Signs LQG has the right redefinition (or wrong?)

[marcus]

Another sign: LQG defined this way turns out to be a generalized topological quantum field theory (TQFT).

==quote page 2 section III "TQFT on manifolds with defects" ==
...
If C is a two-complex bounded by the (possibly disconnected) graph Γ then (4) defines a state in H_Γ which satisfies the TQFT composition axioms [27]. Thus the model formulated above defines a generalized TQFT in the sense of Atiyah.
==endquote==

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

 

[marcus]

Continuing to hit the key points of http://arxiv.org/abs/1010.1939
The hilbertspace H_Γ of LQG is essentially squareintegrable complexvalued functions on the L-fold cartesian product SU(2)^L.
Now a generic L-tuple of SU(2) elements is what I was writing h. And the equation (4) defines a function Z_C of h.

The spin networks form a basis for the quantum states H_Γ. To have sufficient understanding of the subject matter, I should be able to write any spin network also as a function of h. See equation (15) on page 3 of the paper. I'll try typing what a spin network
{Γ, j_l, i_n: l=1,...,L and n=1,...,N}
looks like as a complexvalued function of h

Here it is (following equation 15)

⟨⊗_ld_jlD^j_l(h_l) | ⊗_ni_n⟩_Γ

"where D^j_l (h_l) is the Wigner matrix in the spin-j representation and ⟨·|·⟩_Γ indicates the pattern of index contraction between the indices of the matrix elements and those of the intertwiners given by the structure of the graph. A G-intertwiner, where G is a Lie group, is an element of a (fixed) basis of the G-invariant subspace of the tensor product ⊗_lH_jl of irreducible G-representations —here those associated to the links l bounded by n. Since the Area is the SU2 Casimir, the spin j_l is easily recognized as the Area quantum number and in is the Volume quantum number."

αβγδεζηθικλμνξοπρσςτυφχψωΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√±←↓→↑↔ ~≈≠≡ ≤≥½∞ ⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅∈ ⊗ ⊕
⊂ ⟨·|·⟩

 

[marcus]

I've listed ten* indications that the current LQG formulation is the right one. No one seems able to provide countervailing evidence.

I also get the impression that the LQG research community has swung over to the new version, or if not entirely yet is not putting up much resistance. (e.g. look at the makeup of the QG school that starts one month from now at Zakopane.)

https://www.physicsforums.com/showthread.php?p=3110549#post3110549

*see posts #70 and #71

=============================
Hi Atyy, thanks for your opinion!

The indication of a de Sitter universe is just that, an indication. Physicists are always doing calculations to first order approx and then gradually improving the accuracy. It's great they got deSitter at first order. The day is young on that one.

I don't see how you can say "probably" divergent. Are you such a great expert that you can put probability measures on the future of research. The arguments in the literature are that the theory is NOT UV divergent. As Tom has said, the prospect of IR divergence doesn't worry him much. It's a common ailment that other theories have learned to live with.

It's not a high priority to address the IR divergence issue, I think. But ways to fix that have been proposed as well. Someone will get around to studying that eventually.

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

Meanwhile, Atyy, doesn't it seem as if the string community is casting around for 4D/nonstring alternatives?

Horava's 4D skew gravity
Verlinde's kinky polymer vision of entropic gravity
Nima's quantum polytopes (his Pirsa talk was about scattering but he hinted at work on gravity in progress)

It wouldn't surprise me if Nima comes up with something on quantum polytope geometry/gravity that is 4D, non-supersymmetric, and looks like a cousin of Rovelli and Rivasseau reformulation of LQG GFT, where quantum polytopes have been coming up frequently as well!
==================

Careful, your information is out of date. There has been an abrupt increase of interest, research activity, and number of researchers just in the past 3 years. Also the formulation has changed radically. You may not know what is going on because you are interested in your own ideas and wish to dismiss the QG realworld.
==================

Atyy, that's interesting! What is the "X" divergence (your name for it). I need a page and paragraph reference so I can see what you are quoting of Rovelli in context. Eyes get tired scanning over page after page looking for quotes. Point me to it and I will be glad to look!

 

[atyy]

It is based on probably divergent series, and the indication of a de Sitter universe removes the higher order terms by ignoring them.

 

[Careful]

I've listed ten* indications that the current LQG formulation is the right one. No one seems able to provide countervailing evidence.

I think it is more accurate to say that nobody really cares anymore after 25 years.

