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

Abstract

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré 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.

30 pages (standard 12pt). v2: version accepted for publication in Int. J. Geom. Meth. Mod. Phys. Some emphasis and a clarification in Sect. 2.1. The Appendix now proves that the complex tangent bundle is a spinor bundle according to precisely the definition given in Sect. 2.1. Proof of the main Theorem 2 made easier to follow

References in corpus (2)

• Dirac-type equations in a gravitational field, with vector wave function
• General reference frames and their associated space manifolds

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