Coexistence of LQG and String Theory?

[soothsayer]

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?

 

[francesca]

This has already been discussed in PF, see this tread:
https://www.physicsforums.com/showthread.php?t=600660&highlight=string+loopIf 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.

 

[soothsayer]

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?? )

 

[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?

 

[francesca]

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?? )

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

 

[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?

 

[Mark M]

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.

 

[francesca]

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
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!

 

[soothsayer]

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]

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
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 don't exist.

 

[francesca]

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

 

[francesca]

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.

 

[francesca]

Too bad experimentalists are so incompetent nowadays - still stuck in 1964

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

 

[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.

 

[julian]

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?

 

[francesca]

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?

 

[francesca]

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.

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

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.

 

[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 U_A(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?

 

[francesca]

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 U_A(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]

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.

Perhaps it's more a a matter of taste, but from my background in gauge theory I don't like expansions; they are valid in certain regimes, you may lose many properties (related to gauge fixing like Gribov ambiguities, topology & instantons, ..., phase transitions), in QCD you don't have a valid expansion in the non-perturbative regime (1/N ?), you don't understand the true nature of anomalies (Atiyah-Singer index theorem, Fujikawa's method) etc.

I don't have myself a favorite one, but you have in Erlangen the experts to ask about this

you are right, 15 minutes by car!

 

[w4k4b4lool4]

Is it clear that LQG can coexist with the Standard Model of particle physics?

 

[w4k4b4lool4]

What are the observables in LQG?

 

[w4k4b4lool4]

What's loopy about LQG anyway?

 

[w4k4b4lool4]

And finally, the big question, why should I work on LQG as opposed to String Theory?

 

[marcus]

Is it clear that LQG can coexist with the Standard Model of particle physics?

First read http://arxiv.org/abs/1102.3660 so you know what today's LQG is. There is a section on putting in matter. There is a reference to a paper on fermions in LQG.
Also there have been a number of papers exploring a way to implement the SM in the quantum spacetime of LQG by Grimstrup's method. Do an arxiv search on the author's name: Grimstrup.

To me, it is NOT clear and this is perhaps the most exciting research opportunity. How to put all kinds of matter into the Loop quantum geometry and run standard particle physics.

What are the observables in LQG?

Length, area, volume, angle. The spectra of the area and volume operators have been derived. Seth Major has a paper on angle.

What's loopy about LQG anyway?

In the early 1990s loops were replaced by spin networks (kind of labeled graph) as the basic object of the theory. Think of a network as a more complicated loop, or as many loops joined together. The spin network is a quantum state of geometry. These labeled networks form a basic for the Hilbert space of states of geometry. Node labels denote volume measurements, link labels indicate area measurements where adjacent chunks of volume meet. LQG is called that for historical reasons. It is networky, not loopy. In a reformed language it could be called "network quantum geometry".

And finally, the big question, why should I work on LQG ...?

You should work on Loop QG only if you are very smart, have a lot of originality and boldness, think deeply about what General Relativity is trying to tell us (general covariance, with no prior geometry), and you should also be excited to learn more about how geometry and matter interact: i.e. what the fundamental degrees of freedom are at the level where geometry and matter are the same. This is what I have noticed about the young people I have seen joining the Loop program and they seem to be having a good time. From a practical standpoint, being European seems to improve your chances. Another positive factor is cosmology (that's the empirical testing ground.) I don't think I would suggest Loop to an American student unless that person already showed (in addition to other qualifications) considerable interest in and familiarity with early universe cosmology. When someone asks that kind of "why should I...?" question you have to think about that person's chance of success and happiness in the program based on their qualifications. Loop already has an influx of talented young people so the point is not to unselectively recruit. (In a fast moving research environment it's not true that "more is better.") But that's just my personal reading of the situation.

 

[w4k4b4lool4]

First read http://arxiv.org/abs/1102.3660 so you know what today's LQG is. There is a section on putting in matter. There is a reference to a paper on fermions in LQG.
Also there have been a number of papers exploring a way to implement the SM in the quantum spacetime of LQG by Grimstrup's method. Do an arxiv search on the author's name: Grimstrup.

Great, thank you very much for the reference. In the meanwhile I cannot resist following on from your response to ask the following questions.

Length, area, volume, angle. The spectra of the area and volume operators have been derived. Seth Major has a paper on angle.

Lets say an observer wants to measure one of these observables. How does he/she proceed? Does it make sense to compute a scattering cross-section, or a decay rate? Does one need to introduce matter to measure these observables?

