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Furey models with Division Algebras

  1. Jul 10, 2015 #1


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    Just noticed a new article of C Furey not arxived, but readable in SCOAP

    Phys.Lett. B742 (2015)
    Charge quantization from a number operator

    Well, it is not new but I have not found a thread mentioning it. It seems to continue the quest from her previous papers

    http://arxiv.org/abs/1405.4601 Generations: Three Prints, in Colour
    http://arxiv.org/abs/1002.1497 Unified Theory of Ideals

    Basically it plays again the Fermion cube, now considering how the octonions and clifford algebra can be used to build a charge operator layering the cube. There is also an intriguing announcement about a generalisation to Pati-Salam.
    Last edited: Jul 10, 2015
  2. jcsd
  3. Jul 16, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
  4. Feb 13, 2017 #3


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    Has anybody took time to look at Furey's papers, specially from the point of view of group representation theory? Her work on generations extract from the octonions a Clifford algebra that seems complex Cl6, is it?
  5. Feb 13, 2017 #4
    Yes, I have red the three papers today. There is only one main and obsessional idea behind the three papers.

    But for me, the most interesting, also the strangest and the most unconventional, idea is exposed in the inspire paper (see résumé).

    If the intention of the author wouldn't have been to look for the reason why the standard model has its actually well-accepted mathematical structure, I suppose that that kind of paper would never have been published!

    But it has been... and that's really ... (citation): "a new vantage point, electrons and quarks are simply excitations from the neutrino, which formally plays the role of a vacuum state...(end of the citation)".

    My crazy questions:
    - "Do you really think it is a realistic/serious hypo-thesis: neutrinos as fundamental stones of the vacuum?" I have some doubts because if I believe what I have seen in diverse lectures that I have made on the topic, the neutrinos cannot completely explain the (dark) matter we are looking for.
    - "Do you think neutrinos can really be obtained by a Lamb-Retherford effect in vacuum?"
  6. Feb 14, 2017 #5


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    Thinking aloud and naively, first thing I wonder about is what group do we get from the complex clifford algebra of six components. I'd expect SO(6) so SU(4). But the whole Clifford algebra has 64 elements, are we to assume that they partition over SO(6) as 1,6,15,20,15,6,1? Not as sum of spinor irreps? Or perhaps the history is about SO(12) or SO(14)?

    The point is, ok, lets assume that we are seeing the article on color+generations is seeing the branching of SU(4) down to SU(3) x U(1).
    It could be it is only seeing branchings of one 15 and four 4,
    4 = (1)(−3) + (3)(1)
    15 = (1)(0) + (3)(4) + (3)(−4) + (8)(0)
    plus one singlet. Neglecting the question of the U(1) charge, it looks as three generations indeed.

    Question is, what representations is the math seeing here and how to obtain them from a GUT theory?

    For reference:
    4 × 4 = 1 + 15
    4 × 4 = 6 + 10
    6 = (3)(−2) + (3)(2)
    10 = (1)(−6) + (3)(−2) + (6)(2)

    The most elemental attempt could be get 64 out of squaring 4+4 but then we get su(3) sextets from the 10:
    (4+4)x(4+4)=6+10+ 1+ 15+ 15+ 1 + 10 + 6
    Last edited: Feb 14, 2017
  7. Feb 15, 2017 #6


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    :smile: Today an alternative view of Cl(6) in the arxiv:

    The Standard Model Algebra by Ovidiu Cristinel Stoica

    Same criticism: I miss some links with the theory of irreducible representations of Spin(6)

    even without su(4), only with su(3) and u(1), I'd expect the clifford aka exterior algebra to show things as:
    (1+3) x (1 + 3) = 1 + 1 + 3 + 1 +3 + 3 x 3 = 1 + 3 + 3 + 8
    (1+3) x (1 + 3) = 1 + 3 + 3 + 6 + 3
    (1+3 + 1 + 3) x (1+3 + 1 + 3)= 6*1 + 5*3 + 5*3 + 2*8 + 2*6
    How do the sextets move to make triplet+antitriplet plus six singlets?

    (and yes, it could be a good thing if the sextet were the top quark plus the neutrinos. Alternatively, a mechanism breaking 6 into 3+3 and some triplets into neutrinos in a way compatible with the previous post, which also divides the quarks in five plus one)
    Last edited: Feb 15, 2017
  8. Feb 17, 2017 #7


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    For reference, the code of the multiple SU(3) representation over the 32 elements that Furey locates in this paper. As I said, the 5+1 structure of the quark sector looks interesting, as it opens a possibility to explain why the top is peculiar.

    My trouble here is if some of the triplets could be higher dimensional representations in disguise, and we are not seeing them because we are using a representation of the ladder operators only valid for SU(2) doublets. On other hand, does it matter? Not sure.

