  A number of physical systems exhibit a particular form of asymptotic
conformal invariance: within a particular range of distances, they are
characterized by a long-range conformal interaction (inverse square potential),
the absence of dimensional scales, and an SO(2,1) symmetry algebra. Examples
from molecular physics to black holes are provided and discussed within a
unified treatment. When such systems are physically realized in the appropriate
strong-coupling regime,the occurrence of quantum symmetry breaking is possible.
This anomaly is revealed by the failure of the symmetry generators to close the
algebra in a manner shown to be independent of the renormalization procedure.
