What exactly is loop quantum gravity?

[nomisrosen]

I don't quite understand the properties of loop quantum gravity. I have searched around and have not come up with anything very helpful. I'm pretty good when it comes to understanding quantum mechanics and string theory, but please, NO MATH! I'm only in high school!

thanks

 

[tom.stoer]

It's difficult w/o math.

The problem with quantum gravity is that naive mechanisms to quantize gravity (which have been applied successfully to other fields) fail for gravity. That means that something fundamental has to be changed for quantum gravity.

There are different approaches to solve these problems, e.g.
a) string theory
b) asymptotic safety
c) loop quantum gravity (LQG)

I don't want to comment on a) and b) here.

Essentially LQG does the following: it introduces new variables which replace the (in GR) well-known metric that describes spacetime + curvature. This is pure math, so I don't want to go into details here, but what happens is that these new variables are rather close to fields that we know from gauge theory like QED and QCD. Indeed in a certain sense gravity looks rather similar to QCD, but there is one additional property of gravity that allows one to apply a second mathematical trick which essentially replaces the fundamental fields with something like "fluxes through surfaces" or "fluxes along circles". These surfaces and circles are embedded into spacetime.

The next step is again rather technical and it becomes possible due to so-called diffeomorphism invariance: one can get rid of the the embedding of circles and surfaces into spacetime. Instead one replaces these entities with a so-called spin network, i.e. a graph with nodes and links between nodes where each link and each node carries some numbers which represent abstract entities from which certain properies of spacetime can be reconstructed. You can think about spacetime as made of cells (I will soon tell you that you can't :-); each cell has a certain volume carried by a node; each cell has certain surfaces and the link between different nodes (sitting inside these cells) carry the areas of the surfcaes.

The problem with this picture is that one might think about these cells as sitting in spacetime - but this is fundamentally wrong: this picture is only due to the construction, but basically there is no spacetime anymore; all there is are nodes and links (and certain numbers attributed to nodes and links). Spacetime is no longer fundamental but becomes an entity emerging from the more fundamental graphs with their nodes and links. The graphs are called spin networks b/c the numbers they are carrying have properties well-known from spins. But this is a mathematical property only, it does not means that there are real spinning objects.

Compare this emerging spacetime to a water surface of a lake. We know that it consists of atoms, and as soon as we get this picture it is clear that there is no water between the atoms; the surface is only an emerging phenomenon, the true fundamental objects are the atoms. In the same sense the spin networks are the fundametal entities from which spacetime, surfaces etc. and their properties like volume, area, curvature etc. can be constructed. Dynamics of spacetime (which was curvature, gravitational waves etc. in GR) is replaced by dynamics of spin networks: within a given graph new nodes with new links can appear (there are mathematical rules, but I don't want to go into detail here).

The last puzzle I have for you is the fact that such a spin network is not a mechanical object which "is" spacetime. Instead quantized spacetime is a superposition of (infinitly many) spin networks. This is well-known in quantum mechanics; there is no reason why an atom should be in a certain state; we can achieve that via preparation or measurement, but in principle a single atom can be in an arbitrary complex quantum state which is a superposition of "an atom sitting here, an atom moving in a certain direction over there, an atom moving in this or that direction, ...".

So classical spacetime is recovered by two averaging process: first there seems to be a regime were this superposition of spin networks is peaked around a single classical spacetime, i.e. were one networks dominates the superposition of infinitly many spin networks; second from this single spin network one can reconstruct spacetime in the same sense as one can reconstruct the water surface from the individual atoms. But there may be different regimes (e.g. in black holes or closed to the big bang) where is classical picture and this averaging does no longer work. It may be that in these regimes all there is are spin networks w/o any classical property like smooth spacetime, areas, volume etc. It's like looking at a single atom: there is no water surface anymore.

