A new integrable system on the sphere and conformally equivariant quantization

A new in tegrable system on the sphere and conformally equiv arian t quan tization C. DUV AL ‡ Cen tre de Physique Th ´ eoriqu e, CNRS, Luminy , Case 907 F-13288 Marseil l e Cedex 9 (F rance) § G. V ALENT ¶ Lab ora toir e de P h ysique Th ´ eorique et des Hautes Energi es, 2, Place Jussieu F-75251 P ari s Cedex 5 (F rance) k 14 September 201 0 Abstract T aking full adv a nt age of t w o indep end en t pro jectiv ely equiv alen t metrics on the ellipsoid le ading to Liouville int egrabilit y of the geodesic flo w via the w ell-kno wn Jacobi-Moser system, w e d isclose a no v el int egrable sys tem on the spher e S n , namely the dual Moser system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlen b ec k systems, into the ca tegory of (lo- cally) St¨ ac k el systems. Moreo v er, it is pro v ed that quan tum integrabilit y of b oth Neumann-Uhlenbec k and dual Moser systems is insu red b y means of th e conformally equiv arian t quantiz ation pro cedu r e. Preprin t: CPT-P051-2010 Keyw ords: Classical integrabilit y , pro jective ly equiv alent metrics, St¨ ac k el systems, conformally equiv ariant quantization, quantum in tegrability . ‡ mailto:duv al@cpt.univ- mrs.fr § UMR 6207 du CNRS as s o c i ´ ee aux Universit´ es d’Aix- Marseille I et I I et Universit ´ e du Sud T oulon-V ar; Lab ora toire affili´ e ` a la FRU MAM-FR2291 ¶ mailto:v alent@lpthe.jussieu.fr k UMR 758 9 du CNRS ass o c i ´ ee ` a l’Universit´ e Paris VI Con ten ts 1 In tro duction 2 1.1 Prolegomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Plan o f the article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 A nov el integrable system on the sphere: t he dual Moser system 6 2.1 Pro jectiv ely equiv alen t metrics and conserv a t io n laws . . . . . . . . . 6 2.2 The example of t w o pro jectiv ely equiv alen t geo desic spra ys for the n -sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 The Jacobi-Moser system . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 The dual Moser system . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 The general construction . . . . . . . . . . . . . . . . . . . . . 11 2.4.2 Liouville in tegrabilit y of the unconstrained system . . . . . . . 13 2.4.3 The Dira c brack ets . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.4 The constrained inte grable system as a St¨ ac k el system . . . . 1 5 2.5 Three St¨ ac k el systems . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.1 The Neumann-Uhlen b ec k system . . . . . . . . . . . . . . . . 19 2.5.2 A syn thetic presen t a tion . . . . . . . . . . . . . . . . . . . . . 20 3 Quan tum in tegrabilit y 20 3.1 Conformally equiv ariant quantization . . . . . . . . . . . . . . . . . . 21 3.2 Quan tum comm utato r s . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 The quan tum Neumann-Uhlen b ec k system . . . . . . . . . . . . . . . 26 3.4 The quan tum dual Moser system . . . . . . . . . . . . . . . . . . . . 2 6 3.5 The quan tum Jacobi-Moser system . . . . . . . . . . . . . . . . . . . 29 4 Conclusion and outlo ok 30 1 In tro duc tion In the wak e o f the celebrated results of Moser [19] concerning the classical in te- grabilit y of the geo desic flo w on t he ellipsoid (first pro v ed b y Jacobi in the three- dimensional case), and of the Neumann-Uhlen b ec k problem on the n -dimensional spheres, w e will presen t a (to our kno wledge) new integrable system whic h relies on a preferred conformally flat metric o n S n . This in tegrable system is actually dual (in the sense of pro jectiv e equiv alence [25, 18]) to the ab o v e-men tio ned Jacobi-Moser system. 2 Among the tec hniques found in the literature, w e will men tion tw o differen t constructs that pro duce inte grable systems together with t heir P oisson-comm ut ing first integrals: • Bihamiltonian systems initiated by Magri [17] and Benen ti [3], and further elab orated by , e.g., Ib ort, Magri, and Marmo [13], F alqui and P edroni [12]. • Pro jectiv ely equiv alen t sy stems disco ve red b y Levi-Civita [16], and geometrical- ly dev elop ed by T abac hnik ov [25, 26 ], T opalov and Matve ev [27]. The equiv alence of these t wo theories has b een pro v ed b y Bo lsino v and Matve ev [6]. W e hav e chos en to w ork using T abachnik o v’s appro a c h. With the help of his general construction [25, 2 0] pro viding, e.g., the Jacobi-Moser first integrals, and, adopting a “dual” approac h, w e deriv e the P oisson-comm uting first in t egr a ls of the new geo desic flow, whic h w e call the dual Moser system. This enables us to provide a glob al expression for these new first in tegrals. Our dual Moser system turns out to b e lo cally St¨ ac k el (in ellipsoidal co ordi- nates); it sho ws up as a “mirror image” of Jacobi-Moser relativ ely to Neumann- Uhlen b eck (see T able 1). In contradistinction to the Jacobi-Moser system, conformal flatness of the new system is a fundamen t al input at the classical, and at the quan tum lev el a s w ell. T o deal with quan tum inte grability of these systems, w e will resort to its com- monly accepted definition, namely that the quantize d first inte grals should still b e in inv olution. This, of course, leav es op en the choice of an adapted quantiz ation pro cedure. It has b een shown [8] that St¨ ac k el systems, using Carter’s quantization pres- cription [7], do remain in tegrable at the quan tum lev el pro vided Rob ertson’s con- dition holds [23]. It is therefore w orth while to study quantum in tegrabilit y making use of a genuine quan tization theory that takes in t o accoun t the conformal geometry underlying the dual Moser system: the confor mally equiv ariant quan tization [10, 9]. One striking discov ery is that the dual Moser system passes the quantum test using the conformally equiv ariant quan tization. W e note that the same is true for the Neumann-Uhlen b eck system. 1.1 Prole gomena Let us recall that t wo indep end e nt metrics on a R iemannian manifold are said to b e pro jectiv ely equiv alent if they ha v e the same unp ar ametrize d geo desics. A shown in, e.g., [2 7, 6] this en tails that the asso ciated g eo desic flo ws a re Liouville-in tegra ble. W e will resort to this path breaking result in the sp ecific, historical, y et fundamen tal example o f the geo desic flo w of the ellipsoid. 