ELKO, Flagpole and Flag-Dipole Spinor Fields, and the Instanton Hopf Fibration

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 → 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 Cℓ1,3 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.

The Theory of Local Mass Dimension One Fermions of Spin One Half

“…The Brazilian-Italian group of Roldão da Rocha, Hoff da Silva, Bueno Rogerio, Cavalcanti, Bonora, Fabbri, Silva-Neto, Bernardini, J. G. Pereira, and Coronado Villalobos, besides examining such important topics as Hawking radiation of mass dimension one fermions [73], continue to develop mathematical physics underlying Elko [73][74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89]. Of these we draw particular attention to mass-dimension transmuting operators considered in [75,77].…”

On the spinor representation

“…The multivector as expressed in Eq. (16) is a boomerang [16]. The condition Z 2 = 4σ Z yields Z 2 = 0, for singular spinors have the property that, namely, the Fierz aggregate is nilpotent.…”

An identification of the Dirac operator with the parity operator

“…Elko spinor fields have been investigated in the last decade [8][9][10][11][12][13][14][15][16][17][18] and has been further applied to gravity, cosmology, and field theory. [8][9][10][11][12][13][14][15][16][17][18] We recall that its starting point, in Ref.…”

“…[8][9][10][11][12][13][14][15][16][17][18] We recall that its starting point, in Ref. [2], is to take eigenspinors of the charge conjugation operator instead of the parity operator.…”