SPIN, STATISTICS, AND GEOMETRY OF RANDOM-WALKS

We develop and unify two complementary descriptions of propagation of spinning particles: the directed random walk representation and the "spin factor" approach. Working in an arbitrary number of dimensions D, we first represent the Dirac propagator in terms of a directed random walk. We then derive the general and explicit form of the gauge connection describing parallel transport of spin and investigate the resulting quantum-mechanical problem of a particle moving on a sphere in the field of a nonabelian SO(D - 1) monopole. This construction, generalizing Polyakov's results, enables us to prove the equivalence of the random walk and path-integral (spin factor) representation. As an alternative, we construct and discuss various Wess-Zumino-Witten forms of the spin factor. We clarify the role played by the coupling between the particle's spin and translational degrees of freedom in establishing the geometrical properties of particle's paths in spacetime. To this end, we carefully define and evaluate Hausdorff dimensions of bosonic and fermionic sample paths, in the covariant as well as nonrelativistic formulations. Finally, as an application of the developed formalism, we give an intuitive spacetime interpretation of chiral anomalies in terms of the geometry of fermion trajectories. © 1991.