized jet mass as the relevant jet-substructure observable,
\begin{equation}
e\equiv \Big{(}\frac{m_J}{Q}\Big{)}^2\ .
\label{jetMass}
\end{equation}
We shall see that imposing this constraint still allows us to capture a majority of events and hence does not significantly impact the cross-section.
\subsection{Soft-drop: a brief review}
\label{sec:review}
The grooming procedure that we use is the soft-drop algorithm. We give here a brief review of the soft-drop groomer and eventually discuss the various hierarchies, the relevant modes and the factorization of the cross section in the next sections.
Soft-drop grooming \cite{Larkoski:2014wba} removes contaminating soft radiation from the jet by constructing an angular ordered tree of the jet, and removing the branches at the widest angles which fail an energy requirement. The angular ordering of the jet is constructed through the Cambridge/Aachen (C/A) clustering algorithm \cite{Ellis:1993tq,Catani:1993hr,Dokshitzer:1997in,Wobisch:1998wt,Wobisch:2000dk}. As soon as a branch is found that passes the test, it is declared the groomed jet, and all the constituents of the branch are the groomed constituents. At the end of the grooming procedure only the narrow energetic core remains from the original jet. Since at large angles all collinear energetic radiation is to be found at the center of the jet, no cone is actually imposed to enclose this core. One simply finds the branch whose daughters are sufficiently energetic. Formally the daughters could have any opening angle, though their most likely configuration is collinear.
The strict definition of the algorithm is as follows. Given an ungroomed jet (which itself is identified first using a suitable algorithm such as the anti-$k_T$, \cite{Cacciari:2008gp}), first we build the clustering history by starting with a list of particles in the jet. At each stage we merge the two particles within the list that are closest in angle\footnote{This merging is usually taken to be summing the momenta of the particles, though one could use winner-take-all schemes \cite{Salam:WTAUnpublished, Bertolini:2013iqa,Larkoski:2014uqa}.}. This gives a pseudo-particle, and we remove the two daughters from the current list of particles, replacing them with the merged pseudo-particle. This is repeated until all particles are merged into a single parent. Then we open the tree back up working backwards so that at each stage of the declustering, we have two branches available, label them $i$ and $j$. We require:
\begin{align}
\label{SD:condition}
\frac{\text{min}\{E_i,E_j\}}{E_i+E_j}> z_{\text{cut}} \Big{(} \frac{\theta_{ij}}{R} \Big{)}^{\beta},
\end{align}
where $z_{\text{cut}}$ is the modified mass drop parameter, $\beta$ is the parameter which controls the angularities, $\theta_{ij}$ is the angle between $i^{th}$ and $j^{th}$ particle, $R$ is the jet radius and $E_i$ is the energy of the branch $i$. If the two branches fail this requirement, the softer branch is removed from the jet, and we decluster the harder branch, once again testing eq.~(\ref{SD:condition}) within the hard branch. The pruning continues until we have a branch that when declustered passes the condition eq.~(\ref{SD:condition}). All particles contained within this branch whose daughters are sufficiently energetic constitute the groomed jet. Intuitively we have identified the first genuine collinear splitting.
For a hadron-hadron collision, one uses the transverse momentum $(p_T)$ with respect to the beam for the condition of eq.~(\ref{SD:condition}),
\begin{align}\label{SD:condition_pp}
\frac{\text{min}\{p_{Ti},p_{Tj}\}}{p_{Ti}+p_{Tj}}> z_{\text{cut}} \Big{(} \frac{\theta_{ij}}{R} \Big{)} ^{\beta}.
\end{align}
We formally adopt the power counting $z_{\text{cut}} \ll 1$, though typically one chooses $z_{\text{cut}} \sim 0.1$. See \cite{Marzani:2017mva} for a study on the magnitude of the power corrections with respect to $z_{\text{cut}}$ for jet mass distributions. To be specific, in this paper we consider only the case $\beta = 0$. Note that for $\beta =0$ the energy of the groomed jet constituents is a collinear unsafe observable~\cite{Larkoski:2014wba, Larkoski:2015lea}, however, the additional constraint of the measured transverse momentum $\bmat{q}_T$ provides a physical collinear cutoff in a similar way a jet shape measurement does. For detailed discussion on this we refer to appendix~\ref{app:IRCsafe}.
\subsection{Hierarchies, modes, and factorization}
\label{sec:hierarchy}
In order to compute the transverse momentum de-correlation $\bmat{q}_T$, defined in eq.~(\ref{observable}), for two groomed jets in di-jet events in $e^+ e^-$ annihilation (fig.~\ref{fig:measurment} (c))
we are going to impose a normalized jet mass measurement as defined in
eq.~(\ref{jetMass}) on both jets. The other parameters that enter our cross section are the soft-drop parameters $z_{\text{cut}} \sim 0.1$, $\beta=0$. Ultimately we are going to integrate over the jet mass measurement up to an appropriate (but still small) cut-off value $e_{\text{cut}}$. \\
We have a rich spectrum of possible hierarchies of momenta, which are all consistent with maintaining $ q_T/Q , e_{\text{cut}}$, $z_{\text{cut}}\ll 1 $. We have that $ q_T/Q , e_{\text{cut}}$, $z_{\text{cut}} $ are now expansion parameters in the effective field theory (EFT), and they should be taken into account in the factorization of the process. We first list and briefly discuss these hierarchies and the corresponding factorization theorems within an EFT. The general modes that we will consider will fall into three classes. Modes that explicitly pass soft drop (usually the highly energetic collinear modes), modes that explicitly fail soft-drop (the global soft function modes) and finally those which can live on the border and need to be tested, as to whether they pass or fail. Only the modes that pass soft-drop will contribute to $e$, while $\bmat{q}_T$ receives contributions from all radiation that fails soft-drop.
The first regime in which we are interested is
$Q\gg Q z_{\text{cut}} \gg q_T \gtrsim Q\sqrt{e} \gg Q\sqrt{e z_{\text{cut}}}$. Here we have low values of $q_T$ which are of the order of $Q\sqrt{e}$. We identify the following modes to be relevant to the cross section:
\begin{align}
\text{soft:}& \;\;\; p_{s}^{\mu} \sim q_T(1,1,1); \nonumber\\
\text{collinear:}& \;\;\; p_{c}^{\mu} \sim Q(\lambda_c^{2},1,\lambda_c) ,\; \lambda_c = \sqrt{e},
\end{align}
and the factorization of the cross section in this region is schematically
\begin{equation}
\label{eq:fact-II}
\frac{d\sigma}{ de_1 de_2 d\pmb{q}_{T} } = H^{ij}_2(Q;\mu) \times S(\pmb{q}_T) \otimes \mathcal{J}^{\perp}_{i}(e_1, Q,z_{\text{cut}},\pmb{q}_T) \otimes \mathcal{J}^{\perp}_{j}(e_2, Q,z_{\text{cut}},\pmb{q}_T).
\end{equation}
Apart from the hard factor $H$ all the other terms in this equation are affected by rapidity divergences.
The global soft function $S$ that appears in the factorization theorem in eq.~(\ref{eq:fact-II}) (and later in the SIDIS case eq.~(\ref{eq:fact_ep})) is the universal function that is also present in the factorization theorem of
Drell-Yan, di-hadron production in electron-positron annihilation, and semi-inclusive DIS with TMDs. The operator definition of the soft function (see refs.~\cite{Collins:2011zzd,GarciaEchevarria:2011rb,Chiu:2012ir}) is given by
\begin{equation}
\label{eq:soft}
S(\pmb{q}_T) = \frac{1}{N_R} \text{tr} \;{\langle} [ S_n^{\dag} S_{\bar{n}}](0) \delta^{(2)}(\pmb{q}_T - \pmb{\mathcal{P}}_{\perp}) [ S_{\bar{n}}^{\dag} S_{n}](0) {\rangle}\;,
\end{equation}
where $N_R = N_c$ for $S_{n/\bar{n}}$ in the fundamental and $N_c^2 -1$ for the adjoint representation of $SU(N_c)$.
This function has been calculated at NNLO in \cite{Echevarria:2015byo}. This function is responsible for the TMD evolution which is actually known up to third order
\cite{Li:2016ctv,Vladimirov:2016dll}. The power corrections to the evolution have been studied in \cite{Scimemi:2016ffw}. Because of the universality of this soft function the non-perturbative corrections that it generates in the TMD-evolution factor are process independent \cite{Collins:2011zzd,GarciaEchevarria:2011rb,Scimemi:2016ffw}.
The soft factor provides finally a rapidity renormalization factor for the jets which is totally analogous to the TMD case, see ref.~\cite{Echevarria:2016scs},
so that in this sense we can re-write eq.~(\ref{eq:fact-II}) as
\begin{equation}
\label{eq:fact-IIa}
\frac{d\sigma}{ de_1 de_2 d\pmb{q}_{T} } = H^{ij}_2(Q;\mu)\times \mathcal{J}^{\perp}_{i}(e_1, Q,z_{\text{cut}},\pmb{q}_T;\mu,\zeta_A) \otimes \mathcal{J}^{\perp}_{j}(e_2, Q,z_{\text{cut}},\pmb{q}_T;\mu,\zeta_B)\;,
\end{equation}
with $\zeta_A\zeta_B=Q^4z_\mathrm{cut}^4$ , which recalls clearly the all-order factorization for the di-hadron fragmentation case using TMD. The hadronization corrections to eq.~(\ref{eq:fact-II}-\ref{eq:fact-IIa}) are discussed in more detail in sec.~\ref{hadronization}.
The jet-TMD of eq.~(\ref{eq:fact-IIa}) can be re-factorized depending on the relative magnitudes of the effective scales which define it so that one can identify the more complete set of modes
\begin{align}
\text{soft:}& \;\;\; p_{s}^{\mu} \sim q_T(1,1,1); \nonumber\\
\text{collinear:}& \;\;\; p_{c}^{\mu} \sim Q(\lambda_c^{2},1,\lambda_c) ,\; \lambda_c = \sqrt{e};\nonumber\\
\text{soft-collinear:}& \;\;\; p_{sc}^{\mu} \sim Qz_{\text{cut}}(\lambda_{sc}^{2},1,\lambda_{sc}),\; \lambda_{sc} = q_T /(Qz_{\text{cut}}); \nonumber\\
\text{collinear-soft:}& \;\;\; p_{cs}^{\mu} \sim Qz_{\text{cut}}(\lambda_{cs}^{2},1,\lambda_{cs}),\; \lambda_{cs} = \sqrt{e / z_{\text{cut}}}
\end{align}
and we illustrate this in fig.~\ref{fig:regions}.
We start considering the limit $q_T \gtrsim Q\sqrt{e} \gg Q\sqrt{e z_{\text{cut}}}$, which corresponds to region II in fig.~\ref{fig:regions},
when
the unintegrated and unsubtracted jet function, $\mathcal{J}^{\perp}_{i}$, in eq.~(\ref{eq:fact-II}) can be re-factorized into three terms,
\begin{equation}
\label{eq:jet-fact-II}
\mathcal{J}^{\perp}_{i}(e, Q,z_{\text{cut}},\pmb{q}_T) = S_{sc,i}^{\perp}(Qz_{\text{cut}},\pmb{q}_{T}) \times \int de'\; S_{cs,i}(e-e',Qz_{\text{cut}}) J_i(e',Q)
\end{equation}
where
all the rapidity divergent part and transverse momentum dependence is contained in the calculable $S_{sc,i}^{\perp}$. The subtracted and unsubtracted jet-TMD are related by
\begin{align} \label{eq:jj}
\mathcal{J}^{\perp}_{i}(e, Q,z_{\text{cut}},\pmb{b},\mu,\zeta)=\sqrt{S(\pmb{b})} \mathcal{J}^{\perp}_{i}(e, Q,z_{\text{cut}},\pmb{b})
\end{align}
where we have expressed all the subtraction in $\pmb{b}$-space. \footnote{Throughout the paper we will interchange between $\pmb{q}_T, \pmb{b}$ spaces for the transverse spectrum and between $e, s$ spaces for the jet mass. We use the same symbol for any function in either space. The variable we are working in should be clear from the argument of the function.}
\begin{figure}[t!]
\centerline{\includegraphics[width = \textwidth]{regions}}
\caption{Three possible hierarchies for $q_T$. Shaded region is one that fails Soft-Drop. (I) Largest $q_T \sim Qz_{\text{cut}}$. The cross section is factorized into 3 function s, cs and c. (II) The soft function s splits into two s and sc.(III) The sc function merges with the cs function.}
\label{fig:regions}
\end{figure}
For smaller values of $q_T$: $Q\gg Q z_{\text{cut}} \gtrsim Q\sqrt{e} \gg q_{T} \sim Q\sqrt{e z_{\text{cut}}}$, the collinear-soft and soft-collinear merge into the same mode which we still refer to as collinear-soft. The soft and collinear modes remain unchanged in their scaling compared to region II. The form of factorization theorem in eq.~(\ref{eq:fact-II}) does not change but now the corresponding jet TMDs are re-factorized as (see region III in fig.~\ref{fig:regions}),
\begin{equation}
\label{eq:jet-fact-III}
\mathcal{J}^{\perp}_{i}(e, Q,z_{\text{cut}},\pmb{q}_{T}) = \int de'\; S_{cs,i}^{\perp}(e-e',Qz_{\text{cut}},\pmb{q}_T) J_i(e',Q).
\end{equation}
Several of the parameters in the differential cross-secion in eq.~(\ref{eq:fact-IIa}) are in practice integrated in experiments, so that it is convenient to explicitly write the cumulant (or partially integrated) distribution
\begin{equation}
\label{eq:int-ds}
\frac{d\sigma}{d\pmb{q}_T} (e_{\text{cut}})= \int_0^{e_{\text{cut}}} de_1 de_2 \frac{d\sigma}{ de_1 de_2 d\pmb{q}_{T} } \; .
\end{equation}
For this cross section we work with the integrated jet function which depends on $e_{\text{cut}}$ rather than $e$,
\begin{equation}
\mathcal{J}^{\perp}_{j}(e_{\text{cut}}, Q,z_{\text{cut}}, \bmat{q}_T;\mu,\zeta) = \int_0^{e_{\text{cut}}} de \; \mathcal{J}^{\perp}_{j}(e, Q,z_{\text{cut}}, \bmat{q}_T;\mu,\zeta)\;.
\end{equation}
and the factorization theorem for electron-positron annihilation is
\begin{equation}
\label{eq:fact_ee}
\frac{d\sigma}{d\pmb{q}_T} (e_{\text{cut}})= H^{ij}_2(Q;\mu) \int \frac{d \pmb{b}}{4\pi} e^{i \pmb {b} \cdot \pmb{q}_T} \mathcal{J}^{\perp}_{i}(e_{\text{cut}}, Q,z_{\text{cut}},\pmb b;\mu,\zeta) \mathcal{J}^{\perp}_{j}(e_{\text{cut}}, Q,z_{\text{cut}},\pmb b;\mu,\zeta)\;.
\end{equation}
The resummation of logarithms inside the jet-TMD implied by eq.~(\ref{eq:jet-fact-II}-\ref{eq:jet-fact-III}) is taken into account defining
the cumulant jet function as
\begin{align}
\label{eq:cum-jet0}
\mathcal{J}^{\perp}_{i}(e_{\text{cut}}, Q,z_{\text{cut}},\pmb{b};\mu,\zeta)&=\sqrt{S(\pmb{b})} \mathcal{J}^{\perp}_{i}(e_{\text{cut}}, Q,z_{\text{cut}},\pmb{b})\ ,
\\
\mathcal{J}^{\perp}_{i}(e_{\text{cut}}, Q,z_{\text{cut}},\pmb{b}) &= S_{sc,i}^{\perp}(Qz_{\text{cut}},\pmb{b}) \mathcal{J}_{i}(e_{\text{cut}}, Q,z_{\text{cut}};\mu) ,
\\
\label{eq:cum-jet}
\mathcal{J}_{i}(e_{\text{cut}}, Q,z_{\text{cut}};\mu) &= \int_0^{e_{\text{cut}}} de \int de'\; S_{cs,i}(e-e',Q,z_{\text{cut}};\mu) J_i(e',Q;\mu)
\end{align}
and we recall that the rapidity divergences are present only in $S$ and $S_{sc,i}^{\perp}$, canceling in their product in eq.~(\ref{eq:cum-jet0}). With the exception of the soft-collinear function, $S_{sc}^{\perp}$, all other ingredients of the factorization are already known at least up to NLO accuracy. In app.~\ref{oneloop} we report the defining matrix elements of each function, we summarize the NLO results and we perform the NLO calculation of $S_{sc}^{\perp}$. We have performed the calculation using rapidity regulator. The connection between rapidity regulator and $\zeta$-parameter is outlined in app.~\ref{sec:nu-zeta}.
Finally we observe that using monte-carlo simulations (particularly \textsc{Pythia} 8~\cite{Sjostrand:2006za,Sjostrand:2007gs}) most of the events fall in the kinematic regime
\begin{equation} \label{eq:hierarchy}
Q\gg Q z_{\text{cut}} \gg q_T \sim Q \sqrt{e_{\text{cut}}} \;.
\end{equation}
An important consequence of the jet function refactorization in~eq.~(\ref{eq:jet-fact-II}) is that the transverse momentum dependent elements decouple from the jet mass elements. This suggests that, as long as we remain within the hierarchy of eq.~(\ref{eq:hierarchy}), then the exact mass cutoff on the invariant mass will only influence the overall normalization and not the shape of the TMD distribution. We test this observation against the monte-carlo simulations by comparing the normalized TMD distributions for various values of $e_{\text{cut}}$. We show the results in fig.~\ref{fig:var_ecut} (left). The jet algorithm is implemented through \textsc{FastJet}-3~\cite{Cacciari:2011ma}.
\begin{figure}[t!]
\centerline{\includegraphics[width = \textwidth]{var_ecut}}
\caption{\textsc{Left}: The normalized cross sections for different values of the jet mass cutoff parameter $e_{\text{cut}}$. We also include the corresponding ratios with respect to the case $e_{\text{cut}} =0.01$. \textsc{Right}: The \emph{relatively} normalized cross section for fixed $e_{\text{cut}} =0.01$ and for different value of the jet radius $R$. The corresponding ratios are with respect to $R=1$. }
\label{fig:var_ecut}
\end{figure}
In addition we note that as long as we measure $q_T \ll Qz_{\text{cut}}$ and for $R \sim 1$ the shape and normalization of the cross section is independent of the choice of $R$. We also demonstrate this with the help of simulations. We simulate events at $Q = 50$ GeV and we analyze them for different values of $R\gtrsim 1$. We show the resulting distributions in fig.~\ref{fig:var_ecut} (right). Note that for that plot we preserve the relative normalizations of the curves.
\subsection{Renormalization group evolution}
The two main quantities involved in the factorization procedure carried out in previous section are the subtracted jet-TMD for which we have
\begin{align}
\label{eq:dsevTMDs}
&\mu\frac{d}{d\mu} \mathcal{J}^{\perp}(e,Q,z_{\rm cut},\boldsymbol b, \mu, \zeta)=\gamma_F^q(\mu,\zeta) \mathcal{J}^{\perp}(e,Q,z_{\rm cut},\boldsymbol b, \mu, \zeta),\\
\label{eq:dsevTMDs-z}
&\zeta \frac{d}{d\zeta} \mathcal{J}^{\perp}(e,Q,z_{\rm cut},\boldsymbol b, \mu, \zeta)=-\mathcal{D}^q(\mu,\boldsymbol b) \mathcal{J}^{\perp}(e,Q,z_{\rm cut},\boldsymbol b, \mu, \zeta),
\end{align}
where on the r.h.s. we have considered just quark initiated jets and we have Fourier transformed with respect to $\bmat{q}_T$ the jet functions appearing in eq.~(\ref{eq:fact-IIa}). Of course this result recalls literally the standard TMD case.
However, because of the re-factorization of $ \mathcal{J}^{\perp}$ (see eq.~(\ref{eq:jet-fact-II}-\ref{eq:jet-fact-III})) this resummation is not complete and large logarithms can still spoil the convergence of the perturbative series. Defining $s$ as the variable conjugate to $e$ in Laplace space (see app.~\ref{transforms}) and
\begin{align}
G\in\left\{S_{sc}^{sub}(Qz_{\rm cut},\pmb{b}),\, S_{cs}(s,Q z_{cut}),\, J(s,Q)\right\};\quad S_{sc}^{sub}(Qz_{\rm cut},\pmb{b})=\sqrt{S(\pmb b)} S_{sc}(Qz_{\rm cut},\pmb{b})\;,
\end{align}
we have
\begin{equation}
\label{eq:unmeasRG}
\mu \frac{d}{d \mu}G= \gamma^{G} (\mu, \alpha_s) G= \left( \Gamma^{G} [\alpha_S] \textbf{l}_{m^2_{G}} + \Delta \gamma^{G}[\alpha_S]\right) G,
\end{equation}
which are formally similar to the TMD case and the values of $m_G$ are reported in the appendix in tab.~\ref{tb:evolution}. The only function in $G$ which has a rapidity evolution equation is $S_{sc}^{sub}$ and it scales like $ \mathcal{J}^{\perp}$ in eq.~(\ref{eq:dsevTMDs-z}).
The cusp part of eq.~(\ref{eq:unmeasRG}) is proportional to the standard cusp anomalous dimension
\begin{equation}
\label{eq:G}
\Gamma^{G}_{\mu}[\alpha_s] = \frac{\Gamma^G_0}{\Gamma^{\text{cusp}}_0} \Gamma^{\text{cusp}} = \frac{\Gamma^G_0}{\Gamma^{\text{cusp}}_0} \sum_{n=0}^{\infty} \left(\frac{\alpha_s}{4 \pi} \right)^{1+n} \Gamma^{\text{cusp}}_n,
\end{equation}
For the non-cusp part we have also a perturbative expansion
\begin{equation}
\label{eq:g}
\Delta \gamma^{G}[\alpha_S] = \sum_{n=0}^{\infty} \left(\frac{\alpha_s}{4 \pi} \right)^{1+n} \gamma^{G}_n.
\end{equation}
The anomalous dimensions that enter in the calculations for each case are given in app.~\ref{app:evolution}. The evolution in rapidity and factorization scales of all quantities can be implemented using the the $\zeta$-prescription whose general framework can be found in ref.~\cite{Scimemi:2018xaf}. We provide some details for the present case in the appendix.
The resummation of potentially large logarithms inside the jet-TMD is done performing the evolution in Laplace space and then integrating such that we get the cumulant before we take the inverse transform.
In this way we resum logarithms which are associated to $e_{\text{cut}}$. All this works as follows.
Starting from eq.~(\ref{eq:cum-jet}), then taking the Laplace and consecutively the inverse transform with respect to $e$ we find
\begin{equation}
\mathcal{J}_{i}(e_{\text{cut}}, Q,z_{\text{cut}};\mu) = \frac{1}{2\pi i} \int_{\gamma-i \infty}^{\gamma+i\infty} ds \frac{\exp(s e_{\text{cut}}) -1}{s} S_{cs,i}(s,Q,z_{\text{cut}};\mu) J_i(s,Q;\mu)\ .
\end{equation}
Then solving the RGE equations for the collinear-soft and jet function as described in app.~\ref{sec:RGEs}, and performing the last remaining integral over the Laplace conjugate variable $s$ we get
\begin{multline}
\mathcal{J}_{i}(e_{\text{cut}}, Q,z_{\text{cut}};\mu) = \exp \Big{(} K_{cs}(\mu,\mu_{cs})+K_{J}(\mu,\mu_{J}) \Big{)} S_{cs,i} (L_{cs} \to \partial_{\omega_{cs}};\mu_{cs}) J_i (L_{J} \to \partial_{\omega_{J}};\mu_J) \\
\Big{(} \frac{\mu_{cs}}{ Q \sqrt{z_{\text{cut}} e_{\text{cut}}}} \Big{)}^{2\omega_{cs}(\mu,\mu_{cs})} \Big{(} \frac{\mu_{J}}{ Q \sqrt{e_{\text{cut}}}} \Big{)}^{2\omega_{J}(\mu,\mu_{J})} \frac{\exp(\gamma_E(\omega_{cs}(\mu,\mu_{cs})+\omega_{J}(\mu,\mu_{J})))}{\Gamma(1-\omega_{cs}(\mu,\mu_{cs})-\omega_{J}(\mu,\mu_{J}))} \ .
\end{multline}
This is our final result for the resummed cumulant jet function. The order of logarithmic accuracy is then determined by the order of which the kernels $K_F$, $\omega_F$, and the fixed order collinear-soft and jet functions are evaluated. At this stage of the calculation the canonical scales, $\mu_{cs}$ and $\mu_J$, are not yet fixed. This allows us to choose the scales such that potentially large logarithms are minimized in momentum space. From the above is clear that the canonical choice of scales such as the fixed order logarithms are minimized are,
\begin{align}
\mu_{cs} &= Q \sqrt{z_{\text{cut}} e_{\text{cut}}}\;, & \mu_J &= Q \sqrt{e_{\text{cut}}} \;.
\end{align}
In numerical applications one needs to perform variations around these scales in order to obtain an estimate of the theoretical uncertainty.
\subsection{Numerical results for $e^+e^-$}
\label{sec:numerics-ee}
In this section, we provide the results of our calculation for $e^+e^-\to \text{2\;jets}$ computed up to NNLL accuracy. The implementation necessarily needs a choice for the rapidity scales and we have done it using the $\zeta$-prescription as described in ref.~\cite{Scimemi:2018xaf} and adapting the code \texttt{artemide} to the present case.
This consisted of performing the evolution of the transverse momentum dependent components within the \texttt{artemide} framework, while
for all other scales not involved in the rapidity evolution,
i.e., the hard and jet functions, see app.~\ref{sec:RGEs}.
There are some important modifications to the $\zeta$-prescription framework for our case which affect the numerics.
One of this is that now
$\zeta_A \zeta_B \sim Q^4 z_{\text{cut}}^4$ compared to the di-hadron decorrelation case where $ \zeta_A \zeta_B \sim Q^4 $.
This means that the effective hard scale to which the distributions are sensitive is lower. Because the TMD factorization is valid when $q_T$ is much lower than the hard scale of the process,
one needs that the product $Q z_\mathrm{cut}$ be sufficiently high. In our plots we have considered the case $q_T\lesssim Q z_\mathrm{cut}$. Then the evolution of the jet-TMD given in eq.~(\ref{eq:unmeasRG}) is also slightly different from the standard hadron TMD, although the changes are implemented easily in the \texttt{artemide} code. A one-loop check of all anomalous dimensions is provided in app.~\ref{oneloop}.
\begin{figure}[h!]
\centerline{\includegraphics[width = \textwidth]{MCvsNLL}}
\caption{Comparison of the NLL result against the partonic shower of \textsc{Pythia} 8 for $R=1$ and $e_{\text{cut}} =0.01$ for two different center of mass energies, \textsc{Left}: 50 GeV, \textsc{Right}: 100 GeV.}
\label{fig:e+e-MC}
\end{figure}
In fig.~\ref{fig:e+e-MC} we compare our analytic result for NLL cross section (normalized) against \textsc{Pythia} simulations for $Q=50$ and 100 GeV. For the purposes of comparison we turn hadronization off in the simulation and we compare against our purely perturbative result. The perturbative calculation depends on the parameter $B_{\text{NP}}$ which in practice implements a cutoff in the inverse Laplace transform such that the soft scale, that behaves as $1/b$, does not hit the Landau pole. As long as we choose this parameter such that convergence of the integral is reached before the cutoff, then the perturbative result is not much sensitive to the value of $B_{\text{NP}}$. Although, as we now discuss, the theoretical uncertainty of the cross section for these energies at NLL is quite large, we find very good agreement with the simulations for the canonical choice of scales (i.e., central line in fig.~\ref{fig:e+e-MC}).
In fig.~\ref{fig:e+e-Zmass} we give the NNLL results including a theoretical uncertainty band. We compare against the NLL cross section and although the error bands seem to be larger than what is typically expected we can clearly see that the result convergences and the theory error decreases by approximately factor of two. To estimate the theoretical uncertainty we first vary all the factorization scales of a factor 2 (0.5) around their canonical value, then we separately take the envelope of the variations involved in rapidity evolution, $\mu,\mu_{sc}$, and of the ones involved only in the virtuality evolution of the jet function, $\mu_{cs},\mu_J$. The final error bands we show are the quadrature of the two contributions. The reason for this prescription is that rapidity and virtuality evolutions are in principle uncorrelated. The uncertainty is somewhat larger than what one might expect for a NNLL calculation, and is practically dominated by the variations in the jet function. This is attributed to the small values of the collinear-soft scale, $\mu_{cs}\sim Q\sqrt{e_{\text{cut}} z_{\text{cut}}}$, which approaches the non-perturbative regime even for values of $Q \sim m_Z$. One might attempt to reduce the uncertainty by increasing either $e_{\text{cut}}$ or $z_{\text{cut}}$, but caution is needed not to invalidate the corresponding hierarchy. We will see later that when only the mass of one jet is measured (e.g., in DIS or hadron-jet decorrelation) then the error band decreases significantly.
\begin{figure}[t!]
\centerline{\includegraphics[width = 0.517\textwidth]{di-lepton}}
\caption{Transverse momentum de-correlation for $e^+e^-\rightarrow$ dijets with center of mass energy at the Z mass. }
\label{fig:e+e-Zmass}
\end{figure}
\section{Jets in DIS}
\label{DIS-jet}
The advent of new colliders like EIC and LHeC makes the measurement of jets interesting also in semi-inclusive deep inelastic scattering (SIDIS) experiments.
Actually we want to explore the possibility of using jets to study the TMDPDF.
For the present case we demand that the hard scattering of the lepton on the proton produces a single jet.
In the Breit frame we measure the transverse component, $\pmb{q}_{T}$, of the transferred momentum, $q^{\mu} = {k'}^{\mu}- {k}^{\mu}$ with respect to the single groomed jet. As before, we impose a jet mass cut-off $e_{\text{cut}}$ and the grooming parameter $z_{\text{cut}}$. In this framework the initial state proton is moving along the $-z$ direction and the final state jet is moving in the opposite $+z$-direction, so that we can assign the directions $n$ and $\bar{n}$ to the beam and jet definition. The contribution to this transverse momentum measurement comes from the initial state radiation which forms part of the TMDPDF and the radiation that fails soft-drop in the final state jet. We demand that there is a single energetic jet with $E_J \sim Q/2 =\sqrt{-q^2} /2$ with accompanying soft radiation.
It is instructive to setup some of the notation that we are using for describing the kinematics in the Breit frame. The virtual photon is assumed to be completely space-like and it has only the $z$ component of the momentum. Defining our light-cone directions $n^{\mu} =(1, 0, 0, +1)$ and $\bar{n}^{\mu} = (1,0,0,-1)$, the resulting photon momentum $q^{\mu}$ is
\begin{equation}
q^{\mu} = \frac{Q}{2} (n^{\mu} - \bar{n}^{\mu})\ ,
\end{equation}
where $Q^2 = - q^2$ is a positive quantity. We assume that at the partonic level, a single quark carrying $x$ fraction of the proton longitudinal momentum undergoes a hard interaction with the virtual photon.
In this frame, the proton is moving along the $-z$ direction and its momentum can be written as:
\begin{equation}
P^{\mu} = \frac{Q}{2 x} \bar{n}^{\mu}\ .
\end{equation}
At tree level and by momentum conservation the final state parton will carry momentum
\begin{equation}
xP^\mu + q^\mu = \frac{Q}{2}(n^{\mu} - \bar{n}^{\mu})+ \frac{Q}{2}\bar{n}^{\mu}= \frac{Q}{2} n^{\mu}\;,
\end{equation}
which is exactly opposite in direction to the incoming beam. Of course this will be modified beyond tree-level when initial and final state radiation is included.
\subsection{Schematics for factorization}
Since we are working with two back-to-back directions, our usual definition of the soft function holds: in other words the change from future pointing to past pointing Wilson lines does not affect its value~\cite{Collins:2011zzd,GarciaEchevarria:2011rb,Echevarria:2014rua,Gaunt:2014ska,Vladimirov:2017ksc}.
Since we still impose the same jet mass measurement on the final state jet, we have all the modes that we had in the $e^+e^-$ case. The main difference is that now the initial hadronic state is a TMDPDF.
The form of the factorized cross section follows again the hierarchy $Q \gg Q z_{\text{cut}} \gg q_{T}, \; R\sim 1$ and
\begin{equation}
\label{eq:fact_ep}
\frac{d\sigma}{dx dQ^2 d\pmb{q}_{T}} = \mathcal{N}(x,Q) H_2(Q,\mu) \times S(\pmb{q}_T) \otimes B_{i\leftarrow h}(x,Q,\pmb{q}_T) \otimes \mathcal{J}_{j}^\perp(e_{\text{cut}}, Q,z_{\text{cut}}, \pmb{q}_T)\ ,
\end{equation}
where $ x = - q^2/(2P \cdot k)$, $k$ is the momentum of the incoming electron, and $\mathcal{N}(x, Q)$ is the over-all normalization which we give later in this section. The un-subtracted TMDPDF is $B_{i\leftarrow h}$.
In our rapidity regularization scheme the (subtracted) TMDPDF is defined as
\begin{align}
F_{i\leftarrow h}(x,\boldsymbol b; \mu, \zeta)=\sqrt{S(\pmb{b})}B_{i\leftarrow h}(x,Q,\pmb{b}) .
\end{align}
At perturbative values of $q_{T}$, the $F_{i\leftarrow h}$ can be matched onto the collinear PDF. The matching coefficients at NNLO are evaluated in~\cite{Gehrmann:2014yya,Echevarria:2016scs}
and in the appendix we review some one-loop results.
Once the subtracted quantities are included we can write
\begin{equation}
\label{eq:fact_ep_II}
\frac{d\sigma}{dx dQ^2 d\pmb{q}_{T}} = \mathcal{N}(x,Q) H_2(Q,\mu) \int \frac{d \pmb{b}}{4\pi^2} e^{i \pmb{b} \cdot \pmb{q}_{T} } F_{i\leftarrow h}(x,Q,\pmb{b},\mu,\zeta_A) \mathcal{J}_{j}^\perp(e_{\text{cut}}, Q,z_{\text{cut}}, \pmb{b};\mu,\zeta_B)\ .
\end{equation}
The evolution under renormalization group equations for the TMDPDF is widely known (see e.g.~\cite{Scimemi:2018xaf, DAlesio:2014mrz,Echevarria:2012pw,Chiu:2011qc}) and we recall a few characteristics here. One has
\begin{align}
&\mu\frac{d}{d\mu} F_{f\leftarrow f'}(x,\boldsymbol b, \mu, \zeta)=\gamma_F^f(\mu,\zeta) F_{f\leftarrow f'}(x,\boldsymbol b, \mu, \zeta),\nonumber\\
&\zeta \frac{d}{d\zeta} F_{f\leftarrow f'}(x,\boldsymbol b, \mu, \zeta)=-\mathcal{D}^f(\mu,\boldsymbol b) F_{f\leftarrow f'}(x,\boldsymbol b, \mu, \zeta),
\end{align}
where $\mathcal{D}_f$ and $\gamma_F^f$ are the rapidity and UV anomalous dimensions, respectively.
The integrability requirement of this couple of equation results in
\begin{align}
\mu \frac{d}{d\mu} \left(-\mathcal{D}^f(\mu,\boldsymbol b)\right)=\zeta \frac{d}{d\zeta} \gamma^f_F(\mu,\zeta)=-\Gamma^{\rm cusp}_f
\end{align}
where $\Gamma^{\rm cusp}_f$ is the cusp anomalous dimension. The UV anomalous dimension is written in these terms as
\begin{align}
\gamma_F^f=\Gamma^{\rm cusp}_f\textbf{l}_\zeta-\gamma_V^f,
\end{align}
$\gamma_V^f$ being the non-cusp part of the anomalous dimension and $\textbf{l}_\zeta=\ln \left(\mu^2/\zeta\right)$. The $\gamma_V$ and $\mathcal{D}$ anomalous dimensions are known up to $\mathcal{O}(a_s^3)$ \cite{Moch:2004pa,Moch:2005tm,Baikov:2009bg,Vladimirov:2016dll,Li:2016ctv}. A numerical calculation for the four-loop cusp anomalous dimension was recently given in \cite{Vogt:2018miu}. All the evolution equations are the same for the case of TMD fragmentation functions, and we do not discuss them any more here.
\subsection{Derivation of the factorized cross section using jets}
In this section we provide some details for the factorization of the SIDIS cross section in eq.~(\ref{eq:fact_ep}, \ref{eq:fact_ep_II}).
The scattering amplitude for the process $e p \to e f$ where $f$ is the final state is given by:
\begin{equation}
i M(ep \to ef ) = (-ie^2)\bar{u}(k') \gamma_\mu u(k) \frac{1}{q^2} {\langle} f {\vert} J^{\mu}(0) {\vert} p(P) {\rangle}\;,
\end{equation}
and thus the corresponding cross section is given by
\begin{multline}
d\sigma (ep \to ef) = \frac{e^4}{4(s-m^2)} \int \frac{d^3 k'}{2 (2\pi)^3 E_{k'}} \; \text{tr}\Big{\lbrack} \slashed{k}\gamma_{\mu} \slashed{k'} \gamma_{\nu}\Big{\rbrack} \\
\sum_f \int d \Pi_f \;{\langle} p(P) {\vert} J^{\dag \mu}(0) {\vert} f {\rangle} {\langle} f {\vert} J^{\nu}(0) {\vert} p(P) {\rangle} (2\pi)^4 \delta^{(4)} (q + P -p_f)\;,
\end{multline}
where $q= k'-k$. We can use the standard parametrization of the final electron phase-space to write:
\begin{equation}
\int \frac{d^3 k'}{2 (2\pi)^3 E_{k'}} = dxdy \frac{y s}{(4\pi)^2}\;,
\end{equation}
where $y = (2P \cdot q)/(2P \cdot k)$ and $s$ is the hadronic Mandelstam variable. We then get,
\begin{equation}
\frac{d\sigma}{dxdy} (ep \to ef) = L_{\mu\nu}(k,k') \sum_f \int d^4 r e^{i q \cdot r}\int d \Pi_f \;{\langle} p(P) {\vert} J^{\dag \mu}(0) {\vert} f {\rangle} {\langle} f {\vert} J^{\nu}(x) {\vert} p(P) {\rangle}\;,
\end{equation}
where $r^{\mu}$ is Fourier conjugate of the momenta $q^{\mu}$ and $L^{\mu\nu}$ is the leptonic tensor,
\begin{equation}
L_{\mu\nu}(k,k') \equiv \frac{\alpha^2 y s}{4(s-m^2)} \text{tr}\Big{\lbrack} \slashed{k}\gamma_{\mu} \slashed{k'} \gamma_{\nu}\Big{\rbrack}\;.
\end{equation}
The next step is to project the hadronic final state $\vert f \rangle $ onto the one that corresponds to the measurement that we are proposing, i.e.,
\begin{equation}
\int d\Pi_f {\vert} f {\rangle} {\langle} f {\vert}\;\; \to \int d\pmb{q}_{T} z dz \int d\Pi_{f[\text{g-jet}(z \pmb{q}_{T}, z)]} {\vert} f {\rangle} {\langle} f {\vert}\;.
\end{equation}
We can now match the full theory hadronic current $J_{\mu}(x)$ onto the SCET$_+$~\cite{Bauer:2011uc} current working in the Breit frame,
\begin{equation}
J^{\mu}(x) = C^{\mu \nu}(Q) \Big{\lbrack} \bar{\chi}_{n,Q} S_{n}^{\dag} W_t^{\dag} U_n\gamma_{\nu} S_{\bar{n}}\chi_{\bar{n},Q} \Big{\rbrack} + \mathcal{O}(\lambda)\;,
\end{equation}
where $\lambda$ is the power counting parameter of our EFT which will turn out to be $q_{\perp}/Q \sim e_{\text{cut}}/Q$. Note that in the same step, through BPS field redefinition, we decoupled the collinear soft modes from the collinear modes and hence the presence of the $U_n$ Wilson lines. In the matching we also have the soft Wilson lines $S_n$. From the kinematic constraints of the measurement and since all the modes that are present in the projected final state are decoupled from each other at the level of the Lagrangian, (we assume that contributions from Glauber gluon exchanges cancel) it is possible to factorize the final state as follows,
\begin{equation}
{\vert} f {\rangle} \;\;\to\;\; {\vert} X_{\bar{n}} {\rangle} {\vert} X_{n} {\rangle} {\vert} X_{s} {\rangle} {\vert} X_{sc} {\rangle} \;,
\end{equation}
where we have included in $X_n$ all possible modes that contribute to the invariant mass measurement. Refactorization of the $n-$collinear sector follows from the same steps as in the case of electron-positron annihilation presented in ref.~\cite{Frye:2016aiz}. We are now ready to factorize the cross section into individual SCET matrix elements. In the final result one needs to be careful regarding all the index contractions and the tensor structures. This was carefully considered in ref.~\cite{Kang:2013nha}. In addition we are considering the case where the frame we are working is rotated such that the transverse momentum of the groomed jet is zero. After all rearrangements we get,
\begin{multline}
\frac{d\sigma}{dxdydzd\pmb{q}_{T}} (ep \to ef) = \sigma_0 (x,Q) \times H_2 (Q) \int d^4r e^{i q \cdot r} \\
\frac{1}{N_c}\sum_{X_s} {\langle} 0 {\vert} S_{n} S_{\bar{n}}^{\dag}(r_{\perp}) {\vert} X_s {\rangle} {\langle} X_s{\vert} S_{\bar{n}} S_{n}^{\dag}(0) {\vert} 0 {\rangle} \\
\times \sum_{X_{\bar{n}}} {\langle} p(P) {\vert} \bar{\chi}_{\bar{n}}(r^{+},r_{\perp}) \frac{\gamma^{+}}{2} {\vert} X_{\bar{n}} {\rangle} {\langle} X_{\bar{n}} {\vert} \chi_{\bar{n}}(0) {\vert} p(P) {\rangle} \\
\times \frac{1}{N_c} \sum_{X_{sc}} {\langle} 0 {\vert} U_n^{\dag} W_t (r_{\perp}) {\vert} X_{sc} {\rangle} {\langle} X_{sc}{\vert} W_t^{\dag} U_n(0) {\vert} 0 {\rangle} \\
\times \frac{z}{2N_c} \text{tr}\sum_{X_{n}} {\langle} 0 {\vert} \frac{\gamma^{-}}{2} \chi_{n}(r^{-},r_{\perp}) {\vert} X_{n} {\rangle} {\langle} X_{n} {\vert} \bar{\chi}_{n}(0) {\vert} 0 {\rangle} {\vert}_{p_{\perp}^{X_n} = 0}\;.
\end{multline}
The hard matching coefficient in general has two Lorentz structures, given the two types of currents, vector and axial. For the case of photon with vector current, we simply have $H^{\mu\nu} \sim g^{\mu\nu}_{\perp}$. We have also multipole-expanded the final result. To proceed with the factorization theorem in momentum space, we remove $r_\perp$ dependence from the various EFT matrix elements by acting the corresponding fields on the final states. This gives us
\begin{multline}
\frac{d\sigma}{dxdydzd\pmb{q}_{T}} (ep \to ef) = \sigma_0 (x,Q) \times H_2 (Q) \int d^4r e^{i q \cdot r+i(\pmb{p}^{X^{R}_{\bar{n}}}_{\perp} + \pmb{p}_{\perp}^{S}) \cdot \pmb{r}_{\perp}} \\
\frac{1}{N_c} \sum_{X_s} {\langle} 0 {\vert} S_{n} S_{\bar{n}}^{\dag}(0) {\vert} X_s {\rangle} {\langle} X_s{\vert} S_{\bar{n}} S_{n}^{\dag}(0) {\vert} 0 {\rangle} \\
\times \sum_{X_{\bar{n}}} {\langle} p(P) {\vert} \bar{\chi}_{\bar{n}}(r^{+}, 0_{\perp}) \frac{\gamma^{+}}{2} {\vert} X_{\bar{n}} {\rangle} {\langle} X_{\bar{n}} {\vert} \chi_{\bar{n}}(0) {\vert} p(P) {\rangle} \\
\times\frac{1}{N_c} \sum_{X_{sc}} {\langle} 0 {\vert} U_n^{\dag} W_t (0) {\vert} X_{sc} {\rangle} {\langle} X_{sc}{\vert} W_t^{\dag} U_n(0) {\vert} 0 {\rangle} \\
\times \frac{z}{2N_{c}} \; \text{tr}\sum_{X_{n}} {\langle} 0 {\vert} \frac{\gamma^{-}}{2} \chi_{n}(r^{-},0_{\perp}) {\vert} X_{n} {\rangle} {\langle} X_{n} {\vert} \bar{\chi}_{n}(0) {\vert} 0 {\rangle} {\vert}_{p_{\perp}^{X_n} = 0}\;,
\end{multline}
where
\begin{equation}
p_{\perp}^{X_{\bar{n}}^{R}} {\vert}_{p_{\perp}^{\text{g-jet}} = 0} = p_{\perp}^{X_{\bar{n}}} - P_{\perp} {\vert}_{p_{\perp}^{\text{g-jet}} = 0} = p_{\perp}^{X_{\bar{n}}} {\vert}_{P_{\perp} = 0} \Big{(} 1 + \mathcal{O}(\lambda)\Big{)}\;,
\end{equation}
is the difference in the transverse momentum of the recoiling initial state collinear radiation and the proton with respect to the hadrons direction, which up to power-corrections of order $\mathcal{O}(\lambda)$ is simply the transverse momentum of the recoiling radiation with respect to the proton. Performing the integral over $d^4r$ we get:
\begin{multline}
\frac{d\sigma}{dxdydzd\pmb{q}_{T}} (ep \to ef) = \sigma_0 (x,Q) \times H_2 (Q) \delta^{(2)} (\pmb{q}_{T}+ \pmb{p}^{X^{R}_{\bar{n}}}_{\perp} + \pmb{p}_{\perp}^{X_s} + \pmb{p}_{\perp}^{X_{sc}} ) \\
\frac{1}{N_c} \sum_{X_s} {\langle} 0 {\vert} S_{n} S_{\bar{n}}^{\dag}(0) {\vert} X_s {\rangle} {\langle} X_s{\vert} S_{\bar{n}} S_{n}^{\dag}(0) {\vert} 0 {\rangle} \\
\times \sum_{X_{\bar{n}}} {\langle} p(P) {\vert} \bar{\chi}_{\bar{n}}(0) \frac{\gamma^{+}}{2} \delta(q^{-} - p_{X_{\bar{n}}}^{-}){\vert} X_{\bar{n}} {\rangle} {\langle} X_{\bar{n}} {\vert} \chi_{\bar{n}}(0) {\vert} p(P) {\rangle} \\
\times \frac{1}{N_c} \sum_{X_{sc}} {\langle} 0 {\vert} U_n^{\dag} W_t (0) {\vert} X_{sc} {\rangle} {\langle} X_{sc}{\vert} W_t^{\dag} U_n(0) {\vert} 0 {\rangle} \\
\times \frac{z}{2N_{c}} \; \text{tr}\sum_{X_{n}} {\langle} 0 {\vert} \frac{\gamma^{-}}{2} \chi_{n}(0) \delta(q^{+} - p_{ X_{n}}^{+}) {\vert} X_{n} {\rangle} {\langle} X_{n} {\vert} \bar{\chi}_{n}(0) {\vert} 0 {\rangle} {\vert}_{p_{\perp}^{X_n} = 0}\;.
\end{multline}
In order to simplify our result further we introduce ``measurement'' delta functions for the soft and initial state matrix elements. This will allow us to absorb the $p_{\perp}^{X_i}$ into the corresponding matrix elements and use
\begin{equation}
\bmat{1}_i = \sum_{X_i} {\vert} X_i {\rangle} {\langle} X_i {\vert}\;,
\end{equation}
to further simplify the form of EFT matrix elements. We also perform a type-I RPI transformation in order to rewrite the proton matrix elements as function of fields with respect to the initial state proton axis. We thus get
\begin{multline}
\frac{d\sigma}{dxdydzd\pmb{q}_{T}} (ep \to ef) = \sigma_0 (x,Q) \times H_2 (Q) \int d\pmb{p}_{\perp}^{\;s} d\pmb{p}_{\perp}^{\;sc} d\pmb{p}_{\perp}^{\;c} \;\;\delta^{(2)} ( \pmb{q}_{T}+\pmb{p}^{c}_{\perp} + \pmb{p}_{\perp}^{s} +\pmb{p}_{\perp}^{sc} ) \\
\frac{1}{N_c} {\langle} 0 {\vert} T\Big{(} S_{n} S_{\bar{n}}^{\dag}(0)\Big{)} \delta^{(2)}(\pmb{p}_{\perp}^{s} - \pmb{\mathcal{P}}_{\perp}) \bar{T} \Big{(} S_{\bar{n}} S_{n}^{\dag}(0) \Big{)} {\vert} 0 {\rangle} \\
\times {\langle} p(P) {\vert} \bar{\chi}_{\bar{n}}(0) \frac{\gamma^{+}}{2} \delta(q^{-} - \mathcal{P}^{-}) \delta^{(2)}(\pmb{p}_{\perp}^{c} - \pmb{\mathcal{P}}_{\perp}) \chi_{\bar{n}}(0) {\vert} p(P) {\rangle} {\vert}_{P_{\perp} = 0} \\
\times \frac{1}{N_c} {\langle} 0 {\vert} T\Big{(} U_n^{\dag} W_t (0) \Big{)} \mathcal{M}_\perp^{SD} \bar{T} \Big{(} W_t^{\dag} U_n(0) \Big{)} {\vert} 0 {\rangle} \\
\times \frac{z}{2N_{c}} \; \text{tr}\sum_{X_{n}} {\langle} 0 {\vert} \frac{\gamma^{-}}{2} \chi_{n}(0) \delta(q^{+} - \mathcal{P}^{+}) {\vert} X_{n} {\rangle} {\langle} X_{n} {\vert} \bar{\chi}_{n}(0) {\vert} 0 {\rangle} {\vert}_{p_{\perp}^{X_{n}} = 0}\;,
\end{multline}
where $\mathcal{M}_\perp^{SD}$ is the measurement function given in eq.~(\ref{eq:meas_sc}). Since we are considering only large radius jets with $R\gtrsim 1$ we may trivially perform the integration of the energy fraction $z$ using $Q\simeq p^{X_n}_+$ up to power corrections. Also performing change of integration variables,
\begin{equation}
dx dy = \frac{dx dQ^2}{xs} \;,
\end{equation}
we get eq.~(\ref{eq:fact_ep}) with
\begin{equation}
\mathcal{N}(x,Q) = \frac{\sigma_{0}(x,Q)}{xs}\;,
\end{equation}
and the matrix elements involved in the functions $S$, $B$, and $\mathcal{J}$ are given in the appendix. For the case of groomed jets with invariant mass cutoff it is possible to refactorize the jet function. This is done in ref.~\cite{Frye:2016aiz} and thus we do not demonstrate it here. Then integrating over $e \in (0,e_{\text{cut}})$ gives the dependence of the jet function in the parameter $e_{\text{cut}}$. This is identical to the analysis in the previous section on $e^+e^-$. This is our final result for the factorization theorem in DIS.
\subsection{Numerical results for DIS}
\label{sec:numerics-DIS}
In this section we use the factorization theorem in eq.~(\ref{eq:fact_ep}) to obtain numerical results for the TMD spectrum of groomed jets in DIS process. Our analysis is done for two center-of-mass energies, EIC: $\sqrt{s} = 100$ GeV and HERA: $318$ GeV. For both energies we integrate over $y = Q^2/(xs) $ and $Q = \sqrt{-q^2}$ in the regions $0.01< y<0.95$ and $40< Q<50$ GeV. For the TMDPDFs we use the fits obtained from Drell-Yan data~\cite{Bertone:2019nxa} with the use of $\zeta$-prescription.
In fig.~\ref{fig:scales} we show our results for NLL and NNLL accuracies for the two center of mass choices, including theoretical uncertainties. We estimate the theoretical scale variations as described in sec.~\ref{sec:numerics-ee}. The groomed jet parameters that we choose are the same as in the di-lepton case: $\beta =0$, $z_{\text{cut}} =0.2$, and $e_{\text{cut}} = 0.01$. As before we find good convergence between the NLL and NNLL result. The absolute value of theoretical scale variation is improvable with higher logarithmic accuracy (NNLL-prime or perhaps N$^3$LL), which needs the explicit calculation of several jet hadronic matrix elements at two loops.
\begin{figure}[h!]
\centerline{\includegraphics[width = \textwidth]{dis_scales}}
\caption{The NLL and NNLL TMD spectra for groomed jets in DIS for EIC (left: $\sqrt{100}$ GeV) and HERA (right: $\sqrt{s} = 318$ GeV) kinematics. The cross section are integrated in $y = Q^2/(xs)$ and $Q = \sqrt{-q^2}$ (see details in the main text). }
\label{fig:scales}
\end{figure}
We further investigate the size of the uncertainty due to the hadronic initial state and the non-perturbative effects induced by TMD evolution. We do that by varying the model parameters as constrained by the phenomenological analysis in ref. ~\cite{Bertone:2019nxa} for our NNLL result. The results are shown in figure~\ref{fig:NP}. We consider both variable and fixed $B_{\text{NP}}=2.5$ GeV$^{-1}$ (for details on the difference of the two schemes see~\cite{Bertone:2019nxa}). We find that the effects (for our kinematics) are particularly small, of the order of $\sim 5\%$, which is much smaller than the theoretical uncertainties. This suggests that we need a better control over the theoretical uncertainties in order to further constrain TMD distributions from groomed jets in DIS. As mentioned earlier the uncertainty can be mitigated with higher logarithmic accuracy or by choosing larger values of $e_{\text{cut}}$, still compatible with factorization. This, will require to treat the region III shown in fig.~\ref{fig:regions}. For this reason it is interesting to investigate the range of values of $e_{\text{cut}}$ for which the energetic wide angle radiation is avoided.
\begin{figure}[t!]
\centerline{\includegraphics[width = \textwidth]{dis_NP}}
\caption{The NNLL cross-section including modeling of the initial hadronic state effects fitted from Derll-Yan processes using two different scenes: fixed and variable $B_{\text{NP}}$.}
\label{fig:NP}
\end{figure}
\section{Hadronization effects }
\label{hadronization}
One of the goals of the paper is to study the non-perturbative effects associated with TMD distributions, in this case the TMDPDF. Usually in any experiment, there are multiple sources of non-perturbative corrections associated with both the initial and final states. To have access to a specific source of corrections, its therefore necessary to separate out the pieces of interest from the uninteresting ones, which in this case constitute the final state hadronization corrections. To access the TMD then, we must already have a good extraction of the rest of the non-perturbative effects. This is the reason why we consider distinct experiments in this paper. The idea, as we shall demonstrate, is that the final state hadronization corrections are exactly the same in the two experiments. The $e^+e^- \to \text{2\;jets}$ case can be used to extract out all the final state hadronization corrections, which can then be used for DIS.
For the $e^+e^-$ observable, the factorization takes the form in eq.~(\ref{eq:fact-IIa}), we can then study the non-perturbative corrections for each collinear object $\mathcal{J}^{\perp}_{i}$, which by symmetry, are the same for the two objects. If we now look at the factorization for DIS, eq.~(\ref{eq:fact_ep_II}), the key point to note is that $\mathcal{J}_{j}^\perp(e_{\text{cut}}, Q,z_{\text{cut}}, \pmb{b};\mu,\zeta_B)$ is the same object that appears in the the case of $e^+e^-$, while $ F_{i\leftarrow h} $ is just the TMDPDF. Thus it now becomes possible to exclusively access the complete TMDPDF. We now wish to systematically list the sources of the non-perturbative corrections associated with each factorized function that appear in our cross section.
In order to use jets it is important to consider all the non-perturbative effects for the case of our observables and in particular the ones coming from the implementation of (groomed) jets. In fig.~\ref{fig:hadronization} we have shown that such corrections are expected to be particularly small and we provide here a discussion about their origin from a theory perspective. We have two measurements on the jet: the jet mass, which is ultimately integrated over some interval and acts as a normalization, and the transverse momentum ($p_{\perp}$) of the radiation that is groomed away. Since we are interested in the shape of the $q_T$ spectrum, we will only consider the non-perturbative effects in cross sections sensitive to it. As was explained in sec.~\ref{sec:main-1}, we are working in the region II of EFT and we are going to discuss how non-perturbative effect arise when we increase the value of $q_T$ (that is, we discuss here the non-perturbative corrections in the small-$b$ limit, where $b \equiv \vert \pmb{b} \vert$). Our factorization theorem has four functions in the IR, the collinear, the global soft, the collinear-soft, the soft-collinear functions, see eq.~(\ref{eq:jet-fact-II}-\ref{eq:jj}), and all of them can potentially contribute to non-perturbative power corrections. Even though the collinear and collinear-soft functions do not contribute to $\pmb{q}_T$ perturbatively, they can still give a non-perturbative power correction to the $\bmat{q}_T$ spectrum\footnote{There are also power corrections of similar magnitude in this region due to the factorization of the $sc$ function from the $cs$, but they are perturbative in nature and can be handled by making a smooth transition to region III.}.
There are two types of non-perturbative corrections that we will consider here. We call \emph{shift} non-perturbative effects the ones which are not altered by the pass and fail procedure of the grooming conditions. An example is the global soft function that is independent of the grooming procedure and it is common to other TMD analysis. We refer to this kind of correction as \emph{shift} non-perturbative effects since, as we will see later, in the simplest case it generates a shift in the TMD spectrum. The second correction instead is related to the grooming procedure with $cs$ and $sc$ soft functions and the jet shape function. In this case non-perturbative effects are driven by the so called ``non-perturbative particles'' and it is obviously only possible when perturbative modes are on the boundary of passing and failing soft-drop. We refer to these contributions as \emph{boundary} non-perturbative effects.
\subsection{\emph{Shift} non-perturbative correction}
For the case of shift correction, we assume that the soft-drop condition remains unaltered by any non-perturbative emissions. Now consider the contribution to the shift correction by each function in turn.
The non-perturbative part of the global soft function defined in eq.~(\ref{eq:soft}) has been studied in the literature in several frameworks \cite{Beneke:1995pq,Korchemsky:1994is,Korchemsky:1997sy,Beneke:1997sr,Becher:2013iya,Scimemi:2016ffw}. Up to ${\cal O}(b^4)$ terms it can be written as
\begin{eqnarray}
\langle 0| T[S_nS^{\dagger}_{\bar n} (\pmb{b})] \bar{T}[ S_{\bar n}S_{n}^{\dagger}(0)] \vert 0 \rangle
= \tilde S( b) +b^2 \; \bar{C}^{(s)}_i( b) \langle 0| O^{i}|0 \rangle \;,
\end{eqnarray}
where $O^i$ is the complete set of local operators that have the same quantum numbers as the soft function. Summation over $i$ is implied. Here $\tilde{S}$ is the perturbative calculable part of the soft function and it contains rapidity and UV divergences as well as the rest of other terms in the equation. We can pull this out as a common factor to write
\begin{eqnarray}
&\langle 0| T[S_nS^{\dagger}_{\bar n} (\pmb{b})] \bar{T}[ S_{\bar n}S_{n}^{\dagger}(0)] |0\rangle
&= \tilde S( b) \Big{(} 1+ b^2\; C^{(s)}_i( b) \langle0| O^{i}|0\rangle \Big{)} \;.
\end{eqnarray}
To maintain the UV scale invariance of the cross section, we need that the second term in the brackets be independent of UV divergences. However additional rapidity divergences may be present in the non-perturbative matrix element on the r.h.s. that cancel with the corresponding rapidity divergence arising in the non-perturbative power corrections to the collinear or soft-collinear functions. This is related to the origin of the non-perturbative correction to the rapidity anomalous dimension and it is usually included also in TMD analysis.
We can perform a similar analysis for the soft-collinear ($sc$) function. When an $sc$ (perturbative) mode passes soft-drop, then it does not contribute to $q_T$ since it becomes part of the groomed jet. But since it has a large + component, it drives the groomed jet mass outside the region of measurement and hence such events are dropped. Therefore, we only need to consider the case when the $sc$ mode fails soft drop. In this case the non-perturbative emission contributes to the $q_T$ measurement if it lies outside the groomed jet. Given the angular scaling of this mode, which is much larger than the collinear-soft ($cs$) and collinear modes that form the groomed jet, the phase space region available is effectively unconstrained (this is also the reason why we ignore any phase space constraints on the soft non-perturbative emissions). Hence the correction in this case will also be a simple shift type and is implemented in the same manner as in the case of the global soft function. As before, we can pull out a common perturbative factor (that includes the perturbative soft drop condition), and write
\begin{eqnarray}
\tilde{S}_{sc}^{\perp}(b,z_{\text{cut}}) {\vert}_{\text{hadr.}} = \tilde{S}_{sc}^{\perp}( b, Qz_{\text{cut}} ) \Big{(} 1 + b^2 C^{(sc)}_i( b,z_{\text{cut}}) \; \langle 0| O^i |0\rangle \Big{)} .
\end{eqnarray}
Notice that now all the $z_\mathrm{cut}$ dependence of the power correction is included in the perturbative calculable coefficient $C^{(sc)}( b,z_{\text{cut}})$, which multiplies the same non-perturbative power correction present also in the global soft function case. The calculation of $C^{(s)},\; C^{(sc)}$ is doable perturbatively, although this consideration goes beyond the present work.
We can then combine all shift corrections that have an unconstrained phase space for non-perturbative emissions together so that in $b$ space we have a multiplicative correction to the perturbative cross section of the form
\begin{eqnarray}
S S_{sc}^{\perp} {\vert}_{\text{hadr.}} = (1+b^2 (\Omega_s+\Omega_{sc}) )S S_{sc}^{\perp} {\vert}_{\text{pert.}} \;,
\end{eqnarray}
where $\Omega_s$ is the same as the TMD case and $\Omega_s$ is a single parameter to be fitted from $e^+e^-$ experiments. It is clear that, in the event of non-trivial $C^{\{(s),\, (sc)\}}$, $\Omega_{s,\, sc}$ can have a mild (logarithmic) dependence on $q_T$ so that this model will work well over a limited range of $q_T$ which may be sufficient for most cases.
We now consider the shift corrections coming from the collinear-soft and the collinear functions. Since these modes determine the region of the groomed jet, we can consider two possible scenarios which give a non-trivial power correction.
\begin{enumerate}
\item{Collinear-soft ($cs$) particles pass soft-drop:}\\
If the $cs$ particles pass the soft-drop (for phase space see figure~\ref{fig:shift}(a)) then any non-perturbative emission scaling as the $cs$ mode can contribute to $q_T$ when it lies outside the groomed jet. In this case, we need to calculate the catchment area of the groomed jet that is determined by the angular distance of the $cs$ subject that passed soft-drop. As was pointed out in \cite{Hoang:2019ceu}, it is possible at NLL, using a coherent branching formalism, to factorize a purely non-perturbative function from all the calculable perturbative effects (including grooming). A detailed analysis of these corrections will be presented in a future work.
\begin{figure}[h!]
\centerline{\scalebox{1}{\includegraphics{Shift}}}
\caption{(a) When the collinear-soft ($cs$) function passes soft drop, the non-perturbative (NP) emissions, with the angular scaling of the $cs$ mode , with a virtuality $\Lambda_{QCD}$ must fall in the phase space shown by the blue shaded area in order to contribute to $q_T$. (b) When the $cs$ function fails soft drop, the NP emission with the angular scaling of the collinear modes must not be clustered with the collinear sub-jet in order to contribute to $q_T$.}
\label{fig:shift}
\end{figure}
\item{Collinear-soft particles fail soft-drop:}\\
In this case collinear modes are the only ones that pass soft-drop (for phase space see figure~\ref{fig:shift}(b)), so that any non-perturbative mode scaling as $cs$ has an unconstrained phase space, by the same logic as for the soft and the $sc$ functions, so that we get a simple shift correction of the same form as the soft, $sc$ and TMD collinear functions.\footnote{Technically in this case the perturbative value of $p_{\perp cs}$ would give a larger correction. However, this correction can eventually be handled by transitioning to a new EFT in which the $sc$ and $cs$ functions merge together. For now we will ignore them and only keep track of the other non-perturbative corrections.} There is another possible interesting correction that will come from the collinear NP emission that lies outside the catchment region that is now determined by the collinear modes alone.
In this case there are two ways of approaching the problem. In one, we consider separating out the non-perturbative corrections before factorizing the $cs$ and collinear modes. The other way is to realize that in the case where $cs$ fails soft-drop, the entire groomed jet mass measurement comes from the jet function alone and using this condition we can define a catchment area for the collinear non-perturbative emissions \textit{without explicitly accessing any information from the $cs$ function}, so that the factorization between the collinear and $cs$ modes is maintained. In this case, we can do a diagrammatic analysis, similar to \cite{Hoang:2019ceu}, for the collinear function, to check if it is possible to factorize the non-perturbative effects from the perturbative. We leave this work for the future.
\end{enumerate}
\subsection{Boundary corrections}
We now consider boundary corrections that leave the $q_T$ measurement function unchanged but only require an expansion of the soft-drop condition in $q^-/Q$. The functions that do not explicitly have a soft-drop condition can then be ignored, which leaves us with only the $sc$ and $cs$ functions. We can follow the same line of reasoning as in \cite{Hoang:2019ceu}.
\begin{figure}[h!]
\centerline{\scalebox{1}{\includegraphics{Boundary}}}
\caption{(a) The case where the sc subjet loses an NP emission (b) The case when the sc subjet gains an NP emission}
\label{fig:Boundary}
\end{figure}
\begin{enumerate}
\item{$sc$ emissions}\\
In this case we demand that either an addition or removal of the non-perturbative emission cause the soft-collinear function to fail soft-drop. Otherwise it will drive up the jet mass outside the measured range. If we consider a non-perturbative emission $q^{\mu}$ along with a perturbative momentum $p^{\mu}$, then we can expand out the soft-drop condition in the non-perturbative momentum.
We can write the complete measurement function as
\begin{eqnarray}
\Theta^{p \pm q} = \Theta\left( \frac{p+q}{E_J}-z_{\text{cut}}\right) \delta^2(\pmb{p}_{\perp sc}- \pmb{p}_{\perp}\mp \pmb{q}_T)\;,
\end{eqnarray}
where $p$ is the momentum of the perturbative $sc$ sub-jet while $q^{\mu}$ is the momentum of the non-perturbative emission. The $\pm$ signs indicate whether the perturbative $cs$ subject gains or loses a non-perturbative momentum after hadronization.
In the case where the $sc$ sub-jet gains a non-perturbative emission, the measurement expanded to leading order looks like
\begin{eqnarray}
\Theta^{p +q} \approx \Theta_{sd}^{p}\delta^2(\pmb{p}_{\perp sc}-\pmb{p}_{\perp})+\frac{q^-}{E_J} \Theta^{\text{b.c.}}( \theta_q,\theta_p, \Delta \phi) \delta^p_{sd}\Big[\delta^2(\pmb{p}_{\perp sc}- \pmb{p}_{\perp})\Big] \;,
\end{eqnarray}
with
\begin{align}
\Theta^{p}_{sd} &\equiv \Theta\left( \frac{p}{E_J}-z_{\text{cut}}\right)\;, & \delta^{p}_{sd} &\equiv \delta\left( \frac{p}{E_J}-z_{\text{cut}}\right)\;.
\end{align}
In this case, the non-perturbative emission $q^{\mu}$ gets clustered with the $sc$ subject. Note that we have expanded $q_i$ from the $\pmb{p}_{\perp}$ measurement since we are working at leading order. The phase-space constraint, $\Theta^{\text{b.c.}}$ (see figure~\ref{fig:Boundary}(a)), , gives the condition that ensures $q^{\mu}$ gets clustered with the $sc$ part.
The second case is when $q^{\mu}$ is emitted off $p^\mu$ but it is not clustered with the $sc$ jet. The short distance condition now acts on $p-q$, which can then be expanded out to give
\begin{eqnarray}
\Theta^{p -q} \approx \Theta_{sd}^{p}\delta^2(\pmb{p}_{\perp sc}- \pmb{p}_{\perp})-\frac{q^-}{E_J} \bar{\Theta}^{\text{b.c.}}( \theta_q,\theta_p, \Delta \phi) \delta^p_{sd}\Big[\delta^2(\pmb{p}_{\perp sc}- \pmb{p}_{\perp})\Big]\;,
\end{eqnarray}
$\bar \Theta^{\text{b.c.}}$ (see figure~\ref{fig:Boundary}(b)), is the phase space region for $q^{\mu}$ so that it falls outside the sc subjet. We can see that the leading power correction scales as $q^-/E_J$, which, given the angular scaling of the $sc$ mode, scales as $q_Tz_{cut}/Q$. Given a typical value of $z_{cut} \sim 0.1$, this factor is then comparable to the $q_T^2/Q^2$ correction that we get from the shift terms.
\item{Soft -Collinear function}\\
We expect that since perturbatively this function does not contribute to $q_T$, the boundary correction should have no effect on the $q_T$ measurement.
\end{enumerate}
We now have listed out all the possible NP corrections to the transverse momentum measurement.
\section{Conclusions}
\label{conclusion}
In this paper, we have presented the computation of the transverse momentum de-correlation observable for fat jets groomed using the Soft-Drop algorithm. We consider two scattering experiments: $e^+e^- \rightarrow$ di-jets and semi-inclusive DIS. In the former, we measure the transverse momentum imbalance between the two groomed jets. We impose a jet mass constraint on our jets in order to ensure collimated jet configurations. Simulation using PYTHIA show that grooming greatly reduces the impact of underlying events as well as final state hadronization. We show that the factorization theorem for this observable involves the universal soft function which also appears in the traditional definition of TMDs. We propose that this observable can be used as a probe of the non-perturbative rapidity anomalous dimension, which is a universal parameter for TMD distributions. We prove within our EFT that the cumulant jet mass constraint only adds to the overall normalization of the perturbative cross section and hence does not impact the shape of the transverse momentum distribution although it does contribute to the uncertainty. We gather or compute all the ingredients necessary to evaluate the cross section to NNLL accuracy and a numerical study for the cases of interest. In the implementation we have used the \texttt{artemide} code \cite{web,Scimemi:2017etj,Bertone:2019nxa} which contains the most recent extraction TMDPDF at higher perturbative orders. As part of the numerical analysis we have used the $\zeta$-prescription \cite{Scimemi:2018xaf} which allows a complete disentanglement of non-perturbative effects of rapidity evolution from the rest. An uncertainty analysis gives us an error band of approximately $\pm$ 10 $\%$. The main ingredient of this error is the perturbative uncertainty which can be systematically improved. As shown in fig.~\ref{fig:hadronization} the hadronization corrections at low $q_{T}$ are significantly smaller than the case of a standard jet axis and it is therefore one of the major advantage of using grooming. These effects are expected to be the same in $e^+e^-$ and SIDIS because of the factorization of the cross section. In the case of $e^+e^-$ these corrections constitute all of non-perturbative effects and they are associated with the final state shower. In order to do a meaningful extraction of non-perturbative parameters in this case, it is therefore necessary to improve the uncertainty from perturbative physics to be better than 5\%. This can be achieved by moving to a higher order in resummation accuracy (N3LL). This is something we leave as a follow up to this paper.
In the SIDIS case we measure the transverse momentum imbalance between the groomed jet and the recoiling lepton. Once again we demand a jet mass measurement in order to ensure sensitivity to collinear physics only. A large part of the contribution to this comes from the soft and collinear radiation that lies outside the jet and, for low transverse momentum, probes the complete TMDPDF. The cross section is again presented to NNLL accuracy and involve much of the same ingredients as in the case of $e^+e^- \rightarrow$ dijets. A higher order perturbative calculation is expected to reduce significatevely errors also in this case.
Concerning the hadronization effects we observe that grooming the jet allows us to have a wide angle jet, which is preferred in low energy experiments, while still being free from non-global logarithms, which are non-factorizable and they are usually present in un-groomed jets. Nevertheless it is possible to measure directly the hadronization effects due to grooming. The idea is to parametrize and extract all of the non-perturbative effects from $e^+e^- \rightarrow$ dijets and use them in SIDIS since they contain all the same matrix elements (in addition to the TMDPDF) as explained in sec.~\ref{hadronization}.
This gives us a robust way to access the TMDPDF while maintaining control over all other uninteresting non-perturbative effects.
\section*{Acknowledgements}
The authors would like to thank Aditya Pathak, Iain W. Stewart, and Wouter J. Waalewijn for useful discussions. I.S. likes to acknowledge the support of Los Alamos National Lab for his visit, during which part of the work was done. D.G.R. and I.S. are supported by the Spanish MECD grant FPA2016-75654-C2-2-P. This project has received funding from the European Union Horizon 2020 research and innovation program under grant agreement No 824093 (STRONG-2020). D.G.R. acknowledges the support of the Universidad Complutense de Madrid through the predoctoral grant CT17/17-CT18/17. Y.M. and V.V. are supported by the U.S. Department of Energy through the Office of Science, Office of Nuclear Physics under Contract DE-AC52-06NA25396 and by an Early Career Research Award, through the LANL/LDRD Program. V.V. is also supported within the framework of the TMD Topical Collaboration. L.Z. is supported by ERC grant ERC- STG-2015-677323.
|
2,869,038,155,724 | arxiv | \section{Introduction}
The Quark-Gluon Plasma (QGP) phase produced in relativistic nucleus-nucleus collisions is expected to have a transient existence and the information about the space-time evolution and dynamics this hot and dense matter are obtained from the various final state observables. Photons (real as well as virtual) are emitted from the entire lifetime of the matter evolution and the weak gauge bosons are produced from the initial hard scatterings. These electroweak probes do not suffer strong interaction with the medium, carry undistorted information from their production point to the detector and thus are regarded as efficient probes to study the initial state and the properties of the strongly interacting matter~\cite{mclaurren, gojkohp, zvi}.
\section{Direct Photons}
Electromagnetic radiations are known as the thermometer of the medium from early days of heavy ion collisions and the direct photons were initially studied to get the temperature of the system formed in these collisions~\cite{dks_qm}.
The experimentally measured inclusive photon spectrum contains a huge background that originates (mostly) from the 2-$\gamma$ decay of $\pi^0$ and $\eta$ mesons. The direct photon spectrum is obtained by subtracting this decay background from the inclusive photon spectrum. Photons produced from the various stages of the matter evolution contribute to the direct photon spectrum.
The prompt photons are produced from the initial hard scatterings and the pre-equilibrium photons are emitted before the medium gets thermalized. These photons dominate the high $p_T$ region of the direct photon spectrum. The thermal photons are radiated from the QGP as well as from the hot hadronic matter and populate the $p_T < 4$ GeV region of the transverse momentum spectrum. The jet-conversion process also contributes significantly to the direct photon spectrum.
There has been significant advancement in the decay background subtraction methods in last couple of decades. The WA98 Collaboration first used the invariant mass analysis method to subtract the decay background from the inclusive photon spectrum~\cite{wa98}. In recent times the PHENIX Collaboration at RHIC and the ALICE Collaboration at the LHC have used several sophisticated methods for the subtraction of the decay background.
The experimental data at RHIC and at the LHC have shown an excess of direct photon yield from heavy ion collisions compared to the (scaled) photon yield from p+p collisions in the region $p_T < 4$ GeV. This excess yield is considered as thermal radiation from the hot and dense medium~\cite{ph_g, al_g}. The direct photon anisotropic flow has been measured at RHIC and LHC energies at different centrality bins~\cite{v2_ex, v2_ex1}. It is well known that the thermal radiation completely dominates the photon anisotropic flow parameter~\cite{phot_v2} and the contributions from other (non-thermal) sources to photon $v_n$ are negligible. However, the theoretical model calculations which explain the charged particle spectra and anisotropic flow successfully, have been found to underpredict the experimental data of photon anisotropic flow by a significant margin~\cite{puzzle}. This is known as direct photon puzzle.
Direct photon data from a number of collision systems have been reported by the PHENIX Collaboration at RHIC in recent times. Apart from the Au+Au and Cu+Cu collisions at 200A GeV, photons from 62.4 and 39A GeV Au+Au collisions have been measured by PHENIX~\cite{veronika, phot_scaling}. Photon data from p+p, p+Au, and d+Au collisions at 200 A GeV are also available now. The 0--5\% p+Au collisions have indicated a thermal production from the medium among the small systems at RHIC.
The PHENIX Collaboration has also reported a new analysis method for the direct photon measurement where the double ratio $R_\gamma$ is calculated using external conversion (that leads to better cancellation of systematics). The results from the new analysis are found to be consistent with the $R_\gamma$ obtained from the internal conversion, virtual, as well as calorimeter methods~\cite{veronika}
One important observation in recent time is the universal scaling behaviour of the low $p_T$ photon yield which scales with the charged particle multiplicity as $(dN_{\rm ch}/d\eta)^\alpha$~\cite{phot_scaling}. The scale factor $\alpha$ (is about 1.25) indicates that the photon multiplicity grows faster than the charged particle multiplicity. The scaling behaviour is found to be independent of the energy, centrality, and system size (see Fig. 1). The STAR data also show a similar scaling however, the magnitude is smaller than PHENIX.
It is to be noted that a theory calculation by Cleymans ${\it et \ al.}$~\cite{dks} predicted similar scaling behaviour between photons and charged particle multiplicity long before where they estimated the scaling coefficient to be about 1.2, close to the $\alpha$ value by PHENIX.
New prompt photon data at LHC have been reported by the ALICE Collaboration. Isolated prompt photons from 7 TeV p+p collisions are available in the range 10 $< \, p_T \, < 60$ GeV where an isolation cut is used to reduce the fragmentation photons. The data is found to be in good agreement with the JETPHOX NLO pQCD calculations~\cite{Dhrub}.
\begin{figure}
\centerline{\includegraphics*[width=11.0 cm, clip=true]{1.ps}}
\caption{(Color online) Integrated direct photon yield ($1.0 < p_T < 5.0$ GeV) scales with the charged particle multiplicity $dN_{\rm ch}/d\eta$. The figure is from Ref.~\cite{phot_scaling}.}
\label{fig3}
\end{figure}
There has been significant advancement in the theory calculation of direct photon spectra and anisotropic flow from relativistic nuclear collisions in recent years. Paquet ${\it {et \ al.}}$~\cite{kompost} have shown that the radiation from pre-hydro phase can be a substantial contribution to the photon and hadronic observables. They introduced a pre-hydro phase KOMPOST between the CGC initial state and the hydrodynamical model evolution to study the effect on photon spectra and elliptic flow parameter. The inclusion of this new phase enhances the photon production significantly which is reflected in the $p_T>$ 3 GeV region of the spectrum. However,
the anisotropic flow parameter is found to change only marginally by the inclusion of the pre hydro phase.
The direct photon puzzle still remains as one of the interesting unsolved problems in relativistic heavy ion collisions. The experimental data from a number of small and large systems as well as from different beam energies have opened up the possibility of understanding the initial state as well as the photon puzzle better. Additionally, photon production from asymmetric collisions (C+Au)~\cite{phot_v1} and also from collisions of deformed nuclei (U+U)~\cite{uu} can play an important role in this regard.
The fully overlapping U+U collisions can lead to different collision geometry depending on the orientation of the colliding nuclei. A recent study using hydrodynamical model calculation has shown that the photon production from tip-tip configuration of U+U collisions is comparable to the production from most central Au+Au collisions at RHIC~\cite{uu}. On the other hand, the elliptic flow from body-body configuration of uranium nuclei is found to be close to the photon $v_2$ from mid-central Au+Au collisions.
The directed flow of photons from a hydrodynamical model calculation has been found to be significantly large and it shows different behaviour compared to the elliptic and triangular flow parameters~\cite{phot_v1}. The $v_1(p_T)$ is found to be negative at smaller $p_T$ values unlike the higher order flow coefficients and it is also found to be dominated by the QGP radiation in the entire $p_T$ region.
It has been shown in a recent study that the initial state nucleon shadowing in the Monte Carlo Glauber model increases the anisotropic flow of photons significantly. The effect of this initial state nucleon shadowing is found to be more prominent for photon observables compared to the hadronic observables~\cite{shadow}.
A radiative recombination model is used by Nonaka $\it {et \ al.}$~\cite{nonaka} to study photon production from heavy ion collisions. In a recombination model, the hadrons are formed by coalescence of valence quarks. The model is modified to allow photon emission and processes such as $q \bar{q} \rightarrow \pi^0 \gamma$ are considered when the QGP hadronizes. They obtain an exponential $p_T$ distribution for the photons similar to a thermal distribution with an effective temperature given by the (blue-shifted) recombination temperature.
The photon-jet correlations can play an important role in understanding the initial state produced in relativistic nuclear collisions. The photon jet transverse momentum imbalance and azimuthal correlations have been studied using JETSCAPE framework in p+p and heavy ion collisions at LHC energy by Sirimanna ${\it et \ al.}$~\cite{sirimanna} . The JETSCAPE is a multistage framework which uses several modules to simulate different stages of jet propagation through the QGP medium. A significantly improved agreement with the photon data has been observed compared to the earlier calculations.
The photon HBT interferometry can be useful to get information about the spatio-temporal evolution of the system produced in relativistic nuclear collisions~\cite{dks_hbt}.
The feasibility of such study at LHC energy has been estimated by Garcia-Montero $\it {et \ al.}$~\cite{garcia_hbt} where they show that the HBT correlation at low $k_T$ ($<1$ GeV) can be statistically significant.
\begin{figure}
\centerline{\includegraphics*[width=9.0 cm,clip=true]{3.ps}}
\caption{(Color online) Elliptic flow of thermal dileptons is sensitive to the shear viscosity parameter $\eta/s(T)$. The figure is from Ref.~\cite{gojko}.}
\label{fig2}
\end{figure}
\section{Dileptons}
The virtual photons or dileptons are also produced from all stages of the fireball evolution similar to the real photons and are considered as a potential probe to study the properties of the strongly interacting matter produced in relativistic nuclear collisions~\cite{wong, rapp}. The dileptons are massive unlike the the real photons. The invariant mass $M_{\rm {ll}}$ of the dilepton pair and the transverse momentum $p_T$, both are tuned to access the different stages of the matter evolution.
In a dilepton mass spectrum, the region below $\phi$ mass is known as the low mass region, $\phi$ mass to $J/\Psi$ mass is known as intermediate mass region and above that is called the high mass region. The dilepton pairs having large $M_{\rm {ll}}$ and $p_T$ are mostly emitted from the hot and dense early stages of the collisions and those having relatively smaller $p_T$ and $M_{\rm ll}$ are emitted from the later stages of the evolving system when the flow is strong.
It has been shown that dileptons with $M_{\rm ll} >$ 1 GeV are dominated by QGP radiation whereas, $M_{\rm ll} <$ 1 GeV dileptons are mostly due to hadronic matter radiation and are important to study the chiral symmetry breaking/restoration. The anisotropic flow of dileptons using a hydrodynamical model calculation shows rich structure as a function of $p_T$ as well as invariant mass due to the interplay of the emission from fluid elements at different temperatures with varying radial flow pattern~\cite{rupa_dil_v2}.
Recent studies have shown that Bayesian analysis simulating the soft hadronic observables from various stages of heavy ion collisions can be useful to constrain the transport coefficients. The preliminary results by JETSCAPE simulation group show that the high temperature behaviour of the $\eta/s$ and $\zeta/s$ are not very clear and the anisotropic flow of dileptons can be used to constrain these shear and bulk viscosity coefficients~\cite{gojkohp}.
Transport models such as coarse grained URQMD~\cite{urqmd} and SMASH~\cite{smash} are used to study the dilepton production at lower beam energies. Vujanovic ${\it et \ al.}$~\cite{gojko} have shown that the dilepton $v_2$ at large invariant masses can be a probe for QGP and the results are sensitive to the relaxation time.
The new results of dilepton production from photon-photon interactions have gathered a lot of attention in recent times~\cite{gg}.
The intense electromagnetic field surrounding a highly charged heavy nucleus in relativistic collisions provides a flux of quasi-real photons where the photon flux increases with (square of) nuclear charge. The dileptons produced through the $\gamma \gamma \rightarrow l^+ l^-$ process are measured in ultra-peripheral collisions.
The STAR and ATLAS experiments have also measured the dilepton production from photon interactions in hadronic collisions complementing the results from ultra-peripheral collisions.
The new low mass preliminary STAR dielectron data from Au+Au collisions at 27 and 54.4 GeV significantly enhance the precision of the in medium rho modification measurements compared to the STAR BES-I results~\cite{star_ee}. The ALICE Collaboration has reported high statistical precision dielectron spectra from Pb+Pb and p+p collisions at 5.02A TeV at LHC analyzing the 2018 data and also soft dielectrons from p+p collisions at 13 TeV with B=0.2 T~\cite{alice_ee}.
The STAR experiment at RHIC has reported the invariant mass and yield distribution of inclusive dimuons in the low $p_T$ region for different centralities of Au+Au collisions. The measurement is done in the mass range 3.2 to 10 GeV/$c^2$ utilizing the Muon Telescope Detector. They observe a significant enhancement with respect to the cocktail in the 60--80\% centrality bin and data are found to be consistent with theoretical calculations~\cite{star_mu}.
Interesting new data on the impact parameter dependence of dimuon acoplanarity in ultra-peripheral Pb+Pb collisions have been reported by the CMS Collaboration. The acoplanarity is basically the relative angular deflection of the dimuon pair. They show that the centrality dependent $\gamma \gamma \rightarrow \mu^+ \mu^-$ production provide valuable insight about the origin of observed broadening of lepton pairs produced from $\gamma \gamma$ scatterings in hadronic collisions~\cite{cms_mu}.
The yield and distributions of dimuons from $\gamma \gamma \rightarrow \mu^+ \mu^-$ processes in Pb+Pb collisions have been measured by ATLAS collaboration in ultra-peripheral as well as non-ultra peripheral collisions. The distribution of acoplanarity and $k_T$ show significant centrality dependence in the preliminary ATLAS data~\cite{atlas_mu}.
\section{W{$^\pm$} and $Z$ bosons}
\begin{figure}
\centerline{\includegraphics*[width=9.0 cm,clip=true]{5.ps}}
\caption{(Color online) The nuclear modification factor $R_{\rm PbPb}$ as a function of rapidity calculated using nuclear-suppressed $\sigma^{\rm inel}_{\rm nn}=$41.5 mb from Ref.~\cite{kari}. ATLAS data points are from~\cite{at1, at2}.}
\label{fig3}
\end{figure}
The massive vector bosons are produced from the initial state in heavy ion collisions before the QGP is formed and are considered as valuable probes to study the nuclear modifications of the parton distribution functions (PDF).
The effect of nuclear shadowing on the inelastic nucleon nucleon cross-section $\sigma_{\rm {NN}}$ has been studied by Eskola $\it {et. al.}$~\cite{kari} in an interesting work using the high precision ATLAS $W^\pm$ and $Z$ data from Pb+Pb collisions at 5.02A TeV~\cite{at1, at2}. The Glauber model formalism is widely used in centrality dependent calculations in heavy ion collisions where the beam energy dependence is incorporated using the $\sigma_{\rm {NN}}$ obtained from the p+p measurements. Thus, a suppression in the value of $\sigma_{\rm {NN}}$ is expected due to shadowing and saturation phenomena at small $x$ values for larger nuclei compared to protons. The NNLO QCD calculations and nuclear PDFs are used to estimate the $\sigma_{\rm NN}$ in~\cite{kari} and a significantly suppressed value (about 41.5 mb down from 71 mb at 5.02A TeV) is obtained which is expected to affect the experimental analysis of the nuclear modification factor.
Gauge boson associated with jet production can provide valuable information about jet-quenching and parton energy loss in the hot and dense medium. The $W^\pm$+jets production at LHC energy has been studied by Zhang ${\it et \ al.}$~\cite{sherpa} using a Monte Carlo event generator SHERPA (a sharp and smooth algorithm), the Linear Boltzmann transport model and a parton shower. The $W^\pm$+jets, which are dominated by quark jets are expected to provide complementary information on jet quenching along with Z+jets and $\gamma$-jets.
There has been significant development in the $W^\pm$ and $Z$ boson measurements from p+p, p+Pb, and Pb+Pb collisions at different LHC energies. The ATLAS Collaboration has measured the $W^\pm$ and Z bosons from p+p and Pb+Pb collisions at 5.02A TeV where the production yields are observed via their leptonic decay channels~\cite{atlas_w}. The p+p data are found to be in good agreement with the NNPDF3.1 and the Pb+Pb data match well with all the theory predictions. They also show that the isospin effect plays an important role in the analysis of $W^\pm$ data.
The CMS Collaboration has recently reported preliminary $Z$ boson data from Drell-Yan process at 8.16A TeV p+Pb collisions at LHC and the new measurement is extended to a lower mass region where the data are found to be explained well by EPPS16 with shadowing compared to free nucleon PDFs~\cite{cms_w}. The Z boson yield at various centrality bins are compared to the HG-PYTHIA model. Additionally, a high precision $Z$ boson azimuthal anisotropy measurement from Pb+Pb collisions by CMS Collaboration shows that the $v_2$ is consistent with zero.
The production of $Z$ boson from 8.16A TeV p+Pb collision is also reported by LHCb experiment where the predictions at the forward and backward rapidities are found to be sensitive to the nPDFs in a unique kinematic domain~\cite{lhcb}. The structure of the nucleus can be studied in a complementary fashion from the LHCb results.
The ALICE W$^\pm$ data from Pb+Pb collisions at 5.02A TeV show that the $R_{\rm {AA}}$ does not depend on the collision centrality significantly. The $Z$ boson $R_{\rm {AA}}$ by ALICE is found to be consistent with theoretical calculations and only at large rapidities the value deviates from 1~\cite{alice_w}.
All these new high precision data and the sophisticated theory calculations promise a bright future for the electroweak probes in relativistic nuclear collisions.
|
2,869,038,155,725 | arxiv | \section{\normalsize Introduction}
In this paper we propose an inexact augmented Lagrangian algorithm~(ALCC)
for solving conic convex problems of the form
\begin{equation}
\label{eq:conic_problem}
(P): \min \big\{\rho(x) + \gamma(x) : Ax-b\in\mathcal{K},~ x \in \chi\big\},
\end{equation}
where
$\rho:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}$,
$\gamma:\mathbb{R}^n\rightarrow\mathbb{R}$ are proper, closed, convex
functions, and $\gamma$ has a Lipschitz continuous gradient $\nabla
\gamma$ with the Lipschitz constant $L_{\gamma}$, $A\in\mathbb{R}^{m\times
n}$, $\mathcal{K}\subset\mathbb{R}^m$ is a
nonempty, closed, convex cone, and $\chi\subset\mathop{\bf dom}(\rho)$ is a ``simple''
compact set in the sense that the optimization problems of the form
\begin{equation}
\label{eq:nonsmooth-operation}
\min_{ x \in \chi} \left\{\rho(x)+\norm{x-\bar{x}}_2^2\right\}
\end{equation}
can be efficiently solved for any $\bar{x}\in\mathbb{R}^n$. Note
that we do not require $A\in\mathbb{R}^{m\times n}$ to satisfy any
additional regularity properties.
For notational convenience, we set
\[
p(x) := \rho(x) + \gamma(x).
\]
In some problems, the compact set $\chi$ is explicitly present. For example, in a zero-sum game the decision $x$ represents a mixed strategy and the set $\chi$ is a simplex. In others, $\chi$ may not be explicitly present, but one can formulate an equivalent problem where the vector of decision variables can be constrained to lie in a bounded feasible set without any loss of generality. For example, if $\gamma$ is strongly convex, or if $\rho$ is a norm and $\gamma(\cdot)\geq 0$, then the decision vector $x$ can be restricted to lie in a appropriately defined norm ball centered at any feasible solution.
We assume that the following constraint qualification holds for $(P)$.
\begin{assumption}
\label{asp:KKT}
The problem $(P)$ in \eqref{eq:conic_problem} has a Karush-Kuhn-Tucker~(KKT) point,
i.e., there exists $y^*\in\mathcal{K}^*$ such that
$g_0(y^*):=\inf\{p(x)-\fprod{y^*,~Ax-b}:\ x\in\chi\} =p^* > -\infty$,
where $p^\ast$ denotes the optimal value of $(P)$ and $\mathcal{K}^\ast$
denotes the dual cone corresponding to $\mathcal{K}$, i.e., $\mathcal{K}^\ast:=\{y\in\mathbb{R}^m:\ \fprod{y,x}\geq 0\ \forall x\in\mathcal{K}\}$.
\end{assumption}\\
Assumption~\ref{asp:KKT} clearly holds whenever there exists
$\tilde{x}\in\mathop{\bf rel int}(\chi)$ such that $A\tilde{x}-b\in
\mathbf{int}(\mathcal{K})$~\cite{Boyd04_1B}.
\subsection{Special cases}
Many important optimization problems are special cases of
\eqref{eq:conic_problem}. Below, we briefly discuss some
examples.
\textbf{\emph{Min-max games with convex loss function:}}
This problem is a generalization of the matrix game discussed in
\cite{Nesterov05}. The decision maker can choose from $n$ possible
actions. Let $x \in \mathbb{R}^n_{+}$ denote a mixed strategy over the set of
actions, i.e., $x\in\chi := \{x: \sum_{j=1}^n x_j = 1, x \geq \mathbf{0}\}$. Suppose the
mixed strategy $x$ must
satisfy constraints of the form $Ax - b \in \mathcal{K}$. These constraints
could be modeling average cost constraints. For example, one may
have constraints of the form $Ax \leq b$, where $A \in \mathbb{R}^{m\times n}$ and $A_{ij}$ denotes
amount of resource $i$ consumed by action $j$. One may also have
constraints that restrict the total probability weight of some given subsets
of actions.
The adversary has $p$ possible actions. The expected loss to decision maker when she
chooses the mixed strategy $x\in\mathbb{R}^n$ and the adversary chooses the mixed
strategy $y\in \mathbb{R}^p$ is given by
\[
\rho(x) + y^T Cx - \phi(y),
\]
where $\rho$ is a convex function, and $\phi$ is a strongly convex
function. Then the decision maker's optimization problem that minimizes the expected worst case loss is given by
\begin{equation}
\label{eq:min-max-strat}
\min \left\{ \rho(x) + \gamma(x):\ Ax - b \in \mathcal{K},\ x \in \chi\right\},
\end{equation}
where
\begin{equation}
\label{eq:matrix-game}
\gamma(x) = \max\bigg\{ y^T Cx -\phi(y):\ \sum_{k=1}^py_k = 1,\ y \geq \mathbf{0}\bigg\}.
\end{equation}
From Danskin's theorem, it follows that
$\nabla\gamma(x)=C^Ty(x)$, where $y(x)$ denotes the unique minimizer in
\eqref{eq:matrix-game} for a given $x$.
In \cite{Nesterov05}, Nesterov
showed that $\nabla\gamma$ is Lipschitz continuous with Lipschitz
constant $\sigma_{\max}(C)^2/\tau$, where $\tau$ denotes the convexity
parameter for the strongly convex function $\phi$. Thus, it follows that
the minimax optimization problem~\eqref{eq:min-max-strat} is a special
case of \eqref{eq:conic_problem}.
\textbf{\emph{Problems with semidefinite constraints:}}
Let $\mathcal{S}^m$ denote the set of $m \times m$ symmetric matrices, and
let $\mathcal{S}^m_+$ denote the closed convex cone of $m \times m$ symmetric positive
semidefinite matrices.
A convex optimization problem with a linear matrix inequality constraint is of the form
\begin{equation}
\label{eq:cvx-sdp}
\min \bigg\{ \rho(x): \sum_{j=1}^n A_j x_j + B \in \mathcal{S}^m_+\bigg\},
\end{equation}
where $\rho$ is a convex function, $B \in \mathcal{S}^m$, and $A_j \in \mathcal{S}^m$
for $j = 1, \ldots, n$.
Convex problems of the form~\eqref{eq:cvx-sdp} can model many applications in
engineering, statistics
and combinatorial optimization
\cite{Boyd04_1B}. In most of these applications, either the
constraints imply that the decision vector $x$ is bounded, or one can often
establish that the optimal solution lies in a norm-ball. In such cases,
\eqref{eq:cvx-sdp} is a special case of \eqref{eq:conic_problem}.
Consider the $\ell_1$-minimization problem of the form
\begin{equation}
\label{eq:min-l1-LMI}
\min\bigg\{\norm{x}_1: \sum_{j=1}^n A_j x_j + B \in \mathcal{S}^m_+\bigg\}.
\end{equation}
Suppose a feasible solution $x_0$ for this problem is known.
Then~\eqref{eq:min-l1-LMI} is a special case of \eqref{eq:conic_problem}
with $\rho(x)=\norm{x}_1$,
$\gamma(\cdot)= 0$, $\mathcal{K}=\mathcal{S}^m_+$ and
$\chi=\{x\in\mathbb{R}^n:\norm{x}_1\leq\norm{x_0}_1\}$.
The main bottleneck step in solving this problem using the ALCC algorithm\
reduces to the ``shrinkage'' problem of the form
$\min\{\lambda\norm{x}_1+\norm{x-\bar{x}}_2^2:\ \norm{x}_1\leq
\norm{x_0}_1\}$
that can be solved very efficiently for any given $\bar{x}\in\mathbb{R}^n$ and $\lambda>0$.
\subsection{Notation}
Let $S\subset\mathbb{R}^m$ be a nonempty, closed, convex
set. Let $d_S:\mathbb{R}^m\rightarrow\mathbb{R}_+$
denote the function
\begin{equation}
\label{eq:dist_func}
d_S(\bar{x}):=\min_{x \in S} \norm{x-\bar{x}}_2,
\end{equation}
i.e., $d_{S}(\bar{x})$ denotes the $\ell_2$-distance of the vector
$\bar{x}\in\mathbb{R}^m$ to the set $S$. Let
\begin{equation}
\label{eq:Kproj}
\Pi_S(\bar{x}):=\mathop{\rm argmin}\{\norm{x-\bar{x}}_2:\ x\in S\},
\end{equation}
denote the $\ell_2$-projection of the vector $\bar{x}\in\mathbb{R}^m$ onto the set
$S$. Since $S\subset\mathbb{R}^m$ is a nonempty, closed, convex set,
$\Pi_S(\cdot)$ is well defined. Moreover, $d_{S}(\bar{x}) =
\norm{\bar{x}-\Pi_{S}(\bar{x})}_2$.
\subsection{New results}
The main results of this paper are as follows:
\begin{enumerate}[(a)]
\item Every limit point of the sequence of ALCC primal iterates $\{x_k\}$
is an optimal solution of \eqref{eq:conic_problem}.
\item The sequence of ALCC dual iterates $\{y_k\}$ converges to a KKT point of
\eqref{eq:conic_problem}.
\item For \emph{all} $\epsilon>0$, the primal ALCC iterates $x_k$ are
$\epsilon$-feasible, i.e., $x_k\in\chi$ and $d_\mathcal{K}(Ax_k-b)\leq\epsilon$,
and $\epsilon$-optimal, i.e., $|p(x_k)-p^*|\leq\epsilon$ after at most
$\mathcal{O}\left(\log\left(\epsilon^{-1}\right)\right)$ ALCC
iterations that require solving at most
$\mathcal{O}(\epsilon^{-1}\log(\epsilon^{-1}))$ problems of the
form \eqref{eq:nonsmooth-operation}.
\end{enumerate}
\smallskip
\noindent Since \eqref{eq:conic_problem} is a conic convex programming
problem, many special cases of \eqref{eq:conic_problem}
can be solved in polynomial time, at least in theory, using interior point
methods. However, in practice, the interior point methods are not able
to solve very large
instances of~\eqref{eq:conic_problem} because the computational
complexity of a matrix
factorization step, which is essential in these methods, becomes prohibitive.
On the other hand, the computational bottleneck in the ALCC algorithm\ is the
projection \eqref{eq:nonsmooth-operation}. In many optimization
problems that arise in applications, this projection can be solved
very efficiently as is the case with noisy compressed
sensing and matrix completion problems discussed in \cite{Ser10_1J}, and the convex
optimization problems with semidefinite constraints discussed above.
The convergence results above imply that the ALCC algorithm\ can solve very large instances of
\eqref{eq:conic_problem} very efficiently provided the corresponding
projection~\eqref{eq:nonsmooth-operation} can be solved
efficiently. The numerical results reported in~\cite{Aybat11_1J,Ser10_1J} for a
special case of ALCC algorithm\ provide evidence that our proposed algorithm
can be scaled to solve very large instances of the conic
problem~\eqref{eq:conic_problem}.
\subsection{Previous work}
\label{sec:previous}
Rockafellar~\cite{Roc73_1J} proposed an inexact augmented Lagrangian
method to solve problems of the form
\begin{equation}
\label{eq:rockafellar_problem}
p^*=\min \big\{p(x) : f(x)\geq 0,~x \in \chi\big\},
\end{equation}
where $\chi\subset\mathbb{R}^n$ is a closed convex set,
$p:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}$ is a convex function and
$f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ such that each
component $f_i(x)$ of $f = (f_1, \ldots, f_m)$ is a
concave function for $i=1,\ldots,m$.
Rockafellar~\cite{Roc73_1J} defined the ``penalty'' Lagrangian
\begin{equation}
\label{eq:Lagrangian_rock}
\tilde{\mathcal{L}}_\mu(x,y):=p(x)+\frac{\mu}{2}~\Big\|\Big(\frac{y}{\mu}-f(x)\Big)_+\Big\|_2^2
-\frac{\norm{y}_2^2}{2\mu},
\end{equation}
where $(\cdot)_+:=\max\{\cdot,\mathbf{0}\}$ and $\max\{\cdot,\cdot\}$ are
componentwise operators, and $\mu$ is a fixed penalty parameter.
Rockafellar~\cite{Roc73_1J} established that given $y_0 \in \mathbb{R}^m$,
the primal-dual iterates sequences
$\{x_k, y_k\} \subset \chi\times \mathbb{R}^m$
computed according to
\begin{align}
\tilde{\mathcal{L}}_\mu(x_k,y_k)&\leq\inf_{x\in\chi}\tilde{\mathcal{L}}_\mu(x,y_k)
+\alpha_k, \label{eq:x-update_rock}\\
y_{k+1}&=\left(y_k+\mu f(x_k)\right)_+, \label{eq:y-update_rock}
\end{align}
satisfy $\lim_{k\in\mathbb{Z}_+} p(x_k)=\bar{p}$ and $\limsup_{k\in\mathbb{Z}_+}
f(x_k)\leq \mathbf{0}$ when
\eqref{eq:rockafellar_problem} has a KKT point and the parameter sequence $\{\alpha_k\}$
satisfies the summability condition
$\sum_{k=1}^{\infty}\sqrt{\mu~\alpha_k}<\infty$. Martinet~\cite{Mar78_1J} later
showed that the summability condition on
parameter sequence $\{\alpha_k\}$ is not necessary. However, in both
\cite{Mar78_1J,Roc73_1J} no iteration complexity result was given for
the algorithm \eqref{eq:x-update_rock}--\eqref{eq:y-update_rock} when $p$
was not continuously twice differentiable.
In this paper we show convergence rate results for an
augmented Lagrangian algorithm where we allow
penalty parameter $\mu$ to be a non-decreasing positive sequence $\{\mu_k\}$. After we
had independently
established these results, which are extensions of our previous results
in~\cite{Ser10_1J}, we became aware of a previous work by
Rockafellar~\cite{Rockafellar1976} where he proposed several different
variants of the algorithm in
\eqref{eq:x-update_rock}--\eqref{eq:y-update_rock} where $\mu$
could be updated between iterations. Rockafellar~\cite{Rockafellar1976}
established that for all non-decreasing positive multiplier sequences
$\{\mu_k\}$ satisfying the summability condition
$\sum_{k=1}^{\infty}\sqrt{\mu_k~\alpha_k}<\infty$, $\{y_k\}$ is bounded
and any limit point of $\{x_k\}$ is optimal to
\eqref{eq:rockafellar_problem}; moreover,
\begin{align}
\label{eq:rock-rate}
\max_{i=1,\ldots, m}\{f_i(x_k)\}\leq
\frac{\norm{y_{k+1}-y_k}_2}{\mu_k},
\quad p(x_k)-p^* \leq
\frac{1}{2\mu_k}(\alpha_k+\norm{y_k}_2^2).
\end{align}
Note that the results in \cite{Rockafellar1976} only provide an
\emph{upper bound} on the sub-optimality; no lower bound is
provided. Since the iterates $\{x_k\}$ are only feasible in the limit,
it is possible that $p(x_k) \ll p^\ast$ and establishing a lower bound on the
sub-optimality is critical. Moreover,
Rockafellar~\cite{Rockafellar1976} does not discuss how to compute
iterates satisfying~\eqref{eq:x-update_rock} and assumes that a
black-box oracle produces such iterates; consequently, there are no
basic operation level complexity bounds in~\cite{Rockafellar1976}.
In this paper, we extend
\eqref{eq:rockafellar_problem}
to a conic convex program where $f(x)=Ax-b$, and $\mathcal{K}$ is a closed,
convex cone. We show that primal ALCC iterates
$\{x_k\}\subset\chi$ satisfies $d_\mathcal{K}(Ax_k-b)\leq\mathcal{O}(\mu_k^{-1})$ and
$|p(x_k)-p^*|\leq\mathcal{O}(\mu_k^{-1})$, i.e. we provide \emph{both} an upper and a
lower bound, using an inexact stopping condition that is an extension of
\eqref{eq:x-update_rock}. ALCC algorithm\ calls an optimal first order method, such as
FISTA~\cite{Beck09_1J}, to compute an iterate
$x_k$ satisfying a stopping condition similar to
\eqref{eq:x-update_rock}. By
carefully selecting the sub-optimality parameter sequence $\{\alpha_k\}$ and the
penalty parameter sequence $\{\mu_k\}$, we are able to establish a bound
on the
number of generalized
projections of the form \eqref{eq:nonsmooth-operation} required
to obtain an $\epsilon$-feasible and $\epsilon$-optimal solution to
\eqref{eq:conic_problem}, and also provide an operation level
complexity bound.
In \cite{Rockafellar1976}, Rockafellar also provides an iteration complexity result for a different inexact augmented Lagrangian method. Given a non-increasing sequence $\{\alpha_k\}$ and a non-decreasing
sequence $\{\mu_k\}$ such that $\sum^{\infty}_{k=1}\sqrt{\mu_k~\alpha_k}<\infty$,
the infeasiblity and suboptimality can be \emph{upper} bounded (see~\eqref{eq:rock-rate})
when the duals $\{y_k\}$ are updated according to
\eqref{eq:y-update_rock} and the primal iterates $\{x_k\}$
satisfy
\begin{align}
\inf\{\norm{s}_2:\
s\in\partial\phi_k(x_k)\}\leq\sqrt{\frac{\alpha_k}{\mu_k}},
\label{eq:rock-dist}
\end{align}
where $\phi_k(x):=
\tilde{\mathcal{L}}_{\mu_k}(x,y_k)+\mathbf{1}_\chi(x)+\frac{1}{2\mu_k}\norm{x-x_{k-1}}_2^2$,
$\tilde{\mathcal{L}}_{\mu_k}$ is defined in \eqref{eq:Lagrangian_rock} and $\mathbf{1}_\chi$ is the indicator function of the closed convex set $\chi$.
With this new stopping condition, Rockafellar~\cite{Rockafellar1976} was
able to establish a \emph{lower} bound $p(x_k)-p^*\geq
-\mathcal{O}(\mu_k^{-1})$. Note that the stopping condition \eqref{eq:rock-dist}
is much stronger than \eqref{eq:x-update_rock} -- in this paper we establish
the lower bound using the weaker stopping condition \eqref{eq:x-update_rock}.
First order methods for minimizing
functions with Lipschitz continuous
gradients~\cite{Nesterov04,Nesterov05} (and also the non-smooth
variants~\cite{Beck09_1J,Tseng08}) can only guarantee convergence in
function values; therefore, the subgradient condition~(\ref{eq:rock-dist})
has to be re-stated
in terms of function values in order to use a first-order algorithm to
compute the iterates. This is impossible when the objective function is
non-smooth. Therefore, one cannot establish operational level
complexity results for a method that uses the gradient stopping condition \eqref{eq:rock-dist}
with first order methods.
Next, consider the case where $p$ is smooth, i.e. $\rho(\cdot)
=0$. Suppose $\chi=\mathbb{R}^n$,
$\nabla\gamma$ is Lipschitz continuous with constant $L_\gamma$ and
$f(x)=Ax-b$. Then, it is easy to establish that $\nabla \phi_k$ is also
Lipschitz continuous with Lipschitz constant
$L_\phi=L_\gamma+\mu_k\sigma_{\max}^2(A)+\mu_k^{-1}=\mathcal{O}(\mu_k)$.
Since
$\phi_k(x_k)-\inf_{x\in\mathbb{R}^n}\phi_k(x)\leq\xi$ implies that
$\norm{\nabla \phi_k(x_k)}_2\leq\sqrt{2L_\phi\xi}$, in order to
ensure \eqref{eq:rock-dist} one has to set
$\xi\leq\frac{1}{2\sigma_{\max}^2(A)}\frac{\alpha_k}{\mu_k^2}$. Thus,
the complexity of computing each iterate $x_k$ satisfying \eqref{eq:rock-dist} will be significantly
higher than the complexity of computing $x_k$ satisfying \eqref{eq:x-update_rock}, which is the one used in the ALCC algorithm.
Therefore,
although Rockafellar's method using \eqref{eq:rock-dist} has the same iteration complexity with
ALCC algorithm, the operational level complexity of a first-order algorithm based on
the gradient stopping criterion~\eqref{eq:rock-dist} will be
significantly higher than the complexity of the ALCC algorithm\
where $\xi=\alpha_k$. In summary, Rockafellar~\cite{Rockafellar1976} is
only able to show an
upper bound on sub-optimality of iterates for the stopping
criterion~\eqref{eq:x-update_rock} that leads to an efficient algorithm;
whereas the subgradient stopping criterion~\eqref{eq:rock-dist} that
results in a lower bound is not practical for a first-order algorithm.
In \cite{Lan11_1J}, Lan, Lu and Monteiro
consider problems of the form
\begin{equation}
\label{eq:lan-et-al-11}
\min\{\fprod{c,x}:~Ax=b,~x\in\mathcal{K}\},
\end{equation}
where $\mathcal{K}$ is a closed convex cone.
They proposed
computing an approximate solution for \eqref{eq:lan-et-al-11} by minimizing the Euclidean
distance to the set of KKT points using Nesterov's
accelerated proximal gradient algorithm~(APG)
\cite{Nesterov04,Nesterov05}. They show that at most
$\mathcal{O}\left(\epsilon^{-1}\right)$ iterations of Nesterov's APG
algorithm~\cite{Nesterov04,Nesterov05} suffice to compute a point
whose distance to the set of KKT points is at most $\epsilon>0$. In
\cite{Lan12_1J}, Lan and Monteiro proposed a first-order penalty
method to solve the following more general problem
\begin{equation}
\label{eq:conic_problem_lan}
\min\{\gamma(x):~Ax-b\in\mathcal{K},~x\in\chi\},
\end{equation}
where $\gamma$ is a convex function with Lipschitz continuous
gradient, $\mathcal{K}$ is a closed, convex cone, $\chi$ is a simple convex
compact set and $A\in\mathbb{R}^{m\times n}$. In order to solve
\eqref{eq:conic_problem_lan}, they used Nesterov's APG algorithm on
the
perturbed penalty problem
\begin{equation*}
\min\{\gamma(x)+\xi\norm{x-x_0}_2^2+\frac{\mu}{2}~d_{\mathcal{K}}(Ax-b)^2:~x\in\chi\},
\end{equation*}
where $x_0\in\chi$, $d_\mathcal{K}$ is as defined in \eqref{eq:dist_func}, and
$\xi>0$, $\mu>0$ are fixed perturbation and penalty parameters. They
showed that Nesterov's APG algorithm can compute a primal-dual
solution $(\tilde{x},\tilde{y})\in\chi\times\mathcal{K}^*$ satisfying
$\epsilon$-perturbed KKT conditions
\begin{equation}
\label{eq:pertubed-KKT}
\fprod{\tilde{y},~\Pi_\mathcal{K}(A\tilde{x}-b)}=0,\quad
d_\mathcal{K}(A\tilde{x}-b)\leq\epsilon,\quad \nabla
\gamma(\tilde{x})-A^T\tilde{y}\in-\mathcal{N}_\chi(\tilde{x})+\mathcal{B}(\epsilon),
\end{equation}
using $\mathcal{O}\left(\epsilon^{-1}\log\left(\epsilon^{-1}\right)\right)$
projections onto $\mathcal{K}$ and $\chi$, where
$\mathcal{N}_\chi(\tilde{x}):=\{s\in\mathbb{R}^n:~\fprod{s, x-\tilde{x}}\leq
0,~\forall x\in\chi\}$ and
$\mathcal{B}(\epsilon):=\{x\in\mathbb{R}^n:~\norm{x}_2\leq\epsilon\}$. Note that
since $\xi$ and $\mu$ are fixed, additional iterations of the
Nesterov's APG algorithm will not improve the quality of the
solution.
The optimization problem \eqref{eq:conic_problem_lan} is a special case of
\eqref{eq:conic_problem} with $\rho(\cdot)= 0$.
Thus, ALCC can solve \eqref{eq:conic_problem_lan}. We show that every
limit point of the ALCC iterates are optimal for
\eqref{eq:conic_problem_lan}. Furthermore, for \emph{any} $\epsilon >0$,
ALCC iterates are $\epsilon$-optimal,
and $\epsilon$-feasible for \eqref{eq:conic_problem_lan}
within $\mathcal{O}\left(\epsilon^{-1}\log\left(\epsilon^{-1}\right)\right)$
projections onto $\mathcal{K}$ and $\chi$ as
is the case with the algorithm proposed in \cite{Lan12_1J}.
Lan and Monteiro~\cite{Lan12_2J} proposed an inexact augmented
Lagrangian method to solve a special case of \eqref{eq:conic_problem}
with $\mathcal{K}=\{\mathbf{0}\}$ and $\rho(\cdot)=0$; and showed that
Nesterov's APG algorithm can compute a primal-dual solution
$(\tilde{x},\tilde{y})\in\chi\times\mathbb{R}^m$ satisfying
\eqref{eq:pertubed-KKT} using
$\mathcal{O}\left(\epsilon^{-1}\left(\log\left(\epsilon^{-1}\right)\right)^{\frac{3}{4}}
\log\log\left(\epsilon^{-1}\right)\right)$
projections onto $\chi$ and $\mathcal{K}$.
Aybat and Iyengar~\cite{Ser10_1J} proposed an inexact augmented
Lanrangian algorithm (FALC) to solve the
composite norm minimization problem
\begin{equation}
\label{eq:composite-norm}
\min_{X\in\mathbb{R}^{m\times
n}}\{\mu_1\norm{\sigma(\mathcal{F}(X)-G)}_\alpha+\mu_2\norm{\mathcal{C}(X)-d}_\beta+\gamma(X):\
\mathcal{A}(X)-b\in\mathcal{Q}\},
\end{equation}
where the function $\sigma(\cdot)$ returns the singular values of its
argument; $\alpha$ and $\beta\in\{1,2,\infty\}$; $\mathcal{A},\mathcal{C},\mathcal{F}$ are
linear operators such that either $\mathcal{C}$ or $\mathcal{F}$ is injective, and
$\mathcal{A}$ is surjective; $\gamma$ is a convex function with a Lipschitz
continuous gradient and $\mathcal{Q}$ is a closed convex set. It was shown
that any limit point of the FALC iterates is an optimal solution of
the composite norm minimization problem~\eqref{eq:composite-norm}; and
for all $\epsilon>0$, the FALC iterates are $\epsilon$-feasible and
$\epsilon$-optimal after
$\mathcal{O}\left(\log\left(\epsilon^{-1}\right)\right)$ FALC iterations,
which require $\mathcal{O}\left(\epsilon^{-1}\right)$ shrinkage type
operations and Euclidean projection onto the set $\mathcal{Q}$. The limitation
of FALC is that it requires $\mathcal{A}$ to be a surjective mapping.
Consider a feasible set of the form
\begin{equation}
\label{eq:product-cone}
\{x\in\mathbb{R}^n:~A_1x-b_1\in\mathcal{K}_1,\ A_2x-b_2\in\mathcal{K}_2,\ x\in\chi\},
\end{equation}
where $\mathcal{K}_i$ is a closed convex cone, $A_i\in\mathbb{R}^{m_i\times n}$ and
$b_i\in\mathbb{R}^{m_i}$ for $i=1,2$. The set in \eqref{eq:product-cone} can
be reformulated as the feasible set in \eqref{eq:conic_problem} by
choosing
$A=\left(
\begin{array}{c}
A_1 \\
A_2 \\
\end{array}
\right)$ and $\mathcal{K}=\mathcal{K}_1\times\mathcal{K}_2$, where $m=m_1+m_2$. FALC can work with such a set
only if $A$ has linearly independent rows, i.e., $\mathop{\bf rank}(A)=m_1+m_2$. This
is a severe limitation for the
practical problem. On the other hand, the ALCC algorithm~works for the feasible sets
of the form \eqref{eq:product-cone} without any additional
assumption. Thus, ALCC can be used to solve much larger class of
optimization problems.
In our opinion the ALCC algorithm~proposed in this paper unifies all the previous work on fast first-order penalty and/or augmented Lagrangian algorithms for solving optimization problems that are special cases of \eqref{eq:conic_problem}. We do not impose any regularity conditions on the constraint matrix $A$ and the projection step \eqref{eq:nonsmooth-operation} is the natural extension of the gradient projection step. We believe that this unified treatment will spur further research in understanding the limits of performance of the first order algorithms for general conic problems.
\section{Preliminaries}
\label{sec:preliminaries}
In Section~\ref{sec:apg}, first we briefly discuss a variant of Nesterov's APG
algorithm~\cite{Nesterov04,Nesterov05} to solve
\eqref{eq:conic_problem} without conic constraints. Next, we introduce
a dual function for the conic problem in \eqref{eq:conic_problem} and
establish some of its properties in Section~\ref{sec:prelims}. The definitions and the results of Section~\ref{sec:prelims}
are extensions of the corresponding definitions and results
in \cite{Rockafellar-73,Roc73_1J},
to the case where $\mathcal{K}\subset\mathbb{R}^m$ is a general closed, convex cone.
\subsection{Accelerated Proximal Gradient~(APG) algorithm}
\label{sec:apg}
\begin{figure}[ht]
\rule[0in]{6.5in}{1pt}\\
\textbf{Algorithm APG}$(\bar{\rho}, \bar{\gamma}, \chi, x_0,\textsc{stop})$\\
\rule[0.125in]{6.5in}{0.1mm}
\vspace{-0.25in}
{\small
\begin{algorithmic}[1]
\STATE $x_0^{(1)}\gets x_0$, $x_1^{(2)} \gets x_0$, $t_1\gets 1$, $\ell\gets 0$
\WHILE{\textsc{stop} is \FALSE} \label{algeq:stop}
\STATE $\ell \gets \ell + 1$
\STATE $x_\ell^{(1)} \gets \mathop{\rm argmin}\left\{\bar{\rho}(x)+\fprod{\nabla
\bar{\gamma}\left(x_\ell^{(2)}\right),~x-x_\ell^{(2)}} +
\frac{L_{\bar{\gamma}}}{2}~\norm{x-x_\ell^{(2)}}_2^2: x\in
\chi\right\}$ \label{algeq:tseng_z}
\STATE $t_{\ell+1}\gets \left(1+\sqrt{1+4~t^2_{\ell}}\right)/2$
\STATE $x_{\ell+1}^{(2)} \gets x_\ell^{(1)} +
\left(\frac{t_{\ell}-1}{t_{\ell+1}}\right)\left(x_\ell^{(1)}-x_{\ell-1}^{(1)}\right)$
\ENDWHILE
\end{algorithmic}
\rule[0.25in]{6.5in}{0.1mm}
}
\vspace{-0.5in}
\caption{Accelerated Proximal Gradient Algorithm}\label{alg:apg}
\vspace{-0.25in}
\end{figure}
In this section we state and briefly discuss the details of a particular
implementation of Fast Iterative Shrinkage-Thresholding
Algorithm~\cite{Beck09_1J} (FISTA), which extends Nesterov's
accelerated proximal gradient algorithm~\cite{Nesterov04,Nesterov05}
for minimizing smooth convex functions over simple convex sets, to
solve non-smooth convex minimization problems.
FISTA computes an $\epsilon$-optimal solution to
$\min\{\bar{\rho}(x)+\bar{\gamma}(x):\ x\in\mathbb{R}^n\}$
in $\mathcal{O}\left(\epsilon^{-\frac{1}{2}}\right)$ iterations, where
$\bar{\rho}:\mathbb{R}^n \rightarrow\mathbb{R}$ and $\bar{\gamma}:\mathbb{R}^n
\rightarrow\mathbb{R}$ are continuous convex functions such that $\nabla
\bar{\gamma}$ is Lipschitz continuous on $\mathbb{R}^n$ with constant
$L_{\bar{\gamma}}$. Tseng~\cite{Tseng08} showed that this rate
result for FISTA also holds when $\bar{\rho}:\mathbb{R}^n \rightarrow
(-\infty, +\infty]$ and $\bar{\gamma}:\mathbb{R}^n \rightarrow
(-\infty, +\infty]$ are proper, lower semicontinuous, and convex
functions such that $\mathop{\bf dom} \bar{\rho}$ is closed and $\nabla
\bar{\gamma}$ is Lipschitz
continuous on $\mathbb{R}^n$.
This extended version of FISTA is displayed in Figure~\ref{alg:apg} as APG algorithm. Hence, FISTA can solve constrained
problems of the form
\begin{align}
\label{eq:tseng_problem}
\min\{\bar{\rho}(x)+\bar{\gamma}(x):\ x\in\chi\},
\end{align}
where $\chi\subset\mathbb{R}^n$ is a simple closed convex set.
The APG algorithm~displayed in Figure~\ref{alg:apg} takes as
input the functions $\bar{\rho}$ and $\bar{\gamma}$, the simple closed
convex set $\chi\subset\mathbb{R}^n$, an initial iterate
$x^{(0)}\in\chi$ and a stopping criterion \textsc{stop}.
Lemma~\ref{lem:tseng_corollary} gives the iteration complexity of
the APG algorithm.
\begin{lemma}
\label{lem:tseng_corollary}
Let $\bar{\rho}$ and $\bar{\gamma}$ be a proper, closed,
convex functions
such that $\mathop{\bf dom} \bar{\rho}$ is closed and $\nabla \bar{\gamma}$ is
Lipschitz continuous on
$\mathbb{R}^n$ with constant $L_{\bar{\gamma}}$. Fix $\epsilon>0$ and let
$\{x_\ell^{(1)},x_\ell^{(2)}\}$ denote the sequence of iterates
computed by the APG algorithm\ when $\textsc{stop}$
is disabled. Then
$\bar{\rho}\big(x_\ell^{(1)}\big)+\bar{\gamma}\big(x_\ell^{(1)}\big)\leq
\min\{\bar{\rho}(x)+\bar{\gamma}(x):\ x\in\chi\}+\epsilon$ whenever
$\ell\geq\sqrt{\frac{2L_{\bar{\gamma}}}{\epsilon}}~\norm{x^\ast-x_0}_2-1$,
where $x^\ast
\in \mathop{\rm argmin}\{\bar{\rho}(x) + \bar{\gamma}(x):\ x\in\chi\}$.
\end{lemma}
\begin{proof}
See Corollary~3 in \cite{Tseng08} and Theorem 4.4 in \cite{Beck09_1J} for
the details of proof.\end{proof}
\subsection{A dual function for conic convex programs and its properties}
\label{sec:prelims}
For all $\mu\geq 0$, optimization problem $(P)$ in
\eqref{eq:conic_problem} is equivalent to
\begin{equation}
\label{eq:penalty_problem}
\min \left\{ p(x)+\frac{\mu}{2}\norm{Ax-s-b}_2^2 : Ax-s=b,~x \in \chi,~s
\in \mathcal{K}\right\}.
\end{equation}
Let $y\in\mathbb{R}^m$ denote a Lagrangian dual variable corresponding to
the equality constraint in \eqref{eq:penalty_problem}, and let
\begin{equation}
\label{eq:Lmu_def}
\mathcal{L}_\mu(x,y):=\min_{s\in\mathcal{K}} \left\{ p(x
-\fprod{y, Ax-s-b}+\frac{\mu}{2}\norm{Ax-s-b}_2^2\right\}
\end{equation}
denote the ``penalty'' Lagrangian function for
\eqref{eq:penalty_problem} with $\mathop{\bf dom} \mathcal{L}_\mu=\chi \times \mathbb{R}^m$.
For $\mu>0$,
\begin{align}
\mathcal{L}_\mu(x,y)&=p(x
+\frac{\mu}{2}~\left ( \min_{s\in\mathcal{K}}
\left\|Ax-s-b-\frac{y}{\mu}\right\|_2^2-\frac{\norm{y}_2^2}{\mu^2}
\right),
\nonumber\\
&=p(x
+\frac{\mu}{2}~d_\mathcal{K}\left(Ax-b-\frac{y}{\mu}\right)^2
-\frac{\norm{y}_2^2}{2\mu}, \label{eq:L_mu}
\end{align}
where $d_{\mathcal{K}}(\cdot)$ is the distance function defined
in~\eqref{eq:dist_func}.
When $\mu = 0$, the definition in \eqref{eq:Lmu_def} implies that
\begin{equation}
\label{eq:lagrangian_0}
\mathcal{L}_0(x,y)=\left\{
\begin{array}{ll}
p(x)-\fprod{y, Ax-b}, & y\in\mathcal{K}^*, \\
-\infty, & \mbox{otherwise.}
\end{array}
\right.
\end{equation}
For $\mu \geq 0$, we define a dual function $g_\mu:\mathbb{R}^m\rightarrow\mathbb{R}$ for \eqref{eq:conic_problem} such that
\begin{equation}
\label{eq:g_mu}
g_\mu(y):=\inf_{x\in\chi}\mathcal{L}_\mu(x, y).
\end{equation}
Note that from \eqref{eq:lagrangian_0} it follows that $g_0$ is the
Lagrangian dual function of $(P)$.
The definitions above and the results detailed below are immediate extensions of
corresponding definitions and results in~\cite{Rockafellar-73}, given for $\mathcal{K}=\mathbb{R}^m_+$, to the case where $\mathcal{K}$ is a general closed convex cone.
We state and prove the extensions here for the sake of completeness.
These results are used in Section~\ref{sec:alcc} to establish the convergence properties of ALCC iterate sequence.
\begin{lemma}
\label{lem:L-cvx}
For all $\mu \geq 0$, $x\in\chi$ and $y\in\mathbb{R}^m$, $\mathcal{L}_\mu$ defined in \eqref{eq:Lmu_def} satisfies
\begin{align}
\label{eq:LmuF}
\mathcal{L}_\mu(x,y)=\inf_{u\in\mathbb{R}^m} \left\{F_\mu(x,u)-\fprod{y,u}\right\},
\end{align}
where $F_\mu:\chi\times \mathbb{R}^m\rightarrow\mathbb{R}\cup\{+\infty\}$ is defined as follows
\begin{equation}
\label{eq:Fmu}
F_\mu(x,u):=\left\{
\begin{array}{ll}
p(x)+\frac{\mu}{2}\norm{u}_2^2, & \hbox{if $Ax-b\in\mathcal{K}+u$}, \\
+\infty, & \hbox{otherwise.}
\end{array}
\right.
\end{equation}
Hence, $\mathcal{L}_\mu(x,y)$ is convex in $x\in\chi$ and concave in
$y\in\mathbb{R}^m$, and $g_\mu(y)$ defined in
\eqref{eq:g_mu} is concave in $y\in\mathbb{R}^m$.
\end{lemma}
\begin{proof}
The representation in \eqref{eq:LmuF} trivially follows from the definition of $F_\mu$
in \eqref{eq:Fmu}. For a fixed $x\in\chi$, \eqref{eq:Lmu_def} implies that $\mathcal{L}_\mu(x,y)$ is the
infimum of affine functions of $y$, hence $\mathcal{L}_\mu(x, y)$ is concave in $y$. Hence, $g_\mu$ defined in
\eqref{eq:g_mu} is the infimum of concave functions; therefore, it is also concave. For a fixed $y\in\mathbb{R}^m$, when $\mu>0$, convexity of $\mathcal{L}_\mu(x,y)$ in $x$ follows from \eqref{eq:L_mu} and the fact that $p(\cdot)$ and $d_\mathcal{K}(\cdot)$ are convex functions; otherwise, when $\mu=0$, it trivially follows from \eqref{eq:lagrangian_0}.
\end{proof}
\begin{lemma}
\label{lem:prox}
Let $g:\mathbb{R}^m\rightarrow\mathbb{R}\cup\{+\infty\}$ be a proper closed convex
function. For $\mu > 0$, let
\[
\psi_\mu(y)=\min_{z\in\mathbb{R}^m}\Big\{g(z)+\frac{1}{2\mu}\norm{z-y}_2^2\Big\}, \quad \pi_\mu(y) = \mathop{\rm argmin}_{z\in\mathbb{R}^m}\Big\{g(z)+\frac{1}{2\mu}\norm{z-y}_2^2\Big\}
\]
denote the Moreau regularization of and the proximal map corresponding to $g$, respectively. Then, for all $y_1,y_2\in\mathbb{R}^m$,
\begin{equation}
\label{eq:prox}
\norm{\pi_\mu(y_1)-\pi_\mu(y_2)}_2^2+\norm{\pi^c_\mu(y_1)-\pi^c_\mu(y_2)}_2^2\leq\norm{y_1-y_2}_2^2,
\end{equation}
where $\pi^c_\mu(y) := y-\pi_\mu(y)$ for all $z\in\mathbb{R}^m$. Moreover,
$\psi_\mu:\mathbb{R}^m\rightarrow\mathbb{R}$ is an everywhere finite, differentiable convex function such that
\begin{equation}
\label{eq:moreau-grad}
\nabla\psi_\mu(y)=\frac{1}{\mu}~(y-\pi_\mu(y))=\frac{1}{\mu}~\pi^c_\mu(y),
\end{equation}
is Lipschitz continuous with constant $\frac{1}{\mu}$.
\end{lemma}
\begin{proof}
The proof of \eqref{eq:prox} is given in \cite{Rockafellar-76} and the rest of the claims including \eqref{eq:moreau-grad} are shown in \cite{Hiriart-Urruty-Lemarechal-1993}.
\end{proof}
\begin{theorem}
\label{thm:moreau}
Suppose Assumption~\ref{asp:KKT} holds.
Then, for any $\mu>0$, $g_\mu$ is an everywhere finite, continuously differentiable concave function and $g_\mu$ achieves its maximum value at any KKT
point. Moreover,
\begin{equation}
\label{eq:g_moreau}
g_\mu(y)=\max_{z\in\mathbb{R}^m}
\left\{g_0(z)-\frac{1}{2\mu}\norm{z-y}_2^2 \right\},
\end{equation}
and
\begin{equation}
\label{eq:grad_gk}
\nabla g_\mu(y)=-\frac{1}{\mu}(y-\pi_\mu(y)),
\end{equation}
is Lipschitz continuous with Lipschitz constant equal to $\frac{1}{\mu}$, where $\pi_\mu(y)\in\mathcal{K}^*$ denotes the unique maximizer in \eqref{eq:g_moreau}.
\end{theorem}
\begin{proof}
Fix $\mu \geq 0$, define
\begin{equation}
\label{eq:h_mu}
h_\mu(u):=\inf_{x\in\chi}F_\mu(x,u).
\end{equation}
Note that $F_{\mu}(x,u) = p(x)+\frac{\mu}{2}\norm{u}_2^2 + \mathbf 1_{\mathcal{K}}(Ax-b-u)$, where $\mathbf 1_{\mathcal{K}}(\cdot)$ denotes the indicator function of the set $\mathcal{K}$; therefore, $F_{\mu}(x,u)$ is convex in $(x,u)$. Since $F_\mu$ is convex in $(x,u)$, $\chi$ is a convex set and $h_{\mu}(0) = \inf_{x \in \chi} \{ p(x)+
\mathbf 1_{\mathcal{K}}(Ax-b)\} = p^\ast > -\infty$, it follows that $h_{\mu}$ is a convex
function such that $h_\mu(\cdot)>-\infty$ \cite{Boyd04_1B}. From the definition of $F_{\mu}$, it
follows that for all $u\in\mathbb{R}^m$,
\begin{equation*}
h_\mu(u)= h_0(u)+ \mu~\omega(u),
\end{equation*}
where $\omega(u) := \frac{1}{2}\norm{u}_2^2$. Substituting \eqref{eq:LmuF}
in \eqref{eq:g_mu}, for all $\mu \geq 0$, we get
\begin{equation*}
g_\mu(y)=\inf_{u\in\mathbb{R}^m} \left\{h_\mu(u)-\fprod{y,u} \right\}=-h_\mu^*(y),
\end{equation*}
where $h_\mu^*$ denotes the conjugate of the convex function $h_\mu$.
