The observable algebra of lattice QCD

Quantum chromodynamics (QCD) is studied on a finite latticewithin the Hamiltonian approach. First, the field algebra AΛ as comprising a gluonic part, with basic building block being the crossed product C*-algebra C(G) ⊗αG, and a fermionic (CAR-algebra) part generated by the quark fields, is discussed. By classical arguments, AΛ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra OΛi 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 insidewith the “rest of the world”, OΛi is tensorized with the algebra of gauge invariant operators at infinity. This way, the full observable algebra OΛ is constructed. It turns out that its irreducible representations are labelled by the ℤ3-valued global gluonic boundary flux, leading to three inequivalent charge superselection sectors. By the global Gauss law, these can be labelled in terms of the global colour charge carried by quark fields.