 

[atyy]

I say probably divergent because Rovelli says so.

There are 3 sorts of divergences in Rovelli's classification.

1) UV - not present
2) IR - present but not a problem
3) X (my nomenclature) - probably present, and probably a problem.

 

[marcus]

I say probably divergent because Rovelli says so.

3) X (my nomenclature) - probably present, and probably a problem.

I asked for a page reference in my initial response https://www.physicsforums.com/showpost.php?p=3111122&postcount=73 to this post, and you have not offered one.
I assume this is because you cannot find anywhere that Rovelli says "probably present and probably a problem" about some kind of divergence.

So far, if we cannot get a handle on it and discuss it, this "X" is just a mystifying "Atyyism"
Please give some concrete substance to your comment!

 

[atyy]

I asked for a page reference in my initial response https://www.physicsforums.com/showpost.php?p=3111122&postcount=73 to this post, and you have not offered one.
I assume this is because you cannot find anywhere that Rovelli says "probably present and probably a problem" about some kind of divergence.

So far, if we cannot get a handle on it and discuss it, this "X" is just a mystifying "Atyyism"
Please give some concrete substance to your comment!

Please quote the page request explicitly.

 

[marcus]

I've listed ten* indications that the current LQG formulation is the right one. No one seems able to provide countervailing evidence.

I also get the impression that the LQG research community has swung over to the new version, or if not entirely yet is not putting up much resistance. (e.g. look at the makeup of the QG school that starts one month from now at Zakopane.)

https://www.physicsforums.com/showthread.php?p=3110549#post3110549

*see posts #70 and #71

I say probably divergent because Rovelli says so.

There are 3 sorts of divergences in Rovelli's classification.

1) UV - not present
2) IR - present but not a problem
3) X (my nomenclature) - probably present, and probably a problem.

Atyy, that's interesting! What is the "X" divergence (your name for it). I need a page and paragraph reference so I can see what you are quoting of Rovelli in context. Eyes get tired scanning over page after page looking for quotes. Point me to it and I will be glad to look!

I asked for a page reference in my initial response https://www.physicsforums.com/showpost.php?p=3111122&postcount=73 to this post, and you have not offered one.
I assume this is because you cannot find anywhere that Rovelli says "probably present and probably a problem" about some kind of divergence.

So far, if we cannot get a handle on it and discuss it, this "X" is just a mystifying "Atyyism"
Please give some concrete substance to your comment!

Please quote the page request explicitly.

OK, done. I can't tell whether you are just playing games or whether you are really confused about a type of very large-scale (cosmological) divergence that R. mentioned.

If I knew exactly what you meant by "X" divergence, maybe I could help clarify.

 

[atyy]

The request appears to be after my post mentioning X, not before.

 

[marcus]

The request appears to be after my post mentioning X, not before.

I've asked you for page refs several times. It's an ongoing problem. Not giving pointer can (in some people) be associated with inaccurate paraphrase or quotes out of context that seem to mean something else. You must surely be aware of this. In this case I did ask for specific pointer AFTER your comment about "X" divergence.

Lets not quibble over trivia. I'm interested to know what you think is this X that Rovelli says "probably divergent and probably a problem" about. Or if he actually did not say that then what is this X that YOU think is probable and probably a problem?

I'm interested to know! It could be a type of divergence which might arise if you include the whole universe (with no cosmological event horizon) in the analysis. So if the universe is infinite you get bigger and bigger spinnetworks, growing in size without limit. That would be interesting to discuss and to think of how it might be handled. But since you don't say what you mean by "X" I am unable to be sure what you think is a problem!

 

[atyy]

I've asked you for page refs several times. It's an ongoing problem. Not giving pointer can (in some people) be associated with inaccurate paraphrase or quotes out of context that seem to mean something else. You must surely be aware of this. In this case I did ask for specific pointer AFTER your comment about "X" divergence.

Good. And it appeared in a post preceding my mention of X. That's ok. But in that case, if I don't provide the page reference, it's because I haven't seen it, not because it doesn't exist.

http://arxiv.org/abs/1010.1939 p6

UV "There are no ultraviolet divergences, be cause there are no trans-Planckian degrees of freedom.

