Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion

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.

113 pages; preprint of a book published by Springer Nature

Cited by in corpus (7)

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• Stochastic dark matter: Covariant Brownian motion from Planckian discreteness
• Simulating Nelsonian Quantum Field Theory
• Earthquake Quantization
• Torsion-Driven Nonlinearity in Spinless Quantum Mechanics