Quantum SU(2 1) supersymmetric Calogero Moser spinning systems

Transcription

1 Quntum SU 1 supersymmetrc Clogero Moser spnnng systems rxv: v hep-th 9 Apr 018 SergeyFedoru, EvgenyIvnov, Olf Lechtenfeld b, StepnSdorov Bogolubov Lbortory of Theoretcl Physcs, JINR, Dubn, Moscow regon, Russ fedoru@theor.jnr.ru, evnov@theor.jnr.ru, sdorovstepn88@gml.com b Insttut für Theoretsche Phys nd Remnn Center for Geometry nd Physcs Lebnz Unverstät Hnnover, Appelstrsse, Hnnover, Germny lechtenf@tp.un-hnnover.de Abstrct SU 1 supersymmetrc mult-prtcle quntum mechncs wth ddtonl sem-dynmcl spn degrees of freedom s consdered. In prtculr, we provde n N = 4 supersymmetrzton of the quntum U spn Clogero Moser model, wth n ntrnsc mss prmeter comng from the centrlly-extended superlgebr ŝu 1. The full system dmts n SU 1 covrnt seprton nto the center-of-mss sector nd the quotent. We derve explct expressons for the clsscl nd quntum SU 1 genertors n both sectors s well s for the totl system, nd we determne the relevnt energy spectr, degenerces, nd the sets of physcl sttes. PACS: w, Pb, 1.60.Jv, Ds Keywords: supersymmetry, superfelds, deformton, supersymmetrc mechncs

2 1 Introducton The mny-prtcle Clogero-Moser systems 1,, 3, 4, 5 nd ther generlztons occupy dstngushed plce n the contemporry theoretcl nd mthemtcl physcs. Aprt from such notble mthemtcl propertes, s the clsscl nd quntum ntegrblty, these systems possess wde rnge of physcl pplctons whch re hrd to enumerte. Among these pplctons, t s worth to menton, e.g., close connecton between the lgebr of observbles n the Clogero system nd the hgher-spn lgebr, s ws ponted out n 6, 7. In the sme ppers, there ws reveled n mportnt role of Clogero-le models for descrbng prtcles wth frctonl sttstcs. Another wdely nown pplctons of the Clogero-Moser systems concern the blc hole physcs. It ws suggested 8 tht the Clogero-Moser systems cn provde mcroscopc descrpton of the extreme Ressner-Nordström blc hole n the nerhorzon lmt. It ws rgued tht, from the M-theory perspectve, n mportnt role n ths correspondence should be plyed by vrous N = 4 supersymmetrc extensons of the Clogero- Moser models. Supersymmetrc Clogero-Moser systems hve lso further pplctons n strng theory see, for exmple, 9 nd N=4 super Yng-Mlls theory 10, 11. Keepng n mnd these physcl nd mthemtcl motvtons, t seems of gret nterest to construct nd study new versons of supersymmetrc Clogero-type systems. In recent pper 1, there ws proposed the superfeld mtrx model of SU 1 supersymmetrc mechncs 1 s new N=4 extenson of d = 1 Clogero Moser mult-prtcle system. Ths mtrx model s mssve generlzton of the multprtcle N = 4 model constructed nd studed n 1,. It s nturlly formulted n d = 1 hrmonc superspce 3 nd s descrbed by the followng set of N=4 hrmonc superfelds: n commutng generl superfelds X b = X b,,b = 1,...,n combned nto n hermtn n n-mtrx superfeld X = X b whch trnsform n djont representton of Un nd represent off-shell SU 1 multplets 1, 4, 3; n commutng nlytc complex superfelds Z + formng Un spnor Z + = Z +, Z+ = Z + nd representng off-shell SU 1 multplets 4,4,0; n non-propgtng nlytc topologcl guge superfelds V ++ = V ++b, Ṽ ++b = V ++ b. The mtrx superfeld X = Xζ H s defned on the SU 1 hrmonc superspce ζ H ta,θ ±, θ ±,w ± = 1,, whle the nlytc superfelds Z +, Z+ nd V ++ on the nlytc hrmonc subspce ζ A = t A, θ +,θ +,w ± ζh. The relevnt superfeld cton s wrtten s S mtrx = 1 µ H Tr X + 1 µ A 4 V Z 0 + Z + + c µ A TrV ++, 1.1 where the nvrnt ntegrton mesures re wrtten s µ H = dwdt A d θ dθ d θ + dθ + 1+mθ + θ mθ θ+, µ A = dwdt A d θ + dθ ThsndofdeformedN=4supersymmetrcmechncswsntroducedndstudedn13,14,15,16,17,18. Some of the SU 1 models cn be derved by dmensonl reducton from N=1 Lgrngns on the curved d = 4 mnfold R S 3 19, 0. 1

3 The mss-dmenson prmeter m s encoded n the centrlly-extended superlgebr ŝu 1 s the contrcton prmeter to the flt N=4,d = 1 superlgebr. It does not explctly pper n 1.1 but comes out n the component cton from the mesure µ H nd the θ-expnson of the superfelds X s result of solvng the pproprte SU 1 covrnt constrnts for more detls, see 1, 16.The locl Un trnsformtons of the nvolved superfelds re gven by X = e λ Xe λ, Z + = e λ Z +, V ++ = e λ V ++ e λ e λ D ++ e λ. 1.3 The superfeld V 0 ζ A s prepotentl for the snglet prt TrX of the mtrx superfeld X see detls n 1. The constnt c n 1.1 s prmeter of the model. After quntzton, t specfes externl SU spns of the physcl sttes, c s+1 Z >0, whch mples tht the set of these sttes splts nto rreducble SU multplets. The mtrx d = 1 superfeld X hs the physcl component felds X = X b = X, Ψ = Ψ b nd uxlry bosonc component felds, the superfelds Z +, Z + hve the bosonc components Z = Z, Z = Z = Z nd uxlry fermonc felds. Choosng the WZ guge V ++ = θ + θ+ At A, elmntng uxlry felds nd redefnng the spnor felds s Z Z /TrX1/, we obtn from 1.1 the on-shell component cton S mtrx = S b +S f, 1.4 S b = 1 Tr dt X X m X c dttra + dt Z Z Z Z + dt S S, 1.5 4X 0 S f = 1 Tr dt Ψ Ψ Ψ Ψ +m Ψ Ψ dt Ψ 0 Ψ 0 S. 1.6 X 0 Here, X 0 := 1 n TrX, Ψ 0 := 1 n TrΨ, Ψ 0 := 1 n Tr Ψ, S := Z Z := Z Z, 1.7 nd Z Z := Z Z. The Un guge-covrnt dervtves n 1.5, 1.6 re defned by X = Ẋ +A,X, Ψ = Ψ +A,Ψ, Ψ = Ψ +A, Ψ. 1.8 Z = Ż +AZ, Z = Z Z A. 1.9 The bsc novel feture of the cton 1.4 s compred to the more conventonl ctons of supersymmetrc mechncs s the presence of the sem-dynmcl spn vrbles Z 1,3 whch hs drstc mpct on the structure of the relevnt spce of quntum sttes. These vrbles Itwsshownn17,thtthecentrllyextendedsuperlgebrŝu 1cnberepresentedssem-drectsum of su 1 nd n extr R-symmetry genertor: ŝu 1 su 1+ u1. The centrl chrge s combnton of the R-symmetry genertor nd the nternl U1 genertor of su 1. In the models under consderton the centrl chrge opertor s dentfed wth the cnoncl Hmltonn. 3 The netc term of the vrbles Z n the cton 1.5 s of the frst-order n the tme dervtves, n contrst to the dynmcl vrble X b wth the second-order netc term. Just for ths reson we cll Z, Z sem-dynmcl vrbles. In the Hmltonn see below, they pper only n the ntercton terms nd enter through the SU current S.

