Abstract
Nelson's theory of universal Brownian motion is generalized to manifold-valued processes with Hamiltonian of the form H=(p−A) 2+φ. It is shown that a spin model of Bopp & Haag is such a process. We show that, as the radius of the sphere of this model approaches zero, we recover the Pauli equation. We analyze further the case of no external field, singling out a continuous random variable which we call the angular momentum and showing that this random variable has the quantum mechanical expectation values. We also prove an ergodic theorem to the effect that the average value of the angular momentum equals its time average along trajectories.
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Communicated by M. Kac
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Dankel, T.G. Mechanics on manifolds and the incorporation of spin into Nelson's Stochastic mechanics. Arch. Rational Mech. Anal. 37, 192–221 (1970). https://doi.org/10.1007/BF00281477
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DOI: https://doi.org/10.1007/BF00281477