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Sundance Bilson-Thompson ribbons in LQG soldiers on

  1. Feb 9, 2010 #1
    http://arxiv.org/abs/1002.1462

    Embedding the Bilson-Thompson model in an LQG-like framework

    Deepak Vaid
    (Submitted on 8 Feb 2010)
    We argue that the Quadratic Spinor Lagrangian approach allows us to approach the problem of forming a geometrical condensate of spinorial tetrads in a natural manner. This, along with considerations involving the discrete symmetries of lattice triangulations, lead us to discover that the quasiparticles of such a condensate are tetrahedra with braids attached to its faces and that these braid attachments correspond to the preons in Bilson-Thompson's model of elementary particles. These "spatoms" can then be put together in a tiling to form more complex structures which encode both geometry and matter in a natural manner. We conclude with some speculations on the relation between this picture and the computational universe hypothesis.
     
  2. jcsd
  3. Feb 9, 2010 #2

    MTd2

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    There are no hyperbolic tessellations with finite above 4 dimensions with finite, 10 different types in 5 dimensions with infinite sides, and no more hyperbolic tessalations with more.

    http://en.wikipedia.org/wiki/List_of_regular_polytopes#Tessellations_of_hyperbolic_4-space

    If Deepak Vaid is right, I guess it is finally possible to come up with a theory which naturally gives the correct number of dimensions, if one wants to recover both GR and SM at large scale. Of course, that is not the one he presented, but it would work as a simplified model.
     
  4. Feb 9, 2010 #3
    Fascinating insight, esp given it turns string theories 30-year run of 10/11/26 epicycle dimensions on its head. Briane Greene, Michio Kaku and Stephan Hawking loudly crowed how no other theory of physics predicts space time dimensionality as string theory does. Why not write a paper?
     
    Last edited: Feb 9, 2010
  5. Feb 10, 2010 #4

    MTd2

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    Write a paper? Who?
     
  6. Feb 10, 2010 #5

    arivero

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    Nope. Tiling theory is one of the typical sources of extra dimensional thinking in mathematics. Even the (25,1) dimensions of string theory are suspected to be related to the 24 dimensions of the Leech lattice.

    OT: Actually, I am not sure if the theory of epicycles was used even for 30 years. Ptolemy prefers to use eccentricity of the circle orbit instead of epicycles. Of course you need one epicycle for the planets, but this one is real, centered in the sun orbits. It is there, and you can provide a simpler shape to the theory by putting the sun in the center but the calculations are exactly the same. A "theory of epicycles" is about to add a second and even a third epicycle to correct the small deviations of the original theory. But such corrections, if I recall correctly, are needed only after Tycho Brahe, and his disciple Kepler already did the ellipse trick. So not sure if they were really used in calculations, not even if proposed as a published theory, were them?

    I suppose the popular use of the word "epicycle" is to mean "Ad-Hoc", is it? But the extra dimensions are not Ad-Hoc.
     
  7. Feb 10, 2010 #6

    MTd2

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    There is just 1 kind of (regular) tessellation above dimension 5, and they can be just euclidean. This is not the case of a (25,1) space and neither of the leech lattice. That is not compatible with the choice of an unique quantum "atom" of geometry.
     
  8. Feb 10, 2010 #7

    I was suggesting you
     
  9. Feb 10, 2010 #8
    How about dimensional reduction?
     
  10. Feb 10, 2010 #9

    MTd2

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    Yes, the choice of compactification does look like "epicycles", or "ad-Hoc", for me. On one hand, you know that the theory is unique, but this cool part ends, at least for me, when one has to appeal to very complicated geometries to create particle content. And there is a huge degeneracy when it comes to find SM schemes, there is, you can find theories close SM or extensions of it, by different compactifications.

    So, it "sounds like" that it doesn't matter how the particle content is changed, you can always rearrange geometry, to get the correct particle theory, just like it doesn't matter how the orbital system is changed, you can always change epicycles, to get the correct orbital theory...
     
  11. Feb 22, 2010 #10
    Hi,

    Thanks for the comments on my paper. It is nice to know that someone actually took the time to read it. BTW I would appreciate any and all feedback - no matter how brutal, since that is the best way to see what needs work.

    As for whether or not this line of thought can lead to a prediction for the "natural" number of large-scale (coarse-grained) space-time dimensions ...

    That is something I have thought about, though haven't yet found any clear resolution. Of course, I belong to the school of thought which feels that strings should be emergent structures in any reasonably complete theory of quantum geometry, a la X G Wen, and therefore have never found the arguments for various dimensions (10, 26 etc.) in string theory particularly compelling.

    Also, some of you might be interested in the following article on "Wang Tiles" which are a concrete example of doing computation with tiles:

    http://en.wikipedia.org/wiki/Wang_tile

    Cheers,

    Deepak
     
  12. Mar 4, 2010 #11

    MTd2

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    What are the consequences of having a grid that is not regular? We should ask ourselves that to exclude higher dimensions.
     
  13. Mar 4, 2010 #12
    1. The grid or graph or whatever you wish to call the "exoskeleton" of spacetime, is definitely not regular. Of course, to be precise you have to define what you mean by "regular" etc. etc. I'll assume you're talking about the regularity of a cubic or square lattice. The problem with such graphs is that they cannot account for the scaling behavior of the structure of large-scale geometry that we observe (see for instance: Weygaert et al : http://arxiv.org/abs/astro-ph/0404397).

    2. From the perspective of information processing, a hierarchical lattice is much more efficient in distributing information than a regular one (like the cubic lattice).

    3. Assuming that matter is described by local topological defects, then the presence of matter implies that the grid cannot be regular.

    Ultimately I think the answer to this question (why 3+1 dimensions) is going to be determined by considerations of the efficiency of information processing. Something along the line of Goldilocks' tale: 2+1 dim. is too hot, 4+1 is too cold and 3+1 is just right :)
     
  14. Mar 4, 2010 #13

    MTd2

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    What I mean by regular is a regular is a tessellation by regular polytopes :)

    http://en.wikipedia.org/wiki/List_of_regular_polytopes#Overview

    If we are not talking about regular polytopes, there are extremely interesting lattices, for example, in 24 dimensions, the leech lattice, which is the densest lattice in 24 dimensions. But that is not formed by regular polytopes, which have the property of the interesting bounding property.

    Well, as you can see in those tables, cubic lattices are very boring:
    "The hypercube honeycomb is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge."
    So, the best ones for are not the cubics, for example:

    A lattice of 24-cells in euclidean 4 space: (probably also the lattice of highest density in 4 euclidean dimensions). Also, there are no analogues of 24 cell polytopes in any other dimension.
    http://en.wikipedia.org/wiki/Icositetrachoric_honeycomb

    A lattice of dodecahedrons in 3-hyperbolic space:

    http://en.wikipedia.org/wiki/Order-5_dodecahedral_honeycomb

    Also note that of 3 out of the 5 4-hyperbolic tesselations, do not contain any cubic symmetry in their structure, and in fact there are 2 with 24-cells on their vertices.
     
    Last edited: Mar 4, 2010
  15. Mar 4, 2010 #14

    MTd2

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    1. I don't understand why... I am talking about tiny things...
    2. Why? And BTW, what is an hierarchical lattice?
    3. I was thinking on the most stable configuration, the vacuum state. Disturb that, and you get matter.

    Looking at here http://www.sbfisica.org.br/bjp/files/v27_567.pdf , it seems that and hierarchical lattice is one made of IFS fractal. Is that it?
     
    Last edited: Mar 4, 2010
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