Exploring Connes' Finite Noncommutative Geometry Model

[MTd2]

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

...

 

[MTd2]

I wonder what is the largest symmetry group that can be consistent with Conne`s Spectral Geometry, with and without embedding the Standard Model.

 

[marcus]

As you can tell from the mention of kool-aid the post you quoted was not a serious description. It's more a parody than anything else. Somebody at the forum should write a concise faithful account of NCG standard model. Maybe Arivero could.

I wonder what is the largest symmetry group that can be consistent with Conne`s Spectral Geometry, with and without embedding the Standard Model.

I guess you saw, already in the abstract of 1004.0464, the mention of SU(2)xSU(2)xSU(4):

==quote==
The reduction from the natural symmetry group SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that the two layers of space-time are...
==endquote==

Somebody else needs to answer your question about the "largest" symmetry group. I would guess that this depends on the choice of the finite space F, or the associated algebra A_F. But don't understand the relationship.

there is some detail about that in 1004.0464. In section 9.6 on page 34

 

[arivero]

It is a really good question, of course it fixes the finite space, and then I guess (albet it is a long time since I look at it) it also fixes the dimension of the finite space. So a better question could be which groups can be built for each dimension.

 

[marcus]

It is a really good question, of course it fixes the finite space, and then I guess (albet it is a long time since I look at it) it also fixes the dimension of the finite space. So a better question could be which groups can be built for each dimension.

Arivero,

What is your perspective on what Urs says here:
http://www.math.columbia.edu/~woit/wordpress/?p=3292&cpage=2#comment-69896
is he overly optimistic about what Connes has achieved?
Or what further development might achieve?

 

[tom.stoer]

Can anybody explain to me what the reasons for a specific F are? Does Connes simply check some Fs and takes the one that allows for the SM? Is F singled out by something else? Are there more choices for F possible? How many?

I mean, if he takes one specific F out of many possibilities, isn't that a re-writing of the SM - perhaps w/o any additional benefit?

 

[arivero]

it does not seem exceptionally optimistic :-) It is true that Connes's model has advanced a lot, and his current interest (I hear from third hand nowadays, sorry) on Pati-Salam is a big jump too, as B-L is, IMHO, a needed piece.

The most optimistic comment by Urs is that it is already linked to susy. To me, this link with susy is a must; D=10 is singled out by susy (which in turn is singled out, arguably, by division algebras). It is puzzling that Alain has got into D=10 (mod 8) without finding a clue of susy. This should be the "further development". But note that there is almost nobody working in the NCG SM.

In fact, if you quotient any measure of the results between the number of persons in the field, NCG is the most productive Beyond Standard Model theory :D

 

[marcus]

Thanks Arivero!
I want to highlight Tom Stoer's question. Hope someone (you?) will answer.

Can anybody explain to me what the reasons for a specific F are? Does Connes simply check some Fs and takes the one that allows for the SM? Is F singled out by something else? Are there more choices for F possible? How many?

Also since someone else might be reading who has not read Urs Schreiber's comment on Connes spectral geometry Standard Model, I will just paste in. Urs is a former John Baez co-author, respected guy in Hamburg's math department, one of those people I try especially to understand what they are saying.

==quote Urs==
I suppose you have followed Alain Connes’ construction (here is a survey and links) of the standard model by a Kaluza-Klein compactification in spectral geometry. It unifies all standard model gauge fields, gravity as well as the Higgs as components of a single spin connection. Connes finds a remarkably simple characterizaiton of the vector bundle over the compactification space such that its sections produce precisely the standard model particle spectrum, three chiral generations and all.

Alain Connes had computed the Higgs mass in this model under the big-desert hypothesis to a value that was in a rather remarkable chain of events experimentally ruled out shortly afterwards by the Tevatron. But the big desert is a big assumption and people got over the shock and are making better assumptions now. We’ll see.

Apart from being a nice geometrical unification of gravity and the other forces (credits ought to go all the way back to Kaluza and Klein, but in spectral geometry their orginal idea works out better) Connes’ model has some other striking features:

the total dimension of the compactified spacetime in the model as seen by K-theory is and has to be, as they showed, to produce exactly the standard model spectrum plus gravity: D= 4+6.

