Four-vector vs. four-scalar representation of the Dirac wave function

TL;DR

This paper compares scalar and vector representations of the Dirac wave function in curved spacetime, establishing a unified framework and showing their equivalence in describing covariant Dirac equations.

Contribution

It introduces a unified framework using vector bundle theory to relate scalar and vector representations of the Dirac wave function in curved spacetime.

Findings

Both representations can describe a variety of covariant Dirac equations.

The standard Dirac equation in curved spacetime is equivalent to a covariant four-vector formulation.

Abstract

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincar\'e group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.