Hestenes' Tetrad and Spin Connections

TL;DR

This paper demonstrates that Hestenes' tetrad approach to spin connections in curved space-time can be reformulated as a Yang-Mills gauge theory, providing new insights into non-Riemannian geometries and form invariance of Dirac's equation.

Contribution

It introduces a Yang-Mills formulation of the Dirac Lagrangian using Hestenes' tetrad, extending previous Minkowski space results to Riemannian and non-Riemannian geometries.

Findings

Yang-Mills formulation of Dirac in curved space

Derivation of non-Riemannian spin connections from linear connections

Identification of spin connections preserving Dirac's form invariance

Abstract

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field is mapped to a set of Yang-Mills gauge potentials and a complex scalar field. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on Hestenes' tetrad and scalar fields. We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.