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.