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The LQG vertex amplitude (Rovelli, Engle, Pereira)

by marcus
Tags: amplitude, engle, pereira, rovelli, vertex
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marcus
#1
May16-07, 08:04 PM
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this just out

http://arxiv.org/abs/0705.2388
The loop-quantum-gravity vertex-amplitude
Jonathan Engle, Roberto Pereira, Carlo Rovelli
6 pages

"Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy n-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity."

important paper

==exerpt==
However, the suspicion that something is wrong with the BC model has long been agitated. Its boundary state space is similar, but does not exactly match, that of loop quantum gravity; in particular the volume operator is ill-defined. Worse, recent calculations appear to indicate that some n-point functions fail to yield the correct low-energy limit [13]. All these problems are related to the way the intertwiner quantum numbers (associated to the operators measuring angles between the faces bounding the elementary quanta of space) are treated: These quantum numbers are fully constrained in the BC model by imposing the simplicity constraints as strong operator equations (Cn psi = 0). But these constraints are second class and imposing such constraints strongly may lead to the incorrect elimination of physical degrees of freedom[14].

It is therefore natural to try to implement in 4d the general picture discussed above by correcting the BC model[7, 15]. In this letter we show that this is possible, by properly imposing some of the constraints weakly (<phi, Cn psi> = 0), and that the resulting theory has remarkable features.

First, its boundary quantum state space matches exactly the one of SO(3) loop quantum gravity: no degrees of freedom are lost.

Second, as the degrees of freedom missing in BC are recovered, the vertex may yield the correct low-energy n-point functions.

Third, the vertex can be seen as a vertex over SO(3) spin networks or SO(4) spin networks, and is both SO(3) and SO(4) covariant.

Finally, the theory can be obtained as a bona fide quantization of a discretization of euclidean GR on a Regge triangulation...
==endquote==

EDIT to reply to Jal without needing an extra post:
Hi Jal, I agree with your point about these papers being stimulating reading. Not sure I understand what you mean about the 12 tetrahedra. However in any case the main thing is probably that 5 tetrahedra fit together to make a jacket around a foursimplex. The 3d surface of a foursimplex, in other words, consists of five tets. And 9 tets would form the surface for two adjacent foursimplices. Maybe the 12 tets you mention could be the surface tets of three adjacent 4d simplices.

For me the exciting thing is that they may be on the way to fixing the BC vertex amplitude formula. People have been using the Barrett-Crane vertex formula for some 10 years as the provisional basis for SPINFOAM DYNAMICS. A spinfoam is like a "Feynman diagram" for the geometry of spacetime and just like a regular Feynman diagram for electrons and light you need a formula for calculating the amplitude at each vertex (like the probability that whatever the vertex says to happen actually happens). And the BC spinfoam approach has been working pretty well but there were signs that the formula for calculating the amplitude at any given vertex might not be quite right. So Rovelli and friends may have FIXED the formula so it works better.
If this is true it would make this the QG Paper of the Year for 2007.
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jal
#2
May17-07, 05:59 PM
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Hi Marcus!
If people do not have their creative side stimulated when reading these paper then they are missing the biggest part of the fun.
When they have 12 tetra they will have a 3d model.
jal
kneemo
#3
May18-07, 03:39 PM
P: 116
Interesting stuff, Marcus.

I've been intrigued by SO(3) LQG after reading Olaf Dreyer's gr-qc/0404055. Who knows, maybe SO(3) LQG will turn out to be dual to Dan Freedman's SO(3)-Invariant Extended Supergravity.

marcus
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May18-07, 04:40 PM
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The LQG vertex amplitude (Rovelli, Engle, Pereira)

Quote Quote by kneemo View Post
Interesting stuff, Marcus.

I've been intrigued by SO(3) LQG after reading Olaf Dreyer's gr-qc/0404055...
it's good to have different people see these things from different angles---I know you and Marni, Carl, Riofrio and others discuss a lot and you may come up with a good way to look at this result

I see it differently from you, "merely" as a step along the way to a Lorentz or deSitter version. I see this paper as (in effect if everything checks out in SO(4) euclidean version) firing off the starting gun of a race to do the same thing for a different group---and a different set of labels.

If it all checks out it is a very important result (this Engle Rovelli Pereira thing) but it is not the end of the trail.
since I'm not sufficiently knowledgeable about this please see what others say about it.
marcus
#5
May21-07, 07:49 PM
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Well Rovelli will be giving one of the invited talks at the big international conference in Mexico----Loops '07----in a month or so.

