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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
This systematic algebraic approach covers statistical mechanics, quantum field theory, and mathematical physics. In addition to a careful formulation of the problems' physical motivations, it features self-contained descriptions of the mathematical methods for arriving at solutions. Readers thus learn the vocabulary and physical bases of the theory. 1972 edition.
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