4 defne extr SU symmetres wth the genertors 1.7, wth respect to whch the physcl sttes crry ddtonl spn quntum numbers nd so form the pproprte SU multplets. The dgonl su lgebr s n essentl prt of the nternl lgebr su ŝu 1. Also, note the presence of the osclltor-type terms n 1.5 nd 1.6, wth the ntrnsc prmeter m s the relevnt frequency. The smplest one-prtcle n=1 cse of the system 1.4 ws quntzed n recent pper 4. Here we consder the quntum verson of the system 1.4 for n rbtrry n. As shown n 1, t the clsscl level the system 1.4 descrbes n SU 1 supersymmetrc extenson of the U-spn Clogero Moser model 5, 6, 7, 8, 9, 30 generlzng the Clogero Moser system of refs. 1,, 3, 4, 5 to the cse wth ddtonl nternl spn degrees of freedom. Therefore, the bsc purpose of the present pper cn be formulted s constructon of new quntum mult-prtcle spnnng Clogero Moser type system wth deformed N=4,d = 1 supersymmetry. The quntzton of the Clogero-type mult-prtcle systems cn be ccomplshed by the two methods, bsclly ledng to the sme result. One method 31, 6, 7, 8, 7, 30 s bsed on the constructon of the Dunl opertors for gven system. Usng such opertors mes t possble to represent multprtcle system s n osclltor-le system for whch the Dunl opertors ply the role of generlzed momentum opertors. Another wy of quntzng multprtcle systems s bsed on consderng mtrx systems wth ddtonl guge symmetres 34, 7, 8, 35, 36, 9, 30. The elmnton of some degrees of freedom n such mtrx systems results n the stndrd mult-prtcle Clogero-type systems. Due to the osclltor nture of mtrx opertors, the quntzton of mtrx systems s smpler nd the mn ts of ths pproch conssts n fndng solutons of the constrnts genertng guge symmetres. In ths pper, we wll mnly stc to the second method. We wll present the explct expressons of the mult-prtcle opertors of deformed N = 4 supersymmetry, n the mtrx cse nd for the reduced system. The pln of the pper s s follows. In Secton we construct the Hmltonn formlsm for the mtrx system1.4 nd show tht the model ndeed descrbes SU 1 supersymmetrzton of the U-spn Clogero Moser model 5, 6, 7, 8, 9, 30. In Secton 3 we fnd, by Noether procedure, the superchrges of the underlyng ŝu 1 superlgebr, n mtrx cse nd for system wth the reduced phse vrbles spce. In the ltter cse ŝu 1 s closed up to the constrnts genertng some resdul guge nvrnces. In Secton 4 we construct quntum relzton of the deformed N = 4, d = 1 superlgebr ŝu 1 for the mult-prtcle Clogero Moser system. In the cse of the reduced system wth n bosonc poston coordntes such superlgebr s closed up to the genertors of the U1 n guge symmetry, le n the clsscl cse. Ths ŝu 1 superlgebr s represented s sum of two ŝu 1 superlgebrs. One ŝu 1 cts n the center-of-mss sector, wheres the other opertes only on the supervrbles prmetrzng the quotent over ths sector. The spn opertors re common for both these superlgebrs. In Sectons 5-7 we nlyze the energy spectrum n ll cses: for the center-of-mss subsystem, for the system wth reltve supercoordntes nd n the generl cse, when ll poston opertors re ncluded. The lst Secton 8 contns Summry nd outloo. 3

5 Hmltonn nlyss nd guge fxng The cton 1.4 yelds the cnoncl Hmltonn where H totl = H mtrx TrAG,.1 H mtrx = 1 Tr P +m X m Ψ Ψ S S + Ψ 0 Ψ 0 S.. 4X 0 X 0 The Hmltonn.1 nvolves the mtrx momentum P b X b nd nother mtrx quntty G b X,P b + Ψ,Ψ } b +Z Z b cδ b..3 The cton 1.4 lso produces the prmry constrnts P Z + Z 0, P Z Z 0,.4 P Ψ b Ψ b 0, P Ψ b Ψ b 0,.5 P A b 0..6 The constrnts.4,.5 re second clss nd so we ntroduce Drc brcets for them. As the result, we elmnte the moment P Z, P b Ψ nd ther c.c. The resdul vrbles obey the Drc brcets b X,P } d c = δ d δc, b Z, Z } l b = δ l δ, b Ψ b, Ψ } d lc = δ l δδ d c. b.7 Requrng the constrnts.6 to be preserved by the Hmltonn.1 genertes secondry constrnts G b 0..8 Despte the presence of the constnt c n.3 these constrnts re frst clss: wth respect to the Drc brcets.7 they form un lgebr, b G,G } d c = δ d G b c δb c Gd,.9 nd so produce the Un nvrnce of the cton 1.4 X = e α Xe α, Ψ = e α Ψ e α, Z = e α Z, A = e α Ae α e α t e α,.10 where α b t un re d=1 guge prmeters. In the frst-order formulton, the system 1.4 s represented by the cton S mtrx = dtl mtrx,.11 L mtrx = Tr PẊ + Ψ Tr Ψ Ψ Ψ + Z Z Z Ż H mtrx +TrAG,.1 where H mtrx ws defned n.. 4

6 Let us fx prtl guge for the trnsformtons.10. To ths end, we ntroduce the followng notton for the mtrx entres of X nd P: x := X, p := P no summton over, x b := X b, p b := P b for b, x := 0, p := 0 no summton over,.13.e., X b = x δ b +x b, P b = p δ b +p b nd X 0 = 1 n x. Note tht n =1 TrP = p p + b p b p b, TrXP = x p + b x b p b. In the notton.13 the constrnts.8 te the form for b nd G b = x x b p b p p b x b +x c p c b p c x c b +T b 0.14 G = x c p c p c x c +T c 0 no summton over.15 for the dgonl elements of G, wth T b := Z Z b + Ψ,Ψ } b..16 Provded tht the Clogero-le condtons x x b re fulflled, we cn mpose the guge x b 0, b,.17 for the constrnts.14. Then we ntroduce Drc brcets for the constrnts.14,.17 nd elmnte x b by.17 nd p b by.14: p b = T b x x b, b..18 Due to the resolved form of guge-fxng condtons, new Drc brcets for the remnng vrbles concde wth.7: x,p b } = δ b, Z, Z } l b = δ l δ, b Ψ b, Ψ lc d } = δ l δ d δ b c..19 In the guge.17, the constrnts.15 become T c := T c = Z Z + Ψ,Ψ } c 0 no summton over.0 nd they generte locl U1 n trnsformtons of Z nd Ψ b wth b. Preservton of the condtons.17, ẋ b = x b,h totl } = 0, llows one to express T b A b = p b =, b..1 x x b x x b 5

7 Insertng.17,.14 nd.1 nto., we rrve t the reduced totl Hmltonn H red = H C M A T c,. where A = A no summton over nd the generlzed Clogero Moser Hmltonn s defned s H C M = 1 p p +m x x + 1 S S + Ψ 0 Ψ 0 S. 4X 0 X 0 b T b T b x x b mtr Ψ Ψ.3 The sme fnl result cn be ttned n dfferent wy. Elmntng A b, b, by the equtons of moton A b = T b /x x b we obtn tht the cton 1.4 n the guge.17 tes the form S C M = dt 1 ẋ ẋ m x x +Tr Ψ Ψ Ψ Ψ +m Ψ Ψ 1 b Z Ż Z Z + T b T b x x b + S S Ψ 0 Ψ 0 S 4X 0 X 0 } A T c..4 The cton.4 produces the Hmltonn.3, the constrnts.0 nd the brcets.19. The mportnt ngredents of the cton.4 re blner combntons of Z nd Z wth the externl SU ndces S j := Z Z j no summton over, S j := S j..5 Wth respect to the Drc brcets.7 the objects S j for ech ndex form u lgebrs S j,s b l } = δb δ j S l δ l S j..6 The object S j forms the dgonl u lgebr n the product of bove ones S j,s l } = δ j S l δ l S j..7 The trplets of the qunttes.5 see lso 1.7 S j := Z Z j, Sj := S j.8 generte su lgebrs } S j,s l b = δb ε S jl +ε jl S,.9 6

8 S j,s l} = ε S jl +ε jl S..30 Below we wll lso use the brcets } Sj,Z = δ Z j, S j, Z } = δ Z j..31 One more mtrx present n the cton.4 s T b defned n.16. These qunttes form un lgebr.9 wth respect to the Drc brcets: T b,t c d } = δ d T c b δ b c T d..3 The odd mtrx vrbles re trnsformed by djont un representton: T b,ψ c d} = δ d Ψ c b δ b c Ψ d, } b T,Ψ 0 = 0,.33 T b, Ψ c d} = δ d Ψ c b δ b c Ψ d, b T, Ψ } 0 = These un trnsformtons commute wth u trnsformtons generted by S j : T b,s j } = Let us consder the bosonc core of the system.4 nd demonstrte tht t corresponds just to the spn Clogero Moser model. Omttng terms wth fermonc vrbles, we fnd S bose C M = 1 dt ẋ ẋ m x x Z Ż Z Z + } A ZZ c 1 TrS S b x x b + S S,.36 4X 0 where b TrS S b := S j S bj nd S j re defned n.5. The nlogous reducton of the Hmltonn.3 yelds H C M = 1 p p +m x x + 1 b.37 TrS S b x x b S S 4X The Hmltonn.38 contns potentl n the center-of-mss sector wth the coordnte X 0 the lst term n.38. Modulo ths extr potentl, the bosonc lmt of the system constructed s none other thn the U-spn Clogero Moser model whch s mssve generlzton of the U-spn Clogero model 5, 6, 8, 9, 30. Thus the system.4 wth the Hmltonn.3 descrbes SU 1 supersymmetrc extenson of the U-spn Clogero Moser model. 3 Superchrges In ths Secton we wll fnd the clsscl expressons for the genertors of the deformed N=4 supersymmetry SU 1 supersymmetry for the n-prtcle systems, both n the mtrx formulton nd n the cse of the reduced system wth n poston coordntes. 7

