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Marcolli Pierpaoli: Early Universe models from Noncommutative Geometry

  1. Aug 26, 2009 #1


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    In case anyone wants to comment, I thought I'd start a thread on it.

    Early Universe models from Noncommutative Geometry
    Matilde Marcolli (Caltech), Elena Pierpaoli (USC)
    49 pages, 26 figures
    (Submitted on 25 Aug 2009)
    "We investigate cosmological predictions on the early universe based on the noncommutative geometry models of gravity coupled to matter. Using the renormalization group analysis for the Standard Model with right handed neutrinos and Majorana mass terms, which is the particle physics content of the most recent noncommutative geometry models, we analyze the behavior of the coefficients of the gravitational and cosmological terms in the Lagrangian derived from the asymptotic expansion of the spectral action functional of noncommutative geometry. We find emergent Hoyle-Narlikar and conformal gravity at the see-saw scales and a running effective gravitational constant, which affects the propagation of gravitational waves and the evaporation law of primordial black holes and provides Linde models of negative gravity in the early universe. The same renormalization group analysis also governs the running of the effective cosmological constant of the model. The model also provides a Higgs based slow-roll inflationary mechanism, for which one can explicitly compute the slow-roll parameters. The particle physics content allows for dark matter models based on sterile neutrinos with Majorana mass terms."

    Marcolli is full professor in the math department at Caltech.
    She taught at MIT, then later was associate prof with tenure at the Max Planck Institute for Mathematics at Bonn (2003-2008).
    She was born near the end of 1969 and got her PhD in 1997. Whole bunch of honors.
    Last edited: Aug 27, 2009
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  3. Aug 28, 2009 #2


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    I see that Marcolli has been invited to speak at the next International Conference of Mathematicians (ICM).

    http://www.icm2010.org.in/speakers.php [Broken]

    The ICM is held every four years (like the Olympics :biggrin:) and on the order of 1000 mathematicians participate. The main prizes and honors are awarded. There are parallel sessions in all the different subfields.

    There is a section called Mathematical Physics with about 10 invited speakers. Marcolli is one of those 10. It's a significant recognition, I think.

    You may know a couple of the others who are on that list of 10 in Math Physics. Gary Gibbons and Niklas Beisert.

    Germany has 3 on the list of ten. The UK has 2. The USA has 2 (one of which is Marcolli.)
    If this had taken place in 2008, then Marcolli would still have been at the Max Planck Institute for Math and Germany would have 4, with only 1 for the USA. Excuse the nationalistic silliness, but this is like Olympic gold medals except it is in Math Phys. I woud like my country to be doing better, making a better showing at the ICM in 2010, especially in Math Phys. You can see from the photograph that Marcolli has a good build for mathematical physics. http://www.its.caltech.edu/~matilde/ She would not be doing so well in the pole vault. But Math Physics is a lot more important than the pole vault. :biggrin:
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  4. Aug 28, 2009 #3


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    Running G and Lambda is also how Steven Weinberg is proposing to explain inflation and the graceful exit from inflation, according to his 6 July talk at CERN. It's barely possibe this lady could be one jump ahead of Weinberg. There is some discussion of running constants on pages 12 and 13 of the paper. And then applied to the cosmological constant, and inflation, around page 40 and 41:
    4.12. Variable effective cosmological constant.
    The relation between particle physics and the cosmological constant, through the contribution of the quantum vacua of fields, is well known since the seminal work of Zeldovich [49]. The cosmological constant problem is the question of reconciling a very large value predicted by particle physics with a near zero value that conforms to the observations of cosmology. Among the proposed solutions to this problem are various models, starting with [49], with a varying effective cosmological constant, which would allow for a large cosmological constant in the very early universe, whose effect of negative pressure can overcome the attractive nature of gravity and result in accelerated expansion, and then a decay of the cosmological constant to zero (see also [40] for a more recent treatment of variable cosmological constant models). Often the effective cosmological constant is produced via a non-minimal coupling of gravity to another field, as in [25], similarly to what one does in the case of an effective gravitational constant.

    In the present model, one can recover the same mechanism of [25] via the non-minimal coupling to the Higgs field, but additionally one has a running of the effective cosmological constant γ0 (Λ) which already by itself may produce the desired effect of decaying cosmological constant. We illustrate in this section an example of how different choices of the parameter f4 , for a fixed choice of f2 , generate different possible decay behaviors of the cosmological constant. These can then be combined with the effect produced by the non-minimal coupling with the Higgs field, which behaves differently here than in the case originally analyzed by [25].

