Worldline instantons for the momentum spectrum of Schwinger pair production in spacetime dependent fields

We show how to use the worldline-instanton formalism to calculate the momentum spectrum of the electron-positron pairs produced by an electric field that depends on both space and time. Using the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula with a worldline representation for the propagator in a spacetime field, we make use of the saddle-point method to obtain a semiclassical approximation of the pair -production spectrum. To check the final result, we integrate the spectrum and compare with the results obtained using a previous instanton method for the imaginary part of the effective action.