Now “as seen by K-theory” was shown by Stolz and Teichner and students to mean in a precise sense: as seen by quantum superparticles (here is some link — you can ask me for a better link). In fact what they consider is almost exactly the spectral triples that Connes considers, with some slight variation and from a slightly different angle. For the relation see the nLab entry on spectral triple (ask me to expand that entry…).

As also indicated at that entry: there is a decent theory of how to obtain a spectral triple as the point particle limit of a superconformal 2-dimensional CFT. Yan Soibelmal will have an article on that in our book Precisely because Connes’ model turns out to have real K-theory dimension 4+6 does it have a chance to be the point particle limit of a critical 2D SCFT. That would even give it the UV-completion — as they say — that would make its quantization consistent (which, remember, contains gravity).

I think there is some impressive progress here. It is not coming out of the physics departments, though, but out of the math departmens. For some reason.
==endquote==

[EDIT: short deletion] I think in some of these papers he explains how the finite space was arrived at--I don't understand the process and would like to know the answer to your question.

 

[tom.stoer]

It can't only be that because of deriving predictions. I think in some of these papers he explains how the finite space was arrived at--I don't understand the process and would like to know the answer to your question.

Yeah, I was too sloppy. I'll delete my last sentence but stress the follwowing questions:

Can anybody explain to me what the reasons for a specific F are? Does Connes simply check some Fs and takes the one that allows for the SM? Is F singled out by something else? Are there more choices for F possible? How many?

 

[suprised]

Apart from F, there is an arbitray function f which defines higher corrections and which is unspecified, so the theory is to be considered as a classical, effective theory still in need of an UV completion in order to make sense quantum mechanically; the usual questions and problems thus remain. I wonder whether it can be embedded in string theory; perhaps I'll ask a student to try.

Its virtue is an efficient encoding of the standard model in a mathematical structure, and it is unclear to me how many other choices would be there in principle. This reminds me of the exceptional groups which too naturally encode the standard model representations (which has been known since ages), but whose significance remains unclear.

 

[MTd2]

This reminds me of the exceptional groups which too naturally encode the standard model representations (which has been known since ages), but whose significance remains unclear.

I thought the only exceptional group that contained the SM was E6

 

[S.Daedalus]

I wonder whether it can be embedded in string theory; perhaps I'll ask a student to try.

There's some ideas on that on http://ncatlab.org/nlab/show/higher+category+theory+and+physics#SpecStandModAndGravity":

Connes’ spectral triple whose particle spectrum reproduces the standard model of particle physics has a chance of being the point particle limit or decategorification of the kind of 2-spectral triple – a 2-dimensional superconformal field theory – of the kind that is considered in string theory.

There's also a paper by Chamseddine on the arXiv: http://arxiv.org/abs/hep-th/9705153"

 

[unusualname]

I can't answer tom.stoer's question about F, and I suspect Connes hasn't got a clear answer either.

Connes has obviously realized that some group theoretic structure is required to describe the microscopic and concocted a particularly obscure model to partially fit.

If his model ever does improve on anything already known, it will surely be reduced to a much simpler description that physicists will be bothered to consider and probably be found to be equivalent to alternate constructions in any case.

Sorry, but he can't just keep tinkering with an already overly abstract and complex model anytime experiments show it isn't correct and expect to be taken seriously.

 

[suprised]

I thought the only exceptional group that contained the SM was E6

Well above E6 the reps are vector-like and contain in a sense "too many copies" of the wanted reps. Still the type of reps is allright.

More precisely, there is a sequence of exceptional groups (related by deleting nodes of their Dynkin diagrams), and the decompositions contain reps in fundamental reps of the types and charges which appears in the SM. What is often not realized that the sequence goes on below E6, because those "exceptional" groups just coincide with classical groups:

E8 -> E7 -> E6 -> SO(10) -> SU(5) -> SU(3)xSU(2)

(U(1)'s omitted).
So the SM gauge group is in this sense nothing but "E3", with correct reps (3,2) etc.

