  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.
