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

TL;DR

This paper derives a system of equations for electron-positron pair production in space- and time-dependent electric fields using the Wigner formalism, connecting it with quantum kinetic theory and providing exact solutions for specific field configurations.

Contribution

It establishes a formal connection between the Wigner formalism and quantum kinetic theory for inhomogeneous fields and provides exact solutions for certain electric field profiles.

Findings

Spatial variations significantly affect Wigner function components at high momenta.

Exact solutions are obtained for constant and Sauter-type electric fields.

The formalism reduces to quantum kinetic theory in the homogeneous limit.

Abstract

The non-perturbative electron-positron pair production (Schwinger effect) is considered for space- and time-dependent electric fields $\vec{E}(\vec{x},t)$. Based on the Dirac-Heisenberg-Wigner (DHW), formalism we derive a system of partial differential equations of infinite order for the sixteen irreducible components of the Wigner function. In the limit of spatially homogeneous fields the Vlasov equation of quantum kinetic theory (QKT) 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 $E(t)=E_0\operatorname{sech}^2(t/\tau)$. These analytic solutions translate into corresponding expressions within the DHW 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.