Advertisement

Double gauge invariance and covariantly-constant vector fields in Weyl geometry

Research Article

Abstract

The wave equation and equations of covariantly-constant vector fields (CCVF) in spaces with Weyl nonmetricity turn out to possess, in addition to the canonical conformal-gauge, a gauge invariance of another type. On a Minkowski metric background, the CCVF system alone allows us to pin down the Weyl 4-metricity vector, identified herein with the electromagnetic potential. The fundamental solution is given by the ordinary Lienard–Wiechert field, in particular, by the Coulomb distribution for a charge at rest. Unlike the latter, however, the magnitude of charge is necessarily unity, “elementary”, and charges of opposite signs correspond to retarded and advanced potentials respectively, thus establishing a direct connection between the particle/antiparticle asymmetry and the “arrow of time”.

Keywords

Geometrization of electromagnetism Conformal invariance  Lienard–Wiechert field Charge quantization 

Notes

Acknowledgments

We would like to thank Profs. D.V. Alexeevski, A.Ya. Burinskii, B. N. Frolov and A. S. Rabinowitch for friendly discussions and valuable comments. The authors are also indebted to the referees for critical remarks and advices which helped to improve the paper and stimulated subsequent work. One of the authors (V. K.) wants to express his particular gratitude to Prof. E.T. Newman for the long-term support and kind attention.

References

  1. 1.
    Weyl, H.: Ann. Phys. (Lpz.) 59, 101 (1919)ADSCrossRefGoogle Scholar
  2. 2.
    Rosen, N.: Found. Phys. 12, 213 (1982)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Pauli, W.: Theory of Relativity. Pergamon Press, Oxford (1958)MATHGoogle Scholar
  4. 4.
    Eddington, A.S.: Proc. R. Soc. Lond. A 99, 104 (1921)ADSCrossRefGoogle Scholar
  5. 5.
    Dirac, P.: Proc. R. Soc. Lond. A 333, 403 (1973)ADSCrossRefGoogle Scholar
  6. 6.
    Schmidt, H.-J.: Int. J. Geom. Meth. Mod. Phys. 4, 209 (2007). arXiv:gr-qc/0602017 ADSCrossRefGoogle Scholar
  7. 7.
    Vizgin, V.: NTM-Schriftenr. Leipzig 21, 23 (1984)MathSciNetGoogle Scholar
  8. 8.
    Filippov, A.T.: On Einstein-Weyl unified model of dark energy and dark matter. arXiv:0812.2616 [gr-qc]
  9. 9.
    London, F.: Z. Phys. 42, 375 (1927)ADSCrossRefGoogle Scholar
  10. 10.
    Weyl, H.: Z. Phys. 56, 330 (1929)ADSCrossRefGoogle Scholar
  11. 11.
    Gorbatenko, M.V., Pushkin, A.V., Schmidt, H.-J.: Gen. Relativ. Gravit. 34, 9 (2002). arXiv:gr-qc/0106025 CrossRefGoogle Scholar
  12. 12.
    Gorbatenko, M.V., Pushkin, A.V.: Gen. Relativ. Gravit. 34, 175 (2002)Google Scholar
  13. 13.
    Gorbatenko, M.V., Pushkin, A.V.: Gen. Relativ. Gravit. 34, 1131 (2002)Google Scholar
  14. 14.
    Rabinowitch, A.S.: Class. Quantum Gravit. 20, 1389 (2003)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kassandrov, V.V.: Gravit. Cosmol. 8, 57 (2002). arXiv:math-ph/0311006
  16. 16.
    Barut, A.O., Haugen, R.: Ann. Phys. 71, 519 (1970)ADSCrossRefGoogle Scholar
  17. 17.
    Hall, G.S.: J. Math. Phys. 32, 181 (1991)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Hall, G.S.: J. Math. Phys. 33, 2663 (1992)ADSGoogle Scholar
  19. 19.
    Kassandrov, V.V.: Gravit. Cosmol. 1, 216 (1995). arXiv:gr-qc/0007027 ADSGoogle Scholar
  20. 20.
    Kassandrov, V.V., Rizcallah, J.A.: Twistor and “weak” gauge structures in the framework of quaternionic analysis, arXiv:gr-qc/0012109
  21. 21.
    Stephani, Hans, et al.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar
  22. 22.
    Faddeev, L., Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (2007)MATHGoogle Scholar
  23. 23.
    Einstein, A.: Weyl, H., Sitzungsber. d. Berl. Akad. 465 (1918)Google Scholar
  24. 24.
    Hall, G.S., Haddow, B.M., Pulham, J.R.: Gravit. Cosmol. 3, 175 (1997)ADSGoogle Scholar
  25. 25.
    Kassandrov, V.V., Rizcallah, J.A.: Proceedings of Inter Conference “Geometrization of Physics II” in memory of A.Z. Petrov, ed. V.I. Bashkov, p. 137. Kazan State University Press, Kazan (1995)Google Scholar
  26. 26.
    Kassandrov, V.V., Rizcallah, J.A.: Proceedings of the International School-Seminar “Foundation of Gravitation & Cosmology”, Odessa, p. 98 (1995)Google Scholar
  27. 27.
    Kassandrov, V.V.: Acta Applic. Math. 50, 197 (1998)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kassandrov, V.V.: Algebraic Structure of Space-Time and Algebrodynamics. People’s Friendship University Press, Moscow (1992). (in Russian)MATHGoogle Scholar
  29. 29.
    Rizcallah, J.A.: Master’s Thesis. People’s Friendship University Press, Moscow (1995). (in Russian)Google Scholar
  30. 30.
    Bonnor, W. B., Vadyia, P. C.: General Relativity (papers in honor of J.L. Synge) ed. O’Raifeartaigh, p. 119. Clarendon Press, Oxford (1972)Google Scholar
  31. 31.
    Kassandrov, V.V., Khasanov, ISh: J. Phys. A Math. Theor. 46, 175206 (2013). arXiv:1211.7002 [physics.gen-ph]ADSCrossRefGoogle Scholar
  32. 32.
    Einstein, A.: Bietet die Feldtheorie Mglichkeiten zur Lsung des Quantenproblems? Sitzungsber. Preuss. Akad. Wiss, 359 (1923)Google Scholar
  33. 33.
    Vizgin, V. P., Barbour, J.B.: Unified Field Theories: In the First Third of the 20th Century. Modern Birkhauser Classics, p. 208 (2011)Google Scholar
  34. 34.
    Costakis, S., Miritzis, J., Querella, L.: J. Math. Phys. 40, 3063 (1999)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of Gravitation and CosmologyPeoples’ Friendship University of RussiaMoscowRussia
  2. 2.School of EducationLebanese UniversityBeirutLebanon

Personalised recommendations