diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzharc" "b/data_all_eng_slimpj/shuffled/split2/finalzzharc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzharc" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \nLocalization is from a mathematical and physical point of view considered to be an unresolved issue in a relativistic context, see for example \\cite{NW49}, \\cite{F1}, \\cite{H2}, \\cite{H1} and references therein. The concept is well-defined from a non-relativistic quantum-mechanical (QM) point of view. However, by combining concepts of QM and relativity, from which the frame-work of quantum field theory (QFT) emerged, the localization issue was never completely resolved. This is not merely an issue of construction but points to a more fundamental question about the existence of localizable particles.\n\\\\\\\\\nThe authors in \\cite{NW49} tackled the problem by demanding certain requirements from the states of a coordinate operator, which are used to localize elementary systems. Independently, among other operators the author in \\cite{PR48} gave as well a definition of an appropriate coordinate operator, which is equivalent to the one obtained by \\cite{NW49}. Although the operator is unique for massive particles with arbitrary spin and for every massless system with spin $0,\\,1\/2$, it has two major physical setbacks. First, eigenstates of the Newton-Wigner-Pryce (NWP) operator propagate superluminally and moreover the states are not covariant w.r.t. general Lorentz-transformations. Hence, there are two different opinions on this subject; either such localizable particles do not exist in a relativistic context or one can still make physical sense of the Newton-Wigner-Pryce operator and its respective states. \n\\\\\\\\\nIn this work, we take the second point of view and give, in our opinion, convincing arguments why the NWP sates are of physical interest. In particular, the importance of these states relies on the fact that they can be used to calculate the probability amplitude of finding a particle at a certain spatial-position at time $t$. Moreover, the second quantization of this operator allows us to calculate the probability amplitude of finding $k$-particles at certain spatial-positions at time $t$.\n\\\\\\\\\nIn order to have a richer example of the NWP-space in a quantum field theoretical context (than the massive scalar field \\cite{Muc7}), we work with the free \\textbf{massless} scalar field. In particular, we show that the localizability of particles, in the massless case, is not a lost cause. In general, the covariance group in the massless case is more than solely the Poincar\\'e group, it is given by the conformal group. Hence, we investigate the covariant transformation property of NWP-states under conformal transformations. The importance therein has a physical reasoning. It is first of all done in order to prove that although the NWP-states are non-covariant w.r.t. to the whole conformal group there exist transformations of a subgroup of the conformal group that preserve their covariant character. Furthermore, given two observers measuring, for a given covariant state, the probability of finding $k$-particles at $k$-spatial-positions. If these two observers are in two relatively to each other transformed frames, for example one is rotated, space-time translated or dilated to the other then they can agree on the observation of the probability of spatial positions, by using the covariant transformation rules.\n\\\\\\\\\nBesides calculations of position-probability amplitudes of particles, the NWP-operator has an additional physical relevance. In this work we prove that by solely using the coordinate and relativistic momentum operator, one is able to define all generators of the conformal group. Hence, in some sense, the NWP-operator can be seen as essential for implementing relativistic symmetries. \n\\\\\\\\\nAnother motivation besides understanding localization to translate the conformal group into the coordinate space, is non-commutative (NC) geometry applied to quantum field theory (QFT). In particular the geometry that is quantized in NCQFT is the geometry of the NWP-space. Hence, in order to understand the outcome and the action of a deformation of QFT using, for example (see \\cite{MUc1}) the conformal group, we need a better understanding of the commutative case.\\\\\\\\\nThe paper is organized as follows; Section two gives a quick introduction to the tools needed in this paper, where we define the Fock space of the free massless scalar field, the Newton-Wigner-Pryce operator and the algebra of the Fourier-transformed ladder operators. In section four, the conformal algebra for a massless scalar field is transformed into the NWP-space.\nSection five investigates the covariance of the NWP-space under the Conformal group.\n\n\n\n\n \\begin{convention}\n \tWe use $d=n+1$, for $n\\in\\mathbb{N}$ and the Greek letters are split into $\\mu, \\,\\nu=0,\\dots,n$. Moreover, we use Latin letters for the spatial components which run from $1,\\dots,n$ and we choose the following convention for the Minkowski scalar product of $d $-dimensional vectors, $a\\cdot b=a_0b^0+a_kb^k=a_0b^0- \\vec{a}\\cdot\\vec{b}$. Furthermore, we use the common symbol $\\mathscr{S}(\\mathbb{R}^n)$ for the Schwartz-space.\n \\end{convention} \n\\section{Preliminaries}\n\\subsection{Bosonic Fock-space and Fourier-transformation}\nWe briefly define the Bosonic Fock \nspace for a free scalar field \n $\\phi$ with mass $m=0$ on the $n+1$-dimensional Minkowski spacetime. In particular a particle with momentum $\\mathbf{p} \\in \\mathbb{R}^n$ has \n the energy $\\omega_{\\mathbf{p} }$ given by $\\omega_{\\mathbf{p} }=|\\mathbf{p}|$. Another quantity needed for the definition of the Fock-space is the Lorentz-invariant measure: $d^n\\mu(\\mathbf{p} )=d^n\\mathbf{p}\/( {2\\omega_{\\mathbf{p}}})$.\n \\begin{definition} The Bosonic Fock space $\\mathscr{H}^{+}$ is defined \n \tas in \\cite{Fr,S}:\n \t\\begin{equation*}\n \t\\mathscr{H}^{+}=\\bigoplus_{k=0}^{\\infty}\\mathscr{H}_{k}^{+}\n \t\\end{equation*}\n \twhere the $k$ particle subspaces are given as\n \t\\begin{align*}\n \t\\mathscr{H}_{k}^{+}&=\\{\\Psi_{k}: \\partial V_{+} \\times \\dots \\times \n \t\\partial V_{+} \\rightarrow \\mathbb{C}\\quad \\mathrm{symmetric}\n \t|\\\\ &\\left\\Vert \\Psi_k \\right\\Vert^2 =\\int \n \td^n\\mu(\\mathbf{p_1})\\dots\\int d^n\\mu(\\mathbf{p_k})\n \t|\\Psi_{k}(\\mathbf{p_1},\\dots,\\mathbf{p_k})|^2<\\infty\\},\n \t\\end{align*}\n \twith\n \t\\begin{equation*}\n \t\\partial V_{+}:=\\{p\\in \\mathbb{R}^{n+1}|p^2=0,p_0>0\\}.\n \t\\end{equation*}\n \\end{definition}\n The particle annihilation and creation operators can be defined by their \n action on $k$-particle wave functions\n \\begin{align*}\n (a_c(f)\\Psi)_k(\\mathbf{p_1},\\dots,\\mathbf{p_k})&=\\sqrt{k+1}\\int \n d^n\\mu(\\mathbf{p})\\overline{f(\\mathbf{p})}\n \\Psi_{k+1}(\\mathbf{p},\\mathbf{p_1},\\dots,\\mathbf{p_k})\\\\\n (a_c(f)^{*}\\Psi)_k( \\mathbf{p },\\mathbf{p_1},\\dots,\\mathbf{p_k})&= \\left\\{\n \\begin{array} {cc}\n 0, \\qquad &k=0 \\\\ \\frac{1}{\\sqrt{k}}\\sum\\limits_{l=1}^{k} f(\\mathbf{p_l})\n \\Psi_{k-1}(\\mathbf{p_1},\\dots,\\mathbf{p_{l-1}},\\mathbf{p_{l+1}},\\dots,\\mathbf{p_k}),\\quad \n &k>0\n \\end{array} \\right.\n \\end{align*}\n with $f \\in \\mathscr{H}_{1} $ and $\\Psi_k \\in \\mathscr{H}_{k}^{+}$ . The commutation relations of the smeared annihilation and creation operators $a_c(f), \n a_c(f)^{*}$ follow from their action on the functions $\\Psi$ and are given as follows\n \\begin{align*}\n [a_c(f), a_c(g)^{*}]=\\langle f,g\\rangle=\\int d^n\\mu(\\mathbf{p}) \\overline{f(\\mathbf{p})} \n g(\\mathbf{p}), \\qquad\n [a_c(f), a_c(g)]=0=[a_c^{*}(f), a_c^{*}(g)].\n \\end{align*}\nThe ladder operators with sharp momentum are \n introduced as operator valued distributions in the following\n \\begin{align*}\n a_c(f)=\\int d^n\\mu(\\mathbf{p}) \\overline{f(\\mathbf{p})}a_c(\\mathbf{p}), \n \\qquad a_c(f)^{*}=\\int d^n\\mu(\\mathbf{p}) {f(\\mathbf{p})}a_c^{*}(\\mathbf{p}),\n \\end{align*}\n where the ladder operators with sharp \n momentum satisfy the well-known commutator relations\n \\begin{align*}\n [a_c(\\mathbf{p}), a_c(\\mathbf{q})^{*}]=2\\omega_{\\mathbf{p} \n }\\delta^n(\\mathbf{p}-\\mathbf{q}), \\qquad\n [a_c(\\mathbf{p}), a_c(\\mathbf{q})]=0=[a_c^{*}(\\mathbf{p}), a_c^{*}(\\mathbf{q})].\n \\end{align*}\n In the following sections, we use the non-covariant normalization given by, \n \n \\begin{align}\n {a}(\\mathbf{p})= \\frac{{a}_c(\\mathbf{p})}{\\sqrt{2\\omega_{\\mathbf{p} \n \t}}},\\qquad {a}^{*}(\\mathbf{p})= \\frac{{a}^{*}_c(\\mathbf{p})}{\\sqrt{2\\omega_{\\mathbf{p} \n }}}.\n \\end{align}\n Next, we define the Fourier-transformation of the ladder operators that were given in the former expressions. These transformations are of physical importance in regards to the following sections. \n \\begin{definition}\\label{sec2.2}\\textbf{Fourier-transformation}\\\\\\\\\n \tIn order to change from momentum space to the NWP-space we use explicit expressions for the Fourier-transformed creation and annihilation operators which are given by,\n \t\\begin{align*}\n \t{a}(\\mathbf{p})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{x}\\, e^{ip_kx^k} \\tilde{a}(\\mathbf{x}),\\qquad {a}^{*}(\\mathbf{p})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{x}\\, e^{-ip_kx^k} \\tilde{a}^*(\\mathbf{x}).\n \t\\end{align*}\n \tThe commutation relation between the ladder operators in momentum space gives us directly the relations for ladder operators of the NWP space, \n \t\\begin{align*}\n \t\\delta^n(\\mathbf{p}-\\mathbf{q})&=[{a}(\\mathbf{p}),{a}^*(\\mathbf{q})]=(2\\pi)^{-n} \n \t\\iint\n \td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, e^{ip_kx^k}\\, e^{-iq_ky^k}\\underbrace{ [\\tilde{a}(\\mathbf{x}),\\tilde{a}^*(\\mathbf{y})]}_{=\\delta^n(\\mathbf{x}-\\mathbf{y})}.\n \t\\end{align*} \n Inverse Fourier-transformations of momentum space operators to the NWP-space ladder operators are given by,\n \t\\begin{align}\\label{inf}\n \t\\tilde{a}(\\mathbf{x})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{p}\\, e^{-ip_kx^k}{a}(\\mathbf{p}),\\qquad \n \t\\tilde{a}^*(\\mathbf{x})=(2\\pi)^{-n\/2} \\int\n \td^n \\mathbf{p}\\, e^{ ip_kx^k}{a}^*(\\mathbf{p}).\n \t\\end{align}\n \t\n \t\n \t\n \t\n \t\n \t\n \t\n \t\n \t\n \\end{definition} \n \n\\subsection{Massless NWP-operator}\nThe (spatial)-QFT-position operator for a massive free scalar field is defined by the Newton-Wigner-Pryce operator, \\cite{PR48} and \\cite{NW49}. On a one particle wave-function it acts as follows, \\cite[Chapter 3c, Equation 35]{Sch}\n \\begin{equation}\\label{NWP}\n (X_{j} \\varphi)(\\mathbf{p})=-i \\left( \\frac{p_j}{2\\omega_{m,\\mathbf{p}}^2}\n + \\frac{\\partial}{\\partial p^j } \n \\right)\\varphi(\\mathbf{p}),\n \\end{equation}\n where $\\omega_{m,\\mathbf{p}}$ is the relativistic-energy for a particle with mass $m$. \n In order to prove that the NWP-operator is in the massless case equivalently represented as in the massive case, we can proceed in different ways. First, we can take Equation (\\ref{NWP}) and perform the massless limit. Since, the energy $\\omega_{m,\\mathbf{p}}$ is the only term that is affected by the limit, the massless NWP-operator is given by exchanging in Formula (\\ref{NWP}) the massive energy with $\\omega_{\\mathbf{p}}$. Therefore, the coordinate operator for a massless QFT has the same form as in the massive case. Another possible path to proceed is to define the coordinate operator as unitary equivalent to the second-quantized spatial-component of the relativistic momentum operator. In terms of the Fock space operators the relativistic momentum operator is given as,\n \\begin{align}\\label{mop}\n P_{\\mu}&= \\int d^n \\mathbf{p}\\, p_{ \\mu}\\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p})\n .\n \\end{align} \n The unitary equivalence of the coordinate to the spatial momentum operator is given by the unitary map represented by the Fourier-transformation. In particular, it was proven that the NWP-operator can be represented as a self-adjoint operator on the domain, $\\bigotimes_{i=1}^k \\mathscr{S}(\\mathbb{R}^n)$, with $\\mathscr{S}(\\mathbb{R}^n)$ denoting the Schwartz space, for details see \\cite{Muc2} and \\cite{Muc3}. The explicit Fock-space representation of the spatial-coordinate operator is given by,\n \\begin{align*}\n X_j&=-i\\int d^n \\mathbf{p}\\, {a}^*(\\textbf{p}) \\frac{\\partial}{\\partial p^j} {a}(\\textbf{p}).\n \\end{align*} \n Since the spatial-momentum operator has the same form in the massless case as in the massive one, the NWP-operator, defined through the unitary equivalence, takes the same form as in the massive case. This is stems from the fact that ladder operators do not explicitly depend on the mass. The commutation relations between the spatial-momentum operator and the NWP-operator are given by, see \\cite{SS} or \\cite{Muc3},\n \\begin{align}\\label{ccr}\n [X_j,P_k]&=-i\\eta_{jk}N,\n \\end{align} \n where $N$ is the particle-number operator represented in Fock-space as\n \\begin{align}\\label{pn}\tN= \\int\n d^n \\mathbf{p}\\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p}).\n \\end{align}\n Next, we want to give the physical interpretation and relevance of this operator. The eigenfunctions of the Newton-Wigner-Pryce operator, simultaneously representing the localized wave functions at time $x_{0}=0$, are given by, \\cite[Chapter 3, Equation 38]{Sch}\n\\begin{align*} \n\\Psi_{\\mathbf{x},0}(\\mathbf{p})=(2\\pi)^{-n\/2}\\,e^{-i\\mathbf{p} \\cdot \\mathbf{x}}\\,(2\\omega_{p})^{1\/2}.\n\\end{align*} \n Let a free massless scalar field be in a state $\\Phi(\\mathbf{p})$ at time $t=0$. Then, the probability amplitude of finding a particle at the position $\\mathbf{x}$ is given by, \\cite[Chapter 3, Equation 44]{Sch}\n\\begin{align*} \n\\langle \\overline{\\Psi}_{\\mathbf{x}}, \\Phi\\rangle=\\int d^n\\mu(\\mathbf{p})\\, \\Psi_{\\mathbf{x},0}(\\mathbf{p}) \\Phi(\\mathbf{p}).\n\\end{align*} \nIn order to extend this quantity to $k$-particles to calculate the probability amplitude for finding $k$-particle at positions $\\mathbf{x}_{1} \\cdots \\mathbf{x}_{k}$ at time $x_{0}={x}_{10}= \\cdots={x}_{k0}$, the following operator is introduced, \\cite[Chapter 7, Equation 99]{Sch},\n \\begin{align}\\label{copes}\n \\phi_1( {x}) =\\int d^n\\mu(\\mathbf{p}) \\overline{\\Psi_{\\mathbf{x},x_{0}}(\\mathbf{p})} a_c(\\mathbf{p}) =(2\\pi)^{-n\/2}\\,\n \\int d^n \\mathbf{p} \\, e^{-i {p} \\cdot {x}}\\, a (\\mathbf{p}),\n \\end{align}\n and we apply $k$-times $\\phi_1$ on $\\vert\\Phi^1 \\rangle$ and the vacuum as follows,\n \\begin{align*} \n \\Phi^1( x_1 ,\\cdots, x_k ) = (k!)^{-1\/2} \\langle 0\\vert \\phi_1(x_{1})\\cdots\\phi_1(x_{k})\n \\vert\\Phi^1 \\rangle.\n \\end{align*}\n It is important to point out that the operator $\\phi_1(x)$ at time $x^{0}=0$ is nothing else than the Fourier-transformed annihilation operator, see Equation (\\ref{inf}). Moreover, \tthe Newton-Wigner-Pryce operator has the following coordinate space representation (for proof see \\cite[Lemma 3.1]{Muc7}),\n \\begin{align*}\n X_j&= \\int d^n \\mathbf{x}\\, x_j \\, \\tilde{a}^*(\\textbf{x}) \\tilde{a}(\\textbf{x})\n .\n \\end{align*} \n\\subsection{Constantly Used Integrals}\nTo make this work self-contained, we give in this section a general formula for certain Fourier-transformed functions. In particular, these formulas can be found in \\cite[Chapter III, Section 2.6-2.8]{GS1},\n\\begin{align}\\label{f1}\n\\tilde{P}^{\\lambda}&= \\int\\,d^n\\mathbf{p}\\, \\vert\\vec{p}\\vert^{2\\lambda} \\exp({-i\\,p_k z^k})\\\\\\nonumber\n&= 2^{2\\lambda+n}\\pi^{\\frac{1}{2}n}\\,\\frac{\\Gamma(\\lambda+\\frac{1}{2}n)}{\\Gamma(-\\lambda)}\\left(z_1^2+\\cdots +z_n^2\\right)^{-\\lambda-\\frac{1}{2}n},\n\\end{align}\nwhere $\\Gamma$ denotes the Gamma-function.\n \n\\section{Conformal Group in the NWP-space}\nIn this section we change the basis of the conformal group, which is given in terms of Fock-space operators in the momentum space, to the coordinate space. The base changes are mathematically well-defined on the domain $\\bigotimes_{i=1}^k \\mathscr{S}(\\mathbb{R}^n)$. This conclusion follows from the fact that we use the Fourier-transformation for the base change and this transformation acts as a linear automorphism on the Schwartz-space. Hence, we conclude that the base change is well-defined.\n\\\\\\\\However, an exception to the rule is given by the Lorentz generators generating boosts. They do not map the Schwartz-space to itself since the coefficient functions of the differential operators are\nnot differentiable at $p=0$. Hence, if one intends to calculate the adjoint action w.r.t. the Lorentz-boosts one has to choose the space of analytic functions on $S^2 \\times \\R$, more specifically of analytic sections, which is the light-cone without the tip, (see \\cite[ Chapter 8.6]{ND}).\\footnote[1]{I am indebted to Prof. N. Dragon for this remark.}\n\\\\\\\\\nFrom a quantum field theoretical point of view, the conformal group is of great interest. \nIn particular, the group is defined as the set of all conformal transformations. The definition of a conformal transformation is an \ninvertible mapping $\\mathbf{x}'\\rightarrow \\mathbf{x}$, leaving a\n$d$-dimensional metric $g$ invariant. The invariance holds, modulo a scale factor, \\cite{DMS}:\n\\begin{equation} \\label{cond1}\ng'_{\\mu\\nu}(x')=F(x)g_{\\mu\\nu}(x).\n\\end{equation}\nConformal mappings can be essentially summarized as Lorentz \ntransformations, translations, dilations and special \nconformal transformations. The generators of these transformations are given by the \noperators $M_{\\mu\\nu}$, $P_{\\rho}$, $D$, $K_{\\sigma}$.\n\\\\\\\\\nThe conformal algebra is defined by the commutation relations of the \ngenerators and is given as follows:\n\\begin{equation}\n[M_{\\mu\\nu},M_{\\rho\\sigma}]\n=i\\left(\\eta_{\\mu\\sigma}M_{\\nu\\rho}+\\eta_{\\nu\\rho}M_{\\mu\\sigma}-\\eta_{\\mu\\rho}M_{\\nu\\sigma}-\n\\eta_{\\nu\\sigma}M_{\\mu\\rho},\n\\right)\n\\end{equation}\n\\begin{equation}\n[P_{\\rho},M_{\\mu\\nu}]=i\\left(\\eta_{\\rho\\mu}P_{\\nu}-\\eta_{\\rho\\nu}P_{\\mu}\\right),\n\\qquad \n[K_{\\rho},M_{\\mu\\nu}]=i\\left(\\eta_{\\rho\\mu}K_{\\nu}-\\eta_{\\rho\\nu}K_{\\mu}\\right),\n\\end{equation}\n\\begin{equation}\\label{f11}\n[P_{\\rho},D]=iP_{\\rho}, \\qquad [K_{\\rho},D]=-iK_{\\rho},\n\\end{equation}\n\\begin{equation}\\label{f12}\n[P_{\\rho},K_{\\mu}]= 2i\\left(\\eta_{\\rho\\mu}D-M_{\\rho\\mu}\\right),\n\\end{equation}\nwhere all other commutators are equal to zero.\n\\\\\\\\ \nTo transform the momentum and Lorentz operators in to the NWP-space we can take the massive expressions (see \\cite{Muc7}) and perform a massless limit, i.e. $m\\rightarrow0$. However, in order to make this work self-contained we calculate most expressions explicitly. \n\\begin{lemma}\n\tThe zero component in the massless case is given by,\n\t\\begin{align*} P_{0}\n\t= \n\t\\int\n\td^n \\mathbf{x}\\,\n\t\\tilde{a}^*(\\mathbf{x}) \\left(\\tilde{\\omega} \\ast\n\t\\tilde{a} \\right)(\\mathbf{x} ),\n\t\\end{align*} \n\twith function $\\tilde{\\omega}(x):=-\\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2}) |\\mathbf{x} |^{- ( n+1)}$ and $\\ast$ denoting the convolution.\\\\\\\\\n\tThe massless spatial momentum operator takes the same form in coordinate space as in the massive case, \n\t\\begin{align*} \n\tP_j&=\n\t- i\n\t\\int\n\td^n \\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x}) \n\t\\frac{\\partial}{\\partial x^{j}}\\tilde{a} (\\mathbf{x} ).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tFor the proof we take the expression of the massless energy operator (see Equation (\\ref{mop})) and transform, by explicit Fourier-transformation, into the coordinate space,\n\t\\begin{align*} \n\tP_{0}&= \\int\n\td^n \\mathbf{p}\\, \\omega_{\\mathbf{p}} \\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p})\\\\&=(2\\pi)^{-n} \n\t\\int\n\td^n \\mathbf{p}\\, |\\mathbf{p}| \\int\n\td^n \\mathbf{x}\\, e^{ip_kx^k} \\tilde{a}^*(\\mathbf{x}) \\int\n\td^n \\mathbf{y} \\,e^{-ip_ly^l} \\tilde{a} (\\mathbf{y} ) \\\\&=(2\\pi)^{-n} \n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, \\left( \\int\n\td^n \\mathbf{p}\\, {|\\mathbf{p}| } \\, e^{ip_k(x-y)^k} \\right)\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} )\t \\\\ &= -\\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2})\n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, \n\t\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ {n+1}} } \n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} ),\n\t\\end{align*}\n\twhere the Fourier-transformation can be found in \\cite[Chapter III, Section 2.6]{GS1} or see Equation (\\ref{f1}). The spatial momentum operator does not depend explicitly on the mass term, hence its representation in the massless is equivalent to the massive case, (see \\cite{Muc7}).