Removal of self-interactions in the Dirac-Maxwell equations in one spatial dimension

We propose a theoretical framework that permits us to eliminate the unphysical self-repulsion that occurs if a spatially localized charged particle interacts with its own electric field. As an example of this framework, we study the time-resolved interaction between an electronic and positronic wave packet by solving the coupled set of two-particle Dirac-Maxwell equations. The restriction of the dynamics to only one spatial dimension permits us to neglect the magnetic field and therefore any effects due to retardation are absent. Here the unwanted self-repulsion can be removed by separating the total electric field into two portions, each of which is generated by only one particle and is evolved independently of the other. For example, the Maxwell equation for the electronic field has only the electronic charge density as a source term and only this field is coupled to the positron in the two-particle Dirac equation.