  The Wilsonian renormalization group implies that an arbitrary four
dimensional field theory with an ultraviolet cutoff is equivalent to a theory
which is renormalizable by power counting at energy scales much below the
cutoff. This applies to any theory including those with non-renormalizable
interactions as long as we fine-tune the mass parameters. We analyze two simple
models with current-current interactions but without elementary gauge fields
from this viewpoint. We show how to tune the parameters of the models so that
they become equivalent to QED at energies much below the cutoff.
