Charge Superselection Sectors for QCD on the Lattice

TL;DR

This paper analyzes the structure of charge superselection sectors in lattice QCD, showing that global boundary fluxes classify inequivalent representations and relate to the color charge of quarks.

Contribution

It introduces a detailed algebraic framework for lattice QCD, revealing how boundary fluxes determine superselection sectors and connect to global color charge.

Findings

Irreducible representations are labeled by ${f Z}_3$ boundary fluxes.

Charge sectors correspond to global color charge (triality).

The observable algebra is characterized by generators and relations.

Abstract

We study quantum chromodynamics (QCD) on a finite lattice $\Lambda$ in the Hamiltonian approach. First, we present the field algebra ${\mathfrak A}_{\Lambda}$ as comprising a gluonic part, with basic building block being the crossed product $C^*$-algebra $C(G) \otimes_{\alpha} G$, and a fermionic (CAR-algebra) part generated by the quark fields. By classical arguments, ${\mathfrak A}_{\Lambda}$ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra ${\mathfrak O}^i_{\Lambda}$ of internal observables is defined as the algebra of gauge invariant fields, satisfying the Gauss law. In order to take into account correlations of field degrees of freedom inside $\Lambda$ with the ``rest of the world'', we have to extend ${\mathfrak O}^i_{\Lambda}$ by tensorizing with the algebra of gauge invariant operators at infinity. This way we construct the full observable algebra ${\mathfrak O}_{\Lambda} .$ It is proved that its irreducible representations are labelled by ${\mathbb Z}_3$-valued boundary flux distributions. Then, it is shown that there exist unitary operators (charge carrying fields), which intertwine between irreducible sectors leading to a classification of irreducible representations in terms of the ${\mathbb Z}_3$-valued global boundary flux. By the global Gauss law, these 3 inequivalent charge superselection sectors can be labeled in terms of the global colour charge (triality) carried by quark fields. Finally, ${\mathfrak O}_{\Lambda}$ is discussed in terms of generators and relations.