Uniform Logarithmic Sobolev Inequalities and the Spectral Mass Gap in Euclidean Wilson Lattice Gauge Theory

This paper develops a mathematical and physical framework connecting a uniformlogarithmic Sobolev inequality, abbreviated as LSI, for the Euclidean Wilson latticemeasure with a non-vanishing spectral mass gap in the continuum limit.Let G be a compact gauge group, such as SU(N), and let Ue ∈ G denote theparallel transporter associated with a lattice edge e. The Wilson probability measureis defined bydµa,Λ(U) = Z−1a,Λexp−SW,a,Λ(U)dU,where a is the lattice spacing, Λ is a finite lattice volume, dU is the product Haarmeasure, and SW,a,Λ is the Wilson action.The central hypothesis is the existence of a constant ϱa > 0, uniform with respectto the lattice spacing, such thatEntµa,Λ(f2) ≤2ϱa Ea,Λ(f, f), ϱa ≥ ϱ0 > 0.Here,Entµ(f2) = Zf2log(f2) dµ −Zf2 dµlog Zf2 dµand Ea,Λ is the associated Dirichlet form.The logarithmic Sobolev inequality implies a Poincaré inequality and, therefore,a lower bound on the spectral gap of the reversible diffusion generator. However,a mathematically correct continuum statement requires an important scaling distinction. A gap that remains positive in lattice units is not automatically a finitepositive physical gap as a → 0. In Hamiltonian units, the dimensionless lattice gapmust scale proportionally to a:∆lat a ∼ a mphys, so that∆phys a =∆lataa−→ mphys > 0.The paper discusses the principal analytical mechanisms capable of producinguniform inequalities, including Bakry–Émery curvature estimates, perturbative convexity, Dobrushin-type criteria, block decomposition, gradient propagation, reflectionpositivity, exponential clustering, and Osterwalder–Schrader reconstruction.