Poisson Structures of Calogero-Moser and Ruijsenaars-Schneider Models

arXiv:091 2.3468 P oisson Structures of Calogero-Moser and Ruijsenaa rs-Schneider Mo dels Inˆ es Aniceto, l Jean Av a n c and An tal Jevic ki p l CAMGSD, Dep artamento de Matem´ atic a, Instituto Sup erior T ´ ecnic o, Av. R ovisc o Pais, 1049-001 Lisb o a, Portugal c LPTM, Universite de Cer gy-Pontoise (CNRS UMR 8089), Saint- Martin2 2 avenue A dolphe Chauvin, F-95302 Cer gy-Pontoise Ce dex, F r anc e p Dep artment of Physics, Br own Un iversity, Box 1843, Pr ovidenc e, R I 02912, US A E-mails: ianic eto@math.ist.utl.pt, Je an.Ava n@ u-c er gy.fr & Antal Jevicki@br own.e du Abstrac t: W e exa mine the Hamiltonian structures of s ome Caloge r o-Moser and Ruijsenaar s- Schn e ider N -b o dy int e grable mo dels. W e prop ose explicit formulations of the bihamiltonian structures for the discre te models , and field-theore tica l r ealizations of these structures. W e discuss the relev ance of thes e realizations as co lle ctiv e - field theory for the discrete models . 1 Intro ductio n Bihamiltonian structur es for N -bo dy dynamical s y stems can b e seen a s a dual for m ula tion of integra- bilit y , in the sense that they substitute a hierarch y of compatible P o isson structure s to a hier arch y of commuting Hamiltonia ns, to establish Liouville int e g rability of a given sys tem [ 1 , 2 ] . Our sp ecific int er est for this formulation stems here from the conjecture that, in the case o f N -bo dy Ruijsenaars- Schn e ider (RS) mo del [ 3 ], its higher Hamiltonian structur es may b e the relev ant framework to describ e the dynamics of some magnon-type solutions of string theor y [ 4 , 5 ]. Relev ance o f higher Poisson str uc - tures were demonstrated in the as so ciated Sine- Gordon theory in [ 6 ]. This leads us to a genera l questioning of the bihamiltonia n structures for related integrable discr ete N -bo dy systems, and co ntin uo us realizatio ns ther e of. Mor e sp ecifically: The explicit realiz a tion of the bihamiltonia n str uc tur e for the ratio na l Calog ero-Mose r (CM) mo del [ 1 , 7 , 2 ] 1 is the basis for o ur construction, leading us tow ards our current prop osition of a bihamiltonian structure for A n rational Ruijsenaars-Schneider mo dels, and trigo no metric Calog ero-Mose r mo dels. A t this p oint we wish to make an imp ortant remark: a realiza tion of a bihamiltonian structure was prop osed long ag o for the Relativis tic N-b o dy T o da mo del (see e.g. [ 8 , 9 ]) which is a long-r a nge limit 1 W e would l ik e to thank one of the referees for p oin ting out [ 7 ]. of Ruijsenaars-Schneider dynamics. How ever the difficulty in the full C a logero -Moser a nd Ruijsenaar s- Schn e ider c a se lies (in technical terms) in the dynamical nature of the r -ma trix s tructure which pre- cludes the use of canonical definitio ns a la Sklyanin (such as discussed in [ 8 ]) of the seco nd Hamiltonian structure as a dir ect “qua dratization” of the first Hamiltonian structure encapsula ted in a n y linear r - matrix structure (see also [ 10 , 11 ] and reference s therein). In paralle l we prop ose a r ealization of thes e three bihamiltonian structures in terms of contin uo us field theories, which can b e iden tified, at least in the tw o Ca logero- Moser cases, with the collec tiv e- field contin uous limit of the discrete systems. Identifi c ation in the RS cas e is more questionable and sha ll be a ccordingly dealt with in a further study; we shall only give some co mmen ts ab out it. W e shall successively descr ibe the results for the r ational CM, trig onometric CM and rational RS mo dels. W e denote in the discrete case ” bihamiltonian str uctures” only those pair s of compatible Poisson brack ets ob eying in addition the hiera rch y equatio n: { h n , O } 1 = { h n − 1 , O } 2 , where h n are the tow er of commuting Hamiltonia ns, and O is any obser v able. The biha milto nian str ucture for the discrete r ational CM mo del is describ ed in [ 1 ] and further justified in [ 2 ] b y explicit cons truction of the cor resp onding deformation of the canonica l 1 -form by a Nijenh uis- to rsion free tensor. W e give her e an explicit re a lization of the first tw o Poisson s tructures in terms of a collective field α ( x ). The fir s t one