Calculating propagators for relativistic wave equations

Here we examine the propagation of relativistic fields in spacetime using the viewpoint applied to derive the Rayleigh--Sommerfeld diffraction integral in three--dimensional space. We use this theory to find the propagators for both the Klein--Gordon equation and the Dirac equation. Based on these results, we highlight some apparently overlooked aspects of the Feynman checkerboard model as it is usually applied to the Dirac equation.