A mathematical framework for the completely positive semigroup coupling bet
ween classical and quantum systems is proposed. The coupling ensures a flow
of information from the quantum system to the classical one and the influe
nce of the classics on the dynamics of the quantum system in a dissipative
way. The classical evolution on average is modified by the expectation valu
e of some quantum operator. Examples of a classical particle moving along a
geodesic line in a curved space interacting with the quantum system, and t
he coupling of a two state quantum system to all pure states, are discussed
. (C) 1999 American Institute of Physics. [S0022-2488(99)00803-8].