On Haag's theorem and renormalization ambiguities

TL;DR

This paper examines Haag's theorem's implications for quantum field theory, highlighting its role as a no-go theorem that challenges the construction of a non-ambiguous, non-perturbative renormalization framework.

Contribution

It clarifies the impact of Haag's theorem on renormalization, especially its limitations for non-perturbative approaches in quantum field theory.

Findings

Perturbative renormalization diminishes Haag's theorem's impact for small couplings.

Non-perturbative renormalization cannot be constructed with free-field interaction references.

Haag's theorem acts as a no-go theorem for analytic continuation from perturbative to non-perturbative QFT.

Abstract

We revisit the implications of Haag's theorem in the light of the renormalization group. There is still some lack of discussion in the literature about the possible impact of the theorem on the standard (as opposite of axiomatic) quantum field theory, and we try to shed light in this direction. Our discussion then deals with the interplay between Haag's theorem and renormalization. While we clarify how perturbative renormalization (for the sub-class of interactions that are renormalizable) marginalizes the its impact when the coupling is formally small, we argue that a non-perturbative and non-ambiguous renormalization cannot be built if there is any reference to the interaction picture with free fields. In other words, Haag's theorem should be regarded as a no-go theorem for the existence of a non-ambiguous analytic continuation from perturbative to non-perturbative QFT.