Signs LQG has the right redefinition (or wrong?)

marcus

LQG was redefined in 2010. The essentials are summarized in two December papers,
http://arxiv.org/abs/1012.4707
http://arxiv.org/abs/1012.4719
What indications do you see that this was the right (or wrong) move?
How do you understand the 2010 reformulation? How would you characterize it?
What underlying motivation(s) do you see?

marcus

As a footnote to that, there will be the 2011 Zakopane QG school the first two weeks of March. Rovelli has 10 hours of lecture, presumably to present his current understanding of the theory at a level for advanced PhD students and postdocs wanting to get into LQG research. This will be, I guess, the live definitive version.

People who coming fresh to this subject should realize that the LQG redefintion relies heavily on analogies with QED and QCD---Feynman diagram QED and lattice gauge QCD. N chunks of space instead of N particles. The graph truncation. The 2-complex ("foam") analog of the 4D lattice.

Also that the formulation does not depend on a smooth manifold or any such spacetime continuum. The graph need not embedded in a continuum (although it may optionally be so at times to accomplish some mathematical construction). To me, the graph represents a restriction of our geometric information----symbolically to a finite set of instruments/readings, or a finite set of chunks of space that we know about.

Or when talking about much smaller scale, a finite set of geometric elements we can infer something about (if not directly probe with macro instuments.)

A 2-complex ("foam") is just the one-higher-dim. analog of a graph. Instead of being a combination of 0 and 1-dimensional nodes and links, a 2-complex is the analogous combination of 0 and 1 and 2-dimensional vertexes, edges, and faces.

A graph can serve as a boundary of a 2-complex (or "foam"). If the graph comes in two separate components, the initial and the final. Then the foam can describe a possible way that the initial graph component evolves into the final. Presumably one of many possible evolutionary paths or histories.

What we are talking about is the evolution of geometric information. Probably the simplest way of talking about this that one can imagine.

There is no smooth manifold in the picture, in part simply to establish that a smooth manifold exists would require an uncountable infinity of physical measurements. It is too great an assumption to make about the world. The spirit of quantum mechanics is to concentrate on what we can actually observe and measure---the interrelationships between pieces of information. and how these evolve.

This is probably the reason that QG has gradually settled down to a manifoldless definition. In the redefined LQG there is no spacetime (in and of itself) there is only "what we can say about it"--a web of geometric info. Some measured or inferred volumes, areas, angles...

Now QED, for instance, needs to be redefined on this web of geometry---no longer should it be defined on a manifold. Information should be located on information, and by information. What we can say, not what "is".

It is this redefinition which one sees beginning to happen in the other December paper I mentioned, called "Spinfoam Fermions" http://arxiv.org/abs/1012.4719

MTd2

Do you think LQG will require a principle of relative locality? Do you think this

http://arxiv.org/abs/1101.3524

has anything to do with that?

marcus

I think the job of the theorist is to develop testable theories which are possible to be right.

Put yourself in Freidel's place. It is not his job to "believe" theories (whatever that means.)

The January "Rel. Loc." paper argues that Rel. Loc. is testable. It can be falsified if one finds that the momentum algebra is flat. It is very interesting. Extremely.

Also LQG has changed enormously in the past year, or several years, and is extremely interesting. It is also falsifiable.

Trying now to reconfigure LQG so that it would fit into the Rel Loc philosophy is AFAICS premature speculation. What makes sense to me, now, is develop and test them both so that we have a better idea of how reality is structured. Maybe one or the other can be falsified!

have to go

MTd2

So, it was just a coincidence that he used a restriction in the phase space, in the new paper, right?

marcus

Please help me out with more specifics. Page references even! You must be talking about Freidel and the Rel Loc paper. I haven't studied it. Point me to a paragraph on some page, or to some equation in the Rel. Loc. paper.

My eyes get tired looking thru stuff to find what somebody is talking about.

