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Prog. Theor. Phys. Vol. 29 No. 2 (1963) pp. 296-320

[ Full Text PDF : FREE ACCESS (1407K) ]

Quantization of the Gravitational Field in the Expanding Universe

Hidekazu Nariai and Toshiei Kimura

Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima

(Received July 12, 1962)

Abstract:

An attempt is made to quantize, in the linear approximation, the gravitational field in the expanding universe, from the standpoint of our canonical formalism worked out in a previous paper. By using a transformation function and decomposing suitably the field operator into positive and negative frequency parts, explicit forms of the four-dimensional commutation function and the propagation function are derived in the case of three special world models. There appears the difficulty of divergence term in the propagation function, with the exception of one model. Our propagation function, however, satisfies the relation between Hadamard's elementary solution and the propagation function postulated by De Witt and Brehme. In addition, a connection between the propagation function and the time-ordered product of field operators is established by introducing a state of quasi-vacuum in parallel with our decomposition mentioned above. Finally, detailed discussions of several difficulties are given in connection with the problem of causality. It is suggested from our analysis that the quantum theory and the general relativity cannot rigorously be compatible, so far as we stand on the hypothesis of the expanding universe.


URL : http://ptp.ipap.jp/link?PTP/29/296/
DOI : 10.1143/PTP.29.296

[ Full Text PDF : FREE ACCESS (1407K) ] Citation:

References:

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Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 29 No. 6 (1963) pp. 915-932 :
    Electrodynamics in the Expanding Universe
    Hidekazu Nariai and Toshiei Kimura
  2. Progress of Theoretical Physics Vol. 31 No. 6 (1964) pp. 1138-1152 :
    Quantum Theory of Fields in an Expanding Space. I
    Hidekazu Nariai and Toshiei Kimura
  3. Progress of Theoretical Physics Vol. 37 No. 5 (1967) pp. 831-846 :
    Non-Linear Theory of Gravitational Instability in the Expanding Universe
    Kenji Tomita
  4. Progress of Theoretical Physics Vol. 40 No. 1 (1968) pp. 49-59 :
    On the Green's Function in an Expanding Universe and Its Role in the Problem of Mach's Principle
    Hidekazu Nariai
  5. Progress of Theoretical Physics Vol. 40 No. 3 (1968) pp. 657-667 :
    The Behavior of a Massless Scalar Field in a Closed Universe
    Hidekazu Nariai
  6. Progress of Theoretical Physics Vol. 46 No. 2 (1971) pp. 433-438 :
    On the Removal of Initial Singularity in a Big-Bang Universe in Terms of a Renormalized Theory of Gravitation. I
    Hidekazu Nariai
  7. Progress of Theoretical Physics Vol. 47 No. 6 (1972) pp. 1824-1843 :
    Hamiltonian Approach to the Dynamics of Expanding Homogeneous Universes in the Brans-Dicke Cosmology
    Hidekazu Nariai
  8. Progress of Theoretical Physics Vol. 50 No. 2 (1973) pp. 472-487 :
    On the Generatin of Sound from the Primordial Turbulence in an Expanding Universe
    Hidekazu Nariai
  9. Progress of Theoretical Physics Vol. 51 No. 5 (1974) pp. 1368-1376 :
    On the Generation of Sound from the Primordial Turbulence in an Expanding Universe. II
    Hidekazu Nariai
  10. Progress of Theoretical Physics Vol. 55 No. 4 (1976) pp. 1116-1132 :
    Propagators for a Scalar Field in a Homogeneous Expanding Universe. I
    Hidekazu Nariai and Kenji Tanabe
  11. Progress of Theoretical Physics Vol. 57 No. 1 (1977) pp. 67-81 :
    Propagators for a Scalar Field in Some Bianchi-Type I Universe
    Hidekazu Nariai
  12. Progress of Theoretical Physics Vol. 58 No. 2 (1977) pp. 560-574 :
    On a Quantized Scalar Field in Some Bianchi-Type I Universe
    Hidekazu Nariai
  13. Progress of Theoretical Physics Vol. 59 No. 5 (1978) pp. 1532-1542 :
    On the Creation of Scalar Particles in Some Anisotropic Universe
    Hidekazu Nariai
  14. Progress of Theoretical Physics Vol. 60 No. 3 (1978) pp. 739-746 :
    On a Quantized Scalar Field in an Isotropic Closed Universe
    Hidekazu Nariai
  15. Progress of Theoretical Physics Vol. 64 No. 4 (1980) pp. 1280-1293 :
    On the Definition of a Positive Frequency Part for a Quantized Scalar Field in Several Robertson-Walker Universes
    Hidekazu Nariai and Takehiro Azuma

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