The meaning of renormalizability

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 $ℒeff$ 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 $Λ$ is not sent to infinity, and even though $ℒe ff$ 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 $ℏ = c = 1$ ).
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 $Λ$ 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 $ℒ$ 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.