\n\\end{proof}\nOther important expressions in the NWP-context are the velocity and the particle number operator. One way to calculate the velocity operator is by using the Heisenberg equation of motion, \n\\begin{align}\\label{heq}\n[P_{0},X_{j}]=-iV_j.\n\\end{align}\nSince, we represented the relativistic energy and the NWP-operator in coordinate space, the calculation of the commutator gives us the velocity directly in coordinate space. In order to represent the particle number operator in the NWP-space we apply a Fourier-transformation on the expression in momentum space (see Equation (\\ref{pn})).\n\\begin{lemma}\n\tThe velocity operator is given, in the massless case, as follows\n\t\\begin{align}\n\tV_j= -i\\int d^n \\mathbf{x} \\, \n\t\\tilde{a}^*(\\mathbf{x}) \\left(\\tilde{\\omega} _j\\ast\n\t\\tilde{a}\\right) (\\mathbf{x} ),\n\t\\end{align}\n\twith function $\\tilde{\\omega} _j(\\mathbf{x}):=- \\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2}) \\, |\\mathbf{x} |^{- ( n+1)}\\,x_j$ and $\\ast$ denoting the convolution. \\\\\\\\ Moreover, the particle number operator $N$ is in the coordinate space given by,\n\t\\begin{align}\n\tN= \\int d^n \\mathbf{x} \\, \n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{x} ).\n\t\\end{align}\n\t\n\n\\end{lemma}\n\\begin{proof}\nWe calculate the velocity by taking the commutator of the spatial coordinate space and the zero component of the momentum (see Heisenberg-Equation (\\ref{heq})), which is given in the former theorem, i.e. \n\t\\begin{align*}\n\t[P_0,X_j]&=\\iint d^n \\mathbf{x} \\, d^n \\mathbf{z} \\, z_j\\, [\n\t\\tilde{a}^*(\\mathbf{x}) \\left(\\tilde{\\omega}\\ast\n\t\\tilde{a}\\right) (\\mathbf{x} ),\\tilde{a}^*(\\mathbf{z}) \\tilde{a} (\\mathbf{z}) ]\\\\&=\n\t\\iiint d^n \\mathbf{x} \\, d^n \\mathbf{y}\\,d^n \\mathbf{z}\\,z_j \\, \\tilde{\\omega}(\\mathbf{x}-\\mathbf{y})\\underbrace{[\n\t\t\\tilde{a}^*(\\mathbf{x}) \n\t\t\\tilde{a} (\\mathbf{y} ),\\tilde{a}^*(\\mathbf{z}) \\tilde{a} (\\mathbf{z}) ]}_{-\\delta(\\mathbf{x}-\\mathbf{z}) \\tilde{a}^*(\\mathbf{z})\\tilde{a} (\\mathbf{y})+\\delta(\\mathbf{y}-\\mathbf{z}) \\tilde{a}^*(\\mathbf{x})\\tilde{a} (\\mathbf{z})}\\\\&=\n\t-\\iint d^n \\mathbf{x} \\, d^n \\mathbf{y}\\, \\,(x-y)_j \\, \\tilde{\\omega}(\\mathbf{x}-\\mathbf{y}) \\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y}).\n\t\\end{align*}\n\tFor the particle number operator we use the Fourier-transformation as in the proof of the former lemma,\n\t\t\\begin{align*} \n\tN&= \\int\n\t\td^n \\mathbf{p}\\, {a}^*(\\mathbf{p}) {a}(\\mathbf{p})\\\\&=(2\\pi)^{-n} \n\t\t\\int\n\t\td^n \\mathbf{p}\\, \\int\n\t\td^n \\mathbf{x}\\, e^{ip_kx^k} \\tilde{a}^*(\\mathbf{x}) \\int\n\t\td^n \\mathbf{y} \\,e^{-ip_ly^l} \\tilde{a} (\\mathbf{y} ) \\\\&=(2\\pi)^{-n} \n\t\t\\iint\n\t\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\,\\underbrace{ \\left( \\int\n\t\td^n \\mathbf{p}\\, e^{ip_k(x-y)^k} \\right)}_{(2\\pi)^{n}\\delta(\\mathbf{x}-\\mathbf{y}) }\\tilde{a}^*(\\mathbf{x}) \n\t\t\\tilde{a} (\\mathbf{y} ) .\n\t\\end{align*}\n\\end{proof}\nNext, we turn our attention to the expressions of Lorentz generators. They are given as in the massive case by (see \\cite[Equation\n3.54]{IZ} and in this context see also \\cite[Appendix]{SS}),\n\\begin{align}\\label{lbcaop1}\nM_{j0}&= i\\int d^n\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \n\\left( \\frac{p_j}{2\\omega_{\\textbf{p}}}-\\omega_{\\textbf{p}}\\frac{\\partial}{\\partial p^j } \\right) a(\\textbf{p}),\n\\\\ \\label{lbcaop2}\nM_{ik}&=i \\int d^n\\mathbf{p}\\, {a}^{*}(\\textbf{p}) \n\\left(p_i \\frac{\\partial}{\\partial p^k }-p_k\\frac{\\partial}{\\partial p^i }\\right)\na(\\textbf{p}).\n\\end{align}\n\tIt is interesting and important to note that operators of boost and rotations can be given by the second quantization (denoted by $d\\Gamma(\\cdot)$ see \\cite[Chapter X.7]{RS2}) of symmetric or skew-symmetric products of the momentum and coordinate operator.\n\\begin{theorem}For the massless scalar field the generators of the proper orthochronous Lorentz-group $\\mathscr{L}^{\\uparrow}_{+}$ can be written in terms of the second quantization of products of the NWP-operator with relativistic four-momentum as,\n\t\\begin{align}\\label{p1}\n\tM_{0j}=\\frac{1}{2}\\big( d\\Gamma(X_jP_0)+d\\Gamma(P_0X_j)\\big) ,\n\\qquad \n\t\tM_{ik} = d\\Gamma(X_iP_k)-d\\Gamma(X_{k}P_i).\n\t\t\\end{align}\n\t\\end{theorem}\n\t\\begin{proof}\n\tFor the boosts we have,\n\t\\begin{align*}\n\tM_{0j}&=\\frac{1}{2}\\big( d\\Gamma(X_jP_0)+d\\Gamma(P_0X_j)\\big) =\\frac{1}{2} d\\Gamma([X_j,P_0])+d\\Gamma(P_0X_j) \n\t \\\\ &=\\frac{i}{2} d\\Gamma(V_j)+d\\Gamma(P_0X_j) ,\n\t\\end{align*}\n\twhere in the last lines we used the Heisenberg equation of motion. By comparing the last line with Formula (\\ref{lbcaop1}) the proof follows. \t For rotations, the representation follows trivially. \n\t\\end{proof}\n\tThe interesting fact about the former Theorem is that in principle one can define the generators of the Lorentz-group in a $k$-particle space by solely using the translation group and the NWP-operator. This resembles the proof of the Maxwell-equations by using the canonical commutation relations of Feynman (published by Dyson, \\cite{FD}). In particular, the existence of the NWP-operator and the translation group are sufficient to build the Poincar\\'e group, i.e. the group responsible for the implications of special relativity.\n\tHence, the commutation relations with the addition of relativistic energy can be used to implement relativistic principles. This fact, is, in our opinion, an additional argument for the physical sense of the NWP-operator. \\\\\\\\\nNext, we turn our attention to the explicit expressions of Lorentz generators in the coordinate space.\n\\begin{lemma}\n\tThe boost generators of the Lorentz group expressed in the terms of ladder operators of the free massless scalar field are given in the coordinate space as\n\t\\begin{align*}\n\tM_{0j}=\\frac{1}{2}\t\\iint d^n \\mathbf{x} \\, d^n \\mathbf{y}\\,(x+y)_j\\, \\tilde{\\omega}(\\mathbf{x}-\\mathbf{y}) \\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y}) .\n\t\\end{align*}\n\tThe operator of rotations takes the, from a quantum mechanical point of view, familiar and to the massive case equivalent form, \n\t\\begin{align*}\n\tM_{ik}=\t i \\int\n\td^n \\mathbf{x} \\,\n\t\\tilde{a}^*(\\mathbf{x})\n\t\\left(x_i \\frac{\\partial}{\\partial x^k }-x_k\\frac{\\partial}{\\partial x^i }\\right) \\tilde{a} (\\mathbf{x}).\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}Since in the proof of \\cite[Lemma 4.3]{Muc7} we did not explicitly encounter the mass when we calculated the representation of rotations in the coordinate space, we conclude that they have the same form.\tNext, we turn to the proof of the representation of Lorentz boosts in coordinate space. The first term is simply the massless velocity operator times $\\frac{i}{2}$ (see Equation (\\ref{p1})). Hence, we focus here only on the second part,\n\t\\begin{align*}\n\t&- i\\int d^n\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\, \\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^j} a(\\textbf{p})\\\\&=\n\t- i\\int d^n\\mathbf{p}\\,\\int\n\td^n \\mathbf{x}\\, e^{-ip_rx^r} \\,\\tilde{a}^*(\\mathbf{x}) \\, \\omega_{\\textbf{p}}\\frac{\\partial}{\\partial p^j} \\int\n\td^n \\mathbf{y}\\, e^{ip_sy^s} \\tilde{a}(\\mathbf{y}) \n\t\\\\&= (2\\pi)^{-n}\\iint\td^n \\mathbf{x}\\,\td^n \\mathbf{y}\n\t\\left( \\int d^n\\mathbf{p}\\, |\\mathbf{p}| \\, e^{-ip_k(x-y)^k} \\right)\\,y_j\\,\\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y}) \n\t\\\\&=-\\pi^{-\\frac{n+1}{2}}\\Gamma(\\frac{n+1}{2})\n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, y_j\\, \n\t\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ {n+1}} } \n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} )\n\t\\\\&= \n\t\\iint\n\td^n \\mathbf{x}\\,d^n \\mathbf{y}\\, y_j\\,\\tilde{\\omega}(\\mathbf{x}-\\mathbf{y})\n\t\\tilde{a}^*(\\mathbf{x}) \n\t\\tilde{a} (\\mathbf{y} )\n\t,\n\t\\end{align*} \n\\end{proof}\nIn order to translate the special conformal and the dilatation operator into the coordinate space, we first write the momentum space representation, see \\cite{SS}. Moreover, we restrict this part to the physical space-time dimension four. The dilatation operator is given in momentum space as \n\\begin{align}\nD=-i\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{2}+p^{l}\\frac{\\partial}{\\partial p^{l}}\\right)\n{a}(\\textbf{p}),\n\\end{align}\nand for the special conformal operators we have,\n\\begin{align}\nK_{0}&=-\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{4 \\,\\omega_{\\mathbf{p}}}+\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}-\\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\na(\\textbf{p}),\\\\ \nK_{j}&= -\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\n\\frac{p_j}{4\\,\\omega_{\\mathbf{p}}^2}+3 \\frac{\\partial}{\\partial p^{j}}+2p^l\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p^{j}}-p_{j}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\na(\\textbf{p}).\n\\end{align}\n\\begin{lemma}\n\tThe four-dimensional dilatation operator is given in the coordinate representation as follows,\n\t\\begin{align}\n\tD=i\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\left(\\frac{3}{2}+x^{l}\\frac{\\partial}{\\partial x^{l}}\n\t\\right)\\tilde{a}(\\mathbf{x}).\n\t\\end{align}\n\\end{lemma}\n\\begin{proof}\n\tThe Fourier-transformation is straight-forward in this case, e.g.\n\t\\begin{align*}\n\tD&=-i\t\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{2}+p^{l}\\frac{\\partial}{\\partial p^{l}}\\right)\n\t{a}(\\textbf{p})\n\t\\\\&=-i(2\\pi)^{-3} \n\t\\int\n\td^3 \\mathbf{p}\\, \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x}) \\left(\\frac{3}{2}+p^{l}\\frac{\\partial}{\\partial p^{l}}\\right)\\int\n\td^3 \\mathbf{y} \\,e^{ ip_ky^k} \\tilde{a} (\\mathbf{y} ) \t\\\\&=-\n\t\\frac{3i}{2}\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\tilde{a}(\\mathbf{x})\n\t-i(2\\pi)^{-3} \n\t\\iint\n\td^3 \\mathbf{p}\\, \n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x}) \\int\n\td^3 \\mathbf{y} \\,\\left(y^{l}\\frac{\\partial}{\\partial y^{l}}e^{ ip_ky^k}\\right) \\tilde{a} (\\mathbf{y} ) \\\\&=\n\t \t\\frac{3i}{2}\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\tilde{a}(\\mathbf{x})\n\t+i(2\\pi)^{-3} \n\t\\iint\n\td^3 \\mathbf{p}\\, \n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x}) \\int\n\td^3 \\mathbf{y} \\,e^{ ip_ky^k} y^{l}\\frac{\\partial}{\\partial y^{l}}\\tilde{a} (\\mathbf{y} ) \\\\&=\n\ti\t\\int d^3\\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\\left(\\frac{3}{2}+ x^{l}\\frac{\\partial}{\\partial x^{l}}\\right)\\tilde{a}(\\mathbf{x}) .\n\t\\end{align*}\n\\end{proof}\nThe outcome of the dilatation operator is interesting. In essence, it is equivalent to its momentum representation modulo a sign. This is what one might expect, since this operator performs dilatations in a complimentary manner, with regard to the momentum and coordinate space. \\\\\\\\\nNext, we transform the special conformal operators into the coordinate space. From a calculational point of view these operators are the most difficult ones. \n\\begin{lemma}\n\tThe zero component of the special conformal operators has the following form in coordinate space \n\t\\begin{align*}\n\tK_{0}&= \\pi^{-2}\t\\iint d^3 \\mathbf{x}\\,d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x}) \\,\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2} \\left( \\frac{9}{8}-\n\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2}\\vert\\mathbf{y}\\vert^2\n\t+ \\frac{y^{l}}{2} \\frac{\\partial}{\\partial y^{l}}\n\t\\right) \\tilde{a} (\\mathbf{y} ).\n\t\\end{align*}\n\t\n\t\n\\end{lemma}\n\n\n\n\\begin{proof}\n\tIn order to make this calculation more readable we separate the zero component into three parts as follows,\n\t\\begin{align*}\n\tK_{0}&=-\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\\frac{3}{4 \\,\\omega_{\\mathbf{p}}}+\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}-\\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\n\ta(\\textbf{p})\\\\&=:\n\tK_{0}^1+K_{0}^2+K_{0}^3.\n\t\\end{align*}\n\tLet us take a look at the first part,\n\t\\begin{align*}\n\tK^1_{0}&=-\\frac{3}{4}\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\,\\omega_{\\mathbf{p}}^{-1}\n\ta(\\textbf{p})\n\t\\\\&=-\\frac{3}{4}(2\\pi)^{-3} \n\t\\iint d^3 \\mathbf{x}\\,d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x})\n\t\\left( \\int d^3 \\mathbf{p} \n\t\\, e^{-ip_k(x-y)^k } \\,\\omega_{\\mathbf{p}}^{-1}\\right)\n\t\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= -\\frac{3}{8} \\pi^{-2}\t\\iint d^3 \\mathbf{x}\\,d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x}) \\,\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2} \\tilde{a} (\\mathbf{y} ) .\n\t\\end{align*}\n\tNext we turn our attention to the second term of the zero component of the special conformal operator, \n\t\\begin{align*}\n\tK^2_{0}&=-\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\,\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}\n\ta(\\textbf{p})\n\t\\\\&=- (2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(\\frac{p^{l}}{\\omega_{\\mathbf{p}}}\\frac{\\partial}{\\partial p^{l}}e^{ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= - (2\\pi)^{-3} \\int\n\td^3 \\mathbf{x}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\int d^3\\mathbf{p}\\,\\frac{1}{\\omega_{\\mathbf{p}}} e^{-ip_rx^r}\n\t\\int d^3 \\mathbf{y} \\,\\left( y^{l}\\frac{\\partial}{\\partial y^{l}}e^{ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= (2\\pi)^{-3} \\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(\\int d^3\\mathbf{p}\\,\\frac{1}{\\omega_{\\mathbf{p}}} e^{-ip_k(x-y)^k}\\right)\n\t\\, \\left(3+y^{l} \\frac{\\partial}{\\partial y^{l}}\\right)\\tilde{a} (\\mathbf{y} )\n\t\\\\&= (2)^{-1} \\pi^{-2} \\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\left( \\vert \\mathbf{x}-\\mathbf{y}\\vert^{-2} \\right)\n\t\\, \\left(3+y^{l} \\frac{\\partial}{\\partial y^{l}}\\right)\\tilde{a} (\\mathbf{y} ),\n\t\\end{align*}\n\twhere in the last lines we performed a partial integration and used Formula (\\ref{f1}). The last term of the zero component is given by\n\t\\begin{align*}\n\tK^3_{0}&=\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\, \\omega_{\\mathbf{p}}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\n\ta(\\textbf{p})\\\\&\n\t=(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left( \n\t\\omega_{\\mathbf{p}}\\,\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\n\te^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \n\t\\\\&= (2\\pi)^{-3}\\iint\n\td^3 \\mathbf{x}\\,\t d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\\left(\n\t\\int d^3\\mathbf{p}\\, {\\omega_{\\mathbf{p}}}\\, e^{-ip_k(x-y)^k}\\right) \\,\\vert\\mathbf{y}\\vert^2 \\tilde{a} (\\mathbf{y} ) \n\t\\\\&= -( \\pi)^{-2}\\iint\n\td^3 \\mathbf{x}\\, d^3 \\mathbf{y} \\,\\tilde{a}^*(\\mathbf{x})\\left(\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-4} \n\t\\right)\t\\vert\\mathbf{y}\\vert^2 \\tilde{a} (\\mathbf{y} ) , \n\t\\end{align*}\n\twhere in the last lines we used the action of the differential operators w.r.t. $\\mathbf{p}$ on the exponential function and used as before Equation (\\ref{f1}).\n\t\n\\end{proof}\nNext, we give the base change of the spatial part of the special conformal operator. \n\\begin{lemma}\n\tThe spatial components of the special conformal operators have the following form in coordinate space,\n\t\\begin{align}\n\tK_{j} = -\\frac{i}{16}\\pi^{-1}\\iint&\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x}) \n\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-1} \n\t\\, \\frac{\\partial}{\\partial y^{j}}\\tilde{a} (\\mathbf{y} )\\\\&\\nonumber+\n\ti \\int\n\td^3 \\mathbf{x} \\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(3x_j+ 2x_{j}x^{l} \\frac{\\partial}{\\partial x^{l}}- x_{l}x^{l} \\frac{\\partial}{\\partial x^{j}} \\right) \\tilde{a} (\\mathbf{x} )\n\t\\end{align}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\tLet us first recall and define the expression of the special conformal operators in the momentum space, i.e. \n\t\\begin{align*}\n\tK_{j}&= -\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\n\t\\frac{p_j}{4\\,\\omega_{\\mathbf{p}}^2}+3 \\frac{\\partial}{\\partial p^{j}}+2p_l\\frac{\\partial}{\\partial p_{l}}\\frac{\\partial}{\\partial p^{j}}-p_{j}\\frac{\\partial}{\\partial p^{l}}\\frac{\\partial}{\\partial p_{l}}\\right)\n\ta(\\textbf{p})\\\\&=\n\tK^{1}_{j}-3i\tX_{j}+\tK^{2}_{j}+\tK^{3}_{j}.\n\t\\end{align*}\n\tThe second term in the spatial conformal operator is simply the coordinate operator times a constant. This fact was used in \\cite{SE}. Hence, the remaining terms of interest are $K^{1}_{j},\\,K^{2}_{j},\\,K^{3}_{j}$. Let us start with first object, \n\t\\begin{align*}\n\tK^{1}_{j}&= -\n\t\\frac{1}{4}\n\t\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(\n\t\\frac{p_j}{\\,\\omega_{\\mathbf{p}}^2} \\right)\n\ta(\\textbf{p})\\\\&= -\n\t\\frac{1}{4}\n\t(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_rx^r} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(\t\\frac{p_j}{\\,\\omega_{\\mathbf{p}}^2} e^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} ) \\\\&= -\n\t\\frac{i}{4}\n\t(2\\pi)^{-3} \\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(\\int d^3\\mathbf{p}\\,\\frac{1}{\\omega_{\\mathbf{p}}^2} e^{-ip_k(x-y)^k}\\right)\n\t\\, \\frac{\\partial}{\\partial y^{j}}\\tilde{a} (\\mathbf{y} )\n\t\\\\&= - \\frac{i}{16}\\pi^{-1}\\iint\n\td^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\,\\tilde{a}^*(\\mathbf{x}) \n\t\\vert \\mathbf{x}-\\mathbf{y}\\vert^{-1} \n\t\\, \\frac{\\partial}{\\partial y^{j}}\\tilde{a} (\\mathbf{y} ),\n\t\\end{align*}\n\twhere in the last lines we expressed the momentum as a derivative, performed a partial integration and used the Fourier-transformation given in Formula (\\ref{f1}) for $\\lambda=-1$. A general term which includes, after taking the corresponding indices, the second and third operator of the spatial special conformal object is given by,\n\t\\begin{align*}\n\tO_{j}&= \n\t\\int d^3\\mathbf{p}\\,{a}^{*}(\\textbf{p}) \\left(p_{r}\\frac{\\partial}{\\partial p_{l}}\\frac{\\partial}{\\partial p^{j}}\\right)\n\ta(\\textbf{p})\\\\&= \n\t(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_sx^s} \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(\tp_r\\frac{\\partial}{\\partial p_{l}}\\frac{\\partial}{\\partial p^{j}}e^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} )\n\t\\\\&= i\n\t(2\\pi)^{-3}\\int d^3\\mathbf{p} \\int\n\td^3 \\mathbf{x}\\, e^{-ip_sx^s } \\tilde{a}^*(\\mathbf{x})\n\t\\int\n\td^3 \\mathbf{y} \\,\\left(y_{j}y^{l} \\frac{\\partial}{\\partial y^{r}} e^{ ip_ky^k} \\right)\\tilde{a} (\\mathbf{y} )\\\\&= - i \\int\n\td^3 \\mathbf{x} \\,\\tilde{a}^*(\\mathbf{x})\n\t\\left(\\eta_{rj}x^l+\\eta_{r}^{\\,\\,l}x_{j}+ x_{j}x^{l} \\frac{\\partial}{\\partial x^ {r}} \\right) \\tilde{a} (\\mathbf{x} ),\n\t\\end{align*}\n\twhere in the last lines we applied the derivatives to the exponential, expressed the coordinate $y$\tas a derivative and performed a partial integration. \n\\end{proof}\n \\begin{remark}\n \tAs for Lorentz generators, dilatation and special conformal operators can be written in terms of symmetric second-quantized products of the momentum and NWP-operator. For the dilatation operator it is straight forward,\n \t\\begin{align*}\n \tD=\\frac{1}{2}\\big(d\\Gamma(P_jX^j) +d\\Gamma(X^jP_j)\\big).\n \t\\end{align*}\n \twhile the special-conformal operators are a bit more involved. For the zero component we have, \n \t\t\\begin{align*}\n \t\tK_0&=-\\frac{3}{4}d\\Gamma(P_0^{-1})-id\\Gamma(V^{l}X_l)+d\\Gamma(P_0X_lX^l) \t\\end{align*}\t\nwhile the spatial components in terms of the NWP and the momentum operator read, \n\\begin{align*}\n \t\tK_j&=-\\frac{1}{4}d\\Gamma(P_0^{-1}V_j)-3id\\Gamma(X_j)+2d\\Gamma(P^lX_lX_j)-d\\Gamma(P_jX_lX^l) .