Classical Limit in LQG: Has GR Been Shown?

[Physics Monkey]

Hi everyone,

I would like to ask a very simple and direct question: has the classical limit of any of the various loopy approaches to quantum gravity been shown to be GR? Perturbative fluctuations around flat space, for example?

I've seen what appear to me to be conflicting claims on this board and in other settings.

I have a few polite requests that I hope people are willing to agree to.
1. Please no history or long lists of articles.
2. Please no comparing LQG to string theory, we have plenty of that already.
3. Please no long introduction articles or general summaries, I know quite a bit of the theory.
4. I reserve the right to add other polite requests should the need arise.

I just want to know the state of the art from the loop people on the semi-classical limit. I would be happy to talk about this, as well as what we might want from such a limit, prospects for getting there if we're not already, etc.

Thanks!

 

[marcus]

Hi everyone,

I would like to ask a very simple and direct question: has the classical limit of any of the various loopy approaches to quantum gravity been shown to be GR?

As far as I know, it has not been shown with complete rigor. There are a number of pieces of evidence that suggest that Loop has the right limit, and these are listed in one of the recent survey papers. I'll get a link.

The suggestive evidence includes things like the graviton propagator (in the approximately flat case) coming out right.
And the application to cosmology coming out right (conventional FRW universe soon after bounce).
No rigorous proof as yet though, AFAIK.

One link would be http://arxiv.org/abs/1010.1939
Look on page 5, Section V "Relation with GR"
You will see a list of 6 pieces of evidence. Point 5 is the graviton propagator work. Point 6 is the cosmology result.
======================
You could also look at point 1: classical limit (hbar --> 0) shown to be the same as large distance limit, and this has been studied in the simplest interesting case (a 4-simplex, or 5-valent spinfoam vertex) and shown to be correct.
Still more work to be done, generalizing result to more cases.

 

[marcus]

Loop gravity kinematics goes back before 2007, but I should probably mention that the dynamics of today's theory was arrived at in 2007. That was an exciting year in which the theory was radically revised by at least two different groups working in parallel. Efforts to prove stuff like classical limit would have started at square one around 2008, after the dust settled.

So in today's form, with both kinematics and dynamics, the theory is about 3 years old.
What I tend to go by is the rate of progress that has been made since 2007 on things like verifying the classical limit, and linking up with Loop cosmology. There is considerable momentum in the right direction.

I don't expect them to have dotted all the i's and crossed all the t's

 

[negru]

I really don't mean to troll this thread, but I distinctly remember some conversations on this forum maybe a year or two ago in which Marcus and others were saying "who cares that string theory has gravitons, they don't even exist". It is good to see that they finally came to their senses now that LQG is apparently making positive contact with gravitons.

(and the same thing with matter in LQG)

 

[marcus]

2. Please no comparing LQG to string theory, we have plenty of that already.

..."who cares that string theory has gravitons, they don't even exist". It is good to see that they finally came to their senses now that LQG is apparently making positive contact with gravitons...

Graviton is a useful analytical concept when you have a fixed flat background or some other fixed prior geometry. Simply having a graviton does not show a theory recovers GR where the geometry is highly curved and/or dynamically changing.
But although graviton is not universally applicable in describing gravity, essentially does not exist in all situations, it is certainly one thing on the Loop agenda. Work started in 2006 to artificially restrict the theory to a flat geometry configuration so that one could have a "graviton propagator" similar to what you get in flat perturbative theories, and check if it came out right.

I don't think anyone "finally came to their senses" in 2006 when the work on N-point functions and propagator started. It was always in the cards one would have to deal with the approx. flat case and see if you recover Newton etc.

You probably misunderstood the import of the previous discussion. Get a link to it if you want.

I think the key thing is that no one in the Loop community would imagine that merely having gravitons in some flat, or fixed curved, geometric setup would suffice to prove a theory yields gravity in the full GR sense. If you like, you could translate that as "who cares?" But it is not that simple. One still wants to be able to cover that base and deal with the flat geometry case, so one does care.

 

[atyy]

The key papers are http://arxiv.org/abs/0809.2280 (FK model) and http://arxiv.org/abs/0902.1170 , http://arxiv.org/abs/0907.2440 (EPRL model).

I think the exposition is clearest in http://arxiv.org/abs/1004.4550 . Using its equation numbering, my rough understanding is that the "in-out" transition is given by a sum over terms like W(h_l) (Eq 3, 4). After a change of variable from h_l to H_l (Table 1), we get W(H_l)~exp(iS_Regge) (Eq 56, 57, 58).

