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A Exceptional Quantum Geometry and Particle Physics

  1. Aug 29, 2018 #1
    What do you think about the article

    Exceptional Quantum Geometry and Particle Physics
    Posted by John Baez


    and the following discussion? It also includes links to the original papers.

    Exceptional quantum geometry and particle physics


    Exceptional quantum geometry and particle physics II


    It talks about how they get the gauge group of the Standard Model from the symmetry group of the exceptional Jordan algebra. They are talking about SU(3) x SU(2) x U(1)/Z/6 instead of SU(3) x SU(2) x U(1).

  2. jcsd
  3. Aug 29, 2018 #2
    Baez writes

    "The exceptional Jordan algebra contains a lot of copies of 4-dimensional Minkowski spacetime. The symmetries of the exceptional Jordan algebra that preserve any one of these copies form a group…. which happens to be exactly the gauge group of the Standard Model!"

    I think the appropriate response might be - so what? OK, they found that group. But it's not functioning as a gauge group, and I don't see a way to augment the relationship so that it gives rise to gauge fields on one of those "copies of 4-dimensional Minkowski spacetime".

    There are numerous relationships between exceptional algebraic objects and groups appearing in physics. For example, SU(5) is a simple group containing the SM gauge group, and E8 contains SU(5) x SU(5). You can probably find some subobject of E8 that will pick out the SM group as its complement.
  4. Aug 30, 2018 #3
    In more detail, the idea rests on the series of group inclusions F4 (SU(3)xSU(3))/Z3 (SU(3)xSU(2)xU(1))/Z6.

    F4 is the automorphism group (group of mappings to itself) of h3(O), the algebra of 3x3 octonion-valued matrices.

    (SU(3)xSU(3))/Z3 is the subgroup of F4 which maps the subalgebra h2(O) to itself.

    And (SU(3)xSU(2)xU(1))/Z6 is a subgroup of that, which maps a subsubalgebra h2(C) to itself.

    ((edit: That was wrong, that's not how it works... See this tweet by Baez. Basically,

    (SU(3)xSU(3))/Z3 is the symmetry group for "h3(C+C^3)", h3(O) with a preferred imaginary unit for each octonion,

    and then (SU(3)xSU(2)xU(1))/Z6 is a subgroup of that which picks out an h2(O) within h3(O), but a preferred imaginary unit makes that "h2(C+C^3)", and so there is a preferred h2(C) within the h2(O).))

    Further, h3(O) has 27 dimensions, h2(O) has 10 dimensions and can be given a Minkowski metric, and h2(C) has 4 dimensions and can be given a Minkowski metric.

    So you could say this is reminiscent of string theory. Bosonic string theory has 26 dimensions (and the speculated "bosonic M-theory" would have 27 dimensions), superstring theory has 10 dimensions, and the observable world has 4 dimensions.

    There is a proposal (Pierre Ramond, Hisham Sati) that there might be a 27-dimensional theory consisting of a 16-dimensional space fibered over 11-dimensional M-theory. The 16-dimensional space is OP2, the octonionic projective plane or Cayley plane. It also has the automorphism group F4, and there are various ways to obtain OP2 as a quotient of h3(O).

    So as usual, one way to try to give this idea more content, is to make it into string theory.
    Last edited: Aug 30, 2018
  5. Aug 30, 2018 #4

    Urs Schreiber

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    Science Advisor
    Gold Member

    Right, and in fact it doesn't. I had made that point here.

    I wish people were more careful with claims about octonions in fundamental physics. After all, they do underly some good physics, notably the existence of the M2-brane cocycle in 11d supergavity. (neatly reviewed in arxiv.org/abs/1003.3436).
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