Global Charge, Gauss Law and Superselection Rules for QCD on the Lattice

Quantum chromodynamics (QCD) on a finite lattice $\mathrm{\Lambda }$ in the Hamiltonian approach is analyzed. First, we present the field algebra ${\mathfrak{A}}_{\mathrm{\Lambda }}$ as comprising a gluonic part, with basic building block being the crossed product $C^{\ast }$-algebra $C(G){\otimes }_{\alpha }G$, and a fermionic (CAR-algebra) part generated by quark fields. By classical arguments, ${\mathfrak{A}}_{\mathrm{\Lambda }}$ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra ${\mathfrak{O}}_{\mathrm{\Lambda }}^i$ 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 $\mathrm{\Lambda }$ with the "rest of the world", we have to extend ${\mathfrak{O}}_{\mathrm{\Lambda }}^i$ by tensorizing with the algebra of external gauge invariant operators. This way we construct the full observable algebra ${\mathfrak{O}}_{\mathrm{\Lambda }}$. We prove 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, ${\mathfrak{O}}_{\mathrm{\Lambda }}$ is discussed in terms of generators and relations.