Generalized Quantum Statistics and Lie (Super)Algebras

TL;DR

This paper explores how generalized quantum statistics relate to Lie (super)algebras, specifically type B, and examines their physical properties and implications for quantum mechanics.

Contribution

It establishes a connection between generalized quantum statistics and Lie (super)algebras of type B, including examples involving the superalgebra B(1|1)=osp(3|2).

Findings

Fock spaces correspond to irreducible representations of Lie (super)algebras.

Generalized statistics exhibit noncommutative coordinates.

Links between quantum equations of motion and commutation relations are discussed.

Abstract

Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1)=osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?