Noncommutative Geometry and Physics

This week the Newton Institute in Cambridge is running (with funding from the Templeton Foundation) a workshop on the topic of Noncommutative Geometry and Physics: Fundamental Structure of Space and Time. The program is here, some of the talks are online here, and Paul Cook is blogging here (with truly scary pictures of a Newton Institute restroom). Thursday evening they will be having a public panel discussion, entitled The Nature of Space and Time: An Evening of Speculation.

For someone interested not in quantum gravity but in particle physics, the most interesting of these talks is doubtless that of Alain Connes, entitled Noncommutative Geometry and the standard model with neutrino mixing. He has a new paper out, with the same title, with more details promised in a forthcoming paper with Chamseddine. I’ve been carrying the paper around for a while now, hoping to understand exactly what he’s doing, but it’s rather dense and some of the calculations are involved and don’t carry much of an explanation. I really wish I’d been at the talk to hear his exposition of what he’s up to here.

Among the other talks at the conference I would like to have heard would be that of Samson Shatashvili, who is doing some very interesting things with 2d gauge theories. He has a new paper out (with Gerasimov) which looks quite readable, entitled Higgs Bundles, Gauge Theories and Quantum Groups.

Another conference going on that is finishing up this week is the Erice “International School of Subnuclear Physics”, this year entitled The Logic of Nature, Complexity and New Physics, and dedicated to Richard Dalitz. Back during the sixties, seventies and eighties, the Erice School was an important yearly event, featuring the best theorists around giving expository lectures on the latest ideas, often including spectacularly beautiful lectures by Sidney Coleman. This year’s school just looks profoundly weird, with an interesting and reasonable set of lectures on the experimental side, but the theoretical side mostly devoted to “complexity and the Landscape”, featuring an opening lecture and mini-course about the Landscape by Susskind, a mini-course about the Landscape and its computational complexity (i.e. why it is hopeless to ever use it to predict anything) by Denef and Douglas, and more complexity from Zichichi, Beck, Gell-Mann and Tsallis. Many of the lectures are available here. Steve Hsu is blogging from the conference, and he reports that Susskind says the Landscape program is science since it now gives exactly one bit of information about the universe (the sign of the spatial curvature k), although he expects Andrei Linde to be able to make that bit disappear if he wants to.

Update: Urs Schreiber has an excellent discussion of the Connes program here and here.

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41 Responses to Noncommutative Geometry and Physics

  1. PPCook says:

    Hi Peter, thanks for the link. The conference here is very interesting, but what has been very tough is the realisation that to understand the Connes paper it does not help to hear him speak. It is, of course, a pleasure to hear him speak. And he is an excellent speaker. It’s just to benefit one really must hear him speak while holding all the ideas behind the approach in your head. I think there is much promise in his approach, but it does seem to be very constructive. It seems the most important point to start learning about the Connes approach would be his paper with Chamseddine The Spectral Action Principle. Perhaps after spending years staring at the standard model one might gain the same feeling as Connes and one might feel that the starting points are intuitive, although perhaps not. One must bear in mind Alain Connes’ own words when assessing his approach to physics:

    “There are two fundamental sources of ‘bare’ facts for the mathematician. These are, on the one hand the physical world which is the source of geometry, and on the other hand the arithmetic of numbers which is the source of number theory. Any theory concerning either of these subjects can be tested by performing experiments either in the physical world or with numbers. That is, there are some real things out there to which we can confront our understanding.”

    Treating experimental data on the same footing as number theory seems to remove the need for motivating certain assumptions. The standard model just is – we have observed it, that is good enough. At the end of his talk he described string theorists who look for more than is apparent as living in “a dreamworld”. I think this is understandable, but the dream is alluring. It is also very natural for scientists to seek explanations, but yet it is proper to respect experimental evidence. Nevertheless the formulation of Connes is extremely compact, and one hopes it is not just exquisite engineering.

  2. John Baez says:

    Connes surely doesn’t get the right dimensionless constants in the Standard Model Lagrangian – if he did, folks at the Newton Institute would be drinking champagne and dancing naked in the streets. So, how does he manage to come so close yet not that far? And, what is his attitude towards these constants? Did his audience press him on this point?

