Update: I revised my research about the first undeleted comment below this article. Now I think that it was written by Paul Ginsparg, not Sheldon Glashow. The clichés "Dark Ages", "theologians" have the same explanation as before (the article by Ginsparg and Glashow) but the sentence "the competition is fierce because the stakes are so low" is a typical Ginsparg's phrase.Lee Smolin has just sent an e-mail about the paper by Nicolai et al. to Jacques Distler and me. Recall that Nicolai, Peeters, and Zamaklar recently submitted the most meaningful article about
loop quantum gravity (LQG) published since 1997, to say the least. They showed the ideas and techniques of LQG in detail, and they focused on the open problems that, so far, prevent LQG from becoming a serious candidate for a theory of anything - especially the infinitely ambiguous Hamiltonian constraint. Nicolai, Peeters, and Zamaklar have abruptly become the world's leading LQG experts. I've discussed their paper at
Well, it's not easy to write down a technical paper that would solve the problems mentioned by Nicolai et al., or at least downplay the importance of their arguments - especially because their arguments simply are correct and serious. It's easier to write a verbal e-mail, and here is one:
- Dear Jacques and Lubos,
Since you are posting dismissive comments on LQG based on Herman and friennd's review you might be interested in what we experts think. My message to them is below, comments welcome. As before, if you would like to post this, you have my permission.
Thanks,
Lee
L.M.: Note the bizarre formulation "we experts". Is this the kind of bold language that Mr. Lazaridis is impressed by? Lee's mail continues with a series of jokes (I did not edit lots of Lee's typos to keep his text authentic):
- ps You can criticize a theory, but please do not call us "true believers". Most of us have always been quite careful and precise in making claims for LQG. We have always made it clear there are open issues, always mentioning them in reviews and review talks. This is differnt from string theory where its hard to find a review or review talk giving a correct account of the precise state of open issues or a precise statement of the extent to which key conjectures are supported by present evidence. (I challenge you to find a published general review of string theory that gives a precise statement of what is known at the time of publication regarding the evidence for conjectures such as perturbative finiteness, or S-duality of does a careful job of parsing which versions of the AdS-CFT correspondence are supported by present evidence.)
L.M.: The review of AdS/CFT hep-th/9905111 has 260+ pages (and 1100+ citations). I am not sure whether 260 pages are enough for Lee. For other reviews of AdS/CFT, see hep-th/0009139, hep-th/9912164, hep-th/0209067, hep-th/0309246, hep-th/0310119 and others. For evidence on various versions of the AdS/CFT correspondence, see the 3333+ papers at Spires (citations of Maldacena). I hope that everyone understands that I can't list the whole literature about all topics of string theory - not even all the major papers that prove various points Lee mentioned.
- As you can check, there was no issue raised in the review that was not discussed and well understood in the field ten years ago. At the same time, they ignore most work done over the last 10 years which is motivated by the problemms they discuss. Imagine someone claims to write a pedagogical and general review of string theory and limits it to a technical discussion of the problem showing worldsheet perturbation theory is finite and consistent. You would call it unfair, even if you agreed it was accurate on that issue. That is roughly the situation their review presents us with.
L.M.: I don't exactly understand what's unfair about the fact that the stringy perturbative expansion is finite and consistent. A review of perturbative string theory that limits to a technical discussion of stringy perturbative expansions looks completely fair to me - is not it a tautology? A more up-to-date review should also discuss nonperturbative physics. Well, I am sure that the people who prefer weird philosophical speculations that contradict our knowledge about physics won't be satisfied with a technical review, but this fact certainly does not make such a review less meaningful.
Perhaps, Lee did not mean a real review but rather a non-technical article that would claim that something may be wrong with string theory, without doing any meaningful calculations. OK, I would not call this a "review" - perhaps a silliness.
Lee's statement that Nicolai et al. neglect the LQG literature in the last 10 years is simply not true as everyone can easily check; let's avoid stronger words. Finally, let me use Lee's permission and post his letter to Nicolai et al. A version of this mail without my comments is available on Peter Woit's blog.
Smolin writes to Nicolai et al.
