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Covariant Loop Gravity and Livine's Thesis

  1. Oct 26, 2003 #1

    marcus

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    A spin foam is a "mousse de spin"
    On Friday Rovelli is giving a symposium talk on spin foams and he was Etera Livine's thesis director.
    My uninformed guess is that Rovelli will talk about Livine's thesis and in particular chapter 8 (Covariant loop gravity) which reflects potentially important work by Alexandrov, some of which Livine co-authored. This work extends the symmetry of loop gravity to SL(2,C) and seems to get rid of the Immirzi parameter (!). The thesis is at http://arxiv.org/gr-qc/0309028

    Quite recent, another reason it would seem like a good thing to bring up at the Strings Meets Loops symposium this Friday. Here is a quote from the abstact, giving links to a paper of Alexandrov/Livine among other things:

    ------------quote from abstract--------

    'Thu, 4 Sep 2003
    Boucles et Mousses de Spin en Gravite Quantique
    Etera R. Livine
    165 pages, in French; PhD Thesis 2003, Centre de Physique Theorique CNRS-UPR 7061 (France)

    I review the formalism of loop quantum gravity, in both its real and complex formulations, and spin foam theory which is its path integral counterpart. Spin networks for non-compact groups are introduced (following hep-th/0205268) to deal with gauge invariant structures based on the Lorentz group. The whole formalism is studied in details in three dimensions in both its canonical formulation (loop gravity) and its spin foam formulation. The main output (following gr-qc/0212077) is the discreteness of timelike intervals and the continuous character of spacelike distances even at the quantum level. Then it is explained how to extend these considerations to the 4-dimensional case. I review the Barrett-Crane model, its geometrical interpretation, its link with general relativity and the role of causality. It is shown to be the history formulation of a covariant canonical formulation of loop gravity (following gr-qc/0209105), whose link with standard loop quantum gravity is discussed. Similarly to the 3d case, spacelike areas turn out continuous. Finally, ways of extracting informations from the non-perturbative spin foam structures are discussed."
    ------------------end quote---------

    BTW Livine seems to think it matters that time is discrete but space is continuous---a result from Freidel/Livine/Rovelli
    http://arxiv.org/gr-qc/0212077
    Why should this result be so important? It is admittedly odd.
     
    Last edited: Oct 26, 2003
  2. jcsd
  3. Oct 27, 2003 #2
    Marcus, if I may it is the New Physics and interpretation of 'Spacetime', spacetime is infact discrete, Galaxies are where spacetime resides. The MINKOWSKI SPACE VACUUM that intervines between Galaxies is continuous, spacetimes are coupled to matter in Galaxies, where as, Space-Vacuum has no observed matter, but is an Energy field.

    My money is going to be on the re-classification of Space energy, and Spacetime energies, seperated into distinct Horizons, Time-depandant and NonTime-dependant.

    E= Mc2...space energy(non-mass), converts to Spacetime energy (mass) :wink:
     
  4. Oct 27, 2003 #3

    marcus

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    Re: Re: Covariant Loop Gravity and Livine's Thesis

    I hope your bets and mine are in play money---monopoly money--because real mathematics is happening this time. There is an equation like eee-equals-em-cee-square that tells the dark energy that empty space contains but you and I do not know this equation and, what is more interesting, real mathematicians (who are characterized by uncanny ability to intuit and guess theoretical results---this is how they know what to try to prove logically) these people who are professionally selected for almost prophetic intuition about exactly this "space and time" kind of stuff---THEY ALSO do not know. We are in a dark room and there is something in the room with us that we cannot see and have never seen before and not even the people with very good eyes and ears can picture it. This is an amusing situation.

    On usenet spr you can hear turmoil and cries of pain and frustration because the most articulate branches of physical theory seem unable to predict the positive cosmological constant, or even to ACCOMODATE it in any graceful way. the cosmological constant is the density of dark energy----observed to be 0.6 joules per cubic kilometer.

    Instead of guessing, I propose we relax and enjoy a historical moment in science
     
    Last edited: Oct 27, 2003
  5. Oct 27, 2003 #4

    marcus

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    Livine generalized on Abhay Ashtekar

    In his thesis Livine does a remarkable thing in 10 clearly written pages----he generalizes the Ashtekar-Lewandowski measure to the non-compact case. Yes this sounds like a mouthful of dry sand----something of purely technical interest.

    But if the rest of the community checks this out and it is correct, and if this has not been done before (which I dont think it has but I could be mistaken) then it is a notable event. Like when somebody crossed the English Channel in a pedalpowered aircraft---they have you know. It goes in the edition of Guinness Book of Records that has a picture of Gauss on the cover.

    Well maybe I'm mistaken but I will explain why anyway. So far Rovelli/Ashtekar/Lewandowski and all have only done loop gravity in the case where the group of motions or symmetries is limited in size----doesnt stretch off to infinity. Spin networks are labeled with integer numbers (or half integers) because the group of symmetries is so bounded and controllable, so-called "compact", that all the ways of physically representing it are like countable and labeled by "spin" numbers. Now for about 100 years if you were a mathematician and you proved something in the compact case you ALWAYS tried to extend the result to the more difficult less controllable non-compact case. this is a kneejerk professional reflex learned by generation after generation of graduate students. But Ashtekar and Lewandowski did not extend their construction to the non-compact case evidently because they COULD not----and that was where the group was the Lorentz group: wooooooo special relativity symmetries! important physically as well as mathematically.
    But Livine does this in 10 pages somewhere (pages 47-57) in the middle of his PhD thesis at the University of Marseille.

    So what is going on? What does this mean to us? Well to quantize anything you have to make a hilbertspace which means that you have to be able to integrate functions defined on the space of all possibilities.
    In loop gravity the basic set of possibilities is all possible geometries on the particular manifold you're looking at.
    this set of all possible geometries is denoted A.

    You have to be able to define wavefunctions---just number-valued functions on this A. And you have to be able to INTEGRATE their squares because the square of the amplitude is the probability and you have to be able to sum it up under the integral sign. the whole thing depends on being able to write integrals and sum up functions defined on the configuration space or space of possibilities. So Ashtekar and Lewandowski defined the "measure" which is what the "means of integration" is called. You could call it the Ashtekar/Lewandowski "tool for integrating functions on A", or their technique for building the basic loop gravity hilbertspace of squareintegrable wavefunctions.

    The hilbertspace of squareintegrable functions is fundamental and you cant get it unless you can integrate functions defined on the configs and you cant integrate unless you have a "measure" on the configs.

    Its like having a surface measure or a volume element to integrate with in more familiar situations.

    In a certain sense, progress was held up a while because there was no measure in a certain important version of the theory and Livine just constructed one-----at least it looks like it based on my imperfect knowledge of the situation.

    I will try to talk about this and say how the 10 pages go.
     
    Last edited: Oct 27, 2003
  6. Oct 27, 2003 #5

    selfAdjoint

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    This morning I've been reading the Friedel-Livine paper, Spin Networks for Non-Compact Groups, which you cited above as hep-th/0205268. In this they do the math behind Livine's definition. It's a marvelous paper, written with limpid clarity, and shows how they apply familiar theorems from algebraic geometry, plus a good deal of ingenuity, to define the crucial measure that enables the whole structure. This is exciting and new, and has that feel of "why wasn't this discovered ages ago" that you associate with important results.
     
  7. Oct 27, 2003 #6
    Marcus, you say:There is an equation like eee-equals-em-cee-square that tells the dark energy that empty space contains but you and I do not know this equation and, what is more interesting, real mathematicians (who are characterized by uncanny ability to intuit and guess theoretical results---this is how they know what to try to prove logically) these people who are professionally selected for almost prophetic intuition about exactly this "space and time" kind of stuff---THEY ALSO do not know.

    And not wanting to continue or hijack your posts, I have to beg to differ, for I believe I KNOW!

    Just to ask a little one thing, are fields three dimensional?, and why do all 2-dimensional fields surround matter?:wink:

    Huh mmmm...Contraction something to do with it maybe?
     
  8. Oct 27, 2003 #7

    marcus

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    Yes selfAdjoint I too have been reading the Freidel-Livine paper this morning! It perches on my knees as I lean over to type because there is no place to put it down on the cluttered surfaces around here. Great paper. also in ENGLISH so easier to quote

    yes the columbus-egg feel of important results

    it would be terrible not to have someone to share this sense of discovery with---a kind of lonliness---at this time of day and so I am very glad for your post on the subject
     
  9. Oct 27, 2003 #8

    marcus

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    the idea of taking a maximal tree in any graph and contracting
    that tree to a point and having a new graph consisting of one
    vertex from which extend a number of loops like the petals of a daisy, nice and visual

    it may be a standard proceedure in some other mathematical venue that they just imported and adapted to loop gravity

    and then the use of algebraic geometry
     
  10. Oct 27, 2003 #9

    marcus

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    that must be a great feeling, which most people only get
    the first time they fall in love

    you must be feeling kind of wired, with extra energy and alertness

    don't mention hijacking threads since your excursions so far are completely friendly
     
  11. Oct 27, 2003 #10

    selfAdjoint

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    I really enjoyed the way they used the basic tree property (no loops) again and again in proving their theorems. All this time LQG people have been saying network-network, and never did anyone see the maximal tree in it before.

    And then as you say, the old theorems on varieties applied to the reductive groups. BTW I found a little tutorial on them: http://www.stieltjes.org/archief/rep9899/node11.html which is a nice complement to the paper.
     
  12. Oct 28, 2003 #11

    marcus

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    bet you will visit some campuses while on this road trip to Illinois with yr offspring, good time of year to do this

    the F/L article "spin networks for non-compact groups" is majorly fantastic

    a fock space (a direct sum of L2 spaces) with creation/annihil
    of spinnetworks corresponding to adding/subtracting a loop

    Rovelli has signed on. There is a significant paragraph on middle page 4 of Freidel/Livine/Rovelli "Spectra of Length and Area in
    (2+1) Lorentzian Loop Quantum Gravity"
    http://arxiv.org/gr-qc/0212077
    "....we call a connection on a graph Γ the assignment of group elements..."
    and the footnote on the same page "AS A CONSEQUENCE THE FULL SPACE OF QUANTUM STATES OF THE GEOMETRY...CAN NOT BE OBTAINED AS A PROJECTIVE LIMIT ANY MORE...similar to a Fock space."

    This strikes me as Rovelli's assimilation of a fundamental change in the theory he founded (with Smolin and Ashtekar) some ten years ago. The quantumstates of the geometry is now to be obtained by gluing lots of L2 spaces together and will have creation annihilation operators. As far as I can see it just gets better :)

    this morning did not even think to turn on the computer, just picked up F/L's and F/L/R's papers and resumed reading
    after an hour it occurred to me you might be as well
     
    Last edited: Oct 28, 2003
  13. Oct 28, 2003 #12

    selfAdjoint

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    Livine's action on an edge

    This formula for the basic action, which in occurs in all these Livine papers, puzzles me. k-1s(e)ge(A)kc(e). The part in the middle is the holonomy of connection A along edge e, resulting in group element g, as we have seen before. My problem is with the k's. Apparently from his notation, k denotes an operator from the Lie Algebra of G, and the subscripts indicate they are evaluated at the two vertices at the ends of e, but where did Lie(G) get mapped into points of Σ? I haven't been able to find an explanation of this

    Has anybody got any ideas?
     
  14. Oct 28, 2003 #13

    marcus

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    selfAdjoint, David Louapre may answer your question, if not I will try. Incidentally he says that Laurent Freidel will be giving the talk on spin foams this Friday, in Rovelli's place. See his PF post:

    https://www.physicsforums.com/showthread.php?s=&postid=88141#post88141

    But Freidel is the person whose papers (with Livine, and Rovelli) we have been reading. Even if the talk is only very general for a broad audience it seems like it could be interesting to hear Freidel's point of view. I hope we can get a link to the talk as soon as one is available.
     
    Last edited: Oct 28, 2003
  15. Oct 28, 2003 #14

    jeff

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    Re: Livine's action on an edge

    I'm not sure I understand your confusion. This just gives the behaviour of the holonomy ge(A) under gauge transformations A → Ak = k-1Ak + k-1dk:

    ge(Ak) ≡ k-1s(e)ge(A)kc(e).

    See p41 of livine's thesis.
     
  16. Oct 28, 2003 #15

    selfAdjoint

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    Re: Re: Livine's action on an edge

    Yes I was at that page when I posted. I understand the gauge transformation of the connection form, but how did the Lie elemnts get evaluated at the two ends of the edge? So far the only source of Lie algebra elements is the connection itself - through its form.

    I have been all over google looking for a development of this - none of my books have it - and nothing useful has turned up. I tried to work it out with stoke's theorem, treating the two points as boundary of the edge, but that didn't come out in the right form either. So I'm stumped.
     
  17. Oct 28, 2003 #16
    Re: Re: Re: Livine's action on an edge

    I'm not sure I understand your puzzle but let me try to answer it. The gauge transformation of the connection is parametrized by k which is a field over the manifold, with values in the group G. So k in

    k^{-1}Ak + k^[-1} dk

    is actually k(x) for x point of the manifold.

    Now take a connection field A and g(A) the holonomy of A along an edge. Apply a gauge transformation (parametrized by a G-valued field k(x)) to A and look what happens to g(A). It appears that the way g(A) is modified depends only of the value of k(x) for x=start of the edge and x=end of the edge. So call k_{s(e)} and k_{t(e)} this values, write the transformation of g(A) and forget about the value of k(x) for every other x !

     
  18. Oct 28, 2003 #17

    jeff

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    Re: Livine's action on an edge

    The mapping is from Σ to the lie algebra, as on p25.

    I thought by "action" you'd meant one obtained from a lagrangian density.
     
  19. Oct 28, 2003 #18

    marcus

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    hi selfAdjoint, I will add a bit of detail to what David L. just said.
    I've been busy elsewhere or would have replied earlier.
    When you write down a holonomy along an edge it is an integral and I guess you can approximated it by a Riemmann sum
    And all along the way the k and k-1 and k will cancel.
    So only the first and the last k-1 and k will not be canceled and will appear. Something like that. I will try to find an online paper where this is spelled out.

    the holonomy on an edge (gauge-transformed A) =
    k-1(start) X holonomy on edge (original A) X k(end)
    I am sure I have seen this proved maybe in some basic LQG
    survey, somewhere anyway.

    It looks like Louapre, alas, was just passing thru :(
     
    Last edited: Oct 28, 2003
  20. Oct 28, 2003 #19

    selfAdjoint

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    OK folks, thanks to all of you I get it now. I feel really dumb but what can you do. I do appreciate the answers, and they do completely enlighten me on the subject. Onward and upward.
     
  21. Oct 28, 2003 #20

    marcus

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    I looked in my basic beginner's textbook of LQG which is by Marcus Gaul and Carlo Rovelli and even THERE the thing is not proven. they just say on page 13

    "despite the inhomogeneous transf. rule (19) of the connection
    (which I think we already have written in this thread) under gauge...the holonomy TURNS OUT to transform homogeneously like"

    and they write the same thing that you and I just wrote that just brackets the original holonomy by the endpoint k-inverse and k.

    well that is Gaul/Rovelli
    http://arxiv.org/gr-qc/9910079

    its one of those times when the prof says IT IS EASY TO SHOW, I guess it is a Riemmann sum with a lot of cancelation, an integral on the interval [0,1], some exercise in bookkeeping
    maybe it is just a fun thing to do at the blackboard so no one puts it in books, or they all do it in basic gauge theory for QFT years before
     
    Last edited: Oct 28, 2003
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