Classical interactions in quantum field theory

TL;DR

The paper develops a formalism to constrain a field to propagate only through tree diagrams by introducing a linear Lagrange-multiplier term, enabling a reorganization of perturbation theory that preserves linear constraints. It derives Feynman rules with mixed $\lambda$- and $\phi$-propagators, and shows how a classical field can interact with quantum fields within a consistent framework, demonstrated in an $O(N)$ model with cubic couplings in $d=6-\epsilon$. One-loop renormalization-group analysis yields explicit $\beta$-functions and anomalous dimensions, revealing IR-stable fixed points for sufficiently large $N$ and demonstrating radiative symmetry breaking with dimensional transmutation and a metastable Coleman–Weinberg-type vacuum. The results generalize to multiple fields and suggest connections to classical-quantum hybrids in measurement contexts or emergent interactions, while offering a structured path to explore classical limits within quantum field theory and their physical implications.

Abstract

I review the formalism, Feynman rules, and combinatorics that constrain a field to propagate ``classically", strictly in tree diagrams, either by itself, or interacting with other, purely quantum fields. The perturbation theory is reorganized by virtue of the linear terms that introduce the constraints via Lagrange multipliers, generalizing and giving results that cannot be obtained with the standard procedures which start at the quadratic terms. I apply the formalism to a theory of an $O(N)$-symmetric quantum field interacting with a ``classical" scalar field via cubic interactions in six spacetime dimensions. Using the renormalization group, I examine the effective potential, symmetry breaking with radiative corrections, the fixed points in $d=6-ε$ dimensions, and compare with other works. Other possible generalizations and applications of the formalism are also discussed.

Figures (4)

• Figure 1: The top diagram is one of the many with the same line structure that can be formed with the Feynman rules described in the text. An odd number, with alternating signs, can be formed, and their sum gives the bottom tree diagram of the "classical" theory (solid lines denote $\phi$ and wavy lines denote $\lambda$).
• Figure 2: A diagram of the interacting quantum-classical theory, with the classical field propagating in tree diagrams, with arbitrary loops of the quantum field (wavy lines). The classical tree skeleton is formed with summations of mixed propagators as in Fig. 1.
• Figure 3: The signs and zeros for the beta functions, for $N=80$.
• Figure 4: The signs and zeros of the beta functions for $N=40$.