Quantum mechanics for three versions of the Dirac equation in a curved spacetime

TL;DR

This paper explores three formulations of the Dirac equation in curved spacetime, analyzing their mathematical properties, conservation laws, and hermiticity conditions to deepen understanding of quantum mechanics in gravitational fields.

Contribution

It introduces two alternative tensor-based versions of the Dirac equation in curved spacetime and examines their hermiticity and conservation properties compared to the standard form.

Findings

Probability current conservation leads to specific conditions on coefficient fields.

Hermiticity of the Dirac Hamiltonian varies between formulations.

Standard Dirac equation's hermiticity is not invariant under all coefficient transformations.

Abstract

We present a recent work on the Dirac equation in a curved spacetime. In addition to the standard equation, two alternative versions are considered, derived from wave mechanics, and based on the tensor representation of the Dirac field. The latter considers the Dirac wave function as a spacetime vector and the set of the Dirac matrices as a third-order tensor. Having the probability current conserved for any solution of the Dirac equation gives an equation to be satisfied by the coefficient fields. A positive definite scalar product is defined and a hermiticity condition for the Dirac Hamiltonian is derived for a general coordinate system in a general curved spacetime. For the standard equation, the hermiticity of the Dirac Hamiltonian is not preserved under all admissible changes of the coefficient fields.