Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension

In this paper we prove that the static solution of the Cauchy problem for a massless real scalar field that is sourced by a point charge in 1 + 1 dimensions is asymptotically stable under perturbation by compactly-supported radiation. This behavior is due to the process of back-reaction. Taking the approach of Kiessling, we rigorously derive the expression for the force on the particle from the principle of total energy-momentum conservation. We provide a simple, closed form for the particle's self-action, and show that it is restorative in this model, i.e. proportional to negative velocity, and causes the charge to return to rest after the radiation passes through. We establish these results by studying the joint evolution problem for the particle-scalar field system, and proving its global well-posedness and the claimed asymptotic behavior.