Time-Slicing Path-integral in Curved Space

TL;DR

This paper develops rigorous, covariant time-slicing path integral formulations for quantum and stochastic dynamics in curved space, resolving longstanding issues with covariance and representation equivalence.

Contribution

It constructs a family of equivalent, covariant path integral actions for curved space dynamics, clarifying their asymptotic equivalence and transformation properties.

Findings

Established a criterion for correct time-slice actions.

Proved all actions are asymptotically equivalent to Gaussian form.

Demonstrated covariance under nonlinear variable transformations.

Abstract

Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables. In this work, we construct rigorous and covariant formulations of time-slicing path integrals for quantum and classical stochastic dynamics in curved space. We first establish a rigorous criterion for correct time-slice actions of path integrals (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are asymptotically equivalent to a Gaussian (Lemma 2). Using these results, we further construct a continuous family of equivalent actions parameterized by an interpolation parameter $\alpha \in [0,1]$ (Lemma 3). The action generically contains a spurious drift term linear in $\Delta \boldsymbol x$, whose concrete form depends on $\alpha$. Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under nonlinear transform of variables. The $\alpha = 0$ representation of time-slice action is particularly convenient because it is Gaussian and invariant, as long as $\Delta \boldsymbol x$ transforms according to Ito's formula.