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

[atyy]

Strictly speaking yes, you are right, but there is no finished theory yet is there? So intuition is still important (and really, I mean, how can you throw away the intuition that led to the proof, even after the finished theory!)

On the other hand, I've never understood intuitively why large spin should be the semi-classical limit, so maybe that intuition will be a red herring.

 

[marcus]

Of course this is work in progress Atyy. I've been watching it and have a sense of the people and the momentum. You may have a different feel. Either way we both know parts definitely still have to be nailed down!

Just for nuance, I will quote Rovelli's section on page 12 where he cites the good work of Barrett group:

== http://arxiv.org/pdf/1004.1780 ==
The analysis of the vertex (49) as well as that of its euclidean analog (55) in this limit has been carried out in great detail for the 5-valent vertex, by the Nottingham group [26, 27, 45, 46]. The remarkable result of this analysis is that in this limit the vertex behaves as

W_v ∼ e ^iS_Regge

where S_Regge is a function of the boundary variables given by the Regge action, under the identifications of these with variables describing a Regge geometry. The Regge action codes the Einstein equations’ dynamics. Therefore this is an indication that the vertex can yield general relativity in the large distance limit. More correctly, this result supports the expectation that the boundary amplitude reduces to the exponential of the Hamilton function of the classical theory.
==endquote==

Supports, does not yet prove.

And we are still just looking at a 5-valence vertex. Which BTW is in line with your mention of the triangulated manifold picture, because a 4-simplex has 5 sides (the dual replaces it with a vertex and replaces each of its 5 sides by an edge). My hunch is that graduate students can extend the result to higher-valence vertices. It's how I'm used to seeing things go, but who knows? You think not?

 

[marcus]

In fact someone WAS working on a missing detail Rovelli describes right after what I quoted on page 12. The two terms in the vertex amplitude. A Marseille postdoc and a couple of grad students. They posted in April soon after the survey paper appeared.
http://arxiv.org/abs/1004.4550
"We show how the peakedness on the extrinsic geometry selects a single exponential of the Regge action in the semiclassical large-scale asymptotics of the spinfoam vertex."

Barrett's group left it with both a +iRegge and a -iRegge. One wanted to get rid of or suppress the negative exponential, and just have a single exponential term. So Bianchi et al took care of that.

There's been a kind of stampede of results in the past 6 months or year, bringing us closer to what appears may be a satisfactory conclusion.

 

[atyy]

My hunch is that graduate students can extend the result to higher-valence vertices. It's how I'm used to seeing things go, but who knows? You think not?

I don't knoow - what I would like to see aesthetically is that it's a GFT, and that GFT renormalization is essential, and that matter must somehow come along automatically. But Barrett et al's, and also Conrady and Freidel's are the most intriguing results I have seen from the manifold point of view. But in which case, I think there must be Asymptotic Safety somehow, and a link via what Dittrich et al are saying.

 

[marcus]

As further motivation for the move towards manifoldless QG+M, I should quote (again) that passage from Marcolli's May 2010 paper. Marcolli mentions the view of Chamseddine and Connes. This is section 8.2 page 45.

==quote http://arxiv.org/abs/1005.1057==
8.2. Spectral triples and loop quantum gravity.

The Noncommutative Standard Model, despite its success, still produces an essentially classical conception of gravity, as seen by the Einstein–Hilbert action embedded in eq. (8.2). Indeed, the authors of [36] comment on this directly in the context of their discussion of the mass scale Λ, noting that they do not worry about the presence of a tachyon pole near the Planck mass since, in their view, “at the Planck energy the manifold structure of spacetime will break down and one must have a completely finite theory.”

Such a view is precisely that embodied by theories of quantum gravity, including of course loop quantum gravity—a setting in which spin networks and spin foams find their home. The hope would be to incorporate such existing work toward quantizing gravity into the spectral triple formalism by replacing the “commutative part” of our theory’s spectral triple with something representing discretized spacetime.

Seen from another point of view, if we can find a way of phrasing loop quantum gravity in the language of noncommutative geometry, then the spectral triple formalism provides a promising approach toward naturally integrating gravity and matter into one unified theory.
==endquote==

More discussion of the Marcolli May 2010 paper in this thread:
https://www.physicsforums.com/showthread.php?t=402234

 

[marcus]

I guess one way to put the point is to observe that LQG is amphibious.

The graph description of geometry (nodes of elemental volume, linked to neighbors by elemental area) can live embedded in a manifold and also out on its own---as combinatoric data structure.

In the April 2010 status report and survey of LQG, the main version presented is the manifoldless formulation which Rovelli calls "combinatorial". But he also, in small print, describes earlier manifoldy formulations using embedded graphs. In my view those are useful transitional formulations. They can be used to transfer concepts and to prove large limits and to relate to classical GR. Stepping stones, bridges, scaffolding.

It's not unusual to prove things in two or more stages--first prove the result for an intermediate or restricted case, then show you can remove the restriction. But as I see it, the manifoldless version is the real McCoy.

 

[atyy]

So what's the manifoldless take on renormalization?

In the manifoldy view it has to happen somewhere, since one started with a triangulation of the manifold.

 

[marcus]

...In the manifoldy view it has to happen somewhere, since one started with a triangulation of the manifold.

Historically, the LQG of the 1990s did not start with a triangulation of a manifold. It started with loops, which were superseded by slightly more complicated objects: spin networks. These have nothing to do with triangulations.

Spin networks can be embedded in a manifold. But the matter fields, if they enter the picture, are defined on the spin network---by labeling the nodes and links.

So what's the manifoldless take on renormalization?

Nodes carry fermions. Links carry Yang-Mills fields. Geometry is purely relational. The basic description is a labeled graph. The graph carries matter fields and there are no infinities.
See the statement of problem #17 on page 14 of the April paper. This points to what i think is now the main outstanding problem---going from QG to QG+M---including dynamics in what is (so far at best) a kinematic description of matter and geometry.

