Post-Newtonian Hamiltonian description of an atom in a weak gravitational field

TL;DR

This paper develops a relativistic Hamiltonian framework for an atom in a weak gravitational field, providing a systematic derivation of gravity coupling and discussing foundational issues related to the Equivalence Principle in quantum mechanics.

Contribution

It extends previous Hamiltonian models to include weak post-Newtonian gravity using the parametrized post-Newtonian metric, offering a systematic derivation of gravitational coupling.

Findings

Derived a Hamiltonian including weak gravitational effects.

Clarified the implementation of gravity coupling in quantum systems.

Discussed implications for the Equivalence Principle in quantum mechanics.

Abstract

We extend the systematic calculation of an approximately relativistic Hamiltonian for centre of mass and internal dynamics of an electromagnetically bound two-particle system by Sonnleitner and Barnett [1] to the case including a weak post-Newtonian gravitational background field, described by the Eddington--Robertson parametrised post-Newtonian metric. Starting from a proper relativistic description of the situation, this approach allows to systematically derive the coupling of the model system to gravity, instead of `guessing' it by means of classical notions of relativistic effects. We embed this technical result into a critical discussion concerning the problem of implementing and interpreting general couplings to the gravitational field and the connected problem of how to properly address the question concerning the validity of the Equivalence Principle in Quantum Mechanics.