Euclidean quantum mechanics and stochastic integrals

Graham, Robert

Paul Lescot, Jean-Claude Zambrini

Abstract The study of Riemannian stochastic differential geometry is almost as old as the theory of stochastic differential equations themselves. But almost nothing is known about stochastic symplectic geometry. Using only the geometrical content of the heat equation (in a sense inspired by E. Cartan) and its relations with quantum dynamics, we present here the first steps toward such a stochastic version of symplectic geometry.

Sheldon Goldstein

M. S. Wang

A relation between the kernel of a quantum system in Feynman's path-integral theory and the transition probability of the corresponding Markov process, with the diffusion coefficient rotated to the imaginary axis, is obtained in the framework of stochastic mechanics. If a stationary state is considered, the relation reduces to a form relating a Euclidean (i.e., imaginary time) quantum theory to a Markov process.

Aubrey Truman, Ian M Davies

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.