I think Freidel is great and I am waiting to hear his online March 1 seminar talk about Relative Locality.
And since the International LQG Seminar connects a halfdozen places around the world and they can all ask questions I am waiting to hear what questions Freidel gets from people in PennState, Perimeter, Marseille, Nottingham, Warsaw....

The only trouble is 1 March is also the first day of the Zakopane school and both Ashtekar and Rovelli are scheduled to give 2 hour lectures on that same day. So there is a huge time conflict. The school is important and also Freidel's talk is. How they ever managed to schedule it like that is beyond my comprehension.

MTd2

Look at the abstract of

http://arxiv.org/abs/1101.3524

"We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. ... This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity."

http://arxiv.org/abs/1101.0931

p.2
"Physics takes place in phase space and there is no invariant
global projection that gives a description of processes in
spacetime. From their measurements local observers can
construct descriptions of particles moving and interacting
in a spacetime, but different observers construct different
spacetimes, which are observer-dependent slices of phase
space."

Sounds like that LQG makes sense only with relative locality, sort of, at least in 3d.

marcus

I see what you are driving at. Thanks for the detailed reference. I'm not going to agree or disagree yet because I don't understand Relative Locality well enough. The way I imagined it, in the Rel Loc paper the momentum space that was critical was that of matter particles. The crucial question was whether or not material momenta added in a flat vectorspace way. Was the matter momentum space curved or not? The phase space that was at issue in Rel Loc included matter. That is how I was thinking.
In the 3D paper you cited, the topic is pure gravity, no matter. Or am I missing something?
The connection is too tenuous for me to follow, at this point. Maybe someone else can respond more helpfully.

MTd2

Yes, no matter. But you were talking about redefinitions, I took it as being general redefinitions about the fundamentals of the theory! Lol, I guess I went off topic. I don't know, maybe a new thread is required? I don't know how to put it.

marcus

I suppose what you are asking about is relevant (at least eventually). I am simply not prepared to respond in any useful way. It's natural to ask in what way is LQG compatible with the Rel. Loc. principle? Is it even compatible at all? Or are they in spirit quite close? It seems to me natural that people would be asking such questions at the ILQGS on March 1, if indeed Freidel gives the scheduled talk, and if the others are available to listen and comment.

If you can't get a satisfactory discussion here and now, then you or I must start a thread about this issue after Freidel's talk (presumably 1 March).

Ultmately it comes down to empirical tests. Rel Loc is testable by testing the addition of particle momenta and suchlike stuff. LQG is testable because of its robust prediction of a cosmological bounce, some bearing on inflation, related features of Cmb. But although neither can be assumed a priori true, their mathematical (in)compatibility is surely an interesting question.
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Instead of talking about Rel Loc now, what I want to do is quote some of the 4707 paper where he points out analogies with QED and QCD. He gives the full definition of LQG in three equations and half a page, and then he starts with some motivation:

This is the theory. It is Lorentz invariant [18]. It can be coupled to fermions and Yang-Mills fields [19], and to a cosmological constant [20, 21], but I will not enter into this here. The conjecture is that this mathematics describes the quantum properties of spacetime, reduces to the Einstein equation in the classical limit, and has no ultraviolet divergences. I now explain more in detail what the above means.

A. Quantum geometry: quanta of space
A key step in constructing any interactive quantum field theory is always a finite truncation of the dynamical degrees of freedom. In weakly coupled theories, such as low-energy QED or high-energy QCD, we rely on the particle structure of the free field and consider virtual processes involving finitely many, say N, particles, described by Feynman diagrams. These processes involve only the Hilbert space H_N = ⊕_n=1,N H_n , where H_n is the n-particle state space.

In strongly coupled theories, such as confining QCD, we resort to a non-perturbative truncation, such as a finite lattice approximation. In both cases (the relevant effect of the remaining degrees of freedom can be subsumed under a dressing of the coupling constants and) the full theory is formally defined by a limit where all the degrees of freedom are recovered.

