On the geometry and quantum theory of regular and singular spinors

Abstract: We relate the Lounesto classification of regular and singular spinors to the orbits of the $Spin(3,1)$ 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 $C$. 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.