Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On symbol correspondences for quark systems II: Asymptotics

P. A. S. Alcântara, P. de M. Rios

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.

On symbol correspondences for quark systems II: Asymptotics

Abstract

We study the semiclassical asymptotics of twisted algebras induced by symbol correspondences for quark systems (-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 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 , defining Magoo spheres, and studying their asymptotic limits. We end by highlighting the possible generalizations from to other compact symmetry groups, specially compact simply connected semisimple Lie groups, commenting on some peculiarities from our treatment for deserving further investigations.
Paper Structure (20 sections, 58 theorems, 307 equations, 1 figure)

This paper contains 20 sections, 58 theorems, 307 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic framework and preliminary results
  3. The smooth and the coarse Poisson spheres
  4. The symplectic foliation of the smooth Poisson sphere
  5. The coarse Poisson sphere
  6. Main results for $C^\ast$-algebras on (co)adjoint orbits
  7. Preliminary considerations for semiclassical asymptotics
  8. PBW Theorem and Poisson algebras of harmonic functions
  9. Universal correspondences for general quark systems
  10. Asymptotic analysis for general quark systems
  11. Rays of universal correspondences: fuzzy orbits
  12. On the asymptotics of Berezin fuzzy orbits
  13. First criterion for Poisson: convergence of symbols
  14. Second criterion for Poisson: characteristic matrices
  15. Universal correspondences on the coarse Poisson sphere
  16. ...and 5 more sections

Key Result

Proposition 2.2

The polynomial $\tau:\mathfrak{sl}(3)\to\mathbb C$ given by is $SU(3)$-invariant and separates the points of $\overline{\mathcal{F}}$.

Figures (1)

  • Figure 1: GT basis for $\mathfrak{sl}(3)$, cf. Definition I.2.1.

Theorems & Definitions (133)

  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9
  • Remark 2.10
  • Theorem 2.11
  • proof
  • ...and 123 more