Abstract
Villars's equations, first derived from the adiabatic limit of time-dependent Hartree-Fock theory, are presented here as a set of exact conditions for the decoupling (in the adiabatic limit) of a sub-system from a classical Hamiltonian system. It is emphasized that these conditions do not incorporate all the required decoupling conditions. By combining the additional requirements with Villars's equations, we obtain a mathematically complete system, which will yield exact solutions where such exist, but which can also be applied to cases of approximate decoupling.
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