-----------------------------------------------------
S8h. Nonrenormalizable theories as effective theories
-----------------------------------------------------
The difference between renormalizable and unrenormalizable theories is
that the former are specified by a (small) finite number of parameters
while the latter are specified by an infinite number of parameters.
In a renormalizable quantum field theory, only few counterterms
must be added to the action in order to get a consistent
finite perturbative expansion at all orders. This means that a few
parameters suffice to get a consistent theory which will be correct
at the energies of interest (which should be essentially independent
of what happens at the inaccessible large energies).
In a nonrenormalizable quantum field theory, infinitely many
counterterms must be added to the action in order to get a consistent
finite perturbative expansion at all orders. This means that with a few
parameters one can only get an effective low order theory, which may,
however, still be good enough at the energies of interest.
But for better approximation, one needs to determine more and
more parameters...
In both cases, it is possible to extract approximate results from
computations, and the parameters can be tuned to fit the experimental
results. This gives a consistent procedure for predictions. Indeed,
many nonrenormalizable theories are in use as effective field theories.
(See hep-ph/0308266 for a recent survey on effective field theories.)
People who dislike nonrenormalizable theories do this on the basis of
a claim that their predictive value is nil because of the infinitely
many constants. But this is as unfounded as saying that thermodynamics
is not predictive because it depends on a function (the expression for
the free energy, say) that requires an infinite number of degrees of
freedom for its complete specification. Clearly, in the latter case,
the widespread use of finitely parameterized imperfect free energies
does not hamper the usefulness of thermodynamics. The same can be
said about nonrenormalizable field theories. It only implies that to
extract arbitrarily precise predictions one needs correspondingly
much information as input. We know that this is the case already for
many simpler phenomena in physics.
See also
J.Gegelia, G.Japaridze, N.Kiknadze, K.Turashvili
"Renormalization" Of Non Renormalizable Theories
hep-th/9507067.
J Gegelia, G Japaridze
Perturbative Approach to Non-renormalizable Theories
hep-th/9804189,