Abstract
We reintroduce the Riemann-Cartan-Weyl (RCW) spacetime geometries of quantum mechanics [Rapoport (1996),Int. J. Theor. Phys. 35(2)] in two novel ways: first, through the covariant formulation of the Fokker-Planck operator of the quantum motions defined by these geometries, and second, by stochastic extension of Cartan's development method. The latter is a gauge-theoretic formulation of nonlinear diffusions in spacetime in terms of the stochastic differential geometry associated to the RCW geometries with Weyl torsion. The Weyl torsion plays the fundamental role of describing the first moment (incorporating also the fluctuations due to the second moment) of the stochastic diffusion processes. In this article we present the most general expression of the Weyl torsion one-form given in terms of its de Rham decomposition into the exact component associated with the 0-spin field ϕ and two electromagnetic potentials, one the codifferential of a 2-form and the other a harmonic 1-form. We thus give an original description of the Maxwell theory and its relation to torsion. We associate these electromagnetic potentials with the irreversibility of the diffusions. In an Appendix, we give a self-contained presentation of the theory of diffusions on manifolds and the stochastic calculi as a basis for the Cartan stochastic copying method.
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Rapoport, D.L. Riemann-Cartan-Weyl quantum geometry. II Cartan stochastic copying method, Fokker-Planck operator and Maxwell-de Rham equations. Int J Theor Phys 36, 2115–2152 (1997). https://doi.org/10.1007/BF02435948
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DOI: https://doi.org/10.1007/BF02435948