Puppet_Sock:

The big deal in QED is that you can show that all of those loops
can be swept under one rug. Or rather, under a finite number of
rugs. What's that buzz phrase? Dyson-Schwinger equations? Not sure.
Been a looooong time. But the idea is, at each order of alpha,
you can show that by modifying the "bare" values of just the
mass and charge of the electron, plus the mass of the photon,
you can "hide" the infinities. So, a finite set of measurements
fixes the entire theory.

What you get is that the divergences cancel order by order,
so that the number you get at each order is finite (ward identities).
That doesnt guarantee that the series converges in the limit of an
infinite number of terms, and indeed it most likely does not.

The question is then, whther qed is a theory which is independent of
anything else. It's quite likely that it isn't, since the only interacting
theories which are permissible in a relativistic theory are those which
are asymptotically free. But, since qed is just the low energy limit of
a theory which is believed to be asymptotically free, qed need only be
finite up to an energy known as the landau pole with the landau pole
being large (relative to the point that qed is no longer the correct
description of the phenomena). I recall a crude estimate of something
like 10^250 GeV, which exceeds the energy of the universe and is well
above the point at which qcd is already needed to explain the muon magnetic
moment).