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Does anyone know of good essays/articles that critique the various approaches to canonical quantum gravity (as opposed to string theory)? I just want to get a sense of what all has been tried, and what people have succeeded and failed at. I do not want detailed technical discussions of the model itself.

In particular, I want to know what crucial ingredients are needed in the canonical approach to incorporate the holographic bound on the entropy? Also, has there been a successful black-hole entropy count using canonical approaches? Does any canonical approach hold promise to reconcile quantum physics in the frames of an observer falling into a black hole and an observer that stays outside?

I know of the following books, which are very detailed, but I want more of a bird's eye view:

  1. Steven Carlip, Quantum Gravity in 2+1 dimensions
  2. Carlo Rovelli, Quantum Gravity
  3. Daniele Oriti, Approaches to Quantum Gravity

I am not looking for books, but essays and/or overview articles.

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  • $\begingroup$ What do you mean by "success"? There has been no success in quantum gravity by any measure of scientific enterprise. The better question is why people still expect to see success with a force description when nature keeps screaming "gravity is not a force" at them? $\endgroup$ – CuriousOne Sep 22 '15 at 22:00
  • $\begingroup$ For LQG, see e.g. arxiv.org/abs/hep-th/0501114 $\endgroup$ – Qmechanic Sep 22 '15 at 22:27
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To start with, there is no canonical approach to quantum gravity, rather, there are different quantisation procedures (as for all the other interactions), which do or do not fail for this or that other reason without having to go to black hole entropy.

The standard quantisation procedure by means of the path integral fails simply because it is not renormalisable: everything works (somehow) fine but the coupling constant does not have the right dimensions required for a theory to be renormalisable perturbatively. There is no critique, it is just what it is and you can find reasonable walkthroughs in any textbook on QFT (Ryder, for instance).

As a workaround, since the path integral terms do not seem to be polite enough, some additional models have been proposed. Loop quantum gravity and spin foam models (many papers by Oriti, see for example an introduction here, or by Perez, see here and references therein) happen to be very promising, because they reduce the complexity of the infinite degrees of freedom discretising the topology of the space time through triangulations: states and functionals are then defined on the knots networks and scattering amplitudes can be computed in terms of irreducible representations of some Lie groups. Unfortunately, in four dimensions such representations happen to be infinite dimensional and thus problems of renormalisability still occur. A branch named Group Field Theory is being developed only using Lie groups representations to write down all the observable quantities of interest (for instance Krajewski) but still at some point they fail upon some infinities (mostly due to Lie groups being non-compact).

Other theories try to only quantise some linear terms on the right hand side of the Einstein's equations, but besides being very particular effective theories, they too have problems of renormalisability in correspondence of some divergences.

Quantisation procedure via Fock space would not work, being the equation of motion non-linear.

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  • $\begingroup$ A very naive question, but why do we think gravity has to quantize directly, at all? What if it is a mesoscopic thermodynamic average of something else, instead, like water waves? We know how to describe the quantum theory of water reasonably well, but it doesn't carry trough to the level of macroscopic waves. $\endgroup$ – CuriousOne Sep 22 '15 at 22:29
  • $\begingroup$ Well, that might in principle be. The common belief is that since everything else in the universe in quantised (and the standard model does work), we reasonably expect gravity to be quantised as well. $\endgroup$ – gented Sep 22 '15 at 22:39
  • $\begingroup$ I have no doubts about the ultimate quantization, the difficulties in doing it just gives me this feeling as if we are missing something between the phenomenology of gravity and the quantum level, but I haven't seen any papers that would explore the question what that missing piece may be or a (simple?) reason why it's not needed, after all. $\endgroup$ – CuriousOne Sep 22 '15 at 22:44
  • $\begingroup$ "Canonical" quantum gravity refers to the process of canonical quantization via Hamiltonians and defining operators from Poisson brackets (leading to the de Witt equation), not a standard method. $\endgroup$ – Slereah Sep 25 '15 at 8:31

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