The Bundles of Algebraic and Dirac-Hestenes Spinors Fields

The main objective of this paper is to clarify the ontology of DiracHestenes spinor fields (DHSF ) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime M=(M, g,∇, τg, ↑) admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE ) when M is a Lorentzian spacetime. 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 Dirac-Hestenes spinor fields (both fields are sections of CllSpine 1,3 (M)) is clarified. We study in details the theory of the covariant derivatives of Clifford and left and right spin-Clifford fields. Moreover, we find (for the first time) a consistent Dirac equation for a DHSF Ψ ∈ sec CllSpine 1,3 (M) (denoted DECl) on a Lorentzian spacetime. We succeeded also in obtaining 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 and its representatives on the Clifford bundle we provide a consistent theory for the covariant derivatives of Clifford e-mail: [email protected] e-mail: [email protected] or [email protected]