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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

    https://golem.ph.utexas.edu/category/2018/08/exceptional_quantum_geometry_a.html


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

    Exceptional quantum geometry and particle physics

    https://arxiv.org/abs/1604.01247

    Exceptional quantum geometry and particle physics II

    https://arxiv.org/abs/1808.08110

    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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    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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