Eventually this is why one started with this stuff: the classical picture of spacetime seems to become inconsistent when one tries to quantize it, i.e. when one defines these superpositins etc. These inconsistencies do not bother us as long as we talk about spacetime here, in the solar system etc. But they become a pain in the a... when we talk about spacetime near a singularity like a black hole or like the big bang. In order to understand these new non-classical regimes of spacetime a fundamentally new picture is required. This is what LQG (and other approaches) are aiming for: construct a new mathematical model from which well-known classical spacetime (like in GR) can be reconstrcuted, but which does not break down in certain regimes but remains well-defined and consistent.

He must so to speak throw away the ladder, after he has climbed up on it - Ludwig Wittgenstein

 

[marcus]

This is a really good introductory summary. I suggest proposing it for a BTSM forum FAQ. The only minor "typo" is that the usual term is "node" where you have said "knot".
(The two words come from the same etymological root and originally meant the same, but in normal technical language one says that a spin network consists of nodes and links.)
Otherwise this is great. Clearly and concisely giving just the essentials.

 

[tom.stoer]

Thanks a lot - and sorry for the "knots", of course it must read "node"!

If you like we can discuss and improve the text, so instead of writing new posts I can change the original text (new color).

 

[marcus]

... (new color).

I would suggest you not bother with new color. It's a minor typo of no consequence. If the mentors like the idea of FAQ for BTSM they will (I expect) do a final edit and move the text to wherever it needs to be.

As far as substantive changes, I can't think of anything to add or alter! We are lucky to have something both careful, relaxed, and readable like this is.

Say! maybe I will move the text to SA forum so people can discuss quietly.

 

[Kevin_Axion]

Nice introduction but what I really want to know is how close LQG is to becoming fully quantized? Does it even have a semi-classical description yet?

 

[tom.stoer]

Aren't these two completely different questions?

Of course LQG is fully quantized ... Hilbert space, operator algebra, inner product, path integral ... what is missing?

Yes, there are attempts for semiclassical / smooth / large-j limit and construction of coherent states; there are results regarding the graviton propagator; ...; but this is (afaik) work in progress and (at least for me) it seems to be too early to present physical results.

 

[Kevin_Axion]

Aha, yes they are different questions. That was my fault. Thanks for the response. What else is left for LQG to do?

 

[tom.stoer]

semiclassical limit for reasonable smooth spacetimes like BHs, FRW, ...
n-point functions for gravitons (on top of these spacetimes)
absence of infrared divergence
renormalization group
equivalence of canonical and covariant / spin foam quantization
matter degrees of freedom (gauge fields)
nature of cosmological constant (and Immirzi parameter)
phenomenology, experimental tests (*)

Everything but (*) is currently under investigation and I think for all topics the different research groups make good progress. There seems to be no major obstacle, no road block.

For (*) only very indirect results are to be expected, but I think LQG shares this with all other QG approaches, unfortunately.

 

[Kevin_Axion]

So they're not even close.

 

[tom.stoer]

close to what?

 

[Kevin_Axion]

To having a full and complete description of LQG.

 

[atyy]

semiclassical limit for reasonable smooth spacetimes like BHs, FRW, ...
n-point functions for gravitons (on top of these spacetimes)
absence of infrared divergence
renormalization group
equivalence of canonical and covariant / spin foam quantization
matter degrees of freedom (gauge fields)
nature of cosmological constant (and Immirzi parameter)
phenomenology, experimental tests (*)

Everything but (*) is currently under investigation and I think for all topics the different research groups make good progress. There seems to be no major obstacle, no road block.

For (*) only very indirect results are to be expected, but I think LQG shares this with all other QG approaches, unfortunately.

Also:

consistency of the infinite refinement limit (discussed eg. in http://arxiv.org/abs/1010.5437 and http://arxiv.org/abs/1010.5437 )

 

[atyy]

To having a full and complete description of LQG.

They could be close. String theory says that theories without gravity have gravity, so if LQG doesn't have gravity it could have gravity. (More seriously, some lattice gauge theories have a dual formulation as spin foams, and string theory famously says some gauge theories contain gravity).

