The Dirac oscillator, generalised parastatistics and colour Lie superalgebras

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 $\mathfrak{p}\mathfrak{s}\mathfrak{o}(3\mathrm{\mid }2)$, $\mathfrak{p}\mathfrak{s}\mathfrak{o}(3\mathrm{\mid }4)$ and ${\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{01}(1\mathrm{\mid }2)\oplus {\mathfrak{s}\mathfrak{l}}_{10}(1\mathrm{\mid }1)$. 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.