9 3.1 Mtrx system The odd SU 1 trnsformtons of the component mtrx felds enterng 1.4 re s follows 4 δψ = ǫ X +mx+ ǫ js j δx = ǫ Ψ + ǫ Ψ, X 0 1, δ Ψ = ǫ X mx+ ǫj S j X 0 1, 3.1 wheres the supertrnsltons of the spn felds re represented by the SU rottons wth the composte prmeters δz = ω j Z j, δ Z = ω j Zj 3. ω j = ǫ Ψ j 0 + ǫ Ψj 0. X 0 Under the trnsformtons 3.1, 3. nd δa = 0 the cton 1.4 trnsforms s δs mtrx = dt Λ 1, Λ 1 = ǫ Tr X +mxψ + ǫ Tr X mx Ψ ωj S j. 3.3 Usng 3.1, 3. nd 3.3 we obtn the followng expressons for Noether superchrges: Q = Tr P mx Ψ + Sj Ψ 0j, X 0 Q = Tr P +mx Ψ S Ψ j j 0, X 0 where P = X. The genertors 3.4 consttute n ŝu 1 superlgebr wth respect to the Drc brcets.7 Q, Q } = δ H m I δ F, Q,Q } = 0, Q, Q } = Here, H = H mtrx, where H mtrx ws defned n., nd lso the su nd u1 genertors re present: I = ε j S j +Tr Ψ Ψj, 3.6 F = 1 Tr Ψ Ψ. 3.7 The Hmltonn H commutes wth ll other genertors nd so cn be dentfed wth the centrl chrge opertor of ŝu 1. The rest of Drc brcets mong the genertors., 3.4, 3.6, 3.7 s gven by the reltons } H,Q } = H, Q = } H,I = H,F} = 0, These trnsformtons re sum of the ntl lner supertrnsltons plus extr compenstng guge trnsformtons 1.3 wth λ = θ+ ǫ θ + ǫ A whch re requred for preservng WZ guge for V ++. 8

10 F,Q } = Q, F, Q } = Q, F,I } = 0, 3.9 I,Q j} = δ j Q +ε j Q, I, Q } j = δ j Q +ε j Q, 3.10 } I,I j l = δ l I j δj I l Note tht the frst-order cton.11 s nvrnt, up to the surfce term δs mtrx = dt Λ 1 wth the substtuton X = P n Λ 1, under the trnsformtons 3.1, 3., δa = 0 nd δp = m ǫ Ψ + ǫ Ψ ǫ S j Ψ 0j + ǫ S j Ψj 0 X It s worth pontng out tht δh = 0 nd δg b = 0 under these trnsformtons. 3. Reduced system n the stndrd Clogero Moser representton Let us compute the ŝu 1 chrges for the reduced system.4 whch follows from the mtrx formulton fter mposng the guge.17. On the pttern of.13, we ntroduce the followng notton for the entres of Ψ nd Ψ : ψ := Ψ, ψ := Ψ no summton over, ψ b := Ψ b, ψ b := Ψ b for b Note tht TrPΨ = p ψ + p b ψ b nd Ψ 0 = 1 n ψ b n, Ψ 0 = 1 n ψ =1 n. =1 In the guge.17, supertrnsltons re sum of the trnsformtons 3.1, 3. nd the ddtonl compenstng guge trnsformtons.10 wth the composte prmeters α b = ǫ ψ b ǫ ψ b x x b for b, α b = 0 for = b These trnsformtons preserve the condtons.17 nd hve the followng explct form δx = ǫ ψ + ǫ ψ, 3.15 δψ = ǫ ẋ +mx + ǫ js j + b α ψ b ψ b α b, X 0 b δ ψ = ǫ ẋ mx + ǫj S j + α b ψb X ψ b α b, 0 b δψ b = ǫ b T α b ψ x x ψ b + c α ψ c b ψ c α b c, b δ ψ b = ǫ T b x x b α b ψ ψ b + δz = ωj Z j + b α b Z b, c c α c ψc b ψ c α c b, δ Z = ω j Z j b Z b α b,

11 δa = b α b T b +T b α b x x b An mportnt property s tht the constrnts.0 re nvrnt wth respect to these supersymmetry trnsformtons, δt = 0. Also, δ A T c = 0. The vrton of the cton.4 under the supersymmetry trnsformtons reds δs C M = dt Λ, where Λ = ǫ ẋ +mx ψ + ǫ ẋ mx ψ + 1 α b T b ωj S j.3.0 b The correspondng Noether superchrges re found to be Q = Q = p mx ψ + b p +mx ψ + b T b ψ b x x b + Sj Ψ 0j X 0, T b ψb x x b S j Ψ j 0 X 0, 3.1 where p = ẋ. These expressons cn be lso obtned by nsertng.17,.18 nto 3.4 nd turnng to the nottons.13, Wth respect to the Drc brcets.19, the genertors 3.1 form, up to the resdul U1 n guge trnsformtons generted by.0, the followng ŝu 1 superlgebr Q, Q } = δ H mi δ F+ b Q,Q } = b Q, Q } = b ψ b ψ b x x b T T b, ψ b ψb x x b T T b. ψ b ψb x x b T T b, 3. Here we used tht the lst relton n.19, beng cst n the notton.13, 3.13, mounts to the reltons ψ, ψ } b = δ δ b, ψ b, ψ } d c = δ δd δb c, In 3., H = H C M, wth H C M defned by.3, nd the genertors I, F were defned n 3.6, 3.7. The Hmltonn.3 commutes wth the superchrges 3.1 modulo the frst-clss constrnts.0: Q,H } = b T b ψ b x x b 3 T T b, Q,H } = b T b ψb x x b 3 T T b. 3.3 The genertors I, F stsfy the sme Drc brcets s n 3.8, 3.9, 3.10, nd

12 4 Quntum mult-prtcle ŝu 1 superlgebr Quntum su 1 superlgebr obtned by quntzng the Drc brcets 3.5, 3.8, 3.9, 3.10, 3.11, s formed by the followng non-vnshng ntcommuttors : Q, Q } = δ H+m I δ F, F,Q = 1 1 Q, F, Q = Q, I,Q j = 1 δ j Q +ε j Q, I, Q 1 j = δ j Q +ε j Q, I,I j l = δ j I l δli j. 4.1 The second- nd thrd-order Csmr opertors of su 1 re defned by the expressons 18 C = C 3 = 1 m H F 1 I I + 1 Q, 4m Q, 4. C m H F Q δ j 8m m H F I} j, Q j. 4.3 In ths Secton we wll present the explct form of ths deformed N =4 supersymmetry lgebr for multprtcle system constructed n the prevous Sectons. We wll do t for the mtrx formulton of ths system nd for the reduced system wth n poston coordntes. 4.1 Mtrx formulton Superchrges of the ŝu 1 superlgebr In the mtrx formulton, the n-prtcle system s descrbed by quntum opertors X b, P b ; Ψ b, Ψ b ; Z, Z b whch stsfy the quntum counterprt of the Drc brcets lgebr.7: X b,p c d = δ d δ b c, Z, Z b j = δ j δ, b Ψ b, Ψ jc d } = δ jδ d δ b c. 4.4 The quntum superchrges re unquely restored by the clsscl expressons 3.4: Q = Tr P mxψ + Sj Ψ 0j, X 0 Q = Tr P+mX Ψ S Ψ j j 0, X where the su genertors re S = Z Z. 4.6 These genertors form the quntum lgebr of the correspondng dgonl externl lgebr.30. The closure of the genertors 4.5 s the full ŝu 1 superlgebr 4.1 wth the 11

