Continuum limit of lattice gauge theories in the context of renormalized perturbation theory

It is shown that in spite of the modifications introduced by Wilson and Polyakov, the gauge theory on a lattice in the Abelian case in the limit of zero lattice spacing has the same renormalized $S$ matrix as quantum electrodynamics, to all orders in the renormalized coupling constant. Apparently nonrenormalizable vertices contained in the lattice Lagrangian contribute to mass, wave-function, and coupling-constant renormalizations, but do not contribute to the "finite parts" as a result of being multiplied by additional powers of lattice spacing. It is crucial for this renormalizability that the lattice theory respects local gauge symmetry and discrete symmetries and has the correct "classical continuum limit." The fact that in a renormalizable field theory divergences are contained in the first few terms of the Taylor series expansion of the Green's functions about the external momenta, and that these divergences are mild, play an important role in our proof. Umklapp processes characteristic of the lattice regularization do not have any observable consequences in the continuum limit. Thus Wilson's lattice action is well suited for nonperturbative considerations of gauge theories.