I guess it will be the rock in the stream: the defining talk of that conference. There will be a lot of good stuff besides that.

Of course he will be talking about this paper. The abstract is already posted.

Here's about the Loops '07 conference
http://physicsforums.com/showthread.php?t=164403

Here's the website:
http://www.matmor.unam.mx/eventos/loops07/index.html

143 people registered so far. Here's Rovelli's talk abstract:

"Carlo Rovelli: Vertex amplitude and propagator in loop quantum gravity

It has recently been possible to begin the computation of n-points functions in loop quantum gravity. I review the basic ideas and the present state of these calculations. The Barrett-Crane vertex appears to yield some n-point functions with the correct low-energy limit, but there are also indications of a wrong behavior. The problem can be traced to the way intertwiner quantum numbers are treated in the Barrett-Crane model. This is also the source of the general discrepancy between the Barrett-Crane vertex and LQG. I present a new vertex for loop quantum gravity, introduced in collaboration with Jonathan Engle and Roberto Pereira, which may correct the problem and is fully consistent with the LQG kinematics."

There is a lot of NEWNESS about the Loops'07, compared say with what you get in many fields where there are basically the same old papers every year.

Besides the new results there are new people giving plenary talks----like Jim Hartle.
=================

For me, a cliffhanger is what will Jan Ambjorn say. Here is the abstract of his invited talk:
"Jan Ambjørn: 4d quantum gravity as a sum over histories

In this plenary talk I will review the attempts to formulate 4d quantum gravity as a sum over histories in such a way that computer simulations can be performed. I will report on computer simulations of a quantum universe with a positive cosmological constant as well as a quantum universe where test matter is included."

We don't know how Loll's program, with her group at Utrecht, has been going. The last reports of 4D computer simulations (of universes that make themselves out of 4-simplex building blocks differently each time) were about a year ago. Apparently they have continued running Monte Carlo models of simplex universes, adding features like matter. So what has happened? We don't know: they could have met with terrible frustration (as has happened before in simplicial gravity models) or they could have interesting successes. Probably Ambjorn will tell about both sorts of outcomes in their work.
marcus
#6
May21-07, 08:18 PM
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When end of June comes we will have a MIP list of nominations to choose from.
Let's see what the list would look like if we were starting the second quarter MIP poll today. How about this? Would you suggest changes?

http://arxiv.org/abs/0705.2388
The loop-quantum-gravity vertex-amplitude
Jonathan Engle, Roberto Pereira, Carlo Rovelli
6 pages

"Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy n-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity."


http://arxiv.org/abs/0705.2222
Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
Abhay Ashtekar
21 pages, to appear in the Proceedings of the 11th Marcel Grossmann Conference

"As per organizers' request, my talk at the 11th Marcel Grossmann Conference consisted of two parts. In the first, I illustrated recent advances in loop quantum gravity through examples. In the second, I presented an overall assessment of the status of the program by addressing some frequently asked questions. This account is addressed primarily to researchers outside the loop quantum gravity community."


http://arxiv.org/abs/0704.3214
(2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
D. Benedetti, R. Loll, F. Zamponi
38 pages, 13 figures

"We perform a non-perturbative sum over geometries in a (2+1)-dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and opens the way for a better understanding of scaling and renormalization issues."


http://arxiv.org/abs/0705.0674
A new spinfoam vertex for quantum gravity
Etera R. Livine, Simone Speziale
25 pages 6 figures

"We introduce a new spinfoam vertex to be used in models of 4d quantum gravity ... We study the condition under which these states can be considered semiclassical, and we show that the semiclassical ones dominate the evaluation of quantum correlations..."


http://arxiv.org/abs/gr-qc/0703144
Dynamical coherent states and physical solutions of quantum cosmological bounces
Martin Bojowald
30 pages, 3 figures

"A new model is studied which describes the quantum behavior of transitions through an isotropic quantum cosmological bounce in loop quantum cosmology sourced by a free and massless scalar field. ... an exactly solvable model ... basis for a systematic perturbation theory of loop quantum gravity... remarkably different from what is known for harmonic oscillator coherent states. ... the quantum state ... evolve[s] deterministically through the bounce."
Dcase
#7
May21-07, 10:51 PM
P: 122
Hi Marcus,

For what it is worth, I speculate that these papers will eventually become MIP:

