Existence of LQG: Investigating Limits of Spin Foam Models

[atyy]

In http://arxiv.org/abs/1010.1939, Eq 26 & 27, Rovelli used 2 limits to define the current spin foam models. But he doesn't know if those limits exist.

In http://arxiv.org/abs/1010.5437, Rovelli and Smerlak further elaborated properties of the limits, assuming their existence.

Frank Hellmann commented on them in http://arxiv.org/abs/1105.1334
"The refinement limit considered in [RS10] envisions going to an “infinite complete 2-complex” and thus is of a significantly different nature. ... We would also expect that in the case of gravity we will not be able to define the refinement limit exactly, but only approximately ... In the case of general spin foam models Bahr has suggested a set of cylindrical consistency conditions [Bah]."

A new comment has appeared in Kisielowski, Lewandowski, and Puchta's http://arxiv.org/abs/1107.5185
"One of the open problems of the spin foam approaches to the 4D gravity is definition of the total amplitude that takes into account all the foams. A recent breakthrough in this issue is Rovelli-Smerlak’s projective limit definition [28]. How do our diagrams fit in this limit? Those questions will be answered soon either by us or by the readers."

I've bolded the statements above which refer to unpublished work. I await these developments with bated breath

 

[atyy]

Perhaps a complete tangent, but http://arxiv.org/abs/1102.5524" .

 

[marcus]

Thanks for pointing me to that Guifre Vidal paper. To give any newcomer an idea of who G Vidal is, here are his 43 papers in Condensed Matter that arxiv lists:
http://arxiv.org/find/cond-mat/1/au:+Vidal_G/0/1/0/all/0/1

And here is the paper you pointed to as a LQG-like development in cond-matt tensor networks.
http://arxiv.org/abs/0907.2994
Tensor network decompositions in the presence of a global symmetry
Sukhwinder Singh, Robert N. C. Pfeifer, Guifre Vidal
(... last revised 10 Nov 2010)
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance also in the context of tensor network algorithms, thus setting the stage for cross-fertilization between these two areas of research.
4 pages, 4 figures

 

[marcus]

BTW Atyy, since the thread is about the existence of LQG, which I think for you means the current status of work on the refinement limit of the transition amplitude, you might be interested in looking at section III-D of the new Zakopane Lectures (starting on page 13)

The Lectures have been expanded and extensively revised as of this week, 23 July, and in a sense they supersede what you cited in your post #1, and take into account the Frank Hellmann paper of May 2011.

See also the last paragraph of section III-C, which cites Frank's paper. This is the section where the transition amplitude is defined.

There is also the overview section I, which briefly summarizes the situation on page 3 in the paragraph containing equation (3). The whole treatment has been streamlined by focusing on foams built on the duals to cellular decompositions of spacetime. It is is a natural restriction which limits the class of 2-complexes C that enter as foams. The "summing" limit was side-lined, possibly as extra baggage, and in this concentrated treatment of LQG one only looks at the C --> ∞ limit which was associated with "refining" the foam, subject to its boundary condition network.

It is interesting to see how things have clarified and developed since February when the first version of the Lectures came out.

If anyone is coming to this new, the July version of the Zakopane Lectures on LQG is at
http://arxiv.org/abs/1102.3660
This is the currently definitive version of LQG presented in a 34-page paper that is partly a short introductory
"textbook" developing the theory from math basics, and partly a review article sketching the field's antecedents and reporting on its status at present.

 

[marcus]

I've bolded the statements above which refer to unpublished work. I await these developments with bated breath

Anyway, when you read the July 2011 version of Zako Lectures, you see that the picture has jelled. There is no need for Atyy or anyone to be awaiting with bated breath

The indications are pretty clear that the theory exists and you can do calculations with it.

 

[atyy]

Anyway, when you read the July 2011 version of Zako Lectures, you see that the picture has jelled. There is no need for Atyy or anyone to be awaiting with bated breath

The indications are pretty clear that the theory exists and you can do calculations with it.