    This still does not resolve the fine tuning problem, of course, because we are trading the fine tuning of the cosmological constant for the tuning of the parameters f2 and f4 of the model, but the fact that these parameters have a geometric meaning in terms of the spectral action functional may suggest geometric constraints.
    Last edited: Aug 28, 2009
  5. Aug 28, 2009 #4


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    I'd like to learn more about Connes NC geometry, but it seems even more abstract than string theory!
  6. Aug 28, 2009 #5


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    I know what you mean about abstract, but having said that, the fields are very different. Thousands of people have worked on string and the field has made little progress during the past ten years.

    Moreover some of the top, most talented people seem to be drifting out---developing other interests on a part or full-time basis. The field has lost focus.

    By contrast Connes NC geometry has been worked on by only a half dozen or so and has made great progress in the past 10 years. It is only about 10 years old.

    The derivation of the Standard Model was only very recent, in 2006. So what we are talking about now is essentially stuff that happened in the past 3 ore 4 years.

    It is a very focused field. They don't have a lot of different variations. The model that Marcolli is using is essentially the same as what John Barrett presented in 2006 and what, at the same time, Connes and Chamseddine presented.

    BTW this is the same John Barrett who is giving the series of lectures on spin foam at the Corfu QG school starting about a week from now. It is impressive to me that he is a Loop QG master who has acquired a sideline in NC geometry---one of the very first to successfully formulate the Standard Model using Connes geometry. His paper and Connes' came out in the same week.

    Marcolli cites Barrett as her reference [4]
    "[4] J. Barrett, Lorentzian version of the noncommutative geometry of the Standard Model
    of particle physics. J. Math. Phys. 48 (2007)..."

    In case anyone is curious, here's the preprint:

    A Lorentzian version of the non-commutative geometry of the standard model of particle physics
    John W. Barrett
    14 pages
    (Submitted on 31 Aug 2006)
    "A formulation of the non-commutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes' internal space geometry so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the see-saw mechanism of neutrino mass generation."
    Last edited: Aug 28, 2009
  7. Aug 28, 2009 #6


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    The point is, I don't even know what is the basis of Connes' non commutative geometry, besides the buzzword "non commutative geometry".
  8. Aug 28, 2009 #7


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    If you really don't know anything beyond the "buzzword", then to start a gradual process of absorption or assimilation you should get familiar with the algebraic object called a ring. (two operations + and x, addition and multiplication)
    And become familiar with the idea that inside a ring R there can be some "sticky" subsets S
    such that if you take any element s in S, and multiply it by a general r in R, that the result rs is still in S.

    S is called an "ideal" and the stickiness of it is summed up by saying for any r, the set rS is contained in S.

    That means that once you are in S you cannot get out by multiplying. (By adding yes OK but multiplying by any other element in the ring will not get you out of S.)

    This is an extremely simple, idiotically simple, idea and it just happens to also be extremely powerful. As I will illustrate, and that will be enough for a start.

    A good example of a ring is C(M) where M is some compact manifold and C(M) is all the complexvalued functions on M. Or realvalued, would be just as good for now, but say complexvalued. So f(m) is a complex number.

    You can add two elements f and g of C(M), and you can multiply them in the obvious way.

    (f+g)(m) = f(m) + g(m) and (fg)(m) = f(m)g(m)

    It turns out that if you pick a point mo in the manifold, and consider the set of all functions f which are zero at that point, that set is an ideal. It is sticky. If a function is in that set then f(mo) = 0 and no matter what other function you multiply it by the result will always be zero at mo.

    This also is a very obvious idea. And that ideal {f: f(mo) = 0} is maximal in the sense that there is no larger ideal subset of C(M) which contains it.

    It turns out that this describes all the maximal ideals in C(M) and that there is a one-to-one correspondence between the maximal ideals of the ring C(M) and the points of M.

    Indeed if somebody secretly chooses a manifold (say a torus) and constructs the ring of functions on it as an algebraic object, and then, without telling you that M is a torus, they describe the ring of functions on it---by some algebraic catalog proceedure. Then you could deduce what the manifold is. All the information about the manifold is contained already in the ring of functions defined on the manifold. As algebraic relations.

    This is the step that is hard to accomodate. It is somebody's theorem, a Russian perhaps.

    But once you believe that given any manifold you can have a ring R = C(M) which contains all the information, algebraically somehow, about the manifold, then you have an obvious way to generalize what a manifold is.

