Generalized stochastic mechanics on Riemannian manifolds

Daniela Dohrn, Francesco Guerra

Ian M Davies

Laura M. Morato

In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one-to-one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.

Ettore Aldrovandi, Daniela Dohrn, Francesco Guerra

Michele Pavon

Robert Graham, D. Roekaerts

In the zero-fermion and n-fermion sectors of Euclidean supersymmetric quantum theories on an n-dimensional manifold a transformation is known which reduces the functional integrals to Gaussian functional integrals. The transformation, often called a Nicolai map, can be interpreted as a Markovian stochastic process. We investigate two different possibilities to generalize this result to an arbitrary p-fermion sector (0\ensuremath{\le}p\ensuremath{\le}n), and find that only one of them is viable, in general. We give a general representation of the solution of the Schr\"odinger equation in the p-fermion sector in terms of a Markovian stochastic process which extends the stochastic process of the zero- and n-fermion sectors.