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 spinstatistics relation. We discuss a class of associative algebras arising in the construction of the propagators and wave functions. These (momentumspace) wave functions are sections of bundles over (complexified) spheres, constructed in analogy to the YangMills instanton bundles. We then derive propertime and ramdomwalk representations of the propagators. The latter are conveniently expressed in terms of matrixvalued 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 halfinteger and integer spins.
 Publication:

Annals of Physics
 Pub Date:
 June 1992
 DOI:
 10.1016/00034916(92)90176M
 Bibcode:
 1992AnPhy.216..226J