\n \t\t\\end{align*}\n \t\t \n \\end{remark}\n \n In this section we expressed all generators of the conformal group in terms of the ladder operators of the NWP space for $x_0=0$. However, the operator $\\phi_1$ (see Equation (\\ref{copes})) responsible for calculating the probability of finding $k$-particle at a certain position has an explicit time dependence. Hence, in order to connect our results with this time-dependency we give the following proposition.\n \\begin{proposition}\\label{sf}\nThe operator $\\phi_1$ that has the following form,\n \t\\begin{align*} \n \t\\phi_1( {x}) =(2\\pi)^{-n\/2}\\,\n \t\\int d^n \\mathbf{p}\\, e^{-i {p} \\cdot {x}}\\,a(\\mathbf{p}),\n \t\\end{align*}\n can be obtained by performing a time-translation on the annihilation operator $\\tilde{a}$ of the NWP-space\n \tas follows\n \t\\begin{align*} \n \t\\phi_1( x^{0}, \\mathbf{x}) = e^{ix^{0}P_{0}} \\tilde{a}(\t \\mathbf{x})e^{-ix^{0}P_{0}}, \\qquad x^0\\in\\mathbb{R}.\n \t\\end{align*}\n \t\n \\end{proposition}\t\\begin{proof}\n By using the inverse Fourier-transformation of $\\tilde{a}$ and the transformation of the annihilation operator in momentum space under time-translation, the proof is completed, i.e. \n \\begin{align*} \n e^{ix^{0}P_{0}} \\tilde{a}(\t \\mathbf{x})e^{-ix^{0}P_{0}}&=(2\\pi)^{-n\/2} \\int\n d^n \\mathbf{p}\\, e^{-ip_kx^k}e^{ix^{0}P_{0}}{a}(\\mathbf{p})e^{-iy^{0}P_{0}}\\\\&=(2\\pi)^{-n\/2} \\int\n d^n \\mathbf{p}\\, e^{-ip_kx^k}\\,e^{-ip_{0}x^{0} }{a}(\\mathbf{p}) .\n \\end{align*}\n\\end{proof}Hence, we can simply time-translate all the operators of the conformal group in the NWP-space, by using the same time $x^0$, in order to obtain the expressions for a general time and in terms of the operator $\\phi_1$. Since the zero component of the momentum commutes with the spatial momentum and the rotations, we only have to calculate the time-translation for the remaining generators by using the algebra.\n\n\n\n\\begin{lemma}\n\tThe dilatation transforms under time-translations as follows,\n\t\\begin{align*}\n\te^{ix^{0}P_{0}}D e^{-ix^{0}P_{0}} \t= D-x^{0}P_{0}, \n\t\\end{align*}\n\twhere $x^0\\in\\mathbb{R}$. Under time-translations the special conformal operator transforms as,\n\t\\begin{align*}\n\te^{ix^{0}P_{0}}K_{\\mu} e^{-ix^{0}P_{0}}&=\n\tK_{\\mu}-2x^{0}\\left(\n\t\\eta_{0\\mu}D-M_{0\\mu}\n\t\\right)-(x^{0})^2\n\t\\left( P_{\\mu}\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tThe proof solely uses the conformal algebra (see Formulas (\\ref{f11}) and (\\ref{f12})). For the dilatations we have,\n\t\\begin{align*}\n\te^{ix^{0}P_{0}}D e^{-ix^{0}P_{0}} =\n\tD-x^{0}P_{0} +\\frac{i^2}{2!}(x^{0})^2\n\t\\underbrace{[P_{0},[P_{0},D]]+ \\cdots}_{=0}= D-x^{0}P_{0},\n\t\\end{align*}\n\twhere in the last lines we used the Backer-Hausdorff formula.\tA more complex expression is the transformation of the special-conformal operator, i.e. \n\t\\begin{align*}\n\te^{ix^{0}P_{0}}K_{\\mu} e^{-ix^{0}P_{0}}&=K_{\\mu}+ix^{0}[P_{0},K_{\\mu}]+\\frac{i^2}{2!}(x^{0})^2\n\t[P_{0},[P_{0},K_{\\mu}]]+\\underbrace{\\cdots}_{=0}\n\t\\\\&= K_{\\mu}-2x^{0}\\left(\n\t\\eta_{0\\mu}D-M_{0\\mu}\n\t\\right)-(x^{0})^2\n\t\\left( \\eta_{00}P_{\\mu}\\right) ,\n\t\\end{align*}\n\twhere we used the Baker-Campbell-Hausdorff formula and the specific commutator relations of the conformal algebra.\n\t\n\\end{proof}\nSince, we have all expressions in terms of generators of the NWP-space we can equivalently give the conformal algebra expressed by operators $\\phi^*_1(x_0,\\mathbf{x})$ and $\\phi_1(x_0,\\mathbf{y})$. Note that neither the coordinate operator nor its respective eigenstates are covariant w.r.t. general Lorentz-transformations. Nevertheless, the generators of the conformal group obey the covariant transformation property independent of their representation. \tHence, even though we use non-covariant eigenstates, operators of the conformal group represented in the NWP-space respect the Poincar\\'e symmetry. \n\\section{Conformal-Transformations of NWP-States} \nIn this section we calculate the explicit adjoint action of the conformal group on the Fourier-transformed ladder operators. In some cases this is done by taking the massless limit of the massive theory. \\\\\\\\\n\t The main physical motivation to calculate the adjoint action w.r.t. conformal transformations, in the coordinate space, is given by the fact that we use operator $\\phi_1(x)$ to calculate the probability of localizing $k$-particles at $k$-spatial positions. Therefore, we intend to investigate under which subgroup of the conformal group the probability amplitudes of localization are covariant. \n\t\\\\\\\\ In order find the covariance group, we first define the unitary operator generating transformations of the orthochronous proper Poincar\\'e group $\\mathscr{P}^{\\uparrow}_{+}=\\mathscr{L}^{\\uparrow}_{+}\\ltimes\\mathbb{R}^4$ denoted by $U(y,\\Lambda)$. It transforms creation and annihilation operators in the following fashion, \\cite[Chapter 7]{Sch},\n\t\\begin{align}\\label{traf}\n\tU(y,\\Lambda) a(\\mathbf{p})U(y,\\Lambda)^{-1}&= \\sqrt{\\frac{\t\\omega_{\\Lambda \\mathbf{p}}}{\t\\omega_{ \\mathbf{p}}}}e^{-i(\\Lambda p)_{\\mu}y^{\\mu}}{a}(\\Lambda\\mathbf{p}) ,\\qquad (y,\\Lambda) \\in \\mathscr{P}^{\\uparrow}_{+},\\\\\\label{traf1}\n\tU(y,\\Lambda) a^{*}(\\mathbf{p})U(y,\\Lambda)^{-1}&= \\sqrt{\\frac{\t\\omega_{\\Lambda \\mathbf{p}}}{\t\\omega_{ \\mathbf{p}}}}e^{ i(\\Lambda p)_{\\mu}y^{\\mu}}{a}^{*}(\\Lambda\\mathbf{p}) ,\\qquad (y,\\Lambda) \\in \\mathscr{P}^{\\uparrow}_{+}\n\t.\n\t\\end{align} \n\tBy using the former equations we first calculate the adjoint action of translations on the NWP-ladder operators.\n\t\\begin{lemma}\\label{l63}\n\t\tThe Fourier-transformed ladder operator $\\tilde{a}$ of the massless field transforms under translations as follows, \n\t\t\\begin{align}\\label{eq51}\n\t\tU({y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U({y},\\mathbb{I}) ^{-1} = c_{y } \\left(\n\t\t\\frac{\t 1}{\\vert \\mathbf{x+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a} (\\mathbf{x+y}),\n\t\t\\end{align} \n\t\twhere $c_{y }= \\frac{i y^{0}}{ \\pi^2}$, $\\ast$ denotes the convolution and $y\\in\\mathbb{R}^d$.\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tThe proof for spatial translations can be done as in the massive case (see \\cite[Lemma 5.1]{Muc7}) since for this proof there is no explicit mass dependency. In particular they act accordingly as spatial translations in coordinate space, i.e.\n\t\t\t\\begin{align*}\t\tU( \\mathbf{y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U( \\mathbf{y},\\mathbb{I}) ^{-1} \n\t\t\t &=\n\t\t\t(2\\pi)^{3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} U(\\mathbf{y},\\mathbb{I}) {a}(\\mathbf{p}) U(\\mathbf{y},\\mathbb{I}) ^{-1}\\\\\n\t\t\t&=(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_k(x+y)^k} {a}(\\mathbf{p})= \\tilde{a}(\t \\mathbf{x}+\\mathbf{y}).\n\t\t\t\t\\end{align*} \n\t\t For time translations, we take the massless limit of the massive case (see \\cite[Lemma 5]{Muc7}) where we had the integral, \n\t\t\t\\begin{align*}(2\\pi)^{-3}\\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} e^{-iy^{0} \\omega_{m,\\mathbf{p}}} =\n\t\t\\frac{i\ty^{0}}{ 2\\pi^2} \\frac{ m^2K_{2}(m\\sqrt{\\vert \\mathbf{x}-\\mathbf{z}\\vert^{2}-(y^{0})^2})}{\\vert \\mathbf{x}-\\mathbf{z}\\vert^{2}-(y^{0})^2},\\end{align*} \n\t\t with $\\omega_{m,\\mathbf{p}}$ being the energy for a particle with mass $m$. By using the following limit, \n\t\t\\begin{align}\\label{ml0}\n\t\t\\lim\\limits_{m\\rightarrow 0} m^2\\, K_{2}(m f)=\\frac{2}{f^2},\n\t\t\\end{align} \n\t the explicit form of the time translation can be given in the massless case as follows, \n\t\t\t \\begin{align*} U(y^{0},\\mathbb{I}) \\tilde{a}(\t \\mathbf{x}) U(y^{0},\\mathbb{I})^{-1} &=e^{iy^{0}P_{0}} \\tilde{a}(\t \\mathbf{x})e^{-iy^{0}P_{0}}\\\\\n\t\t\t &=(2\\pi)^{-3\/2} \\int\n\t\t\t d^{3} \\mathbf{p}\\, e^{-ip_kx^k} e^{-iy^{0} \\omega_{\\mathbf{p}}} {a}(\\mathbf{p}) \\\\\n\t\t\t &=(2\\pi)^{-3 }\\int\n\t\t\t d^{3} \\mathbf{z} {\\left(\\int\n\t\t\t \td^{3} \\mathbf{p}\\, e^{-ip_k(x-z)^k} e^{-i \\omega_{\\mathbf{p}}y^{0}}\\right)}_{\n\t\t\t } \\tilde{a}(\\mathbf{z})\\\\&=\\frac{i\ty^{0}}{ \\pi^2}\n\t\t\t \\int\n\t\t\t d^{3} \\mathbf{z} \\left(\n\t\t\t \\frac{\t 1}{\\vert \\mathbf{x-z} \\vert^{2}-(y^{0})^2 } \\right)^2\\tilde{a}(\\mathbf{z}).\n\t\t\t \\end{align*}\n\t\t\t By taking the commutativity of the components of the momentum operator into account, we obtain the general transformation behavior under translations by using the former two results, i.e.\n\t\t\t \t\\begin{align} \tU( {y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U( {y},\\mathbb{I}) ^{-1} =\n\t\t U(y^{0},\\mathbb{I}) \tU( \\mathbf{y},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U( \\mathbf{y},\\mathbb{I}) ^{-1} U(y^{0},\\mathbb{I})^{-1}.\n\t\t\t \t\\end{align} \n\t\t\t \n\t\\end{proof}\nBefore we turn to the Lorentz transformations in the massless case, we define the rotation group. \tIn particular we denote matrices of the Lorentz group representing pure rotations as $\\Lambda_R$. They are given by matrices of the following form,\n\\[\\Lambda_R=\\left(\n\\begin{matrix} \n1 & 0\\\\\n0 & R\t \n\\end{matrix}\\right), \\qquad R\\in SO(3).\n\\]\n\t\\begin{lemma}\\label{l622}Under pure rotations the Fourier-transformed ladder operators $\\tilde{a}$ transforms in a covariant manner as follows, \n\t\t\\begin{align*}\n\t\tU(0,\\Lambda_R) \\tilde{a} (\t \\mathbf{x}) U(0,\\Lambda_R) ^{-1} =\n\t\t\\tilde{a} (\t \\mathbf{R x }) .\n\t\t\\end{align*}\n\t \n\t\\end{lemma}\n\t\\begin{proof}\n\t\tThe proof is done analog to the proof of the former Lemma. However, we could take the results of \\cite[Lemma 5.2]{Muc7} and perform the massless limit by using Equation (\\ref{ml0}). First, we calculate the action of pure spatial rotations on the Fourier-transformed annihilation operator, \n\t\t\t\\begin{align*}\n\t\t\tU(0,\\Lambda_R) \\tilde{a}^{\\,}(\t \\mathbf{x}) \tU(0,\\Lambda_R) ^{-1} &=\n\t\t\t(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} U(0,\\Lambda_R){a}(\\mathbf{p}) U(0,\\Lambda_R)^{-1}\\\\&=\n\t\t\t(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-ip_kx^k} \\sqrt{\\frac{\t\\omega_{R\\mathbf{p}}}{\t\\omega_{ \\mathbf{p}}}}{a}(R\\mathbf{p}) \n\t\t\t\\\\&=\n\t\t\t(2\\pi)^{-3\/2} \\int\n\t\t\td^{3} \\mathbf{p}\\, e^{-i p_k(R x)^k} {a}( \\mathbf{p}),\n\t\t\t\\end{align*}\n\t\t\tin the last lines we used the transformation behavior of the non-covariant momentum ladder operators (see \\cite[Chapter 7, Equation 57]{Sch}) and the orthogonality of $\\Lambda_{R}$ for pure spatial rotations. \n\t\\end{proof}\t\nThe NWP ladder operators transform non-covariantly w.r.t. the Lorentz-boosts. This is to be expected, since the NWP-operator is non-covariant w.r.t. the boosts. However, we see in the next theorem that the operator $\\phi_1(x)$ is at least covariant w.r.t. space-time translations and pure rotations.\n\\begin{theorem}\\label{t2}\n\tThe NWP-ladder operators transform under space-time translations and pure rotations, i.e. \twith $(y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}$, as follows,\n\\begin{align}\n\tU(y,\\Lambda_R) \\tilde{a}(\\mathbf{x}) U(y,\\Lambda_R)^{-1}&=\n\t \\frac{i y^{0}}{ \\pi^2} \\left(\n\t \\frac{\t 1}{\\vert \\mathbf{Rx+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a} (\\mathbf{Rx+y}) \\\\\n\t \tU(y,\\Lambda_R) \\tilde{a}^{*}(\\mathbf{x}) U(y,\\Lambda_R)^{-1}&=-\n\t \t\\frac{i y^{0}}{ \\pi^2} \\left(\n\t \t\\frac{\t 1}{\\vert \\mathbf{Rx+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a}^{*} (\\mathbf{Rx+y}).\n\\end{align}\nMoreover, the operator used to calculate the probabilities, i.e. $\\phi_1(x)$ transforms covariantly,\n\\begin{align}\\label{t2e2}\nU(y,\\Lambda_R) \\phi_1(x) U(y,\\Lambda_R)^{-1}= \\phi_1(\\Lambda_Rx+y)\\qquad (y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}.\n\\end{align}\n\t\t\n\n\\end{theorem}\n\t\n \n\\begin{proof}\n\tBy using Lemmas \\ref{l63} and \\ref{l622} we deduce,\n\t\t\\begin{align*} \tU(y,\\Lambda_R) \\tilde{a}(\\mathbf{x}) U(y,\\Lambda_R)^{-1} &= \tU(y,\\mathbb{I})\tU(0,\\Lambda_R)\\tilde{a}(\\mathbf{x})U(0,\\Lambda_R)^{-1}\tU(y,\\mathbb{I})^{-1}\t\t\n\t\\\\&=\t\t \t \tU(y,\\mathbb{I}) \\tilde{a}(R\\mathbf{x})\n\t\tU(y,\\mathbb{I})^{-1}\t \n\t\t\\\\&= \\frac{i y^{0}}{ \\pi^2} \\left(\n\t\t\\frac{\t 1}{\\vert \\mathbf{Rx+y} \\vert^{2}-(y^{0})^2 } \\right)^2\\ast\\tilde{a} (\\mathbf{Rx+y}) ,\t\n\t\t\\end{align*} \n\t\twhere in the last lines we used the fact that a general Poincar\\'e transformation can be split in first Lorentz-transformation and then space-time translation. An analog proof can be done for the creation operator $\\tilde{a}^{*}$ or we can use the fact that we have unitary operators generating the transformations and take the adjoint of the former equality. \\\\\\\\ In order to calculate the adjoint action of the operator $\\phi_1(x)$, we use the fact that time translations commute with the unitary operator of the Poincar\\'e group for pure rotations, i.e.\n\t\t\\begin{align*}\n\t\t\tU(y,\\Lambda_R) \\phi_1(x) U(y,\\Lambda_R)^{-1}& =\n\t\t\t\tU(y,\\Lambda_R)U(x_0,\\mathbb{I}) \\tilde{a}(\\mathbf{x}) U(x_0,\\mathbb{I})^{-1}U(y,\\Lambda_R)^{-1}\\\\&=U(x_0,\\mathbb{I})\n\t\t\t\tU(y,\\Lambda_R) \\tilde{a}(\\mathbf{x}) U(y,\\Lambda_R)^{-1}U(x_0,\\mathbb{I})^{-1}\\\\&=\n\t\t\t\tU(x_0+y_0,\\mathbb{I})\n\t\t\t\tU(\\mathbf{y},\\mathbb{I}) \\tilde{a}(R\\mathbf{x}) U(\\mathbf{y},\\mathbb{I})^{-1}U(x_0+y_0,\\mathbb{I})^{-1}\n\t\t\t\t\\\\&=\n\t\t\t\tU(x_0+y_0,\\mathbb{I}) \\tilde{a}(R\\mathbf{x+y}) U(x_0+y_0,\\mathbb{I})^{-1}= \\phi_1(\\Lambda_Rx+y).\n\t\t\\end{align*}\n\t\t\n\t\t\n\n\t\t\n\\end{proof}\nNext, we investigate the adjoint action w.r.t. the dilatations. \n\tIn order to calculate the adjoint action of dilatations on the NWP-space we first calculate the commutator of the dilatation operator with the coordinate operator. From that we can deduce the adjoint action of the dilatations acting on the coordinate ladder operators. This construction is analog to the one in \\cite[Chapter 9.4]{SU}.\n\t\\begin{lemma}\nThe adjoint action w.r.t. the dilatations acting on the NWP-operator is given as follows,\n\t\\begin{align*}\n\te^{i\\alpha D} X_j e^{-i\\alpha D}=e^{-\\alpha}X_j,\n\t\\end{align*}\nwhere in order to perform the calculations of the adjoint action the following commutator relation was used,\n\t\\begin{align*}\n\t[D,X_{j}]=iX_{j}.\n\t\\end{align*} \nFrom the former equation it follows that $X_{j}e^{-i\\alpha D}|\\mathbf{x}\\rangle$ is an eigenvector of \n$X_{j}$ with eigenvalue $e^{-\\alpha}x_{j}$. Therefore, one concludes that the transformation behavior of the operator $\\tilde{a}(\\mathbf{x})$ under dilatations is\n\t\\begin{align*}\n\te^{i\\alpha D}\\tilde{a}(\\mathbf{x})\te^{-i\\alpha D}=\ne^{ \\frac{3}{2}\\alpha}\\,\\tilde{a}(e^{ \\alpha}\\mathbf{x}).\n\t\\end{align*}\n \n\t\\end{lemma}\n\t\n\t\n\t\\begin{proof}\n\t\tIn order to proof the transformational behavior we first calculate the commutator of the dilatation operator with the NWP-operator, \n\t\t\\begin{align*}\n\t\t[D,X_{j}]&=\n\t\ti\t\\iint d^3\\mathbf{x}\\,d^3\\mathbf{y}\\,y_j\\,[\\tilde{a}^*(\\mathbf{x})\\left(\\frac{3}{2}+ x^{l}\\frac{\\partial}{\\partial x^{l}}\\right)\\tilde{a}(\\mathbf{x}) , \\tilde{a}^*(\\mathbf{y}) \\tilde{a}(\\mathbf{y})]\\\\&=\n\t\ti\t\\iint d^3\\mathbf{x}\\,d^3\\mathbf{y}\\,y_j\\,x^{l}\\,\\underbrace{[\\tilde{a}^*(\\mathbf{x}) \\frac{\\partial}{\\partial x^{l}} \\tilde{a}(\\mathbf{x}) , \\tilde{a}^*(\\mathbf{y}) \\tilde{a}(\\mathbf{y})]}_{-\\delta(\\mathbf{x}-\\mathbf{y})\\tilde{a}^*(\\mathbf{y})\\frac{\\partial}{\\partial x^{l}} \\tilde{a}(\\mathbf{x})+\\tilde{a}^*(\\mathbf{x}) \\tilde{a}(\\mathbf{y})\\frac{\\partial}{\\partial x^{l}}\\delta(\\mathbf{x}-\\mathbf{y})} \t \n\t\t\t\\\\&\t=-i\\int d^3x \\left(x_{j}\\,x^l \\tilde{a}^*(\\mathbf{x})\\frac{\\partial}{\\partial x^{l}}\\tilde{a}(\\mathbf{x})+\n\t\t\t \\,x_j\\,\\left(\\frac{\\partial}{\\partial x^{l}}\\left(x^l\\tilde{a}^*(\\mathbf{x})\\right) \\right) \\tilde{a}(\\mathbf{x})\\right)\n\t\t\t\t\\\\&=\t\t-i\\int d^3x \\left(x_{j}\\,x^l \\tilde{a}^*(\\mathbf{x})\\frac{\\partial}{\\partial x^{l}}\\tilde{a}(\\mathbf{x})-x^l\\tilde{a}^*(\\mathbf{x})\n\t\t\t\\left(\\frac{\\partial}{\\partial x^{l}}\\left(\t x_j\\, \\tilde{a}(\\mathbf{x})\\right) \\right)\\right)\n\t\t\t\t\\\\&=\t i\\int d^3x \\,x_j\\, \t\\tilde{a}^*(\\mathbf{x})\\tilde{a}(\\mathbf{x}),\n\t\t\\end{align*}\n\t\twhere in the last lines we used the vanishing commutator of the particle number operator with the coordinate operator. Moreover, we integrated over delta distributions and performed partial integrations. \\\\\\\\\t\tFrom the former commutator relation we obtain for the adjoint action of the NWP-operator w.r.t. dilatations the following, \n\t\t\\begin{align*}\n\t\t e^{i\\alpha D}X_{j}e^{-i\\alpha D}&=X_{j}+i\\alpha [D,X_{j}]+\\frac{i^2}{2!}\\alpha^2 [D,[D,X_{j}]]+\\cdots\\\\\n\t\t&= X_{j}+i^2\\alpha\\,X_{j}+\\frac{i^4}{2!}\\alpha^2X_{j}+\\cdots\n\t\t\\\\&= \\sum\\limits_{n=0}^{\\infty}\\frac{(i^2\\alpha)^n}{n!}X_{j}=e^{-\\alpha}X_j,\n\t\t\\end{align*}\n\t\twhere in the last lines we used the commutator relation between the dilatation and the NWP-operator and the Baker-Campbell-Hausdorff formula. As in \\cite[Chapter 9.4]{SU} we look at the following expression,\n\t\t\t\\begin{align*}\n\t\t\t X_{j}\\,e^{-i\\alpha D}|\\mathbf{x}\\rangle &=e^{-i\\alpha D}e^{i\\alpha D} X_{j}\\,e^{-i\\alpha D}|\\mathbf{x}\\rangle\\\\&= e^{-i\\alpha D}e^{-\\alpha}X_j|\\mathbf{x}\\rangle=e^{-\\alpha}x_j\\,e^{-i\\alpha D}|\\mathbf{x}\\rangle.\n\t\t\t\\end{align*}\n\t\tTherefore $X_{j}e^{-i\\alpha D}|\\vec{x}\\rangle$ is an eigenvector of $X_{j}$ for\n\t\tthe eigenvalue $e^{-\\alpha}x_j$, \n\t\tthus contained in the\n\t\teigenspace $\\mathcal{H}_{e^{-\\alpha}\\vec{x} }$ of $X_{j}$. Hence, the transformation on the creation operator of the NWP-space has to have the following form, \n\t\t\\begin{align}\\label{daf}\n\t\te^{i\\alpha D}\\tilde{a}^*(\\mathbf{x})\te^{-i\\alpha D}=\n\t\tf(\\alpha)\\,\\tilde{a}^*(e^{ \\alpha}\\mathbf{x}),\n\t\t\\end{align}\n\t\twhere $f(\\alpha)$ is a real-valued function of the transformation parameter $\\alpha$. To find the concrete form of the function $f$ we calculate the adjoint action of the coordinate operator\n\t\tw.r.t. the dilatations but in the Fock representation of the NWP-space, i.e. \n\t\\begin{align*}\n\te^{i\\alpha D}X_j\te^{-i\\alpha D}&= e^{i\\alpha D}\\,\\int d^3 \\mathbf{x}\\,x_j\\,\n\t\\tilde{a}^*( \\mathbf{x})\t\\tilde{a} ( \\mathbf{x})\te^{-i\\alpha D}\\\\&=\n\t\\int d^3 \\mathbf{x}\\,x_j\\,\n\tf(\\alpha)^2\\tilde{a}^*(e^{ \\alpha}\\mathbf{x})\\tilde{a} (e^{ \\alpha}\\mathbf{x})\\\\&=e^{-\\alpha}\n\t\\int d^3 \\mathbf{x}\\,e^{ -3\\alpha }\\,x_j\\,\n\tf(\\alpha)^2\\tilde{a}^*( \\mathbf{x})\\tilde{a} ( \\mathbf{x}), \n\t\\end{align*}\n\t\twhere in order for the NWP-operator to transform correctly the function $f$ has to be chosen as $f(\\alpha)= e^{ \\frac{3}{2}\\alpha }$. In order to solidify our approach we apply the adjoint action on the particle number operator in the coordinate space, \n\t\t\\begin{align*}\n e^{i\\alpha D}N\te^{-i\\alpha D}&= e^{i\\alpha D}\\,\\int d^3 \\mathbf{x}\\,\n\t\\tilde{a}^*( \\mathbf{x})\t\\tilde{a} ( \\mathbf{x})\te^{-i\\alpha D}\\\\&=\n\t \\int d^3 \\mathbf{x}\\,\n\t\tf(\\alpha)^2\\tilde{a}^*(e^{ \\alpha}\\mathbf{x})\\tilde{a} (e^{ \\alpha}\\mathbf{x})\\\\&=\n\t\t\\int d^3 \\mathbf{x}\\,e^{ -3\\alpha }\n\t\tf(\\alpha)^2\\tilde{a}^*( \\mathbf{x})\\tilde{a} ( \\mathbf{x})=N, \n\t\t\\end{align*}\n\twhere in the last lines a variable substitution was performed and the commutator relation $[D,N]=0$ is used to argue that the adjoint action leaves the particle number operator invariant and therefore the Ansatz (\\ref{daf}) with explicit form of $f$ gives us the right transformational behavior. \n\t\\end{proof}\n\t To calculate the adjoint action of dilatations on the ladder operators of the NWP-space we could have, as well, taken the well-known transformation property of the ladder operators in momentum space. After doing so we could have performed an inverse Fourier-transformation. However, one of the purposes of this work is to demonstrate to the reader that all calculations can as well be done in the NWP-space. In particular, the NWP-space is on an equal footing with the momentum space. \\\\\\\\\nSince we are interested in transformations of the time translated expression of the ladder operators we need to take the commutation relations between the momentum and dilatation operators into account. \n\\begin{theorem}\\label{t3}\n\tThe operator $\\phi_1(x)$ that is responsible for calculating probabilities of finding $k$-particles in $k$-spatial positions transforms under dilatations as follows, \n\t\\begin{align*}\ne^{i\\alpha D}\\phi_1(x)e^{-i\\alpha D}=e^{\\frac{3}{2} \\alpha}\\phi_1(e^{\\alpha}x).