diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzoyla" "b/data_all_eng_slimpj/shuffled/split2/finalzzoyla" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzoyla" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}In this paper, we study the following Navier-Stokes equations with time-fractional derivative in a bounded subset of $\\Omega\\subset \\mathbf{R}^{2}$ with a smooth boundary $\\partial \\Omega$:\n\\begin{align*}\n\\begin{cases}\n^{C}D_{t}^{\\alpha}u+(u\\cdot \\nabla)u-\\nu\\Delta u+\\nabla p=f,~~~\\mathrm{div}~u=0,~\\forall~(x,t)\\in \\Omega\\times(0,T],\\\\\n~u(x,t)|_{\\partial \\Omega}=0,~t\\in (0,T],\\\\\n~u(x,0)=u_{0},~x\\in \\Omega,\\\\\n\\end{cases} \\tag{1.1}\n\\end{align*}\nwhere $^{C}D_{t}^{\\alpha}$ represents the Caputo-type fractional derivative of order $\\alpha\\in(0,1)$, $u=(u_{1}(x,t),$ $ u_{2}(x,t))$ denotes the velocity field at a point $x\\in\\Omega$ and $t\\in[0,T]$, $\\nu>0$ is viscosity coefficient, $p=p(x,t)$ represents the pressure field, $f=f(x,t)$ is the external force and $u_{0}=u_{0}(x)$ is the initial velocity.\n\nNotice that the problem (1.1) reduces to the classical Navier-Stokes equations (NSEs) for $\\alpha=1$. The existence and non-existence of solutions for the NSEs have been discussed in [1].\nChemin et al.[2] studied the global regularity for the large solutions to the NSEs. Miura [3] focused on the uniqueness of mild solutions to the NSEs. Germain [4] presented the uniqueness criteria for the solutions of the Cauchy problem associated to the NSEs. The existence of global weak solutions for supercritical NSEs was discussed [5]. The lower bounds on blow up solutions for the NSEs in homogeneous Sobolev spaces were studied in [6]. The numerical methods for solving the NSEs have been investigated by many authors [7,8,9,10,11,12,23]. The study of time-fractional Navier-Stokes equations (TFNSEs) has become a hot topic of research due to its significant role in simulating the anomalous diffusion in fractal media. There are also some analytical methods available for solving the TFNSEs. Momani and Odibat [13] applied Adomian decomposition method to obtain the analytical solution of the TFNSEs. In [14, 15], the homotopy perturbation (transform) method was used to find the analytical solution of the TFNSEs. Wang and Liu [16] solved TFNSEs by applying the transform methods. Concerning the existence of global and local mild solutions to TFNSEs, see Carvalho-Neto and Gabriela [17], Zhou and Peng [18]. Moreover, Zhou and Peng [19] investigated the existence of weak solutions and optimal control for TFNSEs, while Peng et al.[20] presented the rigorous exposition of local solutions of TFNSEs in Sobolev space. However, one can notice that there are only a few works related to the numerical solution of the TFNSEs. The details of meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the TFNSEs can be found in the literature [21]. The purpose of this paper is to present finite difference\/element method to obtain the numerical solution of TFNSEs.\n\nThe rest of the paper is arranged as follows. In Section 2, we give some notations and preliminaries. Section 3 deals with a semi-discrete scheme for the TFNSEs, which is based on a mixed finite element method in space. We also discuss the stability and error estimates of this semi-discrete scheme. In Section 4, we use a finite difference approximation to discrete time direction to obtain the fully discrete scheme. The stability and error estimates for the discrete schemes are also found. In Section 5, numerical results are discussed to confirm our theoretical analysis. Conclusions are given in the final section.\n\\bigskip\n\\section{Notations and preliminaries}\n\nIn this section, we present some preliminary concepts of the functional spaces. Firstly, we introduce the following Hilbert spaces:\n\\begin{align*}\nX=H_{0}^{1}(\\Omega)^{2},~Y=L^{2}(\\Omega)^{2},~M=L_{0}^{2}(\\Omega)=\\{v\\in L^{2}(\\Omega);\\int_{\\Omega}v dx=0\\},\n\\end{align*}\nwhere the space $L^{2}(\\Omega)$ is associated with the usual inner product $(\\cdot,\\cdot)$ and the norm $\\|\\cdot\\|$. The space $X$ is associated with the following inner product and equivalent norm:\n\\begin{align*}\n((u,v))=(\\nabla u,\\nabla v),~\\|u\\|_{X}=\\|\\nabla u\\|_{0}=\\| u\\|_{1}.\n\\end{align*}\n\nDenote by $V$ and $H$ the closed subsets of $X$ and $Y$ respectively, which are given by\n\\begin{align*}\nV=\\{v\\in X;\\mathrm{div}~ v=0\\},~H=\\{v\\in Y;\\mathrm{div}~ v=0,v\\cdot n|_{\\partial\\Omega}=0\\}.\n\\end{align*}\n\nWe denote the Stokes operator by $A=-P\\triangle$, in which $P$ is the $L^{2}$-orthogonal projection of $Y$ onto $H$. The domain of $A$ is $\\mathcal{D}(A)=H^{2}(\\Omega)^{2}\\cap V$ and let $H^{s}=\\mathcal{D}(A^{\\frac{s}{2}})$ with the norm $\\|v\\|_{s}=\\|A^{\\frac{s}{2}} v\\|$. Observe that $H^{2}=\\mathcal{D}(A)$, $H^{1}=V$ and $H^{0}=H$.\n\nNext, we define the Riemann-Liouville fractional integral operator of order $\\beta$ ($\\beta\\geq0$) as (see [22])\n\\begin{align*}\nI_{t}^{\\beta}g(t)=\\frac{1}{\\Gamma(\\beta)}\\int_{0}^{t}(t-s)^{\\beta-1}g(s)ds,t>0, \\tag{2.1}\n\\end{align*}\nwith $I_{t}^{0}g(t)=g(t)$.\n\nThe Caputo-type derivative of order $\\alpha \\in (0,1]$, $^{C}D_{t}^{\\alpha}$ in (1.1) is defined by\n\\begin{align*}\n^{C}D_{t}^{\\alpha}g(t)=\\frac{d}{dt}\\{I_{t}^{1-\\alpha}[g(t)-g(0)]\\}=\\frac{d}{dt}\\{\\frac{1}{\\Gamma(1-\\alpha)}\\int_{0}^{t}(t-s)^{-\\alpha}[g(t)-g(0)]ds\\}.\\tag{2.2}\n\\end{align*}\nFurther, the operator $^{C}D_{t}^{-\\alpha}$ is defined as\n\\begin{align*}\n^{C}D_{t}^{-\\alpha}g(t)=I_{t}^{\\alpha}g(t)=\\frac{1}{\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}g(s)ds,t>0.\\tag{2.3}\n\\end{align*}\nwhere $\\Gamma(\\cdot)$ stands for the gamma function $\\Gamma(x)=\\int_{0}^{\\infty}t^{x-1}e^{-t}dt$.\n\nNext we introduce the following continuous bilinear forms $a(\\cdot,\\cdot)$ and $d(\\cdot,\\cdot)$ on $X\\times X$ and $X\\times M$ respectively as follows:\n\\begin{align*}\na(u,v)=\\nu(\\nabla u,\\nabla v),~u,v\\in X, ~d(v,q)=(q,\\mathrm{div}v),~v\\in X, q \\in M,\n\\end{align*}\nand the trilinear form $b(u,v,w)$ on $X\\times X\\times X$ is given by\n\\begin{align*}\nb(u,v,w)=((u\\cdot \\nabla)v+\\frac{1}{2}(\\mathrm{div}u)v,w)=\\frac{1}{2}((u\\cdot \\nabla)v,w)-\\frac{1}{2}((u\\cdot \\nabla)w,v),~u,v,w\\in X.\n\\end{align*}\n\nIt is well-known that the trilinear form $b(u,v,w)$ has the following properties:\n\\begin{align*}\n&b(u,v,w)=-b(u,w,v),~b(u,v,v)=0,~u,v,w\\in X,\\\\\n&|b(u,v,w)|\\leq \\mu_{0}\\|u\\|_{1}\\|v\\|_{1}\\|w\\|_{1},~u,v,w\\in X.\n\\end{align*}\n\nIn terms of the above notations, the weak formulation of problem (1.1) is as follows: find $(u,p)\\in(X,M)$ for all $t\\in [0,T]$ such that for all $(v,q)\\in(X,M)$:\n\\begin{align*}\n\\begin{cases}\n^{C}D_{t}^{\\alpha}(u,v)+a(u,v)+b(u,u,v)-d(v,p)+d(u,q)=(f,v),\\\\\nu(0)=u_{0}.\n\\end{cases} \\tag{2.4}\n\\end{align*}\n\nIn [19], Zhou and Peng discussed the existence and uniqueness of weak solutions for the problem (2.4). The objective of the present work is to obtain the numerical solution of the problem at hand.\n\\section{Finite element method for space discretization}\n\nLet $T^{h}(\\Omega)=\\{K\\}$ be a mesh of $\\Omega$ with a mesh size function $h(x)$, which is the diameter $h_{K}$ of element $K$ containing $x$.\nAssuming $h=h_{\\Omega}=\\max\\limits_{x\\in\\Omega}h(x)$ be the largest mesh size of $T^{h}(\\Omega)$, we introduce the mixed finite element subspace $(X_{h},M_{h})$ of $(X,M)$ and define the subspace $V_{h}$ of $X_{h}$ as\n\\begin{align*}\nV_{h}=\\{v_{h}\\in X_{h};d(v_{h},q_{h})=0,~\\forall q_{h}\\in M_{h}\\}.\n\\end{align*}\n\nLet $P_{h}:Y\\rightarrow V_{h}$ denote the $L^{2}$-orthogonal projection defined by\n\\begin{align*}\n(P_{h}v,v_{h})=(v,v_{h}),~\\forall v\\in Y,~v_{h}\\in V_{h}\\}.\n\\end{align*}\n\nWith the above notations, we need some further basic assumptions on the mixed finite element spaces (Refs.[8,10,12]).\n\n(A1) \\textit{Approximation}. For each $(v,q)\\in (\\mathcal{D}(A),M\\cap H^{1}(\\Omega))$, there exist approximations $(\\pi_{h}v,\\rho_{h}q)\\in (X_{h},M_{h})$ such that\n\\begin{align*}\n\\|v-\\pi_{h}v\\|_{1}\\leq Ch\\|A v\\|,~\\|q-\\rho_{h}q\\|\\leq Ch\\|q\\|_{1}.\\tag{3.1}\n\\end{align*}\n\n(A2) \\textit{Inverse~estimate}. For any $(v,q)\\in (X_{h},M_{h})$, the following relations hold:\n\\begin{align*}\n\\|\\nabla v\\|\\leq Ch^{-1}\\| v\\|,~\\|q\\|\\leq Ch^{-1}\\|q\\|_{-1}.\\tag{3.2}\n\\end{align*}\n\n(A3) \\textit{Stability~property}. For any $(v,q)\\in (X_{h},M_{h})$, the well-known inf-sup condition holds:\n\\begin{align*}\n\\sup\\limits_{v\\in X_{h}}\\frac{d(v,q)}{\\|v\\|_{1}}\\geq \\lambda \\|q\\|,\\tag{3.3}\n\\end{align*}\nwhere $\\lambda>0$ is a constant.\n\nFurther, the following classical properties hold:\n\\begin{align*}\n&\\|v-P_{h}v\\|+h\\|\\nabla (v-P_{h}v)\\|\\leq Ch^{2}\\|A v\\|,v\\in\\mathcal{D}(A);\\tag{3.4}\\\\\n&\\|v-P_{h}v\\|\\leq Ch\\|\\nabla (v-P_{h}v)\\|,v\\in X.\\tag{3.5}\n\\end{align*}\n\nThe standard finite element Galerkin approximation for (2.4) holds as follows: Find $(u_{h},p_{h})\\in(X_{h},M_{h})$ for all $t\\in [0,T]$ such that for all $(v_{h},q_{h})\\in(X_{h},M_{h})$, we have\n\\begin{align*}\n\\begin{cases}\n^{C}D_{t}^{\\alpha}(u_{h},v_{h})+a(u_{h},v_{h})+b(u_{h},u_{h},v_{h})-d(v_{h},p_{h})+d(u_{h},q_{h})=(f,v_{h}),\\\\\nu_{h}(0)=u_{0h}=P_{h}u_{0}.\n\\end{cases} \\tag{3.6}\n\\end{align*}\n\nWith the above semi-discrete approximation, a discrete analogue of the Stokes operator $A$ is defined as $A_{h}=-P_{h}\\triangle_{h}$ via the condition $(-\\triangle_{h}u_{h},v_{h})=((u_{h},v_{h}))$ for all $u_{h},v_{h}\\in X_{h}$. The trilinear form $b(u_{h},v_{h},w_{h})$ satisfies the following properties:\n\\begin{align*}\n&b(u_{h},v_{h},w_{h})=-b(u_{h},w_{h},v_{h}),~b(u_{h},v_{h},v_{h})=0,~u_{h},v_{h},w_{h}\\in X_{h};\\tag{3.7}\\\\\n&|b(u_{h},v_{h},w_{h})|\\leq \\mu_{0}\\|u_{h}\\|_{1}\\|v_{h}\\|_{1}\\|w_{h}\\|_{1},~u_{h},v_{h},w_{h}\\in X_{h}.\\tag{3.8}\n\\end{align*}\n\n\\textbf{Theorem 3.1.} For any $t\\in [0,T]$ and $0<\\alpha < 1$, let $u_{h}$ be the solution of equation (3.6). Then there exists a positive constant $C$ such that\n\\begin{align*}\n\\|u_{h}\\|^{2}+\\nu_{1}\\int_{0}^{t}\\|u_{h}\\|_{1}^{2}ds\\leq \\|u_{0h}\\|^{2}+\\frac{(1-\\alpha_{1})T^{1+\\beta}}{2\\nu(1+\\beta)\\Gamma(\\alpha)}+\\frac{\\alpha_{1}T}{2\\nu\\Gamma(\\alpha)}\\max\\limits_{t\\in [0,T]}\\|f\\|_{-1}^{\\frac{2}{\\alpha_{1}}},\n\\end{align*}\nwhere $\\nu_{1}=\\frac{\\nu T^{\\alpha-1}}{2\\Gamma(\\alpha)}>0$ is constant.\n\n\\textbf{Proof.} Taking $v_{h}=u_{h}$, $q_{h}=p_{h}$ in (3.6) and using the Young's inequality, we get\n\\begin{align*}\n^{C}D_{t}^{\\alpha}\\|u_{h}\\|^{2}+\\nu\\|u_{h}\\|_{1}^{2}=(f,u_{h}) \\leq \\|f\\|_{-1}\\|u_{h}\\|_{1}\n \\leq \\frac{1}{2\\nu}\\|f\\|_{-1}^{2}+\\frac{\\nu}{2}\\|u_{h}\\|_{1}^{2},\\tag{3.9}\n\\end{align*}\nwhere $\\|f\\|_{-1}=\\|A^{-\\frac{1}{2}}f\\|$ denotes the dual operator in $\\mathcal{D}(A^{-\\frac{1}{2}})$.\n\nApplying the integral operator (2.3) to both sides of (3.9) and using the Young's inequality, we obtain\n\\begin{align*}\n&\\|u_{h}\\|^{2}+\\frac{\\nu}{2\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\|u_{h}\\|_{1}^{2}ds\\leq\\|u_{0h}\\|^{2}+\\frac{1}{2\\nu\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\|f\\|_{-1}^{2}ds\\\\\n&\\leq \\|u_{0h}\\|^{2}+\\frac{1-\\alpha_{1}}{2\\nu\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\frac{\\alpha-1}{1-\\alpha_{1}}}ds+\\frac{\\alpha_{1}}{2\\nu\\Gamma(\\alpha)}\\int_{0}^{t}\\|f\\|_{-1}^{\\frac{2}{\\alpha_{1}}}ds\\\\\n&\\leq \\|u_{0h}\\|^{2}+\\frac{(1-\\alpha_{1})T^{1+\\beta}}{2\\nu(1+\\beta)\\Gamma(\\alpha)}+\\frac{\\alpha_{1}T}{2\\nu\\Gamma(\\alpha)}\\max\\limits_{t\\in [0,T]}\\|f\\|_{-1}^{\\frac{2}{\\alpha_{1}}},\n\\end{align*}\nwhere $\\beta=\\frac{\\alpha-1}{1-\\alpha_{1}}$ with $0<\\alpha_{1} < 1$.\n\nIn view of the inequality\n\\begin{align*}\n\\frac{\\nu}{2\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\|u_{h}\\|_{1}^{2}ds\\geq\\frac{\\nu T^{\\alpha-1}}{2\\Gamma(\\alpha)}\\int_{0}^{t}\\|u_{h}\\|_{1}^{2}ds,\n\\end{align*}\nit follows that\n\\begin{align*}\n\\|u_{h}\\|^{2}+\\frac{\\nu T^{\\alpha-1}}{2\\Gamma(\\alpha)}\\int_{0}^{t}\\|u_{h}\\|_{1}^{2}ds\\leq \\|u_{0h}\\|^{2}+\\frac{(1-\\alpha_{1})T^{1+\\beta}}{2\\nu(1+\\beta)\\Gamma(\\alpha)}+\\frac{\\alpha_{1}T}{2\\nu\\Gamma(\\alpha)}\\max\\limits_{t\\in [0,T]}\\|f\\|_{-1}^{\\frac{2}{\\alpha_{1}}}.\n\\end{align*}\n\nThis completes the proof.\n\n\\textbf{Theorem 3.2.} For any $t\\in [0,T]$, $0<\\alpha < 1$, let $(u,p)$ and $(u_{h},p_{h})$ be the solutions of equations (2.4) and (3.6) respectively. Then there exists a positive constant $C$ such that\n\\begin{align*}\n\\|u-u_{h}\\|\\leq Ch^{2},~\\|p-p_{h}\\|\\leq Ch. \\tag{3.10}\n\\end{align*}\n\n\n\\textbf{Proof.} Setting $(\\xi,\\eta)=(u-u_{h},p-p_{h})$, we deduce from (2.4) and (3.6) that\n\\begin{align*}\n\\begin{cases}\n^{C}D_{t}^{\\alpha}(\\xi,v)+a(\\xi,v)+b(\\xi,u_{h},v)+b(u_{h},\\xi,v)+b(\\xi,\\xi,v)-d(v,\\eta)+d(\\xi,q)=0,\\\\\n\\xi_{0}=u_{0}-P_{h}u_{0}.\n\\end{cases} \\tag{3.11}\n\\end{align*}\n\nTaking $v=\\xi$ and $q=\\eta$ in (3.11), we get\n\\begin{align*}\n^{C}D_{t}^{\\alpha}\\|\\xi\\|^{2}+\\nu\\|\\xi\\|_{1}^{2}+b(\\xi,u_{h},\\xi)=0.\n\\end{align*}\n\nUsing the properties of $b(u_{h},v_{h},w_{h})$ together with Young's inequality, we obtain\n\\begin{align*}\n^{C}D_{t}^{\\alpha}\\|\\xi\\|^{2}+\\nu\\|\\xi\\|_{1}^{2}&=b(\\xi,\\xi,u_{h})\\\\\n&\\leq C_{0}\\|\\xi\\|\\|\\xi\\|_{1}\\|u_{h}\\|_{1}\\\\\n&\\leq \\nu\\|\\xi\\|_{1}^{2}+C_{1}\\|\\xi\\|^{2}\\|u_{h}\\|_{1}^{2}.\\tag{3.12}\n\\end{align*}\n\nApplying (2.3) to both sides of (3.12), we have\n\\begin{align*}\n\\|\\xi\\|^{2}\\leq \\|\\xi_{0}\\|^{2}+\\frac{C_{1}}{\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\|\\xi\\|^{2}\\|u_{h}\\|_{1}^{2}ds.\\tag{3.13}\n\\end{align*}\n\nBy means of the generalized integral version of Gronwall's lemma [24], we get\n\\begin{align*}\n\\|\\xi\\|^{2}&\\leq \\|\\xi_{0}\\|^{2}\\exp(\\frac{C_{1}}{\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\|u_{h}\\|_{1}^{2}ds)\\\\\n&\\leq \\|u_{0}-P_{h}u_{0}\\|^{2}\\exp[C_{2}(\\|u_{0h}\\|^{2}+\\frac{(1-\\alpha_{1})T^{1+\\beta}}{2\\nu(1+\\beta)\\Gamma(\\alpha)}+\\frac{\\alpha_{1}T}{2\\nu\\Gamma(\\alpha)}\\max\\limits_{t\\in [0,T]}\\|f\\|_{-1}^{\\frac{2}{\\alpha_{1}}})]\\\\\n&\\leq Ch^{4}.\\tag{3.14}\n\\end{align*}\n\n\nFurthermore, setting $v=\\xi$ and $q=0$ in (3.11), and using inf-sup condition (3.3), combining (3.1)-(3.5),(3.8),(3.14) and using the integral operator (2.3) in (3.15), we conclude that\n\\begin{align*}\n\\|\\eta\\| &\\leq \\sup\\limits_{V_{h}}\\frac{|d(\\xi,\\eta)|}{\\lambda\\|\\xi\\|_{1}}\\\\\n&=\\sup\\limits_{V_{h}}\\frac{|^{C}D_{t}^{\\alpha}\\|\\xi\\|^{2}+\\nu\\|\\xi\\|_{1}^{2}+b(\\xi,u_{h},\\xi)|}{\\lambda\\|\\xi\\|_{1}}\\\\\n&\\leq Ch. \\tag{3.15}\n\\end{align*}\n\nThis completes the proof.\n\n\\section{Finite difference method for time discretization}\n\nThe discretization of time-fractional derivative can be found in [25-32] and references therein.\nHere, we will introduce a uniform grid by discretizing the temporal domain $[0,T]$ given by the points: $t_{n}=n\\tau$ for $n=0,1,\\ldots,N$, with the time-step size $\\tau=T\/N$. Hence, the Riemann-Liouville fractional integral operator of order $\\alpha$ can be discretized as follows:\n\\begin{align*}\nI_{t}^{\\alpha}g(t_{n})&=\\frac{1}{\\Gamma(\\alpha)}\\sum\\limits_{k=1}^{n}\\int_{t_{k-1}}^{t_{k}}(t_{n}-s)^{\\alpha-1}g(s)ds\\\\\n&=\\frac{1}{\\Gamma(\\alpha)}\\sum\\limits_{k=1}^{n}(\\int_{t_{k-1}}^{t_{k}}(t_{n}-s)^{\\alpha-1}g(t_{k})ds)+\\gamma_{\\alpha}^{n}\\\\\n&=\\frac{\\tau^{\\alpha}}{\\Gamma(\\alpha+1)}\\sum\\limits_{k=1}^{n}g(t_{k})[(n-k+1)^{\\alpha}-(n-k)^{\\alpha}]+\\gamma_{\\alpha}^{n}\\\\\n&=\\frac{\\tau^{\\alpha}}{\\Gamma(\\alpha+1)}\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}g(t_{n-k})+\\gamma_{\\alpha}^{n},\\tag{4.1}\n\\end{align*}\nwhere $w_{k}^{\\alpha}=(k+1)^{\\alpha}-k^{\\alpha}$ and the truncation error $\\gamma_{\\alpha}^{n}$ is given by\n\\begin{align*}\n\\gamma_{\\alpha}^{n}&=\\frac{1}{\\Gamma(\\alpha)}\\sum\\limits_{k=1}^{n}\\int_{t_{k-1}}^{t_{k}}(t_{n}-s)^{\\alpha-1}[g(s)-g(t_{k})]ds\\\\\n&=\\frac{1}{\\Gamma(\\alpha)}\\sum\\limits_{k=1}^{n}\\int_{t_{k-1}}^{t_{k}}(t_{n}-s)^{\\alpha-1}g^{\\prime}(\\zeta)(s-t_{k})ds,~s<\\zeta0$, let the coefficient $w_{k}^{\\alpha}$ be given by (4.1). Then\n\\begin{align*}\n&(\\mathrm{i})~ w_{0}^{\\alpha}=1,w_{k}^{\\alpha}>0,k=0,1,2,\\cdots;\\\\\n&(\\mathrm{ii})~ w_{k}^{\\alpha}>w_{k+1}^{\\alpha},k=0,1,2,\\cdots;\\\\\n&(\\mathrm{iii})\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}=n^{\\alpha}\\leq N^{\\alpha}.\n\\end{align*}\n\n\nApplying the integral operator (2.3) to both sides of (3.6), we obtain\n\\begin{align*}\n&(u_{h},v_{h})+\\frac{1}{\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}[a(u_{h},v_{h})+b(u_{h},u_{h},v_{h})-d(v_{h},p_{h})+d(u_{h},q_{h})]ds\\\\\n&=(u_{0h},v_{h})+\\frac{1}{\\Gamma(\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}(f,v_{h})ds. \\tag{4.3}\n\\end{align*}\n\nLet $u_{h}^{n}$ and $p_{h}^{n}$ be the numerical solutions of $u_{h}(t)$ and $p_{h}(t)$ at $t=t_{n}$ respectively. By (4.2) and (4.3), our full discrete scheme of equation (2.4) can be defined by seeking\n$(u_{h}^{n},p_{h}^{n})\\in(X_{h},M_{h})$ such that for all $(v_{h},q_{h})\\in(X_{h},M_{h})$:\n\\begin{align*}\n&(u_{h}^{n},v_{h})+\\beta_{0}\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}[a(u_{h}^{n-k},v_{h})+b(u_{h}^{n-k},u_{h}^{n-k},v_{h})-d(v_{h},p_{h}^{n-k})+d(u_{h}^{n-k},q_{h})]\\\\\n&=(u_{h}^{0},v_{h})+\\beta_{0}\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}(f^{n-k},v_{h}), \\tag{4.4}\n\\end{align*}\nwhere $\\beta_{0}=\\frac{\\tau^{\\alpha}}{\\Gamma(\\alpha+1)}$.\n\n\n\\textbf{Theorem 4.1.} For any $0<\\tau0$ be the viscosity coefficient and that\n\\begin{align*}\n\\|u_{h}^{0}\\|^{2}+\\sum\\limits_{k=0}^{n}\\|f^{k}\\|_{-1}^{2}\\leq \\frac{\\beta_{1}\\nu}{C^{\\dag}\\mu_{0}}. \\tag{4.7}\n\\end{align*}\nThen\n\\begin{align*}\na(v_{h},v_{h})+b(v_{h},u_{h}^{n},v_{h})\\geq0, \\tag{4.8}\n\\end{align*}\nwhere $\\mu_{0}>0$ is defined by (3.8) and $C^{\\dag}>0$ is constant.\n\n\n\\textbf{Proof.} Making use of (4.6) and (4.7), we get\n\\begin{align*}\n\\|u_{h}^{n}\\|_{1}^{2}\\leq \\frac{\\nu}{\\mu_{0}},\n\\end{align*}\nthat is,\n\\begin{align*}\n\\nu-\\mu_{0}\\|u_{h}^{n}\\|_{1}^{2}\\geq0.\n\\end{align*}\n\nBy the property of $b(v_{h},u_{h}^{n},v_{h})$, we get\n\\begin{align*}\na(v_{h},v_{h})+b(v_{h},u_{h}^{n},v_{h})\\geq (\\nu-\\mu_{0}\\|u_{h}^{n}\\|_{1}^{2})\\|v_{h}\\|_{1}^{2}\\geq 0.\n\\end{align*}\n\nThe proof of the lemma is completed.\n\n\\textbf{Theorem 4.2.} For $0<\\alpha < 1$, let $(u_{h}(t_{n}),p_{h}(t_{n}))$ and $(u_{h}^{n},p_{h}^{n})$ be the solutions of equations (3.6) and (4.4) respectively. There exists a constant $C$ such that\n\\begin{align*}\n\\|u_{h}(t_{n})-u_{h}^{n}\\|\\leq C \\tau^{\\alpha+1},~\\|p_{h}(t_{n})-p_{h}^{n}\\|\\leq C \\tau^{\\alpha+1}. \\tag{4.8}\n\\end{align*}\n\n\\textbf{Proof.} Let $\\xi^{n}=u_{h}(t_{n})-u_{h}^{n}$ and $\\eta^{n}=p_{h}(t_{n})-p_{h}^{n}$. Using (4.2)-(4.4) and noting $\\xi^{0}=0$, we deduce\n\\begin{align*}\n&(\\xi^{n},v_{h})+\\beta_{0}\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}[a(\\xi^{n-k},v_{h})+b(\\xi^{n-k},u_{h}^{n-k},v_{h})-d(v_{h},\\eta^{n-k})+d(\\xi^{n-k},q_{h})]\\\\\n&=(\\gamma_{\\alpha}^{n},v_{h}). \\tag{4.9}\n\\end{align*}\n\nFor $n=1$, taking $v_{h}=\\xi^{1}$ and $q_{h}=\\eta^{1}$ in (4.9), we have\n\\begin{align*}\n(\\xi^{1},\\xi^{1})+\\beta_{0}[a(\\xi^{1},\\xi^{1})+b(\\xi^{1},u_{h}^{1},\\xi^{1})]=(\\gamma_{\\alpha}^{1},\\xi^{1}).\n\\end{align*}\n\nBy Cauchy-Schwarz inequality and Lemma 4.3, we get\n\\begin{align*}\n\\|\\xi^{1}\\|\\leq \\|\\gamma_{\\alpha}^{1}\\|\\leq C \\tau^{\\alpha+1}. \\tag{4.10}\n\\end{align*}\n\nLet us assume that $\\|\\xi^{m}\\|\\leq C \\tau^{\\alpha+1}$ for $m=2,3,\\ldots,n-1$. In order to show that the first inequality in (4.8) holds for $m=n$, we set $v_{h}=\\xi^{n-k}$ and $q_{h}=\\eta^{n-k}$ in (4.9). Then\n\\begin{align*}\n(\\xi^{n},\\xi^{n-k})+\\beta_{0}\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}[a(\\xi^{n-k},\\xi^{n-k})+b(\\xi^{n-k},u_{h}^{n-k},\\xi^{n-k})]=(\\gamma_{\\alpha}^{n},\\xi^{n-k}).\n\\end{align*}\n\nIn view of the elementary identity $ab=\\frac{1}{2}(a^{2}+b^{2})-\\frac{1}{2}(a-b)^{2}$, Young's inequality and Lemma 4.3, we get\n\\begin{align*}\n\\frac{1}{2}[\\|\\xi^{n}\\|^{2}+\\|\\xi^{n-k}\\|^{2}]\\leq \\frac{1}{2}\\|\\xi^{n}-\\xi^{n-k}\\|^{2}+\\frac{1}{2}[\\|\\xi^{n-k}\\|^{2}+\\|\\gamma_{\\alpha}^{n}\\|^{2}], \\tag{4.11}\n\\end{align*}\nwhich implies that\n\\begin{align*}\n\\|\\xi^{n}\\|^{2}\\leq C \\tau^{2\\alpha+2}. \\tag{4.12}\n\\end{align*}\n\nBy inverse estimate (3.2) together with (4.12), we have\n\\begin{align*}\n\\|\\xi^{n}\\|_{1}\\leq Ch^{-1}\\|\\xi^{n}\\| \\leq C \\tau^{\\alpha+1}. \\tag{4.13}\n\\end{align*}\n\n\nOn the other hand, setting $v_{h}=\\xi^{1}$ and $q_{h}=0$ for $n=1$ in (4.9) and making use of Cauchy-Schwarz inequality together with (3.1)-(3.4), (3.8) and (4.13), we get\n\\begin{align*}\n\\|\\eta^{1}\\|&\\leq \\sup\\limits_{V_{h}}\\frac{|d(\\xi^{1},\\eta^{1})|}{\\lambda\\|\\xi^{1}\\|_{1}}=\\sup\\limits_{V_{h}}\\frac{|\\|\\xi^{1}\\|^{2}+\\beta_{0}[\\nu\\|\\xi^{1}\\|_{1}^{2}+b(\\xi^{1},u_{h}^{1},\\xi^{1})]-(\\gamma_{\\alpha}^{1},\\xi^{1})|}{\\lambda\\|\\xi^{1}\\|_{1}}\\\\\n&\\leq C \\tau^{\\alpha+1}. \\tag{4.14}\n\\end{align*}\n\nUsing the assumption $\\|\\eta^{m}\\|\\leq C \\tau^{\\alpha+1}$ for $m=2,3,\\ldots,n-1$ and taking $v_{h}=\\xi^{n-k}$ and $q_{h}=0$ in (4.9), by (4.12) and (4.13), similar to the derivation of (4.14) for $m=n$, we obtain\n\\begin{align*}\n\\|\\eta^{n}\\|&\\leq \\sup\\limits_{V_{h}}\\frac{|d(\\xi^{n},\\eta^{n})|}{\\lambda\\|\\xi^{n}\\|_{1}}\\leq \\sup\\limits_{V_{h}}\\frac{|\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}d(\\xi^{n-k},\\eta^{n-k})|}{\\lambda\\|\\xi^{n}\\|_{1}}\\\\\n&=\\sup\\limits_{V_{h}}\\frac{|(\\xi^{n},\\xi^{n-k})+\\beta_{0}\\sum\\limits_{k=0}^{n-1}w_{k}^{\\alpha}[\\nu\\|\\xi^{n-k}\\|_{1}^{2}+b(\\xi^{n-k},u_{h}^{n-k},\\xi^{n-k})]-(\\gamma_{\\alpha}^{n},\\xi^{n-k})|}{\\lambda\\|\\xi^{n}\\|_{1}}\\\\\n&\\leq C \\tau^{\\alpha+1}.\n\\end{align*}\n\n\nThis completes the proof.\n\nNext we give the error estimate for fully discrete scheme.\n\n\\textbf{Theorem 4.3.} For $0<\\alpha < 1$, let $(u(t_{n}),p(t_{n}))$ and $(u_{h}^{n},p_{h}^{n})$ be the solutions of equations (2.4) and (4.4) respectively. Then there exists a positive constant $C$ such that\n\\begin{align*}\n\\|u(t_{n})-u_{h}^{n}\\|\\leq C(h^{2}+\\tau^{\\alpha+1}),~\\|p(t_{n})-p_{h}^{n}\\|\\leq C(h+\\tau^{\\alpha+1}). \\tag{4.15}\n\\end{align*}\n\n\\textbf{Proof.} It is easy to show that (4.15) follows from Theorem 3.2 and Theorem 4.2 via triangle inequality.\n\n\n\\section{Numerical example}\n\nIn this section, we demonstrate the effectiveness of our numerical methods with the aid of examples. We use mixed finite element method for the discretization of spatial direction and finite difference approximation for time discretization. The convergence rates of numerical solutions with respect to space step $h$ and time step $\\tau$ are discussed. We consider the regular (uniform) domain $\\Omega=(0,1)\\times(0,1)$ and the time interval is chosen to be $[0,1]$ with the viscosity coefficient $\\nu=1.5$.\nFor an appropriate body force $f$, the analytical solution $(u,p)=((u_{1},u_{2}),p)$ of the unstable flow problem with homogeneous boundary conditions becomes\n\\begin{align*}\n&u_{1}=2x^{2}(x-1)^{2}y(y-1)(2y-1)e^{-t},~u_{2}=-2y^{2}(y-1)^{2}x(x-1)(2x-1)e^{-t},\\\\\n&p=(x^{2}-y^{2})e^{-t},\n\\end{align*}\nwhich automatically satisfy the initial and boundary conditions.\n\nThe errors $\\|e^{n}\\|$ are computed in $L^{2}$-discrete norm. The results of numerical experiments are compared with analytical solution by the rates of the convergence, which are approximately by\n\\begin{align*}\n\\mathrm{Rate }= |\\frac{\\mathrm{ln}(\\|e_{f}^{n}\\|\/\\|e_{c}^{n}\\|)}{\\mathrm{ln}(N_{f}\/N_{c})}|,\n\\end{align*}\nwhere $\\|e_{f}^{n}\\|$ and $\\|e_{c}^{n}\\|$ denote the error on finer grid and coarser grid, $N_{f}$ and $N_{c}$ represent the numbers of meshes on finer grid and coarser grid, respectively.\n\nThe spatial convergence rates for the components of velocity $(u_{1},u_{2})$ and pressure $p$ with fixed time step $\\tau=1\/8$ with different values of $\\alpha$ are shown in Fig.1. The convergence rates of velocity $(u_{1},u_{2})$ are in accordance with spatial convergence order $\\mathcal{O}(h^{2})$ and the pressure $p$ are closer to order $\\mathcal{O}(h)$. Fig.2 give the temporal convergence rates for the components of velocity and pressure with fixed spatial step $h=1\/15$ with different values of $\\alpha$. We can see that the rates of convergence are closer to the theoretical convergence order $\\mathcal{O}(\\tau^{\\alpha+1})$.\n\n\nFig.3 depicts the numerical solutions of the components of velocity $(u_{1},u_{2})$ and pressure $p$, with $h=1\/15$ and $\\tau=0.1$, when $\\alpha=0.4$ and $\\alpha=0.8$, respectively. It is not difficult to find that a pair of warm- and cold-core eddies emerge in the velocity field.\n\n\n\n\n\n\n\n\n\\bigskip\n\\section{Conclusion}\n\nIn this study, the finite difference\/element method is presented to solve the TFNSEs and the convergence error estimates for the discrete schemes in $L^{2}$-norm are obtained. We present the numerical experiment to illustrate the accuracy of schemes, and the result fully verify the convergence theory. The numerical examples also confirm the thesis [18,19,20] that in\nprocedure of citing and novelty of the obtained results. Furthermore, the presented methods and analytical techniques in this work can also be extended to other nonlinear time-fractional partial differential equations.\n\n\n\n\n\n\n\\section{Acknowledgment}\nGuang-an Zou is supported by National Nature Science Foundation of China (Grant No. 11626085), Yong Zhou is supported by National Nature Science Foundation of China (Grant No. 11671339).\n\n\\section*{References}\n\n[1] Lemari\\'{e}-Rieusset, P.G. Recent developments in the Navier-Stokes problem, Chapman Hall\/CRC Research Notes in Mathematics, 431. Chapman Hall\/CRC, Boca Raton, FL, 2002, 395 p.\n\n[2] Chemin, J.Y., Gallagher, I., Paicu, M. Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. of Math. (2), V.173, N.2, 2011, pp.983-1012.\n\n\n[3] Miura, H. Remark on uniqueness of mild solutions to the Navier-Stokes equations, J. Funct. Anal., V.218, N.1, 2005, pp.110-129.\n\n[4] Germain, P. Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations, J. Differential Equations, V.226, N.2, 2006, pp.373-428.\n\n[5] Nahmod, A.R., Pavlovic N., Staffilani, G. Almost sure existence of global weak solutions for supercritical Navier-Stokes equations, SIAM J. Math. Anal. V.45, N.6, 2013, pp.3431-3452.\n\n[6] Robinson, J.C., Sadowski, W., Silva, R.P. Lower bounds on blow up solutions of the three-dimensional Navier-Stokes equations in homogeneous Sobolev spaces, J. Math. Phys., V.53, N.11, 2012, 115618, 15 pp.\n\n[7] Ingram, R. A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations, Math. Comp., V.82, N.284, 2013, pp.1953-1973.\n\n[8] Bernardi, C., Raugel, G.A. conforming finite element method for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., V.22, N.3, 1985, pp.455-473.\n\n[9] He, Y., Sun, W. Stability and convergence of the Crank-Nicolson\/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., V.45, N.2, 2007, pp.837-869.\n\n[10] Girault, V., Raviart, P.A. Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer-Verlag, Berlin, 1986, 374 p.\n\n\n[11] Kaya, S., Rivi\\`{e}re, B. A discontinuous subgrid eddy viscosity method for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., V.43, N.4, 2005, pp.1572-1595.\n\n[12] Shan, L., Hou, Y. A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations, Appl. Math. Comput., V.215, N.1, 2009, pp.85-99.\n\n[13] Momani, S., Odibat, Z. Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., V.177, N.2, 2006, pp.488-494.\n\n[14] Ganji, Z.Z., Ganji, D.D., Ganji, Ammar D., Rostamian, M. Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method, Numer. Methods Partial Differential Equations, V.26, N.1, 2010, pp.117-124.\n\n[15] Kumar, D., Singh, J., Kumar, S. A fractional model of Navier-Stokes equation arising in unsteady flow of a viscous fluid, J. Ass. Arab Univ. Basic Appl. Sci., V.17, 2015, pp.14-19.\n\n[16] Wang, K., Liu, S. Analytical study of time-fractional Navier-Stokes equation by using transform methods, Adv. Differential Equ., N.61, 2016, pp.12.\n\n[17] De Carvalho-Neto, P.M., Gabriela, P. Mild solutions to the time fractional Navier-Stokes equations in $R^{N}$, J. Differential Equations, V.259, N.7, pp.2948-2980.\n\n[18] Zhou, Y., Peng, L. On the time-fractional Navier-Stokes equations, Comput. Math. Appl., V.73, N.6, 2017, pp.874-891.\n\n[19] Zhou, Y., Peng, L. Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl., V.73, N.6, 2017, pp.1016-1027.\n\n[20] Peng, L., Zhou, Y., Ahmad, B., Alsaedi, A. The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces, Chaos Solitons Fractals, V.102, 2017, pp.218-228.\n\n[21] Thamareerat, N., Luadsong, A., Aschariyaphotha, N. The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations, SpringerPlus, V.5, N.417, 2016, pp.19.\n\n[22] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and applications of fractional differential equations, Elsevier, 2006, 523 p.\n\n[23] Layton, W.J., Labovschii, A., Manica, C.C., Neda, M., Rebholz, L. G. The stabilized, extrapolated\ntrapezoidal finite element method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., V.198, 2009, pp.958-974.\n\n[24] Kruse R. Strong and weak approximation of semilinear stochastic evolution equations, Springer, Cham, 2014, 177 p.\n\n[25] Zeng, F., Li, C., Liu, F., Turner, I. The use of finite difference\/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., V.35, N.6, 2013, pp.A2976-A3000.\n\n[26] Stynes, M., O'Riordan, E., Gracia, J.L. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., V.55, N.2, pp.2017, 1057-1079.\n\n[27] Stynes, M., Gracia, J.L. Preprocessing schemes for fractional-derivative problems to improve their convergence rates, Appl. Math. Lett., V.74, 2017, pp.187-192.\n\n[28] Kopteva, N., Stynes, M. Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem, Adv. Comput. Math., V.43, N.1, 2017, 77-99.\n\n[29] Zeng, F., Li, C., Liu, F., Turner, I. Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput., V.37, N.1, 2015, pp.A55-A78.\n\n[30] Zheng, M., Liu, F., Liu, Q., Burrage, K., Simpson, M.J. Numerical solution of the time fractional reaction-diffusion equation with a moving boundary, J. Comput. Phys., V.338, 2017, pp.493-510.\n\n[31] Cui, M.R. Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, J. Comput. Phys., V.231, N.6, 2012, pp.2621-2633.\n\n[32] Jiang, Y.J., Ma, J.T. High-order finite element methods for time-fractional partial differential\nequations, J. Comput. Appl. Math., V.235, N.11, 2011, 3285-3290.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Overview}\n\n[This document will be expanded when \\BibTeX\\ version 1.00 comes out.\nPlease report typos, omissions, inaccuracies,\nand especially unclear explanations\nto {\\tt biblio@tug.org} ({\\tt http:\/\/lists.tug.org\/biblio}).\nSuggestions for improvements are wanted and welcome.]\n\nThis documentation, for \\BibTeX\\ version 0.99b,\nis meant for general \\BibTeX\\ users;\nbibliography-style designers should read this document\nand then read ``Designing \\BibTeX\\ Styles''~\\cite{btxhak},\nwhich is meant for just them.\n\nThis document has three parts:\nSection~\\ref{differences}\ndescribes the differences between versions 0.98i and 0.99b\nof \\BibTeX\\ and between the corresponding versions of the standard styles;\nSection~\\ref{latex-appendix}\nupdates Appendix~B.2 of the \\LaTeX\\ book~\\cite{latex};\nand Section~\\ref{odds-and-ends}\ngives some general and specific tips\nthat aren't documented elsewhere.