The Hilbert space of loop gravity is constructed in a way that shares features with both these strategies. The analog of the state space H_N in loop gravity is the space

H_Γ = L₂[SU(2)^L/SU(2)^N]. (4)

where the states ψ(h_l) live. Γ is an abstract (combinatorial) graph, namely a collection of links l and nodes n, and a “boundary” relation that associates an ordered couple of nodes (s_l, t_l) (called source and target), to each link l. (See the left panel in Figure 1.) L is the number of links in the graph, N the number of nodes, and the L₂ structure is the one defined by the the Haar measure. The denominator means that the states in H_Γ are invariant under the local SU(2) gauge transformation on the nodes

ψ(U_l)→ψ(V_sl U_lV_tl⁻¹), V ∈ SU(2), (5)

the same gauge transformation as in lattice Yang-Mills theory.

States in H_Γ represent quantum excitation of space formed by (at most) N “quanta of space”. The notion of “quantum of space” is basic in loop gravity. It indicates a quantum excitation of the gravitational field, in the same sense in which a photon is a quantum excitation of the electromagnetic field. But there is an essential difference between the two cases, which reflects the difference between the electromagnetic field in Maxwell theory, and the gravitational field in general relativity: while the former lives over a fixed (Minkowski) metric spacetime, the second represents itself spacetime.

Accordingly, a photon is a particle that “lives in space”, that is, it carries a momentum quantum number ⃗k, or equivalently a positions quantum number ⃗x, determining the spatial localization of the photon with respect to the underlying metric space. The quanta of loop gravity, instead, are not localized in space. Rather, they define space themselves, as do classical solutions of the Einstein equations.

More precisely, the N “quanta of space” are only localized with respect to one another: the links of the graph indicates “who is next to whom”, that is, the adjacency link that defines the spatial relations among the N quanta. (See the right panel in Figure 1.) Thus, these quanta carry no quantum number such as momentum ⃗k or position ⃗x.

Rather, they carry quantum numbers that define a quantized geometry, namely a quantized version of the information contained in a classical (three- dimensional) metric. The way this happens is elegant, and follows from a key theorem due to Roger Penrose, called the spin-geometry theorem, which is at the root of loop gravity [22]. I give here an extended version of this theorem,...

tom.stoer

I don't think that LQG has been redefined.

Rovelli states that it is time to make the next step from the construction of the theory to the derivation of results. Nevertheless the construction is still not complete as long as certain pieces are missing. Therefore e.g. Thiemann's work regarding the Hamiltonian approach (which is not yet completed and for which the relation to spin foams is still not entirely understood) must still back up other programs

There are still open issues to be solved:
- construction, regularization and uniqueness of the Hamiltonian H
- meaning of "anomaly-free constraint algebra" in the canonical approach
- relation between H and SF (not only kinematical)
- coarse-graining of spin networks, renormalization group approach
- nature and value of the Immirzi parameter
- nature and value of the cosmological constant
- nature of matter and gauge fields (on top, emergent, ...); yes, gauge fields!
And last but not least: If a reformulation is required (which would indicate that the canonical formalism is a dead end), then one must understand why it is a dead end! We don't know yet.

My impression that Rovelli's new formulation does not address all these issue. His aim is more to develop calculational tools to derive physical results in certain sectors of the theory.

Let's look at QCD: there are several formulations of QCD (PI, canonical, lattice, ...), every approach with its own specific benefits and drawbacks. But nobody would ever claim that QCD has been reformulated (which sounds as if certain approaches would be out-dated). All approaches are still valid and are heavily used to understand to understand QCD vacuum, confinement, hadron spectroscopy, QGP, ... There is not one single formulation of QCD.

So my conclusion is that a new formulation of LQG has been constructed, but not that LQG has been reformulated.

oldman

To inject a shallow note into this deep thread:

What, in this context, can we say about "Nothing at all" --- the Vacuum, about which Peacock made the comment (in his " Cosmological Physics):

"It is perhaps just as well that the average taxpayer, who funds research in physics, is unaware of the trouble we have in understanding even nothing at all" ?