1 - A new spinfoam vertex for quantum gravity
Etera R. Livine, Simone Speziale
25 pages 6 figures

and

2 - Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
Abhay Ashtekar
21 pages, to appear in the Proceedings of the 11th Marcel Grossmann Conference

because

#1 - section 3.2 Saddle Point Analysis
This links to mathematical game theory through the von Neumann Minimax Theorem 1928 - although:
a - Archimedes used a similar technique when attempting to find the value of PI through maximum interior polygons and minimum exterior polygons
b - Newton also used a similar technique when extending ballistics to mechanics to find the elliptical equilibrium of planetary orbits [loop planetary gravity] by finding the minimal velocity to prevent the return of a projectile to Earth and the maximum velocity to prevent escape of a projectile from Earth

- NOTE - spinfoam will likely be extended to twistfoam [both based on Penrose terminology] with a Calabi-Yau Manifolds likely becoming the ideal twistfoam since the imaginary unit "i" has been known to have a strong correlation with electromagnetism since the 1890s work of Charles Proteus Steinmetz, IEEE and applied mathematician.

#2 - section A, paragraph 2, sentence 2:
"The von Neumann theorem ensures us that, under suitably physically motivated assumptions, the algebra admits a unique irreducible representation, namely the standard Schroedinger one."

- SEE - John von Neumann biographies
http://www-groups.dcs.st-and.ac.uk/~...n_Neumann.html
and / or
http://en.wikipedia.org/wiki/John_von_Neumann

The latter has a great discussion of how von Neumann reconciled the Schroedinger Wave with the Heisenberg Matrix Mechanics in the section ‘Quantum mechanics’
kneemo
#8
May22-07, 10:12 PM
P: 116
Quote Quote by marcus View Post
I see it differently from you, "merely" as a step along the way to a Lorentz or deSitter version. I see this paper as (in effect if everything checks out in SO(4) euclidean version) firing off the starting gun of a race to do the same thing for a different group---and a different set of labels.
Yup, I still remember Baez's Cartan geometry examples where G/H=SO(3)/SO(2). I agree that SO(3) is just a baby example, opening the way to richer groups like SO(4,1) and even E8(8).
marcus
#9
Jun8-07, 08:50 PM
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Many potentially important research papers have come out this April-May-June quarter. When end of June comes we will have a MIP list of second-quarter QG research papers to choose from. It may be more difficult than usual to make a choice.
Let's see what the list would look like if we were starting the second quarter MIP poll today. Would you suggest additions or changes?

http://arxiv.org/abs/0705.2388
The loop-quantum-gravity vertex-amplitude
Jonathan Engle, Roberto Pereira, Carlo Rovelli
6 pages

"Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy n-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity."


http://arxiv.org/abs/0705.2222
Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
Abhay Ashtekar
21 pages, to appear in the Proceedings of the 11th Marcel Grossmann Conference

"As per organizers' request, my talk at the 11th Marcel Grossmann Conference consisted of two parts. In the first, I illustrated recent advances in loop quantum gravity through examples. In the second, I presented an overall assessment of the status of the program by addressing some frequently asked questions. This account is addressed primarily to researchers outside the loop quantum gravity community."


http://arxiv.org/abs/0704.3214
(2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
D. Benedetti, R. Loll, F. Zamponi
38 pages, 13 figures

"We perform a non-perturbative sum over geometries in a (2+1)-dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and opens the way for a better understanding of scaling and renormalization issues."


http://arxiv.org/abs/0705.0674
A new spinfoam vertex for quantum gravity
Etera R. Livine, Simone Speziale
25 pages 6 figures

"We introduce a new spinfoam vertex to be used in models of 4d quantum gravity ... We study the condition under which these states can be considered semiclassical, and we show that the semiclassical ones dominate the evaluation of quantum correlations..."


http://arxiv.org/abs/gr-qc/0703144
Dynamical coherent states and physical solutions of quantum cosmological bounces
Martin Bojowald
30 pages, 3 figures

"A new model is studied which describes the quantum behavior of transitions through an isotropic quantum cosmological bounce in loop quantum cosmology sourced by a free and massless scalar field. ... an exactly solvable model ... basis for a systematic perturbation theory of loop quantum gravity... remarkably different from what is known for harmonic oscillator coherent states. ... the quantum state ... evolve[s] deterministically through the bounce."