13 followng even genertors H = H bose +H ferm, 4.7 H bose = 1 Tr P +m X ns S 4X 0, 4.8 H ferm = m Tr Ψ, Ψ Ψ + 0 Ψ 0 S, X I = ε j S j +Tr Ψ Ψj, 4.10 F = 1 4 Tr Ψ, Ψ The set of physcl sttes of the mtrx system s sngled out by the n constrnts G b = X,P b + Ψ,Ψ } b +Z Z b q +1 δ b 0, 4.1 whch re quntum counterprts of the clsscl constrnts.8 the subscrpt W denotes Weyl-orderng nd should be mposed on the wve functons. The constnt q + 1 present n 4.1 dffers from the clsscl constnt c due to orderng mbgutes. The opertors 4.1 form un lgebr G b,g c d = δ c b G d δ d G c b It s mportnt tht ll constnts pperng n the dgonl prt of G b,.e. t =b, re equl to q +1. A corollry of 4.1 s tht u1 genertor G = Z Z nq ncludes spn Z-opertors only. As we wll see below, the un constrnts 4.1 hve trnsprent menng: The physcl sttes re sun snglets. The constrnt 4.14 fxes the homogenety degree of the physcl sttes wth respect to spn vrbles, whence q Z > Seprton of the center-of-mss sector Let us splt the mtrx qunttes s wth X b = 1 n δ b X 0 + ˆX b, P b = 1 n δ b P 0 + ˆP b, Ψ b = 1 n δ b Ψ 0 + ˆΨ b, Ψ b = 1 n δ b Ψ 0 + ˆ Ψ b, 4.15 X 0 = 1 X, P 0 = 1 P, Ψ n n 0 = 1 Ψ, Ψ0 = 1 n n W Ψ

14 beng the center-of-mss opertors nd ˆX b = X b 1 n δ b X 0, ˆP b = P b 1 n δ b P 0, ˆΨ b = Ψ b 1 n δ b Ψ 0, ˆ Ψ b = Ψ b 1 n δ b Ψ the trceless prts of mtrx opertors. In terms of the vrbles 4.16, 4.17 the superchrges 4.5 re represented s where Q = Q 0 + ˆQ, Q = Q 0 + ˆ Q, 4.18 Q 0 = P 0 mx 0 Ψ 0 + Sj Ψ 0j X 0, Q0 = P 0 +mx 0 Ψ 0 S j Ψ j 0 X nvolve only the center-of-mss opertors 4.16 nd spn vrbles, wheres ˆQ = TrˆP m ˆX ˆΨ, ˆ Q = TrˆP+m ˆX ˆ Ψ 4.0 depend on the trceless prts The even opertors 4.7, 4.8, 4.9, 4.10, 4.11 dmt smlr splttng Here, H = H 0 +Ĥ, I = I 0 +Î, F = F 0 + ˆF. 4.1 H 0 = 1 P0 +m X 0 + m I 0 = ε j S j +Ψ 0 Ψ 0, Ψ 0 S S 4X 0 + S Ψ 0 Ψ 0 X 0, 4. j Ψ 0, 4.3 F 0 = 1 4 Ψ 0, Ψ 0, 4.4 nd Ĥ = 1 ˆP Tr +m + ˆX m ˆΨ Tr, ˆ Ψ, 4.5 Î = ε j Tr ˆΨˆ Ψj, 4.6 ˆF = 1 ˆΨ 4 Tr, ˆ Ψ. 4.7 The sets Q 0, Q0, H 0, I 0, F 0 nd ˆQ, ˆ Q, Ĥ, Î, ˆF form ŝu 1 superlgebrs 4.1 on ther own, wth the vnshng mutul ntcommuttors: Q 0, ˆQ } = Q 0, } ˆ Q = 0, etc. Thus, we hve sngled out the center-of-mss sector from the totl system. Note tht the ŝu 1 genertors ˆQ, ˆ Q, Ĥ, Î, ˆF hve no cton on the spn opertors Z whch n fct remn n the center-of-mss sector. 13

15 It s of mportnce tht the constrnts 4.1 nvolve n fct only the trceless prts 4.17 of the mtrx opertors prt from the spn vrble opertors. Indeed, they cn be rewrtten n the form G b = ˆX, ˆP b + ˆ Ψ, ˆΨ } b +Z Z b q +n 1 δ b n However, due to the presence of the sme spn vrbles n the center-of-mss sector, these constrnts re pplcble lso to the correspondng quntum sttes nd so ccomplsh ln between the two sectors. 4. Quntum lgebr for SU 1 spnnng Clogero Moser system 4..1 ŝu 1 superlgebr wth n dynmcl bosons The quntum counterprt of the multprtcle system from Sect. 3. s descrbed by the quntum opertors x, p ; ψ, ψ ; ψ b, ψ b, b; Z, Z whch stsfy the lgebr x,p b = δ b, Z, Z b j = δ j δ b, ψ, ψ } jb = δ j δ b, ψ b, ψ } d jc = δj δd δb c b,c d. 4.9 Performng the Weyl-orderng n the quntum counterprt of 3.1, we obtn the quntum superchrges: where Q = Q = T b = Z Z b + ψ ψ b p mx ψ + Sj Ψ 0j ψ ψ b + X 0 x x b b b ψ ψ b p +mx ψ S j Ψ j 0 X 0 b x x b ψ b + ψ ψ b ψ b + re quntum counterprts of.16 t b nd X 0 = 1 x, Ψ n 0 = 1 n c, c b + b T b ψ b x x b, T b ψb x x b, 4.30 ψ c ψc b + ψ c ψ c b 4.31 ψ, Ψ0 = 1 n Computng the ntcommuttors of the superchrges 4.30, ψ. 4.3 Q, Q } = δ H n +m I δf ψ b ψb x b x b T T b, 4.33 Q,Q } = ψ b ψ b x b x b T T b, 4.34 Q, Q } b ψ ψb = x x b T T b 4.35 b 14

16 we fnd the explct form of the quntum even genertors H = 1 p p +m x x + m ψ, ψ + m ψ b, ψ b S S 4X 0 + S Ψ 0 Ψ 0 X T b T b x b x b, 4.36 I = ε j S j + ψ ψ j + ψ b ψj b, 4.37 b b F = 1 4 ψ, ψ ψ b, ψ b b The commuttors of the genertor H wth odd genertors Q, Q re quntum generlzton of 3.3. The remnng genertors I, F obey the sme commutton reltons s n 4.1. From the ntcommuttors obtned we observe tht the genertors 4.30, 4.36, 4.37, 4.38 form the ŝu 1 superlgebr 4.1 up to the dfferences T T b. However, recllng the constrnts.0, ths reduced system s specfed lso by the condtons T q n 1 = Z Z + c ψ c ψc ψ c ψ c q 0, 4.39 whch must be supermposed on the physcl sttes. Therefore, the dfferences T T b re vnshng on the physcl sttes, nd the physcl sector of the relevnt Hlbert spce s closed under ŜU 1 symmetry. It s mportnt tht the quntum constrnts 4.39 commute wth the ŝu 1 genertors: T,Q = T, Q = In ddton, the qunttes 4.31 stsfy the lgebr T b,t c d = δ b c T d δ d T c b, 4.41 where T = T t fxed. It s nstructve to be convnced tht the numertor n the lst term n 4.36 s ndeed reduced to tht for U spn Clogero Moser system 30, when ppled to the bosonc wve functons Φ bos defned by the condtons It s esy to chec tht n ths cse ψ Φ bos = ψ b Φ bos = 0. nd the constrnt 4.39 s reduced to T b Z Z b +n 1δ b Z Z q 0. 15

