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

[sheaf]

If the boundary state is conceived as the boundary of 4D spacetime, is this still manifoldless?

I thought by "manifoldless" he meant that the spin networks were no longer thought of as being embedded in a three-space as they were originally so-conceived to "feel out" the three-geometry.

The "boundary state" is then some superposition of spin network states so is a quantum object. Manifolds (if we mean smooth manifolds) then only arise when we do the semi classical coherent state extraction process.

I think.

ETA Marcus beat me to it !

 

[marcus]

I like your answer, Sheaf. It's concise and quite possibly more helpful to Atyy.

 

[atyy]

Marcus and Sheaf - I'll buy that - technically. What I feel uneasy with is that can you really start from the "new" view which is not that new. In the "old" spin foam view, one started with a discretiztion of a manifold - and in that sense the smooth manifold disappeared right away. So is the new view really new? And isn't where the discrete manifold view where the theory came from still shown up in that the semi-classical limit only gets some bit of the Regge action, not the Einstein-Hilbert action?

BTW, what happened to Kaminski et al, is Rovelli not buying their manifold?

 

[marcus]

...
BTW, what happened to Kaminski et al, is Rovelli not buying their manifold?

On the contrary, Rovelli is highlighting that "Kaminski et al" paper in both of his key papers this year. In both the March 1003.3483 and the April 1004.1780 papers he makes it clear that the result in that paper is one of the three recent advances that his new presentation of LQG rests on.

"Kaminski et al" main author is Lewandowski so I think of it as Lewandowski et al. It does not force us to use manifolds. Instead, it serves as a bridge between the new LQG way and the earlier development that in fact did use manifolds.

So Rovelli makes a point of using Lewandowski's 2009 form of the spinfoam vertex, in his manifoldless presentation. It appeared at just the right time, so to speak.

If anybody is unfamiliar with the recent literature, the Lewandowski paper is
"Spinfoams for all LQG"
Earlier spinfoam vertex formulas were hampered by some restrictive assumptions and did not thoroughly connect with the old canonical LQG which Lewandowski in collaboration with Ashtekar contributed significantly to developing. He was the natural person to make the connection and assure continuity. I will get the link
http://arxiv.org/abs/0909.0939

To put 0909.0939 in perspective, here is what Bianchi Rovelli Vidotto say about it in the March paper:

==quote "Towards Spinfoam Cosmology" 1003.3483==
The dynamics of loop quantum gravity (LQG) can be given in covariant form by using the spinfoam formalism. In this paper we apply this formalism to cosmology. In other words, we introduce a spinfoam formulation of quantum cosmology, or a “spinfoam cosmology”.

We obtain two results. The first is that physical transition amplitudes can be computed, in an appropriate expansion. We compute explicitly the transition amplitude between homogeneous isotropic coherent states, at first order.

The second and main result is that this amplitude is in the kernel of an operator C, and the classical limit of C turns out to be precisely the Hamiltonian constraint of
the Friedmann dynamics of homogeneous isotropic cosmology. In other words, we show that LQG yields the Friedmann equation in a suitable limit.

LQG has seen momentous developments in the last few years. We make use of several of these developments here, combining them together. The first ingredient we utilize is the “new” spinfoam vertex[1–5].

The second is the Kaminski-Kisielowski-Lewandowski extension of this to vertices of arbitrary-valence[6].

The third ingredient is the coherent state technology[7–20], and in particular the holomorphic coherent states discussed in detail in [21]. These states define a holomorphic representation of LQG[8, 22], and we work here in this representation.
==endquote==

 

[atyy]

KKL and a Bahr paper that studies KKL mentions embedding in a 4D manifold all over:
http://arxiv.org/abs/0909.0939
http://arxiv.org/abs/0912.0540
http://arxiv.org/abs/1006.0700

 

[marcus]

Benjamin Bahr's paper is interesting
http://arxiv.org/abs/1006.0700

I wouldn't say that the mention of manifolds was interesting, since that's been par for the course for most of the past 15 years----typical of LQG from say 1994 to 2009. Typical treatment embedded graphs in manifolds.

Now that the new formulation is getting away from embedding graphs in manifolds, you can expect to see papers like Bahr's supporting the idea that it doesn't make much, if any, essential difference.

That, for example, spin-network knots that might have happened in the embedded case (but not now) do not matter, or get undone, or are not involved in the physical Hilbert space.

You might like to take a look at the Bahr paper. That is one of the main results. The absence of spin-network knot classes in the physical hilbert.

This makes it more likely that embedding would make any physical difference in the theory, which was my intuitive take (that embedding is irrelevant or undesirable) but it is nice to see a result proven along those lines.

Not sure what you point is, with those particular links, but thanks in any case!

 

[atyy]

Benjamin Bahr's paper is interesting
http://arxiv.org/abs/1006.0700

I think so too.

This makes it more likely that embedding would make any physical difference in the theory, which was my intuitive take (that embedding is irrelevant or undesirable) but it is nice to see a result proven along those lines.

Counter-intuitively, he says in his discussion "Although the physical Hilbert space does not contain any knotting information of the graphs, it should be emphasized that this does not mean that the theory is insensitive to knotting within the space-time four-manifold M = Sigma × [0, 1]!"

 

[marcus]

That will probably be a separate issue for a separate paper.

 

[marcus]

Sheaf offered an interesting thought in another thread that relates to section E of the April paper---about holomorphic coherent states in LQG, where the spin-network states can be labeled with elements of SL(2,C) rather than with SU(2) irreps.

Interesting discussion.

I wonder if you started with an G - spin network, where G is some bigger group having SU(2) as a subgroup, then performed the semiclassical coherent state approximation technique referred to in the New Look paper, what dimensionality of manifold you would end up with...

This of course is assuming you could define such a spin network consistently.

I want to think about that some, and maybe eventually comment. But will do it here so as not to get off-topic in the other thread.

 

[marcus]

The way I see it, what increasingly stands out is that the spin-network is the natural/correct way to represent states of geometry.
But then the question immediately arises how to think of a spin-network?.

And the answer that comes to mind is that a spin-network is nothing else than specific type of numerical-valued function defined on a group manifold.

It is a certain kind of device for getting ordinary complex numbers from "tuples" of SU(2) group elements. And the graph places a symmetry restriction on those functions from the group manifold.

As I recall, when you look at the coherent states discussed by Bianchi Magliaro Perini, they have generalized the LABELS to be elements of SL(2,C). But their state is still a function defined on "tuples" of SU(2).
=====================
So *bang* I'm stuck. People seem interested in how this might be generalized. Do you generalize the group manifold, to be tuples of some larger G? Or do you generalize the labels (as in the BMP case)? I draw a blank. My reaction is not satisfactory, for now at least.
=====================
So for now I will merely back up and say why a spin-network should be thought of as a function from the L-fold cartesian product SU(2)^L to the complex numbers. We've talked about it before, but it won't hurt to try to say it better.

 

[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.