Non-Relativistic Approximation of Dirac Equation for Slow Fermions Coupled to the Chameleon and Torsion Fields in the Gravitational Field of the Earth

TL;DR

This paper derives a non-relativistic approximation of the Dirac equation for slow fermions in Earth's gravitational field, including effects of chameleon and torsion fields, using Foldy–Wouthuysen transformations.

Contribution

It introduces a novel low-energy effective potential for fermions coupled to gravity, chameleon, and torsion fields in Earth's weak gravitational field.

Findings

Derived effective gravitational potentials for slow fermions

Included effects of chameleon and torsion fields in the approximation

Extended previous models to incorporate non-minimal couplings

Abstract

We analyse a non-relativistic approximation of the Dirac equation for slow fermions, coupled to the chameleon field and torsion in the spacetime with the Schwarzschild metric, taken in the weak gravitational field of the Earth approximation. We follow the analysis of the Dirac equation in the curved spacetime with torsion, proposed by Kostelecky (Phys. Rev. D69, 105009 (2004)), and apply the Foldy--Wouthuysen transformations. We derive the effective low-energy gravitational potentials for slow fermions, coupled to the gravitational field of the Earth, the chameleon field and to torsion with minimal and non-minimal couplings.