New action is key to Loop gains (the Dittrich-Speziale action)

[marcus]

The D-S action was first described just one year ago in arXiv0802.0864.

Page 3 "...The reader might wonder at this point whether (6) is a discretization of a continuum action for GR, like Regge’s is a discretization of the Einstein-Hilbert action ∫√gR. We argue that this is the case, the continuum avatar of (6) being Plebanski’s action [8]."

The most recent use of the new action was in December 2008, in the third of a series of papers by Rovelli and others,
LQG propagator: III. The new vertex, arXiv0812.5018.

Page 4 "...S (j, n) is the Regge action [19] of such a geometrical
4-simplex (more precisely, the Dittrich-Speziale action [20]), divided by 8π$\hbar$G_Newton ."

More about this later.

Speziale was one of Rovelli's students and then postdoc at Perimeter for several years ( http://www.perimeterinstitute.ca/personal/sspeziale/ ) and now fills a permanent faculty position at Marseille.
Dittrich did her Diplomarbeit with Loll, was at Berlin or Potsdam for a while, postdoc at Perimeter, and is now at Utrecht.

Trivia: since we say tetrahedron for 3-simplex, it would be correct to call the 4-simplex a pentachoron.

 

[marcus]

One way to appreciate the significance of the D-S action (also the new "area-angle" variables) is to see it in a narrative context. Here is the story as told by Speziale in this review
http://arxiv.org/abs/0810.1978
It is an invited review of the definition of n-point functions in LQG---the construction of background-free graviton propagator---intended for a special issue of Advanced Science Letters edited by Martin Bojowald.

Here's a quote from page 11
"...Recently Dittrich and myself proposed an alternative solution: instead of constraining the areas, we added angles between triangles as variables to the action. In [63] we showed that these extra variables solve naturally the problem of defining a discrete metric uniquely, when satisfying constraints which are local and easy to write explicitly. Using these constraints we wrote an action which is completely equivalent to Regge’s, and thus to general relativity in the continuum limit.

If our action emerges from a spin foam model, then the same argument that we run in the previous Section for the Ponzano-Regge model can be used, and support the idea that with the correct boundary state the model has the right low-energy physics, at least on a fixed triangulation. Due to the extra angle variables, our action can arise only from spin foam models which sum over both spins and intertwiners, not only spins like in Barrett-Crane. In turn, precisely the lack of intertwiner degrees of freedom was taken by Rovelli and his group as the fundamental problem with the Barrett-Crane model. This shows up explicitly when trying to compute the angle correlations, which fail to have even just the correct 1/j₀ scaling [59].

Following this line of thought, an improved spin foam model was proposed in [64], which at least naively has the right matching with spin network states on the boundary. Not long afterwards, Livine and myself realized [65] that this model can also be obtained starting from the coherent states we had previously introduced in [66], a result independently found also by Krasnov and Freidel [67].

Further developments include [27, 68, 69, 70], and excitement has arisen around the possibility that this new model might indeed cure the problems of Barrett-Crane’s and give general relativity in the semiclassical limit."

What I bolded has, I believe, been shown. See section 3 of the most recent Rovelli et al paper.

 

[Coin]

Hi Marcus,

This might be a little over my head. Do I have this much right?: What this group is trying to do is study the graviton propagator for LQG, but they initially found that the graviton propagator behaves inconsistently with what we expect of GR. The advantage of the new action is that it produces a graviton propagator which gives correct GR behavior?

 

[marcus]

Hi Marcus,

This might be a little over my head. Do I have this much right?: What this group is trying to do is study the graviton propagator for LQG, but they initially found that the graviton propagator behaves inconsistently with what we expect of GR. The advantage of the new action is that it produces a graviton propagator which gives correct GR behavior?

In any case it is more correct than what they had before. I'm still trying to understand but I'm pretty sure that's right.
Rovelli and Alesci have a series of papers which form the backbone
LQG propagator I ( http://arxiv.org/abs/0708.0883 )
LQG propagator II ( http://arxiv.org/abs/0711.1284 )
LQG propagator III ( http://arxiv.org/abs/0812.5018 )
In each case they are working with the covariant form of LQG---spinfoam---where the dynamics is determined by the formula for calculating an amplitude at a vertex, the spinfoam vertex formula. Their aim is to find the graviton propagator, or the n-point correlation functions.

