Asymptotic estimates in quantum electrodynamics

We discuss various aspects of the estimate of large orders of perturbation theory in quantum electrodynamics. Gauge-invariant subclasses of diagrams corresponding to a fixed number of fermion loops have a behavior dominated by solutions to the coupled Maxwell and Dirac equations for complex values of the charge. We present numerical evidence for the existence of such solutions. The complete theory involving an arbitrary number of fermion loops is expected to exhibit strong cancellations. We show the relation of this problem to the Thomas-Fermi approximation and raise some related mathematical questions.