Christine Dantas has tried to construct a list of top ten loop quantum gravity (and spin foam) results that would be somewhat analogous to the list for string theory described in the previous article.
I think that her list is an unsubstantiated outcome of sloppy thinking - thinking based primarily on manipulation with words rather than formulae and mathematical concepts; comparing any point in this list (or their union) to AdS/CFT is exuberantly irrational; and let me explain why.
1) The states of the theory are known precisely. The kinematical Hilbert space has been rigorously constructed. The Hilbert space of spatially diffeomorphism invariant (Hdiffeo) and gauge invariant states of a gauge field on a manifold Sigma has an orthonormal basis whose elements are in one to one correspondence with the diffeomorphism equivalence classes of embeddings of spin network into Sigma. In the case of pure general relativity in 3 + 1 dimensions, with vanishing cosmological constant, the gauge group is SU(2). In this case the labels on the edges are given by ordinary SU(2) spins. [gr-qc/0504147]
Every quantum theory that claims to exist must have a Hilbert space, but having a Hilbert space is not yet a result of anything. All physically usable quantum theories have separable Hilbert spaces and all separable (infinite-dimensional) Hilbert spaces are unitarily equivalent to each other. This implies that at Christine's level of rigor, we have known all separable Hilbert spaces in the world since 1928 or so. Clearly, physics only starts once we have some observables, and their discussion starts with the point #2.
Of course, an additional problem with loop quantum gravity is that its kinematical Hilbert space is not even separable - it seems as the most inconsistent Hilbert space that was ever proposed to be relevant for anything in the history of physics. Literature also seems to imply that it is not unique. Moreover, various states in this Hilbert space that should correspond to the vacuum are known to have unphysical properties and be unusable for physics, see e.g. this paper by Witten about the Kodama state, also discussed in #9.
Much like in the cases below, I would not call #1 a "result" and certainly not an "interesting result". One just randomly chooses a set of degrees of freedom - there are trillions of other choices - and quantizes them according to random, unprincipled rules. He does not get anything better than what he started with: garbage in, garbage out. There is nothing here that should be called a "result" and that could suddenly change the opinion of a rational scientist. It's like a priest who forces a believer to fabricate evidence for one verse of the Bible or another, and when the task is completed, they consider the problem solved. In Christine's case, the Bible is called "An Invitation to Loop Quantum Gravity", as she makes more than obvious. This is no science; it is evangelism.
I think that an opinion can only be called a "scientific result" if its full character was not inserted into the definition of the task: if there is at least something unexpected about it. If there is nothing unexpected about anything, we don't talk about results but about a work with religious dogmas and about spreading of pre-determined propaganda. Unlike string theory where all the major results have been surprising in many ways, there is nothing surprising in loop quantum gravity - just the generic preconceptions and inconsistent math that we started with - and these ten points demonstrate this observation very clearly.
2) Certain spatially diffeomorphism invariant observables have been constructed. In the case of general relativity and supergravity, these include the volume of the universe, the area of the boundary of the universe, or of any surface defined by the values of matter fields. These operators all preserve the diffeomorphism invariance of the states. The area, volume and length operators have discrete, finite spectra, valued in terms of the Planck length. There is hence a smallest possible volume, a smallest possible area, and a smallest possible length, each of Planck scale. The spectra have been computed. The area and volume operators can be promoted to genuine physical observables, by gauge fixing the time gauge so that at least locally time is measured by a physical field. The discrete spectra remain for such physical observables, hence the spectra of area and volume constitute genuine physical predictions of the quantum theory of gravity.
These observables cannot be operationally defined or measured; they're unphysical. Indeed, if the spectrum of distances of areas is universally discrete with a certain Planckian spacing, you will never find an apparatus that could measure these quantities with a better resolution and test whether the predicted spectrum is correct. You would first have to translate these bizarre statements about the spectrum to some truly physical observables such as the graviton scattering amplitudes.
More technically, it is also false that the spectra of the length and volume operators are well-behaved or even uniquely determined (see e.g. page 27 of Nicolai et al.). Only the area operators have a well-defined spectrum because the observables were constructed in such a way that exactly these operators are well-behaved. Moreover, the discreteness of the area is not a derived result that follows from quantum gravity only: it is an equivalent description of the "new variables" that replace the metric by the gauge fields, but this map is not one-to-one and imposes new quantization rules which are exactly the "discrete area" rules. For a mathematically thinking person, these two things - the new variables and the area spectrum - are transparently equivalent, and the derived "result" was simply put in. There is no result: it is just a translation of the assumptions to a different language.
Finally, it is trivial to see that the discreteness of proper areas, lengths, and/or volumes violates the Lorentz symmetry unless the dominant spin networks or spin foams are singular. A discrete spin foam is always a new kind of aether, something that we have known to be fundamentally wrong at least since the 1905 discovery of special relativity.
