On the semi-classical approximation of the solution of the Heisenberg equation with spin

We consider Hamiltonians describing the motion of some charged particles with spin in an electromagnetic field. Let U-h(t,s) be its propagator and F-h an observable. Then the solution of the Heisenberg equation with F-h at t = s is given by U-h(t,s)*FhUh(t,s). In this paper we compute the semi-classical approximation of U-h(t,s)*FhUh(t,s) in terms of pseudo-differential operators. From this formula we get the classical limit as h --> 0 of the time evolution of the mean value of F-h for initial states centered suitably in classical phase space. Then the relation between quantum and classical mechanics can be shown.