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Exact quantum states for the diagonal Bianchi type IX model with negative cosmological constant

Robert Paternoga·Robert Graham
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Abstract

Quantum states of the diagonal Bianchi type IX model with negative cosmological constant $Λ$ are obtained by transforming the Chern-Simons solution in Ashtekar's variables to the metric representation. We apply our method developed earlier for $Λ>0$ and obtain five linearly independent solutions by using the complete set of topologically inequivalent integration contours in the required generalized Fourier-transformation. A caustic in minisuperspace separates two Euclidean regimes at small and large values of the scale parameter from a single classically interpretable Lorentzian regime in between, corresponding to the fact that classically these model-Universes recollapse. Just one particular solution out of the five we find gives a normalizable probability distribution on both branches of the caustic. However, in contrast to the case of positive cosmological constant, this particular solution neither satisfies the semi-classical no-boundary condition, nor does the special initial condition it picks out for $\hbar \to 0$ evolve into a classically interpretable Universe.

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Robert Paternoga, Robert Graham. 1996-05-17. Exact quantum states for the diagonal Bianchi type IX model with negative cosmological constant. https://doi.org/10.1103/physrevd.54.4805

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