Lee Smolin wrote an interesting responses to comments in the comment section of my posting about his Physics Today piece entitled “Why No New Einstein?”. I’m reposting it here.

“Dear Peter and colleagues,

I am grateful for the attention given to my essay. I only want to emphasize a few points here. The main thing is that the essay is carefully written. It does not advocate more funds to LQG or any other program. It explicitly advocates more support and positions for young, ambitious theorists pursuing their own research programs who are unaffiliated with any larger program. Several proposals are made for how to accomplish this. I would hope that the focus of the discussion could be on these proposals.

-String theory is criticized in the essay mainly because it is currently sociologically dominant, and so subject to the problems mentioned. It was necessary to do so as many readers of physics today will be unfortunately unaware that there are any problems with string theory, or any viable alternatives. Anyone with a long enough memory will know that the sociological issues in high energy theory predate string theory, and have hurt physics in the past, i.e. in the case of S-Matrix theory.

-I hope I don’t have to say that I am not anti-string theory. My current last paper on the ArXiv is a technical paper in string theory, and I have 14 more in past years, plus 8 papers on related topics such as the landscape. I wouldn’t have written these papers if I didn’t think there was a good chance string theory is relevant to nature. The fact that someone like me who contributes sometimes, but not exclusively, to string theory, is not considered “a string theorist” is part of the sociological problems my essay criticizes. Similarly, the fact that one can elicit angry responses, and be called “anti-string” for carefully and correctly recounting the actual status of various conjectures is a sign of an unhealthy sociology. No one calls someone anti-LQG or anti-QCD when they do a similarly honest summary of what is known and not known in those fields.

-I would claim that the sociological issues mentioned in the essay have hurt string theory even more than they have hurt the alternative programs, because they greatly limit the range of ideas worked on, and because people with a lot of imagination and intellectual independence are either selected out or choose themselves to work within communities which are more friendly to diversity and imagination. As a result, key issues such as the question of a background dependent formulation, or perturbative finiteness, don’t get a lot of attention, in spite of their centrality for the whole program.

-I was grateful that someone noted the range of subjects at the LQG meetings. This was not planned, it is a natural outcome of the more open and curious atmosphere among people who work on the subject. We don’t believe we should have a meeting without inviting people from alternative and rival programs to report to us what they are doing, as well as to serve as critics. At the meeting in Marseille last May we even invited a persistent critic of LQG-Ted Jacobson-an early contributor who is now very critical of the subject-to give a talk to lay out his criticisms. I think it would be very good for string theory if the organizers of their meetings took a similar attitude.

-Someone asked for a blanket term for LQG, CDT, causal sets etc. We use background independent approaches to quantum gravity. There is a lot of interchange of ideas, techniques and people among these programs, and many of us have contributed to more than one. There is a very different intellectual climate, in which diversity, creativity and independence are strongly encouraged.

-Someone is asking for what is “LQG proper?” But the fact is that a lot of different things are now going on roughly under the name of or related to LQG. After all, this is now a community of > 100 people and there is no orthodoxy and no one trying to control what people work on. We agree generally on what has been achieved and what problems remain open, but not much beyond that. There is a healthy variety of approaches and attitudes towards the open problems. If there is one thing we all agree on it is that no approach is likely to achieve the right theory that is not background independent at its foundations. Come to the meeting and see what is happening.

-While the point of my essay was not to advocate more funding to any particular direction, if you ask me I will of course say that I think that people working on background independent approaches to quantum gravity deserve much more support. Among them are Loll and Freidel, that I am glad someone mentioned, but there are many others.

-I did not, as Lubos implies, advocate funding a large number of people who do nothing but think about the foundations of quantum theory. What I do advocate is much more support for the kind of person who might be inclined to work on foundational issues. These are deep and independent thinkers who believe that the road to progress in physics is confronting the hard problems directly. But there is no need to argue about whether more funding for foundations of quantum mechanics would be fruitful. The experiment has been done. For decades there was no support at all, and slow progress. Then, because of the possibility that quantum computers could break codes, there has been a lot of support for the last few years. And a lot of progress has been made, both experimentally and theoretically on aspects of foundations of QM.

-Although this essay was not written to advocate LQG, since it is attacked in response I should try to clear some things up. Someone asks for an accounting of the present status of the field. I among others, have given one in hep-th/0408048, shortly to be updated.

As to the issue of anomalies, i.e. the claim that we ignore the established knowledge that “INFINITE-DIMENSIONAL CONSTRAINT ALGEBRAS generically acquire anomalies on the quantum level…” is simply false. It is contradicted by rigorous existence and uniqueness theorems in LQG. As a few people do nevertheless take this seriously let me start from a point we can agree about and see if we can clear this up for good. I would hope we can all agree that:

1) The approach to quantization of constrained systems is different in string theory and LQG. The former approach depends on a gauge fixing that refers to a fixed background metric. It results in the construction of a Fock space. The latter is background independent and involves no background metric, no gauge fixing and results in a state space unitarily inequivalent to a Fock space.

2) There is a body of rigorous results that support each kinds of quantization. Hence it cannot be a question of which is correct mathematically. Both are correct, within their contexts. It is a question only of which construction is appropriate for which theories and which describes nature.

3) The treatment of constraints in string theory depends on certain technical features of 1+1 dimensional theories, particularly the fact that there is a gauge in which L_0 plays the role of a Hamiltonian and therefore should, in that gauge, be quantized so as to have a positive spectrum. The anomalies are not generic, as asserted above, rather they depend on the additional condition that L_0 should be a positive operator. There are other reps of Diff(S^1 ) that are non-anomalous but in which L_0 is not positive. So a choice is made in the standard quantization of string theory, which his motivated by the physics. This does not mean it is the right choice for all physical theories.

4) Conversely the existence and uniqueness theorems which support the LQG quantization work only in 2+1 dimensions and above for the reason that gauge fields don’t have local degrees of freedom in 1+1 dimensions. The existence theorems tell us that there are quantizations in 2+1 and higher of diffeo invariant gauge theories that have unitary, anomaly free realizations of diffeo invariance. The uniqueness theorem tells us that the resulting state space we use in LQG is unique.

5) Now it is true that Starodubstev and Thiemann have found it an interesting exercise to apply the LQG techniques to free string theory. Not surprisingly they get a theory that is unitarily inequivalent to the usual one. This does not mean that the usual quantization of string theory is wrong, nor does it mean that the LQG techniques are wrong when applied to other problems, where the existence and uniqueness theorems together with a large number of results prove their worth. All we learn is that the two quantizations are inequivalent, which was to have been expected.

6) With regard to the non-standard quantization, in which holonomies, but not local field operators are well defined, it is of course true that when applied to standard systems this leads to inequivalent results. “This apparently leads to unphysical consequences, such as an unbounded spectrum for the harmonic oscillator.” But, give me a break, do you really think someone is proposing to replace the standard quantization of the harmonic oscillator with the alternative one? What is being proposed is that the quantization used in LQG is well suited to the quantization of diffeo invariant gauge theories.

In case it is not obvious, let me emphasize that harmonic oscillators are not relevent here, and can play no role in a background independent quantum theory, precisely because the division of a field into harmonic modes requires a fixed background metric. Thus, the physics of the problem REQUIRES an alternative quantization.

The detailed motivation is, I think, well argued in the papers, and are supported by the results as well as the existence and uniqueness theorems. First, is well known that a complete coordinatization of the gauge invariant configuration space for a non-Abelian gauge theory requires the holonomies. Second, using them gives rise to the unitary non-anomolous reps of the spatial diffeomorphisms.

Nor is anyone proposing using non-seperable Hilbert spaces for the full theory, the point is that when one mods out by the piecewise smooth spatial diffeos one is left with a seperable Hilbert space.

