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Quaternionic-valued Gravitation in 8D, Grand Unification and Finsler Geometry

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Abstract

A unification model of 4D gravity and SU(3)×SU(2)×U(1) Yang-Mills theory is presented. It is obtained from a Kaluza-Klein compactification of 8D quaternionic gravity on an internal CP 2=SU(3)/U(2) symmetric space. We proceed to explore the nonlinear connection \(A^{a}_{\mu}( \textbf{x}, \textbf{y} ) \) formalism used in Finsler geometry to show how ordinary gravity in D=4+2 dimensions has enough degrees of freedom to encode a 4D gravitational and SU(5) Yang-Mills theory. This occurs when the internal two-dim space is a sphere S 2. This is an appealing result because SU(5) is one of the candidate GUT groups. We conclude by discussing how the nonlinear connection formalism of Finsler geometry provides an infinite hierarchical extension of the Standard Model within a six dimensional gravitational theory due to the embedding of SU(3)×SU(2)×U(1)⊂SU(5)⊂SU(∞).

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References

  1. Jordan, P., von Neumann, J., Wigner, E.: Ann. Math. 35, 2964 (1934)

    Article  Google Scholar 

  2. MacCrimmon, K.: A Taste of Jordan Algebras. Springer, New York (2003)

    Google Scholar 

  3. Freudenthal, H.: Ned. Akad. Wet. Proc. Ser. 57A, 218 (1954)

    MathSciNet  Google Scholar 

  4. Tits, J.: Ned. Akad. Wet. Proc. Ser. 65A, 530 (1962)

    MathSciNet  Google Scholar 

  5. Springer, T.: Ned. Akad. Wet. Proc. Ser. 65A, 259 (1962)

    MathSciNet  Google Scholar 

  6. Adams, J.: Lectures on Exceptional Lie Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1996)

    MATH  Google Scholar 

  7. Schafer, R.: An Introduction to Nonassociative Algebras. Academic Press, New York (1966)

    MATH  Google Scholar 

  8. Tze, C.H., Gursey, F.: On the Role of Division, Jordan and Related Algebras in Particle Physics. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  9. Okubo, S.: Introduction to Octonion and Other Nonassociative Algebras in Physics. Cambridge University Press, Cambridge (2005)

    Google Scholar 

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

    MATH  Google Scholar 

  11. Dixon, G.: J. Math. Phys. 45(10), 3678 (2004)

    Article  Google Scholar 

  12. Castro, C.: Int. J. Geom. Methods Mod. Phys. 4(8), 1239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Castro, C.: Int. J. Geom. Methods Mod. Phys. 6(6), 911 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Einstein, A.: Ann. Math. 46, 578 (1945)

    Article  MATH  Google Scholar 

  15. Einstein, A., Strauss, E.: Ann. Math. 47, 731 (1946)

    Article  MATH  Google Scholar 

  16. Moffat, J., Boal, D.: Phys. Rev. D 11, 1375 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  17. Castro, C.: Phys. Lett. B 668, 442 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  18. Borchsenius, K.: Phys. Rev. D 13, 2707 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  19. Marques, S., Oliveira, C.: J. Math. Phys. 26, 3131 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Marques, S., Oliveira, C.: Phys. Rev. D 36, 1716 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  21. Castro, C.: J. Math. Phys. 48(7), 073517 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  22. Castro, C.: Int. J. Geom. Methods Mod. Phys. 9(3), 1250021 (2012)

    Article  Google Scholar 

  23. Castro, C.: Adv. Appl. Clifford Algebras 22(1), 1 (2012)

    Article  MATH  Google Scholar 

  24. Beil, R.: Int. J. Theor. Phys. 32(6), 1021 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Asanov, G., Koselev, M.: Rep. Math. Phys. 26(3), 401 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Miron, R., Hrimiuc, D., Shimada, H., Sabau, S.: The Geometry of Hamilton and Lagrange Spaces. Kluwer Academic, Dordrecht (2001)

    MATH  Google Scholar 

  27. Vacaru, S.: Finsler-Lagrange Geometries and standard theories in physics: new methods in Einstein and string gravity. arXiv:0707.1524

  28. Vacaru, S., Stavrinos, P., Gaburov, E., Gonta, D.: Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity, p. 693. Geometry Balkan Press

  29. Brandt, H.: Found. Phys. Lett. 4(6), 523 (1991)

    Article  MathSciNet  Google Scholar 

  30. Brandt, H.: Found. Phys. Lett. 5(1), 43 (1992)

    Article  MathSciNet  Google Scholar 

  31. Brandt, H.: Found. Phys. Lett. 5(4), 315 (1992)

    Article  MathSciNet  Google Scholar 

  32. Vacaru, S.: SIGMA 4, 071 (2008)

    MathSciNet  Google Scholar 

  33. Eisenhart, L.: Proc. Natl. Acad. Sci. USA 37, 311 (1951)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Eisenhart, L.: Proc. Natl. Acad. Sci. USA 38, 505 (1952)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Randybar-Daemi, S., Percacci, R.: Phys. Lett. B 117, 41 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  36. Pope, C.: Lectures in Kaluza-Klein theory (2002). http://faculty.physics.tamu.edu/pope/ihplec.pd

  37. Batakis, N.: Class. Quantum Gravity 3, L99 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Batakis, N.: The gravitational and electroweak interactions unified as a gauge theory of the de Sitter group. hep-th/9605217

  39. Witten, E.: Nucl. Phys. B 186, 412 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  40. Cho, Y., Soh, K., Park, Q., Yoon, J.: Phys. Lett. B 286, 251 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  41. Yoon, J.: Phys. Lett. B 308, 240 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  42. Yoon, J.: Phys. Lett. A 292, 166 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Yoon, J.: Class. Quantum Gravity 18, 1863 (1999)

    Article  ADS  Google Scholar 

  44. Hoppe, J.: Quantum theory of a massless relativistic surface and a two dimensional bound state problem. PhD thesis, MIT (1982)

  45. Smith, F.T.: Int. J. Theor. Phys. 24, 155 (1985)

    Article  ADS  Google Scholar 

  46. Smith, F.T.: Int. J. Theor. Phys. 25, 355 (1985)

    Article  ADS  Google Scholar 

  47. Smith, F.T.: From sets to quarks. hep-ph/9708379

  48. Smith, F.T.: E 6, strings, branes and the standard model. CERN CDS EXT-2004-031

  49. Pavsic, M.: Int. J. Mod. Phys. A 21, 5905 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Pavsic, M.: Phys. Lett. B 614, 85 (2005)

    Article  MathSciNet  ADS  Google Scholar 

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Acknowledgements

We thank M. Bowers for her assistance and to Sergiu Vacaru for many discussions on Finsler geometry.

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  1. Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA, 30314, USA

    Carlos Castro

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  1. Carlos Castro
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Correspondence to Carlos Castro.

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Castro, C. Quaternionic-valued Gravitation in 8D, Grand Unification and Finsler Geometry. Int J Theor Phys 51, 3318–3329 (2012). https://doi.org/10.1007/s10773-012-1212-9

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