On the Structure of the Observable Algebra of QCD on the Lattice

TL;DR

This paper analyzes the algebraic structure of observable operators in lattice QCD, revealing a detailed classification of the hadronic component and its relation to Lie superalgebra representations.

Contribution

It provides a detailed mathematical description of the observable algebra in lattice QCD, including the classification of irreducible representations by triality and its relation to Lie superalgebras.

Findings

The gluonic algebra is isomorphic to a CCR algebra over SU(3).

Hadronic algebra representations are classified by triality.

Hadronic algebra is isomorphic to the commutant of the triality operator.

Abstract

The structure of the observable algebra ${\mathfrak O}_{\Lambda}$ of lattice QCD in the Hamiltonian approach is investigated. As was shown earlier, ${\mathfrak O}_{\Lambda}$ is isomorphic to the tensor product of a gluonic $C^{*}$-subalgebra, built from gauge fields and a hadronic subalgebra constructed from gauge invariant combinations of quark fields. The gluonic component is isomorphic to a standard CCR algebra over the group manifold SU(3). The structure of the hadronic part, as presented in terms of a number of generators and relations, is studied in detail. It is shown that its irreducible representations are classified by triality. Using this, it is proved that the hadronic algebra is isomorphic to the commutant of the triality operator in the enveloping algebra of the Lie super algebra ${\rm sl(1/n)}$ (factorized by a certain ideal).