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Conformal and affine Hamiltonian dynamics of general relativity

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Abstract

The Hamiltonian approach to the General Relativity is formulated as a joint nonlinear realization of conformal and affine symmetries by means of the Dirac scalar dilaton and the Maurer–Cartan forms. The dominance of the Casimir vacuum energy of physical fields provides a good description of the type Ia supernova luminosity distance—redshift relation. Introducing the uncertainty principle at the Planck’s epoch within our model, we obtain the hierarchy of the Universe energy scales, which is supported by the observational data. We found that the invariance of the Maurer–Cartan forms with respect to the general coordinate transformation yields a single-component strong gravitational waves. The Hamiltonian dynamics of the model describes the effect of an intensive vacuum creation of gravitons and the minimal coupling scalar (Higgs) bosons in the Early Universe.

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References

  1. Glashow S.L.: Partial symmetries of weak interactions. Nucl. Phys. 22, 579 (1961)

    Article  Google Scholar 

  2. Weinberg S.: A model of leptons. Phys. Rev. Lett. 19, 1264 (1967)

    Article  ADS  Google Scholar 

  3. Salam, A.: The standard model. In: Svartholm N. (ed.) Elementary Particle Theory, p. 367, Almqvist and Wikdells, Stockholm (1969)

  4. Bogoliubov N.N., Logunov A.A., Oksak A.I., Todorov I.T.: General Principles of Quantum Field Theory. Springer, Berlin (1989)

    Google Scholar 

  5. Giovannini, M.: Theoretical tools for the physics of CMB anisotropies. Int. J. Mod. Phys. D 14, 363 (2005). [arXiv:astro-ph/0412601]

  6. Mukhanov V.: Physical Foundations of Cosmology. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  7. Gorbunov D.S., Rubakov V.A.: Introduction to the Theory of the Early Universe: Hot Big Bang Theory. World Scientific, Singapore (2010)

    Google Scholar 

  8. Fock V.: Geometrization of Dirac’s theory of the electron (in German). Z. Phys. 57, 261 (1929)

    Article  MATH  ADS  Google Scholar 

  9. Ogievetsky V.I.: Infinite-dimensional algebra of general covariance group as the closure of finite-dimensional algebras of conformal and linear groups. Lett. Nuovo Cim. 8, 988 (1973)

    Article  MathSciNet  Google Scholar 

  10. Einstein A.: Approximative integration of the field equations of gravitation. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916, 688 (1916)

    Google Scholar 

  11. Einstein A.: Über Gravitationswellen (in German). ibid 1918, 154 (1918)

    Google Scholar 

  12. Deser S.: Scale invariance and gravitational coupling. Ann. Phys. 59, 248 (1970)

    Article  MathSciNet  ADS  Google Scholar 

  13. Dirac P.A.M.: Long range forces and broken symmetries. Proc. Roy. Soc. Lond. A 333, 403 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  14. Hilbert D.: Nachrichten von der Kön. Ges. der Wissenschaften zu Göttingen, (in German). Math. Phys. Kl. 3, 395 (1915)

    Google Scholar 

  15. Borisov, A.B., Ogievetsky, V.I.: Theory of dynamical affine and conformal symmetries as gravity theory. Theor. Math. Phys. 21, 1179 (1975). [Teor. Mat. Fiz. 21, 329 (1974)]

  16. Coleman S.R., Wess J., Zumino B.: Structure of phenomenological Lagrangians 1. Phys. Rev. 177, 2239 (1969)

    Article  ADS  Google Scholar 

  17. Volkov, D.V.: Phenomenological lagrangians. Fiz. Elem. Chast. Atom. Yadra 4, 3 (1973). [Preprint ITF-69-73, Kiev, 1969]

  18. Behnke, D., Blaschke, D.B., Pervushin, V.N., Proskurin, D.: Description of supernova data in conformal cosmology without cosmological constant. Phys. Lett. B 530, 20 (2002). [arXiv:gr-qc/0102039]