\nIt's assumed throughout that you're familiar with\nthe relevant sections of the \\LaTeX\\ book.\n\nThis documentation also serves as sample input to help\n\\BibTeX\\ implementors get it running.\nFor most documents, this one included, you produce the reference list by:\nrunning \\LaTeX\\ on the document (to produce the {\\tt aux} file(s)),\nthen running \\BibTeX\\ (to produce the {\\tt bbl} file),\nthen \\LaTeX\\ twice more (first to find the information in the {\\tt bbl} file\nand then to get the forward references correct).\nIn very rare circumstances you may need an extra \\BibTeX\/\\LaTeX\\ run.\n\n\\BibTeX\\ version 0.99b should be used with \\LaTeX\\ version 2.09,\nfor which the closed bibliography format is the default;\nto get the open format, use the optional document style {\\tt openbib}\n(in an open format there's a line break between major blocks of a\nreference-list entry; in a closed format the blocks run together).]\n\nNote: \\BibTeX\\ 0.99b is not compatible with the old style files;\nnor is \\BibTeX\\ 0.98i compatible with the new ones\n(the new \\BibTeX, however, is compatible with old database files).\n\nNote for implementors: \\BibTeX\\ provides logical-area names\n\\hbox{\\tt TEXINPUTS:} for bibliography-style files and\n\\hbox{\\tt TEXBIB:} for database files it can't otherwise find.\n\n\n\\section{Changes}\n\\label{differences}\n\nThis section describes the differences between\n\\BibTeX\\ versions 0.98i and 0.99b, and also between\nthe corresponding standard styles.\nThere were a lot of differences;\nthere will be a lot fewer between 0.99 and 1.00.\n\n\n\\subsection{New \\BibTeX\\ features}\n\\label{features}\n\nThe following list explains \\BibTeX's new features and how to use them.\n\\begin{enumerate}\n\n\\item\nWith the single command `\\hbox{\\verb|\\nocite{*}|}'\nyou can now include in the reference list\nevery entry in the database files, without having to explicitly\n\\verb|\\cite| or \\hbox{\\verb|\\nocite|} each entry.\nGiving this command, in essence,\n\\hbox{\\verb|\\nocite|}s\nall the enties in the database, in database order,\nat the very spot in your document\nwhere you give the command.\n\n\\item\n\\label{concat}\nYou can now have as a field value (or an {\\tt @STRING} definition)\nthe concatenation of several strings.\nFor example if you've defined\n\\begin{verbatim}\n @STRING( WGA = \" World Gnus Almanac\" )\n\\end{verbatim}\nthen it's easy to produce nearly-identical\n{\\tt title} fields for different entries:\n\\begin{verbatim}\n @BOOK(almanac-66,\n title = 1966 # WGA,\n . . .\n @BOOK(almanac-67,\n title = 1967 # WGA,\n\\end{verbatim}\nand so on. Or, you could have a field like\n\\begin{verbatim}\n month = \"1~\" # jan,\n\\end{verbatim}\nwhich would come out something like\n`\\hbox{\\verb|1~January|}' or `\\hbox{\\verb|1~Jan.|}' in the {\\tt bbl} file,\ndepending on how your bibliography style defines\nthe {\\tt jan} abbreviation.\nYou may concatenate as many strings as you like\n(except that there's a limit to the overall length\nof the resulting field);\njust be sure to put the concatenation character `{\\tt\\#}'$\\!$,\nsurrounded by optional spaces or newlines,\nbetween each successive pair of strings.\n\n\\item\n\\BibTeX\\ has a new cross-referencing feature,\nexplained by an example.\nSuppose you say \\hbox{\\verb|\\cite{no-gnats}|} in your document,\nand suppose you have these two entries in your database file:\n\\begin{verbatim}\n @INPROCEEDINGS(no-gnats,\n crossref = \"gg-proceedings\",\n author = \"Rocky Gneisser\",\n title = \"No Gnats Are Taken for Granite\",\n pages = \"133-139\")\n . . .\n @PROCEEDINGS(gg-proceedings,\n editor = \"Gerald Ford and Jimmy Carter\",\n title = \"The Gnats and Gnus 1988 Proceedings\",\n booktitle = \"The Gnats and Gnus 1988 Proceedings\")\n\\end{verbatim}\nTwo things happen.\nFirst, the special \\hbox{\\tt crossref} field tells \\BibTeX\\\nthat the \\hbox{\\tt no-gnats} entry should inherit\nany fields it's missing from\nthe entry it cross references, \\hbox{\\tt gg-proceedings}.\nIn this case it in inherits the two fields\n\\hbox{\\tt editor} and \\hbox{\\tt booktitle}.\nNote that, in the standard styles at least,\nthe \\hbox{\\tt booktitle} field is irrelevant\nfor the \\hbox{\\tt PROCEEDINGS} entry type.\nThe \\hbox{\\tt booktitle} field appears here\nin the \\hbox{\\tt gg-proceedings} entry\nonly so that the entries that cross reference it\nmay inherit the field.\nNo matter how many papers from this meeting exist in the database,\nthis \\hbox{\\tt booktitle} field need only appear once.\n\nThe second thing that happens:\n\\BibTeX\\ automatically puts the entry \\hbox{\\tt gg-proceedings}\ninto the reference list if it's cross\nreferenced by two or more entries that you\n\\verb|\\cite| or \\hbox{\\verb|\\nocite|},\neven if you don't \\verb|\\cite| or \\hbox{\\verb|\\nocite|}\nthe \\hbox{\\tt gg-proceedings} entry itself.\nSo \\hbox{\\tt gg-proceedings} will automatically appear\non the reference list if one other entry\nbesides \\hbox{\\tt no-gnats} cross references it.\n\nTo guarantee that this scheme works, however,\na cross-referenced entry must occur later in the database files\nthan every entry that cross-references it.\nThus, putting all cross-referenced entries at the end makes sense.\n(Moreover, you may not reliably nest cross references;\nthat is, a cross-referenced entry may\nnot itself reliably cross reference an entry.\nThis is almost certainly not something you'd\nwant to do, though.)\n\nOne final note:\nThis cross-referencing feature is completely unrelated\nto the old \\BibTeX's cross referencing,\nwhich is still allowed.\nThus, having a field like\n\\begin{verbatim}\n note = \"Jones \\cite{jones-proof} improves the result\"\n\\end{verbatim}\nis not affected by the new feature.\n\n\\item\n\\BibTeX\\ now handles accented characters.\nFor example if you have an entry with the two fields\n\\begin{verbatim}\n author = \"Kurt G{\\\"o}del\",\n year = 1931,\n\\end{verbatim}\nand if you're using the \\hbox{\\tt alpha} bibliography style,\nthen \\BibTeX\\ will construct the label\n\\hbox{[G{\\\"o}d31]} for this entry, which is what you'd want.\nTo get this feature to work you must place the entire accented\ncharacter in braces;\nin this case either \\hbox{\\verb|{\\\"o}|}\nor \\hbox{\\verb|{\\\"{o}}|} will do.\nFurthermore these braces must not themselves be\nenclosed in braces (other than the ones that might delimit\nthe entire field or the entire entry);\nand there must be a backslash\nas the very first character inside the braces.\nThus neither \\hbox{\\verb|{G{\\\"{o}}del}|}\nnor \\hbox{\\verb|{G\\\"{o}del}|} will work for this example.\n\nThis feature handles all the accented characters and\nall but the nonbackslashed foreign symbols found in Tables\n3.1 and~3.2 of the \\LaTeX\\ book.\nThis feature behaves similarly for ``accents'' you might define;\nwe'll see an example shortly.\nFor the purposes of counting letters in labels,\n\\BibTeX\\ considers everything contained inside the braces\nas a single letter.\n\n\\item\n\\BibTeX\\ also handles hyphenated names.\nFor example if you have an entry with\n\\begin{verbatim}\n author = \"Jean-Paul Sartre\",\n\\end{verbatim}\nand if you're using the \\hbox{\\tt abbrv} style,\nthen the result is `J.-P. Sartre'$\\!$.\n\n\\item\n\\label{preamble}\nThere's now an \\hbox{\\verb|@PREAMBLE|} command\nfor the database files.\nThis command's syntax is just like \\hbox{\\verb|@STRING|}'s,\nexcept that there is no name or equals-sign, just the string.\nHere's an example:\n\\begin{verbatim}\n @PREAMBLE{ \"\\newcommand{\\noopsort}[1]{} \"\n # \"\\newcommand{\\singleletter}[1]{#1} \" }\n\\end{verbatim}\n(note the use of concatenation here, too).\nThe standard styles output whatever information you give this command\n(\\LaTeX\\ macros most likely) directly to the {\\tt bbl} file.\nWe'll look at one possible use of this command,\nbased on the \\hbox{\\verb|\\noopsort|} command just defined.\n\nThe issue here is sorting (alphabetizing).\n\\BibTeX\\ does a pretty good job,\nbut occasionally weird circumstances conspire to confuse \\BibTeX:\nSuppose that you have entries in your database for\nthe two books in a two-volume set by the same author,\nand that you'd like volume~1 to appear\njust before volume~2 in your reference list.\nFurther suppose that there's now a second edition of volume~1,\nwhich came out in 1973, say,\nbut that there's still just one edition of volume~2,\nwhich came out in 1971.\nSince the {\\tt plain} standard style\nsorts by author and then year,\nit will place volume~2 first\n(because its edition came out two years earlier)\nunless you help \\BibTeX.\nYou can do this by using the {\\tt year} fields below\nfor the two volumes:\n\\begin{verbatim}\n year = \"{\\noopsort{a}}1973\"\n . . .\n year = \"{\\noopsort{b}}1971\"\n\\end{verbatim}\nAccording to the definition of \\hbox{\\verb|\\noopsort|},\n\\LaTeX\\ will print nothing but the true year for these fields.\nBut \\BibTeX\\ will be perfectly happy pretending that\n\\hbox{\\verb|\\noopsort|} specifies some fancy accent\nthat's supposed to adorn the `a' and the~`b';\nthus when \\BibTeX\\ sorts it will pretend that\n`a1973' and `b1971' are the real years,\nand since `a' comes before~`b'$\\!$, it will place volume~1 before volume~2,\njust what you wanted.\nBy the way, if this author has any other works included\nin your database, you'd probably want to use instead something like\n\\hbox{\\verb|{\\noopsort{1968a}}1973|} and\n\\hbox{\\verb|{\\noopsort{1968b}}1971|},\nso that these two books would come out in a reasonable spot\nrelative to the author's other works\n(this assumes that 1968 results in a reasonable spot,\nsay because that's when the first edition of volume~1 appeared).\n\nThere is a limit to the number of \\hbox{\\verb|@PREAMBLE|} commands\nyou may use, but you'll never exceed this limit if\nyou restrict yourself to one per database file;\nthis is not a serious restriction,\ngiven the concatenation feature (item~\\ref{concat}).\n\n\\item\n\\BibTeX's sorting algorithm is now stable.\nThis means that if two entries have identical sort keys,\nthose two entries will appear in citation order.\n(The bibliography styles construct these sort keys---%\nusually the author information followed by the year and the title.)\n\n\\item\n\\BibTeX\\ no longer does case conversion for file names;\nthis will make \\BibTeX\\ easier to install on Unix systems, for example.\n\n\\item\nIt's now easier to add code for processing a\ncommand-line {\\tt aux}-file name.\n\n\\end{enumerate}\n\n\n\\subsection{Changes to the standard styles}\n\nThis section describes changes to the standard styles\n({\\tt plain}, {\\tt unsrt}, {\\tt alpha}, {\\tt abbrv})\nthat affect ordinary users.\nChanges that affect style designers appear in\nthe document ``Designing \\BibTeX\\ Styles''~\\cite{btxhak}.\n\\begin{enumerate}\n\n\\item\nIn general, sorting is now by ``author''$\\!$, then year, then title---%\nthe old versions didn't use the year field.\n(The {\\tt alpha} style, however, sorts first by label,\nthen ``author''$\\!$, year, and title.)\nThe quotes around author mean that some entry types\nmight use something besides the author, like the editor or organization.\n\n\\item\nMany unnecessary ties (\\verb|~|) have been removed.\n\\LaTeX\\ thus will produce slightly fewer\n`\\hbox{\\tt Underfull} \\verb|\\hbox|' messages\nwhen it's formatting the reference list.\n\n\\item\nEmphasizing (\\hbox{\\verb|{\\em ...}|})\nhas replaced italicizing (\\hbox{\\verb|{\\it ...}|}).\nThis will almost never result in a difference\nbetween the old output and the new.\n\n\\item\nThe {\\tt alpha} style now uses a superscripted~`$^{+}$' instead of a~`*'\nto represent names omitted in constructing the label.\nIf you really liked it the way it was, however,\nor if you want to omit the character entirely,\nyou don't have to modify the style file---%\nyou can override the~`$^{+}$' by\nredefining the \\hbox{\\verb|\\etalchar|} command\nthat the {\\tt alpha} style writes onto the {\\tt bbl} file\n(just preceding the \\hbox{\\verb|\\thebibliography|} environment);\nuse \\LaTeX's \\hbox{\\verb|\\renewcommand|} inside\na database \\hbox{\\tt @PREAMBLE} command,\ndescribed in the previous subsection's item~\\ref{preamble}.\n\n\\item\nThe {\\tt abbrv} style now uses `Mar.' and `Sept.'\\\nfor those months rather than `March' and `Sep.'\n\n\\item\nThe standard styles use \\BibTeX's new cross-referencing feature\nby giving a \\verb|\\cite| of the cross-referenced entry and by\nomitting from the cross-referencing entry\n(most of the) information that appears\nin the cross-referenced entry.\nThese styles do this when\na titled thing (the cross-referencing entry)\nis part of a larger titled thing (the cross-referenced entry).\nThere are five such situations:\nwhen (1)~an \\hbox{\\tt INPROCEEDINGS}\n(or \\hbox{\\tt CONFERENCE}, which is the same)\ncross references a \\hbox{\\tt PROCEEDINGS};\nwhen (2)~a {\\tt BOOK}, (3)~an \\hbox{\\tt INBOOK},\nor (4)~an \\hbox{\\tt INCOLLECTION}\ncross references a {\\tt BOOK}\n(in these cases, the cross-referencing entry is a single\nvolume in a multi-volume work);\nand when (5)~an \\hbox{\\tt ARTICLE}\ncross references an \\hbox{\\tt ARTICLE}\n(in this case, the cross-referenced entry is really a journal,\nbut there's no \\hbox{\\tt JOURNAL} entry type;\nthis will result in warning messages about\nan empty \\hbox{\\tt author} and \\hbox{\\tt title} for the journal---%\nyou should just ignore these warnings).\n\n\\item\nThe \\hbox{\\tt MASTERSTHESIS} and \\hbox{\\tt PHDTHESIS}\nentry types now take an optional {\\tt type} field.\nFor example you can get the standard styles to\ncall your reference a `Ph.D.\\ dissertation'\ninstead of the default `PhD thesis' by including a\n\\begin{verbatim}\n type = \"{Ph.D.} dissertation\"\n\\end{verbatim}\nin your database entry.\n\n\\item\nSimilarly, the \\hbox{\\tt INBOOK} and \\hbox{\\tt INCOLLECTION}\nentry types now take an optional {\\tt type} field,\nallowing `section~1.2' instead of the default `chapter~1.2'$\\!$.\nYou get this by putting\n\\begin{verbatim}\n chapter = \"1.2\",\n type = \"Section\"\n\\end{verbatim}\nin your database entry.\n\n\\item\nThe \\hbox{\\tt BOOKLET}, \\hbox{\\tt MASTERSTHESIS},\nand \\hbox{\\tt TECHREPORT} entry types now format\ntheir \\hbox{\\tt title} fields as if they were\n\\hbox{\\tt ARTICLE} \\hbox{\\tt title}s\nrather than \\hbox{\\tt BOOK} \\hbox{\\tt title}s.\n\n\\item\nThe \\hbox{\\tt PROCEEDINGS} and \\hbox{\\tt INPROCEEDINGS}\nentry types now use the \\hbox{\\tt address} field\nto tell where a conference was held,\nrather than to give the address\nof the publisher or organization.\nIf you want to include the\npublisher's or organization's address,\nput it in the \\hbox{\\tt publisher}\nor \\hbox{\\tt organization} field.\n\n\\item\nThe \\hbox{\\tt BOOK}, \\hbox{\\tt INBOOK}, \\hbox{\\tt INCOLLECTION},\nand \\hbox{\\tt PROCEEDINGS} entry types now allow either\n\\hbox{\\tt volume} or \\hbox{\\tt number} (but not both),\nrather than just \\hbox{\\tt volume}.\n\n\\item\nThe \\hbox{\\tt INCOLLECTION} entry type now allows\na \\hbox{\\tt series} and an \\hbox{\\tt edition} field.\n\n\\item\nThe \\hbox{\\tt INPROCEEDINGS} and \\hbox{\\tt PROCEEDINGS}\nentry types now allow either a \\hbox{\\tt volume} or \\hbox{\\tt number},\nand also a \\hbox{\\tt series} field.\n\n\\item\nThe \\hbox{\\tt UNPUBLISHED} entry type now outputs,\nin one block, the \\hbox{\\tt note} field\nfollowed by the date information.\n\n\\item\nThe \\hbox{\\tt MANUAL} entry type now prints out\nthe \\hbox{\\tt organization} in the first block\nif the \\hbox{\\tt author} field is empty.\n\n\\item\nThe {\\tt MISC} entry type now issues a warning\nif all the optional fields are empty\n(that is, if the entire entry is empty).\n\n\\end{enumerate}\n\n\n\\section{The Entries}\n\\label{latex-appendix}\n\nThis section is simply a corrected version of\nAppendix~B.2 of the \\LaTeX\\ book~\\cite{latex},\n\\copyright~1986, by Addison-Wesley.\nThe basic scheme is the same, only a few details have changed.\n\n\n\\subsection{Entry Types}\n\nWhen entering a reference in the database, the first thing to decide\nis what type of entry it is. No fixed classification scheme can be\ncomplete, but \\BibTeX\\ provides enough entry types to handle almost\nany reference reasonably well.\n\nReferences to different types of publications contain different\ninformation; a reference to a journal article might include the volume\nand number of the journal, which is usually not meaningful for a book.\nTherefore, database entries of different types have different fields.\nFor each entry type, the fields are divided into three classes:\n\\begin{description}\n\n\\item[required]\nOmitting the field will produce a warning message\nand, rarely, a badly formatted bibliography entry.\nIf the required information is not meaningful,\nyou are using the wrong entry type.\nHowever, if the required information is meaningful\nbut, say, already included is some other field,\nsimply ignore the warning.\n\n\\item[optional]\nThe field's information will be used if present,\nbut can be omitted without causing any formatting problems.\nYou should include the optional field if it will help the reader.\n\n\\item[ignored]\nThe field is ignored.\n\\BibTeX\\ ignores any field that is not required or optional, so you can include\nany fields you want in a \\hbox{\\tt bib} file entry. It's a good idea\nto put all relevant information about\na reference in its \\hbox{\\tt bib} file entry---even information that\nmay never appear in the bibliography. For example, if you want to\nkeep an abstract of a paper in a computer file, put it in an \\hbox{\\tt\nabstract} field in the paper's \\hbox{\\tt bib} file entry. The\n\\hbox{\\tt bib} file is likely to be as good a place as any for the\nabstract, and it is possible to design a bibliography style for\nprinting selected abstracts.\nNote: Misspelling a field name will\nresult in its being ignored,\nso watch out for typos\n(especially for optional fields,\nsince \\BibTeX\\ won't warn you when those are missing).\n\n\\end{description}\n\nThe following are the standard entry types, along with their required\nand optional fields, that are used by the standard bibliography styles.\nThe fields within each class (required or optional)\nare listed in order of occurrence in the output,\nexcept that a few entry types may perturb the order slightly,\ndepending on what fields are missing.\nThese entry types are similar to those adapted by Brian Reid\nfrom the classification scheme of van~Leunen~\\cite{van-leunen}\nfor use in the {\\em Scribe\\\/} system.\nThe meanings of the individual fields are explained in the next section.\nSome nonstandard bibliography styles may ignore some optional fields\nin creating the reference.\nRemember that, when used in the \\hbox{\\tt bib}\nfile, the entry-type name is preceded by an \\hbox{\\tt @} character.\n\n\\begin{description}\n\\sloppy\n\n\\item[article\\hfill] An article from a journal or magazine.\nRequired fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt journal},\n\\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt volume}, \\hbox{\\tt number},\n\\hbox{\\tt pages}, \\hbox{\\tt month}, \\hbox{\\tt note}.\n\n\\item[book\\hfill] A book with an explicit publisher.\nRequired fields: \\hbox{\\tt author} or \\hbox{\\tt editor},\n\\hbox{\\tt title}, \\hbox{\\tt publisher}, \\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt volume} or \\hbox{\\tt number}, \\hbox{\\tt series},\n\\hbox{\\tt address}, \\hbox{\\tt edition}, \\hbox{\\tt month},\n\\hbox{\\tt note}.\n\n\\item[booklet\\hfill] A work that is printed and bound,\nbut without a named publisher or sponsoring institution.\nRequired field: \\hbox{\\tt title}.\nOptional fields: \\hbox{\\tt author}, \\hbox{\\tt howpublished},\n\\hbox{\\tt address}, \\hbox{\\tt month}, \\hbox{\\tt year}, \\hbox{\\tt note}.\n\n\\item[conference\\hfill] The same as {\\tt INPROCEEDINGS},\nincluded for {\\em Scribe\\\/} compatibility.\n\n\\item[inbook\\hfill] A part of a book,\nwhich may be a chapter (or section or whatever) and\/or a range of pages.\nRequired fields: \\hbox{\\tt author} or \\hbox{\\tt editor}, \\hbox{\\tt title},\n\\hbox{\\tt chapter} and\/or \\hbox{\\tt pages}, \\hbox{\\tt publisher},\n\\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt volume} or \\hbox{\\tt number}, \\hbox{\\tt series},\n\\hbox{\\tt type}, \\hbox{\\tt address},\n\\hbox{\\tt edition}, \\hbox{\\tt month}, \\hbox{\\tt note}.\n\n\\item[incollection\\hfill] A part of a book having its own title.\nRequired fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt booktitle},\n\\hbox{\\tt publisher}, \\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt editor}, \\hbox{\\tt volume} or \\hbox{\\tt number},\n\\hbox{\\tt series}, \\hbox{\\tt type}, \\hbox{\\tt chapter}, \\hbox{\\tt pages},\n\\hbox{\\tt address}, \\hbox{\\tt edition}, \\hbox{\\tt month}, \\hbox{\\tt note}.\n\n\\item[inproceedings\\hfill] An article in a conference proceedings.\nRequired fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt booktitle},\n\\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt editor}, \\hbox{\\tt volume} or \\hbox{\\tt number},\n\\hbox{\\tt series}, \\hbox{\\tt pages}, \\hbox{\\tt address}, \\hbox{\\tt month},\n\\hbox{\\tt organization}, \\hbox{\\tt publisher}, \\hbox{\\tt note}.\n\n\\item[manual\\hfill] Technical documentation. Required field: \\hbox{\\tt title}.\nOptional fields: \\hbox{\\tt author}, \\hbox{\\tt organization},\n\\hbox{\\tt address}, \\hbox{\\tt edition}, \\hbox{\\tt month}, \\hbox{\\tt year},\n\\hbox{\\tt note}.\n\n\\item[mastersthesis\\hfill] A Master's thesis.\nRequired fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt school},\n\\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt type}, \\hbox{\\tt address}, \\hbox{\\tt month},\n\\hbox{\\tt note}.\n\n\\item[misc\\hfill] Use this type when nothing else fits.\nRequired fields: none.\nOptional fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt howpublished},\n\\hbox{\\tt month}, \\hbox{\\tt year}, \\hbox{\\tt note}.\n\n\\item[phdthesis\\hfill] A PhD thesis.\nRequired fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt school},\n\\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt type}, \\hbox{\\tt address}, \\hbox{\\tt month},\n\\hbox{\\tt note}.\n\n\\item[proceedings\\hfill] The proceedings of a conference.\nRequired fields: \\hbox{\\tt title}, \\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt editor}, \\hbox{\\tt volume} or \\hbox{\\tt number},\n\\hbox{\\tt series}, \\hbox{\\tt address}, \\hbox{\\tt month},\n\\hbox{\\tt organization}, \\hbox{\\tt publisher}, \\hbox{\\tt note}.\n\n\n\\item[techreport\\hfill] A report published by a school or other institution,\nusually numbered within a series.\nRequired fields: \\hbox{\\tt author},\n\\hbox{\\tt title}, \\hbox{\\tt institution}, \\hbox{\\tt year}.\nOptional fields: \\hbox{\\tt type}, \\hbox{\\tt number}, \\hbox{\\tt address},\n\\hbox{\\tt month}, \\hbox{\\tt note}.\n\n\\item[unpublished\\hfill] A document having an author and title,\nbut not formally published.\nRequired fields: \\hbox{\\tt author}, \\hbox{\\tt title}, \\hbox{\\tt note}.\nOptional fields: \\hbox{\\tt month}, \\hbox{\\tt year}.\n\n\\end{description}\n\nIn addition to the fields listed above, each entry type also has an\noptional \\hbox{\\tt key} field, used in some styles\nfor alphabetizing, for cross referencing,\nor for forming a \\hbox{\\verb|\\bibitem|} label.\nYou should include a \\hbox{\\tt key} field for any entry whose\n``author'' information is missing;\nthe ``author'' information is usually the \\hbox{\\tt author} field,\nbut for some entry types it can be the \\hbox{\\tt editor}\nor even the \\hbox{\\tt organization} field\n(Section~\\ref{odds-and-ends} describes this in more detail).\nDo not confuse the \\hbox{\\tt key} field with the key that appears in the\n\\hbox{\\verb|\\cite|} command and at the beginning of the database entry;\nthis field is named ``key'' only for compatibility with {\\it Scribe}.\n\n\n\\subsection{Fields}\n\nBelow is a description of all fields\nrecognized by the standard bibliography styles.\nAn entry can also contain other fields, which are ignored by those styles.\n\\begin{description}\n\n\\item[address\\hfill]\nUsually the address of the \\hbox{\\tt publisher} or other type\nof institution.\nFor major publishing houses,\nvan~Leunen recommends omitting the information entirely.\nFor small publishers, on the other hand, you can help the\nreader by giving the complete address.\n\n\\item[annote\\hfill]\nAn annotation.\nIt is not used by the standard bibliography styles,\nbut may be used by others that produce an annotated bibliography.\n\n\\item[author\\hfill]\nThe name(s) of the author(s),\nin the format described in the \\LaTeX\\ book.\n\n\\item[booktitle\\hfill]\nTitle of a book, part of which is being cited.\nSee the \\LaTeX\\ book for how to type titles.\nFor book entries, use the \\hbox{\\tt title} field instead.\n\n\\item[chapter\\hfill]\nA chapter (or section or whatever) number.\n\n\\item[crossref\\hfill]\nThe database key of the entry being cross referenced.\n\n\\item[edition\\hfill]\nThe edition of a book---for example, ``Second''$\\!$.\nThis should be an ordinal, and\nshould have the first letter capitalized, as shown here;\nthe standard styles convert to lower case when necessary.\n\n\\item[editor\\hfill]\nName(s) of editor(s), typed as indicated in the \\LaTeX\\ book.\nIf there is also an \\hbox{\\tt author} field, then\nthe \\hbox{\\tt editor} field gives the editor of the book or collection\nin which the reference appears.\n\n\\item[howpublished\\hfill]\nHow something strange has been published.\nThe first word should be capitalized.\n\n\\item[institution\\hfill]\nThe sponsoring institution of a technical report.\n\n\\item[journal\\hfill]\nA journal name.\nAbbreviations are provided for many journals; see the {\\it Local Guide}.\n\n\\item[key\\hfill]\nUsed for alphabetizing, cross referencing, and creating a label when\nthe ``author'' information\n(described in Section~\\ref{odds-and-ends}) is missing.\nThis field should not be confused with the key that appears in the\n\\hbox{\\verb|\\cite|} command and at the beginning of the database entry.\n\n\\item[month\\hfill]\nThe month in which the work was\npublished or, for an unpublished work, in which it was written.\nYou should use the standard three-letter abbreviation,\nas described in Appendix B.1.3 of the \\LaTeX\\ book.\n\n\\item[note\\hfill]\nAny additional information that can help the reader.\nThe first word should be capitalized.\n\n\\item[number\\hfill]\nThe number of a journal, magazine, technical report,\nor of a work in a series.\nAn issue of a journal or magazine is usually\nidentified by its volume and number;\nthe organization that issues a\ntechnical report usually gives it a number;\nand sometimes books are given numbers in a named series.\n\n\\item[organization\\hfill]\nThe organization that sponsors a conference or that publishes a \\hbox{manual}.\n\n\\item[pages\\hfill]\nOne or more page numbers or range of numbers,\nsuch as \\hbox{\\tt 42--111} or \\hbox{\\tt 7,41,73--97} or \\hbox{\\tt 43+}\n(the `{\\tt +}' in this last example indicates pages following\nthat don't form a simple range).\nTo make it easier to maintain {\\em Scribe\\\/}-compatible databases,\nthe standard styles convert a single dash (as in \\hbox{\\tt 7-33})\nto the double dash used in \\TeX\\ to denote number ranges\n(as in \\hbox{\\tt 7--33}).\n\n\\item[publisher\\hfill]\nThe publisher's name.\n\n\\item[school\\hfill]\nThe name of the school where a thesis was written.\n\n\\item[series\\hfill]\nThe name of a series or set of books.\nWhen citing an entire book, the the \\hbox{\\tt title} field\ngives its title and an optional \\hbox{\\tt series} field gives the\nname of a series or multi-volume set\nin which the book is published.\n\n\\item[title\\hfill]\nThe work's title, typed as explained in the \\LaTeX\\ book.\n\n\\item[type\\hfill]\nThe type of a technical report---for example,\n``Research Note''$\\!$.\n\n\\item[volume\\hfill]\nThe volume of a journal or multivolume book.\n\n\\item[year\\hfill]\nThe year of publication or, for\nan unpublished work, the year it was written.\nGenerally it should consist of four numerals, such as {\\tt 1984},\nalthough the standard styles can handle any {\\tt year} whose\nlast four nonpunctuation characters are numerals,\nsuch as `\\hbox{(about 1984)}'$\\!$.\n\n\\end{description}\n\n\n\\section{Helpful Hints}\n\\label{odds-and-ends}\n\nThis section gives some random tips\nthat aren't documented elsewhere,\nat least not in this detail.\nThey are, roughly, in order\nof least esoteric to most.\nFirst, however, a brief spiel.\n\nI understand that there's often little choice in choosing\na bibliography style---journal~$X$ says you must use style~$Y$\nand that's that.\nIf you have a choice, however, I strongly recommend that you\nchoose something like the {\\tt plain} standard style.\nSuch a style, van~Leunen~\\cite{van-leunen} argues convincingly,\nencourages better writing than the alternatives---%\nmore concrete, more vivid.\n\n{\\em The Chicago Manual of Style\\\/}~\\cite{chicago},\non the other hand,\nespouse the author-date system,\nin which the citation might appear in the text as `(Jones, 1986)'$\\!$.\nI argue that this system,\nbesides cluttering up the\ntext with information that may or may not be relevant,\nencourages the passive voice and vague writing.\nFurthermore the strongest arguments for\nusing the author-date system---like ``it's the most practical''---%\nfall flat on their face with the advent\nof computer-typesetting technology.\nFor instance the {\\em Chicago Manual\\\/} contains,\nright in the middle of page~401, this anachronism:\n``The chief disadvantage of [a style like {\\tt plain}] is that additions\nor deletions cannot be made after the manuscript is typed without changing\nnumbers in both text references and list.''\n\\LaTeX, obviously, sidesteps the disadvantage.\n\nFinally, the logical deficiencies of the author-date style\nare quite evident once you've written a program to implement it.\nFor example, in a large bibliography,\nusing the standard alphabetizing scheme,\nthe entry for `(Aho et~al., 1983b)'\nmight be half a page later than the one for `(Aho et~al., 1983a)'$\\!$.\nFixing this problem results in even worse ones.\nWhat a mess.\n(I have, unfortunately, programmed such a style,\nand if you're saddled with an unenlightened publisher\nor if you don't buy my propaganda,\nit's available from the Rochester style collection.)\n\nOk, so the spiel wasn't very brief;\nbut it made me feel better,\nand now my blood pressure is back to normal.\nHere are the tips for using \\BibTeX\\\nwith the standard styles\n(although many of them hold for nonstandard styles, too).\n\\begin{enumerate}\n\n\\item\nWith \\BibTeX's style-designing language\nyou can program general database manipulations,\nin addition to bibliography styles.\nFor example it's a fairly easy task for someone familiar with the language\nto produce a database-key\/author index of all the entries in a database.\nConsult the {\\em Local Guide\\\/} to see\nwhat tools are available on your system.\n\n\\item\nThe standard style's thirteen entry types\ndo reasonably well at formatting most entries,\nbut no scheme with just thirteen formats\ncan do everything perfectly.\nThus, you should feel free to be creative\nin how you use these entry types\n(but if you have to be too creative,\nthere's a good chance you're using the wrong entry type).\n\n\\item\nDon't take the field names too seriously.