Equality of the Inertial and the Gravitational Masses for a Quantum Particle

TL;DR

This paper proves that the inertial and gravitational masses are equal for a non-relativistic quantum particle by deriving the wave equation in curved spacetime from broad quantum assumptions and covariance, strengthening previous results.

Contribution

It provides a new proof of mass equality in quantum mechanics without relying on the equivalence principle or classical equations of motion.

Findings

Inertial and gravitational masses are equal in the derived wave equation.

The proof does not depend on classical motion assumptions.

Strengthens previous theoretical results on mass equivalence in quantum particles.

Abstract

We investigate the interaction of the gravitational field with a quantum particle. We derive the wave equation in the curved galilean spacetime from the very broad Quantum mechanical assumptions and from covariance under the Milne group. The inertial and gravitational masses are equal in that equation. So, we give the proof of the equality for the non-relativistic quantum particle, without applying the equivalence principle to the Schr\"odinger equation and witout imposing any relation to the classical equations of motion. This result constitutes a substantial strengthening of the previous result obtained by Herdegen and the author.