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.

中文翻译：

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一维空间中的洛伦兹协变相互作用电子-光子系统

波动方程的洛伦兹协变系统是为一个空间维度中的量子力学双体系统制定的，由一个电子和一个光子组成。Manifest Lorentz 协方差是使用 Dirac 的多时间波函数形式化实现的，即波函数 $\Psi(\mathbf{x}_{ph},\mathbf{x}_{el})$ 其中 $\mathbf{x }_{el},\mathbf{x}_{ph}$ 分别是电子和光子的通用时空事件。它们的相互作用是通过符合子流形 $\{\mathbf{x}_{el}=\mathbf{x}_{ph}\}$ 的洛伦兹不变路径无交叉边界条件实现的，与粒子电流守恒。相应的初始边界值问题被证明是适定的。电子和光子轨迹在超曲面玻姆-狄拉克理论中显示为全局存在，对于典型的粒子初始条件。还介绍了一些数值实验的结果，这些实验说明了康普顿散射以及一种新现象：电子捕获和释放光子。