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+{"text":"\\section*{Supplemental Material}\n\nThis supplemental material conveys additional results on the weak coupling regime, the effects of a harmonic trap, correlation functions and mutual information in the strong coupling phase separated regime, and a derivation of the spinful $PXP$ Hamiltonian at half and full filling.\n\n\n\n\\section{Weak coupling Rydberg dressed regime}\n\nTo obtain the weak-coupling Hamiltonian, see Eq. \\eqref{eqn:cos} and above it, we linearize the tight-binding dispersion around $\\pm k_F$. Without interactions, the bosonized Hamiltonian takes the form:\n\\begin{equation}\n    H^{nonint}_\\eta = \\frac{1}{2\\pi} \\int dx \\left[ v (\\pi \\Pi_\\eta (x))^2 + v (\\nabla \\phi_\\eta(x))^2 \\right],\n\\end{equation}\nwhere $v_F= \\frac{d \\varepsilon}{dk}_{k=k_F}=\\frac{2t}{a}\\sin(k_F a)$, where $a$ is the lattice constant of the order of $\\alpha$. Next we transform the lattice interaction Eq.~\\eqref{eq:rydbergint} to the momentum space:\n\\begin{equation}\n  V_d(q) = \\frac{1}{N}\\sum_n \n  e^{-i q a n } V_d(a n).\n\\end{equation}\nOne can write $V_d(a n) = \\int \\frac{d k}{2\\pi} e^{i k a n} V_C(k) = \\frac{1}{L}\\sum_{k=\\frac{2\\pi l}{L}} e^{i k a n} V_C(k)|_{L\\to\\infty}$, where $V_C(k) = \\int dx V_d(x) e^{ikx}$ is defined in the continuum (C) limit $a\\rightarrow 0$. Using the Poisson summation formula $\\sum_n e^{i n g} = N \\sum_m \\delta_{g-2\\pi m}$ one obtains:\n\\begin{equation}\n  V_d(q) = \\frac{1}{L}\\sum_m V_C\\left(q+\\frac{2\\pi m}{a}\\right).\n\\end{equation}\nAs $V_C(k)$ decays as $e^{-|k|r_c\/2}$ for $r_c\\gg a$ the $m\\neq0$ terms can be neglected and $V_d(q)\\approx \\frac{1}{L} V_c(q)$ used in the main text is obtained.\n\n\n\n\n\n\nApart from the spin-spin correlations shown in Fig.~\\ref{fig:weak}(a), we also calculate the density-density correlation \n$\\langle n_{L\/2 \\pm x} n_{L\/2} \\rangle$\nfor the two (positive and negative backscattering) phases in Fig.~\\ref{fig:rhorho}. \nAs the charge sector is always gapless the density-density correlation decays as power-law in both cases.\nThe density-density and spin-spin power-law exponents in the repulsive backscattering region are both $-(K_\\rho + K_\\sigma)$. \nWe extracted the exponent for $N_p = 120$ from the spin-spin correlator (dashed line in Fig.~\\ref{fig:weak}(a)) and plotted a dashed line with the same slope in Fig.~\\ref{fig:rhorho}. \nOne can confirm that the power (slope) of the two matches well in the large $x$ limit.\n\n\\begin{figure}[b!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{rhocorr4.png}\n    \\caption{The density-density correlation function for $N_p = 70$ (blue) and $N_p = 120$ (yellow) particles in a $L = 200$ chain.\n    All parameters are identical to that of Fig.~\\ref{fig:weak}.\n    The density-density correlation follows a power-law scaling on both cases. \n    }\n    \\label{fig:rhorho}\n\\end{figure}\n\n\\section{Effect of trap potential}\n\n\nThe calculations in the main text focus on an optical lattice but ignores the presence of the harmonic trap potential.\nIn real experiments, the harmonic trap is inevitable and will affect the previously shown results. While we expect this to have a weak effect on the weak coupling phases, we need to confirm that the phase separated regime in Fig.~\\ref{fig:strong}(b) survives. We include a harmonic trap by adding to the Hamiltonian a term $\\sum_{r,\\sigma} V_r c_{r\\sigma}^{\\dag}c_{r\\sigma}$ with $V_r = \\frac{1}{2} m\\omega^2 r^2$.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.49\\linewidth]{trapN1.png}\n    \\includegraphics[width=0.49\\linewidth]{trapEE1.png}\n    \\includegraphics[width=0.49\\linewidth]{trapN2.png}\n    \\includegraphics[width=0.49\\linewidth]{trapEE2.png}\n    \\caption{\n    Left accumulated particle $\\mathcal{N}_L(x)$ as a function of position and the bipartite entanglement entropy in a $L = 400$ chain in the presence of a harmonic trap potential.\n    (a) $N_p = 167$ and $\\frac{1}{2} m\\omega^2 x_{\\textrm max}^2 = t$, (b) $N_p = 163$ and $\\frac{1}{2} m\\omega^2 x_{\\textrm max}^2 = 2t$.\n    The yellow dashed lines are the linear fit near the center ($x = 200$) and the deviation from this line indicates different filling (and thus different phase) away from the center.\n    }\n    \\label{fig:trap}\n\\end{figure}\n\nHere, we take the experimental setup in Ref.~\\onlinecite{Guardado-Sanchez2021} with $t = h \\times 1.7$ kHz and $a = 752$ nm. \nFig.~\\ref{fig:trap} is the case of $N_p = 167, 163$ particles in a $L=400$ chain with a trap potential $\\frac{1}{2} m\\omega^2 x_{\\textrm max}^2 = t, 2t$, respectively with $x_{\\textrm max}=L\/2$.\nThis corresponds to a trap frequency of order $10^2$Hz, which is safely larger than the typical harmonic trap. \n\nIt is clear that the slope (average particle density) at the center is different from that of the two boundaries. \nThe bipartite entanglement entropy also shows distinct behavior between center and boundaries, and the entanglement is low at the center regions, following the trend seen in Fig.~\\ref{fig:strong}(c). Here, the center region is the magnetic domain wall and the charge like puddles are to its side. \nFrom these results, while the overall structure of the phase separated state is modified by the trap \nwe conclude that the phase separated regime of the model will not be prohibited by the harmonic trap in experiments and we expect can be directly measured. \n\n\n\\section{Correlations in the phase separated regime}\n\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=0.32\\linewidth]{PS_rhorho.png}\n    \\includegraphics[width=0.325\\linewidth]{PS_sigsig.png}\n    \\includegraphics[width=0.30\\linewidth]{MI.png}\n    \\caption{\n    Correlation functions for the same parameters in Fig.~\\ref{fig:strong}(b,c). \n    ($L=400$, $N_p=170$ in the strong coupling regime) \n    (a) The density-density correlation function where $x_0 = 87$ [blue], $212$ [yellow], $324$ [green] which are the centers of the $c=1$ puddles in Fig.~\\ref{fig:strong}(c).\n    (b) The spin-spin correlation function where $x_0 = 154$ [blue], $263$ [yellow] which are in between the $c=1$ puddles.\n    (c) The mutual information (MI) between two points $x$, $y$ in the system.\n    The MI shows three puddles which corresponds to the three $c=1$ regions, and the information spread is blocked by the magnetized region. \n    }\n    \\label{fig:MI}\n\\end{figure*}\n\nIn this section, we further investigate the phase separated state in the context of correlations to provide additional results that imply the presence of Luttinger liquid puddles.\nIn the main text Fig.~\\ref{fig:strong}(c), we claimed that the three regions with $\\bullet$$\\bullet$$\\circ$$\\circ$$\\circ$ patterns have a central charge of $c=1$ which results from the gapless charge degrees of freedom and is inherited from the weak coupling limit of the problem. \nWe check this directly calculating the density-density correlation function and observing its functional form. \n\nIn Fig.~\\ref{fig:MI}(a) we plot the density-density correlation function for the same parameters in Fig.~\\ref{fig:strong}(b,c), where the center $x_0$ is chosen as the center of the three $c=1$ regions. \nOne can observe slow power-law decay of the correlation, although the exponent is hard to extract due to the small size of each region.\nIn contrast, the spin-spin correlation function decays exponentially, even in the magnetized region. \nIn Fig.~\\ref{fig:MI}(b) shows the fast decay of such correlation, where $x_0$ is the center of the magnetized region (in between the $c=1$ regions).\n\nFinally, we plot the mutual information (MI) between two points $x$, $y$ in the system. \nThe MI between two subsets $A$ and $B$ of the system is defined as~\\cite{Amico08}:\n\\begin{align}\n    \\mathcal{I}(A, B) = \\mathcal{S}(A) + \\mathcal{S}(B) - \\mathcal{S}(A\\cup B),\n\\end{align}\nwhere $\\mathcal{S}(A)$ is the entanglement entropy (or von Neumann entropy) introduced in the main text.\nFig.~\\ref{fig:MI}(c) plots the MI between all pair of points in the system, and shows that the MI is large only between points within the same $c=1$ region. \nThis shows the natural result that mutual information is large in gapless regions, and also the interesting feature that the gapped regions serve as a barrier of information spread between the gapless $c=1$ puddles.\n\n\\section{Bosonization analysis at strong coupling}\n\n\nWe now present details of the bosonization discussion regarding phases for arbitrary commensurate filling. For a filling equal to $\\frac{p}{q}$, $p,q$ being mutually prime integers, an additional (umklapp) term appears in the low-energy theory, corresponding to simultaneous scattering of $q$ fermions from $-k_F$ to $k_F$. This can be written as \\cite{schulz_mott} \n\\begin{equation}\n    H_{um}=U_{um}\\int dx \\cos(q\\sqrt{2} \\varphi_\\rho) \\cos^q(\\sqrt{2}\\varphi_\\sigma)\n\\end{equation}, where for weak coupling $U_{um}\\sim U (U\/t)^{q-2}$ \\cite{schulz_mott,Giamarchi}. For even $q$, the most relevant part of $\\cos^q(\\sqrt{2}\\varphi_\\sigma)$ is just a constant, such that $U_{um}$ is relevant for $K_\\rho<4\/q^2$, leading to a charge gap in that case. Apart from half filling, this implies that the density degree of freedom can remain gapless below a threshold value of $U$. For odd $q$, $\\cos(\\sqrt{2}\\varphi_\\sigma)$ has to be kept. At weak coupling, if $V_C(2k_F)<0$ this term is fixed $\\cos(\\sqrt{2}\\varphi_\\sigma)=\\pm1$ and the same condition for $K_\\rho$ is valid , while for $V_C(2k_F)>0$ one can use the fixed-point value $K_\\sigma=1$ \\cite{schulz_mott} such that $K_\\rho< 3\/q^2$ is required. In the latter case, a gap will open simultaneously in the density and spin excitations. The regions of $1\/5$ filling of Fig. \\ref{fig:strong} (b,c) actually conform to weak coupling expectations. In that case $k_F r_c\\approx 5$ such that $V_C(2k_F)\/V_0 \\approx - 0.16$ and a spin gap is expected, while the density gap requires strong interactions $K_\\rho<0.16$. Note that the observed strong modulation of density occurs at $2k_F$, same as for a weak-coupling density wave. The \n$\\odot$%\n$\\odot$%\n$\\circ$%\n$\\circ$%\n$\\circ$%\n$\\circ$ regions, on the other hand, has filling $1\/4$ with a less stringent requirement for the charge gap $K_\\rho<0.25$, allowing to understand the development of a charge gap there. While in weak coupling, a spin gap would have been also expected due to $V_C(2k_F)\/V_0\\approx-0.026$, its absence points to inapplicability of the continuum weak coupling analysis for $1\/4$ filling here, possibly due to extremely small value of  $V_C(2k_F)\/V_0$ and finite size effects. The above arguments also suggest that for 1\/6 filling of Fig.~\\ref{fig:strong}(a), there will be a spin gap $V_C(2k_F)\/V_0\\approx-0.2$ and no charge gap ($K_\\rho<0.11$ is required), consistent with the numerical result. The charge modulation has the period $6$ (see Fig.~\\ref{fig:strong2}(a)), corresponding to the usual $2k_F=2\\pi\/6$ Luttinger liquid correlations \\cite{Giamarchi}, in contrast to the period $3$ ($4k_F$) modulation expected in the insulating state \\cite{schulz_mott,Giamarchi}.\nThe bipartite entanglement entropy (see Fig.~\\ref{fig:strong2}(b)) also shows clear $c=1$ scaling. \n\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.49\\linewidth]{rhocorr5.png}\n    \\includegraphics[width=0.49\\linewidth]{PS_EE3.png}\n    \\caption{\n    (a) The power-law scaling density-density correlation for the system in Fig.~\\ref{fig:strong}(a), $N_p = 140$. (b) The bipartite entanglement entropy as a function of the cut position. The solid line is the CFT scaling form (Eq.~\\eqref{eq:ee}) with $c=1$. \n    }\n    \\label{fig:strong2}\n\\end{figure}\n\n\\section{Optical tweezers and the $PXP$ limit}\n\nIn the limit of no tunneling (i.e. focusing on the optical tweezer set up as opposed to an optical lattice) and strong coupling we determine the effective many-body Hamiltonian, recovering the $PXP$ model in the limit of a fully filled Fermi sea and identify a magnetic $PXP$ model with a direct exchange interaction in the limit of half-filling.\n\n\nThe $PXP$ limit assumes a strong mixing of ground and Rydberg state $\\Omega \\gg \\Delta$. In this case, the dominant interaction is the repulsion between adjacent Rydberg atoms, that is also larger than $\\Omega$. Essentially, the interaction forbids atoms in Rydberg state to occupy neighboring sites. For spinless bosons, Rydberg Hamiltonian reduces then to the $PXP$ model, $H = \\sum_i P_{i-1}X_i P_{i+1}$~\\cite{sun2008,olmos2012,turner2018}. \n$P_i = (1-Z_i)\/2$ is a projector to the ground state. \nThe typical $PXP$ model studied intensively in the context of quantum many-body scars and slow dynamics focused on bosonic Rydberg atoms.\nHere, we consider fermionic Rydberg atoms which introduces the spin degrees of freedom to the original problem. \n\nFirst, consider a chain of Rydberg atoms with two particles (spin up and down) per site. \nThe $X_i$ operator connects the $|g\\uparrow, g\\downarrow\\rangle$ state with the $(|R\\uparrow,g\\downarrow\\rangle + |g\\uparrow,R\\downarrow\\rangle)\/\\sqrt{2}$ state and thus this model reduces to the original $PXP$ model.\nThis is reasonable in the sense that each site has two fermions which can be considered as a bosonic degree of freedom. \n\nNext, we consider a Rydberg atom chain with the filling of one particle per site. Neglecting the coupling of the laser to the spin degree of freedom (which is appropriate for $s$-like states), the Hamiltonian reduces to the original $PXP$ model with additional spin degeneracy, i.e.\n\\begin{align}\n    H_{PXP}^{spin} = \\sum_i P_{i-1} X_i P_{i+1},\n\\end{align}\nwhere $X = | R \\uparrow\\rangle\\langle g\\uparrow| + | R \\downarrow\\rangle\\langle g\\downarrow| + \\textrm{h.c.}$ is the Pauli matrix in the $eg$-space, $P = | g \\uparrow\\rangle\\langle g\\uparrow| + | g \\downarrow\\rangle\\langle g\\downarrow|$ is the projection to the ground state.\n\n\n\nThe spin degeneracy, however, can be lifted by interactions due to the exchange process. \nBoth exchange and supexchange interactions depend on wavefunction overlap \\cite{auerbach2012interacting} and can be of the same order. Therefore, one can expect that: (1) they decrease with distance fast (2) they are larger for the Rydberg state.\nAs the Rydberg interaction energy prohibits nearest neighbor atoms to be both in the Rydberg state, the next possibility is to either have spin-spin interactions with a neighboring ground-state atom, or a next-neighbor Rydberg atom. Combining both contributions, the spin exchange term can be written as:\n\\begin{equation}\n\\begin{gathered}\n    H_{exch}^{spin} = \\sum_i  J_{Rg} \\vec{S}_i \\cdot \\vec{S}_{i+1} (P_i Q_{i+1} + Q_i P_{i+1}) \n    \\\\\n    + J_{RR} \\vec{S}_i \\cdot \\vec{S}_{i+2} Q_i Q_{i+2},\n    \\end{gathered}\n\\end{equation}\nwhere $Q = 1 - P$.\n\n\nFor the direct exchange mechanism one can provide the following estimates for the couplings $J_{Rg}$ and $J_{RR}$. The Rydberg-Rydberg exchange coupling takes the form\\cite{auerbach2012interacting} :\n\\begin{equation}\n\\begin{gathered}\n     J_{RR} = \\frac{1}{2} \\int d^3 {\\bf r} d^3{\\bf r}' V_R(|{\\bf r} - {\\bf r}'|) \\psi^*_R({\\bf r}) \\psi_R({\\bf r}')\\times \n     \\\\\n     \\times\\psi_R^*({\\bf r}'+2 a \\hat{x} \\bf) \\psi_R({\\bf r}+2 a \\hat{x} \\bf),\n     \\end{gathered}\n\\end{equation}\nwhere $\\hat{x}$ is a unit vector along the chain direction and $V_R(|{\\bf r} - {\\bf r}'|) \\propto 1\/|{\\bf r} - {\\bf r}'|^6$, and $\\psi_{g,R}({\\bf r})$ correspond to ground and excited state atomic wavefunctions. One observes then that the integral is dominated by ${\\bf r} \\approx {\\bf r}'$ and is determined by the overlap between Rydberg wavefunctions displaced by $2a$. As the size of the Rydberg wavefunction scales with principal quantum number as $n^2$, the integral can be estimated to scale (taking the integrand value in the middle between two atoms) as $\\sim \\exp\\left( - 2 \\frac{2a}{a_B^* n^2}\\right)$, where $a_B^*$ is the Bohr radius for the atom. \n\nThe Rydberg-ground exchange coupling is determined by:\n\\begin{equation}\n\\begin{gathered}\n     J_{Rg} = \\frac{1}{2} \\int d^3 {\\bf r} d^3{\\bf r}' V_{Rg}(|{\\bf r} - {\\bf r}'|) \\psi^*_R({\\bf r}) \\psi_R({\\bf r}')\\times\n     \\\\\n     \\times\\psi_g^*({\\bf r}'+a \\hat{x} \\bf) \\psi_g({\\bf r}+ a \\hat{x} \\bf),\n     \\end{gathered}\n\\end{equation}\nwhere $V_{Rg}(|{\\bf r} - {\\bf r}'|)$ is the interaction between Rydberg and ground state atom, that is likely to be much weaker than the Rydberg interaction. The scaling of this integral with $n$ can be estimated analogously to be $\\sim \\exp\\left( - \\frac{a}{a_B^* n^2}- \\frac{a}{a_B^*}\\right)$. \nUnfortunately,  in typical tweezer setups \\cite{bernien2017} $a\\gg a_B^* n^2$, suggesting that the exchange interactions are likely quite small for the current set ups.\n\n\n\n\\end{appendix}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGiven a pointed space $X$, denote by $\\mathcal E(X)$ the group of\n(based) self homotopy equivalences, i.e., the group of\nautomorphisms of $X$ in the pointed homotopy category. From now on\nwe shall consider connected complexes of finite type $X$ which are\neither finite or with finitely many non trivial homotopy groups.\nWe denote by $\\dim X=N$ its topological or homotopical dimension.\nUnless explicitly stated otherwise, all  spaces will be of this\nkind.\n\nAlthough the computation of  $\\mathcal E(X)$ is known to be a hard task,\nthere are two classical and key results that impose to this group\nimportant structural constraints:\n\nOn one hand, a theorem of Sullivan  \\cite[Theorem 10.3]{su} and\nWilkerson   \\cite[Theorem B]{wi} states that $\\mathcal E(X)$ is finitely\npresented. This was originally proved for simply connected spaces\nand later on generalized to virtually nilpotent spaces by Dror,\nDwyer and Kan \\cite[Theorem 1.1]{ddk}. The main step in the proof\nis to show that $\\mathcal E(X_\\mathbb Q)$ is an algebraic group and that\n$\\mathcal E(X)$ is commensurable with an arithmetic subgroup of\n$\\mathcal E(X_\\mathbb Q)$. As a consequence, it can be shown that there exists\na finite bound for the finite orders of elements of $\\mathcal E(X)$.\n\nOn the other hand we have the following theorem  due to Dror and\nZabrodsky:\n\n\\begin{theorem}\\label{principal}{\\em \\cite[Theorem B]{d-z}} Let $G$ be a subgroup of\n$\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le N}(X)$. Then $G$ is\nitself nilpotent. In particular, $\\mathcal E_{ \\sharp }^m(X)$ is nilpotent.\n\\end{theorem}\n\nRecall that, for $0\\le m\\le\\infty$, $\\mathcal E_{ \\sharp }^m(X)$ is the\ndistinguished subgroup   of $\\mathcal E(X)$ formed by those classes\ninducing the identity on the homotopy groups up to $m$. In other\nwords, $$ \\mathcal E_{ \\sharp }^m(X)=\\text{ker}\\bigl(\\mathcal E(X)\\longrightarrow\n\\Pi_{i\\le m}\\text{aut}\\,\\pi_iX\\bigr). $$  If $\\dim X=N$ we shall\ndenote  $\\mathcal E_{ \\sharp }^N(X)$ simply by  $\\mathcal E_{ \\sharp }(X)$.\n\n\n  Here, we present a slightly different proof for this well known result\nin which we use a broader study of self homotopy equivalences in\nthe homotopy category ${\\cal L}^*$ of (based) spaces with local\ncoefficients. Recall (see \\cite[Chap.VI]{white} for instance) that\nobjects in this category are pairs $(X,{\\cal M})$ in which $X$ is a\n(based) topological space and ${\\cal M}=\\{M_x\\}_{x\\in X}$ is a local\ncoefficient system in $X$. On the other hand, a morphism\n$(f,\\Theta)\\colon (X,{\\cal M})\\to(Y,{\\cal H})$ is a pair formed by a\nbased map $f\\colon X\\to Y$ and a morphism $\\Theta\\colon\nf^*{\\cal H}\\to{\\cal M}$  of local coefficient. By $f^*{\\cal H}$ we denote,\nas usual, the local coefficient system on $X$ induced by $f$,\ni.e., $(f^*{\\cal H})_x=H_{f(x)}$. For each $x\\in X$ we shall denote\nby $\\Theta_x\\colon H_{f(x)}\\to M_x$ the corresponding group\nmorphism at $x$. After considering the appropriate homotopy\nnotion, one obtains the homotopy category   ${\\cal L}^*$. The\ngroup of self homotopy equivalences of an object $(X,{\\cal M})\\in\n{\\cal L}^*$ shall be denoted by   ${\\mathcal E}(X;\\calm)$. Then, we prove:\n\n\n\n\n\n\n\\begin{theorem}\\label{local} Let $X$ be a finite Postnikov piece\nand let  $G\\subset {\\mathcal E}(X;\\calm)$ be a subgroup which acts nilpotently\non both $\\pi_*(X)$ and  ${\\cal M}$. Then, $G$ acts nilpotently on $H^*(X;{\\cal M})$.\n\\end{theorem}\n\n\nAt the sight of Theorem \\ref{principal}, and taking into account\nthe bound of finite orders of elements of $\\mathcal E(X)$ plus the\nexistence of a ``fracture lemma\" for this group (see \\cite[Theorem\n8.2]{Rutter}), it has been of interest to study whether  $\\mathcal E_{ \\sharp }(X)$\nsatisfies the same structural restrictions when taking\n$p$-localization, $p$-completion or considering $\\mathcal E_{ \\sharp p }(X)$. This\ndenotes the subgroup of $\\mathcal E(X)$ formed by those classes which\ninduce the identity on the homotopy groups of $X$ with\ncoefficients on $\\mathbb Z\/p$, up to the dimension of $X$, i.e., $$\n\\mathcal E_{ \\sharp p }(X)=\\text{ker}\\bigl(\\mathcal E(X)\\longrightarrow \\Pi_{\\le\nN}\\text{aut}\\,\\pi_i(X;\\mathbb Z\/p)\\bigr). $$ As examples of this, we\nmention two interesting results for a given nilpotent space $X$:\nMaruyama proved \\cite[Theorem 0.1]{ma} that\n$\\mathcal E_{ \\sharp }(X)_{(p)}=\\mathcal E_{ \\sharp }^N(X_{(p)})$ while, on the other hand,  M{\\o}ller\nshowed \\cite[Theorem 4.3]{mo} that\n$$\\mathcal E_{ \\sharp }(X_{\\mathbb Z_p})=Ext(\\mathbb Z\/p^\\infty,\\mathcal E_{ \\sharp }(X))=\n\\mathcal E_{ \\sharp }(X_p{\\displaystyle\\hat{}}). $$  Here and henceforth,\n$(-)_{\\mathbb Z_p}$ denotes $H_*(-;\\mathbb Z\/p)$-localization while\n$(-)_{(p)}$ and $(-)_p{\\displaystyle\\hat{}}$ are the classical\nlocalization and completion on the prime $p$.\n\nIn this paper we plan to continue this investigation extending\nTheorem \\ref{principal} above, considering a subgroup of $\\mathcal E(X)$\nwhich acts nilpotently in the homotopy groups of $X$ localized,\ncompleted or with coefficients in $\\mathbb Z\/p$. Concerning this purpose\nwe prove:\n\n\n\n\\begin{theorem}\\label{dos} Assume that $\\pi_1(X)$ is a nilpotent group and  let $G$ be a subgroup of $\\mathcal E(X)$ which acts\nnilpotently on $\\pi_{\\le N}(X)_{(p)}$, for $p$ any prime number\nand $0$. If the nilpotency orders of all these actions are bounded\nby a fixed integer, then G is nilpotent.\n\\end{theorem}\n\\begin{remark} {\\em Observe that in the theorem above the condition of $\\pi_1(X)$ being nilpotent is essential. Otherwise,\nchoose any finite simple group $G$ which is known to be\ngenerically trivial, i.e.,  $G_{(p)}=\\{1\\}$ for $p$ any prime\nnumber or zero. On the other hand observe that the map $G\\to \\text{aut}\\,\nG$ given by inner automorphisms is a monomorphism. Indeed, its\nkernel is the center of $G$ which is trivial since $G$ is simple.\nThis inclusion renders the non nilpotent group $G$ as a subgroup\nof $\\mathcal E\\bigl(K(G,1)\\bigr)$ which acts nilpotently in the\nlocalized homotopy group.