Report on the Quantum Gravity School: the lectures (2 – Asymptotic Safety)
Let me attempt a brief summary of the Asymptotic Safety approach to quantum gravity, as I understood it. (There may very well be inaccuracies and even gross mistakes; I expect you to point them out if you see them). The cornestone of the program is the hypothesis, first proposed by Weinberg, that the renormalization group flow for gravity might have a non-Gaussian fixed point when examined nonperturbatively. That would mean that the quantum theory would be well-defined despite not being perturbatively renormalizable.
Consider a “theory space” formed by all possible diffeomorphism invariant action functionals of the spacetime metric. You can “coordinatize” it by the values of dimensionless coupling constants of different terms, where the dimensionless couplings are constructed from the dimensional ones dividing them by the energy scale k at which you are thinking the theory (to the relevant power). For example, you would have the Einstein-Hilbert action with couplings G and Lambda (suitably rendered dimensionless), plus terms with any power of the curvature scalar, and of the square of the Ricci tensor, and so on. You can define on this space the Exact Renormalization Group Equation (ERGE) which determines the flow of the effective action for gravity in this space. The effective action is the action from which all interactions at scale k can be calculated accurately at tree level. Varying the scale k, the couplings start “running” and some may be turned on or off. If the flow of the effective action, for k going to infinity, reaches a fixed point at which the couplings are not all zero, this is a non-Gaussian UV fixed point and the theory is said to be assymptotically safe. If in addition the flow towards the fixed point is attractive in only a finite number n of dimensions in theory space, the “bare action” you find at infinite k will have only n free parameters, and the exact quantum theory will be as predictive as a perturbatively renormalizable theory with n adjustable parameters is.
Of course, solving the ERGE exactly is out of the question; it is an infinite system of coupled differential equation. The strategy Reuter uses is “truncation” –arbitrarily decide to consider only actions with a given number of terms of a given kind. For example the first and most brutal truncation is the Einstein-Hilbert one: consider only the flow of the two terms of the EH action, with couplings G and L (the L is supposed to be read as "Lambda" and represent the consmological constant). It would be an abuse of language to call this an “approximation”, because a priori there is no reason to believe that the results of the exact flow will be close to those of the truncated flow, or that taking more terms will yield better and better approximations, as long as there are still an infinite amount of neglected terms. To a skeptical mind, this renders the whole program worthless. But Reuter managed to convinced many of us nevertheless that the program was being highly successful. I will explain now the results with which he archivied this effect.
First: the flow of the EH-truncated renormalization group does have a non-Gaussian UV fixed point. The trajectories flowing back from it with decreasing k spend a lot of “time” in the regime where the dimensionless couplings are small. In these region the flow looks “classical”: the dimensionful G and L are constant.
Second: There is a cutoff scheme involved in the calculation. For the exact ERGE, the results are independent of the cutoff scheme, but this is not guaranteed to happen in the truncated calculation. However, Reuter and his collaborators find that the results they have obtained are in fact independent to a high degree of the cutoff used. They take this as partial evidence that a similar fixed point exists in the full, nontruncated theory.
Third: The “next order” of including in the action a third term, proportional to R^2, has been carried out. It must be stressed that there is no reason at all to suppose that the results with this term included would resemble those without it. Instead of two coupled differential equations we have three now, so the situation is much more complicated. However, surprisingly enough, essentially the same fixed point is found! The values of dimensionless G, L on it are almost exactly the same than those at the fixed point found in the EH-truncation, and the coupling of the added term is very small at the critical point.
Fourth: There is very recent work which extends the previous results to actions containing all powers of R up to R^6. The system of 7 coupled diffential equations is now hugely complicated, and if the exact theory did not have a fixed point it would be “magical” that the flow leads to the same point as the previous truncations did. But it does! The GL-projection of the flow near the fixed point is still essentially the same as the one found with the original EH-truncation. Moreover, the dimension of the “attractive hypersurface” is only 3, which means that the bare action has only 3 independent parameters instead of 7 within this truncation. This gives hope that the exact theory may be predictive.
These results had us all very excited! Reuter also talked about some implications for cosmology that arise if we assume the EH-truncated flow to be a good approximation. One important one is that, if the RG trajectory realised in Nature has a “long” classical regime at all, then the physical cosmological constant L is automatically constrained to be much smaller than the physical Newton’s constant G. Thus the smallness of L poses no extra “naturalness” problem beyond the mere existence of a classical regime. Another one is that Reuter expects the truncation to break down as an approximation in the infrared, at length scales much larger than the “classical” regime. Nonlocal terms would presumably begin to act there. A dimensional argument shows that the scale at which this should happen is the scale of the physical cosmological constant! Reuter therefore makes a (very tentative) prediction of new physics at the Hubble scale, and even speculates on a relation to alternative MOND-like theories to dark matter.
Much more work would be needed to see if there is anything substantial in this last speculation, and in the approach as a whole. The exciting thing is that it is a little explored path, which despite being fully "background independent" uses techniques familiar from ordinary QFT, and may be able to make contact with it more easily that models which introduce discrete physics like LQG or spin foams. Of course, even if the exact renormalization group flow has a fixed point with all the required properties and quantum gravity exists as a theory by its own right, this doesn’t mean that this is realised in Nature! String theory provides a very different UV completion to gravity, and if it is true I guess it would render the UV fixed point of “pure gravity” physically irrelevant. But if one of the main motivations for string theory in the first place is the non-renormalizability of pure gravity, then Reuter’s results are making this motivation rather shaky.