Hamiltonian reductions of free particles under polar actions of compact Lie groups

Hamiltonian reductio ns of free partic l es u nder p olar actions o f c o mpact Lie gro u ps L. FEH ´ ER a and B.G. PUSZT AI b a Departmen t of Theoretical Ph ysics, MT A KFKI RMKI 1525 Budap est 114, P .O.B. 49, Hungary , and Departmen t of Theoretical Ph ysic s, Univ ersit y of Szeged Tisza La jos krt 84-86, H-6720 Sz eged, Hungary e-mail: lfeher@rmki.kfki.h u b Cen tre de rec herc h es mat h´ ematiques, Univ ersit ´ e de Montr ´ eal C.P . 6128, succ. cen tre ville, Mon tr ´ eal, Qu´ ebec, Canada H3C 3J7, and Departmen t of Mathematics and Statistics, Concordia Univ ersit y 1455 de Maisonneuv e Blvd. W est, Mon tr ´ eal, Qu´ ebec, Canada H3G 1M8 e-mail: pusztai@CRM.UMon treal.CA Abstract Classical and quantum Hamiltonian reductions of fr ee geodesic systems of complete Riemannian manifolds a re inv estig ated. The redu ced systems are describ ed under the assumption that the underlying co mpact symmetry group acts in a p olar manner in the sense that th ere exist regularly em b ed d ed, clo sed, connected submanifolds meeting all orbits o rthogonally in the configuration sp ace. Hyp erp olar actions on Lie group s and on symmetric spaces l ead to fa milies of in tegrable systems of s p in Calogero-Sutherland t yp e. 1 In tro du ction In the theory of integrable systems one of the basic facts is tha t man y interes ting mo dels arise as Hamiltonian reductions of certain canonical ‘free’ systems that can b e integrated ‘o b viously’ due to their large symmetries. F or example, the cele brated Sutherland mo del of interacting particles on the line, defined classically b y t he Hamiltonian H ( q , p ) = 1 2 n X k =1 p 2 k + ν ( ν − 1) X 1 ≤ i 0. It is easy to see that the res triction of functions on Y to Σ giv es rise to a n injectiv e map 3 C ∞ ( Y , V ) G − → C ∞ (Σ , V K ) W . (4.5) Note also that an y function F ∈ C ∞ ( Y , V ) G is uniquely determine d b y its restriction to a connected, op en comp onen t ˇ Σ ⊂ Σ. Then in tro duce the linear space F un( ˇ Σ , V K ) := { f ∈ C ∞ ( ˇ Σ , V K ) | ∃F ∈ C ∞ c ( Y , V ) G , f = F | ˇ Σ } . (4.6) 2 See e.g. [23], para graph I I.3.7, a nd refer e nces therein. W e do not include the Ricci s c alar in the quantum Hamiltonian, but its inclusion w ould no t cause any extra difficulty in our a rguments. 3 This map is k nown to b e sur jective [1 4] if dim( V ) = 1. It would be in teresting to generalize this r esult, and another imp ortant q ue s tion is to find all cases for which dim( V ) > dim( V K ) = 1 like in the e x amples in [2, 9]. 7 By its isomorphism with C ∞ c ( Y , V ) G ⊂ L 2 ( Y , V , dµ Y ), F un( ˇ Σ , V K ) b ecomes a pre- Hilb ert space and w e denote its closure b y F un( ˇ Σ , V K ). It is not difficult to v erify the natural isometric isomorphism F un( ˇ Σ , V K ) ≃ L 2 ( Y , V , dµ Y ) G , (4.7) and it is also w orth remarking t hat F un( ˇ Σ , V K ) con tains C ∞ c ( ˇ Σ , V K ). Because of (4.7), a natural quantum mec hanical analogue of the classical Ha milto nian reduc- tion is obtained b y taking the reduced Hilb ert space to b e F un( ˇ Σ , V K ). The reduced Ha milton op erator results from ∆ 0 Y on accoun t of the following simple observ ation. There exists a unique linear o p erator ∆ eff : F un( ˇ Σ , V K ) → F un( ˇ Σ , V K ) (4.8) defined by the pr o p ert y ∆ eff f = (∆ Y F ) | ˇ Σ , for f = F | ˇ Σ , F ∈ C ∞ c ( Y , V ) G . (4.9) In other w ords, the ‘effectiv e Laplace-Beltrami op erator’ ∆ eff is the restriction of ∆ Y to C ∞ c ( Y , V ) G , whic h is w ell-defined b ecause the metric η is G -inv ariant. Next w e presen t the explicit f o rm ula o f ∆ eff . 