of thes e is the alrea dy known collective-field for mu la tion of the rational CM [ 12 – 14 ]. The forma lis m was recognized as b eing suitable for a useful re pr esentation of higher conse rved charges a nd symmetries of the N-b o dy s ystem [ 14 , 15 ] F or thee second Poisso n bra cket one requir es a defor mation of the Poisson brackets o f α ( x ) to g ether with a change in the r ealization of the v ar iables, under sto o d from the change in the phase-space v o lume elemen t in the co lle ctiv e field formulation, precisely related to the differing Poisson structure. W e then discuss the case o f the trigono metric CM mo del. Based on the identification b etw een the second Poisson structur e s of the ratio nal CM and first P ois son s tr uctures of the trig onometric CM, we prop ose a second Poisson s tructure for the trigono metric CM. A consis ten t formulation in the framework of a contin uous field theory is prop osed in ter ms of a co llective field α ( x ) . The v a lidit y of the hier arch y e q uation for the c orresp onding tw o brack ets is co njectured in the dis c rete c a se from consistency chec ks on the contin uous rea liz ation. W e finally address the case of ratio nal RS mo del. The first Poisson structure on discrete observ ables was der ived rece ntly [ 16 ]; we prop os e her e a direct formulation from the Lax matr ix Poisson structure and its key r , s -matrix formulation. Once again the iden tification of this Poisson s tructure with the second Poisso n structure of rational CM model allows us to prop ose a second Poisson structure for the ra tional RS mo del, with the hiera rch y pro per t y . W e then constr uct a field-theor e tical r ealization of this bihamiltonian structure. Its relev ance as a collective field theory for ra tional RS is, as we have indicated, a delicate issue, essentially p ostp oned until further s tudies. All matrix indices throughout this pap er a re taken to v ary b etw een 1 a nd N for N a given finite int eg er. 2 2 Bihamil tonian Structure for Rational C alogero -Moser This was derived in [ 1 , 2 ]. It is expresse d directly in terms of o bserv ables, r e s pectively I k ≡ 1 k tr  L k  and J ℓ = tr  L ℓ − 1 Q  , where L is the La x matrix and Q is the p osition matrix : L ij = p i δ ij + g ( q i − q j ) (1 − δ ij ) , Q = diag ( q i ) . (1) F rom the firs t canonica l Poisson brack et { p i , q j } 1 = δ ij one g ets the first Poisson br ack et expressio n for the in v ar iant v ariables I k , J ℓ : { I k , I m } 1 = 0 , { I k , J ℓ } 1 = − ( k + ℓ − 2) I k + ℓ − 2 , (2) { J k , J ℓ } 1 = ( ℓ − k ) J k + ℓ − 2 . The second bra ck et is obtained directly b y exploiting the reduction scheme y ielding L and Q from the original matrix v ariables, and the constructio n of an explicit Nijenh uis -torsion fr e e tensor yielding the second Poisson bracket o f T ∗ gl ( n ). It re ads: { I k , I m } 2 = 0 , { I k , J ℓ } 2 = − ( k + ℓ − 1) I k + ℓ − 1 , (3) { J k , J ℓ } 2 = ( ℓ − k ) J k + ℓ − 1 . It is not easy to expr ess { , } 2 in ter ms o f the p, q v ariables , a lthough it may b e a very useful alternative in view of the extension to the trigonometric CM or rational RS mo dels. Remark It is ea sy to chec k (directly) that thes e tw o compatible Poisson br ack et structures are in fact one pair amongst any o ne c ho sen in the following set: { I k , I m } a = 0 , { I k , J ℓ } a = − ( k + ℓ − 2 + a )  1 + λ a k  I k + ℓ − 2+ a , (4) { J k , J ℓ } a = ( ℓ − k ) J k + ℓ − 2+ a , where a is any integer in Z and λ a an arbitrar y c -num b er. Indeed one has: Theorem 1 Any linear combination { , } a + x { , } a ′ with a 6 = a ′ , x ∈ C , yields a skew-symmetric asso ciative Poisson bra ck et. One ha s here a one-par ameter ( λ a ) multihamiltonian str ucture when a ∈ Z . Mo re gener al mixed brack ets { I k , J ℓ } a may b e derived but we have not solved the gene r al cob oundary eq ua tion asso cia ted to it. It will b e imp ortant so on to sp ecify the third Hamiltonia n structure o f the hier arch y sta rting with { , } 0 and { , } 1 . It can b e dir ectly co mputed using the explicit recursio n op er a tor in [ 2 ]. It is unambiguously found to b e given by { , } a with a = 3 and λ 3 = 0. 