# Coexistence of LQG and String Theory?

Can Loop Quantum Gravity and String Theory coexist? What if String theory was correct in its description of matter and 3 of the 4 forces, with the only difference from modern M theory being that there is simply no real particle called the graviton, and instead, gravity is as is described by LQG: curvature of the space-time continuum, which is composed of spin networks?

Probably a bit unlikely that BOTH of theories are partially correct, but could they theoretically be unified?

## Answers and Replies

This has already been discussed in PF, see this tread:

If I can add just a small word: yes, you can combine them, there are of course interesting mathematical tool that you can use from string to LQG and from LQG to string. But in physics you should carefully consider the question that you want to answer. If you want to unify forces, there is string theory and LQG has few to say, maybe can be the quantum theory of gravity that you want to unify with the other quantum theories via string theory. If you want to quantize gravity, you do LQG, and for the moment string theory has not proven to have something to say about quantum gravity (if you do AdS/CFT, you can use a gravitational system to say something about a quantum theory, but people have not yet been able to do the viceversa).
Also, string theory uses perturbation theory, while LQG is a non-pertubative theory of gravity, so they simply address different regimes. By the way, you can have the graviton in LQG, of course: it appears in the perturbative regime!
If by string theory you have in mind just extra dimensions, of course there is such a mathematical construction also in LQG (but then, to do physics, I would like to describe our usual 4 dimensional spacetime).
If by LQG you have in mind just the quantization based on Wilson loop, of course this is also present in QFT and therefore could maybe be implemented in string theory.

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Thanks for the reference! I'll check it out.

I guess I want to quantize gravity. I don't particularly care if it is unified with the other three forces or not. It seems to me that it could be possible to quantize gravity using LQG, or something like it, but that String Theory could still be the correct description for all particles in the universe.

Though I think, in that case, having 11 dimensions would be a little silly, as that was mainly motivated by the question of why gravity is so much weaker than the other forces; an obstacle to unification which is not needed in LQG. I guess to unify the two theories, String theory would have to be modified in this regard. But I wonder whether the unification of gravity into QFT via graviton follows NECESSARILY from the basic foundation of String Theory, or whether it was added in as a way to nicely unify the forces, and could just as easily be left out. My assumption is the latter, but I don't know enough about string theory (who does?? :tongue:)

atyy
But what is the evidence that LQG has gravity?

In http://arxiv.org/abs/1109.0499, the authors conclude "The present work gives explicitly the critical congurations of the spinfoam amplitude and their geometrical interpretations. However we didn't answer the question such as whether or not the nondegenerate critical congurations are dominating the large-j asymptotic behavior, although we expect the Lorentzian nondegenerate congurations are dominating when the Barbero-Immirzi parameter is small.", so it seems that it is still unknown if LQG can produce the right semiclassical limit.

Is there an update, or am I interpreting Han and Zhang wrongly?

Also, isn't it still unknown if LQG is a reasonable quantum theory?

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But what is the evidence that LQG has gravity?
Is there an update, or am I interpreting Han and Zhang wrongly?

Immirzi parameter small means smaller than one. If, as it seems, the Immirzi parameter works like the theta parameter in QCD, it is reasonable to expect that it should be between 0 and 1. But no, you are right, there is not a final word on the convergence of the amplitude. There are some positive signs, for which please allow people like me to be optimistic :-)
Also, isn't it still unknown if LQG is a reasonable quantum theory?

I think that it is pretty acknowledged that LQG is a theory of quantum gravity at this point, more or less reasonable depending on your taste. On the other hand, nobody would claim that this is the theory of quantum gravity that Nature has chosen. I agree that this is totally open.

When we say reasonable, satisfactory, viable... all this kind of things, it does not mean that the theory is in a final form, but that is structured enough to provide a framework on which we can work, built on it and do some physically interesting calculation. That's all.

