Gravitation and Cosmology

, Volume 21, Issue 3, pp 191–199 | Cite as

Quantum cosmology with scalar fields: Self-adjointness and cosmological scenarios

  • Carla R. Almeida
  • Antonio B. Batista
  • Júlio C. Fabris
  • Paulo R. L. V. Moniz


We discuss the issue of unitarity in particular quantum cosmological models with scalar fields. The time variable is recovered, in this context, by using the Schutz formalism for a radiative fluid. Two cases are considered: a phantom scalar field and an ordinary scalar field. For the first case, it is shown that the evolution is unitary provided a convenient factor ordering and inner product measure are chosen; the same happens for the ordinary scalar field, except for some special cases in which the Hamiltonian is not self-adjoint but admits a self-adjoint extension. In all cases, even in the cases not exhibiting a unitary evolution, a formal computation of the expectation value of the scale factor indicates a nonsingular bounce. The importance of unitary evolution in quantum cosmology is briefly discussed.


Wave Packet Hamiltonian Operator Unitary Evolution Quantum Cosmology Cosmological Scenario 
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  1. 1.
    B. S. DeWitt, Phys. Rev. 160, 1113 (1967).zbMATHCrossRefADSGoogle Scholar
  2. 2.
    C. W. Misner, Phys. Rev. 186, 1319 (1969).zbMATHCrossRefADSGoogle Scholar
  3. 3.
    C. W. Misner, Phys. Rev. 186, 1328 (1969).zbMATHCrossRefADSGoogle Scholar
  4. 4.
    N. Pinto-Neto, “Quantum cosmology,” in Cosmology and Gravitation, Ed. M. Novello (Editions Frontieres, Gif-sur-Yvette, 1996).Google Scholar
  5. 5.
    K. Kuchar, “Time and interpretation of quantum gravity,” in Proc. 4th Canadian Conference on General Relativity and Relativistic Astrophyscis (World Scientific, Singapore, 1992).Google Scholar
  6. 6.
    R. Colistete Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 57, 4707 (1998).MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    B. F. Schutz, Phys. Rev. D 2, 2762 (1970).zbMATHMathSciNetCrossRefADSGoogle Scholar
  8. 8.
    B. F. Schutz, Phys. Rev. D 4, 3559 (1971).CrossRefADSGoogle Scholar
  9. 9.
    V. G. Lapchinskii and V. A. Rubakov, Theor. Math. Phys. 33, 1076 (1977).CrossRefGoogle Scholar
  10. 10.
    F. Tipler, Phys. Rep. 137, 231 (1986).MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    M. Bojowald, Class. Quantum Grav. 29, 213001 (2012).MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    P. R. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1983).Google Scholar
  13. 13.
    N. Pinto-Neto and J. C. Fabris, Class. Quantum Grav. 30, 143001 (2013).MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    F. G. Alvarenga, J. C. Fabris, N. A. Lemos, and G. A. Monerat, Gen. Rel. Grav. 34, 651 (2002).CrossRefGoogle Scholar
  15. 15.
    F. G. Alvarenga, A. B. Batista, J. C. Fabris, and S. V. B. Gonc¸ alves, Gen. Rel. Grav. 35, 1659 (2003).zbMATHCrossRefADSGoogle Scholar
  16. 16.
    S. Pal and N. Banerjee, Phys. Rev. D 90, 104001 (2014).CrossRefADSGoogle Scholar
  17. 17.
    F. G. Alvarenga, A. B. Batista, and J. C. Fabris, Int. J.Mod. Phys. D 14, 291 (2005).zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    J. C. Fabris, F. T. Falciano, J. Marto, N. Pinto-Neto, and P. Vargas Moniz, Braz. J. Phys. 42, 475 (2012).CrossRefADSGoogle Scholar
  19. 19.
    B. Vakili, Phys. Lett. B 718, 34 (2012).CrossRefADSGoogle Scholar
  20. 20.
    C. Brans and R. H. Dicke, Phys.Rev. 124, 925 (1961).zbMATHMathSciNetCrossRefADSGoogle Scholar
  21. 21.
    J. E. Lidsey, D. Wands, and E. J. Copeland, Phys. Rep. 337, 343 (2000).MathSciNetCrossRefADSGoogle Scholar
  22. 22.
    E. Alvarez and J. Conde, Mod. Phys. Lett. A 17, 413 (2002).zbMATHMathSciNetCrossRefADSGoogle Scholar
  23. 23.
    I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, San Diego, 2007).Google Scholar
  24. 24.
    M. Reed and B. Simon, Methods of ModernMathematical Physics (Academic Press, New York, 1975), Vol. 2.Google Scholar
  25. 25.
    A. M. Essin and D. J. Griffiths, Am. J. Phys. 74, 109 (2006).CrossRefADSGoogle Scholar
  26. 26.
    R. Colistete Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev. D 57, 4707 (1998).MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    J. C. Fabris, N. Pinto-Neto, and A. Velasco, Class. Quantum Grav. 16, 3807 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Carla R. Almeida
    • 1
  • Antonio B. Batista
    • 1
  • Júlio C. Fabris
    • 1
  • Paulo R. L. V. Moniz
    • 2
    • 3
  1. 1.DF–UFESVitória, ESBrazil
  2. 2.Centro de Matemática e Aplicações-UBICovilhãPortugal
  3. 3.Departmento de FisicaUniversidade da Beira InteriorCovilhãPortugal

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