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. 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 $\Delta \mathbf{x}$, 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 $\Delta \mathbf{x}$ transforms according to Ito's formula.

20pages, 2 figures

References in corpus (6)

• State-dependent diffusion: thermodynamic consistency and its path integral formulation
• Path-integral analysis of fluctuation theorems for general Langevin processes
• Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization
• Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems
• Covariant Formulation of Non-linear Langevin Theory with Multiplicative Gaussian White Noises
• Covariant Non-equilibrium Thermodynamics from Ito-Langevin Dynamics

Cited by in corpus (2)

• Path integrals and stochastic calculus
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts