Structure of Noncommutative Fock space

TL;DR

This paper explores the structure of Fock space for bosons in noncommutative phase spaces, revealing unique properties like non-commuting creation and annihilation operators and alternative basis constructions.

Contribution

It introduces the concept of noncommutative Fock space with non-commuting operators and constructs explicit coherent states within this framework.

Findings

Noncommutative Fock space differs from traditional Fock space.

Eigenvectors of commuting Hermitian operators form a basis.

Explicit two-dimensional coherent state is constructed.

Abstract

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose-Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which lead to noncommutative Fock space. By this we mean that creation and annihilation operators corresponding to different degrees of freedom of the bosons do not commute each other. The main character of the noncommutative Fock space is there are no ordinary number representations because of the non-commutativity between different number operators. However, eigenvectors of several pairs of commuting Hermitian operators are obtained which can also be served as bases in this Fock space. As a simple example, an explicit form of two-dimensional canonical coherent state in this noncommutative Fock space is constructed and its properties are discussed.