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New baez article, BF

  1. Mar 21, 2006 #1


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  2. jcsd
  3. Mar 21, 2006 #2


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    back now with time to take a closer look

    Exotic Statistics for Loops in 4d BF Theory
    John C. Baez, Derek K. Wise, Alissa S. Crans
    40 pages, many figures

    "After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that loop-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two loops just as one would normally switch point particles, but also a set of generators that switch two loops by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the 'braid permutation group' of Fenn, Rimanyi and Rourke. In the context 4d BF theory this group naturally acts on the moduli space of flat G-bundles on the complement of a collection of unlinked unknotted circles in R^3. When G is unimodular, this gives a unitary representation of the loop braid group. We also discuss 'quandle field theory', in which the gauge group G is replaced by a quandle."

    exerpts from introduction, page 2 and following
    Exotic statistics also arise naturally in the context of 3d quantum gravity. As we ‘turn on gravity’, letting Newton’s gravitational constant kappa become nonzero, ordinary quantum field theory on 3d Minkowski spacetime deforms into a theory where the Poincaré group goes over to a quantum group called the kappa-Poincaré group. Moreover, if we begin with a field theory of bosons, their statistics become exotic as we turn on gravity. For a thorough treatment of these fascinating phenomena, see the papers by Freidel and collaborators [11, 12], the paper by Krasnov [20], and the many references therein.

    So, the exotic statistics reduce to ordinary bosonic statistics in the limit where Newton’s constant goes to zero. They also reduce to bosonic statistics in the limit where the particles are at rest relative to each other, since then p1 and p2 become proportional and their commutator vanishes.
    The corrections to the usual law for addition of energy-momenta are interesting in themselves. Like the exotic statistics, these corrections become negligible in the limit kappa ->0. Under the name of ‘doubly special relativity’, modified laws for adding energy-momentum have already been studied by many authors. The paper by Freidel, Kowalski-Glikman and Smolin [12] gives a good account of how these modifications arise in 3d quantum gravity; their paper also explains more of the history of this subject.
    It would be wonderful to generalize all the above results to 4d gravity, but for now all we can handle is a simpler theory: 4d BF theory. This may eventually be relevant to gravity, since one can describe general relativity in 4 dimensions either as the result of constraining 4d BF theory with a certain gauge group, or perturbing around 4d BF theory with some other gauge group. The first approach goes back to Plebanski [28], and it underlies a great deal of work on spin foam models of quantum gravity [1, 25, 27], especially the Barrett–Crane model. The second approach goes back to MacDowell and Mansouri [22], and has recently been explored by Freidel and Starodubtsev [13].

    kappa is just a convenient multiple of G, like 8 pi G

    Here's an assortment of references, some of many.

    [11] L. Freidel and D. Louapre, Ponzano-Regge model revisited I: gauge fixing, observables and interacting spinning particles, Class. Quant. Grav. 21 (2004), 5685-5726. Also available as hep-th/0401076.

    L. Freidel and D. Louapre, Ponzano-Regge model revisited II: equivalence with Chern-Simons, available as gr-qc/0410141 L. Freidel and E. R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and effective field theory, available as hep-th/0502106.

    [12] L. Freidel, J. Kowalski-Glikman and L. Smolin, 2+1 gravity and doubly special relativity, Phys. Rev. D69 (2004), 044001. Also available as hep-th/0307085.

    [13] L. Freidel and A. Starodubtsev, Quantum gravity in terms of topological observables, available as hep-th/0501191.

    [20] K. Krasnov, Quantum gravity with matter via group field theory, available as hep-th/0505174.

    [21] Xiao-Song Lin, The motion group of the unlink and its representations, preprint, 2005.

    [22] S. W. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38 (1977), 739–742. Erratum, ibid. 38 (1977), 1376.

    [25] D. Oriti, Spin Foam Models of Quantum Spacetime, Ph.D. thesis, University of Cambridge. Also available as gr-qc/0311066.

    random comments
    Derek Wise is a student of Baez at UC Riverside. He gave a talk at the October Loops '05 conference.

    there are some new (to me) symmetry groups, fascinating pictures, the article draws quite a bit on unpublished "loop braid group" work by XiaoSong Lin, could be a help to Freidel in extending his 3D results (gravity and matter) to 4D.
    I suspect paper will be important in physics, but can't tell for sure: hope other people look at it.
    It is amazingly understandable. For the first time I get some inkling of why DSR happens.

