Breakdown of the Equivalence between Passive Gravitational Mass and Energy for a Quantum Body

TL;DR

This paper demonstrates that while the expectation values of passive gravitational mass and energy are equivalent macroscopically for stationary quantum states, their operators do not commute microscopically, leading to potential observable effects.

Contribution

It reveals a fundamental breakdown of the equivalence principle at the quantum operator level for hydrogen atoms in gravitational fields, with implications for quantum gravity tests.

Findings

Operators do not commute in the post-Newtonian approximation.

Expectation values remain equivalent for stationary states.

Possible detection via electromagnetic radiation emitted by moving atoms.

Abstract

It is shown that passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of the general relativity does not commute with its energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive at a macroscopic level for stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level for stationary quantum states can be experimentally detected by studying unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity using spacecraft or satellite.