The probability distribution of the number of electron-positron pairs produced in a uniform electric field

TL;DR

This paper derives the probability distribution and key statistical measures for electron-positron pair production in a uniform electric field, revealing divergence issues at supercritical fields and restoring positivity through discrete level summation.

Contribution

It introduces a recursive formula for calculating pair production probabilities and analyzes the distribution's behavior, especially at supercritical field strengths.

Findings

Probability-generating function constructed and analyzed.

Divergence occurs at supercritical fields, indicating continuum limit failure.

Summation over discrete levels restores positive definiteness.

Abstract

The probability-generating function of the number of electron-positron pairs produced in a uniform electric field is constructed. The mean and variance of the numbers of pairs are calculated, and analytical expressions for the probability of low numbers of electron-positron pairs are given. A recursive formula is derived for evaluating the probability of any number of pairs. In electric fields of supercritical strength |eE| > \pi m^2/ \ln 2, where e is the electron charge, E is the electric field, and m is the electron mass, a branch-point singularity of the probability-generating function penetrates the unit circle |z| = 1, which leads to the asymptotic divergence of the cumulative probability. This divergence indicates a failure of the continuum limit approximation. In the continuum limit and for any field strength, the positive definiteness of the probability is violated in the tail of the distribution. Analyticity, convergence, and positive definiteness are restored upon the summation over discrete levels of electrons in the normalization volume. Numerical examples illustrating the field strength dependence of the asymptotic behavior of the probability distribution are presented.