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Von Neumann’s quantization of general relativity

  • Elementary Particles and Fields Theory
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Abstract

Von Neumann’s procedure is applied to quantizing general relativity. Initial data for dynamical variables in the Planck epoch, where the Hubble parameter value coincided with the Planck mass are quantized. These initial data are defined in terms of the Fock orthogonal simplex in the tangent Minkowski spacetime and the Dirac conformal interval. The Einstein cosmological principle is used to average the logarithm of the determinant of the spatial metric over the spatial volume of the visible Universe. The splitting of general coordinate transformations into diffeomorphisms and transformations of the initial data is introduced. In accordance with von Neumann’s procedure, the vacuum state is treated is a quantum ensemble that is degenerate in quantum numbers of nonvacuum states. The distribution of the vacuum state leads to the Casimir effect in gravidynamics in just the same way as in electrodynamics. The generating functional for perturbation theory in gravidynamics is found by solving the quantum energy constraint. The applicability range of gravidynamics is discussed along with the possibility of employing this theory to interpret modern observational data.

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References

  1. A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 44, 778 (1915); Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) n48, 844 (1915).

    Google Scholar 

  2. D. Hilbert, Nachr. Kön. Ges. Wiss. Göttingen, Math. -Phys. Kl. 3, 395 (1915).

    Google Scholar 

  3. E. Noether, Nachr. Kön. Ges. Wiss. Göttingen, Math. -Phys. Kl. 235 (1918).

  4. V. Fock, Z. Phys. 57, 261 (1929).

    Article  ADS  Google Scholar 

  5. P. A. M. Dirac, Can. J. Math. 2, 129 (1950); Proc. R. Soc. London, Ser. A 246, 326, 333 (1958; Phys. Rev. 114, 924 (1959); The Principles of Quantum Mechanics (Oxford Univ. Press, New York,1982).

    Article  Google Scholar 

  6. R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: An Introduction to Current Research, Ed. by L. Witten (Wiley, New York, 1962), p. 227.

  7. A. L. Zel’manov and V. G. Agakov, Elements of the General Theory of Relativity (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  8. B. S. DeWitt, Phys. Rev. 160, 1113 (1967).

    Article  ADS  Google Scholar 

  9. J. A. Wheeler, in Battelle Rencontres: Lectures in Mathematics and Physics, 1967, Ed. by C. M. De-Witt and J. A. Wheeler (Benjamin, New York, 1968), p. 242.

  10. C. W. Misner, Phys. Rev. 186, 1319 (1969).

    Article  ADS  Google Scholar 

  11. A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 142 (1917).

  12. P. A. M. Dirac, Proc. R. Soc. London, Ser. A 333, 403 (1973).

    Article  ADS  Google Scholar 

  13. S. Deser, Ann. Phys. (N. Y.) 59, 248 (1970).

    Article  ADS  Google Scholar 

  14. A. B. Borisov and V. I. Ogievetskii, Theor. Math. Phys. 21, 1179 (1974).

    Article  Google Scholar 

  15. S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177, 2239 (1969)

    Article  ADS  Google Scholar 

  16. D. V. Volkov, Sov. J. Part. Nucl. 4, 1 (1973); PreprintNo. 69-73, ITPh (Inst. Theor. Phys., Kiev,1969).

    Google Scholar 

  17. B. M. Barbashov et al., Phys. Lett. B 633, 458 (2006).

    Article  ADS  Google Scholar 

  18. A. B. Arbuzov et al., Phys. Lett. B 691, 230 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  19. V. N. Pervushin et al., Gen. Relativ. Grav. 44, 2745 (2012).