 

[marcus]

Thiemann uses an old-fashioned version of LQG here, but it gives the general idea:

http://arxiv.org/abs/gr-qc/9705019
QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories
Thomas Thiemann
(Submitted on 10 May 1997)
"It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum. Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized."

 

[atyy]

Historically, the LQG of the 1990s did not start with a triangulation of a manifold. It started with loops, which were superseded by slightly more complicated objects: spin networks. These have nothing to do with triangulations.

But aren't we talking about spin foams?

Also, if we take the Barrett result seriously, they only get to something like the Regge action. That needs a continuum limit to look like GR - that's why Loll et al - who started with the Regge! - try to link to Asymptotic Safety or some hopefully well defined theory in the continuum limit.

 

[marcus]

But aren't we talking about spin foams?

Not [EDIT: exclusively] as far as I know, Atyy. I was being careful to say "nodes" to indicate that I was talking about spin-networks.

Also, if we take the Barrett result seriously,

I hope you are not mistaking Barrett et al for the final word. They only considered vertices of valence 5. And Bianchi-Magliaro-Perini have already improved on them. What we are talking about is work in (rapid) progress. So it is something of a moving target of discussion.

As a general philosophical point, we have no indication that spacetime exists (George Ellis has given forceful arguments that it does not.) The spacetime manifold is a particular kind of interpolation device. (Like the smooth trajectory of a particle, which QM says does not exist.)
Since the 4D continuum does not exist we do not need to triangulate it and in fact spinfoams should not be viewed as embedded in or as triangulating a 4D continuum. They are histories depicting how an unembedded spin-network could evolve. Each spinfoam gives one possible evolutionary history.

Like the huge set of possible paths in a Feynman path integral.

Also a general spinfoam could not possibly correspond to a triangulation (you must realize this since you have, yourself, cited the Lewandowski 2009 paper "Spinfoams for all LQG")
So let's stop referring to spinfoams as dual to triangulations of some mythical 4D continuum

Fields live on graphs and they evolve on foams, as labels or colorings of those graphs and foams. That's the premise in the context of this discussion, and on which the LQG program will succeed or fail. We don't know which of course because it is in progress right now.

 

[atyy]

Rovelli cites Barrett, and Barrett is talking about spin foams. Of course Barrett is not the final word, but where is is the indication that this is a reasonable line of research at all?

And Bianchi-Magliaro-Perini have already improved on them.

That too is a spin foam paper.

Edit: I missed an "else" above - ie. "where else is this" not "where is this"

 

[marcus]

...

Edit: I missed an "else" above - ie. "where else is this" not "where is this"

I don't see where, but maybe it doesn't matter. In which post?
And I missed an "exclusively".

Basically you can't talk about foams without talking about networks and vice-versa. One is a path history by which the other might evolve. Or the foam is a possible bulk filling for a boundary network state.

What I suggested we stop talking about, and move on from, is foams that are dual to triangulations and foams which are embedded. Those are both too restrictive.

 

[atyy]

OK, I see Rovelli has listed what I'm asking about as his open problem #6, where he refers to further studies along the lines of http://arxiv.org/abs/0810.1714, whose preamble goes "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."

BTW, thanks for pointing out the Bianchi-Magliaro-Perini (BMP) paper - it helps me makes sense of what Barrett is doing by taking the large j limit as semiclassical - I always thought that should be the hbar zero limit - which is what BMP do.

So do you think one should take the N infinity limit first followed by hbar, or the other way? Would you like to guess now - and see in a couple of months, or however fast those guys are going to work - as to whether the Barrett result will hold up if the N infinity limit is taken first?

 

[marcus]

Atyy, let me highlight the main issue we are discussing. I think it is the manifoldless formulation of LQG. You seem reluctant to accept the idea that this is what Rovelli is presenting in the April paper (the subject of this thread.)

As further motivation for the move towards manifoldless QG+M, I should quote (again) that passage from Marcolli's May 2010 paper. Marcolli mentions the view of Chamseddine and Connes. This is section 8.2 page 45.

==quote http://arxiv.org/abs/1005.1057==
8.2. Spectral triples and loop quantum gravity.
...
“at the Planck energy the manifold structure of spacetime will break down and one must have a completely finite theory.”

Such a view is precisely that embodied by theories of quantum gravity, including of course loop quantum gravity—a setting in which spin networks and spin foams find their home. ..
==endquote==

I guess one way to put the point is to observe that LQG is amphibious.

The graph description of geometry (nodes of elemental volume, linked to neighbors by elemental area) can live embedded in a manifold and also out on its own---as combinatoric data structure.

In the April 2010 status report and survey of LQG, the main version presented is the manifoldless formulation which Rovelli calls "combinatorial". But he also, in small print, describes earlier manifoldy formulations using embedded graphs. In my view those are useful transitional formulations. They can be used to transfer concepts and to prove large limits and to relate to classical GR. Stepping stones, bridges, scaffolding...

So what's the manifoldless take on renormalization?
In the manifoldy view it has to happen somewhere, since one started with a triangulation of the manifold.

Maybe you are not, but you seem to have been stuck on the idea that because in some papers the type of spinfoam was restricted to be dual to a triangulation of a 4D manifold that somehow ALL spinfoams must not only live in manifolds (which is not true) but even must be dual to triangulations! This is far from the reality. As a convenience, to prove something, one can restrict to special cases like that (the preamble of a paper may give some indication of what special case is in play in that paper.)

...Nodes carry fermions. Links carry Yang-Mills fields. Geometry is purely relational. The basic description is a labeled graph. The graph carries matter fields and there are no infinities.
See the statement of problem #17 on page 14 of the April paper. This points to what i think is now the main outstanding problem---going from QG to QG+M---including dynamics in what is (so far at best) a kinematic description of matter and geometry.

Just from reading the April paper you can see (but you already know) that the way dynamics is handled is as a "path integral" over all possible spinfoams that fit the boundary.
So if nodes carry fermions and links carry Y-M fields, then when we go over to dynamics this means fermions travel along edges, Y-M fields along faces, and interactions occur at vertices.

