Path integrals for spinning particles, stationary phase and the Duistermaat-Heckmann theorem

We examine the problem of the evaluation of both the propagator and of the partition function of a spinning particle in an external field at the classical as well as the quantum level, in connection with the asserted exactness of the stationary phase approximation. At the classical level we argue that exactness of this approximation stems from the fact that the dynamics (on the two-sphere S2) of a spinning particle in a magnetic field is the reduction from R4 to S2 of a linear dynamical system on R4. At the quantum level, however, and within the path integral approach, the restriction, inherent to the use of the stationary phase approximation, to regular paths clashes with the fact that no regulators are present in the action that enters the path integral. This is shown to lead to a prefactor for the path integral that is strictly divergent, except in the classical limit. A critical comparison is made with the various approaches that have been presented in the literature. The validity of a formula given in literature for the spin propagator is extended to the case of motion in an arbitrary magnetic field.