\nSometimes, for instance, you might have to include\nthe publisher's address along with the publisher's name\nin the \\hbox{\\tt publisher} field,\nrather than putting it in the \\hbox{\\tt address} field.\nOr sometimes, difficult entries work best when you\nmake judicious use of the {\\tt note} field.\n\n\\item\nDon't take the warning messages too seriously.\nSometimes, for instance, the year appears in the title,\nas in {\\em The 1966 World Gnus Almanac}.\nIn this case it's best to omit the {\\tt year} field\nand to ignore \\BibTeX's warning message.\n\n\\item\nIf you have too many names to list in an\n\\hbox{\\tt author} or \\hbox{\\tt editor} field,\nyou can end the list with ``and others'';\nthe standard styles appropriately append an ``et~al.''\n\n\\item\nIn general, if you want to keep \\BibTeX\\ from changing\nsomething to lower case, you enclose it in braces.\nYou might not get the effect you want, however,\nif the very first character after the left brace is a backslash.\nThe ``special characters'' item later in this section explains.\n\n\\item\nFor {\\em Scribe\\\/} compatibility, the database files\nallow an \\hbox{\\tt @COMMENT} command; it's not really\nneeded because \\BibTeX\\ allows in the database files\nany comment that's not within an entry.\nIf you want to comment out an entry,\nsimply remove the `{\\tt @}' character preceding the entry type.\n\n\\item\nThe standard styles have journal abbreviations that are\ncomputer-science oriented;\nthese are in the style files primarily for the example.\nIf you have a different set of journal abbreviations,\nit's sensible to put them in \\hbox{\\tt @STRING} commands\nin their own database file and to list this database file\nas an argument to \\LaTeX's \\hbox{\\verb|\\bibliography|} command\n(but you should list this argument before the ones that\nspecify real database entries).\n\n\\item\nIt's best to use the three-letter abbreviations for the month,\nrather than spelling out the month yourself.\nThis lets the bibliography style be consistent.\nAnd if you want to include information for the day of the month,\nthe {\\tt month} field is usually the best place.\nFor example\n\\begin{verbatim}\n month = jul # \"~4,\"\n\\end{verbatim}\nwill probably produce just what you want.\n\n\\item\nIf you're using the \\hbox{\\tt unsrt} style\n(references are listed in order of citation)\nalong with the \\hbox{\\verb|\\nocite{*}|} feature\n(all entries in the database are included),\nthe placement of the \\hbox{\\verb|\\nocite{*}|} command\nwithin your document file will determine the reference order.\nAccording to the rule given in Section~\\ref{features}:\nIf the command is placed at the beginning of the document,\nthe entries will be listed in exactly the order\nthey occur in the database;\nif it's placed at the end,\nthe entries that you explicitly\n\\hbox{\\verb|\\cite|} or \\hbox{\\verb|\\nocite|}\nwill occur in citation order,\nand the remaining database entries will be in database order.\n\n\\item\nFor theses, van Leunen recommends not giving\nthe school's department after the name of the degree,\nsince schools, not departments, issue degrees.\nIf you really think that giving the department information\nwill help the reader find the thesis,\nput that information in the \\hbox{\\tt address} field.\n\n\\item\nThe \\hbox{\\tt MASTERSTHESIS} and \\hbox{\\tt PHDTHESIS} entry types\nare so named for {\\em Scribe\\\/} compatibility;\n\\hbox{\\tt MINORTHESIS} and \\hbox{\\tt MAJORTHESIS}\nprobably would have been better names.\nKeep this in mind when trying to classify\na non-U.S.\\ thesis.\n\n\\item\nHere's yet another suggestion for what to do when an author's\nname appears slightly differently in two publications.\nSuppose, for example, two journals articles use these fields.\n\\begin{verbatim}\n author = \"Donald E. Knuth\"\n . . .\n author = \"D. E. Knuth\"\n\\end{verbatim}\nThere are two possibilities.\nYou could (1)~simply leave them as is,\nor (2)~assuming you know for sure that\nthese authors are one and the same person,\nyou could list both in the form that the author prefers\n(say, `Donald~E.\\ Knuth').\nIn the first case, the entries might be alphabetized incorrectly,\nand in the second, the slightly altered name might\nfoul up somebody's electronic library search.\nBut there's a third possibility, which is the one I prefer.\nYou could convert the second journal's field to\n\\begin{verbatim}\n author = \"D[onald] E. Knuth\"\n\\end{verbatim}\nThis avoids the pitfalls of the previous two solutions,\nsince \\BibTeX\\ alphabetizes this as if the brackets weren't there,\nand since the brackets clue the reader in that a full first name\nwas missing from the original.\nOf course it introduces another pitfall---`D[onald]~E.\\ Knuth' looks ugly---%\nbut in this case I think the increase in accuracy outweighs\nthe loss in aesthetics.\n\n\\item\n\\LaTeX's comment character `{\\tt\\%}' is not a comment character\nin the database files.\n\n\\item\nHere's a more complete description of\nthe ``author'' information referred to in previous sections.\nFor most entry types the ``author'' information\nis simply the \\hbox{\\tt author} field.\nHowever:\nFor the \\hbox{\\tt BOOK} and \\hbox{\\tt INBOOK} entry types\nit's the \\hbox{\\tt author} field, but if there's no author\nthen it's the \\hbox{\\tt editor} field;\nfor the \\hbox{\\tt MANUAL} entry type\nit's the \\hbox{\\tt author} field, but if there's no author\nthen it's the \\hbox{\\tt organization} field;\nand for the \\hbox{\\tt PROCEEDINGS} entry type\nit's the \\hbox{\\tt editor} field, but if there's no editor\nthen it's the \\hbox{\\tt organization} field.\n\n\\item\nWhen creating a label,\nthe \\hbox{\\tt alpha} style uses the ``author'' information described above,\nbut with a slight change---%\nfor the \\hbox{\\tt MANUAL} and \\hbox{\\tt PROCEEDINGS} entry types,\nthe {\\tt key} field takes precedence over the \\hbox{\\tt organization} field.\nHere's a situation where this is useful.\n\\begin{verbatim}\n organization = \"The Association for Computing Machinery\",\n key = \"ACM\"\n\\end{verbatim}\nWithout the {\\tt key} field, the \\hbox{\\tt alpha} style\nwould make a label from the first three letters of information\nin the \\hbox{\\tt organization} field;\n\\hbox{\\tt alpha} knows to strip off the `\\hbox{\\tt The }'$\\!$,\nbut it would still form a label like `\\hbox{[Ass86]}'$\\!$,\nwhich, however intriguing, is uninformative.\nIncluding the {\\tt key} field, as above,\nwould yield the better label `\\hbox{[ACM86]}'$\\!$.\n\nYou won't always need the {\\tt key} field to override the\n\\hbox{\\tt organization}, though:\nWith\n\\begin{verbatim}\n organization = \"Unilogic, Ltd.\",\n\\end{verbatim}\nfor instance, the \\hbox{\\tt alpha} style would\nform the perfectly reasonable label `\\hbox{[Uni86]}'$\\!$.\n\n\\item\nSection~\\ref{features} discusses accented characters.\nTo \\BibTeX, an accented character is really a special case\nof a ``special character''$\\!$,\nwhich consists of everything from a left brace at the top-most level,\nimmediately followed by a backslash,\nup through the matching right brace.\nFor example in the field\n\\begin{verbatim}\n author = \"\\AA{ke} {Jos{\\'{e}} {\\'{E}douard} G{\\\"o}del\"\n\\end{verbatim}\nthere are just two special characters,\n`\\hbox{\\verb|{\\'{E}douard}|}' and `\\hbox{\\verb|{\\\"o}|}'\n(the same would be true if the pair of double quotes\ndelimiting the field were braces instead).\nIn general, \\BibTeX\\ will not do any processing\nof a \\TeX\\ or \\LaTeX\\ control sequence inside a special character,\nbut it {\\em will\\\/} process other characters.\nThus a style that converts all titles to lower case\nwould convert\n\\begin{verbatim}\n The {\\TeX BOOK\\NOOP} Experience\n\\end{verbatim}\nto\n\\begin{verbatim}\n The {\\TeX book\\NOOP} experience\n\\end{verbatim}\n(the `{\\tt The}' is still capitalized\nbecause it's the first word of the title).\n\nThis special-character scheme is useful for handling accented characters,\nfor getting \\BibTeX's alphabetizing to do what you want,\nand, since \\BibTeX\\ counts an entire special character as just one letter,\nfor stuffing extra characters inside labels.\nThe file \\hbox{\\tt XAMPL.BIB} distributed with \\BibTeX\\\ngives examples of all three uses.\n\n\\item\nThis final item of the section describes \\BibTeX's names\n(which appear in the \\hbox{\\tt author} or \\hbox{\\tt editor} field)\nin slightly more detail than what\nappears in Appendix~B of the \\LaTeX\\ book.\nIn what follows, a ``name'' corresponds to a person.\n(Recall that you separate multiple names in a single field\nwith the word ``and''$\\!$, surrounded by spaces,\nand not enclosed in braces.\nThis item concerns itself with the structure of a single name.)\n\nEach name consists of four parts: First, von, Last, and~Jr;\neach part consists of a (possibly empty) list of name-tokens.\nThe Last part will be nonempty if any part is,\nso if there's just one token, it's always a Last token.\n\nRecall that Per Brinch~Hansen's name should be typed\n\\begin{verbatim}\n \"Brinch Hansen, Per\"\n\\end{verbatim}\nThe First part of his name has the single token ``Per'';\nthe Last part has two tokens, ``Brinch'' and ``Hansen'';\nand the von and Jr parts are empty.\nIf you had typed\n\\begin{verbatim}\n \"Per Brinch Hansen\"\n\\end{verbatim}\ninstead, \\BibTeX\\ would (erroneously) think ``Brinch'' were a First-part token,\njust as ``Paul'' is a First-part token in ``John~Paul Jones''$\\!$,\nso this erroneous form would have two First tokens and one Last token.\n\nHere's another example:\n\\begin{verbatim}\n \"Charles Louis Xavier Joseph de la Vall{\\'e}e Poussin\"\n\\end{verbatim}\nThis name has four tokens in the First part, two in the von, and\ntwo in the Last.\nHere \\BibTeX\\ knows where one part ends and the other begins because\nthe tokens in the von part begin with lower-case letters.\n\nIn general, it's a von token if the first letter at brace-level~0\nis in lower case.\nSince technically everything\nin a ``special character'' is at brace-level~0,\nyou can trick \\BibTeX\\ into thinking that\na token is or is not a von token by prepending a dummy\nspecial character whose first letter past the \\TeX\\ control sequence\nis in the desired case, upper or lower.\n\nTo summarize, \\BibTeX\\ allows three possible forms for the name:\n\\begin{verbatim}\n \"First von Last\"\n \"von Last, First\"\n \"von Last, Jr, First\"\n\\end{verbatim}\nYou may almost always use the first form;\nyou shouldn't if either there's a Jr part,\nor the Last part has multiple tokens but there's no von part.\n\n\\end{enumerate}\n\n\n\\section{Bibliography-style hacking}\n\\label{style}\n\nThis document starts (and ends) with Section~\\ref{style},\nbecause in reality it is the final section of ``\\BibTeX ing''~\\cite{btxdoc},\nthe general documentation for \\BibTeX.\nBut that document was meant for all \\BibTeX\\ users,\nwhile this one is just for style designers,\nso the two are physically separate.\nStill, you should be completely familiar with ``\\BibTeX ing''$\\!$,\nand all references in this document\nto sections and section numbers\nassume that the two documents are one.\n\nThis section,\nalong with the standard-style documentation file \\hbox{\\tt btxbst.doc},\nshould explain how to modify\nexisting style files and to produce new ones.\nIf you're a serious style hacker you should be familiar\nwith van~Leunen~\\cite{van-leunen} for points of style,\nwith Lamport~\\cite{latex} and Knuth~\\cite{texbook} for formatting matters,\nand perhaps with {\\em Scribe\\\/}~\\cite{scribe} for compatibility details.\nAnd while you're at it, if you don't read the great little book by Strunk and\nWhite~\\cite{strunk-and-white}, you should at least look at its\nentries in the database and the reference list\nto see how \\BibTeX\\ handles multiple names.\n\nTo create a new style,\nit's best to start with an existing style that's close to yours,\nand then modify that.\nThis is true even if you're simply updating an old style\nfor \\BibTeX\\ version 0.99\n(I've updated four nonstandard styles,\nso I say this with some experience).\nIf you want to insert into a new style\nsome function you'd written for an old (version 0.98i) style,\nkeep in mind that the order of the arguments to\nthe assignment ({\\tt :=}) function has been reversed.\nWhen you're finished with your style,\nyou may want to try running it on the entire \\hbox{\\tt XAMPL.BIB} database\nto make sure it handles all the standard entry types.\n\nIf you find any bugs in the standard styles,\nor if there are things you'd like to do\nwith bibliography-style files but can't,\nplease complain to Oren Patashnik.\n\n\n\\subsection{General description}\n\nYou write bibliography styles in a postfix stack language. It's\nnot too hard to figure out how by looking at the standard-style documentation,\nbut this description fills in a few details (it will fill in more\ndetails if there's a demand for it).\n\nBasically the style file is a program, written in an unnamed language, that\ntells \\BibTeX\\ how to format the entries that will go in the reference list\n(henceforth ``the entries'' will be ``the entry list''\nor simply ``the list''$\\!$, context permitting).\nThis programming language has ten commands, described in the next subsection.\nThese commands manipulate the language's objects:\nconstants, variables, functions, the stack, and the entry list.\n(Warning: The terminology in this documentation,\nchosen for ease of explanation, is slightly different from \\BibTeX's.\nFor example, this documentation's ``variables'' and ``functions''\nare both ``functions'' to \\BibTeX.\nKeep this in mind when interpreting \\BibTeX's error messages.)\n\nThere are two types of functions: {\\it built-in\\\/} ones that \\BibTeX\\ provides\n(these are described in Section~\\ref{built-in-fns}), and ones you define\nusing either the \\hbox{\\tt MACRO} or \\hbox{\\tt FUNCTION} command.\n\nYour most time-consuming task, as a style designer,\nwill be creating or modifying functions\nusing the \\hbox{\\tt FUNCTION} command\n(actually, becoming familiar with the references listed above will be\nmore time consuming, but assume for the moment that that's done).\n\nLet's look at a sample function fragment.\nSuppose you have a string variable named \\hbox{\\tt label}\nand an integer variable named \\hbox{\\tt lab.width},\nand suppose you want to append the character `{\\tt a}' to \\hbox{\\tt label}\nand to increment \\hbox{\\tt lab.width}:\n\\begin{verbatim}\n . . .\n label \"a\" * 'label := \n lab.width #1 + 'lab.width := \n . . .\n\\end{verbatim}\nIn the first line,\n\\hbox{\\tt label} pushes that variable's value onto the stack.\nNext, the {\\tt \"a\"} pushes the string constant `{\\tt a}' onto the stack.\nThen the built-in function {\\tt *} pops the top two strings and\npushes their concatenation.\nThe \\hbox{\\tt 'label} pushes that variable's name onto the stack.\nAnd finally, the built-in function {\\tt :=} pops\nthe variable name and the concatenation and performs the assignment.\n\\BibTeX\\ treats the stuff following the {\\tt \\%} as a comment\nin the style file.\nThe second line is similar except that it uses {\\tt \\#1},\nwith no spaces intervening between the `{\\tt \\#}' and the `{\\tt 1}'$\\!$,\nto push this integer constant.\n\nThe nonnull spacing here is arbitrary: multiple spaces, tabs, or newlines\nare equivalent to a single one (except that you're probably better off\nnot having blank lines within commands, as explained shortly).\n\nFor string constants, absolutely any printing character\nis legal between two consecutive double quotes, but \\BibTeX\\ here\n(and only here) treats upper- and lower-case equivalents as different.\nFurthermore, spacing {\\em is\\\/} relevant within a string constant,\nand you mustn't split a string constant across lines\n(that is, the beginning and ending double quotes must be on the same line).\n\nVariable and function names may not begin with a numeral and\nmay not contain any of the ten restricted characters\non page~143 of the \\LaTeX\\ book,\nbut may otherwise contain any printing characters.\nAlso, \\BibTeX\\ considers upper- and lower-case equivalents to be the same.\n\nIntegers and strings are the only value types for constants and variables\n(booleans are implemented simply as 0-or-1 integers).\nThere are three kinds of variables:\n\\begin{description}\n\n\\item[global variables\\hfill] These are either integer- or string-valued,\ndeclared using an \\hbox{\\tt INTEGERS} or \\hbox{\\tt STRINGS} command.\n\n\\item[entry variables\\hfill] These are either integer- or string-valued,\ndeclared using the \\hbox{\\tt ENTRY} command.\nEach has a value for each entry on the list\n(example: a variable \\hbox{\\tt label} might store\nthe label string you'll use for the entry).\n\n\\item[fields\\hfill] These are string-valued, read-only variables\nthat store the information from the database file;\ntheir values are set by the \\hbox{\\tt READ} command.\nAs with entry variables, each has a value for each entry.\n\\end{description}\n\n\n\\subsection{Commands}\n\nThere are ten style-file commands:\nFive (\\hbox{\\tt ENTRY}, \\hbox{\\tt FUNCTION}, \\hbox{\\tt INTEGERS},\n\\hbox{\\tt MACRO}, and \\hbox{\\tt STRINGS})\ndeclare and define variables and functions;\none (\\hbox{\\tt READ}) reads in the database information;\nand four (\\hbox{\\tt EXECUTE}, \\hbox{\\tt ITERATE}, \\hbox{\\tt REVERSE},\nand \\hbox{\\tt SORT}) manipulate the entries and produce output.\nAlthough the command names appear here in upper case,\n\\BibTeX\\ ignores case differences.\n\nSome restrictions:\nThere must be exactly one \\hbox{\\tt ENTRY} and one \\hbox{\\tt READ} command;\nthe \\hbox{\\tt ENTRY} command, all \\hbox{\\tt MACRO} commands,\nand certain \\hbox{\\tt FUNCTION} commands\n(see next subsection's description of \\hbox{\\tt call.type\\$})\nmust precede the \\hbox{\\tt READ} command;\nand the \\hbox{\\tt READ} command must precede the four that\nmanipulate the entries and produce output.\n\nAlso it's best (but not essential) to leave at least one blank line\nbetween commands and to leave no blank lines within a command;\nthis helps \\BibTeX\\ recover from any syntax errors you make.\n\nYou must enclose each argument of every command in braces.\nLook at the standard-style documentation\nfor syntactic issues not described in this section.\nHere are the ten commands:\n\\begin{description}\n\n\\item[\\hbox{\\tt ENTRY}\\hfill]\nDeclares the fields and entry variables.\nIt has three arguments, each a (possibly empty) list of variable names.\nThe three lists are of:\nfields, integer entry variables, and string entry variables.\nThere is an additional field that \\BibTeX\\ automatically\ndeclares, \\hbox{\\tt crossref}, used for cross referencing.\nAnd there is an additional string entry variable automatically declared,\n\\hbox{\\tt sort.key\\$}, used by the \\hbox{\\tt SORT} command.\nEach of these variables has a value for each entry on the list.\n\n\\item[\\hbox{\\tt EXECUTE}\\hfill]\nExecutes a single function.\nIt has one argument, the function name.\n\n\\item[\\hbox{\\tt FUNCTION}\\hfill]\nDefines a new function.\nIt has two arguments; the first is the function's name and the\nsecond is its definition.\nYou must define a function before using it;\nrecursive functions are thus illegal.\n\n\\item[\\hbox{\\tt INTEGERS}\\hfill]\nDeclares global integer variables.\nIt has one argument, a list of variable names.\nThere are two such automatically-declared variables,\n\\hbox{\\tt entry.max\\$} and \\hbox{\\tt global.max\\$},\nused for limiting the lengths of string variables.\nYou may have any number of these commands, but a variable's declaration\nmust precede its use.\n\n\\item[\\hbox{\\tt ITERATE}\\hfill]\nExecutes a single function, once\nfor each entry in the list, in the list's current order\n(initially the list is in citation order, but the \\hbox{\\tt SORT}\ncommand may change this).\nIt has one argument, the function name.\n\n\\item[\\hbox{\\tt MACRO}\\hfill]\nDefines a string macro.\nIt has two arguments; the first is the macro's name, which is treated like\nany other variable or function name,\nand the second is its definition, which must be double-quote-delimited.\nYou must have one for each three-letter month abbreviation;\nin addition, you should have one for common journal names.\nThe user's database may override any definition you define using this command.\nIf you want to define a string the user can't touch,\nuse the \\hbox{\\tt FUNCTION} command, which has a compatible syntax.\n\n\\item[\\hbox{\\tt READ}\\hfill]\nDredges up from the database file\nthe field values for each entry in the list.\nIt has no arguments.\nIf a database entry doesn't have a value for a field\n(and probably no database entry will have a value for every field),\nthat field variable is marked as missing for the entry.\n\n\\item[\\hbox{\\tt REVERSE}\\hfill]\nExactly the same as the\n\\hbox{\\tt ITERATE} command except that it executes the function\non the entry list in reverse order.\n\n\\item[\\hbox{\\tt SORT}\\hfill]\nSorts the entry list using\nthe values of the string entry variable \\hbox{\\tt sort.key\\$}.\nIt has no arguments.\n\n\\item[\\hbox{\\tt STRINGS}\\hfill]\nDeclares global string variables.\nIt has one argument, a list of variable names.\nYou may have any number of these commands, but a variable's declaration\nmust precede its use.\n\\end{description}\n\n\n\\subsection{The built-in functions}\n\\label{built-in-fns}\n\nBefore we get to the built-in functions,\na few words about some other built-in objects.\nThere is one built-in string entry variable, \\hbox{\\tt sort.key\\$},\nwhich the style program must set if the style is to do sorting.\nThere is one built-in field, \\hbox{\\tt crossref},\nused for the cross referencing feature\ndescribed in Section~4.\nAnd there are two built-in integer global variables,\n\\hbox{\\tt entry.max\\$} and \\hbox{\\tt global.max\\$},\nwhich are set by default to some internal \\BibTeX\\ constants;\nyou should truncate strings to these lengths before\nyou assign to string variables,\nso as to not generate any \\BibTeX\\ warning messages.\n\nThere are currently 37 built-in functions.\nEvery built-in function with a letter in its name ends with a `{\\tt \\$}'$\\!$.\nIn what follows, ``first''$\\!$, ``second''$\\!$,\nand so on refer to the order popped.\nA ``literal'' is an element on the stack, and it will be either\nan integer value, a string value, a variable or function name,\nor a special value denoting a missing field.\nIf any popped literal has an incorrect type, \\BibTeX\\ complains and pushes\nthe integer 0 or the null string, depending on whether the function\nwas supposed to push an integer or string.\n\\begin{description}\n\n\\item[\\hbox{\\tt >}\\hfill]\nPops the top two (integer) literals,\ncompares them, and pushes the integer 1 if the second is greater than\nthe first, 0 otherwise.\n\n\\item[\\hbox{\\tt <}\\hfill]\nAnalogous.\n\n\\item[\\hbox{\\tt =}\\hfill]\nPops the top two (both integer or both string) literals,\ncompares them,\nand pushes the integer 1 if they're equal, 0 otherwise.\n\n\\item[\\hbox{\\tt +}\\hfill]\nPops the top two (integer) literals and pushes their sum.\n\n\\item[\\hbox{\\tt -}\\hfill]\nPops the top two (integer) literals and pushes their difference\n(the first subtracted from the second).\n\n\\item[\\hbox{\\tt *}\\hfill]\nPops the top two (string) literals,\nconcatenates them (in reverse order, that is, the order in which\npushed), and pushes the resulting string.\n\n\\item[\\hbox{\\tt :=}\\hfill]\nPops the top two literals and assigns\nto the first (which must be a global or entry variable)\nthe value of the second.\n\n\\item[\\hbox{\\tt add.period\\$}\\hfill]\nPops the top (string) literal,\nadds a `{\\tt .}' to it if the last non`{\\tt \\}}' character\nisn't a `{\\tt .}'$\\!$, `{\\tt ?}', or `{\\tt !}'$\\!$,\nand pushes this resulting string.\n\n\\item[\\hbox{\\tt call.type\\$}\\hfill]\nExecutes the function whose name is the entry type of an entry.\nFor example if an entry is of type {\\tt book}, this function executes\nthe {\\tt book} function.\nWhen given as an argument to the \\hbox{\\tt ITERATE} command,\n\\hbox{\\tt call.type\\$} actually produces the output for the entries.\nFor an entry with an unknown type,\nit executes the function \\hbox{\\tt default.type}.\nThus you should define (before the \\hbox{\\tt READ} command) one function\nfor each standard entry type as well as a \\hbox{\\tt default.type} function.\n\n\\item[\\hbox{\\tt change.case\\$}\\hfill]\nPops the top two (string) literals;\nit changes the case of the second according to the\nspecifications of the first, as follows. (Note: The word `letters' in\nthe next sentence refers only to those at brace-level~0, the top-most\nbrace level; no other characters are changed, except perhaps for\n``special characters''$\\!$, described in Section~4.)\nIf the first literal is the\nstring~`{\\tt t}'$\\!$, it converts to lower case all letters except the very\nfirst character in the string, which it leaves alone, and except the\nfirst character following any colon and then nonnull white space,\nwhich it also leaves alone; if it's the string~`{\\tt l}'$\\!$, it converts all\nletters to lower case; and if it's the string~`{\\tt u}'$\\!$, it converts all\nletters to upper case.\nIt then pushes this resulting string. If either\ntype is incorrect, it complains and pushes the null string; however,\nif both types are correct but the specification string (i.e., the\nfirst string) isn't one of the legal ones, it merely pushes the second\nback onto the stack, after complaining. (Another note: It ignores\ncase differences in the specification string; for example, the strings\n{\\tt t} and {\\tt T} are equivalent for the purposes of this built-in\nfunction.)\n\n\\item[\\hbox{\\tt chr.to.int\\$}\\hfill]\nPops the top (string) literal,\nmakes sure it's a single character, converts it to the\ncorresponding ASCII integer, and pushes this integer.\n\n\\item[\\hbox{\\tt cite\\$}\\hfill]\nPushes the string that was the\n\\hbox{\\verb|\\cite|}-command argument for this entry.\n\n\\item[\\hbox{\\tt duplicate\\$}\\hfill]\nPops the top literal from the stack and pushes two copies of it.\n\n\\item[\\hbox{\\tt empty\\$}\\hfill]\nPops the top literal and pushes\nthe integer 1 if it's a missing field or a string having no\nnon-white-space characters, 0 otherwise.\n\n\\item[\\hbox{\\tt format.name\\$}\\hfill]\nPops the top three literals\n(they are a string, an integer, and a string literal).\nThe last string literal represents a name list (each name\ncorresponding to a person), the integer literal specifies which name\nto pick from this list, and the first string literal specifies how to\nformat this name, as explained in the next subsection.\nFinally, this function pushes the formatted name.\n\n\\item[\\hbox{\\tt if\\$}\\hfill]\nPops the top three literals (they\nare two function literals and an integer literal, in that order);\nif the integer is greater than 0, it executes the second literal,\nelse it executes the first.\n\n\\item[\\hbox{\\tt int.to.chr\\$}\\hfill]\nPops the top (integer) literal,\ninterpreted as the ASCII integer value of a single character,\nconverts it to the corresponding single-character string, and pushes\nthis string.\n\n\\item[\\hbox{\\tt int.to.str\\$}\\hfill]\nPops the top (integer) literal,\nconverts it to its (unique) string equivalent, and pushes this string.\n\n\\item[\\hbox{\\tt missing\\$}\\hfill]\nPops the top literal and\npushes the integer 1 if it's a missing field, 0~otherwise.\n\n\\item[\\hbox{\\tt newline\\$}\\hfill]\nWrites onto the {\\tt bbl} file\nwhat's accumulated in the output buffer.\nIt writes a blank line if and only if the output buffer is empty.\nSince \\hbox{\\tt write\\$} does reasonable line breaking, you should use\nthis function only when you want a blank line or an explicit line break.\n\n\\item[\\hbox{\\tt num.names\\$}\\hfill]\nPops the top (string) literal\nand pushes the number of names the string represents---one plus\nthe number of occurrences of the substring ``and'' (ignoring case differences)\nsurrounded by nonnull white-space at the top brace level.\n\n\\item[\\hbox{\\tt pop\\$}\\hfill]\nPops the top of the stack but\ndoesn't print it; this gets rid of an unwanted stack literal.\n\n\\item[\\hbox{\\tt preamble\\$}\\hfill]\nPushes onto the stack the concatenation of all the\n\\hbox{\\tt @PREAMBLE} strings read from the database files.\n\n\\item[\\hbox{\\tt purify\\$}\\hfill]\nPops the top (string) literal,\nremoves nonalphanumeric characters except for white-space characters and\nhyphens and ties (these all get converted to a space), removes\ncertain alphabetic characters contained in the control sequences\nassociated with a ``special character''$\\!$, and pushes the resulting string.\n\n\\item[\\hbox{\\tt quote\\$}\\hfill]\nPushes the string consisting of the double-quote character.\n\n\\item[\\hbox{\\tt skip\\$}\\hfill]\nIs a no-op.\n\n\\item[\\hbox{\\tt stack\\$}\\hfill]\nPops and prints the whole stack;\nit's meant to be used for style designers while debugging.\n\n\\item[\\hbox{\\tt substring\\$}\\hfill]\nPops the top three literals\n(they are the two integers literals {\\it len\\\/} and {\\it start}, and a\nstring literal, in that order).\nIt pushes the substring of the (at most) {\\it len\\\/} consecutive characters\nstarting at the {\\it start\\\/}th character (assuming 1-based indexing)\nif {\\it start\\\/} is positive, and ending at the $-${\\it start\\\/}th character\nfrom the end if {\\it start\\\/} is negative\n(where the first character from the end is the last character).\n\n\\item[\\hbox{\\tt swap\\$}\\hfill]\nSwaps the top two literals on the stack.\n\n\\item[\\hbox{\\tt text.length\\$}\\hfill]\nPops the top (string) literal,\nand pushes the number of text characters it contains, where an\naccented character (more precisely, a ``special character''$\\!$,\ndefined in Section~4)\ncounts as a single text character, even if it's missing\nits matching right brace, and where braces don't count as\ntext characters.\n\n\\item[\\hbox{\\tt text.prefix\\$}\\hfill]\nPops the top two literals\n(the integer literal {\\it len\\\/} and a string literal, in that order).\nIt pushes the substring of the (at most) {\\it len\\\/} consecutive text\ncharacters starting from the beginning of the string. This function\nis similar to \\hbox{\\tt substring\\$}, but this one considers\na ``special character''$\\!$, even if\nit's missing its matching right brace, to be a single text character\n(rather than however many ASCII characters it actually comprises),\nand this function doesn't consider braces to be text characters;\nfurthermore, this function appends any needed matching right braces.\n\n\\item[\\hbox{\\tt top\\$}\\hfill]\nPops and prints the top of the stack on the terminal and log file.\nIt's useful for debugging.\n\n\\item[\\hbox{\\tt type\\$}\\hfill]\nPushes the current entry's type (book, article, etc.),\nbut pushes the null string\nif the type is either unknown or undefined.\n\n\\item[\\hbox{\\tt warning\\$}\\hfill]\nPops the top (string) literal\nand prints it following a warning message.\nThis also increments a count of the number of warning messages issued.\n\n\\item[\\hbox{\\tt while\\$}\\hfill]\nPops the top two (function) literals,\nand keeps executing the second as long as the (integer)\nliteral left on the stack by executing the first is greater than 0.\n\n\\item[\\hbox{\\tt width\\$}\\hfill]\nPops the top (string) literal\nand pushes the integer that represents its width in some relative units\n(currently, hundredths of a point, as specified by the June 1987 version\nof the $cmr10$ font; the only white-space character with nonzero width\nis the space).