Snowmass White Paper: The Quest to Define QFT

TL;DR

This review discusses the long-standing challenge of rigorously defining Quantum Field Theory, highlighting the gaps in current axiomatic systems and emphasizing the importance of resolving this foundational issue for physics and mathematics.

Contribution

It provides a comprehensive overview of seven decades of research on QFT definitions, identifying the limitations of existing axiomatic frameworks and underscoring the open problem of a complete formulation.

Findings

Existing QFT axioms do not cover all physical situations

No current system fully captures the scope of QFT

Formulating a complete QFT definition remains an open problem

Abstract

This article provides a review of the literature on rigorous definitions and constructions in Quantum Field Theory, spanning the period of seven decades. Comparing with the ideas and constructions found in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians.