Hermiticity of the Dirac Hamiltonian in Curved Spacetime

TL;DR

This paper investigates the Hermiticity of the Dirac Hamiltonian in curved spacetime, revealing that non-Hermiticity arises from time-dependent position eigenstates and providing a corrected Hermitian form for general metrics.

Contribution

It derives a Hermitian Dirac Hamiltonian in curved spacetime accounting for non-negligible metric time dependence, resolving a paradox in previous formulations.

Findings

Non-Hermiticity is due to time-dependent position eigenstates.

A corrected Hermitian Hamiltonian expression is provided.

The work clarifies the conditions for Hermiticity in curved spacetime quantum mechanics.

Abstract

In previous work on the quantum mechanics of an atom freely falling in a general curved background spacetime, the metric was taken to be sufficiently slowly varying on time scales relevant to atomic transitions that time derivatives of the metric in the vicinity of the atom could be neglected. However, when the time-dependence of the metric cannot be neglected, it was shown that the Hamiltonian used there was not Hermitian with respect to the conserved scalar product. This Hamiltonian was obtained directly from the Dirac equation in curved spacetime. This raises the paradox of how it is possible for this Hamiltonian to be non-hermitian. Here, we show that this non-hermiticity results from a time dependence of the position eigenstates that enter into the Schrodinger wave function, and we write the expression for the Hamiltonian that is Hermitian for a general metric when the time-dependence of the metric is not neglected.