In Loop Quantum Gravity abstract graphs are often sketched of vertices (drawn as dots) connected by edges (drawn as lines) that represent "what we can say" about the dimensional circumstances we live in. An example is Fig. 1 of Rovelli's Christmas review that was linked to in the original post of this thread.

The simplest thing we can say about the vacuum seems to be that it is quite symmetric; here is the same as there, and now is no different from then, as far as the vacuum is concerned. That's why we expect the laws of physics to be covariant in what we call spacetime.

Yet abstract graphs that are drawn, like Rovelli's, show no symmetry at all. They're lopsided and skew, as well they might be when gravitating matter or interacting fermions are involved. If they were drawn to represent the Vacuum (or perhaps a time average of it) wouldn't these graphs be more symmetric, perhaps even lattice-like? Lots of symmetries to explore then. Which brings me to ask: if this is so, what is it that makes or keeps the Vacuum so symmetric and, in the absence of localised mass/energy, spatially flat? Non-localised energy that can't be detected? Or something else that everybody except me understands?

marcus

I think I see now the distinction you are making between a new formulation
and a re formulation.

Personally I do not suspect that the Hamiltonian approach is a dead end. We cannot know the future of research, but my expectation is that people will continue to work on completing the Hamiltonian approach and it will ultimately prove equivalent.

It might (at that future point in history) look different, of course. There might, for example, be no smooth manifold, no continuum, the spinnetworks (if they remain in the Hamiltonian formulation) might not be embedded. Or they might be. I don't see us as able to predict how the various versions of the theory will look.

But as an immediate sign that the Ham. approach is not yet a dead end, there is the Freidel paper that was just posted two days ago.

Plus, some of the other things you mentioned remain interesting and important open problems (in whatever formulation one confronts them), such as:

marcus

I suppose that one reason for the power of General Rel is that it is general. One can have solutions with no recognizable symmetry at all.

To be a satisfactory quantum version of GR, Loop must imitate that basic feature.

Of course it is technically possible to confine LQG to an approximately flat sector. This has been done in the "graviton propagator papers" circa 2007.
====================

Had to leave abruptly to take care of something else, before finishing. Back now.
The thing about your post is that it raises intriguing questions.

BTW you mentioned the Christmas review paper. That gives one formulation of the theory, in 3 equations. He says clearly there are other formulations and he is just giving his understanding of what LQG is---so in that sense he seems to agree with Tom Stoer. Indeed the paper goes over OTHER formulations in a later section, fairly extensively----BF theory, GFT, canonical Hamiltonian style, versions using manifolds and so on.

But I find it makes discussion simpler to focus on the one current formulation. Which you may have in mind since you mentioned the recent review paper (1012.4707).

In that case one should observe that the graphs are purely combinatorial. It doesn't matter how they are drawn---with long curly lines or short wiggly lines---or lopsided with all the nodes but one off by themselves in a corner. The visual characteristics of the graph are for the most part inconsequential.

I guess the important things to communicate is that a graph is purely combinatorial and quite general. It could have 2 nodes and 4 links, or it could have billions of nodes and billions of links. It has no special symmetry. The way of treating it mathematically is supposed to be the same whether it has 2 nodes or a trillion nodes.

Combinatorial means it consists of two finite sets and two functions.
NODES = {1,2,3,...N}
LINKS = {1,2,3,...L}
s: LINKS ->NODES
t: LINKS -> NODES

The auxilliary functions s and t are the source and target functions that, for each link, tell you where that link starts from and where it ends up.
For a given link l, the two nodes that link connects are s(l) and t(l).

It's like the minimum math info that could define an oriented graph. The symbol for that simple combinatorial info is gamma Γ.

What i think is the great thing about it is that it allows you to define a Hilbertspace H_Γ and do non-trivial stuff. The Hilbertspace has gauge symmetries specified by Γ

Remember that gauge symmetries are symmetries in our information, how it is presented, they are not real material symmetries of a physical situation.

The graph Γ is very much about how we sample the geometric reality of nature (or so I think anyway). It is about what degrees of geometric freedom we capture. (and which others we perhaps overlook.) My interpretation could be quite wrong---it is certainly not authoritative.

