Semiclassical analysis of Dirac fields on curved spacetime

TL;DR

This paper develops a semiclassical framework for Dirac fields in curved spacetime, incorporating spin effects and electromagnetic interactions, leading to effective equations describing phenomena like the gravitational spin Hall effect.

Contribution

It introduces a Wentzel-Kramers-Brillouin approximation for Dirac fields on curved backgrounds, deriving spin-dependent ray equations and connecting them to the Mathisson-Papapetrou equations.

Findings

Derived effective ray equations with spin-dependent terms.

Showed the role of Berry connection and curvature in semiclassical dynamics.

Connected gravitational spin Hall effect to classical spinning particle equations.

Abstract

We present a semiclassical analysis for Dirac fields on an arbitrary spacetime background and in the presence of a fixed electromagnetic field. Our approach is based on a Wentzel-Kramers-Brillouin approximation, and the results are analyzed at leading and next-to-leading order in the small expansion parameter $\hbar$. Taking into account the spin-orbit coupling between the internal and external degrees of freedom of wave packets, we derive effective ray equations with spin-dependent terms. These equations describe the gravitational spin Hall effect of localized Dirac wave packets. We treat both massive and massless Dirac fields and show how a covariantly defined Berry connection and the associated Berry curvature govern the semiclassical dynamics. The gravitational spin Hall equations are shown to be particular cases of the Mathisson-Papapetrou equations for spinning objects.