QG five principles: superpos. locality diff-inv. cross-sym. Lorentz-inv.

[sheaf]

Interestingly, in one of his original http://math.ucr.edu/home/baez/penrose/" Penrose says

One might ask whether corresponding rules might be invented which lead to other dimensional schemes. I don't in fact see a priori why one shouldn't be able to invent rules, similar to the ones I use, for spaces of other dimensionality. But I'm not quite sure how one would do this. Also it's not obvious that the whole scheme for getting the space out in the end would still work. The rules I use are derived from irreducible representations of SO(3). These have some rather unique features

Of course SO(3) (forgetting about the double cover) is essentially SU(2), so SU(2) was integral to the original spin network idea. I haven't studied in detail how Penrose extracts three-space from the spin network, but it's interesting that he also considered the idea of going to higher dimensions. I wonder what the "unique features" were that he was referring to...

ETA: I'd be interested to see if Penrose's methods for extracting directions from a spin network have any relationship with the coherent state approaches in http://arxiv.org/abs/1004.1780

 

[atyy]

Spin networks of arbitrary groups arise in
http://arxiv.org/abs/0907.2994
http://arxiv.org/abs/1008.4774

Another approach for generalization is group field theory, which is related to spin foams, not spin networks, though some spin foams are related to spin networks.
http://arxiv.org/abs/0903.3475

 

[sheaf]

Spin networks of arbitrary groups arise in
http://arxiv.org/abs/0907.2994
http://arxiv.org/abs/1008.4774

Another approach for generalization is group field theory, which is related to spin foams, not spin networks, though some spin foams are related to spin networks.
http://arxiv.org/abs/0903.3475

Thanks for the references !

 

[atyy]

Thanks for the references !

Related, perhaps, to the work from Vidal's group is http://arxiv.org/abs/cond-mat/0407140

"Remarkably, it appears that the theory of loop quantum gravity can be reformulated in terms of a particular kind of string-net, where the strings are labeled by positive integers."

"String-nets with positive integer labeling were first introduced by Penrose (Penrose, 1971), and are known as “spin networks” in the loop quantum gravity community. More recently, researchers in this field considered the generalization to arbitrary labelings (Kauffman and Lins, 1994; Turaev, 1994)."

 

[sheaf]

I came across a nice presentation of Benjamin Bahr on coherent states (not sure if it's been posted before) :

http://www.fuw.edu.pl/~jpa/qgqg3/BenjaminBahr.pdf"

 

[marcus]

Sheaf, thanks for the link. Bahr did a nice job of presentation.

I think the QGQG3 talks are not available video or audio, just the slides PDF. In Bahr's case the slides are so complete and careful that they are useful by themselves.
http://www.fuw.edu.pl/~jpa/qgqg3/schedule.html

Since this thread is about the "new look" way of formulating LQG that we got this spring, I should mention that Eugenio Bianchi is giving a talk at Perimeter on 3 November. We may get video of that.
In the April paper Rovelli partially attributes the reformulation to him. Also the Bahr slides just mentioned cite the coherent LQG states work by Bianchi Magliaro Perini. The November talk could be about any of a number of topics. To give an idea of Bianchi's current research interests I will list his recent papers. I think he got his PhD around 2008 and is still on first postdoc, but has already done a bunch of things.1. http://arxiv.org/abs/1005.0764
Face amplitude of spinfoam quantum gravity
Eugenio Bianchi, Daniele Regoli, Carlo Rovelli
Comments: 5 pages, 2 figures

2. http://arxiv.org/abs/1004.4550
Spinfoams in the holomorphic representation
Eugenio Bianchi, Elena Magliaro, Claudio Perini
Comments: 9 pages

3. http://arxiv.org/abs/1003.3483
Towards Spinfoam Cosmology
Eugenio Bianchi, Carlo Rovelli, Francesca Vidotto
Comments: 8 pages

4. http://arxiv.org/abs/1002.3966
Why all these prejudices against a constant?
Eugenio Bianchi, Carlo Rovelli
Comments: 9 pages, 4 figures

5. http://arxiv.org/abs/0912.4054
Coherent spin-networks
Eugenio Bianchi, Elena Magliaro, Claudio Perini
Comments: 15 pages, appendix added

6. http://arxiv.org/abs/0907.4388
Loop Quantum Gravity a la Aharonov-Bohm
Eugenio Bianchi
Comments: 19 pages, 1 figure

7. http://arxiv.org/abs/0905.4082
LQG propagator from the new spin foams
Eugenio Bianchi, Elena Magliaro, Claudio Perini
Comments: 28 pages
Journal-ref: Nucl.Phys.B822:245-269,2009

8. http://arxiv.org/abs/0812.5018
LQG propagator: III. The new vertex
Emanuele Alesci, Eugenio Bianchi, Carlo Rovelli
Comments: 9 pages
Journal-ref: Class.Quant.Grav.26:215001,2009

9. http://arxiv.org/abs/0809.3718
Asymptotics of LQG fusion coefficients
Emanuele Alesci, Eugenio Bianchi, Elena Magliaro, Claudio Perini
Comments: 14 pages, minor changes

10. http://arxiv.org/abs/0808.1971
Intertwiner dynamics in the flipped vertex
Emanuele Alesci, Eugenio Bianchi, Elena Magliaro, Claudio Perini
Comments: 12 pages, 7 figures
Journal-ref: Class.Quant.Grav.26:185003,2009

11. http://arxiv.org/abs/0808.1107
Semiclassical regime of Regge calculus and spin foams
Eugenio Bianchi, Alejandro Satz
Comments: 30 pages, no figures. Updated version with minor corrections, one reference added
Journal-ref: Nucl.Phys.B808:546-568,2009

12. http://arxiv.org/abs/0806.4710
The length operator in Loop Quantum Gravity
Eugenio Bianchi
Comments: 33 pages, 12 figures; NPB version
Journal-ref: Nucl.Phys.B807:591-624,2009

13. http://arxiv.org/abs/0709.2051
The perturbative Regge-calculus regime of Loop Quantum Gravity
Eugenio Bianchi, Leonardo Modesto
Comments: 43 pages, typos corrected, version accepted by Nucl.Phys.B
Journal-ref: Nucl.Phys.B796:581-621,2008

 

[marcus]

...
Since this thread is about the "new look" way of formulating LQG that we got this spring, I should mention that Eugenio Bianchi is giving a talk at Perimeter on 3 November. We may get video of that.
In the April paper Rovelli partially attributes the reformulation to him...

Bianchi gave a talk(s) on "new look" formulation of LQG at the SIGRAV conference at Pisa in September. So far I don't know of an online source. Rovelli cited the SIGRAV lectures in a paper he just posted, in which one section (section IV) parallels Bianchi's SIGRAV talk(s).

The paper is an extremely interesting one, and constitutes another "new LQG" chapter:
http://arxiv.org/abs/1010.1939
Simple model for quantum general relativity from loop quantum gravity
Carlo Rovelli
8 pages, 3 figures
(Submitted on 10 Oct 2010)
"New progress in loop gravity has lead to a simple model of 'general-covariant quantum field theory'. I sum up the definition of the model in self-contained form, in terms accessible to those outside the subfield. I emphasize its formulation as a generalized topological quantum field theory with an infinite number of degrees of freedom, and its relation to lattice theory. I list the indications supporting the conjecture that the model is related to general relativity and UV finite."

It is uncanny how Feynman-like and how much like QED this Loop approach is beginning to look.

The indications that GR is recovered are now increasingly strong. See the 6 items in section V of the paper, starting on page 5.

The "new look" version has been born from the convergence of a remarkably diverse collection of approaches to QG:

==quote introduction==
A simple model has recently emerged in the context of loop quantum gravity. It has the structure of a generalized topological quantum field theory (TQFT), with an infinite number of degrees of freedom, local in the sense of classical general relativity (GR). It can be viewed as an example of a “general-covariant quantum field theory”. It is defined as a function of two-complexes and may have mathematical interest in itself. I present the model here in concise and self-contained form.

The model has emerged from the unexpected convergence of many lines of investigation, including canonical quantization of GR in Ashtekar variables [1–5], Ooguri’s [6] 4d generalization of matrix models [7–11], covariant quantization of GR on a Regge-like lattice [12–14], quantization of geometrical “shapes” [15–18] and Penrose spin-geometry theorem [19]. The corresponding literature is intricate and long to penetrate. Here I skip all ‘derivations’ from GR, and, instead, list the elements of evidence supporting the conjectures that the transition amplitudes are finite and the classical limit is GR.

The model’s dynamics is defined in Sec. II. States and operators in Sec. III and IV. Sec. V reviews the evidence relating the model to GR, and some of its properties.

==endquote==

The concise and self-contained presentation is, in fact really concise! It is accomplished in HALF A PAGE! right at the start. By stating four QG "Feynman rules". See at the bottom of page 1 where he says "This completes the definition of the model."

Reference [29] in Rovelli's paper is to:
E. Bianchi, “Loop Quantum Gravity, Lectures at the XIX SIGRAV Conference on General Relativity and Gravitational Physics. Scuola Normale Superiore-Pisa.” 9/2010.
http://www.sigrav.org/Announcements/Pisa2010/ProgramPT.pdf
http://www.sigrav.org/index.it.php

BTW this side-comment caught my attention. It may be related to the conversations at Kharkov with Andrey Losev that are mentioned in the Acknowledgments section.
==quote==
It can be viewed as an example of a “general-covariant quantum field theory”. It is defined as a function of two-complexes and may have mathematical interest in itself.
==endquote==

I already got the sense that the April paper http://arxiv.org/abs/1004.1780 was digging up stuff that might have inherent mathematical interest. The use of graphs to define "graph Hilbert spaces", operators and gauge transformations. The use of graphs to grade complexity in systems of approximation--the graph itself becomes a kind of "renormalization" order-parameter. Equipped with the obvious partial ordering on the set of graphs. Intriguing.

These two-complexes are purely combinatorial objects (just graphs raised up one level).

 

[marcus]

There is a lot of substance in the October paper. Probably 2010 is going to count as an important year for the Loop program.
==quote starting at bottom of page 4==
...
The running of the Newton between the Planck scale and low-energy can modify this relation.
...
When Γ is disconnected, for instance if it is formed by two connected components, expression (20) defines transition amplitudes between the connected components. This transition amplitude can be interpreted as a quantum mechanical sum over histories.

Slicing a two-complex, we obtain a history of spin networks, in steps where the graph changes at the vertices. The sum (20) can therefore be viewed either as a Feynman sum over histories of 3-geometries, or as a sum over 4-geometries.

This is what connects the two intuitive physical pictures mentioned in Section II: the particular geometries summed over can also be viewed as histories of interactions of quanta of space.

The amplitude of the individual histories is local, in the sense of being the product of face and vertex amplitudes. It is locally Lorentz invariant at each vertex, in the sense that the vertex amplitude (21) is SL2C invariant: if we choose a different SU 2 subgroup of SL2C (in physical terms, if we perform a local Lorentz transformation), the amplitude does not change.

The entire theory is background independent, in the sense that no fixed metric structure is introduced in any step of the definition of the model. The metric emerges only via the expectation value (or the eigenvalues) of the Penrose metric operator.
==endquote==

 

[marcus]

More on the two alternative interpretations of this form of LQG
==quote starting middle of page 2==
There are two related but distinct physical interpretations of the above equations, that can be considered. The first is as a concrete implementation of Misner-Hawking intuitive “sum over geometries”

Z = ∫_{Metrics/Diff} Dg_µν e^{(i/h)S[g_µν]}

(6)​

As we shall see, indeed, the integration variables in (5) have a natural interpretation as 4d geometries (Sect. IV B), and the integrand approximates the exponential of the Einstein-Hilbert action S[g_µν ] in the semiclassical limit (Sect.V). Therefore (5) gives a family of approximations of (6) as the two-complex is refined.

But there is a second interpretation, compatible with the first but more interesting: the transition amplitudes (4), formally obtained sandwiching the sum over geometries (6) between appropriate boundary states, can be interpreted as terms in a generalized perturbative Feynman expansion for the dynamics of quanta of space (Sect. IV A).

In particular, (4) implicitly associates a vertex amplitude (given explicitly below in (21)) to each vertex v: this is the general-covariant analog for GR of the QED vertex amplitude

[single vertex QED Feynm. diagr. here] = e γ^AB_µ δ(p₁+p₂+k).

(7)​

Therefore the transition amplitudes (4) are a general covariant and background independent analog of the Feynman graphs. These remarks about interpretation should become more clear in the last section.

==endquote==

 

[atyy]

What do you think prevents Rovelli from saying that the classical limit is GR? My understanding is that he only gets Regge solutions, with presumably one free parameter, whereas one would hope for the full Einstein-Hilbert action when h approaches zero. But I'm not sure this is the reservation he has in mind.

 

[marcus]

My experience of him is that he is careful and thorough---doesn't assert flatly what he is not doubly sure of---qualifies with reservations as appropriate.
So I would not expect him to make assertive leaps.

And for what purpose? As long as progress towards the goal is clearly being made.

BTW Atyy, it seems to me that Jerzy Lewandowski actually has gone ahead of Rovelli in claiming LQG recovers GR. I would have to check his most recent paper to be sure. Do you recall? It is better when other people declare success.

Have a look at Lewandowski et al Gravity Quantized and see how close you think they come to outright claiming the limit.

http://arxiv.org/abs/1009.2445

Also let's remember that Rovelli's goal is not merely Pure Gravity.

He has always said the goal was a general covariant quantum field theory with matter. At least that is how I remember it as of, like, 2003 in a draft of his book.

You can see Jerzy L. already angling in the direction of matter. He says the way is to proceed gradually, first a massless scalar field, then gradually more complicated matter. It is not time for anybody to blow any trumpets, even if they have, or almost have, pure gravity.

Those are just my personal thoughts about it. I can't tell what these researcher think or guess what will actually happen.

 

[atyy]

My experience of him is that he is careful and thorough---doesn't assert flatly what he is not doubly sure of---qualifies with reservations as appropriate.
So I would not expect him to make assertive leaps.

And for what purpose? As long as progress towards the goal is clearly being made.

BTW Atyy, it seems to me that Jerzy Lewandowski actually has gone ahead of Rovelli in claiming LQG recovers GR. I would have to check his most recent paper to be sure. Do you recall? It is better when other people declare success.

Yes, but what is the reason for the reservation? I have my guess, but he doesn't seem to state it.

No, I don't recall Lewandowksi claiming such a thing - hmmm, maybe you are thinking of http://arxiv.org/abs/1009.2445 ?

 

[marcus]

Yes, but what is the reason for the reservation? I have my guess, but he doesn't seem to state it.

No, I don't recall Lewandowksi claiming such a thing - hmmm, maybe you are thinking of http://arxiv.org/abs/1009.2445 ?

Yes I was thinking of the September paper 1009.2445 called Gravity Quantized.

 

[atyy]

Yes I was thinking of the September paper 1009.2445 called Gravity Quantized.

The abstract here was more easily understandable to me. I think it's the same sort procedure. http://arxiv.org/abs/0711.0119

 

[atyy]

There is some possibility that the N infinity limit is not needed. Ashtekar et al found in a very particular case that "Thus, the physical inner product of the timeless framework and the transition amplitude in the deparameterized framework can each be expressed as a discrete sum without the need of a ‘continuum limit’: A countable number of vertices suffices; the number of volume transitions does not have to become continuously infinite.' http://arxiv.org/abs/1001.5147 This is one of the most confusing things I find.

I see Rovelli and Smerlak are going to address this soon!

 

[marcus]

There is some possibility that the N infinity limit is not needed. Ashtekar et al found in a very particular case that "Thus, the physical inner product of the timeless framework and the transition amplitude in the deparameterized framework can each be expressed as a discrete sum without the need of a ‘continuum limit’: A countable number of vertices suffices; the number of volume transitions does not have to become continuously infinite.' http://arxiv.org/abs/1001.5147 This is one of the most confusing things I find.

I see Rovelli and Smerlak are going to address this soon!

You are talking about reference [68] in the October paper 1010.1939 and the top right corner of page 6 where [68] is cited. This looks like it might be exciting. I will copy some material here so we can both look at it with less risk of referential uncertainty.

 

[marcus]

Atyy let's lay it out (what you mentioned) and have a look. Here's the October "Simple Model" paper http://arxiv.org/abs/1010.1939

Here's the reference to the forthcoming paper that you mentioned
[68] C. Rovelli and M. Smerlak, “Summing over triangulations or refining the triangulation?” To appear.

Here is the passage where the paper is cited. The all-important concept here is the concept of a projective limit. I first encountered this in an upper division math course in pointset topology, taking the limit where the index is not the natural numbers but is a partially ordered set---like subsets ordered by inclusion or like vector subspaces. We were using a Bourbaki book and John Kelley's topology text.

==quote page 6==
A. Physical amplitudes, expansion and divergences

Physical amplitudes.

Consider the subspace of H_Γ where the spins j_l vanish on a subset of links. States in this subspace can be naturally identified with states in H_Γ′ , where Γ′ is the subgraph of Γ where j_f ≠ 0. Hence the family of Hilbert spaces H_Γ has a projective structure and the projective limit
H = lim_Γ→∞ H_Γ is well defined.

H is the full Hilbert space of states of the theory. It describes an infinite number of degrees of freedom.⁵

In the same manner, two-complexes are partially ordered by inclusion: we write C ′ ≤ C if C has a sub-complex isomorphic to C ′ ...
==endquote==

These two-complexes C, analogously to graphs Γ, are purely combinatorial objects (connectivity and adjacency relations described on abstract sets.) He's got a partial order now on two things---both the graph hilbertspaces and the unlabeled spinfoam frameworks (two-complexes, the bare plot-outlines of a story).

Now he's going to explain that you get the same result taking the projective limit (expanding or "refining" the graphs) as you do by summing all the possible foam histories.

==continued quote==
The transition amplitudes Z (h_l ) are defined on H.

These same transition amplitudes can be defined summing over all two-complexes bounded by Γ.


In spite of the apparent difference, these two definitions are equivalent [68], since the reorganization of the sum (26) in terms of the sub-complexes where j_f ≠ 0 gives (27). The sum (27) can be viewed as the analog of the sum over all Feynman graphs in conventional QFT. Thus, the amplitudes (4) are families of approximations to the physical amplitudes (26).
==endquote==

αβγδεζηθικλμνξοπρσςτυφχψωΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√±←↓→↑↔~≈≠≡ ≤≥½∞(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[atyy]

1. "in the classical limit the vertex amplitude goes to the Regge action of large simplices. This indicates that the regime where the expansion is effective is around at space; this is the hypothesis on which the calculations in items 5 and 6 above are based."

So I guess the reservation about relating to GR is not only that one gets Regge instead of EH, but also that it's valid only near flat space.

2. "correspondingly to the fact that the presence of a cosmological constant sets a maximal distance and effectively puts the system in a box".

?

3. "The second source of divergences is given by the limit (26)."

I wonder why he doesn't call the potential divergences here UV divergences. Is it simply because technically there is no metric in the UV, so no UV?

 

[marcus]

So I guess the reservation about relating to GR is not only that one gets Regge instead of EH, but also that it's valid only near flat space.

However, there is nothing in what Rovelli says that suggests this, Atyy. You have to realize the context of "item 5". It is about the graviton calculations done by Rovelli and others starting around 2006.

The concept of graviton is perturbative, primarily meaningful as a small perturbation around flat (or other fixed) geometry. In order to calculate about such things in LQG one must, in practice, constrain or force the theory into an approximately flat sector. This was the challenge. It was done by imposing boundary conditions. And was ultimately successful.

In items 1 thru 6 he in no way suggests that LQG relates properly to GR only in flat case! The hints are that the relationship is general. He says in the flat case too.

If you restrict to the approximately flat case, as in items 5 and its continuation 6, then he says LQG behaves as it should in that flat case---roughly speaking one sees inverse-square fall-off of the graviton propagator---Newton law behavior.
========================

In the passage you quoted he is talking about an expansion. A tool for calculation.
A given expansion will have limits of validity. He says that the given means of calculation happens to be valid around the flat case. That is a different topic---you are quoting from a different section: Section 5A "expansion and divergences".

That is not the section where he discusses the various indications that LQG relates properly to GR. That part came earlier.

If he meant to say that the proper relation to GR was only in the flat case he would certainly have said that , but in fact he didn't.

 

[atyy]

However, there is nothing in what Rovelli says that suggests this, Atyy. You have to realize the context of "item 5". It is about the graviton calculations done by Rovelli and others starting around 2006.