\nThis function takes the literal literally;\nthat is, it assumes each character in the string is to be printed as\nis, regardless of whether the character has a special meaning to \\TeX,\nexcept that ``special characters'' (even without their right braces) are\nhandled specially.\nThis is meant to be used for comparing widths of label strings.\n\n\\item[\\hbox{\\tt write\\$}\\hfill]\nPops the top (string) literal\nand writes it on the output buffer (which will result in\nstuff being written onto the {\\tt bbl} file when the buffer fills up).\n\n\\end{description}\n\nNote that the built-in functions \\hbox{\\tt while\\$} and \\hbox{\\tt if\\$}\nrequire two function literals on the stack.\nYou get them there either by immediately preceding the name of a function\nby a single quote, or, if you don't feel like defining a new function with\nthe \\hbox{\\tt FUNCTION} command,\nby simply giving its definition (that is, giving what would be the second\nargument to the \\hbox{\\tt FUNCTION} command, including the surrounding braces).\nFor example the following function fragment appends the character `{\\tt a}'\nif the string variable named \\hbox{\\tt label} is nonnull:\n\\begin{verbatim}\n . . .\n label \"\" =\n 'skip$\n { label \"a\" * 'label := }\n if$\n . . .\n\\end{verbatim}\nA function whose name you quote needn't be built in\nlike \\hbox{\\tt skip\\$} above---it may, for example,\nbe a field name or a function you've defined earlier.\n\n\n\\subsection{Name formatting}\n\nWhat's in a name?\nSection~4 pretty much describes this.\nEach name consists of four parts: First, von, Last, and Jr;\neach consists of a list of name-tokens,\nand any list but Last's may be empty for a nonnull name.\nThis subsection describes the format string you must supply to\nthe built-in function \\hbox{\\tt format.name\\$}.\n\nLet's look at an example of a very long name.\nSuppose a database entry~\\cite{prime-number-theorem} has the field\n\\begin{verbatim}\n author = \"Charles Louis Xavier Joseph de la Vall{\\'e}e Poussin\"\n\\end{verbatim}\nand suppose you want this formatted ``last name comma initials''$\\!$.\nIf you use the format string\n\\begin{verbatim}\n \"{vv~}{ll}{, jj}{, f}?\"\n\\end{verbatim}\n\\BibTeX\\ will produce\n\\begin{verbatim}\n de~la Vall{\\'e}e~Poussin, C.~L. X.~J?\n\\end{verbatim}\nas the formatted string.\n\nLet's look at this example in detail.\nThere are four brace-level~1 {\\em pieces\\\/} to this format string,\none for each part of a name.\nIf the corresponding part of a name isn't present (the Jr part for this name),\neverything in that piece is ignored.\nAnything at brace-level~0 is output verbatim\n(the presumed typo `{\\tt ?}' for this name is at brace-level~0),\nbut you probably won't use this feature much.\n\nWithin each piece a double letter tells \\BibTeX\\ to use whole tokens, and\na single letter, to abbreviate them (these letters must be at brace-level~1);\neverything else within the piece is used verbatim\n(well, almost everything---read on).\nThe tie at the end of the von part (in \\hbox{\\verb|{vv~}|})\nis a discretionary tie---\\BibTeX\\ will output a tie at that point\nif it thinks there's a need for one;\notherwise it will output a space.\nIf you really, really, want a tie there,\nregardless of what \\BibTeX\\ thinks, use two of them\n(only one will be output); that is, use \\hbox{\\verb|{vv~~}|}.\nA tie is discretionary only if it's the last character of the piece;\nanywhere else it's treated as an ordinary character.\n\n\\BibTeX\\ puts default strings {\\em between\\\/} tokens of a name part:\nFor whole tokens it uses either a space or a tie,\ndepending on which one it thinks is best,\nand for abbreviated tokens it uses a period followed by\neither a space or a tie.\nHowever it doesn't use this default string after the last token in a list;\nhence there's no period following the `J' for our example.\nYou should have used\n\\begin{verbatim}\n \"{vv~}{ll}{, jj}{, f.}\"\n\\end{verbatim}\nto get \\BibTeX\\ to produce the same formatted string but with the question\nmark replaced by a period.\nNote that the period should go inside the First-name piece,\nrather than where the question mark was, in case a name has no First part.\n\nIf you want to override \\BibTeX's default between-token strings, you\nmust explicitly specify a string.\nFor example suppose you want a label to contain the first letter from each\ntoken in the von and Last parts, with no spaces;\nyou should use the format string\n\\begin{verbatim}\n \"{v{}}{l{}}\"\n\\end{verbatim}\nso that \\BibTeX\\ will produce `{\\tt dlVP}' as the formatted string.\nYou must give a string for each piece whose default you want overridden\n(the example here uses the null string for both pieces), and this string\nmust immediately follow either the single or double letter for the piece.\nYou may not have any other letters at brace-level~1 in the format string.\n\n\n\\section{Introduction}\n\nMulti-pass amplifiers are used for the generation of the currently highest laser pulse energies and average powers~\\cite{Keppler2013,Koerner2011,Korner2016, Jung2016, Siebold2016, Negel:15}.\nOscillators, regenerative amplifiers and multi-pass amplifiers are limited in pulse energy by the damage of optical components induced by the high intensity.\nMulti pass amplifiers possess a higher damage threshold compared to laser oscillators and regenerative amplifiers due to the absence of intra cavity enhancement and non-linear crystals~\\cite{Butze:04, Martinez1998}.\nOptical damage can be avoided by increasing the laser beam size, but this typically increases the sensitivity of the beam propagation and output beam characteristics (size and divergence) to the active medium thermal lens.\nThe active medium design and material choice can be optimized to decrease the thermal lens, e.g., in thin-disk lasers~\\cite{Giesen1994, Brauch:95, Stewen2000, Peters2009}. \nStill, the higher pump powers and the increased beam size needed for energy and power scaling, make the thermal lens in the active medium the most severe limitation to be overcome~\\cite{Chvykov:16, Negel:15,Vaupel:13}.\nFor these reasons an optical layout has to be chosen that minimizes the changes of the beam propagation for variations of the thermal lens of the active medium.\n\n\nCommonly, relay imaging (4f-imaging) from active medium to active medium is used to realize a multi-pass amplifier where the output power is independent of the thermal lens of the active medium because the beam is imaged from pass to pass~\\cite{Hunt:78}.\nThe 4f-design exhibits various advantages: It sustains equal beam size at each pass on the active medium independently of the dioptric power of the active medium and independently of the beam size at the active medium. \nMoreover, numerous passes can be realized with only a few optical elements~\\cite{Georges:91, Wojtkiewicz:04, Kaksis:16, Plaessmann:93, Lundquist2010, Plaessmann1992, Erhard2000,Scott:01, Papadopoulos:11, Banerjee:12, Neuhaus:08}.\nYet, this design does not represent the optimal solution for a multi-pass amplifier because the phase front curvature of the beam after propagation through this optical system depends strongly on variations of the active medium dioptric power.\n\nIn this paper we present a novel multi-pass architecture whose propagation from pass to pass is given by a succession of Fourier transforms and short propagations. \nThe working principle, the practical implementation and the advantages in term of stability to variations of dioptric power (thermal lens) and aperture effects of the active medium are detailed.\n\n\n\\section{Thermal lens and aperture effects}\n\\label{Thermal lens and aperture effects}\n\nIn this section, we briefly review well-known facts and various definitions \nneeded\nin the following sections.\n\nWhen a laser beam crosses the pumped active medium, it is amplified and it experiences a position-dependent optical phase delay (OPD) that distorts its phase front curvature.\nWhen simulating the propagation of the laser beam, this distortion can be mostly accounted for by approximating the active medium with a spherical lens~\\cite{Osterink68, Koechner:70} that can be easily described within the ABCD-matrix formalism~\\cite{Kogelnik:66}.\n\n\nThe position-dependent gain (and absorption in the unpumped region) can be approximated by an average gain per pass and a position-dependent transmission function that shows a gradual transition from maximal transmission (including gain) on the optical axis to zero transmission far from the axis.\nThis position dependent gain is commonly referred to as \"soft aperture\".\n In this study we consider only Gaussian beams in the fundamental mode and we approximate the super-Gaussian aperture of the pumped active medium as Gaussian aperture.\n %\n This approximation is justified because the Fourier transform (that underlies the here presented designs) converts the high-frequency spatial distortions of the beam caused by the super-Gaussian aperture into a halo of the fundamental mode that is eliminated in the next pass at the active medium. \nTherefore, the beam propagating in the amplifier shows only minor higher-order distortions from the fundamental mode so that the description of the super-Gaussian aperture can be effectively accomplished using a Gaussian aperture~\\cite{PHDSchuhmann}. \nIn fact, a Gaussian aperture transforms a Gaussian input beam into a Gaussian output beam~\\cite{siegman1986lasers}.\nThe relation between the $1\/e^2$-radii of the input ($w_\\mathrm{in}$) and output ($w_\\mathrm{out}$) beams takes the simple form\n\\begin{equation}\n w^{-2}_\\mathrm{out}=w^{-2}_\\mathrm{in}+W^{-2} ,\n \\label{eq:aperture-decrease}\n\\end{equation}\nwhere $W^2$ is the $1\/e^2$-radius of the intensity transmission function of the Gaussian aperture~\\cite{siegman1986lasers,Eggleston:81}.\nThis simple behavior of a Gaussian aperture acting on a Gaussian beam (TEM00-mode) can be captured into an ABCD-matrix for Gaussian optics as~\\cite{Kogelnik:65,siegman1986lasers,Bowers:92,Andrews:95,Kalashnikov:97,PHDSchuhmann}:\n\\begin{eqnarray}\n M_\\mathrm{aperture}=\\left[\n \\begin{array}{cc} 1 & 0 \\\\ \n -i\\frac{\\lambda}{\\pi W^2} & 1\n \\end{array}\n\\right]\\ ,\n\\end{eqnarray}\nwhere $\\lambda$ is the wavelength of the laser beam.\n\nAn active medium (AM) usually exhibits both a lens effect (position-dependent OPD) and an aperture effect (position-dependent gain).\nThese can be merged into a single ABCD-matrix describing the active medium $M_\\mathrm{AM}=M_\\mathrm{lens} \\cdot M_\\mathrm{aperture}$, where $M_\\mathrm{lens}$ is the ABCD-matrix of a thin spherical lens.\nThe product of these two ABCD-matrices becomes\n\\begin{eqnarray}\n M_\\mathrm{AM}=\n \\left[\n \\begin{array}{cc} 1 & 0 \\\\ \n \\hspace{-2mm}-V & 1\n \\end{array}\n \\right] \\hspace{-1mm}\\cdot\\hspace{-1mm}\n \\left[\n \\begin{array}{cc} 1 & 0 \\\\ \n \\hspace{-2mm}-i\\frac{\\lambda}{\\pi W^2} & 1\n \\end{array}\n\\right]= \\left[\n \\begin{array}{cc} 1 & 0 \\\\ \n \\hspace{-2mm}-V-i\\frac{\\lambda}{\\pi W^2} & 1\n \\end{array}\n \\right],\n \\label{eq:matrix-2}\n\\end{eqnarray}\nwhere $V$ is the dioptric power of the active medium.\nIn Eq.~(\\ref{eq:matrix-2}) we have assumed that the length of the active medium is negligible, i.e., that the length of the active medium is much smaller than the Rayleigh length of the beam.\nThis assumption is generally well justified for high-power lasers given the large beam waists they have at the active medium.\n\nThe similarity of $M_\\mathrm{AM}$ and $M_\\mathrm{lens}$ allows us to define a complex dioptric power $\\tilde{V}$ ($\\tilde{V} \\in \\mathbb{C}$) of the active medium as\n\\begin{equation}\n \\tilde{V} =V+i\\frac{\\lambda }{\\pi W^2}.\n\\end{equation}\nIts real part $V$ is the ``standard'' dioptric power (focusing-defocusing effects), while the imaginary part accounts for the Gaussian aperture.\nThe real part can be articulated as a sum of various contributions\n\\begin{equation}\n V = V_\\mathrm{unpumped} + V_\\mathrm{thermal}\n = V_\\mathrm{unpumped} + V_\\mathrm{thermal}^\\mathrm{avg} + \\Delta V \\ ,\n\\end{equation}\nwhere $V_\\mathrm{unpumped}$ is the dioptric power of the unpumped active medium and $V_\\mathrm{thermal}$ the dioptric power related to the pumping of the active medium (thermal-lens effect).\nThe latter can be expressed by a known average a value $V_\\mathrm{thermal}^\\mathrm{avg}$ and an unknown deviation $\\Delta V$ from this average value that might depend among others on running conditions and imperfection of the active medium due to the manufacture process.\n\nIn Secs.~\\ref{sec:state-of-the-art}-\\ref{sec:many-passes}, for simplicity, but without loss of generality, we assume that $V_\\mathrm{unpumped}+V_\\mathrm{thermal}^\\mathrm{avg}=0$.\nIn Sec.~\\ref{sec:implementation}, when realistic implementations of multi-pass amplifiers are presented, we relax this assumption.\nMoreover, in the various plots presented in this paper we have fixed the value of the aperture size to be $W = 4w_\\mathrm{in}$.\nFor comparison, in some figures we also present results for $W = \\infty$, to highlight the aperture effects.\n \nA laser mode size of $w_\\mathrm{in} \\approx 0.7R_p$, where $R_p$ is the radius of the super-Gaussian pump spot, is typically used in the thin-disk laser community for efficient laser operation in the fundamental mode.\nThe above choice of a Gaussian aperture with $W=4w_\\mathrm{in}$ rest upon the fact that it produces a similar reduction of the fundamental mode size and a similar reduction of the fundamental mode transmission compared to a super-Gaussian aperture fulfilling the relation $w_\\mathrm{in} \\approx 0.7R_p$ ~\\cite{PHDSchuhmann}.\n \n\n\\section{State-of-the-art multi-pass amplifiers}\n\\label{sec:state-of-the-art}\n\nThe commonly used state-of-the-art multi-pass amplifiers are based on 4f relay imaging (4f) from active medium (AM) to active medium~\\cite{Georges:91, Wojtkiewicz:04, Kaksis:16, Plaessmann:93, Lundquist2010, Plaessmann1992, Erhard2000,Papadopoulos:11, Banerjee:12,Scott:01, Hunt:78}, so that in these amplifiers the beam proceeds as a succession of propagations\n\\begin{equation*}\n\\noindent\\hspace*{5mm} \\cdots AM-4f-AM-4f-AM-4f-AM-4f-AM\\cdots .\n\\end{equation*}\nThe geometrical layout and the corresponding propagation of a laser beam through a 4f segment is given in Fig.~\\ref{fig:4f-segment-a}.\nBy concatenating for example six of such propagations, the six-pass amplifier of Fig.~\\ref{fig:4f-segment-b} can be realized.\n\\begin{figure}[t!]\n\\centering\n \\begin{subfigure}[t]{.9\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{4f_prop.eps} \n \\caption{} \\label{fig:4f-segment-a}\n \\end{subfigure}\n \\begin{subfigure}[t]{.9\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{8Pass_4ff.eps} \n \\caption{} \\label{fig:4f-segment-b}\n \\end{subfigure}\n\\caption{(a) Scheme of the laser beam propagation from active medium to active medium through a 4f-imaging segment. \nThe 4f-segment is given by a free propagation of distance f, a focusing optics with focal length f, a free propagation with length 2f, a second focusing element with focal length f, and a free propagation of length f. \nThe vertical magenta lines represent the position of the active medium; the vertical black lines indicate the position of the optic with focal length f. \nThe black curves represent the beam size evolution along the optical axis $z$. \n(b) 6-pass amplifier and corresponding beam size evolution along the propagation axis realized by concatenating five 4f-imaging stages and assuming a Gaussian aperture at the active medium of $W=4w_\\mathrm{in}$. \nThe black curves show the beam size evolution for an active medium dioptric power having the design value, i.e. $V = 0$. \nThe red and blue curves show the propagations (beam size evolution) for $V=\\pm\\frac{1}{40 f}$. The gray curves represent the beam evolution for $V=0$ and $W=\\infty$.\nThe discontinuity of the waist size at the gain medium is caused by the (Gaussian) aperture effect of the finite pump region.}\n\\label{fig:4f-segment}\n\\end{figure}\n\n\nIt can be demonstrated that the 4f-imaging warrants the relation $w_\\mathrm{in, \\, N}=w_\\mathrm{out, \\, N-1}$~\\cite{Hunt:78}.\nHere, we defined $w_\\mathrm{in,\\, N}$ as the size of the beam incident on the $N$-th pass at the active medium, and $w_\\mathrm{out,\\, N}$ as the size of the beam departing from the $N$-th pass at the active medium after amplification.\nIn the absence of aperture effects ($W=\\infty$) in the active medium $w_\\mathrm{out, \\, N}=w_\\mathrm{in, \\, N}$. \nHence, in the absence of aperture effects the beam size is identically reproduced at each pass on the active medium independently of the size of the laser beam.\nThis property, together with the simplicity of the implementation (many passes using few optical elements), are the main advantages of the 4f-based multi-pass architecture.\nHowever, for non-vanishing aperture effects ($W \\neq \\infty$) \nthe beam size is decreased when passing the active medium $w_\\mathrm{out, \\, N}w_\\mathrm{in, \\, 1}$ (see~Fig.~\\ref{Fourier}).\nAt the second pass through the active medium, this larger beam is reduced by the aperture: $w_\\mathrm{out, \\, 2}0$), i.e. the tunneling energy \nbetween nearest neighbor sites, and \n\\begin{equation} \nU = {g\\over \\pi a_z} \\int dx\\, dy \\, |W_{{\\bf i}}(x,y)|^4 \n\\end{equation}\nis the on-site strength of the \ninter-atomic interaction. ${\\hat n}_{{\\bf i}\\sigma}= \n{\\hat c}_{{\\bf i}\\sigma}^+{\\hat c}_{{\\bf i}\\sigma}$ \nis the number operator which describes the number \nof atoms with spin $\\sigma$ at the site ${\\bf i}$, and consequently \nthe total number operator reads \n\\begin{equation} \n{\\hat N}= \\sum_{{\\bf i} \\, \\sigma} \\, {\\hat n}_{{\\bf i}\\sigma} \\; . \n\\end{equation}\nNotice that Eq. (7) holds under the conditions $|a_s|\\ll a_z$ and $|a_s|\\ll a$, \nwhich ensure the absence of confinement induced resonance \\cite{cir} and \nno distorsion of Cooper pairs due to neighbor valleys of the \noptical confinement. \nIn the Hubbard-like Hamiltonian (\\ref{ham}) we have not included the \ntunneling energies between sites which are not nearest neighbor \nbecause they are exponentially suppressed. We have also \nassumed the on-site one-body energies to be the same on all sites \nand therfore dropped them as irrelevant \\cite{stoof}. \n\n\\section{Mean-field BCS equations}\n\nIt is well known that the BCS state appears only \nin the case of an attractive strength, i.e. $U<0$ \\cite{stoof}. \nIn the past the negative-$U$ Hubbard Hamiltonian has been investigated \nby various authors \\cite{review} as a model for \nhigh-$T_c$ superconductivity. More recently, it has been used to study \nthe BCS-BEC crossover on 2D and 3D lattices \nboth at zero \\cite{andrenacci,kujawa,capone} \nand finite temperature \\cite{dupuis,tamaki}. \nAs stressed in the introduction, motivated by recent \ntheoretical and experimental achievement with ultracold atoms \nin optical lattices, here we reconsider the 2D negative-$U$ Hubbard \nHamiltonian to investigate the pair condensation, and in particular \nthe condensate fraction of Fermi atoms in the 2D \nlattice at zero temperature. Note that he condensate fraction has been \ncalculated by Kujawa \\cite{kujawa} in the 3D square lattice \nwith a generalized Hubbard model, but only in the special case $|U|\/t=\\infty$ . \nIn the following sections we calculate, as a function of $|U|\/t$ \nand of the filling factor, the energy gap and \ncondensate fraction in the 2D square lattice, analyzing also \nthe pair binding energy, which is always finite \nin the 2D BCS-BEC crossover. \n\nWe start by decoupling the interaction Hamiltonian of Eq. (\\ref{ham}) in both \nnormal and anomalous channels \\cite{privitera}\n\\begin{eqnarray} \n{\\hat n}_{{\\bf i}\\uparrow}{\\hat n}_{{\\bf i}\\downarrow} &\\simeq& \n\\langle {\\hat n}_{{\\bf i}\\uparrow}\\rangle \n{\\hat n}_{{\\bf i}\\downarrow} + \n{\\hat n}_{{\\bf i}\\uparrow} \\langle {\\hat n}_{{\\bf i}\\downarrow}\\rangle \n- \\langle {\\hat c}_{{\\bf i}\\uparrow}^+{\\hat c}_{{\\bf i}\\downarrow}^+\\rangle \n{\\hat c}_{{\\bf i}\\uparrow}{\\hat c}_{{\\bf i}\\downarrow}\n\\\\\n&-& {\\hat c}_{{\\bf i}\\uparrow}^+{\\hat c}_{{\\bf i}\\downarrow}^+ \n\\langle {\\hat c}_{{\\bf i}\\uparrow}{\\hat c}_{{\\bf i}\\downarrow} \\rangle \n- \\langle {\\hat n}_{{\\bf i}\\uparrow}\\rangle \n\\langle {\\hat n}_{{\\bf i}\\downarrow}\\rangle \n+ \n\\langle {\\hat c}_{{\\bf i}\\uparrow}^+{\\hat c}_{{\\bf i}\\downarrow}^+\\rangle \n\\langle {\\hat c}_{{\\bf i}\\uparrow}{\\hat c}_{{\\bf i}\\downarrow} \\rangle \\; . \n\\nonumber\n\\end{eqnarray}\nWe also assume \n\\begin{equation} \n{n\\over 2} = \n\\langle {\\hat n}_{{\\bf i}\\uparrow}\\rangle = \n\\langle {\\hat n}_{{\\bf i}\\uparrow}\\rangle \n\\end{equation}\nand introduce the (real) mean-field, site-independent, \ngap order parameter \n\\begin{equation} \n\\Delta = - U \n\\langle {\\hat c}_{{\\bf i}\\uparrow}^+{\\hat c}_{{\\bf i}\\downarrow}^+\\rangle \n= - U \\langle {\\hat c}_{{\\bf i}\\downarrow}{\\hat c}_{{\\bf i}\\uparrow}\\rangle \\; . \n\\end{equation} \nIn this way we obtain the mean-field Hamiltonian \n\\begin{eqnarray} \n{\\hat H}_{MF} &=& -t \\sum_{\\langle {\\bf i}{\\bf j} \\rangle \\, \\sigma} \n{\\hat c}^+_{{\\bf i}\\sigma} {\\hat c}_{{\\bf j}\\sigma} \n+ {U n\\over 2}\\sum_{\\bf i} \\left( {\\hat n}_{{\\bf i}\\uparrow} \n+ {\\hat n}_{{\\bf i}\\downarrow} \\right) \n\\\\\n&+& \\Delta \\sum_{\\bf i} \n\\left( {\\hat c}_{{\\bf i}\\uparrow}{\\hat c}_{{\\bf i}\\downarrow} \n+ {\\hat c}_{{\\bf i}\\downarrow}^+{\\hat c}_{{\\bf i}\\uparrow}^+ \\right) \n- {Un^2\\over 4} N_s + {\\Delta^2\\over U}N_s \\; , \n\\nonumber\n\\label{ham-mf} \n\\end{eqnarray}\nwhere $N_s$ is the number of lattice sites. \n\nIn the dual space of wavevectors ${\\bf k}=(k_x,k_y)$, \nsetting \n\\begin{equation} \n{\\hat c}_{{\\bf i}\\sigma} = \\sum_{\\bf k} {\\hat c}_{{\\bf k}\\sigma} \n\\ {e^{i{\\bf k}\\cdot {\\bf R}_{\\bf i}}\\over \\sqrt{N_s}} \\; , \n\\end{equation}\nwhere ${\\hat c}_{{\\bf k}\\sigma}$ destroys an atom of \nspin $\\sigma$ and wavevector ${\\bf k}$, \nthe mean-field Hamiltonian (\\ref{ham-mf}) becomes \n\\begin{eqnarray} \n{\\hat H}_{MF} &=& \\sum_{\\bf k} \n\\left( \\epsilon_{\\bf k} + {Un\\over 2} \\right) \n{\\hat c}_{{\\bf k}\\sigma}^+ \n{\\hat c}_{{\\bf k}\\sigma} \n\\\\\n&+& \\Delta \\sum_{{\\bf k}} \\left( \n{\\hat c}_{{\\bf k}\\uparrow} \n{\\hat c}_{-{\\bf k}\\downarrow}\n+ {\\hat c}_{-{\\bf k}\\downarrow}^+\n{\\hat c}_{{\\bf k}\\uparrow}^+ \\right) \n- {Un^2\\over 4} N_s + {\\Delta^2\\over U}N_s \\; , \n\\nonumber\n\\end{eqnarray}\nwhere \n\\begin{equation} \n\\epsilon_{\\bf k} = - 2 t \n\\left( \\cos{(k_x a)} + \\cos{(k_y a)} \\right) \\; , \n\\label{single}\n\\end{equation}\nis the single-particle energy. \nWe stress that we are considering only the lowest Bloch band. \nThis single-band approximation for the BCS-BEC crossover is \nreliable since the crossover occours at magnetic fields that are \nrelatively far away from the Feshbach resonance underlying it \n\\cite{stoof2}. Moreover the approximation is reliable under \nthe following conditions \\cite{stoof2,georges}: \ni) there are no more than two fermions per site; \nii) the two lowest bands do not overlap, implying that \n$V_0\\gg E_r$, which means $8t\\ll E_r$, \nand $|U| \\ll E_g$. $E_r=\\hbar^2k_L^2\/(2m)$ is the \nrecoil energy with $k_L=2\\pi\/a$ the wavevector of the 2D optical \nlattice, and $E_g$ is the energy gap between the first and \nthe second Bloch band.\n\nWe calculate the thermodynamic potential\n\\begin{equation}\n\\Omega = \\langle {\\hat H}_{MF}\\rangle - \\mu \\, \\langle \\hat{N} \\rangle \\; ,\n\\label{ther}\n\\end{equation}\nwhere $\\mu$ is the chemical potential which determines the\naverage number $N=\\langle{\\hat N}\\rangle$ of fermions, \nby introducing the Bogoliubov canonical transformation: \n\\begin{equation} \n{\\hat \\alpha}_{\\bf k} = u_{\\bf k} \\, {\\hat c}_{{\\bf k}\\uparrow} \n- v_{\\bf k} \\, {\\hat c}_{-{\\bf k}\\downarrow}^+ \\; , \n\\quad\\quad \n{\\hat \\beta}_{\\bf k} = u_{\\bf k} \\, {\\hat c}_{-{\\bf k}\\downarrow} \n+ v_{\\bf k} \\, {\\hat c}_{{\\bf k}\\uparrow}^+ \\; , \n\\end{equation}\nwhere $u_{\\bf k}$ and $v_{\\bf k}$ are real and \n$u_{\\bf k}^2+v_{\\bf k}^2=1$. After the mimimization \nof $\\Omega$ with respect to $\\mu$ and $\\Delta$ \nwe recover the standard BCS equation \\cite{stoof,privitera}\nfor the average number of particles per site \n\\begin{equation}\nn = 2 {1 \\over N_s} \\sum_{\\bf k} v_{\\bf k}^2 \\; ,\n\\label{bcs1}\n\\end{equation}\nand the familiar BCS gap equation\n\\begin{equation}\n{1\\over |U|} = {1 \\over N_s} \\sum_{\\bf k} {1\\over 2 E_{\\bf k}} \\; , \n\\label{bcsGap}\n\\end{equation}\nwhere the quasi-particle \namplitudes $u_{\\bf k}$ and $v_{\\bf k}$ are given by \n\\begin{equation} \n\\label{vk}\nv_{\\bf k}^2 = {1\\over 2} \\left( 1 - \\frac{\\epsilon_{\\bf k} \n- h }{E_{\\bf k}} \\right) \\, , \n\\end{equation} \nand $u_{\\bf k}^2=1 - v_{\\bf k}^2$. Here the Bogoliubov energy reads \n\\begin{equation}\nE_{\\bf k}=\\left[\\left(\\epsilon_{\\bf k}-h \\right)^2 \n+ \\Delta^2\\right]^{1\/2} \n\\end{equation}\nwhere \n\\begin{equation} \nh = \\mu - {Un\\over 2} \\; , \n\\end{equation}\nis the effective chemical potential which takes into account \nthe Hartree interaction. \nThe effective chemical potential $h$ and the gap energy $\\Delta$ \nare obtained by solving equations (\\ref{bcs1}) and (\\ref{bcsGap}). \nIn the continuum limit $\\sum_{\\bf k}\\to a^2 N_s \\int_{\\cal BZ} \nd^2{\\bf k}\/(2\\pi)^2$ \nand introducing the density of states (DOS) per site \n\\begin{eqnarray} \n{\\cal D}(\\epsilon) &=& a^2 \\int_{\\cal BZ} {d^2{\\bf k}\\over (2\\pi)^2} \\; \n\\delta(\\epsilon_{\\bf k}-\\epsilon) \n\\label{DoS-exact}\n\\\\\n&=& {1\\over 2\\pi^2 t} \nK\\left( \\sqrt{1-{\\epsilon^2\\over 16t^2}} \\right) \\, \n\\Theta\\left(1-{\\epsilon^2\\over 16t^2}\\right) \\; , \n\\nonumber \n\\end{eqnarray}\nwhere ${\\cal BZ}=[-\\pi\/2,\\pi\/a]\\times [-\\pi\/a,\\pi\/2]$ is the first \nBrillouin zone, $K(x)$ is the complete Elliptic integral of the first kind \nand $\\Theta(x)$ is the step function, \nthe number equation (\\ref{bcs1}) and the gap equation (\\ref{bcsGap}) \ncan be written as \n\\begin{eqnarray}\nn &=& \\int_{-4t}^{4t} d\\epsilon \\, {\\cal D}(\\epsilon) \\, \n\\left(1 - {\\epsilon-h\\over \\sqrt{(\\epsilon-h)^2+\\Delta^2}}\\right) \\; , \n\\label{exact1}\n\\\\\n{1\\over |U|} &=& \\int_{-4t}^{4t} d\\epsilon \\, {\\cal D}(\\epsilon) \\, \n{1\\over 2\\sqrt{(\\epsilon-h)^2+\\Delta^2}} \\; . \n\\label{exact2}\n\\end{eqnarray}\n\n\\begin{figure}\n\\centerline{\\epsfig{file=compare-dos.eps,width=7.4cm,clip=}}\n\\caption{(Color online). Scaled binding energy $E_B\/t$ as a function\nof the scaled interaction strength $|U|\/t$, with $t$ the\ntunneling energy. Solid lines are the results obtained with the \nexact density of states (exact DOS) given by Eq. (\\ref{DoS-exact}), while \ndashed lines are the results obtained with the approximate\ndensity of states (approx DOS) given by Eq. (\\ref{DoS-approx}).}\n\\label{fig1}\n\\end{figure}\n\nAs discussed in \\cite{review}, quite generally in two dimensions \na bound-state energy $E_B$ exists for any value \nof the negative interaction strength $U$. For the contact potential \nthe bound-state equation in the lattice is \n\\begin{equation} \n{1 \\over |U|} = \\int_{-4t}^{4t} d\\epsilon \\, {\\cal D}(\\epsilon) \\, \n{1\\over 2(\\epsilon - \\epsilon_{\\bf 0}) + E_B} \\, , \n\\label{binding}\n\\end{equation} \nwhere $\\epsilon_{\\bf 0}=-4t$ is the lower value of the single-particle \nenergy $\\epsilon_{\\bf k}$, occurring at ${\\bf k}={\\bf 0}$. \nIf we approximate the true DOS with a constant value \nin the interval $[-4t,4t]$, i.e. \n\\begin{equation} \n{\\cal D}(\\epsilon)\\simeq {1\\over 8t} \\, \n\\Theta\\left(1-{\\epsilon^2\\over 16t^2}\\right) \\; ,\n\\label{DoS-approx}\n\\end{equation} \nthat ensures the normalization \n\\begin{equation} \n\\int_{-4t}^{4t} d\\epsilon \\, {\\cal D}(\\epsilon) =1 \\, ,\n\\end{equation} \nthe bound-state equation can be solved analytically giving \n\\begin{equation} \n{1 \\over |U|} = {1\\over 16 t} \n\\ln{\\left|{E_B+16t\\over E_B}\\right|} \\; . \n\\end{equation}\n\nIn Fig. \\ref{fig1} we plot the \nbinding energy $E_B\/t$ as a function\nof the interaction strength $|U|$ obtained with this approximate \nformula (dashed line). For comparison we plot also the \nexact result (solid line), obtained by numerically \nsolving Eq. (\\ref{binding}). The figure shows that the agreement between \nthe two curves is extremely good. The BCS-BEC crossover is governed \nby the adimensional parameter $|U|\/t$ or equivalently \nby the scaled binding energy $E_B\/t$. The limit of large \ntunneling and small interaction $|U|\/t\\ll 1$ corresponds \nto the BCS regime where $E_B\/t$ is close to zero. \nInstead the limit of strong \nlocalization and large interaction $|U|\/t\\gg 1$ \ncorresponds to the BEC regime where $E_B\/t$ is large. \n\n\\begin{figure}\n\\centerline{\\epsfig{file=echem.eps,width=7.4cm,clip=}}\n\\caption{(Color online). Scaled effective chemical potential $h\/t$ (upper \npanel) and scaled chemical potential $\\mu\/t$ (lower panel) as a function\nof the scaled interaction strength $|U|\/t$, with $t$ the\ntunneling energy. Results obtained \nfor three values of the filling factor $x=n\/2$.\nFilled circles are the results obtained with the \nexact density of states \ngiven by Eq. (\\ref{DoS-exact}), while lines are the results \nobtained with the approximate\ndensity of states given by Eq. (\\ref{DoS-approx}).}\n\\label{fig2}\n\\end{figure}\n\nThe quite good agreement between the solid curve and the dashed curve \nof Fig. \\ref{fig1} suggests that one could \nuse the approximate \nDOS to study various ground-state properties of the system in the BCS-BEC. \nWithin the approximation of a constant DOS in the band, \ni.e. Eq. (\\ref{DoS-approx}), \nthe number density equation and the gap equation read \n\\begin{eqnarray}\nn &=& {1\\over 8t} \n\\Big( 8t - \\sqrt{(4t-h)^2+\\Delta^2} \n\\label{approx1}\n\\\\\n&+& \\sqrt{(4t+h)^2+\\Delta^2} \\Big) \\; ,\n\\nonumber \n\\\\\n{1\\over |U|} &=& {1\\over 16t} \\, \n\\ln{\\left|{h+\\sqrt{h^2+\\Delta^2}\\over h-8t+\n\\sqrt{(h-8t)^2+\\Delta^2}}\\right|} \\; .\n\\label{approx2}\n\\end{eqnarray}\nIt is then straightforward to plot (see Fig. \\ref{fig2}) \nthe effective chemical potential $h$ \n(upper panel) and the chemical potential $\\mu$ (lower panel) \nas a function of the scaled \ninteraction strength $|U|\/t$, \nfor different values of the filling factor $x=n\/2$ \n($0\\leq x \\leq 1$). In the figure the lines \nare obtained by using Eqs. (\\ref{approx1}) and (\\ref{approx1}) \nbased on the approximate DOS of Eq. (\\ref{DoS-approx}), while \nthe filled circles are obtained by using Eqs. (\\ref{exact1}) \nand (\\ref{exact2}) with the exact DOS of Eq. (\\ref{DoS-exact}). \n\nFig. \\ref{fig2} shows that \nat half filling ($x=0.5$) the \neffective chemical potential $h$ remains always \nconstant and equal to zero, \nand the corresponding chemical potential $\\mu$ follows the \nsimple law $\\mu =-|U|\/2$. Moreover, the lower panel of Fig. \\ref{fig2}\nshows that, at fixed filling factor $x$, the chemical \npotential $\\mu$ as a function \nof $U$ is close to a straight line (it is true straight line \nonly for $x=0.5$) and approaches $\\mu\\simeq -|U|\/2$ for large $|U|$. \n\n\\begin{figure}\n\\centerline{\\epsfig{file=delta.eps,width=7.4cm,clip=}}\n\\centerline{\\epsfig{file=chi-delta.eps,width=7.4cm,clip=}}\n\\caption{(Color online). Upper panel: Scaled energy gap $\\Delta\/t$ \nas a function of scaled interaction strength $|U|\/t$\nwith $t$ the tunneling energy. The three curves correspond \nto five different values of the filling factor $x=n\/2$.