# Quantum Cosmology — The Story So Far

Chapter

## Abstract

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.

## Keywords

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

- 1.L. Smolin (1979), What is the problem of quantum gravity?, preprint based on part 1 of the thesis (Harvard University).Google Scholar
- 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.E.g. G.’ t Hooft (1973),
*Nucl. Phys*.**B62**, 444; S. Deser and P. Van Nieuwenhuizen (1974),*Phys. Rev*.**D10**, 401, 411.ADSCrossRefGoogle Scholar - 4.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 - 5.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 - 6.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.For classical and quantum description of homogeneous cosmologies, see M. Ryan,
*Hamiltonian Cosmology*(Springer, 1972); M. A. H. MacCallum in*General Relativity*—*An Einstein Survey*(Cambridge, 1979); C. W. Misner in*Magic without Magic*(eds. J. Klauder; Freeman, 1972).Google Scholar - 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.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.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.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.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.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.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.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.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.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.This prescription is discussed for, e.g., in ref. [16].Google Scholar
- 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 - 20.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.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 - 22.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 - 23.
- 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.
- 26.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.See for, e.g., the papers in ref. [14].Google Scholar
- 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.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 - 30.T. Padmanabhan (1983),
*Gen. Rel. Grav*.**15**, 435;*Int. J. Theo. Phys*. (1983),**22**, 1023.MathSciNetADSCrossRefGoogle Scholar - 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.
- 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.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.T. Padmanabhan (1985),
*Ann. Phys*.**165**, 38(1985);*Current Science*(1985),**54**, 912.MathSciNetADSCrossRefGoogle Scholar - 36.J. Hartle and S. W. Hawking (1983),
*Phys. Rev*.**D28**, 2960; S. W. Hawking (1984),*Nucl. Phys*.**B239**, 257.MathSciNetADSGoogle Scholar - 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.

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© Kluwer Academic Publishers 1989