Fix $\mu>0$, since $h_\mu$ is a sum of two convex functions, it follows from Theorem~16.4 in \cite{Rockafellar-book-70} that
\begin{equation}
\label{eq:conjugate_sum}
g_\mu(y)=-(h_0 + \mu \omega)^*(y)
=-\min_{z\in\mathbb{R}^m} \left\{h^*_0(z)+\mu~\omega^\ast\left(\frac{y-z}{\mu}\right)\right\}.
\end{equation}
Since $h^*_0=-g_0$ and $\omega^*=\omega$, the result
\eqref{eq:g_moreau} immediately follows from \eqref{eq:conjugate_sum}.
Note that \eqref{eq:g_moreau} shows that $-g_\mu$ is the Moreau regularization of $-g_0$. Therefore, Lemma~\ref{lem:prox} and \eqref{eq:g_moreau} imply that $g_\mu$ is everywhere finite, differentiable concave function such that $\nabla g_\mu$ is given in \eqref{eq:grad_gk}.
Let $y^*$ be a KKT point of \eqref{eq:conic_problem}. Note that $\pi_\mu(y^*)=y^*$. Hence $\nabla g_\mu(y^*)=\mathbf{0}$. Concavity of $g_\mu$ implies that $y^*\in\mathop{\rm argmax} g_\mu(y)$ for any KKT point $y^*$.
\end{proof}
\begin{theorem}
\label{thm:grad_diff}
Fix $\mu>0$ and $\bar{y}\in\mathbb{R}^m$. Suppose $\bar{x}\in\chi$ is an
$\xi$-optimal solution to $\min_{x\in\chi}L_{\mu}(x,\bar{y})$, i.e.
$
L_{\mu}(\bar{x},\bar{y}) \leq \min\{L_{\mu}(x,\bar{y}):\ x\in\chi\}
+ \xi = g_{\mu}(\bar{y}) + \xi.
$
Then
\begin{equation}
\label{eq:grad_diff}
\mu~\norm{\nabla_y \mathcal{L}_\mu(\bar{x},\bar{y})-\nabla g_\mu(\bar{y})}_2^2\leq 2\xi.
\end{equation}
\end{theorem}
\begin{proof}
For $\mu>0$, $g_\mu$ is concave and $\nabla g_\mu$ is Lipschitz continuous with Lipschitz constant equal to $\frac{1}{\mu}$; therefore,
\begin{equation}
\label{eq:lipschitz-g}
g_\mu(y)\geq g_\mu(\bar{y})+\fprod{\nabla g_\mu(\bar{y}),
y-\bar{y}}-\frac{1}{2\mu}\norm{y-\bar{y}}_2^2,
\end{equation}
for all $y\in\mathbb{R}^m$. Moreover, since for every $x\in\chi$,
$\mathcal{L}_\mu(x,y)$ is concave in $y$, it follows that for all $y\in\mathbb{R}^m$
\begin{equation}
\label{eq:lowerboundL}
\mathcal{L}_\mu(\bar{x}, \bar{y})+\fprod{\nabla_y
\mathcal{L}_\mu(\bar{x},\bar{y}),~y-\bar{y}}\geq \mathcal{L}_\mu(\bar{x},y)\geq
g_\mu(y).
\end{equation}
Combining \eqref{eq:lipschitz-g}, \eqref{eq:lowerboundL} and the fact
that $\bar{x}$ is $\xi$-optimal and $y$ is arbitrary, we get
\[
\xi\ge
\sup_{y\in\mathbb{R}^m}\left\{\fprod{\nabla
g_\mu(\bar{y})-\nabla_y \mathcal{L}_\mu(\bar{x},\bar{y}),~
y-\bar{y}}-\frac{1}{2\mu}~\norm{y-\bar{y}}_2^2\right\} =
\frac{\mu}{2} \norm{\nabla
g_\mu(\bar{y})-\nabla_y \mathcal{L}_\mu(\bar{x},\bar{y})}_2^2.
\]
\end{proof}
\section{ALCC Algorithm}
\label{sec:alcc}
In order to solve $(P)$ given in \eqref{eq:conic_problem}, we
inexactly solve the sequence of sub-problems:
\begin{equation}
\label{eq:subproblem}
(SP_k):\ \min_{x\in\chi} P_k(x,y_k),
\end{equation}
where
\[
P_k(x,y) :=\frac{1}{\mu_k}~\mathcal{L}_{\mu_k}(x,y) = \frac{1}{\mu_k}~p(x) +
\frac{1}{2}~d_\mathcal{K}\left(Ax-b-\frac{y}{\mu_k}\right)^2.
\]
For notational convenience, we define
\[
f_k(x,y) :=\frac{1}{2}~d_\mathcal{K}\left(Ax-b-\frac{y}{\mu_k}\right)^2.
\]
Therefore, $P_k(x,y) = \frac{1}{\mu_k}~p(x) + f_k(x,y)$.
The specific choice of penalty parameter and Lagrangian dual sequences, $\{\mu_k\}$ and $\{y_k\}$, are discussed later in this section.
\begin{lemma}
\label{lem:lipschitz}
For all $k\geq 1$ and $y\in\mathbb{R}^m$, $f_k(x,y)$ is convex in $x$. Moreover,
\begin{equation}
\label{eq:gradfk}
\nabla_x f_k(x,y)=A^T\left(Ax-b-\frac{y}{\mu_k}-\Pi_\mathcal{K}\left(Ax-b-\frac{y}{\mu_k}\right)\right),
\end{equation}
and $\nabla_x f_k(x,y)$ is Lipschitz continuous in $x$ with constant $L=\sigma^2_{\max}(A)$.
\end{lemma}
\begin{proof}
See appendix for the proof.
\end{proof}
\begin{figure}[t]
\rule[0in]{6.5in}{1pt}\\
\textbf{Algorithm ALCC}~$(x_0,~\{\alpha_k,~\eta_k,~\mu_k\})$\\
\rule[0.125in]{6.5in}{0.1mm}
\vspace{-0.25in}
{\small
\begin{algorithmic}[1]
\STATE $y_1\gets \mathbf{0}$, $k\gets 1$
\WHILE{$k\geq 1$}
\STATE $x_k\gets\mbox{\scshape{Oracle}}(P_k, y_k, \alpha_k, \eta_k, \mu_k)$ /*~\emph{See Section~\ref{sec:oracle} for \mbox{\scshape{Oracle}}} ~*/
\STATE $y_{k+1}\gets\mu_k\left[\Pi_\mathcal{K}\left(Ax_k-b-\frac{y_k}{\mu_k}\right)-\left(Ax_k-b-\frac{y_k}{\mu_k}\right)\right]$ \label{algeq:y_update}
\STATE $k\gets k+1$
\ENDWHILE
\end{algorithmic}
\rule[0.25in]{6.5in}{0.1mm}
}
\vspace{-0.5in}
\caption{Augmented Lagrangian Algorithm for Conic Convex Programming} \label{alg:alcc}
\vspace{-0.25in}
\end{figure}
The ALCC algorithm\ is displayed in Figure~\ref{alg:alcc}. The inputs to ALCC are
an initial point $x_0\in\chi$
and a parameter sequence
$\{\alpha_k,~\eta_k,~\mu_k\}$ such that
\begin{equation}
\label{eq:params}
\alpha_k\searrow 0, \quad \eta_k\searrow 0, \quad 0<\mu_k\nearrow \infty.
\end{equation}
\subsection{Oracle}
\label{sec:oracle}
The subroutine $\mbox{\scshape{Oracle}} (P, \bar{y}, \alpha, \eta, \mu)$ returns
$\bar{x} \in \chi$ such that $\bar{x}$ satisfies one of the following
two conditions:
\begin{align}
&0\leq P(\bar{x},\bar{y})-\inf_{x\in\chi}P(x,\bar{y})\leq \frac{\alpha}{\mu}, \label{eq:eps_opt}\\
&\exists q\in\partial_x P(\bar{x},\bar{y})+\partial_x \mathbf{1}_\chi(\bar{x})~ \mbox{ s.t. }
\norm{q}_2\leq\frac{\eta}{\mu}, \label{eq:grad_opt}
\end{align}
where $\mathbf{1}_\chi(\cdot)$ denotes the indicator function of the set $\chi$.
Let $\bar{\rho}_k(x):=\frac{1}{\mu_k}~\rho(x)$ and
$\bar{\gamma}_k(x):=\frac{1}{\mu_k}~\gamma(x)+f_k(x,y_k)$. Then
$\nabla \bar{\gamma}_k$ exists and is Lipschitz continuous with
Lipschitz constant
\begin{equation}
\label{eq:lipschitz-constant}
L_{\bar{\gamma}_k}:= \frac{1}{\mu_k}~L_\gamma+\sigma^2_{\max}(A).
\end{equation}
Let
\begin{equation}
\label{eq:chi-opt-k}
\chi\supset\chi_k^*:=\mathop{\rm argmin}_{x\in \chi}P_k(x,y_k)
\end{equation}
denote the set of optimal solutions to $(SP_k)$. Then, Lemma~\ref{lem:tseng_corollary}
guarantees that the APG algorithm~with the initial iterate $x_{k-1}\in\chi$
requires at most
\begin{equation}
\label{eq:lmax}
\ell_{\max}(k):=\sqrt{\frac{2\mu_k L_{\bar{\gamma}_k}}{\alpha_k}}~d_{\chi_k^*}(x_{k-1})
\end{equation}
iterations to compute
$\frac{\alpha_k}{\mu_k}$-optimal solution to the $k$-th subproblem
$(SP_k)$ in \eqref{eq:subproblem}. Thus, setting the stopping
criterion $\textsc{stop} = \{l
\geq \ell_{\max}(k)\}$ ensures that the output of the APG algorithm\ satisfies
\eqref{eq:eps_opt}. Thus, we have
shown that
there exists a subroutine
$\mbox{\scshape{Oracle}}(P_k, y_k, \alpha_k, \eta_k, \mu_k)$ that can compute
$x_k$ satisfying either \eqref{eq:eps_opt} or \eqref{eq:grad_opt}.
As indicated earlier,
the computational complexity of each
iteration in the APG algorithm\ is
dominated by the complexity of computing the solution to~\eqref{eq:nonsmooth-operation}.
\subsection{Convergence properties of ALCC algorithm}
In this section we investigate the convergence rate of ALCC algorithm.
\begin{lemma}
\label{lem:cone}
Let $\mathcal{K}\subset\mathbb{R}^n$ denote a closed, convex cone and
$\bar{x}\in\mathbb{R}^n$. Then $\bar{x}-\Pi_\mathcal{K}(\bar{x})\in-\mathcal{K}^*$ and
$\fprod{\bar{x}-\Pi_\mathcal{K}(\bar{x}),~\Pi_\mathcal{K}(\bar{x})}=0$, where
$\mathcal{K}^*=\{s\in\mathbb{R}^n:\ \fprod{s, x}\geq 0\; \forall
x\in\mathcal{K}\}$. Finally, if $x\in-\mathcal{K}^*$, then $\Pi_\mathcal{K}(x)=\mathbf{0}$.
\end{lemma}
\begin{proof}
See appendix for the proof.
\end{proof}
From Lemma~\ref{lem:cone}, it follows that the dual variable $y_{k+1}$ computed in
Line~\ref{algeq:y_update} of ALCC algorithm~ satisfies $y_{k+1}\in\mathcal{K}^*$. Also note
that for all $k\geq 1$,
\begin{equation}
\label{eq:ykp}
y_{k+1}=y_k+\mu_k\nabla_y \mathcal{L}_{\mu_k}(x_k, y_k).
\end{equation}
Next, we establish that the sequence of dual variables $\{y_k\}$ generated by ALCC algorithm~is bounded for an appropriately chosen parameter
sequence.
\begin{lemma}
\label{lem:inexact-opt}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ be the sequence of primal-dual
ALCC iterates for a given input parameter sequence
$\{\alpha_k,~\eta_k,~\mu_k\}$ satisfying \eqref{eq:params}. Then, for all
$k\geq 1$,
\begin{equation}
0\leq \mathcal{L}_{\mu_k}(x_k,y_k)-g_{\mu_k}(y_k)\leq \xi^{(k)}, \label{eq:eps_opt_L}
\end{equation}
where
\begin{equation}
\label{eq:xik}
\xi^{(k)} = \max\{\alpha_k,~\eta_k~d_{\chi_k^*}(x_{k})\},
\end{equation}
and $\chi_k^*\subset\chi$ is defined in \eqref{eq:chi-opt-k}.
\end{lemma}
\begin{proof}
Fix $k\geq 1$.
Suppose $x_k =\mbox{\scshape{Oracle}}(P_k, y_k, \alpha_k, \eta_k, \mu_k)$ satisfies
\eqref{eq:eps_opt}. Then we
have
\begin{equation}
\label{eq:inexact-subopt}
P_k(x_k, y_k)\leq \inf_{x\in\chi}P_k(x, y_k)+ \frac{\alpha_k}{\mu_k}=
\frac{g_{\mu_k}(y_k)+\alpha_k}{\mu_k}.
\end{equation}
Suppose instead that $x_k = \mbox{\scshape{Oracle}}(P_k, y_k, \alpha_k, \eta_k, \mu_k)$
satisfies \eqref{eq:grad_opt}. Then, there exists
$q_k\in\partial_x P_k(x_k,y_k)+\partial \mathbf{1}_{\chi}(x_k)$ such
that $\norm{q_k}_2\leq\frac{\eta_k}{\mu_k}$. Since
$P_k(x,y_k)+\mathbf{1}_{\chi}(x)$ is convex in $x$, it follows that
\begin{equation}
\label{eq:inexact-grad}
P_k(x_k,y_k)\leq \inf_{\bar{x}\in\chi_k^*}P_k(\bar{x},y_k)+\fprod{q_k, x_k-\bar{x}}\leq
\frac{g_{\mu_k}(y_k)+\eta_k~d_{\chi_k^*}(x_{k})}{\mu_k}.
\end{equation}
Since $P_k(x,y)=\frac{1}{\mu_k}\mathcal{L}_{\mu_k}(x,y)$, the desired result
follows from \eqref{eq:inexact-subopt} and \eqref{eq:inexact-grad}.
\end{proof}\\
The following result was originally established in \cite{Roc73_1J} for $\mathcal{K} = \mathbb{R}^m_+$. We state and prove the
extension to general convex cones for completeness.
\begin{theorem}
\label{thm:bounded-y}
Suppose $B:=\sum_{k=1}^{\infty}\sqrt{2~\xi^{(k)}\mu_k}<\infty$,
where $\xi^{(k)}$ is defined in \eqref{eq:xik}. Then, for all $k \geq 1$, $\norm{y_k}_2\leq
B+\norm{y^*}_2$ where $y^*$ is any KKT
point of $(P) $.
\end{theorem}
\begin{proof}
Lemma~\ref{lem:inexact-opt} and
Theorem~\ref{thm:grad_diff} imply that $\sqrt{2~\xi^{(k)}\mu_k} \geq\norm{\mu_k\nabla_y\mathcal{L}_{\mu_k}(x_k,y_k)-\mu_k\nabla g_{\mu_k}(y_k)}_2.$
Next, adding and subtracting $y_k$, and using \eqref{eq:grad_gk} and
\eqref{eq:ykp}, we get
\begin{equation}
\sqrt{2~\xi^{(k)}\mu_k} \geq \norm{\mu_k\nabla_y\mathcal{L}_{\mu_k}(x_k,y_k) + y_k - (y_k
+\mu_k\nabla g_{\mu_k}(y_k))}_2 =
\norm{y_{k+1}-\pi_{\mu_k}(y_k)}_2, \label{eq:y-My}
\end{equation}
Since $\sum_{k=1}^{\infty}\sqrt{2~\xi^{(k)}\mu_k}<\infty$, it follows that $\xi^{(k)}\mu_k\rightarrow 0$. Thus,
$\lim_{k\in\mathbb{Z}_+} \big(y_{k+1}-\pi_{\mu_k}(y_k)\big)=0$.
Assumption~\ref{asp:KKT} guarantees that a KKT point $y^*\in\mathcal{K}^*$
exists. Since
$y^*\in\mathop{\rm argmax}_{y\in\mathbb{R}^m} g_0(y)$,
Theorem~\ref{thm:moreau} implies that
$y^*\in\mathop{\rm argmax}_{y\in\mathbb{R}^m} g_{\mu_k}(y)$ for all $k \geq 1$. Therefore, $\nabla
g_{\mu_k}(y^*)=0$, and consequently, by \eqref{eq:grad_gk},
$y^*=\pi_{\mu_k}(y^*)$. Since $\pi_{\mu_k}$ is non-expansive, it follows
that
\begin{equation*}
\norm{\pi_{\mu_k}(y_k)-y^*}_2=\norm{\pi_{\mu_k}(y_k)-\pi_{\mu_k}(y^*)}_2\leq\norm{y_k-y^*}_2.
\end{equation*}
Hence,
\begin{eqnarray}
\norm{y_{k+1}-y^*}_2&\leq
&\norm{y_{k+1}-\pi_{\mu_k}(y_k)}_2+\norm{\pi_{\mu_k}(y_k)-y^*}_2,
\nonumber\\
&\leq &\norm{y_{k+1}-\pi_{\mu_k}(y_k)}_2+\norm{y_k-y^*}_2,\nonumber\\
&\leq & \sqrt{2~\xi^{(k)}\mu_k}+\norm{y_k-y^*}_2. \label{eq:y-induction-eps}
\end{eqnarray}
Since $y_1=\mathbf{0}$, the desired result is obtained by summing the
above inequality over $k$.
\end{proof}
In the rest of this section we investigate the convergence properties
of ALCC for the multiplier sequence $\{\alpha_k,\eta_k,\mu_k\}$ defined as follows
\begin{equation}
\label{eq:mult-seq}
\mu_k = \beta^k~\mu_0,\quad \alpha_k = \frac{1}{k^{2(1+c)}~\beta^k}~\alpha_0,\quad \eta_k = \frac{1}{k^{2(1+c)}~\beta^k}~\eta_0,
\end{equation}
for all $k\geq 1$, where $\beta>1$, $c, \alpha_0, \eta_0$ and
$\mu_0$ are all strictly positive. Thus, $\alpha_k\searrow0$, $\eta_k\searrow0$ and $\mu_k\nearrow\infty$.
Let $\infty>\Delta_\chi:=\max_{x\in\chi}\max_{x'\in\chi}\norm{x-x'}_2$ denote the diameter of the compact set $\chi$. Clearly, $d_{\chi_k^*}(x_{k})\leq\Delta_\chi$ for all $k\geq 1$, where $\chi_k^*\subset\chi$ is defined in \eqref{eq:chi-opt-k}.
Hence, from the definition of $\xi_k$ in \eqref{eq:xik}, it follows that
\begin{equation}
\sqrt{\xi^{(k)}\mu_k}\leq
\frac{1}{k^{1+c}}~\sqrt{\mu_0\max\{\alpha_0,~\eta_0\Delta_\chi\}},\quad
\forall k\geq 1,
\end{equation}
and $\sum_{k=1}^{\infty}\sqrt{\xi^{(k)}\mu_k}<\infty$ as
required by Theorem~\ref{thm:bounded-y}. First, we lower bound the
sub-optimality as a function of primal infeasibility of the iterates.
\begin{theorem}
\label{thm:subopt-lower}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ be the sequence of primal-dual
ALCC iterates corresponding to a parameter sequence
$\{\alpha_k, \eta_k, \mu_k\}$ satisfying \eqref{eq:params}. Then
\begin{equation*}
p(x_k)-p^*\geq-\norm{y^*}_2~d_\mathcal{K}
\left(Ax_k-b-\frac{y_k}{\mu_k}\right)+\frac{1}{\mu_k}\fprod{y_k,y^*},
\end{equation*}
where $y^*\in\mathcal{K}^*$ denotes any KKT point of $(P)$ and $p^*$ denotes the
optimal value of $(P)$ given in \eqref{eq:conic_problem}.
\end{theorem}
\begin{proof}
The dual function $g_0(y)=-\infty$ when $y\not\in\mathcal{K}^*$; and for all
$y\in\mathcal{K}^*$, the dual function $g_0$ of $(P)$ can be equivalently
written as
\begin{eqnarray*}
g_0(y)&= &\fprod{b,y}+\inf_{x\in\mathbb{R}^n} \left\{p(x)+\mathbf{1}_{\chi}(x)-\fprod{A^Ty,x}\right\},\\
&=& \fprod{b,y}-(p+\mathbf{1}_\chi)^*(A^Ty).
\end{eqnarray*}
Hence, the dual of $(P)$
is
\begin{equation}
\label{eq:dualproblem}
(D): \quad \max_{y\in\mathcal{K}^*}\fprod{b,y}-(p+\mathbf{1}_\chi)^*(A^Ty).
\end{equation}
Any KKT point $y^*\in\mathcal{K}^*$ is
an optimal solution of \eqref{eq:dualproblem}.
Let $b_k:=b+\frac{y_k}{\mu_k}$ for all $k\geq 1$. For $\kappa>0$, define
\begin{eqnarray*}
(\mathcal{P}_k): \quad\lefteqn{\min_{x\in\chi} \left\{p(x)+\kappa~d_{\mathcal{K}}(Ax-b_k)\right\},}\\
&=& \min_{x\in\mathbb{R}^n, s\in\mathcal{K}} \left\{ p(x)
+\mathbf{1}_{\chi}(x)+\kappa~\norm{Ax-b_k-s}_2 \right\},\\
&=& \max_{\norm{w}_2\leq\kappa}\ \min_{x\in\mathbb{R}^n, s\in\mathcal{K}} \left\{p(x)
+\mathbf{1}_{\chi}(x)+\fprod{w,~Ax-b_k-s}\right\},\\
&= & \max_{\norm{w}_2\leq\kappa} \left\{ -\fprod{b_k, w} +\inf_{s\in\mathcal{K}}\fprod{-w,s} -\sup_{x\in\mathbb{R}^n}
\left\{\fprod{-A^Tw,x}-\left(p(x)+\mathbf{1}_{\chi}(x)\right)\right\}\right\}.
\end{eqnarray*}
Since $\inf_{s\in\mathcal{K}}\fprod{-w, s} > -\infty$, only if $-w \in
\mathcal{K}^\ast$; by setting $y=-w$, we obtain the following dual problem
$(\mathcal{D}_k)$ of $(\mathcal{P}_k)$:
\begin{equation*}
(D_k): \quad \max_{\norm{y}_2\leq\tau,~y\in\mathcal{K}^*}
\left\{\fprod{b_k,y}-(p+\mathbf{1}_\chi)^*(A^Ty)\right\}.
\end{equation*}
Since $y^*\in\mathcal{K}^*$ is feasible to $(\mathcal{D}_k)$ for
$\kappa=\norm{y^*}_2$, and $x_k\in\chi$ is feasible to $(P_k)$, weak
duality implies that
\[
p(x_k)+\norm{y^*}_2~d_\mathcal{K}(Ax_k-b_k)\geq
\fprod{b,y^*}-(p+\mathbf{1}_\chi)^*(A^Ty^*)+\frac{1}{\mu_k}\fprod{y_k,
y^*} =p^*+\frac{1}{\mu_k}\fprod{y_k, y^*},
\]
where the equality follows from strong duality between $(P)$ and $(D)$.
\end{proof}
Next, we upper bound the suboptimality.
\begin{theorem}
\label{thm:subopt-upper}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ be the sequence of primal-dual
ALCC iterates corresponding to a parameter sequence
$\{\alpha_k, \eta_k, \mu_k\}$ satisfying
\eqref{eq:params}. Let $p^*$ denote the optimal value of $(P)$.
Then
\begin{equation}
\label{eq:subopt-upper}
P_k(x_k,
y_k) -
\frac{1}{\mu_k}~p^*\leq\frac{1}{\mu_k}
~\xi_k^\ast+\frac{1}{2\mu_k^2}~\norm{y_k}_2^2,
\end{equation}
where $\xi_k^\ast = \max\{\alpha_k, \eta_k~d_{\chi^*}(x_k)\}$
and $\chi^*$ denote the set of optimal solutions to $(P)$.
\end{theorem}
\begin{proof}
Fix $k\geq 1$ and let $x^*\in\chi^*$.
Suppose that $x_k = \mbox{\scshape{Oracle}}(P_k, y_k, \alpha_k, \eta_k, \mu_k)$
satisfies \eqref{eq:eps_opt}. Then, since $x^*\in\chi$, from
\eqref{eq:inexact-subopt}, it follows that
\begin{equation}
P_k(x_k,y_k)\leq \inf_{x\in\chi}P_k(x, y_k)+\frac{\alpha_k}{\mu_k
\leq P_k(x^*, y_k)+\frac{\alpha_k}{\mu_k}. \label{eq:subopt-upper-inexact}
\end{equation}
Next, suppose that $x_k = \mbox{\scshape{Oracle}}(P_k, y_k, \alpha_k, \eta_k, \mu_k)$
satisfies \eqref{eq:grad_opt}. Then, since
$P_k(x,y_k)+\mathbf{1}_{\chi}(x)$ is convex in $x$ for all $k\geq 1$, it follows that
\begin{equation}
\label{eq:subopt-upper-grad}
P_k(x_k,y_k)\leq P_k(x^*,y_k)+\fprod{q_k, x_k-x^*}\leq
P_k(x^*,y_k)+\frac{\eta_k~\norm{x_k-x^*}_2}{\mu_k}.
\end{equation}
From \eqref{eq:subopt-upper-inexact} and \eqref{eq:subopt-upper-grad},
it follows that
\begin{equation}
P_k(x_k,y_k)-\frac{1}{\mu_k}~p^*\leq
\frac{1}{2}
~d_\mathcal{K}\left(Ax^*-b-\frac{y_k}{\mu_k}\right)^2
+\frac{\max\{\alpha_k,~\eta_k~\norm{x_k-x^*}_2\}}{\mu_k}. \label{eq:subopt-upper-crude}
\end{equation}
Since $Ax^*-b\in\mathcal{K}$, Lemma~\ref{lem:infeasibility} implies that $d_\mathcal{K}\left(Ax^*-b-\frac{y_k}{\mu_k}\right)\leq\frac{\norm{y_k}_2}{\mu_k}$. Moreover, since $x^*\in\chi^*$ is arbitrary, from \eqref{eq:subopt-upper-crude} it follows that
\begin{equation}
P_k(x_k,y_k)-\frac{1}{\mu_k}~p^*\leq\frac{\norm{y_k}^2_2}{2\mu_k}
+\frac{\max\{\alpha_k,~\eta_k~\inf_{x^*\in\chi^*}\norm{x_k-x^*}_2\}}{\mu_k}. \label{eq:subopt-upper1}
\end{equation}
\end{proof}
Note that since $f_k(\cdot)\geq 0$, we have
$P_k(x_k,y_k)\geq\frac{1}{\mu_k}~p(x_k)$ for all $k\geq 1$. Hence,
\begin{equation}
\label{eq:subopt-upper2}
p(x_k)-p^*\leq \xi_k^\ast+\frac{1}{2\mu_k}~\norm{y_k}_2^2.
\end{equation}
Now, we establish a bound on the infeasibility of the primal ALCC iterate sequence.
\begin{theorem}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ denote the sequence of primal-dual
ALCC iterates for a parameter sequence
$\{\alpha_k, \eta_k, \mu_k\}$ satisfying \eqref{eq:params} and $y^*\in\mathcal{K}^*$ be a KKT point of $(P)$. Then
\begin{equation}
\label{eq:penalty}
0\leq d_\mathcal{K}\left(Ax_k-b\right)\leq \frac{\norm{y_k}_2+\norm{y_{k+1}-y_k}_2}{\mu_k}
\end{equation}
for all $k\geq 1$, where $\xi_k^\ast = \max\{\alpha_k, \eta_k~d_{\chi^*}(x_k)\}$
and $\chi^*$ denote the set of optimal solutions to $(P)$.
\end{theorem}
\begin{proof}
From Step~\ref{algeq:y_update} in ALCC algorithm, it follows that
\begin{align*}
\frac{y_{k+1}-y_k}{\mu_k}&=\Pi_\mathcal{K}\left(Ax_k-b-\frac{y_k}{\mu_k}\right)-(Ax_k-b),\\
&=\Pi_\mathcal{K}\left(Ax_k-b-\frac{y_k}{\mu_k}\right)-\Pi_\mathcal{K}(Ax_k-b)+\Pi_\mathcal{K}(Ax_k-b)-(Ax_k-b).
\end{align*}
Hence,
\begin{equation*}
d_\mathcal{K}(Ax_k-b)\leq\frac{\norm{y_{k+1}-y_k}_2}{\mu_k}+\left\|\Pi_\mathcal{K}\left(Ax_k-b-\frac{y_k}{\mu_k}\right)-\Pi_\mathcal{K}(Ax_k-b)\right\|_2.
\end{equation*}
The result now follows from the fact that $\Pi_\mathcal{K}$ is non-expansive.
\end{proof}
In the next theorem we establish the convergence rate of ALCC algorithm.
\begin{theorem}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ denote the sequence of primal-dual
ALCC iterates for a parameter sequence
$\{\alpha_k, \eta_k, \mu_k\}$ satisfying \eqref{eq:mult-seq}. Then for
all $\epsilon>0$, $d_\mathcal{K}(Ax_k-b)\leq\epsilon$ and
$|p(x_k)-p^*|\leq\epsilon$ within
$\mathcal{O}\left(\log\left(\epsilon^{-1}\right)\right)$ \mbox{\scshape{Oracle}}\ calls, which
require solving at most
$\mathcal{O}\left(\epsilon^{-1}\log\left(\epsilon^{-1}\right)\right)$ problems
of the form \eqref{eq:nonsmooth-operation}.
\end{theorem}
\begin{proof}
To simplify the notation, let $\alpha_0=\eta_0=\mu_0=1$, and, without
loss of generality, assume that $1\leq \mathcal{D}$, where $\mathcal{D}:=\max_{x\in\chi}d_{\chi^*}(x)\leq\Delta_\chi<\infty$. Then, clearly $d_{\chi^*}(x_k)\leq \mathcal{D}$ for all $k\geq 1$.
First, \eqref{eq:penalty} implies that
\begin{equation}
\label{eq:infeasibility}
d_\mathcal{K}(Ax_k-b)\leq \frac{1}{\beta^k}\left(\norm{y_k}_2+\norm{y_{k+1}-y_k}_2\right).
\end{equation}
Moreover, from Step~\ref{algeq:y_update} of ALCC algorithm, it follows that
\begin{equation}
\label{eq:penalty-explicit}
d_\mathcal{K}\left(Ax_k-b-\frac{y_k}{\mu_k}\right)\leq \frac{\norm{y_{k+1}}_2}{\mu_k}=
\frac{1}{\beta^k}\norm{y_{k+1}}_2.
\end{equation}
Now, Theorem~\ref{thm:subopt-lower}, \eqref{eq:subopt-upper2} and \eqref{eq:penalty-explicit} together imply that
\begin{equation}
\label{eq:subopt}
|p(x_k)-p^*|\leq
\frac{1}{\beta^k}\max\left\{\norm{y^*}_2\left(\norm{y_{k+1}}_2+\norm{y_k}_2\right),
~\frac{\mathcal{D}}{k^{2(1+c)}}+\frac{\norm{y_k}_2^2}{2}\right\}
\end{equation}
Theorem~\ref{thm:bounded-y} shows that $\{y_k\}$ is a bounded
sequence. Hence, from \eqref{eq:infeasibility} and \eqref{eq:subopt},
we have
\begin{equation}
\label{eq:outer-iter-complexity}
d_\mathcal{K}(Ax_k-b)=\mathcal{O}\left(\frac{1}{\beta^k}\right), \quad \quad
|p(x_k)-p^*|=\mathcal{O}\left(\frac{1}{\beta^k}\right).
\end{equation}
Hence, \eqref{eq:outer-iter-complexity} implies that for all $\epsilon>0$,
an $\epsilon$-optimal and $\epsilon$-feasible solution to
$(P)$ can be computed within $\mathcal{O}\left(\log\left(\epsilon^{-1}\right)\right)$ iterations of ALCC algorithm.
The values of $L_{\bar{\gamma}_k}$, $\alpha_k$ and $\mu_k$ are given respectively in \eqref{eq:lipschitz-constant} and \eqref{eq:mult-seq}. Substituting them in the expression for $\ell_{\max}(k)$ in \eqref{eq:lmax} and using the fact that $d_{\chi_k^*}(x_{k-1})\leq\Delta_\chi$, we obtain
\begin{equation}
\label{eq:inner-iter-complexity}
\ell_{\max}(k)\leq\sqrt{\frac{2L_\gamma}{\beta^k}+2\sigma^2_{\max}(A)}~d_{\chi_k^*}(x_{k-1})~\beta^k k^{1+c}=\mathcal{O}\left(\beta^k k^{1+c}\right).
\end{equation}
Hence, \eqref{eq:inner-iter-complexity} imply that at most $\mathcal{O}\left(\epsilon^{-1} \log(\epsilon^{-1})\right)$
problems of the form \eqref{eq:nonsmooth-operation} are solved during $\mathcal{O}\left(\log\left(\epsilon^{-1}\right)\right)$ iterations of ALCC algorithm. Indeed, let $N_\epsilon\in\mathbb{Z}_+$ denote total number of problems of the form \eqref{eq:nonsmooth-operation} solved to compute an $\epsilon$-optimal and $\epsilon$-feasible solution to $(P)$. From \eqref{eq:outer-iter-complexity} and \eqref{eq:inner-iter-complexity}, it follows that there exists $c_1>0$ and $c_2>0$ such that
\begin{equation*}
N_\epsilon\leq\sum_{k=1}^{\log_\beta\left(\frac{c_1}{\epsilon}\right)}\ell_{\max}(k)\leq\sum_{k=1}^{\log_\beta\left(\frac{c_1}{\epsilon}\right)}c_2\beta^k k^{1+c}\leq \frac{\beta}{\beta-1}\left(\frac{c_1}{\epsilon}-1\right)\left(\log_\beta\left(\frac{c_1}{\epsilon}\right)\right)^{1+c}.
\end{equation*}
\end{proof}
\begin{corollary}
\label{cor:opt}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ denote the sequence of
primal-dual ALCC iterates for a parameter sequence $\{\alpha_k,~\eta_k,~\mu_k\}$ satisfying \eqref{eq:mult-seq}. Then $\lim_{k\in\mathbb{Z}_+}p(x_k)=p^*$
and $\lim_{k\in\mathbb{Z}_+}d_\mathcal{K}(Ax_k-b)=0$. Moreover, for all $\mathcal{S}\subset\mathbb{Z}_+$ such that $\bar{x}=\lim_{k\in\mathcal{S}}x_k$, $\bar{x}$ is an optimal solution to $(P)$.
\end{corollary}
\begin{proof}
Since $\chi$ is compact,
Bolzano--Weierstrass theorem implies that there exists a subsequence
$\mathcal{S}\subset\mathbb{Z}_+$ such that $\bar{x}=\lim_{k\in\mathcal{S}}x_k$ exists. Moreover,
taking the limit of both sides of \eqref{eq:infeasibility} and
\eqref{eq:subopt}, we have $\lim_{k\in\mathbb{Z}_+}d_\mathcal{K}(Ax_k-b)=0$ and $\lim_{k\in\mathbb{Z}_+}p(x_k)=p^*$. Hence, $\lim_{k\in\mathcal{S}}d_\mathcal{K}(Ax_k-b)=0$ and $\lim_{k\in\mathcal{S}}p(x_k)=p^*$.
\end{proof}\\
Note that even though $p(x_k) \rightarrow p^\ast$, the primal iterates themselves may not converge.
Rockafellar~\cite{Roc73_1J} proved that the dual iterate sequence
$\{y_k\}$ computed via
\eqref{eq:x-update_rock}--\eqref{eq:y-update_rock}, converges to a KKT
point of \eqref{eq:rockafellar_problem}. We want to extend this
result to the case where $\mathcal{K}$ is a general convex cone.
The
proof in \cite{Roc73_1J}
uses the fact that the penalty multiplier $\mu$
is fixed in \eqref{eq:x-update_rock}--\eqref{eq:y-update_rock} and it
is not immediately clear how to extend this result to the setting with $\{\mu_k\}$ such that
$\mu_k\rightarrow\infty$. In Theorem~\ref{thm:dual-limit}, we extend
Rockafellar's result in \cite{Roc73_1J}
to arbitrary convex cones $\mathcal{K}$ when $f(x)=Ax-b$ and the penalty multipliers $\mu_k\rightarrow\infty$. After we independently proved~Theorem~\ref{thm:dual-limit}, we became
aware of an earlier work of Rockafellar~\cite{Rockafellar1976}
where he also extends the dual convergence result in \cite{Roc73_1J} to
the setting where $\{\mu_k\}$ is an increasing sequence. See
Section~\ref{sec:previous} for a detailed
discussion of our contribution in relation to this earlier work by
Rockafellar.
\begin{theorem}
\label{thm:dual-limit}
Let $\{x_k,y_k\}\in\chi\times\mathcal{K}^*$ denote the sequence of primal-dual
ALCC iterates corresponding to a parameter sequence
$\{\alpha_k, \eta_k, \mu_k\}$ satisfying \eqref{eq:mult-seq}.
Then $\bar{y}:=\lim_{k\in\mathbb{Z}_+}y_k$ exists and $\bar{y}$ is a KKT
point of $(P)$ in \eqref{eq:conic_problem}.
\end{theorem}
\begin{proof}
It follows from~\eqref{eq:y-My} that for all $k\geq 1$ we have
\begin{equation}
\label{eq:y-My-limit}
\lim_{k\in\mathbb{Z}_+}\norm{y_{k+1}-\pi_{\mu_k}(y_k)}_2\leq\lim_{k\in\mathbb{Z}_+}
\sqrt{2\xi^{(k)}\mu_k}=0,
\end{equation}
where $\xi^{(k)}$ is defined in \eqref{eq:xik}. Moreover,
Theorem~\ref{thm:bounded-y} shows that $\{y_k\}$ is a
bounded sequence. Hence, \eqref{eq:y-My-limit} implies that
$\{\pi_{\mu_k}(y_k)\}$ is also a bounded sequence.
From \eqref{eq:g_moreau}, it follows that
$g_{\mu_k}(y_k)=g_0(\pi_{\mu_k}(y_k))-\frac{1}{2\mu_k}~\norm{\pi_{\mu_k}(y_k)-y_k}_2^2$
and $g_{\mu_k}(y_k)\geq g_0(y^*)-\frac{1}{2\mu_k}~\norm{y^*-y_k}_2^2$
for any KKT point $y^*$. Since $g_0(y^*)=p^*$, we have that
\begin{equation}
\label{eq:g0-lowerbound}
g_0(\pi_{\mu_k}(y_k))\geq p^*-\frac{1}{2\mu_k}~\norm{y^*-y_k}_2^2.
\end{equation}
Since $\{y_k\}$ is bounded, taking the limit inferior of both sides of
\eqref{eq:g0-lowerbound} we obtain
\begin{equation}
\liminf_{k\in\mathbb{Z}_+}g_0(\pi_{\mu_k}(y_k))\geq p^*-\lim_{k\in\mathbb{Z}_+}\frac{1}{2\mu_k}~\norm{y^*-y_k}_2^2=p^*.\label{eq:liminf}
\end{equation}
Moreover, since $\pi_{\mu_k}(y_k)\in\mathcal{K}^*$ for all $k\geq 1$, weak
duality implies that $\limsup_{k\in\mathbb{Z}_+}g_0(\pi_{\mu_k}(y_k))\leq
p^*$. Thus, using \eqref{eq:liminf}, we have that
\begin{equation}
\label{eq:lim-g0}
\lim_{k\in\mathbb{Z}_+}g_0(\pi_{\mu_k}(y_k))=p^*.
\end{equation}
Since $\{\pi_{\mu_k}(y_k)\}$ is bounded, there exists
$\mathcal{S}\subset\mathbb{Z}_+$ and $\bar{y}\in\mathcal{K}^*$ such that
\begin{equation}
\label{eq:y-lim}
\bar{y}:=\lim_{k\in\mathcal{S}}\pi_{\mu_k}(y_k)=\lim_{k\in\mathcal{S}}y_{k+1},
\end{equation}
where the last equality follows from \eqref{eq:y-My-limit}.
From \eqref{eq:Lmu_def} and \eqref{eq:g_mu}, it follows that
\begin{equation*}
g_0(y)=\inf_{x\in\chi,s\in\mathcal{K}}\big\{p(x)-\fprod{y,Ax-s-b}\big\}.
\end{equation*}
Hence, $-g_0$ is a pointwise supremum of linear functions, which are
always closed. Lemma~3.1.11 in \cite{Nesterov04} establishes that $-g_0$ is
a closed convex function. Since a closed convex function is always
lower semicontinous, we can conclude that $-g_0$ is lower
semicontinuous, or equivalently, $g_0$ is
an upper semicontinuous function. Hence, \eqref{eq:lim-g0} and
\eqref{eq:y-lim} imply that
\begin{equation*}
p^*=\lim_{k\in\mathbb{Z}_+}g_0(\pi_{\mu_k}(y_k))=\limsup_{k\in\mathcal{S}}g_0(\pi_{\mu_k}(y_k))\leq
g_0(\bar{y})\leq p^*,
\end{equation*}
where the first inequality is due to upper semicontinuity of $g_0$ and
the last one is due to weak duality and the fact that $\bar{y}\in\mathcal{K}^*$. Thus, we have
\begin{equation}
\label{eq:KKT-y}
g_0(\bar{y})=\lim_{k\in\mathbb{Z}_+}g_0(\pi_{\mu_k}(y_k))=p^*,
\end{equation}
which implies that $\bar{y}\in\mathcal{K}^*$ is a KKT point of \eqref{eq:conic_problem}.
Moreover, since \eqref{eq:y-induction-eps} holds for any KKT point, we can
substitute $\bar{y}$ for $y^*$ in the expression. Thus, we have
\begin{equation}
\label{eq:convergent-y}
\norm{y_{\ell}-\bar{y}}_2\leq\norm{y_k-\bar{y}}_2+\sum_{t\geq
k}\sqrt{2\xi_t\mu_t},\quad \forall \ell> k.
\end{equation}
Fix $\epsilon>0$. Since the sequence $\{\sqrt{\xi^{(k)}\mu_k}\}$ is summable, it follows
that there exists $N_1\in\mathbb{Z}_+$ such that
$\sum_{t=k}^{\infty}\sqrt{2\xi_t\mu_t}\leq\frac{\epsilon}{2}$ for all $k > N_1$. Moreover, since the $\{y_k\}_{k \in \mathcal{S}}$ converges to $\bar{y}$, it follows that there exists $N_2 \in\mathcal{S}$ such that $N_2\geq N_1$ and $\norm{y_{N_2}-\bar{y}}_2\leq\frac{\epsilon}{2}$.
Hence, \eqref{eq:convergent-y} implies that
$\norm{y_\ell-\bar{y}}_2\leq\epsilon$ for all $\ell> N_2$. Therefore, $\lim_{k\in\mathbb{Z}_+}y_k=\bar{y}$.
\end{proof}
\section{Conclusion}
In this paper we build on previously known augmented Lagrangian
algorithms for convex
problems with standard inequality
constraints~\cite{Rockafellar-73,Roc73_1J} to develop the ALCC algorithm\ that
solves convex problems with conic constraints.
In each iteration of the ALCC algorithm, a sequence of ``penalty'' Lagrangians---see \eqref{eq:L_mu}---are
inexactly minimized
over a ``simple'' closed convex set.
We show that
recent results on optimal first-order
algorithms~\cite{Beck09_1J,Tseng08}~(see also \cite{Nesterov04,Nesterov05}), can be used to
bound the number of basic operations needed in each iteration to
inexactly minimize the ``penalty'' Lagrangian sub-problem.
By carefully controlling the growth of the penalty parameter $\mu_k$ that
controls the iteration complexity of ALCC algorithm, and the decay of
parameter $\alpha_k$ that controls the suboptimality of each
sub-problem,
we show that ALCC algorithm\ is a theoretically efficient
first-order, inexact augmented Lagrangian algorithm for structured
non-smooth conic convex programming.
\bibliographystyle{siam}
|
2,869,038,155,726 | arxiv | \section*{Acknowledgements}
The author thanks Christian Fleischhack for numerous discussions
and many helpful comments on several drafts of this dissertation. He thanks Johannes Aastrup, Martin Laubinger and Benjamin Schwarz for continued support, in particular, at the beginning of his doctoral studies.
He is grateful for discussions with various members of the math faculty of the University of Paderborn. In particular, with Joachim Hilgert, Bernhard Kr\"otz, Alexander Schmeding and Andreas Schmied. He thanks Gerd Rudolph for general discussions and comments on a first draft of the paper \cite{InvConn}, and the anonymous SIGMA referees for several helpful comments and suggestions. He also thanks Benjamin Bahr and Alexander Stottmeister for providing him with physical background
as well as Jonathan Engle, Christian Fleischhack, Hanno Sahlmann and Stefan Waldmann for supporting him on his academic path.
Finally, he expresses his deep gratitude to all the beloved people who have encouraged him during his time in Paderborn both close and from distance.
The author has been supported by the Emmy-Noether-Programm of
the Deutsche Forschungsgemeinschaft under grant FL~622/1-1.
\newpage
\tableofcontents
\newpage
\section{Introduction}
\subsection{Quantum Gravity}
One of the most challenging problems of modern physics is the embedding of quantum mechanics and general relativity into a superordinated (and mathematically substantiated) physical theory. Such a unified description is expected to play a role whenever massive objects are concentrated on small spaces being the case, e.g., for big bang scenarios or black holes. There, one would wish such a theory to resolve the singularities that appear when one describes these phenomena
in the classical framework of general relativity.
Serious difficulties in combining quantum mechanics with general relativity arise from the conceptual differences of these two theories and from the lack of physical experiments hinting to some kind of quantum gravity effects. Such effects, however, one would expect from a theory unifying gravitational and quantum nature of matter.
So, at this point, unification can only be done on a purely theoretical level, and here one basically has the choice between the following two strategies. First, one can try to construct a completely new theory which, in the appropriate physical limits, reproduces general relativity and quantum mechanics. Second, one can try to quantize general relativity directly, hoping to end up with a unified theory or, to be more realistic, to get some hints on how such a theory should look like.
Following these philosophies, promising candidates are string theory, a pertubative approach to the construction of a superordinated theory, and the loop quantum gravity approach we will follow in this thesis.
\subsubsection{Loop Quantum Gravity}
Being a non-pertubative and background independent approach, loop quantum gravity (\gls{LQG}) \cite{BackLA, Thiemann}
seems to be appropriate for understanding quantum gravity effects near classical singularities where curvature is by no means small. Indeed, within this context it was possible to derive the Bekenstein-Hawking area law for a large class of black holes. \cite{blackbeken, Doma}
Now, being a canonical quantization of gravity, LQG is based on a splitting of space-time into time and space, entailing that the four-dimensional covariance of general relativity is no longer manifest. \cite{Thiemann, Hanno} In particular, the 4-dimensional diffeomorphism constraint splits up into a spatial and the Hamiltonian constraint, the latter one defining the dynamics of LQG.
Unfortunately, the quantization of the Hamiltonian of the full theory turns out to be difficult and is, at this point, not completely understood. \cite{Thiemann0} Here, symmetry reduced versions of LQG like loop quantum cosmology \cite{Bojolivrev} can help to better understand this quantization for the full theory.
In addition to that,
mathematical developments like the
reduction concept to be developed in this work carry over to a bigger class of gauge field theories, so that LQG is not an isolated field of research but an approach to quantum gravity whose developments enhance other areas of theoretical physics and even mathematics.
\subsubsection{Challenges}
Although symmetry reduced versions of loop quantum gravity exists, no conceptually satisfying reduction concept has been developed so far. Indeed,
to this point reduction has been done on a rather intuitional level, so that the connection to the full theory was usually not manifest but had to be established in a laborious way. Standard homogeneous isotropic loop quantum cosmology (\gls{LQC}) may serve as a prime example for this. However, since symmetric situations in nature occur, for a physical theory one would expect to have a reasonable reduction concept which allows to get rid of superfluous degrees of freedom.
Now, being an Ashtekar approach to quantum field theories, LQG is based on functional integrals. So, besides the definition of a reduced quantum configuration space, a reduction theory here should include the construction of corresponding Radon measures which (as for the full theory) allow to integrate over field configurations.
In addition to that, appropriately reduced holonomy-flux algebras (and Hamiltonians) have to be represented on the respective kinematical Hilbert spaces of square integrable functions. Then, states on the reduced algebras have to be embedded into respective symmetric sectors of LQG. \cite{BojoKa}
Once such a concept has been established, the consideration of highly symmetric systems might allow to make verifiable predictions that can help to advance the full theory.
\subsection{Mathematical Context}
The basic mathematical objects studied in this thesis are spaces of connections on principal fibre bundles, spectra of $\Cstar$-algebras of bounded functions and normalized Radon measures. Since we are investigating the problem of symmetry reduction, also left actions of (Lie) groups will play an important role.
\subsubsection{Invariant and Generalized Connections}
In its simplest form, the configuration space\footnote{Indeed, actually the quotient $\Con\slash \GAG$ of $\Con$ w.r.t.\ the set $\GAG$ of gauge transformations of $P$
is considered as physically relevant configuration space. However, to keep it simple, in this work we concentrate on the space $\Con$ and its quantum analogue $\A$ (see below). Here, the main reason is that we rather expect technical than conceptual difficulties when carrying over the developments of this work to the ``up to gauge''-case, i.e., to the quotient $\A\slash \ovl{\mathcal{G}}$ of $\A$ w.r.t.\ the compact group $\ovl{\mathcal{G}}$ of generalized gauge transformations. See, e.g., the outlook section for some more details.}
of a classical gauge field theory is formed by the set $\Con$ of smooth connections on a principal fibre bundle $(P,\pi,M,S)$.\footnote{In the LQG approach we usually have $P=\Sigma\times \SU$ for a 3-dimensional Cauchy surface $\Sigma$ of a space-time
$M$.}
Symmetries are realized by Lie groups of automorphisms $(G,\Phi)$, and symmetry reduction
just means to calculate the set
\begin{align}
\label{eq:invconnPhi}
\AR:=\{\w\in \Con\:|\: \Phi_g^*\w=\w\text{ for all }g\in G\}
\end{align}
of smooth connections invariant under pullbacks by all
symmetry group elements. \cite{Wang, HarSni}
Then, in the Ashtekar approach to quantum gauge field theories, the quantum configuration space $\ovl{\Con}$ of the full theory is formed by the spectrum of a separating $C^*$-algebra $\PaC$ of bounded functions on $\Con$.
Here, the $C^*$-algebra of cylindrical functions $\PaC\subseteq B(\Con)$ is generated by matrix entries\footnote{With respect to some faithful matrix representation of the structure group $S$.} of parallel transports along the elements of a fixed set $\Pa$ of curves in the base manifold of $P$. Here, the main reason for switching from $\Con$ to $\A$ is that (in contrast to $\Con$) there exists a natural measure on $\A$, the Ashtekar-Lewandowski one. This measure allows to integrate over field configurations and defines the $L^2$ Hilbert space on which the unique representation \cite{UniqLew, ChUn} of the holonomy-flux algebra is realized.
The separation property of $\PaC$ here guarantees that the space $\Con$ is canonically embedded\footnote{As we have not chosen any topology on $\Con$, this just means that $\iota_\Con$ is injective.} via
\begin{align*}
\iota_\Con \colon \Con\rightarrow \A,\quad \w \mapsto [f\mapsto f(\w)].
\end{align*}
Since
$\iota_\Con(A)\subseteq\A$ is even dense \cite{Rendall}, the space $\A$ can be seen as some kind of compactification of $\Con$ (provided that the structure group is compact, as then $\PaC$ is unital).
\subsubsection{Symmetry Reduction}
Traditionally, symmetry reduction in LQG has been
done by calculating the spectrum of a $\Cstar$-algebra of the form $\ovl{\PaC'|_{\AR}}\subseteq B(\AR)$. This is just the closure of
\begin{align*}
\PaC'|_{\AR}:=\left\{f|_\AR\:\big|\: f\in \PaC'\right\}
\end{align*}
in $B(\AR)$, where $\PaC'$ denotes the $\Cstar$-algebra of cylindrical functions that corresponds to some set $\Pa'$ of curves in $M$.
Choosing $\Pa\neq \Pa'$ then usually offers the problem that the reduced space cannot be naturally embedded into $\A$. Indeed, this is exactly the case for homogeneous isotropic LQC ($P=\RR^3\times \SU$) where originally the set of linear curves was used for $\Pa'$. \cite{Brunnhack} In particular, in view of the embedding strategy for states proposed in \cite{BojoKa} this is disadvantageous. To fix this problem, in \cite{ChrisSymmLQG} the space $\ARQ=\mathrm{Spec}\big(\ovl{\PaC|_{\AR}}\big)$ was introduced. Indeed, $\ARQ$ is naturally embedded in $\A$, even homeomorphic to the closure of $\iota_\Con(\AR)$ in $\A$.
Now, $\ARQ$ arises from a quantization of the reduced classical space $\AR$ and not from a reduction of the quantum space $\A$.
So, in this thesis we will follow the second, conceptual more satisfying approach which has not been investigated so far.
This means that we will perform a symmetry reduction directly on the quantum level just by extending the left action
$\cw\colon G\times \Con \rightarrow \Con$,
$(g,\w)\mapsto \Phi_{g^{-1}}^*\w$
which determines
\begin{align*}
\AR=\{\w\in \Con\:|\: \cw(g,\w)=\w\:\: \forall\:g\in G \}
\end{align*}
to a left action $\specw\colon G\times \A\rightarrow \A$. This will provide us with the quantum-reduced configuration space
\begin{align*}
\AQR:=\{\ovl{\w}\in \A\:|\: \specw(g,\ovl{\w})=\ovl{\w}\:\: \forall\: g\in G \}
\end{align*}
which, as we will see, always contains the quantized reduced classical space $\ARQ$. In several situations, this inclusion even turns out to be proper so that in our context quantization and reduction usually do not commute. This is also the main reason why it is much easier to construct measures on $\AQR$ than on $\ARQ$.\footnote{For the case of homogeneous isotropic LQC we will construct measures on $\ARQ$ by hand, see Section \ref{sec:HomIsoCo}.} Indeed, for
$\AQR$ so-called modification techniques are available, which only work with limitations for $\ARQ$. This is just because they do not leave this space
invariant.
To get a rough idea what modification of a generalized connection here means, it is important to know that for compact and connected structure groups (the typical LQG case) one can identify $\A$ with the space of homomorphisms $\TRHOM$. Such homomorphisms map curves in $\Pa$ to structure group elements, and
modification then just means to change the values of such a map along some fixed curves in a specific way. The same techniques then will also allow to construct normalized Radon measures by means of projective structures on $\AQR$.
\subsubsection{Projective Structures and Normalized Radon Measures}
A practicable way to construct normalized Radon measures on compact Hausdorff spaces (and the usual way to do it for quantum configuration spaces in LQG)
is to identify the space $X$ of interest as a projective limit of a reasonable family of compact Hausdorff spaces $\{X_\alpha\}_{\alpha\in I}$.
Here, reasonable means that each of these spaces carries a natural normalized Radon measure, and that all these measures are consistent in the way described below.
Here, the basic idea is that for $X$ high-dimensional in a certain sense, each $X_\alpha$ catches finitely many degrees of freedom of this space.
Now, to be a projective limit basically means that\footnote{This definition differs slightly from the standard one \cite{ProjTechAL}, but is equivalent to it.}
\begin{enumerate}
\item
\label{akakakaka}
There exist continuous surjective projection maps $\pi_\alpha\colon X\rightarrow X_\alpha$ which separate the points in $X$
\item
\label{bkakakaka}
For each two $\alpha_1,\alpha_2\in I$ ($I$ is a directed set) with $\alpha_1\leq \alpha_2$ there exists a transition map $\pi^{\alpha_2}_{\alpha_1}\colon X_{\alpha_2}\rightarrow X_{\alpha_1}$ with $\pi^{\alpha_2}_{\alpha_1}\cp \pi_{\alpha_2}=\pi_{\alpha_1}$.
\end{enumerate}
A Riesz-Markov argument shows that the normalized Radon measures on $X$ are in bijection with the consistent families $\{\mu_\alpha\}_{\alpha\in I}$ of normalized Radon measures $\mu_\alpha$ on $X_\alpha$. Here, consistency just means that for $\alpha_1,\alpha_2\in I$ with $\alpha_1\leq \alpha_2$ the equality $\pi^{\alpha_2}_{\alpha_1}(\mu_{\alpha_2})=\mu_{\alpha_1}$ holds.
Then, for $\A\cong \TRHOM$ one usually chooses $I$ to consist of certain finite tuples $\alpha=(\gamma_1,\dots,\gamma_k)$ of curves $\gamma_1,\dots,\gamma_k\in \Pa$, and defines
\begin{align*}
\pi_\alpha(\homm):=(\homm(\gamma_1),\dots,\homm(\gamma_k))\in S^k
\end{align*}
for $S^k$ the $k$-fold product of the structure group. Choosing these tuples appropriately then ensures that $\pi_\alpha$ is surjective, and that the Haar measure on $S$ can be used to define a respective consistent family of normalized Radon measures.
Now, we will see that under the identification $\A\cong \TRHOM$ the quantum-reduced space $\AQR\subseteq \A$ corresponds to a subset $\ITRHOM \subseteq \TRHOM$ of homomorphisms that fulfil certain invariance properties. These properties give non-trivial restrictions to the images of the maps $\pi_\alpha$, so that we will be forced to adapt the whole projective structure in order to obtain a non-trivial measure on $\ITRHOM$.
For the spaces $\ARQ$ the situation is even more difficult, just because the subsets $\pi_\alpha\big(\ARQ\big)\subseteq \pi_\alpha\big(\AQR\big)$ are usually much more complicated.
\subsection{Aims and Organization}
Although this dissertation is rather motivated by the framework of LQG, its main goal is to provide general tools and concepts that allow to perform a mathematically rigorous symmetry reduction also in other (quantum) gauge field theories. Here, we will focus on the reduction of the quantum configuration space and the definition of reasonable Radon measures thereon.
We also attack the question whether, in our context, quantization and reduction commute or not. In course of this, we will prove a general characterization theorem for invariant smooth connections on principal fibre bundles which generalizes the classical results of Wang \cite{Wang} and Harnad, Shnider and Vinet \cite{HarSni}.
The definition of representations of respective reduced holonomy-flux algebras on the constructed $L^2$-Hilbert spaces
is left as a future task.
This work is organized as follows:
\begingroup
\setlength{\leftmargini}{12pt}
\begin{itemize}
\item
The preliminaries in Section \ref{sec:prel} contain the notations, as well as the basic definitions, conventions and facts concerning principal fibre bundles, projective structures and Radon measures.
\item
In the first part of Section \ref{sec:specmathback},
we present the LQC relevant cases which will serve as prime examples during this work. In addition to that, we discuss the properties of $\SU$ being relevant for our later calculations. In the second part, we will collect the facts on spectra of $\Cstar$-algebras of bounded functions which build the mathematical backbone of the reduction concept introduced in \cite{ChrisSymmLQG} and that one, we will develop in Section \ref{sec:SPecExtGr}.
In the last part, we highlight the most crucial properties of the Bohr compactification of a locally compact abelian group. This is of relevance because the Bohr compactification $\RB$ of $\RR$ plays a key role in standard homogeneous isotropic LQC (see Section \ref{sec:HomIsoCo}), and also in the constructions in Subsection \ref{sec:ConSp}. We close Section \ref{sec:specmathback} with a characterization of the continuous abelian group structures on spectra of certain unital $\Cstar$-algebras.
\item
In Section \ref{sec:SPecExtGr}, we are going to lift Lie groups $(G,\Phi)$ of automorphisms of principal fibre bundles $P$ to spectra $\A$ of $\Cstar$-algebras of cylindrical functions $\PaC\subseteq B(\Con)$. Here, $\Con$ denotes the set of smooth connections on the respective bundle and $B(\Con)$ the set of bounded functions on $\Con$.
This will provide us with the notion of an invariant generalized connection, the quantum analogue of an invariant classical (smooth) one.
In the first step, we will use the concept of a $\Cstar$-dynamical system in order to extend an action $\text{\gls{CCW}}\colon G\times X\rightarrow X$ (of a group $G$ on a set $X$)
to an action $\specw\colon G\times \mathrm{Spec}(\aA)\rightarrow \mathrm{Spec}(\aA)$ on the spectrum of a $\Cstar$-subalgebra $\aA\subseteq B(X)$.
Then, we adapt this to the case where $X$ equals the set of smooth connections on a principal fibre bundle. Here, it will be crucial that
the set of paths $\Pa$, used for the definition of $\PaC$, is invariant in the sense that
\begin{align*}
\gamma\in \Pa\qquad \Longrightarrow \qquad \wm_g\cp \gamma \in \Pa\qquad\forall\: g\in G
\end{align*}
with $\wm\colon G\times M\rightarrow M$ the action induced by $\Phi$ on the base manifold $M$ of $P$.
Finally, we will consider the case where the structure group $S$ of $P$ is compact, and where the set of curves has the additional property of independence.\footnote{This is the case, e.g.\, if $\Pa$ is the set of embedded analytic curves and $S$ is connected.} In this situation, it will be possible to identify the quantum-reduced configuration spaces $\AQR$ with spaces of so-called invariant homomorphisms. This will later be important for the investigations of the inclusion relations between $\ARQ$ and $\AQR$, as well as the construction of measures on $\AQR$. At the end of Section \ref{sec:SPecExtGr}, we will show that the spaces $\ARQ$ and $\AQR$ are usually of measure zero w.r.t.\ the Ashtekar-Lewandowski measure (LQG standard measure) on $\A$.
\item
In Section \ref{susec:LieALgGenC}, we will develop modification techniques for invariant homomorphisms in the event that the induced action $\wm$ on $M$ is analytic and pointwise
proper.
In the first part, we collect the relevant facts and definitions concerning analytic and Lie algebra generated curves, whereby
$\gamma$ is called Lie algebra generated iff it is (up to parametrization) of the form $\gamma\colon t\mapsto \wm_x(\exp(t\cdot\g))$ for some $x\in M$ and $\g\in \mg\backslash \mg_x$. Here, $\mg_x$ denotes the Lie algebra of the $\wm$-stabilizer $G_x$ of $x$.
In Subsection \ref{sec:ModifLAGC}, we will modify invariant homomorphisms along such Lie algebra generated curves, and in Subsection \ref{sec:inclrel} we apply this in order to show that the inclusion $\ARQ\subseteq \AQR$ is proper in several situations. In particular, we conclude that quantization and reduction do not commute in \mbox{(semi-)homogeneous} LQC. For homogeneous isotropic LQC this will be shown in Section \ref{sec:HomIsoCo}.
In the last part of Section \ref{susec:LieALgGenC}, we will prove an analogue of the modification result from Subsection \ref{sec:ModifLAGC}, but now for free
curves. These are embedded analytic curves $\gamma$ containing a subcurve $\delta$ (free segment) for which $\wm_g\cp \delta =\delta$ holds whenever $\mathrm{im}[\wm_g\cp \delta] \cap \mathrm{im}[\delta]$ is infinite for some $g\in G$. We will show that, under the condition that $\wm$ is analytic and pointwise proper, each free curve $\gamma$ is discretely generated by the symmetry group.\footnote{This means that we find a (maximal) free segment $\delta$ of $\gamma$ such that $\gamma$ admits a decomposition into finitely many subcurves, each being (up to parametrization) an initial or final segment of $\wm_g\cp \delta$ for some $g\in G$. Here, each of these subcurves which is not an initial or final segment of $\gamma$, then even equals (up to parametrization) the full segment $\wm_g\cp\delta$ for the respective $g\in G$.}
Moreover, we will see that if each $\wm$-stabilizer is even a normal subgroup and $\wm$ is in addition transitive or proper, then each embedded analytic curve is either free or (up to parametrizations) Lie algebra generated.
For instance, this is the case in \mbox{(semi-)homogeneous} LQC, where it will us to construct a normalized Radon measure on $\AQR$ if the structure group is $\SU$, see Section \ref{sec:MOQRCS}.
\item
In Section \ref{sec:MOQRCS}, we will construct normalized Radon measures on certain quantum-reduced configuration spaces for the case that $\wm$ is analytic and pointwise proper. Here, the main idea is to split up the set $\Paw$ of embedded analytic curves into suitable subsets $\Pa_\alpha$, $\alpha\in I$, each being closed under decomposition and inversion of its elements. In fact, then (Subsection \ref{subsub:InvHoms})
\begin{align*}
\AQRw\cong\prod_{\alpha\in I}\AQRInd{\alpha}
\end{align*}
holds and, provided that we have constructed normalized Radon measures on each of the spaces $\AQRInd{\alpha}$, we obtain a normalized Radon measure on $\AQRw$, just by taking the Radon product one.
For instance, if $\wm$ is proper and free, we have $\AQRw\cong\AQRInd{\mg}\times \AQRInd{\mathrm{F}}$ where $\AQRInd{\mg}$ corresponds to the set of Lie algebra generated and $\AQRInd{\mathrm{F}}$ to the set of the free curves.
So, in this case it suffices to construct normalized Radon measures on these two spaces, which is exactly the content of Section \ref{sec:MOQRCS}.
In fact, in the first part of this section,
we will construct a normalized Radon measure on $\AQRFNS$ for the case that $S$ is compact and connected. Here, $\AQRFNS$ corresponds to the set of such free curves whose stabilizer\footnote{As we will see, this is a well-behaving quantity if $\wm$ is analytic and pointwise proper.} is trivial, whereby
in the above situation
$\AQRFNS=\AQRInd{\mathrm{F}}$ holds, just because
there
$\wm$ was assumed to be free.
In the second part of Section \ref{sec:MOQRCS}, we will construct a normalized Radon measure $\mLAS$ on $\AQRInd{\mg}$, exemplarily, for the most LQC relevant case that $S=\SU$ (and for each $n$-torus). Here, we will require some additional conditions on the $\wm$-stabilizers, which appear to hold, e.g.\, in spherically symmetric, \mbox{(semi-)homogeneous} and homogeneous isotropic LQC.
Then, if $\wm$ is in addition transitive (such as in homogeneous and homogeneous isotropic LQC), we have the reasonable kinematical Hilbert space $L^2(\AQRLA,\mLAS)$. Indeed, in the transitive case $\AInd{\mg}$, hence $\AQRLA$ is a physically meaningful candidates for a quantum(-reduced) configuration space because
there $\iota\colon \Con \rightarrow \AInd{\mg}$ is injective as, in this situation, the cylindrical functions that correspond to $\Pags$ separate the points in $\Con$.
\item
In Section \ref{sec:HomIsoCo}, we will focus on homogeneous isotropic LQC. We show that quantization and reduction do not commute and investigate the measure theoretical aspects of the classically reduced quantum configuration space $\ARQ$. This space corresponds to the set of embedded analytic curves in $M=\RR^3$ and is
homeomorphic to the compact Hausdorff space\footnote{The topology on $\RR\sqcup \RB$ is quite tricky. Details will be given in Section \ref{sec:HomIsoCo}.} $\RR\sqcup \RB$. \cite{ChrisSymmLQG}
In contrast to the LQC standard approach (where the reduced quantum space is defined by all linear curves in $\RR^3$ and is homeomorphic to the compact abelian group $\RB$) on $\RR\sqcup \RB$ no Haar measure is available.
This will be shown in the first part of Section \ref{sec:HomIsoCo}, where we prove that no continuous group structure can exist on this space.
Then, changing the focus from Haar to normalized Radon measures, we will show that $\muB$ is the unique normalized Radon measure which is invariant under
the canonical extension $\Transl\colon \RR\times \RB \rightarrow \RB$ of the additive group action \RPLUS$\colon \RR \times \AR \rightarrow \AR$ of $\RR$ on $\AR\cong \RR$.
Moreover, we will prove that the same invariance condition singles out a normalized Radon measure on
$\ARQ\cong \RR\sqcup \RB$. This measure even
defines the same kinematical Hilbert space $\Lzw{\RB}{\muB}$ as we have in standard LQC, supporting this approach from the mathematical side.
In the last part of Section \ref{sec:HomIsoCo}, we will use projective structures in order to construct further normalized Radon measures on $\RR\sqcup \RB$ and, finally, compare the respective Hilbert spaces of square integrable functions thereon.
\item
In Section \ref{CHarinvconn}, we will prove a characterization theorem for invariant connection on principal fibre bundles which generalizes the classical results of Wang \cite{Wang} and Harnad, Shnider and Vinet \cite{HarSni}. We consider several special situations such as (almost) fibre transitivity (Case \ref{scase:slicegleichredcluster} and \ref{th:wang}), Lie groups of gauge transformations (Case \ref{scase:GaugeTransf}), trivial bundles (Case \ref{scase:trivbundle}) and the gauge fixing situation (Case \ref{scase:OneSlice}). Along the way, we give applications to loop quantum gravity. In particular, we will calculate the \mbox{(semi-)homogeneous} and spherically symmetric connections already introduced in Example \ref{ex:LQC} and which we will use in Subsection \ref{sec:inclrel} in order to show that the inclusion $\ARQ\subseteq \AQR$ is proper in \mbox{(semi-)homogeneous} and spherically symmetric LQC. We also show that the set of invariant connections depends crucially on the explicit lift of an action $\wm\colon G\times M\rightarrow M$ to $P$, see Remark \ref{rem:liftuntersch}.
\end{itemize}
\endgroup
\noindent
Each of the sections \ref{sec:SPecExtGr} -- \ref{CHarinvconn} closes with a short summary of its most relevant results.
\section{Preliminaries}
\label{sec:prel}
In this brief section we fix the notations and collect some facts and definitions which are more or less standard, but crucial for this thesis.
\subsection{Notations}
\label{sec:notations}
Manifolds are always assumed to be smooth or analytic. If $M, N$ are manifolds and $f\colon M\rightarrow N$ differentiable, then $\dd f\colon TM\rightarrow TN$ denotes the differential map between their tangent bundles. The map $f$ is said to be an immersion iff for each $x\in M$ the restriction $\dd_xf:=\dd f|_{T_xM}\colon T_xM\rightarrow T_{f(x)}N$ is injective. Elements of tangent spaces usually are written with arrows, such as $\vec{v}_x\in T_xM$. Here, we subscript the base point $x$ whenever it helps to clarify the calculations. In particular, in Section \ref{CHarinvconn} this will be helpful to keep the track of the calculations.
Let $V$ be a finite dimensional vector space. A $V$-valued 1-form $\w$ on the manifold $N$ is a smooth map $\w\colon TN\rightarrow V$ whose restriction $\w_y:=\w|_{T_yN}$ is linear for all $y\in N$. The pullback of $\w$ by $f$ is the $V$-valued 1-form $f^*\w\colon TM\rightarrow V$, $\vec{v}_x\rightarrow \w_{f(x)}(\dd_xf(\vec{v}_x))$. If it is clear which tangent space $\vec{v}$ belongs to, we usually do not subscript $\w$ by the base point, e.g., we write $\w(\vec{v}_x)$ instead of $\w_x(\vec{v}_x)$.
Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. For $g\in G$ we define the corresponding conjugation map by $\text{\gls{CONJ}}\colon G\rightarrow G$, $h\mapsto g h g^{-1}$. Its differential $\dd_e\Co{g}\colon \mathfrak{g}\rightarrow \mathfrak{g}$ at the unit element $e\in G$ is denoted by $\Add{g}$, and by \gls{ADD} we will denote the left action $\Ad\colon G\times \mg\rightarrow \mg$.
Let $\Psi$ be a (left) action of the Lie group $G$ on the manifold $M$. For $g\in G$ and $x\in M$ we define
$\Psi_g\colon M\rightarrow M$, $y\mapsto \Psi(g,y)$ and $\Psi_x\colon G\rightarrow M$, $h\mapsto\Psi(h,x)$, respectively.
If it is clear which action is meant,
we will often write $L_g$ instead of $\Psi_g$ as well as $g\cdot x$ or $g x$ instead of $\Psi_g(x)$.
For $\vec{g}\in \mathfrak{g}$ and $x\in M$ the map
\begin{align}
\label{eq:fundvf}
\wt{g}(x):=\dttB{t}{0}\: \Psi_x(\exp(t\vec{g}))
\end{align}
is called the \emph{fundamental vector field w.r.t.\ $\vec{g}$}.
The Lie subgroup $G_x:=\left\{g\in G\: \big| \: g\cdot x=x\right\}$ is called the \emph{stabilizer} of $x\in M$ (w.r.t.\ $\Psi$), and its Lie algebra $\mathfrak{g}_x$ equals $\ker[\dd_x\Psi]$, see e.g.\ \cite{DuisKolk}.
The \emph{orbit} of $x$ under $G$ is the set $Gx:=\mathrm{im}[\Psi_x]$. $\Psi$ is said to be \emph{transitive} iff $Gx=M$ holds for one (and then each) $x\in M$. The action $\Psi$ is called \emph{proper at $x$} iff for each net
$\{g_\alpha\}_{\alpha\in I}\subseteq G$ the convergence of
$\{\Psi(g_\alpha,x)\}_{\alpha\in I}\subseteq M$ implies the existence of a convergent subnet\footnote{This is a net $\{h_\beta\}_{\beta\in J}\subseteq G$ together with a map $\iota\colon J\rightarrow I$ such that $h_\beta=g_{\iota(\beta)}$ for all $\beta\in J$. Moreover, for each $\alpha\in I$ we find $\beta \in J$ such that $\iota(\beta')\geq \alpha$ holds for all $\beta'\geq \beta$.} of $\{g_\alpha\}_{\alpha\in I}$. This is equivalent to require that $\wm_x^{-1}(K)\subseteq G$ is compact whenever $K\subseteq M$ is compact, i.e., that $\wm_x$ is a proper map. Then, $\wm$ is called \emph{pointwise proper} iff it is proper at $x$ for all $x\in M$.
Finally, $\Psi$ is said to be \emph{proper} iff the convergences of $\{\Psi(g_\alpha,x_\alpha)\}_{\alpha\in I}\subseteq M$ and $\{x_\alpha\}_{\alpha\in I}\subseteq M$ imply the existence of a convergent subnet of $\{g_\alpha\}_{\alpha\in I}$. Analogous conventions hold for right actions.
A curve is a continuous map $\gamma\colon D\rightarrow M$. Here $D\subseteq \mathbb{R}$ is an interval, i.e., a set of the form $(a,b]$, $[a,b)$ or $[a,b]$ with $a<b$. The curve $\gamma$ is said to be of class\footnote{We allow $k\in \{\mathbb{N}_{\geq 1},\infty,\omega\}$, where $\omega$ means analytic.} $\CC{k}$ iff there is a $\Ck$-curve (in the sense of $\CC{k}$ maps between manifolds) $\gamma'\colon I\rightarrow M$ such that $\gamma'|_{D}=\gamma$. Here, $I$ is an open interval that contains $D$, and $\gamma'$ is called an extension of $\gamma$ in this case. The same conventions hold for diffeomorphisms $\adif\colon D\rightarrow D'\subseteq\RR$.
The $\Ck$-curve $\gamma\colon D\rightarrow M$ is called an embedding iff we find an extension $\gamma'\colon I\rightarrow M$ which is an injective immersion and a homeomorphism onto its image equipped with the relative topology. If $k=\w$, we say that $\gamma$ is an embedded analytic curve.
A curve $\gamma$ is called \emph{piecewise} (embedded) $\CC{k}$ or (embedded) \emph{$\CC{k}$-path} iff there are real numbers $a=\tau_0 <\ldots <\tau_k=b$ such that for each $0\leq i\leq k-1$ the restriction $\gamma|_{[\tau_i,\tau_{i+1}]}$ is an (embedded) curve of class $\CC{k}$. For the case that $\gamma$ is $C^k$ for some $k\in \NN_{\geq 1}$ or if $t$ is not contained in the interior of $D$, we will define the tangent vector $\dot\gamma(t) \in T_{\gamma(t)}M$
in the canonical way.\footnote{Recall that there occur some technical difficulties if one just treats $M$ as $\CC{k}$-manifold. This is because only for $k=\infty$ the algebraic definition of a tangent vector coincides with the geometric one. So, for $k\neq \infty$, i.e, if $\gamma$ is a $C^k$-curve in the smooth manifold $M$, we use some smooth chart $(U,\phi)$ around $\gamma(t)$ in order to obtain a smooth curve $\delta$ through $\gamma(t)$ with $\dttB{s}{t}(\phi\cp\delta)(t)=\dttB{s}{t}(\phi\cp\gamma)(t)$. Then, we define the tangent vector of $\gamma$ at $t$ to be the equivalence class $[\delta] \in T_{\gamma}(t)M$. Now, for the case that $t$ is not contained in the interior of $D$, we just use an extension of $\gamma$ in order to define $\dot\gamma(t)$.}
In the following, $I$ and $K$ will usually denote open and compact intervals, respectively, whereby $I$ also will occur as index set if it is not in conflict with our notations.
If $W$ is a set and $U\subseteq W$ a subset, then $U^c=W\backslash U$ denotes the complement of $U$ in $W$.
If $W$ is a topological space, by $\ovl{U}$ we usually denote the closure of $U$ in $W$. A different convention holds for subsets $\bB\subseteq B(Z)$ of $\Cstar$-algebras of bounded functions. Here, $\ovl{\bB}$ denotes the $\Cstar$-subalgebra of $B(Z)$ which is generated by $\bB$, see also Convention \ref{conv:Boundedfunc}.
If $Y$ is a locally compact Hausdorff space, then $\Cinf(Y)$ denotes the set of complex-valued, continuous functions on $Y$ that \emph{vanish at infinity}. This is that for each $f\in \Cinf(Y)$ and $\epsilon >0$ there is a compact subset $K_\epsilon\subseteq Y$ such that $|f|\leq \epsilon$ on $Y\backslash K_\epsilon$. If $\aA$ is a Banach algebra, then $\mathrm{Spec}(\aA)$ denotes the set of all non-zero, multiplicative, $\mathbb{C}$-valued functionals on $\aA$.
Then, by $\mathcal{G} \colon \aA \rightarrow \Cinf(\mathrm{Spec}(\aA))$, $a\mapsto \left[\hat{a}\colon f\mapsto f(a)\right]$ we will denote the Gelfand transformation.
Recall that the Gelfand-Naimark theorem states that $\mathcal{G}$ is an isometric $^*$-isomorphism if $\aA$ is a $\Cstar$-algebra.
\subsection{Principal Fibre Bundles}
\label{subsec:InvConn}
Let $\pi\colon P\rightarrow M$ be a smooth map between the manifolds $P$ and $M$, and denote by $\text{\gls{FX}}:=\pi^{-1}(x)\subseteq P$ the fibre over $x\in M$ in $P$. Moreover, let $S$ be a Lie group that acts via $R\colon P\times S\rightarrow P$ from the right on $P$.
If there is an open covering $\{U_\alpha\}_{\alpha\in I}$ of $M$ and a family $\{\phi_\alpha\}_{\alpha\in I}$ of diffeomorphisms
$\phi_\alpha\colon \pi^{-1}(U_\alpha)\rightarrow U_\alpha\times S$ with
\vspace{-3pt}
\begin{align}
\label{eq:bundlemaps}
\phi_\alpha(p\cdot s)=\big(\pi(p),[\pr_2\cp\phi_\alpha](p)\cdot s\big) \qquad\quad \forall\:p\in \pi^{-1}(U_\alpha),\forall\: s\in S,
\end{align}
\newline
\vspace{-28pt}
\newline
then \gls{PMS} is called \emph{principal fibre bundle} with total space $P$, projection map $\pi$, base manifold $M$ and structure group $S$. Here, $\pr_2$ denotes the projection onto the second factor $S$. It follows from \eqref{eq:bundlemaps} that $\pi$ is surjective, and that:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\itspacecc
$R_s(F_x)\subseteq F_x$ for all $x\in M$ and all $s\in S$.
\item
\itspacec
If $x\in M$ and $p,p'\in F_x$, then $p'=p\cdot s$ for a unique element $s\in S$.
\end{itemize}
\endgroup
\itspacec
\noindent
Then, for $p,p'\in F_x$ contained in the same fibre, we will denote by $\text{\gls{DIFF}}(p,p')\in S$ the unique element for which $p'=p\cdot s$ holds.
The subspace $Tv_pP:=\ker[d_p\pi]\subseteq T_pP$ is called \emph{vertical tangent space} at $p\in P$ and
\vspace{-3pt}
\begin{align*}
\wt{s}(p):=\dttB{t}{0}\: p\cdot \exp(t\vec{s})\in Tv_pP\qquad\quad \forall\: p\in P
\end{align*}
\newline
\vspace{-30pt}
\newline
denotes the fundamental vector field of $\vec{s}$ w.r.t.\ the right action of $S$ on $P$. Recall that the map $\mathfrak{s}\ni\vec{s}\rightarrow \wt{s}(p)\in Tv_pP$ is a vector space isomorphism for all $p\in P$.
Complementary to that, a (smooth) \emph{connection} $\w$ is an $\mathfrak{s}$-valued 1-form on $P$ with
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\itspacecc
$R_s^*\w= \Add{s^{-1}}\cp\: \w$ for all $s\in S$,
\item
\itspacec
$\w_p(\wt{s}(p))=\vec{s}$ for all $\vec{s}\in \mathfrak{s}$.
\end{itemize}
\endgroup
\itspacecc
\noindent
The subspace $Th_pP:=\ker[\w_p]\subseteq T_pP$ is called the \emph{horizontal tangent space} at $p$ (w.r.t.\ $\w$). We have $\dd R_s(Th_pP)=Th_{p\cdot s}P$ for all $s\in S$ and one can show that $T_pP= Tv_pP\oplus Th_pP$ holds for all $p\in P$. The set of smooth connections on $P$ is denoted by \gls{Con} in the following.
\subsubsection{Parallel Transports}
Let $\gamma\colon [a,b]\rightarrow M$ be a $\CC{1}$-curve in $M$ and $\w$ a connection on $P$. Then, for each $p\in F_{\gamma(a)}$ there is a unique $\CC{1}$-curve $\gamma_p^\w\colon [a,b]\rightarrow P$ with
$\pi\cp\gamma_p^\w=\gamma$, $\gamma_p^\w(a)=p$ as well as $\dot\gamma_p^\w(t)\in Th_{\gamma_p^\w(t)}P$, i.e.,
$\w_{\gamma_p^\w(t)}(\dot\gamma_p^\w(t))=0$
for all $t\in [a,b]$. \cite{KobNomiz}
This curve is called \emph{horizontal lift} of $\gamma$ w.r.t.\ $\omega$ in $p$ and
\begin{align*}
\text{\gls{PATRA}}\colon F_{\gamma(a)}&\rightarrow F_{\gamma(b)}\\
p&\mapsto \gamma_p^\w(b)
\end{align*}
is called \emph{parallel transport} along $\gamma$ w.r.t.\ $\omega$. The map $\parall{\gamma}{\omega}$ is a morphism, i.e., $\parall{\gamma}{\omega}(p\cdot s)=\parall{\gamma}{\omega}(p)\cdot s$ holds for all $p\in F_{\pi(p)}$ and all $s\in S$.
For a $\CC{1}$-path $\gamma$ one defines the parallel transport by $\parall{\gamma}{\omega}:= \parall{\gamma_0}{\omega}\cp\dots\cp\parall{\gamma_{k-1}}{\omega}$ where $\gamma_i$ are $\CC{1}$-curves with $\gamma_i=\gamma|_{[\tau_i,\tau_{i+1}]}$ for $0\leq i\leq k-1$ and $a=\tau_0<\ldots< \tau_k=b$. It is straightforward to see that this definition is independent of the explicit decomposition of $\gamma$.
\subsubsection{Automorphisms and Invariant Connections}
\label{subsub:invconn}
A diffeomorphism $\kappa\colon P\rightarrow P$ is said to be an \emph{automorphism} iff
$\kappa(p\cdot s)=\kappa(p)\cdot s$ holds for all $p\in P$ and all $s\in S$. It is straightforward to see that an $\mathfrak{s}$-valued 1-form $\w$ on $P$ is a connection iff this is true for the pullback $\kappa^*\w$. A \emph{Lie group \gls{GPHI} of automorphisms of $P$} is a Lie group $G$ together with a left action $\Phi$ of $G$ on $P$
such that the map $\Phi_g$ is an automorphism for each $g\in G$. This is equivalent to say that $\Phi(g,p\cdot s)=\Phi(g,p)\cdot s$ holds for all $p\in P$, $g\in G$ and all $s\in S$. In this situation, we will often write $gps$ instead of $(g\cdot p)\cdot s=g\cdot(p\cdot s)$. Each such left action $\Phi$ gives rise to three further important left actions:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\itspacecc
The action $\wm$ induced on the base manifold is defined by
\begin{align}
\label{eq:INDA}
\begin{split}
\text{\gls{WM}}\colon \quad G\times M &\rightarrow M\\
(g,m)&\mapsto (\pi\cp\Phi)(g, p_m)
\end{split}
\end{align}
where $p_m\in F_m$ is arbitrary. Then, $\wm$ is smooth because for $s_0\colon U_0\rightarrow P$ a smooth local section with $U_0\subseteq M$ open we have $\wm|_{G\times U_0}=\Phi(\cdot,s_0)$. Then, $\Phi$ is called \emph{fibre transitive} iff $\wm$ is transitive.
\item
\itspacec
We equip $\text{\gls{Q}}=G\times S$ with the canonical Lie group structure and define \cite{Wang}
\begin{align}
\label{eq:THETA}
\begin{split}
\text{\gls{THA}}\colon \qquad Q \times P&\rightarrow P\\
((g,s),p)&\mapsto \Phi\left(g, p\cdot s^{-1}\right).
\end{split}
\end{align}
\item
\itspacec
The action $\cw$ induced on the set $\Con$ of smooth connections is defined by
\begin{align}
\label{eq:connact}
\begin{split}
\text{\gls{CW}}\colon\quad G\times \Con&\rightarrow \Con\\
(g,\w)&\mapsto \Phi_{g^{-1}}^*\w.
\end{split}
\end{align}
\end{itemize}
\endgroup
\vspace{4pt}
\begin{definition}[Invariant Connection]
\label{def:Invconn}
A connection $\w$ is called $\Phi$-invariant iff $\Phi_g^*\w=\w$ holds for all $g\in G$.
\end{definition}
This definition is equivalent to require that for each $p\in P$ and $g\in G$ the differential $\dd_pL_g$ induces an isomorphism between the horizontal tangent spaces $Th_pP$ and $Th_{gp}P$. In literature sometimes this condition is used to define $\Phi$-invariance of connections.
We conclude this subsection with the following straightforward facts, see also \cite{Wang}:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\itspacec
Consider the representation $\text{\gls{QREP}}\colon Q \rightarrow \Aut(\mathfrak{s})$, $(g,s)\mapsto \Add{s}$. Then it is straightforward to see that each $\Phi$-invariant connection $\w$ is of type $\rho$, i.e., $\w$ is an $\mathfrak{s}$-valued 1-form on $P$ with $L_{q}^*\w=\rho(q)\cp \w$ for all $q\in Q$.
\item
\itspace
An $\mathfrak{s}$-valued 1-form $\w$ on $P$ with $\w(\wt{s}(p))=\vec{s}$ for all $\vec{s}\in \mathfrak{s}$ is a $\Phi$-invariant connection iff it is of type $\rho$.
\item
\itspace
Let \gls{QP} denote the stabilizer of $p\in P$ w.r.t.\ $\THA$ and $G_{\pi(p)}$ the stabilizer of $\pi(p)$ w.r.t.\ $\wm$. Then
$G_{\pi(p)}=\left\{h\in G\: | \:L_h\colon F_p\rightarrow F_p \right\}$ and we have the Lie group homomorphism
\begin{align}
\label{eq:phip}
\text{\gls{FIBAP}}\colon G_{\pi(p)}\rightarrow S\quad \text{by requiring that}\quad \Phi(h,p)=p\cdot\fiba_p(h)\quad \text{for all}\quad h\in G_{\pi(p)}.
\end{align}
If $\mathfrak{q}_p$ and $\mathfrak{g}_{\pi(p)}$ denote the Lie algebras of $Q_p$ and $G_{\pi(p)}$, respectively, then
\vspace{-6pt}
\begin{align}
\label{eq:staoQ}
Q_p=\{(h,\fiba_p(h))\:|\:h\in G_{\pi(p)}\}\qquad\text{and}\qquad \mathfrak{q}_p=\big\{\big(\hspace{1pt}\raisebox{-1pt}{$\vec{h}$},\dd_e\fiba_p\big(\hspace{1pt}\raisebox{-1pt}{$\vec{h}$}\hspace{1.5pt}\big)\big)\:\big|\:\raisebox{-1pt}{$\vec{h}$}\in \mathfrak{g}_{\pi(p)}\big\}.
\end{align}
\end{itemize}
\endgroup
\subsection{Projective Structures and Radon Measures}
\label{subsec:ProjStruc}
In this subsection, we will collect the necessary facts on projective structures and Radon measures. Here, our conventions concerning Radon measures are the same as in \cite{Elstrodt}, see Definition \ref{def:Mass}.
We start with the following non-standard definition of a projective limit:
\begin{definition}[Projective Limit]
\label{def:ProjLim}
Let $\{X_\alpha\}_{\alpha\in I}$ be a family of compact Hausdorff spaces where $(I,\leq)$ is a directed set. Recall that this means that $\leq$ is a reflexive and transitive relation on $I$, and that for each two $\alpha,\alpha'\in I$ we find some $\alpha''\in I$ with $\alpha,\alpha'\leq \alpha''$. A compact Hausdorff space $X$ is called projective limit of the family $\{X_\alpha\}_{\alpha\in I}$ iff
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item
\label{def:ProjLim1}
\itspacec
For each $\alpha\in I$ there is a continuous, surjective map $\pi_\alpha\colon X\rightarrow X_\alpha$.
\item
\label{def:ProjLim2}
\itspacec
For $\alpha_1,\alpha_2 \in I$ with $\alpha_1\leq \alpha_2$ there is a continuous map $\pi^{\alpha_2}_{\alpha_1}\colon X_{\alpha_2}\rightarrow X_{\alpha_1}$ for which $\pi^{\alpha_2}_{\alpha_1}\cp \pi_{\alpha_2}=\pi_{\alpha_1}$ holds.
It follows that each of these maps is surjective and that $\pi^{\alpha_2}_{\alpha_1}\cp\pi^{\alpha_3}_{\alpha_2}=\pi^{\alpha_3}_{\alpha_1}$ holds if $\alpha_1\leq \alpha_2\leq\alpha_3$ for $\alpha_1,\alpha_2,\alpha_3\in I$.
\item
\label{def:ProjLim3}
\itspacec
If $x,y\in X$ with $x\neq y$, then there is some $\alpha\in I$ with $\pi_\alpha(x)\neq \pi_\alpha(y)$.
\end{enumerate}
\endgroup
\end{definition}
It is proven in Lemma \ref{lemma:equivalence} that the above definition of a projective limit is equivalent to the usual definition \cite{ProjTechAL} as a subset
\begin{align*}
\widehat{X}=\left\{\hx \in \textstyle\prod_{\alpha\in I}X_\alpha\: \:\big|\:\: \pi_{\alpha_1}^{\alpha_2}(x_{\alpha_2})=x_{\alpha_1}\:\: \forall\:\alpha_2\geq \alpha_1\right\}
\end{align*}
of the Tychonoff product $\prod_{\alpha \in I}X_\alpha$. In particular, each two projective limits of a fixed family of compact Hausdorff spaces are homeomorphic if the same transition maps are used.
Anyhow, in this thesis the main reason for writing a compact Hausdorff $X$ as a projective limit is due to Lemma \ref{lemma:normRM}. This states that we obtain a normalized Radon measures on $X$ if we define a consistent family (see next definition) of normalized Radon measures on the spaces $X_\alpha$. Using the standard Tychonoff product approach, here we always had to take care of the identification of $X$ with the space $\widehat{X}$.
For this reason, Definition \ref{def:ProjLim} is much more convenient for the purpose to construct normalized Radon measures on $X$.
\begin{Definition}[Borel, Radon Measures]
\label{def:Mass}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item
\label{def:Mass1}
A Borel measure $\mu$ on a Hausdorff space $Y$ is a locally finite\footnote{This means that for each $y\in Y$ we find $U\subseteq Y$ open with $y\in U$ and $\mu(U)<\infty$.} measure $\mu\colon \mathfrak{B}(Y)\rightarrow [0,\infty]$, where $\mathfrak{B}(Y)$ denotes the Borel $\sigma$-algebra of $Y$. It is said to be normalized if $\|\mu\|:=\mu(Y)=1$.
\item
\label{def:Mass2}
\itspace
A Borel measure $\mu$ is called inner regular iff for each $A\in \mathfrak{B}(Y)$ we have
\begin{align*}
\mu(A)=\sup\{\mu(K):K \text{ is compact and }K\subseteq A\}.
\end{align*}
\item
\label{def:Mass3}
\itspace
A Radon measure $\mu$ is an inner regular Borel measure. It is called finite if
$\mu(Y)<\infty$ holds. Recall that each finite Radon measure is outer regular, i.e., for each $A\in \mathfrak{B}(Y)$ we have
\begin{align*}
\mu(A)=\inf\{\mu(U):U \text{ is open and }A\subseteq U\}.
\end{align*}
\item
\itspace
Assume that we are in the situation of Definition \ref{def:ProjLim} and $\{\mu_\alpha\}_{\alpha\in I}$ is a family of Radon measures $\mu_\alpha\colon \mathfrak{B}(X_\alpha)\rightarrow [0,\infty]$. Then, $\{\mu_\alpha\}_{\alpha\in I}$ is called consistent iff $\mu_{\alpha_1}$ equals the push forward measure $\pi^{\alpha_2}_{\alpha_1}(\mu_2)$ whenever $\alpha_1\leq \alpha_2$ for $\alpha_1,\alpha_2\in I$.
\end{enumerate}
\endgroup
\end{Definition}
\begin{lemma}
\label{lemma:normRM}
\begin{enumerate}
\item
\label{lemma:normRM1}
Let $\mu$ be a finite Radon measure on $X$ and $f\colon X\rightarrow Y$ a continuous map. Then, the push forward measure $f(\mu)$ is a finite Radon measure on $Y$.
\item
\label{lemma:normRM2}
Let $X$ and $\{X_\alpha\}_{\alpha\in I}$ be as in Definition \ref{def:ProjLim}. Then, the normalized Radon measures on $X$ are in bijection with the consistent families of normalized Radon measures on $\{X_\alpha\}_{\alpha\in I}$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Since $\mu$ is finite, $f(\mu)$ is a finite Borel measure. Then, inner regularity of $f(\mu)$ is straightforward from inner regularity of $\mu$.
\item
See Lemma \ref{lemma:ConstMeas}.
\end{enumerate}
\end{proof}
\end{lemma}
In Subsection \ref{sec:ConSp}, we will construct measures on Tychonoff products $X=X_1\times \dots\times X_k$ from measures $\mu_1,\dots,\mu_k$ on topological spaces $X_1,\dots,X_k$ that are not second countable. In this case, the Borel $\sigma$-algebra $\mathfrak{B}(X)$ is usually larger than the product $\sigma$-algebra $\mathfrak{B}(X_1)\otimes \dots\otimes \mathfrak{B}(X_k)$. For this reason, we cannot use the standard product measure approach for $ \sigma$-finite measures (see, e.g., \cite[Chap.V, \S 1]{Elstrodt}). Now, if the spaces $X_i$ are locally compact Hausdorff and $\mu_i$ are finite Radon measures, then we have the notion of the Radon product measure, being defined on $\mathfrak{B}(X)$. Since such Radon product measures seem to occur in standard literature usually only by means of existence statements without a concrete definition, we decided to provide such a definition at this point. This is just to clarify what exactly we mean by Radon product measures in the following.
\begin{lemdef}[Radon Product Measure]
\label{def:ProductMa}
For a topological space $Z$ we denote by $C_c(Z)$ the set of continuous functions having compact support in $Z$.
\begin{enumerate}
\item
\label{def:ProductMa1}
Let $\mu$ and $\nu$ be normalized Radon measures on the locally compact Hausdorff spaces $X$ and $Y$, and denote by $\III$ and $\JJJ$ the corresponding normalized\footnote{We have $\|\III\|=\mu(X)=1$ as well as $\|\JJJ\|=\mu(Y)=1$ by 2.8 Satz in \cite[Chap.VIII, \S 2]{Elstrodt}.} positive linear functionals on $C_c(X)$ and $C_c(Y)$, respectively.
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
By Theorem (13.2) in \cite{Hewitt} we obtain a well-defined positive linear functional
\begin{align*}
\III\times \JJJ\colon C_c(X\times Y)\rightarrow \mathbb{C}
\end{align*}
just by
\begin{align}
\label{eq:wellfunc}
(\III\times \JJJ) (f):=\III\left[x\mapsto \JJJ(f(x,\cdot))\right]=\JJJ\left[y\mapsto \III(f(\cdot,y))\right]\qquad \forall\: f\in C_c(X\times Y).
\end{align}
In particular, this means that $f_y\colon x\mapsto \III(f(\cdot,y))\in C_c(X)$ and $f_x\colon y\mapsto \JJJ(f(x,\cdot))\in C_c(Y)$. Observe that $\III\times \JJJ$ is normalized because:
\begingroup
\setlength{\leftmarginiii}{13pt}
\begin{itemize}
\item[$\triangleright$]
\vspace{5pt}
$\|\III\times \JJJ\|\leq 1$ since for each $f\in C_c(X\times Y)$ we have
\begin{align*}
|(\III\times \JJJ)(f)|\stackrel{\eqref{eq:wellfunc}}{\leq}\|\III\|\|x\mapsto \JJJ(f(x,\cdot))\|_\infty\leq \|\III\|\|\JJJ\|\sup_{x\in X}\|f(x,\cdot)\|_\infty \leq \|f\|_\infty
\end{align*}
\item[$\triangleright$]
$\|\III\times \JJJ\|\geq 1-\epsilon$ for each $\epsilon\in (0,1)$ since
we find $f_1\in C_c(X)$ and $f_2\in C_c(Y)$ with
\begin{align*}
|\III(f_1)|\geq\sqrt{1-\epsilon}\cdot \|f_1\|_\infty\qquad \text{ as well as}\qquad |\III(f_2)|\geq \sqrt{1-\epsilon}\cdot \|f_2\|_\infty,
\end{align*}
so that for $f_1\otimes f_2 \colon (x,y)\mapsto f_1(x)\cdot f_2(y)$ we have
\begin{align*}
|(\III\times \JJJ)(f_1\otimes f_2)|\stackrel{\eqref{eq:wellfunc}}{=}|\III(f_1)\cdot \JJJ(f_2)|\geq |1-\epsilon|\cdot \|f_1\|_\infty\cdot \|f_2\|_\infty=|1-\epsilon|\cdot \|f_1\otimes f_2\|_{\infty}.
\end{align*}
\end{itemize}
\endgroup
\item
Let $\mu\times \nu$ denote the respective normalized Radon measure on $\mathfrak{B}(X\times Y)$ defined by the Riesz-Markov theorem in the form 2.5 Satz in \cite[Chap.VIII, \S 2]{Elstrodt}.
Then, by \eqref{eq:wellfunc} for all $f\in C_c(X\times Y)$ Fubini's formula holds
\begin{align*}
\int_{X\times Y}f\: \dd (\mu\times \nu) = \int_X\left(\int_Yf(x,y)\:\dd\nu(y)\right)\dd\mu(x)=\int_Y\left(\int_Xf(x,y)\:\dd\mu(x)\right)\dd\nu(y).
\end{align*}
\end{itemize}
\endgroup
\item
\label{def:ProductMa2}
For $i=1,\dots,n$ let $\mu_i$ be a normalized Radon measure on the locally compact Hausdorff space $X_i$, and let $\III_i$ denote the corresponding normalized positive linear functional on $C_c(X_i)$. Then, it is straightforward from Part \ref{def:ProductMa1}) that the linear functional
\begin{align}
\label{eq:wellfuncs}
\III_n(f):=\III_{\sigma(1)}\left[x_{\sigma(1)}\mapsto \III_{\sigma(2)}\left[\cdots x_{\sigma(n-1)}\mapsto \III_{\sigma(n)}(x_{\sigma(n-1)}\mapsto f(x_1\dots,x_n) )\cdots \right]\right]
\end{align}
for $f\in C_c(X_1\times{\dots} \times X_n)$ is well defined, normalized, positive and independent of $\sigma \in S_n$. In particular, the corresponding Radon measure $\mu_n$ on $\mathfrak{B}(X_1\times {\dots} \times X_n)$ is normalized, and
due to \eqref{eq:wellfuncs} a respective Fubini's formula holds.
\item
\label{def:ProductMa3}
Let $\{X_\iota\}_{\iota\in I}$ be a family of compact Hausdorff spaces and $\{\mu_\iota\}_{\iota \in I}$ a family of respective normalized Radon measures. We consider the compact Hausdorff space $X:=\prod_{\iota\in I}X_\iota$ and define a normalized Radon measure $\mu_I$ on $X$ as follows.
Let $\J$ denote the set of all finite tuples $J=(\iota_1,\dots,\iota_k)$ of mutually different elements of $I$. Define $X_J:=X_{\iota_1}\times {\dots} \times X_{\iota_k}$, $\mu_J:=\mu_{\iota_1}\times {\dots} \times \mu_{\iota_k}$ and the respective projection map by
\begin{align*}
\pi_J\colon X\rightarrow X_J,\:\:
\prod_{\iota\in I}x_\iota \mapsto (x_ {\iota_1},\dots,x_ {\iota_k}).
\end{align*}
We write $J\leq J'$ for $J,J'\in \JJ$ with $J=(\iota_1,\dots,\iota_k)$ and $J'=(\iota'_1,\dots,\iota'_{k'})$ iff there exists an injection $\sigma\colon \{\iota_1,\dots,\iota_k\}\rightarrow \{\iota'_1,\dots,\iota'_{k'}\}$. Finally, we consider the transition maps
\begin{align*}
\pi^{J'}_J\colon X_{J'}\rightarrow X_J,\:\:\big(x_{\iota'_1},\dots,x_{\iota'_{k'}}\big)\mapsto \big(x_{\sigma(\iota_1)},\dots,x_{\sigma(\iota_{k})}\big).
\end{align*}
Obviously, $(\J,\leq)$ is a directed set, and $X$ a projective limit of $\{X_J\}_{J\in \JJ}$. Moreover, it is straightforward from \eqref{eq:wellfuncs} that $\{\mu_J\}_{J\in \JJ}$ is a respective consistent family of normalized Radon measures. We define $\mu_I$ to be the corresponding normalized Radon measure on $X$ provided by Lemma \ref{lemma:normRM}.
\hspace*{\fill} $\lozenge$
\end{enumerate}
\end{lemdef}
\begin{remark}[Fubini]
\label{rem:unednlichfubini}
In the situation of Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa3} assume that $I=I_1\sqcup I_2$ for $I_1,I_2\neq \emptyset$. Moreover, let $\mu_1$ and $\mu_2$ denote the corresponding Radon product measures on $X_1=\prod_{\iota\in I_1} X_\iota$ and $X_2=\prod_{\iota\in I_2}X_\iota$, respectively. Then, it is easy to see that $\mu=\mu_1\times \mu_2$ holds.
In fact, by continuity of the corresponding linear functionals $\III,\III_1,\III_2$ one only has to check that $\III=\III_1\times \III_2$ holds on the dense $^*$-subalgebra $\mathrm{Cyl}(X)\subseteq C(X)$ consisting of such continuous function which can be written in the form $f\cp \pi_J$ with $f\in C(X_J)$ for some $J\in \J$, see also proof of Lemma \ref{lemma:normRM}.\ref{lemma:normRM2}. This, however, is straightforward from the definitions.
\end{remark}
\section{Special Mathematical Background}
\label{sec:specmathback}
This section essentially collects some background material we could not find in the standard literature in this form.
In the first part, we will fix some conventions concerning certain symmetric situations that appear in loop quantum gravity (\gls{LQG}) and will serve as prime examples during this work. In particular, we will specify sets of invariant connections that belong to these situations. Since the corresponding calculations are rather technical than illustrative, we have decided to shift them to Section \ref{CHarinvconn} and the Appendix, and
only to provide the relevant information at this point.
The necessary techniques
will be developed in Section \ref{CHarinvconn}, where we attack the characterization problem of invariant connections in full generality.
In the second part, we will prove some straightforward facts on maps on spectra and then apply them to the case where the $\Cstar$-algebra consists of certain bounded functions on a set. This is important for the investigations in Section \ref{sec:SPecExtGr} where we consider $\Cstar$-subalgebras of bounded functions on sets of smooth connections.
In the last part, we will give a brief introduction into the Bohr compactification of a locally compact abelian group. This will be used
to characterize the abelian continuous group structures on spectra of $\Cstar$-subalgebras of bounded functions by families of quasi-characters, see Definition \ref{def:quasichar}.
\subsection{Loop Quantum Gravity Case}
Usually, in loop quantum gravity (LQG) the structure group is $\SU$, and the principal fibre bundle is of the form $P=\Sigma \times \SU$ for a 3-dimensional manifold (Cauchy surface) $\Sigma$. Although the results of this dissertation apply to a much larger class of principal bundles and structure groups, in the LQG relevant examples of this work the bundle will be just of the form $\RR^3\times \SU$. We now collect some facts, notations and conventions concerning the Lie group $\SU$, the elements of $\RR^n$ and sets of curves in $\RR^3$.
\begin{convention}
\label{conv:sutwo1}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item
\label{conv:sutwo111}
By $\text{\gls{VR}}\colon \SU \rightarrow \SOD$ we denote the universal covering map.
Then, for $\ssigma\in \SU$ we have
\begin{align}
\label{eq:covm}
\ssigma(x):=\varrho(\ssigma)(x)=\murs^{-1}(\Add{\ssigma}(\murs(x)))
\end{align}
for the linear isomorphism $\text{\gls{MURS}}\colon \RR^3 \rightarrow \mathfrak{su}(2)$, $\sum_{i=1}^3 x^i \vec{e}_i\mapsto \sum_{i=1}^3 x^i \tau_i$ with
\begin{align*}
\tau_1:=\begin{pmatrix} 0 & -\I \\ -\I & 0 \end{pmatrix}\qquad\qquad \tau_2:=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\qquad\qquad \tau_3:=\begin{pmatrix} -\I & 0 \\ 0 & \I \end{pmatrix}.
\end{align*}
Although this identification of $\RR^3$ with $\su$ is standard, we decided to introduce the map $\murs$ at this point. The main reason is that in our applications we will need this identification permanently, as it simplifies the formulas and calculations drastically, see, e.g., \eqref{eq:rotinvconn}, \eqref{eq:homis} and the Appendices \ref{subsec:InvIsoHomWang} and \ref{subsec:IsotrConn}.
The alternative would be to calculate in coordinates which we want to avoid whenever it is possible.
Recall that\footnote{Observe that $\exp$ is surjective because $\SU$ is compact and connected.} each $\ssigma\in \SU$ can be written in the form
\begin{align}
\label{eq:expSU2}
\ssigma= \cos(\alpha/2)\cdot \me + \sin(\alpha/2)\cdot \murs(\vec{n})=\exp\big(\alpha/2\cdot\murs(\vec{n})\big)
\end{align}
for some $\vec{n}\in \RR^3$ with $\|\vec{n}\|=1$ and $\alpha\in [0,2\pi)$. In this case $\uberl(\ssigma)$ rotates a point $x$ by the angle $\alpha$ w.r.t.\ the axis $\vec{n}$.
\item
In the sequel, $\RR^n$ will occur as vector space (tangent space), as base manifold of a principal fibre bundle, and as a symmetry group. In the first case we write its elements with arrows such as $\vec{v}$, and the same will be done if an element of $\RR^n$ occurs as a normal vector, rotation axis or as the traversing direction of a linear curve, see next point. In the second case, i.e., if $\RR^n$ is considered as a base manifold, we usually write $x$, and in the last case we will write $v$. Nevertheless, in all three situations we will make free use of the vector space operations in $\RR^n$. For instance, if $G=\RR^n$ acts via addition on $M=\RR^n$, we will write $v+x$. If $G=\RR_{\neq 0}$ acts via multiplication on $M=\RR^n$, we will write $\lambda \cdot x$. In the same way, if $n=3$, we also apply the above map $\murs$ to all these elements, i.e., $\murs(\vec{v})$, $\murs(x)$ and $\murs(v)$.
\item
\label{conv:sutwo2}
Whenever $M=\Sigma=\RR^3$, the sets $\Pall,\Pal,\Paln,\Pacirc$ are defined as follows:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item
\label{conv:lincirccurves1}
According to the notations introduced in the previous point,
by $\Pall$ we will denote the set of curves of the form
$x+\gamma_{\vec{v},l}$ for $x, \vec{v}\in \mathbb{R}^3$ with $\|\vec{v}\|=1$ and
$\gamma_{\vec{v},l}\colon [0,l]\rightarrow \mathbb{R}$, $t\mapsto t \vec{v}$.
By $\Pal$ we will denote the set of all linear curves, i.e., the set of all embedded analytic curves with $\mathrm{im}[\gamma]$ contained in a line through some fixed point $x\in \RR^3$.
Then, $\Pal$ consists of all embedded analytic curves $\gamma$ equivalent to an element of $\Pall$, i.e.,
\begin{align*}
\gamma = \delta\cp \adif\quad\text{for some}\quad \delta\in \Pall\quad\text{and}\quad \adif \colon I\rightarrow \RR\quad\text{an analytic diffeomorphism}.
\end{align*}
Here, $I$ denotes some open interval which contains $\operatorname{\mathrm{dom}}[\gamma]$.
Finally, by $\Paln\subseteq \Pal$ we will denote the subset of all linear curves traversing through the origin.
\item
\label{conv:lincirccurves2}
Let $\Pacirc$ consist of all circular curves, i.e., all curves of the form
\begin{align*}
\gc{n}{r}{x}{\tau} \colon [0,\tau]&\rightarrow \RR^3\\
t&\mapsto x + \cos(t)\:\vec{r} + \sin(t) \:\vec{n}\times \vec{r}
\end{align*}
for $\vec{n},\vec{r},x\in \RR^3$ with $\|\vec{n}\|=1$ as well as $0< \tau< 2\pi$.
\end{enumerate}
\endgroup
\item
For $\s\in \su$ and $\vec{v},x,v \in \RR^3$ we define the corresponding maximal tori in $\SU$ by
\begin{align}
\label{eq:tori}
\text{\gls{Tor1}}:=\{\exp(t\s)\:|\: t\in \RR\}\qquad \text{\gls{Tor2}}:=H_{\murs(\vec{v})}\qquad \text{\gls{Tor3}}:=H_{\murs(x)}\qquad \text{\gls{Tor4}}:=H_{\murs(v)}.
\end{align}
Hence, we suppress the map $\murs$ in the last three cases.
\hspace*{\fill}$\lozenge$
\end{enumerate}
\endgroup
\end{convention}
In Section \ref{sec:MOQRCS} we will be concerned with certain equivariant maps to $\SU$.
Inter alia, the next lemma then will be essential for calculating spaces of such maps.
\begin{lemma}
\label{lemma:torus}
\begin{enumerate}
\item
\label{lemma:torus1}
If $s s'=s's$ for $s,s'\in \SU$, then $s,s'\in H_{\vec{n}}$ for some $\vec{n}\in\RR^3\backslash \{0\}$. This torus is uniquely determined if $s\neq \pm \me$ or $s'\neq \pm \me$.
\item
\label{lemma:torus2}
Let $s,s'\in H_{\vec{n}}$ be different from $\pm\me$. If $s'=\alpha_h(s)$ holds for some $h\in \SU$, then we have the following two possibilities:
\begingroup
\setlength{\leftmarginii}{17pt}
\begin{enumerate}
\item[a)]
\vspace{-7pt}
$s=s'$ and $h\in H_{\vec{n}}$
\item[b)]
$s'=s^{-1}$ and $h=\exp\left(\textstyle\frac{\pi}{2}\cdot\murs(\vec{m})\right)$ for some $\mm\in \RR^3$ orthogonal to $\vec{n}$ with $\|\mm\|=1$ and uniquely determined up to a sign.
In particular, $\alpha_h(s_0)=s_0^{-1}$ holds for $\pm \me\neq s_0\in H_{\vec{n}'}$ iff $\langle\vec{m},\vec{n}'\rangle=0$.
\end{enumerate}
\endgroup
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
This follows by straightforward calculation involving \eqref{eq:expSU2}, or from the general theory of compact and connected Lie groups as $\{-\me,\me\}$ are the irregular elements of $\SU$.
\item
We write $s=\exp(t\cdot\murs(\vec{n}))$ and $s'=\exp(t'\murs(\vec{n}))$ for unique reals $t,t'\in [0,2\pi)$, cf.\ \eqref{eq:expSU2}.
Then
\begin{align}
\label{eq:bla}
\exp\big(t\cdot \murs(\varrho(h)(\vec{n}))\big)=\alpha_h(s)=s'=\exp(t'\murs(\vec{n})).
\end{align}
If $\varrho(h)(\vec{n})=\vec{n}$, then $t=t'$ and $s=s'$, hence, $h\in H_{\vec{n}}$ by Part \ref{lemma:torus1}), so that case \textit{a)} holds.
If $\varrho(h)(\vec{n})\neq\vec{n}$ for $\vec{v}:=\varrho(h)(\vec{n})$, then from \eqref{eq:bla}, \eqref{eq:expSU2} and $\|\vec{v}\|=1$ we obtain
\begin{align*}
\cos(t)\cdot \me + \sin(t)\cdot \murs(\vec{v})=\cos(t')\cdot \me + \sin(t')\cdot \murs(\vec{n}).
\end{align*}
Then $\cos(t)=\cos(t')$, hence $\sin(t)=\pm \sin(t')$, i.e., $\vec{n}=-\vec{v}$ because $\vec{v}\neq \vec{n}$ and since $\sin(t),\sin(t')\neq 0$ as $s,s'\neq \pm \me$.
Consequently, $\varrho(h)$ is a rotation by $\pi$ around an axis $\vec{m}$ orthogonal to $\vec{n}$. So, if $s_0\in H_{\vec{n}'}$ with $\langle\vec{m},\vec{n}'\rangle=0$, then
\begin{align*}
\alpha_h(s_0)=\exp\!\big(t_0\cdot \murs(\varrho(h)(\vec{n}' ))\big)=\exp(-t_0\cdot \murs(\vec{n}'))=s_0^{-1}.
\end{align*}
Finally, that this equality only holds for such $s_0\neq \pm \me$ with $s_0\in H_{\vec{n}'}$ for $\langle\vec{m},\vec{n}'\rangle=0$, is clear from the fact that $\mm$ is unique up to a sign.
\end{enumerate}
\end{proof}
\end{lemma}
The next example collects the most relevant loop quantum cosmology (\gls{LQC}) cases to be discussed in this work. The proofs concerning invariant connections can be found in the Appendix \ref{subsec:InvIsoHomWang} and Section \ref{CHarinvconn}. The reason for providing these results already at this point is that we will need them during the following sections.
\begin{example}[Loop Quantum Cosmology]
\label{ex:LQC}
Let $P=\RR^3\times \SU$. We consider the following symmetry groups and actions:\footnote{Observe that $\Pe(g,p)=g\cdot_\varrho p$ for $\cdot_\varrho$ the group multiplication in $\Gee$.}
$$
\begin{tabular}{rlrll}
\gls{GE}\!\!\!\!\!&$:=\Gee$ \qquad\qquad & \quad\qquad $\Pe((v,\sigma),(x,s))$\!\!\!\!\!&$:=(v+\text{\gls{VR}}(\sigma)(x),\sigma s)$\\
\gls{GI}\!\!\!\!\!&$:=\SU$ \qquad\qquad & \quad\qquad $\Pii(\sigma,(x,s))$\!\!\!\!\!&$:=(\varrho(\sigma)(x),\sigma s)$\\
\gls{GH}\!\!\!\!\!&$:=\RR^3$ \qquad\qquad & \quad\qquad $\Ph(v,(x,s))$\!\!\!\!\!&$:=(v+x,s)$
\end{tabular}$$
For each of these actions the corresponding induced action is pointwise proper. In the second case, this is clear from compactness of $\SU$, and in the last case this holds because $\wm$ is just the addition in $\RR^3$. In the first case ,we have $\Ge=P$ so that even $\Pe$ is (pointwise) proper as topological groups always act properly on themselves.
\par
\begingroup
\leftskip=8pt
\vspace{8pt}
\noindent
{\bf\textit{Homogeneous Isotropic LQC:}}
\vspace{2pt}
\noindent
The Lie group $\Ge$ resembles the Euclidean group $\RR^3 \rtimes \SOD$ in the sense that
the action $\wm$ induced by $\Pe$ on $\RR^3$ gives rise to the same orbits as the canonical action of the euclidean group does.
The $\Pe$-invariant connections are of the form
\vspace{-3pt}
\begin{align}
\label{eq:homis}
\w^c(\vec{v}_x,\vec{\sigma}_s)= c \Add{s^{-1}}[\murs(\vec{v}_x)]+s^{-1}\vec{\sigma}_s \qquad \forall\:(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P,
\end{align}
where $c$ runs over $\mathbb{R}$. This follows from Wang's original theorem \cite{Wang} and the explicit calculations can be found in Appendix \ref{subsec:InvIsoHomWang}.
\vspace{8pt}
\noindent
{\bf\textit{Spherically Symmetric LQC:}}
\vspace{2pt}
\noindent
It is proven in Example \ref{bsp:Rotats} that the spherically symmetric connections, i.e., the $\Pii$-invariant ones are of the form
\begin{align}
\label{eq:rotinvconnn}
\begin{split}
\w^{abc}(\vec{v}_x,\vec{\sigma}_s):= \Add{s^{-1}}\!\big[&a(x)\hspace{1pt}\murs(\vec{v}_x)+ b(x)[\hspace{1pt}\murs(x),\murs(\vec{v}_x)]
+c(x)[\:\murs(x),[\hspace{1pt}\murs(x),\murs(\vec{v}_x)]]\big]+ s^{-1}\vec{\sigma}_s
\end{split}
\end{align}
for $(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P$. Here, $a,b,c\colon \mathbb{R}^3\rightarrow \mathbb{R}$ are rotation invariant maps that can be written in the form
$a(x)=f\big(\|x\|^2\big)$, $b(x)=g\big(\|x\|^2\big)$, $c(x)=h\big(\|x\|^2\big)$
for smooth functions $f,g,h\colon (-\epsilon,\infty)\rightarrow \RR$ and some $\epsilon>0$.
\vspace{8pt}
\noindent
{\bf\textit{(Semi-)Homogeneous LQC:}}
\vspace{2pt}
\noindent
This is a special case of Example \ref{ex:transinv}, where by semi-homogeneity we mean that $G=G_{SH}$ is a two-dimensional linear subspace of $\RR^3$ and the corresponding action is $\Phi_{SH}(v,(x,s)):=(v+x,s)$.
Here, for $W$ an algebraic complement of $G$ in $\RR^3$, the $\Phi_{SH}$-invariant connections are parametrized by the smooth maps $\psi \colon \mg\times TW\rightarrow \su$ whose restrictions $\psi|_{\mg\times T_xW}$ are linear for all $x\in W$, hence by the smooth maps $\psi \colon \RR^2\times T\RR\rightarrow \su$ whose restrictions $\psi|_{\RR^2\times T_t\RR}$ are linear for all $t\in \RR$.
In the homogeneous case, i.e., if $\Gh=\RR^3$, the $\Ph$-invariant connections are in bijection with the linear maps $L\colon \RR^3\rightarrow \su$, see also Example \ref{ex:eukl}. \hspace*{\fill}$\lozenge$
\endgroup
\par
\end{example}
\subsection{Bounded Functions and Maps on Spectra}
\label{subsec:GrAcOnSp}
In the first part, we relate automorphisms of abelian $\Cstar$-algebras with homeomorphisms of their spectra. In the second part, we will apply this to $\Cstar$-algebras of bounded functions, being the relevant quantities in the framework of loop quantum gravity. We collect some basic facts on denseness \cite{Rendall,ChrisSymmLQG} of a set $X$ in the spectrum of a $\Cstar$-subalgebra of bounded functions thereon as well as on embeddability of the spectrum of a restriction $\Cstar$-algebra into the spectrum of the full one. In particular, this will provide us with the mathematical backbone of traditional reduction approach in the framework of Loop Quantum Gravity. \cite{ChrisSymmLQG}
\subsubsection{Maps on Spectra}
\begin{definition}
For a Banach algebra $\aA$, let $\|\cdot\|_\aA$ denote the corresponding norm. For $a\in \aA$ denote by $\|\cdot\|_a\colon \aA'\rightarrow \RR_{\geq 0}$ the seminorm $\|\chi\|_a:=|\chi(a)|$ for $\aA'$ the topological dual of $\aA$.
\end{definition}
\begin{Lemma}
\label{lemma:homzuspec}
\begin{enumerate}
\item
\label{lemma:homzuspec1}
If $\lambda\colon \aA\rightarrow \bB$ is a homomorphism of abelian Banach algebras $\aA$ and $\mathfrak{B}$, then the map
\begin{align*}
\ovl{\lambda}\colon \mathrm{Spec}(\bB)\rightarrow \mathrm{Spec}(\aA),\quad
\chi \mapsto \chi \cp \lambda
\end{align*}
is continuous if it is well defined. In particular, this is the case if $\lambda$ is surjective or unital.\footnote{More precisely, if $\aA$ and $\bB$ are unital and $\lambda(1_\aA)=1_\bB$.}
\item
\label{lemma:homzuspec2}
If $\aA$ is an abelian $\Cstar$-algebra, then $\eta\colon \Aut(\aA)\rightarrow \Homeo(\mathrm{Spec}(\aA))$, $\lambda\mapsto\ovl{\lambda}$
is a group antiisomorphism.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
The image of $\ovl{\lambda}$ consists of homomorphisms. So, the only case in which
well-definedness fails is when $\ovl{\lambda}(\chi)=0$ for some $\chi\in \mathrm{Spec}(\bB)$.
If $\lambda$ is unital, then $\ovl{\lambda}(\chi)(1_{\aA})=(\chi\cp\lambda)(1_{\aA})=\chi(1_\bB)=1\neq 0$ for all $\chi\in \mathrm{Spec}(\bB)$, so that $\ovl{\lambda}$ is well defined in this case. If $\lambda$ is surjective and $\chi\in \mathrm{Spec}(\bB)$, then $\chi(b)\neq 0$ for some $b\in \bB$ and we find $a\in\aA$ with $\lambda(a)=b$. Hence, $\ovl{\lambda}(\chi)(a)=\chi(b)\neq 0$ for all $\chi \in \mathrm{Spec}(\bB)$ also in this case. For continuity let $\mathrm{Spec}(\bB)\supseteq\{\chi_\alpha\}_{\alpha\in I}\rightarrow \chi$ be a converging net. Then for $a\in \aA$ and $\epsilon> 0$ we find $\alpha_\epsilon\in I$ such that $\|\chi_\alpha-\chi\|_{\lambda(a)}\leq \epsilon$ for all $\alpha\geq \alpha_\epsilon$. But for $\alpha\geq \alpha_\epsilon$ we have
\begin{align*}
\big\|\ovl{\lambda}(\chi_\alpha)-\ovl{\lambda}(\chi)\big\|_a=|\chi_\alpha(\lambda(a))-\chi(\lambda(a))|=\|\chi_\alpha-\chi\|_{\lambda(a)}\leq \epsilon
\end{align*}
so that
$\mathrm{Spec}(\aA)\supseteq\{\ovl{\lambda}(\chi_\alpha)\}_{\alpha\in I}\rightarrow \ovl{\lambda}(\chi)$ shows continuity of $\ovl{\lambda}$.
\item
Each $\lambda\in \Aut(\aA)$ is a surjective homomorphism so that the image of $\eta$ consists of well defined and continuous maps. Then $\eta$ is an antihomomorphism as
\begin{align*}
\eta(\lambda\cp \lambda')(\chi)=(\chi \cp \lambda)\cp \lambda'=\eta(\lambda')(\chi\cp \lambda)=\eta(\lambda')(\eta(\lambda)(\chi)).
\end{align*}
Now, $\lambda^{-1}\in \Aut(\aA)$ exists so that $\eta(\lambda)^{-1}=\eta(\lambda^{-1})$ is continuous as well. This means that the image of $\eta$ consists of homeomorphisms and shows well-definedness of this map. To verify injectivity assume that $\eta(\lambda)=\eta(\lambda')$ for $\lambda,\lambda'\in \Aut(\aA)$. Then for
$a\in\aA$ and $\chi\in \mathrm{Spec}(\aA)$ we have
\begin{align*}
\mathcal{G}(\lambda(a))(\chi)&=(\chi\cp\lambda)(a)=\eta(\lambda)(\chi)(a)\\
&=\eta(\lambda')(\chi)(a)=(\chi\cp\lambda')(a)
=\mathcal{G}(\lambda'(a))(\chi)
\end{align*}
so that $\mathcal{G}(\lambda(a))=\mathcal{G}(\lambda'(a))$. Then injectivity of $\mathcal{G}$ implies $\lambda(a)=\lambda'(a)$ for all $a\in \aA$, hence $\lambda=\lambda'$. For surjectivity of $\eta$ define
\begin{align}
\label{eq:tautau}
\begin{split}
\tau\colon\Homeo(\mathrm{Spec}(\aA))&\rightarrow\Aut(\aA)\\
h&\mapsto \left[a\mapsto \mathcal{G}^{-1}\left[\mathcal{G}(a)\cp h\right]\right].
\end{split}
\end{align}
This is well defined because $\mathcal{G}(a)\cp h\in \mathrm{C}_0(\mathrm{Spec}(\aA))$ for each $a\in \aA$ so that $\tau(h)\colon \aA\rightarrow \aA$ is well defined for all $h\in \Homeo(\mathrm{Spec}(\aA))$. Moreover, $\tau(h)$ is a homomorphism since $\mathcal{G}$ and $\mathcal{G}^{-1}$ are.
Finally,
\begin{align*}
\left[\tau(h^{-1})\cp \tau(h)\right](a)&=\mathcal{G}^{-1}\left[\mathcal{G}(\tau(h)(a))\cp h^{-1}\right]
=\mathcal{G}^{-1}\left[\mathcal{G}(a)\cp h \cp h^{-1}\right]=a
\end{align*}
so that $\tau(h)\in \Aut(\aA)$.
Then for $\chi\in \mathrm{Spec}(\aA)$ and all $a\in \aA$ we have
\begin{align*}
\eta(\tau(h))(\chi)(a)&=\ovl{\tau(h)}(\chi)(a)=\chi(\tau(h)(a))
=\chi\big(\mathcal{G}^{-1}[\mathcal{G}(a)\cp h]\big)\\
&=[\mathcal{G}(a)\cp h](\chi)=\mathcal{G}(a)(h(\chi))=h(\chi)(a),
\end{align*}
hence $\eta\cp \tau = \id_{\Homeo(\mathrm{Spec}(\aA))}$, which shows surjectivity of $\eta$.
\end{enumerate}
\end{proof}
\end{Lemma}
\subsubsection{Bounded Functions}
\label{subsec:boundedfun}
For a set $\ZX$ let $\Cb(\ZX):=\{f\colon \ZX\rightarrow \mathbb{C}\:|\:\|f\|_{\infty}< \infty\}$ denote the set of bounded, complex-valued functions on $\ZX$. Then, $\Cb(\ZX)$ is an abelian $\Cstar$-algebra w.r.t.\ the supremum norm $\|\cdot\|_\infty$.
\begin{convention}
\label{conv:Boundedfunc}
Let $\aA\subseteq \Cb(\ZX)$ be some fixed $\Cstar$-subalgebra and $\upsilon\colon \SX\rightarrow \ZX$ a map with $\SX$ some further set.
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\label{conv:Boundedfunc1}
\itspacec
Let $Z$ be a set, $\bB\subseteq B(Z)$ a subset and $\Gen$ the $^*$-algebra generated by $\bB$. Then, by $\ovl{\bB}$ we will denote the closure of $\mathfrak{G}$ in $B(Z)$.
\item
\label{conv:Boundedfunc2}
\itspace
The spectrum of $\aA\subseteq B(\ZX)$ is denoted by $\QZX$ in the following. This is motivated by the first part of the next lemma. Since $\ZX$ is not assumed to carry any topology, this will not be in conflict with the notation concerning closures of subsets of topological spaces introduced in Subsection \ref{sec:notations}.
\item
\label{conv:Boundedfunc3}
\itspacec
The pullback of $\aA$ by $\upsilon$ is the $^*$-algebra $\upsilon^*(\aA):=\{f\cp \upsilon\:|\: f\in \aA\}\subseteq B(\SX)$. The closure $\rR_\upsilon:=\ovl{\upsilon^*(\aA)}\subseteq B(\SX)$ is called restriction of $\aA$ w.r.t.\ $\upsilon$. Its spectrum is denoted by $\XNR$ in the following.\footnote{In view of the second point, it would be logical to denote $\mathrm{Spec}(\rR_\upsilon)$ by $\ovl{Y^\upsilon}$. However, we will write $\XNR$ because the set $Y$ is already encoded in the map $\upsilon$. Moreover, our notation suggests that we are dealing with the spectrum of a \emph{restriction} $\Cstar$-algebra. Then, if $\aA$ is unital, we have $\XNR\cong \YNR$ by Lemma \ref{lemma:dicht}.\ref{lemma:dicht3}, where the latter space (defined in the last point of this Convention \ref{conv:Boundedfunc}.) is a closed subset of $\X$. For this reason, it makes sense to use the letter $X$ for both spaces.}
\item
\label{conv:Boundedfunc4}
\itspace
Let $\ZX_\aA$ denote the set of all $x\in \ZX$ for which the map
\begin{align*}
\text{\gls{IOTAY}}\colon \ZX &\rightarrow \mathrm{Hom}(\aA,\mathbb{C})\\
x&\mapsto [f\mapsto f(x)]
\end{align*}
is non-zero, i.e., $\ZX_\aA=\{x\in \ZX\: |\:\exists\: f\in \aA : f(x)\neq 0\}$. Hence, $x\in \ZX_\aA$ iff $\iota_\ZX(x)\in \mathrm{Spec}(\aA)$.
\item
\label{conv:Boundedfunc5}
\itspacec
The set $\YNR\subseteq \QZX$ is defined to be the closure
\begin{align*}
\YNR:=\ovl{\iota_\ZX\big(\ZX_\aA\cap\upsilon(\SX)\big)}\subseteq \mathrm{Spec}(\aA)
\end{align*}
of $\iota_\ZX\big(\ZX_\aA\cap\upsilon(\SX)\big)$ in $\mathrm{Spec}(\aA)$.
The third part of the next lemma then shows that
\begin{align*}
\YNR\cong \XNR=\mathrm{Spec}(\rR_\upsilon)
\end{align*}
holds if $\aA$ is unital.\hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\end{convention}
The first part of the following lemma is a slight variation of Proposition 2.1 in \cite{ChrisSymmLQG} which, in turn, originates from \cite{Rendall}. The second part can also be derived from Corollary 2.19 in \cite{ChrisSymmLQG}.
\begin{lemma}
\label{lemma:dicht}
\begin{enumerate}
\item
\label{lemma:dicht1}
If $\ZX$ is a set and $\aA\subseteq \Cb(\ZX)$ a $\Cstar$-subalgebra, then $\iota_X(\ZX_\aA)\subseteq \mathrm{Spec}(\aA)$ is dense, i.e., $\QZX=\ovl{\iota_\ZX(\ZX_\aA)}$. The map $\iota_\ZX$ is injective iff $\aA$ separates the points in $\ZX_\aA$.
\item
\label{lemma:dicht2}
Let $\aA$ be unital, $\SX$ a set and $\upsilon\colon \SX \rightarrow \ZX$ a map.
Then, $\ovl{\upsilon^*}\colon \XNR\rightarrow \QZX$ is
the unique continuous map which extends $\upsilon$ in the sense that the following diagram is commutative:
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
\XNR \ar@{->}[r]^-{\ovl{\upsilon^*}} & \QZX \\
\SX\ar@{.>}[u]^{\iota_{\SX}}\ar@{->}[r]^-{\upsilon} & \ZX \ar@{.>}[u]^{\iota_\ZX}
}
\end{xy}
}
\end{center}
The map $\ovl{\upsilon^*}$ is an embedding.\footnote{This means that $\ovl{\upsilon^*}$ is a homeomorphism to its image equipped with the relative topology.}
\item
\label{lemma:dicht3}
In the situation of Part \ref{lemma:dicht2}) we have
\begin{align*}
\mathrm{im}\!\left[\ovl{\upsilon^*}\right]=\ovl{\upsilon^*}(\XNR)\stackrel{!}{=}\ovl{\iota_X(\upsilon(Y))}=\YNR,
\end{align*}
i.e., $\XNR\cong \YNR$ by the embedding property of $\ovl{\upsilon^*}$.
