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 Z2×Z2-graded Lie superalgebras pso(3|2), pso(3|4) and osp01(1|2)⊕sl10(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.