Quantum cosmology of openR×${\mathit{S}}^2$×${\mathit{S}}^1$

We examine the classical and quantum cosmology of a Kantowski-Sachs spacetime manifold with a topology openR×${\mathit{S}}^2$×${\mathit{S}}^1$ and a nonzero cosmological constant, within the framework of canonical quantum gravity. The classical trajectories are analyzed, and it is shown that the classical problem can be reduced to that of free particles. Both the Hartle-Hawking ‘‘no-boundary’’ proposal and the Vilenkin ‘‘outgoing flux’’ proposal are examined. First, the Hartle-Hawking proposal is generalized to canonical quantum gravity by imposing generic initial conditions on the solutions to the Wheeler-DeWitt equation at small scale factors. The resulting wave function has essentially the same leading exponential behavior as the semiclassical approximation to the ‘‘no-boundary’’ Euclidean functional integral in the nonoscillatory region. It is further shown, by calculating overlap integrals with semiclassical wave functions, that the wave function diverges too quickly in this region to be interpreted probabilistically. An analogue of the Baum-Hawking factor appears, and is interpreted as indicating that, given a small Λ, the wave function is highly peaked around a class of configurations in which the ${\mathit{S}}^2$ radius is R=[1/(2G${\mathrm{\Lambda }}^{1/2}$)]. These configurations are classically unstable. Second, given a complete set of solutions to the Wheeler-DeWitt equation, the outgoing flux proposal is found to be insufficient to select a unique wave function. A source is found for the Wheeler-DeWitt current of the above solutions. No analogue of the Baum-Hawking factor appears in the outgoing flux model.