The Bundles of Algebraic and Dirac-Hestenes Spinor Fields

The main objective of this paper is to clarify the ontology of DiracHestenes 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 (CllSpine 1,3 (M)) and right (ClrSpine 1,3 (M)) spin-Clifford bundles on the spin manifold (M, g). The relation between left ideal algebraic spinor fields (LIASF) and DiracHestenes spinor fields (both fields are sections of CllSpine 1,3 (M)) is clarified. We study in details the theory of covariant derivatives of Clifford fields as well as that of left and right spin-Clifford fields. A consistent Dirac equation for a DHSF Ψ ∈ sec CllSpine 1,3 (M) (denoted DECl) on a Lorentzian spacetime is found. We also obtain a representation of the DECl 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 CllSpine 1,3 (M) and Ξ ∈ secPSpine 1,3 (M), there is a well defined sum of even multivector fields ψΞ ∈ sec Cl(M, g) (EMFS) associated with Ψ. Such a 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 e-mail: [email protected] e-mail: [email protected] or [email protected]