This systematic algebraic approach concerns problems involving a large number of degrees of freedom. It extends the traditional formalism of quantum mechanics, and it eliminates conceptual and mathematical difficulties common to the development of statistical mechanics and quantum field theory. Further, the approach is linked to research in applied and pure mathematics, offering a reflection of the interplay between formulation of physical motivations and self-contained descriptions of the mathematical methods.
The four-part treatment begins with a survey of algebraic approaches to certain physical problems and the requisite tools. Succeeding chapters explore applications of the algebraic methods to representations of the CCR/CAR and quasi-local theories. Each chapter features an introduction that briefly describes specific motivations, mathematical methods, and results. Explicit proofs, chosen on the basis of their didactic value and importance in applications, appear throughout the text. An excellent text for advanced undergraduates and graduate students of mathematical physics, applied mathematics, statistical mechanics, and quantum theory of fields, this volume is also a valuable resource for theoretical chemists and biologists.
Gerard G Emch
Country of Publication:
Series: Dover Books on Physics
01 June 2009
GENERAL MOTIVATION 1. Why Not Stay in Fock Space? a. Quantum Mechanics b. Scattering Theory c. Fock Space d. The Relativistic, Free, Scalar-Meson Field e. A Prototype for Quantum Field Theory: the van Hove Model f. A Prototype for Statistical Mechanics: the BCS Model g. Outlook 2. The Emergence of the Algebraic Approach a. The Jordan Algebra of Observables in Traditional Quantum Mechanics b. Structure Axioms 1 to 5 (composition laws of observables) c. Structure Axiom 6 d. Structure Axioms 7 and 8 e. Proposition-Calculus f. Structure Axiom 9 and GNS Construction g. Structure Axiom 10 (uncertainty principle) GLOBAL THEORIES 1. Basic Facts About Representations a. Definition of a Representation b. Irreducible Representations and Pure States c. Examples d. Weak Topologies and Physical Equivalence of Representations e. von Neumann Algebras and Quasi-Equivalence of Representations f. Traces and Types g. S*-Algebras and Connections with Other Approaches 2. Symmetries and Symmetry Groups a. Definition of a Symmetry b. Symmetry Groups c. Amendable Groups d. Invariant and Extremal Invariant States, Asymptotic Abelianness e. The KMS Condition f. Decomposition Theory CANONICAL COMMUTATION AND ANTICOMMUTATION RELATIONS 1. Canonical Commutation Relations a. Properties of the Schrodinger Representation b. Uniqueness Theorems c. The C*-Algebra of the Canonical Commutation Relations d. Haag Theorem e. C*-Inductive Limit and IDPS f. Representations Associated to Product States 2. Canonical Anticommutation Relations QUASI-LOCAL THEORIES 1. General Theory of Local Systems a. Quasi-Local Algebras and Locally Normal States b. First Consequences of the Postulates 2. Some Simple Models of Statistical Mechanics a. Quantum Lattice Systems b. Free Quantum Gases BIBLIOGRAPHY INDEX