I believe that this program is important and it would be out of place to criticize its physics too much. Here are what makes it really informative for me:
- The counterterm structure of gravity in the t'Hooft Veltman formalism is hard to get intuition about. Any method of understanding the physical content of a counterterm is physically important, regardless of any overreach in the physical interpretation.
- The conformal structure of GR has never been properly formulated or understood. We can solve classical electrostatic problems using conformal invariance, but we can't solve GR problems using conformal invariance. Why not? The gravitational field has scale invariance in the same way Maxwell's equations do, and it is just as local, but something global breaks down when you try to do inversion. This suggests there is something classical we still don't understand.
- The PT program is still young, and there are a host of new insights that are to be gained by exploring every action that was traditionally rejected because of some sickness but is saved by PT. The no-ghost theorem by itself is important, and must be internalized. Higher derivative unitary theories seemed impossible until now. The classical theory is impossible as formulated--- it doesn't have a decent initial value formulation. This is probably solved by the analytic continuation methods in PT theories.
When you have a new idea, you are acting as its advocate, you have to present the most far-reaching conclusions that you think might hold. This means that you sometimes overreach. But you asked for a critique--- here is an ill-informed one with no new ideas, (unlike the paper):
- Conformal gravity can't solve quantum gravity, because it is based on path integrals, and not string theory. It will fail for the same reason N=8 SUGRA fails. Not because it is not perturbatively renormalizable or unitary, but because the physics is wrong at short distances. It needs to be made holographically consistent, at the very least, and the quantum gravity naive path integral isn't. The authors claim the theory has no black holes, but there are black hole vacuum solutions in the theory, since these are empty-space solutions, and there should be extremal black holes too, because the extra form fields won't break conformal invariance any worse than vacuum. This is a feeling, not a calculation.
- Vacuum Conformal gravity solutions include Einstein vacua, and it is a higher derivative form. If you recover only Einstein gravity solutions, and no exponentially wild ones (as suggested by the proof of unitarity), one would expect that barring a symmetry requirement, you would recover Einstein gravity at long distances. There might be a conformal symmetry requirement that forces the Einstein term to vanish, and requires conformal gravity in some conformally invariant theory (maybe, I don't know), but at low energies, I can't imagine how you could require conformal gravity when you have nonconformal stuff like QCD breaking any putative additional symmetry by an explicit confinement scale? The argument in the paper is that this breaking is spontaneous and not explicit, but this is not true in QCD--- the conformal breaking is from the running coupling, not from a VEV.
- It seems unlikely that dark-matter and dark-energy can be described well by this approach, because the standard cosmological model feels pretty good to me.
- Bender's PT methods require hard analytic continuations, and I find it difficult to verify the results, because they are so sensitive to nuances of analytic continuation. Also, the results are sometimes easier to express by using a more traditional quantum mechanical formulation, rather than the PT one. This is entirely my fault, but still, I have to say it so you can take the rest of this with a grain of salt. The proof of unitarity is the key point. This is a huge surprise, and I wouldn't trust it until I verified it for myself.
But my criticisms, although sincere, are deplorable in this context. This mathematical theory is full of new and beautiful ideas, regardless of whether the more speculative physical claims turn out to be true.
This answer is my own preliminary reaction. I found this paper truly moving, thank you for bringing it up.