P. A. S. Alcântara | Searcharxiv

arXiv2026

On symbol correspondences for quark systems II: Asymptotics

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.

math-ph↗

arXiv2026

On symbol correspondences for quark systems I: Characterizations

We present the characterizations of symbol correspondences for mechanical systems that are symmetric by $SU(3)$, which we refer to as \emph{quark systems}. The quantum quark systems are the unitary irreducible representations of $SU(3)$ of class $(p,q)$, $p,q\in\mathbb N_0$, together with their operator algebras. We study symbol correspondences from quantum operators to smooth functions on the phase space of a classical quark system, when such a phase space is a (co)adjoint orbit: either the complex projective plane $\mathbb CP^2$ or the flag manifold that is the total space of a fiber bundle $\mathbb CP^1\hookrightarrow \mathcal E\to \mathbb CP^2$. In the first case, we refer to pure-quark systems and the characterization of their correspondences is given in terms of characteristic numbers, similarly to the case of spin systems, cf. [26]. In the second case, we refer to general quark systems, particularly mixed-quark systems, and the characterization of their correspondences is given in terms of characteristic matrices, which introduces various novel features. Furthermore, we present the $SU(3)$ decomposition of the product of quantum operators and their corresponding twisted products of classical functions, for both pure and mixed quark systems.

math-ph↗