Quantum-classical interactions through the path integral

I consider the case of two interacting scalar fields, $\varphi$ and $\psi$, 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 $\lambda$ that is used to enforce the constraint. These terms and their relation to the confining properties of the theory are discussed.

Figure 1

An example of a diagram used for the calculation of $U(\lambda )$ and the derivation of Eqs. (21, 22). The wiggly lines denote $A_i$, $A_j$ fields. The curly lines denote the $\lambda$ field and the solid lines denote $A_0$. The solid lines with a dot denote the modified $A_0-A_0$ propagator, ${\tilde{G}}_{00}$, derived in the text, that does not contain the Coulomb interaction; the $\lambda -A_0$ and $\lambda -\lambda$ lines that carry the Coulomb propagator cannot be also added in its place, because the vertices are only linear in $\lambda$.