Exact Heisenberg operator solutions for multi-particle quantum mechanics
aa r X i v : . [ qu a n t - ph ] J un Yukawa Institute Kyoto
DPSU-07-1YITP-07-26 arXiv:0706.0768
June 2007
Exact Heisenberg operator solutions for multi-particlequantum mechanics
Satoru Odake a and Ryu Sasaki b a Department of Physics, Shinshu University,Matsumoto 390-8621, Japan b Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan
Abstract
Exact Heisenberg operator solutions for independent ‘sinusoidal coordinates’ asmany as the degree of freedom are derived for typical exactly solvable multi-particlequantum mechanical systems, the Calogero systems based on any root system. TheseHeisenberg operator solutions also present the explicit forms of the annihilation-creationoperators for various quanta in the interacting multi-particle systems. At the same timethey can be interpreted as multi-variable generalisation of the three term recursion re-lations for multi-variable orthogonal polynomials constituting the eigenfunctions.
Modern quantum physics is virtually unthinkable without annihilation-creation operators,which are defined as the positive/negative energy parts of the
Heisenberg field operatorsolutions of a free field theory, an infinite collection of independent harmonic oscillators.These annihilation-creation operators map an eigenstate of a free Hamiltonian into another,but not connecting those of a full theory. Our knowledge of the Heisenberg operator solutionsof a full interacting theory, on the other hand, is quite limited in spite of the central roleplayed by the Heisenberg operator solutions in field theory in general. In the so-called exactlyolvable quantum field theories, factorised S -matrices and some of the correlation (Green’s)functions are the highest achieved points up to now.Following our embryonic work [1] on the construction of exact Heisenberg operator solu-tions for various degree one quantum mechanics, we present in this paper a modest first stepin the quest of deriving exact Heisenberg operator solutions for a family of interacting multi-particle dynamics. These are the Calogero systems [2], which are integrable multi-particledynamics based on root systems. For the theories based on the classical root systems, the A , B , C and D series, the number of particles can be as large as wanted, but not infiniteas in field theories. A complete set of exact Heisenberg solutions for ‘sinusoidal coordinates’[1, 3], as many as the degree of freedom, is derived elementarily in terms of the universalLax pair [4, 5, 6], which is a well-established solution mechanism for classical and quantumCalogero-Sutherland-Moser systems [2, 7, 8] based on any root system [9]. Explicit formsof various annihilation-creation operators are obtained as the positive/negative energy partsof the Heisenberg operator solutions. They map an eigenvector of the full Hamiltonian intoanother. These sinusoidal coordinates and the corresponding annihilation-creation operatorsprovide multi-variable generalisation of the three term recursion relations [1] of orthogonalpolynomials constituting the eigenfunctions of the Calogero Hamiltonian.This paper is organised as follows. In section two, the rudimentary facts and notationof Calogero systems based on any root system are recapitulated together with the universalLax pair matrices. In section three the complete set of exact Heisenberg operator solutionsis derived quite elementarily based on generating functions constructed from the universalLax matrices. Remarks on the special features of the D -type theories are given at the endof the section. The final section is for a summary and comments for further research. TheAppendix gives the list of the preferred sets of weight vectors for explicit representations ofLax matrices based on the exceptional and the non-crystallographic root systems. In this paper we will derive exact Heisenberg operator solutions for the Calogero systems .They are one-dimensional multi-particle dynamics with inverse (distance) potential inside aharmonic confining potential. They have a remarkable property that they are exactly solvableat the classical [10] and quantum [2, 6] levels. The exact quantum solvability has been shownin the Schr¨odinger picture , as the entire energy spectrum is known and the corresponding2igenfunctions can be constructed explicitly by a finite number of algebraic processes (thelower triangularity of the Hamiltonian in certain basis) [6]. The exact quantum solvabilityin the
Heisenberg picture of the Calogero systems will be demonstrated in the next sectionby constructing the explicit Heisenberg operator solutions for the independent ‘sinusoidalcoordinates’ as many as the degree of freedom. Let us briefly recapitulate the essence of thequantum Calogero systems [2, 6] together with appropriate notation necessary in this paper.