\n\t\\end{align*}\n\\end{theorem}\n\\begin{proof}\nWe can rewrite the operator $\\phi_1(x)$ as \n\t\\begin{align*}\n\\phi_1(x)= \tU(x^{0},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U(-x^{0},\\mathbb{I}).\n\t\\end{align*}\nSince we know the adjoint action on the ladder operators in the NWP-space we can directly calculate the transformation on all expressions. First we calculate the adjoint action of the dilatations on the zero component of the momentum operator,\n\t\\begin{align*}\ne^{i\\alpha D}\\,P_0\\,e^{-i\\alpha D}&= -\\pi^{-2}\\Gamma(2)\n\\iint\nd^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\, \n\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ 4} } \ne^{i\\alpha D}\\,\\tilde{a}^*(\\mathbf{x}) \\, e^{i\\alpha D}e^{-i\\alpha D}\\,\n\\tilde{a} (\\mathbf{y} )\\,e^{-i\\alpha D}\\\\&\n= -\\pi^{-2}\\Gamma(2)\n\\iint\nd^3 \\mathbf{x}\\,d^3 \\mathbf{y}\\, \n\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{ 4} } \\, e^{ 3\\alpha}\\,\n \\tilde{a}^*(e^{ \\alpha}\\mathbf{x}) \\,\n\\tilde{a} (e^{ \\alpha}\\mathbf{y} ) \\\\&\n= e^{\\alpha} P_{0},\n\t\\end{align*}\nwhere in the last lines we used the explicit transformation given in Equation (\\ref{daf}) and performed a variable substitution. Next, we apply the adjoint action of dilatations on this operator and use the fact that we are dealing with unitary transformations, i.e.\n\\begin{align*}\ne^{i\\alpha D}U(x^{0},\\mathbb{I}) \\tilde{a} (\t \\mathbf{x}) U(-x^{0},\\mathbb{I})e^{-i\\alpha D} &=e^{\\frac{3}{2} \\alpha}\nU(e^{\\alpha}x^{0},\\mathbb{I}) \\tilde{a} (e^{ \\alpha} \\mathbf{x}) U(-e^{\\alpha}x^{0},\\mathbb{I})\\\\&= \\frac{i\te^{\\frac{5}{2} \\alpha} x^{0}}{2\\pi^2}\n\\int\nd^{3} \\mathbf{z} \\left(\n\\frac{\t 1}{\\vert e^{ \\alpha}\\mathbf{x-z} \\vert^{2}-(e^{\\alpha}x^{0})^2 } \\right)^2\\tilde{a}(\\mathbf{z}).\n\\end{align*}\n\\end{proof}\nBy using the former lemmas we calculate the adjoint action of translations, rotations and dilatations on $\\phi_1(x)$. This is in particular important since the operator $\\phi_1(x)$ is used to calculate probability amplitudes of finding $k$-particles in $k$-spatial positions. Hence, by knowing the transformational behavior, two frames that are connected via translations, rotations and\/or dilatations can calculate the same probability by taking the proper transformation into account.\n\t\\begin{theorem}\n\t\tThe operator\t $\\phi_1(x)$ transforms covariantly under space-\\textbf{time} translations, pure rotations and dilatations, i.e. \n\t\t\\begin{align*}&\n\te^{i\\alpha D}\tU(y,\\Lambda_R) \\phi_1(x)U(y,\\Lambda_R)^{-1}\te^{-i\\alpha D}\n\t\t= e^{\\frac{3}{2} \\alpha}\\phi_1(\te^{\\alpha}(\\Lambda_R x+ y))\n ,\n\t\t\\end{align*} \n\t\twith $(y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}$ and $\\alpha\\in\\mathbb{R}$.\n\t\tMoreover, the explicit transformation of the Fourier-transformed operator $\\tilde{a}$ under the action of the subgroup $(y,\\Lambda_{R}) \\in \\mathscr{P}^{\\uparrow}_{+}$ is in coordinate space given as, \n\t\t\\begin{align*}&\n\t\te^{i\\alpha D}\t\tU(y,\\Lambda_R)\\tilde{a}(\\mathbf{x})U(y,\\Lambda_R)^{-1}e^{-i\\alpha D}\n\t\t= c_{y' } \\left(\n\t\t\\frac{\t 1}{\\vert \\mathbf{Rx}+ \\mathbf{y} \\vert^{2}-( y^{0})^2 } \\right)^2\\ast\\tilde{a} (e^{\\alpha}(\\mathbf{Rx}+ \\mathbf{y}) ),\n\t\t\\end{align*} \n\t\twhere $c_{y' }:= \\frac{i e^{-\\frac{3}{2}\\alpha} y^{0}}{ \\pi^2}$, $\\ast$ denotes the convolution and $y\\in\\mathbb{R}^4$.\n \n\t\\end{theorem}\n\\begin{proof}\nIn order to prove the first part of the theorem we use \tthe former Lemmas and the algebra of the Poincar\\'e group. First, we examine how the operator $\\phi_1(x)$ transforms under space-time translations and pure rotations,\n\t\\begin{align*}\n\t& e^{i\\alpha D}\t\tU(y,\\Lambda_R) \\phi_1(x)U(y,\\Lambda_R)^{-1}\te^{-i\\alpha D}\\\\ & =\te^{i\\alpha D}\t\\phi_1(\\Lambda_R x+y )\te^{-i\\alpha D} \\\\& =\n\te^{\\frac{3}{2}\\alpha}\\phi_1(e^{\\alpha}\\Lambda_R x+e^{\\alpha}y )\n\\\\&=\t \ne^{\\frac{3}{2}\\alpha}U(e^{\\alpha}(x_0+ y_0),\\mathbb{I}) \\tilde{a}(e^{\\alpha}(\\mathbf{Rx+ y}))U(-e^{\\alpha}(x_0+ y_0),\\mathbb{I}) \\\\&=e^{\\frac{3}{2}\\alpha}\nc_{y^{''} } \\left(\n\\frac{\t 1}{\\vert e^{\\alpha}(\\mathbf{Rx+ y}) \\vert^{2}-(e^{\\alpha}(x_0+ y_0))^2 } \\right)^2\\ast\\tilde{a} (e^{\\alpha}(\\mathbf{Rx+ y})),\n\\end{align*}\nwhere $c_{y^{''} }:=\\frac{i e^{\\alpha}(x_0+ y_0)}{ \\pi^2}$ and \n\twhere for the calculation we used the transformation property given in Theorem \\ref{t3}, the dilatation action Equation (\\ref{t2e2})\n and the explicit time translation given in Equation (\\ref{eq51}).\n\\end{proof}\n \n\\section{Conclusion and Outlook}\nIn this work we supported the point of view, that the NWP operator and its respective states have a physical meaning. In particular, we defined the second-quantized version of the NWP-operator for a massless scalar field and wrote the operator in its respective eigenbase. The meaning of the second-quantized operator is given by its respective eigenfunctions. They are used to calculate the probability amplitude of finding particles at certain spatial positions for a fixed time. Furthermore, by transforming the whole conformal group into the NWP-eigenbase we showed that this space can be treated and worked on at equal footing as the momentum space. \n\\\\\\\\\nAlthough eigenstates of this operator are non-covariant w.r.t. the Poincar\\'e group we showed indirectly that combinations of these states are covariant. This is proven by the fact that covariant operators remain covariant, independently of their respective representation. Hence, even though we work with non-covariant eigenstates, symmetries of the theory remain untouched. This is, in our opinion, another strong argument why the coordinate space, i.e. the NWP-space, is equally entitled from a physical point of view as the momentum space. \n\\\\\\\\\nMoreover, we showed that the operator generating the NWP-space, i.e. $\\phi_1(x)$ transforms in a covariant manner under the subgroup of space-time translations, rotations and dilatations of the conformal group, namely \n\t\\begin{align*}&\n\te^{i\\alpha D}\tU(y,\\Lambda_R) \\phi_1(x)U(y,\\Lambda_R)^{-1}\te^{-i\\alpha D}\n\t= e^{\\frac{3}{2} \\alpha}\\phi_1(\te^{\\alpha}(\\Lambda_R x+ y))\n\t.\n\t\\end{align*}\nHence, given two observers that are related by space-time translations, rotations or\/and dilatations, they both measure the same probability amplitude, by taking the respective covariant transformation into account. \\\\\\\\\t\nFurthermore, we were able to supply an additional argument to point out the physical relevance of the NWP-operator. Namely, it is possible, even in a second-quantized context, to express the generators \nof the conformal group by using the NWP-operator and the relativistic momentum. In our opinion this supplies a strong argument for the physical sense of the NWP-operator, since the conformal group can be written in terms of products of the coordinate and momentum operators. \\\\\\\\\n\tSince we formulated the whole framework of a free massless scalar field, its respective transformations and symmetries into the NWP-space, the next step in progress is to understand localization from a non-commutative point of view. In particular, by imposing a space-space non-commutativity the subject of interest is to be able to calculate the probability of finding a particle at a certain mean value of a spatial position. In our opinion, the ground work for such investigations was presented in this paper. \n\n\t\\section*{Aknowledgments}\n\tThe author is indebted to Prof. K. Sibold for initiating deep conceptual questions of this work. Furthermore, we would like to thank Prof. K. Sibold and S. Pottel for many fruitful discussions during different stages of this paper. With regards to the domains of the Lorentz-boosts am indebted to Prof. Norbert Dragon. The corrections of Dr. Z. Much are thankfully acknowledged.\n\t\n \n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \n If $k$ and $N$ are positive integers, let $S_k(N)$ be the rational vector space of cusp forms of weight $k$ on $\\Gamma_0(N)$ with rational Fourier coefficients.\n These forms have a Fourier expansion at $\\infty$ of the form\n\\[\n f(z)= \\sum_{n=n_0}^{\\infty} a(n) q^{n} \\text{ }\\text{ with }\\text{ }a(n_0) \\neq 0,\n\\]\nand we define $\\operatorname{ord}_{\\infty}(f) := n_0$. Let $g(N)=\\dim(S_2(N))$ be the genus of $X_0(N)$. We say that $\\infty$ is a Weierstrass point on the modular curve $X_0(N)$ if there exists $0 \\neq f \\in S_2(N)$ such that $\\operatorname{ord}_{\\infty}(f) > g(N)$. Ogg \\cite{Ogg} proved the following theorem.\n\\begin{theorem}\\label{thm:Ogg}\n If $p$ is a prime such that $p \\nmid N$, and if $g(N)=0$, then $ \\infty $ is not a Weierstrass point on $X_0(pN)$.\n\\end{theorem}\n A non-geometric proof of Theorem 1.1 was given in [AMR09] (previously, certain cases of level $p\\ell$ for distinct primes $p$ and $\\ell$ were considered in \\cite{Kohnen}, \\cite{Kilger}). To state our first result, when $N$ is a positive integer and $p$ is a prime such that $p \\nmid N$, we require the Atkin-Lehner operator $W_{p}^{pN}$ on $S_{k}(pN)$ defined in \\eqref{eqn:Atkin-Lehner}. Furthermore, if \n\\[\nf(z)= \\sum_{n=n_0}^{\\infty} a(n) q^{n} \\in S_k(N),\n\\] \ndefine \n\\[\n v_p(f):=\\inf\\{v_p(a(n))\\}.\n\\]\n With this notation, we prove the following theorem.\n\\begin{theorem}\\label{thm:main}\n Let $N$ be a positive integer and $k$ be a positive even integer. Let \n $p$ be a prime with $p \\geq \\max(5,k+1)$ and $p \\nmid N$. Suppose that $0 \\neq f \\in S_k(pN)$ satisfies \n\\[\n v_p(f)=0, \\text{ }\\text{ }v_p(f|_{k}W^{pN}_{p}) \\geq 1-k\/2.\n\\]\n Then \n\\[\n \\operatorname{ord}_{\\infty}(f) \\leq \\dim(S_k(pN)).\n\\]\n\\end{theorem} \n As a corollary, we prove an analogue of Ogg's theorem.\n\\begin{corollary}\\label{thm:analogue}\n Suppose that $N$ is a positive integer, that $k$ is a positive even integer, and that $p$ is a prime with $p \\geq \\max(5,k+1)$ and $p \\nmid N$. If $S_{k}(N)= \\{0\\}$, then for $0 \\neq f \\in S_{k}(pN)$, we have \n\\[\n \\operatorname{ord}_{\\infty}(f) \\leq \\dim(S_k(pN)).\n\\]\n\\end{corollary}\n There is a finite list of $N$ and $k$ for which $S_k(N)=\\{0\\}$. For $k=2$, there are $15$ such values of $N$ \\cite[pg. 110]{Ono}. For $k \\geq 4$, the rest are\n\\begin{align*}\n k&=4: N= 1, 2, 3, 4 \\\\\n k&=6: N= 1, 2 \\\\\n k&=8,10,14: N= 1. \n\\end{align*}\n \n It is natural to seek to understand the subspace of forms $f \\in S_{k}(N)$ which vanish to order greater than the dimension.\n If $N$ is a positive integer and $k$ is a positive even integer, define the subspace\n\\[\n W_{k}(N):=\\{f \\in S_k(N): \\operatorname{ord}_{\\infty}(f) > \\dim(S_k(N))\\}.\n\\]\n With this notation, we have $W_{2}(N)=\\{0\\}$ if and only if $\\infty$ is not a Weierstrass point on $X_0(N)$. As a corollary of Theorem~\\ref{thm:main}, we obtain a bound for $\\dim(W_{k}(pN))$.\n\\begin{corollary}\\label{thm:Subspace}\n Suppose that $p \\geq \\max(5,k+1)$ is a prime satisfying $p \\nmid N$. Then we have \n\\[\n\\dim(W_{k}(pN)) \\leq \\dim(S_k(N)).\n\\]\n \\end{corollary}\nNote that this implies Theorem~\\ref{thm:Ogg} in the case $k=2$. It is interesting to note that the bound in Corollary~\\ref{thm:Subspace} is independent of $p$. Thus, for fixed $N$, the spaces $W_{k}(pN)$ have uniformly bounded dimension as $p \\rightarrow \\infty$.\n\\begin{remark}\nFor squarefree $N$, Arakawa and B\\\"{o}cherer \\cite{Arakawa-Bocherer} study the space\n\\[\nS_k(N)^{*}:= \\{ f \\in S_k(N): f\\big|_k W_{p}^{pN}+p^{1-\\frac{k}{2}}\\sl U_p=0 \\ \\ \\text{for all $p \\mid N$} \\}.\n\\]\nWe will use a similar subspace to prove Corollary~\\ref{thm:Subspace}.\n\\end{remark}\nThe following examples, which we computed with Magma, illustrate Corollary~\\ref{thm:Subspace} for small values of $N$.\n\\begin{example}\n For an example which is sharp, set $N=1$, $p=19$, and $k=16$. Here, we have $\\dim(S_k(pN))=24$ and $\\dim(S_k(N))=1$. In this case, there is a form $f \\in S_k(pN)$ with $f=q^{25}+\\cdots$.\n \\end{example}\n \\begin{example}\n To get an example which is sharp and for which $pN$ is not prime, set $N=2$, $p=23$, and $k=12$. Here, $\\dim(S_k(pN))=64$ and $\\dim(S_k(N))=2$. In this case, there are forms $f$ and $g$ with $f=q^{67}+\\cdots$ and $g=q^{68}+\\cdots$.\n \\end{example}\n \\begin{example}\n Corollary~\\ref{thm:Subspace} is not always sharp. For example, set $N=1$, $p=29$, and $k=28$. Here, $\\dim(S_k(pN))=67$ and $\\dim(S_k(N))=3$. In this case, there is no non-zero $f \\in S_k(N)$ satisfying $\\operatorname{ord}(f) > 67$.\n \\end{example}\n The paper is organized as follows. Section $2$ contains the background necessary to prove these results. Section $3$ contains the proof of Theorem~\\ref{thm:main}, which uses results from \\cite{Ahlgren-Masri-Rouse}. Finally, Section $4$ contains the proofs of Corollary~\\ref{thm:analogue} and Corollary~\\ref{thm:Subspace}. \n\\section{Preliminaries on Modular Forms}\nThe definitions and facts given here can be found in \\cite{Diamond-Shurman} and \\cite{Ahlgren-Masri-Rouse}. Let $N$ and $k$ be positive integers.\nLet $\\varepsilon_{\\infty}(N)$ denote the number of cusps on $X_0(N)$, let $g(N)$ denote its genus, and let $\\varepsilon_{2}(N)$, $\\varepsilon_{3}(N)$ denote the numbers of elliptic points of orders $2$ and $3$, respectively. Then we have\n\\begin{equation}\\label{genus}\ng(N)=\\mfrac{[\\operatorname{SL}_2(\\mathbb{Z}):\\Gamma_0(N)]}{12}-\\mfrac12\\varepsilon_{\\infty}(N)-\\mfrac14\\varepsilon_{2}(N)-\\mfrac13\\varepsilon_{3}(N)+1,\n\\end{equation}\n\\[ \\varepsilon_{2}(N)= \\begin{cases} \n0 & \\text{if } 4 \\mid N, \\\\\n\\displaystyle\\prod_{p \\mid N} \\(1+\\(\\mfrac{-4}{p}\\)\\) & \\text{otherwise,} \n\\end{cases}\n\\] \n\\[ \\varepsilon_{3}(N)= \\begin{cases} \n0 & \\text{if } 9 \\mid N, \\\\\n\\displaystyle\\prod_{p \\mid N} \\(1+\\(\\mfrac{-3}{p}\\)\\) & \\text{otherwise.} \n\\end{cases}\n\\]\nWe have the well-known formula\n\\[\n[\\operatorname{SL}_{2}(\\mathbb{Z}):\\Gamma_0(N)]=N\\prod_{p \\mid N}\\(1+\\mfrac{1}{p}\\).\n\\] \nFor weights $k \\geq 4$, we have\n\\begin{equation}\\label{*}\n\\dim(S_k(N))=(k-1)(g(N)-1)+\\floor{\\mfrac{k}{4}}\\varepsilon_{2}(N)+\\floor{\\mfrac{k}{3}}\\varepsilon_{3}(N)+\\(\\mfrac{k}{2}-1\\)\\varepsilon_{\\infty}(N).\n\\end{equation} \n \n A form in $S_k(N)$ may have forced vanishing at the elliptic points. As in \\cite{Ahlgren-Masri-Rouse}, let $\\alpha_{2}(N,k)$ and $\\alpha_{3}(N,k)$ count the number of forced complex zeroes of a form $f \\in S_k(N)$ at the elliptic points of order $2$ and $3$, respectively. These are given by \n\\begin{equation}\\label{eq:table}\n(\\alpha_{2}(N,k),\\alpha_{3}(N,k))= \\begin{cases} \n(\\varepsilon_{2}(N),2\\varepsilon_{3}(N)) & \\text{if } k \\equiv 2 \\pmod{12}, \\\\\n(0,\\varepsilon_{3}(N)) & \\text{if } k \\equiv 4 \\pmod{12}, \\\\\n(\\varepsilon_{2}(N),0) & \\text{if } k \\equiv 6 \\pmod{12}, \\\\\n(0,2\\varepsilon_{3}(N)) & \\text{if } k \\equiv 8 \\pmod{12}, \\\\\n(\\varepsilon_{2}(N),\\varepsilon_{3}(N)) & \\text{if } k \\equiv 10 \\pmod{12}, \\\\\n(0,0) & \\text{if } k \\equiv 0 \\pmod{12}.\n\\end{cases}\n\\end{equation} \n\n If $d=\\dim(S_k(N)),$ then $S_k(N)$ has a basis $\\{f_1,...,f_d\\}$ with integer coefficients with the property \n\\begin{equation}\\label{eqn:INTEGRALITY}\nf_i(z)=a_iq^{c_i}+O(q^{c_{i}+1}), \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1 \\leq i \\leq d,\n\\end{equation}\nwhere $a_i \\neq 0$ and $c_10$. \nThe equations for $\\nu$ are \n\\begin{equation}\n\\begin{split}\n\\frac{2}{\\rho}\\,\\partial_z \\nu \\= & \\sum_{\\Lambda=0}^3 \\left[ \\partial_\\rho \\log W_\\Lambda \\,\\partial_z \\log W_\\Lambda + a_\\Lambda^2\\, \\partial_\\rho L_\\Lambda \\,\\partial_z L_\\Lambda \\right], \\\\\n\\frac{4}{\\rho}\\,\\partial_\\rho \\nu \\=& \\sum_{\\Lambda=0}^3 \\left[ \\left(\\partial_\\rho \\log W_\\Lambda\\right)^2 -\\left( \\partial_z \\log W_\\Lambda\\right)^2+ a_\\Lambda^2\\, \\left( \\left(\\partial_\\rho L_\\Lambda\\right)^2 - \\left(\\partial_z L_\\Lambda\\right)^2 \\right)\\right]\\,.\\label{eq:nuEq}\n\\end{split}\n\\end{equation}\nThe integrability is guaranteed by the harmonicity of the functions on the right-hand sides. These equations can be integrated in a case-by-case manner depending on the choice of sources for the harmonic functions.\n\\footnotetext{ More generically, the scalars are given by\n\\begin{equation}\nZ_\\Lambda = {\\cal G}^{(\\Lambda)}_\\ell\\left(L_\\Lambda\\right) \\,,\\qquad \\star_3 d\\left(H_0\\,d\\phi\\right) = dL_0\\,, \\qquad dT_I = \\frac{dL_I}{ {\\cal G}^{(I)}_\\ell(L_I)^2} \\,,\n\\label{eq:WGF&Hfoot}\n\\end{equation}\nwhere ${\\cal G}^{(\\Lambda)}_\\ell$ can be freely chosen among the following five generating functions of one variable:\n\\begin{align}\n{\\cal G}^{(\\Lambda)}_1 (x) &= \\frac{\\sinh(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda} \\,,\\qquad {\\cal G}^{(\\Lambda)}_2 (x) = i\\,\\frac{\\cosh(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda}\\,,\\qquad {\\cal G}^{(\\Lambda)}_3 (x) = x +b_\\Lambda\\,,\\nonumber \\\\\n{\\cal G}^{(\\Lambda)}_4 (x) &= \\frac{\\sin(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda} \\,,\\qquad {\\cal G}^{(\\Lambda)}_5(x) =\\frac{\\cos(a_\\Lambda x+b_\\Lambda)}{a_\\Lambda}\\,,\\qquad a_\\Lambda \\in \\mathbb{R}_+, \\quad b_\\Lambda\\in \\mathbb{R} \\,.\n\\label{eq:DefFi}\n\\end{align}\nHowever, we did not find regular solutions for the branches $\\ell=2,4,5$ (see \\cite{Bah:2020pdz,Bah:2021owp,Bah:2021rki} for more details). Therefore, we consider only the branch $\\ell=1$ from which the function $\\ell=3$ can be obtained with \\eqref{eq:BPSlimit}.}\n\nIt is important to point out that $W_\\Lambda$ are ``vacuum'' warp factors, as they are not coupled to any gauge potentials, whereas $Z_\\Lambda$ are generated and induced by the gauge potentials. However, one can take a neutral limit by sending all gauge potentials to zero while keeping $Z_\\Lambda$ non-trivial, which then satisfy the vacuum equation $\\Delta \\log Z_\\Lambda =0$. The neutral limit is given by\n\\begin{equation}\n(b_\\Lambda, a_\\Lambda) \\to \\infty \\qquad\\text{with}\\quad e^{b_\\Lambda}\/a_\\Lambda \\quad\\text{and}\\quad a_\\Lambda L_\\Lambda\\quad \\text{held fixed.}\n\\label{eq:neutrallimit}\n\\end{equation}\nIt shows that the present ansatz is generically in the non-BPS regime of the M2-M2-M2-KKm system.\n\nMoreover, at the other side of the parameter space,\n\\begin{equation}\n(b_\\Lambda, a_\\Lambda) \\to 0 \\qquad\\text{with}\\quad b_\\Lambda\/a_\\Lambda \\quad \\text{held fixed,}\n\\label{eq:BPSlimit}\n\\end{equation}\nthe solutions converge to the linear branch $Z_\\Lambda = L_\\Lambda+b_\\Lambda\/a_\\Lambda$. As we will see in the next section, this leads to the BPS equations for the M2-M2-M2-KKm system. Therefore, the ansatz admits a BPS limit considering the above transformation.\n\nIt is remarkable that one can deviate significantly from the BPS regime while maintaining the linearity of the Einstein-Maxwell equations. Moreover, it only changes the warp factors from a linear function (BPS regime) to a $\\sinh$ function (non-BPS regime) in terms of harmonic functions. As we will see later, the price to pay for having a linear ansatz will be to use non-BPS sources that have a fixed charge-to-mass ratio, but not necessarily 1.\n\n\n\\subsection{BPS regime}\n\n\nWe consider that the two-tori and six-torus are rigid, $W_\\Lambda=1$. Moreover, we take the limit \\eqref{eq:BPSlimit} for the scalars $(Z_\\Lambda, H_0,T_I)$ \\eqref{eq:WGF&H}, which leads to $L_\\Lambda \\= Z_\\Lambda - b_\\Lambda\/a_\\Lambda$. Therefore, the solutions can be better written such that $Z_\\Lambda$ are harmonic functions and the gauge potentials are determined by\n\\begin{equation}\n\\Delta Z_\\Lambda \\= 0\\,,\\qquad \\star_3 d (H_0\\,d\\phi) \\= dZ_0\\,,\\qquad T_I \\= - \\frac{1}{Z_I}\\,.\n\\end{equation}\nWe recognize the BPS equations of motion one can obtain from supersymmetric static M2-M2-M2-KKm solutions. With such a choice, $\\nu=0$ \\eqref{eq:nuEq}, and the base is flat $\\mathbb{R}^3$ as expected. Physical solutions are necessarily sourced by point particles on the $z$-axis:\n\\begin{equation}\nZ_\\Lambda \\= \\sum_i^n \\frac{q_i^{(\\Lambda)}}{\\sqrt{\\rho^2+(z-z_i)^2}}\\+ c_\\Lambda\\,, \\qquad H_0 \\= \\sum_i^n \\frac{q_i^{(0)}\\,(z-z_i)}{\\sqrt{\\rho^2+(z-z_i)^2}}\\+ \\bar{c}_0\\,.\n\\end{equation}\nThe sources at $(\\rho=0,z=z_i)$ are either static extremal four-charge black holes if $q_i^{(\\Lambda)} \\neq 0$ or smooth Gibbons-Hawking centers if $q_i^{(I)}=0$. \n\nTherefore, the linear non-BPS floating-brane ansatz contains the BPS floating brane ansatz for the static M2-M2-M2-KKm system.\n\n\n\n\\subsection{Non-BPS regime}\n\\label{sec:GenWeylSol}\n\nGeneric non-BPS solutions are constructed for non-zero $(a_\\Lambda,b_\\Lambda)$ in \\eqref{eq:WGF&H}. One can check that sourcing the harmonic functions by point sources lead to naked singularities. We therefore consider that the harmonic functions are sourced by \\emph{rods}, that is segment sources. \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.23\\textwidth]{Rods.pdf}\n\\caption{Schematic description of axisymmetric rod sources.}\n\\label{fig:RodsSources}\n\\end{figure}\n\nIn this section, we establish our formalism and highlight the basic ingredients that can be added linearly through the ansatz. We save the construction of the solutions for later. \n\n\n We consider $n$ distinct rods of length $M_i>0$ along the $z$-axis centered around $z=z_i$. Without loss of generality, we can order them as $z_i < z_j$ for $i 0$ \\eqref{eq:WarpfactorsRods}, which requires, if $b_\\Lambda \\neq +\\infty$, \n\\begin{equation}\nb_\\Lambda \\geq 0 \\,,\\quad a_\\Lambda \\,P_i^{(\\Lambda)}\\, \\geq\\, 0 \\,,\\qquad \\Lambda=0,...,3, \\,\\, i=1,...,n.\n\\label{eq:RegPositivity}\n\\end{equation}\nWith this condition, the solutions are regular out of the $z$-axis. \n\\item[\u2022] \\textbf{Regularity at the rods on the $z$-axis:}\n\n\nWe consider the local spherical coordinates centered around the $i^\\text{th}$ rod, \\begin{equation}\n\\rho= \\sqrt{r_{i}\\left(r_{i}+M_{i}\\right)} \\sin \\theta_i,\\, \\qquad z\\= \\left(r_{i}+\\frac{M_{i}}{2}\\right) \\cos \\theta_{i}+z_{i}. \n\\label{eq:localcoordrod}\n\\end{equation}\n At the rod, $r_i \\to 0$, we have\n\\begin{equation}\n\\frac{R_+^{(i)}}{R_-^{(i)}} \\propto \\frac{1}{r_i}\\,,\\qquad \\frac{E_{+-}^{(i,i)}E_{-+}^{(i,i)}}{E_{++}^{(i,i)}E_{--}^{(i,i)}} \\propto r_i^2,\n\\end{equation}\nwhile all other quantities are well-behaved. From \\eqref{eq:WarpfactorsRods}, we have\\footnote{Since $a_\\Lambda P_i^{(\\Lambda)}\\geq 0$ is required if $b_\\Lambda \\neq \\infty$, $Z_\\Lambda$ can vanish at the rod only if one takes $b_\\Lambda = \\infty$, that is the neutral limit \\eqref{eq:neutrallimit} for the pair $(Z_\\Lambda,T_\\Lambda)$. In that case, $Z_\\Lambda \\,\\propto\\, r_i^{- a_\\Lambda P_i^{(\\Lambda)}}$ where $a_\\Lambda P_i^{(\\Lambda)}$ can be taken negative.}\n\\begin{equation}\nZ_\\Lambda \\,\\propto\\, r_i^{- a_\\Lambda P_i^{(\\Lambda)} }\\,, \\qquad W_\\Lambda \\,\\propto\\, r_i^{-G_i^{(\\Lambda)}}\\,,\\qquad e^{2\\nu} \\,\\propto \\, r_i^{\\alpha_{ii}}\\,,\n\\end{equation}\nand therefore, the metric \\eqref{eq:nonBPSfloatingRemind} is singular at the rod, except for specific choices of weights that will characterize the physical rods of the non-BPS floating brane ansatz in M-theory. It will be convenient to use the following aspect ratios, $d_i$,\n\\begin{equation}\n\\begin{split}\nd_1 &\\,\\equiv\\, 1\\,,\\qquad d_i \\,\\equiv\\, \\prod_{j=1}^{i-1} \\prod_{k=i}^n \\left(\\dfrac{(z_k^- - z_j^+)(z_k^+ - z_j^-)}{(z_k^+ - z_j^+)(z_k^- - z_j^-)} \\right)^{\\alpha_{jk}}\\quad \\text{when } i=2,\\ldots n\\,.\n\\end{split}\n\\label{eq:dialphaDefcharged}\n\\end{equation}\nThe regularity at each rod requires that the sources are in one of these categories (see appendix \\ref{App:AttheRod} for more details):\n\\begin{itemize}\n\\item[-] \\underline{\\emph{Black rods:}} the $y_a$-fibers and the $\\phi$-circle have a finite size at the $i^\\text{th}$ rod if\n\\begin{equation}\nG^{(\\Lambda)}_i \\= 0 \\,,\\qquad P^{(\\Lambda)}_i \\= \\frac{1}{2 \\, a_\\Lambda}\\,.\n\\label{eq:blackrodWeight}\n\\end{equation}\nOne can check that $\\alpha_{ii}=1$ and the timelike Killing vector $\\partial_t$ shrinks at the rod which therefore defines the locus of a horizon. The topology of the horizon is either T$^7\\times$S$^2$ or T$^6\\times$S$^3$ depending on the close environment around the rod (see Fig.\\ref{fig:RodCategories} and \\ref{fig:TouchingRods}).\\footnote{\\label{footnote1}The horizon topology depends on the close environment of the black rod. If the rod is connected on one side as in Fig.\\ref{fig:TouchingRods}, it corresponds to a T$^6\\times$S$^3$ horizon. Otherwise, it is a T$^7\\times$S$^2$ horizon as in Fig.\\ref{fig:RodCategories}.} \n\nMoreover, the surface gravity and horizon area associated to the black hole give\\footnote{The surface gravity can be read from the local geometry such as $ds^2 \\propto d\\rho_i^2 - \\rho_i^2 \\kappa_i^2 dt^2 + \\ldots$ with $\\rho_i\\to 0$, and is associated to the temperature of the black hole, ${\\cal T}_i = \\kappa_i\/(2\\pi )$. Moreover, note that if the rod is connected to another one, for instance $z_{i-1}^+=z_i^-$, the surface gravity is still finite since $(z_{i-1}^+-z_i^-)\/d_i=\\text{cst}$.}\n\\begin{equation}\n\\begin{aligned}\n\\kappa_{i} & \\equiv \\frac{2}{d_{i} M_{i}} \\prod_{\\Lambda=0}^4 \\sqrt{\\frac{a_{\\Lambda}}{e^{b_{\\Lambda}}}}\\, \\prod_{j \\neq i}\\left(\\frac{z_{j}^{+}-z_{i}^{-}}{z_{j}^{-}-z_{i}^{-}}\\right)^{\\operatorname{sign}(i-j) \\alpha_{i j}},\\\\\nA_{i} &=\\frac{(2 \\pi)^8\\,M_{i} }{\\kappa_{i}}\\,\\prod_{a=1}^7 R_{y_a}.\n\\end{aligned}\n\\label{eq:Area&SurfaceGrav}\n\\end{equation}\nThe rod on its own corresponds to the static limit of a four-charge non-extremal black hole \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca}.\\footnote{Since all $P_i^{(\\Lambda)}$ are non-zero, the rod carries non-zero M2-M2-M2-KKm charges given by \\eqref{eq:individualMcharges}.} However, in the present ansatz, one can stack linearly multiple black holes on a line and their interactions can be studied.\n\\item[-] \\underline{\\emph{Bubble rods:}} One of the T$^6$ directions, for instance $y_1$ or $y_2$, shrinks at the $i^\\text{th}$ rod while all other fibers are finite by taking\n\\begin{equation}\n\\pm G_i^{(1)} =-G_i^{(0)} = a_0 P_i^{(0)} = a_1 P_i^{(1)} =\\frac{1}{2} , \\quad G_i^{(2)}=G_i^{(3)}=P_i^{(2)}=P_i^{(3)}= 0,\n\\label{eq:bubblerodWeight1}\n\\end{equation}\nwhere the ``$+$'' corresponds to a shrinking $y_1$-circle and the ``$-$'' corresponds to a shrinking $y_2$-circle. The weights for which another T$^6$ direction degenerates can be obtained by permuting the $I=1,2,3$ indexes. Similarly, the $y_7$ circle shrinks while all other fibers remain finite by considering\n\\begin{equation}\na_0\\,P_i^{(0)}\\= 1 , \\qquad G_i^{(\\Lambda)} \\= P_i^{(I)} \\= 0\\,.\n\\label{eq:bubblerodWeight2}\n\\end{equation}\nFor each possibility, the $(r_i, y_a)$-subspace corresponds to an origin of an $\\mathbb{R}^2$ defining the locus of a bubble with a T$^6\\times$S$^2$ or T$^5\\times$S$^3$ (see Fig.\\ref{fig:RodCategories} and \\ref{fig:TouchingRods}).\\footnote{\\label{footnote2}As for a black rod, the local topology of the bubble depends if the bubble rod is disconnected (Fig.\\ref{fig:RodCategories}) or connected (Fig.\\ref{fig:TouchingRods}).} It corresponds to a smooth coordinate degeneracy if an algebraic equation, which we can denote as a \\emph{bubble equation}, is satisfied. For a rod where the $y_1$ or $y_2$ circle degenerates smoothly, the bubble equation relates the asymptotic radius to the internal parameters as follows\\footnote{If the rod is connected to another one, e.g. $z_{i-1}^+=z_i^-$, the constraint is always finite since $(z_{i-1}^+-z_i^-)\/d_i=\\text{cst}$.}\n\\begin{equation}\nk_{i}\\,R_{y_{a}} \\= d_{i} M_{i}\\,\\sqrt{\\frac{e^{b_{1}+b_0}}{a_{1}a_{0}}}\\, \\prod_{j \\neq i}\\left(\\frac{z_{j}^{+}-z_{i}^{-}}{z_{j}^{-}-z_{i}^{-}}\\right)^{\\operatorname{sign}(j-i) \\alpha_{i j}}, \\quad a=1 \\text{ or }2,\n\\label{eq:RegCondBu}\n\\end{equation}\nwhere $k_i$ is an orbifold parameter $k_i\\in \\mathbb{N}$ that corresponds to the order of the conical deficit on the $\\mathbb{R}^2$. The constraint corresponding to the degeneracy of another T$^6$ direction or the $y_7$-circle is obtained by permuting $\\frac{e^{b_1}}{a_1} \\to \\frac{e^{b_I}}{a_I}$ or $\\frac{e^{b_1}}{a_1} \\to \\frac{e^{b_0}}{a_0}$ respectively. Moreover, one can check that the gauge field is regular at the rod (see Appendix \\ref{App:AttheRod}).\n\nThe smooth bubble where one of the T$^6$ directions degenerates has nonzero $P_i^{(0)}$ and $P_i^{(I)}$. Therefore, it carries only a M2 brane charge and a KKm charge, given by \\eqref{eq:individualMcharges}. Similarly, the smooth bubble that shrinks the $y_7$ direction carries only a KKm charge. Moreover, these charges can be freely reduced to zero without changing the topology of the bubble by taking the neutral limit \\eqref{eq:neutrallimit}, and the resulting objects will be smooth vacuum bubbles in M-theory.\n\nTherefore, \\emph{seven types of physical rod sources} correspond to a smooth coordinate degeneracy of one the T$^6\\times$S$^1$ directions defined by an origin of an $\\mathbb{R}^2$ where a smooth bubble is sitting.\n\\end{itemize}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{RodSpecies.pdf}\n\\caption{Schematic description of three physical rods and the behavior of the circles on the $z$-axis when the rod are disconnected. The black rod sources a static M2-M2-M2-KKm black hole with a T$^7\\times$S$^2$ horizon. The species-1 bubble rod corresponds to the degeneracy of the $y_1$-circle inducing a T$^6\\times$S$^2$ M2-KKm bubble The species-7 bubble rod induces the degeneracy of the $y_7$-circle and a T$^6\\times$S$^2$ KKm bubble. Each section in between the sources has a conical excess, defining a strut.}\n\\label{fig:RodCategories}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{ConnectedRods.pdf}\n\\caption{Schematic description of connected rods by making the rods of Fig.\\ref{fig:RodCategories} touch. The $\\phi$-circle does not shrink along the rod configuration, and the solution is strut-free.}\n\\label{fig:TouchingRods}\n\\end{figure}\n\n\nThere are 8 categories of sources that can be used to generate physical asymptotically-$\\mathbb{R}^{1,3}$ solutions in M-theory: a non-extremal M2-M2-M2-KKm black hole, 6 non-BPS M2-KKm bubbles and a non-BPS KKm bubble. Remarkably, we have for any choices:\n\\begin{equation}\n\\begin{split}\n\\alpha_{jk}& \\= \\begin{cases} \n1 \\qquad &\\text{if the }j^\\text{th}\\text{ and }k^\\text{th}\\text{ rods are of the same nature,} \\\\\n\\frac{1}{2} \\qquad &\\text{otherwise,}\n\\end{cases} \n\\end{split}\n\\label{eq:dialphaDefcharged2}\n\\end{equation}\nwhich means that the three-dimensional base does not depend on the specific nature of the sources.\n\nFinally, one can think of other types of sources that are singular in M-theory but regular in other duality frameworks. This will be clarified when discussing the ansatz in other string theory frameworks in the section \\ref{sec:dualFrame}. \n\n\nThe effective mass induced by the $i^\\text{th}$ rod, ${\\cal M}_i$, can be derived from \\eqref{eq:ADMmass4d} by isolating the contribution of the rod. We find\n\\begin{equation}\n{\\cal M}_i \\= \\frac{1}{4 G_4} \\left(a_0 \\coth b_0\\, Q_{\\text{KKm},i}+ \\sum_{I=1}^3 a_I \\coth b_I\\, Q_{\\text{M2},i}^I\\right)\\,,\n\\end{equation}\nwhere the brane charges carried by the rod are given in \\eqref{eq:individualMcharges}. Therefore, the charge-to-mass ratios induced by the branes are determined by $a_\\Lambda \\coth b_\\Lambda$, and are the same for all the rods. It is a consequence for having a linear ansatz. If it still allows to remarkably deviate from the BPS regime, one cannot have two rods with opposite charges for instance.\n\n\\newpage\n\\item[\u2022] \\textbf{Regularity on the $z$-axis out of the rods:}\n\nAt $\\rho=0$ and $z \\notin [ z_i^-,z_i^+]$, the functions $R_+^{(i)}\/R_-^{(i)}$ and $E_{\\pm \\pm }^{(i,j)}$ are finite and non-zero. Therefore, the warp factors are well-defined and the $\\phi$-circle degenerates as the cylindrical coordinate singularity. The analysis reduces to the regularity of the three-dimensional base, $ds_3^2 = e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2$, and the warp factor $e^{2\\nu}$ can induce conical singularities. More precisely, we have\n\\begin{equation}\ne^{2 \\nu} \\sim \\begin{cases}\n~~ 1 &\\text{if } z < z_{1}^- \\text{ and } z > z_{n}^+,\\\\\n~~ d_i^2\\qquad &\\text{if } z \\in \\, ]z_{i-1}^+, z_i^- [\\,,\\quad i=2,..,n,\n\\end{cases}\n\\end{equation}\nwhere we remind that $d_i$ are the aspect ratios \\eqref{eq:dialphaDefcharged}. Therefore, the base metric above and below the rod configuration, $z < z_{1}^-$ and $z > z_{n}^+$, has no conical singularity and is smooth there. However, in between the $(i-1)^\\text{th}$ and $i^\\text{th}$ rods, we have $ds_3^2 \\sim d_i^2 \\left(d\\rho^2 + dz^2 +\\frac{\\rho^2}{d_i^2} d\\phi^2 \\right)$. Since $d_i <1$, the segment has a conical excess and describes a strut, i.e. a string with a negative tension (see Fig.\\ref{fig:RodCategories}). The strut is a singularity that accounts for the lack of repulsion between the $(i-1)^\\text{th}$ and $i^\\text{th}$ sources. It can be associated with a curvature singularity and an energy as done in \\cite{Costa:2000kf} and reviewed in \\cite{Bah:2021owp}.\n\nThe existence of a strut between two rods requires that they are disconnected. Therefore, by connecting the rods, which implies that they are of different nature, the solutions are free of struts and conical excesses \\cite{Elvang:2002br,Bah:2021owp,Bah:2021rki}. These solutions consist in connecting bubbles and possibly non-extremal black holes on a line (see Fig.\\ref{fig:TouchingRods}). For such configurations, the sources do not require a singular mechanism to be prevented from collapsing and are kept apart by topological bubbles \\cite{Bah:2021owp}. This provides a new paradigm in the microstate geometry program for constructing smooth bubbling geometries without the aid of supersymmetry. The standard lore of supersymmetry is that gravitational attraction is counterbalanced by electromagnetic repulsion. In the present ansatz, the solutions are generically non-BPS with arbitrarily low electromagnetic potentials, and the lack of repulsion is balanced by the inherent pressure of squeezed bubbles \\cite{Bah:2021owp}. \n\\end{itemize}\n\nIn conclusion, the non-BPS floating brane ansatz contains a large phase space of axisymmetric regular static solutions that are asymptotic to four-dimensional flat space plus compactification circles. Typical solutions consist of non-extremal four-charge black holes and 7 types of smooth bubbles on a line, which are kept apart by struts if disconnected. In addition, smooth bubbling geometries can be constructed by considering chains of connected bubbles of different species. We will construct explicit solutions in section \\ref{sec:Sol4d}.\n\n\n\n\n\n\\subsection{Five-dimensional solutions}\n\\label{sec:5dNonBPSGen}\n\nTo construct five-dimensional non-BPS solutions, the four-dimensional base, $(\\rho,z,\\phi,y_7)$, must be considered as a four-dimensional infinite space. That is, we take $y_7$ to be an angle of periodicity $2\\pi$. As detailed in the appendix \\ref{App:Reg5d}, one must also consider the neutral limit \\eqref{eq:neutrallimit} for the pair $(Z_0,H_0)$ and source $Z_0$ by a semi-infinite rod.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.22\\textwidth]{Rods5d.pdf}\n\\caption{Schematic description of axisymmetric rod sources with a semi-infinite rod in $Z_0$.}\n\\label{fig:RodsSources5d}\n\\end{figure}\n\nGeneric five-dimensional solutions are therefore given by \\eqref{eq:nonBPSfloatingRemind}, sourced by $n$ finite rod of length $M_i$ and a $(n+1)^\\text{th}$ semi-infinite rod (see Fig.\\ref{fig:RodsSources5d}). We will consider that the $n^\\text{th}$ and $(n+1)^\\text{th}$ rods are connected to avoid a conical excess between them, $z_n^+=z_{n+1}^-$. The warp factors and gauge potentials are similar to the ones given for four-dimensional solutions \\eqref{eq:WarpfactorsRods} with the neutral limit \\eqref{eq:neutrallimit} for $(Z_0,H_0)$ and additional terms corresponding to the semi-infinite rod, that is\n\\begin{align}\n&Z_I= \\frac{1}{2a_I} \\left[e^{b_I} \\prod_{i=1}^n \\left( \\frac{R_+^{(i)}}{R_-^{(i)}}\\right)^{a_I P^{(I)}_i}-e^{-b_I} \\prod_{i=1}^n \\left( \\frac{R_-^{(i)}}{R_+^{(i)}}\\right)^{a_I P^{(I)}_i} \\right]\\,,\\quad T_I\\=-\\frac{1}{Z_I}\\,\\sqrt{1+a_I^2\\,Z_I^2} \\,,\\nonumber\\\\\n& Z_0 \\= \\left(r_+^{(n)}-(z-z_n^+) \\right)^{-G_{n+1}}\\,\\prod_{i=1}^n \\left( \\frac{R_+^{(i)}}{R_-^{(i)}}\\right)^{G_i} \\,,\\quad H_0 \\=0\\,,\\quad W_\\Lambda = \\prod_{i=1}^n \\left( \\frac{R_+^{(i)}}{R_-^{(i)}}\\right)^{G^{(\\Lambda)}_i} \\, , \\label{eq:WarpfactorsRods5d} \n\\end{align}\n\\begin{align}\n&e^{2\\nu} \\= \\frac{1}{2} \\left(\\frac{r_+^{(n)}-(z-z_n^+)}{r_+^{(n)}} \\right)^{{G_{n+1}}^2}\\,\\prod_{i=1}^n \\left(\\frac{E_{++}^{(n,i)}\\,R_-^{(i)}}{E_{+-}^{(n,i)}\\,R_+^{(i)}} \\right)^{G_{n+1}G_i }\\,\\prod_{i,j=1}^n\\, \\left( \\frac{E_{+-}^{(i,j)}E_{-+}^{(i,j)}}{E_{++}^{(i,j)}E_{--}^{(i,j)}}\\right)^{\\frac{1}{2}\\,\\alpha_{ij}} \\,, \\nonumber\n\\end{align}\nwhere the main functions are given in \\eqref{eq:RpmdefRemind} and the exponents, $\\alpha_{ij}$, are now given by\\footnote{The exponents are slightly different from \\eqref{eq:Defalpha} due to the neutral limit \\eqref{eq:neutrallimit} taken for the pair $(Z_0,H_0)$. More precisely, we have taken $(a_0,b_0) \\to \\infty$ with $G_i = a_0 P^{(0)}_i$ finite in \\eqref{eq:HarmFuncGen} and \\eqref{eq:HarmFuncGen2}.}\n\\begin{align}\n\\alpha_{ij} &\\,\\equiv\\, G_i G_j\\+ \\sum_{\\Lambda =0}^3 G_i^{(\\Lambda)}G_j^{(\\Lambda)} \\+ \\sum_{I=1}^3 a_I^2\\,P_i^{(I)}P_j^{(I)}\\,. \\label{eq:alpha5d}\n\\end{align}\n\nNote that the solutions have no KKm charges along $y_7$ because $H_0=0$. We have therefore defined a family of non-BPS M2-M2-M2 solutions given by $10 n + 8$ parameters. \n\n\\subsubsection{Regularity constraints}\n\\label{sec:regM5d}\n\nWe will be brief in the regularity analysis since it is similar to the discussion for four-dimensional solutions. We refer to the appendix \\ref{App:Reg5d} for more details.\n\nThe five-dimensional metric \\eqref{eq:5dframework} is asymptotically flat if \n\\begin{equation}\n \\prod_{I=1}^3 \\frac{\\sinh b_I}{a_I} \\= 1\\,.\n\\label{eq:condonasymp5d}\n\\end{equation}\nThe solution is regular everywhere out of the finite rods, especially at the semi-infinite rod where the $y_7$ degenerates smoothly as an origin of a $\\mathbb{R}^2$ if and only if\n\\begin{equation}\nG_{n+1} \\= 1\\,.\n\\label{eq:CondGn+1}\n\\end{equation}\nMoreover, the solution is regular at the finite $i^\\text{th}$ rod if its weights fall into one of these eight categories (see appendix \\ref{App:AttheRod2} for more details):\n\\begin{itemize}\n\\item[-] \\underline{\\emph{Black rods:}} \nWe consider that\n\\begin{equation}\nG^{(\\Lambda)}_i \\=0\\,,\\qquad G_i\\= a_I P^{(I)}_i\\= \\frac{1}{2},\n\\label{eq:blackrodWeight5d}\n\\end{equation}\nand the rod corresponds to a horizon of a black hole where the timelike Killing vector $\\partial_t$ shrinks. The surface gravity, temperature and area of the black hole are given by\n\\begin{align}\n\\kappa_i &\\,\\equiv\\, \\frac{2 \\sqrt{z_n^+-z_i^+}}{d_i\\,M_i}\\, \\sqrt{\\frac{a_1 a_2 a_3}{e^{b_1+b_2+b_3}}}\\,\\prod_{j=1}^i \\left( \\frac{z_n^+ - z_j^-}{z_n^+ - z_j^+}\\right)^{ G_j} \\,\\prod_{j\\neq i} \\left(\\frac{z_j^+ - z_i^-}{z_j^- -z_i^-} \\right)^{\\text{sign}(i-j)\\, \\alpha_{ij}}\\,, \\nonumber\\\\\n{\\cal T}_i &\\= \\frac{\\kappa_i}{2\\pi}\\,,\\qquad A_i \\= \\frac{(2\\pi)^8\\,M_i}{\\kappa_i}\\,\\prod_{a=1}^6 R_{y_a}\\,,\n\\label{eq:surfaceGrav5d}\n\\end{align}\nwhere $d_i$ is still given by \\eqref{eq:dialphaDefcharged}. Because the three $P_i^{(I)}$ are finite, the black hole carries generically three M2 charges given by \\eqref{eq:individualMcharges}. Each charge can be taken to be zero by considering the neutral limit \\eqref{eq:neutrallimit}, which corresponds to $a_I = \\sinh b_I\\to \\infty$. \n\nMore precisely, the rod corresponds to the horizon of a three-charge static non-extremal black hole for which the single-center solutions have been derived in \\cite{Cvetic:1996xz,Cvetic:1997uw,Cvetic:1998xh}. \n\n\\item[-] \\underline{\\emph{Bubble rods:}} The $y_1$ or $y_2$ circle shrinks at the $i^\\text{th}$ rod while all other fibers are finite if we consider\n\\begin{equation}\n \\pm G_i^{(1)} = -G_i^{(0)} =G_i = a_1 P_i^{(1)} =\\frac{1}{2} , \\quad G_i^{(2)}=G_i^{(3)}=P_i^{(2)}=P_i^{(3)}= 0,\n\\label{eq:bubblerodWeight15d}\n\\end{equation}\nwhere the ``$+$'' corresponds to a shrinking $y_1$-circle and the ``$-$'' corresponds to a shrinking $y_2$-circle. One can obtain the weights for another degenerating T$^6$ direction by permuting the $I=1,2,3$ indexes. Similarly, on can make the $y_7$ circle degenerate by considering\n\\begin{equation}\nG_i \\= 1 , \\qquad G_i^{(\\Lambda)} \\= P_i^{(I)} \\= 0\\,.\n\\label{eq:bubblerodWeight25d}\n\\end{equation}\nFor each possibility, the local geometry corresponds to a bubble sitting at a smooth origin of a $\\mathbb{R}^2$ if a bubble equation is satisfied. For a rod that makes the circle $y_1$ or $y_2$ degenerate smoothly, the bubble equation constrains the internal parameters as a function of the extra-dimension radius such as\n\\begin{equation}\nk_i \\,R_{y_a} \\= \\frac{d_{i} M_{i}}{\\sqrt{z_n^+-z_i^+}}\\,\\sqrt{\\frac{e^{b_{1}}}{a_{1}}}\\, \\prod_{j=1}^i \\left( \\frac{z_n^+ - z_j^+}{z_n^+ - z_j^-}\\right)^{ G_j} \\, \\prod_{j \\neq i}\\left(\\frac{z_{j}^{+}-z_{i}^{-}}{z_{j}^{-}-z_{i}^{-}}\\right)^{\\operatorname{sign}(j-i) \\alpha_{i j}}, \\quad a=1 \\text{ or }2,\n\\label{eq:RegCondBu5d}\n\\end{equation}\nwhere $k_i$ is an orbifold parameter, $k_i\\in \\mathbb{N}$, that corresponds to the order of the conical deficit on the $\\mathbb{R}^2$. The bubble equation corresponding to another degenerating T$^6$ direction or the $y_7$-circle is obtained by permuting $\\frac{e^{b_1}}{a_1} \\to \\frac{e^{b_I}}{a_I}$ or $\\frac{e^{b_1}}{a_1} \\to 1$ respectively. \n\nThe smooth bubble where one of the T$^6$ direction degenerates carries only a M2 brane charge, given by \\eqref{eq:individualMcharges}, because only one $P^{(I)}_i$ is nonzero \\eqref{eq:bubblerodWeight15d}. Similarly, the smooth bubble where the $y_7$ direction shrinks is purely topological and carries no charge. We have therefore six types of M2 bubbles and one type of vacuum bubble that can smoothly source the solutions.\n\\end{itemize}\n\nThe mass induced by each rod, ${\\cal M}_i$, can be derived from \\eqref{eq:ADMmass5d} and related to the brane charges \\eqref{eq:individualMcharges} such as\n\\begin{equation}\n{\\cal M}_i \\= \\frac{\\pi}{2 G_5} \\,\\sum_{I=1}^3 a_I \\coth b_I \\,Q_{\\text{M2},i}^I\\,.\n\\end{equation}\nThe charge-to-mass ratios for each stack of M2 branes are fixed by $a_I \\coth b_I$ for all rods.\n\nAs for four-dimensional solutions, a strut is separating each pair of disconnected rods. More precisely, if the $(i-1)^\\text{th}$ and $i^\\text{th}$ rods are not connected, the $\\phi$-circle degenerates in between them with a conical excess of order $d_i^{-1}>1$ \\eqref{eq:dialphaDefcharged}. This singularity can be bypassed by considering connected rods. For such configurations, the struts disappear and the non-BPS sources are held apart by pure topology. \n\n\nTo conclude, the family of solutions, given by \\eqref{eq:WarpfactorsRods5d}, contains a large phase space of regular non-BPS axisymmetric static solutions that are asymptotic to five-dimensional flat space plus compactification circles. Typical solutions consist of non-BPS non-extremal three-charge black holes and 7 types of smooth bubbles on a line, which are held apart by struts if disconnected. Moreover, smooth bubbling geometries can be constructed by considering chains of connected bubbles of different species. We will construct explicit solutions in section \\ref{sec:Sol5d}.\n\n\\subsection{Other duality frames}\n\\label{sec:dualFrame}\n\nSo far we have limited the discussion to the M-theory frame, but the construction can be dualized to different string theory frames through dimensional reduction and T-dualities. Each frame will have new types of regular rod sources that are singular in M-theory. To illustrate this property, we will focus on dualization in the D1-D5-P-KKm frame, but we refer the interested reader to the appendix \\ref{App:dualFrame} where we have dualized the solutions in six different frames, from the D2-D2-D2-D6 to the P-M5-M2-KKm frame. In addition, the study of the D1-D5-P-KKm frame will provide a link to the author's previous constructions of smooth bubbling geometries \\cite{Bah:2020pdz,Bah:2021owp,Bah:2021rki}.\n\n\n\nAfter reduction along $y_5$ and a series of T-dualities along $y_1$, $y_2$ and $y_6$, the M2-M2-M2-KKm solutions \\eqref{eq:nonBPSfloatingRemind} transform to D1-D5-P-KKm solutions where the common direction for the momentum P charge, the D1 and D5 branes is $y_6$. The metric, the R-R gauge fields $C^{(p)}$, the NS-NS two-form gauge field $B_2$ and the dilaton in type IIB are given by \n\\begin{align}\nds_{10}^2 \\= &- \\frac{dt^2}{Z_3 \\sqrt{W_3W_0 Z_1 Z_2}} \\+ \\sqrt{\\frac{ Z_1 Z_2}{W_3 W_0}} \\left[\\frac{1}{Z_0} \\left(dy_7 +H_0 \\,d\\phi\\right)^2 \\+ Z_0\\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right)\\right] \\nonumber\\\\\n& +\\sqrt{\\frac{Z_1}{Z_2}}\\left[\\sqrt{\\frac{W_3}{W_0}} \\,\\left(W_1\\,dy_1^2+ \\frac{dy_2^2}{W_1}\\right) +\\sqrt{\\frac{W_0}{W_3}} \\left(\\frac{dy_3^2}{W_2} + W_2 \\,dy_4^2 \\right) \\right] \\label{eq:D1D5PKKm} \\\\\n& + \\frac{Z_3}{\\sqrt{W_3 W_0 Z_1 Z_2}}\\,\\left(dy_6+T_3\\, dt \\right)^2\\,, \\qquad \\Phi \\= \\frac{1}{2} \\log \\frac{Z_1}{W_0 W_3 Z_2}\\,,\\qquad B_2 \\= 0\\,, \\nonumber \\\\\nC^{(0)} \\=& 0 \\,,\\qquad C^{(2)}\\= H_2 \\,d\\phi \\wedge dy_7 \\- T_1 \\,dt \\wedge dy_6\\,,\\qquad C^{(4)} \\= 0\\,,\\nonumber \n\\end{align}\nwhere $H_2$ is the electromagnetic dual of $T_2$, and is given by the same expression as $H_0$ in \\eqref{eq:WarpfactorsRods} but with $P_i^{(0)} \\to P_i^{(2)}$ (see appendix \\ref{App:dualFrame}).\n\nOne can recognize the type IIB frame of topological stars and five-dimensional charged Weyl solutions of \\cite{Bah:2020pdz} by taking\n\\begin{equation}\nW_0 \\= W_1\\= W_2\\= W_3 \\= 1 \\,,\\qquad Z_0 \\=Z_1 \\=Z_2 \\= Z \\,,\\qquad T_3 \\= 0 \\,,\\qquad Z_3 \\= W\\,,\n\\end{equation}\nTaking $T_3=0$ while keeping a non-trivial $Z_3=W$ is possible by considering the neutral limit for the pair $(Z_3,T_3)$ \\eqref{eq:neutrallimit}, that is no P-charges. If we now consider $Z_0$ to be different from $Z_1=Z_2$, we retrieve the framework of \\cite{Bah:2021owp,Bah:2021rki} where six-dimensional smooth bubbling solutions, as bubble bag ends \\cite{Bah:2021rki}, have been constructed.\n\n\n\nThe warp factors and gauge potentials are governed by the linear system of equations obtained from the non-BPS floating brane ansatz of section \\ref{sec:WeylAns}.\n\n\nFor the non-BPS solutions, the physical rod sources are different from the eighth found in sections \\ref{sec:regM4d} and \\ref{sec:regM5d}. For instance, the $y_6$-circle degenerates at the $i^\\text{th}$ rod for a solutions given by \\eqref{eq:WarpfactorsRods} defining the locus of a bubble if we consider its associated weights to be \n\\begin{equation}\nG^{(\\Lambda)}_i \\= 0 \\,,\\qquad a_0 P^{(0)}_i \\= a_1 P^{(1)}_i \\= a_2 P^{(2)}_i \\= -a_3 P^{(3)}_i \\=\\frac{1}{2}\\,.\n\\label{eq:bubblerodWeight4}\n\\end{equation}\nAs a black rod \\eqref{eq:blackrodWeight}, they do not require to turn on the weights of the torus warp factors $W_\\Lambda$, and therefore the T$^4$ is rigid as for BPS solutions. The only difference is that $a_3 P_i^{(3)}=-\\frac{1}{2}$. However, taking $a_3 P_i^{(3)}$ negative is compatible with the regularity condition \\eqref{eq:RegPositivity} only if one takes the neutral limit, $(a_3,b_3)\\to \\infty$ \\eqref{eq:neutrallimit}. It requires to take the gauge potential for the P charge to be zero and the solutions have no momentum charges as in \\cite{Bah:2020pdz,Bah:2021owp,Bah:2021rki}. A rod given by \\eqref{eq:bubblerodWeight4} corresponds to a smooth D1-D5-KKm bubble in type IIB.\\footnote{Smoothness requires a bubble equation of a similar type to those obtained previously.} However, it is singular in the M-theory frame \\eqref{eq:nonBPSfloatingRemind}. Therefore, there are specific species of bubble rods that correspond to smooth loci in a unique string theory frame.\n\n\\section{Explicit four-dimensional solutions}\n\\label{sec:Sol4d}\n\nIn this section, we derive explicit solutions that are asymptotic to $\\mathbb{R}^{1,3}$ with extra compact dimensions using the non-BPS floating brane ansatz in different string theory frames. There is a very large number of configurations that one can think about. First, we will construct chain of non-extremal four-charge black holes, either separated by struts or smooth bubbles, and discuss their physics. Second, we will construct smooth bubbling geometries in the manner of \\cite{Bah:2021owp,Bah:2021rki}, that have the same mass and charges as non-extremal four-charge black holes.\n\n\n\n\\subsection{Chain of static non-extremal black holes}\n\nWe first study solutions for which the main ingredients are non-extremal four-charge black holes on a line.\n\n\\subsubsection{Black holes separated by struts}\n\nWe consider $n$ finite rods of length $M_i$ on the $z$-axis with weights corresponding to black rods given by \\eqref{eq:blackrodWeight} (see Fig.\\ref{fig:ChainofBHStruts}). It is convenient to divide the warp factors, $Z_\\Lambda$ \\eqref{eq:WarpfactorsRods}, into two parts: one will encode changes in topology and the other will correspond to a well-behaved flux decoration such as\n\\begin{equation}\nZ_\\Lambda \\= \\frac{{\\cal Z}_\\Lambda}{\\sqrt{U_t}}\\,,\\qquad U_t \\,\\equiv\\, \\prod_{i=1}^n\n\\left(1-\\frac{2 M_i}{R_+^{(i)}} \\right)\\,,\\qquad {\\cal Z}_\\Lambda \\,\\equiv\\, \\frac{e^{b_\\Lambda}-e^{-b_\\Lambda}\\,U_t}{2 a_\\Lambda}\n\\end{equation}\nThe remaining functions \\eqref{eq:WarpfactorsRods} give\n\\begin{align}\n& W_\\Lambda = 1,\\quad e^{2\\nu} \\=\\prod_{i,j=1}^n\\, \\sqrt{ \\frac{E_{+-}^{(i,j)}E_{-+}^{(i,j)}}{E_{++}^{(i,j)}E_{--}^{(i,j)}}},\\quad H_0 \\= \\frac{1}{2 a_0}\\sum_{i=1}^n \\left(r_-^{(i)}-r_+^{(i)} \\right),\\quad T_I\\=-\\sqrt{a_I^2+\\frac{U_t}{{\\cal Z}_I^2}}, \\nonumber \n\\end{align}\nwhere the generating functions $R_\\pm^{(i)}$ and $E_{\\pm \\pm}^{(i,j)}$ are given in \\eqref{eq:RpmdefRemind}. The metric and gauge potential in M-theory are given in \\eqref{eq:nonBPSfloatingRemind}, while the four-dimensional reduction along T$^6\\times$S$^1$ is given by \\eqref{eq:4dSol}:\n\\begin{align}\nds_4^2 &\\= -\\frac{U_t\\,dt^2}{\\sqrt{{\\cal Z}_0 {\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3}} \\+ \\frac{\\sqrt{{\\cal Z}_0 {\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3}}{U_t} \\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right), \\nonumber \\\\\nz^I &\\= i\\,\\frac{|\\epsilon_{IJK}|}{2} \\sqrt{\\frac{{\\cal Z}_J {\\cal Z}_K}{{\\cal Z}_0\\,{\\cal Z}_I} },\\quad \\Phi_\\Lambda \\=0 \\,,\\quad \\bar{F}^{0} \\= -dH_0 \\wedge d\\phi \\,,\\quad \\bar{F}^{I} \\= dT_I \\wedge dt\\,.\n\\label{eq:4dprofilechainofBH}\n\\end{align}\nThe function $U_t$ behaves as a product of ``Schwarzschild factors'' that vanishes at each rod and makes the timelike Killing vector shrink. It induces the topology of the solutions and forms the horizons. The ${\\cal Z}_\\Lambda$ depend only on the gauge field parameters, and are always positive and finite. In the neutral limit \\eqref{eq:neutrallimit}, all ${\\cal Z}_\\lambda$ converge to 1 while $U_t$ remains unchanged. We retrieved the solutions of Schwarzschild black holes on a line \\cite{Israel1964,Gibbons:1979nf}.\n\n\nAt large distance $r$, with $(\\rho,z)=(r\\sin\\theta,r\\cos\\theta)$, the geometry is asymptotically flat if \\eqref{eq:condonasymp} is satisfied, and one can read the ADM mass from \\eqref{eq:ADMmass4d}:\n\\begin{equation}\n{\\cal M} \\= \\frac{\\sum_{\\Lambda=0}^3 \\coth b_\\Lambda}{8 G_4}\\,\\sum_{i=1}^n M_i\\,.\n\\label{eq:MassBHchain}\n\\end{equation}\nMoreover, the solutions carry four charges that are M2-M2-M2-KKm charges in M-theory. They are given by the sum of the rod charges carried by each rod \\eqref{eq:individualMcharges}: \n\\begin{equation}\nQ^I_{\\text{M2}} \\= \\frac{1}{2a_I}\\,\\sum_{i=1}^n M_i \\,,\\qquad Q_{\\text{KKm}} \\= \\frac{1}{2a_0} \\,\\sum_{i=1}^n M_i.\n\\end{equation}\n\\begin{itemize}\n\\item[\u2022] \\underline{A unique rod: map to the four-charge non-extremal black hole \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca}.}\n\\end{itemize} \nWe consider a unique rod, $n=1$, and the spherical coordinates $$(\\rho,\\,z)= \\left(\\sqrt{r \\left(r -M_{1}\\right)} \\sin \\theta,\\, \\left(r-\\frac{M_{1}}{2}\\right) \\cos \\theta +z_{1}\\right). $$ The four-dimensional metric, gauge fields and scalars \\eqref{eq:4dprofilechainofBH} simplify to\n\\begin{align}\nds_4^2 &\\= -\\frac{r(r-M_1)\\,dt^2}{\\sqrt{r_0 r_1 r_2 r_3}} \\+ \\sqrt{r_0 r_1 r_2 r_3} \\left(\\frac{dr^2}{r(r-M_1)}+d\\theta^2 +\\sin^2\\theta d\\phi^2\\right),\\quad \\Phi_\\Lambda \\=0, \\label{eq:4chargeBH} \\\\\nz^I &\\= i\\,\\frac{a_0 a_I\\,|\\epsilon_{IJK}|}{2\\sinh b_0 \\sinh b_I}\\sqrt{\\frac{r_J r_K}{r_0\\,r_I} } \\,,\\quad \\bar{F}^{0} \\= \\frac{M_1}{2 \\sinh b_0}\\sin \\theta \\,d\\theta \\wedge d\\phi \\,,\\quad \\bar{F}^{I} \\= \\frac{M_1}{2 \\sinh b_I\\,r_I^2}dt\\wedge dr\\,, \\nonumber\n\\end{align}\nwhere we have defined $r_\\Lambda \\,\\equiv\\, r+ \\frac{e^{-b_\\Lambda}}{2 \\sinh b_\\Lambda}\\,M_1$.\\footnote{Note that the metric no longer depend on $a_\\Lambda$. In four dimensions, the dependence in $a_\\Lambda$ for regular solutions is a function of $\\prod_\\Lambda a_\\Lambda$ which is fixed in terms of $b_\\Lambda$ to have asymptotically flat solutions \\eqref{eq:condonasymp}.} We retrieve the static solutions of a four-charge non-extremal black hole \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca} by relating the boost parameters in these papers, $\\delta_\\Lambda$, to our parameters, $b_\\Lambda$, such as $\\cosh 2\\delta_\\Lambda = \\coth b_\\Lambda$.\n\\begin{itemize}\n\\item[\u2022] \\underline{Multiple four-charge black holes on a line.}\n\\end{itemize}\nBy considering $n>1$, we perform the multicentric generalization of the static solutions in \\cite{Cvetic:1995kv,Chong:2004na,Chow:2014cca}.\nIn the IR, the geometry is made of a chain of four-charge static non-extremal black holes on the $z$-axis held apart by struts (see Fig.\\ref{fig:ChainofBHStruts}). At each rod, the timelike Killing vector $\\partial_t$ vanishes defining the locus of a horizon with area \\eqref{eq:Area&SurfaceGrav}. If one wants the black holes to be in thermal equilibrium, their surface gravity must be equal \\eqref{eq:Area&SurfaceGrav}, which non-trivially constrains their positions and sizes. Moreover, in between each rod the $\\phi$-circle shrinks with a conical excess of order $d_i^{-1} >1$ \\eqref{eq:dialphaDefcharged} defining the loci of the struts.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.38\\textwidth]{ChainofBHStruts.pdf}\n\\caption{Description of a chain of static four-charge non-extremal black holes in the four-dimensional framework \\eqref{eq:4dprofilechainofBH} and the behavior of the $\\phi$-circle on the $z$-axis.}\n\\label{fig:ChainofBHStruts}\n\\end{figure}\n\nThe struts carry an energy that counterbalances the attraction between the black holes. This energy corresponds to the interaction between the black holes in string theory. It can be derived using the method of \\cite{Costa:2000kf,Bah:2021owp}. More precisely, the energy of the strut that separates the $(i-1)^\\text{th}$ and $i^\\text{th}$ black holes on the chain carries an energy given by\n\\begin{equation}\nE_i \\= - \\frac{1-d_i}{4 G_4}\\,(z_i^- - z_{i-1}^+)\\,, \\qquad G_4 = \\frac{G_{11}}{(2\\pi)^{7} \\prod_{i=1}^{7} R_{y_i}}.\n\\end{equation}\nFor a configuration of two identical black holes separated by a distance $\\ell$, the energy of the strut in between them is given by\n\\begin{equation}\nE \\= - \\frac{M^2 \\ell}{4 G_4 (M+\\ell)^2},\n\\end{equation}\nwhere $M$ is the length of each rod. When the separation is large, $\\ell \\gg M$, the energy of the strut approximates the Newtonian potential between two particles of mass, $\\frac{M}{2G_4}$, in four dimensions. From this perspective, the strut measures the binding energy between the two black holes, or rather the potential energy needed to keep the two black holes from collapsing on each other. An important observation is that the effective ADM masses of the black holes \\eqref{eq:MassBHchain}, from the Newtonian point of view, depend on $b_\\Lambda$, which are associated to the four charges of the black holes \\eqref{eq:4chargeBH} with the regularity condition \\eqref{eq:condonasymp}. This implies that the binding energy as measured by the strut also accounts for effects due to the electromagnetic fields of the non-extremal black holes.\n\nFor finite separation between the two black holes, the gravitational potential between them deviates significantly from that of the Newtonian limit. In particular, the gravitational potential between two black holes vanish when $\\ell \\to 0$. In this limit, the two rod sources merge and the two-body configuration becomes a single non-extremal black hole. \n\n\n\n\n\\subsubsection{Black holes separated by smooth bubbles}\n\\label{sec:chainBHbu}\n\nStruts are singularities that do not have a consistent UV description in string theory as they correspond to cosmic strings with negative tension. Therefore, it is crucial to find a mechanism to resolve them and to obtain more relevant configurations in string theory. The non-BPS regime will require a new mechanism for this task. In the analysis of \\cite{Costa:2000kf} and continued in \\cite{Bah:2021owp}, it was shown that struts can be classically resolved into smooth bubbles. More precisely, one can consider the same black hole systems as before but sourcing each segment between them by a bubble rod. The resulting configuration will consist in a chain of connected rods which would be a succession of black holes and bubbles without struts. The condition for having connected rods fixes their centers in terms of their size:\n\\begin{equation}\nz_i^+ \\= z_{i+1}^- \\,,\\quad \\forall i \\qquad \\Rightarrow \\qquad z_i \\= z_n^+ + \\frac{M_i}{2} - \\sum_{j=i}^n M_j\\,.\n\\label{eq:connectedcond}\n\\end{equation}\nOne can also freely choose the origin of the $z$-axis so that $z_n^+=0$. \n\nFor simplicity, we consider the bubble rods to be smooth KKm bubbles given by the weights \\eqref{eq:bubblerodWeight2}, but one could a priori take any of the seven species of bubble rods detailed in section \\ref{sec:regM4d}. This allows a faithful description of the solutions in the five-dimensional framework \\eqref{eq:5dframework}. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{ChainofBHBub.pdf}\n\\caption{Description of a chain of static four-charge non-extremal black holes and KKm bubbles in the five-dimensional framework \\eqref{eq:5dprofilechainofBH} and the behavior of the $\\phi$ and $y_7$ circles on the $z$-axis.}\n\\label{fig:ChainofBHBub}\n\\end{figure}\nTherefore, we consider $n=2N+1$ rods of lengths $M_i$ where the $N+1$ odd rods correspond to black rods \\eqref{eq:blackrodWeight} and the $N$ even rods correspond to KKm bubble rods \\eqref{eq:bubblerodWeight2} (see Fig.\\ref{fig:ChainofBHBub}). As in the previous section, we divide the warp factors $Z_\\Lambda$ \\eqref{eq:WarpfactorsRods} in meaningful pieces such as\n\\begin{equation}\n\\begin{split}\nZ_0 &\\= \\frac{{\\cal Z}_0}{U_{y_7}\\sqrt{U_t}}, \\qquad Z_I \\= \\frac{{\\cal Z}_I}{\\sqrt{U_t}}\\,,\\\\\nU_t &\\,\\equiv\\, \\prod_{i=1}^{N+1} \\left(1-\\frac{2 M_{2i-1}}{R_+^{(2i-1)}} \\right)\\,, \\qquad U_{y_7} \\,\\equiv\\, \\prod_{i=1}^N \\left(1-\\frac{2 M_{2i}}{R_+^{(2i)}} \\right)\\,, \\\\\n{\\cal Z}_0 &\\,\\equiv\\, \\frac{e^{b_0}-e^{-b_0}\\,U_t U_{y_7}^2}{2 a_0} \\,,\\qquad {\\cal Z}_I \\,\\equiv\\, \\frac{e^{b_I}-e^{-b_I}\\,U_t}{2 a_I}.\n\\end{split}\n\\end{equation}\n\nThe remaining functions \\eqref{eq:WarpfactorsRods} give\n\\begin{align}\n& W_\\Lambda = 1,\\qquad H_0 \\= \\frac{1}{2 a_0} \\left[r_-^{(1)}- r_+^{(n)}+\\sum_{i=1}^N \\left(r_-^{(2i)}-r_+^{(2i)} \\right) \\right],\\qquad T_I\\=-\\sqrt{a_I^2+\\frac{U_t}{{\\cal Z}_I^2}}, \\nonumber \\\\\n& e^{2 \\nu}=\\frac{E_{-+}^{(1, n)}}{\\sqrt{E_{++}^{(n, n)} E_{--}^{(1,1)}}} \\prod_{i=1}^{N} \\prod_{j=1}^{N+1} \\sqrt{\\frac{E_{--}^{(2 i, 2 j-1)} E_{++}^{(2 i, 2 j-1)}}{E_{-+}^{(2 i, 2 j-1)} E_{+-}^{(2 i, 2 j-1)}}}\\,,\n\\end{align}\nwhere $R_\\pm^{(i)}$ and $E_{\\pm \\pm}^{(i,j)}$ are given in \\eqref{eq:RpmdefRemind}. The metric and gauge potential in M-theory are given in \\eqref{eq:nonBPSfloatingRemind}, while the five-dimensional reduction along T$^6$ is given by \\eqref{eq:5dframework}\n\\begin{align}\nds_5^2 &\\= - \\frac{U_t\\,dt^2}{\\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{2}{3}}} \\+ \\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{1}{3}} \\left[\\frac{U_{y_7}}{{\\cal Z}_0} \\left(dy_7 +H_0 \\,d\\phi\\right)^2 \\+ \\frac{{\\cal Z}_0}{U_t U_{y_7}} \\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right)\\right], \\nonumber \\\\\nX^I &\\= \\frac{|\\epsilon_{IJK}|}{2} \\left(\\frac{{\\cal Z}_J {\\cal Z}_K}{{\\cal Z}_I^2} \\right)^\\frac{1}{3},\\qquad \\Phi_\\Lambda \\= 0 \\,,\\qquad F^{(I)} \\= dT_I \\wedge dt\\,.