Dear Friends,
Thanks very much for all the time and work you put into your review. While I disagree with a number of your assertions, both in point of detail and of attitude, what is certainly very much appreciated is your evident willingness to "get your hands dirty," learn the technicalities and attack key problems. It is very good that you do this, as indeed too few of us loop people have taken the time to try to learn the details and attack problems in string theory.
L.M.: Well, Lee is definitely right that no one in LQG has learned string theory at the technical level. An open question is whether this is possible, at least in principle.
Some points you raise have been underappreciated. The issue of what happens to the chiral anomaly, and whether there is fermion doubling in LQG is one I have suggested to many graduate students and postdocs over the years, but so far no one takes it up. It would be good to know if LQG forces us to believe in a vector model of weak interactions.
L.M.: This formulation is really cute. On one hand, Lee does not want others to call him a true LQG believer. On the other hand, he's ready to let LQG "force him to believe a vector model of weak interactions" - as opposed to the V-A model (left-handed chiral couplings) incorporated in the Standard Model. A vector, non-chiral model of the weak interactions has been falsified for 50 years or so, and the thinking of a person who could be "forced to believe it" - by a set of tools that have absolutely no justification whatsoever - is not a scientific one.
At the same time, the major difficulties you raise were underestood to be there more than ten years ago. This is especially true with respect to issues concerning the hamiltonian constraint such as the algebra and ultralocality.
What is missing from your "review" is an appreciation of how the work doneover the last ten years addresses these difficulties. Indeed the fact that much work in the field has been on spin foam models is exactly because the problems you worry about do not arise in spin foam models. I will explain this below. Other work, such as Thiemann's master constraint approach, also is motivated by a possible resolution of these problems.
L.M.: Note that Lee's answer is getting tougher right now, as the quotation marks around the word "review" help to show. Lee wants Nicolai et al. to appreciate something that does not really exist. Incidentally, Nicolai et al. noticed that some LQG people have partially abandoned the canonical LQG and jumped at the spin foam models - whose equivalence to the old picture is very unclear. The spin foam models don't solve the problems with the infinite ambiguities either, as Nicolai et al. explain in one of their sections.
As you will appreciate, like any active community of 100+ people there is a range of opinions about the key unsolved problems. I have the sense that you are aware of only one out of several influencial points of view.
The view your concerns reflect is what one might call the "orthodox hamiltonian" point of view towards LQG. According to this, the aim of work in lqg is not so much to find the quantum theory of gravity as to work through the excercise of quantizing a particular classical theory, which is Einstein's. From this point of view, the program would fail if it turned out that there was not a consistent canonical quantization of the Einstein's equations.
While I will refer to my own views so as not to implicate anyone else, you should beware that this is not necessarily the dominant view in the field. It is a respectable view, and I have the greatest respect for my friends who hold it. But, were it to fail, many of us would still believe that loop quantum gravity is the most promising approach to quantum gravity.
L.M.: Even if the Ashtekar program fails, we will think that LQG is the most promising approach - but you should not call us "true believers". ;-) Weird.
This is not avoidence of hard problems, there are good physical reasons for this assertion, which I'd like to explain.
What I and others have taken as most important about Ashtekar's great advance is the discovery that GR can be writen as a diffeomorphism invariant gauge theory, where the configuration space is that of a connection on a manifold Sigma, mod gauge transformations and Diff(Sigma). This turns out to be true not only of Einstein's theory in 4d but of all the classical gravity theory we know, in all dimentions, including supergravity, up to d=11, and coupled to a variety of matter fields.
L.M.: Note that the subsequent research could not find a single piece of evidence for Lee's speculations that gravity can be written in this way. Also note that "Ashtekar's advance" is equivalent to the ideas that were assumed to be failed in the first place. If there is no Hamiltonian realization of Ashtekar's picture, there won't be any other realization either.
This is a kinematical observation and it leads to a hypothesis at the kinematical level, which is that the quantum theory of gravity, whatever it is, is to be written in terms of states which come from the quantization of this configuration space.
L.M.: I find these sentences entertainingly self-contradictory. How can someone say "Whatever quantum gravity is, it must be this particular naive 19th century model"? The probability that the correct theory of quantum gravity is exactly what Lee says is something like 10^{-1700}. The belief in this kind of model is nothing more than a random guess.