However, for LQG as pure gravity, they are far. Eg. even the same authors can't decide how to interpret the semiclassical limit. Contrast http://arxiv.org/abs/1004.4550 and http://arxiv.org/abs/1105.0216v5 - these are not just any authors, but people who established the state of the art on the interpretation of the semiclassical limit http://arxiv.org/abs/0905.4082

 

[Kevin_Axion]

Yes but doesn't AdS/CFT require that the gauge theory be an N = 4 supersymmetric Yang-Mills theory i.e LQG isn't supersymmetric.

 

[tom.stoer]

Do 'they' have a full and complete description of QCD?

I mean they have a complete and consistent theory at hand, though much work has to be done to complete the details, to consolidate the proofs, to work out new tools. They address their open issues - that's what science is.

In QCD color confinement has opposed a theoretical justification, but hints and indications are available; nevertheless we believe in QCD as the correct theory of strong interactions.

The main issue is a lack of (*) b/c afaik no prediction (not post-diction) of LQG is available that could falsify LQG or decide between LQG and other QG approaches.

 

[tom.stoer]

Yes but doesn't AdS/CFT require that the gauge theory be an N = 4 supersymmetric Yang-Mills theory i.e LQG isn't supersymmetric.

In the string / m-theory context this is a rather striking result, but LQG uses a completely different quantization procedure and therefore you can't conclude that AdS/CFT results must apply to LQG; it could very well be that the asymptotic safety program tells us that even standard but non-perturbative quantization of GR is UV complete and fully consistent; it could even be that in a couple of years we have more than obe fully consistent theory of QG at hand.

 

[atyy]

Yes but doesn't AdS/CFT require that the gauge theory be an N = 4 supersymmetric Yang-Mills theory i.e LQG isn't supersymmetric.

AdS/CFT is thought to be braoder than that. However, that is the most celebrated case. Supersymmetric spin foams are studied in http://arxiv.org/abs/0710.3540 and http://arxiv.org/abs/1004.0672 .

 

[atyy]

In the string / m-theory context this is a rather striking result, but LQG uses a completely different quantization procedure and therefore you can't conclude that AdS/CFT results must apply to LQG; it could very well be that the asymptotic safety program tells us that even standard but non-perturbative quantization of GR is UV complete and fully consistent; it could even be that in a couple of years we have more than obe fully consistent theory of QG at hand.

Yes, but how about the duality between spin foams and lattice gauge theory? I don't know the exact restrictions for such a duality, but I know there's some correspondence.

Also, LQG is capable of quantizing 3D lattice BF theory (in some cases with additional stuff) , which reminds me of the 3D Chern-Simons (with supersymmetry and matter) ABJM case of AdS/CFT.

 

[Kevin_Axion]

AdS/CFT is thought to be braoder than that. However, that is the most celebrated case. Supersymmetric spin foams are studied in http://arxiv.org/abs/0710.3540 and http://arxiv.org/abs/1004.0672 .

That doesn't seem natural in terms of LQG. LQG is generally celebrated for not containing SUSY and other new parameters i.e extra-dimensions. So what is this Asymptotic Safety you and tom.stoer are talking about, has it had any reasonable progress?

 

[atyy]

That doesn't seem natural in terms of LQG. LQG is generally celebrated for not containing SUSY and other new parameters i.e extra-dimensions. So what is this Asymptotic Safety you and tom.stoer are talking about, has it had any reasonable progress?

It could be natural - eg. http://arxiv.org/abs/1103.6264 mentions emergent gravity.

I don't know about Asymptotic Safety. The state of the art is, I believe, still http://arxiv.org/abs/0805.2909 and the beautiful results of CDT. It is of course unclear what exactly CDT is connected to, except for gut feeling. I think the recent directions that are interesting are http://arxiv.org/abs/1012.3081 and http://arxiv.org/abs/1012.4280

 

[tom.stoer]

I think we are far away from understanding the relation between AdS/CFT / strings and LQG / spin networks / foams. Strings are both broader (matter degrees of freedom, ...) and narrower (gravity requires strings / swampland - no b/c LQG comes w/o strings but with gravity).

Thiemann published some new papers regarding "SUGRA-LQG"; I just started reading his n-dim LQG papers but had no time to check the SUGRA stuff.