I wonder whether the unification of gravity into QFT via graviton follows NECESSARILY from the basic foundation of String Theory, or whether it was added in as a way to nicely unify the forces, and could just as easily be left out. My assumption is the latter, but I don't know enough about string theory (who does?? :tongue:)
If the unification you want is just to describe gravity by a quantum field theory, that LQG provide a possible theory for this: spinfoam theory, the path integral formulation of LQG, is a QFT with some new feature coming from the fact of being general covariant. But the unification of string theory is more ambitious... but I let the experts of strings talk about it :tongue:

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atyy
I think that it is pretty acknowledged that LQG is a theory of quantum gravity at this point, more or less reasonable depending on your taste. On the other hand, nobody would claim that this is the theory of quantum gravity that Nature has chosen. I agree that this is totally open.

When we say reasonable, satisfactory, viable... all this kind of things, it does not mean that the theory is in a final form, but that is structured enough to provide a framework on which we can work, built on it and do some physically interesting calculation. That's all.

Thanks very much for the replies! It's great to have some professionals answer questions of laymen (I'm a biologist) like me!

What I understand very poorly is - what are the criteria for saying that a spin foam theory is a quantum theory? Eg. satisfies some criterion that is analogous to say, unitarity? I don't know what that criterion is in spin foams. I thought maybe the criterion should be that it solves the hamiltonian constraint of the canonical formalism - but isn't the link between spin foams and the canonical formalism still murky?

Also, it seems that Rovelli wants to do something like a "summing=refining" step to define full quantum gravity. In the Zakopane lectures, on p21, he indicates the C→∞ limit is full quantum gravity. So without that limit, spin foams are not a theory of quantum gravity. However, it seems the limit may not exist?

What if String theory was correct in its description of matter and 3 of the 4 forces, with the only difference from modern M theory being that there is simply no real particle called the graviton?

That wouldn't work, string theory always comes with a graviton. It's one of it's defining features.
Though I think, in that case, having 11 dimensions would be a little silly, as that was mainly motivated by the question of why gravity is so much weaker than the other forces;

No, that's not why string theory has extra dimensions. It requires extra dimensions to be mathematically consistent. Otherwise, it has negative norm states. In order to rid the theory of these negative norm states, you introduce terms called Virasoro operators. These require that Bosonic string theories have 26 spacetine dimensions. For superstring theories, it's only 10. M-theory needs one more.

Thanks very much for the replies! It's great to have some professionals answer questions of laymen (I'm a biologist) like me!
You welcome. It's a pleasure to know that there so good biologists around
Do you have any advice for "bioforums" so that I can try the flux in the opposite direction?
What I understand very poorly is - what are the criteria for saying that a spin foam theory is a quantum theory?
The EPRL vertex has been constructed so that it satisfies these 3 criteria (credits Eugenio Bianchi):
1. locality (each term of the vertex expansion acts locally)
2. Lorentz invariance (yes, the theory is locally Lorentz invariant)
3. the Hamiltonian should be a density
Eg. satisfies some criterion that is analogous to say, unitarity?
Be careful, unitary could mean different things... In LQG, the theory is unitary in the sense that it is built with unitary representations of the Lorentz group. But there are other sense, in which it is not unitary...
I don't know what that criterion is in spin foams. I thought maybe the criterion should be that it solves the hamiltonian constraint of the canonical formalism - but isn't the link between spin foams and the canonical formalism still murky?
There are indications, not a final prove. It's a tremendous task, because of the non-linearity of the theory. But if such a final prove would exist, this would be equivalent to solve the full theory! In fact, one should not prove it just for one vertex, but for al the vertex of the full amplitude... but people never work with withe the full amplitude, in LQG as in QED, see answer below.
Rovelli ... indicates the C→∞ limit is full quantum gravity. So without that limit, spin foams are not a theory of quantum gravity. However, it seems the limit may not exist?
This limit correspond to consider all the order in the vertex amplitude. This limit exists in the same sense that it exists the limit for a QED with all imaginable Feynman diagrams... but who cares about this limit? If you compute QED at some order bigger than the 5th - I may be wrong to the 5th, it could be the 8th, but in every case: - the theory is already more accurate in its prediction than our possibilities to check it: the theory works. The same spirit should be assumed for quantum gravity. This does not mean that the limit should not be better understood, it has to, but I feel it is more a job for mathematicians than for physicists

But we are a bit out of the topic of the tread, so I stop here :tongue:
Cheers,
Frances

tom.stoer
... But if such a final prove would exist, this would be equivalent to solve the full theory! In fact, one should not prove it just for one vertex, but for al the vertex of the full amplitude... but people never work with withe the full amplitude, in LQG as in QED, see answer below.
This limit correspond to consider all the order in the vertex amplitude. This limit exists in the same sense that it exists the limit for a QED with all imaginable Feynman diagrams...
This is dangerous what you are saying!