http://arxiv.org/abs/0705.4398
The Dark Side of a Patchwork Universe
Martin Bojowald
24 pages, 2 figures, Contribution to the special issue on Dark Energy by Gen. Rel. Grav
(Submitted on 30 May 2007)

"While observational cosmology has recently progressed fast, it revealed a serious dilemma called dark energy: an unknown source of exotic energy with negative pressure driving a current accelerating phase of the universe. All attempts so far to find a convincing theoretical explanation have failed, so that one of the last hopes is the yet to be developed quantum theory of gravity. In this article, loop quantum gravity is considered as a candidate, with an emphasis on properties which might play a role for the dark energy problem. Its basic feature is the discrete structure of space, often associated with quantum theories of gravity on general grounds. This gives rise to well-defined matter Hamiltonian operators and thus sheds light on conceptual questions related to the cosmological constant problem. It also implies typical quantum geometry effects which, from a more phenomenological point of view, may result in dark energy. In particular the latter scenario allows several non-trivial tests which can be made more precise by detailed observations in combination with a quantitative study of numerical quantum gravity. If the speculative possibility of a loop quantum gravitational origin of dark energy turns out to be realized, a program as outlined here will help to hammer out our ideas for a quantum theory of gravity, and at the same time allow predictions for the distant future of our universe."
marcus
#10
Jun18-07, 02:56 PM
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P: 23,108
There were a lot of potentially important research papers that appeared this April-May-June quarter. It may be more difficult than usual to make a choice. I'm considering having the MIP poll early. Any other June papers that come out could be included in the third quarter poll, if they are of special interest.

To keep the list short I've removed some good ones: one by Livine and Speziale, one by Speziale solo, one by Loll, Benedetti, Zamponi. One by Bojowald (Dynamical coherent states and physical solutions of quantum cosmological bounces) didn't get on the first quarter poll because it appeared right at the end of March, and was carried over.

Here's the list, in case anyone would like to comment or suggest changes.

http://arxiv.org/abs/0705.2388
The loop-quantum-gravity vertex-amplitude
Jonathan Engle, Roberto Pereira, Carlo Rovelli
6 pages

"Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy n-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity."


http://arxiv.org/abs/0705.2222
Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
Abhay Ashtekar
21 pages, to appear in the Proceedings of the 11th Marcel Grossmann Conference

"As per organizers' request, my talk at the 11th Marcel Grossmann Conference consisted of two parts. In the first, I illustrated recent advances in loop quantum gravity through examples. In the second, I presented an overall assessment of the status of the program by addressing some frequently asked questions. This account is addressed primarily to researchers outside the loop quantum gravity community."


http://arxiv.org/abs/gr-qc/0703144
Dynamical coherent states and physical solutions of quantum cosmological bounces
Martin Bojowald
30 pages, 3 figures

"A new model is studied which describes the quantum behavior of transitions through an isotropic quantum cosmological bounce in loop quantum cosmology sourced by a free and massless scalar field. ... an exactly solvable model ... basis for a systematic perturbation theory of loop quantum gravity... remarkably different from what is known for harmonic oscillator coherent states. ... the quantum state ... evolve[s] deterministically through the bounce."

http://arxiv.org/abs/0705.4398
The Dark Side of a Patchwork Universe
Martin Bojowald
24 pages, 2 figures, Contribution to the special issue on Dark Energy by Gen. Rel. Grav
(Submitted on 30 May 2007)

"While observational cosmology has recently progressed fast, it revealed a serious dilemma called dark energy: an unknown source of exotic energy with negative pressure driving a current accelerating phase of the universe. All attempts so far to find a convincing theoretical explanation have failed, so that one of the last hopes is the yet to be developed quantum theory of gravity. In this article, loop quantum gravity is considered as a candidate, with an emphasis on properties which might play a role for the dark energy problem. Its basic feature is the discrete structure of space, often associated with quantum theories of gravity on general grounds. This gives rise to well-defined matter Hamiltonian operators and thus sheds light on conceptual questions related to the cosmological constant problem. It also implies typical quantum geometry effects which, from a more phenomenological point of view, may result in dark energy. In particular the latter scenario allows several non-trivial tests which can be made more precise by detailed observations in combination with a quantitative study of numerical quantum gravity. If the speculative possibility of a loop quantum gravitational origin of dark energy turns out to be realized, a program as outlined here will help to hammer out our ideas for a quantum theory of gravity, and at the same time allow predictions for the distant future of our universe."


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