Operator algebras in statistical mechanics and noncommutative probability theory
type: Item
instance of: scholarly article
title: Operator algebras in statistical mechanics and noncommutative probability theory
zbMATH Open document ID: 0619.46056
DOI: 10.1007/BF01103857
author: M. Sh. Gol'dshtejn
author: Vadim V. Anshelevich
published in: Journal of Soviet Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The paper is a survey of various results concerning some applications of the theory of operator algebras, especially in quantum mechanics and noncommutative probability theory. It is designed to be in a sense complementary to \textit{O. Bratteli}'s and \textit{D. W. Robinson}'s book ''Operator Algebras and Quantum Statistical Mechanics I, II'', (1979; Zbl 0421.46048) and (1981; Zbl 0463.46052), and includes mainly topics not dealt with in the book. Among these are: a description of the algebra of quasi-local observables for a quantum spin system, some properties of KMS-states, a connection between invariant states and first integrals, and first integrals of the multidimensional isotropic Heisenberg model. All this constitutes the more physically oriented part of the article. The other part is devoted to noncommutative probability theory on von Neumann algebras and presents noncommutative analogues of two classical theorems: the central limit theorem for random variables satisfying Rosenblatt's condition and the individual ergodic theorem. Somewhere between these two parts is placed a central limit theorem for the distribution of eigenvalues of the multi-particle Schrödinger operator. Most of the results in the paper have appeared before, however, some of them are presented with complete proofs and for some others explanatory comments and remarks are provided.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 81T05
zbMATH DE Number: 4004947
zbMATH Keywords: noncommutative probability theory
zbMATH Keywords: algebra of quasi-local observables for a quantum spin system
zbMATH Keywords: KMS-states
zbMATH Keywords: invariant states
zbMATH Keywords: first integrals of the multidimensional isotropic Heisenberg model
zbMATH Keywords: central limit theorem for random variables satisfying Rosenblatt's condition
zbMATH Keywords: individual ergodic theorem
zbMATH Keywords: distribution of eigenvalues of the multi-particle Schrödinger operator
reviewed by: Andrzej Łuczak
MaRDI profile type: Publication
cites work: Q3854185
cites work: Complex structures on a manifold with boundary
cites work: ON INVARIANT MEASURES OF DYNAMICAL SYSTEMS OF ONE-DIMENSIONAL STATISTICAL MECHANICS
cites work: Q5535438
cites work: Q4742751
cites work: Q4085018
cites work: Q5653193
cites work: SOME PROBLEMS OF NON-COMMUTATIVE ERGODIC THEORY
cites work: Description of differentiations in algebras of the type of locally observable spin system algebras
cites work: Q3895950
cites work: Multiple time analyticity of a quantum statistical state satisfying the {${\rm KMS}$} boundary condition
cites work: Gibbs states of a one dimensional quantum lattice
cites work: On the equivalence of the KMS condition and the variational principle for quantum lattice systems
cites work: On uniqueness of KmMS states of one-dimensional quantum lattice systems
cites work: On the XY-model on two-sided infinite chain
cites work: On the dynamics and ergodic properties of the XY model
cites work: On KMS boundary condition
cites work: Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas
cites work: Q3913911
cites work: Existence of phase transitions for quantum lattice systems
cites work: Stationary solutions of the Bogoliubov hierarchy equations in classical statistical mechanics. II
cites work: On the equilibrium states in quantum statistical mechanics
cites work: Ergodic theorems for convex sets and operator algebras
cites work: Statistica mechanics of quantum spin systems. III
cites work: Integral Representations of Invariant States on B* Algebras
cites work: Two soluble models of an antiferromagnetic chain
cites work: Q3319265
cites work: Quasi-Uniform Ergodic Theorems in von Neumann Algebras
cites work: Representations of uniformly hyperfinite algebras and their associated von Neumann rings
cites work: Statistical mechanics of quantum spin systems
cites work: Statistical mechanics of quantum spin systems. II
cites work: Return to equilibrium
cites work: Constraints imposed upon a state of a system that satisfies the K.M.S. boundary condition
cites work: Q3876492
cites work: Ergodic theorems for dynamical semi-groups on operator algebras
cites work: Ergodic Theorems for Semifinite Von Neumann Algebras-I
full work available at URL: https://doi.org/10.1007/bf01103857
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description: scientific article; zbMATH DE number 4004947