We are talking about the Hamiltonian H; and we agree that one has to prove that H annihilates physical states. This is not just an eigenvalue equation like (H-E)|phys> = 0, it is a remnant of the fundamental symmetry, i.e. the local constraint algebra of the theory. This constraint algebra must hold in any circumstances.

Compare it to QED or QCD: unitarity is guarantueed in all order of perturbation theory; it's not that unitarity violations converge to zero with αn; the same holds for gauge the symmetry. There are canonical formulations of QED and QCD which are completely gauge fixed and which are therefore gauge invariant; no approximation (of H) can ever spoil gauge invariance b/c it's formulated in physical d.o.f.; the difference is that gauge inv. in QED and QCD comes from Gauss law only, whereas in LQG the constraint algebra contains H as well.

Now if you say that implementing H as a constraint on physical states approximately may indeed affect the consistency of the constraint algebra then this brings the whole LQG project in danger!

That wouldn't work, string theory always comes with a graviton. It's one of it's defining features.

Thanks, Mark. I've been wondering that.

julian
Gold Member
This limit correspond to consider all the order in the vertex amplitude. This limit exists in the same sense that it exists the limit for a QED with all imaginable Feynman diagrams... but who cares about this limit? If you compute QED at some order bigger than the 5th - I may be wrong to the 5th, it could be the 8th, but in every case: - the theory is already more accurate in its prediction than our possibilities to check it: the theory works. The same spirit should be assumed for quantum gravity. This does not mean that the limit should not be better understood, it has to, but I feel it is more a job for mathematicians than for physicists

But we are a bit out of the topic of the tread, so I stop here :tongue:
Cheers,
Frances

I know that at a certain order in peturbative QED things start to go wrong and what were assumed to be small corrections turn out to be huge - apparently this happens early on in peturbative QCD. Plus I'd be careful of comapring other theories to background dependent field theories like QED and QCD which mathematically dont exist.

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This is dangerous what you are saying!

We are talking about the Hamiltonian H; and we agree that one has to prove that H annihilates physical states. This is not just an eigenvalue equation like (H-E)|phys> = 0, it is a remnant of the fundamental symmetry, i.e. the local constraint algebra of the theory. This constraint algebra must hold in any circumstances.

Compare it to QED or QCD: unitarity is guarantueed in all order of perturbation theory; it's not that unitarity violations converge to zero with αn; the same holds for gauge the symmetry. There are canonical formulations of QED and QCD which are completely gauge fixed and which are therefore gauge invariant; no approximation (of H) can ever spoil gauge invariance b/c it's formulated in physical d.o.f.; the difference is that gauge inv. in QED and QCD comes from Gauss law only, whereas in LQG the constraint algebra contains H as well.

Now if you say that implementing H as a constraint on physical states approximately may indeed affect the consistency of the constraint algebra then this brings the whole LQG project in danger!

Consider a simple quantum mechanical system, and try to describe it in the covariant formalism. The simplest thing could be a hydrogen atom, but this is a special case because it can be solved in a closed form. So, just to fix the ideas, think of an helium atom. In this case one would have a space of "non-physical" states given by the wave functions $\psi(x,t)$ that satisfy the analog of the Wheeler-deWit equation $C \Psi=0$, where $C$ is the constraint $C=i \hbar\,d/dt - H(q_i, i \hbar\, d/dq_i)$. If we would "agree that one has to prove that H annihilates physical states", then we should also believe that the quantum mechanics of the helium atom can be formulated if and only if we can prove that such operator annihilates the physical states. But the physical states, thought in this way, are not known: nobody knows the exact solutions of the quantum dynamics of the interaction between the electrons and between the electrons and the nucleus. Therefore even usual quantum mechanics of an helium atom would be in danger if we follow this kind of logic. On the other hand, we can treat the helium atom by considering a perturbative expansion, calculating order by order the transitions amplitudes. Note that in this case the constraint express a fundamental symmetry of the system: this is an invariance under reparametrization of the covariant formulation of the dynamics of the helium atom.

