On quantum and parallel transport in a Hilbert bundle over spacetime

TL;DR

This paper analyzes the Hilbert bundle approach to stochastic quantum mechanics in curved spacetime, proving the quantum transport law reduces to parallel transport in the flat space limit and establishing gauge invariance of the propagator.

Contribution

It provides a detailed proof linking quantum transport to parallel transport and demonstrates gauge invariance of the quantum-geometric propagator in curved spacetime.

Findings

Quantum transport coincides with parallel transport in flat space limit.

The quantum-geometric propagator is gauge covariant with a contact point measure.

The framework confirms and extends Prugovečki's earlier proposals.

Abstract

We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugove\v{c}ki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugove\v{c}ki. We furthermore show that the quantum-geometric propagator in curved spacetime proposed by Prugove\v{c}ki, yielding a Feynman path integral-like formula involving integrations over intermediate phase space variables, is Poincar\'e gauge covariant (i.e.$\!$ is gauge invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a ``contact point measure'' in the soldered stochastic phase space bundle raised over curved spacetime.