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A Weyl-Cartan space-time model based on the gauge principle

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Abstract

Based on the requirement that the gauge invariance principle for the Poincaré-Weyl group be satisfied for the space-time manifold, we construct a model of space-time with the geometric structure of a Weyl-Cartan space. We show that three types of fields must then be introduced as the gauge (“compensating”) fields: Lorentz, translational, and dilatational. Tetrad coefficients then become functions of these gauge fields. We propose a geometric interpretation of the Dirac scalar field. We obtain general equations for the gauge fields, whose sources can be the energy-momentum tensor, the total momentum, and the total dilatation current of an external field. We consider the example of a direct coupling of the gauge field to the orbital momentum of the spinor field. We propose a gravitational field Lagrangian with gauge-invariant transformations of the Poincaré-Weyl group.

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References

  1. 1.

    E. Cartan, Enseignement, 26, 200–225 (1928).

  2. 2.

    J. Schouten and D. Struik, Einfuhrung in die neueren Methoden der Differentialgeometrie, Noordhoff, Groningen (1935).

  3. 3.

    A. A. Slavnov and L. D. Faddeev, Introduction to the Quantum Theory of Gauge Fields [in Russian], Nauka, Moscow (1988); English transl.: Gauge Fields: Introduction to Quantum Theory (Front. Phys., Vol. 83), Addison-Wesley, Redwood City, Calif. (1991).

  4. 4.

    A. A. Sokolov, I. M. Ternov, V. Ch. Zhukovskiui, and A. V. Borisov, Gauge Fields [in Russian], Moskov. Gos. Univ., Moscow (1986).

  5. 5.

    R. Utiyama, Phys. Rev., 101, 1597–1607 (1956).

  6. 6.

    D. Ivanenko, ed., Fundamental Particles and Compensating Fields [in Russian], Mir, Moscow (1964).

  7. 7.

    A. M. Brodskij, D. Ivanenko, and G. A. Sokolik, Sov. Phys. JETP, 14, 930–931 (1962).

  8. 8.

    T. W. B. Kibble, J. Math. Phys., 2, 212–221 (1961).

  9. 9.

    B. N. Frolov, Vestn. MGU, Ser. Fiz., Astron., No. 6, 48–58 (1963).

  10. 10.

    B. N. Frolov, “Contemporary problems of gravitation [in Russian],” in: Collected Works 2nd Contemp. Grav. Conf., Tbilisi Univ. Press, Tbilisi (1967), pp. 270–278.

  11. 11.

    Y. M. Cho, Phys. Rev. D, 14, 3335–3340 (1976).

  12. 12.

    F.W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Modern Phys., 48, 393–416 (1976).

  13. 13.

    F. G. Basombrio, Gen. Relativity Gravitation, 12, 109–136 (1980).

  14. 14.

    D. Ivanenko and G. Sardanashvily, Phys. Rep., 94, 1–45 (1983).

  15. 15.

    F.W. Hehl, J. L. McCrea, E.W. Mielke, and Yu. Ne’eman, Phys. Rep., 258, 1–171 (1995).

  16. 16.

    G. Sardanashvily, “On the geometric foundation of classical gauge gravitation theory,” arXiv:gr-qc/0201074v1 (2002).

  17. 17.

    R. T. Hammond, Rep. Progr. Phys., 65, 599–649 (2002).

  18. 18.

    B. N. Frolov, “Problems of the theory of gravitation with a quadratic Lagrangian in spaces with torsion and nonmetricity [in Russian],” Doctoral dissertation, MSU, Moscow (1999).

  19. 19.

    Yu. N. Obukhov, Int. J. Geom. Meth. Mod. Phys., 3, 95–137 (2006); arXiv:gr-qc/0601090v1 (2006).

  20. 20.

    B. N. Frolov, Poincare Gauge Gravitational Theory [in Russian], MPGU, Moscow (2003).

  21. 21.

