A unified treatment of dynamics and scattering in classical and quantum statistical mechanics

The common formal features of classical and quantum statistical mechanics are investigated at three separate levels: at the level of L2 spaces of wave-packets on Γ-space, of Liouville spaces B2 consisting of density operators constructed from such wave-packets, and of phase-space representation spaces P of Γ-distribution functions. It is shown that at the last level the formal similarities become so outstanding that all key quantities in P-spaces, such as Liouville operators, Hamilton functions, position and momentum observables, etc., are represented by expressions which to the zeroth order in ħ coincide in the classical and quantum case, and in some instances coincide completely. Scattering theory on the B2 Liouville spaces takes on the same formal appearance for classical and quantum statistical mechanics, and to the zeroth order in ħ it coincides in both cases. This makes possible the formulation of a classical approximation to quantum scattering, and of a computational scheme for determining ϱout from ϱin for successive orders of ħ.