A Lorentz-Covariant Interacting Electron-Photon System in One Space Dimension

Abstract

A Lorenz-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 $\\Psi(\\mathbf{x}_{ph},\\mathbf{x}_{el})$ where $\\mathbf{x}_{el},\\mathbf{x}_{ph}$ 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 $\\{\\mathbf{x}_{el}=\\mathbf{x}_{ph}\\}$, 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.