The SU(3) Algebra in a Cyclic Basis

TL;DR

This paper identifies special cyclic bases for the su(3) algebra where all gluons interact equally, potentially extending to other su(n) algebras, with implications for understanding gluon interactions in QCD.

Contribution

The authors discover specific cyclic bases for su(3) algebra through computer search and propose their generalization to other su(n) algebras.

Findings

Found particular cyclic bases for su(3) via computational methods.

Proposed that all cyclic bases for su(3) can be derived from these.

Suggested cyclic bases may exist for su(n), n>3.

Abstract

With the couplings between the eight gluons constrained by the structure constants of the su(3) algebra in QCD, one would expect that there should exist a special basis (or set of bases) for the algebra wherein, unlike in a Cartan-Weyl basis, {\em all} gluons interact identically (cyclically) with each other, explicitly on an equal footing. We report here particular such bases, which we have found in a computer search, and we indicate associated $3 \times 3$ representations. We conjecture that essentially all cyclic bases for su(3) may be obtained from these making appropriate circulant transformations,and that cyclic bases may also exist for other su(n), n>3.