ar X iv : h ep - t h / 06 07 21 0 v 2 2 6 Ju l 2 00 6 Quantum - classical interactions through the path integral

I consider the case of two interacting scalar fields, φ and ψ, and use the path integral formalism in order to treat the first classically and the second quantum-mechanically. I derive the Feynman rules and the resulting equation of motion for the classical field which should be an improvement of the usual semi-classical procedure. As an application, I use this method in order to enforce Gauss's law as a classical equation in a non-abelian gauge theory, and derive the corresponding Feynman rules.