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

TL;DR

This paper develops a simplified four-dimensional path integral approach for the massive Dirac propagator, avoiding extra dimensions and revealing supersymmetry, with applications to nonrelativistic limits and quenched QED.

Contribution

It introduces a systematic, five-dimensional extension-free path integral representation for the Dirac propagator, highlighting supersymmetry and gauge properties.

Findings

Derived two equivalent path integral representations without five-dimensional extension

Connected relativistic propagator to a three-dimensional Pauli Hamiltonian

Obtained gauge transformation properties of the Green function in quenched QED

Abstract

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.