Multiloop Euler-Heisenberg Lagrangians, Schwinger pair creation, and the QED N - photon amplitudes

Abstract: An update is given on our long-term effort to perform a three-loop check on the Affleck-Alvarez-Manton/Lebedev-Ritus exponentiation conjecture for the imaginary part of the Euler-Heisenberg Lagrangian, using 1+1 dimensional QED as a toy model. After reviewing the history and significance of the conjecture, we present trigonometric integral representations for the single electron loop contributions to the three-loop Lagrangian, and develop a symmetry-based method for the calculation of their weak-field expansion coefficients.