The English Flower Garden: Design, Arrangement and Plans Followed by a Description of All the Best Plants for it and Their Culture and the Positions Fitted for Them


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Dec 31, 1895
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In this chapter, and the following one, we examine various applications of C*-algebras and their states to statistical mechanics. Principally we analyze the structural properties of the equilibrium states of quantum systems con sisting of a large number of particles. In Chapter 1 we argued that this leads to the study of states of infinite-particle systems as an initial approximation. There are two approaches to this study which are to a large extent comple mentary. The first approach begins with the specific description of finite systems and their equilibrium states provided by quantum statistical mechanics. One then rephrases this description in an algebraic language which identifies the equilibrium states as states over a quasi-local C*-algebra generated by sub algebras corresponding to the observables of spatial subsystems. Finally, one attempts to calculate an approximation of these states by taking their limit as the volume of the system tends to infinity, the so-called thermodynamic limit. The infinite-volume equilibrium states obtained in this manner provide the data for the calculation of bulk properties of the matter under considera tion as functions of the thermodynamic variables. By this we mean properties such as the particle density, or specific heat, as functions of the temperature and chemical potential, etc. In fact, the infinite-volume data provides a much more detailed, even microscopic, description of the equilibrium phenomena although one is only generally interested in the bulk properties and their fluctuations.
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.
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