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

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

May 1989

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989phrvd..39.2924p&link_type=abstract

Physical Review D (Particles and Fields), Volume 39, Issue 10, 15 May 1989, pp.2924-2932

Physics

78

Scientific paper

We construct the quantum gravitational density matrix ρ(gαβ,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[gαβ,qk(matter)]. In the adiabatic approximation, ρ can be expressed as exp(-l2) where l2(gαβ,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.

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