  3. steve says:

    Peter,

    I never thought I’d be defending string theory, but I think your remark above is too negative.

    Lenny didn’t claim victory because there is a single robust prediction from the Landscape. He’s seems disturbed that most low-energy observables are unpredictable, even in an anthropic framework.

    However, it does mean the Landscape is falsifiable — if Planck measures a positive curvature it will strongly disfavor the scenario.

  4. Peter Shor says:

    John,

    It looked to me from his abstract that the dimensionless constants were parameters that Connes could put in his model, and the only actual predictions (so far) came at unification scale? I don’t understand any of this, so could somebody verify whether this is right?

    Anyway, it seems to me that if the string theorists got this close (for whatever values of close he got), there clearly would be dancing in the streets.

  5. A says:

    Following your link, I tried to read the Susskind lectures, but they contain a few words, almost no equations, plenty of cartoons.
    My child got interested.

  6. amanda says:

    “However, it does mean the Landscape is falsifiable — if Planck measures a positive curvature it will strongly disfavor the scenario”

    It will disfavor the *particular* scenario that LS pushes, with Coleman-de Luccia instantons. Of course, LS has the bad habit of pretending that his way of doing things is the only way — cf his repeated declarations that black hole complementarity, which is nothing but the wildest of wild speculations, is a “law of nature”!

  7. MathPhys says:

    amanda,

    What do you mean by “black hole complementarity” more precisely?

  8. steve says:

    Amanda,

    I’m under the impression that if there are many metastable vacua (almost all with much larger vacuum energy than our own), then it is highly likely that our universe must have originated in a tunneling (bubble nucleation) event. If so, the curvature has to be negative. It so happens that Coleman-Deluccia worked out the bubble form, but I don’t see that Lenny is making a nontrivial assumption.

    Am I missing something?

  9. D R Lunsford says:

    Has anyone improved on Brout’s paper as an exposition for physicists?

    -drl

  10. Peter Woit says:

    steve,

    I didn’t say he was claiming victory, just that he was using this to answer certain people who argue that this is not science. It’s good to hear that he’s disturbed by not having any low-energy predictions, but the problem with the Landscape is not just at low-energy, it doesn’t give predictions at any energy.

    As for the single bit of info here, I remember a time when string theorists were going on about no-go theorems that showed that you couldn’t have string theory in deSitter space, i.e. with a positive cosmological constant. When a positive cosmological constant was found, they seem to have come up with a way of dealing with that problem. If the spatial curvature comes out positive, I’m sure they’ll come up with something.

  11. urs says:

    John Baez said

    Connes surely doesn’t get the right dimensionless constants in the Standard Model Lagrangian – if he did, folks at the Newton Institute would be drinking champagne and dancing naked in the streets. So, how does he manage to come so close yet not that far? And, what is his attitude towards these constants? Did his audience press him on this point?

    I don’t think the point of this quest for the spectral action of the standard model is to predict the standard model’s properties.

    I think the main point is first of all to understand which spectral triple precisely is the one whose spectral geometry describes the standard model.

    It would of course certainly be a nice side effect if some properties of the standard model were derivable this way, maybe in the sense that they might turn out to be forced to have a certain value to admit a spectral description at all.

    So I think the point is that if you want to understand something deeply, you should first try to find its most elegant/powerful/compact description. And Connes rightly points out that encoding the entire standard model into a spectral triple does achieve such a description.

    And we learn by that, for instance, that we observe a world of metric dimension 4 and KO-dimension 4+6 mod 8.

    While not a prediction, that looks like a remarkable insight.

    I am going to say more about that at the n-Café. So far there is an introductory entry.

  12. wolfgang says:

    I did not see much discussion on the quantization of the theory in the latest paper of Alain Connes. Is renormalization etc. already taken care of automatically in this approach (or described in another paper) ?

  13. urs says:

    I did not see much discussion on the quantization of the theory in the latest paper of Alain Connes. Is renormalization etc. already taken care of automatically in this approach (or described in another paper) ?

    As far as I can tell, so far this is just a way to rewrite the standard action functional in a different form.

    See section 6 of Connes’ latest paper for remarks on what might be expected as a “UV completion” of this spectral action.