In the early 1990s loops were replaced by spin networks (kind of labeled graph) as the basic object of the theory. Think of a network as a more complicated loop, or as many loops joined together. The spin network is a quantum state of geometry. These labeled networks form a basic for the Hilbert space of states of geometry. Node labels denote volume measurements, link labels indicate area measurements where adjacent chunks of volume meet. LQG is called that for historical reasons. It is networky, not loopy. In a reformed language it could be called "network quantum geometry".

How does the notion of `spin' come into this in this very intuitive and clear picture you have described? (Am I to imagine that this spin network is "real", i.e. made up from some kind of distribution of energy, or is it just a mathematical construction?)

Thank you Marcus for the very thought-proviking and interesting response.
Wakabaloola

 

[marcus]

Does one need to introduce matter to measure these observables?

My answer is YES.
I should say that I'm just an interested onlooker from the sidelines. You should really have someone who is actively doing Loop research to answer your questions.

If you want to measure the area of a surface, that surface has to be defined somehow. a black hole horizon, the top of your work table. Space has no meaning apart from relationship with matter or with features of the gravitational field (like a horizon) defined and observed through matter. I think this is a really important issue and wish I were more able to give an informed answer.

How does the notion of 'spin' come into this in this very intuitive and clear picture you have described? (Am I to imagine that this spin network is "real", i.e. made up from some kind of distribution of energy, or is it just a mathematical construction?)

Just a partial answer for now. The way I think of it, Loop is not about what Nature is made of, it's about how she responds to measurement. A state summarizes past (geometric + ideally matter) measurements. You want a model of how the state evolves so you can predict other measurements. First we have to say how states are going to be presented in math terms.

A GRAPH Γ (made of nodes and links) is not a physical thing, it is just a way of truncating to get a finite number of geometric degrees of freedom. Only such and such measurements are under consideration (a finite web of measurements).

If you look at the first few pages of 1102.3660 you will see how H_Γ the graph Hilbert space is defined.

Think of ordinary quantum mechanics in the simplest case of a particle in a onedimensional "box" which is just the interval [0,1] of the real line. The configurations are just position numbers from 0 to 1, and the state is just a complex valued function defined on that interval. States are square-integrable functions: L₂[0,1].

You will see in 1102.3660 that they do the analogous thing. CONFIGURATIONS are technically "connections" (a diffy geom. term) that assign a GROUP ELEMENT to every leg of the graph.*
A connection tells you how a parallel transport vector swings and sways and rolls around as you move along that leg. So the config. space of (finitized) connections is just the cartesian product G^L where L is the number of links in the graph. And by the way G is SU(2), we are focusing on rotations in 3D.
In diffy geom. a connection describes a configuration of geometry at the classical level, and in our case we are talking about SU(2) connections.

You want a SQUARE INTEGRABLE FUNCTION ON THE SPACE OF CONNECTIONS in direct analogy with the particle in the box. That means L₂[G^L ]

Now how do you get a basis for the vectorspace of complex valued functions on G? or the cartesian product? The Peter-Weyl theorem say to use the REPRESENTATIONS of the group. And you know the representations of SU(2) are labeled by half-integers.

So you label the graph with half-integers and presto you have a machine which can eat "configurations" and give you back a complex number. It can eat an L-tuple of group elements belonging to G^L (which is our finitized or truncated "connection") and chew it up and give a number.
And functions of that form constitute a BASIS of the whole vectorspace L₂[G^L]

You might want to look at 1102.3660, or somebody might be able to suggest something better as an introduction. This what I've supplied here is just a start.

*In classical differential geometry a connection tells you about parallel transport along all possible paths in the manifold. That's too much information. That's why we truncate the information we are dealing with and just consider moving along the legs of a finite graph. And then we recover by considering more and more complicated graphs. But first the theory is constructed on one particular finite graph.

 

[w4k4b4lool4]

Lets say an observer wants to measure one of these observables ... Does one need to introduce matter to measure these observables?

My answer is YES.
I should say that I'm just an interested onlooker from the sidelines. You should really have someone who is actively doing Loop research to answer your questions.

If you want to measure the area of a surface, that surface has to be defined somehow. a black hole horizon, the top of your work table. Space has no meaning apart from relationship with matter or with features of the gravitational field (like a horizon) defined and observed through matter. I think this is a really important issue and wish I were more able to give an informed answer.

