Time-Slicing Path-integral in Curved Space

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 (NTV). In this work, we construct a rigorous and covariant formulation of time-slicing path integrals for dynamics in curved space. We first establish a rigorous criterion for equivalence of $\textit{time-slice Green's function}$ (TSGF) in the continuum limit (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 equivalent to a Gaussian (Lemma 2). We further construct a continuous family of equivalent path-integral actions parameterized by an interpolation parameter $\alpha \in [0,1]$ (Lemma 3). The action generically contains 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 NTV. The $\alpha = 0$ representation of time-slice action is particularly convenient because it is Gaussian and transforms as a scalar, as long as $\Delta \boldsymbol x$ transforms according to $\textit{Ito's formula}$.