    In fact, his collaborator Chamseddine once began trying to understand if the spectral action obtained from a 1-Dirac operator could be the limit of a UV-complete one obtained from a 2-Dirac operator

  14. Alejandro Rivero says:

    wolfgang, one intriguing thing of Connes approach is that some details that are supposed to come from quantisation appear here as a consequence of the axions. Paricularly, Poincare duality imposes anomaly cancellation.

  15. A says:

    Actually the prediction of negative curvature has nothing to do with string theory: as shown by Coleman-deLuccia, a negative curvature arises from vacuum decay, for any potential studied so far. Next, anthropic arguments are used to argue that inflation maybe does not need to suppress the curvature down to unobservably small values.

  16. PPCook says:

    John,
    No one in the audience asked about the dimensionless constants…
    Best wishes,
    Paul

  17. urs says:

    The Yukawa coupling constants are encoded in the “metric” of the internal (d_KO = 6)-dimensional space, namely in the Dirac operator associated with it. See equation (1.21) of http://arxiv.org/abs/hep-th/9606001

    I talk about that here.

  18. urs says:

    Furthermore, apparently the bare gauge coupling is related to the volume of spacetime by equations (2.20) and (2.29) of the review. More details are here

    http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_1.html#c004520

    So, in a word, Connes does not predict the parameters of the standard model. What he does instead is to identify the “geometry of a non-geometric KK compactification” that does reproduce the standard model (including the dimensionless constants).

  19. steve says:

    A: Yes, but as I mentioned in a Landscape scenario our universe likely originated in a negative curvature bubble nucleation, so there is a definite sign prediction for k. Inflation might have flattened things out, but it would be hard to explain a positive measurement of k.

    It isn’t that negative k, or a nucleation origin, require string theory, but rather the converse.

  20. TheGraduate says:

    Hi,

    I know people are in the process of figuring out what Connes’ paper means but I was wondering if anybody could summarize in a sentence or two of nontechnical language what issues are in play. So far I understand the following:

    1. This paper says something about the number of dimensions being equal mod 8 … and since (26 = 10 ) mod 8 … that says something about string theory … perhaps

    2. It also makes progress in a group theoretic description of the standard model …

    I apologize for my abyssmal ignorance and would greatly appreciate some breakdown.

  21. Andy Neitzke says:

    `What he does instead is to identify the “geometry of a non-geometric KK compactification” that does reproduce the standard model (including the dimensionless constants).’

    Does he explain what theory one has to “compactify” on this nongeometric background to get the Standard Model?

  22. urs says:

    First of all, many thanks to “anon” for pointing out a stupidity I said. The corrected statement is here

    http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_1.html#c004536

    Does he explain what theory one has to “compactify” on this nongeometric background to get the Standard Model?

    Well, his theory is that given by the action functional which you cook up from the spectral triple.

    So, given any spectral geometry, Connes considers the action functional obtained from its generalized “heat kernel expansion”.

    That’s his way to formulate things. There are indications, though, that, if applied for instance to the Dirac-Ramond operator, this generalized “heat kernel expansion” reproduces precisely the effective string theory equations of motion.

    That’s at least what Ali Chamseddine claims to have shown. See p. 10 of

    http://arxiv.org/abs/hep-th/9701096

    as well as

    http://arxiv.org/abs/hep-th/9705153 .

  23. urs says:

    summarize in a sentence or two of nontechnical language what issues are in play.

    The idea is this:

    To any given ordinary Riemannian manifold X, we may associate the Einstein-Hilbert Action. That’s gravity.

    We know from Kaluza-Klein, that if X looks like Y x S^1 that gravity on X looks like gravity coupled to electromagnetism on Y.

    Now, Connes comes along annd says that there is something like a “generalized” Riemannian manifold. And he provides a way to define a notion of action for that, such that for X an ordinary Riemannian manifold that action reduces to the ordinary Einstein-Hilbert action (plus correction terms of higher order).

    So, he says, assume our spacetime is not an ordinary Riemannian manifold with extra forces in it, but just some generalized Riemannian manifold with only gravity.