There were updates also to http://arxiv.org/abs/1010.5437v3 and http://arxiv.org/abs/1010.1939 (19 July 2011), showing the same open issues.

 

[marcus]

I don't see that Rovelli's 1010.1939 has been extensively re-written and brought up to date as a status report.

It still has the same message as it did back in October 2010, which is appropriate---it is not a review article or status report---more a brief specialized outreach in the direction of mathematicians.

It is still an October 2010 article---I don't know what minor correctins/changes were made in the version 2---typos? references?

I think you can instead actually go by the July version of 1102.3660. It parallels Rovelli's video talk from the Madrid Loops conference, and is even more up to date. It is clearly intended as a current review article, status report, for the field.
The July version is extensively re-written. Clearly no longer the February 2011 paper. A lot has happened.

So it seems a bit legalistic to quibble about whatever Rovelli said in October. Why don't we simply read what he says NOW? Give the guy a break and really listen

 

[atyy]

v1: I am still correcting and updating the lectures. Comments and corrections very welcome. 24 pages. 10 figures
v2: I am still correcting and updating the lectures. Comments and corrections very welcome. 24 pages. 10 figures
v3: Still correcting and updating the lectures. Comments and corrections very welcome. 24 pages. 10 figures. Last update, homeworks added
v4: This is a largely restructured and expanded version of the lectures. In particular, I have added a substantial introductory and orientation Section, a discussion on the justification for the vertex expansion, and more details on the applications. Comments and corrections always very welcome. 34 pages. Many figures

So what's the difference in scientific content since v1?

 

[f-h]

The indications are pretty clear that the theory exists and you can do calculations with it.

I think that is wildly optimistic, to say the least, I am not sure anybody really expects the limits to exist as such, and while it's possible to calculate with the theory the physical meaning of these calculations is very much subject to a lively debate in the community. I wrote the paper on the expansions exactly to let others know of some of the arguments that are being exchanged behind closed doors, especially where they maybe contradict the most optimistic readings.

 

[marcus]

And as I recall Rovelli in his current review of the field cites your paper and makes (I think several, maybe one) reference(s) in the text.

It's great to have a chance of some input from you. Could you summarize for us what you are saying in your paper?

Maybe in a more layman-style way that makes it easier for a wider readership to understand?

 

[marcus]

If anyone wants to do some reading ahead of a possible intuitive explanation by f-h, one paper to look at would be:
http://arxiv.org/abs/1105.1334

One change in the July 2011 version of Rovelli's Z-Lectures is (I think at least) that he several times mentions the possible need to restrict the spinfoams of the theory to be dual to cellular decompositions
and as I recall when he speculates that this might turn out to be necessary he mentions an f-h paper---either the PhD thesis or the May paper I mentioned.

I also saw reference to restricting spinfoams to be dual to triangulations but that is more something which he refers to other people doing. His own position seems at this point to be on the fence---undecided between general 2cell-complex versus dual-to-choppings of spacetime.

Correct me if I'm wrong---it's busy here, company all yesterday (which was great!) and no time to study up on every detail.

I'd like very much to hear more about this:

while it's possible to calculate with the theory the physical meaning of these calculations is very much subject to a lively debate in the community​

When you have a theory that is crystallizing into a definite shape, and that you can calculate with, then I think the "physical meaning" is what you calculate.
Then the next question is whether that physical meaning agrees with observation. Does it correctly describe nature.

Maybe you have some different idea of "physical meaning"?

 

[f-h]

It is certainly necessary to restrict the class of 2-complexes. Otherwise even at the 1-vertex level you can construct arbitrarily many divergent 2-complexes. The most plausible such restriction I've seen this far is this:

http://arxiv.org/abs/1107.5185

As for your second question, you need a mathematical framework and a consistent way to associate physics to the numbers you calculate. In the rough we're all agreed that the Oeckl-Rovelli boundary proposal is the way to go, but in practice there are a lot more open questions than settled ones.

