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Noncommutative geometry and the standard model

  1. Oct 19, 2003 #1

    marcus

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    I hate to risk additional distraction when we have so much thoughtful discussion---and add an additional thread when there are several good ones going, but I just came across a 15 October post at arxiv which I want to pass along, should anyone be interested.

    It is an 11-page article contributed to the "Encyclopedia of Mathematical Physics" and called "Noncommutative geometry and the standard model".

    From the introduction: "The aim of this contribution is to explain how Connes derives the standard model of electro-magnetic, weak and strong forces from noncommutative geometry..."

    http://arxiv.org/hep-th/0310145

    The author is at Marseille (Center for Theoretical Physics)---same place as Rovelli and a number of folk recently writing loop gravity articles. His name is Thomas Schuecker, doesnt mean anything to me but may to you. Seems to be a lot of ferment of all kinds at the Marseille CPT.

    The writing style impressed me as straightforward and concise
     
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  3. Oct 19, 2003 #2

    selfAdjoint

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    I've read the paper. It gives a good account of the ideas behind noncommutative geometry, and a really great recap of standard model with everything you have to match laid out. And then he (a) doesn't show you the derivation, and (b) says the derivation doesn't really work. For irreducible triples (the basic unit of nc spacetime) you get only one generation of quarks. And no matter how you fiddle the triples, you can't escape massless neutrinos. Shades of Thiemann! Shades of the tachyon!

    Nevertheless, this is about as simple an introduction to the hot topic of noncommutative geometry that you're going to find, and I highly enjoyed it.
     
  4. Oct 24, 2003 #3
    What about non-commutativity and LQG?

    Has anyone read anything about it?

    If string theory and the Connes' approach both have it, is there anything non-trivial that the CQG peaple have to say?
     
  5. Oct 24, 2003 #4

    selfAdjoint

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    Both have what?
     
  6. Oct 24, 2003 #5

    marcus

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    I cant check this right now but I think quantum groups or hopf algebras are used nontrivially in for example Smolin
    "Quantum Gravity with a Positive Cosmological Constant"

    http://arxiv.org/hep-th/0209079

    I could be misremembering so I will try to check this in time to edit
    and find a better lead if I am wrong. You might be the person to explicate if essential use is made
     
  7. Oct 24, 2003 #6
    What I meant is that both approaches have a picture of geometry being non-commutative at some level. In the case of Connes' it is somewhat there from the begining (also in all this new fuzzy physics). In the case of string theory, I know that they get in some limits a non-commutative field theory as the low energy limit, but I ignore the details.

    In the case of LQG, it is known that some geometric operators do not commute (http://arxiv.org/abs/gr-qc/9806041), but not much more has been written.
     
    Last edited: Oct 24, 2003
  8. Oct 24, 2003 #7

    marcus

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    Yes, if by nontrivial use of "noncommutative" algebra you would include the use of a quantum group or hopf algebra then that paper certainly qualifies. But you may not mean this. In any case here is the abstract for "Quantum Gravity With a Positive Cosmological Constant":

    -----------begin quote--------

    "A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter space-
    time. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime.

    Furthermore, one may derive directly from the Wheeler-deWitt equation corrections to the energy-momentum relations for matter fields of the form E2 = p2 +m2+αlPlanckE3 +... where α is a computable dimensionless constant. This may lead in the next few years to experimental tests of the theory.

    To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation.

    The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout."
    -------------end quote------------------

    I am used to associating "noncommutative geometry" with the idea of deformed groups or quantum groups, where (if for example there is a matrix representation) the matrices and matrix-operations are all "deformed" by a parameter which one can imagine is very small, which if it were zero, would give us the familiar algebraic situation back again. Things like this have been in use in LQG since around 1995 when Smolin posted a precursor to the paper I mentioned:

    "Linking topological quantum field theory and nonperturbative quantum gravity"

    http://arxiv.org/gr-qc/9505028

    However a seemingly more thorough and satisfactory treatment is in the recent paper "Quantum gravity with a positive cosmological constant."

    I hope that this is not completely off target and corresponds to your associations with "noncommutative geometry" also.
     
