Geometric derivation of the Schrödinger equation from classical mechanics in curved Weyl spaces

A theory physically equivalent to traditional nonrelativistic quantum mechanics is presented, in which both dynamical and probabilistic concepts enter in a classical way. Particle trajectories are deterministically governed by classical mechanics, only the initial position being at random. Quantum effects are supposed to arise from a modification of the geometry of space, due to the presence of matter. However, unlike gravitational forces, which are related to the metric of space-time, quantum-mechanical forces are proved to be related to the transplantation law of vectors. The resulting geometry of space, in the nonrelativistic limit, is found to be Weyl's geometry. Both particle motion and geometry of space are obtained from a unique averaged least-action principle.