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We present an analytic calculation of the semiclassical electron-positronpair creation rate by time-dependent electrical fields. We use two methods,first the imaginary time method in the WKB-approximation and second theworld-line instanton approach. The analytic tools for both methods aregeneralized to time-dependent electric fields with more than one component. Forthe WKB method an expansion of the momentum spectrum of produced pairs aroundthe canonical momentum $\vec{P}=0$ is presented which simplifies thecomputation of the pair creation rate. We argue that the world-line instantonmethod of [Dunne et al., Phys. Rev. D73, 065028 (2006)] implicitly performsthis expansion of the momentum spectrum around $\vec{P}=0$. Accordingly thegeneralization to more than one component is shown to agree with the WKB resultobtained via this expansion. However the expansion is only a good approximationfor the cases where the momentum spectrum is peaked around $\vec{P}=0$. Thusthe 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 hereare only applicable in these cases. We study the two component case of arotating electric field and find a new analytic closed form for the momentumspectrum using the generalized WKB method. The momentum spectrum for this fieldis not peaked around $\vec{P}=0$.

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