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Birkhäuser Boston eBooks

Linear and Related Models of Time Evolution in Quantum Statistical Mechanics

A. G. Shuhov, Yu. M. Suhov

Published Jan 1, 1985• 4 citations
Publisher

Abstract

The aim of this paper is to give a review of some recent results concerning the problem of constructing and studying time evolution for quantum systems with infinitely many degrees of freedom. The first rigorous results in this direction are due to O. E. Lanford and D. W. Robinson [17–20], see also [13], Ch. 5.3. An approach to this problem has been proposed by O. Bratteli and D. W. Robinson [12], see also [13], Ch. 6.3. Among recent papers we refer to [1–3], [10–11], [16], [24]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4899-6653-7_6

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The aim of this paper is to give a review of some recent results concerning the problem of constructing and studying time evolution for quantum systems with infinitely many degrees of freedom. The first rigorous results in this direction are due to O. E. Lanford and D. W. Robinson [17–20], see also [13], Ch. 5.3. An approach to this problem has been proposed by O. Bratteli and D. W. Robinson [12], see also [13], Ch. 6.3. Among recent papers we refer to [1–3], [10–11], [16], [24]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The language of operator algebras is of great help for the formulation of questions and answers in quantum statistical mechanics. In Chapter 1 we present a minimal mathematical introduction to operator algebras, with physical applications in mind. In Chapter 2 we study some questions related to the quantum statistical mechanics of spin systems, with particular attention to the time evolution of infinite systems. The basic reference for these two chapters is Bratteli-Robinson: Operator algebras and quantum statistical mechanics I, II. In Chapter 3 we discuss the nonequilibrium statistical mechanics of quantum spin systems, as it is currently being developped.
The language of this first section is quite heuristic, as it only aims at a broad presentation of the problems we want to turn our attention to. A rigorous formulation of some of these questions, and of their answers, is reviewed in section 2. A few illustrative models are mentioned briefly in Section 3. Section 4 is entirely devoted to the question raised by the very existence of crystals in a world where the fundamental interactions are believed to be covariant at least under the group of rigid motions in IR 3 . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In 1932, E.Wigner [1] introduced a phase-space method of computing physical observables in quantum (statistical) mechanics; some surveys have been published recently [2], [3]. See also a review article by Carruthers [4]. Here we give a brief synopsis of the method to make this paper self-contained, referring the reader to the references for details, including proofs. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.