    Google Scholar 

  19. Zakharov, A.F., Pervushin, V.N.: Conformal cosmological model parameters with distant SNe Ia data: ‘gold’ and ‘silver’. Int. J. Mod. Phys. D 19, 1875 (2010). [arXiv:1006.4745 [gr-qc

    Google Scholar 

  20. Barbashov, B.M., Pervushin, V.N., Zakharov, A.F., Zinchuk, V.A.: Hamiltonian cosmological perturbation theory. Phys. Lett. B 633, 458 (2006). [arXiv:hep-th/0501242]

    Google Scholar 

  21. Arbuzov, A.B., Barbashov, B.M., Nazmitdinov, R.G., Pervushin, V., Borowiec, A., Pichugin, K.N., Zakharov, A.F.: Conformal Hamiltonian dynamics of general relativity. Phys. Lett. B 691, 230 (2010). [arXiv:1007.0293 [gr-qc

    Google Scholar 

  22. Blas, D., Shaposhnikov, M., Zenhausern, D.: Scale-invariant alternatives to general relativity. Phys. Rev. D 84, 044001 (2011). [arXiv:1104.1392 [hep-th

  23. Garcia-Bellido, J., Rubio, J., Shaposhnikov, M., Zenhausern, D.: Phys. Rev. D 84, 123504 (2011). [arXiv:1107.2163 [hep-ph

  24. Penrose R., MacCallum M.A.H.: Twistor theory: an approach to the quantization of fields and space-time. Phys. Rept. 6C, 241–316 (1972)

    MathSciNet  ADS  Google Scholar 

  25. Cartan E.: Lecons Sur la Geometric des Espaces de Riemann. Gauthier-Villars, Paris (1946)

    Google Scholar 

  26. Pervushin, V.N.: Dynamical affine symmetry and covariant perturbation theory for gravity. Theor. Math. Phys. 27, 330 (1976). [Teor. Mat. Fiz. 27, 16 (1976)]

  27. Landau L.D., Lifshitz E.M.: Classical Theory of Fields. Pergamon Press, New York (1975)

    Google Scholar 

  28. Dirac P.A.M.: The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. Lond. A 246, 333 (1958)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  29. Dirac P.A.M.: Fixation of coordinates in the Hamiltonian theory of gravitation. Phys. Rev. 114, 924 (1959)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  30. Weyl, H.: Gravitation und Elektrizität. Sitzungsber. d. Berl. Akademie 465 (1918)

  31. Arnowitt R., Deser S., Misner C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, pp. 227–265. Wiley, New York (1962)

    Google Scholar 

  32. Lichnerowicz A.: L’integration des equations de la gravitation relativiste et le probleme des n corps (in French). J. Math. Pures Appl. B 37, 23 (1944)

    MathSciNet  Google Scholar 

  33. York J.W.: Gravitational degrees of freedom and the initial-value problem. Phys. Rev. Lett. 26, 1656 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  34. Kuchar K.: A bubble-time canonical formalism for geometrodynamics. J. Math. Phys. 13, 768 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  35. Zelmanov A.L.: Chronometric invariants and accompanying coordinates in General Relativity (in Russian). Dokl. AN USSR 107, 315 (1956)

    MathSciNet  Google Scholar 

  36. Zelmanov A.L.: Kinemetric invariants and their relation to the chronometric invariants of Einstein’s theory of gravity (in Russian). ibid 209, 822 (1973)

    MathSciNet  ADS  Google Scholar 

  37. Einstein A.: Cosmological considerations in the general theory of relativity. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1917, 142 (1917)

    Google Scholar 

  38. Friedmann, A.: Über die Krümmung des Raumes (in German). Z. Phys. 10, 377 (1922). [Gen. Relativ. Gravit. 31, 1991 (1999)]

  39. Friedmann A.: Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes, (in German). ibid 21, 306 (1924)

    MathSciNet  Google Scholar 

  40. DeWitt B.S.: Quantum theory of gravity 1. The Canonical Theory. Phys. Rev. 160, 1113 (1967)

    Article  MATH  ADS  Google Scholar 

  41. Wheeler, J.A.: In Batelle Rencontres: 1967, Lectures in Mathematics and Physics. DeWitt, C., Wheeler, J.A. (eds.) New York (1968)

