Exact plane-wave solutions of the coupled Maxwell–Klein–Gordon equations

Abstract

Solutions of the classical Maxwell–Klein–Gordon equations are investigated for which the Klein–Gordon field is assumed to be ψ(x)=αeipμxμ. It is shown that for this class the exponential factor can be ‘‘gauged away’’ and the resulting system of equations can be reduced to a single (complicated) nonlinear equation. Furthermore, the electromagnetic four-potential field becomes massive ‘‘absorbing scalar particles.’’ The steady-state (or stationary) subclass of the resulting system of equations is examined. It is proved that in absence of any magnetic field, the steady-state system does not have a solution. In the simple case for which four-potential components Aμ depend on one spatial coordinate, the equations are completely solved and explicitly analyzed.