I think this is still some ways from showing an ok classical limit, since the summation of (Eq 3, 4) must still be done. Dynamical triangulations started form a similar point, but it didn't give anything sensible until it was changed to Causal Dynamical Triangulations, so that still has to be checked. As Conrady and Freidel say in their concluding remarks "That is, we have demonstrated the proper semiclassicality for certain histories that one should sum over in computing amplitudes. What we are ultimately interested in is the semiclassical property of the sum over amplitudes. Given a boundary spin network, we would like to sum over all spins in the interior compatible with the boundary spin network and show that the resulting amplitude gives an object that can be interpreted as the exponential of the Hamilton–Jacobi functional of a gravity action. Our result is a necessary condition for this to happen, but we have not shown that this is sufficient."

Bojowald's LQC has very nice results, but I don't know the connection to any of the fuller formalisms like canonical LQG or spinfoams.

 

[marcus]

The key papers are http://arxiv.org/abs/0809.2280 (FK model) and http://arxiv.org/abs/0902.1170 , http://arxiv.org/abs/0907.2440 (EPRL model).

I think the exposition is clearest in http://arxiv.org/abs/1004.4550 ...

Bojowald's LQC has very nice results, but I don't know the connection to any of the fuller formalisms like canonical LQG or spinfoams.

Bojowald's LQC was replaced by Ashtekar's LQC in 2006 which gets those results and went on to even nicer ones And Ashtekar has been working on the connection of LQC with spinfoam. Papers by himself and through collaborators now in Marseille. The essential equation has been recovered (a quantum corrected form of the basic Friedmann eqn of cosmology). So the connection of Loop Cosmo with the full theory is getting pretty strong.
Ashtekar's 2010 review "The Big Bang and the Quantum" is currently the best status report.

Atyy it sounds in your post as if you think that Regge action does not give Einstein action in the limit as lengths go to zero. I think basically Regge calculus just IS General Relativity restricted to the piecewise flat manifolds. And the PL manifolds are dense. So any theory that recovers Regge GR very likely recovers ordinary GR in the same breath. Am I missing something?

What you said about Causal Dynamical Triangulations does not seem relevant since CDT uses identical building blocks (of two types) all the same size with the same edge lengths.
Regge GR uses freely variable edge lengths. It is a different animal from CDT.

BTW thanks for the links! I'll take a look at 1004.4550 , which is most recent one you mentioned. I see that two of the three authors of that paper are Rovelli PhDs who just took postdocs with Ashtekar at Penn State.

 

[atyy]

Atyy it sounds in your post as if you think that Regge action does not give Einstein action in the limit as lengths go to zero. I think basically Regge calculus just IS General Relativity restricted to the piecewise flat manifolds. And the PL manifolds are dense. So any theory that recovers Regge GR very likely recovers ordinary GR in the same breath. Am I missing something?

What you said about Causal Dynamical Triangulations does not seem relevant since CDT uses identical building blocks (of two types) all the same size with the same edge lengths.
Regge GR uses freely variable edge lengths. It is a different animal from CDT.

As Rovelli says in your link "in the semiclassical regime the sum (5) truly reduces to a sum over geometries weighted by the exponential of the GR action". My understanding is that what we really need is the sum over geometries to reproduce GR, not just elements of the sum. DT starts from this point, and shows that there's still plenty of room for things to go wrong.

 

[marcus]

As Rovelli says in your link "in the semiclassical regime the sum (5) truly reduces to a sum over geometries weighted by the exponential of the GR action". My understanding is that what we really need is the sum over geometries to reproduce GR, not just elements of the sum. DT starts from this point, and shows that there's still plenty of room for things to go wrong.

I don't understand. You may have a point but I don't see it. In what sense are we limited to "just elements of the sum"?

Maybe, since you are quoting from page 5 section V "Relation with GR" of the paper
1010.1939 that I mentioned, I should copy the whole paragraph. More context can make something clearer.
==quote 1010.1939==
V. RELATION WITH GR

A number of elements of evidence support the conjecture that the model is related to GR:

1. The classical limit of the theory is given sending h ̄ → 0 at fixed value of boundary geometry. Since geometrical quantities are defined by spins j multiplied by powers of (17), the limit is the “large quantum numbers” j → ∞ limit, as always in quantum theory. In other words, the classical limit of pure quantum gravity is also the large distance limit, as expected.

The asymptotic expansion of the vertex (21) for high quantum numbers has been studied in detail and computed explicitly for five-valent vertices [51–54]. The result is that it gives the Regge approximation of the Hamilton function of the spacetime region bounded by the 3-geometry determined by the spin network surrounding v.