\n\nIn a similar way, we may even produce an example of a solvable,\nnon nilpotent group, of homotopy equivalences acting nilpotently\nin the localized homotopy groups of the space. Consider the\nsymmetric group $\\Sigma_3$ and observe that\n${\\Sigma_3}_{(2)}=\\mathbb Z\/2$ while ${\\Sigma_3}_{(p)}=1$ for $p\\not=2$.\nAgain, $\\Sigma_3\\subset\\mathcal E\\bigl(K(\\Sigma_3,1)\\bigr)$ is a\nsolvable non nilpotent group acting nilpotently on any\n${\\Sigma_3}_{(p)}$.}\n\\end{remark}\n\nA more subtle and slightly  different situation is given when\nconsidering nilpotent actions of subgroups of self homotopy\nequivalences on the Frattini factor of the homotopy groups. Recall\nthat given a group $G$, the Frattini subgroup $\\Phi(G)$ is the\nintersection of all maximal proper subgroups of $G$. The quotient\n$G\/\\Phi(G)$ is called the Frattini factor.\n\n\\begin{theorem}\\label{uno} Assume that\n$\\pi_{\\le N}(X)$ is a finite nilpotent group and let $G$ be a\nsubgroup of $\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le\nN}(X)\/\\Phi\\bigl(\\pi_{\\le N}(X)\\bigr)$. Then, G is nilpotent.\n\\end{theorem}\n\nIn particular, taking into account that for an abelian $p$-group\n$G$ its Frattini factor is precisely $G\\otimes\\mathbb Z\/p$, we obtain\nthe following:\n\n\\begin{corollary}\\label{nuevo} Assume that\n$\\pi_{\\le N}(X)$ is a finite abelian group and let $G$ be a\nsubgroup of $\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le\nN}(X)\\otimes\\mathbb Z\/p$ for any prime $p$. Then, G is\nnilpotent.\\hfill$\\square$\n\\end{corollary}\n\nNotice that, by the Universal Coefficients Theorem for homotopy,\n$\\pi_*X\\otimes\\mathbb Z\/p=\\text{Ext}(\\mathbb Z\/p,\\pi_*X)$ is a subgroup of\n$\\pi_*(X;\\mathbb Z\/p)$. Hence, as an immediate consequence of Corollary\n\\ref{nuevo} above we get:\n\n\\begin{corollary}\\label{coeficientes} Assume that\n$\\pi_{\\le N}(X)$ is a finite abelian group and let $G$ be a\nsubgroup of $\\mathcal E(X)$ which acts nilpotently on $\\pi_{\\le\nN}(X;\\mathbb Z\/p)$ for  any prime $p$. Then, G is\nnilpotent.\\hfill$\\square$\n\\end{corollary}\n\n\n\n\nHaving studied the nilpotency of a general subgroup of $\\mathcal E(X)$,\nwe  now focus on the group $\\mathcal E_{ \\sharp p }(X)$ and give necessary conditions\nfor it to be nilpotent.\n\n\\begin{theorem}\\label{tres}  Let $X$ be a space for which $\\pi_{\\le N}(X)$ is a finite abelian $p$-group. Then\n$\\mathcal E_{ \\sharp p }(X)$ is nilpotent and $\\mathcal E_{ \\sharp p }(X)\/\\mathcal E_{ \\sharp }(X)$ is a finite $p$-group.\n\n\\end{theorem}\n\n\\begin{theorem}\\label{cuatro}\nLet $X$ be a space for which $\\pi_{\\le N}(X)$ is a finitely\ngenerated abelian group. Then $\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$ is\nnilpotent.\n\\end{theorem}\n\n\\begin{remark}\\label{importante}  {\\em Observe that in general\n$\\mathcal E_{ \\sharp p }(X)$ is bigger than $\\mathcal E_{ \\sharp }(X)$. For instance consider\n$X=K(\\mathbb Z\/{p^r},n)$, $r,n\\ge 2$. Obviously $\\mathcal E_{ \\sharp }(X)=\\{1\\}$, while\nthe automorphism $\\rho$ of $\\mathbb Z\/{p^r}$ given by\n$\\rho(1)=p^{r-1}+1$ induces a non trivial element of $\\mathcal E_{ \\sharp p }(X)$.\nIndeed, by the Universal Coefficients Theorem for homotopy, $$\n\\pi_*(X,\\mathbb Z\/p)=\\pi_n(X,\\mathbb Z\/p)\\oplus \\pi_{n-1}(X,\\mathbb Z\/p), $$  in\nwhich\\hfill\\break\n\n\\medskip\n\\noindent$\\pi_n(X,\\mathbb Z\/p)=\\hom(\\mathbb Z\/p,\\mathbb Z\/{p^r})$ {and}\n$\\pi_{n-1}(X,\\mathbb Z\/p)=\\text{Ext}(\\mathbb Z\/p,\\mathbb Z\/{p^r})=\\mathbb Z\/p.$\n\n\\medskip\n\\noindent Trivially $\\rho$ induces the identity on both. Note that\nthis example also shows that even\n$\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$  can be bigger than $\\mathcal E_{ \\sharp }(X)$.}\n\\end{remark}\n\nThe paper is organized as follows: in the next section we collect\nthe results we shall need from group theory and from which\nTheorems \\ref{dos} and \\ref{uno} are immediately deduced. Theorem\n\\ref{local} and \\ref{principal} are proved in section \\S3.\nFinally, in  section \\S4 we establish Theorems \\ref{tres} and\n\\ref{cuatro}.\n\n\\section{From group theory}\n\nWe begin by recalling some basic facts. If $G$ is a group acting\non another group $A$ (i.e., $A$ is a $G$-group), the $n$-th  $G$-commutator subgroup\n$\\Gamma^n_G(A) \\subset A$ is the group generated by\n$\\{(ga^{-1})a\\,|\\, g\\in G,\\, a\\in\\Gamma_G^{n-1}(A)\\}$, being\n$\\Gamma_G^0(A)=A$. The action is then nilpotent of nilpotency\norder $r$, $\\text{nil}\\,_GA=r$, if this is the smallest integer for which\n$\\Gamma^r_G(A)=\\{1 \\}$. The group $G$ also acts in each\n$\\Gamma_G^n(A)$ and\n$\\Gamma_G^m\\bigl(\\Gamma_G^n(A)\\bigr)=\\Gamma_G^{m+n}(A)$.\n\nStatements of next sections shall heavily rely in the following\nresults:\n\n\\begin{lemma}\\label{lema1}  Let $A$ be a $G$-group. Then:\n\\begin{itemize}\n\\item[(i)] $\\Gamma_G^1(A)$ is a normal subgroup of $A$ and the $G$-action induced on $A\/\\Gamma_G^1(A)$ is trivial.\n\\item[(ii)] The quotient morphism  $A{\\stackrel{q}{\\longrightarrow}} A\/\\Gamma_G^1(A)$ is  equivariant and initial with\nrespect to trivial actions, i.e., every equivariant morphism\n$A{\\stackrel{f}{\\longrightarrow}}H$, in which the $G$-action on\n$H$ is trivial, factors uniquely through $q$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} (i) is trivial. For (ii) observe that, for any $f$ as in the lemma, $\\Gamma_G^1(A)\\subset\\ker f $.\n\\end{proof}\n\n\\begin{lemma}\\label{lema2}  Let $A$ be a $G$-group. If $A$ is nilpotent then, for any $m$, $\\Gamma_G^m(A)_{(p)}=\\Gamma_G^m(A_{(p)})$.\n\\end{lemma}\n\n\\begin{proof} Since $\\Gamma_G^m(A)=\\Gamma_G^{1}\\bigl(\\Gamma_G^{m-1}(A)\\bigr)$, once we show that\n$\\Gamma_G^1(A)_{(p)}=\\Gamma^1_G(A_{(p)})$ an easy induction proves\nthe lemma. As localization  is an exact functor in the category of\nnilpotent groups, the localization morphism $f:A\\to A_{(p)}$\nrestricts to $f:\\Gamma_G^1(A)\\to \\Gamma_G^1(A)_{(p)}$. Hence, we\nmay consider $\\Gamma_G^1(A)_{(p)}$, as well as\n$\\Gamma_G^1(A_{(p)})$, as subgroups of $A_{(p)}$. Then, for any\n$g\\in G$ and $a\\in A$, the trivial identity\n$\\bigl(gf(a)^{-1}\\bigr)f(a)=f\\bigl((ga^{-1})a\\bigr)$ shows\nequality of both subgroups.\n\\end{proof}\n\n\\begin{proposition}\\label{uf} The group $G$ acts nilpotently on the nilpotent group $A$ if and only if $G$ acts\nnilpotently on $A_{(p)}$ for $p$ any prime number or zero and all\nthese nilpotency orders are bounded.\n\\end{proposition}\n\n\\begin{proof} Assume $G$ acts nilpotently on $A$, i.e., $\\Gamma_G^m(A)=\\{1\\}$ for some $m$. Hence, by Lemma \\ref{lema2}\nand for any $p$, $\\Gamma_G^m(A_{(p)})=\\{1\\}$.\n\nConversely, assume $\\text{nil}\\,_GA_{(p)}\\le m$, for all $p$ ($p$ a prime\nnumber or $0$), and let $a$ be an element of $\\Gamma_G^m(A)$. If\n$a$ has finite order,  say it is a $q$-element, then it obviously\nsurvives under the $q$-localization morphism\n$\\Gamma_G^m\\bigl(A)\\to\\Gamma^m_G(A)_{(q)}$. For a general group,\nelements of infinite order are not guaranteed to survive under\nrationalization (for instance, the rationalization of the free\nproduct of two finite groups is trivial while it contains elements\nof infinite order). However for a nilpotent group, which is our\ncase, one can easily show by induction on the nilpotency order of\nthe group, that any element of infinite order is not sent to zero\nunder rationalization.  Taking into account, again by Lemma\n\\ref{lema2}, that $\\Gamma^m_G(A)_{(p)}=\\Gamma^m_G(A_{(p)})=\\{1\\}$,\nit follows that $a=1$ and the proof is complete.\n\\end{proof}\n\n\n\\begin{proposition}\\label{nuevolema} Let $G$ be a group acting on a finite nilpotent group $A$ in such a way that the\ninduced action on the Frattini factor $A\/\\Phi(A)$ is nilpotent.\nThen, the $G$-action on $A$ is also nilpotent.\n\\end{proposition}\n\n\\begin{proof} Recall \\cite[5.1]{Gorestein} that the Frattini subgroup of a group $A$, $\\Phi(A)$, is defined to be the intersection of all its maximal proper subgroups. The Frattini factor of $A$ is $A\/\\Phi(A)$.\nObserve in the first place that, since $\\Phi(A)$ is a characteristic subgroup of $A$, i.e., it is invariant under any automorphism of $A$, $G$ in fact induces a\nnatural  action on the Frattini factor $A\/\\Phi(A)$ which, by\nhypothesis, is nilpotent. Hence, since $A\/\\Phi(A)$ is nilpotent,\nthe induced action on\n$\\bigl(A\/\\Phi(A)\\bigr)_{(p)}=A_{(p)}\/\\Phi(A)_{(p)}$ is also\nnilpotent by Lemma \\ref{lema2}. Next, observe that for any finite\ngroup $A$, $\\Phi(A)_{(p)}=\\Phi(A_{(p)})$. Indeed, this is\nimmediate from the definition taking into account that\nlocalization commutes with limits,  in particular, with\nintersections (see for instance \\cite{hmr}). Therefore, we conclude that $G$ acts nilpotently on\n$A_{(p)}\/\\Phi(A_{(p)})$. Considering $\\varphi\\colon\nG\\to\\text{aut}\\,\\bigl(A_{(p)}\/\\Phi(A_{(p)})\\bigr)$ via this action, and\ntaking into account that $A_{(p)}\/\\Phi(A_{(p)})$ is a finite\n$p$-group, we may apply \\cite[Corollary 5.3.3]{Gorestein} to\nobtain that $\\varphi(G)$ is also a $p$-group. But the action of a\n$p$-group on another $p$-group is always nilpotent, and therefore\n$G$ acts nilpotently on $A_{(p)}$. Since this is the case for any\n$p$ and $A$ is finite we may apply  Proposition \\ref{uf} and the\nproposition follows.\n\\end{proof}\n\nFrom these results we immediately deduce:\n\n\\bigskip\n\\noindent {\\it Proof of Theorems \\ref{dos} and \\ref{uno}.} Apply\ndirectly Propositions \\ref{uf} and \\ref{nuevolema} above to the\nsubgroup $G$ of $\\mathcal E (X)$ to obtain that $G$ acts nilpotently on\n$\\pi_{\\le N}(X)$. Then, the result follows from Theorem\n\\ref{principal}. \\hfill$\\square$\n\\bigskip\n\n\n\n\nClosely related to Proposition \\ref{nuevolema}, we have the\nfollowing:\n\n\\begin{proposition}\\label{propodos} Let $G$ be a group acting on an abelian $p$-group $A$ which has an exponent $p^n$. If\n$G$ acts nilpotently on $A\\otimes\\mathbb Z\/p$, then it does so on $A$\nand $$ \\text{nil}\\,_GA\\le n\\cdot\\text{nil}\\,_GA\\otimes\\mathbb Z\/p. $$\n\\end{proposition}\n\\begin{proof} Call $r=\\text{nil}\\,_GA\\otimes\\mathbb Z\/p$ and observe that $\\Gamma_G^m(A\\otimes\\mathbb Z\/p)=\\Gamma_G^m(A)\\otimes\\mathbb Z\/p$ for any\n$m$. Therefore, since $\\Gamma_G^r(A\\otimes\\mathbb Z\/p) =0$,\n$\\Gamma_G^r(A)\\subset pA$. Assume, as induction hypothesis, that\n$\\Gamma_G^{kr}(A)\\subset p^kA$, for $k<n$. Hence, $$\n\\Gamma_G^{nr}(A)=\\Gamma_G^{(n-1)r}\\bigl(\\Gamma_G^r(A)\\bigr)\\subset\n\\Gamma_G^{(n-1)r}(pA)= p\\Gamma_G^{(n-1)r}(A)\\subset p^nA. $$ Since\n$A$ has $p^n$ as exponent, the proposition follows.\n\\end{proof}\n\n\\begin{proposition}\\label{propouno} Let $A$ be a finite abelian\n$p$-group and let $G\\subset\\text{aut}\\,(A)$ be such that $\\sigma\\otimes\n{\\mathbb Z\/p}=1_{A\\otimes\\mathbb Z\/p}$ for each $\\sigma\\in G$. Then $G$ is a\n$p$-group.\n\\end{proposition}\n\n\\begin{proof}  As $A$ is a finite  abelian $p$-group,\nthe Frattini factor $A\/\\Phi(A)$ (respec. the projection\n$A\\rightarrow A\/\\Phi(A)$) is naturally  identified with\n$A\\otimes\\mathbb Z\/p$ (respec.\\ the map $A\\rightarrow A\\otimes\\mathbb Z\/p$).\nNow, if $G$ is  not a $p$-group, there exists a non trivial\n$p^\\prime$-automorphism $\\sigma\\in G$ which, by hypothesis and\nusing the identification above,  induces the identity on the\nFrattini factor of $A$. But according to \\cite[Theorem\n5.1.4]{Gorestein}, the only $p^\\prime$-automorphism that induces\nthe identity on the Frattini factor of a $p$-group is the\nidentity.  Thus $G$ must be a $p$-group.\n\\end{proof}\n\nAs an immediate consequence we get:\n\n\\begin{corollary}\\label{corolario} In the conditions of the proposition above, the action of $G$ on $A$ is nilpotent.\n\\end{corollary}\n\n\\begin{proof} Indeed, recall that the action of a\n$p$-group $H$ on another $p$-group is always nilpotent.\n\\end{proof}\n\nIn which follows $G$ is a group acting on another group $A$. Given\n$g,h\\in G$ and $a\\in A$, we use the following usual notation: $$\n[a,g]=a^{-1}(ga),\\quad [g,a]=(ga^{-1})a,\\quad\n[g,h]=g^{-1}h^{-1}gh. $$ Hence, the following, which can be\nconsidered as a variation of the Witt-Hall identity \\cite[Theorem\n5.1]{magnus}, is obtained by direct calculation.\n\n\\begin{lemma}\\label{jo}\nFor any $f,g\\in G$ and $b\\in A$, the following identity holds:\n$$\\big[[f^{-1},g^{-1}],gb\\big]b^{-1}\\big[[g,b^{-1}], f\\big]\nb\\big[[f,b],fgf^{-1}\\big]=1.$$\n\\end{lemma}\n\n\\begin{lemma}\\label{jo2} Let $H$ be a subgroup of $G$ and $K$ a normal subgroup of $H$. Then,\n$$\n\\bigr[[H,K],A\\bigr]\\subset\\langle\\big[K,[H,A]\\bigr],\\bigl[H,[K,A]\\bigr]\\rangle.\n$$\n\\end{lemma}\n\n\\begin{proof}\nMaking $f^{-1}=h$, $g^{-1}=k$ and $gb=a$ in Lemma \\ref{jo},  it\nfollows that $$\n\\bigl[[h,k],a\\bigr]=\\bigl[h^{-1}k^{-1}h,[h^{-1},g^{-1}a]\\bigr]\ng^{-1}a^{-1}\\bigl[h^{-1},[k^{-1},g^{-1}a]\\bigr]g^{-1}a. $$ As $K$\nis normal in $H$, $\\bigl[h^{-1}k^{-1}h,[h^{-1},g^{-1}a]\\bigr]\\in\n\\big[K,[H,A]\\bigr]$. On the other hand, as commutators are normal\nsubgroups,\n$$g^{-1}a^{-1}\\bigl[h^{-1},[k^{-1},g^{-1}a]\\bigr]g^{-1}a\\in\ng^{-1}a^{-1} \\bigl[H,[K,A]\\bigr]g^{-1}a=\\bigl[H,[K,A]\\bigr], $$\nand the lemma follows.\n\\end{proof}\n\n\\begin{lemma}\\label{jo3} If the action of $G$ on $A$ is nilpotent, then\nfor each $n,m\\ge 0$, $$ [\\Gamma^n(G),\\Gamma_G^m(A)]\\subset\n\\Gamma_G^{n+m+1}(A). $$ In particular,\n$\n[\\Gamma^n (G), A]\\subset \\Gamma_G^{n+1}(A).\n$\n\\end{lemma}\n\n\\begin{proof} Set $\\text{nil}\\,_GA=r$. If $m\\ge r$ the assertion is obvious. Assume the lemma holds for all $n$ and $m\\le 1$ and let us prove it for $m=0$ by\ninduction on $n$:\n\nTrivially,\n$\n[\\Gamma^0(G),\\Gamma_G^0(A)]=[G,A]= \\Gamma_G^{1}(A)$. Finally, $$\n\\begin{aligned}\n{[\\Gamma^n(G),A]}&=\\bigl[[G,\\Gamma^{n-1}(G)], A\\bigr]\\subset\n\\text{(By Lemma \\ref{jo2})}\\\\ &\\subset \\langle\n\\bigl[\\Gamma^{n-1}(G),[G,A]\\bigr],\n\\bigl[G,[\\Gamma^{n-1}(G),A]\\bigr]\\rangle\\subset \\text{(By\ninduction)}\\\\ &\\subset\\langle [\\Gamma^{n-1}(G),\\Gamma^1_G(A)],\n[G,\\Gamma^n_G(A)]\\rangle\\subset \\text{(Again by induction)}\\\\\n&\\subset\\Gamma_G^{n+1}(A).\\\\\n\\end{aligned}\n$$\n\\end{proof}\n\n\\begin{proposition}\\label{jo4}\nLet $G$ be a subgroup of $\\text{aut}\\,(A)$. Then, $\\text{nil}\\, G\\le \\text{nil}\\,_GA-1$.\n\\end{proposition}\n\n\\begin{proof}\nAssume $\\text{nil}\\,_GA=r$. By Lemma \\ref{jo3},\n$[\\Gamma^{r-1}(G),A]\\subset \\Gamma_G^r(A)=\\{1\\}$, and therefore\n$\\Gamma^{r-1}(G)=\\{1\\}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Self homotopy equivalence of spaces with local coefficients}\n\nAs stated in the Introduction, and following the notation and\napproach of the standard reference \\cite[Chap. VI.2]{white}, in\nthis section we consider self homotopy equivalences in the\nhomotopy category ${\\cal L}^*$ of based spaces with local\ncoefficients. Observe that a self homotopy equivalence of an\nobject $(X,{\\cal M})\\in {\\cal L}^*$ is given by  $(f,\\Theta)\\colon\n(X,{\\cal M})\\to (X,{\\cal M})$ in which $f\\colon X\\to X$ is a based\nhomotopy equivalence and $\\Theta\\colon {\\cal M}\\to{\\cal M} $ is an\nisomorphism of the coefficient system ${\\cal M}$. Note that such a\nself equivalence $(f,\\Theta)$ acts  in $\\pi_*(X)$ by $\\pi_*f$, in\n${\\cal M}$ by $\\Theta$, and in $H^*(X;{\\cal M})$ by $H^*(f,\\Theta)$.\n\nIt is also convenient to  recall how cohomology classes with local\ncoefficients are represented by maps into the ``twisted\nEilenberg-MacLane space'' (see \\cite[Chapter 5.2]{baues},\n\\cite{Gitler}, \\cite{moller2} or  \\cite{siegel} for precise\ndetails). Let $K({\\cal M},n)$ be a fixed realization of the\nEilenberg-MacLane space of type $(M_{x_0},n)$ being $M_{x_0}$ the\ngroup of the system ${\\cal M}$ at the base point. On the other hand,\ndenote by $L({\\cal M},n)$ the space obtained by applying the Borel\nconstruction  to the universal fibration $\\pi_1(X)\\to\\widetilde\nK\\stackrel{q}{\\to} K(\\pi_1(X),1)$ and the space $K({\\cal M},n)$,\ni.e.,\n\n$$ L({\\cal M},n)=\\widetilde K\\times_{\\pi_1(X)} K({\\cal M},n), $$\nand it fits into the fibration $$ K({\\cal M},n)\\longrightarrow\nL({\\cal M},n)\\stackrel{p}{\\longrightarrow}K(\\pi_1(X),1),\\quad\np(a,b)=q(a). $$ Then, for a given space $X$, $H^n(X;{\\cal M})$ is\nin one to one correspondence with the set $[X,L({\\cal\nM},n)]_{K(\\pi_1(X),1)}$ of homotopy classes of maps over\n$K(\\pi_1(X),1)$ from $X$ to $L({\\cal M},n)$.\n\n\\bigskip\n\n\n\\noindent {\\it Proof of Theorem \\ref{local}.} To avoid excessive\nnotation we shall not distinguish between a homotopy class and a\nmap which represents it. First, observe that, if $(f,\\Theta)$ is a\nself homotopy equivalence of $(X,{\\cal M})$ and $\\alpha\\colon X\\to\nL({\\cal M},n)$ represents a class of $H^*(X;{\\cal M})$,\n$H^n(f,\\Theta)(\\alpha)$ is represented by the map $$\nX\\stackrel{f}{\\longrightarrow}\nX\\stackrel{\\alpha}{\\longrightarrow}L({\\cal M},n)\\stackrel{\\xi}{\\longrightarrow}L({\\cal M},n)\n$$ in which $\\xi$ is defined by the action of $\\Theta_{x_0}$ on\n$M_{x_0}$. Explicitly, for  $(a,b)\\in L({\\cal M},n)$,\n$\\xi(a,b)=(a,\\overline{\\Theta}_{x_0}b)$ where and\n$\\overline{\\Theta}_{x_0}$ is the realization of $\\Theta_{x_0}$.\nObserve that $\\xi$ is well defined as $\\Theta\\colon {\\cal M}\\to{\\cal M}$\nis a morphism of local coefficient systems.\n\nMoreover, if $\\alpha,\\beta\\colon X\\to L({\\cal M},n)$ are in\n$H^n(X;{\\cal M})$, they coincides after composing with the fibration\n$p\\colon L({\\cal M},n)\\to K(\\pi(X),1)$, and therefore, for each $x\\in\nX$, $\\alpha(x),\\beta(x)$ live in the same fiber $K({\\cal M},n)$ of\n$p$. Hence, $\\alpha$ and $\\beta$ can be added up on $K({\\cal M},n)$\nand the resulting map $\\alpha+\\beta$ represents precisely their\nsum as cohomology classes with twisted coefficients.\n\n\nWe shall prove the theorem by induction on the length of the\nPostnikov decomposition of $X$. Assume  $X=K(\\pi,m)$ and let\n$(f,\\Theta)\\colon (X,{\\cal M})\\to(X,{\\cal M})$ be a self equivalence.\nThen, in view of the above, for any $n$-cohomology class\n$\\alpha\\colon K(\\pi,m)\\to L({\\cal M},n)$,\n$H^n(f,\\Theta)(\\alpha)-\\alpha$ is represented by the map\n$K(\\pi,m)\\to L({\\cal M},n)$ which, fiberwise, is\n$\\overline\\Theta_{x_0}\\alpha f-\\alpha$. Writing\n$\\overline\\Theta_{x_0}\\alpha f-\\alpha= \\overline\\Theta_{x_0}\\alpha\nf- \\overline\\Theta_{x_0}\\alpha + \\overline\\Theta_{x_0}\\alpha\n-\\alpha$ it is straightforward, using the nilpotency hypothesis,\nto show that the $s$-th commutator of the action of $G$ on\n$H^*(X;{\\cal M})$ vanishes as long  as\n$s\\le\\max\\{\\text{nil}\\,_G\\pi_*(X),\\text{nil}\\,_G{\\cal M}\\}$.\n\nAssume the theorem holds for $X=X^{r-1}$ and let $X=X^r$ be a\n$r$-dimensional Postnikov piece.  Consider the Serre spectral\nsequence with local coefficients on ${\\cal M}$ associated to the\nfibration $$ K(\\pi_r(X),r)\\to X\\to X^{r-1}. $$ whose $E_2$-term is\n$$ E_2^{*,*}=H^*\\bigl(X^{r-1};{\\cal H}^*(K(\\pi_r(X),r);{\\cal\nM})\\bigr). $$ Note that $G$ acts naturally in the base, total\nspace and fiber of this fibration, and hence, it does so in all\nthe terms of the spectral sequence.  The same argument used for\n$r=1$ shows that  $G$ acts nilpotently on the local coefficient\nsystem ${\\cal H}^*(K(\\pi_r(X),r);{\\cal M})$ and therefore, by\ninduction hypothesis, $G$ acts nilpotently on\n$H^*\\bigl(X^{r-1};{\\cal H}^*(K(\\pi_r(X),r);{\\cal M})\\bigr)$.\n\nAs the spectral sequence converges, the action of $G$ on the\nassociated graded module of  $H^*(X;{\\cal M})$ is nilpotent. Finally,\nreasoning by induction on the filtration degree we deduce that the\n$G$-action on  $H^*(X;{\\cal M})$ is also nilpotent.\\hfill$\\square$\n\n\nIn particular, for any space $X$ and any $j$ we may consider the\nlocal coefficient system given by $\\pi_jX$. In this case, any self\nhomotopy equivalence $f\\in\\mathcal E(X)$ can be seen as a self homotopy\nequivalence $(f,\\pi_jf^{-1})\\in\\mathcal E(X;\\pi_jX)$. Hence, any\nsubgroup of $\\mathcal E(X)$ may be considered as a subgroup of\n$\\mathcal E(X,\\pi_jX)$ which then acts naturally on $H^*(X;\\pi_jX)$ when\nconsidering local coefficients. In this context,  the theorem\nabove reads:\n\n\n\\begin{corollary}\\label{noveas} Let $X$ be a finite Postnikov piece and  let $G$ be a subgroup of\n$\\mathcal E(X)$ which acts nilpotently on $\\pi_*(X)$. Then, for any $j$,\n$G$ acts nilpotently on $H^*(X;\\pi_j)$.\\hfill$\\square$\n\\end{corollary} This result is used in the proof of Theorem \\ref{principal}  that we now present:\n\n\\bigskip\n\\noindent {\\it Proof of Theorem \\ref{principal}.} Consider the\nrestriction to $G$ of the exact sequence $\n{1}\\to\\mathcal E_{ \\sharp }(X)\\to\\mathcal E(X)\\to  \\Pi_{i\\le N}\\text{aut}\\,\\pi_i(X) $: $$\n{1}\\to\\mathcal E_{ \\sharp }(X)\\cap G\\to G\\to \\Pi_{i\\le N}\\text{aut}\\,\\pi_i(X). $$\nThe image  of $G$ under this morphism, call it $\\widetilde G$, is\na subgroup of automorphism of the group $\\pi_{\\le N}(X)$ in which\n$G$ acts nilpotently by hypothesis. Then, by Proposition\n\\ref{jo4}, $\\widetilde G$ is itself nilpotent and $\\text{nil}\\,\\widetilde\nG< \\text{nil}\\,_G\\pi_{\\le N}(X)$. Therefore, if we prove that $G$ acts\nnilpotently  on $\\mathcal E_{ \\sharp }(X)$, then (see for instance \\cite[Proposition\n4.1]{hmr}) $G$ would be nilpotent and\n\\begin{equation}\\label{nose1}\n\\text{nil}\\, G< \\text{nil}\\,_G\\mathcal E_{ \\sharp }(X)+\\text{nil}\\,_G\\pi_{\\le N}(X).\n\\end{equation} For that, observe in the first place that $[X,X]\\cong[X^N,X^N]$, where\n$X^N$ denotes the $N$-th Postnikov stage of $X$, and this\nbijection restricts to an isomorphism $\\mathcal E_{ \\sharp }(X)\\cong\\mathcal E_{ \\sharp }(X^N)$. On\nthe other hand, consider the exact sequence\n\\begin{equation}\\label{nose2} {1}\\to A_j\\to \\mathcal E_{ \\sharp }(X^j)\\to \\mathcal E_{ \\sharp }(X^{j-1})\n\\end{equation} where $\\mathcal E_{ \\sharp }(X^j)\\to\n\\mathcal E_{ \\sharp }(X^{j-1})$ is just the obvious restriction and $A_j$ its\nkernel. Since $G$ acts on any $\\mathcal E_{ \\sharp }(X^j)$ and $\\mathcal E_{ \\sharp }(X^1)={1}$, it\nwill be enough to show that $G$ acts nilpotently on every $A_j$ to\nconclude, by an easy induction, that it does so on\n$\\mathcal E_{ \\sharp }(X^N)=\\mathcal E_{ \\sharp }(X)$.\n\nBy classical obstruction theory of liftings (see \\cite[Chapter\n6.6]{white}) recall that, for $j\\ge 2$, there is a bijection\n$\\varphi\\colon B_j\\to H^j(X^j;\\pi_j)$ where\n\n\\begin{itemize}\n\\item\nThe cohomology is taken with local coefficients.\n\n\\item $B_j$ is the set of homotopy classes of $[X^j,X^j]$ which restrict to  the identity on $X^{j-1}$, i.e., homotopy\nclasses of liftings of $X^j\\to X^{j-1}$ to $X^j$.\n\n\\item $\\varphi(g)=\\delta(g,1)$ is the difference cochain of degree $j$ between $g$ and the identity on $X^j$.\n   \\end{itemize}\nRecall also that, in general,\n$\\delta(g,f)=\\delta(g,1)+\\delta(1,f)$ and that\n$\\delta(gh,fh)=H^j(h)\\bigl(\\delta(g,f)\\bigr)$. Moreover, if\n$h\\in\\mathcal E_{ \\sharp }(X^j)$, $\\delta(hg,hf)$ is the image of $\\delta(g,f)$\nunder the map $H^j(X^j;\\pi_j)\\to H^j(X^j;\\pi_j)$ induced by $h$ on\n$\\pi_j$.