Hermitian Dirac Hamiltonian in time dependent gravitational field

TL;DR

This paper demonstrates that the standard Dirac Hamiltonian in a time-dependent gravitational field is non-hermitian under covariant inner product, and proposes a hermitian alternative linked to the canonical field Hamiltonian.

Contribution

It introduces a hermitian Hamiltonian for Dirac particles in time-dependent gravitational fields, resolving non-hermiticity issues present in the standard formulation.

Findings

Standard Dirac Hamiltonian is non-hermitian in time-dependent metrics

A new hermitian Hamiltonian is derived

The new Hamiltonian aligns with quantum field theory concepts

Abstract

It is shown by a straightforward argument that the Hamiltonian generating the time evolution of the Dirac wave function in relativistic quantum mechanics is not hermitian with respect to the covariantly defined inner product whenever the background metric is time dependent. An alternative, hermitian, Hamiltonian is found and is shown to be directly related to the canonical field Hamiltonian used in quantum field theory.