The Bundles of Algebraic and Dirac-Hestenes Spinor Fields

TL;DR

This paper clarifies the mathematical structure of Dirac-Hestenes spinor fields on Riemann-Cartan spacetimes, deriving a consistent Dirac equation and exploring their relation to multivector fields and gauge invariances.

Contribution

It provides a rigorous geometric framework for Dirac-Hestenes spinor fields, deriving the Dirac-Hestenes equation and analyzing their covariant derivatives and invariance properties.

Findings

Derived a consistent Dirac equation for DHSF on Lorentzian spacetime.

Represented the Dirac-Hestenes equation within the Clifford bundle framework.

Identified the gauge invariance properties of the Dirac-Hestenes equation.

Abstract

The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) when spacetime is a Lorentzian. To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (Cl_{Spin_{1,3}^{e}}^{l}(M)) and right (Cl_{Spin_{1,3}^{e}}^{r}(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 Cl_{Spin_{1,3}^{e}}^{l}(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 a consistent Dirac equation for a DHSF (denoted DECl^{l}) on a Lorentzian spacetime. We succeeded also in obtaining a representation of the DECl^{l} in the Clifford bundle Cl(M,g). It is such equation that we call the DHE and it is satisfied by even multivector fields (EMFS). Of course, such a EMFS is not a spinor field. We provide moreover a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DECl^{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.