Path integral quantization of the supersymmetric model of the Dirac particle

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Abstract
We investigate in detail the representation of the Dirac propagator as a path integral over virtual trajectories in the phase space with anticommuting variables. Using the reparametrization and supergauge invariant action integral proposed by Berezin and Marinov [Ann. Phys. 104, 336 (1977)], we analyze the relation of causality to reparametrization invariance, and we construct the Faddeev–Popov measure for the symbol of the evolution operator. We present a precise definition of the path integral as a limit of finite-dimensional integrals, and we explicitly perform the integration obtaining the familiar result. We also analyze another approach, where reparametrization and supergauge invariance of the action is preserved with the help of independent einbein variables. This method, as opposed to Faddeev–Popov technique, dispenses with the gauge fixing problem and allows us to construct the correct functional measure by explicit factorization of the volume of reparametrization group and supergauge group; the former is infinite, the latter identically vanishes. A sequence of finite-dimensional approximations to the functional measure is given also in this case together with explicit calculations.