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

Abstract

Goncalves et al. [Phys. Rev. D 75, 124023 (2007)] derived a nonrelativistic limit of the generalized 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 artifact of a flawed application of the exact Foldy-Wouthuysen transformation. In this paper we derive the correct nonrelativistic limit of the generalized 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.