3 3 Realization of the B ihamilto nian Structure: Col lective Field Theory The collective field theo ry des cribing the N → ∞ co n tinuous limit of the N -site CM mo del was describ ed in [ 17 ]. It is obtained as the r esult of a phase-space in tegra l, ov er the contin uous v e r sion of v ariables p and q , r eplacing the discrete trac e s of p oly no mials of the Lax matrix L (substituted consistently by p ( x )) and p osition matrix Q (substituted by q ( x )). The dyna mical v ariables α ± are ident ified with the end-p oints of the p -int e g ration. Their Poisson brack et structure m us t b e determined by consistency with the orig inal Poisson bra ck et s tructure of the discrete tr aces, precisely I k and J ℓ . The phase space integration how ever implies a subtle redefinition o f the o bserv ables, when higher Hamiltonian s tr uctures are to b e represented, since the inv a riant phase-s pace volume is accor dingly redefined. The first Poisson structur e is describ ed b y [ 14 ] : I k = ´ α dp dq p k k ≡ ´ dx α k +1 k ( k +1) , J ℓ = ´ α dp dq q · p ℓ − 1 ≡ ´ dx x α ℓ ℓ , (5) with the Poisson bra c ket struc tur e for α given by the fir st Poisson structure in KdV: { α ( x ) , α ( y ) } 1 = − δ ′ ( x − y ) (6) It is immediate to chec k that it yie lds pr ecisely the Poisson brack ets { , } 1 . T o obtain the rea lization of the seco nd Poisson structure in ter ms of ” collective” fields we assume that the collective v ar ia bles I k and J ℓ are obtained by a similar integration, o ver a mo dified phase space volume, ta k ing into a c c ount the change in Poisson bra ckets of the same densities I k , J ℓ in terms of p and q . In par ticula r we assume that the deg r ee in p of the densit y yielding resp ectively I k and J k again differ b y one unit. W e a re thus lead to the following ge ne r al for m for the observ ables . I k = ˆ dx α ( x ) k + a f ( k ) , J ℓ = ˆ dx x α ( x ) ℓ + a − 1 g ( ℓ ) (7) and Poisson structure for α , a ssumed to be p olynomial symmetric in α : { α ( x ) , α ( y ) } = − α ( x ) c/ 2 α ( y ) c/ 2 δ ′ ( x − y ) (8) Determination of the num b ers a, c, f ( k ) and g ( ℓ ) follows stra ightf o rwardly from plugging ( 7 ) and ( 8 ) int o ( 3 ), yielding the following results up to a n ov e rall normalizatio n of all k -indexed o bserv ables by a factor λ k − 1 with arbitra ry λ (corr esp onding to a n arbitrary renormalisa tion of α ). The second Poisson str ucture { , } 2 is rea liz ed by: I k = ´ α p − 1 dp dq ≡ ´ dx α k k 2 , J ℓ = ´ α p − 1 dp dq q p ℓ − 1 ≡ ´ dx x α ℓ − 1 ℓ − 1 , with the following Poisson br a ck ets fo r α : { α ( x ) , α ( y ) } 2 = α ( x ) α ( y ) δ ′ ( x − y ) . 4 Notice that the result for the co ntin uo us observ ables is indeed obtained by a change in the phas e space volume dp dq → p − 1 dp dq . Accordingly the canonica l discrete v ar iable b e comes now ln p , and one consistently finds that it is now ln α ( x ) which (in the contin uous limit) has a canonical Poisson bra ck et structure. This Poisson bracket is the third in the KdV hierarch y . It thus seems that the second Poisson brack et of KdV { α ( x ) , α ( y ) } ∼ α ( x ) 1 / 2 α ( y ) 1 / 2 δ ′ ( x − y ) do es not play a role in the CM fra mew o rk. 2 Also notice that a lthough the second discrete Poisson bra ck et do es r ealize the hierarchy pro per t y and is therefor e cor rectly identified as THE second Poisson br a ck et in the rationa l CM biha milto nia n hierarch y the ” second” contin uous Poisson bracket is not so, since not o nly the field br ack et but also the definition of the o bserv ables has to b e changed. If for consistency one compute the Poisson br ack et of the same v ariables in terms of contin uo us fields, it yields instead  h n , ˆ dx x α ℓ − 1 ℓ − 1  2 ≡  h n +2 , ˆ dx x α ℓ − 1 ℓ − 1  1 , exhibiting a shift o f 2 in the degree of Hamilto nian, from which one inescapa bly concludes that the contin uous r ealization of the second discrete Hamiltonian structure fo r r ational CM is in fact a third Hamiltonian s tructure fo r the co lle ctiv e field theor y . The s econd Hamiltonia n structure of the latter corres p onds obviously to the second KdV bra cket, and is see mingly (as we hav e said) not manifest in the discrete CM frame. 