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

TL;DR

This paper demonstrates that in certain quantum field theories with infinitely many couplings, higher derivative terms do not appear in propagators when the spacetime has constant curvature, simplifying the theory's structure and aiding quantum gravity research.

Contribution

It proves the absence of higher derivative terms in propagators for theories with constant curvature manifolds and clarifies the structure of non-renormalizable theories with infinitely many couplings.

Findings

Higher derivatives are absent in propagators on constant curvature manifolds.

Certain lagrangian terms remain ungenerated by renormalization if absent at tree level.

The metric of constant curvature is a suitable perturbative vacuum in quantum gravity.

Abstract

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.