A Lorentz-covariant interacting electron–photon system in one space dimension

A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions Ψ(2)(xph,xel) where xel,xph are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold {xel=xph}, compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a hypersurface Bohm–Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.