4 T rigono metric Cal ogero-M oser Mo del An alg ebra o f obser v ables for the discrete CM trigono metric mo del is written [ 18 ] in ter ms o f the co ordinate matrix 3 K = P j exp( q j ) e j j and Lax matrix L = P i p i e ii + P i 6 = j g cos( q i − q j ) sin( q i − q j ) e ij using the first canonical Poisson structure { p i , q j } = δ ij . The ov erc o mplete set of o bserv ables  W mn = T r L m e nQ , m ≥ 0 , n ≥ 0  is ea s ily shown to r ealize a W 1+ ∞ algebra (alb eit in a very deg enerate re pr esentation due to the existence o f algebra ic relations b etw een the W mn issuing from their realiz a tion as N xN matrices): { W mn , W pq } 1 = ( mq − n p ) W m + p − 1 ,n + q + lower-order terms. In order to define a second Poisson structure, following the prev io us deriv atio n, we shall use as independent v ariables not the ( p i , q j ) but a subset of algebr aically indep endent observ ables from the set { W mn } such that the change of v a r iables b e bijective (at least from a given W ey l cham b er for the p osition a nd momen ta v ariables, since the disc rete p ermutation o ver indices is facto r ed out by the use o f inv ar iant traces). Guided by the discus s ion in [ 2 ] we see that either { W m 0 , W m 1 , m ≤ N } or { W 0 m , W 1 m , m ≤ N } pr ovide such a subset. Using the firs t subset seems a pr iori natura l since it contains the Hamiltonians W l, 0 . Howev er { W m 1 , W p 1 } 1 = ( mq − np ) W m + p − 1 , 2 + low er-or der terms.. It is in pr inciple p o ssible to r e - express W m + p − 1 , 2 in terms o f W k, 1 and W l, 0 since these second-index 0 and 1 observ ables provide an alg e braically complete set o f new v ariables. How ever this re- expression is exp ected to be quite cumbersome: in particular it will certainly yield no n-linear expr essions, sug gesting that a consistent guess o f compa tible s econd Poisson brack et will b e difficult to for m ula te. 2 Here we are referring to the long wa ve l ength li mit of the second Poisson br ac ket, not the full Poisson br ac ket. 3 W e should int r oduce some notation here. The matrix e ij is a matrix whi c h has all element s zero except the element ij . 5 The second set how ever clos e s linear ly a nd e xplicitly under the first Poisson bracket and it is thus this one which we c ho o se to define the Poisson hiera r ch y . It is also crucial to no te that no lower-order term app ears in its Poisson brackets. It then turns out by simple insp ection that the first Poisson structure for trigo nometric CM expr e ssed in terms of v ar iables W 0 , 1; m is isomo r phic to the seco nd Poisson structure for ra tional CM. It thus seems natural to pro p os e as a second Poisson s tructure for trigonometric CM the third Poisson str ucture o f ra tional CM. In terms of W - v ariables it ea sily r eads: { W im , W j n } 2 = ( in − j m ) W i + j − 1 ,m + n +1 . (9) This character iz es { , } 1 and { , } 2 as a pair of compatible Poisson structur e s for tr igonometric CM mo del. How ever, in order to further characterize { , } 1 and { , } 2 as a bihamiltonia n structure for the trigonometric CM mo del, we need to prov e that it realize s the hier arch y equality for evolution o f observ ables: { W m 0 , W in } 2 ≡ { W m +1 , 0 , W in } 1 , i = 0 , 1 , m ≤ N . This is not easy since it implies tha t one is able to compute the seco nd Poisson bracket ot the v ariables W m 0 , once again a difficult task given that they are redunda n t v a riables and we do not cont r ol the low er-o rder terms. W e s hall now use the co llective field description o f the co n tinuous limit to at lea st establish the consistency of this sta tement. 5 Realization: Co ntinuous T rigono metric Calo gero-Mo ser Mo del It is k nown that for a particular v alue of the coupling constant the trig o nometric CM mo del is equiv- alent, a t the contin uum level, to a fre e fermion on a circle [ 1 7 ]. This sugges ts that the collec tiv e field theory for trigono metric CM should aga in b e expr essed as a phase space int e g ral, this time ov er a circle in the q v aria ble, yielding the realization o f the first Poisson struc tur e as: W 0 m = T r e mQ bec omes W 0 m = ´ dx e mx α ( x ) , W 1 m = T r e mQ L becomes W 1 m = ´ dx e mx α ( x ) 2 2 , and generically : W nm = T r e mQ L n bec omes W nm = ˆ dx e mx α ( x ) n +1 n + 1 with the Poisson bra c ket { α ( x ) , α ( y ) } 1 = δ ′ ( x − y ). This set o f int eg rated