    You can take any ring at all (even one where the multiplication is not commutative! as it is in the ring C(M) of functions on a manifold) and you can consider the maximal ideals of that ring as a new kind of manifold.

    Now that is a bit too much, too general. Connes only wants to consider NICE rings, with a few extra tools and structure, as accessories to make working with the ring a good experience.
    There are some axioms. The ring should have a norm, some absolute value or size type measure looking like ||f||. The ring should have a special operator defined on it. Things like that.
    But the basic tactic is to generalize the idea of a manifold by studying the maximal ideals of a ring that might or might not be the ring of functions defined on a manifold. Perhaps the manifold was never there! Perhaps the ring just came into existence but you treat it as if it were the ring of functions on some unknown manifold.

    Why is this interesting?

    Because Bernie Riemann invented the diff manifold in 1850 and we haven't had a new concept of a continuum for over 150 years.

    What is string theory defined on? A vintage-1850 manifold. What are branes? Vintage-1850 manifolds. In whatever branch of physics, our usual idea of a continuum is this very ancient idea of Riemann.

    Connes comes along and says suppose that spacetime is the set of maximal ideals of a certain ring, and suppose the damned ring is not even entirely commutative! Maybe it is mostly commutative but it has a small discrete noncommutative piece. He calls this "nearly commutative" or "almost commutative". I forget which.

    The attraction of this kind of things is that it is the first fundamentally new model of a continuum in a long long time.

    Of course Loop uses networks, and simplicial complexes---combinatorial objects---but these are not a continuum, although they might represent a quantum state of geometry.

    Because it is such a new idea of the continuum, people are attracted to study it and try to learn what the new possibilities are.

    This is a very brief introduction, only enough to take you past what you refer to as the "buzzword" level. You will gradually absorb the rest, just from the atmosphere, if you are patient. :biggrin:
    Last edited: Aug 29, 2009
  9. Aug 29, 2009 #8


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    Marcus, would like to post this on my blog? Not a just like a regular poster, but just post interesting things, like this one. You could inaugurate it! Right now, I just tested its posting ability. I invited Lee Smoling to post there, when his article with Garrett is uploaded. I guess he accepted.

    Last edited: Aug 29, 2009
  10. Aug 29, 2009 #9


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    does anybody know a good review article as introduction to non-commutative geometry? I'd like to understand the basics and especially the SM derivation, of course.
  11. Aug 29, 2009 #10
    I would suggest to browse Alain's web site. He has a lot of downloadable stuff. In particular, there is a quick overview for physicist.

    If I can state my personal opinion : back in the beginning of the 1930's, we physicists decided to go with Dirac's practical manipulations. It was not a mistake, because we needed to be able to test quantum mechanics and apply it right away. But we should have taken von Neumann seriously, and too few of us explored the rigorous way. Why should we expect any fundamental progress from models whose mathematics are not well-defined ? Another example of what Alain & co have in store for us is a full understanding of the renormalisation procedure of Feynman diagrams. Most textbooks we grew up with would need to be re-written in this regards.
  12. Aug 29, 2009 #11
    "Introduction to Noncommutative Spaces and their Geometry" by Giovanni Landi, arxiv hep-th/9701078. Lots of examples and a large appendix with results from functional analysis and topology for background info.

    Lots of articles and two pdf books on alainconnes.org.

    "Noncommutative Geometry" by Alain Connes.

    "Noncommutative Geometry, Quantum Fields and Motives" by Connes and Marcolli. Very detailed presentation of Quantum Field Theory, Renormalization, Standard Model and the NC geometry presentation of the Standard Model. All that is in chapter 1 (all 340 pages of it).

    A more abbreviated presentation of the NC geometry construction of the Standard Model is in arxiv hep-th/0901.0577 by Ali Chamseddine. In the article he mentions that he and Connes are preparing a long article in a language more accessible to physicists. He also outlines their refined model of the renormalization cutoff function which they hope will give a better estimate of the Higgs mass (better than their 170GeV estimate which has been experimentally excluded).

    I am no expert on NC geometry but I wonder if there might be a NCG model of spacetime which would exhibit the energy dependent dimensional reduction, 4--->2, in Causal Dynamical Triangulations and other approaches. See: https://www.physicsforums.com/showthread.php?t=323417 That would be really cool!

    Cheers, Skippy
  13. Aug 29, 2009 #12


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    It is an honor to be invited to contribute that post to your new blog. But you see what good company we have here---stoer has asked an appropriate question and humanino has answered*. I am happy with PF, have no desire for another place.

