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Reports on Mathematical Physics

Quantum stochastic processes and the Feynman path integral for a single spinless particle

W. Garczyński

Published Mar 1, 1973• 7 citations
Publisher
DOI
10.1016/0034-4877(73)90011-6

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We give here a mathematically rigorous form to an earlier work by Barchielli, Lanz, and Prosperi, in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle. With the help of Albeverio and Ho/egh-Krohn’s theory of Feynman path integrals, we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization, continuity, and positive definiteness. An explicit calculation of the Feynman path integral which defines the functional allows an analysis of the process to be made.
We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.