This as you know, leads directly to the diffeo classes of spin net states. Furthermore, given the recent uniqueness theorems, that hilbert space is unique for spacetime dimension 3 or greater.
L.M.: Well, all infinite-dimensional separable Hilbert spaces are isomorphic. But that's far from creating a physical theory which must also have a unique or almost unique Hamiltonian or another observable that describes its dynamics.
Thus, o long as the object is to construct a theory based on diffeomorphism invariant states, it cannot be avoided.
L.M.: This is such an obviously incorrect statement that I'm not sure whether Lee really believes it, or he just believes that others will believe him. If one looks at all the different descriptions of various vacua in string theory, all of them prove that Lee's point is wrong - and there are many different types of loopholes that make Lee's conclusion extremely easy to avoid.
The main physical hypothesis of LQG is not that the quantum Einstein equations describe nature. It is that the hilbert space of diffeo classes of spin nets, extended as needed for matter, p-form fields, supersymmetry etc, is the correct arena for quantum gravitational physics. Given that the theorems show that this hilbert space exists rigorously, this is a well defined hypothesis about physics. It may hold whether or not the Einstein equations quantized give the correct dynamics.
L.M.: Right, that's one of the ways how to formulate the "big" hypothesis of LQG. A subtlety is that every time such a hypothesis becomes a little bit more concrete, one can falsify it in a few minutes.
A lot already follows from this hypothesis. It gives us states, discreteness of some geometric diffeo invariant observers, a physical interpretation in terms of discrete quantum geometry etc.
L.M.: Well, yes, a lot of contradictions with physics of our Universe - such as the Lorentz symmetry breaking of order one - already follows from that "innocent" assumption.
But there is also a lot of freedom. We are free to pick the dimension, topology, and algebra whose reps and intertwiners label the spin networks. This then gives us a large class of diffeo invariant quantum gauge theories, of which the choices that come from GR in d=4 are only one example. These are possible kinematics for consistent background independent quantum field theories.
L.M.: Lee obviously seems to think that the more freedom a theory gives him, the better.
Now let us come to dynamics. I believe the most important observation for an understanding of quantum dyannics in this class of theories is that all gravitational theories we know, in all dimensions, super or not, are constrained topological field theories.
L.M.: One can hardly ever get a theory with local degrees of freedom from a theory without local degrees of freedom. Even if something like that were possible for a mysterious reason, extraordinary claims require extraordinary evidence.
(See my latest review, hep-th/0408048, for details and references for all assertions here.)
L.M.: You can also see most of the assertions on my blog; the existence of written assertions itself does not make the statements serious.
This means they are related to BF theories by non-derivative constraints, quadratic in the B fields.
A lot follows from this very general observation. It allows a direct construction of spin foam models, by imposing the quadratic constraints in the measure of the path integral for BF theory. This was the path pioneered by Barrett and Crane. The construction of the Barrett Crane and other spin foam models does not depend on the existence of a well defined hamiltonian constraint.
L.M.: Well, so if this is true, then it proves that the Hamiltonian LQG is not equivalent to the spin foams.
The properties that have been proven for it, such as certain convergence results, also do not depend on any dynamical results from the hamiltonian theory.
The relation to topological field theory is also sufficient to determinethe basic form of fields and states on boundaries. In 4d these give the role of Chern-Simons theory in horizon and other boundary states. Thus, it gives the basic quantum geometry of horizons.
L.M.: That's an uncontrollable approach to ideas. A black hole is a finite energy object in spacetime, and a quantum theory of gravity must describe it as a generic state. Inventing a new kind of explanation and dynamics for the horizons is unjustified. Note that the main idea of LQG is to construct space from "atoms" and "links". Black hole is another example of "space", and already for the black holes, the original "atoms" and "links" are not enough. So one invents a new type of physics for the horizons, and argues that there is the Immirzi parameter that solves the discrepancies. All the predictions of the Immirzi parameter are falsified by explicit calculations, but it does not matter.