\item
\label{lemma:dicht4}
If $\rho \colon \ZX\rightarrow \ZX$ is a map, then
$\aA\subseteq \rho^*(\aA)$ implies $\rho(\ZX_\aA)\subseteq \ZX_\aA$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Assume there is $\chi\in U:=\mathrm{Spec}(\aA)\backslash \ovl{\iota_\ZX(\ZX_\aA)}$. Then $U$ is an open neighbourhood of $\chi$. Since the space $\mathrm{Spec}(\aA)$ is locally compact Hausdorff, by Urysohn's lemma we find a continuous function $\hat{f}\colon \mathrm{Spec}(\aA)\rightarrow [0,1]$ such that $\hat{f}(\chi)=1$ and $\hat{f}$ has compact support contained in $U$.
Then $\hat{f}\in \Cinf(\mathrm{Spec}(\aA))$ so that $\hat{f}=\mathcal{G}(f)$ for some $f\in \aA$. Consequently, $f(x)=\iota_\ZX(x)(f)=\hat{f}(\iota_\ZX(x))=0$ for all $x\in \ZX_\aA$ by construction, and $f(x)=0$ for all $x\in \ZX\backslash \ZX_\aA$ by definition of $\ZX_\aA$. Consequently, $f=0$ and $\hat{f}=\mathcal{G}(f)=0$, which contradicts $\hat{f}(\chi)=1$.
The injectivity statement is immediate from the definitions.
\item
Since $\aA$ and $\rR_\upsilon$ are unital, we have $\ZX_\aA=\ZX$ and $\SX_{\rR_\upsilon}=\SX$.
Then $\upsilon^*\colon \aA \rightarrow \rR_\upsilon$ is a unital algebra homomorphism, so that by Lemma \ref{lemma:homzuspec}.\ref{lemma:homzuspec1} the map $\ovl{\upsilon^*}\colon \mathrm{Spec}(\rR_\upsilon)\rightarrow \mathrm{Spec}(\aA)$ is continuous and well defined.\footnote{Observe that there is no ad hoc reason for $\upsilon^*(\aA)$ to be closed, i.e., $\upsilon^*$ is not necessarily surjective. For this reason we are forced to assume unitality in order to guarantee well-definedness of $\ovl{\upsilon^*}$.} Obviously, $\ovl{\upsilon^*}\cp \iota_\SX = \iota_\ZX\cp \upsilon$ so that uniqueness follows from denseness of $\mathrm{im}[\iota_\SX]$ in $\XNR=\mathrm{Spec}(\rR_\upsilon)$ and continuity of $\ovl{\upsilon^*}$.
Now, assume that $\ovl{\upsilon^*}(\chi)=\ovl{\upsilon^*}(\chi')$ for $\chi,\chi'\in \mathrm{Spec}(\rR_\upsilon)$. Then
\begin{align*}
\chi(\upsilon^*(f))=\ovl{\upsilon^*}(\chi)(f)=\ovl{\upsilon^*}(\chi')(f)=\chi'(\upsilon^*(f))\qquad\forall\:f\in \aA,
\end{align*}
hence
$\chi|_{\upsilon^*(\aA)}=\chi'|_{\upsilon^*(\aA)}$. Since $\upsilon^*(\aA)\subseteq\rR_\upsilon$ is dense and $\chi,\chi'$ are continuous, $\chi=\chi'$ follows.
Now, since $\XNR$ is compact and $\mathrm{im}\!\left[\ovl{\upsilon^*}\right]$ is a Hausdorff space, the bijective continuous map $\ovl{\upsilon^*}\colon \XNR\rightarrow \mathrm{im}\!\left[\ovl{\upsilon^*}\right]$ is a homeomorphism.
\item
By \ref{lemma:dicht2}) we have $\ovl{\upsilon^*}(\iota_\SX(\SX))=\iota_\ZX(\upsilon(\SX)) $, hence
\begin{align*}
\ovl{\upsilon^*}\!\left(\QSX\right)\stackrel{\ref{lemma:dicht1})}{=}\ovl{\upsilon^*}\Big(\ovl{\iota_\SX(\SX)}\Big)\subseteq \ovl{\ovl{\upsilon^*}(\iota_\SX(\SX))}=\ovl{\iota_\ZX(\upsilon(\SX))}.
\end{align*}
Here, for the inclusion in the third step we have used continuity of $\ovl{\upsilon^*}$. This shows
$ \ovl{\upsilon^*}\!\left(\QSX\right)\subseteq \ovl{\iota_\ZX(\upsilon(\SX))}$. For the converse inclusion we calculate
\begin{align*}
\iota_\ZX(\upsilon(\SX))\subseteq\ovl{\upsilon^*}\left(\ovl{\iota_\SX(\SX)}\right)\stackrel{\ref{lemma:dicht1})}{=}\ovl{\upsilon^*}\left(\QSX\right),
\end{align*}
hence $\ovl{\iota_\ZX(\upsilon(\SX))}\subseteq \ovl{\upsilon^*}\!\left(\QSX\right)$
since $ \ovl{\upsilon^*}\!\left(\QSX\right)$
is compact.
\item
If $x\in \ZX_\aA$, then $f(x)\neq 0$ for some $f\in \aA$. Since by assumption we have $f=g\cp \rho$ for some $g\in \aA$, we obtain $g(\rho(x))\neq 0$, hence $\rho(x)\in \ZX_\aA$.
\end{enumerate}
\end{proof}
\end{lemma}
\subsection{The Bohr Compactification}
\label{subsec:Bohrcomp}
We start with some basic definitions and facts, cf.\ \cite{RudinFourier}. Then we show that the Bohr compactification of a locally compact abelian group $G$ equals the spectrum of the almost periodic functions $\CAP(G)$ on $G$. Finally, we characterize the continuous abelian group structures on the spectrum of a unital $\Cstar$-subalgebra of the bounded functions on a set.
\subsubsection{Basic Definitions}
\label{sss:Basicdefs}
The statements not proven here can be found, e.g., in \cite{RudinFourier}:
\vspace{5pt}
\noindent
If $G$ is an LCA group, then the dual group $\DG$ of $G$ is the set of continuous homomorphisms $\chi\colon G\rightarrow S^1$, for $S^1\subseteq \mathbb{C}$ the unit circle, endowed with the group structure
\begin{align*}
(\chi*\chi')(g):=\chi(g)\cdot\chi'(g),\qquad\quad \chi^{-1}(g):=\ovl{\chi(g)},\qquad\quad 1_\DG\colon g\mapsto 1 \qquad\quad \forall\:g\in G.
\end{align*}
The elements of $\DG$ always separate the points in $G$ and if $G$ is compact, then $\int_{G}\chi\: \dd \muH=0$ iff $\DG\ni \chi\neq 1$ for $\muH$ the Haar measure on $G$.
The group $\DG$ becomes an LCA group when equipped with the topology generated by the sets
\begin{align*}
B_{K,\epsilon}(\chi):=\{\chi'\in \DG\:|\: |\chi(g)-\chi'(g)|< \epsilon\text{ for all }g\in K\}.
\end{align*}
Here $K\subseteq G$ is compact, $\chi\in \DG$ and $\epsilon>0$. If $\widehat{\DG}$ denotes the dual of $\DG$, Pontryagin duality states that the map $j\colon G\rightarrow \widehat{\DG}$, $j(g)\colon \chi\mapsto\chi(g)$ is an isomorphism and a homeomorphism.
Now, if we equip $\DG$ with the
discrete topology, then we obtain a further LCA group $\DG_d$. The Bohr compactification $\GB$ of $G$ is defined to be the dual of $\DG_d$. $\GB$ is compact since for duals of discrete LCA groups this is always the case. Moreover, we have $\widehat{\DG}\subseteq \GB$ because $\GB$ equals the set of all homomorphisms $\psi\colon \DG\rightarrow S^1$ whereas $\widehat{\DG}$ consists of the continuous (w.r.t.\ the topology on $\DG$, not w.r.t.\ the discrete topology on $\DG_d$) ones.
One can show that the map $i_{\mathrm{B}}\colon G\rightarrow \GB$, defined as $j$ above, is a continuous isomorphism to the dense subgroup $\iB(G)\subseteq \GB$. Since $\DG_d$ is discrete, each compact set is finite. Consequently, the topology on $\GB$ is generated by the sets
\begin{align}
\label{eq:bohrtop}
B_{\chi,\epsilon}(\psi):=\{\psi '\in \GB\:|\: |\psi(\chi)-\psi'(\chi)|< \epsilon\}\qquad\text{for}\quad\epsilon>0 \quad\text{and}\quad\chi\in \DG_d.
\end{align}
\begin{lemma}
\label{lemma:Bohriso}
Let $\text{\gls{CAP}}\subseteq B(G)$ denote the $\Cstar$-algebra generated by the elements of $\DG$. Then, the restriction map $\res\colon \mathrm{Spec}(\CAP(G))\rightarrow \GB$, $\psi\mapsto \psi|_{\DG}$ is a homeomorphism.
\begin{proof}
Obviously, $\res$ is well defined and continuous. Moreover, it is injective because $\CAP(G)$ is generated by $\DG$.
So, the crucial part is to show surjectivity of $\res$. For this, let $\widehat{\psi}\in\GB$ and define $\psi$ on the $^*$-span $\Gen$ of $\DG$ in $\CAP(G)$ by
\begin{align*}
\psi(f):=\textstyle\sum_{i=1}^k \beta_i\cdot \widehat{\psi}(\chi_i) \qquad\text{for}\qquad \Gen\ni f=\sum_{i=1}^k \beta_i \cdot\chi_i.
\end{align*}
This map is well defined and linear because $\DG\subseteq \CAP(G)$ is a linearly independent subset. In fact, let $\Gamma_B$ denote the dual group of $\GB$ and $j_\Gamma\colon \Gamma_d \rightarrow \Gamma_B$ the canonical map $j_\Gamma(\chi)\colon \psi \mapsto \psi(\chi)$ defined as the map $j$ above. Then $j_\Gamma(\Gamma)\subseteq C(\GB)$
is contained in the dual group of $\GB$, so that
$\int_{\GB}j_\Gamma(\chi)\: \dd \muH=0$ iff $\chi\neq 1$ for $\muH$ the Haar measure on $\GB$ and $\chi \in \DG$. Now, it is straightforward to see that
\begin{align*}
\psi(\ovl{f}\cdot f')=\ovl{\psi(f)}\cdot\psi(f')\qquad\text{holds for all}\qquad f,f'\in \Gen.
\end{align*}
Then $\psi$ extends (by linearity and continuity) to a well-defined element\footnote{Multiplicativity on $\CAP(G)$ follows from continuity and multiplicativity on $\Gen$.} of $\mathrm{Spec}(\CAP(G))$ if we can show that $\psi$ is continuous on $\Gen$. For this, let $\Gen\ni f=\sum_{i=1}^k \beta_i \cdot\chi_i$ be as above and $\epsilon>0$. We choose $g\in G$ such that $\big|\widehat{\psi}(\chi_i)-\iB(g)(\chi_i)\big|\leq \frac{\epsilon}{k\max(|\beta_1|,\dots,|\beta_k|)}$ for $1\leq i\leq k$ and obtain
\begin{align*}
|\psi(f)|&\leq \big|\textstyle\sum_{i=1}^k\beta_i\cdot \big[ \widehat{\psi}(\chi_i)-\iB(g)(\chi_i)\big]\big|+ \big|\textstyle\sum_{i=1}^k \beta_i\cdot \iB(g)(\chi_i)\big|\\
& \leq \max(|\beta_1|,\dots,|\beta_k|)\cdot\textstyle\sum_{i=1}^k \frac{\epsilon}{k\max(|\beta_1|,\dots,|\beta_k|)} + \|f\|_\infty,
\end{align*}
hence $|\psi(f)|\leq \|f\|_\infty +\epsilon$ for all $\epsilon>0$. This shows $|\psi(f)|\leq \|f\|_\infty$ for all $f\in \CAP(G)$, hence continuity of $\psi$.
\end{proof}
\end{lemma}
So, in the following we tacitly identify $\GB$ with $\mathrm{Spec}(\CAP(G))$ where we carry over the group structure and the corresponding Haar measure $\muH$ from $\GB$ to $\mathrm{Spec}(\CAP(G))$, i.e., we define
\begin{align*}
\psi_1 + \psi_2 := \res^{-1}(\res(\psi_1)+ \res(\psi_2))\qquad \psi^{-1}:=\res^{-1}\big(\raisebox{-0pt}{$\ovl{\res(\psi)}$}\big)\qquad e:=\res^{-1}(e)\qquad \muB:=\res^{-1}(\muH).
\end{align*}
A more concrete description of this group structure is given in the proof of Proposition \ref{prop:Specgroup}. There we show that abelian group structures on spectra of unital $\Cstar$-subalgebras of bounded functions can be encoded by families of quasi-characters on the underlying set, see Definition \ref{def:quasichar}.
\begin{lemconv}[Bohr Compactification of $\RR$]
\label{lemconv:RBMOD}
\begin{enumerate}
\item
\label{lemconv:RBMOD1}
If $G=\RR$, then $\GR$ consists exactly of the functions of the form
$\chi_l\colon x\mapsto e^{\I l x}$ for $l\in \RR$. \cite{RudinFourier}. Since we do not consider Bohr compactifications of other locally compact abelian groups in the following, by \gls{muB} we will always denote the Haar measure on $\RB$.
\item
\label{prop:Bohrmod24}
Let $\psi \in \RB$, $L=\{l_\alpha\}_{\alpha \in I}\subseteq \RR$ a collection of $\mathbb{Q}$-independent reals, and $L^\perp\subseteq \mathbb{R}$ a subset for which $L \sqcup L^\perp$ is a $\mathbb{Q}$-base of $\RR$. Then, for $\{q_\alpha\}_{\alpha\in I}\subseteq \mathbb{Q}$ and $\{s_\alpha\}_{\alpha\in I}\subseteq S^1$ we find $\psi'\in \RB$ with
\begin{align*}
\psi'(\chi_{q_\alpha\cdot l_\alpha})=s_\alpha\quad \forall\:\alpha\in I\qquad\quad\text{and}\qquad\quad \psi'(\chi_{l})=\psi(\chi_{l})\quad\forall\: l \in \Span_{\mathbb{Q}}(L^\perp).
\end{align*}
In fact, we choose $\{x_\alpha\}_{\alpha\in I}\subseteq \RR$ such that $\chi_{q_\alpha\cdot l_\alpha}(x_\alpha)=s_\alpha$ holds for all $\alpha\in I$, and define
\begin{align*}
\zeta\left(\chi_{q\cdot l_\alpha}\right):=\chi_{q\cdot l_\alpha}(x_\alpha)\: \text{ if }\:\alpha\in I \qquad\quad\text{as well as}\qquad\quad \zeta\left(\chi_{q\cdot l}\right):=\psi(\chi_{q\cdot l})\: \text{ if }\:l\in L^\perp
\end{align*}
for all $q\in \mathbb{Q}$.
Then, for $l\in \RR$ arbitrary we have a unique representation of the form
\begin{align*}
l=\textstyle\sum_{i=1}^k q_i\: l_{\alpha_i} + \textstyle\sum_{j=1}^{k'} q_j'\: l'_j .
\end{align*}
with $l'_j \in L^\perp$ and $q_i,q'_j \in \mathbb{Q}$ for $1\leq i\leq k$, $1\leq j\leq k'$. Here, we define
\begin{align*}
\psi'\!\left(\chi_{l}\right):=\textstyle\prod_{i=1}^k\zeta\big(\raisebox{1pt}{$\chi_{q_i\cdot l_{\alpha_i}}$}\big) \cdot \textstyle\prod_{j=1}^q\zeta\big(\raisebox{1pt}{$\chi_{q'_j\cdot l'_j}$}\big)
\end{align*}
as well as $\psi(1):=1$. Then, it is straightforward to see that $\psi'\colon \Gamma\rightarrow S^1$ is a homomorphism with the desired properties.
\item
\label{prop:Bohrmod21}
Let $\Per$ denote the set of maps $\phi\colon \RR_{>0}\rightarrow [0,2\pi)$ with
\begin{align*}
\phi(l+l')=\phi(l)+\phi(l')\:\bmod \: 2\pi\qquad \forall\:l,l'\in \RR_{>0}.
\end{align*}
Then $\RB\cong \Per$.
In fact, if $\phi\in \Per$, then $\psi\colon \chi_l\mapsto \e^{\I \sign(l) \phi(|l|)}$ for $l\neq 0$ and $\psi(1):=1$ is a well-defined element of $\RB$ because
\begin{align*}
\psi(\chi_{l}\cdot\chi_{l'})
=\e^{\I \sign(l+l')\: \phi(|l+l'|)}
=\e^{\I \sign(l)\:\phi(|l|)} \e^{\I \sign(l')\:\phi(|l'|)}
=\psi(\chi_{l})\cdot\psi(\chi_{l'}).
\end{align*}
Here, the second equality is clear for $\sign(l)=\sign(l')$ and
follows from
\begin{align*}
\phi(l)=\phi(l-l'+l')=\phi(l-l')+\phi(l') +2\pi n \quad\text{for}\quad l>l'>0
\end{align*}
in the other cases.
Now, if $\psi\in \RB$, then $\psi(\chi_l)\in S^1$, hence $\psi(\chi_l)=\e^{\I \phi'(l)}$ for $\phi'(l)\in [0,2\pi)$ uniquely determined. This defines a map $\phi'\colon \RR \rightarrow [0,2\pi)$ whose restriction $\phi:=\phi'|_{\RR_{>0}}$ has the desired properties.
\item
\label{rem:RBOHR}
In the following, by the Bohr compactification
\gls{RB}
of $\RR$ we will understand both
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
In Subsection \ref{sec:ConSp}, the dual group of $\GR_d$ (see Convention \ref{conv:sammel}).
\item
In Section \ref{sec:HomIsoCo}, the spectrum of the $\Cstar$-algebra $\CAP(\RR)$ generated by the set $\GR$ of continuous characters on $\RR$.
\end{itemize}
\endgroup
\noindent
In both cases we will refer to the modifications result in Part \ref{prop:Bohrmod24} which obviously also applies to the $\mathrm{Spec}(\CAP(\RR))$ case.
\end{enumerate}
\end{lemconv}
\subsubsection{Abelian Group Structures and Quasi-Characters}
\begin{definition}
\label{def:quasichar}
Let $X$ be a set and $\aA\subseteq B(X)$ some unital $C^*$-subalgebra. A subset $\uU\subseteq \aA$ with\footnote{Recall Convention \ref{conv:Boundedfunc} for the closure of a subset of a $\Cstar$-subalgebra of bounded functions.} $\ovl{\uU}=\aA$ is called family of quasi-characters on $X$ iff
\begingroup
\setlength{\leftmargini}{15pt}
\begin{enumerate}
\item
\label{def:quasichar1}
$\mathrm{im}[f]\subseteq S^1$ for all $f\in \uU$.
\item
\label{def:quasichar2}
$\uU$ is closed under pointwise multiplication and complex conjugation.
\item
\label{def:quasichar3}
The elements of $\uU$ are linearly independent.
\item
\label{def:quasichar4}
For $x,y\in X$ there is a net $\{z_\alpha\}_{\alpha\in I}\subseteq X$ such that $f(x)f(y)=\lim_\alpha f(z_\alpha)$ for all $f\in \uU$. Moreover, there is a net $\{e_\alpha\}_{\alpha\in J}$ such that $\lim_\alpha f(e_\alpha)=1$ for all $f\in \uU$.
\end{enumerate}
\endgroup
\noindent
If $X$ carries an abelian group structure, then $f\in \aA$ is said to be a character iff $\mathrm{im}[f]\subseteq S^1$, and
\begin{align*}
f(x+ y)=f(x)\cdot f(y)\qquad\text{as well as}\qquad f(x^{-1})=\ovl{f}(x)
\end{align*}
holds for all $x,y\in X$.
\end{definition}
\begin{proposition}
\label{prop:Specgroup}
\begin{enumerate}
\item
Let $X$ be a set and $\aA\subseteq B(X)$ a unital $C^*$-algebra. Then, the families of quasi-characters are in bijection with the continuous abelian group structures on $\X$.
\item
If $X$ carries an abelian group structure, then a continuous abelian group structure on $\X$ is compatible in the sense that
\begin{align*}
\iota_X(x)+ \iota_X(y)=\iota_X(x + y)\qquad \forall\:x,y\in X_\aA
\end{align*}
iff the respective family $\uU$ consists of characters.
\end{enumerate}
\end{proposition}
The technical details of the proof can be found in Proposition \ref{prop:SpecgroupA}.
\begin{proof}
Each family of quasi-characters $\uU$ gives rise to a continuous abelian group structure on $\X$ just by
\begin{align*}
(\psi_1+\psi_2)(f):= \textstyle\sum_{i=1}^n \beta_i\:\psi_1(f_{\alpha_i})\psi_2(f_{\alpha_i})\qquad \psi^{-1}(f):=\textstyle\sum_{i=1}^n \beta_i\:\ovl{\psi_1(f_{\alpha_i})}\qquad e(f):=\sum_{i=1}^n\beta_i
\end{align*}
for $\psi_1,\psi_2,\psi \in \X$ and $f:=\textstyle\sum_{i=1}^n\beta_i f_{\alpha_i}$ with $f_{\alpha_i}\in \uU$, $\beta_i\in \mathbb{C}$ for $1\leq i\leq n$.
Here, one has to show that the above maps extend by continuity to $\aA$ being the closure of the $^*$-algebra generated by $\uU$.
Conversely, if $\X$ carries a continuous abelian group structure and $\Gamma$ denotes its dual group, then $\uU:=\mathcal{G}^{-1}(\Gamma)$ is the desired family of quasi-characters for $\mathcal{G}\colon \aA\rightarrow C(\X)$ the Gelfand transform.
\end{proof}
Of course, if $G$ is a compact abelian group, then $\DG\subseteq \CAP(G)$ is the family of (quasi-)characters that corresponds to the continuous group structure on $\GB\cong \mathrm{Spec}(\CAP(G))$.
\section{Spectral Extensions of Group Actions}
\label{sec:SPecExtGr}
In this section, we develop a reduction concept which, applied to the framework of loop quantum gravity, allows to perform a symmetry reduction directly on the quantum level.
We will consider the general situation where the symmetry
is represented by a Lie group of automorphisms of the principal fibre bundle of interest. The basic idea
then is to lift this symmetry
to a left action on the quantum configuration space of LQG. In analogy to the classical situation, where the reduced configuration space is formed by the set $\AR$ of invariant connections on $P$, the quantum-reduced configuration space then will be formed by such elements which are invariant under the whole symmetry group. In contrast to that, reduction in LQG traditionally means to quantize the reduced classical space $\AR$, and we will see that the resulting space is always contained in our quantum-reduced one. Moreover, in the next section we will show that this inclusion is even proper in several situations, so that in this context quantization and reduction usually do not commute.
In a first step, we will use the concept of a $\Cstar$-dynamical system in order to extend a left action $\cw\colon G\times X\rightarrow X$ of a group $G$ on a set $X$ to the spectrum of a $\Cstar$-subalgebra $\aA\subseteq B(X)$.
Then, we adapt this to the case where $X$ equals the set of smooth connections on a principal fibre bundle, and where $\aA$ is generated by parallel transports along
a distinguished set of curves in its base manifold.
Finally, we consider the case where the structure group is compact, and where the set of curves has an additional independence property. In this situation, it will be possible to identify the quantum configuration space of LQG with a space of homomorphisms of paths. The quantum-reduced configuration space then will be formed by the invariant ones. In the next section, this will allow us to investigate the inclusion relations between quantized reduced classical and respective quantum-reduced configuration spaces in much more detail.
We now start with some general statements concerning group actions on spectra of abelian $\Cstar$-algebras.
\subsection{Group Actions on Spectra}
\label{subsec:graonsp}
Recall \cite{0922.46050} that a $\Cstar$-dynamical system is a triple $(\aA,G,\ah)$ consisting of a $\Cstar$-algebra $\aA$, a group $G$ and an antihomomorphism $\ah\colon G\rightarrow \Aut(\aA)$. If $G$ is a topological group, then $\ah$ is said to be continuous iff for each $a\in \aA$ the map $\ah(\cdot)(f)\colon G\rightarrow \aA$, $g\mapsto \ah(g)(a)$ is continuous. In \cite{0922.46050} it is shown that each $\Cstar$-dynamical system with $G$ locally compact\footnote{The proof there also works if $G$ is an arbitrary topological group.} and $\ah$ continuous, gives rise to a continuous action $\specw$ of $G$ on $\mathrm{Spec}(\aA)$.
In the next lemma we discuss this assignment for the abelian case. In the first part, we will drop the continuity assumptions and show that the assignment $\ah \mapsto \specw$ is bijective.
In the second part, we then investigate the continuity properties of the respective maps. For instance, if
$\aA$ is unital, it turns out that $\specw$ is continuous iff $\ah$ is. This will provide us with a necessary condition for continuity of $\specw$.
\begin{lemma}
\label{lemma:leftactionsvsautos}
Let $\aA$ be an abelian $\Cstar$-algebra.
\begin{enumerate}
\item
\label{lemma:leftactionsvsautos1}
The $\Cstar$-dynamical systems $(\aA,G,\ah)$ are in bijection with the left actions $\specw\colon G\times \mathrm{Spec}(\aA)\rightarrow \mathrm{Spec}(\aA)$ for which $\specw_g$ is continuous for all $g\in G$.
\item
\label{lemma:leftactionsvsautos2}
If $G$ is a topological group, then continuity
of $\ah$ implies continuity of $\specw$. The converse implication holds if $\aA$ is unital.
\end{enumerate}
\end{lemma}
Recall that $\specw_g\colon \mathrm{Spec}(\aA)\rightarrow \mathrm{Spec}(\aA)$ is defined by $\specw_g(\chi):=\specw(g,\chi)$.
\begin{proof}
\begin{enumerate}
\item
Let $\ah\colon G\rightarrow \Aut(\aA)$ be given and define the corresponding left action by $\specw(g,\chi):=\eta(\ah(g))(\chi)$. Then, $\specw_g$ is well defined and continuous by Lemma \ref{lemma:homzuspec}.\ref{lemma:homzuspec2}.
Moreover, since $\eta$ and $\ah$ are antihomomorphisms, the left action property follows from
\begin{align*}
\specw(gh,\chi)&=\eta(\ah(gh))(\chi)=\eta(\ah(h)\cp\ah(g))(\chi)\\
&=\eta(\ah(g))\left(\eta(\ah(h))(\chi)\right)
=\specw(g,\eta(\ah(h))(\chi))=\specw(g,\specw(h,\chi)).
\end{align*}
Conversely, if $\specw\colon G\times \mathrm{Spec}(\aA)\rightarrow \mathrm{Spec}(\aA)$ is a left action, then $\specw_g\in \mathrm{Homeo}(\mathrm{Spec}(\aA))$ for each $g\in G$ so that $\ah(g):=\tau(\specw_g)$ is an element of $\Aut(\aA)$. Here, $\tau\colon \mathrm{Homeo}(\mathrm{Spec}(\aA))\rightarrow \Aut(\aA)$ denotes the map \eqref{eq:tautau}, which is just the inverse of $\eta$. Since $\eta$ is an antiisomorphism, the same is true for $\tau=\eta^{-1}$, so that
\begin{align*}
\ah(gh)=\tau(\specw_{gh})=\tau(\specw_g\cp \specw_h)
=\tau(\specw_h)\cp \tau(\specw_g)=\ah(h)\cp \ah(g).
\end{align*}
\item
Assume that $\ah$ is continuous and let $G\times \mathrm{Spec}(\aA)\supseteq \{(g_\alpha,\chi_\alpha)\}_{\alpha\in I}\rightarrow (g,\chi)\in G\times \mathrm{Spec}(\aA)$ be a converging net. Then $\{g_\alpha\}_{\alpha\in I}\rightarrow g$ and $\{\chi_\alpha\}_{\alpha\in I}\rightarrow \chi$ are converging nets as well. By continuity of $\ah$ for each $a\in \aA$ and each $\epsilon>0$ we find $\alpha_\epsilon\in I$ such that $\|\ah(g)(a)-\ah(g_\alpha)(a)\|_\aA\leq \frac{\epsilon}{2}$ for all $\alpha \geq \alpha_\epsilon$. Then, since $\|\chi_\alpha\|_{\mathrm{op}}\leq 1$ for all $\alpha\in I$, we obtain
\begin{align*}
|\chi_\alpha\big(\ah(g)(a)-\ah(g_\alpha)(a)\big)|\leq \|\ah(g)(a)-\ah(g_\alpha)(a)\|_\aA\leq\frac{\epsilon}{2}\qquad \forall\:\alpha \geq \alpha_\epsilon.
\end{align*}
Moreover, since $\{\chi_\alpha\}_{\alpha\in I}\rightarrow \chi$, we find $\alpha'_{\epsilon}\in I$ with
$\|\chi-\chi_\alpha\|_{\ah(g)(a)}\leq \frac{\epsilon}{2}$
for all $\alpha\geq \alpha'_\epsilon$.
Then for $\alpha\in I$ with $\alpha\geq \alpha_\epsilon,\alpha'_\epsilon$ we have
\begin{align*}
\|\specw(g,\chi)-\specw(g_\alpha,\chi_\alpha)\|_a
&=\|\eta(\ah(g))(\chi)-\eta(\ah(g_\alpha))(\chi_\alpha)\|_{a}\\
&=|\chi(\ah(g)(a))-\chi_\alpha(\ah(g_\alpha)(a))|\\
&\leq \|\chi-\chi_\alpha\|_{\ah(g)(a)} + |\chi_\alpha\big(\ah(g)(a)-\ah(g_\alpha)(a)\big)|\leq \epsilon,
\end{align*}
which shows the first part.
Now, let $\aA$ be unital. For fixed $a\in \aA$ we consider the continuous function
\begin{align*}
h\left((g,\chi),(g',\chi')\right):= (\mathcal{G}(a)\cp \specw)(g,\chi)-(\mathcal{G}(a)\cp \specw)(g',\chi'),
\end{align*}
where $(g,\chi),(g',\chi')\in G\times \mathrm{Spec}(\aA)$. Then $h^{-1}(B_\epsilon(0))$ is open and contains $((g,\chi),(g,\chi))$ for all $(g,\chi)\in G\times \mathrm{Spec}(\aA)$. Let $g\in G$ be fixed. Then for each $\chi\in \mathrm{Spec}(\aA)$ we find open neighbourhoods $B_\chi\subseteq G$, $U_\chi \subseteq \mathrm{Spec}(\aA)$ of $g$ and $\chi$, respectively, such that
\begin{align*}
B_\chi\times U_\chi \times B_\chi\times U_\chi\subseteq h^{-1}(B_\epsilon(0))
\end{align*}
holds.
By compactness of $\mathrm{Spec}(\aA)$
there are $\chi_1,\dots, \chi_n\in \mathrm{Spec}(\aA)$ such that the corresponding sets $U_{\chi_j}$ cover $\mathrm{Spec}(\aA)$. Then $B_g:= B_{\chi_1} \cap \dots \cap B_{\chi_n}$ is an open neighbourhood of $g$ and we obtain
\begin{align*}
\left|(\mathcal{G}(a)\cp\specw)(g,\chi)-(\mathcal{G}(a)\cp\specw)(h,\chi)\right|< \epsilon \qquad \forall\: \chi \in \mathrm{Spec}(\aA), h\in B_g.
\end{align*}
Consequently, $\left\|\mathcal{G}(a)\cp\specw_g -\mathcal{G}(a)\cp\specw_h\right\|_{\infty}\leq \epsilon$, so that
\begin{align*}
\|\ah(g)(a)-\ah(h)(a)\|_{\aA}&= \|\tau(\specw_g)(a)-\tau(\specw_h)(a)\|_{\aA}\\
&=\|\mathcal{G}^{-1}[\mathcal{G}(a)\cp\specw_g]-\mathcal{G}^{-1}[\mathcal{G}(a)\cp\specw_g]\|_{\aA}\\
&=\|\mathcal{G}^{-1}[\mathcal{G}(a)\cp\specw_g-\mathcal{G}(a)\cp\specw_g]\|_{\aA}\\
&=\left\|\mathcal{G}(a)\cp\specw_g-\mathcal{G}(a)\cp\specw_h\right\|_{\infty}\\
&\leq \epsilon
\end{align*}
for all $h\in B_g$. This shows continuity of $\ah(\cdot)(a)$ at $g\in G$.
\end{enumerate}
\end{proof}
\begin{definition}[$\specw$-Invariance]
\label{def:ixschlange}
Let $\cw$ be a left action of the group $G$ on the set $X$ and $\aA\subseteq \Cb(X)$.
\begin{enumerate}
\item
\label{def:ixschlange1}
$\aA$ is called $\cw$-invariant iff $\cw_g^*(\aA)\subseteq \aA$ for all $g\in G$, i.e., if $(\aA,G,\ah)$ is a $\Cstar$-dynamical system with $\ah(g)(f):=\cw_g^*(f)$ for all $g\in G$ and all $f\in \aA$.
\item
\label{def:ixschlange2}
We define the set
\begin{align*}
\text{\gls{XR}}:=\left\{x\in X\:|\:\cw(g,x)=x\text{ for all } g\in G \right\}
\end{align*}
of invariant elements, and denote the spectrum of the $\Cstar$-algebra $\ovl{\aA|_{\XR}}$ by \gls{XRRQ}.
\item
\label{def:ixschlange3}
We define
$\text{\gls{XRQ}}:=
\ovl{\iota_X(X_\aA\cap \XR)}$ to be the closure of $\iota_X(X_\aA\cap \XR)$ in $\X$.
\end{enumerate}
\end{definition}
\begin{remark}
Let $Y:=\XR$ and $\upsilon \colon Y\rightarrow X$ the canonical inclusion map. Moreover, let $\XNR$, $\YNR$ denote the corresponding spaces introduced in Convention \ref{conv:Boundedfunc}. Then
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
$\XRRQ=\XNR$ because $\ovl{\aA|_{\XR}}=\rR_\upsilon=\ovl{\upsilon^*(\aA)}$,
\item
$\XRQ=\YNR$ because $\YNR=\ovl{\iota_X(X_\aA\cap \upsilon(\XR))}=
\ovl{\iota_X(X_\aA\cap \XR)}$.
\hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\end{remark}
\begin{Proposition}
\label{prop:autspec}
Let $\cw$ be a left action of the group $G$ on the set $X$ and $\text{\gls{aA}}\subseteq \Cb(X)$ a $\cw$-invariant
$\Cstar$-algebra.
\begin{enumerate}
\item
\label{prop:autspec1}
There is a unique left action $\text{\gls{SPECW}}\colon G\times \X \rightarrow \X$
such that:
\begin{enumerate}
\item[$\mathrm{(a)}$]
$\specw_g$ is continuous for all $g\in G$,
\item[$\mathrm{(b)}$]
$\specw$ extends $\cw$ in the sense that on $X_\aA$ we have
\begin{align}
\label{eq:extens}
\specw_g\cp\iota_X=\iota_X\cp\cw_g\qquad\forall\: g\in G.
\end{align}
\end{enumerate}
$\specw$ is explicitly given by $\specw\big(g,\x\big)= \x\cp \cw_g^*$.
\item
\label{prop:autspec2}
If $G$ is a topological group, then $\specw$ is continuous if $\cw^*_\bullet f\colon G\rightarrow \aA$, $g \mapsto f\cp \cw_g$ is continuous for each $f\in \aA$. The converse implication holds if $\aA$ is unital.
\item
\label{prop:autspec3}
The set of invariant elements
\begin{align*}
\XQR:=\left\{\x \in \X\:\big|\: \specw(g,\x)=\x\text{ for all }g\in G\right\}
\end{align*}
is closed in $\X$, and we have $\XRQ\subseteq \XQR$.
\item
\label{prop:autspec4}
If $\aA$ is unital, then
$\XRRQ\cong \XRQ$ via $\ovl{i_{\XR}^*}\colon \XRRQ\rightarrow \XRQ\subseteq \X$.
The following diagram is commutative
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
\XRRQ\: \ar@{->}[r]^-{\ovl{i_{\XR}^*}}_-{\cong} & \: \XRQ\: \ar@{->}[r]^-{\subseteq} & \:\XQR\: \ar@{->}[r]^-{\subseteq} & \:\X \\
\XR\: \ar@{.>}[u]^{\iota_{\XR}} \ar@{^{(}->}[r]^-{i_{\XR}}& \:i_{\XR}(\XR)\: \ar@{.>}[u]^{\iota_X} \ar@{->}[rr]^{\subseteq} & & \: X \ar@{.>}[u]^{\iota_{X}}.
}
\end{xy}
}
\end{center}
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
First observe that \eqref{eq:extens} makes sense because $\cw_{g^{-1}}^*(\aA)\subseteq \aA$ implies $\aA = \cw_{g}^*(\cw_{g^{-1}}^*(\aA))\subseteq \cw_{g}^*(\aA)$, hence $\cw_g\colon X_\aA\rightarrow X_\aA$ by Lemma \ref{lemma:dicht}.\ref{lemma:dicht4}.
For uniqueness, let $\specw$ and $\specw'$ be two such extensions of $\cw$.
Then, by \eqref{eq:extens} we have
$\specw'_g|_{\iota_X(X_\aA)}=\specw_g|_{\iota_X(X_\aA)}$ for all $g\in G$ so that $\specw'_g=\specw_g$ by $\mathrm{(a)}$ and denseness of $\iota_X(X_\aA)$ in $\mathrm{Spec}(\aA)$. For existence, consider the $\Cstar$-dynamical system $(\aA,G,\ah)$ for $\ah(g):= \cw_g^*$.
Then, Lemma \ref{lemma:leftactionsvsautos}.\ref{lemma:leftactionsvsautos1} provides us with a corresponding left action $\specw\colon G\times \mathrm{Spec}(\aA)\rightarrow \mathrm{Spec}(\aA)$ such that $\specw_g$ is continuous for all $g\in G$. This action is
given by
\begin{align*}
\specw(g,\ovl{x})=\eta(\ah(g))(\ovl{x})=\ovl{x}\cp \ah(g)=\ovl{x}\cp \cw^*_g,
\end{align*}
so that for $x\in X_\aA$ and all $f\in \aA$ we have
\begin{align*}
\specw_g(\iota_X(x))(f)
=\iota_X(x)(\cw_g^*f)=f(\cw_g(x))=(\iota_X\cp\cw_g)(x)(f).
\end{align*}
\item
We have $\cw_\bullet^*f=\ah(\cdot)(f)$, so that the continuity statement is clear from Lemma \ref{lemma:leftactionsvsautos}.\ref{lemma:leftactionsvsautos2}.
\item
Let $\XQR\supseteq \{\x_\alpha\}_{\alpha\in I}\rightarrow \x\in \X$ be a converging net. Then for all $g\in G$ it follows from continuity of $\specw_g$ that
$\specw(g,\x)=\specw\left(g,\lim_\alpha\x_\alpha\right)=\lim_\alpha\specw(g,\x_\alpha)=\lim_\alpha \x_\alpha=\x$,
hence closedness of $\XQR$. Moreover, $\iota_X(\XR\cap X_\aA)\subseteq \XQR$ by \eqref{eq:extens} so that $\ovl{\iota_X(\XR\cap X_\aA)}\subseteq \XQR$.
\item
This follows from the parts \ref{lemma:dicht2}), \ref{lemma:dicht3}) of Lemma \ref{lemma:dicht} if we define $\upsilon:=i_{\XR}$.
\end{enumerate}
\end{proof}
\end{Proposition}
\begin{remark}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
In the following sections, we will see that the details of the inclusion relations between the sets $\XRQ$ and $\XQR$ usually are far from being trivial.
\item
If $\aA$ is unital, then Part \ref{prop:autspec1}) can also be derived from Corollary 2.19 in \cite{ChrisSymmLQG} by extending each $\specw_g^*\colon \aA\rightarrow \aA$ uniquely to a continuous map $\specw_g\colon \mathrm{Spec}(\aA)\rightarrow \mathrm{Spec}(\aA)$. In fact, it follows from the uniqueness property of these maps that $\specw_{gh}=\specw_g\cp \specw_h$ holds for all $g,h\in G$. Then, $\specw\colon (g,\ovl{x})\mapsto \specw_g(\ovl{x})$ is a well-defined group action with the properties from Proposition \ref{prop:autspec}.\ref{prop:autspec1}.
\item
In Subsection \ref{sec:InvGenConnes} we will use Proposition \ref{prop:autspec} in order to perform a symmetry reduction on quantum level in the framework of LQG. Then, in Section \ref{sec:HomIsoCo}, we will use this proposition in order to derive some uniqueness statements concerning normalized Radon measures on cosmological quantum configuration spaces occurring in LQG, see Proposition \ref{lemma:bohrmassdichttrans} and Corollary \ref{cor:eindbohr}.
\end{itemize}
\endgroup
\end{remark}
\subsection{Invariant and Generalized Connections}
\label{sec:InvGenConnes}
We now adapt the previous subsection to the situation where the action $\cw$ comes from a Lie group of automorphisms $(G,\Phi)$ on a principal fibre bundle \gls{PMS}. This means that $\cw$ is given by \eqref{eq:connact}, i.e., acts on the set $\Con$ of smooth connections on $P$ by
\begin{align*}
\text{\gls{CW}}\colon G\times \Con\rightarrow \Con,\quad
(g,\w)\mapsto \Phi_{g^{-1}}^*\w.
\end{align*}
Moreover, the $\Cstar$-algebra $\text{\gls{aA}}\subseteq B(\Con)$ is generated by parallel transports along suitable curves in $M$ in this case.
More precisely, let \gls{Pa} be a fixed set of $\Ck$-paths in the base manifold $M$ and $\text{\gls{NU}}=\{\nu_x\}_{x\in M}\subseteq P$ a fixed family of elements with $\nu_x\in F_x$ for all $x\in M$. By $\cC$ we will denote the set of all bounded functions on $\Con$ which are of the form
\begin{align}
\label{eq:deltadef}
h_\gamma^\nu\colon \w\mapsto f\cp \text{\gls{DIFF}}\left(\nu_{\gamma(b)},\text{\gls{PATRA}}\big(\nu_{\gamma(a)}\big)\right)
\end{align}
for $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ and $f\in C_0(S)$. Recall that $\parall{\gamma}{\w}\colon F_{\gamma(a)}\rightarrow F_{\gamma(b)}$ denotes the parallel transport along $\gamma$ w.r.t.\ $\w$ as well as $\diff(p,p')\in S$ the unique element with $p'=p\cdot \diff(p,p')$ for $p,p'$ contained in the same fibre over $M$.
The corresponding unital $\Cstar$-algebra of cylindrical functions is defined by $\text{\gls{PaC}}:=\ovl{\cC}\subseteq B(\Con)$. According to Convention \ref{conv:Boundedfunc}, here $\ovl{\cC}$ denotes the closure of the $^*$-algebra generated by $\cC$ in $B(\Con)$.
This definition is independent of the choice of $\nu$, as for $\nu'=\{\nu'_x\}_{x\in M}\subseteq P$ another such family and $\Pa\ni \gamma\colon [a,b]\rightarrow M$ we have
\begin{align}
\label{eq:indep}
\diff\big(\nu'_{\gamma(b)},\parall{\gamma}{\w}\big(\nu'_{\gamma(a)}\big)\big)=
\Delta\big(\nu'_{\gamma(b)},\nu_{\gamma(b)}\big)\cdot\diff\big(\nu_{\gamma(b)},\parall{\gamma}{\w}\big(\nu_{\gamma(a)}\big)\big)\cdot\Delta\big(\nu_{\gamma(a)},\nu'_{\gamma(a)}\big).
\end{align}
\begin{convention}
\label{hgamma}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
In the following, let $\text{\gls{PSIX}}(p):=\diff(\nu_x,p)$ for all $p\in F_x$ and all $x\in M$, and define
\begin{align*}
h_\gamma^\nu(\w):=\diff\left(\nu_{\gamma(b)},\parall{\gamma}{\w}\big(\nu_{\gamma(a)}\big)\right)=\psi_{\gamma(b)}\left(\parall{\gamma}{\w}\big(\nu_{\gamma(a)}\big)\right)\qquad\forall \:\w\in \Con.
\end{align*}
\item
If $S$ is compact, then there exists \cite{BroeD} a faithful matrix representation $\rho\colon S\rightarrow \GLNC$ of $S$, which we will fix in the following. Then, since the matrix entries of $\rho$ separate the points in $S$, the functions
\begin{equation}
\label{eq:gniij}
\:[h_\gamma^\nu]_{ij}:=\rho_{ij}\cp h_\gamma^\nu,
\end{equation}
together with the unit, generate $\PaC$.
\item
We will denote the spectrum of $\PaC$ by \gls{AQ}. If it is necessary to avoid confusion, we will write \gls{AALPHA} with an index $\alpha$ referring to the involved set of curves \gls{PALPHA}. The respective $\Cstar$-algebra of cylindrical functions then is denoted by \gls{PACALPHA}.
\end{itemize}
\endgroup
\end{convention}
The elements $\ovl{\w}\in \A$ are called generalized connections and form the quantum configuration space of LQG. Here, the main reason for replacing $\Con$ by $\A$ is that the latter space is compact and can be equipped with a normalized Radon measure in a canonical way, see Subsection \ref{subsub:InvHoms}. Moreover, in the relevant cases $\Con$ is canonically embedded\footnote{Here, this just means that $\iota_\Con$ is injective.} via the map $\iota_\Con$ (see Convention \ref{conv:Boundedfunc}) as the next lemma shows. More concretely, it suffices, e.g., that each tangent vector of $M$ is realized as a final tangent vector of a curve in $\Pa$ for which each final segment is an element of $\Pa$ as well. Thus, if $M=\RR^3$, it suffices that $\Pa$ contains all linear curves:
\begin{lemma}
\label{lemma:separating}
The map $\iota_\Con\colon \Con \rightarrow \A$ is injective if for each $\vec{v}\in TM$ there is $\gamma\in \Pa$ and $s\in\operatorname{\mathrm{dom}}[\gamma]=[a,b]$, such that $\dot\gamma(s)=\vec{v}$ and $\gamma|_{[a,t]}\in \Pa$ for all $t\in (a,s]$ or $\gamma|_{[t,b]}\in \Pa$ for all $t\in [s,b)$.
\begin{proof}
This is a straightforward generalization of Appendix A in \cite{ChrisSymmLQG}, see Lemma \ref{lemma:separatingA}.
\end{proof}
\end{lemma}
According to Definition \ref{def:ixschlange}.\ref{def:ixschlange1} we have
\begin{align*}
\text{\gls{AR}}=\{\w\in\Con\:|\: \cw(g,\w)=\w\text{ for all }g\in G\}= \{\w\in\Con\:|\: \Phi_g^*\w=\w\text{ for all }g\in G\},
\end{align*}
so that $\AR$ equals the set of $\Phi$-invariant connections on $P$, see Definition \ref{def:Invconn}.
Then, traditionally, in LQG the space $\text{\gls{ARRQ}}:=\mathrm{Spec}\big(\raisebox{0pt}{$\ovl{\PaC_\alpha|_{\AR}}$}\big)$ (cf.\ Definition \ref{def:ixschlange}.\ref{def:ixschlange2})
is used as reduced quantum configuration space, see, e.g., \cite{MathStrucLQG}. Here, the index $\alpha$ hints to the fact that usually not the same set of curves as for the full theory is used to define $\PaC_\alpha$.
To summarize, we have made the following specifications:
\begin{align*}
\cw \qquad \text{--} \qquad &\text{left action } \cw\colon G\times \Con\rightarrow \Con,\:(g,\w)\mapsto \Phi_{g^{-1}}^*\w\text{ for }(G,\Phi)\\[-6pt]
&\text{a Lie group of automorphisms of }P\\
\aA \qquad \text{--} \qquad &\Cstar\text{-algebra }\PaC\text{ generated by the parallel transport functions }h_\gamma^\nu\text{ for }\gamma\\[-6pt]
&\text{contained in the fixed set of $\Ck$-paths $\Pa$ in $M$.}\\%[4pt]
\XR \qquad \text{--} \qquad &\text{set $\AR$ of invariant connections on $P$.}\\[2pt]
\XRRQ \qquad \text{--} \qquad &\text{spectrum $\ARRQ=\mathrm{Spec}\big(\ovl{\PaC|_{\AR}}\big)$.}\\[2pt]
\XRQ \qquad \text{--} \qquad &\text{closure $\ARQ=\ovl{\iota_\Con(\Con_\PaC \cap \AR)}\subseteq \A$.}
\end{align*}
Now, in order to perform a reduction on quantum level, i.e, to obtain a well-defined extension $\specw$ of $\cw$ to $\A$ providing us with the space $\AQR$ of invariant generalized connections, we first have to investigate under which assumptions the $\Cstar$-algebra $\PaC$ is $\cw$-invariant. It turns out, to be sufficient that $\Pa$ fulfils the following invariance property.
\begin{definition}
\label{def:InvPfade}
A set $\Pa$ of $\CC{k}$-paths in $M$ is said to be $\Phi$-invariant iff $\wm_g \cp \gamma\in \Pa$ holds for all $g\in G$ and all $\gamma \in \Pa$. Here, \gls{WM} denotes the left action induced by $\Phi$ on $M$, see \eqref{eq:INDA}. \hspace*{\fill}$\lozenge$
\end{definition}
Of course, if $\Pa$ is a collection of $\CC{k}$-paths in $M$ and $\wm$ is of class $\CC{k}$, the set $\langle \Pa\rangle$ of all compositions $\wm_g\cp \gamma$ with $g\in G$ and $\gamma \in \Pa$ is $\Phi$-invariant and consists of $\CC{k}$-paths as well.
\begin{lemma}
\label{lemma:CylSpecActionvorb}
If $\Pa$ is $\Phi$-invariant, then $\PaC$ is $\cw$-invariant.
\end{lemma}
\begin{proof}
This is just a straightforward consequence of the fact that if $\gamma_p^\w$ is the horizontal lift of $\gamma\colon [a,b]\rightarrow M$ in $p\in F_{\gamma(a)}$ w.r.t.\ the connection $\w$, then $\Phi_g\cp \gamma_p^\w$ is the horizontal lift of $\wm_g\cp \gamma$ in the starting point $\Phi_g(p)$ w.r.t.\ the connection $\cw(g,\w)$. The technical details can be found in Lemma \ref{lemma:CylSpecActionvorbA}.
\end{proof}
We now are able transfer Proposition \ref{prop:autspec} to generalized connections, where we only consider the case where $S$ is compact, i.e., where $\PaC$ is unital. This is basically because for our further considerations we will need that $S$ is compact anyway. For instance, we will identify $\A$ with a space of homomorphisms of paths, and for this compactness will be necessary. In addition to that, the homeomorphism $\ARRQ\cong \ARQ$ which we have (see next corollary) if $S$ is compact will be crucial for the investigations of the inclusion relations between $\ARRQ$ and $\AQR$.
\begin{corollary}
\label{cor:CylSpecAction}
Let $\Pa$ be $\Phi$-invariant and $S$ be compact. Then
\begin{align*}
\specw\colon G\times \A&\rightarrow \A, \quad
(g,\ovl{\w})\mapsto \ovl{\w}\cp \cw_g^*
\end{align*}
is the unique left action such that:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
$\specw_g$ is continuous for all $g\in G$,
\item
$\specw$ extends $\cw$ in the sense that on $\Con$ we have
\begin{align*}
\specw_g\cp\iota_\Con=\iota_\Con\cp\cw_g\qquad\forall\: g\in G.
\end{align*}
\end{itemize}
\endgroup
\noindent
The quantum-reduced space $\text{\gls{AQR}}=\left\{\w \in \A\:\big|\: \specw(g,\w)=\w\text{ for all }g\in G\right\}$ is compact and the following diagram is commutative:
\begin{align}
\label{eq:inclusionsdiag}
\begin{split}
\makebox[0pt]{
\begin{xy}
\xymatrix{
\text{\gls{ARRQ}}\: \ar@{->}[r]^-{\ovl{i_{\AR}^*}}_-{\cong} & \:\text{\gls{ARQ}}\: \ar@{->}[r]^-{\subseteq} & \:\text{\gls{AQR}}\:\ar@{->}[r]^-{\subseteq} &\:\text{\gls{AQ}} \\
\ar@{.>}[u]^{\iota_{\AR}} \text{\gls{AR}} \: \ar@{^{(}->}[r]^-{i_{\AR}} & \:i_{\AR}(\AR)\: \ar@{.>}[u]^{\iota_\Con} \ar@{^{(}->}[rr]^{\subseteq} & &\:\:\text{\gls{Con}} \ar@{.>}[u]^{\iota_{\Con}}.
}
\end{xy}
}
\end{split}
\end{align}
Here, $\ovl{i_{\AR}^*}\colon \ARRQ\rightarrow \ARQ$ is a homeomorphism.
The action $\specw$ is continuous iff $\theta_\bullet^*f\colon G\rightarrow \PaC$, $g \mapsto f\cp \theta_{g^{-1}}^*$ is continuous for all $f$ of the form $\rho_{ij}\cp h_\gamma^\nu$.
\begin{proof}
By Lemma \ref{lemma:CylSpecActionvorb} we have $\cw_g^*(\PaC)\subseteq \PaC$ for all $g\in G$. Moreover,
$\PaC$ is unital and $\cw_\bullet^*f\colon G\rightarrow\PaC$, $g\mapsto \cw_g^*f$, is continuous for all $f\in \PaC$ iff\footnote{For this, observe that $\cw_\bullet^*1=1$ and that $\cw_\bullet^*[\lambda f+ \ovl{\mu g}\hspace{1pt}]=\lambda\: \cw_\bullet^*f+ \mu\: \ovl{\cw_\bullet^*g}$ for $\lambda,\mu\in \mathbb{C}$ and $f,g\in \PaC$.} this is the case for all generators $f=\rho_{ij}\cp h^\nu_\gamma$.
Consequently, the claim follows from Proposition \ref{prop:autspec}.
\end{proof}
\end{corollary}
\begin{remark}
\label{rem:subspt}
The elements of $\AQR$ are called invariant generalized connections and the space $\AQR$ will be equipped with its canonical subspace topology in the following..
\end{remark}
The next example shows that the action $\specw$ is usually not continuous in the standard cosmological applications.
\begin{example}[Discontinuous Spectral Action]
\label{ex:Eukl}
Let $P=\RR^3 \times \SU$, $\Ge=\Gee$ and $\Pe$
be as in Example \ref{ex:LQC}, define $\nu:=\{(x,\me)\}_{x\in \mathbb{R}^3}$ and let $\Pa$ contain all linear curves in $\mathbb{R}^3$. The following arguments also apply to the homogeneous case, i.e., where the group is $\Gh=\RR^3$ and $\Ph$ is as in the same Example.
\vspace{6pt}
\noindent
We show that for the linear curve $\gamma_0\colon [0,1]\rightarrow \RR^3$, $t\mapsto t\vec{e}_1$ and $f:=\rho_{11}\cp h_{\gamma_0}^\nu$ the map $\theta_\bullet^*f\colon G\rightarrow \PaC$, $g\mapsto f\cp \cw_g$ is not continuous at $g=e\in \Ge$. For this, we define
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
smooth connections $\w^r$ for $r\in \RR$ and
\item
group elements $g_\lambda \in \Ge$ for $\lambda\in \RR$ with $g_\lambda \rightarrow e\in \Ge$ for $\lambda \rightarrow 0$,
\end{itemize}
\endgroup
\noindent
such that for each $\lambda >0$ we find $r>0$ with $|\theta_{e}^*f(\w^r)- (\theta_{g_\lambda}^*f )(\w^r)|=1$, i.e., $\|\theta_{e}^*f- (\theta_{g_\lambda}^*f )\|\geq 1$. From this it is clear that $\theta_\bullet^*f\colon G\rightarrow \PaC$ is not continuous at $g=e\in \Ge$, so that $\specw$ is not continuous by Corollary \ref{cor:CylSpecAction}.
\begingroup
\setlength{\leftmargini}{25pt}
\begin{enumerate}
\item[(a)]
For $r\in \RR$ we define the connection $\omega^r$ by the right invariant geometric distribution specified by the following smooth sections $\mathcal{E}^r_i\colon P \rightarrow TP$ for $1\leq i\leq 3$:
\begin{align*}
\mathcal{E}^r_1(x,s)&:=(\vec{e}_1,r x_2\:\tau_2\cdot s)\in T_{(x,s)}P \qquad\text{ for }\qquad x=(x_1,x_2,x_3)\in \RR^3
\\
\mathcal{E}^r_{i}(x,s)&:=(\vec{e}_{i},\vec{0})\hspace{109.5pt}\text{for }\qquad i=2,3.
\end{align*}
Here, $\tau_2\cdot s:=\dd_\me R_s(\tau_2)\in T_s\SU$.
\item[(b)]
Let $\gamma_y\colon[0,1]\rightarrow \mathbb{R}^3$ denote the linear curve that starts on the $\vec{e}_2$-axis at $y$ and traverses in $\vec{e}_1$-direction with constant velocity $\vec{e}_1$, i.e., $\gamma_y(t)=y\cdot\vec{e}_2 + t\cdot\vec{e}_1$.
Then, its horizontal lift $\gamma_y^{r}\colon [0,1]\rightarrow \mathbb{R}^3\times \SU$ w.r.t.\ $\w^r$ in $(y\cdot \vec{e}_2,\me)$ is given by
$\gamma_y^{r}(t)=\left(\gamma_y(t),\exp(try\cdot \tau_2)\right)$ because
$\pi \cp \gamma_y^{r}=\gamma_y$ and
\begin{align*}
\dot\gamma_y^{r}(t)=(\vec{e}_1,ry\tau_2\cdot \exp(try\tau_2))=\mathcal{E}^r_1\left(\gamma_y(t),\exp(try\cdot \tau_2)\right)=\mathcal{E}^r_1\left(\gamma_y^{r}(t)\right).
\end{align*}
\item[(c)]
By the choice of $\nu$ we have
\begin{align*}
h_{\gamma_y}^\nu(\w^r)=\pr_2\cp \gamma_y^{r}(1)=\exp(ry\cdot \tau_2)\stackrel{\eqref{eq:expSU2}}{=}\begin{pmatrix} \cos(ry) & -\sin(ry) \\ \sin(ry) & \cos(ry)\end{pmatrix}.
\end{align*}
Then, for $g=(-\lambda \vec{e}_2,\me)$ we obtain\footnote{See, e.g., proof of Lemma \ref{lemma:CylSpecActionvorb}, i.e., \eqref{eq:trafogenerators} in Lemma \ref{lemma:CylSpecActionvorbA}, where $\gamma'=\wm_{g^{-1}}\cp\gamma$.}
$\cw_g^*h^\nu_{\gamma_y}=h^\nu_{\gamma_{\lambda +y}}$, hence
\begin{align}
\label{eq:Absch}
\begin{split}
\big\|\cw_{e}^*\big[\rho_{11}\cp h^\nu_{\gamma_0}\big]-\cw_{g}^*\big[\rho_{11}\cp h^\nu_{\gamma_0}\big]\big\|_{\infty}
&\geq \big|\left(\rho_{11}\cp h^\nu_{\gamma_0}-\cw_g^*\big[\rho_{11}\cp h^\nu_{\gamma_0}\big]\right)(\w^r)\big|\\
&=\big|\big(\rho_{11}\cp h^\nu_{\gamma_0}-\rho_{11}\cp h^\nu_{\gamma_{\lambda }}\big)(\w^r)\big|\\
&=|1-\cos(\lambda r)|.
\end{split}
\end{align}
This expression equals $1$ for $r=\pi/(2\lambda)$ and implies discontinuity of the action $\specw$ as we have explained above.
\hspace*{\fill} {$\lozenge$}
\end{enumerate}
\endgroup
\end{example}
\subsection{Homomorphisms of Paths and Invariance}
\label{subsec:InvHoms}
To this point, we have seen that a Lie group of automorphisms $(G,\Phi)$ on $P$ and a set of $\Ck$-paths $\Pa$ provides us with the quantized reduced classical spaces $\ARRQ\cong\ARQ$. Under the condition that $\Pa$ is $\Phi$-invariant, we even obtain the quantum-reduced configuration space $\AQR$.
In order to investigate the inclusion properties between the spaces $\AQR$ and $\ARQ$, and to construct reasonable measures thereon,
we now are going to identify the elements of $\A$ with homomorphisms of paths, so-called generalized parallel transports. These are maps assigning to a curve $\gamma\in \Pa$ an equivariant mapping $F_{\gamma(a)}\rightarrow F_{\gamma(b)}$ where $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$. This identification will be possible under the assumption that $S$ is compact and that the set $\Pa$ has some additional independence property. Under this assignment $\AQR$ then occurs as a subspace of homomorphisms being invariant in a natural sense.
The big advantage of considering $\A$ as a space of homomorphisms is
the possibility to apply geometrical techniques like modification of (in this case invariant) homomorphisms to be developed in Section \ref{susec:LieALgGenC}. These modifications are crucial for investigations concerning the inclusion relations between $\ARQ$ and $\AQR$ (cf.\ Subsection \ref{sec:inclrel}). Indeed, they will allow us to construct invariant elements that cannot be contained in $\ARQ$ or, more precisely, that cannot be approximated by classical (smooth) invariant connections.
In addition to that,
modification turns out to be a key tool for the construction
of measures on $\AQR$, cf.\ Remark \ref{rem:euklrem} or Section \ref{sec:MOQRCS}.
For the rest of this section, let $\PMS$ be a principal fibre bundle with compact structure group $S$, and $\nu=\{\nu_x\}_{x\in M}\subseteq P$ a family with $\nu_x\in F_x$ for all $x\in M$. Finally, recall the map $\psi_x(p)=\diff(\nu_x,p)$ for $p\in F_x$ from Convention \ref{hgamma}.
\subsubsection{Homomorphisms of Paths}
\label{sec:hompaths}
We start our considerations with a short
introduction into homomorphisms of paths, and then highlight their relation to the space $\A$.
\begin{definition}
\begin{enumerate}
\item
Let $\gamma\colon [a,b]\rightarrow M$ be a curve.
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\vspace{-4pt}
The inverse of $\gamma$ is defined by $\gamma^{-1}\colon [a,b]\ni t\mapsto \gamma(b+a-t)$.
\item
\vspace{2pt}
A decomposition of $\gamma$ is a family of curves $\{\gamma_i\}_{0\leq i\leq k-1}$
with $\gamma|_{[\tau_i,\tau_{i+1}]}=\gamma_i$ for $0\leq i\leq k-1$ and real numbers $a=\tau_0<{\dots}<\tau_k=b$.
\end{itemize}
\endgroup
\noindent
\item
A set of $\Ck$-paths $\Pa$ is said to be stable under decomposition and inversion iff
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\vspace{-4pt}
$\gamma\in \Pa$ implies $\gamma^{-1}\in \Pa$,
\item
\vspace{2pt}
for each decomposition $\{\gamma_i\}_{0\leq i\leq k-1}$ of $\gamma$ we have $\gamma_i\in \Pa$ for all $0\leq i\leq k-1$.
\end{itemize}
\endgroup
\end{enumerate}
\end{definition}
The space $\HOM$ of homomorphisms of paths (generalized parallel transports) is defined as follows.
\begin{definition}
\label{def:hompaths}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
For $x,x'\in M$ let $\Iso(x,x')$ denote the set of equivariant mappings $\phi \colon F_x\rightarrow F_{x'}$, and define
\begin{align*}
\AF:=\bigsqcup_{x,x'\in M}\Iso(x,x').
\end{align*}
Of course, here equivariance means that $\phi \cp R_s=R_s\cp \phi$ holds for all $s\in S$.
\item
\itspacecc
Define the equivalence relation\footnote{$\gamma_1$ and $\gamma_2$ are called holonomy equivalent in this case.} \gls{CSIM} on $\Pa$ by $\gamma\csim \gamma'$
iff $\parall{\gamma}{\w}=\parall{\gamma'}{\w}$ holds for all $\w\in \Con$. Observe that by definition we have that $\gamma\not\csim \gamma'$ if the start and end points of $\gamma$ and $\gamma'$ do not coincide.
\item
\itspacecc
Let \gls{HOM} denote the set of all maps $\homm\colon \Pa \rightarrow \AF$ such that for $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ we have:
\begin{enumerate}
\itspace
\item[\textrm{(a)}]
$\homm(\gamma)\in \Iso(\gamma(a),\gamma(b))$
and $\homm(\gamma)=\id_{F_{\gamma(a)}}$ if $\gamma$ is constant,
\item[\textrm{(b)}]
$\homm(\gamma)= \homm(\gamma_{k-1})\cp \dots\cp \homm(\gamma_{0})$ if $\{\gamma_i\}_{0\leq i\leq k-1}\subseteq \Pa$ is a decomposition of $\gamma$,
\item[\textrm{(c)}]
$\homm(\gamma^{-1})=\homm(\gamma)^{-1}$,
\item[\textrm{(d)}]
$\homm(\gamma)=\homm(\gamma')$ if
$\gamma\csim \gamma'$.
\end{enumerate}
\end{itemize}
\endgroup
\end{definition}
\begin{remark}
\label{rem:homomorphbed}
\begin{enumerate}
\item
Obviously, $\homm\in \HOM$ mimics the algebraic properties of the map $\gamma\mapsto \parall{\gamma}{\w}$ for $\w\in \Con$. Consequently, $\Con$ can canonically be identified with a subset of $\HOM$. Then, one may ask whether there is a natural topology on $\HOM$ for which this subset is even dense.\footnote{This means that for each $\homm\in \HOM$ we find a net $\{\w_\alpha\}_{\alpha\in I}\subseteq \Con$ with $\homm(\gamma)=\lim_\alpha \parall{\gamma}{\w_\alpha}$ for each $\gamma\in \Pa$ in a reasonable sense.} We will see that for compact $S$ the above identification of $\Con$ with a subset of $\HOM$ extends to the spectrum $\A$ in a canonical way, i.e., $\A$ can be identified with a subset of $\HOM$ as well. If the set $\Pa$ has the additional property of independence, see Definition \ref{def:indepref}, this identification even turns out to be bijective, so that in this case $\A\cong \HOM$ holds. Hence, carrying over the topology on $\A$ to $\HOM$, $\Con$ can be considered as a dense subset of $\HOM$ just because $ \iota_\Con(\Con)$ is dense in $\A$. A concrete description of this topology is given in
Definition \ref{def:indepref}.\ref{conv:muetc}.
\item
\label{rem:euklrem5}
Our definition of $\HOM$ differs from the traditional one \cite{Ashtekar2008} in the following two points:
{\bf Decompositions instead of concatenations:}
\noindent
We require $\homm$ to be compatible w.r.t.\ decompositions and not w.r.t.\ concatenations of paths. This avoids unnecessary technicalities as, in the following sections, it will allow us to restrict to embedded analytic curves instead of considering all the piecewise ones. For this observe that both sets give rise to the same $\Cstar$-algebra of cylindrical functions. Hence, define the same quantum configuration space $\A$.
In particular, we get rid of unnecessary redundancies such as the occurrence of curves
$\gamma\colon [0,t]\rightarrow M$ with\footnote{In other words, $\gamma$ is holonomical equivalent to a concatenation of a curve with its inverse.}
\begin{align}
\label{eq:stichstr}
\gamma|_{[s,t]}\csim \big[\gamma|_{[0,s]}\big]^{-1}\text{ for some }s\in (0,t),
\end{align}
i.e., of curves
holonomy equivalent to the constant curve $[0,1]\ni t\mapsto \gamma(0)$. Without going to much into detail, we state that
if $\Pa$ consists of embedded analytic curves and $S$ is connected, then (cf.\ Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt2}):
$\gamma_1 \csim \gamma_2$ for $\gamma_1,\gamma_2\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma_i]=[a_i,b_i]$ for $i=1,2$
{\bf iff} $\gamma_1\isim \gamma_2$, i.e.,
\begin{align*}
\mathrm{im}[\gamma_1]=\mathrm{im}[\gamma_2]\qquad \text{and}\qquad \gamma_1(a_1)=\gamma_2(a_2),\: \gamma_1(b_1)=\gamma_2(b_2)
\end{align*}
{\bf iff}
we find an analytic diffeomorphism $\adif\colon \operatorname{\mathrm{dom}}[\gamma_1] \rightarrow \operatorname{\mathrm{dom}}[\gamma_2]$ with $\gamma_1=\gamma_2\cp \rho$ (we write $\gamma_1 \text{\gls{PSIM}} \gamma_2$ in this case).
In the piecewise embedded analytic situation things are more complicated as there $\csim$ and $\isim$ cannot longer coincide.
Indeed, let $\delta \colon [0,1]\rightarrow M$ be piecewise embedded analytic with $\delta(0)=\delta(1)$ and $\parall{\delta}{\w} \neq \id_{F_{\delta(0)}}$ for some $\w\in \Con$.\footnote{Apply, e.g., Proposition A.1 in \cite{ParallTranspInWebs} in order to construct such a connection.}
\vspace{-25pt}
\hspace{140pt}
\begin{tikzpicture}
\draw[->,line width=1pt] (0,0) .. controls (0,2.5) and (-2.5,0.5) .. (-0.1,0);
\draw (-1.5,1) node {\(\delta\)};
\draw[->,line width=1pt] (0,0) .. controls (0,2.5) and (2.5,0.5) .. (0.1,0);
\draw (1.5,1) node {\(\delta'\)};
\filldraw[black] (0,0) circle (1.7pt);
\end{tikzpicture}
\vspace{9pt}
We choose a curve $\gamma\colon[0,2]\rightarrow M$ with $\gamma|_{[0,1]}=[\gamma|_{[1,2]}]^{-1}=\delta$ and obviously have $\gamma\isim \delta$, but $\parall{\gamma}{\w}=\id_{F_{\delta(0)}}\neq \parall{\delta}{\w}$.
Moreover, even if we first identified piecewise embedded analytic curves which only differ by an insertion or a rejection of a segment $\gamma$ fulfilling \eqref{eq:stichstr}, we also would need non-commutativity of the structure group if we want to replace $\csim$ by some piecewise version of $\psim$. Indeed, let
$\delta$ and $\delta'$ be piecewise embedded analytic as sketched in the picture above. Then, for $\gamma_1,\gamma_2$
piecewise embedded analytic such that $\gamma_1$ first runs through $\delta$, then through $\delta'$, and $\gamma_2$ does it the other way round,
we cannot even find a continuous map $\adif \colon \operatorname{\mathrm{dom}}[\gamma_1]\rightarrow \operatorname{\mathrm{dom}}[\gamma_2]$ such that $\gamma_1=\gamma_2\cp \adif$ holds. Consequently, $\gamma_1\nsim_{\mathrm{par}} \gamma_2$, but we have $\gamma_1\csim \gamma_2$ for abelian $S$ because then
\begin{align*}
\parall{\gamma_1}{\w}=\parall{\delta}{\w}\cp \parall{\delta'}{\w}=\parall{\delta'}{\w}\cp \parall{\delta}{\w}=\parall{\gamma_2}{\w}.
\end{align*}
\vspace{10pt}
{\bf Values in $\boldsymbol{\IsoF}$ instead in the structure group $\boldsymbol{S}$:}
\noindent
If $\nu=\{\nu_x\}_{x\in M}$ is a choice of elements $\nu_x \in F_x$ as described in the beginning of this subsection, then $\HOM$ can be identified with the set $\TRHOM$ of all maps $\hommm\colon \Pa \rightarrow S$ that fulfil
the algebraic properties \textrm{(b)} -- \textrm{(d)} from Definition \ref{def:hompaths}. The corresponding bijection
\begin{align*}
\Omega_\nu\colon \HOM\rightarrow \TRHOM
\end{align*}
is just given by
\begin{align}
\label{eq:ON}
\Omega_\nu(\homm)(\gamma)=(\psi_{\nu_{\gamma(b)}}\cp \homm(\gamma))(\nu_{\gamma(a)})\qquad \text{for}\qquad \operatorname{\mathrm{dom}}[\gamma]=[a,b].
\end{align}
Its inverse is
\begin{align*}
\Omega_\nu^{-1}(\hommm)(\gamma)(p):=\nu_{\gamma(b)}\cdot \hommm(\gamma)\cdot \psi_{\nu_{\gamma(a)}}(p)\qquad \forall\:p\in F_{\gamma(a)}.
\end{align*}
This identification is especially convenient if $P$ is trivial, i.e., if $P=M\times S$. indeed, then we have the canonical choice $\nu_x:=(x,e)$ for all $x\in M$ whereby $\psi_x=\pr_2$.
So, for such a trivial bundle we define
\begin{align}
\label{eq:tricBHoms}
\Omega:=\Omega_\nu\qquad\quad\text{and have}\qquad\quad \TRHOM=\Omega(\HOM),
\end{align}
Hence, $\Omega(\homm)(\gamma)=\pr_2(\homm(\gamma)(\gamma(a),e))$ for $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$.
In the following, we only refer to the spaces $\TRHOM$ in specific applications. This is because the general formulas (and the proofs) are
usually less technical if we work with the spaces $\HOM$.\footnote{The reader who might not believe is encouraged to compare the formulas \eqref{eq:kappadef} and \eqref{eq:algpro}. Moreover, he might write down \eqref{eq:invprop} (and the respective proof) in terms of the space $\Hom(\Pa,S)$ for an arbitrary choice $\{\nu_x\}_{x\in M}$, and then does the same for the projection maps $\pi_p$ we will introduce in Lemma and Definition \ref{def:topo}.\ref{def:topo3}.} In addition to that, we do not need to refer to any choice $\{\nu_x\}_{x\in M}$ in this case.
\hspace*{\fill}$\lozenge$
\end{enumerate}
\end{remark}
We now come to the definitions we need to identify the spaces $\A$ and $\HOM$.
\begin{definition}
\label{def:indepref}
\begin{enumerate}
\item
\label{conv:muetc}
We define the topology $\TOHO$ on $\HOM$ to be generated by the sets of the form
\begin{align}
\label{eq:opensets}
U^{p_1,\dots,p_k}_{\gamma_1,\dots,\gamma_k}(\homm):=\left\{\homm'\in \HOM\:|\: \homm'(\gamma_i)(p_i)\in \homm(\gamma_i)(p_i)\cdot U\quad\forall \: 1\leq i\leq k \right\}.
\end{align}
Here $U$ denotes a neighbourhood of $e\in S$, $\homm\in \HOM$, $k\in \mathbb{N}_{>0}$ and $\Pa \ni \gamma_i\colon [a_i,b_i] \rightarrow M$, $p_i \in F_{\gamma_i(a_i)}$ for $1\leq i\leq k$.
\item
\label{refnbff}
We define independence of a set of curves $\Pa$ as follows.
\begingroup
\setlength{\leftmarginii}{14pt}
\begin{itemize}
\item
\vspace{-8pt}
A refinement of a finite subset $\{\gamma_1,\dots,\gamma_l\}\subseteq \Pa$ is a finite collection $\{\delta_1,\dots,\delta_n\}\subseteq \Pa$ such that for each path $\gamma_j$ we find a decomposition $\{(\gamma_j)_i\}_{1\leq i\leq k_j}$ such that each subcurve $(\gamma_j)_i$ is equivalent to one of the paths $\delta_r$ or $\delta_r^{-1}$ for $1\leq r\leq n$.
\item
\itspacecc
A family $\{\delta_1,\dots,\delta_n\}\subseteq \Pa$ is said to be independent\footnote{Our notion of independence coincides with that of weakly independence introduced in \cite{Ashtekar2008}.} iff for each collection $\{s_1,\dots,s_n\}\subseteq S$
there is $\w\in \Con$ such that (recall \eqref{eq:deltadef}) $h^\nu_{\delta_i}(\w)= s_i$ for all $1\leq i\leq n$. Due to \eqref{eq:indep} this definition does not depend on the explicit choice of $\nu$.
\item
\itspacecc
$\Pa$ is said to be independent iff each finite collection $\{\gamma_1,\dots,\gamma_l\}\subseteq \Pa$
admits an independent refinement.\footnote{In particular, $\Pa$ cannot contain constant curves.}
\end{itemize}
\endgroup
\item
\label{def:mapkappa}
Let $S$ be compact. For $\qw\in \A$ and $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ we let
\begin{align}
\label{eq:dfdcccccf}
\parall{\gamma}{\qw}(p):=\lim_\alpha \parall{\gamma}{\w_\alpha}(p) \qquad \forall\: p\in F_{\gamma(a)}
\end{align}
for $\Con\supseteq \{\w_\alpha\}_{\alpha\in I}\rightarrow \qw$ an approximating net, and define
\begin{align}
\begin{split}
\label{eq:kappadef}
\text{\gls{KAPPA}}\colon \ovl{\Con} &\rightarrow \HOM\\
\ovl{\w}&\mapsto \left[\gamma \mapsto \parall{\gamma}{\qw}\right].
\end{split}
\end{align}
\end{enumerate}
\end{definition}
\begin{remark}
Obviously, \eqref{eq:dfdcccccf} just means to define the generalized parallel transport function that corresponds to $\ovl{\w}$ as a limit of the classical parallel transports w.r.t.\ the elements $\w_\alpha$ approximating $\ovl{\w}$. This is the coordinate free description of the map $\kappa$. A more concrete formula involving a choice of $\{\nu_x\}_{x\in M}$ as well as a choice of a faithful matrix representation $\rho$ of $S$ is provided in the next lemma adapting the standard results \cite{Ashtekar2008} to our framework and shows that \eqref{eq:kappadef}, hence \eqref{eq:dfdcccccf} is well defined.
\end{remark}
\begin{lemma}
\label{lemma:homeotopo}
Let $S$ be compact and $\rho$ a faithful matrix representation of $S$. Moreover, let $\{\nu_x\}_{x\in M}\subseteq P$ be a family of elements with $\nu_x\in F_x$ for all $x\in M$.
\begin{enumerate}
\item
\label{lemma:SpeczuHommm}
The map \eqref{eq:kappadef} is well defined and injective. It is surjective if $\Pa$ is independent. Moreover, the following formula holds for $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ and $[h_\gamma^\nu]_{ij}=\rho_{ij}\cp h_\gamma^\nu$ as in \eqref{eq:gniij}:
\begin{align}
\label{eq:algpro}
\kappa(\ovl{\w})(\gamma)(p)=\nu_{\gamma(b)}\cdot \rho^{-1}\left(\left(\ovl{\w}\left([h^\nu_\gamma]_{ij}\right)\right)_{ij}\right)\cdot \psi_{\gamma(a)}(p)\qquad \forall \: p\in F_{\gamma(a)}.
\end{align}
\item
\label{it:ddd}
If $\Pa$ is independent, then $\TOHO$ is the unique topology making $\kappa$ a homeomorphism.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
$\rho$ is a homeomorphism to $B:=\mathrm{im}[\rho]\subseteq \GLNC$ since $S$ is compact and $B$ is Hausdorff.
Then, by compactness of $B$ we have
\begin{align*}
\lim_\alpha\big(\iota_\Con(w_\alpha)([h^\nu_\gamma]_{ij})\big)_{ij}=(\ovl{\w}([h^\nu_\gamma]_{ij})_{ij}\in B
\end{align*}
For this, observe that $I$ is directed and that by the definition of the Gelfand topology on $\A$ for each $\epsilon>0$ and all $1\leq i,j\leq n$ we find $\alpha_{ij}\in I$ with
\begin{align*}
\big\|\qw\left([h^\nu_\gamma]_{ij}\right)-[h^\nu_\gamma]_{ij}(\w_\alpha)\big\|<\epsilon\qquad \forall\: \alpha\geq \alpha_{ij}.
\end{align*}
So, using continuity of $\rho^{-1}$ and that of the right multiplication in the bundle $P$, we obtain
\begin{align*}
\begin{split}
\kappa(\ovl{\w})(p)&=
\parall{\gamma}{\qw}(p)=\lim_\alpha \parall{\gamma}{\w_\alpha}(p)\\
& =\lim_\alpha \nu_{\gamma(b)}\cdot \psi_{\gamma(b)}\left(\parall{\gamma}{\w_\alpha}(\nu_{\gamma(a)})\right)\cdot \psi_{\gamma(a)}(p)\\
&=\lim_\alpha \nu_{\gamma(b)}\cdot h_\gamma^\nu(\w_\alpha)\cdot \psi_{\gamma(a)}(p)\\
&=\lim_\alpha\nu_{\gamma(b)}\cdot \rho^{-1}\left(\left(\iota_\Con(\w_\alpha)\left([h^\nu_\gamma]_{ij}\right)\right)_{ij}\right)\cdot \psi_{\gamma(a)}(p)\\
&=\nu_{\gamma(b)}\cdot \rho^{-1}\left(\left(\ovl{\w}\left([h^\nu_\gamma]_{ij}\right)\right)_{ij}\right)\cdot \psi_{\gamma(a)}(p),
\end{split}
\end{align*}
hence \eqref{eq:algpro}.
In particular, this shows that the limit $\lim_\alpha \parall{\gamma}{\w_\alpha}(p)$ exists and is independent of the choice of the net $\{\w_\alpha\}_{\alpha\in I}$.
Using the same continuity arguments, we conclude that $\kappa(\ovl{\w})\in \HOM$ from $\kappa\left(\mathrm{im}[\iota_\Con]\right)\subseteq \HOM$.
Let $\dD\subseteq \PaC$ denote the dense unital $^*$-subalgebra of $\PaC$ generated by the functions $[h^\nu_\gamma]_{ij}$.
If $\kappa(\ovl{\w}_1)=\kappa(\ovl{\w}_2)$ for $\ovl{\w}_1,\ovl{\w}_2\in \mathrm{Spec}(\PaC)$, then $\ovl{\w}_1|_\dD=\ovl{\w}_2|_\dD$ by \eqref{eq:algpro}, hence $\ovl{\w}_1=\ovl{\w}_2$ by continuity of this maps. This shows injectivity of $\kappa$.
Now, if $\Pa$ is independent and $\homm\in \HOM$, we define the corresponding preimage $\qw_\homm\in \A$ as follows. Let $[h^\nu_\gamma]_{ij}$ denote the complex conjugate of the function $[h^\nu_\gamma]_{ij}$ and define
\begin{align*}
\qw_\homm(1):=1\qquad\quad
\qw_\homm([h^\nu_\gamma]_{ij}):=\rho_{ij}\cp \psi_{\nu_{\gamma(b)}}\cp\homm(\gamma)(\nu_{\gamma(a)}) \quad\qquad \qw_\homm([h^\nu_\gamma]^*_{ij}):=\ovl{\qw_\homm([h^\nu_{\gamma}]_{ji})}.
\end{align*}
For $f\in \dD$ choose a representation as a sum of products of the form
\begin{align*}
F=\sum_l \lambda_l\cdot h_{l,1}\cdot {\dots}\cdot h_{l,n(l)}
\end{align*}
where $\lambda_l \in \mathbb{C}$ and each
$h^l_{k}$ equals $1$, a generator $[h^\nu_\gamma]_{ij}$ or the complex conjugate $[h^\nu_\gamma]^*_{ij}$ of a generator. We assign to $f$ the value
\begin{align*}
\qw_\homm(F):=\sum_l\lambda_l\cdot \qw_\homm(h_{l,1})\cdot {\dots}\cdot \qw_\homm(h_{l,n(l)}).
\end{align*}
Obviously, $\qw_\homm$ is a $^*$-homomorphism if it is well defined and, it extends (by linearity) to an element of $\mathrm{Spec}(\PaC)$ if it is continuous. For well-definedness assume that $F_1$,$F_2$ are two representations of $f$,
and denote by $[\gamma_1],\dots, [\gamma_m]$ the equivalence (holonomy equivalence) classes of the paths that occur in both expressions.
Let $\{\delta_1,\dots,\delta_n\}\subseteq \Pa$ be a refinement
of $\{\gamma_1,\dots,\gamma_m\}$ and $\w\in\Con$ with\footnote{For simplicity, assume that $\operatorname{\mathrm{dom}}[\delta_r]=[0,1]$ for $1\leq r\leq n$.} $h^\nu_{\delta_r}(\w)=\psi_{\nu_{\delta_r(1)}}\homm(\delta_r)(\nu_{\delta_r(0)})$ for $1\leq r\leq m$. Then it follows from the algebraic properties of parallel transports and $\homm$ that
\begin{align*}
\qw_\homm(F_1)=F_1(\w)=f(\w)=F_2(\w)=\qw_\homm(F_2).
\end{align*}
This shows well-definedness, and continuity now follows from
$|\qw_\homm(f)|=|f(\w)|\leq \|f\|_\infty$.
By construction we have $\kappa(\qw_\homm)=\homm$, which shows surjectivity of $\kappa$.
\item
The space $\HOM$ equipped with $\TOHO$ is obviously Hausdorff. So, since $\A$ is compact and $\kappa$ is bijective, we only have to show that $\kappa$ is continuous. For this, let $\A \supseteq\{\ovl{\w}_\alpha\}_{\alpha\in I}\rightarrow \qw \in\A$ be a converging net, $\Pa\ni\gamma\colon [a,b]\rightarrow M$, $p\in F_{\gamma(a)}$ and $U\subseteq S$ a neighbourhood of $e\in S$. Then, by \eqref{eq:algpro} we have
\begin{align*}
F_{\gamma(b)}\ni \kappa(\ovl{\w}_\alpha)(\gamma)(p) &\stackrel{\eqref{eq:algpro}}{=} \nu_{\gamma(b)}\cdot \rho^{-1}\left(\left(\ovl{\w}_\alpha\left([h^\nu_\gamma]_{ij}\right)\right)_{ij}\right)\cdot \psi_{\gamma(a)}(p)\\
& \hspace{3.2pt}\rightarrow \:\:\nu_{\gamma(b)}\cdot \rho^{-1}\left(\left(\ovl{\w}\left([h^\nu_\gamma]_{ij}\right)\right)_{ij}\right)\cdot \psi_{\gamma(a)}(p)
= \kappa(\ovl{\w})(\gamma)(p)\in F_{\gamma(b)},
\end{align*}
so that $\{\kappa(\ovl{\w}_\alpha)(\gamma)(p)\}_{\alpha\in I}$ converges in $F_{\gamma(b)}$ to $\kappa(\ovl{\w})(\gamma)(p)\in F_{\gamma(b)}$. Consequently, we find $\alpha_0 \in I$ such that
\begin{align*}
\kappa(\ovl{\w}_\alpha)(\gamma)(p)\in \kappa(\ovl{\w})(\gamma)(p)\cdot U\qquad \forall\: \alpha\geq \alpha_0,
\end{align*}
hence $\kappa(\ovl{\w}_\alpha)\in U_{\gamma}^p(\kappa(\ovl{\w}))$ for all $\alpha\geq \alpha_0$.
Since $U_{\gamma_1,\dots,\gamma_k}^{p_1,\dots,p_k}(\homm)=U_{\gamma_1}^{p_1}(\homm)\cap \dots \cap U_{\gamma_k}^{p_k}(\homm)$ and $I$ is directed, the claim follows.
\end{enumerate}
\end{proof}
\subsubsection{Invariant Homomorphisms}
\label{subsub:InvHoms}
We now use the identification of $\A$ with $\HOM$ in order to obtain a more concrete description of the space $\AQR$.
\begin{definition}[Invariant Homomorphisms]
\label{def:invhomm}
Let $\Pa$ be $\Phi$-invariant and independent.
We define the space of $\Phi$-invariant homomorphisms by
$\text{\gls{IHOM}}:=\kappa\big(\AQR\big)$ and equip it with the subspace topology w.r.t.\ $\HOM$.\footnote{Since $\kappa$ is a homeomorphisms, this is just the topology carried over from $\AQR$ by $\kappa$, cf. Remark \ref{rem:subspt}.}
\end{definition}
\begin{lemma}
\label{lemma:ShapeInvHomms}
Let $\Pa$ be $\Phi$-invariant and independent and $\homm\in \HOM$.
Then $\homm\in \IHOM$ iff
\begin{align}
\label{eq:invprop}
\homm(\wm_g\cp \gamma)(\Phi_g(p))=\Phi_g( \homm(\gamma)(p))\qquad \forall\: g\in G,\forall\:\gamma\in \Pa,\forall\:p\in F_{\gamma(a)},\;
\end{align}
where $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$.
\begin{proof}
Observe that $\homm\in \IHOM$ iff $\kappa(\Theta_g(\kappa^{-1}(\homm)))=\homm$
for all $g\in G$, i.e., iff
\begin{align*}
\kappa(\Theta_g(\kappa^{-1}(\homm)))(\gamma)(p)=\homm(\gamma)(p)\qquad \forall\: \gamma\in \Pa,\:\forall\:g\in G.
\end{align*}
So, let $\{\w_\alpha\}_{\alpha\in I}\subseteq \Con$ be a net with $\{\iota_\Con(\w_\alpha)\}_{\alpha\in I}\rightarrow \kappa^{-1}(\homm)$.
Then from
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item[a)]
continuity of $\specw_g$ by Corollary \ref{cor:CylSpecAction},
\item[b)]
$\specw_g\cp \iota_\Con = \iota_\Con \cp \cw_g$ by Corollary \ref{cor:CylSpecAction},
\item[c)]
$\parall{\gamma}{\cw(g,\w)}(p)=\Phi_g \!\left(\parall{g^{-1}\gamma}{\w}(\Phi_{g^{-1}}(p))\right)$ for $g^{-1}\gamma:=\wm_{g^{-1}}\cp \gamma$,\footnote{See, e.g., \eqref{eq:patra} in the proof of Lemma \ref{lemma:CylSpecActionvorb}.}
\item[d)]
continuity of $\Phi$ and
\item[e)]
the definition of $\kappa$
\end{itemize}
\endgroup
\noindent
we obtain
that
\begin{align}
\label{eq:InvHomTau}
\begin{split}
\kappa\big((\specw_g\cp \kappa^{-1})(\homm) \big)(\gamma)(p) &\stackrel{\mathrm{a)}}{=}\kappa\left(\lim_\alpha \big(\Theta_g\cp \iota_\Con\big)(\w_\alpha)\right)(\gamma)(p)\\%\nonumber\\
&\stackrel{\mathrm{b)}}{=}\kappa\left(\lim_\alpha (\iota_\Con\cp\cw_g)(\w_\alpha)\right)(\gamma)(p)\stackrel{\mathrm{e)}}{=}\lim_\alpha \parall{\gamma}{\cw_g(\w_\alpha)}(p)\\
&\stackrel{\mathrm{c)}}{=}\lim_\alpha \Phi_g\!\left(\parall{g^{-1}\gamma}{\w_\alpha}\left(\Phi_{g^{-1}}(p)\right)\right)\stackrel{\mathrm{d)}}{=}\Phi_g\!\left(\lim_\alpha\parall{g^{-1}\gamma}{\w_\alpha}\left(\Phi_{g^{-1}}(p)\right)\right)\\%\nonumber\\
&\stackrel{\mathrm{e)}}{=}\Phi_g\Big(\homm\!\left(g^{-1}\gamma\right)\left(\Phi_{g^{-1}}(p)\right)\!\Big)
\end{split}
\end{align}
Replacing $g$ by $g^{-1}$, we see that $\homm\in \IHOM$ iff for all $g\in G$, $\gamma\in \Pa$ and all $p\in F_{\gamma(a)}$ we have $\Phi_g(\homm(\gamma)(p))=\homm(\wm_g\cp \gamma)(\Phi_g(p))$.
\end{proof}
\end{lemma}
In the following let $\Pa$ be $\Phi$-invariant and independent.
\begin{remark}[Invariance up to Gauge Transformations]
\label{rem:homaction}
\begin{enumerate}
\item
\label{rem:euklrem4}
\vspace{6pt}
Instead of $\AQR\cong\IHOM$, one also can consider the space $\RedGauge$ of generalized connections that are invariant up to gauge transformations. This is the set of elements $\homm\in \HOM$ which fulfil that for each $g\in G$ there is a generalized gauge transformation\footnote{This just means that $\pi \cp\sigma =\pi$ and $\sigma(p\cdot s)=\sigma(p)\cdot s$ for all $\in S$.} $\sigma\colon P\rightarrow P$ with $\specw_g(\homm)=\sigma^*(\homm)$, i.e.,
\begin{align*}
\specw_g(\homm)(\gamma)(p)=(\sigma \cp \homm(\gamma))(\sigma^{-1}(p))\qquad \forall\: \gamma\in \Pa, \forall\:p\in F_{\gamma(a)}
\end{align*}
with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$. Obviously, $\AQR\cong \IHOM\subseteq \RedGauge$, and usually $\RedGauge$ would be seen as the more physical space. However, in this thesis we want to concentrate on the spaces $\AQR$ and $\ARQ$ just because we rather expect technical than conceptual difficulties when carrying over the current developments to the ``up to gauge'' case.
\item
\label{rem:euklrem55}
In Remark \ref{rem:homomorphbed}.\ref{rem:euklrem5} we have seen that each family $\nu=\{\nu_x\}_{x\in M}\subseteq P$ of elements with $\nu_x\in F_x$ for all $x\in M$ allows to identify the spaces $\HOM$ and $\TRHOM$ via the bijection
\begin{align*}
\Omega_\nu\colon \HOM\rightarrow \TRHOM
\end{align*}
defined by \eqref{eq:ON}.
Then, composing $\Omega_\nu$ with the bijection $\kappa \colon \A\rightarrow \HOM$ from Definition \ref{def:indepref}.\ref{def:mapkappa}, we obtain from \eqref{eq:ON} and \eqref{eq:algpro} that
\begin{align}
\label{eq:hilfseq}
\begin{split}
(\Omega_\nu\cp \kappa)(\ovl{\w})(\gamma)&=\rho^{-1}\left(\left(\ovl{\w}([h_\gamma^\nu]_{ij})\right)_{ij}\right)
\end{split}
\end{align}
for $\rho$ is a faithful matrix representation of $S$.
\item
For $P$ a trivial principal fibre bundle we extend \eqref{eq:tricBHoms} to
\begin{align*}
\Omega:=\Omega_\nu\quad\:\: \TRHOM=\Omega(\HOM) \quad\:\: \ITRHOM:=\Omega(\IHOM)
\end{align*}
for $\nu$ the canonical choice $\nu_x=(x,e)$ for all $x\in M$. Recall that then (see Remark \ref{rem:homomorphbed}.\ref{rem:euklrem5})
\begin{align}
\label{eq:trivp}
\Omega(\homm)(\gamma)=\pr_2(\homm(\gamma)(\gamma(a),e))\qquad\text{for $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ holds}.
\end{align}
The spaces $\ITRHOM$ will only occur in concrete examples in the following.\hspace*{\fill}$\lozenge$
\end{enumerate}
\end{remark}
Fixing $\nu$, one can always try to evaluate the invariance condition \eqref{eq:invprop} in terms of the spaces $\TRHOM$. However, if $\Phi$ is too complicated, even an appropriate choice of $\nu$ will give rise to cumbersome conditions. This is one of the reasons why we prefer to deal with the space $\HOM$ instead of $\TRHOM$ in this work.
Nevertheless, the next example shows that in our LQC prime examples (Example \ref{ex:LQC}) the action $\Phi$ is simple enough to give rise to very natural invariance conditions in terms of the space $\Hom(\Pa,\SU)$. Basically, this is the case because for each of the groups $\Ge$, $\Gi$, $\Gh$ we have that
$\pr_2(g\cdot (x,\me))$ does not depend on the base point $x$, i.e., the effect of a group element on the fibres is constant in a certain sense.
\begin{example}[Loop Quantum Cosmology]
\label{rem:euklrem6}
Let $P=\RR^3\times \SU$, and $G:=\Ge$, $\Phi:=\Pe$ be defined as in Example \ref{ex:LQC}. Moreover, assume that $\Pa$ is independent, $\Pe$-invariant and closed under decomposition and inversion of its elements.
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\label{rem:euklrem61}
We have $g^{-1}=(-\uberll{\sigma^{-1}}{v},\sigma^{-1})$ for $g=(v,\sigma)\in \Ge$.
Moreover, if $\gamma\in \Pa$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ and $\homm\in \IHOM$, then \eqref{eq:invprop} and \eqref{eq:trivp} give
\begin{align*}
\Omega(\homm)(\wm_g\cp\gamma)& \stackrel{\eqref{eq:trivp}}{=} \pr_2(\homm(\wm_g\cp \gamma)((\wm_g(\gamma(a)),\me)))\stackrel{\eqref{eq:invprop}}{=}(\pr_2\cp\Phi_g)\big(\homm(\gamma)(\Phi_{g^{-1}}((\wm_g(\gamma(a)),\me)))\big).
\end{align*}
The left hand side equals $\Omega(\homm)(v+\sigma(\gamma))$ and for the right hand side we obtain
\begin{align*}
(\pr_2\cp\Phi_g)\big(\homm(\gamma)(\Phi_{g^{-1}}((\wm_g(\gamma(a)),\me)))\big)&=(\pr_2\cp\Phi_g)\big(\homm(\gamma)\big(\big(\gamma(a),\sigma^{-1}\big)\big)\\
&=\sigma\cdot \pr_2\big(\homm(\gamma)\big(\big(\gamma(a),\sigma^{-1}\big)\big)\\
&=\sigma\cdot \pr_2\big(\homm(\gamma)\big(\big(\gamma(a),\me\big)\big)\cdot \sigma^{-1}\\
&= \alpha_\sigma(\Omega(\homm)(\gamma)).
\end{align*}
It follows that $\hommm \in \Hom_\red(\Pa,\SU)$ iff
\begin{align}
\label{eq:InvGenConnRel}
\hommm(v+\uberll{\sigma}{\gamma)}=(\Co{\sigma}\cp \hommm)(\gamma)\qquad \forall\:(v,\sigma)\in \Ge, \forall\:\gamma \in \Pa
\end{align}
holds.
In particular, this means that the value of $\hommm$ is independent of the starting point of the path $\gamma$. In the same way, we see that for the spherically symmetric and \mbox{(semi-)homogeneous} cases we have
\begin{align}
\begin{array}{lrclcl}
\label{eq:algrels}
\Gi \colon & \hommm(\uberll{\sigma}{\gamma})\!\!\!&=&\!\!\!\Co{\sigma}\cp \hommm(\gamma) && \forall\:\sigma\in \SU,\forall\:\gamma \in \Pa, \\
\Gh\colon& \hommm(v+\gamma)\!\!\! & =&\!\!\! \hommm(\gamma)&& \forall\:v\in \RR^3,\forall\:\gamma \in \Pa \\
G_{SH}\colon& \hommm(L(v)+\gamma)\!\!\! & =&\!\!\! \hommm(\gamma)&& \forall\:v\in \RR^2\subseteq \RR^3,\forall\:\gamma \in \Pa
\end{array}
\end{align}
for $\Gi$, $\Gh$ and $G_{SH}$ (semi-homogeneous) defined as in Example \ref{ex:LQC}, and $L\colon \RR^2 \rightarrow \RR^3$ an injective linear map.
\item
\label{rem:euklrem62}
It already follows from \eqref{eq:InvGenConnRel} that for $\Ge$ we have $\hommm(x+\gamma_{\vec{v},l})\in H_{\vec{v}}$ for all $\hommm\in \Hom_\red(\Pa,\SU)$ and the straight lines
$x+ \gamma_{\vec{v},l}\in \Pal$ defined as in Convention \ref{conv:sutwo1}.\ref{conv:sutwo2}.
In fact, obviously $\sigma(\gamma_{\vec{v},l})=\gamma_{\vec{v},l}$ holds for each $\sigma\in H_{\vec{v}}$ just because $\sigma$ rotates around the velocity vector $\vec{v}$ of the linear curve $\gamma_{\vec{v},l}$. Then, \eqref{eq:InvGenConnRel} shows
\begin{align*}
\hommm(x+\gamma_{\vec{v},l})=\hommm(x+\sigma(\gamma_{\vec{v},l}))\stackrel{\eqref{eq:InvGenConnRel}}{=}\Co{\sigma}\cp \hommm(\gamma_{\vec{v},l})\stackrel{\eqref{eq:InvGenConnRel}}{=}\Co{\sigma}\cp \hommm(x+\gamma_{\vec{v},l})\qquad \forall\: \sigma\in H_{\vec{v}},
\end{align*}
so that, choosing $\sigma\neq \pm\me$, Lemma \ref{lemma:torus}.\ref{lemma:torus1} implies that $\hommm(x+ \gamma_{\vec{v},l})\in H_{\vec{v}}$.
Obviously, then for $\Gi$ the same statement holds if $x\in \Span_\RR(\vec{v})$. \hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\end{example}
The next remark collects some straightforward consequences of the case where $\Pa$ naturally decomposes into independent, $\Phi$-invariant subsets being closed under decomposition and inversion of their elements. For this, we will need
\begin{convention}
\label{conv:invhomm}
If the set of curves \gls{PALPHA} carries the index $\alpha$, then (in reference to Convention \ref{hgamma})
we denote the respective quantum spaces by
\begin{align*}
\text{\gls{ARRQALPHA}}\qquad\qquad\quad \text{\gls{ARQALPHA}} \qquad\qquad\quad \text{\gls{AQRALPHA}}\qquad\qquad\quad \text{\gls{AALPHA}}.
\end{align*}
Moreover, we denote the corresponding bijection from Definition \ref{def:indepref}.\ref{def:mapkappa} by
\begin{align*}
\text{\gls{KAPPAALPHA}}\colon \A_\alpha\rightarrow \text{\gls{HOMALPHA}}\qquad\text{and define}\qquad\text{\gls{IHOMALPHA}}:=\kappa_\alpha\big(\AQRInd{\alpha}\big).
\end{align*}
\end{convention}
\begin{remark}[Restricting Homomorphisms]
\label{rem:restriction}
Let $\Pay \subseteq \Pax$ both be independent, $\Phi$-invariant and closed under decomposition and inversion. Then we have the following commutative diagram:
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
&\mathrm{Spec}\big(\raisebox{0pt}{$\ovl{\PaCx|_{\AR}}$}\big)=\hspace{-25pt} &\ARRQInd{0}\ar@{->}[r]^-{\ovl{i^{*}_{\AR}}}_{\cong}\ar@{->}[d]_-{\ovl{\upsilon_{\alpha 0}}} & \ARQInd{0}\ar@{->}[r]^-{\subseteq}& \AQRInd{0} \ar@{->}[d]^-{\res'_{0\alpha}} \ar@{->}[r]^-{\subseteq} & \AInd{0}\ar@{->}[d]^-{\res'_{0\alpha}} \ar@{->}[r]^-{\kappa_0}_-{\cong} & \HOMInd{0}\ar@{->}[d]^-{\res_{0\alpha}}\\
&\mathrm{Spec}\big(\raisebox{0pt}{$\ovl{\PaCy|_{\AR}}$}\big)=\hspace{-25pt}&\ARRQInd{\alpha}\ar@{->}[r]^-{\ovl{i^*_{\AR}}}_{\cong} & \ARQInd{\alpha}\ar@{->}[r]^-{\subseteq}& \AQRInd{\alpha}\ar@{->}[r]^-{\subseteq}&\AInd{\alpha}\ar@{->}[r]^-{\kappa_\alpha}_-{\cong}& \HOMInd{\alpha},
}
\end{xy}
}
\end{center}
where we have omitted the indices $0$ and $\alpha$ at the maps $\ovl{i^*_{\AR}}$. We have used the following maps:
\vspace{5pt}
\begin{tabular}{rrlll}
$\upsilon_{\alpha 0}\colon$&\hspace{-12pt}$\ovl{\PaCy|_{\AR}}$ &\hspace{-8pt} $\hookrightarrow$ & \hspace{-8pt} $\ovl{\PaCx|_{\AR}}$ \quad & \hspace{-12pt} \text{--}\:\: The canonical inclusion. \\
$\ovl{\upsilon_{\alpha0}}\colon$&\hspace{-12pt}$\ARRQInd{0}$ &\hspace{-8pt} $\rightarrow$ & \hspace{-8pt} $\ARRQInd{\alpha}$\quad & \hspace{-12pt} \text{--}\:\: The corresponding map from Lemma \ref{lemma:homzuspec}.\ref{lemma:homzuspec1}. \\
$\res_{0\alpha}\colon$&\hspace{-12pt}$\homm$ & \hspace{-8pt} $\mapsto$ & \hspace{-8pt} $\homm|_{\Pay}$\quad & \hspace{-12pt} \text{--}\:\: The restriction of the elements of $\HOMInd{0}$ to $\Pay$.
\end{tabular}
\vspace{5pt}
\noindent
Moreover, $\res'_{0\alpha}:= \kappa_\alpha^{-1}\cp \res_{0\alpha} \cp \kappa_0$ is
$\res'_{0\alpha}$ is continuous because $\kappa_0$ is a homeomorphism by Lemma \ref{lemma:homeotopo}.\ref{it:ddd}, and since $\res_{0\alpha}$ is continuous just by the definition of the topologies on $\HOMInd{0}$ and $\HOMInd{\alpha}$.
Commutativity of the above diagram now follows from the definitions of $\kappa_0$ and $\kappa_\alpha$, \eqref{eq:inclusionsdiag}, as well as the fact that $\ovl{\upsilon_{\alpha0}}$ just assigns to $\psi\in \ARRQInd{0}$ the restriction $\psi|_{\ovl{\PaCy|_{\AR}}}$. Indeed, the last two points give
\begin{align*}
\iota_{\Con}\cp i_{\AR}\:\stackrel{\star}{=}\:\ovl{i^*_\AR}\cp \iota_{\AR}\qquad\text{and} \qquad \iota^\alpha_{\AR}\:\stackrel{\star\star}{=}\:\ovl{\upsilon_{\alpha0}}\cp \iota^0_{\AR},
\end{align*}
respectively, and then commutativity is
easily checked on the dense subset $\iota_{\AR}(\AR)\subseteq \ARRQInd{0}$. For this, let $\Pay\ni \gamma\colon [a,b]\rightarrow M$ and $p\in F_{\gamma(a)}$. Then, for $\w\in \AR$ we have
\begin{align*}
\Big(\res_{0\alpha}\cp \kappa_0\cp \ovl{i^*_\AR}\Big)&\Big(\iota^0_{\AR}(\w)\Big)(\gamma)(p)\\
& \stackrel{\eqref{eq:inclusionsdiag}}{=}\kappa_0\big(\big(\iota^0_\Con\cp i_{\AR}\big)(\w)\big)(\gamma)(p)\stackrel{\eqref{eq:kappadef}}{=}\parall{\gamma}{i_{\AR}(\w)}(p)
\stackrel{\eqref{eq:kappadef}}{=}\kappa_\alpha\big(\big(\iota^\alpha_\Con\cp i_{\AR}\big)(\w)\big)(\gamma)(p)\\
&
\stackrel{\star}{=}
\kappa_\alpha\big(\big(\ovl{i^*_\AR}\cp\iota^\alpha_{\AR}\big)(\w)\big)(\gamma)(p)
\stackrel{\star\star}{=}\kappa_\alpha\big(\big(\ovl{i^*_\AR}\cp \ovl{\upsilon_{\alpha0}} \cp\iota^0_{\AR}\big)(\w)\big)(\gamma)(p)\\
&\hspace{3pt}=\:\hspace{1pt}
\big(\kappa_\alpha\cp \ovl{i^*_\AR}\cp \ovl{\upsilon_{\alpha0}} \big)\big(\iota^0_{\AR}(\w)\big)(\gamma)(p).
\end{align*}
Here, the superscripts $0$ and $\alpha$ hint to the fact that $\iota_\Con^0$ and $\iota_\Con^\alpha$ map into different spectra.
\hspace*{\fill}$\lozenge$
\end{remark}
Remark \ref{rem:restriction} provides us with the following
\begin{proposition}
\label{rem:euklrem2b}
Assume that we have $\Pax=\bigsqcup_{\alpha\in I}\Pa_\alpha$ for non-empty sets $\Pa_\alpha$ which are independent, $\Phi$-invariant and closed under decomposition and inversion. Then, $\IHOMInd{0}\cong \prod_{\alpha\in I}\IHOMInd{\alpha}$
via the map
\begin{align*}
\Xi^0_{I}\colon \IHOMInd{0}&\rightarrow \prod_{\alpha\in I}\IHOMInd{\alpha},\\
\qw &\mapsto \prod_\alpha \res_{0\alpha}(\qw).
\end{align*}
In particular, if each $\IHOMInd{\alpha}$ carries a normalized Radon measures $\mu_\alpha$, then we obtain a normalized Radon measure $\mu$ on $\IHOMInd{0}$ just by
\begin{align*}
\mu:=\big(\Xi^0_{I}\big)^{-1}\!\left(\textstyle\prod_{\alpha\in I}\mu_\alpha\right)
\end{align*}
for $\prod_{\alpha\in I}\mu_\alpha$ the Radon product measure on $\prod_{\alpha\in I}\IHOMInd{\alpha}$ from Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa3}.
\end{proposition}
\begin{proof}
The map $\Xi^0_{I}$ is obviously continuous and injective. Moreover, it is surjective because if $\homm_\alpha\in \IHOMInd{\alpha}$ for all $\alpha\in I$,
then $\homm(\gamma):= \homm_\alpha(\gamma)$ for $\gamma\in \Pa_\alpha$
is a well-defined element of $\IHOMInd{0}$, just by the properties of the sets $\Pa_\alpha$.
\end{proof}
We close this section with some investigations concerning
the measure theoretical aspects of the reduced spaces $\ARQ$ and $\AQR$.
The final remark then contains an outlook of the next sections.
Assume that $M$ is analytic, $\wm$ is an analytic action, $S$ is compact and connected and \gls{PAW} the set of embedded analytic curves in $M$. Moreover, assume that $\Pa\subseteq \Paw$ is $\Phi$-invariant\footnote{Since $\wm$ is analytic, one can always choose $\Pa=\Paw$.} and closed under decomposition and inversion of its elements. As we will see in Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt3}, then $\Pa$ is independent. We recall the
Ashtekar-Lewandowski measure\footnote{This will be a special case of the construction in Subsection \ref{sec:FreeM}, see Definition \ref{def:AshLewx}.} \cite{ProjTechAL} \gls{mAL}
on $\A$
being characterized by the following property:
\vspace{5pt}
\noindent
Let $\alpha=(\gamma_1,\dots,\gamma_k)$ for $\gamma_1,\dots,\gamma_k \in \Pa_\w$ be a finite subset such that\footnote{We write $\gamma \text{\gls{CPSIM}} \gamma'$ iff there are open intervals $I\subseteq \operatorname{\mathrm{dom}}[\gamma]$ and $I'\subseteq \operatorname{\mathrm{dom}}[\gamma']$ such that $\gamma(I)=\gamma'(I')$ holds, see Definition \ref{def:analytLieAlgBD}.\ref{def:analytLieAlgBD1}.}
$\gamma_i \nsim_\cp \gamma_j$ holds for all $1\leq i\neq j\leq k$, and where
$\operatorname{\mathrm{dom}}[\gamma_i]=[a_i,b_i]$ for $i=1,\dots,k$.
Then, the push forward of $\mAL$ by the map
\begin{align}
\label{projm}
\begin{split}
\pi_\alpha\colon \A&\rightarrow S^k\\
\ovl{\w}&\mapsto \big(\psi_{\gamma_1(b_1)}\big(\kappa(\ovl{\w})(\gamma_1)(\nu_{\gamma_1(a_1)})\big),\dots,\psi_{\gamma_k(b_k)}\big(\kappa(\ovl{\w})(\gamma_k)(\nu_{\gamma_k(a_k)})\big)\big)
\end{split}
\end{align}
equals the Haar measure on $S^{|\alpha|}$.
\vspace{5pt}
\noindent
A closer look at the invariance property \eqref{eq:invprop} gives
\begin{lemma}
\label{cor:AshtLew}
If $\dim[S]\geq 1$ and $\gamma\nsim_\cp \wm_g\cp\gamma$ holds for some $\gamma\in \Paw$ and $g\in G$, then the Borel sets $\AQR, \ARQ\subseteq \A$ are of measure zero w.r.t. $\mAL$.
\begin{proof}
The subsets $\AQR, \ARQ$ are Borel sets as they are compact by Corollary \ref{cor:CylSpecAction}. Since by the same Corollary we have $\ARQ\subseteq \AQR$, it suffices to show that $\mAL(\AQR)=0$. Now,
\begin{align*}
\mAL\!\left(\AQR\right)\leq \mAL\!\left(\pi_\alpha^{-1}\big(\pi_\alpha\big(\AQR\big)\big)\right)=\mu_\alpha(\pi_\alpha\big(\AQR\big))\qquad \forall\:\alpha\in I,
\end{align*}
and for $\alpha:=(\gamma,\wm_g\cp \gamma)$ we find $d,b\in S$ such that $B:=\pi_\alpha\big(\AQR\big)=\{(s,c\cdot s\cdot d)\:|\: s\in S\}$ holds.
In fact, for $\qw\in \AQR$ and $\homm:=\kappa(\qw)$ we have
\begin{align}
\label{eq:invpi}
\begin{split}
\pi_{\wm_g\cp \gamma}(\qw)&=\psi_{\wm_g(\gamma(b))}\big(\homm(\wm_g\cp\gamma)(\nu_{\wm_g(\gamma(a))})\big)\stackrel{\eqref{eq:invprop}}{=}\psi_{\wm_g(\gamma(b))}\big(\Phi_g\cp \homm(\gamma)\big(\Phi_{g^{-1}}(\nu_{\wm_g(\gamma(a))})\big)\big)\\
&=\underbrace{\psi_{\wm_g(\gamma(b))}(\Phi_g(\nu_{\gamma(b)}))}_{c}\: \cdot\: \psi_{\nu_{\gamma(b)}}(\homm(\gamma)(\nu_{\gamma(a)}))\cdot \underbrace{\psi_{\gamma(a)}\big(\Phi_{g^{-1}}(\nu_{\wm_g(\gamma(a))})\big)}_{d},
\end{split}
\end{align}
which equals $c \cdot \pi_\gamma(\qw)\cdot d$.
Now, since $\dim[S]\geq 1$, we find $\{s_n\}_{n\in\NN}\subseteq S$ with $s_n\neq s_m$ for $n\neq m$, hence
\begin{align*}
B\cdot (e,s_n) \cap B\cdot (e,s_m)=\emptyset\qquad \text{ for } n\neq m.
\end{align*}
Then, if $\mu_\alpha(B)> 0$, $\sigma$-additivity and translation invariance of $\mu_\alpha$ would imply that
\begin{align*}
\mu_\alpha(S\times S)\geq \sum_{n\in \NN}\mu_\alpha(B\cdot (e,s_n))=\sum_{n\in \NN}\mu_\alpha(B)=\infty
\end{align*}
holds. This, however, contradicts that $\mu_\alpha$ is normalized.
\end{proof}
\end{lemma}
We conclude this section with the following remark, providing an outlook of the next sections.
\begin{remark}
\label{rem:euklrem}
\begin{enumerate}
\item
\label{rem:euklrem0}
If $\wm$ is analytic and pointwise proper, then
$\AQR, \ARQ\subseteq \A$ are of measure zero w.r.t. $\mAL$ whenever $\dim[S]\geq 1$ and $G_x\neq \{e\}$ for some $x\in M$.
In fact, we will see in Lemma \ref{lemma:sim}.\ref{lemma:sim2} that for such an action $\wm$ and $\g\in \mg\backslash\mg_x$ for $x\in M$ we always find $l>0$ such that the restriction of the curve $\delta \colon t\mapsto \wm_x(\exp(t\cdot\g))$ to $[0,l]$ is embedded analytic. Then, for $g:=\exp(l/3\cdot \g)$ and $\gamma:=\delta|_{[0,l/3]}$ the requirement $\gamma \nsim_\cp \wm_g\cp \gamma$ of Corollary \ref{cor:AshtLew} is obviously fulfilled.
\item
\label{rem:euklrem1}
Corollary \ref{cor:AshtLew} already shows that the invariance properties of the elements of $\AQR$ give non-trivial restrictions to the images of the projection maps $\pi_\alpha$. In particular, this is the case if the curves forming the index $\alpha=(\gamma_1,\dots,\gamma_k)$ are related in the sense that $\wm_g\cp\gamma_i\csim \gamma_j$ holds for some $g\in G$ and some $1\leq i\neq j\leq k$.
Now, non-trivial restrictions can also occur if we have $\alpha=\gamma$ for a single curve $\gamma\in \Pa$, namely if $\gamma$ is invariant under some symmetry group element $g\in G\backslash\{e\}$, i.e., $\phi_g\cp\gamma =\gamma$. Indeed, then by \eqref{eq:invpi} we have $\pi_\gamma(\qw)=\pi_{\wm_g\cp \gamma}(\qw)=c\cdot \pi_\gamma(\qw)\cdot d$, and for $S$ compact and connected and the case that $c=d^{-1}\neq \pm \me$, such a relation can already restrict $\mathrm{im}[\pi_\gamma]$ to a maximal torus as Lemma \ref{lemma:torus} shows.\footnote{See also the second point in Example \ref{rem:euklrem6}} So, when constructing measures on the spaces $\ARQ$ and $\AQR$ by means of such projection maps, one has to take these invariance properties into account.
\item
As we will see in Subsection \ref{sec:inclrel}, the inclusion $\ARQ\subseteq \AQR$ is usually proper. Consequently, there are further highly non-trivial restrictions to the image of $\pi_\alpha$ when restricting to $\ARQ$. For this reason, it seems to be very hard to provide a general notion of a reduced measure on these spaces. Anyhow,
for the case of homogeneous isotropic LQC, in Section \ref{sec:HomIsoCo}, we will discuss the measure theoretical aspects of the space $\ARQ$ in detail.
\item
\label{rem:dsdfdf}
For the space $\AQR$ we will investigate (next two sections) the case where $\Pa= \Paw$ holds and where $\wm$ is analytic and pointwise proper. This will allow us to follow the lines of Proposition \ref{rem:euklrem2b}, as, in this case, $\Paw$ splits up into
free and continuously generated curves. We denote the respective sets by $\Paf$ and $\Pac$. The set $\Paf$ consists of all $\gamma\in \Paw$ which contain a subcurve $\delta$ (free segment) such that
\begin{align*}
\wm_g\cp \delta \cpsim\delta \:\:\:\text{ for }\:\:\: g\in G \qquad\Longrightarrow \qquad \wm_g\cp \delta = \delta,
\end{align*}
and then $\Pac$ is just its complement in $\Paw$. We will see in Proposition \ref{prop:freeseg}.\ref{prop:freeseg1} that each $\gamma\in \Paf$ is discretely generated by the symmetry group, i.e., is build up finitely many translates of initial and final segments\footnote{Here, translates of initial and final segments of $\delta$ can only occur as initial and final segments of $\gamma$.} of one of its maximal free segments $\delta$. Now,
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\vspace{-3pt}
If $\wm$ is transitive or proper and admits only stabilizers which are normal subgroups, then $\Pac$ equals the set $\Pags$ of Lie algebra generated curves, i.e., of embedded analytic curves equivalent to a curve of the form $[0,l]\ni t\mapsto \wm(\exp(t\cdot\g),x)$ for some $x\in M$, $\g\in \mg\backslash \mg_x$ and $l>0$. Then, in special situations (such as if $\wm$ acts free), we will be able to construct a normalized Radon measure on $\IHOMLAS$ for the case that $S=\SU$. This will be done in Subsection \ref{sec:ConSp}.
\item
Let $\Pafns\subseteq \Paf$ denote the subset of non-symmetric curves, i.e., of curves for which $\wm_g\cp \gamma \neq \gamma$ holds for all $g\in G\backslash\{e\}$.\footnote{In other words, the stabilizer of $\gamma$ (a well-behaving and important quantity in the situation where $\wm$ is analytic and pointwise proper) is trivial.}
We will see in Subsection \ref{sec:FreeM} that $\HOMFNS$ carries a natural normalized Radon measure whenever the structure group of the bundle is compact and connected. This measure even specializes to the Ashtekar-Lewandowski measure on $\HOMW$ if $G=\{e\}$. This is because, according to our definitions (see Definition \ref{def:freeSeg}.\ref{def:freeSeg32}), then $\Pafns=\Paw$ holds.
\item
If $\wm$ is proper and free, then $\Pac=\Pags$ and $\Paf=\Pafns$ holds. Here, both sets are independent, $\Phi$-invariant and closed under decomposition and inversion of their elements. So, for $S=\SU$ we will obtain a normalized Radon measure on
\begin{align*}
\IHOM\cong\IHOMLAS \times \IHOMFNS
\end{align*}
just by taking the Radon product of the measures from the first two points, cf.\ Proposition \ref{rem:euklrem2b}.
\end{itemize}
\endgroup
\item
\label{it:sdsdds}
The set $\Pags$ will be of particular interest because:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\vspace{-3pt}
Lie algebra generated curves turn out to be very useful for investigations concerning the inclusion relations between the spaces $\ARQ$ and $\AQR$, as they often allow to decide the inclusion problem on the level of the Lie algebras of the symmetry and the structure group, see Subsection \ref{sec:inclrel}.
\item
Whenever $\wm$ acts transitively, the cylindrical functions that correspond to $\Pags$ separate the points in $\Con$, so that
$\iota\colon \Con \rightarrow \AInd{\mg}$ is injective. This is clear from surjectivity of $\dd_e\wm_x \colon \mg \rightarrow T_xM$ and Lemma \ref{lemma:separating}. So, in this case not only $\AInd{\mg}\cong \HOMLAS$ but also $\AQRLA\cong \IHOMLAS$ is a physically meaningful candidate for a (reduced) quantum configuration space. Indeed, there are approaches to LQG which only use linear curves to define $\A$ \cite{JonEng3}. Moreover, originally only linear curves were used to define the quantum configuration space of LQC and also in the reduction paper \cite{BojoHomoCosmo} only Lie algebra generated curves were taken into account.\footnote{In this rather heuristic paper the author restricts to the transitive situation with $G_x=\{e\}$, and quantizes (unspecified) sets containing $\AR$ by using parallel transport functions along Lie algebra generated curves.}
\end{itemize}
\endgroup
\end{enumerate}
\end{remark}
\subsection{Summary}
\label{concl:Actionlift}
\begin{enumerate}
\item
\label{concl:Actionlift1}
In the first part of this section we have seen that it is always possible to extend a left action
\begin{align*}
\cw\colon G\times X\rightarrow X
\end{align*}
of a group $G$ on a set $X$ uniquely to a left action
\begin{align*}
\specw\colon G\times \X\rightarrow \X
\end{align*}
on the spectrum $\X$ of a $\Cstar$-algebra $\aA\subseteq B(X)$. The action $\specw$ is always continuous in $\X$, and continuous if $G\ni g\mapsto \cw_g^*f\in \aA$ is continuous for all $f\in \aA$. For $\aA$ unital, here even the iff statement holds.
Continuity of $\specw_g$ and the extension property\footnote{Recall that $X_\aA$ just denotes the set of elements $x\in X$ for which $\aA\rightarrow \mathbb{C}$, $f\mapsto f(x)$ is not the zero functional.}
\begin{align*}
\specw_g\cp \iota_X|_{X_\aA}=\iota_X\cp \cw_g|_{X_\aA}\qquad \forall\: g\in G
\end{align*}
always imply that $\XRQ\subseteq \XQR$ holds. Here, $\XRQ$ denotes the closure in $\X$ of
\begin{align*}
\XR\cap X_\aA=\{x\in X_\aA\:|\: \cw(g,x)=x\quad\forall\: g\in G\}
\end{align*}
and $\XQR=\{\x\in \X\:|\: \specw(g,\x)=\x\quad\forall\: g\in G\}$ is the closed subset of $\specw$-invariant elements.
\item
\label{concl:Actionlift2}
In the second part, we have applied the first one to the quantum gauge field situation
where $X$ equals the set $\Con$ of smooth connections on a principal fibre bundle $\PMS$, and $\cw$ comes from a Lie group $(G,\Phi)$ of automorphisms of $P$, i.e.,
\begin{align*}
\cw(g,\w)=\Phi_{g^{-1}}^*\w\qquad\forall\: g\in G.
\end{align*}
In this situation $\aA$ is the $\Cstar$-algebra $\PaC$ of cylindrical functions that corresponds to any set $\Pa$ of $\CC{k}$-paths in $M$ which is invariant under pullbacks by the translations $\wm_g$ ($\Pa$ is $\Phi$-invariant). Here, $\wm$ denotes the action induced by $\Phi$ on $M$. In particular, we have seen that the set $\ARQ$ of quantized invariant (classical) connections is always contained in the compact set $\AQR$ of invariant generalized (quantum) connections.
\item
\label{concl:Actionlift3}
In the last part of this section, we have identified the quantum spaces $\A$ with spaces $\HOM$ of homomorphisms of paths. We have seen that this is always possible if the set $\Pa$ is independent.\footnote{This is the case, e.g., if $M$ is analytic, $S$ is compact and connected, and $\Pa$ is the set $\Paw$ of embedded analytic curves in $M$, cf.\ Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt3}.} Under the further assumption that $\Pa$ is $\Phi$-invariant, this has allowed us to identify $\AQR$ with the subspace $\IHOM\subseteq \HOM$ of invariant homomorphisms.
Moreover, using invariance, we have shown that in the analytic (LQG) situation, i.e., $M$ and $\wm$ are analytic, $\wm$ is pointwise proper and $\Pa=\Paw$, the sets $\ARQ$ and $\AQR$ are of measure zero w.r.t.\ the Ashtekar-Lewandowski measure $\mAL$ (standard measure in LQG) on $\A$ provided that $\dim[S]\geq 1$, $G_x\neq \{e\}$ holds for some $x\in M$, and $S$ is compact and connected.
Moreover, we have illustrated that defining a normalized Radon measures on $\AQR\cong \IHOMW$ can be done by decomposing $\Paw$ into elements continuously,
and into elements discretely generated (see Remark \ref{rem:euklrem}.\ref{rem:dsdfdf}) by the symmetry group. This decomposition will be discussed in much more detail in the next section.
\end{enumerate}
\section{Modification of Invariant Homomorphisms}
\label{susec:LieALgGenC}
Let $\PMS$ be a principal fibre bundle and $(G,\Phi)$ a Lie group of automorphisms thereon. In the previous section, we have introduced the space $\AQR$ of invariant generalized connections that corresponds to a $\Phi$-invariant set $\Pa$ of $\CC{k}$-curves in $M$. For the case that $\Pa$ is in addition independent and $S$ is compact, we have identified $\AQR$ with the subset $\IHOM \subseteq \HOM$ of homomorphisms having additional invariance properties.
We now switch to the analytic situation, i.e., we consider embedded analytic curves and actions $\Phi$ for which the induced action $\wm$ on $M$ is analytic and pointwise proper. We will prove certain modification results for invariant homomorphisms which are crucial for both our investigations of the inclusion relations between $\ARQ$ and $\AQR$, and the construction of normalized Radon measures on $\AQR$ in Section \ref{sec:MOQRCS}. In analogy to Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod24}, modification here just means to change the value of a given invariant homomorphism on some distinguished set of curves in a specific way.
In the following, $\Paw$ will always denote the set of embedded analytic curves in $M$, and $S$ is assumed to be compact and connected. Recall that compactness of $S$ guarantees well-definedness of the map $\kappa\colon \A\rightarrow \HOM$, $\qw\mapsto\big[\gamma \mapsto \parall{\gamma}{\qw}\hspace{1pt}\big]$. Connectedness is crucial because:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
As we will see in Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt3}, a subset $\Pa\subseteq \Paw$ closed under decomposition and inversion of it elements is independent if $S$ is connected. Recall that this ensures that $\kappa$ is bijective, i.e., a homeomorphism w.r.t.\ the topology $\TOHO$ from Definition \ref{def:indepref}.\ref{conv:muetc}.
\item
If $\dim[S]\geq 1$, the equivalence relation $\csim$ equals the (elsewise coarser) equivalence relation $\psim$ that will occur in the fundamental statements of this section.
Here, $\gamma_1\psim\gamma_2$ just means that both curves coincide up to parametrization, i.e.,
\begin{align*}
\gamma_1\text{\gls{PSIM}}\gamma_2 \:\: \Longleftrightarrow \:\: \gamma_1=\gamma_2\cp \adif|_{\operatorname{\mathrm{dom}}[\gamma_1]}\: \text{ for }\: \adif \colon I\rightarrow \RR \:\text{ analytic diffeomorpism with }\dot\adif>0.
\end{align*}
\end{itemize}
\endgroup
\noindent
In Subsection \ref{subsec:LaGC}, we will collect the relevant facts and definitions concerning analytic and Lie algebra generated curves. In Subsection \ref{sec:ModifLAGC}, we modify invariant homomorphisms along such Lie algebra generated curves, and in Subsection \ref{sec:inclrel} we will apply this in order to obtain some general conditions which allow to decide whether the inclusion
$\ARQ\subseteq\AQR$
is proper. Basically, this will be done by constructing elements of $\AQR$ that cannot be approximated by classical (smooth) invariant connections.
In particular, we conclude that quantization and reduction do not commute in \mbox{(semi-)homogeneous} loop quantum cosmology. For homogeneous isotropic LQC, this will be shown in Section \ref{sec:HomIsoCo}. Finally, in the last part of this section, we will prove an analogue of the modification result from Subsection \ref{sec:ModifLAGC}, now for free
curves, cf.\ Remark \ref{rem:euklrem}.\ref{rem:dsdfdf}. We first show that each analytic embedded curve which contains a free segment
is discretely generated
by the symmetry group. Then, we use this in order to modify homomorphism along such free segments.
\subsection{Analytic and Lie Algebra Generated Curves}
\label{subsec:LaGC}
This subsection basically collects the properties of analytic and Lie algebra generated curves that will be relevant for our later considerations. We will start with some elementary facts on analytic curves. Then,
we highlight the most important properties of the Lie algebra generated ones, in particular, for the case that the induced action $\wm$ is analytic and pointwise proper.
\subsubsection{Basic Properties of Analytic Curves}
We start with a lemma collecting some standard properties of (embedded) analytic curves. For this, we will need
\begin{definition}
\label{remdef:eqrels}
Let $\gamma_1,\gamma_2\in \Paw$ with $\operatorname{\mathrm{dom}}[\gamma_i]=[a_i,b_i]$ for $i=1,2$. We define the equivalence relations $\psim$ and $\isim$ on $\Paw$ by
\begin{align*}
\gamma_1\hspace{2.5pt}\text{\gls{PSIM}}\hspace{1pt}\gamma_2 \quad \Longleftrightarrow \quad &
\gamma_1=\gamma_2\cp \adif|_{\operatorname{\mathrm{dom}}[\gamma_1]}\: \text{ for }\: \adif \colon I\rightarrow \RR \:\text{ analytic diffeomorpism with }\dot\adif>0.\\
\gamma_1\isim\gamma_2 \quad \Longleftrightarrow \quad & \mathrm{im}[\gamma_1]=\mathrm{im}[\gamma_2]\: \text{ and }\: \gamma_1(a_1)=\gamma_2(a_2)\: \text{ as well as }\: \gamma_1(b_1)=\gamma_2(b_2).
\end{align*}
Recall that this means that $I\subseteq \RR$ is open and that $\operatorname{\mathrm{dom}}[\gamma_1]\subseteq I$ holds.
\end{definition}
\begin{convention}
\label{conv:extensaccum}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
In the sequel, by an accumulation point of a topological space $X$ we will understand an element $x\in X$ for which we find a net $\{x_\alpha\}_{\alpha\in I}\subseteq X\backslash\{x\}$ with $\lim_\alpha x_\alpha=x$.
\item
According to the notations subsection, domains of curves
are always assumed to be intervals (non-empty interior).
\item
An analytic (immersive) curve $\gamma$ is called extendible iff we find an analytic (immersive) curve $\delta$ with open domain, such that $\gamma=\delta|_{\operatorname{\mathrm{dom}}[\gamma]}$ holds and $\operatorname{\mathrm{dom}}[\gamma]\subseteq \operatorname{\mathrm{dom}}[\delta]$ is properly contained.
It
is called maximal (or inextendible) iff no extension exists. The same conventions hold for analytic diffeomorphisms $\adif\colon I\rightarrow \RR$. (Observe that the domain of a maximal analytic (immersive) curve is necessarily open.)
\end{itemize}
\endgroup
\end{convention}
\begin{lemma}
\label{lemma:BasicAnalyt}
\begin{enumerate}
\item
\label{lemma:BasicAnalyt1}
Let $\gamma_i\colon (a_i,b_i)\rightarrow M$ be an analytic embedding for $i=1,2$ and $x$ an accumulation point of $\mathrm{im}[\gamma_1]\cap \mathrm{im}[\gamma_2]$ (w.r.t.\ the subspace topology inherited from $M$). Then $\gamma_1(I_1)=\gamma_2(I_2)$ for open intervals $I_i\subseteq (a_i,b_i)$ with $x\in \gamma_i(I_i)$ for $i=1,2$.
\item
\label{maximalextension}
Each (immersive) analytic curve has a maximal extension.
\item
\label{lemma:BasicAnalyt2}
We have
\begin{align*}
\gamma_1\isim \gamma_2\quad \Longleftrightarrow\quad \gamma_1\psim \gamma_2 \quad
\Longrightarrow \quad \gamma_1\csim \gamma_2.
\end{align*}
If $\dim[S]\geq 1$, then
\begin{align*}
\gamma_1\isim \gamma_2\quad \Longleftrightarrow\quad \gamma_1\psim \gamma_2 \quad \Longleftrightarrow\quad \gamma_1\csim \gamma_2.
