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

• URL: http://arxiv.org/abs/2312.06019v2
• Date: Tue, 12 Nov 2024 18:44:16 GMT
• Title: On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions
• Authors: Lawrence Frolov, Samuel E. Leigh, A. Shadi Tahvildar-Zadeh,
• Abstract summary: 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. Manifest covariance is achieved using Dirac's formalism of multi-time wave functions. The corresponding initial 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.
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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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