A Calogero system is a Hamiltonian dynamics associated with a root system [4, 9] ∆ of rank r , which is a set of vectors in R r with its standard inner product. Its dynamical variables arethe coordinates { q j } and their canonically conjugate momenta { p j } , satisfying the canonicalcommutation relations :[ q j , p k ] = iδ jk , [ q j , q k ] = [ p j , p k ] = 0 , j, k = 1 , . . . , r. (2.1)These will be denoted by vectors in R r , q def = ( q , . . . , q r ) , p def = ( p , . . . , p r ) , p · q def = r X j =1 p j q j , p = r X j =1 p j , q = r X j =1 q j . (2.2)The momentum operator p j acts as a differential operator p j = − i ∂∂q j , j = 1 , . . . , r. The ‘factorised’ Hamiltonian is H ( p, q ) def = 12 r X j =1 (cid:16) p j − i ∂W∂q j (cid:17)(cid:16) p j + i ∂W∂q j (cid:17) , (2.3)= 12 p + ω q + 12 X ρ ∈ ∆ + g | ρ | ( g | ρ | − | ρ | ( ρ · q ) − E , (2.4) W ( q ) def = − ω q + X ρ ∈ ∆ + g | ρ | log | ρ · q | , g | ρ | > , ω > . (2.5)The summation is over the set of positive roots ∆ + , with ∆ = ∆ + ∪ ( − ∆ + ). The real positive coupling constants g | ρ | are defined on orbits of the corresponding Coxeter group, i.e. For the A -type theory, it is customary to consider A r − and to embed all the roots in R r . This isaccompanied by the introduction of one more degree of freedom, q r and p r . The genuine A r − theorycorresponds to the relative coordinates and their momenta, and the extra degree of freedom is the center ofmass coordinate and its momentum. H ( s α ( p ) , s α ( q )) = H ( p, q ) , ∀ α ∈ ∆ , (2.6)with the reflection s α defined by s α ( x ) def = x − ( α ∨ · x ) α, ∀ x ∈ R r , α ∨ def = 2 α/ | α | . (2.7)Obviously the Heisenberg equations of motion for the coordinates { q j } are trivial i [ H , q j ] = ddt q j = p j , j = 1 , . . . , r, (2.8)whereas those for the canonical momenta { p j } have the same form as the Newton equations: d dt q j = ddt p j = i [ H , p j ] = − ω q j + X ρ ∈ ∆ + g | ρ | ( g | ρ | − | ρ | ρ j ( ρ · q ) , j = 1 , . . . , r. (2.9)The hard repulsive potential ∼ / ( ρ · q ) near the reflection hyperplane H ρ def = { q ∈ R r | ρ · q =0 } is insurmountable at the quantum level as well as the classical. Thus the motion is alwaysconfined within one Weyl chamber. This feature allows us to constrain the configurationspace to the principal Weyl chamber (Π: set of simple roots) P W def = { q ∈ R r | α · q > , α ∈ Π } , (2.10)without loss of generality.The positive semi-definite form of the factorised Hamiltonian (2.3) simply allows thedetermination of the ground state wavefunction: H φ ( q ) = 0 , φ ( q ) def = e W ( q ) = Y ρ ∈ ∆ + | ρ · q | g | ρ | · e − ω q , (2.11)which is real and obviously square integrable Z P W φ ( q ) d r q < ∞ . (2.12)The constant part E of the Hamiltonian (2.3) is usually called the ground state energy E = ω (cid:16) r X ρ ∈ ∆ + g | ρ | (cid:17) . (2.13)4he excited energy spectrum is integer spaced and is independent of the coupling constants { g | ρ | } [6]: H φ n ( q ) = E n φ n ( q ) , n def = ( n , . . . , n r ) , n j ∈ N def = Z ≥ , (2.14) E n def = ωN n , N n def = r X j =1 n j f j . (2.15)Here { f j } are the integers related to the exponents { e j } of ∆: f j def = 1 + e j . (2.16)They indicate the degrees where independent Coxeter invariant polynomials exist. The setof integers F ∆ def = { f , f , . . . , f r } (2.17)is shown in Table I for each root system ∆.