Supersymmetry, exact Foldy-Wouthuysen transformation, and gravity

Abstract

Gravitational effects in relativistic quantum mechanics are investigated. The Dirac particle in general space-time metrics is considered and the Dirac Hamiltonian is constructed in terms of the Newman-Penrose formalism. To discuss the physical meaning of the Dirac Hamiltonian, it is necessary to perform the Foldy-Wouthuysen transformation. In most cases this transformation exists only in an approximate form. In this paper we show that for supersymmetric Dirac Hamiltonians not depending explicitly on time the exact Foldy-Wouthuysen transformation can always be constructed. Further, we derive criteria for spin coefficients for which the accompanying Dirac Hamiltonian is supersymmetric. These criteria are fulfilled by the class of static axisymmetric space-time metrics. For the subclass of stationary metrics, the exact Foldy-Wouthuysen transformation is calculated and the transformed Dirac Hamiltonian is derived. Recently, Obukhov constructed a different exact Foldy-Wouthuysen transformation for that class of space-time metrics and calculated the Dirac Hamiltonian in the Foldy-Wouthuysen representation. We show that the expansion series in orders of $1/m{c}^{2}$ of our and Obukhov's Dirac Hamiltonians coincide.