  I study some aspects of the renormalization of quantum field theories with
infinitely many couplings in arbitrary space-time dimensions. I prove that when
the space-time manifold admits a metric of constant curvature the propagator is
not affected by terms with higher derivatives. More generally, certain
lagrangian terms are not turned on by renormalization, if they are absent at
the tree level. This restricts the form of the action of a non-renormalizable
theory, and has applications to quantum gravity. The new action contains
infinitely many couplings, but not all of the ones that might have been
expected. In quantum gravity, the metric of constant curvature is an extremal,
but not a minimum, of the complete action. Nonetheless, it appears to be the
right perturbative vacuum, at least when the curvature is negative, suggesting
that the quantum vacuum has a negative asymptotically constant curvature. The
results of this paper give also a set of rules for a more economical use of
effective quantum field theories and suggest that it might be possible to give
mathematical sense to theories with infinitely many couplings at high energies,
to search for physical predictions.
