Livine's Feb 17 talk, if you only watch one Perimeter talk this year

[marcus]

http://pirsa.org/10020079/
Great talk about cool results. Perimeter has videos of a lot of excellent valuable lectures on line but if you would only watch one, be sure it's this

 

[marcus]

The talk is both a survey and a report on new results. It places a series of results obtained over the past five years in a particular perspective---incidentally it is in part an "under-30" point of view (that could mean something, I don't know for sure if it does) since Livine will turn 28 this year. I will make a topic outline of the talk, extracted from the slides PDF, because the talk covers many topics and organizes the connections between them.

I think many of us remember the 2005 papers of Freidel Livine which dealt with 3d spinfoam (Ponzano-Regge) gravity. They found that Feynman diagrams appeared automatically, they found that some type of DSR was inevitable, matter occurred as topological defects. One wondered if any of this would carry over to 4d. It appears that it has, but the appearance of matter is not in the same simple "defect" way as before. Particles do not appear as something as simple as a conical defect singularity. Yet they do appear (or at least scalar fields, with ongoing work to uncover more complex forms.) So it seems that the initiative of 2005 is being carried out. It will be interesting if this program succeeds.
This is one reason I want to make an outline of the talk, so that we will have a kind of rough plan or overview of this program. Many of the topics are stated by Livine in a relaxed informal language (an advantage of the seminar talk compared with the journal article style.)

 

[atyy]

The link between GFT and NCFT is very interesting. I hope they develop it more. From NCFT you can get "emergent gravity" http://arxiv.org/abs/hep-th/0212262 , and from GFT you can get "emergent matter" http://arxiv.org/abs/0903.3475 (with the caveats that the NCFT in GFT is defined using a different non-commutative product, and that the gravity in NCFT is not necessarily general relativistic). That's why a whole bunch of people who worked on NCFT renormalization are now working on GFT renormalization.

 

[marcus]

The link between GFT and NCFT is very interesting.

d'accord!

I hope they develop it more.

moi aussi! it is, as the French say, "cool". :-D

From NCFT you can get "emergent gravity" http://arxiv.org/abs/hep-th/0212262 , and from GFT you can get "emergent matter" http://arxiv.org/abs/0903.3475 (with the caveats that the NCFT in GFT is defined using a different non-commutative product, and that the gravity in NCFT is not necessarily general relativistic). That's why a whole bunch of people who worked on NCFT renormalization are now working on GFT renormalization.

Like Vincent Rivasseau, yes?

I'd like to make this video lecture into a kind of Michelin guide to help me navigate through the next few months of research papers. I printed out the PDF of Livine's slides.
There are duplications because to explain the links he jumps back from a slide about GFT to an earlier SF slide and then forwards to a NCGFT slide and so on. Here is a nonrepeating sequence of slides from the PDF, if anyone wants to save paper in printing:

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,
23,24,
29,30,31,32,
37,38,39,40,
45,46,47,48,49,50

In other words the initial series of slides #1 thru #18 is without repeats
and then he skips around back and forth, repeating some, because he is drawing a connection between one formalism and another.
And then there is another series without repeats and so on...

No big deal, just a way to save printer ink if you want---there are about 31 non-repeating slides out of about 51 total.
I think it is worthwhile having the printout of the PDF because then one can scrutinize more critically and say what does this actually mean and how far have they actually gotten with this. Livine is obviously very good, so it is imperative to read critically. Also a lot is difficult, I find. Some of the ideas are unfamiliar.