  9. Oct 24, 2003 #8
    I agree that deformations also lie in the category of what is done in non-commutativity. The q-deformed spin networks go back to 1995 I think in works by Smolin and Major. What I mean is more physically related questions:

    If I want to measure the volume of a region and the area of the closed surface that bounds it, and these operator do not commute, then I need to worry about an uncertainty principle. Do we get something like [x^i,x^j] different from zero? For x^i coordinates on the manifold.
     
  10. Oct 24, 2003 #9
    Marcus..here is a "sort of explanation" given by Ed Witten some time ago:http://arxiv.org/abs/gr-qc/0306083
     
  11. Oct 24, 2003 #10
    First significant experimental results might be already there. See e.g.:
    http://arxiv.org/abs/astro-ph/0304527
     
  12. Oct 24, 2003 #11

    marcus

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    this is in reply to nonunitary's post, a few posts back:
    -------------
    " I agree that deformations also lie in the category of what is done in non-commutativity. The q-deformed spin networks go back to 1995 I think in works by Smolin and Major. What I mean is more physically related questions:

    If I want to measure the volume of a region and the area of the closed surface that bounds it, and these operator do not commute, then I need to worry about an uncertainty principle. Do we get something like [x^i,x^j] different from zero? For x^i coordinates on the manifold."
    -------------
    :smile:
    Of course you mustnt take me for a quantum gravity expert speaking for that field of research. I am struggling to get some
    perspective on it---and assume that several people here (perhaps including yourself) are more knowledgeable at least in some details of the subject.

    But in my reading I have seen quite a few LQG "observables" defined---like area and volume operators----and these are operators on hilbertspace and (I would assume) do not generally commute.

    If one has two surfaces defined by some matter and they happen to intersect then I would imagine that the two area operators do not commute! It would be par for the course.

    But as a footnote to what you said. I think in GR the coordinates have no physical meaning (this is "the problem of the meaning of the coordinates" which worried Einstein between 1912 and 1915). Indeed points in the manifold have no physical meaning and are not observables in the classic situation. And this is carried over to the LQG quantum version of GR----so there would not be operators on the hilbertspace corresponding to the x-coordinate or the y-coordinate of some arbitrary coordinate "patch". AT LEAST SO I THINK, its really something you need to determine, but i've never seen anything like such observables in my loop gravity reading.

    To get points of reference one would presumably introduce matter and then looking at positions and momenta etc relative to something phyical one would have more or less the same commutativity and noncommutativity as in basic quantum mechanics, or exactly the same. I cant think why it would be different. Same Heisenberg principle as always. Just no absolute space and time coordinates! But I could be wrong. I wish we had an authority to ask.
     
    Last edited: Oct 24, 2003
  13. Oct 24, 2003 #12
    Marcus,

    I absolutely agree that simply coordinates on the manifold have no physical meaning. However, one could imagine definig some observables like the ones given by Rovelli
    http://arxiv.org/abs/gr-qc/0110003
    that are built out of matter or the like and that have the non-commutativity property.

    We should certainly ask an expert!
     
  14. Oct 24, 2003 #13

    marcus

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    Ranyart yes! That paper by Witten is partly in response to Smolin's "Quantum Gravity With a Positive Cosmological Constant".

    Indeed Witten has only 4 references at the end and reference #3 is to this Smolin paper. Here is a quote from the main part of Witten's paper:

    ------quote from Witten------
    "Some authors have proposed the Kodama wavefunction as a starting point for understanding the real universe; for a
    review and references, see [3]. Our discussion here will make it clear how the Kodama state should be interpreted. For example, in the Fock space that one can build (see [3]) in expanding around the Kodama state, gravitons of one helicity will have positive energy and those of the opposite helicity will have negative energy."
    ---------------

    So Witten features this Smolin paper prominently and it sort of gives him the occasion for writing his paper. "since other people are using Kodama state, here is what I have to say about it". The Witten paper does not seem to raise any serious obstacle to Smolin and in fact this summer Smolin and a guy in the HEP group at SLAC named Stephon Alexander came out with a follow-on development which uses that model of quantum gravity (with positive Lambda) to predict inflation without any special "inflaton" or special stuff put in by hand----an early universe inflationary epoch just from quantizing the gravitational field. It may be a good thing that Smolin is onto. Really too early to say. here is the paper:

