The Schwinger mechanism revisited

TL;DR

This paper clarifies the difference between the vacuum persistence probability rate and the actual fermion-antifermion pair production rate in an electric field, providing exact expressions and physical insights.

Contribution

It derives exact formulas for both the vacuum persistence rate and the pair production rate, showing they differ and explaining the physical reasons for this discrepancy.

Findings

The vacuum persistence rate w is given by the standard Schwinger series.

The pair production rate Γ is the first term of the series for w.

w and Γ differ physically and mathematically.

Abstract

The vacuum persistence probability, $P_{vac}(t)$, for a system of charged fermions in a fixed, external, and spatially homogeneous electric field, was derived long ago by Schwinger; $w = -log[P_{vac}(t)]/ (V t)$ has often been identified as the rate at which fermion-antifermion pairs are produced per unit volume due to the electric field. In this paper, we separately compute exact expressions for both $w$ and for the rate of fermion-antifermion pair production per unit volume, $\Gamma$, and show that they differ. While $w$ is given by the standard Schwinger mechanism result $w$, an infinite series, the pair production rate, $\Gamma$, is just the first term of that series. Our calculation is done for a system with periodic boundary conditions in the $A_0=0$ gauge but the result should hold for any consistent set of boundary conditions. We discuss, the physical reason why the rates $w$ and $\Gamma$ differ.