Since, in turn, the Regge action is known to be the Einstein-Hilbert action S[g_μν] of a Regge geometry, we have that

[I omit LaTex for eqn.22 which says that the LQG spinfoam vertex amplitude W_v is proportional to exp(iS[g_μν]) the exponential of the Einstein-Hilbert action]

Accordingly, in the semiclassical regime the sum (5) truly reduces to a sum over geometries weighted by the exponential of the GR action, as in (6).

==endquote==

 

[atyy]

The basic problem is that Rovelli gets a sum over geometries, but GR only has geometry - no sum.

If you sum over geometries, you may run into problems, like DT. Or you may get nice results like CDT. It remains to be seen which of these happens in EPRL/FK.

 

[Physics Monkey]

Thanks for the replies so far.

To summarize, there is no complete demonstration for spinfoams, etc. that the semiclassical limit is GR, although there is plenty of evidence that such a hypothesis is not crazy. There are also some remaining issues to be worked out which may be subtle i.e. DT vs CDT, poor behavior of polymer-like stat mech models, etc.

I am satisfied, thanks.

Let me know ask a different question. Why is it so hard to demonstrate the semi-classical limit in loopy approaches? I will break my own polite request and make a brief comparison to string theory. In that case the semiclassical limit almost falls into your lap, indeed historically as far as I know, it wasn't even looked for.

I would offer a tentative guess. In the case of more abstract spin foam type models, these seem to be simplest precisely in the deeply quantum limit, so I might expect it to be harder show semi-classical properties. However, it seems to me that the old "quantizing GR" approach should have long ago been able to produce a semi-classical limit. If I can be just a little bit incendiary, it seems at this time still a bit premature to call loopy approaches "quantum GR".

 

[ensabah6]

Let me know ask a different question. Why is it so hard to demonstrate the semi-classical limit in loopy approaches? I will break my own polite request and make a brief comparison to string theory. In that case the semiclassical limit almost falls into your lap, indeed historically as far as I know, it wasn't even looked for.
.

You get the wrong dimensions and particle content. Solutions of GR involving broken SUSY symmetry and strongly varying change are also not recovered.

 

[atyy]

Hmmm, I don't know - but maybe they are like the small N cases of AdS/CFT where gravity is said to be quantum? Is it the case that in both large and small N there we have Einstein-Hilbert like terms (otherwise why would it be called gravity?), with large N being dominated by one configuration, while in small N we sum over multiple configurations?

The other problem in a discretized geometry approach is that even for a single "path", the pieces may not fit together to form a manifold-like thing.

 

[marcus]

Thanks for the replies so far.

To summarize, there is no complete demonstration for spinfoams, etc. that the semiclassical limit is GR, although there is plenty of evidence that such a hypothesis is not crazy. There are also some remaining issues to be worked out which may be subtle i.e. DT vs CDT, poor behavior of polymer-like stat mech models, etc.
...

Let me ask a question now PM. As you know LQG has been developing fast in the past 3 years or so, and differs greatly from what it was a few years back. I don't know how one can discuss LQG at this point without reading the two most recent summaries, which define the theory.
One is six pages (plus references) : 1010.1939
One is 22 pages (plus references): 1102.3660
Or at the very least, the more recent of those two.

My sense of who you are a physics grad student doing research in string/condensed matter. Perhaps some stringy approach to a condensed matter problem. Please correct me if I've misremembered. Anyway, as a busy string+condensed matter physicist, perhaps working on your PhD thesis, do you have time to read a 22 page paper?

Have you in fact already read 1102.3660? I hope so. This, the February paper, is the more pedagogical and makes things easier to understand. The October paper has some interesting additional material but is too abbreviated to be a good introduction.

 

[tom.stoer]

I think this http://arxiv.org/abs/1103.4602 is another paper indidacing that LQG reproduces GR in a certain limit.

Regarding the general discussion: LQG is designed for background independence. Looking at GR (Einstein-Hilbert action, Einstein field equations) one finds that one has to invest some time to derive the limits of flat-space / Newtonian gravity / gravitational waves. The same applies to LQG as well - it was not designed for perturbative calculations, therefore it's not amazing that it's harder to construct the graviton propagator.

Look at lattice QCD: nobody is able to do perturbative scattering calculations on the lattice, nevertheless everybody believes that it's a valid formulation of QCD.

 

[Physics Monkey]

I think this http://arxiv.org/abs/1103.4602 is another paper indidacing that LQG reproduces GR in a certain limit.

