Back reaction in light cone QED

We consider the back reaction of quantum electrodynamics upon an electric field ${E(x}_{+})=-A_{-}^{\prime }{(x}_{+})$ which is parallel to $x^3$ and depends only on the light cone coordinate $x_{+}{=(x}^0{+x}^3)/\sqrt{2}.$ Novel features are that the mode functions have simple expressions for arbitrary $A_{-}{(x}_{+})$ and that one cannot ignore the usual light cone ambiguity at zero $+$ momentum. Each mode of definite canonical momentum $k_{+}$ experiences pair creation at the instant when its kinetic momentum $p_{+}{=k}_{+}-{\mathrm{eA}}_{-}{(x}_{+})$ vanishes, at which point operators from the surface at $x_{-}=-\infty$ play a crucial role. Our formalism permits a more explicit and complete derivation of the rate of particle production than is usually given. We show that the system can be understood as the infinite boost limit of the analogous problem of an electric field which is homogeneous on surfaces of constant $x^0.$