标题： 狄拉克振子，广义拟统计和颜色李超代数 显示英文标题

标题： The Dirac oscillator, generalised parastatistics and colour Lie superalgebras

摘要： 我们研究了一维、二维和三维空间中的狄拉克振子，表明相应的升降算符实现了$ \mathbb{Z}_2\times\mathbb{Z}_2 $-分次李超代数$ \mathfrak{pso}(3|2) $、$ \mathfrak{pso}(3|4) $和$ \mathfrak{osp}_{01}(1|2) \oplus \mathfrak{sl}_{10}(1|1)$。 这些代数结构与分数统计及其福克空间相关。 我们证明这些颜色代数和福克空间对于分析狄拉克振子及其本征空间是有用的，特别是在$ (1+3) $-维情况下。 除了这项工作外，据我们所知，最近由Ito和Nago（arXiv:2501.07311）发表的文章是唯一另一篇在相对论设置中使用$ \mathbb{Z}_2\times \mathbb{Z}_2 $分次颜色李超代数的工作。 显示英文摘要

摘要： 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.