\end{align*}
\item
\label{lemma:BasicAnalyt4}
Let $\delta \colon [0,k] \rightarrow M$ and $\delta' \colon [0,k'] \rightarrow M$ be analytic embeddings.
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item
\label{dddd}
If $\delta$ and $\delta'$ share an initial segment, i.e.,
\vspace{-16pt}
\begin{align*}
\delta|_{[0,s]}\isim \delta'|_{[0,s']} \qquad \text{for}\qquad s\in (0,k],\: s'\in (0,k'],
\hspace{98.25pt}
\begin{tikzpicture}
\filldraw[black] (0,0) circle (2pt);
\draw[->,line width=0.7pt,color=red] (0.25,0) .. controls (1,0) and (1,0.5) .. (0.5,0.5);
\draw[->,line width=1.5pt] (0,0) -- (1.5,0);
\draw (1.3,0.3) node {\(\delta\)};
\draw[color=red] (0.25,0.5) node {\(\delta'\)};
\draw[-,line width=1pt] (0.37,0) -- (0.37,0.15);
\end{tikzpicture}
\end{align*}
then either
\begingroup
\setlength{\leftmarginiii}{15pt}
\begin{itemize}
\item
\vspace{-5pt}
\hspace{18pt}$\delta \psim \delta'|_{[0,t']}$\quad for\quad $t'\in (0,k']$ \qquad or\hspace*{\fill}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\filldraw[black] (0,0) circle (2pt);
\draw[->,line width=1.05pt,color=red] (0,0) -- (2,0);
\draw[->,line width=1.5pt] (0,0) -- (1.5,0);
\draw (1,0.2) node {\(\delta\)};
\draw[color=red] (1.8,0.2) node {\(\delta'\)};
\end{tikzpicture}
\item
\vspace{-5pt}
$\delta|_{[0,t]}\psim \delta'$\quad\hspace{17pt} for\quad $t\phantom{'}\in (0,k)$.\hspace*{\fill}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\filldraw[black] (0,0) circle (2pt);
\draw[->,line width=1.1pt,color=red] (0,0) -- (1,0);
\draw[->,line width=1.5pt] (0,0) -- (1.5,0);
\draw[color=red] (0.7,0.2) node {\(\delta'\)};
\draw (1.7,0.2) node {\(\delta\)};
\end{tikzpicture}
\end{itemize}
\endgroup
\item
\vspace{5pt}
If $\delta$ and $\delta'^{-1}$ share an initial segment, i.e.,
\vspace{-15pt}
\begin{align*}
\delta|_{[0,s]}\isim \big[\delta'|_{[s',k']}\big]^{-1} \qquad \text{for}\qquad s\in (s,k],\: s'\in[0,k'),
\hspace{73pt}
\begin{tikzpicture}
\filldraw[red] (0.5,0.5) circle (1.5pt);
\draw[-,line width=0.6pt,color=red] (0.25,0) .. controls (1,0) and (1,0.5) .. (0.5,0.5);
\draw[->,line width=1.5pt] (0,0) -- (1.5,0);
\draw[->,line width=1.05pt] (0.25,0) -- (0,0);
\draw (1.3,0.3) node {\(\delta\)};
\draw[color=red] (0.25,0.5) node {\(\delta'\)};
\draw[-,line width=1.3pt] (0.4,0) -- (0.4,0.15);
\end{tikzpicture}
\end{align*}
then either
\begingroup
\setlength{\leftmarginiii}{15pt}
\begin{itemize}
\item
\vspace{-5pt}
\hspace{18pt}$\delta \psim \big[\delta'|_{[t',k']}\big]^{-1}$\quad for\quad $t'\in [0,k')$ \qquad or\hspace*{\fill}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\filldraw[red] (2,0) circle (1.5pt);
\draw[->,line width=1.0pt,color=red] (2,0) -- (0,0);
\draw[->,line width=1.5pt] (0,0) -- (1.5,0);
\draw (1,0.2) node {\(\delta\)};
\draw[color=red] (1.8,0.25) node {\(\delta'\)};
\end{tikzpicture}
\item
\vspace{-5pt}
$\delta|_{[0,t]}\psim \delta'^{-1}$\quad\hspace{29.5pt} for\quad $t\phantom{'}\in (0,k)$.\hspace*{\fill}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\filldraw[red] (1,0) circle (1.5pt);
\draw[->,line width=1.1pt,color=red] (1,0) -- (0,0);
\draw[->,line width=1.5pt] (0,0) -- (1.5,0);
\draw[color=red] (0.7,0.2) node {\(\delta'\)};
\draw (1.7,0.2) node {\(\delta\)};
\end{tikzpicture}
\end{itemize}
\endgroup
\end{enumerate}
\endgroup
\item
\label{Basanalyt}
Let $\gamma_i\colon [0,k_i]\rightarrow M$ be an analytic embedding for $i=1,2$. Then, for $i=1,2$ the set $\gamma_i^{-1}(\mathrm{im}[\gamma_1]\cap\mathrm{im}[\gamma_2])$ is the disjoint union of finitely many isolated points and $0\leq m\leq 2$ disjoint compact
intervals $\{L_i^p\}_{1\leq p\leq m}$ with
$\gamma_1|_{L_1^p}\isim \big[\gamma_2|_{L_2^p}\big]^{\pm 1}$ for $1\leq p\leq m$.
If $m=1$, then either
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\vspace{-5pt}
$L_i^m=[0,k_i]$\:\: for some \:\:$i\in \{1,2\}$\qquad or\hspace*{\fill}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\draw[<->,line width=1.0pt,color=red] (0.5,0) -- (1.2,0);
\draw[<->,line width=1.5pt] (0,0) -- (2,0);
\draw[color=red] (0.85,0.2) node {\(\gamma_i\)};
\draw (1.7,0.2) node {\(\gamma_j\)};
\end{tikzpicture}
\item
\vspace{-5pt}
$L_i^m$ is of the form $[l_i,k_i]$ or $[0,l_i]$ with $0<l_i<k_i$ for $i=1,2$. \hspace*{\fill}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\draw[<->,line width=1.0pt,color=red] (0.9,0) -- (2,0);
\draw[<->,line width=1.5pt] (0,0) -- (1.5,0);
\draw (0.5,0.2) node {\(\gamma_j\)};
\draw[color=red] (1.8,0.25) node {\(\gamma_i\)};
\end{tikzpicture}
\end{itemize}
\endgroup
If $m=2$, then for $i=1,2$ we hav
\vspace{-10pt}
\begin{align*}
\quad\:\: L_i^p=[0,l^p_i]\:\:\text{ and }\:\: L_i^q=[l^q_i,k_i]\:\:\text{ for some }\:\: l_i^p\neq l_i^q,\: 1\leq p\neq q\leq 2.
%
\hspace{50pt}
\begin{tikzpicture}
\filldraw[white] (2,0) circle (2pt);
\draw[<-,line width=1.0pt,color=red] (1.15,0) -- (1.6,0);
\draw[->,line width=1.0pt,color=red] (0.4,0) -- (0.85,0);
\draw[-,line width=1pt,color=red] (1.6,0) .. controls (1.8,0) and (2,0.5) .. (1,0.5);
\draw[-,line width=1pt,color=red] (0.4,0) .. controls (0.2,0) and (0,0.5)
.. (1,0.5);
\draw[<->,line width=1.5pt] (0.4,0) -- (1.6,0);
\draw (1,0.2) node {\(\gamma_j\)};
\draw[color=red] (0,0.4) node {\(\gamma_i\)};
\end{tikzpicture}
\end{align*}
\item
\label{lemma:BasicAnalyt3}
If $S$ is connected with $\dim[S]\geq 1$, and
$\Pas\subseteq \Paw$ is closed under decomposition and inversion of its elements, then $\Pas$ is independent.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
Let $x=\gamma_i(\tau_i)$ with $\tau_i\in (a_i,b_i)$ for $i=1,2$. Moreover, let $(U,\phi)$ be an analytic submanifold chart of $\mathrm{im}[\gamma_1]$ which is centered at $x$ and maps $\mathrm{im}[\gamma_1]\cap U$ into the $x$-axis. Choose $\epsilon>0$ such that $\gamma_2(B_{\epsilon}(\tau_2))\subseteq U$ and consider the analytic functions $f_{\zd}:=\phi^{\zd}\cp\gamma_2|_{B_{\epsilon}(\tau_2)}$ for $k=2,\dots,\dim[M]$. Then $\tau_2$ is an accumulation point of zeroes of $f_{\zd}$, so that $f_{\zd}=0$ by analyticity of $f_{\zd}$. This shows $\phi(\gamma_2(B_{\epsilon}(\tau_2))\subseteq \phi(U\cap\mathrm{im}[\gamma_1])$, hence $\gamma_2(B_{\epsilon}(\tau_2))\subseteq \mathrm{im}[\gamma_1]$.
The claim then holds for $I_2:=B_{\epsilon}(\tau_2)$ and $I_1:=\gamma_1^{-1}( \gamma_2(I_2))$.
\item
Let $\gamma\colon D\rightarrow M$ be an analytic curve, denote by $\EE$ the set of all extensions of $\gamma$ with open domain and define its maximal extension
\begin{align*}
\gamma_0\colon I:= \textstyle\bigcup_{\delta\in \EE}\operatorname{\mathrm{dom}}[\delta]\rightarrow M
\end{align*}
by $\gamma_0(x):=\delta(x)$ for $\delta\in \EE$ with $x\in \operatorname{\mathrm{dom}}[\delta]$. Then, $\gamma_0$ is well defined because for $\delta'\in \EE$ with $x\in \operatorname{\mathrm{dom}}[\delta']$ we necessarily have $\delta'(x)=\delta(x)$. In fact, since $\delta$ and $\delta'$ coincide on $D$, by analyticity they coincide on $\operatorname{\mathrm{dom}}[\delta]\cap\operatorname{\mathrm{dom}}[\delta']\ni x$.
If $\gamma$ is in addition immersive, we restrict $\gamma_0$ to the maximal (necessarily open) interval containing $D$ on which $\gamma_0$ is an immersion.
\item
The implication $\psim\: \Rightarrow\: \isim$ is obvious and $\psim\: \Rightarrow\: \csim$ is clear because parallel transports are invariant under reparametrization by orientation preserving diffeomorphisms.
Now, assume that $\gamma_1\sim_{\mathrm{im}} \gamma_2$ and let $\gamma_i'\colon (a_i',b_i')\rightarrow M$ be an analytic embedding with $[a_i,b_i]\subseteq (a_i',b_i')$ and $\gamma_i'|_{[a_i,b_i]}=\gamma_i$ for $i=1,2$. Then $\gamma_i(a_i)$ and $\gamma_i(b_i)$ are accumulation points of $\mathrm{im}[\gamma'_1]\cap\mathrm{im}[\gamma'_2]$, so that
by Part \ref{lemma:analytCurvesIndepetc1}) we can arrange that $N:=\mathrm{im}[\gamma_1']= \mathrm{im}[\gamma_2']$ just by shrinking the intervals $(a'_i,b_i')$ in a suitable way. Now, $N$ is a real analytic submanifold in the topological sense and the maps $\gamma_i'$ are diffeomorphisms w.r.t.\ its analytic structure.
Consequently, $\adif:=\gamma_2'^{-1}\cp \gamma_1'$ is the desired diffeomorphism.
Now, let $\dim[S]>1$ and $\nu=\{\nu_x\}_{x\in M}$ a choice of elements with $\nu_x\in F_x$ for all $x\in M$. If $\gamma_1\csim \gamma_2$, then $\gamma_1(a_1)=\gamma_2(a_2)$ and $\gamma_1(b_1)=\gamma_2(b_2)$ by definition, and we have to show that
$\mathrm{im}[\gamma_1]= \mathrm{im}[\gamma_2]$. Now, if this is not true, then we find $\tau_1\in (a_1,b_1)$ or $\tau_2\in (a_2,b_2)$ such that $\gamma_1(\tau_1)\notin \mathrm{im}[\gamma_2]$ or $\gamma_2(\tau_2)\notin \mathrm{im}[\gamma_1]$, respectively. If $\gamma_1(\tau_1)\notin \mathrm{im}[\gamma_2]$, by compactness of $\mathrm{im}[\gamma_2]$ we find a neighbourhood $U$ of $\gamma_1(\tau_1)$ in $M$ with $U\cap \mathrm{im}[\gamma_2]=\emptyset$. Let $\w\in \Con$ be fixed and $s:=h_{\gamma_2}^\nu(\w)$. If we choose $s'\neq s$, then by Proposition A.1 in \cite{ParallTranspInWebs} we find $\w'\in\Con$ such that $\w'$ equals $\w$ outside $U$ and $h_{\gamma_1}^\nu(\w')=s'$. But, $h_{\gamma_2}^\nu(\w)=h_{\gamma_2}^\nu(\w')$ since $\mathrm{im}[\gamma_2]\subseteq M\backslash U$, so that
\begin{align*}
h_{\gamma_2}^\nu(\w')=h_{\gamma_2}^\nu(\w)=s\neq s'= h_{\gamma_1}^\nu(\w')
\end{align*}
contradicts that $\gamma_1\csim \gamma_2$.
\item
It suffices to show \textit{(a)} because \textit{(b)} follows from \textit{(a)} if we replace $\delta'$ by $\delta'^{-1}$. Let
\begin{align*}
t:=\sup\big\{s\in [0,k]\:\big|\:\exists\: s'\in [0,k'] \colon \delta|_{[0,s]}\isim \delta'|_{[0,s']} \big\}.
\end{align*}
By assumption we have $t>0$, so that $\delta|_{[0,t]}\isim \delta'|_{[0,t']}$ for some $t'\in (0,k']$ by continuity, hence $\delta|_{[0,t]}\psim \delta'|_{[0,t']}$ by Part \ref{lemma:BasicAnalyt2}).
\begingroup
\setlength{\leftmarginii}{17pt}
\begin{itemize}
\item
If $t=k$, then $\delta\psim \delta'|_{[0,t']}$ and we have done (the same for $t'=k'$).
\item
If $t<k$ and
$t'<k'$, then $\delta(t)=\delta'(t')$ is an accumulation point of
\begin{align*}
\delta((0,t+\epsilon))\cap \delta'((0,t'+\epsilon'))\quad \text{for} \quad\epsilon,\epsilon'>0 \quad\text{suitable small}.
\end{align*}
Then, $\delta|_{[0,t_0]}\isim \delta'|_{[0,t'_0]}$ for some $t_0>t$, $t'_0> t'$ by Part \ref{lemma:BasicAnalyt1}),
contradicting the choice of $t$.
\end{itemize}
\endgroup
\item
The statement is clear if $T:=\mathrm{im}[\gamma_1]\cap \mathrm{im}[\gamma_2]$ is finite. In the other case there exists an
accumulation point of $T$, just by compactness of $T$. Let
$\gamma_i'\colon (a_i',b_i')\rightarrow M$ be an extension of $\gamma_i$ for $i=1,2$.
Part \ref{lemma:analytCurvesIndepetc1}) shows that we find $[s_i,t_i]\subseteq [0,k_i]$ for $i=1,2$ with $\gamma_1|_{[s_1,t_1]}\isim [\gamma_2|_{[s_2,t_2]}]^{\pm 1}$. The claim now follows by repeated application of Part \ref{lemma:BasicAnalyt4}).
In fact, replacing one of the curves by its inverse if necessary, we can assume that $\gamma_1|_{[s_1,t_1]}\isim \gamma_2|_{[s_2,t_2]}$ holds. By Part \ref{lemma:BasicAnalyt4}) we can assume\footnote{The case $t_2=k_2$ follows analogously.} that $t_1=k_1$, i.e., $\gamma_1|_{[s_1,k_1]}\isim \gamma_2|_{[s_2,t_2]}$, and then the same part shows that one of the following two cases holds:
\begin{align*}
\gamma_1 &\isim \gamma_2|_{[r_2,t_2]}\quad \text{ for }\quad 0\leq r_2\leq s_2 \qquad \text{or}\\
\gamma_1|_{[r_1,k_1]} &\isim \gamma_2|_{[0,t_2]}\quad\hspace{3.8pt} \text{ for }\quad0<r_1\leq s_1.
\end{align*}
In the first case the claim is clear, and in the second one it is clear if
$T':=\gamma_1([0,r_1])\cap \gamma_2([t_2,k_2])$ if finite.
In the other case, $T'$ admits an accumulation point just by compactness.
Then, applying Part \ref{lemma:analytCurvesIndepetc1}) we find $[x_1,y_1]\subseteq [0,r_1)$ and $[x_2,y_2]\subseteq (t_2,k_2]$ with
\begin{align*}
\gamma_1|_{[x_1,y_1]}\isim \big[\gamma_2|_{[x_2,y_2]}\big]^{\pm 1}.
\hspace{40pt}
\begin{tikzpicture}
\draw[<-,line width=1.0pt,color=red] (1.1,0) -- (1.6,0);
\draw[-,line width=1pt,color=red] (1.6,0) .. controls (2,0.6) and (0.3,0.6) .. (0.7,0);
\draw[->,line width=1pt,color=red] (0.7,0) .. controls (0.9,0) and (0.9,0) .. (0.95,0.28);
\draw[<->,line width=1.5pt] (0.3,0) -- (1.6,0);
\draw (1.27,0.22) node {\(+\)};
\draw[color=red] (1.9,0.25) node {\(\gamma_2\)};
\draw (0.2,0.2) node {\(\gamma_1\)};
\end{tikzpicture}
\qquad\quad
\begin{tikzpicture}
\draw[<-,line width=1.0pt,color=red] (1,0) -- (1.6,0);
\draw[-,line width=1pt,color=red] (1.6,0) .. controls (1.9,0.3) and (1,0.7) .. (0.9,0.3);
\draw[-,line width=1pt,color=red] (0.9,0.3) .. controls (0.9,0) and (0.8,0) .. (0.7,0);
\draw[->,line width=1pt,color=red] (0.7,0) .. controls (0.6,0) and (0.6,0.2) .. (0.6,0.4);
\draw[<->,line width=1.5pt] (0.3,0) -- (1.6,0);
\draw (1.25,0.22) node {\(-\)};
\draw[color=red] (1.85,0.3) node {\(\gamma_2\)};
\draw (0.2,0.2) node {\(\gamma_1\)};
\end{tikzpicture}
\end{align*}
Combining Part \ref{lemma:BasicAnalyt4}) with injectivity of $\gamma_1$ and $\gamma_2$, we see that $\gamma_1|_{[x_1,y_1]}\isim [\gamma_2|_{[x_2,y_2]}]^{- 1}$ is not possible, and that
$\gamma_1|_{[0,r_1']}\isim \gamma_2|_{[t_2',k_2]}$ for $r_1'\in (0,r_1)$ and $t'_2\in (t_2,k_2)$.
\item
The full proof can be found in Lemma \ref{lemma:analytCurvesIndepetc}.\ref{lemma:analytCurvesIndepetc5}.
Basically, one applies Part \ref{Basanalyt}) in order to show that each finite collection $\{\gamma_1,\dots,\gamma_k\}$ admits a refinement $\{\delta_1,\dots,\delta_m\}$ such that $\mathrm{im}[\delta_i]\cap \mathrm{im}[\delta_j]$ is finite for $i\neq j$. The rest then follows from compactness of $\mathrm{im}[\delta_i]$ and Proposition A.1 in \cite{ParallTranspInWebs}.
\end{enumerate}
\end{proof}
\subsubsection{Basic Properties of Lie Algebra Generated Curves}
In this subsection, we collect the relevant properties of Lie algebra generated curves.
First, we provide the basic definitions and highlight their most important properties. Then, we show that for $\wm$ analytic and pointwise proper the set $\Pags$ of Lie algebra generated curves as well as its complement in $\Paw$ both are closed under decomposition and inversion of their elements. This will be extended to an decomposition of $\Paw$ into four natural subsets (invariant under inversions and decompositions as well) in Corollary \ref{rem:freinichtLiealg}. Recall that, as we have shown in Proposition \ref{rem:euklrem2b}, such a decomposition gives rise to a respective factorization of the space $\AQRw$.
\begin{definition}
\label{def:analytLieAlgBD}
\begin{enumerate}
\item
\label{def:analytLieAlgBD1}
We say that the two curves $\gamma_1,\gamma_2$ in the manifold $M$ share an open segment and write $\gamma_1\text{\gls{CPSIM}} \gamma_2$ iff we find open intervals $I_i\subseteq \operatorname{\mathrm{dom}}[\gamma_i]$ for $i=1,2$ with $\gamma_1(I_1)=\gamma_2(I_2)$.
\item
\label{def:analytLieAlgBD2}
For $x\in M$ and $\vec{g}\in \mathfrak{g}\backslash\mathfrak{g}_x$ we define the curve
\begin{align*}
\gamma^x_{\vec{g}}\colon \RR &\rightarrow M\\
t&\mapsto \wm(\exp(t\cdot\vec{g}),x).
\end{align*}
\item
\label{def:analytLieAlgBD3}
\itspacec
Let $x\in M$ and $\vec{g},\vec{g}'\in \mathfrak{g}\backslash\mathfrak{g}_x$. We say that $\vec{g}$ and $\vec{g}'$ are related and write $\vec{g}\text{\gls{XSIM}}\vec{g}'$ iff there is some $g\in G$ for which $\gamma^x_{\vec{g}} \cpsim \wm_g\cp\gamma^x_{\vec{g}'}$ holds.
\item
\label{def:analytLieAlgBD4}
For $x\in M$ and $\vec{g}\in \mathfrak{g}\backslash\mathfrak{g}_x$ we define the subgroup $G_{[\g]}^x\subseteq G_x$ by
\begin{align}
\text{\gls{ADSTRGXSTAB}}:=\{h\in G_x\:|\: \Ad_h(\g)\in \spann_\RR(\g)\}.
\end{align}
Here, the brackets in the subscript $[\g]$ refer to the fact that $G_{[\g]}^x=G_{[\g']}^x$ holds if $\g'= \lambda \cdot\g$ for some $\lambda\neq 0$, see also Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve1}.
\end{enumerate}
\end{definition}
\begin{remdef}
\label{rem:ppropercurve}
\begingroup
\setlength{\leftmargini}{23pt}
\begin{enumerate}
\item
\label{rem:ppropercurve0}
In the following, we will tacitly use that
\begin{align*}
\wm_h\cp \gamma_\g^x(t)&=\wm\big(h\cdot \exp(t\cdot \g)\cdot h^{-1},x\big)\\
&=\wm\big(\exp\big(t\cdot \Ad_h(\g)\big),x\big)\\
&=\gamma_{\Ad_h(\g)}^x(t)
\end{align*}
for all $t\in \RR$ and all $h\in G_x$.
\item
\label{rem:ppropercurve1}
Let \gls{SPMG} denote the projective space\footnote{This is the set $\mg\backslash\{0\}$ modulo the equivalence relation $\g'\simpr \g$\quad $\Longleftrightarrow$\quad $\g'\in \spann_\RR(\g)$.} that corresponds to $\mg$, and
let $\text{\gls{PRMG}}\colon \mg\rightarrow \Sp\mg$ denote
the corresponding projection map. Moreover, for $\g\in \mg\backslash\mg_x$ let $[\g]$ be the class of $\g$ in $\pr_\mg(\mg\backslash \mg_x)$, i.e., $[\g]:=\pr_\mg(\g)$.
Then, the subgroup $G_{[\g]}^x$ from Definition \ref{def:analytLieAlgBD}.\ref{def:analytLieAlgBD4} can also be characterized as follows.
Since $\Ad(h)$ is linear for each $h\in G_x$, the map
\begin{align*}
\text{\gls{ADDSTRICH}}\colon G_x\times \pr_\mg(\mg\backslash \mg_x)&\rightarrow \pr_\mg(\mg\backslash \mg_x),\quad
(h,[\g])\mapsto \big[\!\Ad_h(\g)\big]
\end{align*}
is a well-defined left action, and then $\text{\gls{ADSTRGXSTAB}}\subseteq G_x$ is just the stabilizer of $[\g]$ w.r.t.\ $\Ad'$. The relevance of this action and its orbits will be discussed in Remark \ref{lemremmgohnermgx}.
\item
\label{rem:ppropercurve2}
The importance of the group $G_{[\g]}^x$ comes from the condition in the next Lemma \ref{lemma:sim}.\ref{lemma:sim3} which, for the special case that $V=\spann_\RR(\g)$ is one-dimensional and $\wm$ is analytic, can more naturally be written in the form
(see Definition \ref{def:stable} and Lemma and Remark \ref{rem:dfggfg}.\ref{rem:dfggfg1})
\begin{align}
\label{eq:staby}
\gamma_{\g}^x|_{[0,l]} \psim \gamma_{\pm\Ad_h(\g)}^x|_{[0,l']}\quad\text{for }\quad h\in G_x \qquad\Longrightarrow \qquad h\in G_{[\g]}^x,
\end{align}
whereby $l,l'>0$ are such that both curves are embedded analytic.\footnote{Due to Lemma \ref{lemma:sim}.\ref{lemma:sim2} such reals always exist, see also Part \ref{rem:ppropercurve4}) of this remark.}
As we will see in the next subsection, this is the key condition (besides analyticity and pointwise properness of $\wm$) which will make it possible to modify invariant homomorphisms along Lie algebra generated curves that correspond to $\g\in \mg\backslash \mg_x$. Basically, this will be done by replacing the last factor in the general parallel transport formula \eqref{eq:trivpar} (for invariant connections along Lie algebra generated curves) by certain equivariant mappings $\Psi\colon \spann_\RR(\g)\rightarrow S$. The point here is that if the invariant homomorphism comes from an invariant connection, the left hand side of \eqref{eq:staby} already implies that the value of this homomorphism on both curves coincide, just because parallel transports are invariant under reparametrizations. In the purely algebraical setting, however, we will need \eqref{eq:staby} in order to guarantee this.
\item
\label{rem:ppropercurve3}
Observe that $\gamma^x_{\vec{g}}$ is analytic if $\wm$ is analytic because the exponential map of $G$ is analytic. Part \ref{lemma:sim5} of the next lemma then states that if $\wm$ is in addition pointwise proper, each Lie algebra generated curve ${\gamma_{\g}^x}$ is maximal in the following sense:
An analytic immersion $\gamma$ sharing an open segment with ${\gamma_{\g}^x}$, i.e., $\gamma\cpsim {\gamma_{\g}^x}$ is already a subcurve of ${\gamma_{\g}^x}$. This will be important for our modifications in Subsection \ref{sec:ModifLAGC} and
is usually not true if $\wm$ is not pointwise proper:
Indeed, let $\wm\colon \RR_{>0}\times \RR^n \rightarrow \RR^n$, $(\lambda,y)\mapsto \lambda\cdot y$, $0\neq x\in\RR^n$ and $\g=1\in T_1\RR_{>0}$. Then ${\gamma_{\g}^x}(t)=\e^{t}\cdot x$ since the exponential map $\exp\colon \mg \rightarrow \RR_{>0}$ is just given by $\lambda\mapsto \e^{\lambda}$. However, obviously $\gamma\colon \RR \rightarrow \RR$, $t\mapsto t\cdot x$ is an immersion with $\gamma\cpsim {\gamma_{\g}^x}$ but $\mathrm{im}\big[{\gamma_{\g}^x}\big]=\RR_{>0}\cdot x$ is properly contained in $\mathrm{im}[\gamma]=\RR \cdot x$.\hspace*{\fill}$\dagger$
In addition to that, pointwise properness will guarantee that $\xsim$ even defines an equivalence relation on $\mg\backslash \mg_x$ as we will see in the last Part of the next lemma.
\item
\label{rem:ppropercurve4}
The second part of Lemma \ref{lemma:sim} shows that Lie algebra generated curves are always immersions and that for each $\g\in \mg\backslash \mg_x$ there is a unique number $\tau_\g\in \RR_{>0}\sqcup{ \infty}$ such that ${\gamma_{\g}^x}|_{[a,a+l]}$ is embedded iff $l<\tau_\g$. It is a very useful observation that this already implies that $\Ad_h(\g)=\pm \g$ holds for each $h\in G_{[\g]}^x$ provided that $\wm$ is pointwise proper.
In fact, if $\lambda\neq \pm 1$, then $\lambda\neq 0$ because $\Ad_{h^{-1}}\cp\Ad_h=\id_\mg$. In particular, replacing $h$ by $h^{-1}$, we can assume that $|\lambda|<1$ holds. Since $\Ad_h$ is linear, scaling $\g$ if necessary, by the above statements, we also can assume that $y:=\wm(\exp(\g), x)\neq x$. Now, the sequence defined by
\begin{align*}
\qquad\qquad\qquad\quad\wm(h^n,y)&=\wm(h^n\cdot \exp(\g)\cdot h^{-n},x)\qquad\qquad\qquad\qquad\qquad\qquad (h\in G^x_{[\g]}\subseteq G_x)\\
&=\wm(\exp(\Ad_{h^{n}}(\g)),x)=\wm(\exp(\lambda^n\cdot \g),x)
\end{align*}
has the same limit as $\{\wm(h^n,x)\}_{n\in \NN}$, namely $x$. Then, pointwise properness of $\wm$ implies that $g\cdot y =g \cdot x$ holds for some $g\in G$. Hence, $y=x$, which contradicts the choice of $y$.
\hspace*{\fill}$\lozenge$
\end{enumerate}
\endgroup
\end{remdef}
\begin{lemma}[Lie Algebra Generated Curves]
\label{lemma:sim}
Let $x\in M$ and $\vec{g},\vec{g}'\in \mathfrak{g}\backslash \mathfrak{g}_x$.
\begin{enumerate}
\item
\label{lemma:sim1}
The horizontal lift of $\gamma:=\wm_{g'}\cp \gamma^x_{\vec{g}}|_{[0,l]}$
w.r.t.\ the invariant connection $\w$ in the point $p\in F_x$ is given by (recall that $\wt{g}$ denotes the fundamental vector field \eqref{eq:fundvf} which corresponds to $\g$)
\begin{align}
\label{eq:trivpar}
\wt{\gamma}(t):=g'\cdot \exp(t\cdot\vec{g})\cdot p\cdot \exp(-t \cdot \w(\wt{g}(p)))\qquad \forall\: t\in [0,l].
\end{align}
\item
\label{lemma:sim2}
The curve $\gamma:=\gamma^x_{\vec{g}}$ is an immersion.
\begingroup
\setlength{\leftmarginii}{17pt}
\begin{itemize}
\vspace{-4pt}
\item
If $\gamma$ is injective, then $\gamma|_{[a,b]}$ is embedded for all $[a,b]$ with $a<b$.
\item
\vspace{3pt}
If $\gamma$ is not injective, then it is cyclic in the sense that there is $\tau\in \RR_{>0}$ uniquely determined, such that
\begin{align*}
\gamma(t)=\gamma(t')\qquad\Longleftrightarrow \qquad t=t' + n\tau \quad\text{for some}\quad n\in \mathbb{Z}.
\end{align*}
Hence, the curves $\gamma|_{[a,b]}$ are embedded iff $b\in (a,a+\tau)$ for $a,b\in \RR$.
\end{itemize}
\endgroup
\item
\label{lemma:sim3}
Let $V\subseteq \mg$ be an $\Add{G_x}$-invariant linear subspace with $V\cap \mg_x =\{0\}$.
Then,\footnote{Here, $\psim$ is defined as in the analytic case, whereby the respective diffeomorphism $\adif$ is assumed to be smooth instead of analytic.}
\begin{align*}
\gamma_{\g'}^x|_{[0,l']} \psim \wm_h\cp \gamma_{\g}^x|_{[0,l]}\quad\text{for }\quad h\in G_x,\: \g,\g'\in V \qquad\Longrightarrow \qquad \textstyle\frac{l}{l'} \vec{g}=\Add{h^{-1}}(\g').
\end{align*}
\item
\label{lemma:sim4}
If $\vec{g}\xsim \vec{g}'$, then
$\lambda \vec{g}-\Add{h^{-1}}(\vec{g}')\in \mathfrak{g}_x$
for some $h\in G_x$ and $\lambda\in \RR_{\neq 0}$. In particular, if $G_x=\{e\}$, then $\vec{g}\nsim_x \vec{g}'$ whenever $\g,\g'$ are linearly independent.
\item
\label{lemma:sim5}
Let $\wm$ be analytic and pointwise proper.
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{enumerate}
\item
\label{sim5a}
\vspace{-4pt}
If $\gamma\colon D \rightarrow M$ is an analytic immersion with $\gamma\cpsim\wm_g \cp {\gamma_{\g}^x}$, then $\gamma=\wm_g \cp{\gamma_{\g}^x}\cp\adif$ for some analytic diffeomorphism $\adif\colon I\rightarrow \RR$ ($I$ open).
\vspace{2pt}
\item
\label{sim5b}
If
$\delta\colon K \rightarrow M$ is an embedded analytic curve with $\delta \cpsim \wm_g\cp\gamma_{\g}^x$, then
$\delta \psim \wm_g\cp\gamma_{\pm\g}^x|_{K}$
for $K \subseteq \RR$ a compact interval.
\end{enumerate}
\endgroup
\item
\label{lemma:sim444}
If $\wm$ is analytic and pointwise proper, then $\xsim$ defines an equivalence relation on $\mg\backslash \mg_x$.
\end{enumerate}
\end{lemma}
Obviously, claim $\textit{(b)}$ in Part \ref{lemma:sim5} is immediate from claim $\textit{(a)}$ in Part \ref{lemma:sim5}, and it also can be proven by the techniques from Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt4}. We here decided to show the more general statement as it makes some of our further proofs more elegant, and because the only additional notion which we need is that of a maximal extension of an analytic (immersive) curve. At this point, the author expresses his gratitude to Christian Fleischhack for advising him to introduce maximal extensions in order to prove the more general statement.
\begin{proof}
\begin{enumerate}
\item
Let $\xi:= \w(\wt{g}(p))$.
Then, $\pi(\wt{\gamma}(t))=\wm_{g'}\cp\wm(\exp(t\vec{g}),x)=\gamma(t)$ and
\begin{align*}
\w\big(\raisebox{-1pt}{$\dot{\wt{\gamma}}(t)$}\big
&\stackrel{\eqref{eq:trivpar}}{=} \w\Big((\dd R_{\exp(-t\xi)}\cp \dd L_{\exp(t \vec{g})})(\wt{g}(p))\Big)
- \w\left(\wt{\xi}\big(R_{\exp(-t\xi)}\big(\Phi(\exp(t\vec{g}), p)\big)\big)\right)\\
&\:\hspace{1pt}= (\Add{\exp(t\xi)}\cp\: \w)(\wt{g}(p)) - \xi=\Add{\exp(t\xi)}(\xi) -\xi =0,
\end{align*}
where we have used $\Phi$-invariance of $\w$ in the first two steps.
\item
First observe that $\gamma$ is an immersion because\footnote{Observe that $\Ad_g\colon \mg_x \rightarrow \mg_y$ is an isomorphism for $y=\wm(g,x)$. In fact, if $\Ad_{{g}^{-1}}(\g)\in \mg_x$ for $\g\in \mg$, then $\g=\Ad_{g}({\vec{c}})$ for ${\vec{c}}\in \mg_x$, hence $\wm(\exp(t\g),y)=\wm_{g}\cp \wm(\exp\big(t{\vec{c}}), x)=y$ for all $t\in \RR$, i.e., $\g\in \mg_y$.}
\begin{align*}
\dot \gamma (t)=\dd_e\wm_{\wm(\exp(t\vec{g}), x)}(\vec{g})=0\qquad \Longleftrightarrow\qquad \vec{g}\in \mathfrak{g}_{\wm(\exp(t\vec{g}), x)}=\Add{\exp(t\vec{g})}(\mathfrak{g}_x).
\end{align*}
In fact, then $\dot \gamma (t)=0$ implies $\vec{g}=\Add{\exp(t\vec{g})}\big(\vec{h}\big)$ for some $\vec{h}\in \mathfrak{g}_x$, hence $\vec{h}=\Add{\exp(-t\vec{g})}(\vec{g})=\vec{g}$, which contradicts the choice of $\vec{g}$.
If $\gamma$ is injective and $[a,b]\subseteq \RR$, then for $\epsilon>0$ the curve $\gamma|_{[a-\epsilon,b+\epsilon]}$ is an embedding just by compactness of $[a-\epsilon,b+\epsilon]$. Of course, $\gamma|_{(a-\epsilon,a+\epsilon)}$ then is an embedding as well.
Now, assume that $\gamma$ is not injective and observe that
\begin{align}
\label{eq:cycl}
\gamma(t')=\gamma(t)\qquad \Longleftrightarrow\qquad\gamma(t'+a)=\gamma(t+a)\text{ for all }a\in \RR .
\end{align}
Let $\tau$ denote the infimum of $\{t\in \RR_{>0}\:|\: \gamma(0)=\gamma(t)\}\subseteq \RR$.
Then $\gamma(\tau)=\gamma(0)$ by continuity of $\gamma$, and $\tau>0$ because $\gamma$ is locally injective as it is an immersion.
So, if $\gamma(t')=\gamma(t)$ for $t,t'\in \RR$, then $\gamma(0)=\gamma(t-t')$ by \eqref{eq:cycl}, and we have
$0\leq (t-t')+n \tau \leq \tau$ for some $n\in \mathbb{Z}$. Then, minimality of $\tau$ shows that $t-t' + n\tau \in \{0,\tau\}$, hence $t-t'=n' \tau$ for some $n'\in \mathbb{Z}$.
\item
By assumption, we find a diffeomorphism $\adif\colon I'\rightarrow \RR$ with $\gamma_{\g'}^x|_{[0,l']}=\wm_h\cp \gamma_{\g}^x\cp \adif|_{[0,l']}$ where $\adif(0)=0$ and $\adif(l')=l$. Then
\begin{align*}
G_x\ni H(t):=\exp(-t\cdot \mathrm{Ad}_{h^{-1}}(\g'))\cdot \exp(\rho(t)\cdot \g) \qquad \forall\: t\in [0,l']
\end{align*}
and $H(0)=e$.
Hence, $\mg_x\ni\dot H(0)=\dot\rho(0)\cdot\g -\Add{h^{-1}}(\g')$, so that by $\Add{G_{x}}$-invariance of $V$ and since $\g,\g'\in V$, $V\cap \mg_x=\{0\}$, we have
\begin{align}
\label{eq:vor}
\Add{h^{-1}}(\g')=\dot\rho(0)\cdot\g.
\end{align}
Then
$H(t)=\exp(-t \dot\adif(0)\cdot\g)\cdot \exp(\adif(t)\cdot\g)=\exp([\adif(t)-t \dot\adif(0)]\cdot \g)$,
hence
\begin{align*}
\mg_x\ni \dd L_{H(t)^{-1}}\dot H(t)=[\dot\adif(t)-\dot\adif(0)]\cdot\g.
\end{align*}
This shows $\dot\adif(t)=\dot\adif(0)$, hence $\adif(t)=\lambda t$ for $\lambda=\dot\adif(0)$. Then $\lambda=\frac{l}{l'}$ because $l=\adif(l')=\lambda l'$,
so that the rest is clear from \eqref{eq:vor}.
\item
By assumption, we find $g\in G$ and $I,I' \subseteq \RR$ open intervals with
$\wm_g\cp \gamma^x_{\vec{g}}(I)=\gamma^x_{\vec{g}'}(I')$.
Since both curves are embeddings, $\adif\colon I'\rightarrow I$ defined by
\begin{align*}
\adif:=\left(\wm_g\cp \gamma^x_{\vec{g}}\big|_{I}\right)^{-1}\cp \gamma^x_{\vec{g}'}|_{I'}
\end{align*}
is a diffeomorphism
for which we can assume\footnote{In fact, elsewise we replace $\g$ by $-\g$. For this, observe that $\gamma_\g^x|^{-1}_{[0,l]}= \wm_{\exp(l\g)}\cp\gamma_{-\g}^x|_{[0,l]}$.} that $\dot\adif>0$.
Now,
\begin{align*}
G_x\ni \exp(-t'\g')\cdot g\cdot \exp(\adif(t')\:\g)\qquad\forall\:t'\in I'
\end{align*}
and for $t_0'\in I'$ fixed we define
$G_x\ni h:=\exp(-t'_0\vec{g}')\cdot g\cdot\exp(\adif(t'_0)\:\vec{g})$.
Then, for
\begin{align*}
\delta(t'):=h^{-1}\cdot\exp(-t'\g')\cdot g\cdot\exp(\adif(t')\:\g)\qquad\forall\:t'\in I'
\end{align*}
we have $\delta(t'_0)=e$,
and a simple calculation shows that
\begin{align*}
\mathfrak{g}_x\ni\dot\delta(t'_0)= \lambda\vec{g}-\Add{h^{-1}}(\vec{g}')\qquad\text{ for }\qquad\lambda := \dot\adif(t'_0)\neq 0.
\end{align*}
\item
It suffices to consider the case where $g=e$ because $\wm$ is analytic, and $\gamma$ and $\wm_g\cp \gamma_{\g}^x$ share an open segment iff $\wm_{g^{-1}}\cp \gamma$, $\gamma_{\g}^x$ do so.
Then, substituting $\g$ by $-\g$ if necessary we find an analytic diffeomorphism $\adif \colon I\rightarrow \RR$ with $\gamma|_I={\gamma_{\g}^x}\cp \adif$ and $\dot\adif>0$, see also Part \ref{lemma:sim4}). Let $\gamma$ be maximal immersive and denote by $\adif'\colon I'\rightarrow \RR$ the maximal immersive extension of $\adif$. Then $\gamma|_{I'}={\gamma_{\g}^x} \cp \adif'$ because $\gamma|_{I'}$ and ${\gamma_{\g}^x} \cp \adif'$ coincide on $I$. We now have to show that $I'=\operatorname{\mathrm{dom}}[\gamma]$. For this, let $I'=(r',s')$, $\operatorname{\mathrm{dom}}[\gamma]=(r,s)$ and assume that $s'<s$.
We choose a monotonous increasing sequence $\{s'_n\}_{n\in \mathbb{N}}\subseteq I'$ with $\lim_n s'_n= s'$:
\begingroup
\setlength{\leftmarginii}{17pt}
\begin{itemize}
\item
\vspace{-3pt}
If $\lim_n \adif(s'_n)=t$ exists, then $\gamma(s')$ is an accumulation point of $\mathrm{im}[\gamma]\cap \mathrm{im}[{\gamma_{\g}^x}]$, so that
Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt1} provides us with
open intervals $J'\ni s'$ and $J\ni t$,
and an analytic diffeomorphism $\adif_0\colon J'\rightarrow J$ with $\gamma|_{J'}={\gamma_{\g}^x}\cp \adif_0$. Then, glueing together $\adif'$ and $\adif_0$ provides us with a contradiction to maximality of $\adif'$.
\item
\vspace{2pt}
If $\lim_n \adif(s'_n) =\infty$, let\footnote{We define $\tau_\g:=\infty$ iff ${\gamma_{\g}^x}$ is injective, see also next definition.} $0<d<\tau_\g$ and choose $s'_n$ in such a way that
\begin{align*}
\adif(s'_n)=\adif(s'_0)+n\cdot d\qquad \forall\: n\in \NN_{\geq 1}.
\end{align*}
Then, for $x_1:=\wm_{\exp(s'_0\cdot \g)}(x)$, $x_2:=\wm_{\exp(d\cdot \g)}(x_1)\neq x_1$ and $g_n:=\exp(n d\cdot \g)$ we have
\begin{align*}
\lim_n \wm(g_n,x_1)=\gamma(s')=\lim_n \wm(g_n,x_2).
\end{align*}
Hence, $\wm(g,x_1)=\wm(g,x_2)$ for some $g\in G$ by pointwise properness of $\wm$. This implies $x_1=x_2$ and contradicts the choices.
\end{itemize}
Consequently, $s'=s$, and in the same way we see that $r=r'$ holds.
\endgroup
\item
This is a straightforward consequence of Part \ref{lemma:sim5}. In fact, if $\g\xsim\g'$ and $\g'\xsim \g''$ with
\begin{align*}
\wm_g\cp {\gamma_{\g}^x} \cpsim \gamma_{g'}^x \qquad\quad \text{as well as}\qquad\quad \gamma_{g'}^x \cpsim \wm_{g''}\cp \gamma_{\g''}^x,
\end{align*}
then for $K'$ a compact interval by Part \ref{lemma:sim5} we find open intervals $K$ and $K''$ such that
\begin{align*}
\wm_g\cp \gamma^x_{\pm\g}|_{K}\psim \gamma^x_{\g'}|_{K'}\psim \wm_{g''}\cp\gamma^x_{\pm\g''}|_{K''}
\end{align*}
holds, hence $\g\xsim \g''$.
\end{enumerate}
\end{proof}
So, for the case that $\wm$ is analytic and pointwise proper, the above lemma provides us with the following notions:
\begin{definition}
\label{def:liegeneq}
Let $\wm$ be analytic and pointwise proper.
\begin{enumerate}
\item
\label{def:liegeneq144}
For $\g\in\mg \backslash \mg_x$, we denote by $\tau_\g\in \RR_{>0}\sqcup {\infty}$ the period of ${\gamma_{\g}^x}$ introduced in Lemma \ref{lemma:sim}.\ref{lemma:sim2}, where we define $\tau_\g:=\infty$ if ${\gamma_{\g}^x}$ is injective. Recall that ${\gamma_{\g}^x}|_{[a,b]}$ is an embedding iff $b-a<\tau_\g$.
\item
\label{def:liegeneq23}
By \gls{XSIMORB} we will denote the set $(\mg\backslash \mg_x) \slash_{\xsim}$ of equivalence classes w.r.t.\ $\xsim$.
\item
\label{def:liegeneq1}
By $\text{\gls{PAG}}\subseteq \Paw$ we denote the set of all embedded analytic curves of the form $\wm_g \cp {\gamma_{\g}^x}|_K$ for $x\in M$, $\g\in \mg\backslash \mg_x$, $g\in G$ and $K\subseteq \RR$ compact.\footnote{Of course, for $K=[k_0,k_1]$ this means that $k_1-k_0<\tau_\g$ for $\tau_\g$ the period of ${\gamma_{\g}^x}$.}
\item
\label{def:liegeneq2}
By $\text{\gls{PAGS}} \subseteq \Paw$ we will denote the set of all curves equivalent ($\psim$) to a curve in $\Pag$. By the second part of Lemma \ref{lemma:sim}.\ref{lemma:sim5}, this is exactly the set of all $\gamma\in \Paw$ with $\gamma \cpsim \delta$ for some $\delta\in \Pag$.
\item
\label{def:liegeneq11}
For $x\in M$, $g\in G$ and $y:=\wm_g(x)$, we define $\wt{\Add{g}}\colon \xsimorbb_x\rightarrow \xsimorbb_y$, $[\g]\mapsto [\Add{g}(\g)]$. This map is well defined because
\begin{align*}
\g\xsim \g'&\quad\Longrightarrow \quad \wm_{g_0}\cp {\gamma_{\g}^x} \cpsim \gamma^x_{\g'}\:\text{ for some }\: g_0\in G_x\\
& \quad\Longrightarrow \quad \wm_{g\cdot g_0\cdot g^{-1}}\cp \gamma^{y}_{\Ad_g(\g)} \cpsim \gamma^y_{\Ad_g(\g')}\\
&\quad\Longrightarrow \quad \Ad_{g}(\g)\sim_y \Ad_{g}(\g').
\end{align*}
Then, $\wt{\Ad}_{g}$ is even bijective as its inverse is just given by $\wt{\Ad}_{g^{-1}}\colon \xsimorbb_y\rightarrow \xsimorbb_x$.
\end{enumerate}
\end{definition}
We close our considerations by stating that in the analytic and pointwise proper case $\Pag, \Pags$ and $\Paw\backslash \Pags$ are closed under decomposition and inversion of their elements. In particular, by Proposition \ref{rem:euklrem2b} we have the splitting\footnote{Provided, of course, that $\Pag$ and $\Paw\backslash \Pags$ are not empty. Elsewise, the respective factor just has to be dropped.} $\AQRw\cong \AQRInd{\mg}\times \AQRInd{\mg^c}$ where the latter factor is the quantum-reduced space which corresponds to the set $\Paw\backslash \Pags$.
\begin{corollary}
\label{cor:decompo}
Let $\wm$ be analytic and pointwise proper.
Then, $\Pag, \Pags$ and $\Paw\backslash \Pags$ are closed under decomposition and inversion of their elements.
\begin{proof}
For $\Pag$ and $\Pags$ the statement is clear from the definitions and
\begin{align}
\label{eq:hom}
\big[\wm_g\cp{\gamma_{\g}^x}|_{[a,b]}\big]^{-1}= \wm_{g }\cp \gamma_{-\g}^{y}|_{[a,b]}\qquad\text{for}\qquad y:=\wm(\exp([a+b]\cdot\g),x).
\end{align}
Now, if $\gamma$ is not equivalent to an element of $\Pag$, i.e., if $\gamma\in \Paw\backslash \Pags$, then
the same must be true for $\gamma^{-1}$, just because $\Pags$ is closed under inversions and
\begin{align*}
\gamma\psim \delta \qquad \Longleftrightarrow \qquad\gamma^{-1}\psim \delta^{-1}.
\end{align*}
Moreover, by Lemma \ref{lemma:sim}.\ref{lemma:sim5}
there cannot exist a subcurve $\gamma'=\gamma|_K$ of $\gamma$ which is equivalent to an element of $\Pag$, i.e., which is contained in $\Pags$. Consequently, $\Paw\backslash \Pags$ is closed under decompositions and inversions as well.
\end{proof}
\end{corollary}
\subsection{Modifications along Lie Algebra Generated Curves}
\label{sec:ModifLAGC}
In the sequel, let $\wm$ be always analytic and pointwise proper and $\dim[S]\geq 1$.\footnote{Recall that this ensures that the equivalence relations $\csim$ and $\psim$ coincide, see Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt2}.} Moreover, let $\Pagw\subseteq \Paw$ be a $\Phi$-invariant subset which is closed under decompositions and inversions and that contains all Lie algebra generated curves, i.e., the set $\Pag$ from Definition \ref{def:liegeneq}.\ref{def:liegeneq1}. Then, according to Convention \ref{conv:invhomm}, we will denote by $\ARQLAI$ the respective quantized reduced classical configuration space, and
by $\text{\gls{IHOMLAI}}\cong\text{\gls{KAPPALAI}}\big(\text{\gls{AQRLAI}}\big)$ the respective quantum-reduced one.
We now are going to modify invariant homomorphisms, i.e., elements of $\IHOMLAI$ along Lie algebra generated curves. Here, we will make use of the fact that for each $p\in F_x$ the action $\Phi$ provides us with a canonical lift of ${\gamma_{\g}^x}$, namely $t\mapsto \Phi_{\exp(t\g)}(p)$. Basically, then we will mimic \eqref{eq:trivpar} by replacing the last factor
\begin{align*}
\Psi_\w \colon \g \mapsto \exp(-\w(\wt{g}(p)))
\end{align*}
by maps
$\Psi\colon V\rightarrow S$ (with $V$ a special linear subspace of $\mg$)
having the correct equivariance property. For this, observe that by invariance of $\w$
we have
\begin{align*}
\Psi_\w \cp\Ad_h =\alpha_{\phi_p(h)}\cp\:\Psi_\w\qquad\forall\:h\in G_{\pi(p)}
\end{align*}
with $\text{\gls{FIBAP}}\colon G_{\pi(p)}\rightarrow S$ the unique Lie group homomorphisms which fulfils (cf.\ \eqref{eq:phip})
\begin{align*}
\Phi(h,p)=p\cdot \fiba_p(h)\qquad \forall\:h\in G_{\pi(p)}.
\end{align*}
For instance,
let $V\subseteq \mg$ be an $\Add{G_{x}}$-invariant linear subspace with $V\cap \mg_{x}=\{0\}$. Then, a linear map $L\colon V\rightarrow \ms$ is called $\Add{G_{x}}^p$-equivariant iff
\begin{align*}
\Ad_{\fiba_p(h)}\cp \: L = L\cp \Ad_{h}\qquad \forall\: h\in G_{x},
\end{align*}
and in the second part of Proposition \ref{th:invhomm} we will modify invariant homomorphism by means of such maps. This will be helpful, in particular, for our investigations concerning the inclusion relations between the spaces $\ARQLAI$ and $\AQRLAI$ in Subsection \ref{sec:inclrel}. However, for our investigations in Subsection \ref{sec:ConSp} (construction of measures), we will need a slightly different version of this:
\begin{definition}
\label{def:eqmaps}
A map $\Psi\colon \spann_\RR(\g) \rightarrow S$ with
\begin{align}
\label{eq:equii}
\Psi((\lambda+\mu)\cdot\g)=\Psi(\lambda\cdot \g)\cdot \Psi(\mu\cdot \g)\qquad \forall\:\lambda,\mu\in \RR
\end{align}
is said to be $\Add{G_{[\g]}}^p$-equivariant iff
\begin{align}
\label{eq:equiii}
\alpha_{\fiba_p(h)}\cp\:\Psi=\Psi\cp \Ad_h\qquad\forall\:h\in G^x_{[\g]}
\end{align}
holds. \hspace*{\fill}$\lozenge$
\end{definition}
Then, in order to obtain well-defined homomorphisms when modifying by means of such maps, we will be forced to require $\g$ to have the additional property of stability. This generalizes the condition in Lemma \ref{lemma:sim}.\ref{lemma:sim3} which automatically holds for $\Add{G_{x}}$-invariant linear subspaces as discussed above.
\begin{definition}[Independent and Complete Families]
\label{def:stable}
\begin{enumerate}
\item
\label{def:stable2}
An element $\g\in \mg \backslash\mg_x$ is called \emph{stable} iff
\begin{align}
\label{eq:stab1}
\gamma^x_{\g}|_{[0,l]} \psim \gamma^x_{\pm\Ad_h(\g)}|_{[0,l']}\quad\text{for}\quad h\in G_x\qquad \Longrightarrow\qquad h\in G_{[\g]}^x
\end{align}
\item
\label{eq:iindepg3}
\itspacec
A family of stable elements $\{\g_{\alpha}\}_{\alpha\in I_x}\subseteq \mg\backslash \mg_x$ is said to be \emph{independent} iff $\g_\alpha \nsim_x \g_\beta$ holds for all $\alpha,\beta \in I_x$ with $\alpha\neq \beta$. It is called \emph{complete} iff for each $\mathfrak{r}\in \xsimorb$ we have $\g_\alpha \in \mathfrak{r}$ for some $\alpha\in I_x$.\footnote{Recall that $\xsimorb$ denotes the classes in $\mg\backslash \mg_x$ w.r.t.\ $\xsim$.}
\end{enumerate}
\end{definition}
\begin{lemrem}[Stability]
\label{rem:dfggfg}
\begin{enumerate}
\item
\label{rem:dfggfg0}
It already follows from stability \eqref{eq:stab1} of $\g \in \mg\backslash \mg_x$ (and analyticity) that
\begin{align*}
\pm\Ad_h(\g)=\g\qquad\text{and}\qquad l'=l\qquad \text{holds},
\end{align*}
hence $\gamma^x_{\g} = \gamma^x_{\pm\Ad_h(\g)}$.
In fact, by Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve4}, we have $\Ad_h(\g)=q\cdot \g$ for $q\in\{-1,1\}$, hence $\gamma^x_{\g}|_{[0,l]} = \gamma^x_{\pm q\cdot\g}\cp \adif|_{[0,l]}$ for $\adif\colon [0,l]\rightarrow [0,l']$ a diffeomorphism with $\dot\adif>0$. Then,
\begin{align*}
H(t):=\exp(-t\cdot \g)\cdot \exp(\pm q \adif(t)\cdot \g)\in G_x \qquad \forall\: t\in [0,l],
\end{align*}
so that $\pm q=1$ and $\dot\adif=1$ because
\begin{align*}
\mg_x\ni \dd L_{H(t)^{-1}}\dot H(t)=[\pm q\dot\adif(t)-1]\cdot \g\qquad \forall \: t\in [0,l].
\end{align*}
This shows $\pm\Ad_h(\g)=\pm q\cdot \g=\g$ as well as $\adif(t)=t$, hence $l'=\adif(l)=l$.
\item
\label{rem:dfggfg1}
As already mentioned in Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve2}, Condition \eqref{eq:stab1} generalizes
the property from Lemma \ref{lemma:sim}.\ref{lemma:sim3} in a specific way.
More concretely, $\g_0\in \mg\backslash \mg_x$ is stable iff
\begin{align}
\label{eq:stab2}
\gamma^x_{\g'}|_{[0,l']} \psim \gamma^x_{\Ad_h(\g)}|_{[0,l]}\quad\text{for}\quad h\in G_x,\:\g,\g'\in [\g_0]\qquad \Longrightarrow\qquad \frac{l}{l'}\g=\Ad_{h^{-1}}(\g').
\end{align}
In fact, obviously \eqref{eq:stab2} implies \eqref{eq:stab1}. Now, if \eqref{eq:stab1} holds for $\g_0$, and if we have
\begin{align}
\label{eq:stab2gfgf}
\gamma^x_{\g'}|_{[0,k']} \psim \gamma^x_{\Ad_h(\g)}|_{[0,k]}\quad\text{for}\quad h\in G_x,\:\g,\g'\in [\g_0],
\end{align}
then $\g=q\lambda\cdot \g_0$, $\g'=q'\lambda'\cdot \g_0$ for some $\lambda,\lambda'>0$ and $q,q'\in \{-1,1\}$, hence
\begin{align*}
\gamma^x_{q'\cdot \g_0}\big|_{[0,\lambda'k']} \psim \gamma^x_{q\cdot \Ad_h(\g_0)}\big|_{[0,\lambda k]}.
\end{align*}
Then, the group property of $G^x_{[\g_0]}$ and \eqref{eq:stab1} show that $h\in G^x_{[\g_0]}$ if $q'=1$ or $q=1$. Moreover, if $q,q'=-1$, by analyticity of the involved curves we have
\begin{align*}
\gamma^x_{ \g_0}|_{[0,r']} \psim \gamma^x_{ \Ad_h(\g_0)}|_{[0,r]}
\end{align*}
for some $r',r>0$, so that \eqref{eq:stab1} shows $h\in G^x_{[\g_0]}$ as well. Consequently,
\begin{align*}
\Ad_h(\g)=\pm \g=\lambda \cdot \g'\qquad\text{for some}\qquad \lambda\neq 0,
\end{align*}
and then the arguments from Part \ref{rem:dfggfg0}) applied to \eqref{eq:stab2gfgf} show the right hand side of \eqref{eq:stab2}.
\item
\label{rem:dfggfg2}
It is clear from \eqref{eq:stab2} and Lemma \ref{lemma:sim}.\ref{lemma:sim3} that $\g$ is stable whenever $\spann_\RR(\g)$ is contained in an $\Ad_{G_x}$-invariant linear subspace $V$ of $\mg$ with $V\cap \mg_x=\{0\}$. For instance, this is always true if $G_x=\{e\}$ holds.
\item
Let
$L\colon V\rightarrow \ms$ be an $\Ad_{G_x}^p$-equivariant linear map and $0\neq \g\in V$. Then, it is an interesting observation that\footnote{Here, the minus sign is just to show the resemblance to \eqref{eq:trivpar}.}
\begin{align*}
\Psi\colon \spann_\RR(\g)\rightarrow S,\quad \g\mapsto \exp(- L(\g))
\end{align*}
is $\Add{G_{[\g]}}^p$-equivariant. Indeed, examples for such $\Ad_{G_x}^p$-equivariant linear maps $L\colon \mg\rightarrow \ms$ are given by $(\Phi_p^*\w)|_{\mg}$ for $\w$ a $\Phi$-invariant connection on $P$.
\item
In Proposition \ref{th:invhomm}, we will modify elements $\homm\in \IHOMLAI$, first, by specifying their values on the curves $\gamma_{\g_0}^x|_{[0,l]}$. Second, by extending them to the curves $\wm_g\cp \gamma_{\g_0}^x|_{[0,l]}$ in such a way that the resulting homomorphism $\homm'$ is invariant.
Then, in order to guarantee that $\homm'$ is well defined, we will need condition \eqref{eq:stab1}, i.e., stability of $\g$. This condition guarantees that $\homm'$ takes the same values on each two curves being related as on the left hand side of \eqref{eq:stab2}. Basically, this is just because the equivariances of the involved maps then will cancel out each other in the correct way, see \eqref{eq:clacu0}.
\item
\label{rem:dfggfg44}
The stability property of an element $\g\in \mg\backslash\mg_x$ usually has to be checked by hand.
Indeed, in Subsection \ref{sec:ConSp} (for the case that $S=\SU$) we will discuss the situation where each $\wm$-orbit $\m$ admits an independent and complete family $\{\g_\alpha\}_{\alpha \in I_x}\subseteq \mg\backslash\mg_x$ (of stable elements) for some $x\in\m$. We will show that then
$\IHOMLAS$ is homeomorphic to a Tychonoff product of compact Hausdorff spaces on each of which a natural Radon measure exists.
This will allow us to a define a normalized Radon measure on $\IHOMLAS$ just by taking the Radon product one. In particular, this will be possible for the cases of \mbox{(semi)-homogeneous}, spherically symmetric and homogeneous isotropic loop quantum cosmology as for these situations we will show by hand that such independent and complete families of stable elements exist.
\hspace*{\fill}$\lozenge$
\end{enumerate}
\end{lemrem}
The next lemma collects some elementary properties of independent and complete families that will become relevant for our considerations in Subsection \ref{sec:ConSp}.
Recall that there we use such families in order to parametrize the space $\IHOMLAS$. The last point of the next lemma then will show injectivity of this parametrization.
The first two points are just to switch between such different ones.
\begin{lemma}
\label{lemma:completee}
Let $y=\wm_g(x)$ for $x\in M$ and $g\in G$. Moreover, let
$\{\g_{\alpha}\}_{\alpha\in I_x}\subseteq \mg\backslash \mg_x$ and $\{\g'_{\beta}\}_{\beta\in I_y}\subseteq \mg\backslash \mg_y$ both be independent and complete.
\begin{enumerate}
\item
\label{lemma:completee1}
There is a unique bijection $\tau_{xy}\colon I_x\rightarrow I_y$ with\footnote{Recall Definition \ref{def:liegeneq}.\ref{def:liegeneq11} for the definition of $\wt{\Ad}$.}
$\wt{\Ad}_g([\g_\alpha])=[\g_{\tau_{xy}(\alpha)}]$ for all $\alpha\in I_x$.
\item
\label{it:completee1}
Let $\wt{\Ad}_g([\g])=[\g']$ for $[\g] \in \xsimorb$ and $[\g']\in \xsimorbb_y$.
Then, we find $g_0\in G$ and an analytic diffeomorphism $\adif\colon \RR\rightarrow \RR$ with $\adif(0)=0$ and $\adif(\tau_\g)=\tau_{\g'}$, such that
\begin{align}
\label{eq:simcon}
\gamma^y_{\g'}= \wm_{g_0}\cp \gamma^x_{\pm \g}\cp \adif.
\end{align}
In particular, if $\g\xsim \g'$ for $\g,\g'\in \mg\backslash \mg_x$, then we find $h\in G_x$ with
\begin{align}
\label{eq:simconfestfuss}
\gamma^x_{\g'}= \wm_h\cp \gamma^x_{\pm \g}\cp \adif =\gamma^x_{\pm \Ad_h(\g)}\cp \adif.
\end{align}
\item
\label{lemma:homzueq33}
The values of $\homm\in \IHOMLAI$ on the curves
\begin{align*}
\wm_g\cp \gamma^y_{\g'}|_{[0,l']}\qquad \text{for all }g\in G,\:\g'\in \mg\backslash\mg_y\text{ and } l'\in (0,\tau_{\g'})
\end{align*}
are completely determined by its values on the curves $\gamma^x_{\g_\alpha}|_{[0,l]}$ for all $\alpha\in I_x$ and $l\in (0,\tau_{\g_\alpha})$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
This is clear from bijectivity of $\wt{\Ad}_g$.
\item
Since $\Ad_g(\g)\xsim \g'$, we find $g'\in G$ with $\gamma_{\Ad_g(\g)}^y\cpsim \wm_{g'}\cp \gamma^y_{\g'}$. Hence,
\begin{align*}
\wm_g\cp \gamma_{\g}^x=\gamma_{\Ad_g(\g)}^y\cpsim \wm_{g'}\cp \gamma^y_{\g'}\qquad&\Longrightarrow\qquad
\wm_{g^{-1} g'}\cp \gamma^y_{\g'}\cpsim{\gamma_{\g}^x} \\
\qquad&\Longrightarrow\qquad \wm_{g^{-1} g'}\cp\gamma^y_{\g'}= \gamma^x_{\pm\g}\cp \adif\hspace{50pt} (\text{Lemma } \ref{lemma:sim}.\ref{lemma:sim5})
\\ \qquad&\Longrightarrow\qquad \gamma^y_{\g'}= \wm_{g'^{-1} g}\cp\gamma^x_{\pm\g}\cp \adif
\end{align*}
for $\adif\colon \RR\rightarrow \RR$ an analytic diffeomorphism.
\item
By \eqref{eq:simcon} we find $\alpha\in I_x$ with $\wm_g\cp \gamma^y_{\g'}|_{[0,l]}\psim \wm_{g\cdot g_0}\cp \gamma_{\pm\g_\alpha}^x|_{[0,\adif(l)]}$, hence
\begin{align*}
\wm_g\cp \gamma^y_{\g'}|_{[0,l]}&\psim \wm_{g\cdot g_0}\cp \gamma_{\g_\alpha}^x|_{[0,\adif(l)]}\qquad\quad \text{or}\\
\wm_g\cp \gamma^y_{\g'}|_{[0,l]}&\psim \wm_{g\cdot g_0}\cp \gamma_{-\g_\alpha}^x|_{[0,\adif(l)]}\psim \wm_{g\cdot g_0}\cp \wm_{\exp(-\adif(l)\cdot \g_\alpha)}\cp \big[\gamma_{\g_\alpha}^x|_{[0,\adif(l)]}\big]^{-1}.
\end{align*}
The claim now follows from the invariance and homomorphism properties of $\homm$.
\end{enumerate}
\end{proof}
\end{lemma}
Before we come to the desired modification result, we want to show how the action
\begin{align*}
\text{\gls{ADDSTRICH}}\colon G_x\times \pr_\mg(\mg\backslash \mg_x)\rightarrow \pr_\mg(\mg\backslash \mg_x),\quad
(h,[\g])\mapsto \big[\!\Ad_h(\g)\big]
\end{align*}
introduced in Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve1} can help to find independent and complete families as discussed above.
\begin{remark}[$\Ad'$-Orbits]
\label{lemremmgohnermgx}
Let $\madgx:=\pr_\mg(\mathfrak{g}\backslash \mg_x)\slash G_x$ denote the set of $\Ad'$-orbits in $\pr_\mg(\mathfrak{g}\backslash \mg_x)$ and choose
$\oo \in \madgx$ as well as $[\g],[\g']\in \oo$. Then, $\g'=\lambda \Ad_h(\g)$ for some $\lambda\neq 0$ and $h\in G_x$. Hence,
\begin{align}
\label{eq:compladorb}
\gamma^x_{\g'}=\gamma^x_{\lambda \Ad_h(\g)} = \wm_h\cp \gamma_{\lambda \g}^x \cpsim \wm_h\cp \gamma_{\g}^x,
\end{align}
which shows that $\g\xsim\g'$.
So, in order to obtain a complete and independent family $\{\g_\alpha\}_{\alpha\in I}\subseteq \mg\backslash \mg_x$ as in Definition \ref{def:stable}.\ref{eq:iindepg3}, one can proceed as follows:
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item[--]
\vspace{-2pt}
First, one identifies two orbits $\oo,\oo' \in \madgx$ iff there are $[\g]\in \oo$ and $[\g']\in \oo'$ with $\g\xsim \g'$.
\item[--]
Second, if possible, one chooses a stable element $[\g_\alpha]$ in each of the ``remaining'' orbits.
\end{itemize}
\endgroup
\end{remark}
The first part of the next proposition shows, how one can modify invariant homomorphisms along Lie algebra generated curves that correspond to stable elements $\g\in \mg\backslash \mg_x$. The second part is adapted to the requirements of the next subsection, where we will use it in order to construct invariant generalized connections that cannot be approximated by the classical (smooth) invariant ones. Hence, cannot be contained in $\ARQLAI$.
\begin{proposition}
\label{th:invhomm}
Let $\wm$ be analytic and pointwise proper, assume that $\IHOMLAI\neq \emptyset$, fix $p\in P$ and define $x:=\pi(p)$.
\begin{enumerate}
\item
\label{th:invhomm1}
Let $\{\g_\alpha\}_{\alpha \in I_x}\subseteq \mg\backslash\mg_x$ be independent, and $\{\Psi_\alpha\}_{\alpha\in I_x}$ a family of $\Add{G_{[\g_\alpha]}}^p$-equivariant maps $\Psi_\alpha\colon \spann_\RR(\g_\alpha)\rightarrow S$.
For $\alpha\in I_x$, we define
\begin{align}
\label{eq:ihdef}
\Upsilon_{\pm, \alpha,l}\colon F_x\ni p'\mapsto
\Phi_{p}\left(\exp(\pm l\cdot \g_\alpha)\right)\cdot \Psi_\alpha(\pm l\cdot \g_\alpha) \cdot \Delta(p,p').
\end{align}
Then, for each $\homm'\in \IHOMLAI$ the map
\begin{align}
\label{eq:homdeff}
\homm(\gamma)(p'):=
\begin{cases}
(\Phi_g\cp \Upsilon_{\pm, \alpha,l})(\Phi_{g^{-1}}(p')) &\mbox{if } \gamma \csim \wm_g\cp \gamma_{\pm\g_\alpha}^x|_{[0,l]} \:\text{ for }\:\alpha \in I_x,\:g\in G\\
\homm'(\gamma)(p') & \mbox{else}
\end{cases}
\end{align}
is a well-defined element of $\IHOMLAI$.
\item
\label{th:invhomm2}
Let $\{V_\alpha\}_{\alpha\in I}$ be non-trivial $\Add{G_{x}}$-invariant linear subspaces of $\mg$ for which the sums $V_\alpha \oplus \mg_x$ (for all $\alpha \in I$) and $V_\alpha\oplus V_\beta \oplus \mg_x$ (for $\alpha\neq \beta$) are direct. Moreover, for each $\alpha\in I$ let $L_\alpha\colon V_\alpha \rightarrow \ms$ be a non-trivial $\Add{G_{x}}$-equivariant linear map.
For $\g_\alpha \in V_\alpha\backslash\{0\}$, let
\begin{align}
\label{eq:defmodh}
\Upsilon_{\g_\alpha,l}\colon F_x\ni p'\mapsto \Phi_{p}\left(\exp(l\cdot \g_\alpha)\right)\cdot \exp\left(-l L_\alpha(\g_\alpha)\right) \cdot \Delta(p,p').
\end{align}
Then, for each $\homm'\in \IHOMLAI$ the map
\begin{align}
\label{eq:homdef}
\homm(\gamma)(p'):=
\begin{cases}
(\Phi_g\cp \Upsilon_{\g_\alpha,l})(\Phi_{g^{-1}}(p')) &\mbox{if } \gamma \csim \wm_g\cp \gamma_{\g_\alpha}^x|_{[0,l]} \text{ for }\g \in V_\alpha, g\in G\\
\homm'(\gamma)(p') & \mbox{else}
\end{cases}
\end{align}
is a well-defined element of $\IHOMLAI$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
The crucial part is well-definedness, the rest are just straightforward calculations. For well-definedness, let $\gamma\in \Pagw$, $\alpha,\beta \in I_x$, $\g=\pm\g_\alpha$, $\g'=\pm\g_\beta$ and $g,g'\in G$ with
\begin{align*}
\gamma \csim \wm_{g}\cp \gamma_{\g}^x|_{[0,l]}\qquad\qquad\text{and}\qquad\qquad \gamma \csim\wm_{g'}\cp \gamma_{\g'}^x|_{[0,l']}.
\end{align*}
Then, $\wm_{g}\cp \gamma_{\g}^x|_{[0,l]} \csim\wm_{g'}\cp \gamma_{\g'}^x|_{[0,l']}$, and we have to verify that $\homm(\wm_g\cp \gamma_{\g}^x|_{[0,l]})=\homm(\wm_{g'}\cp \gamma_{\g'}^x|_{[0,l']})$ holds. Applying the definitions, we see that this is equivalent to show that
\begin{align*}
\homm(\gamma_{\g'}^x|_{[0,l']})= \homm(\wm_{g'^{-1} g}\cp \gamma_{\g}^x|_{[0,l]})
\end{align*}
holds. Since $g'^{-1}g\in G_x$ and $\gamma_{\g'}^x|_{[0,l']}\csim \wm_{g'^{-1}g}\cp \gamma_{\g}^x|_{[0,l]}$,
it suffices to show that
\begin{align*}
\gamma_{\g'}^x|_{[0,l']}\csim \wm_h\cp \gamma_{\g}^x|_{[0,l]} \text{ for }h\in G_x \qquad \Longrightarrow\qquad \homm(\gamma_{\g'}^x|_{[0,l']})=\homm(\wm_h\cp \gamma_{\g}^x|_{[0,l]}).
\end{align*}
However, in the above situation we have $\g_\alpha\xsim \g_\beta$, hence $\alpha=\beta$ by independence of $\{\g_\alpha\}_{\alpha\in I_x}$.
Then $\g,\g'\in [\g_\alpha]$, so that by stability of $\g_\alpha$ and \eqref{eq:stab2}
we have
$\lambda \g=\Ad_{h^{-1}}(\g')$ for
$\lambda=\frac{l}{l'}$,
hence
\begin{align}
\label{eq:clacu0}
\begin{split}
\homm(\wm_h\cp \gamma_{\g}^x|_{[0,l]})(p') &=\Phi_p(h\cdot\exp(l'\lambda\hspace{1pt} \g))\cdot \Psi_\alpha( l\cdot \g)\cdot \Delta(p,\Phi_{h^{-1}}(p'))\\
&=\Phi(h\cdot h^{-1}\exp(l'\g')\cdot h, p)\cdot\Psi_\alpha(l \cdot \g)\cdot \fiba_p(h)^{-1}\cdot \Delta(p,p')\\
&=\Phi(\exp(l' \g'),p)\cdot\Psi_\alpha\big(l\Ad_{h}(\g)\big)\cdot \Delta(p,p')\\
&=\Phi(\exp(l' \g'),p)\cdot \Psi_\alpha(l'\g')\cdot \Delta(p,p')\\
&= \homm\big(\gamma_{\g'}^x|_{[0,l']}\big)(p').
\end{split}
\end{align}
Now, it is immediate from the definitions that
$\homm(\gamma)(p'\cdot s)=\homm(\gamma)(p')\cdot s$ for all $s\in S$ and that
\begin{align*}
\homm(\wm_g\cp \gamma)(\Phi_g(p))=(\Phi_g\cp\homm)(\gamma)(p)\qquad \forall\: g\in G.
\end{align*}
So, in order to show
that $\homm\in \IHOMLAI$ it remains to show the homomorphism properties of $\homm$. Here, it suffices to verify these properties for such curves that are equivalent to one of the curves $\wm_g\cp\gamma^x_{\g}|_{[0,l]}$ for $\g=\pm\g_\alpha$ with $\alpha\in I_x$. In fact, if $\gamma$ is not equivalent to one of these curves, then by Corollary \ref{cor:decompo} the same is true for its inverse and all its subcurves $\gamma|_{K'}$ for $K'\subseteq \operatorname{\mathrm{dom}}[\gamma]$ so that the homomorphism properties here are clear from $\homm'\in \IHOMLAI$.
Now, since
\begin{align*}
\wm_g\cp
{\gamma_{\g}^x}|_{[a,b]}\csim\wm_{g\cdot \exp(a \g)}\cp \gamma_{\g}^x|_{[0,b-a]} \qquad\text{as well as}\qquad \big[{\gamma_{\g}^x}|_{[a,b]}\big]^{-1}\csim\gamma_{-\g}^x|_{[-b,-a]}
\end{align*}
for $\gamma\csim \wm_g\cp\gamma^x_{\g}|_{[0,l]}$ we have $\gamma^{-1} \csim \wm_{g\cdot \exp(l\vec{g})}\cp \gamma_{-\g}^x|_{[0,l]}$. Hence, for $h:=g\cdot \exp(l\cdot\vec{g})$ and $\g=\pm \g_\alpha$ we obtain
\begin{align*}
\homm\big(\gamma^{-1})(q
%
&=(\Phi_{h}\cp\Upsilon_{\mp,\alpha,l})(\Phi_{h^{-1}}(q))\\
%
%
&=\Phi_h\cp \Phi_{p}\left(\exp(-l\cdot\vec{g})\right)\cdot \Psi_\alpha^{-1}(l \g)\cdot \Delta(p,\left(\Phi_{h^{-1}}(q)\right)).
\end{align*}
Then, for $q:=\homm(\gamma)(p')=(\Phi_g\cp\Upsilon_{\pm,\alpha,l})(\Phi_{g^{-1}}(p'))$ we have
\begin{align*}
\Delta\left(p,\Phi_{h^{-1}}(q)\right)&=\Delta\left(p,(\Phi_{\exp(-l\cdot\g)}\cp\Upsilon_{\pm,\alpha,l})(\Phi_{g^{-1}}(p'))\right) \\
&=\Delta\left(p,p\cdot\Psi_\alpha(l \cdot\g)\cdot \Delta(p,\Phi_{g^{-1}}(p'))\right)\\
&=\Psi_\alpha(l \cdot\g)\cdot \Delta\left(p,\Phi_{g^{-1}}(p')\right).
\end{align*}
Since $\Phi_h\cp \Phi_{p}\left(\exp(-l\vec{g})\right)=\Phi\left(g\cdot \exp(l\g)\cdot \exp(-l\g) ,p\right)=\Phi(g,p)$, we get
\begin{align*}
\homm\big(\gamma^{-1}\big)\big(\homm(\gamma)\big(p'\big)\big)&= \Phi_g(p)\cdot \Delta\left(p,\Phi_{g^{-1}}(p')\right)\\
&=\Phi\left(g,p\cdot \Delta\left(p,\Phi_{g^{-1}}(p')\right)\right)=\Phi(g, \Phi(g^{-1},p'))=p',
\end{align*}
hence $\homm\big(\gamma^{-1}\big)\cp \homm(\gamma)=\id_{F_{\gamma(0)}}$.
Finally, let $\gamma\csim \wm_g\cp \gamma_{\g}^x|_{[0,l]}$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ and $s\in (a,b)$. We choose $l'\in (0,l)$ with $\gamma(s)=\wm_g\cp \gamma_{\g}^x(l')$ and define $\gamma_1:=\gamma|_{[a,s]}$ as well as $\gamma_2:=\gamma|_{[s,b]}$. Then
\begin{align*}
\gamma_1& \csim \wm_g\cp \gamma_{\g}^x|_{[0,l']}\hspace{73pt}\qquad\text{and}\\
\gamma_2& \csim \wm_g\cp \gamma_{\g}^x|_{[l',l]}\csim \wm_{h}\cp \gamma_{\g}^x|_{[0,l-l']}\quad \text{for}\quad h:=g\cdot \exp(l'\vec{g}).
\end{align*}
Now, $\Phi_{h^{-1}}(\homm(\gamma_1)(p'))=p\cdot \Psi_\alpha(l' \g)\cdot \Delta(p,\Phi_{g^{-1}}(p'))$ and
\begin{align*}
\homm(\gamma_2)(q)&=(\Phi_h\cp\Upsilon_{\pm,\alpha,l-l'})(\Phi_{h^{-1}}(q))\\
&=\Phi\left(g\cdot \exp(l'\g)\cdot \exp([l-l']\hspace{1pt} \g), p\right)\cdot \Psi_\alpha([l-l']\hspace{1pt} \g)\cdot \Delta(p,\Phi_{h^{-1}}(q))\\
&=\Phi\left(g\cdot \exp(l\cdot\g), p\right)\cdot \Psi_\alpha([l-l']\hspace{1pt} \g)\cdot \Delta(p,\Phi_{h^{-1}}(q)),
\end{align*}
so that
$\homm(\gamma_2)\cp\homm(\gamma_1)(p')=\Phi\left(g\cdot \exp(l\cdot\g), p\right)\cdot \Psi_\alpha(l \g)\cdot \Delta(p,\Phi_{g^{-1}}(p'))
=\homm(\gamma)(p')$.
\item
The crucial part is well-definedness, as the rest follows analogously to Part \ref{th:invhomm1}). Now, as in Part \ref{th:invhomm1}) it suffices to show that for $\g\in V_\alpha$, $\g'\in V_\beta$, $h\in G_x$
\begin{align*}
\gamma_{\g'}^x|_{[0,l']}\csim \wm_h\cp\gamma_{\g}^x|_{[0,l]}\qquad\Longrightarrow \qquad \homm(\gamma_{\g'}^x|_{[0,l']})=\homm(\wm_h\cp\gamma_{\g}^x|_{[0,l]}).
\end{align*}
Now, in the above situation we have $\alpha=\beta$ just by Lemma \ref{lemma:sim}.\ref{lemma:sim4} and the direct sum property $V_\alpha \oplus V_\beta \oplus \mg_x$. By Part \ref{lemma:sim3}) of the same lemma we even have $\textstyle\frac{l}{l'} \g=\Ad_{h^{-1}}(\g')$, so that for
\begin{align*}
\Psi_\alpha(l\cdot \g):=\exp(-l L_\alpha(\g))\qquad \forall\:l>0,\:\forall\: \g\in V_\alpha\text{ and } \alpha \in I
\end{align*}
the calculation \eqref{eq:clacu0} shows the claim.
\end{enumerate}
\end{proof}
\end{proposition}
\subsection{Inclusion Relations}
\label{sec:inclrel}
In this brief subsection, we will use the modification results from the previous part, in order to derive some general conditions which allow to decide whether
the inclusion
$\ARQLAI\subseteq\AQRLAI$
is proper. This will be done in Proposition \ref{prop:incl}, where we construct elements of $\IHOMLAI\cong \AQRLAI$ that cannot be approximated by the elements of $\iota_\Con(\AR)\subseteq \AQRLAI$, i.e., by classical (smooth) invariant connections. In the first part of this proposition, we will provide a criterion which can be applied whenever the set $\AR$ of invariant connections is explicitly know.
Then, in the last two parts of the same proposition, we will basically use that due to formula \eqref{eq:trivpar} (and linearity of the involved maps) the parallel transports along Lie algebra generated curves which correspond to linearly dependent Lie algebra elements are related in a certain way. In particular, this will allow us to show that quantization and reduction do not commute in \mbox{(semi-)homogeneous} LQC as well.
In the following, we still assume that $S$ is compact and connected with $\dim[S]\geq 1$, and that $\wm$ is analytic and pointwise proper. We fix $p\in P$, define $x:=\pi(p)$, and let $V, V_1,V_2,V_3$ be non-trivial $\Add{G_x}$-invariant linear subspaces of $\mg$ such that $V_i\oplus V_j\oplus \mg_x$ is direct for all $1\leq i\neq j\leq 3$ and
\begin{align*}
V\cap \mg_x=\{0\}\qquad V_1\cap \mg_x=\{0\} \qquad V_2\cap \mg_x=\{0\}\qquad V_3\cap \mg_x=\{0\}.
\end{align*}
Then, by $L,L_1,L_2,L_3\neq 0$ we will denote respective non-trivial $\Add{G_x}^p$-equivariant linear maps.\footnote{Observe that
each linear map $L\colon V\rightarrow \ms$ is $\Add{G_x}^p$-equivariant if $S$ is commutative.}
\begin{proposition}
\label{prop:incl}
If
$\IHOMLAI\neq \emptyset$,
then
we have $\ARQLAI\subsetneq \AQRLAI$ if:
\begin{enumerate}
\item
\label{prop:incl1}
We find $\g\in \mg\backslash\mg_x$ stable and $\Psi\colon \spann_\RR(\g)\rightarrow S$ an $\Add{G_{[\g]}}^p$-equivariant map, such that
\begin{align}
\label{eq:cccccccc}
\Psi(\g)\notin\ovl{C}\qquad\quad\text{for}\qquad\quad C:=\bigcup_{\w\in \AR}\exp\left(\w\!\left(\hspace{0.2pt}-\dd_e\Phi_p(\g)\right)\right).
\end{align}
\item
\label{prop:incl2}
We have $V\oplus \mg_x \subsetneq \mg$ and $S=S^1$.
\item
\label{prop:incl3}
We have $S=\SU$ and there are $\g_i\in V_i$ for $i=1,2,3$, such that $L_1(\g_1),L_2(\g_2),L_3(\g_3)$ are linearly independent and $\g_3\in\spann_\RR(\g_1,\g_2)$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Define $\homm$ by \eqref{eq:homdeff} for $|I_x|=1$, $\g_\alpha=\g$ and $\Psi_\alpha=\Psi$, and assume that
$\homm \in \kappa_{\mg'}(\ARQLAI)$. Then, we find a net $\{\w_\alpha\}_{\alpha\in I}\subseteq \AR$ with $\{\kappa_{\mg'}(\iota_\Con(\w_\alpha))\}_{\alpha\in I}\rightarrow \homm$, hence $\homm(\gamma)(p)=\lim_\alpha\parall{\gamma}{\w_\alpha}(p)$ for
$\gamma:={\gamma_{\g}^x}|_{[0,1]}$. Consequently,
\begin{align*}
\Phi_p(\exp(\g))\cdot \Psi(\g)&=
\homm(\gamma)(p)= \lim_\alpha\parall{\gamma}{\w_\alpha}(p)\\
&\hspace{-3pt}\stackrel{\eqref{eq:trivpar}}{=}\hspace{-1pt}
\lim_\alpha \Phi_p(\exp(\g))\cdot \exp\left(\w_\alpha\!\left(-\wt{g}(p)\right)\right)\\
&=
\lim_\alpha \Phi_p(\exp(\g))\cdot \exp\left(\w_\alpha\!\left(-\dd_e\Phi_p(\g)\right)\right) \in \Phi_p(\exp(\g))\cdot \ovl{C},
\end{align*}
which contradicts the choice of $\g$.
\item
We here only sketch the proof. The details can be found in Appendix \ref{app:incl}.
\begingroup
\setlength{\leftmarginii}{25pt}
\begin{itemize}
\item[(a)]
\vspace{-4pt}
We choose $0\neq \g_3\in V$, $\g_1\in \mg\backslash [V\oplus \mg_x]$ and define $\g_2:=\g_3+\g_1$. Then, for $i=1,2,3$ we let $\gamma_i:=\gamma_{\g_i}^x|_{[0,l]}$ for some $l>0$ with $l<\tau_{\g_1},\tau_{\g_2},\tau_{\g_3}$.
\item[(b)]
\vspace{2pt}
It follows that $\g_3 \nsim_x \g_1,\g_2$, so that modifying (Proposition \ref{th:invhomm}.\ref{th:invhomm2}) the homomorphism $\homm'\in \IHOMLAI$ along $\gamma_{\g_3}^x|_{[0,l]}$ does not change its values on $\gamma_{\g_i}^x|_{[0,l]}$ for $i=1,2$.
\item[(c)]
\vspace{2pt}
We fix $\homm'\in \IHOMLAI$ and define $\homm_\mu$ by Proposition \ref{th:invhomm}.\ref{th:invhomm2} for $|I_x|=1$, $\g_\alpha:=\g_3$, $V_\alpha:=V$ and $L_\alpha:=\mu L$ for $\mu\in \RR$, i.e., we only modify along $\g_3$.
\item[(d)]
\vspace{2pt}
We fix $p\in F_x$. Then, for $\w\in \AR$ the value
\begin{align*}
\kappa_{\mg'}(\iota_\Con(\w))(\gamma_i)(p)=\parall{\gamma_i}{\w}(p)
\end{align*}
is given by \eqref{eq:trivpar} for $i=1,2,3$. In particular, the value $\kappa_{\mg'}(\iota_\Con(\w))(\gamma_3)(p)$ is related to the values $\kappa_{\mg'}(\iota_\Con(\w))(\gamma_1)(p)$ and $\kappa_{\mg'}(\iota_\Con(\w))(\gamma_2)(p)$ just because $\g_3=\g_1-\g_2$.
\item[(e)]
\vspace{2pt}
Recall the open subsets \eqref{eq:opensets} and choose $U:=\exp\hspace{1pt}(\hspace{1pt}\I (-\epsilon,\epsilon))$ for $\epsilon<\textstyle\frac{\pi}{4}$. It follows that there exists $\mu \in \RR$ such that\footnote{Here, it is important that $U^{p,p}_{\gamma_1,\gamma_2}(\homm_\mu)=U^{p,p}_{\gamma_1,\gamma_2}(\homm')$ is the same neighbourhood for all $\mu\in \RR$, just by point (b).}
\begin{align*}
\kappa_{\mg'}(\iota_\Con(\w))\in U^{p,p}_{\gamma_1,\gamma_2}(\homm_\mu)\qquad\Longrightarrow \qquad \kappa_{\mg'}(\iota_\Con(\w))\notin U^{p}_{\gamma_3}(\homm_\mu),
\end{align*}
hence $\kappa_{\mg'}(\iota_\Con(\AR))\cap U^{p,p,p}_{\gamma_1,\gamma_2,\gamma_3}(\homm_\mu)=\emptyset$. Then, $\homm_\mu \notin \kappa_{\mg'}\big(\ARQ\big)=\kappa_{\mg'}\Big(\ovl{\iota_\Con(\AR)}\Big)$, hence $\AQR\ni\kappa_{\mg'}^{-1}(\homm_\mu)\notin \ARQ$, which shows the claim.
\end{itemize}
\endgroup
\item
We choose any $\epsilon'\in \IHOMLAI$, being non-empty by assumption, and define $\homm\in \IHOMLAI$ by \eqref{eq:homdef} w.r.t.\ $I_x:=\{1,2,3\}$. Moreover, we let $\gamma_i:=\gamma^x_{\g_i}|_{[0,1]}$ for $i=1,2,3$.
We choose a neighbourhood $W$ of $0$ in $\su$ such that $\exp|_{W}$ is a diffeomorphism. Then, scaling the $\g_i$, we can assume that $\s_i:=L_i(\g_i)\in W$ for $i=1,2,3$.
We now show that $\homm$ cannot be approximated by the elements of $\kappa_{\mg'}(\iota_\Con(\AR))$, i.e., by classical (smooth) invariant connections.
For this, let $\|\cdot\|_\murs$ denote the euclidean norm on $\RR^3$ carried over to $\su$ by $\murs\colon \RR^3\rightarrow \su$, and choose
$\epsilon >0$ such that
\begin{align*}
B_\epsilon(\s_i):=\{\s\in \su\:|\: \|\s_i-\s\|_\murs<\epsilon\}\subseteq W\qquad \text{for }i=1,2,3.
\end{align*}
Then, $\exp(B_\epsilon(\s_i))$ is open
in $\SU$, and since $\pm \me\notin \exp(B_\epsilon(\s_i))$, we have
\begin{align*}
U_\epsilon^{i}:=\exp^{-1}(\exp(B_\epsilon(\s_i)))=\left\{\s + 2\pi n\frac{\s}{\|\s\|}\:\bigg|\: \s\in B_\epsilon(\s_i), n\in \mathbb{Z}\right\}
\end{align*}
by formula \eqref{eq:expSU2}, i.e., by the periodicity property of the exponential map of $\SU$.
Since $B_\epsilon(\s_i)$ does not contain the origin ($\exp(0)=\me$), there is $\alpha \in (0,2\pi)$ such that\footnote{Here, $\measuredangle\left(\vec{v},\vec{w}\right)$ means the minimum of the angles between $\vec{v}$, $\vec{w}$ and $-\vec{v}$, $\vec{w}$, so that $C^i_\alpha$ is a double cone.}
\begin{align*}
U_\epsilon^{i}\subseteq C^i_\alpha:=\{\s \in \su\:|\: \measuredangle\left(\murs^{-1}(\s_i),\murs^{-1}(\s)\right)< \alpha)\}
\end{align*}
for $C_\alpha^i$ the double cone in $\su$ determined by the axis $\s_i$ and the opening angle $\alpha$.
Conversely, for each $\alpha\in (0,2\pi)$ we also find some $\epsilon(\alpha)> 0$ such that $U_{\epsilon(\alpha)}^{i}\subseteq C^i_\alpha$ holds.
Let $U\subseteq \SU$ be an open neighbourhood of $\me$ with $\exp(-\s_i)\cdot U \subseteq \exp(B_\epsilon(\s_i))^{-1}$ for $i=1,2,3$.
Then, if $\kappa_{\mg'}(\iota_\Con(\w))\in U_{\gamma_1,\gamma_2,\gamma_3}^{p,p,p}(\homm)$, by \eqref{eq:trivpar} and the definition of $\homm$ we have
\begin{align*}
\exp(-\w(\wt{g}_i(p)))\in \exp(-L_i(\g_i))\cdot U= \exp(-\s_i)\cdot U\subseteq \exp(B_\epsilon(\s_i))^{-1},
\end{align*}
i.e., $\exp(\w(\wt{g}_i(p)))\in \exp(B_\epsilon(\s_i))$, hence $\w(\wt{g}_i(p))\in C^i_\alpha$ for $i=1,2,3$. Now, by the choice of $\g_3$, we have $\w(\wt{g}_3(p))\in \spann_\RR(\w(\wt{g}_1(p)),\w(\wt{g}_2(p)))$,
being a subset of
\begin{align*}
C^1_\alpha+C^2_\alpha:=\{v+w\:|\: v\in C^1_\alpha,w\in C^2_\alpha\}.
\end{align*}
We now show that, for $\alpha$ suitable small, $\left[C^1_\alpha+C^2_\alpha\right]\cap C^3_\alpha=\{0\}$ holds. This then contradicts that $\w(\wt{g}_3(p))\in U_{\epsilon(\alpha)}^{3}\subseteq C^3_\alpha\backslash\{0\}$ and shows the claim.
Since the (pre)image of a cone of the above form under a linear isomorphism contains a cone of the above form, it suffices to consider the case where $\vec{n}_i=\vec{e}_i$ for $i=1,2,3$. Here, we have to show that the equation\footnote{The single expressions parametrize the cones around the $y$-, $z$- and $x$-axis, respectively.}
\begin{align*}
\begin{pmatrix} s y \cos(\theta)\\ y \\ s y \sin(\theta) \end{pmatrix}+\begin{pmatrix} r z \cos(\phi)\\ r z \sin(\phi)\\ z \end{pmatrix}= \begin{pmatrix} x \\ t x \cos(\eta)\\ t x \sin(\eta) \end{pmatrix}
\end{align*}
has no solution for $0< r,s,t \leq \epsilon$ provided that $\epsilon$ is suitable small. But, this is clear since the determinant of
\begin{align*}
\begin{pmatrix} x\\ y \\ z \end{pmatrix}\mapsto
\begin{pmatrix}
-1 & s \cos(\theta) & r \cos(\phi) \\
- t \cos(\eta) & 1 & r\sin(\phi)\\
- t \sin(\eta) & s \sin(\theta) & 1
\end{pmatrix} \cdot
\begin{pmatrix} x\\ y \\ z \end{pmatrix}
\end{align*}
tends to $-1$ for $\epsilon \rightarrow 0$.
\end{enumerate}
\end{proof}
\end{proposition}
\begin{corollary}
\label{cor:incl}
Let $(P,\pi,M,S)$ be a principal fibre bundle and $(G,\Phi)$ a Lie group of automorphisms of $P$. Moreover, let the induced action $\varphi$ be analytic and pointwise proper. Then, in the following situations we have $\ARQLAI\subsetneq \AQRLAI\colon$
\begin{enumerate}
\item
\label{cor:incl1}
Let $S=S^1$
and $\dim[G]\geq2$:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\label{cor:incl2}
\vspace{-4pt}
\itspacec
$\IHOMLAI\neq\emptyset$ and there is $x\in M$ such that $\dim[G]-\dim[G_x]\geq 2$. In addition to that, we find an $\Add{G_x}$-invariant vector $\g\in \mg\backslash \mg_x$.
\item
$\varphi$ acts transitively and free.\footnote{Observe that $\wm$ then is automatically pointwise proper because $M\cong G$.}
\end{itemize}
\endgroup
\item
\label{cor:incl22}
Let $S=\SU$, $\dim[G]\geq 2$ and $G_x=\{e\}$ for some $x\in M \colon$
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\itspacec
\vspace{-4pt}
$\IHOMLAI\neq \emptyset$.
\item
$\varphi$ is transitive.
\end{itemize}
\endgroup
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
If $\g\in \mg\backslash \mg_x$ is $\Add{G_x}$-invariant, let $V:=\Span_\RR(\g)$ and $L\colon \lambda \g \mapsto \lambda \s$ for some fixed $\s\in \ms\backslash\{0\}$. This map is linear and, by commutativity of $S^1$, $\Add{G_x}^p$-equivariant for each $p\in \pi^{-1}(x)$. Moreover, since $\dim[G]-\dim[G_x]\geq 2$, we have $V\oplus \mg_x\subsetneq \mg$ so that the claim is clear from from Proposition \ref{prop:incl}.\ref{prop:incl2}.
If $G_x=\{e\}$ and $\varphi$ is transitive, then Wang's theorem \cite{Wang} (see also Case \ref{th:wang}) shows $\AR\cong \Hom_\RR(\mg,\ms)$, so that $\kappa_{\mg'}(\iota_\Con(\AR))\subseteq\IHOMLAI\neq \emptyset$. Moreover, since each $\g\in \mg\backslash\mg_x$ is $\Add{G_x}$-invariant,
the requirements of the first case are fulfilled.
\item
We choose $\g_1,\g_2\in\mg$ linearly independent and $\g_3\in \Span_\RR(\g_1,\g_2)$ neither contained in $V_1:=\Span_\RR(\g_1)$ nor contained in $V_2:=\Span_\RR(\g_2)$. Let $V_3:=\Span_\RR(\g_3)$ and $\s_i:=\tau_i$ for $i=1,2,3$. Then the maps $L_i\colon \lambda \g_i\mapsto \lambda \s_i$ are linear and $\Add{G_x}^p$-equivariant for each $p\in \pi^{-1}(x)$ so that the claim follows from Proposition \ref{prop:incl}.\ref{prop:incl3}. The second part is clear.
\end{enumerate}
\end{proof}
\end{corollary}
We close this subsection with an application of Corollary \ref{cor:incl} to loop quantum cosmology, by showing that
$\ARQLAI\subsetneq \AQRLAI$ holds (quantization and reduction do not commute) in
\mbox{(semi-)homogeneous} LQC.
In Subsection \ref{subsec:QuantvsRed}, we will see that this is also true in the
homogeneous isotropic case.
\begin{example}[(Semi-)Homogeneous Loop Quantum Cosmology]
\label{ex:LQCInc}
Let $P=\RR^3\times \SU$ and $\Ph$, $\Phi_{SH}$ be defined as in Example \ref{ex:LQC}.
We claim that quantization and reduction do not commute in \mbox{(semi-)homogeneous} loop quantum cosmology. In fact,
\begingroup
\setlength{\leftmargini}{14pt}
\begin{itemize}
\item
\vspace{-4pt}
In the homogeneous case $(\Gh,\Ph)$ this follows from the second part of Corollary \ref{cor:incl}.\ref{cor:incl22}.
\item
In the semi-homogeneous case $(G_{SH},\Phi_{SH})$, the action
$\wm$ is pointwise proper because $\Ph$ is proper and and each linear subspace of $\RR^3$ is closed. Since $\dim[G_{HS}]\geq 2$, by the first part of Corollary \ref{cor:incl}.\ref{cor:incl2} it suffices to show that $\IHOMLAI\neq \emptyset$. But, this is clear because (as already stated in Example \ref{ex:LQC}) the set $\AR$ is in bijection with the smooth maps $\psi \colon \RR^2\times T\RR\rightarrow \su$ for which the restrictions $\psi|_{\RR^2\times T_x\RR}$ are linear for all $x\in \RR$.\hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\end{example}
\subsection{Modifications along Free Segments}
\label{sec:ModifreeSeg}
Complementary to our investigations of Lie algebra, i.e., continuously generated curves, we now are going to study the set $\Paf$ of free curves in $M$. This is the set of all embedded analytic curves that contain a segment which, in a certain sense, does not overlap with its translates by the symmetry group.\footnote{The precise definition will given below, cf.\ also Remark \ref{rem:euklrem}.\ref{rem:dsdfdf}.} We will show that each such curve is covered by finitely many translates (of initial and final segments) of one of its maximal free segments. This will provide us with a canonical decomposition of such free curves by means of the symmetry group. In course of this, we will split up the set $\Paw\backslash \Pags$ into three subsets $\Pacs$, $\Pafs$ and $\Pafns$
being closed under inversions and decompositions as well. Recall that Proposition \ref{rem:euklrem2b} then provides us with a respective factorization
\begin{align}
\label{eq:dsfs444ffff}
\AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{CNL}}
\times \AQRFNS\times \AQRInd{\mathrm{FS}}.
\end{align}
Here, we always have $\Paf=\Pafs\sqcup \Pafns$, whereby $\Pafns$ consists of all free curves whose stabilizer is trivial,\footnote{See Definition \ref{def:freeSegg}.} so that $\Pafs$ consists of all free curves for which this is not the case.
Then, the second space $\AQRInd{\mathrm{CNL}}$
is the least accessible one, just because the set $\Pacs$ of continuously but not Lie algebra generated curves is so.
However, we will show that $\Pacs=\emptyset$ holds, i.e.,
that we have $\Paw=\Pags\sqcup \Paf$ whenever $\wm$ is transitive or proper and admits only stabilizers which are normal subgroups of $G$. So, if $\wm$ is in addition free, we even have $\Paw=\Pags\sqcup \Pafns$, hence
$\AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{FN}}$, so that in this case it suffices to define normalized Radon measure on these two factors in order to define a normalized Radon measure on $\AQRw$.
Now, in the second part of Section \ref{sec:MOQRCS}, we will use Proposition \ref{th:invhomm}.\ref{th:invhomm1} in order to construct a normalized Radon measure $\mLAS$ on $\AQRInd{\mg}$, e.g., for the case that $\wm$ is free and $S$ equals $\SU$ or an $n$-torus. Moreover, in the last part of the present Section, we will prove an analogue of this proposition for free curves, which we then use in Subsection \ref{sec:FreeM}
in order to construct a normalized Radon measure $\mFNS$ on $\AQRInd{\mathrm{FN}}$. So, together these two measures give rise to a normalized Radon measure
\begin{align*}
\mLAS\times \mFNS\qquad \qquad\text{on}\qquad\qquad \AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{FN}}
\end{align*}
whenever the respective requirements hold. This will be the case, e.g.\, in \mbox{(semi-)homogeneous} LQC.
\begin{definition}
\label{def:freeSegg}
For $\gamma$ an analytic curve, we define the group\footnote{The group property is easily verified, and for $\gamma$ embedded analytic also clear from $\textit{(a)}$ in Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma1}.}
\begin{align*}
\text{\gls{GGAMMA}}:=\{g\in G \:|\: \wm_g\cp \gamma = \gamma\cp \adif \text{ for } \adif\colon \operatorname{\mathrm{dom}}[\gamma]\rightarrow \operatorname{\mathrm{dom}}[\gamma]\text{ an analytic diffeomorphism with }\dot\adif>0\}
\end{align*}
as well as the equivalence relation \gls{SEGSIM} on $G$ by
\begin{align*}
g\sim_\gamma g'\qquad \Longleftrightarrow \qquad g^{-1}g'\in G_\gamma.
\end{align*}
Then,
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
$\gamma$ is called symmetric iff $G_\gamma\neq \{e\}$.
\item
$\gamma$ is called a free segment iff
\begin{align*}
\gamma\cpsim\wm_g\cp \gamma\quad\text{for}\quad g\in G\qquad\Longrightarrow \qquad g\in G_\gamma.
\end{align*}
\item
$\gamma$ is called free iff $\gamma|_K$ is a free segment for some compact interval $K\subseteq \operatorname{\mathrm{dom}}[\gamma]$.
\item
For $\delta$ an analytic curve let
\begin{align*}
\text{\gls{HGAMMADELTA}}:=\{g\in G \:|\: \wm_g\cp \delta\cpsim \gamma \}\qquad\quad\text{as well as}\qquad\quad H_{[\gamma,\delta]}:=H_{\gamma,\delta}\hspace{1pt} \slash \!\sim_\delta.
\end{align*}
\end{itemize}
\endgroup
\end{definition}
The next lemma collects the relevant properties of the above quantities for the embedded analytic case.
\begin{lemma}
\label{lemma:curvestablemma}
Let $\wm$ be analytic and pointwise proper, and $\Paw\ni \gamma\colon [a,b]\rightarrow M$. Moreover, let $\Paw\ni \delta\colon L\rightarrow M$ be a free segment.
\begin{enumerate}
\item
\label{lemma:curvestablemma1}
We have $\wm_h\cp \gamma = \gamma$ for all $h\in G_\gamma$. Hence,
\begin{enumerate}
\item[(a)]
\label{lemma:curvestablemma1aa}
\vspace{-3pt}
$\gamma\cpsim\wm_g\cp \gamma \qquad \Longrightarrow \qquad\wm_g\cp \gamma = \gamma$
\item[(b)]
\label{lemma:curvestablemma1a}
\vspace{3pt}
$G_\gamma=\bigcap_{t\in \operatorname{\mathrm{dom}}[\gamma]}G_{\gamma(t)}$ is a closed subgroup of $G$,
\item[(c)]
\label{lemma:curvestablemma1b}
\vspace{2.5pt}
$G_{\gamma|_D}=G_\gamma$ for each interval $D\subseteq \operatorname{\mathrm{dom}}[\gamma]$.
\end{enumerate}
\item
\label{lemma:curvestablemma2}
Assume that $\gamma$ is not free, and let $t\in [a,b]$.
Then, for each interval $D\subseteq [a,b]$ with $t\in K$ we find $g\in G\backslash G_{\gamma(t)}$ with $\gamma|_{K}\cpsim \wm_g\cp \gamma|_K$.
\item
\label{lemma:curvestablemma3}
Let $\gamma$ be free and $\{h_\alpha\}_{\alpha\in I}\subseteq H_{\gamma,\delta}$ a family of representatives of $H_{[\gamma,\delta]}$. Then,
\begin{enumerate}
\item
\label{it:p1}
\vspace{-3pt}
$H_{[\gamma,\delta]}$ is finite.
\item
\label{it:p2}
\vspace{2pt}
There exists a unique decomposition $a=k_0<k_1<\dots<k_n=b$ of $\operatorname{\mathrm{dom}}[\gamma]$ into compact intervals $K_i=[k_i,k_{i+1}]$ for $0\leq i\leq n-1$, such that either
\begin{align}
\label{eq:bbbb}
\begin{split}
\gamma|_{K_i} &\:\:\nsim_{\cp} \:\hspace{1pt}\wm_{h_\alpha}\cp \delta \qquad\qquad\:\:\forall\:\alpha\in I \qquad\text{or}\\
\gamma|_{K_i}&\psim \wm_{h_{\alpha_i}}\cp [\delta|_{L_i}]^{\pm 1}\quad\hspace{5.5pt}\text{for } \alpha_i\in I \text{ and } L_i\subseteq L\text{ uniquely determined}.
\end{split}
\end{align}
Here, $L_i=L=[l_1,l_2]$ if $1\leq i\leq k-2$ and $L_0,L_{n-1}$ are both of the form
\begin{align}
\label{eq:oftheform}
\:[m,l_2] \quad\text{ for }\quad m\in [l_1,l_2)\qquad \qquad\text{or}\qquad\qquad [l_1,m]\quad \text{ for }\quad m\in (l_1,l_2].
\end{align}
\vspace{-15pt}
\begin{align}
\label{eq:decompo}
\raisebox{-30pt}{
\begin{tikzpicture}
\draw[-,line width=1.5pt] (-0.5,0) -- (4,0);
\draw[->,line width=1.5pt,dotted] (4,0) -- (6,0);
\draw[-,line width=1.5pt] (-0.5,-0.2) -- (-0.5,0.1);
\draw (6.3,0) node {\(\gamma\)};
\draw[-,line width=1.25pt] (1.5,1.25) -- (2.5,1.25);
\draw[-,line width=1pt] (1.5,1.16) -- (1.5,1.39);
\draw[-,line width=1pt] (2.5,1.16) -- (2.5,1.39);
\draw (2.7,1.3) node {\(\delta\)};
\draw[-,line width=1.25pt,color=red] (1.515,1.35) -- (1.97,1.35);
\draw[color=red] (1.75,1.55) node {\(_{L_0}\)};
\draw[-,line width=1.25pt,color=olive,dotted] (2,1.35) -- (2.48,1.35);
\draw[-,line width=1.25pt,dotted,color=olive] (-1,0.15) -- (-0.5,0.15);
\draw[-,line width=1pt] (-1,0.05) -- (-1,0.25);
\draw[-,line width=1pt] (0,-0.15) -- (0,0.25);
\draw (0,-0.3) node {\(_{k_1}\)};
\draw (-0.5,-0.3) node {\(_{k_0}\)};
\draw[color=red] (-0.25,0.35) node {\(_{L_0}\)};
\draw[-,line width=1.25pt,color=red] (-0.5,0.15) -- (-0.02,0.15);
\draw[color=blue] (-0.25,-0.6) node {\(_{K_0}\)};
\draw[-,line width=1.5pt,color=blue] (-0.47,0) -- (-0.02,0);
\draw[-,line width=1pt] (0.5,-0.15) -- (0.5,0.15);
\draw (0.5,-0.3) node {\(_{k_2}\)};
\draw[color=blue] (1.1,-0.6) node {\(_{K_2}\)};
\draw[-,line width=1.5pt,color=blue] (0.51,0) -- (1.48,0);
\draw[-,line width=1pt] (1.5,-0.15) -- (1.5,0.15);
\draw (1.5,-0.3) node {\(_{k_3}\)};
\draw[-,line width=1pt] (0.485,0.15) -- (1.515,0.15);
\draw[-,line width=1pt] (1.9,-0.15) -- (1.9,0.15);
\draw (1.9,-0.3) node {\(_{k_4}\)};
\draw[-,line width=1pt] (2.9,-0.15) -- (2.9,0.15);
\draw (2.9,-0.3) node {\(_{k_5}\)};
\draw[-,line width=1pt] (1.885,0.15) -- (2.915,0.15);
\draw[color=blue] (3.25,-0.6) node {\(_{K_5}\)};
\draw[-,line width=1.5pt,color=blue] (2.91,0) -- (3.58,0);
\draw[-,line width=1pt] (3.6,-0.15) -- (3.6,0.15);
\draw (3.6,-0.3) node {\(_{k_6}\)};
\draw[-,line width=1pt] (3.585,0.15) -- (4,0.15);
\draw[-,line width=1pt,dotted] (4,0.15) -- (4.6,0.15);
\draw[->,line width=0.8pt,dotted] (2.3,1.15) .. controls (2.3,0.8) and (3.7,0.3) .. (3.8,0.3);
\draw (3.45,0.65) node {\(_{h_{\alpha_6}}\)};
\draw[->,line width=0.8pt,dotted] (2.1,1.15) .. controls (2.1,0.8) and (2.5,0.4) .. (2.6,0.3);
\draw (2.15,0.4) node {\(_{h_{\alpha_4}}\)};
\draw[->,line width=0.8pt,dotted] (1.8,1.15) .. controls (1.8,0.8) and (1.4,0.4) .. (1.3,0.3);
\draw (1,0.35) node {\(_{h_{\alpha_2}}\)};
\draw[->,line width=0.8pt,dotted,color=red] (1.65,1.15) .. controls (1.65,0.8) and (0.4,0.45) .. (0.08,0.3);
\draw (0.7,0.9) node {\(_{h_{\alpha_0}}\)};
\end{tikzpicture}}
\end{align}
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
The implications $\textit{(a)}$ and $\textit{(b)}$ are clear, and the implication $\textit{(c)}$ follows from the analyticities of $\wm_h\cp \gamma$ and $\gamma$ just by
\begin{align*}
h\in G_{\gamma|_D}\qquad\Longrightarrow\qquad (\wm_h\cp \gamma)|_D=\gamma|_D
\qquad\Longrightarrow\qquad \wm_h\cp \gamma =\gamma.
\end{align*}
Now, to show that $\wm_h\cp \gamma=\gamma$ holds for $h\in G_\gamma$, we assume that $\wm_h(\gamma(t_0))\neq \gamma(t_0)$ for $t_0\in (a,b)$.\footnote{Observe that the end points of $\gamma$ are necessarily fixed by $\wm_h$.} Then, we have $\wm_h(\gamma(t_0))=\gamma(t_1)$ for some $t_1\in (a,b)$, and we may assume that $t_0<t_1$ holds. Let $t_0<s_0<t_1$. Then $\wm_h(\gamma(s_0))=\gamma(s_1)$ for some $t_1< s_1$ because $\gamma^{-1}\cp \wm_{h}\cp \gamma$ is a homeomorphism, hence monotonous. Applying $\wm_h$ inductively,
we obtain sequences $\{t_n\}_{n\in \NN}, \{s_n\}_{n\in \NN}\subseteq (a,b)$ with $t_n <s_n< t_{n+1} < s_{n+1}$ as well as $\gamma(t_n)=\wm_{h^n}(\gamma(t_0))$ and $\gamma(s_n)=\wm_{h^n}(\gamma(s_0))$ for all $n\in \NN$.
Obviously, $\lim_n s_n =\lim_n t_n\in[a,b]$ exists, so that
\begin{align*}
\lim_n \wm_{h^{n}}(\gamma(t_0))=\lim_n \gamma(t_n)=\lim_n \gamma(s_n)=\lim_n \wm_{h^{n}}(\gamma(s_0)).
\end{align*}
Pointwise properness of $\wm$ now implies $\gamma(t_0)=\gamma(s_0)$, which
contradicts that $t_0\neq s_0$ and injectivity of $\gamma$.
\item
It suffices to show the claim for $D=K$ compact. Moreover,
we only consider the case where $t<b$ holds because the case $a<t$ follows analogously. Then, switching from $\gamma$ to $\gamma^{-1}$ and reparametrizing if necessary, we can assume that $K=[0,r]$ with $0\leq t< r$.
Now, assume that the statement is wrong, i.e., that for each $g\in G\backslash G_{\gamma}$ with $\wm_g\cp \gamma|_{K}\cpsim \gamma|_K$ we have $g\in G_{\gamma(t)}$.\footnote{Observe that we find such a group element since elsewise $\gamma|_K$ would be a free segment.}
Then, to derive a contradiction, it suffices to show that:
{\bf Claim:} For $t<k_0<k<b$ we find $g\in G_{\gamma(t)}$ with $\wm_g(\gamma(k))=\gamma(k')$ for $t< k'<k_0$.
In fact, let $n_0\geq 1$ with $t+ \textstyle\frac{1}{n_0}< r$. Then, for each $n\geq n_0$ we find $g_n\in G_{\gamma(t)}$ such that
\begin{align*}
\wm_{g_n}(\gamma(r))=\gamma(t_n)\qquad \text{for}\qquad t< t_n<t+ \textstyle\frac{1}{n}.
\end{align*}
Consequently, $\lim_n \wm_{g_n}(\gamma(r))=\gamma(t)=\lim_n \wm_{g_n}(\gamma(t))$, hence $\gamma(r)=\gamma(t)$ by pointwise properness of $\wm$. Since $t\neq r$, this contradicts injectivity of $\gamma$.
\vspace{3pt}
{\bf Proof of the Claim:}
\vspace{-3pt}
Since $\gamma|_{[t,k_0]}$ is not a free segment ($\gamma$ is not free), we find $g\in G\backslash G_\gamma$ with $\wm_g\cp \gamma|_{[t,k_0]} \cpsim \gamma|_{[t,k_0]}$,
and by assumption we have $g\in G_{\gamma(t)}$. The two possible configurations are
\begin{align*}
\hspace{40pt}
\begin{tikzpicture}
\draw[-,line width=1pt,color=red] (0,0) .. controls (0.1,0.6) and (0.7,0.4) .. (1,0.1);
\draw[-,line width=1pt,color=red] (1,0.1) .. controls (1.1,0) and (1.2,0) .. (1.3,0);
\draw[->,line width=1pt,color=red] (1.4,0) .. controls (1.5,0) and (1.7,0.2) .. (1.7,0.4);
\draw[->,line width=1.5pt] (0,0) -- (2,0);
\filldraw[black] (0,0) circle (2pt);
\draw[color=red] (1.5,0.5) node {\(_{k_0}\)};
\draw (2,-0.3) node {\(_{k_0}\)};
\draw (-0.2,-0.02) node {\(_t\)};
\draw (2.2,0) node {\(\gamma\)};
\draw[color=red] (2.35,0.5) node {\(\wm_g\cp\gamma\)};
\end{tikzpicture}
%
\qquad\qquad\raisebox{20pt}{$\text{and}$} \qquad\qquad\quad
\begin{tikzpicture}
\draw[-,line width=1pt,color=red] (0,0) .. controls (0,1.2) and (2,0.4) .. (1.4,0);
\draw[->,line width=1pt,color=red] (1.1,0) .. controls (1,0) and (0.7,0) .. (0.6,0.25);
\draw[->,line width=0.8pt,color=red] (1.1,0) .. controls (0.8,0) and (0.7,0) .. (0.5,0.45);
\draw[->,line width=1.5pt] (0,0) -- (2,0);
\filldraw[black] (0,0) circle (2pt);
\draw[color=red] (0.4,0.17) node {\(_{k_0}\)};
\draw[color=red] (0.72,0.43) node {\(_{k}\)};
\draw (2,-0.3) node {\(_{k_0}\)};
\draw (-0.2,-0.02) node {\(_t\)};
\draw (2.25,0) node {\(\gamma.\)};
\draw[color=red] (2,0.5) node {\(\wm_g\cp\gamma\)};
\end{tikzpicture}
\end{align*}
Then, by injectivity of $\gamma$ and Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt4}, one of the following situations holds:
\begin{align*}
\hspace{36.5pt}
&\raisebox{20pt}{
\begin{tikzpicture}
\draw[->,line width=1pt,color=red] (0,0) -- (1.5,0);
\draw[->,line width=1.5pt] (0,0) -- (2,0);
\filldraw[black] (0,0) circle (2pt);
\draw[color=red] (1.45,0.25) node {\(_{k_0}\)};
\draw (2,-0.3) node {\(_{k_0}\)};
\draw (-0.2,-0.02) node {\(_t\)};
\draw (2.2,0) node {\(\gamma\)};
\draw[color=red] (0.6,0.25) node {\(\wm_g\cp\gamma\)};
\end{tikzpicture}}%
\qquad\qquad\qquad\raisebox{20pt}{$\text{or}$} \qquad\qquad\quad\:
\begin{tikzpicture}
\draw[-,line width=1pt,color=red] (0,0) .. controls (0,0.8) and (2,0.4) .. (1.4,0);
\draw[->,line width=1pt,color=red] (1.1,0) -- (0.85,0);
\draw[->,line width=0.8pt,color=red] (1.1,0) -- (0.4,0);
\draw[->,line width=1.5pt] (0,0) -- (2,0);
\filldraw[black] (0,0) circle (2pt);
\draw[color=red] (0.9,0.25) node {\(_{k_0}\)};
\draw[color=red] (0.5,0.23) node {\(_{k}\)};
\draw (2,-0.3) node {\(_{k_0}\)};
\draw (-0.2,-0.02) node {\(_t\)};
\draw (2.25,0) node {\(\gamma.\)};
\draw[color=red] (2,0.5) node {\(\wm_g\cp\gamma\)};
\end{tikzpicture} \\[-30pt]
&\raisebox{0pt}{
\begin{tikzpicture}
\draw[->,line width=1pt,color=red] (0,0) -- (2.5,0);
\draw[->,line width=1.5pt] (0,0) -- (2,0);
\filldraw[black] (0,0) circle (2pt);
\draw (1.9,0.3) node {\(_{k_0}\)};
\draw[color=red] (2.5,-0.3) node {\(_{k_0}\)};
\draw (-0.2,-0.02) node {\(_t\)};
\draw[color=red] (2.7,0.25) node {\(\wm_g\cp \gamma\)};
\draw (1,0.25) node {\(\gamma\)};
\end{tikzpicture}}
\end{align*}
More precisely,
\begingroup
\setlength{\leftmarginii}{25pt}
\begin{itemize}
\item[(a)]
In the first case, we have
\begin{align*}
\gamma|_{[t,s_0]}&\psim \wm_g\cp \gamma|_{[t,k_0]} \:\text{ for some }\: s_0\in (t,k_0]\qquad\text{or}\\
\gamma|_{[t,k_0]}&\psim \wm_g\cp \gamma|_{[t,s_0]} \hspace{3pt}\text{ for some }\: s_0\in (t,k_0].
\end{align*}
In both situations, for $s_0=k_0$ we would have $g\in G_\gamma$, so that $s_0<k_0$ holds just because $g\notin G_\gamma$ by assumption.
Moreover, replacing $g$ by $g^{-1}$ if necessary, we can assume that the first relation holds.
Then $\wm_g(\gamma)(k_0)=\gamma(s_0)$ for some $t<s_0<k_0$, and applying $\wm_g$ once more, we find that $\wm_g(\gamma(s_0))=\gamma(s_1)$ for some $t<s_1<s_0$ just by monotonicity of $\gamma^{-1}\cp \wm_h\cp \gamma$. Inductively, we obtain $\{s_n\}_{n\in \NN}\subseteq (t,k_0)$ with $s_{n+1}< s_n$ and $\gamma(s_n)=\wm_{g^n}(\gamma(s_0))$ for all $n\in \NN$.
Let $s:=\lim_n s_n$. Then $\gamma(s)=\lim_n \wm_{g^{n}}(\gamma(s_0))$, hence $\wm_g(\gamma(s))=\gamma(s)$ by continuity of $\wm_g$. Consequently, $\lim_n \wm_{g^n}(\gamma(s))=\gamma(s)=\mathrm{im}_n\wm_{g^n}(\gamma(s_0))$, so that by pointwise properness of $\wm$ we have $\gamma(s)=\gamma(s_0)$. This, however contradicts $s<s_0$ and injectivity of $\gamma$, showing that case (a) cannot occur.
\item[(b)]
In the second case, we have
$\wm_g\cp \gamma|_{[s,k_0]}\psim \big[\gamma|_{[s',k_0']}\big]^{-1}$ for some $t<s<k_0$ and some $t<s'<k_0'<k_0$.
Then, Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt4} and injectivity of $\wm_g\cp \gamma|_{[t,k]}$ show that
\begin{align*}
\wm_g\cp \gamma|_{[s,k]}\psim \big[\gamma|_{[k',k_0']}\big]^{-1}
\end{align*}
holds for some $t<k'< s'< k_0'<k_0$, hence $\wm_g(\gamma(k))=\gamma(k')$ for $1<k'<k_0$.
\end{itemize}
\endgroup
\item
If $I$ is finite, then $\textit{(b)}$ is clear from Lemma \ref{lemma:BasicAnalyt}.\ref{Basanalyt}. However, if $I$ is infinite, then we find $\{\alpha_{n_i}\}_{i\in \mathbb{N}}\subseteq I$ mutually different with $\gamma|_{K_{n_i}}\psim \delta^{\pm 1}$ for intervals $K_{n_i}\subseteq K$ which mutually can only share start and end points.
Let $k\neq k'$ be contained in $K_{n_0}$
and define sequences $\{x_i\}_{i\in \NN}, \{x'_i\}_{i\in \NN} \subseteq \operatorname{\mathrm{dom}}[\gamma]$ by
\begin{align*}
x_i:=\gamma^{-1}\!\left(\wm\big(h_{\alpha_{n_i}}\cdot h^{-1}_ {\alpha_{n_0}},\gamma(k)\big)\right)\qquad\text{as well as}\qquad x'_i:=\gamma^{-1}\!\left(\wm\big(h_{\alpha_{n_i}}\cdot h^{-1}_ {\alpha_{n_0}},\gamma(k')\big)\right).
\end{align*}
By compactness of $\operatorname{\mathrm{dom}}[\gamma]$, we can assume that $\lim_i x_i =x\in \operatorname{\mathrm{dom}}[\gamma]$ exists. Then, since
the intervals $K_{n_i}$ can only share start and end points, it follows that $\lim_i x'_{i}=x$ holds.
\hspace{150pt}
\begin{tikzpicture}
\draw[->,line width=1.5pt] (0,0) -- (4,0);
\filldraw[black] (0,0) circle (2pt);
\draw (4.3,0) node {\(\gamma\)};
\draw[-,line width=3pt] (0.5,0) -- (1.2,0);
\draw (0.7,0.3) node {\(_{K_n}\)};
\draw[-,line width=1pt,color=blue] (1,-0.068) -- (1,0.4);
\draw[color=blue] (1,0.5) node {\(_{x_n}\)};
\draw[-,line width=1pt] (0.6,0.068) -- (0.6,-0.4);
\draw (0.7,-0.5) node {\(_{x'_n}\)};
\draw[-,line width=3pt] (2.45,0) -- (3.2,0);
\draw (3.3,0.28) node {\(_{K_{n+1}}\)};
\draw[-,line width=1pt,color=blue] (2.6,-0.068) -- (2.6,0.4);
\draw[color=blue] (2.6,0.5) node {\(_{x_{n+1}}\)};
\draw[-,line width=1pt] (3.1,0.068) -- (3.1,-0.4);
\draw (3.1,-0.5) node {\(_{x'_{n+1}}\)};
\draw[-,line width=0.8pt,dashed,color=red] (0.9,-0.25) -- (2.7,-0.25);
\draw[color=red] (1.8,-0.55) node {\(_{B_\epsilon(x)}\)};
\draw[color=red] (0.85,-0.25) node {\(_(\)};
\draw[color=red] (2.75,-0.25) node {\(_)\)};
\draw[-,line width=3pt] (1.5,0) -- (2,0);
\draw (1.9,0.3) node {\(_{K_{m>n+1}}\)};
\end{tikzpicture}
Hence, for $g_i:=h_{\alpha_{n_i}}\cdot h^{-1}_ {\alpha_{n_0}}$ we have
\begin{align*}
\lim_i \wm_{g_i}(\gamma(k))=\lim_i \gamma(x_i)=\gamma(x)= \lim_i \gamma(x'_i)=\lim_i \wm_{g_i}(\gamma(k')),
\end{align*}
so that $\gamma(k)=\gamma(k')$ by pointwise properness of $\wm$. This contradicts the choices and shows that $I$ is finite.
\end{enumerate}
\end{proof}
\begin{definition}
\label{def:freeSeg}
\begin{enumerate}
\item
\label{def:freeSeg3}
By $\text{\gls{PAF}}\subseteq \Paw$ we will denote the set of all free embedded analytic curves.
\item
\label{def:freeSeg32}
We define the subsets $\Pafs,\Pafns\subseteq \Paw$ of free symmetric and free non-symmetric curves by
\begin{align*}
\text{\gls{PAFS}}:=\{\gamma\in \Paf\:|\: G_\gamma\neq\{e\}\}\quad\qquad \text{and}\qquad\qquad \text{\gls{PAFNS}}:=\{\gamma\in \Paf\:|\: G_\gamma=\{e\}\},
\end{align*}
respectively.
\item
\label{def:freeSeg4}
The set of continuously generated curves is defined by $\text{\gls{PAC}}:=\Paw\backslash \Paf$. Then, by $\text{\gls{PACS}}:=\Pac\backslash \Pags$ we will denote the set of
continuously but not Lie algebra generated curves.
\end{enumerate}
\end{definition}
\begin{remark}
\label{decrem}
It is immediate from the definition of a Lie algebra generated curve that $\Pags \cap \Paf=\emptyset$, hence $\Pags\subseteq \Paw \backslash \Paf=\Pac$ holds. Consequently, we have $\Pac=\Pags\sqcup \Pacs$ which shows that
\begin{align}
\label{eq:decompospw}
\Paw= \Pags\sqcup\Pacs \sqcup \Pafs\sqcup \Pafns
\end{align}
holds.
\hspace*{\fill} $\lozenge$
\end{remark}
The first part of (the next) Proposition \ref{prop:freeseg} shows that for $\wm$ analytic and pointwise proper, a free curve is discretely generated by the symmetry group.
This means that for each free curve $\gamma$ we find a free segment $\delta$ such that $\gamma$ can be decomposed into subcurves, each being equivalent to a translate of an initial or final segment of $\delta$. Here, each of these subcurves which is not an initial or final segment of $\gamma$ then even equals the translate the full segment $\delta$.
This decomposition will be inalienable for the construction\footnote{More precisely, for the separation property of the projective structure introduced there, as well as surjectivity of the involved projection maps.} of the normalized Radon measure on $\AQRInd{\mathrm{FN}}\cong \IHOMFNS$ in Subsection \ref{sec:FreeM}.
The second part of Proposition \ref{prop:freeseg} shows that $\Paw=\Pags\sqcup \Paf$ holds whenever $\wm$ is transitive and only admits stabilizers which are normal subgroups. Obviously, then we even have $\Paw=\Pags\sqcup \Pafns$ if $\wm$ is in addition free.
Similarly, in the third part, we will prove that each continuously generated curve is contained in a $\wm$-orbit if $\wm$ is proper. Then, as in the second part, we will show that $\Paw=\Pags\sqcup \Paf$ holds if $\wm$ admits only normal stabilizers, hence $\Paw=\Pags\sqcup \Pafns$ if $\wm$ acts in addition freely. So, for $\wm$ non-trivial we have:
\renewcommand*{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{c|c|c|c|c}
$\wm$ & $\Pags$ & $\Pacs$ & $\Pafns$ & $\Pafs$\\[3pt] \hline
free & many& ? & ?& $\emptyset$\\
transitive $+$ normal stabilizers & many & $\emptyset$ & ?& ?\\
\hspace{10pt} proper $+$ normal stabilizers & many & $\emptyset$ & ?& ?
\end{tabular}
\end{center}
\renewcommand*{\arraystretch}{1}
Anyhow, before we come to the proofs, we first want to give some straightforward applications of these results.
\begin{example}[Free Curves in LQC]
\label{eq:invelements}
We now apply the results from Proposition \ref{prop:freeseg} to the \mbox{(semi-)homogeneous}, homogeneous isotropic and spherically symmetric LQC case.
\par
\begingroup
\leftskip=8pt
\vspace{8pt}
\noindent
{\bf\textit{(Semi-)Homogeneous LQC:}}
\vspace{2pt}
\noindent
In the homogeneous case, $\wm$ is transitive and free, so that $\Paw=\Pags\sqcup \Pafns$ holds by Proposition \ref{prop:freeseg}.\ref{prop:freeseg2}. In the semi-homogeneous case (see Example \ref{ex:LQC}) $\wm$ is proper because the additive action of $\RR^3$ on $\RR^3$ is proper and each linear subspace of $\RR^3$ is closed. Since $\wm$ acts freely, Proposition \ref{prop:freeseg}.\ref{prop:freeseg3} shows that $\Paw=\Pags\sqcup \Pafns$ holds as well.
\par
\endgroup
\noindent
\par
\begingroup
\leftskip=8pt
\vspace{8pt}
\noindent
{\bf\textit{Homogeneous Isotropic LQC:}}
\vspace{2pt}
\noindent
Unfortunately, $G_0=\SU \subseteq \Gee$ is not a normal subgroup, so that we cannot conclude that $\Pac=\emptyset$, i.e., $\Paw=\Pags\sqcup \Paf$ holds.
However, we have $\Paw=\Pags\sqcup\Pacs\sqcup \Pafns$ because $\Pafs=\emptyset$.
In fact, if $\gamma\colon [0,k]\rightarrow \RR^3$ is symmetric, then we can assume that $\gamma(0)=0$ just by transitivity, and because $\wm_{g\cdot h\cdot g^{-1}}\cp (\wm_g\cp \gamma)=\wm_g\cp \gamma$ for $e\neq h\in G_\gamma$. So, let $\gamma(0)=0$ and $h\in G_\gamma$. Then $\wm_h(\gamma(0))=\gamma(0)$ shows $h\in \SU$, and since $\wm_h(\gamma(k))=\gamma(k)\neq 0$, $h$ corresponds to a rotation by some angle $\alpha\in (0,2\pi)$ around the axis (through $0$) determined by $\gamma(k)$. Since by Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma1} we have $\wm_h(\gamma(t))=\gamma(t)$ for all $t\in [0,k]$, $\mathrm{im}[\gamma]$ is completely contained in this axis determined by $\gamma(k)$. It is now obvious\footnote{We have $\gamma \cpsim \gamma^0_{(x,0)}$ if we consider $x$ as vector in $\RR^3$, i.e., $(x,0)\in \RR^3\times \su$.} from Lemma \ref{lemma:sim}.\ref{lemma:sim5} that $\gamma\in \Pags$. This shows $\Pafs=\emptyset$.
\vspace{8pt}
\noindent
{\bf \textit{Spherically Symmetric LQC:}}
\vspace{2pt}
\noindent
By the same arguments as in the previous case, we see that a symmetric curve necessarily has to be contained in an axis through the origin. Since in the spherically symmetric situation linear curves are not Lie algebra generated, we have $\Pafs=\Paln$.\footnote{Recall (a) in Convention \ref{conv:sutwo1}.\ref{conv:sutwo2} for the definition of $\Paln$.} Moreover, each continuously generated curve has to be contained in a sphere by Proposition \ref{prop:freeseg}.\ref{prop:freeseg3}.
