On symbol correspondences for quark systems II: Asymptotics

ON SYMBOL CORRESPONDENCES F OR QUARK SYSTEMS I I: ASYMPTOTICS P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Abstract. W e study the semiclassical asymptotics of twisted algebras in- duced by sym b ol corresp ondences for quark systems ( S U (3)-symmetric me- chanical systems) as deﬁned in our previous paper [3]. The linear span of harmonic functions on (co)adjoint orbits is identiﬁed with the space of p oly- nomials on su (3) restricted to these orbits, and we ﬁnd tw o equiv alent criteria for the asymptotic emergence of Poisson algebras from sequences of twisted algebras of harmonic functions on (co)adjoin t orbits which are induced from sequences of sym bol corresp ondences (the fuzzy orbits). Then, w e proceed by “gluing” the fuzzy orbits along the unit sphere S 7 ⊂ su (3), deﬁning Mago o spheres, and studying their asymptotic limits. W e end b y highlighting the possible generalizations from S U (3) to other compact symmetry groups, sp e- cially compact simply connected semisimple Lie groups, commen ting on some peculiarities from our treatment for S U (3) deserving further in v estigations. Contents 1. In tro duction 2 2. Basic framework and preliminary results 3 2.1. The smo oth and the coarse Poisson spheres 5 2.2. Main results for C ∗ -algebras on (co)adjoint orbits 8 2.3. Preliminary considerations for semiclassical asymptotics 12 2.4. PBW Theorem and Poisson algebras of harmonic functions 14 2.5. Univ ersal corresp ondences for general quark systems 19 3. Asymptotic analysis for general quark systems 21 3.1. Ra ys of universal correspondences: fuzzy orbits 21 3.2. On the asymptotics of Berezin fuzzy orbits 22 3.3. First criterion for Poisson: conv ergence of sym b ols 25 3.4. Second criterion for Poisson: characteristic matrices 27 4. Univ ersal corresp ondences on the coarse Poisson sphere 31 4.1. P encils of corresp ondence rays: Mago o spheres 31 4.2. On the asymptotics of the Berezin Mago o sphere 38 5. Concluding remarks 43 References 45 App endix A. A pro of of Prop osition 3.22 46 App endix B. Alternative pro of of Corollary 3.23 48 2020 Mathematics Subje ct Classiﬁc ation. 17B08, 20C35, 22E46, 22E70, 41A60, 43A85, 53D99, 81Q20, 81S10, 81S30. Key wor ds and phr ases. Dequantization, Quantization, Symmetric mechanical systems, Symbol correspondences, Quark systems ( S U (3)-symmetric systems), Semiclassical asymptotics. This w ork was supported in part b y Co ordena¸ c˜ ao de Aperfei¸ coamento de P essoal de N ´ ıvel Superior (CAPES), Brasil - Finance Code 001. 1 2 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS 1. Introduction This is Paper I I of tw o serial works on corresp ondences for quark systems, i.e. mec hanical systems with S U (3)-symmetry . Here w e presen t the asymptotic analysis of t wisted pro ducts induced b y sym bol corresp ondences ov er symplectic (co)adjoin t orbits, as deﬁned in [3] (henceforth referred to as P aper I), and address the question of how such twisted pro ducts can b e extended to the unit sphere S 7 ⊂ su (3). Throughout this pap er, we shall often recall and refer to the results of Paper I. Th us, excerpts of Paper I are cited by adding an “I” to the num b er; for instance, (I.2.71) means equation (2.71) of Paper I, lik ewise for Prop osition I.3.5, etc. While w e w ork ed with abstract orbits C P 2 and E in Paper I, in this text we adopt a sp ecial family of actual (co)adjoin t orbits in S 7 ⊂ su (3), the orbits that are equiv alent b y rescaling to an orbit in su (3) of highest weigh t ( p, q ), p, q ∈ N , which shall be called r ational orbits , so that this family deﬁnes a r ational c o arsening of the orbit foliation of S 7 , cf. Deﬁnitions 2.4 and 2.7. F or each of these rational orbits, the sequences of corresp ondences suitable for semiclassical asymptotic analysis are r ays of c orr esp ondenc es deﬁned on sequences of quantum systems determined by ra ys in the lattice of dominan t weigh ts given b y the orbits themselves. That is, for eac h rational orbit O ξ ⊂ S 7 if ω ( p 0 ,q 0 ) is the ﬁrst highest weigh t whose orbit is equiv alent to O ξ , w e consider the sequence of highest w eights ( ω ( sp 0 ,sq 0 ) ) s ∈ N , with its sequence of symbol corresp ondences to functions on O ξ , cf. Deﬁnition 3.1. Then, for each irrep ( sp 0 , sq 0 ), a symbol corresp ondence deﬁnes a t wisted algebra on a d 2 -dimensional subspace of C ∞ C ( O ξ ), where d = dim( sp 0 , sq 0 ), which is isomorphic to the matrix algebra M C ( d ). Hence, each ξ -ra y of corresp ondences deﬁnes a sequence of twisted algebras of functions on O ξ , also called a fuzzy orbit . The necessit y of working with sequences of (increasing) ﬁnite dimensional t wisted algebras of functions on O ξ and inv estigating if/when/ho w their asymptotic limits coincide with the classical Poisson algebra of smo oth functions on O ξ , stems from some results for S U (3)-inv ariant unital C ⋆ -algebra structures on C ∞ ( O ξ ), whic h w e state and prov e, cf. Theorem 2.11, Prop osition 2.16 and Corollary 2.19. Ho wev er, while the metho d used in [17] for studying such asymptotic limits can b e generalized from spin systems to pure-quark systems, alb eit with greater diﬃ- cult y , its generalization to mixed-quark systems seems hop eless, so in this pap er w e dev elop a new method using the univ ersal env eloping algebra. Also, in [17] the criterion for recov ering the P oisson algebra of harmonic functions on S 2 as an asymptotic limit of spin t wisted algebras is more clearly seen b y comparison to a suitable sequence of Stratonivich-W eyl corresp ondences for s pin systems. Ho w- ev er, for quark systems, sp eciﬁcally mixed quark systems, the characterization of Stratonivic h-W eyl correspondences is quite cumbersome, cf. Remark I.5.26, so here w e adopt as paradigm the sequences of (highest weigh t) Berezin corresp ondences. Karab ego v [12] has shown in a quite general setting that such Berezin corre- sp ondences satisfy a v ersion of the so called c orr esp ondenc e principle , which we en unciate in the context of quark system as an asymptotic ( s → ∞ ) Poisson typ e prop ert y , cf. Deﬁnition 3.4. Then, we apply the results for Berezin corresp ondences to derive a classiﬁcation of ξ -rays of corresp ondences for quark systems which are of Poisson type, stated in tw o diﬀerent w ays, cf. Theorems 3.17 and 3.21. Thereafter, given ξ -ra ys of correspondences with their induced sequences of t wisted algebras, deﬁned for each and ev ery rational orbit O ξ ⊂ S 7 , we pro ceed by “gluing” all these fuzzy orbits together along the rational coarsening of S 7 , thus ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 3 deﬁning a Mago o spher e , cf. Deﬁnition 4.3. W e do so by ﬁrst deﬁning a chain of nested subsets of rational orbits as a sequence indexed by n ∈ N which con v erges to the full set of rational orbits in S 7 . This leads to the deﬁnition of a Magoo sphere as a bi-sequence of twisted algebras and, b y ﬁrst taking the asymptotic limit s → ∞ and then the chain limit n → ∞ , w e arrive at the deﬁnition of Mago o spheres of Poisson type, cf. Deﬁnition 4.7, and Theorem 4.8 shows that this prop erty is satisﬁed for a Mago o sphere if and only if ev ery fuzzy orbit is of P oisson type. In verting the order of the limits for a Mago o sphere, taking n → ∞ ﬁrst and then s → ∞ , leads to the deﬁnition of Mago o spheres of uniform Poisson typ e , cf. Deﬁnition 4.15 and Prop osition 4.16. Thus, w e end by studying if this prop erty is satisﬁed for the Berezin Mago o sphere, and Theorem 4.19 states that this is so if w e restrict to an y compact “cylinder” S 7 | K ⊂ S 7 whic h do es not con tain neigh b orho ods of the nongeneric orbits. On the other hand, in Prop osition 4.24 w e presen t an example of Mago o sphere of Poisson type for which the uniform Poisson prop ert y do es not hold even in any suc h a “cylinder”, sho wing that the Berezin Mago o sphere is special, in this sense. How ev er, we hav e not yet been able to prov e or disprov e the uniform Poisson prop ert y for the whole Berezin Mago o sphere. This pap er is organized as follows. In section 2 we stablish some basic to ols and results used throughout the pap er. W e describ e the symplectic foliation of su (3) and its unit sphere S 7 b y (co)adjoint orbits, and in tro duce the coarse Poisson sphere as the countable collection of ratio- nal orbits in S 7 . Then we state and prov e some results on C ⋆ -algebras and discuss ho w they imply the necessity to work with sequences of ﬁnite-dimensional twisted algebras to study the semiclassical asymptotic limit. W e also describ e harmonic functions on orbits and on S 7 as p olynomial functions, resorting to an isomorphism from the univ ersal env eloping algebra U ( sl (3)) to P ol y ( su (3)) in order to describ e the Poisson algebra of polynomials. Then we use the pullbac k of symbol corre- sp ondences to U ( sl (3)) so that w e can deal with corresp ondences deﬁned on a ﬁxed domain, which makes it easier to take asymptotic limits. In section 3, we develop the semiclassical analysis of twisted algebras of func- tions on orbits (fuzzy orbits). First, we repro duce some general results of [12] in the sp eciﬁc setting of quark systems, and w e use them to obtain t w o equiv alent conditions for a ray of corresp ondences to b e of Poisson type. The ﬁrst criterion is a comparison b etw een limits of symbols and polynomials, and the second one is by means of the characteristic matrices deﬁned in P ap er I. Section 4 is dev oted to “gluing” the fuzzy orbits along the coarse P oisson sphere, deﬁning the Mago o spheres, and studying their asymptotic limits. Then, in the last section 5 we discuss how most of the results of b oth pap ers I and I I can b e generalized to other compact symmetry groups, sp ecially to general compact simply connected semisimple Lie groups, and ﬁnish with last comments on p eculiarities from our treatment of S U (3) that deserve further inv estigations. Finally , in Appendix A w e presen t a pro of of Proposition 3.22, an d in Appendix B w e summarize the Clebsch-Gordan approac h to the asymptotics of twisted algebras for pure-quark systems, which is presented in full in [1]. 2. Basic framework and preliminar y resul ts W e will work with sym b ol corresp ondences for functions on concrete adjoint or- bits O ⊂ su (3) rather than the abstract ones, C P 2 or E , as indicated in Remark 4 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS I.3.3. Our approach shall be based on the quite general metho d that Karab ego v ap- plied to Berezin corresp ondences in [12]. W e b egin b y establishing some deﬁnitions and notations. W e refer to Paper I, Section 2, for details. Recall that { E j = iλ j / √ 2 : j = 1 , ..., 8 } , cf. (I.2.1), is an orthonormal basis of su (3) w.r.t. the standard inner pro duct (I.2.8), and the fundamen tal weigh ts are (2.1)  1 = 1 √ 2 E 3 + 1 √ 6 E 8 ,  2 = r 2 3 E 8 , cf. (I.2.65), so that dominant w eigh ts are of the form (2.2) ω p = p  1 + q  2 , p = ( p, q ) ∈ N 0 × N 0 , and w e iden tify an irreducible represen tation with highest w eigh t ω p b y the pair p = ( p, q ), the case p = (0 , 0) b eing the trivial representation which is often discarded. No w, by the Stone-W eierstrass Theorem, (2.3) P oly ( O ) := { f | O : f ∈ P ol y ( su (3)) } is uniformly dense in C ∞ C ( O ) for ev ery orbit O ⊂ su (3). Since the space P oly d ( su (3)) of complex homogeneous p olynomials on su (3) of degree d ∈ N is an in v ariant sub- space for the S U (3)-action, the linear span of harmonic functions on O is precisely P ol y ( O ). How ev er, although P oly d ( su (3)) pro vide a grading for the algebra of p olynomials P ol y ( su (3)), its restriction to an orbit O ⊂ su (3), (2.4) P ol y d ( O ) := { f | O : f ∈ P ol y d ( su (3)) } , do es not provide a grading of P ol y ( O ) b ecause the restriction of p olynomials of diﬀeren t degrees from su (3) to O ma y coincide. F or instance, if ( x 1 , ..., x 8 ) are co ordinates on su (3) w.r.t. the orthonormal basis { E j } 1 ≤ j ≤ 8 , then (2.5) 8 X j =1 x 2 j   O ≡ 1 ∀ O ⊂ S 7 , where S 7 ⊂ su (3) is the unitary sphere. In the same v ein, there is a homogeneous cubic p olynomial, asso ciated to the cubic Casimir of S U (3), that is constant along eac h orbit O ⊂ su (3), cf. Prop osition 2.2, further b elow. Even so, we will still mak e use of P ol y d ( O ), as w ell as (2.6) P ol y ≤ d ( O ) = d M m =0 P ol y m ( O ) . Lik ewise, for the unitary sphere S 7 ⊂ su (3), (2.7) P oly ( S 7 ) := { f | S 7 : f ∈ P ol y ( su (3)) } is uniformly dense in C ∞ C ( S 7 ), and we will also make use of the spaces (2.8) P ol y d ( S 7 ) := { f | S 7 : f ∈ P ol y d ( su (3)) } , P ol y ≤ d ( S 7 ) = d M m =0 P ol y m ( S 7 ) . ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 5 2.1. The smo oth and the coarse Poisson spheres. W e shall b e interested in algebras of functions on S 7 ⊂ su (3) or on orbits O ⊂ S 7 . Notation 2.1. L et F b e the ar c of cir cumfer enc e given by the interse ction of the unitary spher e S 7 ⊂ su (3) with the close d princip al Weyl chamb er, so that F is the subset obtaine d by r emoving the endp oints. We c an write the p oints of F as ξ ( x,y ) := r 3 2 ( x  1 + y  2 ) ≡ i √ 6 2 x + y 0 0 0 − x + y 0 0 0 − x − 2 y ! ∈ F , (2.9) wher e (   ξ ( x,y )   2 = x 2 + xy + y 2 = 1 x, y ≥ 0 , (2.10) with strict ine quality in (2.10) for ξ ( x,y ) ∈ F . Given ξ ( x,y ) = ξ ∈ F , we write O ( x,y ) = O ξ ⊂ S 7 for its orbit, identifying F with the set of unitary orbits, (2.11) F ∋ ξ ↔ O ξ ⊂ S 7 ⊂ su (3) . F or functions on O ξ , we denote the supr emum norm by ∥ ∥ ξ wher e as on S 7 we denote the supr emum norm by ∥ ∥ ∞ . In addition, we use the left-invariant inte gr al on O ξ induc e d by the Haar me asur e of S U (3) to deﬁne the inner pr o duct ⟨·|·⟩ ξ as (2.12) ⟨ f 1 | f 2 ⟩ ξ = Z O ξ f 1 ( ς ) f 2 ( ς ) d ς for f 1 , f 2 ∈ L 2 ( O ξ ) w.r.t. the inner-pr o duct norm ∥ f ∥ ξ, 2 = q ⟨ f | f ⟩ ξ . 2.1.1. The symple ctic foliation of the smo oth Poisson spher e. W e recall that the collection of all unitary adjoint orbits O ξ ⊂ S 7 deﬁnes a symplectic foliation of the smo oth P oisson manifold ( S 7 , b Π g ), where b Π g = Π g | S 7 for Π g the KAKS Poisson bi-v ector on g = su (3) given by (2.13) Π g = X j,k,l c l kj x l ∂ j ⊗ ∂ k , where c l k,j are the constant structures of su (3) in the basis { E 1 , ..., E 8 } and likewise for ( x 1 , ..., x 8 ) b eing co ordinates in this basis, see [13]. W e denote this foliation by (2.14) [ ξ ∈F ( O ξ , Π g | O ξ ) = ( S 7 , b Π g ) , b Π g = Π g | S 7 , ( O ξ , Π g | O ξ ) ≡ ( O ξ , Ω ξ ) , Ω ξ = Π g | O ξ symplectic . The orbits for ξ ( x,y ) ∈ F are the lea ves O ξ ( x,y ) ≃ E of the regular part of this foliation, with the t w o closing orbits O ξ (1 , 0) ≃ O ξ (0 , 1) ≃ C P 2 comprising the singular lea ves. W e now describ e this singular foliation in more detail. Recall parametrization (2.10) of F . F or x ≥ 1 / √ 3, we ha ve y ≤ 1 / √ 3 and we consider the orbit O ( x,y ) as a S 2 bundle ov er the base S U (3) /H , where each ﬁb er S 2 is generated b y the action of H ≃ U (2). In this manner, as ξ ( x,y ) approac hes ξ (1 , 0) , whose isotropic subgroup is H , the 2-spheres given b y the action of H on ξ ( x,y ) m ust collapse. More explicitly , via the parametrization of H by Euler angles, (2.15) R U ( α, β , γ ) = exp( − iαU 3 ) exp  − β 2 ( U + − U − )  exp( − iγ U 3 ) , 6 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS w e get the following parametrization of the ﬁb er that con tains ξ ( x,y ) : (2.16) √ 3 4 { 2 x + y (1 − cos( β )) } E 3 + √ 3 2 y sin( β ) cos( α ) E 6 + √ 3 2 y sin( β ) sin( α ) E 7 + 1 4 { 2 x + y (1 + 3 cos( β )) } E 8 = i 2 √ 6 4 x + 2 y 0 0 0 − 2 x − y 0 0 0 − 2 x − y ! + iy 2 r 3 2 0 0 0 0 cos( β ) e − iα sin( β ) 0 e iα sin( β ) − cos( β ) ! . This is a 2-sphere centered at the diagonal matrix (2.17) √ 3 4 (2 x + y ) E 3 + 1 4 (2 x + y ) E 8 = i 2 √ 6 (2 x + y )  2 , − 1 , − 1  in the aﬃne 3-dimensional space given by translations b y E 6 , E 7 , E 3 − √ 3 E 8 , and the radius of the sphere is (2.18)  ( y ) = √ 3 2 y → 0 , as y → 0 . P arameterizing the solutions of (2.10) by y ∈ [0 , 1], we get (2.19) x ( y ) = − y + p 4 − 3 y 2 2 = ⇒ ξ ( x,y ) = ξ ( x ( y ) ,y ) =: ζ y , so that we ha v e F = { ζ y : y ∈ [0 , 1] } and F = { ζ y : y ∈ (0 , 1) } , and w e set (2.20) F ≤ := { ζ y : y ∈ [0 , 1 / √ 3] } , F ≤ := { ζ y : y ∈ (0 , 1 / √ 3] } . Th us each leaf O ( x,y ) = O ζ y of the symplectic foliation in a neighborho o d of O (1 , 0) = O ζ 0 in S 7 is parametrized by y ∈ [0 , 1 / √ 3] and (2.21) f : F ≤ → R + , y 7→ f ( y ) = √ 3 4 y 2 , is a Morse function for the Morse-Bott singularit y at y = 0. Analogously , for x ≤ 1 / √ 3, y ≥ 1 / √ 3, we consider the orbit O ( x,y ) as an S 2 bundle ov er S U (3) / q H and obtain equations (2.18)-(2.21) with x ↔ y in terchanged, describing the foliation in a neighborho o d of the Bott-Morse singular orbit O ξ (0 , 1) . F urthermore, the tw o closed neigh b orho ods { 0 ≤ y ≤ 1 / √ 3 } and { 1 ≥ y ≥ 1 / √ 3 } are glued together at the mesonic orbit O ( x,y ) with x = y = 1 / √ 3. Th us, the singular foliation of ( S 7 , b Π g ) b y (co)adjoin t orbits, with singularities of Morse-Bott type, is analogous to the singular foliation of S 2 b y circles of constant latitude, with singularities of Morse t ype, except that no w w e hav e isolated singular orbits (isomorphic to C P 2 ), instead of isolated singular p oints. But for our purp oses, it will also b e useful to construct the foliation via the sp ecial p olynomial function b elow. Again, let ( x 1 , ..., x 8 ) b e co ordinates on sl (3) in the basis { E 1 , ..., E 8 } and recall the parametrization ξ ( x,y ) ∈ F , cf. (2.9)-(2.10). Prop osition 2.2. The p olynomial τ : sl (3) → C given by (2.22) τ = 6( x 2 1 + x 2 2 + x 2 3 ) x 8 − 2 x 3 8 + 6 √ 3( x 1 ( x 4 x 6 + x 5 x 7 ) − x 2 ( x 4 x 7 − x 5 x 6 )) − 3( x 2 4 + x 2 5 + x 2 6 + x 2 7 ) x 8 + 3 √ 3 x 3 ( x 2 4 + x 2 5 − x 2 6 − x 2 7 ) is S U (3) -invariant and sep ar ates t he p oints of F . ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 7 The proof of this proposition is deferred to after Proposition 2.25, since the latter will b e used for this pro of. Remark 2.3. Thus, for e ach ξ ∈ F , the orbit O ξ ⊂ S 7 is exactly the pr eimage by τ | S 7 of the r e al numb er (2.23) χ ξ := τ ( ξ ) . In addition, the p olynomial which is the c omplement of the r estriction τ | O ξ , (2.24) ˇ τ ξ := τ | S 7 − χ ξ ∈ P oly ( S 7 ) , is an S U (3) -invariant p olynomial vanishing on O ξ and only on this orbit of S 7 . 2.1.2. The c o arse Poisson spher e. How ev er, we shall not concern ourselv es with functions on all unitary orbits, but only on a countable family identiﬁed as follows. Consider the equiv alence relation ∼ on orbits of su (3), which is given by rescaling: (2.25) O ∼ O ′ ⇐ ⇒ ∃ α > 0 s.t. v 7→ αv is a bijection O → O ′ . Deﬁnition 2.4. An integral orbit is the orbit in su (3) of a dominant weight. A rational orbit is an orbit in S 7 ⊂ su (3) e quivalent to some inte gr al orbit. Notation 2.5. We shal l denote by Q ⊂ F the subset of r ational orbits, and by Q the r esp e ctive subset of F . Deﬁnition 2.6. F or e ach ξ ∈ Q , its integral radius is (2.26) r ( ξ ) := min { R > 0 : R ξ is a dominant weight } and its ﬁrst dominant w eigh t is (2.27) ω ξ := r ( ξ ) ξ . In other words, for each ξ ∈ Q , (2.28) ( r ( ξ )) 2 = ∥ ω ξ ∥ 2 = 2 3 ( p 2 1 + p 1 q 1 + q 2 1 ) , ω ξ = r ( ξ ) ξ = ω ( p 1 ,q 1 ) , where ω ξ = ω ( p 1 ,q 1 ) is the ﬁrst nonzero dominan t weigh t prop ortional to ξ ∈ Q , that is, the dominant weigh t ω ( p 1 ,q 1 ) ∝ ξ with the smallest 1 nonzero norm in su (3), whic h is by deﬁnition the in tegral radius r ( ξ ) of ξ . Note that for ξ ∈ ( Q \ Q ) = ( F \ F ), we hav e r ( ξ ) = p 2 / 3 and the ﬁrst dominan t weigh t is either ω (1 , 0) =  1 , for the deﬁning representation of S U (3), or ω (0 , 1) =  2 for its dual, cf. (2.1). On the other hand, for any ξ ( x,y ) ∈ F , we hav e that ξ ( x,y ) ∈ Q if and only if x/y ∈ Q (hence Deﬁnition 2.4 and Notation 2.5), thus the set of rational orbits is dense in the set of all adjoin t unitary orbits. Therefore, the collection of all rational orbits provides a coun tably dense sym- plectic foliation of the Poisson manifold ( S 7 , b Π g ) which includes the singular leav es O (1 , 0) ≃ O (0 , 1) ≃ C P 2 of foliation (2.14). W e denote this b y (2.29) [ ξ ∈Q ( O ξ , Π g | O ξ ) =: {S 7 , b Π g } ⊂ ( S 7 , b Π g ) . Deﬁnition 2.7. We shal l r efer to {S 7 , b Π g } as the rational coarsening of ( S 7 , b Π g ) , or simply r efer to {S 7 , b Π g } as the coarse Poisson sphere . 2 1 Clearly , if ω ( p 1 ,q 1 ) ∝ ξ , then ω ( sp 1 ,sq 1 ) ∝ ξ ∀ s ∈ N , with   ω ( sp 1 ,sq 1 )   = s   ω ( p 1 ,q 1 )   . 2 In implicit contrast to the smooth Poisson sphere ( S 7 , b Π g ). 8 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Remark 2.8. As emphasize d, {S 7 , b Π g } is the dense subset of ( S 7 , b Π g ) wher e we have a wel l deﬁne d function (2.30) r : {S 7 , b Π g } → R + , O ξ 7→ r ( ξ ) , for r ( ξ ) the inte gr al r adius 3 of O ξ , cf. (2.26) -(2.28). This function r , as deﬁne d by (2.26) -(2.30), has minimum e qual to p 2 / 3 , which is the inte gr al r adius of the two singular orbits in S 7 , but r has no upp er b ound b e c ause we c an have ξ ( x,y ) ∝ ω ( p 1 ,q 1 ) for p 1 and q 1 without c ommon divisors and as lar ge as we want. In fact, the argument in Remark 2.8 actually implies: Prop osition 2.9. The inte gr al r adius function r : Q → R + , cf. (2.26) - (2.30) , is unb ounde d on any neighb orho o d of any ξ ∈ Q . Remark 2.10. The e quivalenc e r elation (2.25) c omp ensates, up to a p oint, for the fact that we wil l b e working with actual adjoint orbits emb e dde d in su (3) , r ather than the abstr act orbits C P 2 or E . A b onus for this setting is that we shal l later b e able to investigate how the twiste d algebr as deﬁne d for e ach O ξ ∈ {S 7 , b Π g } c an or c annot b e “glue d” along the r ational c o arsening of Poisson manifold ( S 7 , b Π g ) , for appr opriate families of symb ol c orr esp ondenc e se quenc es, in an asymptotic limit. 2.2. Main results for C ∗ -algebras on (co)adjoin t orbits. W e now state and pro ve the main results for C ∗ -algebras on (co)adjoint orbits of S U (3) that will b e relev ant for our considerations on asymptotics of quark systems. 4 First, for the particular cases of pure-quark systems, we hav e the analogous of the no-go theorem for spin systems, that is, w e hav e the theorem below whic h is just the translation for the pair ( S U (3) , C P 2 ) of the theorem prov ed by Rieﬀel in [16] for the pair ( S U (2) , C P 1 ). 5 Theorem 2.11. Any S U (3) -e quivariant unital C ∗ -algebr a structur e on C ∞ C ( C P 2 ) is c ommutative. Pr o of. W e shall follo w closely to Rieﬀel’s pro of for ( S U (2) , C P 1 ), making the nec- essary adaptations for ( S U (3) , C P 2 ). The main idea is to show that the pro duct of linear p olynomials is commutativ e and generates the en tire algebra for C ∞ C ( C P 2 ). Let A ⋆ = ( A, , ∗ , ∥ ∥ ) denote a S U (3)-equiv arian t unital C ∗ -algebra structure on A = C ∞ C ( C P 2 ), where  , ∗ and ∥ ∥ are the pro duct, inv olution and C ∗ -norm, resp ectiv ely . W e know that A decomp oses as a sum of irreps ( n, n ), for every non negativ e in teger n , and each such irrep app ears just once, cf. Prop osition I.4.2 and Deﬁnition I.4.3. Let A n ⊂ A b e the inv arian t subspace where S U (3) acts via the irrep ( n, n ), so that (2.31) A = A 0 ⊕ A 1 ⊕ A 2 ⊕ · · · A n ⊕ A n +1 ⊕ · · · Lemma 2.12. A 0 is the line ar sp an of the identity in A ⋆ = ( A, , ∗ , ∥ ∥ ) . 3 W e emphasize, for clarity , that the integral radius r of a rational orbit in {S 7 , b Π g } is not the radius ϱ of the t wo-sphere that ﬁb ers ov er C P 2 for a generic orbit, cf. (2.18). 4 It is not yet known to us whether (some of ) the results presented b elow hav e b een stated or prov ed b efore, therefore we do so here. 5 In [16], Rieﬀel actually stated his theorem with respect to S O (3), but since the action of S U (2) on S 2 ≃ C P 1 is eﬀectively an action of S O (3), the tw o statements are equiv alent. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 9 Pr o of. Let e ∈ A b e the identit y in A ⋆ = ( A, , ∗ , ∥ ∥ ). Then (2.32) e g a = ( ea g − 1 ) g = a = ( a g − 1 e ) g = ae g , hence e ∈ A 0 . □ The next lemma is the crucial part of the pro of of the theorem. Lemma 2.13. The pr o duct on A 1 ⊂ A is c ommutative, that is, (2.33) a  b = b  a , ∀ a, b ∈ A 1 . Pr o of. Consider the commutator on A ⋆ : (2.34) [ a, b ] ⋆ := a  b − b  a , ∀ a, b ∈ A . Since A ⋆ is S U (3)-equiv arian t, the map A 1 × A 1 ∋ ( a, b ) 7→ [ a, b ] ⋆ ∈ A factors through an equiv ariant map A 1 ∧ A 1 ∋ a ∧ b 7→ [ a, b ] ⋆ ∈ A . First, one can easily v erify that (2.35) (1 , 1) ∧ (1 , 1) = (1 , 1) ⊕ (3 , 0) ⊕ (0 , 3) . Then, by straightforw ard computations, we obtain that the highest weigh t vectors for each resp ectiv e summand in (2.35) are as follows (cf. Deﬁnition I.2.1): (2.36) e (1 , 1) > = r 3 2 e ((1 , 1); 0 1 , 0) − 1 √ 2 e ((1 , 1); 0 1 , 1) ! ∧ e ((1 , 1); (210) , 1 / 2) + e ((1 , 1); (120) , 1) ∧ e ((1 , 1); (201) , 1 / 2) , e (3 , 0) > = e ((1 , 1); (201) , 1 / 2) ∧ e ((1 , 1); (210) , 1 / 2) , e (0 , 3) > = e ((1 , 1); (120) , 1) ∧ e ((1 , 1); (210) , 1 / 2) . By Sch ur’s Lemma and the decomp osition of A into irreps ( n, n ), cf. (2.31), we conclude that the in v ariant subspace of A 1 ∧ A 1 corresp onding to (3 , 0) ⊕ (0 , 3) is in the k ernel of the induced commutator map, whereas the restriction of such map to the inv ariant subspace corresp onding to (1 , 1) is either an isomorphism or the null map, hence [ A 1 , A 1 ] ⋆ is either A 1 or 0. Supp ose that [ A 1 , A 1 ] ⋆ = A 1 . Then (2.37) e (1 , 1) > + T − ( e (3 , 0) > ) = √ 6 e (1;  1 , 0) ∧ e (1; (210) , 1 / 2) is mapp ed into a highest weigh t vector of A 1 b y the induced commutator map on A 1 ∧ A 1 . So we can choose a 0 , a > ∈ A 1 , where a 0 is self adjoint and a > is a highest w eight vector, suc h that [ a 0 , a > ] ⋆ = a > . Let B k b e the space spanned b y the  - pro duct of at most k elements of A 1 ⊂ A ⋆ , and B ⋆ the algebra generated b y A 1 . Since the pro duct map A 1 × A n → A that sends ( a 1 , a ) ∈ A 1 × A n to a 1  a is a bilinear map, it factors through A 1 ⊗ A n . Then, by the equiv ariance of A ⋆ and the Clebsc h-Gordan series of (1 , 1) ⊗ ( n, n ), we get that B k ⊂ A 1 ⊕ ... ⊕ A k . By the Leibniz rule and induction on k , we hav e (2.38) [ a 0 , ( a > ) k ] ⋆ = a >  [ a 0 , ( a > ) k − 1 ] ⋆ + [ a 0 , a > ]  ( a > ) k − 1 = k ( a > ) k for every k ∈ N . Hence B 1 ⊂ ... ⊂ B k ⊂ ... even tually stabilizes, otherwise [ a 0 , · ] ⋆ w ould b e an unbounded operator, from (2.38), contradicting the fact the we hav e a C ∗ -algebra. So B ⋆ is ﬁnite dimensional and there is some k such that C ⋆ = A 0 ⊕ B ⋆ is a ﬁnite dimensional unital subalgebra whose underlying space decomp oses as (2.39) C = A 0 ⊕ A 1 ⊕ ... ⊕ A k . 10 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS On the other hand, b ecause C ⋆ is a ﬁnite dimensional C ∗ -algebra, in principle it w ould b e a direct sum of full matrix algebras, say (2.40) C ⋆ ≃ n M j =1 M C ( d j ) . But b y the assumption [ A 1 , A 1 ] ⋆ = A 1 and Sc hur’s Lemma, we hav e a S U (3)- equiv ariant homomorphism φ : su (3) → A 1 inducing an inner action α ϕ of su (3) on C ⋆ that coincides with the inﬁnitesimal action induced b y the natural S U (3)-action. Then eac h identit y 1 j ∈ M C ( d j ) is in the center of C ⋆ , which means eac h 1 j is ﬁxed b y S U (3). How ev er, C has only one copy of the trivial irrep of S U (3), namely on A 0 , cf. (2.39), hence [ A 1 , A 1 ] ⋆ = A 1 implies that C ⋆ is a S U (3)-equiv arian t unital subalgebra of A ⋆ isomorphic to a full matrix algebra, (2.41) C ⋆ ≃ M C ( d ) , and furthermore implies that we ha v e a S U (3)-equiv ariant isomorphism (2.42) A ⋆ ≃ C ⋆ ⊗ C ′ ⋆ , where C ′ ⋆ is the comm utan t of C ⋆ in A ⋆ , whose underlying space C ′ is also in v ariant b y the action of S U (3). Therefore, from (2.31) and (2.39), either C ′ = A 0 , in which case A ⋆ ≃ C ⋆ is isomorphic to a matrix algebra, cf. (2.41), in contradiction to A = C ∞ C ( C P 2 ), or for each non trivial A n ⊂ C ′ , A 1 ⊗ A n has tw o copies of A n , another contradiction, cf. (2.31). Th us [ A 1 , A 1 ] cannot b e A 1 and hence [ A 1 , A 1 ] = 0. □ T o ﬁnish the pro of of the theorem, let again C ⋆ b e the C ∗ -subalgebra generated b y A 0 ⊕ A 1 , and let C k b e the linear span of the pro duct of at most k elements in A 0 ⊕ A 1 , for k ∈ N . As already argued in the pro of of the previous lemma, C k ⊂ A 0 ⊕ A 1 ⊕ ... ⊕ A k . Suppose that the c hain C 1 ⊂ C 2 ⊂ ... ⊂ C k ⊂ ... ev entually stabilizes, whic h means A n 0  A 1 ⊂ A n 0 − 1 ⊕ A n 0 for some n 0 ∈ N , and C ⋆ is a ﬁnite-dimensional C ∗ -algebra. Again, C ⋆ w ould in principle b e a direct sum of full matrix algebras, cf. (2.41). But since C ⋆ is comm utative due to Lemmas 2.12 and 2.13, we could at most ha v e (2.43) C ⋆ ≃ dim C M j =1 C j , C j ≃ C ∀ j ∈ { 1 , ..., dim C } . Let 1 j ∈ C j b e its iden tit y , so that the primitiv e sp ectrum of C ⋆ is a ﬁnite discrete space Prim( C ⋆ ) = { ker( π 1 ) , ..., k er( π dim C ) } , where each π j is multiplication by 1 j , whic h works as a pro jection onto C j . By S U (3)-equiv ariance of C ⋆ , we hav e an induced contin uous action of S U (3) on Prim( C ⋆ ). Since S U (3) is connected, this action is trivial, implying that each 1 j is ﬁxed by S U (3). But since C carries only one copy of (0 , 0), namely the subspace A 0 , cf. (2.39), C ⋆ m ust b e isomorphic to C , whic h con tradicts the fact that A 1 ⊂ C . Therefore, for ev ery k ∈ N , C k is a prop er subspace of C k +1 and C = A = ⇒ C ⋆ = A ⋆ is commutativ e. □ Then, the following corollary is immediate from Theorem 2.11 and its pro of. Corollary 2.14. L et O ≃ C P 2 b e a nongeneric (c o)adjoint orbit of S U (3) . Then, ther e is no S U (3) -e quivariant unital C ∗ -algebr a structur e A ⋆ =  A, , ∗ , ∥ ∥  for A = C ∞ C ( O ) with a nontrivial S U (3) -e quivariant homomorphism φ : su (3) → A ⋆ as (2.44) φ : su (3) ∋ X 7→ a X ∈ A , a [ X,Y ] = [ a X , a Y ] ⋆ , ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 11 wher e [ · , · ] ⋆ is the c ommutator in A ⋆ . F urthermor e, if A ⋆ =  A, , ∗ , ∥ ∥  is a S U (3) - e quivariant unital C ∗ -algebr a structur e for an invariant subsp ac e A ⊂ C ∞ C ( O ) , then a nontrivial S U (3) -e quivariant homomorphism φ : su (3) → A ⋆ as in (2.44) exists only if A is ﬁnite dimensional, in which c ase A ⋆ is isomorphic to a ful l matrix algebr a with a φ -induc e d inner action (2.45) α ϕ : su (3) × A ⋆ → A ⋆ , ( X, a ) 7→ [ a X , a ] ⋆ , which c oincides with the natur al action 6 of su (3) on A . No w, for a generic (co)adjoint orbit O ≃ E of S U (3), an analogous of Theorem 2.11 is not known to us. But we can state a weak ened version of Corollary 2.14. Deﬁnition 2.15. A S U (3) -e quivariant unital C ∗ -algebr a A ⋆ =  A, , ∗ , ∥ ∥  is a b ona-ﬁde S U (3) - C ∗ - algebra if ther e is a nontrivial S U (3) -e quivariant homomor- phism φ : su (3) → A ⋆ as in (2.44) inducing a nontrivial inner action α ϕ of su (3) on A ⋆ as in (2.45) . In this c ase, we denote the algebr a by A ϕ ⋆ =  A, , ∗ , ∥ ∥ , φ  . Prop osition 2.16. L et O ≃ E b e a generic (c o)adjoint orbit of S U (3) and assume that A ϕ ⋆ =  A, , ∗ , ∥ ∥ , φ  is a b ona-ﬁde S U (3) - C ∗ -algebr a for A = C ∞ C ( O ) . Then, the C ∗ -algebr a gener ate d by φ ( su (3)) is a ﬁnite-dimensional b ona-ﬁde S U (3) - C ∗ - sub algebr a C ϕ ⋆ ⊂ A ϕ ⋆ which is isomorphic to the algebr a of op er ators on an irr ep of S U (3) and we have the S U (3) -e quivariant isomorphism (2.46) A ϕ ⋆ ≃ C ϕ ⋆ ⊗ C ′ ⋆ , wher e C ′ ⋆ is the c ommutant of C ϕ ⋆ in A ϕ ⋆ . F urthermor e, the φ -induc e d inner action α ϕ of su (3) on C ϕ ⋆ c oincides with the natur al su (3) -action on the underlying sp ac e C ⊂ A , but α ϕ vanishes on C ′ ⋆ . Pr o of. The pro of follo ws closely to most of the pro of of Lemma 2.13. Denote by A 1 ⊂ A the complex linear span of the image of φ . 7 Then, similar to what we did in the pro of of Lemma 2.13, for each k ∈ N , let B k b e the linear span of pro ducts of at most k elements of A 1 . Eac h B k is an S U (3)-inv arian t subspace of A for which the natural su (3)-action coincides with the induced inner action as in (2.45), that is, for ev ery X ∈ su (3) there is a X ∈ A 1 suc h that the natural action of X on B k is of the form B k ∋ b 7→ [ a X , b ] ⋆ ∈ B k . Again, we claim that the chain B 1 ⊂ ... ⊂ B k ⊂ ... stabilizes. Supp ose it do esn’t. Then, there is a sequence ( D k ) k ≥ 2 suc h that each D k ⊂ B k \ B k − 1 is a S U (3)-in v ariant subspace of A carrying a representation a k with (2.47) lim k →∞ | a k | = ∞ , cf. Notation I.2.3. Th us, we can take a X 0 = φ ( X 0 ) ∈ A 1 for X 0 = 2 i ( T 3 + U 3 ) and normalized highest weigh t vectors e k > ∈ D k , so that (2.48)   [ a X 0 , e k > ] ⋆   = | a k | → ∞ , whic h is absurd, since [ a X 0 , · ] ⋆ m ust b e a b ounded op erator. Therefore, the C ∗ -algebra generated b y A 1 = Span C ( φ ( su (3))) is a ﬁnite dimen- sional C ∗ -subalgebra C ϕ ⋆ ⊂ A ϕ ⋆ with a closed nontrivial inner action of su (3), (2.49) α ϕ : su (3) × C ϕ ⋆ → C ϕ ⋆ , ( X, c ) 7→ [ a X , c ] ⋆ , 6 S U (3) acts on the space A of A ⋆ and this induces the natural inﬁnitesimal action of su (3). 7 The Lie algebra su (3) is a r eal vector space and the homomorphism (2.44) is a real linear map. 12 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS whic h coincides with the natural su (3)-action on the underlying space C ⊂ A . In complete analogy to Lemma 2.12, the subspace A 0 ⊂ A of in v ariant elements is unidimensional and is generated b y the identit y of A ϕ ⋆ . Using a suitable Casimir op erator, cf. (I.B.3), the morphism of su (3) in to C ϕ ⋆ creates a non trivial inv ariant elemen t in C ϕ ⋆ , th us A 0 ⊂ C ϕ ⋆ so that C ϕ ⋆ is also unital, hence it is a bona-ﬁde S U (3)- C ∗ -subalgebra of A ϕ ⋆ , and in the same vein as was shown in Lemma 2.13, C ϕ ⋆ m ust b e isomorphic to a full matrix algebra, (2.50) C ϕ ⋆ ≃ M C ( d ) . In particular, the comp osition of φ with the ab o ve isomorphism giv es a represen- tation of su (3) on C d , whic h is the inﬁnitesimal action induced by a representation of S U (3) since the group is simply connected. Suc h S U (3)-representation on C d is irreducible b ecause a pro jection on any inv arian t subspace of C d spans a trivial irrep of S U (3) within C ϕ ⋆ , but C ϕ ⋆ carries only one copy of the trivial irrep, namely A 0 . Thus, C ϕ ⋆ is isomorphic to the algebra of op erators on an irrep of S U (3), and w e hav e the global S U (3)-equiv arian t isomorphism (2.46) with α ϕ v anishing on C ′ ⋆ , the commutan t of the C ∗ -algebra generated by φ ( su (3)). □ In view of the ab ov e, we in tro duce: Deﬁnition 2.17. A b ona-ﬁde S U (3) - C ∗ -algebr a A ϕ ⋆ =  A, , ∗ , ∥ ∥ , φ  is a faithful S U (3) - C ∗ -algebr a if the inner su (3) -action α ϕ c oincides with the natur al su (3) - action on the underlying sp ac e A . Deﬁnition 2.18. L et A ϕ ⋆ =  A, , ∗ , ∥ ∥ , φ  b e a b ona-ﬁde S U (3) - C ∗ -algebr a. If A ϕ ⋆ de c omp oses as in (2.46) , wher e C ϕ ⋆ is a faithful S U (3) - C ∗ -sub algebr a and α ϕ vanishes on C ′ ⋆ , then C ϕ ⋆ is the S U (3) - core of A ϕ ⋆ . Th us, a b ona-ﬁde S U (3)- C ∗ -algebra A ϕ ⋆ =  A, , ∗ , ∥ ∥ , φ  is faithful if and only if A ϕ ⋆ = C ϕ ⋆ ( C ′ = A 0 in (2.46)), and we can restate the previous results as: Corollary 2.19. L et O b e any (c o)adjoint orbit of S U (3) and A ⊆ C ∞ C ( O ) an invariant subsp ac e. If A ϕ ⋆ =  A, , ∗ , ∥ ∥ , φ  is a faithful S U (3) - C ∗ -algebr a, then A is ﬁnite dimensional. Mor e gener al ly, if A ϕ ⋆ is a b ona-ﬁde S U (3) - C ∗ -algebr a, then A ϕ ⋆ has a ﬁnite-dimensional S U (3) -c or e C ϕ ⋆ isomorphic to the algebr a of op er ators on an irr ep of S U (3) deﬁne d by φ . In p articular, if O ≃ C P 2 , then A ϕ ⋆ = C ϕ ⋆ . 