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Journal of Mathematical Physics

Time Translations in the Algebraic Formulation of Statistical Mechanics

Daniel A. Dubin, Geoffrey L. Sewell

Published Oct 1, 1970• 80 citations
Publisher

Abstract

We present a new description of time translations in the C*-algebraic formulation of statistical mechanics. This description is based on weaker assumptions than the hitherto accepted ones, due to Haag, Hugenholtz, and Winnink (HHW) [Commun. Math. Phys. 5, 215 (1967)]. It is shown that these weaker assumptions still lead to the principal results of HHW for Gibbs states and, further, that our assumptions, unlike those of HHW, are valid for the ideal Bose gas and strong-coupling BCS models.

DOI
10.1063/1.1665084

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We present a new description of time translations in the C*-algebraic formulation of statistical mechanics. This description is based on weaker assumptions than the hitherto accepted ones, due to Haag, Hugenholtz, and Winnink (HHW) [Commun. Math. Phys. 5, 215 (1967)]. It is shown that these weaker assumptions still lead to the principal results of HHW for Gibbs states and, further, that our assumptions, unlike those of HHW, are valid for the ideal Bose gas and strong-coupling BCS models.
Forty years ago, Haag and Kastler [27] introduced the algebraic approach to Quantum Field Theory, and soon their method was extended to statistical mechanics. The rationale was to free the theory from particular realizations, linked to specific models, and to focus on the basic structure, namely, the algebra of observables and states on it. This more abstract language, exploiting the deep mathematical theories of C*-algebras and von Neumann algebras, soon became the standard approach to the mathematically rigorous description of physical systems with infinitely many degrees of freedom. Textbooks flourished and, among many authors, Gérard Emch argued forcefully for the new approach [6].
The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*-algebra U of quasilocal bounded observables in Fock space HF, an arbitrary, locally normal state φ on U, and a corresponding Gel'fand-Naimark-Segal (GNS) representation Rφ of U in a Hilbert space Hφ. We then construct a set QL of local, closed operators whose domains are dense in HF, such that QL includes all the local observables of the system. The representation Rφ is then extended so as to provide a *-homomorphism of HL into the closed, densely defined operators in Hφ. Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo-Martin-Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space-correlation functions, and the temporally ergodic properties of time-correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of QL are thermodynamical limits of the corresponding correlations for finite systems.
Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described.As for the facts presented in QFT, they have essentially been established in a previous paper (A.Arai, Rev. Math.Phys.17(2005), 10711109). §1. Introduction: Strong Time Operators and Their FundamentalProperties This paper is concerned with time operators in algebraic quantum mechanics (including quantum statistical mechanics) and quantum field theory.The concept of time operators itself is a general one and there are some types on them [1, 2, 5, 9].In the present paper we concentrate our attention on strong time operators as defined below and their realizations in algebraic quantum mechanics and quantum field theory.Let \ ma t h f r a k { H} be a complex Hilbert space.We denote the inner product and the norm of \ ma t h f r a k { H} by \l a ng l e \c dot , \ r a n g l e _ { \ m a t h f r a k { H } } (anti-linear in the first variable) and \ V e r t \ c d o t \ V e r t _ { \ ma t h f r a k { H} } respectively.If there is no danger of confusion, then the subscript \ ma t h f r a k { H} in \l a ng l e \c do t , \ r a n g l e _ { \ m a t h f r a k { H } } and \ operator A on a Hilbert space, \mat hrm{D}(A) (resp. Ran (A)) denotes the domain (resp. range) of A . IfA is closable, then we denote its closure by \ov e r l i ne { A} .Let H be a self-adjoint operator on \ ma t h f r a k { H} and T be a symmetric operator on \ ma t h f r a k { H} .If there exists a bounded self-adjoint operator C\neq 0 on \ ma t h f r a k { H} such that \mathrm{D}(C)=\mathf rak{H}, (1.1) e^{-i tH}\mathrm{D}(T)\subset \mathrm{D}(T) , \foral l t\i n \mathbb{R},
We construct a C*-algebraic formulation of the dynamics of a pair of mutually interacting quantum-mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin-Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time-dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation-dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time-dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.
Measurements distributed in time provide information about a system at more than one time; they cannot be described in terms of the conventional language of a system quantum state evolving in time. This paper, the second in a series, explores connections among various ways of formulating a quantum-mechanical description of time-distributed measurements. The natural formulation, involving a ``sum over histories,'' arises directly from Feynman's rules for combining probability amplitudes. One equivalent formulation uses a standard measurement model, in which the system is coupled to a set of ``measuring apparatuses.'' A second equivalent formulation uses the language of ``effects'' and ``operations.'' Still a third formulation attempts to create a new language of multiple-time states and multiple-time eigenstates.