The parity operator for parafermions and parabosons

• URL: http://arxiv.org/abs/2604.12393v1
• Date: Tue, 14 Apr 2026 07:27:50 GMT
• Title: The parity operator for parafermions and parabosons
• Abstract summary: We reexamine the definition of parafermions and parabosons by means of Green's triple relations.<n>We extend these relations by including a parity operator $P$ which is also determined by means of triple relations.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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