Some remarks on quantum mechanics in a curved spacetime, especially for a Dirac particle

TL;DR

This paper clarifies the definition and gauge-dependence of the Hamiltonian and energy operators in quantum mechanics for Dirac particles in curved spacetime, emphasizing the importance of proper gauge choices.

Contribution

It provides detailed analysis of the Hamiltonian's gauge dependence and clarifies the conditions for well-defined energy in quantum mechanics within curved spacetime.

Findings

Hamiltonian operator depends on gauge choice in curved spacetime.

Proper gauge fixing is essential for meaningful energy definitions.

Explicit analysis for Dirac equation in Schwinger gauge.

Abstract

Some precisions are given about the definition of the Hamiltonian operator H and its transformation properties, for a linear wave equation in a general spacetime. In the presence of time-dependent unitary gauge transformations, H as an operator depends on the gauge choice. The other observables of QM and their rates also become gauge-dependent unless a proper account for the gauge choice is done in their definition. We show the explicit effect of these non-uniqueness issues in the case of the Dirac equation in a general spacetime with the Schwinger gauge. We show also in detail why, the meaning of the energy in QM being inherited from classical Hamiltonian mechanics, the energy operator and its mean values ought to be well defined in a general spacetime.