2.3. Preliminary considerations for semiclassical asymptotics. W e no w re- ﬂect on the semiclassical asymptotics for quark systems, in light of the results of the previous subsection. First, w e look at the program of deformation quan tization. Since ev ery (co)adjoin t orbit O of S U (3) is a Hamiltonian S U (3)-space [13], the S U (3)-inv arian t symplectic form on O , cf. (2.14), deﬁnes the classical algebra of observ ables, whic h is the Poisson algebra A P =  A, · , {· , ·}  , where · is the p oin t wise pro duct on A = C ∞ C ( O ), with resp ect to which the Poisson brack et {· , ·} is a deriv ation in b oth entries. F urthermore, we ha v e a nontrivial equiv ariant homomorphism b φ from su (3) to A P , (2.51) b φ : su (3) → A , X 7→ b a X , s.t. b a [ X,Y ] = { b a X , b a Y } , whic h induces a nontrivial action b α of su (3) on A P , given by (2.52) b α : su (3) × A P → A P , ( X, f ) 7→ { b a X , f } . ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 13 In this setting, the program of deformation quan tization amounts to deforming the point wise product · on A P to a noncomm utativ e product  ℏ on A [[ ℏ ]], the ring of formal pow er series in the deformation parameter ℏ with coeﬃcients in A = C ∞ C ( O ), suc h that, for any f = P ∞ k =0 f k ℏ k ∈ A [[ ℏ ]] and g = P ∞ k =0 g k ℏ k ∈ A [[ ℏ ]], 8 (2.53) lim ℏ → 0 f  ℏ g = f 0 g 0 , lim ℏ → 0  ℏ − 1 ( f  ℏ g − g  ℏ f )  = i { f 0 , g 0 } . Suc h a formal algebra A ℏ = ( A [[ ℏ ]] ,  ℏ ) w ould be though t of as a ‘quan tum” algebra deforming the classical algebra A P and this is often called a quantization of O . But since S U (3) is the symmetry group of O , any true quantum algebra which resp ects the S U (3)-equiv ariance of A is a S U (3)-equiv arian t unital C ∗ -algebra. Ho wev er, from Theorem 2.11, for A = C ∞ C ( C P 2 ) it is impossible for suc h a formally deformed algebra A ℏ to conv erge 9 to a S U (3)-equiv ariant unital C ∗ -algebra A ⋆ suc h that its commutator tends to the Poisson brac ket in some limit of elemen ts in A . F urthermore, any S U (3)-symmetric quantum algebra worth y of its name must ha ve an inner action of su (3), that is, quantum op erators generating the symmetry group. Ho w ev er, from Prop osition 2.16, for A = C ∞ C ( E ) it is imp ossible for such a formally deformed algebra A ℏ to conv erge to a b ona-ﬁde S U (3)- C ∗ -algebra A ϕ ⋆ suc h that its commutator approaches the Poisson brack et in some limit, since the su (3)-action (2.52) is trivial only on the subspace of constant functions, but the inner su (3)-action on A ϕ ⋆ is only nontrivial on a ﬁnite dimensional subspace of A . On the other hand, from Corollary 2.19, for an y (co)adjoint orbit O of S U (3), if w e ask for a nontrivial S U (3)-equiv arian t homomorphism from su (3) to a S U (3)- equiv ariant unital C ∗ -algebra structure A ⋆ on some in v arian t subspace A ⊂ C ∞ C ( O ), w e end up with a S U (3)-core that is isomorphic to the algebra of operators on some irrep of S U (3), that is, a quantum quark system in the sense of Deﬁnition I.5.6. It follows from these previous results that, just as for spin systems and func- tions on C P 1 , in order to prop erly approach the asymptotic limit of noncomm u- tativ e pro ducts of functions on a S U (3)-(co)adjoint orbit O , we must w ork with sequences of symbol corresp ondences from quantum quark systems, in other words, sequences of twisted algebras deﬁned on increasing ﬁnite-dimensional subspaces of C ∞ C ( O ) which are induced from sym b ol corresp ondence sequences, and then study the asymptotic limit of such sequences as the dimension tends to inﬁnity . Th us, the ﬁrst problem w e must deal with is the identiﬁcation of sequences of quan tum quark systems that are suitable for semiclassical asymptotic analysis. F or pure-quark systems, the classical phase space is the orbit O ≃ C P 2 , and eac h symbol corresp ondence is an isomorphism b et w een the algebra of δ ( p ) × δ ( p ) complex matrices M C ( δ ( p )) and the corresp onding t wisted algebra (cf. Deﬁnition I.3.21) on a δ ( p ) 2 -dimensional subspace of C ∞ C ( C P 2 ), where δ ( p ) is the dimension of an S U (3)-irrep p = ( p, 0), or q p = (0 , p ), which is given b y (2.54) δ ( p ) = dim( p ) = dim( q p ) = ( p + 1)( p + 2) 2 , so that, p 7→ p + 1 = ⇒ δ ( p ) 7→ δ ( p + 1) = δ ( p ) + p + 2. In this scenario, we must consider sequences of quantum pure-quark systems (( p, 0)) p ∈ N or ((0 , p )) p ∈ N . 8 F or explicit y constructions of deformation quan tizations of coadjoin t orbits of compact semisimple Lie groups, we refer to [7, 14]. 9 F or instance, b y treating ℏ as a constan t, as it is in Ph ysics (and whic h for an appropriate choice of units can b e set ℏ = 1), reinterpreting the limits in (2.53) accordingly (semiclassical limit of high energies, high momen ta, high quantum num bers, high exp ectation v alues etc.). 14 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS The asymptotic analysis of suc h sequences of symbol correspondences and twisted algebras for pure-quark systems can b e work ed out in a wa y that, although quite more cum b ersome, is somewhat analogous to the treatment developed in [17] for spin systems. In Appendix B we summarize the steps and results of this approach . There we show the conditions (on the characteristic num b ers) for the sequence of symbol corresp ondences and their twisted algebras to b e of Poisson type, that is, for the sequence of twisted pro ducts (  p ) p ∈ N to b e suc h that, in some sense, 10 (2.55) lim p →∞ f  p g = f g , lim p →∞ p [ f , g ] ⋆ p = i { f , g } , ∀ f , g ∈ P ol y ( C P 2 ) . Ho wev er, the choice of sequences for pure-quark systems needs to b e b etter justiﬁed with a principle that can b e extended to mixed quark systems, where the classical phase space is a generic orbit O ≃ E and the matrix algebras of quantum quark systems are indexed not by single in tegers, but by pairs ( p, q ) of integers. Suc h a generalized principle shall lead to the deﬁnition of “rays” of corresp on- dences for each (co)adjoin t orbit O ξ in the coarse Poisson sphere {S 7 , b Π g } , cf. (2.29), as presented in the next section (cf. Deﬁnition 3.1). With this deﬁnition, w e shall be able to make sense of limits similar to the ones in (2.55) and thus study the conditions for such rays of corresp ondences to b e of Poisson t yp e. But even with such identiﬁcation of the sequences of general quantum quark systems suitable for semiclassical asymptotic analysis, the approach presented in App endix B is not easily generalized to the asymptotic analysis of mixed quark systems. So we shall dev elop a new framework using the PBW Theorem for the univ ersal env eloping algebra of sl (3), as presented in the next subsections. 2.4. PBW Theorem and P oisson algebras of harmonic functions. W e con- sider general orbits O ≃ O ξ ⊂ S 7 ⊂ su (3) and, in what follo ws, in v ok e the P oincar ´ e- Birkhoﬀ-Witt (PBW) Theorem to describ e the Poisson algebra on P ol y ( O ). First, we take sl (3) as the complexiﬁcation of su (3), (2.56) Span R { E j : j = 1 , ..., 8 } = su (3) ⊂ sl (3) = Span C { E j : j = 1 , ..., 8 } . Note that the restriction of complex p olynomials provides identiﬁcation P ol y ( sl (3)) ≡ P oly ( su (3)) . F urthermore, on sl (3) we hav e the bilinear form (2.57) ( X, Y ) = tr( X Y ) ∀ X , Y ∈ sl (3) , whic h is just a renormalization of the Killing form (and naturally restricts to su (3)), and which deﬁnes the standard inner pro duct (cf. (I.2.8)) (2.58) ⟨ X | Y ⟩ = ( X † , Y ) = tr  X † Y  ∀ X, Y ∈ sl (3) . W e consider the GT basis of sl (3), with adjoin t representation (1 , 1) of S U (3), as depicted on Figure 2.4. W e also imp ose an ordering on this orthonormal basis s.t. { e 1 , e 2 , e 3 } are annihilation op erators, { e 4 , e 5 , e 6 } are creation op erators and { e 7 , e 8 } are Cartan operators. Sp eciﬁcally , we shall c ho ose the ordered basis v ectors (2.59) e 1 = e (0 , 2 , 1)1 / 2 = T − , e 2 = e (1 , 0 , 2)1 = − U − , e 3 = e (0 , 1 , 2)1 / 2 = V − , e 4 = e (2 , 0 , 1)1 / 2 = − T + , e 5 = e (1 , 2 , 0)1 = U + , e 6 = e (2 , 1 , 0)1 / 2 = V + , e 7 = e  11 = − √ 2 U 3 , e 8 = e  10 = p 2 / 3( T 3 + V 3 ) = p 2 / 3(2 T 3 + U 3 ) , 10 The precise sense for these limits is presen ted in Section 3 and Appendix B. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 15 e (2 , 0 , 1) , 1 / 2 = − T + e (0 , 2 , 1) , 1 / 2 = T − e (2 , 1 , 0) , 1 / 2 = V + e (1 , 2 , 0) , 1 = U + e 0 1 , 1 = e  11 = − √ 2 U 3 e 0 1 , 0 = e  10 = r 2 3 ( T 3 + V 3 ) e (0 , 1 , 2) , 1 / 2 = V − e (1 , 0 , 2) , 1 = − U − Figure 1. GT basis for sl (3), cf. Deﬁnition I.2.1. cf. Deﬁnition I.2.1, and we denote this choice of ordered basis for sl (3) by (2.60) B 1 = { e 1 , ..., e 8 } . By PBW Theorem [11], the universal env eloping algebra U ( sl (3)) has a basis (2.61) B ∞ = [ d ∈ N 0 B d , B d = { e j 1 ...e j d : 1 ≤ j 1 ≤ ... ≤ j d ≤ 8 } , where the empty pro duct ( d = 0) is the unit y 1 and where e 1 , · · · , e 8 satisfy the comm utation relations of su (3) (but not any sp eciﬁc nilp otency relation apriori, that is, not represen ted by matrices of a sp eciﬁc dimension apriori). Thus, for eac h arbitrary d ∈ N , each basis vector in the ordered basis B d is an ordered pro duct of d elements of B 1 , this ordered pro duct induced by the order in B 1 . F or instance, (2.62) B 2 = { e 2 1 , e 1 e 2 , e 1 e 3 , · · · e 1 e 8 , e 2 2 , e 2 e 3 , · · · e 2 e 8 , e 2 3 , e 3 e 4 , · · · e 2 8 } . Deﬁnition 2.20 (PBW) . The univ ersal env eloping algebra of sl (3) , (2.63) U ( sl (3)) := Span C ( B ∞ ) , cf. (2.60) - (2.61) , is deﬁne d by the su (3) -c ommutation r elations for B 1 and the fact that c ommutation is a derivation. It is a gr ade d algebr a wher e e ach subsp ac e (2.64) U d ( sl (3)) := Span C ( B d ) is the sp ac e of elements of homogeneous degree d . On the other hand, the degree of a gener al u ∈ U ( sl (3)) is given by (2.65) deg( u ) := min { d ∈ N 0 : u ∈ U ≤ d ( sl (3)) } , wher e (2.66) U ≤ d ( sl (3)) := d M m =0 U m ( sl (3)) . Th us, for instance, e 2 e 1 is not homogeneous of degree 2, that is e 2 e 1 / ∈ U 2 ( sl (3)), but e 2 e 1 has degree 2 since e 2 e 1 = e 1 e 2 − [ e 1 , e 2 ] = e 1 e 2 − e 3 ∈ U ≤ 2 ( sl (3)). Therefore, once chosen the ordering (2.59)-(2.61) deﬁning B ∞ , the linear map (2.67) β : U ( sl (3)) → P ol y ( sl (3)) deﬁned in the basis B ∞ b y (2.68) β [ e j 1 ...e j d ]( X ) = ( e j 1 , X ) ... ( e j d , X ) ∀ X ∈ sl (3) , 16 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS is an isomorphism of vector spaces 11 , which breaks down into isomorphisms 12 (2.69) β | U d ( sl (3)) : U d ( sl (3)) → P oly d ( sl (3)) , ∀ d ∈ N 0 . In particular, for ω ( p,q ) as in (2.2), we hav e (2.70) ( β [2 T 3 ]( − iω ( p,q ) ) = p β [2 U 3 ]( − iω ( p,q ) ) = q . No w, lo oking at the (extension of the) adjoin t action (2.71) S U (3) × U ( sl (3)) ∋ ( g , u ) 7→ Ad g ( u ) ≡ g ug − 1 ∈ U ( sl (3)) , w e ha v e that U d ( sl (3)) is not S U (3)-inv arian t b ecause in general the action of g ∈ S U (3) on u ∈ U d ( sl (3)) adds monomials of low er degrees. Ho wev er, this action nev er adds monomials of higher degrees, hence U ≤ d ( sl (3)) is S U (3)-inv arian t. Prop osition 2.21. The line ar map β , cf. (2.67) - (2.68) , is not S U (3) -e quivariant. Pr o of. β gives an isomorphism b et w een U d ( sl (3)) and P ol y d ( sl (3)), cf. (2.69), and P ol y d ( sl (3)) is S U (3)-inv arian t but U d ( sl (3)) is not. □ On the other hand, deﬁning the natural pro jection (2.72) π d : U ( sl (3)) → U d ( sl (3)) , u 7→ π d ( u ) , w e hav e the follo wing prop osition. Prop osition 2.22. F or e ach d ∈ N 0 , the map (2.73) β d : U ≤ d ( sl (3)) → P oly d ( sl (3)) , u 7→ β d [ u ] := β [ π d ( u )] , is a line ar S U (3) -e quivariant surje ction. Pr o of. The statement is trivial for d = 0. F or d > 0, linearity and surjectivity are immediate, so we pro v e only equiv ariance. Note that, for an y u ∈ sl (3) ≡ U 1 ( sl (3)), (2.74) β 1 [ Ad g ( u )] = ( Ad g ( u ) , · ) = ( u, Ad g − 1 ( · )) = ( β 1 [ u ]) g , ∀ g ∈ S U (3) . F or d > 1, w e ha ve already argued ab o v e that U ≤ d − 1 ( sl (3)) is inv ariant, thus β d [ Ad g ( u )] = 0, ∀ u ∈ U ≤ d − 1 ( sl (3)), ∀ g ∈ S U (3). On the other hand, if u = e j 1 ...e j d is an element of B d , we hav e (2.75) Ad g ( u ) = X k 1 ,...,k d D (1 , 1) k 1 ,j 1 ( g ) ...D (1 , 1) k d ,j d ( g ) e k 1 ...e k d , where D (1 , 1) k,j are Wigner D -functions in the basis B 1 , cf. Deﬁnition I.2.6. In general, the indices k 1 , ..., k d are not necessarily in increasing order, so the rewriting of e k 1 ...e k d in the basis B ∞ , by applying commutation relations, splits in t wo parts: (2.76) e k 1 ...e k d = e k f k 1 ,...,k d (1) ...e k f k 1 ,...,k d ( d ) + v k 1 ,...,k d , 11 This is not a canonical isomorphism U ( sl (3)) → P oly ( sl (3)) since it depends on a given but not canonical choice of basis for U ( sl (3)), and is obviously not an algebra homomorphism. 12 Looking at the inv erses of (2.68) and (2.69), these are given by a choice of or der e d basis for each B d whose elemen ts are or der e d pro ducts of elements in B 1 . In the con text of aﬃne systems, where B 1 = { x i , ∂ /∂ x i } , this is also referred to as the ordering problem in quan tization. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 17 where f k 1 ,...,k d ∈ S d is some permutation that places the indices k 1 , ..., k d in in- creasing order, and v k 1 ,...,k d ∈ U ≤ d − 1 ( sl (3)). Therefore (2.77) π d ( Ad g ( u )) = X k 1 ,...,k d D (1 , 1) k 1 ,j 1 ( g ) ...D (1 , 1) k d ,j d ( g ) e k f k 1 ,...,k d (1) ...e k f k 1 ,...,k d ( d ) . Using the fact that the pro duct of p olynomials is commutativ e, the application of β on π d ( Ad g ( u )) allows us to leav e out the p ermutations f k 1 ,...,k d , giving (2.78) β d [ Ad g ( u )] = X k 1 ,...,k d D (1 , 1) k 1 ,j 1 ( g ) ...D (1 , 1) k d ,j d ( g ) β 1 [ e k 1 ] ...β 1 [ e k d ] = X k 1 D (1 , 1) k 1 ,j 1 ( g ) β 1 [ e k 1 ] ! ... X k d D (1 , 1) k d ,j d ( g ) β 1 [ e k d ] ! . W e hav e already prov ed that β 1 is equiv arian t, so (2.79) β d [ Ad g ( u )] = β 1 [ e j 1 ] g ...β 1 [ e j d ] g = ( β 1 [ e j 1 ] ...β 1 [ e j d ]) g = β d [ u ] g , whic h prov es the equiv ariance of β d . □ Using the commutation relations, one can easily v erify the next prop osition. Prop osition 2.23. The p ointwise pr o duct of elements of P ol y ( su (3)) satisﬁes (2.80) β deg( u )+deg( v ) [ uv ] = β deg( u ) [ u ] β deg( v ) [ v ] for every u, v ∈ U ( sl (3)) . F or the P oisson brack et, we ha v e the following c haracterization. Prop osition 2.24. The Poisson bive ctor Π g deﬁnes a Poisson br acket {· , ·} on P ol y ( su (3)) satisfying (2.81) n β deg( u ) [ u ] , β deg( v ) [ v ] o = β deg( u )+deg( v ) − 1 [ uv − vu ] for every u, v ∈ U ( sl (3)) . Pr o of. It is immediate that (2.81) is sk ew-symmetric. W e will sho w no w that it is a bideriv ation. F or any u, ˜ u, v ∈ U ( sl (3)), let d = deg( u ) + deg( ˜ u ) + deg( v ). By (2.80), we hav e (2.82) β deg( u ) [ u ] β deg( ˜ u ) [ ˜ u ] = β deg( u )+deg( ˜ u ) [ u ˜ u ] = ⇒  β deg( u ) [ u ] β deg( ˜ u ) [ ˜ u ] , β deg( v ) [ v ]  =  β deg( u )+deg( ˜ u ) [ u ˜ u ] , β deg( v ) [ v ]  = β d − 1 [ u ˜ uv − v u ˜ u ] = β d − 1 [ u ( ˜ uv − v ˜ u )] + β d − 1 [( uv − vu ) ˜ u ] , and again using (2.80), (2.83) β d − 1 [ u ( ˜ uv − v ˜ u )] = β deg( u ) [ u ] β deg( ˜ u )+deg( v ) − 1 [ ˜ uv − v ˜ u ] = β deg( u ) [ u ]  β deg( ˜ u ) [ ˜ u ] , β deg( v ) [ v ]  , (2.84) β d − 1 [( uv − vu ) ˜ u ] = β deg( u )+deg( v ) − 1 [ uv − vu ] β deg( ˜ u ) [ ˜ u ] =  β deg( u ) [ u ] , β deg( v ) [ v ]  β deg( ˜ u ) [ ˜ u ] , th us (2.81) is a deriv ation in the ﬁrst co ordinate. Since it is sk ew-symmetric, it is a bideriv ation. 18 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS T o ﬁnish, w e will v erify that (2.81) matches the Poisson brack et of Π g for linear p olynomials, and the bideriv ation prop ert y will imply equalit y for p olynomials of an y degree. F or the linear co ordinates ( x 1 , ..., x 8 ) in the basis { E 1 , ..., E 8 } , we hav e x j = tr  E † j ·  = − tr( E j · ) = − β 1 [ E j ] = ⇒ { x j , x k } = β 1 [ E j E k − E k E j ] = 8 X l =1 c l j k β 1 [ E l ] = 8 X l =1 c l kj x l = Π g ( x j , x k ) . Therefore, (2.85) { f , h } = Π g ( f , h ) , for every f , h ∈ P ol y ( su (3)). □ Finally , we shall also make use of the symmetrization linear map (2.86) S : P ol y ( sl (3)) → U ( sl (3)) , S ( β 1 [ e j 1 ] ...β 1 [ e j d ]) = 1 d ! X f ∈ S d e j f (1) ...e j f ( d ) , where S d is the symmetric group. Prop osition 2.25. The symmetrization map S is e quivariant. A lso, for every p olynomial f ∈ P ol y d ( sl (3)) , we have S ( f ) ∈ U ≤ d ( sl (3)) and β d [ S ( f )] = f . Pr o of. F rom Prop osition 2.22, (2.87) ( β 1 [ e j 1 ] ...β 1 [ e j d ]) g = X k 1 ,...,k d D (1 , 1) k 1 ,j 1 ( g ) ...D (1 , 1) k d ,j d ( g ) β 1 [ e k 1 ] ...β 1 [ e k d ] . Applying S , we obtain (2.88) S (( β 1 [ e j 1 ] ...β 1 [ e j d ]) g ) = 1 d ! X k 1 ,...,k d X f ∈ S d D (1 , 1) k 1 ,j 1 ( g ) ...D (1 , 1) k d ,j d ( g ) e k f (1) ...e k f ( d ) . The pro duct of Wigner D -functions is ob viously commutativ e, so (2.89) X k 1 ,...,k d D (1 , 1) k 1 ,j 1 ( g ) ...D (1 , 1) k d ,j d ( g ) e k f (1) ...e k f ( d ) = X k 1 ,...,k d  D (1 , 1) k f (1) ,j f (1) ( g ) e k f (1)  ...  D (1 , 1) k f ( d ) ,j f ( d ) ( g ) e k f ( d )  = Ad g ( e j f (1) ) ...Ad g ( e j f ( d ) ) = Ad g ( e j f (1) ...e j f ( d ) ) . Therefore, (2.90) S (( β 1 [ e j 1 ] ...β 1 [ e d ]) g ) = 1 d ! X f ∈ S d Ad g ( e j f (1) ...e j f ( d ) ) = Ad g ( S ( β 1 [ e j 1 ] ...