Regarding Non-renormalizatibility of GR

Question

I've been doing some reading trying to get to a better understanding of some renormalization issues with the Einstein-Hilbert action. But, something odd came into mind that I'm hoping some users may have the background to comment on it. Specifically, most of the sources I have seen are based on expanding the Einstein Hilbert action about a flat background to achieve a Lagrangian of the form $$(\partial h)^2+{1\over{k}}(\partial h)^2h+{1\over k^2}(\partial h)^2h^2+...$$ with terms that become more and more divergent the further out one goes. But, for some reason I find I'm a little pre-occupied with the notion that perhaps the particular expansion itself has issues with convergence that may or may not fundamentally have to do with the E-H action in its non-expanded form. We commonly entertain the notion of a cut-off energy range to deal with the renormalization issues in the 'effective field theory' speak. How do we know that the expansion of the E-H itself isn't only valid as a low energy approximation though? Do we know that there isn't an expansion of an alternate form which has convergent terms for a range of energies where the first runs into problems?

Answer 1

I think this basically sums up the program for what quantum gravity is. The modern viewpoint is that general relativity (and really just about any quantum field theory) is an effective field theory, and the full theory of quantum gravity must provide an ultraviolet completion. As explained in the Donoghue review suggested by bechira (another good review is the Living Review by Cliff Burgess), the effective field theory viewpoint suggests that the EH action should be supplemented with higher curvature corrections (terms like $R^2$, $R_{\mu\nu} R^{\mu\nu}$, etc.), suppressed by appropriate powers of the Planck scale. This makes the effects of these terms difficult to detect, and in general the coefficients in front of these terms will depend on the details of the UV completion. For example, in string theory these terms can be computed using matching calculations to a low energy effective action (which includes in addition to the graviton a scalar dilaton field and a 2-form $B_{\mu\nu}$ field).

Another idea that seems in line with what you are asking is the Asymptotic Safety program in quantum gravity. In that scenario, they are looking for a fixed point that the theory flows to in the UV that is different from the free theory (i.e. perturbations around flat space). I think not much is known about the UV theory, and most research consists of proving that it actually exists. But, if it did exist, this would probably be the alternate expansion that you are looking for.