Generalized Fock Spaces, New Forms of Quantum Statistics and their Algebras

TL;DR

This paper develops a unified theoretical framework for various quantum statistics using generalized Fock spaces, enabling the construction of new quantum statistics and operator algebras.

Contribution

It introduces a comprehensive formalism that unifies existing quantum statistics and creates new types of quantum statistics and operator algebras.

Findings

Unified formalism for quantum statistics and algebras

Construction of new quantum statistics models

Development of novel creation and annihilation operator algebras

Abstract

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.