  Following Feynman's successful treatment of the polaron problem we apply the
same variational principle to quenched QED in the worldline formulation. New
features arise from the description of fermions by Grassmann trajectories, the
supersymmetry between bosonic and fermionic variables and the much more
singular structure of a renormalizable gauge theory like QED in 3+1 dimensions.
We take as trial action a general retarded quadratic action both for the
bosonic and fermionic degrees of freedom and derive the variational equations
for the corresponding retardation functions. We find a simple analytic,
non-perturbative, solution for the anomalous mass dimension gamma_m(alpha) in
the MS scheme. For small couplings we compare our result with recent four-loop
perturbative calculations while at large couplings we find that gamma_m(alpha)
becomes proportional to (alpha)^(1/2). The anomalous mass dimension shows no
obvious sign of the chiral symmetry breaking observed in calculations based on
the use of Dyson-Schwinger equations, however we find that a perturbative
expansion of gamma_m(alpha) diverges for alpha > 0.7934. Finally, we
investigate the behaviour of gamma_m(alpha) at large orders in perturbation
theory.
