Approaches to Quantum Gravity

112 R. PercacciIn section 8.2 I introduce the general idea of asymptotic safety; the reader isreferred to [46] for a more detailed discussion. In section 8.3 I describe some peculiaritiesof the gravitational RG, which derive from the dual character of the metricas a dynamical field and as definition of lengths. Recent evidence for a FP, comingmainly from the ERGE, is reviewed in section 8.4. Some relations to otherapproaches to Quantum Gravity are briefly mentioned in section 8.5.8.2 The general notion of asymptotic safetyThe techniques of effective QFT have been recognized as being of great generalityand are now quite pervasive in particle physics. An effective field theory isdescribed by an effective action Ɣ k which can be thought of as the result of havingintegrated out all fluctuations of the fields with momenta larger than k. Weneed not specify here the physical meaning of k: for each application of the theoryone will have to identify the physically relevant variable acting as k (in particlephysics it is usually some external momentum). One convenient definition of Ɣ kthat we shall use here is as follows. We start from a (“bare”) action S[φ A ] formultiplets of quantum fields φ A , describing physics at an energy scale k 0 .Weaddto it a term S k [φ A ], quadratic in the φ A , which in Fourier space has the form:S k [φ] = ∫ d d qφ A RkAB (q 2 )φ B .ThekernelRkAB (q 2 ), henceforth called the cutofffunction, is chosen in such a way that the propagation of field modes φ A (q) withmomenta q < k is suppressed, while field modes with momenta k < q < k 0 areunaffected. We formally define a k-dependent generating functional of connectedGreen functions∫ (∫ )W k [J A ]=−log (dφ A ) exp −S[φ A ]−S k [φ A ]− J A φ A (8.1)and a modified k-dependent Legendre transform∫Ɣ k [φ A ]=W k [J A ]− J A φ A − S k [φ A ] , (8.2)where S k has been subtracted. The “classical fields” δW kare denoted again φδ J AAfor notational simplicity. This functional interpolates continuously between S, fork = k 0 , and the usual effective action Ɣ[φ A ], the generating functional of oneparticleirreducible Green functions, for k = 0. It is similar in spirit, but distinctfrom, the Wilsonian effective action. In the following we will always use thisdefinition of Ɣ k , but much of what will be said should be true also with otherdefinitions.In the case of gauge theories there are complications due to the fact that the cutoffinterferes with gauge invariance. One can use a background gauge condition, whichcircumvents these problems by defining a functional of two fields, the background