Yang–Mills and Mass Gap proof

Title: Clay Institute Version: Existence of Quantum Yang–Mills Theory and Mass Gap on ℝ⁴

Description:This paper presents a rigorous mathematical proof of the existence of a non-trivial quantum Yang–Mills theory on four-dimensional Euclidean space ℝ⁴ with a compact gauge group, and establishes the existence of a strictly positive mass gap. The analysis is conducted within the Osterwalder–Schrader framework, using reflection positivity, Euclidean invariance, and regularity of the Schwinger functions to construct a Wightman quantum field theory.

The spectral analysis of the reconstructed Hilbert space shows that the vacuum is isolated and that the spectrum of the Hamiltonian has a positive lower bound, confirming the existence of a mass gap. The approach utilizes lattice regularization, functional integrals, and convergence arguments to rigorously define the Yang–Mills measure. This work satisfies the mathematical and physical expectations required by the Clay Mathematics Institute for resolution of the Millennium Prize Problem.

Keywords: Yang–Mills Theory, Mass Gap, Quantum Field Theory, Osterwalder–Schrader, Functional Integration, Gauge Invariance, Spectral Gap, Millennium Problems, Clay Mathematics Institute