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, $\delta (\mathbf{p}+{\mathbf{p}}^{\mathrm{\prime }})$, fixes the positron momentum, ${\mathbf{p}}^{\mathrm{\prime }}$, in terms of the electron momentum, $\mathbf{p}$. For $E(t,z)$, on the other hand, $p_z$ and $p_z^{\mathrm{\prime }}$ are independent. However, previous exact-numerical studies have considered only the probability as a function of a single momentum variable, $P(p_z)$, $P(p_z^{\mathrm{\prime }})$ or $P(p_z^{\mathrm{\prime }}-p_z)$, but not the correlation $P(p_z,p_z^{\mathrm{\prime }})$. In this paper, we show how to obtain $P(p_z,p_z^{\mathrm{\prime }})$ by solving the Dirac equation numerically. To do so, we split the wave function into a background and a scattered wave, $\psi (t,\mathbf{x})={\psi }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{.}}(t,\mathbf{x})+{\psi }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{.}}(t,\mathbf{x})$, where ${\psi }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{.}}\propto \exp (\pm ipx+\text{gauge term})$. ${\psi }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{.}}$ vanishes outside a past light cone and is obtained by solving $(i{\gamma }^{\mu }{D}_{\mu }-m){\psi }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{.}}=-(i{\gamma }^{\mu }{D}_{\mu }-m){\psi }_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{.}}$ backwards in time starting with ${\psi }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{.}}(t\to +\mathrm{\infty },\mathbf{x})=0$.