Quantum Cosmology — The Story So Far

  • T. Padmanabhan
Part of the Fundamental Theories of Physics book series (FTPH, volume 29)


While commenting about the state of quantum gravity, Lee Smolin has said in 1979:1 “... while there has been a lot of interesting and imaginative work ... nothing which could be definitely called progress has been accomplished in this time — say, two decades ...”. In the course of this chapter, I will try to convince you that the situation remains just as bad today.


Quantum Gravity Schrodinger Equation Quantum Fluctuation Quantum Cosmology Stable Ground State 
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Notes and References

  1. 1.
    L. Smolin (1979), What is the problem of quantum gravity?, preprint based on part 1 of the thesis (Harvard University).Google Scholar
  2. 2.
    These questions, of course, are discussed in literature. Some good reviews of quantum gravity are by C. J. Isham, in Quantum Gravity I & II (eds. C. J. Isham, R. Penrose and D. W. Sciama; C.U.P., 1975,1980); J. A. Wheeler in Relativity, Groups and Topology (eds. B. S. De Witt and C. De Witt; Gordon and Breach, 1964); J. Hartle in The Very Early Universe (eds. G. W. Gibbons, S. W. Hawking, S. T. C. Silkos; Cambridge, 1983) and the various articles in Quantum Theory of Gravity (ed. S. Christensen, Adam Hilger, 1984).Google Scholar
  3. 3.
    E.g. G.’ t Hooft (1973), Nucl. Phys. B62, 444; S. Deser and P. Van Nieuwenhuizen (1974), Phys. Rev. D10, 401, 411.ADSCrossRefGoogle Scholar
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    T. Padmanabhan, T. R. Seshadri and T. P. Singh (1985) J. Mod. Phys. (to appear); K. Eppeley and E. Hannah (1977), Found. Phys. 7, 51; D. N. Page and C. D. Geilker (1981), Phys. Rev. Letts. 47, 979.Google Scholar
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    There are many good reviews on these subjects; for example, see P. Van Nieuwenhuizen (1983) Relativity, Groups and Topology, II’ (eds. B. S. De Witt and R. Stora; North-Holland); J. Scherk (1975), Rev. Mod. Phys. 47, 123; J. H. Schwarz (1982), Phys. Rep. 89, 223.Google Scholar
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    Some limited application of nonperturbative techniques in quantum gravity can be found in T. E. Tomboulis, Phys. Letts. 70B(1977), 361; 97B (1980), 77; L. Smolin, Nucl. Phys. B208(1982), 439; S. Weinberg (1979) in General Relativity — An Einstein Centenary Survey (eds. S. W. Hawking and W. Israel; Cambridge).Google Scholar
  7. 7.
    For classical and quantum description of homogeneous cosmologies, see M. Ryan, Hamiltonian Cosmology (Springer, 1972); M. A. H. MacCallum in General RelativityAn Einstein Survey (Cambridge, 1979); C. W. Misner in Magic without Magic (eds. J. Klauder; Freeman, 1972).Google Scholar
  8. 8.
    For detailed discussion of ‘minisuperspace’, see C. W. Misner, op. cit (ref. [7]) and B. S. De Witt (1967) Phys. Rev. 162, 1195.Google Scholar
  9. 9.
    This approach was initiated by J. V. Narlikar in (1979), M. N. Roy. Astron. Soc. 183, 159; Gen. Rel. Grav. (1979), 10, 883 and was developed further by the author; T. Padmanabhan (1982), PhD thesis.ADSGoogle Scholar
  10. 10.
    It is easy to show that null geodesies remain null geodesies under conformal transformations, making the light cones conformally invariant. The converse is somewhat more difficult to prove but can be done.Google Scholar
  11. 11.
    Coordinate transformations expressing k = ± 1 FRW Universes in the conformally flat form are given in, e.g., F. Hoyle and J. V. Narlikar (1974) Action at a Distance in Physics and Cosmology (Freeman).Google Scholar
  12. 12.
    A down-to-earth discussion of path integrals can be found in R. P. Feynman and A. R. Gibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965). A flavour of more advanced topics can be found in L. S. Shulman, Techniques and Applications of Path Integration (Wiley, 1981).Google Scholar
  13. 13.
    For many of the details in this section and the next two, see J. V. Narlikar and T. Padmanabhan (1983) Phys. Repts. 100, 152 and T. Padmanabhan (1983) Phys. Rev. D28, 745.MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    An excellent discussion of this and related issues can be found in K. Kuchar, Canonical quantisation of gravity, in Relativity, Astrophysics and Cosmology (ed. W. Israel; Reidel, 1973); also see J. Hartle and K. Kuchar in Quantum Theory of Gravity, op. cit. (ref. [2]). The problem of separating temporal information from dynamical information is stressed in Kuchar’s article. For a more recent discussion, see T. Banks (1985), Nucl. Phys. B249, 332.Google Scholar
  15. 15.
    The ‘usual’ conclusion drawn is that “time is a semi-classical concept which cannot be extended... into the domain of quantum gravity”. See, e.g., T. Banks op. cit. [14], p. 336.Google Scholar
  16. 16.
    For previous discussion of this topic, see Hawking’s contribution in General Relativity — An Einstein Survey, op. cit. (e.g. ref. [6]).Google Scholar
  17. 17.
    This should be clear from the Schrödinger approach to field quantization described in Appendix 1. Also see J. Greensite and M. B. Halpern, Nucl. Phys. B242 (1984) 167.MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    This prescription is discussed for, e.g., in ref. [16].Google Scholar
  19. 19.
    The Wheeler-De Witt equation is discussed in many places. Particularly useful articles are the ones by Wheeler (1964; cited in ref. [2]), Kuchar (cited in ref. [14]) and D. R. Brill and R. H. Gowdy, Rep. Prog. Phys. (1970), 33, 413.ADSCrossRefGoogle Scholar
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    The role of observer in quantum cosmology is not clear. See, e.g., W. Patton and J. A. Wheeler in Quantum Gravity I, op. cit. (ref. [2]); T. Banks, op. cit. (ref. [14]).Google Scholar
  21. 21.
    D. R. Brill and R. H. Gowdy, Rep. Prog. Phys. (1970), 33, 413; C. W. Misner, op. cit. (ref. [7]); K. Kuchar, op. cit. (ref. [14]).ADSCrossRefGoogle Scholar
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    S. Fulling (1973), Phys. Rev. D7, 2850; P. C. W. Davies (1975), J. Phys. 8, 365; W. G. Unruh (1976), Phys. Rev. D14, 870; T. Padmanabhan (1982), Ap. Sp. Sci. 83, 247; Class. Q. Grav. (1984), 2, 117.ADSGoogle Scholar
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    See papers cited in ref. [13], and Phys. Letts. (1982), 87A, 226; (1983) 15, 435.Google Scholar
  24. 24.
    The comparison between the singular behaviour of hydrogen atom and the universe was first suggested by Wheeler. It is explored systematically in papers cited above (ref. [23]).Google Scholar
  25. 25.
    See, e.g., J. V. Narlikar (1979), M. N. Roy. Astron. Soc. 183, 159.ADSGoogle Scholar
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    J. V. Narlikar (1984), Found. Phys. 14, 443; J. V. Narlikar (1981), Found. Phys., 11, 473; T. Padmanabhan and J. V. Narlikar (1982), Nature 295, 677.ADSCrossRefGoogle Scholar
  27. 27.
    See for, e.g., the papers in ref. [14].Google Scholar
  28. 28.
    The application of QSG is not limited to FRW models; for the application of QSG to other cases, see T. Padmanabhan (1982), Gen. Rel. Grav. 14, 549; Int. J. Theo. Phys. (1983), 22,1023 and Class. Q. Grav. (1984), 1, 149.MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    This semiclassical approximation is developed in T. Padmanabhan (1983), Phys. Rev. D28, 745; Detailed application to cosmology can be found in T. Padmanabhan (1983), Phys. Rev. D28, 756.MathSciNetADSGoogle Scholar
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    T. Padmanabhan (1983), Gen. Rel. Grav. 15, 435; Int. J. Theo. Phys. (1983), 22, 1023.MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    The analysis here is based on T. Padmanabhan (1983) Phys. Letts. 93A, 116. Creating the universe is a very popular pastime of quantum cosmologists, see, e.g., R. Brout et. al., Nucl. Phys. (1980), 170, 228; L. Lindley, Nature (1981), 291, 391; A. Vilenkin (1983), Phys. Rev. D27, 2848.ADSCrossRefGoogle Scholar
  32. 32.
    T. Padmanabhan (1984), Phys. Letts. 104A, 196.MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    This is similar to the vacuum functional in electromagnetism and is derived in T. Padmanabhan (1983) Phys. Letts. 96A, 110. For a detailed discussion of Equation (105), see Gen. Rel. Grav. (1985), 17, 215.ADSCrossRefGoogle Scholar
  34. 34.
    E.g., see B. De Witt, Phys. Rev. Letts. (1964), 13, 114; C. J. Isham, A. Salam, and J. Strathdee (1971), Phys. Rev. D3, 1805.ADSCrossRefGoogle Scholar
  35. 35.
    T. Padmanabhan (1985), Ann. Phys. 165, 38(1985); Current Science (1985), 54, 912.MathSciNetADSCrossRefGoogle Scholar
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    J. Hartle and S. W. Hawking (1983), Phys. Rev. D28, 2960; S. W. Hawking (1984), Nucl. Phys. B239, 257.MathSciNetADSGoogle Scholar
  37. 37.
    For example, the ‘wave function of the inflationary Universe’ can be obtained as a E = 0 solution of GWD equation for the action in Equation (95). T. Padmanabhan (1985), TIFR preprint.Google Scholar
  38. 38.
    E. Baum, Phys. Letts. 133B (1983), 185.MathSciNetADSGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • T. Padmanabhan
    • 1
  1. 1.Theoretical Astrophysics GroupTata Institute of Fundamental ResearchBombayIndia

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