3 The k ey p oin t of our approac h to Liouville/quan tum in tegr a bilit y of the geo desic flo w of the ellipsoid, E n , lies in the fact that E n = { Q ∈ R n +1 | P n α =0 ( Q α ) 2 /a α = 1 } with metric g 1 = P n α =0 ( dQ α ) 2 | E n admits, as a matter of fact, another indep enden t, and pro jectiv ely equiv alen t Riemannian metric, g 2 , that w e will in t r o duce shortly . Put Q α = q α √ a α for all α = 0 , . . . , n , so that w e ha v e q · q = P n α =0 ( q α ) 2 = 1. The mapping Q 7→ q : E n → S n is a diffeomorphism and the metric on S n , induced from the Euclidean am bien t metric, reads now 1 g 1 = g | S n where g = n X α =0 a α dq 2 α (1.1) where g is, at the momen t, view ed as a (flat ) metric on R n +1 . The equations o f the geo desics o f the ellipsoid are w ell-know n, and retain the form ¨ q α + Γ α β γ ˙ q β ˙ q γ = 0 , (1.2) for all α = 0 , . . . , n , where the Christoffel sym b o ls o f ( S n , g 1 ) are giv en b y Γ α β γ = q α a α δ β γ P q 2 λ /a λ (1.3) for all α , β , γ = 0 , . . . , n . (See also Equation (2.12) yielding the a ssociated geo desic spra y .) W e note that the constraint equation ˙ q 2 + q · ¨ q = 0 (1.4) is indeed satisfied b y (1.2). F ollo wing [25, 27], let us define, on S n , the conformally flat metric g 2 = g | S n with g = 1 P q 2 β /a β n X α =0 dq 2 α (1.5) where t he conformally flat metric g is defined o n R n +1 \ { 0 } . The equations of the geo desics for the latter metric are readily found b y using the expression of the Christoffel sym b ols of g, namely Γ α β γ = 1 P q 2 λ /a λ  δ β γ q α a α − δ α β q γ a γ − δ α γ q β a β  (1.6) for all α, β , γ = 0 . . . , n . One obtains the equations of the g eo desic of ( R n +1 \{ 0 } , g), viz., ¨ q α + q α a α P ˙ q 2 β P q 2 γ /a γ = 2 ˙ q α P q β ˙ q β /a β P q 2 γ /a γ (1.7) 1 T o av oid clutter, we will o ften times write q α ≡ q α , as no co nfusion o ccurs in Euc lidea n space . 4 for all α = 0 , . . . , n . The latter, suitably restrained to S n , is precisely Equation (1.2) with a differen t parametrization (again, the constraint (1.4) is duly preserv ed b y Equation (1 .7)); the metrics g 1 and g 2 are pro jectiv ely equiv alen t. W e will put this fact in broader p erspective within Section 2. Remark 1.1. The Γ α β γ and Γ α β γ , give n b y (1.3) and (1 .6) resp ectiv ely , may b e view ed as the comp onen ts of t w o pro jectiv ely equiv alent linear connections ∇ and ∇ o n R n +1 \ { 0 } . While ∇ is clearly the Levi-Civita connection of the metric g , the con- nection ∇ is, instead, a Newton-Cartan connection ( see, e.g., [14]). Th is means that ∇ is a symmetric linear connection that parallel-transp orts a (spacelik e) con- tra v ariant symmetric 2-tensor γ = P n α,β =0 γ αβ ∂ q α ⊗ ∂ q β and a (timelik e) 1- form θ = P n α =0 θ α dq α spanning k er ( γ ). Here, t he degenerate “metric” is g iv en by γ αβ = 1 a α δ αβ − q α q β a α a β 1 P q 2 λ /a λ and θ α = q α . (1.8) 1.2 Main results The main results of our art icle can b e summarized as follows . Theorem 1.2. The ge o d esic flow on ( T ∗ S n , P α dp α ∧ dq α ) a b ove the c onf o rmal ly flat m a nifold ( S n , g 2 ) is Liouvil le-inte gr a ble, and admits the fol lowin g set of Pois s o n- c ommuting first inte gr als F α = q 2 α n X β =0 a β p 2 β + X β 6 = α ( a α p α q β − a β p β q α ) 2 a α − a β (1.9) with α = 0 , . . . , n . We wil l c al l the system ( F 0 , . . . , F n ) the dual Moser system. This theorem follows directly from Prop ositions 2 .7 , 2.8, and 2 .10. Let us call D 1 2 , 1 2 ( S n ) the space of differen tial op erator s on S n with argumen ts and v alues in the space o f 1 2 -densities of t he sphere S n ; w e, lik ewise, denote b y P ol( T ∗ S n ) the space of fib erwise p olynomial functions on T ∗ M . It has b een prov ed [10] tha t there exists a unique in v ertible linear mapping Q 1 2 , 1 2 : P ol( T ∗ S n ) → D 1 2 , 1 2 ( S n ) that (i) intert wines the actio n of the conformal group O( n + 1 , 1) a nd (ii) preserv es t he principal sym b ol: w e call it the c onformal ly e quivari a nt quantization mapping. Theorem 1.3. Q uan tum inte gr ab ility of the dual Moser s ystem hold s true in terms of the c onformal ly e quiva riant quantization Q 1 2 , 1 2 . This last result stems from Theorem 3.4, Prop ositions 3.6, a nd 3.1 2. 5 1.3 Pla n of the article The pap er is organized as follows. Section 2 giv es us the opp ortunit y to in tro duce three distinguished classically in tegrable systems o n the sphere, namely , the Jacobi- Moser system, its dual coun ter- part, and the Neumann-Uhlen b eck system. The construction of the set of m utually P o isson-comm uting first in tegrals is review ed and sp ecialized to the case of the dual Moser system. T he resulting system is show n to b e St¨ ac kel, t he ellipsoidal coordinates b eing the separating ones. In Section 3 w e address the quan tum in tegrability issue o f these systems, in terms of the confo rmally equiv aria n t quan tization. W e prov e that the Neumann- Uhlen b eck and the dual Moser systems are, indeed, quan tum in tegrable using this quan tizat io n metho d, w ell adapted to the conformal flatness of configuration space. Section 4, pro vides a conclusion to t he presen t article, and gathers some p er- sp ectiv es fo r future w ork. 2 A nov e l integrable syst em on the sp here: the dual Moser system 2.1 Pro jectiv ely equ iv alen t metrics and conserv ation la ws Let us recall, a lmost verb atim , T abach nik o v’s construction [25] of a maximal set of indep enden t P oisson-comm uting first integrals for a sp ecial Lio uville-in tegrable system, namely a bi-Hamiltonian system asso ciated with tw o pro jective ly equiv alent metrics, g 1 and g 2 , on a configuration manifold M . 2 See also [18, 6 ] for an alternative construction. W e start with tw o Riemannian manifolds ( M , g 1 ) and ( M , g 2 ) of dimension n . The tangen t bundle T M is endo w ed with tw o distinguished 1-f o rms λ 1 and λ 2 , namely λ N = g ∗ N θ , where θ is the canonical 1-form of T ∗ M , and g N : T M → T ∗ M is view ed a s a bundle isomorphism. W e then write, lo cally , λ N = g N ij u i dx j where g N = g N ij ( x ) dx i ⊗ dx j for N = 1 , 2 . Like wise, the Lagrangia n functions to consider are the fib erwise quadratic p olynomials L N = 1 2 g N ij ( x ) u i u j . Denote b y ω N = d λ N the cor r esp onding symplectic 2-forms of T M , and also by X N = X L N the a ssociat ed geo desic spra ys. W e hav e λ N ( X N ) = 2 L N and ω N ( X N ) = − dL N (2.1) for all N = 1 , 2. The (maximal) in tegral curve s of the v ector fields X N on T M pro ject o n to configura t io n space as the geo desics of ( M , g N ). 2 The formalism can be easily extended to the cas e o f Finsler structures [2 5]; here, we will no t need such a g e ne r ality . 6 In t r o duce then the diffeomorphism φ : T M → T M defined b y φ ( x, u ) = ( x, ˜ u ) where ˜ u = u p L 1 ( x, u ) /L 2 ( x, u ). 3 This diffeomorphism is, indeed, designed to relate the tw o Lagrangia ns, viz., L 1 = φ ∗ L 2 . (2.2) Clearly , the t w o metrics g 1 and g 2 ha v e the same unparametrized geo desics (w e write g 1 ∼ g 2 ) iff φ ∗ ( X 1 ) ∧ X 2 = 0 (2.3) i.e., iff the the push-forw a rd φ ∗ ( X 1 ) and X 2 are functionally dep enden t. The metho d, to obtain a g enerating function for the conserv ed quan tities in in v olutio n, consists then in singling o ut, apart fro m ω 1 , a preferred X 1 -in v arian t 2-form constructed in terms of ω 2 . Prop osition 2.1. Supp o s e that g 1 ∼ g 2 , and define ω ′ 2 = d ( L − 1 2 2 λ 2 ) , then L X 1 ( φ ∗ ω ′ 2 ) = 0 . (2.4) Pr o of. W e hav e L X 1 ( φ ∗ ω ′ 2 ) = d (( φ ∗ ω ′ 2 )( X 1 )) = φ ∗ d ( ω ′ 2 ( φ ∗ X 1 )) = φ ∗ d ( h ω ′ 2 ( X 2 )), for some function h (see (2.3)). The definition of ω ′ 2 then readily yields L X 1 ( φ ∗ ω ′ 2 ) = φ ∗ d ( h d ( L − 1 2 2 λ 2 )( X 2 )) = φ ∗ d ( h ( − 1 2 L − 3 2 2 ( dL 2 ∧ λ 2 )( X 2 ) + L − 1 2 2 ω 2 ( X 2 ))) = 0 in view of (2.1 ). So, the sough t extra X 1 -in v arian t 2-for m is φ ∗ ω ′ 2 . (Note that L X 1 ( φ ∗ ω 2 ) 6 = 0.) It enters naturally in to the definition of a “g enerating function” f t ∈ C ∞ ( T M , R ) of first-integrals g iven b elo w. Corollary 2.2. The function f t = ( t − 1 ω 1 + φ ∗ ω ′ 2 ) n ω n 1 (2.5) is X 1 -invariant when ever t 6 = 0 . 