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Osterwalder–Schrader axioms for scalar QED3_3 in the Coulomb phase

Primarily AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.MP — Mathematical Physicsmath.PR — Probability

Contributed by Mark Goldstein ↗, Alfredo Guevara ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

King constructed the continuum, infinite-volume limit of three-dimensional scalar quantum electrodynamics, the U(1)U(1) (abelian) Higgs model, along Neumann–Dirichlet cubes. For his state at fixed photon mass we prove the Osterwalder–Schrader axioms: regularity E0′′\mathrm{E0}'' with absolute constants, invariance under the full Euclidean group, reflection positivity about every plane, symmetry, and the measure with its ergodic decomposition, on whose components the cluster axiom holds in all its forms. Rotation invariance is proved by comparing King's cubes with periodic tori carrying a smooth mass wall on two lattices whose orientations differ by arctan⁡(3/4)\arctan(3/4). With real scalar sources, the joint field of the curvature and the renormalised Wick square satisfies, on a tube of small Wick amplitudes, the axioms: regularity, invariance under translations and the lattice point group, Euclidean invariance of the joint field outside an exceptional set of couplings, reflection positivity for Wick amplitudes below a threshold, symmetry, and the measure with the cluster axiom in its weak form; above the threshold the Euclidean clause is open. The curvature results hold on King's whole coupling range; the Wick results hold under a one-sided sign clause of the Wick leg, which we prove on explicit coupling regions. The limit of vanishing photon mass is treated in a subsequent paper.

Author comments: Companion paper: Osterwalder–Schrader axioms for scalar QED$_3$ in the Higgs phase (hexagon:2611.00321). Project: https://github.com/marikgoldstein/scalar-qed-existence

Provenance statement

This paper and its companion represent verified output from our scalar-qed-existence project (https://github.com/marikgoldstein/scalar-qed-existence), generated via agent orchestration. The purpose of the project is to finalize King's treatment of the theory, which already provides insight into the proof. We are currently in the process of digesting the results.

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Version history

  1. v1 · 291 pagesSubmitted by Mark GoldsteinCompanion paper: Osterwalder–Schrader axioms for scalar QED$_3$ in the Higgs phase (hexagon:2611.00321). Project: https://github.com/marikgoldstein/scalar-qed-existenceCurrentNov 12, 2026