I see. This is very interesting, but it is also puzzling at the same time. If LQG is a quantum theory of gravity, which thus far has not incorporated matter, but nevertheless requires matter in order for one to access and measure its observables, then in what sense can we address physical questions in this context? (And I suppose I'm referring to the truly quantum degrees of freedom of the theory, as opposed to the effective classical limit.)

Also, if the quantum gravity theory is defined in this very precise manner (which you've described in a great way by the way Marcus!), then, intuitively, wouldn't we expect to have to carry out a similar procedure for the matter contribution? (Of course, I know the same techniques don't work for matter .. unless, one thinks along the lines of asymptotic safety, maybe); but what I mean is that, although Einstein's equation is classical, it seems to be hinting towards a strong correlation between matter and gravity .. so that I would interpret that as a strong hint that we should be treating both matter and gravity on an equal footing. This elaborate setup that you have described should somehow be, I don't know, let's say a "mirror image" of the matter theory! I suppose AdS/CFT (or maybe I should say gauge/gravity duality) is kind-of thinking along these lines, and what an indirect way to do so, what an indirect way to seal this strong correlation between gravity and matter at the quantum level! Wow! (Of course, the latter is also speculative at present, but, having passed numerous checks, there is a large community that has been convinced that it is true.)

I am a string theorist at heart, but I'm continuously trying to learn about new ideas. After all, the largest breakthroughs have been made by bringing tools from one area of science to another (spontaneous symmetry breaking being a good example I suppose).

Just a partial answer for now. The way I think of it, Loop is not about what Nature is made of, it's about how she responds to measurement. A state summarizes past (geometric + ideally matter) measurements. You want a model of how the state evolves so you can predict other measurements. First we have to say how states are going to be presented in math terms.

A GRAPH Γ (made of nodes and links) is not a physical thing, it is just a way of truncating to get a finite number of geometric degrees of freedom. Only such and such measurements are under consideration (a finite web of measurements).

If you look at the first few pages of 1102.3660 you will see how H_Γ the graph Hilbert space is defined.

Think of ordinary quantum mechanics in the simplest case of a particle in a onedimensional "box" which is just the interval [0,1] of the real line. The configurations are just position numbers from 0 to 1, and the state is just a complex valued function defined on that interval. States are square-integrable functions: L₂[0,1].

You will see in 1102.3660 that they do the analogous thing. CONFIGURATIONS are technically "connections" (a diffy geom. term) that assign a GROUP ELEMENT to every leg of the graph.*
A connection tells you how a parallel transport vector swings and sways and rolls around as you move along that leg. So the config. space of (finitized) connections is just the cartesian product G^L where L is the number of links in the graph. And by the way G is SU(2), we are focusing on rotations in 3D.
In diffy geom. a connection describes a configuration of geometry at the classical level, and in our case we are talking about SU(2) connections.

I see. So I suppose the analogy with the QM above is:

connections (or g\in G) ~ x\in[0,1] interval

and

SQUARE INTEGRABLE FUNCTION ON THE SPACE OF CONNECTIONS ~ ψ(x) of particle in box

By the way, is the full 4D diffeomorphism group still intact?
And why are you assuming D=4 dimensions? If you start with a higher dimensionality do interesting things happen (I don't know .. like enhanced symmetries or something of the like) for specific values of D? I suppose, ideally, one would like to PREDICT D=4 ...

You want a SQUARE INTEGRABLE FUNCTION ON THE SPACE OF CONNECTIONS in direct analogy with the particle in the box. That means L₂[G^L ]

Now how do you get a basis for the vectorspace of complex valued functions on G? or the cartesian product? The Peter-Weyl theorem say to use the REPRESENTATIONS of the group. And you know the representations of SU(2) are labeled by half-integers.

So you label the graph with half-integers and presto you have a machine which can eat "configurations" and give you back a complex number. It can eat an L-tuple of group elements belonging to G^L (which is our finitized or truncated "connection") and chew it up and give a number.
And functions of that form constitute a BASIS of the whole vectorspace L₂[G^L]

That's very clear and enlightening!

You might want to look at 1102.3660, or somebody might be able to suggest something better as an introduction. This what I've supplied here is just a start.

*In classical differential geometry a connection tells you about parallel transport along all possible paths in the manifold. That's too much information. That's why we truncate the information we are dealing with and just consider moving along the legs of a finite graph. And then we recover by considering more and more complicated graphs. But first the theory is constructed on one particular finite graph.

I see. Have any alternatives (that lack the discreteness you are referring to) been suggested?
Might this discreteness be physical after all?

