Charge Superselection Sectors for QCD on the Lattice

Abstract

We study quantum chromodynamics (QCD) on a finite lattice $Λ$ in the Hamiltonian approach. First, we present the field algebra ${\mathfrak A}_Λ$ as comprising a gluonic part, with basic building block being the crossed product $C^*$-algebra $C(G) \otimes_α G$, and a fermionic (CAR-algebra) part generated by the quark fields. By classical arguments, ${\mathfrak A}_Λ$ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra ${\mathfrak O}^i_Λ$ 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 $Λ$ with the ``rest of the world'', we have to extend ${\mathfrak O}^i_Λ$ by tensorizing with the algebra of gauge invariant operators at infinity. This way we construct the full observable algebra ${\mathfrak O}_Λ .$ 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, ${\mathfrak O}_Λ$ is discussed in terms of generators and relations.