On bound states of Dirac particles in gravitational fields

TL;DR

This paper analyzes the quantum bound states of Dirac particles in curved spacetime geometries, extending non-relativistic results and exploring magnetic field effects, with implications for experiments like GRANIT.

Contribution

It provides analytical solutions for Dirac particles in various curved spacetimes, incorporating boundary conditions, spin effects, and magnetic fields, extending previous Schrödinger-based results.

Findings

Bound states depend on geometry and boundary conditions.

Relativistic corrections influence energy spectra and spin contributions.

Weak magnetic fields could produce observable effects in experiments.

Abstract

We investigate the quantum motion of a neutral Dirac particle bouncing on a mirror in curved spacetime. We consider different geometries: Rindler, Kasner-Taub and Schwarzschild, and show how to solve the Dirac equation by using geometrical methods. We discuss, in a first-quantized framework, the implementation of appropriate boundary conditions. This leads us to consider a Robin boundary condition that gives the quantization of the energy, the existence of bound states and of critical heights at which the Dirac particle bounces, extending the well-known results established from the Schrodinger equation. We also allow for a nonminimal coupling to a weak magnetic field. The problem is solved in an analytical way on the Rindler spacetime. In the other cases, we compute the energy spectrum up to the first relativistic corrections, exhibiting the contributions brought by both the geometry and the spin. These calculations are done in two different ways. On the one hand, using a relativistic expansion and, on the other hand, with Foldy-Wouthuysen transformations. Contrary to what is sometimes claimed in the literature, both methods are in agreement, as expected. Finally, we make contact with the GRANIT experiment. Relativistic effects and effects that go beyond the equivalence principle escape the sensitivity of such an experiment. However, we show that the influence of a weak magnetic field could lead to observable phenomena.