Foldy-Wouthuysen transformation of the generalised Dirac Hamiltonian in a gravitational-wave background

TL;DR

This paper corrects previous claims about spin-precession caused by gravitational waves by applying proper gauge-invariant transformations, showing that the effect is an artefact and does not physically occur.

Contribution

It provides the correct non-relativistic limit of the Dirac Hamiltonian in gravitational waves using both transformations, clarifying the non-existence of the proposed spin-precession effect.

Findings

Previous spin-precession effect was a gauge artefact.

Correct transformations yield gauge-invariant Hamiltonian.

Spin-precession effect does not occur in physical reality.

Abstract

Goncalves et al. derived a non-relativistic limit of the generalised Dirac Hamiltonian in the presence of a gravitational wave, using the exact Foldy-Wouthuysen transformation. This gave rise to the intriguing notion that spin-precession may occur even in the absence of a magnetic field. We argue that this effect is not physical as it is the result of a gauge-variant term that was an artefact of a flawed application of the exact Foldy-Wouthuysen transformation. In this paper we derive the correct non-relativistic limit of the generalised Dirac Hamiltonian in the presence of a gravitational wave, using both the exact and standard Foldy-Wouthuysen transformation. We show that both transformations consistently produce a Hamiltonian where all terms are gauge-invariant. Unfortunately however, we also show that this means the novel spin-precession effect does not exist.