All what we can say is that there seems to be a miraculous relation between classical gravity + QFT, LQG and strings. All these theories predict a fundametal relation regarding black holes, but none of them is complete (classical gravity + QFT is inconsistent, LQG cannot fix the Immirz parameter, strings only talk about BPS BHs).

The main problem for all these theories is that they lack experimental input and falsifiability (in practice, not in principle). We live in an age confronted with a paradigm shift in theoretical physics.

 

[tom.stoer]

It seems that both string theory and LQG will turn into theory-producing machineries (instead of theories). There seems to be no reason why not
- construct LQG-like theories in n dimensions (n>2 arbitrary)
- incorporate SUSY / SUGRA / gauge fields
- use higher spin / SU(n) groups
- use qantum deformation

The main difference is that all these theory-producing machines in string theory (they call it 'vacua') are related via dualities and relations like that, whereas in LQG you would do "ordinary construction of theories" w/o any relation between them.

 

[Kevin_Axion]

I see, thanks for the input. I really like LQG for it's simple geometrical solution but dislike it because of the Immirizi parameter (kind of seems like cheating). I also like string theory for it's success but I believe it's tackling way too many problems at once, I mean we don't even know the full extent of physics and String Theory doesn't really go far beyond the SM to be a TOE let alone a QG theory. How does string theory treat space-time anyways? Does it just add new dualities and degrees of freedom to construct a theory of QG?

 

[atyy]

The main problem for all these theories is that they lack experimental input and falsifiability (in practice, not in principle). We live in an age confronted with a paradigm shift in theoretical physics.

Oh, I hope not. That would be the end of physics. I think there's plenty of theory outside of quantum gravity, and also LQG already applies to condensed matter, and there is hope that strings will too. After all the renormalization group came via high energy, was truly understood in condensed matter, and then fed back into HEP for no longer being worried about sweeping infinities under the rug, and ideas of asymptotic safety.

 

[tom.stoer]

OK; I'll change my wording:

The main problem for all these theories is that they lack experimental input and falsifiability (in practice, not in principle). We live in an age confronted with a paradigm shift in (certain areas of) theoretical physics.

 

[nomisrosen]

I didn't understand a thing...thanks for trying!

 

[Kevin_Axion]

We can't leave you unsatisfied! What didn't you understand?

 

[nomisrosen]

the whole thing...im no physicist haha just an average high school student who happens to enjoy reading about physics!

 

[Kevin_Axion]

Well, do you firstly know concepts in Quantum Mechanics/Quantum Field Theory or General Relativity? These are essential to understanding Quantum Gravity i.e loop quantum gravity or string theory

 

[tom.stoer]

@nomisrosen: I reported this thread (my last post) b/c I think we hijacked this thread and lost you from the very beginning; I hope they can split this thread such that we can focus on your initial question and my response, post #2. Forget about everything else.

So what is unclear about #2? And what could be a good starting point based on your existing knowledge?

 

[marcus]

Tom, I agree the thread got off track. Here is post #6 by Kevin for example:

Nice introduction but what I really want to know is how close LQG is to becoming fully quantized? Does it even have a semi-classical description yet?

We got away from concentrating on the most basic explanation of what LQG is---we veered off into more technical evaluations of it.

So suppose we re-focus on math-less explanation for Rosen or some hypothetical high school student.

I will make an attempt similar to yours.

I will say to Rosen that that Einstein showed us 100 years ago that gravity is really geometry. Gravity is caused by the changing shape of space.

"Quantum" carries the idea of uncertainty, indefiniteness. Nature's resistance to being precisely pinned down. Nature's ability to be in a mixture or "superposition" of different states----so to speak one on top of the other.

So "quantum gravity" = uncertain geometry.

LOOP quantum gravity is basically an approach using spin networks to describe the uncertain geometry of the universe.

They started out using loops but soon found that a kind of spiderweb network worked better than simple loops. So loop QG is a bad name---they don't use loops. they should call it spin network QG. The name refers to the historical beginnings.

To understand LQG you have to concentrate on understanding spin networks.

