On symbol correspondences for quark systems I: Characterizations

TL;DR

This paper characterizes symbol correspondences for $SU(3)$-symmetric quantum systems called quark systems, analyzing their phase space representations and operator products, with new insights into pure and mixed systems.

Contribution

It introduces novel characterizations of symbol correspondences for $SU(3)$ quark systems, including characteristic numbers and matrices, extending the understanding of phase space mappings.

Findings

Characterization of correspondences for pure-quark systems via characteristic numbers.

Introduction of characteristic matrices for mixed-quark systems.

Decomposition of operator products into twisted products of classical functions.

Abstract

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.