Noncommutative Geometry and Stochastic Processes

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 Schrödinger equation. Indeed, by a direct numerical check, one can recover the kernel of the Schrödinger equation starting by an ordinary Brownian motion. This class of stochastic processes needs a Clifford algebra to exist.