Identify NN-FTs that satisfy the Osterwalder–Schrader axioms

Determine which neural network field theory representations—specified by an architecture φ_θ and a parameter measure P(θ) that generate Euclidean Green’s functions—satisfy the Osterwalder–Schrader axioms (including reflection positivity) in dimensions d ≥ 2, thereby ensuring analytic continuation to unitary Lorentzian quantum field theories.

Background

The paper proves a universality theorem showing that any Euclidean quantum field theory, modeled as a probability distribution on tempered distributions S'(R^d), admits a neural network representation with countably many parameters (or formally with a single parameter). This provides existence but not physicality: Euclidean QFTs must satisfy the Osterwalder–Schrader (OS) axioms to guarantee analytic continuation to a unitary Lorentzian theory.

The authors note that in one dimension (quantum mechanics), mechanisms to engineer reflection positivity—one of the most subtle OS axioms—are known within neural network frameworks. However, they highlight that extending such constructions and verifying which higher-dimensional NN-FTs actually obey the OS axioms remains unresolved. Clarifying this would delineate the physically admissible subset of NN-FTs among the broad class guaranteed by the universality result.