As for my paper, the central question I ask is what is the physical meaning of the expansions done. That is, given how we want to interpret the theory, what physical regime do these calculations reflect. My proposal is that the approximation calculated by them is on a spacetime of form B^4 u B^4, that is, the 1 vertex level approximation leads to disjoint spacetimes, they argue that it leads to cosmological spacetimes (too), so who is right?

My observation in the paper is that the 2-point correlation function between any observables on the two components of the boundary vanishes at the 1-vertex level exactly (that is, at any level of the graph truncation, for any boundary state, and without going to the asymptotics). Thus if you want to keep the interpretation of the 2-point functions that underlies the graviton propagator, this has to correspond to a space time topology which completely prevents any propagation, that is, one that is disjoint.

Obviously they disagree. :) I'll page Francesca to see if she wants to respond to this.

 

[atyy]

In the rough we're all agreed that the Oeckl-Rovelli boundary proposal is the way to go, but in practice there are a lot more open questions than settled ones.

I have a very basic question about the boundary - is one free to choose any boundary? Or does it have to satisfy some constraint or dynamical equations?

 

[f-h]

Good question. The idea is that we allow any boundary data, possibly subject to some constraints that are considered "kinematical". The dynamics then implement the rest of the constraints, giving the overlap of the boundary state with the physical states. By conditioning this amplitude appropriately this can then be seen as a sort of transition amplitude/probability: Given that I see ABC on that part of the boundary what is the probability that out of all possible things compatible with ABC I will see exactly XYZ on the right part of the boundary.

 

[francesca]

Obviously they disagree. :) I'll page Francesca to see if she wants to respond to this.

Obviously I disagree we have been chatting about this a while...

It is certainly necessary to restrict the class of 2-complexes. Otherwise even at the 1-vertex level you can construct arbitrarily many divergent 2-complexes. The most plausible such restriction I've seen this far is this:
http://arxiv.org/abs/1107.5185

It is likely that there could be a restriction, even if I would not say that it's "certainly necessary". I agree that this is something that has to be studied, thank you to point to the new paper by the Polishes, it seems very interesting!

As for my paper, the central question I ask is what is the physical meaning of the expansions done. That is, given how we want to interpret the theory, what physical regime do these calculations reflect.

This has been the focus of my work of the last few years. A brief summary can be found in http://arxiv.org/abs/1107.2633" .

My proposal is that the approximation calculated by them is on a spacetime of form B^4 u B^4, that is, the 1 vertex level approximation leads to disjoint spacetimes, they argue that it leads to cosmological spacetimes (too), so who is right?

We are both right! There is a factorization and there is a cosmological interpretatation, why not?! This is in the same spirit of what has been done in covariant quantum cosmology so far, I think in particular to the no-boundary proposal of Hawking, but also to Vilenkin's wave function and so on...
The factorization of the transition amplitude, that leads to the 2 disjoint spacetimes, is nothing surprising: it is inherited from the classical theory and the usual expression of the transition amplitude. In fact in http://arxiv.org/abs/1107.2633" you can read

The classical Hamilton function of a homogeneous isotropic cosmology is the difference between two boundary terms. With the cosmological constant $\Lambda$ it gives
$S_H= \int {\mathrm{d}}t\, (a\dot a^2 + \frac\Lambda3 a^3)\Big|_{\dot a =\pm \sqrt{\frac\Lambda3} a}= \frac23 \sqrt{\frac\Lambda3} (a^3_{fin}-a^3_{in})$
where $a$ is the scale factor and $\dot a$ its time derivative. Therefore at the first order in $\hbar$ the quantum transition amplitude factorizes:
$W(a_{fin},a_{in})=e^{\frac i \hbar \,S_H(a_{fin},a_{in})}=W(a_{fin})\overline{W(a_{in})}$ .
The same happens for the spinfoam amplitude
$\langle W |\psi_{H_\ell(z_{fin},z_{in})}\rangle=W(z_{fin},z_{in}) = W(z_{fin})\, \overline {W(z_{in})}$ .

In simple words, this is saying that at the first order the probability to go from a state 1 to a state 2 is given by the probability of 1 to exist times the probability of 2 to exist. And, this is remarkable, the distribution of probability for a state to exist is peaked when the state, labelled by the scale factor $a$ and the extrinsic curvature (related to $\dot a$), satisfies the right relation between $a$ and $\dot a$, namely the Friedmann equation $\left(\dot{a}/a\right)^2=\Lambda/3$ (in absence of matter, matter can be easily added in this spinfoam framework in effective way, in the same manner we have added the cosmological constant in http://arxiv.org/abs/1101.4049" ).

Said so, I'm of couse interested in studying the next orders. Spinfoam Cosmology moves together with the new results in the full theory: a year ago only the fist order were available, now we have Mingjy's computation of the 3-point function. Therefore now we have more technologies to study the correlation between 2 cosmological states, even if the actual computation is cumbersome and it has not been done yet. Nonetheless, we have so much to learn from the first order!

My observation in the paper is that the 2-point correlation function between any observables on the two components of the boundary vanishes at the 1-vertex level exactly (that is, at any level of the graph truncation, for any boundary state, and without going to the asymptotics). Thus if you want to keep the interpretation of the 2-point functions that underlies the graviton propagator, this has to correspond to a space time topology which completely prevents any propagation, that is, one that is disjoint.

Again, this seems just to indicate that the correlations will appear at the next order, I totally agree and I have pointed to this me too. See http://arxiv.org/abs/1107.2633" .

The main open issue remains the computation of quantum corrections. Higher order quantum correction can come by considering more than one vertex in the spinfoam. We are not interested in a mere sequence of edges and vertex, because it has to be equivalent to a single vertex [http://arXiv.org/abs/1010.4787" ]. We would like to have instead spinfoam faces spanning from the initial to the final states and carrying the correlations between the two states (see FIG.4).

 

[francesca]

I don't see that Rovelli's 1010.1939 has been extensively re-written [...] It is still an October 2010 article---I don't know what minor correctins/changes were made in the version 2---typos? references?

The only difference is a change of Kharkov into Moscow due to movie policies about the set location

 

[marcus]

The only difference is a change of Kharkov into Moscow due to movie policies about the set location

Did Carlo actually make an appearance in that film about Landau? Or did the director invite him to join in a panel discussion of the physics and events of that era (e.g. around 1942)? I don't know anything about this, and have only seen hints in 1010.1939.

Ilya Khrzhanovksky the director of "Dau" is young and somewhat original---I could not predict what he would do or not do. Any involvement whatever with making that film could have been fun. Also Landau is an intrinsically interesting film character.

 

[f-h]

So I will try to keep this brief, with just a few counterpoints...

We are both right! There is a factorization and there is a cosmological interpretatation, why not?! This is in the same spirit of what has been done in covariant quantum cosmology so far, I think in particular to the no-boundary proposal of Hawking, but also to Vilenkin's wave function and so on...

These proposals only give us a framework to discuss cosmology, we do indeed agree on the framework (and my paper explicitly uses the no-boundary proposal) but within that framework I still must insist on the very physical question of what solutions, quantum and classical, you are picking up.

The factorization of the transition amplitude, that leads to the 2 disjoint spacetimes, is nothing surprising: it is inherited from the classical theory and the usual expression of the transition amplitude.

Very good, but in the classical theory it is only true for the homogeneous sector of the classical theory. Whereas in the 1-vertex approximation it is true for all states for all quantum number regimes and for all boundary graphs. So the factorisation is much stronger than can be explained from the classical theory, and in fact persists where the classical theory would predict correlations (e.g. if the boundary data on one side of the space time includes a gravitational wave).

