Covariant formalism for the Berry connection due to gravity

TL;DR

This paper develops a covariant formalism for Berry connection induced by gravity, extending previous electromagnetic results, and explores its implications for massless particles and effects like Aharonov-Bohm and Pancharatnam-Berry phases.

Contribution

It introduces a covariant framework for gravitational Berry connection using WKB approximation, applicable to massive and massless Dirac particles, and links it to known quantum effects.

Findings

Berry connection splits into two parts with distinct physical implications.

One part vanishes in spherically symmetric metrics, relating to Aharonov-Bohm effect.

The other part is associated with Pancharatnam-Berry phase.

Abstract

It is well-known that Dirac particles gain geometric phase, namely Berry phase, while moving in an electromagnetic field. Researchers have already shown covariant formalism for the Berry connection due to an electromagnetic field. A similar effect is expected to happen due to the presence of Gravity. We use WKB approximation to develop a covariant formalism of Berry-like connection in the presence of Einstein gravity, which can be further used to describe the Berry-like phase or simply Berry phase. We also extend this formalism for massless Dirac particles (Weyl particles).Then we further show that this connection can be split into two parts, one of which vanishes when the metric is spherically symmetric and thus can be linked to the Aharonov-Bohm-like effect in the 3 + 1 formalism. At the same time, the other term can be related to the Pancharatnam-Berry like effect.