Worldline path integral for the massive Dirac propagator: A four-dimensional approach

We simplify and generalize an approach proposed by Di Vecchia and Ravndal to describe a massive Dirac particle in external vector and scalar fields. Two different path integral representations for the propagator are derived systematically without the usual five-dimensional extension and shown to be equivalent due to the supersymmetry of the action. They correspond to a projection on the mass of the particle either continuously or at the end of the time evolution. It is shown that the supersymmetry transformations are generated by shifting and scaling the supertimes and the invariant difference of two supertimes is given for the general case. A nonrelativistic reduction of the relativistic propagator leads to a three-dimensional path integral with the usual Pauli Hamiltonian. By integrating out the photons we obtain the effective action for quenched QED and use it to derive the gauge-transformation properties of the general Green function of the theory.