On the relativistic quantum mechanics of a photon between two electrons in \(1+1\) dimensions

A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions \(\Psi ({\textbf {x}}_{\text {ph}},{\textbf {x}}_{\text {e}_1},{\textbf {x}}_{\text {e}_2})\) where \({\textbf {x}}_{\text {ph}},{\textbf {x}}_{\text {e}_1},{\textbf {x}}_{\text {e}_2}\) are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds \(\{{\textbf {x}}_{\text {ph}}={\textbf {x}}_{\text {e}_1}\}\) and \(\{{\textbf {x}}_{\text {ph}}={\textbf {x}}_{\text {e}_2}\}\) compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.