So, as Julian was stressing, it is important to understand the expansion we are working with. But I would not ask to my quantum gravity theory something that I would not even ask for a simple quantum mechanical system.

atyy
Consider a simple quantum mechanical system, and try to describe it in the covariant formalism. The simplest thing could be a hydrogen atom, but this is a special case because it can be solved in a closed form. So, just to fix the ideas, think of an helium atom. In this case one would have a space of "non-physical" states given by the wave functions $\psi(x,t)$ that satisfy the analog of the Wheeler-deWit equation $C \Psi=0$, where $C$ is the constraint $C=i \hbar\,d/dt - H(q_i, i \hbar\, d/dq_i)$. If we would "agree that one has to prove that H annihilates physical states", then we should also believe that the quantum mechanics of the helium atom can be formulated if and only if we can prove that such operator annihilates the physical states. But the physical states, thought in this way, are not known: nobody knows the exact solutions of the quantum dynamics of the interaction between the electrons and between the electrons and the nucleus. Therefore even usual quantum mechanics of an helium atom would be in danger if we follow this kind of logic. On the other hand, we can treat the helium atom by considering a perturbative expansion, calculating order by order the transitions amplitudes. Note that in this case the constraint express a fundamental symmetry of the system: this is an invariance under reparametrization of the covariant formulation of the dynamics of the helium atom.

So, as Julian was stressing, it is important to understand the expansion we are working with. But I would not ask to my quantum gravity theory something that I would not even ask for a simple quantum mechanical system.

I don't think the comparison with helium is the same. If the full quantum gravity does not exist, then there is no diffeomorphism invariance, and canonical LQG does not exist. So it is not a matter of the theory existing, but we can only calculate approximately. I understand that physics has historically not proceeded rigourously, but that's because there have been observations and experiments to help theorists out. Too bad experimentalists are so incompetent nowadays - still stuck in 1964 :tongue:

I don't think the comparison with helium is the same. If the full quantum gravity does not exist, then there is no diffeomorphism invariance, and canonical LQG does not exist. So it is not a matter of the theory existing, but we can only calculate approximately.
Physics is always a matter of calculating approximately, from the spherical cows to the helium atom to quantum gravity. Indipendently of what would be your favorite theory about human access to Nature, a theory does not exist if it satisfy some abstract laws that we have assumed as rules of the game (so quantum gravity works even if we have not solved the full theory, as tom.stoer was asking): rather, a theory exists if we can write down a coherent formulation that captures physical reality and we can compute. In this sense, what we have in quantum gravity follows what we often have in the physical quantum theories that we know. Maybe canonical quantum gravity does not exist, I do not know. But covariant quantum gravity exists and is defined by the amplitudes.
I understand that physics has historically not proceeded rigourously, but that's because there have been observations and experiments to help theorists out.
I agree that historically the new step in physics has not be done starting from rigorous mathematics, otherwise we would not have Einstein's general relativity or Feynman's path integral. But I disagree that this happened because there were experiments, on the contrary. From Copernicus to Newton, to Einstein, we have had big steps in the understanding of Nature with no new experiments to support them, but solving a teoretical problem, like combining previous successful theories. In fact, search for rigor might be misleading. When Richard Feynman went to a general relativity conference, the 1957 Chapel Hill Conference on General Relativity, he famously said to the relativists (about quantum gravity!): "Don't be so rigorous or you will not succeed!" Feynman had an amazing grasp on how Nature works, so better keep in mind his advice But careful: the fact that one does an approximation, does not mean that "it is not rigorous" in the sense that there is no physical guidance of what we do, it is exactly the contrary.

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Too bad experimentalists are so incompetent nowadays - still stuck in 1964 :tongue:
ps: experimentalists are not so bad! it's great time for people at Cern, it's nice to see all this excitation... I would not blame them if we don't have yet experiments in quantum gravity, actually it's a theoretician's job to propose some phenomenology to observe... Higgs was crying saying that he would not have believed to see such a day in his life, I would be happy even being less lucky than him :tongue:

tom.stoer
francesca,

this is not really my point.