    You are welcome to copy that post to your blog, just give a link back to PF as acknowledgement.

    The Garden of Forking Paths is an elegant reference to Borges. Beautiful name for a blog especially one about fundamental physics, the most Borgesian of the sciences. The graphic header for your blog is also an unusually fine choice. Whenever anyone mentions Borges I am compelled to show what an excellent translator Richard Wilbur is. His version of this sonnet could be the best translation of Spanish verse into English that I know.
    I quote from memory and so do not have the punctuation right and may make a few mistakes:


    One thing does not exist, Oblivion.
    God saves the metal, and he saves the dross,
    and his prophetic memory guards from loss
    the moons to come, and those of evenings gone.

    Everything is: The shadows in the glass
    which, in between the day's two twilights, you
    have scattered by the thousands, or shall strew,
    henceforward in the mirrors that you pass.

    And everything is part of that diverse
    crystalline memory, the universe.
    Whoever through its endless mazes wanders,
    hears door on door click shut behind his stride,
    and only from the sunset's farther side
    shall view at last the Archetypes and Splendors.

    Daniel, tom, humanino, the crystalline memory is the (4D block) universe and the archetypes and splendors are the true laws of physics which, I hope, we will never discover and so may continue always to investigate. "God", I suppose, is a figure of speech. The "endless mazes" must be the garden of forking paths afterwhich Daniel MTd2 has named his new blog. Good luck with blogging!

    *now I see also skippy has replied.
    Last edited: Aug 29, 2009
  14. Aug 29, 2009 #13


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    I'm starting a blog because I want to moderate and control my topics :). This will increase the probability of specialists not fearing offenses.

    The name is not The Garden of Forking Paths, but the "The Forking Pahts of the Blue Tiger Library". :)

    It is a reference to 3 stories of borges and 1 of Kafka.

    http://en.wikipedia.org/wiki/Blue_Tigers -> fundamental observational issues

    http://en.wikipedia.org/wiki/The_Garden_of_Forking_Paths -> you said it, but I thought of the interplay of determinism and randomness in our world.

    http://en.wikipedia.org/wiki/The_Library_of_Babel -> the seek for the truth

    The picture has a castle behind the labyrinth, so it is a reference to The Castle, from Kafka. A reference for the unsurmountable imaginary obstacles of the world. I found that picture here, when searching on google images. If I find a better picture, with a castle beyond a maze, I will put there.

    Marcus, can I copy any of your posts that I think are interesting, perhaps with some modifications, provided that I provide a source?
    Last edited: Aug 29, 2009
  15. Aug 29, 2009 #14


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    There is a kind of compatibility between the research fields of Noncommutative Geometry/Quantum Geometry and the Borges story of Blue Tigers. Wiki has an excellent summary which is, itself, enjoyable to read:

    I suspect that most new physics blogs evaporate because there are so many already, but Daniel's might take hold and grow if only because of the fine name:
    Forking Paths of the Blue Tiger Library.

    Of all writers, Borges was the most accurately aware of modern physics and I mentioned before my inexpert opinion that fundamental physics is in some sense Borgesian, so it seems appropriate to have a reference to "The Garden of Forking Paths" and "Blue Tigers".
    Unlikely things are possible---the blog might succeed.

    I don't want another place to post my reflections on the physics scene, but it's tempting.
    One thing is, I wish Daniel would write email to Matilde Marcolli. I think that if you look at her picture:
    you see at once someone whose judgement is to be trusted, and someone who might appreciate Borges. She also has nice hands, dresses completely in black, and has made a good choice of hiking boots. But the main thing is, you can see from her eyes that she is entirely to be trusted, as concerns the forefront of fundamental physics.

    My impulse, if I were to post on Daniel's blog, would be to post Matilde's picture
    http://www.its.caltech.edu/~matilde/ (in case you missed the link the first time)
    and perhaps also the abstract of her recent arxiv post

    Daniel, if you want to quote from here I would say yes provisionally, permission revocable at any time, keeping in mind some things:
    1. Give a link to the PF thread. My primary "loyalty" is to PF and this would be good advertising for PF. On the chance that your blog succeeds and gets eyeballs it would help to send some more eyeballs over here, and potential posters.
    2. Edit if you want, but mention that you are editing: "I have made minor revisions in Marcus' post." Or something like that. "Edited with author's permission."
    3. If I get embarrassed for some reason, I will ask you to stop. Otherwise I am granting open-ended permission.
    Last edited: Aug 29, 2009
  16. Aug 29, 2009 #15


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