Elko Spinor Fields and Massive Magnetic Like Monopoles

Abstract

In this paper we recall that by construction Elko spinor fields of λ and ρ types satisfy a coupled system of first order partial differential equations (csfopde) that once interacted leads to Klein-Gordon equations for the λ and ρ type fields. Since the csfopde is the basic one and since the Klein-Gordon equations for λ and ρ possess solutions that are not solutions of the csfopde for λ and ρ we infer that it is legitimate to attribute to those fields mass dimension 3/2 (as is the case of Dirac spinor fields) and not mass dimension 1 as previously suggested in recent literature (see list of references). A proof of this fact is offered by deriving the csfopde for the λ and ρ from a Lagrangian where these fields have indeed mass dimension 3/2. Taking seriously the view that Elko spinor fields due to its special properties given by their bilinear invariants may be the description of some kind of particles in the real world a question then arises: what is the physical meaning of these fields? Here we proposed that the fields λ and ρ serve the purpose of building the fields $\mathcal {K}\in \mathcal {C}\ell ^{0} (M,\eta )\otimes {\mathbb {R}}_{1,3}^{0}$ and $\mathcal {M}\in \sec \mathcal {C}\ell ^{0}(M,\eta ) \otimes {\mathbb {R}}_{1,3}^{0}$ (see (37)). These fields are electrically neutral but carry magnetic like charges which permit them to couple to a $su(2)\simeq {\textit {spin}}_{3,0}\subset {\mathbb {R}}_{3,0}^{0}$ valued potential $\mathcal {A}\in \sec {\textstyle \bigwedge \nolimits ^{1}} T^{\ast }M\otimes {\mathbb {R}}_{3,0}^{0}$ . If the field $\mathcal {A}$ is of short range the particles described by the $\mathcal {K}$ and $\mathcal {M}$ fields may be interacting and forming condensates of zero spin particles analogous to dark matter, in the sense that they do not couple with the electromagnetic field (generated by charged particles) and are thus invisible. Also, since according to our view the Elko spinor fields as well as the $\mathcal {K}$ and $\mathcal {M}$ fields are of mass dimension 3/2 we show how to calculate the correct propagators for the $\mathcal {K}$ and $\mathcal {M}$ fields. We discuss also the main difference between Elko and Majorana spinor fields, which are kindred since both belong to class five in Lounesto classification of spinor fields. Most of our presentation uses the representation of spinor fields in the Clifford bundle formalism, which makes very clear the meaning of all calculations.