Approaches to Quantum Gravity

Asymptotic safety 123larger, and it is easy to find theories where a polynomial potential in φ is renormalizableand asymptotically free. Thus, gravity seems to provide a solution to theso-called triviality problem of scalar field theory.It is tempting to speculate with Fradkin & Tseytlin [16] that in the presenceof gravity all matter interactions are asymptotically free. One-loop calculationsreportedin[8; 39] indicate that this may be the case also for gauge and Yukawainteractions. Then, in studying the FP, it would be consistent to neglect matterinteractions, as we did in the 1/N expansion. If this is the case, it may becomepossible to show asymptotic safety for realistic unified theories including gravityand the SM.For the time being, the Gravitational FP has been found with a number ofdifferent approximations: the 2 + ɛ expansion, the 1/N expansion, polynomialtruncations with a variety of cutoffs and gauges, the two Killing vector reductionand the most general four-derivative gravity theory at one loop. The fact that allthese methods yield broadly consistent results should leave little doubt about theexistence of a nontrivial FP with the desired properties.8.5 Other approaches and applicationsIn this final section we briefly comment on the relation of asymptotic safety toother approaches and results in Quantum Gravity.Gravity with the Einstein–Hilbert action has been shown by Goroff & Sagnotti[18] andvandeVen[45] to be perturbatively nonrenormalizable at two loops.Stelle [42] proved that the theory with action (8.11) and = 0 is perturbativelyrenormalizable: all divergences can be absorbed into redefinitions of the couplings.In general, asymptotic safety does not imply that in the UV limit only a finitenumber of terms in (8.10) survive: there could be infinitely many terms, but therewould be relations between their coefficients in such a way that only a finite numberof parameters would be left free. At one loop or in the large-N limit, the ERGEpredicts that the UV critical surface can be parametrized by the four couplings ˜,˜G, λ and ξ, the first two being nonzero at the FP and UV-relevant, the latter twobeing asymptotically free and marginal. Thus, at least in some approximations,asymptotic safety implies that near the FP quantum corrections to the action (8.11)will not generate new terms when one takes the UV limit. This is very similar to theresult of Stelle. The main difference lies therein, that the perturbative proof holdsat the Gaussian FP while the statement of asymptotic safety holds near the non-Gaussian one. According to the ERGE, the Gaussian FP is unstable, and movingby an infinitesimal amount towards positive ˜G (even with ˜ = 0) would cause thesystem to be dragged in the direction of the repulsive eigenvector towards the non-Gaussian FP (see fig. 8.1). It is unclear whether in a more accurate description it