collective-field densities realizes indeed the leading (linear) order o f the Poisson brack et algebra for the discrete W mn generator s under the first Poisson br ack et. No te that a similar pro per t y already held in the ratio na l case , when one extended the Poisson a lgebra to the redundant discrete genera tors T r L m Q n , realize d in the contin uum limit a s ´ dx x m α n +1 n +1 . Realization of the seco nd Poisso n structure is, stric tly s pea king, only av aila ble at this stage for the generator s . W 0 m , W 1 m W e ass ume as a generic fo r m for this rea lization the following monomial int eg rals: W im = ˆ dx e ( m + a ) x α ( x ) i +1+ b i + 1 + b . Indeed this is the o nly wa y to guara n tee that the separate additivity (up to a cons tan t!) o f the indice s i and m will b e preser ved in the formulation o f the Poisson a lgebra. The Poisson str ucture for the field 6 α is taken to b e the most generic symmetric monomial expre s sion in α and e x { α ( x ) , α ( y ) } = e c 2 ( x + y ) ( α ( x ) α ( y )) d/ 2 δ ′ ( x − y ) . Plugging these ans ¨ atze for W 0 m , W 1 m int o the exp ected algebr aic str ucture yields a unique ans w er : a = − 1 , c = 2 , b = 0 , d = 0 . In par ticular o ne re ma rks that it is the new ˜ α ( x ) ≡ e − x α ( x ) which now realizes a c anonical Poisson brack et { ˜ α ( x ) , ˜ α ( y ) } 2 = δ ′ ( x − y ). Because this rea lization is unique, and completely determined b y the Poisson bracket s o f the inde- pendent genera tors W 0 m , W 1 m , it see ms acceptable to conjecture that it will entail a similar realiza tion for the redundant higher- order generator s W nm , n ≥ 2 . F r om o ur previo us conjecture they a r e repre - sented as: W nm = ˆ dx e ( m − 1) x α ( x ) n +1 n + 1 . W e can now compute at lea st the leading o r der o f the actual Ha miltonian ac tio n on these conjectured contin uous observ ables, implied by the second Poisson structur e: { W n 0 , W im } contin uou s (2) = nm W n + i − 1 ,m + n +1 . If a s we have conjectured, this representation is indeed the contin uous representation of the seco nd Poisson structur e o n a ll the o bs erv ables of the trigo no metric CM mo del, this equatio n g uarantees that, at the discrete level, we hav e: { W n 0 , W im } discrete (2) = nm W n + i − 1 ,m + n +1 = { W n +1 , 0 , W im } discrete (1) up to lower-order terms, which are in a n y case not a ccessible to the contin uous representation. Therefore is is not inco nsistent to characterize { , } discrete (2) as a second Hamiltonian structur e in a multihamiltonian hierarch y for the trigono metric CM. 6 Bihamil tonian Structure for Rational Rui jsenaa rs-Schnei der M o del A consistent construction of a bihamiltonia n structure ca n b e formulated on the following lines . a. The c a nonical Poisson structure in terms of the ba sic v aria bles p and q is ag ain r e-expressed as a Poisson structure for the following v ariables: I k = T r L k k , J ℓ = T r Q L ℓ − 1 , (10) where L is the Lax ma trix for ra tional RS and Q = dia g ( q i ) a s b efor e. Direct deriv ation o f the Poisson structure for these observ ables now follows fro m the r matrix structure of the rationa l RS Lax ma tr ix L . It is given by L = N X k,j =1 γ q k − q j + γ b j e kj , b k = e p k Y j 6 = k  1 − γ 2 ( q k − q j ) 2  1 / 2 . (11) 7 The matrix e kj is the N × N matrix with a ll c ompo nent s ze r o except the k j co mpo nen t, which is one. The canonica l Poisson bra ck et in the canonical v ariables q k , p j : { p k , p j } 0 = { q k , q j } 0 = 0 , { q j , p k } 0 = δ kj . (12) bec omes, in the q k , b j v ariables: { q k , q j } = 0 , { q k , b j } = b k δ kj , (13) { b k , b j } =  1 q j − q k + γ − 1 q k − q j + γ + 2(1 − δ kj ) q k − q j  b k b j . This Poisson bra c ket is quadr atic in the Lax matrix [ 19 ] L :  L 1 ⊗ , L 2  = a 12 L 1 L 2 − L 1 L 2 d 12 − L 1 s 12 L 2 + L 2 s 21 L 1 . (14) where: d 12 = − a C M 12 − w , a 12 = − a C M 12 − s C M 12 + s C M 21 + w, (15) s 21 = s C M 12 − w , s 12 = s C M 21 + w , The following tensors alr eady existed in the Caloger o-Moser case [ 20 ]: 4 a C M 12 = − X k 6 = j 1 q j − q k e j k ⊗ e kj , (17) s C M 12 = X k 6 = j 1 q j − q k e j k ⊗ e kk . They a ctually also define [ 19 ] the famo us no n- skew symmetric dynamica l r - matrix of the ra