Quantum decoherence in a four-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 dimensions. This divergence is related to a previously-found logarithmic divergence in the entropy of the scalar field, which cannot be absorbed into a renormalization of the Hawking-Bekenstein entropy of the black hole. As the latter decays, the logarithmic divergence induces a non-commutator term \nd{\delta H}\rho in the quantum Liouville equation for the density matrix \rho of the scalar field, leading to quantum decoherence. The order of magnitude of \nd{\delta H} is \mu^2/M_P, where \mu is the mass of the scalar particle.