Derivation of Feynman Propagator Using Cauchy's Residue Theorem of a Dirac Delta Driven Wave Equation.

Abstract

The Green’s Function of a Dirac  Driven Wave Equation was obtained after applying Cauchy’s Residue Theorem to a contour integral. Solutions were obtained for initial conditions t < 0 and t > 0 by Fourier transforming the Green’s function, obtaining preferred spatial variables of the Fourier transform, and inverse Fourier transforming back to a Green’s function. Contour integrals were set up, prior to their evaluation the simple poles of the complex function were translated off the Real axis. Two contour integrals were evaluated, and the derived retarded Green’s function, from initial condition t < 0 and advanced Green’s function, from initial condition t > 0 represent a Feynman Propagator, an amplitude probability function used frequently in Quantum Field Theory.