Figure 1: L. Riofrio, our blogging colleague from the Bush Street and the author of another Lorentz-breaking alternative to string theory, according to which the speed of light is changing via the formula "GM=tc^3" (recent paper at SLAC). ;-) This is another contribution to the discussion whether physics PhDs must inevitably understand basics of modern physics. Well, they may have other virtues, of course. ;-)
3) Loop quantum gravity leads to a detailed microscopic picture of the quantum geometry of a black hole or cosmological horizon. This picture reproduces completely and explains the results on the thermodynamic and quantum properties of horizons from the work of Bekenstein, Hawking and Unruh. This picture is completely general and applies to all black hole and cosmological horizons.
These assertions are completely absurd and contradict everything we know about the black holes. Loop quantum gravity explains absolutely nothing about the black hole puzzles. Is the information lost? Is the interior related to the exterior by the complementarity principle? Loop quantum gravity can't answer such questions.
One must mutate the theory in yet crazier and more ambiguous ways and, for example, allow the space to artificially terminate at the horizon, in order to even allow for the existence of the black holes; simply pretend that the interior - and its entropy - do not exist. What a surprise that one can then heuristically argue that the entropy will be associated with the are of the horizon.
However, all quantitative predictions of this picture have been shown to be wrong. The numerical multiplicative constant controlled by the "Immirzi parameter" came out incorrectly. People attempted to allow a renormalization of the overall area and connect the value of the Immirzi parameter with the quasinormal modes. All these hypotheses have been falsified. The numerical constant in the entropy calculation in loop quantum gravity has been found erroneous about 6 times, and the new predicted results have nothing to do with anything else, certainly not with the quasinormal modes.
In the case of quasinormal modes, the relevant predictions for their behavior was found to be incorrect, too. Ln(3), a value whose relevance for Schwarzschild's highly-damped modes I proved with Andy Neitzke, was abruptly shown to be non-universal, and so on, and so on. The black hole mess in loop quantum gravity is a textbook example of the lethal disasters that one inevitably encounters if she does not care about the mathematical consistency of the picture of quantum gravity.
4) Among the operators that have been constructed and found to be finite on Hdiffeo is the Hamiltonian constraint (or, as it is often called, the Wheeler de Witt equation). Not only can the Wheeler deWitt equation be precisely defined, it can be solved exactly. Several infinite sets of solutions have been constructed, as certain superpositions of the spin network basis states, for all values of the cosmological constant. These are exact, physical states of quantum general relativity. If one fixes a physical time coordinate, in terms of the values of some physical fields, one can also define the Hamiltonian for evolution in that physical time coordinate and it is also given by a finite operator on a suitable extension of Hdiffeo including matter fields.
What a crazy sequence of untrue statements addressed to an audience that must be thoroughly unfamiliar with the messy question of dynamics in loop quantum gravity. The widely cited paper of Nicolai et al. has shown that all the existing attempts to define the Hamiltonian in quantum gravity suffer from virtually all problems that one could imagine: infinite ambiguities of all the coefficients, failure to close the constraints off-shell, remaining UV singularities even after infinitely many choices are made, ultralocality that prevents signals from propagating from one point to another point, and others.
It is fair to say that dynamics in loop quantum gravity is entirely unknown, and what is known seems unphysical, undetermined, and singular. There has not been a single good news in loop quantum gravity so far - something that would indicate that things "fit together". Because of all these facts, there is no actual Hamiltonian or Wheeler-deWitt equation in loop quantum gravity, and Christine's point #4 is meaningless.
5) Evolution amplitudes corresponding to the quantization of the Einstein equations in 3 + 1 dimensions, are known precisely for vanishing and non-vanishing values of the cosmological constant, and for both the Euclidean and Lorentzian theories.
This is a complete nonsense, too. It is not even known whether anything that looks like "space" or "spacetime" exists in either of the hypothetical descriptions of quantum gravity, and everything we have - both general conceptual results as well as detailed results - indicate that the answer is that nothing that looks like space exists within these frameworks. This fundamental problem (combined with the unknown Hamiltonian etc., see above) of course makes it impossible to study any dynamical amplitudes that could be physically interpreted as something relevant for the events in spacetime. Some time ago, we have also explained the irrational character of the attempts to derive the graviton propagator.
6) Spin foam models with matter have been extensively studied in 2+1 dimensions. At least in 2 + 1 dimensions the scattering of matter coupled to quantum gravity is described by a version of deformed special relativity.
Gravity in 2+1 dimensions is very different from gravity in higher dimensions because it has no local dynamical degrees of freedom. The Ricci tensor and the Riemann tensor both have 6 components. Ricci flatness therefore implies flatness. More generally, the matter distribution completely determines the metric. There are no gravity waves or gravitons here. Virtually all characteristic features of higher-dimensional quantum gravity are absent in 2+1 dimensions.
