Partition function for (2+1)-dimensional Einstein gravity

Abstract

Taking (2+1)-dimensional pure Einstein gravity for arbitrary genus $g>~1$ as a model, we investigate the relation between the partition function formally defined 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 gauge-fixing conditions in a form suitable for our purpose. Then the gauge-fixing procedure is applied to the partition function $Z$ for (2+1)-dimensional gravity, formally defined on the entire phase space. We show that basically it reduces to a partition function defined for the reduced system, whose dynamical variables are $({\ensuremath{\tau}}^{A}{,p}_{A})$. (Here the ${\ensuremath{\tau}}^{A}$'s are the Teichm\"uller parameters and the ${p}_{A}$'s are their conjugate momenta.) As for the case of $g=1$, we find out that $Z$ is also related with another reduced form, whose dynamical variables are not only $({\ensuremath{\tau}}^{A}{,p}_{A})$, but also $(V,\ensuremath{\sigma})$. [Here $\ensuremath{\sigma}$ is a conjugate momentum to the two-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 Faddeev-Popov determinant associated with the time-reparametrization 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 differential operator ${P}_{1}$ can appear in the path-integral measure in the reduced representation of $Z$. It depends on how to define the path-integral domain for the shift vector ${N}_{a}$ in $Z$: If it is defined to include ker ${P}_{1}$, the nontrivial factor does not appear. On the other hand, if the integral domain is defined to exclude ker ${P}_{1}$, the factor appears in the measure. This factor can depend on the dynamical variables, typically as a function of $V$, and can influence the semiclassical dynamics of the (2+1)-dimensional spacetime. These results shall be significant from the viewpoint of quantum gravity and quantum cosmology.