∆ F ∆ ∆ F ∆ A r − { , , . . . , r, } E { , , , , , , , } B r { , , , . . . , r } F { , , , } C r { , , , . . . , r } G { , } D r { , , . . . , r − , r } I ( m ) { , m } E { , , , , , } H { , , } E { , , , , , , } H { , , , } Table I: The set of integers F ∆ = { f , f , . . . , f r } for which independentCoxeter invariant polynomials exist. Excited states eigenfunctions have the following general structure: φ n ( q ) = φ ( q ) P n ( q ) , (2.18)in which P n ( q ) is a Coxeter invariant polynomial in { q j } of degree N n . For A r − root system, f r = 1 corresponds to the degree of freedom for the center of mass coordinate.The B r and C r Calogero systems are equivalent. .2 Quantum Lax Pair The derivation of the Heisenberg operator solutions depends heavily on the universal Laxpair which applies to any root system. The universal Lax pair operators [5, 6] are L ( p, q ) def = p · ˆ H + X ( q ) , X ( q ) def = i X ρ ∈ ∆ + g | ρ | ( ρ · ˆ H ) ρ · q ˆ s ρ , (2.19) M ( q ) def = − i X ρ ∈ ∆ + g | ρ | | ρ | ( ρ · q ) ˆ s ρ + i X ρ ∈ ∆ + g | ρ | | ρ | ( ρ · q ) × I, (2.20)in which I is the identity operator and { ˆ s α : α ∈ ∆ } are the reflection operators of the rootsystem. They act on a set of R r vectors R def = { µ ( k ) ∈ R r | k = 1 , . . . , d } , permuting themunder the action of the reflection group. The vectors in R form a basis for the representationspace V of dimension d . The operator M satisfies the relation [5, 6] X µ ∈R M µν = X ν ∈R M µν = 0 , (2.21)which is essential for deriving quantum conserved quantities and annihilation-creation op-erators. The matrix elements of the operators { ˆ s α : α ∈ ∆ } and { ˆ H j : j = 1 , . . . , r } aredefined as follows:(ˆ s ρ ) µν def = δ µ,s ρ ( ν ) = δ ν,s ρ ( µ ) , ( ˆ H j ) µν def = µ j δ µν , ρ ∈ ∆ , µ, ν ∈ R . (2.22)The Lax equation i [ H , L ] = ddt L = [ L, M ] (2.23)is equivalent to the Heisenberg equation of motion for { q j } and { p j } for the Hamiltonian(2.4) without the harmonic confining potential, i.e, ω = 0. It should be emphasised that thel.h.s. of (2.23) is a quantum commutator, whereas the r.h.s. is a matrix commutator as wellas quantum. The full Heisenberg equations of motion with the harmonic confining potentialread i [ H , L ± ] = ddt L ± = [ L ± , M ] ± iωL ± , (2.24)in which M is the same as before (2.20), and L ± and Q are defined by L ± def = L ± iωQ, Q def = q · ˆ H, ( L + ) † = L − , (2.25)6ith L , ˆ H as earlier (2.19), (2.22). One direct consequence of the Lax pair equation (2.24)is the existence of a wide variety of quantum conserved quantities : ddt Ts( L ǫ L ǫ · · · L ǫ k ) = 0 , ǫ j ∈ { + , −} , ∀ k ∈ N , k X j =1 ǫ j = 0 , (2.26)in which Ts( A ) denotes the total sum of a matrix A [5, 6] with suffices µ, ν ∈ R :Ts( A ) def = X µ,ν ∈R A µ ν . (2.27)This is a simple outcome of the property (2.21) of the matrix M [5, 6, 11, 12]. The quantumconserved quantities are the simplest example of more general results: i [ H , Ts( L ǫ L ǫ · · · L ǫ k )] = ddt Ts( L ǫ L ǫ · · · L ǫ k ) , ∀ k ∈ N , = iωm Ts( L ǫ L ǫ · · · L ǫ k ) , k X j =1 ǫ j = m. (2.28)In other words, Ts( L ǫ L ǫ · · · L ǫ k ) shifts the eigenvalue of H by m unit (or, ωm ) when it actson any eigenstate of H . Namely such operators are all candidates of annihilation-creationoperators for H : e i H t Ts( L ǫ L ǫ · · · L ǫ k ) e − i H t = Ts( L ǫ L ǫ · · · L ǫ k ) e iωmt , k X j =1 ǫ j = m. (2.29)Note that the order of L + and L − is immaterial.The results and statements in this subsection are universal , i.e. they are valid for anyroot system ∆ and for any choice of the Lax pair matrices, i.e. the choice of R . For derivingthe explicit forms of the Heisenberg operator solutions, however, it is convenient to chooseas R the set of the least dimensions for each ∆. For the classical root systems, A , B and D ,they are the set of vector weights : A r − : R def = { e , e , . . . , e r } , d def = R = r, (2.30) B r , D r : R def = {± e , ± e , . . . , ± e r } , d def = R = 2 r, (2.31)in which { e j } are the orthonormal basis of R r , e j · e k = δ j k . We list in the Appendix thepreferred choice of R for the exceptional and the non-crystallographic root systems. To be more precise, for B r they should be called the set of short roots. Exact Heisenberg Solutions
