Semiclassical pair production rate for time-dependent electrical fields with more than one component: -WKB-approach and world-line instantons

We present an analytic calculation of the semiclassical electron-positron pair creation rate by time-dependent electrical fields. We use two methods, first the imaginary time method in the WKB-approximation and second the world-line instanton approach. The analytic tools for both methods are generalized to time-dependent electric fields with more than one component. For the WKB method an expansion of the momentum spectrum of produced pairs around the canonical momentum $\vec{P}=0$ is presented which simplifies the computation of the pair creation rate. We argue that the world-line instanton method of [Dunne et al., Phys. Rev. D73, 065028 (2006)] implicitly performs this expansion of the momentum spectrum around $\vec{P}=0$. Accordingly the generalization to more than one component is shown to agree with the WKB result obtained via this expansion. However the expansion is only a good approximation for the cases where the momentum spectrum is peaked around $\vec{P}=0$. Thus the expanded WKB result and the world-line instanton method of [Dunne et al., Phys. Rev. D73, 065028 (2006)] as well as the generalized method presented here are only applicable in these cases. We study the two component case of a rotating electric field and find a new analytic closed form for the momentum spectrum using the generalized WKB method. The momentum spectrum for this field is not peaked around $\vec{P}=0$.