We relate the Lounesto classification of regular and singular spinors to the orbits of the group in the space of Dirac spinors. We find that regular spinors are associated with the principal orbits of the spin group while singular spinors are associated with special orbits whose isotropy group is . We use this to clarify some aspects of the classical and quantum theory of spinors restricted to a class in this classification. In particular, we show that the degrees of freedom of an ELKO field, which has been proposed as a candidate for dark matter, can be reexpressed as a Dirac field preserving locality. Alternatively after introducing the ELKO dual, it can be re-interpreted as four anticommuting Lorentz scalar fields with internal symmetry the spin representation of the Lorentz group. We also propose an interacting Lagrangian which can consistently describe all 6 classes of regular and singular spinors.

中文翻译：

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论正则旋量和奇异旋量的几何和量子理论


我们将规则旋量和奇异旋量的 Lounesto 分类与狄拉克旋量空间中的群轨道联系起来。我们发现规则旋量与自旋群的主轨道相关，而奇异旋量与各向同性群为 的特殊轨道相关。我们用它来澄清仅限于该分类中的一类的旋量的经典和量子理论的某些方面。特别是，我们证明了 ELKO 场的自由度（被提议作为暗物质的候选者）可以重新表示为保留局域性的狄拉克场。或者，在引入 ELKO 对偶之后，可以将其重新解释为具有内部对称性的四个反交换洛伦兹标量场（洛伦兹群的自旋表示）。我们还提出了一个相互作用的拉格朗日量，它可以一致地描述所有 6 类正则和奇异旋量。