 

[marcus]

So there are about 30 slides. I want to make a kind of outline showing the structure of the talk. Here is Etera Livine's abstract:
A review of Spinfoams and Group Field Theory
February 17, 2010
Abstract: We will review the definitions of spin foam models for quantum gravity and the recent advances in this field, such as the "graviton propagator", the definition of coherent states of geometry and the derivation of non-commutative field theories as describing the effective dynamics of matter coupled to quantum gravity. I will insist on the role of group field theories as providing a non-perturbative definition of spinfoams and their intricate relation with non-commutative geometry and matrix models.

http://pirsa.org/10020079/

The name Etera, I learned recently, is a South Pacific form of Ezra, the zed consonant being unavailable in some languages. For example a Maori speaker would not say "Ezra" he would instead say "Etra", and because it is not customary to run consonants together, he would insert a light vowel sound between the t and the r, making it E-te-ra. So this is simply the polynesian way to pronounce a name that is common with us and we are all used to.

 

[marcus]

I printed out the 30 slides, and made a kind of book to study away from the computer.

A key idea here is BF theory. I'd be interested in what people's intuition is about that.
It is "almost" a theory of gravity. It becomes a theory of gravity if one adds a term to the Lagrangian.

It does not require a background geometry, just a manifold, a basic shapeless limp continuum to work with, where coordinates and differential forms can be defined. So it is independent of background geometry.

To me the field B looks like a "Lagrange multiplier" which has the job of making the other field F (the curvature of a connection) come out to zero.
In the simple case, the equations of motion say that F = 0.

That is without any additional term. And there turn out to be no local degrees of freedom.

Intuitively, the GR equation says that in the absence of matter the geometry "wants" to be flat. If you force there to be some centers of curvature it will try to adapt in the most relaxed way, to be nearly flat, or what it thinks is the flattest it can get.

So BF theory has this extremely simple Lagrangian which carries out this idea of a connection A, and its curvature F, and the connection wants to be as flat, or relaxed, as possible. So there is this multiplier B, which in the absence of anything else will force F = 0.

And that represents in some way the natural state of "nothing" a null-world. So then you come in and tweak B a little bit, so it can't do it's job so perfectly any more. You add a "constraint" term in the Lagrangian where you hamper B. Then local degrees of freedom appear. The system starts to respond to something besides merely the topology of the manifold.

Does anyone have a better initial intuition about BF theory? It is real basic here. Anyone want to give their basic perspective on it?

 

[marcus]

MTd2 spotted this and added it to our bibliography. As I recall this work by Baratin Oriti is cited by Etera Livine in his Feb 17 video.
BF theory is central to this paper.

http://arxiv.org/abs/1002.4723
Group field theory in non-commutative metric variables
Aristide Baratin, Daniele Oriti
6 pages
(Submitted on 25 Feb 2010)
"We introduce a dual formulation of group field theories, making them a type of non-commutative field theories. In this formulation, the variables of the field are Lie algebra variables with a clear interpretation in terms of simplicial geometry. For Ooguri-type models, the Feynman amplitudes are simplicial path integrals for BF theories. This formulation suggests ways to impose the simplicity constraints involved in BF formulations of 4d gravity directly at the level of the group field theory action. We illustrate this by giving a new GFT definition of the Barrett-Crane model."

They define a kind of Fourier transform between functions defined on the Lie group and functions defined on the associated Lie algebra. This was also discussed in the 17 February video talk. This Fourier transform allows to establish the duality between certain GFT and certain NCQFT (noncommutative quantum field theories.)
I want to quote a short passage from the conclusions:

==quote==
We have introduced a new non-commutative representation of group field theories, based on the quantum group Fourier transform, turning them into non-local and noncommutative field theories on Lie algebras. We have shown that the resulting Lie algebra variables correspond to the B variables of simplicial BF theory, in any dimension, and that the corresponding Feynman amplitudes for arbitrary simplicial complex have the form of simplicial BF path integral. This realizes an explicit GFT duality between spin foam models and simplicial gravity path integrals, and allows to understand clearly how simplicial geometry is encoded in the GFT formalism...

==endquote==

Now this is the first time I have heard someone tell a physical meaning of the B variable of the BF theory.
This suggests that my original intuitive handle on the B variable was in error, or at least inadequate.

