Decoherence in the density matrix describing quantum three-geometries and the emergence of classical spacetime

We construct the quantum gravitational density matrix ρ($g_{\alpha \beta }$,$g_{\alpha \beta }^{\mathcal{’}}$) for compact three-geometries by integrating out a set of unobserved matter degrees of freedom from a solution to the Wheeler-DeWitt equation Ψ[$g_{\alpha \beta }$,$q_{k\mathrm{}\left(\mathrm{matter}\right)\mathrm{}]}$. In the adiabatic approximation, ρ can be expressed as exp(-$l^2$) where $l^2$($g_{\alpha \beta }$,$g_{\alpha \beta }^{\mathcal{’}}$) is a specific ‘‘distance’’ measure in the space of three-geometries. This measure depends on the volumes of the three-geometries and the eigenvalues of the Laplacian constructed from the three-metrics. The three-geometries which are ‘‘close together’’ ($l^2$≪1) interfere quantum mechanically; those which are ‘‘far apart’’ ($l^2$≫1) are suppressed exponentially and hence contribute decoherently to ρ. Such a suppression of ‘‘off-diagonal’’ elements in the density matrix signals classical behavior of the system. In particular, three-geometries which have the same intrinsic metric but differ in size contribute decoherently to the density matrix. This analysis provides a possible interpretation for the semiclassical limit of the wave function of the Universe.