Nonlinear trident using WKB and worldline instantons

We consider nonlinear trident, $e^{\scriptscriptstyle -}\to e^{\scriptscriptstyle -} e^{\scriptscriptstyle -} e^{\scriptscriptstyle +}$, in various electric background fields. This process has so far been studied for plane-wave backgrounds, using Volkov solutions. Here we first use WKB for trident in time-dependent electric fields, and then for fields which vary slowly in space. Then we show how to use worldline instantons for more general fields which depend on both time and space. For time-dependent fields the WKB approach is at least as simple to use as the worldline approach, but already the relatively modest step of including a slow spatial dependence makes the worldline approach much more efficient.