Dirac-Hestenes spinor fields in Riemann-Cartan spacetime

TL;DR

This paper develops a rigorous mathematical framework for Dirac-Hestenes spinor fields in Riemann-Cartan spacetimes, establishing their relation to other spinor types and generalizing the Dirac equation beyond Minkowski space.

Contribution

It introduces the Spin-Clifford bundle, clarifies the equivalence of different spinor fields, and generalizes the Dirac-Hestenes equation to Riemann-Cartan spacetimes.

Findings

Defined Dirac-Hestenes spinor fields as sections of the Spin-Clifford bundle.

Proved the equivalence of various spinor field types in Riemann-Cartan spacetime.

Generalized the Dirac-Hestenes equation to curved spacetime with torsion.

Abstract

In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-K\"ahler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.