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Abstract

We examine the discrete Ponzano-Regge formulation of (2+1)-dimensional gravity in the context of a consistent histories approach to quantum cosmology. We consider 2- dimensional boundaries of a 3-dimensional spacetime. The 2-dimensional boundaries are tessellated as the surface of a single tetrahedron. Two classes of the tetrahedral tessellation are considered—the completely isotropic tetrahedron and the two-parameter anisotropic tetrahedron. Using Ponzano-Regge wavefunctions, we calculate expectation values and uncertainties for the edge lengths of these tetrahedra. In doing so, we observe finite size effects in the expectation values and uncertainties when the calculations fail to constrain the space of histories accessible to the system. There is, however, an indication that the geometries of the tetrahedra (as quantified by the ratios of their edge lengths) freeze out with increasing cutoff. Conversely, cutoff invariance is observed in our calculations provided the space of histories is constrained by an appropriate fixing of the tetrahedral edge lengths. It is thus suggested that physically meaningful results regarding the early state of our universe can be obtained providing we formulate the problem in a careful manner. A few of the difficulties inherent in quantum cosmology are thereby addressed in this study of an exactly calculable theory.