A network is made of nodes and links. Nodes are the junctions where links meet. Nodes are labeled to represent a bit of volume and links are labeled by area numbers. Networks can be mixed or superimposed, so we can get vague chunks of volume glued together at blurry bits of area.

A network with enough nodes and links can represent quite a lot of geometry---the nodes provide places where particles can be located and the links indicate possible moves the particles can make. A labeled network represents a kind of simplified world, or the geometry thereof. Nodes and links have no specified location in some conventional space. They ARE location themselves. There is no conventional standardized space.

Another name for network is "graph". A spin network is a labeled graph where the node labels refer to vol and the link labels refer to area where adjacent volumes meet.

I think if I had to explain LQG to a high schooler I would begin by drawing examples of graphs on the blackboard, or scratch paper.
I'm not sure this explanatory tactic would work. It may be a dud (failure). I'll see what NomisRosen says.

 

[nomisrosen]

Marcus, thank you so much! That really helped. The uncertain geometry did it for me. But now, what are these nodes and spin networks made of..? Do they operate at the Planck scale?

Also, is there some sort of wave function of probability to know how this geometry might behave in a certain situation?

What determines how much space a node can give rise too? And of course, what is "outside the node"

...sorry for all the questions, answer what you

 

[atyy]

In quantum mechanics, the basic wavefunction is a position. An arbitary wavefunction is a superposition of positions. Additionally, observable quantities like position or momentum correspond to position or momentum operators. The probability of observing a particular momentum is given by the "product" of the wavefunction (superposition of positions) and the operator (momentum operator).

In LQG, the wavefunction is the spin-network, and the spin-network is in some sense geometry. So an arbitrary wavefunction is a superposition of geometries. An observable quantity like volume corresponds to a volume operator. The probability of observing a particular geometrical quantity like volume is given by a "product" of the wavefunction (superposition of geometries) and the operator (volume operator).

Here geometry means spatial geometry. However, gravity is spacetime geometry or the time evolution of spatial geometry. It is unknown how to describe the time evolution of spatial geometry in LQG, which remains a major problem.

I think the lesson of string theory is that not every geometrical object in the theory is necessarily spatial or spacetime geometry. So it has from time to time been suggested that the geometry of LQG represents not just space or spacetime, but possibly also matter. It is unknown which, if any, interpretations of the same mathematics as physical quantities will eventually work.

 

[tom.stoer]

However, gravity is spacetime geometry or the time evolution of spatial geometry. It is unknown how to describe the time evolution of spatial geometry in LQG, which remains a major problem.

That's only partially true. There are proposals for a time-evolution operator or Hamiltonian H which look consistent. In addition the spin foam formulation seems to avoid this problem completely; in addition they are working on a harmonization of these two formulations, canonical spin networks and spin foams.

So it has from time to time been suggested that the geometry of LQG represents not just space or spacetime, but possibly also matter. It is unknown which, if any, interpretations of the same mathematics as physical quantities will eventually work.

This is still a rather speculative idea - but of course it would be a highly appreciated major breakthrough

 

[ensabah6]

That's only partially true. There are proposals for a time-evolution operator or Hamiltonian H which look consistent. In addition the spin foam formulation seems to avoid this problem completely; in addition they are working on a harmonization of these two formulations, canonical spin networks and spin foams.


This is still a rather speculative idea - but of course it would be a highly appreciated major breakthrough

what do you think of the current work of including a spectral triple with LQG quantization? i.e NCG+LQG?

 

[tom.stoer]

what do you think of the current work of including a spectral triple with LQG quantization? i.e NCG+LQG?

Do you really mean NCG a la Connes or simply q-Deformation?

The latter one seems to be natural (forget about the reason for the SL(2,C) = 4-dim spacetime, take any symmetry group and study its spin foams w/o any reference to dimension; this includes SU, SO, SP?, E? and q-deformation)

Regarding NCG: I do not know enough about it, but it seems that it could spoil the simple picture of LQG; the Connes approach is rather special and seems to explain nothing (it simply replaces the standard model with a special NC geometry, but it can't explain why THIS gemometry, not something else); I would prefer to see matter and the cc emering from q-defomed / framed spin networks - but this could be wishful thinking ...