\n\nFrom now on, as in Corollary \\ref{noveas}, any $f\\in\\mathcal E(X)$, and\nthus in $G$, shall be considered as a self homotopy class\n$(f,\\pi_jf^{-1})\\in\\mathcal E(X,\\pi_j)$. Hence, restricting $\\varphi$ to\n$A_j$ we obtain a map $\\varphi\\colon A_j\\hookrightarrow\nH^j(X^j;\\pi_j)$ which is a $G$-map with respect to the action\n$g\\cdot f=g^{-1}fg$, $g\\in G$, $f\\in A_j$,  and the usual action\non $H^j(X^j;\\pi_j)$: if $\\alpha\\in H^j(X_j,\\pi_j)$, and $g\\in G$,\n$g\\cdot\\alpha$ is the cohomology class represented by the map\n\n$$ X^j{\\stackrel{g}{\\longrightarrow}}\nX^j{\\stackrel{\\alpha}{\\longrightarrow}}\nL(\\pi_j,j){\\stackrel{\\xi}{\\longrightarrow}} L(\\pi_j,j),\\qquad\n\\alpha\\in H^j(X^j;\\pi_j), $$ with $\\xi\\colon L(\\pi_j,j)\\to\nL(\\pi_j,j)$ induced by $\\pi_*(g^{-1})$.\n\nMoreover, this restriction is a group morphism. Indeed, given\n$f,h\\in A_j$,\n$\\varphi(fh)=\\delta(fh,1)=\\delta(f^{-1}fh,f^{-1})=\\delta(h,f^{-1})=\\delta(h,1)+\\delta(1,f^{-1})=\\delta(h,1)+\\delta(f,1)=\n\\varphi(f)+\\varphi(h)$. As an immediate consequence we then obtain\nthat $A_j$ is an (abelian!) subgroup of $H^j(X^j;\\pi_j)$.\n\nFinally, by Corollary \\ref{noveas}, $G$ acts nilpotently on\n$H^j(X^j;\\pi_j)$ for any $j$. Hence, it does so on $A_j$ and $$\n\\text{nil}\\,_G\\,A_j\\le \\text{nil}\\,_G\\,H^j(X^j;\\pi_j). $$ Thus, by induction on\n$j$ using repeatedly \\cite[Proposition 4.1]{hmr}, one easily sees\nin view of (\\ref{nose2}) that\n$\\text{nil}\\,_G\\,\\mathcal E_{ \\sharp }(X)=\\text{nil}\\,_G\\,\\mathcal E_{ \\sharp }(X^N)\\le\\sum_{j=2}^N\\text{nil}\\,_G\\,A_j\\le\\sum_{j=2}^N\\text{nil}\\,_G\\,H^j(X^j;\\pi_j)$\nand the theorem follows.\\hfill$\\square$\n\n\n\n\n\n\\section{Groups which fix the homotopy groups}\nIn this section we establish Theorems \\ref{tres} and \\ref{cuatro}.\n\n\\bigskip\n\\noindent {\\it Proof of Theorem \\ref{tres}.}   Let\n$\\alpha\\in\\mathcal E_{ \\sharp p }(X)$. Then, for each $i\\le\\dim X$, the morphism\n$\\pi_i(\\alpha;\\mathbb Z\/p)\\colon\\pi_i(X,\\mathbb Z\/p)\\to\\pi_i(X,\\mathbb Z\/p)$ is just\nthe identity. On the other hand, the Universal Coefficients\nTheorem for homotopy yields the following split short exact\nsequence $$\n0\\longrightarrow\\text{Ext}(\\mathbb Z\/p,\\pi_{i+1}X)\\longrightarrow\\pi_i(X;\\mathbb Z\/p)\\longrightarrow\n\\hom(\\mathbb Z\/p;\\pi_iX)\\longrightarrow 0. $$ Thus, both\n$\\text{Ext}(\\mathbb Z\/p,\\pi_{i+1}\\alpha)$ and $\\hom(\\mathbb Z\/p;\\pi_i\\alpha)$\nare the identity. But observe that\n$\\text{Ext}(\\mathbb Z\/p,\\pi_{i+1}X)=\\pi_{i+1}X\\otimes\\mathbb Z\/p$ so that\n$\\pi_{i+1}\\alpha\\otimes\\mathbb Z\/p=1_{\\pi_{i+1}X}$.\n\nHence, by Proposition \\ref{propouno}, for each $i\\le\\dim X$, the\nimage of $\\mathcal E_{ \\sharp p }(X)$ in $\\text{aut}\\,(\\pi_iX)$ is a $p$-group and then, by\nCorollary \\ref{corolario}, the action of $\\mathcal E_{ \\sharp p }(X)$ on $\\pi_iX$ is\nnilpotent. Thus, by Theorem \\ref{principal}, $\\mathcal E_{ \\sharp p }(X)$ is\nnilpotent. On the other hand, notice that $\\mathcal E_{ \\sharp }(X)$ is precisely\nthe kernel of the obvious map $\\mathcal E_{ \\sharp p }(X)\\to\\Pi_{i\\le\\dim\nX}\\text{aut}\\pi_i(X)$. Hence, as we just proved that the image of\nthis map is a $p$-group, $\\mathcal E_{ \\sharp p }(X)\/\\mathcal E_{ \\sharp }(X)$ is a finite $p$-group,\nand the proof is complete. \\hfill$\\square$\n\n\\bigskip\n\n\\noindent {\\it Proof of Theorem \\ref{cuatro}.} Write\n$\\pi_iX=\\mathbb Z^{n_i}\\oplus\\bigl(\\oplus_{p\\,\\,\\text{prime}}T_p(\\pi_iX)\\bigr)$\nin which $T_p(\\pi_iX)$ is the group of $p$-torsion elements in\n$\\pi_iX$. Now, if $\\alpha\\in\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$, then\nfor each $i\\le\\dim X$,\n$\\pi_i(\\alpha)|_{T_p(\\pi_iX)}\\in\\text{aut}\\,\\bigl(T_p(\\pi_iX)\\bigr)$.  Let\n$z_1,\\ldots,z_{n_i}$ be generators of $\\mathbb Z^{n_i}\\subset\\pi_iX$.\nThen $\\pi_i(\\alpha)(z_k)=\\sum_{j=1}^1{n_i}a_{k,j}z_j+\\omega$,\nwhere $\\omega$ is the torsion part. But this element has to\ncoincide with $z_k$ mod $p$, for all prime $p$. Therefore, also\nfor any $p$, $ a_{k,j}=0(\\text{mod}\\,p) $ for $k\\not=j$ and\n$a_{k,k}=1(\\text{mod}\\, p)$, for $1\\le k\\le n_i$. The only\npossible solution is $a_{k,j}=0$, $k\\not=j$, and $a_{k,k}=1$. In\nother words, $\\pi_i(\\alpha)(z_k)=z_k+\\omega$, in which $\\omega$ is\na torsion element.\n\nAdding up, for any element $\\gamma\\in\\pi_iX$,\n$\\pi_i(\\alpha)\\gamma-\\gamma$ is a torsion element in $\\pi_iX$.\nThis is equivalent to say that the $1$-commutators of the action\nof $\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$ on $\\pi_iX$ live in the\ntorsion part of $\\pi_iX$. However, by Corollary \\ref{corolario},\nthe action of $\\cap_{p\\,\\,\\text{prime}}\\mathcal E_{ \\sharp p }(X)$ on the torsion\npart is nilpotent and therefore,  the action on $\\pi_iX$ is also\nnilpotent. Apply Theorem \\ref{principal} and the proof is\ncomplete.\\hfill$\\square$\n\n\\begin{remark} {\\em We end up by noting that the hypothesis of Theorem \\ref{tres} are necessary. Indeed, consider\n$X=K\\bigl((\\mathbb Z\/2)^2,n\\bigr)$ and observe that, for a prime $p$\ndifferent from $2$, $\\mathcal E_{ \\sharp p }(X)=\\mathcal E(X)=GL_2(\\mathbb Z\/2)\\cong \\Sigma_3$\nwhich is not nilpotent.\n\nOn the other hand, take $X=K(\\mathbb Z^2,n)$ for which\n$\\mathcal E(X)=GL_2(\\mathbb Z)$. In this case $\\mathcal E_{ \\sharp }(X)=\\{1\\}$ and, for any\nprime $p$, $\\mathcal E_{ \\sharp p }(X)$ fits in the following short exact sequence $$\n\\{1\\}\\to\\mathcal E_{ \\sharp p }(X)\\to GL_2(\\mathbb Z)\\to GL_2(\\mathbb Z\/p)\\to\\{1\\} $$ where the\nsurjection $GL_2(\\mathbb Z)\\to GL_2(\\mathbb Z\/p)$ is just the mod-$p$\nreduction. Hence $\\mathcal E_{ \\sharp p }(X)=\\mathcal E_{ \\sharp p }(X)\/\\mathcal E_{ \\sharp }(X)$ is an infinite, non\nnilpotent group.}\n\\end{remark}\n\n\n\\nocite{*}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA wide range of minimally invasive therapies have been developed for cancer treatment, additionally to open surgery \\cite{Ahmed.2014,mauri2017technical,tomasian2018percutaneous}. One of these methods is the use of microwave ablation (MWA). Especially for smaller tumors, MWA shows promising results for treatment \\cite{tehrani2020use}. As the minimal ablative margin (MAM) is crucial for the local tumor progression (LTP), it is of greatest importance to assess if the malignancy has been adequately and completely treated, regardless of the etiology. For each millimeter increase of the MAM, a 30\\% reduction of the relative risk for LTP was found. The MAM is especially important as the only significant independent predictor of LTP (p = 0.036) \\cite{laimer2020minimal}. During the intervention, magnetic resonance (MR) imaging offers several advantages like a good soft-tissue contrast without the need of contrast agent, the free orientation and positioning of single slice scans and the possibility to accurately track changes in the temperature inside the tissue \\cite{Gorny.2019,Kagebein.2018c,Rieke.2008,Senneville.2007}.\\\\\n\\\\\n\\textbf{Contribution.} In this work, we propose a novel approach for the creation of a volumetric thermometry map without the development of a fully 3D sequence. The introduced 2.5D thermometry method utilizes any common 2D gradient-echo (GRE) sequences. Therefore, possible temporal limitations are less restricting than for the 3D sequences and images with higher resolution may be acquired offering standard thermometry accuracy of around $1^\\circ$C deviation while being more robust towards MR inhomogeneities \\cite{Gorny.2019}. We will show that our method is well-suited to reconstruct the actual coagulation zone after thermal ablation.\\\\\n\\\\\n\\textbf{Related Work.} Zhang et al.\\cite{zhang2019variable} propose a golden-angle\u2010ordered 3D stack\u2010of\u2010radial multi-echo spoiled gradient\u2010echo sequence with a variable flip angle. The image reconstruction is performed offline offering a temporal resolution between 2s-5s. Jiang et al.\\cite{Jiang.2020} use an accelerated 3D echo-shifted sequence and the Gadgetron framework for image reconstruction. Temporal resolution lies at around 3s with a temperature error of less than $0.65^\\circ$C. Quah et al.\\cite{quah2020simultaneous} are aiming at an increased volume coverage for thermometry without multiple receive coils. An extended k\u2010space hybrid reconstruction was used, yielding an error of $<1^\\circ$C and an acquisition time of 3.5s for each image. Fielden et al.\\cite{Fielden.2018} present a comparison study between cartesian, spiral-out and retraced spiral-in\/out (RIO) trajectories. Using the 3D RIO sequence, they achieved a true temporal resolution of 5.8s with a temporal standard deviation of $1.32^\\circ$C. Marx et al.\\cite{Marx.2014} introduced the MASTER sequence for volumetric MR thermometry acquisition, acquiring six slices in around 5s. In a later work \\cite{Marx.2017} they use optimized multiple-echo spiral thermometry sequences, which yield a better precision than the usual 2D Fourier transform thermometry. Image acquisition takes between 7s-11s. Svedin et al.\\cite{svedin2018multiecho} make use of a multi-echo pseudo-golden angle stack-of-stars sequence and offline image reconstruction using MATLAB. They achieved a temporal resolution of around 2s and a spatial average of the standard deviation through a time of $0.3-1.0^\\circ$C. Od\u00e9en et al.\\cite{odeen2016mr} propose the use of a 3D gradient recalled echo pulse sequence with segmented EPI readout. To estimate the temperature change, they also integrate a bioheat equation. They achieved a temperature root mean square error of $1.1^\\circ$C. Golkar et al.\\cite{golkar2018fast} introduce a fast GPU based simulation approach for cryoablation monitoring. The reconstruction takes 110s and the final result shows a Dice coefficient of 0.82. A summarize of the related work in comparison to our method is shown in Table \\ref{tab:RelatedWork}.\n\\begin{table}[ht]\n\\centering\n\\caption{Overview about the related work in comparison to this work. Every work has been observed according to the following: 1) The kind of image sequence used. 2) The online or offline capability of the reconstruction framework. 3) The temporal resolution of the whole image acquisition in seconds. 4) The temperature accuracy in \u00b0C. 5) The resulting Dice Score similarity measurement if available.}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n                       & \\begin{tabular}[c]{@{}c@{}}Image \\\\ Sequence\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Reconstruction \\\\ Framework\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Temporal \\\\ Resolution {[}s{]}\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Temperature \\\\ Accuracy {[}\u00b0C{]}\\end{tabular} & Dice Score \\\\ \\hline\nZhang et al.{[}20{]}   & 3D                                                        & offline                                                             & 2 - 5                                                                  & ---                                                                      & ---                                                   \\\\ \\hline\nJiang et al.{[}6{]}    & 3D                                                        & \\begin{tabular}[c]{@{}c@{}}online \\\\ (Gadgetron)\\end{tabular}       & 3                                                                      & \\textless 0.65                                                           & ---                                                   \\\\ \\hline\nQuah et al.{[}13{]}    & Stack of 2D                                               & hybrid                                                              & 3.5                                                                    & \\textless 1                                                              & ---                                                   \\\\ \\hline\nFielden et al.{[}3{]}  & 3D                                                        & online                                                                 & 5.8                                                              & 1.32                                                                     & ---                                                   \\\\ \\hline\nMarx et al.{[}10{]}    & Stack of 2D                                               & ---                                                                 & 5                                                                      & 1.3                                                                      & ---                                                   \\\\ \\hline\nMarx et al.{[}9{]}     & Stack of 2D                                               & online                                                                 & 7 - 11                                                                 & 0.29 - 0.65                                                                      & ---                                                   \\\\ \\hline\nSvedin et al. {[}17{]} & 3D                                                        & \\begin{tabular}[c]{@{}c@{}}offline \\\\ (MATLAB)\\end{tabular}         & 2                                                                      & 0.3 - 1.0                                                                & ---                                                   \\\\ \\hline\nOdeen et al.{[}12{]}  & Stack of 2D                                               & \\begin{tabular}[c]{@{}c@{}}offline \\\\ (MATLAB)\\end{tabular}                                                                 & 2.4 - 4.8                                                                    & \\textless 1.1                                                                      & ---                                                   \\\\ \\hline\nGolkar et al.{[}4{]}   & 3D                                                        & ---                                                                 & 110                                                                    & ---                                                                      & 0.82                                                  \\\\ \\hline\nThis work             & Single 2D                                               & online                                                              & variable                                                               & 1                                                                        & 0.75                                                  \\\\ \\hline\n\\end{tabular}\n\\label{tab:RelatedWork}\n\\end{table}\n\\section{Material and Method}\n\\label{sec:MaterialAndMethod}\n\\subsection{Image Acquisition}\n\\label{subsec:ImageAcquisition}\nThe proposed 2.5D thermometry relies on sampling the volume of interest (VOI) using a common 2D GRE sequence and [1,...,n] different orientations. The GRE sequence can directly reconstruct magnitude and phase images simultaneously. To ensure a proper sampling of the VOI in our setup the GRE sequence is rotated by $22.5^\\circ$ around the applicator's main axis. This results in an evenly distributed sample of eight different orientations. To increase the spatial resolution, the angles between the acquired scans should be as high as possible, resulting in the following acquisition order: $0 ^\\circ$, $90 ^\\circ$, $45 ^\\circ$, $135 ^\\circ$, $22.5 ^\\circ$, $112.5 ^\\circ$, $67.5 ^\\circ$, and $157.5 ^\\circ$. To reduce the delay between image acquisition and visualization of the volumetric thermometry map, the SIEMENS Healthineers Access-I Framework was integrated. The framework allows for fetching the image data directly from SIEMENS MR devices without an intermediate imaging archive system.\n\\subsection{2.5D Thermometry Reconstruction}\n\\label{subsec:25DThermometryReconstruction}\nBefore treatment starts, reference phase images are acquired for each of the eight orientations. Each newly acquired phase image will start computing the up-to-date 2D thermometry map for the current orientation during the treatment. To do so, the proton resonance frequency shift (PRFS) method is used as described by Rieke et al. \\cite{Rieke.2008}. The temperature $T$ based on the PRFS is computed using the following Equation\n\\begin{equation}\n\\label{eq:PRFS}\n    T = \\frac{\\phi(t) - \\phi (t_0)}{\\gamma \\alpha B_0 TE} + T_0\n\\end{equation}\nwith $\\phi(t) - \\phi (t_0)$ defining the phase difference between the current time point $\\phi(i)$ and the reference timepoint $\\phi(i_0)$, $\\gamma = 42,576\\frac{MHz}{T}$ representing the gyromagnetic ratio of hydrogen protons, $\\alpha = 0.01\\frac{ppm}{\\Delta T}$ representing the proton resonance frequency change coefficient, $B_0$ representing the used magnetic field strength and $TE$ representing the used echo time. The constant $T_0$ needs to be added to the temperature since Equation \\ref{eq:PRFS} otherwise only computes the temperature change, neglecting the tissue's base temperature. The Access-I integration and 2D thermometry computation were implemented as modules using MeVisLab 3.4.1 \\cite{ritter2011medical}. \n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{MICCAI2021Workflow_v6.PNG}\n    \\caption{Schematic overview of the proposed method.}\n    \\label{fig:MethodOverview}\n\\end{figure}\nThe 2.5D thermometry reconstruction itself was implemented using C++. A schematic overview of the method can be seen in Figure \\ref{fig:MethodOverview}. To handle the voxel values during slice rotation every cartesian coordinate was mapped to the corresponding cylindrical coordinate representation using Equation \\ref{eq:CartesianToPolar},\n\\begin{align}\n\\label{eq:CartesianToPolar}\n    P_r(x,y,z) &= P_c(r, \\theta, z)\\\\\n    r &= \\sqrt{\\left(x-x_c \\right)^2 + \\left(y-y_c\\right)^2}\\nonumber \\\\\n    \\theta &= atan2\\left(\\frac{x-x_c}{y-y_c}\\right)\\nonumber\n\\end{align}\nwhere $x,y$ represents the Cartesian coordinates of the current voxel and $x_c,y_c$ represents the Cartesian coordinates of the centerline corresponding to the applicator's axis for every slice $z$ in the reconstructed volume. Upon acquisition of the reference images, a multi-dimensional population map is created. For each voxel $(x_i,y_i,z_i)$ in the reconstructed volume, this population map holds information about the radius $r$ and angle $\\theta$ of the cylindrical coordinates, the general interpolation weight $I_w$, the adjacent interpolation partner coordinates $IP_{left}(x,y)$ and $IP_{right}(x,y)$ in the 2D live data as Cartesian representation and the weights $w_1$ and $w_2$ of those interpolation partners. The weights may be acquired using Equation \\ref{eq:Weights},\n\\begin{align}\n\\label{eq:Weights}\n    w_{1} &= \\left\\vert\\frac{\\theta_{IP_{left}} - \\theta_{i}}{\\theta_{IP_{left}} - \\theta_{IP_{right}}}\\right\\vert\\\\\n    w_2 &= 1 - w_1\\nonumber\n\\end{align}\nwith $\\theta_{i}$ representing the cylindric angle of the current Voxel $i$ and $\\theta_{IP_{left}}$, $\\theta_{IP_{right}}$ representing the orientation angles of the left and right interpolation partners, respectively. The 2D population map can be applied to every slice of the final 3D output volume, reducing the computational power needed. During the intervention, every acquired live image triggers the reconstruction of the up-to-date 2.5D thermometry map. Here, the heat value for each voxel is reconstructed using Equation \\ref{eq:Reconstruction},\n\\begin{equation}\n\\label{eq:Reconstruction}\n    T_i = I_w \\cdot (w_{1} \\cdot T_{IP_{left}} + w_{2} \\cdot T_{IP_{right}})\n\\end{equation}\nwith $T_i$ representing the temperature of the current voxel $i$ and $T_{IP_{left}}$, $T_{IP_{right}}$ representing the temperature of the adjacent interpolation partners. Occurring vessels or other structures, which cause a heat sink effect are segmented during the intervention planning. Subsequently, the segmented structure is saved as an additional Look-Up Volume. Here, each voxel can be checked if it is part of a heat sink structure. Using this knowledge, the interpolation weight $I_w$, which ranges between $[0,1]$, may be adjusted. Figure \\ref{fig:Reconstruction} shows a single dimension of the population map for parameter weighting, a reconstructed heat map, a coagulation estimation based on an empirically defined threshold and the corresponding ground truth segmentation. The source code is available for download at \\url{https:\/\/github.com\/jalpers\/2.5DThermometryReconstruction}.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{ImageReconstruction_v3.png}\n    \\caption{A) Example population map for output weights color coded on gray scale. B) Reconstructed volumetric heat map. C) Estimated coagulation necrosis based on a threshold of $57^\\circ$C. D) Manually segmented ground truth.}\n    \\label{fig:Reconstruction}\n\\end{figure}\n\\subsection{Evaluation}\n\\textbf{Phantom design.} To create a first proof of concept, a pilot study was conducted to evaluate the 2.5D thermometry reconstruction using 13 bio protein phantoms as described by Bu Lin et al.\\cite{BuLin.2008}. The coagulation zone's visibility in the post-treatment data sets increased by adding a contrast agent ($0,5\\mu mol\/L$ Dotarem) to the phantoms. For six phantoms, additional polyvinyl chloride (PVC) tubes with a diameter of $5mm$ and a wall thickness of $1mm$ were integrated into the phantoms (three single-tubes, three double-tubes) to simulate a possible heat sink (HS) effect.\\\\\n\\\\\n\\textbf{Experimental setup.} The applicator of the permittivity feedback control MWA system (MedWaves Avecure, Medwaves, San Diego, CA, USA, 14G) was placed inside the phantom by sight and secured in position. Subsequently, the phantoms were placed inside a $1,5T$ MR scanner (Siemens Avanto, Siemens Healthineers, Germany). The coaxial cables connected to the applicator and MW generator were led through a waveguide. Chokes and electrical grounding measures were added as described by Gorny et al.\\cite{Gorny.2019} to reduce radio frequency interference. In the case of the perfusion phantoms, the PVC tubes were led through the wave guide. They were connected to a diaphragm pump and a water reservoir outside the scanning room. A flow meter (SM6000, ifm electronic, Essen, Germany) was interposed between the reservoir and the pump, providing a flow rate of $800mL\/min$. Observations showed a moderate HS effect using this setup with a maximum antenna power of 36W. Additionally, temperature sensors were inserted in two phantoms to experimentally verify the temperature accuracy of $1 ^\\circ C$.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{experimental_setup_v1.2.jpg}\n    \\caption{Experimental evaluation setup. Flexible tubes (blue) lead the water (a) through a flow meter (b), a diaphragm pump (c) and the bio protein phantom (e). The coaxial cables (red) connect the applicator with the MW generator d).