Unfortunately, we cannot conclude that $\Pacs=\emptyset$ holds, i.e., that we have $\Paw=\Pags\sqcup \Pafns \sqcup \Paln$ because for $x\neq 0$ the stabilizer $G_x$ equals the maximal torus $H_x\subseteq \SU$ which is not a normal subgroup as well.
\hspace*{\fill}$\lozenge$
\par
\endgroup
\end{example}
We now come to the desired
\begin{proposition}
\label{prop:freeseg}
Let $\wm$ be analytic and pointwise proper.
\begin{enumerate}
\item
\label{prop:freeseg1}
If $\gamma\colon [a,b]\rightarrow M$ is free, then we find a free segment $\delta\colon L\rightarrow M$ and points $a=t_0<{\dots}<t_n=b$ such that
\begin{align*}
\gamma|_{[t_i,t_{i+1}]}\psim \wm_{g_i}\cp [\delta|_{L_i}]^{\pm 1}\:\text{ for some }\: g_i\in G\qquad \forall\: 0\leq i\leq n-1.
\end{align*}
Here $L_i=L=[l_1,l_2]$ for $1\leq i\leq k-2$ and $L_1,L_{n-1}$ are both of the form
\begin{align*}
\:[m,l_2] \quad\text{ for }\quad m\in [l_1,l_2)\qquad \qquad\text{or}\qquad\qquad [l_1,m]\quad \text{ for }\quad m\in (l_1,l_2].
\end{align*}
\hspace{100pt}
\begin{tikzpicture}
\draw[-,line width=1.5pt] (-0.5,0) -- (4,0);
\draw[->,line width=1.5pt,dotted] (4,0) -- (6,0);
\draw[-,line width=1.5pt] (-0.5,-0.2) -- (-0.5,0.1);
\draw (6.2,0) node {\(\gamma.\)};
\draw[-,line width=1.25pt] (1.5,1.25) -- (2.5,1.25);
\draw[-,line width=1pt] (1.5,1.16) -- (1.5,1.39);
\draw[-,line width=1pt] (2.5,1.16) -- (2.5,1.39);
\draw (2.7,1.3) node {\(\delta\)};
\draw[-,line width=1.25pt,color=red] (1.515,1.35) -- (1.97,1.35);
\draw[color=red] (1.75,1.55) node {\(_{L_0}\)};
\draw[-,line width=1.25pt,color=olive,dotted] (2,1.35) -- (2.48,1.35);
\draw[-,line width=1.25pt,dotted,color=olive] (-1,0.15) -- (-0.5,0.15);
\draw[-,line width=1pt] (-1,0.05) -- (-1,0.25);
\draw[-,line width=1pt] (0,-0.15) -- (0,0.25);
\draw (0,-0.3) node {\(_{k_1}\)};
\draw (-0.5,-0.3) node {\(_{k_0}\)};
\draw[color=red] (-0.25,0.35) node {\(_{L_0}\)};
\draw[-,line width=1.25pt,color=red] (-0.5,0.15) -- (-0.02,0.15);
\draw[-,line width=1pt] (1,-0.15) -- (1,0.15);
\draw (1,-0.3) node {\(_{k_2}\)};
\draw[-,line width=1pt] (1.9,-0.15) -- (1.9,0.15);
\draw (1.9,-0.3) node {\(_{k_3}\)};
\draw[-,line width=1pt] (2.8,-0.15) -- (2.8,0.15);
\draw (2.8,-0.3) node {\(_{k_4}\)};
\draw[-,line width=1pt] (3.6,-0.15) -- (3.6,0.15);
\draw (3.6,-0.3) node {\(_{k_5}\)};
\draw[-,line width=1pt,dotted] (4.6,-0.15) -- (4.6,0.15);
\draw (4.6,-0.3) node {\(_{k_6}\)};
\draw (3.4,0.6) node {\(_{g_5}\)};
\draw[->,line width=0.8pt,dotted] (2.4,1.15) .. controls (2.5,0.8) and (3.5,0.3) .. (3.95,0.1);
\draw (3,0.15) node {\(_{g_{4}}\)};
\draw[->,line width=0.8pt,dotted] (2.27,1.15) .. controls (2.3,0.8) and (3.1,0.3) .. (3.45,0.1);
\draw (2.3,0.15) node {\(_{g_{3}}\)};
\draw[->,line width=0.8pt,dotted] (2.15,1.15) .. controls (2.15,0.9) and (2.4,0.3) .. (2.6,0.1);
\draw (1.7,0.15) node {\(_{g_{2}}\)};
\draw[->,line width=0.8pt,dotted] (2,1.15) .. controls (2,1) and (1.5,0.4) .. (1.2,0.1);
\draw[->,line width=0.8pt,dotted] (1.8,1.15) .. controls (1.8,0.8) and (0.7,0.2) .. (0.6,0.1);
\draw (0.4,0.15) node {\(_{g_{1}}\)};
\draw[->,line width=0.8pt,dotted,color=red] (1.65,1.15) .. controls (1.65,0.8) and (0.4,0.45) .. (0.08,0.3);
\draw (0.7,0.8) node {\(_{g_{0}}\)};
\end{tikzpicture}
\item
\label{prop:freeseg2}
If $\wm$ is transitive and $G_x$ is a normal subgroup for (one and then each) $x\in M$, then an element of $\Paw$ is either free or contained in $\Pags$, i.e.,
$\Paw=\Pags\sqcup \Paf$. If $\wm$ acts even free, then $\Paw=\Pags\sqcup \Pafns$ holds.
\item
\label{prop:freeseg3}
If $\wm$ is proper, then each $\gamma\in \Pac$ is contained in a $\wm$-orbit.
If $G_x$ is in addition a normal subgroup for all $x\in M$, then $\Paw=\Pags\sqcup \Paf$ holds.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Basically, we have to show that a suitable choice of $\delta$ fills the gaps in \eqref{eq:decompo}.
Since $\gamma$ is free, we find $K\subseteq [a,b]$ compact such that $\gamma|_{K}$ is a free segment. Let ${\mathfrak{K}}$ denote the set of all such intervals and let ${\mathfrak{K}}$ be ordered by inclusion. Then each chain ${\mathfrak{L}}$ in ${\mathfrak{K}}$ has an upper bound, namely the closure $L$ of $\bigcup_{L'\in {\mathfrak{L}}} L'$. In fact, $\delta_L:=\gamma|_L$
is a free segment because $\delta_L \cpsim \wm_g\cp \delta_L$ implies $\gamma|_{L'} \cpsim \wm_g\cp \gamma|_{L'}$ for some $L'\in {\mathfrak{L}}$, hence $\gamma|_{L''} \cpsim \wm_g\cp \gamma|_{L''}$ for all ${\mathfrak{L}}\ni L''\geq L'$. Consequently, $\gamma|_{L''}\psim \wm_g\cp \gamma|_{L''}$ for all $L''\geq L'$, hence $\gamma|_L\isim \wm_g\cp \gamma|_L$ by continuity. Then, $\gamma|_L\psim \wm_g\cp \gamma|_L$ by Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt2}, so that $L$ is an upper bound of ${\mathfrak{L}}$. Consequently, by Zorn's lemma the set ${\MK_{\mathrm{m}}}\subseteq {\mathfrak{K}}$ of maximal elements is non-empty.
Let $L=[l_1,l_2]\in {\MK_{\mathrm{m}}}$, $\delta:=\gamma|_L$ and $a=k_0<\dots<k_n=b$ the respective decomposition from Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma3}. Let $\{h_\alpha\}_{\alpha \in I}\subseteq H_{[\gamma,\delta]}$ be a family of representatives and $0\leq i_1<\dots<i_m\leq n-1$ the indices for which we find $\alpha_p \in I$ with $\gamma|_{K_{i_p}}\psim \wm_{h_{\alpha_p}}\cp [\delta|_{L_{i_p}}]^{\pm 1}$ holds, see Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma3}. We now proceed in two steps:
\begingroup
\setlength{\leftmarginii}{22pt}
\begin{enumerate}
\item[\textbf{A)}]
We first show that each $K_{i_p}$ can be assumed to be maximal, i.e., $K_{i_p}\in {\MK_{\mathrm{m}}}$ for all $1\leq p\leq m$.
\item[\textbf{B)}]
Then we show that under this assumption for each $1\leq p\leq m$ we have:
\begingroup
\setlength{\leftmarginiii}{25pt}
\begin{enumerate}
\item[\textbf{i.)}]
$a<k_{i_p}$ \quad\hspace{10pt}$\Longrightarrow$\quad $\exists\: \alpha\in I\backslash\{\alpha_p\} \colon (\wm_{h_\alpha}\cp \delta)(l_j)=\gamma(k_{i_p})$\hspace{10pt} for some $j\in \{1,2\}$.
\item[\textbf{ii.)}]
\vspace{3pt}
$k_{i_p+1}< b$ \quad$\Longrightarrow$\quad $\exists\: \alpha\in I\backslash\{\alpha_p\} \colon (\wm_{h_\alpha}\cp \delta)(l_j)=\gamma(k_{i_p+1})$ for some $j\in \{1,2\}$.
\end{enumerate}
\endgroup
\end{enumerate}
\endgroup
\noindent
Since \textbf{B)} implies that $(i_1,\dots,i_m)=(1,\dots,n-1)$, the claim then follows.
\vspace{2pt}
{\bf Part A:}
First observe that $\gamma|_{K_{i_p}}$ is a free segment for $1\leq p\leq m$ because
\begin{align*}
\wm_g\cp \gamma|_{K_{i_p}} \cpsim \gamma|_{K_{i_p}}\qquad&\Longrightarrow \qquad \wm_{h_{\alpha_p}^{-1}\cdot g\cdot h_{\alpha_p}}\cp \delta \cpsim \delta\\
\qquad &\Longrightarrow \qquad \wm_{h_{\alpha_p}^{-1}\cdot g\cdot h_{\alpha_p}}\cp \delta =\delta\\
\qquad &\Longrightarrow \qquad \wm_g\cp \gamma|_{K_{i_p}}=\gamma|_{K_{i_p}},
\end{align*}
hence $K_{i_p}\in {\mathfrak{K}}$. Now, it may happen that $K_{i_p}\notin {\MK_{\mathrm{m}}}$.
\hspace{25pt}
\begin{tikzpicture}
\draw[-,line width=1.5pt] (0,0) -- (4,0);
\draw[->,line width=1.5pt,dotted] (4,0) -- (5,0);
\draw (5.3,0) node {\(\gamma\)};
\draw (9.2,0) node {\(\Longrightarrow \:\:\:\:\text{ replace } L \text{ by }L':=[k_2,t]\in{\MK_{\mathrm{m}}}\)};
\draw[-,line width=1.8pt,color=blue] (0,0) -- (0.9,0);
\draw[-,line width=1.8pt,dotted,color=olive] (-0.4,-0) -- (0,0);
\draw[-,line width=1.5pt] (0,-0.2) -- (0,0.2);
\draw (0,-0.35) node {\(_{k_0}\)};
\draw[color=blue] (0.45,-0.7) node {\(_{L=K_0}\)};
\draw[-,line width=1pt] (0.9,-0.15) -- (0.9,0.15);
\draw (0.9,-0.35) node {\(_{k_1}\)};
\draw[-,line width=1.8pt,color=blue] (1.4,0) -- (2.2,0);
\draw[-,line width=3pt,color=white] (2.2,0) -- (2.6,0);
\draw[-,line width=1.8pt,dotted,color=olive] (2.2,-0) -- (2.6,0);
\draw[-,line width=1pt] (1.4,-0.15) -- (1.4,0.15);
\draw (1.4,-0.35) node {\(_{k_2}\)};
\draw[color=blue] (1.8,-0.7) node {\(_{K_{i_p}=K_2}\)};
\draw[-,line width=1pt] (2.2,-0.15) -- (2.2,0.15);
\draw (2.2,-0.35) node {\(_{k_3}\)};
\draw[-,line width=1pt] (2.6,-0.15) -- (2.6,0.15);
\draw (2.6,-0.35) node {\(_{t}\)};
\draw (3.4,-0.35) node {\(_{k_4}\)};
\draw[-,line width=3pt,color=white] (3.38,0) -- (3,0);
\draw[-,line width=1.8pt,dotted,color=olive] (3.38,0) -- (3,0);
\draw[-,line width=1.8pt,color=blue] (3.4,0) -- (4.2,0);
\draw[-,line width=1pt] (3.4,0.15) -- (4.2,0.15);
\draw[-,line width=1pt,dotted] (3.05,0.15) -- (3.4,0.15);
\draw[-,line width=1pt] (3.4,-0.15) -- (3.4,0.15);
\draw[-,line width=1pt] (4.2,-0.15) -- (4.2,0.15);
\draw (4.2,-0.35) node {\(_{k_5}\)};
\draw[color=blue] (3.82,-0.7) node {\(_{K_4}\)};
\draw (1.2,0.9) node {\(_{\wm_{h_{\alpha_p}}}\)};
\draw[->,line width=0.8pt,dotted,color=black] (0.4,0.25) .. controls (0.8,0.8) and (1.4,0.8) .. (1.8,0.25);
\draw[-,line width=1pt] (0,0.15) -- (0.9,0.15);
\draw[-,line width=1pt,dotted] (-0.4,0.15) -- (0,0.15);
\draw[-,line width=1pt] (1.4,0.15) -- (2.2,0.15);
\draw[-,line width=1pt,dotted] (2.2,0.15) -- (2.6,0.15);
\draw[->,line width=0.8pt,dotted,color=black] (0.5,0.25) .. controls (1,0.7) and (3.6,0.5) .. (3.8,0.25);
\draw (3,0.8) node {\(_{\wm_{h_{\alpha_{q}}}}\)};
\end{tikzpicture}
In this case, let $K_{i_p}\subseteq L'\in {\MK_{\mathrm{m}}}$ and define $\delta':=\gamma|_{L'}$. Then,
\begin{align*}
\delta=\gamma|_L \psim \wm_{h_{\alpha_p}^{-1}}\cp \delta'|_{K_{i_p}}
\end{align*}
so that maximality of $L$ shows that either $L=K_0$ or $L=K_{n-1}$ holds.
Assume that $L=K_0$ and let $a=k'_0<\dots < k'_{n'}=b$ be the decomposition of $\operatorname{\mathrm{dom}}[\gamma]$ w.r.t.\ $\delta'$. Then $K_0'=K_0=L\neq L'$, so that we can assume that $L\neq K_0$ from the beginning.
By the same argument we also can assume that $L\neq K_{n-1}$. Then, there cannot exist $1\leq p\leq m$ with $K_{i_p}\notin {\MK_{\mathrm{m}}}$ since this (as we have shown) would contradict that $L\neq K_0,K_{n-1}$.
\vspace{6pt}
{\bf Part B:}
We only discuss the second case as the first one follows analogously. Moreover, to simplify the notations, we assume that $K_{i_p}=[k_{i_p},k_{i_p+1}]=[0,k]$ and define $K:=K_{i_p}$.
Recall that $\delta':=\gamma|_K$ is a maximal free segment just because $K\in {\MK_{\mathrm{m}}}$ by {\bf Part A}. Finally,
let $n_0\in \mathbb{N}$ with $k+\frac{1}{n_0}\leq b$.
It follows that $K_n:=\big[0,k+\frac{1}{n}\big]\notin {\mathfrak{K}}$ for each $n\geq n_0$ so that we find $g_n\in G\backslash G_{\gamma|_{K_n}}= G\backslash G_{\delta'}$
(see $\textit{(b)}$ in Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma1}) with
$\gamma|_{K_n}\cpsim \wm_{g_n}\cp \gamma|_{K_n}$.
We now proceed in two steps:
\begingroup
\setlength{\leftmarginii}{17pt}
\begin{enumerate}
\item[\textbf{I}]
First, we show that the claim follows if there exists $h\in G\backslash G_{\delta'}$ with $g_{n} \segsims h$ for infinitely many $n\geq n_0$.
\item[\textbf{II}]
Second, we show that such an element $h$ exists.
\end{enumerate}
\endgroup
\vspace{2pt}
{\bf Step I:}
We can assume that $g_n\segsims h$ holds for all $n\geq n_0$, hence
\begin{align*}
\gamma|_{K_n}\cpsim \wm_{h}\cp \gamma|_{K_n} \qquad \forall\: n\geq n_0.
\end{align*}
Then, we find sequences
\begin{align*}
K_0\supseteq \{x_m\}_{m\in \NN} \rightarrow k' \in K\qquad\text{and}\qquad K_0\supseteq \{y_m\}_{m\in\NN} \rightarrow k'' \in K
\end{align*}
with $\gamma(x_m)=\wm_{h}(\gamma(y_m))$ for all $m\in \NN$, hence
$\gamma(k')=\wm_{h}(\gamma(k''))$. Moreover, since $\gamma(k')$ is an accumulation point of
$\gamma(K_0)\cap (\wm_h\cp \gamma)(K_0)$, for $\gamma'$ an extension of
$\gamma$ by Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt1} we find open intervals $I',I''\subseteq \operatorname{\mathrm{dom}}[\gamma']$ with $k'\in I'$, $k''\in I''$ and
\begin{align}
\label{eq:inters}
\gamma'(I')=\wm_h(\gamma'(I'')).
\end{align}
Then, $k',k'' \in \{0,k\}$, since elsewise we would have $\delta'\cpsim \wm_h\cp \delta'$, hence $h\in G_{\delta'}$ which contradicts the choice of $h$. However, we even can assume that $k'$ and $k''$ are not both zero.
In fact, since $\gamma|_{K}\nsim_\cp \wm_{h}\cp \gamma|_{K}$, we can arrange that $\{x_m\}_{m\in \NN}\subseteq [k,k+1/n_0]$ (then $k'=k$) or $\{y_m\}_{m\in \NN}\subseteq [k,k+1/n_0]$ (then $k''=k$).
Combining this with \eqref{eq:inters}, we see that there are $\epsilon,\epsilon'>0$ such that one of the following situations holds:
\vspace{10pt}
\quad
\begin{tikzpicture}
\draw (0.6,1.05) node {\(_{\boldsymbol{k',\: k''=k}}\)};
\draw[-,line width=1.5pt,dotted] (0,0) -- (1,0);
\draw[-,line width=1.5pt] (1,0) -- (3.5,0);
\draw[->,line width=1.5pt,dotted] (3.5,0) -- (4,0);
\draw (4.3,0) node {\(\gamma\)};
\filldraw[black] (0,0) circle (2pt);
\draw[-,line width=1.5pt] (2,-0.15) -- (2,0.15);
\draw (2,-0.4) node {\(_{k}\)};
\draw[-,line width=1.5pt] (1,-0.15) -- (1,0.15);
\draw (1,-0.4) node {\(_{0}\)};
\draw[color=blue] (1.2,0.5) node {\(_{\delta'}\)};
\draw[->,line width=0.8pt, dotted,color=blue] (1,0.3) -- (1.9,0.3);
\draw[<-,line width=0.8pt, dotted] (2.1,0.3) -- (3,0.3);
\draw[->,line width=0.8pt, dotted,color=red] (1.6,0.3) -- (1.9,0.3);
\draw[<-,line width=0.8pt, dotted,color=orange] (2.1,0.3) -- (2.4,0.3);
\draw[line width=0.8pt] (2,0.3) circle (1pt);
\draw[->,line width=0.8pt,dotted,color=black] (1.5,0.4) .. controls (1.7,1) and (2.3,1) .. (2.5,0.4);
\draw (2,0.6) node {\(_{\wm_h}\)};
\draw (2,1.05) node {\(_{\text{flip}}\)};
\draw (3.25,0.7) node {\(_{h\in G_{\delta'(k)}}\)};
\draw[-,line width=1.5pt] (3,-0.15) -- (3,0.15);
\draw[color=orange] (2.37,0) node {\()\)};
\draw[color=orange] (2.5,-0.42) node {\(_{k+\epsilon}\)};
\draw[color=red] (1.63,0) node {\((\)};
\draw[color=red] (1.5,-0.42) node {\(_{k-\epsilon'}\)};
\end{tikzpicture}
\begin{tikzpicture}
\draw (0.32,1) node {\(_{\boldsymbol{k'=k}}\)};
\draw (0.32,0.75) node {\(_{\boldsymbol{k''=0}}\)};
\draw[-,line width=1.5pt,dotted] (0,0) -- (1,0);
\draw[-,line width=1.5pt] (1,0) -- (3.5,0);
\draw[->,line width=1.5pt,dotted] (3.5,0) -- (4,0);
\draw (4.3,0) node {\(\gamma\)};
\filldraw[black] (0,0) circle (2pt);
\draw[-,line width=1.5pt] (2,-0.15) -- (2,0.15);
\draw (2,-0.4) node {\(_{k}\)};
\draw[-,line width=1.5pt] (1,-0.15) -- (1,0.15);
\draw (1,-0.4) node {\(_{0}\)};
\draw[color=blue] (1.2,0.5) node {\(_{\delta'}\)};
\draw[->,line width=0.8pt, dotted,color=blue] (1,0.3) -- (1.95,0.3);
\draw[->,line width=0.8pt, dotted] (2.05,0.3) -- (3,0.3);
\draw[-,line width=0.8pt, dotted,color=red] (1.0,0.3) -- (1.3,0.3);
\draw[-,line width=0.8pt, dotted,color=orange] (2.05,0.3) -- (2.3,0.3);
\draw[->,line width=0.8pt,dotted,color=black] (1.5,0.4) .. controls (1.7,1) and (2.3,1) .. (2.5,0.4);
\draw (2,0.6) node {\(_{\wm_h}\)};
\draw (2,1.05) node {\(_{\text{shift}}\)};
\draw[-,line width=1.5pt] (3,-0.15) -- (3,0.15);
\draw[color=orange] (2.25,0) node {\()\)};
\draw[color=orange] (2.5,-0.42) node {\(_{k+\epsilon}\)};
\draw[color=red] (1.25,0) node {\()\)};
\draw[color=red] (1.25,-0.42) node {\(_{\epsilon'}\)};
\end{tikzpicture}
\begin{tikzpicture}
\draw (0.32,1) node {\(_{\boldsymbol{k'=0}}\)};
\draw (0.32,0.75) node {\(_{\boldsymbol{k''=k}}\)};
\draw[-,line width=1.5pt,dotted] (0,0) -- (1,0);
\draw[-,line width=1.5pt] (1,0) -- (3.5,0);
\draw[->,line width=1.5pt,dotted] (3.5,0) -- (4,0);
\draw (4.3,0) node {\(\gamma\)};
\filldraw[black] (0,0) circle (2pt);
\draw[-,line width=1.5pt] (2,-0.15) -- (2,0.15);
\draw (2,-0.4) node {\(_{0}\)};
\draw[-,line width=1.5pt] (1,-0.15) -- (1,0.15);
\draw[->,line width=0.8pt, dotted] (1,0.3) -- (1.95,0.3);
\draw[->,line width=0.8pt, dotted,color=blue] (2.05,0.3) -- (3,0.3);
\draw[-,line width=0.8pt, dotted,color=red] (3.05,0.3) -- (3.3,0.3);
\draw[-,line width=0.8pt, dotted,color=orange] (2.05,0.3) -- (2.3,0.3);
\draw[color=blue] (2.8,0.5) node {\(_{\delta'}\)};
\draw[<-,line width=0.8pt,dotted,color=black] (1.5,0.4) .. controls (1.7,1) and (2.3,1) .. (2.5,0.4);
\draw (2,0.6) node {\(_{\wm_h}\)};
\draw (2,1.05) node {\(_{\text{shift}}\)};
\draw[-,line width=1.5pt] (3,-0.15) -- (3,0.15);
\draw (3,-0.4) node {\(_{k}\)};
\draw[color=orange] (2.25,0) node {\()\)};
\draw[color=orange] (2.3,-0.42) node {\(_{\epsilon}\)};
\draw[color=red] (3.25,0) node {\()\)};
\draw[color=red] (3.5,-0.42) node {\(_{k+\epsilon'}\)};
\end{tikzpicture}
\vspace{10pt}
\begin{tabular}{lrcrclcl}
\:\: &$\boldsymbol{k',\: k''=k\colon}$ &\hspace{-10pt} & $\gamma([k,k+\epsilon))$&$=$&$(\wm_h\cp\delta')((k-\epsilon',k])$ &\hspace{34pt}& $h\in H_{\gamma,\delta'}$\\
&$\boldsymbol{k'=k,\: k''=0\colon}$ & & $\gamma([k,k+\epsilon))$&$=$&$(\wm_h\cp\delta')([0,\epsilon'))$ && $h\in H_{\gamma,\delta'}$\\
&$\boldsymbol{k'=0,\: k''=k\colon}$ & & $\delta'([0,\epsilon))$&$=$&$(\wm_h\cp \gamma)([k,k+\epsilon'))$ && $h^{-1}\in H_{\gamma,\delta'}$.
\end{tabular}
\vspace{6pt}
Hence, in the same order
\begin{align*}
\gamma(k_{i_p+1})\:=\:\gamma(k)&\:=\:(\wm_h\cp\delta')(k)\hspace{9.5pt}\:=\:(\wm_{h\cdot h_{\alpha_p}}\cp \delta)(l)\qquad \hspace{9.5pt}\text{for some}\quad l\in [l_1,l_2]\\
\gamma(k_{i_p+1})\:=\:\gamma(k)&\:=\:(\wm_h\cp\delta')(0)\hspace{10pt}\:=\:(\wm_{h\cdot h_{\alpha_p}}\cp \delta)(l)\qquad\hspace{9.5pt} \text{for some}\quad l\in [l_1,l_2]\\ \gamma(k_{i_p+1})\:=\:\gamma(k)&\:=\:(\wm_{h^{-1}}\cp\delta')(0)\:=\:(\wm_{h^{-1}\cdot h_{\alpha_p}}\cp \delta)(l)\qquad \text{for some}\quad l\in [l_1,l_2].
\end{align*}
Moreover, in the first two cases we have $h\in H_{\gamma,\delta'}$, hence
\begin{align*}
h\cdot h_{\alpha_{p}}\in H_{\gamma,\delta}\qquad \Longrightarrow\qquad h\cdot h_{\alpha_{p}}\segsim h_{\alpha_q} \text{ for some }I\ni \alpha_q \neq \alpha_p.
\end{align*}
For this, observe that $\alpha=\alpha_p$ would imply that $h\in G_\delta$, as a straightforward calculation shows.
Then, by the same arguments we see that
$h^{-1}\cdot h_{\alpha_p}\segsim h_{\alpha_q}$ for some $I\ni {\alpha_q} \neq \alpha_p$ in the third case.
Consequently,
$\textbf{B.ii.)}$ follows if we show that even $l\in \{l_1,l_2\}$ holds. This, however, follows easily from the maximality of the interval $K_{i_q}$ that corresponds to $\alpha_q$.
\vspace{2pt}
{\bf Step II:}
We assume that an element $h\in G\backslash G_ {\delta'}$ as in Step I does not exist. Then, replacing $\{g_n\}_{n\in \NN}$ by a subsequence we can arrange that $g_n \nsim_{\delta'} g_m$ whenever $m>n$. Moreover, we can assume that $g_n\notin H_{\gamma,\delta'}$ for all $n\geq n_0$. In fact, if for each $n\geq n_0$ we find $\wt{n}\geq n$ with $g_{\wt{n}}\in H_{\gamma,\delta'}$, then finiteness of $I$ shows that there must be $h\in H_{\gamma,\delta'}$ with the properties from Step I.
Thus, we can assume that
\begin{align*}
\gamma|_{K_n}\cpsim \wm_{g_n}\cp \gamma|_{[k,k+1/n]}\qquad \forall\:n\geq n_0 \qquad \text{as well as}\qquad g_n\nsim_{\delta'} g_m\qquad\forall\: n\neq m,
\end{align*}
and we now show that this is impossible.
For this, we write\footnote{Basically, this just means that $\gamma_1$ and $\gamma_2$ traverse into the same direction.} $\gamma_1 \!\upharpoonleft\! \upharpoonright\! \gamma_2$ for $\gamma_1,\gamma_2 \in \Paw$ iff $\gamma_1|_J=\gamma_2 \cp \adif$ for $\adif\colon J\rightarrow J'$ an analytic diffeomorphism with $\dot\adif >0$. Analogously, write $\gamma_1 \!\upharpoonleft\! \downharpoonright\!\gamma_2$ if $\gamma_1|_J=\gamma_2 \cp \adif$ for $\adif\colon J\rightarrow J'$ an analytic diffeomorphism with $\dot\adif <0$.
We proceed in two steps:
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item[i.)]
Let $\gamma_n:=\wm_{g_n}\cp \gamma|_{[k,k+1/n]}$ for all $n\geq n_0$ and assume that $\gamma_n \!\upharpoonleft\! \upharpoonright\!\gamma_m$.
Then, it follows from Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt4} and $g_n\nsim_{\delta'} g_m$ that either
\vspace{5pt}
\begin{align*}
(\wm_{g_n}\cp \gamma)(K)\subseteq \gamma([k,k+1/m])\qquad \text{or} \qquad (\wm_{g_m}\cp \gamma)(K)\subseteq \gamma([k,k+1/n])\qquad \text{holds}.
\end{align*}
\hspace{15pt}
\begin{tikzpicture}
\draw[-,line width=0.8pt] (0.8,-0.12) -- (0.8,0.12);
\draw (0.8,-0.4) node {\(_{k+\frac{1}{m}}\)};
\draw[-,line width=0.8pt] (1.6,-0.12) -- (1.6,0.12);
\draw (1.6,-0.4) node {\(_{k}\)};
\draw[-,line width=0.8pt] (2.4,-0.12) -- (2.4,0.12);
\draw (2.4,-0.4) node {\(_{0}\)};
\draw[-,line width=0.8pt,color=orange] (2.2,0.5) .. controls (1.5,0.5) and (1.4,0.7) .. (1.3,0.3);
\draw[-,line width=0.8pt,color=orange] (1.3,0.3) .. controls (1.3,0) and (1.2,0) .. (1.1,0);
\draw[-,line width=0.8pt,color=orange] (1.1,0) .. controls (1,0) and (1,0.2) .. (1,0.35);
\draw[-,line width=1pt,color=orange] (1.08,0.35) -- (0.92,0.35);
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\draw[-,line width=0.9pt,color=orange] (1.38,0.29) -- (1.22,0.31);
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\draw[-,line width=0.9pt,color=orange] (2.2,0.42) -- (2.2,0.58);
\draw[color=orange] (2.4,0.5) node {\(_{0}\)};
\draw[color=orange] (1.3,0.8) node {\(\gamma_m\)};
\draw[-,line width=1.2pt] (0.8,0) -- (2.4,0);
\draw (2.8,0) node {\(\gamma_n\)};
\end{tikzpicture}
\quad\:\:\raisebox{20pt}{$\boldsymbol{\Longrightarrow}$}\qquad
\begin{tikzpicture}
\draw[-,line width=0.8pt] (0.8,-0.1) -- (0.8,0.12);
\draw (0.8,-0.4) node {\(_{k+\frac{1}{m}}\)};
\draw[-,line width=0.8pt] (1.6,-0.1) -- (1.6,0.12);
\draw (1.6,-0.4) node {\(_{k}\)};
\draw[-,line width=0.8pt] (2.4,-0.1) -- (2.4,0.12);
\draw (2.4,-0.4) node {\(_{0}\)};
\draw[-,line width=1.2pt,color=orange] (1.1,0.03) -- (3.3,0.03);
\draw[-,line width=1pt,color=orange] (1.08,0.39) -- (0.92,0.39);
\draw (0.5,0.39)[color=orange] node {\(_{k+\frac{1}{n}}\)};
\draw[-,line width=0.8pt,color=orange] (1.1,0.04) .. controls (1,0.04) and (1,0.24) .. (1,0.39);
\draw[-,line width=1pt,color=orange] (2.6,-0.08) -- (2.6,0.17);
\draw (2.6,0.35)[color=orange] node {\(_{k}\)};
\draw[-,line width=1pt,color=orange] (3.3,-0.08) -- (3.3,0.17);
\draw[color=orange] (3.3,0.35) node {\(_{0}\)};
\draw[-,line width=1.2pt] (0.8,0) -- (2.4,0);
\end{tikzpicture}
\qquad \raisebox{20pt}{or}\qquad
\begin{tikzpicture}
\draw[-,line width=0.8pt] (0.8,-0.1) -- (0.8,0.12);
\draw (0.8,-0.4) node {\(_{k+\frac{1}{m}}\)};
\draw[-,line width=0.8pt] (1.6,-0.1) -- (1.6,0.12);
\draw (1.6,-0.4) node {\(_{k}\)};
\draw[-,line width=0.8pt] (2.4,-0.1) -- (2.4,0.12);
\draw (2.4,-0.4) node {\(_{0}\)};
\draw[-,line width=1.2pt,color=orange] (1.1,0.03) -- (1.5,0.03);
\draw[-,line width=1pt,color=orange] (1.08,0.39) -- (0.92,0.39);
\draw (0.5,0.39)[color=orange] node {\(_{k+\frac{1}{n}}\)};
\draw[-,line width=0.8pt,color=orange] (1.1,0.04) .. controls (1,0.04) and (1,0.24) .. (1,0.39);
\draw[-,line width=1pt,color=orange] (1.2,-0.08) -- (1.2,0.17);
\draw (1.2,0.35)[color=orange] node {\(_{k}\)};
\draw[-,line width=1pt,color=orange] (1.5,-0.08) -- (1.5,0.17);
\draw[color=orange] (1.5,0.35) node {\(_{0}\)};
\draw[-,line width=1.2pt] (0.8,0) -- (2.4,0);
\end{tikzpicture}
In fact, elsewise we would have
\begin{align*}
\wm_{g_n}\cp \gamma|_{K} \cpsim \wm_{g_m}\cp \gamma|_{K}&\quad \Longrightarrow\quad \delta'\cpsim \wm_{g_n^{-1} g_m}\cp \delta'
\quad \Longrightarrow\quad g_n^{-1} g_m \in G_{\delta'}\\ &\quad \Longrightarrow\quad g_n\sim_{\delta'} g_m.
\end{align*}
In particular, it cannot happen that for infinitely many $n\geq n_0$ we find $n'>n$ with
$\gamma_n \!\upharpoonleft\! \upharpoonright\! \gamma_{n'}$.
In fact, then for each $\epsilon>0$ we would find
$n>n_0$ and $h_n\in G$ with
$\wm_{h_n}(\gamma(K))\subseteq \gamma([k,k+\epsilon))$.
Hence, we would find a sequence $\{n_i\}_{i\in \NN}\subseteq \NN$ such that $\wm_{h_{n_i}}(\gamma(0))$ and $\wm_{h_{n_i}}(\gamma(k))$ both converge to $\gamma(k)$ so that $\gamma(0)=\gamma(k)$ by pointwise properness of $\wm$. Of course, this contradicts injectivity of $\gamma$.
\item[ii.)]
By assumption, we have
\begin{align*}
\hspace{-18.5pt}\gamma|_{K_{n_0}}\!\upharpoonleft\! \downharpoonright\! \gamma_n
\qquad\qquad \text{or}\qquad\qquad \gamma|_{K_{n_0}}\!\upharpoonleft\! \upharpoonright\! \gamma_n
\end{align*}
for infinitely many $n\geq n_0$.
\hspace{25pt}
\begin{tikzpicture}
\draw[-,line width=1.2pt] (-0.2,-0.12) -- (-0.2,0.12);
\draw (-0.2,-0.4) node {\(_{0}\)};
\draw[-,line width=1.2pt] (0.8,-0.12) -- (0.8,0.12);
\draw (0.8,-0.4) node {\(_{k}\)};
\draw[-,line width=1.2pt] (1.4,-0.12) -- (1.4,0.12);
\draw (1.6,-0.478) node {\(_{k+\frac{1}{n_0}}\)};
\draw[-,line width=0.8pt,color=orange] (1.9,0.5) .. controls (1.2,0.5) and (1.1,0.7) .. (1,0.3);
\draw[-,line width=0.8pt,color=orange] (1,0.3) .. controls (1,0) and (0.9,0) .. (0.8,0);
\draw[-,line width=0.8pt,color=orange] (0.7,0) .. controls (0.6,0) and (0.6,0.2) .. (0.6,0.35);
\draw[-,line width=1.1pt,color=blue] (0.68,0.35) -- (0.52,0.35);
\draw (0.2,0.35)[color=blue] node {\(_{k+\frac{1}{n}}\)};
\draw[-,line width=0.9pt,color=blue] (1.08,0.29) -- (0.92,0.31);
\draw (1.25,0.33)[color=blue] node {\(_{k}\)};
\draw[-,line width=0.9pt,color=blue] (1.9,0.42) -- (1.9,0.58);
\draw[color=blue] (2.1,0.5) node {\(_{0}\)};
\draw[color=orange] (1.2,0.8) node {\(_{\wm_{g_{n}}\: \cp\: \gamma|_{K_n}}\)};
\draw[->,line width=1.2pt,dotted] (2.2,0) -- (2.8,0);
\draw (3.1,0) node {\(\gamma\)};
\filldraw[black] (-0.7,0) circle (1.5pt);
\draw[-,line width=1.2pt,dotted] (-0.7,0) -- (-0.2,0);
\draw[-,line width=1.2pt] (-0.2,0) -- (2.2,0);
\draw[-,line width=0.8pt, color=olive,dotted] (-0.2,-0.14) -- (0.8,-0.14);
\draw[color=olive] (0.4,-0.35) node {\(_{\delta'}\)};
\draw(-0.6,0.6) node {\(\!\upharpoonleft\! \downharpoonright\!\)};
\end{tikzpicture}
%
%
\qquad\qquad\raisebox{16pt}{or}
\qquad\quad\qquad
\begin{tikzpicture}
\draw[-,line width=1.2pt] (-0.2,-0.12) -- (-0.2,0.12);
\draw (-0.2,-0.4) node {\(_{0}\)};
\draw[-,line width=1.2pt] (0.8,-0.12) -- (0.8,0.12);
\draw (0.8,-0.4) node {\(_{k}\)};
\draw[-,line width=1.2pt] (1.4,-0.12) -- (1.4,0.12);
\draw (1.6,-0.478) node {\(_{k+\frac{1}{n_0}}\)};
\draw[-,line width=0.8pt,color=orange] (0.6,0.3) .. controls (0.5,0.7) and (0.4,0.5) .. (-0.3,0.5);
\draw[-,line width=0.8pt,color=orange] (0.8,0) .. controls (0.7,0) and (0.6,0) .. (0.6,0.3);
\draw[-,line width=0.8pt,color=orange] (1,0.35) .. controls (1,0) and (0.9,0) .. (0.9,0);
\draw[-,line width=1.1pt,color=blue] (0.92,0.35) -- (1.08,0.35);
\draw (1.5,0.3)[color=blue] node {\(_{k+\frac{1}{n}}\)};
\draw[-,line width=0.9pt,color=blue] (0.68,0.31) -- (0.52,0.29);%
\draw (0.4,0.31)[color=blue] node {\(_{k}\)};
\draw[-,line width=0.9pt,color=blue] (-0.3,0.42) -- (-0.3,0.58);
\draw[color=blue] (-0.45,0.5) node {\(_{0}\)};
\draw[color=orange] (0.4,0.8) node {\(_{\wm_{g_{n}}\: \cp\: \gamma|_{K_n}}\)};
\draw[->,line width=1.2pt,dotted] (2.2,0) -- (2.8,0);
\draw (3.1,0) node {\(\gamma\)};
\filldraw[black] (-0.9,0) circle (1.5pt);
\draw[-,line width=1.2pt,dotted] (-0.9,0) -- (-0.2,0);
\draw[-,line width=1.2pt] (-0.2,0) -- (2.2,0);
\draw[-,line width=0.8pt, color=olive,dotted] (-0.2,-0.14) -- (0.8,-0.14);
\draw[color=olive] (0.4,-0.35) node {\(_{\delta'}\)};
\draw(-0.8,0.5) node {\(\!\upharpoonleft\! \upharpoonright\!\)};
\end{tikzpicture}
In the first case, Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt4} in combination with $g_n\notin H_{\gamma,\delta'}$ shows that
\hspace{25pt}
\begin{tikzpicture}
\draw[-,line width=1.2pt] (-0.2,-0.12) -- (-0.2,0.12);
\draw (-0.2,-0.4) node {\(_{0}\)};
\draw[-,line width=1.2pt] (0.8,-0.12) -- (0.8,0.12);
\draw (0.8,-0.4) node {\(_{k}\)};
\draw[-,line width=1.2pt] (1.4,-0.12) -- (1.4,0.12);
\draw (1.6,-0.478) node {\(_{k+\frac{1}{n_0}}\)};
\draw[-,line width=0.8pt,color=orange] (1.9,0.5) .. controls (1.2,0.5) and (1.1,0.7) .. (1,0.3);
\draw[-,line width=0.8pt,color=orange] (1,0.3) .. controls (1,0) and (0.9,0) .. (0.8,0);
\draw[-,line width=0.8pt,color=orange] (0.7,0) .. controls (0.6,0) and (0.6,0.2) .. (0.6,0.35);
\draw[-,line width=1.1pt,color=blue] (0.68,0.35) -- (0.52,0.35);
\draw (0.2,0.35)[color=blue] node {\(_{k+\frac{1}{n}}\)};
\draw[-,line width=0.9pt,color=blue] (1.08,0.29) -- (0.92,0.31);
\draw (1.25,0.33)[color=blue] node {\(_{k}\)};
\draw[-,line width=0.9pt,color=blue] (1.9,0.42) -- (1.9,0.58);
\draw[color=blue] (2.1,0.5) node {\(_{0}\)};
\draw[color=orange] (1.2,0.8) node {\(_{\wm_{g_{n}}\: \cp\: \gamma|_{K_n}}\)};
\draw[->,line width=1.2pt,dotted] (2.2,0) -- (2.8,0);
\draw (3.1,0) node {\(\gamma\)};
\filldraw[black] (-0.7,0) circle (1.5pt);
\draw[-,line width=1.2pt,dotted] (-0.7,0) -- (-0.2,0);
\draw[-,line width=1.2pt] (-0.2,0) -- (2.2,0);
\draw[-,line width=0.8pt, color=olive,dotted] (-0.2,-0.14) -- (0.8,-0.14);
\draw[color=olive] (0.4,-0.35) node {\(_{\delta'}\)};
\end{tikzpicture}
\qquad\qquad\raisebox{16pt}{$\boldsymbol{\Longrightarrow}$}\qquad\quad\qquad
\begin{tikzpicture}
\draw[-,line width=1.3pt,color=orange] (1.1,0.03) --(3.4,0.03);
\draw[-,line width=1.2pt,dotted] (-0.3,0) -- (0.2,0);
\draw[-,line width=1.2pt] (0.2,0) -- (2,0);
\draw[->,line width=1.2pt,dotted] (2,0) -- (2.5,0);
\filldraw[black] (-0.3,0) circle (1.5pt);
\draw[color=orange] (2.5,0.4) node {\(_{\wm_{g_{n}}\: \cp\: \gamma|_{K_n}}\)};
\draw[-,line width=1.2pt] (0.2,-0.12) -- (0.2,0.12);
\draw (0.2,-0.4) node {\(_{0}\)};
\draw[-,line width=1.2pt] (1.2,-0.12) -- (1.2,0.12);
\draw (1.2,-0.4) node {\(_{k}\)};
\draw[-,line width=1.2pt] (1.8,-0.12) -- (1.8,0.12);
\draw (1.8,-0.478) node {\(_{k+\frac{1}{n_0}}\)};
\draw[-,line width=0.8pt,color=orange] (1.1,0.03) .. controls (1,0.03) and (1,0.2) .. (1,0.35);
\draw[-,line width=1pt,color=blue] (1.08,0.35) -- (0.92,0.35);
\draw (0.5,0.35)[color=blue] node {\(_{k+\frac{1}{n}}\)};
\draw[-,line width=0.9pt,color=blue] (2.7,-0.12) -- (2.7,0.12);
\draw (2.7,-0.4)[color=blue] node {\(_{k}\)};
\draw[-,line width=0.9pt,color=blue] (3.4,-0.12) -- (3.4,0.12);
\draw (3.4,-0.4)[color=blue] node {\(_{0}\)};
\end{tikzpicture}
\vspace{-20pt}
\begin{align*}
\gamma([k,k+1/n_0])\subseteq \mathrm{im}[\gamma_n]\qquad \forall n\geq n_0.
\end{align*}
Hence, $\gamma_{m}\!\upharpoonleft\! \upharpoonright\! \gamma_n$ for all $m,n\geq n_0$, which is impossible by i.). This shows that the first case cannot occur. In the same way, we deduce that in the second case we have $\gamma(K)\subseteq \mathrm{im}[\gamma_n]$ for all $n\geq n_0$, hence
$\gamma_{m}\!\upharpoonleft\! \upharpoonright\! \gamma_n$ for all $m,n\geq n_0$, contradicting i.) as well.
\end{enumerate}
\endgroup
\item
It is straightforward to see that $\delta\in \Paw$ is not free iff
$\wm_g\cp \delta$ is not free for all $g\in G$.
Consequently, we can restrict to curves starting at a fixed point $x\in M$ in the following.
Moreover, since $\wm$ is transitive and $G_x$ is a closed normal subgroup, we can identify $M$ with the (analytic) Lie group $G\slash G_x$ via $\phi\colon G\slash G_x\rightarrow M$, $[g]\mapsto \wm_x(g)$. Let $L_{[g]}\colon G\slash G_x\rightarrow G\slash G_x$, $[h]\mapsto [g\cdot h]$ denote the left translation w.r.t.\ the group structure on $G\slash G_x$ and observe that
$L_{[g]}\cp \phi^{-1}=\phi^{-1}\cp L_g$ just because $\phi\cp L_{[g]}=L_g\cp \phi$. We define $\gamma':=\phi^{-1}\cp \gamma$.
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item
\label{it:aa}
Then, $\gamma'$ is analytic embedded. Moreover, $\gamma'$ is not free if this is the case for $\gamma$.
For this, assume that $\gamma'|_K$ is a free segment for some $K\subseteq \operatorname{\mathrm{dom}}[\gamma]$. Then $\wm_g \cp \gamma|_K \cpsim \gamma|_K$ implies
\begin{align*}
L_{[g]}\cp \gamma'|_K=\phi^{-1}\cp L_g \cp \gamma|_K \cpsim \phi^{-1}\cp \gamma|_K=\gamma'|_K,
\end{align*}
hence $L_{[g]}\cp \gamma'|_K = \gamma'|_K$. Then $\wm_g\cp \gamma|_K = \gamma|_K$, showing that $\gamma$ is free as well.
\item
\label{it:bb}
Assume there is $g\in G$ and an element $\g'$ of the Lie algebra of $G\slash G_x$ such that
\begin{align*}
\gamma'\cpsim \gamma'_0 \colon L &\rightarrow G\slash G_x\\
t&\mapsto L_{[g]}\cp \exp(t\g')
\end{align*}
holds for $L\subseteq \RR$ compact. Then, since the canonical projection $\pi \colon G \rightarrow G\slash G_x$ is a submersion, we find $\g\in \mg$ with $\dd_e \pi(\g)=\g'$. Hence, for all $t\in \operatorname{\mathrm{dom}}[\gamma]$ we have
\begin{align*}
\gamma=\phi \cp \gamma'\cpsim &\big[L\ni t\mapsto L_g\cp \phi(\exp(t\g'))\big]\\
=&\big[L\ni t \mapsto L_g\cp \phi(\pi(\exp(t\g)))\big]\\
=&\big[L\ni t\mapsto L_g\cp \wm_x(\exp(t\g))\big]
\end{align*}
because $\pi$ is a Lie group homomorphism. In combination with Lemma \ref{lemma:sim}.\ref{lemma:sim5} this implies $\gamma\in \Pags$.
\end{enumerate}
\endgroup
So, in order to show the claim, we only have to consider the situation where $M=G$ and $g\colon [-a,a]\rightarrow G$ is an embedded analytic curve with $g(0)=e$ and $a>0$. Here, it suffices to show that there is an open interval $I\subseteq [-a,a]$ and $h\colon I\rightarrow \RR$ smooth
with
\begin{align}
\label{eq:sdsd}
\dot g(t)= h(t)\cdot\dd L_{g(t)} \dot g(0)\qquad \forall\: t\in I.
\end{align}
In fact, then for $t_0\in [i_1,i_2]\subseteq I$ fixed and $\g:=\dot g(0)$, the unique\footnote{For $t_0\in I=(i_1,i_2)$ define $v\colon [0,\min(1,i_2-t_0)]\rightarrow \mg$, $t\mapsto h(t+t_0)\cdot \g$ and apply Satz 1.10 in \cite{HelgaBaum} in order to obtain $\ovl{g}\colon [0,i_2-t_0] \rightarrow G$ uniquely determined by $\ovl{g}(0)=e$ and $\dot{\ovl{g}}(t)=\dd L_{\ovl{g}(t)}h(t+t_0)\cdot \g$. Then $\ovl{g}(t)=g(t_0)^{-1}g(t+t_0)$ because \eqref{eq:sdsd} shows that the right hand side fulfils these conditions. Consequently, $g$ is uniquely determined by \eqref{eq:sdsd}.} solution of \eqref{eq:sdsd} with $g(t_0)=e$ is given by
\begin{align*}
g(t)=g(t_0)\cdot \exp\left(\left[\int_{t_0}^t h(s)\:\dd s \right]\!\g\right)\qquad \forall\: t\in I,
\end{align*}
so that
$g \cpsim L_{g(t_0)}\cp \exp(\cdot\: \g)$.
Now, to show \eqref{eq:sdsd} let $K_n:=[-\textstyle\frac{1}{n},\textstyle\frac{1}{n}]$ and choose $g_n\in G\backslash\{e\}$ with $g|_{K_n}\cpsim L_{g_n}\cp g|_{K_n}$ for each $n\in \NNge$. This is possible because $g$ is not free.
Then, $\lim_n g_n =e$ as for each $n\in \NNge$ there is $x_n\in g(K_n)$ with $g_n\cdot x_n \in g(K_n)$, hence
\begin{align*}
\lim_n \:[g_n\cdot x_n], \lim_n x_n = g(0)=e\qquad\Longrightarrow\qquad \lim_n g_n =\lim_n x_n^{-1}=e.
\end{align*}
Since $g$ is an embedding, we find a closed neighbourhood $U$ of $e$ in $G$ and $K\subseteq (-a,a)$ compact with $U\cap \mathrm{im}[g]=g(K)$.
By continuity\footnote{Let $V\subseteq G$ be a neighbourhood of $e$ with $V^2\subseteq U$ and choose $n_0>0$ such that $g_n\in V$ and $g(K_n)\subseteq V$ for all $n\geq n_0$.} of $g$ and that of the group multiplication in $G$, we find $n_0>0$ such that $K_n\subseteq K$ and $g_n\cdot g(K_n)\subseteq U$ for all $n\geq n_0$.
\vspace{9pt}
\hspace{50pt}
\begin{tikzpicture}
\draw[->,line width=2.5pt] (0,0) .. controls (1,0.8) and (2,-0.3) .. (3,1);
\filldraw[black] (0,0) circle (2.5pt);
\filldraw[white] (1.5,0.3) circle (30pt);
\draw[-,line width=0.8pt,color=blue] (1.7,0.32) .. controls (1.5,0.31) and (1.5,0.31) .. (1.5,0.7);
\draw[-,line width=0.8pt,color=blue] (1.9,0.33) .. controls (2,0.33) and (2,0.33) .. (2,0.6);
\draw[-,line width=0.8pt,color=blue] (1.42,0.7) -- (1.58,0.7);
\draw[-,line width=0.8pt,color=blue] (1.92,0.6) -- (2.08,0.6);
\draw[-,line width=1pt] (0,0) .. controls (1,0.8) and (2,-0.3) .. (3,1);
\draw[black,line width=0.8pt, dashed] (1.5,0.3) circle (30pt);
\draw (0.4,1.1) node {\(U\)};
\draw (3.2,1.2) node {\(g\)};
\draw (0.8,0.1) node {\(_K\)};
\draw[-,line width=1pt] (1.4,0.24) -- (1.4,0.36);
\draw (1.4,-0.1) node {\(_{g|_{K_n}}\)};
\draw[-,line width=0.6pt] (1.1,0.18) -- (1.7,0.18);
\draw (1.1,0.3) node {\(_{[}\)};
\draw (1.7,0.3) node {\(_]\)};
\draw (1.3,0.45) node {\(_e\)};
\draw[color=blue] (1.55,0.95) node {\(_{\wm_{g_n}\cp\hspace{1pt}g|_{K_n}}\)};
\end{tikzpicture}
\qquad\qquad\raisebox{25pt}{\LARGE{$\boldsymbol{\Longrightarrow}$}}\qquad\qquad\quad
\begin{tikzpicture}
\draw[->,line width=2.5pt] (0,0) .. controls (1,0.8) and (2,-0.3) .. (3,1);
\filldraw[black] (0,0) circle (2.5pt);
\filldraw[white] (1.5,0.3) circle (30pt);
\draw[-,line width=0.8pt,color=blue] (0.8,0.2) -- (0.8,0.6);
\draw[-,line width=0.8pt,color=blue] (2.3,0.3) -- (2.3,0.6);
\draw[-,line width=0.8pt,color=blue] (0.8,0.6) -- (2.3,0.6);
\draw[-,line width=1pt] (0,0) .. controls (1,0.8) and (2,-0.3) .. (3,1);
\draw[black,line width=0.8pt, dashed] (1.5,0.3) circle (30pt);
\draw (0.4,1.1) node {\(U\)};
\draw (3.2,1.2) node {\(g\)};
\draw (0.8,0) node {\(_K\)};
\draw[-,line width=1pt] (1.4,0.24) -- (1.4,0.36);
\draw (1.4,-0.1) node {\(_{g|_{K_n}}\)};
\draw[-,line width=0.6pt] (1.1,0.18) -- (1.7,0.18);
\draw (1.1,0.3) node {\(_{[}\)};
\draw (1.7,0.3) node {\(_]\)};
\draw (1.3,0.45) node {\(_e\)};
\draw[color=blue] (1.55,0.85) node {\(_{\wm_{g_n}\cp\hspace{1pt}g|_{K_n}}\)};
\end{tikzpicture}
\vspace{7pt}
Now, $g|_{K_n}\cpsim L_{g_n}\cp g|_{K_n}$ implies $g\cpsim L_{g_n}\cp g|_{K_n}$ so that Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt4} provides us with compact intervals $K'\subseteq K_n$ and $K''\subseteq [-a,a]$ which are maximal w.r.t.\ the property that $g(K'')=(L_{g_n}\cp g)(K')$ holds. Since $g_n\cdot g(K_n)\subseteq U$ and $U\cap \mathrm{im}[g]=g(K)$, we have $K''\subseteq K$. Then, $K'=K_n$ because $K''$ neither contains $a$ nor $-a$. This shows that the curve $g_n\cdot g|_{K_n}$ is completely contained in $g|_K$.
Consequently, we find open intervals $J_n\subseteq K$ and $I_n\subseteq K_n$ with $0\in I_n$, as well as diffeomorphisms ${\adif}_n\colon I_n\rightarrow J_n$ such that
\begin{align}
\label{eq:hhhlgdfgrgf}
L_{g_n}\cp g|_{I_n}= g|_{J_n}\cp \adif_n\qquad\:\forall\:n\geq n_0.
\end{align}
Then $g_n=g_n\cdot g(0)=g(\adif_n(0))$, and since everything is smooth, for $t_n:=\adif_n(0)\in K$ and $\g:=\dot g(0)$ we obtain
\begin{align*}
\dd L_{g(t_n)} \g = \dd L_{g_n}\: \dot g(0)\stackrel{\eqref{eq:hhhlgdfgrgf}}{=} \dot\adif_n(0)\: \dot g(\adif_n(0))=\dot\adif_n(0)\: \dot g(t_n),
\end{align*}
hence, $\dd L_{g(t_n)^{-1}}\: \dot g(t_n) \in \Span_\RR(\g)$ for all $n\in \NNge$.
Let $\wt{g}\colon K\rightarrow \mg$, $t\mapsto \dd L_{g(t)^{-1}}\: \dot g(t)$ and denote by $s$ the supremum of $\|\wt{g}(t)\|$ for $t\in K$ and $\|\cdot\|$ a fixed norm in $\mg$. Then, $s$ is finite because $\wt{g}$ is continuous.
Now, $\wt{g}$ is even analytic and intersects the image of the analytic embedded curve
\begin{align*}
\delta\colon [-s\slash \|\g\|,s\slash \|\g\|]\rightarrow \mg,\quad t\mapsto t\cdot \g
\end{align*}
in infinitely many points.
Then, $\mathrm{im}[\delta]\cap \mathrm{im}[\wt{g}]$ contains an accumulation point, and it follows as in the
proof of Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt1} that we find an
open interval $I\subseteq K$ for which $\wt{g}(I)\subseteq \Span_\RR(\g)$ holds. This shows that
\begin{align*}
\dd L_{g(t)^{-1}} \dot g(t)=\wt{g}(t) = h(t)\cdot \g\qquad\forall\: t\in I
\end{align*}
holds for some smooth map $h\colon I\rightarrow \RR$, hence \eqref{eq:sdsd}.
\item
We fix $t\in (a,b)$ and choose compact neighbourhoods $K \subseteq (a,b)$ of $t$ and $U\subseteq M$ of $x:=\gamma(t)$ with $\gamma(K)=\mathrm{im}[\gamma]\cap U$.
We first show that we find $\{g'_n\}_{n\in \NN}\subseteq G\backslash G_x$ with $\wm(g'_n,x)\in \gamma(K)$ and $\lim_n g_n=e$.
\begingroup
\setlength{\leftmarginii}{14pt}
\begin{itemize}
\item
\vspace{-4pt}
Let $n_0\in \NNge$ be such that $K_n:=\big[t- \frac{1}{n},t+\frac{1}{n}\big]\subseteq K$ for all $n>n_0$ and use Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma2} in order to fix some $g_n\in G\backslash G_x$ with $\wm_{g_n}\cp \gamma|_{K_n} \cpsim \gamma|_{K_n}$ for each $n\geq n_0$.
Then, there exist $t_n, s_n\in K_n$ with $\gamma(t_n)=\wm(g_n, \gamma(s_n))$ for all $n\geq n_0$, hence
\begin{align}
\label{eq:sdfdfsfsd}
\lim_n \wm(g_n,\gamma(s_n))=\lim_n\gamma(t_n)=\gamma(t)=x
\end{align}
Then, by properness of $\wm$ we find a subnet of
$\{g_n\}_{n\in \NN_{\geq n_0}}$ which converges to an element $h\in G$, and since manifolds are first countable, we even can assume $\lim_n g_n =h$ from the beginning. So, by continuity of $\wm$ and \eqref{eq:sdfdfsfsd} we have
\begin{align*}
\wm(h,x)=\wm(h,\gamma(t))=\lim_n \wm(g_n,\gamma(s_n))=x\qquad \Longrightarrow \qquad h\in G_x,
\end{align*}
where in the second step we have used that $\lim_n \gamma(s_n)=\gamma(t)$.
\item
If we can prove that $\wm(g_n,x)\in \gamma(K)$ holds for infinitely many $n\geq n_0$, we just have to replace $g_n$ by $g_n\cdot h^{-1}$ in order to get the desired sequence $\{g'_n\}_{n\in \NN}\subseteq G\backslash G_x$.
To this end, let $V\subseteq G$ and $W\subseteq M$ be neighbourhoods of $h$ and $x$, respectively, with $\wm(V,W)\subseteq U$.
\vspace{3pt}
We choose $n_0' \geq n_0$ such that $g_n\in V$ and $\gamma(K_n)\subseteq W$ for all $n\geq n_0'$. Then $g_n \cdot \gamma(K_n)\subseteq U$ for all $n\geq n_0'$, and since $\wm_{g_n}\cp \gamma|_{K_n}\cpsim \gamma|_K$, the same arguments as in Part \ref{prop:freeseg2} show that $(\wm_{g_n}\cp \gamma)(K_n)\subseteq \gamma(K)$, hence $\wm(g_n,x)\in \gamma(K)$ for all $n\geq n_0'$. \hspace*{\fill}$\dagger$
\end{itemize}
\endgroup
Let $\mg=\mg_0\oplus \mg_x$ and $U\subseteq \mg_0$, $V\subseteq \mg_x$ be neighbourhoods of zero, such that
\begin{align*}
h\colon U\times V\rightarrow W,\quad (u,v)\mapsto \exp(u)\cdot \exp(v)
\end{align*}
is a diffeomorphism to an open subset $W\subseteq G$. Since the differential $\dd_e f=\dd_e\wm_x|_{TU}$ of $f:=\wm_x\cp\exp|_U$ is injective, shrinking $U$ we can assume that $f$ is an analytic embedding.
We define $O:=f(U)$ and choose an analytic submanifold chart $(\phi_0,U_0)$ of $O$ around $x$. Then, we find $I\subseteq \operatorname{\mathrm{dom}}[\gamma]$ open with $t\in I$ and $\gamma(I)\subseteq U_0$. Since $\lim_n g_n'= e$, we find $n'\in \NN$ such that for each $n\geq n'$ we have $g'_n\in W$, hence $g_n'=h(u_n',v_n')$ for some $(u_n',v_n')\in U\times V$. Consequently,
\begin{align*}
\wm(g_n',x)=(\wm_x\cp h)(u_n',v_n')=f(u_n')\in O\qquad \forall\:n\geq n',
\end{align*}
and since $\wm(g_n',x)\in \gamma(K)$,
by the embedding property of $\gamma$ we find $n''\geq n'$ such that $\wm(g_n',x)\in \gamma(I)$ holds for all $n\geq n''$.
Consequently, $\phi_0(x)$ is an accumulation point of $(\phi_0\cp \gamma)(I) \cap \phi_0(O\cap U_0)$ in $\phi_0(O\cap U_0)$. Then, by analyticity of the components of $\phi_0\cp \gamma$ we find $J\subseteq I$ open with $t\in J$ and $\gamma(J)\subseteq O\subseteq Gx$.
Now, if we choose an analytic embedding $\gamma'\colon [a-\epsilon,b+\epsilon]$ with $\gamma'|_{[a,b]}=\gamma$, then Part \ref{prop:freeseg1}) shows that $\gamma'$ is not free as well. Then, the above arguments show that for each $t\in [a,b]$ we find an open interval $J\subseteq [a-\epsilon,b+\epsilon]$ with $t\in J$ and $\gamma'(J)\subseteq G \gamma(t)$. Since finitely many of such intervals $J_1,\dots,J_k$ cover $[a,b]$, and since orbits are disjoint, $\mathrm{im}[\gamma]$ must be contained in $G x$. This shows the first part.
For the second part,
let $L\subseteq J$ be compact, such that $\gamma(L)\subseteq O$. Then,
\begin{align*}
\gamma':=\pi \cp\exp\cp f^{-1}\cp \gamma|_L
\end{align*}
is an embedded analytic curve in $G\slash G_x$ which is not free. In fact, for analyticity observe that $(V',\phi')$ with $V':=\pi(\exp(U))$ and $\phi':=(\pi\cp \exp|_U)^{-1}$ is an analytic chart of $G\slash G_x$ around $[e]$, and that $f\colon U\rightarrow O$ is an analytic diffeomorphism. Moreover, that $\gamma'$ is not free follows by the same arguments as in the proof of Part \ref{prop:freeseg2}), cf.\ (a). Consequently, $\gamma'$ is Lie algebra generated by Part \ref{prop:freeseg2}), and since $\dd_e \pi$ is a submersion and a Lie group homomorphism, we conclude that $\gamma|_L$, hence $\gamma$, is Lie algebra generated as well, cf.\ (b) in Part \ref{prop:freeseg2}).
\end{enumerate}
\end{proof}
\end{proposition}
Using the first part of the above proposition, we now obtain that each of the sets $\Pafs$, $\Pafns$, $\Pags$ and $\Pacs$ are closed under decomposition and inversion of its elements, hence that the factorization \eqref{eq:dsfs444ffff} holds.
\begin{corollary}
\label{rem:freinichtLiealg}
The sets $\Pafs$, $\Pafns$, $\Pags$ and $\Pacs$ are closed under decomposition and inversion of their elements.
\begin{proof}
By Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma1} $\Paf$ is closed under inversions, and Proposition \ref{prop:freeseg}.\ref{prop:freeseg1} shows that it is also closed under decompositions.
Then, Part $\textit{(b)}$ Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma1} shows
that the sets $\Pafns$ and $\Pafs$ are closed under decomposition and inversion as well.
Finally, by Corollary \ref{cor:decompo} the set $\Paw\backslash \Pags=\Pacs \sqcup \Paf$ is invariant under decomposition and inversion, so that the inverse or a subcurve of an element of $\Pacs$ must be contained in $\Pacs\sqcup\Paf$. However, the inverse of a free segment is a free segment, and freeness of a subcurve of $\gamma\in \Pacs$ already would imply freeness of $\gamma$.
\end{proof}
\end{corollary}
We close this section with the following (discrete) analogue to Proposition \ref{th:invhomm}.\ref{th:invhomm1}. There, we modify invariant homomorphisms along free segment $\delta$ by means of $G_\delta$-invariant maps:
\begin{definition}
Let $\delta\colon[k_1,k_2]\rightarrow M$ be a free segment and $p\in F_{\delta(k_1)}$. Then, by $\MPD$ we will denote the set of all maps $\Psi\colon \RR\rightarrow S$
which are invariant under $G_\delta$ in the sense that $\alpha_{\fiba_p(h)}\cp \Psi=\Psi$ holds for all $h\in G_\delta$, and that fulfil
\begin{align*}
\Psi(\lambda-\lambda')=\Psi(\lambda)\cdot \Psi(\lambda')^{-1} \qquad\forall\: \lambda,\lambda'\in\RR.
\end{align*}
\end{definition}
\begin{remark}
If $G_\delta=\{e\}$, examples for such invariant maps $\Psi\colon \RR \rightarrow S$ are just given by $\Psi\colon \lambda\mapsto \exp(\lambda\cdot\s)$ for $\s\in \ms$. So, if $S$ is compact and connected ($\exp$ is surjective), then for each $s\in S$ and for each $\lambda\neq 0$ we find $\Psi \in \MPD$ with $\Psi(\lambda)=s$.
However, if $G_\delta$ is non-trivial, one has to decide from case to case whether such non-trivial maps exist.
\hspace*{\fill} $\lozenge$
\end{remark}
\begin{proposition}
\label{lemma:freemodify}
Let $\Pa\subseteq \Paw$ be closed under decompositions and inversions, $\homm'\in \IHOM$, $S$ compact and connected with $\dim[S]\geq 1$,\footnote{Recall that this ensures that the equivalence relations $\csim$ and $\psim$ coincide, see Lemma \ref{lemma:BasicAnalyt}.\ref{lemma:BasicAnalyt2}.} and $\delta\colon L=[l_1,l_2]\rightarrow M$ a free segment. Moreover, let $t\in L$, $p\in F_{\delta(t)}$ and $\Psi\in \MPD$.
\vspace{6pt}
\noindent
We define $\homm \in \IHOM$ as follows.
For $[s_1,s_2]\subseteq L$ and $s\in L$ let
\begin{align*}
\delta|_{[s_2,s_1]}:=[\delta|_{[s_1,s_2]}]^{-1}\qquad\text{as well as}\qquad\homm'(\delta|_{[s,s]})(q):=q
\end{align*}
and define
\begin{align*}
\wt{\Psi}\!\left(\delta|_{[a,b]}\right)(p'):=
\homm'\left(\delta|_{[t,b]}\right)(p)\cdot \Psi(b-a) \cdot \diff\big(\homm'\!\left(\delta|_{[t,a]}\right)(p),p'\big)\qquad \forall\:p'\in F_{\delta(a)}.
\end{align*}
Then, for $\gamma\in \Paw$ we choose a family of representatives $\{h_\alpha\}_{\alpha\in I}$ of $H_{[\gamma,\delta]}$ and let $k_0<\dots < k_n$ denote the respective unique decomposition from Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma3} of $\operatorname{\mathrm{dom}}[\gamma]=[k_0,k_n]$ into compact intervals $K_i=[k_i,k_{i+1}]$ for $0\leq i \leq n-1$. Recall that then either
\begin{align*}
\qquad\gamma|_{K_i}\nsim_\cp \wm_{h_\alpha}\cp \delta\qquad \forall\: \alpha\in I\qquad\qquad\text{(we define $\alpha(i):=0\notin I$ and $h_{\alpha(i)}:=e$ in this case)}
\end{align*}
or\footnote{Recall that $L_i=L$ for $1\leq i\leq n-2$ and $L_0,L_{n-1}\subseteq L$ are of the form $[l_1,l]$ or $[l,l_2]$ for $l_1<l\leq l_2$ or $l_1\leq l<l_2$, respectively.} $\gamma|_{K_i} \psim \wm_{h_{\alpha(i)}}\cp [\delta|_{L_i}]^{p_i}$ holds for $\alpha(i) \in I$, $p_i\in\{-1,1\}$ and $L_i\subseteq L$ uniquely determined. Let
\begin{align}
\label{eq:dkdfjs<afjfdsf}
\wt{\homm}(\gamma|_{K_i}):=
\begin{cases}
\homm'(\gamma|_{K_i}) &\mbox{if } \alpha(i)=0,\\
\Phi_{h_{\alpha(i)}}\cp \wt{\Psi}([\delta|_{L_i}]^{p_i})\cp \Phi_{h^{-1}_{\alpha(i)}} &\mbox{if } \alpha(i)\in I
\end{cases}
\end{align}
\noindent
as well as $\homm(\gamma):= \wt{\homm}(\gamma|_{K_0})\cp \dots \cp \wt{\homm}(\gamma|_{K_{n-1}})$.
Then, $\homm$ is a well-defined element of $\IHOM$.
\begin{proof}
First assume that $\homm$ is well defined. Then $\homm(\gamma)(p'\cdot s)=\homm(\gamma)(p')\cdot s$ for $s\in S$ is immediate from the definitions. For invariance of $\homm$ let $g\in G$. Then, for $\wm_g\cp \gamma$ we can use the same index set and the same decomposition as for $\gamma$, provided that we define $h'_{\alpha}:=g\cdot h_{\alpha}$ for all $\alpha\in I$. We obtain
\begin{align*}
\homm(\wm_g\cp \gamma)\cp \Phi_g &= \wt{\homm}(\wm_g\cp \gamma|_{K_0})\cp \dots \cp \wt{\homm}(\wm_g\cp \gamma|_{K_{n-1}})\cp \Phi_g\\
& \hspace{-3pt}\stackrel{\eqref{eq:dkdfjs<afjfdsf}}{=} \Phi_{g}\cp\wt{\homm}(\gamma|_{K_0})\cp \Phi_{g^{-1}}\cp \dots \cp \Phi_{g}\cp\wt{\homm}(\gamma|_{K_{n-1}})\cp \Phi_{g^{-1}}\cp \Phi_g
= \Phi_g\cp \homm(\gamma).
\end{align*}
Now, multiplicativity of $\homm$ is clear if $\gamma$ is splitted at some point contained in an interval $K_i$ with $\alpha(i)=0$. For the other case, it suffices to show that
$\wt{\Psi}\left(\delta|_{[s,b]}\right)\cp \wt{\Psi}\left(\delta|_{[a,s]}\right)=\wt{\Psi}\left(\delta|_{[a,b]}\right)$ holds for all $s,a,b\in L$ with $a<s<b$ or $b<s<a$. Now,
\begin{align*}
\wt{\Psi}\!\left(\delta|_{[a,s]}\right)(p')=\homm'(\delta|_{[t,s]})(p)\cdot \Psi(s-a)\cdot \diff(\homm'\left(\delta|_{[t,a]}\right)(p),p')
\end{align*}
and $\wt{\Psi}\!\left(\delta|_{[s,b]}\right)(q)=\homm'(\delta|_{[t,b]})(p)\cdot \Psi(b-s)\cdot \diff\!\left(\homm'\!\left(\delta|_{[t,s]}\right)(p),q\right)$, hence
\begin{align*}
\wt{\Psi}\!\left(\delta|_{[s,b]}\right)\big(\wt{\Psi}\!\left(\delta|_{[a,s]}\right)(p')\big)
&=\homm'(\delta|_{[t,b]})(p)\cdot \Psi(b-s)\cdot \diff\!\left(\homm'\!\left(\delta|_{[t,s]}\right)(p),\homm'(\delta|_{[t,s]})(p)\right)\\
&\hspace{1.8pt}\qquad\qquad\qquad\cdot \Psi(s-a)\cdot \diff\!\left(\homm'\!\left(\delta|_{[t,a]}\right)(p),p'\right)\\
&=\homm'(\delta|_{[t,b]})(p)\cdot \Psi(b-a)\cdot \diff\!\left(\homm'\!\left(\delta|_{[t,a]}\right)(p),p'\right)\\
&=\wt{\Psi}\left(\delta|_{[a,b]}\right)(p').
\end{align*}
For the inversion property observe that
\begin{align*}
\homm(\gamma^{-1})= \wt{\homm}\big([\gamma|_{K_{n-1}}]^{-1}\big)\cp \dots \cp \wt{\homm}\big([\gamma|_{K_0}]^{-1}\big).
\end{align*}
Then, $\homm(\gamma^{-1})\cp \homm(\gamma)=\id_{F_{\gamma(k_0)}}$ and $\homm(\gamma)\cp \homm(\gamma^{-1})=\id_{F_{\gamma(k_n)}}$ are clear if we can show that for all $0\leq i\leq n-1$ we have
\begin{align*}
\wt{\homm}\big([\gamma|_{K_i}]^{-1}\big)\cp \wt{\homm}([\gamma|_{K_i}])=\id_{F_{\gamma(k_i)}}\qquad\text{and}\:\:\qquad\wt{\homm}([\gamma|_{K_i}])\cp \wt{\homm}\big([\gamma|_{K_i}]^{-1}\big)=\id_{F_{\gamma(k_{i+1})}},
\end{align*}
respectively. Again, this is clear if $\alpha(i)=0$, and in the other case we have $[\delta|_{[a,b]}]^{-1}=\delta|_{[b,a]}$, hence
$\wt{\Psi}\left([\delta|_{[a,b]}]^{-1}\right)(q)=\homm'(\delta|_{[t,a]})(p)\cdot \Psi(a-b)\cdot \diff\left(\homm'\!\left(\delta|_{[t,b]}\right)(p),q\right)$. Then
\begin{align*}
\wt{\Psi}\left([\delta|_{[a,b]}]^{-1}\right)(\wt{\Psi}\left(\delta|_{[a,b]}\right)(p'))=\homm'(\delta|_{[t,a]})(p)\cdot \diff\!\left(\homm'\!\left(\delta|_{[t,a]}\right)(p),p'\right)=p'
\end{align*}
as well as
\begin{align*}
\wt{\Psi}\left(\delta|_{[a,b]}\right)(\wt{\Psi}\left([\delta|_{[a,b]}]^{-1}\right)(p'))=\homm'(\delta|_{[t,b]})(p)\cdot \diff\!\left(\homm'\!\left(\delta|_{[t,b]}\right)(p),p'\right)=p',
\end{align*}
which shows that
\begin{align*}
\wt{\Psi}\left([\delta|_{[a,b]}]^{-p_i}\right)(\wt{\Psi}\left([\delta|_{[a,b]}]^{p_i}\right)(p'))=p'\qquad\text{and}\qquad\wt{\Psi}\left([\delta|_{[a,b]}]^{p_i}\right)(\wt{\Psi}\left([\delta|_{[a,b]}]^{-p_i}\right)(p'))=p'
\end{align*}
holds. From this, the inversion property is clear.
\vspace{6pt}
\noindent
It remains to show that $\homm$ is well defined. Obviously, $\homm(\gamma)=\homm(\gamma')$ if $\gamma\csim \gamma'$, so that we only have to discuss what happens if we choose another family $\{h'_\alpha\}_{\alpha\in I'}$ of representatives of $H_{[\gamma,\delta]}$. Now, we can assume that $I=I'$, and that the decompositions of $\gamma$ into intervals $K_i$ w.r.t.\ both families coincide. In fact, this is clear from $h_\alpha \segsim h'_\alpha$ and the uniqueness of these decompositions.
Obviously, it suffices to show that
\begin{align}
\label{eq:gdfdg}
\Phi_{h_{\alpha(i)}}\cp \wt{\Psi}([\delta|_{L_i}]^{p_i})\cp \Phi_{h^{-1}_{\alpha(i)}}=\Phi_{h'_{\alpha(i)}}\cp \wt{\Psi}([\delta|_{L_i}]^{p'_i})\cp \Phi_{{h'_{\alpha(i)}}^{\hspace{-12pt}-1}}\qquad \forall\: \alpha(i)\neq 0.
\end{align}
Since $h:=h^{-1}_{\alpha(i)}h'_{\alpha(i)} \in G_\delta$ we have $p_i=p_i'$. Then, if we write
$[\delta|_{L_i}]^{p_i}=\delta|_{[a,b]}$ for $a,b\in \RR$, we obtain
\begin{align*}
\big(\Phi_{h}\cp \wt{\Psi}([\delta|_{L_i}]^{p_i}) \cp \Phi_{h^{-1}}\big)(p')&=(\Phi_{h}\cp \homm')\left(\delta|_{[t,b]}\right)(p)\cdot \Psi(b-a) \cdot \diff\big(\homm'\!\left(\delta|_{[t,a]}\right)(p),\Phi_{h^{-1}}(p')\big) \\
&=\homm'\!\left(\wm_h\cp \delta|_{[t,b]}\right)(\Phi_{h}(p))\cdot \Psi(b-a) \cdot \diff\big((\Phi_h\cp\homm')\!\left(\delta|_{[t,a]}\right)(p),p'\big)\\
&=\homm'\!\left( \delta|_{[t,b]}\right)(\Phi_{h}(p))\cdot \Psi(b-a) \cdot \diff\big(\homm'\!\left(\wm_h\cp\delta|_{[t,a]}\right)(\Phi_h(p)),p'\big)\\
&=\homm'\!\left(\delta|_{[t,b]}\right)(p)\cdot \fiba_p(h)\cdot \Psi(b-a) \cdot \diff\big(\homm'\!\left(\delta|_{[t,a]}\right)(p)\cdot \fiba_p(h),p'\big)\\
&=\homm'\!\left(\delta|_{[t,b]}\right)(p)\cdot \fiba_p(h) \cdot \Psi(b-a) \cdot \fiba_p(h)^{-1}\cdot\diff\big(\homm'\!\left(\delta|_{[t,a]}\right)(p),p'\big)\\
&=\homm'\!\left(\delta|_{[t,b]}\right)(p)\cdot \alpha_{\fiba_p(h)}(\Psi(b-a)) \cdot\diff\big( \homm'\!\left(\delta|_{[t,a]}\right)(p),p'\big)\\
&=\homm'\!\left(\delta|_{[t,b]}\right)(p)\cdot \Psi(b-a) \cdot\diff\big(\homm'\!\left(\delta|_{[t,a]}\right)(p),p'\big)\\
&=\wt{\Psi}([\delta|_{L_i}]^{p_i})(p'),
\end{align*}
hence \eqref{eq:gdfdg}. Here, in the second step, we have used \eqref{eq:invprop} for the first factor, as well as
\begin{align*}
q\cdot \diff\big(q,\Phi_{h^{-1}}(p')\big)=\Phi_{h^{-1}}(p')\qquad &\Longrightarrow\qquad \Phi_h(q)\cdot \diff\big(q,\Phi_{h^{-1}}(p')\big)=p'\\
&\Longrightarrow\qquad \diff\big(q,\Phi_{h^{-1}}(p'\big)\big)=\diff\big(\Phi_h(q),p'\big)
\end{align*}
with $q,p'\in F_{\delta(a)}$ for the last one. In the third step, we have used $\wm_h\cp\delta=\delta$ in the first factor and \eqref{eq:invprop} in the last one. The fourth step is clear, and in the fifth one, we have use that
\begin{align*}
q\cdot \fiba_p(h)\cdot \diff\big(q\cdot \fiba_p(h),p'\big)=p'\qquad &\Longrightarrow\qquad \fiba_p(h)\cdot \diff\big(q\cdot \fiba_p(h),p'\big)=\diff(q,p')\\
&\Longrightarrow\qquad \diff\big(q\cdot \fiba_p(h),p'\big) = \fiba_p(h)^{-1}\cdot\diff(q,p').
\end{align*}
Finally, in the seventh step, we have used the invariance property of $\Psi\in \MPD$.
\end{proof}
\end{proposition}
\subsection{Summary}
\label{concl:modify}
In this Section, we basically have discussed the situation where the action $\wm$ induced by the symmetry\footnote{This means a Lie group of automorphisms on the principal fibre bundle $\PMS$ under consideration.} on the base manifold is analytic and pointwise proper. In the first part,
we have modified invariant homomorphisms along Lie algebra generated curves, and in the
second one
we have applied this to the LQG setting in order to show that quantization and reduction do not commute in several situations. In particular, we have shown that this is the case
in \mbox{(semi-)homogeneous}
LQC. In the last part, we have proven that
free curves are discretely generated by the symmetry group, and that we have the decomposition
\begin{align*}
\Paw=\Pags\sqcup \Pacs\sqcup \Pafns\sqcup \Pafs
\end{align*}
of $\Paw$
into subsets, each being closed under inversions and decomposition. Hence, the factorization
\begin{align*}
\AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{CNL}}
\times \AQRFNS\times \AQRInd{\mathrm{FS}}
\end{align*}
by Proposition \ref{rem:euklrem2b}.
Moreover, we have shown that even $\Pacs=\emptyset$, hence
$\Paw = \Pags\sqcup \Paf$ holds if $\wm$ admits only normal stabilizers and is transitive or proper. Recall that $\Paf=\Pafs\sqcup \Pafns$ denotes the set of free curves, whereby $\Pafns$ consists of such free curves whose stabilizer is trivial and $\Pafns$ of those having a non-trivial one. Consequently,
\begin{align*}
\Paw=\Pags\sqcup \Pafns\qquad\Longrightarrow \qquad \AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{FS}}
\end{align*}
holds if $\wm$ is in addition free. This is the case, e.g., in \mbox{(semi-)homogeneous} LQC and also important in view of the next section. There, we will construct normalized Radon measures on each of these two factors, providing us with the respective Radon product measures on $\AQRw$ in this case.
For the measure on $\AQRInd{\mathrm{FS}}$, there we will use the modification result for
free segments which we have proven in Proposition \ref{lemma:freemodify}.
\section{Measures on Quantum-Reduced Configuration Spaces}
\label{sec:MOQRCS}
In the previous section, we have seen that $\Paw=\Pags\sqcup \Pacs\sqcup \Pafns\sqcup \Pafs$ holds if $\wm$ analytic and pointwise proper, whereby each of the occurring sets is closed under decomposition and inversion of its elements, hence\footnote{If one the sets of curves is empty, one just has to remove the respective factor in the product \eqref{eq:dsfs444}.}
\begin{align}
\label{eq:dsfs444}
\AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{CNL}}
\times \AQRFNS\times \AQRInd{\mathrm{FS}}
\end{align}
by Proposition \ref{rem:euklrem2b}.
So, in order to define a normalized Radon measure on $\AQRw$, it suffices to construct respective normalized Radon measures on each of the factors occurring on the right hand side of \eqref{eq:dsfs444}.
In this section, we will use the developments of the previous one in order to provide general constructions for the spaces
\begin{align*}
\AQRInd{\mg^\sim}\cong \IHOMLAS
\qquad\qquad\text{and}\qquad\qquad \AQRInd{\mathrm{FN}}\cong\IHOMFNS.
\end{align*}
Recall that these spaces correspond to the sets of Lie algebra generated and free non-symmetric curves, respectively. So, in particular, our constructions will provide us with a normalized Radon measure on $\AQRw$ whenever $S=\SU$ and $\wm$ acts properly and free. Indeed, then we have $\Paw=\Pags \sqcup\Pafs$, hence $\AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{FS}}$ and all requirements of Subsection \ref{sec:ConSp} are fulfilled.
More precisely, in Subsection \ref{sec:FreeM} we will construct a normalized Radon measure on $\AQRInd{\mathrm{FN}}$ for the general case that $S$ is compact and connected. We will
define a projective structure on $\AQRInd{\mathrm{FN}}$ which generalizes that one, used for the construction of the Ashtekar-Lewandowski measure on $\A_\w$.
Roughly speaking, the projection maps will be defined by \eqref{projm}, whereby in the respective indices now only free segments will occur. Proposition \ref{prop:freeseg}.\ref{prop:freeseg1} then guarantees that $\AQRInd{\mathrm{FN}}$ is separated by these maps, and their surjectivity will be established by Proposition \ref{lemma:freemodify}. Then, as in the Ashtekar-Lewandowki case, the Haar measure on $S$ can be used to define a consistent family of normalized Radon measure, providing us with a normalized Radon measure on $\IHOMFNS$. In specific circumstances, the same construction can also be used to define a normalized Radon measure on $\AQRInd{\mathrm{FS}}$. This will be demonstrated for the case of spherically symmetric LQC where the set $\Pafs$ just consists of all linear curves traversing through the origin. There, the respective projection maps can be chosen in such a way that their images are just products of maximal tori, each carrying a Haar measure by itself. However, there is no ad hoc reason why there should be such a convenient choice in the general case. So, in specific situations one has to investigate the sets $\Pafs$ and $\AQRInd{\mathrm{FS}}$ explicitly if one wants to use the developed techniques in order to construct a measures on $\AQRInd{\mathrm{FS}}$.
Now, the mentioned restriction to the structure group comes from our constructions for the space $\AQRInd{\mg^\sim}$ (Subsection \ref{sec:ConSp}). These we will exemplarily do for the most LQG-relevant case where $S=\SU$. Analogous constructions appear to be possible also for other compact and connected structure groups, in any case for the abelian ones (tori).
Indeed, for our considerations we will assume that each $\wm$-orbit $\m=[x_\m]\subseteq M$ with $\mg\backslash \mg_{x_\m}\neq \emptyset$ (we will denote the set of all such orbits by $\Mm$) admits an independent and complete family $\{\g_\alpha\}_{\alpha\in I_\m}\subseteq \mg\backslash \mg_{x_\m}$ as in Definition \ref{def:stable}\ref{eq:iindepg3}.\footnote{Due to Lemma and Remark \ref{rem:dfggfg}.\ref{rem:dfggfg2}
this is always the case, e.g., if $\wm$ is free.} Then, as we will show in the first part of Subsection \ref{sec:ConSp}, $\AQRInd{\mg^\sim}$ is homeomorphic to the Tychonoff product
\begin{align*}
Y=\prod_{\m\in \Mm,\alpha\in I_\m} Y_{\m,\alpha}
\end{align*}
for $Y_{\m,\alpha}$ the set of $\Ad_{G_{[\g_\alpha]}}^{p_\m}$-equivariant mappings $\Psi\colon \spann_\RR(\g_\alpha)\rightarrow S$ (cf.\ Definition \ref{def:eqmaps}) equipped with a suitable topology. Here,
$p_\m\in F_{x_\m}$ is a fixed choice for each $\m\in \Mm$.
The above homeomorphism is just a straightforward consequence of Proposition \ref{th:invhomm}.\ref{th:invhomm1} where, except for compactness and connectedness, no further requirements on the structure group $S$ have to be done.
Then, it is the non-trivial part to calculate these factors explicitly, to define reasonable measures thereon, and, finally, to show that the measure on $\AQRInd{\mg^\sim}\cong Y$ induced by the respective Radon product measure on $Y$ does not depend on any choices one has done. Then, following the arguments of Subsection \ref{sec:ConSp}, one immediately sees that for $S$ the n-torus, each of the above factors is either homeomorphic to the $n$-fold product of $\RB$ by
\begin{align*}
[\hspace{1pt}\RB\hspace{1pt}]^n&\rightarrow Y_{\m,\alpha}\\
(\psi_1,\dots,\psi_n)&\mapsto \big[\lambda\cdot \g_{\m,\alpha} \mapsto \big(\psi_1(\chi_\lambda),\dots,\psi_n(\chi_\lambda)\big)\big],
\end{align*}
or consists of the trivial map $\Psi\colon \lambda\cdot \g_\alpha \mapsto e$. This is just because by Remark \ref{rem:ppropercurve}.\ref{rem:ppropercurve4} and commutativity of $S$, equivariance of $\Psi\in Y_{\m,\alpha}$ is either a trivial condition or reads $\Psi(\g)=\psi(-\g)$, hence $\Psi(\g)=e$ for all $ \g\in \spann_\RR(\g_{\m,\alpha})$.
Then, in both cases we have a canonical Radon measure on $Y_{\m,\alpha}$, providing us with a respective Radon product measure on $Y$.
However, for $\SU$ (and any other non-commutative structure group) equivariance gives more complicated conditions which have to be investigated carefully. This is the content of the second part of this section.
In both subsections, we will discuss the \mbox{(semi-)homogeneous}, homogeneous isotropic as well as the spherically symmetric LQC case.
\subsection{The Reduced Ashtekar-Lewandowski Measure}
\label{sec:FreeM}
As already mentioned above, we now are going to equip $\AQRInd{\mathrm{FN}}\cong\IHOMFNS$ with a projective structure, which we then use to construct a normalized Radon measure thereon. This will be done in analogy to the construction of the Ashtekar-Lewandowski measure \cite{ProjTechAL} on $\A_\w\cong \HOMW$ which we even get back
if $G=\{e\}$ holds. This is just because due to our definitions then $\Pafns=\Paw$, hence $\IHOMFNS=\HOMW$ holds.
In order to avoid trivialities, we will assume that $\IHOMFNS$ is not empty. Moreover, we let
$\nu=\{\nu_x\}_{x\in M}\subseteq P$ be a fixed family with $x\in F_x$ for all $x\in M$ and $\psi_x(p):=\diff(\nu_x,p)$ for all $p\in F_x$ as in Convention \ref{hgamma}. Finally, we will assume that $S$ is compact and connected with $\dim[S]\geq 1$.
We start our investigations with the definition of the directed set:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\itspace
Let $\grF$ denote the set of all finite tuples $(\gamma_1,\dots,\gamma_k)$ with $\gamma_1,\dots,\gamma_k \in \Pafns$ free segments such that we have
\begin{align*}
\wm_g\cp \gamma_i \nsim_\cp \gamma_j\qquad \forall\: g\in G,\: 1\leq i\neq j\leq k.
\end{align*}
\item
\itspace
For $(\gamma_1,\dots,\gamma_k), (\gamma'_1,\dots,\gamma'_p)\in \grF$ write $(\gamma_1,\dots,\gamma_k)\leq (\gamma'_1,\dots,\gamma'_p)$
iff each $\gamma_i$ admits a decomposition $\{(\gamma_i)_j\}_{1\leq j\leq n_i}$ such that each restriction $(\gamma_j)_i$ is equivalent to one of the curves $\wm_g\cp \gamma'_j$ or its inverses for some $g\in G$ and $1\leq j\leq p$.
\end{itemize}
\endgroup
\begin{lemma}[Directed Set]
\label{lemma:projdefx}
The pair $(\grF,\leq)$ is a directed set.
\end{lemma}
\begin{proof}
Let $(\gamma_1,\dots,\gamma_k),(\gamma'_1,\dots,\gamma'_p)\in \grF$.
We inductively construct an upper bound in $\grF$ as follows.
For each $1\leq j\leq p$ let $a_j=k_0^j<\dots<k^j_{n_j}=b_j$ denote the decomposition from Lemma \ref{lemma:curvestablemma}.\ref{lemma:curvestablemma3} of
$\operatorname{\mathrm{dom}}[\gamma'_j]= [a_j,b_j]$ w.r.t.\ $\gamma_1\colon [l_1,l_2]\rightarrow M$
into intervals $K_i^j=\big[k^j_i,k^j_{i+1}\big]$.
We first remove all segments $\gamma'_j|_{K_i^j}$ that are equivalent to a translate of $\gamma_1$ or its inverse.
\vspace{4pt}
\noindent
More precisely, let $\gamma'_{j,1},\dots, \gamma'_{j,m_j}$
denote the segments $\gamma'_j|_{K_i^j}$ of $\gamma_j'$ with
\begin{align*}
\gamma'_j|_{K_i^j}\nsim_\cp \wm_g \cp \gamma_1\qquad\forall\: g\in G
\end{align*}
and replace $(\gamma'_1,{\dots},\gamma'_p)$ by $(\gamma_{1,1}',\dots,\gamma'_{1,m_1},\dots,\gamma_{k,1}',\dots,\gamma'_{k,m_k})$.
Here, it may happen that $\gamma'_j|_{K_0}\csim \wm_g\cp [\gamma_1|_{L'}]^{\pm 1}$ or $\gamma'_j|_{K_{n-1}}\csim \wm_g\cp [\gamma_1|_{L'}]^{\pm 1}$ holds with $L'=[l_1,l]$ for $l<l_2$ or $L'=[l,l_2]$ for $l_1<l$. In this case, we split $\gamma_1$ at these (at most $2p$) points and obtain curves $\gamma_{1,1},\dots,\gamma_{1,n_1}$.
We now apply the same procedure to $\gamma_2$ and $(\gamma_{1,1}',\dots,\gamma'_{1,m_1},\dots,\gamma_{k,1}',\dots,\gamma'_{k,m_k})$, and inductively end up with finitely many free segments $\gamma_{1,1},\dots,\gamma_{1,q_1},\dots, \gamma_{1,p},\dots,\gamma_{1,q_p}$ and $\gamma''_1,\dots,\gamma''_m$ that fulfil
\begin{align*}
\wm_g\cp \gamma_{r,s}\nsim_\cp \gamma_{r',s'}&\:\text{ for }\:(r,s)\neq (r',s') \:\text{ because }\:(\gamma_1,\dots,\gamma_k)\in \grF \qquad\text{as well as}\\
\wm_g\cp \gamma''_{q}\nsim_\cp \gamma''_{q'}&\:\text{ for }\:q\neq q' \hspace{41.5pt}\text{ because }\:(\gamma'_1,\dots,\gamma'_p)\in \grF
\end{align*}
for all $g\in G\backslash\{e\}$. For this, observe that each of these curves
is a subcurve of one of the free segments $\gamma_1,\dots,\gamma_k, \gamma'_1,\dots,\gamma'_p$.
Moreover, by construction we have
\begin{align*}
\gamma_{r,s}\nsim_\cp \wm_g\cp \gamma''_{q}\qquad\forall\: g\in G\backslash\{e\},\: 1\leq r\leq k,\:1\leq s\leq q_r,\:1\leq q\leq m
\end{align*}
as well as
\begin{align*}
(\gamma_1,\dots,\gamma_k),(\gamma'_1,\dots,\gamma'_p)\leq (\gamma_{1,1},\dots,\gamma_{1,q_1},\dots, \gamma_{1,l},\dots,\gamma_{1,q_p},\gamma''_1,\dots,\gamma''_p).
\end{align*}
\end{proof}
We now are able to define the projection and transition Maps:
\begin{lemdef}[Projection and Transition Maps]
\label{def:ProjLimALfreex}
\begin{enumerate}
\item
\label{hompropers}
For $\gamma\in \Pafns$ with $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ let
\begin{align}
\label{eq:projmapdef}
\pi_\gamma(\homm):=\psi_{\gamma(b)}\big(\homm(\gamma)(\nu_{\gamma(a)})\big)\qquad\forall\: \homm\in \HOMFNS.
\end{align}
Then, it is immediate to see that for each $\homm\in \HOMFNS$ we have
\begin{align*}
\pi_{\gamma^{-1}}(\homm)=\pi_{\gamma}(\homm)^{-1}\qquad\text{as well as}\qquad \pi_\gamma(\homm)=\pi_{\gamma|_{[a,t]}}(\homm) \cdot \pi_{\gamma|_{[t,b]}}(\homm)\quad \forall\:t\in (a,b).
\end{align*}
Moreover, for each $g\in G$ and $\homm\in \IHOMFNS$ we have (cf. \eqref{eq:invpi})
\begin{align}
\label{eq:invpii}
\pi_{\wm_g\cp \gamma}(\homm)&=
\underbrace{\psi_{\wm_g(\gamma(b))}(\Phi_g(\nu_{\gamma(b)}))}_{c}\: \cdot\: \psi_{\nu_{\gamma(b)}}(\homm(\gamma)(\nu_{\gamma(a)}))\cdot \underbrace{\psi_{\gamma(a)}\big(\Phi_{g^{-1}}(\nu_{\wm_g(\gamma(a))})\big)}_{d}
.
\end{align}
\item
For $\alpha=(\gamma_1,\dots,\gamma_k)\in \grF$ with $\operatorname{\mathrm{dom}}[\gamma_i]=[a_i,b_i]$ for $i=1,\dots,k$ we define the map $\pi_{\alpha}\colon \IHOMFNS\rightarrow X_\alpha:=S^{|\alpha|}$ by
\begin{align}
\label{eq:dfdssssttt}
\pi_\alpha(\homm):= \big(\psi_{\gamma_1(b_1)}\big(\homm(\gamma_1)(\nu_{\gamma_1(a_1)})\big),\dots,\psi_{\gamma_k(b_k)}\big(\homm(\gamma_k)(\nu_{\gamma_k(a_k)})\big)\big).
\end{align}
This map is surjective by (the next) Proposition \ref{lemma:consFamx}.\ref{lemma:consFamx1}.
\item
Let $\alpha=(\gamma_1,\dots,\gamma_k)\leq (\gamma'_1,\dots,\gamma'_{k'})=\alpha'$ and $\{(\gamma_i)_j\}_{1\leq j\leq n_i}$ be the corresponding decomposition of $\gamma_i$ for $1\leq i\leq k$.
\begingroup
\setlength{\leftmarginii}{12pt}
\begin{itemize}
\item
We find $p_{ij}\in\{1,-1\}$, $1\leq m_{ij} \leq k'$ and $g_{ij}\in G$
with
\begin{align}
\label{eq:fdfdfdf}
(\gamma_i)_j^{p_{ij}}\csim \wm_{g_{ij}}\cp \gamma'_{m_{ij}}.
\end{align}
Here, for all $1\leq i\leq k$ we have
$m_{ij}\neq m_{ij'}$ for $1\leq j\neq j'\leq n_i$ just by injectivity of $\gamma_i$. In addition to that, we denote by $c_{ij}$ and $d_{ij}$ the structure group elements $c$ and $d$ from \eqref{eq:invpii} that correspond to the curve $\wm_{g_{ij}}\cp \gamma'_{m_{ij}}$.
Observe that the above quantities are even uniquely determined because if \eqref{eq:fdfdfdf} in addition holds for $p'_{ij}\in\{1,-1\}$, $1\leq m'_{ij} \leq k'$ and $g'_{ij}\in G$, then
\begin{align*}
\gamma'_{m_{ij}}\csim \wm_{g_{ij}^{-1}g'_{ij}}\cp \Big[\gamma'_{m'_{ij}}\Big]^{p_{ij}\cdot p'_{ij}} \qquad&\Longrightarrow\qquad \gamma'_{m_{ij}}\cpsim \wm_{g_{ij}^{-1}g'_{ij}}\cp \gamma'_{m'_{ij}}\\[-2pt]
&\Longrightarrow\qquad m_{ij}=m'_{ij}\\
&\Longrightarrow\qquad g_{ij}^{-1}g'_{ij}=e\\
&\Longrightarrow\qquad p_{ij}=p_{ij}',
\end{align*}
where the second step is clear from $(\gamma_1',\dots,\gamma_k')\in \grF$ and the third one from freeness of $\gamma'_{m_{ij}}$ and $G_{\gamma'_{j}}=e$.
\item
We obtain a well-defined and continuous map $\pi^{\alpha'}_\alpha\colon X_{\alpha'}\rightarrow X_\alpha$ if we define
\begin{align}
\label{eq:alphaalphsastrichx}
\pi^{\alpha'}_\alpha\!\left(x_1,\dots,x_{k'}\right):=\Bigg(\prod_{j=1}^{n_1}\left(c_{1j}\cdot x_{m_{1j}}\cdot d_{1j}\right)^{p_{1j}},\dots,\prod_{j=1}^{n_k}\left(c_{kj}\cdot x_{m_{kj}}\cdot d_{kj}\right)^{p_{kj}}\Bigg).
\end{align}
In fact, $\pi^{\alpha'}_\alpha\cp \pi_{\alpha'}=\pi_\alpha$ is immediate from Part \ref{hompropers}), and due to surjectivity of $\pi_{\alpha'}$ (proven in Proposition \ref{lemma:consFamx}.\ref{lemma:consFamx1}))
this definition then cannot depend on any choices.
\end{itemize}
\endgroup
\item
Let $\mu_k$ denote the Haar measure on $S^{k}$ for $k\in \mathbb{N}_{\geq 1}$ and define $\mu_\alpha:=\mu_{|\alpha|}$ for $\alpha\in \grF$. Observe that $\mu_k$ equals the $k$-fold product $\mu^k$ of the Haar measure $\muH$ on $S$ (cf.\ Definition \ref{def:ProductMa}.\ref{def:ProductMa2}) as this normalized Radon measure is translation invariant. In fact, using a Riesz-Markov argument, translation invariance is straightforward from
Fubini's formula.
\end{enumerate}
\end{lemdef}
The next proposition shows that $\IHOMFNS$ is indeed a projective limit of $\{X_\alpha\}_{\alpha\in \grF}$ w.r.t.\ the above projection and transition maps. Moreover, it shows that $\{\mu_{\alpha}\}_{\alpha\in \grF}$ is a respective family of normalized Radon measures defining a normalized Radon measure on $\IHOMFNS$.
\begin{proposition}
\label{lemma:consFamx}
\begin{enumerate}
\item
\label{lemma:consFamx1}
The maps $\pi_\alpha\colon \IHOMFNS\rightarrow S^{|\alpha|}$ are surjective, and together they separate the elements of $\IHOMFNS$.
\item
\label{lemma:consFamx2}
$\IHOMFNS$ is a projective limit of $\{X_\alpha\}_{\alpha\in \grF}$ w.r.t.\ the maps from Lemma and Definition \ref{def:ProjLimALfreex}.
\item
\label{lemma:consFamx3}
$\{\mu_{\alpha}\}_{\alpha\in \grF}$ is a consistent family of normalized Radon measures w.r.t.\ $\{X_\alpha\}_{\alpha\in \grF}$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
For surjectivity, let $\homm_0\in \IHOMFNS\neq \emptyset$, $\alpha=(\gamma_1,\dots,\gamma_k)\in\grF$ with $\operatorname{\mathrm{dom}}[\gamma_i]=[a_i,b_i]$ and $s'_i:=\psi_{\gamma_i(b_i)}\big(\homm_0(\gamma_i)(\nu_{\gamma_i(a_i)})\big)$ for $1\leq i\leq k$. Since $S$ is compact and connected, the exponential map of $S$ is surjective. Hence, for $(s_1,\dots,s_k)\in S^k$ we find $\s_1,\dots,\s_k\in \ms$ such that $s_i=s_i'\cdot\exp([b_i-a_i]\cdot \s_i)$ holds for all $1\leq i\leq k$.
Now, in the situation of Proposition \ref{lemma:freemodify} let $\delta:=\gamma_1$, $\Psi\colon r\mapsto \exp(r \cdot \s_1)$, $t:=a_1$ and $p:=\nu_{\gamma_1(a_1)}$. Moreover, denote by $\homm_1$ the respective modified homomorphism. Then,
\begin{align*}
\pi_{\gamma_1}(\homm_1)&=\psi_{\gamma_1(b_1)}\left(\homm_1(\gamma_1)(\nu_{\gamma_1(a_1)})\right)\\
&=\psi_{\gamma_1(b_1)}\left(\homm_0(\gamma_1)(\nu_{\gamma_1(a_1)}) \cdot\exp([b_1-a_1]\hspace{1pt} \s_1\right)\\
&=s'_1 \cdot\exp([b_1-a_1]\hspace{1pt} \s_1) =s_1,
\end{align*}
and applying the same argument inductively, we obtain $\homm_k\in \IHOMFNS$ with $\pi_{\gamma_i}(\homm_k)=s_i$ for $1\leq i\leq k$, hence $\pi_\alpha(\homm_k)=(\pi_{\gamma_1}(\homm_k),\dots, \pi_{\gamma_k}(\homm_k))=(s_1,\dots,s_k)$. For this observe that modifying $\homm_i$ along $\gamma_{i+1}$ does not change its values on the curves $\gamma_j$ for $1\leq j\leq i$. This is clear because $H_{[\gamma_i,\gamma_j]}=\emptyset$ holds for $1\leq i\neq j\leq k$, so that surjectivity of $\pi_\alpha$ follows.
Now, for the separation property let $\homm,\homm'\in \IHOMFNS$ be different, i.e., $\homm(\gamma)\neq \homm'(\gamma)$ for some $\gamma\in \Pafns$. By Proposition \ref{prop:freeseg}.\ref{prop:freeseg1}, we find a maximal free segment $\delta=\gamma|_{K}$, a decomposition $t_0<\dots< t_n$ of $\operatorname{\mathrm{dom}}[\gamma]=[t_0,t_n]$ and elements $g_0,\dots,g_{n-1} \in G$, such that
\begin{align*}
\gamma|_{[t_i,t_{i+1}]}\csim \wm_{g_i}\cp \delta^{\pm 1}\qquad\forall\: 1\leq i\leq n-2
\end{align*}
as well as
\begin{align*}
\gamma|_{[t_0,t_{1}]}\csim \wm_{g_0}\cp [\delta|_{L}]^{\pm 1}\qquad \text{and}\qquad \gamma|_{[t_{n-1},t_n]}\csim \wm_{g_{n-1}}\cp [\delta|_{L'}]^{\pm 1}
\end{align*}
holds for $L,L'$ both of the form $[l,l_2]$ with $l_1\leq l<l_2$ or $[l_1,l]$ with $l_1<l\leq l_2$. Then, splitting $\delta$ at the respective (at most two) points $\delta(l)$, we obtain a decomposition of $\delta$ into at most three subcurves which define an index $\alpha\in \grF$ for which $\pi_\alpha(\homm)\neq \pi_\alpha(\homm')$ holds.
\item
It remains to show continuity of the maps $\pi_\alpha$. Here, it suffices to consider the case where the index $\alpha$ consists of one single free segment $\gamma$. Since the topology on $\IHOMFNS$ is the subspace topology inherited from $\HOMFNS$,\footnote{See Definition \ref{def:invhomm}.} $\pi_\gamma$ is continuous iff
\begin{align*}
\pi_\gamma\cp \kappa_{\mathrm{FN}}|_{\AQRFNS}
\end{align*}
is continuous w.r.t.\ to the subspace topology on $\AQRFNS$ inherited from $\A_{\mathrm{FN}}=\mathrm{Spec}(\PaC_{\mathrm{FN}})$. This, however, is the case if $\pi_\gamma\cp \kappa_{\mathrm{FN}}$ is continuous w.r.t.\ the topology on $\A_{\mathrm{FN}}$.
Here, $\kappa_{\mathrm{FN}}\colon \A_{\mathrm{FN}}\rightarrow \HOMFNS$ denotes the respective homeomorphism from Definition \ref{def:indepref}.\ref{def:mapkappa}. However, $\pi_\gamma\cp \kappa_{\mathrm{FN}}\colon \A_{\mathrm{FN}}\rightarrow S$ is continuous by \eqref{eq:algpro}, just because
$\left[h_\gamma\right]_{ij}\in \PaC_{\mathrm{FN}}$.
\item
We have to show that $\pi^{{\alpha'}}_{\alpha}(\mu_{{\alpha'}})=\mu_{\alpha}$ holds if $\alpha\leq {\alpha'}$, and by the Riesz-Markov theorem
it suffices to verify that
\begin{align*}
\int_{X_\alpha}f \:\dd\mu_\alpha=\int_{X_\alpha}f \:\dd\pi^{{\alpha'}}_{\alpha}(\mu_{{\alpha'}})\qquad\forall\: f\in C(X_\alpha).
\end{align*}
Now, if $g\in C(X_{\alpha'})$, $x=(x_1,\dots,x_{k'})\in S^{k'}$ and $1\leq i\leq k'$, Fubini's formula reads
\begin{align}
\label{eq:Fubinix}
\int_{X_{\alpha'}}g \:\dd\mu_{\alpha'}= \int_{S^{k'-1}} \left(\int_S g(x) \:\dd\mu_1(x_i)\right)\dd\mu_{k'-1}(\raisebox{-0.2ex}{$x^i$}),
\end{align}
where $S^{k'-1}\ni x^i:=(x_1,\dots,x_{i-1},x_{i+1},\dots,x_{k'})$. Hence, for $f\in C(X_\alpha)$ we have
\begin{align*}
\begin{split}
\int_{X_\alpha}f \:\dd\pi^{{\alpha'}}_{\alpha}(\mu_{{\alpha'}})&=\int_{X_{\alpha'}}\Big(f\cp \pi^{{\alpha'}}_{\alpha}\Big) \:\dd\mu_{{\alpha'}}
=\int_{S^{k'-1}}\left(\int_S\Big(f\cp \pi^{{\alpha'}}_{\alpha}\Big)(x) \:\dd\mu_1(x_i)\right)\dd\mu_{k'-1}(\raisebox{-0.2ex}{$x^i$}).
\end{split}
\end{align*}
By the definition of $\grF$, each of the variables $x_1,\dots,x_{k'}$ occurs in exactly one of the products on the right hand side of \eqref{eq:alphaalphsastrichx}. Consequently, the components $\big[\raisebox{-0.2ex}{$\pi^{{\alpha'}}_{\alpha}$}\big]_i$ of $\pi^{\alpha'}_\alpha$ mutually depend on different variables $\ovl{x}_i=\big(x_{m_{i,1}},\dots,x_{m_{i,n(i)}}\big)$.
Then, by left-, right- and inversion invariance of $\mu_1$, for $x\in S^{k'}$ we have
\begin{align*}
\int_S\Big(f\cp \pi^{{\alpha'}}_{\alpha}\Big)(x) \:\dd\mu_1(x_{m_{k,1}})
& =\int_S f\Big(\big[\raisebox{-0.2ex}{$\pi^{{\alpha'}}_{\alpha}$}\big]_1(\raisebox{-0.1ex}{$\ovl{x}_1$}),\dots,\big[\raisebox{-0.2ex}{$\pi^{{\alpha'}}_{\alpha}$}\big]_{k-1}(\raisebox{-0.1ex}{$\ovl{x}_{k-1}$}),\big[\raisebox{-0.2ex}{$\pi^{{\alpha'}}_{\alpha}$}\big]_{k}(\raisebox{-0.1ex}{$\ovl{x}_{k}$})\Big) \:\dd\mu_1(x_{m_{k,1}})
\\
& =\int_S f\Big(\big[\raisebox{-0.2ex}{$\pi^{{\alpha'}}_{\alpha}$}\big]_1(\raisebox{-0.1ex}{$\ovl{x}_1$}),\dots,\big[\raisebox{-0.2ex}{$\pi^{{\alpha'}}_{\alpha}$}\big]_{k-1}(\raisebox{-0.1ex}{$\ovl{x}_{k-1}$}),x_{m_{k,1}}\Big) \:\dd\mu_1(x_{m_{k,1}}).
\end{align*}
Then, applying the same argument inductively
we obtain
\begin{equation*}
\int_{X_\alpha}f \:\dd\pi^{{\alpha'}}_{\alpha}(\mu_{{\alpha'}}) =\int_{S^{k'}} f(x_{m_{1,1}},\dots,x_{m_{k,1}})\:\dd\mu_{\alpha'}(x)=\int_{X_\alpha} f\:\dd\mu_{\alpha},
\end{equation*}
where the last step follows inductively from \eqref{eq:Fubinix} and $\muH(S)=1$.
\end{enumerate}
\end{proof}
\end{proposition}
\begin{lemrem}[Independence from the Choices]
By Proposition \ref{lemma:consFamx}.\ref{lemma:consFamx2}, there is a unique normalized Radon measure $\mu$ on $\IHOMFNS$ which corresponds to the family $\{\mu_\alpha\}_{\alpha \in \grF}$ for $\mu_\alpha$ the Haar measure on $S^{|\alpha|}$.
Now, we obtain different projection maps $\pi'_\alpha$ if we choose another family $\nu'=\{\nu'_x\}_{x\in M}\subseteq P$ with $\nu'_x\in F_x$ for all $x\in M$. This family, however, gives rise to the same measure on $\IHOMFNS$ because:
It follows from \eqref{eq:indep} that for each $\alpha\in \grF$ we find $d_1,b_1,\dots,d_k,b_k \in S$ such that
$\pi_\alpha(\homm)=(s_1,\dots,s_k)$ implies $\pi'_\alpha(\homm)=(d_1\cdot s_1 \cdot b_1,\dots,d_k\cdot s_k \cdot b_k)$ for all $(s_1,\dots,s_k)\in X_\alpha$.
Consequently, $\pi_\alpha'=\mathrm{d}\cdot \pi_\alpha \cdot \mathrm{b}$ holds for $\mathrm{d}:=(d_1,\dots,d_k)$ and $\mathrm{b}:=(b_1,\dots,b_k)$, so that for $B\in \mathfrak{B}(X_\alpha)$ we have
\begin{align*}
\homm\in \pi_\alpha'^{-1}(B)\quad\Longleftrightarrow \quad\pi'_\alpha(\homm)\in B
\quad &\Longleftrightarrow \quad\pi_\alpha(\homm)\in \mathrm{d}^{-1}\cdot B\cdot\mathrm{b}^{-1}\\
&\Longleftrightarrow \quad\homm\in \pi_\alpha^{-1}(\mathrm{d}^{-1}\cdot B\cdot\mathrm{b}^{-1}).
\end{align*}
Hence,
\begin{align*}
\pi_\alpha'(\mu)(B)=\mu\left(\pi_\alpha'^{-1}(B)\right)=\mu\left(\pi_\alpha^{-1}\left(\mathrm{d}^{-1}\cdot B\cdot\mathrm{b}^{-1}\right)\right)=\mu_\alpha\left(\mathrm{d}^{-1}\cdot B\cdot\mathrm{b}^{-1}\right)=\mu_\alpha(B)
\end{align*}
by invariance of $\mu_\alpha$.\hspace*{\fill}$\lozenge$
\end{lemrem}
\begin{definition}[The Reduced Ashtekar-Lewandowski Measure]
\label{def:AshLewx}
The normalized Radon measure on $\IHOMFNS$ that corresponds to the consistent family of normalized Radon measures $\{\mu_\alpha\}_{\alpha\in \grF}$ is denoted by \gls{MFNS} in the following. For the special case that $G=\{e\}$, this measure is called Ashtekar-Lewandowski measure on $\HOMW$ and is denoted by \gls{mAL}.
\end{definition}
\begin{remark}[Loop Quantum Cosmology]
\label{ex:Fullmeas}
In the next subsection, we are going to define a normalized Radon measure $\mLAS$ on $\AQRLA\cong \IHOMLAS$, in particular, for the case of (semi-)homogeneous, spherically symmetric and homogeneous isotropic LQC. As already mentioned in Remark \ref{rem:euklrem}.\ref{it:sdsdds}, in the two transitive cases (homogeneous isotropic and homogeneous LQC) the space $\AQRLA$ is already a reasonable quantum-reduced configuration space by itself. So, there we have the reasonable kinematical Hilbert space $\Lzw{\AQRLA}{\mLAS}$.
\begin{enumerate}
\item
\label{ex:Fullmeas1}
Anyhow, in (semi-)homogeneous LQC we even have the normalized Radon measure $\mu_\red:=\mLAS \times \mFNS$ on $\AQRw\cong \IHOMW$ because $\Paw=\Pags\sqcup \Pafns$, hence
\begin{align*}
\AQRw\cong \AQRInd{\mg}\times \AQRFNS
\end{align*}
holds by Example \ref{eq:invelements}.
\item
\label{ex:Fullmeas2}
In the homogeneous isotropic and spherically symmetric case we unfortunately do not know whether $\Pacs=\emptyset$ holds. In the homogeneous isotropic case, we elsewise could define $\mu_\red=\mLAS \times \mFNS$ as well, just because there $\Pafns=\emptyset$ holds by Example \ref{eq:invelements}.
In the spherically symmetric case, we additionally had to define a measure $\mu_{\mathrm{FS}}$ on
\begin{align*}
\AQRFS \cong \Hom_\red(\Pafs,\IsoF)=\Hom_\red(\Paln,\IsoF)
\end{align*}
for $\Paln$ the set of linear curves in $\RR^3$ traversing through the origin.\footnote{See (a) in Convention \ref{conv:sutwo1}.\ref{conv:sutwo2} for the definition of $\Paln$ as well as Example \ref{eq:invelements} for the equality $\Pafs=\Paln$.}
This, however, is easy as we can use the same techniques as for the construction of $\mFNS$:
\begingroup
\setlength{\leftmarginii}{13pt}
\begin{itemize}
\item
Instead of $\grF$, we consider the set $\grl$ of tuples $(\gamma_1,\dots,\gamma_k)$ with $\gamma_i\in \Paln$ and $\mathrm{im}[\gamma_i]\subseteq \Span_\RR(\vec{e}_1)$ for $1\leq i\leq k$. Hence, we restrict to such elements of $\Paln$ which completely traverse in the $\vec{e}_1$-axis. Obviously, each other curve in $\Pafs=\Paln$ can be obtained by rotating such a curve around a suitable axis.
\item
We choose $\nu_x=(x,\me)$, i.e., $\psi_x=\pr_2\colon P\ni(x,s)\mapsto s\in \SU$ for all $x\in \RR^3$, and define the projection and transition maps exactly as in Lemma and Definition \ref{def:ProjLimALfreex}. Then,
\begin{align}
\label{eq:dssdds}
\pi_\gamma(\homm)\stackrel{\eqref{eq:projmapdef}}{=}\psi_{\gamma(b)}\big(\homm(\gamma)(\nu_{\gamma(a)})\big)\stackrel{\eqref{eq:ON}}{=}\Omega_\nu(\homm)(\gamma)
\stackrel{\eqref{eq:tricBHoms}}{=} \Omega(\homm)(\gamma)=\hommm(\gamma)
\end{align}
for $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$ and $\hommm:=\Omega(\homm)$. Since for $\sigma\in H_{\vec{e}_1}=H_{\tau_1}$ we have $\gamma=\sigma(\gamma)$, this gives
\begin{align*}
\pi_\gamma(\homm)=\pi_{\sigma(\gamma)}(\homm)\stackrel{\eqref{eq:dssdds}}{=}\hommm(\sigma(\gamma))\stackrel{\eqref{eq:algrels}}{=}\alpha_\sigma(\hommm(\gamma))=\alpha_\sigma(\pi_\gamma(\homm)).
\end{align*}
Then, Lemma \ref{lemma:torus}.\ref{lemma:torus1} shows $\mathrm{im}[\pi_\gamma]\subseteq H_{\tau_1}\cong S^1$,
and by Proposition \ref{lemma:freemodify}, we even have $\mathrm{im}[\pi_\alpha]\cong S^{|\alpha|}$. This follows by the same arguments as in the proof of Proposition \ref{lemma:consFamx}.\ref{lemma:consFamx1} because $\Psi_\mu\colon r\mapsto \exp(r\cdot [\mu\cdot\tau_1])$ for $\mu\in\RR$ is contained in $\Mult_{p,\gamma}$ for $p=(\gamma(a),\me)$. In fact, this is clear from $G_\gamma=H_{\tau_1}$, and that $\phi_p(\sigma)=\sigma$ holds for all $\sigma\in G_\gamma$ as for $p=(x,\me)$ we have (see Example \ref{ex:LQC} for the definiton of $\Pii\colon \SU \times P\rightarrow P$)
\begin{align*}
\Pii(\sigma,p)=(x,\sigma)\qquad\Longrightarrow\qquad \Pii(\sigma,p)= p\cdot \sigma \qquad\Longrightarrow\qquad \phi_p(\sigma)=\sigma.
\end{align*}
Then, using the Haar measure on $H_{\tau_1}\cong S^1$ instead of $\SU$, by the same arguments as in the $\Pafns$ case, we obtain a normalized Radon measure $\mu_{\mathrm{FS}}$ on $\Hom_\red(\Paln,\IsoF)$.
\end{itemize}
\endgroup
\end{enumerate}
\end{remark}
\subsection{Lie Algebra Generated Configuration Spaces}
\label{sec:ConSp}
In the previous part, we have constructed a normalized Radon measure on $\IHOMFNS$, whereby we have used the developments of Subsection \ref{sec:ModifreeSeg}.
In this part, we are going to write the space $\IHOMLAS$ as a Tychonoff product of compact Hausdorff spaces which we determine exemplarily for the most LQC relevant case that $S=\SU$. Then, we define normalized Radon measures on each of these spaces, providing us with a normalized Radon measures on $\IHOMLAS$.
For the first step, we will need that each $\wm$-orbit which contains some Lie algebra generated curve admits an independent and complete family of stable Lie algebra elements. We show that, under these circumstances, $\IHOMFNS$ can be written as a Tychonoff product of compact Hausdorff spaces which are just given
by the sets of equivariant maps that correspond to the elements of the above independent and complete families. Here, we will only need that the structure group is compact and connected.
In the second part, we will determine these sets of equivariant mappings explicitly for the case that $S=\SU$. We define normalized Radon measures on each of them and show that the corresponding Radon product measure gives rise to a normalized Radon measure on $\IHOMFNS$ which does not depend on any choices we have made.\footnote{For instance, the above families of stable elements.} As already sketched in the beginning of this section, these constructions also work for the abelian case and are even easier there. For non-abelian $S$ one basically has to investigate to which maximal tori equi\-variance can restrict the image of a map $\Psi\colon \spann_\RR(\g)\rightarrow S$ having the homomorphism properties \eqref{eq:equii}. Indeed, for $S$ compact and connected, multiplicativity already restricts such an image to a unique maximal torus if $\Psi(\lambda\cdot \g)$ is regular for some $\lambda\in \RR$,\footnote{This is because $\Psi(\mu\cdot \g)\cdot \Psi(\lambda\cdot \g)=\Psi([\mu+\lambda]\cdot \g)=\Psi(\lambda\cdot \g)\cdot \Psi(\mu\cdot \g)$ for all $\mu\in \RR$.} and obviously this issue is trivial in the abelian case.
Finally, the set of occurring tori has to be equipped with a normalized Radon measure having certain invariance properties which make the whole construction independent of any choices at the end.
Using general theory of compact and connected Lie groups, then one might obtain some generalizations of the construction we will work out in detail for $\SU$.
So, in the next part, the structure group will be compact and connected, as well as $\SU$ in the subsequent ones.\footnote{Anyhow, since it simplifies the notations, we often will write $\ms$ instead of $\su$.} Moreover, as in the previous section, $\wm$ is always assumed to be analytic and pointwise proper.
By $\Mm$ we will denote the set of $\varphi$-orbits $\m$ in $M$ with $\mg\backslash \mg_x\neq \emptyset$ for one (and then each) $x\in \m$. We fix a family $\{p_\m\}_{\m\in \Mm}\subseteq P$ of elements with $x_\m:=\pi(p_\m)\in \m\in \Mm$ for all $\m\in \Mm$, where for each such $\m$ we let $\{\g_{\m,\alpha}\}_{\alpha \in I_\m}\subseteq \mg\backslash\mg_{x_\m}$ be independent and complete. Of course, we only will consider the situation where $\IHOMLAS\neq \emptyset$ holds.\footnote{Observe that $\IHOMLAS=\emptyset$ already implies that $\IHOMW=\emptyset$ holds just because each element of the latter space restricts to an element of the former one.}
\subsubsection{Tychonoff Products}
We now are going to write $\IHOMLAS$ as a Tychonoff product of compact Hausdorff spaces
for the general case that $S$ is compact and connected. The first step towards this is performed in Lemma and Definition \ref{def:topo}, where the strategy is basically the following:
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
In the Parts \ref{def:topo1}) and \ref{def:topo3}), for $x\in M$ and $p\in F_x$, we assign to $\epsilon \in \IHOMW$ an $\Ad_{G_{x}}^p$-equivariant (in the first factor) map $\Psi\colon \mg\backslash \mg_{x} \times \RR_{>0}\rightarrow S$ just by
\begin{align}
\label{eq:mappodappo}
\Psi(\g,l) := \Delta\big(\Phi_{\exp(l\cdot \g)}(p),\homm\big(\gamma_\g^{x}|_{[0,l]}\big)(p)\big)
\end{align}
Recall that $t\mapsto\Phi(\exp(t\cdot \g),p)$ is the canonical lift of ${\gamma_{\g}^x}$ in $p\in F_x$ which we already have used in Proposition \ref{th:invhomm}.\ref{th:invhomm1} for modifying invariant homomorphisms along such Lie algebra generated curves ${\gamma_{\g}^x}$. Moreover, as usual, for $p,p'$ contained in the same fibre, here $\diff(p,p')$ denotes the unique element $s\in S$ for which $p'=p\cdot s$ holds.
\item
In Part \ref{def:topo11}), we will split up the map \eqref{eq:mappodappo} into several
maps $\spann_\RR(\g_\alpha)\rightarrow S$ being equivariant in the sense of Definition \ref{def:eqmaps}. Here, $\alpha$ runs over some index set $I_x$ for $\{\g_\alpha\}_{\alpha\in I_x}\subseteq \mg\backslash\mg_x$ an independent and complete family of stable elements.
\item
Then, in Part \ref{def:topo2}), we will define the relevant topologies on the occurring spaces, and in the last part we will provide the desired homeomorphism between $\IHOMLAS$ and the mentioned Tychonoff product..
\end{itemize}
\endgroup
\begin{lemdef}
\label{def:topo}
Let $p\in P$, $x:=\pi(p)$ and assume that $\mg\backslash \mg_x\neq \emptyset$ holds.
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item
\label{def:topo1}
By \gls{EQP} we denote the set of $\Add{G_{x}}^p$-equivariant maps $\Psi\colon \mg\backslash \mg_{x} \times \RR_{>0}\rightarrow S$ that fulfil\footnote{The reason for introducing the factor $\RR_{>0}$ will become clear in Part \ref{def:topo3}).}
\begin{align}
\label{eq:Equi0}
\Psi(\lambda\cdot \g,l)=\Psi(\g,|\lambda| l)^{\sign(\lambda)}\qquad\forall\: \lambda\in \RR_{\neq 0},\: l>0
\end{align}
and
\begin{align}
\label{eq:Equi}
\Psi(\g,l+ l')=\Psi(\g,l)\cdot\Psi(\g,l')\qquad \forall\: l,l'\geq 0.
\end{align}
Here, equivariance means that
\begin{align*}
\Psi(\Ad_h(\g),\cdot)=\alpha_{\fiba_p(h)}\cp \Psi(\g,\cdot) \qquad \forall\: h\in G_{x}, \forall\:\g\in \mg\backslash \mg_{x}.
\end{align*}
\item
\label{def:topo11}
Recall the quantities we have considered in Subsection \ref{sec:ModifLAGC}.
Then, since we have $G_{[\g]}^x\subseteq G_x$, for each $\Psi\in \Eq_p$ and each $\g\in \mg\backslash \mg_x$ the map $\Psi_\g\colon \Span_\RR(\g)\rightarrow S$ defined by
\begin{align*}
\Psi_\g(0):=\me \qquad\qquad\text{and}\qquad\qquad \Psi_\g(\lambda\cdot \g):=\Psi(\lambda\cdot\g,1)\:\text{ for }\: \lambda \in \RR_{\neq 0}
\end{align*}
is $\Ad^p_{G_{[\g]}}$-equivariant in the sense of Definition \ref{def:eqmaps}. We will denote by $Y^p_\g$ the set of all such $\Ad^p_{G_{[\g]}}$-equivariant maps and define
\begin{align*}
\res_\g^p\colon \Eq_{p}\rightarrow Y_\g^p,\quad
\Psi \mapsto \Psi_{\g}.
\end{align*}
\item
\label{def:topo2}
We equip $\Eq_p$ with the topology
generated by the sets
\begin{align*}
U_{\g,l}(\Psi):=\{\Psi'\in \Eq_p\:|\: \Psi'(l\cdot \g)\in \Psi(l\cdot \g)\cdot U\}
\end{align*}
for $\Psi\in \Eq_p$, $U\subseteq S$ open, $\g\in \mg\backslash\mg_x$ and $l>0$. Similarly, we equip the spaces $Y^p_\g$ with the topologies generated by the sets
\begin{align*}
U_{\lambda}(\Psi):=\{\Psi'\in Y^p_\g\:|\: \Psi'(\lambda \cdot \g)\in \Psi(\lambda\cdot \g)\cdot U\},
\end{align*}
for $\Psi\in Y_\g^p$, $U\subseteq S$ open and $\lambda\in \RR$.
It is easy to see that $\res_\g^p$ is continuous w.r.t.\ these topologies.
\item
\label{def:topo3}
We define $\pi_p\colon \mg\backslash \mg_{x}\times \RR_{>0}\times \IHOMLAS \rightarrow S$ by
\begin{align*}
\pi_p(\g,l,\homm) := \Delta\big(\Phi_{\exp(l\g)}(p),\homm\big(\gamma_\g^{x}|_{[0,l]}\big)(p)\big)\qquad\forall\: 0<l<\tau_\g
\end{align*}
as well as
\begin{align*}
\pi_p(\g,l,\homm):=\prod_{i=1}^k\pi_p(\g,l_i,\homm)\quad\text{for}\quad 0<l_1,\dots,l_k<\tau_\g \quad\text{with}\quad l=l_1+\dots+l_k.
\end{align*}
Then, straightforward calculations, cf.\ Appendix \ref{app:Bohrmodl}, similar to that in the proofs of Proposition \ref{th:invhomm}.\ref{th:invhomm1} and Proposition \ref{lemma:freemodify} show that the map
\begin{align}
\label{eq:pip}
\pip_p \colon \IHOMLAS\rightarrow \Eq_p,\quad
\homm\mapsto \pi_p(\cdot,\cdot,\homm)
\end{align}
is well defined and continuous. Consequently, $\pi_p$ is well defined as well. In particular, the topological space $Y_\g^p$ is compact. In fact, $\IHOMLAS$ is compact and $\res_\g^p\cp \pip_p$ is continuous, as well as surjective by the next corollary.
Obviously, we have $\pi_p=\alpha_{\Delta(p',p)}\cp\pi_{p'}$ if $\pi(p)=\pi(p')$ holds. Moreover, if $g\in G$, $s\in S$, $\g\in \mg\backslash\mg_y$ and $h\in G_{y}$ for $y:=\pi(g\cdot p)$, a straightforward calculation shows that we have, cf.\ Appendix \ref{app:Bohrmodl}
\begin{align}
\label{eq:verkn}
\pi_{g\cdot p\cdot s}(\lambda\Add{h}(\g),l,\homm)=\alpha_{s^{-1} \fiba_{g\cdot p}(h)}\cp \pi_{p}\big(\Add{g^{-1}}(\g),|\lambda| l,\homm\big)^{\sign(\lambda)}\qquad \forall\:\lambda\in \RR_{\neq 0}, \:l>0.
\end{align}
This equation will be relevant for our considerations in the final part of this subsection. There, we will show that the measure we construct on $\IHOMLAS$ does not depend on any choices such as
the independent and complete families we have fixed in the beginning of this subsection.
\item
\label{def:topo5}
To simplify the notations, let
\begin{align*}
Y_{\m,\alpha}:=Y_{\g_{\m,\alpha}}^{p_\m}\qquad\quad\text{as well as}\qquad\quad\res_{\m,\alpha}:=\res_{\g_{\m,\alpha}}^{p_\m}\qquad\forall\: \m\in \Mm, \forall\: \alpha\in I_\m.
\end{align*}
We equip $Y:=\prod_{\m\in \Mm,\alpha\in I_\m}Y_{\m,\alpha}$ with the Tychonoff topology and define
\begin{align*}
\Pi_Y\colon\IHOMLAS \rightarrow Y\qquad\text{by}\qquad \Pi_{Y}:=\prod_{\m\in \Mm}\hspace{1pt}\big[\res_{\m}\cp \pip_{p_\m}\big]
\end{align*}
for the natural map $\res_{\m}:=\prod_{\alpha\in I_\m}\res_{\m,\alpha}\colon \Eq_{p_\m}\rightarrow \prod_{\alpha\in I_\m}Y_{\m,\alpha}$.
\end{enumerate}
\endgroup
\end{lemdef}
The homeomorphism property of $\Pi_Y$ is a straightforward consequence of Lemma \ref{lemma:completee}.\ref{lemma:homzueq33} and Proposition \ref{th:invhomm}.\ref{th:invhomm1}, as we now show in
\begin{corollary}
The map $\Pi_Y\colon \IHOMLAS \rightarrow Y$ is a homeomorphism.
