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arXiv · 2312.06019

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

Lawrence Frolov·Samuel E. Leigh·A. Shadi Tahvildar-Zadeh
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Abstract

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.

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Lawrence Frolov, Samuel E. Leigh, A. Shadi Tahvildar-Zadeh. 2023-12-10. On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions. https://arxiv.org/abs/2312.06019

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