\nLower panel: Scaled energy gap $\\Delta\/t$ \nas a function of filling factor $x=n\/2$, where \nthe three curves correspond to three different values of the \nscaled interaction strength $|U|\/t$, with $t$ the tunneling energy.\nFilled circles are the results obtained with the \nexact density of states given by Eq. (\\ref{DoS-exact}), \nwhile lines are the results obtained with the approximate\ndensity of states given by Eq. (\\ref{DoS-approx}).} \n\\label{fig3}\n\\end{figure} \n\nIn Fig. \\ref{fig3} we plot the energy gap $\\Delta$ vs \ninteraction strength $|U|$ (upper panel) and vs \nfilling factor $x$ (lower panel). \nThe upper panel shows that, at fixed filling factor $x$, \nthe energy gap $\\Delta$ grows by increasing the \nscaled interaction strength $|U|\/t$, \nthat is by increasing the localization. Instead, the lower \npanel shows that, at fixed \nscaled interaction strength $|U|\/t$, the scaled energy gap \n$\\Delta\/t$ reaches its maximum value at half filling $x=1\/2$, \ni.e. when on the average there is one atom per site. \nThis effect is clearly seen in the lower panel of Fig. \\ref{fig3} \nwhere we consider three values of $|U|\/t$. Notice \nthat the behavior of $\\Delta$ as a function of $x$ is perfectly \nsymmetric with respect to $x=1\/2$ (half filling). \nAlso in Fig. \\ref{fig3} the agreement between the results \nobtained with the exact DOS and the ones calculated with the \napproximate DOS is quite good, and it improves by increasing $|U|\/t$. \nMotivated by this finding, in the remaining part of the paper we use \nthe approximate DOS, which is much simpler for numerical \ncomputations and produces analytical results. \n\n\\section{Condensate fraction}\n\nThe main task of the paper is to analyze the condensate \nfraction of fermions. \nAs shown by Yang \\cite{yang}, the BCS state \nguarantees the off-diagonal-long-range-order \\cite{penrose} \nof the Fermi gas, namely that, in the limit wherein\nboth unprimed coordinates approach an infinite distance from the primed \ncoordinates, the two-body density matrix factorizes as follows: \n\\begin{eqnarray}\n\\label{2bodydm}\n\\langle \n{\\hat \\psi}^+_{\\uparrow}({\\bf r}_1') \n{\\hat \\psi}^+_{\\downarrow}({\\bf r}_2') \n{\\hat \\psi}_{\\downarrow}({\\bf r}_1) \n{\\hat \\psi}_{\\uparrow}({\\bf r}_2) \n\\rangle \n\\\\\n= \\langle \n{\\hat \\psi}^+_{\\uparrow}({\\bf r}_1') \n{\\hat \\psi}^+_{\\downarrow}({\\bf r}_2') \n\\rangle \\langle \n{\\hat \\psi}_{\\downarrow}({\\bf r}_1) \n{\\hat \\psi}_{\\uparrow}({\\bf r}_2)\n\\rangle \\, . \n\\nonumber \n\\end{eqnarray}\nThe largest eigenvalue of the two-body density matrix (\\ref{2bodydm})\ngives the number of pairs in the lowest two-particle state, \ni.e. the condensate number of Fermi pairs \\cite{leggett,yang,campbell}. \nIn this way, the number $N_0$ of condensed fermions is given by \n\\begin{equation}\n\\label{ODLRO:def}\nN_0 = 2 \\int d^3{\\bf r} \\, d^3{\\bf r}' \\; | \\langle\n{\\hat \\psi}_{\\downarrow}({\\bf r})\n{\\hat \\psi}_{\\uparrow}({\\bf r}') \\rangle |^2 \n= 2 \\sum_{{\\bf i}{\\bf j}} \\; | \\langle \n{\\hat c}_{{\\bf i}\\downarrow} \n{\\hat c}_{{\\bf j}\\uparrow}\\rangle |^2 . \n\\end{equation} \nNotice that, as said above, $N_0$ counts the number of \ncondensed fermions: $0\\leq N_0\\leq N$ \n\\cite{sala-odlro3} and not of condensed pairs. \nIt is then quite easy to show that the the condensate \nnumber of atoms per site is \n\\begin{equation} \nn_0 = 2 {1\\over N_s} \\sum_{\\bf k} u_{\\bf k}^2 v_{\\bf k}^2 \\; . \n\\end{equation} \nWith the help of Eqs. (\\ref{vk}) this number is thus given by :\n\\begin{equation} \nn_0 = {\\Delta^2\\over 2} \\int_{-4t}^{4t} d\\epsilon \\, {\\cal D}(\\epsilon) \\, \n{1\\over (\\epsilon-h)^2+\\Delta^2} \\; , \n\\end{equation}\nand using the approximate DOS of Eq. (\\ref{DoS-approx}) it reads: \n\\begin{equation} \nn_0 = {\\Delta\\over 16 t} \n\\left( \\arctan{\\left({4t-h\\over \\Delta}\\right)} + \n\\arctan{\\left({4t+h\\over \\Delta}\\right)} \\right) \\; . \n\\label{approx3}\n\\end{equation}\n\n\\begin{figure}\n\\centerline{\\epsfig{file=cond.eps,width=7.cm,clip=}}\n\\caption{(Color online). Condensate fraction $n_0\/n$ \nas a function of scaled interaction strength $|U|\/t$\nwith $t$ the tunneling energy. The three curves correspond \nto three different values of the filling factor $x=n\/2$.} \n\\label{fig4}\n\\end{figure} \n\nFig. \\ref{fig4} shows the condensate fraction $n_0\/n$ of fermions, \ncalculated with Eqs. (\\ref{approx1}), (\\ref{approx2}) and (\\ref{approx3}), \nas a function of scaled interaction strength $|U|\/t$ \nfor three values of the filling factor $x$. We have verified \nthat the plotted results are in good agreement with the \nones obtained by using the exact DOS, \nexcept in the case of very small values of $|U|\/t$. In any case, \nthe condensate fraction $n_0\/n$ vanishes when \nthe scaled interaction strength $|U|\/t$ goes to zero. Moreover, \nas shown in the figure, the condensed fraction \ngrows very fast for values of the scaled \ninteraction strength $|U|\/t\\leq 8$, it shows a shoulder, \nand then it reaches its \nasymptotic value $n_0\/n \\simeq 1-x$ rather slowly. \n\n\\begin{figure}\n\\vskip 0.4cm\n\\centerline{\\epsfig{file=chi-cond.eps,width=7.cm,clip=}}\n\\centerline{\\epsfig{file=chi-n0.eps,width=7.cm,clip=}}\n\\caption{(Color online). Upper panel: Condensate fraction $n_0\/n$ \nas a function of the filling factor $x=n\/2$. \nLower panel: Number $n_0$ of condensed atoms per site. \nThe curves correspond to three different values of the \nscaled interaction strength $|U|\/t$, \nwith $t$ the tunneling energy.} \n\\label{fig5}\n\\end{figure} \n\nThis result is confirmed in the upper panel of Fig. \\ref{fig5}, \nwhere we report the condensate fraction $n_0\/n$ \nas a function of the filling factor $x$ at fixed \nscaled interaction strength $|U|\/t$. \nThe figure clearly shows that $n_0\/n$ ranges from one \nto zero, being extremely close to one for $x \\ll 1$ and \napproaching zero as $x$ goes to $1$. \nThis means that there is a full BEC-BCS crossover by increasing $x$ \nat constant scaled interaction strength $|U|\/t$. \nMoreover, if the scaled interaction strength $|U|\/t$ is large, \nthe condensate fraction $n_o\/n$ follows a straight line \nduring the BEC-BCS crossover. \nFor the sake of completeness, in the lower panel of Fig. \\ref{fig5} \nwe plot also the number $n_0$ of condensed atoms per site \nas a function of the filling factor $x$. \nThe results show that the curves of $n_o$ vs $x$ have a behavior \nsimilar to those of $\\Delta$ vs $x$ (see Fig. \\ref{fig3}). \nIn the limit $|U|\/t\\to \\infty$ one finds that $n_0=(1-x)2x$ \nand consequently $n_0\/n=(1-x)$. \n\nFinally, we observe that, after a simple rescaling of the chemical \npotential, namely ${\\tilde h}=h+4t$, in the limit $t\\to \\infty$ \nwith $ta^2\\to \\pi\\hbar^2\/m$, Eq. (\\ref{approx3}) becomes \nthe condensate number equation found in Ref. \\cite{sala-odlro2} \nfor the 2D uniform superfluid Fermi gas. \n\n\\section{Conclusions} \n\nBy using the mean-field extended BCS theory and \nthe concept of off-diagonal long-range order, that is \nthe existence of a macroscopic eigenvalue of the two-body \ndensity matrix, we have investigated the condensate fraction \nof fermionic pairs in a uniform 2D Fermi gas. \nWe have shown that the condensate number $n_0$ of fermi atoms per site \nis extremely useful \nto characterize the BCS-BEC crossover, that is induced by changing the \nadimensional ratio $|U|\/t$ between the interaction energy $|U|$ \nand the tunneling energy $t$. In particular, we have found that \nboth the scaled binding energy $E_B\/t$ \nof atomic pairs and the condensate fraction $n_0\/n$ \ngrow by increasing the ratio $|U|\/t$ at fixed filling \nfactor $x=n\/2$ (with $n$ the average number of fermions per site). \nIn addition, our results suggest that \nfixing the ratio $|U|\/t$, or equivalently the scaled \nbinding energy $E_B\/t$, there is a full BEC-BCS crossover \nby increasing the filling factor from zero to one. \nFinally, we have found that the analytical results obtained \nby using an approximate density of states are in quite \ngood agreement with the numerical ones deduced from the exact density \nof states. In our calculations we have used the mean-field theory and \nit is important to stress that recent \nMonte Carlo simulations have shown that, at zero-temperature, \nbeyond-mean-field effects\nare negligible in the BCS side of the BCS-BEC crossover while they become\nrelevant in the deep BEC side \\cite{astrakharchik,bertaina}. \nIn any case, we think that our mean-field results, and the \nreliable analytical formulas we have obtained, can be of interest \nfor near future experiments with degenerate gases \nmade of alkali-metal atoms confined in quasi-2D optical lattices. \n\nIn this paper we have investigated zero-temperature pair condensation. \nAccording to the Mermin-Wagner theorem \\cite{stoof}, for an infinite \n2D system there is condensation (off-diagonal-long-range order) only \nat zero temperature. However, for a finite 2D system \ncondensation could be possible also at non-zero temperature. \nThe investigation of this issue, which requires a beyond mean-field approach,\nfor 2D fermions in a lattice is in progress. \nAnother puzzling issue is the filling of the second Bloch band: \nwe plan to investigate its effect on pair condensation \nby analyzing a multi-band version of the present theory. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nQuantum theory displays many counterintuitive features which are in stark contrast to our everyday experiences in the macroscopic world. Possibly the most extreme of these is the collapse of the wavefunction due to measurement; its contentious interpretation has given rise to the \\emph{measurement problem}. Obviously, the only possibility to observe and study wavefunction collapse and its entailments is to conduct measurements on the collapsed wavefunction. Therefore, in order to gain a better understanding of what the collapse means and how it occurs, one has to study \\emph{repeated} measurements on the same quantum system, both from a theoretical and from an experimental perspective. This should be seen as motivation for our work on temporal quantum correlations. In theories different from orthodox quantum mechanics, for example when wavefunction collapse is not absolutely instanteous~\\cite{Pearle}, the properties of temporal correlations are likely to be different from those presented here.\n\nQuantum correlations have mostly been investigated for scenarios of several spacelike separated parties sharing some nonlocal correlations. The simplest situation one can consider here is the Clauser-Horne-Shimony-Holt (CHSH)~\\cite{CHSH} scenario: two parties, commonly dubbed \\emph{Alice} and \\emph{Bob}, each operate with a physical system of their own on which they respectively conduct one of two dichotomic (i.e. two-valued) measurements. Then, on the one hand, quantum theory entails phenomena that cannot be achieved classically: many quantum states that have the property of being \\emph{entangled} let Alice and Bob observe correlations between their measurements which cannot be explained by classical models defined in terms of local hidden variables; this non-classicality can be detected by observing violations of the \\emph{CHSH inequalities}. These inequalities precisely characterize those correlations having local hidden variable models. Furthermore, \\emph{Hardy's nonlocality paradox}~\\cite{Har} shows that this feature is not solely a quantitative trait of the joint outcome probabilities: it proves that there also exists a qualitative difference between quantum correlations and the realm of local hidden variable models. On the other hand, it has been found out that there are nevertheless strict limitations on which correlations can be observed with quantum-mechanical systems. The \\emph{Popescu-Rohrlich box} (PR-box) is a joint probability distribution that is consistent with the causality principle of \\emph{no-signaling}, but yet such a PR box cannot be constructed in a quantum-mechanical world. This can be seen most directly from the \\emph{Tsirelson bound}, which specifies the maximal quantum violation of the CHSH inequalities.\n\nIn this paper, we study a \\emph{temporal} version of the CHSH scenario. We may imagine a single physical system in a laboratory, on which the two experimentalists Alice and Bob can conduct their measurements. However it so happens that their work shifts do not intersect, and Alice leaves the lab before Bob arrives. Now it is known that Alice, during her shift, has measured one of the two $\\pm 1$-valued observables $a_1$ or $a_2$, and likewise, Bob will measure one of the two $\\pm 1$-valued observables $b_1$ or $b_2$. It is crucial to assume that Alice only conducts one of the two projective measurements $a_1$ and $a_2$, so that she cannot disturb the system and its natural dynamics in any other way. Then which joint probability distributions for the measurement outcomes can possibly arise in this way? In the following sections we answer certain aspects of this question. Just like in the spatial case, we find both fundamental possibilities achievable by such quantum correlations, and fundamental limitations on these quantum correlations. There are analogues of all the spatial phenomena mentioned in the previous paragraph: impossibility of hidden variable models---following~\\cite{Lap}, no locality or non-invasiveness assumption is actually needed---, a version of Hardy's paradox which turns out to be stronger than in the spatial scenario, the possibility of signaling in a limited form, impossibility of the PR-box, and the Tsirelson bound. Moreover, although the set of joint probabilities realizable by spatial quantum correlations is strictly contained in the set of joint probabilities realizable by temporal quantum correlations, we find that the set of realizable $\\emph{correlators}$ is the same in the temporal case as in the spatial case.\n\nThere has been a considerable amount of previous work on the properties of temporal quantum correlations. In particular, the Leggett-Garg inequalities~\\cite{LG} characterize the probabilistic hidden variable models for the scenario that one measures two-time correlators between three $\\pm 1$-valued observables\\footnote{In the standard scenario, these three observables are actually a single observable measured at three different times, but this assumption is not relevant to the argument.}, and it is known that these can be violated quantum-mechanically. In contrast to spacelike separated situations, it is not necessary here to have more than one observable for each ``party'', i.e. at each point in time, since the observables between the different points in time need not commute, leading to specifically quantum phenomena. Very recently, Avis, Hayden and Wilde~\\cite{AHW} have classified all tight Leggett-Garg inequalities for the two-time correlators between any number of dichotomic observables as precisely the facets of the cut polytope. Some other relevant references include~\\cite{Lap} and~\\cite{Bru}.\n\n\n\\section{Joint probabilities in the temporal CHSH scenario}\n\\label{temporalCHSH}\n\nWe start with several statements about temporal correlations between projective quantum measurements of $\\pm 1$-valued observables. Then we describe the temporal CHSH scenario, which has been outlined in the introduction, in a little more detail.\n\n\\paragraph{Setting the stage.}\n\\label{stage}\nConsider a single quantum system with an underlying Hilbert space $\\mathcal{H}$ and dynamics described by the Hamiltonian $H$. Furthermore, we have $\\pm 1$-valued, i.e. dichotomic, observables $a$ and $b$, which are hermitian operators on $\\mathcal{H}$ with the property\n\\begin{displaymath}\na^2=\\mathbbm{1},\\qquad b^2=\\mathbbm{1}.\n\\end{displaymath}\nNote that we can bring any pair of two-valued observables into this form by relabelling the outcomes as $+1$ and $-1$. Now Alice measures $a$ at time $t_A$ and Bob measures $b$ at time $t_B$. Both measurements are assumed to be perfect projective von Neumann measurements, so that the state collapses to an eigenstate of the corresponding observable upon the measurement. This assumption is relevant for Alice since it limits the way in which her measurement $a$ can influence the system; we will see in paragraph~\\ref{CHSHpar} that if we would allow arbitrary generalized measurements (L\\\"uders measurements) for Alice, then any set of joint outcome probabilities without signaling from Bob to Alice could be modelled even with commuting Kraus operators, i.e. with a classical probabilistic system.\n\nHowever for Bob, the assumption of projective measurements is not essential: since his post-measurement state does not get measured, this post-measurement state is irrelevant and only his outcome probabilities matter. And concerning these, we can always enlarge the Hilbert space to turn any POVM into a projective measurement while preserving the outcome probabilities.\n\nWe take the system to be in the pure initial state $|\\psi\\rangle$ just before Alice's measurement at time $t_A$. The assumption of a pure initial state is merely for notational convenience, and all following calculations would also apply mutatis mutandis to the case of a mixed initial state. Note also that in case of a mixed initial state described by a density operator $\\rho$ on $\\mathcal{H}$, we can replace it by a purification $|\\psi\\rangle$ on $\\mathcal{H}\\otimes\\mathcal{H}'$ for some $\\mathcal{H}'$, while replacing the observables $a$ and $b$ by $a\\otimes\\mathbbm{1}$ and $b\\otimes\\mathbbm{1}$. This retains all joint outcome probabilities.\n\nWhen working in the Heisenberg picture, the unitary evolution of the state is trivial, while Bob's observables evolve according to\n\\begin{displaymath}\nb'\\equiv e^{-iH(t_B-t_A)}b\\, e^{iH(t_B-t_A)}.\n\\end{displaymath}\nSince the observable $b$ was arbitrary, the evolved observable $b'$ is also just an arbitrary $\\pm 1$-valued observable on $\\mathcal{H}$. Hence as far as the existence of quantum-mechanical models for joint probabilities is concerned, the dynamics is irrelevant. In particular, we will choose $H=0$ for simplicity, so that $b'=b$. Then wavefunction collapse is the only ``dynamics'' present in our formalism.\n\n\\paragraph{Joint probabilities and correlators.} Now we calculate the joint probabilities in terms of $a$, $b$ and $|\\psi\\rangle$. For the $\\pm 1$-valued observable $a$, the projection operator onto the $+1$-eigenspace and the projection operator onto the $-1$-eigenspace are given by, respectively,\n\\begin{displaymath}\n\\frac{\\mathbbm{1}+a}{2},\\qquad \\frac{\\mathbbm{1}-a}{2}.\n\\end{displaymath}\nand in the same way for $b$. Using the Born rule together with the projection postulate shows that the joint probability for Alice to get the outcome $r\\in\\{-1,+1\\}$ and for Bob to get the outcome $s\\in\\{-1,+1\\}$ reads as\n\\begin{align}\n\\begin{split}\n\\label{jointprobs}\nP(r,s)&=\\left\\langle\\psi\\left|\\frac{\\mathbbm{1}+ra}{2}\\cdot\\frac{\\mathbbm{1}+sb}{2}\\cdot\\frac{\\mathbbm{1}+ra}{2}\\right|\\psi\\right\\rangle\\\\[1pc]\n&=\\tfrac{1}{4}+\\tfrac{1}{4}r\\langle\\psi|a|\\psi\\rangle+\\tfrac{1}{8}s\\langle\\psi|b|\\psi\\rangle+\\tfrac{1}{8}rs\\langle\\psi|\\{a,b\\}|\\psi\\rangle+\\tfrac{1}{8}s\\langle\\psi|aba|\\psi\\rangle.\n\\end{split}\n\\end{align}\nIn this expression, $\\{\\cdot,\\cdot\\}$ denotes the anticommutator of two operators. $P(r,s)$ is the probability that Alice observes the outcome $r$, multiplied by the probability that Bob gets the outcome $s$ upon measuring the state of the system after state collapse due to Alice's outcome being $r$.\n\nWe also consider \\emph{correlators}, which are defined as\n\\begin{align}\n\\begin{split}\n\\label{correlation}\nC&\\equiv\\sum_{r,s}rs\\: P(r,s)\\\\\n&=P(+1,+1)+P(-1,-1)-P(-1,+1)-P(+1,-1).\n\\end{split}\n\\end{align}\nUsing\\eq{jointprobs}, the correlator can be expressed as\n\\begin{equation}\n\\label{corrs}\nC=\\tfrac{1}{2}\\langle\\psi|\\{a,b\\}|\\psi\\rangle\n\\end{equation}\nwhich is intuitive since only the $rs$-term in equation\\eq{jointprobs} suggests any kind of correlation between the outcomes. So strangely, even though our scenario has a clear temporal order, the correlators do not depend on who measures first! As far as we can see, this curious property does not generalize to observables with more than two outcomes or to scenarios with more than two parties. \n\nNote that when we use the term ``correlation'', we simply mean ``specification of joint outcome probabilities for all allowed choices of observables'', while the notion of ``correlator'' refers only to the quantity\\eq{corrs}.\n\n\\paragraph{The CHSH scenario.}\n\\label{CHSHpar}\nIn the CHSH scenario, Alice and Bob both have an independent choice between two observables. While Alice can select either the observable $a_1$ or the observable $a_2$, Bob has the freedom to measure either $b_1$ or $b_2$. For each of the resulting four choices, we obtain a distribution of joint probabilities of the form\\eq{jointprobs}. We will use the notation\n\\begin{equation}\n\\label{probs}\nP(r,s|k,l)\n\\end{equation}\nto denote the probability that Alice gets the outcome $r\\in\\{-1,+1\\}$ and Bob gets the outcome $s\\in\\{-1,+1\\}$, given that Alice measures $a_k$ and Bob measures $b_l$. Finally, we will use the notation $C_{kl}$ for the correlator between $a_k$ and $b_l$.\n\nAs announced in paragraph~\\ref{stage}, it will now be proven that any set of probabilities\\eq{probs} has a quantum-mechanical representation in terms of generalized measurements (L\\\"uders measurements) for Alice, under the assumption that these probabilities satisfy causality in the sense that there is no backward signaling from Bob to Alice. This intuitive condition means that the joint probabilities can be factorized as\n\\begin{equation}\n\\label{factor}\nP(r,s|k,l)=P_B(s|r;k,l)P_A(r|k)\n\\end{equation}\nwhere $P_A(s|k)$ designates the outcome probabilities for Alice's measurement alone, and these are assumed to be independent of Bob's data $l$ and $s$. On the other hand, Bob's conditional outcome probabilities $P_B(s|r;k,l)$ may well depend on Alice's data in an arbitrary way. Condition\\eq{factor} is necessary for the existence of a representation via generalized measurements, since the product representation\\eq{factor} is essentially how one would typically calculate the joint probabilities starting from the quantum-mechanical data: first determine Alice's outcome probabilities $P_A(r|k)$ given the initial state $|\\psi\\rangle$, then calculate Bob's outcome probabilities $P_B(s|r;k,l)$, and finally multiply these two probabilities to obtain the desired result. Bob's probabilities $P_B(s|r;k,l)$ depend on the system's quantum state after Alice's measurement, and this state in turn is determined by $k$, $r$ and $|\\psi\\rangle$.\n\nConversely, in order to find a quantum-mechanical representation for an arbitrary such set of probabilities, consider a five-dimensional Hilbert space with orthonormal basis $\\{|0\\rangle,|1^+\\rangle,|1^-\\rangle,|2^+\\rangle,|2^-\\rangle\\}$. We take the initial state of the system to be $|\\psi\\rangle=|0\\rangle$. There exist generalized measurements such that the state after Alice's measurement is $|1^+\\rangle$ if she measured $a_1$ and obtained a $+1$ outcome, and it is $|1^-\\rangle$ if she measured $a_1$ and obtained a $-1$ outcome, and similarly for $|2^+\\rangle$ and $|2^-\\rangle$. Concretely, one can implement such measurements for example by using the Kraus operators\n\\begin{displaymath}\nV_k^r\\equiv \\sqrt{P_A(r|k)}\\:\\left|k^r\\rangle\\langle 0\\right|+\\sum_{k',r'}\\tfrac{1}{\\sqrt{2}}\\:\\big|k'^{r'}\\big\\rangle\\big\\langle k'^{r'}\\big|,\\quad r\\in\\{-1,+1\\}\n\\end{displaymath}\nas describing the measurement of $a_k$. The first term guarantees that the post-measurement state of $V_k^r$ is the desired $|k^r\\rangle$ and that the given measurement statistics are reproduced, both on the initial state $|\\psi\\rangle=|0\\rangle$. (The other terms are merely needed for satisfaction of the completeness relation $\\sum_{r}V_k^{r\\dagger}V_k^r=\\mathbbm{1}$.) For Bob, we can choose the two POVMs $\\left\\{\\Pi_1^+,\\Pi_1^-\\right\\}$, $\\left\\{\\Pi_2^+,\\Pi_2^-\\right\\}$ with\n\\begin{displaymath}\n\\Pi_l^s\\equiv\\mathrm{diag}\\left(\\tfrac{1}{2},\\:P(s|+1;1,l),\\:P(s|-1;1,l),\\:P(s|+1;2,l),\\:P(s|-1;2,l)\\right)\\\\\n\\end{displaymath}\nas representing the measurements $b_1$ and $b_2$; since Bob's post-measurement state does not get observed, we do not have to specify any Kraus operators implementing these POVMs. By construction, these POVMs reproduce the desired outcome probabilities $P_B(s|r;k,l)$ on the corresponding states $|k^r\\rangle\\in\\{|1^+\\rangle, |1^-\\rangle, |2^+\\rangle, |2^-\\rangle\\}$. This ends the construction of a quantum-mechanical model with generalized measurements for\\eq{factor}. Some final remarks: since neither the initial state nor any post-measurement state is a superposition of basis states, this construction effectively yields a classical stochastic system. The trick in the construction is that Alice's post-measurement state keeps track of both her measurement setting and her outcome. This conditional state collapse to mutually orthogonal states would not be possible if we would only allow projective measurements for Alice.\n\n\\paragraph{Temporal hidden variable models.}\n\nUsing the assumption of what they called ``macroscopic realism'' and ``non-invasiveness'', Leggett and Garg~\\cite{LG} derived an inequality satisfied by temporal correlations in hidden variable models which is violated by certain temporal quantum correlations. Macroscopic realism is the assumption that the system is, at each instant in time, definitely situated in one of several distinct states. This system state determines all measurement outcomes exactly; in this sense, all observables possess preexisting definite values. This is thought to apply to macroscopic objects in particular, hence the name ``macroscopic realism'', or more succinctly ``macrorealism''.\n\nThe crucial assumption now is non-invasiveness: this postulates that a measurement does not disturb the state of the system. There is an additional hidden assumption which has been made explicit and dubbed ``induction'' by Leggett~\\cite{Real}: it is understood that the state of the system at time $t$ is sufficient information to calculate the outcomes of all future measurements. (In other words, causality only propagates forward in time.) All of these assumptions seem rather natural when dealing with macroscopic systems. In a manner analogous to the spatical case, one can now use these premises to derive (see~\\cite{Real}, compare~\\cite{Bru}) the temporal CHSH inequality:\n\\begin{equation}\n\\label{CHSH}\nS_{\\textrm{CHSH}}\\equiv C_{11}+C_{12}+C_{21}-C_{22}\\leq 2.\n\\end{equation}\nOn the other hand, it is known that this inequality can be violated by certain quantum correlations~\\cite{Bru}. This is an exciting area due to promising prospects of using such results for testing the applicability of quantum theory in the macroscopic domain~\\cite{exp}.\n\nWe will get back to hidden variable models in section~\\ref{hardysection}.\n\n\\paragraph{Comparison to the spatial scenario.}\nIn general, the non-invasiveness assumption for hidden variable models is the exact analogue of locality in the spatial case. In both cases, the distribution of joint measurement outcomes is a probabilistic combination (i.e. a convex combination) of a collection of realistic models; a realistic model in turn is described by a hidden variable $\\lambda$, constant over space and time, which determines all the outcomes of all possible measurements in a definite way. Therefore, there is absolutely no difference between local hidden variable models in spatial scenarios, and non-invasive hidden variable models in temporal scenarios. \n\nSo the reason that one considers inequalities characterizing hidden variable models for temporal scenarios which are different from those in the spatial case is not that the hidden variable models are different --- they are the same. The reason is that the quantum-mechanical correlations are very different and strongly depend on whether one considers a spatial scenario or a temporal scenario. Although the Leggett-Garg inequality is perfectly valid as a spatial Bell inequality in a three-party scenario, it is not interesting in this case: since there is only one observable per party, no quantum violations are possible, and likewise no violations by more general no-signaling theories.\n\nLet us also note that any set of joint outcome probabilities for a spatial Bell test can also appear in the temporal scenario. Mathematically, this follows from the fact that we recover exactly the spatial joint probabilities by taking $a$ and $b$ in\\eq{jointprobs} to operate on separate tensor factors. Physically, this is clear since we can just think of Alice's and Bob's spatially separated quantum systems as a single quantum system, and then simply imagine that Alice conducts her measurement first, with Bob's measurement operating at a later time.\n\nTo end the comparison with spatial scenarios, let us recast\\eq{corrs} in the following form:\n\n\\begin{prop}\nWhile a spatial correlator is given by the expectation value of the tensor products of the observables, a temporal correlator is given by half the expectation value of the anticommutator of the observables:\n\\begin{displaymath}\n\\textrm{spatial: }\\: C=\\langle\\psi|a\\otimes b|\\psi\\rangle \\quad\\longrightarrow\\quad \\textrm{temporal: }\\: C=\\tfrac{1}{2}\\langle\\psi|\\{a,b\\}|\\psi\\rangle\n\\end{displaymath}\n\\end{prop}\n\n\\paragraph{The qubit case and beyond.}\n\nAs a first example of temporal quantum correlations, we consider a single qubit in the Bloch sphere picture. This case has also been treated in~\\cite{Bru}.\n\nLet the system have an initial state given in terms of the Bloch vector $\\vec v$. A dichotomic observable is described by a unit vector $\\vec{a}\\in\\mathbb{R}^3$, such that the probability for getting the outcome $r\\in\\{-1,+1\\}$ on the state $\\vec{v}$ is given by\n\\begin{equation}\n\\label{qubitAlice}\n\\tfrac{1}{2}(1+r\\,\\vec{a}\\cdot\\vec{v})\n\\end{equation}\nAnd in case that the outcome $r$ has been observed, the state has collapsed to $r\\,\\vec{a}$.\n\nThe dynamics of the qubit between $t_1$ and $t_2$ in this representation is specified by a rotation matrix $R\\in SO(3)$, such that the state prior to Bob's measurement is $R(r\\,\\vec{a})=r\\,R(\\vec{a})$. Then given that Alice obtained the outcome $r$, the probability for Bob to get the outcome $s$ is consequently\n\\begin{equation}\n\\label{qubitBob}\n\\tfrac{1}{2}(1+rs\\,\\vec{b}\\cdot R(\\vec{a})).\n\\end{equation}\nAfter multiplying the two expressions\\eq{qubitAlice} and\\eq{qubitBob} to get the joint probability and summing over $r$ and $s$ with the appropriate sign, the correlator explicitly reads according to the definition\\eq{correlation}\n\\begin{equation}\n\\begin{split}\nC & = \\tfrac{1}{2}(1 + \\vec a \\cdot \\vec v) \\tfrac{1}{2}(1 + \\vec b \\cdot R(\\vec a)) \\\\\n& + \\tfrac{1}{2}(1 - \\vec a \\cdot \\vec v) \\tfrac{1}{2}(1 + \\vec b \\cdot R(\\vec a)) \\\\\n& - \\tfrac{1}{2}(1 + \\vec a \\cdot \\vec v) \\tfrac{1}{2}(1 - \\vec b \\cdot R(\\vec a)) \\\\\n& - \\tfrac{1}{2}(1 - \\vec a \\cdot \\vec v) \\tfrac{1}{2}(1 - \\vec b \\cdot R(\\vec a)) \\\\\n& = R(\\vec a) \\cdot \\vec b.\n\\end{split}\n\\end{equation}\nSo remarkably, this correlator does not depend on the initial state,\nwhich is due to the collapse after Alice has measured, and the structure of the correlator as a particular linear combination of joint probabilities.