 

[marcus]

Let's add Rovelli's March 2010 Zakopane talk to this, and try to see how it fits together with Livine's February Perimeter talk.
http://www.fuw.edu.pl/~jpa/qgqg3/CarloRovelli.pdf

 

[marcus]

Livine's Perimeter talk is a Spinfoam survey and roadmap. It maps out a certain program.

An important piece of that program is a paper which was not finished in February when Livine gave the talk. But is finished now, and posted. So another piece of what Livine was talking about just fell into place.
http://arxiv.org/abs/1005.2090 , noted by MTd2 yesterday:

http://arxiv.org/abs/1005.2090

U(N) Coherent States for Loop Quantum Gravity

Laurent Freidel, Etera R. Livine
(Submitted on 12 May 2010)
We investigate the geometry of the space of N-valent SU(2)-intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian Gr(N,2), they possesses a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the well-known coherent intertwiners.

My hunch is that it is hard to understand the significance of this latest paper unless you look back at the February seminar talk roadmap. The U(N) paper is not self-explanatory. Indeed Laurent Freidel is typically in too much of a hurry to be didactic. Other people have more pedagogical style. He normally just does what he has to do and doesn't give a hoot if people understand. It works for him.

 

[marcus]

Some of what I've said in this thread has been inadequately thought out. There is a new line of development taking shape, that I want to report on, one that seems to go back to a 2005 paper of Girelli and Livine. An important role is played by the group U(N). Unitary nxn complex matrices. Some notation, for instance, in this wiki article on SU(N)
http://en.wikipedia.org/wiki/Special_unitary_group
It is not clear to me why the group U(N) should suddenly become prominent in Lqg, but all I can do is list the papers in this new line of development--stare at them, and try to understand the direction that Livine and the others are exploring.

The most recent appeared yesterday:
http://arxiv.org/abs/1006.2451
Dynamics for a 2-vertex Quantum Gravity Model
Enrique F. Borja, Jacobo Diaz-Polo, Iñaki Garay, Etera R. Livine
28 pages
(Submitted on 12 Jun 2010)
"We use the recently introduced U(N) framework for loop quantum gravity to study the dynamics of spin network states on the simplest class of graphs: two vertices linked with an arbitrary number N of edges. Such graphs represent two regions, in and out, separated by a boundary surface. We study the algebraic structure of the Hilbert space of spin networks from the U(N) perspective. In particular, we describe the algebra of operators acting on that space and discuss their relation to the standard holonomy operator of loop quantum gravity. Furthermore, we show that it is possible to make the restriction to the isotropic/homogeneous sector of the model by imposing the invariance under a global U(N) symmetry. We then propose a U(N) invariant Hamiltonian operator and study the induced dynamics. Finally, we explore the analogies between this model and loop quantum cosmology and sketch some possible generalizations of it."

I am guessing that the root paper in this line of development is the 2005 Girelli Livine:
http://arxiv.org/abs/gr-qc/0501075
Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators
Florian Girelli, Etera R. Livine
16 pages, 3 figures
(Submitted on 25 Jan 2005)
"Loop Quantum Gravity defines the quantum states of space geometry as spin networks and describes their evolution in time. We reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models. This allow us to make a link with non-commutative geometry and to look at the issue of the semi-classical limit of LQG from a new perspective..."

To me this was a bit mind-boggling. How can one see a spin network as a collection of harmonic oscillators? At each node there is a set of N links connected at that node. Are these incoming or outgoing links to be seen as oscillators? Back in 2005, when this paper appeared, it seemed a little too off-beat---but now it appears that something has come of the idea. It has begun to bother me that I don't have intuition about this.

