Generalized Gelfand-Dikii equation for fermionic Schwinger pair production

TL;DR

This paper extends the Gelfand-Dikii formalism to spinor QED to analyze fermionic Schwinger pair production, providing new solvable models and numerical evidence for fermionic cases based on scalar solutions.

Contribution

It generalizes the Gelfand-Dikii equation for fermionic QED and demonstrates how solitonic solutions can be adapted to fermionic pair production scenarios.

Findings

Development of a generalized Gelfand-Dikii equation for spinor QED

Identification of solvable electric field configurations for fermions

Numerical evidence supporting the adaptation of scalar solutions to fermionic cases

Abstract

Schwinger pair creation in a purely time-dependent electric field can be reduced to an effective quantum mechanical problem using a variety of formalisms. Here we develop an approach based on the Gelfand-Dikii equation for scalar QED, and on a generalization of that equation for spinor QED. We discuss a number of solvable special cases from this point of view. In previous work, two of the authors had shown for the scalar case how to use the well-known solitonic solutions of the KdV equation to construct P\"oschl-Teller like electric fields that do not pair create at some fixed but arbitrary momentum. Here, we present numerical evidence that this construction can be adapted to the fermionic case by a mere change of parameters.