}\n    \\label{fig:StudyDesignSketch}\n\\end{figure}\nRight before treatment, ten reference phase images were acquired and averaged for each orientation to compensate for static noise. The MWA duration was set to 15 minutes with a temperature limit of $90^\\circ$C. The GRE sequence offers a slice thickness of $5mm$, a field of view (FOV) of $256mm * 256mm$, a matrix of $256 * 256$, and a bandwidth of $260 Hz\/Px$. Image acquisition took around $1.1s$ with a $5s$ break to simulate the temporal resolution for a breathing patient. The TE was $3.69ms$, the TR $7.5ms$, and the flip angle $7 ^\\circ$. For post-treatment observation a 3D turbo spin echo (TSE) sequence (TE = $156ms$, TR = $11780ms$, flip angle = $180 ^\\circ$, matrix = $256 * 256$, FOV = $256mm * 256mm$, bandwidth = $40 Hz\/Px$, slice thickness = $1mm$) was used. The 3D TSE allows for proper visualization of the real coagulation zone due to a very high tissue contrast. Extraction of the coagulation ground truth was done manually by a clinical expert using MEVIS draw (Fraunhofer MEVIS, Bremen, Germany). All data sets used are available for download at \\url{http:\/\/open-science.ub.ovgu.de\/xmlui\/handle\/684882692\/89}.\\\\\n\\\\\n\\textbf{Statistical evaluation.} Final evaluation of the acquired data was performed using the dice similarity coefficient (DSC) as explained in Equation \\ref{eq:DSC}\n\\begin{equation}\n    \\label{eq:DSC}\n    DSC = \\frac{2*TP}{2*TP + FP + FN}\n\\end{equation}\nwith $TP$ representing the true positives, $FP$ the false positives and $FN$ the false negatives. Additionally, the standard error of the mean (SEM) was computed at a confidence level of 95\\% (p = 0.05) using Equation \\ref{eq:SEM}\n\\begin{align}\n    \\label{eq:SEM}\n    \\sigma &= \\sqrt{\\frac{\\sum(x_i-\\bar{x})^2}{N-1}}\\\\\n    SEM &= \\frac{\\sigma}{\\sqrt{N}} * 1.96\\nonumber\n\\end{align}\nwith $\\sigma$ representing the standard deviation, $x_i$ the current sample, $\\bar{x}$ the mean value and $N$ the sample size. To compute the SEM at a confidence level of 95\\% it has to be multiplied by 1.96, which is the approximated value of the 97.5 percentile of the standard normal distribution.\n\\section{Results}\nSummarized evaluation results can be seen in Figure \\ref{fig:SummarizedResults}. Empirically determined coagulation thresholds were set between $51^\\circ$C and $61^\\circ$C depending on each phantom's pH value.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{DiceSensitivityResults_v4.png}\n    \\caption{Summarized evaluation results for phantoms without HS effect, phantoms with HS effect and the overall results. Note that the data range [0,0.3] was left out because no data points are present in that range.}\n    \\label{fig:SummarizedResults}\n\\end{figure}\nIt is noticeable that the DSCs for HS phantoms show a very high SEM with $0.70\\pm0.15 (\\pm21.25\\%)$ and $0.74\\pm0.06 (\\pm8.49\\%)$ regarding the sensitivity. The high range results from a corrupted dataset due to heavy artifacts within the image data. Leaving the corrupted dataset out of the evaluation, the SEM shows a significantly lower deviation of $0.76\\pm0.062 (\\pm8.07\\%)$ and $0.77\\pm0.048 (\\pm6.25\\%)$ for the DSC and sensitivity, respectively. Observations show a slightly higher DSC and sensitivity for phantoms without any HS effect. Here, the values range from $0.79\\pm0.04 (\\pm 4.53\\%)$ and $0.79\\pm0.04 (\\pm 5.55\\%)$, respectively. Evaluation showed an overall SEM for the DSC of $0.75\\pm0.07 (\\pm9.76\\%)$ and a SEM for sensitivity of $0.77\\pm0.04 (\\pm4.99\\%)$. To evaluate the computational effort, every major step was performed 100 times. The creation of the population map and the heat sink look up volume took $25.53ms\\pm 3.33ms$ and $3.91s\\pm 0.59s$, respectively. These two steps need to be done just once before start of the treatment. The reconstruction of the 2.5D thermometry map was performed in $18.02ms\\pm 5.91ms$ on a customary workstation (Intel( R) Core(TM) i5-6200U CPU, double-core 2.30GHz, 8GB RAM, Intel(R) HD Graphics 520). This reconstruction will be performed every time a new image is acquired during treatment.\n\n\\section{Discussion and Conclusion}\nThe aim of this work is to develop a volumetric thermometry map, which can be applied to a wide variety of clinical setups. Therefore, our work heavily relies on the up-to-date standard 2D GRE sequence for image acquisition. This allows for the standard accuracy of the thermometry up to $1.0^\\circ$C. Nonetheless, the sampling of the 3D volume also results in some disadvantages, which need to be addressed in the future. First, the diffusion of the heat inside the tissue is not linear over time. Therefore, it would be necessary to include an adaptive temporal and spatial resolution depending on the current intervention time. A new study should be conducted to identify an optimal sequence protocol for this 2.5D thermometry approach. Second, we found that the reconstruction sometimes shows stair-case artifacts. Because only one image is acquired every few seconds, the time difference between adjacent orientations may be very high. The temperature difference for each voxel dependent on the applicator's radius may be computed and applied to the corresponding voxel on every other out-of-date data to compensate for this error. This transfer of the heat gradient may improve the reconstruction accuracy. Another approach may be the use of a model-based reconstruction to take different tissue characteristics into account. To pseudo-increase the temporal resolution, bio heat transfer simulations may also be included during reconstruction. The acquired live data may be able to adjust the simulation parameters to increase the simulation accuracy. Finally, our study only performs on bio protein phantoms. Results show a proof of concept for the proposed method, but it still has to be evaluated in real tissue and a more realistic clinical environment. Therefore, perfused ex vivo livers may be a way to go in the future. Additionally, we currently assume a breath-holding state or at least a breath-triggered image acquisition. Research shows that a wide range of interventional registration methods is available, but further investigations in this area still need to be done to create an applicable method. The last issues arise because of the MR inhomogeneity during image acquisition. The slightest disturbances may result in heavy image artifacts. Proper shielding of the MW generator is needed to reduce the SNR loss over time thus increasing the thermometry and reconstruction accuracy. \\\\\n\\\\\nIn conclusion, we proposed a novel method for 2.5D thermometry map reconstruction based on common GRE sequences rotated around the applicator's main axis. A pilot study was conducted using bio protein phantoms to simulate cases with possible heat sink effects and without. The evaluation shows promising results regarding the DSC of the reconstructed 2.5D thermometry map and a manually defined ground truth. Future work should address the reconstruction method's improvement by integrating further apriori knowledge like the estimated shape of the heat distribution. Furthermore, a more realistic study should be conducted with bigger sample size and real tissue. In sum, the method shows a high potential to improve the clinical success rate of minimally invasive ablation procedures without necessarily hampering the standard clinical workflow of the individual clinician.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe inflationary paradigm in the early Universe lays a strong foundation for successful description of the CMB and large scale structure (LSS) of the Universe so far  \\cite{Starobinsky:1980te,Starobinsky:1979ty,Akrami:2018odb} with the  predictions of spectral index $n_s=0.9649\\pm 0.0042$ for $50-60$ e-foldings of quasi-de Sitter expansion. Besides its apparent success, some CMB anomalies seem to arise and challenge any {current} model of inflation. \nUnderstanding the observed anomalies in the latest Planck's CMB data could further elucidate some fundamental aspects of the quantum nature of the fluctuations which are believed to seed the large scale structure (LSS) of the Universe. Concretely, one of the main CMB anomalies is the presence of a hemispherical power asymmetry (HPA) at large angular scales \\cite{Akrami:2019bkn} which has been seen from the NASA's Wilkinson Microwave Anisotropy Probe (WMAP) to the latest Planck data and its significance is standing now at $3.3\\sigma$ \\cite{Akrami:2014eta}. \n In short, the HPA is an asymmetry in the two point temperature correlations on opposite hemispheres of the sky, with slightly higher power in the southern ecliptic hemisphere and slightly lower power in the northern ecliptic hemisphere \\cite{Creswell:2021eqi}. This is a significant challenge to our standard model of (inflationary) cosmology, which predicts isotropy \\cite{Yeung:2022smn,Aluri:2022hzs}.  \n This anomaly seems to appear significantly at low-multipoles $\\ell =2-64$ \\cite{Schwarz:2015cma}. The HPA is phenomenologically represented by the following form of the position dependent primordial power spectrum parametrized as \\cite{Lyth:2013vha,Lyth:2014mga}\n\\begin{equation}\n\t\\mathcal{P}_\\mathcal{R}\\left( k,\\,\\hat{\\boldsymbol{n}}\\right) \\simeq \\mathcal{P}_{\\mathcal{R}\\:\\: iso} (k)\\left( 1+2A(k) \\hat{\\textbf{p}}\\cdot\\hat{\\textbf{n}}\\right) \\,,\n\t\\label{pwa}\n\\end{equation}\nwhere $\\mathcal{P}_{\\mathcal{R}\\:\\: iso} (k)$ is the statistically homogeneous and isotropic power spectrum, $A(k)$ is the amplitude of the observed dipolar asymmetry which is constrained as $\\vert A \\vert = 0.066\\pm 0.021$ ($3.3\\sigma$) at the large angular scales $\\ell <64$ or at the wave numbers  $k \\lesssim 0.0045\\textrm{ Mpc}^{-1}$. The HPA seems to be present even up to the multipoles $~600$ as seen by the Planck 2013, Hansen \\emph{et al.} and Aiola  \\emph{et al.} \\cite{Planck:2013lks,Hansen:2008ym,Aiola:2015rqa}. \n\n Here  $\\hat{\\boldsymbol{p}}$ is the direction of maximal symmetry and $\\hat{\\textbf{n}}=\\frac{\\boldsymbol{x}}{x_{\\text{ls}}}$ is the line of sight from earth and $x_{\\text{ls}} = 14,000 \\textrm{Mpc}^{-1}$ is the co-moving distance to the surface of last scattering. \nFollowing \\eqref{pwa} we can derive \\cite{Lyth:2013vha}\n\\begin{equation}\n\tA(k) = \\frac{\\mathcal{P}_\\mathcal{R}\\left( k,\\,\\hat{\\textbf{n}} \\right) - \\mathcal{P}_\\mathcal{R}\\left( k,\\,-\\hat{\\textbf{n}} \\right) }{4\\mathcal{P}_{\\mathcal{R}\\:\\: iso}}\\,. \n\t\\label{Akg}\n\\end{equation}\nThe best explanation  for the HPA, so far, invoked an additional modulated superhorizon fluctuation with a wavelength larger than the last scattering surface, whose interactions  with the short wavelength  curvature perturbations introduces large scale dependent primordial non-Gaussianities \\cite{Schmidt:2012ky,Erickcek:2008sm,Erickcek:2008jp,Byrnes:2015asa,Byrnes:2015dub,Namjoo:2014nra}. It was first noted in \\cite{Lyth:2013vha,Lyth:2014mga} that both the HPA and and the power suppression (PS) at low-multipoles \\cite{Akrami:2019bkn} could be addressed in a single framework in the context of scale dependent primordial non-Gaussianities and curvaton-type scenarios. This later has been generalized and phenomenologically studied in model independent way \\cite{Adhikari:2015yya,Byrnes:2016zxb}.  In addition, the HPA has been addressed via a spatial dependent parametrization of the $\\alpha$-vacua Bogoliubov coefficients of de~Sitter space \\cite{Ashoorioon:2015pia}, in the context of anisotropic inflationary models involving gauge fields \\cite{Maleknejad:2012fw} and also with spatial dependent initial conditions for the inflaton field \\cite{Donoghue:2007ze}. So far all the studies ascertained that the HPA cannot be explained by the standard single-field slow-roll inflationary scenario \\cite{Akrami:2014eta,Hansen:2008ym}. \n\nDespite the above mentioned had hoc explanations for the HPA \\cite{Akrami:2019bkn}, Planck data actually hints at it for both 2-point and 3-point correlations (even the equilateral shape for which three wave numbers of the modes are equal). This means that the anomaly seems to be there in all the $n$-point correlations, which begs for a fundamental explanation. Therefore, the answer must be there somewhere in the description of inflationary quantum fluctuations.\n\nThe main goal of this paper is to investigate whether the HPA can be explained through a new understanding of the quantization of the inflationary perturbations without introducing any new parameters and any new degrees of freedom. Thus we only focus on single field slow-roll inflation and propose a new vacuum for inflationary quantum fluctuations. Being more precise, we pay careful attention to the discrete symmetries such as $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ which is expected to be spontaneously broken in a dynamical spacetime. Our scheme of quantization exactly exploits this last aspect leading to a promising explanation for the HPA anomaly, with additional prediction for primordial in gravitational wave spectrum as well.  \n\nThroughout the paper, we use the metric signature $\\left( -,\\,+,\\,+,\\,+ \\right)$, overdot and $^\\prime$ denotes derivative with respect to cosmic time (t) and conformal time ($ \\tau$) respectively, overbar denotes background values for flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) spacetime described by the metric $ds^2= -dt^2+a^2(t)d\\textbf{x}^2$, $H=\\frac{\\dot{a}}{a}$ denotes the Hubble parameter. \n Four-dimensional indices are labelled by small Greek letters and three dimensional quantities are denoted by $i,j =1,2,3$. We set $\\hbar=c=1$ and $M_p = \\frac{1}{\\sqrt{8\\pi G}}=1$. \n\n\\section{Brief review of inflationary quantum fluctuations in single field inflation}\n\\label{sec:review}\nIn this section, we review standard inflationary quantum fluctuations and the corresponding derivation of observables \\cite{Martin:2015dha,Baumann:2018muz}. In standard single field inflation, we start with a classical action for a canonical scalar field with a suitable choice of potential.\\footnote{Note that inflationary scenarios like Starobinsky and Higgs inflation can be brought to this form via a conformal transformation \\cite{Kehagias:2013mya}.}\n\\begin{equation}\\label{action-S}\n\tS= \\int d^4x\\sqrt{-g} \\left[ \\frac{M_p^2}{2} R- \\frac{1}{2} (\\partial_\\mu \\phi)( \\partial^\\mu \\phi  )-V\\left( \\phi \\right)\\right]  \\,, \n\\end{equation}\nTo implement inflationary framework of expansion, we first solve the equations of motion of \\eqref{action-S} in flat FLRW spacetime and obtain a quasi-de Sitter solution described by the following conditions on the slow-roll parameters:\n \\begin{equation}\n \t\\epsilon = -\\frac{\\dot{H}}{H^2}\\ll 1,\\quad \\eta = \\frac{\\dot{\\epsilon}}{H\n \t\\epsilon} \\ll 1\\,\\text{,}\n \\label{sldef}\n \\end{equation}\nwhich can be computed from the Friedmann and Raychaudhuri equations given by \n\\begin{equation}\n\tH^2 = \\frac{1}{3}\\left( \\frac{\\dot{\\phi}^2}{2}+V(\\phi) \\right),\\quad \\dot{H} = -\\frac{\\dot{\\phi}^2}{2}\\,. \n\t\\label{Freq}\n\\end{equation}\nThe next step is to study the scalar and tensor perturbations around the obtained inflationary background solution. When it comes to the scalar part we have perturbations arising from metric fluctuations ($g_{\\mu\\nu}=\\bar{g}_{\\mu\\nu}+\\delta g_{\\mu\\nu}$) and also the scalar field fluctuations ($\\phi = \\bar{\\phi}+\\delta\\phi$). The metric fluctuations can be represented by the Arnowitt-Deser-Misner (ADM) metric of the form \n\\begin{equation}\\label{ADMmetric}\n\tds^2 = - a^2\\left( \\tau \\right) \\Big(N^2d\\tau^2+  \\gamma_{ij} \\left( dx^i+N^i d\\tau \\right) \\left( dx^j+N^j d\\tau \\right) \\Big)\\,.\n\\end{equation}\nwhere $d\\tau = \\frac{dt}{a}$ is conformal time, $ N$ and $N_i$ are the lapse and shift functions respectively.  In the unitary gauge, we fix \n$\\delta\\phi=0$ then we linearly expand the ADM metric  as\n\\begin{equation}\\label{eq:ADMfluctuation}\n\tN= 1+\\delta N,\\quad N_i = a\\partial_i\\chi,\\quad \\gamma_{ij} =  \\left( 1+2\\zeta \\right)\\delta_{ij}+h_{ij}\\,, \n\\end{equation}\nwhere $\\chi$ is a scalar function of spacetime, $\\zeta $ is the curvature perturbation , $h_{ij}$ is the transverse and traceless spin-2 fluctuation. From the linear perturbed equations of motion we obtain the following constraints \n\\begin{equation}\n\t\\begin{aligned}\n\\delta N & = \\frac{\\dot{\\zeta}}{H} \\\\ \n\\partial^2\\chi & = -\\frac{\\partial^2 \\zeta}{H}+ \\epsilon a^2\\dot{\\zeta}\n\\end{aligned}\n\\end{equation}\n{Substituting \\eqref{ADMmetric} into the action \\eqref{action-S} and expanding to second order in the fluctuations, we find the second order action for the scalar perturbation}:\n\\begin{equation}\\label{scalar}\n\t\\delta^{(2)}S_{s} = \\frac{1}{2}\\int d\\tau d^3x a^2\\frac{\\phi^{\\prime 2}}{\\mathcal{H}^2} \\Bigg[ \\zeta^{\\prime 2} -\\left( \\partial\\zeta \\right)^2 \\Bigg]\n\t\\,\\text{,}\n\\end{equation}\nwhere $\\mathcal{H} = \\frac{a^\\prime}{a}= aH$. \nSimilarly we obtain the following second order action for tensor fluctuations:\n\\begin{equation}\n\t\\delta^{(2)}S_h = \\frac{1}{8} \\int d\\tau d^3x a^2\\Bigg[  h_{ij}^\\prime h_{ij}^\\prime -\\partial_k h_{ij}\\partial^k h^{ij}   \\Bigg]\n\t\\,\\text{.}\n\\end{equation}\nWhat we have so far is totally classical and we haven't promoted any fields and conjugate momenta to operators. To quantize the fluctuations we first redefine the fields as \n\\begin{equation}\n\tv = a\\frac{\\dot{\\phi}}{H}\\zeta,\\quad u_{ij} = \\frac{a}{2} h_{ij} \\,,\n\t\\label{rescaleeq}\n\\end{equation}\nThe second order actions in these new variables (which are famously known as Mukhanov-Sasaki variables) become \n\\begin{equation}\n\t\\delta^{(2)}S_s = \\frac{1}{2} \\int d\\tau d^3x \\Big[v^{\\prime 2}-\\left(\\partial_iv\\right)^2-\\left( \\frac{\\nu_s^2-\\frac{1}{4}}{\\tau^2} \\right) v^2\\Big],\n\t\\label{screfa}\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\nu_s \\approx \\frac{3}{2}+\\epsilon+\\frac{\\eta}{2}\n\\end{equation}\n\\begin{equation}\n\t\\delta^{(2)}S_s = \\int d\\tau d^3x  \\Big[u_{ij}^{\\prime 2}-\\left(\\partial_iu_{ij}\\right)^2-\\left( \\frac{\\nu_t^2-\\frac{1}{4}}{\\tau^2} \\right) u_{ij}^2\\Big],\n\t\\label{tensor-mode}\n\\end{equation}\nand\n\\begin{equation}\n\t\\nu_t \\approx \\frac{3}{2}+\\epsilon\\,. \n\\end{equation}\nWe now promote the fields $v,\\, u_{ij}$ and their corresponding conjugate momenta $\\pi_s,\\, \\pi_{ij}$ to operators and define the canonical commutation relations as \n\\begin{equation}\n\t\\begin{aligned}\n\t\\Big[\\hat{v}\\left( \\tau,\\,\\textbf{x} \\right),\\, \\hat{\\pi}_s\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\Big] & = i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right) \\\\ \n\t\t\\Big[\\hat{u}_{ij}\\left( \\tau,\\,\\textbf{x} \\right),\\, \\hat{\\pi}_{ij}\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\Big] & = i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right)\n\t\t\\label{cancom}\n\t\\end{aligned}\n\\end{equation}\nWe expand the field $\\hat{v}\\left( \\tau,\\, \\textbf{x} \\right)$ in terms of creation and annihilation operators  $ a_\\textbf{k},\\, a^\\dagger_{\\textbf{k}} $ as\n\\begin{equation}\n\t\\begin{aligned}\n\t\\hat{v}\\left( \\tau,\\, \\textbf{x} \\right) & = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d^3k \\hat{v}_\\textbf{k}\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} \\\\ \n\t& = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int  d^3k\\Bigg[ a_\\textbf{k} {v}_k\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} + a_\\textbf{k}^\\dagger {v}_k^\\ast e^{-i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\\, \n\t\\end{aligned}\n\\end{equation}\nwhere $v_k$ is the mode function which satisfies Mukhanov-Sasaki (MS) equation \n\\begin{equation}\n\tv_k^{\\prime\\prime} + \\left( k^2-\\frac{v_s^2-\\frac{1}{4}}{\\tau^2}  \\right) v_k = 0\\,. \n\t\\label{MS-equation}\n\\end{equation}\nand $\\left[ a_\\textbf{k},\\, a^\\dagger_{\\textbf{k}^\\prime}\\right]  = \\delta\\left( \\textbf{k}-{\\textbf{k}}^\\prime \\right)$\nThe generic solution of \\eqref{MS-equation} by treating $\\nu_s\\approx\\, {\\rm const}$  is \n\\begin{equation}\n\t\\begin{aligned}\n&\tv_k\\left( \\tau \\right)   \\\\ &= \\dfrac{\\sqrt{\\pi}}{2} \\left(  -\\tau  \\right)^{1\/2} e^{i\\left( 2\\nu_s+1\\right) \\pi}\\Bigg[A_k H^{(1)}_{\\nu_s}\\left( -k \\tau \\right)+ B_k H^{(2)}_{\\nu_s}\\left( -k  \\tau \\right)\\Bigg].\n\t\\end{aligned}\n\\end{equation}\nwhere $A_k$ and $B_k$ are the Bogoliubov coefficients. \nDue to the commutation relations \\eqref{cancom}, the mode functions must satisfy \n\\begin{equation}\nv_kv^{\\ast\\prime}_k - v_k^\\ast v^{\\prime}_k = i\n\\end{equation}\nwhich implies\n\\begin{equation}\n\t\\vert A_k\\vert^2 - \\vert B_k\\vert^2 =1\\,.\n\t\\label{wrAB}\n\\end{equation}\nFrom the above relation we can carry a parametrization $A_k = \\cosh\\alpha$ and $B_k =\\sinh\\alpha$. The case $\\alpha\\neq 0$ corresponds the well-known $\\alpha-$ vacuum in the limit $\\left( \\epsilon,\\,\\eta \\right)\\to 0$ \\cite{Allen:1985ux}. \nThe case $\\alpha=0$ corresponds to the Bunch-Davies (BD) vacuum in which case the mode functions behave like Minkowski-one in the limit $\\tau\\to -\\infty$ when they are deep inside the horizon i.e., when $k\\gg aH$ the mode function of the BD vacuum becomes \n\\begin{equation}\n\tv_k \\simeq \\sqrt{\\frac{1}{2k}}e^{-ik\\tau}\n\\end{equation}\nThus the BD vacuum is given by\n\\begin{equation}\na_\\textbf{k} \\vert 0\\rangle =0\\,.\n\\end{equation}\nIn the limit $\\left( \\epsilon,\\, \\eta \\right)\\to 0$ the mode functions can be approximated as \n\\begin{equation}\n\tv_k\\left( \\tau \\right) = \\sqrt{\\frac{1}{2k}} e^{-ik\\tau}\\left( 1-\\frac{i}{k\\tau} \\right)\n\\end{equation} \nThe two point correlations of the fields can be computed as \n\\begin{equation}\n\t\\langle 0 \\vert \\hat{v}\\left( \\textbf{x},\\, \\tau \\right)  \\hat{v}\\left( \\textbf{x}+\\boldsymbol{\\xi}, \\, \\tau \\right)\\vert 0\\rangle = \\frac{4\\pi}{\\left( 2\\pi \\right)^3} \\int  \\frac{dk}{k} \\frac{\\sin k \\xi}{k\\xi} k^3 \\vert v_k\\vert^2\\,.\n\\end{equation}\n The power spectrum of the canonical scalar field $v$ is defined as \n\\begin{equation}\n\\begin{aligned}\n\t\\langle 0\\vert \\hat{v}_\\textbf{k}\\hat{v}_{\\textbf{k}^\\prime}\\vert 0\\rangle & = \\vert v_k\\vert^2 \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\\, \\\\ \n\t& = P_v\\left( k \\right) \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\n\t\\end{aligned}\n\\end{equation}\nThe power spectrum of curvature perturbation is obtained by pulling back the rescaling we have performed in \\eqref{rescaleeq}\n\\begin{equation}\n\t\\langle \\zeta_\\textbf{k}\\zeta_{\\textbf{k}^\\prime}\\rangle  = \\frac{1}{2a^2\\epsilon} \\times \\langle v_\\textbf{k} v_{\\textbf{k}^\\prime}\\rangle\\, = \\frac{2\\pi^2}{k^3} P_\\zeta \\left( k \\right) \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\\,. \n\\end{equation}\nThe curvature perturbation $\\zeta$ freezes out on super-horizon scales and therefore we evaluate the spectrum of curvature perturbation at the time of horizon exit $k = a H$. \nThus, the dimensionless power spectrum of curvature perturbation becomes\\footnote{Since the curvature perturbation is conversed on super-horizon scales the (scale dependent) magnitude of power spectrum is given by \\eqref{Amppr} which is evaluated at Horizon exit.}\n\\begin{equation}\n\t\\begin{aligned}\n\tP_\\zeta\\left( k \\right) \\Big\\vert_{k=aH} & = \\frac{k^3}{2\\pi^2} \\frac{1}{2a^2\\epsilon}P_v\\\\ \n\t& = \\frac{H^2}{8\\pi^2\\epsilon} \\Big\\vert_{k =a H}\\,.\n\t\\end{aligned}\n\\label{Amppr}\n\\end{equation}\nIn the above derivation we substituted $\\tau \\approx -\\frac{1}{aH}$. \nThe tilt of the power spectrum $P_\\zeta$ can be computed as \n\\begin{equation}\n\tn_s -1 = \\frac{d\\ln P_\\zeta}{d\\ln k} \\approx -\\frac{d\\ln P_\\zeta}{dN} \\approx -2\\epsilon-\\eta\\,. \n\t\\label{speci}\n\\end{equation}\nNote that the above result \\eqref{Amppr} and \\eqref{speci} remains the same even without performing the approximation $\\nu_s\\approx 3\/2\\implies \\left( \\epsilon,\\, \\eta \\right)\\to 0$ \\cite{Kinney:2009vz}. Since the power spectrum is obtained by evaluating it at $k =aH$, the small corrections to the parameter $\\nu_s$ give subdominant results. In addition, the spectral index $n_s$ and $n_t $ are often written in terms of $\\nu_s,\\, \\nu_t$, but such definitions are not precisely correct as it is explained in  \\cite{Kinney:2009vz}. One must indeed evaluate the power spectrum at $k = aH$ because beyond this point curvature perturbation is frozen. So, the super-horizon amplitude of the power spectrum can be written in terms of quantities evaluated at the horizon exit \\cite{Leach:2000yw} by fixing the normalization factors \\cite{Kinney:2009vz}. \nSimilarly,  we can work out the mode function for the tensor fluctuations and imposing the BD vacuum we obtain \n\\begin{equation}\n\t\\begin{aligned}\n\t& u_{ij}  = e_{ij} \\dfrac{\\sqrt{\\pi}}{2} \\left(  -\\tau  \\right)^{1\/2} e^{i\\left( 2\\nu_t+1\\right) \\pi} H^{(1)}_{\\nu_t}\\left( -k \\tau \\right)\n\t\\end{aligned}\n\\end{equation}\nWe can perform similar calculations for the tensor power spectrum and tilt and as a result we obtain \n\\begin{equation}\n\tP_T = \\frac{2H^2}{\\pi^2}\\Bigg\\vert_{k= a H},\\quad n_t = -\\frac{d\\ln P_T}{d\\ln N} \\Bigg\\vert_{k= a H} = -2\\epsilon\\,. \n\\end{equation}\n\n\n\\section{New scheme of quantizing inflationary power spectra} \n\\label{sec:questions}\nIn the previous section, we have discussed the standard quantization scheme of inflationary fluctuations. In this section, we enlist some open questions in the context of quantization of inflationary  fluctuations. Based on these, we propose a new scheme of quantization of the inflationary fluctuations, which not only target some of the open questions, but also leads to some observable features in the CMB and primordial gravitational wave spectra. Below we discuss point by point some of the aspects of standard calculations of inflationary quantum fluctuations in relation with our understanding of spacetime in quantum theory and general relativity.  \n\\begin{itemize}\n\t\\item By quantizing inflationary fluctuations we actually deal with quantum field theory (QFT) in curved spacetime. But in standard QFT we have particles that propagate forward and backward in time (anti-particle). A natural question is what happens to particles and anti-particle states in a curved spacetime. This implies one should consistently define the role of time reversal in a curved spacetime. \n\t\\item QFT in Minkowski spacetime  is mainly built on discrete symmetries such as $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ or $\\mathcal{P}\\mathcal{T}$ (in the context of real scalar field), but in curved spacetime we often do not pay attention to the notion of discrete symmetries. The open question here is whether $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ holds or not in curved spacetime? \n   \\item\n   In classical GR time is a coordinate and in quantum theory time is a parameter \\cite{Donoghue:2017pgk,Donoghue:2019ecz}. In the case of inflationary cosmology we usually quantize both gravitational and matter degrees of freedom \\cite{Martin:2004um,Baumann:2018muz,Kinney:2009vz} taking a classical notion of time. However, the concept of time in quantum theory is very different from the classical one \\cite{Rovelli:2004tv}.\n   \\item Since we quantize gravitational degrees of freedom, we can interpret inflationary quantum fluctuations as features of (linearized) quantum gravity \\cite{Martin:2004um}. Note that, in the canonical quantum gravity that emerges through Wheeler-de Witt equation, time does not appear explicitly which indicates a difficulty to define positive\/negative frequencies, like we usually do following the standard Schr\\\"{o}dinger equation. This is famously known as the problem of time in quantum cosmology \\cite{Kiefer:2007ria,Rovelli:2004tv}.\n  \n   \\item Quantum theory is always time symmetric. The arrow of time emerges only after we specify initial and final states \\cite{Hartle:2013tm}. This implies that we could first formulate QFT in curved spacetime, in a time symmetric way, and then impose initial conditions. \n\\end{itemize}\nOur new formalism for the inflationary quantum fluctuations is highly motivated from the above questions. Before we go into details, we write down here few guidelines\n\n\\begin{itemize}\n\t\\item We identify the expansion or contraction of Universe with the shrinking or growing size of the co-moving horizon (in the context of de Sitter and quasi de Sitter). Change of size of the horizon defines our classical or thermodynamical arrow of time.\n\t\\item We build the quantum theory based on discrete transformations of spacetime within the co-moving horizon. We completely detach ourselves from the notions of observers, Penrose diagrams and the corresponding regions of spacetime when we do quantization because all such concepts are not well-defined according to quantum theory. In fact, classical notion of region of spacetime cannot have any consistent meaning from the quantum mechanical point of view \\cite{Donoghue:2021meq}. \n\t\\item \n\tThe way forward is to first understand quantum mechanics in a dynamical spacetime and then take the classical limit. To achieve this goal, we take a small step by distinguishing classical notion of time with the quantum mechanical notion of time. This small step allow us to create all possible quantum states in the given spacetime geometry.\n\t\\item In our formalism, quantum fluctuations always come in pair from two different vacua described by $\\mathcal{P}\\mathcal{T}$ or $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ transformations. By this scheme we have the quantum fluctuations that propagate forward in time as well as backward in time and both can exist within the horizon. As the horizon changes, both of these type of fluctuations either exit or enter the horizon depending on whether the spacetime is expanding or contracting.\n\t\\item These pairs of quantum fluctuations within the horizon evolve independently of each other which means the commutator of corresponding operators always vanish. This means even though we can create quantum states non-locally we still do not violate locality and causality in our formulation of QFT.   \n\\end{itemize}\n\n\\section{Quantum fluctuations in an expanding de Sitter spacetime}\n\\label{sec:dsspace}\nIn this section, we focus on elucidating our proposal in the context of exact de Sitter (dS) spacetime in FLRW co-ordinates. Since inflationary spacetime is near dS there are several concepts we can take forward from the understanding of dS. To make our analysis simpler we consider quantization of a massless scalar field in dS spacetime ignoring the issues with IR singularities and defining dS invariant vacuum \\cite{Higuchi:2009ew,Allen:1985ux}.\n\nThe dS spacetime in FLRW coordinates looks like\n\\begin{equation}\n\t\\begin{aligned}\n\tds^2 & = -dt^2 + a(t)^2d\\textbf{x}^2\n\t            & = \\frac{1}{H^2\\tau^2}\\left( -d\\tau^2+ d\\textbf{x}^2  \\right)\\,. \n\t\\end{aligned}\n\t\\label{dsmetric}\n\\end{equation}\nwhere $d\\tau= \\frac{dt}{a}$ is the conformal time  and the\nscale factor is given by\n\\begin{equation}\n\ta(t) = e^{Ht},\\quad H^2 = \\left(\\frac{1}{a}\\frac{da}{dt}\\right)^2={\\rm const}\\,.\n\\end{equation}\nNote that dS metric \\eqref{dsmetric} is $\\mathcal{P}\\mathcal{T}: \\tau\\to -\\tau,\\, \\textbf{x}\\to -\\textbf{x}$ symmetric. \nThe scalar curvature of dS spacetime is $R=12H^2$ which does not tell whether the Hubble parameter is positive or negative. Each point in dS is surrounded by a co-moving horizon given by the radius\n\\begin{equation}\n\tr_H = \\Big\\vert  \\frac{1}{aH} \\Big\\vert\n\t\\label{horizonds}\n\\end{equation}\nOne very simple observation we can make from \\eqref{dsmetric} is that the spatial size of the Universe $a(t)^2$ grows with coordinate time $t$ when $H>0$. We then interpret the metric \\eqref{dsmetric} as modelling an expanding universe. \nIn this case the scale factor grows with time $t$ and the size of the horizon shrinks. \nWe note here that this observation stems from the convention that $t$ runs from $-\\infty $ to $+\\infty$ (which means conformal time $-\\infty<\\tau< 0$). Defining $\\bar{t}= -t$, which corresponds to \\say{$t$ running from $+\\infty$ to $-\\infty$ or $+\\infty>\\tau> 0$} we find the scale factor $a(\\tau)$ growing with $\\bar{t}$ for $H<0$, corresponding to expansion. Therefore, the expanding dS space can only characterized by shrinking horizon rather than fixing the notion of time being positive or negative. Since time is a parameter in quantum theory, we can have quantum states that are labeled by $\\tau<0$ and\/or $\\tau>0$. In the case of standard QFT in dS one usually make a biased choice of either $\\tau<0$ or $\\tau>0$ which actually breaks the time reversal nature of dS spacetime and also standard scheme of quantization completely ignores existence of states that evolve backward in time. \nTherefore, \n in our scheme of quantization in dS spacetime, we can have quantum states that propagate forward in time $t: -\\infty \\to \\infty$ with $H > 0$ and states that propagate backward in time following an arrow of time $t: \\infty \\to -\\infty$ with $H< 0$. Note that both type of quantum states can co-exist in a single expanding Universe. We stress again, the expanding dS Universe is defined by the following arrows of time\n\\begin{equation}\n\t\\tau:  \\pm \\infty \\to 0\n\\end{equation}\nwhere $-$ sign corresponds to a state propagating forward in time and $+$sign  corresponds to a state propagating backward in time. In our new picture quantization of a (massless) scalar field in dS spacetime leads to a pair of  quantum fields that are produced out of two vacua. Being more precise we define total vacuum as the direct sum of two vacua\n\\begin{equation}\n\t\\vert 0\\rangle = \\frac{1}{\\sqrt{2}} \\Bigg(\\vert 0\\rangle_{\\rm I}\\oplus \\vert 0\\rangle_{\\rm II} \\Bigg)\\,. \n\\end{equation}\nIn the vacuum $\\vert 0\\rangle_{\\rm I}$ we create quantum fields at the position $ \\textbf{x} $ that evolve forward in time ($\\tau <0 $) and in the vacuum $\\vert 0\\rangle_{\\rm II}$ we create quantum fields at $ -\\textbf{x}$ which evolve backward in time ($\\tau >0$).\nThe action of a massless scalar field in dS space is given \n\\begin{equation}\nS_{\\phi} = -\\frac{1}{2}\\int d\\tau d^3x a^2\t\\phi\\left( \\partial_\\tau^2+2\\mathcal{H} \\partial_\\tau + k^2 \\right) \\phi\\,. \n\\end{equation}\nFirst thing we can notice here is that the above action is invariant under $\\mathcal{P}\\mathcal{T}$ which means  $\\textbf{x}\\to -\\textbf{x}$ and \n\\begin{equation}\\label{timeref}\n\tt\\to -t\\quad  \\left(-\\tau \\to \\tau\\right) \\implies H\\to -H \\quad \\left( \\mathcal{H}\\to -\\mathcal{H} \\right)\n\\end{equation}\nRescaling the field $\\phi\\to a \\phi$  gives the following action which leads to Mukhanov-Sasaki equation as\n\\begin{equation}\n\tS_\\phi = \\frac{1}{2} \\int d\\tau d^3x \\Big[ \\phi^{\\prime 2} - \\left( \\partial_i\\phi \\right)^2  -  \\frac{2}{\\tau^2}  \\phi^2 \\Big]\\,. \n\t\\label{msaction}\n\\end{equation}\nThe above Lagrangian is again symmetric under $\\mathcal{P}\\mathcal{T}: \\textbf{x}\\to -\\textbf{x},\\, \\tau\\to -\\tau$. When we quantize this scalar field\nwe propose that the scalar field operator $\\hat{\\phi}\\left( \\tau,\\, \\textbf{x} \\right)$ is the direct sum of the pair of fields as\n\\begin{equation}\n\t\\hat{\\phi}\\left( \\tau,\\, \\textbf{x} \\right) =  \\frac{1}{\\sqrt{2}}\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus \t\\frac{1}{\\sqrt{2}}\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right) \\,. \n\\end{equation}\nwhich can be expanded in terms of different creation and annihilation operators in the following way\n\\begin{equation}\n\t\\begin{aligned}\n\t&\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right)  \\\\ & = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ a_\\textbf{k} {\\varphi}_{\\rm I\\,k}\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} + a_\\textbf{k}^\\dagger {\\varphi}^\\ast_{\\rm I\\,k}\\left( \\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\\, \n\t\\end{aligned}\n\\label{vdfield}\n\\end{equation}\nwhere the creation and annihilation operators $a_\\textbf{k},\\, a^\\dagger_\\textbf{k}$ satisfy the canonical commutation relations  and \n \\begin{equation}\na_\\textbf{k}\\vert 0 \\rangle_{\\rm I} = 0\\,. \n\\end{equation}\nHere the mode function $\\varphi_{\\rm I\\,,k}$ is the solution of Mukhanov-Sasaki equation (that comes from varying \\eqref{msaction}) given by \n\\begin{equation}\n\t\\varphi_{\\rm I,\\,k}  = \\alpha_{1k} \\frac{e^{-ik\\tau}}{\\sqrt{2k}}\\left( 1-\\frac{i}{k\\tau} \\right) +\\beta_{1k} \\frac{e^{ik\\tau}}{\\sqrt{2k}}\\left( 1+\\frac{i}{k\\tau} \\right)\\,, \n\t\\label{f1}\n\\end{equation}\nwhere the Bogoliubov coefficients can be fixed as $\\left( \\alpha_{1,\\,k},\\, \\beta_{1,\\,k}\\right) = \\left( 1,\\,0 \\right)$, which is compatible with the Wronskian condition $\t\\varphi_{\\rm I,\\,k} \t\\varphi^{\\prime\\ast}_{\\rm I,\\,k}-\t\\varphi^\\ast_{\\rm I,\\,k}\t\\varphi^\\prime_{\\rm I,\\,k} = i   $ that corresponds to the canonical commutation relation\n\\begin{equation} \n\\big[ \\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right),\\, \\hat{\\pi}_{\\rm I}\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\big] = i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right). \n\\end{equation}\nThe choice $\\left( \\alpha_{\\rm I,\\,k},\\, \\beta_{\\rm I,\\,k}\\right) = \\left( 1,\\,0 \\right)$ defines the vacuum and $\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\vert 0\\rangle $ corresponds to the positive frequency modes in the limit $\\tau\\to -\\infty$. Quantum mechanically $\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\vert 0\\rangle $ is the postive energy state that propagate forward in time at \\textbf{x}. \nSimilarly,  we can expand the second field operator as \n\\begin{equation}\n\t\\begin{aligned}\n\t\t&\t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)  \\\\ & = \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ b_\\textbf{k} \\varphi_{\\rm II\\,k}\\left( -\\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} + b_\\textbf{k}^\\dagger {\\varphi}^\\ast_{\\rm II\\,k}\\left( -\\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\\, \n\t\\end{aligned}\n\t\\label{vdfield2}\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\varphi_{\\rm II,\\,k}  = \\alpha_{2,\\,k} \\frac{e^{ik\\tau}}{\\sqrt{2k}}\\left( 1+\\frac{i}{k\\tau} \\right) +\\beta_{2,\\,k} \\frac{e^{-ik\\tau}}{\\sqrt{2k}}\\left( 1-\\frac{i}{k\\tau} \\right)\\,, \n\t\\label{f2}\n\\end{equation}\nwhere the Bogoliubov coefficients can be fixed as $\\left( \\alpha_{2,\\,k},\\, \\beta_{2,\\,k}\\right) = \\left( 1,\\,0 \\right)$ which is compatible with the Wronskian condition $\t\\varphi_{\\rm II,\\,k} \t\\varphi^{\\prime\\ast}_{\\rm II,\\,k}-\t\\varphi^\\ast_{\\rm II,\\,k}\t\\varphi^\\prime_{\\rm II,\\,k} = -i   $ that corresponds to the canonical commutation relation \\cite{Donoghue:2019ecz}\n\\begin{equation} \n\t\\big[ \\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right),\\, \\hat{\\pi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x}^\\prime \\right)\\big] = -i\\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right). \n\\end{equation}\nwhich describe the quantum fields that propagate backward in time. \nHere the second vacuum is defined by \n\\begin{equation}\n\tb_\\textbf{k}\\vert 0 \\rangle_{\\rm II} = 0\\,. \n\\end{equation}\nHere  $\t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)\\vert 0\\rangle_{\\rm II}$ corresponds to positive energy state that evolves backward in time at -\\textbf{x}. \nWe demand that these two quantum fluctuations evolve independently. This is manifest by\n\\begin{equation}\n\t[\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right),\\, \t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)   ] =0\\,. \n\\end{equation}\nwhich implies that the respective creation and annihilation operators commute\n\\begin{equation}\n\t\\big[a_\\textbf{k},\\, b_{\\textbf{k}^\\prime}\\big] = 0,\\quad \\big[a^\\dagger_\\textbf{k},\\, b^\\dagger_{\\textbf{k}^\\prime}\\big] = 0 \n\\end{equation}\nSince dS spacetime is perfectly $\\mathcal{P}\\mathcal{T}$ symmetric we have that the quantum fields  $\t\\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right)\\vert 0\\rangle_{\\rm I}$ and  $\t\\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)\\vert 0\\rangle_{\\rm II}$ behave identically, which can be seen from the fact that their equal time correlations are the same\n\\begin{equation}\n\t\\begin{aligned}\n& \\frac{1}{a^2}{}_{\\rm I}\\langle 0\\vert \\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\hat{\\varphi}_{\\rm I}\\left( \\tau,\\, \\textbf{x}^\\prime \\right)\\vert 0\\rangle_{\\rm I} = \\\\ & \\frac{1}{a^2} {}_{\\rm II}\\langle 0\\vert \\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right) \\hat{\\varphi}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x}^\\prime \\right)\\vert 0\\rangle_{\\rm II} = \\frac{H^2}{4\\pi^2k^3}\\,. \n\t\\end{aligned}\n\\label{eqcorr}\n\\end{equation}\nThis implies the correlations of quantum fields related by $\\mathcal{P}\\mathcal{T}$ transformations are identical in the case of dS spacetime.  \nThis indicates our formulation of QFT in dS spacetime might be $\\mathcal{P}\\mathcal{T}$ or $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ (if we include charged fields) symmetric.\\footnote{We note that the result in \\eqref{eqcorr} is similar to what was derived in the context of elliptic dS spacetime \\cite{SANCHEZ19871111,Schrodinger1956}. However, in our scheme of quantization we do not make any (classical) antipodal identification of spacetime and therefore we significantly deviate conceptually from the proposal of elliptic dS made by Erwin Schr\\\"{o}dinger \\cite{Schrodinger1956}.  }\n\nHowever, this may not be true in a spacetime with less symmetries such as quasi-dS. Indeed, in the context of inflationary quantum fluctuations, it is often assumed, in the effective field theory (EFT) approach, that Lorentz symmetry (in other words $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$) must be spontaneously broken \\cite{Cheung:2007st}. However, it is not known how $\\mathcal{C}\\mathcal{P}\\mathcal{T}$ symmetry can be broken in a curved spacetime. \n\t In the next section, we provide a framework of quantization where we can explicitly see that $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ is spontaneously broken in the sense that the qunatum fields defined by our new notion of $\\mathcal{P}\\mathcal{T}$ transformations evolve differently. \n\n\\section{Double vacuum inflationary spacetime and the power spectra}\nIn the previous section, we have studied the quantization of a massless scalar field in dS. Here we quantize the inflationary quantum fluctuations. There is a crucial difference here. In the previous section we quantized an arbitrary scalar field in dS spacetime which is completely fixed but in the context of inflation we quantize metric and matter degrees of freedom in order to find an effective quantum correction to the classical quasi-dS (inflationary) spacetime. To be more precise, the classical inflationary spacetime is given by an arrow of time and the background initial conditions are \n\\begin{equation}\nt: -\\infty \\to \\infty \\implies \\left( \\epsilon,\\, \\eta \\right): \\left( \\approx 0,\\, \\approx 0 \\right) \\to \\left( \\sim 1,\\, \\sim 1 \\right)\\,. \n\\end{equation}\nWhich means the slow-roll parameters are very small and positive during inflation and inflation ends when the slow-roll parameters evolve to the order of unity, which happens at some time $t_{\\rm end}\\gg 1$. The inflationary quantum fluctuations are produced in the phase when $\\left( \\epsilon,\\, \\eta \\right) \\ll 1$. As we discussed earlier, ``time'' in quantum theory has totally different role and one must be very careful in quantizing inflationary fluctuations. We emphasize that the quantum fluctuations do not necessarily follow arrow of time prescribed by the initial conditions in classical theory. \n\n Unlike dS spacetime the quasi-dS metric does not have $\\mathcal{P}\\mathcal{T}$-symmetry, therefore naturally one would expect the $\\mathcal{P}\\mathcal{T}$-symmetry to be spontaneously broken at the quantum level. To see this, we apply the scheme of quantization similar to what we have done in the context of exact dS. Let us first focus on inflationary scalar fluctuations which are described by the action \\eqref{scalar}. After the field redefinition $\\hat{v}$ is the canonical variable which we need to quantize \\eqref{screfa}. As we postulated before quantum fluctuations always come as pair from two different vacua related by $\\mathcal{P}\\mathcal{T}$ transformations. This means that unlike the standard quantization of inflationary fluctuations, where only one kind of quantum states are considered which only evolve forward in time, here we describe a pair of quantum fluctuations that evolve both forward and backward in time. \n\n Similar to the exact dS case we write the canonical field operator as a direct sum of two quantum fields \n\\begin{equation}\n\t\\hat{v}\\left( \\tau,\\, \\textbf{x} \\right) = \\frac{1}{\\sqrt{2}} \\hat{v}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus  \\frac{1}{\\sqrt{2}} \\hat{v}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right)\\,. \n\\end{equation}\nwhich emerges from two different vacua. The total vacuum is the direct sum of these two which can be written as\n\\begin{equation}\n\t\\vert 0\\rangle_{\\rm qdS} = \\vert 0 \\rangle_{\\rm qdS_{I}} \\oplus \\vert 0\\rangle_{\\rm qdS_{II}}\\,. \n\\end{equation}\nThe quantum field operators acting on the vacua $\\hat{v}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\vert 0\\rangle_{\\rm qdS_{II}} $ and $\\hat{v}_{\\rm I}\\left( -\\tau,\\, -\\textbf{x} \\right)\\vert 0\\rangle_{\\rm qdS_{II}}$ leads to the creation of the pair of quantum fluctuations at $ \\textbf{x} $ and at $ -\\textbf{x}  $ which propagate forward in time and back ward in time respectively. We will make it clear shortly the meaning of quantum fluctuations that evolve forward and backward in time in the context of quasi-dS.