Now let us proceed to derive the explicit Heisenberg operator solutions for various ‘sinusoidalcoordinates’ [1, 3] as many as the degree of freedom. For simplicity of presentation, let usfix the angular frequency of the harmonic confining potential as unity, ω = 1 hereafter. Wehave already known one exact Heisenberg operator solution. In [1] it was shown that thequadratic invariant η (1) ∝ q = r X j =1 q j (3.1)is the simplest sinusoidal coordinate:[ H , [ H , η (1) ]] = 4( η (1) − c H − c ) , c , c : const , (3.2)where c is given by η (1) = c q and c = c E . As shown in [1] the exact Heisenberg solution e i H t η (1) e − i H t (3.3)is easily evaluated. It should be noted that any root system has a degree 2 Coxeter invariant2 ∈ F ∆ (Table I), which is proportional to η (1) . In fact, η (1) is a sinusoidal coordinate [1] fora Hamiltonian H ge more general than the Calogero system; a general homogeneous potentialof degree − H ge = 12 r X j =1 ( p j + q j ) + V ( q ) , r X j =1 q j ∂∂q j V ( q ) = − V ( q ) , (3.4)which satisfies [ H ge , [ H ge , q ]] = 4( q − H ge ).For the Calogero system based on any root system ∆, we will derive the explicit formsof the Heisenberg operator solutions: e i H t η ( j ) e − i H t , j = 1 , . . . , r, (3.5)for a complete set of sinusoidal coordinates defined by { η (1) , η (2) , . . . , η ( r ) } , η ( j ) def = Ts( Q f j ) = Tr( Q f j ) , f j ∈ F ∆ . (3.6)In (3.6), the matrix Q (2.25) is diagonal therefore its total sum (Ts) is the same as thetrace (Tr). Let us note that any Coxeter invariant polynomial in { q j } can be expressed as8 polynomial in { η (1) , η (2) , . . . , η ( r ) } . It is easy to verify η (1) ∝ q for an root system. Thehigher sinusoidal coordinates for the classical root systems are: A r − : η ( k ) ∝ r X j =1 q k +1 j , k = 1 , . . . , r − η ( r ) ∝ r X j =1 q j , (3.7) B r , D r : η ( k ) ∝ r X j =1 q kj , k = 1 , . . . , r, (3.8)except for η ( r ) for D r which reads η ( r ) ∝ r Y j =1 q j . (3.9)See the remark at the end of the section. As shown in Table I, all the integers { f j } are even for B r and D r , except for f r = r for odd r of D r . This is also related to the fact thatTr( Q l +1 ) = r X j =1 q l +1 j + r X j =1 ( − q j ) l +1 = 0 , for the Lax operators based on the vector weights (2.31).The sinusoidal coordinates take various different forms for the exceptional and non-crystallographic root systems. Note that the overall normalisation of { η ( k ) } is immaterial.The derivation of exact Heisenberg operator solutions is quite elementary. Let us in-troduce a generating function of the total sum of the homogeneous polynomials in L + and L − : G ( j ) ( s ) def = Ts (cid:0) ( L + + s L − ) f j (cid:1) , s ∈ C , f j ∈ F ∆ , (3.10)which is a polynomial in s of degree f j G ( j ) ( s ) = f j X l =0 b f j ; f j − l s l , b † f j ; l = b f j ; − l . (3.11)The coefficient b f j ; f j − l is the total sum of a completely symmetric product consisting of f j − l times L + and l times L − and it can be explicitly evaluated. As shown in (2.29), we obtain e i H t b f j ; f j − l e − i H t = b f j ; f j − l e i ( f j − l ) t . (3.12)Namely b f j ; f j − l is either an annihilation operator ( l > f j /
2) being the positive energy part or a creation operator ( l < f j /
2) being the negative energy part or a conserved quantity( l = f j / b f j ; 0 , b f k ; 0 ] = 0 , j = k. (3.13)9n the other hand, for s = − L + − L − = 2 iQ and G ( j ) ( −