    Using his generalized notion of the Einstein-Hilbert action, we may check for each such generalized Riemannian manifold what its “gravitational” action looks like.

    In a similar fashion, fermionic matter propagating on an ordinary Riemannian spin manifold (= coupled to gravity) may be generalized to fermionic matter propagating on a generalized Riemannian manifold.

    So, Connes says, what I want to do is to find a generalized Riemannian manifold X of the form Y x Z, such that my generalized theory of gravity coupled to fermions on X looks like gravity coupled to our standard model on Y.

    He plays around a little and finally finds Z such that this works.

  24. anon. says:

    urs wrote:

    There are indications, though, that, if applied for instance to the Dirac-Ramond operator, this generalized “heat kernel expansion” reproduces precisely the effective string theory equations of motion.

    Is this a leading-order statement in both g_string and alpha’, or are higher terms in the “heat kernel expansion” that should reproduce some or all of the corrections?

  25. urs says:

    Is this a leading-order statement in both g_string and alpha’, or are higher terms in the “heat kernel expansion” that should reproduce some or all of the corrections?

    Yes, Chamseddine checks leading order only.

    I don’t know if the higher orders “should” agree. But if they do, we’d get a nicely coherent picture of what is going on.

    Maybe to emphasize that, allow me to reformulate Connes’ approach somewhat suggestively this way:

    He is considering the theory of a superparticle (worldline susy) in d_KO = 10 dimensions, determined by a Dirac operator D.

    He takes the interactions of this superparticle to be such that the effective target space action is Tr(f(F)) + (psi,D psi).

    Then he looks for compactifications

    D = D_0 + D_F

    such that this effective target space action agrees (to lowest order) with the standard model coupled to gravity.

    That’s not precisely how Connes states it. And maybe it’s wrong. But maybe it’s right.

  26. PPCook says:

    I Just noticed the paper A Lorentzian version of the non-commutative geometry of the standard model of particle physics by John Barrett today (who has also spoken here at the NCGW, but not on this topic). It seems strange that, to my knowledge, it hasn’t been mentioned amongst the online discussion of the Connes paper even though it appeared on the same day as the Connes paper and in fact even a few entries earlier on the archive. The results appear at first to be identical.

  27. Chris W. says:

    From Connes (hep-th/0608226):

    8. Acknowledgements
    The detailed computations and extension of this work to the left-right model will appear in a joint work with Ali Chamseddine and Matilde Marcolli. The need to have independence between the KO-dimension and the metric dimension already emerged in the work of P Dabrowski and L., Sitarz on Podle´s quantum spheres [10]. The results of this work were announced in a talk at the Newton Institute in July 2006, and the fear of a numerical error in the above computations delayed the present publication. It is a pleasure to acknowledge the independent preprint by John Barret (A Lorentzian version of the non-commutative geometry of the standard model of particle physics) with a similar solution of the fermion doubling problem which accelerated the present publication.

  28. urs says:

    Coming back to the question on what, if anything, the spectral thing predicts, there are a couple of intersting remarks in

    hep-th/9605001

    (which is in general a pretty good account of the details involved).

    In the introduction, the authors cite a couple of papers that argued that “most” models of Yang-Mills-Higgs type that one could write down are not obtainable by spectral action.

    In the concluding section, it furthermore says that

    – assuming a Higgs, the spectral stuff predicts a minimum of two generations

    – including also the measured top mass then also predicts a maximum of 5 generations

    – existence of the Higgs forces the electroweak sector to be chiral

    – “the choice of possible gauge groups is very much restricted” it says

    – finally: the huge span of fermion masses can not be explained.

    So, there are a couple of internal consistency conditions, but the compactification metric (hence the set of dimensionless parameters) is essentially free input.

    I have tried to expand on the superparticle point of view mentioned above:

    http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_2.html

  29. anon. says:

    – existence of the Higgs forces the electroweak sector to be chiral

    !

    So the claim is that any time there’s a spectral geometry formulation of your theory, you can only Higgs a gauge group with chiral fermions? There’s no way to e.g. add a Higgs for SU(3)_color to this model?

    Is there a heuristic explanation of why this should be true? It seems very strange.