\nThis correlator is very similar to the correlator known from maximally entangled two-qubit states\nand therefore we can now find the maximal qubit value using simple techniques.\nThe CHSH quantity then reads:\n\\begin{align*}\nS_{\\textrm{CHSH}}^{\\textrm{qubit}} &= C_{11}+C_{12}+C_{21}-C_{22}\\\\\n&= R(\\vec a_1) \\cdot (\\vec b_1 + \\vec b_2) + R(\\vec a_2) \\cdot (\\vec b_1 - \\vec b_2)\n\\end{align*}\nFor finding its maximum, note that since the vectors $\\vec b$ are normalized, the vectors in the brackets are orthogonal.\nMoreover, $|\\vec b_1 + \\vec b_2|^2 + |\\vec b_1 - \\vec b_2|^2 = 4$ and so we can introduce\ntwo new orthogonal normalized vectors $\\vec b_+$ and $\\vec b_-$ such that\n$\\vec b_1 + \\vec b_2 = 2 \\cos \\alpha \\, b_+$ and $\\vec b_1 - \\vec b_2 = 2 \\sin \\alpha \\, b_-$ for some angle $\\alpha$.\nPlugging this into the expression for $S_{\\textrm{qubit}}$ and optimizing over the $R(\\vec{a}_i)$, which are also normalized vectors, yields the Tsirelson bound of $2\\sqrt{2}$, which is therefore the maximal value achievable with a qubit. In particular, this violates the bound\\eq{CHSH}, confirming that quantum theory cannot be equivalent to a probabilistic hidden variable theory with preexisting values for all observables and repeatable measurements.\n\nAll the concrete examples of temporal quantum correlations which we will consider in the following sections are modelled on qubits. So here let us quickly demonstrate that not all quantum correlations in the temporal CHSH scenario can arise from qubit data. Consider a qutrit system with orthonormal basis $\\{|0\\rangle,|1\\rangle,|2\\rangle\\}$, and the following prescriptions:\n\\begin{itemize}\n\\item the initial state $|\\psi\\rangle=|0\\rangle$,\n\\item $a_1$ measures if the system is in the state $|0\\rangle+|1\\rangle$,\n\\item $a_2$ measures if the system is in the state $|0\\rangle+|2\\rangle$,\n\\item $b_1$ measures if the system is in the state $|2\\rangle$,\n\\item $b_2$ is any dichotomic observable.\n\\end{itemize}\nThis system has the following properties: Alice's outcomes both have probability $1\/2$, independent of whether she chooses $a_1$ or $a_2$. But her choice drastically affects Bob's prospects upon measuring $b_1$: when Alice chooses $a_1$, he will definitely observe a $-1$ outcome; however when Alice chooses $a_2$, his outcome will be uniformly random and independent of hers. Such behavior is impossible in a qubit system: one would necessarily need to have $b_1=-\\mathbbm{1}$, otherwise Bob's outcome could not be definite after Alice's non-trivial measurement of $a_1$. But then obviously his outcome would also have to be a definite $-1$ when Alice measured $a_2$, which it is not allowed to be. It would be interesting to try and turn this into a dimension witness in the sense of~\\cite{DW}.\n\n\\section{Correlator space and the Tsirelson bound}\n\nWe may ask whether the temporal correlators satisfy the Tsirelson bound generally, or whether this just holds for the case of a qubit system. From the qubit case we know that the Tsirelson bound can be attained; but a priori, some temporal quantum correlations may in principle be so strong that even the Tsirelson inequality\n\\begin{equation}\n\\label{tsirelson_bound}\nS_{\\textrm{CHSH}}=C_{11}+C_{12}+C_{21}-C_{22}\\leq 2\\sqrt{2}\n\\end{equation}\nis violated.\n\nWhat we mean here by \\emph{correlator space} is the set of quadruples\n\\begin{displaymath}\n(C_{11},C_{12},C_{21},C_{22})\n\\end{displaymath}\nwhich can appear as correlators between Alice's and Bob's measurements in a quantum-mechanical world. Recall that the correlators are defined as\n\\begin{equation}\n\\label{correlation2}\nC_{kl}\\equiv \\sum_{r,s\\in\\{-1,+1\\}}rs\\,P(r,s|k,l)\n\\end{equation}\nso that there is a linear map from probability space down to correlator space. Obviously, taking the projection of a point from probability space down to correlator space throws away some data, so specifying the four correlators is not sufficient for knowing the full set of joint probabilities. Yet the correlators contain precious information about the system, for example the maximal violation of the CHSH inequality, and they are also related to the possibility of producing PR-box behavior (see section~\\ref{PRboxsection}).\n\nFor the remaining part of this section, we will consider the scenario in which Alice has a choice between $m\\in\\mathbb{N}$ dichotomic observables, while Bob has a choice betweeen $n\\in\\mathbb{N}$ dichotomic observables. Even in this generality, it is not hard to use the techniques of Tsirelson for showing that, in correlator space, the temporal quantum region coincides with the spatial quantum region. Tsirelson has proven in his paper~\\cite{Tsi} that the following three statements are equivalent, for any given matrix of correlators $\\left(C_{kl}\\right)_{k=1,\\ldots,m}^{l=1,\\ldots,n}$:\n\\begin{enumerate}\n\\item There exists a $C^*$-algebra $\\mathcal{A}$ with identity, hermitian elements $a_1,\\ldots,a_m,b_1,\\ldots,b_n$ and a state $f$ on $\\mathcal{A}$ such that for any $k, l$, we have \n\\begin{displaymath}\na_kb_l=b_la_k,\n\\end{displaymath}\\begin{displaymath}\n-\\mathbbm{1}\\leq a_k\\leq\\mathbbm{1};\\qquad -\\mathbbm{1}\\leq b_l\\leq\\mathbbm{1},\n\\end{displaymath}\\begin{displaymath}\nf(a_kb_l)=C_{kl}.\n\\end{displaymath}\n\\item There exist Hilbert spaces $\\mathcal{H}_a$ and $\\mathcal{H}_b$ together with Hermitian operators $a_1,\\ldots,a_m\\in\\mathcal{B}(\\mathcal{H}_a)$, $b_1,\\ldots,b_n\\in\\mathcal{B}(\\mathcal{H}_b)$ and a density matrix $\\rho$ on $\\mathcal{H}_a\\otimes\\mathcal{H}_b$ such that\n\\begin{displaymath}\na_k^2=\\mathbbm{1};\\qquad b_l^2=\\mathbbm{1}\n\\end{displaymath}\\begin{displaymath}\n\\mathrm{tr}\\left(\\rho(a_k\\otimes b_l)\\right)=C_{kl}\n\\end{displaymath}\n\\item In the Euclidean space of dimension $\\min(m,n)$, there exist vectors $x_1,\\ldots,x_m,y_1,\\ldots,y_n$ such that\n\\begin{displaymath}\n|x_k|\\leq 1;\\qquad |y_l|\\leq 1;\\qquad \\langle x_k,y_l\\rangle=C_{kl}\\quad\\forall k,l\n\\end{displaymath}\n\\end{enumerate}\n\n\\begin{prop}\nThese conditions are also equivalent to the following two:\n\\begin{enumerate}\n\\item[(a')] There exists a $C^*$-algebra $\\mathcal{A}$ with identity, hermitian elements $a_1,\\ldots,a_m,b_1,\\ldots,b_n$ and a state $f$ on $\\mathcal{A}$ such that for any $k, l$ we have\n\\begin{displaymath}\n-\\mathbbm{1}\\leq a_k\\leq\\mathbbm{1};\\qquad -\\mathbbm{1}\\leq b_l\\leq\\mathbbm{1}\n\\end{displaymath}\\begin{displaymath}\nf\\left(\\tfrac{1}{2}\\left\\{a_k,b_l\\right\\}\\right)=C_{kl}\n\\end{displaymath}\n\\item[(b')] There exists a Hilbert space $\\mathcal{H}$ together with Hermitian operators $a_1,\\ldots,a_m,b_1,\\ldots,b_n\\in\\mathcal{B}(\\mathcal{H})$ and a density matrix $\\rho$ on $\\mathcal{H}$ such that\n\\begin{displaymath}\na_k^2=\\mathbbm{1};\\qquad b_l^2=\\mathbbm{1}\n\\end{displaymath}\\begin{displaymath}\n\\mathrm{tr}\\left(\\rho\\cdot\\tfrac{1}{2}\\left\\{a_k,b_l\\right\\}\\right)=C_{kl}\n\\end{displaymath}\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nWe first show that (b)$\\Rightarrow$(b'). Given the data as in (b), it is clear that they also satisfy (b') if we take $\\mathcal{H}=\\mathcal{H}_a\\otimes\\mathcal{H}_b$ and rename $a_k\\otimes\\mathbbm{1}$ to $a_k$ and $\\mathbbm{1}\\otimes b_l$ to $b_l$.\n\nThe implication (b')$\\Rightarrow$(a') easily follows by choosing $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})$, and $f(x)\\equiv\\mathrm{tr}(\\rho\\cdot x)$.\n\nTo close the circle of implications, we will now check that (a')$\\Rightarrow$(c). But this works in exactly the same way as Tsirelson's own proof~\\cite{Tsi} that (a)$\\Rightarrow$(c): start with the finite-dimensional vector space defined to be the $\\mathbb{R}$-linear span of the $a_k$ and the $b_l$. This vector space carries an inner product, possibly degenerate, which is defined as\n\\begin{displaymath}\n\\langle x,y\\rangle\\equiv f\\left(\\tfrac{1}{2}\\left\\{x,y^*\\right\\}\\right)=\\mathrm{Re}\\:f(y^*x)\n\\end{displaymath}\nAfter quotiening out the null space, this inner product becomes positive definite and produces a Euclidean space such that\n\\begin{displaymath}\n|a_k|^2=\\langle a_k,a_k\\rangle=f(a_k^2)\\leq 1,\\qquad |b_l|^2=\\langle b_l,b_l\\rangle=f(b_l^2)\\leq 1,\\qquad \\langle a_k,b_l\\rangle=C_{kl}\n\\end{displaymath}\nas required. Now just as in~\\cite{Tsi}, all the requirements of (c) are satisfied, except that the dimension of the space has to be at most $\\min(m,n)$. This can also be easily achieved by orthogonal projection of the vectors $x_k\\equiv a_k$ onto the subspace spanned by the vectors $y_l\\equiv b_l$, or in the other way around.\n\\end{proof}\n\nBy\\eq{corrs}, we have therefore proven the following result:\n\n\\begin{thm}\n\\label{spatiotemporal}\nA matrix of correlators $\\left(C_{kl}\\right)_{k=1,\\ldots,m}^{l=1,\\ldots,n}$ can appear as temporal correlations between dichotomic projective measurements on the same system if and only if it can appear as spatial correlations between dichotomic measurements on two spatially separated entangled systems.\n\\end{thm}\n\nIn particular, this implies that the Tsirelson bound\\eq{tsirelson_bound} is indeed generally valid in our temporal setting.\n\n\\section{Impossibility of PR-box correlations}\n\\label{PRboxsection}\n\nWe say that a \\emph{PR-box correlation} is a set of joint probabilities $P(r,s|k,l)$ which has the property that the outcomes $r$ and $s$ are equal if and only if $k=l=2$. This property is equivalent to the requirement that the four correlations\\eq{corrs} are given by\n\\begin{equation}\n\\label{PRbox}\nC_{11}=C_{12}=C_{21}=-1,\\qquad C_{22}=+1.\n\\end{equation}\nCorrelations of this form could be used e.g. to achieve optimal better-than-quantum performance in two-party XOR games (see e.g.~\\cite{CHTW}). When the joint probabilities $P(r,s|k,l)$ are assumed to be no-signaling, then this requirement actually fixes all values for the probabilities uniquely; however this does not apply here as our temporal scenario allows signaling from Alice to Bob.\n\nStarting from\\eq{corrs}, we now determine when a correlator $C_{kl}$ can have a value of $\\pm 1$,\n\\begin{displaymath}\nC_{kl}=\\tfrac{1}{2}\\langle\\psi|\\{a_k,b_l\\}|\\psi\\rangle\\stackrel{!}{=}\\pm 1,\n\\end{displaymath}\nwhich is equivalent to\n\\begin{displaymath}\n\\langle\\psi|a_kb_l|\\psi\\rangle+\\langle\\psi|b_la_k|\\psi\\rangle=\\pm 2.\n\\end{displaymath}\nBut now since the absolute value of each term is $\\leq\\! 1$, and becomes $1$ if and only if $|\\psi\\rangle$ is an eigenstate of the respective operator, it follows that PR-box behavior requires $|\\psi\\rangle$ to be an eigenstate of the following form:\n\\begin{displaymath}\nb_ka_l|\\psi\\rangle=a_kb_l|\\psi\\rangle=(-1)^{(k-1)(l-1)}|\\psi\\rangle\n\\end{displaymath}\nBut these equations imply\n\\begin{displaymath}\n\\langle\\psi|\\psi\\rangle=\\langle\\psi|a_1b_1b_1a_2|\\psi\\rangle=\\langle\\psi|a_1a_2|\\psi\\rangle=\\langle\\psi|a_1b_2b_2a_2|\\psi\\rangle=-\\langle\\psi|\\psi\\rangle\n\\end{displaymath}\nwhich is impossible for any $|\\psi\\rangle\\neq 0$. Therefore, PR-box behavior is impossible even for the temporal quantum correlations which we consider here. We could also have concluded this from theorem~\\ref{spatiotemporal}.\n\n\\section{Strength of signaling}\n\nIn our Bell-test scenario, the backward no-signaling equations\n\\begin{displaymath}\nP(r,-1|k,1)+P(r,+1|k,1)=P(r,-1|k,2)+P(r,+1|k,2)\\qquad \\forall r,k\n\\end{displaymath}\nare still true: the marginal probability governing Alice's measurement cannot possibly depend on the measurement setting of Bob. However the forward no-signaling equations\n\\begin{equation}\n\\label{nosig}\nP(-1,s|1,l)+P(+1,s|1,l)=P(-1,s|2,l)+P(+1,s|2,l)\\qquad \\forall s,l\n\\end{equation}\nare typically violated, since the choice of measurement for Alice influences the system state after her measurement, and therefore changes the outcome probabilities for Bob. Effectively, what Bob sees is not exactly the initial state $|\\psi\\rangle$, but $|\\psi\\rangle$ after undergoing decoherence due to Alice's measurement. It is an interesting question to ask how much the no-signaling equations\\eq{nosig} can be violated by our quantum-mechanical setup. This is why we want to look at the deviations from\\eq{nosig} and determine how large they can possibly be in a quantum theory. Since each of these four possible quantities involve only one fixed measurement setting $l$ of Bob, we will disregard Bob's choice for the rest of this section, and assume that he simply measures any dichotomic observable $b$. The joint probabilities we then consider are of the form $P(r,s|k)$. Then the two signaling quantities are\n\\begin{equation}\n\\begin{split}\nS_{+}\\equiv& P(+1,+1|1)+P(-1,+1|1)-P(+1,+1|2)-P(-1,+1|2)\\\\\nS_{-}\\equiv& P(+1,-1|1)+P(-1,-1|1)-P(+1,-1|2)-P(-1,-1|2).\n\\end{split}\n\\end{equation}\nDue to the total outcome probability for each choice of measurement being $1$, it necessarily holds that $S_++S_-=0$, independent of whether the system is quantum or not. Therefore the interesting question now is, which values of $S_+$ are achievable by quantum mechanics? This is what we are going to answer here.\n\nA priori, $S_+$ can be expected to attain all the values in the interval $[-1,+1]$. The extreme values of $-1$ and $+1$ correspond to perfect signaling in the sense that Bob can definitely tell which measurement Alice had chosen. This can be interpreted as a classical communication channel with a capacity of $1$ bit.\n\n\\begin{thm}\nA signaling level $S_{+}\\in [-1,+1]$ is quantum-mechanical if and only if $|S_+|\\leq\\frac{1}{2}$.\n\\end{thm}\n\n\\begin{proof}\nBy\\eq{jointprobs}, the signaling quantity $S_+$ can be expressed in terms of the observables and the initial state as\n\\begin{equation}\n\\label{Ss}\nS_{+}=+\\tfrac{1}{4}\\langle\\psi|b|\\psi\\rangle+\\tfrac{1}{8}\\langle\\psi|(a_1ba_1-a_2ba_2)|\\psi\\rangle\\\\\n\\end{equation}\nwhere most terms have in fact dropped out. This equation implies\n\\begin{displaymath}\n|S_+|\\leq\\tfrac{1}{4}|\\langle\\psi|b|\\psi\\rangle|+\\tfrac{1}{8}|\\langle\\psi|a_1ba_1|\\psi\\rangle|+\\tfrac{1}{8}|\\langle\\psi|a_2ba_2|\\psi\\rangle|\n\\end{displaymath}\nEach term within the absolute value brackets in turn can be bounded by $1$, since it is the expectation value of a $\\pm 1$-valued observable, so that the bound $|S_+|\\leq 1\/2$ follows.\n\nConversely, since the set of allowed for $S_+$ needs to be convex, it is sufficient to show that the values $+1\/2$ and $-1\/2$ can be attained. For attaining the value $+1\/2$, we can choose\n\\begin{equation}\n\\label{protocol}\n|\\psi\\rangle=|x+\\rangle,\\qquad a_1=\\sigma_x,\\qquad a_2=\\sigma_y,\\qquad b=\\sigma_x,\n\\end{equation}\nwhere a direct calculation shows that this indeed has the required property.\n\\end{proof}\n\nAs was already mentioned briefly, we may also consider the signaling strength in terms of the information which Bob's measurement outcome contains about Alice's choice of setting. This is encoded in the two probabilities\n\\begin{align}\n\\begin{split}\nP(s|k)&\\equiv P(-1,s|k)+P(+1,s|k)\\\\\n&=\\tfrac{1}{2}+\\tfrac{1}{4}s\\langle\\psi|b|\\psi\\rangle+\\tfrac{1}{4}s\\langle\\psi|a_kba_k|\\psi\\rangle\n\\end{split}\n\\end{align}\nwhich define a classical communication channel on the input alphabet $k\\in\\{1,2\\}$ to the output alphabet $s\\in\\{-1,+1\\}$. Since Bob's outcome is only dichotomic, we can equivalently consider the expectation value of his measurement,\n\\begin{displaymath}\nE(s|k)\\equiv P(+1|k)-P(-1|k),\\qquad k\\in\\{1,2\\}\n\\end{displaymath}\nand the question then is, which pairs $(E(s|1),E(s|2))$ can occur quantum-mechanically, and how does this bound the classical capacity by which Alice can use her measurements in order to send information to Bob? The answer to this question is given in the following theorem:\n\n\\begin{thm}\nA pair $(E(s|1),E(s|2))$ can occur quantum-mechanically if and only if\n\\begin{displaymath}\n|E(s|1)+E(s|2)|\\leq 1.\n\\end{displaymath}\nThe maximal communication capacity is $\\log_2\\left(5\/4\\right)\\approx 0.32\\:\\mathrm{bits}$, which can be achieved using the qubit protocol\\eq{protocol}.\n\\end{thm}\n\nThis result is illustrated in figure~\\ref{figEs}.\n\n\\begin{figure}\n\\psset{unit=130pt}\n\\centering{\\begin{pspicture}(-.2,-.2)(1.2,1.2)\n\\psaxes{->}(.5,.5)(-.2,-.2)(1.2,1.2)\n\\psframe[linestyle=dashed](0,0)(1,1)\n\\rput(1.25,.42){$E(s|1)$}\n\\rput(.38,1.2){$E(s|2)$}\n\\psset{fillstyle=crosshatch,fillcolor=lightgray,hatchcolor=gray,hatchangle=20}\n\\pscustom{\n\t\\psline(0,.5)(.5,0)\n\t\\psline(.5,0)(1,0)\n\t\\psline(1,0)(1,.5)\n\t\\psline(1,.5)(.5,1)\n\t\\psline(.5,1)(0,1)\n\t\\psline(0,1)(0,.5)}\n\\psline{->}(1.2,.65)(1,.5)\n\\rput(1.25,.7){(\\ref{protocol})}\n\\end{pspicture}}\n\\caption{Possible pairs of $E(s|1)$ and $E(s|2)$ as they can appear in quantum theory. The whole square-shaped box is the whole region of principally allowed values $-1\\leq E(s|1),E(s|2)\\leq 1$.}\n\\label{figEs}\n\\end{figure}\n\n\\begin{proof}\nSince $E(s|1)+E(s|2)=2S_+$, the constraint $|E(s|1)+E(s|2)|\\leq 1$ immediately follows. On the other hand, the qubit protocol\\eq{protocol} achieves $E(s|1)=1$, $E(s|2)=0$, which is one of the four non-trivial vertices of the convex quadrangle shown if figure~\\ref{figEs}. The other three vertices can be attained by the same protocol after possibly switching $s\\leftrightarrow -s$ and $a_1\\leftrightarrow a_2$. Now since the quantum region has to be convex, and the quadrangle is the smallest convex set containing its vertices, it follows that $|E(s|1)+E(s|2)|\\leq 1$ is also sufficient for the existence of a quantum-mechanical model.\n\nNow we get to the capacity statement. Since classical communication capacity is a convex function of the transition probabilities, we know that the maximal capacity is attained at the quadrangle's vertices. Since the four vertices are all simple permutations of the protocol\\eq{protocol}, the corresponding channels have equal capacity, and it is sufficient to calculate the capacity achievable by the data\\eq{protocol}. A direct calculation shows that the optimal input distribution is a relative frequency of $3\/5$ for $a_1$ and $2\/5$ for $a_2$, resulting in a mutual information of $\\log_2(5\/4)\\approx 0.32$ bits.\n\\end{proof}\n\n\n\\section{A temporal version of Hardy's nonlocality paradox}\n\\label{hardysection}\n\nHardy's paradox~\\cite{Mer} occurs when the joint probabilities have the following properties:\n\\begin{equation}\n\\label{hardyparadox}\n\\begin{split}\nP(+1,+1|1,1)=0\\\\\nP(-1,+1|1,2)=0\\\\\nP(+1,-1|2,1)=0\\\\\nP(+1,+1|2,2)>0\n\\end{split}\n\\end{equation}\nThis is impossible in any realistic theory where Alice's measurements are non-invasive. We note that the only relevant information contained in hidden variables lies in the preexisting values of all relevant observables. Hence any (stochastic) hidden variable model is given by a statistical mixture of the 16 realistic states\n\\begin{equation}\n\\label{realistic}\na_1^\\pm a_2^\\pm b_1^\\pm b_2^\\pm\n\\end{equation}\nwhere in this notation (from~\\cite{Lap}), each sign stands for the corresponding measurement outcome it determines with certainty, and the four signs are independent of each other. By the assumption $P(+1,+1|2,2)>0$, we know that this statistical mixture contains at least one state of the form\n\\begin{displaymath}\na_1^\\pm a_2^+ b_1^\\pm b_2^+.\n\\end{displaymath}\nBut now due to $P(-1,+1|1,2)=0$, this cannot be one of the two states $a_1^- a_2^+ b_1^\\pm b_2^+$. Likewise by $P(+1,-1|2,1)=0$, it cannot be one of the two states $a_1^\\pm a_2^+ b_1^- b_2^+$. Therefore, the statistical mixture of realistic states necessarily contains the state\n\\begin{displaymath}\na_1^+ a_2^+ b_1^+ b_2^+\n\\end{displaymath}\nbut now this contradicts the assumption $P(+1,+1|1,1)=0$\\:! Therefore, the existence of joint probabilities with the property\\eq{hardyparadox} exhibits a rather strong form of contextuality. Note that this kind of reasoning applies to a spatial as well as to a temporal Bell test scenario.\n\nIn fact, \\eq{hardyparadox} is indeed realizable in quantum theory, and it is known that the maximal value for $P(+1,+1|2,2)$ in a spatial scenario is approximately $0.09$~\\cite{Mer}. Here we would like to determine the maximal possible value of $P(+1,+1|2,2)$ in the temporal CHSH scenario. Again, since joint probabilities for the temporal case comprise those of the spatial case, the maximal temporally realizable value of $P(+1,+1|2,2)$ has to be at least $0.09$. We will now proceed to show that one can achieve a substantially higher value than this. This shows again that temporal quantum correlations are often stronger than spatial quantum correlations.\n\n\\begin{thm}\nThe maximal value for $P(+1,+1|2,2)$ in the temporal Hardy paradox is $1\/4$.\n\\end{thm}\n\n\\begin{proof}\nIn order for a probability $P(r,s|k,l)$ like\\eq{jointprobs} to vanish, one needs that\n\\begin{itemize}\n\\item either Alice's outcome $r$ by itself is already impossible to occur, i.e. the other outcome $-r$ occurs with certainty. This means that the initial state is a $-r$-eigenstate of $a_k$, $a_k|\\psi\\rangle=-r|\\psi\\rangle$.\n\\item or Alice's post-measurement state $\\frac{\\mathbbm{1}+ra_k}{2}|\\psi\\rangle$ (unnormalized) is such that Bob's outcome is impossible, i.e. it is a $-s$ eigenstate of $b_l$,\n\\begin{displaymath}\nb_l(\\mathbbm{1}+ra_k)|\\psi\\rangle=-s(\\mathbbm{1}+ra_k)|\\psi\\rangle\n\\end{displaymath}\n\\end{itemize}\nHence, vanishing joint probability is equivalent to\n\\begin{equation}\n\\label{jointvanish}\n(\\mathbbm{1}+sb_l)(\\mathbbm{1}+ra_k)|\\psi\\rangle=0\n\\end{equation}\nwhich can be interpreted as a vanishing amplitude for the two outcomes to occur together. So the vanishing constraints from\\eq{hardyparadox} are equivalent to\n\\begin{align}\n&(\\mathbbm{1}+b_1)(\\mathbbm{1}+a_1)|\\psi\\rangle=0\\label{hardyb1a1},\\\\\n&(\\mathbbm{1}+b_2)(\\mathbbm{1}-a_1)|\\psi\\rangle=0\\label{hardyb2a1},\\\\\n&(\\mathbbm{1}-b_1)(\\mathbbm{1}+a_2)|\\psi\\rangle=0\\label{hardyb1a2}.\n\\end{align}\nThe qubit protocol\n\\begin{displaymath}\n\\label{hardymax}\n|\\psi\\rangle=|x+\\rangle,\\qquad a_1=-\\sigma_x,\\qquad a_2=\\sigma_y,\\qquad b_1=\\sigma_y,\\qquad b_2=-\\sigma_x\n\\end{displaymath}\ndoes indeed satisfy all of these constraints, and it achieves a value of $P(+1,+1|2,2)=1\/4$ as promised. The remaining part of the proof is dedicated to showing that this value is optimal.\n\nFor $b_1$, the equations\\eq{hardyb1a1} to\\eq{hardyb1a2} mean that $(\\mathbbm{1}+a_1)|\\psi\\rangle$ has to lie in the $-1$-eigenspace of $b_1$ (since this vector has zero projection onto the $+1$-eigenspace), and similarly that $(\\mathbbm{1}+a_2)|\\psi\\rangle$ has to lie in the $+1$-eigenspace of $b_1$. These eigenspaces are necessarily orthogonal since $b_1$ is hermitian. Hence, given any initial state $|\\psi\\rangle$ together with $\\pm 1$-valued observables $a_1,a_2,b_2$ which satisfy\\eq{hardyb2a1}, we can find an observable $b_1$ which also satisfies\\eq{hardyb1a1} and\\eq{hardyb1a2} if and only if\n\\begin{equation}\n\\label{hardya1a2}\n(\\mathbbm{1}+a_1)|\\psi\\rangle\\perp(\\mathbbm{1}+a_2)|\\psi\\rangle,\\qquad\\textrm{i.e.}\\qquad\\langle\\psi|(\\mathbbm{1}+a_1)(\\mathbbm{1}+a_2)|\\psi\\rangle=0,\n\\end{equation}\nholds. So this condition is equivalent to\\eq{hardyb1a1} and\\eq{hardyb1a2} together and also comprises the case that $|\\psi\\rangle$ is any eigenstate of $a_1$ or $a_2$.\n\nNow imagine that we have $|\\psi\\rangle$, $a_1$ and $a_2$ such that\\eq{hardya1a2} is satisfied. Then what can we choose for $b_2$ in order to also satisfy\\eq{hardyb2a1}? Equation\\eq{hardyb2a1} means exactly that $(\\mathbbm{1}-a_1)|\\psi\\rangle$ is contained in the $-1$-eigenspace of $b_2$. When $p$ stands for the projection operator onto the vector $(\\mathbbm{1}-a_1)|\\psi\\rangle$, this means exactly that\n\\begin{displaymath}\np\\leq\\frac{\\mathbbm{1}-b_2}{2}.\n\\end{displaymath}\n(Here, the partial order ``$\\leq$'' is the usual partial order on the set of hermitian operators\\footnote{recall that $x\\leq y$ in this order is defined to mean that $y-x$ is positive semi-definite.}.) On the other hand, we have\n\\begin{displaymath}\n(\\mathbbm{1}-a_1)|\\psi\\rangle\\langle\\psi|(\\mathbbm{1}-a_1)\\leq p\n\\end{displaymath}\nsince the norm of $(\\mathbbm{1}-a_1)|\\psi\\rangle$ is at most $1$. Hence, we can conclude from these two inequalities that\n\\begin{displaymath}\n\\mathbbm{1}+b_2\\leq \\mathbbm{2}-2p\\leq \\mathbbm{2}-2(\\mathbbm{1}-a_1)|\\psi\\rangle\\langle\\psi|(\\mathbbm{1}-a_1).\n\\end{displaymath}\nWhen plugging this result into the expression for the ``paradoxical'' probability $P(+1,+1|2,2)$, we obtain\n\\begin{align*}\nP(+1,+1|2,2)&=\\tfrac{1}{8}\\langle\\psi|(\\mathbbm{1}+a_2)(\\mathbbm{1}+b_2)(\\mathbbm{1}+a_2)|\\psi\\rangle\\\\\n&\\leq\\tfrac{1}{2}\\left\\langle\\psi\\left|\\left(\\mathbbm{1}+a_2\\right)\\right|\\psi\\right\\rangle-\\tfrac{1}{4}\\langle\\psi|(\\mathbbm{1}+a_2)(\\mathbbm{1}-a_1)|\\psi\\rangle\\langle\\psi|(\\mathbbm{1}-a_1)(\\mathbbm{1}+a_2)|\\psi\\rangle,\n\\end{align*}\nwhere it has been used that $(\\mathbbm{1}+a_2)^2=2(\\mathbbm{1}+a_2)$. But now\\eq{hardya1a2} can be applied to evaluate the second term by using\n\\begin{align*}\n\\langle\\psi|(\\mathbbm{1}+a_2)(\\mathbbm{1}-a_1)|\\psi\\rangle=& 2\\langle\\psi|(\\mathbbm{1}+a_2)|\\psi\\rangle-\\langle\\psi|(\\mathbbm{1}+a_2)(\\mathbbm{1}+a_1)|\\psi\\rangle\\\\\n\\stackrel{(\\ref{hardya1a2})}{=}& 2\\langle\\psi|(\\mathbbm{1}+a_2)|\\psi\\rangle.\n\\end{align*}\nHence we finally end up with\n\\begin{displaymath}\nP(+1,+1|2,2)\\leq\\tfrac{1}{2}\\langle\\psi|(\\mathbbm{1}+a_2)|\\psi\\rangle-\\tfrac{1}{4}\\langle\\psi|(\\mathbbm{1}+a_2)|\\psi\\rangle^2\n\\end{displaymath}\nwhich is of the form $x-x^2$ for $x=\\frac{1}{2}\\langle\\psi|(\\mathbbm{1}+a_2)|\\psi\\rangle$. The maximal value of this function is $\\frac{1}{4}$, hence the claim is proven.\n\\end{proof}\n\n\\paragraph*{Acknowledgements.} This work was initiated in joint discussions with Sibasish Ghosh and Tomasz Paterek, who therefore have contributed substantially to the present results. It all happened thanks to an invitation by Andreas Winter for the author to visit the Centre for Quantum Technologies in Singapore. Ramon Lapiedra has kindly provided useful comments on an earlier version of this manuscript and endured a long discussion over his work~\\cite{Lap}. Finally, this work would not have been possible without the excellent research conditions within the IMPRS graduate program at the Max Planck Institute.\n\n\\bibliographystyle{halpha.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\t\nObservations of black hole exteriors constitute one of the most successful confirmation of general-relativity predictions~\\cite{Abbott:2016blz}. On the other hand, black hole interiors are still puzzling and lack a consistent theory capable of describing them due to the presence of a singularity where Einstein's theory is expected to break down~\\cite{Penrose:1964wq}. \nAccording to classical general relativity, this would not be a problem for outside observers as a black hole is endowed with a horizon that hides the singularity. However, at the semiclassical level -- i.e. with matter fields quantized in a classical background -- black holes turn out to be less dark than their classical counterpart, indeed they evaporate by emitting radiation whose spectrum is approximately thermal~\\cite{Hawking:1974sw}. \nThis feature has puzzled physicists for a long time by raising ``apparent'' contradictions. The most famous is the ``information loss paradox'' according to which an initial pure state describing collapsing matter would evolve into a final mixed state after black hole evaporation, thus giving rise to an inconsistency between general relativity and quantum mechanics as \\textit{no} unitary operator exists that can evolve pure states into mixed states~\\cite{Hawking:1976ra}; see also Ref.~\\cite{Mathur:2009hf} for a pedagogical review.\n\nIt should be emphasized that this problem appears because one tries to extend the validity of the semiclassical approach up to the Planck scale, and extrapolate physical predictions in regimes where quantum-gravity effects should not be negligible. In this respect, there is \\textit{not} really a paradox but it could be just our ignorance about Planck-scale physics that prevents us from concluding anything definite about what happens to the information during black hole evaporation~\\cite{Unruh:2017uaw}.\n \n \nA different version of the paradox involving time scales shorter than black hole life time was formulated. Indeed, if one makes the ``apparently'' natural assumption that as seen from the outside a black hole behaves like a quantum system whose number of degrees of freedom is given by the horizon area, then one can show that after some time the von Neumann entropy of Hawking radiation would exceed the thermodynamic entropy of black hole, which is impossible. This gives rise to a contradiction even before reaching the end point of the evaporation when the semiclassical approach should still be valid. Thus, it was argued that the von Neumann entropy of the Hawking radiation should start decreasing at roughly half of the black hole life time, and follow the so-called Page curve consistently with the unitary quantum evolution~\\cite{Page:1993df,Page:1993wv,Page:1993up}. \n\nIn the past decades both the aforementioned starting assumption and the corresponding alternative version of the information loss problem have been taken very seriously by a large part of the theoretical physics community. To emphasize the relevance of the hypothesis on the black hole degrees of freedom as seen from an outside observer, some authors even referred to it with the expression ``central dogma''~\\cite{Almheiri:2020cfm}. Indeed, lots of efforts both to support the central dogma and to find a resolution to the induced entropy problem have been made. \nIn particular, recently a novel computation~\\cite{Penington:2019npb,Penington:2019kki,Almheiri:2019psf,Almheiri:2019hni,Almheiri:2019qdq,Marolf:2020rpm,Bousso:2021sji} of the von Neumann entropy of the Hawking radiation by using the Ryu-Takayanagi entropy formula~\\cite{Ryu:2006bv} was shown to be consistent with a unitarity quantum evolution, and was claimed to be a strong motivation in support of the central dogma.\n \nIn this paper we adopt an open-minded and critical point of view in discussing whether the current formulation of the information loss paradox is well-posed. We aim at carefully inspecting all the assumptions that are usually made and that are claimed to lead to the information loss problem(s). In particular, in Sec.~\\ref{Sec:info paradox} we review various formulations of the problem by emphasizing some of the details that are often overlooked, and that are necessary for the subsequent analysis. In Sec.~\\ref{Sec:arguments} we examine the standard arguments in favour of the central dogma. In Sec.~\\ref{Sec:counterex}, we present a thought experiment which unveils an incompatibility between the validity of semiclassical gravity to describe infalling matter far from the Planckian regime and the central dogma. This allows us to argue that as long as the low-energy effective field theory holds true, i.e. far from the Planck scale, such that the horizon can be described as a smooth surface, then there exists \\textit{no} information loss paradox. Moreover, in Sec.~\\ref{Sec:micr} we comment on the microscopic interpretation of the Bekenstein-Hawking entropy; while in Sec.