OK, I see Rovelli has listed what I'm asking about as his open problem #6, where he refers to further studies along the lines of...

If you look at problem #6, you will see it is about equation (52). If you look at (52) you will see that manifolds are not involved. Unembedded spinfoams are involved.
He is asking about possible infrared divergences in equation (52) which is a manifoldless equation. Infrared means large j limit. The spin labels get big. That is, large volumes and areas. And check out equations (6-8): area and volume operators are also defined in a manifold-free way! The very concept of area is manifoldless. That's on page 2.

Because LQG tools are "amphibious" as I said, if somebody wants to prove something they can always restrict to some special case or consider embedded foams and networks as a help---getting a preliminary result. And indeed Rovelli refers to some 2008 work, on a preliminary result about large j divergences, that used a manifold. But you should be careful not to conclude that therefore problem #6 involves manifolds or embedded foams. It doesn't follow.

Indeed equation (52) and the whole core formulation is manifoldless---it is just supporting results that are drawn from alternative older formulations and stuff brought in for comparison (showing the convergence of different lines of development) as in section II-F.

 

[marcus]

...
BTW, thanks for pointing out the Bianchi-Magliaro-Perini (BMP) paper - it helps me makes sense of what Barrett is doing by taking the large j limit as semiclassical - I always thought that should be the hbar zero limit - which is what BMP do.

So do you think one should take the N infinity limit first followed by hbar, or the other way? Would you like to guess now - and see in a couple of months, or however fast those guys are going to work - as to whether the Barrett result will hold up if the N infinity limit is taken first?

That's an intriguing proposal! As usual you are thinking way ahead of me. It sounds like you have visualized a way that they might proceed towards proving that both the largescale, and the semiclassical limits are OK.
At the moment I am not clear enough on how it might be done. And I have absolutely no idea about the timetable. I will take a look at the BMP paper and see if I can get some notion.

Do we measure time in months, or in generations of graduate students? Maybe in generations Will it be one of Rovelli's PhDs (e.g. Bianchi) or might it be a PhD of a PhD (e.g. someone advised by Bianchi). I find it bizarre to look into the future.
One thing they know how to do in LQG is attract and train smart people. And the effort is really focused---with a clear philosophy.

About philosophy, did you notice that Rovelli never showed any interest in the braid representation of matter? (Sundance B-T, Perimeter people, you remember.) Can you think of a reason? How can spin-network links be braided or have any kind of knots? To knot the links you must have it embedded in a manifold. But at short distances the manifold structure dissolves! Rovelli explained this in a series of slides at Strings 2008, depicting how a tangle can untangle. While mathematically appealing, the braid-matter idea was philosophically inconsistent with the program's main (manifoldless) direction---none of the Marseille alumni went for it.

 

[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.

 

[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.

Atyy, thanks for pointing me to this Ashtekar paper. I found what I think is the passage, on page 4:
==quote Ashtekar 1001.5147 ==
In LQC one can arrive at a sum over histories starting from a fully controlled Hamiltonian theory. We will find that this sum bears out the ideas and conjectures that drive the spin foam paradigm. Specifically, we will show that: i) the physical inner product in the timeless framework equals the transition amplitude in the theory that is deparameterized using relational time; ii) this quantity admits a vertex expansion a la SFMs in which the M -th term refers just to M volume transitions, without any reference to the time at which the transition takes place; iii) the exact physical inner product is obtained by summing over just the discrete geometries; no ‘continuum limit’ is involved; and, iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of GFT, where, moreover, the GFT coupling constant λ is closely related to the cosmological constant Λ. These results
were reported in the brief communication [1]. Here we provide the detailed arguments and proofs. Because the Hilbert space theory is fully under control in this example, we will be able to avoid formal manipulations and pin-point the one technical assumption that is necessary to obtain the desired vertex expansion: one can interchange the group averaging integral and a convergent but infinite sum defining the gravitational contribution to the vertex expansion(see discussion at the end of section III A). In addition, this analysis will shed light on some long standing issues in SFMs such as the role of orientation in the spin
foam histories [49], the somewhat puzzling fact that spin foam amplitudes are real rather than complex [31], and the emergence of the cosine cos S_EH of the Einstein action —rather than e^iS_EH— in the classical limit [32, 33].
==endquote==

It's later now and I've had a chance to take a leisurely look. I didn't realize the interest of this paper before. It's going to be helpful to me, so am extra glad to have it pointed out. I can not address your remark right away but will read around in the paper and aim for a general understanding. Bringing LQC on board spinfoams is fairly new. I'll try to respond tomorrow.

 

[Fra]

Marcus, thanks for your further comments. I've again been away for a few days and just got back.

It is interesting that you are thinking in terms of what, in Computer Science, are called "data structures" used for storage and retrieval. Just for explicitness, I will mention that some examples of data structures are graphs, trees, linked lists, stacks, heaps. Not something I know much about. It is also intriguing that you mention a type of Fourier transform (the FFT).
====================

I think that primarily for pragmatic reasons the game (in QG) is now to find SOMETHING that works. Not necessarily the most perfect or complete, but simply some solution to the problem of a manifold-less quantum theory of geometry and matter.

As for my own perspective, and it's association to LQG - I seek a NEW intrinsic measurement theory that is also built on an intrinsic information theory, where information is subjective and evolving transformations between observers, (rather than just relational with a structural realism view of the transformations as equivalence classes).

So key points to mer are

1. An INTRINSIC representation of information (ie. "memory" STORAGE)

2. Datacompression (different amounts of "information" can be stored in the same amount of memory, depending on the choice of compression - I suggest the compression algorithms are a result of evolution; the laws of physics "encode" compression algorithms of histories of intrinsic data).