I am frankly puzzled why someone who claims to know the literature well would throw up examples like the harmonic oscillator up in this context. I can try to understand their point of view, but it certainly reads as if they either are choosing to ignore the basic point, which is that background independent quantizations cannot use fock space, or they are looking to make debating points to impress ignorant outsiders. They must know comments like this are not going to influence experts, because they are, after all, taken from our own papers, written precisely because we wanted to clarify the difference between the new and standard quantizations and the limits of the applicability of each.

With regard to the sociology of the string-loop division, “Roughly speaking, string theorists are fundamentally particle theorists with a strong understanding of quantum theory, whereas loop people are gravitists with a background in GR”, this is a myth. Rovelli, myself and many other people in LQG were trained as particle physicists, myself at Harvard in the late 70’s. Most of the physical motivation for LQG comes directly from ideas about formulating gauge theories in terms of loops that were studied by Polyakov, Wilson, Migdal, Mandelstam, Neilsen and others. LQG is squarely an outgrowth of their intellectual tradition. The only thing we added was to correctly treat the diffeomorphism invariance exactly in the quantum theory. This led to new results just as the exact treatment of gauge invariance in lattice gauge theory led to new results. I would claim that we made progress in LQG precisely because we had a very good grounding in QFT.

String theory, as it is practiced, makes much more contact with the general relativity tradition, especially the once discredited tradition of extending general relativity to add dimensions and degrees of freedom in the search for a unified field theory. You are much more likely to read a paper which studies solutions to a generalizationsof the Einstein equations, with hbar=0, by a string theorist than by someone working on a background independent approach to quantum gravity.

This of course does not mean that string theory is wrong. But I believe it does mean that by enforcing a narrowly restrictive notion of what constitutes good work, the community of string theorists has hampered progress in string theory by excluding from consideration the lessons learned by attempts to do what string theory must do eventually if it is to be a real theory: which is to find a background independent formulation of a quantum theory of spacetime.”

Actually, I wasn’t talking about spatial diffeos, but rather the group of 4-diffeos, which is the full constraint algebra in covariant formulations. The Dirac algebra is an artefact of the foliation of spacetime which nobody likes, but phase space is intrinsically a covariant concept. This observation apparently goes back to Lagrange and has been emphasized e.g. by Witten, Ashtekar and Rovelli. My suggestion for how one can use cohomological methods to construct the covariant phase space and use it for quantization can be found on the arxiv.

The reps do not depend on a reference metric or foliation. They do, however, depend on a privileged curve, which I call the observer’s trajectory. The key idea is to expand all fields around it before quantization. The trajectory is not a background structure, however, because must be quantized together with the fields.

I recently tried to initiate a discussion about this on spr. Since I have already abused Peter’s hospitality by promoting my own ideas, maybe the discussion should be moved.

Re the last comment, on recovering ordinary QFT from LQG, let me stress again that there are explicit known semiclassical states, and ordinary QFT is recovered at long wavelengths by studying excitations of them. Hence, we know that the physics of flat, or DeSitter spacetime is in the theory.

The problem is to go beyond these results to

i) show that the ground state, subject to some appropriate boundary or asymptotic conditions, is such a state, ii) show whether classical spacetime emerges from a generic physical state and iii) show whether lorentz invariance is preserved, broken or deformed by Planck scale corrections in the ground state, and hence predict what should be seen in AUGER, GLAST, ICECUBE and other upcoming experiments.

In recent papers, Freidel et al show how deformed Poincare invariance arises as the limit of LQG coupled to matter in 2+1. See also hep-th/0501091 for an admittedly heuristic argument that this is true also in 3+1.

For these see sections 4.4 and 5 of hep-th/0408048 and the references provided there.

As to whether it might be easier to obtain certain results in a different formulation with anomalous reps of the spatial diffeo’s, perhaps, and this could be worth trying, but only if one does not put in what is to be shown, which would be the case if those reps are constructed with reference to a background metric.

“But surely, the new quantization should in some way resemble a Fock type quantization in the limiting case where there is a classical metric. So some limit of the new quantization should recover the familiar harmonic oscillator.”

Well isn’t thats the big unanswered question though in LQG, that they can’t seem to recover minkowski space? In many ways it makes sense if you think about it, the anomalous reps constrain ones freedom considerably, whereas the generic formalism they use doesn’t contain that structure, or rather it subsumes it so completely it washes it out. Finding a mechanism to retrieve that as some sort of limit strikes me as a *hard* problem in asymptotic behaviour. Everyone who does QG knows how hard that can be, in one guise or another.

The results are very preliminary. I don’t think it’s in anyone’s best interest to make sweeping pronouncements.good caution. I will try to avoid doing that

thank you Aaron, your words are partially reassuring. I remember the parameters which they discuss tuning—-essentially G (or the inverse kappa) and an asymmetry parameter Delta (or the related alpha) which intuitively tells the “squatness” of 4-simplex (its foreshortening of timelike edges).Those are the parameters are inputs to the model. They don’t tune them so much as describe the phase diagram of their model in the plane of those two parameters. They find one phase which seems to describe a macroscopic universe.

This is a different issue than the continuum limit where, as you take the spacing to zero, you tune various parameters to achieve a phase transition.

Also, I never said that the continuum limit of this theory (if such a thing exists) is something like only lives on S^3 x S^1. Who knows what will happen. They have some interesting ideas on the effective dimension as a function of length scales, but this doesn’t necessarily mean some bizarre spacetime topology. Rather, it could describe some sort of fuzzy graviton, or something else. Beats me.

It’s not even clear that these models describe anything really. The results are very preliminary. I don’t think it’s in anyone’s best interest to make sweeping pronouncements.

Aaron:

You can’t just take the lattice size to zero to get a continuum limit. Things are generally more complicated than that and involve tuning various couplings so as to achieve a phase transition in the limit. Ambjorn et al comment on this on pp. 12-3.thank you Aaron, your words are partially reassuring. I remember the parameters which they discuss tuning—-essentially G (or the inverse kappa) and an asymmetry parameter Delta (or the related alpha) which intuitively tells the “squatness” of 4-simplex (its foreshortening of timelike edges).

but I am still not fully confident that even with the correct tuning of the parameters for a “continuum limit” —obtained by computing the path integral for simplexes of size ‘a’ repeatedly as ‘a’ goes to zero—-I am still not confident that in the limit there is a continuum with topology of R x S3

Or, as you say, S1 x S3, since they sometimes make it periodic so it fits in the computer and replace R by the circle S1, in which case what we are talking about may really be S1xS3.

I appreciate the clarification, but I still have a suspicion that the spacetime of Renate Loll is not topologically either S1xS3 or RxS3 but may instead be a mathematical novelty.

(my doubts arose partly because the dimension varies with scale and tends to be quite a bit less than 4 at close quarters)

the basic CDT spacetime is the set R x S3, but I am not sure that the topology is R x S3. In fact I am unsure how to define the topology on it, except as the cumulative effect of many histories, or in other words as a somewhat uncertain topology.The topology of each element in their path integral is always the same, usually S^1 x S^3.

the CDT continuum is constructed as the limit of PL 4-manifolds as the size of the simplex goes to zero. Any PL 4-manifold is a topological 4-manifold. In this case each of the approximating PL 4-manifolds is topologically R x S3.You can’t just take the lattice size to zero to get a continuum limit. Things are generally more complicated than that and involve tuning various couplings so as to achieve a phase transition in the limit. Ambjorn

et alcomment on this on pp. 12-3.Chris W:

From Quantum general relativity and the classification of smooth manifolds (Hendryk Pfeiffer):…The diffeomorphism invariance of the classical observables then implies in the language of the triangulations that all physical quantities computed from the path integral, are independent of which triangulation is chosen. The discrete formulation on some particular triangulation therefore amounts to a complete fixing of the gauge freedom under space-time diffeomorphisms…(I’ll leave it at that. It’s been raining in Vermont for several days. I’ve spent a good portion of the past 36 hours whitewater kayaking, and could use a good night’s sleep.)–

Chris, that was a very interesting paper by Hendryk Pfeiffer. I have a glaring point of mathematical ignorance that perhaps you might help me with.

the basic CDT spacetime is the set R x S3, but I am not sure that the topology is R x S3. In fact I am unsure how to define the topology on it, except as the cumulative effect of many histories, or in other words as a somewhat uncertain topology.

the CDT continuum is constructed as the limit of PL 4-manifolds as the size of the simplex goes to zero. Any PL 4-manifold is a topological 4-manifold. In this case each of the approximating PL 4-manifolds is topologically R x S3.

But I am not sure in what sense, if at all, the limit of such things has to be a topological 4-manifold. This is Hendryk Pfeiffer’s term—-locally homeomorphic to R4. In a topological 4-manifold, as Pfeiffer says, the transition (i.e. coordinate change) functions are C0 continuous but not necessarily differentiable.

So I am not sure that Pfeiffers paper would apply to the CDT spacetime, for instance.

BTW I notice he posted the first version 21 April 2004. A couple of weeks later Pfeiffer was at the “Loops 04″ conference that Carlo Rovelli organized at Marseille, and talked about it. On the second day of the conference, Tuesday 4 May, Renate Loll gave a talk on CDT in the morning and Hendryk Pfeiffer spoke that afternoon. It would have been an appropriate time to ask this kind of question, I suppose.

There may be some embarrassingly obvious answer to my doubts about this. If you are aware of one please let me know after you have rested up from kayaking.

this is not intended to elicit a response from Lee Smolin (if he happens to be still reading the thread) but simply to acknowledge that I stand corrected by him on the subject of LQG and spin foams spacetime discrete structure. Accordingly I remain undecided on how to view the CDT lack of discreteness declared by Loll et al for instance on page 2 of hep-th/0505113. They say their probing by Monte Carlo-style simulation has so far not uncovered evidence of “fundamental discreteness…[or]…a minimal length scale.”

1. I can, on the one hand, set aside notions of reconciliation and consider LQG and CDT to be incompatible models, each consistent and supporting calculation, of quantum spacetime dynamics. Presumably, each is able to make testable predictions (though CDT is at a somewhat earlier stage of its development) and empirical observation will eventually distinguish between them.

2. On the other hand, I can surmise that Loll et al have not probed sufficiently and that further computer simulation or analysis might uncover discrete spacetime structure in their model. For instance, if I understand correctly, no one yet has constructed an operator in CDT corresponding to an observer measuring a physical area or volume. If this is eventually done, further along in CDT development, I cannot see how to rule out the possibility that such an operator have discrete spectrum. Thus some discrete structure might appear at a later stage.

sorry for misstating Smolin’s role in PI, back a ways in this thread—he cleared that up

Some remarks,

For the history of early “eigenvolumes” and the use of quantums of volume (in the same sense that quantums of energy) see

Length Scale for the Constant Pressure Ensemble: Application to Small Systems and Relation to Einstein Fluctuation Theory. J. Phys. Chem. 1996, 100, 422-432.

One can prove that those “quantum of volume” and related to quantum of area of standard BH. One can also prove that in the limit quantumA –> 0 the BH is stationary (constant total A).

You have claimed in several articles that departures from standard (string theory)

E^2 = p^2 + m^2

predicted from LQG could be “directly” verified in the next generation of acellerators (2007?). Have you considered an “indirect” verification of quantum of volume at macroscopic scales? I am working in that.

What is the status of Lorentz invariance and frame independence in today LQG?

You claim a consistent classical limit. Can LQG obtain departures from GR at the extragalactical scale (e.g. TF law for anomalous galaxies)?

Lee Smolin,

Regarding discrete structure of spacetime, i think that can be proved indirectly with usual experiment.

In fact, some well proven mathematical theorems show that if area and volume are really zero one obtains wrong answers for macroscopic questions. It is an usual error to believe that classical gravitation is based in a pure diferential manifold. It is a error conected with some wrong asumptions of usual calculus (non standard analysis and hiperreal numbers solve that partially)

I can prove that if quantum of area –> 0 then there is no Hawking radiation, there is no dissipative effects at astro or cosmo scale, etc. Therefore there is quantum sure!

The quantum of area/volume is related to departures from standard (string theory)

E^2 = p^2 + m^2

Moreover, the existence of quantum of volume is needed in the computation of partition functions, since that many standard computed PF have dimension of volume. That is well known. In fact the first “quantums” of volume were named eigenvolumes by Guggenheim.

I think that you are not replied M query. I am sure that one cannot obtain a pure consistent classical state from LQG, due to mathematical incongruencies. Someone derived GR from LQG?

I am not talking about obtain certain spectra that like how. The same situation arises in usual limits of QM. Somewhat like one need “decoherence” effects (h^2 terms to Schrodinguer like in standard Calderia-Legget equation) we need add news terms to usual LQG. This also solves the old problem of time in HQG; Wald recent proposal is unnecesary.

“So the attitude is rather different from other approaches. Some string theorists admit they do not know what string theory is, but they nevertheless are sure it is right.”

Great!!! One day Witten admits that nobody know that is really string or M theory, and another day he claims that it beatiful and elegant!

Lee: I realize that what I wrote could be construed as a personal attack, and I apologize for that. My excuse is that I feel a great sense of frustration for not being noticed (by physicists, mathematicians were always much more open-minded). I started to work on multi-dimensional generalizations of the Virasoro algebra back in 1987 and published the first paper in 1989. Although not not explicitly stated, it was always obvious to me that this algebra must have applications to quantum gravity, for the same reason that the ordinary Virasoro algebra is relevant to string theory.

My funding ran out in 1993, which made sense at that time because my program was stuck. But after the key obstacle was removed by Rao and Moody a few years later, it was possible to develop a representation theory, which may be regarded as the quantum analogue of tensor calculus (tensor fields carry classical reps of the diffeo group). It was totally obvious to me that this must be to quantum gravity like tensor calculus is to classical gravity, i.e. very important. The complete lack of interest was extremely frustrating in that situation.

Thus, I behaved nicely throughout the 1990s, and it didn’t do me any good. After more than a decade, one can start to lose patience. I only started to receive feedback after I criticized string theory in math-ph/0103013 (from a quite original viewpoint). This taught me that being nice is something that the physics community simply does not award. So be it.

Given my present age and family situation, I am not really personally interested in funding anymore; not unless it comes in the form of a permanent position in Stockholm without teaching duties anyway. But I would very much like to see an influx of people thinking along these lines, because I believe it is a promising idea, but I myself am stuck at this time.

On a different note, I think that your proposals have some serious problems. In what way would your Einstein fellowships differ from a MacArthur genius grant except that string theorists are excluded? Who would decide who should get these grants – you, Ed Witten or maybe Lubos Motl? It seems to me that funding must ultimately be decided by tenured professors, i.e. the same people who decide about funding today. Finally, should a paper like hep-th/0412325 be regarded as string theory or not?

” So long as the resulting theory is well defined, I don’t see the force of an argument from a priori grounds. that experience shows Fock type quantizations must be right in all cases because they only work when there is a fixed background metric, while the whole point of the new quantization is that it provides an answer to the question of how to construct a well defined QFT in the absence of any background metric.”But surely, the new quantization should in some way resemble a Fock type quantization in the limiting case where there is a classical metric. So some limit of the new quantization should recover the familiar harmonic oscillator.