  30. John Baez says:

    Connes writes:

    It is a pleasure to acknowledge the independent preprint by John Barret (A Lorentzian version of the non-commutative geometry of the standard model of particle physics) with a similar solution of the fermion doubling problem which accelerated the present publication.

    Heh. The French never, never spell John Barrett’s name right.

    But, it’s great that he scooped Connes. He’d been working on this for quite a while – he was quite excited about it when I saw him this April in Marseille, for the thesis defense of Alejandro Perez.

  31. urs says:

    Is there a heuristic explanation of why this should be true?

    I am not entirely sure. What I know is this:

    the Higgs arises as the “internal” component of the connection, A_int.

    By the general logic of the approach, this term is given by the commutator of the internal Dirac operator with an algebra element a

    A_int = a [D_int ,a’] .

    The algebra and its representation is such that projected on the anti-particle sector [D_int,a’] = 0, so we just get a contribution on the particle sector.

    This is computed for instance on p. 26 of hep-th/960353 .

    After the computation, you impose the self-adjointness condition and find that the term A_int is exactly given by a quaternion-valued function, which is to be interpreted as the Higgs.

    A similar discussion must be in hep-th/9605001 somewhere, I guess.

  32. nontrad says:

    At the risk of disrupting the ongoing discussion, I flipped through Douglas’ 08.31.06 Erice lecture on ‘computational complexity and fundamental physics’.

    It all seemed mostly innocuous until I arrived at the slide inwhich Douglas’ began to discuss protein folding, and that made me pause.

    Douglas makes some comments about evolution and the random order of amino acids in protein chains, which don’t strike as anything other then a crude cartoon of the reality of proteins.
    How and why such a cartoon-ish sketch of proteins appears in a talk on cosmological questions is then troubling to me since it suggests a situation of cartoons built on cartoons. Or analogies built on analogies. Or a castle made of sand.

    Douglas then refers to Smolin’s ‘cosmological natural selection’ and call’s it ‘bizarre’, which immediately made me realize that perhaps Douglas is unaware of how his own thesis appears just as bizarre as those Douglas perceives as being bizarre. The pot calling the kettle black typically indicates bad news for someone.

    Susskind’s talk similarly features loops of DNA appearing more then once.

    Hard to say that it isn’t troubling to see what appears to be a tower of cartoons based on cartoons.

    Who knows, perhaps all of these folks are super geniuses who have really figured is all out and are on the verge of a tottally complete and unified theory of everything.

    I guess I’m caught up on the fact that their road to that end seems so fantastically unreasonable!

  33. Chris W. says:

    Reiterating PPCook’s report, as a follow-on to ‘nontrad’ and ‘cox’:

    At the end of his talk he [Connes] described string theorists who look for more than is apparent as living in “a dreamworld”.

    Indeed. And they talk about this dreamworld as though it was as solidly grounded in reality as the Atlantic Ocean, DNA, proteins, or the universe outside our solar system. To paraphrase Descartes, “I speculate that it exists, therefore it does”.

  34. Alejandro Rivero says:

    appear here as a consequence of the axions.

    Er… axioms

  35. Have you boys forgotten what anniversary September 8 is? I haven’t.

  36. D R Lunsford says:

    Urs – you might like this:

    D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Principle of general Q-covariance, Journal of Mathematical Physics 4, 788-796 (1963)

    -drl

  37. kristo says:

    Is there somewhere an account available of the ‘discussion’ “The Nature of Space and Time: An Evening of Speculation” on Thursday?

  38. Alejandro Rivero says:

    kristo, sorry there was at least three bloggers+PhysicsForums dwellers and nobody of us has taken time about it, perhaps swept by the main happennings. The evening run smoothly without surprises, but with some anecdote. Hawkings computer reboot I have described in dorigo his blog. Amazing also the transmutation of Heller from a quantum theoretist in the morning to a teologist in the evening, these guys at vaticano have a good hiring board.

  39. Alejandro Rivero says:

    Baez says: Heh. The French never, never spell John Barrett’s name right.

    Actually I am having a hard time trying to distinghish if he thanks John Barret or John Baez in the first minutes of his talk :-DDD

    (between minutes 20…21 of the MP3 file. Not at the end, where Barret is clearly named)

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