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

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• Dirac equation
• Nonperturbative effects in field theory
• Schwinger effect