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

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 $C^{\ast }$-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 $\mathrm{sl}(1/n)$ (factorized by a certain ideal).

arXiv comment33 pages

References in corpus (5)

• On the Gauss Law and Global Charge for QCD
• Center vortices and confinement vs. screening
• Polynomial super-gl(n) algebras
• Charge Superselection Sectors for Scalar QED on the Lattice
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• The Hilbert space costratification for the orbit type strata of SU(2)-lattice gauge theory
• Quasicharacters, recoupling calculus and costratifications of lattice quantum gauge theory