What I am saying is "that implementing H as a constraint on physical states approximately may indeed affect the consistency of the constraint algebra then this brings the whole LQG project in danger!"

In QCD I have something like

(H-E)|proton> = 0

If H is completely gauge fixed this is a physical Hamiltonian H. What I do not have is the solution |proton>. Now I can make an approximation H'; this affects |proton>, so the new eigenstate |proton'> will be an approximation as well. But b/c H is gauge fixed I can make any approximation H' I like! The |proton'> will somewhow be wrong (or a bad approx.) but I cannot break gauge invariance.

But if H in LQG cannot be fully 'gauge fixed' and if I now make an approx. H' this does not only affect the state (the solution) but the consistency condition H~0 which could introduce an inconsistency.

What I do not fully understand is the problem with anomalies. If both Gauss and Diffeo-constraints are implemented H is the only constraint left. If H is invariant w.r.t. to Gauss and Diffeo I am in the same situation as described for QCD and I can't break any symmetry - besides H~0 itself (I can't introduce any anomaly b/c Gauss and Diffeo do no longer exist in physical variables).

But if some gauge symmetry is still left I cannot expect to make an approx H' w/o introducing an anomaly.

So what I don't understand is at which point an anomaly may enter LQG.

Perhaps the main problem is the step-wise fixing of constraints.

btw.: my feeling is that covariant LQG does exist but that it cannot be related to canonical LQG - and that this may be an even bigger problem b/c of the simplicity constraints not being taken into account appropriately; I am with Alexandrov that
1) either you use Dirac brackets which modify the symplectic structure classically such that nobody is able to quantize the theory (constrint algebra with structure functions, ...)
2) or that you use Poisson brackets + simplicity constraints w/o changing to Dirac brackets which is not allowed b/c the simplicity constraints are second-class so the resulting theory is not defined in a sound manner. I don't think that this problem has been fully addressed, neither in canonical LQG where H is not known, nor in spin foams where the measure cannot be fixed and where the amplitudes should be related to some well-defined H.

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julian
Gold Member
I personaly think that mathematical rigour is the guiding force in the absence of experimental results.

But, as a tentative example of the force of physics+maths! - the infrared catastrophe of QED says that the renormalised charge will tend to zero regardless of what the bare charge is assumed to be. Landau et al concluded that QED is wrong!! But the actual resolution is Electro-Weak theory. You need a higgs type QFT to get out of this physical prediction. One of my favorate quotes of theoretic physics..."Veltman: I do not care what or how, but we must have is at least one renormalizable theory with massive charged vector bosons, and whether that looks like nature is of no concern, those are details that will be fixed later by some model freak...
t'Hoof't: I can do that.
Veltman: What did you say?
't'Hooft:I can do that."

Apparently some people have found the Higgs particle underneath the sofa.

tom.stoer
what's your conclusion for LQG?

What I am saying is "that implementing H as a constraint on physical states approximately may indeed affect the consistency of the constraint algebra then this brings the whole LQG project in danger!"
I do not think that this is correct. There is nothing wrong in solving equations approximately. This is also true for the gauge. For example, in doing cosmological perturbation theory one has to solve constraints, of course, but does so only order by order in the expansion.
In QCD I have something like...
QCD is different than gravity. In QCD, the hard part is the dynamics, which is a priori something separated from the constraints. In gravity, the dynamics is is intertwined with the gauge symmetries. It is a good idea to copy techniques from QCD, but it is a bad idea to oblige oneself to do exactly as in QCD.
my feeling is that covariant LQG does exist but that it cannot be related to canonical LQG
This is possible. If this is the case, we should follow the path that leads to a computable theory, and forget the other. It is always good to follow different paths, and see which one works.
1) either you use Dirac brackets which modify the symplectic structure classically such that nobody is able to quantize the theory (constrint algebra with structure functions, ...)
2) or that you use Poisson brackets + simplicity constraints w/o changing to Dirac brackets which is not allowed b/c the simplicity constraints are second-class so the resulting theory is not defined in a sound manner. I don't think that this problem has been fully addressed, neither in canonical LQG where H is not known, nor in spin foams where the measure cannot be fixed and where the amplitudes should be related to some well-defined H.
Here there is some confusion between canonical and covariant. It is not true that in the canonical theory H is not known. And it is not true that in the covariant theory the measure cannot be fixed. There exist a well defined Hamiltonian, and there exist a well defined measure, which is fixed by locality requirements (see for instance http://arxiv.org/abs/1005.0764). Of course maybe these theories can be improved, but for the moment they work, and no inconsistency has been found. Regarding "the amplitudes should be related to some well-defined H", I think I disagree with the "should": it would be great if somebody succeeded in doing this, but if one of the two versions of the theory is correct, it does not need the other to be interesting and possibly physically viable. Everybody is free to have fun quibbling about what would be good having in a perfect world (me too, possibly with some good whisky) but then as physicists we should come back to this world and find a theory that works...