Once we have the basic form of spin foam models, which follow from the general relation to BF theories, we can consider the problem of dyanmics in the following light. Given the choices made above, the spin foam amplitudes are chosen from the invariants of the algebra which labels the spin networks. There is then a large class of theories, differing by the choice of the spin foam amplitudes. Each is a well defined spin foam model, which gives amplitudes to propgate the spin network states based onthe chosen dimension and algebra.
L.M.: Well, this is exactly why Nicolai et al. say that the spin foam models in their current form are not a well-defined theory. If the amplitude of any basic process is undetermined and there is an infinite amount of such unknown but relevant things, then we know nothing. Moreover, it seems obvious that the infinite ambiguity is not just a temporary state of affairs but a very basic defining feature of LQG.
The lack of uniqueness is unaviodable, because there is a general class of theories, just like there is a general class of lattice gauge theories.
L.M.: That's a completely wrong comparison - pure gauge theories have no dimensionless parameters at all if they're asymptotically free. The main problem is not a couple of discrete choices - such as the spacetime dimension. The main problem is the ambiguity of the details of the Hamiltonian - or the amplitudes of the microscopic spin foam processes. I am afraid that Lee is not interested in these "details".
These theories exist, and the general program of LQG as some of us understand it, is to study them.
L.M.: Lee obviously uses the words "theory" and "exist" with a different meaning than the rest of us. If a system of ideas can't predict quantitative results after a finite number of measurements, then it does not exist as a physical theory. A non-physicist can construct a class of theories in which anything can happen in the Universe, depending on God's decisions. Does it mean that he has found a real physical theory?
From a modern, renormalization group point of view, the first phsyical question to be answered is which of these theories lead to evolution that is sensible, i.e. which spin foam ampltidues are convergent in some approrpiate sense. The second physical question is to classify the universality classes of the spin foam models and, having done this, learn which classes of theories have a good low energy behaivor that reproduces classical GR and QFT.
L.M.: The religious person can say the same things. From a modern, genetic point of view, the first physical question is which God's decision lead to finite answers. The second question is to classify the universality classes of God's decisions and, having done this, learn which classes of God's decisions reproduce the fossils that the evolutionary heretics have claimed to diminish the importance of creation. ... Well, one can always say these words and define some questions, but it does not mean that they're good questions. This is a point that Lee does not want to understand - that a whole idea or approach is identified as less valuable if it leads to no explanations of known facts and relations between them and no predictions. For Lee, the discrete structure is a dogma, and independently of the number of decades in which the research shows that it is a weak idea, it must be studied.
It is of course of interest to ask whether some of these theories follow from quantizing classical theories like GR and supergravity, by various methods. But no one should mind if the most successful spin foam model, in terms of both matheamtical elegance and physical results, was not the quantization of a classical theory, but only reproduced the classical theory in the low energy limit. How could one object from a physics point of view, were this true?
L.M.: The reason why this question is meaningless in reality is that there are no successful spin foam models, and most likely, there never will be any.
This is the point of view from which many of us view the problems with the hamiltonian constraint you describe.
The next thing to be emphasized is that there is no evidence that a successful spin foam model must have a corresponding quantum hamiltonain constraint. There are even arguments that it should not. These have not pursuaded everyone in the community, and this is proper, for the healthiest situation is to have differing views about open problems. But it has persuaded many of us, which is why many people in the field turned to the study of spin foam models after the difficulties you describe were understood, more than ten years ago.
L.M.: I wonder why they think that they have a "theory". What they have and can agree about is the religious - and most likely, falsified - assumption that the spacetime looks like some kind of LEGO. The situation of every other, more detailed question is fuzzy. No question can be ever answered if physics is approached in this way - and it is probably not even their goal to answer a question. Moreover, it is not the LQG people, but Democritus (and Maxwell who designed aether and FitzGerald who constructed an actual model) who should be credited for these (wrong) ideas.
For example, Fotini Markopoulou argued that, as the generators of infinitesimal spatial diffeos do not exist in the kinematnical hilbert space, while generators of finte spatial diffeos do exist, the same should be true for time evolution. This implies that there should only be amplitudes for finite evolutions, from which she proposed one could construct causal spin foam models.