tio nal CM model by r C M 12 = a C M 12 + s C M 12 . (18) The tensor w in ( 15 ) only app ears in the RS mo del, it is defined by w = X k 6 = j 1 q k − q j e kk ⊗ e j j . Finally one can see that the tensor s in ( 15 ) ob ey the class ic al co ns istency r e la tion a 12 − d 12 + s 21 − s 12 = 0 . (19) 4 F or the r ational CM model we hav e the Lax matrix L r = N X k =1 p k e kk + N X k 6 = j γ q k − q j e kj , (16) 8 which we shall see to be nece s sary for Poisson-commutation of the tr a ces. W e also deter mine the Poisson brack ets of the Lax o per ator with the po sition op erator Q , defined as Q = X k q k e kk , (20) obtaining  L 1 ⊗ , Q 2  = X i,j,k { L ij , q k } e ij ⊗ e kk = X i,j,k γ q i − q j + γ { b j , q k } e ij ⊗ e kk = − X i,j L ij e ij ⊗ e j j = − L r 1 · X j e j j ⊗ e j j . (21) W e now r e - write the ab ov e Poisson brack ets using as basic v aria bles the following traces: W m n = tr ( L n Q m ) , m = 0 , 1 . (22) The simplest of the Poisson bra ck ets is :  W 0 n , W 0 m  = tr 1 , 2 n,m X i,j =1  L 1 ⊗ , L 2  L n − 1 1 L m − 1 2 = m n tr 1 , 2 (( a 12 − d 12 − s 12 + s 21 ) L n 1 L m 2 ) = 0 , where we used the key consistency relation ( 19 ). The next Poisson br ack et to b e determined is  W 0 n , W 1 m  = tr 1 , 2   X i,j  L 1 ⊗ , L 2  L n − 1 1 L j − 1 2 Q 2 L m − j 2   | {z } A 01 + tr 1 , 2 n X i =1 L m 2 L n − i 1  L 1 ⊗ , Q 2  L i − 1 1 ! | {z } B 01 . The first term is re-wr itten as A 01 = tr 1 , 2    n ( a 12 − s 12 ) L n 1 [ L m 2 , Q 2 ] + m X j =1 L j 2 Q 2 L m − j 2 [ s 12 , L n 1 ]    . W e o nce again ma de use of the cyclicity o f the trace and of the r elation ( 19 ). If we now use the explicit formulas ( 15 ), we find that a 12 − s 12 = − a C M 12 − s C M 12 = − r C M 12 , where the sup e r script C M cor resp onds to the Caloge ro-Moser mo del. Then we wr ite tr 1 , 2   m X j =1 L j 2 Q 2 L m − j 2 [ s 12 , L n 1 ]   = m X j =1  L j QL m − j  lk ( s 12 ) i ′ j ′ kl L n mn ( δ j ′ m δ i ′ n − δ i ′ n δ j ′ m ) = 0 . This allows us to simplify A 01 even further: A 01 = tr 1 , 2  n ( − r C M 12 ) L n 1 [ L m 2 , Q 2 ]  = − n  L n j i ( r C M 12 ) ij m l L m lm ( Q mm − Q ll )  . 9 Using the express ion for r C M 12 in comp onents we fina lly find A 01 = − n X m 6 = l L n ml L m lm = − n tr  L m + n  + n X k L n kk L m kk . Let us now turn to the sec ond term of the Poisson bra ck ets B 01 : B 01 = tr 1 , 2  n L m 2  L 1 ⊗ , Q 2  L n − 1 1  = − n X j L n j j L m j j . The final result for this Poisson bra ck et is just  W 0 n , W 1 m  = − n tr  L m + n  = − n W 0 m + n . ( 2 3) The final Poisson bracket to determine is  W 1 n , W 1 m  = tr 1 , 2   n,m X i,j =1  L 1 ⊗ , L 2  L i − 1 1 Q 1 L n − i 1 L j − 1 2 Q 2 L m − j 2   | {z } A 11 + + tr 1 , 2   m X j =1 L n 1  Q 1 ⊗ , L 2  L j − 1 2 Q 2 L m − j 2   | {z } B 11 + tr 1 , 2 n X i =1 L m 2  L 1 ⊗ , Q 2  L i − 1 1 Q 1 L n − i 1 ! | {z } C 11 . First of all A 11 is A 11 = tr 1 , 2   n X i =1 ( a 12 − s 12 ) L i 1 Q 1 L n − i 1 [ L m 2 , Q 2 ] + m X j =1 ( a 12 + s 21 ) [ L n 1 , Q 1 ] L j 2 Q 2 L m − j 2   + + tr 1 , 2 (([ Q 1 , d 12 ] Q 2 + [ Q 2 , d 12 ] Q 1 + Q 2 [ Q 1 , s 12 ] − Q 1 [ Q 2 , s 21 ]) L n 1 L m 2 ) . T o simplify this express ion, we ne e d a few extra results. The first one is tr 1 , 2 (([ Q 1 , d 12 ] Q 2 + [ Q 2 , d 12 ] Q 1 ) = 0 due to the cyclicity of the tra ce. The second one is tr 1 , 2 ( Q 2 [ Q 1 , s 12 ] L n 1 L m 2 ) = ( L m Q ) ij ( s 12 ) klj i L n lk ( Q kk − Q ll ) = 0 , where we hav e used that s 12 = s C M 21 + w from ( 15 ): the w contribution is zero, b ecause this tenso r is diagona l on b oth spaces 1 and 2; the co ntribution fro m s C M 21 is a ls o zer o due to this tensor b eing diagonal in the first space 1 . A very similar res ult ca n be obtained for : tr 1 , 2 ( Q 1 [ Q 2 , s 21 ]) L n 1 L m 2 ) = 0 , but in this case one would need to use s 21 = s C M 12 − w fro m ( 1 5 ). 