Formalization of QFT

TL;DR

This paper formalizes a key result in constructive quantum field theory within the Lean 4 theorem prover, demonstrating the potential of AI-assisted formalization for mathematical physics.

Contribution

It provides the first machine-checked formalization of the free bosonic quantum field theory satisfying the Glimm-Jaffe axioms in Lean 4.

Findings

Formalization achieved in Lean 4 with minimal assumptions

Demonstrated feasibility of AI-assisted formalization in mathematical physics

Future potential for machine-verified proofs in theoretical physics

Abstract

A foundational result in constructive quantum field theory is the construction of the free bosonic quantum field theory in four-dimensional Euclidean spacetime and the proof that it satisfies the Glimm-Jaffe axioms, a variant of the Osterwalder-Schrader axioms. We present a formalization of this result in the Lean 4 interactive theorem prover. The project is intended as a proof of concept that extended arguments in mathematical physics can be translated into machine-checked proofs using existing AI tools. We begin by introducing interactive theorem proving and constructive quantum field theory, then describe our formalization and the design decisions that shaped it. We also explain the methods we used, including coding assistants, and conclude by considering how AI assisted formalization may influence the future of theoretical physics. Our original release assumed three results, Minlos' theorem, the nuclear property of Schwartz space, and Goursat's theorem. In subsequent releases from our group and from contributors from the Lean community, these assumptions have been proven (or avoided), so that the OS/GJ axioms are now proven using only Lean and its library Mathlib.