    I am up to page 6. It explains where the different classes of particles come from in 3D physics. a little "standard model" for a lowerdimension world.
    Last edited: Mar 22, 2006
  4. Mar 22, 2006 #3


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    Another article on 4D BF theory just appeared. rather than make a separate thread, since it is about 4D BF, I will include it here, in case there is interest.

    this was mentioned by f-h on Dantas blog:
    Alternative symplectic structures for SO(3,1) and SO(4) four-dimensional BF theories
    Merced Montesinos
    13 pages
    Class. Quantum Grav. 23 (2006) 2267-2278

    "The most general action, quadratic in the B fields as well as in the curvature F, having SO(3,1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed. It is shown that the space of solutions to the equations of motion for the BF theory can be endowed with symplectic structures alternative to the usual one. The analysis also includes topological terms and cosmological constant. The implications of this fact for gravity are briefly discussed."

    He points out several open problems in BF theory. If there is substantial interest in discussing Montesinos paper we can initiate a separate thread.


    Returning to the main topic: the new Baez paper, here is a short exerpt from the conclusions section at the end. Baez et al point out a number of open questions to be investigated. I will skip over the ones directed at mathematicians---there are some interesting mathematical research problems---and only quote what they say about physics-related ones.

    "...in detail the workings of the exotic statistics. It would also be interesting to study how, in the kappa -> 0 limit, the exotic statistics approach ordinary bosonic statistics.

    For the physicist, one interesting project would be to study interactions of the loop-like topological defects we have described here. When G is a finite group this should be relatively easy. For G a general Lie group there are problems with divergences, but it is possible that these can be handled just as in 3d gravity [11]. A more ambitious project would be to generalize our results from collections of unlinked unknotted circles to arbitrary embedded graphs, and perhaps relate the resulting theory to the usual theory of spin networks. An even more ambitious project would be to use these ideas as part of a perturbative expansion of MacDowell–Mansouri gravity about 4d BF theory, as proposed by Freidel and Starodubtsev [13]. "
    Last edited: Mar 22, 2006
  5. Mar 22, 2006 #4


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    Has anyone else besides me had a look at the new Baez paper?

    I'm quite curious to know what some other PF-ers think of it.
  6. Apr 25, 2006 #5


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    Baez says that he and Alejandro Perez (Rovelli's group at Marseille) are doing a paper on the dynamics of that 4D BF theory with exotic statistics

    It sounds like we might get it soon:
    "I'm just finishing up a paper with Alejandro Perez..."

    Here is a citation to the work in progress:
    [3] J. C. Baez and A. Perez, Quantization of strings and branes coupled to BF theory, to appear.

    Urs Schreiber has discussed this paper at his blog
    Urs:"I was pointed to

    J. Baez, D. Wise, & A. Crans
    Exotic Statistics for Loops in 4d BF theory

    Also notice that Baez has sometimes REPLACED "LOOPS" WITH "STRINGS" in the title, even though it is much the same paper. Perhaps he is tired of hearing people argue and put the other people down, maybe he wants to stress the similiarity. Or he has some more technical reason---not yet clear to me.

    the original title on arxiv said "loops", and the post here was:
    But if you want to look at the paper NOT AT ARXIV but at Baez site then
    you will see the title there is
    Exotic Statistics for Strings in 4d BF Theory

    and the follow-on article that he is doing with Alejandro Perez (whom we know as spinfoam expert) is provisionally at least titled in string/brane terms.

    Maybe the lesson is to look at 4D BF THEORY because that is in both titles and maybe 4D BF theory is big enough to handle both loop/spinfoam and string/brane
    and who knows maybe does not even need extra dimensions to do it!
    otherwise why say "4D"?
    Last edited: Apr 25, 2006
  7. May 15, 2006 #6
    Actually, we changed "loops" to "strings" because the paper originally used the term "loops" to mean two distinct things -- loop-like topological defects (now called a strings in the paper), and based loops around which the holonomy of a connection is taken. It is the latter of these that correspond to "loops" in loop quantum gravity. So, the paper deals with both loops and strings, but is now clearer about doing so.

    -Derek Wise
  8. May 15, 2006 #7


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    Derek Wise! pleasure to hear from you.

    As you must be well aware that is a fascinating paper---the baez crans wise.

    It gets to my imagination. what if the world was full of THOSE particles predicted by your geometry, instead of the ones we have

    also it is curious that you all should have been investigating "loop braid groups" at the same time that Perimeter people were incorporating Sundance topological preons===the world seems to be getting all twisty and braidy.
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