\end{corollary}
\begin{proof}
$\Pi_Y$ is continuous because the maps $\res_{\m,\alpha}\cp \pip_{p_\m}$ are so. Moreover, $Y$ is Hausdorff because the spaces $Y_{\m,\alpha}$ are Hausdorff. So, since $\IHOMLAS$ is compact, the claim follows if we show that $\Pi_Y$ is bijective. Now, $\Pi_Y$ is injective by Lemma \ref{lemma:completee}.\ref{lemma:homzueq33} and
\begin{align*}
\big(\res_{\m,\alpha}\cp \pip_{p_\m}\big)(\homm)(\lambda\cdot \g_{\m,\alpha})&=\pip_{p_\m}(\homm)(\lambda \cdot \g_{\m,\alpha},1)\\
&=\diff\big(\Phi_{\exp(\lambda \cdot \g_{\m,\alpha})}(p_\m),\homm\big(\gamma_{\lambda\cdot \g_{\m,\alpha}}^{x_\m}|_{[0,1]}\big)(p_\m)\big)\\
&=\diff\big(\Phi_{\exp(\lambda \cdot \g_{\m,\alpha})}(p_\m),\homm\big(\gamma_{\g_{\m,\alpha}}^{x_\m}|_{[0,\lambda]}\big)(p_\m)\big),
\end{align*}
and surjective by Proposition \ref{th:invhomm}.\ref{th:invhomm1}. In fact, if $\Psi_{\m,\alpha}\in Y_{\m,\alpha}$ for all $\m\in \Mm$ and all $\alpha\in I_{\m}$, then for each $\m\in \Mm$ we can modify an element $\homm'\in \IHOMLAS$ w.r.t.\ the family $\{\Psi_{\m,\alpha}\}_{\alpha\in I_\m}$ in order to obtain $\homm\in \IHOMLAS$ with
\begin{align*}
\big(\res_{\m,\alpha}\cp \pip_{p_\m}\big)(\homm)=Y_{\m,\alpha}\qquad \forall\: \alpha\in I_\m.
\end{align*}
Since orbits are disjoint, i.e., $\mathrm{im}\big[\wm_g\cp \gamma_{\g}^{x_\m}\big]\cap \mathrm{im}\big[\wm_{\g'}\cp \gamma_{\g'}^{x_{\m'}}\big]=\emptyset$ if $\m\neq \m'$, it is clear that we can apply Proposition \ref{th:invhomm}.\ref{th:invhomm1} simultaneously for all $\m\in \Mm$.
\end{proof}
\subsubsection{Normalized Radon Measures}
In the previous part, we have seen that $\IHOMW$ is homeomorphic to the Tychonoff product of the compact Hausdorff spaces $Y_{\m,\alpha}$, each of them consisting of certain equivariant maps. So, in order to define a normalized Radon measure on $\IHOMW$, it suffices to define such measures on each of the factors $Y_{\m,\alpha}$. For this, we now calculate these spaces explicitly where we will restrict to the case where $S=\SU$.
We start by recalling some facts and fixing some notations.
\begin{convention}
\label{conv:sammel}
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
By $S^1$ we denote the unit circle in $\mathbb{C}$. This will not be in conflict with our notations as in the sequel no products of the structure group $S=\SU$ will occur.
\item
\itspacec
In the following, let $\DG$ denote the group of all characters on $\RR$, i.e., of all functions of the form $\chi_l\colon x\mapsto \e^{\I l x}$.
By the Bohr compactification $\RB$ of $\RR$ we will understand (cf.\ Lemma and Convention \ref{lemconv:RBMOD}.\ref{rem:RBOHR}) the set of unital homomorphisms $\psi\colon \DG \rightarrow S^1$ equipped with the topology generated by the sets, cf.\ \eqref{eq:bohrtop}
\begin{align*}
V_{l}(\psi):=\{\psi'\in \RB\:|\: \psi'(\chi_l)\in \psi(\chi_l)\cdot V\}
\end{align*}
for $\psi\in \RB$, $V\subseteq S^1$ open and $l>0$. Recall that $\RB$ is a compact abelian group when equipped with the group structure
\begin{align*}
\big(\psi +\psi'\big)(\chi_l):=\psi(\chi_l)\cdot \psi'(\chi_l)\qquad \psi^{-1}(\chi_l):=\psi(\chi_{-l})\qquad 1_{\mathrm{Bohr}}(\chi_l):=1
\end{align*}
for all $l\in \RR$. Moreover, as we have seen in Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod21},
$\RB$ is parametrized the set $\Per$ of all maps $\phi\colon \RR_{>0}\rightarrow [0,2\pi)$ with
\begin{align*}
\phi(l+l')=\phi(l)+\phi(l')\:\bmod \: 2\pi\qquad \forall\:l,l'\in \RR_{>0}.
\end{align*}
\item
Let \gls{SPMSU} denote the projective space\footnote{This is the set $\su\backslash\{0\}$ modulo the equivalence relation $\s'\sim \s$ iff $\s'\in \spann_\RR(\s)$.} that corresponds to $\su$ as well as \gls{PRMSU} the corresponding projection map.
\item
Recall the map $\text{\gls{MURS}}\colon \RR^3 \rightarrow \mathfrak{su}(2)$ from Convention \ref{conv:sutwo1}.\ref{conv:sutwo111} and that $H_{\s}=\{\exp(t\cdot\s)\:|\: t\in \RR\}$.
\item
In the following, we will consider $\Sp\ms$ as an index set, and choose a fixed representative $\s_\beta$ with $\|\text{\gls{MURS}}^{-1}(\s_\beta)\|=1$ in each equivalence class $\beta\in \Sp\ms$.
\item
If $\beta =[\s\hspace{1pt}]\in \Sp\ms$, we will always mean that $\|\murs^{-1}(\s)\|=1$ holds
and define $H_{\beta}:=H_\s$.
\end{itemize}
\endgroup
\end{convention}
In order to determine the spaces $Y_\g^p$,
we now have to investigate the images of the equivariant maps these spaces consist of. Due to the first part of Lemma \ref{prop:Bohrmod2}, this means to identify the maximal tori which such an equivariant map is allowed to map to. Indeed, there we show that the subset of all non-trivial maps\footnote{Observe that if $\Psi(\lambda\cdot \g)=-\me$, then $\Psi(\lambda/2\cdot \g)\neq \pm \me$.} in $Y_\g^p$ which map to the same maximal torus is parametrized by $\RB$. This means that $Y_\g^p$ is either singleton or given by the product of $\RB$ with the set off all occurring tori. The latter set will we determined in the second part of Lemma \ref{prop:Bohrmod2}. There, we show that the
following situations can occur:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item[{\bf 1)}]
$Y_\g^p$ consists of one single element, namely the trivial map.
\item[{\bf 2)}]
All elements of $Y_\g^p$ map into the same maximal torus.
\item[{\bf 3)}]
Each element of $Y_\g^p$ maps into a maximal torus $H_{\vec{n}}$ with $\vec{n}$ contained in a fixed plain through the origin. Moreover, each such torus occurs.
\item[{\bf 4)}]
Each element of $Y_\g^p$ maps into some maximal torus, whereby all maximal tori occur.
\end{itemize}
\endgroup
\noindent
The relevant notions are provided in
\begin{definition}
\label{def:betadef}
Let $p\in P$, $x:=\pi(p)$ and
\begin{align*}
J(\g,p):=\bigcup_{\Psi\in Y^p_\g}\beta(\Psi)\subseteq \{0\}\sqcup\Sp\ms
\end{align*}
for the quantity $\beta(\Psi)$ defined as follows:
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
\vspace{-4pt}
If $\Psi(\lambda\cdot\g)=\me$ for all $\lambda\in \RR$, we let $\beta(\Psi),\s_\beta(\Psi):=0\in \su$.
\item
\vspace{2pt}
In the other case, it follows from
\begin{align*}
\Psi(\lambda\cdot \g)\cdot \Psi(\mu\cdot \g)=\Psi([\lambda+\mu]\cdot \g)=\Psi(\mu\cdot \g)\cdot \Psi(\lambda\cdot \g)\qquad\forall\:\lambda,\mu \in \RR
\end{align*}
and Lemma \ref{lemma:torus}.\ref{lemma:torus1} that $\mathrm{im}[\Psi]\subseteq H_{\beta(\Psi)}$ holds for $\beta(\Psi)\in \Sp\ms$ uniquely determined. \hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\end{definition}
\begin{lemma}
\label{prop:Bohrmod2}
Let $p\in P$, $x:=\pi(p)$ and $\g\in \mg\backslash \mg_{x}$.
\begin{enumerate}
\item
\label{prop:Bohrmod22}
For each $\beta\in J(\g,p)$ and $\phi\in \Per$ we find $\Psi'\in Y_\g^p$ with
\begin{align*}
\Psi'(\lambda\cdot \g)=\exp(\phi(\lambda )\cdot \s_{\beta}) \qquad\forall\: \lambda > 0.
\end{align*}
\item
\label{prop:Bohrmod23}
Exactly one of the following cases holds:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{enumerate}
\item[{\rm \textbf{1})}]
$J(\g,p)=\{0\}$,
\item[{\rm \textbf{2})}]
$J(\g,p)=\{0,\beta\}$ for $\beta \in \Sp\ms$ uniquely determined,
\item[{\rm \textbf{3})}]
$J(\g,p)= \{0\}\sqcup \bigcup_{\vec{n}\in \RR^3\backslash\{0\} \colon\langle\vec{n},\vec{m}\rangle=0}\:[\hspace{1.5pt}\murs(\vec{n})\hspace{0.5pt}]$ for some $\vec{m}\in \RR^3\backslash\{0\}$,
\item[{\rm \textbf{4})}]
$J(\g,p)=\{0\}\sqcup \Sp\ms$.
\end{enumerate}
\endgroup
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
By assumption, we find $\Psi\in Y_\g^p$ with $\beta(\Psi)=\beta$. Now, let
\begin{align*}
\Psi'(\lambda\cdot \g):=\exp\hspace{-1pt}\big(\hspace{-1pt}\sign(\lambda)\phi(|\lambda|)\cdot \s_\beta\big)\qquad\forall\:\lambda\in \RR.
\end{align*}
Then, $\Psi'$ fulfills \eqref{eq:equii}, just by the same arguments as in Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod21}, so that it remains to show \eqref{eq:equiii}.
If $\Psi=\me$, then $\Psi'=\me$, and we have nothing to show. In the other case, we find $\lambda>0$ with\footnote{If $\Psi(\mu \cdot \g)=-\me$, then $\Psi(|\mu| \cdot \g)=-\me$ and we choose $\lambda:=\mu/2$.} $\Psi(\lambda\cdot \g)\neq \pm \me$. Let $\Ad_h(\g)=\mu\cdot \g$ for $h\in G^x_{[\g]}$ and $\mu\in \RR$. Then, $|\mu|=1$ by Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve4}, so that
\begin{align}
\label{eq:dfdf}
\alpha_{\fiba_p(h)}\big(\Psi(\lambda\cdot \g)\big)=\Psi(\lambda\mu \cdot \g)=\Psi(\lambda \cdot \g)^{\sign(\mu)}.
\end{align}
By Lemma \ref{lemma:torus}.\ref{lemma:torus2} we now have the following two possibilities:
\begin{enumerate}
\item[1.)]
\vspace{-4pt}
$\mu=1$ \qquad$\Longrightarrow$\qquad $\fiba_p(h)\in H_{\beta}$ \qquad$\Longrightarrow$
\begin{align*}
\alpha_{\fiba_p(h)}\cp \Psi'(\lambda\cdot \g)
&=\alpha_{\fiba_p(h)}\cp\exp\hspace{-1pt}\big(\hspace{-1pt}\sign(\lambda)\phi(|\lambda|)\cdot\s_{\beta}\big)
= \exp\hspace{-1pt}\big(\hspace{-1pt}\sign(\lambda)\phi(|\lambda|)\cdot\s_{\beta}\big)\\
&=\Psi'(\lambda\cdot\g)=(\Psi'\cp \Ad_h)(\lambda\cdot\g).
\end{align*}
\item[2.)]
\vspace{2pt}
$\mu=-1$ \qquad$\Longrightarrow$\qquad $\alpha_{\fiba_p(h)}(s)=s^{-1}$ for all $s\in H_{\beta}$ \qquad$\Longrightarrow$
\begin{align*}
\alpha_{\fiba_p(h)}\cp \Psi'(\lambda\cdot \g)
&=\alpha_{\fiba_p(h)}\cp\exp\hspace{-1pt}\big(\hspace{-1pt}\sign(\lambda)\phi(|\lambda|)\cdot\s_{\beta}\big)
= \exp\hspace{-1pt}\big(\hspace{-1pt}\sign(\lambda)\phi(|\lambda|)\cdot\s_{\beta}\big)^{-1}\\
&=\Psi'(-\lambda\cdot \g)=(\Psi'\cp \Ad_h)(\lambda\cdot\g).
\end{align*}
\end{enumerate}
\item
Basically, this follows as in Part \ref{prop:Bohrmod22}) by repeated application of Lemma \ref{lemma:torus}.\ref{lemma:torus2} involving a case differentiation. The details of the (not complicated but long) proof can be found in Appendix \ref{app:Bohrmod}.
\end{enumerate}
\end{proof}
\end{lemma}
We now are ready to determine the spaces $Y_\g^p$, and to define normalized Radon measures thereon. Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa3} then provides us with a normalized Radon measure $\mLAS$ on
\begin{align*}
\IHOMLAS\cong Y=\prod_{\m\in \Mm,\:\alpha\in I_\m}Y_{\m,\alpha},
\end{align*}
which, in the final part of this subsection, we show to be independent of any choices. However, before we come to this, we first will give some applications to loop quantum cosmology collected in Example \ref{ex:cosmoliealgmaasse} and Remark \ref{StanLQC}.
Now, in the first part of (the next) Lemma and Definition \ref{lemdef:projspaces}, we will define the four different spaces which $Y_\g^p$ can be homeomorphic to.
Except for the Hausdorff property of the defined topologies, here no difficulties will arise.
In the second and the third part, we define the corresponding bijections and establish their homeomorphism properties. Here, the bijections are easily defined, but for their homeomorphism property we will have some hard work to do. In the last part, we provides the measures on the spaces defined in the first part.
\begin{lemdef}
\label{lemdef:projspaces}
Let $p\in P$, $x:=\pi(p)$ and $\g\in \mg\backslash\mg_x$.
\begin{enumerate}
\item
\label{lemdef:projspaces1}
We define
\begin{align*}
X_{\g}^p:=
\begin{cases}
\{0_{\mathrm{Bohr}}\} &\mbox{if Case } {\bf 1)} \text{ holds for } J(\g,p)\qquad\textbf{Type 1},\\
\RB &\mbox{if Case } {\bf 2)} \text{ holds for } J(\g,p)\qquad\textbf{Type 2},\\
\RB\wti S^1 &\mbox{if Case } {\bf 3)} \text{ holds for } J(\g,p)\qquad\textbf{Type 3},\\
\RB\wti S^2 &\mbox{if Case } {\bf 4)} \text{ holds for } J(\g,p)\qquad\textbf{Type 4}.
\end{cases}
\end{align*}
Here, $S^2\subseteq\RR^3$ denotes the unit sphere and $S^1\subseteq \mathbb{C}$ the unit circle. Moreover, the products denote the quotient spaces
\begin{align*}
\RB\wti S^i:= \big[\RB\times S^i\big]\slash \sim
\end{align*}
w.r.t.\ the equivalence relation $\sim$ defined by
\begin{align*}
(\psi,v)\sim (\psi',v')\qquad \Longleftrightarrow \qquad (\psi',v')= (\psi^{-1},-v)\quad\text{or} \quad\psi,\psi'=0_{\mathrm{Bohr}}
\end{align*}
for $i=1,2$.
In both cases, we will denote the respective projection map by $\pr_0$.
We equip each of the above spaces with its natural topology. Hence,
$\RB\times S^i$ with the product topology and $\RB\wti S^i$ with the respective quotient topology for $i=1,2$. Obviously, $\RB\wti S^i$ is
compact for $i=1,2$, and the Hausdorff property is proven below.
\vspace{5pt}
{\bf Remark:} These definitions might seem quite artificial at a first sight. However, they have the big advantage that the canonical Radon measures on $\RB$, $S^2$ and $S^1$ (see Part \ref{lemdef:projspaces5})) can be used to define a normalized Radon measure on the above spaces in a straightforward way. In addition to that, it should be intuitively clear already at this point that the trivial map will be assigned to the class $[(\NB,v)]$, and that the factors $S^2$ and $S^1$ will label the occurring maximal tori in the respective cases. Here, we will need the identification of $(\psi,v)$ with $(\psi^{-1},-v)$ since, due to our further definitions, $v$ and $-v$ will refer to the same maximal torus in $\SU$.
\item
\label{lemdef:projspaces3}
For each plane\footnote{This means $\mm\in \RR^3\backslash\{0\}$ and $[\mm]=[\mm']$ if $\mm'=\lambda\cdot\mm$ for $\lambda\neq 0$.} $[\mm]:=\{\vec{v}\in \RR^3\:|\: \langle\vec{v},\mm\rangle=0\}\subseteq \RR^3$,
we fix an angle-preserving\footnote{If $\LM,\LSM$ are two such maps, then $\LM\cp {\LSM}^{-1}\in O(2)$.} map (homeomorphism) $\LM$ between the great circle on $S^2$ cutted out by $[\mm]$ and the unit circle $S^1\subseteq \mathbb{C}$.
Then, if Case {\bf 3)} holds for $J(\g,p)$, i.e., if
\begin{align}
\label{eq:hmnejhedrffds}
J(\g,p)\backslash\{0\}= \bigcup_{\vec{n}\in \RR^3\backslash\{0\} \colon\langle\vec{n},\vec{m}\rangle=0}[\hspace{1.5pt}\murs(\vec{n})\hspace{0.5pt}]\quad\text{ for some } \quad\vec{m}\in \RR^3\backslash\{0\},
\end{align}
then we define $\xi_{\g}^p\colon J(\g,p)\rightarrow S^1$ by (recall that due to Convention \ref{conv:sammel} $\|\murs^{-1}(\s_\beta)\|=1$ holds)
\begin{align*}
\xi_{\g}^p(\beta):=
\begin{cases}
v^1_0 &\mbox{if } \beta=0\\
\LM\big(\murs^{-1}(\s_\beta)\big) & \mbox{else}
\end{cases}
\end{align*}
for $v^1_0\in S^1$ some fixed element.
Similarly, in Case {\bf 4)}, we define $\xi_\g^p\colon J(\g,p) \rightarrow S^2$ by
\begin{align*}
\xi_{\g}^p(\beta):=
\begin{cases}
v^2_0 &\mbox{if } \beta=0\\
\murs^{-1}(\s_\beta) & \mbox{else}
\end{cases}
\end{align*}
for $v^2_0\in S^2$ some fixed element.
\item
\label{lemdef:projspaces4}
Using the above maps, we construct our bijection $\tau^p_\g\colon Y_\g^p\rightarrow X_\g^p$ as follows:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\vspace{-5pt}
For $\Psi\in Y_\g^p$ with $\beta(\Psi)\neq 0$ we find $\phi\in \Per$ uniquely determined by
\begin{align*}
\Psi(\lambda\cdot \g)=\exp\left(\phi(\lambda)\cdot \s_{\beta(\Psi)}\right)\qquad \forall\: \lambda\in \RR,
\end{align*}
and denote by $\psi$ the corresponding element of $\RB$ from Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod21}.
\item
\vspace{2pt}
We define $\tau_\g^p\colon Y_\g^p\rightarrow X_\g^p$ by
\begin{align*}
\tau_\g^p(\Psi):=
\begin{cases}
0_{\mathrm{Bohr}} &\mbox{if Case } {\bf 1)} \text{ holds for } J(\g,p),\\
\psi &\mbox{if Case } {\bf 2)} \text{ holds for } J(\g,p),\\
\big[\big(\psi,\xi_\g^p\big(\beta(\Psi)\big)\big)\big]
&\mbox{if Case } {\bf 3)} \text{ or Case } {\bf 4)} \text{ holds for } J(\g,p).
\end{cases}
\end{align*}
\end{itemize}
\endgroup
The map $\tau_\g^p$ is a well-defined homeomorphism which is independent of the explicit choice of $\s_\beta\in \beta$ we have made in Convention \ref{conv:sammel}. In fact, the independence is immediate from the definitions, and the homeomorphism property is shown below.
\item
\label{lemdef:projspaces5}
\itspacec
We define normalized Radon measures $\mu_\g^p$ on the spaces $X_\g^p$ as follows. If $X_\g^p$ is of \textbf{Type 1}, there is only one possibility. If $X_\g^p$ is of \textbf{Type 2}, we define $\mu_\g^p:=\mu_{\mathrm{Bohr}}$.
For $X_\g^p$ of \textbf{Type 3} and \textbf{Type 4} we proceed as follows:
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{itemize}
\item[a)]
\vspace{-2pt}
We equip $S^1$ with the Haar measure $\mu_1$ as well as
$S^2$ with the canonical Radon measure $\mu_2$ induced by the Haar measure on $\SOD$.\footnote{Let $\mu_R$ denote the Haar measure on $\SOD$ and $\Omega \colon \SOD\times S^2 \rightarrow S^2$ the canonical left action. Let $\vec{n}\in S^2$ be fixed and denote by $G_{\vec{n}}$ the $\Omega$-stabilizer of $\vec{n}$. Then, $\SOD\slash G_{\vec{n}}\cong S^2$ (by the map $[g]\mapsto \Omega(g,\vec{n})$), and we equip the quotient with the push forward of $\mu_R$ by the corresponding projection map.
It is straightforward to see that the measure $\mu_{2}$ induced on $S^2$ by this diffeomorphism does not depend on the explicit choice of $\vec{n}\in S^2$.}
Observe that this measure is invariant under the action of $\SOD$ on $S^2$, i.e., $\mu_{2}(R_3(A))=\mu_{2}(A)$ holds for all $A\in \mathfrak{B}(S^2)$ and all $R_3\in \SOD$.
\item[b)]
We equip $\RB\times S^1$ and $\RB\times S^2$ with the respective Radon product measures
\begin{align*}
\mu_{1\times}:=\mu_{\mathrm{Bohr}}\times \mu_1\quad\qquad\text{and} \quad\qquad \mu_{2\times}:=\mu_{\mathrm{Bohr}}\times \mu_2
\end{align*}
from Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa1}.
\item[c)]
We define the measures on $\RB\wti S^1$ and $\RB\wti S^2$ by the push forwards of $\mu_{1\times}$ and $\mu_{2
\times}$ by the respective projection maps.
\end{itemize}
\endgroup
\end{enumerate}
\end{lemdef}
\begin{proof}
\begin{enumerate}
\item[1)]
For $i=1,2$ we have:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
If $\psi\neq 0_{\mathrm{Bohr}}$ and $v\in S^i$, then an open neighbourhood of $[(\psi,v)]$, e.g., is given by $\pr_0(U\times V)$ for $U\subseteq \RB$ an open neighbourhood of $\psi$ with $\NB\notin U$ and $V\subseteq S^i$ an open neighbourhood of $v$. This is clear from
\begin{align}
\label{eq:dsvvasdr}
\pr_0^{-1}(\pr_0(U\times V))=U\times V \:\cup \: U^{-1}\times -V.
\end{align}
\item
If $U_0\subseteq \RB$ is a symmetric open neighbourhood of $0_{\mathrm{Bohr}}$, then $\pr_0(U_0\times S^i)$ is an open neighbourhood of $[(0_{\mathrm{Bohr}},v)]$ in $\RB\wti S^i$, just because
\begin{align}
\label{eq:sfsfdsfsdf}
\pr_0^{-1}\!\left(\pr_0\!\left(U_0\times S^i\right)\right)\stackrel{\eqref{eq:dsvvasdr}}{=}U_0\times S^i
\end{align}
is open.
\item
So, choosing $U_0$ and $U$ as above, such that in addition $U_0\cap U^{\pm 1}=\emptyset$ holds, we can separate $[(\NB,v_0)]$ and $[(\psi,v)]$ for $\psi \neq \NB$ and $v,v_0\in S^i$ by $\pr_0(U_0\times S^i)$ and $\pr_0(U\times V)$. In fact, by construction we even have
\begin{align*}
\pr_0^{-1}(\pr_0(U_0\times S^1))\cap \pr_0^{-1}(\pr_0(U\times V))=\emptyset.
\end{align*}
\item
The remaining cases follow from \eqref{eq:dsvvasdr} and the fact that for $\psi,\psi'\neq 0_{\mathrm{Bohr}}$ with $\psi\neq \psi'$ as well as $v,v'\in S^i$ with $v\neq v'$ we find neighbourhoods $U, U'$ of $\psi, \psi'$ as well as neighbourhoods $V,V'$ of $v,v'$, such that
\begin{align*}
\emptyset&=U\cap U', U\cap U^{-1}, U'\cap {U'}^{-1}, U^{-1}\cap U'\qquad\text{as well as} \\
\emptyset&= V\cap V', V\cap -V, V'\cap -V', -V\cap V'
\end{align*}
holds, respectively.
\end{itemize}
\endgroup
\item[3)]
Obviously, $\tau_\g^p$ is well defined and injective, and
its surjectivity follows from Lemma \ref{prop:Bohrmod2}.\ref{prop:Bohrmod22} and Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod21}. So, since $Y_\g^p$ is compact and $X_\g^p$ is Hausdorff by Part \ref{lemdef:projspaces1}), we only have to prove continuity of $\tau_\g^p$ in order to show its homeomorphism property.
{\bf Strategy:} Basically, here the difficulty is to show that the convergence of a net $Y_\g^p\supseteq \{\Psi_\iota\}_{\iota\in J}\rightarrow \Psi\in Y_\g^p$ already implies the ``convergence'' of the corresponding ``maximal tori'' $\beta(\Psi_\iota)$ to the ``maximal torus'' $\beta(\Psi)$ of the limes map. This is clear if $Y_\g^p$ is of \textbf{Type 2} (or \textbf{Type 1}) as there $\beta(\Psi_\iota)=\beta(\Psi)$ holds for all $\iota\in J$. For $Y_\g^p$ of \textbf{Type 3} or \textbf{Type 4}, we have to consider such $\lambda>0$ for which $\Psi(\lambda\cdot \g)$ is regular, i.e., different from $\pm \me$. here, we have to use the local diffeomorphism property of the map
\begin{align*}
S^i\times [ (0,\pi)\sqcup (\pi,2\pi)]\ni(v,t)\mapsto \exp(t\cdot \murs(v)),
\end{align*}
whose local inverses give back the two possible values $(v_\lambda,t_\lambda)$ and $(-v_\lambda,2\pi-t_\lambda)$ in $[0,2\pi]$ for which
\begin{align*}
\exp( t_\lambda\cdot \murs( v_\lambda))=\Psi(\lambda\cdot \g)=\exp( (2\pi-t_\lambda)\cdot \murs( -v_\lambda))
\end{align*}
holds.\hspace*{\fill}$\dagger$
\begingroup
\setlength{\leftmarginii}{13pt}
\begin{itemize}
\item[$\triangleright$]
If $X_\g^p$ is of \textbf{Type 1}, we have nothing to show because $\mathrm{im}\big[\tau_\g^p\big]=0_{\mathrm{Bohr}}$ holds in this case.
\end{itemize}
\endgroup
For the other types, let $Y_\g^p\supseteq \{\Psi_\iota\}_{\iota\in J}\rightarrow \Psi\in Y_\g^p$ be a converging net.
We choose $\phi\in \Per$, $\{\phi_\iota\}_{\iota\in J}\subseteq \Per$ as well as $\beta\in \{0\}\sqcup\Sp\ms$, $\{\beta_\iota\}_{\iota\in J}\subseteq \{0\}\sqcup\Sp\ms$ such that for each $\lambda>0$ we have
\begin{align}
\label{eq:covbbb}
\Psi(\lambda\cdot \g)&=\exp(\phi(\lambda)\cdot\s_\beta)\qquad\text{as well as}\qquad
\Psi_\iota(\lambda\cdot \g)=\exp(\phi_\iota(\lambda)\cdot\s_{\beta_\iota})\quad \forall\: \iota\in J.
\end{align}
Let $\psi\in \RB$ and $\{\psi_\iota\}_{\iota\in J}\subseteq \RB$ denote the elements of $\RB$ from Lemma and Convention \ref{prop:Bohrmod2}.\ref{prop:Bohrmod21} that correspond to $\phi$ and $\{\phi_\iota\}_{\iota\in J}$, respectively. Finally, define
\begin{align*}
f\colon S^2\times [0,2\pi)&\rightarrow \SU\\
(v,t)&\mapsto \exp(t\cdot\murs(v)),
\end{align*}
i.e., $f(v,t)=\cos(t)\cdot \me+\sin(t)\cdot\murs(v)$ by \eqref{eq:expSU2}.
\begingroup
\setlength{\leftmarginii}{13pt}
\begin{itemize}
\item[$\triangleright$]
If $Y_\g^p$ is of \textbf{Type 2}, then $\beta_\iota=\beta$ holds for all $\iota\in J$. Now,
$g\colon S^1\rightarrow H_\beta$, $\e^{\I t}\mapsto \exp(t\cdot s_\beta)$ is a homeomorphism, and for all $\lambda\in \RR$ we have
\begin{align*}
\psi(\chi_\lambda)=g^{-1}\cp \Psi(\lambda\cdot \g)\qquad\text{and}\qquad \psi_\iota(\chi_\lambda)= g^{-1}\cp \Psi_\iota(\lambda\cdot \g)\quad\forall\:\iota\in J.
\end{align*}
Consequently, the convergence $\{\Psi_\iota\}_{\iota\in J}\rightarrow \Psi$ implies $\{\psi_\iota\}_{\iota\in J}\rightarrow \psi$ just by the definition of the topologies on $Y_\g^p$ and $\RB$.
\end{itemize}
\endgroup
$\triangleright$ Let $Y_\g^p$ be of \textbf{Type 3} or of \textbf{Type 4}.
\vspace{3pt}
We first assume that $\Psi=\me$, hence $\beta=0$:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item[--]
\vspace{-4pt}
Then, $\lim_\iota \Psi_\iota(\lambda\cdot \g)=\Psi(\lambda\cdot \g)=\me$ for all $\lambda\in \RR$, so that for each such $\lambda$ and each $n\in\mathbb{N}_{>0}$ we find $\iota_0\in J$ with
\begin{align}
\label{eq:jhgsd}
\phi_\iota(\lambda)\in [0,1/n)\sqcup (2\pi-1/n,2\pi)\qquad \forall\:\iota\geq \iota_0.
\end{align}
In fact, if $C:=[0,1/n)\sqcup (2\pi-1/n, 2\pi)$ and $U:=f\big(S^2\times C\big)$, then (cf.\ formula \eqref{eq:expSU2}) $U=\exp\big([0,1/n) \cdot \murs(S^2)\big)$ is an open neighbourhood of $\me$ in $\SU$ provided that $n$ is suitable large. Since $f^{-1}(U)=S^2\times C$, the claim is clear from equation \eqref{eq:covbbb} and the definition of $f$.
\item[--]
Then, \eqref{eq:jhgsd} shows that
\begin{align*}
\lim_\iota \psi_\iota(\chi_\lambda)=\lim_\iota\e^{\I \sign(\lambda)\phi_\iota(|\lambda|)}\stackrel{\eqref{eq:jhgsd}}{=} 1\qquad\forall\: \lambda>0,
\end{align*}
hence $\lim_\iota\psi_\iota = 0_{\mathrm{Bohr}}$. Consequently,
\begin{align*}
\lim_\iota\tau_\g^p(\Psi_\iota)=[(0_{\mathrm{Bohr}}, v_0^i)]
\end{align*}
for $i\in\{1,2\}$ because each neighbourhood of $[(0_{\mathrm{Bohr}},v_0^i)]$ in $\RB\wti S^i$ contains an open neighbourhood of the form $\pr_0\! \left(U_{0}\times S^i\right)$ for $U_0$ an symmetric open neighbourhood of $0_{\mathrm{Bohr}}$ in $\RB$.
\item[--]
In fact, by the definition of the quotient topology,
$W\subseteq \RB\wti S^i$ with $[(0_{\mathrm{Bohr}},v_0^i)]\in W$ is an open neighbourhood of $[(0_{\mathrm{Bohr}},v_0^i)]$ iff $\pr_0^{-1}(W)$ is an open subset of $\RB\times S^i$, hence an open neighbourhood of $(0_{\mathrm{Bohr}},v)$ for all $v\in S^i$. Then, we have
\begin{align*}
\pr_0^{-1}(W)\supseteq \textstyle\bigcup_{\nu}U^\nu\times V^\nu
\end{align*}
for open neighbourhoods $U^\nu\subseteq \RB$ of $0_{\mathrm{Bohr}}$ as well as $V^\nu\subseteq S^i$ open subsets with $S^i=\bigcup_{\nu}V^\nu$. Since $S^i$ is compact, we even have $S^i=V^{\nu_1}\cup\dots\cup V^{\nu_l}$ for finitely many indices, so that by \eqref{eq:sfsfdsfsdf} the statement holds for $U_0=U\cap U^{-1}$ with $U:=U^{\nu_1}\cap\dots\cap U^{\nu_l}$.
\end{itemize}
\endgroup
\vspace{3pt}
We now assume that $\Psi(\lambda\cdot \g)\neq\me$ holds for some $\lambda>0$, hence $\beta\neq 0$:
\begingroup
\setlength{\leftmarginii}{13pt}
\begin{itemize}
\item[--]
\vspace{-4pt}
We even can assume that $\Psi(\lambda\cdot \g)\neq\pm\me$ since elsewise we replace $\lambda$ by $\lambda/2$.
Then, for $v:=\murs^{-1}(\s_\beta)$ we find open neighbourhoods
\begin{align*}
V\subseteq S^2\:\text{ with }\:v\in V \qquad J\subseteq (0,\pi)\sqcup (\pi,2\pi)\:\text{ with }\:\phi(\lambda) \in
\end{align*}
as well as $ U\subseteq \SU$ with
$\Psi(\lambda\cdot \g)\in U$,
such that $f':=f|_{V\times J}$ is a diffeomorphism to $U$.
In fact, since $t:=\phi(\lambda)\in (0,\pi)\sqcup (\pi,2\pi)$, i.e., $\sin(t)\neq 0$,
\begin{align*}
0=\dd_{(t,v)} f((\Delta t,\Delta v) )=\Delta t \:[-\sin(t)\cdot\me+\cos(t)\cdot\murs(v)] +\sin(t)\cdot \murs(\Delta v)
\end{align*}
implies $\Delta t=0$, hence $\Delta v=0$.
\item[--]
Let $\iota_0\in J$ be such that $\Psi_\iota(\lambda)\in U$ holds for all $\iota\geq \iota_0$. Then, for such $\iota\geq \iota_0$ we have
\begin{align*}
\s_{\beta_\iota}= q_\iota\cdot \big(\:\murs\cp \pr_1\cp{f'}^{-1}\big)(\Psi_\iota(\lambda\cdot \g)) \qquad\phi_\iota(\lambda)= 2\pi p_\iota + q_\iota\cdot \big(\hspace{1pt}\pr_2\cp{f'}^{-1}\big)(\Psi_\iota(\lambda\cdot \g))
\end{align*}
for $q_\iota\in \{-1,1\}$ uniquely determined and $p_\iota=0$ for $q_\iota=1$ as well as $p_\iota=1$ for $q_\iota=-1$.
Since the occurring maps are continuous and
\begin{align*}
\s_{\beta}= \big(\murs\cp \pr_1\cp{f'}^{-1}\big)(\Psi(\lambda\cdot \g)) \qquad\qquad \phi(\lambda)=\big(\pr_2\cp{f'}^{-1}\big)(\Psi(\lambda\cdot \g))
\end{align*}
holds,
we have
\begin{align*}
\s_\beta= \lim_\iota q_\iota\cdot \s_{\beta_\iota}\qquad\text{and}\qquad \phi(\lambda)=\lim_\iota q_\iota \phi_\iota(\lambda)+ 2\pi p_\iota.
\end{align*}
Consequently, in the \textbf{Type 3} case (analogously in the \textbf{Type 4} case) with $\mm$ as in \eqref{eq:hmnejhedrffds}, we obtain
\begin{align}
\label{eq:xiconv}
\begin{split}
\xi_\g^p(\beta)&=\LM \big(\murs^{-1}(s_\beta)\big)=\LM\Big(\murs^{-1}\Big(\lim_\iota q_\iota\cdot s_{\beta_\iota}\Big)\Big)\\
&=\lim_\iota q_\iota \cdot \LM \big(\murs^{-1}(s_{\beta_\iota})\big)=\lim_\iota q_\iota\cdot\xi_\g^p(\beta_\iota).
\end{split}
\end{align}
\end{itemize}
\endgroup
\noindent
Then, for each further $\lambda'>0$ with $\Psi(\lambda'\cdot \g)\neq \pm \me$ we have $\s_\beta= \lim_\iota q'_\iota\cdot \s_{\beta_\iota}$
for $q_\iota'\in \{-1,1\}$ uniquely determined as well. So, since $\s_\beta= \lim_\iota q_\iota\cdot \s_{\beta_\iota}$, we find $\iota_0\in J$ such that $|q_\iota-q_\iota'|<2$ holds for all $\iota\geq \iota_0$, hence $q_i=q_i'$ for all $\iota\geq \iota_0$.
Consequently,
\begin{align}
\begin{split}
\label{eq:psiconv}
\psi(\chi_{\lambda'})&=\e^{\I \phi(|\lambda'|)}= \e^{\I \lim_\iota [q_\iota \phi_\iota(|\lambda'|)+2\pi p_\iota]}\\
&=\lim_\iota\e^{\I\: [q_\iota\!\phi_\iota(|\lambda'|)+2\pi p_\iota]}=\lim_\iota\e^{\I\: q_\iota\!\phi_\iota(|\lambda'|)}=\lim_\iota \psi_\iota(\chi_{\lambda'})^{ q_\iota}.
\end{split}
\end{align}
Moreover, if $\Psi(\lambda')=-\me$, then $\Psi(\lambda'/2)\neq \pm \me$ and
\begin{align*}
\lim_\iota\psi_\iota(\chi_{\lambda'})^{q_\iota}=\big[\lim_\iota\psi_\iota(\chi_{\lambda'/2})^{q_\iota}\big]^{2}\stackrel{\eqref{eq:psiconv}}{=}\psi(\chi_{\lambda'}).
\end{align*}
Finally, if $\Psi(\lambda')=\me$, then the same arguments as in the $\Psi=\me$ case show that $\lim_\iota \psi_\iota(\lambda')= 1=\psi(\lambda')$, hence $\lim_\iota \psi_\iota(\lambda')^{q_\iota}= 1=\psi(\lambda')$ holds as well.
Consequently, $\psi(\chi_{\lambda'})=\lim_\iota \psi_\iota(\chi_{\lambda'})^{ q_\iota}$ holds for all $\lambda'>0$, so that
by the definition of the topology on $\RB$, we have $\lim_\iota \psi_\iota^{q_\iota}= \psi$. Hence,
\begin{align*}
\lim_\iota \tau_\g^p(\Psi_\iota)&=\lim_\iota\left[\left(\psi_\iota, \xi_\g^p(\beta_\iota)\right)\right]=\lim_\iota\left[\left(\psi_\iota^{q_\iota},q_\iota \cdot\xi_\g^p(\beta_\iota)\right)\right]\\
&=\left[\lim_\iota\left(\psi_\iota^{q_\iota},q_\iota \cdot\xi_\g^p(\beta_\iota)\right)\right]\stackrel{\eqref{eq:xiconv}}{=}\big[(\psi,\xi_\g^p(\beta))\big],
\end{align*}
where in the third step we have used continuity of the projection map $\pr_0$ and in the second one that $(\psi,v)\sim (\psi^{-1},-v)$ holds.
\end{enumerate}
\end{proof}
To this point, we have identified the spaces $Y_\g^p$ with the spaces $X_\g^p$, on each of which we have defined normalized Radon measure with suitable invariance properties. So, we now are ready to identify $\IHOMLAS$ with a respective Tychonoff product, and to define a normalized Radon measure on this space. To this end, we simplify the notations by defining
\begin{align*}
\mu_{\m,\alpha}:=\mu_{\g_{\m,\alpha}}^{p_\m}\qquad\qquad X_{\m,\alpha}:=X_{\g_{\m,\alpha}}^{p_\m}\qquad\qquad\tau_{\m,\alpha}:=\tau_{\g_{\m,\alpha}}^{p_\m
\end{align*}
for all $\m\in \Mm$ and all $\alpha\in I_\m$.
\begin{definition}[The normalized Radon measure $\mLAS$]
\begingroup
\setlength{\leftmargini}{17pt}
\begin{itemize}
\item
We equip
\begin{align*}
X:=\prod_{\m\in \Mm,\alpha\in I_\m}X_{\m,\alpha
\end{align*}
with the Radon product
(see Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa3}) of the normalized Radon measures $\mu_{\m,\alpha}$
from Lemma and Definition \ref{lemdef:projspaces}.\ref{lemdef:projspaces5}.
\item
By \gls{mLAS} we denote the normalized Radon measure on $\IHOMLAS$ carried over from $X$ by the homeomorphism
\begin{align*}
\eta_X:=\Xi_X \cp \Pi_Y\colon \IHOMLAS\rightarrow X
\end{align*}
with
\begin{align*}
\Xi_X:=\prod_{\m\in \Mm,\alpha\in I_\m}\tau_{\m,\alpha}\colon Y\rightarrow X
\end{align*}
the product of the maps $\tau_{\m,\alpha}\colon Y_{\m,\alpha}\rightarrow X_{\m,\alpha}$ from Lemma and Definition \ref{lemdef:projspaces}.\ref{lemdef:projspaces4}.\hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\end{definition}
Before we come to the independence of $\mLAS$ from the choices we have made, we now calculate the space $\IHOMLAS$ for our three standard LQC situations from Example \ref{ex:LQC}.
\begin{example}[Loop Quantum Cosmology]
\label{ex:cosmoliealgmaasse}
Assume that we are in the situation of Example \ref{ex:LQC}, where $P=\RR^3\times \SU$. We now determine the space $X$ for the case of (semi-)homogeneous, spherically symmetric and homogeneous isotropic LQC. We start with
\par
\begingroup
\leftskip=8pt
\vspace{8pt}
\noindent
{\bf\textit{(Semi)-Homogeneous LQC:}}
\vspace{2pt}
\noindent
We even can assume that $P$ is an arbitrary $\SU$-bundle and that
$\wm$ acts transitively and free in homogeneous case, or just free in the semi-homogeneous one.
\begingroup
\setlength{\leftmargini}{25pt}
\begin{itemize}
\item
If we are in the homogeneous case,
$\Mm$ is singleton and $G_x=\{e\}$ for all $\g\in \mg$. Consequently, each such $\g\in \mg\backslash \mg_x$ is stable by Lemma and Remark \ref{rem:dfggfg}.\ref{rem:dfggfg2}.
Let $I:=I_\m:=\Sp\mg$ and $\g_\alpha\in \alpha$ for each $\alpha\in I$. Then, $\{\g_\alpha\}_{\alpha\in I}$ is obviously complete, and independent by Lemma \ref{lemma:sim}.\ref{lemma:sim4} (see also Remark \ref{lemremmgohnermgx}).
It follows that
\begin{align*}
\IHOMLAS\cong Y=\prod_{\alpha\in I}Y_{\m,\alpha} \cong X=\Big[\RB\wti S^2\Big]^{|\Sp\mg|}
\end{align*}
because $J(\g,p)=\{0\}\sqcup \Sp\ms$ holds for all $p\in P$ and all $0\neq \g\in \mg$. This is clear from $G^x_{[\g]}=\{e\}$ since then $\Psi\colon \lambda \cdot \g \mapsto \exp(\lambda\cdot \s)$ is an element of $Y_\g^p$ for all $\s\in \su$.
\item
In the semi-homogeneous case, i.e., if $\wm$ acts not transitively but free, the same arguments show that we have
\begin{align*}
\IHOMLAS\cong \Big[\RB\wti S^2\Big]^{|\Mm\times \Sp\mg|}.
\end{align*}
\end{itemize}
\endgroup
\vspace{8pt}
\noindent
{\bf\textit{Spherically Symmetric LQC:}}
\vspace{2pt}
\noindent
We now consider the second case in Example \ref{ex:LQC}. Here,
$\Mm$ can be parametrized by the positive $x$-axis $\vec{A}=\{\lambda \cdot\vec{e}_1 \:|\: \lambda>0\}$ as we can ignore the origin\footnote{This is in contrast to the classical (smooth) situation, where it usually makes a big difference whether one takes singular orbits (such as here the origin) into account or not.
Indeed, as we will see in Example \ref{bsp:Rotats} (where we calculate the set of smooth spherically symmetric connections explicitly) this is exactly the case in spherically symmetric LQC.
} because its stabilizer is the whole group, i.e. $\mg\backslash \mg_0=\emptyset$.
Then, $G_{x}=H_{\tau_1}$ and $\mg_x=\Span_\RR(\tau_1)$ holds for all $x\in \vec{A}$. For each $\m \in \Mm$ we denote by $x_\m\in \m$ the unique representative contained $\vec{A}$, i.e., the only element of $\m\cap \vec{A}$. Finally, we let $p_\m:=(x_\m,e)$ for all $\m\in \Mm$.
\vspace{6pt}
\noindent
We claim that for each $x\in \vec{A}$ the space $\mathfrak{G}_{x}$ is parametrized by the angles in $(0,\pi/2]$:
\begingroup
\setlength{\leftmargini}{25pt}
\begin{itemize}
\item
For each $\g\in \mg\backslash \mg_x$, $\mathrm{im}\big[\gamma_\g^x\big]$ is the circle in $\RR^3$ which arises from rotating $x$ around the axis through the origin determined by $\murs^{-1}(\g)$. In particular, if $\g'\in \mg\backslash \mg_x$ is a further element, then we either have
\begin{align*}
\mathrm{im}\big[\gamma_\g^x\big]=\mathrm{im}\big[\gamma_{\g'}^x\big]\qquad\Longrightarrow\qquad \g'=\pm\g
\end{align*}
or $\mathrm{im}\big[\gamma_\g^x\big]\cap\mathrm{im}\big[\gamma_{\g'}^x\big]$ consists of at most two points.
\item
We consider the family $\{\g_{\alpha}\}_{\alpha\in (0,\pi)}\subseteq \mg\backslash\mg_x=\su\backslash \Span_\RR(\tau_1)$ of elements
\begin{align*}
\g_{\alpha}:=\cos(\alpha)\cdot \tau_1 +\sin(\alpha)\cdot\tau_2\quad\text{for}\quad 0<\alpha<\pi.
\end{align*}
These elements are stable because by the first point
$\gamma_{\g_\alpha}^x|_{[0,l]}\psim\gamma_{\pm\Ad_h(\g_\alpha)}^x|_{[0,l']}$ already implies that $\Ad_h(\g_\alpha)=\pm \g_\alpha$ holds.
\item
Since $\Ad\colon G_{x}\times \mg\backslash \mg_x\rightarrow \mg\backslash \mg_x$ for $G_x=H_{\tau_1}$ acts on $\mg=\su\cong \RR^3$ via rotations around the $\tau_1$-axis, the above family is complete. In fact, for each $\g\in \mg\backslash \mg_x=\su\backslash \spann_\RR(\tau_1)$ we find $\alpha\in (0,\pi)$, $\lambda \neq 0$ and $h\in H_{\tau_1}$ such that $\g= \lambda\Ad_h(\g_\alpha)$, i.e., $\g\xsim \g_\alpha$ holds.
\item
The family $\{\g_{\alpha}\}_{\alpha\in (0,\pi)}$ is not independent because
\begin{align*}
\Ad_h(\g_{\pi/2+ \epsilon})=-\g_{\pi/2- \epsilon} \qquad\Longrightarrow\qquad\g_{\pi/2+ \epsilon} \xsim \g_{\pi/2-\epsilon}\qquad\forall\: 0< \epsilon < \pi/2
\end{align*}
if $h\in H_{\tau_1}$ corresponds to a rotation by the angle $\pi$.
\item
So, replacing the above family by $\{\g_{\alpha}\}_{\alpha\in (0,\pi/2]}$ does not change completeness, and we even have independence. In fact, if $\alpha,\beta \in (0,\pi/2]$ with $\g_\alpha \xsim \g_\beta$, then Lemma \ref{lemma:completee}.\ref{it:completee1} shows
\begin{align*}
\gamma_{\g_\alpha}^x|_{[0,l]}\psim \gamma_{\pm \Ad_h(\g_\beta)}^x|_{[0,l']}\qquad\text{for some}\qquad h\in G_x.
\end{align*}
Hence, $\g_\alpha = \pm \Ad_h(\g_\beta)$ by the first point, just because $\mathrm{im}\big[\gamma_{\g_\alpha}^x\big]\cap\mathrm{im}\big[\gamma_{\pm \Ad_h(\g_\beta)}^x\big]$ is infinite. So, since $\Ad_h$ rotates in $\su\cong \RR^3$ around the $\tau_1$-axis, it is clear by construction
that $h=e$ and $\g_\alpha=\g_\beta$ must hold.
\end{itemize}
\endgroup
\noindent
It remains to determine the type of $J(\g_\alpha,p_\m)$ for all $\alpha\in (0,\pi/2]$ and all $\m\in \Mm$. We will show that $J(\g_\alpha,p_\m)$ is of {\bf Type 4} if $\alpha\in (0,\pi/2)$, and of {\bf Type 3} if $\alpha=\pi/2$, so that
\begin{align*}
\IHOMLAS\cong \Big[\RB\wti S^1\Big]^{|\RR_{>0}|}\times \Big[\RB\wti S^2\Big]^{|(0,\pi/2)\times \RR_{>0}|}
\end{align*}
holds. In fact, let $x\in \vec{A}$ be as above. Then, since the non-trivial elements $h\in G_{x}=H_{\tau_1}$ rotate $\g_\alpha$ in $\su\cong \RR^3$ w.r.t.\ to an angle in $(0,2\pi)$ around the $\tau_1$-axis, it is clear that $\Ad_h(\g_\alpha)$ can only be equal to $\pm \g_\alpha$ if $h=e$ or
\begin{align*}
\alpha=\pi/2\qquad \text{and}\qquad h=\pm\exp(\textstyle\frac{\pi}{2}\tau_1)\qquad \text{with}\qquad\Ad_h(\g_{\pi/2})=-\g_{\pi/2},
\end{align*}
whereby $h$ corresponds to a rotation by the angle $\pi$.
Consequently,
\begingroup
\setlength{\leftmargini}{25pt}
\begin{itemize}
\item
We have $G_{[\g_\alpha]}^{x}=\{e\}$ if $\alpha\in (0,\pi/2)$, so that in this case $J(\g_\alpha,p_\m)$ is of {\bf Type 4} for all $\m\in \Mm$.
\item
We have $G_{[\g_{\pi/2}]}^x=\{e,\pm\exp(\textstyle\frac{\pi}{2}\tau_1)\}$. So, since the map $\fiba_{p_\m}\colon \SU \rightarrow \SU$ equals $\id_\SU$, for each $\m\in \Mm$ and $h=\pm\exp(\textstyle\frac{\pi}{2}\tau_1)$ the equivariance of a map $\Psi\colon \spann_\RR(\g_{\pi/2})\rightarrow \SU$ just reads (for $h=e$ equivariance gives no conditions on $\mathrm{im}[\Psi]$)
\begin{align*}
\Psi\big(\lambda\cdot\g_{\pi/2}\big)^{-1}=\Psi(\lambda\cdot\Ad_h(\g_{\pi/2}))=\alpha_{\fiba_{p_\m}(h)}(\Psi(\lambda\cdot\g_{\pi/2}))=\alpha_h(\Psi(\lambda\cdot\g_{\pi/2}))\qquad\forall\: \lambda\in \RR.
\end{align*}
So, by Lemma \ref{lemma:torus}.\ref{lemma:torus2} it should be clear that\footnote{The maps $\Psi_{\vec{n}}\colon \lambda \cdot \g_{\pi/2}\mapsto \exp(\lambda\cdot \murs(\vec{n}))$ for $\vec{n}\in \RR^3\backslash \{0\}$ with $\langle\vec{n},\vec{e}_1\rangle=0$ are all non-trivial and $\Ad_{G_{[\g_{\pi/2}]}}^{p_\m}$-equivariant, see e.g.\ Lemma \ref{lemma:torus}.\ref{lemma:torus2}.}
\begin{align*}
J(\g_{\pi/2},p_\m)=\{0\}\sqcup \textstyle\bigcup_{\vec{n}\in \RR^3\backslash\{0\} \colon\langle\vec{n},\vec{e}_1\rangle=0}\:[\hspace{1.5pt}\murs(\vec{n})\hspace{0.5pt}]
\end{align*}
is of {\bf Type 3} for all $\m\in \Mm$.
\end{itemize}
\endgroup
\noindent
\vspace{8pt}
\noindent
{\bf\textit{Homogeneous Isotropic LQC:}}
\vspace{2pt}
\noindent
We now consider the first case in Example \ref{ex:LQC}. Since $\wm$ is transitive, $\Mm$ is singleton, and we choose $p=(0,\me)$ so that $x=0$ as well as $\{0\}\times \SU=G_x\subseteq \Gee$ holds.
Observe that for $(\vec{v},\s)\in \mg= \RR^3\times\su$ and $h=(0,\sigma)\in \{0\}\times \SU=G_x$, we have
\begin{align}
\begin{split}
\label{conjuuuuu}
\Ad_h((\vec{v},\s))&=\dttB{t}{0}\alpha_h((t \vec{v},\exp(t \s)))\\
& =\dttB{t}{0} (0,\sigma)\cdot_\varrho (t \vec{v},\exp(t \s)) \cdot_\varrho (0,\sigma)^{-1}\\
&= \dttB{t}{0} (t \varrho(\sigma)(\vec{v}),\sigma\cdot \exp(t\cdot \s))
\cdot_\varrho (0,\sigma^{-1})\\
&=\dttB{t}{0} (t\varrho(\sigma)(\vec{v}),\alpha_\sigma(\exp(t\cdot \s)))\\
&=(\varrho(\sigma)(\vec{v}),\Ad_\sigma(\s))
\end{split}
\end{align}
for $\Ad_\sigma$ the differential of the conjugation by $\sigma$ in $\SU$.
\vspace{5pt}
\noindent
We obtain an independent and complete family of elements of
\begin{align*}
\mg\backslash \mg_x=\big[\RR^3\times \su\big]\backslash \big[\{0\}\times \su\big]=\big[\RR^3\backslash\{0\}\big] \times \su
\end{align*}
as follows.
We fix $\vec{v},\vec{v}_{\perp}\in \RR^3\backslash\{0\}$ orthogonal to each other and normalized, and define $\s_0:=\murs(\vec{v})$ as well as $\s_{\perp}:=\murs(\vec{v}_\perp)$. Moreover, let
\begin{align*}
\Span_\RR(\s_0,\s_\perp)\supseteq E_\geq&:=\{\lambda_1 \s_0 +\lambda_2 \s_\perp\: |\: \lambda_1 \in \RR,\lambda_2 \geq 0\}\\
\Span_\RR(\s_0,\s_\perp)\supseteq E_>&:=\{\lambda_1 \s_0 +\lambda_2 \s_\perp\: |\: \lambda_1 \in \RR,\lambda_2 > 0\}\\
\Span_\RR(\s_0,\s_\perp)\supseteq E_0\hspace{2pt}&:=E_> \sqcup \{0\}
\end{align*}
denote the upper and the positive upper half plane determined by $\s_0$ and $\s_\perp$, as well as
$E_0$ the union of the positive upper half plane with the origin. It is shown in Appendix \ref{subsec:Aclacula} that the family $\{(\vec{v},\s)\}_{\s\in E_0}$ is
independent and complete (and consists of stable elements).
\vspace{6pt}
\noindent
We now calculate $J((\vec{v},\s),p)$ for all $\s\in E_0$. For simplicity, here we will assume that $\vec{v}=\vec{e}_1$ and $\vec{v}_\perp=\vec{e}_2$ holds, hence $\s_0=\tau_1$ and $\s_\perp=\tau_2$.
\begingroup
\setlength{\leftmargini}{25pt}
\begin{itemize}
\item
Let $\Psi\colon \spann_\RR((\vec{v},\s))\rightarrow \SU$ be $\Ad^p_{G_{[(\vec{v},\s)]}}$-equivariant and
\begin{align*}
h=(0,\sigma)\in G^x_{[(\vec{v},\s)]}\subseteq G_x = \{0\}\times \SU.
\end{align*}
Then, since $\fiba_{p}\colon \Gee \rightarrow \SU$ is just given by $\pr_{\SU}$, equivariance of $\Psi$ reads
\begin{align}
\label{equivariancebbbb}
\Psi(\lambda\cdot(\varrho(\sigma)(\vec{v}),\Ad_\sigma(\s)))\stackrel{\eqref{conjuuuuu}}{=}\Psi(\Ad_h(\lambda\cdot(\vec{v},\s)))
=\alpha_\sigma(\Psi(\lambda\cdot(\vec{v},\s)))\qquad\forall\:\lambda\in \RR.
\end{align}
\item
If $\s=0$, then $\{0\}\times H_{\vec{v}}\subseteq G^x_{[(\vec{v},0)]}$
because
\begin{align*}
\Ad_h((\vec{v},0))\stackrel{\eqref{conjuuuuu}}{=}(\varrho(\sigma)(\vec{v}),0)=(\vec{v},0)\qquad \forall\: h=(0,\sigma)\in \{0\}\times H_{\vec{v}}.
\end{align*}
Combining this with \eqref{equivariancebbbb}, we obtain $\Psi((\vec{v},0))=\alpha_{\sigma}(\Psi(\vec{v},0))$ for all $\sigma\in H_{\vec{v}}$,
hence $\mathrm{im}[\Psi]\subseteq H_{\vec{v}}$
by Lemma \ref{lemma:torus}.\ref{lemma:torus1}. This already shows that $J((\vec{v},0),p)$ is either of {\bf Type 1} or of {\bf Type 2}.
However, $J((\vec{v},0),p)=\{0,[\murs(\vec{v})]\}=\{0,[\s_0]\}$ is of {\bf Type 2} because $\Psi\colon \lambda\cdot( \vec{v},0)\mapsto \exp(\lambda\cdot \murs(\vec{v}))$ is $\Ad^p_{G_{[(\vec{v},0)]}}$-equivariant and non-trivial. In fact, if $h=(0,\sigma)\in G^x_{[(\vec{v},0)]}$ is arbitrary, then
\begin{align*}
(\varrho(\sigma)(\vec{v}),0)\stackrel{\eqref{conjuuuuu}}{=}\Ad_h((\vec{v},0))=(\pm \vec{v},0)
\end{align*}
by Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve4}, so that $\varrho(\sigma)(\vec{v})=\pm \vec{v}$ holds. Consequently, for each $\lambda \in \RR$ we have
\begin{align*}
\Psi(\Ad_h(\lambda\cdot( \vec{v},0)))&=\Psi((\pm\lambda\cdot \vec{v},0))=\exp(\lambda\cdot\murs(\varrho(\sigma)(\vec{v})))\\
&=\exp(\lambda\cdot\Ad_\sigma(\murs(\vec{v})))=\alpha_\sigma(\Psi(\lambda\cdot (\vec{v},0)))=\alpha_{\fiba_p(h)}(\Psi(\lambda\cdot (\vec{v},0))).
\end{align*}
\item
If $\s\neq 0$, then $G^x_{[(\vec{v},\s)]}=\{e,\pm \exp(\frac{\pi}{2} \tau_3)\}$. In fact, by Remark and Definition \ref{rem:ppropercurve}.\ref{rem:ppropercurve4} for
\begin{align*}
h=(0,\sigma)\in G^x_{[(\vec{v},\s)]}\subseteq G_x=\{0\}\times \SU
\end{align*}
we have $\Ad_h((\vec{v},\s))=\pm (\vec{v},\s)$, whereby the positive case, i.e., $\varrho(\sigma)(\vec{v})=\vec{v}$
can only occur if $\sigma=e$. This is because here we must have $\sigma\in H_{\vec{v}}=H_{\tau_1}$, so that $\Ad_\sigma$ rotates in $\su\cong \RR^3$ around the $\tau_1$-axis. By the definition of the set $E_>$, then it is clear that $\Ad_\sigma(\s)=\s$ can only hold for $\sigma=e$. Now, if $\Ad_h((\vec{v},\s))=-(\vec{v},\s)$, then
$\alpha_\sigma(\exp(\lambda\cdot\s))=\exp(\lambda\cdot\Ad_\sigma(\s))=\exp(\lambda\cdot\s)^{-1}$ for all $\lambda\in \RR$,
so that Lemma \ref{lemma:torus}.\ref{lemma:torus2} already shows that $\sigma=\pm \exp(\frac{\pi}{2} \tau_3)$ holds. This is also in line with $\varrho(\sigma)(\vec{v})=-\vec{v}$, and as
in the previous point it is now clear that
\begin{align*}
J((\vec{v},\s),p)=\{0\}\sqcup \textstyle\bigcup_{\vec{n}\in \RR^3\backslash\{0\} \colon\langle\vec{n},\vec{e}_3\rangle=0}\:[\hspace{1.5pt}\murs(\vec{n})\hspace{0.5pt}]
\end{align*}
is of {\bf Type 3}.
\end{itemize}
\endgroup
\noindent
It follows that
\begin{align}
\label{eq:wichgle}
\IHOMLAS\cong \RB \times \Big[\RB \wti S^1\Big]^{|E_>|}
\end{align}
Here the first factor $\RB$ corresponds to $(\vec{v},0)\in \RR^3\times \su$ and determines the image of an invariant homomorphism on the set of linear curves $\Pal$. For this, recall that
each such curve is equivalent to $\wm_g\cp\gamma^x_{(\vec{v},0)}|_{[0,l]}$ for some $l>0$ and $g\in G$. Then,
\begingroup
\setlength{\leftmargini}{13pt}
\begin{itemize}
\item[$\triangleright$]
Performing the constructions of this subsection for the set $\Pal$ instead of $\Pags$ provides us with the homeomorphism
\begin{align*}
\eta_\lin \colon \IHOMLL\rightarrow \RB
\end{align*}
for which $\eta_\lin(\mu_\lin)=\muB$ holds with $\mu_\lin$ the respective Radon measure on $\IHOMLL$. In particular, using Remark \ref{rem:unednlichfubini}, it is not hard to see that then $\mu_\lin$ is the push forward of $\mLAS$ by the restriction map (see Remark \ref{rem:restriction})
\begin{align*}
\res_{\mg\lin}\colon \IHOMLAS\rightarrow \IHOMLL,\quad
\homm \mapsto \homm|_{\Pal}.
\end{align*}
This has also consequences for standard homogeneous isotropic LQC being discussed in Remark \ref{StanLQC}.
\hspace*{\fill}$\lozenge$
\end{itemize}
\endgroup
\endgroup
\par
\end{example}
\begin{remark}[Standard Homogeneous Isotropic LQC]
\label{StanLQC}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item
\label{StanLQC1}
The standard quantum configuration space of homogeneous isotropic LQC \cite{MathStrucLQG} is given by $\ARRQLL$, i.e., the spectrum of the restriction $\Cstar$-algebra $\ovl{\PaC_\lin|_{\AR}}$. Here, $\PaC_\lin$ denotes the $\Cstar$-algebra of cylindrical functions that correspond to the set $\Pal$ of linear curves in $\RR^3$. By \eqref{eq:homis} we have $\AR \cong \RR$, and by Lemma \ref{lemma:separating} the map $\iota_\Con\colon \Con \rightarrow \A_\lin$ is injective because the functions in $\PaC_\lin$ separate the points in $\Con$. The latter statement means that $\A_\lin$ is a reasonable quantum configuration space because $\Con$ is naturally embedded therein. Now, $\Pal$ is obviously $\Pe$-invariant, so that by Corollary \ref{cor:CylSpecAction} the quantum-reduced space $\AQRL\subseteq \A_\lin$ exists, is homeomorphic to $\IHOMLL$, and is physically meaningful as well.
Now, we have $\ovl{\PaC_\lin|_{\AR}}=\CAP(\RR)$, hence $\ARRQLL=\RB$.
In fact, for $\g:=(\vec{v},0)\in \RR^3 \times \su$ we have $\exp(t\cdot \g)=(t\cdot \vec{v},\me)$, so that by \eqref{eq:trivpar} the horizontal lift of $\gamma_l:=\gamma_\g^x|_{[0,l]}$ in $p=(x,e)$ w.r.t.\ $\w^c$ is just given by
\begin{align}
\label{eq:horliftlin}
\begin{split}
\wt{\gamma}_l(t)&=\Pe((t\cdot\vec{v},\me),(x,\me))\cdot \exp(- t\cdot \w^c(\:\wt{g}((x,\me))))\\
&=(x+t\cdot \vec{v},\me)\cdot \exp(-c t \cdot\murs(\vec{v}))=(x+t\cdot\vec{v}, \exp(-c t \cdot\murs(\vec{v}))).
\end{split}
\end{align}
Consequently, for $\nu$ the standard choice $\nu_x=(x,\me)$ for all $x\in M$, we have
\begin{align}
\label{eq:patralinc}
h_\gamma^\nu(\w^c)= \exp(-c l \cdot\murs(\vec{v}))\stackrel{\eqref{eq:expSU2}}{=} \cos(cl\|\vec{v}\|)\cdot\me - \sin(cl\|\vec{v}\|) \cdot\murs(\vec{v}/\|\vec{v}\|),
\end{align}
so that by \eqref{eq:expSU2} the $\Cstar$-algebra $\PaC_\lin$ is generated by the constant function $1$ and the functions $t\mapsto \sin(l t)$, $t\mapsto \cos(l t)$ for all $l\neq 0$. Hence, by the characters $\chi_l$ for $l\in \RR$. Consequently, $\PaC_\lin=\CAP(\RR)$ and $\ARRQLL=\mathrm{Spec}(\CAP(\RR))=\RB$.
\item
\label{StanLQC2}
As we have seen in the end of Example \ref{ex:cosmoliealgmaasse},
$\AQRL\cong \IHOMLL$ is homeomorphic to $\RB$ as well.
A straightforward calculation now shows that the composition
\begin{align*}
\eta_\lin \cp \kappa_\lin \cp \ovl{i^*_\AR}\colon \RB\cong\ARRQLL\rightarrow \RB
\end{align*}
is even the identity $\id_{\RB}$ on $\RB$.
\begin{figure}[h]
\begin{minipage}[h]{\textwidth}
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
\RB \cong \ARRQLL \ar@{->}[r]^-{\ovl{i^*_\AR}}_-{\cong} & \: \ARQLL \: \ar@{->}[r]^-{\subseteq} & \AQRL\ar@{->}[r]_-{\cong}^-{\kappa_\lin} &\IHOMLL \ar@{->}[r]_-{\cong}^-{\eta_\lin}& \RB.
}
\end{xy}}
\end{center}
\end{minipage}
\end{figure}
\FloatBarrier
Here,
$\ovl{i^*_\AR}$
is the identification from \eqref{eq:inclusionsdiag} and $\kappa_\lin$
the respective homeomorphism from Subsection \ref{subsec:InvHoms}.
In particular, this means that $\ARRQLL\cong \AQRL$ holds, i.e., that in the homogeneous isotropic case quantization and reduction commutes if one restricts to linear curves.
Since our construction provide us with the Haar measure $\muB$ on $\RB$, being used for the definition of the standard LQC kinematical Hilbert space $L^2(\RB,\muB)$ \cite{MathStrucLQG}, we have reproduced this Hilbert space by performing a reduction on quantum level.
Finally, observe that we now can easily embed the traditional LQC configuration space
\begin{align*}
\ARRQLL\cong \ARQLL\cong \IHOMLL \stackrel{(*)}{\cong} \ITRHOML
\end{align*}
into the quantum-reduced space $\IHOMW\stackrel{(**)}{\cong} \ITRHOMW$ via the map
\begin{align*}
\iota_\lin\colon \ITRHOML&\rightarrow \ITRHOMW,
\end{align*}
defined by
\begin{align*}
\iota_\lin(\hommm)(\gamma):=\homm(\gamma)\:\text{ if }\:\gamma\in \Pal\qquad\text{and}\qquad\iota_\lin(\hommm)(\gamma):=\me\:\text{ for }\:\gamma\in \Paw\backslash \Pal.
\end{align*}
If we use the standard choice $\nu_x=(x,\me)$ for all $x\in \RR^3$ for the identifications $(*)$ and $(**)$, on the level of the spaces $\IHOMLL$ and $\IHOMW$ this just mean to assign to $\homm\in \ITRHOML$ the element $\homm'\in \IHOMW$ with $\homm'(\gamma)=\homm(\gamma)$ for $\in \Pal$ and
\begin{align*}
\homm'(\gamma)((\gamma(a),s)):=(\gamma(b),s)\qquad \forall\:s\in \SU\quad\text{and}\quad \Paw\backslash \Pal\ni \gamma\colon [a,b]\rightarrow \RR^3.
\end{align*}
So, following this approach, there are many ways to embed $\ARRQLL$ into $\IHOMW$, whereby (at least from the mathematical point of view) using the standard choice of $\nu$ seems to be the most natural one. Its physical relevance, however, might first become clear once the dynamics of the quantum reduced theory has been successfully established.
\item
Using the identification of $\IHOMLL$ with $\ITRHOML$ from Remark \ref{rem:homomorphbed}.\ref{rem:euklrem5} via the standard choice $\nu_x=(x,e)$ for all $x\in \RR^3$, one finds that the continuous group structure on $\RB\cong \ITRHOML$ corresponds to the group structure on $\ITRHOML$ defined by
\begin{align*}
(\hommm_1*\hommm_2)(\gamma):=\hommm_1(\gamma) \hommm_2(\gamma)\qquad\qquad \hommm^{-1}(\gamma):=\hommm(\gamma)^{-1}\qquad\qquad e(\gamma):=\me
\end{align*}
for all $\gamma\in \Pal$.
These operations are well defined because for each fixed $\vec{v}\in \RR^3\backslash\{0\}$ we have\footnote{For the definition of the curves $\gamma_{\vec{v},l}$ See (a) in Convention \ref{conv:sutwo1}.\ref{conv:sutwo2}.} $\hommm(x+\gamma_{\vec{v},l})\in H_{\vec{v}}$ for all $\hommm \in \ITRHOML$, i.e., $[\hommm_1(\gamma),\hommm_2(\gamma)]=0$ for all $\gamma\in \Pal$ and all $\hommm_1,\hommm_2 \in\ITRHOML$. This is clear from Lemma \ref{lemma:torus}.\ref{lemma:torus1} because
\begin{align*}
\hommm(x+\gamma_{\vec{v},l})=\hommm(\gamma_{\vec{v},l})=\hommm(\sigma(\gamma_{\vec{v},l}))=\alpha_\sigma(\hommm(\gamma_{\vec{v},l}))=\alpha_\sigma(\hommm(x+\gamma_{\vec{v},l}))\qquad \forall\: \sigma\in H_{\vec{v}}
\end{align*}
by \eqref{eq:algrels}. However, $[\hommm_1(\gamma),\hommm_2(\gamma)]=0$ usually\footnote{Choose, e.g., $\gamma=\gamma_\g^0$ for $\g= (\vec{v},\s)$ with $\s\notin \Span_\RR(\murs(\vec{v}))$ and use that $J(\g,(0,\me))$ is of {\bf Type 3} in this case, see third part of Example \ref{ex:cosmoliealgmaasse}.} does not hold for all $\hommm_1,\hommm_2\in \ITRHOMW$ if $\gamma\notin \Pal$, so that one cannot define the same group structure, e.g., on $\ITRHOMW$.
\end{enumerate}
\endgroup
\end{remark}
\subsubsection{Independence from the Choices}
In this final subsection, we show that the definition of the measure $\mLAS$ does not depend on any choices we have made. We start with simplifying the notations.
We consider the index set $I:=\{[\m,\alpha]\:|\: \m\in \Mm, \alpha\in I_\m\}$ and define for each $\iota=[\m,\alpha]\in I$:\footnote{Recall Lemma and Definition \ref{def:topo} as well as Lemma and Definition \ref{lemdef:projspaces}.}
\begin{align*}
\g_\iota:=\g_{\m,\alpha}\qquad\qquad \Eq_\iota:=\Eq_{p_\m}\qquad\qquad Y_\iota:=Y_{\g_{\m,\alpha}}^{p_\m}\qquad\qquad X_\iota:=X_{\g_{\m,\alpha}}^{p_\m},
\end{align*}
the measure $\mu_\iota:=\mu_{\g_{\m,\alpha}}^{p_\m}$,
as well as the maps
\begin{align*}
\pip_\iota&:=\pip_{p_\m}\colon \IHOMLAS\rightarrow \Eq_\iota\\
\res_\iota &:=\res_{\g_{\m,\alpha}}^{p_\m}\colon \Eq_\iota\rightarrow Y_\iota\\
\tau_\iota &:=\tau_{\g_{\m,\alpha}}^{p_\m}\colon Y_\iota\rightarrow X_\iota\\
\hspace{-90pt}\text{(if $Y_\iota$ of respective type)}\hspace{40pt} \xi_\iota&:=\xi^{p_\m}_{\g_{\m,\alpha}}\colon J(\g_{\m,\alpha},p_\m) \rightarrow S^i \quad \text{for}\quad i\in \{1,2\}\\
\eta_\iota&:= \tau_\iota\cp\res_\iota\cp\pip_\iota\colon \IHOMLAS\rightarrow X_\iota
\end{align*}
and $\eta=\prod_{\iota\in I}\eta_\iota\colon \IHOMLAS\rightarrow X=\prod_{\iota\in I}X_\iota$.
Moreover, as in Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa3}:
\begingroup
\setlength{\leftmargini}{17pt}
\begin{itemize}
\item
Let $\J$ denote the set of all finite tuples $J=(\iota_1,\dots,\iota_k)$ of mutually different elements of $I$.
\item
Define $X_J:=X_{\iota_1}\times {\dots} \times X_{\iota_k}$, $\mu_J:=\mu_{\iota_1}\times {\dots} \times \mu_{\iota_k}$, and the respective projection by
\begin{align*}
\pi_J\colon X\rightarrow X_J,\:\:
\textstyle\prod_{\iota\in I}x_\iota \mapsto (x_ {\iota_1},\dots,x_ {\iota_k}).
\end{align*}
\item
Write $J\leq J'$ for $J,J'\in \JJ$ with $J=(\iota_1,\dots,\iota_k)$ and $J'=(\iota'_1,\dots,\iota'_{k'})$ iff there exists an injection $\sigma\colon \{\iota_1,\dots,\iota_k\}\rightarrow \{\iota'_1,\dots,\iota'_{k'}\}$, and define the maps
\begin{align*}
\pi^{J'}_J\colon X_{J'}\rightarrow X_J,\:\:\big(x_{\iota'_1},\dots,x_{\iota'_{k'}}\big)\mapsto \big(x_{\sigma(\iota_1)},\dots,x_{\sigma(\iota_{k})}\big).
\end{align*}
\item
Let $\mu_I$ denote the corresponding normalized Radon measure from Lemma \ref{lemma:normRM} on $X$.
\end{itemize}
\endgroup
\noindent
Finally, let $\widehat{\pi}_J:=\pi_J\cp\eta$ for $J\in \JJ$. Then,
the measure $\mLAS=\eta^{-1}(\mu_I)$ is uniquely determined by the property that
$\widehat{\pi}_J(\mLAS)=\mu_J$ holds for all $J\in \JJ$.
\vspace{10pt}
\noindent
We now choose a further selection $\{p'_\m\}_{\m\in \Mm}\subseteq P$ of elements with $x'_\m:=\pi(p'_\m)\in \m\in\Mm$, as well as $\{\g'_{\m,\alpha}\}_{\alpha\in I'_\m}\subseteq \mg\backslash \mg_{x'_\m}$ a respective independent and complete family for each $\m\in \Mm$. Moreover, for each plane $[\mm]$ in $\RR^3$ through $0$ we choose a further map $\LSM$ as in Lemma and Definition \ref{lemdef:projspaces}.\ref{lemdef:projspaces3}.
Then,
\begingroup
\setlength{\leftmargini}{17pt}
\begin{itemize}
\item
For each $\m\in \Mm$ we fix $g_\m\in G$ with $x'_\m=\wm(g_\m,x_\m)$.
\item
By Lemma \ref{lemma:completee}.\ref{lemma:completee1},
we can assume that $I'_\m=I_\m$ as well as $[\g'_{\m,\alpha}]=[\Ad_{g_{\m}}(\g_{\m,\alpha})]$
holds for all $\m\in \Mm$ and all $\alpha\in I_\m$. We define $g_\iota:=g_{\m}$ for $\iota={[\m,\alpha]}$ and denote by $s_\iota$ the unique element $s\in \SU$ for which $p'_\m=g_\iota\cdot p_\m\cdot s$ holds.
\item
Then, for $I'$ and $\g'_\iota$ defined as $I$ and $\g_\iota$ above, we can assume that $I'=I$ as well as $[\g'_\iota]=[\Ad_{g_\iota}(\g_\iota)]$ holds
for all $\iota\in I$.
\item
We define the corresponding spaces $\Eq'_\iota, Y_\iota', X'_\iota, X', X'_J$ and maps $\pip'_\iota,\res'_\iota, \tau'_\iota,\xi'_\iota,\eta_\iota',\eta',\widehat{\pi}'_J$ exactly as above, and denote the respective measures by $\mu'_\iota, \mu'_J$ and $\mLAS'$.
\end{itemize}
\endgroup
\noindent
Then, in order to show $\mLAS=\mLAS'$, it suffices to verify that $\widehat{\pi}'_J(\mLAS)=\mu'_J$ holds for all $J\in \JJ$.\footnote{This is immediate from the fact that by definition of $\mLAS'$, $\eta'^{-1}(\mLAS')$ is the Radon product measure on $X'$ w.r.t.\ the Radon measures $\mu'_\iota$ on $X'_\iota$.} This, however, follows if we prove that (this is done in Proposition \ref{prop:invmasss}.\ref{prop:invmasss2})
\begin{align}
\label{eq:dfffd}
\mu'_{J}(\widehat{\pi}'_{J}(E))=\mu_J(\widehat{\pi}_J(E))\quad\text{for}\quad\widehat{\pi}'_{J}(E) \quad\text{measurable},
\end{align}
and that for $\homm,\homm'\in \IHOMLAS$, $\iota\in I$ the implication (done in Proposition \ref{prop:invmasss}.\ref{prop:invmasss1})
\begin{align}
\label{eq:dfffdd}
\eta_\iota(\homm)=\eta_\iota(\homm')\qquad\Longrightarrow \qquad\eta'_\iota(\homm)=\eta'_\iota(\homm'
\end{align}
holds.
In fact, then for $E:=\widehat{\pi}'^{-1}_{J}(A')$ with $A'\in \mathfrak{B}(X'_J)$ we have
\begin{align}
\label{eq:sdfdfhd4355}
\mu'_{J}(A')&=\mu'_{J}\big(\widehat{\pi}'_{J}(E)\big)\stackrel{\eqref{eq:dfffd}}{=}\mu_J\big(\widehat{\pi}_{J}(E)\big)
=\widehat{\pi}_{J}(\mLAS)\big(\widehat{\pi}_J(E)\big)\\
&=\mLAS\big(\widehat{\pi}_{J}^{-1}\big(\widehat{\pi}_J(E)\big)\big)=\mLAS(E)=\widehat{\pi}'_{J}(\mLAS)(A'),
\end{align}
where in the fifth step we have used that $\widehat{\pi}_{J}^{-1}\big(\widehat{\pi}_J(E)\big)=E$ holds. Here, $E\subseteq\widehat{\pi}_{J}^{-1}\big(\widehat{\pi}_J(E)\big)$ is clear, and obviously we have $\homm\in \widehat{\pi}_{J}^{-1}\big(\widehat{\pi}_J(E)\big)$ iff $\widehat{\pi}_J(\homm)\in \widehat{\pi}_J(E)$. So, if we can show that the latter condition implies $\homm\in E$, then $\widehat{\pi}_{J}^{-1}\big(\widehat{\pi}_J(E)\big)= E$ follows. Now, $\widehat{\pi}_J(\homm)\in \widehat{\pi}_J(E)$ means that we find $\homm'\in E$ with
$\kla_{\iota_i}(\homm)=\kla_{\iota_i}(\homm')$ for all $1\leq i\leq k$, hence
\begin{align}
\label{eq:szbvvuzdefg}
\kla'_{\iota_i}(\homm)=\kla'_{\iota_i}(\homm')\qquad\text{ for all }\:\qquad 1\leq i\leq k
\end{align}
by \eqref{eq:dfffdd}. Consequently,
\begin{align*}
\widehat{\pi}'_{J}(\homm)&\hspace{3.2pt}=(\pi'_{J}\cp \kla')(\homm)=\big(\kla'_{\iota_1}(\homm),\dots,\kla'_{\iota_k}(\homm)\big)\\
&\stackrel{\eqref{eq:szbvvuzdefg}}{=}\big(\kla'_{\iota_1}(\homm'),\dots,\kla'_{\iota_k}(\homm')\big)=\widehat{\pi}'_{J}(\homm')\in \widehat{\pi}'_{J}(E)= A',
\end{align*}
hence $\homm\in E$. This shows \eqref{eq:sdfdfhd4355}, so that $\widehat{\pi}'_J(\mLAS)=\mu'_J$ holds for all $J\in \JJ$, hence $\mLAS=\mLAS'$.
Thus, the following proposition establishes independence of $\mLAS$ from any choices we have made.
\begin{proposition}
\label{prop:invmasss}
\begin{enumerate}
\item
\label{prop:invmasss1}
$X_\iota$ and $X'_\iota$ are of the same type for all $\iota\in I$. Moreover, for each such $\iota$ we find a homeomorphism $\Omega\colon \RB\rightarrow \RB$ with $\Omega^{-1}(\muB)=\muB$, as well as $R_2\in O(2)$ or $R_3\in \SOD$ such that
\begin{align}
\label{eq:beh}
\:\kla'_\iota(\homm)=
\begin{cases}
\kla_\iota(\homm)=0_{\mathrm{Bohr}} &\mbox{if } X_\iota \text{ is of } \:{\bf Type 1},\\
(\Omega \cp \kla_\iota)(\homm) &\mbox{if } X_\iota \text{ is of } \:{\bf Type 2},\\
\big[\big(\Omega(\psi), R_2(v)\big)\big] &\mbox{if } X_\iota \text{ is of } \:{\bf Type 3} \text{ and } \kla_\iota(\homm)= [(\psi, v)], \\
\big[\big(\Omega(\psi), R_3(v)\big)\big] &\mbox{if } X_\iota \text{ is of } \:{\bf Type 4} \text{ and } \kla_\iota(\homm)= [(\psi, v)].
\end{cases}
\end{align}
In the {\bf Type 3} and {\bf Type 4} case $\Omega$ is an unital homomorphism, so that the respective expressions are well defined.
In particular, $\eta_\iota(\homm)=\eta_\iota(\homm')$ for $\homm,\homm' \in\IHOMLAS$ implies $\eta'_\iota(\homm)=\eta'_\iota(\homm')$, showing \eqref{eq:dfffdd}.
\item
\label{prop:invmasss2}
Condition \eqref{eq:dfffd} holds, i.e., we have
\begin{align*}
\mu_J(\widehat{\pi}_J(E))=\mu'_{J}(\widehat{\pi}'_{J}(E))
\end{align*}
if $\widehat{\pi}'_{J}(E)$ or $\widehat{\pi}_{J}(E)$ is measurable for $E\subseteq \IHOMLAS$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item[2.)]
This is straightforward from Riesz-Markov theorem and Fubini's formula \eqref{eq:wellfuncs} if we can show that the statement holds for $J=\iota$, i.e., that we have $\mu_\iota(\eta_\iota(E))=\mu'_{\iota}(\eta'_\iota(E))$ if $\eta_\iota(E)$ or $\eta'_\iota(E)$ is measurable for $E\subseteq \IHOMLAS$. For this, observe that by Part \ref{prop:invmasss1}) $\eta_\iota(E)$ is measurable iff $\eta'_\iota(E)$ is so.
Now, using \eqref{eq:beh}, the statement is clear if $X_\iota$ is of \textbf{Type 1} or of \textbf{Type 2}.
In the \textbf{Type 3} case we calculate
\begin{align*}
\mu'_\iota(\kla'_\iota(E))&=\pr_0(\mu_{1\times})\big(\kla'_\iota(E)\big)=(\muB\times \mu_{1})\left(\pr_0^{-1}\big(\kla'_\iota(E)\right)\big)\\
&\hspace{-3.5pt}\stackrel{\eqref{eq:beh}}{=}(\muB\times \mu_{1})\left((\Omega\times R_2) \cp \pr_0^{-1}\left(\eta_\iota(E)\right)\right)\\
&=(\Omega^{-1}\times R_2^{-1})(\muB\times \mu_{1})\left(\pr_0^{-1}\left(\eta_\iota(E)\right)\right)\\
&=\pr_0\!\left((\Omega^{-1}\times R_2^{-1})(\mu_{1\times})\right)\left(\eta_\iota(E)\right),
\end{align*}
and similarly we obtain $\mu'_\iota(\kla'_\iota(E))=\pr_0\!\left((\Omega^{-1}\times R_3^{-1})(\mu_{2\times})\right)\left(\eta_\iota(E)\right)$ in the \textbf{Type 4} case. Since $\Omega^{-1}(\muB)=\muB$, $R_2^{-1}(\mu_1)=\mu_1$ and $R_3^{-1}(\mu_2)=\mu_2$, the Riesz-Markov theorem and Fubini's formula show that
\begin{align*}
\big(\Omega^{-1}\times R_2^{-1}\big)(\mu_{1\times})=\mu_{1\times}\qquad\quad\text{and}\qquad\quad\big(\Omega^{-1}\times R_3^{-1}\big)(\mu_{2\times})=\mu_{2\times},
\end{align*}
respectively, hence the claim.
\item[1.)]
We proceed in two steps.
\vspace{4pt}
\noindent
{\bf Step 1:}
We first assume that $\g'_\iota=\Ad_{g_\iota}(\g_\iota)$, where we have
\begin{align}
\label{eq:transf}
\begin{split}
\pip'_\iota(\homm)(\g'_\iota,l)&=\pi_{p'_\iota}(\g'_\iota,l,\homm)=\pi_{g_\iota\cdot p_\iota\cdot s_\iota}(\Ad_{g_\iota}(\g_\iota),l,\homm)\\
&\!\!\stackrel{\eqref{eq:verkn}}{=}\alpha_{s_\iota^{-1}}\cp \pi_{p_\iota}(\g_\iota,l,\homm)=\alpha_{s_\iota^{-1}}\cp \pip_\iota(\homm)(\g_\iota,l).
\end{split}
\end{align}
Consequently, \eqref{eq:beh} is clear if $X_\iota$ is of \textbf{Type 1}, as then $\pip_\iota(\homm)(\g_\iota,\cdot)=\me$ holds for all $\homm\in \IHOMLAS$ so that the same is true for $\pip'_\iota(\g'_\iota,\cdot)$.
For the other types, let $\pip_\iota(\homm)(\g_\iota,\cdot)\neq \me$ for $\homm\in \IHOMLAS$ and write
\begin{align}
\label{eq:phiB}
\pip_\iota(\homm)(\g_\iota,l)=\exp\big(\phi(l)\cdot\s_{\beta(\homm)}\big)\qquad\forall\:l>0
\end{align}
for $\phi\in \Per$, and $\beta(\homm):=\beta(\Psi)$ the element from
Definition \ref{def:betadef} that correspond to $\Psi:=\res_\iota(\pip_\iota(\homm))\in Y_\iota$. Then, \eqref{eq:transf} shows that
\begin{align*}
\pip'_\iota(\homm)(\g'_\iota,l)=\exp\big(\phi(l)\cdot \Ad_{s^{-1}_\iota}(\s_{\beta(\homm)})\big)\qquad\forall\:l>0.
\end{align*}
Now, since the definition of $\tau'_\iota$ does not depend on the choice of $s_\beta\in \beta$ which we have made in Convention \ref{conv:sammel} (see also Lemma and Definition \ref{lemdef:projspaces}.\ref{lemdef:projspaces4}), we can assume that $\tau'_\iota$ is defined by using the elements $\Ad_{s^{-1}_\iota}(\s_{\beta})$ for $\beta\in \Sp\ms$. Then $\Omega=\id_{\RB}$,
and since $\murs^{-1}\cp \Ad_{s^{-1}}\cp\: \murs$ is a rotation in $\RR^3$, it is clear that $X_\iota$ and $X'_\iota$ are of the same type. Then, the claim is obvious in the \textbf{Type 2} case, as there $\beta(\homm)$ is independent of $\homm\in \IHOMLAS$. In the \textbf{Type 4} case, let $v:=\xi_\iota(\beta(\homm))=\murs^{-1}(\s_{\beta(\homm)})$. Then \eqref{eq:beh} is clear from
\begin{align*}
v':&=
\xi'_\iota([\Ad_{s_\iota^{-1}}(\s_{\beta(\homm)})])=\murs^{-1}( \Ad_{s_\iota^{-1}}(\s_{\beta(\homm)}))\\
&=\big(\murs^{-1}\cp \Add{s_\iota^{-1}} \cp \:\murs \big)(\murs^{-1}(\s_{\beta(\homm)}))=R_3(v).
\end{align*}
Finally, in the \textbf{Type 3} case, let $\mm,\mm'\in \RR^3\backslash\{0\}$ be vectors as in \eqref{eq:hmnejhedrffds} which correspond to $X_\iota$ and $X'_\iota$, respectively. Then, $\vec{m}'=\lambda\cdot (\murs^{-1}\cp \Add{s_\iota}\cp\: \murs)(\vec{m})=R_3(\vec{m})$ for some $\lambda\neq 0$, so that for $v:=\xi_\iota(\beta(\homm))=\LM\big(\murs^{-1}(\s_{\beta(\homm)})\big)$ we have
\begin{align*}
v':&=\xi'_\iota([\Ad_{s_\iota^{-1}}(\s_{\beta(\homm)})])=\LSMS\big(\murs^{-1}(\Ad_{s_\iota^{-1}}(\s_{\beta(\homm)}))\big)
=\big(\LSMS\cp R_3\big)\big(\murs^{-1}(\s_{\beta(\homm)})\big)\\
&=\big(\underbrace{\LSMS\cp \LMS^{-1}}_{\in O(2)}\cp \underbrace{\LMS\cp R_3\cp \LM^{-1}}_{\in O(2)}\big)\big(\LM\big(\murs^{-1}(\s_{\beta(\homm)})\big)\big)
=R_2\big(v\big)
\end{align*}
for $R_3:=\murs^{-1}\cp \Add{s_\iota^{-1}} \cp \:\murs\in O(3)$ and $R_2=\LSMS\cp R_3\cp \LM^{-1}\in O(2)$.
\vspace{2pt}
\noindent
{\bf Step 2:}
To complete the proof, we have to treat the situation where $p'_\m=p_\m$, $g_\iota=e$, $s_\iota=e$ and $x=y$ holds, in full generality. This means that we have to consider the case where we have
$[\g'_\iota]=[\g_\iota]$,
i.e., $\g'_\iota \xsim \g_\iota$ but not necessarily $\g'_\iota = \g_\iota$.
Now, by Lemma \ref{lemma:completee}.\ref{it:completee1} we find $h\in G_x$ such that we have
\begin{align*}
\gamma^x_{\g'_\iota}= \wm_{h}\cp \gamma^x_{\pm \g_\iota}\cp \adif
\end{align*}
for $\adif\colon \RR\rightarrow \RR$ an analytic diffeomorphism with $\adif(0)=0$ and $\adif(\tau_{\g'_\iota})=\tau_{\g_\iota}$. We define
\begin{align*}
s_\alpha(l):=\Delta\big(\Phi_{h^{-1}}\cp\Phi_{\exp(l\cdot \g'_\iota)}\big(\Phi_{h}(p_\m)\big),\Phi_{\exp(\pm \adif(l)\cdot\g_\iota)}\big(p_\m\big)\big)
\end{align*}
as well as $s_0:=\fiba_{p_\m}(h)$, and obtain for $\homm\in \IHOMLAS$
\begin{align}
\label{eq:smult}
\begin{split}
\pip'_{\iota}(\homm)&(\g'_\iota,l)=\Delta\!\left(\Phi_{\exp(l\cdot \g'_\iota)}\big(p_\m\big), \homm\left(\gamma^x_{\g'_\iota}|_{[0,l]}\right)\left(p_\m\right)\right)\\
&=\Delta\!\left(\Phi_{\exp(l\cdot \g'_\iota)}\big(p_\m\big), \homm\left(\wm_{h}\cp\gamma^x_{\wg}|_{[0,\adif(l)]}\right)\left(p_\m\right)\right)\\
&=\Delta\!\left(\Phi_{\exp(l\cdot \g'_\iota)}\big(p_\m\big), \Phi_{h}\cp \homm\left(\gamma^x_{\wg}|_{[0,\adif(l)]}\right)\big(\Phi_{{h}^{-1}}(p_\m)\big)\right)\\
&=\Delta\!\left(\Phi_{{h}^{-1}}\cp\Phi_{\exp(l\cdot \g'_\iota)}\big(p_\m\big), \homm\left(\gamma^x_{\wg}|_{[0,\adif(l)]}\right)\big(\Phi_{{h}^{-1}}(p_\m)\big)\right)\\
&=\Delta\!\left(\Phi_{{h}^{-1}}\cp\Phi_{\exp(l\cdot \g'_\iota)}\big(\Phi_{h}(p_\m)\big)\cdot \fiba_{p_\m}(h)^{-1}, \homm\left(\gamma^x_{\wg}|_{[0,\adif(l)]}\right)\left(p_\m\right)\cdot \fiba_{p_\m}(h)^{-1}\right)\\
&=\big(\alpha_{\fiba_{p_\m}(h)}\cp \Delta\big)\!\left(\Phi_{{h}^{-1}}\cp\Phi_{\exp(l\cdot \g'_\iota)}\big(\Phi_{h}(p_\m)\big), \homm\left(\gamma^x_{\wg}|_{[0,\adif(l)]}\right)\left(p_\m\right)\right)\\
&=\alpha_{s_0}\left(s_\alpha(l)\cdot \Delta\!\left(\Phi_{\exp(\pm \adif(l)\cdot \g_\iota)}\big(p_\m\big), \homm\left(\gamma^x_{\wg}|_{[0,\adif(l)]}\right)\left(p_\m\right)\right)\right)\\
&=\alpha_{s_0}(s_\alpha(l))\cdot\alpha_{s_0}\big( \pip_\iota(\homm)\big(\wg,\adif(l)\big)\big)\\
&=\alpha_{s_0}(s_\alpha(l))\cdot\alpha_{s_0}\big( \pip_\iota(\homm)\big(\g_\iota,\adif(l)\big)\big)^{\pm 1}.
\end{split}
\end{align}
{\bf Case A:} Assume that $s_\alpha=\me$ and write, see \eqref{eq:phiB}
\begin{align*}
\pip_\iota(\homm)(\g_\iota,l)=\exp\big(\phi(l)\cdot\s_{\beta(\homm)}\big)\qquad\forall\:l>0
\end{align*}
as in {\bf Step 1}. Then, \eqref{eq:smult} shows
\begin{align*}
\pip'(\homm)(\g'_\iota,l)=\exp\big( \phi(\adif(l))\cdot \pm\Add{s_0}\big(\s_{\beta(\homm)}\big)\big)\qquad\forall\: 0<l<\tau_{\g}.
\end{align*}
Now, by the same arguments as in {\bf Step 1}, we can assume that $\tau'_\iota$ is defined by using the elements $\pm\Add{s_0}(\s_{\beta})$ for $\beta\in \Sp\ms$. Consequently,
\begin{align*}
\Omega(\psi)(\chi_l)=\psi(\chi_{\adif(l)})\qquad \forall\:\psi\in \RB, \forall\:0<l<\tau_{\g'},
\end{align*}
hence $\Omega(\psi)(\chi_l)=\psi(\chi_{\adif(l_1)+ \dots +\adif(l_k)})$
if $l=l_1+\dots+l_k$ for $0<l_1,\dots,l_k<\tau_{\g'}$.
Then, $\Omega$ is a bijective\footnote{This is because $\adif|_{(0,\tau_{\g_\iota})}$ is bijective and $\psi \in \RB$ is uniquely determined by its values on each subset of $\DG$ of the form $\{\chi_l\:|\: l\in (0,\tau)\}$ for $\tau>0$.} and unital homomorphism which is continuous because we have
\begin{align*}
\|\Omega(\psi)\|_{\chi_{l}}=\|\psi\|_{\chi_{\adif(l_1)+ \dots +\adif(l_k)}}
\end{align*}
for the seminorm $\|\psi\|_{\chi_{l}}:=\|\psi(\chi_l)\|$, by the definition of the topology on $\RB$. Since the normalized Radon measure $\Omega(\muB)$ is translation invariant, it equals $\muB$. Hence, the same is true for $\Omega^{-1}(\muB)$. The rest now follows as in {\bf Step 1}.
\vspace{4pt}
{\bf Case B:} Assume that $s_\alpha\neq \me$ and let $s'_\alpha:=\alpha_{s_0}\cp s_\alpha$. We choose $\homm_0\in \IHOMLAS$ with $\pip_\iota(\homm_0)(\g_\iota,\cdot)=\me$, hence $s'_\alpha=\pip'_\iota(\homm_0)(\g'_\iota,\cdot)$ by \eqref{eq:smult}. In particular, $\mathrm{im}[s'_\alpha] \subseteq H_{\beta'(\homm_0)}$ for $\beta'(\homm_0)$ the element from Definition \ref{def:betadef} that correspond to $\Psi':=\res'_\iota(\pip'_\iota(\homm_0))\in Y'_\iota$.
{\bf We claim that:}
\begin{align*}
\mathrm{im}\big[\pip'_\iota(\homm)(\g'_\iota,\cdot)\big]\subseteq H_{\beta'(\homm_0)}\qquad\forall\: \homm\in \IHOMLAS,
\end{align*}
hence $\mathrm{im}[\pip_\iota(\homm)(\g_\iota,\cdot)]\subseteq \alpha_{s_0^{-1}}(H_{\beta'(\homm_0)}$) for all $\homm\in \IHOMLAS$ because $\mathrm{im}[s'_\alpha]\subseteq H_{\beta'(\homm_0)}$.
{\bf Proof:}
In fact,
elsewise we find $\homm_1\in \IHOMLAS$ with $0\neq \beta'(\homm_1)\neq \beta'(\homm_0)$. We choose $\pm \me\neq s_1,s_2\in H_{\beta'(\homm_0)}$ and $0<l_1,l_2< \tau_{\g'_\iota}$ $\mathbb{Z}$-independent with $s'_\alpha(l_i)=s_i$ for $i=1,2$. This is always possible because $s'_\alpha$ is continuous, $s'_\alpha(0)=\me$, and $s'_\alpha\neq\me$ as $s_\alpha\neq \me$ by assumption.
Combining Proposition \ref{th:invhomm}.\ref{th:invhomm1} with Lemma \ref{prop:Bohrmod2}.\ref{prop:Bohrmod22} and the Parts \ref{prop:Bohrmod24}), \ref{prop:Bohrmod21}) of Lemma and Convention \ref{lemconv:RBMOD}, we find $\homm_1'\in \IHOMLAS$ with $\beta'(\homm'_1)=\beta'(\homm_1)$ as well as
\begin{align}
\label{eq:dfdsvc77v7676vs}
\pip'_\iota(\homm_1')(\g'_\iota,l_1)=\me\qquad\quad\text{and}\qquad\quad
\pip'_\iota(\homm_1')(\g'_\iota,l_2)=:b\neq \pm \me.
\end{align}
Let $d_i:=\alpha_{s^{-1}_0}(\pip_\iota(\homm_1')\big(\g_\iota,\adif(l_i))\big)^{\pm 1}$ (confer \eqref{eq:smult}) for $i=1,2$.
Then \eqref{eq:smult} yields
\begin{align*}
\me\stackrel{\eqref{eq:dfdsvc77v7676vs}}{=}\pip'_\iota(\homm_1')(\g'_\iota,l_1)\stackrel{\eqref{eq:smult}}{=} s_\alpha'(l_1)\cdot \alpha_{s_0}(\pip_\iota(\homm_1')\big(\g_\iota,\adif(l_i))\big)^{\pm 1}& =s_1\cdot d_1\quad\:\:(s_1\in H_{\beta'(\homm_0)}\backslash \{-\me,\me\})\\%\quad
&\Longrightarrow \hspace{8.5pt} d_1\in H_{\beta'(\homm_0)}\backslash \{-\me,\me\}.
\end{align*}
Hence, $\pm\me\neq b=s_2\cdot d_2 \in H_{\beta'(\homm_0)}$ because $d_1$ and $d_2$ are contained in the same maximal torus, namely $H_{\beta'(\homm_0)}$. This contradicts that $b\in H_{\beta'(\homm_1)}\backslash \{-\me,\me\}$. \hspace*{\fill}$\boldsymbol{\dagger}$
\vspace{5pt}
Thus, for $\beta:=\big[\Ad_{s_0^{-1}}(\s_{\beta'(\homm_0)})\big]$ we have $\beta(\homm)\in \{0,\beta\}$ for all $\homm\in \IHOMLAS$ with $\beta(\homm)$ defined as in {\bf Step 1}. Similarly, we have $\beta'(\homm)\in \{0,\beta'(\homm_0)\}$ for all $\homm\in \IHOMLAS$.
Now, $Y'_\iota$ cannot be of {\bf Type 1} since this would imply that $s'_\alpha=\me$ holds. Moreover, if $Y_\iota$ would be of {\bf Type 1}, then \eqref{eq:smult} would give that $\pip'(\homm)(\g'_\iota,\cdot)=s_\alpha'$ holds for all $\homm\in \IHOMLAS$, which is only possible if $Y_\iota'$ is of {\bf Type 1}, just by Lemma \ref{prop:Bohrmod2}.\ref{prop:Bohrmod22} and Proposition \ref{th:invhomm}.\ref{th:invhomm1}. Consequently, $Y_\iota$ and $Y'_\iota$ are both of {\bf Type 2}.
So, to finish the proof, we write
\begin{align*}
\begin{tabular}{rlrl}
$\pip'_\iota(\homm_0)(\g'_\iota,l)\hspace{-9pt}$&$=\exp\big(\phi'_0(l)\cdot \s_{\beta'(\homm_0)}\big)$&for& $\phi'_0\in \Per,\forall\: 0<l<\tau_{\g'_\iota}$\\[3pt]
$\pip'_\iota(\homm)(\g'_\iota,l)\hspace{-9pt}$&$=\exp\big(\phi'_\homm(l)\cdot \s_{\beta'(\homm)}\big)$&for& $\phi'_\homm\in \Per,\forall\: 0<l<\tau_{\g'_\iota}$\\[3pt]
$\pip_\iota(\homm)(\g_\iota,l)\hspace{-9pt}$&$=\exp\big(\phi_\homm(l)\cdot \s_{\beta(\homm)}\big)$&for& $\phi_\homm\in \Per,\forall\: 0<l<\tau_{\g_\iota}$
\end{tabular}
\end{align*}
whereby $\pip'_\iota(\homm_0)(\g'_\iota,l)=s'_\alpha(l)$.
Then, \eqref{eq:smult} and $\beta'(\homm)=\beta'(\homm_0)$ show that
\begin{align*}
\exp\big(\phi'_\homm(l)\cdot \s_{\beta'(\homm)}\big)=\exp\big(\phi'_0(l)\cdot \s_{\beta'(\homm_0)}\pm \phi_\homm(\adif(l))\cdot \s_{\beta'(\homm)}\big)\qquad\forall\:0<l<\tau_{\g'_\iota}.
\end{align*}
Hence,
\begin{align*}
\Omega(\psi)(\chi_l)&= \e^{\I\hspace{1pt}\phi'_\homm(l)}=\e^{\I\hspace{1pt}[\phi'_0(l)\pm \phi_\homm(\adif(l))]}\\
&=\e^{\I\hspace{1pt}\phi'_0(l)}\cdot \e^{\pm \I\hspace{1pt}\phi_\homm(\adif(l))}
=\psi'_0(\chi_l)\cdot \psi\big(\chi_{\adif(l)}\big)^{\pm 1}=[\psi_0\pm \Omega_\adif(\psi)](\chi_l),
\end{align*}
where $\Omega_\adif(\psi)(\chi_l):=\psi\big(\chi_{\adif(l)}\big)$
for all $0<l<\tau_{\g'_\iota}$. Of course, here $\psi'_0\in \RB$ denotes the element that corresponds to $\phi'_0\in \Per$. The rest now follows as in {\bf Step 1} because $\psi\mapsto \psi'_0\pm \psi$ is a homeomorphism which preserves $\muB$.
\end{enumerate}
\end{proof}
\subsection{Summary}
\label{conclmeasures}
\begin{enumerate}
\item
\label{conclmeasures1}
In the first part of this section, we have defined the normalized Radon measure $\mFNS$ on $\AQRInd{\mathrm{FN}}$
for the case that $S$ is compact and connected. This measure specializes to the Ashtekar-Lewandowski measure $\mAL$ on $\AQRw$ if the symmetry group is trivial. In the second part, we have constructed the normalized Radon measure $\mLAS$ on $\AQRInd{\mg}$ for the case that $S=\SU$ and that each $\wm$-orbit $\m$ admits an independent and complete family $\{\g_{\m,\alpha}\}_{\alpha\in I_\m}\subseteq \mg\backslash \mg_x$ for some $x\in \m$.
So, if in this situation additionally $\Paw=\Pags\sqcup \Pafns$ holds, we have the normalized Radon measure $\mLAS\times \mFNS$ on
\begin{align*}
\AQRw\cong \AQRInd{\mg}\times \AQRFNS,
\end{align*}
which defines a kinematical $L^2$-Hilbert space for the corresponding reduced theory.
In particular, this is the case in (semi)-homogeneous LQC as we have discussed in Remark \ref{ex:Fullmeas}.\ref{ex:Fullmeas1}. Recall that for all of our constructions we have assumed that the action $\wm$ induced on $M$ is analytic and pointwise proper.
\item
As already mentioned in the beginning of this subsection, the constructions there are also possible in the abelian case ($S$ is an $n$-torus) and are even easier.\footnote{The respective spaces $Y_\g^p$ of $\Ad_{G_{[\g]}}^p$-equivariant maps
$\Psi\colon \spann_\RR(\g)\rightarrow S$ are either $\{e\}$ or $[\RB]^n$.} Then, for an arbitrary compact and connected structure group $S$ one has to equip the respective spaces $Y_\g^p$
of equivariant maps
$\Psi\colon \spann_\RR(\g)\rightarrow S$
with suitable Radon measures. Here, suitable means that these measure have to fulfil certain invariance properties which make the whole construction independent of any choices one has to do. For this observe that we have used fixed families of stable Lie algebra elements in order to parametrize the space $\AQRInd{\mg}$. Using the theory of compact and connected Lie groups, here one might obtain some generalizations of the constructions we have worked out for $\SU$.
\item
In Example \ref{ex:cosmoliealgmaasse} we have shown that the measure $\mLAS$ is also available in the spherically symmetric and in the homogeneous isotropic case. Unfortunately, there we have no measure on $\AQRw$ at this point, just because we have not determined the set $\Pacs$ (of continuously but not Lie algebra generated curves) so far.
More concretely, if this set were empty, in both cases we would have a normalized Radon measure on $\AQRw$ because:
\begingroup
\setlength{\leftmarginii}{13pt}
\begin{itemize}
\item
Then, $\AQRw\cong\AQRInd{\mg}\times \AQRFNS$ would also hold in the homogeneous isotropic case, since we have shown in Remark \ref{ex:Fullmeas}.\ref{ex:Fullmeas2} that the set ($\Pafs$ of free curves having non-trivial stabilizer) is empty there.
\item
In the spherically symmetric case we would have
\begin{align*}
\AQRw\cong\AQRInd{\mg}\times \AQRFNS\times \AQRInd{\mathrm{FS}},
\end{align*}
where we already have constructed a normalized Radon measure on $\AQRInd{\mathrm{FS}}$ by hand, see Remark \ref{ex:Fullmeas}.\ref{ex:Fullmeas2}.
\end{itemize}
\endgroup
\item
As already mentioned in Remark \ref{rem:euklrem}.\ref{it:sdsdds}, in the homogeneous isotropic case where $\wm$ is transitive, $\AQRLA$ is a physically meaningful quantum-reduced configuration space by itself. This is because the cylindrical functions that correspond to $\Pags$ then separate the points in $\Con$, so that
$\Con $ is canonically embedded via $\iota\colon \Con \rightarrow \AInd{\mg}$.
Our constructions here provide us with the reduced kinematical Hilbert space $\Lzw{\AQRLA}{\mLAS}$, which even specializes to the standard LQC kinematical Hilbert space $\Lzw{\RB}{\muB}$ if we apply the same constructions for the subset $\Pal\subseteq \Pags$, see end of Example \ref{ex:cosmoliealgmaasse} and Remark \ref{StanLQC}.\ref{StanLQC2}.
More precisely, we have seen that each of the maps in
\begin{figure}[h]
\begin{minipage}[h]{\textwidth}
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
\RB \cong \ARRQLL \ar@{->}[r]^-{\ovl{i^*_\AR}}_-{\cong} & \: \ARQLL \: \ar@{->}[r]^-{\subseteq} & \AQRL\ar@{->}[r]_-{\cong}^-{\kappa_\lin} &\IHOMLL \ar@{->}[r]_-{\cong}^-{\eta_\lin}& \RB
}
\end{xy}}
\end{center}
\end{minipage}
\end{figure}
\FloatBarrier
\noindent
is a homeomorphism and that their concatenation is just the identity on $\RB$. In particular, this means that $\ARRQLL\cong \AQRL$ holds, i.e., that quantization and reduction commute in homogeneous isotropic LQC if one only uses linear curves in order to define the reduced spaces. Moreover, in Remark \ref{StanLQC}.\ref{StanLQC2} we have seen that, due to this fact, the standard quantum configuration space $\ARRQLL\cong \IHOMLL$ can be embedded into the full quantum-reduced space $\AQRw\cong \IHOMW$
just by the simple map
$\iota_\lin\colon \IHOMLL\rightarrow \IHOMW$ defined by
\begin{align*}
\iota_\lin(\homm)(\gamma):=\homm(\gamma)\:\text{ if }\:\gamma\in \Pal\qquad\text{and}\qquad\iota_\lin(\homm)(\gamma):=\me\:\text{ for }\:\gamma\in \Paw\backslash \Pal.
\end{align*}
\end{enumerate}
\section{Homogeneous Isotropic LQC}
\label{sec:HomIsoCo}
The traditional way to do symmetry reduction in LQC is to quantize the set
$\AR$ of
connections invariant under the given
symmetry group. In a first step, this means to calculate the spectrum $\ARRQInd{\alpha}$ of a separating $\Cstar$-algebra of the form $\rR=\ovl{\PaC_\alpha|_{\AR}}\subseteq B(\AR)$ \cite{MathStrucLQG,ChrisSymmLQG}.
Here, $\PaC_\alpha$ denotes the $\Cstar$-algebra of cylindrical functions which is generated by some distinguished set
$\Pa_\alpha$ of curves in base manifold.
Now, in \cite{ChrisSymmLQG} it was shown that $\ARRQInd{\alpha}$ can be compatibly embedded into the quantum configuration space\footnote{Recall that this is the spectrum of the $\Cstar$-algebra of cylindrical functions that correspond to the set $\Paw$ of embedded analytic curves in $\RR^3$.} $\A_\w$ of LQG iff
\begin{align}
\label{eq:inkldsakhfdfd898}
\PaC_\w|_{\AR}\subseteq \PaC_\alpha|_{\AR}
\end{align}
holds. Here, compatibly means that there exist an embedding of $\ARRQInd{\alpha}$ into $\A_\w$ which extends the inclusion map $\AR\hookrightarrow \Con$ in the sense of Lemma \ref{lemma:dicht}.\ref{lemma:dicht2}.
Now, in standard homogeneous isotropic LQC \cite{MathStrucLQG}, see also Remark \ref{StanLQC}, the set $\Pal$ of linear curves is used to define the reduced configuration space. It was shown in \cite{Brunnhack} that here \eqref{eq:inkldsakhfdfd898} does not hold, i.e., that no compatible embedding of $\ARRQInd{\lin}\cong \RB$ into $\A_\w$ exists.
In particular, for the embedding strategy proposed for states in \cite{BojoKa} this is disadvantageous.
So, to fix this problem, in \cite{ChrisSymmLQG} the reduced space
$\ARRQInd{\w}$ was introduced. It was shown that $\ARRQInd{\w}$ is compatibly embedded into $\A_\w$ via the map $\ovl{i_{\AR}^*}$
(see, e.g., \eqref{eq:inclusionsdiag}), and that it is homeomorphic to the compact Hausdorff space $\RR\sqcup \RB$.\footnote{The topology on $\RR\sqcup \RB$ is quite tricky. Details will be given in Subsection \ref{GAM}.}
In order to define the dynamics of the reduced theory,
it is
reasonable to construct natural measures on the reduced quantum configuration spaces.
Indeed, such measures usually define $L^2$-Hilbert spaces on which
representations of respective reduced holonomy-flux algebras can be defined.
Now, being a compact abelian group, $\ARRQLL\cong \RB$ admits the Haar measure $\muB$, which defines the standard kinematical Hilbert space $\Lzw{\RB}{\muB}$ of homogeneous isotropic LQC.
However, for
\begin{align*}
\RR\sqcup \RB\cong \text{\gls{qRR}}:=\ARRQInd{\w}
\end{align*}
the situation is much more difficult as there no Haar measure can exist. This will be shown in Proposition \ref{prop:noGroupStruc} where we prove that it is impossible to equip $\RR\sqcup \RB$ with a group structure continuous w.r.t.\ its Gelfand topology. Basically, this is because existence of such a structure would imply that $|\RB|=|\RR|$ holds. This, however, contradicts that for the cardinality of $\RB$ we have $|\RB|\geq |2^\RR|$. Then, changing the focus from Haar to normalized Radon measures, it is a crucial observation that $\muB$ is the unique normalized Radon measure $\mu$ on $\RB$ for which the pullbacks
\begin{align*}
{\Transl_v\!}^* \colon \Lzw{\RB}{\mu}\rightarrow \Lzw{\RB}{\mu},\quad f\mapsto f\cp \Transl_v,
\end{align*}
are unitary operators (even form a strongly continuous one-parameter group), cf.\ Proposition \ref{lemma:bohrmassdichttrans}. Here,
$\Transl\colon \RR\times \RB \rightarrow \RB$ denotes the unique extension (cf.\ Proposition \ref{prop:autspec}) of the additive group action \RPLUS$\colon \RR \times \AR \rightarrow \AR$
of $\RR$ on $\AR \cong\RR$. This fact is important because the exponentiated reduced fluxes (``momentum operators'') are represented in this way.
Indeed, one now might ask whether the same condition
singles out a normalized Radon measure on $\RR\sqcup\RB$. We will show that this is the case (Corollary \ref{cor:noext}), and that this measure is given by $\muB$ as well. In particular, the respective kinematical Hilbert space equals the standard one.
At this point, it is important to know that the Borel $\sigma$-algebra of $\RR\sqcup \RB$ is just given by $\mathfrak{B}(\RR)\sqcup \mathfrak{B}(\RB)$ (the topology on $\RR\sqcup \RB$ has not such an easy decomposition), so that
\begin{align*}
\mu(A_1\sqcup A_2):=\muB(A_2)\quad \text{for}\quad A_1\in \mathfrak{B}(\RR)\quad\text{and}\quad A_2\in \mathfrak{B}(\RB)
\end{align*}
is a well-defined normalized Radon measure on $\RR\sqcup \RB$, cf. Lemma \ref{lemma:Radon}. In the last part, we will equip this space with a projective structure in order to construct further normalized Radon measures thereon. We then close
this section with a brief discussion of the corresponding $L^2$-Hilbert spaces they define.
On the way, we will show that quantization and reduction do not commute in homogeneous isotropic LQC. As in Subsection \ref{sec:inclrel}, this will be done by constructing elements of $\AQRw$ which cannot be contained in $\ARQw\cong \ARRQw\cong \RR\sqcup \RB$.
\subsection{Setting}
In the following, let $P:=\RR^3\times \SU$, $G:=\Ge=\Gee$ ($\mg=\RR^3\times \su$) and $\Phi:=\Pe$, see Example \ref{ex:LQC}. Moreover, let $\nu_x:=(x,\me)$ for all $x\in M$ and recall (see \eqref{eq:homis}) that the corresponding set $\AR$ of $\Phi$-invariant connections consists exactly of the elements of the form ($c$ runs over $\RR$)
\begin{align*}
\w^c_{(x,s)}(\vec{v}_x,\vec{\sigma}_x)= c \Add{s^{-1}}[\murs(\vec{v}_x)]+s^{-1}\vec{\sigma}_s \qquad \forall\:(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P.
\end{align*}
So, in the sequel we will always identify $\AR$ with $\RR$.
In contrast to Subsection \ref{sec:ConSp} (see Convention \ref{conv:sammel}), by $\RB$ in the following we will understand the spectrum of the $\Cstar$-algebra $\CAP(\RR)$. This has just conceptual reasons and makes mathematically no difference as Lemma \ref{lemma:Bohriso} shows. Finally, $\rR$ will always denote the $\Cstar$-algebra (vector space direct sum)
\begin{align*}
\rR:=C_0(\RR)\oplus \CAP(\RR)\subseteq B(\RR).
\end{align*}
As shown in \cite{ChrisSymmLQG}, this is exactly the restriction $\Cstar$-algebra $\ovl{\PaC_\w|_{\AR}}$ (under the identification $\AR\cong \RR$), so that
$\ARRQw=\mathrm{Spec}(C_0(\RR)\oplus \CAP(\RR))$ holds.
\subsection{Group Structures, Actions and some Measure Theoretical Aspects}
\label{GAM}
In this subsection, we show that the Haar measure on $\RB$ is the unique normalized Radon measure which is invariant under the spectral extension
$\Transl\colon \RR\times \RB \rightarrow \RB$ of the additive group action \RPLUS$\colon \RR \times \RR \rightarrow \RR$, $(v,t)\mapsto v+t$. In particular, we will identify $\muB$ as the unique normalized Radon measure $\mu$ on $\RB$ for which the maps ${\Transl_v\!}^* \colon \Lzw{\RB}{\mu}\rightarrow \Lzw{\RB}{\mu}$, $f\mapsto f\cp \Transl_v$ are unitary operators for all $v\in \RR$, see Proposition \ref{lemma:bohrmassdichttrans}. In particular, we prove that the one-parameter group $\{{\Transl_v\!}^*\}_{v\in \RR}$ then is strongly continuous. We show that the same invariance condition\footnote{Namely, that the translations w.r.t.\ the spectral extension $\Transw\colon \RR\times \qRR \rightarrow \qRR$ of {\scriptsize$\Sigma_\RR$}$\colon \RR \times \RR \rightarrow \RR$ act as unitary operators. Here, it is equivalent to require that this measure is invariant under the translations $\Transw_v$ for all $v\in \RR$.} singles out a normalized Radon measure on $\qRR$, which then even defines the same Hilbert space $\Lzw{\RB}{\muB}$ as in the traditional approach. This reinforces the standard LQC approach from the mathematical side.
On the way, we verify that, in contrast to $\RB$, there exists no continuous group structure, hence no Haar measure on $\qRR$.
Since the group structure on $\RB$ canonically extends\footnote{We have $\iota'_\RR(x)+\iota'_\RR(y)=\iota'_\RR(x+y)$ for $\iota'_\RR\colon \RR \rightarrow\RB$ the canonical embedding from Subsection \ref{subsec:boundedfun}.} the additive group structure on $\RR$, we might start by clarifying whether there is also such an extension of the additive group structure on $\RR$ to $\qRR=\mathrm{Spec}(C_0(\RR)\oplus \CAP(\RR))$.
Indeed, this would provide us with a Haar measure on $\qRR$, in particular, invariant under the action $\Transw\colon \RR \times \qRR\rightarrow \qRR$, $(v,\x)\mapsto \iota_\RR(v)+\x$. This is because $\Transw$ would be the unique extension of \RPLUS\ in this case just because it fulfils the respective conditions from Proposition \ref{prop:autspec}.\ref{prop:autspec1}.
However, already Proposition \ref{prop:Specgroup} shows that such an extension of the additive group structure on $\RR$ cannot exist.
\begin{corollary}
\label{cor:noext}
There is no continuous group structure on $\qRR=\mathrm{Spec}(\rR)$ which is compatible with the additive group structure on $\mathbb{R}$.
\begin{proof}
Assume there is such a group structure. Then it is necessarily abelian as it is so on a dense subset of $\mathrm{Spec}(\rR)$. So, by Proposition \ref{prop:Specgroup} there is a set $\uU\subseteq \rR$ of characters on $\RR$ generating $\rR$. Since $\uU\subseteq \rR$, these characters are continuous, hence contained in $\CAP(\RR)$. For this, recall that $\CAP(\RR)$ is generated by all continuous characters on $\RR$. Since $\CAP(\RR)$ is closed, the $\Cstar$-subalgebra generated by $\uU$ must be contained in $\CAP(\RR)$. This contradicts that $\uU$ generates $\rR$ just because $C_{0}(\mathbb{R})\neq \{0\}$.
\end{proof}
\end{corollary}
Anyhow, using Proposition \ref{prop:autspec} we obtain the following left actions on $\qRR$ and $\RB$.
\begin{corollary}
\label{cor:chrisleftact}
There are unique left actions
$\Multw\colon \RR_{\neq 0}\times \qRR \rightarrow \qRR$ and $\Transw\colon \RR \times \qRR\rightarrow \qRR$,
separately continuous in $\qRR$, that extend the multiplication $\cdot \colon \RR_{\neq 0}\times \RR \rightarrow \RR$, $(\lambda,t)\mapsto \lambda \cdot t$ and the translation $+\colon \RR\times \RR \rightarrow \RR$, $(v,t)\mapsto v+ t$, respectively, in the sense of \eqref{eq:extens}. The action $\Transw$ is continuous and $\Multw$ is not so. The same statements hold for $\RB$ instead of $\qRR$ where we denote the respective actions by $\Multl$ and $\Transl$ in the following. For $\Transl$ we have
\begin{align}
\label{eq:Bohrwirk}
\Transl(v,\psi)=\iota'_\RR(v)+\psi\qquad \forall\: v\in \RR,\forall\:\psi\in \RB
\end{align}
with $\iota'_\RR\colon \RR \rightarrow\RB$ the canonical embedding and $+$ the addition in $\RB$.
\begin{proof}
Obviously, $C_0(\RR) \oplus \CAP(\mathbb{R})$ is invariant under pullback by $\cdot_\lambda$ and $+_v$ for all $\lambda\neq 0$ and all $v\in \RR$. This is obvious for $C_0(\RR)$ and follows for $\CAP(\mathbb{R})$ from $\cdot_\lambda^*(\chi_l)=\chi_{\lambda l}$ and $+_v^*(\chi_l)=\e^{\I l v}\cdot \chi_l$
for all $l\in \RR$. Consequently, Proposition \ref{prop:autspec}.\ref{prop:autspec1} provides us with the unique left actions $\Multw$ and $\Transw$.
Obviously $\cdot_\bullet ^*\colon \lambda \mapsto [t\mapsto \e^{\I \lambda l t}]$ is not continuous (w.r.t.\ the supremum norm) for $l\neq 0$, so that $\Multw$ is not continuous by unitality of $\rR$ and Proposition \ref{prop:autspec}.\ref{prop:autspec2}. In contrast to that, for each $f\in \rR$ the map $+_\bullet^*(f)\colon v\mapsto [t\mapsto f(v+t)]$ is continuous, so that Proposition \ref{prop:autspec}.\ref{prop:autspec2} shows continuity of $\Transw$. Here continuity is clear if $f=\chi_l$ for some $l\in \RR$ because
\begin{align*}
\|+_v^*(\chi_l)-+_{v'}^*(\chi_l)\|_\infty=\big\|\big[\e^{\I lv}-\e^{\I lv'}\big]\chi_l\big\|_\infty=\big|\e^{\I lv}-\e^{\I lv'}\big|\cdot \|\chi_l\|_\infty=\big|\e^{\I lv}-\e^{\I lv'}\big|.
\end{align*}
It follows for $f\in C_0(\RR)$ from
equicontinuity of $f|_K$ for each compact subset $K\subseteq \RR$,
and that for each $\epsilon$ we find $K\subseteq \RR$ compact with $|f(t)|<\epsilon/ 2$ for all $t\in \RR\backslash K$.
In fact, let $K':= K + [-\delta,\delta]$ for $\delta >0$. Then for all $t\in \RR \backslash K'$ and all $v\in [-\delta, \delta]$ we have $t, v+t\in \RR\backslash K$, hence
\begin{align}
\label{eq:compl}
|f(t)-f(v+t)|\leq |f(t)|+|f(v+t)|< \epsilon.
\end{align}
Moreover, by equicontinuity of $f|_{K'}$ we find $W\subseteq \RR$ a neighbourhood of $0\in \RR$, such that
\begin{align}
\label{eq:compll}
|f(t)-f(t+v)|<\epsilon \qquad\:\forall\: t\in K',\:\forall\: v\in W.
\end{align}
So, combining \eqref{eq:compl} and \eqref{eq:compll} we obtain
\begin{align*}
|f(t)-f(t+v)|<\epsilon \qquad\:\forall\: t\in \RR,\:\forall\: v\in W\cap [-\delta,\delta],
\end{align*}
hence $\|\!+_v^*(f)-f \|_\infty<\epsilon$ for all $v\in W\cap [-\delta,\delta]$. This show continuity of $+^*_\bullet$ at $v=0$. Then, replacing $f$ by $+^*_{v'}(f)$, we deduce continuity of $+^*_{v'}$ for all $v'\in \RR$.
Finally, \eqref{eq:Bohrwirk} holds because the left action $\RR\times \RB \rightarrow \RB$, $(v,\psi)\mapsto \iota'_\RR(v)+\psi$ fulfils the condition from Proposition \ref{prop:autspec}.\ref{prop:autspec1}, hence equals the unique left action $\Transl$.
\end{proof}
\end{corollary}
At the end of this subsection, the action $\Transw$ will provide us with a uniqueness statement concerning normalized Radon measures on $\qRR$. The first step toward this is performed in the following
\begin{proposition}
\label{lemma:bohrmassdichttrans}
\begin{enumerate}
\item
\label{lemma:bohrmassdichttrans1}
If $\mu$ is a normalized Radon measure on $\RB$ with $\Transl_v(\mu)=\mu$ for all $v\in \RR$, then $\mu=\muB$. In particular, $\muB$ is the unique normalized Radon measure $\mu$ on $\RB$ for which the translations ${\Transl_v\!}^* \colon \Lzw{\RB}{\mu}\rightarrow \Lzw{\RB}{\mu}$, $f\mapsto f\cp \Transl_v$ are unitary operators.
\item
\label{lemma:bohrmassdichttrans2}
The one-parameter group $\{{\Transl_v\!}^*\}_{v\in \RR}$ of unitary operators
\begin{align*}
{\Transl_v\!}^* \colon \Lzw{\RB}{\muB}\rightarrow \Lzw{\RB}{\muB},\quad f\mapsto f\cp \Transl_v
\end{align*}
is strongly continuous.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
We have to show that $\mu(\psi+A)=\mu(A)$ holds for all $\psi\in \RB$ and all $A\in \mathfrak{B}(\RB)$. Then, by inner regularity, it suffices to show the case were $A$ is compact.
So, let $K\subseteq \RB$ be compact.
Since $\mu$ is finite, it is outer regular. Hence, for each $n\in \NNge$ we find $U_n\subseteq \RB$ open with $K\subseteq U_n$ and $\mu(U_n)-\mu(K)\leq\textstyle\frac{1}{n}$. By continuity of the addition in $\RB$, for each $\psi'\in K$ the preimage $+^{-1}(U_n)$ contains a set $V'\times W'$ with $V',W'\subseteq \RB$ open, $\NB\in V'$ and $\psi'\in W'\subseteq U_n$. By compactness of $K$ there are finitely many such quantities $\NB\in V_1',\dots,V'_k \subseteq \RB$, $W_1',\dots,W_k'\subseteq U_n$ with $K\subseteq W_1'\cup \dots\cup W'_k=:W'$ and $V_i' + W_i' \subseteq U_n$ for $1\leq i\leq k$.
Let $V'\subseteq V_1'\cap\dots\cap V_k'$ be a neighbourhood of $\NB$.
By denseness of $\iota'_\RR(\RR)$ in $\RB$ we find $x'\in \RR$ with $\psi-\iota'_\RR(x')\in V'$. Then, $\psi-\iota'_\RR(x') +K\subseteq V' +W' \subseteq U_n$, so that for each $n\in \NNge$ we obtain from \eqref{eq:Bohrwirk} that
\begin{align*}
\mu(\psi +K)-\mu(K)\stackrel{\ref{eq:Bohrwirk}}{=}\mu\big(\psi-\iota'_\RR(x') +K\big) -\mu(K)\leq \mu\big(U_n\big) -\mu(K)\leq \textstyle\frac{1}{n},
\end{align*}
hence $\mu(K)\geq \mu(\psi +K)$. Then, applying the same argument to the compact set $K':=\psi+K$ and $\psi':=-\psi$, we see that $\mu(\psi +K)=\mu(K')\geq \mu(\psi'+K')=\mu(K)$.
For the last statement assume that ${\Transl_v\!}^*$ is unitary for each $v\in \RR$. Then, for $K\subseteq \RB$ compact, $\chi_K$ the characteristic function of $K$ and $v\in \RR$
we have
\begin{align*}
\mu(K)&=\int_{\RB}\chi_{K}\:\dd \mu=\int_{\RB} |\chi_{K}|^2\:\dd \mu=\langle \chi_K,\chi_K \rangle
=\big\langle {\Transl_{v}\!}^*(\chi_K),{\Transl_{v}\!}^*(\chi_K)\big\rangle\\
&= \int_{\RB} |\chi_{\iota'_\RR(v)+K}|^2\dd \mu =\int_{\RB}\chi_{\iota'_\RR(v)+K}\dd \mu=\mu\big(\iota'_\RR(v)+K\big).
\end{align*}
By the first part this already implies the translation invariance of $\mu$.
\item
We have to show that w.r.t.\ the $L^2$-norm $\|\cdot\|_2$ we have
\begin{align*}
\lim_{v'\rightarrow v}{\Transl_{v'}\!}^*(f)=f\qquad\forall\: f\in \Lzw{\RB}{\muB},\:\forall\: v\in \RR.
\end{align*}
Since $\muB$ is regular, $C(\RB)$ is dense in $\Lzw{\RB}{\muB}$. Moreover, since ${\Transl_{v}\!}^*(f)(\psi)=f(\iota'_\RR(v)+\psi)$, for each $f\in C(\RB)$ and each $\epsilon>0$ we find $\delta>0$ such that
\begin{align*}
\|{\Transl_{v}\!}^*(f)- {\Transl_{v'}\!}^*(f)\|_\infty<\epsilon \qquad\forall\: v'\in B_\delta(v).
\end{align*}
In fact, by 3.8 Satz in \cite{Elstrodt}, for $\epsilon>0$ we find a neighbourhood $U$ of $0_{\mathrm{Bohr}}\in \RB$ with
\begin{align*}
\psi-\psi'\in U\qquad \Longrightarrow \qquad |f(\psi)-f(\psi')|<\epsilon.
\end{align*}
Now, for $\delta>0$ suitable small we have $\iota'_\RR(B_\delta(0))\subseteq U$. This is just because $\iota'_\RR$ is continuous as $\CAP(\RR)$ consists of continuous functions on $\RR$, see Proposition 2.1 in \cite{ChrisSymmLQG}. Thus, for all $v'\in B_\delta(v)$ we have $\iota_\RR'(v)-\iota_\RR'(v')=\iota_\RR'(v-v')\in U$, hence
\begin{align*}
\|{\Transl_{v}\!}^*(f)- {\Transl_{v'}\!}^*(f)\|_\infty&=\sup_{\psi\in \RB}|f(\psi + \iota_\RR'(v))-f(\psi+\iota_\RR'(v'))|\leq \epsilo
\end{align*}
because $\psi+\iota'_\RR(v)-\psi+\iota'_\RR(v')=\iota'_\RR(v)-\iota'_\RR(v')\in U$.
So, let $f\in \Lzw{\RB}{\muB}$ and $v\in \RR$ be fixed. Then, for $\epsilon>0$ by denseness of $C(\RB)$ in $\Lzw{\RB}{\muB}$ we find $f_\epsilon\in C(\RB)$ with $\|f-f_\epsilon\|_{2}\leq \frac{\epsilon}{3}$. Moreover, by the above arguments we find $\delta>0$ such that $\|{\Transl_{v}\!}^*(f_\epsilon)- {\Transl_{v'}\!}^*(f_\epsilon)\|_\infty\leq \frac{\epsilon}{3}$ for all $v'\in B_\delta(v)$. Consequently, by translation invariance of $\muB$ for all such $v'$ we have
\begin{align*}
\|{\Transl_{v}\!}^*(f)-{\Transl_{v'}\!}^*(f)\|_2&\leq
\|{\Transl_{v}\!}^*(f)-{\Transl_{v}\!}^*(f_\epsilon)\|_2+\|{\Transl_{v}\!}^*(f_\epsilon)-{\Transl_{v'}\!}^*(f_\epsilon)\|_2+\|{\Transl_{v'}\!}^*(f_\epsilon)-{\Transl_{v'}\!}^*(f)\|_2 \\
&= \|f-f_\epsilon\|_2+\|{\Transl_{v}\!}^*(f_\epsilon)-{\Transl_{v'}\!}^*(f_\epsilon)\|_2+\|f_\epsilon-f\|_2
\leq \epsilon.