    Stephon Alexander, Justin Malecki, Lee Smolin
    "Quantum Gravity and Inflation"
    http://arxiv.org/abs/hep-th/0309045

    their abstract refers to the "normalizability" the Kodama state---good to be cautious: it must be an open question whether this approach to quantum gravity can be applied generally:

    --------quote from abstract--------
    Using the Ashtekar-Sen variables of loop quantum gravity, a new class of exact solutions to the equations of quantum cosmology is found for gravity coupled to a scalar field, that corresponds to inflating universes. The scalar field, which has an arbitrary potential, is treated as a time variable, reducing the hamiltonian constraint to a time-dependent Schroedinger equation. When reduced to the homogeneous and isotropic case, this is solved exactly by a set of solutions that extend the Kodama state, taking into account the time dependence of the vacuum energy. Each quantum state corresponds to a classical solution of the Hamiltonian-Jacobi equation. The study of the latter shows evidence for an attractor, suggesting a universality in the phenomena of inflation. Finally, wavepackets can be constructed by superposing solutions with different ratios of kinetic to potential scalar field energy, resolving, at least in this case, the issue of normalizability of the Kodama state.

    ----------end quote-------
     
    Last edited: Oct 24, 2003
  15. Oct 24, 2003 #14

    selfAdjoint

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    Is Kodaima well defined?

    Marcus if you read down further in the paper you see that Witten regards the Chern-Simons state of Yang-Mills theory, and by extension the Kodaima solution in QG as purely formal, and for a lot of reasons he gives, not physical. we could discuss this more, I suppose.
     
  16. Oct 24, 2003 #15

    marcus

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    Re: Is Kodama well defined?

    Indeed Field-Medalist Witten suggests that it is "unphysical" and he works on strings and branes. By contrast, Smolin and Malecki and Alexander do not consider it necessarily unphysical and get results with it. There could be something interesting here.

    I don't propose to drag you and me into a discussion of the the hairy Kodama state. If Stephon Alexander is a young guy from Imperial College London, and Brown who recently was promoted to HEP group SLAC and Witten wants to warn him "Stop! consider the unphysicality! dont fall for the blandishments of Smolin and waste your promising carreer on a vain pursuit of the Kodama state", well you and I can not do any good by discussing it.

    Strictly from the sideline find it fascinating that Witten apparently wrote that 11 page paper in reaction to Smolin's paper. Kodama was way back in 1988. But Witten didnt write then, he waited till right after Smolin's use of Kodama.

    Then in reply Smolin/Alexander/Malecki CITE Witten, of course and say---this Kodama business has some issues: "see what Witten said and ALSO see the forthcoming paper by Freidel and Smolin". So Smolin has gotten Laurent Freidel to help him find out more about it. (Freidel is someone whose papers I watch out for)

    I'm interested to see how the Kodama story plays out. Maybe something new will come out of it. But at the moment, our PF poster just asked did LQG have anything like noncommutative geometry---which for me means quantum groups and the q-deformation of representations and stuff. And so I am answering yes ever since 1995 or so some LQG work has that and it connects to non-zero Lambda. For better or worse Smolin got started monkeying with a non-zero cosmological constant some years before the 1998 discovery that there was one and, for better or worse, got started doing LQG with quantum groups. It is curious and not anything that I could have a hunch about either way.
     
    Last edited: Oct 24, 2003
  17. Oct 24, 2003 #16

    selfAdjoint

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    To me, Witten's action in waiting from 1988 till Smolin's paper to put out his warning is logical. Notice he says that the state, although unphysical, is interesting to study. There's nothing wrong with it as math, is his line, just don't try to build physics with it. So there would be no reason for a warning until Smolin takes it up and proposes to build physics.

    I am not going to get involved in this either. But a Witten paper I can digest at one siting is something I couldn't pass up.
     
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