Nonrelativistic Limit of the Dirac-Schwarzschild Hamiltonian: Gravitational Zitterbewegung and Gravitational Spin-Orbit Coupling

TL;DR

This paper derives the nonrelativistic limit of the Dirac equation in a gravitational field, revealing gravitational zitterbewegung and spin-orbit effects, and confirms particle-antiparticle symmetry without spin-gravity coupling.

Contribution

It provides a detailed analysis of the gravitational corrections to the Dirac equation using Foldy-Wouthuysen transformation, highlighting physical interpretations and symmetry properties.

Findings

Identification of gravitational zitterbewegung effects.

Absence of direct spin-gravity coupling, preserving parity.

Verification of particle-antiparticle symmetry in gravitational corrections.

Abstract

We investigate the nonrelativistic limit of the gravitationally coupled Dirac Equation via a Foldy-Wouthuysen transformation. The relativistic correction terms have immediate and obvious physical interpretations in terms of a gravitational zitterbewegung, and a gravitational spin-orbit coupling. We find no direct coupling of the spin vector to the gravitational force, which would otherwise violate parity. The particle-antiparticle symmetry described recently by one of us in [Phys.Rev.A 87 (2013) 032101] is verified on the level of the perturbative corrections accessed by the Foldy-Wouthuysen transformation. The gravitational corrections to the electromagnetic transition current are calculated.