Analytic Continuation of Stochastic Mechanics

TL;DR

This paper explores the analytic continuation of stochastic processes on complexified manifolds, connecting real and imaginary quadratic variations to classical Brownian motion and quantum stochastic mechanics.

Contribution

It introduces a unified framework for stochastic processes on complex manifolds, linking Nelson's stochastic quantization with analytic continuation techniques.

Findings

Real quadratic variation yields classical Brownian motion and Feynman-Kac formula.

Imaginary quadratic variation leads to stochastic mechanics for quantum particles.

Provides a bridge between classical and quantum stochastic descriptions.

Abstract

We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. Using Nelson's stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the process has a purely real quadratic variation, we obtain the one-sided Wiener process that is encountered in the theory of Brownian motion. In this case, the result coincides with the Feyman-Kac formula. On the other hand, for a purely imaginary quadratic variation, we obtain the two-sided Wiener process that is encountered in stochastic mechanics, which provides a stochastic description of a quantum particle on a curved spacetime.