I. in the first paper they discovered a shortcoming of the socalled Barrett-Crane vertex that had been in use up to that point.
II. in the second paper they determined an asymptotic behavior that a new vertex formula would have to have in order to avoid the BC failure.
III. in the third paper they report that a new vertex formula (which happens to correspond to Dittrich-Speziale action) has the desired asymptotic behavior.

This does not yet show that this new vertex will yield the correct low energy limit. It doesn't fail in the way the BC vertex did, but further investigation may show it fails some other way.

Perhaps the clearest account of what has happened in the past two years in this regard is provided by the third paper in that series.
It is concise, only 6 pages main text (plus biblio). I will give the title and link again:
LQG propagator III. The new vertex
http://arxiv.org/abs/0812.5018
I'll try to say more later.

 

[marcus]

Another piece of the new LQG picture came out today. MTd2 flagged the paper.
It is by Barrett, Fairbairn and others including an occasional poster at PF.

http://arxiv.org/abs/0902.1170
Asymptotic analysis of the EPRL four-simplex amplitude
Authors: John W. Barrett, Richard J. Dowdall, Winston J. Fairbairn, Henrique Gomes, Frank Hellmann
(Submitted on 6 Feb 2009)

Abstract: An asymptotic formula for a certain 4d Euclidean spin foam 4-simplex amplitude is given for the limit of large spins. The analysis covers the model with Immirzi parameter less than one defined separately by Engle, Livine, Pereira and Rovelli (EPRL) and Freidel and Krasnov (FK). We are also able to analyse the EPRL model with Immirzi parameter greater than one. The asymptotic formula has one term which is proportional to the cosine of the Regge action for gravity, and it is shown that this term is present whenever the boundary data determines a non-degenerate Euclidean geometry for the 4-simplex. A scheme for resolving the phase ambiguity of the boundary data in these cases is also presented.

 

[marcus]

MTd2 has flagged yet another key paper related to the rapidly developing proof of LQG low-energy limit. This seems to be going by way of the Dittrich-Speziale action.

It seems that we may be going to call this the AREA-ANGLE REGGE action. We need to conform with common usage in the research community. If they like that terminology, we conform.

In his seminal 1960 paper "GR without coordinates", Tulio Regge created a simplex version of GR using edge-lengths as his variables. Other people have tried other forms of Regge action substituting areas of triangles for the edge-lengths. Dittrich-Speziale were first to try a combination of areas and dihedral angles. The signs are this is the right way.

It's fairly clear that some type of Regge setup is a good way to approach low-energy limit questions in quantum gravity. People have had 60 years of playing around with Regge and know it as a reasonable facsimile of continuum GR.

So now MTd2 flags this new paper where a Rovelli Phd student named Valentin Bonzom connects area-angle Regge with BF theory.

It's another piece of the puzzle. Things seem to me to be approaching some kind of tipping point.

http://arxiv.org/abs/0903.0267
From lattice BF gauge theory to area-angle Regge calculus
Valentin Bonzom
18 pages, 2 figures
(Submitted on 2 Mar 2009)
"We consider Riemannian 4d BF lattice gauge theory, on a triangulation of spacetime. Introducing the simplicity constraints which turn BF theory into simplicial gravity, some geometric quantities of Regge calculus, areas, and 3d and 4d dihedral angles, are identified. The parallel transport conditions are taken care of to ensure a consistent gluing of simplices. We show that these gluing relations, together with the simplicity constraints, contain the constraints of area-angle Regge calculus in a simple way, via the group structure of the underlying BF gauge theory. This provides a precise road from constrained BF theory to area-angle Regge calculus. Doing so, a framework combining variables of lattice BF theory and Regge calculus is built. The action takes a form à la Regge and includes the contribution of the Immirzi parameter. In the absence of simplicity constraints, the standard spin foam model for BF theory is recovered. Insertions of local observables are investigated, leading to Casimir insertions for areas and 6j-symbols for 3d angles. The present formulation is argued to be suitable for deriving spin foam models from discrete path integrals."

