Approaches to Quantum Gravity

8Asymptotic safetyR. PERCACCI8.1 IntroductionThe problems of perturbative Quantum Field Theory (QFT) in relation to the UVbehaviour of gravity have led to widespread pessimism about the possibility ofconstructing a fundamental QFT of gravity. Instead, we have become accustomedto thinking of General Relativity (GR) as an effective field theory, which only givesan accurate description of gravitational physics at low energies. The formalism ofeffective field theories provides a coherent framework in which quantum calculationscan be performed even if the theory is not renormalizable. For example,quantum corrections to the gravitational potential have been discussed by severalauthors; see [5] and references therein. This continuum QFT description is widelyexpected to break down at very short distances and to be replaced by somethingdramatically different beyond the Planck scale. There is, however, no proof thatcontinuum QFT will fail, and the current situation may just be the result of the lackof suitable technical tools. Weinberg [46] described a generalized, nonperturbativenotion of renormalizability called “asymptotic safety” and suggested that GRmay satisfy this condition, making it a consistent QFT at all energies. The essentialingredient of this approach is the existence of a Fixed Point (FP) in the RenormalizationGroup (RG) flow of gravitational couplings. Several calculations wereperformed using the ɛ-expansion around d = 2 dimensions, supporting the viewthat gravity is asymptotically safe [17; 10; 20]. However, the continuation to fourdimensions (ɛ → 2) was questionable and this line of research slowed down forsome time. It was revived by Reuter [36] who calculated the gravitational beta functionsdirectly in d = 4 dimensions, using a truncation of an Exact RenormalizationGroup Equation (ERGE). Matter couplings were considered by Dou & Percacci[13]; then Souma [41] found that these beta functions admit a non-Gaussian FP.Further work [22; 23; 34; 11] strongly supports the view that this FP is not a mereartifact of the approximations made. An extensive review of this subject can befound in [29].Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.Published by Cambridge University Press. c○ Cambridge University Press 2009.