β 1 [ e d ])) . This prov es the equiv ariance of S . The remaining of the statement follo ws straight- forw ardly from the deﬁnition. □ W e can no w prov e Prop osition 2.2. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 19 Pr o of of Pr op osition 2.2. Since τ is homogeneous of degree 3, w e ha v e τ = β 3 [ S ( τ )] from Prop osition 2.25. F rom equiv ariance of S and β 3 , cf. Propositions 2.22 and 2.25, τ is ﬁxed by S U (3) if and only if S ( τ ) is ﬁxed by S U (3). But S ( τ ) is (pro- p ortional to) the cubic Casimir op erator of sl (3), cf. (I.B.3) or e.g. [9, eq. (7.31)], so this completes the pro of of S U (3)-inv ariance for τ . F or the separation of orbits, note that (2.91) τ ( ξ ( x,y ) ) = 2 x 3 + 3 x 2 y − 3 xy 2 − 2 y 3 for every ξ ( x,y ) ∈ F . T aking (2.92) f ( x, y ) = 2 x 3 + 3 x 2 y − 3 xy 2 − 2 y 3 , h ( x, y ) = x 2 + xy + y 2 , the critical p oints of τ | F are the critical p oints of the restriction of f to the ellipse h = 1 in the ﬁrst quadrant, cf. (2.10). By the metho d of Lagrange multipliers, we w ant to solve for λ ∈ R and x, y ≥ 0, the system: (2.93) ( 2 x 2 + 2 xy − y 2 = λ (2 x + y ) x 2 − 2 xy − 2 y 2 = λ ( x + 2 y ) = ⇒ x 2 − y 2 = λ ( x + y ) . There are t w o kind of solutions: x + y = 0, which lies outside the ﬁrst quadran t, and x + y  = 0, which implies xy = 0, meaning the critical p oin t must b e an endpoint of F . Therefore, τ | F is injective. □ Remark 2.26. As a homo gene ous cubic p olynomial, τ is an o dd function, so (2.94) τ ( − ξ ( x,y ) ) = − τ ( ξ ( x,y ) ) . In p articular, τ vanishes on the mesonic orbit, cf. (I.2.76). This is aligne d with the fact that the cubic Casimir op er ator S ( τ ) assumes the form C ( p, q ) 1 in the r epr esentation ( p, q ) with C ( p, q ) = − C ( q , p ) . 2.5. Univ ersal corresp ondences for general quark systems. F or a dominan t w eight ω of su (3), let H ω b e a quark system with highest w eight ω . 13 If ω is prop ortional to ξ ∈ Q , that is, if ω = ∥ ω ∥ ξ , then the quantum quark system H ω admits symbol corresp ondences to O ξ , cf. Theorems I.4.8 and I.5.9. As suggested b y the previous subsection, it will b e useful to work on the universal algebra, so w e pullbac k sym b ol corresp ondences to U ( sl (3)) via the irreducible representation ρ ω : U ( sl (3)) → B ( H ω ) of the universal en v eloping algebra on H ω whic h is induced b y the irreducible representation of S U (3) on H ω in the natural wa y . Deﬁnition 2.27. Given a dominant weight ω = ∥ ω ∥ ξ , with ξ ∈ Q , a universal corresp ondence for ω , or simply a universal corresp ondence , is a map (2.95) w : U ( sl (3)) → P ol y ( O ξ ) : u 7→ w [ u ] that factors thr ough a symb ol c orr esp ondenc e W ω : B ( H ω ) → P oly ( O ξ ) and the irr ep ρ ω of U ( sl (3)) on H ω , as shown in the diagr am b elow: (2.96) U ( sl (3)) B ( H ω ) P oly ( O ξ ) ρ ω w W ω 13 F or simplicity , w e shall often denote a quantum quark system with highest weigh t ω = ω ( p,q ) , cf. Deﬁnitions I.4.6 and I.5.6, simply by its Hilb ert space H ω . 20 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Remark 2.28. Sinc e U ( sl (3)) is inﬁnite dimensional and B ( H ω ) is ﬁnite dimen- sional, e ach ρ ω (and henc e also w ) has an inﬁnite dimensional kernel, which is a primitive ide al of U ( sl (3)) by deﬁnition. Thus, universal c orr esp ondenc es ar e p ar- ticular instanc es of e quivariant line ar maps U ( sl (3)) → P oly ( O ξ ) whose kernels ar e elements of Prim U ( sl (3)) . Remark 2.29. In this way, ac c or ding to Deﬁnition 2.27, for a classic al mixe d- quark system O ξ ≃ E , ξ ∈ Q , we only c onsider c orr esp ondenc es fr om mixe d-quark systems H ω , with ω = ω ( p,q ) satisfying pq  = 0 , cf. Deﬁnition I.5.6. A family of corresp ondences of particular interest to us is the Berezin family . The pro jection Π > ∈ B ( H ω ) onto the highest weigh t subspace is an operator k ernel that gives a Berezin corresp ondence B ω : B ( H ω ) → P oly ( O ξ ), A 7→ B ω A , s.t. (2.97) B ω A ( Ad g ( ξ )) = tr( A Π g > ) , for ω = ∥ ω ∥ ξ , cf. Prop osition I.4.19, Remark I.4.20 and Theorem I.5.24. Deﬁnition 2.30. Given a dominant weight ω = ∥ ω ∥ ξ , ξ ∈ Q , the universal Berezin corresp ondence for ω is the universal c orr esp ondenc e b : U ( sl (3)) → P oly ( O ξ ) obtaine d fr om B ω given by (2.97) ac c or ding to Deﬁnition 2.27. The map β obtained from PBW theorem is very p ertinen t to describ e universal Berezin corresp ondences. Prop osition 2.31. The universal Ber ezin c orr esp ondenc e b : U ( sl (3)) → P oly ( O ξ ) for ω = ∥ ω ∥ ξ , ξ ∈ Q , is given by b [ u ]( Ad g ( ξ )) = β  Ad g − 1 ( u )  ( − iω ) for every u ∈ U ( sl (3)) and g ∈ S U (3) . Pr o of. Let e > ∈ H ω b e a highest weigh t unit vector. By deﬁnition, (2.98) b [ u ]( Ad g ( ξ )) =  e >   ρ ω  Ad g − 1 ( u )  e >  = β  Ad g − 1 ( u )  ( − iω ) , where the last equation comes from decomposing Ad g − 1 ( u ) in the basis B ∞ , cf. (2.61), and using (2.70). □ By construction, a universal corresp ondence w : U ( sl (3)) → P oly ( O ξ ) for the w eight ω = ∥ ω ∥ ξ , ξ ∈ Q , induces a twisted pro duct  on the image of w by (2.99) w [ u ]  w [ v ] = w [ uv ] for every u, v ∈ U ( sl (3)), so that, recalling Remark 2.28, this is the same pro duct induced by the symbol corresp ondence W : B ( H ω ) → P ol y ( O ξ ) that generates w . With that in mind, w e also import the notion of Stratono vic h-W eyl correspondences for the universal ones. Deﬁnition 2.32. A universal c orr esp ondenc e w : U ( sl (3)) → P ol y ( O ξ ) is of typ e Stratono vich-W eyl if, for every u, v ∈ U ( sl (3)) , (2.100) Z O ξ w [ uv ]( ς ) d ς = Z O ξ w [ u ]( ς ) w [ v ]( ς ) d ς . Th us, Prop osition I.3.13 translates for universal corresp ondences as: Theorem 2.33. No universal Ber ezin c orr esp ondenc e is Str atonovich-Weyl. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 21 3. Asymptotic anal ysis for general quark systems The ﬁrst problem we face in order to work out semiclassical analysis for quark systems is the iden tiﬁcation of p ertinent sequences of quantum quark systems 14 . W e need to ﬁnd some principle that recov ers the case of spin systems, where this problem do es not exist at all, cf. [17]. Suc h a principle should align with the fact that the orbits O (1 , 0) and O (0 , 1) , b eing isomorphic to C P 2 , corresp ond to classical pure-quark systems and only admit corresp ondences from quantum pure- quark systems ( p, 0) and (0 , p ). This restriction, together with Deﬁnition 2.27, p oin ts to a reasonable principle: giv en ξ ∈ Q so that O ξ ⊂ {S 7 , b Π g } , we shall lo ok at the ray from the origin in the direction of ξ , in the lattice of dominan t weigh ts; or in other words, we shall lo ok at the sequence of dominant weigh ts ( sω ξ ) s ∈ N . 3.1. Ra ys of univ ersal corresp ondences: fuzzy orbits. Deﬁnition 3.1. Given ξ ∈ Q so that O ξ ⊂ {S 7 , b Π g } , a ray of symbol corresp on- dences attached to ξ , or in short, a ξ -ra y of symbol corresp ondences is a se quenc e of symb ol c orr esp ondenc es  W sω ξ : B ( H sω ξ ) → P oly ( O ξ )  s ∈ N , cf. Deﬁnition 2.6. A c c or dingly, a ξ -ra y of univ ersal corresp ondences is a se quenc e  w s ξ : U ( sl (3)) → P ol y ( O ξ )  s ∈ N , wher e e ach w s ξ is an universal c orr esp ondenc e for s ω ξ ac c or ding to Deﬁnition 2.27. If ( W s ξ ) s ∈ N denotes the se quenc e of images of ( W sω ξ ) s ∈ N or ( w s ξ ) s ∈ N , we have the induc e d ξ -ra y of twisted products (  s ξ ) s ∈ N , wher e e ach  s ξ : W s ξ × W s ξ → W s ξ is given by (3.1) w s ξ [ u ]  s ξ w s ξ [ v ] = w s ξ [ uv ] , ∀ u, v ∈ U ( sl (3)) . Then, the p air se quenc e  W s ξ ,  s ξ  s ∈ N shal l b e c al le d a ξ -ray of twisted algebras , or in short a fuzzy ξ -orbit , denote d (3.2) W ( O ξ ) =  W s ξ ,  s ξ  s ∈ N . The restriction stated in Remark 2.29, that we do not consider corresp ondences from quantum pure-quark systems to classical mixed-quark systems, no w reverber- ates in the fact that the sequence of images of a ξ -ra y of corresp ondences, ( W s ξ ) s ∈ N , asymptotically cov ers P ol y ( O ξ ), as shown in Lemma 3.3 b elow. Notation 3.2. F or a line ar sp ac e V c arrying a r epr esentation of S U (3) , we denote by V a the maximal invariant subsp ac e of V wher e S U (3) acts via (c opies of ) the irr ep a . Lemma 3.3. Given ξ ∈ Q and f ∈ P ol y ( O ξ ) , ther e exists s 0 ∈ N such that, for every fuzzy ξ -orbit, we have f ∈ W s ξ , ∀ s ≥ s 0 . Pr o of. Without loss of generality , we assume f ∈ P ol y ( O ξ ) ( a,b ) . If ξ ∈ Q , recall (2.28), hence ( sp 1 , sq 1 ) ≡ s ω ξ . Then, w e can conclude f is in the image of w s ξ if (3.3) m ( sp 1 , sq 1 ; a, b ) = m ( a, b ) , 14 Recall that each quan tum quark system is iden tiﬁed by a pair of natural num bers, ( p, q ), so in principle we could ha ve bi-sequences of suc h systems, and this will b e explored in the next section. 22 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS cf. Notation I.5.8. F rom (I.5.17)-(I.5.18), it is suﬃcient to hav e (3.4) s min { p 1 , q 1 } ≥ max { a, b } . Therefore, it is suﬃcient to take (3.5) s 0 = max { a, b } , to ha v e f in the image of w s ξ for ev ery s ≥ s 0 . Now, if ξ ∈ Q \ Q , then f is nonzero only if a = b . Since ( a, a ) is m ultiplicity free in P oly ( O ξ ), and (3.6) ( s, 0) ⊗ (0 , s ) = s M n =0 ( n, n ) , f is in the image of w s ξ for every s ≥ max { a, b } , cf. (3.5). □ In view of the previous lemma, we in tro duce the follo wing. Deﬁnition 3.4. A ξ -r ay of (symb ol or universal) c orr esp ondenc es is of Poisson t yp e , or e quivalently a fuzzy ξ -orbit is of Poisson type , if the ξ -r ay of twiste d pr o ducts (  s ξ ) s ∈ N satisﬁes lim s →∞   f 1  s ξ f 2 − f 1 f 2   ξ = 0 , (3.7) lim s →∞    sr ( ξ )[ f 1 , f 2 ] ⋆ s ξ − i { f 1 , f 2 }    ξ = 0 , (3.8) for every f 1 , f 2 ∈ P ol y ( O ξ ) , wher e [ · , · ] ⋆ s ξ is the c ommutator of  s ξ and {· , ·} is the Poisson br acket deﬁne d by Ω ξ = b Π g | O ξ , cf. (2.14) . In this c ase, we denote (3.9) W ( O ξ ) ∼ − − → P oly ( O ξ , Ω ξ ) , cf. (3.2) , wher e the r.h.s. denotes the Poisson algebr a of p olynomials on O ξ . Remark 3.5. The eﬀe ctive asymptotic p ar ameter for the c ommutator [ · , · ] ⋆ s ξ , for e ach r ational orbit O ξ ⊂ {S 7 , b Π g } , is actual ly sr ( ξ ) , cf. (3.8) , wher e s ∈ N and r ( ξ ) is the inte gr al r adius of ξ , that is, ω ξ = r ( ξ ) ξ , cf. Deﬁnition 2.6. However, sinc e r ( ξ ) is ﬁxe d, for e ach ξ -r ay of c orr esp ondenc es ( w s ξ ) s ∈ N , we c an c onsider s ∈ N as the single asymptotic p ar ameter in (3.8) , ∀ ξ ∈ Q , sinc e s ∈ N is also the single asymptotic p ar ameter for the pr o duct, ∀ ξ ∈ Q , cf. (3.7) . This helps to r e ad al l asymptotic limits studie d as limits of se quenc es just indexe d by s ∈ N , ∀ ξ ∈ Q . But in this way, when c onsidering every ξ ∈ Q to gether, in Se ction 4, we shal l have to r esort to a ξ -dep endent r esc aling of the c ommutator of the twiste d pr o duct, as in Deﬁnition 4.2, further b elow. With the aim of verifying conditions for P oisson, w e ﬁrst explore rays of Berezin corresp ondences, reproducing some results of [12] for the particular setting of S U (3). This will mak e clear a suﬃcien t condition for a ξ -ra y of univ ersal cor- resp ondences to b e of Poisson type, and then we will prov e this condition is also necessary . 3.2. On the asymptotics of Berezin fuzzy orbits. Deﬁnition 3.6. F or any orbit O ξ ⊂ S 7 , its ξ -r ay of twiste d algebr as deﬁne d by the ξ -r ay of universal Ber ezin c orr esp ondenc es ( b s ξ ) s ∈ N , cf. Deﬁnition 2.30 and Pr op o- sition 2.31, and Deﬁnition 3.1, is c al le d the Berezin fuzzy ξ -orbit , denote d B ( O ξ ) . ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 23 Then, for B ( O ξ ), ξ ∈ Q , and every s ∈ N , consider the err or map (3.10) ε s ξ : U ( sl (3)) → P ol y ( O ξ ) , u 7→ ε s ξ [ u ] , ε s ξ [ u ] := ( sr ( ξ )) − deg( u ) b s ξ [ u ] − ( − i ) deg( u ) β deg( u ) [ u ]   O ξ . Prop osition 3.7. Every err or map ε s ξ is S U (3) -e quivariant. Also, given u ∈ U ( sl (3)) , ther e exists M ( u ) ≥ 0 such that (3.11)   ε s ξ [ u ]   ξ ≤ M ( u ) sr ( ξ ) ≤ 1 s r 3 2 M ( u ) , for every ξ ∈ Q and 1 < s ∈ N . Pr o of. The equiv ariance is immediate from the equiv ariance of the maps in the r.h.s. of (3.10). F or the upp er b ound, we use Prop osition 2.31 to get (3.12) ε s ξ [ u ]( Ad g ( ξ )) = deg( u ) − 1 X j =1 ( − i ) j ( sr ( ξ )) j − deg( u ) β [ π j ( Ad g − 1 ( u ))]( ξ ) for every g ∈ S U (3). Let ψ j : U ≤ deg( u ) ( sl (3)) → P oly ( S 7 ) b e giv en by (3.13) ψ j [ v ] = β [ π j ( v )]   S 7 . T ake an equiv arian t inner product on U ≤ deg( u ) ( sl (3)) so that u has norm ∥ u ∥ and ψ j has op erator norm ∥ ψ j ∥ . Hence   ψ j [ Ad g − 1 ( u )]   ∞ ≤ ∥ ψ j ∥∥ u ∥ for every g ∈ S U (3). Using triangular inequality and setting (3.14) M ( u ) = (deg( u ) − 1) max {∥ ψ j ∥∥ u ∥ : j = 1 , ..., deg( u ) − 1 } , w e get what we wan t, since sr ( ξ ) > 1, ∀ ξ ∈ Q , ∀ s ≥ 2. □ W e can rewrite (3.10) as (3.15) ( sr ( ξ )) − deg( u ) b s ξ [ u ] = ( − i ) deg( u ) β deg( u ) [ u ]   O ξ + ε s ξ [ u ] , so that we get immediately from Prop osition 3.7, the follo wing: Corollary 3.8. F or every u ∈ U ( sl (3)) , we have (3.16) lim s →∞ ( sr ( ξ )) − deg( u ) b s ξ [ u ] = ( − i ) deg( u ) β deg( u ) [ u ]   O ξ . W e shall now use the symmetrization map S given by (2.86). Restrictions of elemen ts in P ol y ( su (3)) generate P oly ( S 7 ) and P oly ( O ξ ), and likewise, restrictions of P ol y d ( su (3)) a generate P ol y d ( S 7 ) a and P ol y d ( O ξ ) a , cf. Notation 3.2. On the other hand, for f either in P ol y d ( O ξ ) a or P oly d ( S 7 ) a , we know there exists e f ∈ P ol y d ( su (3)) that restricts to f . If e f has any comp onen t in P ol y d ( su (3)) b for b  = a , we can subtract it and the restriction still matc hes f , so such f is alwa ys the restriction of some element of P ol y d ( su (3)) a . Belo w, we will apply the symmetrization map S on f ∈ P ol y d ( O ξ ) a (ev entually also on f ∈ P ol y d ( S 7 ) a ), which will b e a little abuse of notation for the application of S on the resp ective e f ∈ P ol y d ( sl (3)) a ≡ P oly d ( su (3)) a that restricts to f . No w, by adjoining basis of P oly d ( su (3)) a for each d ∈ N , w e pro duce a basis of P oly ( su (3)) a comprised only by homogeneous p olynomials. Restricting the ele- men ts of such basis to O ξ , we obtain a generating set for P oly ( O ξ ) a , from which w e can extract a basis { h 1 , ..., h m } with h j ∈ P oly d ( j ) ( O ξ ) a . 24 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS F rom the ab o v e considerations and Prop osition 2.25, we can deﬁne (3.17) u j = ( − i ) − d ( j ) S ( h j ) ∈ U ≤ d ( j ) ( sl (3)) a , h j = ( − i ) d ( j ) β d ( j ) [ u j ]   O ξ ∈ P oly d ( j ) ( O ξ ) a . Let u 1 , ..., u m ∈ U ( sl (3)) a b e chosen in this wa y . Lemma 3.9. Ther e exists s 0 ∈ N such that (3.18) n ( sr ( ξ )) − d (1) b s ξ [ u 1 ] , ..., ( sr ( ξ )) − d ( m ) b s ξ [ u m ] o is a b asis of P oly ( O ξ ) a for every s ≥ s 0 . Pr o of. Since { h 1 , ..., h m } is l.i., there are ς 1 , ..., ς m ∈ O ξ suc h that the matrix H with entries ( H ) j,k = h k ( ς j ) is non singular. Let B ( s ) b e the matrix with entries ( B ( s )) j,k = ( sr ( ξ )) − d ( k ) b s ξ [ u k ]( ς j ). F rom Corollary 3.8, B ( s ) conv erges to H , so there exists s 0 ∈ N suc h that B ( s ) is non singular for every s ≥ s 0 , in other words, the set (3.18) is l.i. for s ≥ s 0 . □ The ab o ve Lemma guaran tees that we can decompose any elemen t of P ol y ( O ξ ) a as a linear com bination of symbols of ﬁxed elements of the universal algebra, and this simpliﬁes the writing of twisted pro ducts. Prop osition 3.10. If f ∈ P ol y ( O ξ ) a , then ther e ar e s 0 ∈ N and α j ( s ) ∈ C for j ∈ { 1 , ..., m } and s ≥ s 0 such that (3.19) f = m X j =1 α j ( s )( sr ( ξ )) − d ( j ) b s ξ [ u j ] = m X j =1 α ∞ j h j , α ∞ j = lim s →∞ α j ( s ) ∈ C . Pr o of. Of course, f is a linear com bination of an y basis of P ol y ( O ξ ) a , thus we can write (3.19) b y Lemma 3.9 and the construction of { h 1 , ..., h m } . Recalling the pro of of Lemma 3.9, let A ( s ) = B ( s ) − 1 for s ≥ s 0 , and F = H − 1 , so that (3.20) α j ( s ) = m X k =1 ( A ( s )) j,k f ( ς k ) , α ∞ j = m X k =1 ( F ) j,k f ( ς k ) . By con tinuit y of the in version map, we ha v e A ( s ) → F , implying α j ( s ) → α ∞ j . □ Theorem 3.11. F or ξ ∈ Q , let (  s ξ ) s b e the se quenc e of twiste d pr o ducts of the Ber ezin fuzzy ξ -orbit B ( O ξ ) , cf. Deﬁnition 3.6. As s → ∞ , the uniform c onver- genc es f 1  s ξ f 2 → f 1 f 2 , (3.21) sr ( ξ )[ f 1 , f 2 ] ⋆ s ξ → i  f 1 , f 2  , (3.22) cf. (3.7) - (3.8) , hold for every f 1 , f 2 ∈ P oly ( O ξ ) . Pr o of. W e start by proving (3.21). By bilinearity of the pro ducts, it is suﬃcient to sho w the conv ergence for f j ∈ P oly ( O ξ ) a j . Now, w e use Prop osition 3.10 to write (3.23) f j = m j X k =1 α j k ( s )( sr ( ξ )) − d j ( k ) b s ξ [ u j k ] = m j X k =1 α j k h j k , for s ≥ s 0 ∈ N , where lim s →∞ α j k ( s ) = ( α j k ) ∞ ≡ α j k , (3.24) ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 25 lim s →∞ ( sr ( ξ )) − d j ( k ) b s ξ [ u j k ] = ( − i ) d j ( k ) β d j ( k ) [ u j k ] = h j k , (3.25) cf. Corollary 3.8. Therefore, (3.26) lim s →∞ f 1  s ξ f 2 = lim s →∞ m 1 X j =1 m 2 X k =1 α 1 j ( s ) α 2 k ( s )( sr ( ξ )) − ( d 1 ( j )+ d 2 ( k )) b s ξ  u 1 j u 2 k  = m 1 X j =1 m 2 X k =1 α 1 j α 2 k h 1 j h 2 k = m 1 X j =1 α 1 j h 1 j ! m 2 X k =1 α 2 k h 2 k ! = f 1 f 2 , where we hav e used Prop osition 2.23. Similarly for proving (3.22), we ha ve that (3.27) lim s →∞ sr ( ξ ) [ f 1 , f 2 ] ⋆ s ξ = lim s →∞ m 1 X j =1 m 2 X k =1 α 1 j ( s ) α 2 k ( s ) × ( sr ( ξ )) − ( d 1 ( j )+ d 2 ( k ) − 1) b s ξ  u 1 j u 2 k − u 2 k u 1 j  = i m 1 X j =1 m 2 X k =1 α 1 j α 2 k  h 1 j , h 2 k  = i n f 1 , f 2 o . where we hav e used Prop osition 2.24. □ Therefore, according to Deﬁnition 3.4, we hav e: Corollary 3.12. F or any ξ ∈ Q , the Ber ezin fuzzy ξ -orbit is of Poisson typ e, (3.28) B ( O ξ ) ∼ − − → P oly ( O ξ , Ω ξ ) , or in other wor ds, the ξ -r ay of Ber ezin c orr esp ondenc es is of P oisson typ e. 3.3. First criterion for Poisson: conv ergence of symbols. The thread from Corollary 3.8 to Prop osition 3.10 mak es it clear that the pro of of Theorem 3.11 dep ends solely on Corollary 3.8 of Prop osition 3.7, therefore any ξ -ray of univ ersal corresp ondence satisfying Corollary 3.8 is of P oisson type. That is, w e already ha ve: Prop osition 3.13. F or ξ ∈ Q , a ξ -r ay of universal c orr esp ondenc es ( w s ξ ) s ∈ N is of Poisson typ e if (3.29) lim s →∞ ( sr ( ξ )) − deg( u ) w s ξ [ u ] = ( − i ) deg( u ) β deg( u ) [ u ]   O ξ , ∀ u ∈ U ( sl (3)) . W e now sho w that (3.29) is also a necessary condition for Poisson t yp e. Lemma 3.14. If a ξ -r ay ( w s ξ ) s ∈ N of universal c orr esp ondenc es is of Poisson typ e, then (( sr ( ξ )) − deg( u ) w s ξ [ u ]) s ∈ N is a b ounde d se quenc e in P ol y ( O ξ ) , ∀ u ∈ U ( sl (3)) . Pr o of. W e pro v e it b y induction on the degree of u . First, tak e u ∈ sl (3) ≡ U 1 ( sl (3)) non null and let (3.30) N s =   w s ξ [ u ]   − 1 ξ , so the sequence ( h s ), with h s = N s w s ξ [ u ], is in the unit sphere of P ol y ( O ξ ) (1 , 1) . F or an y v ∈ U 1 ( sl (3)) with (3.31) e v = uv − vu  = 0 , w e hav e that e v ∈ U 1 ( sl (3)) as well and, by Sc h ur’s Lemma, the sequences ( f s ) and ( e f s ) given by f s = N s w s ξ [ v ] and e f s = N s w s ξ [ e v ] are nev er zero and b ounded. 