~\\ref{Sec:Comp-fir} we discuss the relevance of our conclusions in comparison to the black hole complementarity and the firewall paradigm. In Sec.~\\ref{Sec:discussions}, we summarize our results and discuss the power of our conclusions. \n \nThroughout the work we adopt the units $c=\\hbar=k_{\\rm B}=1.$ We will use different symbols for several entropies: $S_{\\rm BH}$ for the Bekenstein-Hawking entropy; $S_{\\rm bh}$ for the total black hole entropy that does not necessarily coincide with $S_{\\rm BH}$; $S_{\\rm th}$ for the thermodynamic entropy; $S_{\\rm vN}$ for the von Neumann entropy; $S_{\\rm rad}$ for the von Neumann entropy of the Hawking radiation; $S_{\\rm m}$ for the von Neumann entropy of additional matter.\n \n \n\n\t\\section{Is there an information loss paradox?}\\label{Sec:info paradox}\n\t\n\tIt is often stated that the thermal nature of black holes eventually leads to an incompatibility between semiclassical general relativity and one of the pillar of quantum mechanics, i.e. the unitarity of the $S$-matrix, thus leading to the well-known information loss paradox~\\cite{Hawking:1976ra}. However, we strongly believe that the assumptions on which \n\tthe paradoxical implications rely are often overlooked, and deserve a deeper and thorough inspection.\n\tTherefore, it is crucial to reformulate the problem by highlighting and critically revising all the basic assumptions that are usually made. In what follows we carefully analyze three different scenarios that are normally considered.\n\t\n\t\n\t\\subsection{Case A: unitarity problem}\n\t\n\tThe simplest formulation of the information loss problem can be stated as the incompatibility between the two following assumptions.\n\t%\n\t\\begin{enumerate}\n\t\t\\item[A1.] Quantum states evolve in a unitary way. In particular, pure states evolve into pure states;\n\t\t\n\t\t\\item[A2.] Semiclassical general relativity is a valid low-energy effective field theory to describe black hole physics during the entire evaporation process:\n\t\tblack holes evaporate completely emitting thermal radiation and end up leaving a regular spacetime (see Fig.~\\ref{fig1a}). \n\t\n \\end{enumerate}\n\t%\n\tThe hypothesis A1 implies that there exists a unitarity $S$-matrix operator that describes the evolution from ingoing collapsing matter to outgoing radiation during the entire black hole formation and evaporation. Whereas A2 means that the semiclassical approach, according to which fields are quantized in a classical curved background, is always valid even at the end point of the evaporation.\n\t\n\tThese two assumptions are clearly incompatible. If the emitted radiation is in a pure thermal state (as suggested by Hawking's calculation~\\cite{Hawking:1974sw}), then after the black hole evaporation the final state will be thermal and mixed. This scenario is illustrated in Fig.~\\ref{fig1a}, in which it is clear that before reaching the end point of the evaporation we can draw Cauchy surfaces, e.g. $\\Sigma_1$ and $\\Sigma_2,$ on which the state of the joint system (black hole plus radiation) is pure; whereas after the evaporation the state on each Cauchy slice turns out to be mixed, e.g. on $\\Sigma_3.$\n\tTherefore, if the matter that formed the black hole was initially in a pure state, the previous argument would imply a breakdown of the unitary condition on the evolution operator in quantum mechanics as pure states cannot evolve unitarily into mixed states, thus contradicting A1.\n\n\tObviously, there is no reason to trust the semiclassical picture up to the end of the black hole evaporation, indeed it is quite reasonable to believe that at least in the latest stages quantum-gravity effects cannot be neglected and should be consistently taken into account. In fact, it is \\textit{not} clear what the final state would be. For instance, it could be given by a naked timelike singularity~\\cite{Hawking:1974sw} as depicted in Fig.~\\ref{fig1b}. \n\tIf this is the case, we would not be able to conclude that the final state is mixed. Indeed, $\\Sigma_3$ in Fig.~\\ref{fig1b} is not a Cauchy hypersurface, meaning that we would need a description of the singularity in order to predict the final quantum state.\n\t\n\t\n\tFurthermore, let us note that both the regular spacetime and the naked singularity spacetime represent only some arbitrary extrapolation of semiclassical gravity beyond its regime of validity. In fact, these spacetimes are not globally hyperbolic and contain a Cauchy horizon $\\mathcal{C}$. The shaded orange regions in Figs.~\\ref{fig1a} and~\\ref{fig1b} are in the causal future of the singularity, and it is not possible to evolve the initial value problem from the hypersurface $\\Sigma_2$ to $\\Sigma_3$ without providing a description of the singularity which is outside the realm of semiclassical gravity. Semiclassical gravity can only study the maximally Cauchy development \\cite{Wald:1984rg} depicted in Fig.~\\ref{fig1c}. In this portion of spacetime the initial value problem is well defined. However, the spacetime is not geodesically complete, therefore in principle we should not expect a pure state on $\\Sigma_2$ to evolve into a pure state on $\\Sigma_3.$ Although in a different language, similar considerations were presented in \\cite{Unruh:2017uaw}.\n\t\n\n\tHence, this weak formulation of the information loss problem would not be particularly worrisome. On the other hand, it is well known that one can formulate a stronger version of the paradox according to which problems seem to arise even when the black hole mass is much larger than Planck mass.\n\n\n\n\t\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Subfigure 1 list of figures text][]{\n\t\t\\includegraphics[scale=0.42]{penrose-evap-regular}\\label{fig1a}}\\qquad\n\t\\subfloat[Subfigure 2 list of figures text][]{\n\t\t\\includegraphics[scale=0.42]{penrose-evap-singu}\\label{fig1b}}\\qquad\n\t\t\\subfloat[Subfigure 3 list of figures text][]{\n\t\t\\includegraphics[scale=0.42]{penrose-evap-cauchy.png}\\label{fig1c}}\n\t\\protect\\caption{Penrose diagrams of an evaporating black hole with (a) regular and (b) singular spacetime (naked timelike singularity) as the final state. In both cases, the spacetime has a Cauchy horizon labeled by $\\mathcal{C}$. If the final state is singular or we only consider the maximal Cauchy development (c), a unitary operator does \\textbf{not} necessarily evolve a pure state on $\\Sigma_2$ into a pure state on $\\Sigma_3$. The blue dotted line represents the ($\\text{inner}\\,\\cup\\, \\text{outer}$) trapping horizon.}\n\\end{figure} \n\t\n\n\n\n\t\\subsection{Case B: entropy problem}\\label{CaseB}\n\t\n\t\n\tWe now present a stronger formulation of the information problem originally due to Page~\\cite{Page:1993df,Page:1993wv,Page:1993up}, that it is sometimes referred to as the ``entropy problem''; see also Ref.~\\cite{Almheiri:2020cfm}. It consists in the incompatibility among the following assumptions.\n\t%\n\t\\begin{enumerate}\n\t\t\n\t\t\\item[B1.] Quantum states evolve in a unitary way. In particular, pure states evolve into pure states;\n\t\t\n\t\t\\item[B2.] Semiclassical general relativity is a valid low-energy effective field theory to describe black hole physics far from the Planckian regime;\n\t\t\n\t\t\\item[B3.] As seen from the outside, a black hole behaves like a quantum system whose number of degrees of freedom is given by $A\/4G,$ with $A$ being the apparent-horizon area.\n\t\n\t\t\n\t\\end{enumerate}\n\t%\n\tIt is clear that in this formulation the first assumption is left unchanged ($\\text{A1}=\\text{B1}$), whereas the second one has been significantly weakened ($\\text{B2}\\subset \\text{A2}$). Indeed, B2 means that the semiclassical approach can be trusted only for black holes whose mass is sufficiently larger than Planck mass, and the horizon is still assumed to be a smooth surface, i.e. an infalling observer would experience nothing special when crossing the horizon, consistently with the equivalence principle.\n\t\n Furthermore, it is important to note that, if the mass of the black hole is sufficiently larger than Planck mass, it is possible to foliate the spacetime in a way that each Cauchy hypersurface intersects all the infalling matter and outgoing Hawking radiation, but avoids regions with high curvature. Also, the extrinsic curvature of the hypersurfaces remains small. This foliation is usually referred to as ``nice slicing''. As a consequence of the nice slicing, if B2 holds, the semiclassical description will be valid also for the matter and Hawking particles falling into the black hole.\n\t\n\tThe price to pay in this formulation of the information problem is the introduction of the third hypothesis B3 that is sometime referred to as ``central dogma''~\\cite{Almheiri:2020cfm}, according to which the black hole entropy\\footnote{Here, we consider the black hole entropy as the number of physical degrees of freedom of a black hole as seen from the outside.} $S_{\\rm bh}$ simply coincides with the Bekenstein-Hawking entropy $S_{\\rm BH}=A\/4G,$ i.e. $S_{\\rm bh}=S_{\\rm BH}$~\\cite{Bekenstein:1972tm,Bekenstein:1973ur}; in other words the dimension of the black hole Hilbert space is assumed to be $e^{S_{\\rm BH}}.$ We will discuss the motivations and plausibility of the central dogma in the next section. Before that, let us quickly repeat the standard argument~\\cite{Page:1993wv,Page:1993up,Almheiri:2020cfm} according to which requiring the simultaneous validity of assumptions B1, B2, B3 leads to a paradoxical conclusion.\n\n\n\nLet $\\mathcal{H}_{\\rm bh}$ and $\\mathcal{H}_{\\rm rad}$ be the Hilbert spaces of the black hole and of the radiation, respectively. B1 implies that, if the initial state before black hole formation was pure, then the joint state of black hole plus radiation, i.e. $\\left|\\psi \\right\\rangle \\in \\mathcal{H}_{\\rm bh}\\otimes \\mathcal{H}_{\\rm rad},$ must also be pure. By tracing over the black hole degrees of freedom, we can obtain the density matrix for the radiation subsystem and its von Neumann entropy $S_{\\rm rad}$ (which coincides with the von Neumann entropy of the black hole subsystem). From B3 it follows that during the evaporation process the thermodynamic black hole entropy $S_{\\rm bh}$ decreases as the horizon area decreases. Whereas the entropy $S_{\\rm rad}$ of the outgoing radiation tends to increase according to Hawking semiclassical computation which can be trusted as long as B2 is valid. This means that there exists a time scale\\footnote{Note that different notions of time are possible. We will use the time function orthogonal to the Cauchy foliation $\\Sigma_t$ (each Cauchy hypersurface is a constant-time hypersurface).} $t_{\\rm Page}$ -- known as Page time -- after which $S_{\\rm rad}>A\/4G=S_{\\rm BH}=S_{\\rm bh};$ see Fig.~\\ref{fig3} for an illustration. \nThe last inequality states that the von Neumann entropy of the Hawking radiation (or, equivalently, of the black hole subsystem) becomes larger than the number of available degrees of freedom in the black hole, so that for times $t>t_{\\rm Page}$ \nthe joint state of the black hole coupled to the radiation becomes mixed, thus contradicting the hypothesis B1 and giving rise to the paradox.\n\n\nTo avoid this problem and preserve unitarity, \nit was claimed that the entropy of the Hawking radiation should start decreasing at relatively early times when $t\\sim t_{\\rm Page}$ and vanish at the end of the evaporation~\\cite{Page:1993wv,Page:1993up,Almheiri:2020cfm}. It is also sometimes claimed~\\cite{Page:1993wv,Dvali:2015aja} that this might happen thanks to non-negligible non-thermal corrections arising for times larger than $t_{\\rm Page}.$ In this way the von Neumann entropy of the radiation would purify consistently with unitarity and follow the so-called Page curve; see Fig.~\\ref{fig3}.\nA price to pay in this scenario is the need for new physics far from the Planckian regime, i.e. abandoning of B2. \n\n\n\\begin{figure}[t!]\n\t\\includegraphics[scale=0.55]{page-curve}\n\t\\centering\n\t\\protect\\caption{This figure shows the behaviors of Bekenstein-Hawking entropy $A\/4G$ (red dashed line), and of the von Neumann entropy of the Hawking radiation $S_{\\rm rad}$ (yellow dashed line) as functions of the time $t$ defining the foliation of Cauchy hypersurfaces. If the thermodynamic entropy of black hole $S_{\\rm bh}$ is equal to $A\/4G$ (i.e. if B3 holds), then for times larger than the Page time $t_{\\rm Page}$ the entropy of radiation exceeds the thermodynamic one and leads to the information loss problem. In this scenario, the issue would be solved if and only if the entropy of radiation would follow the so-called Page curve (blue solid line).}\n\t\\label{fig3}\n\\end{figure}\n\t\n\\subsection{Case C: No paradox}\\label{case-C}\n\nIn the Case B we noticed that the assumption B3 plays a crucial role in giving rise to paradoxical conclusions even when quantum-gravity effects are still negligible, thus implying either the lack of a unitary evolution or the need for new physics already at early times when the effective-field-theory description should still be reliable. \n\nWe now discuss a third scenario that is usually less appreciated, in which the assumption B3 is dropped and the only hypothesises are the following:\n\n\\begin{enumerate}\n\t\n\t\\item[C1.] Quantum states evolve in a unitary way. In particular, pure states evolves into pure states;\n\t\n\t\\item[C2.] Semiclassical general relativity is a valid low-energy effective field theory to describe black hole physics far from the Planckian regime.\n\t\n\\end{enumerate}\t\nThese two assumptions do \\textbf{not} necessarily lead to any information loss paradox! Since the thermodynamic entropy of the black hole is not constrained to be given by the horizon area, the outgoing Hawking quanta can always be entangled with the ingoing ones. In other words, the (negative energy) ingoing flux of the radiation contributes to the increase of the thermodynamic black hole entropy, so that although the entropy of the radiation can become larger than the Bekenstein-Hawking entropy, i.e. $S_{\\rm rad}>S_{\\rm BH},$ it will never exceed the total black hole entropy $S_{\\rm bh}$, and {\\it no} entropy paradox arises.\n\n\nIt is worth mentioning that in analog gravity models for black holes~\\cite{Barcelo2005} correlations between ingoing and outgoing quanta can be experimentally measured~\\cite{Steinhauer:2015saa,Weinfurtner:2019zyc}.\nFurthermore, results towards the resolution of the information problem in this context seem to suggest that the outgoing Hawking radiation is entangled even after the disappearance of the analog black hole~\\cite{Liberati:2019}. The extrapolation of these partial results to gravitational black holes seems to agree with this third scenario for which B3 does not hold. While it is true that the analogy is far from perfect, it is also true that the lessons drawn in this context should not be completely ignored as analog systems represent the closest we can get to the experimental detection of Hawking radiation.\n\n\nGiven the validity of the assumptions C1 and C2, we can surely state that the whole dynamics remains unitary as long as time scales shorter than the black hole life time are considered. Instead, when approaching the end point of the evaporation process quantum-gravity effects must be taken consistently into account, and to do so a full theory of quantum gravity is necessary. \n\nTherefore, it is important to stress that the validity of semiclassical general relativity before the black hole reaches Planckian size is \\textbf{not} necessarily incompatible with the unitarity of quantum mechanics. It is important to stress that the information may or may not be lost in the final stages of the evaporation depending on the physics describing that regime, but this would depend on the full theory of quantum gravity and goes beyond the scope of semiclassical gravity. While this should definitely be a well-known fact, the information loss paradox is often stated as the incompatibility between C1 and C2, whereas the hypothesis B3 is usually implicitly assumed and taken for granted. \n\nIn fact, it should be emphasized that the information loss problem related to the growing of the von Neumann entropy is just a consequence of imposing B3 as a hypothesis. In other words, the imposition of the central dogma implies the emergence of a paradox which, otherwise, would not arise.\nWith this in mind, in the next section we carefully and critically inspect the motivations that are usually given in support of the central dogma.\n\n\n\n\n\n\n\\section{Standard arguments for the central dogma}\n\\label{Sec:arguments}\n\nIn the previous section we have shown that the requirement of the central dogma in the Case B, i.e. the imposition of the assumption B3, directly leads to a paradoxical conclusion. Therefore, we believe it is of crucial importance to better understand the motivations supporting this hypothesis. \n\n\\begin{enumerate}\n\n\n\\item[(i)] To our understanding, the main motivation comes from the fact that black holes obey standard thermodynamic laws~\\cite{Bardeen:1973gs} with an entropy given by the Bekenstein-Hawking formula~\\cite{Bekenstein:1972tm,Bekenstein:1973ur}\n\\begin{equation}\nS_{\\rm bh}=S_{\\rm BH}=\\frac{A}{4G}.\n\\end{equation} \nFrom this observation, and from the fact that for a generic system the thermodynamic entropy $S_{\\rm th}$ imposes an upper bound on the von Neumann entropy $S_{\\rm vN},$ i.e.\n\\begin{equation}\\label{eq:S-comp}\n S_{\\rm th}\\geq S_{\\rm vN},\n\\end{equation}\n\none would expect that the apparent-horizon area $A=16\\pi G^2M^2$ has an intrinsic statistical nature and the total number of internal states might be bounded by $e^{S_{\\rm BH}}.$\n\n\\item[(ii)] This picture is reinforced by the imposition of the Bekenstein bound~\\cite{Bekenstein:1980jp}, which states that the entropy of any quantum system localized in a region of circumferential radius $R$ and of total energy $E$ is bounded as \n\\begin{equation}\n S\\leq 2\\pi ER\\,.\\label{bek-bound}\n\\end{equation}\nIn the case of a Schwarzschild black hole we have $R=2GM$ and $E=M,$ so that the previous inequality reads\n\\begin{equation}\n S_{\\rm bh}\\leq \\frac{A}{4G}=S_{\\rm BH}\\,.\\label{bek-bound-BH}\n\\end{equation}\nTherefore, if we assume the Bekenstein bound to be valid, then the Bekenstein-Hawking entropy is the maximal possible entropy for a black hole as seen from the outside. \n\nThe Bekenstein bound seems to be not applicable when self-gravity effects are not negligible, e.g. for instance during a gravitational collapse and for sufficiently large cosmological regions. For this reason, Bousso proposed the so-called covariant entropy bound~\\cite{Bousso:1999xy} whose range of applicability is wider than Bekenstein bound and reduces to the latter when self-gravity effects are negligible. For a Schwarzschild black hole the Bousso bound coincides with the one in Eq.~\\eqref{bek-bound-BH}.\n\n\\item[(iii)] The previous bounds on the entropy seem to suggest an intrinsic connection between geometry and information. This has brought many people to believe in the existence of a holographic principle~\\cite{tHooft:1993dmi,Susskind:1994vu} according to which the area of any hypersurface poses a limit on the information that can be stored within the adjacent spacetime regions. It is often claimed that holography should be a property of a consistent theory of quantum gravity~\\cite{Bousso:2002ju}, and it can be considered the main modern motivation in support of the central dogma.\n\n\n\\item[(iv)] A more recent argument related to the previous ones comes from a computation of Hawking radiation that involves the use of the Ryu-Takayanagi entropy formula~\\cite{Ryu:2006bv,Lewkowycz:2013nqa} in the context of holography and AdS\/CFT correspondence~\\cite{Maldacena:1997re}. Through such a computation one can reproduce a behavior for the von Neumann entropy of the radiation that is compatible with the Page curve. This result was interpreted as a strong indication towards the validity of the central dogma~\\cite{Almheiri:2020cfm}. \nIt must be also emphasized that this result has been rigorously obtained in simplified settings like 2D black hole spacetimes~\\cite{Penington:2019npb,Penington:2019kki,Almheiri:2019psf,Almheiri:2019hni,Almheiri:2019qdq}.\n\n\\end{enumerate}\n\n\nIn the next section we will present a thought experiment illustrating a simple physical configuration in which the central dogma does \\textit{not} hold; in particular, we will comment on the limitations of the previous arguments.\n\n\n\n\\section{General relativity or central dogma?}\n\\label{Sec:counterex}\n\n\\subsection{Physical configuration}\n\nLe us assume the hypothesis B2 to be valid and consider a physical setup in which a spherically symmetric solar mass black hole is radiating Hawking quanta and, at the same time, also accreting matter. The type of matter falling into the black hole is in principle completely arbitrary, but we can surely assume that its quantum state is entangled with a second component, so that the total state of such additional matter degrees of freedom is pure\\footnote{For a different type of matter, e.g. in a mixed state, the following considerations will be more involved but they can be repeated and shown to be still valid.}. For instance, we can imagine a device that generates two fluxes of matter entangled with each other: one falls into the black hole, whereas the other reaches $\\mathscr{I}^+.$ Therefore, while the total state of matter is still pure, an observer at infinity would only observe the outgoing flux of matter and measure a mixed state.\n\nWe work in the regime in which the outgoing flux of Hawking radiation is carrying an energy per unit of time much smaller than the mass of the black hole, so that the adiabatic approximation is valid and we can implement Hawking computation to obtain the temperature of the black hole:\n\\begin{equation}\nT=\\frac{1}{8\\pi G M}\\,.\n\\end{equation}\nThen, the mass loss is given by the Stefan\u2013Boltzmann law,\n\\begin{equation}\n\\frac{{\\rm d}M}{{\\rm d}v}=- \\sigma_{\\rm SB} T^4 A=-\\sigma_{\\rm SB}\\frac{16\\pi}{\\left(8\\pi \\right)^4}\\frac{1}{G^2M^2}\\,,\n\\end{equation}\nwhere $\\sigma_{\\rm SB}$ is the Stefan\u2013Boltzmann constant and $v$ is the ingoing Eddington\u2013Finkelstein coordinate which provides a notion of time for an observer on $\\mathscr{I}^-$~\\cite{Wald:1984rg}.\n\n\\begin{figure}[t]\n\t\\includegraphics[scale=0.45]{penrose-matter.png}\n\t\\centering\n\t\\protect\\caption{Penrose diagram of a black hole whose mass is much larger than Planck mass and that emits thermal radiation according to semiclassical general relativity. The blue solid line is a timelike hypersurface on which one places a device that emits two entangled fluxes of matter (red solid arrows): one falling into the black hole, while the other reaching $\\mathscr{I}^+$. The black hole mass remains constant in time as it is assumed that the additional energy of the ingoing flux compensates the energy loss due to Hawking radiation.}\n\t\\label{fig4}\n\\end{figure}\n\nFurthermore, the ingoing flux of additional matter can be chosen arbitrarily but, for simplicity, we assume that it has the same energy as the outgoing flux of Hawking radiation. In this way, the mass of the black hole remains constant in time and, as a consequence, the horizon area (or, in other words, the Bekenstein-Hawking entropy $S_{\\rm BH}=A\/4G$) remains constant too.\nThe Penrose diagram for this physical setup is shown in Fig.~\\ref{fig4}. \n\nWe can now consider the entropy of the matter degrees of freedom for an asymptotic observer. If the semiclassical approach holds true in the regime under investigation (B2), then the horizon is smooth and we would not expect any kind of interaction between the outgoing Hawking radiation and the infalling flux of matter. Thus, the behavior of the von Neumann entropy $S_{\\rm m}$ associated to the additional outgoing (or ingoing) matter is \\textbf{independent} of whether $S_{\\rm rad}$ follows Hawking computation or the Page curve. In fact, in both cases the entropy $S_{\\rm m}$ is a monotonically increasing function of time; see Fig.~\\ref{fig5} for a schematic illustration. This means that we do not need to require B1 to elaborate our argument.\n\nLet us also point out that the states associated to the additional matter degrees of freedom are prepared outside the black hole, and therefore they can be engineered in such a way to be and stay orthogonal. This means that there is \\textit{no} dependency relation among the matter states that could even in principle decrease the value of the entropy $S_{\\rm m}.$\n\n\n\\subsection{Contradiction between B2 and B3 }\n\\label{subsec:contradiction-B2-B3}\n\nThe important point to notice in the above physical scenario is the following. If we assume the central dogma to be true, i.e. B3, then after some time $S_{\\rm m}$ would exceed the Bekenstein-Hawking entropy of the black hole and we would conclude that the total state of the additional matter is \\textit{not} pure anymore. It is not surprising that we would obtain a similar conclusion as in the case of the entropy-problem version of the information loss paradox (i.e. Case B in Sec.~\\ref{Sec:info paradox}), where in that case the role of $S_{\\rm m}$ is played by $S_{\\rm rad}.$\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\subfloat[Subfigure 1 list of figures text][]{\n\t\t\\includegraphics[scale=0.415]{curv-matter-1.png}\\label{fig5a}}\\qquad\n\t\\subfloat[Subfigure 2 list of figures text][]{\n\t\t\\includegraphics[scale=0.415]{curv-matter-2.png}\\label{fig5b}}\n\t\\protect\\caption{Schematic representations of the von Neumann entropies of the outgoing Hawking radiation $S_{\\rm rad}$ (yellow solid line) and of the additional matter degrees of freedom $S_{\\rm m}$ (blue solid line), in comparison with the gravitational Bekenstein-Hawking entropy $A\/4G$ (red dashed line) which is assumed to be constant in our physical setup. In the plot (a) $S_{\\rm rad}$ is determined by Hawking semiclassical computation; while in (b) $S_{\\rm rad}$ follows the Page curve. In both cases, if we assume the central dogma to be true, then the number of degrees of freedom in the black hole is \\textit{not} enough to maintain the entanglement between ingoing and outgoing matter fluxes. At the same time, the Hawking radiation and the additional matter cannot be entangled with each other if we assume general relativity to hold at the horizon scale.}\n\t\\label{fig5}\n\\end{figure} \n\nHowever, in the setup just described everything is under control. Indeed, we know for sure that the matter degrees of freedom inside the black hole are entangled with the degrees of freedom outside. In particular, since the pair of (ingoing and outgoing) fluxes is in an entangled pure state, at any instant of time (or, equivalently, on any Cauchy hypersurface) the degrees of freedom of matter (delivered by the ingoing flux) inside the black hole are the same number as the ones of matter (delivered by the outgoing flux) outside of it, and they will eventually exceed the maximum limit allowed by the central dogma, i.e. $S_{\\rm m}>A\/4G.$ This implies that the central dogma cannot be satisfied in our physical configuration. \n\nRemarkably, this conclusion does not depend on whether the evaporation process is unitary or not. Indeed our argument applies to both cases of Figs.~\\ref{fig5a} and~\\ref{fig5b}. Thus, without requiring B1, in our thought experiment we showed the existence of a contradiction between B2 and B3. \n\n\nWe strongly believe that the following fact should be highly emphasized.\nThe information loss problem is usually stated as an incompatibility between quantum mechanics and general relativity. However, the apparent issue only arises because of the imposition of the additional hypothesis B3 that is usually taken for granted; indeed, the usually claimed inconsistency is between B1 and $\\text{B2}\\cup\\text{B3}$~\\cite{Page:1993wv,Almheiri:2020cfm}. \n\nIn fact, the information\/entropy problem in Case B should be formulated as the incompatibility between the validity of semiclassical general relativity as a low-energy effective field theory far from the Planckian regime (implying a smooth horizon) and the central dogma. Therefore, we need to abandon either B2 or B3. In this respect, there exists \\textit{no} information loss paradox since either $\\text{B1}\\cup\\text{B2}$ (Case C) or $\\text{B1}\\cup\\text{B3}$ does not necessarily lead to paradox.\nIn our opinion the assumption B3 should be critically revisited, and indeed the conclusion drawn from our thought experiment can be also phrased as a \\textit{counterexample} to the central dogma if B2 is assumed to be the valid one.\n\n\n\nObviously, our argument would fail if some new physics at the horizon scale is invoked. However, since we expect quantum-gravity effects to emerge only when approaching the end of the evaporation, i.e. at Planck scales, then the effective-field-theory description is completely valid in the regime we considered. This is exactly the point that should be stressed. Contrary to what is usually stated, if \\textit{no} new physics is assumed at the horizon scale then surely one needs to abandon B3 (i.e. to consider Case C) and then there is no information loss problem before reaching the latest stages of black hole evaporation where, instead, a consistent theory of quantum gravity is needed to make any prediction.\n\nFinally, let us note that a physical setup with infalling additional matter similar to the one considered in our thought experiment was studied in \\cite{Lowe1999}. The author argues that this configuration does not need to represent a counterexample to the central dogma if the degrees of freedom of the matter reach the singularity and are non-locally transferred to the radiation. This would lead to entanglement between the matter outside the black hole and the outgoing Hawking radiation. While this is certainly a possibility, it must be noted that the singularity is in the future of any observer. Therefore, the mechanism that would be necessary to transfer information from the singularity to the exterior would not only require interaction to propagate in a spacelike way (as pointed out in~\\cite{Lowe1999}), but also backward in time. Furthermore, in our scenario this would require the outgoing matter to be maximally entangled simultaneously with the ingoing matter and the outgoing Hawking radiation, thus violating the monogamy of entanglement theorem~\\cite{Coffman:1999jd}.