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(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.