Representations of the Dirac wave function in a curved spacetime

TL;DR

This paper explores different mathematical representations of the Dirac wave function in curved spacetime, establishing their equivalences and relations through a unified geometric framework.

Contribution

It introduces a common geometric framework for scalar and vector representations of the Dirac wave function and proves their equivalence under various connection choices.

Findings

Scalar and vector representations are mathematically equivalent in curved spacetime.

The standard Dirac equation with any tetrad choice is equivalent to a vector wave function formulation.

The paper provides theorems relating different representations and connection choices.

Abstract

The Dirac wave function in a curved spacetime is usually defined as a quadruplet of scalar fields. It can alternatively be defined as a four-vector field. We describe these two representations in a common geometrical framework and we prove theorems that relate together the different representations and the different choices of connections. In particular, the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the Dirac equation for a vector wave function, in the same spacetime.