\n\nGoing into the details\nwe expand the fields in terms of creation and annhilation operators as explained below. \n\\begin{equation}\n\t\\begin{aligned}\n& \\hat{v}_{\\rm I}\\left( \\tau,\\, \\textbf{x} \\right) \\\\ &  =\t \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ c_\\textbf{k} {v}_{\\rm I,\\,k}\\left( \\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} + c_\\textbf{k}^\\dagger {v}_{\\rm I,\\,k}^\\ast\\left( \\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\n\\end{aligned}\n\\label{vid}\n\\end{equation}\nwith $c_\\textbf{k},\\, c_\\textbf{k}^\\dagger$ being the creation and annihilation operators of quasi-dS vacuum-I defined by\n\\begin{equation}\n\tc_\\textbf{k}\\vert 0\\rangle_{\\rm qdS_I} = 0\\,, \n\t\\label{qds1}\n\\end{equation}\nand $ {v}_{\\rm I,\\,k}$ is the mode function obtained by solving MS-equation for $v_{\\rm I}\\left(\\tau \\right)$ \n\\begin{equation}\n\tv_{\\rm I,\\,k}^{\\prime\\prime}+ \\left( k^2-\\frac{{\\nu}_s^2-\\frac{1}{4}}{\\tau^2} \\right) v_{\\rm I,\\,k}^2 =0\\,.\n\\end{equation}\nHere $\\tau: -\\infty \\to 0$ and \n the quasi-dS vacuum $\\vert 0\\rangle_{\\rm qdS_I}$ corresponds to the creation of positive frequency modes in the limit $\\tau\\to -\\infty$. \nNotice that unlike the dS case the MS-equation \\eqref{MS-equation} is not symmetric under time reversal because there is an additional time dependence enters through the nearly constant variable $\\nu_s$ which contains the slow-roll parameters $\\left( \\epsilon,\\, \\eta \\right)$  \\eqref{sldef}. Assuming $0<\\left( \\epsilon,\\, \\eta\\right) \\ll 1$ and constant we obtain  \n\\begin{equation}\n\t\\begin{aligned}\n&\t{v}_{\\rm I,\\,k}  \\\\ & = \\frac{\\sqrt{\\pi}}{2} \\left( -\\tau \\right)^{1\/2} e^{\\left( i\\nu_s+1\\right)} \\Bigg[C_k H^{(1)}_{\\nu_s}\\left( -k \\tau \\right)+ D_k H^{(2)}_{\\nu_s}\\left( -k  \\tau \\right)\\Bigg]\\,. \n\t\\label{new-vac1}\n\t\\end{aligned}\n\\end{equation}\nThe above mode function corresponds to the creation of positive frequency modes in the limit $\\tau\\to -\\infty$ for the case $\\left( \\mathcal{C}_k,\\, D_k \\right) = \\left( 1,\\,0 \\right) $ which corresponds to the standard BD state that satisfies the Wronskian $v_{I,k}v_{I,k}^{\\prime\\ast}-v_{I,k}^\\ast v_{I,k}^{\\prime}=i  (\\implies \\vert C_k\\vert^2-\\vert D_k\\vert^2=1)$. We can expand the mode function for the case of $\\left( \\mathcal{C}_k,\\, D_k \\right) = \\left( 1,\\,0 \\right) $ in the order of slow-roll as\n\\begin{equation}\n\t\\begin{aligned}\n\t{v}_{\\rm I,\\,k} &  \\approx \\frac{\\sqrt{\\pi}}{2} \\left( -\\tau \\right)^{1\/2} e^{\\left( i\\nu_s+1\\right)}  H^{(1)}_{\\nu_s}\\left( -k \\tau \\right)\\,\\\\ \n\t& \\approx \\sqrt{\\frac{1}{2k}} e^{-ik\\tau}\\left( 1-\\frac{i}{k\\tau} \\right) \\\\ & + \\left( \\epsilon+\\frac{\\eta}{2} \\right) \\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{\\frac{k}{aH}} \\frac{\\partial H^{(1)}_{\\nu_s}\\left( \\frac{k}{aH}\\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\n\\label{new-vac1BD}\n\\end{aligned}\n\\end{equation}\n\nThe second field $\\hat{v}_{II\\,,\\textbf{k}}\\left( -\\tau,\\, -\\textbf{x} \\right)$ is expanded as \n\\begin{equation}\n\t\\begin{aligned}\n&\t\\hat{v}_{\\rm II}\\left( -\\tau,\\, -\\textbf{x} \\right) \\\\ & =\t \\frac{1}{\\left( 2\\pi \\right)^{3\/2}}\\int d\\tau d^3k\\Bigg[ d_\\textbf{k} {v}_{\\rm II,\\,k}\\left( -\\tau \\right) e^{-i\\textbf{k}\\cdot \\textbf{x}} + d_\\textbf{k}^\\dagger {v}_{\\rm I,\\,k}^\\ast\\left( -\\tau \\right) e^{i\\textbf{k}\\cdot \\textbf{x}} \\Bigg]\n\t\\end{aligned}\n\\label{v2d}\n\\end{equation}\nwhere $d_\\textbf{k},\\ , d_\\textbf{k}^\\dagger$ are the creation and annihilation operators that satisfy the canonical commutation relations and the second vacuum is defined by\\footnote{We note that $\n\t\t\\big[c_\\textbf{k},\\, d_{\\textbf{k}^\\prime}\\big] = 0,\\quad \\big[c^\\dagger_\\textbf{k},\\, d^\\dagger_{\\textbf{k}^\\prime}\\big] = 0 $ similar to the dS case which means the pair of quantum fluctuations are not causally connected.}\n\\begin{equation}\nd_\\textbf{k}\\vert 0\\rangle_{\\rm qdS_{II}}=0\\,.\n\\end{equation}\nThe evolution of the mode function $v_{\\rm II,\\,k}$ is determined by solving the following time reversed MS-equation \n\\begin{equation}\n\tv_{\\rm II,\\,k}^{\\prime\\prime}+ \\left( k^2-\\frac{\\bar{\\nu}_s^2-\\frac{1}{4}}{\\tau^2} \\right) v_{\\rm II,\\,k}^2 =0\\,.\n\\end{equation}\nHere $\\tau: +\\infty \\to 0$ and\n\\begin{equation}\n\t\\bar{\\nu}_s  \\approx \\frac{3}{2}-\\epsilon-\\frac{\\eta}{2}\n\\end{equation}\nHere the time reversal transformations are \n\\begin{equation}\n\tt\\to -t \\implies H\\to -H,\\quad \\epsilon\\to -\\epsilon,\\quad \\eta\\to -\\eta\\,. \n\t\\label{timerevqds}\n\\end{equation}\nwhich can be understood in the following sense. Our goal here is to define how the quantum fluctuations propagate backward in time in an expanding Universe. To achieve this goal we need to formulate the meaning of time reversal operation. Logically, if the fluctuation propagates forward in time in a slow-roll background, the fluctuation that goes backward in time experience spacetime as a \"slow-climb\"\\footnote{In the standard slow-roll we have scalar field rolling down the potential whose time reversal can be understood as a phantom field that is slowly climbing the potential \\cite{Piao:2004tq}. } which is given by reversing the signs of the parameters $\\left( \\epsilon,\\, \\eta \\right)$ as stated in \\eqref{timerevqds}. We impose this time reversal operation in a completely quantum mechanical sense and this has no classical meaning i.e., our background (classical) dynamics are completely determined by Friedmann equations \\eqref{Freq} and we do not apply at all time reversal to the classical background. Since we treat time differently at quantum level we restrain ourselves from any intuition from classical physics.\\footnote{Notion of time is a very non-trivial concept in physics and its meaning varies in different contexts. We suggest the reader \\cite{Rovelli:2004tv} for an extended physical discussion. Our statement, that quantum states evolving backward in time has no classical analog, is deeply rooted in the quantum gravity concept time as it is presented in p. 184 of \\cite{Rovelli:2004tv}.} Since dynamics of quantum fields emerge from MS-equation \\eqref{MS-equation} the functions $\\left( \\epsilon,\\, \\eta \\right)$ are now treated along with time in as parameters to specify the nature of quantum state. This would encode a subtle difference between quantum fluctuations propagating forward and backward in time.\n\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\t&\tv_{\\rm II,\\,k}\\left( -\\tau \\right)   \\\\ &= \\dfrac{\\sqrt{\\pi}}{2} \\left(  \\tau  \\right)^{1\/2} e^{i\\left( 2\\bar{\\nu}_s+1\\right) \\pi}\\Bigg[\\bar{C}_k H^{(1)}_{\\bar{\\nu}_s}\\left( k \\tau \\right)+ \\bar{D}_k H^{(2)}_{\\bar{\\nu}_s}\\left( k  \\tau \\right)\\Bigg].\n\t\t\t\\end{aligned}\n\t\t\\label{v2sol}\n\\end{equation}\nNow imposing the Wronskian condition $v_{\\rm II,\\,k}v_{\\rm II,\\,k}^{\\prime\\ast}-v_{\\rm II,\\,k}^\\ast v_{\\rm II,\\,k}^\\prime = -i$ which corresponds to the canonical commutation relation for a reversed arrow of time \\cite{Donoghue:2019ecz}\n\\begin{equation}\n\t\\Big[ \\hat{v}_{\\rm II}\\left( -\\tau,\\,-\\textbf{x} \\right),\\,  \\hat{\\Pi}_{\\rm II}\\left( -\\tau,\\,-\\textbf{x}^\\prime \\right) \\Big] = -i \\delta\\left( \\textbf{x}-\\textbf{x}^\\prime \\right)\\,. \n\\end{equation}\nwe obtain $\\vert \\bar{C}_k\\vert^2-\\vert \\bar{D}_k\\vert^2 = 1$. We consider $\\left(\\bar{C},\\, \\bar{D}_k\\right) =0$ which corresponds postive energy states in the limit $\\tau \\to \\infty$. Expanding \\eqref{v2sol} up to the leading order for $\\left(\\bar{C},\\, \\bar{D}_k\\right) =0$ we get\n\\begin{equation}\n\t\\begin{aligned}\n\t\t{v}_{\\rm II,\\,k} &  \\approx \\frac{\\sqrt{\\pi}}{2} \\left( \\tau \\right)^{1\/2} e^{\\left( i\\nu_s+1\\right)}  H^{(1)}_{\\nu_s}\\left( k \\tau \\right)\\,\\\\ \n\t\t& \\approx \\sqrt{\\frac{1}{2k}} e^{ik\\tau}\\left( 1+\\frac{i}{k\\tau} \\right) \\\\ & -\\left( \\epsilon+\\frac{\\eta}{2} \\right) \\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{k\\tau} \\frac{\\partial H^{(1)}_{\\nu_s}\\left( k\\tau \\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\n\t\t\\label{new-vac2BD}\n\t\\end{aligned}\n\\end{equation}\nComparing \\eqref{new-vac1BD} and \\eqref{new-vac2BD} we can deduce that the two mode functions get different slow-roll (quantum) corrections. \nThis is because the parameter $\\nu_s$ that enters in the MS-equation \\eqref{MS-equation} contains a tiny time dependence. In the standard framework of quantization this is treated it to be nearly constant there is only one type of quantum fluctuation (see Sec.~\\ref{sec:review}). But the fact that MS-equation has time dependence in terms of $\\epsilon,\\, \\eta$ is crucial to probe the nature of quantum fluctuations. These quantities change sign under discrete spacetime transformation and this has to be understood in a completely quantum mechanical sense. Our scheme of quantization implies a pair of quantum fluctuations: those that propagate forward in time ($\\tau: -\\infty \\to 0$)  at spatial position $\\textbf{x}$ which behave differently from those fluctuations that propagate backward in time ($\\tau: +\\infty \\to 0$) at the spatial position $-\\textbf{x}$.  These pairs of fluctuations exit the horizon in the two opposite directions and now we observe the modes as they re-enter. \n\nSo far we applied our quantization method for the scalar mode $v\\left( \\tau,\\,\\textbf{x} \\right)$. We can apply the similar method of quantization for the tensor modes \\eqref{tensor-mode} by writing the tensor fluctuation as direct some fluctuations that are related by $\\mathcal{P}\\mathcal{T}$ transformations \n\\begin{equation}\n\t\\hat{u}_{ij} \\left( \\tau,\\, \\textbf{x} \\right)= \\frac{1}{\\sqrt{2}} \\hat{u}^{\\rm I}_{ij}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus  \\frac{1}{\\sqrt{2}}\\hat{u}^{\\rm II}_{ij}\\left( -\\tau,\\,-\\textbf{x} \\right)\n\\end{equation}\nThe above fields can be written in terms of creation and annihilation operators similar to what we have done for \\eqref{vid} and \\eqref{v2d}. The corresponding mode functions tensor modes can be straight forwardly derived as\n\\begin{equation}\n\t\\begin{aligned}\n\t\t{u}^{\\rm I}_{ij,\\,k} \n\t\t& \\approx e_{ij}\\sqrt{\\frac{1}{2k}} e^{-ik\\tau}\\left( 1-\\frac{i}{k\\tau} \\right) \\\\ & + e_{ij} \\epsilon\\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{-k\\tau} \\frac{\\partial H^{(1)}_{\\nu_t}\\left( -k\\tau \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\n\t\\end{aligned}\n\t\t\\label{new-vac1TBD}\n\\end{equation}\nand \n\\begin{equation}\n\t\\begin{aligned}\n\t\t{u}^{\\rm II}_{ij,\\,k} \n\t\t& \\approx e_{ij}\\sqrt{\\frac{1}{2k}} e^{ik\\tau}\\left( 1+\\frac{i}{k\\tau} \\right) \\\\ & - e_{ij} \\epsilon\\frac{\\sqrt{\\pi}}{2\\sqrt{k}} \\sqrt{k\\tau} \\frac{\\partial H^{(1)}_{\\nu_t}\\left( k\\tau \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\n\t\\end{aligned}\n\t\t\\label{new-vac2TBD}\n\\end{equation}\nwhere $e_{ij}$ denotes the polarization tensor. \nIn the next section, we will compute the inflationary power spectra for scalar and tensor modes.\n\n\\section{Inflationary power spectra and predictions for hemispherical asymmetry}\n\nAs we learned in the previous section, inflationary quantum fluctuations now come in pair which exit the horizon on two opposite sides. The two point correlations of these fluctuations can be computed as \n\\begin{equation}\n\t\\begin{aligned}\n\t\t&\t{}_{\\rm qdS_{I}}\\langle 0 \\vert \\hat{v}_{\\rm I}\\left( \\textbf{x},\\, \\tau \\right)  \\hat{v}_{\\rm I}\\left( \\textbf{x}+\\boldsymbol{\\xi}, \\, \\tau \\right)\\vert 0\\rangle_{\\rm qdS_{I}}  = \\\\ & \\frac{4\\pi}{\\left( 2\\pi \\right)^3} \\int \\frac{dk}{k} \\frac{\\sin k \\xi}{k\\xi} k^3 \\vert v_{\\rm I,\\,k}\\vert^2\\, \\\\ \n\t\t&\t{}_{\\rm qdS_{II}}\\langle 0 \\vert \\hat{v}_{\\rm II}\\left( -\\textbf{x},\\, -\\tau \\right)  \\hat{v}_{\\rm II}\\left( -\\textbf{x}-\\boldsymbol{\\xi}, \\, -\\tau \\right)\\vert 0\\rangle_{\\rm qdS_{II}}  = \\\\ &\\frac{4\\pi}{\\left( 2\\pi \\right)^3} \\int  \\frac{dk}{k} \\frac{\\sin k \\xi}{k\\xi} k^3 \\vert v_{\\rm II,\\,k}\\vert^2\\,. \n\t\\end{aligned}\n\\label{power-spectrav}\n\\end{equation}\nSubstituting the expressions \\eqref{new-vac1BD} and \\eqref{new-vac2TBD} in the above expressions we learn that the above two correlations are not equal. In the case of exact dS we obtained the correlations to be equal (see \\eqref{eqcorr}) but in the case of quasi-dS we obtained an asymmetry because the background spacetime is not $\\mathcal{P}\\mathcal{T}$ symmetric. \nAs we know that the curvature perturbation is frozen on super-horizon scales. \nTo calculate the two point correlations of curvature perturbation on super-horizon scales we re scale the canonical fields with the classical background quantities \\eqref{rescaleeq}. This implies \n\\begin{equation}\n\t\\begin{aligned}\n{}_{\\rm qdS}\\langle 0 \\vert \\zeta_\\textbf{k} \\zeta_{\\textbf{k}^\\prime} \\vert 0\\rangle_{\\rm qdS} & =  \\left( \\frac{1}{2a^2\\epsilon}\\right)\\Bigg\\vert_{\\rm classical}  \\Big[\\frac{1}{2}{}_{\\rm qdS_I}\\langle 0 \\vert \\hat{v}_{\\rm I\\,\\textbf{k}} \\hat{v}_{\\rm I\\,\\textbf{k}^\\prime} \\vert 0\\rangle_{\\rm qdS_I}\\\\ & \\quad +\\frac{1}{2}{}_{\\rm qdS_{II}}\\langle 0 \\vert \\hat{v}_{\\rm {II}\\,\\textbf{k}} \\hat{v}_{\\rm {II}\\,\\textbf{k}^\\prime} \\vert 0\\rangle_{\\rm qdS_{II}}\\Big] \\\\ \n& = \\frac{2\\pi^2}{k^3}\\left( P_{\\zeta_1}+P_{\\zeta_2} \\right) \\delta\\left( \\textbf{k}+\\textbf{k}^\\prime \\right)\\,, \n\\end{aligned}\n\\end{equation}\nwhere $\\zeta_1,\\,\\zeta_2$ are the curvature perturbations that exit the horizon on two opposite directions and they become frozen on super-horizon scales which follows from \\eqref{power-spectrav}. With appropriate normalization we evaluate the power spectrum of these curvature perturbations at the moment of horizon exit. Computing the two power spectra at the horizon exit we obtain\n\\begin{equation}\n\t\\begin{aligned}\n\tP_{\\zeta1} & = \\frac{k^3}{2\\pi^2}\\frac{1}{2a^2\\epsilon} \\vert v_{\\rm I,\\,k}\\vert^2\\Bigg\\vert_{\\tau = -\\frac{1}{aH}} \\\\ \n\t& \\approx \\frac{H^2}{8\\pi^2\\epsilon}  \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right) \\\\ &\\quad + \\frac{H^2}{4\\pi\\epsilon} \\left( \\epsilon+\\frac{\\eta}{2} \\right) \\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_s}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\\, \\\\\n\t\tP_{\\zeta2} & = \\frac{k^3}{2\\pi^2}\\frac{1}{2a^2\\epsilon} \\vert v_{\\rm II,\\,k}\\vert^2\\Bigg\\vert_{\\tau_\\ast = \\frac{1}{aH}} \\\\ \n\t& \\approx \\frac{H^2}{8\\pi^2\\epsilon}  \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right) \\\\ & - \\frac{H^2}{4\\pi\\epsilon} \\left( \\epsilon+\\frac{\\eta}{2} \\right) \\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_s}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_s}\\Big\\vert_{\\nu_s=3\/2}\\,. \n\t\\end{aligned}\n\\label{pw12}\n\\end{equation}\nIf we combine the above two expressions we obtain \n\\begin{equation}\n\tP_\\zeta = \\frac{1}{2}P_{\\zeta1}+ \\frac{1}{2} P_{\\zeta2}  = \\frac{H^2}{8\\pi^2\\epsilon}\\Bigg\\vert_{k=aH}\n\\end{equation}\nwhich is the expression in the context of standard inflation (see Sec.~\\ref{sec:review}).  From \\eqref{power-spectrav} we can deduce that the power spectrum $P_{\\zeta 1}$ can be mapped to the two-point correlations in the direction $\\hat{\\textbf{n}}$ and $P_{\\zeta 2}$ can be mapped to the two-point correlations in the opposite direction $-\\hat{\\textbf{n}}$ of the CMB sky. The difference between the two power spectra gives us the non-zero scale dependent contribution. Our formalism of producing inflationary correlations in pair naturally explains the hemispherical asymmetry and this can also be interpreted  as an indication of spontaneous breaking of $\\mathcal{P}\\mathcal{T}$ symmetry in the inflationary background. In our context we can define the \nthe amplitude of dipolar modulation as (according to Eq.~\\eqref{Akg})\n\\begin{equation}\n\tA(k) = \\frac{P_{\\zeta1}-P_{\\zeta2}}{4P_\\zeta}\n\t\\label{Ako}\n\\end{equation}\nFrom \\eqref{pw12} we can notice that the two power spectra differ only by a small scale dependent correction of the order of slow-roll parameters. In addition, through CMB we can only probe very limited range of $k$ corresponding to initial $7-8$ e-foldings centered around the pivot scale \\cite{Martin:2004um}. \n At the leading order the first terms in the two power spectra dominate which gives us the tilt of the two power spectra nearly the same in the leading order in slow-roll approximation \n\\begin{equation}\n\t\\frac{d\\ln P_{\\zeta 1}}{d\\ln k}\\approx \t\\frac{d\\ln P_{\\zeta 2}}{d\\ln k} \\approx n_s-1 \\approx -2\\epsilon-\\eta\\,.\n\t\\label{tilts2}\n\\end{equation}\nThe above result is in-line with the data from the Planck satellite \\cite{Mukherjee:2015mma,Axelsson:2013mva}. \n\nIn Fig.~\\ref{fig:fig1} we depict the power asymmetry amplitude of the scalar power spectra for $n_s = 0.963$. This plot remains the same for any single-field slow-roll inflation.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{Fig1}\n\t\\caption{In the above plot we depict the amplitude of CMB dipolar modulation or hemispherical asymmetry of scalar power spectra obtained from \\eqref{Ako}. In the plot we choose the pivot scale $k_\\ast=a_\\ast H_\\ast=0.05 {\\rm Mpc}^{-1}$ and fix $n_s=0.963$. The blue dot with error bar in the plot corresponds to the observational constraint $\\vert A \\vert = 0.066\\pm 0.021$ on HPA at large angular scales or at low-$\\ell-2-64$ or $k\\lesssim 10^{-1}k_\\ast$. We can notice from the plot that the HPA ceases to exist for the small angular scales or high-$\\ell$ which is compatible with the Planck satellite observations \\cite{Planck:2013lks,Akrami:2014eta,Aiola:2015rqa}.  }\n\t\\label{fig:fig1}\n\\end{figure}\nSimilar to the scalar perturbations we propose the tensor perturbations to be direct sum of two kinds of fluctuations related by our definition of $\\mathcal{P}\\mathcal{T}$ transformation. \n\\begin{equation}\n\t\\hat{u}_{ij}\\left( \\tau,\\, \\textbf{x} \\right) = \\frac{1}{\\sqrt{2}}\t\\hat{u}_{\\rm I,\\, ij}\\left( \\tau,\\, \\textbf{x} \\right) \\oplus \\frac{1}{\\sqrt{2}} \\hat{u}_{\\rm II,\\, ij}\\left( -\\tau,\\, -\\textbf{x} \\right)\\,. \n\\end{equation}\nSimilar to the scalar power spectra we obtain two tensor power spectra which describe two point tensor correlations in the direction $\\hat{\\textbf{n}}$ and $-\\hat{\\textbf{n}}$ respectively. The two power spectra of tensor correlations are computed as \n\\begin{equation}\n\t\\begin{aligned}\n\t\tP_{h1} & = \\frac{k^3}{2\\pi^2}\\frac{4}{a^2} \\vert u_{\\rm I\\,k}\\vert^2\\Bigg\\vert_{\\tau = -\\frac{1}{aH}} \\\\ \n\t\t& \\approx \\frac{2H^2}{\\pi^2} \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right) \\\\ &\\quad + \\frac{H^2}{\\pi}\\epsilon\\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_t}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\\, \\\\\n\t\tP_{h2} & = \\frac{k^3}{2\\pi^2}\\frac{4}{a^2} \\vert u_{\\rm II,\\,k}\\vert^2\\Bigg\\vert_{\\tau_\\ast = \\frac{1}{aH}} \\\\ \n\t\t& \\approx \\frac{2H^2}{\\pi^2} \\left( 1+\\left( \\frac{k}{aH} \\right)^2 \\right)\\\\ &\\quad - \\frac{H^2}{\\pi} \\epsilon \\left(\\frac{k}{aH}\\right)^3 H_{3\/2}^{(1)} \\left( \\frac{k}{aH} \\right) \\frac{\\partial H^{(1)}_{\\nu_t}\\left( \\frac{k}{aH} \\right)}{\\partial \\nu_t}\\Big\\vert_{\\nu_t=3\/2}\\,. \n\t\\end{aligned}\n\t\\label{pwt12}\n\\end{equation}\nThe hemispherical power asymmetry of the tensor-power spectrum can be defined as \n\\begin{equation}\n\tT(k) = \\frac{P_{h1}-P_{h2}}{4P_h}\n\t\\label{Tko}\n\\end{equation}\nTo quantify the above amplitude $T(k)$ we choose Starobinsky or Higgs inflationary potential in Einstein frame \n\\begin{equation}\n\tV(\\phi) = \\frac{\\lambda}{4} \\left( 1- e^{-\\sqrt{\\frac{2}{3}}\\phi} \\right)^2\\,.\n\t\\label{staro-in}\n\\end{equation}\nwhich gives the slow-roll parameters in terms of e-folds as \\cite{Kehagias:2013mya}\n\\begin{equation}\n\t\\epsilon = \\frac{3}{4N^2},\\quad \\eta= \\frac{2}{N}\n\\end{equation}\nIn Fig.~\\ref{fig:fig2} we depict HPA amplitude of the tensor power spectra for the case of Starobinsky potential \\eqref{staro-in}.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{Fig2}\n\t\\caption{In the above plot we depict the amplitude of dipolar modulation for primordial gravitational waves (PGWs) or HPA of tensor power spectra obtained from \\eqref{Tko}. The above plot is for the case of Starobinsky or Higgs inflation \\eqref{staro-in} for $N=55$ number of e-folds corresponding to the pivot scale $k_\\ast =0.05 {\\rm Mpc}^{-1}$.}\n\t\\label{fig:fig2}\n\\end{figure}\nSimilar the case of the tilt of scalar power spectra \\eqref{tilts2} we obtain the tilt of the tensor power spectra almost the same in the two opposite directions of the sky\n\\begin{equation}\n\t\\frac{d\\ln P_{h 1}}{d\\ln k}\\approx \t\\frac{d\\ln P_{h 2}}{d\\ln k} \\approx n_t \\approx -2\\epsilon\\,.\n\t\\label{tilts3}\n\\end{equation}\nFrom Fig.~\\ref{fig:fig1} and Fig.~\\ref{fig:fig2} we can notice that the HPA decreases as we increase the wave-number. This is a physically expected behavior because the short wavelength modes do not see the curvature of spacetime therefore the asymmetry should decrease with increasing $k$. In other words we also see it as how $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ symmetry is broken in a curved spacetime as a function of length scales.  \n\n\n\\section{Conclusions}\n\nThe success of inflationary cosmology is its mapping of quantum fluctuations to the temperature fluctuations in the CMB. The HPA seems to pose a significant challenge to the standard understanding of inflationary quantum fluctuations because there is an asymmetry in the temperature correlations in the northern and southern hemispheres of the CMB sky. In this paper, we investigated whether the power asymmetry in the CMB can be originated from re-formulating our understanding of the quantization of inflationary fluctuations. Inspired from several open questions in the context of QFT in curved spacetime and quantum cosmology we proposed that the inflationary quantum fluctuations are generated in pairs which are related by discrete transformations of spacetime. Our approach is inspired from the fact that QFT in Minkowski spacetime is built on discrete symmetries such as $\\mathcal{C}\\mathcal{P}\\mathcal{T}$. In the case of curved spacetime i.e., \nwhen it involves gravity one must be careful in understanding spacetime reflection symmetries because gravity introduces dynamics. In our framework, we divide the notions of time into classical and quantum mechanical aspects. In the context of inflationary cosmology the standard procedure is to find a background geometry and quantize gravitational and matter degrees of freedom respecting the classical arrow of time \\cite{Martin:2004um}. However, we develop a new framework of quantization in canonical single field inflation where at the quantum level we allow the existence of quantum states that propagate both forward and backward in time based on the $\\mathcal{P}\\mathcal{T}$ transformations (since we only deal with chargeless fields we omit $\\mathcal{C}$ transformation) of the spacetime. Both of these quantum states in our framework co-exist within the co-moving horizon and they exit the horizon in the two opposite directions. We explicitly showed that these two type of quantum states even though created non-locally still do not violate any causality and locality principles in our formulation of QFT.