A Klein operator for paraparticles

TL;DR

This paper demonstrates how two distinct algebraic structures for paraparticles, known as relative parafermion and paraboson types, are interconnected through a Klein transformation, clarifying their relationship.

Contribution

It reveals the connection between two fundamental types of paraparticle commutation relations via a Klein transformation.

Findings

The two types of relative commutation relations are related by a Klein transformation.

This connection clarifies the algebraic structure of paraparticles.

The work provides a unified view of parafermions and parabosons.

Abstract

It has been known for a long time that there are two non-trivial possibilities for the relative commutation relations between a set of $m$ parafermions and a set of $n$ parabosons. These two choices are known as ``relative parafermion type'' and ``relative paraboson type'', and correspond to quite different underlying algebraic structures. In this short note we show how the two possibilities are related by a so-called Klein transformation.