Can Loop Quantum Gravity connect in any way with string theory? The one difficulty I see with LQG is that it requires an enormous number of degrees of freedom, e.g. these spin variables in the net.  This is in contrast to stringy holographic theory where the fields in a space are equivalent to fields on a boundary or a horizon of one dimension lower.  In this setting entropy of a black hole is the entanglement entropy of states interior and exterior to the black hole.  This reduces the amount of data, and thus entropy, required.
Are there suggestions, conjectures or maybe serious theory which attempts to describe the spin variables of LQG according to such entanglements in string-brane theory?
 A: At a fundamental level there seems to be an incompatibility.  String theory requires an infinite amount of other fields in addition to the simple metric tensor.  LQG meanwhile pretends to merely be a humble theory of gravity.
But string theory would be inconsistent if it was just about gravity, so at a basic level there seems to be a clash.  I believe the only way around that would be if LQG were to add extra structure on top of their theory, and I believe the state of the art is far from being able to manage much if any matter degrees of freedom.
I'm sure others will bring up other salient incompatibilities, but I am a little skeptical about those simply b/c it seems that LQG is still in a very raw and preliminary stage of development and the literature seem to change the tone and their predictions every year or so.  For instance, not long ago we were led to believe that Lorentz breaking was an element of LQG, but no apparently that is not a correct statement.  
I suppose the correct answer would be to say that even posing the question is premature. 
A: It's a very good question and I can't match the expert knowledge of those who have already answered, but I also can't resist making a few points.
In LQG it is hard to calculate anything because it does not give a classical limit that looks like smooth spacetime. If such a limit existed we would expect to be able to look at small deviations from flat spacetime and it is hard to see how that could lead anywhere except to a perturbative theory of gravitons. From the particle physics world we know that the supergravity/superstring route is probably the only way to do that. Some people might say that there could be another unknown way or that LQG would somehow avoid such a perturbative limit but let's assume otherwise until there is some good explanation of how that would work.
In that case LQG could only work if it included matter as string theory does. Matter might be emergent but that would mean that LQG has to work as it is with a classical limit and that does not seem to be the case, so probably LQG needs matter put in as some additional degrees of freedom. I think Lee Smolin tried to generalise LQG to look more like string theory in the early days before he gave up and became more dismissive of string theory. For example, he and others looked for higher dimensional version and supersymmetric version of LQG but there was nothing very promising. I think it would be wrong for a younger generation to assume that no progress can be made with such a connection just because others could not find it.
On the string theory side the fundamental issue is that it's degrees of freedom and underlying principles are not known or fully understood. LQG has spins and knots. Spin half entities are qubits which also arise in string theory. This does not mean there is a connection because such entities arise everywhere as representations of symmetries. However, LQG and string theory share similar origins out of gauge theories and they share some mathematical structures. The areas where they are least well understood are also the areas where we might expect to see connections if there are any.
Personally I think that theorists have to take Dirac's advise to look for elegant mathematical structures and be driven by them. When they find them in relation to one approach it suggests that the approach has some promise. Both string theory and LQG passed this test when trying to solve the same problem but failed so far to make contact with experiment. I think you have to keep a broad view and look for connections between the mathematics of these and other approaches that look interesting. Unfortunately sociological issues driven by the way funding is allocated seems to discourage people from looking at the bigger picture.  
Of course some experimental input would also help.
A: I would say that the it is the other way around. The number of degrees of freedom in LQG is not "enormous", and it is much less, and not much more that in strings.  The spin variables on the net are just the same variables as in classical general relativity, namely the metric, except that there is a cut off at small scale. So, there are infinitely many less variables than in classical general relativity.  because the degrees of freedom with wavelenght smaller than the Planck length exist in classical GR but not in LQG. It is true that in strings one might expect an holographic principle that reduces the degrees of freedom to the boundary theory, but this is not a boundary a theory in our physical spacetime. It is a boundary theory in a spacetime of higher dimensions. So, the boundary theory has still more dimensions than our space, and therefore the number of degrees of freedom is still much bigger that in LQG.
About the stage of development, certainly both loops and strings are very preliminary, and not fully understood, but I would definitely say that LQG is far better understood (it is a simpler theory than strings). In LQG we know the fundamental degrees of freedom and we can write the basic equations of the full theory in a compact form in a few equations.  See for instance my recent review paper http://fr.arxiv.org/abs/1012.4707.  In strings we do not yet know the fundamental degrees of freedom, and we only know certain "corners" of the theory, with many indications that these different corners fit into a single scheme. But the actual single scheme we do not know yet.  So, the basic theoretical situation in LQG is simple and clear, not so in strings.
Finally, no the predictions in LQG have not been changing. In fact, I wish there were solid predictions. There is none for the moment, like for strings. What exists in both cases is suggestion of possibilities. For instance strings suggested that perhaps the gravitational force could change at a measurable distance because of the extra dimensions, that supersymmetric particles be seen at lower energy, that black holes be formed at CERN. Nothing of this has been true so far, but this does not invalidate strings, because the theory is perfectly compatible with these effects existing but also compatible with these effects not existing. In very much the same way, it had been suggested that LQG might be combatible with violations of Lorentz invariance, and these seem to have been quite ruled out now by observations. But LQG is perfectly compatible with these Lorentz violation not being there. In fact, personally I have always been very skeptical at he suggestions that LQG might lead to Lorentz violation. If you read my old papers, I have always insisted that the theory is perfectly compatible with Lorentz invariance, and I could not see a source of Lorentz violation. These papers, for instance arXiv:gr-qc/0205108, which we wrote almost 10 years ago, indicate that LQG is locally Lorentz invariant, and were written long before the recent indications against Lorentz violation.
Regarding the compatibility between Loops and Strings, I really do not know.  It is true that all the incompatibilities that Columbia indicates are there.  But on the other hand we do not yet know the fundamental degrees of freedom of string theory.  If there is a fundamental description of strings, this should be background independent; perhaps could it resemble somehow LQG?  And in LQG there is no hint at the solution of the unification problem. Would it lead towards something more resembling strings?  If I had to bet, I would say no, the two paths are really different, but I would not rule out the possibility a priori.  I agree that the question is a bit premature, but I would say that the reason is not hat LQG is in a raw stage. LQG is a well and clearly defined theory, what is hard is to compute out of it. The reason is more because strings are in a raw stage, since we do not know a fundamental formulation of the theory, its basic degrees of freedom.
carlo rovelli 
A: Dear Lawrence, an equivalence between LQG and string theory - or an LQG-like description of string theory physics - has surely been an attractive idea for many physicists (myself included) but it is impossible because of fundamental differences in virtually all general features and predictions of both frameworks.


