REGGE CALCULUS IN ANISOTROPIC QUANTUM COSMOLOGY

We investigate the no-boundary path integral prescription in a Regge calculus model in three spacetime dimensions. The simplicial manifold is topologically the 'doughnut' DBAR x S1, where DBAR is the two-dimensional closed disc. The discretization and the edge lengths are chosen symmetrically so that the boundary 2-metric is completely described by two independent edge lengths, and the interior 3-metric is completely described by these plus two further independent edge lengths. We pass to a limit in which the discretization on the boundary becomes infinitely dense. The classical solutions in the resulting semi-continuum model are in qualitative agreement with the solutions to the continuum Einstein equations with the same boundary data. Convergent contours for the no-boundary integral in the semi-continuum model are sought in analogy with the conformal rotation prescription of Gibbons et al and using steepest-descent techniques to analyse the contours for the conformal factor. A convergent prescription is found in the case of a vanishing cosmological constant, and the resulting wavefunction is shown to have the expected semiclassical behaviour. Possible reasons for our not having found a similar prescription with a non-vanishing cosmological constant are discussed.