  We show that a transformed Caldeira-Leggett Hamltonian has two distinct
families of fixed points, rather than a single unique fixed point as often
conjectured based on its connection to the anisotropic Kondo model. The two
families are distinguished by a sharp qualitative difference in their quantum
coherence properties and we argue that this distinction is best understood as
the result of a transition in the model between degeneracy and non-degeneracy
in the spectral function of the ``spin-flip'' operator.
