ELKO, flagpole and flag-dipole spinor fields, and the instanton Hopf fibration

TL;DR

This paper explores the mathematical structure of ELKO and flag-dipole spinor fields, their relation to instantons, and how they can be derived from Majorana and Weyl spinors, extending the understanding of spinor classifications.

Contribution

It demonstrates that the mapping between Dirac and ELKO spinors prevents ELKO from describing instantons and shows how various spinor types are interconnected through algebraic projections.

Findings

ELKO spinor fields cannot describe instantons due to their topological mapping.

Type-(4) and ELKO spinor fields can be derived from Majorana and Weyl spinors.

Explicit computation of bilinear covariants for ELKO and flag-dipole spinors.

Abstract

In a previous paper we explicitly constructed a mapping that leads Dirac spinor fields to the dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields). ELKO spinor fields are prime candidates for describing dark matter, and belong to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification, based on the relations and values taken by their associated bilinear covariants. Such a mapping between Dirac and ELKO spinor fields was obtained in an attempt to extend the Standard Model in order to encompass dark matter. Now we prove that such a mapping, analogous to the instanton Hopf fibration map $S^3... S^7\to S^4$, prevents ELKO to describe the instanton, giving a suitable physical interpretation to ELKO. We review ELKO spinor fields as type-(5) spinor fields under the Lounesto spinor field classification, explicitly computing the associated bilinear covariants. This paper is also devoted to investigate some formal aspects of the flag-dipole spinor fields, which correspond to the class-(4) under the Lounesto spinor field classification. In addition, we prove that type-(4) spinor fields (corresponding to flag-dipoles) and ELKO spinor fields (corresponding to flagpoles) can also be entirely described in terms of the Majorana and Weyl spinor fields. After all, by choosing a projection endomorphism of the spacetime algebra Cl(1,3) it is shown how to obtain ELKO, flagpole, Majorana and Weyl spinor fields, respectively corresponding to type-(5) and -(6) spinor fields, uniquely from limiting cases of a type-(4) (flag-dipole) spinor field, in a similar result obtained by Lounesto.