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Hindustan Book Agency

Contributions in Mathematical Physics

S. Twareque Ali, Kalyan B. Sinha

Published Jan 1, 2007• 8 citations
Publisher

Abstract

Professor Gerard G. Emch has been one of the pioneers of the C-algebraic approach to quantum and classical statistical mechanics. In a prolific scientific career, spanning nearly five decades, Profess
DOI
10.1007/978-93-86279-33-0

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This unique volume provides a comprehensive overview of exactly solved models in statistical mechanics by looking at the scientific achievements of F Y Wu in this and related fields, which span four decades of his career. The book is organized into topics ranging from lattice models in condensed matter physics to graph theory in mathematics, and includes the author's pioneering contributions. Through insightful commentaries, the author presents an overview of each of the topics and an insider's look at how crucial developments emerged. With the inclusion of important pedagogical review article
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].
In 1972, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory” was published by Gerard Emch [8], giving the axioms for a physical system in an algebraic setting using the language of Irving Segal [19], and then continuing to obtain the C*-algebraic formalism for a physical system. Of these axioms, only the fifth contained an assumption that was questionable in its physical content. Bearing in mind that for each observable A and state φ , one obtains a distribution of values for the observed results of measurement, we have: Axiom 5: For any element A in the set of observables A\mathfrak{A}A and any non-negative integer n , there is at least one element, denoted A n , in A\mathfrak{A}A such that (i) the set of dispersion-free states for A n is contained in the set of dispersion-free states for A, (ii) < φ ; A n > = < φ ; A > n for all φ in the set of dispersion-free states for A . (Here < φ ; B > is the expectation of B in state φ .)