\n\\label{eq:5dprofilechainofBH}\n\\end{align}\nThe warp factors $U_t$ and $U_{y_7}$ force the degeneracy of the $t$ and $y_7$ fibers at the rods. In the regions where $U_{y_7}$ vanishes, that is at the even rods, the $y_7$ direction shrinks forming a smooth origin of an $\\mathbb{R}^2$ (with potential conical defects) if we have \\eqref{eq:RegCondBu}\n\\begin{equation}\nk_{i}\\,R_{y_7} \\,\\equiv\\, \\,\\frac{d_{2i} M_{2i}\\,e^{b_{0}}}{a_{0}}\\, \\prod_{j \\neq i}^N\\left(\\frac{z_{2j}^{+}-z_{2i}^{-}}{z_{2j}^{-}-z_{2i}^{-}}\\right)^{\\operatorname{sign}(j-i)} \\,\\prod_{j =1}^{N+1}\\left(\\frac{z_{2j-1}^{+}-z_{2i}^{-}}{z_{2j-1}^{-}-z_{2i}^{-}}\\right)^{\\frac{\\operatorname{sign}(j-i-1\/2)}{2}}, \\quad \\forall i=1,...,N,\n\\end{equation}\nwhere we remind that $z_j^\\pm$ are the rod endpoint coordinates on the $z$-axis \\eqref{eq:RpmdefRemind} and $d_j$ are the aspect ratios \\eqref{eq:dialphaDefcharged}. These $N$ ``bubble equations'' fix the lengths of the bubble rods, $M_{2i}$, according to the other parameters. Moreover, if one wants the black holes to be in thermal equilibrium, one needs in addition to impose that their surface gravities \\eqref{eq:Area&SurfaceGrav} are equal, which also constrains the length of the black rods. \n\nThe $\\phi$-circle does not shrink anymore along the rod configuration and the four-charge non-extremal black holes are held apart by KKm bubbles. The struts have been then successfully replaced by smooth bubbles. This mechanism has been explored in \\cite{Costa:2000kf,Bah:2021owp}. In one word, vacuum bubbles are reluctant to be squeezed as it wants to expand \\cite{Witten:1981gj}, and provide the necessary pressure between two non-BPS objects \\cite{Costa:2000kf}. If the electromagnetic fluxes stabilize the bubbles \\cite{Stotyn:2011tv}, their reluctance to be squeezed remains \\cite{Bah:2021owp}.\n\nFinally, the solutions are asymptotic to $\\mathbb{R}^{1,3}\\times$S$^1$ if \\eqref{eq:condonasymp} is satisfied. The ADM mass is given by \\eqref{eq:ADMmass4d}\n\\begin{equation}\n{\\cal M} \\= \\frac{\\sum_{\\Lambda=0}^3 \\coth b_\\Lambda\\,\\sum_{i=1}^{N+1} M_{2i-1}+2 \\coth b_0\\,\\sum_{i=1}^{N} M_{2i}}{8 G_4}\\,.\n\\end{equation}\nMoreover, the M2-M2-M2-KKm charges of the solutions in M-theory can be obtained from the sum of the charges carried by each rod \\eqref{eq:individualMcharges}: \n\\begin{equation}\nQ^I_{\\text{M2}} \\= \\frac{1}{2a_I}\\,\\sum_{i=1}^{N+1} M_{2i-1}\\,,\\qquad Q_{\\text{KKm}} \\= \\frac{\\sum_{i=1}^{2N+1} M_{i}+\\sum_{i=1}^{N} M_{2i}}{2a_0}.\n\\end{equation}\n\n\\subsection{Smooth bubbling geometries}\n\nWe now seek to construct smooth non-BPS bubbling solutions without black hole sources and struts. This consists in building a chain of connected bubble rods of different species. In M-theory alone, there are 7 species of smooth bubbles that can be used, and there are several others in different duality frameworks as described in section \\ref{sec:dualFrame}. This induces a wide variety of configurations possible. In this section, we will construct an explicit M2-M2-M2-KKm configuration and a D1-D5-KKm configuration in type IIB. \n\n\\subsubsection{In M-theory}\n\\label{sec:BuSolMtheory}\n\nAs described in section \\ref{sec:regM4d}, among the seven species of physical bubble rods in M-theory, six correspond to the smooth degeneracy of one of the T$^6$ direction and carry a unique M2 charge and a KKm charge. To construct a smooth configuration with M2-M2-M2-KKm charges, one needs at least three species of these kinds. \n\n\nTherefore, we will consider a chain of $n=3N$ connected bubble rods of length $M_i$ which will successively make the $y_1$, $y_3$ and $y_5$ circles shrink (see Fig.\\ref{fig:MtheoryBubbles}). Having connected rods constrains the rod centers in terms of their sizes as in \\eqref{eq:connectedcond}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{ChainofBuMtheory.pdf}\n\\caption{Description of a chain of smooth M2-KKm bubbles in M-theory and the behavior of the $\\phi$, $y_1$, $y_3$ and $y_5$ circles on the $z$-axis (the other fibers have finite size everywhere). The species-$k$ bubble rod corresponds to the degeneracy of the $y_k$-circle as an origin of a $\\mathbb{R}^2$.}\n\\label{fig:MtheoryBubbles}\n\\end{figure}\n\nThe expression of the warp factors and gauge potentials are given by \\eqref{eq:WarpfactorsRods} with the following choices of weights, obtained from \\eqref{eq:bubblerodWeight1},\n\\begin{equation}\na_0 P_i^{(0)}=-G_i^{(0)}= \\frac{1}{2}\\,,\\quad \\begin{array}{l} G_{3i-2}^{(1)} = a_1 P_{3i-2}^{(1)} =\\frac{1}{2} , \\quad G_{3i-2}^{(2)}=G_{3i-2}^{(3)}=P_{3i-2}^{(2)}=P_{3i-2}^{(3)}= 0, \\\\\nG_{3i-1}^{(2)} = a_2 P_{3i-1}^{(2)} =\\frac{1}{2} , \\quad G_{3i-1}^{(1)}=G_{3i-1}^{(3)}=P_{3i-1}^{(1)}=P_{3i-1}^{(3)}= 0, \\\\\nG_{3i}^{(3)} = a_3 P_{3i}^{(3)} =\\frac{1}{2} , \\quad G_{3i}^{(1)}=G_{3i}^{(2)}=P_{3i}^{(1)}=P_{3i}^{(2)}= 0.\n\\end{array}\n\\end{equation}\nAs in the previous constructions, it is convenient to split the warp factors $Z_\\Lambda$ such as\n\\begin{align}\nZ_0 &\\= \\frac{{\\cal Z}_0}{\\sqrt{U_1 U_2 U_3}}, \\qquad Z_I \\= \\frac{{\\cal Z}_I}{\\sqrt{U_I}}\\,,\\label{eq:WarpfactorMtheoryBu1} \\\\\nU_I &\\,\\equiv\\, \\prod_{i=0}^{N-1} \\left(1-\\frac{2 M_{3i+I}}{R_+^{(3i+I)}} \\right)\\,, \\qquad {\\cal Z}_0 \\,\\equiv\\, \\frac{e^{b_0}-e^{-b_0}\\,U_1 U_2 U_3}{2 a_0} \\,,\\qquad {\\cal Z}_I \\,\\equiv\\, \\frac{e^{b_I}-e^{-b_I}\\,U_I}{2 a_I}. \\nonumber\n\\end{align}\nThe functions $U_I$ are products of ``Schwarzschild factors'' that vanish at all the $(3i+I)^\\text{th}$ rods while the ${\\cal Z}_\\Lambda$ are non-zero and finite. The remaining functions \\eqref{eq:WarpfactorsRods} give\n\\begin{align}\n& W_0 = \\left( U_1 U_2 U_3\\right)^{\\frac{1}{2}},\\quad W_I \\= U_I^{-\\frac{1}{2}} ,\\quad H_0 \\= \\frac{1}{2 a_0} \\left( r_-^{(1)}- r_+^{(n)} \\right),\\quad T_I\\=-\\sqrt{a_I^2+\\frac{U_I}{{\\cal Z}_I^2}}, \\nonumber \\\\\n& e^{2 \\nu}=\\frac{E_{-+}^{(1, n)}}{\\sqrt{E_{++}^{(n, n)} E_{--}^{(1,1)}}} \\prod_{i,j=0}^{N-1} \\,\\prod_{\\substack{I,J=1\\\\I1$ consists therefore in a chain of static three-charge black holes in five dimensions. The black holes are separated by struts, that are segments where the $\\phi$-circle degenerates with a conical excess of order $d_i^{-1} >1$ \\eqref{eq:dialphaDefcharged}. The struts account for the lack of repulsion between the non-extremal black holes and encodes the binding energy of the system.\n\n\nThe singular struts can be replaced by regular bubbles in a manner similar to that used for the chain of four-dimensional black holes in the section \\ref{sec:chainBHbu}. Specifically, one can source the M-theory solutions with bubble rods at each segment between the black holes. The bubble rods can either be vacuum bubbles that make the $y_7$ circle degenerate smoothly \\eqref{eq:bubblerodWeight25d}, or M2 bubbles that correspond to the degeneracy of one of the T$^6$ directions \\eqref{eq:bubblerodWeight15d}.\n\n\n\\subsection{Smooth bubbling geometries}\n\nWe now construct smooth non-BPS bubbling solutions that consist in a chain of connected bubble rods of different species. We will restrict to the five-dimensional analogs of the M-theory solutions constructed in section \\ref{sec:BuSolMtheory}. One could have also built asymptotically-$\\mathbb{R}^{1,4}$ D1-D5 non-BPS bubbling solutions by taking the equivalent of the four-dimensional solutions of section \\ref{sec:BuSolIIB}.\n\nWe consider a chain of $n=3N$ connected bubble rods of length $M_i$ where the circles $y_1$, $y_3$ and $y_5$ successively degenerate. Having connected rods constrains the rod centers in terms of their sizes as in \\eqref{eq:connectedcond}, and we also assume that the origin of the $z$-axis is at $z_n^+=0$. The warp factors and gauge potential are given by \\eqref{eq:WarpfactorsRods5d} with the following choices of weights, obtained from \\eqref{eq:bubblerodWeight15d},\n\\begin{equation}\nG_i =-G_i^{(0)}= \\frac{1}{2}\\,,\\quad \\begin{array}{l} G_{3i-2}^{(1)} = a_1 P_{3i-2}^{(1)} =\\frac{1}{2} , \\quad G_{3i-2}^{(2)}=G_{3i-2}^{(3)}=P_{3i-2}^{(2)}=P_{3i-2}^{(3)}= 0, \\\\\nG_{3i-1}^{(2)} = a_2 P_{3i-1}^{(2)} =\\frac{1}{2} , \\quad G_{3i-1}^{(1)}=G_{3i-1}^{(3)}=P_{3i-1}^{(1)}=P_{3i-1}^{(3)}= 0, \\\\\nG_{3i}^{(3)} = a_3 P_{3i}^{(3)} =\\frac{1}{2} , \\quad G_{3i}^{(1)}=G_{3i}^{(2)}=P_{3i}^{(1)}=P_{3i}^{(2)}= 0.\n\\end{array}\n\\end{equation}\nIt is also convenient to divide the warp factors $Z_\\Lambda$ such as\n\\begin{align}\nZ_0 &\\= \\frac{1}{\\left(r_+^{(n)}-z\\right)\\,\\sqrt{U_1 U_2 U_3}}, \\qquad Z_I \\= \\frac{{\\cal Z}_I}{\\sqrt{U_I}}\\,,\\label{eq:WarpfactorMtheoryBu15d} \\\\\nU_I &\\,\\equiv\\, \\prod_{i=0}^{N-1} \\left(1-\\frac{2 M_{3i+I}}{R_+^{(3i+I)}} \\right)\\,, \\qquad {\\cal Z}_I \\,\\equiv\\, \\frac{e^{b_I}-e^{-b_I}\\,U_I}{2 a_I}\\,. \\nonumber\n\\end{align}\nThe remaining functions \\eqref{eq:WarpfactorsRods5d} give\n\\begin{align}\n& W_0 = \\left( U_1 U_2 U_3\\right)^{\\frac{1}{2}},\\qquad W_I \\= U_I^{-\\frac{1}{2}} ,\\qquad H_0 \\= 0 ,\\qquad T_I\\=-\\sqrt{a_I^2+\\frac{U_I}{{\\cal Z}_I^2}}, \\nonumber \\\\\n& e^{2 \\nu}=\\frac{1}{Z_0}\\,\\sqrt{\\frac{E_{-+}^{(1, n)}}{2E_{++}^{(n, n)} E_{--}^{(1,1)}}} \\prod_{i,j=0}^{N-1} \\,\\prod_{\\substack{I,J=1\\\\Iz_n^+=0$, corresponds to a coordinate degeneracy since $r_+^{(n)}-z=0$. The $y_7$-circle reduces to zero size as a smooth origin of $\\mathbb{R}^2$ (see appendix \\ref{App:AttheRod2} for more details).\n\nAt the finite rods, the $y_1$, $y_3$ and $y_5$ fibers shrink successively to zero size since $U_1$, $U_2$ and $U_3$ vanish alternatively. These loci correspond to smooth origin of $\\mathbb{R}^2$ with potential conical defects if $3N$ bubble equations are satisfied (see Appendix \\ref{App:AttheRod2} for more details). These bubble equations arise from the regularity condition \\eqref{eq:RegCondBu5d} at each rod. They are non-trivial multivariate polynomials that fix all rod lengths, $M_i$. \n\nEach species of bubble rod carries a M2 charge as discussed in section \\ref{sec:regM5d}. Therefore, by combining the three species, the solutions have M2-M2-M2 charges. They are given by\n\\begin{equation}\nQ^I_{\\text{M2}} \\= \\frac{1}{2a_I}\\,\\sum_{i=0}^N M_{3i+I}.\n\\label{eq:ChargeMtheoBu5d}\n\\end{equation}\nIn the five-dimensional frame after reduction on the T$^6$ \\eqref{eq:5dframework}, the solutions are \n\\begin{align}\nds_5^2 &\\= \\left(U_1 U_2 U_3 \\right)^{\\frac{1}{3}} \\left[- \\frac{dt^2}{\\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{2}{3}}} \\+ \\left({\\cal Z}_1 {\\cal Z}_2 {\\cal Z}_3 \\right)^{\\frac{1}{3}}\\, \\Biggl[\\left(r_+^{(n)}-z \\right) \\,dy_7^2 \\right. \\\\\n& \\left. \\hspace{5.5cm}\\+ \\frac{1}{\\left(r_+^{(n)}-z \\right)U_1 U_2 U_3} \\left( e^{2\\nu} \\left(d\\rho^2 + dz^2 \\right) +\\rho^2 d\\phi^2\\right)\\Biggr] \\right] \\nonumber \\\\\n\\Phi_0 &= \\frac{1}{2}\\log (U_1 U_2 U_3) \\,,\\quad \\Phi_I = -\\frac{1}{2}\\log U_I \\,,\\quad X^I =\\frac{|\\epsilon_{IJK}|}{2} \\left(\\frac{U_I\\,{\\cal Z}_J {\\cal Z}_K}{\\sqrt{U_J U_K}\\,{\\cal Z}_I} \\right)^\\frac{1}{3},\\quad F^{I} = dT_I \\wedge dt\\,. \\nonumber\n\\end{align}\nTherefore, they are singular at the rods in five dimensions, and these singularities are resolved in M-theory as the degeneracy of the extra dimensions. Moreover, the solutions are asymptotic to $\\mathbb{R}^{1,4}$ if \\eqref{eq:condonasymp5d} is satisfied. The ADM mass is given by \\eqref{eq:ADMmass5d}\n\\begin{equation}\n{\\cal M} \\= \\frac{\\pi}{4G_5} \\sum_{I=1}^3 \\left( \\coth b_I \\sum_{i=0}^{N-1} M_{3i+I}\\right)\\,.\n\\label{eq:MassMtheoBu5d}\n\\end{equation}\nFurthermore, since the extra scalars, $\\Phi_\\Lambda$, are turned on, the five-dimensional solutions are not solutions of the STU Lagrangian as detailed in section \\ref{sec:5dreduction}. Nevertheless, they have the same conserved charges as a non-extremal three-charge static black hole in five dimensions, but they are horizonless and terminate the spacetime as a chain of non-BPS bubbles in M-theory.\n\n\\begin{itemize}\n\\item[\u2022] \\underline{BPS limit:}\n\\end{itemize}\n\nAs for the four-dimensional bubbling geometries constructed in section \\ref{sec:BuSolMtheory}, one can derive the BPS limit of the present five-dimensional solutions. This consists of taking the gauge field parameters to zero, $(b_I, a_I) \\to 0 $. For simplicity, we consider $b_I= a_I=\\lambda$ with $\\lambda \\to 0$. First, from the bubble equations \\eqref{eq:RegCondBu5d}, we have \n\\begin{equation}\nM_{i} \\underset{\\lambda\\to 0}{\\sim} \\lambda \\,q_{i},\\qquad i=0,...,3N,\n\\end{equation}\nwhere $q_i$ are finite constants as $\\lambda \\to 0$, and the whole structure reduces to a point. However, it is clear that the charges \\eqref{eq:ChargeMtheoBu5d} and the mass \\eqref{eq:MassMtheoBu5d} do not vanish:\n\\begin{equation}\nQ^I_{\\text{M2}} \\sim \\frac{1}{2}\\,\\sum_{i=0}^N q_{3i+I} , \\qquad {\\cal M} \\sim \\frac{\\pi}{4 G_5}\\left(Q^1_{\\text{M2}}+Q^2_{\\text{M2}}+Q^3_{\\text{M2}} \\right),\n\\end{equation}\nThe warp factors and gauge potentials, \\eqref{eq:WarpfactorMtheoryBu15d} and \\eqref{eq:WarpfactorMtheoryBu25d}, behave as\n\\begin{equation}\n\\begin{split}\nU_I &\\sim1\\,, \\qquad {\\cal Z}_I \\sim 1+ \\frac{Q_\\text{M2}^I}{r^2}, \\qquad r_+^{(n)}-z \\sim r^2 \\sin^2 \\theta \\,, \\qquad T_I \\sim - \\frac{1}{{\\cal Z}_I}\\,,\\qquad e^{2\\nu} \\sim 1\\,.\n\\end{split}\n\\end{equation}\nwhere $(r,\\theta)$ are the five-dimensional spherical coordinates $(\\rho,z)=\\frac{1}{2}(r^2\\sin2\\theta,r^2\\cos 2 \\theta)$. Therefore, the solutions approach a three-charge static BPS black hole in five dimensions.\n\nThe BPS black hole horizon is thus smoothly resolved into smooth non-BPS bubbles wrapped by flux by adding a small amount of non-extremity. At small $(a_I,b_I)$, the solutions are almost indistinguishable from the BPS black hole. They must develop an AdS$_2$ throat that does not end in a horizon but as non-BPS bubbles.\n\n\n\\section{Discussion}\n\n\nIn this paper, we have shown that non-BPS linear ansatz that allow non-trivial matter fields and topology can be directly derived from Maxwell-Einstein equations in string theory. In M-theory on T$^6\\times$S$^1$, our ansatz enables four gauge potentials corresponding to three stacks of M2 branes and a KKm vector, and it can be dualized to other string frames such as the D1-D5-P-KKm frame. Focusing on solutions that are asymptotic to four- and five-dimensional flat space, we derived families of four-charge non-extreme black holes on a line and non-BPS bubbling geometries. The latter can have the same charges and mass as non-extremal black holes but are horizonless and smooth. We have highlighted examples that are almost indistinguishable from the BPS black hole when they are slightly non-BPS but terminate the spacetime smoothly as a chain of non-BPS bubbles.\n\n\n\nWhile the present non-BPS floating brane ansatz opens a new door to the study of non-BPS solitons in string theory, several questions need to be explored in future work. First, although the smooth charged bubble is known to be a meta-stable vacuum \\cite{Stotyn:2011tv,StabilityPaper}, the stability of chains of such objects remains to be studied. Second, one can think of constructing new families of solutions with different boundaries than the flat asymptotic. Third, it will also be interesting to add brane degrees of freedom to the ansatz in M-theory, such as M5 or P brane charges. This will enable Chern-Simons interactions to be turned on as in the BPS floating brane ansatz for multicenter solutions. In addition, to construct more astrophysically-interesting solutions, a rotational degree of freedom will need to be added. \n\nFurthermore, understanding the origin of the solutions as bound states of strings and branes will require careful analysis of the geometric transition that occurs. For BPS solutions, for example, the geometric transition that gives rise to smooth geometries is well understood \\cite{Bena:2013dka}. The present one uses very different mechanisms because our solutions are far from being the non-BPS extensions of known BPS smooth bubbling solutions as their BPS limit suggests, and they are based on non-trivial topology changes of compact tori. They should therefore be very illuminating on constructions of string and brane boundary states in the non-supersymmetric regime, but one should not expect these bound states to use transitions similar to those of BPS bound states.\n\nFinally, it would be interesting to have a better geometrical and physical understanding of the smooth bubbling geometries. One can compare them with the non-extremal black hole with the same mass and charges as in \\cite{Bah:2021rki}. Moreover, one can derive their multipole moments, probe them by light geodesics or compute their quasi-normal modes using technologies developed for similar geometries \\cite{Bena:2020uup,Bianchi:2020miz,Bah:2021jno,Mayerson:2020tpn,Bacchini:2021fig,Ikeda:2021uvc,Bena:2020yii}. One could also being interested in performing M2-brane probe computation in the smooth bubbling solutions in M-theory. If we do not expect the branes to feel an entirely flat potential that allows them to ``float'' everywhere in space, our non-linear ansatz suggest special flat regions where probe can be trapped.\n\n\\section*{Acknowledgments}\nI am grateful to Ibrahima Bah for his rich advice, and to Iosif Bena, Nejc Ceplak, Anthony Houppe and Nick Warner for interesting discussions. This work is supported in part by NSF grant PHY-1820784.\n\n\n\\vspace{1cm}\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\n\nExploring various phases of quark and nuclear matter\nis one of the primary goals in high-energy heavy-ion collision physics.\nCompared to the QCD phase transition at high $T$,\nless is known for the cold and dense matter,\npartly because the lattice Monte-Carlo simulations are difficult\nat large $\\mu$.\nOne of the most instructive\napproaches to compressed baryonic matter is to consider\nthe strong coupling limit (SCL) of lattice \nQCD~\\cite{Kawamoto,Faldt1986,Bilic1992,Nishida2004}.\nIn fact, effective free energies at finite $T$ and $\\mu$\nhave been analytically derived in SCL,\nand it is successfully applied to nuclear many-body problems~\\cite{Tsubakihara}.\nWhile baryon effects would be important in dense matter,\nthey have been ignored in finite $T$ treatments\nthrough the $1\/d$ expansion ($d$ is the spatial dimension).\n\nIn this proceedings, first we derive an expression of the effective free energy\nat finite $T$ and $\\mu$ including baryon effects~\\cite{KMOO_2007}.\nWe find the second and first order phase transition at high and low $T$,\nseparated by the tricritical point (TCP) in the chiral limit. \nWith finite quark masses,\nthese second order phase boundary and TCP\nbecome the cross over and the critical end point,\nthus the obtained phase diagram seems to show essential qualitative features\nof that in the real world.\n\nOne of the problems in SCL\nis in the ratio, $R_{\\mu T} = \\mu_c(T=0)\/T_c(\\mu=0)$;\nthis ratio would be larger than two in the real world,\nbut it is less than one half in all of the works based on SCL.\nIn the second part of this proceedings, \nwe demonstrate that\nthe ratio increases to $R_{\\mu T} \\sim 1.8$ at $g \\sim 1$\nwith $1\/g^2$ corrections.\nWith finite bare quark mass and baryon effects,\nit may be possible to understand the shape of the actual phase diagram.\n\n\\section{Effective free energy in the strong coupling limit of lattice QCD}\n\\label{Sec:Model}\n\nIn SCL, we ignore pure gluonic action ($\\propto 1\/g^2$),\nand we obtain the following staggered fermion effective action\nafter integrating spatial links,\n\\begin{equation}\n{\\cal Z}=\\int {\\cal D}[\\chi,{\\bar{\\chi}},U_0]\n\\exp\\left[-S_F^{(0)}\n\t+\\frac12 (M, V_M M)\n\t+(\\bar{B},V_B B)\n\\right]\\,,\n\\label{Eq:ActionA}\n\\end{equation}\nwhere\n$(A,B)=\\sum_x A_xB_x$,\nand \nthe mesonic and baryonic composites and their propagators are\n$M_x={\\bar{\\chi}}^a_x\\chi^a_x$, \n$B_x=\\varepsilon_{abc}\\chi^a_x\\chi^b_x\\chi^c_x\/6$,\n$V_M(x,y)$\nand\n$V_B(x,y)$.