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Octonionic Non-Abelian Gauge Theory

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Abstract

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) e ×U(1) m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) e ×SU(2) m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.

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References

  1. Dirac, P.A.M.: Quantized singularities in the electromagnetic field. Proc. R. Soc. Lond. A 133, 60 (1931)

    Article  ADS  Google Scholar 

  2. Saha, M.N.: Note on Dirac’s theory of magnetic poles. Phys. Rev. 75, 1968 (1949)

    Article  ADS  Google Scholar 

  3. Malkus, W.: The interaction of the Dirac magnetic monopole with matter. Phys. Rev. 83, 899 (1951)

    Article  ADS  MATH  Google Scholar 

  4. Goto, E.: Expected behaviour of the Dirac monopole in the cosmic space. Prog. Theor. Phys. 30, 700 (1963)

    Article  ADS  Google Scholar 

  5. Rosenbaum, D.: Direct action and magnetic monopoles. Phys. Rev. 147, 891 (1966)

    Article  ADS  Google Scholar 

  6. Zwanziger, D.: A regular theory of magnetic monopoles. Phys. Rev. B 137, 647 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  7. Goldhaber, A.S.: On the monopole-dyon system in non-Abelian Gauge theories. Phys. Rev. B 140, 1407 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  8. Schwinger, J.: A magnetic model of matter. Phys. Rev. 144, 1087 (1966); [151, 1055 (1966)]

    Article  MathSciNet  ADS  Google Scholar 

  9. Goldhaber, A.S.: Spin and statistics connection for charge monopole composites. Phys. Rev. Lett. 36, 1122 (1976)

    Article  ADS  Google Scholar 

  10. Zwanzinger, D.: Quantum field theory of particles with both electric and magnetic charges. Phys. Rev. 176, 1489 (1968)

    Article  ADS  Google Scholar 

  11. Soper, A.: Multi-monopoles close together. Nucl. Phys. B 199, 290 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  12. Wu, T.T., Yang, C.N.: Dirac monopole without strings: monopole harmonics. Nucl. Phys. B 107, 365 (1976)

    Article  ADS  Google Scholar 

  13. Price, P.B.: Evidence for detection of a moving magnetic monopole. Phys. Rev. Lett. 35, 487 (1975)

    Article  ADS  Google Scholar 

  14. Cabibbo, N., Ferrari, E.: Quantum electrodynamics with Dirac monopoles. Nuovo Cimento 23, 1147 (1962)

    Article  MathSciNet  Google Scholar 

  15. ’t Hooft, G.: Magnetic monopoles in unified gauge theories. Nucl. Phys. B 79, 276 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  16. Polyakov, A.M.: Particle spectrum in quantum field theory. JETP Lett. 20, 194 (1974)

    ADS  Google Scholar 

  17. Julia, B., Zee, A.: Poles with both magnetic and electric charges in non-Abelian gauge theory. Phys. Rev. D 11, 2227 (1975)

    Article  ADS  Google Scholar 

  18. Prasad, M.K.: Yang-Mills-Higgs monopole solutions of arbitrary topological charge. Commun. Math. Phys. 80, 137 (1981)

    Article  ADS  Google Scholar 

  19. Prasad, M.K., Sommerfield, C.M.: Exact classical solution for the ’t Hooft monopole and the Julia-Zee dyon. Phys. Rev. Lett. 35, 760 (1975)

    Article  ADS  Google Scholar 

  20. Bogomolny, E.B.: Stability of classical solutions. Sov. J. Nucl. Phys. 24, 449 (1976)

    Google Scholar 

  21. Dickson, L.E.: On quaternions and their generalization and the history of the eight square theorem. Ann. Math. 20, 155 (1919)

    Article  MATH  Google Scholar 

  22. Pais, A.: Remark on the algebra of interactions. Phys. Rev. Lett. 7, 291 (1961)

    Article  ADS  Google Scholar 

  23. Koplinger, J.: Dirac equation on hyperbolic octonions. Appl. Math. Comput. 182, 443 (2006)

    Article  MathSciNet  Google Scholar 

  24. Gogberashvili, M.: Octonionic version of Dirac equations. Int. J. Mod. Phys. A 21, 3513 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Gogberashvili, M.: Octonionic electrodynamics. J. Phys. A, Math. Gen. 39, 7099 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Nurowski, P.: Split octonions and Maxwell equations. Acta Phys. Pol. A 116, 992 (2009)

    Google Scholar 

  27. Dzhunushaliev, V.: Colorless operators in a non-associative quantum theory. Phys. Lett. A 355, 298 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Dzhunushaliev, V.: Non-associativity, supersymmetry and hidden variables. J. Math. Phys. 49, 042108 (2008)

    Article  MathSciNet  Google Scholar 

  29. Nash, P.L.: Second gravity. J. Math. Phys. 51, 042501 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  30. Demir, S.: Hyperbolic octonion formulation of gravitational field equations. Int. J. Theor. Phys. 52, 105 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bisht, P.S., Pandey, B., Negi, O.P.S.: Interpretations of octonion wave equations. Fizika B 17, 405 (2008)

    ADS  Google Scholar 

  32. Chanyal, B.C., Bisht, P.S., Negi, O.P.S.: Generalized octonion electrodynamics. Int. J. Theor. Phys. 49, 1333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Chanyal, B.C., Bisht, P.S., Negi, O.P.S.: Generalized split-octonion electrodynamics. Int. J. Theor. Phys. 50, 1919 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dangwal, S., Bisht, P.S., Negi, O.P.S.: Octonionic gauge formulation for Dyonic fields (2006). arXiv:hep-th/0608061

  35. Negi, O.P.S., Dehnen, H.: Gauge formulation for two potential theory of dyons. Int. J. Theor. Phys. 50, 2446 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Günaydin, M., Gürsey, F.: Quark structure and octonions. J. Math. Phys. 14, 1651 (1973)

    Article  MATH  Google Scholar 

  37. Günaydin, M.: Octonionic Hilbert spaces, the Poincaré group and SU(3). J. Math. Phys. 17, 1875 (1976)

    Article  ADS  Google Scholar 

  38. Catto, S.: Exceptional projective geometries and internal symmetries (2003). hep-th/0302079v1

  39. Chanyal, B.C., Bisht, P.S., Li, T., Negi, O.P.S.: Octonion quantum chromodynamics. Int. J. Theor. Phys. 51, 3410 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Figueroa, J.M., Farrill, O.: Electromagnetic duality for children. www.maths.ed.ac.uk. Teaching-lectures (1998)

  41. Olive, D.I.: Exact electromagnetic duality (1995). hep-th/9508089

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Authors and Affiliations

  1. Department of Physics, Kumaun University, S.S.J. Campus, Almora, 263601, U.K., India

    B. C. Chanyal, P. S. Bisht & O. P. S. Negi

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  1. B. C. Chanyal
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  2. P. S. Bisht
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  3. O. P. S. Negi
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Correspondence to B. C. Chanyal.

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Chanyal, B.C., Bisht, P.S. & Negi, O.P.S. Octonionic Non-Abelian Gauge Theory. Int J Theor Phys 52, 3522–3533 (2013). https://doi.org/10.1007/s10773-013-1655-7

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