General Relativity and Gravitation

, Volume 27, Issue 1, pp 35–53 | Cite as

Quantization of closed mini-superspace models as bound states

  • J. H. Kung


The Wheeler-DeWitt equation is applied to closedk>0 Friedmann-Robertson-Walker metric with various combination of cosmological constant and matter (e.g., radiation or pressureless gas). It is shown that if the universe ends in the matter dominated era (e.g., radiation or pressureless gas) with zero cosmological constant, then the resulting Wheeler-DeWitt equation describes a bound state problem. As solutions of a nondegenerate bound state system, the eigen-wave functions are real (Hartle-Hawking). Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the various energy densities of the universe. If we assume that our universe is closed, then the quantum number of our universe isN∼(Gk)−1∼10122. The largeness of this quantum number is naturally explained by an early inflationary phase which resulted in a flat universe we observe today. It is also shown that if there is a cosmological constant Λ>0 in our universe that persists for all time, then the resulting Wheeler-DeWitt equation describes a non-bound state system, regardless of the magnitude of the cosmological constant. As a consequence, the wave functions are in general complex (Vilenkin).


Radiation Wave Function Energy Density Quantization Condition Quantum Number 
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  1. 1.
    Dirac, P. A. M. (1964).Lectures on Quantum Mechanics (Belfer Graduate School of Science Monographs 2, Yeshiva University, New York).Google Scholar
  2. 2.
    Wald, R. M. (1984).General Relativity (University of Chicago Press, Chicago).Google Scholar
  3. 3.
    Arnowitt, R., Deser, S., and Misner, C. W. (1963). InGravitation: An Introduction to Current Research (Wiley, New York).Google Scholar
  4. 4.
    DeWitt, B. S. (1967).Phys. Rev. 160, 1113.CrossRefGoogle Scholar
  5. 5.
    Wheeler, J. A. (1968). InBattelle Rencontres: 1967 Lectures in Mathematics and Physics, C. M. DeWitt and J. A. Wheeler, eds. (Benjamin, New York).Google Scholar
  6. 6.
    Misner, C. W. (1972). InMagic Without Magic: John Archibald Wheeler Festschrift, J. R. Klauder, ed. (Freeman, San Francisco).Google Scholar
  7. 7.
    Halliwell, J. J. (1988).Phys. Rev. D 38, 2468.Google Scholar
  8. 8.
    Misner, C. W. (1970). InRelativity, M. Carmeli, S. I. Fickler, and L. Witten, eds. (Plenum, New York).Google Scholar
  9. 9.
    Hawking, S. W., and Page, D. N. (1986).Nucl. Phys. B 264, 185.Google Scholar
  10. 10.
    Blyth, W. F., and Isham, C. J. (1975).Phys. Rev. D 11, 768.Google Scholar
  11. 11.
    Christodoulakis, T., and Zanelli, J. (1984).Phys. Lett. 102A, 227.Google Scholar
  12. 12.
    Esposito, G., and Platania, G. (1988).Class. Quant. Grav. 5, 937.Google Scholar
  13. 13.
    Gibbons, G. W., and Grishchuk, L. P. (1988).Nucl. Phys. B 313, 736.CrossRefGoogle Scholar
  14. 14.
    Hartle, J. B., and Hawking, S. W. (1983).Phys. Rev. D 28, 2960.Google Scholar
  15. 15.
    Hawking, S. W. (1984).Nucl. Phys. B 239, 257.CrossRefGoogle Scholar
  16. 16.
    Vilenkin, A. (1986).Phys. Rev. D 33, 3560.Google Scholar
  17. 17.
    Vilenkin, A. (1986).Phys. Rev. D 37, 888.Google Scholar
  18. 18.
    Narlikar, J., Padmanabhan, T. (1986).Gravity, Gauge Theories and Quantum Cosmology (Reidel, Dordrecht).Google Scholar
  19. 19.
    Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973).Gravitation (W. H. Freeman, San Francisco).Google Scholar
  20. 20.
    Ringwood, G. A. (1976).J. Phys. A: Math. Gen. 9, 1253.Google Scholar
  21. 21.
    Cheng, K. S. (1972).J. Math. Phys. 13, 1723.Google Scholar
  22. 22.
    Peleg, Y. (199?). Preprint, Brandeis University, BRX-TX-342.Google Scholar
  23. 23.
    Arfken, G. (1970).Mathematical Methods for Physicists (Academic, New York).Google Scholar
  24. 24.
    Gradshteyn, I. S., and Ryzhik, I. M. (1965).Table of Integral, Series, and Products (Academic, New York).Google Scholar
  25. 25.
    Merzbacher, E. (1970).Quantum Mechanics (Wiley, New York).Google Scholar
  26. 26.
    Guth, A. (1981).Phys. Rev. D 23, 347.Google Scholar
  27. 27.
    Kolb, E. W., and Turner, M. S. (1990).The Early Universe (Addison-Wesley, New York).Google Scholar
  28. 28.
    Linde, A. D. (1984).Sov. Phys. JETP 60, 211.Google Scholar
  29. 29.
    Vilenkin, A. (1984).Phys. Rev. D 30, 549.Google Scholar
  30. 30.
    Rubakov, V. A. (1984).Phys. Lett. 148B, 280.Google Scholar
  31. 31.
    Zel'dovich, Ya. B., and Starobinskii, A. A. (1984).Sov. Astron. Lett. 10, 135.Google Scholar
  32. 32.
    Albrecht, A., and Brandenberger, R. (1985).Phys. Rev. D 31, 1225.Google Scholar
  33. 33.
    Kung, J. H., and Brandenberger, R. (1989).Phys. Rev. D 40, 2532.Google Scholar
  34. 34.
    Kasner, E. (1921).Am. J. Math. 43, 217.Google Scholar
  35. 35.
    Schücking, E., and Heckmann, O. (1958). InOnzieme Conseil de Physique Solvay (Editions Stoops, Brussels).Google Scholar
  36. 36.
    Misner, C. W. (1969).Phys. Rev. Lett. 22, 1071.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • J. H. Kung
    • 1
  1. 1.Harvard-Smithsonian Center for AstrophysicsCambridgeUSA

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