Analytic Continuation of Stochastic Mechanics

• URL: http://arxiv.org/abs/2109.10710v2
• Date: Mon, 21 Mar 2022 18:29:37 GMT
• Title: Analytic Continuation of Stochastic Mechanics
• Authors: Folkert Kuipers
• Abstract summary: We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. If the process has a purely real variation, we obtain the one-sided Wiener process encountered in the theory of Brownian motion. For a purely imaginary quadratic variation, we obtain a description of a quantum particle on a curved spacetime.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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