Non-Relativistic Approximation of the Dirac Equation for Slow Fermions in Static Metric Spacetimes

TL;DR

This paper derives a general non-relativistic Hamiltonian for slow fermions in static spacetimes, incorporating effects of Earth's gravity and chameleon fields, using a Foldy-Wouthuysen transformation.

Contribution

It presents a comprehensive derivation of the effective gravitational potential for slow fermions in static metrics, including torsion and chameleon field effects, using a standard Foldy-Wouthuysen approach.

Findings

Derived the most general effective gravitational potential for slow fermions.

Included effects of torsion and chameleon fields in the non-relativistic limit.

Provided a Hamiltonian suitable for experimental tests of gravity and new fields.

Abstract

We analyse the non-relativistic approximation of the Dirac equation for slow fermions moving in spacetimes with a static metric, caused by the weak gravitational field of the Earth and a chameleon field, and derive the most general effective gravitational potential, induced by a static metric of spacetime. The derivation of the non-relativistic Hamilton operator of the Dirac equation is carried out by using a standard Foldy-Wouthuysen (SFW) transformation. We discuss the chameleon field as source of a torsion field and torsion-matter interactions.