IR "However, there are potential large-volume divergences, coming from the sum over j"

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

 

[marcus]

To keep on track, since we have a new page, I will copy the "business part" of my last substantive post.
==quote==
As I see it, the QG goal is to replace the live dynamic manifold geometry of GR with a quantum field you can put matter on. The title of Dan Oriti's QG anthology said "towards a new understanding of space time and matter" That is one way of saying what the QG researchers's goal is. A new understanding of space and time, and maybe laying out matter on a new representation of space and time will reveal a new way to understand matter (no longer fields on a fixed geometry).

Sources on the 2010 redefinition of LQG are
introductory overview: http://arxiv.org/abs/1012.4707
concise rigorous formulation: http://arxiv.org/abs/1010.1939
phenomenology (testability): http://arxiv.org/abs/1011.1811
adding matter: http://arxiv.org/abs/1012.4719

Among alternative QGs, the LQG stands out for several reasons---some I already indicated---which I think are signs that the 2010 reformulation will prove a good one:

• testable (phenomenologists like Aurelien Barrau and Wen Zhao seem to think it is falsifiable)
  
• analytical (you can state LQG in a few equations, or Feynman rules, you can calculate and prove symbolically, massive numerical simulations are possible but not required)
  
• similar to QED and lattice GCD (the cited papers show remarkable similarities---the two-complex works both as a Feynman diagram and as a lattice)
  
• looks increasingly like a reasonable way to set up a background independent quantum field theory.
  
• an explicitly Lorentz covariant version of LQG has been exhibited
  
• matter added
  
• a couple of different ways to include the cosmological constant
  
• indications that you recover the classic deSitter universe.
  
• LQG defined this way turns out to be a generalized topological quantum field theory (see TQFT axioms introduced by Atiyah).
  
• sudden speed-up in the rate of progress, more researchers, more papers

These are just signs---the 2010 reformulation might be right---or to put it differently, there may be good reason for us to understand the theory, as presented in brief by the October paper http://arxiv.org/abs/1010.1939.
...
...
[To expand on the point that in 1010.1939 form] it "looks like" QED and QCD, except that it is background independent and about geometry, instead of being about particles of matter living in fixed background. Somehow it manages to look like earlier field theories. The presentation on the first page uses "Feynman rules".

These Feynman rules focus on an amplitude Z_C(h)
where C is a two-complex with L boundary or "surface" edges, and h is a generic element of SU(2) and h is (h₁, h₂,...,h_L), namely a generic element of SU(2)^L

The two-complex C is the "diagram". The boundary edges are the "input and output" of the diagram---think of the boundary as consisting of two separate (initial and final) components so that Z becomes a transition amplitude. ...

The central quantity in the theory is the complex number Z_C(h) and one can think of that number as saying a quantum probability, a transition amplitude:

Z_roadmap(boundary conditions)

==endquote==
==quote http://arxiv.org/abs/1010.1939 page 2 section III "TQFT on manifolds with defects" ==
...
If C is a two-complex bounded by the (possibly disconnected) graph Γ then (4) defines a state in H_Γ which satisfies the TQFT composition axioms [27]. Thus the model formulated above defines a generalized TQFT in the sense of Atiyah.
==endquote==

αβγδεζηθικλμνξοπρσςτυφχψωΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√±←↓→↑↔~≈≠≡≤≥½∞ ⇐⇑⇒⇓⇔∃ℝℤℕℂ∈⊗⊕⊂ ⟨·|·⟩

 

[marcus]

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

That problem goes away if the universe you are modeling has a finite size.
Would you like to have that explained?

 

[atyy]

That problem goes away if the universe you are modeling has a finite size.
Would you like to have that explained?

Sure.

Rovelli says that for the IR divergence, but not for X.

IR "This is consistent with the fact that q-deformed amplitudes are suppressed for large spins, correspondingly to the fact that the presence of a cosmological constant sets a maximal distance and effectively puts the system in a box"."

X "Less is known in this regard, but it is tempting to conjecture that this sum could be regularized by the quantum deformation as well."

 

[marcus]

That problem goes away if the universe you are modeling has a finite size.
Would you like to have that explained?

Sure.

we don't have to speculate about "quantum deformation". Sure R. mentioned it and it is interesting to think how it might affect the picture. But (26) is already not a problem if the U simply has finite size.

That is because LQG has a UV cutoff, effectively. It has a limit how fine resolution, how small you can measure. The "cell size" does not shrink below some scale.

(26) is about considering larger and larger foams, ordered by inclusion. U finite implies that process must terminate. So limit exists. That's all I was saying.