10 With these results, A 11 bo ils down to A 11 = tr 1 , 2   − n X i =1 r C M 12 L i 1 Q 1 L n − i 1 [ L m 2 , Q 2 ] + m X j =1 r C M 21 [ L n 1 , Q 1 ] L j 2 Q 2 L m − j 2   . In this last expressio n we a gain us e d the relations directly derived fro m ( 15 ) a 12 − s 12 = − r C M 12 , a 12 + s 21 = r C M 21 . The t wo terms in A 11 are further simplified by the use of r C M 12 in comp onents: − n X i =1 tr 1 , 2  r C M 12 L i 1 Q 1 L n − i 1 [ L m 2 , Q 2 ]  = − n X i =1 X k 6 = l  L i QL n − i  lk L m kl = − n tr  QL m + n  + n X i =1 X k  L i QL n − i  kk L m kk , and likewise m X j =1 tr 1 , 2  r C M 21 [ L n 1 , Q 1 ] L j 2 Q 2 L m − j 2  = m tr  QL m + n  − m X j =1 X k  L j QL m − j  kk L n kk . Finally A 11 bec omes simply A 11 = ( m − n ) tr  QL m + n  + n X i =1 X k  L i QL n − i  kk L m kk − m X j =1 X k  L j QL m − j  kk L n kk . W e still have to determine the other terms B 11 and C 11 . Let us pro cee d with B 11 : B 11 = m X j =1 tr 1 , 2 L n 1 L 2 · X k e kk ⊗ e kk · L j − 1 2 Q 2 L m − j 2 ! = m X j =1 X k L n kk  L j QL m − j  kk . In order to obtain the la s t line, we hav e used the fact that X k L n kk ( QL m ) kk − X k L n kk ( L m Q ) kk = X k L n kk Q kk L m kk − X k L n kk L m kk Q kk = 0 . T urning to C 11 one similarly obtains C 11 = − n X i =1 X k L m kk  L i QL n − i  kk . W e finally write the r esult for the Poisson bracket:  W 1 n , W 1 m  = A 11 + B 11 + C 11 = ( m − n ) tr  QL m + n  = ( m − n ) W 1 m + n . 11 Summarizing the results obtained for the Poisson brack ets of the traces, we hav e for the rationa l RS mo del:  W 0 n , W 0 m  1 = 0 ,  W 0 n , W 1 m  1 = − nW 0 m + n , (24)  W 1 n , W 1 m  1 = ( m − n ) W 1 m + n . Renormalizing the v ariables W 0 , 1 n to our v ariables I k , J ℓ , by I k = 1 k W 0 k , J ℓ = W 1 ℓ − 1 . we get: { I k , I ℓ } 1 = 0 , { J ℓ , I k } 1 = ( k + ℓ − 1) I k + ℓ − 1 , (25) { J ℓ , J m } 1 = ( m − ℓ ) J m + ℓ − 1 . Another deriv ation of these Poisso n structure was recently given [ 16 ], using the realiza tio n of the RS model by KKS reductio n [ 2 1 ], ther e by b y passing the explicit use of r matrix str ucture. The key rema r k here is tha t this canonica l (firs t) bracket for the rational RS is isomorphic to the second br ack et { , } 2 (with λ 2 = 0) for the r ational CM. This is c o nsistent with the remar k in [ 2 ] on the formal equality of the ca nonical symplectic form o n T ∗ GL ( n, C ) yielding the first Poisson structure o f trigonometric CM mo del, with the relev ant symplectic fo r m yielding the second brack e t fo r the ra tional CM mo del; together with the well-known Ruijsenaa rs duality b etw een trigonometr ic CM and rational RS, certainly v alid at least when the firs t Poisson s tr uctures are considered in b oth formulations. b. Even though a direct computation o f the new symplectic for m deformed by a Nijenh uis- torsion free tensor (i.e. the new ca nonical 1 - form) is no t av ailable for rational RS (la cking an obvious choice of such Nijenhu is -torsion free tensor), we how ever prov e , in view of the explicit computations of Section 2, that the natura l second Poisson bra c kets for the ratio na l RS hie r arch y are expressed in terms of the observ ables I k , J ℓ , by the for m o f the third Poisson brackets for rational CM written there. Precisely: { I k , J ℓ } 2 = 0 , { J ℓ , I k } 2 = ( k + ℓ ) I k + ℓ , ( 2 6) { J ℓ , J m } 2 = ( m − ℓ ) J m + ℓ +1 . Pro of: 1. { , } 2 is compatible with { , } 1 as a Poisson bra ck et structur e for the observ ables I k , J ℓ of RS since the Jacobi identit y equatio ns for { , } 2 + x { , } 1 are the same as for { , } CM 3 + x { , } CM 2 . 2. W e hav e the following r e la tion { J k , I ℓ } 2 ≡ d (2) dt ℓ J k = { J k , I ℓ +1 } 1 = d (1) dt ℓ +1 J k 12 which now characterizes { , } 1 , { , } 2 as a b ona fide bihamiltonian struc tur e for the RS hierar ch y , defined by the set of Hamiltonians { I ℓ } . 7 Field-theo r etical Realization of the Ruijsenaars-Schneider Structures W e now prop ose from first principle s a field-theor etical r ealization of the tw o (bihamiltonian) Poisson structures previously computed for the ra tional RS mo dels. The first bracket is r ealized a s: I k = ˆ dq e kα k 2 , J ℓ = ˆ dq q e ( ℓ − 1) α ℓ − 1 , with Poisson br ack et { α ( x ) , α ( y ) } = δ ′ ( x − y ) . The exp onential re pr esentation in α is motiv ated by the existence of the Ruijsenaa rs duality b etw een rationa l RS and trigonometr ic CM [ 22 ] under exchange of the v ariables p , q . Acco r dingly , it app ears consis ten t to as sume a dualized ( x ↔ α ) r epresentation in the contin uum case for the Poisson str ucture. The s econd Poisson structur e is now realized in the contin uum, following a similar scheme