Then a Freidel Livine paper appeared:
http://arxiv.org/abs/0911.3553
The Fine Structure of SU(2) Intertwiners from U(N) Representations
Laurent Freidel, Etera R. Livine
21 pages
(Submitted on 18 Nov 2009)
"In this work we study the Hilbert space space of N-valent SU(2) intertwiners with fixed total spin, which can be identified, at the classical level, with a space of convex polyhedra with N face and fixed total boundary area. We show that this Hilbert space provides, quite remarkably, an irreducible representation of the U(N) group. This gives us therefore a precise identification of U(N) as a group of area preserving diffeomorphism of polyhedral spheres. We use this results to get new closed formulae for the black hole entropy in loop quantum gravity."

WHOAHH! Picture, around some point in space a blur consisting of all the N-faced convex polyhedra that can surround that point (normalized to all have the same total area). Imagine the uncertain flickery fuzzy blur of all those geometric possibilities, at that point. Now I can breathe a little easier, it begins to make a bit more sense.

And then, soon after, there came the next Freidel Livine, the one MTd2 spotted and added to the biblio.

http://arxiv.org/abs/1005.2130
U(N) Coherent States for Loop Quantum Gravity
Laurent Freidel, Etera R. Livine
(Submitted on 12 May 2010)
"We investigate the geometry of the space of N-valent SU(2)-intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian Gr(N,2), they possesses a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the well-known coherent intertwiners."

So now it doesn't seem so tough to understand. At every point in the network we have a blur of uncertain geometry (all possible convex polyhedra distributing possible bits of area and angle in all directions). Now all we have to do is fit together the separate pictures of uncertain geometry at the various nodes. At each node in the network there is a picture and these must be fused into a larger picture. Then we will have a "living" spin network.

Now Livine does the obvious thing which every good mathematician would know to do. He gets some friends together and they examine the simplest version of the problem, where you only have TWO of these blurs, only TWO chimaera nodes, that have to be fitted together. And he considers the symmetries of the problem...

And that is where we came in. Yesterday's paper. Wake up. If you can put two together then experience suggests that then you can assemble more than two. Just to repeat for clarity a bit of the abstract:
http://arxiv.org/abs/1006.2451
Dynamics for a 2-vertex Quantum Gravity Model
Enrique F. Borja, Jacobo Diaz-Polo, Iñaki Garay, Etera R. Livine
28 pages
(Submitted on 12 Jun 2010)
"We use the recently introduced U(N) framework for loop quantum gravity to study the dynamics of spin network states on the simplest class of graphs: two vertices linked with an arbitrary number N of edges..."

 

[marcus]

Now we should look to see what followup papers this group of researchers has in preparation.

[12] E.Borja, J.Díaz-Polo, I.Garay and E.R.Livine, Matrix Models for LQG Intertwiners, in preparation

[34] E.F. Borja, J. Díaz-Polo, I. Garay and E.R. Livine, Quantum gravity dynamics on the 3+N vertex graph and black hole evaporation, in preparation

[35] E.F. Borja, J. Díaz-Polo, I. Garay and E.R. Livine, Loop Quantum Cosmology from the U(N) Framework, in preparation

These are just the "in prep" references from yesterday's paper.
I suppose Freidel could have something along this line in prep also.
It is interesting that several of the authors are at least part-time at Erlangen. Thiemann recently moved from AEI-Golm to Erlangen and set up his own LQG program there. Some Rovelli PhDs have gone there as postdocs or on some other basis.

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

Going back 7 years, I recall when Livine's PhD thesis came out. For epigraph there was an ironical quote from the Babylonian Talmud (compiled c. 500 AD) http://www.arxiv.org/abs/gr-qc/0309028

It is never advisable for anyone to speculate on these 4 questions:

What is there above?
What is there below?
What was before the world?
What will be after?

Whoever does, it would be better for him that he was never born.


Actually the quote used was in French:
Quiconque s’est jamais avisé de spéculer sur ces 4 questions :

– Qu’y a-t-il au-dessus ?
– Qu’y a-t-il en-dessous ?
– Qu’y avait-il avant le monde ?
– Qu’y aura-t-il après ?

Il aurait mieux valu pour lui qu’il ne fut jamais né.