Momentum correlation in pair production by spacetime dependent fields from scattered wave functions

We consider Sauter-Schwinger pair production by electric fields that depend on both time and space, $E(t,z)$ and $E(t,x,y)$. For space-independent fields, $E(t)$, momentum conservation, $\delta({\bf p}+{\bf p}')$, fixes the positron momentum, ${\bf p}'$, in terms of the electron momentum, ${\bf p}$. For $E(t,z)$, on the other hand, $p_z$ and $p'_z$ are independent. However, previous exact-numerical studies have considered only the probability as a function of a single momentum variable, $P(p_z)$, $P(p'_z)$ or $P(p'_z-p_z)$, but not the correlation $P(p_z,p'_z)$. In this paper, we show how to obtain $P(p_z,p'_z)$ by solving the Dirac equation numerically. To do so, we split the wave function into a background and a scattered wave, $\psi(t,{\bf x})=\psi_{\rm back.}(t,{\bf x})+\psi_{\rm scat.}(t,{\bf x})$, where $\psi_{\rm back.}\propto\exp(\pm ipx+\text{gauge term})$. $\psi_{\rm scat.}$ vanishes outside a past light cone and is obtained by solving $(i\gamma^\mu D_\mu-m)\psi_{\rm scat.}=-(i\gamma^\mu D_\mu-m)\psi_{\rm back.}$ backwards in time starting with $\psi_{\rm scat.}(t\to+\infty,{\bf x})=0$.