Magnetic Like Particles and Elko Spinor Fields

Abstract

This chapter scrutinizes the theory of the so-called Elko spinor fields (in Minkowski spacetime) which always appears in pairs and which from the algebraic point of view are in class five in Lounesto classification of spinor fields. We show how these fields differs from Majorana fields (which also are in class five in Lounesto classification of spinor fields) and that Elko spinor fields (as it is the case for Majorana fields) do not satisfy the Dirac equation. We discuss the class of generalized Majorana spinor fields (objects that are spinor with “components” with take values in a Grassmann algebra) that satisfy Dirac equations, clarifying some obscure presentations of that theory appearing in the literature. More important, we show that the original presentation of the theory of Elko spinor fields as having mass dimension 1 leads to breakdown of Lorentz and rotational invariance by a simple choice of the spatial axes in an inertial reference frame. We then present a Lagrangian field theory for Elko spinor fields where these fields (as it is the case of Dirac spinor fields) have mass dimension 3∕2. We explicitly demonstrate that Elko spinor fields cannot couple to the electromagnetic field, that they describe pairs of “magnetic” like particles which are coupled to a short range su(2) gauge potential. Thus they eventually can serve to model dark matter. The causal propagator for the 3∕2 mass dimension Elko spinor is explicitly calculated with the Clifford bundle of (multivector) fields. Taking the opportunity given by the formalism developed in our theory we present a very nice representation of the parity operator acting on Dirac-Hestenes spinor fields.