Decoherence in the density matrix describing the quantum 3-geometries and the emergence of classical spacetime

Abstract

We construct the quantum gravitational density matrix ρ(gαβ,gαβ’) for compact three-geometries by integrating out a set of unobserved matter degrees of freedom from a solution to the Wheeler-DeWitt equation Ψ[gαβ,qk(matter)]. In the adiabatic approximation, ρ can be expressed as exp(-l2) where l2(gαβ,gαβ’) 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.