Noncommutative geometry and stochastic processes

Marco Frasca

doi:10.1007/978-3-319-68445-1_57

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arXiv · 1412.4693

Noncommutative geometry and stochastic processes

Marco Frasca

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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.

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Marco Frasca. 2017-07-14. Noncommutative geometry and stochastic processes. https://doi.org/10.1007/978-3-319-68445-1_57

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