On symbol correspondences for quark systems II: Asymptotics

TL;DR

This paper investigates the semiclassical limits of twisted algebras related to quark systems, focusing on asymptotic behaviors of harmonic functions on coadjoint orbits and introducing Magoo spheres through a novel gluing process.

Contribution

It provides new criteria for the emergence of Poisson algebras from sequences of symbol correspondences and extends the analysis to the construction of Magoo spheres.

Findings

Identified criteria for Poisson algebra emergence from twisted algebras.

Developed the concept of Magoo spheres by gluing fuzzy orbits.

Discussed potential generalizations to other compact Lie groups.

Abstract

We study the semiclassical asymptotics of twisted algebras induced by symbol correspondences for quark systems ($SU(3)$-symmetric mechanical systems) as defined in our previous paper [3]. The linear span of harmonic functions on (co)adjoint orbits is identified with the space of polynomials on $\mathfrak{su}(3)$ restricted to these orbits, and we find two equivalent criteria for the asymptotic emergence of Poisson algebras from sequences of twisted algebras of harmonic functions on (co)adjoint orbits which are induced from sequences of symbol correspondences (the fuzzy orbits). Then, we proceed by "gluing" the fuzzy orbits along the unit sphere $\mathcal S^7\subset \mathfrak{su}(3)$, defining Magoo spheres, and studying their asymptotic limits. We end by highlighting the possible generalizations from $SU(3)$ to other compact symmetry groups, specially compact simply connected semisimple Lie groups, commenting on some peculiarities from our treatment for $SU(3)$ deserving further investigations.