Stochastic processes in conformal Riemann-Cartan-Weyl gravitation

Abstract

We construct the Riemann-Cartan geometries with torsion generated by the action of the conformal Weyl group. We study the wave operators associated to these structures, which, in addition to the usual Laplace-Beltrami operator, have a term which is a gradient vector field conjugate to the one-form given by the torsion potential derived from the Weyl group, and which we associate with a relativistic extension of the drift vector field in Nelson's construction of stochastic mechanics. In fact, our construction is valid for configuration spaces of any dimension. We sketch the construction of the stochastic processes on space-time generated by these operators, where the invariant measure is found to be defined by the conformal structure. We discuss briefly the relation with the theory of Dirichlet forms and D. Bohm's quantum potential in the theory of hidden variables, which in this setting acquire a gauge-geometric status previously unknown.