\end{align*}
\end{enumerate}
\end{proof}
\end{proposition}
Although Corollary \ref{cor:noext} states that no extension of the addition in $\RR$ to $\qRR$ exists, one should clarify whether there is any other continuous group structure on this space. Indeed, such a structure would provide us with a canonical measure on $\qRR$. In addition to that we want to prove an analogue to Proposition \ref{lemma:bohrmassdichttrans}. For this and the considerations in the last subsection, it is comfortable to use the following description of $\qRR$ proven in \cite{ChrisSymmLQG}.
\begin{lemdef}
\label{remdefchris}
Let $\emptyset\neq Y\subseteq \RR$ be open and $C_{0,Y}(\RR)$ the set of continuous function vanishing at infinity and outside $Y$. Then Corollary B.2 in \cite{ChrisSymmLQG} shows $C_{0,Y}(\RR)\cap \CAP(\RR)=\{0\}$ and that $\aA_Y:=C_{0,Y}(\RR)\oplus \CAP(\RR)\subseteq B(\RR)$ is closed. Moreover, if we
equip $Y \sqcup \RB$ with the topology generated by the sets of the following types:\cite{ChrisSymmLQG}
\begin{align*}
\begin{array}{lcrclcl}
\textbf{Type 1:} && V & \!\!\!\sqcup\!\!\! & \emptyset
&& \text{with open $V \subseteq Y$} \\
\textbf{Type 2:} && K^c & \!\!\!\sqcup\!\!\! & \RB
&& \text{with compact $K \subseteq Y$} \\[-1.5pt]
\textbf{Type 3:} && f^{-1}(U) & \!\!\!\sqcup\!\!\! & \mathcal{G}(f)^{-1}(U)
&& \text{with open $U \subseteq \mathbb{C}$ and $f \in \CAP(\RR)$},
\end{array}
\end{align*}
then
Proposition 3.4 in \cite{ChrisSymmLQG} states that $\mathrm{Spec}(\aA_Y)\cong Y\sqcup \RB$ holds via the homeomorphism $\xi\colon Y\sqcup \RB\rightarrow \mathrm{Spec}(\aA_Y)$ defined by
\begin{equation}
\label{eq:Ksiii}
\xi(\x) :=
\begin{cases}
f\mapsto f(\x) &\mbox{if } \x\in Y\\
f_0\oplus f_{\mathrm{AP}}\mapsto \x(f_{\mathrm{AP}}) & \mbox{if } \x\in \RB.
\end{cases}
\end{equation}
Here, $f_0\in C_0(\RR)$ and $f_{\mathrm{AP}}\in \CAP(\RR)$.
It is straightforward to see that the subspace topologies of $Y$ and $\RB$ w.r.t.\ the above topology coincide with their usual ones. In abuse of notation\footnote{Indeed, in order to be in line with the notations in \eqref{eq:inclusionsdiag} we rather should define $\qR:=\ARQw$ and introduce another symbol for $\RR\sqcup \RB$. However, in the following we will rather deal with the space $\RR\sqcup \RB$, so that it is much more convenient to use the memorable symbol $\qR$ for this space. Besides this, the spaces $\RR\sqcup \RB$, $\ARRQw$ and $\ARQw$ are homeomorphic, and using $\qR$ for $\RR\sqcup \RB$ our notations are in line with \cite{ChrisSymmLQG}.} we define $\qRY:=Y\sqcup \RB$ as well as $\text{\gls{qR}}:=\RR\sqcup \RB$, and equip these spaces with the above topology.
\hspace*{\fill}{\scriptsize$\blacksquare$}
\end{lemdef}
Except for Proposition \ref{prop:noGroupStruc}, where we show that the spaces $\qRY$ cannot be equipped with group structures continuous w.r.t.\ the above topologies, we will only deal with the space $\qR$ in the sequel. Observe that it is immediate from the above definitions that $\RR$ is a dense open subset of $\qR$.
\begin{lemrem}
\label{remtrnas}
\begin{enumerate}
\item
\label{remtrnas1}
In contrast to the standard LQC case, for $\qR$ the canonical embedding $\iota_\RR\colon \RR\rightarrow \qRR$ does not map $\RR$ into the $\RB$ part, but canonically onto the $\RR$ part, i.e.,
\begin{align}
\label{eq:rmap}
\xi^{-1}(\iota_\RR(x))=x\in\RR\subseteq \qR.
\end{align}
This is because $\xi\big(\xi^{-1}(\iota_\RR(x))\big)(f)=f(x)=\xi(x)(f)$ for all $f\in \rR$.
\item
\label{remtrnas11}
Since $\xi$ is a homeomorphism, the action $\Transw$ from Corollary \ref{cor:chrisleftact} transfers via $\xi$ to an action $\TTransw\colon \RR\times \qR\rightarrow \qR$ on $\qR$, i.e., $\TTransw(v,\x):=\xi^{-1}\big(\Transw(v,\xi(\x))\big)$. Then, for all $v\in \RR$ and $x\in\RR \subseteq \qR$ we have
\begin{align}
\label{eq:sigmaquer}
\begin{split}
\TTransw(v,x)&\stackrel{\eqref{eq:rmap}}{=}\xi^{-1}(\Transw(v,\iota_\RR(x)))=\xi^{-1}(\iota_\RR(v+x))=\iota_\RR(v+x)\in\RR\subseteq \qR.
\end{split}
\end{align}
So, for $\x\in \RB\subseteq \qR$ and $\{x_\alpha\}_{\alpha\in I}\subseteq \RR\subseteq \qR$ a net with $\lim_\alpha x_\alpha=\x$,\footnote{Here we mean the limit in $\qR$} we have
\begin{align}
\label{eq:sigmaquerbohr}
\TTransw(v,\x)&=\lim_\alpha \TTransw(v,x_\alpha)\stackrel{\eqref{eq:sigmaquer}}{=}\lim_\alpha\iota_\RR(v+x_\alpha)=\iota'_\RR(v)+ \x\in \RB \subseteq \qR.
\end{align}
Here, $\iota'_\RR\colon \RR \rightarrow \RB$ denotes the canonical embedding of $\RR$ into $\RB$, so that in the last term by $+$ we mean the addition in $\RB$.
The last step follows from
{\bf Claim 1:} $\{\iota_\RR(v+x_\alpha)\}_{\alpha \in I}$ and $\{\iota'_\RR(v) + \iota'_\RR(x_\alpha)\}_{\alpha \in I}$ (considered as nets in $\qR$) have the same limit in $\qR$, i.e.,
\begin{align}
\label{eq:limgl}
\lim_\alpha \iota_\RR(v+x_\alpha) = \lim_\alpha \big(\iota'_\RR(v) + \iota'_\RR(x_\alpha)\big).
\end{align}
In fact, using {\bf Claim 1}, Equation \eqref{eq:sigmaquerbohr} is clear from $\lim_\alpha \iota'_\RR(x_\alpha)=\x$.
This, in turn, holds because by the definition of the topology on $\qR$ for each $l\in \RR$ and each open subset $U\subseteq \mathbb{C}$ with $\x(\chi_l)\in U$ we find
$\alpha_0\in I$ such that $x_\alpha \in \chi^{-1}_l(U)$ for all $\alpha\geq \alpha_0$, hence $\iota'_\RR(x_\alpha)\in \mathcal{G}(\chi_l)^{-1}(U)$ for all $\alpha\geq \alpha_0$. This shows $\lim_\alpha \iota'_\RR(x_\alpha)=\x$ because the Gelfand topology on $\RB$ equals the initial topology w.r.t.\ the Gelfand transforms of the functions $\chi_l$ for $l \in \mathbb{\RR}$, see e.g.\ Subsection 2.3 in \cite{ChrisSymmLQG}.
{\bf Proof of Claim 1:}
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
For each compact $K\subseteq \RR$ we find $\alpha_0\in I$ such that $x_\alpha \in [-v+K]^c$, hence $v+x_\alpha \in K^c$ for all $\alpha\geq \alpha_0$. This is clear from $\lim_\alpha x_\alpha=\x$ because $\x\in \RB\subseteq \qR$ and $[-v+K]^c\sqcup \RB$ is an open neighbourhood of $\x$.
\item
Hence, since we know that $\{\iota_\RR(v+x_\alpha)\}_{\alpha\in I}$ converges to some element of $\qR$, it must converge to some $\ovl{z}\in \RB\subseteq \qR$.
\item
In order to show $\qR\supseteq \{\iota'_\RR(v) + \iota'_\RR(x_\alpha)\}_{\alpha\in I}\rightarrow \ovl{z}\in \qR$, it suffices to verify that we find $\alpha_0\in I$ with $\iota'_\RR(v) + \iota'_\RR(x_\alpha) \in \mathcal{G}(\chi_l)^{-1}(U)$ for all $\alpha \geq \alpha_0$. Here $l\in \RR$ and $U\subseteq \mathbb{C}$ is open with $\z(\chi_l)\in U$.
In fact, then the above net converges in $\RB$ to $\ovl{z}\in \RB$, hence in $\qR$ because the subspace topology of $\RB$ w.r.t.\ the topology on $\qR$ equals its usual one.
\item
Now,
\begin{align*}
\iota'_\RR(v) + \iota'_\RR(x_\alpha) \in \mathcal{G}(\chi_l)^{-1}(U)\quad &\Longleftrightarrow\quad (\iota'_\RR(v) + \iota'_\RR(x_\alpha))(\chi_l) \in U\\
\quad &\Longleftrightarrow\quad\chi_l(v)\cdot \chi_l(x_\alpha)\in U\\
\quad &\Longleftrightarrow\quad\iota_\RR(v+x_\alpha) \in \chi_l^{-1}(U),
\end{align*}
and we find $\alpha_0\in I$ such that $\iota_\RR(v+x_\alpha) \in \chi_l^{-1}(U)$ for all $\alpha\geq \alpha_0$. The last statement is clear from $\lim_\alpha \iota_\RR(v+x_\alpha)=\ovl{z}$ and that a base of neighbourhoods of $\z$ in $\qR$ is formed by finite intersections of sets of \textbf{Type 2} and \textbf{Type 3}.
\hspace*{\fill}{\footnotesize$\dagger$}
\end{itemize}
\endgroup
\item
\label{lemma:nullBohr}
Observe that $\xi(\ovl{x})(\chi_l)=1$ for all $l\in \mathbb{R}$ iff $\ovl{x}\in \{0_\mathbb{R},0_{\mathrm{Bohr}}\}$. In fact,
of course we have $\xi(\ovl{x})(\chi_l)=1$ for $\ovl{x}\in \{0_\mathbb{R},0_{\mathrm{Bohr}}\}$. Conversely, if $\xi(\ovl{x})(\chi_l)=1$ for all $l\in \mathbb{R}$ and
$\ovl{x}=y\in \mathbb{R}$, then $y= 0$ because $\xi(\ovl{x})(\chi_{\pi/2y})=\I$ if $y\neq 0$. Similarly, if $\ovl{x}\in \RB$, then $\ovl{x}(\chi_l)=1=0_{\mathrm{Bohr}}(\chi_\tau)$ for all $l \in \mathbb{R}$. Hence, $\ovl{x}=0_{\mathrm{Bohr}}$ as the functions $\chi_l$ generate $\CAP(\RR)$.
\hspace*{\fill}{$\lozenge$}
\end{enumerate}
\end{lemrem}
The next proposition shows that $\qRR\cong \qR$ cannot be equipped with a group structure continuous w.r.t.\ its canonical Gelfand topology.
\begin{proposition}
\label{prop:noGroupStruc}
There is no continuous group structure on $\qRY$.
\begin{proof}
Assume there is such a group structure with multiplication $\star$ and unit element $e$.
In a first step we show that there is some $\ovl{x}\in \qRY$ for which the continuous\footnote{Recall that the relative topology of $\RB$ w.r.t.\ that on $\qRY$ equals its usual one.}
restriction
$\star[\ovl{x},\cdot\:]|_{\RB}$
takes at least one value in $\oRR$.
So, assume that this is not the case. Then $\star[\ovl{x},\RB]\subseteq \RB$ for each $\ovl{x} \in \qRY$ so that for $\psi\in \RB\neq \emptyset$ we obtain $e=\star\big[\psi^{-1},\psi\big]\in\RB$. It follows that $\qRY=\star\big[\qRY,e\big]\subseteq \RB$, which is impossible.
Consequently, we find $\ovl{x} \in \qRY$ and $\psi\in \RB$ such that $\ovl{x}\star \psi\in \oRR$.
Then, the preimage $U$ of $Y$ under the continuous map $\star[\ovl{x},\cdot\:]|_{\RB}$ is a non-empty open subset of $\RB$.
Since $\RB$ is compact, finitely many translates of the form $\psi+ U$ cover $\RB$, so that for the cardinality of $\RB$ we obtain
\begin{align*}
| \RB |&=\left|\bigcup_{i=0}^n\psi_i+U\right|\leq \left|\bigsqcup_{i=0}^n U\right|
=\left|\bigsqcup_{i=0}^n \star[\ovl{x},U]\right|\leq n|\mathbb{R}|=|\mathbb{R}|.
\end{align*}
But, this is impossible because $|\RB|> |\mathbb{R}|$.
In fact, by Lemma \ref{lemma:Bohriso} $\RB=\mathrm{Spec}(\CAP(\RR))$ is in bijection with the set $\RB'$ of all (not necessarily continuous) unital homomorphisms $\Gamma\rightarrow S^1$ for $\Gamma$ the dual\footnote{The set of all continuous characters on $\RR$, i.e., $\gamma=\{\chi_l\:|\: l\in \RR\}$.} group of $\RR$. Consequently, it suffices to show that $|\RB'|> |\mathbb{R}|$.
For this, let $\{\tau_\alpha\}_{\alpha\in I}\subseteq \mathbb{R}$ be a $\mathbb{Q}$-base of $\mathbb{R}$, then\footnote{$\mathbb{R}$ equals the set $F$ of all finite subsets of $\mathbb{Q}\times I$. Then $|F|=|\mathbb{Q}\times I|$ since $\mathbb{Q}\times I$ is infinite. Similarly, one has $|\mathbb{Q}\times I|=|I|$ because $I$ is infinite.}
$|I|=|\mathbb{R}|$ and we obtain an injective map $\iota\colon 2^I\rightarrow \RB$ as follows.
For $J\subseteq I$ let
\begin{align*}
\delta_J(\alpha):=\left\{
\begin{array}{ll}
0 & \mbox{if } \alpha\in J \\
\frac{2\pi}{\tau_\alpha} & \mbox{if } \alpha\notin J
\end{array}\right.
\end{align*}
and define $\psi(J)\colon \Gamma\rightarrow S^1$ by $\psi(J)(\chi_0):=1$ as well as
\begin{align*}
\psi(J)\left(\chi_{\tau}\right):=\prod_{i=1}^n \chi_{q_i \cdot \tau_{\alpha_i}}(\delta_J(\alpha_i))\quad \text{for } \tau=\sum_{i=1}^l q_i\cdot \tau_{\alpha_i} \text{ with } q_1,\dots,q_l \in \mathbb{Q}.
\end{align*}
Then $\iota\colon J\mapsto \psi(J)$ is injective because
$\psi(J)(\chi_{q\cdot \tau_\alpha})=1$ for all $q\in \mathbb{Q}$ iff $\alpha\in J$, hence
$|\RB|=|\RB'|\geq |2^I|=|2^\RR|>|\mathbb{R}|$.
\end{proof}
\end{proposition}
So, since it is not possible to equip $\qR$, i.e., $\qRR$ with a canonical Haar measure, we now concentrate on some general properties of normalized Radon measures on $\qR$.
\begin{lemma}
\label{lemma:Radon}
Let $\Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$, $\mathfrak{B}(\mathbb{R})$ and $\mathfrak{B}(\RB)$ denote the Borel $\sigma$-algebras of the topological spaces $\qR$, $\RR$ and $\RB$, respectively.
\begin{enumerate}
\item
\label{lemma:Radon1}
We have $\Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}=\mathfrak{B}(\mathbb{R})\sqcup\mathfrak{B}(\RB)$.
\item
\label{lemma:Radon2}
If $\mu$ is a finite Radon measure on $\Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$, then so are $\mu|_{\mathfrak{B}(\mathbb{R})}$ and $\mu|_{\mathfrak{B}(\RB)}$. Conversely, if $\mu_1$, $\mu_2$ are finite Radon measures on $\mathfrak{B}(\mathbb{R})$ and $\mathfrak{B}(\RB)$, respectively, then
\begin{align}
\label{eq:RadonMeasures}
\mu(A):=\mu_1(A\cap \mathbb{R})+ \mu_2(A\cap \RB)\quad \text{for}\quad A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}
\end{align}
is a finite Radon measure on $\Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
First observe that the right hand side is a $\sigma$-algebra and remember that the relative topologies on $\mathbb{R}$ and $\RB$ w.r.t.\ the topology on $\qR$ coincide with their usual ones. So, if
$U\subseteq \qR$ is open, then $U \cap \mathbb{R}$, $U \cap \RB$ are open in $\mathbb{R}$ and $\RB$, respectively. Hence, $U=(U \cap \mathbb{R}) \sqcup (U \cap \RB) \subseteq \mathfrak{B}(\mathbb{R})\sqcup\mathfrak{B}(\RB)$, showing that $\Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}\subseteq \mathfrak{B}(\mathbb{R})\sqcup\mathfrak{B}(\RB)$. Conversely, if $U\subseteq \mathbb{R}$ is open, then $U$ is open in $\qR$, hence $\mathfrak{B}(\mathbb{R})\subseteq \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$. Finally, if $A\subseteq \RB$ is closed, then $A$ is compact in $\RB$. This implies compactness of $A$ in $\qR$, so that $A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$ for each closed $A\subseteq \RB$. Since $\mathfrak{B}(\RB)$ is generated by all such closed subsets, we have $\mathfrak{B}(\RB)\subseteq \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$ as well.
\item
The measures $\mu|_{\mathfrak{B}(\mathbb{R})}$ and $\mu|_{\mathfrak{B}(\RB)}$ are well-defined by \ref{lemma:Radon1}) and obviously finite. So, it remains to show their inner regularity. But inner regularity is clear because a subset of $\mathbb{R}$ or $\RB$ is compact iff it is so w.r.t.\ topology on $\qR$.
For the second statement, let $\mu$ be defined by \eqref{eq:RadonMeasures}. Then $\mu$ is a finite Borel measure by \ref{lemma:Radon1}) and its inner regularity follows by a simple $\epsilon\slash 2$ argument from the inner regularities of $\mu_1$ and $\mu_2$.
\end{enumerate}
\end{proof}
\end{lemma}
So, by the above lemma each normalized Radon measure on $\qR$ can be written in the form
\begin{align}
\label{eq:mutetc}
\mu(A)=t\:\mu_1(A\cap \mathbb{R})+ (1-t)\:\mu_2(A\cap \RB) \qquad \forall\: A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}
\end{align}
for $t\in [0,1]$ and normalized\footnote{Of course, if $t=0$ or $t=1$, it doesn't matter which normalized Radon measure we choose on $\RR$ or $\RB$, respectively. So, in these cases we allow $\mu_1=0$ and $\mu_2=0$, respectively.} Radon measures $\mu_1$ and $\mu_2$ on $ \mathfrak{B}(\mathbb{R})$ and $\mathfrak{B}(\RB)$, respectively. So, the crucial step is to fix the measures $\mu_1, \mu_2$ and the parameter $t$.
For the dependence of the induced Hilbert space structure on the parameter $t$ observe that for $\mu_1,\mu_2$ fixed, $t_1,t_2\in (0,1)$ and $\mu_{t_1},\mu_{t_2}$ the respective measures defined by \eqref{eq:mutetc},
the spaces $\Lzw{\qR}{\mu_{t_1}}$ and $\Lzw{\qR}{\mu_{t_2}}$ are isometrically isomorphic.
In fact, for $A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$ let $\chi_A$ denote the respective characteristic function and define the map
\begin{align*}
\phi \colon \Lzw{\qR}{\mu_{t_1}}&\rightarrow \Lzw{\qR}{\mu_{t_2}}\\
f & \mapsto \sqrt{\frac{t_1}{t_2}} \:\chi_{\RR}\cdot f + \sqrt{\frac{(1-t_1)}{(1-t_2)}}\:\chi_{\RB}\cdot f.
\end{align*}
Then, $\phi$ is an isometric isomorphism just by the general transformation formula, so that the parameter $t$ gives rise to at most $3$ different (not canonically isomorphic) Hilbert space structures, see also Lemma \ref{lemma:techlemma}.
Anyhow, the next corollary to Proposition \ref{lemma:bohrmassdichttrans} and Lemma and Remark \ref{remtrnas}.\ref{remtrnas11} shows that invariance of $\mu$ under the translations $\TTransw_v\colon \qR\rightarrow \qR$ already forces $t=0$ and $\mu_2=\muB$.
\begin{corollary}
\label{cor:eindbohr}
\begin{enumerate}
\item
If $\mu$ is a normalized Radon measure on $\qR$ with $\TTransw_v(\mu)=\mu$ for all $v\in\RR$, then $\mu=\muB$, i.e., $t=0$. In particular, if one wants $\TTransw_v^* \colon \Lzw{\qR}{\mu}\rightarrow \Lzw{\qR}{\mu}$, $f\mapsto f\cp \Transw_v$ to be unitary for each $v\in \RR$, then $t$ has necessarily to be zero and $\mu_2$ has to equal $\muB$.
\item
The one-parameter group $\big\{\hspace{-1pt}\TTransw_v^*\big\}_{v\in \RR}$ of unitary operators
\begin{align*}
\TTransw_v^* \colon \Lzw{\qR}{\muB}\rightarrow \Lzw{\qR}{\muB},\quad f\mapsto f\cp \TTransw_v
\end{align*}
is strongly continuous.\footnote{Of course, here we mean
$\muB(A):=\muB(A\cap \RB)$ for all $A\in \mathfrak{B}(\RR\sqcup \RB)$.}
\end{enumerate}
\end{corollary}
\begin{proof}
\begin{enumerate}
\item
Assume that $t>0$ and that $\mu_1$ is finite. If $\mu(K)>0$ for $K\subseteq \RR$ compact, then $\mu_1(\RR)=\infty$ just by $\sigma$-additivity and \eqref{eq:sigmaquer}. Consequently, $\mu_1(K)=0$ for all compact subsets $K\subseteq \RR$, hence $\mu_1=0$. Now \eqref{eq:sigmaquerbohr} shows that $\mu_2$ fulfils the requirements of Proposition \ref{lemma:bohrmassdichttrans}.\ref{lemma:bohrmassdichttrans1}, so that $\mu_2=\muB$ follows.
Finally, unitality of the maps $\TTransw_v$ implies $\mu(\TTransw(v,K))=\mu(K)$ for each compact subset $K\subseteq \qR$ and each $v\in \RR$, hence $\TTransw_v(\mu)=\mu$ for each $v\in \RR$ by inner regularity. Consequently, the last part is clear from the first one.
\item
This is clear from Proposition \ref{lemma:bohrmassdichttrans}.\ref{lemma:bohrmassdichttrans2} since by \eqref{eq:sigmaquer} and \eqref{eq:sigmaquerbohr} we have
\begin{align*}
\|\TTransw^*_v(f)-\TTransw^*_{v'}(f)\|_2=\|\Transl^*_v(f|_{\RB})-\Transl^*_{v'}(f|_{\RB})\|_2\qquad \forall\: f\in \Lzw{\qR}{\muB}.
\end{align*}
\end{enumerate}
\end{proof}
Even if Corollary \ref{cor:eindbohr} singles out the measure $\muB$ (and support the standard LQC approach from the mathematical side), in the following subsections (except for the next short one), we
will be concerned with the construction of a projective structure and consistent families of normalized Radon measures on $\qR$. Basically, this will be done in analogy to the construction in \cite{oai:arXiv.org:0704.2397} for the standard LQC configuration space $\RB$, and we will end up with singling out the measures of the form
\begin{align}
\label{eq:fammeas}
\mu(A)=t\:\adif(\lambda)(A\cap \mathbb{R})+ (1-t)\:\muB(A\cap \RB) \qquad \forall\: A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}
\end{align}
for $t\in [0,1]$. Here,
$\adif(\lambda)$ denotes the push forward of the restriction of the Lebesgue measure to $\mathfrak{B}((0,1))$ by a homeomorphism $\adif\colon (0,1)\rightarrow \RR$. At least the $\RB$ part of these measures then is in line with the above Corollary.
The main intention for our investigations is rather to provide a mathematically satisfying derivation of these choices than a physically justification for this.
However, before we start constructing measures, we first investigate the inclusion relations between the spaces $\A_\w$ and $\ARQw\cong \ARRQw= \qRR\cong \qR$.
\subsection{Quantization vs. Reduction}
\label{subsec:QuantvsRed}
As we have seen in Remark \ref{StanLQC}.\ref{StanLQC2}, $\ARRQLL\cong \AQRL$ holds, i.e.,
quantization and reduction commute if one only uses linear curves for both the full quantum configuration space $\A_\lin$ and the quantized reduced classical space $\ARRQLL$.
In this short subsection we show that this is no longer true if one takes all embedded analytic curves into account.
We start with the following diagram sketching the relevant spaces and their relations
\begin{figure}[h]
\begin{minipage}[h]{\textwidth}
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
\qR \ar@{->}[r]^-{\xi}_-{\cong} &\qRR \ar@{->}[r]^-{\ovl{i^*_\AR}}_-{\cong} & \: \ARQw \: \ar@{->}[r]^-{\subseteq} & \AQRw\ar@{->}[r]_-{\cong}^-{\kappa} &\IHOMW \ar@{->}[r]^-{\Omega}_-{\cong} & \ITRHOMW \\
& \mathbb{R} \ar@{->}[u]^-{\iota_\mathbb{R}} \ar@{->}[r]^-{\cong} &\: \AR\: \ar@{->}[u]^-{\iota_{\AR}}, &&}
\end{xy}}
\caption*{\hspace{-120pt}\textbf{Embedding: }$\varsigma:=\Omega\cp \kappa \cp \ovl{i^*_\AR}\cp \xi\colon \qR\hookrightarrow \ITRHOMW$}
\end{center}
\end{minipage}
\end{figure}
\FloatBarrier
\noindent
and denote the concatenation of all these maps by $\varsigma\colon \qR\rightarrow \IHOMW$. Here, $\Omega=\Omega_\nu$ for $\nu$ the canonical choice $\nu_x=(x,\me)$ for all $x\in M$, cf.\ last two parts of Remark \ref{rem:homaction}.
To better understand what this map $\varsigma$ actually does, observe that:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item[a)]
For $\rho$ the standard representation of $\SU$, we obtain from \eqref{eq:hilfseq} that
\begin{align*}
(\Omega\cp\kappa)(\ovl{\w})(\gamma)=\big(\ovl{\w}([h_\gamma^\nu]_{ij})\big)_{ij}\qquad \forall\:\gamma\in \Paw,\:\forall\:\ovl{\w}\in \A.
\end{align*}
\item[b)]
Applying the definitions, we find that
\begin{align*}
\ovl{i^*_\AR}(\ux)([h_\gamma^\nu]_{ij})=\ux(i_\AR^*([h_\gamma^\nu]_{ij}))=\ux\big([h_\gamma^\nu]_{ij}\cp i_\AR\big)=\ux\big([h_\gamma^\nu]_{ij}|_\AR\big)\qquad \forall\: \ux\in \qRR.
\end{align*}
\item[c)]
For a linear curve $\gamma\colon [0,l]\rightarrow \RR^3$, $t\mapsto t\cdot \vec{v}$ for $\vec{v}\in \RR^3$ with $\|\vec{v}\|=1$ we have
\begin{align*}
[h_\gamma^\nu]_{ij}(\w^c)=\big(\pr_2\cp \parall{\gamma}{\w^c}\big)(\gamma(0),\me)= \exp(-cl \cdot\murs(\vec{v}))\stackrel{\eqref{eq:expSU2}}{=} \cos(-cl)\cdot \me + \sin(-cl) \cdot\murs(\vec{v}).
\end{align*}
Here, the first step is clear from the definitions, and for the second step one might apply \eqref{eq:trivpar} to the fact that $\gamma(t)=\wm(\exp(t\vec{v}),0)$ is Lie algebra generated (cf.\ Definition \ref{def:analytLieAlgBD}.\ref{def:analytLieAlgBD2}), i.e., $\gamma=\gamma_{\vec{v}}^0|_{[0,l]}$ for $0\in \RR^3$ and $\vec{v} \in \RR^3\times \su$ the Lie algebra of $\Ge=\Gee$.
\end{itemize}
\endgroup
\noindent
Using this, we obtain the following Corollary to Lemma and Definition \ref{remdefchris}.\ref{lemma:nullBohr}.
\begin{corollary}
\label{cor:nullbohr}
If $\varsigma(\ovl{x})(\gamma)=\me$ for all $\gamma\in \Pal$,
then $\ovl{x}\in\{0_{\mathrm{Bohr}},0_\mathbb{R}\}$.
\begin{proof}
It suffices to show the claim for curves as in c) with $\vec{v}=\vec{e}_1$ and $l>0$. Then
\begin{align*}
\begin{split}
\me&=\varsigma(\ovl{x})(\gamma)
=(\Omega\cp \kappa)\Big(\ovl{i^*_\AR}(\xi(\ovl{x}))\Big)(\gamma)\\
\!\!&\stackrel{\text{a)}}{=}\left(\Big(\ovl{i^*_\AR}(\xi(\ovl{x}))\Big)\big([h_\gamma^\nu]_{ij}\big)\right)_{ij}
\stackrel{\text{b)}}{=}\left(\xi(\ovl{x})\left([h_\gamma^\nu]_{ij}|_\AR\right)\right)_{ij}\\
\!\!&\stackrel{\text{c})}{=}\begin{pmatrix} \xi(\ovl{x})(c\mapsto \cos(-l c)) & \mathrm{i}\: \xi(\ovl{x})(c\mapsto \sin(-l c)) \\ \mathrm{i}\: \xi(\ovl{x})(c\mapsto \sin(-l c)) & \xi(\ovl{x})(c\mapsto \cos(-l c)) \end{pmatrix},
\end{split}
\end{align*}
hence $\xi(\ovl{x})(\chi_{l})=1$ for all $l\in \mathbb{R}$. So, the claim is clear from
Lemma and Definition \ref{remdefchris}.\ref{lemma:nullBohr}.
\end{proof}
\end{corollary}
\begin{proposition}
\label{lemma:propincl}
We have $\ARQw\subsetneq \AQRw$.
\end{proposition}
\begin{proof}
Let
$\tilde{\hommm} \colon \mathbb{R}_{>0}\times (0,2\pi)\rightarrow \mathbb{R}$ be a function with
\begin{align*}
\tilde{\hommm}(r,x+y)=\tilde{\hommm}(r,x)+\tilde{\hommm}(r,y) \:\bmod \: 2\pi
\end{align*}
whenever $r\in \mathbb{R}_{>0}$ and $x,y,x+y\in (0,2\pi)$.
Then, we obtain an element $\hommm \in \ITRHOMW$
if we define \big(cf.\ (b) in Convention \ref{conv:sutwo1}.\ref{conv:sutwo2} for the definition of $\Pacirc$ and $\gc{n}{r}{x}{\tau}$\big)
\[
\hommm(\gamma):=
\begin{cases}
\exp\big(\tilde{\hommm}(\|\vec{r}\|,\tau)\: \murs(\vec{n})\big) &\mbox{if } \gamma \csim \gc{n}{r}{x}{\tau}\in \Pacirc\\
\me & \mbox{else}.
\end{cases}
\]
Here, well-definedness follows from analyticity and Lemma \ref{lemma:sim}.\ref{lemma:sim5}.\footnote{Observe that $\gc{n}{r}{x}{\tau}\psim x + \gamma^{\vec{r}}_{(0,\murs({\vec{n}}))}\big|_{[0,\tau/2]}$ for ${\vec{r}}$ considered as point in $\RR^3$ and $(0,\murs({\vec{n}})\in \RR^3\times \su=\mg$, see also beginning of the next subsection.} Moreover, invariance is straightforward to see. Here, $0_\RR$ corresponds to the choice $\wt{\hommm}=0$ ($\w_0$ is the trivial connection) but the homomorphism that corresponds to $\NB$ is not so easy to compute. However, it is easily checked that choosing different maps $\wt{\hommm}$ we obtain more than two different invariant homomorphisms with $\hommm(\gamma)=\me$ for all $\gamma\in \Pal$. Since by Corollary \ref{cor:nullbohr} that cannot all come from an element of $\qR\cong \ARQw$, the claim follows.
\end{proof}
For the rest of this section we will be concerned with the construction of a projective structure (and consistent families of normalized Radon measures) on the space $\qR\cong\qRR$. The next subsection (except for Lemma \ref{lemma:WeierErzeuger}.\ref{lemma:WeierErzeuger2}) then serves as a motivation for our later constructions. Basically, there we discuss which problems occur if one tries to use projection maps involving the identification of $\qRR$ with a subset of $\IHOMW$. However, some of the investigations are quite technical and exhausting, and the reader not so interested in these difficulties may just take a look at the second part of Lemma \ref{lemma:WeierErzeuger} in order to proceed with Subsection \ref{subsec:ProjStrucon}.
\subsection{Motivation of the Construction}
\label{subsec:Motivation}
Recall that the $\Cstar$-algebra $\rR=C_0(\RR)\oplus \CAP(\RR)$ is already generated by parallel transports along linear and circular curves \cite{ChrisSymmLQG}, i.e., that the $\Cstar$-algebras $\rR=\ovl{\PaC_{\w}|_{\AR}}$ and $\rR_{\lin\mc}:=\ovl{\PaC_{\lin\mc}|_{\AR}}$ coincide. Here, $\PaC_{\lin\mc}$ denotes the
$\Cstar$-algebra of cylindrical functions that corresponds to the set of curves $\Pa_{\lin\mc}=\Pall \sqcup \Pacirc$ for $\Pall$ and $\Pacirc$ defined as in Convention \ref{conv:sutwo1}.\ref{conv:sutwo2}. This means that an element of $\qRR$ is completely determined by its values on the generators of $\PaC_{\lin\mc}$, i.e., on the matrix entries of the parallel transport functions $\RR \ni c\mapsto \parall{\gamma}{\w^c}$ for $\gamma\in \Pa_{\lin\mc}$.
It follows that, using the natural identification of $\qRR$ with a subset of $\IHOMW$ via $\varsigma'$
\begin{figure}[h]
\begin{minipage}[h]{\textwidth}
\begin{center}
\makebox[0pt]{
\begin{xy}
\xymatrix{
&\qRR \ar@{->}[r]^-{\ovl{i^*_\AR}}_-{\cong} & \: \ARQw \: \ar@{->}[r]^-{\subseteq} & \AQRw\ar@{->}[r]_-{\cong}^-{\kappa} &\IHOMW,
}
\end{xy}}
\caption*{\hspace{-120pt}\textbf{Embedding: }$\varsigma':=\kappa \cp \ovl{i^*_\AR}\colon \qRR\hookrightarrow \IHOMW$}
\end{center}
\end{minipage}
\end{figure}
\FloatBarrier
\noindent
the values of the homomorphism $\varsigma'(\ux)$ that corresponds to $\ux\in \qRR$ are completely determined by its values on the elements of $\Pa_{\lin\mc}$. Moreover, due to the invariance of $\varsigma'(\ux)$ it suffices to consider elements of $\Pa_{\lin\mc}$ which are of the form
\begin{align}
\label{eq:kkurven}
\gamma_l:=\gamma_{\vec{e}_1,l}\qquad\forall\: l>0\qquad\qquad\text{and}\qquad\qquad\gamma_{\tau,r}:=\gamma_{\vec{e}_3,r\vec{e}_1}^{0,\tau}\qquad\forall\: r>0,\:\forall\: \tau\in (0,2\pi),
\end{align}
i.e., we can fix a traversing direction for linear curves, and a midpoint together with a traversing plain for the circular ones.
Now, since all these curves are Lie algebra generated, it makes sense to use the maps $\pi_p$ (see also b) below) from Lemma and Definition \ref{def:topo} which we have introduced Subsection \ref{sec:ConSp} in order to investigate the space $\IHOMLAS$. For this observe that
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item[a)]
The elements of $\Pa_{\lin\mc}$ are Lie algebra generated because
\begin{align}
\label{eq:kurven}
x+\gamma_{\vec{v},l}=\gamma^x_{(\vec{v},0)}\big|_{[0,l]}\qquad\qquad\text{and}\qquad\qquad\gc{n}{r}{x}{\tau}\psim x + \gamma^{\vec{r}}_{(0,\murs({\vec{n}}))}\big|_{[0,\tau/2]}
\end{align}
where the second equivalence is clear from the explanations to \eqref{eq:expSU2} in Convention \ref{conv:sutwo1}.\ref{conv:sutwo111}.
\item[b)]
For $p\in P$, $x=\pi(p)$ and $\g\in \mg\backslash \mg_x$ we have
\begin{align*}
\pi_p(\g,l,\homm) = \Delta\big(\Phi_{\exp(l\g)}(p),\homm\big(\gamma_\g^{x}|_{[0,l]}\big)(p)\big)\qquad \forall\: l<\tau_\g.
\end{align*}
Hence,
$\pi_p(\g,l,\cdot)\colon \IHOM\rightarrow \SU$
assigns to $\homm\in \IHOM$ the difference in $F_{{\gamma_{\g}^x}(l)}$ between $\Phi_{\exp(l\g)}(p)$ and $\homm\big(\gamma_\g^{x}|_{[0,l]}\big)(p)\big)$. Obviously, $\pi_p(\g,l,\cdot)$ is continuous.
\item[c)]
The big advantage of using the maps $\pi_p$ is that for $\g\in \mg\backslash \mg_x$ and $\homm\in \IHOM$ fixed, $\mathrm{im}[\pi_p(\g,\cdot,\homm)]$ is contained in a maximal torus in $\SU$. This is clear from Lemma \ref{lemma:torus}.\ref{lemma:torus1} and (combine \eqref{eq:Equi} with \eqref{eq:pip})
\begin{align*}
\pi_p(\g,l+l',\homm)=\pi_p(\g,l,\homm)\cdot \pi_p(\g,l',\homm)\qquad \text{for }l,l',l+l'<\tau_\g.
\end{align*}
\end{itemize}
\endgroup
\noindent
It follows that $\qRR$ is separated by the maps
\begin{align}
\label{eq:projmaps}
\begin{split}
\pi_l&\colon \ux \mapsto \pi_{(0,\me)}\big((\vec{e}_1,0),l,\varsigma'(\ux)\big)\qquad \forall\: l>0\\
\pi_{\tau,r}&\colon \ux \mapsto \pi_{(r\cdot \vec{e}_1,\me)}\big((0,\tau_3),\textstyle\frac{\tau}{2},\varsigma'(\ux)\big)\quad \forall\:r>0\:,\forall\: \tau\in (0,2\pi).
\end{split}
\end{align}
Here, $\pi_l$ and $\pi_{\tau,r}$ correspond to the choices, see \eqref{eq:kkurven} and \eqref{eq:kurven}
\begin{align*}
\gamma_l=\gamma^0_{(\vec{e}_1,0)}|_{[0,l]},\: p=(0,\me)\quad\qquad\text{and}\quad\qquad \gamma_{\tau,r}=\gamma^{r\vec{e}_1}_{(0,\tau_3)}|_{[0,\tau/2]},\: p=(r\cdot\vec{e}_1,\me),
\end{align*}
respectively.
The question whether we can use these projection maps in order to define reasonable measures on $\qRR$ then depends crucially on in which way the maximal tori (see c)) the maps
\begin{align*}
\pi_{\bullet}(\ux)\colon \RR_{>0}\rightarrow \SU,\: l \mapsto \pi_l(\ux)\qquad\quad\text{and}\quad\qquad \pi_{\bullet,r}(\ux)\colon (0,2\pi)\rightarrow \SU,\: \tau \mapsto \pi_{\tau,r}(\ux)
\end{align*}
are mapping to depend on $\ux\in \qRR$.
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
For $\pi_\bullet$ this question is immediately answered since a straightforward calculation shows that
\begin{align}
\label{eq:patralll}
\pi_l(\iota_\RR(c))&=h_{\gamma_l}^{\nu}(\w^c)
=\begin{pmatrix} \cos(lc) &\I\sin(lc) \\ \I\sin(lc) & \cos(lc) \end{pmatrix}
=\exp(-lc \tau_1)\in H_{\tau_1}\qquad \forall\: c\in \RR,\forall\: l>0
\end{align}
for $\iota_\RR \colon \AR \cong \RR \rightarrow \qRR$ the canonical embedding of $\RR$ into $\mathrm{Spec}(\CAP(\RR)\oplus C_0(\RR))$. Since $\mathrm{im}[\iota_\RR]$ is dense in $\qRR$, since $\pi_l$ is continuous and since $H_{\tau_1}$ is closed, we even have $\pi_l(\qRR)=H_{\tau_1}$ for all $l>0$. So, using $\pi_l$ as projection maps, we could take the Haar measure on $H_{\tau_1}\cong S^1$ in order to define a respective consistent family of normalized Radon measures.
\item
For $\pi_{\bullet,r}$ this dependence is more complicated. Here, a straightforward calculation shows that
\begin{align}
\label{eq:patrall}
\begin{split}
\pi_{\tau,r}(\iota_\RR(c))=\exp\big(\textstyle\frac{\tau}{2}\tau_3\big)^{-1}\cdot h_{\gamma_{\tau,r}}^\nu(\w^c)
&=\exp\hspace{-1pt}\big(\hspace{-1pt}-\textstyle\frac{\tau}{2}\cdot[2r c\cdot\tau_2+\tau_3]\big
\end{split}
\end{align}
holds for all $c\in \RR$, $r>0$ and all $\tau\in (0, 2\pi)$. Hence, $\pi_{\tau,r}(\iota_\RR(c))\in H_{[2rc\cdot\tau_2+\tau_3]}$ for all $\tau\in (0,2\pi)$, and we will see in Lemma \ref{lemma:BildCirc}.\ref{lemma:BildCirc3} that $\pi_{\tau,r}(\qRR\backslash \mathrm{im}[\iota_\RR])= H_{\tau_2}$ holds for all $\tau\in (0,2\pi)$ and all $r>0$. Further details of the set $\mathrm{im}[\pi_{\tau,r}]$ are provided in Lemma \ref{lemma:BildCirc}.
Then, applying \eqref{eq:expSU2} to \eqref{eq:patrall} we immediately see that for $\pc:= \textstyle\sqrt{c^2r^2+\frac{1}{4}}$ we have
\begin{align}
\label{eq:matrixentr}
\pi_{\tau,r}(\iota_\RR(c))\stackrel{\eqref{eq:expSU2}}{=}\begin{pmatrix} \cos(\pc \tau)+\frac{\I}{2\pc}\sin(\pc \tau) &\frac{cr}{\pc} \sin(\pc \tau) \\ -\frac{cr}{\pc}\sin(\pc \tau) & \cos(\pc \tau)-\frac{\I}{2\pc}\sin(\pc \tau) \end{pmatrix}.
\end{align}
\end{itemize}
\endgroup
\noindent
From this, we conclude the third part of the following
\begin{lemma}
\label{lemma:WeierErzeuger}
\begin{enumerate}
\item
\label{lemma:WeierErzeuger1}
Let $f\in C_0(\RR)$ vanishes nowhere. Then, the functions $\{f\}\sqcup \{\chi_l\}_{l\in \mathbb{R}}$ generate a dense $^*$-subalgebra of $C_0(\mathbb{R})\oplus \CAP(\RR)$. If $f$ is in addition injective,\footnote{Recall that $\mathrm{im}[f]\subseteq \mathbb{C}$.} then $f$ generates a dense $^*$-subalgebra of $C_0(\RR)$.
\item
\label{lemma:WeierErzeuger2}
Each nowhere vanishing injective $f\in C_0(\RR)$ is a homeomorphism onto its image.
\item
\label{lemma:WeierErzeuger3}
The matrix entries of parallel transports along all linear curves (with some fixed traversing direction) and one single circular curve already generate $C_0(\mathbb{R})\oplus \CAP(\RR)$. In particular, the projection maps $\{\pi_l\}_{l>0}$ and $\pi_{\tau,r}$ for some fixed reals $\tau,r>0$ separate the points in $\qRR$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Since $f(x)\neq 0$ for all $x\in \mathbb{R}$, the $^*$-algebra $\gG$ generated by $\{f \cdot\chi_l\}_{l \in \mathbb{R}}\subseteq C_0(\mathbb{R})$ separates the points in $\mathbb{R}$ and vanishes nowhere. Consequently, $\gG$ is dense in $C_0(\mathbb{R})$ by the complex Stone-Weierstrass theorem for locally compact Hausdorff spaces. Since $\CAP(\RR)$ is generated by the functions $\{\chi_l\}_{l \in \mathbb{R}}$, the first claim follows. If $f$ is in addition injective, then the $^*$-algebra generated by $f$ is dense in $C_0(\mathbb{R})$ because $f$ separates the points in $\mathbb{R}$ and vanishes nowhere.
\item
Let $\RR\sqcup \{\infty\}$ denote the one point compactification of $\RR$. Then $\ovl{f}\colon \RR\sqcup \{\infty\} \rightarrow \CCC$ defined by $\ovl{f}(\infty):=0$ and $\ovl{f}|_\RR:=f$ is continuous and injective, hence a homeomorphism onto its image $\mathrm{im}\big[\ovl{f}\hspace{1.5pt}\big]=\mathrm{im}[f]\sqcup\{0\}$. Consequently, $f^{-1}=\ovl{f}^{-1}|_{\mathrm{im}[f]}$ is continuous as well.
\item
If $l=\tau r$, then \eqref{eq:patralll} and \eqref{eq:matrixentr} show that
\begin{align}
\label{eq:funcco}
f(c):=\big[\pi_{\tau,r}(\iota_\RR(c))\big]_{11}-\left[\pi_l(\iota_\RR(c))\right]_{11}=\cos(\pc \tau)+\textstyle\frac{\I}{2\pc}\sin(\pc \tau)-\cos(cr \tau)
\end{align}
for all $c\in \RR$,
hence
$f\in C_0(\RR)$. Now, if $f(c)$ is zero, then $\sin(\pc \tau)$ must be $0$, hence $\cos(\pc \tau)$ must be $\pm 1$. Then $\cos(cr\tau)$ must be $\pm 1$ as well, showing that $c=\frac{\pi n}{r\tau}$ for some $n\in \mathbb{Z}\backslash\{0\}$. In fact, $n=0$ means $\sin(\tau/2)=0$, which contradicts that $\tau\in (0,2\pi)$. Then
\begin{align*}
\cos(\pc \tau)=\cos\left(\pi n \textstyle\sqrt{1+\textstyle\frac{\tau^2}{4 \pi^2n^2}}\right),
\end{align*}
but $n \sqrt{1+\textstyle\frac{\tau^2}{4 \pi^2n^2}}\notin \mathbb{Z}$ because $0<\tau <2\pi$. Consequently, $\cos(\pc \tau)\neq 1$, which contradicts that $\cos(\pc \tau)-\cos(cr\tau)=0$. This shows that $f$ vanishes nowhere. Now, since
\begin{align*}
h_{\gamma_{l}}^\nu(\w^c) \stackrel{\eqref{eq:patralll}}{=}\pi_{l}(\iota_\RR(c)) \qquad\text{as well as}\qquad
h_{\gamma_{\tau,r}}^\nu(\w^c) \stackrel{\eqref{eq:patrall}}{=}\exp(\textstyle\frac{\tau}{2}\tau_3)\cdot\pi_{\tau,r}(\iota_\RR(c)),
\end{align*}
$f$ is also contained in the $^*$-algebra generated by the parallel transports along $\gamma_{\tau,r}$ and $\{\gamma_l\}_{l>0}$. Since by \eqref{eq:patralll} this algebra also contains all characters $\chi_l$,
the first statement follows from Part \ref{lemma:WeierErzeuger1}).
Now, applying the definitions we find that
\begin{align}
\label{eq:pilrtau}
\begin{split}
\pi_l(\ux)&=\left(\ux\left([h_{\gamma_l}^\nu]_{ij}|_{\AR}\right)\right)_{ij}\\
\pi_{\tau,r}(\ux)&= \exp\left(\textstyle\frac{\tau}{2}\tau_3\right)^{-1}\cdot\big(\ux\big([h_{\gamma_{\tau,r}}^\nu]_{ij}|_\AR\big)\big)_{ij}.
\end{split}
\end{align}
So, since we have shown that the functions $\{[h_{\gamma_l}^\nu]_{ij}|_\AR\}_{l>0}$ and $[h_{\gamma_{\tau,r}}^\nu]_{ij}|_\AR$ generate $C_0(\RR)\oplus \CAP(\RR)$, and since $\ux\in \mathrm{Spec}(C_0(\RR)\oplus \CAP(\RR))$, the claim follows.
\end{enumerate}
\end{beweis}
\end{lemma}
\begin{remdef}
\label{rem:bohrproj}
\begin{enumerate}
\item
\label{rem:bohrproj1}
In the following, for $k\in \NN_{\geq 1}$ by $S^k$ we understand $k$-fold product of the unit circle $S^1$, i.e., $S^k:=\big(S^1\big)^k$. Since in the beginning of this section we have fixed the structure group of our principal fibre bundle to be $\SU$, this will not be in conflict with our notations.
\item
\label{rem:bohrproj2}
As already stated in Remark \ref{StanLQC}, in standard homogeneous isotropic LQC the quantum configuration space is given by $\RB$ and homeomorphic to $\ITRHOML$.\footnote{The homeomorphism was basically due to the fact that invariance restricts the value of $\hommm\in \ITRHOML$ on a linear curve traversing into the direction $\vec{v}$ to the maximal torus $H_{\vec{v}}$ in $\SU$.}
There, a projective structure and a consistent family of normalized Radon measures can be defined as follows: \cite{oai:arXiv.org:0704.2397}
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
One considers the set $I$ of all $\ZZ$-independent tuples $L=(l_1,\dots,l_k)$ with $l_1\dots,l_k\in \RR$, and defines $(l_1\dots,l_k) \leqZ (l'_1\dots,l'_{k'})$ iff $l_i\in \Span_\ZZ(l'_1\dots,l'_{k'})$ for all $1\leq i\leq k$.
\item
One defines the projection maps by $\pi_L\colon \RB\rightarrow S^{|L|}=:X_L$, $\psi \mapsto(\psi(\chi_{l_1}),\dots, \psi(\chi_{l_k}))$ as well as the transition maps by
\begin{align*}
\pi^{L'}_L\colon S^{|L'|}&\rightarrow S^{|L|}\\
(s_1,\dots ,s_{k'})&\mapsto \textstyle\left(\prod_{i=1}^{k'}s_i^{n_{i1}},\dots,\prod_{i=1}^{k'}s_i^{n_{ik}}\right),
\end{align*}
where $l_i=\sum_{j=1}^{k'}n_{ij} l'_j$ for $1\leq i\leq k$. Surjectivity of $\pi_L$ then is clear from Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod24}, and continuity is immediate from the definitions of the Gelfand topology on $\RB$. The remaining properties of a projective limit then are easily verified.
\item
One fixes the Haar measures $\mu_{|L|}$ on $S^{|L|}=X_L$. This gives rise to a consistent family of normalized Radon measures which exactly corresponds to the Haar measure on $\RB$, i.e., $\pi_L(\muB)=\mu_{|L|}
, see, e.g., proof of Lemma \ref{lemma:Projm}.
\end{itemize}
\endgroup
Equivalent to this is to define the projection maps (cf.\ \eqref{eq:tori} for the definition of $H_{\vec{e}_1}$)
\begin{align*}
\pi'_L\colon \ITRHOML&\rightarrow [H_{\vec{e}_1}]^{|L|}=:X'_L\\
\homm &\mapsto (\homm(\gamma_{\vec{e}_1,l_1}),\dots,\homm(\gamma_{\vec{e}_1,l_k}))
\end{align*}
(and respective transition maps),
and $\mu'_L$ the Haar measure on $[H_{\vec{e}_1}]^{|L|}\cong S^{|L|}$. \hspace*{\fill}$\lozenge$
\end{enumerate}
\end{remdef}
The third part of Lemma \ref{lemma:WeierErzeuger} shows that for the separation property from Definition \ref{def:ProjLim}.\ref{def:ProjLim3} it suffices to consider the projection maps $\pi_l$ for all $l>0$ and $\pi_{\tau,r}$ for some fixed reals $\tau,r>0$. Let
\begin{align*}
\pi'_l:=\pi_l\cp \xi \qquad\qquad \text{and}\qquad\qquad \pi'_{\tau,r}:=\pi_{\tau,r}\cp \xi.
\end{align*}
Then, for $L\in I$ as above and
\begin{align*}
\pi''_L\colon \qR &\rightarrow [H_{\vec{e}_1}]^{|L|}\\
\x&\mapsto \big(\pi'_{l_1}(\x),\dots,\pi'_{l_k}(\x)\big)
\end{align*}
we also have $\mathrm{im}[\pi''_L]=[H_{\vec{e}_1}]^{|L|}$.\footnote{We even have $\pi''_L(\RB)=[H_{\vec{e}_1}]^{|L|}$ (for $\RB\subseteq \qR=\RR\sqcup \RB$). This follows from Lemma and Convention \ref{lemconv:RBMOD}.} So, using the projection maps $\pi''_L$ we can use the Haar measure on $[H_{\vec{e}_1}]^{|L|}$, and the crucial question then is
whether such a canonical measure also exists on the image of $\pi'_{\tau,r}$. In addition to that, a suitable directed set and corresponding transition maps have to be defined. For these reasons we now investigate the
image of $\pi'_{\tau,r}$ in more detail.
\begin{lemma}
\label{lemma:BildCirc}
Let $\tau,r>0$ be fixed.
\begin{enumerate}
\item
\label{lemma:BildCirc1}
$\pi'_{\tau,r}\big(\qR\big)$ is of measure zero w.r.t.\ the Haar measure on $\SU$.
\item
\label{lemma:BildCirc2}
There is no proper Lie subgroup $H\subsetneq \SU$ that contains $\pi'_{\tau,r}\big(\qR\big)$.
\item
\label{lemma:BildCirc3}
We have $\pi'_{\tau,r}\big(\qR\big)=\pi'_{\tau,r}\big(\RR\big)\cup H_{\tau_2}$ with
\begin{align*}
\pi'_{\tau,r}(\RB)=H_{\tau_2}\qquad\quad \text{and}\qquad\quad\pi'_{\tau,r}(\RR)\:\cap\: \pi'_{\tau,r}(\RB)= \{\pm\me\}.
\end{align*}
\item
\label{lemma:BildCirc4}
Let $a_n:=\frac{\sign(n)}{r}\sqrt{\frac{n^2\pi^2}{\tau^2}-\frac{1}{4}}$ for $n\in \mathbb{Z}_{\neq 0}$ and
\begin{align*}
A_0:=(a_{-1},a_1)\qquad A_n:=(a_n,a_{n+1})\:\text{ for }\:n\geq 1\qquad A_n:=(a_{n-1}, a_n)\:\text{ for }\:n\leq -1.
\end{align*}
Then, $\pi'_{\tau,r}|_{A_n}$ is injective for all $n\in \ZZ$,
\begin{align*}
\pi'_{\tau,r}(a_{n})=\me\:\text{ iff }\:|n|\text{ is even}\qquad\qquad \pi'_{\tau,r}(a_{n})=-\me\:\text{ iff }\:|n|\text{ is odd}
\end{align*}
as well as $\pi'_{\tau,r}\big(A_m\big)\cap \pi'_{\tau,r}\big(A_n\big)=\emptyset$ for all $m,n\in \ZZ$ with $m\neq n$.
For increasing $|n|$, the sets
\begin{align*}
B_n:=[a_{2n},a_{2(n+1)}]\:\text{ for }\:n\geq 1\qquad\qquad B_n:=[a_{2(n-1)}, a_{2n}]\:\text{ for }\:n\leq -1
\end{align*}
merge to $H_{\tau_2}$ in the following sense. For each
$\epsilon > 0$ we find $n_\epsilon\in \mathbb{N}_{\geq1}$ such that for $|n| \geq n_\epsilon$ we have
\begin{align*}
\forall\: s\in B_n:\exists\: s' \in H_{\tau_2}:\|s-s'\|_{\mathrm{op}}\leq \epsilon.
\end{align*}
\end{enumerate}
\begin{proof}
The proof can be found in Appendix \ref{sec:ProofOfLemmaImCirc}.
\end{proof}
\end{lemma}
The first and the second part of the above lemma already show that it is hard to equip $\mathrm{im}[\pi'_{\tau,r}]$ with a reasonable measure.
In addition to that, it is difficult to define a reasonable ordering, i.e., a directed set labeling the projection spaces.
Indeed, the first thing one might try is to define $\pi'_{\tau,r}\leq \pi'_l$ or $\pi'_l\leq \pi'_{\tau,r}$. However, then one has to define reasonable transition maps between $\mathrm{im}[\pi'_{\tau,r}]$ and $\mathrm{im}[\pi'_l]$, i.e., maps
\begin{align*}
\TT_1\colon \mathrm{im}[\pi'_{\tau,r}]\rightarrow \mathrm{im}[\pi'_l] \quad\qquad\text{or}\quad \qquad \TT_2\colon \mathrm{im}[\pi'_l]&\rightarrow \mathrm{im}[\pi'_{\tau,r}]
\end{align*}
with $\TT_1\cp \pi'_{\tau,r}=\pi'_l$ or $\TT_2\cp \pi'_l=\pi'_{\tau,r}$, respectively. This, however, is difficult and even impossible if $\tau r =l$:
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
\itspace
Let $0, \frac{2\pi}{l}\in \RR\subseteq \qR$, i.e., $0=\xi^{-1}(\iota_\RR(0))$ and $\frac{2\pi}{l}=\xi^{-1}\big(\iota_\RR\big(\frac{2\pi}{l}\big)\big)$, cf.\ \eqref{eq:rmap}. Then
\begin{align*}
\pi'_l(0)=\pi_l(\iota_\RR(0))\stackrel{\eqref{eq:patralll}}{=}\me=\pi'_l\big(\textstyle\frac{2\pi}{l}\big)\qquad\text{but}\qquad \pi'_{\tau,r}(0)\stackrel{\eqref{eq:patrall}}{=}\exp(-\frac{\tau}{2}\tau_3)\neq \pi'_{\tau,r}\big(\frac{2\pi}{l}\big).
\end{align*}
Here, the inequality on the right hand side is clear because by Lemma \ref{lemma:BildCirc}.\ref{lemma:BildCirc4} we have that $\pi'_{\tau,r}(x)=\pi'_{\tau,r}(y)$ for $x\neq y$ enforces $\pi'_{\tau,r}(x)=\pm \me$. Consequently, there cannot exist a transition map $\TT_2\colon \mathrm{im}[\pi'_l]\rightarrow \mathrm{im}[\pi'_{\tau,r}]$ as then
\begin{align*}
\pi'_{\tau,r}(0)=(\TT_2\cp \pi'_l)(0)=(\TT_2\cp \pi'_l)\big(\textstyle\frac{2\pi}{l}\big)=\pi'_{\tau,r}\big(\textstyle\frac{2\pi}{l}\big)
\end{align*}
would hold.
\item
\itspace
We have $\pi'_{\tau,r}(a_{2n})=\me$ for all $n\in \mathbb{Z}_{\neq 0}$, so that for a transition map $\TT_1\colon \mathrm{im}[\pi'_{\tau,r}]\rightarrow \mathrm{im}[\pi'_{l}]$ we would have
\begin{align*}
\TT_1(\me)=\left(\TT_1\cp\pi'_{\tau,r}\right)(a_{2n})=\pi'_l( a_{2n}) \qquad\forall \: n\in \ZZ_{\neq 0}.
\end{align*}
Then, for $\epsilon >0$ we find $n_\epsilon \in \mathbb{N}_{\geq 0}$ such that\footnote{It is clear that $\lim_n [l a_{2n}- 2\pi n] =0$, and that $\lim_n [l a_{2n}- l a_{2(n+1)}]= 2\pi$. Hence, we have to show that we find $n_0\in \NN_{\geq 1}$ such that $l a_{2n}- l a_{2(n+1)} \neq 2\pi$ for all $n\geq n_0$. Now, $l a_{2n}- l a_{2(n+1)}=2n\pi\left[\sqrt{1-\textstyle\frac{\tau^2}{4(2n+2)^2\pi^2}}-\sqrt{1-\textstyle\frac{\tau^2}{4(2n)^2\pi^2}}\right] + 2\pi \sqrt{1-\textstyle\frac{\tau^2}{4(2n+2)^2\pi^2}}$, where the first summand tends to zero for $n\rightarrow \infty$ and is negative. Since the second summand is smaller than $2\pi$, the whole expression is smaller than $2\pi$ for $n$ suitable large.} $l a_{2n}- l a_{2(n+1)}\in B_\epsilon(2\pi)\backslash\{2\pi\}$ for all $n\geq n_\epsilon$. But, $\pi'_{l}(a_{2n})=\pi'_{l}(\raisebox{0pt}{$a_{2(n+1)}$})$ implies $la_{2(n+1)}-la_{2n} =k_n2\pi$ for some $k_n \in \mathbb{Z}$,
so that we get a contradiction if we choose $\epsilon< 2\pi$.
\end{itemize}
\endgroup
\noindent
However, by Lemma \ref{lemma:WeierErzeuger}.\ref{lemma:WeierErzeuger3}) it suffices to
take one fixed circular curve $\gamma_{\tau,r}$ into account. So,
we can circumvent the above transition map problem by sticking to the directed set $I$ from Remark and Definition \ref{rem:bohrproj}. More precisely, we can incorporate the map $\pi'_{\tau,r}$ into each of the projection maps as follows:
\begin{enumerate}
\ite
\itspace
For $I\ni L=(l_1,\dots,l_k)$ we can define
\begin{align*}
\pi'_L\colon \qR &\rightarrow \SU^{k+1}\\
\x&\mapsto \big(\pi'_{l_1}(\x),\dots,\pi'_{l_k}(\x),\pi'_{\tau,r}(\x)\big).
\end{align*}
But, then $\mathrm{im}[\pi'_L]$ crucially depends on the $\mathbb{Z}$-independence of $l_1,\dots,l_k,r\tau$, as, e.g., we have $\pi_L(\RB)=[H_{\vec{e}_1}]^k \times H_{\vec{e}_2}$ in the $\mathbb{Z}$-independent case and $\pi_L(\RB)\cong S^1$ if $l_1,\dots,l_k,r\tau$ are multiples of the same real number. For this, observe that we cannot restrict to independent tuples without adapting $\leq$. This is because for $(l',r\tau)$, $(l'',r\tau)$ independent and $(l_1,\dots,l_k)\in I$ an upper bound of $L':=l',L'':=l''$, the tuple $(l_1,\dots,l_k,r\tau)$
does not need to be independent as well. In fact, for $l'=l-r\tau$ this cannot be true for any such $L$.
All this makes it difficult to find transition maps and suitable consistent families of measures for these spaces.
\ite
\itspace
Basically, Lemma \ref{lemma:WeierErzeuger}.\ref{lemma:WeierErzeuger3}) is due to the fact that the $C_0(\RR)$-part
of the function
$a\colon \RR\ni c\mapsto (\pi'_{\tau,r}(c))_{11}$ (given by the \eqref{eq:funcco})
vanishes nowhere. Now, we can try to find some analytic curve $\gamma$ and a projection map $\pi_\gamma\colon \qR\rightarrow \SU$
such that for one of the entries $(\pi_{\gamma}(\cdot))_{ij}$, $1\leq i,j\leq 2$ the $\CAP(\RR)$-part is zero and the $C_0(\RR)$-part vanishes nowhere and is injective. Then, condition \ref{def:ProjLim3}) from Definition \ref{def:ProjLim} would hold for the projection maps $\wt{\pi}_L\colon \qR \rightarrow \mathrm{im}[\pi_\gamma] \sqcup [H_{\vec{e}_1}]^k$
\begin{equation}
\label{eq:PiLMuster}
\wt{\pi}_L(\ovl{x}) :=
\begin{cases}
\pi_\gamma(\x) & \mbox{if } \ovl{x}\in \mathbb{R}\\
\pi'_L(\x) &\mbox{if } \ovl{x}\in \RB
\end{cases}
\end{equation}
and we could define the transition maps and measures on $\mathrm{im}[\pi_\gamma]$ and $[H_{\vec{e}_1}]^k$ separately. However, even if such a curve $\gamma$ exists, it is not to be expected that it is easier to find reasonable measures on $\mathrm{im}[\pi_\gamma]$ than on $\mathrm{im}[\pi'_{\tau,r}]$.
\end{enumerate}
In the next subsection we will follow the philosophy of the second approach. Here, we use distinguished generators of $C_0(\RR)\oplus \CAP(\RR)$
in order to define projective structures on $\qR$ in a more direct way. This will allow us to circumvent the image problem we have for the projection maps $\pi'_{\tau,r}$ and $\pi_\gamma$. So, the crucial part will not be to define the projective structure, but to determine the respective consistent families of normalized Radon measures. Here, the main difficulties will arise from determining the Borel $\sigma$-algebras of the projection spaces.
\subsection{Projective Structures on $\qR$}
\label{subsec:ProjStrucon}
In this subsection, we will use the characters $\{\chi_l\}_{l\in \RR}$ and an injective nowhere vanishing $f\in C_0(\RR)$ in order to define a projective structure on $\qR$. This will be done in analogy to the definition of the projective structure on $\RB$ presented in Remark and Definition \ref{rem:bohrproj}, cf.\ \cite{oai:arXiv.org:0704.2397}. In the last part, we will use this construction in order to fix the normalized Radon measures \eqref{eq:fammeas}.
We start with the following definitions, resetting (and collecting) some of the notations we have introduced in the previous subsection.
\begin{definition}
\label{def:ProjLimit}
Assume that $f\in C_0(\mathbb{R})$ is injective and $f(x)\neq 0$ for all $x\in \mathbb{R}$.
\begin{enumerate}
\item
Let $I$ denote the set of all finite tuples $L=(l_1,\dots,l_k)$ consisting of $\mathbb{Z}$-independent real numbers $l_1,\dots,l_k$. Moreover, let $|L|$ denote the length $k$ of the tuple $L$.
\item
For $L,L'\in I$ define $L\leqZ L'$ iff $l_i \in \spann_{\mathbb{Z}}(l'_1,\dots,l'_{k'})$ for all $1\leq i\leq k$.
\item
For $L\in I$ and $k:=|L|$ define $\pi_L\colon \qR \rightarrow \prfl{f}{L}=: X_L$ by
\begin{equation}
\label{eq:PiL}
\pi_L(\ovl{x}) :=
\begin{cases}
f(\ovl{x}) & \mbox{if } \ovl{x}\in \mathbb{R}\\
\big(\ovl{x}(\chi_{l_1}),\dots,\ovl{x}(\chi_{l_k})\big) &\mbox{if } \ovl{x}\in \RB,
\end{cases}
\end{equation}
and equip $X_L$ with the final topology $\T_F$ w.r.t.\ this map. Recall that here and in the following $S^k$ just denotes the $k$-fold product of the unit circle $S^1$.
\item
For $L,L'\in I$ with $L\leqZ L'$ define $\pi_L^{L'}\colon X_{L'}\rightarrow X_L$ by $\pi_L^{L'}(y):=y$ if $y\in \mathrm{im}[f]$ and
\begin{align}
\label{eq:transit}
\pi^{L'}_{L} (s_1,\dots,s_{k'}):= \left(\prod_{i=1}^{k'}{s_i}^{n^i_1} ,\dots, \prod_{i=1}^{k'}{s_i}^{n^i_{j}} \right)
\quad\text{ if }\quad l_j=\sum_{i=1}^{k'} n^i_jl'_i
\end{align}
with $n_j^i\in \mathbb{Z}$ for $1\leq j\leq k=|L|$, $1\leq i\leq k'=|L'|$ and $(s_1,\dots, s_{k'})\in S^{|L'|}$.
\end{enumerate}
\end{definition}
We now show
that $\qR$ is indeed a projective limit of $\{X_L\}_{L\in I}$. Moreover, we determine the Borel $\sigma$-algebras of the spaces $X_L$. This will lead to an analogous decomposition of finite Radon measures as for the space $\qR$. Here, the crucial point is to show that the subspace topologies of $\mathrm{im}[f]$ and $S^{|L|}$ w.r.t.\ the final topology on $X_L$ are just their canonical ones.
For this, we will need the following definitions and facts:
\begingroup
\setlength{\leftmargini}{17pt}
\begin{itemize}
\item
\itspacecc
Let $\T_f$ and $\T_{L}$ denote the standard topologies
on $\mathrm{im}[f]$ and $S^{|L|}$, respectively, i.e.,
the subspace topology on $\mathrm{im}[f]$ inherited from $\mathbb{R}$ and the product topology on $S^{|L|}$.
\item
\itspacecc
For $L\in I$ let $\widehat{\pi}_L \colon \RB \rightarrow S^{|L|}$ denote the restriction of $\pi_L$ to $\RB$.
\item
\itspacecc
For $L,L'\in I$ with $L\leqZ L'$ let $\widehat{\pi}_L^{L'} \colon S^{|L'|} \rightarrow S^{|L|}$ denote the restriction of $\pi^{L'}_{L}$ to $S^{|L'|}$.
\item
\itspacecc
For $q\in \mathbb{Q}_{\neq 0}$ define $\chi_{l,q}:=\chi_{l/q}$ as well as $\widehat{\chi}_l:=\mathcal{G}(\chi_l)$ for $l\in \RR$.
\item
\itspacecc
Since each $L\in I$ consists of
$\mathbb{Q}$-independent reals, we find (and fix) a subset $L^\perp\subseteq \mathbb{R}$ for which $\ovl{L}:=L\sqcup L^\perp$ is a $\mathbb{Q}$-base of $\mathbb{R}$. It is clear that, together with the constant function $1=\chi_{0}$, the functions $\left\{\chi_{l,n}\right\}_{(l,n) \in \ovl{L} \times \NN_{>0}}$ generate a dense $^*$-subalgebra
of $\CAP(\RR)$.
\item
\itspacecc
For $p\in \mathbb{N}_{\geq 1}$ and $A\subseteq S^1$ let $\hat{p}\colon S^1\rightarrow S^1$, $s\mapsto s^p$ and define
\begin{align*}
\sqrt[p]{A}:=\{s\in S^1\:|\: s^p\in A\}\qquad\text{as well as}\qquad A^p:=\{s^p\:|\: s\in A\}.
\end{align*}
If $\mathcal{O}\subseteq S^1$ is open, then $\mathcal{O}^p$ and $\sqrt[p]{\mathcal{O}}=\hat{p}^{-1}(\mathcal{O})$ are open as well. This is because $\hat{p}$ is open (inverse function theorem) and continuous.
\item
\itspacecc
For $A\subseteq S^1$ and $m\in \mathbb{Z}\backslash\{0\}$ we define
\begin{equation*}
A^{\sign(m)}:=
\begin{cases}
A & \mbox{if }m >0\\
\{\ovl{z}\:|\: z\in A\} &\mbox{if } m <0.
\end{cases}
\end{equation*}
\end{itemize}
\endgroup
\noindent
The next lemma highlights the relevant properties of the maps $\widehat{\pi}_L$.
\begin{lemma}
\label{lemma:OpenMapp}
Let $L=(l_1,\dots,l_k)\in I$.
\begin{enumerate}
\item
\label{lemma:OpenMapp1}
Let $\psi \in \RB$, $q_i\in \mathbb{Q}$ and $s_i\in S^1$ for $1\leq i\leq k$. Then we find $\psi'\in \RB$ with
\begin{align*}
\psi'(\chi_{l_i,q_i})=s_i\quad \forall\:1\leq i\leq k\qquad\quad\text{and}\qquad\quad \psi'(\chi_{l})=\psi(\chi_{l})\quad\forall\: l \in \Span_{\mathbb{Q}}(L^\perp).
\end{align*}
\item
\label{lemma:OpenMapp2}
Let $l\in \RR$, $m_i\in \mathbb{Z}\backslash\{0\}$ and $\mathcal{O}_i\subseteq S^1$ open for $1\leq i\leq n$. Then, for $m:=|m_1\cdot {\dots} \cdot m_n|$ and $p_i:=|\frac{m}{m_i}|$, we have
\begin{align}
\label{eq:WurzelUrbild}
\bigcap_{i=1}^n \widehat{\chi}^{-1}_{l,m_i}(\mathcal{O}_i)=\bigcap_{i=1}^n \widehat{\chi}^{-1}_{l,m}\left(\!\raisebox{0.0ex}{$\left[\!\sqrt[p_i]{\mathcal{O}_i}\:\right]^{\sign(m_i)}\!$}\:\right)
=\widehat{\chi}^{-1}_{l,m}\left(\mathcal{O}\right)
\end{align}
for the open subset $\mathcal{O}=\left[\!\sqrt[p_1]{\mathcal{O}_1}\:\right]^{\sign(m_1)} \cap \dots \cap \left[\!\sqrt[p_n]{\mathcal{O}_n}\:\right]^{\sign(m_n)}\subseteq S^1$.
\item
\label{lemma:OpenMapp3}
Let $A_i, B_j\subseteq S^1$ for $1\leq i\leq k$, $1\leq j\leq q$, with $B_1,\dots,B_q\neq \emptyset$. Moreover, let $h_1,\dots,h_q \in \mathbb{R}$ such that $l_1,\dots,l_k$, $h_1,\dots,h_q$ are $\mathbb{Z}$-independent. For $m_1,\dots,m_k, n_1,\dots,n_q \in \mathbb{Z}\backslash\{0\}$ let
\begin{align}
\label{eq:BaseRbohr}
\!\!\!\!W:=\underbrace{\widehat{\chi}_{l_1,m_1}^{-1}(A_1)\cap\dots \cap \widehat{\chi}_{l_k,m_k}^{-1}(A_k)}_{U}\:\cap\: \underbrace{\widehat{\chi}_{h_1,n_1}^{-1}(B_1)\cap\dots \cap \widehat{\chi}_{h_q,n_q}^{-1}(B_q)}_{U'}.
\end{align}
Then $\widehat{\pi}_L(W)= A_1^{m_1} \times \dots \times A_k^{m_k}$.
\item
\label{lemma:OpenMapp4}
The map $\widehat{\pi}_L$ is surjective, continuous and open.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
This is clear from Lemma and Convention \ref{lemconv:RBMOD}.\ref{prop:Bohrmod24}.
\item
Obviously, $\mathcal{O}$ is open, and since
the second equality in \eqref{eq:WurzelUrbild} is clear, it suffices to show that
\begin{align*}
\widehat{\chi}^{-1}_{l,m}(A)=\widehat{\chi}^{-1}_{l,p\cdot |m|}\left(\big[\!\raisebox{-0.3ex}{$\sqrt[p]{A}$}\:\big]^{\sign(m)}\right)
\end{align*}
holds
for $A\subseteq S^1$, $l\in \mathbb{R}$, $p \in \mathbb{N}_{\geq 1}$ and $m\in \mathbb{Z}\backslash\{0\}$.
To show the inclusion $\supseteq$, let
$\psi \in \widehat{\chi}^{-1}_{l,p\cdot |m|}\left(\big[\!\raisebox{-0.3ex}{$\sqrt[p]{A}$}\:\big]^{\sign(m)}\right)$. Then
\begin{align*}
\psi(\chi_{l,p\cdot |m|}) \in \big[\!\raisebox{-0.3ex}{$\sqrt[p]{A}$}\:\big]^{\sign(m)}\qquad\Longrightarrow\qquad \psi(\chi_{l, m})=\big[\raisebox{-0.15ex}{$\psi(\chi_{l,p\cdot |m|})^{p}$}\big]^{\sign(m)}\in A.
\end{align*}
For the converse inclusion let $\psi \in\widehat{\chi}_{l,m}^{-1}(A)$. Then
\begin{align*}
\big[\raisebox{-0.2ex}{$\psi(\chi_{l,p\cdot |m|})^{p}$}\big]^{\sign(m)}=\psi(\chi_{l,m})\in A \qquad\Longrightarrow\qquad \psi(\chi_{l,p\cdot |m|})\in \big[\!\raisebox{-0.15ex}{$\sqrt[p]{A}$}\:\big]^{\sign(m)}.
\end{align*}
\item
We proceed in two steps:
\begingroup
\setlength{\leftmarginii}{15pt}
\begin{itemize}
\item
\itspace
We show that $\widehat{\pi}_L(W)=\widehat{\pi}_L(U)$. For this, it suffices to verify that $\widehat{\pi}_L(U)\subseteq \widehat{\pi}_L(W)$ because the converse inclusion is clear from $W\subseteq U$.
So, for $\psi \in U$ we have to show that $\widehat{\pi}_L(\psi) \in \widehat{\pi}_L(W)$. Since $B_j\neq \emptyset$, we find $z_j \in B_{j}$
for all $1\leq j\leq q$.
By \ref{lemma:OpenMapp1}) we find $\psi'\in \RB$ with $\psi'(\chi_{l_i,m_i})=\psi(\chi_{l_i,m_i})\in A_i$ for all $1\leq i\leq k$ and $\psi'(\chi_{h_j,n_j})=z_j\in B_j$ for all $1\leq j\leq q$. This shows $\psi'\in U\cap U'=W$, hence $\widehat{\pi}_L(\psi')\in \widehat{\pi}_L(W)$. Consequently,
\begin{align*}
\widehat{\pi}_L(\psi)&=(\psi(\chi_{l_1,m_1}),\dots,\psi(\chi_{l_k,m_k}))
=\big(\psi'(\chi_{l_1,m_1}),\dots,\psi'(\chi_{l_k,m_k})\big)=\widehat{\pi}_L(\psi')\in \widehat{\pi}_L(W).
\end{align*}
\item
We show $\widehat{\pi}_L(U)=A^{m_1}_1 \times \dots \times A^{m_k}_k$. For this, it suffices to verify the inclusion $\supseteq$ as the opposite inclusion is clear from the definitions. To this end, fix $\psi\in \RB$ and let $s_i \in A_i^{m_i}$ for $1\leq i\leq k$ be chosen freely. Then, we find $z_i\in A_i$ with $z_i^{m_i}=s_i$, and
\ref{lemma:OpenMapp1}) provides us with some $\psi'\in \RB$ with $\psi'(\chi_{l_i,m_i})=z_i\in A_i$ for $1\leq i\leq k$. Then $\psi'\in U$ and $\psi'(\chi_{l_i})=z_i^{m_i}=s_i \in A_i^{m_i}$ for all $1\leq i\leq k$.
\end{itemize}
\endgroup
\item
Continuity of $\widehat{\pi}_L$ is clear from
\begin{align*}
\widehat{\pi}_L^{-1}(A_1,\dots,A_k)= \widehat{\chi}_{l_1}^{-1}(A_1)\cap\dots \cap \widehat{\chi}_{l_k}^{-1}(A_k)\qquad\forall\:A_1,\dots A_k \subseteq S^1,
\end{align*}
and surjectivity is clear from Part \ref{lemma:OpenMapp3}) if we choose $A_1,\dots,A_k=S^1$.
For openness observe that the $^*$-algebra generated by $1$ and $\{\chi_{l,m}\}_{(l,m) \in \ovl{L} \times \mathbb{Z}\backslash\{0\}}$ is dense in $\CAP(\RR)$ as it equals the $^*$-algebra generated by the all characters $\chi_l$. Then, the subsets of the form $\chi_{l,m}^{-1}(\mathcal{O})$ with $\mathcal{O}\subseteq S^1$ open and $(l,m) \in \ovl{L}\times \mathbb{Z}\backslash\{0\}$
provide a subbasis for the topology of $\RB$.\footnote{This is because the Gelfand topology on $\RB$ equals the initial topology w.r.t.\ the Gelfand transforms of the elements of each subset $\bB\subseteq \CAP(\RR)$ that generates a dense subset $\dD$ of $\CAP(\RR)$, see e.g.\ Subsection 2.3 in \cite{ChrisSymmLQG}. Consequently, the Gelfand topology on $\RB$ equals the initial topology w.r.t.\ the functions $1$ and $\chi_{l,m}$ for $(l,m) \in \ovl{L}\times \mathbb{Z}\backslash\{0\}$.
Since the preimage of a subset of $\mathbb{C}$ under $\mathcal{G}(1)$ is either empty or $\RB$, the claim is clear.}
So, a base of this topology is given by all finite intersections of such subsets. Then Part \ref{lemma:OpenMapp2}) shows that, in order to obtain a base for the topology of $\RB$, it suffices to consider intersections of the form \eqref{eq:BaseRbohr} with $A_1,\dots,A_k,B_1,\dots,B_q$ open in $S^1$. For this, observe that since $\widehat{\chi}^{-1}_{l,m}(S^1)=\RB$, we can assume that all $l_1,\dots,l_k$ occur in each of these intersections.
Then, since $A_i^{m_i}$ is open if $A_i$ is open, Part \ref{lemma:OpenMapp3}) shows that $\widehat{\pi}_L$ is an open map.
\end{enumerate}
\end{beweis}
\end{lemma}
The next lemma
highlights the crucial properties of the final topology of the spaces $X_L$. In addition to that,
the Borel $\sigma$-algebras of these spaces are determined.
\begin{lemma}
\label{lemma:reltop}
Let $L=(l_1,\dots,l_k)\in I$.
\begin{enumerate}
\item
\label{lemma:reltop1}
The subspace topologies of $\mathrm{im}[f]$ and $S^{|L|}$ w.r.t.\ the final topology $\T_F$ on $X_L$ are given by
$\T_f$ and $\T_L$, respectively.
For a subset $U\subseteq \mathrm{im}[f]$ we have $U\in \T_f$ iff $U$ is open in $X_L$.
\item
\label{lemma:reltop2}
$X_L$ is a compact Hausdorff space.
\item
\label{lemma:reltop3}
We have $\mathfrak{B}\!\left(X_L\right)=\mathfrak{B}(\mathrm{im}[f])\sqcup\BTK$ and
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item
\itspacec
If $\mu$ is a finite Radon measure on $\mathfrak{B}\!\left(X_L\right)$, then $\mu|_{\mathfrak{B}(\mathrm{im}[f])}$ and $\mu|_{\mathfrak{B}(S^{|L|})}$ are finite Radon measures as well.
\item
If $\mu_{f}\colon \mathfrak{B}(\mathrm{im}[f]) \rightarrow [0,\infty)$ and $\mu_{S}\colon \BTK \rightarrow [0,\infty)$ are finite Radon measures, then
\begin{align}
\label{eq:xh}
\mu(A):=\mu_{f}(A\cap \mathrm{im}[f])+ \mu_{S}\big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big)
\qquad\forall\: A\in \mathfrak{B}\!\left(X_L\right)
\end{align}
is a finite Radon measure on $\mathfrak{B}\big(X_L\big)$.
\end{enumerate}
\endgroup
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
We first collect the following facts we have already proven during this section:
\begin{enumerate}
\item[(a)]
\itspacecc
The topology on $\qR$ induces the standard topologies on $\RR$ and $\RB$.
\item[(b)]
$U\in \T_F$ iff $\pi_L^{-1}(U)$ is open in $\qR$.
\item[(c)]
\vspace{-0.8pt}
$W\subseteq \RR$ is open in $\qR$ iff $W$ is open in $\RR$.
\item[(d)]
\vspace{-1pt}
If $B\subseteq \RB$ is open, then there is $U\subseteq \mathrm{im}[f]$ such that $f^{-1}(U)\sqcup B$ is open in $\qR$.\footnote{By (e) this is equivalent to show that we find an open subset $W\subseteq \RR$ such that $W\sqcup B$ is open in $\qR$. But, this is clear if $B=\widehat{\chi}_l^{-1}(\mathcal{O})$ for some open subset $\mathcal{O}\subseteq S^1$ and $l\in \RR$ (see Type 3 sets defined in Lemma and Remark \ref{remdefchris}). Since the sets of the form $\widehat{\chi}_l^{-1}(\mathcal{O})$ provide a subbasis for the topology on $\RB$, the claim follows.}
\item[(e)]
\itspacecc
$f\colon \RR\rightarrow \mathrm{im}[f]$ is a homeomorphism.
\item[(f)]
\itspacecc
$\widehat{\pi}_L\colon \RB\rightarrow S^{|L|}$ is continuous and open.
\end{enumerate}
We start with the statements concerning the subspace topologies:
\vspace{1ex}
$\boldsymbol{\mathrm{im}[f]}$:
Let $U\subseteq \mathrm{im}[f]$. Then:
\qquad\quad\hspace{9pt}
\:$U$ is open w.r.t.\ the topology inherited from $X_L$
\qquad$\Longleftrightarrow$ $\:\exists\: V\subseteq S^{|L|}$ such that $U\sqcup V$ is open in $X_L$
\qquad$\Longleftrightarrow$ $\:\exists\: V\subseteq S^{|L|}$ such that $\pi_L^{-1}(U\sqcup V)$ is open in $\qR$\hspace*{\fill}{(b)}
\qquad$\Longleftrightarrow$ $\:\exists\: V\subseteq S^{|L|}$ such that $f^{-1}(U)\sqcup \widehat{\pi}_L^{-1}(V)$ is open in $\qR$
\qquad$\Longleftrightarrow$ $\:f^{-1}(U)$ is open in $\RR$\hspace*{\fill}{(c)
\qquad$\Longleftrightarrow$ $\:U\in \T_f$ \hspace*{\fill}(e)
\vspace{2ex}
$\boldsymbol{S^{|L|}}$:
Let $V\subseteq S^{|L|}$. Then:
\qquad\quad\hspace{9pt}
$V$ is open w.r.t.\ the topology inherited from $X_L$
\qquad$\Longleftrightarrow$ $\:\exists\: U\subseteq \mathrm{im}[f]$ such that $U\sqcup V$ is open in $X_L$
\qquad$\Longleftrightarrow$ $\:\exists\: U\subseteq \mathrm{im}[f]$ such that $\pi_L^{-1}(U\sqcup V)$ is open in $\qR$\hspace*{\fill}{(b)}
\qquad$\Longleftrightarrow$ $\:\exists\: U\subseteq \mathrm{im}[f]$ such that $f^{-1}(U)\sqcup \widehat{\pi}_L^{-1}(V)$ is open in $\qR$
\qquad$\Longleftrightarrow$ $\:\widehat{\pi}_L^{-1}(V)$ is open in $\RB$\hspace*{\fill}{(a),(d)}
\qquad$\Longleftrightarrow$ $\:V\in \T_L$ \hspace*{\fill}(f)
\vspace{1ex}
Finally, observe that $\mathrm{im}[f]$ is open in $X_L$ because $\pi_L^{-1}(\mathrm{im}[f])=\RR$ is open in $\qR$. Then $U\subseteq \mathrm{im}[f]$ is open in $X_L$ iff $U$ is open w.r.t.\ the topology on $\mathrm{im}[f]$ inherited from $X_L$. Since this topology equals $\T_f$, the claim follows.
\item
The spaces $X_L$ are compact by compactness of $\qR$ and continuity of $\pi_L$. For the Hausdorff property observe that $\T_F$ contains all sets of the following types:
\begin{align*}
\begin{array}{lcrclcl}
\textbf{\textit{Type 1':}} && f(V) & \!\!\!\sqcup\!\!\! & \emptyset
&& \text{with open $V \subseteq \RR$,} \\
\textbf{\textit{Type 2':}} && f(K^c) & \!\!\!\sqcup\!\!\! & S^{|L|}
&& \text{with compact $K \subseteq \RR$,} \\
\textbf{\textit{Type 3':}} && f\big(\chi_{l_i}^{-1}(\mathcal{O})\big) & \!\!\!\sqcup\!\!\! & \pr_i^{-1}(\mathcal{O})
&& \text{with $\mathcal{O}\subseteq S^1$ open and $1\leq i\leq k$.}
\end{array}
\end{align*}
Here $\pr_i\colon S^{|L|}\rightarrow S^1$, $(s_1,\dots,s_k)\mapsto s_i$ denotes the canonical projection. In fact,
the preimage of a set of \textbf{\textit{Type m'}} under $\pi_L$ is a subset of $\qR$ of \textbf{\textit{Type m}}, cf.\ Lemma and Definition \ref{remdefchris}.
Then, by injectivity of $f$ the elements of $\mathrm{im}[f]$ are separated by sets of \textbf{\textit{Type 1'}}. Moreover, if $x\in \mathrm{im}[f]$ and $(s_1,\dots,s_k)\in S^{|L|}$, then we can choose a relatively compact neighbourhood $W$ of $f^{-1}(x)$ in $\RR$ and define $U:= f(W)$ and $V:=f\raisebox{1pt}{$\big($}\raisebox{-1pt}{${\ovl{W}\hspace{0.9pt}}^c$}\raisebox{1pt}{$\big)$}\sqcup S^{|L|}$. Finally, if $(s_1,\dots,s_k),(s'_1,\dots,s'_k)\in S^{|L|}$ are different elements, then $s_i\neq s'_i$ for some $1\leq i\leq k$. Then, for open neighbourhoods $\mathcal{O},\mathcal{O}' \subseteq S^1$ of $s_i$ and $s_i'$, respectively, with $\mathcal{O} \cap \mathcal{O}' =\emptyset$ we have
$\big[f\big(\chi_{l_i}^{-1}(\mathcal{O})\big) \sqcup \pr_i^{-1}(\mathcal{O})\big]\cap \big[f\big(\chi_{l_i}^{-1}(\mathcal{O}')\big) \sqcup \pr_i^{-1}(\mathcal{O}')\big]=\emptyset$.
\item
We repeat the arguments from the proof of Lemma \ref{lemma:Radon}.
If $U\subseteq X_L$ is open, then $U\cap \mathrm{im}[f]\in \T_f$ and $U\cap S^{|L|}\in \T_L$
by Part \ref{lemma:reltop1}). This shows $U\in \mathfrak{B}(\mathrm{im}[f])\sqcup\BTK$, i.e., $\mathfrak{B}\left(X_L\right)\subseteq\mathfrak{B}(\mathrm{im}[f])\sqcup\BTK$ as the right hand side is a $\sigma$-algebra. For the converse inclusion recall that $U\in \T_f$ iff $U$ is open in $X_L$, again by the first part, hence $\mathfrak{B}(\mathrm{im}[f])\subseteq \mathfrak{B}(X_L)$. Finally, if $A\subseteq S^{|L|}$ is closed, then $A$ is compact w.r.t.\ $\T_L$. This means that $A$ is compact w.r.t.\ the subspace topology inherited from $X_L$, implying that $A$ is compact as a subset of $X_L$. Then $A$ is closed by the Hausdorff property of $X_L$ so that $A\in \mathfrak{B}\!\left(X_L\right)$, hence $\BTK\subseteq \mathfrak{B}\!\left(X_L\right)$. Now,
\begin{enumerate}
\item[\textit{(a)}]
\itspacecc
The measures $\mu|_{\mathfrak{B}(\mathrm{im}[f])}$ and $\mu|_{\mathfrak{B}(S^{|L|})}$ are well defined and obviously finite. Their inner regularities follow from the fact that subsets of $\mathrm{im}[f]$ and $S^{|L|}$ are compact w.r.t.\ $\T_f$ and $\T_L$, respectively, iff they are so w.r.t.\ the topology on $X_L$, just by by Part \ref{lemma:reltop1}).
\item[\textit{(b)}]
\itspacecc
If $\mu$ is defined by \eqref{eq:xh},
then $\mu$ is a finite Borel measure and its inner regularity follows by a simple $\epsilon\slash 2$ argument from the inner regularities of $\mu_{f}$ and $\mu_{S}$.
\end{enumerate}
\end{enumerate}
\end{beweis}
\end{lemma}
Combining the Lemmata \ref{lemma:OpenMapp} and \ref{lemma:reltop} we obtain
\begin{proposition}
\label{th:projlim}
\begin{enumerate}
\item
$\qR$ is a projective limit of $\{X_L\}_{L\in I}$.
\item
A family $\{\mu_L\}_{L \in I}$ of measures $\mu_L$ on $X_L$ is a consistent family of normalized Radon measures w.r.t.\ $\{X_L\}_{L\in I}$ iff the following holds:
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item
There is $t\in [0,1]$ such that for each $L\in I$ and $A\in \mathfrak{B}\!\left(X_L\right)$ we have
\begin{align*}
\mu_L(A)=t\: \mu_f(A\cap \mathrm{im}[f])+ (1-t)\: \mu_{S,L}\big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big)
\end{align*}
for $\mu_f$ and $\mu_{S,L}$ normalized\footnote{If $t$ equals $0$ or $1$, we allow $\mu_f=0$ or $\mu_{S,L}=0$, respectively.} Radon measure on $\mathrm{im}[f]$ and $S^{|L|}$, respectively.
\item
For all $L,L'\in I$ with $L\leqZ L'$ we have $\widehat{\pi}^{L'}_{L}(\mu_{S,L'})=\mu_{S,L}$.
\end{enumerate}
\endgroup
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
The spaces $X_L$ are compact and Hausdorff by Lemma \ref{lemma:reltop}.\ref{lemma:reltop2}.
Moreover, each $\pi_L$ is surjective by Lemma \ref{lemma:OpenMapp}.\ref{lemma:OpenMapp4}.
If $L,L'\in I$ with $L\leqZ L'$, then continuity of the maps $\pi^{L'}_{L}$ is clear from
$\pi^{L'}_{L}\cp \pi_{L'}=\pi_{L}$ which, in turn, is immediate from multiplicativity of the functions $\chi_l$. Finally, condition \ref{def:ProjLim3}) from
Definition \ref{def:ProjLim} follows from injectivity of $f$ and the fact that the functions $\{\chi_l\}_{l\in \mathbb{R}}$ generate $\CAP(\RR)$.
\item
Let $\{\mu_L\}_{L\in I}$ be a consistent family of normalized Radon measures w.r.t.\ $\{X_L\}_{L\in I}$. Then Lemma \ref{lemma:reltop}.\ref{lemma:reltop3} shows that for each $L\in I$ we have
\begin{align*}
\mu_L(A)= \mmu_{f,L}(A\cap \mathrm{im}[f])+ \mmu_{S,L}\big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big)\qquad \forall\: A\in \mathfrak{B}\!\left(X_L\right)
\end{align*}
for $\mmu_{f,L}$ and $\mmu_{S,L}$ finite Radon measures on $\mathrm{im}[f]$ and $S^{|L|}$, respectively. Then, consistency forces that $\mmu_{f,L}=\mmu_{f,L'}$ for all $L,L'\in I$. In fact, by Lemma \ref{lemma:normRM} there is a unique normalized Radon measure $\mu$ on $\qR$ for which $\mu_L=\pi_L(\mu)$ holds for all $L\in I$. Consequently, for each $A\in \mathfrak{B}(\mathrm{im}[f])$ and all $L\in I$ we have
\begin{align*}
\mmu_{f,L}(A)=\mu_{L}(A)=\pi_L(\mu)(A)= \mu\big(f^{-1}(A)\big)=:\mmu_f(A).
\end{align*}
By the same arguments,\textit{(b)} follows from consistency of the measures $\{\mu_L\}_{L\in I}$.
Finally, if $t:= \mmu_f(\mathrm{im}[f])\in (0,1)$, then \textit{(a)} holds for $\mu_f:=\frac{1}{t}\mmu_f$ and $\mu_{S,L}:=\frac{1}{1-t}\mmu_{S,L}$ for $L\in I$.
If $t=0$, we define $\mu_{S,L}:=\mmu_{S,L}$ for all $L\in I$ and if $t=1$, we define $\mu_f:=\mmu_f$.
For the converse implication let
$\{\mu_L\}_{L\in I}$ be a family of measures $\mu_L$ on $X_L$ such that \textit{(a)} and \textit{(b)} hold. Then, Lemma \ref{lemma:reltop}.\ref{lemma:reltop3} shows that each $\mu_L$ is a finite Radon measure, and obviously we have $\mu_L(X_L)=1$.
Finally, from \textit{(b)} for $A\in \mathfrak{B}(X_L)$ we obtain
\begin{align*}
\pi^{L'}_{L}(\mu_{L'})(A)&=\mu_{L'}\left(\pillstr^{-1}(A)\right)\\[-4pt]
&= t\: \mu_f\left(\pillstr^{-1}(A)\cap \mathrm{im}[f]\right)+(1-t)\:\mu_{S,L'} \left(\pillstr^{-1}(A)\cap S^{|L'|}\right)\\[-4pt]
&=t\: \mu_f\left(A\cap \mathrm{im}[f]\right)+ (1-t)\:\mu_{S,L'} \Big(\!\left(\raisebox{-0.1ex}{$\widehat{\pi}^{L'}_{L}$}\right)^{-1}\left(\raisebox{-0.1ex}{$A\cap S^{|L|}$}\right)\!\Big)\\[-4pt]
&=t\: \mu_f\left(A\cap \mathrm{im}[f]\right)+(1-t)\:\widehat{\pi}^{L'}_{L}(\mu_{S,L'}) \big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big)\\[-1pt]
&=t\: \mu_f\left(A\cap \mathrm{im}[f]\right)+(1-t)\:\mu_{S,L} \big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big)\\[-1pt]
&=\mu_{L}(A).
\end{align*}
\end{enumerate}
\end{beweis}
\end{proposition}
\subsection{Radon Measures on $\qR$}
\label{subsec:CylMeas}
In this final subsection we use the results of the previous part in order to fix normalized Radon measures on $\qR$. Due to
Proposition \ref{th:projlim}
this can be done as follows:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item[{\bf 1}]
\itspace
Determine a family of normalized Radon measures $\{\mu_{S,L}\}_{L\in I}$ on $S^{|L|}$ that fulfil condition \textit{(b)}.
\item[{\bf 2}]
\itspace
Fix an injective and nowhere vanishing element $f\in C_0(\RR)$ with suitable image together with a normalized Radon measure $\mu_f$ on $\mathrm{im}[f]$.
\item[{\bf 3}]
\itspace
Adjust $t\in [0,1]$.
\end{enumerate}
\endgroup
\noindent
In the following let $\lambda$ denote the Lebesgue measure on $\mathfrak{B}(\RR)$. Moreover, for $B\in \mathfrak{B}(\RR)$ and $\eta \colon B\rightarrow \RR$ measurable let $\eta(\lambda):=\eta\big(\lambda|_{\mathfrak{B}(B)}\big)$.
\vspace{10pt}
\noindent
{\bf Step 1}
\newline
\vspace{-2.3ex}
\newline
We choose
$\mu_{S,L}$ to be the Haar measure $\mu_{|L|}$ on $S^{|L|}$ because:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\itspace
This is canonical from the mathematical point of view, and these
measures fulfil the required compatibility conditions as the next lemma shows.
\item
\itspace
This is in analogy to the case $\RB$ \cite{{oai:arXiv.org:0704.2397}}, where this choice
results in the usual Haar measure on this space, see Remark and Definition \ref{rem:bohrproj}.
\item
\itspace
These measures will suggest a natural choice of $f$ and $\mu_f$ in {\bf Step 2}.
\end{itemize}
\endgroup
\begin{lemma}
\label{lemma:Projm}
Let $\mu_f\colon \mathfrak{B}(\mathrm{im}[f])\rightarrow [0,1]$ be a normalized Radon measure and $t\in[0,1]$. For each $L\in I$ let
\begin{align*}
\mu_L(A):= t\: \mu_f(A\cap \mathrm{im}[f]) +(1-t)\: \mu_{|L|}\raisebox{0pt}{$\big($}\raisebox{-1pt}{$A\cap S^{|L|}$}\big) \qquad \forall\: A\in \mathfrak{B}(X_L).
\end{align*}
Then $\{\mu_L\}_{L\in I}$ is a consistent family of normalized Radon measures and the corresponding normalized Radon measure $\mu$ on $\qR$ is given by
\begin{align}
\label{eq:Radmeas}
\mu(A)=t\: f^{-1}(\mu_f)(A\cap \mathbb{R}) +(1-t)\: \mu_{\mathrm{Bohr}}(A\cap \RB)\qquad \forall\:A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}.
\end{align}
\end{lemma}
\begin{beweis}
Let $L\in I$, $A \in \mathfrak{B}(X_L)$ and $\mu$ be defined by \eqref{eq:Radmeas}. Then
\begin{align*}
\pi_L(\mu)(A)&=t\: f^{-1}(\mu_f)\big(f^{-1}(A\cap \mathrm{im}[f])\big) + (1-t)\: \mu_{\mathrm{Bohr}}\big(\widehat{\pi}_L^{-1}\big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big)\big)\\
&= t\:\mu_f(A\cap \mathrm{im}[f])+ (1-t)\:\widehat{\pi}_L(\mu_{\mathrm{Bohr}})\big(\raisebox{-1pt}{$A\cap S^{|L|}$}\big).
\end{align*}
So, if we know that
$\widehat{\pi}_L(\mu_{\mathrm{Bohr}})=\mu_{|L|}$ holds for all $L\in I$, the claim follows. In fact, consistency of $\{\mu_L\}_{L\in I}$ then is automatically fulfilled because $\mu$ is a well-defined normalized Radon measure.
Now, in order to show $\widehat{\pi}_L(\mu_{\mathrm{Bohr}})=\mu_{|L|}$, it suffices to show translation invariance of the normalized Radon measure $\widehat{\pi}_L(\mu_{\mathrm{Bohr}})$.
For this, let $\tau \in S^{|L|}$. Then, by surjectivity of $\widehat{\pi}_L$ we find $\psi \in \RB$ with $\widehat{\pi}_L(\psi)=\tau$. Since $\widehat{\pi}_L$ is a homomorphism w.r.t.\ the group structure\footnote{Confer Subsection \ref{subsec:Bohrcomp}.} on $\RB$, for $A\subseteq S^{|L|}$ we have
\begin{align*}
\widehat{\pi}_L\left(\psi+ \widehat{\pi}_L^{-1}(A)\right)=\widehat{\pi}_L(\psi)\cdot \widehat{\pi}_L\left(\widehat{\pi}_L^{-1}(A)\right)=\tau \cdot A.
\end{align*}
Applying $\widehat{\pi}_L^{-1}$ to both sides gives $\psi+ \widehat{\pi}_L^{-1}(A)\subseteq \widehat{\pi}_L^{-1}(\tau \cdot A)$. For the opposite inclusion let $\psi'\in \widehat{\pi}_L^{-1}(\tau\cdot A)$. Then $\psi'- \psi \in \widehat{\pi}_L^{-1}(A)$ because
$\widehat{\pi}_L(\psi'- \psi)=\tau^{-1}\cdot \widehat{\pi}_L(\psi')\in A$. Consequently, $\psi'\in \psi+ \widehat{\pi}_L^{-1}(A)$, hence $\widehat{\pi}_L^{-1}(\tau\cdot A)\subseteq \psi+ \widehat{\pi}_L^{-1}(A)$, i.e., $\psi+ \widehat{\pi}_L^{-1}(A)=\widehat{\pi}_L^{-1}(\tau\cdot A)$. Then
\begin{align*}
\widehat{\pi}_L(\mu_{\mathrm{Bohr}})(\tau\cdot A)&=\mu_{\mathrm{Bohr}}\left(\widehat{\pi}_L^{-1}(\tau\cdot A)\right)=\mu_{\mathrm{Bohr}}\left(\psi +\widehat{\pi}_L^{-1}(A)\right)\\
&= \mu_{\mathrm{Bohr}}\left(\widehat{\pi}_L^{-1}(A)\right)=\widehat{\pi}_L(\mu_{\mathrm{Bohr}})(A)
\end{align*}
for all $A\in \BTK$. This shows that $\widehat{\pi}_L(\mu_{\mathrm{Bohr}})$ is translation invariant.
\end{beweis}
\vspace{-2pt}
{\bf Step 2}
\newline
\vspace{-2.3ex}
\newline
If $f,f'\in C_0(\mathbb{R})$ both are injective and vanish nowhere, then the respective projective structures from Definition \ref{def:ProjLimit} are equivalent in the sense that the corresponding spaces $X_L, X'_L$ are homeomorphic via the maps $\Omega_L\colon X_L\rightarrow X'_L$ defined by $\Omega_L|_{S^{|L|}}:=\id_{S^{|L|}}$ and $\Omega_L|_{\mathrm{im}[f]}:=f'\cp f^{-1}$. Moreover, if $\mu_f$ is a normalized Radon measure on $\mathrm{im}[f]$, then $\mu_{f'}:=\big(f'\cp f^{-1}\big)(\mu_f)$ is a normalized Radon measure on $\mathrm{im}[f']$, and it is clear from \eqref{eq:Radmeas} that the corresponding Radon measures $\mu,\mu'$ on $\qR$
from Lemma \ref{lemma:Projm} coincide. All this makes sense because,
in contrast to $\CAP(\RR)$ where we have the canonical generators $\{\chi_l\}_{l\in \mathbb{R}}$, in $C_0(\mathbb{R})$ there is no distinguished nowhere vanishing, injective generator $f\in C_0(\RR)$. But, this also means that we can restrict to functions with a reasonable image such as the ``shifted'' circle $S^1_{\mathrm{s}}:=1+ S^1\backslash\{-1\}\subseteq \mathbb{C}$. In fact, here the analogy to $S^{|L|}$ suggests to use the Haar measure $\mu_1$ on $S^1$. So, in the following we will restrict to the elements of the subset
\begin{align*}
\F:=\{f\in C_0(\RR)\:|\: \mathrm{im}[f]=S^1_{\mathrm{s}}\},
\end{align*}
where for each $f\in \F$ we define $\mu_f:=\mu_{\mathrm{s}}\colon \mathfrak{B}\big(S^1_{\mathrm{s}}\big)\rightarrow [0,1]$. Here,
\begin{align*}
\mu_{\mathrm{s}}(A):=\mu_1(A-1)=+_1(\mu_1)\qquad\forall\: A\in \mathfrak{B}\big(S^1_{\mathrm{s}}\big)
\end{align*}
with $+_1\colon S^1\backslash\{-1\}\ni z\mapsto z+1\in S^1_{\mathrm{s}}$.
It follows that
\begin{align}
\label{eq:LebesgueHomeo}
\big\{f^{-1}(\mu_f)\:\big|\: f\in \F\big\}=\big\{\rho(\lambda)\:\big|\: \rho\in \HHH\big\}
\end{align}
for \gls{HHH} the set of homeomorphisms $\rho\colon (0,1)\rightarrow \RR$.
\newline
\vspace{-8pt}
\newline
{\bf Proof of \eqref{eq:LebesgueHomeo}:}
We consider the function $h\colon(0,1]\ni t\mapsto e^{\I\hspace{1pt} 2\pi[t-1\slash 2]} \in S^1$. Then
$\mu_1=h(\lambda)|_{\mathfrak{B}(S^1)}$ and for $f\in \F$ we have $f^{-1}(\mu_f)= \rho(\lambda)$ for $\HHH\ni \rho:=f^{-1}\cp+_1\cp h|_{(0,1)}$. Conversely, if $\rho \in \HHH$, then $\rho(\lambda)=f^{-1}(\mu_{\mathrm{s}})$ for $\F\ni f:=+_1\cp h\cp \rho^{-1}$.\hspace*{\fill}{\scriptsize$\blacksquare$}
\newline
\vspace{-4pt}
\newline
So, if we restrict to projective structures arising from elements $f\in \F$, then Lemma \ref{lemma:Projm} and \eqref{eq:LebesgueHomeo} select the normalized Radon measures of the form
\begin{align}
\label{eq:murhot}
\mu_{\rho,t}(A):=t\:\rho(\lambda)(A\cap \mathbb{R})+ (1-t)\:\mu_{\mathrm{Bohr}}(A\cap \RB)\qquad\forall\:A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}
\end{align}
for $\rho\colon (0,1)\rightarrow \mathbb{R}$ a homeomorphism and $t\in [0,1]$.
\vspace{1ex}
\newline
\vspace{-2ex}
\newline{\bf Step 3}
\newline
\vspace{-2.3ex}
\newline
To adjust the parameter $t\in[0,1]$ we now take a look at the Hilbert spaces $\Hil_{\rho,t}:=\Lzw{\qR}{\mu_{\rho,t}}$.
\begin{lemma}
\label{lemma:techlemma}
For $A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$}$ let $\chi_A$ denote the corresponding characteristic function.
\begin{enumerate}
\item
\label{lemma:techlemma1}
If $\rho_1,\rho_2\colon (0,1)\rightarrow \RR$ are homeomorphisms and $t_1,t_2 \in (0,1)$, then
\begin{align*}
\begin{split}
\varphi \colon \Lzw{\qR}{\mu_{\rho_1,t_1}}&\rightarrow \Lzw{\qR}{\mu_{\rho_2, t_2}}\\
\psi & \mapsto \sqrt{\frac{t_1}{t_2}} \:(\chi_{\RR}\cdot \psi)\cp \big(\rho_1\cp \rho_2^{-1}\big) + \sqrt{\frac{(1-t_1)}{(1-t_2)}}\:\chi_{\RB}\cdot \psi
\end{split}
\end{align*}
is an isometric isomorphism. The same is true for
\begin{align*}
&\varphi \colon \Hil_{\rho_1,1}\rightarrow \Hil_{\rho_2,1},\: \psi \mapsto (\chi_{\RR}\cdot \psi)\cp \big(\rho_1\cp \rho_2^{-1}\big),\\
&\varphi \colon \Hil_{\rho_1,0}\rightarrow \Hil_{\rho_2,0},\: \psi \mapsto \psi.
\end{align*}
\item
\label{lemma:techlemma2}
If $t=1$, then $\Hil_{\rho,1}\cong \Lzw{\RR}{\rho(\lambda)}\cong \Lzw{\RR}{\lambda}$ for each $\rho\in H$. Here $\cong$ means canonically isometrically isomorphic.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
This is immediate from the general transformation formula.
\item
The first isomorphism is just because $\mu_{\rho,0}(\RB)=0$. Then, by the first part it suffices to specify the second isomorphism for the case that $\rho$ is a diffeomorphism. But, in this case we have $\rho(\lambda)=\frac{1}{|\dot\rho|}\lambda$, so that for the isomorphism
\begin{align*}
\varphi\colon \Lzw{\RR}{\rho(\lambda)}\rightarrow \Lzw{\RR}{\lambda},\quad \psi \mapsto \frac{1}{\sqrt{|\dot \rho|}} \psi
\end{align*} we obtain
\begin{align*}
\langle \varphi(\psi_1),\varphi(\psi_2)\rangle_\lambda&=\int_{\RR} \psi_1 \ovl{\psi_2} \:\:\frac{1}{|\dot \rho|} \dd\lambda= \int_{\RR} \psi_1 \ovl{\psi_2} \:\: \dd\rho(\lambda)
=\langle \psi_1,\psi_2\rangle_{\rho(\lambda)}.
\end{align*}
\end{enumerate}
\end{proof}
\subsection{Summary}
\label{Finrem}
\begin{enumerate}
\item
\label{Finrem4}
In Subsection \ref{subsec:QuantvsRed} we have shown that quantization and reduction do not commute in homogeneous isotropic LQC, i.e., that
the inclusion $\ARRQw\cong \ARQw\subsetneq \AQRw$ is indeed proper in this case, see also Example \ref{ex:LQCInc}.
\item
\label{Finrem3}
In Proposition \ref{lemma:bohrmassdichttrans} we have seen that the Haar measure on $\RB$ is uniquely determined by the condition that the translations w.r.t.\ the spectral extension $\Transl\colon \RR\times \RB \rightarrow \RB$ of the additive action \RPLUS$\colon \RR \times \RR \rightarrow \RR$, $(v,t)\mapsto v+t$ act as unitary operators on the respective Hilbert space of square integrable functions. In addition to that, we have shown that the one-parameter group $\{{\Transl_v\!}^*\}_{v\in \RR}$ of unitary operators ${\Transl_v\!}^*\colon \Lzw{\RB}{\muB}\rightarrow \Lzw{\RB}{\muB}$ is strongly continuous.
Corollary \ref{cor:eindbohr} then states that the same unitality condition forces $\mu=\muB$ also for the space $\RR\sqcup\RB=\qR\cong \qRR=\ARRQw$, and that the respective family of unitary operators is strongly continuous as well. Consequently, if one wants to represent the exponentiated reduced fluxes (``momentum'' operators) by translations w.r.t.\ the respective spectral extension $\TTransw \colon \RR \times \qR \rightarrow \qR$ of \RPLUS$\colon \RR \times \RR \rightarrow \RR$, there is only the measure $\muB$ which can be used. So, following these lines, one ends up with the same kinematical Hilbert space as used in standard homogeneous isotropic LQC approach, namely $\Lzw{\RB}{\muB}$.
\item
\label{Finrem1}
In the last three subsections we have established a projective structure on $\qR\cong \qRR=\ARRQw$ in order to construct further normalized Radon measures thereon. Here, using Haar measures on tori we have derived the normalized Radon measure
, cf. \eqref{eq:murhot}
\begin{align*}
\mu_{\rho,t}(A):=t\:\rho(\lambda)(A\cap \mathbb{R})+ (1-t)\:\mu_{\mathrm{Bohr}}(A\cap \RB)\qquad\forall\:A\in \Borel\raisebox{0.2ex}{$($}\raisebox{-0.1ex}{$\qR$}\raisebox{0.2ex}{$)$},
\end{align*}
with $t\in [0,1]$, $\adif\colon (0,1)\rightarrow \RR$ a homeomorphism and $\lambda$ the restriction of the Lebesgue measure to $\mathfrak{B}((0,1))$.
Then,
Lemma \ref{lemma:techlemma} shows that up to \emph{canonical} isometrical isomorphisms the parameters $\rho$ and $t$ give rise to the following three Hilbert spaces:
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item[1)]
$\Hil_{\rho,1}\cong \Lzw{\RR}{\lambda} \cong \Lzw{\RR}{\rho(\lambda)}$ for all $\rho\in \text{\gls{HHH}}$\hspace*{\fill}{(Lemma \ref{lemma:techlemma}.\ref{lemma:techlemma2})}
\item[2)]
$\Hil_{\rho,t}\cong L^2\big(\hspace{1pt}\raisebox{-0.1ex}{$\qR$},\mu_{\rho_0,t_0}\big)$ for all $\rho\in \HHH$, $t\in (0,1)$\hspace*{\fill}{(Lemma \ref{lemma:techlemma}.\ref{lemma:techlemma1})}
\item[3)]
$\Hil_{\rho,0}\cong \Lzw{\RB}{\mu_{\mathrm{Bohr}}}$ for all $\rho\in \HHH$ \hspace*{\fill}{($\RR$ is of measure zero)}
\end{enumerate}
\endgroup
\noindent
Here, the Hilbert spaces in $2)$ and $3)$ are isometrically isomorphic just because their Hilbert space dimensions coincide. In contrast to that,
the cases $1)$ and $3)$ cannot be isometrically isomorphic because $\Lzw{\RR}{\lambda}$ is separable and $\Lzw{\RB}{\mu_{\mathrm{Bohr}}}$ is not so.
Anyhow, even if $L^2\big(\hspace{1pt}\raisebox{-0.1ex}{$\qR$},\mu_{\rho,t}\big)$ for $t\in(0,1)$ and $\Lzw{\RB}{\mu_{\mathrm{Bohr}}}$ are isometrically isomorphic, there may exist representations of the reduced holonomy-flux algebra (reduced algebra of observables) on the former space being not unitarily equivalent to the standard representation \cite{MathStrucLQG} on $\Lzw{\RB}{\mu_{\mathrm{Bohr}}}$. So, the next step towards physics might be to construct such representations on $L^2\big(\hspace{1pt}\raisebox{-0.1ex}{$\qR$},\mu_{\rho,t}\big)$.
\end{enumerate}
In the previous sections we have discussed the problem of symmetry reduction in quantum gauge field theories, in particular, in the framework of loop quantum gravity. The problem of determining respective sets of invariant classical connections forming the reduced configuration spaces of the corresponding classical theories has been left open so far. This is the content of the final Section \ref{CHarinvconn}. There we prove a general characterization theorem for invariant connections on principal fibre bundles and, in particular, calculate the sets of invariant connections used in Subsection \ref{sec:inclrel} in order to show that (in the situations discussed there) quantization and reduction do not commute.
\section{A Characterization of Invariant Connections}
\label{CHarinvconn}
The set of connections on a principal fibre bundle $(P,\pi,M,S)$ is closed under pullback by automorphisms, and it is natural to search for connections that do not change under this operation. Especially, connections invariant under a Lie group $(G,\Phi)$ of automorphisms are of particular interest as they reflect the symmetry of the whole group and, for this reason, find their applications in the symmetry reduction of (quantum) gauge field theories. \cite{MathStrucLQG, ChrisSymmLQG}
The first classification theorem for such connections was given by Wang \cite{Wang}, cf.\ Case \ref{th:wang}. This applies to the situation where the induced action $\varphi$ acts transitively on the base manifold and states that each point in the bundle gives rise to a bijection between the set of $\Phi$-invariant connections and certain linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$. In \cite{HarSni} the authors generalize this to the situation where $\varphi$ admits only one orbit type. More precisely, they discuss a variation\footnote{Amongst others, they assume the $\varphi$-stabilizer of $\pi(p_0)$ to be the same for all $p_0\in P_0$.} of the case where the bundle admits a submanifold $P_0$ with $\pi(P_0)$ intersecting each $\varphi$-orbit in a unique point, see Case \ref{scase:OneSlice} and Example \ref{example:SCHSV}.
Here, the $\Phi$-invariant connections are in bijection with such smooth maps $\psi\colon \mathfrak{g}\times P_0\rightarrow \mathfrak{s}$ for which the restrictions $\psi|_{\mathfrak{g}\times T_{p_0}P_0}$ are linear for all $p_0\in P_0$ and that fulfil additional consistency conditions.
Now, in the general case we consider \emph{$\Phi$-coverings} of $P$. These are families $\{P_\alpha\}_{\alpha\in I}$ of immersed submanifolds\footnote{At the moment assume that $P_\alpha\subseteq P$ is a subset which, at the same time, is a manifold such that the inclusion map $\iota_\alpha \colon P_\alpha \rightarrow P$ is an immersion. Here, we tacitly identify $T_{p_\alpha}P_\alpha$ with $\mathrm{im}[\dd_{p_\alpha}\iota_\alpha]$. Note that we do not require $P_\alpha$ to be an embedded submanifold of $P$. Details will be given in Convention \ref{conv:Submnfds}.} $P_\alpha$ of $P$ such that each $\varphi$-orbit has non-empty intersection with $\bigcup_{\alpha\in I}\pi(P_\alpha)$ and for which
\begin{align*}
T_{p}P = T_{p}P_\alpha + \dd_e\Phi_p(\mathfrak{g}) + Tv_{p}P
\end{align*}
holds whenever $p\in P_\alpha$ for some $\alpha\in I$. Here, $Tv_{p}P\subseteq T_pP$ denotes the vertical tangent space at $p\in P$ and $e$ the identity in $G$. Observe that the intersection properties of the sets $\pi(P_\alpha)$ with the $\varphi$-orbits in the base manifold need not to be convenient in any sense. Here one might think of situations in which $\varphi$ admits dense orbits, or of the almost fibre transitive case, cf.\ Case \ref{scase:slicegleichredcluster}.
Let $\THA\colon (G\times S)\times P\rightarrow P$ be defined by $((g,s),p)\mapsto \Phi(g,p)\cdot s^{-1}$ for $(G,\Phi)$ a Lie group of automorphisms of $\PMS$. Then, the main result of this section can be stated as follows:
\begin{satz}
Each $\Phi$-covering $\{P_\alpha\}_{\alpha\in I}$ of $P$ give rise to a bijection between the $\Phi$-invariant connections on $P$ and the families $\{\psi_\alpha\}_{\alpha\in I}$ of smooth maps $\psi_\alpha\colon \mathfrak{g}\times TP_\alpha \rightarrow \mathfrak{s}$ such that ${\psi_\alpha}|_{\mathfrak{g}\times T_{p_\alpha} P_\alpha}$ is linear for all $p_\alpha\in P_\alpha$ and that fulfil the following two (generalized Wang) conditions:
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
\vspace{-1pt}
$\wt{g}(p_\beta) + \vec{w}_{p_\beta}-\wt{s}(p_\beta)=\dd L_q\vec{w}_{p_\alpha}\quad \Longrightarrow\quad \psi_\beta(\vec{g},\vec{w}_{p_\beta})-\vec{s}=\text{\gls{QREP}}(q)\cp\psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)$,
\item
\vspace{-1pt}
$\psi_\beta\big(\Add{q}(\vec{g}),\vec{0}_{p_\beta}\big)=\qrep(q)\cp \psi_\alpha\big(\vec{g},\vec{0}_{p_\alpha}\big)$.
\end{itemize}
\endgroup
\vspace{-4pt}
\noindent
Here, $q\in G\times S$, $p_\alpha\in P_\alpha$, $p_\beta\in P_\beta$ with $p_\beta=q\cdot p_\alpha$ and $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$, $\vec{w}_{p_\beta}\in T_{p_\beta}P_\beta$. Moreover, $\wt{g}$ and $\wt{s}$ denote the fundamental vector fields that correspond to the elements $\vec{g}\in \mathfrak{g}$ and $\vec{s}\in \mathfrak{s}$, respectively, $\rho$ is the map from Definition \ref{def:Invconn} and $\Ad_q(\g):=\Ad_g(\g)$ for $q=(g,s)\in Q$.
\end{satz}
\noindent
Using this theorem, the calculation of invariant connections reduces to identifying a $\Phi$-covering that makes the above conditions as easy as possible. Here, one has to
find the balance between quantity and complexity of these conditions. Of course, the more submanifolds there are, the more conditions we have, so that usually it is convenient to use as few of them as possible. For instance, in the situation where $\varphi$ is transitive it suggests itself to choose a $\Phi$-covering that consists of one single point, which, in turn, has to be chosen appropriately. Also if there is some $m\in M$ contained in the closure of each $\varphi$-orbit, one single submanifold is sufficient, see Case \ref{scase:slicegleichredcluster} and Example \ref{ex:Bruhat}. The same example shows that sometimes pointwise\footnote{Here pointwise means to consider such elements $q\in G\times S$ that are
contained in the $\THA$-stabilizer of some fixed $p_\alpha\in P_\alpha$ for some $\alpha\in I$.} evaluation of the above conditions proves non-existence of $\Phi$-invariant connections.
In any case, one can use the inverse function theorem to construct a $\Phi$-covering $\{P_\alpha\}_{\alpha\in I}$ of $P$ such that the submanifolds $P_\alpha$ have minimal dimension in a certain sense, see Lemma \ref{lemma:suralpha} and Corollary \ref{cor:reductions}. This reproduces the description of connections by means of local 1-forms on $M$ provided that $G$ acts trivially or, more generally, via gauge transformations on $P$, see Case \ref{scase:GaugeTransf}.
Finally, since orbit structures can depend very sensitively on the action or the group, one cannot expect to have a general concept for finding the $\Phi$-covering optimal for calculations.
Indeed, sometimes these calculations become easier if one uses coverings that seem less optimal at a first sight (as, e.g., if they have no minimal dimension, cf.\ calculations in Appendix \ref{subsec:IsotrConn}).
In the first part, we will introduce the notion of a $\Phi$-covering, the central object of this section. In the second part, we prove the main theorem and deduce a slightly more general version of the result from \cite{HarSni}. In the last part, we will show how to construct $\Phi$-coverings to be used in special situations. In particular, we consider the (almost) fibre transitive case, trivial principal fibre bundles and Lie groups of gauge transformations. Along the way we give applications to loop quantum gravity.
\subsection{$\Phi$-Coverings}
\label{sec:phiCoverings}
We will start this subsection with some facts and conventions concerning submanifolds. Then, we provide the definition of a $\Phi$-covering and discuss some its properties.
\begin{convention}
\label{conv:Submnfds}
Let $M$ be a manifold.
\begingroup
\setlength{\leftmargini}{25pt}
\begin{enumerate}
\item
\itspace
A pair $(N,\tau_N)$ consisting of a manifold $N$ and an injective immersion $\tau\colon N\rightarrow M$ is called submanifold of $M$.
\item
\itspace
If $(N,\tau_N)$ is a submanifold of $M$, we tacitly identify $N$ and $TN$ with their images $\tau_N(N)\subseteq M$ and $\dd\tau_N(TN)\subseteq TM$, respectively.
In particular, this means:
\begingroup
\setlength{\leftmarginii}{11pt}
\begin{itemize}
\item
\itspace
If $M'$ is a manifold and $\kappa\colon M\rightarrow M'$ a smooth map, then for $x\in N$ and $\vec{v}\in TN$ we write $\kappa(x)$ and $\dd\kappa(\vec{v})$ instead of $\kappa(\tau_N(x))$ and $\dd\kappa( \dd\tau(\vec{v}))$, respectively.
\item
If $\Psi\colon G\times M\rightarrow M$ is a left action of the Lie group $G$ and $(H,\tau_H)$ a submanifold of $G$, then the restriction of $\Psi$ to $H\times N$ is defined by
\begin{align*}
\Psi|_{H\times N}(h,x):= \Psi(\tau_H(h),\tau_N(x)) \qquad \forall\: (h,x)\in H\times N.
\end{align*}
\item
If $\w\colon TM\rightarrow V$ is a $V$-valued 1-form on $M$, then
\begin{align*}
(\Psi^*\w)|_{TG\times TN}\:(\vec{m},\vec{v}):=(\Psi^*\w)(\vec{m},\dd\tau(\vec{v}))\qquad\quad\forall\: (\vec{m},\vec{v})\in TG\times TN.
\end{align*}
\item
We will not explicitly refer to the maps $\tau_N$ and $\tau_H$ in the following.
\end{itemize}
\endgroup
\item
\itspacec
Open subsets $U\subseteq M$ are equipped with the canonical manifold structure making the inclusion map an embedding.
\item
\itspace
If $L$ is a submanifold of $N$ and $N$ is a submanifold of $M$, we consider $L$ as a submanifold of $M$ in the canonical way.
\hspace*{\fill}{$\Diamond$}
\end{enumerate}
\endgroup
\end{convention}
\begin{definition}
A submanifold $N\subseteq M$ is called $\Psi$-patch iff for each $x\in N$ there is an open neighbourhood $N'\subseteq N$ of $x$ and a submanifold $H$ of $G$ through $e$ such that the restriction $\Psi|_{H\times N'}$ is a diffeomorphism to an open subset $U\subseteq M$.
\end{definition}
\begin{remark}
\label{rem:patch}
\begingroup
\begin{enumerate}
\item
\itspacec
It follows from the inverse function theorem and\footnote{The sums are not necessarily direct.}
\begin{align*}
\dd_{(e,x)}\Psi(\mathfrak{g}\times T_xN)= \dd_e\Psi_x(\mathfrak{g}) + \dd_x\Psi_e(T_xN)= \dd_e\Psi_x(\mathfrak{g})+T_xN\qquad \forall\: x\in N
\end{align*}
that $N$ is a $\Psi$-patch iff for each $x\in N$ we have $T_xM= \dd_e\Psi_x(\mathfrak{g})+T_xN $.\footnote{In fact, let $V\subseteq \dd_e\Psi_x(\mathfrak{g})$ be an algebraic complement of $T_xN$ in $T_xM$ and $V'\subseteq \mathfrak{g}$ a linear subspace with $\dim[V']=\dim[V]$ and $\dd_e\Psi_x(V')=V$. Then, we find a submanifold $H$ of $G$ through $e$ with $T_eH=V'$, so that $\dd_{(e,x)}\Psi\colon T_eH\times T_xN \rightarrow T_xM$ is bijective.}
\item
\itspacec
Open subsets $U\subseteq M$ are always $\Psi$-patches. They are of maximal dimension, which, for instance, is necessary if there is a point in $U$ whose stabilizer equals $G$, see Lemma \ref{lemma:mindimslice}.1.
\item
\itspacec
We allow zero-dimensional patches, i.e., $N=\{x\}$ for $x\in M$. Necessarily, then $\dd_e\Psi_x(\mathfrak{g})=T_xM$ and $\Psi|_{H\times N}=\Psi_x|_{H}$ for each submanifold $H$ of $G$.\hspace*{\fill}{{$\Diamond$}}
\end{enumerate}
\endgroup
\end{remark}
\noindent
The second part of the next elementary lemma equals Lemma 2.1.1 in \cite{DuisKolk}.
\begin{lemma}
\label{lemma:mindimslice}
Let $(G,\Psi)$ be a Lie group that acts on the manifold $M$ and let $x\in M$.
\begin{enumerate}
\item
\label{item:s}
If $N$ is a $\Psi$-patch with $x\in N$, then $\dim[N]\geq \dim[M]-\dim[G]+\dim[G_x]$.
\item
Let $V$ and $W$ be algebraic complements of $\dd_e\Psi_x(\mathfrak{g})$ in $T_xM$ and of $\mathfrak{g}_x$ in $\mathfrak{g}$, respectively. Then, there are submanifolds $N$ of $M$ through $x$ and $H$ of $G$ through $e$ such that $T_xN=V$ and $T_eH=W$. In particular, $N$ is a $\Psi$-patch and $\dim[N]=\dim[M]-\dim[G]+\dim[G_x]$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
By Remark \ref{rem:patch}.1 and since $\ker[\dd_e\Psi_x]=\mathfrak{g}_x$, we have
\begin{align}
\label{eq:leq}
\dim[M]\leq \dim[\dd_e\Psi_x(\mathfrak{g})]+\dim[T_xN]=\dim[G]-\dim[G_x]+\dim[N].
\end{align}
\item
Of course, we find submanifolds $N'$ of $M$ through $x$ and $H'$ of $G$ through $e$ such that $T_xN'=V$ and $T_eH'=W$. So, if $\vec{g}\in \mathfrak{g}$ and $\vec{v}_x\in T_xN'$, then $0=\dd_{(e,x)}\Psi(\vec{g},\vec{v}_x)=\dd_e\Psi_x(\vec{g})+\vec{v}_x$ implies $\dd_e\Psi_x(\vec{g})=0$ and $\vec{v}_x=0$. Hence, $\vec{g}\in \ker[\dd_e\Psi_x]=\mathfrak{g}_x$ so that\footnote{Recall that $\dd_{(e,x)}\Psi|_{T_eH'\times T_eN'}\colon \big(\raisebox{-1pt}{$\vec{h},\vec{v}_x$}\big)\mapsto \dd_{(e,x)}\Psi\big(\raisebox{-1pt}{$\dd_e\tau_H(\vec{h}),\dd_x\tau_N(\vec{v}_x)$}\big)$.} $\dd_{(e,x)}\Psi|_{T_eH'\times T_eN'}$ is injective. It is immediate from the definitions that this map is surjective so that by the inverse function theorem we find open neighbourhoods $N\subseteq N'$ of $x$ and $H\subseteq G$ of $e$ such that $\Psi|_{H\times N}$ is a diffeomorphism to an open subset $U\subseteq M$. Then $N$ is a $\Psi$-patch and since in \eqref{eq:leq} equality holds, also the last claim is clear.
\end{enumerate}
\end{proof}
\end{lemma}
\begin{definition}
\label{def:pmappe}
Let $(G,\Phi)$ be a Lie group of automorphisms of the principal fibre bundle $P$ and recall the actions $\varphi$ and $\THA$ defined by \eqref{eq:INDA} and \eqref{eq:THETA}, respectively.
A family of $\THA$-patches $\{P_\alpha\}_{\alpha\in I}$ is said to be a $\Phi$-covering of $P$ iff each $\varphi$-orbit intersects at least one of the sets $\pi(P_\alpha)$.
\end{definition}
\begin{remark}
\label{bem:Psliceeigensch}
\begin{enumerate}
\item
If $O\subseteq P$ is a $\THA$-patch, then Lemma \ref{lemma:mindimslice}.1 and \eqref{eq:staoQ} yield
\vspace{-1ex}
\begin{align*}
\dim[O]&\geq \dim[P]-\dim[Q]+\dim[Q_p]
\stackrel{\eqref{eq:staoQ}}{=}\dim[M]-\dim[G]+\dim[G_{\pi(p)}].
\end{align*}
\item
\itspace
It follows from Remark \ref{rem:patch}.1 and $\dd_e\THA_p(\mathfrak{q})=\dd_e\Phi_p(\mathfrak{g}) + Tv_pP$ that $O$ is a $\THA$-patch iff
\begin{align}
\label{eq:transv}
T_pP=T_pO+ \dd_e\Phi_p(\mathfrak{g}) + Tv_pP\qquad \forall\: p\in O.