3. The compression algorithms are also information. The coded data is meaningless if the coding systme is unknown.

4. Any given observer, has to evolve and test their own coding system. Only viable observers survive, and these have "fit" coding system. The only way to tell wether a coding system is "good" or "bad" is for the observer to interact with the environment and see wether it is fit enough to stay in business. So there is no objective measure of fitness.

I think that primarily for pragmatic reasons the game (in QG) is now to find SOMETHING that works. Not necessarily the most perfect or complete, but simply some solution to the problem of a manifold-less quantum theory of geometry and matter.

If one could just get one manifold-less quantum theory that reduced to General Relativity in the large limit, that would provide pointers in the right direction---could be improved-on gradually, and so forth.

Yes, that ambition fits my view of Rovelli's way of putting it too. I think he wrote somewhere that if we can just find ANY consistent theory that does the job, it would be a great step.

But I do not share that ambition. I think that acknowledging ALL issues with current models that we can distinguish, will make it easier, rather than harder to find the best next level of understanding.

It's in THIS respect that I do not quite find the abstract network interpretation motivated. The MOTIVATION seems to come from the various triangulations or embedded manifold view. Then afterwards it's true that one can capture the mathematics and forget about the manifold motivation, but then the obvious question is, is this the RIGHT framework we are looking for? I am not convinced. Maybe it's related to it, but I still think, if we acknowledge all the obvious points that there should be a first principle construction of the "abstract view" in terms of intrinsic measurements and notions.

When you say getting rid of the manifold, I see several possible meanings here

a) just get rid of the OBJECTIVE continuum manifold

a') get rid of the subjective continuum because it's unphysical, it's more like an interpolated mathematical continuum abstraction around the physical core.

b) get rid of the notion of objective event index (spacetime is really a kind of indexed set of events) (ie. wether discrete or continous). This is already done in GR - the hole argument etc. Ie. the lack of OBJECTIVE reality to points in the event index (if I allow myself to translade the hole argument to the case of a "discrete manifold")

b') get rid of the notion of subjective event index (since we want the theory of be observer invariant; and only talk about EQUIVALENCE CLASSES of observers)

I think we need to do a + a´+ b , but b´ is not possible since it is the very context in which any inference lives. I think Rovelli tries to do also b´and replace it with structural realism of the equivalence classes.

If you understand my argument and quest for an intrinsic inference, this is a sin and unphysical itself. I'm suggesting that the notion of observer invariant equivalence classes itlsef is "unphysical". (some of the arguements are those of smolin/unger)

But I also think that if we really reduce the discrete set of events to the pure information theoretic abstraction, we also remove the 3D structure. All we have is an index, and how order and dimensional meausres emergets must be described also from first principle selforganising.

So I expect the abstract reconstruction of "pure measurements" to start from a simple distinguishable index, combined with datastructures representing coded information, and communication between such structures (where the communication is what generates the index first as histories, then as recoded compressed structures) (*)

(*) I think this is what is missing. The abstract LQG view, is MOTIVATED from the normal manifold/GR analogy, and therefore it doesn't qualify as a first principle relation between pure measurements in the sense I think we need.

/Fredrik

 

[Fra]

Even when we do reduce the manifold to measurements, you still keep mentioning notions such as area and volume.

But from a first principle reconstruction - what do we really mean by "area" or "volume"?? I find it far from clear. I'd like to see the "geometric notions" (if they are even needed?) should be constructed more purely from information geometry than what is customer.

I think it needs to be rephrased into more abstract things such as capacity, amount of information, or channel bandwith etc. Then we also - automatically - can not distinguish matter and space of particular dimensions etc. This reconstruction seem to still be missing in LQG.

/Fredrik

 

[atyy]

But I also think that if we really reduce the discrete set of events to the pure information theoretic abstraction, we also remove the 3D structure. All we have is an index, and how order and dimensional meausres emergets must be described also from first principle selforganising.

I think GR itself provides some of this. GR is not geometrical. It only is geometrical if you measure spacetime with test partcles and ideal clocks ('observers'). However, neither of those exist in GR, since all you have is the coexistence of various fields (gravitational, electromagnetic etc.). There are no observers, except in certain parts of the universe where they emerge from fields, and are able to approximately isolate themselves and and say here is a test particle and an ideal clock which are not affected by the rest of the universe. What is unclear in classical GR is whether thesde observers can really emerge from the fields.

 

[atyy]

==quote Ashtekar 1001.5147 ==
In LQC one can arrive at a sum over histories starting from a fully controlled Hamiltonian theory. We will find that this sum bears out the ideas and conjectures that drive the spin foam paradigm. Specifically, we will show that: i) the physical inner product in the timeless framework equals the transition amplitude in the theory that is deparameterized using relational time; ii) this quantity admits a vertex expansion a la SFMs in which the M -th term refers just to M volume transitions, without any reference to the time at which the transition takes place; iii) the exact physical inner product is obtained by summing over just the discrete geometries; no ‘continuum limit’ is involved; and, iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of GFT, where, moreover, the GFT coupling constant λ is closely related to the cosmological constant Λ. These results
were reported in the brief communication [1]. Here we provide the detailed arguments and proofs. Because the Hilbert space theory is fully under control in this example, we will be able to avoid formal manipulations and pin-point the one technical assumption that is necessary to obtain the desired vertex expansion: one can interchange the group averaging integral and a convergent but infinite sum defining the gravitational contribution to the vertex expansion(see discussion at the end of section III A). In addition, this analysis will shed light on some long standing issues in SFMs such as the role of orientation in the spin
foam histories [49], the somewhat puzzling fact that spin foam amplitudes are real rather than complex [31], and the emergence of the cosine cos S_EH of the Einstein action —rather than e^iS_EH— in the classical limit [32, 33].
==endquote==

This paper the second in a pair of papers, the first http://arxiv.org/abs/0909.4221 is a conceptual summary, the second http://arxiv.org/abs/1001.5147 explains why certain steps like exchanging order of integration and summation are not cheating in particular cases.