\nWe showed that these pairs of quantum fluctuations behave exactly the same in the case of exact dS spacetime, but they are slightly different in the case of inflationary spacetime, which can be interpreted as the spontaneous breaking of $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ symmetry. This slight deviation seems to produce the power asymmetry hinted in the CMB observations. Our quantization scheme also applied to the case of inflationary tensor power spectrum and we predict a power asymmetry there as well. Notably, the HPA for scalar and tensor power spectrum is significant only for large angular scales or small wave numbers and it decreases for small angular scales or large wave numbers. This is expected since the spontaneous breaking of $(\\mathcal{C})\\mathcal{P}\\mathcal{T}$ symmetry is significant for small $k$, which are more affected by the background curved spacetime, compared to the large $k$.\nWe quantified all our predictions which serves as an observational test of our formalism. If the future observations \\cite{Li:2019bsg,Zhai:2017ibd} confirm these results we will learn significantly about the nature of inflationary quantum fluctuations as well as QFT in curved spacetime. \n \n\n\n\n\n\\begin{acknowledgments}\nKSK acknowledges the support from JSPS and KAKENHI Grant-in-Aid for Scientific Research No. JP20F20320, and thank Mainz U. for hospitality where part of the work has been carried out. J. Marto is supported by the grant UIDB\/MAT\/00212\/2020. We would like to thank Chris Ripken for the useful discussions and suggestions on quantization procedure and for the initial collaboration on the project.  We thank Gerard 't Hooft for his inspiring talks and discussions related to QFT in curved spacetime. We thank Yashar Akrami, Norma G. Sanchez, Masahide Yamaguchi, Alexei A. Starobinsky, Luca Buoninfante, Francesco Di Fillippo, Yasha Neiman, Paolo Gondolo, Martin Reuter, Eiichiro Komatsu, Paulo V. Moniz, Dhiraz Kumar Hazra and L. Sriramkumar for very useful discussions.\n\\end{acknowledgments}\n\n\n\n\n\n\\bibliographystyle{apsrev4-2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThis note reports the Tevatron average top-quark mass obtained by\ncombining the most precise published and preliminary measurements of\nthe top-quark mass. It is an update of the combination presented in\nRef.~\\cite{TeVTopComboPRD}, where further details can be found.\nThe ATLAS and CMS collaborations have also performed a combination of their most recent\ntop quark mass measurements~\\cite{lhccombi}.\n\nThe CDF and D\\O\\ collaborations have performed several direct experimental measurements of the\ntop-quark mass (\\ensuremath{M_{\\mathrm{t}}}) using data collected at the Tevatron\nproton-antiproton collider located at the Fermi National Accelerator\nLaboratory.  These pioneering measurements were first based on approximately\n$0.1~\\ensuremath{\\mathrm{fb}^{-1}}$ of \\hbox{Run\\,I}\\ data\n~\\cite{Mtop1-CDF-di-l-PRLa}-\\cite{Mtop1-D0-allh-PRL}\ncollected from 1992 to 1996,\nand included results from the decay channels \\ensuremath{\\ttbar\\ra\\WbWb\\ra\\had}\\ (alljets),\n\\ensuremath{\\ttbar\\ra\\WbWb\\ra\\ljt}\\ ($\\ell$+jets), and \n\\ensuremath{\\ttbar\\ra\\WbWb\\ra\\dil}\\ ($\\ell\\ell$), where $\\ell=e$ or $\\mu$.  \nDecays with $\\tau \\to e, \\mu$ are included in the direct $W \\to e$ and\n$W \\to \\mu$ channels. \nIn \\hbox{Run\\,II}\\ (2001--2011), many top mass measurements have been performed, and\nthose considered here are the most\nrecent results in these channels, using up to $8.7~\\ensuremath{\\mathrm{fb}^{-1}}$ of data for CDF\n(corresponding to the full CDF Run II dataset)\n~\\cite{\nMtop2-CDF-MEt-new,\nMtop2-CDF-l+jt-pub-new,\nMtop2-CDF-di-l-pub,\nMtop2-CDF-allh-pub},\nand up to $5.4~\\ensuremath{\\mathrm{fb}^{-1}}$ of data for D\\O\\\n~\\cite{ \nMtop2-D0-l+ja-final,\nMtop2-D0-l+jt-new2,\nMtop2-D0-di-l}.\nThe CDF analysis based upon charged particle tracking for exploiting\nthe transverse decay length \nof $b$-tagged jets ($L_{XY}$) and the transverse momentum of\nelectrons and muons from $W$ boson decays ($\\pte$) uses\na data set corresponding to a luminosity of\n1.9~$\\ensuremath{\\mathrm{fb}^{-1}}$~\\cite{Mtop2-CDF-trk}, and\nthere are no plans to update this analysis.\nThe D\\O\\ \\hbox{Run\\,II}\\  measurements presented in this note include the most\nrecent Run II measurement in  \nthe $\\ell\\ell$~\\cite{Mtop2-D0-di-l}  channel using 5.4~fb$^{-1}$ of data \nand in the $\\ell$+jets channel \\cite{Mtop2-D0-l+jt-new2} \nwith 3.6~fb$^{-1}$ of data. Both results are now published.\nSince the combination performed in 2011~\\cite{Mtop-tevewwgSum11}, a new\nfinal state signature was introduced by CDF that requires events to\npossess missing transverse energy ($\\met$) and jets, but no identified\nlepton (``MEt'')~\\cite{Mtop2-CDF-MEt-new,Mtop2-CDF-MEt-published}.  \nThis sample is statistically independent from the previous three CDF\nchannels.\n\nWith respect to the July 2011 combination\\,\\cite{Mtop-tevewwgSum11} and the published version of the \ncombination~\\cite{TeVTopComboPRD}, the \\hbox{Run\\,II}\\ CDF measurement in the \n$\\ell$+jets channel has been updated using 8.7~fb$^{-1}$ \nof data, an improved analysis technique, and improved jet energy resolution \\cite{Mtop2-CDF-l+jt-pub-new}. \nThe CDF measurement in the MEt channel was updated to use the full\n\\hbox{Run\\,II}\\ data set for CDF of  8.7~fb$^{-1}$ of data as \nwell~\\cite{Mtop2-CDF-MEt-new}.\nThe now published \\hbox{Run\\,II}\\ CDF measurements in the $\\ell\\ell$ channel\\,\\cite{Mtop2-CDF-di-l-pub} and alljets \nchannel\\,\\cite{Mtop2-CDF-allh-pub}  are unchanged. The measurement \nbased on charged particle tracking~\\cite{Mtop2-CDF-trk} was\nincorporated as described in the past \ncombinations~\\cite{Mtop-tevewwgSum11}. \nFrom the corresponding analysis only\nthe measurement of the top quark mass using the mean decay length $\\Lxy$ of\n$B$ hadrons in $b$-tagged lepton+jets events has been used. It\nis independent of energy information in the calorimeter, and its main\nsource of systematic uncertainty is uncorrelated with the dominant\nones from the jet energy scale calibration in other measurements. This\nmeasurement of $m_t$ is essentially uncorrelated with the higher\nprecision CDF result from the lepton+jets channel. The overlap between\nthe data samples used for the decay-length method and the lepton+jets\nsample has therefore no effect. \n\nThe Tevatron average top-quark mass is obtained by combining five published\n\\hbox{Run\\,I}\\ measurements~\\cite{Mtop1-CDF-di-l-PRLb, Mtop1-CDF-di-l-PRLb-E,\n  Mtop1-D0-di-l-PRD, Mtop1-CDF-l+jt-PRD, Mtop1-D0-l+jt-new1,\n  Mtop1-CDF-allh-PRL} with four published \\hbox{Run\\,II}\\ CDF \nresults~\\cite{Mtop2-CDF-l+jt-pub-new,Mtop2-CDF-di-l-pub,Mtop2-CDF-allh-pub, \nMtop2-CDF-trk}, one preliminary \\hbox{Run\\,II}\\ CDF\nresult~\\cite{Mtop2-CDF-MEt-new}, and two published \n\\hbox{Run\\,II}\\ D\\O\\ results~\\cite{Mtop2-D0-l+jt-new2,\nMtop2-D0-di-l}.\nThis combination \nsupersedes previous\ncombinations~\\cite{Mtop-tevewwgSum11,Mtop1-tevewwg04,Mtop-tevewwgSum05,\n  Mtop-tevewwgWin06,Mtop-tevewwgSum06, Mtop-tevewwgWin07, Mtop-tevewwgWin08, \n  Mtop-tevewwgSum08, Mtop-tevewwgWin09,Mtop-tevewwgSum10}. \n\nThe definition and evaluation of the systematic uncertainties and the understanding of the\ncorrelations among channels, experiments, and Tevatron runs is the outcome of many years of \njoint work between the CDF and D\\O\\ collaborations and is described in detail\nelsewhere~\\cite{TeVTopComboPRD}.\n\nThe input measurements and uncertainty categories used in the combination are \ndetailed in Sections~\\ref{sec:inputs} and~\\ref{sec:uncertainty}, respectively. \nThe correlations assumed in the combination are discussed in \nSection~\\ref{sec:corltns} and the resulting Tevatron average top-quark mass \nis given in Section~\\ref{sec:results}.  A summary is presented\nin Section~\\ref{sec:summary}.\n \n\\section{Input Measurements}\n\\label{sec:inputs}\n\nTwelve measurements of \\ensuremath{M_{\\mathrm{t}}}\\ used in this combination are shown in Table~\\ref{tab:inputs}.\nThe \\hbox{Run\\,I}\\ measurements all have relatively large statistical\nuncertainties and their systematic uncertainties are dominated by the\ntotal jet energy scale (JES) uncertainty.  In \\hbox{Run\\,II}\\, both CDF and\nD\\O\\ take advantage of the larger \\ensuremath{t\\overline{t}}\\ samples available and employ\nnew analysis techniques to reduce both of these uncertainties.  In\nparticular, the \\hbox{Run\\,II}\\ D\\O\\ analysis in the $\\ell$+jets channel and the \n\\hbox{Run\\,II}\\ CDF analyses in the $\\ell$+jets, alljets, and MEt channels \nconstrain the response of light-quark jets using the kinematic information from $W\\ra\nqq^{\\prime}$ decays (so-called {\\em in situ}\ncalibration)~\\cite{Mtop1-CDF-l+jt-PRD,Abazov:2006bd}. Residual JES uncertainties associated with\n$p_{T}$ and $\\eta$ dependencies as well as uncertainties specific to\nthe response of $b$ jets are treated separately. The\n\\hbox{Run\\,II}\\ D\\O\\ $\\ell\\ell$ measurement uses the JES determined in the\n$\\ell$+jets channel by {\\em in situ} calibration~\\cite{Mtop2-D0-di-l}.\n\n\\vspace*{0.10in}\n\n\\begin{table}[t]\n\\caption[Input measurements]{Summary of the measurements used to determine the\n  Tevatron average \\ensuremath{M_{\\mathrm{t}}}.  Integrated luminosity ($\\int \\mathcal{L}\\;dt$) has units of\n  \\ensuremath{\\mathrm{fb}^{-1}}, and all other numbers are in $\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$.  The uncertainty categories and \n  their correlations are described in Section~\\ref{sec:uncertainty}.  The total systematic uncertainty \n  and the total uncertainty are obtained by adding the relevant contributions \n  in quadrature. ``n\/a'' stands for ``not applicable'', ``n\/e'' for ``not evaluated''.}\n\\label{tab:inputs}\n\\begin{center}\n\n\\renewcommand{\\arraystretch}{1.30}\n{\\tiny\n\\begin{tabular}{l|ccc|cc|cccc|cc|c} \n\\hline \\hline\n       & \\multicolumn{5}{c|}{{\\hbox{Run\\,I}} published} \n       & \\multicolumn{6}{c|}{{\\hbox{Run\\,II}} published} \n       & \\multicolumn{1}{c}{{\\hbox{Run\\,II}} prel.}  \\\\ \n      \n       & \\multicolumn{3}{c|}{ CDF } \n       & \\multicolumn{2}{c}{ D\\O\\ }\n       & \\multicolumn{4}{|c|}{ CDF }\n       & \\multicolumn{2}{c|}{ D\\O\\ }\n       & \\multicolumn{1}{c}{ CDF }\n        \\\\\n\n                      &     $\\ell$+jets   & $\\ell\\ell$     &   alljets  &     $\\ell$+jets &   $\\ell\\ell$  &    $\\ell$+jets     &     $\\ell\\ell$  &    alljets  &     Lxy     &   $\\ell$+jets      &   $\\ell\\ell$    &      MEt \\\\\n\\hline                                                                                                                                                       \n$\\int \\mathcal{L}\\;dt$&       0.1  &     0.1  &    0.1  &     0.1  &     0.1 &      8.7    &       5.6 &     5.8  &         1.9 &         3.6 &       5.3 &      8.7   \\\\\n\\hline                                                                                                                                                       \nResult                &   176.1    & 167.4    &186.0    & 180.1    & 168.4   &      172.85 &    170.28 &  172.47  &      166.90 &      174.94 &    174.00 &    173.95\\\\\n\\hline                                                                                                                                                       \n\\shortstack{ {\\em In situ} light-jet cali- \\\\\nbration (iJES)}                   &        n\/a &      n\/a & n\/a     &      n\/a &      n\/a &       0.49 &      n\/a  &     0.95 &         n\/a &        0.53 &      0.55  &      1.05  \\\\\n\\shortstack{ Response to $b$\/$q$\/$g$ \\\\\njets (aJES)}                          &        n\/a &      n\/a & n\/a     &      0.0 &      0.0 &       0.09 &      0.14 &     0.03 &         n\/a &        0.0  &      0.40 &      0.10 \\\\\n\\shortstack{ Model for $b$ jets  \\\\\n(bJES)}                               &        0.6 &      0.8 & 0.6     &      0.7 &      0.7 &       0.16 &      0.33 &     0.15 &         n\/a &        0.07 &      0.20 &      0.17  \\\\\n \\shortstack{ Out-of-cone correction \\\\\n     (cJES)}                          &        2.7 &      2.6 & 3.0     &      2.0 &      2.0 &       0.21 &      2.13 &     0.24 &        0.36 &        n\/a  &      n\/a  &      0.18  \\\\\n\\shortstack{ Light-jet response (2) \\\\ \n    (dJES)}                               &        0.7 &      0.6 & 0.3     &      2.5 &      1.1 &       0.07 &      0.58 &     0.04 &        0.06 &        0.63 &      0.56 &      0.04  \\\\\n\\shortstack{ Light-jet response (1) \\\\\n (rJES)}                               &        3.4 &      2.7 & 4.0     &      n\/a &      n\/a &       0.48 &      2.01 &     0.38 &        0.24 &        n\/a  &      n\/a  &      0.40  \\\\\n\\shortstack{ Lepton modeling  \\\\\n(LepPt)}                              &        n\/e &      n\/e & n\/e     &      n\/e &      n\/e &       0.03 &      0.27 &     n\/a    &        n\/a  &        0.17 &      0.35 &      n\/a   \\\\\n\\shortstack{ Signal modeling \\\\\n (Signal)}                             &        2.6 &      2.9 & 2.0     &      1.1 &      1.8 &       0.61 &      0.73 &     0.62 &        0.90 &        0.77 &      0.86 &      0.64\\\\\n \\shortstack{ Jet modeling \\\\\n  (DetMod)}                             &        0.0 &      0.0 & 0.0     &      0.0 &      0.0 &       0.0  &      0.0  &     0.0  &        0.0  &        0.36 &      0.50 &      0.0 \\\\\n\\shortstack{ Offset \\hspace{0.5cm} \\\\\n (UN\/MI)}                              &        n\/a &      n\/a &    n\/a  &      1.3 &      1.3 &       n\/a  &      n\/a  &     n\/a  &        n\/a  &        n\/a  &      n\/a  &      n\/a  \\\\\n\\shortstack{ Background from \\\\\ntheory (BGMC)}                        &        1.3 &      0.3 &    1.7  &      1.0 &      1.1 &       0.12 &      0.24 &     0.0  &        0.80 &       0.18  &      0.0  &      0.0   \\\\\n \\shortstack{ Background based on\\\\\ndata (BGData)}                     &        0.0 &      0.0 &    0.0  &      0.0 &      0.0 &      0.16  &      0.14 &     0.56 &        0.20 &       0.23  &      0.20 &      0.12  \\\\\n \\shortstack{ Calibration method\\\\\n(Method)}                      &        0.0 &      0.7 & 0.6     &      0.6 &      1.1 &       0.00 &      0.12 &     0.38&         2.50 &        0.16 &      0.51 &      0.31  \\\\\n \\shortstack{ Multiple interactions\\\\\n model (MHI)}                          &        n\/e &      n\/e &    n\/e  &      n\/e &      n\/e &       0.07 &      0.23 &     0.08&         0.0  &        0.05 &      0.0  &      0.18  \\\\\n\\hline                                                                                                                                                       \n \\shortstack{ Systematic uncertainty\\\\\n (Syst)}                   &        5.3 &      4.9 & 5.7    &       3.9 &      3.6 &       0.98 &      3.09 &     1.49 &        2.90 &        1.24 &      1.44 &      1.35 \\\\\n\\shortstack{ Statistical uncertainty\\\\\n (Stat)}                   &        5.1 &     10.3 &10.0    &       3.6 &     12.3 &       0.52 &      1.95 &     1.43 &        9.00 &        0.83 &      2.36 &      1.26 \\\\\n\\hline                                                                                                                                                       \nTotal uncertainty                     &        7.3 &     11.4 &11.5    &        5.3 &    12.8 &       1.11 &      3.79 &     2.06 &        9.46 &       1.50 &      2.76 &      1.85\\\\ \n\\hline\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\nThe D\\O\\ Run~II $\\ell$+jets analysis uses the JES determined from the\nexternal calibration derived from $\\gamma$+jets events as an\nadditional Gaussian constraint to the {\\em in situ} calibration. Therefore,\nthe total resulting JES uncertainty is split into one part obtained from the \n{\\em in situ} calibration and another part determined from the external calibration.\nTo do this, the measurement without external\nJES constraint has been combined iteratively with a pseudo-measurement\nusing the method of Refs.~\\cite{Lyons:1988, Valassi:2003} \nthat uses only the external calibration in a way that the combination gives the total JES uncertainty. \nThe splitting obtained in this way is used to assess both the statistical part of the JES uncertainty and the part of \nthe JES uncertainty due to the external calibration constraint~\\cite{Mtop2-D0-comb}.\n\nThe $\\Lxy$ technique developed by CDF \nuses the decay length of $B$ mesons from $b$-tagged jets.\nWhile the statistical sensitivity of this analysis is not as good as\nthat of the more\ntraditional methods, this technique has the advantage that it is almost entirely independent of\nJES uncertainties since it\nuses primarily tracking information.\n\\vspace*{0.10in}\n\nThe D\\O\\ \\hbox{Run\\,II}\\ $\\ell$+jets result is a combination of the \npublished Run~IIa (2002--2005) measurement ~\\cite{Mtop2-D0-l+ja-final} with 1~fb$^{-1}$ \nof data and the result obtained with 2.6~fb$^{-1}$ of data from Run~IIb (2006--2007)~\\cite{Mtop2-D0-l+jt-new2}.\nThis analysis includes an additional particle response correction on top of the\nstandard {\\it in situ} calibration.\nThe D\\O\\ \\hbox{Run\\,II}\\ $\\ell\\ell$ result is based on a neutrino weighting technique using 5.4~fb$^{-1}$\nof \\hbox{Run\\,II}\\  data~\\cite{Mtop2-D0-di-l}. \n\\vspace*{0.10in}\n\nTable~\\ref{tab:inputs} lists the individual uncertainties of each result,\nsubdivided into the categories described in the next Section.  The\ncorrelations between the inputs are described in\nSection~\\ref{sec:corltns}.\n\n\n\n\\section{Uncertainty Categories}\n\\label{sec:uncertainty}\n\nWe employ uncertainty categories similar to what was used for the previous Tevatron\naverage~\\cite{TeVTopComboPRD,Mtop-tevewwgSum11}, with small\nmodifications to better account for their correlations.\nThey are divided such that sources of systematic uncertainty that share the same or similar origin are \ncombined as explained in Ref.~\\cite{TeVTopComboPRD}. \nFor example, the {\\it Signal modeling} ({\\it Signal}) category\ndiscussed below includes the uncertainties from  different systematic\nsources that are correlated due to their origin in the modeling of the simulated signal samples.\n\n Some systematic uncertainties have been separated into multiple\ncategories to accommodate specific types of correlations.\nFor example, the jet energy scale (JES) uncertainty is subdivided\ninto six components to more accurately accommodate our\nbest understanding of the relevant correlations between input measurements. \n\nFor this note we use the new systematic naming scheme described in Ref.~\\cite{TeVTopComboPRD}. In parentheses, the \nold names of the  systematic uncertainties are provided.  There is a one-to-one matching between the new and old systematic \ndefinitions of categories.\n\n\\vspace*{0.10in}\n\n\\begin{description}\n  \\item[Statistical uncertainty (Statistics):] The statistical uncertainty associated with the\n    \\ensuremath{M_{\\mathrm{t}}}\\ determination.\n \\item[{\\em In situ} light-jet calibration (iJES):] That part of the\n   JES uncertainty that originates from\n   {\\em in situ} calibration procedures and is uncorrelated among the\n   measurements.  In the combination reported here, it corresponds to\n   the statistical uncertainty associated with the JES determination\n   using the $W\\ra qq^{\\prime}$ invariant mass in the CDF \\hbox{Run\\,II}\\\n   $\\ell$+jets, alljets, and  MEt measurements and the D\\O\\ Run~II\n   $\\ell\\ell$ and $\\ell$+jets\n   measurements. \n   For the D\\O\\ Run~II $\\ell$+jets measurement, it also includes the uncertainty\n   coming from the MC\/data difference in jet response that is uncorrelated\n   with the other D\\O\\ Run~II measurements. \n   Residual JES uncertainties arising from effects\n   not considered in the {\\em in situ} calibration are included in other\n   categories. \n  \\item[Response to \\boldmath{$b\/q\/g$} jets (aJES):] That part of the JES\n    uncertainty that originates from \n    average differences in detector electromagnetic over hadronic ($e\/h$)\n    response for hadrons produced in the fragmentation of $b$-jets and light-quark \n    jets. \n  \\item[Model for \\boldmath{$b$} jets (bJES):] That part of the JES uncertainty that originates from\n    uncertainties specific to the modeling of $b$ jets and that is correlated\n    across all measurements.  For both CDF and D\\O\\ this includes uncertainties \n    arising from \n    variations in the semileptonic branching fractions, $b$-fragmentation \n    modeling, and differences in the color flow between $b$-quark jets and light-quark\n    jets.  These were determined from \\hbox{Run\\,II}\\ studies but back-propagated\n    to the \\hbox{Run\\,I}\\ measurements, whose {\\it Light-jet response (1)}  uncertainties ({\\it rJES}, see below) were \n    then corrected to keep the total JES uncertainty constant.\n  \\item[Out-of-cone correction (cJES):] That part of the JES uncertainty that originates from\n    modeling uncertainties correlated across all measurements.  \n    It specifically includes the modeling uncertainties associated with light-quark \n    fragmentation and out-of-cone corrections. For D\\O\\ \\hbox{Run\\,II}\\ measurements,\n    it is included in the {\\it Light-jet response (2) (dJES)} category.\n  \\item[Light-jet response (1) (rJES):] The remaining part of the JES\n    uncertainty that covers the absolute calibration for CDF's \\hbox{Run\\,I}\\\n    and \\hbox{Run\\,II}\\ measurements. It also includes small contributions\n    from the uncertainties associated with modeling multiple\n    interactions within a single bunch crossing and corrections for\n    the underlying event. \n   \n   \n   \n   \n   \n   \n  \\item[Light-jet response (2) (dJES):] That part of the JES\n    uncertainty that includes D\\O's \\hbox{Run\\,I}\\ and \\hbox{Run\\,II}\\ calibrations of\n    absolute response (energy dependent), the relative response\n    ($\\eta$-dependent), and the out-of-cone showering correction that \n    is a detector effect. This uncertainty term for CDF includes only\n    the small relative response calibration ($\\eta$-dependent) for\n    \\hbox{Run\\,I}\\ and \\hbox{Run\\,II}. \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \\item[Lepton modeling (LepPt):] The systematic uncertainty arising from uncertainties\n    in the scale of lepton transverse momentum measurements. It was not\n    considered as a source of systematic uncertainty in the \\hbox{Run\\,I}\\\n    measurements. \n  \\item[Signal modeling (Signal):] The systematic uncertainty arising from uncertainties\n    in \\ensuremath{t\\overline{t}}\\ modeling that is correlated across all\n    measurements. This includes uncertainties from variations of the amount of initial and \n    final state radiation and from the choice of parton density function used\n    to generate the \\ensuremath{t\\overline{t}}\\ Monte Carlo samples\n    that calibrate each method. For D\\O, it also includes the uncertainty \n    from higher-order corrections evaluated from a comparison of \\ensuremath{t\\overline{t}}\\ samples generated by {\\textsc MC@NLO} ~\\cite{MCNLO} and \n    {\\textsc ALPGEN}~\\cite{ALPGEN}, both interfaced to {\\textsc HERWIG}~\\cite{HERWIG5,HERWIG6} for the simulation of parton showers and hadronization.