Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion

TL;DR

This paper presents a unified mathematical framework connecting Brownian motion and quantum mechanics, extending to relativistic theories on manifolds, and highlighting the role of stochastic calculus in quantum symmetries.

Contribution

It introduces a novel approach to unify non-relativistic and relativistic quantum mechanics using stochastic processes and second order geometry.

Findings

Quantum mechanics can be modeled by a complex-rotated Wiener process.

Relativistic stochastic theories require Ito deformation of symmetries.

Path integral formulations on manifolds involve quadratic variation and affine connection.

Abstract

We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.