4.1 The eff ectiv e Laplace-Beltrami op erator A conv enien t lo cal decomp osition of the Laplace-Beltrami op erator in to ‘radial’ a nd ‘orbita l’ (angular) parts is alw ay s applicable if one has a lo cal, orthogonal ‘cross section’ of the G - orbits on a Riemannian G -manifold [2 4]. Up o n restriction to G -equiv arian t functions, the orbital part can b e calculated explicitly , and in our case w e can a pply this decomposition o v er ˇ Y since w e are dealing with a p olar action. Before describing the r esult, w e need some further notations. Thinking of ˇ Σ as the (smo oth part of the) reduced configuration space, denote the elemen ts of ˇ Σ b y q and cons ider the G orbit G.q thr o ugh any q ∈ ˇ Σ. Both ˇ Σ and G.q are regularly em b edded submanifolds of Y a nd by their em b eddings they inherit Riemannian metrics , η ˇ Σ and η G.q , fr om ( Y , η ). Let ∆ ˇ Σ and ∆ G.q denote the L a place-Beltrami op erators defined on the resp ectiv e Riemannian manifolds ( ˇ Σ , η ˇ Σ ) a nd ( G.q , η G.q ). Intro duce the smo o th densit y function δ : ˇ Σ → R > 0 b y δ ( q ) := v olume of the Riemannian manifold ( G.q , η G.q ) . (4.10) Of course, t he v olume is understo o d with r esp ect to the measure b elonging to η G.q . (By the same form ula, δ can also b e defined as a W - inv ar ian t function on Σ.) Referring to (2.7), let us define t he function J : ˇ Σ → End( K ⊥ ) by J := I | ˇ Σ , (4.11) and no tice that, b ecause of the G -symmetry , the inertia o p erator J ( q ) carries the same info r- mation as the metric η G.q . Denote by { T α } and { T β } some fixed dual bases o f K ⊥ with resp ect to the scalar pro duct B , B ( T α , T β ) = δ β α . (4.12) 8 In fa ct (see also Remark 4.3 b elow), one has δ ( q ) = C | det b α,β ( q ) | 1 2 with b α,β ( q ) = B ( J ( q ) T α , T β ) (4.13) and a q -indep enden t constan t C > 0, whose v alue could b e giv en but is not imp ortan t for us. Prop osition 4.1. On F un( ˇ Σ , V K ) (4.6) the eff e ctive L aplac e -Beltr ami op er ator (4.9) c a n b e expr esse d in the fol lowing form: ∆ eff = δ − 1 2 ◦ ∆ ˇ Σ ◦ δ 1 2 − δ − 1 2 ∆ ˇ Σ ( δ 1 2 ) + b α,β ρ ′ ( T α ) ρ ′ ( T β ) . (4.14) Mor e explicitly, this me ans that for any q ∈ ˇ Σ and f ∈ F un( ˇ Σ , V K ) , one has (∆ eff f )( q ) = δ − 1 2 ( q )(∆ ˇ Σ ( δ 1 2 f ))( q ) − δ − 1 2 ( q )(∆ ˇ Σ δ 1 2 )( q ) f ( q ) + b α,β ( q ) ρ ′ ( T α ) ρ ′ ( T β ) f ( q ) , (4.15) wher e the line ar op er ator on V K in the last term c ontains the inverse b α,β ( q ) = B ( J − 1 ( q ) T α , T β ) of the matrix b α,β ( q ) (4.13). Pro of. T ak e an arbitrary function F ∈ C ∞ ( Y , V ) a nd consider its restrictions f := F | ˇ Σ and F q := F | G.q ∀ q ∈ ˇ Σ . (4.16) Then (∆ Y F )( q ) can b e found with the aid of the following w ell-kno wn formula. Lemma 4.2. Usin g the ab ove no tations, (∆ Y F )( q ) = (∆ rad f )( q ) + (∆ G.q F q )( q ) (4.17) with the r adial p art o f ∆ ˇ Y given by ∆ rad = δ − 1 2 ◦ ∆ ˇ Σ ◦ δ 1 2 − δ − 1 2 ∆ ˇ Σ ( δ 1 2 ) . (4.18) The statemen t of Lemma 4.2 is quite standard (see e.g. [24]), and one can also v erify it in a direct manner b y simply writing out the L a place-Beltrami op erator in suc h lo cal co ordina t es, { y a } = { q i } ∪{ z α } , a round q ∈ ˇ Σ that are comp o sed o f some co ordinates q i on ˇ Σ and co o r dina t es z α around the origin of the coset space G/K according to the G -equiv ariant diffeomorphism ˇ Σ × G/ K ≃ ˇ Y (4.19) defined b y ˇ Σ × G/K ∋ ( q , g K ) 7→ φ g ( q ) ∈ ˇ Y . The co ordinates can b e c hosen b y taking adv an tage of t he lo cal diffeomor phism K ⊥ ∋ z = z α T α 7→ e z K ∈ G/K , (4.20) and the metric on Y is then represen ted by g a,b ( q , z ) := η ( q , z ) ( ∂ y a , ∂ y b ) satisfying g i,j ( q , z ) = g i,j ( q , 0) , g i,α ( q , z ) = 0 , g α,β ( q , 0) = b α,β ( q ) (4.21) 9 with b α,β ( q ) in (4.13). On accoun t o f this blo c k-diagona l structure, the lo cal expression ∆ Y ← → 1 √ g ∂ y a ◦ √ g g ab ◦ ∂ y b , g := | det g a,b | (4.22) separates into the sum of t w o terms, whic h yield the t wo terms in ( 4.17). Since f ∈ F un( ˇ Σ , V K ) corresp onds to F ∈ C ∞ c ( Y , V ) G , to prov e the prop osition it is no w enough to calculate ∆ G.q F q for equiv a rian t functions F q ∈ C ∞ ( G.q , V ) G . The result of this latter calculation can presumably b e a lso found in the literature, but one can also compute it directly b y using the exp onential co ordinates on G.q ≃ G/ K in tr o duced ab o ve. One finds that (∆ G.q F q )( q ) = b α,β ( q ) ρ ′ ( T α ) ρ ′ ( T β ) f ( q ) if F q ∈ C ∞ ( G.q , V ) G , (4.23) whic h completes the pro of of the prop osition. Q.E.D. Remark 4.3. The coset space G/K carries a G -inv ar ia n t Ha a r measure, whic h is unique up to a multiplic ativ e constan t. The Haar measure is a sso ciated with a G -in v ariant differen tial f orm of top degree o n G/ K . This differen tial form is uniquely determined b y its v alue at the origin K ∈ G/K . Upo n the iden tification G/K ≃ G.q , the origin b ecomes q , and the v a lue of the G -in v ariant volume form asso ciated with the metric η G.q giv es a t the origin | det b α,β ( q ) | 1 2 ( dz 1 ∧ dz 2 ∧ · · · ∧ d z m ) q , m := dim( G/K ) . (4.24) F orm ula ( 4 .13) follows easily from this remark. 4.2 The reduced quan tum system The effectiv e Laplace-Beltrami o p erator ∆ eff (4.14) can b e show n to b e essen tially self-adjoin t on the domain F un( ˇ Σ , V K ) ⊂ F un( ˇ Σ , V K ) ≃ L 2 ( Y , V , dµ Y ) G . In order to relate the reduced Hilbert space to the Riemannian metric on the smo oth part of the reduced configuration manifold, ( ˇ Y red , η red ) ≃ ( ˇ Σ , η ˇ Σ ), the following lemma is neede d. Lemma 4.4. The c omplement of the dense, op en submanifold ˇ Y ⊂ Y of princip al orb i t typ e has zer o me asur e with r esp e ct to dµ Y . Pro of. It is kn ow n [20] that the non-principal orbits o f a g iv en ty p e fill lower-dimensional regular submanifolds in Y and at most coun tably many differen t types of orbits can o ccur. Since the measure dµ Y is smo oth, and Y is second coun t a ble, this implies (see e.g. [26], page 529) t ha t Y \ ˇ Y has measure zero. Q.E.D . T o pro ceed furt her, we also need the following in tegration formula: Z Y ( F 1 , F 2 ) V dµ Y = Z ˇ Y ( F 1 , F 2 ) V dµ ˇ Y = Z ˇ Σ ( f 1 , f 2 ) V δ dµ ˇ Σ , F i ∈ C ∞ c ( Y , V ) G , f i = F i | ˇ Σ . (4.25) The first equality is guaranteed b y Lemma 4.4. The second equalit y holds since, as is standard to show , the measure on ˇ Y ≃ ˇ Σ × G/K tak es the pro duct form dµ ˇ Y = ( δ dµ ˇ Σ ) × dµ G/K , (4.26) 10 where dµ G/K is the probabilit y Haar measu re on G/K , dµ ˇ Σ is the measu re on ˇ Σ asso ciated with the Riemannian metric η ˇ Σ , the densit y δ is defined in (4.10); a nd ( F 1 , F 2 ) V is G -in v arian t. The integration form ula and the fa ct that F un( ˇ Σ , V K ) contains C ∞ c ( ˇ Σ , V K ), in asso ciation with C ∞ c ( ˇ Y , V ) G ⊂ C ∞ c ( Y , V ) G , together imply that F un( ˇ Σ , V K ) ≃ L 2 ( ˇ Σ , V K , δ dµ ˇ Σ ) . (4.27) By transforming aw ay the factor δ from the ‘induced measure’ δ dµ ˇ Σ , w e obtain t he final result. Theorem 4.5. Using the ab ove notations, the r e duction of the quantum system d e fine d by the closur e of − 1 2 ∆ Y on C ∞ c ( Y , V ) ⊂ L 2 ( Y , V , dµ Y ) l e ads to the r e duc e d Hamilton op er ator − 1 2 ∆ red given by ∆ red = δ 1 2 ◦ ∆ eff ◦ δ − 1 2 = ∆ ˇ Σ − δ − 1 2 (∆ ˇ Σ δ 1 2 ) + b α,β ρ ′ ( T α ) ρ ′ ( T β ) . (4.28) This op er ator is essential ly se l f - a djoint on the dense domain δ 1 2 F un( ˇ Σ , V K ) in the r e duc e d Hilb ert sp ac e ide n tifie d as L 2 ( ˇ Σ , V K , dµ ˇ Σ ) . Pro of. The multiplication op erator U : f 7→ δ 1 2 f is an isometry from L 2 ( ˇ Σ , V K , δ dµ ˇ Σ ) to L 2 ( ˇ Σ , V K , dµ ˇ Σ ). Plainly , Prop osition 4.1 and Lemma 4.4 imply that ∆ red = U ◦ ∆ eff ◦ U − 1 is a symmetric op erat or on the dense domain U (F un( ˇ Σ , V K )) = δ 1 2 F un( ˇ Σ , V K ) ⊂ L 2 ( ˇ Σ , V K , dµ ˇ Σ ). The essen tial self-adjointness of ∆ red can b e traced back to the essen tial self-adjointness of ∆ Y on C ∞ c ( Y , V ). More details on this last p oin t ar e provided in [25]. Q.E.D. Let us no w compar e the result of the quan tum Ha miltonian reduction giv en b y Theorem 4.5 with the classical r educed system in Theorem 3 .1. First, the classical kinetic energy clearly corresp onds to − 1 2 ∆ ˇ Σ . F ormally , the second term of the classical Hamiltonian H red (3.10) corresp onds the third term of − 1 2 ∆ red (4.28). This t erm can b e in terpreted as p oten tial energy if dim ( V K ) = 1, otherwise it is a ‘spin dependen t p oten tial energy’. As r epresen ted b y the second t erm in (4.28), an extra ‘measure factor’ a pp ears at the quan tum lev el in general, whic h has no trace in H red . This term gives a constant or a non-trivial con tr ibutio n to the p oten tial energy dep ending on the concrete examples. In the quan tum Hamiltonian r eduction w e started from the full configura tion space Y , while classically we hav e restricted our attention to ˇ Y from the b eginning. In some sense, the outcome of the quan tum Hamiltonian reduction can nev ertheless b e view ed as a quantization of the reduced classical system o f Theorem 3.1 b e c ause Y \ ˇ Y has zero measure. How ev er, this delicate corr espo ndence needs further in v estigation (see also [27]). The structure o f the full (singular) reduced phase space P red coming f r o m T ∗ Y should b e explored, t o o, since it is clear that the reduced geo desic flows ma y leav e ˇ P red ⊂ P red in certain cases. It should b e stressed that at the abstract lev el, on accoun t of the natural identifications (4.7) a nd (4.27), the reduced Hilb ert space is simply pro vided b y the G - singlets L 2 ( Y , V , dµ Y ) G . In some cases ( for example if Y is a compact Lie group) ∆ Y p ossesses pure p oint sp ectrum that can b e determined from kno wn results (suc h as the P eter-W eyl theorem) in har monic analysis. In suc h cases finding the spectrum of t he reduced Hamilton opera t o r b ecomes a problem in branc hing rules, since it requires finding the ab ov e G -singlets among the eigensubspaces of ∆ Y . 11 5 Examples related to spin Sutherland t yp e mo dels W e here recall from [17, 18] a class of imp ortant h yp erp olar actions on compact Lie groups. By sub jecting them to the classical a nd quan tum Hamiltonian reduction as described in this pap er, one ma y obtain, and solv e, a large fa mily o f spin Sutherland t yp e in tegra ble mo dels. Details on some of these mo dels will b e rep orted elsewhere. Let Y b e a compact, connected, semisimple Lie group carrying the Riemannian metric induced by a multiple of t he Killing form. T ak e the ‘reduction group’ G to b e an y symm etric sub gr oup of Y × Y . That is to sa y , G is an y subgroup whic h is p oin t wise fixed b y a n inv olutiv e automorphism σ of Y × Y and contains the connected comp onen t of the full subgroup fixed by σ . Consider the follow ing action of G on Y : φ ( a,b ) ( y ) := ay b − 1 ∀ y ∈ Y , ( a, b ) ∈ G ⊂ Y × Y . (5.1) This action is known to b e h yp erp olar (see [17, 18] and the references there). The sections are pro vided b y certain to ri, A ⊂ Y . In fact, A is the exp onen tial of an Ab elian subalgebra A o f the correct dimension lying in the subspace ( T e ( G.e )) ⊥ of T e Y . The underlying reasons b ehind the a pp earance o f Sutherland type mo dels in asso ciation with these a ctio ns are the exp onen tial parametrization of Σ = A tog ether with the decomp o sition of the Lie algebra of Y in to join t eigensubspace s of A . One can illustrate this b y the pa rticular examples to whic h w e now turn. First, consider σ ( y 1 , y 2 ) = ( y 2 , y 1 ). Then G = { ( a, a ) | a ∈ Y } ≃ Y and the action ( 5 .1) is just the adjoint action of Y on itself, for whic h the sections are the maximal tori of Y . The asso ciated spin ( and in exceptional cases spinless) Sutherland mo dels w ere studied in [2, 7]. Second, tak e an y non- t r ivial automorphism θ of Y and set σ ( y 1 , y 2 ) := ( θ ( y 2 ) , θ − 1 ( y 1 )). Now G = { ( θ ( a ) , a ) | a ∈ Y } ≃ Y , (5 .2 ) and (5.1) yields the action of Y on itself by θ -twis ted conjuga t ions, φ ( θ ( a ) ,a ) ( y ) = θ ( a ) y a − 1 . The in teresting cases are when θ corr esp onds to a Dynkin diagra m symme try of Y . Some of the resulting generalized spin Sutherland mo dels hav e b een describ ed in [11, 28]. Third, suppose that θ 1 and θ 2 are t w o in v olutiv e automorphisms of Y and let K 1 and K 2 b e corresp onding sym metric subgroups of Y , i.e., ( Y , K j ) are symmetric pairs f o r j = 1 , 2. By taking σ ( y 1 , y 2 ) := ( θ 1 ( y 1 ) , θ 2 ( y 2 )), one obtains G = K 1 × K 2 ⊂ Y × Y , (5.3) and (5.1) b ecomes the so-called Hermann action on Y . Besides this action, the induced action of K 1 on Y /K 2 is also h yp erp olar. The resulting spin Sutherland t yp e mo dels ha ve not y et been explored systematically , apa rt from the case [12] of the isotropy action o f K 1 on the symmetric space Y /K 1 arising under K 1 = K 2 . Since they include, in fact, in teresting spinless cases, w e shall return to t his class of mo dels elsewhere. It could be also w orth while to in v estigate the reduced systems induced by o ther p olar actions giv en in [17, 18 ]. Because of the compactness of Y , in t he ab o v e cases the corresp onding Sutherland mo dels in v o lve trigonometric p oten tial functions. Hyp erb olic analo gues of these mo dels can b e deriv ed [13] by starting from non- compact semisimple L ie groups Y . This requires a slight extension 12 of the theory of p olar actions, so as to co v er suitable actions on pseudo-Riemannian manifolds. Rational degenerations of the trigonometric mo dels can b e obtained by Hamiltonian reduction in those cases for whic h the G -action has a fix ed p oin t, p , b y using that in those cases the represen tation of G on T p Y inherits the p olar prop ert y of the o r iginal action. It is an imp o rtan t op en problem whether the fo r ma lism of Hamiltonia n reduction under p olar actions ma y be extended in suc h a w a y to incorp orate also the elliptic Calogero- Sutherland t yp e mo dels. Ac kno wledgemen ts. The work of L.F. w as supp o rted in part by the Hungaria n Scien tific Researc h F und (OTKA gra nt T049495) and by the EU netw ork ‘ENIGMA’ (contract n um- b er MR TN-CT-2004-5652). He thanks S. Ho c hgerner and L. V erho czki fo r useful discussions. B.G.P . is gr ateful to J. Harnad for hospitality in Mon treal. References [1] D. K a zhdan, B. Kostant a nd S. Stern b erg, Hamiltonian gr oup actions and dynamic al sys- tems of Calo ger o typ e, Comm. Pure Appl. Math. 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