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The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF ) and its relationship with even multivector fields, on a Riemann-Cartan spacetime (RCST) M =(M, g, ∇, τg, ↑) admitting a spin structure, and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE ) in the case where M is a Lorentzian spacetime (the general case when M is a RCST will be discussed in another publication). To this aim we introduce the Clifford bundle of multivector fields (Cℓ(M, g)) and the left (Cℓ l We also obtain a representation of the DE Cℓ l in the Clifford bundle Cℓ(M, g). It is such equation that we call the DHE and it is satisfied by Clifford fields ψΞ ∈ sec Cℓ(M, g). This means that to each DHSF Ψ ∈ sec Cℓ l Spin e 1,3 (M ) and to each spin frame Ξ ∈ sec P Spin e 1,3 (M ), there is a well-defined sum of even multivector fields ψΞ ∈ sec Cℓ(M, g) (EMFS ) associated with Ψ. Such an EMFS is called a representative of the DHSF on the given spin frame. And, of course, such a EMFS (the representative of the DHSF ) is not a spinor field. With this crucial distinction between a DHSF and its representatives on the Clifford bundle, we provide a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DE Cℓ l and the DHE, although related, are equations of different mathematical natures. We study also the local Lorentz invariance and the electromagnetic gauge invariance and show that only for the DHE such transformations are of the same mathematical nature, thus suggesting a possible link between them. *

This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. We show moreover that algebraic constraints imply that any class 5 spinor field is such that the 2-component spinor fields entering its structure have opposite helicities. The proof of our statement is based on Lounesto general classification of all spinor fields, according to the relations and values taken by their associated bilinear covariants, and can eventually shed some new light on the algebraic investigations concerning dark matter.

Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF ) and Dirac-Hestenes spinor fields (DHSF ) on Minkowski spacetime as some equivalence classes of pairs (Ξu, ψ Ξu ), where Ξu is a spinorial frame field and ψ Ξu is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the 'bilinear covariants' (on Minkowski spacetime) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary RiemannCartan spacetimes. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of

We present the theory, the experimental evidence and fundamental physical consequences concerning the existence of families of undistorted progressive waves (UPWs) of arbitrary speeds 0 ≤ v < ∞, which are solutions of the homogeneous wave equation, Maxwell equations, Dirac, Weyl and KleinGordon equations.