 

[marcus]

... The uncertain geometry did it for me. But now, what are these nodes and spin networks made of..? Do they operate at the Planck scale?

Also, is there some sort of wave function of probability to know how this geometry might behave in a certain situation?

What determines how much space a node can give rise too? And of course, what is "outside the node"

...

I think of spin networks as descriptors used to describe simplified geometry. So a spin network is analogous to a word, or a number. We don't need to ask "what is the number 3 made of?" or "what is the word mass made of?" I guess adding more and more nodes and links to the network is in some way analogous to adding more decimal places to a number---making the description more refined/accurate/realistic.

The bottom line is not "what is it made of?" but rather: does it work as a description? Is it the right way to diagram the uncertain geometric reality?

In answer to your first main question, YES the spin network description is supposed to work at planck scale!
This is, in fact, one of the principal goals of LQG research! To find a description of the world's uncertain geometry that continues to work in extreme circumstances (like extreme density, where classical geometry suffers a "singularity" and fails to make sense.)

Beyond that, and equally important, the aim is to have a description that predicts enough about the early universe to be TESTABLE. To be science (and not just myth or fairytale) it has to predict features that people can look for in the ancient light (the so-called microwave background or CMB). A good description must risk falsification by predicting some observable footprint in the oldest light. Or traces in something else, say neutrinos?, which might have been left over from the extreme density Planck era.

Your second main question was about describing behavior.

We can think of a spin network as describing an instantaneous state of geometry, so then we want to know how that evolves. Eventually we want to be able to talk about how geometry interacts with matter---so there is this general issue of behavior, spelled out in transition probabilities.

The descriptive tool used in LQG to get transition probabilities (from one spin network state to another) is called a spin foam.

A foam is like the moving picture of a changing network. If you picture a network as a spider web, then a foam is sort of like a honeycomb. Both are mathematical objects. In LQG, both have labels.

So your second question points in that direction: is there some sort of mathematical machinery to calculate transition probabilities, from one geometric state to the next? The answer is YES. There are spin foams which are the paths of evolution of spin networks, and techniques have been developed for calculating probability amplitudes.

Also I think at this point you find disagreement. Different LQG researchers have proposed different procedures for calculating transition amplitudes. There are unresolved issues about infinities that have to be ironed out. And some way must be found to TEST. If a theory does not risk falsification by making predictions about something that you can reasonably expect might be observed, then it is empty. Ways to test are beginning to be proposed, but are still controversial. There is a lot of work to do.

This week the biannual LQG conference is being held. To see the list of talks, and get an idea of the topics being researched, go here
http://www.iem.csic.es/loops11/
and click on the menu where it says "scientific program".

 

[atyy]

That's only partially true. There are proposals for a time-evolution operator or Hamiltonian H which look consistent. In addition the spin foam formulation seems to avoid this problem completely; in addition they are working on a harmonization of these two formulations, canonical spin networks and spin foams.


This is still a rather speculative idea - but of course it would be a highly appreciated major breakthrough

Yes indeed to both points. Maybe the problem has already been solved by Thiemann's old proposal, and it just isn't understood how to extract the right classical limit. I've often read that it's thought the Thiemann solution had the wrong classical limit, but I don't know the literature apart from isolated statements here and there in other papers. Do you know any papers that examine the classical/semiclassical limit of Thiemann's solution?

 

[nomisrosen]

I think of spin networks as descriptors used to describe simplified geometry. So a spin network is analogous to a word, or a number. We don't need to ask "what is the number 3 made of?" or "what is the word mass made of?" I guess adding more and more nodes and links to the network is in some way analogous to adding more decimal places to a number---making the description more refined/accurate/realistic.

The bottom line is not "what is it made of?" but rather: does it work as a description? Is it the right way to diagram the uncertain geometric reality?

I don't really understand how the universe's geometry can be something real and yet not be made of anything...

Are the links basically like guidelines of space that nothing can cross (as in the only way to get from one point in space to another)?

Is this a good visualization of how matter interacts with spin foam?


Thanks again for all your clarifications.

