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Three papers about the new LQG vertex (Marseille)
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Oct29-07, 11:25 AM
A detailed introduction of the new LQG spinfoam model---flipped version of the Barrett-Crane vertex amplitude---appeared in August*. Quite a lot of follow-up work has been posted, including three papers that appeared yesterday. Recently posted extensions include numerical work by Rovelli et al to check how the model does in the semiclassical limit
and construction of the Lorentzian version by Pereira. The initial paper only treated the Euclidean case---that being simpler to formulate. It was clear that extending it to the more complicated case was on the agenda---the only question being when.
Several vertex amplitudes were proposed by various people at about the same time----besides Rovelli's Marseille bunch there were papers by Freidel, Krasnov, Livine and Speziale. Here's a new paper by Engle and Pereira which attempts a comparison.
Engle and Pereira are co-authors with Rovelli of the Marseille version. It looks to me as if they are making an effort to understand the competing proposals and sort out the differences.
Coherent states, constraint classes, and area operators in the new spin-foam models
Jonathan Engle, Roberto Pereira
(Submitted on 26 Oct 2007)
"Recently, two new spin-foam models have appeared in the literature, both motivated by a desire to modify the Barrett-Crane model in such a way that the imposition of certain second class constraints, called cross-simplicity constraints, are weakened. We refer to these two models as the FKLS model, and the flipped model. Both of these models are based on a reformulation of the cross-simplicity constraints. This paper has two main parts. First, we clarify the structure of the reformulated cross-simplicity constraints and the nature of their quantum imposition in the new models. In particular we show that in the FKLS model, quantum cross-simplicity implies no restriction on states. The deeper reason for this is that, with the symplectic structure relevant for FKLS, the reformulated cross-simplicity constraints, in a certain relevant sense, are now first class, and this causes the coherent state method of imposing the constraints, key in the FKLS model, to fail to give any restriction on states. Nevertheless, the cross-simplicity can still be seen as implemented via suppression of intertwiner degrees of freedom in the dynamical propagation. In the second part of the paper, we investigate area spectra in the models. The results of these two investigations will highlight how, in the flipped model, the Hilbert space of states, as well as the spectra of area operators exactly match those of loop quantum gravity, whereas in the FKLS (and Barrett-Crane) models, the boundary Hilbert spaces and area spectra are different."
Lorentzian LQG vertex amplitude
(Submitted on 26 Oct 2007)
"We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, the model can be obtained from the Lorentzian Barrett-Crane model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero-Immirzi parameter limit."
Numerical indications on the semiclassical limit of the flipped vertex
Elena Magliaro, Claudio Perini, Carlo Rovelli
4 pages, 8 figures
(Submitted on 26 Oct 2007 (v1), last revised 27 Oct 2007 (this version, v2))
"We introduce a technique for testing the semiclassical limit of a quantum gravity vertex amplitude. The technique is based on the propagation of a semiclassical wave packet. We apply this technique to the newly introduced "flipped" vertex in loop quantum gravity, in order to test the intertwiner dependence of the vertex. Under some drastic simplifications, we find very preliminary, but surprisingly good numerical evidence for the correct classical limit."
*there was plenty of activity before the detailed introduction in August! A 6-page paper about the new vertex was posted in May 2007 http://arxiv.org/abs/0705.2388 and subsequently published in Physical Review Letters (Phys. Rev. Lett.99 161301). Rovelli gave a talk about it at Loops '07 in June. I think the first fairly complete discussion, however, was this:
Flipped spinfoam vertex and loop gravity
Jonathan Engle, Roberto Pereira, Carlo Rovelli
37 pages, 4 figures
(Submitted on 9 Aug 2007)
"We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett-Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett-Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO(3) loop quantum gravity -- the two can be identified as eigenstates of the same physical quantities -- providing a solution to the problem of connecting the covariant SO(4) spinfoam formalism with the canonical SO(3) spin-network one. The vertex amplitude is SO(3) and SO(4)-covariant. It rectifies the triviality of the intertwiner dependence of the Barrett-Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized."
Other references can be found in the paper that Engle and Pereira just posted.
Nov1-07, 09:33 PM
New papers are coming out fast, about the Marseille new vertex.
Today another. This time bringing in the IMMIRZI parameter. This is important in first-generation LQG---called canonical---but until this year Immirzi was not discovered in spinfoam.
Now in 2006 and 2007 there is all this new work in spinfoam, and among other things there is the Marseille new vertex formula----giving new spinfoam dynamics---and new things are coming out.
Recent papers suggest that this dynamics has the good largescale limit, or semiclassical limit----that it is working out. So naturally one wants to know if the immirzi parameter arises in it.
LQG vertex with finite Immirzi parameter
Jonathan Engle, Etera Livine, Roberto Pereira, Carlo Rovelli
(Submitted on 1 Nov 2007)
"We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Immirzi parameter gamma. We cover the euclidean as well as the lorentzian case. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on gamma, and, remarkably, holds in the lorentzian case as well. The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite gamma. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions."
Carlo Rovelli must be feeling lucky now. Things didn't have to work out so nicely. In August, when the defining paper came out (link in preceding post) they weren't doing it for the LORENTZ case, only the Euclidean because the notation is simpler. Now they have control of both cases, so that works. And the BRIDGE between spinfoam and old-style LQG is established. And they get a DISCRETE SPECTRUM area operator in spinfoam. And the IMMIRZI works out right. So somebody hit the piñata, kids, and the candy is on the floor. Get in there and get your share. That's how it looks to me.
The Marseille team has scored big.
I've been waiting for a paper like this for the better part of three years. Puts it all together.
Or if not all, a major piece of it---there are still things to work out like how to realize the standard bag of particles to mention but one (remember that Perimeter ball-and-tube scheme)
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