 

[marcus]

I occasionally try to reflect on how the Marseille network has grown. I recently looked at the list of people who have gotten their PhD with Rovelli. He gets some very good people. The list includes some of the most creative, collectively a massive contribution to the field. One can ask where did Valentin Bonzom come from. Well in 2006 he was a university student at Lyon, at the École Normale. He was doing experimental physics, something with QED and atomic physics. He was one of halfdozen collaborators on an experimental paper.
He is a native French speaker who happens to write unusually clear efficient English style.
(I shouldn't have to belabor the significance of that.)
He is mathematically sophisticated. He can think like a modern mathematician when he needs to. ENS Lyon is where Freidel came from, and it is where Etera Livine is now. Both are former Rovelli PhD students. I would guess that Bonzom happened to meet Livine there. They collaborated on a paper. This somehow led to Bonzom going to Marseille to do his doctorate.
This is his first solo paper. It is worth remarking that the quality is very good.
Something about the French higher-education/research system works. I don't understand it well enough to say more.

I want to share this with you though. Look at the list of PhD supervisions on Rovelli's vita:
==quote http://www.cpt.univ-mrs.fr/~rovelli/vita.pdf ==
Ph.D. Thesis supervised
1. Josè Balduz (Carnegie Mellon University, Pittsburgh)
Completed November 1994
“Dynamical Model of Quantum Measurement”
2. Hugo Morales-Tecotl (SISSA, Trieste)
Completed August 1993
“Fermions in the Loop Representation”
Hugo Morales-Tecotl is Full Professor (Profesor Titular C) at the Physics Department of the Universidad Autonoma Metropolitana Iztapalapay, Mexico city, Mexico.
3. Junichi Iwasaki (University of Pittsburgh)
Completed January 1994
“The Linearization of Quantum gravity”
4. Norbert Grott (University of Pittsburgh)
Completed April 1998
“Moduli spaces in intersecting Knot Theory”
5. Marcelo Bareira (University of Pittsburgh).
“Black hole emission spectra”
Marcelo Bareira is Assistent Professor at the Pontificia Universidade Catolica and Research
Professor at Centro Universitario do Sul, Minas, Brasil.
6. Bill Curry (University of Pittsburgh)
“The relational interpretation of quantum mechanics”
7. Peush Upadhya (University of Pittsburgh)
“Loop quantum gravity”
8. Alejandro Perez (University of Cordoba)
Completed May 2001
“Finiteness of Spin Foam models”
Alejandro Perez has obtained an Assistant Professor postion at the Penn State University, in State College, USA. He has then obtained the position of Maitre de Conference at the Université de la Méditerranée, in Marseille, France.
9. Marcus Gaul (Munich University)
Completed 2001
“Hamiltonian constraint in LQG”
10. Richard Livine (Université de la Méditerranée).
Completed 2002
“Modèles de mousse de spin” (Prix de Thèse 2003 de l’Université de la Méditerranée)
Richard Livine has a permanent CR2 (Chargé de Recherche) position at the Ecole National Superieure de Lyon, France.
11. Daniele Colosi (Université de la Méditerranée et Università di Roma)
Completed Mars 2005
“Dynamique quantique covariante”
Daniele Colosi has obtained a postdoctoral position at the University of Morelia, Mexico.
12. Luisa Doplicher (Università di Roma)
Completed February 2005
“Teoria dei campi quantistica covariante”
Luisa Doplicher has obtained a postdoctoral position at the Sissa, Trieste, Italy.
13. Florian Conrady (Berlin University)
Completed September 2005
“The classical limit of spin foam models”
Florian Conrady has obtained a postdoctoral position at Penn State University, State College, USA.
14. Simone Speziale (Università di Roma)
Completed January 2006
“2d Quantum Gravity”
Simone Speziale has obtained a postdoctoral position at the Perimeter Institute, Toronto.
15. Winston Fairbairn (Université de la Méditerranée)
“Separability in LQG”(Prix de Thèse 2007 de l’Université de la Méditerranée)
Winston Fairbairn has obtained a postdoctoral position in Nottingham, UK.
16. Mauricio Mondragon Lopez (Université de la Méditerranée)
Completed March 2008
“Probability in relativistic quantum mechanics”
17. Emanuele Alesci (Università di Roma III)
Completed January 2008
“Scattering amplitudes in LQG”
18. Elena Magliaro (Università di Roma III)
“Feynman rules in quantum gravity”
19. Claudio Perini (Università di Roma)
“Feynman rules in quantum gravity”
20. Roberto Pereira (Université de la Méditerranée)
“The loop quantum gravity vertex”