Physicists have been trying to explain the "loop quantum gravity" advocates that their statement about the discreteness of areas in their model - something that is often sold as the only result of loop quantum gravity - is meaningless for many years.
There are many ways to see it.
The area of a surface Sigma is not a gauge-invariant quantity. In the context of general relativity, gauge invariance means invariance under coordinate redefinitions. If you change the coordinates, the surface Sigma defined by some locations in a coordinate space changes into a different surface. Consequently, any quantity associated with such a surface in the coordinate space - such as its proper area - fails to be gauge-invariant. It is thus unphysical because only gauge-invariant i.e. coordinate-independent quantities may be measured.
In fact, the scattering S-matrix amplitudes (and its AdS counterparts given by the correlators on the boundary) are the only known "big" gauge-invariant observables in quantum gravity we know. I've been saying this trivial thing for years, for example under "S-matrix" in Objections to LQG where I also write that the discreteness of areas is not measurable because there are no "finer" sticks that could be used to measure areas with a trans-Planckian accuracy (it is actually exactly the same problem as gauge-dependence if you think for a while).
The discreteness of areas is not only gauge-dependent. Even if you decide to forbid all gauge transformations, the discreteness is also an unphysical artifact of a globally invalid change of variables. In loop quantum gravity, one makes such a field redefinition that the proper areas essentially become magnetic fluxes. Proper areas (integrals of an induced sqrt[det(g)] on the surface) are clearly not quantized in a normal quantized theory of gravity based on the metric tensor but magnetic fluxes associated with a gauge field are quantized which proves that the field redefinition can't be one-to-one. But the champions of loop quantum gravity are always ready to promote a lethal bug as a virtue.
Incidentally, the discreteness of proper areas is also the main assumption in the simplest proof that such a theory can't be Lorentz-invariant. If the Lorentz invariance holds, the areas of a highly boosted, nearly light-like surface in spacetime must go to zero with the inverse Lorentz factor as we increase the boost and keep a fixed portion of the surface in the coordinate space. Clearly, this is impossible if a theory simultaneously implies that such an area must have discrete eigenvalues. This problem can't be "small" in any sense. The freedom to perform boosts and make similar arguments is the very essence of special relativity. The violation of its consequences in this thought experiment is maximal which means that LQG can't respect the Lorentz invariance, not even approximately.
Back to gauge invariance.
The argument that generic quantities like that are not gauge-invariant in quantum gravity is an argument that a good student understands in 20 seconds. Nevertheless, it is my pleasure to announce that after 20 years or so, two loop quantum gravity proponents have finally understood it!
Thomas Thiemann is famous for his re-discovery of Feynman's joke called the "Unworldliness function" (the theory of everything is U=0) that he calls "Master Constraint M".Indeed, these two fast thinkers now say that the Planckian discreteness of LQG is an empty statement because the areas are not gauge-invariant. It is not hard to calculate that after 500 years or so, they will also understand more than 1/2 of my blog article about LQG. Congratulations. ;-)
Barrett-Crane vertex is wrong
Another LQG paper. A more general complaint against LQG is that it clearly can't give a correct long-distance limit that would resemble smooth space or even general relativity. There are many ways to see this fact, too. Under the change of variables, the unknown higher-derivative (divergent) terms added to the Einstein-Hilbert action are simply transformed to unknown terms in the Hamiltonian or the spin foam rules - infinitely many unknown terms in the interaction vertices. Their collection is a problem equivalent to non-renormalizability. Also, infinitely many adjustments of the vertex have to be made to allow a smooth space solution at all.
The most popular "named" vertex in LQG was the Barrett-Crane vertex. Carlo Rovelli has been working on his "background-independent graviton propagator", a combination of words that is an oxymoron because every propagator is by definition background-dependent. Finally,
related paper they readjust the vertex so that a few particular wrong terms are canceled. They don't seem to realize that there are infinitely many other components of the correlators that are guaranteed not to work unless an infinite number of adjustments are made.
The only way to show that these problems cancel is to demonstrate that the theory respects the symmetries that are responsible for such cancellations. But LQG clearly doesn't respect any of these symmetries so it is obvious that none of the things that should work will work. Again, this is a 20-second argument but it will take something in between 20 and 500 more years for their coleagues to get this point. They will have to calculate hundreds of additional inconsistencies in hundreds of similar attempted models before they will start to see a pattern. I, for one, prefer to use poweful tools of mathematics and physics to see the same result - and much more general results such as the invalidity of discrete theories of spacetime - in 20 seconds, instead of these pseudomathematical detailed masturbations.
And that's the memo.
Carlo Rovelli disagrees with their "interesting paper" and sketches two different ways to quantize the system. In one of them, the discreteness is manifest, he argues. It's a typical paper about a confusion how formalities should be done. The spectrum of true observables in a well-defined physical theory cannot depend on any subtleties how we deal with a Hamiltonian.
Thiemann and Dittrich don't write these things in the most natural way but at least it is somewhat clear that they have some physically meaningful point in mind. On the other hand, I have no idea what Rovelli really wants to say except for creating fog. Moreover, his whole discussion is classical in character because he always talks about configuration spaces and Poisson brackets. It is not a legitimate framework in which one can safely answer purely quantum questions such as the question about discreteness of some operators. Cannot he just deal directly with quantum mechanics? I find all his papers irritatingly silly.