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Title: On the semi-classical approximation to the wave function of the universe and its stochastic interpretation
Authors: Pollock, M.D.
Issue Date: Sep-1988
Publisher: Elsevier B.V.
Citation: Nuclear Physics B, 1988, Vol.306, p931-945
Abstract: In quantum cosmology, a wave function ψ for a given theory can be obtained by solving the Wheeler-DeWitt equation, using the semi-classical approximation to the path integral over euclidean metrics to impose the boundary condition, as described by Hawking and his collaborators. If the universe is expanding as a quasi-de Sitter space-time, then it is possible to derive a Fokker-Planck equation for the probability distribution P, as shown by Starobinsky. Arguing by analogy with quantum mechanics in flat space-time, one would expect that P ∝ ψψ*. We examine this assertion by reference to the scale-invariant theory Image, whose wave function has been calculated in mini-superspace by Horowitz, and those classical solutions are de Sitter space-times. It appears that deviations from the relation P ∝ ψψ* are attributable to long-wavelength fluctuations δφe ≈ H/2π in the effective inflation field φe = √βR = √12β H. Their existence is taken into account in the derivation of the Fokker-Planck equation, but not in the derivation of ψ when this is restricted to mini-superspace. In the limit β → ∞, we find that δφe/φe → 0 and that P ∝ ψψ*. The scale invariant theory Image) can be similarly analyzed. Inclusion of a kinetic term Image destroys this similarly, which is restored however upon addition of a term Image.
Description: Restricted Access.
ISSN: 0550-3213
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Copyright: 1988 Elsevier B.V.
Appears in Collections:Research Papers (TP)

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