The coupled quantum-classical description

THE COUPLED QUANTUM-CLASSICAL DESCRIPTION 2.1. Partial Wigner transform 

We consider to begin with an isolated system with a Hamiltonian operator H[t). Its state is given by the density operator r t) which satisfies the L-vN equation of motion. 

From this operator it is possible to obtain the quasiclassical phase density 

A partially Wigner transformed operator is similarly defined by Aw(P,P) = Id 5 exp(iP5/ ) P - S/2 A R + S/2) and physical properties are obtained from the trace as 

This can be further developed for given Hamiltonians. Here, we are interested in the dynamics of coupled quantum and quasiclassical variables that follows from a Hamiltonian operator 

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