Rigorous Foundations of Four-Dimensional Euclidean Yang–Mills Theory on a Finite Lattice

Abstract

This paper develops the mathematical and physical foundations of the first phaseof a proposed research program for the rigorous construction of four-dimensionalquantum Yang–Mills theory. The analysis is restricted initially to pure non-AbelianYang–Mills theory with compact gauge group SU(N), no dynamical fermionic matter,and a finite Euclidean lattice. This choice isolates the gauge-field dynamics andavoids the additional analytical difficulties associated with fermionic determinants,chiral symmetry, and ultraviolet regularization.The lattice formulation is constructed from group-valued link variables equippedwith the normalized Haar measure. Gauge invariance is implemented exactly at finitelattice spacing through the Wilson plaquette action. The resulting finite-volumepartition function is a well-defined positive integral over a compact configurationspace. This property allows one to establish finiteness of the partition function andboundedness of a broad class of gauge-invariant observables, including Wilson loopsand normalized correlation functions.The finite-volume result is mathematically rigorous but does not by itself proveconfinement, existence of the infinite-volume theory, or the Yang–Mills mass gap.These stronger conclusions require uniform estimates, thermodynamic limits, continuum scaling, reflection positivity, Osterwalder–Schrader reconstruction, and nonperturbative spectral analysis. The present phase establishes the controlled startingpoint from which the subsequent infinite-volume and continuum problems may beformulated precisely.