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Gravitation and Cosmology

, Volume 19, Issue 3, pp 171–177 | Cite as

Quantum billiards in multidimensional models with fields of forms

  • V. D. IvashchukEmail author
  • V. N. Melnikov
Article

Abstract

A Bianchi type I cosmological model in (n + 1)-dimensional gravity with several forms is considered. When the electric non-composite brane ansatz is adopted, the Wheeler-DeWitt (WDW) equation for the model, written in a conformally covariant form, is analyzed. Under certain restrictions, asymptotic solutions to the WDW equation near the singularity are found, which reduce the problem to the so-called quantum billiard on the (n − 1)-dimensional Lobachevsky space ℍ n−1. Two examples of quantum billiards are considered: a 2-dimensional quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard for an 11D model with 120 4-forms, whichmimics the SM2-brane sector of D = 11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved.

Keywords

Cosmological Model Asymptotic Solution Bianchi Type Multidimensional Model Quantum Cosmology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D. M. Chitre, PhD. thesis (University of Maryland, 1972).Google Scholar
  2. 2.
    V. A. Belinskii, E. M. Lifshitz, and I. M. Khalatnikov, Usp. Fiz. Nauk 102, 463 (1970) [in Russian]; Adv. Phys. 31, 639 (1982).ADSCrossRefGoogle Scholar
  3. 3.
    C.W. Misner, Quantum cosmology, Phys. Rev. 186, 1319 (1969).zbMATHGoogle Scholar
  4. 4.
    C. W. Misner, The Mixmaster cosmological metrics, preprint UMCP PP94-162; gr-qc/9405068.Google Scholar
  5. 5.
    A. A. Kirillov, Sov. Phys. JETP 76, 355 (1993) [Zh. Eksp. Teor. Fiz. 76, 705 (1993), in Russian]; Int. J. Mod. Phys. D 3, 431 (1994).ADSGoogle Scholar
  6. 6.
    V. D. Ivashchuk, A. A. Kirillov, and V. N. Melnikov, On stochastic properties ofmultidimensional cosmological models near the singular point, Izv.Vuzov (Fizika) 11, 107 (1994) (in Russian) [Russ. Phys. J. 37, 1102 (1994)].MathSciNetGoogle Scholar
  7. 7.
    V. D. Ivashchuk, A. A. Kirillov, and V. N. Melnikov, On stochastic behavior of multidimensional cosmological models near the singularity, Pis’ma ZhETF 60(4), 225 (1994) (in Russian) [JETP Lett. 60, 235 (1994)].Google Scholar
  8. 8.
    V. D. Ivashchuk and V. N. Melnikov, Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity, Class. Quantum Grav. 12, 809 (1995); grqc/9407028.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    V. D. Ivashchuk and V. N. Melnikov, Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity, J. Math. Phys. 41, 6341 (2000); hep-th/9904077.MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    T. Damour and M. Henneaux, Chaos in superstring cosmology, Phys. Rev. Lett. 85, 920 (2000); hepth/0003139.MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    T. Damour, M. Henneaux, and H. Nicolai, Cosmological billiards, topical review. Class. Quantum Grav. 20, R145 (2003); hep-th/0212256.MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    A. Kleinschmidt, M. Koehn, and H. Nicolai, Supersymmetric quantum cosmological billiards, Phys. Rev. D 80, 061701 (2009); Arxiv: 0907.3048.MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    A. Kleinschmidt and H. Nicolai, Cosmological quantum billiards, Arxiv: 0912.0854.Google Scholar
  14. 14.
    H. Lü, J. Maharana, S. Mukherji and C. N. Pope, Cosmological solutions, p-branes and the Wheeler-DeWitt equation, Phys. Rev. D 57, 2219 (1997); hep-th/9707182.CrossRefGoogle Scholar
  15. 15.
    V. D. Ivashchuk and V. N. Melnikov, Multidimensional classical and quantum cosmology with intersecting p-branes, J. Math. Phys. 39, 2866 (1998); hep-th/9708157.MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    J.W. York, Role of conformal three-geometry in the dynamics of gravitation, Phys. Rev. Lett. 28(16), 1082 (1972).ADSCrossRefGoogle Scholar
  17. 17.
    G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    V. D. Ivashchuk and V. N. Melnikov, Sigma-model for the generalized composite p-branes, Class. Quantum Grav. 14, 3001 (1997); Corrigendum ibid. 15, 3941 (1998); hep-th/9705036.MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    C. W. Misner, In: Magic without magic: John Archibald Wheeler, ed. J. R. Klauder (Freeman, San Francisko, 1972).Google Scholar
  20. 20.
    J. J. Halliwell, Derivation of the Wheeler-DeWitt equation from a path integral for minisuperspace models, Phys. Rev. D 38, 2468 (1988).MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    V. D. Ivashchuk, V. N. Melnikov, and A. I. Zhuk, On Wheeler-DeWitt equation inmultidimensional cosmology, Nuovo Cim.B 104(5), 575 (1989).MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    V. D. Ivashchuk and V. N. Melnikov, On billiard approach in multidimensional cosmological models, Grav. Cosmol. 15(1), 49 (2009); ArXiv: 0811.2786.MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    E. Cremmer, B. Julia, and J. Scherk, Supergravity theory in eleven dimensions, Phys. Lett. B 76, 409 (1978).ADSCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Center for Gravitation and Fundamental MetrologyVNIIMSMoscowRussia
  2. 2.Institute of Gravitation and Cosmology of Peoples’ Friendship University of RussiaMoscowRussia

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