Generalized Fock Spaces and New Forms of Quantum Statistics

TL;DR

This paper develops a comprehensive framework for generalized Fock spaces that unifies existing quantum statistics and introduces new types, enhancing understanding of particle interactions and algebraic structures in quantum systems.

Contribution

It formulates a three-tiered formalism for generalized Fock spaces, unifying various quantum statistics and enabling the creation of new statistical models and algebras.

Findings

Unified framework for various quantum statistics.

Construction of new quantum statistics models.

Application to particles with singular interactions.

Abstract

The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of statistics, and establish interconnections among them whenever it is possible. We formulate a theory of generalized Fock spaces, which has a three tired structure consisting of Fock space, statistics and algebra. This general formalism unifies various forms of statistics and algebras, which were earlier considered to describe different systems. Besides, the formalism allows us to construct many new kinds of quantum statistics and the associated algebras of creation and destruction operators. Some of these are: orthostatistics, null statistics or statistics of frozen order, quantum group based statistics and its many avatars, and `doubly-infinite' statistics. The emergence of new forms of quantum statistics for particles interacting with singular potential is also highlighted.