“Also see gr-qc/0311055, gr-qc/0407094, and gr-qc/0407093″

Cool – more category theorists.

From

Quantum general relativity and the classification of smooth manifolds(Hendryk Pfeiffer):Also see gr-qc/0311055, gr-qc/0407094, and gr-qc/0407093.

(I’ll leave it at that. It’s been raining in Vermont for several days. I’ve spent a good portion of the past 36 hours whitewater kayaking, and could use a good night’s sleep.)

Thanks again for all the insightful comments. Perhaps I can add something to a few of the threads of discussion.

-On LQG and discrete structure. First, do we agree that even though electrons move in space the spectrum of the hydrogen atom being discrete means that quantum mechanics of the atom has discrete structure? In a very similar sense, since all the geometric observables including volume, area (and yes length) have discrete spectra, corresponding to a discrete basis (of diffeo classes of embeddings of labeled graphs) then the quantum geometry of space has become discrete. The key point is that the discreteness scale-roughly L_Planck, cannot be taken to zero, otherwise black hole entropy comes out wrong, and semiclassical states do not correspond to classical metrics.

-But it is true that if you derive a version of LQG from a strict quantization of GR, there is a fixed background, which is the bare differential manifold. There is no background metric but there is a background topology and differential structure, defining the diffeo classes of embeddings of the spin networks.

-Hence, Markopoulou followed by Freidel and others, proposed dropping the embedding and basing the theory just on combinatorial spin networks. These models are then discrete in a stronger sense. There are some advantages to this (reformulation in terms of a matrix model, cleaner relation to causal sets) but one can non longer claim the theory is a precise result of a quantization of GR. Both frameworks, with and without embeddings, continue to be studied.

-The discreteness of length was shown in T. Thiemann, gr-qc/9606092, J.Math.Phys. 39 (1998) 3372-3392. Angles also have disrete spectra: S. Major, Class.\ Quant.\ Grav.\ {\bf 16}, 3859 (1999) gr-qc/9905019; gr-qc/0101032.

-On spin foam models and discreteness. There are several different spin foam models under study. In all of them a history is a discrete labeled combinatorics structure (for example branched 2-complex.) In some of them the label sets are continuous because they come from the rep theory of Lorentz or Poincare and areas are not discrete. But these have not been shown to correspond to evolution amplitudes for canonical states. Others (Reisenberger, Markopoulou, etc) do give evolution amplitudes for spin networks and have discrete areas.

- M asks, is there a suitable correspondence principle where known physics can be recovered? The answer is yes. There are several results that show that excitations of certain LQG states reproduce, for momenta small in Planck units, the spectra of conventional QFT’s including gravitons, photons etc on flat space or de Sitter spacetime. Some are cited in section 4.4 of my review hep-th/0408048. See also hep-th/0501091. See recent papers by Freidel, Livine and others that show in full detail how standard Feynman perturbation theory emerges from a spin foam model for gravity coupled to matter in 2+1 when G_Newton goes to zero.

As to what people in non-string approaches to quantum gravity are doing, I agree, why not look at the conferences? Here are some recent ones, some with talks available.

http://www.cpt.univ-mrs.fr/%7Erovelli/program2.html

http://www.ws2004.ift.uni.wroc.pl/flash.html

(talks at: http://www.ws2004.ift.uni.wroc.pl/html.html)

http://www.perimeterinstitute.ca/activities/scientific/PI-WORK-2/

-Several of the comments ask, why quantize as in LQG? Why not quantize with another approach (such as one that uses anomalous reps?)

I do not see how there can be an apriori reason to prefer one quantization scheme over another one. Our job is to construct candidate quantum theories of gravity, compare their results and learn from them. In LQG there are existence and uniqueness theorems that prove that the approach exists, and theorems that guarantee uv finiteness. Thus, the approach leads to a structure that mathematically exists and within which computations can be done. Many computations have been done.

We are thus no longer at a stage where it is interesting to ask why do or why not do questions. There are now a different class of questions which include: Does the theory make predictions? How do they compare with experiment? What properties have been shown? What remains to be shown? There are certainly several key open issues to discuss, and we are not shy to discuss them.

No one is claiming that we know LQG is the right theory of nature. We are claiming that it is a well developed approach, that gives an apparently consistent answer to what we think is a necessary question, which is how to construct a diffeo invariant QFT in the absence of a fixed background metric. This gives a rich arena with many open problems and many things to do either to understand it better, make predictions, or as a jumping off point for the invention and study of new theories.

So the attitude is rather different from other approaches. Some string theorists admit they do not know what string theory is, but they nevertheless are sure it is right. In LQG we study well defined theories, which have many good properties, but most of us feel no need to “believe in them” pending experimental confirmation.

So our attitude is if someone like Thomas Larsson has a different approach that’s great. We know what its like to be starting something new other people don’t understand or support, and we will support you, so long as you don’t waste your and our time attacking us on a priori grounds. We suggest you should try to develop your ideas to at least the point where we can compare the results.

For example, someone asks, “What’s wrong with anomalies? Sure, it turns first class constraints into second class constraints, but Dirac showed us how to deal with that.” Fine, we only insist that this is not the only way. The LQG results and theorems show that you can find diffeomorphism invariant states through a different procedure, involving only first class constraints, which is

a. Construct a kinematical Hilbert space, which is a rep of a Poisson algebra that coordinatizes the phase space, which carries a unitary and non-anomalous rep of the spatial diffeo’s.

b. Use that non-anomalous unitary rep to construct explicitly another Hilbert space, which is the space of diffeomorphism invariant states.

c. Compute many observables of interest representing diffeo invariant classical quantities as finite operators on this space, leading to predictions of physical interest, an ultraviolet finite theory etc.

There are by now so many rigorous results supporting this construction that the burden of proof is on the other side: given that this procedure works and leads to a well defined finite physical theory, why not explore its consequences as a possible quantum theory of gravity?

So when Urs says, “It seems to me that the reason to drop weak continuity in the quantization of gravity in 3+1 dimensions is that it makes an otherwise intractable problem tractable – but possibly at the cost of having oversimplified a hard problem,” fine, but lets discuss the results. Does this lead to a space of states with enough physical states and with a well defined dynamics? YES. Are some states interpretable as semiclassical states? YES. Does that dynamics have all the properties we require for a quantum theory of gravity? YES to some questions such as uv finiteness, other questions are still open, such as a proof that the ground state is semiclassical.

-Aaron says, “It is, in fact, a radically different approach to quantization that, when applied to current theories, gives experimentally incorrect answers.” Thomas Larsson argues that “I find it very disturbing that LQG methods yield the wrong result for the harmonic oscillator.” I don’t understand the logic of their arguments at all. Yes, it is a different quantization, i.e. one based on representations of the algebra of Wilson loops and electric flux’s rather than local field operators. Yes, it is unitarily inequivalent to Fock space. That is good, as Fock space knows about a particular fixed background metric. If a background independent Hilbert space, which quantizes the whole space of metrics, were unitarily equivalent to a Fock space based on a single fixed metric, something would be wrong.

The claim is precisely that this is a new class of QFT’s which is available to quantize diffeo invariant gauge theories in 2+1 dimensions and above, and which has novel features and leads to novel results. So long as the resulting theory is well defined, I don’t see the force of an argument from a priori grounds. that experience shows Fock type quantizations must be right in all cases because they only work when there is a fixed background metric, while the whole point of the new quantization is that it provides an answer to the question of how to construct a well defined QFT in the absence of any background metric.

