Necessity and sufficiency of current Osterwalder–Schrader axioms for reconstruction

Establish whether the presently used versions of the Osterwalder–Schrader (OS) axioms provide necessary and sufficient conditions for the reconstruction of Wightman quantum field theories from Euclidean data, and, if they do not, develop and formalize a version of the OS axioms that is both provable within Lean/Mathlib and necessary and sufficient for the reconstruction theorem.

Background

The paper formalizes the free massive scalar field in the Glimm–Jaffe (measure-theoretic) Euclidean framework and discusses next steps, including formalizing the OS reconstruction theorem that connects Euclidean and Minkowski formulations.

The authors note that, ideally, one would use an axiom system both provable in their formal environment and strong enough to guarantee reconstruction via necessary and sufficient conditions. However, they indicate that it is not completely clear whether the currently used OS axioms meet this standard, pointing to recent discussion questioning this point.

Clarifying this would align the formalization program with a foundational bridge between Euclidean correlation data and the Wightman axioms in Minkowski signature.

References

"Ideally this would use a version of the OS axioms which are both provable and state necessary and sufficient conditions for reconstruction, but this is not completely clear for the current axioms (a recent discussion is )."

— Formalization of QFT  (2603.15770 - Douglas et al., 16 Mar 2026) in Going forward, first paragraph (Section 6)

As we will address in the main text, this definition can be elevated to the level of a theorem in the case that $\mathcal{G}$ is a discrete groupoid. In the more general case in which $\mathcal{G}$ may be continuous, this definition should be regarded as a conjecture whose validity rests on an open problem in reconstruction for symmetric multitensor categories Sec. 10.3.

— Toward a Unique Filter for the Gravitational Path Integral  (2609.20697 - Klinger, 17 Sep 2026) in Definition 2 (McNamara-Wang Filter), Introduction