Compatibility between the Brownian metric and the kinetic metric in Nelson stochastic quantization

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Abstract
In the frame of Nelson stochastic quantization for dynamical systems on a manifold, we consider diffusion processes with Brownian covariance given by a Riemannian metric on the manifold. The dynamics is specified through a stochastic variational principle for a generalization of the classical action, with a given kinetic metric. The resulting programming equation, of the Hamilton-Jacobi type, depends on both metrics, the Brownian one and the kinetic one. We introduce a simple notion of compatibility between the two metrics, such that the programming equation and the continuity equation lead to the Schrödinger equation on the manifold. DOI: http://dx.doi.org/10.1103/PhysRevD.31.2521 © 1985 The American Physical Society

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