Generations: three prints, in colour

Abstract

We point out a somewhat mysterious appearance of SUc(3) representations, which exhibit the behaviour of three full generations of standard model particles. These representations are found in the Clifford algebra ℂl(6), arising from the complex octonions. In this paper, we explain how this 64-complex-dimensional space comes about. With the algebra in place, we then identify generators of SU(3) within it. These SU(3) generators then act to partition the remaining part of the 64-dimensional Clifford algebra into six triplets, six singlets, and their antiparticles. That is, the algebra mirrors the chromodynamic structure of exactly three generations of the standard model’s fermions. Passing from particle to antiparticle, or vice versa, requires nothing more than effecting the complex conjugate, ∗: i ↦ − i. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.

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References

  1. [1]

    M. Günaydin and F. Gürsey, Quark Structure and the Octonions, J. Math. Phys. 14 (1973) 1651.

    MathSciNet  Article  Google Scholar 

  2. [2]

    M. Günaydin and F. Gürsey, Quark Statistics and Octonions, Phys. Rev. D 9 (1974) 3387.

    ADS  Google Scholar 

  3. [3]

    G. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Kluwer Academic Publishers (1994).

  4. [4]

    G. Dixon, Division Algebras: Spinors: Idempotents: The Algebraic Structure of Reality, arXiv:1012.1304 [INSPIRE].

  5. [5]

    G.M. Dixon, Division Algebras: Family Replication, J. Math. Phys. 45 (2004) 3878.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    A. Connes and J. Lott, Particle Models and Noncommutative Geometry, Nuc. Phys. B Proc. Suppl. 18B (1990) 29.

    ADS  MathSciNet  MATH  Google Scholar 

  7. [7]

    L. Boyle and S. Farnsworth, Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics, arXiv:1401.5083 [INSPIRE].

  8. [8]

    A. Anastasiou, L. Borsten, M.J. Duff, L.J. Hughes and S. Nagy, An octonionic formulation of the M-theory algebra, arXiv:1402.4649 [INSPIRE].

  9. [9]

    C.A. Manogue and T. Dray, Octonions, E 6 and Particle Physics, J. Phys. Conf. Ser. 254 (2010) 012005 [arXiv:0911.2253] [INSPIRE].

    Article  Google Scholar 

  10. [10]

    C.D. Carone and A. Rastogi, An Exceptional electroweak model, Phys. Rev. D 77 (2008) 035011 [arXiv:0712.1011] [INSPIRE].

    ADS  Google Scholar 

  11. [11]

    G. Cossu, M. D’Elia, A. Di Giacomo, B. Lucini and C. Pica, G 2 gauge theory at finite temperature, JHEP 10 (2007) 100 [arXiv:0709.0669] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    J. Danzer, C. Gattringer and A. Maas, Chiral symmetry and spectral properties of the Dirac operator in G2 Yang-Mills Theory, JHEP 01 (2009) 024 [arXiv:0810.3973] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  13. [13]

    J. Greensite, K. Langfeld, Š. Olejník, H. Reinhardt and T. Tok, Color Screening, Casimir Scaling and Domain Structure in G 2 and SU(N) Gauge Theories, Phys. Rev. D 75 (2007) 034501 [hep-lat/0609050] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    K. Holland, P. Minkowski, M. Pepe and U.J. Wiese, Exceptional confinement in G 2 gauge theory, Nucl. Phys. B 668 (2003) 207 [hep-lat/0302023] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  15. [15]

    L. Liptak and Š. Olejník, Casimir scaling in G2 lattice gauge theory, Phys. Rev. D 78 (2008) 074501 [arXiv:0807.1390] [INSPIRE].

    ADS  Google Scholar 

  16. [16]

    C. Furey, Unified Theory of Ideals, Phys. Rev. D 86 (2012) 025024 [arXiv:1002.1497] [INSPIRE].

    ADS  Google Scholar 

  17. [17]

    J.C. Baez, The Octonions, Bull. Am. Math. Soc. 39 (2002) 145 [math/0105155] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    J.H. Conway and D.A. Smith, On Quaternions and Octonions, Their Geometry, Arithmetic, and Symmetry, Peters (2003).

  19. [19]

    S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, CUP (1995).

  20. [20]

    S. De Leo and K. Abdel-Khalek, Octonionic representations of GL(8,R) and GL(4,C), J. Math. Phys. 38 (1997) 582 [hep-th/9607140] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    A. Sudbery, Division Algebras, (Pseudo)orthogonal Groups and Spinors, J. Phys. A 17 (1984) 939.

    ADS  MathSciNet  MATH  Google Scholar 

Download references

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Correspondence to Cohl Furey.

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ArXiv ePrint: 1405.4601

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Furey, C. Generations: three prints, in colour. J. High Energ. Phys. 2014, 46 (2014). https://doi.org/10.1007/JHEP10(2014)046

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Keywords

  • Beyond Standard Model
  • QCD
  • Gauge Symmetry