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数理解析研究所講究録

Towards Local Temperature States in QFT (Mathematical Aspects of Quantum Information and Quantum Chaos)

Izumi Ojima

Published Apr 1, 2000• 0 citations
OpenAlex

Abstract

Simple example of local temperature stateAiming at a general formulation of non-equilibrium states in the framework of QFT, we start from a simple model-example of local temperature state: Let $\varphi(x)$ be a massless free scalar field (in four dimensions) characterized by(Given a two-point function $\omega^{(2)}(\varphi(x)\varphi(y))$ , we can define a quasi-free state $\omega$ of $\varphi$ through the Wick formula:When a two-point function $\omega^{(2)}$ is chosen consistently with Eqs.$(1, 2)$ asthe corresponding quasi-free state $\omega=\omega_{\beta}$ describes a global thermal equi- librium satisfying the KMS condition for any pair of polynomial fields $A=$ $\varphi(x_{1})\cdots\varphi(x_{r})$ and $B=\varphi(y_{1})\cdots\varphi(y_{s})$ , $\omega_{\beta}(A\alpha_{(i\beta,\mathrm{O})}(B))=\omega_{\beta}(BA)$ , (5) *Talk based on a joint work of D.Buchholz, H.Roos and $\mathrm{I}.\mathrm{O}$ . in progress

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Thermodynamic Properties of Non-Equilibrium States in Quantum Field Theory

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Within the framework of relativistic quantum field theory, a novel method is established which allows to distinguish non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang”. 1

•Jan 1, 2001•47 citations

The Relation Between KMS-states for Different Temperatures

Christian D. Jäkel

Given a thermal field theory for some temperature $β^{-1}$, we construct the theory at an arbitrary temperature $ 1 / β'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region $Ø_\circ \subset \rr^4$, but coincide with the given KMS state in the space-like complement of a slightly larger region $\hatØ$. By a weak*-compactness argument there always exists a convergent subnet of states as the size of $ Ø_\circ$ and $ \hatØ$ tends towards $ \rr^4$. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of $ Ø_\circ$ and $ \hatØ$. We show that this surface energy can be controlled by a generalized cluster condition.

arXiv:hep-th/9803245•Mar 30, 1998•0 citations

Local temperature and correlations in quantum many-body systems

Senaida Hernández Santana

Quantum Mechanics was established as the theory of the microscopic world, which allowed to understand processes in atoms and molecules. Its emergence led to a new scientific paradigm that quickly spread to different research fields. Two relevant examples are Quantum Thermodynamics and Quantum Many-Body Theory, where the former aims to characterize thermodynamic processes in quantum systems and the latter intends to understand the properties of quantum many-body systems. In this thesis, we tackle some of the questions in the overlap between these disciplines, focusing on the concepts of temperature and correlations. Specifically, it contains results on the following topics: locality of temperature, correlations in long-range interacting systems and thermometry at low temperature. The problem of locality of temperature is considered for a system at thermal equilibrium and consists in studying whether it is possible to assign a temperature to any of the subsystems of the global system such that both local and global temperatures are equal. We tackle this problem in two different settings, for generic one-dimensional spin chains and for a bosonic system with a phase transition at non-zero temperature. In the first case, we consider generic one-dimensional translation-invariant spin systems with short-range interactions and prove that it is always possible to assign a local temperature equal to the global one for any temperature, including at criticality. For the second case, we consider a three-dimensional discretized version of the Bose-Einstein model at the grand canonical ensemble for some temperature and particle density, and characterize its non-zero-temperature phase transition. Then, we show that temperature is locally well-defined at any temperature and at any particle density, including at the phase transition. Additionally, we observe a qualitative relation between correlations and locality of temperature in the system. Moving to correlations, we consider fermionic two-site long-range interacting systems at thermal equilibrium. We show that correlations between anti-commutative operators at non-zero temperature are upper bounded by a function that decays polynomially with the distance and with an exponent that is equal to the interaction exponent, which characterizes the interactions in the Hamiltonian. Moreover, we show that our bound is asymptotically tight and that the results extend to density-density correlations as well as other types of correlations for quadratic and fermionic Hamiltonians with long-range interactions. Regarding the results on thermometry, we consider a bosonic model and prove that strong coupling between the probe and the system can boost the thermal sensitivity for low temperature. Furthermore, we provide a feasible measurement scheme capable of producing optimal estimates at the considered regime.

