On symbol correspondences for quark systems

Abstract: We present the characterization of symbol correspondences for mechanical systems that are symmetric under $SU(3)$, which we refer to as quark systems. The quantum systems are the unitary irreducible representations of $SU(3)$, denoted by $Q(p,q)$, $p,q\in\mathbb N_0$, together with their operator algebras. We study the cases when the classical phase space is a coadjoint orbit: either the complex projective plane $\mathbb CP^2$ or the flag manifold that is the total space of 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. [24]. In the second case, we refer to generic 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 generic quark systems. In preparation for asymptotic analysis of these twisted products, we also present the $SU(3)$ decomposition of the pointwise product of classical functions.