2.2 The example of t w o pro jectiv ely equiv alen t geo desic spra ys for the n -sphere W e recall the construction of the first-in tegrals in inv olution yielding the Liouville- in tegrabilit y of t he geo desic flow on T E n ∼ = T S n . Let us parametrize T R n +1 b y the couples q , v ∈ R n +1 . The constraints defining the embedding T S n ֒ → T R n +1 are q 2 := n X α =0 q 2 α = 1 and v · q := n X α =0 v α q α = 0 . (2.6) 3 Note that φ is the identit y on the zer o section of T M . 7 As already men tioned in Section 1, the (unparametrized) geo desics of the el- lipsoid E n with semi-axes 4 a 0 , a 1 , · · · , a n are precisely giv en b y those of the sphere S n = { q ∈ R n +1 | P n α =0 q 2 α = 1 } endo w ed with either pro jectiv ely equiv alen t metrics g 1 = n X α =0 a α dq 2 α    S n & g 2 = 1 B n X α =0 dq 2 α    S n (2.7) where B = n X α =0 q 2 α a α . (2.8) The corresp onding La g rangians on T S n are resp ectiv ely L 1 = 1 2 A & L 2 = 1 2 B n X α =0 v 2 α (2.9) where A = n X α =0 a α v 2 α . (2.10) The asso ciated Cartan 1-f orms then read in this case λ 1 = n X α =0 a α v α dq α & λ 2 = 1 B n X α =0 v α dq α . (2.11) Prop osition 2.3. (i) The ge o desic spr ays fo r the m etrics g N ar e gi v en by the Hamil- tonian ve ctor fields X N = X L N , for N = 1 , 2 , namely X 1 = n X α =0 v α ∂ ∂ q α − v 2 B n X α =0 q α a α ∂ ∂ v α (2.12) and X 2 = n X α =0 v α ∂ ∂ q α − 1 B n X α =0  2 v α n X β =0 v β q β a β − v 2 q α a α  ∂ ∂ v α (2.13) r esp e ctively. (ii) Condition (2.3) holds true, im plying g 1 ∼ g 2 . Pr o of. Using (2.1) together with the constrain ts (2.6), w e th us ha v e to solv e for X 1 , resp. X 2 , the equation ω N ( X N ) + dL N + λ d ( q 2 − 1) + µ d ( v · q ) = 0 where λ and µ are Lagra nge multipliers. The latter are, in fine , completely determined and readily yield (2.12), resp. (2 .13). No w, the diffeomorphism φ : ( q , v ) 7→ ( q , ˜ v ) introduced in Section 2.1 is giv en b y ˜ v = v p AB /v 2 ; routine calculation yields φ ∗ ( ∂ q α ) = ∂ q α + q α / ( a α B ) E with E = P v α ∂ v α the Euler v ector field; also φ ∗ ( ∂ v α ) = p AB /v 2 ( ∂ v α − v α ( v − 2 − a α / A ) E ). This, along with the constrain t P v α q α = 0, helps us pro v e Equation (2.3). 4 W e will, later on, deal with the choice 0 < a 0 < a 1 < · · · < a n . 8 2.3 The Jacobi-Moser system Let us review here, and in some detail, the main result obtained by T abachnik o v [25] via t he general pro cedure of Section 2.1, starting with the geo desic flo w on ( S n , g 1 ). The diffeomorphism φ of T S n , namely φ ( q , v ) = ( q , ˜ v = v p L 1 /L 2 ), is such that ˜ v = v p AB /v 2 where v 2 = P v 2 α ; whence φ ∗ λ 2 = C √ A P v α dq α , where C = 1 √ B v 2 . (2.14) No w, since φ ∗ ω ′ 2 = d ( L − 1 2 1 φ ∗ λ 2 ), easy computation then leads to φ ∗ ω ′ 2 = C √ 2 Ω 1 where Ω 1 = n X α =0 dv α ∧ dq α + dC C ∧ n X α =0 v α dq α . (2.15) The function C defined by (2.14) is a first in tegral of the system, namely X 1 C = 0 . (2.16) This function C is the Jo achimsthal first-in tegral of t he geo desic flow of ( S n , g 1 ). The next step consists in lifting the 1-parameter family of first-integrals (2.5) to T R n +1 b y taking adv antage of the constraints (2.6), and to put f t = ( t − 1 ω 1 + Ω 1 ) n ∧ d ( v .q ) ∧ q · dq ω n 1 ∧ d ( v .q ) ∧ q · dq (2.17) with a sligh t abuse of notation using the constancy of C (see (2.16)) in (2.15). Elemen tary calculation yields f t =  ω n ∗ − ( n/v 2 ) ω n − 1 ∗ ∧ v · dv ∧ v · dq  ∧ q · dv ∧ q · dq ω n 1 ∧ ( dv .q + v · dq ) ∧ q · dq (2.18) where ω ∗ = P b α dv α ∧ dq α − (1 /v 2 ) v · dv ∧ v · dq , together with b α = t − 1 a α + 1, for all α = 0 , . . . , n . The following lemma [25] will b e used to complete the calculation. Lemma 2.4. L et ω = P c α dv α ∧ dq α and ω 0 = P dv α ∧ dq α , then ω n ∧ q · d v ∧ q · dq ω n +1 0 = n ! n Y α =0 c α n X α =0 q 2 α c α ω n − 1 ∧ v · dv ∧ v · dq ∧ q · dv ∧ q · dq ω n +1 0 = ( n − 1)! n Y α =0 c α X α<β ( v α q β − v β q α ) 2 c α c β . 9 Using the tw o ab o v e form ulæ, we find f t = N /D where N = n ! Y b α X q 2 α b α − n ( n − 1)! v 2 Y b α X α<β ( v α q β − v β q α ) 2 b α b β D = n ! Y a α X q 2 α /a α . This en t a ils f t = g t /C (up to a constan t o ve rall factor ) , where C is the Jo ac himsthal first-in tegral, and g t = v 2 n X α =0 q 2 α b α − 1 2 X α 6 = β ( v α q β − v β q α ) 2 b α b β . T a king into accoun t the expression b α = t − 1 a α + 1, and the constrain ts (2.6 ) , w e end up with g t = n X α =0 a α v 2 α a α + t − n X α =0 a α v 2 α a α + t n X α =0 a α q 2 α a α + t + n X α =0 a α v α q α a α + t ! 2 . (2.19) A t la st, the first-in tegrals F α defined by g t = n X α =0 F α a α + t (2.20) are easily found to b e F α = a α v 2 α + X β 6 = α a α a β ( v α q β − v β q α ) 2 a α − a β . (2.21) These are the Jac obi-Moser first-integrals. In terms of the momen ta p α = a α v α (see (2.11)), they read F α = p 2 α a α + X β 6 = α ( p α a β q β − p β a α q α ) 2 a α a β ( a α − a β ) . (2.22) Remark 2.5. W e indeed reco v er the Moser first in tegrals F α = P 2 α + X β 6 = α ( P α Q β − P β Q α ) 2 a α − a β (2.23) b y means of the cano nical transforma t ion ( p, q ) 7→ ( P , Q ) where P α = p α / √ a α , and Q α = q α √ a α . 2.4 The dual M oser system Let us now adopt a “dual” standp oint by exc hanging the rˆ ole of the tw o pro jectiv ely equiv alent metrics on the n -sphere, i.e., by letting g 1 ↔ g 2 in the ab ov e deriv ation. In doing so, we will work out a complete set of comm uting first-in tegrals of the geo desic flo w on the conformally flat manif o ld ( S n , g 2 ). This will turn out to pro vide a new in tegrable system o n the n -sphere. 10 2.4.1 The general construction Let us now apply the general pro cedure o utlined in Section 2.1 starting with the geo desic flow on ( S n , g 2 ), and replacing mutatis mutandis all reference to g 1 b y that of g 2 . W e first need to w ork out the expression of the 2 -form φ ∗ ω ′ 1 in Prop osition 2.1. The diffeomorphism φ of T S n , viz., φ ( q , v ) = ( q , ˜ v = v p L 2 /L 1 ), is ˜ v = v p v 2 / ( AB ) where, again, v 2 = P v 2 α . W e hence get φ ∗ λ 1 = p v 2 / ( AB ) P a α v α dq α . No w, since φ ∗ ω ′ 1 = d ( L − 1 2 2 φ ∗ λ 1 ), thanks to φ ∗ L 1 = L 2 , w e are led t o φ ∗ ω ′ 1 = r 2 A b Ω 2 where b Ω 2 = n X α =0 a α dv α ∧ dq α − dA 2 A ∧ λ 1 , (2.24) with λ 1 = P a α v α dq α (see (2.11 ) ) . W e then find the new Jo achims thal first-in tegra l, J , of the geo desic flow o f ( S n , g 2 ) with the help of the expression ( 2 .13) of the geo desic spray X 2 . In fa ct, easy calculation yields X 2 A = (2 A/B ) X 2 ( B ), hence X 2 J = 0 where J = A B 2 . (2.25) Utilizing t his constant of the motio n, and Equation (2.4), w e obt a in, see (2.24), L X 2 Ω 2 = 0 where Ω 2 = b Ω 2 B . (2.26) Again, a nd to ease the calculation, w e will