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Nonequilibrium Local States in QFT : As a typical example case in generalized sector theory(Quantum Field Theories: Fundamental Problems and Applications)

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素粒子論研究•Aug 20, 2003•0 citations

Mathematical aspects of Bose–Einstein condensation in equilibrium and local equilibrium conditions

Luigi Accardi, Carlo Pandiscia

In the paper4 the notion of local KMS condition, introduced in Ref. 3 and extended in Ref. 2, was shown to open new possibilities in the study of the problem of Bose–Einstein Condensation (BEC). In this paper we analyze the general structure of states on the CCR algebra over a pre-Hilbert space that satisfies the local KMS condition with respect to a free Hamiltonian [Formula: see text] and to a given inverse temperature function [Formula: see text]. The replacement of Hilbert space by a pre-Hilbert space allows one to deal with test functions more singular than those usually considered in the theory of distributions (thus allowing e.g., fractal critical surfaces) and is equivalent (in the language of Weyl algebras) to consider a degenerate symplectic form. It is precisely this degeneracy that allows one to introduce in an intrinsic way the notions of [Formula: see text]-critical subspace (resp. [Formula: see text]-critical surface) and of states exhibiting BEC, independently of infinite volume limits and of boundary conditions. We prove that the covariance of any local KMS state is uniquely determined by the pair [Formula: see text], through a nonlinear extension of the Planck factor. For a large class of such states (including all known examples) the covariance splits into a sum of two mutually singular terms: one corresponding to a regular state, the other one with support on a critical surface (or more generally a critical subspace) uniquely determined by [Formula: see text] and [Formula: see text]. In particular we prove that, if such a state is gauge invariant Gaussian (quasi-free), then the [Formula: see text]-equilibrium condition uniquely determines the regular part of the state, while the singular part is arbitrary.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Dec 1, 2017•0 citations

Conformal defects of general dimensions at finite temperature

Yucheng Li, Haruki Nakayama, 辰磨 西岡

We consider defect conformal field theories (DCFTs) at finite temperature, where a $p$-dimensional conformal defect wraps the thermal circle. Extending the previous study of line defects to defects of general dimensions, we determine the general forms of the thermal one-point functions of bulk scalars, conserved currents and the stress tensor by imposing the residual symmetry and the conservation laws. The low temperature expansion of the thermal one-point functions is shown to be reproduced by the bulk-defect operator expansion, allowing us to read off the thermal one-point coefficients of defect operators. We examine the general results in four classes of examples with free bulk theories: the trivial defect, free scalar and fermion theories with a boundary, the free scalar theory with a localized $ϕ$ deformation in $d=2\,p+2$ dimensions and the free $\mathrm{O}(N)$ model with a localized $ϕ^2$ deformation in $d=p+2-ε$ dimensions. In the last example with $ε= 1$, where the defect becomes an interface, we find that the thermal one-point functions coincide with those of the free scalar theory with the Dirichlet boundary condition, in accordance with the conjectured factorization of an interface CFT into two decoupled boundary CFTs.

