Quantum-classical interactions through the path integral

Abstract

I consider the case of two interacting scalar fields, $\ensuremath{\phi}$ and $\ensuremath{\psi}$, 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 semiclassical 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. I argue that the theory is renormalizable and equivalent to the usual Yang-Mills theory as far as the gauge field terms are concerned. There are additional terms in the effective action that depend on the Lagrange multiplier field $\ensuremath{\lambda}$ that is used to enforce the constraint. These terms and their relation to the confining properties of the theory are discussed.