lift the 1-pa rameter family of first- in tegrals ( 2 .5) to T R n +1 , using the constraints (2 .6 ), a nd put this time f t = ( t ω 2 + Ω 2 ) n ∧ d ( v · q ) ∧ q · dq ω n 2 ∧ d ( v · q ) ∧ q · dq . (2.27) W e trivially get f t = ( t b ω 2 + b Ω 2 ) n ∧ q · d v ∧ q · d q b ω n 2 ∧ q · d v ∧ q · d q (2.28) where b ω 2 = B ω 2 , i.e., b ω 2 = P dv α ∧ dq α − d log B ∧ P v α dq α and b Ω 2 is as in ( 2 .24). Let us men tion the follow ing somewhat techn ical lemma. Lemma 2.6. Up on defi ning b ω ∗ = P b α dv α ∧ dq α , wher e b α = a α + t , for every α = 0 , 1 , . . . , n , we have ( t b ω 2 + b Ω 2 ) k = b ω k ∗ − k b ω k − 1 ∗ ∧  t d log B ∧ v · d q + 1 2 d log A ∧ λ 1  + k ( k − 1 ) b ω k − 2 ∗ ∧ t d lo g B ∧ v · dq ∧ 1 2 d log A ∧ λ 1 11 for al l k = 1 , . . . , n , and b ω n − 1 ∗ = ( n − 1)! X α<β Y γ 6 = α,β b γ dv γ ∧ dq γ A somewhat demanding computation yields f t =  b ω n ∗ − n b ω n − 1 ∗ ∧ dA/ (2 A ) ∧ λ 1  ∧ q · d v ∧ q · d q b ω n 2 ∧ q · dv ∧ q · dq . (2.29) Resorting to Lemmas 2.4 and 2.6 in order to ev aluate f t , w e obt a in the partial result b ω n ∗ ∧ q · d v ∧ q · d q = n ! n Y β =0 b β n X α =0 q 2 α b α ! ω n +1 0 (2.30) where ω 0 = P dv α ∧ dq α . Lik ewise, some more effort is needed to find b ω n − 1 ∗ ∧ 1 2 d log A ∧ λ 1 ∧ q · dv ∧ q · dq = ( n − 1)! n Y γ =0 b γ X α<β ( a α v α q β − a β v β q α ) 2 b α b β ω n +1 0 A . (2.31) W e just hav e to plug Equations (2.30) and (2.31) in to the express ion (2.28) to find 5 f t = n Y γ =0 b γ n X α =0 q 2 α b α − 1 A X α<β ( a α v α q β − a β v β q α ) 2 b α b β ! . (2.32) W e will, again, deal with the rescaled first-in tegral g t = J Q b α f t (2.33) where J is an (2.25 ), as a generating function of the sought conserv ative system F 0 , F 1 , . . . , F n . Using the definition b α = a α + t , and Equation (2.20), w e readily prov e that the geo desic flo w on T S n ab o v e ( S n , g 2 ) admits the follo wing first integrals, viz., F α = 1 B 2 Aq 2 α + X β 6 = α ( a α v α q β − a β v β q α ) 2 a α − a β ! (2.34) with α = 0 , 1 , . . . , n , where A a nd B are giv en b y Equations (2.10) and (2.8), resp ectiv ely . With the help of the bundle isomorphism g − 1 2 : T ∗ S n → T S n pro vided by the metric g 2 , w e can pull-back the previous first in tegrals to the cotangent bundle of S n . Whence the following result. 5 W e hav e b ω n 2 ∧ q · dv ∧ q · dq = n ! ω n +1 0 . 12 Prop osition 2.7. The ge o desi c flow on the c otang e nt bund le of ( S n , g 2 ) admits the fol lowing set of first inte gr als, v i z ., F α = q 2 α n X β =0 a β p 2 β + X β 6 = α ( a α p α q β − a β p β q α ) 2 a α − a β (2.35) with α = 0 , . . . , n . In the next section w e will prov e that these first in tegrals are actually indep en- den t and are m utually P oisson comm uting. 2.4.2 Liouville integrabilit y of the unconstrained system F rom now on, w e c ho ose to work in a purely Hamiltonian f ramew ork whic h will turn out to b e w ell-suited to the quan tization pro cedure that we will examine in the next section. Let us recall that the Hamiltonian of the system is giv en b y H = 1 2 n X α,β =0 g αβ 2 p α p β = 1 2 B n X α =0 p 2 α (2.36) where B = P n α =0 q 2 α /a α (see (2.8)). W e will sho w that the new set (2.35) of first-integrals of motion indeed turns the geo desic flow on the sphere ( S n , g 2 ) in to an integrable system dual to the Jacobi- Moser geo desic flow on the ellipsoid. Prop osition 2.8. T he functions F 0 , . . . , F n of ( T ∗ R n +1 , P n α =0 dp α ∧ dq α ) given by (2.35) ar e in inv olution, namely { F α , F β } = 0 (2.37) for al l α, β = 0 , . . . , n . Mor e over the fol lowing holds true { H , F α } = 2  B a α p 2 α − q 2 α X β p 2 β  X γ p γ q γ . (2.38) Pr o of. W rite F α = A α + B α with A α = q 2 α J, J = n X β =0 a β p 2 β , (2.39) where J is t he Joachim sthal first in tegral (2 .25), and B α = X β 6 = α M 2 αβ a α − a β , M αβ = a α p α q β − a β p β q α . (2.40) 13 One can c hec k the follow ing relatio nships {A α , A β } = − 4 J q α q β M αβ , { A α , B β } = 4 J a α q α q β M αβ a α − a β , { B α , B β } = 0 , for all α , β = 0 , . . . , n , whic h r eadily imply Equation (2.37). Let us furthermore observ e that we hav e t he following P oisson brac k ets, viz., { H , A α } = 2 J B q α p α − 4 q 2 α H B n X β =0 p β q β and { H , B α } = − 2 J B q α p α + 2 B a α p 2 α n X β =0 p β q β , whic h prov es Equation (2.38). The pro of is complete. Remark 2.9. Notice in contradistinc tion to the Jacobi-Moser case, that (i) the metric g giv en b y (1 .5 ) on the am bien t space R n +1 \{ 0 } is no longer flat, and ( ii) the conserv ation relations { H , F α } = 0 for α = 0 , . . . , n are only v alid for the constrained system, see (2.38), where p · q = 0. Let us also men tion the interes ting relations n X α =0 F α = n X α =0 q 2 α n X β =0 a β p 2 β (2.41) n X α =0 F α a α = n X α =0 p α q α ! 2 (2.42) n X α =0 F α a 2 α = − 2 H + 2 n X α =0 p α q α n X β =0 p β q β a β , (2.43) of whic h the last one leads to another pro of of (2.38). 2.4.3 The Dirac brac k ets Our goal is now to deduce from the kno wledge of (2.35) indep enden t quan tities in in v olution I 1 , . . . , I n on ( T ∗ S n , P n i =1 dξ i ∧ dx i ) from the symple ctic em b edding ι : T ∗ S n ֒ → T ∗ R n +1 defined by the constrain ts Z 1 ( p, q ) = n X α =0 q 2 α − 1 = 0 , Z 2 ( p, q ) = n X α =0 p α q α = 0 . (2.44) 14 Prop osition 2.10. Th e r estrictions F α | T ∗ S n = F α ◦ ι of the f unctions (2.35) do Poisson-c omm ute on T ∗ S n . Pr o of. W e get, using the Dir a c brac k ets [1, 21], { F α | T ∗ S n , F β | T ∗ S n } = { F α , F β }| T ∗ S n − 1 { Z 1 , Z 2 } [ { Z 1 , F α }{ Z 2 , F β } − { Z 1 , F β }{ Z 2 , F α } ] | T ∗ S n (2.45) for second-class constrain ts. The denominator { Z 1 , Z 2 } = − 2 P n α =0 q 2 α do es not v anish; one can also chec k tha t { Z 1 , F α } = − 4 a α p α q α (1 + Z 1 ) and that { Z 2 , F α } = 0 for all α = 0 , . . . , n . The fact that { F α , F β } = 0 completes the pro of. 2.4.4 The constr ained in tegrable syst em as a St¨ ack el system In order to pro vide explicit expressions f o r the sough t functions in inv olution I 1 , . . . , I n , w e r esort to Jacobi ellipsoidal co or dina t es x 1 , . . . , x n on S n . Those are defined b y Q λ ( q , q ) = n X α =0 q 2 α a α − λ = − U x ( λ ) V ( λ ) (2.46) where U x ( λ ) = n Y i =1 ( λ − x i ) and V ( λ ) = n Y α =0 ( λ − a α ) (2.47) and are suc h that a 0 < x 1 < a 1 < x 2 < . . . < x n < a n . (2.48) Notice that Equation (2.46) yields the lo cal expressions q 2 α ( x ) = n Y i =1 ( a α − x i ) Y β 6 = α ( a α − a β ) (2.49) for all α = 0 , 1 , . . . , n . Let us men tion the follow ing identit y , deduced from (2.49), viz., ∂ q α ∂ x i = − 1 2 q α a α − x i (2.50) for all i = 1 , . . . , n and α = 0 , 1 , . . . , n It is easy to sho w that the induced metric g 2 = (1 /B ) P n α =0 dq 2 α | S n , see (2.7), is indeed giv en by g 2 = P n i,j =1 g ij ( x ) dx i dx j with g ij ( x ) = 1 4 B n X α =0 q 2 α ( a α − x i )( a α − x j ) = g i ( x ) δ ij (2.51) 15 where, using a result tak en from [19], w e ha v e g i ( x ) = − 1 4 B U ′ x ( x i ) V ( x i ) = − 1 4 B Q j 6 = i ( x i − x j ) Q α ( x i − a α ) . (2.52) This metric is a ctually p ositiv e-definite b ecause o f the inequalities (2 .48). In these ellipsoidal co ordinates, w e obta in B = n X α =0 q 2 α a α = 1 a 0 x 1 · · · x n a 1 · · · a n . Up on defining the constrained “momen ta” ξ i (for i = 1 , . . . , n ), via the induced canonical 1 -form λ | T ∗ S n = P n i =1 ξ i dx i = ι ∗ P n α =0 p α dq α , w e find p α ( ξ , x ) = − q α ( x ) 2 B n X i =1 g i ( x ) ξ i a α − x i . (2.53) W e expres s, for con v enience, the Hamilto nia n (2.36) on ( T ∗ S n , P n i =1 dξ i ∧ dx i ), whic h is then found to b e H = 1 2 n X i =1 g i ( x ) ξ 2 i (2.54) where g i ( x ) = 1 / g i ( x ). Let us no w compute the expres sion of the conserv ed quan tities ( 2 .35) on T ∗ S n . Prop osition 2.11. The dual Moser c onserve d quantities ( F α | T ∗ S n ) α =0 ,...,n r etain the form F α | T ∗ S n = a α q 2 α ( x ) B n X i =1 x i g i ( x ) ξ 2 i a α − x i . (2.55) Pr o of. On the one ha nd, in view of Equations (2.49) and ( 2.53), one g ets, see (2.39), A α | T ∗ S n = q 2 α ( x ) B n X i =1 x i g i ( x ) ξ 2 i , using the iden tities n X α =0 q 2 α a α − x i = 0 for all i = 1 , . . . , n . On the other hand, a similar computation giv es M αβ | T ∗ S n = a α − a β 2 B q α ( x ) q β ( x ) n X i =1 x i g i ( x ) ξ i ( a α − x i )( a β − x i ) , 16 so tha t B α | T ∗ S n = q 2 α ( x ) B n X i =1 ( x i ) 2 g i ( x ) ξ 2 i a α − x i , pro ving that F α | T ∗ S n (where F α = A α + B α ) is, indeed, as in (2.55 ) . Prop osition 2.12. The fol lowing ho l d s on T ∗ S n , viz n X α =0 F α | T ∗ S n = 1 B n X i =1 x i g i ( x ) ξ 2 i = J | T ∗ S n (2.56) n X α =0 F α | T ∗ S n a α = 0 (2.57) n X α =0 F α | T ∗ S n a 2 α = − 2 H. (2.58) Pr o of. The pro of is a direct consequence of Equations (2.4 1), (2.42), and (2.43), together with the constrain ts (2.44). As a preparation to the pro of that o ur system is, indeed St¨ ac k el, let us in tro duce, for conv enience, the symmetric functions, σ k ( x ) a nd σ i k ( x ) with x = ( x 1 , . . . , x n ), that will b e useful in the sequel, namely , U x ( λ ) ≡ n Y j =1 ( λ − x j ) = n X k =0 ( − 1) k λ n − k σ k ( x ) (2.59) U x ( λ ) λ − x i ≡ Y j 6 = i ( λ − x j ) = n X k =1 ( − 1) k − 1 λ n − k σ i k − 1 ( x ) . (2.60) W e will also use σ k ( a ) a nd σ α k ( a ) with a = ( a 0 , a 1 , . . . a n ), whic h are defined similarly . Let us notice that the previous conserv ed quan tities (2.55) can b e written as F α | T ∗ S n = a α G a α ( ξ , x ) Y β 6 = α ( a α − a β ) where G λ ( ξ , x ) = 1 B n X i =1 x i g i ( x ) Y j 6 = i ( λ − x j ) ξ 2 i . (2.61) 17 Prop osition 2.13. L et the functions I 1 , . . . , I n of T ∗ S n b e define d by G λ ( ξ , x ) = n X k =1 ( − 1) k − 1 λ n − k I k ( ξ , x ) . (2.62) Then I k ( ξ , x ) = n X i =1 A i k ( x ) ξ 2 i with A i k ( x ) = 1 B x i g i ( x ) σ i k − 1 ( x ) . (2.63) Pr o of. By plugging the definition (2.60) of the symmetric functions σ i k ( x ) of or der k = 0 , 1 , . . . , n − 1 (in the v ariables ( x 1 , . . . , x n ), with the exclusion of index i ) in to (2.61), o ne gets the desired result. Theorem 2.14. The dual Moser system I 1 , . . . , I n on T ∗ S n , given by (2.63), defines a St¨ ackel system, with St¨ ackel matrix B = A − 1 of the form B i k ( x k ) = ( − 1) i ( x k ) n − i − 1 4 V ( x k ) (2.64) for i, k = 1 , . . . , n . This implies [22] that the functions I 1 , . . . , I n ar e indep ende n t and in invol ution, which entails that the St¨ a c k el c o o r dinates x 1 , . . . , x n ar e se p ar ating for the Hamilton-Jac obi e q uation. Pr o of. It is ob vious from its expression (2 .6 4) that B is a St¨ ac k el matrix [22]. W e just need to prov e that A is the in v erse matrix of B . T o t his aim we first prov e a useful iden tity . Let us consider the integral in the complex plane 1 2 i π Z | z | = R z n − i ( z − λ ) U x ( λ ) U x ( z ) dz . When R → ∞ the previous integral v anishes b ecause the integrand decreases as 1 /R 2 for large R (let us r ecall that i ≥ 1). W e then compute this in t egral using the theorem of residues and get the iden tit y n X k =1 ( x k ) n − i U ′ x ( x k ) Y j 6 = k ( λ − x j ) = λ n − i . (2.65) Equipped with this iden tity let us now prov e that P n k =1 B i k A k j = δ i j . Multiplying this relatio n by ( − 1) j − 1 λ n − j and summing ov er j fr o m 1 to n , w e get the equiv alen t relation n X k =1 B i k n X j =1 ( − 1) j − 1 λ n − j A k j = ( − 1) i − 1 λ n − i , 18 whic h b ecomes, using (2.63) and (2.60): n X k =1 B i k g k ( x ) Y j 6 = k ( λ − x j ) = ( − 1) i − 1 λ n − i . Using the explicit form of g k ( x ) given via (2.52) and of the matrix B , this last relation reduces to the iden tit y (2.65), whic h completes the deriv atio n of (2.64). Remark 2.15. A few remarks are in order. 1. The first in tegral defined b y (2.56) is precisely the Joac himsthal inv ariant (2.25) of the dual Moser system. 2. It should b e emphasized that the bihamiltonian c hara cter of our syste m is ob vious with our c hoice of ellipsoidal co ordinates since, from their v ery defini- tion, the x i are the eigenv alues of the Benen ti (1 , 1)-tensor field ( L j i ) a ssociated with a sp ecial conformal K illing tensor. 3. One can give some simple p oten tials for dual Moser. Denoting by J k the new first integrals, w e ha v e J k = I k − v k , v k = µσ k ( x ) + ν ( σ 1 ( x ) σ k ( x ) − σ k +1 ( x )) (2.66) with k = 1 , . . . , n . Those will pairwise P oisson comm ute (see [22], p. 1 01) if the p oten tial terms can b e written in the form v k = P n i =1 A i k ( x ) f i ( x i ), implying f i ( x i ) = P n k =1 B k i v k . A short computation, using the explicit form (2.64) of the matr ix B and the relation (2.59), indeed giv es f i ( x i ) = ( x i ) n − 1 4 V ( x i ) ( µ + ν x i ) . 2.5 Three St¨ ac k el systems 2.5.1 The Neumann-Uhlen b eck system In additio n to the t w o previously studied in tegrable systems, it ma y b e useful to consider the w ell-know n Neumann-Uhlen b ec k system [29, 30, 19] on the cotangen t bundle of the round sphere S n . It is initially defined on ( T ∗ R n +1 , P n α =0 dp α ∧ dq α ) b y the Hamiltonian H = 1 2 n X α =0  p 2 α + a α q 2 α  (2.67) with the parameters 0 < a 0 < a 1 < . . . < a n . This system is classically in tegrable, with the following comm uting first in tegrals o f the Hamiltonian flo w in T ∗ R n +1 : F α ( p, q ) = q 2 α + X β 6 = α ( p α q β − p β q α ) 2 a α − a β with α = 0 , 1 , . . . , n. (2.68) 19 Under symplectic reduction, with the second class constrain ts (2.44), it b ecomes an integrable system o n ( T ∗ S n , P n i =1 dξ i ∧ dx i ). W rit ing e g = P n i =1 e g i ( x )( dx i ) 2 the induced Euclidean metric on S n with e g i ( x ) = − 1 4 U ′ x ( x i ) /V ( x i ), the indep enden t P o isson-comm uting functions I k ( k = 1 , . . . , n ) a re I k ( ξ , x ) = n X i =1 e g i ( x ) σ i k − 1 ( x ) ξ 2 i − σ k ( x ) with H = 1 2 I 1 , (2.69) where e g i = 1 / e g i . 