Diffraction for the Dirac–Coulomb Propagator

Abstract

The Dirac equation in $$\mathbb {R}^{1,3}$$ with potential $${\textsf{Z}}/r$$ is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is $$1-{\epsilon }$$ derivatives smoother than the main singularities, for all $${\epsilon }>0,$$ and is a conormal singularity.