Thus I maintain that if you dislike the topological interpretation of the vertex expansion you have to come up with a different one that explains all the further peculiar features...

Said so, I'm of couse interested in studying the next orders.

And so you see, Marcus, round and round we go, we've been at this for a while ;) It won't get settled till we have the next orders.

I personally am more interested in the conceptual/mathematical side of this. Can we actually understand whether these expansions are consistent? To do that we need to understand renormalisation, etc... My Marseilles friends don't like me saying this, but in my view there are plenty of conceptual questions to answer before we can claim we know how to calculate physics with the theory. ;)

 

[marcus]

A considerable number of people e.g. Frank's "Marseille friends" and folks in Ashtekar's group at Penn State (see work by Magliaro and Perini) treat LQG as a well-defined theory that one can do calculations with. There seem to be legitimate differences of opinion. Frank says some things have to be worked out before he will accept that the calculations are "physicially meaningful".

I'm interested in keeping the discussion up-to-date. Atyy opened the thread based on an October 2010 paper that was not a review but rather a special purpose paper, and has not been updated to reflect the current situation. It was based on conversations Rovelli had with Russian mathematician Andrei Losev---which he found stimulating and decided to write up. The presentationis primarily for mathematicians and it is very much as of October 2010.

In http://arxiv.org/abs/1010.1939, Eq 26 & 27, Rovelli used 2 limits to define the current spin foam models. But he doesn't know if those limits exist.
...

I don't see that Rovelli's 1010.1939 has been extensively re-written and brought up to date as a status report.

It still has the same message as it did back in October 2010, which is appropriate---it is not a review article or status report---more a brief specialized outreach in the direction of mathematicians.

It is still an October 2010 article---I don't know what minor correctins/changes were made in the version 2---typos? references?

I think you can instead actually go by the July version of 1102.3660. It parallels Rovelli's video talk from the Madrid Loops conference, and is even more up to date. It is clearly intended as a current review article, status report, for the field.
The July version is extensively re-written. Clearly no longer the February 2011 paper. A lot has happened.

So it seems a bit legalistic to quibble about whatever Rovelli said in October. Why don't we simply read what he says NOW? Give the guy a break and really listen

The only difference is a change of Kharkov into Moscow due to movie policies about the set location

Francesca points out that the October 2010 paper that Atyy cited was not updated except to change a place-name as requested by some Russian flim-makers who were Rovelli's and Losev's hosts at the time.

=======================

Whatever the actual situation is now about showing convergence, and determining the classical and continuum limits, we stand to gain by using current sources.

A paper by Magliaro and Perini just appeared on archive this week---specifically about the spinfoam continuum and classical limits. Progress is being made there so we should check it out.
http://arxiv.org/abs/1108.2258

Also the most recent actual review article has been considerably expanded and rewritten. A new version came out in July, and version 5 came out 3 August.
that is http://arxiv.org/abs/1102.3660

The Magliaro Perini August 2011 paper I just mentioned gives a streamlined and updated version of material from their May 2011 paper. In case it might be useful here is the abstract of the May paper:
==quote Magliaro Perini 1105.0216 ==
Regge gravity from spinfoams

We consider spinfoam quantum gravity in the double scaling limit γ→0, j→∞ with γj=const., where γ is the Immirzi parameter, j is the spin and γj gives the physical area in Planck units. We show how in this regime the partition function for a 2-complex takes the form of a path integral over continuous Regge metrics and enforces Einstein equations in the semiclassical regime. The Immirzi parameter must be considered as dynamical in the sense that it runs towards zero when the small wavelengths are integrated out. In addition to quantum corrections which vanish for ℏ→0, we find new corrections due to the discreteness of geometric spectra which is controlled by γ.
==endquote==

 

[marcus]

