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

We construct the quantum gravitational density matrix ρ(g_αβ,g_αβ^’) 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_αβ,q_k(matter)]. In the adiabatic approximation, ρ can be expressed as exp(-l²) where l²(g_αβ,g_αβ^’) 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’’ (l²≪1) interfere quantum mechanically; those which are ‘‘far apart’’ (l²≫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.