I think the interesting comparison is between Ashtekar et al's Eq 3.10 of http://arxiv.org/abs/0909.4221 (same as Eq 3.20 of http://arxiv.org/abs/1001.5147) and Oriti's unnumbered final equation on p5 of http://arxiv.org/abs/gr-qc/0607032, which is the same as Freidel's Eq 11 in http://arxiv.org/abs/hep-th/0505016 .

There are some differences between the proposals, eg. Freidel proposes the physical scalar product to be his Eq 16, which differs from his Eq 11, whereas if you read Oriti's discussion, he is unsure whether it should be Freidel's Eq 11 or 16. It is also interesting to compare Ashtekar's and Oriti's discussions of GFT renormalization.

Edit: I fixed the typo above that marcus pointed out below.

 

[marcus]

...
I think the interesting comparison is between Ashtekar et al's Eq 3.10 of http://arxiv.org/abs/1001.5147 (same as Eq 3.20 of http://arxiv.org/abs/1001.5147) and Oriti's unnumbered final equation on p5 of http://arxiv.org/abs/gr-qc/0607032, which is the same as Freidel's Eq 11 in http://arxiv.org/abs/hep-th/0505016 .
...

Again thanks! I think there was a typo in the post. You may have meant:

I think the interesting comparison is between Ashtekar et al's Eq 3.10 of http://arxiv.org/abs/0909.4221 (same as Eq 3.20 of http://arxiv.org/abs/1001.5147)

And in that case you are of course right---same equation.
=================

My main focus needs to stay on Rovelli's April paper, but I will keep intermittently chewing on the two Ashtekar papers and trying to understand them better. Ashtekar has a different perspective and has been a formative and greatly influential QG figure over the long haul. I have to pay attention especially to his overview of the field. Differences in formal detail can work themselves out---I can probably get along with just Marseille notation. But I have to try to assimilate Ashtekar's vision. Both the papers you pointed to have introduction and conclusion overview sections that I'm finding helpful that way.

 

[marcus]

... GR is not geometrical. It only is geometrical if you measure spacetime with test partcles and ideal clocks ('observers'). However, neither of those exist in GR,...

We may have a slight semantic difference here. When I think of a theory of geometry, I don't expect of it a "theory of everything" that would explain how life might evolve and how conscious beings able to make measurements and construct clocks might arise from the various matter fields.

All I ask from a classical theory of geometry is that it give me what GR gives----geometries.
A geometry is an equivalence class of metrics (with attendant matter) under diffeomorphism.

So for me GR is the paradigm theory of geometry---it more or less defines for me what geometry is. Granted the theory does not provide its own observers, but it is observer-ready in a kind of "plug-and-play" sense.

By itself a metric (with attendant matter distribution) gives the geometric relations among all material "events" (such as particle collisions). And it determines the world-lines of all "particles".

Admittedly the concept of a "particle" is either a bit ad hoc or a bit fuzzy---we must indulge the theory in small ways, allow it a few marbles. It does not explain or predict the existence of marbles. Or some people prefer clouds of dust---then the grains of dust are the marbles.

But that strikes me as a kind of comical quibbling. A theory of geometry does not have to explain how there could be a freely falling grain of dust. All it needs to be is ready for you to insert a marble or a cloud of dust into its picture of geometry---it will take charge from there on.

This may sound a pretty superficial and unphilosophical but that's how I think of classical geometry.

GR does what it needs to---explains what flat means and why geometry is usually nearly flat (because matter is sparse) and how distances to galaxies can expand and how you can get black holes and gravitational redshift and all that basic geometry stuff that we observe.
Anyway that is my simplistic attitude about geometry.

So your expressed reservation about GR seems like a non-reservation

 

[atyy]

So your expressed reservation about GR seems like a non-reservation

It's not a reservation. I take the view that GR is not about geometry, except technically in the sense that all the fields of the standard model are geometrical because of the gauge symmetry. Thus in GR, observable geometry only emerges when one has matter. That, I believe, is the true lesson of GR. The plug and play view is not background independent, because you have test particles that move on a fixed background, without themselves affecting the background.

 

[marcus]

Atyy: "GR is not about geometry."
Marcus: "Geometry is precisely what GR is about. GR is the paradigm or model theory in that department."

No basis for discussion there---beyond sterile semantics. We had best get back to Rovelli's paper.

 

[marcus]

This will respond in part, as well, to Fra's concerns about the QG agenda.

Several of Fra's posts responded to my couching the agenda in negative terms--a manifoldless QG+M.

To put what I see as the main direction is more positive terms, I'll propose this alternative---a more fully relational QG+M.

This notion of a goal to work towards has been around for decades (I don't know how long). The idea is that GR---the paradigm classical theory---only tells us about the web of geometric relations among events.

There is no substantive objective continuum, because of diff-invariance. One can morph the situation around. Points have no definable identity except where marked by some physical event, like an intersection of worldlines---or some identifiable feature of the gravitational field itself which can mark an event.

So if space is anything, it is an insubstantial web of relationships. To pass to a quantum picture basically means to construct a hilbertspace of webs of relationships, and define operators on it. Or? Do you have some more accurate and concise way to put it?

(looking back at Fra's post #69 I think I may have just now said some things that were contained in what Fra said---except that he went quite a bit further in certain directions---the importance of the observer and information-theoretical considerations.)

================
BTW re Atyy's "not about geometry" comment: Actually GR has matter. You can have dust or marbles adrift on the righthand-side of the main GR equation. In that sense it as plenty of observers already (assuming you do not require observers to be conscious and wear conventional timepieces on their wrists and so forth). If a grain of sand can serve as an observer (and I would argue that it can) then you can put in as many observers as you want---the main equation is set up for it. The effect of those observers will be taken account of in the gravitational field. Logically there is no need for "test particles".

 

[Fra]

I don't mean to just provide "negative terms", I actually wanted to drive the discussion in the constructive sense, by providing noting some provocative points with the picture and focus on some foundational issues that exists conflicting between a measurement theory.

