Infinite-Volume and Continuum Limits in Four-Dimensional Quantum Yang–Mills Theory

Abstract

The construction of a mathematically rigorous four-dimensional quantum Yang– Mills theory with a non-zero mass gap remains one of the central problems in mathematical physics.The principal difficulty is not the formal definition of the classical field equations, but the construction of a non-perturbative quantum theory satisfying positivity, locality, gauge invariance, Euclidean covariance, and a strictlypositive spectral gap above the vacuum.This manuscript develops a mathematical and physical framework for the fourthand fifth stages of a proposed Yang–Mills research program: the infinite-volumelimit and the continuum limit. The analysis begins with a finite hypercubic lattice,a compact non-Abelian gauge group G, and the Wilson probability measure. Theinfinite-volume theory is formulated through thermodynamic limits of local gaugeinvariant observables. The continuum limit is then expressed as a controlled limit inwhich the lattice spacing a tends to zero while physical correlation lengths measuredin lattice units diverge.Asymptotic freedom provides the perturbative ultraviolet mechanism for therenormalization-group trajectory. It does not, by itself, prove the existence of thenon-perturbative continuum measure or the mass gap. The proposed programtherefore combines constructive field theory, reflection positivity, finite-volume estimates, renormalization-group interpolation, Schwinger–Dyson identities, functionalrenormalization group equations, and Monte Carlo simulations.The desired conclusion is the existence of a non-trivial continuum theory whosegauge-invariant correlation functions satisfy the Osterwalder–Schrader axioms andwhose Hamiltonian possesses a spectral gap∆ = inf (Spec(H) \ {0}) > 0The present work is formulated as a research program and does not claim thatthe four-dimensional Yang–Mills existence and mass-gap problem has already beensolved.