17 Now, tng nto ccount tht n the numertor n 4.36 b, t s esy to chec tht 1 T b T b 1 Sj S bj +qq +1, b, 4.4 where S j = Z Z j no summton over. The opertors S j re just the quntum verson defned n.8. Foe ech vlue of the ndex they generte su lgebrs nd of S j commute wth ech other for b. The expresson 4.4 concdes wth tht pperng n the rtonl U spn Clogero Moser model, wth q beng the prwse spn couplng constnt Dvson nto subsystems Usng the smple dentty K M = 1 n K b M b + 1 n K K b M M b b whch s vld for rbtrry n-vector opertors K, M, = 1,...,n, nd ntroducng the center-of-mss qunttes 4.3 nd we cn represent the chrges 4.30 s the sums P 0 = 1 p, 4.43 n Q = Q 0 + Q, Q = Q 0 + Q The frst tems Q 0, Q 0 n these sums were defned n 4.19, nd they nvolve only the centrlof-mss supercoordntes, wheres the second tems Q, Q depend only on the dfferences of the supercoordntes: Q = 1 n b ψ p p b mx x b ψ b ψ ψ b + T b ψ b, x x b x x b b b Q = 1 p p b +mx x b ψ n ψ b b ψ ψ b + b T ψb. x x b x x b b b 4.45 Snce S j,t b = 0, Q 0, Q 0 ntcommute wth second Q, Q : Q 0, Q j} = Q 0, Q j } = Q0, Q j} = Q0, Q j } = Followng 30, ths model cn be referred to s the reduced mtrx U spn Clogero Moser model, wth q Z >0. There exsts nother type of Us spn models, the so-clled exchnge-opertor models 31, 3, 30, 33, for whch the spn couplng constnt s n rbtrry number. Our SU 1 supersymmetrc mult-prtcle system yelds just the frst type of U spn models n the bosonc sector. 16

18 Ths mples tht the bosonc genertors 4.36, 4.37, 4.38 cn lso be represented s smlr sums, H = H 0 + H, I = I 0 + I, F = F 0 + F, 4.47 where H 0,I 0 nd F 0 re gven by eqs. 4., 4.3, 4.4 nd so nvolve only the center-ofmss coordntes, whle the rest of opertors s defned by the expressons H = 1 p p b +m x x b + 1 T b T b 4n x b b x b + m ψ ψ b, ψ 4n ψ m b + ψ b, ψ b, 4.48 b b I 1 = ε j ψ ψ ψj b n ψ b j + ψ b ψj b, 4.49 b b F = 1 ψ ψ b, ψ 8n ψ 1 b + ψ b, 4 ψ b b The sets of the genertors Q 0, Q 0, H 0, I 0, F 0 nd Q, Q, H, I, F form two seprte mutully ntcommutng ŝu 1 superlgebrs. Note tht second set genertes n ŝu 1 superlgebr up to the constrnts, s n 4.33, 4.34, Also, note tht the nternl SU genertors 4.49 pperng n the ntcommuttor of superchrges ct on the ndces,j of the fermonc opertors ψ ψ b, ψ ψ b, ψ b, ψ b, b, whle the ndces,j of the spn opertors Z, Z re subject to the cton of the externl SU genertors Subsystems of N =4 supersymmetrc Clogero Moser model We hve found tht the N =4 supersymmetrc n-prtcle Clogero Moser system s drect sum of two subsystems wth dfferent relztons of the ŝu 1 genertors. The genertors Q 0, Q 0, H 0, I 0, F 0 ct n the sector of the center-of-mss opertors X 0, P 0, Ψ 0, Ψ0 nd the spn opertors Z, Z. The second set of the ŝu 1 genertors ˆQ, ˆ Q, Ĥ, Î, ˆF ct, n the mtrx formulton, wthn the sector of the trceless opertors ˆX, ˆP, ˆΨ, ˆ Ψ. Physcl sttes n ths subsystem re specfed lso by the spn opertors Z, Z whch re present n the un constrnts 4.8. These constrnts lso specfy physcl sttes n the center-of-mss sector nvolvng the sme spn opertors. In the reduced formulton, the genertors Q, Q, H, I, F re spnned by the set of opertors x x b, p p b, ψ ψ b, ψ ψ b, ψ b, ψ b, b; Z, Z. It should be ponted out tht the spn opertors Z, Z hve non-zero cton on the physcl sttes wth q 0 for ll subsystems defned bove nd lsted below. The just descrbed structure of the consdered system suggests tht we cn consder three subsystems: I The center-of-mss sector spnned by the quntum opertors X 0, P 0, Ψ 0, Ψ 0, Z, Z nd the symmetry opertors Q 0, Q 0, H 0, I 0, F 0 ; II The pure Clogero Moser mult-prtcle sector wth the center-of-mss sector seprted. It s spnned by the quntum opertors ˆX, ˆP, ˆΨ, ˆ Ψ n the mtrx formulton or by 17 b

19 x x b, p p b, ψ ψ b, ψ ψ b, ψ b, ψ b, b n the reduced formulton. In both formultons, ths subsystem lso nvolves the spn opertors Z, Z. The SU 1 symmetry genertors re ˆQ, ˆ Q, Ĥ, Î, ˆF or Q, Q, H, I, F; III The full Clogero Moser mult-prtcle system whch contns the center-of-mss sector nd so s spnned by the set of ll quntum opertors. The ŜU 1 symmetry genertors re sums of the ŜU 1 genertors ctng n the two prevously defned sectors. Now we re prepred to determne the energy spectrum of ll these systems. 5 Center-of-mss subsystem wth n sets of spn vrbles In ths secton we consder the subsystem I whch descrbes the center-of-mss sector wth the Hmltonn H The center-of-mss supercoordntes 4.3, 4.43 stsfy the followng ntcommutton reltons X 0,P 0 =, Ψ 0, Ψ } 0 = δ, 5.1 whle those for the spn vrbles red Z, Z b = δ δ. b 5. We wll use the followng relzton of the opertor reltons 5.1, 5. X 0 = x 0, P 0 = x 0, Ψ 0 = ψ0, Ψ0 =, 5.3 ψ0 Z = z, Z =, 5.4 z where x 0 s rel commutng vrble, z re complex commutng vrbles nd ψ 0 re complex Grssmnn vrbles. In ths relzton the Hmltonn 4. tes the form H 0 = 1 x +m x 0 +m ψ ψ0 x 0 4 S S +S ψ 0, 5.5 ψ0 where S j = z. Wve functon Φ q x z j 0,z,ψ 0 s subject to the n constrnts orgntng from 4.8: G Φ q = q Φ q = 0, = 1,...,n. 5.6 z z The soluton of eqs. 5.6 s functon whch s homogeneous of degree q wth respect to ech set of spn vrbles z. So the number q tng postve nteger nd hlf-nteger vlues cn be treted s spn ssocted wth every SU group generted by the quntum genertors S j = z, no summton over. 5.7 z j 18

20 The number s wll be ssocted wth the dgonl SU group generted by S j = n =1 S j, 5.8 nd, n wht follows, wll be referred to s SU spn s. Snce Φ q s trnsformed n the drectproductofnspnqsurepresenttons, themxmlexternlsuspnsjusts = nq. It wll be convenent to expnd Φ q nto rreducble multplets of the dgonl SU, wth spns runnng n the ntervls 0,1...nq for nq even or 1/,3/...nq for nq odd. As n llustrton, we dwell on two lower-n cses. n = In ths cse the wve functon Φ 0 x 0,z1,z,ψ 0 s subject to two constrnts T 1 qφ q 0 = z1 q Φ q 0 = 0, T qφ q 0 = z q Φ q 0 = Ther generl soluton s z 1 Φ q 0 = z 1 z q φ+ q s=1 z z 1 z q s z z s 1 z s+1...z s Φ 1... s, 5.10 where z 1 z := z 1 z. The component wve functons φ, Φ 1... s n the expnson 5.10re functons of x 0, ψ 0. They form rreducble SU multplets wth spns s = 0,1,...q. Ther expnsons wth respect to ψ 0 re φ = + +ψ 0 b +ψ 0, 5.11 Φ 1... s = A s +ψ 01 B... s +ψ j 0 C j 1... s +ψ 0 A 1... s. 5.1 All components n these expnsons re functons of x 0 only. In the bosonc wve functon Φ q 0, the felds ±, A ±1... s re bosonc, wheres b, B... s, C j1... s re fermonc. The SU spns of the component wve functons re counted wth respect to the nternl SU wth the genertors 4.3 whch contn, besdes the prt ctng on the bosonc spn vrbles, lso the one ctng on the fermonc vrbles. Let us determne the egenvlues of the Hmltonn 5.5 on the wve functon 5.10,.e. solve the sttonry Schrödnger equton As prerequste, we dduce the followng egenvlue reltons H 0 Φ q,l 0 = E s,l Φ q,l Sj S j z zp 1 z p+1...z s A 1... s = ss+1z zp 1 z p+1...z s A 1... s, 5.14 S j z 1 z = 0, S j S j ψ0 z = S j ψ 0 ψ j ψ0 z = 3 ψ 0 z,

