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MTd2
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...marcus said:Connes' recent paper allows for the finite space F to change at very high energies. I gather that his predictions are about what one can eventually see with LHC and conceivable extensions along the same lines. In that range, where prediction is practical and meaningful, he has already determined what the finite algebra F must be. So the predictions which he lists are based on that.
I would not advise anyone to suppose that Spectral Geometry simply consists of Connes version of it. I don't think that the question in this thread is addressed by focusing on Connes version NCG and imagining that one simply layers that (in its 2010 form) on top of LQG. So it's not clear how talking about Connes NCG specifically is relevant to the topic. But I'm happy to do so!
The current version is defined by three 2010 papers:
http://arxiv.org/abs/1008.3980
Noncommutative Geometric Spaces with Boundary: Spectral Action
Ali H. Chamseddine, Alain Connes
26 pages, J.Geom.Phys.61:317-332,2011
http://arxiv.org/abs/1008.0985
Space-Time from the spectral point of view
Ali H. Chamseddine, Alain Connes
19 pages. To appear in the Proceedings of the 12th Marcel Grossmann meeting
http://arxiv.org/abs/1004.0464
Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
Ali H. Chamseddine, Alain Connes
56 pages, Fortschritte der Physik,58:553-600, 2010
Here are the predictions/postdictions listed in 1004.0464:
==quote Ali and Alain==
...We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions:
(i) The number of fermions per family is 16.
(ii) The symmetry group is U(1)xSU(2)xSU(3).
(iii) There are quarks and leptons in the correct representations.
(iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1).
(v) Top quark mass of 170-175 Gev.
(v) There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent.
We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.
==endquote==
The Connes model is what they call "almost commutative" where the relevant object is the product of a conventional commutative algebra C(M) with a small finite noncommutative F.
The blue highlight suggests that F can change at Planckian energies! This leaves the model open to new physics. It says that the geometry of spacetime can change radically as you increase the magnification.
The red highlight is how Connes recovers from his pre-2008 bad estimate of Higgs mass. He prepares the ground for higher order corrections, but at this time he does not calculate those corrections.
If you think of Connes "almost commutative" space as a sandwich of |F| different colored copies of ordinary 4D space---a finite sandwich of layers determined by F---then as you zoom into Planckian magnification the number of layers and the coloring can change.
The basic object, as I see it, is still an ordinary 4D manifold M, which we treat via the algebra of continuous functions C(M) defined on M. And then drink a little Connes kool-aid and we see that the right algebra is not simply C(M) but is, in fact, C(M) x F,
the cartesian product of the functions on the manifold M, with a little finite matrix algebra.
Pictorially it is as if M has changed to a sandwich of layers each of which looks like M but has an "F-color".
This is a radical oversimplification of course. If you don't like it then make up your own radical oversimplification.
Now Connes, in the next paper, the one presented at the 2009 Paris Marcel Grossmann, takes the bold step of speculating that if you go to REALLY high energies then even C(M) which you thought was the conventional algebra of functions on a classical 4D manifold becomes, itself, a large but finite algebra of matrices! This is something they didn't tell you when you bought your ticket and walked into the crystal palace.
http://www.icra.it/MG/mg12/en/
http://www.icra.it/MG/mg12/en/invited_speakers_details.htm#connes