-If you still want to have an argument on a priori grounds as to why representations of non-canonical algebras will be required to have a background independent quantum theory of gravity, please go back to the papers of Chris Isham from the late 70’s and 80’s where he made a detailed and convincing case for this. These papers, together with the work of Polyakov, Wilson, Midgal etc on formulating quantum gauge theories directly in terms of Wislon loops were the major motivation for LQG. What we did was construct the non-canonical algebras Isham called for from Wilson loops. Also, please note that lattice gauge theory is not based on Fock space.

-Finally, I am not a director of PI, just one of the scientists, so PI is very far from “Smolin’s institute”. Also, when I am defending LQG I try to discuss the whole research program, not my own personal work, which departs in some papers quite a bit from that of many of my friends.

Chris, another thought in connection with what you said earlier:

It should be mentioned that at least some string theorists seem to have a particular antipathy to conjectures about a possible discrete substructure for spacetime, which often play a role in alternative approaches to quantum gravity.Your reference is a bit vague, but you may have heard TALK about a discrete structure of spacetime in canonical-LQG.

In fact the model of spacetime used in that approach is a differentiable manifold, as is the LQG model of 3D space. The basic variables and observables of that theory are constructed on this smooth manifold (not on a lattice or set of discrete points, a mistaken impression easy to get.)

As canonical-LQG theory develops, the spectra of the area and volume operators turn out to be discrete. This means that even though space and spacetime are represented by smooth continuums, when one comes to actually measure some physical area or volume the outcomes of measurement must in theory be confined to a discrete set, which can be calculated in planck units. (The points in the set are very close together, on the order of a planck unit area or unit volume apart, but they are nevertheless separate points). I find this puzzling, and can’t say I quite understand how observation of area and volume can have discrete spectrum. But that is how it turns out.

Not so with length. In canonical-LQG the operator corresponding to measuring a length, if I remember correctly, has not as yet been shown to have discrete spectrum.

Popular accounts of canonical-LQG ordinarily make much of the discrete spectra of area and volume. So also do non-technical survey articles by LQG pioneers such as Ashtekar and Rovelli. That is understandable (the discrete spectra of certain measurement operators are very interesting and have far-reaching consequences) but how far one wants to go towards interpreting that as spacetime’s “discrete structure” is a somewhat matter of taste.

Since the late 1990s the interest of QG researchers has shifted noticeably away from canonical-LQG towards the spin foam approach, whether rightly or wrongly remains to be seen. In the spin foam approach, AFAIK, there is so far no proof that area and volume operators have discrete spectrum.

In my non-expert judgment as an observer, if you put together CDT (e.g. Loll et al) research with Spin Foams (e.g. Freidel et al) you’d get at least 60 percent of current work on spacetime dynamics. And therein would be no discrete spacetime, nor even discrete spectra of area and volume!

But some of the remaining percentage of the work would be canonical-LQG with its discrete spectra albeit constructed on a smooth continuum.

I don’t have statistics on this, only an impression from following the literature, and I am excluding quantum cosmology research (e.g. Bojowald et al) where the model has only a finite number of degrees of freedom. This is a quantum analog of the usual Friedmann equation of classical cosmology—quantum cosmology is symmetry-reduced so it can deal with a finite number of parameters instead of a whole spacetime geometry. Several of the operators in loop quantum cosmology have discrete spectra.

In short, the situation is complicated and it is not at all clear that “QGATS” (quantum gravity alternatives to string) research is moving in the direction you suggest, namely “discrete substructure of spacetime”. Nor is it clear what discrete substructure means, in general.

Probably those you mention as expressing their “particular antipathy” can safely be ignored since they could well be saying more than they actually know about non-string QG. It is no use arguing with them, I should guess, about discreteness, or Lubos Motl’s “aether”, or anything else, for in my experience the antipaths can always find more reasons for antipathy.

[ Chris W. at June 17, 2005 12:36 PM]:

It should be mentioned that at least some string theorists seem to have a particular antipathy to conjectures about a possible discrete substructure for spacetime, which often play a role in alternative approaches to quantum gravity.“often play a role” is vague, Chris, and could be misleading. Aaron Bergman just brought up recent work of Ambjorn and Loll that interested him. This is a prominent example of QG alternatives to string. There is no discrete structure or any suggestion of a minimal length. Here is a quote

http://arxiv.org/hep-th/0505113

Spectral Dimension of the Universe—quote from page 2—

We have recently begun an analysis of the microscopic properties of these quantum spacetimes. As in previous work, their geometry can be probed in a rather direct manner through Monte Carlo simulations and measurements.

At small scales, it exhibits neither fundamental discreteness nor indication of a minimal length scale.—end quote—

Chris, I urge you to keep abreast of the research in QG alternatives to string, which is moving rapidly and does not accord with common “hearsay” that one may get from string experts. I hope you are not relying on hearsay.

The paper which Aaron gave a link to, also recent and by the same authors, is

http://arxiv.org/hep-th/0505154

Reconstructing the UniverseIt would be a good place to get firsthand impressions about this particular non-string development (called CDT, causal dynamical triangulations)

Chris, you say:

Lubos Motl has suggested that such ideas are tantamount to misguided attempts to resurrect the classical aether in the context of quantum gravity.I suspect if we were to go through on a case by case basis we would find that you have received a number of erroneous impressions from Lubos Motl.

Anonymous June 17, 2005 10:16 AM:

It is clear that some of us attach more significance to the harmonic oscillator than you and Smolin do. Fine. But let us at least agree that the Helling-Policastro result deserves to be widely known, just as the fact that string theory is not background independent is well known. Then people can make up their own minds.

It should be mentioned that at least some string theorists seem to have a particular antipathy to conjectures about a possible discrete substructure for spacetime, which often play a role in alternative approaches to quantum gravity. Lubos Motl has suggested that such ideas are tantamount to misguided attempts to resurrect the classical aether in the context of quantum gravity.

It would be interesting to read a thoughtful exposition of the philosophical presuppositions underlying the string theory program (or programs). I haven’t come across one so far, and given the impatience of many string and particle theorists with philosophical discussion of any kind I don’t really expect to see one any time soon.

This intersects in my mind with the issue of how to evaluate and

interpretquantization methods. Here again is a subject for which thoughtful philosophical discussion could be most illuminating. One might wonder why the formulation of a quantum theory should depend on “quantizing” a classical prototype at all. A key notion appears (to me) to be that in our currently accepted understanding of quantum theories, discreteness is a feature ofcertainsolutions, and does not really reside in the fundamental assumptions shared by quantum theories in general. Indeed, quantum field theory is supposed to largely explain the origin of the discrete substructure of matter that was taken as a given* in 19th century and early 20th century physics.–

(* ..with notable exceptions, of course.)

I did not yet get an answer to my question to Urs Schreiber and Robert Helling about rival lines of QG research alternative to string (which I will call “QGATS” for lack of a better term).

So I will repeat the question, just to be clear what i am asking.

The people contributing here (including the string experts Urs Schreiber and Robert Helling) seem to mean several different things by “LQG”, so that confusion gets into this thread of discussion very easily.I would like to know what Schreiber and Helling would identify as the currently active lines of research that are QUANTUM GRAVITY ALTERNATIVES TO STRING.Which people would you gentlemen say are leading QGATS researchers? What papers have been posted recently, say in the past twelve months, that are significant QGATS papers?It would be very helpful to know who you think leading people are, especially the younger crop just getting established.It would be useful to have some specific arXiv numbers of research articles posted in the past twelve months that could serve to typify for us what you think are the main rival directions to string.[Posted by: at June 16, 2005 01:24 PM]

I am grateful to Aaron Bergman for his brief response, with link to a paper by Renate Loll and her co-authors Ambjorn and Jurkiewicz.

But it is unfortunate that Urs and Robert have not yet replied (or perhaps they will choose never to reply to this question.)