21 S j ψ 0 ψ j ψ0 n z zp 1 z p+1...z s A n 1... s = 0 = sψ0 n z zp 1 z p+1...z s A n1... s S j ψ 0 ψ j z zp 1 z p+1...z s ψ 0 1 A... s = = s+1z zp 1 z p+1...z s ψ 01 A... s. They cn be esly checed nd shown to be vld for n rbtrry p s. Due to these reltons, ll the component felds n the expnsons 5.11, 5.1 of the wve functon 5.10 re egensttes of the center-of-mss Hmltonn 5.5 wth the spn s of the dgonl SU group gven by S j. The equton 5.13 mounts to the followng equtons for the component wve functons x x 0 x 0 +m x 0 + ss+1 x 0 +m x x 0 + s+1s+ x 0 0 x 0 +m x 0 + ss 1 x 0 +m x 0 l ± = E 0,l ±m l ±, +m x 0 b l = E 0,l b l, 5.18 A l ± 1... s = E s,l ±ma l ± 1... s, B l 1... s 1 = E s,lb l 1... s 1, C l 1... s+1 = E s,lc l 1... s+1, 5.19 where 5.18 corresponds to s = 0, whle n 5.19 s runs over q 1 vlues, s = 1,...,q. The equtons 5.18 for the felds l ± nd b l hve the form 1 +m x x0 0 f l x 0 = E l f l x nd descrbe the excttons of osclltors. The stndrd solutons of the equton 5.0 re gven v Hermte polynomls H l s f l x 0 = H l x 0 exp mx 0 /, l = 0,1,,..., 5.1 nd hve the energes Thus, the energy spectrum reds E l = ml+1/. 5. E 0,l = m l It s worth pontng out tht the soluton for N = 4 supersymmetrc hrmonc osclltor 5.18 ws orgnlly gven n 13. The equtons 5.19 for the felds A ±1... s, B 1... s 1 nd C 1... s+1 hve the generc form 1 x 0 +m x 0 + γγ 1 x 0 0 f l x 0 = E l fl x 0, 5.4

22 where γ s constnt. It s the well-nown equton descrbng quntum sttes of non-reltvstc prtcle movng n sum of the one-dmensonl osclltor nd conforml nverse-squre potentls, nd t hs the followng generl soluton see, e.g., 1, 3, 37 f l l! x 0 = Γl+γ +1/ x 0 γ L γ 1/ l mx 0 exp mx 0 /, l = 0,1,,..., 5.5 where L γ 1/ l s generlzed Lguerre polynoml. The correspondng energy levels re E γ,l = m l+γ The generl soluton 5.5 ws used n ref. 4 to revel the energy spectrum of the oneprtcle system wth one set of the spn vrbles. Ech equton n the set 5.19 hs the form of 5.4, the prmeter γ beng s+1, s+ nd s, respectvely. Thus, the energy of the sttes of spns s nd s+1/ descrbed by the wve functons A s nd C 1... s+1, s equl to E s,l = ml+s+1/, l = 0,1,,..., s = 1,...,q. The energy of the sttes A 1... s of spn s nd the sttes B 1... s 1 of spn s+1/ s gven only for excted sttes by the sme expresson E s,l = ml+s+1/, l = 1,,3,..., s = 1,...,q. The lowest energy for these sttes corresponds to s = 1, l = 0 nd equls E mn = 3m. 5.7 At q = 1/ we hve the pcture drwn n the Fgure 1. The q = 1 cse encompsses the sme sttes s for q = 1/ s = 1 depcted n Fg. 1, but lso ddtonl sttes wth hgher spns nd hgher energes. The smlr pctures persst t lger q. Summrzng the bove dscusson, we observe the bsc dstncton between the one-prtcle system of ref. 4 nd the center-of-mss sector of n-prtcle system consdered here. In the former cse, the energy spectrum rses s soluton of the egenvlue problems of the type 5.4, wth the Hmltonns nvolvng sum of the osclltor nd the nverse squre potentls. In the ltter cse, the energy spectrum contns s well pure osclltor excttons due to the presence of the egenvlue problems of the type 5.0. n = 3 In ths cse there re three constrnts of the type 5.6 nd for nteger q they led to the followng dependence of the wve functon on the spn vrbles Φ q 0 = z z b z z c z b z c q φ+ z z b q 1 z z c z b z c q z z b Φ b +... b, c,b c +z z q 1 z j 1...z j q z z q 3 Φ 1... q j 1...j q 1... q

23 H 0 11m 9m 7m 5m 3m m m + b A +j B C j A j Fgure 1: The degenercy of energy levels of H 0 for n= nd q=1/. Crcles nd crosses represent bosonc nd fermonc sttes, respectvely. On the left from the dotted vertcl lne the degenercy correspondng to hrmonc osclltor 13, 16 s shown. On the rght sde there s shown sum of SU 1 representtons specfed by ther spn vlues s nd concdng wth those found n 4 for the relevnt spn. For the consdered smplest cse of q = 1/, spn s tes only one vlue, s = 1. The component wve functons n the expnson 5.8 re functons of x 0 nd ψ0, nd they dsply the dependence on ψ0 smlr to tht n 5.1. In the three-spnor cse, the reltons nlogous to 5.14, 5.15, 5.16, 5.17 re lso vld, the dfference s tht now n ddtonl spn vrble z3 ppers n the products. As result, n the energy spectrum we fnd the sme sttes s n the n = cse, though wth bgger multplcty due to extr ndces b n 5.8, s well s the sttes of hgher spns due to the presence of the ddtonl spn vrble z3. In the cse of hlf-nteger q, the n = 3 wve functon hs n expnson n whch the component wve functons crry odd numbers of spnor ndces, s opposed to the expnson 5.8. For exmple, for q = 1/ wve functon s Φ 1 0 = z z 3 z 1 Φ 1 +z 1 z 3 z Φ +z 1 z z 3 Φ The superwve functons Φ x 0,ψ 0 dsply the energy spectrum of the one-prtcle system of ref. 4, ths tme wth the three-fold degenercy. The pctures for hgher n re smlr to those for n = nd n = 3, such tht the number of sttes nd the vlues of dmssble spns re ncresng t ncresng n.

24 6 Clogero Moser system wthout center-of-mss sector As ws mentoned n Introducton, there re two methods of fndng the quntum energy spectrum of multprtcle Clogero-type systems: ether by consderng mtrx models whch produce physclly equvlent Clogero-type systems fter guge-fxng nd the correspondng reducton of phse spce, or through ntroducng Dunl opertors nd pssng to generlzed osclltor system. In ths secton we pply the frst method to quntze the N =4 spn Clogero Moser model under consderton n the mtrx formulton, wth the center-of-mss sector detched Sect The cse of the reduced-phse spce s brefly ddressed n Sect Quntzton n mtrx formulton We consder the quntzton of the mtrx subsystem n whch ŝu 1 superlgebr s formed by the genertors ˆQ, ˆ Q, Ĥ, Î, ˆF defned n 4.0, 4.5, 4.6 nd 4.7. The bsc opertors of ths system re spn opertors Z, Z nd trceless mtrx opertors ˆX b, ˆP b, ˆΨ b, ˆ Ψ b subject to the constrnts 4.8 s usul, ppled to the physcl sttes. Introducng creton nd nnhlton even opertors A b = 1 ˆP b b mˆx, A + b = 1 ˆP b b +mˆx, 6.1 m m TrA = Tr A + = 0, we rewrte the Hmltonn 4.5 n the form Ĥ = m Tr A +,A } + m Tr ˆΨ, ˆ Ψ. 6. In ths notton, the superchrges 4.0 re rewrtten s ˆQ = mtr AˆΨ, ˆ Q = mtr A +ˆ Ψ, 6.3 where quntum ntcommuttors of the nvolved opertors re b A,A + d c = δ d δ b c 1 n δ b δ d c, ˆΨ b, d} ˆ Ψjc = δ d δ b c 1 n δ b d δ c δj. 6.4 The constrnts 4.8 te the form G b = A +,A b + ˆ Ψ, ˆΨ } b +Z Z b q+n 1 δ b n They nvolve the spn opertors, wth the non-vnshng commuttor Z, Z b = δ δ b. 6.6 The opertors Z, A+ b, ˆΨ b form full set of creton opertors. Therefore, the generl structure of the physcl sttes s s follows Z Z 1 1 A + b 1 c 1...A + b c ˆΨj 1d1 e 1... ˆΨ j 3 d 3 e

