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

TL;DR

This paper develops a Lorentz-covariant quantum system for an electron-photon pair in one dimension, demonstrating well-posedness, particle trajectories, and novel phenomena like photon capture and release.

Contribution

It introduces a Lorentz-invariant wave equation framework using multi-time wave functions for electron-photon interactions in one dimension, with proven well-posedness and new physical insights.

Findings

Well-posed initial-boundary-value problem established

Existence of global trajectories in Hypersurface Bohm--Dirac theory

Numerical simulations show Compton scattering and photon capture phenomena

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.