M. Sirugue-Collin

Poisson Processes on Groups and Feynman Path Integrals

We give an expression for the perturbed evolution of a free evolution by gentle, possibly velocity dependent, potential, in terms of the expectation with respect to a Poisson process on a group. Various applications are given in particular to usual quantum mechanics but also to Fermi and spin systems.

A uniqueness theorem for anticommutation relations and commutation relations of quantum spin systems

Under quite natural assumptions we prove that a classical spin lattice has essentially two natural quantum extensions: The quantum spin lattice and the Fermi system. Moreover we derive a transformation from a commuting pair of Fermi systems to a purely anticommuting Fermi system.

On the probabilistic structure of quasi-free states of a Clifford algebra

We prove that the correlation functions of a non-relativistic Fermi field given by a quasi-free state are directly related to the values of the characteristic function of a probability measure over the phase space of a classical spin system.

Stochastic jump processes associated with Dirac equation

We study the stochastic jump processes associated with the Dirac equation where the space derivatives are replaced by discrete approximations.

The Zitterbewegung of a Dirac Electron in a Central Field

For years no probabilistic treatment of time evolution of a particle whose wave function has multicomponents has been available. Compared with the scalar non relativistic situation where the Feynman-Kac formula provided a quite strong tool to study the problem, very few results concerning for instance the Dirac equation were known. However long time ago Feynman and Hibbs [1] mentioned that the free Dirac equation in two space time dimensions allowed for a probabilistic treatment. Also there has been some works by Riazanov [2], Rosen [3] and Daleskii [4], [5].

Probabilistic Expression for the Solution of the Dirac Equation in Fourier Space

Recently there has been a revival of interest for the probabilistic representation of the solution of Dirac equation with the works of Gaveau, Jacobson, Kac and Schulman [1] and the thesis of Jacobson [21 on one hand, and a series of papers by Ichinose [3] on the other hand (for a more complete list of references see [4a,c]).