Unitarity and electron-positron pairs created by strong external fields

Lowest-order perturbative calculations of the electron-positron production probability in relativistic heavy-ion collisions exceed unitarity bounds for the heaviest collision systems at extreme relativistic energies and sufficiently small impact parameters. Starting with the exponential representation of the time-evolution operator in the Furry picture, we derive manifestly unitary and gauge-invariant expressions for transition amplitudes and probabilities associated with the created electron-positron pairs by employing the Magnus expansion to first order. The time-evolved ground state of the electron-positron field around the heavy nuclei is expressed as a superposition of the unperturbed vacuum state and virtual excitation modes consisting of electron-positron pairs.