Confinement, Nonlocality and Haag Duality Violation in the Algebraic Structure of 1+1D QED

In this article, we present a novel formulation of the massless Schwinger model-quantum electrodynamics in $1+1$ dimensions-within the framework of Algebraic Quantum Field Theory (AQFT), emphasizing features that transcend the traditional bosonized treatments. Instead of mapping the model to a free massive scalar field, we construct a net of local observable algebras directly from the gauge-theoretic content, subject to the local ${U(1)}$ gauge symmetry and Gauss's law constraint. We show that local algebras can be consistently defined in terms of gauge-invariant composite operators, while charged fields necessarily fail to be localizable in bounded regions, manifesting confinement as the absence of DHR superselection sectors. Furthermore, we rigorously characterize nonlocal observables such as Wilson line operators within an extended net, and demonstrate the violation of Haag duality as a signature of the nontrivial topological and gauge structure of the theory. We additionally propose a conjecture linking the violation of Haag duality in confining gauge theories to a breakdown in entanglement wedge reconstruction, suggesting that confinement obstructs the local recovery of quantum information. Our approach provides a complete AQFT-based treatment of confinement, Gauss law, and nonlocal gauge-invariant operators in a solvable gauge theory, laying the groundwork for future extensions to non-Abelian models like QCD.