2.5.2 A syn thetic presen t ation Let us observ e that the previous calculation enables us to ha v e a syn thetic viewpoint unifying the Jacobi-Moser, Neumann-Uhlen b ec k, and dual Moser systems. This highligh ts the no velt y of the dual Moser system sp elled o ut in this article. In T able 1, we displa y in eac h row, and for eac h system, the metric, the first in tegrals in inv olution, the Hamiltonian, and the St¨ ack el matrix. (See, e.g., [8] for a deriv a t ion o f the form ulæ in the first tw o columns of this table.). Let us emphasize that in a ll three cases, the metric in the first ro w is indeed the St¨ ac k el metric coming from I 1 , and whic h will b e, later on, in v o lv ed in the quan t izat io n pro cedures. Jacobi-Moser Neumann-Uhlenbec k dua l Moser g i = x i e g i e g i = − U ′ x ( x i ) 4 V ( x i ) g i = 1 x i e g i I k = P i e g i x i σ i k − 1 ξ 2 i I k = P i e g i σ i k − 1 ξ 2 i − σ k ( x ) I k = P i x i e g i σ i k − 1 ξ 2 i H = 1 2 I 1 H = 1 2 I 1 H = 1 2 σ n +1 ( a ) I n x k e B i k ( x k ) e B i k ( x k ) = ( − 1) i ( x k ) n − i 4 V ( x k ) 1 x k e B i k ( x k ) T a ble 1: Three St¨ ac kel systems 3 Quan tu m in tegrability Start with a configuration manifold M of dimension n , and consider the space, S ( M ), of Hamiltonians on T ∗ M that a r e fib erwise po lynomial. A quan t izat io n prescription is a linear isomorphism Q b et w een t his space of symb ol s , S ( M ), and 20 the space, D ( M ), of linear differen tial op erato r s on M ; this iden tification is, in addition, assumed to preserv e the principal sym b ol. It is w ell-known that there is, in general, no uniquely defined quan tization. Ho w eve r, no matter ho w the quan tization is c hosen, w e will adhere to the fo llo wing, usual, definition o f quan tum in tegrability ; see, e.g., [28, 18 , 4, 5]. Definition 3.1. A classic al ly in te g r able system with indep endent, and mutual ly Poisson-c omm uting obse rvables I 1 , . . . I n , is inte gr able at the quantum level iff [ Q ( I k ) , Q ( I ℓ )] = 0 (3.1) for al l k , ℓ = 1 , . . . , n As a consequence, for a giv en integrable classical system, dep ending on the quan tizat io n pro cedure used, quan tum in tegrability may b e achiev ed or not. In what follo ws we will consider and use t w o quantiz ation sch emes for quadratic Hamiltoni- ans: (i) the theory o f conformally equiv ariant quan tization, and (ii) Carter’s minimal prescription. 3.1 Conforma lly equiv arian t quan tization Let us recall that there exists no quan tization mapping that inte rtw ines the action of Diff ( M ). T o b ypass this obstruction, equiv a r ian t quan tization [15, 10] prop oses to further endo w M with a G -structure, and to lo ok under whic h conditions the exis- tence and uniqueness of a G -equiv arian t quan t izatio n can b e guaran teed (the prop er subgroup G ⊂ Diff ( M ) only is assumed t o in tertw ine the quantization mapping Q ). W e recall that the space F λ ( M ) of λ -densities on M , where λ is some complex- v alued w eight, is the space of sections of the complex line bundle | Λ n T ∗ M | λ ⊗ C . If M is orientable, ( M , v ol ), suc h a λ -density can b e, lo cally , cast in to the form φ = f | v ol | λ with f ∈ C ∞ ( M ) ; this en tails t ha t φ tr a nsforms under the a ction of a ∈ D iff ( M ) according to f 7→ a ∗ f | ( a ∗ v o l ) / v ol | λ , or infinitesimally as L λ X ( f ) = X ( f ) + λ Div( X ) f (3.2) for all X ∈ V ect( M ). Remark 3.2. Note that the completion H ( M ) of the space o f compactly supp orted half-densities, F c 1 2 ( M ) , is a Hilb ert space cano nically attac hed to M t ha t will b e used in the sequel. The scalar pro duct of t w o half-densities reads h φ, ψ i = Z M φ ψ where t he bar stands for complex conjugation. 21 W e will denote b y S δ ( M ) = S ( M ) ⊗ F δ ( M ) the graded space o f sym b o ls o f w eight δ . This space is turned in to a V ect( M )-mo dule using the definition (3.2) of the Lie deriv ativ e extended to the canonical lift of V ect( M ) to T ∗ M . Lik ewise, w e will in tro duce the filtered space D λ,µ ( M ) of differential op erators sending F λ ( M ) to F µ ( M ) . A differen tial op erator of order k is, lo cally , written as A = A i 1 ...i k k ( x ) ∂ i 1 . . . ∂ i k + · · · + A i 1 ( x ) ∂ i + A 0 ( x ) (3.3) where A i 1 ...i ℓ ℓ ∈ C ∞ ( M ) for ℓ = 0 , 1 , . . . , k . It is clear that this space of w eigh ted dif- feren tia l op erat ors, D λ,µ ( M ) , b ecomes a V ect( M )-mo dule via the following definition of the Lie deriv a tiv e, namely , L λ,µ X ( A ) = L µ X ◦ A − A ◦ L λ X (3.4) for all X ∈ V ect( M ). F rom now on, we will b e dealing with the case of a conformal (Riemannian) structure, G = SO( n + 1 , 1), with n > 2, dictated b y the conformal flatness of our main example: the dual Moser system. Theorem 3.3 ([10]) . Given a c onform al ly flat Riemannian manifold ( M , g ) , ther e exists (exc ept f o r a discr ete set o f values of δ = µ − λ c al le d r e s o nanc es) a unique c onformal ly-e quiva riant quantization , i.e., a line ar isomorphism Q λ,µ : S δ ( M ) → D λ,µ ( M ) (3.5) that (i) pr eserves the princi p al symb o l , an d ( i i) intertwine s the action s o f the Lie algebr a o( n + 1 , 1) ⊂ V ect( M ) . In the par t icular and piv otal case of sym b ols of degree tw o, at the core o f the presen t study , explicit form ulæ are g iven by the fo llo wing theorem. Theorem 3.4 ([9]) . (i) L et ( M , g) b e a c onformal ly flat Riemannian manifold of dimension n ≥ 3 . Th e c onformal ly e quivariant quantization mapping (3.5) r e s tricte d to symb ols P = P ij 2 ( x ) ξ i ξ j + P i 1 ( x ) ξ i + P 0 ( x ) of de g r e e two is given, for n o n-r esonant values of δ , by Q λ,µ ( P ) = − P ij 2 ◦ ∇ i ◦ ∇ j + i  β 1 ∇ i P ij 2 + β 2 g ij g k ℓ ∇ i P k ℓ 2 + P j 1  ◦ ∇ j + β 3 ∇ i ∇ j ( P ij 2 ) + β 4 g ij g k ℓ ∇ i ∇ j ( P k ℓ 2 ) + β 5 R ij P ij 2 + β 6 R g ij P ij 2 + α ∇ i ( P i 1 ) + P 0 22 wher e ∇ denotes the levi-Civ ita c onne ction, 6 R ij (r esp. R ) the c om p onents of the Ric ci tensor in the cho sen c h art (r esp. the sc alar curvatur e) of the metric g ; the c o efficients α, β 1 , . . . , β 6 dep end on λ, µ , an d n in an expl i c i t fashion. 7 (ii) The quantization map p ing Q λ,µ dep ends o nly on the c onform a l cla s s of g . The a b o v e formu la can b e sp ecialized to the case of half- densit y quan tization of quadratic sym b ols P = P ij ( x ) ξ i ξ j ; one finds 8 Q 1 2 , 1 2 ( P ) = b P + β 3 ∇ i ∇ j ( P ij ) + β 4 g ij g k ℓ ∇ i ∇ j ( P k ℓ ) + β 5 R ij P ij + β 6 R g ij P ij (3.6) where b P = −∇ i ◦ P ij ◦ ∇ j . (3.7) Remark 3.5. The quan tization prescription (3.7), called “minimal” in [8], has b een put fo rw ar d by Carter [7], who dealt with p olynomial sym b ols of degree at most t w o . A great many studies of the quantum sp ectrum for v a rious in tegrable mo dels use naturally Carter’s quan tization [24, 28, 18, 2]. Along with Equation (3.7), the form ulæ f o r the minimal quan tization of low er degree monomials are respective ly c P 0 = P 0 (3.8) c P 1 = i 2  P i 1 ◦ ∇ i + ∇ i ◦ P i 1  (3.9) so tha t c P k = Q 1 2 , 1 2 ( P k ) , ∀ k = 0 , 1 . (3.10) Accordingly , a generalization to cubic monomia ls ha s b een prop osed in [8]: c P 3 = − i 2  ∇ i ◦ P ij k 3 ◦ ∇ j ◦ ∇ k + ∇ i ◦ ∇ j ◦ P ij k 3 ◦ ∇ k  . (3.11) All previously defined o p erators are formally self-a dj o in t on F c 1 2 ( M ) ; see R emark 3 .2. In the case where t he quadratic observ able P = P ij ( x ) ξ i ξ j stems fr om a Killing tensor, 9 i.e., if ∇ ( i P j k ) = 0 fo r all i, j, k = 1 . . . , n , we can rewrite Equation (3.6) as Q 1 2 , 1 2 ( P ) = b P + f ( P ) (3.12) where b P is as in (3 .7 ), a nd the scalar term is giv en b y f ( P ) = c 1 ∆ g T r( P ) + c 2 R ij P ij + c 3 R · T r( P ) (3.13) 6 The cov ariant deriv ative of λ -densities φ = f | vol g | λ , lo ca lly de fined in terms of the Riema nnian density , | vol g | , reads ∇ φ = d f | vol g | λ . 