 

[marcus]

I don't really understand how the universe's geometry can be something real and yet not be made of anything...

This is something in our culture, Simon. It leads to a problem. If something is only real if it is made of something else, where does it stop?

Is this a good visualization of how matter interacts with spin foam?

That is a thoughtprovoking visualization. I wouldn't think of it as literal fact, but as suggestive (a simplified 2D toy universe.) Plus I personally can't vouch. I'm an observer from the sidelines. I am not an expert. I don't do LQG research. It is exciting and interesting so I watch it. Thanks for the link.

Ultimately what matters is HOW NATURE RESPONDS TO MEASUREMENT. How we interact with it. Ultimately you cannot continue to explain nature by saying what it is "made of". There are limits to what we can measure. Certain things lose their operational meaning past Planck scale, if you cannot measure them. Anyway that is what I think.

So I want fundamental descriptions (of interaction and geometric relationship), I do not expect "this made of that" answers.

 

[marcus]

A spin network is not made of anything. It describes a situation you might think of discovering by making measurements. Put a bag around this and these and this and these nodes and your bag will now contain just that much volume. Add up the volume labels contained in the bag---there is one associated with each node.

Plus you can determine the area surrounding the region by adding up the cut links. The bag surrounding that bunch of nodes has to pass through a bunch of links that connect the inside with the outside. Or think of the links as pins puncturing the bag. All those links are labeled, so add up the labels to determine area. I know this seems a bit vague but it actually works pretty well to specify the "skeleton" of a geometry. You can even get angles (as well as areas and vols) from the labels.

The labels are simple enough: whole numbers and half-integers like 1/2, 1, 3/2, 2, 5/2,... They encode the geometry that lives on a particular network or skeleton.

 

[negru]

Only Marcus would find a made up animation based on absolutely nothing to be "thought provoking" ;)

Look he also has a video of color confinement!
http://www.youtube.com/user/DavidMTRutten#p/a/u/0/dvmE1XKZiwI

 

[marcus]

I notice folks still go to this thread, so it might be helpful to bring it up to date. Particularly as regards the new formulation and the OPEN PROBLEMS relating to it that various people have listed.
When I say LQG I mean of course the new formulation that uses spin foam to calculate transition amplitudes between quantum states of geometry.

A quantum state of boundary geometry (e.g. initial and final quantum states) is determined by a network of measurements (e.g. angles, distances, areas...) represented by a labeled graph.

The probability of the implied transition between boundary states is given by an amplitude calculated from the foam (a honeycomb-like "cell complex") enclosed by the boundary. Mathematically a foam is analogous to a graph but at one higher dimension. Instead of merely having nodes and links, it has vertices, edges and faces. Labeled with quantum numbers, a foam describes a possible way that geometry can evolve from one 3D geometric quantum state to another.

I'll refer to the new LQG formulation as the Zakopane formulation. It appeared in a series of 4 papers:
A New Look at LQG... (April 2010) 1004.1780
Geometry of LQG... (May 2010) 1005.2927.
Simple Model... (October 2010) 1010.1939
Zakopane Lectures (February 2011) http://arxiv.org/abs/1102.3660

A remarkable thing about the new formulation developed in these papers is its concise easy-to-understand presentation. A clear definite description of the theory can be given in one page. The fourth paper also gives the math prep needed to appreciate the one--age formulation, making it self-contained. The fourth is basically an improved version of the first and these two papers present a list of OPEN PROBLEMS for researchers to tackle. Several of these are areas where progress is currently being made. The listed problems are primarily conceptual in nature, having to do with the theory itself. There is also current activity in cosmology, investigating ways to test the theory by comparing its predictions with observation.

Another window on interesting conceptual problems, for someone getting into LQG research at graduate or postdoc level, is the October 2011 paper by Freidel Geiller and Ziprick which reveals the classical continuum phase space discretization that Zakopane LQG is the quantization of.
http://arxiv.org/abs/1110.4833
Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
(Submitted on 21 Oct 2011)

In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables...
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For various reasons, the conceptual importance of this paper is hard to overestimate.