*

*As you correctly mentioned, the counting of the degrees of freedom disagrees. String theory respects the holographic principle. It also means that the entropy within a volume is always bounded by the surface in Planck units. On the other hand, LQG admits an arbitrary, volume-extensive, Planckian entropy density - in fact, it predicts a Planckian entropy density of the vacuum (the information about the details of the spin network). For a related class of examples, LQG always predicts a volume-extensive term in the black hole entropy, too. It can only be "cured away" by erasing it (together with the whole black hole interior) by hand and pretending it was never there. It's important to mention that the infinite multiplicity of "string fields" is just an artifact of a formalism - string field theory. One can't add an arbitrary number of excitations of all these kinds into a finite volume (because they would gravitationally collapse). The only truly "physically invariant" measure of the number of degrees of freedom boils down to entropy and ST - as a holographic theory - predicts a much smaller entropy than non-holographic theories such as LQG. In particular, string theory vacua are unique and carry no entropy density.

*LQG breaks the local Lorentz symmetry while string theory exactly preserves is. Because the Fermi satellite has showed that there is no Lorentz violation at the Planck scale, LQG was falsified. (It was falsified in many other ways, too.) String theory remains compatible with the observations. The preservation of Lorentz symmetry in string theory may be seen e.g. perturbatively, by considering strings propagating on a target spacetime. The $SO(d-1,1)$ symmetry of the spacetime directly arises from the $SO(d-1,1)$ global symmetry rotating fields (representing spacetime coordinates) on the world sheet. The violation of Lorentz symmetry in LQG may be seen from the fact that the hypothetical solution - a spin network - picks a privileged reference frame, analogous to the luminiferous aether. In this frame, the entropy density is huge, essentially Planckian, and in all other reference frames, there would be a huge entropy flow in a direction, which would break the rotational symmetry. In the preferred frame, the motion of all objects instantly stops as their kinetic energy is dissipated to the thermal energy of the spin network which is a de facto infinite heat bath.

*String theory implies that the space is smooth and almost flat at long distances. Although there is no fully-general "no-go theorem", all partial models and circumstantial evidence suggest that LQG in any form can never predict a smooth space at long distances. It gets crumbled. For this reason, it doesn't even make sense to ask whether LQG reproduces Einstein's equations at long distances - there are no long distances in LQG.

*LQG implies that there can't be any forces and elementary particles aside from gravity. String theory predicts that gravity has to exist, much like non-gravitational forces and particle species. The absence of other forces in LQG isn't a cosmetic problem that can be fixed. The strength of other forces doesn't go to zero, not even at the Planck scale. In fact, there exist general arguments that gravity has to be the weakest force - just like it is in the real world - so a valid microscopic description can never start by neglecting the non-gravitational forces because it is really gravity that is a correction, not the other way around. String theory predicts the right "draft" of the world with spin-1/2 fermions, spin-1 gauge bosons, potential for gauge anomalies and a nontrivial anomaly cancellation, chiral fermions and chiral interactions, Higgs bosons, Higgs mechanism, confinement of non-Abelian gauge fields, running couplings and other phenomena related to the renormalization group, and so on, while LQG has nothing whatsoever to do with particle physics and is incompatible pretty much with all the basic concepts of particle physics I enumerated. The contrast becomes even stronger if we realize that string theory has also led to (or at least inspired) some of the most novel, explanatory, and important models of beyond-the-standard-model phenomenology such as supersymmetry, models with extra dimensions, deconstructions, and others that are currently studied by a big portion of phenomenologists, even those who don't consider themselves string theorists in any sense.

*String theory includes dualities. They're transformations that totally rearrange the degrees of freedom and change their interpretation. Quantum mechanics is totally crucial for those S-dualities, T-dualities, U-dualities, holographic dualities, and other dualities to work. On the other hand, LQG doesn't imply any dualities. More generally, it doesn't employ quantum mechanics in any deep way. It is just a variation of the ancient Greek models of atoms whose properties are promoted to operators - but this promotion never leads to anything interesting.

*LQG doesn't admit supersymmetry, wants to avoid extra dimensions, strings, extended objects, etc. So LQG is unlikely to be a dual description of any aspect of string theory. It's been established in string theory that supersymmetry is an omnipresent, fundamental symmetry that has to appear in all semi-realistic models at some scale. Extra dimensions are needed for consistency. On the other hand, LQG starts by assuming that none of these things exist, and even though the appearance of extended objects etc. is generic in consistent field theories and vacua of string theory, the LQG research is based on the assumption that they must be avoided. This leads me to a much general point.

*String theory is a natural theory based on objectively important mathematical structures and relationships. Physicists are discovering these features, much like Columbus was discovering America. They are learning new things - and they are identifying the previous errors in their reasoning. On the other hand, LQG is a man-made theory. It is being invented in a similar way as Edison was inventing the light bulb. Preconceptions are what ultimately decides about the shape of the theory. LQG is being constructed step by step. That's why one can never make any solid statements about anything - and one can make no statements that he wouldn't believe at the beginning. This strikingly differs from string theory that implies unique answers to many fundamental questions. For example, it implies that the equivalence principle, local Lorentz symmetry, and constancy of the universal constants have to hold in general. All these issues are permanently open in LQG because someone may always modify the theory in a different way tomorrow. One may learn new conceptual insights about physics - and mathematics - from string theory. That's different from LQG that is designed to depend on no nontrivial mathematics that would be difficult for average undergraduate students. Consequently, one can never learn anything about physics, space, time, or mathematics from LQG. The whole enterprise is meant to find justifications for a predetermined opinion that quantum gravity can be approached in this simple-minded way - a strategy that is not unsimilar to scientists proving the Intelligent Design or geocentrism. So far, however, no justifications have been found.

*The multiplicities of possibilities how the vacuum may look like according to string theory boil down to solutions of objective equations that we pretty much understand: the rules of the game are constant. That's very different from LQG where new models are created at any point by arbitrarily changing the rules of the game. That's related to the previous point that string theory makes some general predictions, even when the right "vacuum" is unknown. LQG can never make any predictions of this kind.

*Information is lost in LQG. Indeed, it is a local theory of a very naive type so even if space and black hole were possible in LQG, one could show that the assumptions of Hawking's original argument are satisfied which implies that the information cannot get out of the black hole for causal reasons even if the black hole could evaporate (which is established in string theory but surely not in LQG). On the other hand, string theory implies that there exist subtle nonlocalities in the bulk spacetime that imply that the information gets out. This answer is known to be valid because there often exist dual descriptions of the stringy physics where unitarity is manifest.

*LQG tries to use ill-defined observables and ignore the well-defined ones. In particular, the area of a surface isn't well-defined at the Planckian accuracy in a theory where measurements cannot measure distances shorter than the Planck scale. This is why all statements of LQG about the "quantization of areas" cannot be operationally or otherwise defined. On the other hand, string theory implies that areas of small surfaces are not well-defined observables and automatically leads us to the physically meaningful observables such as the scattering amplitudes for gravitons - which can't be calculated in LQG. Scattering amplitudes may be measured experimentally and they satisfy important theoretical constraints such as unitarity, too: they're the right way to parameterize "all predictions" of a meaningful relativistic quantum theory. Quite generally, string theory automatically addresses quantities that are important in high-energy physics while LQG is disconnected from all the 20th century physics and its key concepts.
To summarize, the differences between the technical properties of the two frameworks as well as very philosophy what it means to do good science are completely insurmountable.
A: I'm not sure anyone has tried (except possibly Smolin in some of his older papers). But it wouldn't be that hard: take a simple loop-quantum-gravity spin-foam-analog model in 25+1 dimensions (which is going to be rather more complex than the usual 3+1 dimensional spin foam), pick a ground-state-like solution for it that looks something like an extended space-time (preferable one that is flat and large along at least some dimensions, so mostly made up of large graphs that embed well in 25+1-dimensional space), try using that instead of flat 25+1 dimensional space as the background space-time in which a bosonic string world-sheet is embedded, and see what happens. For a start, see if the string still considers 26 to be the critical dimension.
In theory, you should see phenomena suggesting that the string world-sheet is in some sense a quantum of diffeomorphism, and that from the spin-foam viewpoint it can be "gauged away" by moving to a slightly different spin foam solution. At a wild guess, since the string worldsheet propagates along the 2-d spacetime elements in the 25+1-dimensional spin foam, it should have the same the effect as changing the spins on these elements by one unit.