I will explain why I think it is unfortunate.

The discussion of the issues raised in Smolin “New Einstein” essay is clouded by people naively equating

Smolin = LQG = QGATS

So people in this thread dismiss Smolin’s call for more program diversity in US instutions, including QG rivals to string, as self-serving. They assume it would simply result in more support for LQG

whatever that is(none of our critics seem to have an accurate idea of Smolin’s research interests, or what should be called LQG, or what the range is of rival alternative lines of research).What is even more unfortunately misleading is an oversimplification often suggested by posts like those of Schreiber and Helling:

QGATS = LQG = Thomas Thiemann’s January 2004 paper.

Whether or not this is intentional, it has seemed to me that as soon as the possibility is raised that money or attention might be reallocated from string to some alternative QG lines of research, we immediately begin to hear references Thiemann’s Loop-String paper.

I can’t believe the way people are falling all over themselves over this Smolin guy. Why not read Thomas Larsson?…Posted by: D R Lunsford at June 16, 2005 09:49 PM

That is an interesting question, drl. I think the answer in part is that Smolin is a leader in “Quantum Gravity Alternatives to String” research. Call it “QGATS” if you like acronyms

the older term “LQG” is no longer sufficiently inclusive or adequately descriptive.

“QGATS”, or whatever you want to call it, is now a hot group of research lines. Postings on arXiv have been increasing sharply over the past 2 or 3 years, while string-related postings (and citation standings) have stagnated or declined.

Smolin’s institute (PI) is one of the few places in the world where there is a rough balance between research in string and Quantum Gravity alternatives.

Two other places I can think of are Hermann Nicolai’s branch of Max Planck Institute (AEI-Potsdam) and it seems now also Gerard ‘t Hooft’s institute at Utrecht.

Interesting things are happening at all these places. And they are the exception—only a handful of such institutes worldwide. So Smolin is not only a leader in an interesting area, “QGATS” :-), but he is also playing in an interesting league. Accordingly, it is not unusual for someone in touch with current developments in theoretical physics to take note of Smolin’s point of view.

I can’t believe the way people are falling all over themselves over this Smolin guy. Why not read Thomas Larsson? He’s an order of magnitude more interesting. And I’ll bet he’s right.

-drl

From Juan:

You can continue erasing my posts Peter, since that your “arguments” (to say) for erasing them is not consistent i continue to post here because information is important. If your argument is simply “i erase because i want do it”, then please explicit it in “your” blog philosophy. I don’t post my last three or four erased posts, simply the later of minutes ago (with minor modifications).(This was followed by a reposting of a comment that Peter had apparently previously deleted.)Juan,

Your insistence on violating Peter’s guidelines bothers me a lot, and probably many others as well. This blog is Peter’s project, something that takes some of his precious time to build and maintain, and willfully violating his requests and standards so that you can push your own views, your own idea of what is important, is simply improper. A blog is not an unmoderated public forum; it is not your given “right” to say whatever you want just because you personally think it is important. If you think certain things need to be said, then find another forum (e.g., sci.physics.* Usenet groups) where you can raise and discuss those issues. It is important to remember we are all guests here, and should treat the host (in this case, Peter) with the same respect we would want to be treated with if we were the host or hostess.

It should be clear that explicitly violating Peter’s guidelines for keeping comments on topic does not impress others. More likely, it gives you more the appearance of a crank, and makes it much less likely that others will take you seriously. It gives the appearance of a guest who acts rudely to the host or hostess; even worse, the guest apparently complains loudly when the host protests their rude behavior.

So please, Juan, respect Peter’s repeated requests to keep posts on the topic. For an umoderated environment, take advantage of the Usenet groups.

Aaron:

Speaking for myself, I find the recent results of Ambjorn et al intriguing, although I don’t claim to understand them all that well right now.

Since the link you gave in your post is

http://arxiv.org/hep-th/0505154

Reconstructing the universeyou might also be interested by another recent paper by one of the same authors

http://arxiv.org/gr-qc/0506035

Counting a black hole in Lorentzian product triangulations>Hmm Aaron I can imagine in three minutes three arguments to relate d=2 and d=4 in the context of quantum gravity.

- That the curvature tensor is basically a 2-dimensional object,

- That path integrals have the custom of having fractal dimension two.

- That Polyakov action in NonCommutative geometry comes from the four dimensional integration of a two dimensional object (NCG integrates via dixmier trace, you can do strange things).

-that a string has worldsurface d=2

(did I said three arguments? Well, the last one does not score)

Aaron, I just saw your post about the work by Ambjorn, Jurkiewicz and Loll. I don’t have time to change my own post, which follows, to accord with yours but I agree—find the Dynamical Triangulations work intriguing.

It is to be one of the main topics at the Loops 05 conference at AEI in October, and Renate Loll is on the invited speakers list.

—-what I was going to post before I saw Aaron’s was this—

it seems to me that as a first approximation a good map of the current active research in QUANTUM GRAVITY ALTERNATIVES TO STRING is given by the non-string topics posted at the “Loops 05″ conference website

http://loops05.aei.mpg.de/

Background Independent Algebraic QFT

Causal Sets

Dynamical Triangulations

Loop Quantum Gravity

Non-perturbative Path Integrals

A representative list of researchers would be the list of invited speakers for that conference, which has been posted although the conference is still several months away.

http://loops05.aei.mpg.de/index_files/Programme.html

I think that the underlying concern shown by several of the posters here is with MONEY. Does string get to keep it or does it have to share some with its rivals? Therefore we cannot afford to be vague about who the rivals are. Funding for young researchers is an especially important topic in this thread and in the earlier “New Einsteins” thread. Thanks to Smolin for bringing issues of support allocation for young theory people (grad students, postdocs, young faculty) to the fore.

Because rival research lines are at issue, we must be clear about what the main QGAS efforts are. If we include some non-string QG in the physics department of a US university, what kind of research would it be? Name some papers from the past twelve months. Name some young researchers featured in the Loops 05 list of invited speakers.

Speaking for myself, I find the recent results of Ambjorn

et alintriguing, although I don’t claim to understand them all that well right now.The people contributing here (including the string experts Urs Schreiber and Robert Helling) seem to mean several different things by “LQG”, so that confusion gets into this thread of discussion very easily.

I would like to know what Schreiber and Helling would identify as the currently active lines of research that are QUANTUM GRAVITY ALTERNATIVES TO STRING.

Which people would you gentlemen say are leading QGAS researchers? What papers have been posted recently, say in the past twelve months, that are significant QGAS papers?

It would be very helpful to know who you think leading people are, especially the younger crop just getting established.

It would be useful to have some specific arXiv numbers of research articles posted in the past twelve months that could serve to typify for us what you think are the main rival directions to string.

Hi Ummm,

OK, I am not 100% sure what your point is, now.

Fact is that in many discussions about that anomaly issue people get confused by the fact that in string theory there is a gravitational theory on parameter space and one on target space.

In the context of what has become known as the ‘LQG-string’ one is concerned exclusively with the issue of the quantum gravity theory on parameter space. All background dependence of string theory is a red herring for this particular discussion.

Which of course does not mean that there is nothing to discuss concerning the quantization of the target space gravity theory. But that’s not the issue of the ‘LQG-string’.

Best,

Urs

Hi Urs,

I am in agreement that the worldsheet theory is not background dependent. My point is that when an LQG enthusiast gets a bug up their bum about background dependence, they’re talking about strings as perturbation theory against a particular space-time. It’s not necessary to be too technical about this.

To Juan and others:

Do not post here attempts to carry on a discussion about alternatives to GR, QM, etc. This is not a general physics discussion forum. Please do this somewhere else. I will continue to delete all such posts.

