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Moufang plane and octonionic Quantum Mechanics

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Abstract

It is shown that the usual axioms of one-particle Quantum Mechanics can be implemented with projection operators belonging to the exceptional Jordan algebraJ 38 over real octonions. Certain lemmas on these projection operators are proved by elementary means. Use is made of the Moufang projective plane. It is shown that this plane can be orthocomplemented and that there exists a unique probability function. The result of successive, compatible experiments is shown not to depend on the order in which they are performed, in spite of the non-associativity of octonion multiplication. The algebra of observables and the action of the exceptional groupF 4 is studied, as well as a possible relation with the color group SU(3) and quark confinement.

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References

  1. Jordan, P.: Nachr. Ges. Wiss. Göttingen 569–575 (1932); 209–214 (1933); Z. Physik80, 285 (1933)

  2. Jordan, P., Neumann, J. von, Wigner, E.: Ann. Math.36, 29–64 (1934)

    Google Scholar 

  3. Jacobson, N.: Structure and representations of Jordan algebras. Am. Math. Soc. Coll. Publ.39, 356–411 (1968)

    Google Scholar 

  4. Schafer, R.D.: An introduction to non-associative algebras. New York-London: Academic Press 1966

    Google Scholar 

  5. Pais, A.: Phys. Rev. Letters7, 291 (1961)

    Google Scholar 

  6. Gamba, A.: High energy physics and elementary particles (ed. A. Salam). Vienna: IAEA 1965

    Google Scholar 

  7. Goldstine, H. H., Horwitz, L.P.: Proc. Nat. Acad.48, 1134 (1962); Math. Ann.154, 1 (1964);

    Google Scholar 

  8. Horwitz, L.P., Biedenharn, L. C.: Helv. Phys. Acta38, 385 (1965); Exceptional parafermions in a Hilbert space over an associative algebra. Preprint (1977)

    Google Scholar 

  9. Buccella, F., Falcioni, M., Pugliese, A.: Lett. Nuovo Cimento18, 441–446 (1977)

    Google Scholar 

  10. Günaydin, M., Gürsey, F.: Lett. Nuovo Cimento6, 401 (1973)

    Google Scholar 

  11. Günaydin, M., Gürsey, F.: J. Math. Phys.14, 1651 (1973)

    Google Scholar 

  12. Günaydin, M.: Ph. D. Thesis, Yale University (1973) (unpublished)

  13. Günaydin, M., Gürsey, F.: Phys. Rev. D9, 3387 (1974)

    Google Scholar 

  14. Gürsey, F.: Johns Hopkins University workshop on current problems in high energy particle theory. Baltimore, Md. (1974)

  15. Günaydin, M.: J. Math. Phys.17, 1875 (1976)

    Google Scholar 

  16. Fritzsch, H., Gell-Mann, M.: Proceedings of the XVI. International Conference on High Energy Physics, Chicago-Batavia, Ill. (1972), (ed. J. D. Jackson. A. Roberts), Vol. 2, p. 135. Batavia, Ill.: NAL 1973

    Google Scholar 

  17. Piron, C.: Foundations of quantum physics. Reading, Mass.: Benjamin 1976

    Google Scholar 

  18. Emch, G.: Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley-Interscience 1972

    Google Scholar 

  19. Moufang, R.: Abh. Math. Sem. Univ. Hamburg9, 207–222 (1933)

    Google Scholar 

  20. Jordan, P.: Abh. Math. Sem. Univ. Hamburg16, 74–76 (1949)

    Google Scholar 

  21. Gürsey, F.: Yale Report 400-3075-178

  22. This is one of the four possible bilinear products that can be defined over the octonions, see [13]

    Google Scholar 

  23. Freudenthal, H.: Advan. Math.1, 145–190 (1964)

    Google Scholar 

  24. For the literature on the principle of triality, we refer the reader to [4]

    Google Scholar 

  25. Yokota, I.: Below, we shall follow: J. Fac. Sci. Shinshu University3, 35 (1968)

    Google Scholar 

  26. Below, we shall give simple proofs of these lemmas. For more sophisticated and general treatment of related problems see, for example: Springer, T. A., Veldkamp, F. D.: Proc. Koninkl. Akad. Wetenschap A66, 413–451 (1963)

    Google Scholar 

  27. Bumcrot, R. J.: Modern projective geometry, Chapter III. New York: Halt, Rinehart, Winston 1969

    Google Scholar 

  28. Freudenthal, H.: Oktaven, Ausnahmegruppen und Oktavengeometrie. Utrecht (mimeographed) (1951)

  29. Springer, T. A.: Proc. Koninkl. Akad. Wetenschap A63, 74–101 (1960)

    Google Scholar 

  30. Tits, J.: Bull. Acad. Roy. Belg. Sci.39, 309–329 (1953)

    Google Scholar 

  31. Freudenthal, H.: Proc. Koninkl. Ned. Akad. Wetenschap, A58, 151–157 (1955); A62, 165–201 (1959)

    Google Scholar 

  32. Gleason, A. M.: J. Math. Mech.6, 885 (1957)

    Google Scholar 

  33. See p. 68

    Google Scholar 

  34. See § 2.4

    Google Scholar 

  35. Gürsey, F.: New pathways in high energy physics, I (ed. A. Perlmatter). New York: Plenum Press 1976

    Google Scholar 

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Authors and Affiliations

  1. Physikalisches Institut, Universität Bonn, D-5300, Bonn 1, Federal Republic of Germany

    M. Günaydin

  2. Département de Physique Théorique, Université de Genève, CH-1211, Genève 4, Switzerland

    C. Piron & H. Ruegg

Authors
  1. M. Günaydin
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  2. C. Piron
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  3. H. Ruegg
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Additional information

Communicated by R. Haag

Work supported by the Swiss National Science Foundation

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Günaydin, M., Piron, C. & Ruegg, H. Moufang plane and octonionic Quantum Mechanics. Commun.Math. Phys. 61, 69–85 (1978). https://doi.org/10.1007/BF01609468

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  • DOI: https://doi.org/10.1007/BF01609468

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