Quantum Mechanics and the Weak Equivalence Principle

TL;DR

This paper investigates how quantum mechanics interacts with the weak equivalence principle, showing that quantum effects can violate WEP but it is recovered in the classical limit, highlighting fundamental differences between classical and quantum gravity.

Contribution

It derives the Schrödinger equation in accelerated frames using Feynman path integrals and demonstrates quantum violations of WEP depending on mass and initial conditions.

Findings

Quantum wave functions depend on m/ħ, indicating WEP violation.

Probability density becomes mass-independent as ħ approaches zero.

Heavier particles do not necessarily fall faster than lighter ones at the quantum level.

Abstract

We use the Feynman path integral approach to nonrelativistic quantum mechanics twofold. First, we derive the lagrangian for a spinless particle moving in a uniformly but not necessarily constantly accelerated reference frame; then, applying the strong equivalence principle (SEP) we obtain the Schroedinger equation for a particle in an inertial frame and in the presence of a uniform and constant gravity field. Second, using the associated Feynman propagator, we propagate an initial gaussian wave packet, with the final wave function and probability density depending on the ratio m/hbar, where m is the inertial mass of the particle, thus exhibiting the fact that the weak equivalence principle (WEP) is violated by quantum mechanics. Although due to rapid oscillations the wave function does not exist in the classical limit, the probability density is well defined and mass independent when hbar goes to 0, showing the recovery of the WEP. Finally, at the quantum level, a heavier particle does not necessarily falls faster than a lighter one; this depends on the relations between the initial and final common positions and times of the particles.