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The non-perturbative electron-positron pair production (Schwinger effect) isconsidered for space- and time-dependent electric fields $\vec{E}(\vec{x},t)$.Based on the Dirac-Heisenberg-Wigner (DHW), formalism we derive a system ofpartial differential equations of infinite order for the sixteen irreduciblecomponents of the Wigner function. In the limit of spatially homogeneous fieldsthe Vlasov equation of quantum kinetic theory (QKT) is rediscovered. It isshown that the quantum kinetic formalism can be exactly solved in the case of aconstant electric field $E(t)=E_0$ and the Sauter-type electric field$E(t)=E_0\operatorname{sech}^2(t/\tau)$. These analytic solutions translateinto corresponding expressions within the DHW formalism and allow to discussthe effect of higher derivatives. We observe that spatial field variationstypically exert a strong influence on the components of the Wigner function forlarge momenta or for late times.

Schwinger pair production in space- and time-dependent electric fields: Relating the Wigner formalism to quantum kinetic theory