Loop-String goodies in TWF 281 (that MTd2 spotted)

[marcus]

Another installment by John Baez on the September 2008 Corfu QG School.
Thanks to MTd2 for noticing this and supplying the link.
There is a part that is so informative we should have a brief excerpt to look at closely:
==quote Baez==
...
...But now - on to Corfu!

Last time I said a bit about what I learned in Ashtekar and Rovelli's courses. Now I'd like to talk about some other things I learned in Corfu - some things I find even more tantalizing.

In "week232", I explained how gravity in 3d spacetime automatically contains within it a theory of point particles, and how a 4d analogue of 3d gravity automatically contains within it a theory of string-like objects. This 4d theory is called BF theory. Like 3d gravity, it describes a world where spacetime is flat. So, it's boring compared to full-fledged 4d gravity - so boring that we can understand it much better! In particular, unlike 4d gravity, we understand a lot about what happens when you take quantum mechanics into account in 4d BF theory.

But when you remove a surface from spacetime in 4d BF theory, it springs to life! In particular, the surface acts a bit like the worldsheet of a string. It doesn't behave like the strings in ordinary string theory. But Winston Fairbairn has been thinking about this a lot:

17) Winston J. Fairbairn and Alejandro Perez, Extended matter coupled to BF theory, Phys. Rev. D78:024013, 2008. Also available as arXiv:0709.4235.

18) Winston J. Fairbairn, On gravitational defects, particles and strings, JHEP 0809:126, 2008. Also available as arXiv:0807.3188.

19) Winston J. Fairbairn, Karim Noui and Francesco Sardelli, Canonical analysis of algebraic string actions, available as arXiv:0908.0953

And it turns out that if we impose the constraints on BF theory that turn it into general relativity, we obtain the usual Nambu-Goto string, where the action is the area! However, the last of the three papers above shows there are some subtle differences.

I need to think about this a lot more. It was always my hope to reconcile string theory and loop quantum gravity, and this could be the way. Of course, reconciling two things that don't work doesn't necessarily give one that does. A pessimist might say that combining string theory and loop quantum gravity is like combining epicycles and aether. But I'm optimistic. Something interesting is going here.

In a different but possibly related direction, Aristide Baratin gave a talk on recent work he's been doing with Derek Wise and Laurent Freidel. You can get a feel for this work from this paper:

20) Aristide Baratin, Derek K. Wise, 2-Group representations for spin foams, to appear in proceedings of the XXV Max Born Symposium: The Planck Scale, Wroclaw, Poland. Also available as arXiv:0910.1542.

In "week235" I mentioned an amazing paper by Baratin and Freidel called "Hidden quantum gravity in 4d Feynman diagrams: emergence of spin foams". They described a spin foam model that acts just like 4-dimensional flat Minkowski spacetime: couple it to interacting point particles, and you get the usual Feynman diagrams described in a new way!

The big news is that this spin foam model comes from the representations of a 2-group, instead of a group. Namely, the Poincaré 2-group. This is a 2-group I invented which has Lorentz transformations as objects and translations as endomorphisms of any object.

The Poincaré 2-group spin foam model was first studied by Crane, Sheppeard and Yetter. Baratin, Freidel, Wise and I spent a long time developing the theory of infinite-dimensional representations of 2-groups needed to make this model precise - see "week274" for more on all this. Now the details are falling into place, and a beautiful picture is emerging.

I should admit that the paper by Baratin and Wise deals with the Euclidean rather the Lorentzian version of this picture. I hope this is merely because the representation theory of the "Euclidean 2-group" is more tractable than that of the Poincaré 2-group. I hope everything generalizes to the Lorentzian case.

A lot to think about...
==endquote==

More at http://math.ucr.edu/home/baez/week281.html

 

[ccdantas]

A pessimist might say that combining string theory and loop quantum gravity is like combining epicycles and aether. (John Baez)

This is a gem.

 

[ccdantas]

This 4d theory is called BF theory.

He referred to the quantized model, right?

 

[marcus]

Christine, I think there are both classical and quantum stages of this. I am not clear how to reply to your question but I would like us to take a look together at how the program is set out in the intro of the incept paper.
==quote Fairbairn Perez 2007==
...Although conceptually very deep, these results remain three-dimensional.

The next step is to probe all possible extensions of these ideas to higher dimensions. Two ideas have recently been put forward.

The first is to consider that fundamental matter is indeed pointlike and study the coupling of world lines to gravity by using the Cartan geometric framework[6] of the McDowell-Mansouri formulation of gravity as a de-Sitter gaugetheory[7].

The second is to generalize the description of matter as topological defects of spacetime
curvature to higher dimensions. This naturally leads to matter excitations supported by co-dimension two membranes [1], [9].

Before studying the coupling of such sources to quantum gravity, one can consider,
as a first step, the BF theory framework as an immediate generalization of the topological character of three-dimensional gravity to higher dimensions.

