Charge superselection sectors for QCD on the lattice

We study quantum chromodynamics (QCD) on a finite lattice $\Lambda$ in theHamiltonian approach. First, we present the field algebra ${\mathfrakA}_{\Lambda}$ as comprising a gluonic part, with basic building block being thecrossed 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) irreduciblerepresentation. Next, the algebra ${\mathfrak O}^i_{\Lambda}$ of internalobservables is defined as the algebra of gauge invariant fields, satisfying theGauss law. In order to take into account correlations of field degrees offreedom inside $\Lambda$ with the ``rest of the world'', we have to extend${\mathfrak O}^i_{\Lambda}$ by tensorizing with the algebra of gauge invariantoperators at infinity. This way we construct the full observable algebra${\mathfrak O}_{\Lambda} .$ It is proved that its irreducible representationsare labelled by ${\mathbb Z}_3$-valued boundary flux distributions. Then, it isshown that there exist unitary operators (charge carrying fields), whichintertwine between irreducible sectors leading to a classification ofirreducible representations in terms of the ${\mathbb Z}_3$-valued globalboundary flux. By the global Gauss law, these 3 inequivalent chargesuperselection sectors can be labeled in terms of the global colour charge(triality) carried by quark fields. Finally, ${\mathfrak O}_{\Lambda}$ isdiscussed in terms of generators and relations.Comment: 40 page