\nWe keep the timelike link and bare quark mass terms\nunexpanded,\n$S_F^{(0)}=\\sum_{x}[{\\bar{\\chi}}_xe^{\\mu}U_0(x)\\chi_{x+\\hat{0}}\n\t-{\\bar{\\chi}}_{x+\\hat{0}}e^{-\\mu}U^\\dagger_0(x)\\chi_x]\/2\n\t+m_0({\\bar{\\chi}},\\chi)$.\n\nWe decompose the effective action in Eq.~(\\ref{Eq:ActionA}),\ncontaining six fermion terms,\nto the bilinear form in $\\chi$\nin order to perform the quark integral in a finite $T$ treatment.\nDecomposition has been carried out in three steps,\n\\numparts\n\\begin{eqnarray}\n&&e^{(\\bar{B},V_B,B)}\n=\\int {\\cal D}[\\bar{b},b,\\phi,\\phi^\\dagger]\n\te^{\n\t\t-(\\bar{b},V_B^{-1}b)-\\phi_a^\\dagger\\phi_a\n\t\t+\\phi_a^\\dagger D_a+D^\\dagger_a\\phi_a\n\t\t- \\gamma^2 M^2\/2+M\\bar{b}b\/9\\gamma^2\n\t}\n\t\\ ,\n\\label{Eq:DecompA}\n\\\\\n&&e^{M \\bar{b}b\/9\\gamma^2}\n=\\int d[\\omega]\\,e^{\n\t\t-\\omega^2\/2\n\t\t-\\omega(\\alpha M+\\bar{b}b\/9\\alpha\\gamma^2)\n\t\t-\\alpha^2 M^2\/2\n\t\t}\n\\ ,\n\\label{Eq:DecompB}\n\\\\\n&&e^{\\frac12(M,\\widetilde{V}_M M)}\n=\\int d[\\sigma]\\,e^{\n\t\t-(\\sigma,\\widetilde{V}_M\\sigma)\/2\n\t\t-(\\widetilde{V}_M\\sigma,M)\n\t\t}\n\\label{Eq:DecompC}\n\\ .\n\\end{eqnarray}\n\\endnumparts\nIn the first step (\\ref{Eq:DecompA}),\nthe baryonic action is decomposed \nusing the diquark field $\\phi_a$~\\cite{Azcoiti2003},\nwhose expectation value is $\\VEV{\\phi_a}=\\VEV{D_a}$,\nwhere $D_a=\\gamma\\varepsilon_{abc}\\chi^b\\chi^c\/2+\\bar{\\chi}^a b\/3\\gamma$.\nIn the second step (\\ref{Eq:DecompB}),\nthe four fermi interaction term, $M\\bar{b}b$,\nis decomposed utilizing the null potency, $(\\bar{b}b)^2=0$.\nIn the third step (\\ref{Eq:DecompC}),\nchiral condensate $\\sigma$ is introduced\nafter including the remaining four quark interaction terms\nin the propagator,\n$\\widetilde{V}_M=V_M+(\\alpha^2+\\gamma^2)\\delta_{x,y}$.\nThe effective action is now in a bilinear and pfaffian form of $\\chi$,\nand quarks are separated in each spatial point\ncoupling to the auxiliary fields,\n$m_q=\\widetilde{V}_M\\sigma+\\alpha\\omega$,\n\\begin{eqnarray}\n\\fl~~~~~~~~~~~~\nS_F&=&\nS_F^{(0)}\n+({\\bar{\\chi}},m_q\\chi)\n+\\frac12(\\sigma,\\widetilde{V}_M\\sigma)\n+(\\bar{b},\\widetilde{V}_B^{-1} b)\n+\\frac12 (\\omega,\\omega)\n\\nonumber\\\\\n\\fl\n&+&(\\phi^\\dagger,\\phi)\n+{1\\over3\\gamma}\\left[\n\t ({\\bar{\\chi}}^a, \\phi_a^\\dagger b)\n\t+(\\bar{b}\\phi_a,\\chi^a)\n\t\\right]\n+\\frac{\\gamma}{2}\\varepsilon_{cab}\\left[\n\t (\\phi_c^\\dagger, \\chi^a \\chi^b)\n\t+({\\bar{\\chi}}^b{\\bar{\\chi}}^a,\\phi_c)\n\t\\right]\n\\,.\n\\label{Eq:ActionB}\n\\end{eqnarray}\n\nWe can analytically carry out\nMatsubara frequency product of quarks and temporal link integral\nin the mean field approximation\nat zero diquark condensate.\nAt equilibrium, \nan approximate linear relation $\\omega \\propto \\sigma$ holds,\nand the effective free energy is found to be\n\\begin{eqnarray}\n{\\cal F}_\\mathrm{eff}(\\sigma)\n&=&\\frac12b_\\sigma\\sigma^2\n+{\\cal F}_\\mathrm{eff}^{(q)}(b_\\sigma\\sigma)\n+{\\it \\Delta}{\\cal F}_\\mathrm{eff}^{(b)}(g_\\sigma\\sigma)\n\\,,\n\\label{Eq:Feff}\n\\\\\n{\\cal F}_\\mathrm{eff}^{(q)}(m_q)\n&=&-T\\log\\left[C_\\sigma^3-\\frac12C_\\sigma\n+\\frac14\\cosh\\left(\\frac{3\\mu}{T}\\right)\\right]\n\\,,\n\\\\\n{\\it \\Delta}{\\cal F}_\\mathrm{eff}^{(b)}(m_b)\n&\\simeq&-f^{(b)}\\left({\\pi m_b\\over8}\\right)\n\\,, \n\\end{eqnarray}\nwhere $C_\\sigma=\\cosh(\\mathrm{arcsinh}\\,(m_q)\/T)$\nand \n$f^{(b)}(x)=\\log(1+x^2)\/2-[\\arctan{x}-x+x^3\/3]\/x^3-3x^2\/10$.\nIn this effective free energy ${\\cal F}_\\mathrm{eff}$,\nwe have two parameters, $b_\\sigma$ and $g_\\sigma$,\nwhich are related to the decomposition parameters,\n$\\alpha$ and $\\gamma$.\nWithout baryon effects,\nthe first two terms remain in the effective free energy (\\ref{Eq:Feff})\nwith a fixed coefficient $b_\\sigma=d\/2N_c$~\\cite{Nishida2004}.\n\nBaryon effects appear\nin the modification of the coefficient $b_\\sigma$\nand in the additional term ${\\it \\Delta}{\\cal F}_\\mathrm{eff}^{(b)}$\ncoming from the auxiliary baryon determinant.\nWhen we adopt the parameter $\\alpha=0.2$\nwhich approximately maximize the ratio\n$R_{\\mu T}=\\mu_c\/T_c$,\nbaryons are found to have effects\nof the effective free energy gain with respect to $T_c$\nand the extension of the hadronic phase in the larger $\\mu$ direction,\nas shown in Fig.~\\ref{Fig:SCL}~\\cite{KMOO_2007}.\n\n\\begin{figure}\n\\begin{minipage}{\\textwidth}\n\\Psfig{8cm}{QM06-Ohnishi_Fig1a.eps}{}\n\\Psfig{8cm}{QM06-Ohnishi_Fig1b.eps}{}\n\\end{minipage}\n\\caption{Effective free energy as a function of $\\sigma$ (left panel),\nand the phase diagram in the strong coupling limit (right panel).\n}\\label{Fig:SCL}\n\\end{figure}\n\n\\section{Finite coupling correction}\n\\label{Sec:Finite_g}\n\nWhile the SCL seems to show\nqualitative features of the phase diagram in the real world,\nthere are several problems,\nsuch as\nthe parameter dependence and scale modifications~\\cite{KMOO_2007},\nno stable color superconductor (CSC) phase,\nand too small ratio $R_{\\mu T} < 0.5$.\nFinite coupling corrections may solve\nsome of these problems~\\cite{Faldt1986,Bilic1992,Bilic1995,Ipp}.\n\nThe $1\/g^2$ correction on the effective action was\nderived by Faldt and Petersson~\\cite{Faldt1986},\nand we can bosonize the plaquett contributions\nin the mean field approximation as,\n\\begin{eqnarray}\n\\fl~~~~~~~~\n{\\it \\Delta}S_F&=&\n \t\\frac{1}{4N_c^2g^2}\\sum_{x,j>0} \n\t\t(\n\t\t V^\\dagger_x V_{x+\\hat{j}}\n\t\t+V^\\dagger_x V_{x-\\hat{j}}\n\t\t)\n \t- \\frac{1}{8N_c^4g^2}\\sum_{x,k>j>0} \n\t\tM_{x}\n\t\tM_{x+\\hat{j}}\n\t\tM_{x+\\hat{k}}\n\t\tM_{x+\\hat{k}+\\hat{j}}\n\\nonumber\\\\\n\\fl\n&\\simeq&\n\tN_s^3N_\\tau\\left(\n\t\t \\frac{\\beta_t}{4}\\varphi_t^2\n\t\t+\\frac{\\beta_sd}{4}\\varphi_s^2\n\t\\right)\n\t+\\frac{\\beta_t\\varphi_t}{4}\n\t\\sum_x(V_x-V_x^\\dagger)\n\t-\\beta_s\\varphi_s\n\t\\sum_{x,j>0} M_x M_{x+\\hat{j}}\n\t\\,.\n\\label{Eq:ActionC}\n\\end{eqnarray}\nWe have defined $V_x \\equiv {\\bar{\\chi}}_x U_0(x) \\chi_{x+\\hat{0}}$,\n$\\beta_s=(d-1)\/8N_c^4g^2$, and $\\beta_t=d\/2N_c^2g^2$,\nand the auxiliary fields have expectation values of\n$\\VEV{\\varphi_t}=\\VEV{V^\\dagger-V}$\nand \n$\\VEV{\\varphi_s}=2\\VEV{M_xM_{x+\\hat{j}}}$.\nHere we have omitted baryon effects (${\\cal O}(1\/d^{1\/2})$)\nand higher order terms (${\\cal O}(1\/d)$) \nin the $1\/d$ expansion.\n\nThese corrections have a similar structure\nto the SCL effective action (\\ref{Eq:ActionA}),\nand they lead to\nthe modifications of the quark mass and effective chemical potential as \n$\\widetilde{m}_q=\\sigma d(1+4N_c\\beta_s\\varphi_s-\\beta_t\\varphi_t\\cosh\\mu)\/2N_c$\nand\n$\\widetilde{\\mu}=\\mu-\\beta_t\\varphi_t\\sinh\\mu$.\nAt equilibrium, \nwe can put $\\varphi_s=2\\sigma^2 + {\\cal O}(1\/g^2)$,\nand the effective free energy up to ${\\cal O}(1\/g^2)$ is obtained as,\n\\begin{eqnarray}\n{\\cal F}_\\mathrm{eff}=\\frac{d}{4N_c}\\sigma^2+3d\\beta_s\\sigma^4\n\t+\\frac{\\beta_t}{4}\\varphi_t^2-N_c\\beta_t\\varphi_t\\cosh\\mu\n\t+{\\cal F}_\\mathrm{eff}^{(q)}(\\widetilde{m}_q;T,\\widetilde{\\mu})\n\t\\,.\n\t\\label{Eq:FeffCorr}\n\\end{eqnarray}\n\nWith this effective free energy,\n$T_c$ is found to decrease as $g$ decreases,\nwhile $\\mu_c$ stays almost constant.\nAs a result, $R_{\\mu T}$ grows to around 1.8 at $g \\sim 1$\nas shown in Fig.~\\ref{Fig:Finite_g}.\nThe present results are not fully consistent with the previous \nfindings ~\\cite{Bilic1995}.\nThe difference in the bosonization scheme and lattice\nanisotropy~\\cite{Bilic1992,Bilic1995} has to be investigated further.\n\n\\begin{figure}\n\\begin{minipage}{\\textwidth}\n\\Psfig{6cm}{QM06-Ohnishi_Fig2a.eps}{}\n\\Psfig{8cm}{QM06-Ohnishi_Fig2b.eps}{}\n\\end{minipage}\n\\caption{Evolution of the phase diagram with $\\beta=6\/g^2$ (left panel),\nand the $\\beta$ dependence of $T_c$ and $\\mu_c$ (right panel).}\n\\label{Fig:Finite_g}\n\\end{figure}\n\n\n\\section{Summary}\n\nIn this proceedings,\nwe have investigated the phase diagram\nin the strong coupling region of lattice QCD.\nBaryons are found to have effects of extending the hadron phase\nin the larger $\\mu$ direction with respect to $T_c$ up to around\n30 \\%~\\cite{KMOO_2007} in the strong coupling limit.\nWith finite coupling corrections,\nwe have found that the ratio $R_{\\mu T}=\\mu_c\/T_c$ becomes\ncloser to that expected in the real world.\nIt would be interesting to evaluate both of the finite coupling correction\nand baryon effects simultaneously.\n\nThis work is supported in part by the Ministry of Education,\nScience, Sports and Culture,\nGrant-in-Aid for Scientific Research\nunder the grant numbers,\n 13135201,\t\n 15540243,\t\nand 1707005.\t\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Introduction}\nThe search for periodic gravitational wave signals is a stimulating challenge\nfor data analists because of the considerable amount of computing time\nrequired.\n\nFor blind searches, i.e. without any {\\it a priori} knowledge about the source,\na fully coherent analysis can not handle more than a few days of data because\nof the steep dependence of the size of the parameter space on the\nfrequency resolution.\n\nIn \\cite{Astone:2005fn} three data sets, each two days long, from the\nExplorer 1991 run have been coherently studied by means of the \n{\\cal F} statistics method \\cite{Jaranowski:1998qm}\nwhich led to and an upper limit of $1\\cdot 10^{-22}$ on $h$ in\nthe narrow band $921.00$-$921.76$Hz.\n\nA similar technique, applied in \\cite{Abbott:2006vg}\nto the most sensitive 10 hours of the the LIGO S2 run, led\nto an upper limit of $6.6\\cdot 10^{-23}$ for isolated\nneutron stars in the band between $160$ and $728.8$Hz.\n\nWith the widening of the frequency band, due to the advent of\ninterferometers as well as to improvements in the readout of resonant\ndetectors, several incoherent and semi-coherent methods have been\nconceived and employed.\n\nIn \\cite{Abbott:2005pu} the Hough transform technique has been applied to\nthe LIGO S2 data to perform a blind search for isolated neutron stars\non a set of narrow frequency bands in the range $200$-$400$Hz, and a best\nupper limit of $4.43\\cdot 10^{-23}$ has been set.\n\nIn the present work, the simple technique of adding power spectra has been\napplied to the most sensitive $40$Hz band of the 2005 run of the Explorer bar\n\\cite{Astone:2006uf}, resulting in a further improvement on the best upper\nlimit on $h$, which is set to $3.1\\cdot 10^{-23}$ at $920.14$Hz.\n\nAccording to the results reported at the recent Amaldi7 conference,\nthe analysis of the LIGO S4 run \\cite{Abbott:2007td} is leading to a sensible\nimprovement in this direction (about an order of magnitude);\nremarkably, the limit set in the present work is still competitive with\nthe one coming from the S4 data in the same frequency band.\n\n\\subsection{The data set}\nAt the end of April 2005, after a short commissioning break, the resonant\nantenna Explorer was again\n{\\it on air} and operated with the usual, remarkable duty cycle ($86\\%$\nfrom April to December 2005) and a good stability.\n\nThe data stream taken by the bar until the end of 2005\n(after which the sensitivity curve has been modified)\nhas been divided into $25161$ segments, each\nabout 14 minutes long.\nThe most sensitive band of the Fourier transform of these segments,\nnamely $N_f=32178$ frequency bins in the range $885$-$925$Hz,\nhas been selected for the analysis.\n \n\\begin{figure}\n \\centering\n \\includegraphics[width=5in,angle=-90]{noiseexp05.ps} \n \\caption{The noise level of Explorer during the 2005 run.\nThe solid horizontal line is the cut applied to select the best spectra.}\n \\label{fignoise}\n\\end{figure}\n\nA noise cut has been applied to the total power contained in each \nspectrum, with the purpose of discarding the noisy ones thus creating an\nhomogeneous\nset of spectra. This allowed us to apply for the subsequent analysis\nthe simple power addition method, without weighting each spectrum with the\ncorresponding noise level.\n\nThis selection led to the creation of a data set $\\left\\{S_i\\right\\}$\nmade of the $N_1=11749$ cleanest spectra (corresponding to an effective data\ntime of 114 days), and of a second set\ncontaining just the best $N_2=3875$ (used to deal with the critical zones\nof the spectrum, corresponding to the resonant modes of the bar, around 888Hz\nand 920Hz).\n\n\nThe average sensitivity of the second data set is shown in Fig. \\ref{figsensi}\n(the first set is similar except around the resonant modes);\na few noisy lines may be noted, including small 1Hz harmonics on the left part.\n\\begin{figure}\n \\centering\n \\includegraphics[width=6in,angle=0]{spenoshift.eps} \n \\caption{The square root of the average spectrum represents the typical\nExplorer sensitivity curve during the 2005 run.}\n \\label{figsensi}\n\\end{figure}\n\n\n\n\\subsection{The analysis method}\n\nFor a given direction $\\hat{r_j}$ in the sky, the selected spectra have\nbeen deformed according to the Doppler shift formula\n\\begin{eqnarray}\\label{doppler}\nf^{\\rm true}_j\\simeq f_{\\rm exp}\\left(1-\\frac{\\vec{v}_{rel}\n\\cdot\\hat{r_j}}{c}\\right)\\nonumber\n\\end{eqnarray}\nand then summed and renormalized dividing by $N_1$ or by $N_2$:\n\\begin{eqnarray}\\label{sumspe}\n{\\cal S}_j (f^{\\rm true}_j)=\\frac{1}{N_{1,2}}\\sum_{i=1}^{N_{1,2}}\nS_i(f^{\\rm true}_j)\\, .\\nonumber\n\\end{eqnarray}\n\n\nThe speed of the detector relative to the Solar System Baricenter has been\ncomputed thanks to the JPL ephemerides \\cite{JPL}.\nAs the speed of the source have not been taken into account, the search is\nsensitive to isolated neutron stars, and not to those which are part of\nbinary systems.\n\nSpindown has also been neglected: given the frequency resolution of\n$1.2\\cdot 10^{-3}$Hz, this means being sensitive to an average frequency\ndrift up to $6\\cdot10^{-11}$Hz\/s during the observation\nperiod.\\\\\nThe procedure has been repeated for any point of an optimized sky grid\nmade of $N_{\\rm sky}=23927$ possible directions, thus leading to the creation\nof a set $\\left\\{{\\cal S}_j\\right\\}$ containing\n$N_{\\rm sky}$ ``deformed and summed'' spectra.\n\nThe variance of the noise is obtained calculating, for each value of the\nfrequency, the variance of the distribution of the $N_{\\rm sky}$ deformed\nspectra.\nThe result, whose square root is shown in Fig. \\ref{figsigma}, agrees with\nthe general expectation, based on the central limit theorem, that\n\\begin{equation}\n\\sigma\\simeq\\frac{S_h}{\\sqrt{N_{1,2}}}\\, .\n\\end{equation}\n\\begin{figure}\n \\centering\n \\includegraphics[width=5in,angle=0]{sigsky.eps} \n \\caption{Noise variance of the set of deformed and summed spectra.\nTo help comparison with Fig. \\ref{figsensi}, the square root of $\\sigma$ is\nactually shown.}\n \\label{figsigma}\n\\end{figure}\n\nAs expected, the plot shows an anomalous behavior of the noise\nvariance in correspondence of the disturbances of the initial data set.\nThese anomalous zones have not been taken into account for the candidate\nsearch, but have been included in the upper limit determination.\n\nThe detection threshold is fixed by the requirement that the false alarm rate\nshould be less than $1\\%$. According to Poisson statistics one has to impose\n\n\\begin{eqnarray}\\label{poisson}\nP(0,\\lambda)={\\rm e}^{-\\lambda}>.99\\, ,\\nonumber\n\\end{eqnarray}\nwhere the expected number of threshold crossings in absence of signal is\n\\begin{eqnarray}\\label{lambda}\n\\lambda=p\\cdot N_f\\cdot N_{\\rm sky}\\, ,\\nonumber\n\\end{eqnarray}\nbeing $p$ the probability of false detection in a single frequency bin\nand for a single direction in the sky, and $N_f\\cdot N_{\\rm sky}$\nthe trial factor.\n\nOne thus finds the condition $p<1.3\\cdot 10^{-11}$\nwhich, assuming that the $N_{\\rm sky}$ values of the shifted spectra at a\ngiven frequency bin are gaussian distributed, translates to a $7\\sigma$\nthreshold.\n\nIn other words, a detection is claimed if, for some value of\nthe frequency $f$, a given deformed spectrum ${\\cal S}_j$\nsatisfies the condition\n\\begin{eqnarray}\\label{detcond}\n{\\cal S}_j(f)-\\bar{\\cal S}(f)>7 \\sigma(f)\\, ,\n\\end{eqnarray}\nbeing $\\bar{\\cal S}$ the average of the deformed spectra\n$\\left\\{{\\cal S}_j\\right\\}$ as $j$ spans over the $N_{\\rm sky}$\nsky grid points. \n\nSince $h\\propto\\sqrt{S}$, to translate the detection threshold\nin $h$ units one has to multiply the values shown in Fig. \\ref{figsigma}\nby the factor\n\\begin{eqnarray}\\label{factor}\n\\sqrt{\\frac{7}{T}}\\cdot\\sqrt{\\frac{15}{4}}\\, ,\\nonumber\n\\end{eqnarray}\nwhere $T$ is the length of each data segment and the factor $4\/15$\n(the average angular sensitivity of the bar over the solid angle) is\nintroduced to compensate the fact that amplitude modulation has not been taken\ninto account when summing the deformed spectra.\n\n\\subsection{Results}\n\\subsubsection{Cadidate search}\n\nThe threshold is shown in red in Figure \\ref{figthremax},\nwhile the blue line is the maximum value of $h$ found, at any given\nfrequency bin, among the set of the $N_{\\rm sky}$ spectra according to\nthe following formula:\n\\begin{eqnarray}\\label{geth}\nh_{\\rm max}(f)={\\rm max}_j \\left[\\sqrt{\\frac{15}{4}\\frac{{\\cal S}_j(f)-\\bar{\\cal S}(f)}{T}}\\right]\\, .\\nonumber\n\\end{eqnarray}\n\\begin{figure}\n \\centering\n \\includegraphics[width=5in,angle=0]{thremax.eps} \n \\caption{Detection threshold (red, upper curve) and maximal triggers\n (blue, lower). The $1$Hz disturbances on the left have been left only\n for illustrative purpose and cannot be considered as genuine threshold\n crossings.}\n \\label{figthremax}\n\\end{figure}\n\n The anomalous regions of the spectrum have been cut out, since it is not\nreasonable to assume that they follow a gaussian distribution. To illustrate\nthe point, the $1$Hz disturbances have been keft in Figure \\ref{figthremax},\nto show that they would have\nproduced fake candidates, if they had been included in the analysis.\n\nWith these specifications, the maxima are never above the threshold,\nand thus no candidates have been found.\n\n\\subsubsection{Software injections and upper limit}\n\nTo find an upper limit on $h$, a variation of the loudest event method\n\\cite {Brady:2004gt} has been applied.\nThis method allows to determine an upper limit starting from the\nloudest event present in a data stream, irrespectively of the fact that\nsuch an event may be due to noise or to a real signal.\nThe idea is basically that if a strong real signal would have been present\nduring the data taking, it would have produced an event louder than the\nloudest event actually recorded. This idea can be made quantitative by\nstudying the detection efficiency of the experiment, for example by performing\nsoftware injections at various SNR's.\n\nIn our case, we have a loudest event for any frequency bin, and all these\nevents form precisely the curve of maxima depicted in Figure \\ref{figthremax}.\nAs a preliminary step, we had to determine our detection efficiency by\ninjecting fake periodic signals in the Explorer data stream and\nfinding out how hey did loook like after the analysis chain: the left plot of\nFigure \\ref{figinjupp} shows for instance the result of 16 injections\nof signals with $h=3\\cdot 10^{-23}$, equally spaced in frequency by\n$0.1$Hz starting from $919.4$Hz, and coming from randomly chosen directions\nin the sky.\n\\begin{figure}\n \\centering\n \\includegraphics[width=6cm,angle=0]{inje.eps}\n \\includegraphics[width=6cm,angle=0]{upper.eps}\n \\caption{On the left, the outcome of the 16 injections,\n with initial amplitude given by the dashed red line.\n On the right, the upper limt curve.}\n \\label{figinjupp} \n\\end{figure}\nThen, the upper limit at $95\\%$ c.l. at has been determined for each\nfrequency bin as the lowest injected amplitude which had produced a\nsignal larger than the actual maximum at least in $95\\%$ of the cases.\n\nThe right plot on Figure \\ref{figinjupp} shows the result, i.e. the curve\nof $h$ upper limit at $95\\%$ confidence level: the minimum is\n$3.1\\cdot 10^{-23}$ at $920.14$ Hz.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}