There is another interpretation----nodes as "exitations of geometry". N nodes is analogous to a Fock space where there are N particle, say N electrons. In that case the "real" universe would correspond to a graph with a HUGE number of nodes and links. But we develop the math to treat any number. And we deal with examples of small N. You can find that interpretation clearly presented in the Christmas summary paper.

Either way, there is no need for small example graphs to look like anything in particular.
I think they should be, if anything, arbitrary and irregular---to suggest the generality.

oldman

Thanks for this. I guess I was being too fussy about the RHS of figure 1 in Rovelli's paper, with it's superimpsed "grains of space". It reminded me of overinterpreted representations of atoms with whirling electrons trailing smoke. I liked when you earlier said it's all about:'What we can say, not what "is" ' . Just as Niels Bohr believed.

marcus

It still isn't completely clear to me how to think of LQG, but it is getting clearer. I'm glad it is so for you as well. The December review paper is well written, I think.

Here is another enlightening short paragraph. It comes on page 6 after he has finished describing the theory (by stating 3 equations on page 2 and then discussing what they mean, with background etc. Then when that is all done, he says:

This concludes the definition of the theory. I have given here this definition without mentioning how it can be “derived” from a quantization of classical general relativity. This is like defining QED by giving its Hilbert space of free electrons and photons and its Feynman rules, without mentioning either canonical or path integral quantization. A reason for this choice is that I wanted to present the theory compactly. A second reason is that one of the main beauties of the theory is that it can be derived with rather different techniques, that involve different kinds of math and different physical ideas. The credibility of the end result is reinforced by the convergence, but each derivation “misses” some aspects of the whole. In chapter IV below I will briefly summarize the main ones of these derivations. Before this, however, let me discuss what is this theory meant to be good for.

It seems significant to me that no single "derivation" is perfect. The various roads to the present formulation converge but none are complete. The final form of the theory, he seems to be saying, is an educated guess.

Different roads up the mountain, all converging towards the peak...but none quite reaching, so in the end one takes the helicopter. The "derivations" have been valuable to give heuristic guidance, motivation, understanding....but one should not be too tied to the rituals. To repeat a key comparison:

This is like defining QED by giving its Hilbert space of free electrons and photons and its Feynman rules, without mentioning either canonical or path integral quantization.

Well, perhaps that would have been all right! Not only as an essay's expository plan but as an alternative historical line of development. Perhaps the canonical and path integral quantization could have been skipped and then reconstructed after the fact. If by some fluke the Feynman rules had been discovered first. A not entirely serious speculation.

In case anyone is new to the discussion, the recent review of LQG (December 2010) is http://arxiv.org/abs/1012.4707

marcus

Earlier, Tom gave us a good list of unresolved (or only partially resolved) issues in LQG.

I think there are signs that the theory has the right (or a right) redefinition, as given in the December 2010 overview paper http://arxiv.org/abs/1012.4707

I will mention a few of the signs I see of this, but first to mention one very positive sign that just appeared: this is in response to the Lambda issue, the cosmological constant issue, that Tom indicated.

http://arxiv.org/abs/1101.4049
Cosmological constant in spinfoam cosmology
Eugenio Bianchi, Thomas Krajewski, Carlo Rovelli, Francesca Vidotto
4 pages, 2 figures
(Submitted on 20 Jan 2011)
"We consider a simple modification of the amplitude defining the dynamics of loop quantum gravity, corresponding to the introduction of the cosmological constant, and possibly related to the SL(2,C)_q extension of the theory recently considered by Fairbairn-Meusburger and Han. We show that in the context of spinfoam cosmology, this modification yields the de Sitter cosmological solution."

This paper finds a nice natural place for the cosmo constant, and does not resort to the quantum group or q-deformation.

Note that it partly addresses the classical limit issue, since spinfoam cosmology uses the full theory and it is now giving a familiar DeSitter universe as largescale limit.