Quantum Gravity

PATH-INTEGRAL QUANTIZATION 63first place. Genuine non-perturbative approaches are also the canonical methodsdescribed in Chapters 3–6, where established quantization rules are applied toGR. Before we enter the discussion of these approaches, we shall address furthernon-perturbative approaches to quantum general relativity and shall then endwith a brief review of quantum supergravity.2.2.5 Asymptotic safetyAs we have discussed in Sections 2.2.2 and 2.2.3, quantum gravity is perturbativelynon-renormalizable. We have also seen that genuine effects can neverthelessbe calculated at sufficiently low energies if one employs the concept of effectivefield theories. But what happens at large energies? There are two options. Oneoption is that the quantization of GR is impossible and that one has to embarkon a more general framework encompassing all interactions. This is the idea ofstring theory to be discussed in Chapter 9. The other option is that quantumGR is non-perturbatively renormalizable. This may happen in various ways. Oneway is the direct non-perturbative quantization of the Einstein–Hilbert action;this is attempted in the canonical approaches to be discussed in Chapters 4–6and in the path-integral approach via Regge calculus or dynamical triangulationto be discussed in Section 2.2.6. Another way employs the notion of asymptoticsafety, and this is the subject of the present subsection. Strictly speaking, it isnot a direct quantization of GR because more general actions than the Einstein–Hilbert action are used in general. It is, however, close in spirit to it, especiallyto the effective-action approach discussed above.The notion of asymptotic safety was introduced by Weinberg (1979) and isconnected with the fact that coupling parameters in quantum field theory areenergy-dependent due to renormalization. A theory is called asymptotically safeif all essential coupling parameters g i (these are the ones that are invariant underfield redefinitions) approach a non-trivial fixed point for energies k →∞.The‘asymptotic’ thus refers to the limit of large energies, and the ‘safe’ refers to theabsence of singularities in the coupling parameters. A non-trivial fixed point ischaracterized by the fact that at least one of the g i is unequal to zero. We assumethat the g i are made dimensionless, that is,g i (k) =k −di ḡ i (k) ,where ḡ i (k) are the original coupling parameters with mass dimensions d i .A central notion in this approach is played by the ‘theory space’ defined byall action functionals that depend on a given set of fields and contain all termsthat are consistent with a certain symmetry requirement; it is here where onemakes close contact with the effective-theory idea. In the gravity context thesymmetry is, of course, diffeomorphism invariance. One thus considers all actionsS[g µν ,...] with this invariance, that is, actions containing terms R/16πG,c 1 R 2 , c 2 R µν R µν , and so on. There are thus infinitely many coupling parametersḡ i (k) givenbyG, Λ,c 1 ,c 2 ,... (all k-dependent, that is, ‘running’) and their di-