Even if we were satisfied with the 2+1 dimensions and if we decided to flatten our bodies, it is known that the hypothetical link between quantum gravity and loop quantum gravity (Chern-Simons theory, in this case) is invalidated by quantum effects. See e.g. page 4 of Gukov's paper that summarizes some results of Witten etc.; the page starts with the word "Summarizing". The path integrals have different intervals into which the observables belong, some absolute values are missing, and there are other subtle differences.
The 2+1-dimensional gravity is a transparent example showing that the links between Chern-Simons-like gauge theories and gravity are just artifacts of a naive classical reasoning and a rather sloppy manipulation with the notion of an "equal number of degrees of freedom" which is something that always becomes very subtle in any quantum theory because the phase space of a quantum theory is discretized and its dimension therefore can't be fully defined. Reparameterizing variables is unphysical if we are imagining completely different dynamics behind the two sets of degrees of freedom.
SU(2) gauge fields are supposed to behave as d=4 Yang-Mills theory which is something completely different than d=4 quantum gravity, and any conclusions implied by a sloppy relation between the same number of classical fields on both sides is all but guaranteed to lead to nonsensical conclusions at the quantum level, especially at strong coupling. If such a simple classical counting of the degrees of freedom were physically relevant, holography could not work. Misunderstandings of these key differences between classical physics and quantum mechanics are also behind the fact that people as famous as Roger Penrose can't comprehend holography.
7) Coupling to all the standard forms of matter fields are understood, including gauge fields, spinors, scalars and higher p-form gauge fields. Inclusion of matter fields does not affect the finiteness and discreteness of the area and volume observables.
This absurd paragraph is probably based on Lee's typical and favorite quotes of the type "AB, CD, everything else has been fully solved by loop quantum gravity [EF, GH]". When you actually open EF, GH, you only see a lot of nonsense, inconsistencies, and silliness - besides things that have nothing to do with Lee's sentence.
It is known that loop quantum gravity does not predict any extra matter and even the ad hoc inclusion of the realistic matter leads to profound inconsistencies. One cannot include spinors with chiral interactions (required for weak interactions) and probably not even scalars that could lead to the Higgs mechanism. The fact that loop quantum gravity does not imply the existence of other forces and matter is not a mild problem because it is more or less known that in consistent quantum theories of gravity, gravity must be the weakest force and the remaining fields therefore cannot be treated as small additions or perturbations.
8) Spin foam models appropriate for Lorentzian quantum gravity, called causal spin foams, have quantum analogues of all the basic features of general relativistic spacetimes. These include dynamically generated causal structure, light cones and a discrete analogue of multifingered time, which is the freedom to slice the spacetime many different ways into sequences of spatial slices. The spatial slices are spin networks, which are quantum analogues of spatial geometries.
It is known that one can't avoid the pathological domination of the path integral by degenerated, crumpled spacetimes unless one introduces an additional selection rule that prohibits these otherwise generic configurations. All such rules violate unitarity - because they depend on global features of the configuration - and make the theory inconsistent by themselves.
9) For the case of non-vanishing cosmological constant, of either sign, there is an exact physical state, called the Kodama state, which is an exact solution to all of the quantum constraint equations, whose semiclassical limit exists. That limit describes deSitter or anti-deSitter spacetime. Solutions obtained by perturbing around this state, in both gravitational and matter fields, reproduce, at long wavelength, quantum field theory in curved spacetime and the quantum theory of long wave length, free gravitational waves on deSitter or anti-deSitter spacetime. By studying excitations of these states one reproduces conventional quantum field theory, as well corrections to it which may be compared with experiment.
Again, the Kodama state is unphysical because the perturbations around it include negative-energy and even negative-norm (negatively probable) states, as explained by Witten. What Witten politely writes, attempting to paint the Kodama state as an a priori nice one, is almost clearly the last meaningful word that one can say about these superficially intriguing ideas in the context of physics. One can't build physics on states that have as many ghosts as the physical states. Dirac's standards even back in 1927 were certainly higher, and today we know many more constraints, not less.
10) The inverse cosmological constant turns out to be quantized in integral units, so that k = 6pi/G is an integer. This is related to a basic result, which is that in 3 + 1 dimensions, a nonzero cosmological constant implies a quantum deformation of the gauge group whose representations label the edges of the spin networks, where the level which parameterizes the quantum deformation has been calculated for the Euclidean case.
I find the proposal that because of these equations, a quantum deformation of the tangential group could be relevant for a nonzero cosmological constant interesting - and somewhat analogous to some proposals in the context of AdS/CFT and especially dS/CFT. Unfortunately, so far it is not supported by any example of a consistent theory where such a mechanism would be relevant - not even in the dS/CFT case - so these ideas remain wild speculations.
This article is far from having listed all problems of loop quantum gravity. See also the text called "Objections...".