Propagator, sewing rules, and vacuum amplitude for the Polyakov point particle with ghosts

Abstract

We apply techniques developed for strings to the case of the spinless point particle. The Polyakov path integral with ghosts is used to obtain the propagator and one-loop vacuum amplitude. The propagator is shown to correspond to the Green's function for the BRST field theory in Siegel gauge. The reparametrization invariance of the Polyakov path integral is shown to lead automatically to the correct “trace log” result for the one-loop diagram, despite the fact that “naive sewing” of the ends of a propagator would give an incorrect answer. This type of failure of “naive sewing” is identical to that found in the string case. The present treatment provides, in the simplified context of the point particle, a pedagogical introduction to Polyakov path integral methods with and without ghosts.