This paper is dedicated to the second approach, namely the coupling of string-like sources to BF theory in four dimensions. The starting point is the action written in [1] generating a theory of flat connections except at the location of two-dimensional surfaces, where the curvature picks up a singularity, or in other words, where the gauge degrees of freedom become dynamical.

The goal of the paper is two-fold. Firstly, acquire a physical intuition of the algebraic fields involved in the theory which generalize the position and momentum Poincaré coordinates of the particle in three-dimensions. Secondly, provide a complete background independent quantization of the theory in fourdimensions, following the work done in [1].

The organization of the paper is as follows. In section II, we study some classical solutions guided by the three-dimensional example. We show that some specific solutions lead to the interpretation of rigid strings propagating on a flat spacetime. More generally, we prove that the solutions of the theory are in one-to-one correspondence with distributional solutions of general relativity.

In section III, we propose a prescription for computing the physical inner product of the theory. This leads us to an interesting duality between the obtained transition amplitudes and Feynman diagrams coupled to three-dimensional gravity. We finally prove in section IV that the transition amplitudes only depend on the topology of the canonical manifold
and of the spin network graphs.
==endquote Fairbairn Perez==

The layout of the classical BF theory begins around page 3. The quatization begin around page 8.

Reference [1] is to the 2006 Baez Perez paper.

For further possible clarification, I will quote part of the concluding paragraph.

==quote Fairbairn Perez==
In the first part of this paper we have studied the geometrical interpretation of the solutions of the BF theory with string-like conical defects. We showed the link between solutions of our theory and solutions of general relativity of the cosmic string type. We provided a complete geometrical interpretation of the classical string solutions and explained (by analyzing the multiple strings solution) how the presence of strings at different locations induces torsion. In turn torsion can in principle be used to define localization in the theory.

We have achieved the full background independent quantization of the theory introduced in [1]. We showed that the implementation of the dynamical constraints at the quantum level require the introduction of regulators. These regulators are defined by (suitable but otherwise arbitrary) space discretization. Physical amplitudes are independent of the ambiguities associated to the way this regulator is introduced and are hence well defined...
==endquote==

 

[MTd2]

Marcus, I don't know if you agree, but that new Krasnov paper seems to be the most promising one in theoretical physics in the last 40 years. Maybe I say that because I am too tired...

Would you comment that paper?

 

[marcus]

Sure, I am glad to have an invitation to comment on the new Krasnov paper. Morally speaking this is your thread (you brought up Baez discussion of new QG papers in TWF 281)
so if you want to turn the spotlight on to Krasnov's I cannot possibly object.

I think Krasnov now has the problem often faced by an innovator---he has a possibly important new idea and he must communicate it. He already presented this idea 11 months ago in http://arxiv.org/pdf/0811.3147
But the paper is not very readable. It does not capture in brief why the proposal is so interesting. (Creative people often have this problem communicating the interest.)
The November 2008 paper is called
Plebanski gravity without the simplicity constraints
To me, this does not seem to electrify the imagination, although it is an accurate title from a logical point of view.

The next month, Krasnov's friend Laurent Freidel essentially showed him how to communicate the idea. He posted a commentary to it, on arxiv. But may not have submitted it to journal.
http://arxiv.org/abs/0812.3200
Modified gravity without new degrees of freedom
The "payload" point, that sells the idea, is that here is a generalization of Einstein gravity that you can force to be as close to standard Einstein as you want but can also differ from it and it introduces no new degrees of freedom. It is still clean. It generalizes without adding complexity.
There is still just the spin-two graviton---still only the two degrees of freedom we started with. No more.

So now, almost a year later, Krasnov has thought hard how to get this idea across and make it visible. Could it be published in PRL (physical review letters)? So he has put it in a very concise four page version with the more forceful title
http://arxiv.org/abs/0910.4028
Metric Lagrangians with two propagating degrees of freedom

The point of the title is that there are no new degrees of freedom. As a newspaper headline it is still not as good as Freidel's, but probably it is good enough. I wish Krasnov could re-use the title of Freidel's paper, which says it all. Krasnov's title omits the keyword "gravity". The reader is supposed to be intelligent enough to know that "metric" implies it is about gravity.

You asked me to comment. I am just giving you my thoughts on it.
It is radical. It says that maybe the troubles with quantizing GR are because the GR Lagrangian is not quite right. If we can loosen it up and generalize it and allow a slight modification, before we quantize, then maybe things will go better?
And he finds a beautiful way to generalize. Introduce a potential function, get rid of the constraints in the BF formulation.

I would give this a one in five chance of success, and of being important. As I see it, that is a very high probability, because most new physics ideas fail. This is radically new, so one should expect it to fail.

Maybe one in five is too optimistic. But in your post you sound very optimistic, so I will go along with that for now.

Let's look in the paper and say what Krasnov says he is working on and will "appear elsewhere".

