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

TL;DR

This paper investigates the evaluation of propagators and partition functions for spinning particles using path integrals, exploring the exactness of saddle point approximation and its limitations at the quantum level, with implications for magnetic field interactions.

Contribution

It demonstrates the classical exactness of SPA for spinning particles and critically analyzes its quantum limitations, extending existing formulas to arbitrary magnetic fields.

Findings

Classical SPA is exact due to reduction from linear dynamics on ^4 to S^2.

Quantum SPA leads to divergent prefactors without regulators.

The paper extends known spin propagator formulas to arbitrary magnetic fields.

Abstract

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 saddle point approximation (SPA) for this problem. At the classical level we argue that exactness of the SPA stems from the fact that the dynamics (on the two--sphere $S^2$) of a classical spinning particle in a magnetic field is the reduction from $\br^4$ to $S^2$ of a linear dynamical system on $\br^4$. At the quantum level, however, and within the path integral approach, the restriction, inherent to the use of the SPA, 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 to the same problem 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.