in which they review what assumptions and results they consider important about loop quantum gravity, and what do they consider to be the most important questions that would have to be answered before loop quantum gravity might be considered a serious candidate theory to say something about quantum gravity.
If you remember Feynman's lecture about "Cargo Cult Sciences" in which he talks about the experiments with the rats in the 1930s, it may be fair to say that the paper by Nicolai et al. is analogous to the experiment by the guy called Young - because it defines the important conditions and tasks that one must check and master before she can actually do something controllable with loop quantum gravity (or rats). But I am kind of afraid that the LQG community will ignore the paper - despite the fact that it is written in a very friendly tone - and continue their research "as usual".
Although I don't think that Nicolai et al. analyze all problems that one would face if she adopted loop quantum gravity (for example, the appearance of local Lorentz symmetry is not discussed), they are doing a tremendous job in explaining what problems, so far, prevent one from saying that "there is a theory" at all - and they study these problems much more quantitatively and rigorously than I would ever be able to do. Their main points are the following:
- if one has something that can be called a "theory", it should not have infinitely many undetermined parameters, otherwise it is equally unpredictive as a non-renormalizable quantum field theory. I've been emphasizing the point for quite some time, too...
- if one has a dynamical theory, she should know what is the dynamics. If the theory has a Hamiltonian formalism, she should know what the Hamiltonian is, which is not really the case of LQG...
- a theory with local symmetries must have a constraint algebra, and this algebra must close; the non-trivial part is the Hamiltonian constraint whose commutators with itself have not been calculated in an operator form...
- the authors emphasize that it is not enough that the commutator annihilates some states; the commutator must act as another constraint on all states, i.e. off-shell, which is far from being checked in LQG...
- in a theory invariant under four-dimensional diffeomorphisms, it is not enough to study the three-dimensional diffeomorphisms, and therefore the understanding of the Hamiltonian constraint is essential from this viewpoint, too - without a four-dimensional diffeomorphism covariance one can't really talk about "spacetime background independence"...
- even if one adopts the belief that the non-perturbative LQG approach seems to solve the UV divergences, it is very hard to understand where the explicit two-loop divergent counterterm goes; they argue that physics with pre-determined degrees of freedom should be independent on the way how one regularizes it while LQG seems to make the assumption that one particular regularization gives the "right" result...
- concerning the apparently vanishing predictive power, they also talk about the matter couplings where it seems that "anything goes" in LQG; they explain that it not only means that the predictive power is zero, but they also argue that we know enough to say that "anything goes" is probably wrong - various types of matter are essential to regulate the theory and/or cancel anomalies, and if LQG does not seem to confirm that there are potential anomalies, something is probably wrong with its desired long-distance limit; well, I am afraid that most LQG practitioners don't "believe" that a theory can have anomalies...
- they argue that the cosmological models that are connected with LQG in the literature don't seem to follow from LQG, because of things such as the ambiguities of the volume operator and the Barbero-Immirzi parameter that would determine the length of the inflation etc...
- concerning the black hole entropy, they present the usual arguments that I often emphasize, namely that the assumptions about the truncation of the interior and other assumptions are questionable, and the numerical constant 1/4 is not determined...
- concerning spin foams, they're especially worried about the missing proof of the equivalence between the spin foam and the Hamiltonian formalism, which is the reason why the spin foam is not helpful to study the ambiguities...
The section 2 starts with basics about the degrees of freedom in LQG, and the Wheeler-DeWitt equation "H.Psi = 0" that defines dynamics in similar approaches to quantum gravity. They review the problems with the WDW equation as well as the useful path-integral way to generate the solutions. They identify the Hamiltonian constraint as the key problem.
In a friendly way, they sketch the historical motivation for the Ashtekar variables. Mathematically, their presentation is very quantitative and accurate. They explain why the values +i, -i for the Immirzi parameter are natural, and why the usually expected real values are problematic. As far as I see, they don't really discuss the point that the Ashtekar field redefinition is not legitimate globally on the configuration space as it imposes new periodicities and quantization rules that don't follow from the metric itself.
They present the problems with the definition of the volume operator - incidentally, they say in a footnote that Thiemann now believes that its eigenvalues can be arbitrarily close to zero. They also mention the Kodama state (well, a state that was discussed in Yang-Mills theory a long time ago) as a solution to the WDW equation, with references to the problems of this state. A discussion of the Wilson lines follows...
The third section is dedicated to kinematical issues of quantization. They mention that the basic operators are not "weakly continuous", and therefore the Wilson lines must be adopted as the degrees of freedom to quantize, and they also feel uneasy about it. This is also one of the points where they mention the non-separability of the LQG Hilbert space which is normally a big problem, but - as Nicolai et al. argue - this feature can actually also be useful to overcome some problems of geometrodynamics. Nevertheless they realize that the non-separable "polymer" Hilbert space is likely to prevent one from defining any kind of continuous physics.
Another part of the paper explains the role of the Clebsch-Gordan coefficients in the counting of the spin networks, and the special type of functionals. They also write down the usual "natural" scalar product.
