Basic quantum mechanics for three Dirac equations in a curved spacetime

TL;DR

This paper extends quantum mechanics for Dirac equations in curved spacetime, analyzing different formulations and conditions for current conservation, Hermiticity, and the uniqueness of the Hilbert space scalar product.

Contribution

It provides a comprehensive analysis of Dirac equations in curved spacetime, including three formulations, and establishes conditions for current conservation and Hamiltonian Hermiticity.

Findings

Current conservation applies iff the Dirac matrices satisfy a specific PDE.

The Hilbert space scalar product is uniquely determined by quantum axioms in a given frame.

Hermiticity of the Dirac Hamiltonian depends on the choice of gamma matrices.

Abstract

We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, iff the field of Dirac matrices $\gamma ^\mu $ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the $\gamma ^\mu $ matrices. It similarly restricts the choice of the $\gamma ^\mu $ matrices for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. We show that in any given reference frame with minor restrictions on the spacetime metric, the axioms of quantum mechanics impose a unique form for the Hilbert space scalar product. Finally, the condition for the general Dirac Hamiltonian operator to be Hermitian is derived in a general curved spacetime. For DFW, the validity of this hermiticity condition depends on the choice of the $\gamma ^\mu $ matrices.