

2.7 The meaning of
renormalizability
The previous discussion about the cancellation between the cutoffs
on virtual light-particle momenta and the explicit
cutoff-dependence of
is eerily familiar. It
echoes the traditional discussion of the cancellation of the
regularized ultraviolet divergences of loop integrals against the
regularization dependence of the counterterms of the renormalized
Lagrangian. There are, however, the following important
differences:
- The cancellations in the effective theory occur
even though
is not sent to infinity, and even
though
contains arbitrarily many terms which
are not renormalizable in the traditional sense (i.e., terms whose
coupling constants have dimensions of inverse powers of mass in
fundamental units where
).
- Whereas the cancellation of regularization
dependence in the traditional renormalization picture appears ad-hoc and implausible, those in the
effective Lagrangian are sweet reason personified. This is because
they simply express the obvious fact that
only was introduced as an intermediate step in a
calculation, and so cannot survive
uncancelled in the answer.
This resemblance suggests Wilson’s physical
reinterpretation of the renormalization procedure. Rather than
considering a model’s classical Lagrangian, such as
of Equation (1), as something
pristine and fundamental, it is better to think of it also as an
effective Lagrangian obtained by integrating out still more
microscopic degrees of freedom. The cancellation of the ultraviolet
divergences in this interpretation is simply the usual removal of
an intermediate step in a calculation to whose microscopic part we
are not privy.