Another section compares LQG to gauge theories. They start with saying that the Wilson loop quantization of gauge theories has been a failure, and therefore it's not clear why it should be a success for gravity. They show that the physical meaning of the Wilson loops in LQG differs from gauge theories, and the scalar product is much more likely to vanish in LQG - which is associated with its non-separability - while in gauge theory the correlators of Wilson loops are nonzero and continuous functions of the shapes. They are also skeptical about adding extra gauge fields to LQG which should make the correlators behave like in gauge theory, while the natural expectation is that the correlators would behave as badly as in LQG.
The following section explains how the geometric operators (area, volume) act on the spin network states. They complain that one can't really a priori eliminate infinitely large spin networks because this would mean a prescription for the definition how to take the continuum limit. They also show a simple example of a spin network whose area is infinite.
The areas are related to the Bekenstein-Hawking entropy, and they mention that all existing calculations of the Immirzi parameter - or, equivalently, the numerical constant in the black hole entropy - are probably incorrect. Incidentally, tonight there is also a highly original and entertaining :-) paper on black hole entropy on the web:
They improved the format of Rovelli's "dialogue" paper that talked about the physicist Simp and his "maverick" student Sal. The improvement is that now it is a "trialogue"; I kid you not. :-) There is now not one person in the debate who usually says weird things, but two of them. They're called Ted and Carlo and their main point is that the black hole has a much larger entropy than the Bekenstein-Hawking entropy because the extra entropy is hidden anyway. Of course, Ted and Carlo don't care that their proposal violates everything we know about the black holes both from Hawking's calculations as well as from string theory as well as from thermodynamic considerations (the extra entropy is especially bad once the black hole evaporates), and poor Don, the third actor who is now a minority (!), must listen to all this stuff, which is fortunately much easier because of lots of glasses of some fancy drink. ;-)
Let's return to Nicolai et al.
After the black hole discussion, they show some of the serious regularization problems that appear in attempts to define the volume operators - which is much harder than the area operators. They also show that the LQG statements "XY is regularization independent" are simply not true, unless one redefines the meaning of the adjective "regularization independent". The quantities defined from discrete objects are guaranteed to be finite, but they depend on all these choices.
In the section about coupling to the matter field, they mention the loss of predictivity, the surprising fact that the matter fields play no role in solving the problems of quantum gravity (of the Hamiltonian constraint, in this case), and they also mention that the coupling to matter field makes the potentially pretty Ashtekar's theory ugly anyway. They demonstrate that the reality problems encountered for the metric tensor appear for the fermions and other fields, too.
The section 4 is about the constraints at the quantum level: the SU(2) local symmetry and the diffeomorphisms. The normal spin networks are never diffeomorphism invariant, and the LQG people usually "average it over the diffeomorphism group". Nicolai et al. argue that no measure is defined on this "diff group" manifold, and therefore the formal definition is meaningless - but for a particular choice of problems, they show that this problem can be circumvented. They also discuss the ambiguity of the choice of the "physical subspace" of the kinematical Hilbert space. It's not just ambiguity: one often loses the hermiticity of the usual operators. This discussion of the possible choices how to choose the spaces of functions vs. distributions and solve various problems is pretty complex, nearly mathematically rigorous, and detailed.
Of course, the main problem is the Hamiltonian constraint, once again. The naive definition of the Hamiltonian has infinitely many ambiguities. Of course, one could try to pick the right one, but for either choice, no eigenstate is known. They argue that even a much more simple task - the action of the Hamiltonian on a spin network - seems out of reach. They clearly know the LQG technology in detail - much more than I do.
The main problems with the Hamiltonian constraints are
- ambiguities that have already been appreciated
- ambiguities that have not been appreciated yet, i.e. ad hoc choices that the LQG people have been doing
- ultralocality - unlike lattice QCD, the Hamiltonian acts on a vertex in such a way that it only exchanges the intertwiners - they seem to say that this is a general property of all Hamiltonians you can ever propose in LQG, which is pretty bad and potentially prevents any signals from propagating
The section 5 is about the constraint algebra. The situation in LQG does not seem to be any better than in geometrodynamics: there is no known Hilbert space on which the algebra is known to close. They contrast the situation of LQG with the mechanism how the Virasoro algebra works in the bosonic string seamlessly. The non-separability of the LQG Hilbert space also means that one can't define the generators of (infinitesimal) diffeomorphisms, just the finite ones. They emphasize that the verification of the closure requires one to calculate it on the full Hilbert space, not just on some special states.
Their main point here is really that a generic regularization will spoil the closure of the constraint algebra - which more or less means that the regulator is not gauge-invariant - and it's important to see whether it is the case for all solutions in LQG that have been proposed, or whether there can be special nice solutions - a solution includes the definition of the Hilbert space and the Hamiltonian constraint.
If someone is interested in loop quantum gravity, I recommend to read this paper because it is quite certainly the most meaningful paper written on LQG, at least in the last 8 years. The authors understand LQG extremely well, and moreover, they also understand physics very well .
What about the future? I am of course convinced that these problems don't have any solution that could lead to a meaningful theory with a finite number of ambiguities at most. And those who believe that it's OK to live with all these fundamental problems of LQG will continue to live so. ;-)