  42. Grib A.A., Mamaev S.G., Mostepanenko V.M.: Quantum Effects in Strong External Fields. Friedmann Laboratory Publishing, St. Petersburg (1994)

    Google Scholar 

  43. Faddeev, L.D., Popov, V.N.: Covariant quantization of the gravitational field. Sov. Phys. Usp. 16, 777 (1974). [Usp. Fiz. Nauk 111, 427 (1973)]

    Google Scholar 

  44. Goldstone J.: Field theories with superconductor solutions. Nuovo Cimento 19, 154 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  45. Fukugita M., Hogan C.J., Peebles P.J.E.: The cosmic baryon budget. Astrophys. J. 503, 518 (1998)

    Article  ADS  Google Scholar 

  46. Behnke, D.: Conformal Cosmology Approach to the Problem of Dark Matter. PhD Thesis, Rostock Report MPG-VT-UR 248/04 (2004)

  47. Actor A.A.: Scalar quantum fields confined by rectangular boundaries. Fortsch. Phys. 43, 141 (1995)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  48. Bordag M., Klimchitskaya G.L., Mohideen U., Mostepanenko V.M.: Advances in the Casimir Effect. Oxford University Press, New York (2009)

    Book  MATH  Google Scholar 

  49. Wigner E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149 (1939)

    Article  MathSciNet  Google Scholar 

  50. Lifshits, E.M., Khalatnikov, I.M.: Problems of relativistic cosmology. Sov. Phys. Usp. 6, 495 (1964). [Uspekhi fiz. nauk, 80, 391 (1963)]

    Google Scholar 

  51. Lifshits E.M., Khalatnikov I.M.: Investigations in relativistic cosmology. Adv. Phys. 12, 185 (1963)

    Article  ADS  Google Scholar 

  52. Bardeen J.M.: Gauge invariant cosmological perturbations. Phys. Rev. D 22, 1882 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  53. Mukhanov V.F., Feldman H.A., Brandenberger R.H.: Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Phys. Rept. 215, 203 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  54. Tod, K.P.: Three-dimensional Einstein-Weyl geometry. In: Geometry of Low-Dimensional Manyfolds. vol. 1, pp. 237–246, Durham (1989), London Mathematical Society. Lecture Note Series, vol. 150, Cambridge University Press, Cambridge (1990)

  55. Grishchuk L.P.: Gravitational waves in the cosmos and the laboratory. Sov. Phys. Usp. 20, 319 (1977)

    Article  ADS  Google Scholar 

  56. Babak, S.V., Grishchuk, L.P.: The energy momentum tensor for the gravitational field. Phys. Rev. D 61, 024038 (2000). [arXiv:gr-qc/9907027]

  57. Einstein A., Straus E.G.: The influence of the expansion of space on the gravitation fields surrounding the individual stars. Rev. Mod. Phys. 17, 120 (1945)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  58. Flin, P., Gusev, A.A., Pervushin, V.N., Vinitsky, S.I., Zorin, A.G.: Cold dark matter as cosmic evolution of galaxies in relative units. Astrophysics 47, 242 (2004). [arXiv:astro-ph/0301543]

    Google Scholar 

  59. Zakharov A.F., Zinchuk V.A., Pervushin V.N.: Tetrad formalism and frames of references in general relativity. Phys. Part. Nucl. 37, 104 (2006)

    Article  Google Scholar 

  60. Pervushin, V.N., Smirichinski, V.I.: Bogolyubov quasiparticles in constrained systems. J. Phys. A 32, 6191 (1999). [arXiv:hep-th/9902013]

  61. von Neumann J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104, 570 (1931)

    Article  MathSciNet  Google Scholar 

  62. Jordan T.F., Mukunda N., Pepper S.V.: Irreducible representations of generalized oscillator operators. Math. Phys. 4, 1089 (1963)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  63. Andreev A.Y., Kirzhnits D.A.: Tachyons and the instability of physical systems. Phys. Usp. 39, 1071 (1996)

