On L\'{e}on van Hove's 1952 article on the foundations of Quantum Field Theory

TL;DR

This paper revisits Léon van Hove's 1952 work on the foundational issues of Quantum Field Theory, emphasizing divergences, inequivalent representations, and limitations of renormalization, providing modern context and detailed analysis.

Contribution

It offers a detailed reinterpretation of van Hove's original ideas, expanding on his arguments and contrasting them with current QFT approaches.

Findings

Divergences linked to inequivalent Hilbert space representations.

Renormalization fails to describe quantum field time evolution.

Van Hove's insights remain relevant for foundational understanding.

Abstract

In 1952, L\'{e}on van Hove published an article, in French, with the title ``Les difficult\'{e}s de divergences pour um mod\`{e}le particulier de champ quantifi\'{e}''. The article is frequently cited in relation to Haag's theorem and to the issue of the existence of unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. Summarizing in brief, it suggests a link between the appearance of divergences in perturbative Quantum Field Theory and the fact that quantum states belonging to an interacting theory do not belong to the same Hilbert space of the free theory. It also suggests that renormalization fails to provide an accurate description of the time evolution of the quantum field, although it correctly accounts for the S matrix. Due to its relevance, and to the difficulty of finding an English translation, the ideas contained in this article are proposed again here, expanded with derivations and accompanied by a discussion aimed at putting the analysis into context. We highlight the main points from the perspective of a contemporary reader, and underline the differences with the standard approach usually taught in curricular courses in Quantum Field Theory.