Can Equivalence Principle be consistent with the Bohr-Somerfeld-Hansson Theory of the Newtonian Gravity

TL;DR

This paper examines the compatibility of the equivalence principle with the Bohr-Sommerfeld-Hansson quantum theory of Newtonian gravity, revealing that apparent violations are artifacts of classical interpretation and do not occur in the quantum framework.

Contribution

It demonstrates that the perceived breaking of the equivalence principle in the gravitational atom model is due to classical assumptions, not inherent to the quantum theory itself.

Findings

Centrifugal acceleration depends on mass in the classical interpretation.

No violation of the equivalence principle occurs within the quantum framework.

Classical-like interpretation of quantization leads to apparent paradoxes.

Abstract

In this work we consider some consequences of the Bohr-Sommerfeld-Hansson (Old or quasi-classical) quantum theory of the Newtonian gravity, i.e. of the "gravitational atom". We prove that in this case (for gravitational central force and quantized angular momentum) centrifugal acceleration becomes formally-theoretically dependent (proportional to fourth degree) of the mass of "gravitational electron" rotating around "gravitational nucleus" for any quantum number (state). It seemingly leads toward a paradoxical breaking of the relativistic equivalence principle which contradicts to real experimental data. We demonstrate that this equivalence principle breaking does not really appear in the (quasi classical) quantum theory, but that it necessary appears only in a hypothetical extension of the quantum theory that needs a classical like interpretation of the Bohr-Sommerfeld angular momentum quantization postulate. It is, in some sense, similar to Bell-Aspect analysis that points out that a hypothetical deterministic extension of the quantum mechanics, in distinction to usual quantum mechanics, can reproduce experimental data if and only if it is non-local (superluminal) in contradiction with relativistic locality (luminality) principle.