This pattern has fascinated physicists already in the 70's, as everybody is thrilled by "exceptional structures" (this thrill was recently revived in Lisi's attempt). But it was never clear whether this observation had any deeper significance; ie, _why_ an exceptional structure would be chosen.

This is what Connes' model has reminded me of; a cute mathematical structure that efficiently encodes the SM (or part thereof), whose significance is otherwise unclear.

 

[MTd2]

The sequence can go upwards too, until Peter West`s E11. But Lisi`s E8 just have 1 generation with mirror matter... I don`t get why it would have a straightforward physical interest.

This is what Connes' model has reminded me of; a cute mathematical structure that efficiently encodes the SM (or part thereof), whose significance is otherwise unclear.

Apparently, Connes`s model has a close resemblance with spin networks in its origin. One might complete the other. Spin networks giving Conne`s model its quantization and spin network, and thus LQG, acquiring matter fields.

 

[suprised]

The sequence can go upwards too, until Peter West`s E11. But Lisi`s E8 just have 1 generation with mirror matter... I don`t get why it would have a straightforward physical interest.
.

Well indeed the sequence goes up as well (it's not just Peter West's though, there were many ppl involved before him). But I was focussing on GUT-like groups for which the rep content has supposedly something to do with the SM reps.

The higher exceptional algebras En, n>8, are not standard Lie algebras but Kac-Moody and hyperbolic extensions thereof, and as such do not appear as GUT gauge groups, but as symmetry groups of extended supergravities etc. So physically they are on a different footing. In fact there are zillions of other such algebras appearing in string theories, but since there are zillions, each one does not have a particular significance. The old hope was that there would be a unique, mathematically distinguished, structure that "does the job", but this has never worked out.

As for Lisi-E8, I called it "attempt" for good reasons.

 

[arivero]

Can anybody explain to me what the reasons for a specific F are? Does Connes simply check some Fs and takes the one that allows for the SM? Is F singled out by something else? Are there more choices for F possible? How many?

Now, from the historical development of the model, it seems clear that F is not singled out, neither the dimensionality of the space. It started in the eighties as a model of electroweak symmetry breaking, a collaboration with Lott, which later was expanded to include color. Then it was noticed that the two different spaces holding vector and axial-vector interactions could be unified in a single spectral triple F (that was the "reality" axiom), and then ten years later it was noticed that a unwanted duplication of fermions was avoided if the dimension of F was taken to be 6 mod 8 instead of 0. Last, two years ago, it was suggested that a further improvement was inviting to start from a spectral triple for Pati-Salam.

 

[tom.stoer]

Thanks to all; I will check the new papers and come back to you if I have questions

 

[S.Daedalus]

I just recalled that in the paper by Asselmeyer-Maluga and Rose from June of this year, http://arxiv.org/abs/1006.2230" , in the end, they make the following conjecture:

At the end we want to give another interpretation of the Casson handle. Connes
[37] showed that by means of the non-commutative geometry the action of the standard
model can be reproduced. His model is based on the space M ×F were the additiona
space F is ad hoc and has no relation to the spacetime M. In our model the space F
could be interpreted as an expression of the Casson handle and so of the smoothness
of spacetime establishing a deep relation between quantum matter and space.

I was wondering -- how realistic is this? Is there any way in which the Casson handle naturally supplies an F such that it is possible to get the standard model a la Connes from it?

 

[MTd2]

I was wondering -- how realistic is this? Is there any way in which the Casson handle naturally supplies an F such that it is possible to get the standard model a la Connes from it?

There will be an update soon to that paper!

 

[tom.stoer]

I find the idea regarding exotic smoothness very appealing; it introduces no extra structures besides spacetime (in contradisticntion e.g. to Connes F or string theory).

In addition it seems to single out 4-dim. spacetime: only in M⁴ enough extra information encoded in the exotic smooth structure / Casson handels is available to allow for "macroscopic matter distribution" or "many particles". In other dimensions matter would always be "rare".

It would be interesting to speculate regarding a measure on top of the set of "all M's" that does not only count different M's of one dimension but different M's with different dimensions. If the measure is able to distinguish between different differentiable structures (for the same M defined topologically) then the measure would be "peaked" at R⁴ as this is the only case with uncountably many differentiable structures. So dim M = 4 would be preferred statistically.