    Code (Text):

    from sympy import *
    #MV is the multivector from the ga package, not standard anymore in sympy.
    from mv import MV
    e1, e2, e3, e4, e5, e6 = MV.setup('e1 e2 e3 e4 e5 e6', metric='[-1, -1, -1, -1, -1, -1]')

    L4=I/2*( e2*e5+e4*e6)*nu
    L6=I/2*( e1*e6+e2*e3)*nu
    L7=I/2*( e1*e2+e3*e6)*nu
    L8=I/(2*sqrt(3))*( e1*e3+e4*e5-2*e2*e6)*nu


    qB1=( I*e1*e4+e1*e5+e3*e4-I*e3*e5)*nu

    aqR2=( I*e1*e2-e1*e6+e2*e3-I*e3*e6)*nu
    aqG2=( I*e2*e4-e2*e5+e4*e6+I*e5*e6)*nu
    aqR3=( I*e4+e5+e1*e3*e4-I*e1*e3*e5)*nu
    aqG3=(I*e1 + e3 + e1*e2*e6 + e1*e4*e5)*nu
    aqB3=(I*e2 + e6 -e1*e2*e3 - I*e1*e3*e6)*nu

    aqR4=(I*e1 - e3 + e1*e2*e6 - e1*e4*e5)*nu
    aqG4=(-I*e4 + e5 + e1*e3*e4 + I*e1*e3*e5)*nu
    aqB4=(I*e1*e2*e4 - e1*e2*e5 - e1*e4*e6 - I*e1*e5*e6)*nu

    aqR5=(-I*e2 + e6 + e1*e2*e3 - I*e1*e3*e6)*nu
    aqG5=(I*e1*e2*e4 - e1*e2*e5 + e1*e4*e6 + I*e1*e5*e6)*nu
    aqB5=(I*e4 - e5 + e1*e3*e4 + I*e1*e3*e5)*nu

    aqR6=(I*e1*e2*e4 + e1*e2*e5 + e1*e4*e6 - I*e1*e5*e6)*nu
    aqG6=(I*e2 - e6 + e1*e2*e3 - I*e1*e3*e6)*nu
    aqB6=(-I*e1 + e3 + e1*e2*e6 - e1*e4*e5)*nu

    l1=(1 + I*e1*e3 + I*e2*e6 + I*e4*e5)*nu
    l2=(3 - I*e1*e3 - I*e2*e6 - I*e4*e5)*nu
    l3=(-I*e1*e2*e4 - e1*e2*e5 + e1*e4*e6 - I*e1*e5*e6)*nu
    l4=(-I*e1 - e3 + e1*e2*e6 + e1*e4*e5)*nu
    l5=(I*e2 + e6 + e1*e2*e3 + I*e1*e3*e6)*nu
    l6=(I*e4 + e5 - e1*e3*e4 + I*e1*e3*e5)*nu

    for x in (Traise,Tlower,Vraise,Vlower,Uraise,Ulower): print (x*qR1-qR1*x==0)
  9. Feb 25, 2017 #8


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    The other intriguing thing in this family of papers, and in all the octonion-related SM, is that they get ti extract an unified SM with less dimensions than GUT groups.

    I mean, SO(10) is the symmetry group of the 9-sphere. Pati Salam (ok, not a GUT really) is so(6) times so(4), the simmetry group of S5xS3 product of spheres. And SU(5) is the symmetry group of the projective space CP4. Classical group theory GUT seems to live in objects of dimension 8, 9 or higher. Octonions are barely of dimension 8, and most times they are related to the 7-sphere.

    On other hand the standard model, as we know it, seems to have a limit -say, move the higgs to Planck Scale- where it is just SU(3)xU(1), a simpler symmetry which can live in CP2xS1 or in S5 (because SU(4) can). One could think that the Higgs field parametrized some move from d=8 to d=5. And one could hope that non extreme values of these parameters have some mathematical meaning as symmetries in d=7 or d=6. In fact Witten pointed out that d=7 can be obtained by quotienting the Pati-Salam spheres by an U(1) action. But when we try to do calculations in d=7 we are always using either SO(8) over a seven sphere or SU(5) over a product of spheres. I pondered this in a question to MO time ago http://mathoverflow.net/questions/75875/why-su3-is-not-equal-to-so5 Of this three Dynkin diagrams:
    Code (Text):

           o                  o                         o
    o----o    SO(8)    o----o     SU(3)xSO(4)    o====o     SO(5)xSO(4)
           o                  o                         o  
    we want to get the middle one. It seems that octonion tecniques -or some other non commutative alternatives- can pivot between SO(8) and the SM in a way that group theory can not. It is peculiar.
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