The mathematical role of (commutative and noncommutative) infinitesimal random walks over (commutative and noncommutative) riemannian manifolds in Quantum Physics

Abstract

Anderson's nonstandard construction of brownian motion as an infinitesimal random walk on the euclidean line is generalized to an Hausdorff riemannian manifold. A nonstandard Feynman-Kac formula holding on such an Hausdorff riemannian manifold is derived. Indications are given on how these (radically elementary) results could allow to formulate a nonstandard version of Stochastic Mechanics (avoiding both the explicitly discussed bugs of Internal Set Theory as well as the controversial renormalization of the stochastic action). It is anyway remarked how this would contribute to hide the basic feature of Quantum Mechanics, i.e. the noncommutativity of the observables' algebra, whose structure is naturally captured in the language of Noncommutative Probability and Noncommutative Geometry. With this respect some preliminary consideration concerning the notion of infinitesimal quantum random walk on a noncommutative riemannian manifold, the notion obtained by the Sinha-Goswami's definition of quantum brownian motion on a noncommutative riemannian manifold replacing a continuous time interval with an hyperfinite time interval, is presented.