QED in external fields , a functional point of view

A functional partial differential equation is set for the proper graphs generating functional of QED in external electromagnetic fields. This equation leads to the evolution of the proper graphs with the external field amplitude and the external field gauge dependence of the complete fermion propagator and vertex is derived non-perturbativally. The study of QED in the presence of external electromagnetic fields started long ago with the computation of the quantum corrections to the Maxwell Lagrangian [1], for which recent and rich studies can be found in [2]. We find also among the studies of the effect of an external field the dynamical chiral symetry breaking by a magnetic field, the magnetic catalysis [3]. In the studies of magnetic catalysis, the full fermion propagator (including the interaction with the external and the dynamical gauge fields) as well as the full vertex and full photon propagator are used in the Schwinger-Dyson equation. However, usual studies of the Green's functions in an external field take into account the quantum corrections coming from the dynamical gauge field only with pertubative arguments [3]. We will describe here a non-perturbative approach, similar to the background field methods [4], which will lead us to the external field gauge dependence of the full fermion propagator and the full vertex. For this we will set up a functional differential equation showing the evolution of the effective action (Legendre transform of the connected graphs generator functional)