Some remarks on the two-dimensional Dirac equation (Q1188328)

The solutions of the initial value problem for the two-dimensional Dirac equation are studied to exhibit the connection between them and the Poisson-Kac process. The idea of the new proof of the probabilistic formula previously obtained by \textit{Ph. Blanchard}, \textit{Ph. Combe}, \textit{M. Sirugue} and \textit{M. Sirugue-Collin} [Functional integration with emphasis on the Feynman integral, Proc. Workshop, Sherbrooke/Can. 1986, Suppl. Rend. Circ. Math. Palermo, II. Ser. 17, 47-104 (1988; Zbl 0654.60039)] goes back to \textit{J. Kisyński} [Ann. Polon. Math. 29, 259- 272 (1974; Zbl 0287.35061)]. In the method some group-theoretical aspects of the formula are employed and therefore it is natural to use the language of the theory of one-parameter semigroups of linear operators. The initial value problem for the two-dimensional Dirac equation is treated as a special case of perturbation theorem for some class of semigroups. In the second part the Lie-Trotter approximation theorem is employed to show that the formula for the solutions to the two-dimensional Dirac equation with the potential term (obtained by the same authors) is in fact the Feynman-Kac formula related to a Markov process (the Poisson-Kac process) with the values in a Lie group.