L.M.: The fact that the geometrical operations cannot be written in terms of generators is another manifestation of the non-separability of the Hilbert space and ultralocality of any rules that one can construct within the framework. At any rate, it is a sign of an inconsistency. In every working and consistent theory, the generators G of some operation can simply be written as the limit for epsilon going to zero of (T(epsilon)-1)/epsilon where T(epsilon) is the finite transformation by epsilon. If one can't do these things, something is definitely not working with the math.
This was partly motivated by the issue ultralocality. (Btw, you dont emphasize the paper that first raised this worry, which was my gr-qc/9609034). The worry arises because moves such as 2 to 2 moves necessary for propagation do not occur in the forms of the hamiltonian constraint constructed by Thiemann, Rovelli and myself, or Borissov. This is because they involve two nodes connected by a finite edge.
However, the missing moves are there in spin foam models. This concretely confirms Fotini's argument. In fact, as Reisenberger and Rovelli argued, invariance under boosts generated by spacetime diffeo requires that they be there. For one can turn a 1-3 move into a 2->2 (0r 1->4 into 2-> 3) move by slicing the spin foam differently into a sequance of spin networks evolving in time.
L.M.: Well, yes, this is the first part of a proof that one can't ever obtain a Lorentz-invariant (or approximately Lorentz-invariant) theory in this framework. Obviously, our friends are never patient or brave enough to finish the proof.
So we have two arguments that suggest 1) that the problem of ultralocaity comes from requiring infinitesimal timelike diffeos to exist in a theory where infinitesimal spacelike diffeos do not exist and 2) the problem is not present in a path integral approach where there are only amplitudes for finite timelike diffeos.
L.M.: The Lorentz violation has nothing to do with using the Hamiltonian or the path-integral formalism. The existence of a symmetry is a completely physical question, and of course that the spin foams violate the Lorentz symmetry as much as the spin networks with a Hamiltonian.
One can further argue that if there were a regularization of the hamiltonian constraint that produced the amplitudes necesary for propagation and agreed with the spin foam ampltidues, it would have to be derived from a point splitting in time as well as space. This suggests that there is a physical inadquancy of defining dynamics through the hamiltonian constraint, in a formalism where one can regulate only inspace and not in time.
Let me also add that there is good reason to think that the other issues such as the algebra of constraints arise because of the issue of ultralocality. Thiemann's constraints have the right algebra for an ultralocal theory.
It was for these and other reasons that some of us decided ten years ago to put the problems of the hamiltonian constraint to one side and concentrate on spin foam models. That is, we take the canonical methods as having been good enough to give us a kinematical frameowrk for a large class of diffeo invariant gauge theories, but unnecessary and perhaps insufficient for studying dynamics.
L.M.: The scientists should not use these vague words. It's not just that the spin network Hilbert space is insufficient: it's that a description based on it is guaranteed to be wrong. But the LQG practitioners usually do not like to make clear statements. The more vague the picture is, the more one can jump in between different inconsistent statements and avoid the fact that these models - and the basic dogma that underlies them - have been falsified.
At the very least, making a point splitting regularization in both space and time seems a much more difficult problem and hence is less attractive than spin foam methods where one can much more easily get to the physics. Given that the relation to BF theory gives us an independent way to define the dynamics, and path integral methods are more directly connected to many physical questions we want to investigate, there seemed no reason to hold back progress on the chance that the problems of the hamiltonian constraint can be cleanly resolved.
Nothing I've said here means that I am not highly supportive of Thomas's and others efforts to resolve the problems of the hamiltonian dynamics-I am. But it must be said that a "review" of LQG that focues on this issue misses the significance of much of the work done the last ten years.
L.M.: It's easy to miss something if this something is almost equal to zero.
Let me make an analogy. No one has proved perturbative finiteneess of superstring theory past genus two. I could, and have even been tempted to, write a review of the problem, highlighting the heroic work of a few people like d'Hoker and Phong to resolve it.
L.M.: The difference is that there exist hundreds of reasons to think that an inconsistency that would suddenly appear at 3 loops in the superstring expansion is a highly unlikely thing. The integrals over the moduli spaces are free of UV divergences in the bosonic string, and we can prove that the IR divergences disappear in the superstring. Then there are lots of formalisms - Green-Schwarz light-cone gauge string and its non-perturbative completion, namely matrix string theory; Berkovits' pure spinor formalism - that make it pretty insane to believe that something could go wrong. The proof of consistency is almost complete.
