From Euclidean Spectral Inequalities to Minkowski Hamiltonian Gaps

This paper formulates a mathematically controlled route from a Euclideanspectral inequality of the form∆a ≥1/2ϱato a positive Minkowski-space Hamiltonian satisfyingH ≥ ∆ > 0in the continuum limit a → 0. Here a denotes the ultraviolet lattice spacing, ∆a isa finite-scale Euclidean spectral quantity, and ϱa is a scale-dependent coercivity ordecay parameter.The main point is that positivity of ∆a at each finite lattice spacing does not, byitself, imply a nonzero continuum mass gap. A uniform lower bound, tightness ofthe associated probability measures, preservation of Osterwalder–Schrader reflectionpositivity, and convergence of the Euclidean transfer operators are required.Under these hypotheses, the limiting Euclidean theory can be reconstructed as aphysical Hilbert space with a positive self-adjoint Hamiltonian. Iflim infa→0 ϱa > 0,then the reconstructed Hamiltonian satisfiesspec(H) ⊆ {0} ∪ [∆, ∞),where∆ ≥1/2 lim inf a→0 ϱa > 0.The result identifies the Euclidean exponential decay rate with a physical Minkowskispectral gap above the vacuum.