It's nothing new as it's related to the problem of what is an observable in GR and QG, but for some reason the points doesn't seem to get the attention I think it deserves.

So if space is anything, it is an insubstantial web of relationships. To pass to a quantum picture basically means to construct a hilbertspace of webs of relationships, and define operators on it. Or? Do you have some more accurate and concise way to put it?

As far as I understand LQG, this sounds like a good summary of one of it's constructing principles.

But I have an objection to exactly this, but the objection is as much a critique against QM.

My clear conviction is that this is an inappropriate application of QM formalism taking out of context. I suggest that the hilbert space of states of the webs of relations are non-physical as they are not inferrable by an real inside observer. They make sense in the mathematical sense only - and if you accept is as a strucutral realism.

I'm not describing LQG here but I would want put it like something like this (to compromise with your phrasing):

Space, is an insubstantial web of relationships (ie. it's not "material") BUT the information needed to specify this web of relationships is physically coded in matter. Each material system encodes the subjective perspective (up to some horizon).

I further suggest that this picture means that each material observer (matter system) "sees" it's own "hilbert space" (I use quotes as I think this implies a modification of QM as we know it today), and moreoever this hilbert space is not timeless, it evolves with time (where time is just a parameterization of an the entropic flow; which is different to each observer).

Since different observers see different state spaces, that inconsistency is what forms the negotiated consensus and defines the local equivalence classes. So each observers, sees "equivalence classes" of nearby "material observers" whose definition genereally evolve. but one can certainly imagine equilibrium conditions where stable quasi-global classes emerge.

So as I see it the "quantum picture" doesn't involve applying the quantum formalism as is, to the equivalence classes of diff-generated observers, the quantum picture is there from the beginning if we consider the proper discrete measurement theory. What STARTS OUT as a classical measurement theory (ie probability theory, but discrete) gets mixed up by the set of different encoding structures.

The difference as I see it between classical and quantum logic, is that classical logic just uses as simple probability space, where quantum logic uses sets of relates spaces that are related by lossy compressions (such as truncated Fourier transforms). This is why logical operators are different.

I agree this is radical and speculative, and maybe it's optimistic to expect anyone bot buy into this long train of though, but the simple point I have is that:

Quantum theory are we know it, are verified only for what smolin calls subsystems. Which means the cases where the statistics and hilbert spaces can be effectively constructed and encoded in some lab environment before the entire environment has completey evolved into something different.

And some quite simple plausability arguments, and the quest for everything to be inferrable in the inductive rather than deductive sense suggest that the application of normal QM formalism to the equivalence class of GR observers in the suggested way may be the wrong way to approach the entire "QG" problem.

Note sure if that made sense? Because I have also deep concernts about QM foundations, it's not possible to comment on QG without getting into that as well.

/Fredrik

 

[Fra]

To try to make cleaner how we disagree.

"Since different observers see different state spaces, that inconsistency is what forms the negotiated consensus and defines the local equivalence classes."

LQG tries to make a "regular QM theory" to the STATES of the equivalence classes.

I think that we need to find the EVOLUTION of the SYSTEM of interacting observers.

So I guess what I say is that we need to make QM truly relational, like Einstein made SR into GR. Not, try to apply QM as we know it to the classical equivalence classeso GR. I think it's a mistake.

So I think we are seeking "Einsteins equation" for the relational QM. To apply non-relational QM formalism to Einsteins equation is not right.

So I'm suggesting that hte equivalence classes and their symmetries must be evovling, and that this pictures includes ALL interactions. Thus Strong, weak and EM as well. It's not something we can put "ontop" of the pure-gravity quantized. It makes no sense to me.

/Fredrik

 

[atyy]

So I'm suggesting that hte equivalence classes and their symmetries must be evovling, and that this pictures includes ALL interactions. Thus Strong, weak and EM as well. It's not something we can put "ontop" of the pure-gravity quantized. It makes no sense to me.

So this would argue for unification, something like strings? In strings the graviton is sometimes a particle caused by an excitation of a string, but if you change your view it can become part of the background spacetime - and it can go the other way too, the background can become an excited string state about a different background.

 

[Fra]

So this would argue for unification, something like strings? In strings the graviton is sometimes a particle caused by an excitation of a string, but if you change your view it can become part of the background spacetime - and it can go the other way too, the background can become an excited string state about a different background.

Unification yes, and there are some ways for me to relate this construction to ST, but ST has many unsatisfactory traits. An certainly something is missing in the construction principles.

- ST makes use of the continuum, not only the manifolds, but maybe worse the string itself (which I view as a continuum index). This is highly unphysical and doesn't fit into the picture of a physical representation.

- ST have the same simple view of QM. So it does not solve the intrinsic measurement problem and coding of information problem of QM. ST is not the reconstruction of measurement and representation from the combinatorical perspective I think we need.

The second problem, is btw, what forces the higher background dimensions as it's the only way to "encode" all the variety ST wants to. But the problem is then that you do get this landscape that you don understand what it is. Is it real, is it an illusion? And why is there measure on the landscape?

From my point of view, some of the problems of ST might be gone if they replace the string with a more generic "set of sets" in the datacompression sense I mentioned before, that work from discrete indexes. But then, it just isn't string theory anymore.

Not to mention the action of the string, which is basically inherited from classical analogies.

In my view, all actions are generically related to probabilities or information divergences. The "action" is simply the generalized "entropy" in transition space, which is to be maximized. So all action forms should follow in this way (thus beeing inherently entropic).

There is a chance that "string like" structure, prove to be the simplest possible continuum structures in the large complexity limit, but that is still just a possible connection and the logic there is nothing like the logic of the string program.

Somehow, rovelli's reasoning as I've read it, although I object to it, is at least more clear and consistent that the string scheme which I find to be more of toyery.

/Fredrik

 

[Fra]

but if you change your view it can become part of the background spacetime - and it can go the other way too, the background can become an excited string state about a different background.

Generically this makes sense to me, and it would correspond to comparing two different observers. Just like any conditional assessment depends on perspective.

