Operator Algebras and Quantum Statistical Mechanics 1

C- and W-Algebras Symmetry Groups Decomposition of States

verfasst von: Professor Ola Bratteli, Professor Derek W. Robinson

Verlag: Springer Berlin Heidelberg

Buchreihe : Texts and Monographs in Physics

Enthalten in: Professional Book Archive

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Über dieses Buch

In this book we describe the elementary theory of operator algebras and parts of the advanced theory which are of relevance, or potentially of relevance, to mathematical physics. Subsequently we describe various applications to quantum statistical mechanics. At the outset of this project we intended to cover this material in one volume but in the course of develop ment it was realized that this would entail the omission ofvarious interesting topics or details. Consequently the book was split into two volumes, the first devoted to the general theory of operator algebras and the second to the applications. This splitting into theory and applications is conventional but somewhat arbitrary. In the last 15-20 years mathematical physicists have realized the importance of operator algebras and their states and automorphisms for problems of field theory and statistical mechanics. But the theory of 20 years aga was largely developed for the analysis of group representations and it was inadequate for many physical applications. Thus after a short honey moon period in which the new found tools of the extant theory were applied to the most amenable problems a longer and more interesting period ensued in which mathematical physicists were forced to redevelop the theory in relevant directions. New concepts were introduced, e. g. asymptotic abelian ness and KMS states, new techniques applied, e. g. the Choquet theory of barycentric decomposition for states, and new structural results obtained, e. g. the existence of a continuum of nonisomorphic type-three factors.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

The theory of algebra of operators on Hilbert space began in the 1930s with a series of papers by von Neumann, and Murray and von Neumann. The principal motivations of these authors were the theory of unitary group representations and certain aspects of the quantum mechanical formalism. They analyzed in great detail the structure of a family of algebras which are referred to nowadays as von Neumann algebras, or W*-algebras. These algebras have the distinctive property of being closed in the weak operator topology and it was not until 1943 that Gelfand and Naimark characterized and partially analyzed uniformly closed operator algebras, the so-called C*-algebras. Despite Murray and von Neumann’s announced motivations the theory of operator algebras had no significant application to group representations for more than fifteen years and its relevance to quantum mechanical theory was not fully appreciated for more than twenty years. Despite this lapse there has been a subsequent fruitful period of interplay between mathematics and physics which has instigated both interesting structural analysis of operator algebras and significant physical applications, notably to quantum statistical mechanics. We intend to describe this theory and these applications. Although these results have also stimulated further important applications of algebraic theory to group representations and relativistic field theory we will only consider these aspects peripherally.

Ola Bratteli, Derek W. Robinson

C*-Algebras and von Neumann Algebras

Abstract

C*-algebra theory is an abstraction of the structure of certain algebras of bounded operators acting on a Hilbert space and is simultaneously a special case of the theory of Banach algebras. Consequently, the theory can be developed in two different ways. Either one can begin with an abstract description suited to the general analysis of Banach algebras or one may start with a specific representation of the algebra on a Hilbert space. We will follow the first of these approaches.

Ola Bratteli, Derek W. Robinson

Groups, Semigroups, and Generators

Abstract

Physical theories consist essentially of two elements, a kinematical structure describing the instantaneous states and observables of the system, and a dynamical rule describing the change of these states and observables with time. In the classical mechanics of point particles a state is represented by a point in a differentiable manifold and the observables by functions over the manifold. In the quantum mechanics of systems with a finite number of degrees of freedom the states are given by rays in a Hilbert space and the observables by operators acting on the space. For particle systems with an infinite number of degrees of freedom we intend to identify the states with states over appropriate algebras of fields, or operators. In each of these examples the dynamical description of the system is given by a flow, a one-parameter group of automorphisms of the underlying kinematical structure, which represents the motion of the system with time. In classical mechanics one has a group of diffeomorphisms, in quantum mechanics a group of unitary operators on the Hilbert space, and for systems with an infinite number of degrees of freedom a group of automorphisms of the algebra of observables. It is also conventional to describe other symmetries of physical systems by groups of automorphisms of the basic kinematic structure and in this chapter, and Chapter 4, we study various aspects of this group-theoretic description.

Ola Bratteli, Derek W. Robinson

Decomposition Theory

Abstract

The aim of decomposition theory is to express a complex structure as a superposition of simpler components. There is no general rule for what is meant by simpler component and this is determined by the particular application. In an algebraic setting it is usual to examine two complementary forms of decomposition, the decomposition of states and the decomposition of representations. In this chapter we principally describe the theory relating to states, but the intimate connection between states and representations allows us to develop and exploit properties of the representations.

Ola Bratteli, Derek W. Robinson

Backmatter

Titel: Operator Algebras and Quantum Statistical Mechanics 1
verfasst von: Professor Ola Bratteli
Professor Derek W. Robinson
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-02520-8
Print ISBN: 978-3-642-05736-6
DOI: https://doi.org/10.1007/978-3-662-02520-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Introduction

C*-Algebras and von Neumann Algebras

Groups, Semigroups, and Generators

Decomposition Theory

Backmatter