26 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS No w, let (  s ξ ) b e the sequence of t wisted pro ducts induced by ( w s ξ ). The Poisson condition implies that the sequence ( C s ξ ) s of op erators (3.32) C s ξ : P oly ( O ξ ) (1 , 1) ∧ P oly ( O ξ ) (1 , 1) → P oly ( O ξ ) : f ∧ h 7→ sr ( ξ )[ f , h ] ⋆ s ξ con verges p oin t wisely to f ∧ h 7→ i { f , h } . By the Uniform Boundedness Principle, ( C s ξ ) s is uniformly b ounded. Thus (3.33)   C s ξ ( f s ∧ h s )   ξ =   sr ( ξ ) N 2 s w s ξ [ e v ]   ξ = sr ( ξ ) N s    e f s    ξ is b ounded on s . Hence N s ∈ O (1 /s ). This shows the claim for u ∈ U 1 ( sl (3)). T o complete the induction, supp ose the claim holds for every element of the univ ersal env eloping algebra with degree ≤ d . If v j ∈ U ≤ d j ( sl (3)) a j for j ∈ { 1 , 2 } , with 1 ≤ d 1 , d 2 ≤ d , then v 1 v 2 ∈ U ≤ d 1 + d 2 ( sl (3)). Again, the Poisson condition implies that the sequence ( T s ξ ) s of op erators (3.34) T s ξ : P oly ( O ξ ) a 1 ⊗ P oly ( O ξ ) a 2 → P oly ( O ξ ) : f ⊗ h 7→ f  s ξ h con verges p oint wisely to f ⊗ h → f h , so ( T s ξ ) s is uniformly b ounded. Therefore (3.35)    ( sr ( ξ )) − ( d 1 + d 2 ) T s ξ ( w s ξ [ v 1 ] ⊗ w s ξ [ v 2 ])    = ( sr ( ξ )) − ( d 1 + d 2 )   w s ξ [ v 1 v 2 ]   ξ is b ounded on s . By writing u ∈ U ≤ d +1 ( sl (3)) as a linear combination of pro ducts of elements of degrees ≤ d , we conclude that the claim also holds for ev ery element of U ≤ d +1 ( sl (3)). □ Lemma 3.15. If a ξ -r ay ( w s ξ ) s ∈ N of universal c orr esp ondenc es is of Poisson typ e, then (3.36) lim s →∞ ( sr ( ξ )) − 1 w s ξ [ u ] = − iβ 1 [ u ]   O ξ , ∀ u ∈ U ( sl (3)) . Pr o of. Let h s = ( sr ( ξ )) − 1 w s ξ [ u ]. By the previous lemma, ( h s ) is a bounded sequence in P ol y ( O ξ ) (1 , 1) . Let ( e h n ), e h n = h s n , b e an y con vergen t subsequence, e h n → h . W e w ant to prov e that h = − iβ 1 [ u ] | O ξ = − iβ [ u ] | O ξ . T o do so, we will prov e that (3.37) i { h, f } = i {− iβ [ u ] | O ξ , f } for ev ery f ∈ P ol y 1 ( O ξ ), whic h allo ws us to conclude that h and − iβ [ u ] | O ξ ha ve the same Hamiltonian vector ﬁeld, so they diﬀer by a constant 15 ; since b oth functions lies in P oly ( O ξ ) (1 , 1) , whose only constan t function is identically 0, the functions m ust coincide. Th us, let X u b e the vector ﬁeld that represen ts the action of u on C ∞ ( O ξ ), naturally induced by the S U (3)-action, and tak e (3.38) e f = X u ( f ) . Then, let v n ∈ sl (3) ≡ U 1 ( sl (3)) be such that f = w s n ξ [ v n ]. By equiv ariance of w s n ξ , w e hav e that (3.39) e v n = uv n − v n u satisﬁes e f = w s n ξ [ e v n ]. Then (3.40) s n r ( ξ )[ h n , f ] ⋆ s n ξ = w s n ξ [ uv n − v n u ] = w s n ξ [ e v ] = e f , 15 Recall that O ξ is connected. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 27 where  s ξ is the twisted product induced by w s ξ as usual. W e can rewrite (3.40) as (3.41) e f = s n r ( ξ )[ h n − h, f ] ⋆ s n ξ + sr ( ξ )[ h, f ] ⋆ s n ξ . As we argued in the pro of of the previous lemma, the Poisson hypothesis implies that the sequence of op erators ( F n ), (3.42) F n : P oly 1 ( O ξ ) → P oly 1 ( O ξ ) : e h 7→ s n r ( ξ )[ e h, f ] ⋆ s n ξ , is uniformly b ounded, so (3.43) e f = lim n →∞ s n r ( ξ )[ h n − h, f ] ⋆ s n ξ + lim n →∞ s n r ( ξ )[ h, f ] ⋆ s n ξ = i { h, f } . No w, let v = S ( f ) and e v = S ( e f ), so f = β 1 [ v ] | O ξ , e v = uv − v u and (3.44) e f = β 1 [ e v ] | O ξ = β 1 [ uv − vu ] | O ξ , cf. Prop osition 2.25. F rom Prop osition 2.24, we ha v e (3.45) i { h, f } = e f = β 1 [ uv − vu ] | O ξ = { β 1 [ u ] , β 1 [ v ] }| O ξ = i {− iβ [ u ] | O ξ , f } . Therefore, every con vergen t subsequence of the b ounded sequence ( h s ) conv erges to − iβ [ u ] | O ξ , which means the sequence itself conv erges to − iβ [ u ] | O ξ . □ Prop osition 3.16. If a ξ -r ay ( w s ξ ) s ∈ N of universal c orr esp ondenc es is of Poisson typ e, then (3.29) is satisﬁe d. Pr o of. W e prov e by induction on deg ( u ), supp osing it holds ∀ u ∈ U ≤ d ( sl (3)), with the previous Lemma showing it holds for deg( u ) = d = 1. As we did be fore, if v j ∈ U ≤ d j ( sl (3)) a j for j ∈ { 1 , 2 } , with 1 ≤ d 1 , d 2 ≤ d , then v 1 v 2 ∈ U ≤ d 1 + d 2 ( sl (3)). Let ( f s ) and ( e f s ) b e giv en b y f s = ( sr ( ξ )) − d 1 w s ξ [ v 1 ] and e f s = ( sr ( ξ )) − d 2 w s ξ [ v 2 ]. By the h yp othesis of induction, f = ( − i ) d 1 β d 1 [ v 1 ] and e f = ( − i ) d 2 β d 2 [ v 2 ] are the limits of ( f s ) and ( e f s ), resp ectively . Since (3.46)    f s  s ξ e f s − f e f    ξ ≤    f s  s ξ e f s − f  s ξ e f    ξ +    f  s ξ e f − f e f    ξ , w e just need to verify that b oth summands in the r.h.s. conv erge to 0 as s → ∞ . The conv ergence of second summand follo ws straightforw ardly from the P oisson condition. F or the ﬁrst summand, w e use again that the sequence of operators (3.34) is b ounded, so the con v ergences f s → f and e f s → e f imply that the limit of the ﬁrst summand v anishes. □ Therefore, combining Prop ositions 3.13 and 3.16, we ha v e obtained: Theorem 3.17. F or ξ ∈ Q , a ξ -r ay of universal c orr esp ondenc es ( w s ξ ) s ∈ N is of Poisson typ e, so that W ( O ξ ) ∼ − − → P oly ( O ξ , Ω ξ ) , if and only if (3.29) is satisﬁe d. 3.4. Second criterion for Poisson: c haracteristic matrices. As presented in P ap er I, every symbol correspondence for a quark system with dominan t weigh t p  1 + q  2 is uniquely determined by its c haracteristic matrices, cf. Theorems I.4.8 and I.5.9, and Remark I.5.13. Therefore, a natural question is how to write the Poisson condition for a ξ -ray of corresp ondences in terms of the sequence of their characteristic matrices, or characteristic n umbers if ξ = (1 , 0) or (0 , 1). T o answ er this question more clearly , we translate some notation used in this Paper I I to the language of Paper I. 28 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS F or each ξ ∈ Q , let p 1 ξ = ( p 1 , q 1 ) ∈ N 2 0 \ { (0 , 0) } b e the ﬁrst integral pair for ξ , (3.47) ω p 1 ξ = p 1  1 + q 1  2 = ω ξ , cf. Deﬁnition 2.6. Then, ﬁxed ξ , for eac h s ∈ N we denote (3.48) p s ξ = ( sp 1 , sq 1 ) = s p 1 ξ , ω p s ξ ≡ s ω ξ ,    ω p s ξ    = sr ( ξ ) , so that H p s ξ ≡ H sω ξ is the quantum quark system with irrep p s ξ , and ρ p s ξ ≡ ρ sω ξ is the (ﬁnite dimensional, cf. (I.2.17)) representation of U ( sl (3)) on H p s ξ . Also, consider the ξ -ray of symbol correspondences ( W p s ξ ≡ W sω ξ ) s ∈ N generating the ξ -ray of universal corresp ondences ( w s ξ ) s ∈ N according to Deﬁnition 2.27, that is, (3.49) w s ξ = W p s ξ ◦ ρ p s ξ . Finally , denote by C s ξ ( a ) the c haracteristic matrices of W p s ξ ≡ W sω ξ (c haracteristic n umbers as 1 × 1 matrices if p 1 = 0 or q 1 = 0), cf. Deﬁnition I.5.12. Recalling the normalized Hilb ert-Schmidt inner pro duct on each B ( H p s ξ ), (3.50) ⟨ A 1 | A 2 ⟩ p s ξ = 1 dim( p s ξ ) tr( A 1 A 2 ) , cf. (I.3.10), then based on what is known for spin systems, one should exp ect that ξ -rays of corresp ondences of P oisson type tend in some sense to an isometry with resp ect to the inner pro ducts ⟨·|·⟩ p s ξ and ⟨·|·⟩ ξ as s → ∞ , where the latter is the Haar inner pro duct of functions on O ξ . W e now show that this is indeed what happ ens, which will lead to an asymptotic condition for the characteristic matrices. Lemma 3.18. F or any u, v ∈ U ( sl (3)) , we have (3.51) lim s →∞    ω p s ξ    − (deg( u )+deg( v )) D ρ p s ξ ( u )    ρ p s ξ ( v ) E p s ξ = ( − i ) deg( v ) − deg( u )  β deg( u ) [ u ]   β deg( v ) [ v ]  ξ . Pr o of. F or the ξ -ra y ( b s ξ ) s of universal Berezin corresp ondences, (3.52)    ω p s ξ    − (deg( u )+deg( v )) D ρ p s ξ ( u )    ρ p s ξ ( v ) E p s ξ = Z O ξ    ω p s ξ    − (deg( u )+deg( v )) b s ξ [ u ]  b s ξ [ v ]( ς ) d ς . By decomposing U ≤ deg( u ) ( sl (3)) and U ≤ deg( v ) ( sl (3)) in to irreps, it is possible to ﬁnd some d ∈ N suc h that P oly ≤ d ( O ξ ) contains b s ξ [ u ] and b s ξ [ v ] for every s ∈ N . Since P ol y ≤ d ( O ξ ) is ﬁnite dimensional, Corollaries 3.8 and 3.12 imply (3.53) lim s →∞    ω p s ξ    − (deg( u )+deg( v )) b s ξ [ u ]  b s ξ [ v ] = ( − i ) deg( v ) − deg( u ) β deg( u ) [ u ] β deg( v ) [ v ] , cf. (3.48). This con vergence is uniform, so taking the in tegral we get the equation of the statement from (3.52). □ ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 29 Corollary 3.19. If ( W p s ξ ) is of Poisson typ e, then, for every u, v ∈ U ( sl (3)) , (3.54) lim s →∞    ω p s ξ    − (deg( u )+deg( v )) D W p s ξ ◦ ρ p s ξ ( u )    W p s ξ ◦ ρ p s ξ ( v ) E ξ = ( − i ) deg( v ) − deg( u )  β deg( u ) [ u ]   β deg( v ) [ v ]  ξ = lim s →∞    ω p s ξ    − (deg( u )+deg( v )) D ρ p s ξ ( u )    ρ p s ξ ( v ) E p s ξ . Pr o of. It is immediate from Theorem 3.17 and Lemma 3.18. □ Theorem 3.20. If ( W p s ξ ) is of Poisson typ e, then the char acteristic matric es satisfy (3.55) lim s →∞ ( C s ξ ( a )) † C s ξ ( a ) = 1 . Pr o of. Let s b e large enough so that the dimension of the highest weigh t space of (3.56) B ( p s ξ ; a ) = B ( H p s ξ ) a is constant m ≡ m ( a ), cf. Notation I.5.8. T ake u 1 , ..., u m ∈ U ( sl (3)) a as highest w eight vectors of degrees deg ( u γ ) = d ( γ ) such that (3.57) ( − i ) d ( γ 1 ) − d ( γ 2 )  β d ( γ 1 ) [ u γ 1 ]   β d ( γ 2 ) [ u γ 2 ]  = δ γ 1 ,γ 2 . By the previous corollary , for s large enough, the set (3.58)     ω p s ξ    − d (1) ρ p s ξ ( u 1 ) , ...,    ω p s ξ    − d ( m ) ρ p s ξ ( u m )  is a basis of the highest weigh t space of B ( p s ξ ; a ). No w, for σ ∈ { 1 , ..., m } , take (3.59) A s σ = q dim( p s ξ ) e (( a ; σ ); > a ) , where > a stands for the highest weigh t. Denoting (3.60) f s σ = W p s ξ A s σ , the j × k en try of ( C s ξ ( a )) † C s ξ ( a ) is  f s j   f s k  ξ , cf. Deﬁnition I.5.12 and Remark I.5.13, so we w an t to show (3.61) lim s →∞  f s j   f s k  ξ = δ j,k . Let Z ( s ) be the complex square matrix with entries ( Z ( s )) σ,γ = z σ γ ( s ) suc h that (3.62)    ω p s ξ    − d ( γ ) ρ p s ξ ( u γ ) = X σ z σ γ ( s ) A s σ . F rom Corollary 3.19, we hav e that (3.63) lim s →∞ Z ( s ) † Z ( s ) = 1 . Therefore, for C ( s ) = Z ( s ) − 1 , we hav e (3.64) lim s →∞ C ( s ) † C ( s ) = 1 . Also, the entries ( C ( s )) γ ,σ = c γ σ ( s ) are b ounded on s and satisfy (3.65) A s σ = X γ c γ σ ( s )    ω p s ξ    − d ( γ ) ρ p s ξ ( u γ ) . 30 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Hence,  f s j   f s k  ξ is given by (3.66) X γ 1 ,γ 2 c γ 1 j ( s ) c γ 2 k ( s )    ω p s ξ    − ( d ( γ 1 )+ d ( γ 2 )) D W p s ξ ◦ ρ p s ξ ( u γ 1 )    W p s ξ ◦ ρ p s ξ ( u γ 2 ) E ξ , and  A s j   A s k  p s ξ is given by (3.67) X γ 1 ,γ 2 c γ 1 j ( s ) c γ 2 k ( s )    ω p s ξ    − ( d ( γ 1 )+ d ( γ 2 )) D ρ p s ξ ( u γ 1 )    ρ p s ξ ( u γ 2 ) E p s ξ , Since  A s j   A s k  p s ξ = δ j,k , applying Corollary 3.19 on (3.66), we get (3.61). □ The last theorem states that the characteristic matrices of a ξ -ray of sym b ol cor- resp ondences of Poisson type are asymptotically unitary , that is to say , the ξ -ray of symbol corresp ondences needs to satisfy a weak asymptotic Str atonovich-Weyl c ondition to b e of Poisson type. Nonetheless, in order to get a statement of equiv- alence b etw een Poisson prop ert y and the conv ergence of characteristic matrices to sp eciﬁc unitary matrices, in the spirit of what happ ens for spin systems and their c haracteristic num b ers [17], we need to ﬁx a metho d for Clebsch-Gordan decom- p ositions of spaces of op erators. W e can a v oid suc h choice-dependent classiﬁcation b y comparing with Berezin corresp ondences instead. Theorem 3.21. A ξ -r ay of symb ol c orr esp ondenc es ( W p s ξ ) s ∈ N is of Poisson typ e, so that (3.9) is satisﬁe d, if and only if its se quenc e of char acteristic matric es satisfy (3.68) lim s →∞  C s ξ ( a ) − B s ξ ( a )  = 0 , wher e  B s ξ ( a )  s ∈ N is the se quenc e of char acteristic matric es for the ξ -r ay ( B p s ξ ) s ∈ N of Ber ezin symb ol c orr esp ondenc es, cf. (2.97) and Deﬁnition I.5.12. Pr o of. Recalling (3.59), again let f s σ b e the symbol of A s σ via W p s ξ , cf. (3.60), and no w let e f s σ b e the symbol of A s σ via B p s ξ . Since P ol y ( O ξ ) a is ﬁnite dimensional, the limit (3.68) holds if and only if (3.69) lim s →∞    f s σ − e f s σ    ξ = 0 , for every σ ∈ { 1 , ..., m } , which in turn is equiv alen t to (3.70) lim s →∞    ω p s ξ    − deg( u )    W p s ξ ◦ ρ p s ξ ( u ) − B p s ξ ◦ ρ p s ξ ( u )    ξ = 0 for every u ∈ U ( sl (3)) a , cf. Corollary 3.19 and (3.62)-(3.65). Then the statement is a consequence of Corollary 3.12 and Theorem 3.17. □ F or pure-quark systems, with ξ = (1 , 0) or ξ = (0 , 1) and ξ -rays of represen tations (( p, 0)) p ∈ N or ((0 , p )) p ∈ N , resp ectively , the characteristic matrices are 1 × 1 matrices and are just called characteristic n umbers, c p n ∈ R × , cf. Deﬁnition I.5.12 and Remark I.5.13. F rom Theorem 3.20, a ra y of pure-quark corresp ondences is of P oisson type only if (3.71) lim p →∞ | c p n | = 1 , ∀ n ∈ N . Ho wev er, by choosing a decomp osition of the space of op erators in such a wa y that the symmetric Berezin correspondences ha v e only p ositiv e characteristic num- b ers, b p n ∈ R + , Theorem 3.21 provides a ﬁner criterion b y means of the sequence of ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 31 its c haracteristic num bers ( c p n ) p ∈ N for a ray of pure-quark corresp ondences to b e of P oisson type, which is analogous to the criterion for spin systems [17]. W e illustrate this for ξ = (1 , 0) with sequence of represen tations ( p = ( p, 0)) p ∈ N , for which we hav e the following: Prop osition 3.22. F or every p = ( p, 0) , p ∈ N , taking (3.72) e ( n ; (2 n, n, 0); n/ 2) = 1 µ n ( p ) V n + ∈ B ( H p ) , µ n ( p ) = n ! p (2 n + 2)! s ( p + n + 2)! ( p − n )! , the char acteristic numb ers ( b p n ) n ≤ p of the symmetric Ber ezin c orr esp ondenc e ar e al l p ositive and satisfy | b p n − 1 | ∈ O (1 /p ) as p → ∞ , for every ﬁxe d n ∈ N . The proof of this Prop osition is a straightforw ard computation using v arious results and notations from P ap er I, so it is placed in Appendix A. Combining Prop osition 3.22 with Theorem 3.21, we obtain immediately: Corollary 3.23. L et ( W p ) p ∈ N b e a r ay of pur e-quark symb ol c orr esp ondenc es (3.73)  W p : B ( H ( p, 0) ) → P oly ( O (1 , 0) )  p ∈ N , with char acteristic numb ers ( c p n ) n ≤ p , for e ach p ∈ N . Assuming (3.72) , ( W p ) p ∈ N is of Poisson typ e if and only if the char acteristic numb ers satisfy (3.74) lim p →∞ c p n = 1 , ∀ n ∈ N . As men tioned before, in App endix B we pro vide an alternativ e (summary of the) pro of of this corollary using a metho d which, although quite more cumbersome, is analogous to the metho d used in the case of spin systems, cf. [17]. 4. Universal correspondences on the coarse Poisson sphere In this section, we dev elop a metho d for extending the rays of corresp ondences deﬁned for each rational orbit O ξ ⊂ S 7 to a p encil of corresp ondence rays deﬁned on the full coarse Poisson sphere {S 7 , b Π g } , such that an extended S U (3)-inv ariant noncomm utative algebra constructed by this metho d, with pro duct induced from the univ ersal env eloping algebra U ( sl (3)), restricts to a fuzzy ξ -orbit, for each ξ ∈ Q , as in Deﬁnition 3.1. Then we in vestigate if/ho w such extended algebras can reco v er the Poisson algebra of p olynomials on ( S 7 , b Π g ) in some asymptotic limit. 