This idea won't go anywhere without a lot of work. It is still just a radical classical reformulation. To put it into QG service, somebody has to quantize it. I suppose that could mean completely reconstruct spinfoam QG and present a new spinfoam vertex amplitude formula.

Looking to see what further Krasnov papers we may expect, I find this on page 4:
" ...our story may also be applicable to gravity coupled to at least certain types of matter. This will be described elsewhere."

 

[MTd2]

I don't understand why this would be a modification of General Relativity. What I see it is that Kranov thinks that every divergent term in GR is annihilated by a modified GR, which then it becomes GR again. Is it correct?

 

[marcus]

That sounds too good to be true, MTd2. I think you are ahead of me in understanding this paper, so I am listening carefully to what you say---which has motivated me to print out a paper copy and begin studying it.

We should begin by recognizing the limitations. This is a very interesting paper but there are unresolved questions. It is about pure gravity. There is no matter included so far. There is an unresolved question about whether matter can be included in the theory as it stands.

The thought that this new class of Lagrangians is closed under renormalization is still just a hope. He says it is plausible to expect that it is, but he still has to define a renormalization scheme, and prove that it is.

Again, I think that you have gone deeper into the paper than I, and I think your question has to do with the S-matrix. He describes a conjecture on the last page, this is still unproven. I think he conjectures that a certain class of transformations which he is considering does not change graviton scattering. If this were proven it seems to me that the theory might have the correct low energy behavior, and we would "get GR back".
I think this would be an answer to your question about "becoming GR again".
It's an exciting possibility and he indicates that exploring it will be the focus of future work.

I think the paper was written for PRL (physical review letters) which is one of the very top places to publish, and is the right place for time-critical results that are so to speak "news-worthy". The paper opens up some research possibilities that other people could get into and help with. The abstract does not say that he has submitted it to PRL yet. If he does, and if it is accepted, then it will be published soon and other people may be interested and join with him to help prove these conjectures.

Especially I think the conjecture that this class of theories is closed under renormalization.
And before that, they must devise a renormalization scheme. He has some things to say about what that might entail.

 

[MTd2]

That sounds too good to be true, MTd2. I think you are ahead of me in understanding this paper, so I am listening carefully to what you say---which has motivated me to print out a paper copy and begin studying it.

Yes, that's what I was thinking when I said the best paper in the last 40 years. Even if it is pure matter, it doesn't matter. Too good to be true. But that's what I think he was tacitly saying in the 1st paragraph that's his idea is to find a gravitational renormalizable theory like string theory, but without adding extra degrees of freedom. I took that as he is saying he was playing fair in the terms of the games of string theorists, but beating them in simplicity.

But I can try to guess what is the method of renormalization that he will probably find. He is trying to get rid of divergences by trying to show they are related to unphysical theories, right?It kind of remember me asymptotic safety where an infinite number of coupling constants end up being all restricted by a 2-surface. And the use of BF theory, which is about 2 surfaces, lead the divergences to be finite and all BF degenerate into General relativity, which is a point in common to these 2 surfaces. And behold! This closed under renormalization is a discussion on renormalization group, which makes both ideas even closer

 

[MTd2]

So, no comments, doesn't anyone have anything to say? Marcus?

 

[marcus]

... a gravitational renormalizable theory like string theory, but without adding extra degrees of freedom. I took that as he is saying he was playing fair in the terms of the games of string theorists, but beating them in simplicity.

But I can try to guess what is the method of renormalization that he will probably find. ...

It's good that you pointed out what Krasnov is doing, and we should keep this new line of research in mind. As soon as another paper comes out, we should look for signs of progress, and see where the idea is going.

But I have difficulty imagining what is ahead. I am not able to discuss something that seems to be so new and speculative. I cannot guess what method of renormalization he will find.

So for now, discussion seems to be stalled. We just wait now for the next paper.

I will say one thing. Oh, two things.
the first is that at this point in the game one can learn from exploring gravities that one even does not think will succeed. One can explore to find out where the obstacles are and earn something from them.
AsymSafety may be like that. One will be have learned a huge amount. It is definitely worth investigating (as Weinberg repeatedly says) and "worth investigating" does not mean a guarantee of final success. Also Loll CDTriangulations could be like that. Definitely definitely "worth investigating". Much much much has been learned. And yet is is possible that Triagulations might never succeed to model a quantum black hole!

So far I think LoopFoam is the most versatile and has made the most progress, especially in cosmology but in a variety of fronts.

Now this Krasnov thing is definitely worth investigating. I am looking forward. I expect it will teach us things. But this expectation does not translate, in my mind, to an expectation that it will be ultimately right. I am agnostic about that.

The second comment I have to make is that I wish to see Krasnov connect his approach somehow to the Spinfoam approach, if it is possible to do this. If the two can be related, even in a special limited case, it would be extremely interesting.