Quantization = Geometry + Probability

Abstract

If the symplectic geometry of classical mechanics is supplemented by a suitable Riemannian geometry, it becomes possible to define Brownian motion on the classical phase space. It is shown that the integral of a phase factor involving the classical action over a pinned Wiener measure leads, in the limit of diverging diffusion constant, to an intrinsic, coordinate-free characterization of the quantization process for various kinematical operator choices. Natural extensions of this geometric-probabilistic quantization procedure to the case of fields yield proposals for (i) a quantization scheme for a scalar field in an arbitrary background metric, and (ii) a quantization scheme for the gravitational field fully respecting the positivity of the three metric.