Absence of higher derivatives in the renormalization of propagators in quantum field theori.pdf

a r X i v : h e p - t h / 0 2 1 2 0 1 3 v 2 2 8 A p r 2 0 0 3 IFUP-TH/02-44 ABSENCE OF HIGHER DERIVATIVES IN THE RENORMALIZATION OF PROPAGATORS IN QUANTUM FIELD THEORIES WITH INFINITELY MANY COUPLINGS Damiano Anselmi Dipartimento di Fisica E. Fermi, Universita? di Pisa, and INFN 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. 1 1 Introduction The quantization of gravity is still elusive. The removal of divergences of quantum gravity is possible only in the presence of infinitely many independent coupling constants and it is hard, although not impossible in principle, to find computable quantities and make physical predictions about the high-energy behavior of the theory. On the other hand, the strenuous efforts spent in the recent years to search for approaches “beyond quantum field theory” have not produced significant breakthrough