\end{align}
As a consequence
\begingroup
\setlength{\leftmarginii}{13pt}
\begin{itemize}
\item
\itspacec
each $\Phi$-patch is a $\THA$-patch,
\item
$P$ is always a $\Phi$-covering by itself, and if $P=M\times S$ is trivial, then $M\times \{e\}$ is a $\Phi$-covering.
\end{itemize}
\endgroup
\item
\itspace
If $N$ is a $\varphi$-patch and $s_0\colon N \rightarrow P$ a smooth section, i.e., $\pi\cp s_0 =\id_N$, then $O:=s_0(N)$ is a $\THA$-patch by Lemma \ref{lemma:suralpha}.2.\hspace*{\fill}{{$\Diamond$}}
\end{enumerate}
\end{remark}
\begin{lemma}
\label{lemma:suralpha}
Let $(G,\Phi)$ be a Lie group of automorphisms of the principal fibre bundle $(P,\pi,M,S)$.
\begin{enumerate}
\item
If $O\subseteq P$ is a $\THA$-patch, then for each $p\in O$ and $q\in Q$ the differential
$\dd_{(q,p)}\THA\colon T_qQ\times T_{p}O\rightarrow T_{q\cdot p}P$
is surjective.
\item
If $N$ is a $\varphi$-patch and $s_0\colon N \rightarrow P$ a smooth section, then $O:=s_0(N)$ is a $\THA$-patch.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Since $O$ is a $\THA$-patch, the claim is clear for $q=e$. If $q$ is arbitrary, then for each $\kk_q\in T_qQ$ we find some $\vec{q}\in\mathfrak{q}$ such that $\kk_q=\dd L_q\vec{q}$. Consequently, for $\vec{w}_{p}\in T_{p}P$ we have
\begin{align*}
\dd_{(q,p)}\THA\left(\kk_q,\vec{w}_{p}\right)&=\dd_{(q,p)}\THA(\dd L_q\vec{q},\vec{w}_{p})=\dd_{p}L_q\left(\dd_{(e,p)}\THA(\vec{q},\vec{w}_{p})\right).
\end{align*}
So, since left translation w.r.t.\ $\THA$ is a diffeomorphism, $\dd_{p}L_q$ is surjective.
\item
First observe that $O$ is a submanifold of $P$ because $s_0$ is an injective immersion.
By Remark \ref{bem:Psliceeigensch}.2 it suffices to show that
\begin{align*}
\dim\big[T_{s_0(x)}O + \dd_e\Phi_{s_0(x)}(\mathfrak{g})+ Tv_{s_0(x)}P\big]\geq \dim[\hspace{1pt}T_{s_0(x)}P\hspace{1pt}]\qquad \forall\: x\in N.
\end{align*}
For this, let $x\in N$ and $V'\subseteq \mathfrak{g}$ be a linear subspace with $V'\oplus \mathfrak{g}_x$ and $T_xM=T_xN \oplus
\mathrm{d}_e\varphi_x(V')$. Then,
$T_{s_0(x)}O \oplus \dd_e\Phi_{s_0(x)}(V')\oplus Tv_{s_0(x)}P$
because if $\dd_xs_0(\vec{v}_x) +\dd_e\Phi_{s_0(x)}(\vec{g}\os')+ \vec{v}_v=0$ for $\vec{v}_x\in T_xN$, $\vec{g}\os'\in V'$ and $\vec{v}_v\in Tv_{s_0(x)}P$, then
\begin{align*}
0=\dd_{s_0(x)}\pi \big(\dd_xs_0(\vec{v}_x) +\textstyle\dd_e\Phi_{s_0(x)}(\vec{g}\os')+ \vec{v}_v\big)=\vec{v}_x \oplus \dd_e\varphi_x(\vec{g}\os')
\end{align*}
showing that $\vec{v}_x =0$ and $\mathrm{d}_e\phi_{x}(\vec{g}')=0$, hence $\vec{g}'=0$ by the choice of $V'$, i.e., $\vec{v}_v=0$ by assumption.
In particular, $\mathrm{d}_e\phi_{x}(\vec{g}')=0$ if $\mathrm{d}_e\Phi_{s_0(x)}(\vec{g}')=0$, hence $\dim[\mathrm{d}_e\Phi_{s_0(x)}(V')]\geq\dim[\mathrm{d}_e\varphi_x(V')]$, from which we obtain
\begin{align*}
\dim\big[T_{s_0(x)}O + \dd_e\Phi_{s_0(x)}(\mathfrak{g})+ Tv_{s_0(x)}P\big]&\\
& \hspace{-60pt}\geq \dim\big[T_{s_0(x)}O \oplus \dd_e\Phi_{s_0(x)}(V')\oplus Tv_{s_0(x)}P\big] \\
&\hspace{-60pt} =\dim[T_xN] + \dim[\dd_e\Phi_{s_0(x)}(V')] + \dim[S]\\
&\hspace{-60pt} \geq\dim[T_xN] + \dim[\dd_e\varphi_x(V')] + \dim[S]\\
&\hspace{-60pt} =\dim[P].
\end{align*}
\end{enumerate}
\end{proof}
\end{lemma}
\subsection{Characterization of Invariant Connections}
\label{sec:mainth}
In this subsection, we will use $\Phi$-coverings $\{P_\alpha\}_{\alpha\in I}$ of the bundle $P$ in order to characterize the set of $\Phi$-invariant connections by families $\{\psi_\alpha\}_{\alpha\in I}$ of smooth maps $\psi_\alpha\colon \mathfrak{g}\times TP_\alpha \rightarrow \mathfrak{s}$ whose restrictions $\psi_\alpha|_{\mathfrak{g}\times T_{p_\alpha}P_\alpha }$ are linear and that fulfil two additional compatibility conditions. Here, we follow the lines of Wang's original approach, which basically means that we will generalize the proofs from \cite{Wang} to the non-transitive case.
We will proceed in two steps, the first one being performed in the next subsection. There, we show that a $\Phi$-invariant connection gives rise to a consistent family $\{\psi_\alpha\}_{\alpha\in I}$ of smooth maps as described above. We also discuss the situation in \cite{HarSni} in order to make the two conditions more intuitive.
Then, in Subsection \ref{subsec:Reconstruc}, we will verify that such families $\{\psi_\alpha\}_{\alpha\in I}$ glue together to a $\Phi$-invariant connection on $P$.
\subsubsection{Reduction of Invariant Connections}
\label{subsec:RedInvConn}
In the following, let $\{P_\alpha\}_{\alpha\in I}$ be a fixed $\Phi$-covering of $P$ and $\w$ a $\Phi$-invariant connection on $P$. We define $\w_\alpha:=(\THA^*\w)|_{TQ\times TP_\alpha}$ and $\psi_\alpha:=\w_\alpha|_{\mathfrak{g}\times TP_\alpha}$, and for $q'\in Q$ we let $\Co{q'}\colon Q\times P\rightarrow Q\times P$ denote the map $\Co{q'}(q,p):=\left(\Co{q'}(q),p\right)$ for $\Co{q'}\colon Q\mapsto Q$ the conjugation map w.r.t.\ $q'$.
Finally, we let $\Ad_q(\g):=\Ad_g(\g)$ for $q=(g,s)\in Q$ and $\g\in \mg$ in the sequel.
\begin{lemma}
\label{lemma:omegaalpha}
Let $q\in Q$, $p_\alpha\in P_\alpha$, $p_\beta\in P_\beta$ with\footnote{More precisely, $\tau_{P_\beta}(p_\beta)=q\cdot \tau_{P_\alpha}(p_\alpha)$ by Convention \ref{conv:Submnfds}.} $p_\beta=q\cdot p_\alpha$ and $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$. Then
\begin{enumerate}
\item
$\w_\beta(\xii\:)=\qrep(q)\cp \w_\alpha(\vec{0}_\mathfrak{q},\vec{w}_{p_\alpha})$ for all $\xii\in TQ\times TP_\beta$ with $\dd\THA(\xii\:)=\dd L_q\vec{w}_{p_\alpha}$,
\item
$\left(\Co{q}^*\w_\beta\right)\big(\raisebox{-0ex}{$\kk,\vec{0}_{p_\beta}$}\big)=\qrep(q)\cp \w_\alpha\big(\raisebox{-0ex}{$\kk,\vec{0}_{p_\alpha}$}\big)$ for all $\kk\in TQ$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Let $\xii\in T_{q'}Q\times T_pP_\beta$ for $q'\in Q$. Then, since\footnote{See end of Subsection \ref{subsec:InvConn}.} $L_q^*\w = \qrep(q)\cp \w$ for each $q\in Q$ and $q'\cdot p= q\cdot p_\alpha =p_\beta$, we have
\begin{align*}
\w_\beta(\xii\:)&=\w_{q'\cdot p}(\dd_{(q', p)}\THA (\xii\:))=\w_{p_\beta}(\dd L_q\vec{w}_{p_\alpha})
=(L_q^*\w)_{p_\alpha}(\vec{w}_{p_\alpha})\\
&=\qrep(q)\cp \w_{p_\alpha}(\vec{w}_{p_\alpha})=\qrep(q)\cp \w_{p_\alpha}\big(\dd_{(e,p_\alpha)}\THA (\vec{0}_\mathfrak{q},\vec{w}_{p_\alpha})\big)\\
&=\qrep(q)\cp \w_{\alpha}\raisebox{1pt}{$\big($}\vspace{2pt}\vec{0}_\mathfrak{q},\vec{w}_{p_\alpha}\raisebox{1pt}{$\big)$}.
\end{align*}
\item
For $\kk_{q'} \in T_{q'}Q$ let $\gamma \colon (-\epsilon,\epsilon)\rightarrow Q$ be smooth with $\dot\gamma(0)=\kk_{q'}$. Then
\begin{align*}
\big(\Co{q}^*\w_\beta \big)_{(q',p_\beta)}\big(\raisebox{-0ex}{$\kk_{q'},\vec{0}_{p_\beta}$}\big)&={\w_\beta}_{(\Co{q}(q'),p_\beta)}\big(\Add{q}(\kk_{q'}),\vec{0}_{p_\beta}\big)\\
&=\w_{qq'q^{-1}q\cdot p_\alpha }\left(\dttB{t}{0}q\gamma(t) q^{-1}q\cdot p_\alpha\right)\\
& =\left(L_q^*\w\right)_{q'\cdot p_\alpha }\left(\dttB{t}{0}\gamma(t)\cdot p_\alpha\right)\\
& =\qrep(q)\cp \w_{q'\cdot p_\alpha}\left(\dd_{(q',p_\alpha)}\THA\left( \kk_{q'}\right)\right)\\
&=\qrep(q)\cp {\w_\alpha}_{(q',p_\alpha)} \big(\kk_{q'},\vec{0}_{p_\alpha}\big).
\end{align*}
\end{enumerate}
\end{proof}
\end{lemma}
\begin{corollary}
\label{cor:psialpha}
Let $q\in Q$, $p_\alpha\in P_\alpha$, $p_\beta\in P_\beta$ with $p_\beta=q\cdot p_\alpha$ and $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$. Then for
$\vec{w}_{p_\beta}\in T_{p_\beta}P_\beta$, $\vec{g}\in \mathfrak{g}$ and $\vec{s}\in \mathfrak{s}$ we have
\begin{enumerate}
\item[\textit{i.)}]
$\wt{g}(p_\beta) + \vec{w}_{p_\beta}-\wt{s}(p_\beta)=\dd L_q\vec{w}_{p_\alpha}\quad \Longrightarrow\quad \psi_\beta(\vec{g},\vec{w}_{p_\beta})-\vec{s}=\qrep(q)\cp\psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)$,
\item[\textit{ii.)}]
$\psi_\beta\big(\Add{q}(\vec{g}),\vec{0}_{p_\beta}\big)=\qrep(q)\cp \psi_\alpha\big(\vec{g},\vec{0}_{p_\alpha}\big)$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item[\textit{i.)}]
In general, for $\vec{w}_p\in T_pP$, $\vec{g}\in\mathfrak{g}$ and $\vec{s}\in\mathfrak{s}$ we have
\begin{align}
\label{eq:ThetaPhi}
\dd_{(e,p)}\THA((\vec{g},\vec{s}),\vec{w}_p)=\dd_{(e,p)}\Phi(\vec{g},\vec{w}_p)-\wt{s}(p)=\wt{g}(p)+\vec{w}_p-\wt{s}(p)
\end{align}
and, since $\w$ is a connection, for $((\vec{g},\vec{s}),\vec{w}_{p_\alpha})\in \mathfrak{q}\times TP_\alpha$ we obtain
\begin{align}
\label{eq:gugu}
\begin{split}
\w_\alpha((\vec{g},\vec{s}),\vec{w}_{p_\alpha})&=\w\big(\dd_{(e,p_\alpha)}\Phi(\vec{g},\vec{w}_{p_\alpha})-\wt{s}(p_\alpha)\big)\\
&=\w\big(\dd_{(e,p_\alpha)}\Phi(\vec{g},\vec{w}_{p_\alpha})\big)-\vec{s}\\
&=\w_\alpha\left(\vec{g},\vec{w}_{p_\alpha}\right)-\vec{s} =\psi_\alpha\left(\vec{g},\vec{w}_{p_\alpha}\right)-\vec{s}.
\end{split}
\end{align}
Now, assume that $\dd_e\Phi_{p_\beta}(\vec{g}\hspace{1pt})+\vec{w}_{p_\beta}-\wt{s}(p)=\dd L_q\vec{w}_{p_\alpha}$. Then $\dd_{(e,p_\beta)}\THA((\vec{g},\vec{s}),\vec{w}_{p_\beta})=\dd L_q \vec{w}_{p_\alpha}$ by \eqref{eq:ThetaPhi} so that
$\w_\beta((\vec{g},\vec{s}),\vec{w}_{p_\beta})=\qrep(q)\cp \w_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)$ by Lemma \ref{lemma:omegaalpha}.1. Consequently,
\vspace{-8pt}
\begin{align*}
\psi_\beta\left(\vec{g},\vec{w}_{p_\beta}\right)-\vec{s}&\stackrel{\eqref{eq:gugu}}{=}\w_\beta((\vec{g},\vec{s}),\vec{w}_{p_\beta})\\ & \:\: =\qrep(q)\cp \w_\alpha\big(\vec{0}_\mathfrak{q},\vec{w}_{p_\alpha}\big)\stackrel{\eqref{eq:gugu}}{=}\qrep(q)\cp \psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big).
\end{align*}
\item[\textit{ii.)}]
\vspace{-8pt}
Lemma \ref{lemma:omegaalpha}.2 yields
\begin{align*}
\psi_\beta\big(\Add{q}(\vec{g}),\vec{0}_{p_\beta}\big)&=(\Co{q}^*\w_\beta)_{(e,p_\beta)}\big(\vec{g},\vec{0}_{p_\beta}\big)
\\ &=\qrep(q)\cp (\w_\alpha)_{(e,p_\alpha)}\big(\vec{g},\vec{0}_{p_\alpha}\big)
=\qrep(q)\cp \psi_\alpha\big(\vec{g},\vec{0}_{p_\alpha}\big).
\end{align*}
\end{enumerate}
\end{proof}
\end{corollary}
\begin{definition}
A family $\{\psi_\alpha\}_{\alpha\in I}$ of smooth maps $\psi_\alpha\colon \mathfrak{g}\times TP_\alpha \rightarrow \mathfrak{s}$ which are linear in the sense that $\psi_\alpha|_{\mathfrak{g}\times T_{p_\alpha}P_\alpha }$ is linear for all $p_\alpha\in P_\alpha$ is called reduced connection w.r.t.\ $\{P_\alpha\}_{\alpha\in I}$ iff it fulfils the conditions \textit{i.)} and \textit{ii.)} from Corollary \ref{cor:psialpha}.
\end{definition}
\begin{remark}
\label{rem:Psialphaconderkl}
\begingroup
\setlength{\leftmargini}{18pt}
\begin{enumerate}
\item[{\bf 1)}]
In particular, Corollary \ref{cor:psialpha}.\textit{i.)} encodes the following condition
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item[\textit{a.)}]
For all $\beta \in I$, $(\vec{g},\vec{s})\in \mathfrak{q}$ and $\vec{w}_{p_\beta}\in T_{p_\beta}P_\beta$ we have
\begin{align*}
\wt{g}(p_\beta) +\vec{w}_{p_\beta}-\wt{s}(p_\beta)=0\quad \Longrightarrow\quad \psi_\beta(\vec{g},\vec{w}_{p_\beta})-\vec{s}=0.
\end{align*}
\end{enumerate}
\endgroup
\item[{\bf 2)}]
Assume that \textit{a.)} is true and let $q\in Q$, $p_\alpha\in P_\alpha$, $p_\beta \in P_\beta$ with $p_\beta=q\cdot p_\alpha$. Moreover, assume that we find elements $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$ and $((\vec{g},\vec{s}),\vec{w}_{p_\beta})\in \mathfrak{q}\times T_{p_\beta}P_\beta$ such that
\begin{align*}
\dd_{(e,p_\beta)}\THA((\vec{g},\vec{s}),\vec{w}_{p_\beta})=\dd L_q\vec{w}_{p_\alpha}\quad \text{and}\quad \psi_\beta(\vec{g},\vec{w}_{p_\beta})-\vec{s}=\qrep(q)\cp\psi_\alpha(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha})
\end{align*}
holds.
Then
$\psi_\beta\big(\vec{g}\hspace{1pt}',\vec{w}'_{p_\beta}\big)-\vec{s}\hspace{1pt}'=\qrep(q)\cp\psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)$ holds for each element\footnote{Observe that due to surjectivity of $\dd_{(e,p_\beta)}\Phi$ such elements always exist.} $\big((\vec{g}\hspace{1pt}',\vec{s}\hspace{1pt}'),\vec{w}'_{p_\beta}\big)\in \mathfrak{q}\times T_{p_\beta}P_\beta$ with $\dd_{(e,p_\beta)}\THA\big((\vec{g}\hspace{1pt}',\vec{s}\hspace{1pt}'),\vec{w}'_{p_\beta}\big)=\dd L_q\vec{w}_{p_\alpha}$.
In fact, we have
\begin{align*}
\dd_{(e,p_\beta)}\THA\big((\vec{g}-\vec{g}\hspace{1pt}',\vec{s}-\vec{s}\hspace{1pt}'),\vec{w}_{p_\beta}-\vec{w}'_{p_\beta}\big)=0,
\end{align*}
so that by \eqref{eq:ThetaPhi} condition \textit{a.)} gives
\vspace{-4pt}
\begin{align*}
0 & \stackrel{\textit{a.)}}{=}\psi_\beta(\vec{g}-\vec{g}\hspace{1pt}',\vec{w}_{p_\beta}-\vec{w}'_{p_\beta})-(\vec{s}-\vec{s}\hspace{1pt}'))
\hspace{2pt}= [\psi_\beta(\vec{g},\vec{w}_{p_\beta})-\vec{s}\hspace{2pt}]-[\psi_\beta(\vec{g}\hspace{1pt}',\vec{w}'_{p_\beta})-\vec{s}\hspace{1pt}']\\
&= \qrep(q)\cp\psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)-[\psi_\beta(\vec{g}\hspace{1pt}',\vec{w}'_{p_\beta})-\vec{s}\hspace{1pt}'].
\end{align*}
\item[{\bf 3)}]
Assume that $\dd L_q \vec{w}_{p_\alpha}\in T_{p_\beta}P_\beta$ holds for all $q\in Q$, $p_\alpha\in P_\alpha$, $p_\beta \in P_\beta$ with $p_\beta=q\cdot p_\alpha$ and all $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$. Then, $\dd_{(e,p_\beta)}\THA\left(\dd L_q \vec{w}_{p_\alpha}\right)=\dd L_q \vec{w}_{p_\alpha}$, so that it follows from {\bf 2)} that in this case we can substitute
\textit{i.)} by \textit{a.)} and condition
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{enumerate}
\item[\textit{b.)}]
\itspacec
Let $q\in Q$, $p_\alpha\in P_\alpha$, $p_\beta \in P_\beta$ with $p_\beta=q\cdot p_\alpha$. Then
\begin{align*}
\psi_\beta\big(\vec{0}_{\mathfrak{g}},\dd L_q\vec{w}_{p_\alpha}\big)=\qrep(q)\cp\psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)\qquad\quad \forall\:\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha.
\end{align*}
\end{enumerate}
\endgroup
\noindent
Now, \textit{b.)} looks similar to \textit{ii.)} and makes it plausible that the conditions \textit{i.)} and \textit{ii.)} from Corollary \ref{cor:psialpha} together encode the $\qrep$-invariance of the corresponding connection $\w$.
However, usually there is no reason for $\dd L_q \vec{w}_{p_\alpha}$ to be an element of $T_{p_\beta}P_\beta$. Even for $p_\alpha=p_\beta$ and $q\in Q_{p_\alpha}$ this is usually not true. So, typically there is no way to split up \textit{i.)} into parts whose meaning is more intuitive. \hspace*{\fill}{{$\Diamond$}}
\end{enumerate}
\endgroup
\end{remark}
\noindent
Remark \ref{rem:Psialphaconderkl} immediately proves
\begin{scase}[Gauge Fixing]
\label{scase:OneSlice}
Let $P_0$ be a $\THA$-patch of the bundle $P$ such that $\pi(P_0)$ intersects each $\varphi$-orbit in a unique point and $\dd L_q(T_pP_0)\subseteq T_pP_0$ for all $p\in P_0$ and all $q \in Q_p$. Then, a corresponding reduced connection consists of one single smooth map $\psi\colon \mathfrak{g}\times TP_0\rightarrow \mathfrak{s}$, and we have $p=q\cdot p'$ for $q\in Q$, $p,p'\in P_0$ iff $p=p'$ and $q\in Q_p$. So, by Remark \ref{rem:Psialphaconderkl} the two conditions from Corollary \ref{cor:psialpha} are equivalent to:
\newline
\vspace{-10pt}
\newline
Let $p\in P_0$, $q=(h,\phi_p(h))\in Q_p$, $\vec{w}_p\in T_pP_0$ and $\vec{g}\in \mathfrak{g}$, $\vec{s}\in \mathfrak{s}$. Then
\begin{enumerate}
\item[\textit{i'.)}]
$\wt{g}(p) +\vec{w}_{p}-\wt{s}(p)=0\quad \Longrightarrow\quad \psi(\vec{g},\vec{w}_{p})-\vec{s}=0$,
\item[\textit{ii'.)}]
$\psi\big(\vec{0}_{\mathfrak{g}},\dd L_q\vec{w}_{p}\big)=\qrep(q)\cp\psi\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p}\big)$,
\item[\textit{iii'.)}]
$\psi\big(\Add{h}(\vec{g}),\vec{0}_{p}\big)=\Add{\phi_p(h)}\cp\: \psi\big(\vec{g},\vec{0}_{p}\big)$.\hspace*{\fill}{{\scriptsize$\blacksquare$}}
\end{enumerate}
\end{scase}
\noindent
The next example is a slight generalization of Theorem 2 in \cite{HarSni}. There the authors assume that $\varphi$ admits only one orbit type so that $\dim[G_x]=l$ for all $x\in M$. Then, they restrict to the situation where one finds a triple $(U_0,\tau_0,s_0)$ consisting of an open subset $U_0\subseteq \RR^k$ for $k=\dim[M]- [\hspace{1pt}\dim[G]- l\hspace{1pt}]$, an embedding $\tau_0\colon U_0 \rightarrow M$ and a smooth map $s_0\colon U_0\rightarrow P$ with $\pi\cp s_0=\tau_0$ and the addition property that $Q_p$ is the same for all $p\in \mathrm{im}[s_0]$. More precisely, they assume that $G_x$ and the structure group of the bundle are compact. Then they show the non-trivial fact that $s_0$ can be modified in such a way that in addition $Q_p$ is the same for all $p\in \mathrm{im}[s_0]$.
Observe that the authors forgot to require that $\mathrm{im}[\dd_x\tau_0] + \mathrm{im}\big[\dd_e\varphi_{\tau_0(x)}\big]=T_{\tau_0(x)}M$ holds for all $x\in U_0$, i.e., that $\tau_0(U_0)$ is a $\varphi$-patch (so that $s_0(U_0)$ is a $\THA$-patch). Indeed, Example \ref{ex:transinv}.2 shows that this additional condition is crucial. The next example is a slight modification of the result \cite{HarSni} in the sense that we do not assume $G_x$ and the structure group to be compact, but make the ad hoc requirement that $Q_p$ is the same for all $p\in P_0$.
\begin{example}[Harnad, Shnider, Vinet]
\label{example:SCHSV}
Let $P_0$ be a $\THA$-patch of the bundle $P$ such that $\pi(P_0)$ intersects each $\varphi$-orbit in a unique point. Moreover, assume that the $\THA$-stabilizer $L:=Q_{p}$ is the same for all $p\in P_0$. Then, it is clear from \eqref{eq:staoQ} that $H:=G_{\pi(p)}$ and
$\phi:=\phi_{p}\colon H \rightarrow S$ are independent of the choice of $p\in P_0$. Finally, we require that
\begin{align}
\label{eq:dimP}
\begin{split}
\dim[P_0]&=\dim[M]-[\dim[G]-\dim[H]]
=\dim[P]-[\dim[Q]-\dim[H]]
\end{split}
\end{align}
holds.
Now, let $p\in P_0$ and $q=(h,\phi(h))\in Q_p$. Then, for $\vec{w}_{p}\in T_{p}P_0$ we have
\begin{align*}
\dd L_q\vec{w}_{p}&=\dttB{t}{0} \Phi(h,\gamma(t))\cdot \phi^{-1}_{p}(h)
=\dttB{t}{0} [\gamma(t)\cdot \phi_{\gamma(t)}(h)]\cdot \phi^{-1}_{p}(h)\\
&=\dttB{t}{0} [\gamma(t)\cdot \phi_{p}(h)]\cdot \phi^{-1}_{p}(h)=\vec{w}_{p},
\end{align*}
where $\gamma\colon (-\epsilon,\epsilon)\rightarrow P_0$ is some smooth curve with $\dot\gamma(0)=\vec{w}_{p}$. Consequently, $\dd L_q(T_pP_0)\subseteq T_pP_0$ so that we are in the situation of Case \ref{scase:OneSlice}. Here, \textit{ii'.)} now reads $\psi\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p}\big)=\Add{\phi(h)}\cp\:\psi\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p}\big)$ for all $h\in H$ and \textit{iii'.)} does not change. For \textit{i'.)}, observe that the Lie algebra $\mathfrak{l}$ of $L$ is contained in the kernel of $\dd_{(e,p_0)}\THA$.
But $\dd_{(e,p_0)}\THA$ is surjective since $P_0$ is a $\THA$-patch (cf.\ Lemma \ref{lemma:suralpha}.1), so that
\vspace{-1ex}
\begin{align*}
\dim\!\big[\!\ker\!\big[\dd_{(e,p_0)}\THA\big]\big]= \dim[Q] + \dim[P_0]-\dim[P]\stackrel{\eqref{eq:dimP}}{=}\dim[H],
\end{align*}
whereby $\dd_{(e,p_0)}\THA$ means the differential of the restriction of $\THA$ to $Q\times P_0$.
This shows $\ker[\dd_{(e,p)}\THA]=\mathfrak{l}$ for all $p\in P_0$. Altogether, it follows that a reduced connection w.r.t.\ $P_0$ is a smooth, linear\footnote{In the sense that $\psi|_{\mathfrak{g}\times T_pP_0}$ is linear for all $p\in P_0$.} map $\psi\colon \mathfrak{g}\times TP_0\rightarrow \mathfrak{s}$ which fulfils the following three conditions:
\begingroup
\setlength{\leftmargini}{30pt}
\begin{itemize}
\item[\textit{i''.)}]
\vspace{-4pt}
$\psi\big(\raisebox{-0.15ex}{$\vec{h},\vec{0}_{p}$}\big)=\dd_e\phi\big(\raisebox{-0.15ex}{$\vec{h}$}\:\big)\qquad \qquad\quad\hspace{10pt} \qquad \qquad\forall \:\vec{h}\in \mathfrak{h}, \forall\:p\in P_0$,\hspace*{\fill}(use \eqref{eq:ThetaPhi})
\item[\textit{ii''.)}]
\itspace
$\psi\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)=\Add{\phi(h)}\cp \:\psi\big(\vec{0}_{\mathfrak{g}},\vec{w}\big) \qquad\qquad\quad\hspace{6.1pt} \forall\:h\in H, \forall\: \vec{w}\in TP_0$,
\item[\textit{iii''.)}]
\itspace
$\psi\big(\Add{h}(\vec{g}),\vec{0}_{p}\big)=\Add{\phi(h)}\cp\: \psi\big(\raisebox{-0.5pt}{$\vec{g},\vec{0}_{p}$}\big)\quad\qquad\hspace{2.5pt} \forall\: \vec{g}\in\mathfrak{g}, \forall\: h\in H, \forall\: p\in P_0$.
\end{itemize}
\endgroup
\vspace{-4pt}
\noindent
Then, $\mu:= \psi|_{TP_0}$ and $A_{p_0}(\vec{g}):=\psi\big(\vec{g},\vec{0}_{p_0}\big)$ are the maps that are used for the characterization in Theorem 2 in \cite{HarSni}.\hspace*{\fill}{$\Diamond$}
\end{example}
\subsubsection{Reconstruction of Invariant Connections}
\label{subsec:Reconstruc}
Let $\{P_\alpha\}_{\alpha\in I}$ be a $\Phi$-covering of $P$. We now show that each respective reduced connection $\{\psi_\alpha\}_{\alpha\in I}$ gives rise to a unique $\Phi$-invariant connection on $P$. To this end, for each $\alpha\in I$ we define the maps
$\lambda_\alpha\colon \mathfrak{q}\times TP_\alpha\rightarrow \mathfrak{s},
((\vec{g},\vec{s}),\vec{w})\mapsto\psi_\alpha(\vec{g},\vec{w})-\vec{s}$
and
\begin{align*}
\w_\alpha\colon TQ\times TP_\alpha &\rightarrow \mathfrak{s}\\
\big(\kk_q,\vec{w}_{p_\alpha}\big)&\mapsto \qrep(q)\cp \lambda_\alpha\left(\dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha}\right)\!,
\end{align*}
where $\kk_q\in T_qQ$ and $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$.
\begin{lemma}
\label{lemma:lambda}
Let $q\in Q$, $p_\alpha\in P_\alpha$, $p_\beta\in P_\beta$ with $p_\beta=q\cdot p_\alpha$ and $\vec{w}_{p_\alpha}\in T_{p_\alpha}P_\alpha$. Then
\begin{enumerate}
\item
$\lambda_\beta(\xii\:)=\qrep(q)\cp \lambda_\alpha\big(\vec{0}_{\mathfrak{q}},\vec{w}_{p_\alpha}\big)$ for all $\xii\in \mathfrak{q}\times T_{p_\beta}P$ with $\dd\THA_{(e,p_\beta)}(\xii\:)=\dd L_q\vec{w}_{p_\alpha}$,
\item
$\lambda_\beta\big(\Add{q}(\vec{q}\:),\vec{0}_{p_\beta}\big)=\qrep(q)\cp \lambda_\alpha\big(\vec{q},\vec{0}_{p_\alpha}\big)$ for all $\vec{q}\in \mathfrak{q}$.
\end{enumerate}
For all $\alpha \in I$ we have
\begin{enumerate}
\item[3)]
$\ker\!\big[\lambda_\alpha|_{\mathfrak{q}\times T_{p_\alpha}P_\alpha}\big]\subseteq \ker\!\big[\dd_{(e,p_\alpha)}\THA\big]$ for all $p_\alpha\in P_\alpha$,
\item[4)]
the map $\w_\alpha$ is the unique $\mathfrak{s}$-valued 1-form on $Q\times P_\alpha$ that extends $\lambda_\alpha$ and for which
we have $L_q^*\w_\alpha=\qrep(q)\cp \w_\alpha$ for all $q\in Q$.
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item
Write $\xii=((\vec{g},\vec{s}\:),\vec{w}_{p_\beta})$ for $\vec{g}\in\mathfrak{g}$, $\vec{s}\in\mathfrak{s}$ and $\vec{w}_{p_\beta}\in T_{p_\beta}P_\beta$. Then
\vspace{-1ex}
\begin{align*}
\wt{g}(p_\beta) +\vec{w}_{p_\beta}-\wt{s}(p_\beta)\stackrel{\eqref{eq:ThetaPhi}}{=}\dd\THA_{(e,p_\beta)}(\xii\:)=\dd L_q\vec{w}_{p_\alpha}
\end{align*}
so that from condition \textit{i.)} in Corollary \ref{cor:psialpha} we obtain
\begin{align*}
\lambda_\beta(\xii\:)=\psi_\beta(\vec{g},\vec{w}_{p_\beta})-\vec{s}=\qrep(q)\cp\psi_\alpha\big(\vec{0}_{\mathfrak{g}},\vec{w}_{p_\alpha}\big)=\qrep(q)\cp\lambda_\alpha\big(\vec{0}_{\mathfrak{q}}, \vec{w}_{p_\alpha}\big).
\end{align*}
\item
Let $\vec{q}=(\vec{g},\vec{s}\:)$ for $\vec{g}\in\mathfrak{g}$ and $\vec{s}\in \mathfrak{s}$. Then by Corollary \ref{cor:psialpha}.\textit{ii.)} we have
\begin{align*}
\lambda_\beta\big(\!\Add{q}(\vec{q}),\vec{0}_{p_\beta}\big)
&=\psi_\beta\big(\!\Add{q}(\vec{g}),\vec{0}_{p_\beta}\big)- \Add{q}(\vec{s}\:)\\
&=\qrep(q)\cp [\:\psi_\alpha\big(\vec{g},\vec{0}_{p_\alpha}\big)- \vec{s}\:]
=\qrep(q) \cp \lambda_\alpha\big(\raisebox{-0.1ex}{$\vec{q},\vec{0}_{p_\alpha}$}\big).
\end{align*}
\item
This follows from the first part for $\alpha=\beta$, $q=e$ and $\vec{w}_{p_\alpha}=\vec{0}_{p_\alpha}$.
\item
By definition we have $\w_\alpha|_{\mathfrak{q}\times TP_\alpha}=\lambda_\alpha$ and for the pullback property we calculate
\begin{align*}
\left(L_{q'}^*\w_\alpha\right)_{(q,p_\alpha)}\big(\raisebox{-0.1ex}{$\kk_q,\vec{w}_{p_\alpha}$}\big)&={\w_\alpha}_{(q'q,p_\alpha)}\big(\dd L_{q'}\kk_q,\vec{w}_{p_\alpha}\big)\\[-3pt]
&=\qrep\left(q'q\right)\cp\lambda_\alpha\left(\dd L_{q^{-1}q'^{-1}}\dd L_{q'}\kk_q,\vec{w}_{p_\alpha}\right)\\[1pt]
&=\qrep\left(q'\right)\cp\qrep(q)\cp \lambda_\alpha\left(\dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha}\right)\\[1pt]
&=\qrep\left(q'\right)\cp {\w_\alpha}_{(q,p_\alpha)}(\kk_q,\vec{w}_{p_\alpha}),
\end{align*}
where $q,q'\in Q$ and $\kk_q\in T_qQ$.
For uniqueness let $\w$ be another $\mathfrak{s}$-valued 1-form on $Q\times P_\alpha$ whose restriction to $\mathfrak{q}\times TP_\alpha$ is $\lambda_\alpha$ and that fulfils $L_q^*\w=\qrep(q)\cp \w$ for all $q\in Q$. Then
\begin{align*}
\w_{(q,p_\alpha)}\left(\kk_q,\vec{w}_{p_\alpha}\right)&={\w}_{(q,p_\alpha)}\left(\dd L_q\cp \dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha}\right)
=(L_q^*\w)_{(e,p_\alpha)}\left(\dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha}\right)\\
&=\qrep(q)\cp \w_{(e,p_\alpha)}(\dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha})
= \qrep(q)\cp \lambda_\alpha\left(\dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha}\right)\\
&=\w_\alpha(\dd L_{q^{-1}}\kk_q,\vec{w}_{p_\alpha}).
\end{align*}
Finally, smoothness of $\w_\alpha$ is an easy consequence of smoothness of the maps
$\qrep$, $\lambda_\alpha$ and $\mu \colon TQ\rightarrow \mathfrak{q}$, $\kk_q \mapsto \dd L_{q^{-1}}\kk_q$ with $\kk_q\in T_qQ$.
For this, observe that $\mu=\dd\tau \cp \kappa$ for $\tau\colon Q\times Q\rightarrow Q$, $(q,q')\mapsto q^{-1}q'$ and $\kappa\colon TQ\rightarrow TQ\times TQ$, $\kk_q\mapsto \big(\vec{0}_q,\kk_q\big)$ for $\kk_q\in T_qQ$.
\end{enumerate}
\end{proof}
\end{lemma}
\noindent
So far, we have shown that each reduced connection $\{\psi_\alpha\}_{\alpha \in I}$ gives rise to uniquely determined maps $\{\lambda_\alpha\}_{\alpha \in I}$
and $\{\omega_\alpha\}_{\alpha \in I}$. In the final step, we will construct a unique $\Phi$-invariant connection $\w$ out of the data $\{(P_\alpha,\lambda_\alpha)\}_{\alpha\in I}$. Here, uniqueness and smoothness of $\w$ will follow from uniqueness and smoothness of the maps $\w_\alpha$.
\begin{proposition}
\label{prop:reconstr}
There is one and only one $\mathfrak{s}$-valued 1-form $\w$ on $P$ with $\w_\alpha=(\THA^*\w)|_{TQ\times TP_\alpha}$ for all $\alpha \in I$. This 1-form $\w$ is a $\Phi$-invariant connection on $P$.
\begin{proof}
For uniqueness, we have to show that the values of such an $\w$ are uniquely determined by the maps $\w_\alpha$. To this end, let $p\in P$, $\alpha\in I$ and $p_\alpha\in P_\alpha$ such that $p=q\cdot p_\alpha$ for some $q\in Q$. By Lemma \ref{lemma:suralpha}.1 for $\vec{w}_p\in T_pP$ we find some $\xii\in T_qQ\times T_{p_\alpha}P_\alpha$ with $\vec{w}_p=\dd_{(q,p_\alpha)}\THA(\xii\:)$, so that uniqueness follows from
\begin{align*}
\w_p(\vec{w}_p)=\w_{q\cdot p_\alpha}\left(\dd_{(q,p_\alpha)}\THA(\xii\:)\right)
=(\THA^*\w)_{(q,p_\alpha)}(\xii\:)
=\w_\alpha(\xii\:).
\end{align*}
For existence, let $\alpha\in I$ and $p_\alpha\in P_\alpha$. Due to surjectivity of $\dd_{(e,p_\alpha)}\THA$ and Lemma \ref{lemma:lambda}.3 there is a (unique) map $\widehat{\lambda}_{p_\alpha}\colon T_{p_\alpha}P\rightarrow \mathfrak{s}$ with
\begin{align}
\label{eq:lambda}
\widehat{\lambda}_{p_\alpha}\!\cp \dd_{(e,p_\alpha)}\THA=\lambda_\alpha\big|_{\mathfrak{q}\times T_{p_\alpha}P_\alpha}.
\end{align}
Let $\widehat{\lambda}_\alpha \colon \bigsqcup_{p_\alpha\in P_\alpha} T_{p_\alpha} P\rightarrow \mathfrak{s}$
denote the (unique) map whose restriction to $T_{p_\alpha}P$ is $\widehat{\lambda}_{p_\alpha}$ for each $p_\alpha\in P_\alpha$. Then
$\lambda_\alpha=\widehat{\lambda}_\alpha\cp \dd \THA|_{\mathfrak{q}\times TP_\alpha}$ and we construct the connection $\w$ as follows. For $p\in P$ we choose some $\alpha\in I$ and $(q,p_\alpha)\in Q\times P_\alpha$ such that $q\cdot p_\alpha=p$ and define
\begin{align}
\label{eq:defomega}
\w_p\big(\vec{w}_p\big):=\qrep(q)\cp \widehat{\lambda}_\alpha\left(\dd L_{q^{-1}}\big(\vec{w}_p\big)\right)\qquad \forall \:\vec{w}_p\in T_pP.
\end{align}
We have to show that this depends neither on $\alpha\in I$ nor on the choice of $(q,p_\alpha)\in Q\times P_\alpha$.
For this, let $p_\alpha\in P_\alpha$, $p_\beta \in P_\beta$ and $q\in Q$ with $p_\beta=q\cdot p_\alpha$. Then for $\vec{w}\in T_{p_\alpha}P$ we have
$\vec{w}=\dd\THA(\vec{q},\vec{w}_{p_\alpha})$ for some $(\vec{q},\vec{w}_{p_\alpha})\in \mathfrak{q}\times T_{p_\alpha}P_\alpha$, and since $\dd L_q\vec{w}_{p_\alpha}\in T_{p_\beta}P$,
there is $\xii\in \mathfrak{q}\times T_{p_\beta}P_\beta$ such that
$\dd_{(e,p_\beta)}\THA(\xii\:)=\dd L_q\vec{w}_{p_\alpha}$. It follows from the conditions 1) and 2) in Lemma \ref{lemma:lambda}
that
\begin{align}
\label{eq:wohldefs}
\begin{split}
\widehat{\lambda}_\beta(\dd L_q\vec{w})&=\hspace{4.6pt}\widehat{\lambda}_\beta((\dd L_q\cp \dd\THA)(\vec{q},\vec{w}_{p_\alpha}))
=\widehat{\lambda}_\beta\big((\dd L_q\cp \dd\THA)\big(\vec{q},\vec{0}_{p_\alpha}\big)\big)+\widehat{\lambda}_\beta\big(\dd L_q \vec{w}_{p_\alpha}\big)\\[-2pt]
&\hspace{-4.3pt}\stackrel{\eqref{eq:thirdstep}}{=}\widehat{\lambda}_\beta\cp \dd\THA \big( \Add{q}(\vec{q}),\vec{0}_{p_\beta}\big)+\widehat{\lambda}_\beta\cp \dd\THA(\xii\:)\\[-3pt]
&\hspace{-4.3pt}\stackrel{\eqref{eq:lambda}}{=}\lambda_\beta\big(\Add{q}(\vec{q}),\vec{0}_{p_\beta}\big)+\lambda_\beta(\xii\:)\\[1pt]
&=\hspace{4.6pt}\qrep(q)\cp \lambda_\alpha\big(\vec{q},\vec{0}_{p_\alpha}\big)+\qrep(q)\cp \lambda_\alpha\big(\vec{0}_{\mathfrak{q}},\vec{w}_{p_\alpha}\big)\\[1pt]
&=\hspace{4.6pt}\qrep(q)\cp\lambda_\alpha(\vec{q},\vec{w}_{p_\alpha})
=\qrep(q)\cp \widehat{\lambda}_\alpha\cp \dd\THA(\vec{q},\vec{w}_{p_\alpha})
=\qrep(q)\cp \widehat{\lambda}_\alpha(\vec{w}\hspace{1pt}),
\end{split}
\end{align}
where for the third equality we have used that
\begin{align}
\label{eq:thirdstep}
\begin{split}
\left(\dd L_q\cp \dd\THA\right)\big(\vec{q},\vec{0}_{p_\alpha}\big)&=\dttB{t}{0}\:q\cdot(\exp(t\vec{q}\:)\cdot p_\alpha)\\
&=\dttB{t}{0}\:\Co{q}(\exp(t\vec{q}\:))\cdot p_\beta=\dd\THA\big(\Add{q}(\vec{q}\:),\vec{0}_{p_\beta}\big).
\end{split}
\end{align}
Consequently, if $\wt{q}\cdot p_\beta=p$ with $(\wt{q},p_\beta)\in Q\times P_\beta$ for some $\beta \in I$, then $p_\beta=(q^{-1}\wt{q}\:)^{-1}\cdot p_\alpha$ and well-definedness follows from
\begin{align*}
\qrep(\wt{q}\:)\cp \widehat{\lambda}_\beta\left(\dd L_{{\wt{q}}^{-1}}(\vec{w}_p)\right)&=\qrep(q)\cp\qrep\big(q^{-1}\wt{q}\:\big)\cp\widehat{\lambda}_\beta\left(\dd L_{(q^{-1}\wt{q}\:)^{-1}}\big( \dd L_{q^{-1}}\vec{w}_p\big)\right)\\
&=\qrep(q)\cp \widehat{\lambda}_\alpha\big(\dd L_{q^{-1}}\vec{w}_p\hspace{1pt}\big),
\end{align*}
where the last step is due to \eqref{eq:wohldefs} with $\vec{w}=\dd L_{q^{-1}}\vec{w}_p\in T_{p_\alpha}P$.
Next, we show that $\w$ fulfils the pullback property. For this let $(\kk,\vec{w}_{p_\alpha})\in T_qQ\times T_{p_\alpha}P_\alpha$. Then
\begin{align*}
(\THA^*\w)\left(\kk_q,\vec{w}_{p_\alpha}\right)&=\w_{q\cdot p_\alpha}\left(\dd\THA(\kk_q,\vec{w}_{p_\alpha})\right)
\stackrel{\eqref{eq:defomega}}{=}\qrep(q)\cp \widehat{\lambda}_\alpha\!\left(\dd L_{q^{-1}}\dd\THA(\kk_q,\vec{w}_{p_\alpha})\right)\\[-2pt]
&=\qrep(q)\circ \widehat{\lambda}_\alpha\cp \dd\THA\left(\dd L_{q^{-1}} \kk_q,\vec{w}_{p_\alpha}\right)
\stackrel{\eqref{eq:lambda}}{=}\qrep(q)\cp \lambda_\alpha\!\left(\dd L_{q^{-1}} \kk_q,\vec{w}_{p_\alpha}\right)\\[3pt]
&=\w_\alpha(\kk_q,\vec{w}_{p_\alpha}).
\end{align*}
In the third step we have used that $L_{q^{-1}}\cp \THA= \THA(L_{q^{-1}}(\cdot),\cdot)$.
Finally, we have to verify that $\w$ is a $\Phi$-invariant, smooth connection.
For this let $p\in P$ and $(\wt{q},p_\alpha)\in Q\times P_\alpha$ with $p=\wt{q}\cdot p_\alpha$. Then for $q\in Q$ and $\vec{w}_p\in T_pP$ we have
\begin{align*}
\left(L_q^*\w\right)_{p}(\vec{w}_{p})&=\w_{q\cdot p}\left(\dd L_q \vec{w}_{p}\right)=\w_{(q \wt{q})\cdot p_\alpha}\left(\dd L_q \vec{w}_{p}\right)\\
&=\qrep(q)\cp\qrep\left(\wt{q}\hspace{2pt}\right)\cp \widehat{\lambda}_\alpha\left(\dd L_{\wt{q}^{-1}}\vec{w}_{p}\right)
=\qrep(q)\cp \w_{p}(\vec{w}_{p}),
\end{align*}
hence
\begin{align*}
R_s^*\w&=L_{\left(e,s^{-1}\right)}^*\w=\qrep\big(\big(e,s^{-1}\big)\big)\cp \w=\Add{s^{-1}}\cp \w,\\
L_g^*\w& =L_{(g,e)}^*\w=\qrep((g,e))\cp \w=\w.
\end{align*}
So, it remains to show smoothness of $\w$ and that $\w_p(\widetilde{s}(p))=\vec{s}$ holds for all $p\in P$ and all $\vec{s}\in \mathfrak{s}$. For the second property, let $p=q\cdot p_\alpha$ for $(q,p_\alpha)\in Q\times P_\alpha$. Then $q=(g,s)$ for some $g\in G$ and $s\in S$ and we obtain
\begin{align*}
\w_{p}(\widetilde{s}(p))
&=\qrep(q)\cp \widehat{\lambda}_\alpha\left(\dd L_{q^{-1}}\widetilde{s}(q\cdot p_\alpha)\right)\\
&=\qrep(q)\cp \widehat{\lambda}_\alpha\left(\dttB{t}{0}\: p_\alpha\cdot \left(\Co{s^{-1}}(\exp(t\vec{s}\:)\right)\right)\\
&=\qrep(q)\cp\widehat{\lambda}_\alpha\big(\dd\THA\big(\Add{s^{-1}}(\vec{s}\hspace{2pt}),\vec{0}_{p_\alpha}\big)\big)\\
&=\Add{s}\cp\: \lambda_\alpha\big(\Add{s^{-1}}(\vec{s}\hspace{2pt}),\vec{0}_{p_\alpha}\big)
=\Add{s}\cp\Add{s^{-1}}(\vec{s}\hspace{2pt})=\vec{s}.
\end{align*}
For smoothness let $p_\alpha\in P_\alpha$ and choose a submanifold $Q'$ of $Q$ through $e$, an open neighbourhood $P'_\alpha\subseteq P_\alpha$ of $p_\alpha$ and an open subset $U\subseteq P$ such that the restriction
$\widehat{\THA}:=\THA|_{Q' \times P'_\alpha}$
is a diffeomorphism to $U$. Then $p_\alpha \in U$ because $e\in Q'$, hence
\begin{align*}
\w|_U=\widehat{\THA}^{-1}\hspace{-1pt}\raisebox{2pt}{$^*\big[$}\widehat{\THA}\hspace{1pt}\raisebox{2pt}{$^*$} \w\raisebox{2pt}{$\big]$}=\widehat{\THA}^{-1}\hspace{-1pt}\raisebox{0pt}{$^*$}\big[(\THA^*\w)|_{TQ\times TP_\alpha}\big] =\widehat{\THA}^{-1}\hspace{-1pt}\raisebox{2pt}{$^*$}\w_\alpha.
\end{align*}
Since $\w_\alpha$ is smooth and $\widehat{\THA}$ is a diffeomorphism, $\w|_U$ is smooth as well. Finally, if $p=q\cdot p_\alpha$ for $q\in Q$, then $L_q(U)$ is an open neighbourhood of $p$ and
\begin{align*}
\w|_{L_q(U)}=\big(L_{q^{-1}}^*\!\left(L_q^*\hspace{1pt}\w\right)\!\big)\big|_{L_q(U)}= \qrep(q)\cp \big(L_{q^{-1}}^*\w\big)\big|_{L_q(U)} =\qrep(q)\cp L_{q^{-1}}^* \!\left(\w|_U\right)
\end{align*}
is smooth because $\w|_U$ and $L_{q^{-1}}$ are smooth.
\end{proof}
\end{proposition}
\noindent
Corollary \ref{cor:psialpha} and Proposition \ref{prop:reconstr} now prove
\begin{theorem}
\label{th:InvConnes}
Let $G$ be a Lie group of automorphisms of the principal fibre bundle $P$. Then for each $\Phi$-covering $\{P_\alpha\}_{\alpha\in I}$ of $P$ there is a bijection between the corresponding set of reduced connections and the $\Phi$-invariant connections on $P$.
\begin{proof}
Corollary \ref{cor:psialpha} and Proposition \ref{prop:reconstr}
\end{proof}
\end{theorem}
\noindent
As already mentioned in the preliminary remarks to Example \ref{example:SCHSV}, the second part of the next example shows the importance of the transversality condition
\begin{align*}
\mathrm{im}[\dd_x\tau_0] + \mathrm{im}\big[\dd_e\varphi_{\tau_0(x)}\big]=T_{\tau_0(x)}M\qquad\forall\: x\in U_0
\end{align*}
for the formulation in \cite{HarSni}.
\begin{example}[(Semi-)homogeneous Connections]
\label{ex:transinv}
\begin{enumerate}
\item
Let $P=X \times S$ for an $n$-dimensional $\RR$-vector space $X$ and an arbitrary structure group $S$.
Moreover, let $G \subseteq X$ be a linear subspace of dimension $1\leq k\leq n$ acting via
\begin{align*}
\Phi\colon G\times P \rightarrow P,\quad(g,(x,\sigma))\mapsto(g+x,\sigma).
\end{align*}
Let $W$ be an algebraic complement of $G$ in $X$ and
$P_0:=W\times \{e_S\}\subseteq P$. Then, $P_0$ is a $\Phi$-covering because $\THA\colon (G\times S)\times P_0\rightarrow P$ is a diffeomorphism and each $\varphi$-orbit intersects $W$ in a unique point. Consequently, identifying $G$ with its Lie algebra $\mg$, the $\Phi$-invariant connections on $P$ are in bijection with the smooth maps $\psi\colon G\times TW\rightarrow \mathfrak{s}$ such that $\psi_w:=\psi|_{G \times T_wW}$ is linear for all $w\in W$. This is because the conditions \textit{i.)} and \textit{ii.)} from Corollary \ref{cor:psialpha} give no further restrictions in this case. It is straightforward to see that the $\Phi$-invariant connection that corresponds to $\psi$ is explicitly given by
\begin{align}
\label{eq:transinv}
\w^\psi(\vec{v}_x,\vec{\sigma}_s)=\Add{s^{-1}}\!\cp\:\psi_{\pr_W(x)}\big(\pr_G(\vec{v}_x),\pr_W(\vec{v}_x)\big)+\dd L_{s^{-1}}(\vec{\sigma}_s)
\end{align}
for $(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P$.
\item
In the situation of Part 1), let $X=\RR^2$, $G=\Span_\RR(\vec{e}_1)$, $W=\Span_\RR(\vec{e}_2)$ and $P_0=W\times \{e\}$. We fix $0\neq \vec{s}\in \mathfrak{s}$ and define $\psi\colon \mathfrak{g}\times TP_0 \rightarrow \mathfrak{s}$ by
\begin{align*}
\psi_{y}(\lambda \cdot\vec{e}_1,\mu\cdot \vec{e}_2):= \mu \cdot f(y)\cdot\vec{s}\qquad \text{for}\qquad (\lambda \cdot\vec{e}_1,\mu \cdot\vec{e}_2)\in \mathfrak{g}\times T_{(y\cdot\vec{e}_{2},e)}P_0,
\end{align*}
where $f(0):=0$ and $f(y):=1\slash \sqrt[3]{y}$ for $y\neq 0$. Then, $\w^\psi$ defined by \eqref{eq:transinv} is smooth on $P':=Z\times S$ for $Z:= [\hspace{0.5pt} X\backslash \spann_\RR(\vec{e}_1)]$, but not smooth at $((x,0),e)$ because
\begin{align*}
\w^\psi_{((x,y),e)}\big(\big(\vec{0},\vec{e}_2\big),\vec{0}_{\mathfrak{s}}\big)=\psi_y\big(\vec{0},\vec{e}_2\big)=f(y)\cdot \vec{s}\qquad \forall\: y\in \RR.
\end{align*}
In addition to that, there cannot exist any smooth invariant connection $\wt{\w}$ on $P$ which coincides on $P'$ with $\w^\psi$, just because $\lim_{y\rightarrow 0}f(y)\cdot \s$ does not exist.
Now, let $U_0=\RR$, $\tau_0 \colon U_0 \rightarrow \RR^2$, $t\mapsto \big(t,t^3\big)$ and $s_0\colon t \mapsto (\tau_0(t),e)$. Then $(U_0,\tau_0,s_0)$ fulfils the conditions in \cite{HarSni}, but we have $\mathrm{im}[\dd_0\tau_0] + \mathrm{im}\hspace{-1pt}\big[\dd_e\varphi_{\tau_0(0)}\big] = \Span_\RR(\vec{e}_1)\neq T_0X =T_0\RR^2=\RR^2$.\footnote{Then
$\mathrm{im}[\dd_0s_0]+ \mathrm{im}[\dd_e\Phi_{s_0(x)}]+Tv_{s_0(0)}P=\Span_\RR(\vec{e}_1) \oplus Tv_{(0,e)}P\neq\RR^2 \oplus Tv_{(0,e)}P= T_{(0,e)}P$ so that $(U_0,s_0)$ is not a $\THA$-patch as it fails the transversality condition \eqref{eq:transv} from Remark \ref{bem:Psliceeigensch}.2.}
As a consequence, $\ovl{\psi}\colon \mathfrak{g}\times TU_0\rightarrow \mathfrak{s}$ defined by $\ovl{\psi}_t:=(\Phi^*\w^\psi)|_{\mathfrak{g}\times T_tU_0}$ is smooth because for $t\neq 0$ and $r\in T_tU_0 =\RR$ we have
\begin{align*}
\ovl{\psi}_{t}(\lambda\hspace{1pt} \vec{e}_1,r)&=\big(\Phi^*\w^\psi\big)\left(\lambda\hspace{1pt} \vec{e}_1,r\cdot \vec{e}_1 + 3t^2 r\cdot \vec{e}_2\right)
=\w^\psi_{((t,t^3),e)}\!\left(\left(\lambda+r\right)\cdot \vec{e}_1 + 3t^2 r\cdot \vec{e}_2\right)\\
&=\psi_{t^3}\!\left(\left(\lambda+r\right)\cdot \vec{e}_1,3t^2 r\cdot \vec{e}_2\right)= 3 t r\cdot \vec{s},
\end{align*}
as well as $\ovl{\psi}_{0}(\lambda\hspace{1pt} \vec{e}_1,r)=0$ if $t=0$. For the first step, keep in mind that
\begin{align*}
(\Phi^*\w^\psi)|_{\mathfrak{g}\times T_tU_0}(\vec{g},r)=(\Phi^*\w^\psi)(\vec{g},\dd_t s_0(r))
\end{align*}
by Convention \ref{conv:Submnfds}.2.
Then, the maps $\mu:= \ovl{\psi}|_{TU_0}$ and $A_{t_0}(\vec{g}):=\ovl{\psi}\big(\vec{g},\vec{0}_{t_0}\big)$ fulfil the conditions in Theorem 2 in \cite{HarSni} because $\ovl{\psi}$ fulfils the three algebraic conditions in Example \ref{example:SCHSV}, being trivial in this case because $H=\{e\}$.
Now, Theorem 2 in \cite{HarSni} states that $\ovl{\psi}$ can also be obtained by pullbacking and restricting (w.r.t.\ $s_0$) a unique smooth invariant connection $\wt{\w}$ on $P$ (instead of pullbacking and restricting $\w^\psi$). However, such a connection cannot exist as it necessarily has to coincide on $P'$ with $\w^\psi$.
In fact, let $U'_0:=\RR_{\neq 0}$ and $\tau_0'\colon U_0'\rightarrow Z$, $t\mapsto \big(t,t^3\big)$ as well as $s_0'\colon t\mapsto(\tau_0'(t),e)$ be defined as above. Then, $(U_0',s_0')$ is a $\THA$-patch as we have removed the point $0\in U_0$ for which transversality fails. Thus, the restriction of $\ovl{\psi}$ to $\mg\times U_0'$ corresponds to a unique smooth invariant connection $\w'$ on $P'$. This connection must be given by the restriction of $\w^\psi$ to $P'$ because pullbacking and restricting $\w^\psi$ w.r.t.\ $s_0'$ gives rise to the restriction $\ovl{\psi}|_{\mg \times TU_0'}$, just because $s_0'=s_0|_{U_0}$ holds. However, the same is true for $\wt{\w}$, so that both connections coincide on $P'$.
\hspace*{\fill}{{$\Diamond$}}
\end{enumerate}
\end{example}
\subsection{Particular Cases and Applications}
\label{sec:PartCases}
In the first part of this Subsection we consider $\Phi$-coverings of $P$ that arise from the induced action $\varphi$ on the base manifold $M$ of $P$. Then we discuss the case where $\Phi$ acts via gauge transformations on $P$. This leads to a straightforward generalization of the description of connections by consistent families of local 1-forms on $M$. In the second part we discuss the (almost) fibre transitive case and deduce Wang's original theorem \cite{Wang} from Theorem \ref{th:InvConnes}. Finally, we consider the situation where $P$ is trivial and give examples in loop quantum gravity.
\subsubsection{$\Phi$-Coverings and the Induced Action}
\label{subsec:GCovIndAct}
Let $(G,\Phi)$ be a Lie group of automorphisms of the principal fibre bundle $P$. According to Lemma \ref{lemma:mindimslice} for each $x\in M$ there is a $\varphi$-patch (with minimal dimension) $M_x$ with $x\in M$. Consequently, there is an open neighbourhood $M'_x\subseteq M_x$ of $x$ and a local section $s_x\colon U\rightarrow P$ such that $M'_x\subseteq U$ for an open neighbourhood $U\subseteq M$.
Let $I\subseteq M$ be a subset such that\footnote{It is always possible to choose $I=M$.} each $\varphi$-orbit intersects at least one of the sets $M_x$ for some $x\in I$. Then it is immediate from Lemma \ref{lemma:suralpha}.2 that $\{s_x(M'_x)\}_{x\in I}$ is a $\Phi$-covering of $P$. More generally, we have
\begin{corollary}
\label{cor:reductions}
Let $\PMS$ be a principal fibre bundle and $(G,\Phi)$ a Lie group of automorphisms of $P$. Denote by $(M_\alpha,s_\alpha)_{\alpha\in I}$ a family consisting of a collection of $\varphi$-patches $\{M_\alpha\}_{\alpha\in I}$ and smooth sections\footnote{This is that $\pi\cp s_\alpha= \id_{M_\alpha}$.} $s_\alpha\colon M_\alpha\rightarrow P$. Then the sets $P_\alpha:=s_\alpha(M_\alpha)$ are $\THA$-patches. They provide a $\Phi$-covering of $P$ iff each $\varphi$-orbit intersects at least one patch $M_\alpha$.
\begin{proof}
This is immediate from Lemma \ref{lemma:suralpha}.2.
\end{proof}
\end{corollary}
\noindent
We now consider the case where $(G,\Phi)$ is a Lie group of gauge transformations of $P$, i.e., $\varphi_g=\id_M$ for all $g\in G$. Here, we show that Theorem \ref{th:InvConnes} can be seen as a generalization of the description of smooth connections by consistent families of local 1-forms on the base manifold $M$. For this, let $\{U_\alpha\}_{\alpha\in I}$ be an open covering of $M$ and $\{s_\alpha\}_{\alpha\in I}$ a family of smooth sections $s_\alpha\colon U_\alpha \rightarrow P$.
We define $U_{\alpha\beta}:=U_\alpha \cap U_\beta$ and consider the smooth maps $\delta_{\alpha\beta}\colon G\times U_{\alpha\beta}\rightarrow S$ determined by $s_\beta(x)=\Phi(g,s_\alpha(x))\cdot \delta_{\alpha\beta}(g,x)$, and for which we have $\delta_{\alpha\beta}(g,x)=\phi^{-1}_{s_\alpha(x)}(g) \cdot\delta_{\alpha\beta}(e,x)$.
Finally, let $\mu_{\alpha\beta}(g,\vec{v}_x):= \dd L_{\delta^{-1}_{\alpha\beta}(g,x)}\cp \dd_x\delta_{\alpha\beta}(g,\cdot)(\vec{v}_x)$ for $\vec{v}_x\in T_xU_{\alpha\beta}$ and $g\in G$. Then we have
\begin{scase}[Lie Groups of Gauge Transformations]
\label{scase:GaugeTransf}
Let $(G,\Phi)$ be a Lie group of gauge transformations of the principal fibre bundle $(P,\pi,M,S)$. Then, the $\Phi$-invariant connections on $P$ are in bijection with the families $\{\chi_\alpha\}_{\alpha\in I}$ of $\mathfrak{s}$-valued 1-forms $\chi_\alpha\colon U_\alpha\rightarrow\mathfrak{s}$ for which we have
\begin{align}
\label{eq:consistgauge}
\chi_\beta(\vec{v}_x)=\left(\Add{\delta_{\alpha\beta}(g,x)}\cp\: \chi_\alpha\right)(\vec{v}_x)+\mu_{\alpha\beta}(g,\vec{v}_x)\qquad \forall\: \vec{v}_x\in T_xU_{\alpha\beta},\forall\: g\in G.
\end{align}
\begin{proof}
By Corollary \ref{cor:reductions} $\{s_\alpha(U_\alpha)\}_{\alpha\in I }$ is a $\Phi$-covering of $P$. So, let $\{\psi_\alpha\}_{\alpha\in I}$ be a reduced connection w.r.t.\ this covering. We first show that condition \textit{i.)} from Corollary \ref{cor:psialpha} implies
\begin{align}
\label{eq:eq1}
\psi_\beta\big(\raisebox{-0.5pt}{$\vec{g},\vec{0}_{p}$}\big) =\dd_e\phi_p(\vec{g}\hspace{1pt})\qquad \forall\:\vec{g}\in \mathfrak{g}, \forall\:p\in s_\beta(U).
\end{align}
For this
observe that condition \textit{a.)} from Remark \ref{rem:Psialphaconderkl} means that for all $\beta\in I$, $p \in s_\beta(U_\beta)$, $\vec{w}_p\in T_ps_\beta(U_\beta)$ and $\vec{g}\in \mathfrak{g}$, $\vec{s}\in \mathfrak{s}$ we have
\begin{align*}
\dd_e\Phi_p(\vec{g}\hspace{1pt}) + \vec{w}_p -\widetilde{s}(p)=0 \quad \Longrightarrow \quad\psi_\beta(\vec{g},\vec{w}_p)-\vec{s}=0.
\end{align*}
Now, $T_ps_\beta(U_\beta)$
is complementary to $Tv_pP$ and $\mathrm{im}[\dd_e\Phi_p]\subseteq \ker[\dd_p\pi]$ so that \textit{a.)} is the same as
\begin{enumerate}
\item[\textit{a'.)}]
$\dd_e\Phi_p(\vec{g}\hspace{1pt})=\widetilde{s}(p)\quad \Longrightarrow\quad \psi_\beta\big(\raisebox{-0.5pt}{$\vec{g},\vec{0}_p$}\big)=\vec{s}$\quad for\: $\vec{g}\in \mathfrak{g}$, $\vec{s}\in \mathfrak{s}$ \:and all\: $p\in P_\beta$.
\end{enumerate}
But, since $G_x=G$ for all $x\in M$, this just means\footnote{$\dd_e\Phi_p(\vec{g}\hspace{1pt})- \widetilde{s}(p)=0$ iff $(\vec{g},\vec{s}\hspace{1pt})\in \mathfrak{q}_p$ iff $\vec{s}=\dd_e\phi_p(\vec{g}\hspace{1pt})$.} $\psi_\beta\big(\raisebox{-0.5pt}{$\vec{g},\vec{0}_{p}$}\big) =\dd_e\phi_p(\vec{g}\hspace{1pt})$ for all $\vec{g}\in \mathfrak{g}$ and already implies condition \textit{ii.)} from Corollary \ref{cor:psialpha} as $\phi_p$ is a Lie group homomorphism. Consequently, we can ignore this condition in the following.
Now, we have $p_\beta = q\cdot p_\alpha$ for $q\in Q$, $p_\alpha \in P_\alpha$, $p_\beta \in P_\beta$ iff
$\pi(p_\alpha)=\pi(p_\beta)=x\in U_{\alpha\beta}$ and $q=\big(g,\delta^{-1}_{\alpha\beta}(g,x)\big)$. Consequently, the left hand side of condition \textit{i.)} from Corollary \ref{cor:psialpha} reads
\begin{align*}
\wt{g}(s_\beta(x))+\dd_x s_\beta (\vec{v}_\beta) - \wt{s}(s_\beta(x))= \big(\dd L_g\cp \dd R_{\delta_{\alpha\beta}(g,x)}\cp \dd_x s_\alpha\big) (\vec{v}_\alpha),
\end{align*}
\newline
\vspace{-26pt}
\newline
where $\vec{v}_\alpha, \vec{v}_\beta \in T_xM$ and $g\in G$.
This is true for $\vec{v}_\alpha=\vec{v}_\beta=\vec{v}_x$, $\vec{g}=0$ and $\vec{s}=\mu_{\alpha\beta}(g,\vec{v}_x)$, which follows from
\begin{align*}
\dd_x s_\beta (\vec{v}_\beta)&= \dd_x \Big[L_g\cp R_{\delta_{\alpha\beta}(g,\cdot)}\cp s_\alpha\Big] (\vec{v}_x)\\
&=\dd L_g\left[\dd_{s_\alpha(x)}R \big(\dd_x\delta_{\alpha\beta}(g,\cdot)(\vec{v}_x)\big) + \dd R_{\delta_{\alpha\beta}(g,x)}(\dd_x s_\alpha(\vec{v}_x)) \right],\\[4pt]
\wt{s}(s_\beta(x))&=\dttB{t}{0}L_g \cp R_{\delta_{\alpha\beta}(g,x)\cdot \exp(t\vec{s}\:)}(s_\alpha(x))\\[2pt]
&=\dd L_g\left[\dd_{s_\alpha(x)}R \left(\dd L_{\delta_{\alpha\beta}(g,x)}(\vec{s}\:)\right)\right]
=\dd L_g\left[\dd_{s_\alpha(x)}R \big(\dd_x\delta_{\alpha\beta}(g,\cdot)(\vec{v}_x)\big)\right].
\end{align*}
Consequently, by Corollary \ref{cor:psialpha}.\textit{i.)} and for
\begin{align*}
(\psi_\alpha \cp \dd_x s_{\alpha})(\vec{v}_x):= \psi_\alpha\big(\vec{0}_{\mathfrak{g}},\dd_x s_\alpha(\vec{v}_x)\big) \qquad \forall\:\vec{v}_x\in T_xU_{\alpha\beta}.
\end{align*}
we have
\begin{align}
\label{eq:gaugetraf}
\psi_\beta\big(\vec{0}_{\mathfrak{g}},\dd_x s_\beta(\vec{v}_x)\big)=\big(\Add{\delta_{\alpha\beta}(g,x)} \cp\: \psi_\alpha \cp \dd_x s_{\alpha}\big)(\vec{v}_x)+\mu_{\alpha\beta}(g,\vec{v}_x)
\end{align}
for all $g\in G$ and all $\vec{v}_x\in T_xU_{\alpha\beta}$.
Due to part {\bf 2.)} in Remark \ref{rem:Psialphaconderkl} the condition \textit{i.)} from Corollary \ref{cor:psialpha} now gives no further restrictions, so that for $\chi_\beta:=\psi_\beta \cp \dd s_\beta$ we have
\begin{align*}
\psi_\beta(\vec{g}, \dd_xs_\beta(\vec{v}_x))=\dd_e\phi_{s_\beta(x)}(\vec{g})+ \chi_\beta(\vec{v}_x)\qquad \forall\:\vec{g}\in \mathfrak{g},\forall\:\vec{v}_x\in T_xM,\forall\: x\in U_\beta.
\end{align*}
Then, $\psi_\beta$ is uniquely determined by $\chi_\beta$ for each $\beta\in I$, so that \eqref{eq:gaugetraf} yields the consistency condition \eqref{eq:consistgauge} for the maps $\{\chi_\alpha\}_{\alpha\in I}$.
\end{proof}
\end{scase}
\begin{example}[Trivial Action]
If $G$ acts trivially, then for each $x\in U_{\alpha\beta}$ we have
\begin{align*}
\delta_{\alpha\beta}(g,x)=\phi_{s_\alpha(x)}^{-1}(g)\cdot \delta_{\alpha\beta}(e,x)=\delta_{\alpha\beta}(e,x).
\end{align*}
Hence, $\delta_{\alpha\beta}$ is independent of $g\in G$ so that here Case \ref{scase:GaugeTransf} just reproduces the description of smooth connections by means of consistent families of local 1-forms on the base manifold $M$. \hspace*{\fill}$\Diamond$
\end{example}
\subsubsection{(Almost) Fibre Transitivity}
\label{subsec:AlmFTC}
In this subsection, we discuss the situation where $M$ admits an element which is contained in the closure of each $\varphi$-orbit. For instance, this holds for all $x\in M$ if each $\varphi$-orbit is dense in $M$ and, in particular, is true for fibre transitive actions.
\label{subsec:FIbtrcase}
\begin{scase}[Almost Fibre Transitivity]
\label{scase:slicegleichredcluster}
Let $x\in M$ be contained in the closure of each $\varphi$-orbit and let $p\in F_x$. Then, each $\THA$-patch $P_0\subseteq P$ with $p\in P_0$ is a $\Phi$-covering of $P$. Hence, the $\Phi$-invariant connections on $P$ are in bijection with the smooth maps $\psi\colon \mathfrak{g}\times TP_0\rightarrow \mathfrak{s}$ for which $\psi|_{\mathfrak{g}\times T_pP_0}$ is linear for all $p\in P_0$ and that fulfil the two conditions from Corollary \ref{cor:psialpha}.
\begin{proof}
It suffices to show that $\pi\left(P_0\right)$ intersects each $\varphi$-orbit $[o]$. Since $P_0$ is a $\THA$-patch, there is an open neighbourhood $P'\subseteq P_0$ of $p$ and a submanifold $ Q'$ of $Q$ through $(e_G,e_S)$ such that $\THA|_{Q' \times P'}$ is a diffeomorphism to an open subset $U\subseteq P$. Then $\pi(U)$ is an open neighbourhood of $\pi(p)$ and by assumption we have $[o]\cap \pi(U)\neq \emptyset$ for each $[o]\in M\slash G$. Consequently, for $[o]\in M\slash G$ we find $\wt{p}\in U$ with $\pi(\wt{p}\hspace{1.5pt})\in [o]$. Let $\wt{p}=\THA((g',s'),p')$ for $((g',s'),p')\in Q'\times P'$. Then
\begin{align*}
[o]\ni\pi(\wt{p}\hspace{1.5pt})=\pi\left(\Phi(g',p')\cdot s'\right)=\varphi(g',\pi(p'))\in [\pi(p')]
\end{align*}
shows that $[o]=[\pi(p')]$, hence $\pi\left(P_0\right)\cap [o]\neq \emptyset$.
\end{proof}
\end{scase}
\noindent
The next example to Case \ref{scase:slicegleichredcluster} shows that evaluating the conditions \textit{i.)} and \textit{ii.)} from Corollary \ref{cor:psialpha} at one single point can be sufficient to verify non-existence of invariant connections.
\begin{example}[General linear group]
\label{ex:Bruhat}
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
Let $P:=GL(n,\mathbb{R})$ and $G=S=B \subseteq GL(n,\mathbb{R})$ the subgroup of upper triangular matrices. Moreover, let $S_n\subseteq GL(n,\mathbb{R})$ be the group of permutation matrices.
Then $P$ is a principal fibre bundle with base manifold $M:=P\slash S$, structure group $S$ and projection map $\pi\colon P\rightarrow M$, $p\mapsto [p]$. Moreover, $G$ acts via automorphisms on $P$ by $\Phi(g,p):=g\cdot p$ and we have the Bruhat decomposition
\begin{align*}
GL(n,\mathbb{R})=\bigsqcup_{w\in S_n}B w B.
\end{align*}
Then $M=\bigsqcup_{w\in S_n}G\cdot\pi(w)$, $G\cdot \pi(e)=\pi(e)$ and $\pi(e)\in \ovl{G\cdot\pi(w)}$ for all $w\in S_n$. Now, $\mathrm{im}[\dd_e\THA_e]=\mathfrak{g}$, since $\dd_e\THA_e(\vec{g})=\vec{g}$ for all $\vec{g}\in \mathfrak{g}$. Moreover, $\mathfrak{g}={\Span_\mathbb{R}}\{E_{ij} \:|\: 1\leq i\leq j\leq n\}$ so that $V:={\Span_\mathbb{R}}\{E_{ij} \:|\: 1\leq j<i\leq n\}$ is an algebraic complement of $\mathfrak{g}$ in $T_eP=\mathfrak{gl}(n,\mathbb{R})$. By Lemma \ref{lemma:mindimslice}.2 we find a patch $H\subseteq P$ through $e$ with $T_eH=V$ and due to Case \ref{scase:slicegleichredcluster} this is a $\Phi$-covering.
\item
\vspace{-6pt}
A closer look at the point $e\in P$ shows that there cannot exist any $\Phi$-invariant connection on $GL(n,\RR)$. In fact, if $\psi\colon \mathfrak{g}\times TH \rightarrow \mathfrak{s}$ is a reduced connection w.r.t.\ $H$, then for $\vec{w}:=\vec{0}_e$ and $\vec{g}=\vec{s}$ we have
\begin{align*}
\wt{g}(e)+\vec{w} -\widetilde{s}(e)=\vec{g}+\vec{w}-\vec{s}=0.
\end{align*}
So, condition \textit{i.)} from Corollary \ref{cor:psialpha} gives $\psi\big(\vec{g},\vec{0}_e\big)-\vec{g}=0$, hence $\psi\big(\vec{g},\vec{0}_e\big)=\vec{g}$ for all $\vec{g}\in \mathfrak{g}$. Now $q\cdot e =e$ iff $q=(b,b)$ for some $b\in B$. Let
\begin{align*}
V\ni\vec{h}:=E_{n1}\hspace{18pt} B\ni b:= \me + E_{1n}\hspace{18pt} \mathfrak{g}\ni \vec{g}:=E_{11}- E_{1n}- E_{nn}.
\end{align*}
Then, $\wt{g}(e)+\vec{h}=\vec{g}+\vec{h}=b\vec{h} b^{-1}=\dd L_q\vec{h}$,
so that condition \textit{i.)} yields
\begin{align*}
\psi\big(\vec{g},\vec{h}\big)=\qrep(q)\cp \psi\big(\vec{0}_{\mathfrak{g}},\vec{h}\big)=\Add{b}\cp \psi\big(\vec{0}_{\mathfrak{g}},\vec{h}\big),
\end{align*}
hence $\vec{g}+\left[\id-\Add{b}\right]\cp \psi\big(\vec{0}_{\mathfrak{g}},\vec{h}\big)=0$. But $(\vec{g}\hspace{1pt})_{11}=1$ and
\begin{align*}
\big(\raisebox{-0.5pt}{$\psi\hvect-\Add{b}\cp\: \psi \hvect$}\big)_{11}= \big(\raisebox{-1pt}{$\psi\hvect $}\big)_{11} - \big(\raisebox{-0.5pt}{$\psi\hvect $}\big)_{11} =0
\end{align*}
so that $\psi$ cannot exist. \hspace*{\fill}{{$\Diamond$}}
\end{itemize}
\endgroup
\end{example}
\begin{corollary}
\label{cor:SliceFibTran}
If $\Phi$ is fibre transitive, then $\{p\}$ is a $\Phi$-covering for all $p\in P$.
\begin{proof}
It suffices to show that $\{\pi(p)\}$ is a $\varphi$-patch, since then $\{p\}$ is a $\THA$-patch by Corollary \ref{cor:reductions} and a $\Phi$-covering by Case \ref{scase:slicegleichredcluster}. This, however, is clear from Remark \ref{rem:patch}.1. In fact, if $x:=\pi(p)$, then by general theory we know that $M$ is diffeomorphic to $G\slash G_{x}$
via $\phi \colon [g]\mapsto \varphi(g,x)$ and that for each $[g]\in G\slash G_{x}$ we find an open neighbourhood $U\subseteq G\slash G_{x}$ of $[g]$ and a smooth section $s\colon U\rightarrow G$. Then surjectivity of $\dd_e\varphi_x$ is clear from surjectivity of $\dd_{[e]}\phi$ and
\begin{align*}
\dd_e\varphi_{x}\cp \dd_{[e]}s= \dd_{[e]}(\varphi_{x} \cp s)= \dd_{[e]}\varphi(s(\cdot),x)=\dd_{[e]}\phi,
\end{align*}
showing that $T_xM=\dd_e\varphi_x(\mathfrak{g})$.
\end{proof}
\end{corollary}
\noindent
Let $\wm$ be transitive and $p\in P$. Then $\{p\}$ is a $\Phi$-covering by Corollary \ref{cor:SliceFibTran} and $T_p\{p\}$ is the zero vector space. Moreover, we have $p_\alpha=q\cdot p_\beta$ iff $p_\alpha=p_\beta=p$ and $q\in Q_p$. It follows that a reduced connection w.r.t.\ $\{p\}$ can be seen as a linear map $\psi\colon\mathfrak{g}\rightarrow \mathfrak{s}$ that fulfils the following two conditions.
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
$\dd_e\THA_p(\vec{g},\vec{s}\hspace{1pt})=0$\quad $\Longrightarrow$ \quad $\psi(\vec{g}\hspace{1pt})=\vec{s}$\qquad\quad for $\vec{g}\in \mathfrak{g}$, $\vec{s}\in \mathfrak{s}$,
\item
\vspace{-2pt}
$\psi\big(\Add{q}(\vec{g}\hspace{1pt})\big)=\qrep(q)\cp \psi(\vec{g}\hspace{1pt})$\qquad\qquad\quad\hspace{10.7pt} $\forall\: q\in Q_p$, $\forall\:\vec{g}\in \mathfrak{g}$.
\end{itemize}
\endgroup
\noindent
Since $\ker[d_e\THA_p]=\mathfrak{q}_p$, we have shown
\begin{scase}[Hsien-Chung Wang, \cite{Wang}]
\label{th:wang}
Let $(G,\Phi)$ be a fibre transitive Lie group of automorphisms of $P$. Then for each $p\in P$ there is a bijection between the $\Phi$-invariant connections on $P$ and the linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$ that fulfil
\begingroup
\setlength{\leftmargini}{30pt}
\begin{enumerate}
\item[\textrm{a)}]
$\psi\big(\raisebox{-1pt}{$\vec{h}$}\hspace{2pt}\big)=\dd_e\phi_p\big(\raisebox{-1pt}{$\vec{h}$}\hspace{2pt}\big)$ \quad\hspace{34.6pt} $\forall\: \vec{h}\in \mathfrak{g}_{\pi(p)}$,
\item[\textrm{b)}]
\vspace{-2pt}
$\psi\cp\Add{h}=\Add{\phi_p(h)}\cp \:\psi$ \quad\hspace{10pt} $\forall\:h\in G_{\pi(p)}$.
\end{enumerate}
\endgroup
\noindent
This bijection is explicitly given by $\w\mapsto \Phi_p^*\w$.\hspace*{\fill}{{\scriptsize$\blacksquare$}}
\end{scase}
\begin{example}
\label{ex:eukl}
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item \textbf{Homogeneous Connections}
\newline
\vspace{-13pt}
\newline
In the situation of Example \ref{ex:transinv} let $k=n$ and $X=\RR^n$. Then $\Phi$ is fibre transitive and for $p=(0,e)$ we have $G_{\pi(p)}=e$ and $\mathfrak{g}_{\pi(p)}=\{0\}$. Consequently, the reduced connections w.r.t.\ $\{p\}$ are just the linear maps $\psi\colon \RR^n \rightarrow \mathfrak{s}$, and the corresponding homogeneous connections are given by
\vspace{-4pt}
\begin{align*}
{\w^\psi}(\vec{v}_x,\vec{\sigma}_s)=\Add{s^{-1}}\cp\:\psi(\vec{v}_x)+\dd L_{s^{-1}}(\vec{\sigma}_s) \qquad\forall\:(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P.
\end{align*}
\item
\vspace{-8pt}
\textbf{Homogeneous Isotropic Connections}
\newline
\vspace{-13pt}
\newline
As already stated in Example \ref{ex:LQC}, the $\Pe$-invariant connections are of the form
\vspace{-3pt}
\begin{align*}
\w^c(\vec{v}_x,\vec{\sigma}_x)= c \Add{s^{-1}}[\murs(\vec{v}_x)]+s^{-1}\vec{\sigma}_s \qquad \forall\:(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P,
\end{align*}
where $c$ runs over $\mathbb{R}$.
\hspace*{\fill}{$\Diamond$}
\end{itemize}
\endgroup
\end{example}
\noindent
We close this subsection with a remark concerning the relations between sets of invariant connections that correspond to different lifts of the same Lie group action on the base manifold of a principal fibre bundle.
\begin{remark}
\label{rem:liftuntersch}
Let $P$ be a principal fibre bundle and $\Phi,\Phi'\colon G\times P\rightarrow P$ be two Lie groups of automorphisms with $\varphi=\varphi'$. Then the respective sets of invariant connections can differ significantly. In fact, in the situation of the second part of Example \ref{ex:eukl} let
$\Phi'((v,\sigma),(x,s)):=(v+ \varrho(\sigma)(x),s)$. Then $\varphi'=\varphi$ and Appendix \ref{subsec:DifferentLifts} shows that $\w_0(\vec{v}_x,\vec{\sigma}_s):=s^{-1}\vec{\sigma}_s$ for $(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P$ is the only $\Phi'$-invariant connection on $P$.\hspace*{\fill}{{$\Diamond$}}
\end{remark}
\subsubsection{Trivial Bundles -- Applications to LQG}
\label{sec:ApplTrivB}
In this final Subsection we determine the set of spherically symmetric connections on $\RR^3\times \SU$, to be used for the description of spherically symmetric gravitational systems in the framework of loop quantum gravity. To this end, we reformulate Theorem \ref{th:InvConnes} for trivial bundles.
The spherically symmetric connections on $P=\RR^3\times \SU$ are such connections, invariant under the action $\Phi\colon \SU \times P\rightarrow P$, $(\sigma,(x,s))\mapsto (\sigma(x),\sigma s)$. Since $\Phi$ is not fibre transitive, we cannot use Case \ref{th:wang} for the necessary calculations. Moreover, it is not possible to apply the results from \cite{HarSni} (see Example \ref{example:SCHSV}) because the $\varphi$-stabilizer of $x=0$ equals $\SU$ whereas that of each $x\in \RR^3\backslash \{0\}$ is given by the maximal torus $H_x=\{\exp(t \murs(x) \:|\: t\in \RR)\}\subseteq \SU$.
Of course, we could ignore the origin and consider the bundle $\RR^3\backslash \{0\}\times \SU$ together with the $\Phi$-covering $\{\lambda \cdot \vec{e}_1\:|\: \lambda \in \RR_{>0}\}$. This, however, is a different situation because an invariant connection on $\RR^3\backslash \{0\}\times \SU$ is not necessarily extendible to an invariant connection on $\RR^3\times \SU$. This is illustrated in (see also remarks following Example \ref{bsp:Rotats})
\begin{example}
\label{ex:OnePoint}
\begingroup
\setlength{\leftmargini}{18pt}
\begin{itemize}
\item
Let $S$ be a Lie group and $P=\RR^n\times S$. We consider the action $\Phi\colon \RR_{>0}\times P \rightarrow P$, $(\lambda,(x,s))\mapsto (\lambda x,s)$ and claim that the only $\Phi$-invariant connection is given by
\begin{align*}
\w_0(\vec{v}_x,\vec{\sigma}_s):=\dd_sL_{s^{-1}}(\vec{\sigma}_s)\qquad\forall\: (\vec{v}_x,\vec{\sigma}_s)\in T_{(x,e)}P.
\end{align*}
In fact, $P_\infty:=\RR^n\times \{e\}$ is a $\Phi$-covering of $P$ by Corollary \ref{cor:reductions} and it is straightforward to see that condition \textit{i.)} from Corollary \ref{cor:psialpha} is equivalent to the conditions \textit{a.)} and \textit{b.)} from Remark \ref{rem:Psialphaconderkl}.
Let $\psi\colon \mathfrak{g}\times TP_\infty$ be a reduced connection w.r.t.\ $P_{\infty}$ and define $\psi_x:=\psi|_{\mathfrak{g}\times T_{(x,e)}}$.
Since the exponential map $\exp\colon \mathfrak{g}\rightarrow \RR_{>0}$ is just given by $\mu \mapsto \mathrm{e}^{\mu}$ for $\mu \in \RR=\mathfrak{g}$, we have $\wt{g}((x,e))=\vec{g}\cdot x\in T_{(x,e)}P_\infty$ for $\vec{g}\in \mathfrak{g}$. Then for $\vec{w}:=-\vec{g}\cdot x \in T_{(x,e)}P_\infty$ from \textit{a.)} we obtain
\begin{align}
\label{eq:conda}
\psi_x\big(\vec{g},\vec{0}\big)=\psi_x\big(\vec{0}_{\mathfrak{g}},\vec{g}\cdot x\big)\qquad \forall\:\vec{g}\in \mathfrak{g},\forall\: x\in \RR^n.
\end{align}
In particular, $\psi_0\big(\vec{g},\vec{0}\big)=0$, and since $Q_{(0,e)}=\RR_{>0}\times \{e\}$, for $q=(\lambda,e)$ condition \textit{b.)} yields
\begin{align*}
\lambda\:\psi_{0}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)=\psi_{0}\big(\vec{0}_{\mathfrak{g}},\lambda \vec{w}\big)\stackrel{\textit{b.)}}{=}\psi_{0}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)\qquad \forall\:\lambda> 0,\forall\:\vec{w} \in T_{(0,e)}P_\infty,
\end{align*}
hence $\psi_0=0$.
Analogously, for $x\neq 0$, $\vec{w} \in T_{(\lambda x,e)}P_\infty$, $\lambda>0$ and $q=(\lambda,e)$ we obtain
\begin{align*}
\lambda\hspace{1pt}\psi_{\lambda x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)=\psi_{\lambda x}\big(\vec{0}_{\mathfrak{g}},\dd L_q(\vec{w})\big)\stackrel{\textit{b.)}}{=}\qrep(q)\cp \psi_{x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)=
\psi_{x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big),
\end{align*}
i.e., $\psi_{\lambda x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)=\frac{1}{\lambda}\: \psi_{x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)$. Here, in the second step, we have used the canonical identification of the linear spaces $T_{(x,e)}P_\infty$ and $T_{(\lambda x,e)}P_\infty$. Using the same identification, from continuity (smoothness) of $\psi$ and $\psi_{0}=0$ we obtain
\begin{align*}
0=\lim_{\lambda \rightarrow 0}\psi_{\lambda x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)=\lim_{\lambda \rightarrow 0}\frac{1}{\lambda}\: \psi_{x}\big(\vec{0}_{\mathfrak{g}},\vec{w}\big)\qquad\forall\: x\in \RR^n,\forall\:\vec{w}\in T_{(x,e)}P_\infty
\end{align*}
so that $\psi_{x}\big(\vec{0}_{\mathfrak{g}},\cdot\big)=0$ for all $x\in \RR^n$, hence $\psi=0$ by \eqref{eq:conda}.
Finally, it is straightforward to see that $(\Phi^*\w_0)|_{\mathfrak{g}\times TP_\infty}=\psi=0$ holds.
\item
\itspace
Let $P'=\RR^n\backslash \{0\}\times S$ and $\Phi$ be defined as above. Then $K\times\{e\}$, for the unit-sphere $K:=\{x\in \RR^n\:|\: \|x\|=1\}$, is a $\Phi$-covering of $P'$ with the properties from
Example \ref{example:SCHSV}. Evaluating the corresponding conditions \textit{i''.)}, \textit{ii''.)}, \textit{iii''.)} immediately shows that the set of $\Phi$-invariant connections on $P'$ is in bijection with the smooth maps $\psi\colon \RR \times TK\rightarrow\mathfrak{s}$ for which $\psi|_{\RR \times T_kK}$ is linear for all $k\in K$. The corresponding invariant connections are given by
\begin{align*}
\w^\psi(\vec{v}_x,\vec{\sigma}_s)=\psi\left(\textstyle\frac{1}{\|x\|}\pr_{\|}(\vec{v}_x),\pr_{\perp}(\vec{v}_x)\right)+ s^{-1}\vec{\sigma}_s\qquad \forall\:(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P'.
\end{align*}
Here, $\pr_{\|}$ denotes the projection onto the axis defined by $x\in \RR^n$ and $\pr_{\perp}$ the projection onto the corresponding orthogonal complement in $\RR^n$.\hspace*{\fill}{$\Diamond$}
\end{itemize}
\endgroup
\end{example}
\noindent
Also in the spherically symmetric case the $\varphi$-stabilizer of the origin has full dimension, and it turns out to be convenient (cf.\ Appendix \ref{subsec:IsotrConn}) to use the $\Phi$-covering $\RR^3\times \{e\}$ in this situation as well.
Since the choice $P_\infty:=M\times\{e\}$ is always reasonable (cf.\ Lemma \ref{lemma:mindimslice}.1) if there is a point in the base manifold $M$ (of the trivial bundle $M\times S$) whose stabilizer is the whole group, we now adapt Theorem \ref{th:InvConnes} to this situation.
For this, we identify $T_xM$ with $T_{(x,e)}P_\infty$ for each $x\in M$ in the sequel.
\begin{scase}[Trivial Principal Fibre Bundles]
\label{scase:trivbundle}
Let $(G,\Phi)$ be a Lie group of automorphisms of the trivial principal fibre bundle $P=M\times S$. Then the $\Phi$-invariant connections are in bijection with the smooth maps $\psi\colon \mathfrak{g}\times TM\rightarrow \mathfrak{s}$ for which $\psi|_{\mathfrak{g}\times T_xM}$ is linear for all $x\in M$ and that fulfil the following properties.
\newline
\vspace{-8pt}
\newline
Let $\psi^{\pm}\left(\vec{g},\vec{v}_y,\vec{s}\:\right):=\psi\left(\vec{g},\vec{v}_y\right)\pm\vec{s}$ for $((\vec{g},\vec{s}\:),\vec{v}_y)\in \mathfrak{q}\times T_yM$. Then for $q\in Q$, $x\in M$ with $q\cdot (x,e)=(y,e)\in M\times \{e\}$ and all $((\vec{g},\vec{s}\:),\vec{v}_x)\in \mathfrak{q}\times T_xM$ we have
\begin{enumerate}
\item[\textrm{i.)}]
$\wt{g}(x,e) + \vec{v}_x
-\vec{s}=0 \quad \Longrightarrow \quad \psi^{-}(\vec{g},\vec{v}_x,\vec{s}\hspace{1pt})=0$,
\item[\textrm{ii.)}]
$\psi^+(\dd L_q \vec{v}_x)=\qrep(q)\cp \psi\big(\vec{0}_{\mathfrak{g}},\vec{v}_x\big)$\hspace{35pt} $\forall\:\vec{v}_x\in T_xM$,
\item[\textrm{iii.)}]
$\psi\big(\!\Add{q}(\vec{g}),\vec{0}_{y}\big)=\qrep(q)\cp \psi\big(\vec{g},\vec{0}_{x}\big)$ \:\hspace{19.5pt} $\forall\:\vec{g}\in\mathfrak{g}$.
\end{enumerate}
\begin{proof}
The elementary proof can be found in Appendix \ref{subsec:TrivBund}.
\end{proof}
\end{scase}
\begin{example}[Spherically Symmetric Systems in LQG]
\label{bsp:Rotats}
Let $\varrho\colon \SU \rightarrow \SOD$ be the universal covering map and $\sigma(x):=\varrho(\sigma)(x)$ for $x\in \RR^3$. Moreover, let $\text{\gls{MURS}}\colon \RR \rightarrow \mathfrak{su}(2)$ be defined as in Convention \ref{conv:sutwo1}.\ref{conv:sutwo111}. We consider the action of $G=\SU$ on $P=\RR^3\times \SU$ defined by $\Phi(\sigma, (x,s)):=(\varrho(\sigma)(x),\sigma s)$. It is shown in Appendix \ref{subsec:IsotrConn} that the corresponding invariant connections are of the form
\begin{align}
\label{eq:rotinvconn}
\begin{split}
\w^{abc}(\vec{v}_x,\vec{\sigma}_s):= \Add{s^{-1}}\!\big[&a(x)\murs(\vec{v}_x)+ b(x)[\murs(x),\murs(\vec{v}_x)
+c(x)[\murs(x),[\murs(x),\murs(\vec{v}_x)]]\big]+ s^{-1}\vec{\sigma}_s
\end{split}
\end{align}
for $(\vec{v}_x,\vec{\sigma}_s)\in T_{(x,s)}P$ and with rotation invariant maps $a,b,c\colon \mathbb{R}^3\rightarrow \mathbb{R}$ for which the whole expression is a smooth connection.
We claim that the functions $a,b,c$ can be assumed to be smooth as well. More precisely, we show that we can assume that
\begin{align*}
a(x)=f\big(\|x\|^2\big) \qquad\quad b(x)=g\big(\|x\|^2\big)\qquad\quad c(x)=h\big(\|x\|^2\big)
\end{align*}
holds for smooth functions $f,g,h\colon (-\epsilon,\infty)\rightarrow \RR$ with $\epsilon>0$. Then, each pullback of such a spherically symmetric connection by the global section $x\mapsto (x,e)$ can be written in the form
\begin{align*}
\wt{\w}^{abc}(\vec{v}_x)=f'\big(\|x\|^2\big)\:\murs(\vec{v}_x)+g'\big(\|x\|^2\big)\:\murs(x\times \vec{v}_x) +h'\big(\|x\|^2\big)\:\murs\left(x\times (x\times \vec{v}_x)\right)
\end{align*}
for smooth functions $f',g',h'\colon (-\epsilon,\infty)\rightarrow \RR$ with $\epsilon>0$.
\newline
\vspace{-10pt}
\newline
{\bf Proof of the Claim.}
\begingroup
\setlength{\leftmargini}{20pt}
\begin{enumerate}
\item[{\bf 1)}]
\vspace{-6pt}
Smoothness of $\w^{abc}$
implies smoothness of the real functions
\begin{align*}
a_{\vec{n}}(\lambda):= a(\lambda\vec{n})\qquad b_{\vec{n}}(\lambda):=\lambda\: b(\lambda\vec{n})\qquad c_{\vec{n}}(\lambda):= \lambda^2 c(\lambda\vec{n})\qquad \forall\:\lambda\in \mathbb{R}
\end{align*}
for each $\vec{n}\in\mathbb{R}^3\backslash\{0\}$. In fact, $a_{\vec{n}}(\lambda)\cdot \murs(\vec{n})=\w^{abc}_{(\lambda \vec{n},e)}(\vec{n})$ is smooth, so that smoothness of $b_{\vec{n}}$ and $c_{\vec{n}}$ is immediate from smoothness of $\lambda \mapsto \w^{abc}_{(\lambda \vec{e}_1,e)}(\vec{e}_2)$.
\item[{\bf 2)}]
\itspace
Let $\vec{n}$ be fixed. Then
$a_{\vec{n}}$ is even so that $a_{\vec{n}}(\lambda)=f\big(\lambda^2\big)$ for a smooth function $f\colon (-\epsilon_1,\infty)\rightarrow \mathbb{R}$, see \cite{HasslerWhitneyb}.
Moreover, $b_{\vec{n}}$ is smooth and odd so that $b_{\vec{n}}(\lambda) =\lambda\: g\big(\lambda^2\big)$ for some smooth function $g\colon (-\epsilon_2,\infty)\rightarrow \mathbb{R}$, again by \cite{HasslerWhitneyb}. Similarly, $c_{\vec{n}}(\lambda)= l(\lambda^2)$ for a smooth function $l\colon (-\epsilon_3,\infty)\rightarrow \mathbb{R}$. Since $\lambda \mapsto l(\lambda^2)$ is even and $l(0)=0$, for $s\in \mathbb{N}_{>0}$ Taylor's formula yields
\begin{align*}
l\big(x^2\big)&= a_1 x^2 +\dots + a_{s}x^{2s} + x^{2(s+1)}\phi(x)\\
&=x^2 \big(a_1 +\dots + a_{s}x^{2s-2} + x^{2s}\phi(x)\big)=x^2L(x)
\end{align*}
with remainder term $\phi(x):=\frac{1}{(2s+1)!}\frac{1}{x^{2s+2}}\int_{0}^{x}(x-t)l^{(2s+2)}(t) \hspace{1.5pt} \dd t$ for $x\neq 0$ and $\phi(0):=l^{(2s+2)}(0)$. Now, $\phi$ is continuous by Theorem 1 in \cite{HasslerWhitneya} so that $L$ is continuous as well. But $x\mapsto x^2 L(x)$ is smooth so that Corollary 1 in \cite{HasslerWhitneya} shows that $L$ is smooth as well. Now, $L$ is even, hence $L(x)=h(x^2)$ for some smooth function $h\colon (-\epsilon_4,\infty )\rightarrow \mathbb{R}$. Then $c_{\vec{n}}(\lambda)=l\big(\lambda^2\big)=\lambda^2h\big(\lambda^2\big)$
and for $x\neq 0$ we get
\begin{align*}
b(x)&=\|x\|\:b\left(\|x\|\frac{x}{\|x\|}\right)\frac{1}{\|x\|}=
b_{\textstyle{\frac{x}{\|x\|}}}(\|x\|)\frac{1}{\|x\|}
=g\left(\|x\|^2\right),\\
c(x)&=\|x\|^2\:c\left(\|x\|\frac{x}{\|x\|}\right)\frac{1}{\|x\|^2}
=c_{\textstyle\frac{x}{\|x\|}}\left(\|x\|\right)\frac{1}{\|x\|^2}
=h\left(\|x\|^2\right).
\end{align*}
Moreover, for $x=0$ we have
\begin{align*}
b(x)[\murs(x),\murs(\vec{v}_x)]=\:&0=g\big(\|x\|^2\big)[\murs(x),\murs(\vec{v}_x)],\\
c(x)[\murs(x),[\murs(x),\murs(\vec{v}_x)]\big]=\:&0=h(x)[\murs(x),[\murs(x),\murs(\vec{v}_x)]\big]
\end{align*}
so that we can assume $a(x)=f(\|x\|^2)$, $b(x)=g(\|x\|^2)$ and $c(x)=h(\|x\|^2)$ for the smooth functions $f,g,h\colon \left(-\min(\epsilon_1,\dots,\epsilon_4),\infty\right)\rightarrow \RR$.
\end{enumerate}
\endgroup
\noindent
In particular, there are spherically symmetric connections on $\RR^3\backslash \{0\}\times \SU$ which cannot be extended to those on $P$. For instance, if $b=c=0$ and $a(x):=1\slash \|x\|$ for $x\in \RR^3\backslash \{0\}$, then $\w^{abc}$ cannot be extended smoothly to an invariant connection on $\RR^3\times \SU$ since elsewise $a_{\vec{n}}$ could be extended to a continuous (smooth) function on $\RR$. \hspace*{\fill}{$\Diamond$}
\end{example}
\subsection{Summary}
\label{invconnconclusion}
We conclude with a short review of the particular cases that follow from Theorem \ref{th:InvConnes}. For this let $(G,\Phi)$ be a Lie group of automorphisms of the principal fibre bundle $(P,\pi,M,S)$ and $\varphi$ the induced action on $M$.
\begingroup
\setlength{\leftmargini}{20pt}
\begin{itemize}
\item
If $P=M\times S$ is trivial, then $M\times \{e\}$ is a $\Phi$-covering of $P$. As we have demonstrated in the spherically symmetric and scale invariant case (cf.\ Examples \ref{ex:OnePoint} and \ref{bsp:Rotats}), this choice can be useful for calculations if there is a point in $M$ whose $\varphi$-stabilizer is the whole group $G$.
\item
If there is an element $x\in M$ which is contained in the closure of each $\varphi$-orbit,
then each $\THA$-patch that contains some $p\in\pi^{-1}(x)$ is a $\Phi$-covering of $P$, see Example \ref{ex:Bruhat}. If $\varphi$ acts transitively on $M$, then for each $p\in P$ the zero-dimensional submanifold $\{p\}$ is a $\Phi$-covering of $P$ giving back Wang's original theorem, see Case \ref{th:wang} and Example \ref{ex:eukl}.
\item
Let $\Phi$ act via gauge transformations on $P$. In this case, each open covering $\{U_\alpha\}_{\alpha\in I}$ of $M$ together with smooth sections $s_\alpha \colon U_\alpha \rightarrow P$ provides the $\Phi$-covering $\{s_\alpha(U_\alpha)\}_{\alpha\in I}$ of $P$. If $G$ acts trivially, this specializes to the usual description of smooth connections by means of consistent families of local 1-forms on the base manifold $M$.
\item
If we find a $\THA$-patch $P_0$ such that $\pi(P_0)$ intersects each $\varphi$-orbit in a unique point, then $P_0$ is a $\Phi$-covering. If in addition the stabilizer $Q_p$ does not depend on $p\in P_0$, then we are in the situation of \cite{HarSni}, see Example \ref{example:SCHSV}.
\item
Assume that there is a collection of $\varphi$-orbits forming an open subset
$U\subseteq M$. Then $O:=\pi^{-1}(U)$ is a principal fibre bundle and each $\Phi$-invariant connection on $P$ restricts to a $\Phi$-invariant connection on $O$. Conversely, if $U$ is in addition dense in $M$, then one can ask the question whether a $\Phi$-invariant connection on $O$ extends to a $\Phi$-invariant connection on $P$. Since such an extension is necessarily unique (continuity), $\varphi$-orbits not contained in $U$ can be seen as sources of obstructions for the extendibility of invariant connections on $O$ to $P$. Indeed, as the examples in Subsection \ref{sec:ApplTrivB} show, smoothness of these extension gives crucial restrictions. Moreover, by Example \ref{ex:Bruhat}, taking one additional orbit into account can shrink the number of invariant connections to zero.
Of particular interest, in this context, is the case where $G$ is compact as then the orbits of principal type always form a dense and open subset $U$ of $M$ on which the situation of \cite{HarSni} always holds locally \cite{RumSchmBuch}. This gives rise to a canonical $\Phi$-covering of $O$ consisting of convenient patches. So, using the present characterization theorem, there is a realistic chance to get some general classification results in the compact case. These can be used, e.g., to extend the framework of the foundational LQG reduction paper \cite{BojoKa}.
\end{itemize}
\endgroup
\noindent
As Corollary \ref{cor:reductions} shows, in the general situation one can always construct $\Phi$-coverings of $P$ from families of $\varphi$-patches in $M$. In particular, the first three cases arise in this way.
\section{Conclusions and Outlook}
\begingroup
\setlength{\leftmargini}{15pt}
\begin{itemize}
\item
In Subsection \ref{sec:ModifreeSeg}, we have considered the situation where the action $\wm$ induced on the base manifold $M$ is analytic and pointwise proper. We have shown that then the following decomposition of the set $\Paw$ (of embedded analytic curves in $M$) holds
\begin{align*}
\Paw=\Pags\sqcup \Pacs\sqcup \Pafns\sqcup \Pafs,
\end{align*}
and that each of these subsets is invariant under decomposition and inversion of its elements. So, by Proposition \ref{rem:euklrem2b} we have
\begin{align*}
\AQRw \cong \AQRInd{\mg}\times \AQRInd{\mathrm{CNL}}
\times \AQRFNS\times \AQRInd{\mathrm{FS}}
\end{align*}
provided, of course, that each of the above sets of curves is non-empty.\footnote{Elsewise, we just remove the respective factor in the above product.} Recall that
\begingroup
\setlength{\leftmarginii}{12pt}
\begin{itemize}
\item[$\triangleright$]
$\Paf=\Pafns\sqcup \Pafs$ is the set of embedded analytic curves that contain a segment $\delta$ for which $\wm_g\cp \delta=\delta$ holds whenever $\mathrm{im}[\wm_g\cp\delta]\cap \mathrm{im}[\delta]$ is infinite. Here, $\Pafns$ consists of all such curves whose stabilizer (a well-defined quantity in this context) is trivial, so that $\Pafs$ denotes the set of all free curves for which this is not the case. Obviously, $\Pafs=\emptyset$ holds if $\wm$ is free.
\item[$\triangleright$]
$\Pac=\Pags\sqcup \Pacs$ (set of continuously generated curves) consists of all embedded analytic curves which are not free. Here, $\Pags$ consists of such curves which are generated by the Lie algebra of symmetry group and $\Pacs$ is just its complement in $\Pac$. At this point, it is the set $\Pacs$ which makes it hard to define measures on the full space $\A_\w$. However, as we have seen in the last two parts of Proposition \ref{prop:freeseg}, $\Pacs=\emptyset$ holds if $\wm$ admits only normal stabilizers and is transitive or proper.
\end{itemize}
\endgroup
So, for $\wm$ non-trivial we have:
\renewcommand*{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{c|c|c|c|c}
$\wm$ & $\Pags$ & $\Pacs$ & $\Pafns$ & $\Pafs$\\[3pt] \hline
free & many& ? & ?& $\emptyset$\\
transitive $+$ normal stabilizers & many & $\emptyset$ & ?& ?\\
\hspace{10pt} proper $+$ normal stabilizers & many & $\emptyset$ & ?& ?
\end{tabular}
\end{center}
Thus, if $\wm$ is proper and free such as in (semi-)homogeneous LQC (see Example \ref{ex:LQC}), even $\Paw=\Pags\sqcup \Pafns$, hence $\AQRw \cong \AQRInd{\mg}
\times \AQRFNS$ holds.
In Section \ref{sec:MOQRCS}, we have constructed normalized Radon measures $\mLAS$ and $\mFNS$ on $\AQRInd{\mg}$ (for $S=\SU$) and $\AQRFNS$ (for $S$ compact and connected), respectively.
This means that we have the normalized Radon measure $\mLAS\times \mFNS$ on $\AQRw \cong \AQRInd{\mg}\times\AQRFNS$ whenever $\Paw=\Pags\sqcup \Pafns$ holds and $S=\SU$. In particular, this provides us with a normalized Radon measure on $\AQRw$ in \mbox{(semi-)homogeneous} LQC.
Unfortunately, in homogeneous isotropic and spherically symmetric LQC the $\wm$-stabilizers are not normal subgroups. So, we do not know whether $\Pacs=\emptyset$ holds in these cases. More precisely, there we have
\begin{center}
\begin{tabular}{c|c|c|c|c}
LQC & $\Pags$ & $\Pacs$ & $\Pafns$ & $\Pafs$\\[3pt] \hline
homogeneous & many & $\emptyset$ & many& $\emptyset$\\
semi-homogeneous & many & $\emptyset$ & many & $\emptyset$\\
homogeneous isotropic & many & ? & many & $\emptyset$\\
spherically symmetric & many & ?& many& linear curves through origin.
\end{tabular}
\end{center}
Hence, in order to obtain a normalized Radon measure on $\AQRw$, in the last two cases we first had to calculate the set $\Pacs$ by hand.
For this observe that in the spherically symmetric case
it is possible to construct a measure on $\AQRInd{\mathrm{FS}}$ by hand, see Remark \ref{ex:Fullmeas}.\ref{ex:Fullmeas2}.
Consequently, there is still some serious demand for concepts allowing to determine the set $\Pacs$ also in the general case.
\item
Since we have constructed normalized Radon measures on quantum-reduced configuration spaces in LQG, we now can start to reduce the holonomy-flux algebra and try to define reasonable representations on the respective Hilbert spaces of square integrable functions. Then, aiming at some uniqueness results for these representations similar to that one has for the full theory \cite{UniqLew, ChUn}, we should investigate the invariance properties of the constructed measures in detail.
Here, the case of (semi-)homogeneous LQC, where we have the Radon measure $\mLAS\times \mFNS$ on the full quantum-reduced configuration space $\AQRw \cong \AQRInd{\mg}\times\AQRFNS$, may serve as a working example for this.
In addition to that, one also should search for such representations on the Hilbert spaces
we have constructed in Section \ref{sec:HomIsoCo} for the space $\RR\sqcup \RB$.
These representations then can be compared (w.r.t.\ unitary equivalence) with the standard representation on the standard kinematical Hilbert space $\Lzw{\RB}{\muB}$ of homogeneous isotropic LQC.
Complementary to that, here one might use Proposition \ref{lemma:bohrmassdichttrans} and Corollary \ref{cor:eindbohr} in order to prove some kind of uniqueness statement for this standard representation.
\item
Even if the characterization theorem \ref{th:InvConnes} looks quite technical at a first sight, its flexibility makes it a powerful tool for explicit calculations. It simplifies significantly in several situations and provides us, e.g., with the Cases \ref{scase:OneSlice}, \ref{scase:GaugeTransf}, \ref{scase:slicegleichredcluster}, \ref{th:wang} and \ref{scase:trivbundle}. Aiming at some classification results, here the next step will be to investigate such special situations in more detail.
Predestinated for this is the case where $G$ is compact as there
the orbits of principal type always form a dense open subset of $U$ of $M$, defining a dense open subbundle $O$ of $P$ for which the situation of \cite{HarSni} always holds locally \cite{RumSchmBuch}.
This provides us with a $\Phi$-covering of $O$ which consists of very convenient patches. So, using the present characterization theorem there is a realistic chance to get some general classification results which can be used to extend the framework of the foundational LQG reduction paper \cite{BojoKa}.
\item
In gauge field theories, instead of $\Con$ usually the quotient $\Con \slash \GAG$ is considered as the physically relevant configuration space. Here, $\GAG$ denotes the set of gauge transformations\footnote{These are all automorphisms $\sigma$ of $P$ with $\pi\cp \sigma =\pi$.} on the underlying principal fibre bundle and we have $\w\sim_{\GAG} \w'$ for $\w,\w'\in \Con$ iff there is $\sigma\in \GAG$ such that $\w'=\sigma^*\w$ holds.
Similarly, if $(G,\Phi)$ is a Lie group of automorphisms of $P$, instead of $\AR$ also the set
\begin{align*}
\Con_{\red,\GAG}=\{\w\in \Con\:|\: \forall\: g\in G\:\: \exists\:\sigma\in \GAG : \Phi_g^*\w=\sigma^*\w\}
\end{align*}
of connections invariant up to gauge transformations is considered as reduced configuration space.
However, in contrast to the set $\AR$, for $\Con_{\red,\GAG}$ no general characterization theorems seem to exist so far, so that it is potentially harder to compute this space. So, instead of quantizing $\Con_{\red,\GAG}$ (which would also be a classical reduction) one can use the lifted action $\specw$ in order to define ``invariance up to gauge'' directly on the quantum level.
Hence, instead of the space $\AQR\cong \IHOM$ one can consider the space $\RedGauge$ consisting of all
elements $\homm\in \HOM$ for which we have that for each $g\in G$ there is a generalized gauge transformation\footnote{This just means that $\pi\cp \sigma =\pi$ and $\sigma(p\cdot s)=\sigma(p)\cdot s$ holds for all $\in S$.} $\sigma\colon P\rightarrow P$ with $\specw_g(\homm)=\sigma(\homm)$, i.e.,
\begin{align*}
\specw_g(\homm)(\gamma)(p)=(\sigma \cp \homm(\gamma))(\sigma^{-1}(p))\qquad \forall\: \gamma\in \Pa, \forall\:p\in F_{\gamma(a)}
\end{align*}
for $\operatorname{\mathrm{dom}}[\gamma]=[a,b]$. Then, $\IHOM \subseteq \RedGauge$ holds and the concepts of the Sections \ref{susec:LieALgGenC} and \ref{sec:MOQRCS} seem adaptable to the ``up to gauge''-case. Indeed, we rather expect technical than conceptual difficulties at this point.
\item
An alternative construction of the Ashtekar-Lewandowski measure has been presented in \cite{Tlas}. There, the author uses a so-called nicely shrinking net of open neighbourhoods (thickenings) of the closed subset $\HOM\subseteq \mathrm{Maps}(\Pa,\IsoF)$ in order to define the Ashtekar-Lewandowski measure on $\HOM$.\footnote{The author considers the situation where $S$ is compact, connected and semisimple. Moreover, he assumes that $\Pa$ consists of piecewise smooth and immersive loops with fixed base point, i.e., piecewise smooth and immersive curves whose end points equal a fixed point in the base manifold. The arguments, however, also go through if $\Pa=\Paw$ and $S$ is just compact and connected.} Here, the space $\mathrm{Maps}(\Pa,\IsoF)$ (of all maps $\Pa\rightarrow \IsoF$), equipped with the topology defined in analogy to that one in Definition \ref{def:indepref}.\ref{conv:muetc}, is homeomorphic to the Tychonoff product $S^{|\Pa|}$.
More generally, for $X$ a compact Hausdorff space carrying a normalized Radon measure $\mu$ and $C\subseteq X$ a closed subspace, one can define a positive and continuous linear functional
$\III\colon C(C)\rightarrow \mathbb{C}$, i.e., a finite Radon measure on $C$
as follows: (see also \cite{Tlas})
\begingroup
\setlength{\leftmarginii}{20pt}
\begin{itemize}
\item[i)]
Let $\{C_\lambda\}_{\lambda \in \Lambda}$ be a net of neighbourhoods of $C$ such that for each neighbourhood $U$ of $C$ we find $\lambda_0\in \Lambda$ with $C_\lambda\subseteq U$ for all $\lambda\geq \lambda_0$. Such a net is called nicely shrinking.
\item[ii)]
For each $f\in C(C)$ let $\ovl{f}\in C(X)$ be an extension (apply Tietze extension theorem) for which $\big\|\wt{f}\big\|_\infty=\|f\|_\infty$ holds, and define
\begin{align*}
f_\lambda:=\frac{1}{\mu(C_\lambda)}\int_{C_\lambda}\wt{f}\: \dd\mu\qquad \forall\: \lambda\in \Lambda.
\end{align*}
\item[$\triangleright$]
Now, choose $\{C_\lambda\}_{\lambda \in \Lambda}$ in such a way that $\III(f):=\lim_\lambda f_\lambda$ exists for all $f$ contained in a suitable dense subset $\mathfrak{D}\subseteq C(C)$. It is straightforward to see \cite{Tlas} that then $\III(f)$ does not depend on the explicit extension $\wt{f}$ of $f$.
Moreover, it follows that $\III\colon \mathfrak{D}\rightarrow \CCC$ is positive and bounded by 1, hence extends by continuity to a positive (and continuous) linear functional on $C(X)$.
\end{itemize}
\endgroup
Now, the measure used in \cite{Tlas} for $S^{|\Pa|}$ is just the Radon product (cf.\ Lemma and Definition \ref{def:ProductMa}.\ref{def:ProductMa3}) of copies of the Haar measure on $S$, and the net $\{C_\lambda\}_{\lambda \in \Lambda}$ is constructed in a very natural way.
In particular, in view of the complications (non-trivial restrictions to the images of the projection maps) arising from the invariance and inclusion properties of the reduced spaces discussed in this work, the above ``thickening approach'' seems to be predestinated for providing a general notion of a reduced Ashtekar-Lewandowski measure on these spaces. Indeed, it even might help to drop the restriction to the structure group $\SU$ (and tori) we have made in Subsection \ref{sec:ConSp} in order to define the measure on $\AQRInd{\mg}$. The developments of this thesis then should help to find the correct definitions, e.g., by requiring that, when restricting to the set $\Pafns$, we get back the measure constructed in Subsection \ref{sec:FreeM}.
Moreover, the net of thickenings constructed in \cite{Tlas} for $\HOM$ even seems generalizable to the space\footnote{This space is indeed closed as one easily deduces from compactness of the structure group $S$.} $\RedGaugew$. This is basically because this space (under mild assumptions) seems to admit sufficiently many elements in order to make the relevant maps surjective.
\end{itemize}
\endgroup
|
2,869,038,155,727 | arxiv | \section{INTRODUCTION}
NGC\,1241 is a Seyfert\,2 galaxy \citep{veron98} which presents a
complex morphology in the innermost 1.5\,kpc. P$\alpha$ imagery shows
the presence of an emitting circumnuclear ring of star formation (CNR)
with a brightness peak at a radius of $710\pm80$\,pc. It also shows a
0.3\,kpc long bar accompanied by an $m=2$ leading arm both emitting in
P$\alpha$ and centered on the nucleus. Apparently, they do not have
associated absorption features as it might be expected (Reagan \&
Mulchaey 1999, hereafter RM99). The $J$ and $K_s$ images reveal that
the CNR is mounted on a smooth inclined disk with approximately
elliptical isophotes of varying position angle. The major to minor
axis ratio of the outermost isophotes in the $J$ and $K_s$ bands
reveals a disk inclination of 52\arcdeg, consistent with the value
given by Tully (1988) for the large scale disk. Finally, the $K_s$
image shows the presence of a trailing arm ending at the CNR and
centered on the nucleus. These structures (Figure\,1) have been
kinematically studied by D\'{\i}az et al. (2003).
RM99 have found that the $(V-H)$ color map to the southwest of the
line of nodes of NGC\,1241 is redder than in the northeast area, and
reveals an overall dusty morphology consistent with an inclined ring
with a color excess of 1.1\,mag, whose southwest side is the nearest
one. According to these authors, assuming that the dust scale height
is small relative to the scale height of the stars, and that the plane
of the dust is inclined with respect to the bulge stars, dust
absorption might not affect the color of the bulge near the
nucleus. As we will see later, the nucleus of NGC\,1241 is relatively
free of absorption when compared to the color excess of the ring. To
reinforce this picture, none of the absorption features normally
expected near emitting bars are evident near the 300\,pc long bar
found by D\'{\i}az et al. (2003). These authors have also shown that
the ring of dust found by RM99 coincides with the CNR (Figure\,2).
In this paper we examine GEMINI (+QUIRC+Hokupa) near-infrared images
with pixel-photometry. We detected the presence of an azimuthally
symmetric nuclear $(J-K_s)$ color excess with respect to the CNR
($0.82<(J-K_s)<1.15$) which would not be easily explainable in terms
of dust absorption according to the RM99 results
abovementioned. Moreover the $(V-H)$ color map does not show azimutal
symmetry as the ($J-K_s$) one does.
The properties of the $(V-H)$ and ($J-K_s$) in the inner 2\,kpc are
analyzed in terms of the models of Witt, Thronson \& Capuano (1992,
WTC92) ({\it Dust and transfer of Stellar Radiation within Galaxies}),
and stellar population synthesis from 2-MASS NIR color magnitude
diagrams of bar fields in the Large Magellanic Cloud (LMC), where
individual stars are resolved at M$_V\leq-3$ (Nikolaev \& Weinberg,
2000).
The next section of this paper outlines how our observations were made
and discuss the homogenization of Hubble Space Telescope (HST) and GEMINI photometry. In
Section\,3 our results are discussed and the final remarks are given
in the final section.
\section{OBSERVATIONS AND METHODS}
On September 30, 2000, we used the Quick Start service of the Gemini
North 8.1\,m telescope for NIR imaging using Hokupa'a natural guide
star and curvature-sensing adaptive optics system. The latter feeds
the dedicated Quick NIR camera (QUIRC) fitted with a $1024\times1024$
HgCdTe array sensitive to 1-2.5\,$\micron$ radiation providing a final
scale of 0.0197\arcsec\,pixel$^{-1}$. Standard data reduction
procedures were applied to the images. The achieved full width at half
maximum (FWHM) of the Gemini+Hokupa'a system was about 0.4\arcsec\, in
the $J$ band and about 0.3\arcsec\, in the $K_s$ band both measured on the point spread function
of a field star, and estimated on the target galaxy. Image
deconvolution was not applied at this stage because of the photometric
uncertainties that could be introduced by the methods that are
commonly used. This resolution was
good enough to compare the Gemini images with the existing HST-NICMOS3 data (Figure 1).
\paragraph{Photometry.}
HST imagery with F160W and F606W filters and its calibration have been
discussed by RM99. Essentially, the relative fluxes of these frames at
each position are given. A transformation was performed in order to
match colors derived from the HST fluxes $f_{F606W}$ and $f_{F160W}$
to the standard color system of our observations.
For filter F606W, we obtained $m_{F606W}$ from the flux $f_{F606W}$ in
the Vega-mag system according to
\cite{bedinetal05}. Coefficients for the transformation
of $m_{F606W}$ into $V$ of Vega-mag system are provided by
\cite{holtzmanetal95}. $V$ results $\approx0.1$\,mag
brighter than $m_{F606W}$ in agreement with previous
transformation by \cite{malkanetal95} who determined that $V$
would be 0.1\,mag to 0.2\,mag brighter than $m_{F606W}$.
For filter F160W, we calculated $m_{F160W}$ according to
\cite{stephensetal00}. These authors follow two different
procedures to transform $m_{F160W}$ into $H$, each providing
slightly different results. To be coherent with the transformation of
$m_{F606W}$ into $V$, we adopted the procedure based on the Vega-mag
system.
Using homogeneous colors, we carried out pixel photometry to
ascertain whether the morphology seen in $K_s$ band images is
differentially affected by the presence of dust. After separating
all the pixels to the northeast from those to the southwest of the
major axis, we integrated the $K_s$ brightness and $(J-K_s)$ and
$(V-H)$ colors on half-rings of variable radii and plotted them
against the de-projected radius (Figure 3).
Witt, Thronson \& Capuano (1992, hereafter WTC92) models of {\it Dust
and Transfer of Stellar Radiation within Galaxies} have been used to
disentangle color properties due to dust from those due to stellar
population effects. WTC92 models include the effect of light
scattering by dust. Four of the models presented by these authors
constitute plausible scenarios for the region under study: 1) {\it The
Dusty Galaxy}, which considers dust and stars equally distributed
within a sphere. 2) {\it The Cloudy Galaxy}, which considers the
sphere occupied by stars to be larger than that occupied by dust. 3)
{\it The Starburst Galaxy}, in which also stars occupy a larger sphere
than the dust, but follow an $r^{-6}$ distribution. 4) {\it The Dusty
Galactic Nucleus}, in which a sphere of stars is enshrouded in a
cocoon of dust.
Errors in colors for each one of the subsystems quoted in Figure\,3
are directly obtained from fluctuations in pixel photometry and
propagated to the derived quantities according to the transformation
equations of the photometric system.
\section{RESULTS \& DISCUSSION}
Figure\,3 shows that the $(J-K_s)$ reddening increases inwards, as
well as a remarkably similar radial behavior on both sides of the line
of nodes. It also shows a plateau at $(J-K_s)\approx0.8$\,mag at the
position of the CNR, with a maximum of about 1.15 mag in the
nucleus. The $K_s$ band integrated profile is also symmetric. On the
other hand, the upper panel of Figure\,3 shows how dramatically
different are both sides of the circumnuclear ring in the ($V-H$)
color, with an excess $E(V-H)\approx1.0$ in the southwest side with
respect to the northeast one, besides a global mean color excess
$<E(V-H)>\approx0.90$ of the CNR with respect to the disk. The nucleus
is remarkable too, as its ($V-H$) color is similar to that of the disk
outside the CNR, indicating the presence of a transparent window in
that direction as suggested by RM99. This conclusion is corroborated by
the detection of nuclear H$\alpha$ emission (the galaxy is a Sy\,2)
together with P$\alpha$ emission, while the CNR is observed in
P$\alpha$, but obscured in H$\alpha$.
Colors in Table\,1 can be matched to stellar spectral types using
Pickles' (1998) stellar library. The results are quoted in
Table\,2.
We first note that the ($V-H$) colors of all substructures in Table\,1
correspond to younger spectral types than it would be indicated by the
$(J-K_s)$ color, in spite of the stronger reddening in the $V$ band.
\paragraph{The disk.} Its $(J-K_s)$ color is similar to the color in
the foreground of the LMC fields studied by Nikolaev \& Weinberg
(2000) and to the Sagittarius comparison fields studied by Cole
(2001), both obtained from the Two Micron All Sky Survey (2MASS) data.
We obtained similar results by integrating six Milky Way fields
around the LMC, as we will show in the results. Nevertheless, the
disk ($V-H$) color corresponds to a B9V-A2V stellar population, which
leads us to think that star formation occurred not only in the CNR
but also reached the inner disk within its innermost 2\,kpc.
\paragraph{Circumnuclear Ring (CNR).}
Assuming that the difference in the observed ($J-K_s$) between the far
side of the CNR and the disk is caused by extinction, we obtain
$E_{fs_o}(J-K_s)\approx0.30$\,mag and WTC92 {\it Dusty Galaxy} model
leads to an extinction solution with $\tau_V=6.0$ and scattered light
contributing 45$\%$ in $V$ band. This model provides theoretical color
excesses for the CNR far side amounting to
$E_{fs_t}(J-K_s)=0.30$\,mag and $E_{fs_t}(V-H)\approx0.7$, coherent with
the values presented in Table\,1. This source of absorption, probably
diffuse, is intrinsic to the CNR and different from that discussed by
\cite{regan99}, which produces the difference in ($V-H$) between
the CNR near and far sides, attributed to the dusty one-arm clearly
seen in HST F606W filter image by \cite{regan99}. Furthermore, within
the CNR one should add the H$\alpha$ absorbing dust cocoons associated
to each one of the P$\alpha$ emitting blobs (Figure\,1).
\paragraph{Nuclear colors, dust or stellar population?}
Figure 4 shows the NIR color-magnitude diagram $(J-K_s)$ vs. $K_s$ for
0.1\arcsec\, rebinned pixels in the central region of NGC\,1241. The
figure presents a color excess $E_N(J-K_s)\approx0.2$ of the nucleus
with respect to the CNR in a different manner (cf. Figure\,3). It is
not possible to obtain a WTC92 model fitting of both the very red
($J-K_s$) color excess and the very blue nuclear ($V-H$). The
reddening arrow in Figure\,4 points to the same direction where the
nuclear $(J-K_s)$ at the top of the CMD bends. Therefore, it suggests
that dust may affect in a rather subtle way the nuclear $(J-K_s)$
without influencing ($V-H$). Our solution to this rather tricky issue
is that Carbon stars are natural candidates to explain the infrared
excess. Considerations on the LMC infrared Color-Magnitude diagram (CMD) of
Nikolaev \& Weinberg (2000), lead us to propose these substantial
contribution from Carbon stars to the nuclear stellar population.
Following a procedure similar to that of Nikolaev \& Weinberg (2000),
we have used the 2MASS All-Sky Point Source Catalog Statistics Service
facility at {\it http://irsa.ipac.caltech.edu/applications/Stats/} to
obtain an integrated color-magnitude synthesis of the LMC bar and we
have compared it to the NGC\,1241 nucleus.
The 2MASS service provides CMD diagrams and integrated light
photometry in $J$, $H$ and $K$ inside user selected circular regions.
We choose five circular fields along the LMC bar and 2 foreground
regions per bar field, one located 10 degrees North and another 10
degrees South of the bar. All regions and fields were selected with
30' radius. Extractions from the 2MASS Point Source Catalog were
performed. Figure\,5 shows the CMD diagram corresponding to the center
of the LMC bar and to the corresponding comparison field. Then, for
each of the five regions the North and South foreground fields were
averaged, and the mean brightness subtracted from the corresponding
region to correct for foreground contamination. The final integrated
surface magnitudes were obtained by flux averaging the five background
corrected integrated surface magnitudes of the regions. The results
were: $<J>\,=20.38\pm0.07$\,mag/arcsec$^2$,
$<H>\,=19.59\pm0.08$\,mag/arcsec$^2$,
$<K>\,=19.38\pm0.07$\,mag/arcsec$^2$, $<(J-K)>\,=1.00\pm0.10$. The
$<(J-K)>$ color closely agrees with the color of the nucleus of
NGC\,1241. A similar color was obtained by Cole (2001) for the
Sagittarius Dwarf galaxy, after correction for our Galaxy
contamination.
\section{Final Remarks}
We have discussed two dimensional photometry of the central 2\,kpc
of the Sy\,2 galaxy NGC 1241, where a circumnuclear ring of star
formation and the nucleus present peculiar colors when compared to
the underlying disk. HST and GEMINI imagery have been reduced to a
uniform photometric system in order to allow the study of the
photometric properties of these subsystems.
While the dust arm produces the reddening of the CNR near side with
respect to the CNR far side, we propose that an additional source of
diffuse dust obscures uniformly the CNR, thus producing a global
reddening of the CNR compared to the underlying disk. Inside the CNR,
there are cocoons of dust associated to the P${\alpha}$ emitting
condensations.
Finally, the very red $(J-K_S)$ color of the nuclear region together
with the surprising transparency of this region in $(V-H)$, led us to
propose a CMD for the nucleus similar to that of the LMC bar. C-stars
can in fact redden significantly the integrated colors at unresolved
scales, a situation similar to that we are facing in the nuclear
region of NGC\,1241. Carbon stars and asymptotic giant branch
oxygen-rich stars evolve rapidly ($t < 3\times10^4$\,yr) and eject
considerable amounts of dust and gas with velocities low enough
($V_{gas} < 100$\,km s$^{-1}$) to be trapped by the gravitational
potential barrier of the central mass concentration
($M_{kepler}\sim10^9$\,M$_{\odot}$, $r<300$\,pc). The 500
[6\,M$\odot$] C-stars and $2.5\times10^4$ asymptotic giant branch
O-rich stars (according to the LMC bar proportion) inside a radius of
about 50\,pc that are necessary to explain the nuclear colors, would
release material that, gravitationally bounded, could amount to
between $10^{-2}$ and $10^{-1}$ M$_{\odot}$\,yr$^{-1}$ of fuel for the
central engine. The intra-nucleus medium contamination may last during
the lifetime of stars with masses
2\,M$_{\odot}<$\,\,M$_{C-Stars}<6\,$M$_{\odot}$. This scenario may
also explain the systematical increase of the strength of the
optical CN-bands observed in the stellar populations of Sy\,2 galaxy nuclei
(e.g. Gu et al. 2001), and the significant contribution of
intermediate age stars to the optical continuum of low luminosity AGNs
(e.g. Gonzalez-Delgado 2004).
\section{Acknowledgements}
HD thanks the brazilian institutions CNPq and
CAPES. RD thanks the support from Evencio Mediavilla and Romano Corradi.
This research is also partially supported by brazilian grants
MEGALIT/Millennium, and the argentinean Agencia C\'ordoba Ciencia.
JFS thanks to the FAPEMIG Foundation (Minas
Gerais, Brazil).
The 2MASS project is a collaboration between the University
of Massachusetts and the IPAC (JPL/Caltech).
The Gemini 8-meter telescopes is an international partnership
managed by AURA, Inc. under a cooperative agreement with the
NSF (USA), PPARC (UK), NRC (Canada), CONICET (Argentina), ARC (Australia),
CNPq (Brasil) and CONICYT (Chile). The NASA/ESA Hubble Space
Telescope is operated by AURA under NASA contract NAS 5-26555.
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