Algebraic and Statistical Approach to Infinite Quantum Systems

A.K. Mishra, G. Rajasekaran, Taramani Madras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as "infinite", Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, "doubly-infinite" statistics, many representations of orthostatistics, Hubbard statistics and its variations. Key Words : Fock space -- multioscillator algebras -- q-deformation -- quantum groups -- general...

D. A. Slavnov

Nicolaas Marinus Hugenholtz

Infinite systems, particularly infinite quantum systems are considered. It is pointed out that algebraic methods can be applied not only to statistical mechanics but also to quantum field theory. Inequivalent representations of canonical anticommutation relations, states and observables, observable algebras for certain simple cases, the ground state, the equilibrium state at T and O, the Heisenberg model with long-range interaction are covered. (JFP)

Franco Strocchi

David Pierre Ruelle

The language of operator algebras is of great help for the formulation of questions and answers in quantum statistical mechanics. In Chapter 1 we present a minimal mathematical introduction to operator algebras, with physical applications in mind. In Chapter 2 we study some questions related to the quantum statistical mechanics of spin systems, with particular attention to the time evolution of infinite systems. The basic reference for these two chapters is Bratteli-Robinson: Operator algebras and quantum statistical mechanics I, II. In Chapter 3 we discuss the nonequilibrium statistical mechanics of quantum spin systems, as it is currently being developped.

Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk