Smolin on Loop Quantum Gravity

Larry Yaffe’s comments about string theory reflect well mainstream opinion in the particle physics community. On matters of fact I think what he has to say is pretty accurate, but I disagree with some of his statements that reflect not facts but scientific judgements. Of his positive comments about string theory, the ones about its impact on mathematics and about AdS/CFT are right on target. For an interesting talk explaining the status of attempts to use AdS/CFT to say something about QCD, see Larry’s colleague Matt Strassler’s talk this month opening a workshop in Santa Barbara (don’t miss the heated exchange at the end of the talk about whether or not this is all just supergravity).

Larry’s comments about “compelling hints” that there is something “deep and meaningful” to string theory and that it has provided “partial insights” into conceptual problems in quantum gravity are hard to to argue with. But while these hints seem to point in the direction of the existence of an interesting 11 dimensional supersymmetric theory, they provide no evidence that it has anything to do with the standard model. Quite the opposite, the evidence of the “landscape” suggests that any attempts to relate such a theory to the real world produce a framework that is completely vacuous, and can never explain anything (or, equivalently, can explain absolutely anything you choose).

The one place where I think I really disagree with Larry is his claim that, indisputably, “string theory is the most promising framework we have for combining quantum mechanics and gravity”. This “most promising framework” locution has been around now for nearly twenty years. It was justifiable when people were just starting to try and understand the implications of superstring theory, but the failure of twenty years of effort by thousands of very talented physicists has to be taken into account. The fact is that despite all this effort, string theorists still don’t have a consistent theory of 4-dimensional quantum gravity and prospects are not promising that this situation is going to change anytime soon.

As part of this “most promising” comment, Larry has critical things to say about loop quantum gravity. I’m no expert on this myself, but, like many theorists, he seems to me to be holding string theory and loop quantum gravity to quite different standards. Lee Smolin recently wrote to me and Larry to respond to Larry’s comments, he allowed me to reproduce his e-mail here:

“Dear Peter and Larry,

Thanks for the comments, most of which I agree with. But in case either of you are interested, Larry’s comments about loop quantum gravity do not reflect the real results.

A side effect of the sociology of string theory seems to be that there is as much ignorance of the genuine results concerning loop quantum gravity and other approaches to quantum gravity as there is overhype in string theory. It is fascinating that, just there are results that are believed to be true in string theory, despite never having been shown, there are results that have been shown, in some cases rigorously, in lqg, about which many people seem not to have heard about, in spite of being published 5-10 years ago on the archive and in the standard journals.

To combat this I wrote a recent review hep-th/0408048 which I would gently suggest reading before making public pronouncements about the status of the field. There are also good reviews on the rigorous side by Ashtekar and Lewandowski and by Thiemann, as well as two textbooks in press from CUP, one from Rovelli and one more rigorous from Thiemann.

Larry says of LQG that it “has not been shown to have anything to do with gravity. Does it have a large-volume limit? Does it have long distance dynamics…”

Can I mention some of the results that show that lqg quite definitely is a quantum theory of gravity, with details and referenceds in the paper? Larry, if you think any of these results are wrong, please tell us on what step of what calculation or proof someone made an error. Otherwise, we invite you to study the results and the methods by which they were gotten. You might surprise yourself by coming to agree with us, after all this is just quantum gauge theory, but in a diffeomorphism invariant setting rather than on a background manifold.

A key result is the LOST uniqueness theorem which shows that for d >=2 the hilbert space LQG is based on is the UNIQUE quantization of a gauge field that carries a unitary rep of the diffeo group, in which both the wilson loop and non-abelian electric flux operator are well defined operators. (see the paper and references for the precise statement).

Given that GR and supergravity are well understood to have configuration spaces defined as configurations of gauge fields mod diffeos, to which the theorem applies, this implies that the hilbert space used is uniquely suited to the quantization of those theories.

It is further shown that the hamiltonian contraint of GR for d=3+1 is rigorously defined on the hilbert space of diffeo constraints, allowing exact solutions to all the quantum constraints to be constructed.

As far as the path integral is concerned, using the method of spin foam models, based on the observation that GR and supergravity in all dimensions are constrained topological field theory, leads to rigorously defined path integral measures corresponding to the quantization of these theories. There are in addition rigorous UV finiteness results. There are also results that establish correspondences with Regge calculus in various limits.

These results all are quite sufficient to establsih that these theories are precisely the quantization of GR or supergravity. Surely this has something to do with gravity.

Regarding the low energy limit, more explicitly, there are several classes of candidate ground states that have the property that 1) measurements of coarse grained geometrical operators agree with classical flat or deSitter spacetime, up to small fluctuations. 2) small excitations of the gravitational degrees of freedom, which satisfy the constraints to linear order in l_Planck/wavelength have two spin two massless degrees of freedom per momentum mode (i.e. the gravitons are recovered for wavelengths long in planck units, again showing this is a quantization of gravity.) 3) after coupling to any standard matter field, excitations of the ground state yield a cutoff version of the quantum matter field theory on the classical background, cutoff at the planck scale because of the finiteness of quantum geometry.

The classes of states these statements characterize are a) coherent states, b) eigenvalues of coarse grained 3-geometry (sometimes called weave states) and c) for non-zero cosmological constant, the Kodama state.

I would think that finding explicit states with these properties proves that at least linearized gravity and effective field theories correspodning to qft on background manifolds is recovered. Certainly this is again something to do with gravity.

In addition, we can mention 1) the black hole results, which give an exact description of the quantum geometry of an horizon, in agreement with all semiclassical results and 2) the loop quantum cosmology results, which again recover all results of semiclassical quantum cosmology and go beyond them in the context of a rigorously defined framework. Again, some things to do with gravity.

See my paper for complete references.

I’m sorry for the tone, but one loses patience after 15 years. We have always been careful to state results precisely with full qualifications and never to overclaim. By now there are sufficient results that I think the many good people who are working hard in this field deserve to have their results much better known.

As always, I and my (rapdily growing number of) colleagues are happy to talk to anyone and go anywhere to explain the results and the methods by which they were gotten. Indeed, the number of invitations for talks at places that previously expressed no interest previously is growing. I’d certainly be glad to recommend good speakers who could educate your department about the state of art in quantum gravity.



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13 Responses to Smolin on Loop Quantum Gravity

  1. Charles Kilmer says:

    would the resolution of the Poincare Conjecture have any effect on string theory?

    Mathematical Mystery Believed to Have Been Solved
    The Scotsman ^ | Mon 6 Sep 2004 | John von Radowitz

    One of the seven great unsolved mysteries of mathematics may have been cracked by a reclusive Russian who is not remotely interested in the £560,000 prize his solution could win him, it emerged today.

    The Poincare Conjecture involves the study of shapes, spaces and surfaces and makes predictions about the topology of multi-dimensional objects.

    Basically, it says that a three-dimensional sphere can be used in an analogous way to describe higher-dimensional objects that are impossible to visualise.

    Since Henri Poincare suggested the theorem in 1904, some of the greatest mathematicians of the 20th century have struggled to prove it either right or wrong.

    All have failed. But now the world of maths is buzzing with the news that an answer might at long last have been found.

    Dr Grigori Perelman, from the Steklove Institute of Mathematics at the Russian Academy of Sciences in St Petersburg, has published two papers offering a solution to a larger-scale problem called the Geometrization Conjecture.

    This is also concerned with geometry, and experts say that contained within it is proof that the Poincare Conjecture works.

    If Perelman can satisfy his peers that this is the case, he stands to win a one million dollar cash prize from the Clay Mathematics Institute in the United States.

    The Institute is offering million dollar prizes for solutions to each of the mathematical conundrums it calls the Seven Millennium Problems.

    But there is a more fundamental problem the general community of mathematicians needs to solve first. Perelman does not seem to be interested.

    Dr Keith Devlin, a leading mathematician from Stanford University in California, explained: “He´s very reclusive, and won´t talk to anyone. He´s shown no indication of publishing this as a paper, and he´s shown no interest in the prize whatsoever.

    “Has it been proved? We don´t know, but there´s good reason to think it has been. My guess is that in about 12 months people will start to say okay, this is right, but there´s not going to be a golden moment.”

    Dr Perelman published his two papers in November, 2002 and March last year.

    A third is yet to be published.

    By all accounts, Poincare will come out of the first two papers, said Dr Devlin.

    If the conjecture was proved it would have profound ramifications, he told the British Association Festival of Science at the University of Exeter.

    Scientists working on the frontiers of cosmology and physics frequently dealt with hyperdimensions. A solution to the Poincare Conjecture would greatly increase their understanding of the shape of the universe.

    Dr Devlin compared proving Poincare with setting off an avalanche. If you are on top of a mountain, and it is spring, and you jump up and down, a little bit of snow moves. But at the bottom a whole lot of snow comes down.

    “It can´t fail to have enormous implications; it will just be huge.”

    He said solving mathematical problems such as the Poincare Conjecture was more like writing a story than doing a sum, which was why it took so long.

    “It´s just so damn complicated, he said. It really can take two or three years to certify the thing.”

    Proving the Poincare Conjecture would be the first great mathematical breakthrough since Andrew Wiles solved Fermats Last Theorem in 1994.

    This year, Professor Louis de Branges de Bourcia, from Purdue University in the United States, claimed to have proven another of the Millennium Problems called the Riemann Hypothesis.

    The hypothesis is a 150-year-old theory about Prime Numbers – numbers that divide only by one and themselves and are considered the atoms of arithmetic.

    De Branges claimed to have confirmed a conjecture made by the German mathematician Bernhard Riemann in 1859 about the way prime numbers were distributed.

    But, unlike in the case of Poincare Conjecture, the worlds mathematicians are becoming increasingly convinced that he has got it wrong.

    Marcus du Sautoy, Professor of Mathematics at Oxford University, said: “The mathematical community is sceptical whether the methods of Louis de Branges are capable of proving the Riemann Hypothesis.”

    If de Branges turned out to be right, it would have a dramatic impact on both global business and national security.

    Encrypted codes are based on the randomness of prime numbers. If a system could be found that made them predictable, no secret would be safe.

    “What mathematics has been missing is a sort of maths prime spectrometer, like the machine chemists use to tell them what things are made of,” said Prof du Sautoy. “If we had something like that it would bring the world of e-commerce to its knees overnight.”

  2. Alejandro Rivero says:

    Hmm Urs, 1+1 has always appeared to me as pathological. But yep, I can not remember any arguments about the gauge invariance of the area operator, so perhaps you are right about having a problem there. I believe that Rovelli did, in a different focus, a proposal about how general transformations should affect to the eigenvalues and to the mean expected values of an area operator, I can not tell if it applies here.

  3. For something different: I have listened to the talk by Strassler (on what AdS/CFT has to say about QCD) that Peter provided a link to. Very intersting. We had recently discussed that topic here already.

    It is indeed not true that only supergravity plays a role in these calculations. In particular anomalous scaling dimensions of operators on the SYM side is matched to at least first non-sugra order on the string side by a whole small industry. Search the arXive for papers by Tseytlin and by Zarembo (2004), for instance.

  4. Is it clear that area is quantized in LQG?

    I know that the area operator constructed in LQG has discrete spectrum (BTW, which spectrum precisely? Last time I checked there were at least two different ‘proposals’ for this spectrum.)

    But the area operator is not gauge invariant, but has single spin networks as eigenstates. But any state that actually solves all of the constraints, including the Hamiltonian constraint (if it can ever be defined) must be a superposition of spin network states (first of all it must be a knot state (solving the spatial diffeo constraints), but surely even a superposition of knot states) and most probably (nobody knows!) a continuous superposition.

    This means that it is not clear that on physical states the area operator has any eigenstates at all.

    Often LQG properties are argued in terms of spin network states. But these are just a particular choice of basis – of the kinematical Hilbert space. Statement like ‘the universe is a huge spin network’ or things like that make an unjustified identification of a choice of basis with a physical observable.

    BTW, does anyone know about the status of the ‘Phoenix Project/Master Constraint Programme‘ to actually construct the Hamiltonian constraint? As long as this hasn’t been done it seems pretty vain to discuss any properties of LQG.

    BTW, as I discuss here the 1+1D example once again helps to understand the situation: When you decide to restrict to the space of spatially rep invariant functions in 1+1 D gravity and then try to apply the Hamiltonian constraint as an operator on that, you know you cannot succeed. That’s because this amounts to imposing the constraints

    L_n – \bar L_{-n} for all n

    together with

    L_0 + \bar L_0 .

    The first infinite set of constraints can be imposed all right, and even with the methods used in LQG, because these generators have no anomaly among themselves. But then standard results in 1+1d gravity tell you that the Hamiltonian constraint L_0 + \bar L_0 cannot annihilate any of the resulting states.

    I am speculating that this is in fact the reason why Thomas Thiemann in his LQG-string paper decided not to precisely follow the standard LQG procedure (which consists of applying the quantized Hamiltonian constraint as an operator on the space of spatially rep-invariant states) but decided to treat both L_n and \bar L_n by ‘relaxed canonical quantization’.

  5. Alejandro Rivero says:

    Independently of LQG, the interesting issue of having a quantum of area is that is can be used to singularise 4D over any other number of dimensions (we want Newton constant to have units of area). If strings had some way to justify the right compactification, ie to justify 4D, I could understand the criticism of the foamy world. But as it stands, LQG holds a very interesting card in his hand.

  6. Thomas –

    let’s try to avoid bickering and general accusations and just focus on the discussion of some technical facts and problems. I’d be very interested to see Lee Smolin reply here (maybe mediated by Peter Woit).

    You claimed that something is a problem for string theory and I asked you to please demonstrate it. Your reply

    As all string theorists, you consistently require much higher standards from the competition than from yourself.

    leaves me in puzzlement. From past discussions I can guess that you are thinking that somehow results about anomaly cancellation for chiral fermions have to do with anomalies in the canonical quantization of gravity. I frankly admit that I don’t see what you have in mind. But if you can make it precise you should probably do so and show it to somebody who knows more about it than I do.

    It is precisely this problem with discussing full 1+3d nonperturbative quantum gravity that makes me rather wish to discuss a toy example like 1+1d gravity. All open questions of LQG are also non-trivial and unsolved in this context.

  7. Thomas Larsson says:


    I do indeed agree with your claim that for an understanding of a non-perturbative quantization of gravity it should be very helpful to have an idea of the existing quantum reps of the symmetry generated by the classical constraints (though I would not call that alone a quantization of gravity, as you sometimes do).

    I have never claimed (I hope) to have quantized gravity. I do claim, however, that I have succeeded in quantizing the symmetry principle underlying gravity, and I believe that this will prove to be an essential ingredient in quantum gravity itself.

    But I don’t claim that I know how to similarly ‘fix’ the LQG approach in higher dimensions. If you do and if you think that you can make a nonperturbative quantization of the action of closed string field theory and if you think you can deduce some kind of problem for string theory from that please do so and let us know about your result.

    As all string theorists, you consistently require much higher standards from the competition than from yourself. I create (not alone, but anyway), the mathematics that makes it possible to discuss diffeomorphisms on the same footing as conformal transformation, and you say this may be interesting if I also succeed in using this to quantize gravity. The last guy who succeeded in creating both mathematical infrastructure and physics was called Newton, and I think that my results may be valuable even though I am a lesser soul than him, especially since everybody else has failed to make any progress towards on quantum gravity. Not even Einstein created his own math; Ricci and Levi-Civita created tensor calculus for him to use.

    Meanwhile, you seem completely undisturbed by the fact that string theory makes no testable predictions whatsoever and that recent landscape ideas show that it never will, a situation which Witten describes as “[String theory] is also more predictive than conventional quantum field theory.” I don’t believe in LQG, but as long as Smolin does not claim that LQG is more predictive than field theory, he deserves good marks for honesty. Misrepresentation is Fraud’s cousin.

    BTW, did you see our comments on your 2-form gauge theory over here? (Possibly you were on vacation when these were posted.) Since you are only using a 2-form B you have to face the problem that the surface holonomies that you compute are not independent of the ‘parameterization’ of your surface, i.e. of the order in which you multiply the group elements on the plaquettes.

    The amplitudes depend on the triangulation (or quadrangulation, rather); even the number of indices depends on that. This is why it’s only the lattice model, were there is a canonical triangulation, that is well defined. However, there is no order dependence in the sense of non-associativity. Contraction of all index pairs associated with internal links is order independent; finite sums can be performed in any order.

  8. Thomas,

    there is non-perturbative quantization of gravity, where we try to solve the canonical constraints exactly, and there is perturbative quantization, where we compute quantum corrections to classical solutions. The discussion here is about how to do the non-perturbative quantization.

    I do indeed agree with your claim that for an understanding of a non-perturbative quantization of gravity it should be very helpful to have an idea of the existing quantum reps of the symmetry generated by the classical constraints (though I would not call that alone a quantization of gravity, as you sometimes do).

    This is precisely what I claim is missing in the LQG approach, and in the special case of 1+1 dimensions one indeed finds that the idea you are proposing gives us the correct constraint algebra (namely the Virasoro algebra), while LQG misses it.

    But I don’t claim that I know how to similarly ‘fix’ the LQG approach in higher dimensions. If you do and if you think that you can make a nonperturbative quantization of the action of closed string field theory and if you think you can deduce some kind of problem for string theory from that please do so and let us know about your result.

    BTW, did you see our comments on your 2-form gauge theory over here? (Possibly you were on vacation when these were posted.) Since you are only using a 2-form B you have to face the problem that the surface holonomies that you compute are not independent of the ‘parameterization’ of your surface, i.e. of the order in which you multiply the group elements on the plaquettes. To ensure well-defined surface holonomy you’d need to add a 1-form A and the condition B+F=0 and then you’d make contact with Girelli&Pfeiffer’s work.

    BTW, one maybe interesting piece of trivia in the triangle of topics constituted by LQG, strings and non-abelian 2-form field theories is the following:

    If we restrict as in LQG to spatial rep-invariant states we obtain, in 1+1 dimensional gravity, the so-called boundary states. A particularly natural such state is the loop space function which assigns the holonomy of some connection A to a loop. Incidentally, this function is precisely the boundary state which describes a non-abelian 2-form B = -F_A .

    See this entry and hep-th/0408161 for the details.

  9. Thomas Larsson says:


    The following statements are almost identical:

    1. LQG is only quantization in a weaker, nonstandard way. In particular, if the quantization of 3+1D gravity in any way resembles quantization of 1+1D gravity on a world sheet, then the diffeomorphism constraint should acquire an anomaly which cancels against ghosts.

    2. All symmetries, including gauge symmetries like diffeomorphisms, need representations of lowest-energy type.

    Thus, apparently to your great embarrassment, we are saying roughly the same thing (except that ghosts are problematic, because normal ordering doesn’t just ruin nilpotency but makes the BRST operator ill defined). However, spacetime diffeomorphisms do not use lowest-energy representations in string theory neither, although it does treat worldsheet gravity in this correct way.

    So exactly why do you expect LQG people to take your argument seriously, if you don’t take the logical implications seriously yourself?

  10. When quoting results about uniqueness of the Hilbert space in LQG it must be emphasized that the quantization presciption used there is not what is usually called canonical quantization, and by this I mean differences over and above the ambiguities of canonical quantization itself. LQG uses ‘relaxed’ canonical quantization where not both of canonical coordinates and momenta are represented as operators on a Hilbert space.

    To me this is the crucial but hardly ever emphasized assumption in LQG. Some people in LQG that I have talked to didn’t even realize that this is a controversial step. It seems that people expected that it is obviously fine to use this ‘relaxed quantization’. But it is known that when applied to systems whose standard quantization we do understand quite well, like the free particle and even 1+1 dimensional gravity coupled to scalar matter, the ‘relaxed canonical quantization’ gives grossly different results than usual quantum theory (e.g. that obtained from path integral quantization).

    So no matter what one can prove within the LQG framework it is a fact, confirmed for instance by Thomas Thiemann and Josh Willis themselves, that LQG changes the usual rules of quantization.

    I know that the idea is that in any theory with such a ‘relaxed’ canonical quantization a limit exists where the ordinary quantization is reobtained. But apparently the only paper investigating this idea is the ‘Shadow States’ paper by A. Ashtekar, Josh Willis and Fairhurst. But the construction done in that paper explicitly makes recourse to quantum effects of the standard quantum theory, so that this does not prove anything.

    Of course it is true, as Thomas Thiemann has put it, that only experiment can show if standard quantum mechanics still holds at the Planck scale. But

    1) if a propsal for quantum gravity does use non-standard quantum theory this should be made quite clear to everybody, so that people are aware of such a step

    2) it would be nice if there were some kind of hints how the non-standard quantum theory used can flow to the standard theory in some limit. Otherwise even the theoretical motivation to believe in the viability of the non-standard approach is weak. To date no such evidence has been given, and indeed I consider the results given in the ‘Shadow states’ paper as a counterexample.

  11. Aaron says:

    Lee Smolin seems much more sanguine about the state of things than some of his colleagues. This is the first I’ve heard anyone claim that lqg has any semiclassical limit. Can one compute graviton scattering? I’ve also been under the impression that the Hamiltonian constraint remains (as of last May, at least) quite mysterious.

    Finally, as for the black hole results, from the reading I’ve done, they’ve been deeply unimpressive, not the least of which because they don’t actually appear to be done in any theory I’d call lqg. Put another way, I don’t see how any black hole state was constructed in any theory. Instead what seemed to be done was to take a spacetime with a black hole, cut out the black hole, quantize what remains, discard degrees of freedom associated with the bulk and get a result proportional to the area of the horizon. Forgive me if I don’t find that result very impressive.

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