The concept of graviton is perturbative, primarily meaningful in as a small perturbation around flat (or other fixed) geometry. In order to calculate about such things in LQG one must, in practice, constrain or force the theory into an approximately flat sector. This was the challenge. It was done by imposing boundary conditions. And was ultimately successful.

In items 1 thru 6 he in no way suggests that LQG relates properly to GR only in flat case! The hints are that the relationship is general. He says in the flat case too.

If you restrict to the approximately flat case, as in items 5 and its continuation 6, then he says LQG behaves as it should in that flat case---roughly speaking one sees inverse-square fall-off of the graviton propagator---Newton law behavior.

Yes, I see. I mistook section V for item 5.

 

[marcus]

BTW I appreciate your helping me engage with this paper, a lot. Having someone to talk to about it gets me revved up and I read a lot more attentively. Thx!

I'll try to respond to your point #2, if I can. The cosmo constant one, where you said "?"

For sure the cosmo const does introduce a length. Lambda is an inverse area, so you take the sqrt of the reciprocal of the cosmo const and you immediately have a length. As I recall it is around 10-15 billion lightyears, don't remember exactly.

And the cosmo const also causes there to be a cosmic event horizon, which is around 15-16 billion LY.

If there is someone today in a galaxy that far away, we could never send them a message. Even traveling at speed of light it would never get there. And if they waved at us, today, we would never see it even after trillions of years. That's the meaning of the cosmo EH. You may already be quite familiar with it. It exists because of accelerating expansion. Without that, there woud be no EH. It is not the same as the "Hubble radius" which would exist regardless.

We see things today that are much farther than that. The material that emitted the CMB is now about 45 billion LY, so we are seeing stuff that is that far away, but as it was a long time ago and nearer. The cosmo event horizon is a limit on seeing events that happen TODAY.

So there definitely is a length scale associated with Lambda. I don't remember exactly what it is, only approximately.

I haven't figured out what C.R. means by "puts in a box".

EDIT: Probably he means the cosmological event horizon as the box---a maximal distance of things which can affect us. I don't however have a concrete grasp of this as yet.

 

[marcus]

...I am waiting Marcus to open a thread about Rovelli's new paper. It seems to be the best paper of this year, in my opinion. But I want him to give an explanation, to be sure of that.

MTd2 thanks for reminding people of the new Rovelli Smerlak paper! It fills in a detail for the main October paper. You see very clearly how the plan of exposition operates. The two main 2010 papers (at least so far ) are

1004.1780 (which this thread was begun to discuss)
and then 1010.1939 (which repeated the same overview, this time more for mathematicians and physicists outside the Loop community.)

You may remember the concise paragraph in Section IV-A, near the end, which says exactly what the new Rovelli Smerlak say, but in much more detail with all the steps of the argument shown.

The Rovelli Smerlak paper that you call attention to is the expansion of that paragraph at the end of section IV-A in the main October paper 1010.1939.

 

[marcus]

Rovelli's new paper, which MTd2 calls attention to is:

http://arxiv.org/abs/1010.5437
Spinfoams: summing = refining
Carlo Rovelli, Matteo Smerlak
5 pages
(Submitted on 26 Oct 2010)
"In spinfoam quantum gravity, are physical transition amplitudes obtained by summing over foams, or by infinitely refining them? We outline the combinatorial structure of spinfoam models, define their continuum limit, and show that, under general conditions, refining the foams is the same as summing over them. These conditions bear on the cylindrical consistency of the spinfoam amplitudes and on the presence of appropriate combinatorial factors, related to the implementation of diffeomorphisms invariance in the spinfoam sum."

Actually we were already discussing the summing=refining theme earlier in this thread! It was in post #117 and in Atyy's comments leading up to that:

Atyy let's lay it out (what you mentioned) and have a look. Here's the October "Simple Model" paper http://arxiv.org/abs/1010.1939

Here's the reference to the forthcoming paper that you mentioned
[68] C. Rovelli and M. Smerlak, “Summing over triangulations or refining the triangulation?” To appear.

Here is the passage where the paper is cited. The all-important concept here is the concept of a projective limit. I first encountered this in an upper division math course in pointset topology, taking the limit where the index is not the natural numbers but is a partially ordered set---like subsets ordered by inclusion or like vector subspaces. We were using a Bourbaki book and John Kelley's topology text.

==quote page 6==
A. Physical amplitudes, expansion and divergences

Physical amplitudes.

Consider the subspace of H_Γ where the spins j_l vanish on a subset of links. States in this subspace can be naturally identified with states in H_Γ′ , where Γ′ is the subgraph of Γ where j_f ≠ 0. Hence the family of Hilbert spaces H_Γ has a projective structure and the projective limit
H = lim_Γ→∞ H_Γ is well defined.

H is the full Hilbert space of states of the theory. It describes an infinite number of degrees of freedom.⁵

In the same manner, two-complexes are partially ordered by inclusion: we write C ′ ≤ C if C has a sub-complex isomorphic to C ′ ...
==endquote==

These two-complexes C, analogously to graphs Γ, are purely combinatorial objects (connectivity and adjacency relations described on abstract sets.) He's got a partial order now on two things---both the graph hilbertspaces and the unlabeled spinfoam frameworks (two-complexes, the bare plot-outlines of a story).