25 In the holomorphc relzton, Z = z, A + b Z =, z c = â + c b = + c b 1 n δc b + d d, ˆΨ b c = ˆΨ b c = Ψ b c 1 n δc bψ d d, A b c = â + c b = + c b 1 n δc b + d d, ˆ Ψb c = ˆΨ = c b Ψ 1 c b n δ b c Ψ, 6.8 d d we del wth the trceless objects â + nd ˆΨ. Then the physcl sttes 6.7 re rewrtten s z z 1 1 â + b 1 c 1...â + b c ˆΨj 1d1 e 1... ˆΨ j 3 d 3 e The constrnt 6.5 ndctes tht ll physcl sttes re snglets of SUn see 34, 36, 35, 30. Ths s lso drect consequence of vnshng of ll Csmr opertors on the sttes 6.7: G b G b c...g e On the other hnd, the sttes 6.7 belong to rreducble representtons of the group SU wth the genertors S j defned n 4.6 nd the group SU U1 wth the genertors 4.6, 4.7 ctng only on fermonc felds. Egenvlues of the Hmltonn 6. on the sttes 6.7 re specfed by the numbers N A nd N Ψ of the opertors A + b c nd ˆΨ b c : E = m N A +N Ψ n Here we wll bsclly lmt our consderton to the pure bosonc cse, wthout odd opertors ˆΨ b c, ˆ Ψjb c. The set of fermonc sttes cn be generted by cton of the superchrges on the subset of bosonc sttes. Exmples of fermonc sttes wll be constructed below for few smple prtculr cses. The trce prt of the constrnts 6.5 leds to homogenety of the physcl sttes of degree qn wth respect to the spn opertors Z. In ddton, the property tht physcl sttes re the SUn snglets mples the followng structure for them 34, 36, 35, 30 Φ q,s,l Tr A + p Tr A + 3 p 3... Tr A + n pn q 1 r=0 ε 1... n A + l nr+1 Z nr+1 A... + l nr+n Z nr+n 1 n } 0, 6.1 where p,p 3...,p n re rbtrry ntegers nd 0 l nr+1 l nr+...l nr+n < n. The wve functon Φ q,s,l n 6.1 s gven up to the coeffcents C 1... s, where the number s nq cn be nterpreted s SU spn, wth s beng the number of symmetrzed SU ndces of spn vrbles. It s worth pontng out tht for l r < n/ the concdent degrees, l r = l p, re permtted. Besdes, such degree cnnot pper more thn once n the products of monomls ε 1... n A + l nr+1 Z nr+1 A... + l nr+n Z nr+n 1 n 4 }, r = 0,1...q

26 For the hghest spn s = nq, the degrees re gven by l nr+1 = 0, l nr+ = 1,... l nr+n = n 1, r = 0,1...q Degenercy nlyss of bosonc wve functons for the quntum spn Clogero model ws consdered n 33, where t ws notced tht some of possble spn sttes my vnsh. Here we consder the mtrx constructon for the system wth the center-of-mss sector detched, 6 where some of these spn sttes my lso vnsh. Lstng ll dmssble degree numbers l nr+1,l nr+...,l nr+n s rther complcted ts. On the sttes 6.1 the energy 6.11 te the vlues n qn E = m p + l n = The energy s mxml for the choce 6.14: n E s=nq = m p +n 1nq n = The mnml energy corresponds to the choce p = p 3...p n = 0 nd l nr+1 = l nr+ = 0, l nr+3 = l nr+4 = 1, l nr+5 = l nr+6 =, etc: n E mn = m n 1 q n 1, for even n, n 1 q E mn = m n 1, for odd n > The fermonc sttes re constructed wth the help of the opertors ˆΨ b c, on the pttern of 6.1. Such physcl sttes hve ddtonl contrbutons mn Ψ to the energy vlue As ws lredy mentoned, full wve functons cn be generted from the bosonc sttes 6.7 by ctng on them by the superchrges 6.3. Csmr opertors 4., 4. te the followng vlues on the sttes 6.7 nd those produced from 6.7 by SU 1 supersymmetry trnsformton: =1 m C = E + n 1m m 3 C 3 = E + n m E + n 3m, C Csmrs cn te zero egenvlues only for n = t rbtrry q nd for n = 3 t q = 1/ we consder q > 0 n ths pper. The correspondng sets of the quntum sttes belong to typcl representtons of SU 1. For llustrton, we wll consder here these two cses n some detl. n = 6 The constructon of 33 mples reducton to the ngulr spn Clogero model by seprtng the rdl coordnte, whch corresponds n the quntum cse to settng p = 0 n

27 In ths cse the Hmltonn s wrtten s Ĥ = mtr A + A +mtr ˆΨ ˆ Ψ 3m Bosonc wve functons, from whch the full set of the wve functons cn be produced by the superchrges 6.4, re gven by Φ q,s,l = Tr A + l εj ε b Z Z j b q sz 1 1 Z...Z s s s ε b A + s+ b C 1... s 0, 6.0 where C 1... s re coeffcents wth s symmetrc ndces. The wve functons Φ q,s,l re egenfunctons of the Hmltonn 6.19, =1 ĤΦ q,s,l = E s,l Φ q,s,l, s = 0,1...q, 6.1 wth the energy egenvlues E s,l = m l+s The wve functons on whch Csmrs te zero vlues,.e., those belongng to typcl representtons of SU 1, correspond to the choce s = 0, l = 0: Φ q,0,0 = ε j ε b Z Zj b q 0, 6.3 Ths ground stte wve functon s SU 1 snglet, snce t s nnhlted by both superchrges. There s stll nother typcl non-snglet bosonc stte correspondng to s = 1, l = 0: Φ q,1,0 = ε 1 ε c 1c Z 1 c 1 Z q 1ε c b Z 1 Z d A+ d b 0, 6.4 whch gves rse to the fundmentl SU 1 representton. The other two components of ths representton re generted from 6.4 by SU 1 superchrges: ˆQ j Φ q,1,0 = m ε 1 ε c 1c Z 1 c 1 Z q 1ε c b Z 1 Z d ˆΨ j b d 0, etc. 6.5 n = 3, q = 1/ The n = 3 Hmltonn reds Ĥ = m Tr A +,A } + m ˆΨ Tr, ˆ Ψ = mtr A + A +mtr ˆΨ ˆ Ψ 4m. 6.6 For q = 1/, the bosonc wve functons Φ q,s,l s egenfunctons of ths Hmltonn re constructed s Φ 1,1/,l = Tr A + p Tr A + 3 p 3ε 1 3 ε j Z 1 Z j A + Z 3 C 0, Φ 1,1/,l = Tr A + p Tr A + 3 p 3ε 1 3 ε j A + Z 1 A + Z j Z 3 C 0, Φ 1,3/,l = Tr A + p Tr A + 3 p 3ε 1 3 Z 1 A + Z j A + Z C j 0,

28 wth the coeffcents C, C, C j. They hve the followng energy vlues E 1/,l = ml 3, E 1/,l = ml, E 3/,l = ml 1, 6.8 where l = p +3p 3. The mnml energy s cheved on the stte nd t s equl to Φ 1,1/,0 = ε 1 3 ε j Z 1 1 Z A + Z E 1/,0 = 3m Csmr opertors te zero vlues on ths stte. The cton of the SU 1 superchrge, ˆQ j Φ 1,1/,0 = ε 1 3 ε 1 Z 1 1 Z ˆΨj Z 3 0, 6.31 produces n ddtonl fermonc stte whch, together wth 6.9 nd one more bosonc stte generted by further cton of superchrges on 6.31, consttute n typcl fundmentl SU 1 supermultplet. 6. Quntzton of the reduced spnnng Clogero Moser system We brefly dscuss quntzton of the reduced spnnng Clogero Moser mult-prtcle system wthout center-of-mss defned n Sect More explctly, we consder the two-prtcle cse n=. Introducng the holomorphc relzton Z := z, ψ = ψ, Z :=, p z := =, x ψ =, ψ ψ b c = ψ b c, ψb c = ψ, 6.3 c b the dfferentl relzton of Hmltonn 4.48 on physcl sttes s gven by 1 b +m x x b g b + n x x b +const <b Here g b re egenvlues of the quntum opertors 1 T b,t b } < b, nd they correspond to spn couplngs of two nterctng prtcles x nd x b. As ws shown n Sect. 4., one cn represent g b on bosonc sttes v 5.7 s g b = S j S bj +qq +1, < b Ths model s restrcted to postve nteger vlues of q nd s referred to s mtrx model n 30. Arbtrry vlues of q cn be cheved by pplyng the exchnge opertor formlsm nvolvng Dunl opertors see, e.g., 3. As ws lredy mentoned, our SU 1 supersymmetrc system yelds just the U spn mtrx model s ts bosonc core. 7

