Properties of the Schwinger series and pair creation in strong fields

TL;DR

This paper analyzes the mathematical structure of the Schwinger series for pair creation probabilities in strong electric fields, revealing that the series can be summed exactly and that the original formulas are only valid below certain field strengths, with pair production becoming nearly certain at high fields.

Contribution

It provides an exact summation of the Schwinger series and refines the probability formulas for pair creation in strong fields, highlighting limitations of previous approximations.

Findings

Series allows exact summation, with remainder contributions growing rapidly.

Original formulas valid only below 0.0291*E_cr, beyond which probabilities exceed unity.

At 2.95% of E_cr, pair production probability approaches 100%.

Abstract

Probabilities of a pair of fermions and bosons creation in a static and spatially uniform electric field E are represented in the Schwinger formulas by infinite series. It is believed that in weak fields the main contribution to the probability is given by the first term of series, however the size of the remainder apparently was analyzed by nobody. We study the mathematical structure of the Schwinger series by using methods developed during last decades and prove that the given series allows an exact summation and the contribution of remainder growths rapidly with the field strength. As a rule, it is argued that the pair of particles begin to be produced efficiently from the vacuum only in the fields of strength E >= E_cr. However, the direct calculation shows that the Schwinger formula for creation of e+e- pairs is valid only at the field intensities of E < 0.0291*E_cr. At higher fields, the probability of pair production in an unit space-time volume exceeds unity. In this regard, we refine the formula for the probability of pair creation and numerically find that in the field of strength 2.95% of E_cr the pair production probability is almost 100%.