YITP-96-41 gr-qc/9609052 Partition Function for (2+1)-Dimensional Einstein Gravity

Taking (2+1)-dimensional pure Einstein gravity for arbitrary genus g 1 as a model, we investigate the relation between the partition function formally de ned on the entire phase space and the one written in terms of the reduced phase space. The case of g = 1 (torus) is analyzed in detail and it provides us with good lessons for quantum cosmology. We formulate the gaugexing conditions in a form suitable for our purpose. Then the gaugexing procedure is applied to the partition function Z for (2+1)-dimensional gravity, formally de ned on the entire phase space. We show that basically it reduces to a partition function de ned for the reduced system, whose dynamical variables are ( ; pA). [Here the 's are the Teichm uller parameters, and the pA's are their conjugate momenta.] As for the case of g = 1, we nd out that Z is also related with another reduced form, whose dynamical variables are not only ( ; pA), but also (V; ). [Here is a conjugate momentum to the 2-volume (area) V of a spatial section.] A nontrivial factor appears in the measure in terms of this type of reduced form. This factor is understood as a FaddeevPopov determinant associated with the time-reparameterization invariance inherent in this type of formulation. In this manner, the relation between two reduced formulations becomes transparent in the context of quantum theory. As another result for the case of g = 1, one factor originating from the zero-modes of a di erential operator P1 can appear in the path-integral measure in the reduced representation of Z. It depends on how to de ne the path-integral domain for the shift vectors Na in Z: If it is de ned to include kerP1, the nontrivial factor does not appear. On the other hand,