Analytic Continuation of Stochastic Mechanics

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.

15+13 pages. v.2: accepted in J. Math. Phys

References in corpus (4)

• On the Renormalizability of Horava-Lifshitz-type Gravities
• Relating Field Theories via Stochastic Quantization
• Variational Principle of Hydrodynamics and Quantization by Stochastic Process
• On Strongly Coupled Matrix Theory and Stochastic Quantization: A New Approach to Holographic Dualities

Cited by in corpus (1)

• Quantum Theory, Gravity and Second Order Geometry