Now he's going to explain that you get the same result taking the projective limit (expanding or "refining" the graphs) as you do by summing all the possible foam histories.

==continued quote==
The transition amplitudes Z (h_l ) are defined on H.

These same transition amplitudes can be defined summing over all two-complexes bounded by Γ.


In spite of the apparent difference, these two definitions are equivalent [68], since the reorganization of the sum (26) in terms of the sub-complexes where j_f ≠ 0 gives (27). The sum (27) can be viewed as the analog of the sum over all Feynman graphs in conventional QFT. Thus, the amplitudes (4) are families of approximations to the physical amplitudes (26).
==endquote==

It's clear that reference [68], "to appear" is the paper that MTd2 would like us to discuss, but with an earlier title.

I see I was careless in my language here. Refining graphs is analogous to refining 2-complexes (which are graphs-analogs in one higher dimension) but not the same. Rovelli is talking about refining the 2-complexes having the same end result as summing over the 2-complexes.

Let's have a look at the 1010.5437 paper "summing=refining". It does look kind of abruptly illuminating---getting things into focus for us. MTd2, you could be right in your high estimation of it! It makes one realize clearly that a spin network is not a lattice. It is not an approximation as the nonexistent "lattice spacing goes to zero". It IS itself A GEOMETRY, but a geometry with finite complexity. Refinement, then, does not mean to make more continuous, but to allow more geometrical complexity.

I'll print out 1010.5437 and have a closer look, MTd2, as you suggested.

 

[MTd2]

What impressed me it is something very startling, spin foams is now almost kind of CDT! Think about the consequences.

 

[atyy]

Yes, I was waiting for this paper. But now I am completely confused. I had thought that refining was like "renormalization" (group field theory?). But according to Rovelli and Smerlak, refining is like summing, at least under the adjustments in this paper. However, I think some sort of "renormalization" is still needed, since the physical inner product is divergent - but apparently this "renormalization" this is not equivalent to refinement - in which case, what is it?

 

[marcus]

Yes, I was waiting for this paper. But now I am completely confused. I had thought that refining was like "renormalization" (group field theory?). But according to Rovelli and Smerlak, refining is like summing, at least under the adjustments in this paper. However, I think some sort of "renormalization" is still needed, since the physical inner product is divergent - but apparently this "renormalization" this is not equivalent to refinement - in which case, what is it?

Let's think concretely and pictorially about it. 2-complexes (foams) are just the 2D analogs of graphs. So let us think about graphs instead. And try to understand the words in the sense in which Rovelli means them.

To refine a graph means to add more nodes and links. One can refine a graph by various "moves". Break a link and add a node at the break. Add a link connecting two nodes that were not connected before. Take a node where 3 links come together and replace it by a triangle in the obvious way (by 3 new nodes and 3 new links)...and so on.

I am oversimplifying. In the real case we are talking about foams (2-complexes) and there are specified legal moves used to refine them. But I just want to give the rough idea of what refining means. Basically adding nodes: If N is the number of nodes, you can refine so that N --> infinity.

And then there is summing. Where you don't change any graph---you sum over all the graphs. Add up the amplitude for each graph to get the total amplitude (or whatever number is to be computed.)

In one case it's like the limit of a sequence---in the other it's like the sum of a series.
===================

The upshot, unless I'm mistaken, is that refining is not "like renormalization"---I don't quite understand your conception of it. And also refining and summing are different. But you say "refining is like summing". I think the point is not that they are alike, but that the two different procedures give the same answer, give the same amplitude as a result.

Who says "the physical inner product is divergent"? This is a non-perturbative approach. Who says that something like "renormalization" has to play a role here? I thought renormalization is something that comes up in conventional perturbation theory when there are infinities. Lqg is not plagued by infinities AFAIK Finiteness was explained, for instance, in Rovelli's talk to Strings 2008.

 

[atyy]

I see he calls the the potentially divergent radiative corrections at fixed N "renormalization", and the potentially divergent N -> infinity the "continuum limit", whereas I had considered both to be renormalization. OK, let me use his language, since I agree with the substance.

Renormalization:
http://arxiv.org/abs/1010.1939 "radiative corrections renormalize the vertex amplitude."
http://arxiv.org/abs/0810.1714 "Self-energy and vertex radiative corrections in LQG ... At fixed N, the partition function of the theory is given by a sum over spins and intertwiners, which can be interpreted as a version of the Misner-Hawking 'sum over geometries' ... This sum may contain divergent terms. Here we study these terms."

Continuum limit:
http://arxiv.org/abs/0810.1714 "The theory is first cut-off by choosing a 4d triangulation N of spacetime, formed by N 4-simplices; then the continuous theory can be defined by the N -> infinity limit of the expectation values."
http://arxiv.org/abs/1010.5437 "III. THE SPINFOAM CONTINUUM LIMIT ... infinitely refining C, and summing over C. Since the set of foams is discrete, the latter option is easy to define in principle, at least if one disregards convergence issues. But what about the former? It can too ..."

 

[marcus]

From what you said earlier I gathered you were talking about IR divergences---not UV, but those that might arise over large distances. So I'll respond to that. That's an area where there is still stuff to work on! Earlier I didn't realize you meant InfraRed. The status is summarized in the October paper 1010.1939, page 6, section 5A
==quote==

Divergences.

