QUANTUM DECOHERENCE IN A 4-DIMENSIONAL BLACK-HOLE BACKGROUND

We display a logarithmic divergence in the density matrix of a scalar field in the presence of an Einstein-Yang-Mills black hole in four dim ensions. This divergence is related to a previously-found logarithmic divergence in the entropy of the scalar field, which cannot be absorbe d into a renormalization of the Hawking-Bekenstein entropy of the blac k hole. Motivated by the fact that the cutoff in this divergence varie s as the latter decays, by an analysis of black holes in two-dimension al string models and by studies of D-brane dynamics in higher dimensio ns, we propose that the renormalization scale variable be identified w ith time. In this case, the logarithmic divergence we find induces a n on-commutator term delta rho in the quantum Liouville equation for the time evolution of the density matrix rho of the scalar field, leading to quantum decoherence. The order of magnitude of delta H is mu(2)/M rho, where mu is the mass of the scalar particle.