Regarding the general discussion: LQG is designed for background independence. Looking at GR (Einstein-Hilbert action, Einstein field equations) one finds that one has to invest some time to derive the limits of flat-space / Newtonian gravity / gravitational waves. The same applies to LQG as well - it was not designed for perturbative calculations, therefore it's not amazing that it's harder to construct the graviton propagator.

Look at lattice QCD: nobody is able to do perturbative scattering calculations on the lattice, nevertheless everybody believes that it's a valid formulation of QCD.

Thanks for the paper.

I agree that LQG is designed for background independence, but so is GR. And yes, one needs a bit of effort (gauge fixing, etc) to get things like gravitational waves around flat backgrounds from GR, but my impression is that LQG has not yet been definitively shown to contain even the very general background independent formulation of GR. I would be happy if one could get either perturbative semi-classical results around a fixed background or the general background independent classical formulation.

Perhaps the analogy with QCD is telling. Maybe its fairly clear to workers with an intimate knowledge of the field that the semi-classical limit is right, but there is simple a computational barrier to accumulating enough numerical evidence. Certainly this barrier has been overcome in lattice QCD, but quantum gravity is probably more complicated numerically.

 

[tom.stoer]

I can agree to that view.

 

[marcus]

This was an interesting unconfrontational and (I think) objective-minded exchange.

I think this http://arxiv.org/abs/1103.4602 is another paper indidacing that LQG reproduces GR in a certain limit.

Regarding the general discussion: LQG is designed for background independence. Looking at GR (Einstein-Hilbert action, Einstein field equations) one finds that one has to invest some time to derive the limits of flat-space / Newtonian gravity / gravitational waves. The same applies to LQG as well - it was not designed for perturbative calculations, therefore it's not amazing that it's harder to construct the graviton propagator.

Look at lattice QCD: nobody is able to do perturbative scattering calculations on the lattice, nevertheless everybody believes that it's a valid formulation of QCD.

Thanks for the paper.

I agree that LQG is designed for background independence, but so is GR. And yes, one needs a bit of effort (gauge fixing, etc) to get things like gravitational waves around flat backgrounds from GR, but my impression is that LQG has not yet been definitively shown to contain even the very general background independent formulation of GR. I would be happy if one could get either perturbative semi-classical results around a fixed background or the general background independent classical formulation.

Perhaps the analogy with QCD is telling. Maybe its fairly clear to workers with an intimate knowledge of the field that the semi-classical limit is right, but there is simple a computational barrier to accumulating enough numerical evidence. Certainly this barrier has been overcome in lattice QCD, but quantum gravity is probably more complicated numerically.

I can agree to that view.

I wanted to gather it together so I could look at it easily, and learn what I can from it.

 

[atyy]

So what has really been done in these papers? Conrady and Freidel, and Bianchi et al both say they take hbar to zero, which I think would be a classical limit. However, what they all seem to do is take j to infinity, and so maybe hbar is not zero, and that's why they say semiclassical limit?

 

[Fra]

One thing in the reasoning used here strikes me.

It seems now you are comparing non-perturbative QCD and non-perturbative QG (LQG specifically), you also seems to compare the gauge invariance in QCD with the gauge invariance on GR.

Now at least IMHO, there is one very important, big difference that distinguishes these from the inferecial perspective.

In QCD the system is fully satisfying the subsystem requirement, and for all practical purposes any "symmetry" of the system is inferred from the environment, which for all practical purposes has not bound. Meaning any asymptotic observables can be defined without conceptual issues. Ie. in QCD, the observer is EXTERNAL.

In GR, this is very different. The symmetry of the gravitational field is inferred from the inside, because there is no outside (like there is an outside of a baryon). Sure you can consider the gravitational field in a subsystemm but then then the smaller it gets the less significant does it get. Ok, there are some exceptions like hypotetichal microscopic BH but anyway. Also when we mean by gauge fixing in GR, simply means defining the observer. This has physical significance. Something that at least does not compare directly to QCD as far as I understand.

So it's tempting to claims something like this: the lack of perturbative techniques in QCD lacks physical implication (ie we can still make predictions and observations). The lack of perturbative teqchiquie (in the same of "perturbation arond an observer view") OTOH is a really problem, since it means we have nowhere to defined the measurement.

Comment?

/Fredrik

 

[tom.stoer]

Just a question: what do you mean by "gauge invariance of LQG"? The SU(2) gauge invariance of "local rotations in the tangent space of the triads? or diff. inv.? The SU(2) gauge invariance is identical to any other well known gauge invariance (at least when treated in the canonical formalism). The problem arises due to diff. inv. and it is implicitly there already in GR.