My idea is something like

$$\int D(\text{dimension}) \int D(\text{homeomorphism}) \int D(\text{differentiable structure})$$

where the three "path integrals" count different dimensions, different topological manifols and different differential structures on the same topological manifold.

For me this is the first theory I have seen so far that could be able to explain why we live in (exactly) 4 dim. spacetime and how matter does emerge from spacetime itself.

 

[MTd2]

Yes, Tom, this is my favorite too. But, as for dimension, it is quite agnostic for the fundamental number of dimensions, although it justifies why 4 is the one that is picked up to be compatible with our world.

Another cool theory is this one:

http://arxiv.org/abs/1008.1045Quantum Gravity via Manifold Positivity

Michael Freedman
(Submitted on 5 Aug 2010 (v1), last revised 1 Dec 2010 (this version, v3))
The macroscopic dimensions of space should not be input but rather output of a general model for physics. Here, dimensionality arises from a recently discovered mathematical bifurcation: positive versus indefinite manifold pairings. It is used to build an action on a formal chain of combinatorial space-times of arbitrary dimension. The context for such actions is 2-field theory where Feynman integrals are not over classical, but previously quantized configurations. A topologically enforced singularity of the action terminates the dimension at four and, in fact, the final fourth dimension is Lorentzian due to light-like vectors in the four dimensional manifold pairing. Our starting point is the action of causal dynamical triangulations but in a dimension-agnostic setting. It is encouraging that some hint of extra small dimensions emerges from our action.

 

[tom.stoer]

I downloaded this paper some weeks ago but had no time to look at it.

I don't think that exotic smoothness is agnostic about the number of dimensions. If you sum over ALL manifolds and if the measure distinguishes between different dimensions, topologies and differentiable structures then all manifolds != R⁴ form a null set. That means picking at random a manifold out of all manifolds the selected manifold is topologically an R⁴ with probability one.

Btw. the "homotopy" in the integral was nonsense - I changed it to "homeomorphism"

 

[MTd2]

Not a null set because exotic smoothness is a kind of classical limit of some quantum theories

 

[tom.stoer]

Of course a null set

for all n there are countably many (topologically different) manifolds with dim M = n
for all n except 4 there are countably many different differentiable structures
for compact M⁴ there are again countably many different differentiable structures
but for R⁴ there are uncountably many differentiable structures

That means for all n, there are (count all dimensions, all topologically different manifolds and all different differentiable structures) uncountably many differentiable structures. So the differentiable structures for dim M != 4 or dim M = 4 but M != R⁴ are all null set in the set of all differentiable structures.

 

[MTd2]

See the table at p. 14

http://arxiv.org/abs/0904.1276

This is used through this whole paper:

http://arxiv.org/abs/1006.2642

 

[tom.stoer]

I understand; but I am talking about the table on page 4

 

[MTd2]

I see. Well, I just point out that it doesn't rule out physics theories with more than 4 dimensions, but just to what they compactify.

 

[S.Daedalus]

There will be an update soon to that paper!

Ah, thanks for pointing that out, I'll keep an eye open...

My idea is something like

$$\int D(\text{dimension}) \int D(\text{homeomorphism}) \int D(\text{differentiable structure})$$

where the three "path integrals" count different dimensions, different topological manifols and different differential structures on the same topological manifold.

An additional bonus is that there seems to be a sense in which quantum gravity is 'already unified' in this and other geometric-matter theories; if matter's just a part of the geometry, and you sum over all geometries, there's bound to be some of it in there somewhere.

 

[tom.stoer]

An additional bonus is that there seems to be a sense in which quantum gravity is 'already unified' in this and other geometric-matter theories; if matter's just a part of the geometry, and you sum over all geometries, there's bound to be some of it in there somewhere.

I agree, that would be great. Seems that LQG (even with framed graphs) can't do that to the same degree. There is still a "landscape" of different groups one could use for the spin networks + "ad-hoc matter" on top of it. In Connes' approach it seems that F is not singlet out but has to be introduced by hand.

 

[MTd2]

I just finished reading the update paper. Not related to Connes. But it is much better and interesting, actually, if it is really proven true.