Hi Ummm,

no, the issue I commented on is the quantization of the worldsheet theory and how its LQG-like quantization relates to the standard one. This is 2D gravity coupled to scalar fields and it is irrelevant for its discussion whether you want interpret these scalar fields as embedding fields into a target space or not.

So, if it helps, you can forget about the idea of fundamental strings for the moment and just consider the quantization of 2D gravity coupled to scalar fields. This is well understood. In the standard formalism, which uses weakly continuous representations of operators, the ADM constraint algebra of this gravitational system always has an anomaly.

The Thomas mentioned in Robert’s post is Thiemann, not me. Also, I am pretty sure that Stone-von Neumann only applies to QM with finitely many degrees of freedom, so it might not be directly relevant to gravity.

Ummm, the fixed background that Smolin refers to is the target space-time, fixing this is part of the definition of the CFT. Surely you realize that this is an important open problem?

Concerning the different approach towards anomalies in the LQG-like quantization of the string and the ordinary quantization, Lee Smolin wrote:

I guess this refers to the quantization of the Polyakov action after the conformal gauge has been fixed. It is worth pointing out that precisely the same constraint algebra is obtained by taking the Nambu-Goto action or the Polyakov action, regarding them as (background free) gravity on the worldsheet coupled to scalar fields on the worldsheet and compute the ADM constraints of these. One finds a Hamiltonian constraint L + bar L and a diffeomorphisms constraint L – bar L. No gauge has to be fixed at any time.

(This can easily be checked. The calculation is for instance given in Henneaux’s old lecture notes on string theory.)

So the quantization of the string is precisely about the quantization of a background independent gravitational system in two dimensions. And any weakly continuous quantization of the resulting constraint algebra does feature an anomaly.

The resason that ‘oscillators’ make an appearance in this background free quantization is merely due to the special property of the constraint algebra in 2D to have structure constants instead of structure functions. This makes Fourier decomposition in parameter space a useful tool.

It seems to me that the reason to drop weak continuity in the quantization of gravity in 3+1 dimensions is that it makes an otherwise intractable problem tractable – but possibly at the cost of having oversimplified a hard problem.

As Lee goes into some details, I would like to mention our preprint hep-th/0409182 where we discuss Thomas’ approach to the quantization of the string (and also the harmonic oscillator).

There, the upshot is, that the GNS-state of Thomas is not (weakly) continious, a property that is up to discussion but I consider quite suspicious. In the case of the oscillator it leads to a state that is that of an oscillator coupled to a heat bath of infinite temperature. This does not look too promising.

Furthermore, (although this argument might not be rigorous) this treatment suggests to me, that his state is not only independant of diffeomorphisms but also under any map that maps spacetime points bijectively (pointwise, not necessarily even continious) and thus produces a theory that only sees space-time as a set of points and forget completely even about topology.

As I understand, the reason to take such drastic steps are taken is the misconception that one should start from state that is invariant under diffeomorphisms. However, there is no physical motivation for this: It is enough to start with a state that is covariant, i.e. a state in which the action of diffeomorphisms is defined as operators in the corresponding hilbert space. The usual Fock quantization provides exactly this and due to the Stone von Neumann theorem is the only one that does this continiously.

At least in the classical theory, GR is diffeomorphism invariant but any solution (i.e. metric) sponaneously breaks this invariance to the isometry group of that space-time (which is generically trivial). So all the classical states are not invariant but only covariant.

I can not see the problem with the harmonic oscillator. Well, first of all, I missed the preprint number where such problem is discussed. I didn’t know that LQG was already able to support and quantise another force fields jointly with gravity. But in anycase, given that an harmonic oscillator force is not included as one of the forces in the Standard Model, I am happy if it can not be supported under LQG. I would be even happier if the unique supported forces where SU(3)xSU(2)xU(1).

The point about LQG is not that it’s wrong — after all, experiment is the ultimate arbiter of that — but that is not a conservative approach to quantum gravity as is often claimed. It is, in fact, a radically different approach to quantization that, when applied to current theories, gives experimentally incorrect answers.

Thomas Larsson gave a detailed response to Lee’s post in the previous thread. Since it is more likely to elicit a response in this thread than the earlier one, I think it is worth repeating it in full. I will add my own comment at the end, in agreement with one of Thomas’ comments.

* – * – * – * – * – * – * – * – *

Dear Lee,

Let me emphasize that I am not a string theorist – on the contrary, over the last years I have had strong disagreements with Lubos and others, especially over the role of diff anomalies. While initially a statistical physicist, the success of CFT made me interested in Lie algebras, where I discovered how to generalize the Virasoro algebra beyond 1D and developed its representation theory, together with mathematicians like Moody, Rao, Berman and Billig. In particular, the Virasoro algebra in 4D is the anomalous form of the algebra of 4-diffeomorphisms, which is the constraint algebra of GR in covariant formulations (in non-covariant canonical quantization the constraint algebra is modified).

So when I speak about diff anomalies, I do it as someone who has developed new mathematics which has not been absorbed by the physics community. You may wish to keep that in mind.

As to the issue of anomalies, i.e. the claim that we ignore the established knowledge that “INFINITE-DIMENSIONAL CONSTRAINT ALGEBRAS generically acquire anomalies on the quantum level…” is simply false. It is contradicted by rigorous existence and uniqueness theorems in LQG.Whereas I claim that this is true, it is not at all generally accepted. On the contrary, it is widely asserted that there are no pure gravitational anomalies in 4D, see e.g. Weinberg’s QT of F II, ch 22. Nevertheless, the constraint algebra of GR contains many subalgebras isomorphic to the infinite conformal symmetry in 2D, generated by vector fields of the form f(z) d/dz, where e.g. z = x^0 + ix^1 or z = x^2 + ix^3. Upon Fock quantization, these conformal subalgebras will in general acquire anomalies for the usual reason, making the whole shebang anomalous.

The reason why these anomalies cannot be seen in conventional field theory is that the relevant cocycles are functionals of the observer’s trajectory in spacetime. Unless this trajectory is introduced and quantized in conjunction with the fields, the relevant anomalies cannot be formulated. This is IMO the crucial obstruction to the quantization of gravity.

1) The approach to quantization of constrained systems is different in string theory and LQG. The former approach depends on a gauge fixing that refers to a fixed background metric. It results in the construction of a Fock space. The latter is background independent and involves no background metric, no gauge fixing and results in a state space unitarily inequivalent to a Fock space.2) There is a body of rigorous results that support each kinds of quantization. Hence it cannot be a question of which is correct mathematically. Both are correct, within their contexts. It is a question only of which construction is appropriate for which theories and which describes nature.Conventional quantization has turned out to describe nature in other contexts. I think this is a good reason to believe that it is the correct approach. In particular, CFT has been successfully applied to 2D condensed matter, where conformal anomalies have been measured experimentally. This is of course a different context and not directly relevant, but this fact has shaped my basic instinct that anomalies are very real things which cannot depend on the quantization method used.

3) The treatment of constraints in string theory depends on certain technical features of 1+1 dimensional theories, particularly the fact that there is a gauge in which L_0 plays the role of a Hamiltonian and therefore should, in that gauge, be quantized so as to have a positive spectrum. The anomalies are not generic, as asserted above, rather they depend on the additional condition that L_0 should be a positive operator.Yes, this is the crucial point. In any physical theory, there should be some positive operator which can be interpreted as a Hamiltonian; there is a physical requirement that energy be bounded from below. Of course, in GR there is a Hamiltonian constraint rather than a genuine Hamiltonian. This is another reason to introduce the observer’s trajectory; you can define a genuine Hamiltonian as the operator that translates the fields relative to the observer.