There are no ultraviolet divergences, be-
cause there are no trans-Planckian degrees of freedom.
However, there are potential large-volume divergences,
coming from the sum over j . In ordinary Feynman
graphs, momentum conservation at the vertices implies
that the divergences are associated to closed loops. Here
S U 2 invariance at the edges implies that divergences are
associated to “bubbles”, namely subsets of faces forming
a compact surface without boundary [20, 72–75]. Such
large-volume divergences are well known in Regge calcu-
lus, and can be visualized as “spikes” of the 4-geometry.
Spikes are likely to be effectively regulated by going to
the quantum group. It is commonly understood that the
q-deformation amounts to the inclusion of a cosmological
constant. This is consistent with the fact that q-deformed
amplitudes are suppressed for large spins, correspond-
ingly to the fact that the presence of a cosmological con-
stant sets a maximal distance and effectively “puts the
system in a box”. Whether divergent or not, radiative
corrections renormalize the vertex amplitude...
==endquote==

 

[atyy]

Oops I edited my post #127 while you replied. As a matter of fact, the quote of mine you quoted is factually wrong, I have corrected it.

The reason I was calling Rovelli's "continuum limit" renormalization is that I was thinking in terms of what Rivasseau calls "group field theory renormalization". Whereas Rovelli's renormalization is what one might call "spin foam renormalization".

http://arxiv.org/abs/0906.5477
"There are now interesting such spin-foam models [30, 31, 32, 33], hereafter called EPR-FK models, which in four dimension reproduce Regge gravity in a certain semiclassical limit [34, 35]. There are also some glimpses that they might be just renormalizable [36]. These spin-foam models, however, capture only a finite subset of the gravitational degrees of freedom, and the question arises of the existence of a ‘continuum limit’. As the spin-foam amplitude can always be interpreted as a Feynman amplitude of a suitable GFT [37], this question boils down to the problem of renormalization in GFT ..."

 

[marcus]

Oops I edited my post #127 while you replied. As a matter of fact, the quote of mine you quoted is factually wrong, I have corrected it.
...

If OK with you I will just eliminate the quote which you say had an error. Then my post won't seem to make as much sense because the quote that I was responding to won't be there, but that's OK.

 

[atyy]

If OK with you I will just eliminate the quote which you say had an error. Then my post won't seem to make as much sense because the quote that I was responding to won't be there, but that's OK.

Sure, of course - but you can keep it too, since I did write that. I would have put my correction in a later post had I known you were replying already.

 

[marcus]

The reason I was calling Rovelli's "continuum limit" renormalization is that I was thinking in terms of what Rivasseau calls "group field theory renormalization". Whereas Rovelli's renormalization is what one might call "spin foam renormalization".
...

It's a partial analogy with what we know and think of as renormalization. Instead of the "cutoff" being a quantity like a length or an energy, the "cutoff" is a type of complexity.

But it is not like a perturbation series where there are infinities staring you in the face that must be eliminated by actually fiddling with the couplings (the original meaning of "renormalize"). There are no infinities here waiting to bite you when you turn around.

It is just a way of organizing a calculation. Nature always has infinitely many terms to add up and we can only add up a finite number of them. So we add up the N simplest terms and approximate.
Rovelli begins a relevant section by reminding us "There is no physics without approximation." So you add up as many terms as you have time for. There is no fiddling.

If I remember, that was also what Rivasseau was doing. Grading by complexity, and truncating. Simply in order to make a calculation feasible in a finite number of arithmetical steps.

There is a big difference in principle between a calculation which is inherently convergent, so you can do a finite number of terms and it automatically approximates the answer. Versus a calculation which actually blows up and tries to give you infinite quantities, so you have to fiddle.

So I would not like to call what Rovelli is doing with spinfoams "renormalization". It is too likely to mislead people who know the usual meaning of the word. It is more like just taking the approximate limit of a sequence numerically, or numerically summing an ordinary calculus series.

I would call it "complexity graded approximation"---or "simplest-first convergent approximation".

(and so far it is just in UV, as you have reminded me, still plenty to do IR-wise!)

 

[atyy]

No, I don't think so. There are potential divergences that need to be taken care of.

There are the IR divergences which Rovelli talks about in terms of renormalization or radiative corrections - I'll call this spin foam renormalization.

Then there are N -> infinity, continuum limit divergences (I think of these as UV divergences in spirit, although as Rovelli says, technically speaking there are no UV divergences - however, he does agree that the continuum limit divergence potentially exists), which I understand to be related to group field theory renormalization.

BTW, renormalization is not fiddling - historically that's what it was - but not after Wilson. I think the next question is whether the GFT that yields EPRL/FK (or any GFT) is asymptotically safe.

 

[marcus]

Now I was the one who didn't see yours---had to get up for phonecall with the last sentence not done yet :-D
The renormalization group flow does the fiddling for you automatically. So don't call it fiddling if you like (but the coupling constants still run. :-D)

BTW in hardly more than a week Bianchi will give a talk at Perimeter. Do you recall the date? I'll look it up. He has been an important contributor to the formulation presented in the April and October papers.

Also I was impressed by the 60-page "tutorial" on gauge gravity that Randono posted today.

 

[atyy]

I didn't know Bianchi was giving a talk, look forward to seeing it when it's posted.

Yes, I'll have to read the Randono stuff.