 

[atyy]

we don't have to speculate about "quantum deformation". Sure R. mentioned it and it is interesting to think how it might affect the picture. But (26) is already not a problem if the U simply has finite size.

That is because LQG has a UV cutoff, effectively. It has a limit how fine resolution, how small you can measure. The "cell size" does not shrink below some scale.

(26) is about considering larger and larger foams, ordered by inclusion. U finite implies that process must terminate. So limit exists. That's all I was saying.

Then how can "summing = refining"?

http://arxiv.org/abs/1010.5437

 

[marcus]

Then how can "summing = refining"?

http://arxiv.org/abs/1010.5437

Please say explicitly what you think the problem with that is.

You may be confused by the words. "Refining" here does not have a metric scale connotation. All it can mean is to add more cells to the complex.

You have to look directly at the math. What the objects are and how the limits are defined.
You can't just go impressionistically/vaguely by the words. I don't know what your source of confusion is, can only guess---unless you spell out what you are thinking.

But I know that there is no inconsistency between the two types of limit, as defined.
On the one hand summing over cell-complexes and on the other hand taking a cell complex and adding more and more cells to it.

Really it's fine!

 

[atyy]

I'm taking issue with your interpretation that summing = size of the universe.

So a bigger and bigger universe means more and more refining?

The basic result in the summing=refining paper is "We have observed that under certain general conditions, if this limit exist, it can equally be expressed as the sum over foams, by simply restricting the amplitudes to those with nontrivial spins."

Are you saying this limit exists in a finite universe?

 

[marcus]

I'm taking issue with your interpretation that summing = size of the universe.

So a bigger and bigger universe means more and more refining?

Forget the words Atyy, look at the actual math which is the meaning of the "s=r"
paper.

In what I said the U has a finite size. So don't be talking about bigger bigger U.
The U has some size. Say roughly hypersphere w radius of curvature 100 Gly. (a NASA WMAP lower bound estimate from around 2007 as I recall)

Say you start with a dipole spin network like this ([]) labeled to agree with that 100 Gly
(you've surely seen that dipole graph before in R papers, better drawn)
and you start refining. That means adding nodes and links

for the the next twenty gazillion years adding complexity to the graph DOES in fact correspond to the intuitive idea of refining.

But then the process has to terminate, because you got down to where every node has the min vol and every link has the min area.

You run into the finite resolution barrier. smaller is meaningless.

Better to actually look at what the math says than take issue with the words.

Could you be being a wee bit suspicious? and thinking everybody is trying to fool you because you don't understand something? Take it easy. That X is a nonproblem, pragmatically speaking.

 

[marcus]

Are you saying this limit exists in a finite universe?

Abstract math does not work in some given universe. The limit is an interesting abstract question.
Pragmatically, sure. Pragmatically it is a non-problem. In that case.

 

[atyy]

So we fix the boundary. As is done in the summing=refining paper. Your argument is that for fixed boundary, the summing is finite. Since refining is summing, then refining is finite. I don't see that. I think it does mean that summing is a sum over discrete terms, but not necessarily over a finite number of terms "To remove the dependence on C, two options can be envisaged: 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." http://arxiv.org/abs/1010.5437 p2

 

[marcus]

Atyy, we have company this afternoon and evening. I won't be able to answer. Your question is making sense to me and I will need a quiet moment to think about it before replying.

 

[atyy]

Enjoy your company. My answer: this is where GFT renormalization must come in.

 

[atyy]

Other interesting discussions of the convergence of analogous expressions are in Adam Henderson's loop quantum cosmology papers.
http://arxiv.org/abs/0909.4221
http://arxiv.org/abs/1010.0502

Perini has a modification which he thinks may not have a GFT counterpart.
http://arxiv.org/abs/1010.5227

 

[marcus]

Thanks for the pointers to relevant research. I will take a look later today. From the standpoint of abstract math there is no reason to assume the U is finite and it seems ugly to have to appeal to that assumption as a crutch. The question of whether a certain sequence converges is intrinsically interesting!

My observation is practical and non-math, in a sense. IF the universe is finite spatial volume (which we don't know) then it only makes physical sense to consider spin networks with up to N nodes for some large finite N.

So the whole business of taking limits with more and more nodes is moot (from a physical perspective.)