4.1. P encils of corresp ondence ra ys: Mago o spheres. Before starting the construction of pencils of corresp ondence ra ys on the coarse P oisson sphere {S 7 , b Π g } , w e recall that each rational orbit O ξ is a symplectic leaf in a singular foliation of the smo oth Poisson sphere ( S 7 , b Π g ) and that this orbit is the preimage of a ﬁxed n umber χ ξ ∈ R by the restriction of the cubic p olynomial τ to S 7 , cf. (2.22) in Prop osition 2.2, so that, for eac h ξ ∈ Q , we can deﬁne a p olynomial function ˇ τ ξ v anishing on O ξ and only on this orbit in S 7 , cf. (2.24) in Remark 2.3. No w, consider the ideal of p olynomials v anishing on O ξ , (4.1) I ξ := { f ∈ P ol y ( S 7 ) : f | O ξ ≡ 0 } ⊴ P ol y ( S 7 ) , 32 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS in terms of which we can set the natural isomorphism (4.2) P oly ( O ξ ) ≃ P oly ( S 7 ) /I ξ , so that we can write universal corresp ondences as maps (4.3) w ξ : U ( sl (3)) → P ol y ( O ξ ) ≃ P oly ( S 7 ) /I ξ . This can b e extended simultaneously for any ﬁnite set of rational orbits. Let (4.4) P ⊂ Q , ∥P ∥ ∈ N , b e a ﬁnite subset of rational orbits and, for each ξ ∈ P , tak e a universal corre- sp ondence (4.3). In order to “glue together” such corresp ondences, w e inv ok e the in v ariant p olynomial τ | S 7 ∈ P oly ( S 7 ), given b y (2.22), and its restriction comple- men ts ˇ τ ξ ∈ I ξ , given by (2.24). Thus, for each ξ ∈ P , let (4.5) d ξ P := 1 M ξ P Y ξ ′ ∈P \{ ξ } ˇ τ ξ ′ ∈ P oly ( S 7 ) , M ξ P = Y ξ ′ ∈P \{ ξ } ˇ τ ξ ′ ( ξ ) , so that d ξ P w orks like a delta- ξ function on P , that is, for ξ , ξ ′ ∈ P , we hav e (4.6) d ξ P   O ξ ′ ≡ ( 1 , if ξ = ξ ′ 0 , if ξ  = ξ ′ . Then, taking (4.7) I P := \ ξ ∈P I ξ , w e hav e 16 (4.8) P ol y  [ ξ ∈P O ξ  ≡ Y ξ ∈P P ol y ( O ξ ) ≃ P oly ( S 7 ) /I P , and we use d ξ P giv en by (4.5) to “glue” a ﬁnite family of corresp ondences and also a ﬁnite family of corresp ondence rays, as follows. Deﬁnition 4.1. Given a ﬁnite subset P ⊂ Q , a ﬁnite p encil of universal corre- sp ondence ra ys for P , or simply a p encil of corresp ondence ra ys for P , is a se quenc e of maps ( w s P ) s ∈ N , wher e e ach (4.9) w s P : U ( sl (3)) → P ol y ( S 7 ) /I P : u 7→ w s P [ u ] = X ξ ∈P d ξ P w s ξ [ u ] , and wher e, for e ach ξ ∈ P , ( w s ξ ) s ∈ N is a ξ -r ay of universal c orr esp ondenc es, such that w 1 ξ : U ( sl (3)) → P ol y ( O ξ ) is a universal c orr esp ondenc e w.r.t. its ﬁrst domi- nant weight ω ξ = r ( ξ ) ξ , cf. (2.26) - (2.28) . With the ab ov e deﬁnition, we naturally obtain a sequence of twisted pro ducts on the sequence of images of ( w s P ) s ∈ N . Since these pro ducts are not commutativ e, their commutators act as deriv ations on their algebras. But as noted in Section 3, cf. Remark 3.5 w.r.t. (3.8) in Deﬁnition 3.1, a weighte d deriv ation is more suitable for the semiclassical limit. 16 Since P is ﬁnite, one could use the more usual notion of direct sum instead of product. How ev er, we chose the product in anticipation of an inﬁnite product that will take place ahead. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 33 Deﬁnition 4.2. Given a p encil of c orr esp ondenc e r ays ( w s P ) s ∈ N , denote by ( W s P ) s ∈ N its se quenc e of images W s P ⊂ P ol y ( S 7 ) /I P . Then, its induc e d twisted pro duct se- quence (  s P ) s ∈ N on ( W s P ) s ∈ N is given by (4.10) w s P [ u ]  s P w s P [ v ] = w s P [ uv ] = X ξ ∈P d ξ P w s ξ [ u ]  s ξ w s ξ [ v ] , ∀ u, v ∈ U ( sl (3)) , and its r -weigh ted brack et se quenc e  [ · , · ] r ⋆ s P  s ∈ N on ( W s P ,  s P ) s ∈ N is given by (4.11)  w s P [ u ] , w s P [ v ]  r ⋆ s P = X ξ ∈P d ξ P r ( ξ )  w s ξ [ u ] , w s ξ [ v ]  ⋆ s ξ , ∀ u, v ∈ U ( sl (3)) . Ho wev er, any ﬁnite family of leav es is far from suﬃcient to determine the P ois- son algebra of p olynomials on the sphere, as s → ∞ . Thus, we now consider an increasing family of nested ﬁnite subsets P ⊂ Q whose limit is the en tire set of rational orbits Q (so that, in particular, I P → 0). Any c hain of ﬁnite subsets of Q is countable b ecause Q is countable, so we can deﬁne chain sequences of the form (4.12) C = ( P n ) n ∈ N , P n ⊂ Q s.t. |P n | < ∞ , P n ⊊ P n +1 , lim n →∞ P n = Q . F urthermore, on eac h P n as ab ov e, let’s denote, for conv enience, (4.13) d ξ n ≡ d ξ P n , cf. (4.5) , and also, cf. (4.8), (4.14) P n ≡ P oly ( S 7 ) /I P n ≃ Y ξ ∈P n P ol y ( O ξ ) . Deﬁnition 4.3. L et C b e a chain as in (4.12) and, for e ach ξ ∈ Q , let (4.15) n ξ := min { n ∈ N : ξ ∈ P n } . A Mago o p encil of corresp ondence rays for C is a bi-se quenc e (4.16) w C = ( w s n ) n,s ∈ N , with w s n : U ( sl (3)) → P n , u 7→ w s n [ u ] = X ξ ∈P n d ξ n w s ξ [ u ] , wher e ( w s ξ ) s ∈ N is a ξ -r ay of universal c orr esp ondenc es, cf. Deﬁnition 4.1, such that, for every ξ ∈ Q and every u ∈ U ( sl (3)) , (4.17) w s n [ u ] | O ξ = w s n ξ [ u ] | O ξ , ∀ n ≥ n ξ , ∀ s ∈ N . Then, denoting by W C = ( W s n ) n,s ∈ N the bi-se quenc e of images of U ( sl (3)) by w C , its induc e d Mago o pro duct on W C is the bi-se quenc e of pr o ducts (4.18)  C = (  s n ) n,s ∈ N ,  s n ≡  s P n : W s n × W s n → W s n , cf. (4.10) , so that ∀ u, v ∈ U ( sl (3)) , (4.19) w C [ u ]  C w C [ v ] := ( w s n [ u ]  s n w s n [ v ]) n,s ∈ N , and its Mago o brack et on ( W C ,  C ) is the bi-se quenc e of r -weighte d br ackets (4.20) [ · , · ] r ⋆ C =  [ · , · ] r ⋆ s n  n,s ∈ N , [ · , · ] r ⋆ s n ≡ [ · , · ] r ⋆ s P n : W s n × W s n → W s n , 34 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS cf. (4.11) , so that ∀ u, v ∈ U ( sl (3)) , (4.21)  w C [ u ] , w C [ v ]  r ⋆ C =  X ξ ∈P n d ξ n r ( ξ )  w s n [ u ] , w s n [ v ]  ⋆ s n  n,s ∈ N . In this way,  W C ,  C , [ · , · ] r ⋆ C  as ab ove shal l b e c al le d a Mago o sphere , denote d (4.22) W {S 7 , b Π g } =  W C ,  C , [ · , · ] r ⋆ C  . Henceforth, let W {S 7 , b Π g } be a Mago o sphere as just deﬁned ab ov e for the coarse P oisson sphere {S 7 , b Π g } . If we denote the Poisson algebra of complex p olynomials on the smo oth Poisson sphere ( S 7 , b Π g ) by (4.23) P ol y ( S 7 , b Π g ) , w e w an t to study if/when/how W {S 7 , b Π g } conv erges to P ol y ( S 7 , b Π g ) in some as- ymptotic limit. In particular, we are concerned with asymptotics of Mago o product and Mago o brac k et of p olynomials, so the ﬁrst thing to study is if/when the product and the brack et are well deﬁned for general p olynomials on S 7 . Lemma 4.4. Given f ∈ P oly ( S 7 ) , ther e exists s 0 ∈ N such that, for every s ≥ s 0 , f | O ξ is in the image of w s ξ for every ξ ∈ Q . Pr o of. Without loss of generalit y , we can assume that f ∈ P oly ( S 7 ) ( a,b ) , then f | O ξ ∈ P ol y ( O ξ ) ( a,b ) , ∀ ξ ∈ Q (and ∀ ξ ∈ Q when a = b ). Then, we pro ceed as in the pro of of Lemma 3.3, obtaining s 0 giv en by (3.5). □ Lemma 4.5. Given f ∈ P ol y ( S 7 ) , let f | n denote its quotient in P n , cf. (4.14) , and let s 0 ∈ N b e as in L emma 4.4. Then, (4.24) s ≥ s 0 = ⇒ f | n ∈ W s n , ∀ n ∈ N . Pr o of. By h ypothesis, f | O ξ lies in the image of w s ξ for ev ery ξ ∈ Q and ev ery s ≥ s 0 . Th us, for an y ﬁxed n ∈ N and s ≥ s 0 , we need to exhibit u s n ∈ U ( sl (3)) such that f | n = w s n [ u s n ]. F or each ξ ∈ Q , let u s ξ ∈ U ( sl (3)) b e such that f | O ξ = w s ξ [ u s ξ ] for s ≥ s 0 . Since the eigen v alues of the Casimir op erators separate the representations, cf. (I.B.3), there are c s ξ ∈ Z ( U ( sl (3)), ∀ ξ ∈ P n , s.t. (4.25) ∀ ξ , ξ ′ ∈ P n , w s ξ [ c s ξ ′ ] = δ ξ,ξ ′ = ⇒ w s ξ [ c s ξ ′ u s ξ ′ ] = δ ξ,ξ ′ w s ξ [ u s ξ ] . Therefore, from (4.16) and (4.25), (4.26) u s n = X ξ ∈P n c s ξ u s ξ = ⇒ w s n [ u s n ] = X ξ ∈P n d ξ n w s ξ [ u s n ] = X ξ ∈P n d ξ n w s ξ [ u s ξ ] = f | n , and this u s n ∈ U ( sl (3)) is as claimed. □ Remark 4.6. We highlight that s 0 as ab ove dep ends only on f ∈ P oly ( S 7 ) , and it works for any Mago o spher e, obtaine d fr om any p encil of c orr esp ondenc e r ays ( w s n ) n,s ∈ N for any chain C = ( P n ) n ∈ N as in (4.12) , onc e f is ﬁxe d. Thus, in light of L emma 4.5, given f 1 , f 2 ∈ P oly ( S 7 ) , for s 0 = max { s 1 0 , s 2 0 } , with s j 0 as in L emma 4.4 w.r.t. f j , we c an make sense of f 1  C f 2 as a bi-se quenc e ( f 1 | n  s n f 2 | n ) n ∈ N ,s ≥ s 0 , and similarly for the Mago o br acket [ f 1 , f 2 ] r ⋆ C . ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 35 Th us, to explore the asymptotics of W {S 7 , b Π g } , we ﬁrst need to establish the meaning of limits of sequences ( f s ) s ≥ s 0 and ( h n ) n ∈ N , where s is the semiclassical asymptotic parameter and n indexes the neste d ﬁnite subsets P n ⊂ Q in C , cf. (4.12). That these tw o limits are of diﬀerent nature can b e inferred by construction and is implied by (4.17). Hence, to establish these limits, we ﬁrst in v oke (4.27) P = Y ξ ∈Q P ol y ( O ξ ) as an ambien t space, for which w e hav e an inclusion (4.28) P oly ( S 7 )  → P : f 7→ ( f | O ξ ) ξ ∈Q ≡ ( f ξ ) ξ ∈Q , and pro jections (4.29) P → P n : f = ( f ξ ) ξ ∈Q 7→ f | n = ( f ξ ) ξ ∈P n for every n ∈ N , cf. (4.14). 17 Then, we consider tw o distinct t yp es of con vergence: t yp e (i) : for ( f s ) s ≥ s 0 , with each f s = ( f s ξ ) ξ ∈P n ∈ P n and f = ( f ξ ) ξ ∈P n ∈ P n , (4.30) lim s →∞ f s = f ⇐ ⇒ ∀  > 0 , ∃ s ϵ ∈ N : s ≥ s ϵ = ⇒   f s ξ − f ξ   ξ <  ∀ ξ ∈ P n , that is, ( f s ) s ≥ s 0 con verges to f in P n iﬀ f s ξ s →∞ − − − → f ξ uniformly ov er P n . t yp e (ii) : for ( h n ∈ P n ) n ∈ N and h ∈ P , (4.31) lim n →∞ h n = h ⇐ ⇒ h n = h | n ∀ n ∈ N , cf. (4.29), that is, ( h n ∈ P n ) n ∈ N con verges to h ∈ P iﬀ h is a common extension to every h n ∈ P n . Con vergence types (i) and (ii) ab ov e induce tw o diﬀerent kinds of asymptotics for a Mago o sphere, depending on which order of iterated limits we tak e for f 1  C f 2 and [ f 1 , f 2 ] r ⋆ C . W e b egin b y exploring the ordering given b y (i) ﬁrst, then (ii). Deﬁnition 4.7. We say that W {S 7 , b Π g } is of Poisson type if its Mago o pr o duct and Mago o br acket satisfy, for any f 1 , f 2 ∈ P oly ( S 7 ) , (4.32) lim n →∞ lim s →∞ f 1  C f 2 = f 1 f 2 ∈ P ⇐ ⇒ ( f 1 | n  s n f 2 | n ) s ≥ s 0 s →∞ − − − → ( f 1 f 2 ) | n , ∀ n ∈ N , (4.33) lim n →∞ lim s →∞ s [ f 1 , f 2 ] r ⋆ C = i { f 1 , f 2 } ∈ P ⇐ ⇒ ( s [ f 1 | n , f 2 | n ] r ⋆ s n ) s ≥ s 0 s →∞ − − − → i { f 1 , f 2 } | n , ∀ n ∈ N , wher e n indexes P n ∈ C , cf. R emark 4.6 and (4.30) - (4.31) . In this c ase, we write (4.34) W {S 7 , b Π g } n ≺ s − − − → P ol y ( S 7 , b Π g ) , wher e the sup erscript n ≺ s r efers to the or der of the limits in (4.32) - (4.33) . Theorem 4.8. A Mago o spher e is of Poisson typ e if and only if al l of its ξ -r ays of universal c orr esp ondenc es ar e of Poisson typ e. 17 In accordance with notation of Lemma 4.5. 36 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Pr o of. Since the Poisson-t yp e uniform conv ergences for each ξ ∈ Q (cf. (3.7)-(3.8)) trivially extend uniformly ov er ﬁnite sets P n ⊂ Q , the statement follows immedi- ately from the deﬁnitions, cf. (4.30)-(4.31) and (4.32)-(4.33). □ Corollary 4.9. F or j = 1 , 2 , let W j {S 7 , b Π g } b e Mago o spher es c onstructe d fr om the same ξ -r ays of universal c orr esp ondenc es ( w s ξ ) s ∈ N ,ξ ∈Q , but fr om two distinct chains C 1 and C 2 of ﬁnite subsets of Q satisfying (4.12) . Then, (4.35) W 1 {S 7 , b Π g } n ≺ s − − − → P ol y ( S 7 , b Π g ) ⇐ ⇒ W 2 {S 7 , b Π g } n ≺ s − − − → P ol y ( S 7 , b Π g ) . Hence, although there are inﬁnitely many diﬀerent chains C of Q satisfying (4.12), the P oisson condition for a Magoo sphere is independent of their c hoice, th us w e can restrict ourselv es to a canonical c hoice, as follows. Recall there exits a well deﬁned function r : Q → R + , ξ 7→ r ( ξ ), the integral radius of ξ , c f. (2.26) in Deﬁnition 2.6, so we take the index n in (4.12) to b e an increasing function of r , (4.36) n ξ = n ( r ( ξ )) , r ( ξ ) < r ( ξ ′ ) ⇐ ⇒ n ξ < n ξ ′ . Deﬁnition 4.10. The radial c hain C r = ( R n ) n ∈ N is the chain as in (4.12) such that, ∀ ξ , ξ ′ ∈ Q , (4.36) holds. In other words, for any giv en n , R n is the union of all rational orbits whose in tegral radius r ( ξ ) is such that n ξ = n ( r ( ξ )) ≤ n . Thus, R 1 = { ξ (1 , 0) , ξ (0 , 1) } , R 2 = R 1 ∪ { ξ (1 , 1) } , R 3 = R 2 ∪ { ξ (2 , 1) , ξ (1 , 2) } and so on, so that as n increases we add up orbits ξ ∈ Q to R n in increasing order of integral radius. Remark 4.11. A systematic way for determining al l inte gr al orbits of a given r a- dius is as fol lows. 18 L et S 7 ( ρ ) ⊂ su (3) b e the 7 -spher e of r adius √ 2 ρ/ √ 3 c enter e d at the origin, so that the interse ction of S 7 ( ρ ) with the close d princip al Weyl chamb er is given by the p oints (4.37) Ξ ρ ( X,Y ) = X  1 + Y  2 , ( X, Y ≥ 0 X 2 + X Y + Y 2 = ρ 2 . The inte gr al orbits of S 7 ( ρ ) ar e given by the inte ger solutions of (4.38) X 2 + X Y + Y 2 − ρ 2 = 0 , X , Y ∈ N 0 . F or X, Y ∈ Z , the quantity X 2 + X Y + Y 2 is the norm of the Eisenstein inte ger X − Y φ , wher e φ = e i 2 π / 3 , so the pr oblem b e c omes how to factorize ρ 2 in prime factors on Z [ φ ] , which is an UFD with units {± 1 , ± φ, ± φ 2 } . 19 Example 4.12. As an example for R emark 4.11, take ρ 2 = 13 2 · 43 . The inte gr al orbits of S 7 ( ρ ) ar e solutions of (4.39) ( X − Y φ )( X − Y φ ) = 13 2 · 43 in Z [ φ ] . The prime factorization of 13 and 43 in the ring of Eisenstein inte gers ar e, up to the units {± 1 , ± φ, ± φ 2 } , (4.40) 13 = (3 − φ )(4 + φ ) , 43 = (6 − φ )(7 + φ ) , 18 Note that the radius of an in tegral orbit O is alwa ys a natural multiple of the integral radius of O ξ ⊂ S 7 for which O ∼ O ξ , cf. (2.25) and Deﬁnition 2.6. 19 W e refer to [5] for a v ery nice description of the ring Z [ ϕ ]. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 37 thus the set of solutions of (4.39) is (4.41) n α 13(6 − φ ) , α 13(7 + φ ) , α (3 − φ ) 2 (6 − φ ) , α (3 − φ ) 2 (7 + φ ) , α (4 + φ ) 2 (6 − φ ) , α (4 + φ ) 2 (7 + φ ) : α = ± 1 , ± φ, ± φ 2 o . Some solutions r epr esent the same orbit in diﬀer ent Weyl chamb ers. R estricting to the princip al Weyl chamb er, given by X − Y φ with X , Y ≥ 0 , we get the solutions (4.42) n 13 − 78 φ , 78 − 13 φ , 41 − 57 φ , 57 − 41 φ o c orr esp onding to the set of r ational orbits   1 √ 43 , 6 √ 43  ,  6 √ 43 , 1 √ 43  ,  41 13 √ 43 , 57 13 √ 43  ,  57 13 √ 43 , 41 13 √ 43   ⊂ Q . Deﬁnition 4.13. A radial Mago o sphere is a Mago o spher e c onstructe d using the r adial chain C r , cf. Deﬁnitions 4.3 and 4.10. Remark 4.14. However, b e c ause the r adial chain C r = ( R n ) n ∈ N is a c anonic al choic e and in light of Cor ol lary 4.9, fr om now on we shal l always assume this choic e C = C r , by default, when we r efer to a Mago o spher e in gener al. No w, we pro ceed to reverse the order of the iterated limits in Deﬁnition 4.7. Deﬁnition 4.15. We say that a Mago o spher e W {S 7 , b Π g } of Poisson typ e is of uniform Poisson type if its Mago o pr o duct and Mago o br acket satisfy (4.43) lim s →∞ lim n →∞ f 1  C f 2 = f 1 f 2 ∈ P ⇐ ⇒  ( f 1 | O ξ  s ξ f 2 | O ξ ) ξ ∈Q  s ≥ s 0 s →∞ − − − → ( f 1 f 2 | O ξ ) ξ ∈Q , (4.44) lim s →∞ lim n →∞ s [ f 1 , f 2 ] r ⋆ C = i { f 1 , f 2 } ∈ P ⇐ ⇒  ( sr ( ξ )[ f 1 | O ξ , f 2 | O ξ ] ⋆ s ξ ) ξ ∈Q  s ≥ s 0 s →∞ − − − → ( i { f 1 , f 2 }| O ξ ) ξ ∈Q , for any f 1 , f 2 ∈ P oly ( S 7 ) , cf. (4.30) . In this c ase, we write (4.45) W {S 7 , b Π g } ∼ − − → P oly ( S 7 , b Π g ) . The term uniform and notation (4.45) are justiﬁed by the following: Prop osition 4.16. W {S 7 , b Π g } is of uniform Poisson typ e if and only if, for every f 1 , f 2 ∈ P oly ( S 7 ) , we have b oth (4.46)   f 1 | O ξ  s ξ f 2 | O ξ − f 1 f 2 | O ξ   ξ and    sr ( ξ )[ f 1 | O ξ , f 2 | O ξ ] ⋆ s ξ − i { f 1 , f 2 }| O ξ    ξ c onver ging to 0 uniformly over Q , as s → ∞ . Pr o of. This is immediate from the deﬁnitions, cf. (4.30)-(4.31) and (4.43)-(4.44), once we note that, in this case, taking the limit n → ∞ ﬁrst is equiv alen t to replacing (4.30)-(4.31) by just (4.47) lim s →∞ f s = f ⇐ ⇒ ∀  > 0 , ∃ s ϵ ∈ N : ∀ s ≥ s ϵ = ⇒   f s ξ − f ξ   ξ <  ∀ ξ ∈ Q , since lim n →∞ P n = Q and we start with common extensions f 1 , f 2 ∈ P ol y ( S 7 )  → P to f 1 | n , f 2 | n ∈ P n , ∀ n ∈ N . □ 38 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Remark 4.17. F r om Pr op osition 4.16, the uniform Poisson pr op erty only dep ends on the p encil of r ays of universal c orr esp ondenc es, no t on any chain C satisfying (4.12) which is use d to c onstruct the Mago o spher e, just as in Cor ol lary 4.9. So, again, b e c ause C r is a c anonic al choic e we assume C = C r , by default. Ho wev er, the relev ant question is whether there exists any Mago o sphere of uniform Poisson t yp e. In the next subsection, we start inv estigating this question for the paradigmatic Mago o sphere of Poisson t yp e: Deﬁnition 4.18. The Berezin Mago o sphere is the Mago o spher e such that, ∀ ξ ∈ Q , ( w s ξ ) s ∈ N ≡ ( b s ξ ) s ∈ N is the ξ -r ay of Ber ezin universal c orr esp ondenc es, cf. Deﬁni- tion 2.30 and Pr op osition 2.31, and Deﬁnition 4.3. We denote it by (4.48) B {S 7 , b Π g } =  b C ,  C , [ · , · ] r ⋆ C  . 