arXiv:2609.12720•Sep 11, 2026•0 citations
Simple example of local temperature stateAiming at a general formulation of non-equilibrium states in the framework of QFT, we start from a simple model-example of local temperature state: Let φ(x)\varphi(x)φ(x) be a massless free scalar field (in four dimensions) characterized by(Given a two-point function ω(2)(φ(x)φ(y))\omega^{(2)}(\varphi(x)\varphi(y))ω(2)(φ(x)φ(y)) , we can define a quasi-free state ω\omegaω of φ\varphiφ through the Wick formula:When a two-point function ω(2)\omega^{(2)}ω(2) is chosen consistently with Eqs.(1,2)(1, 2)(1,2) asthe corresponding quasi-free state ω=ωβ\omega=\omega_{\beta}ω=ωβ​ describes a global thermal equi- librium satisfying the KMS condition for any pair of polynomial fields A=A=A= φ(x1)⋯φ(xr)\varphi(x_{1})\cdots\varphi(x_{r})φ(x1​)⋯φ(xr​) and B=φ(y1)⋯φ(ys)B=\varphi(y_{1})\cdots\varphi(y_{s})B=φ(y1​)⋯φ(ys​) , ωβ(Aα(iβ,O)(B))=ωβ(BA)\omega_{\beta}(A\alpha_{(i\beta,\mathrm{O})}(B))=\omega_{\beta}(BA)ωβ​(Aα(iβ,O)​(B))=ωβ​(BA) , (5) *Talk based on a joint work of D.Buchholz, H.Roos and I.O\mathrm{I}.\mathrm{O}I.O . in progress
Within the framework of relativistic quantum field theory, a novel method is established which allows to distinguish non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang”. 1
Given a thermal field theory for some temperature β−1β^{-1}β−1, we construct the theory at an arbitrary temperature $ 1 / β'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region $Ø_\circ \subset \rr^4$, but coincide with the given KMS state in the space-like complement of a slightly larger region Ø^\hatØØ^​. By a weak*-compactness argument there always exists a convergent subnet of states as the size of $ Ø_\circ$ and $ \hatØ$ tends towards $ \rr^4$. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of $ Ø_\circ$ and $ \hatØ$. We show that this surface energy can be controlled by a generalized cluster condition.
Quantum Mechanics was established as the theory of the microscopic world, which allowed to understand processes in atoms and molecules. Its emergence led to a new scientific paradigm that quickly spread to different research fields. Two relevant examples are Quantum Thermodynamics and Quantum Many-Body Theory, where the former aims to characterize thermodynamic processes in quantum systems and the latter intends to understand the properties of quantum many-body systems. In this thesis, we tackle some of the questions in the overlap between these disciplines, focusing on the concepts of temperature and correlations. Specifically, it contains results on the following topics: locality of temperature, correlations in long-range interacting systems and thermometry at low temperature. The problem of locality of temperature is considered for a system at thermal equilibrium and consists in studying whether it is possible to assign a temperature to any of the subsystems of the global system such that both local and global temperatures are equal. We tackle this problem in two different settings, for generic one-dimensional spin chains and for a bosonic system with a phase transition at non-zero temperature. In the first case, we consider generic one-dimensional translation-invariant spin systems with short-range interactions and prove that it is always possible to assign a local temperature equal to the global one for any temperature, including at criticality. For the second case, we consider a three-dimensional discretized version of the Bose-Einstein model at the grand canonical ensemble for some temperature and particle density, and characterize its non-zero-temperature phase transition. Then, we show that temperature is locally well-defined at any temperature and at any particle density, including at the phase transition. Additionally, we observe a qualitative relation between correlations and locality of temperature in the system. Moving to correlations, we consider fermionic two-site long-range interacting systems at thermal equilibrium. We show that correlations between anti-commutative operators at non-zero temperature are upper bounded by a function that decays polynomially with the distance and with an exponent that is equal to the interaction exponent, which characterizes the interactions in the Hamiltonian. Moreover, we show that our bound is asymptotically tight and that the results extend to density-density correlations as well as other types of correlations for quadratic and fermionic Hamiltonians with long-range interactions. Regarding the results on thermometry, we consider a bosonic model and prove that strong coupling between the probe and the system can boost the thermal sensitivity for low temperature. Furthermore, we provide a feasible measurement scheme capable of producing optimal estimates at the considered regime.
In the paper4 the notion of local KMS condition, introduced in Ref. 3 and extended in Ref. 2, was shown to open new possibilities in the study of the problem of Bose–Einstein Condensation (BEC). In this paper we analyze the general structure of states on the CCR algebra over a pre-Hilbert space that satisfies the local KMS condition with respect to a free Hamiltonian [Formula: see text] and to a given inverse temperature function [Formula: see text]. The replacement of Hilbert space by a pre-Hilbert space allows one to deal with test functions more singular than those usually considered in the theory of distributions (thus allowing e.g., fractal critical surfaces) and is equivalent (in the language of Weyl algebras) to consider a degenerate symplectic form. It is precisely this degeneracy that allows one to introduce in an intrinsic way the notions of [Formula: see text]-critical subspace (resp. [Formula: see text]-critical surface) and of states exhibiting BEC, independently of infinite volume limits and of boundary conditions. We prove that the covariance of any local KMS state is uniquely determined by the pair [Formula: see text], through a nonlinear extension of the Planck factor. For a large class of such states (including all known examples) the covariance splits into a sum of two mutually singular terms: one corresponding to a regular state, the other one with support on a critical surface (or more generally a critical subspace) uniquely determined by [Formula: see text] and [Formula: see text]. In particular we prove that, if such a state is gauge invariant Gaussian (quasi-free), then the [Formula: see text]-equilibrium condition uniquely determines the regular part of the state, while the singular part is arbitrary.
We consider defect conformal field theories (DCFTs) at finite temperature, where a ppp-dimensional conformal defect wraps the thermal circle. Extending the previous study of line defects to defects of general dimensions, we determine the general forms of the thermal one-point functions of bulk scalars, conserved currents and the stress tensor by imposing the residual symmetry and the conservation laws. The low temperature expansion of the thermal one-point functions is shown to be reproduced by the bulk-defect operator expansion, allowing us to read off the thermal one-point coefficients of defect operators. We examine the general results in four classes of examples with free bulk theories: the trivial defect, free scalar and fermion theories with a boundary, the free scalar theory with a localized ϕϕϕ deformation in d=2 p+2d=2\,p+2d=2p+2 dimensions and the free O(N)\mathrm{O}(N)O(N) model with a localized ϕ2ϕ^2ϕ2 deformation in d=p+2−εd=p+2-εd=p+2−ε dimensions. In the last example with ε=1ε= 1ε=1, where the defect becomes an interface, we find that the thermal one-point functions coincide with those of the free scalar theory with the Dirichlet boundary condition, in accordance with the conjectured factorization of an interface CFT into two decoupled boundary CFTs.