\n    In this combination, the systematic uncertainty arising from a variation of the \n  phenomenological description of color reconnection (CR) between final state  particles \\cite{CR,Skands:2009zm} \n  is included in the {\\it Signal modeling} category.\n  The CR uncertainty is obtained by taking the difference between the {\\textsc PYTHIA}\\,6.4 tune ``Apro\" and the {\\textsc PYTHIA}\\,6.4 tune \n``ACRpro\" that differ only in the CR model. \nThis uncertainty was not evaluated in Run~I since the Monte Carlo\ngenerators available at that time did not allow for variations of the CR model.\nThese measurements therefore do not include this source of systematic uncertainty. Finally, the systematic \nuncertainty associated with variations of the MC generator used to calibrate the mass extraction method is added. \nIt includes variations observed when \n    substituting {\\textsc PYTHIA}\n    \\cite{PYTHIA4,PYTHIA5,PYTHIA6} \n    (\\hbox{Run\\,I}\\ and \\hbox{Run\\,II}) \n    or {\\textsc ISAJET}~\\cite{ISAJET} (\\hbox{Run\\,I}) for {\\textsc HERWIG}~\\cite{HERWIG5,HERWIG6} when \n    modeling the \\ensuremath{t\\overline{t}}\\ signal.  \n  \\item[Jet modeling (DetMod):] The systematic uncertainty arising from uncertainties \nin the modeling of jet interactions in the detector in the MC\nsimulation. For D\\O\\, this includes uncertainties from jet resolution\nand identification. \nApplying jet algorithms to MC events, CDF finds that the resulting\nefficiencies and resolutions closely match those in data. The small\ndifferences propagated to $\\ensuremath{M_{\\mathrm{t}}}$ lead to a negligible uncertainty of\n0.005~GeV, which is then ignored.  \n\n \\item[Background based on data (BGData):] This includes \n    uncertainties associated with the modeling using data of the QCD\n    multijet background in the alljets,\n    MEt, and $\\ell$+jets channels and\n    the Drell-Yan background in the $\\ell\\ell$ channel. \n    This part is uncorrelated between experiments.\n\n  \\item[Background from theory (BGMC):] \nThis systematic uncertainty on the background originating from theory\n(MC) takes into account the    \n uncertainty in modeling the background sources. It is correlated between\n    all measurements in the same channel, and includes uncertainties on the background composition, normalization, and shape of different components, e.g., the \nuncertainties from the modeling of the $W$+jets background in the $\\ell$+jets channel  \nassociated with variations of the factorization scale used to simulate $W$+jets events. \n\n  \\item[Calibration method (Method):] The systematic uncertainty arising from any source specific\n    to a particular fit method, including the finite Monte Carlo statistics \n    available to calibrate each method. \n  \\item[Offset (UN\/MI):] This uncertainty is specific to D\\O\\ and includes the uncertainty\n    arising from uranium noise in the D\\O\\ calorimeter and from the\n    multiple interaction corrections to the JES.  For D\\O\\ \\hbox{Run\\,I}\\ these\n    uncertainties were sizable, while for \\hbox{Run\\,II}, owing to the shorter\n    calorimeter electronics integration time and {\\em in situ} JES calibration, these uncertainties\n    are negligible.\n  \\item[Multiple interactions model (MHI):] The systematic uncertainty arising from a mismodeling of \n  the distribution of the number of collisions per Tevatron bunch crossing owing to the \n  steady increase in the collider instantaneous luminosity during data-taking. \n  This uncertainty has been separated from other sources to account for the fact that \n  it is uncorrelated between experiments.\n\n\\end{description}\nThese categories represent the current preliminary understanding of the\nvarious sources of uncertainty and their correlations.  We expect these to \nevolve as we continue to probe each method's sensitivity to the various \nsystematic sources with improving precision.\n\n\\section{Correlations}\n\\label{sec:corltns}\n\nThe following correlations are used for the combination:\n\\begin{itemize}\n  \\item The uncertainties in the {\\it Statistical uncertainty (Stat)} and\n    {\\it Calibration method (Method)}\n    categories are taken to be uncorrelated among the measurements.\n  \\item The uncertainties in the {\\it In situ light-jet  \n    calibration (iJES)}\n    category are taken to be uncorrelated among the measurements\n    except for D0's $\\ell\\ell$ and $\\ell$+jets measurements, where\n    this uncertainty is taken to be 100\\% correlated since the\n    $\\ell\\ell$ measurement uses the JES calibration  determined in\n    $\\ell$+jets channel.  \n  \\item The uncertainties in the {\\it Response to $b$\/$q$\/$g$ jets (aJES)}, {\\it Light-jet response (2) (dJES)}, {\\it Lepton modeling (LepPt)},\n    and {\\it Multiple interactions model (MHI)} categories are taken\n    to be 100\\% correlated among all \\hbox{Run\\,I}\\ and all \\hbox{Run\\,II}\\ measurements \n    within the same experiment, but uncorrelated between \\hbox{Run\\,I}\\ and \\hbox{Run\\,II}\\\n    and uncorrelated between the experiments.\n  \\item The uncertainties in the {\\it Light-jet response (1) (rJES)}, {\\it Jet modeling (DetMod)}, and {\\it Offset (UN\/MI)} categories are taken\n    to be 100\\% correlated among all measurements within the same experiment \n    but uncorrelated between the experiments.\n  \\item The uncertainties in the {\\it Backgrounds estimated from theory (BGMC)} category are taken to be\n    100\\% correlated among all measurements in the same channel.\n  \\item The uncertainties in the {\\it Backgrounds estimated from data (BGData)} category are taken to be\n    100\\% correlated among all measurements in the same channel and same run period, but uncorrelated between the experiments.\n  \\item The uncertainties in the {\\it Model for $b$ jets (bJES)}, {\\it Out-of-cone correction (cJES)}, and {\\it Signal modeling (Signal)}\n    categories are taken to be 100\\% correlated among all measurements.\n\\end{itemize}\nUsing the inputs from Table~\\ref{tab:inputs} and the correlations specified\nhere, the resulting matrix of total correlation coefficients is given in\nTable~\\ref{tab:coeff}.\n\n\\begin{table}[t]\n\\caption[Global correlations between input measurements]{The matrix of correlation coefficients used to determine the\n  Tevatron average top-quark mass.}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\tiny\n\\begin{tabular}{l|ccc|cc|cccc|cc|c}\n\\hline \\hline\n     & \\multicolumn{5}{c}{{\\hbox{Run\\,I}} published} \n         & \\multicolumn{6}{|c|}{{\\hbox{Run\\,II}} published} \n         & \\multicolumn{1}{c}{{\\hbox{Run\\,II}} preliminary}   \\\\\n      & \\multicolumn{3}{c|}{ CDF } \n         & \\multicolumn{2}{c}{ D\\O\\ }\n         & \\multicolumn{4}{|c|}{ CDF } \n\t & \\multicolumn{2}{c|}{ D\\O\\ }\n         & \\multicolumn{1}{c}{ CDF } \n         \\\\\n\n                &  $\\ell$+jets    &    $\\ell\\ell$ &   alljets &    $\\ell$+jets  &   $\\ell\\ell$   &   $\\ell$+jets  &   $\\ell\\ell$   & alljets   &     $\\Lxy$   &  $\\ell$+jets      & $\\ell\\ell$  &  MEt \\\\\n                                                                                                                  \n\\hline                                                                                                            \nCDF-I $\\ell$+jets    &   1.00  &    0.29  &    0.32  &    0.26  &    0.11  &    0.49  &    0.54  &    0.25  &    0.07  &    0.21  &    0.12  &    0.27 \\\\\nCDF-I $\\ell\\ell$     &   0.29  &    1.00  &    0.19  &    0.15  &    0.08  &    0.29  &    0.32  &    0.15  &    0.04  &    0.13  &    0.08  &    0.17 \\\\\nCDF-I alljets        &   0.32  &    0.19  &    1.00  &    0.14  &    0.07  &    0.30  &    0.38  &    0.15  &    0.04  &    0.09  &    0.06  &    0.16 \\\\\nD\\O-I $\\ell$+jets    &   0.26  &    0.15  &    0.14  &    1.00  &    0.16  &    0.22  &    0.27  &    0.12  &    0.05  &    0.14  &    0.07  &    0.12 \\\\\nD\\O-I $\\ell\\ell$     &   0.11  &    0.08  &    0.07  &    0.16  &    1.00  &    0.11  &    0.13  &    0.07  &    0.02  &    0.07  &    0.05  &    0.07 \\\\\nCDF-II $\\ell$+jets   &   0.49  &    0.29  &    0.30  &    0.22  &    0.11  &    1.00  &    0.48  &    0.29  &    0.08  &    0.30  &    0.18  &    0.33 \\\\\nCDF-II $\\ell\\ell$    &   0.54  &    0.32  &    0.38  &    0.27  &    0.13  &    0.48  &    1.00  &    0.25  &    0.06  &    0.11  &    0.07  &    0.26 \\\\\nCDF-II alljets       &   0.25  &    0.15  &    0.15  &    0.12  &    0.07  &    0.29  &    0.25  &    1.00  &    0.04  &    0.16  &    0.10  &    0.17 \\\\\nCDF-II $\\Lxy$     &   0.07  &    0.04  &    0.04  &    0.05  &    0.02  &    0.08  &    0.06  &    0.04  &    1.00  &    0.06  &    0.03  &    0.04 \\\\\nD\\O-II $\\ell$+jets    &   0.21  &    0.13  &    0.09  &    0.14  &    0.07  &    0.30  &    0.11  &    0.16  &    0.06  &    1.00  &    0.39  &    0.18 \\\\\nD\\O-II $\\ell\\ell$    &   0.12  &    0.08  &    0.06  &    0.07  &    0.05  &    0.18  &    0.07  &    0.10  &    0.03  &    0.39  &    1.00  &    0.11 \\\\\nCDF-II MEt     &   0.27  &    0.17  &    0.16  &    0.12  &    0.07  &    0.33  &    0.26  &    0.17  &    0.04  &    0.18  &    0.11  &    1.00 \\\\\n\n\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:coeff}\n\\end{table}\n\nThe measurements are combined using a program implementing two \nindependent methods: \na numerical $\\chi^2$ minimization and \nthe analytic best linear unbiased estimator (BLUE) method~\\cite{Lyons:1988, Valassi:2003}. \nThe two methods are mathematically equivalent.\nIt has been checked that they give identical results for\nthe combination. The BLUE method yields the decomposition of the uncertainty on the Tevatron $\\ensuremath{M_{\\mathrm{t}}}$ average in \nterms of the uncertainty categories specified for the input measurements~\\cite{Valassi:2003}.\n\n\\section{Results}\n\\label{sec:results}\n\nThe resultant combined value for the top-quark mass is\n\\begin{eqnarray}\n\\nonumber\n\\ensuremath{M_{\\mathrm{t}}}=\\gevcc{\\measStatSyst{173.20}{0.51}{0.71}}. \n\\end{eqnarray}\nAdding the statistical and systematic uncertainties\nin quadrature yields a total uncertainty of $\\gevcc{0.87}$, corresponding to a\nrelative precision of 0.50\\% on the top-quark mass.\nIt has a $\\chi^2$ of 8.5 for 11 degrees of freedom, corresponding to\na probability of 67\\%, indicating good agreement among all input\nmeasurements.  The breakdown of the uncertainties is \nshown in Table\\,\\ref{tab:BLUEuncert}.  The total statistical and systematic  \nuncertainties are reduced relative to the \nSummer 2011 combination \\cite{Mtop-tevewwgSum11} and the published combination~\\cite{TeVTopComboPRD} \ndue to the increase of the CDF data samples  in the $\\ell$+jets and MEt analyses and better \ntreatment of JES corrections in the $\\ell$+jets analysis. \n \nThe pull and weight for each of the inputs, as obtained from the\ncombination with the BLUE method, are listed in Table~\\ref{tab:stat}.\nThe input measurements and the resulting Tevatron average mass of the top \nquark are summarized in Fig.~\\ref{fig:summary}.\n\\vspace*{0.10in}\n\n\\begin{table}[tbh]\n\\caption{\\label{tab:BLUEuncert} \nSummary of the Tevatron combined average $\\ensuremath{M_{\\mathrm{t}}}$. The uncertainty categories are \ndescribed in the text. The total systematic uncertainty and the total\nuncertainty are obtained  \nby adding the relevant contributions in quadrature.}\n\\begin{center}\n\\begin{tabular}{lc} \\hline \\hline\n  & Tevatron combined values (GeV\/$c^2$) \\\\ \\hline\n $\\ensuremath{M_{\\mathrm{t}}}$            & 173.20 \\\\ \\hline\n {\\em In situ} light-jet calibration (iJES)           &  0.36 \\\\\n Response to $b$\/$q$\/$g$ jets (aJES)                  &  0.09  \\\\\n Model for $b$ jets       (bJES)                  &  0.11 \\\\  \n Out-of-cone correction (cJES)                  &  0.01 \\\\ \n Light-jet response (2) (dJES)                  &  0.15 \\\\ \n Light-jet response (1) (rJES)                  &  0.16 \\\\\n Lepton modeling  (LepPt)                       &  0.05 \\\\\n Signal modeling  (Signal)                      &  0.52  \\\\\n Jet modeling (DetMod)                          &  0.08 \\\\\n Offset (UN\/MI)                                 &  0.00 \\\\\n Background from theory (BGMC)                  &  0.06\\\\\n Background based on data (BGData)              &  0.13\\\\\n Calibration method (Method)                    &  0.06 \\\\\n Multiple interactions model (MHI)              &  0.07 \\\\ \\hline\n Systematic uncertainty (syst)                  &  0.71 \\\\ \n Statistical uncertainty (stat)                 &  0.51 \\\\ \\hline\n Total  uncertainty                             &  0.87 \\\\\n   \\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe weights of some of the measurements are negative, which occurs if\nthe correlation between two measurements \nis larger than the ratio of their total uncertainties.\nIn these instances the less precise measurement \nwill acquire a negative weight.  While a weight of zero means that a\nparticular input is effectively ignored in the combination, channels\nwith a negative weight affect the resulting $\\ensuremath{M_{\\mathrm{t}}}$ central value and help reduce the total\nuncertainty~\\cite{Lyons:1988}. \nTo visualize the weight each measurement carries in the combination, Fig.\\,\\ref{fig:Weights} shows the\nabsolute values of the weight of each measurement divided by the sum of the absolute values of the weights\nof all input measurements. Negative weights are represented by bins\nwith a different (grey) color. \nWe note, that due to correlations between the uncertainties the relative weights\nof the different input channels may be significantly different from\nwhat one could expect from the total accuracy of each measurement as represented by\nerror bars in Fig.~\\ref{fig:summary}.\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figures\/TevMtWin13-2digit.pdf}\n\\end{center}\n\\caption[Summary plot for the Tevatron average top-quark mass]\n  {Summary of the input measurements and resulting Tevatron average\n   mass of the top quark.}\n\\label{fig:summary} \n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth]{figures\/TevWeightPlotWin13-neg.pdf}\n\\end{center}\n\\caption{Relative weights of the input measurements in the\n  combination. The relative weights have been obtained by dividing the absolute value of each measurement weight by the sum over all measurements of the absolute values\nof the weights. Negative weights are represented by their absolute\nvalue, but using a grey color.}\n\\label{fig:Weights} \n\\end{figure}\n\n\\begin{table}[t]\n\\caption[Pull and weight of each measurement]{The pull and weight for each of the\n  inputs, as obtained from the combination with the BLUE method to\n  determine the average top quark mass.}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n{\\tiny\n\\begin{tabular}{l|ccccc|cccccc|c}\n\\hline \n\\hline\n       & \\multicolumn{5}{c}{{\\hbox{Run\\,I}} published} \n       & \\multicolumn{6}{|c|}{{\\hbox{Run\\,II}} published} \n       & \\multicolumn{1}{c}{{\\hbox{Run\\,II}} preliminary}  \\\\\n       & \\multicolumn{3}{c}{ CDF } \n       & \\multicolumn{2}{c}{ D\\O\\ }\n       & \\multicolumn{4}{|c}{ CDF } \n       & \\multicolumn{2}{c|}{ D\\O\\ } \n       & \\multicolumn{1}{c}{ CDF } \\\\\n       &  $\\ell$+jets    &    $\\ell\\ell$ &   alljets &    $\\ell$+jets  &   $\\ell\\ell$   &   $\\ell$+jets  &   $\\ell\\ell$   & alljets   &     Lxy   &  $\\ell$+jets      & $\\ell\\ell$  &  MEt \\\\\n\\hline\n\n\n\nPull   & $+0.40$  & $-0.51$  & $+1.11$  & $+1.32$  & $-0.38$  & $-0.51$  & $-0.82$  & $-0.41$  & $-0.67$ & $1.42$ & $+0.30$   & $+0.45$  \\\\\nWeight [\\%]  & $-4.7$ & $-1.1$ & $-0.9$ & $+0.4$ & $-0.2$ & $+62.0$ & $-0.3$ & $+10.5$ & $+0.22$ & $+20.6$ & $+1.4$     & $+11.9$  \\\\\n\\hline \\hline\n\\end{tabular}\n}\n\\end{center}\n\\label{tab:stat} \n\\end{table} \n\nNo input has an anomalously large pull. \nIt is, however, still\ninteresting to determine the top-quark mass separately \nin the alljets, $\\ell$+jets, $\\ell\\ell$, and MEt channels (leaving out\nthe $\\Lxy$ measurement).\nWe use the same methodology,\ninputs, uncertainty categories, and correlations as described above, but fit \nthe four physical observables, \\ensuremath{\\MT^{\\mathrm{alljets}}}, \\ensuremath{\\MT^{\\mathrm{\\ell\\mbox{+}jets}}}, \\ensuremath{\\MT^{\\mathrm{\\ell\\ell}}}, and \\ensuremath{\\MT^{\\mathrm{MEt}}}\\ separately.\nThe results of these combinations are shown in\nFigure~\\ref{fig:three_observables} and Table~\\ref{tab:three_observables}.\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figures\/TevMtWin13-4channels.pdf}\n\\end{center}\n\\caption[Mtop in each channel]{Summary of the combination of the 12\ntop-quark measurements by CDF and D\\O\\ for different final states.}\n\\label{fig:three_observables} \n\\end{figure}\n\n\n\nUsing the results of Table~\\ref{tab:three_observables} \n we calculate the following chi-squared values including correlations:\n$\\chi^{2}(\\ell+{\\rm jets}-\\ell\\ell)=1.30\/1$, $\\chi^{2}(\\ell+{\\rm\n  jets}-{\\rm alljets})=0.07\/1$,  $\\chi^{2}(\\ell+{\\rm jets}-{\\rm MEt})=0.11\/1$, \n$\\chi^{2}(\\ell\\ell-{\\rm alljets})=0.42\/1$,\n  $\\chi^{2}(\\ell\\ell-{\\rm MEt})=1.22\/1$, and $\\chi^{2}({\\rm\n  alljets}-{\\rm MEt})=0.19\/1$.  \nThese correspond to chi-squared probabilities \nof 25\\%, 79\\%, 74\\%, 52\\%, 27\\%, and 66\\%  respectively, indicating\nthat the top-quark mass determined in each decay channel is consistent\nin all cases.\n\n\\begin{table}[t]\n\\caption[Mtop in each channel]{Summary of the combination of the 12\nmeasurements by CDF and D\\O\\ in terms of four physical quantities,\nthe mass of the top quark in the alljets, $\\ell$+jets,  $\\ell\\ell$, and MEt decay channels. }\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\begin{tabular}{ccrrrr}\n\\hline\\hline\n\n\nParameter & Value (\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }) & \\multicolumn{4}{c}{Correlations} \\\\\n               &                                 & $\\ensuremath{\\MT^{\\mathrm{alljets}}}$ &    $\\ensuremath{\\MT^{\\mathrm{\\ell\\mbox{+}jets}}}$  &  $\\ensuremath{\\MT^{\\mathrm{\\ell\\ell}}}$ & $\\ensuremath{\\MT^{\\mathrm{MEt}}}$ \\\\ \\hline\n$\\ensuremath{\\MT^{\\mathrm{alljets}}}$ & $172.7\\pm 1.9$    & 1.00       &                   &      & \\\\\n$\\ensuremath{\\MT^{\\mathrm{\\ell\\mbox{+}jets}}}$ & $173.2\\pm 0.9$    & 0.25       &    1.00       &     &\\\\\n$\\ensuremath{\\MT^{\\mathrm{\\ell\\ell}}}$ & $171.0\\pm 2.1$    & 0.19        &    0.41      & 1.00 & \\\\\n$\\ensuremath{\\MT^{\\mathrm{MEt}}}$& $173.8\\pm 1.8$    & 0.13        &    0.26      & 0.18 & 1.00 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:three_observables}\n\\end{table}\n\nTo test the influence of the choices in modeling the\ncorrelations, we performed a cross-check by changing all non-diagonal\ncorrelation \ncoefficients of the correlation matrix defined in Section~\\ref{sec:corltns} from 100\\% to 50\\% and re-evaluated the combination. \nThe result of this large variation of degree of correlation is a\n$\\gevcc{+0.19}$ shift of the top-quark mass and reduces the total\nuncertainty negligibly.\nThe chosen approach is therefore conservative.\n\n\n\n\nWe also performed separate combinations of all the CDF and D\\O\\ measurements. The results of these combinations are\n$\\gevcc{172.72~\\pm~0.93}$ for CDF and $\\gevcc{174.89~\\pm~1.42}$ for D\\O. Taking all correlations into account, \nwe calculate the chi-square value $\\chi^{2}(CDF-D\\O)=2.25\/1$ corresponding to a probability of 13\\%.\n\n\\section{Summary}\n\\label{sec:summary}\n\nAn update of the combination of measurements of the mass of the top quark\nfrom the Tevatron experiments CDF and D\\O\\ has been presented.  This preliminary\ncombination includes five published \\hbox{Run\\,I}\\ measurements, six published\n\\hbox{Run\\,II}\\ measurements, and  \none preliminary \\hbox{Run\\,II}\\ measurement, but the majority of these\nmeasurements are not yet performed on the full datasets available.  Taking into\naccount the statistical and systematic uncertainties and their\ncorrelations, the preliminary result for the Tevatron average is\n  $\\ensuremath{M_{\\mathrm{t}}}=\\gevcc{\\measStatSyst{173.20}{0.51}{0.71}}$,\nwhere the total uncertainty is obtained assuming Gaussian systematic uncertainties.\nThe central value is 0.02\\,GeV\/$c^2$  higher than our July 2012\naverage~\\cite{TeVTopComboPRD} of \n$\\ensuremath{M_{\\mathrm{t}}}=173.18\\pm0.94$\\,GeV\/$c^2$.\nAdding in quadrature the statistical and systematic uncertainties\nyields a total uncertainty of $\\gevcc{0.87}$ which represents an\nimprovement of $8\\%$.\n\nThe mass of the top quark is now known with a relative precision of\n0.50\\%, limited by the systematic uncertainties, which are dominated by\nthe jet energy scale uncertainty.  \nThis result will be further improved when all analysis channels from  \nCDF and D\\O\\ using the full Run~II data set are finalized.\n \n\n\\section{Acknowledgments}\n\\label{sec:ack}\n\nWe thank the Fermilab staff and the technical staffs of the\nparticipating institutions for their vital contributions. \nThis work was supported by  \nDOE and NSF (USA),\nCONICET and UBACyT (Argentina), \nCNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil),\nCRC Program, CFI, NSERC and WestGrid Project (Canada),\nCAS and CNSF (China),\nColciencias (Colombia),\nMSMT and GACR (Czech Republic),\nAcademy of Finland (Finland),\nCEA and CNRS\/IN2P3 (France),\nBMBF and DFG (Germany),\nMinistry of Education, Culture, Sports, Science and Technology (Japan), \nWorld Class University Program, National Research Foundation (Korea),\nKRF and KOSEF (Korea),\nDAE and DST (India),\nSFI (Ireland),\nINFN (Italy),\nCONACyT (Mexico),\nNSC(Republic of China),\nFASI, Rosatom and RFBR (Russia),\nSlovak R\\&D Agency (Slovakia), \nMinisterio de Ciencia e Innovaci\\'{o}n, and Programa Consolider-Ingenio 2010 (Spain),\nThe Swedish Research Council (Sweden),\nSwiss National Science Foundation (Switzerland), \nFOM (The Netherlands),\nSTFC and the Royal Society (UK),\nand the A.P. Sloan Foundation (USA).\n\n\\clearpage\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}