29 In the smpler cse g b =ϑϑ 1, quntzton ws gven n 31, 6, 7 v Dunl opertors defned s D = + b ϑ x x b 1 K b, 6.35 where K b s permutton opertor, K b x b =x K b. Below we consder the smplest cse n=, where g 1 = ss+1 nd the opertor K 1 becomes Klen-type opertor ctng on the reltve coordnte x 1 x s K 1 x 1 x = x x 1 K 1 = x 1 x K 1, K 1 = Let us consder n detls the two-prtcle system n =. It s descrbed by the lgebr of quntum opertors x,p =, ψ, ψ } j = δ j, Z 1, Z 1 j = Z, Z j = δ j, ψ 1, ψ } 1 j = ψ 1, ψ } j1 = δj, 6.37 where x = 1 x 1 x, p = 1 p 1 p, 6.38 ψ = 1 ψ 1 ψ, ψ = 1 ψ1 ψ Below we use the followng relzton for them x = x, p = x, Z 1 = z 1, Z = z, ψ 1 = ψ 1, ψ 1 = ψ 1, ψ = ψ, ψ = ψ, Z1 j =, Z j =, z j 1 z j ψ1 = ψ 1, ψ 1 = ψ Herex, z ndψ, ψ b, = 1,recomplexcommutngndfermoncntcommutngvrbles, respectvely. The Hmltonn 4.48 wthout center of mss tes the form H = 1 where x +m x +m T 1 T 1 ψ = z 1 = z ψ +ψ 1 ψ 1 +ψ 1 ψ 3 1 z z 1 + T 1,T 1 } 4x ψ ψ ψ 1 1 ψ, ψ ψ + ψ 1 1 ψ. 6.4 The opertors 6.4 ct n the followng wy on the vrbles enterng the wve functon T 1 : z z 1, ψ ψ 1, ψ 1 ψ ; T 1 : z 1 z, ψ ψ 1, ψ 1 ψ

30 nd gve zero, whle ctng on other vrbles. Thus, the opertors T 1 T 1 nd T 1 T 1 trnsform ll components n the expnson of the wve functon nto themselves, wth some coeffcents ncludng the vnshng ones. Therefore, on ll components the Hmltonn 6.41 hs the stndrd form wth the osclltor nd conforml potentls. As the result, we cn fnd ts energy spectrum. Ths system s smlr to the one we hve consdered n Sect. 5, but t hs wder set of fermonc felds. The spn s s ssocted wth the dgonl externl SU group 4.6. In contrst to 4.3, the SU subgroup 4.49 of SU 1 cts only on fermonc felds, whch gves dfferent degenercy pcture for the supergroup SU 1. Let us consder pure bosonc wve functons. They re gven by Ω q = q Ω q,s,l, l s=0 Ω q,s,l x,z1,zj = z q sz 1 z 1 1 z 1...z s 1 z s+1...z s As,l 1... s x, 6.44 nd re subject to the constrnts T 1 Ω q = T Ω q = qω q, q = 1,,3..., T 1 = z1 +ψ z1 1 ψ 1 ψ 1 ψ, 1 T = z ψ 1 ψ 1 +ψ 1 ψ z The egenvlue problem for the Hmltonn 6.41 mounts to the equton 1 x +m x + ss+1 6m A s,l x x 1... s = E s,la s,l 1... s, 6.46 whch s solved s 7 A s,l x 1... s = C 1... s x s+1 L s+1/ l mx exp mx /, l = 0,1,,..., E s,l = m l+s The energy spectrum s consstent wth the energy spectrum 6. clculted n the mtrx formulton. An lterntve constructon of wve functons cn be gven v creton nd nnhlton opertors 31, 6, 7. To solve the equton 6.46, we te the nstz: A s,l 1... s = C 1... s x s+1 Φ s,l +, 6.48 where Φ s,l + s n even functon of x. Introducng the Klen opertor K:=K stsfyng K = 1, Kx = xk, K x = x K, KΦ s,l + = Φ s,l +, As ws dscussed n 4, the equton 6.46 hs n ddtonl soluton whch ws thrown wy for s > 0 due to the presence of sngulrtes t x = 0. Here ths ddtonl soluton must be thrown wy even for the cse s = 0, pplyng the sme resonng to the fermonc expnson of the full wve functon. 9

31 we defne the creton nd nnhlton opertors ± through the Dunl opertor D: ± = D +mx, D = x + s+1 x 1 K, K ± = ± K Then eq s rewrtten s where 1 x +m x + ss+1 6m x A s,l = x s+1 H + Φ s,l +, 6.51 H + = m sk +K 5, H+, ± = ±m ±. 6.5 Tng nto ccount tht KΦ s,l + = Φ s,l +, the functon Φ s,l + s expressed s nd the ssocte energy spectrum s Φ s,l + = + l Φ s,0 +, Φ s,0 + = 0, 6.53 H + Φ s,l + = E s,l Φ s,l +, E s,l = m One cn lso choose n lterntve nstz stsfyng l+s 3, l = 0,1,, A s,l = x s Φ s,l, 6.55 KΦ s,l = Φ s,l The constructon of wve functons v the creton nd nnhlton opertors wll gve the sme soluton for the energy spectrum s We sp detls of ths constructon whch s smlr to the prevous one. 7 Quntzton of the full system center-of-mss plus reltve-coordnte sectors The energy spectrum of theunfed system, whch s sum ofthe center-of-mss sector of Sect.5 nd the reltve coordnte system of Sect.6, cn be found s tensorl product of the spectr of these two subsystems. In the unguged mtrx formulton, the bosonc wve functons re generlzton of 6.1 n the holomorphc relzton 6.8: Φ q,s,l f j1...j s x 0 Tr â + p Tr â + 3 p 3... â+ n pn Tr q 1 r=0 â+ ε 1... n l nr+1 â+ z nr+1... l nr+n z nr+n 1 n Ther generl structure s qute specfed by the three requrements: 30 }

32 The wve functons should be Un nvrnt s consequence of the constrnt 4.1 or ts equvlent form 4.8. Ths mens tht ll Un ndces should be contrcted wth the pproprte nvrnt tensors; They should be of degree qn wth respect to the whole set of spn vrbles n vrtue of the constrnt 4.14; All free SU ndces of the spn vrbles should be symmetrzed nd contrcted wth the ndces of f 1... s x 0. The energy spectrum of dmssble spns of these functons extends from s = 0 to nq for nq even nd from s = 1/ to nq for nq odd. All the fermonc wve functons cn be obtned by cton of the totl superchrges on7.1. The bsc dstnctons of the totl system from the mult-prtcle system of Sect. 6 concern the relztons of the SU symmetry pperng n the nt-commuttors of the superchrges s n nternl subgroup of SU 1. In the system wth the center-of-mss sector detched consdered n Sect.6, ths SU symmetry s gven by 4.49, cts only on the fermonc opertors nd gves rse just to degenercy of the energy spectrum. In the totl system, the nternl SU symmetry cts on the ndces,j,... of ll components of the wve functons 7.1 nd ther fermonc completon. Tng nto ccount the nlyss of the prevous secton, we see tht the problem of descrpton of ll sttes n the unfed cse the opton III n Sect. 4.3 for n rbtrry n s rther complcted. At the sme tme, we cn drectly determne, for ll possble cses, the full energy spectrum smply by pplyng the methods of the prevous sectons. Let us brefly descrbe the energy spectrum for the choce of n =. In ths smplest cse the mtrx system s descrbed by the Hmltonn H = 1 x 0 +m x 0 1 +mψ ψ0 x 0 4 S S +S ψ 0 ψ0 +mtr A + A +mtr Ψ Ψ m. 7. The trceless prt of the generl constrnts 6.5, G b = n X 0,P 0 δ b + A +,A b + Ψ,Ψ } b +Z Z b q +nδ b 0, 7.3 requres wve functons to be SUn sclrs, whle ts trce prt fxes the degree of homogenety wth respect to spn vrbles: G = 0 Z Z 4q = The bosonc wve functons re constructed s Φ q,0,l,l 0 = H l0 x 0 e mx 0 Tr A + l εj ε b Z q b Zj 0, 7.5 Φ q,s,l,l 0 = A l s x 0 Tr A + l q sz εj ε b Z b Zj 1 1 Z...Z s s s ε b A + s+ b 0, s = 1,...q, 7.6 =1 31