tom.stoer
I do not think that this is correct. There is nothing wrong in solving equations approximately. This is also true for the gauge.
No; approximations in gauge theory breaking gauge invariance are disastrous.

This is possible. If this is the case, we should follow the path that leads to a computable theory, and forget the other.
I don't agree; all what I see is that the related problems (simplicity constraints, reality conditions, second class constraints and Dirac brackets, measure, ... H, vertex amplitude, ...) show up in both approaches; there are fundamental problems looking different depending on the perspective (covariant, canonical) but I think they are related.

http://arxiv.org/find/gr-qc/1/au:+Alexandrov_S/0/1/0/all/0/1

Anyway - you are the expert, so I stop insisting; I don't want to be impolite.

Here there is some confusion between canonical and covariant. It is not true that in the canonical theory H is not known.
??? But that's what the experts say. There are candidate hamiltonians suffering from the usual quantization and regularization ambiguities w/o any hint for uniqueness, fundamental principle or something like that. The well-known Thiemann Hamiltonian does not "create volume"; it has been "cured" a couple of times, but I have never seen "the" Hamiltonian. Is there a reference with a construction?

No; approximations in gauge theory breaking gauge invariance are disastrous.
Achtung: we have to be careful with our approximations, but this does not mean that we can not work with them, on the contrary! For instance when you expand around a cosmological solution, you do not ask your variables to be exactly gauge invariant, but only to be gauge invariant up to the order at which you are working.

I don't agree; all what I see is that the related problems (simplicity constraints, reality conditions, second class constraints and Dirac brackets, measure, ... H, vertex amplitude, ...) show up in both approaches; there are fundamental problems looking different depending on the perspective (covariant, canonical) but I think they are related.
The fact that problems of two different approaches are related is correct, but it does not imply that solving a problem in one formalism is the same as solving it in another formalism. There are things that are simple in one language but remain difficult in another. There are problems that can be solved in one language, while are still too cumbersome in another language. For instance QED is clean and beautiful in the covariant language, but much more messy and cumbersome in the canonical language (try to compute a first order QED transition amplitude with canonical methods: it is much more messy... I think you know it well ). We know many problems that have no solution in one language and require to shift language to solve them. For instance, a thermal state for a quantum field is easily written in the algebraic language, but very hard to write in a Hilbert space formalism. Insisting that all problems must be solved in all formalism is not a wise strategy.

Alexandrov has his own point of view about how things should be done. On the other hand, he is the first to admit that in this way he has not succeded in defining a coherent model. He rises a lot of issues, and of course many could turn out to be very useful. One of these are projected spinnetworks, now implemented in some works in spinfoam (Livine, Depuis...). I don't think that his arguments should be taken, as they often sound, as lethal criticisms of other constructions.

There are candidate hamiltonians suffering from the usual quantization and regularization ambiguities w/o any hint for uniqueness, fundamental principle or something like that. The well-known Thiemann Hamiltonian does not "create volume"; it has been "cured" a couple of times, but I have never seen "the" Hamiltonian. Is there a reference with a construction?
It is different to say "the Hamiltonian is not known" and to say "a theory with a well defined Hamiltonian is known, but of course we are not sure is the right one, and some alternatives have been considered, based on potential problems". The confusion is the following. The problem of quantum gravity is not uniqueness. It is existence. "Is there a quantum theory with GR as classical limit?" If we had one working, we would be one important step ahead. Then we would worry about uniqueness, or alternatives. The problem so far as been that no complete consistent theory existed. I think it is more important to study if one such theory exists before start wandering if it is unique.
The "problems" of the Thiemann Hamiltonian, as far as I see, are more hints than else. I think that a shared feeling is that something might and should be changed or it is missing; in this you are right (I am not sure, we should ask Thiemmann, who knows better), but this does not change the fact that it is wrong to say that H is not known. That was the situation 15 years ago: at the time, no well defined Hamiltonian operator was known. Now we are not in that situation anymore.

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tom.stoer
Achtung: we have to be careful with our approximations, but this does not mean that we can not work with them, on the contrary! For instance when you expand around a cosmological solution, you do not ask your variables to be exactly gauge invariant, but only to be gauge invariant up to the order at which you are working.
I don't know how this works in cosmology, in ordinary gauge theory you have to ensure gauge invariance order by order (or, via appropriate implementation like non-perturbatuive gauge fixing) at all orders. When talking about QFT I am *not* talking about perturbation theory, but about writing down a non-perturbative Hamiltonian w/o any approximation. This H must be gauge invariant, otherwise it is rubbish.

There are many points I agree with ...

Alexandrov has his own point of view about how things should be done. On the other hand, he is the first to admit that in this way he has not succeded in defining a coherent model. ...

He rises a lot of issues, and of course many could turn out to be very useful.
This is my point. The issues regarding quantization, consztraint implementation etc. are there; unfortunately there are no solutions in Alexandrov's approaches yet, but the issues remain.

The problem of quantum gravity is not uniqueness. It is existence.
I agree.

"Is there a quantum theory with GR as classical limit?" If we had one working, we would be one important step ahead. Then we would worry about uniqueness, or alternatives.
I agree as well, but ...

... the problems regarding constraint implemenation, ambiguities, measure, anomalies, ... may not shjow up in the semiclassical limit. The quantization anomaly UA(1) is absent in the classical theory. So the existence GR in the semiclassical limit is a major step forward, but no hint regarding a consistent quantization.

Last question: what would be "the Hamiltonian" I should look at today? Where has it been written down?

I don't know how this works in cosmology, in ordinary gauge theory you have to ensure gauge invariance order by order (or, via appropriate implementation like non-perturbatuive gauge fixing) at all orders. When talking about QFT I am *not* talking about perturbation theory, but about writing down a non-perturbative Hamiltonian w/o any approximation. This H must be gauge invariant, otherwise it is rubbish.
Right! and indeed in the covariant approach one does not need to write down the Hamiltonian: one writes the transition amplitudes order by order in an expansion! This is precisely the way some of the problems that worry you are circumvented.
This is my point. The issues regarding quantization, consztraint implementation etc. are there; unfortunately there are no solutions in Alexandrov's approaches yet, but the issues remain.
These issues are not ignored in the EPRL model: they are addressed! There is a criterium for choosing the measure (SU(2) gauge invariance), and the full implementation of the simplicity constraints is implemented by having them imposed at each vertex, namely everywhere in the bulk.
... the problems regarding constraint implemenation, ambiguities, measure, anomalies, ... may not shjow up in the semiclassical limit. The quantization anomaly UA(1) is absent in the classical theory. So the existence GR in the semiclassical limit is a major step forward, but no hint regarding a consistent quantization.
Yes. Of course having the semiclassical limit right is a step forward but not the end of the story. The second major result of the covariant formalism is that the amplitudes are finite at all orders. Actually, this is the main achievement, I think. This was proven by Muxin Han (Marseille), and independently, by Winston Fairbairn and Catherine Meusburger (Erlangen).
Even having that is not the end of the story, still. There are remaining open questions, before claiming that a (tentative) theory of quantum gravity exists and is coherent. But one should discuss the issues that are open in the covariant formalism in terms of the covariant formalism, not in terms of the conundrum of the canonical one. The main open issue in the covariant formalism is whether large radiative corrections spoil the viability of the expansion. This is the main open question at present.
Last question: what would be "the Hamiltonian" I should look at today?
I don't have myself a favorite one, but you have in Erlangen the experts to ask about this

tom.stoer