A somewhat similar observation may apply in the case where we have accelerating expansion (as in a deSitter U or an approximately deS) because then there is a cosmological event horizon. One is in a de facto finite situation. I say MAY apply. I haven't seen that worked out. I feel more confident simply considering the finite U case.

And I'm of course glad if some of the young researchers like the guy you mentioned, Perini, are working on the abstract convergence problem of the "X" sort you mentioned, where you don't assume a finite universe. It will be great if they get a result! And they may, as you suspect, bring GFT method to bear on it.

 

[atyy]

OK, it's fine if we fix a spatial boundary at this stage of the game. What I don't understand then is that I thought LQG has no preferred foliation. And if in LQC there is the forever bouncing universe, then it must be unbounded in time. So what if we took the foliation that way, wouldn't we get a different answer. Or does that mean that there is a preferred foliation? Or are there only a finite number of bounces? (actually I don't believe in the bounce for spinfoams - I think Rovelli is hoping for an outcome like CDT - after performing the full sum - not just the first term - he recovers a finite classical universe - to be fair - CDT has not even discretized down to the Planck scale yet)

 

[marcus]

OK, it's fine if we fix a spatial boundary at this stage of the game. What I don't understand then is that I thought LQG has no preferred foliation. And if in LQC there is the forever bouncing universe, then it must be unbounded in time. So what if we took the foliation that way, wouldn't we get a different answer. Or does that mean that there is a preferred foliation? Or are there only a finite number of bounces? (actually I don't believe in the bounce for spinfoams - I think Rovelli is hoping for an outcome like CDT - after performing the full sum - not just the first term - he recovers a finite classical universe - to be fair - CDT has not even discretized down to the Planck scale yet)

The bounce resolution of the BB singularity is a surprising RESULT that first appeared around 2001 under simplifying assumptions. Since then it has proven rather robust in the sense that they keep improving the theory, and changing the assumptions, and removing restrictions, and running the model over and over, and they keep getting a bounce.

They don't get "forever bouncing". That is not robust. You can for example choose parameters where you just get one bounce (where the BB was). You can't say too much about the prior contracting phase. The theory is not "omniscient" it is just a gradual incremental extension that resolves the singularity in one possible way it could be resolved.

It doesn't say if you get just one bounce, or a finite number, or an infinite number (that depends on choices and cases). It just resolves the one singularity we know about. In a possibly testable way (some phenomenologists think.)

There is more to talk about, in what you say. But I am going to get coffee and straighten up the house a little. Yesterday was fun, in fact, thanks for your good wishes!

============================
Incomplete partial reply to your next post #90. Equation (26) the topic of our discussion has a twocomplex with a boundary graph. But the graph is not labeled with area and volume labels. It is not a spinnetwork. so there is no limit on growth in the picture. one could keep adding nodes forever. So it is not the same as modeling a finite-volume universe. Or so it seems to me---as you well know I'm just an interested observer of the QG research scene, no expert! I'll get back to this later this morning. This is interesting.

 

[atyy]

Also, why isn't a finite universe the same as assuming a spinfoam boundary?

 

[marcus]

Atyy, I like your way of putting the three sorts of possible divergence.

...
1) UV - not present
2) IR - present but not a problem
3) X (my nomenclature) - probably present, and probably a problem.

As I've said, I don't think of your X as a practical problem at all, just an interesting abstract math one that you get when you consider a possibly infinite universe. But your pointer to it has gotten me to read more thoroughly in that Rovelli Smerlak October paper which deals with type X concerns.

As you described it the X question comes up around equation (26) of 1010.1939.
It is helpfully clarified by the Rovelli Smerlak paper, so I'll give the link
http://arxiv.org/abs/1010.5437

Notice that (26) does not have a spin-network in it, or a spinfoam. So one cannot implement the idea of a finite universe in the context of (26). There is nothing to keep one from adding cells to the complex forever.
It is more in the abstract math department. An interesting but not urgent question, as I see it.

What your question just now makes me wonder is how would one implement the idea of surrounding a cellcomplex C with a boundary that you can't stretch? Surrounding it with a fixed labeled spin-network. So that refinement is forced to terminate eventually?

The researchers do not seem to have considered that. Maybe it is a useless problem from their perspective. Perhaps I am missing something and my question is based on misunderstanding. I am trying to think about that while I do the evening chores. Hope to be able to say more later.

 

[marcus]

To remind everybody, including myself, what the main focus of the thread is, since we have a new page I will bring forward the edited topic summary from the preceding page.
==quote==
As I see it, the QG goal is to replace the live dynamic manifold geometry of GR with a quantum field you can put matter on. The title of Dan Oriti's QG anthology said "towards a new understanding of space time and matter" That is one way of saying what the QG researchers's goal is. A new understanding of space and time, and maybe laying out matter on a new representation of space and time will reveal a new way to understand matter (no longer fields on a fixed geometry).

Sources on the 2010 redefinition of LQG are
introductory overview: http://arxiv.org/abs/1012.4707
concise rigorous formulation: http://arxiv.org/abs/1010.1939
phenomenology (testability): http://arxiv.org/abs/1011.1811
adding matter: http://arxiv.org/abs/1012.4719

Among alternative QGs, the LQG stands out for several reasons---some I already indicated---which I think are signs that the 2010 reformulation will prove a good one:

• testable (phenomenologists like Aurelien Barrau and Wen Zhao seem to think it is falsifiable)
  
• analytical (you can state LQG in a few equations, or Feynman rules, you can calculate and prove symbolically, massive numerical simulations are possible but not required)
  
• similar to QED and lattice GCD (the cited papers show remarkable similarities---the two-complex works both as a Feynman diagram and as a lattice)
  
• looks increasingly like a reasonable way to set up a background independent quantum field theory.
  
• an explicitly Lorentz covariant version of LQG has been exhibited
  
• matter added
  
• a couple of different ways to include the cosmological constant
  
• indications that you recover the classic deSitter universe.
  
• LQG defined this way turns out to be a generalized topological quantum field theory (see TQFT axioms introduced by Atiyah).
  
• sudden speed-up in the rate of progress, more researchers, more papers

These are just signs---the 2010 reformulation might be right---or to put it differently, there may be good reason for us to understand the theory, as presented in brief by the October paper http://arxiv.org/abs/1010.1939.
...
...

==endquote==

αβγδεζηθικλμνξοπρσςτυφχψωΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√±←↓→↑↔~≈≠≡≤≥½∞ ⇐⇑⇒⇓⇔∃ℝℤℕℂ∈⊗⊕⊂ ⟨·|·⟩

 

[atyy]

Why do you not read the boundary Γ specified in Eq (26) of http://arxiv.org/abs/1010.1939 as a spin network (or a spin network at two different times)? On the bottom of p4, Rovelli says "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."

 

[marcus]

I don't read the boundary Γ as a spin-network because it is simply a graph. No intertwiners at the nodes or spin labels on the links. These are what give scale to a spin-network (as vol and area).

A mere graph is just adjacency relationship without any idea of scale.

So in (26) the boundary does not constrain the size. It can stretch indefinitely---by billions of lightyears if necessary.

 

[atyy]

I don't read the boundary Γ as a spin-network because it is simply a graph. No intertwiners at the nodes or spin labels on the links. These are what give scale to a spin-network (as vol and area).

A mere graph is just adjacency relationship without any idea of scale.

So in (26) the boundary does not constrain the size. It can stretch indefinitely---by billions of lightyears if necessary.

Eq (26) is the same as (27) according to summing=refining. (27) is in the spin network basis, if you compare to (20), (21). Both (26) and (27) are defined with the same boundary graph.

 

[marcus]

Eq (26) is the same as (27) according to summing=refining. (27) is in the spin network basis, if you compare to (20), (21). Both (26) and (27) are defined with the same boundary graph.

we mustn't confuse 26 and 27!
It is more complicated to get from one to the other than you may think. "s=r" is not a naive equality to be taken literally. You have to do a lot, change what you are working with, define Z*, put the whole thing on a different footing, and introduce multiplicity factors, in order to get from one to the other. I am still trying to figure out how they get from 26 to 27.

anyway the convergence divergence issue you brought up was (26)
It has no spinfoams or spinnetworks in. It has no control on the size of the universe.
It's convergence is an interesting problem without immediate practical physical signif.
============

Would you like to discuss (27) now? as mathematically on a separate footing?

Notice what plugs into the LHS and RHS of 27, the arguments, is something new. It is not the old L-tuple of group elements h1...hL.
It is tuples of halfintegers! (j1...jL) and intertwiners (i1...iN)

those are different mathematical animals from plain old SU(2) elements h1...hL.
And the process of summing is different from the limit.

It will take me a little while to change gears, but I could shift over and look at 27 if you'd like.