    B. N. Frolov, “Physical interpretations of relativity theory,” in: Proc. Intl. Sci. Meeting PIRT-2003, Moscow (2003), pp. 213–219.

  22. 22.

    B. N. Frolov, Gravit. Cosmol., 10, 116–120 (2004).

  23. 23.

    O. V. Babourova, B. N. Frolov, and V. Ch. Zhukovsky, Phys. Rev. D, 74, 064012 (2006); arXiv:hep-th/ 0508088v1 (2005).

  24. 24.

    V. Aldaya and E. Sánchez-Sastre, J. Phys. A, 39, 1729–1742 (2006).

  25. 25.

    Y. Mao, M. Tegmark, A. Guth, and S. Cabi, Phys. Rev. D, 76, 104029 (2007); arXiv:gr-qc/0608121v3 (2006).

  26. 26.

    H. Weyl, Space-Time-Matter (Heidelberger Taschenbücher, Vol. 251), Springer, Berlin (1988).

  27. 27.

    J. M. Charap and W. Tait, Proc. Roy. Soc. Lond. Ser. A, 340, 249–262 (1974).

  28. 28.

    M. Kasuya, Nuovo Cimento B, 28, 127–137 (1975).

  29. 29.

    N. P. Konopleva and V. N. Popov, Gauge Fields [in Russian], Atomizdat, Moscow (1980); English transl., Harwood Academic, Chur (1981).

  30. 30.

    B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979); English transl.: Modern Geometry: Methods and Applications. Part I (Grad. Texts Math., Vol. 93), Springer, New York (1992).

  31. 31.

    D. Gregorash and G. Papini, Nuovo Cimento B, 55, 37–51 (1980); 56, 21–38 (1980).

  32. 32.

    M. Nishioka, Fortschr. Phys., 33, 241–257 (1985).

  33. 33.

    P. A. M. Dirac, Proc. Roy. Soc. Lond. Ser. A, 333, 403–418 (1973).

  34. 34.

    N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields [in Russian], Nauka, Moscow (1984); English transl. prev. ed., Wiley, New York (1980).

  35. 35.

    G. Mack and A. Salam, Ann. Physics, 53, 174–202 (1969).

  36. 36.

    O. V. Babourova and B. N. Frolov, Modern Phys. Lett. A, 12, 2943–2950 (1997); arXiv:gr-qc/9708006v2 (1997).

  37. 37.

    R. W. Tucker and C. Wang, Class. Q. Grav., 15, 933–954 (1998); arXiv:gr-qc/9612019v1 (1996).

  38. 38.

    O. V. Babourova and B. N. Frolov, Class. Q. Grav., 20, 1423–1441 (2003); arXiv:gr-qc/0209077v2 (2002).

  39. 39.

    E. Lubkin, Ann. Physics, 23, 233–283 (1963).

  40. 40.

    R. Utiyama, Progr. Theoret. Phys., 50, 2080–2090 (1973).

  41. 41.

    P. G. O. Freund, Ann. Phys., 84, 440–454 (1974).

  42. 42.

    R. Utiyama, Gen. Relativity Gravitation, 6, 41–47 (1975).

  43. 43.

    B. N. Frolov, “Generalized conformal invariance and gauge theory of gravity,” in: Gravity, Particles, and Space-Time (P. Pronin and G. Sardanashvily, eds.), World Scientific, Singapore (1996), pp. 113–144.

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Correspondence to O. V. Baburova.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 64–78, October, 2008.

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Baburova, O.V., Zhukovsky, V.C. & Frolov, B.N. A Weyl-Cartan space-time model based on the gauge principle. Theor Math Phys 157, 1420–1432 (2008). https://doi.org/10.1007/s11232-008-0117-5

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Keywords

  • gauge field
  • Poincaré-Weyl group
  • Noether theorem
  • Weyl-Cartan space
  • dilatation current