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

Abstract

The nonperturbative electron-positron pair production (Schwinger effect) is considered for space- and time-dependent electric fields $\stackrel{\ensuremath{\rightarrow}}{E}(\stackrel{\ensuremath{\rightarrow}}{x},t)$. Based on the Dirac-Heisenberg-Wigner formalism, we derive a system of partial differential equations of infinite order for the 16 irreducible components of the Wigner function. In the limit of spatially homogeneous fields the Vlasov equation of quantum kinetic theory is rediscovered. It is shown that the quantum kinetic formalism can be exactly solved in the case of a constant electric field $E(t)={E}_{0}$ and the Sauter-type electric field $E(t)={E}_{0}{\mathrm{sech}}^{2}(t/\ensuremath{\tau})$. These analytic solutions translate into corresponding expressions within the Dirac-Heisenberg-Wigner formalism and allow to discuss the effect of higher derivatives. We observe that spatial field variations typically exert a strong influence on the components of the Wigner function for large momenta or for late times.