The Dirac oscillator, generalised parastatistics and colour Lie superalgebras

TL;DR

This paper explores the algebraic structures underlying the Dirac oscillator across various dimensions, revealing connections to colour Lie superalgebras and parastatistics, and demonstrating their utility in analyzing eigenspaces.

Contribution

It identifies and utilizes specific colour Lie superalgebras related to the Dirac oscillator, extending the algebraic framework in relativistic quantum mechanics.

Findings

Ladder operators realize specific $ ext{Z}_2 imes ext{Z}_2$-graded Lie superalgebras.

Colour Lie superalgebras are useful for analyzing Dirac oscillator eigenspaces.

The work extends algebraic analysis of relativistic oscillators in multiple dimensions.

Abstract

We study the Dirac oscillator in one, two and three spatial dimensions, showing that the corresponding ladder operators realise the $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded Lie superalgebras $ \mathfrak{pso}(3|2) $, $ \mathfrak{pso}(3|4) $ and $ \mathfrak{osp}_{01}(1|2) \oplus \mathfrak{sl}_{10}(1|1)$. These algebraic structures are related to parastatistics and their Fock spaces. We demonstrate that these colour algebras and Fock spaces are useful for analysing the Dirac oscillator and its eigenspaces, particularly in $ (1+3) $-dimensions. Apart from this current work, to our knowledge, the recent article by Ito and Nago (arXiv:2501.07311) is the only other such work that makes use of $ \mathbb{Z}_2\times \mathbb{Z}_2 $ graded colour Lie superalgebras in a relativistic setting.