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I study some aspects of the renormalization of quantum field theories withinfinitely many couplings in arbitrary space-time dimensions. I prove that whenthe space-time manifold admits a metric of constant curvature the propagator isnot affected by terms with higher derivatives. More generally, certainlagrangian terms are not turned on by renormalization, if they are absent atthe tree level. This restricts the form of the action of a non-renormalizabletheory, and has applications to quantum gravity. The new action containsinfinitely many couplings, but not all of the ones that might have beenexpected. In quantum gravity, the metric of constant curvature is an extremal,but not a minimum, of the complete action. Nonetheless, it appears to be theright perturbative vacuum, at least when the curvature is negative, suggestingthat the quantum vacuum has a negative asymptotically constant curvature. Theresults of this paper give also a set of rules for a more economical use ofeffective quantum field theories and suggest that it might be possible to givemathematical sense to theories with infinitely many couplings at high energies,to search for physical predictions.Comment: 29 pages; v2 contains minor adjustments and proof corrections - CQG  in pres

Absence of higher derivatives in the renormalization of propagators in quantum field theories with infinitely many couplings