Riemann-Cartan-Weyl quantum geometry,I: Laplacians and supersymmetric systems

In this first article of a series dealing with the geometry of quantum mechanics, we introduce the Riemann-Cartan-Weyl (RCW) geometries of quantum mechanics for spin-0 systems as well as for systems of nonzero spin. The central structure is given by a family of Laplacian (or D'Alembertian) operators on forms of arbitrary degree associated to the RCW geometries. We show that they are conformally equivalent with the Laplacian operators introduced by Witten in topological quantum field theories. We show that the Laplacian RCW operators yield a supersymmetric system, in the sense of Witten, and study the relation between the RCW geometries and the symplectic structure of loop space. The RCW family of Laplacians are the infinitesimal generators of diffusion processes on nondegenerate space-times of systems of arbitrary spin.