Static Electron–Positron Pair Creation in Strong Fields for a Non-linear Dirac Model

We consider the Hartree–Fock approximation of Quantum Electrodynamics, with the exchange term neglected. We prove that the probability of static electron–positron pair creation for the Dirac vacuum polarized by an external field of strength Z behaves as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1-\exp(-\kappa Z^{2/3})}$$\end{document} for Z large enough. Our method involves two steps. First, we estimate the vacuum expectation of general quasi-free states in terms of their total number of particles, which can be of general interest. Then, we study the asymptotics of the Hartree–Fock energy when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z \to+\infty}$$\end{document} which gives the expected bounds.