On the Vlasov equation for Schwinger pair production in a time-dependent electric field

TL;DR

This paper compares two formulations of the quantum Vlasov equation describing Schwinger pair production in time-dependent electric fields, clarifying their relation and analyzing specific field cases.

Contribution

It demonstrates the equivalence of two Vlasov equation versions and relates their asymptotic distributions, providing insights into pair production dynamics.

Findings

The two Vlasov equations are equivalent through a simple relation.

The difference corresponds to 'in-out' and 'in-in' formalisms.

Explicit examples include Sauter and single-soliton fields.

Abstract

Schwinger pair creation in a purely time-dependent electric field can be described through a quantum Vlasov equation describing the time evolution of the single-particle momentum distribution function. This equation exists in two versions, both of which can be derived by a Bogoliubov transformation, but whose equivalence is not obvious. For the spinless case, we show here that the difference between these two evolution equations corresponds to the one between the "in-out" and "in-in" formalisms. We give a simple relation between the asymptotic distribution functions generated by the two Vlasov equations. As examples we discuss the Sauter and single-soliton field cases.