7 See Equations (3.3), (3.4), and (4.4) in [9]. 8 The v alue δ = 0 is non-r esonant [10]. 9 This is the c a se for the integrable systems of St¨ ack el type we ar e studying. 23 where ∆ g is the La place o p erator of ( M , g), and T r( P ) = P ij g ij ; the co efficien ts in ( 3 .13) are resp ectiv ely c 1 = n 2 8( n + 1)( n + 2) , c 2 = n 2 4( n + 1)( n − 2 ) , c 3 = − n 2 2( n 2 − 1)( n 2 − 4) . (3.14) 3.2 Quan tum comm utators In order to implemen t Definition 3.1 of quantum in tegrability , w e will need some preparation regarding the quan tum comm utators of P oisson-comm uting sym b ols. In doing so, w e will opt f o r the conformally equiv ariant quantization Q ≡ Q 1 2 , 1 2 . Prop osition 3.6. L et P and Q b e two, Poisson-c omm uting, quadr atic symb ols on ( T ∗ M , ω = P n i =1 dξ i ∧ dx i ) . T he c ommutator of the two op er ators Q ( P ) and Q ( Q ) , given by (3.12), r etain s the form [ Q ( P ) , Q ( Q )] = i Q ( A P ,Q + V P ,Q ) , (3.15) wher e A P ,Q = − 2 3  ∇ j B j k P ,Q  ξ k (3.16) with 10 B j k P ,Q = P ℓ [ j ∇ ℓ ∇ m Q k ] m + P ℓ [ j R k ] m,nℓ Q mn − ( P ↔ Q ) −∇ ℓ P m [ j ∇ m Q k ] ℓ − P ℓ [ j R ℓm Q k ] m (3.17) and V P ,Q = 2  P j k ∂ j f ( Q ) − Q j k ∂ j f ( P )  ξ k . (3.18) Pr o of. Start with t w o quadrat ic observ ables P and Q . As sho wn in [8], w e hav e − i [ b P , b Q ] = \ { P , Q } + b A P ,Q , where the mo no mial A P ,Q , and the sk ew-symmetric tensor B P ,Q are as in (3.16), and (3 .17), resp ectiv ely . If it is then assumed that { P , Q } = 0 , Equation (3.15) follows directly from the explicit expression (3 .12) of the conformally equiv ariant quan tization mapping, Q , and from Equation (3.1 0). No w, for the L io uville-in tegrable systems considered b elow , all P o isson-com- m ut ing fib erwise p olynomial sym b ols ha v e the form P = P 2 + P 0 , where the indices 0 and 2 refer to the homogeneit y degree. In view of Equation (3.15), and of results obtained in [8], w e find [ Q ( P 2 + P 0 ) , Q ( Q 2 + Q 0 )] = i Q ( A P 2 ,Q 2 + V P 2 ,Q 2 ), whic h means that the zero degree terms P 0 and Q 0 pro duce no quantum corrections. 10 W e use the following co nven tion fo r the Riemann and Ricci tenso rs, na mely , R ℓ i,j k = ∂ j Γ ℓ ik + Γ ℓ sj Γ s ik − ( j ↔ k ), and R ij = R s i,sj . 24 The structure of the quantum corrections (the rig h t hand side of Equation (3.15)) is rather in v olv ed, b ecause of the complexit y of the tensor B P ,Q ; see (3.17). Nev er- theless, for St¨ ac ke l systems ma jo r simplifications o ccur. Indeed, the observ ables I k = I 2 ,k + I 0 ,k with I 2 ,k = P i g i ( x ) σ i k − 1 ( x ) ξ 2 i generate dia gonal Killing tensors. Us- ing the separating co ordinates x i and considering H = 1 2 I 2 , 1 = 1 2 P i g i ( x ) ξ 2 i for the Hamiltonian fixes up the diagonal metric to b e g = P i g i ( x )( dx i ) 2 , with g i = 1 / g i for all i = 1 , . . . , n . Under these assumptions, Prop osition 3.9 in [8] giv es B k ℓ I 2 ,i ,I 2 ,j = − 2 I s [ k 2 ,i R st I ℓ ] t 2 ,j (3.19) for all i, j, k , ℓ = 1 , . . . n , whic h en tails: Prop osition 3.7. A sufficie nt c ondition fo r a St¨ ac k e l system to b e inte gr abl e at the quantum level is R ij = 0 , ∀ i 6 = j (3.20) wher e i, j = 1 , . . . , n , in the sp e cial se p ar ating c o or dinate system ( x i ) . Pr o of. The Killing t ensors I 2 ,i are diagonal, for i = 1 , . . . , n , in the St¨ ac k el co ordina t e system, and the result follo ws f rom (3.19). Remark 3.8. 1. Condition (3.20) is the w ell-kno wn Ro b ertson condition [23], whic h has to hold in the separating co ordinates system. The relation (3.19) was also obtained in [5] b y a direct computation of the comm uta tor in separating co ordinates; ho w ev er the explicit fo r m of the tensor B P ,Q (3.17) w as not giv en there. 2. In Corollary 3.10 o f [8] the Rob ertson condition w as misleadingly claimed to b e a lso necessary . 3. It has b een shown b y Benen ti et al. [4] that the Rob ertson conditio n (3.20) is necessary and sufficien t for t he separabilit y of the Schr¨ odinger equation, comforting the ab ov e definition of quantum integrabilit y . In the next subsec tions w e will examine, successiv ely , quan tum in tegrabilit y for the followin g St¨ ac k el systems: the Neumann-Uhlenbec k, the dual Moser and t he Jacobi-Moser systems. As previously explained, the p oten tial, i.e., zero degree terms in the classical observ ables nev er induce quan tum corrections; they will therefore b e systematically omitted. 25 3.3 The quan tum Neumann-Uhlen b ec k system Let us r ecall that, for the Neumann-Uhlen b ec k system (see T able 1), the St¨ ac k el metric e g = P i e g i ( x )( dx i ) 2 , is e g i ( x ) = − 1 4 U ′ x ( x i ) V ( x i ) = − 1 4 Q j 6 = i ( x i − x j ) Q α ( x i − a α ) (3.21) for i = 1 , . . . , n . If we put e g i = 1 / e g i , the indep enden t and P oisson-comm uting observ ables are give n by I k = n X i =1 e g i ( x ) σ i k − 1 ( x ) ξ 2 i for k = 1 , 2 , . . . , n , and the St¨ ac ke l Hamiltonian is H = 1 2 I 1 = 1 2 P i e g i ( x ) ξ 2 i . Prop osition 3.9. The c onformal ly e quivariant quantization do es pr eserve quantum inte gr ability of the Neumann -Uhlenb e ck system. Pr o of. F rom the fact that ( S n , e g) is the round sphere, w e hav e e R ij = ( n − 1 ) e g i δ ij & e R = n ( n − 1) . (3.22) Straigh tforward computation then leads to f ( I k ) = ( n − k + 1) [ c 4 σ k − 1 ( x ) + 2 ( n − k + 2) c 1 σ k − 1 ( a )] (3.23) for all k = 1 , . . . , n , where c 4 = − 2( n + 1) c 1 + ( n − 1) c 2 + n ( n − 1) c 3 . (3.24) Relations (3.14) readily imply the v anishing o f c 4 . As a consequence, the f ( I k ) are just constant, ensuring that V I k ,I ℓ = 0 (see (3.18)). Equation (3.15 ) and the fact that B I k ,I ℓ = 0 (since the Ricci tensor is diago na l in this co ordinate system) entail that the confor ma lly equiv arian t quantization (whic h coincides, up to a constant term, with Carter’s) preserv es integrabilit y of the system a t the quan tum level. 