Octonionic structures in particle physics

  • Feza Gürsey
Group Theory in Particle Physics
Part of the Lecture Notes in Physics book series (LNP, volume 94)

Abstract

Octonionic extensions of Quantum Mechanics associated with finite exceptional Hilbert spaces are summarized. An interpretation of such spaces as the finite space of internal symmetries allows the introduction of color and flavor degrees of free dom for fundamental particles, color arising from the automorphism group of octonions. As an example the space admitting E symmetry is discussed. The projection operators for various fermion states are constructed and shown to belong to a larger ternary algebra of hermitian operators. A possible application is to the grand unified gauge theory of leptons and quarks based on E6. Another possibility involves the construction of an exceptional superspace by means of octonionic Grassman numbers.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Jordan, J. von Neumann and E. P. Wigner, Ann. Math. 35, 29 (1934).Google Scholar
  2. 2.
    C. Chevalley and R. D. Schafer, Proc. Natl. Acad. Sci. U. S., 36, 137 (1950).Google Scholar
  3. 3.
    For a review see H. Freudenthal, Advances in Math. I., 145 (1965).Google Scholar
  4. 4.
    R. Moufang, Abh. Math. Sem. Univ. Hamburg 9, 207 (1933).Google Scholar
  5. 5.
    J. Pati and A. Salam, Phys. Rev. D8, 1240 (1973; H.Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 433 (1974).Google Scholar
  6. 6.
    H. Georgi, H. Quinn and S. Weinberg, Phys. Rev. Lett. 33, 451 (1974); D. I. D'Jakonov, Leningrad Nucl. Phys. Institute preprint (1977).Google Scholar
  7. 7.
    M. Yoshimura, Phys. Rev. Lett. 41, 381 (1978); S. Dimopoulos, L. Susskind, SLAC-PUB-2126 (1978).Google Scholar
  8. 8.
    D. Toussaint, S. B. Treiman, F. Wilczek and A. Zee, “Matter-Antimatter Accounting, Thermodynamics and Black Hole Radiation,” Princeton preprint (1978); D. Toussaint, F. Wilczek, “Elementary examples of baryon generation,” Princeton preprint (1978).Google Scholar
  9. 9.
    D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13, 3124 (1976); S. Deser and B. Zumino, Phys. Rev. Lett. 62B, 335 (1976).Google Scholar
  10. 10.
    M. Gell-Mann, P. Ramond and J. H. Schwarz, Phys. Lett. 76B, 417 (1978).Google Scholar
  11. 11.
    L. Brink, M. Gell-Mann, P. Ramond and J. Schwartz, Caltech preprint 68-644 (1978).Google Scholar
  12. 12.
    S. MacDowell, Yale preprint COO-3075-201 (1978).Google Scholar
  13. 13.
    F. Gürsey, in Second Workshop on Current Problems in High Energy Particle Theory, ed. G. Domokos and S. Kövesi-Domokos, p. 3 (Johns Hopkins U., Baltimore, Md. 1978).Google Scholar
  14. 14.
    F. Gürsey, “Non Associative Algebras in Quantum Mechanics and Particle Physics,” in Proc. of the 1977 Charlottesville Conference on Non Associative Algebras.Google Scholar
  15. 15.
    For the hyperbolic case see G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, section 19, (Princeton, 1973). See also Ref. 14.Google Scholar
  16. 16.
    N. Jacobson, Structure and Representations of Jordan Algebras, p. 36 (Am. Math. Soc. 1968).Google Scholar
  17. 17.
    M. Günaydin, C. Piron and H. Ruegg, “Moufang Plane and Octonionic Quantum Mechanics,” U. of Genève preprint (1977).Google Scholar
  18. 18.
    A. Albert, Ann. Math. 35, 65 (1934).Google Scholar
  19. 19.
    M. Günaydin and F. Gürsey, Phys. Rev. D9, 3387 (1974); F. Gürsey in “Johns Hopkins Workshop on Current Problems in High Energy Particle Theory” p. 15 (Johns Hopkins U., 1974).Google Scholar
  20. 20.
    Gell-Mann, private communication.Google Scholar
  21. 21.
    R. J. Faulkner, Memoirs of the Am. Math. Soc. No. 104 (Providence, R. I., 1970) nad references therein.Google Scholar
  22. 22.
    I. L. Kantor, Dokl. Akad. Nauk SSSR, 208, No.6 (1973), Translation Soviet Math. Dokl. 14, No.l (1973).Google Scholar
  23. 23.
    I. Bars and M. Günaydin, Harvard preprint (1978).Google Scholar
  24. 24.
    F. Gürsey, in “Int. Symp. on Math. Problems in Th. Phys.” ed. H. Araki, p. 189 (Springer, 1975).Google Scholar
  25. 25.
    F. Gürsey, P. Ramond and P. Sikivie, Phys. Lett. 60B, 177 (1976); F. Gürsey and M. Serdaroğlu, Lett. al Nuovo Cimento, 21, 28 (1978).Google Scholar
  26. 26.
    Y. Achiman and B. Stech, Physics Letters 77B, 389 (1978).Google Scholar
  27. 27.
    For a recent review, see M. L. Perl in Proc. Int. Symp. on Lepton and Photon Int. at High En., Hamburg 1977, ed. F. Gutbrod (Deny, 1977).Google Scholar
  28. 28.
    S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A. Salam in Nobel Symp. No.8, ed. N. Svartholm (Wiley, N. Y. 1969).Google Scholar
  29. 29.
    S. W. Herb et al. Phys. Rev. Lett. 39, 252 (1977).Google Scholar
  30. 30.
    For a review see T. Appelquist, R. M. Barnett and K. Lane “Charm and Beyond” to be published in Ann. Rev. Nucl. and Particle Sci., vol. 28.Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Feza Gürsey
    • 1
  1. 1.Yale UniversityNew Haven

Personalised recommendations