    Article  ADS  Google Scholar 

  64. Riess, A.G. et al.: [Supernova Search Team Collaboration], The farthest known supernova: support for an accelerating universe and a glimpse of the epoch of deceleration. Astrophys. J. 560, 49 (2001). [arXiv:astro-ph/0104455]

    Google Scholar 

  65. Banerjee S.K., Narlikar J.V., Wickramasinghe N.C., Hoyle F., Burbidge G.: Possible interpretations of the magnitude-redshift relation for supernovae of type 1A. Astrophys. J. 119, 2583 (2000)

    ADS  Google Scholar 

  66. Kirzhnits D.A.: The hot universe and the weinberg model. JETP Lett. 15, 529 (1972)

    ADS  Google Scholar 

  67. Bezrukov F.L., Shaposhnikov M.: The standard model Higgs boson as the inflaton. Phys. Lett. B 659, 703 (2008)

    Article  ADS  Google Scholar 

  68. Riess, A.G., et al.: [Supernova Search Team Collaboration], Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998). [arXiv:astro-ph/9805201]

    Google Scholar 

  69. Astier, P., et al.: [The SNLS Collaboration], The supernova legacy survey: measurement of omega(m), omega(lambda) and W from the first year data set. Astron. Astrophys. 447, 31 (2006). [arXiv:astro-ph/0510447]

  70. Cocke W.J., Tifft W.G.: Statistical procedure and the significance of periodicities in double-galaxy redshifts. Astrophys. J. 368, 383 (1991)

    Article  ADS  Google Scholar 

  71. Panagia, N.: High redshift supernovae: cosmological implications. Nuovo Cim. B 120, 667 (2005). [arXiv:astro-ph/0502247]

  72. Zhu, Z.H., Fujimoto, M.K.: Constraints on Cardassian expansion from distant type Ia supernovae. Astrophys. J. 585, 52 (2003). [arXiv:astro-ph/0303021]

    Google Scholar 

  73. Blaschke, D.B., Vinitsky, S.I., Gusev, A.A., Pervushin, V.N., Proskurin, D.V.: Cosmological production of vector bosons and cosmic microwave background radiation. Phys. Atom. Nucl. 67, 1050 (2004). [Yad. Fiz. 67, 1074 (2004)] [arXiv:hep-ph/0504225]

    Google Scholar 

  74. Tegmark, M.: Measuring the metric: a parametrized post Friedmanian approach to the cosmic dark energy problem. Phys. Rev. D 66, 103507 (2002). [arXiv:astro-ph/0101354]

    Google Scholar 

  75. Faulkner T., Tegmark M., Bunn E.F., Mao Y.: Constraining f(R) gravity as a scalar-tensor theory. Phys. Rev. D 76, 063505 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  76. Arbuzov A.B., Barbashov B.M., Borowiec A., Pervushin V.N., Shuvalov S.A., Zakharov A.F.: General relativity and standard model in scale-invariant variables. Gravit. Cosmol. 15, 199 (2009)

    Article  MATH  ADS  Google Scholar 

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Authors and Affiliations

  1. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Victor N. Pervushin, Andrej B. Arbuzov, Boris M. Barbashov, Rashid G. Nazmitdinov & Alexander F. Zakharov

  2. Department of Higher Mathematics, University of Dubna, 141980, Dubna, Russia

    Andrej B. Arbuzov

  3. Department de Física, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain

    Rashid G. Nazmitdinov

  4. Institute of Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9, 50-204, Wrocław, Poland

    Andrzej Borowiec

  5. Kirensky Institute of Physics, 660036, Krasnoyarsk, Russia

    Konstantin N. Pichugin

  6. Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya str. 25, 117259, Moscow, Russia

    Alexander F. Zakharov

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  1. Victor N. Pervushin
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  2. Andrej B. Arbuzov
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  4. Rashid G. Nazmitdinov
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Correspondence to Andrej B. Arbuzov.

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Pervushin, V.N., Arbuzov, A.B., Barbashov, B.M. et al. Conformal and affine Hamiltonian dynamics of general relativity. Gen Relativ Gravit 44, 2745–2783 (2012). https://doi.org/10.1007/s10714-012-1423-7

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