So Matt Smerlak's on the GFT renormalization paper and the N -> infinity, continuum limit paper. It's clear from eg. his GFT renormalization paper or Perini et al's earlier work on radiative corrections that the N -> infinity limit was already regarded intuitively as a continuum limit, and this new paper makes the intuition rigourous. (But at odds with Kaminiski et al's method of defining a continuum limit?)

The N -> infinity limit is potentially divergent, and what does the final sentence of the GFT rernormalization paper http://arxiv.org/abs/0906.5477 say?

"Our method shows how the 'sum over triangulations' in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory."

 

[marcus]

Bianchi's talk will be at this URL:
http://pirsa.org/10110052/
He's scheduled to speak on 9 November. ( I thought earlier but just checked.)

"...and paves the way for the renormalization program in group field theory."
What could they mean by that? When matter is included? IR limit?
Some imprecision of language, perhaps. I won't argue about what Rivasseau means, just wait and see what actually happens.

 

[atyy]

Bianchi's talk will be at this URL:
http://pirsa.org/10110052/
He's scheduled to speak on 9 November. ( I thought earlier but just checked.)

"...and paves the way for the renormalization program in group field theory."
What could they mean by that? When matter is included? IR limit?
Some imprecision of language, perhaps. I won't argue about what Rivasseau means, just wait and see what actually happens.

My interpretation is that no one knows how to renormalize *any* group field theory. If GFT renormalization is worked out, then it will become a class of quantum theories (just as QFTs are another class), perhaps one of which will be a theory of pure gravity, or gravity+matter or gravity unified with matter.

 

[atyy]

wrt http://arxiv.org/abs/1010.5437

There are interesting comments in the final part of this paper about compatibility of the proposed modifications to EPRL and GFT http://arxiv.org/abs/1010.5227

Also interesting is http://arxiv.org/abs/1010.4787

All talk about some sort of cylindrical consistency requirement.

I think it will be interesting to find out if GFT or canonical LQG is fundamental. I think the former would push towards unification, the latter maybe would link up with Asymptotic Safety of pure gravity.

 

[marcus]

Rovelli's April 2010 "New Look" paper was the original basis for this thread. It has now been rewritten, expanded by over 50%, and given a new title: "Lectures on Loop Gravity".

The immediate purpose of "Lectures" is to go along with the 8 hours of lecture Rovelli is scheduled to give at Zakopane March 1 thru 6, this year. The style is in large part pedagogical explaining things that a journal article might assume the reader knows. If "Lectures" continues to be expanded and improved, it could turn into a set of notes on the new formulation of Loop Gravity that could serve as an entry-level textbook.

Interestingly enough, the list of 17 open problems which researchers are invited to tackle is word-for-word unchanged except that it is noted in several cases where progress has occurred, or where a problem has been solved.

The new version is
http://arxiv.org/abs/1102.3660
Lectures on loop gravity
Carlo Rovelli
24 pages 10 figures
(Submitted on 17 Feb 2011)
"This is the first version of the introductory lectures on loop quantum gravity that I will give at the quantum gravity school in Zakopane. The theory is presented in self-contained form, without emphasis on its derivation from classical general relativity. Dynamics is given in the covariant form. The approximations needed to compute physical quantities are discussed. Some applications are described, including the recent derivation of de Sitter cosmology from full quantum gravity."

You may wish to compare the list of open problems given here, in 1102.3660, with the list at the end of the April 2010 paper http://arxiv.org/abs/1004.1780 . It gives an idea of the rate of progress.

 

[marcus]

I think the "Lectures" draft though still rough is a significant advance pedagogically, and we may be able to learn something from its organization. Why? Because Loop Gravity has been transformed in the past two years and has reached a new stage of development--and because some thought has gone into presenting the new formulation of the theory. The field is attracting interest and taking in new researchers---so a problem arises: how do you assimilate the new members into the community, bring them up to speed, and get them started on research in the most efficient way.

I'm going to copy the outline of the various sections of Lectures on L.G. so we can examine and try to understand the thinking behind the way the lectures are organized. It can also give us an idea of what the essential prerequisites are for an entry-level understanding of the subject.

I-Overview, general motivation, where are we in quantum gravity?

II-States and Operators

A-Elementary math: SU(2)
B-Elementary math: Graphs
C-Hilbert spaces
D-Operators
E-Spin network basis
F-Physical picture (this is one of the best sections IMHO)
G-Planck scale
H-Boundary states
​

III-Transition Amplitudes

A-Elementary math: SL(2,C)
B-Elementary math: 2-complexes
C-Transition amplitudes
D-Properties and comments

1. superposition principle
2. locality
3. local Lorentz invariance​

IV-Derivations

A-Dynamics
B-Kinematics
C-Covariant lattice quantization
D-Polyhedral quantum geometry
​

V-Extracting Physics

A-Coherent states and holomorphic representation
B-The euclidean theory
C-Expansions

1. graph expansion
2. vertex expansion
3. large distance expansion​

D-What has already been completed

1. n-point functions
2. cosmology​

VI-Conclusion

Appendix A: Open Problems (1 - 17)

Appendix B: Alternative Forms of the Amplitudes

1. Single equation
2. Feynman rules
3. Using Y explicitly
4. Spin-intertwiner basis
5. Other variants​