So such a general trait is I think sensible.

The Background should be part of the observer. The problem is that the way ST is constructed, the background complexity is not bounded. First of all because it's based on a continuum index, and it becomes highly ambigous IMHO at least how to COUNT and compare evidence in uncountable sets. The choice of limiting procedure becomes crucial. But no care is made about that in ST. The worst part is that the continuum itself is part of the baggage, and already there you have lost control before you've started as the counting procedure (from inference poitn of view) becomes more or less completely ambigous.

/Fredrik

 

[atyy]

The Background should be part of the observer. The problem is that the way ST is constructed, the background complexity is not bounded. First of all because it's based on a continuum index, and it becomes highly ambigous IMHO at least how to COUNT and compare evidence in uncountable sets.

Yes, I can never decide which I'd like better. On the one hand, it'd be nice if we only used integers in the formulation of the most basic theory. On the other hand, there are cases where discreteness emerges from the continuum - say eigenvalues in quantum mechanics - or non-relativistic quantum mechanics of atoms from relativistic quantum field theory.

 

[Fra]

Yes, I can never decide which I'd like better. On the one hand, it'd be nice if we only used integers in the formulation of the most basic theory. On the other hand, there are cases where discreteness emerges from the continuum - say eigenvalues in quantum mechanics - or non-relativistic quantum mechanics of atoms from relativistic quantum field theory.

I see what you mean. I think there is not really a conflict per see between continuum models and discrete ones, it's just that I think it's important to keep in mind from the point of view of inference and counting and rating evidence (inductive reasoning) what the physically distinguishable states are and what is "gauge".

By certain transformations (I'd like to call them datacompression) one can from limiting cases or continuum models compute key parameters that are independent from superficial embeddings or interpolated structures, that can be further used to "index" the continuum structures, maybe even in a countable way.

That's fine as long as we keep track of what the physically distinguishable states are, and what we should count. I prefer to start with the "backbone" and then picture this as indexing a continuum manifold if we need it for comparasion to old models, rather than start with a redundant description, get lost and try to figure out what's physical degrees of freedom and what's just continuum gauge.

For example when you start with a continuum structure, and try to apply inductive inference, construct various entropy or action measures, then it's crucial that we know how and what to count. In a continuum picture, by an ambigous choice of limiting procedure or measure one can pretty much get the results one wants.

This is even more important if one (like I want to) wants to construct also the expected action of this "observer complex", as they way I picture it, the prediction and computation of "probabilities" requires that the state spaces and transitions are countable. Actually finite, or if infinite, at minimum countable and have a well defined limiting procedure. Otherwise the physical measures are not computable.

/Fredrik

 

[marcus]

As we were talking about Rovelli's April paper in some other threads I was impressed by the level of misinformation/misunderstanding.

This is the paper that presents LQG in a manifoldless way giving it a "new look", as Rovelli's title indicates. Of course there is no distinction between canonical LQG and spinfoams here--those approaches were unified earlier. Network and foam are indeed inseparable but that is not what is new.

Someone in another thread stated with great confidence and authority that this version of Lqg had nothing to do with the Einstein-Hilbert action . (The Regge action is the relevant version of E-H, and is derived from the setup.)
Another person flatly stated his conclusion that the April paper merely presented a new spinfoam vertex. We need to get past a wall of ignorance/selective inattention. There is a kind of sea-change in progress---a general shift in the qg picture-- making it more important to be well informed.

In that other thread, Tom responded with a concise and helpful summary of what is happening in the April paper (1004.1780) the topic of this thread, so I'll copy here:

So this new LQG is just a new SF model.

No!

It's about convergence of canonical approach and spin foams; it's about mapping of or identities between certain entities in both frameworks; it's about making LQG accessable for calculations; it's about long-distance limit / semiclassical approximations; it's about consistency of quantization, implementation of constraints, regularization of the Hamiltonian (which is notoriously difficult in old-fashioned LQG) ...

... the more you read the more you will find.

 

[marcus]

Here's some thematic material I want to develop here, taken from another thread about the April 2010 paper on "new look" LQG.

https://www.physicsforums.com/showthread.php?p=2855316#post2855316

marcus:...to better understand what underlies the relation of geometry to matter...

The Loop enterprise is high risk. [But] it does seem to me ... philosophically sound. It gets away from dependence on the manifold. The labeled graph (spin network) is an economical representation of the experimenters' geometrical knowledge (a finite web of volume and area measurements which can also carry particle-detector readings and stuff like that). The program does seem at least to define a clear and reasonable direction.

Rovelli says that recent results provide some indication that they might get the Einstein equation for the simple matterless case. He explains why he thinks they might. That's all, he doesn't say they got it yet.
...​

https://www.physicsforums.com/showthread.php?p=2855561#post2855561

sheaf:...Rightly or wrongly I'm impressed by the convergence of the various approaches. Also, being able to pull the Regge action out of that purely combinatorial framework sounds like good news to me. Even if all this, for the moment, only relates to the vacuum equations, that is an enormous achievement.

So yes, I'm watching all this with a great deal of interest.​

https://www.physicsforums.com/showthread.php?p=2858264#post2858264

ensabah6:So this new LQG is just a new SF model.​

https://www.physicsforums.com/showthread.php?p=2858714#post2858714

tom.stoer:No!

It's about convergence of canonical approach and spin foams; it's about mapping of or identities between certain entities in both frameworks; it's about making LQG accessable for calculations; it's about long-distance limit / semiclassical approximations; it's about consistency of quantization, implementation of constraints, regularization of the Hamiltonian (which is notoriously difficult in old-fashioned LQG) ...

... the more you read the more you will find.​

 

[marcus]

One point to make is that you can look at the spin network graph as a truncation of geometry. Doing physics requires approximation and people habitually think in terms of a truncated series. Some will assume a perturbation series even where there is none(!) and expect to be presented with finite initial segment. But there are other ways to truncate.