    Article  ADS  Google Scholar 

  20. E. P. Wigner, Ann. Math. 40, 149 (1939).

    Article  ADS  MathSciNet  Google Scholar 

  21. V. I. Ogievetsky, Lett. Nuovo Cimento 8, 988 (1973).

    Article  MathSciNet  Google Scholar 

  22. H. B. G. Casimir, Proc. Kön. Nederl. Akad. Wetensch. 51, 793 (1948).

    Google Scholar 

  23. M. Bordag et al., Advances in the Casimir Effect (Oxford Univ. Press, New York, 2009).

    Book  MATH  Google Scholar 

  24. J. von Neumann Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, 1955), p. 418.

    Google Scholar 

  25. D. I. Blokhintsev, J. Phys. (USSR) 2, 71 (1940).

    Google Scholar 

  26. E. Kasner, Am. J. Math. 43, 217 (1921).

    Article  Google Scholar 

  27. S. Perlmutter et al., Astrophys. J. 517, 565 (1999)

    Article  ADS  Google Scholar 

  28. B. P. Schmidt et al., Astrophys. J. 507, 46 (1998).

    Article  ADS  Google Scholar 

  29. A. G. Riess et al., Astrophys. J. 560, 49 (2001); 607, 665 (2004).

    Article  ADS  Google Scholar 

  30. D. Behnke et al., Phys. Lett. B 530, 20 (2002).

    Article  ADS  Google Scholar 

  31. A. F. Zakharov and V. N. Pervushin, Int. J. Mod. Phys. D 19, 1875 (2010).

    Article  ADS  Google Scholar 

  32. M. A. Markov, Prog. Theor. Phys. Suppl. E65, 85 (1965).

    Article  ADS  Google Scholar 

  33. M. A. Markov, Phys. Usp. 37, 57 (1994).

    Article  ADS  Google Scholar 

  34. S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).

    Article  ADS  Google Scholar 

  35. V. Pervushin et al., PoS (Baldin ISHEPP XXI) 023 (2012); arXiv: 1209.4460 [hep-ph]

  36. V. Pervushin et al., PoS (Baldin ISHEPP XXII) 136 (2015); arXiv: 1502.00267 [gr-qc].

  37. A. B. Arbuzov et al., arXiv: 1411.5124 [hep-ph].

  38. A. B. Arbuzov et al., Europhys. Lett. 113, 31001 (2016).

    Article  ADS  Google Scholar 

  39. A. B. Arbuzov et al., Grav. Cosmol. 15, 199 (2009).

    Article  ADS  Google Scholar 

  40. D. J. Cirilo-Lombardo and V. N. Pervushin, Int. J. Geom. Meth. Mod. Phys. 13, 1650113 (2016).

    Article  Google Scholar 

  41. L. B. Litov and V. N. Pervushin, Phys. Lett. B 147, 76 (1984).

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, ul. Joliot-Curie 6, Dubna, Moscow oblast, 141980, Russia

    A. B. Arbuzov, A. Yu. Cherny, D. J. Cirilo-Lombardo, R. G. Nazmitdinov, A. E. Pavlov, V. N. Pervushin & A. F. Zakharov

  2. University Dubna, Universitetskaya ul. 19, Dubna, Moscow oblast, 141982, Russia

    A. B. Arbuzov

  3. National Institute for Plasma Physics (INFIP-CONICET), FCEyN, Universidad de Buenos Aies, Buenos Aires, 1428, Argentina

    D. J. Cirilo-Lombardo

  4. Universitat de les Illes Balears, Cra. de Valldemossa, km 7.5, 07122, Palma (Illes Balears), Spain

    R. G. Nazmitdinov

  5. Hanoi University of Science, Hanoi, Vietnam

    Nguyen Suan Han

  6. Russian State Agrarian University, Timiryazevskaya ul. 49, Moscow, 127550, Russia

    A. E. Pavlov

  7. Institute for Theoretical and Experimental Physics, National Research Center Kurchatov Institute, Bol’shaya Cheremuskinskaya ul. 25, Moscow, 117218, Russia

    A. F. Zakharov

Authors
  1. A. B. Arbuzov
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  2. A. Yu. Cherny
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  3. D. J. Cirilo-Lombardo
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  4. R. G. Nazmitdinov
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  5. Nguyen Suan Han
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  6. A. E. Pavlov
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  7. V. N. Pervushin
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  8. A. F. Zakharov
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Corresponding author

Correspondence to A. B. Arbuzov.

Additional information

Original Russian Text © A.B. Arbuzov, A.Yu. Cherny, D.J. Cirilo-Lombardo, R.G. Nazmitdinov, Nguyen Suan Han, A.E. Pavlov, V.N. Pervushin, A.F. Zakharov, 2017, published in Yadernaya Fizika, 2017, Vol. 80, No. 3, pp. 275–288.

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Arbuzov, A.B., Cherny, A.Y., Cirilo-Lombardo, D.J. et al. Von Neumann’s quantization of general relativity. Phys. Atom. Nuclei 80, 491–504 (2017). https://doi.org/10.1134/S106377881702003X

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  • DOI: https://doi.org/10.1134/S106377881702003X

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