Anyway, in all applications of Lie algebras to physics so far, the reps have been of lowest-weight type. At least for finite-dimensional Lie algebras, all unitary irreps are of this type.

There are other reps of Diff(S^1 ) that are non-anomalous but in which L_0 is not positive.If you consider the restriction to the algebra of polynomial vector fields, generated by L_m with m >= -1, then all irreps have a vacuum vector (or are dual to such a rep).

So a choice is made in the standard quantization of string theory, which his motivated by the physics. This does not mean it is the right choice for all physical theories.OK. I disagree.

4) Conversely the existence and uniqueness theorems which support the LQG quantization work only in 2+1 dimensions and above for the reason that gauge fields don’t have local degrees of freedom in 1+1 dimensions. The existence theorems tell us that there are quantizations in 2+1 and higher of diffeo invariant gauge theories that have unitary, anomaly free realizations of diffeo invariance. The uniqueness theorem tells us that the resulting state space we use in LQG is unique.Contrary to string theorists, I claim that anomaly freedom is not a necessary requirement. To illustrate this point, let me again use the bosonic string as an example and quote from GSW, subsection 2.4: ‘Classical free string theory can be consistently formulated for any spacetime dimension, but quantization with a ghost-free spectrum requires D less than or equal to 26. [...] In the special case of D=26 and a=1 the spectrum is entirely tranverse, with many decoupled zero-norm states.’

Thus, D=26 is special, but D less than 26 is not ruled out by consistency requirements. It is only in 26D that it is possible to pass to the reduced Hilbert space by imposing the physical state condition L_m|phys)=0, but when D less than 26 this is not necessary, because the full, unreduced Hilbert space is already positive-definite.

Thus, my position is that some diff and gauge anomalies are good, making it possible to break diff and gauge symmetry on the quantum level, such as the string in D less than 26 illustrates. This does not mean that all gauge anomalies are good, of course. On the contrary, I recently gave a simple algebraic argument why conventional gauge anomalies, due to chiral fermions and proportional to the third Casimir, indeed are inconsistent. This argument does not apply to observer-dependent anomalies, which are proportional to the second Casimir.

The idea that diff and gauge anomalies may be consistent is of course very controversial.

With regard to the non-standard quantization, in which holonomies, but not local field operators are well defined, it is of course true that when applied to standard systems this leads to inequivalent results. “This apparently leads to unphysical consequences, such as an unbounded spectrum for the harmonic oscillator.” But, give me a break, do you really think someone is proposing to replace the standard quantization of the harmonic oscillator with the alternative one? What is being proposed is that the quantization used in LQG is well suited to the quantization of diffeo invariant gauge theories.In case it is not obvious, let me emphasize that harmonic oscillators are not relevent here, and can play no role in a background independent quantum theory, precisely because the division of a field into harmonic modes requires a fixed background metric. Thus, the physics of the problem REQUIRES an alternative quantization.

Sorry, but here I flatly disagree. I find it very disturbing that LQG methods yield the wrong result for the harmonic oscillator.

I am frankly puzzled why someone who claims to know the literature well would throw up examples like the harmonic oscillator up in this context. I can try to understand their point of view, but it certainly reads as if they either are choosing to ignore the basic point, which is that background independent quantizations cannot use fock space, or they are looking to make debating points to impress ignorant outsiders.I agree that a diff invariant quantization of gravity cannot use Fock space, and I am convinced that such a quantization does not exist. However, a diff covariant Fock space quantization of gravity may very well exist. By this I mean a quantization in analogy with the string for D less than 26: the unreduced Hilbert space is consistent in itself, and diffeomorphisms are promoted to a genuine but anomalous symmetry acting on the full Hilbert space.

A step in this direction was taken in hep-th/0504020. Sure, there are problems: the (manifestly covariant) regularization has not quite been removed, no invariant inner product has been found, and no hard predictions have been extracted. But there is a Hamiltonian which is bounded from below in the regularized theories, the analogous construction for the harmonic oscillator has a spectrum bounded from below (it is not quite right, and I discuss why), and infinities cancel best (though not quite, so I am doing something wrong) in 4D. Most importantly, since phase space variables are promoted to operators in the usual way, this is genuine quantization, which is witnessed by the presence of anomalies.

Finally, I didn’t express myself very well on the sociological issues. I agree with you about the problems with string theory, and I did not mean that funding to LQG should be stopped. However, given what I feel is a major problem (the harmonic oscillator spectrum), and that LQG already is the second biggest player in QG, I cannot really think that it is badly underfunded at present levels.

Posted by Thomas Larsson at June 14, 2005 10:44 PM

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I am not at all qualified to comment on either Lee’s or Thomas’ statements. However, like Thomas, it bothers me a lot that LQG apparently cannot reproduce the known spectrum for the harmonic oscillator in some limit where it should. Historically it has been a requirement for a new theory to reduce to an established earlier theory in the limit of the earlier theory’s applicability, as, say in the well known example that quantum theory gives the classical result in the limit hbar->0, or GR gives the Newtonian result in the flat space limit. Why should LQG be exempt from such a consistency check? Given that there aren’t too many ways to test a theory of quantum gravity, it seems like a failure here should be taken very seriously. It seems like Lee is a little too dismissive when he says,

But, give me a break, do you really think someone is proposing to replace the standard quantization of the harmonic oscillator with the alternative one? What is being proposed is that the quantization used in LQG is well suited to the quantization of diffeo invariant gauge theories.But then again, this is just the perspective of one who has only superficial knowledge about LQG…

here is a footnote to Smolin response relating to something in the original thread:

http://loops05.aei.mpg.de/

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this bears on Smolin’s paragraph 5:

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this is quite possibly a trivial sidecomment but I want to mention a consideration from the standpoint of a “physics watcher” which is that I also benefit from the range of quantumgravity research options that gradstudents and postdocs have at AEI-Potsdam and Perimeter/Waterloo.

when a US grad student who has only the choice string or nothing chooses a research topic this does not tell me anything, I am watching a stock market where investors have only one choice or very limited choice. so I do not benefit from watching his or her behavior.

but by contrast I watch the career of, for instance, Bianca Dittrich at the Albert Einstein Institute with great interest

BECAUSE HERE IS AN INTELLIGENT HIGHLY MOTIVATED PERSON ON THE GROUND who is free to make choices between, say, Thiemann’s Master Constraint canonical approach and Loll’s Triangulation approach (which are on radically different mathematical ground).

I think a lot can be learned by watching intelligent gamblers, or intelligent investors investing precious capital (young research time).

So to me it is very exciting to watch Dittrich’s career because the system she is in gives her choice of programs—-Hermann Nicolai directs AEI “Unified Theories and Quantum Gravity” division in a way that is VERY DIFFERENT FROM VAFA’S HARVARD. Wow, is it different! Nicolai branch of AEI has string but also LQC with bojowald, and canonical-LQG with thiemann and also dynamical triangulations work and a lot of other stuff. If you are there, or at ‘t Hooft”s Utrecht institute, or at Perimeter in Canada then you really have some interesting options. So to me it is more informative to watch what the postdocs do in those places.

And this is not just pure frivolity on my part, I really think you get scientifically valuable information by giving graduate students exposure to various things and a choice of radically different rival approaches, because the scientific enterprise benefits from their special perspective of what attracts them to devote their careers to. We all benefit from their intuitive hunches of what will pay off.

Let us hope that this does not make Artem Starodubtsev and Etera Livine and all of them self conscious knowing that bystanders get a clue of what is happening by seeing what they do.

Quantum gravity is at an intensely exciting stage and a LOT is happening. Onlookers can be glad for the instututes that are run with several alternate programs, and also (in my experience as an observer) they tend to be where the progress is being made.

Yawn.

-drl