A New Perspective on Path Integral Quantum Mechanics in Curved Space-Time

TL;DR

This paper introduces a novel path integral quantum mechanics approach in curved space-time, utilizing local tangent space rotations and Lie transport, revealing new quantum effects related to gravity and the weak equivalence principle.

Contribution

It presents a new method for path integral quantum mechanics in curved space-time using tangent space rotations and Lie transport, highlighting quantum violations of the weak equivalence principle.

Findings

Probability-violating terms relate to quantum violations of the weak equivalence principle.

Remaining probability-conserving terms satisfy the weak equivalence principle.

The propagator includes a curvature-dependent phase interpreted as gravitational Aharonov-Bohm and Berry's phase.

Abstract

A fundamentally different approach to path integral quantum mechanics in curved space-time is presented, as compared to the standard approaches currently available in the literature. Within the context of scalar particle propagation in a locally curved background, such as described by Fermi or Riemann normal co-ordinates, this approach requires use of a constructed operator to rotate the initial, intermediate, and final position ket vectors onto their respective local tangent spaces, defined at each local time step along some arbitrary classical reference worldline. Local time translation is described using a quantum mechanical representation of Lie transport, that while strictly non-unitary in operator form, nevertheless correctly recovers the free-particle Lagrangian in curved space-time, along with new contributions. This propagator yields the prediction that all probability violating terms due to curvature contribute to a quantum violation of the weak equivalence principle, while the remaining terms that conserve probability also correspondingly satisfy the weak equivalence principle, at least to leading-order in the particle's Compton wavelength. Furthermore, this propagator possesses an overall curvature-dependent and gauge-invariant phase factor that can be interpreted as the gravitational Aharonov-Bohm effect and Berry's phase.