Quantum-classical interactions through the path integral

TL;DR

This paper develops a path integral approach to treat one scalar field classically and another quantum-mechanically, deriving new Feynman rules and applying the method to enforce Gauss's law in non-abelian gauge theories.

Contribution

It introduces a novel path integral formalism for mixed classical-quantum interactions and applies it to non-abelian gauge theories, improving semi-classical methods.

Findings

The derived equations of motion improve semi-classical approximations.

The method enforces Gauss's law as a classical constraint in gauge theories.

The theory remains renormalizable and equivalent to standard Yang-Mills for gauge fields.

Abstract

I consider the case of two interacting scalar fields, \phi 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 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. I argue that the theory is renormalizable and equivalent to the usual Yang-Mills 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.