Charge Superselection Sectors for QCD on the Lattice

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 $C^{\ast }$-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 ${\mathfrak{O}}_{\Lambda }^i$ 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 $\Lambda$ with the ``rest of the world'', we have to extend ${\mathfrak{O}}_{\Lambda }^i$ 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.

40 pages

References in corpus (6)

• On the Gauss Law and Global Charge for QCD
• On the Stratified Classical Configuration Space of Lattice QCD
• Polynomial super-gl(n) algebras
• Charge Superselection Sectors for Scalar QED on the Lattice
• On the Gribov Problem for Generalized Connections
• Towards an Effective Field Theory of QED

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