4.2. On the asymptotics of the Berezin Mago o sphere. In this subsection, w e prov e the follo wing result: Theorem 4.19. L et K ⊂ F b e any c omp act, and denote Q K = Q ∩ K . Then, for the Ber ezin Mago o spher e, cf. Deﬁnition 4.18, and for any f 1 , f 2 ∈ P oly ( S 7 ) , (4.49)   f 1 | O ξ  s ξ f 2 | O ξ − f 1 f 2 | O ξ   ξ and    sr ( ξ )[ f 1 | O ξ , f 2 | O ξ ] ⋆ s ξ − i { f 1 , f 2 }| O ξ    ξ c onver ge to 0 uniformly over Q K , as s → ∞ . Pr o of. The pro of uses a series of lemmas. The ﬁrst one is immediate from (3.11) in Prop osition 3.7. Lemma 4.20. F or any u ∈ U ( sl (3)) , the limit (4.50) lim s →∞ ( sr ( ξ )) − deg( u ) b s ξ [ u ] = ( − i ) deg( u ) β deg( u ) [ u ] | O ξ holds uniformly over Q . The following tw o lemmas will b e used to show that w e can ensure the v alidit y of a decomp osition similar to Prop osition 3.10 on all orbits in Q K sim ultaneously . Lemma 4.21. L et m = dim( P ol y ( O ξ 0 ) a ) , ξ 0 ∈ Q . Ther e ar e u 1 , ..., u m ∈ U ( sl (3)) a and s ( ξ 0 ) ∈ N , as wel l as an op en neighb orho o d U ( ξ 0 ) of ξ 0 in F , such that (4.51) n ( − i ) d (1) β d (1) [ u 1 ] , ..., ( − i ) d ( m ) β d ( m ) [ u m ] o , d ( j ) ≡ deg ( u j ) , is a b asis of P oly ( O ξ ) a , for every ξ ∈ U ( ξ 0 ) ⊂ F , and in addition, (4.52) s ≥ s ( ξ 0 ) = ⇒ n ( sr ( ξ )) − ( d (1)) b s ξ [ u 1 ] , ..., ( sr ( ξ )) − ( d ( m )) b s ξ [ u m ] o is also a b asis of P oly ( O ξ ) a , for every ξ ∈ U ( ξ 0 ) ∩ Q =: V ( ξ 0 ) . Pr o of. T ak e { h 1 , ..., h m } ⊂ P oly ( su (3)) a suc h that eac h h j is a homogeneous poly- nomial of degree d ( j ) and { h 1 | O ξ 0 , ..., h m | O ξ 0 } is a basis of P ol y ( O ξ 0 ) a . There are g 1 , ..., g m ∈ S U (3) for whic h the matrix H ( ξ 0 ) with entries (4.53) ( H ( ξ 0 )) j,k = h g j k ( ξ 0 ) is non singular. Consider its extension to the matrix v alued function (4.54) H : t → M C ( m ) , X 7→ ( H ( X )) j,k = h g j k ( X ) , ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 39 from which we hav e the p olynomial ϕ ∈ P ol y d (1)+ ... + d ( m ) ( t ), given by (4.55) ϕ ( X ) = det( H ( X )) . By construction, ϕ ( ξ 0 )  = 0, thus Z = ϕ − 1 (0) ∩ F is ﬁnite, and { h 1 | O ξ , ..., h m | O ξ } is l.i. for every ξ ∈ F \ Z . Since dim( P ol y ( O ξ ) a ) is constant on F , w e conclude that { h 1 | O ξ , ..., h m | O ξ } is a basis of P oly ( O ξ ) a for every ξ ∈ F \ Z . Therefore, there exists  ( ξ 0 ) > 0 such that the closed ball B H ( ξ 0 ) (  ( ξ 0 )) ⊂ M C ( m ) of radius  ( ξ 0 ) centered at H ( ξ 0 ) contains only non singular matrices, that is, (4.56) B H ( ξ 0 ) (  ( ξ 0 )) ⊂ GL m ( C ) , ∃  ( ξ 0 ) > 0 . Then, taking (4.57) U ( ξ 0 ) = { ξ ∈ F : ∥ H ( ξ ) − H ( ξ 0 ) ∥ <  ( ξ 0 ) / 2 } , w e hav e that (4.58) H ( ξ ) ∈ B H ( ξ 0 ) (  ( ξ 0 )) ⊂ GL m ( C ) , ∀ ξ ∈ U ( ξ 0 ) . No w, tak e u j = ( − i ) − d ( j ) S ( h j ) ∈ U ≤ d ( j ) ( sl (3)) a , so that h j = ( − i ) d ( j ) β d ( j ) [ u j ]. By Lemma 4.20, each ( sr ( ξ )) − d ( j ) b s ξ [ u j ] con v erges to h j uniformly on Q as s → ∞ . Th us, for the sequence of matrix-v alued functions B ( s, · ) s ∈ N , where (4.59) B ( s, · ) : Q → M C ( m ) , ξ 7→ ( B ( s, ξ )) j,k = ( sr ( ξ )) − d ( k ) b s ξ [ u k ] g j , w e hav e that B ( s, · ) s ∈ N also conv erges uniformly to H on Q as s → ∞ , that is, (4.60) ∀  > 0 , ∃ s ϵ ∈ N : ∀ s ≥ s ϵ = ⇒ ∥ B ( s, ξ ) − H ( ξ ) ∥ < / 2 , ∀ ξ ∈ Q . Com bining (4.57) and (4.60), ∃ s ( ξ 0 ) ∈ N such that, ∀ ξ ∈ V ( ξ 0 ) , (4.61) s ≥ s ( ξ 0 ) = ⇒ ∥ B ( s, ξ ) − H ( ξ 0 ) ∥ ≤ ∥ B ( s, ξ ) − H ( ξ ) ∥ + ∥ H ( ξ ) − H ( ξ 0 ) ∥ <  ( ξ 0 ) 2 +  ( ξ 0 ) 2 , whic h implies B ( s, ξ ) is non singular to o, that is, (4.62) B ( s, ξ ) ∈ B H ( ξ 0 ) (  ( ξ 0 )) ⊂ GL m ( C ) , ∀ ξ ∈ V ( ξ 0 ) , ∀ s ≥ s ( ξ 0 ) . Since P ol y ( O ξ ) a ≃ P oly ( O ξ 0 ) a , ∀ ξ ∈ F , we conclude from (4.58) that (4.51) is a basis for P ol y ( O ξ ) a , ∀ ξ ∈ U ( ξ 0 ), and from (4.62) that the set in (4.52) is also a basis of P ol y ( O ξ ) a , ∀ ξ ∈ V ( ξ 0 ), ∀ s ≥ s ( ξ 0 ). □ Lemma 4.22. L et u 1 , ..., u m ∈ U ( sl (3)) a , V ( ξ 0 ) and s ( ξ 0 ) b e as in the pr evious lemma. F or f ∈ P ol y ( S 7 ) a , ξ ∈ V ( ξ 0 ) and s ≥ s ( ξ 0 ) , ther e ar e α j ( s, ξ ) ∈ C for j ∈ { 1 , ..., m } , such that (4.63) f = m X j =1 α j ( s, ξ )( sr ( ξ )) − d ( j ) b s ξ [ u j ] = m X j =1 α ∞ j ( ξ )( − i ) d ( j ) β d ( j ) [ u j ] | O ξ , wher e (4.64) lim s →∞ α j ( s, ξ ) = α ∞ j ( ξ ) ∈ C holds uniformly on V ( ξ 0 ) . 40 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Pr o of. F or H ( ξ ) and B ( s, ξ ) as in the previous lemma, let F ( ξ ) = H ( ξ ) − 1 and A ( s, ξ ) = B ( s, ξ ) − 1 , for any ξ ∈ V ( ξ 0 ) and s ≥ s ( ξ 0 ), cf. (4.58) and (4.62). Then, from (4.60), B ( s, ξ ) s →∞ − − − → H ( ξ ) implies (4.65) α j ( s, ξ ) = m X k =1 ( A ( s, ξ )) j,k f g k ( ξ ) s →∞ − − − → α ∞ j ( ξ ) = m X k =1 ( F ( ξ )) j,k f g k ( ξ ) , so it only remains to show that the conv ergence in (4.65) is uniform ov er V ( ξ 0 ). Recall from the pro of of the previous lemma that there is  ( ξ 0 ) > 0 for which B H ( ξ 0 ) (  ( ξ 0 )) ⊂ GL m ( C ) as in (4.56) is compact and b oth H ( ξ ) and B ( s, ξ ) lie in its interior, ∀ ξ ∈ V ( ξ 0 ), ∀ s ≥ s ( ξ 0 ), cf. (4.58) and (4.62). Hence, by con tin uity of the inv ersion map on GL m ( C ), there is C > 0 such that, if s ≥ s ( ξ 0 ) and ξ ∈ V ( ξ 0 ), then ∥ A ( s, ξ ) ∥ and ∥ F ( ξ ) ∥ are b oth b ounded by C , giving (4.66) ∥ A ( s, ξ ) − F ( ξ ) ∥ = ∥ A ( s, ξ )( H ( ξ ) − B ( s, ξ )) F ( ξ ) ∥ ≤ C 2 ∥ H ( ξ ) − B ( s, ξ ) ∥ , with this last line conv erging to 0 uniformly on V ( ξ 0 ), cf. (4.60). Therefore, from (4.67) | α j ( s, ξ ) − α ∞ j ( ξ ) | ≤ m X k =1   ( A ( s, ξ )) j,k − ( F ( ξ )) j,k   | f g k ( ξ ) | , (4.60) and (4.66) imply that the conv ergence in (4.65) is uniform ov er V ( ξ 0 ). □ W e now pro ceed to ﬁnish the pro of of the theorem. Again, by bilinearity of the op erations, it is suﬃcient to sho w the result for f j ∈ P oly ( S 7 ) a j . No w, for any ξ 0 ∈ Q K , and U ( ξ 0 ) ⊂ F as in Lemma 4.21, we ha ve u j 1 , ..., u j m j ∈ U ( sl (3)) a j , with deg( u j k ) = d j ( k ), such that (4.68) { h j 1 | O ξ , ..., h j m j | O ξ } , h j k = ( − i ) d j ( k ) β d j ( k ) [ u j k ] , is a basis of P ol y ( O ξ ) a j for every ξ ∈ U ( ξ 0 ), and also exists s ( ξ 0 ) ∈ N such that (4.69) n ( sr ( ξ )) − ( d j (1)) b s ξ [ u j 1 ] , ..., ( sr ( ξ )) − ( d j ( m j )) b s ξ [ u j m j ] o is a basis of P ol y ( O ξ ) a j for every ξ ∈ V ( ξ 0 ) and s ≥ s ( ξ 0 ). Hence, from Lemma 4.22, there are α j k ( s, ξ ) ∈ C for k ∈ { 1 , ..., m j } such that (4.70) lim s →∞ α j k ( s, ξ ) =: ( α j k ) ∞ ξ ∈ C holds uniformly ov er V ( ξ 0 ), and (4.71) f j | O ξ = m j X k =1 α j k ( s, ξ )( sr ( ξ )) − d j ( k ) b s ξ [ u j k ] = m j X k =1 ( α j k ) ∞ ξ h j k | O ξ , for every ξ ∈ V ( ξ 0 ). Therefore, (4.72) f 1 | O ξ  s ξ f 2 | O ξ = X j,k α 1 j ( s, ξ ) α 2 k ( s, ξ )( sr ( ξ )) − ( d 1 ( j )+ d 2 ( k )) b s ξ [ u 1 j u 2 k ] con verges to (cf. (3.26) in Theorem 3.11) (4.73) X j,k ( α 1 j ) ∞ ξ ( α 2 k ) ∞ ξ h 1 j | O ξ h 2 k | O ξ = f 1 f 2 | O ξ ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 41 uniformly on V ( ξ 0 ). Analogously , (4.74) sr ( ξ )[ f 1 | O ξ , f 2 | O ξ ] ⋆ s ξ = X j,k α 1 j ( s, ξ ) α 2 k ( s, ξ )( sr ( ξ )) − ( d 1 ( j )+ d 2 ( k ) − 1) b s ξ [ u 1 j u 2 k − u 2 k u 1 j ] con verges uniformly on V ( ξ 0 ) to (cf. (3.27) in Theorem 3.11) (4.75) X j,k ( α 1 j ) ∞ ξ ( α 2 k ) ∞ ξ i { h 1 j , h 2 k }| O ξ = i { f 1 , f 2 }| O ξ . T o ﬁnish, by compactness, there exists a ﬁnite set { ξ 1 , ..., ξ k } ⊂ Q K suc h that the open sets U ( ξ 1 ) , ..., U ( ξ k ) ⊂ F (from which w e write the basis (4.68) and (4.69)) co ver K ⊂ F , and therefore V ( ξ 1 ) , ..., V ( ξ k ) ⊂ Q co v er Q K . In the previous para- graph, we hav e prov ed that, for any  > 0, there is s ϵ ( ξ j ) ∈ N such that (4.76) s ≥ s ϵ ( ξ j ) = ⇒       f 1 | O ξ  s ξ f 2 | O ξ − f 1 f 2 | O ξ    ξ <     sr ( ξ )[ f 1 | O ξ , f 2 | O ξ ] ⋆ s ξ − i { f 1 , f 2 }| O ξ    ξ <  , ∀ ξ ∈ V ( ξ j ) . Then, taking (4.77) s ϵ = max { s ϵ ( ξ 1 ) , ..., s ϵ ( ξ k ) } ∈ N , w e get (4.78) s ≥ s ϵ = ⇒       f 1 | O ξ  s ξ f 2 | O ξ − f 1 f 2 | O ξ    ξ <     sr ( ξ )[ f 1 | O ξ , f 2 | O ξ ] ⋆ s ξ − i { f 1 , f 2 }| O ξ    ξ <  , ∀ ξ ∈ Q K . □ Remark 4.23. We emphasize that the uniform c onver genc e establishe d in L emma 4.20 is a sp e cial pr op erty of the Ber ezin Mago o spher e which do es not hold for gener al Mago o spher es of Poisson typ e. F or example, for any ξ ∈ Q , c onsider the ξ -r ay ( w s ξ ) s of universal c orr esp on- denc es given by the fol lowing rule: for every u ∈ U ( sl (3)) ( a,b ) , (4.79) w s ξ [ u ] =     1 + r ( ξ ) s  b s ξ [ u ] if ( a, b )  = (0 , 0) b s ξ [ u ] otherwise . Then, for any u ∈ U ( sl (3)) , we have (4.80) lim s →∞ ( sr ( ξ )) − deg( u ) w s ξ [ u ] = ( − i ) deg( u ) β deg( u ) [ u ] , ∀ ξ ∈ Q , which me ans e ach ξ -r ay ( w s ξ ) is of Poisson typ e. However, if u ∈ U ( sl (3)) lies in any non trivial irr ep, then (4.81) ( sr ( ξ )) − deg( u ) w s ξ [ u ] − ( − i ) deg( u ) β deg( u ) [ u ] = ε s ξ [ u ] + r ( ξ ) s b s ξ [ u ] , wher e ε s ξ is the err or function of b s ξ . Sinc e the inte gr al r adius function r is unb ounde d on any neighb orho o d of any ξ ∈ Q , cf. Pr op osition 2.9, the c onver genc e (4.80) is not uniform anywher e. In view of the previous remark, we ha ve the following: 42 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Prop osition 4.24. F or a gener al Mago o spher e of Poisson typ e, the uniform Pois- son pr op erty may not b e satisﬁe d for any neighb orho o d of any ξ ∈ Q . Pr o of. Since it is enough to sho w this non-uniformity in a single example, w e show it explicitly for a single p olynomial  -pro duct in the example of Remark 4.23. Th us, let u ∈ U 1 ( sl (3)) ≡ sl (3) b e a highest weigh t v ector, so that u 2 ∈ U 2 ( sl (3)) is a highest weigh t vector for a representation (2 , 2). Then, (4.82) f = − iβ 1 [ u ] = ⇒ f | O ξ = ( sr ( ξ )) − 1 b s ξ [ u ] =  sr ( ξ )  1 + r ( ξ ) s  − 1 w s ξ [ u ] , ∀ ξ ∈ Q , and, for the twisted pro duct  s ξ induced by w s ξ , we hav e (4.83) f | O ξ  s ξ f | O ξ = ( sr ( ξ )) − 2  1 + r ( ξ ) s  − 1 b s ξ [ u 2 ] =  1 + r ( ξ ) s  − 1  f 2 | O ξ + ε s ξ [ u 2 ]  . By the triangular inequality , (4.84)   f | O ξ  s ξ f | O ξ − f 2 | O ξ   ξ ≥            1 + r ( ξ ) s  − 1 f 2 | O ξ − f 2 | O ξ      ξ −       1 + r ( ξ ) s  − 1 ε s ξ [ u 2 ]      ξ      . F or the last term in the r.h.s. of (4.84), from Prop osition 3.7, we ha v e (4.85)       1 + r ( ξ ) s  − 1 ε s ξ [ u 2 ]      ξ ≤ M ( u 2 ) r ( ξ )( s + r ( ξ )) , and hence this v anishes uniformly o v er Q . But on the other hand, (4.86)       1 + r ( ξ ) s  − 1 f 2 | O ξ − f 2 | O ξ      ξ = r ( ξ ) s + r ( ξ )   f 2 | O ξ   ξ , and this do es not v anish uniformly anywhere, since r is unbounded on any neigh- b orhoo d of Q , cf. Prop osition 2.9. Hence, although the l.h.s. of (4.84) v anishes as s → ∞ , ∀ ξ ∈ Q , it does not v anish uniformly in an y neigh borho o d of an y ξ ∈ Q . □ Th us, from the bijection F ∋ ξ ↔ O ξ ⊂ S 7 , Theorem 4.19 states that we hav e P oisson uniformity for any compact Berezin Mago o “cylinder”, that is, we hav e (4.87) B {S 7 | K , b Π } ∼ − → P ol y ( S 7 | K , b Π) , cf. (4.45) in Deﬁnition 4.15, where S 7 | K is the compact “cylinder” (4.88) S 7 | K = [ ξ ∈K O ξ ⊂ S 7 . Remark 4.25. However, we haven ’t yet b e en able to pr ove or dispr ove Poisson uniformity of the whole Ber ezin Mago o spher e, that is, for the whole Q . Thus, the question of whether ther e is a Mago o spher e of uniform Poisson typ e r emains op en. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 43 5. Concluding remarks In this series of tw o pap ers on quark systems, we explored the prop erties and results for S U (3) in detail, which allow ed us to paint a clear and detailed picture of quan tum and classical quark systems and their relationship via sym b ol corresp on- dences and semiclassical asymptotics. How ev er, a lot of what has b een done for S U (3) generalizes to other compact symmetry groups. So, here we conclude this series b y highlighting what can b e generalized to other groups and commenting on some p eculiarities of S U (3). W e shall pro ceed by decreasing order of generality , summarizing the main arguments, and refer to [1] for a more complete analysis. In Remark I.3.4, we indicated that the material of section I.3 holds for an y com- pact Lie group. Indeed, let G b e a connected compact Lie group with Lie algebra g . If ρ is a unitary G -irrep on H , then it is ﬁnite dimensional [15], hence, the space B ( H ) of all op erators on H is also ﬁnite dimensional and carries a unitary (with respect to the trace inner product) G -representation. Also, giv en a Hamilton- ian G -space P , we can use the isomorphism P ≃ G/G 0 , where G 0 is the isotropy subgroup of some p oint ς 0 ∈ P , to descend the Haar measure of G to P so that C ∞ C ( P ) ⊂ L 2 ( P ). Thus, deﬁning sym b ol corresp ondences from B ( H ) to C ∞ C ( P ) analogously to Deﬁnition I.3.1, everything done in section I.3 follows. Besides that, the represen tation on B ( H ) is completely reducible b ecause it is a unitary representation on a ﬁnite dimensional space. 20 Also, by the Peter-W eyl Theorem and the already stated isomorphism P ≃ G/G 0 , the space L 2 ( P ) in- herit a decomp osition into irreps from L 2 ( G ), with orthonormal basis comprised by smo oth harmonic functions [8]. These decomp ositions of op erators and functions lead to the characterization of symbol corresp ondences by characteristic matrices (c haracteristic num bers for highest symmetry) in the sense of sections I.4 and I.5. Moreo ver, P cov ers a coadjoint orbit O ⊂ g ∗ via the momentum map, so the coadjoin t orbits are of particular in terest as mo dels of Hamiltonian G -spaces and there are only ﬁnitely many types of them [13]. F or the metho ds of Paper I I, the argumen t used to identify the space of p olynomials on an orbit with the linear span of harmonic functions w orks for general compact Lie groups, so one may reason it’s fairer to restrict the co domain of symbol corresp ondences to space of p olynomials P ol y ( O ) deﬁned as in (2.3), but now replacing su (3) b y g ∗ . Henceforth we make the further assumptions that compact G is semisimple (so the Killing form provides an iden tiﬁcation g ↔ g ∗ and it do esn’t matter whether w e work with coadjoint or adjoin t action [11]) and simply connected (which implies that the irreps of G are all determined b y the Theorem of Highest W eigh t [11] and that the (co)adjoin t G -orbits are simply connected, so they are the unique Hamiltonian G -spaces [4]). Therefore, the irreps obtained from dominant weigh ts and the (co)adjoint orbits exhausts all the p ossibilities of quantum and classical systems, resp ectively , for which there are symbol corresp ondences. A general result due to Wildb erger [6, 19] (that w e sp ecialized for quark systems in Theorem I.5.24) sa ys even more: let ω be a dominant weigh t of g and ξ = ω / ∥ ω ∥ , so that we write H ω for an irrep with highest weigh t ω and O ξ for the orbit of ξ , then the set of symbol corresp ondence from B ( H ω ) to P ol y ( O ξ ) is not empty , it con tains a Berezin corresp ondence (deﬁned via highest w eight ω ). 20 Note that the natural isomorphism B ( H ) ≃ H ⊗ H ∗ allows us to write this representation as the tensor pro duct of ρ with its dual representation, so the decomp osition of B ( H ) into irreps is an instance of Clebsc h-Gordan series. 44 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS F urthermore, the arguments in section II.2, ab out the inadequacy of formally deforming the algebra of C ∞ C ( O ) and pro ceeding instead by looking at series of t wisted algebras of increasing ﬁnite dimensions, these apply in this more general con text, since Proposition 2.16 generalizes to any pair ( G, O ), where G is a compact simply connected semisimple Lie group and O any of its (co)adjoint orbits. Then, similarly to section I I.2, for the complexiﬁcation g C of g , we get an iso- morphism β g C : U ( g C ) → P oly ( g C ) from the PBW Theorem in the same vein of (2.68), and P ol y ( g C ) can b e prop erly iden tiﬁed with P ol y ( g ) so that the p oint wise pro duct and the P oisson brack et on P ol y ( g ) are giv en by expressions analogous to (2.80) and (2.81), resp ectively . Pullbacks of symbol correspondences to the univ er- sal en v eloping algebra are a v ailable as well, so universal Berezin corresp ondences (recall Wildb erger’s argumen t) are given as in Prop osition 2.31, now using β g C . Th us, ev erything p oints to generalizing the deﬁnitions of rays of univ ersal corre- sp ondences, cf. Deﬁnition 3.1, and the ones of Poisson type, cf. Deﬁnition 3.4, in this larger context, wherein the pro of of Theorem 3.11 suits well – we refer again to [12]. Hence, it should b e clear that the criteria in Theorems 3.17, 3.20 and 3.21 hold in the context of any semisimple simply connected compact Lie group. F or the unit sphere S ⊂ g , we still ha v e a countable dense subset of the orbit space S /G comprised by orbits that are equiv alent to highest weigh t orbits in g , in the sense of Deﬁnition 2.4, leading to generalizations of the integral radius and the coarse Poisson sphere, cf. Deﬁnitions 2.6 and 2.7. T o prop erly extend the notion of Mago o sphere, we need inv arian t p olynomials satisfying (4.6), and they can b e constructed using the Harish-Chandra Theorem and the Chev alley Theorem. Then, results analogous to Theorem 4.8 and Corollary 4.9 are av ailable. Besides, a similar version of Theorem 4.19 holds for any compact simply con- nected semisimple group G , that is, the (highest weigh t) Berezin corresp ondences for G satisfy the Poisson prop ert y uniformly on compact sets of the regular stratum of the symplectic foliation of S , b ecause the fundamental premise of such result is the fact that the error maps of Berezin corresp ondences v anish uniformly , as as- serted in Prop osition 3.7, whose statement holds in this greater generality . No w, for some p eculiarities from S U (3). Although not necessary for the main argumen t in subsection I I.2.3, we susp ect that Theorem 2.11 for ( S U ( n ) , C P n − 1 ) can b e generalized from n = 2 , 3 to n > 3, but we still don’t know if this is true. When n = 2 k is even, the action of S U (2 k ) on op erators and orbits is eﬀectiv ely an action of P S U (2 k ) = S U (2 k ) / Z 2 . W e hav e already seen that, in the case of S U (2), there is more freedom for the signs of the characteristic num bers, etc, cf. [17]. It would b e interesting to see if we get more freedom in this resp ect for the characteristic matrices, etc, going from S U (3) to S U (4) and b ey ond to S U (2 k ) and other compact simply connected semisimple Lie groups with center Z 2 . Also, for spin systems the relation b etw een Berezin and Stratono vich-W eyl sym- b ol corresp ondences is rather direct, something we lost for mixed quark systems, cf. Remark I.5.27. But since Stratono vic h-W eyl corresp ondences, and more gen- erally , semi-conformal corresp ondences are also sp ecial, it would b e interesting to in vestigate their relation to Berezin corresp ondences in more detail, still in the case of S U (3), and then see how muc h more complex this relation can get as w e mov e to S U (4) and b eyond to other compact Lie groups. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 45 In particular, a p ertinen t question to b e answered, still in the context of S U (3), is whether there exists a Mago o sphere constructed from Stratonovic h-W eyl corre- sp ondences, or semi-conformal corresp ondences, which is of uniform Poisson type, cf. Deﬁnition 4.15, or at least satisﬁes the uniform P oisson property on compacts of the regular part of the foliation of the unit sphere, as pro ved for the Berezin Mago o sphere in Theorem 4.19. Because, although we hav e not y et answ ered the question of P oisson uniformit y for the whole Berezin Mago o sphere, cf. Remark 4.25, the missing part is the one containing the singularities of the symplectic foliation of S 7 . But moving forw ard to S U (4), and b eyond to S U ( n ), this question could get harder, since the singular foliation of the unit sphere by (co)adjoint orbits is strati- ﬁed and has deep er singularities 21 . So, while for S U (3) the singular foliation of the P oisson unit sphere has only tw o isolated singular orbits and the singularities are of the simplest p ossible type, Morse-Bott type, already in the case of S U (4) the in tersection of the principal W eyl c hamber with the unit sphere is a closed triangle, with its interior mapping to the regular stratum of the symplectic foliation and its edges to the singular strata, wherein the vertices map to the deep er singular orbits which are isomorphic to C P 3 . Thus, it is conceiv able that this more elab- orate singular structure, with qualitativ ely diﬀeren t w a ys of reac hing the deep er singularities starting from the regular stratum, could play a role in the question of P oisson uniformity of Mago o spheres for su (4). And so on for su ( n ). On the other hand, for any compact semisimple Lie group G of rank 2 the sym- plectic foliation of the unit sphere in g is parameterized b y a closed arc of circumfer- ence and the stratiﬁcation of singular orbits is trivial. Besides S U (3), there are tw o other such groups that are simply connected, namely: S U (2) × S U (2) ≃ S pin (4) and S p (2) ≃ S pin (5). 22 In the former case, the generic (co)adjoint orbits are isomorphic to S 2 × S 2 , whereas the degenerate ones are isomorphic to S 2 , with Morse-Bott singularities for the symplectic foliation of S 5 ⊂ su (2) ⊕ su (2) ≃ so (4). Ho wev er, we lack a similar understanding of the orbit foliation in the latter case. It would also b e int eresting to work b oth cases in full details. Finally , it could b e interesting to expand on the inv estigations of asymptotic lo calization, in a general and systematic wa y as was done in [2] for spin systems, no w in the context of quark systems. In the same vein, one could try working out the formalism of sequen tial quantizations, in a complete and detailed w ay as was done for S 2 in [2], now for the (co)adjoint orbits of su (3), and ev entually , p erhaps, joining them together along the coarse Poisson sphere, if p ossible. References [1] P . A. S. Alcˆ antara. On symbol correspondences for systems with compact group of symme- tries. PhD Thesis, Univ ersity of S˜ ao Paulo (in preparation). [2] P . A. S. Alcˆ an tara and P . de M. Rios. Asymptotic lo calization of sym bol correspondences for spin systems and sequential quan tizations of S 2 . A dvanc es in The or etic al and Mathematical Physics , 26:3377–3462, 2022. [3] P . A. S. Alcˆ antara and P . de M. Rios. On symbol corresp ondences for quark systems I: characterizations. arXiv:2203.00660, 2022. 21 W e refer to [10] for a description of the (co)adjoin t orbits and their foliation of su ( n ). 22 F or n ≥ 3, the group S pin ( n ) is the (universal) double cov er of the sp ecial orthogonal group S O ( n ), but for n = 3 , 4 , 5, w e hav e the isomorphisms S pin (3) ≃ S U (2) ≃ S p (1), S pin (4) ≃ S U (2) × S U (2), S pin (5) ≃ S p (2), where S p ( n ) is the group of n × n unitary matrices ov er the quaternions, also called the compact symplectic group. 46 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS [4] M. A. Armstrong. Calculating the fundamental group of an orbit space. Pr o c. Amer. Math. So c. , 84:267–271, 1982. [5] D. A. Cox. Primes of the F orm x 2 + ny 2 . John Wiley and Sons, 1989. [6] H. Figueroa, J. M. Gracia-Bond ´ ıa, and J. C. V arilly . Mo y al quantization with compact sym- metry groups and noncumm utative harmonic analysis. J. Math. Phys. , 31, 1990. [7] R. Fioresi and M. A. Lled´ o. On the deformation quantization of coadjoint orbits of semissimple Lie groups. Paciﬁc J. Math. , 198, 2001. [8] G. B. F olland. A Course in A bstr act Harmonic Analysis . CR C Press, T aylor & F rancis, 2016. [9] W. Greiner and B. M ¨ uller. Quantum Me chanics . Springer, 1994. [10] V. Guillemin, E. Lerman, and S. Sternberg. Symple ctic ﬁbr ations and multiplicity diagr ams . Cambridge University Press, 2010. [11] J. E. Humphreys. Intr o d uction to Lie Algebr as and R epr esentation The ory . Springer New Y ork, 1973. [12] A. V. Karabegov. Berezin’s Quantization on Flag Manifolds and Spherical Modules. T r ans- actions of the Americ an Mathematic al So ciety , 350(4), 1998. [13] A. A. Kirillov. L e ctur es on the Orbit Metho d . American Mathematical So ciety , 2004. [14] M. A. Lled´ o. Deformation quantization of nonregular orbits of Compact Lie groups. L ett. Math. Phys. , 58, 2001. [15] L. Nach bin. On the ﬁnite dimensionality of every irreducible unitary representation of a compact group. Pr o c. Amer. Math. So c. , 12:11–12, 1961. [16] M. Rieﬀel. Deformation quan tization of Heisenberg manifolds. Commun. Math. Phys. , 122:531–562, 1989. [17] P . de M. Rios and E. Straume. Symb ol Corr esp ondenc es for Spin Systems . Birkh¨ auser/Springer, 2014. [18] L. J. Slater. Generalize d Hyp erge ometric F unctions . Cambridge University Press, 1966. [19] N. Wildb erger. On the F ourier transform of a compact semisimple Lie group. J. A ustr al. Math. Soc. , 56:64–116, 1994. Appendix A. A proof of Proposition 3.22 F rom Prop osition I.4.17, (A.1) b p n = ( − 1) p s ( p + 1)( p + 2) 2( n + 1) 3 C ( p, 0) , ( p, 0 , 0) , (0 ,p ) , (0 ,p,p ) , n ( n,n,n ) , 0 , so we just need to compute these CG co eﬃcients. Let a 0 , ..., a n ∈ R b e such that (A.2) T n − ( e ( n ; (2 n, 0 , n ) , n/ 2)) = n X J =0 a J e ( n ; 0 n , J ) . W e know that (A.3) ⟨ e ( p ; ( p, 0 , 0)) ⊗ q e ( q p ; (0 , p, p )) | e ( n ; 0 n , J ) ⟩  = 0 ⇐ ⇒ J = 0 . F rom (I.2.26), w e hav e (A.4) a 0 = p (2 n + 1)! n + 1 . ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 47 Applying U n − to (3.72), we obtain (A.5) e ( n ; (2 n, 0 , n ) , n/ 2) = ( − 1) n µ n ( p ) T n + = ( − 1) p n ! µ n ( p ) s  p n  e ( p ; ( p, 0 , 0)) ⊗ q e ( q p ; ( n, p − n, p )) + X j + k + l = p j  = p c j,k,l e ( p ; ( j, k , l )) ⊗ q e ( q p ; ( p − j + n, p − k − n, p − l )) . Again from (I.2.26), we hav e 23 (A.6) T n − ( q e ( q p ; ( n, p − n, p ))) = n ! s  p n  q e ( p ; (0 , p, p )) , then (A.7) p (2 n + 1)! n + 1 C ( p, 0) , ( p, 0 , 0) , (0 ,p ) , (0 ,p,p ) , n ( n,n,n ) , 0 =  e ( p ; ( p, 0 , 0)) ⊗ q e ( q p ; (0 , p, p ))   T n − ( e ( n ; (2 n, 0 , n ) , n/ 2))  = ( − 1) p µ n ( p ) n ! s  p n  ! 2 . Using the expression for µ n ( p ) in (3.72), we get (A.8) C ( p, 0) , ( p, 0 , 0) , (0 ,p ) , (0 ,p,p ) , n ( n,n,n ) , 0 = ( − 1) p s 2( n + 1) 3 ( p + 1)( p + 2) s  p n   p + n +2 n  . Therefore, (A.9) b p n = s  p n   p + n +2 n  = n Y m =1 s 1 − ( m − 1) /p 1 + ( m + 2) /p > 0 . Since the function (A.10) f ( x ) = n Y m =1 s 1 − ( m − 1) x 1 + ( m + 2) x is analytic around 0, we ha v e that (A.11) lim p →∞ p ( b p n − 1) = f ′ (0) = − n ( n + 2) 2 , that is, | b p n − 1 | ∈ O (1 /p ), ∀ n ∈ N . 23 Note that q e ( q p ; ( j, k , l )) has weight ( j − k ) / 2 for the subrepresentation (2 p − l ) / 2 of t -standard S U (2). 48 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Appendix B. Al terna tive pr oof of Corollar y 3.23 In this app endix, our main goal is to indicate an alternative approach to prov e Corollary 3.23 using the symmetries of Clebsch-Gordan co eﬃcients established by Theorem I.2.16. W e won’t presen t full pro ofs for the statemen ts in this app endix, but we outline all the arguments and refer to [1] for a complete treatment. F or ( x 1 , ..., x 8 ) the coordinates w.r.t. the orthonormal basis { E j : j = 1 , ..., 8 } , w e resort to the following helpful co ordinates: (B.1) t + = x 1 + ix 2 , t − = x 1 − ix 2 , v + = x 4 + ix 5 , v − = x 4 − ix 5 , u + = x 6 + ix 7 , u − = x 6 − ix 7 , t = x 3 , u = ( √ 3 x 8 − x 3 ) / 2 . Indeed, using these co ordinates, we ha v e (B.2) i Π g = √ 2  ∂ t + ⊗ T + + ∂ t − ⊗ T − + ∂ v + ⊗ V + + ∂ v − ⊗ V − + ∂ u + ⊗ U + + ∂ u − ⊗ U − + ∂ t ⊗ T 3 + ∂ u ⊗ U 3  , and, for the harmonic functions, (B.3) X 1 (2 , 1 , 0) , 1 / 2 ≡ 2 v + , X 1 (2 , 0 , 1) , 1 / 2 ≡ − 2 t + , X 1 (1 , 2 , 0) , 1 ≡ 2 u + , X 1 (1 , 0 , 2) , 1 ≡ 2 u − , X 1 (0 , 2 , 1) , 1 / 2 ≡ 2 t − , X 1 (0 , 1 , 2) , 1 / 2 ≡ 2 v − , X 1 0 1 , 1 ≡ − 2 √ 2 u , X 1 0 1 , 0 ≡ 2 r 2 3 (2 t + u ) , so X n ν ,J ∈ P ol y n ( O (1 , 0) ) for every n . Thus, X p = P ol y ≤ p ( O (1 , 0) ) is the image of W p , cf. Corollary I.4.10. F urthermore, we set X = P oly ( O (1 , 0) ). No w, let ( W p ) b e a sequence of symbol corresp ondences as in (3.73), with char- acteristic num bers c p n . Then each W p induces a twisted pro duct  p on X p . The route for the alternative semiclassical analysis is summarized in the following steps: 1. V erify that (B.4) f 1  p f 2 → f 1 f 2 for every f 1 ∈ X 1 and f 2 ∈ X if c p n → 1 as p → ∞ for every n ≥ 1. In addition, P oisson condition and c p 1 → 1 together give that c p n → 1, for every n ≥ 1. 2. Apply induction to conclude that (B.4) holds for every f 1 , f 2 ∈ X if c p n → 1 as p → ∞ , for ev ery n ≥ 1. 3. Sho w that, if c p n → 1 as p → ∞ , for every n ≥ 1, then ∥ [ f 1 , f 2 ] ⋆ p ∥ ∈ O (1 /p ) for ev ery f 1 , f 2 ∈ X . 4. Pro ve that the conv ergence c p 1 → 1 as p → ∞ is equiv alen t to (B.5) p [ f 1 , f 2 ] ⋆ p → i r 3 2 { f 1 , f 2 } for every f 1 ∈ X 1 and every f 2 ∈ X . 5. By induction again, based on the previous tw o steps, show that c p n → 1 as p → ∞ , for ev ery n ≥ 1, also gives (B.5) for every f 1 , f 2 ∈ X . Therefore, if ( W p ) is of Poisson t ype, then Steps 1 and 4 together imply that the c haracteristic num bers satisfy c p n → 1 as p → ∞ for all n ≥ 1; on the other hand if all the c haracteristic n umbers con verge to 1, then Steps 2 and 5 show that ( W p ) is of Poisson type. This prov es Corollary 3.23. ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 49 W e now analyze each of the Steps 1 through 5, as stated ab ov e. Step 1 . F rom Lemma I.2.18 and Theorem I.2.20, the star pro duct of harmonic functions on O (1 , 0) ≃ C P 2 can b e straigh tforw ardly seen to satisfy (B.6) X 1 ν 1 ,J 1  p X n ν 2 ,J 2 = p δ ( p ) n +1 X m = n − 1 X σ X µ ,I ν ,J ( − 1) p c p m c p 1 c p n C 1 , ν 1 J 1 , n, ν 2 J 2 , ( m ; σ ) ν J × C 1 , µ I , n, q µ I , ( m ; σ ) 0 m 0  1 n m µ , I q µ , I 0 m , 0  [ p ] X m ν J , where (B.7) (2 , 1 , 0) = ( n + 1 , n, n − 1) , (2 , 0 , 1) = ( n + 1 , n − 1 , n ) , (1 , 2 , 0) = ( n, n + 1 , n − 1) , (0 , 1 , 2) = ( n − 1 , n, n + 1) , (0 , 2 , 1) = ( n − 1 , n + 1 , n ) , (1 , 0 , 2) = ( n, n − 1 , n + 1) . By determining a prop ortionality (B.8) X µ ,I C 1 , µ I , n, q µ I , ( m ; σ ) 0 m 0  1 n m µ , I q µ , I 0 m , 0  [ p ] ∝ C 1 , 0 1 0 , n, 0 n 0 , ( m ; σ ) 0 m 0  1 n m 0 1 , 0 0 n , 0 0 m , 0  [ p ] , up to order O (1 /p 2 ), we get the following k ey result. Prop osition B.1. F or n ≥ 1 , we have (B.9) X 1 ν 1 ,J 1  p X n ν 2 ,J 2 = n +1 X m = n − 1 X σ X ν ,J c p m c p 1 c p n f n,m ( p ) C 1 , ν 1 J 1 , n, ν 2 J 2 , ( m ; σ ) ν J × C 1 , 0 1 0 , n, 0 n 0 , ( m ; σ ) 0 m 0 X m ν ,J + O (( c 1 p ) − 1 ) , wher e (B.10) f n,n ( p ) = ( − 1) p p δ ( p ) (2 n + 1)(2 n + 3) n ( n + 2)  1 n n 0 1 , 0 0 n , 0 0 n , 0  [ p ] and, for m ∈ { n − 1 , n + 1 } , (B.11) f n,m ( p ) = ( − 1) p p δ ( p ) 4( m + n + 2)( n + 1) 3( m + 1) 2  1 n m 0 1 , 0 0 n , 0 0 m , 0  [ p ] . A lso, the c ontribution O (( c p 1 p ) − 1 ) c omes fr om m = n . Sketch of pr o of. The statemen t follo ws from exhaustive application of ladder op er- ators U − and T − on (B.12) e (( m ; σ ); 0 m , 0) = X µ ,I C 1 , µ I , n, q µ I , ( m ; σ ) 0 m 0 e (1; µ , I ) ⊗ e ( n ; q µ , I ) . This is, how ever, more subtle when m = n , where w e need (B.13) T + ( e ( n ; (021) , 1 / 2)) = − µ 1 ( p ) h e (1; (201) , 1 / 2) , ( e ( n ; (021) , 1 / 2)) i , cf. (3.72), to obtain the contribution of order O (1 /p ). □ 50 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Thereb y we conclude Step 1 if we ev aluate lim p →∞ f n,m ( p ). F or the sak e of read- abilit y , we set x n ≡ x n [ p ] :=  1 n n 0 1 , 0 0 n , 0 0 n , 0  [ p ] , (B.14) y n ≡ y n [ p ] :=  1 n n + 1 0 1 , 0 0 n , 0 0 n +1 , 0  [ p ] . (B.15) By taking Hermitian co jugate, we get (B.16) y n − 1 [ p ] =  1 n n − 1 0 1 , 0 0 n , 0 0 n − 1 , 0  [ p ] , so we only need to determine the v alues of ( x n [ p ]) n

m denote the highest weigh t of ( m, m ), so X 1 > 1 X n > n is a non zero m ultiple of X n +1 > n +1 Theorem B.7. Supp ose the uniform c onver genc e f 1  p f 2 → f 1 f 2 holds for every p air f 1 ∈ X 1 and f 2 ∈ X . If c p 1 → 1 as p → ∞ , then c p n → 1 for al l n ≥ 1 . 52 P . A. S. ALC ˆ ANT ARA AND P . DE M. RIOS Sketch of pr o of. F or n ∈ N , (B.29) X 1 > 1  p X n > n = c n +1 c 1 c n f n,n +1 ( p ) C 1 , > 1 , n, > n , n +1 > n +1 C 1 , 0 1 0 , n, 0 n 0 , n +1 0 n +1 0 X n +1 > n +1 . The statement follows b y induction, using Theorem I.4.5 and Lemma B.4. □ Corollary B.8. If the char acteristic numb ers ( c p n ) deﬁne a se quenc e of c orr esp on- denc es of Poisson typ e and c p 1 → 1 as p → ∞ , then c p n → 1 for every n ≥ 1 . Step 2 . W e’ll proceed by induction from Theorem B.5. Giv en an harmonic function X n ν ,J and p, m ∈ N with p > max { n, m } , let (B.30) L n,m ν ,J [ p ] , R n,m ν ,J [ p ] : X m → X n + m , L n,m ν ,J [ p ]( f ) = X n ν ,J  p f , R n,m ν ,J [ p ]( f ) = f  p X n ν ,J , b e the left and right star pro duct op erators, resp ectiv ely . Lemma B.9. If al l char acteristic numb ers c onver ge to 1 as p → ∞ , then the families of op er ators ( L n,m ν ,J [ p ]) p and ( R n,m ν ,J [ p ]) p ar e uniformly b ounde d for every n, m ≥ 1 . Sketch of pr o of. It follows from Theorem I.C.3 and equation (I.C.6). □ Theorem B.10. If c p n → 1 as p → ∞ for al l n ≥ 1 , then the uniform c onver genc e f 1  p f 2 → f 1 f 2 holds for every p air f 1 , f 2 ∈ X . Sketch of pr o of. Assume that, for n ∈ N , f 1  p f 2 → f 1 f 2 whenev er f 1 ∈ X n and f 2 ∈ X . Every element of X n +1 is a linear combination of an element of X n and p oin t wise pro ducts of the form X n ν ,J X 1 µ ,I , so it is suﬃcient to prov e (B.31) ( X n ν ,J X 1 µ ,I )  p X n ′ ν ′ ,J ′ → X n ν ,J X 1 µ ,I X n ′ ν ′ ,J ′ . The idea is to sum and subtract X n ν ,J  p ( X 1 µ ,I X n ′ ν ′ ,J ′ ) and X n ν ,J  p X 1 µ ,I  p X n ′ ν ′ ,J ′ , then use triangular inequality and Lemma B.9 to conclude what we wan t. □ Step 3 . T o estimate the rate of conv ergence of ∥ [ f 1 , f 2 ] ⋆ p ∥ when the character- istic num bers all go to 1, the symmetric Stratonovic h-W eyl corresp ondence is a suitable reference. So let ( ∗ p S ) b e the twisted pro ducts induced by the symmetric Stratono vich-W eyl corresp ondences. Theorem B.11. F or every f 1 , f 2 ∈ X , we have    [ f 1 , f 2 ] ∗ p S    ∈ O (1 /p ) . Sketch of pr o of. It follows straightforw ardly from Theorem B.6. □ Theorem B.12. If c p n → 1 as p → ∞ for every n ≥ 1 , then ∥ [ f 1 , f 2 ] ⋆ p ∥ ∈ O (1 /p ) for every f 1 , f 2 ∈ X . Sketch of pr o of. F or n 1 , n 2 ∈ N , the idea is to compare    [ X n 1 ν 1 ,J 1 , X n 2 ν 2 ,J 2 ] ∗ p S    and    [ X n 1 ν 1 ,J 1 , X n 2 ν 2 ,J 2 ] ⋆ p    using the norm given by the maximum of co ordinates with resp ect to the basis of harmonic functions as intermediate. Just note that any tw o norms on X n 1 + n 2 are ON SYMBOL CORRESPONDENCES FOR QUARK SYSTEMS I I 53 equiv alent since it is ﬁnite dimensional, and the hypothesis on the c haracteristic n umbers implies that there is C ( n 1 , n 2 ) > 0 such that (B.32)     c p n c p n 1 c p n 2     ≤ C ( n 1 , n 2 ) for every n ≤ n 1 + n 2 . □ Step 4 . The commutator [ X 1 µ ,I , X n ν ,J ] ⋆ p can b e explicitly computed. Prop osition B.13. F or any two C P 2 harmonics X 1 µ ,I , X n ν ,J ∈ X , we have (B.33)  X 1 µ ,I , X n ν ,J  ⋆ p = 1 p p 1 + 3 /p i c p 1 r 3 2 { X 1 µ ,I , X n ν ,J } . In p articular, p [ f 1 , f 2 ] ⋆ p → i p 3 / 2 { f 1 , f 2 } uniformly for every f 1 ∈ X 1 and f 2 ∈ X if and only if c p 1 → 1 as p → ∞ . Sketch of pr o of. Let A = e (1; µ , I ). By deﬁnition of twisted pro duct, and with a little abuse of notation, (B.34)  X 1 µ ,I , X n ν ,J  ⋆ p W p ← → dim( p ) c p 1 c p n µ 1 ( p ) [ A, e ( n ; ν , J )] W p ← → 1 c p 1 2 √ 3 p p ( p + 3) A ( X n ν ,J ) The result follows from (B.2) and (B.3) b y straightforw ard calculation. □ Step 5 . Once more, it go es by induction, where now the base step is Prop osition B.13. The next prop osition con tains the inductive step. Prop osition B.14. Supp ose f  p g → f g uniformly for every f , g ∈ X . F or n ∈ N , if the uniform c onver genc e p [ f , g ] ⋆ p → i p 3 / 2 { f , g } holds for every p air f ∈ X n and g ∈ X , then p [ f , g ] ⋆ p → i p 3 / 2 { f , g } for every f ∈ X n +1 and g ∈ X . Sketch of pr o of. Analogously to Theorem B.10, it is suﬃcient to prov e (B.35) p h X 1 µ ,I X n ν ,J , X n ′ ν ′ ,J ′ i ⋆ p → i r 3 2 n X 1 µ ,I X n ν ,J , X n ′ ν ′ ,J ′ o , and it can b e done by a serial sum and subtraction of suitable terms, resorting to Theorems B.7 and B.12, Lemma B.9 and the uniform b oundedness principle. □ No w, putting the ab ov e proposition together with Theorem B.10 and Prop osition B.13, we ﬁnally obtain: Theorem B.15. If c p n → 1 as p → ∞ for al l n ≥ 1 , then p [ f , g ] ⋆ p → i p 3 / 2 { f , g } uniformly for every f , g ∈ X . Instituto de Ci ˆ encias Ma tem ´ aticas e de Comput ac ¸ ˜ ao, Universidade de S ˜ ao P aulo. S ˜ ao Carlos, SP, Brazil. Email address : pedro.antonio.alcantara@usp.br Email address : prios@icmc.usp.br

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