Brownian Motion, Markov Cosurfaces, Higgs Fields

Abstract

Brownian motion enters quantum theory in many ways, e.g. in the stochastic mechanical formulation of quantum theory, see e.g. ([1],[2],°) in the formulation of quantum field theory as Euclidean field theory, with covariances given by potential operators belonging to Brownian motions ([29], [3],[4]), in the expression of certain field theoretical models as “gases of local times of Brownian motion” ([5] – [7],[26],[33],[36]). In this lecture we shall discuss yet other uses of Brownian motion, and related processes, in the description of the quantum world, more precisely in connection with random group-valued hypersurfaces and gauge fields, leading to a (noncommutative) stochastic analysis with “higher dimensional time”. Let us start with gauge fields. Formally a pure Yang-Mills Euclidean measure gives a “white noise” type of distribution to the curvature 2-form F. Finding the corresponding connection a s.t. F = Da (D covariant derivative) implies then solving a stochastic partial differential equation for Lie algebra-valued one forms. The holonomyoperator is a stochastic one. This is one motivation for developing a suitable theory of stochastic mapping from curves into Lie groups, and extending the study of multiplicative stochastic differential equations to the case of multiplicative stochastic partial differential equations.