Since the Magliaro Perini paper of 10 August is the most recent thing we have bearing on the "Existence" topic, I may as well quote the abstract so we can have something in front of us:
http://arxiv.org/abs/1108.2258
Emergence of gravity from spinfoams
Elena Magliaro, Claudio Perini
(Submitted on 10 Aug 2011)
We find a nontrivial regime of spinfoam quantum gravity that reproduces classical Einstein equations. This is the double scaling limit of small Immirzi parameter (gamma), large spins (j) with physical area (gamma times j) constant. In addition to quantum corrections in the Planck constant, we find new corrections in the Immirzi parameter due to the quantum discreteness of spacetime. The result is a strong evidence that the spinfoam covariant quantization of general relativity possesses the correct classical limit.
9 pages, shorter version of "Regge gravity from spinfoams"
Published in European Physical Letters, vol 95, n 3 (August 2011)

It was marginally interesting that they apparently chose to publish in European Physical Letters, EPL, instead of, say, an American-based journal such as PRL. This puts EPL on the map for me, I want to know a bit more about the journal.
http://epljournal.edpsciences.org/
Hmmm. Some interesting details there. They have an open (free) access policy covering a certain number of their articles.
Here is the TOC for the issue of EPL where the article appeared:
http://epljournal.edpsciences.org/i...ticles/epl/abs/2011/15/contents/contents.html
The EPL online publication date was 20 July, so Magliaro Perini waited until it was published in EPL before they posted their paper on arxiv. And they posted on arxiv in EPL format. Exactly as it appeared in the journal. OK. kind of crisp and snappy. Minor point but stll interesting. No question of revisions with this one---it's the final published version. Btw Magliaro and Perini are a couple and their first kid is due next month. Congratulations and very best wishes to them! Great paper. Ashtekar must be happy to have them at Penn State IGC.
http://arxiv.org/abs/1108.2258

 

[marcus]

The most up-to-date summing up of the status of loop gravity, as relates to the topic of "existence" that Atyy raised is the August 2011 version of the Zakopane lectures.
The theory is defined on page 3 using a single equation (3). The equation (1) referred to here is a formulation of GR.

==quote 1102.3660 page 3==
The main properties of (3) are the following.

i. In a suitable semiclassical limit, (3) approaches (1). A_v(j_e,i_v) approaches the exponential of the Regge action, which in turns approaches the action of general relativity. Therefore (3) is a discretization of the path integral for quantum gravity.

ii. (3) is ultraviolet finite, a property strictly connected to the Planck discreteness of the spin networks. It admits a quantum-deformed version [11, 12] that describes the cosmological constant coupling [13] and is IR finite.⁴ In this version, a theorem assures that (3) is finite.

iii. The amplitude (3) is for pure gravity, but it can be coupled to fermions and Yang Mills fields [14]. The finiteness result continues to hold.
==endquote==

Footnote 4 just observes that in 3D this gives the Turaev-Viro theory.

The UV-finite property stems from the area and volume gap. Below a certain level there is no measurable nonzero area and volume. The geometric operators have discrete spectra. UV-finiteness is not based on the assumption of a metric (as was suggested recently in another thread.)

This point is made on the previous page, at the end of section A1.

==quote 1102.3660 page 2==
Its most remarkable feature is the discreteness of the geometry at the Planck scale, which appears in this context as a rather conventional quantization effect: In GR, the gravitational field determines lengths, areas and volumes. Since the gravitational field is a quantum operator, these quantities are given by quantum operators. Planck scale discreteness follows from the spectral analysis of such operators.

To avoid a common misunderstanding, I emphasize that the discreteness is not given by the fact that the grains of space in Figure 1 are discrete objects. Rather, it is given by the fact that the size of each grain is quantized in discrete steps³, with minimum non-vanishing size at the Planck scale. This is the key result of the theory, which becomes later responsible for the UV finiteness of the transition amplitudes.
==endquote==