Space-time-resolved quantum electrodynamics description of Compton scattering

The standard method for approaching quantum electrodynamic (QED) field theory uses a perturbative $S$-matrix approach. This approach is explicitly nondynamical and provides only a one-time, static map between an initial state to be evolved by the ``full propagator'' of a bona-fide interacting field theory and an asymptotically equivalent effective initial state to be evolved by the ``free propagator'' of the corresponding noninteracting field theory. We provide a detailed derivation of a nonperturbative and dynamical approach to QED that allows one to study the space-time dynamics of electron-photon interactions directly. As an example of this method, we compute the time-resolved dynamics of Compton scattering for a system with a nontrivial spatial structure in only one dimension while restricting to the case of a single electron and at most one photon. This approach retains the massless photon of quantum electrodynamics in contrast to previous approaches that resorted to using massive bosons [T. Cheng, E. R. Gospodarczyk, Q. Su, and R. Grobe, Ann. Phys. 325, 265 (2010)] to represent the photon. The dynamics of Compton scattering are illustrated using joint probability distributions that evolve in time. This information is compared to that provided by the $S$ matrix.