math-ph — Mathematical Physics
Osterwalder–Schrader axioms for scalar QED in the Higgs phase
We study scalar quantum electrodynamics in three dimensions (QED) in its Higgs phase: the three-dimensional lattice abelian Higgs model in unitary gauge, with the Villain action, on lattices whose spacing tends to zero with canonically scaled couplings. At each spacing we construct infinite-volume states, as limit points of the periodic states and as limits of states in finite boxes; they carry the lattice symmetries, reflection positivity and bounds uniform in the volume. For the smeared curvature and bounded functions of the density, the generating functionals converge along a subsequence of spacings to a translation-invariant limit functional with the continuity required for the Osterwalder–Schrader reconstruction. For the limits built on the box states, and on the periodic states along dyadic sequences of sides, the limit curvature field satisfies the axioms OS0–OS3, with invariance under the full Euclidean group. The cluster axiom OS4 is proved in a weak form, on almost every ergodic component. The tools are Gaussian domination of the Villain curvature, Ginibre inequalities and reflection positivity.