Reconstruction of Quantum Yang–Mills Theory from Euclidean Schwinger Functions

Abstract

This paper presents a mathematically controlled research program for Phase Three of the proposed construction of four-dimensional quantum Yang–Mills theory: the reconstruction of the physical Hilbert space, the local observable algebra, and the Hamiltonian from Euclidean Schwinger functions. The formulation begins with a compact gauge group G, a lattice gaugefield represented by group-valued parallel transports, and a gauge-invariant Euclidean measure generated by the Wilson action. The central objective is to establish a continuum family of Schwinger functions satisfying Euclidean covariance, symmetry, regularity, clustering, and Osterwalder–Schrader reflection positivity.Reflection positivity provides the fundamental bridge between Euclidean correlation functions and a physical Hilbert space with positive norm. The quotient-completion procedure then yields the Hilbert space H, while Euclidean time translations induce a positive contraction semigroup. Its generator defines the Hamiltonian H, satisfying H ≥ 0. Gauge-invariant local observables are reconstructed as operators or quadratic forms on H, and the spectral behavior of their correlation functions determines the physical excitation spectrum. In particular,exponential decay of connected Euclidean correlation functions implies a spectral gap above the vacuum.The proposed framework does not claim that the four-dimensional Yang–Mills existenceand mass-gap problem has already been solved. Instead, it specifies the analytical statements that must be proved, the order in which they should be addressed, and the numerical and functional methods that can support, but cannot replace, the required mathematical proofs.