So that's one thing: start seeing a graph as a finite truncation of geometry. I'll give an example using something that anyone reading this probably knows: the 3D hypersphere S³---the 3D analog of S² the familiar 2D surface of a balloon.

For visual warmup I guess we could start with that simpler S² case. Here's a primitive graph for it:

(|)​

consisting of two nodes joined by 3 links, imagined dually as two equilateral triangles glued so as to make the S² surface of a balloon. The two nodes pictured as the North and South poles.

But that's not what I want. I really want a graph used to approximate S³. It could be two nodes ("the point here and the point at infinity") joined by 4 links. Here is bad drawing:

([])​

In a LQG graph the nodes can carry volume and the links represent adjacency and contact-area.
Links can represent area across which neighbor chunks of volume communicate.

So we can imagine this graph dually as two tetrahedra, each with 4 faces, and the faces glued so as to make it topologically the hypersphere.As Rovelli mentions in the April paper, a LQG graph can carry other stuff as well. The nodes carry volume, but can also carry fermions. The links carry area, but can also be labeled with Y-M fields.

Still, their primary job is to carry the most rudimentary basic geometry information.

If you picture a more complicated graph, you can imagine how a surface in manifoldless LQG is defined. You define it as a collection of links (the links which the surface cuts, see equation (6) on page 2).

So an LQG graph is a finite truncation of geometric relationships which in "first order" cases can look like a crude simplification, but can also look naturalistic if you add more nodes and links.

Now let's look at how this graph ([]) is applied in COSMOLOGY. You see its picture on page 4 of the March paper http://arxiv.org/abs/1003.3483

A lot of cosmology involves considering the universe to be spatially the hypersphere S³ so we could expect this. There is section III "The Cosmological Approximation". And then Section III A is about "Graph expansion". (Here "expansion" means analogous to expansion in a power series, not expansion of the universe. But that's coming.)

Now they want to study the expansion of the universe and they want to calculate a transition amplitude between two labeled ([]) graphs, one bigger than the other.
So you look on page 5 and you see a spinfoam connecting two ([]) graphs. The simplest imaginable spinfoam doing that! (Because this is like "first order" truncation.)
And they calculate a spinfoam vertex amplitude because that is how you do dynamics in LQG.
That is section III B about "Vertex expansion".

Actually 1003.3483 is a good companion paper to 1004.1780 because it presents the same manifoldless development of LQG with concrete examples---and without the references, footnotes, and motivating discussion. The March paper gives essentially the same manifoldless treatment of LQG, self-contained, and in some respects easier to learn from.
One should read both.

So a graph (the spin-network with nodes and links) can be a truncation of spatial geometry, but also a spinfoam (the 2-complex analog of a graph, with vertices, edges and faces) can define a truncation as well--of the dynamical evolution. And the authors calculate with it.

They get standard cosmology in the limit. The usual Friedman-Robertson-Walker model that cosmologists use.

 

[Fra]

Marcus, I have a question that perhaps you can answer, since you are well informed about the LQG program and it's neighbourhoods.

Your last post makes me again associate to the way I hoped LQG was before I learned it was not. But maybe there are some published versions or speculative connections to LQG that isn't standard-LQG?

Has anyone considered the following idea: To try to infer matter and matter interactions by considering two different INTERACTING spin-networks? What I mean is to associate the "truncated geometry" with the natural truncation that any observer has due to horizon and information capacity constraints? Then what would the rules be for interacting spin network? And would they possibly reveal non-gravitational interactions? This would be a possible natural link to be put ontop of matter, and there would be two view of it: one view is that somehow matter would be some additional stuff living in the spinnetworks (some additional complexity of some sort) but the other dual view would be simpler: simply that each material particle ENCODES a spinnetwork or a complex of them?

If there is anything like that I would be interested in that. So my this is what I have been "missing". I'm not sure if it exists but maybe you know?

Edit: a good thing with that idea is that LQG would not really be a "pure QG" theory anymore where you have to add manully the other interactions ontop, without matter Encoding the spinnetworks there would be no pure gravity eiter. It's just that if you don't acknowledge that the observer, encoding the relations of the geometry is in fact material and needs somewhere to encode it, it looks like a pure graivty scenario. But the non-gravitational character may possible we encoded in two such views interact. That would be great and it would laso be much close to my own visions. At least someone must have thought of this and aleast tried it and say ran into problems? I'd be interested to review that.

/Fredrk

 

[atyy]

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

 

[marcus]

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

The short answer is yes. The smooth manifold of diff. geom. is a set with complicated specialized structure. It's just one possible way to think of 4D spacetime--not the only one.

Atyy, you realize that Rovelli and the others do not say "manifoldless". The technically correct term for this presentation of LQG is "combinatorial"---that's the word used in the April paper.

I decided to say manifoldless because it gets across the salient point that, when presented this way, the theory has no set which you can identify with the spacetime continuum.

Labeled graphs (dubbed spin-networks) merely represent disembodied finite information.

In this mathematical presentation there is no set corresponding to the points of spacetime, or of space, or of the boundary of any region of space or spacetime. No continua, or continuums, however you say it. Only finite webs of information, which in a rather vague sense one can imagine resulting from a series of measurements (including particle detections) or from the preparation of an "experiment" involving geometry and matter.

The idea of the labeled graph is not to BE spacetime (perhaps with some particles in it) but to represent in a very concise way the state of knowledge---what we might be able to SAY.
Able to say, that is, about the initial and final conditions, or about the boundary conditions, on the basis of some finite bunch of data-taking.

So in this presentation of QG the continuum does not exist. I mean it is not presented as a mathematical object (a set with some structure described by other sets--the usual way math objects, such as for instance smooth manifolds, are described).

I call it a "manifoldless" presentation to emphasize that feature. If it weren't such an awkward mouthful I would say "smoothmanifold-less" because technically it's a smooth manifold that people usually mean when they say manifold and that's the element which has been eliminated from the picture.

 

[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.