Last Thursday Alain Connes gave a talk at CERN TH. Alain is a famous mathematician with important contributions in the areas of operator algebras and non-commutative geometry. He has gathered quite a collection of prestigious awards, including the Fields Medal and the Crafoord Prize. What could bring such a figure to a particle theory seminar? He was sharing his views on the elementary interactions in a talk entitled The Standard Model coupled with gravity, the spectral model and its predictions.

Alain's approach to particle physics is orthogonal to that of the particle physics community. Whereas we try to figure out what sort of new physics could be responsible for the weird structures of the Standard Model, he treats those very structures as an indication of the underlying geometry of space-time. This is certainly original, which has its positives and negatives. One one hand, I find it reassuring that people out there are exploring different ways; in the end, it is conceivable that the standard approach will prove terribly wrong. On the other hand, Alain's language can hardly be understood here at CERN. No, he wasn't speaking french ;-) but I quickly got lost in the forest of KO-theory, metric dimensions and spectral triples. I'm not able to review or evaluate any technical details of his work but I would like to share a few remarks anyway.

His program consists in identifying a structure of space-time could give rise to the Standard Model + gravity. He finds the answer is the product of an ordinary spin manifold by a finite noncommutative discrete space of KO-dimension 6 modulo 8 and of metric dimension 0, whatever it means. The discrete space is responsible for the spectrum, symmetries and the interactions of the Standard Model. Most of the Standard Model parameters correspond to the freedom of parametrizing the internal geometry. There are however three constraints:

- The gauge couplings should be unified at some scale. The unification is rather weird, as there are no exotic gauge bosons, hence no proton decay.
- There is a relation between the sum of the fermion masses squared and the W mass. In practice, this is a constraint on the top mass, which is roughly obeyed in nature.
- Finally, there is a prediction for the higgs quartic coupling, which implies the higgs boson mass of order 170 GeV.

In spite of these objections, I really enjoyed the talk. I think it is due to Alain's manner of speaking: a soft voice full of wonder at the mathematical beauty he perceives in his models. One could think he speaks of autumn trees or little birds in nest, not about scaring non-commutative geometry :-) This sort of enthusiam is rare these days.

The transperiences are not available, as usual. For brave souls, the technical details can be found in the recent paper of Alain and collaborators.

## 7 comments:

A KO-dimension six means you need to add six anticommuting generators to pass from the reduced spectral triple to the unreduced one (See book from Gracia-Bondia et al). Remember that Wilczek-Zee have a trick to produce the standard model group from five generators, and a sixth one can trivially be added to get all the fermions in a same grading.

The tricky part is to fit an horizontal (q-?)symmetry to understand masses. I think recent works of Ma and Zee can have a role here, but no idea of how to fit them.

What is not clear to me whether this is a new kind of "grand unification" or not. On the one hand, Connes said in the talk that the relations between the gauge couplings are fixed (at some GUT like scale) and look like the ones coming from SU(5) grand unification (though without the existence of extra gauge bosons). On the other hand, I fail to see off-hand why one shouldn't be able to rescale the couplings independently, given the direct product SM group. Perhaps I need a closer look.

Moreover I overheard someone saying that the extra three relations between mass parameters that the model "predicts", just correspond to imposing two-loop finiteness (or something like that) and are built-in by hand.

So all-in-all, it is unclear to me whether the model has genuine predictions or is just an efficient and interesting way to parametrize the standard model.

Alejandro, thanks for your explanations. As for me, i'm afraid, you're casting pearls before swine, but i'm sure other, more educated readers may appreciate it.

Anonymous: according to Alain, gauge groups are somehow constrained. Remember, he said that not every gauge group fits in his construction, for example he cannot do it with SU(5). So I would believe in this constraint. For the other two, i also don't know what are the precise assumptions he makes.

I sometimes wonder if Alain Connes is just making it all up. It seems to me in seminars when there is a poor student who has managed to follow some of his work who asks him a question, his eyes open wide and he holds out a hand as if picking a cherry from the air and warbles somethign about non-hausdorff or non-coxeter somethings, seding the student into deeper frustration. It reminds me of star trek when they talk about type 4 quantum singularities. Come on, Connes, you're making it up as you go along, aren't you!

"Come on, Connes, you're making it up as you go along, aren't you!"

There is only one way to find out: redo the calculations. Physicists, who did, might help you to get started. Or you can attend lectures like

http://www.physi.uni-heidelberg.de/

physi/gradkurs/SoSe2007/

index.php

next month to get started. I will also be happy to help you.

If you start now you might be through with the calculation before the LHC produces data and enjoy watching the game.

Thomas Schucker

"Come on, Connes, you're making it up as you go along, aren't you!"

There is only one way to find out: redo the calculations. Physicists, who did, might help you to get started. Or you can attend lectures like

http://www.physi.uni-heidelberg.de/

physi/gradkurs/SoSe2007/

index.php

next month to get started. I will also be happy to help you.

If you start now you might be through with the calculation before the LHC produces data and enjoy watching the game.

Thomas Schucker

I've read the plebian account of the noncommutative standard model and came to wonder if Shape Dynamics' formalism would provide a work around for the issue of Wick rotations from Riemannian to Pseudo-Riemannian manifolds?

What do you think?

Also, and I'll include the link, Shape Dynamics seems to have been tackling its own issues with quantum action of the Shape Dynamics Hamiltonian with Cartan Geometry.

Could Cartan Geometry be the way to extend Connes work and provide a generalization of the Wick rotation?

http://hef.ru.nl/~sgryb/research/shape_dynamics.html

Any insights would be really appreciated.

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