Update: Well, Hiroši Ooguri has kindly pointed out that I should remove the doubts about Berkovits' proof in hep-th/0410079. Hiroši believes that Nathan's proof is complete and simple enough to be understood. The only reason why I did not make a clear statement is my limited ability: I have not been able to understand why Nathan's pure spinor prescription is unitary, or equivalently why its results are equivalent to the RNS prescription. It's the composite character of the "b" ghost that is conceptually difficult for me - but most likely, it's just because I am slower. Hiroši also refers to another proof of perturbative finiteness by Mandelstam that combined the virtues of the RNS and GS formalisms, and argues that Nathan's proof is more straightforward.
On the other hand, Nicolai et al. and others have an almost complete proof that the things can't work in LQG. Of course, if someone thinks that an arbitrarily small uncertainty about anything implies that all bad ideas are equally good as all good ideas, it's hard to explain him that his belief is not rational. In string theory, all the quantitative consistency checks etc. always work. In LQG, they never work, and it makes a difference.
Well, someone can be tempted to write a review claiming that the higher-order superstring amplitudes probably don't work - but only a person who does not care whether others think of him as a good or less-than-good physicist could submit such a review simply because there exist no arguments whatsoever that the expansion should suddenly break down.
I think it would be useful if someone did that, as their work is underappreciated.
L.M.: Well, the "experts" don't share Lee's opinion. It's an interesting piece of math, but it's certainly not one of the most interesting current tasks for string theory. Everyone knows what the answer is, the main answer has been obtained using other means, and most of us are not terribly interested in completely rigorous proofs. We're more interested in big physics questions, not some particular minor tasks associated with a specific formalism, a formalism that does not have to be the most appropriate one to attack a problem.
But it would be very unfair of me to call this a review of, or introduction to, the state of string theory.
L.M.: I don't think it would be unfair; it would be just silly. If someone writes a paper in which he claims that he believes that at 3 loops, stringy perturbative expansion suddenly breaks down because of some mysterious, unspecific, undescribed new effect, he has the full right to express this opinion. But the rest of us have the right to decide whether we think that the author of such a paper is intelligent, and Ginsparg has the right to reject every submission.
Were I to do so, I would rightly be criticized as focusing on a very hard problem that most people in the field have for many years felt was not crucial for the development of the theory. This is not a perfect analogy to what you have done in your "review", but it is pretty close.
L.M.: It is not close because Nicolai et al. show very explicit things that are going wrong if one actually tries to calculate; what Lee is saying is just a promotion of a highly unlikely, unjustified, non-quantitative, and wild hypothesis whose only goal is to support a bizarre piece of ideology.
There are other mis-statments in your review. For example, there are certainly results at the semiclassical level. Otherwise there could not be a lively literature and debate about predictions stemming from LQG for real experiments.
L.M.: Well, it's easy to falsify Lee's "logical" conclusion just by looking at reality. There exist speculative papers that talk about the experimental predictions of "the theory" even though there is no well-defined theory. The reason why this is possible is simply that a piece of paper or hard disk can tolerate anything. ;-)
See my recent hep-th/0501091 for an introduction and references. Of course semiclassical states do not necessarily fit into a rigorous framework-after all, WKB states are typically not normalizable. But I would suggest that it may be too much to require that results in QFT that make experimental predictions be first discovered through rigorous methods. At the standards of particle physics levels of rigor, there are semiclassical results, and these do lead to nontrivial predictions for near term experiments. It is possible that a more rigorouos treatment will in time lead to a rigorous understanding of how classical dynamics emerges-and that is a very important problem. But given that AUGER and GLAST may report within two years, may I suggest that it is reasonable to do what we can do now to draw predictions from the theory.
In closing let me emphasize again that your efforts are very well appreciated. I hope this is the beginning of a dialogue, and that you will be interested to explore other aspects of LQG not covered by or addressed in your review.
Sincerely yours,
Lee Smolin
