Charge Superselection Sectors for Scalar QED on the Lattice

TL;DR

This paper analyzes the algebraic structure of scalar QED on a lattice, demonstrating how charge superselection sectors emerge from the representations of the observable algebra in the Hamiltonian framework.

Contribution

It provides a detailed construction of the observable algebra and classifies its irreducible representations labeled by electric charge, revealing superselection sectors in lattice scalar QED.

Findings

Observable algebra is a $C^*$-algebra generated by gauge-invariant elements.

Representations are labeled by total electric charge.

Hilbert space decomposes into charge superselection sectors.

Abstract

The lattice model of scalar quantum electrodynamics (Maxwell field coupled to a complex scalar field) in the Hamiltonian framework is discussed. It is shown that the algebra of observables ${\cal O}({\Lambda})$ of this model is a $C^*$-algebra, generated by a set of gauge-invariant elements satisfying the Gauss law and some additional relations. Next, the faithful, irreducible and non-degenerate representations of ${\cal O}({\Lambda})$ are found. They are labeled by the value of the total electric charge, leading to a decomposition of the physical Hilbert space into charge superselection sectors. In the Appendices we give a unified description of spinorial and scalar quantum electrodynamics and, as a byproduct, we present an interesting example of weakly commuting operators, which do not commute strongly.