Thank you very much again Marcus!
It's been a great please reading slowly through your every sentence, laying back, and thinking about what you are saying.

Wakabaloola

 

[marcus]

...
Also, if the quantum gravity theory is defined in this very precise manner (which you've described in a great way by the way Marcus!), then, intuitively, wouldn't we expect to have to carry out a similar procedure for the matter contribution? (Of course, I know the same techniques don't work for matter .. unless, one thinks along the lines of asymptotic safety, maybe); but what I mean is that, although Einstein's equation is classical, it seems to be hinting towards a strong correlation between matter and gravity .. so that I would interpret that as a strong hint that we should be treating both matter and gravity on an equal footing. This elaborate setup that you have described should somehow be, I don't know, let's say a "mirror image" of the matter theory! ...
..

Thanks for this interesting and stimulating comment (and for the encouraging words.) I will try to say how LQG at present includes matter. I don't know much about it and will have to check with the current definitive source 1102.3660. As I recall fermion labels go on nodes of the graph and YM field labels go on links. You have to enlarge the graph Hilbert space H_Γ. It is work in progress.
http://arxiv.org/abs/1102.3660

Also in Loop cosmology they use matter a lot, simple generic scalar fields and such. They do it differently. LQG and LQC are both evolving. By this time next year there could be another 30page paper that replaces the 2011 one I'm going to consult

I misremembered, on page 27 the "current standard source" says
==quote==
...Also, I have not covered several recent development, such as the manifest Lorentz invariant formulation of the theory [131], the coupling to fermions and Yang-Mills fields [14, 132], and to a cosmological constant [11, 133], using a quantum group.
==endquote==
So I must look up [14, 132]
[14] http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
(Submitted on 21 Dec 2010)
We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form.
8 pages

[132] http://arxiv.org/abs/1101.3264
Spinfoam Fermions: PCT Symmetry, Dirac Determinant, and Correlation Functions
Muxin Han, Carlo Rovelli
(Submitted on 17 Jan 2011)
We discuss fermion coupling in the framework of spinfoam quantum gravity. We analyze the gravity-fermion spinfoam model and its fermion correlation functions. We show that there is a spinfoam analog of PCT symmetry for the fermion fields on spinfoam model, where a PCT theorem is proved for spinfoam fermion correlation functions. We compute the determinant of the Dirac operator for the fermions, where two presentations of the Dirac determinant are given in terms of diagram expansions. We compute the fermion correlation functions and show that they can be given by Feynman diagrams on the spinfoams, where the Feynman propagators can be represented by a discretized path integral of a world-line action along the edges of the underlying 2-complex.
26 pages, 9 figures

 

[marcus]

My memory of how matter is included, and my understanding of details were not so good. I am still struggling and will take a break, but first let's look on page 5 of reference [14] where it says:
==quote 1012.4719 ==
Now combine this fermionic Fock space with the kinematics of quantum gravity. Fermions must reside on the nodes n of a graph, like in lattice gauge theory. Thus we assign a copy of the Fock space F to each node of the graph. Therefore the states of the gravity+fermion theory live on the space (⊗_lL₂[SU(2)])⊗(⊗_nF), divided by the gauge action of SU(2) at each node. We can write states as Ψ(h_l,ψ_n), where l labels the links of the graph and n the nodes. The spin networks that form a basis of this state are a simple generalization of the pure gravity spin networks.
As before, it is convenient to choose an intertwiner basis at each node n that diagonalizes the volume of the node n, and label it with the volume eigenvalue v_n. That is
|j_l,v_n⟩. In the presence of fermions, spin networks carry an extra quantum number c_n at each node, which labels the basis |c⟩ in the Fock space at the node. That is: |j_l,v_n,c_n⟩. At each v-valent node n bounded by links with spins j₁ , ..., j_v , the intertwiner v_n is an invariant tensor in the tensor product of the v representations j₁, ..., j_v
if c_n =∅ or c_n =2. But it is an invariant tensor in the tensor product of the v + 1 representations j₁, ..., j_v, 1/2 if c_n = ±. In this case, the intertwiner couples the spinor to the gravitational magnetic indices.

==endquote==

So the enlarged graph Hilbert space basis is the usual L₂ of L copies of the group, tensored with N copies of Fock space where N is the number of nodes.

 

[w4k4b4lool4]

Thanks for that Marcus!
Unfortunately, I won't have time to delve deeper into this in the next few days.
I've certainly learned a lot from our conversation,
Wakabaloola