3.4 The quan tum dual Moser system In the basic geometrical construction o f the P oisson-comm ut ing conserv ed quan t it ies I k , w e hav e b een considering the conformally flat metric g 2 giv en b y (2.5 1) and (2.52). No w, in the quantum approa ch to in tegrabilit y , w e c ho ose to use, aga in, the St¨ ac k el metric, g, asso ciated with I 1 . One has g = n X i =1 g i ( x )( dx i ) 2 , with g i ( x ) = 1 x i e g i ( x ) , (3.25) 26 where the Neumann-Uhlen b ec k metric e g is give n b y (3.21), while the first integrals for k = 1 , . . . , n are I k = n X i =1 g i ( x ) σ i k − 1 ( x ) ξ 2 i , g i ( x ) = 1 g i ( x ) . (3.26) Lemma 3.10. T he metric (3.25) has R ic ci tensor R ij =  ( n − 2) x i + n n X k =1 x k − ( n − 1 ) n X α =0 a α  g i δ ij (3.27) and sc ala r curvatur e R = ( n − 1)  ( n + 2 ) n X k =1 x k − n n X α =0 a α  . ( 3 .28) It is c onformal ly flat for n = dim( M ) ≥ 3 . Pr o of. The Ricci tensor can b e computed with the help of classical formulæ for a diagonal metric (see for instance [11], p. 119). The only p ossibly non-v anishing comp onen t s of the Riemann tensor are R ik ,k j , for i 6 = j 6 = k , and R ij,j i , for i 6 = j . Using the relations ∂ i (ln g j ) = 1 x i − x j ( i 6 = j ) , ∂ ij (ln g k ) = 0 ( i 6 = j 6 = k ) one easily gets R ik ,k j = 0, implying R ij = − n X k =1 g k R ik ,k j = 0 , ∀ i 6 = j. The computation o f the remaining comp onents inv olv es a sum whic h is conv enien tly computed using the theorem of residues, giving R ik ,ik = ( x i + x k + n X s =1 x s − n X α =0 a α ) g i g k from whic h one deduces easily the diagonal part of the Ricci tensor, giv en b y (3.27), and the scalar curv ature (3.28). Some extra computation sho ws that the conformal W eyl tensor v anishes in dimension n ≥ 4, and that the Cotton-Y ork tensor v anishes as well for n = 3 . Remark 3.11. Although the metric g 2 giv en b y (2.51) is clearly confo rmally fla t, it is b y no means trivial that the same is true for the St¨ ac k el metric, g, give n by (3.25) on S n . 27 W e are no w in p osition to prov e the following prop osition. Prop osition 3.12. The c onformal ly e quivariant quantization pr o c e dur e (3.12) do es pr eserve quantum inte gr ability of the dual Moser system. Pr o of. Using the definition (3.13) of the scalar term in the formula (3 .1 2) f o r the conformally equiv ariant quantization of the I k , w e find f ( I k ) = [( n − 2) c 2 + k ( c 6 − c 4 )] σ 1 ( x ) σ k ( x ) + [ k c 5 − ( n − 2) c 2 ] σ k +1 ( x ) − k c 4 σ 1 ( a ) σ k ( x ) − 2 k ( n − k + 1) c 1 σ k +1 ( a ) where c 4 w a s already defined in (3.24) and show n to v anish in the pro of of Pro p osi- tion 3.9; we also hav e c 5 = 2( n + 2) c 1 − ( n − 2) c 2 , c 6 = − 2 c 1 + c 2 + 2( n − 1) c 3 . T a king into a ccoun t the relations (3.14) one gets c 5 = c 6 = 0, and we are left, for k = 1 , . . . , n , with f ( I k ) = 2 c 1 h ( n + 2 ) [ σ k ( x ) σ 1 ( x ) − σ k +1 ( x )] − k ( n − k + 1 ) σ k +1 ( a ) i (3.29) where we p osit σ n +1 ( x ) = 0. Let us no w compute V I k ,I ℓ defined b y (3.18) needed to c hec k quantum in tegr a - bilit y via t he comm utator (3.15). In view of (3.26), one finds V I k ,I ℓ = 2 n X i =1 g i  σ i k − 1 ∂ i f ( I ℓ ) − σ i ℓ − 1 ∂ i f ( I k )  ξ i . No w, using the relations [3] ∂ i σ k ( x ) = σ i k − 1 ( x ) , ∀ i, k = 1 , . . . , n, and σ k ( x ) = σ i k ( x ) + x i σ i k − 1 ( x ) , ∀ k = 1 , . . . , n − 1 , as well as σ n ( x ) = x i σ i n − 1 ( x ) , one gets ∂ i f ( I k ) = 2( n + 2) c 1 [ x i + σ 1 ( x )] σ i k − 1 ( x ) whic h ob viously yields V I k ,I ℓ = 0, implying, at last [ Q ( I k ) , Q ( I ℓ )] = 0 ( 3.30) for all k , ℓ = 1 , . . . , n . 28 Remark 3.13. Carter’s (minimal) prescription ( 3.7) also leads to quantum in tegra- bilit y of the system b ecause of the diagonal form (3.2 7) of the Ricci tensor in the separating co ordinates. No w, in con tr adistinction with the Neumann-Uhlen b ec k quan tum syste m, the scalar terms f ( I k ) giv en by (3.29 ) are no longer constan t, yielding quite differen t quan tum observ ables Q ( I k ) and ˆ I k . So , the f a ct that quantum in tegrabilit y is no t only preserv ed b y Carter’s quantum prescription, but a lso b y conformally equiv ariant quantization is a new and notew orth y phenomenon. 3.5 The quan tum Jacobi-Moser system The St¨ a c ke l metric, asso ciated to I 1 is now (see T a ble 1): g = n X i =1 g i ( x )( dx i ) 2 , with g i ( x ) = x i e g i ( x ) , (3.31) where the Neumann-Uhlen b eck metric e g i ( x ) is giv en b y (3.21), a nd the first integrals b y I k = n X i =1 g i ( x ) σ i k − 1 ( x ) ξ 2 i , g i ( x ) = 1 g i ( x ) . (3.32) Lemma 3.14. The Ric ci tenso r and the sc alar curvatur e of the metric (3.31) a r e given by R ij = σ n +1 ( a ) σ 2 n ( x ) σ i n − 2 ( x ) g i δ ij , R = 2 σ n +1 ( a ) σ 2 n ( x ) σ n − 2 ( x ) . (3.33) Pr o of. It is completely similar to the pro of of Lemma 3.10 . This allo ws us to pro v e: Prop osition 3.15. Carter’s p r escription (3.7) p r eserves quantum i n te g r ability of the Jac o b i-Moser system, while the c onformal ly e quivariant quantization do es not. Pr o of. Since the Ricci tensor is diagona l, quantum in tegrability is established for the prescription ( 3 .7). F or the conformally equiv arian t quantization (3.1 2), w e will just g ive a coun ter-example. One has f ( I 1 ) = 2( c 2 + nc 3 ) σ n +1 ( a ) σ n − 2 ( x ) σ 2 n ( x ) and f ( I 2 ) = ( n − 1) c 3 σ n +1 ( a ) σ 2 n ( x ) h 2( n − 1) σ n − 1 ( x ) − n σ 1 ( x ) σ n − 2 ( x ) i − 2 n ( n − 1 ) c 1 . 29 A simple computation giv es V i I 1 ,I 2 = ∂ V I 1 ,I 2 /∂ ξ i = ∂ i f ( I 2 ) − ( σ 1 ( x ) − x i ) ∂ i f ( I 1 ), hence the non-v anishing result V i I 1 ,I 2 = ( − c 3 ) σ n +1 ( a ) x i σ 2 n ( x ) h − 2( σ i 1 ( x ) σ n − 2 ( x ) + σ 1 ( x ) σ i n − 2 ( x )) +( n 2 − 3 n + 4) σ n − 1 ( x ) + n (3 n − 5 ) σ i n − 1 ( x ) i , sho wing t ha t the system lo oses its quan t um integrabilit y via conformally equiv aria n t quan tizat io n. Remark 3.16. Let us mention that quan tum in tegrability of the Neumann-Uhlen- b ec k and Ja cobi- Moser systems has first b een established, in terms of Carter’s quan- tum prescription (3 .7 ), b y T oth [28]. 4 Conclus ion and o u tlo o k T o sum up the main results of the article, let us men tio n tha t we ha v e disclosed a new integrable system on S n , in dualit y with the w ell-kno wn Jacobi-Moser system in terms of pro jectiv e equiv alence. As opp osed to that of the generic ellipsoid, the “dual” metric is conformally flat. This remark able fact enables us to hav e naturally recourse to conformally equiv arian t quantization. The latter turns out to preserv e in tegrabilit y at the quantum lev el. It is, to our kno wledge, the fir st instance o f conformally driv en quantum integrabilit y . This o p ens new p ersp ectiv es related, e.g., to the determination of the conditions under whic h a classically in tegrable system, stemming from second-order Killing tensors on a conformally fla t configura tion manifold, remains quantum-in tegrable via the conformally equiv ariant quan tization. Also, p ossible generalizations o f the Jacobi-Moser system and its dual coun terpart migh t conceiv ably b e put to light in a similar manner. Ac kno wledgemen ts : W e express o ur deep gratitude to Serge T abac hnik ov and V alen tin Ovsienk o for helpful discussions whic h ha v e trig g ered this w ork. W e are also indebted to John Harnad for useful corresp ondence. References [1] R. Abraham, and J.E. Marsden, F oundations of Mec hanics , Second Edition, Addison-W esley Publishing Company , Inc (1987). 30 [2] M. Bellon, M. T alon, T he quantum Neumann m o del: r efine d semiclassic a l r e- sults , Ph ys. Lett. A, 356 (2006 ) 110–1 14, and references therein. [3] S. 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