 

[Fra]

Maybe I was sloppy, anyway what I refer to is the diffeomorphism invariance of GR.

Yes it's there also classically, but that doesn't matter. The point I wanted to make, is that from an inference perspective, there seems to be an ASYMMETRY between the notion of symmetry in QCD and GR; that at least should ask us to question whether what makes sense in QCD makese sense also in QG.

/Fredrik

 

[tom.stoer]

I agree. This asymmetry is just the difference between gauge symmetry an diff. inv.

 

[Fra]

I agree. This asymmetry is just the difference between gauge symmetry an diff. inv.

Yes, but diff inv is also a "gauge symmetry", it's just that they belong to different contexts.

The difference lies on how your infer/measure these symmetries. From pure mathematics, one can easily imagine that you loose this distinction. IMHO, this is exactly the mistake that is made when one considers "quantum states of the universe".

I just think this is a distinction bear in mind when comparing the gauge theories of SM, and GR as a gauge theory. In particular since gauge fixing in GR related to perturbation theory.

/Fredrik

 

[tom.stoer]

Why do you think that gauge fixing is in any way related to perturbatio theory?

I agree that gauge fixing is often presented in the context of perturbation theory, but in principle it has nothing to do with perturbation theory. For gauge theories it means "defining a gauge slice" in a fibre bundle such that every fibre (along which a gauge group acts) is cut exactly once. But if you have a path integral with well-defined measure and with finite total volume before gauge fixing, then gauge fixing is not required.

Simple example: a function a(r) does not depend on phi which is the gauge d.o.f.

$$A = \int d^2r \, a(r) = \int d\phi dr \, a(r) = \text{vol}(\phi) \int dr \, a(r) = 2\pi \int dr \, a(r)$$

You can calculate the physical quantity as

$$A_\text{phys} = \frac{1}{\text{vol}(\phi)}\int d^2r \, a(r) = \int dr \, a(r)$$

Or you can use the FP trick

$$A_\text{phys} = \int d^2r \, \delta(\phi-\phi_0) \, a(r) = \int dr \, a(r)$$

The latter case is used in perturbation theory. Note that if the volume of the gauge group is finite there is no need for gauge fixing; you can simply divide by the volume. The problem arises b/c
a) typically the volume is infinite and therefore the first integral does not exist
b) the cut cannot be defined globally (Gribov ambiguities due to non-trivial geometry of the bundle)

To be honest: I have no idea how to translate this simple picture to diff. inv. - except in special cases.

 

[Fra]

Why do you think that gauge fixing is in any way related to perturbatio theory?

I'll get to this later and try explain.

This is a bit dual IMO.

On one hand I agree that "perturbation theory" in general and gauge theory has nothing whatsoever to do with each other. I think this is what you mean and I agree. But in another sense it has. And it has to do with the context of measurement.

I commented on this in some old thread as well that you can have two views of "perturbation theory" one view is simply as a mathematical solving method. Ie. it has nothing whatsoever to do with physics. It's a pure teqchique. But OTOH, we have a much deeper sense or perturbation in the sense of perturbing an information state. and here things get more complicated.

In short, in GR the OBSERVER is partly a gauge choice; now that is a very weird thing. Not mathematically but conceptually and when you try to understand it in terms of measurment.

I'll try to explain later...

/Fredrik

 

[tom.stoer]

I think I'll get the point ...

 

[atyy]

In these papers, there is mention of hbar being taken to zero, which I thought would be the classical limit. However, what seems to have been taken is the large spin limit. So what is the meaning of the "semiclassical limit" in these papers, ie. what would we recognize as a correct answer? Eg. why do Conrady and Freidel say "Given a boundary spin network, we would like to sum over all spins in the interior compatible with the boundary spin network and show that the resulting amplitude gives an object that can be interpreted as the exponential of the Hamilton–Jacobi functional of a gravity action."?

Apparently the idea is that the classical Hamilton-Jacobi functional of gravity evaluated on a 3d piecewise linear metric gives the Regge action (in some cases), ie. the Regge action is not due to discretization of the theory, but of the particular boundary.
http://relativity.phys.lsu.edu/ilqgs/panel050509.mp3 (he's the third of 3 speakers)
http://relativity.phys.lsu.edu/ilqgs/panel050509.pdf (p 28)

The remarks by Freidel on AdS/CFT near the end of his talk, and the exchange between Ashtekar and Freidel immediately after that is very, very interesting. And Ashtekar's comments on the LQG landscape, following a great question about spinfoam renormalization, are phenomenal!