Geometry of spacetime propagation of spinning particles

Abstract

We develop a unified geometrical description of spacetime propagation of particles of arbitrary spin. We systematize, from the point of view of the theory of induced representations, the various forms of relativistic wave equations. Utilizing Fronsdal's construction of a class of unconstrained wave equations, we obtain closed and explicit forms of the particles' propagators, discuss their analytic properties in the momentum and mass, and express them through subtracted dispersion relations. By requiring unitarity, we then establish the spin-statistics relation. We discuss a class of associative algebras arising in the construction of the propagators and wave functions. These (momentum-space) wave functions are sections of bundles over (complexified) spheres, constructed in analogy to the Yang-Mills instanton bundles. We then derive proper-time and ramdom-walk representations of the propagators. The latter are conveniently expressed in terms of matrix-valued complex “measures” on the spaces of particle's paths. We develop a general method for analyzing fractal properties of sample paths, based on the calculation of ball hitting probabilities (or amplitudes), and find Hausdorff dimensions of paths for half-integer and integer spins.