\nTherefore, we believe that this is far from the conservative scenario as it would surely require new physics beyond the standard semiclassical description, thus violating B2. \n\nLet us also mention that in the context of quantum teleportation the information about a given quantum state can be sent combining a purely classical channel and a purely quantum one~\\cite{Bennett:1992tv} and that the amount of violation of semiclassical causality might be significantly reduced~\\cite{Mukohyama:1998xq}. Although the classical channel can in principle contain very little information, the quantum channel cannot be decoded by its own without the classical channel. Therefore, a mechanism that is able to teleport the quantum portion of the state would not give rise to causality violation. This partially alleviates the problem discussed in this section as all or a part of the information encoded in the quantum channel can leak out from the black hole without giving rise to any causality problem. However, unless at some point the classical channel is also transferred, the information in the quantum channel is useless. Invoking the transfer of the classical channel from the interior of a black hole to the exterior then requires violation of the semiclassical causality. For this reason, the full information can be recovered in accordance with the central dogma only if B2 is violated at least to the extent that allows for the transfer of the minimal amount of information in the classical channel. \n\n\n\n\\subsection{Stronger contradiction}\\label{Sec:stronger_contr}\n\nThrough the thought experiment described in the previous subsection we argued that the entropy problem in Case B should be reformulated as a contradiction between B2 and B3 independently of B1, instead of an incompatibility between B1 and $\\text{B2}\\cup\\text{B3},$ contrarily to what is often claimed. \n\nBefore continuing our analysis, let us note that the assumption B2 can be split in two parts B1a and B2b such that $\\text{B2}=\\text{B2a}\\,\\cup\\,\\text{B2b}:$\\footnote{It is worth mentioning that our assumptions B2a, B2b and B3 coincide with the postulate 2, 4, 3 in~\\cite{Almheiri:2012rt}, respectively; whereas our assumption B1 differs substantially from postulate 1 of~\\cite{Almheiri:2012rt} as we do not assume that the Hawking radiation must purify. Moreover, in~\\cite{Almheiri:2012rt} no contradiction between B2b and B3 was noticed; in fact, B3 was implicitly assumed to be true regardless of the validity of B2b.} \n\\begin{enumerate}\n\t\t\n\t \\item[B2a.] Black holes whose mass is larger than Planck mass emit thermal radiation according to semiclassical general relativity;\n\t\t\n\t\t\\item[B2b.] Infalling matter far from the Planckian regime obeys the laws of general relativity.\n\n\\end{enumerate}\nThe hypothesis B2a means that the semiclassical approach can be trusted outside the horizon for time scales shorter than black hole life time. Whereas, B2b means that in the presence of a black hole emitting radiation, an infalling matter far from the Planckian regime would experience nothing special, e.g. when crossing and leaving the horizon in the past, consistently with the equivalence principle, so that the horizon is assumed to be a smooth surface.\n\nThe contradiction we found in our thought experiment is between B2b and B3, that is, between the laws of general relativity governing the physics of infalling matter and the central dogma; while B2a and B3 can in principle be perfectly compatible. This means that the contradiction is even stronger. \n\nOn the other hand, one could criticize our conclusion by saying that to formulate the paradox in Case~B only B2a is really needed, and the incompatibility is between B1 and $\\text{B2a}\\cup\\text{B3},$ independently of B2b. Although assuming B2a without B2b might require some discontinuous jump in the physics governing the proximity of the horizon, we agree that such a logic can be correct and can lead to an information loss paradox. However, we should immediately notice that this does \\textbf{not} imply an incompatibility between general relativity and quantum mechanics since abandoning B2b already implies something beyond general relativity.\n\nIndeed, in our thought experiment we clearly showed the existence of a contradiction between the assumptions B2b and B3, from which it follows that if B3 is valid, then the infalling matter should experience some new physics whose description goes beyond the laws of general relativity. Remarkably, this implies that the information loss problem in the Case~B (with B1, B2a, B3) is a statement of an inconsistency between quantum mechanics and some new unknown gravitational physics incompatible with general relativity. \nIn other words, this version of Case B does not show any contradiction between general relativity and quantum mechanics. \n\nFurthermore, if we assume B2b,\nthen we must drop B3 and we fall into Case C from which it is clear that \\textit{no} information paradox arises before reaching the Planck scale where, instead, a consistent theory of quantum gravity is needed to make any prediction about the black hole final state. Therefore, we should say that, in the case B, \\textit{the real incompatibility is between general relativity and the information loss paradox itself.} \n\n\n\\subsection{Limitations of the standard arguments}\n\n\nOne way to interpret the conclusion reached in our thought experiment is that we provided a simple working example in which the central dogma does \\textit{not} hold (if B2b is true). Let us now analyze where the usual arguments in favor of the central dogma reviewed in Sec.~\\ref{Sec:arguments} fail.\n\\begin{enumerate}\n\n\n \\item[(i)] The standard thermodynamic argument based on the area-law for the entropy of a black hole fails because it does not take into account the contribution due to the matter fields inside the black hole, e.g. $S_{\\rm m}$ in our thought experiment. Indeed, it should be clarified that the black hole thermodynamic laws~\\cite{Bardeen:1973gs} are intrinsically gravitational in nature. In other words, the Bekenstein-Hawking entropy $S_{\\rm BH}$ takes into account only gravitational degrees of freedom, and in general the total black hole entropy is given by $S_{\\rm bh}=S_{\\rm BH}+S_{\\rm m}.$\n \n To understand why additional matter degrees of freedom must be taken into account for the correct evaluation of the von Neumann entropy, despite they do not appear in the gravitational thermodynamic laws (thus apparently violating the inequality \\eqref{eq:S-comp}), it is useful to consider a system composed by two subsystems ``$A$'' and ``$B$''. If these two subsystems are very far apart, it is possible to study the thermodynamic laws of subsystem $A$ without taking into consideration subsystem $B$. However, this is no longer true if we are interested in the evaluation of the von Neumann entropy, indeed in this case we need to consider both subsystems even if they are very far apart because of quantum correlations due to entanglement. Eq.~\\eqref{eq:S-comp} is only valid if we consider the thermodynamic entropy of the whole system $A\\cup B,$ i.e.\n %\n \\begin{equation}\\label{eq:S-comp2}\n S_{\\rm th}(A\\cup B)\\geq S_{\\rm vN}(A\\cup B)\\,,\n \\end{equation}\n %\n but obviously it does not hold if we consider only the thermodynamic entropy of the subsystem $A$, i.e.\n \n \\begin{equation}\n S_{\\rm th}(A)\\ngeq S_{\\rm vN}(A\\cup B)\\,.\n \\end{equation}\n %\n While this consideration may seem obvious, it is at the root of the contradiction reached in our thought experiment.\n The role of subsystem $A$ is played by the gravitational degrees of freedom, whereas subsystem $B$ is given by the matter degrees of freedom that fall into the black hole. An outside observer cannot be influenced by such matter degrees of freedom, thus they do not need to be included in the description of black hole thermodynamics. However, if these matter degrees of freedom are entangled with degrees of freedom outside, as it happens in our setup and with Hawking radiation, then we cannot discard them when computing the von Neumann entropy. In particular, the fact that at the Page time the entropy of the matter outside (or inside) the black hole exceeds the area is \\textbf{not} in contradiction with Eq.~\\eqref{eq:S-comp2}, but it simply confirms that the apparent-horizon area only accounts for the entropy associated to the gravitational degrees of freedom. Thus, the inequality\n %\n \\begin{equation}\n S_{\\rm vN}\\left( \\text{gravity}\\cup \\text{matter} \\right)>\\frac{A}{4G}\n \\end{equation}\n %\nfor some states does \\textbf{not} lead to any conceptual problem, rather it implies\n %\n \\begin{equation}\n \\frac{A}{4G}=S_{\\rm th}\\left(\\text{gravity} \\right)\\neq S_{\\rm th}\\left(\\text{gravity}\\cup\\text{matter} \\right).\n \\end{equation}\n %\nTherefore, a priori there is no reason why the addition of matter degrees of freedom should not increase the entropy of the black hole, especially if B2 is valid. The same conclusion would also apply to the ingoing Hawking flux which should contribute to an increase of the thermodynamic entropy of the entire black hole system made up of gravitational plus matter (Hawking quanta) degrees of freedom.\n\nThe possibility that the Bekenstein-Hawking entropy only accounts for the degrees of freedom that are accessible (i.e. causally connected) to an outside observer was already suggested in \\cite{Rovelli:2019tbl}. Our thought experiment allow us to conclude that this must be the case if the assumption B2b is satisfied.\n \n This analysis can also easily explain the apparent contradiction between the Bekenstein-Hawking formula and the fact that ``bag of gold'' spacetimes~\\cite{Wheeler:1964qna} can have a huge volume and a very small area. The area law simply does not describe all the bulk degrees of freedom inside a black hole. See also Refs.~\\cite{Christodoulou:2014yia,Rovelli:2017mzl} for a complementary argument against the central dogma.\n \n\n \\item[(ii)] It is often stated that the ingoing flux of Hawking radiation cannot increase the entropy of the black hole otherwise the Bekenstein bound in Eqs.~\\eqref{bek-bound} and~\\eqref{bek-bound-BH} would be violated. However, it must be emphasized that the standard Bekenstein bound cannot be naively applied when both negative and positive energies are involved, and it is not clear whether gravitational corrections can appear when the curved nature of spacetime is taken into account. In fact, a version of such a bound has been rigorously proven only in the context of quantum field theory in flat spacetime~\\cite{Casini:2008cr}. To be more precise, although the proved inequality is applicable to any quantum systems and to any quantum states, it can be interpreted as Bekenstein bound only in special cases.\n\n\n \n From these last observations, we also see no reason why the infalling matter in our thought experiment should not contribute to the increase of the black hole entropy and not violate the Bekenstein bound. \n Moreover, the Bousso entropy bound, that for a black hole implies the one in Eq.~\\eqref{bek-bound-BH}, would also be violated. Indeed, both the statement of the Bousso bound (the covariant entropy conjecture) and its spacelike projection assume the dominant energy condition~\\cite{Bousso:1999xy} that does not hold in quantum field theory in curved spacetime. \n\n \\item[(iii)] As we mentioned in Sec.~\\ref{Sec:arguments}, holography is one of the main motivation that is usually advocated in support of the central dogma. However, our thought experiment simply reveals an incompatibility between B2b and B3, where the latter is normally thought to be a necessity in presence of holography. Therefore, from our argument it follows that holography, if it implies B3, and the validity of semiclassical gravity for matter infalling to a black hole would be incompatible.\n \nWe are not claiming that holography is wrong but argued that it is incompatible with the low-energy effective-field-theory approach according to which the horizon is smooth as long as the black hole mass is larger than Planck mass. For instance, quantum-gravity theories that predict new physics at the horizon scale and that invalidate B2b might be consistent with a holographic principle~\\cite{Bousso:2002ju}. \n However, we do not see any strong reason why quantum gravity should unconditionally incorporate or predict holography. There may be a condition under which holography holds and such a condition may be violated in some cases. Indeed, it is worth mentioning that several recent approaches~\\cite{Tomboulis:2015esa,Anselmi:2017ygm,Anselmi:2018ibi,Donoghue:2018izj,Salvio:2018crh,Holdom:2021hlo,Percacci:2017fkn,Reuter:2019byg,Platania:2018eka,Bonanno:2020bil} aimed at formulating a renonormalizable and unitary quantum field theory of the gravitational interaction seem to be valid proposals for a consistent quantum-gravity theory that does not rely on or have any relation with holography.\n \n \\item[(iv)] Very recently, many interesting works have been done towards the resolution of the entropy problem outlined in Case B~\\cite{Penington:2019npb,Penington:2019kki,Almheiri:2019psf,Almheiri:2019hni,Almheiri:2019qdq}. Although we appreciate that rigorous computations of the Page curve have been made in some specific simplified settings, we have reservations about the claimed interpretation of the result. Let us briefly review the logic that was followed.\n \n \n The computation of the von Neumann entropy of the radiation was obtained by using the Ryu-Takayanagi formula~\\cite{Ryu:2006bv} that can be derived from the gravitational path integral under the assumption of holography~\\cite{Lewkowycz:2013nqa,Harlow:2020bee}. As a result the Page curve was reproduced, and several authors~\\cite{Penington:2019npb,Penington:2019kki,Almheiri:2019psf,Almheiri:2019hni,Almheiri:2019qdq} claimed that the entropy problem (paradox) was solved. Let us also emphasize that the reason why the final state of the radiation purifies in their calculation is that the Ryu-Takayanagi formula for the entropy of the radiation also includes the contribution from the black hole interior. This approach to compute entropy of the radiation is also known as ``island program''~\\cite{Almheiri:2020cfm}.\n \n \n One may now wonder what would be the outcome of the island computation when applied to the von Neumann entropies of radiation and additional matter in our thought experiment. In fact, if one assumes that the Hilbert space of the interior of the black hole (including the additional matter) is no longer independent of the Hilbert space of outgoing radiation, then by using the island formula one could show that eventually the quantity $S_{\\rm rad}+S_{\\rm m}$ will tend to $A\/4G$ and never exceed it. If this was the case, then one would conclude that the central dogma is still respected.\n \n \n We agree with this possibility; however, we believe that such a scenario is completely consistent with the conclusion drawn from our thought experiment. The assumption B3 of central dogma can be satisfied only if B2b is violated. Indeed, in our physical setup the island would correspond to the entire black hole interior, and it so happens that the von Neumann entropy $S_{\\rm m}$ of the infalling matter does not contribute to the von Neumann entropy of the black hole because both ingoing and outgoing fluxes of matter are taken to be part of the same $\\text{island}\\cup\\text{radiation}$ region. This fact implicitly assumes that the additional matter and the radiation get somehow entangled, but this can happen \\textit{if and only if} the matter in free fall interacts with the Hawking radiation, which would be impossible if the laws of general relativity are valid, i.e. if B2b holds.\n \n Therefore, when applied to our physical setup the island program implies that the laws of semiclassical general relativity to describe infalling matter are violated. This might also suggest that new physics beyond general relativity is needed to completely trust and physically interpret the island computation\\footnote{Also, in our opinion it is not totally clear which Euclidean\nsolutions should contribute to the island computation based on the\nreplica method as saddle points, without additional information or\nassumptions.}. \nYet another possible interpretation is that the island formula somehow takes into account the transfer of information through the quantum channel in the language of quantum teleportation. However, as explained in the last paragraph of subsection \\ref{subsec:contradiction-B2-B3}, one still needs to transfer the classical channel from the black hole interior to the exterior and thus the semiclassical general relativity and the central dogma are incompatible with each other. \n\nSee also Ref.~\\cite{Geng:2021hlu} where it was recently argued that the island program\/holography might not be consistent with massless gravitational theories, and Ref.~\\cite{Omiya:2021olc} where it was claimed that the transfer of information based on the island computation requires a nonlocal interaction term in the Hamiltonian. On the other hand, it is also worth mentioning that other authors~\\cite{Harlow:2020bee} claim that holography must be imposed even when performing the standard Euclidean computation~\\cite{Gibbons:1976ue} of the Gibbons-Hawking entropy, or in other words that the Euclidean gravitational path integral is only valid for holographic theories.\n\n\n\n\\end{enumerate}\n\n\n\\section{On the microscopic interpretation of $A\/4G$}\\label{Sec:micr}\n\nLet us now assume that general relativity is a valid theory far from the Planckian regime, so that the horizon is a smooth surface and a free falling observer does not experience anything out of the ordinary before approaching the singularity. Consequently, the central dogma (B3) is violated, and we would fall into Case C discussed in Sec.~\\ref{case-C} according to which no information paradox exists. In such a case, a very natural question to ask is -- what is the microscopic interpretation of the Bekenstein-Hawking entropy $A\/4G?$\n\nMacroscopically, i.e. in a coarse-grained sense, $A\/4G$ can be interpreted as the contribution of a black hole to the total thermodynamic entropy that follows the generalized second law in semiclassical regimes. However, since the central dogma does not hold, then one should give an alternative interpretation of the Bekenstein-Hawking formula at microscopic level, i.e. one should explain which part of the black hole degrees of freedom are described by $A\/4G.$ or instance, in Ref.~\\cite{Frolov:1993ym} it was assumed that the dynamical degrees of freedom of the black hole correspond to thermally excited modes behind the horizon that are invisible to a distant observer, and it was shown that their contribution is indeed proportional to the horizon area.\n\n One expects the microscopic description to be highly model-dependent, and that the correct origin and counting of both gravitational and matter degrees of freedom can be given by a consistent theory of quantum gravity. In what follows we address the above question in three different well-known theories\/scenarios.\n\\paragraph{String theory:}In Ref.~\\cite{Strominger:1996sh} it was shown that the counting of D-brane states correctly reproduces the formula $A\/4G$ for a class of five-dimensional extremal black holes. In particular, no more entropy than $A\/4G$ can be added to the D-brane, which is supposed to correspond to a black hole. Furthermore, such correspondence may be more general and applicable to realistic non-extremal black holes: the gravitational radius of a D-brane increases as the string coupling is raised; and the D-brane becomes a black hole when the gravitational radius becomes as large as the size of a string. If the discontinuity in the mass at the transition is not too large and if the typical D-brane states become the typical black hole states then the entropy of the D-brane correctly accounts for the entropy of the black hole up to a numerical factor of order unity~\\cite{Horowitz:1996nw}. Then, apparently, the black hole entropy cannot significantly exceed the Bekenstein-Hawking value. However, even if the computation can be extended to realistic non-extremal black holes, we should notice that it is not known which degrees of freedom in the black hole spacetime correspond to the D-brane; they may be degrees of freedom localized at the horizon, those localized slightly inside or outside the horizon, those localized near the singularity of the classical black hole solution, or something else. If the D-brane corresponds to degrees of freedom localized or extended outside the horizon in the black hole spacetime then B2b may be violated. Whereas, if the D-brane corresponds to some degrees of freedom localized somewhere strictly inside the horizon (e.g. near the classical singularity), then additional entropy may be stored in a region inside the horizon but away from the place where those degrees of freedom corresponding to the D-brane are localized. In this case, one might argue that the D-brane states would correspond to the gravitational degrees of freedom, i.e. $A\/4G$, and that the additional states would correspond to the matter degrees of freedom, i.e. $S_{\\rm m}.$ Thus in this case, consistently with our thought experiment, the total entropy of the black hole would be given by $S_{\\rm bh}=A\/4G+S_{\\rm m}>A\/4G$. In summary the D-brane picture of black holes in the context of string theory has not yet been fully understood and therefore it is at this stage compatible with various possibilities, such as the one suggested by our thought experiment. \n\n\n\\paragraph{Loop quantum gravity:} In Ref.~\\cite{Ashtekar:1997yu} the Bekenstein-Hawking formula for the entropy was computed in the non-perturbative framework of loop quantum gravity. In particular, by making a quantized phase-space description for the black hole, it was shown that the resulting statistical mechanical entropy is proportional to the horizon area; see also Refs.~\\cite{Rovelli:1996dv,Meissner:2004ju} for related works. The important point to notice in the derivation is that the computed entropy only accounts for the quantum states that describe the horizon geometry, as also pointed out in Ref.~\\cite{Ashtekar:1997yu}. Therefore, also in this approach the horizon area counts purely gravitational degrees of freedom; whereas, matter degrees of freedom are expected to give an additional contribution to the total entropy of the black hole (as seen from outside).\n\n\n\n\\paragraph{Corpuscular gravity:} According to the corpuscular picture~\\cite{Dvali:2011aa,Dvali:2012en,Dvali:2013eja,Dvali:2014ila}, a black hole can be described as a self-sustained system of weakly interacting gravitons. In this case, the number of gravitons $N$ coincides with the Bekenstein-Hawking formula, i.e. $N\\simeq A\/4G.$ Therefore, the gravitons microscopically count the gravitational degrees of freedom of the black hole. Whereas, if we also include matter in the system, then the corresponding degrees of freedom are associated with an additional (non-gravitational) entropy contribution, i.e. $S_{\\rm m},$ so that the total black hole entropy is given by $S_{\\rm bh}\\simeq N+S_{\\rm m}>A\/4G.$ It is worth mentioning that, strictly speaking, for a corpuscular black hole no geometric notion of horizon exists so that, in general, also B2b is violated. However, one can show that in the $N\\gg 1$ limit the corpuscular corrections are negligible, and the values of all relevant physical quantities are compatible with the presence of a smooth horizon~\\cite{Dvali:2011aa}.\n\n\\section{Comments on black hole complementarity and firewall}\\label{Sec:Comp-fir}\n\nLet us now make some comments on black hole complementarity~\\cite{Susskind:1993if} and firewall paradigm~\\cite{Almheiri:2012rt}. In both cases, the central dogma is assumed as a postulate and never questioned.\n\n\\paragraph{Black hole complementarity:} According to black hole complementarity~\\cite{Susskind:1993if,Susskind:1993mu}, an outside observer can replace the black hole in terms of a hot membrane whose surface lays one Planck length above the horizon at the so-called stretched horizon, and whose entropy coincides with the Bekenstein-Hawking entropy $A\/4G$ in agreement with the central dogma. Whereas, an infalling observer would not see any membrane in the proximity of the horizon, and would not experience any deviation from general relativity. \n\nHowever, from the conclusion of our thought experiment, it follows that the entropy of the stretched horizon (i.e. of the black hole) can be bounded by the area only if some new physics beyond general relativity is invoked at the horizon scale; otherwise an outside observer can prepare an experiment and confirm that the actual entropy is larger than $S_{\\rm BH}.$ Therefore, the postulates of black hole complementarity are \\textit{not} compatible with a smooth horizon.\n\n\\paragraph{Firewall:} The firewall paradigm~\\cite{Almheiri:2012rt} also states that black hole complementarity is not compatible with a smooth horizon. However, we should stress that our logic is completely different from the firewall one. Indeed, in the firewall proposal the central dogma was imposed as a postulate and, moreover, the state of Hawking radiation was assumed to purify. In our thought experiment we have not made any assumption on the final state of the Hawking radiation; in fact, our conclusions are more general. We showed the existence of an incompatibility between the validity of general relativity for infalling matter and the central dogma, independently of whether the von Neumann entropy of the Hawking radiation follows the Page curve or not.\n\nFurthermore, regarding the question of whether a firewall exists or not, one cannot really answer without having a consistent theory of quantum gravity which would allow us to predict the final quantum state of black hole evaporation. However, we do not see a strong reason why the state of the Hawking radiation should purify. For instance, if we abandon the central dogma, then it is also very plausible that the final black hole state is a remnant, so that the Hawking radiation remains in a mixed state but the total state of the joint system is pure; see also Ref.~\\cite{Rovelli:2019tbl} for a similar argument.\n\n\n\\section{Discussion}\\label{Sec:discussions}\n\nIn this paper, we critically inspected the assumptions behind the formulation(s) of the information loss problem.\nAlthough it is often stated as an incompatibility between general relativity and the unitarity of quantum mechanics, we have argued that this is true only if the semiclassical gravity description is naively assumed to be valid all the way up to the Planck scale, and if the spacetime is extended beyond the Cauchy horizon in a specific way (Case A). Therefore, this version of the paradox is not particularly worrisome as quantum-gravity effects are expected to become relevant at the final stages of the black hole evaporation.\n\nHowever, an ``apparent'' paradoxical conclusion involving general relativity and quantum mechanics is claimed to arise in regimes where the semiclassical approach is still valid, once the so-called central dogma is taken as an additional hypothesis (Case B). \n\nWe questioned the well-posedness of this second formulation of the paradox. Remarkably, by working in a well-defined physical setup consisting of a black hole whose energy loss due to radiation emission is compensated by an additional infalling flux of matter, \nwe provided a clear working example to show the existence of an incompatibility between the following two assumptions:\n\\begin{itemize}\n\t\t\n\t\t\\item[B2b.] Infalling matter far from the Planckian regime obeys the laws of general relativity;\n\t\t\n\t\t\\item[B3.] As seen from the outside, a black hole behaves like a quantum system whose number of degrees of freedom is given by $A\/4G,$ with $A$ being the apparent-horizon area.\n\n\\end{itemize}\n\nAs a consequence, we argued that Case B does not imply any contradiction between general relativity and quantum mechanics. Indeed, if the entropy problem is formulated as in Sec.~\\ref{CaseB}, then the incompatibility is between the validity of semiclassical gravity far from the Planckian regime and the central dogma, independently of the assumption of unitary evolution, rather than between the assumption of unitary evolution and semiclassical gravity, contrarily to what it is often claimed~\\cite{Almheiri:2020cfm}. \n\nOn the other hand, if the semiclassical gravity assumption B2 is split into B2a and B2b as described in Sec.~\\ref{Sec:stronger_contr}, and if we abandon B2b, then the paradox can be reformulated as an incompatibility among B1, B2a and B3. However, this reformulation must require the violation of B2b, which means that it is\nautomatically inconsistent with the laws of general relativity to describe infalling matter. Therefore, the paradox would consist in an incompatibility between quantum mechanics and some new unknown gravitational physics at the horizon scale. \nFurthermore, it seems at the very least unnatural to assume the validity of semiclassical gravity outside the horizon while simultaneously violating general relativity at the horizon scale.\n\nWe are now in a position to address the two questions raised in the titles of Sec.~\\ref{Sec:info paradox} and Sec.~\\ref{Sec:counterex}.\n\\begin{enumerate}\n\n\\item \\textbf{Is there an information loss paradox?}\\\\\nVery interestingly, our conclusions imply that \\textit{the usually stated information loss paradox has nothing to do with the loss of information}, as quantum mechanics and semiclassical general relativity can be\nperfectly compatible before reaching Planck scales where, instead, a consistent theory of quantum gravity is needed to make any prediction about the final state of black hole evaporation.\nA contradiction only arises once the central dogma (B3) is added to the picture. The main message of this paper is that the problem is \\textit{not} due to an incompatibility between general relativity and quantum mechanics but, instead, it is due to a contradiction between B2b and B3 (and so between B2 and B3) independently of B1.\nTo be fair, we should say that, in the case B, \\textit{the real incompatibility is between general relativity and the information loss paradox itself.}\n\nInformation may or may not be lost due to the formation of an event horizon~\\cite{Visser:2014ypa}. However, here we have shown that there is no reason to argue in any direction within the regime of validity of semiclassical gravity.\n\n\\item \\textbf{General relativity or central dogma?}\\\\\nWhile it is crucial to properly formulate the problem, the question of which of the two assumptions (B2b or B3) should be dropped cannot be answered. In fact, there are a number of approaches that aim to address the process of black hole evaporation; \nfor an incomplete list see~\\cite{Almheiri:2020cfm,Penington:2019npb,Penington:2019kki,Almheiri:2019psf,Almheiri:2019hni,Almheiri:2019qdq,Marolf:2020rpm,Dvali:2011aa,Dvali:2012en,Susskind:1993if,Visser:2014ypa,tHooft:1996rdg,Betzios:2016yaq,Gaddam:2020mwe,Mathur:2005zp,Hayward:2005oet,Giddings:2012gc,Hawking:2014tga,Hawking:2016msc,Frolov:2014jva,Frolov:2017rjz,Bardeen:2014uaa,DAmbrosio:2020mut} and references therein. \nHowever, we should emphasize that it is impossible to understand whether one of these possibilities is indeed physically correct as it is currently impossible to perform an experiment to observe Hawking evaporation.\nTherefore, the reader is free to choose which of the two assumptions to save, but at most one.\n\n\\end{enumerate}\n\n\n\n\n\n\\subsection*{Acknowledgements}\nThe authors would like to thank Gia Dvali, Valeri Frolov, Stefano Liberati, and Tadashi Takayanagi for critical comments and useful suggestions on an initial draft of this manuscript. In particular, they are grateful to Tadashi Takayanagi for enlightening discussions. F.~D.~F.~would like to thank Ibrahim Akal, Ra\\'ul Carballo-Rubio, and Mohammad Ali Gorji for several stimulating discussions. L.~B.~is grateful to YITP for the warm hospitality, where this work was started.\nL.~B.~acknowledges financial support from JSPS and KAKENHI Grant-in-Aid for Scientific Research No.~JP19F19324.\nThe work of F.~D.~F.~was supported by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research No.~17H06359.\nThe work of S.~M.~was supported in part by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No.~17H02890, No.~17H06359, and by World Premier International Research Center Initiative, MEXT, Japan.\n\n\n\\bibliographystyle{utphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}