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Static Electron-Positron Pair Creation in Strong Fields for a Nonlinear 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 $1-\exp(-κZ^{2/3})$ 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 $Z\to+\infty$ which gives the expected bounds.