Noncommutative geometry and stochastic processes

TL;DR

This paper explores how noncommutative geometry supports a stochastic process framework for quantum mechanics, using fractional Wiener processes and Clifford algebra to connect to the Schrödinger and Dirac equations.

Contribution

It introduces a class of fractional Wiener processes on noncommutative geometry that reproduce quantum equations like Schrödinger and Dirac as Fokker-Planck equations.

Findings

Fractional Wiener processes produce complex-valued stochastic processes.

The Fokker-Planck equation of these processes resembles the Schrödinger equation.

Dirac equation in Klein-Gordon form is recovered as a Fokker-Planck equation.

Abstract

The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic processes can be devised, arising as fractional powers of an ordinary Wiener process, that reproduce in a proper way a stochastic process on a noncommutative geometry. These processes are characterized by producing complex values and so, the corresponding Fokker-Planck equation resembles the Schroedinger equation. Indeed, by a direct numerical check, one can recover the kernel of the Schroedinger equation starting by an ordinary Brownian motion. This class of stochastic processes needs a Clifford algebra to exist. In four dimensions, the full set of Dirac matrices is needed and the corresponding stochastic process in a noncommutative geometry is easily recovered as is the Dirac equation in the Klein-Gordon form being it the Fokker--Planck equation of the process.