The parity operator for parafermions and parabosons

TL;DR

This paper extends the algebraic framework of parafermions and parabosons by incorporating a parity operator, revealing connections to orthogonal Lie algebras and superalgebras and analyzing their Fock space representations.

Contribution

It introduces a new algebraic structure with a parity operator for parafermions and parabosons, linking them to orthogonal and orthosymplectic Lie (super)algebras.

Findings

Parafermions with P form the orthogonal Lie algebra so(2n+2).

Parabosons with P form the orthosymplectic Lie superalgebra osp(2|2n).

Spectrum of P relates to Green's order of statistics p.

Abstract

In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator $P$ which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of $n$ parafermions together with $P$ is the orthogonal Lie algebra $so(2n+2)$. The Fock spaces correspond to particular irreducible representations of $so(2n+2)$, and the action of $P$ in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of $n$ parabosons together with $P$ is the orthosymplectic Lie superalgebra $osp(2|2n)$. In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of $osp(2|2n)$. Both for parafermions and parabosons the spectrum of $P$ is closely related to the so-called order of statistics $p$, introduced by Green.