as in the rational and trigo nometric CM case. Assuming that a repres e ntation purely in e α ( x ) will ho ld for the p v ariables, one int r o duces as an ansatz for the obser v ables the gener ic form: I k = ˆ dq e ( k + a ) α k + a , J ℓ = ˆ dq q e ( ℓ + a − 1) α ℓ + a − 1 and similarly for the Poisson bra c ket { α ( x ) , α ( y ) } = e c 2 ( α ( x )+ α ( y )) δ ′ ( x − y ) . The ex po ne ntial form fo r α in the Poisson brack e ts is re q uired by the exp onential fo r m in I k and J ℓ , which must b e preser ved under Poisson br ack et, to yield again I - and J - genera tors. Plugging these ans ¨ atze in the second Poisson bra ck et str ucture unambiguously yields a = − 1 a nd c = 2. F rom c = 2 it is now seen that φ ( x ) ≡ e − α ( x ) is a canonical field, { φ ( x ) , φ ( y ) } = δ ′ ( x − y ). As in the case of ratio nal C M model, this field-theoretical rea lization is b etter in ter pr eted a s a third Poisson brack et for the contin uous theory since one obtains a gain a Hamiltonian evolution with a shift of 2 units in the degree: { h n , O } (2) contin uou s ≡ { h n − 2 , O } (1) contin uou s , setting h n ≡ ´ dq e nα n in both cases, as consistency requires . The is sue is now whether this field-theoretical r ealization can b e obtained directly a s a genuine col- lective field theory for RS mo del. This r equires a re-wr iting of the op era tors I k , J ℓ from a collec tive field theory p ersp ective, and from that a determination of the Poisson structures that arise, thus confirming our ans ¨ atze for these. A collective ex pression for I 1 is known [ 23 ], in terms of qua n tum Ma c Donald op erators , a nd one would like to extend the a na lysis do ne in [ 2 3 ] to higher conserved quantities I k and J ℓ , from their ex pr essions in comp onents found in [ 22 ]. Such a generaliza tion, how ever, is not trivial to obtain, b e cause higher p ow ers of the Lax matr ix make such calculations very cumberso me. Before concluding let us r emark that the c a se of trigonometric RS mo del is muc h more problematic to deal with at this time, due to the difficulty of defining a Poisson-clo sed co mplete subalgebra o f observ ables which could b e used as suitable co or dinates. In this ca se the first Poisson structure o f neither the set { W m 0 , W m 1 , m ≤ N } nor the dual se t { W 0 m , W 1 m , m ≤ N } closes linearly . Indeed one 13 has:  W 1 n , W 1 m  1 = ( m − n ) W 2 m + n + .... and { W n 1 , W m 1 } 1 = ( m − n ) W n + m 2 + .... The difficult y which lead us to eliminate the c hoice of the set { W m 0 , W m 1 , m ≤ N } in the trigonometric CM ca se exists now for b oth sets. 8 Summary T o conclude, we summar iz e the r e sults obtained here and the remaining iss ues r egarding the construc- tion of multihamiltonian structures for the N -b o dy mo dels, their realiz a tion in co n tinuous field theor ies and int er pretation of those as collective field theor ies. A n Calogero-Mos er, rational: Bihamiltonia n structure were a lready known in discrete cas e [ 1 , 2 ]. A collective-field r ealization is prop osed, with consistent “bihamiltonia n” structure and consistently mo dified phase space. A n Calogero-Mos er, trig onometric: Multiple Poisson structures hav e been established. Consistent collective-field r ealizations are prop osed, with consistent “bihamiltonian” structure. The hier arch y equations for the multiple discr ete Poisson structures hav e not b een rig orously established but pass consistency chec ks. A n Ruijsenaars-Sc hnei der, rational: A bihamiltonian structure is established in the discrete ca s e. A contin uum realizatio n is pr op osed, with bihamiltonian structure; ident ific a tion as collective field theory is yet unpr oven. Ackno wl edgements This work was pa rtially funded by CNRS (J.A.) and the Depar tmen t of Energy under contract DE- F G0 2-91E R40688 (I.A. and A.J.); I.A. was a lso pa r tially s upp or ted by the F unda¸ c˜ ao par a a Ciˆ e ncia e a T ecnologia (FCT / Portugal). J.A. wishes to warmly thanks Br own University Physics Department for their hospitality . I.A. would like to tha nk A. Jevicki for the supp or t fo r this work during the summer of 2009. The authors would also like to thank one o f the r eferees and V. 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