1) = (2 i ) f j Ts( Q f j ) = (2 i ) f j η ( j ) = f j X l =0 ( − l b f j ; f j − l ,η ( j ) = (2 i ) − f j f j X l =0 ( − l b f j ; f j − l . (3.14)Thus we arrive at the main result of the paper; the complete set of exact Heisenberg operatorsolutions for the ‘sinusoidal coordinates’ { η ( j ) } : η ( j ) ( t ) def = e i H t η ( j ) e − i H t = (2 i ) − f j f j X l =0 ( − l b f j ; f j − l e i ( f j − l ) t , j = 1 , , . . . , r. (3.15)This clearly shows that η ( j ) ( t ) is a superposition of various sinusoidal motions. The trivialfact that these ‘sinusoidal coordinates’ commute among themselves η ( j ) η ( k ) = η ( k ) η ( j ) , j = k = 1 , . . . , r (3.16)is translated into the commutation relations among the annihilation-creation operators X l,ml + m :fixed [ b f j ; f j − l , b f k ; f k − m ] = 0 . (3.17)Among the annihilation-creation operators belonging to η ( j ) , the two extreme ones corre-sponding to l = 0 and l = f j have a special meaning. They consist of L + ( L − ) only b f j ; f j = Ts (cid:0) ( L + ) f j (cid:1) , b f j ; − f j = Ts (cid:0) ( L − ) f j (cid:1) , b † f j ; f j = b f j ; − f j (3.18)and commute among themselves[ b f j ; f j , b f k ; f k ] = 0 , [ b f j ; − f j , b f k ; − f k ] = 0 , j, k = 1 , . . . , r, (3.19)as is clear from (3.17). These special annihilation-creation operators have been known forsome time [6, 9, 12, 14]. 10ince the l -th term in (3.15) is annihilated by d/dt − i ( f j − l ), we obtain a lineardifferential equation with constant coefficients satisfied by η ( j ) ( t ): f j Y l = − fj step 2 (cid:16) ddt − il (cid:17) · η ( j ) ( t ) = 0 . (3.20)Equivalently this can be rewritten as f j Y l = − fj step 2 (cid:0) ad( H ) − l (cid:1) · η ( j ) = 0 . (3.21)Here ad( H ) denotes a commutator ad( H ) X def = [ H , X ] for any operator X . For even f j case,the factor d/dt − i ( f j − l ) for l = f j / f jf j Y l = − fj, =0step 2 (cid:16) ddt − il (cid:17) · (cid:0) η ( j ) ( t ) − − f j b f j ; 0 (cid:1) = 0 , (3.22) f j Y l = − fj, =0step 2 (cid:0) ad( H ) − l (cid:1) · (cid:0) η ( j ) − − f j b f j ; 0 (cid:1) = 0 , (3.23)instead of f j + 1 for odd f j case. In [1], for a wide class of solvable quantum systems withone degree of freedom, we discussed the ‘closure relation’[ H , [ H , η ]] = ηR ( H ) + [ H , η ] R ( H ) + R − ( H ) (3.24)with R i ( H ) † = R i ( H ). By introducing α ± ( H ) as R ( H ) = α + ( H ) + α − ( H ) and R ( H ) = − α + ( H ) α − ( H ), this closure relation is rewritten as (cid:0) ad( H ) + α + ( H ) (cid:1)(cid:0) ad( H ) + α − ( H ) (cid:1)(cid:0) η + R ( H ) − R − ( H ) (cid:1) = 0 . (3.25)Eqs. (3.21) and (3.23) are multi-particle generalisation of this relation.When a sinusoidal coordinate η ( j ) is multiplied to an eigenvector φ n of H ,(2 i ) f j η ( j ) φ n = f j X l =0 ( − l b f j ; f j − l φ n , (3.26)the l -th term belongs to the eigenspace of H with the eigenvalue E n + f j − l . Thus these f j + 1 terms are all orthogonal to each other. This is a multi-variable generalisation of the11 hree-term recursion relation of the orthogonal polynomials of one variable [1]. Througha similarity transformation in terms of the ground state wavefunction φ ( q ) (2.11), let usdefine ˜ L def = φ − ◦ L ◦ φ , ˜ L ± def = φ − ◦ ˜ L ± ◦ φ = ˜ L ± iQ, (3.27)˜ b f j ; f j − l def = φ − ◦ b f j ; f j − l ◦ φ , j = 1 , . . . , r, l = 0 , , . . . , f j . (3.28)Needless to say the identity φ − ◦ Ts( A n ) ◦ φ = Ts (cid:0) ( φ − ◦ A ◦ φ ) n (cid:1) holds for any matrix A consisting of operators. Then one can present the correspondingresults for the multi-variable orthogonal polynomials { P n ( q ) } (2.18) constituting the eigen-vectors: (2 i ) f j η ( j ) P n ( q ) = f j X l =0 ( − l ˜ b f j ; f j − l P n ( q ) . (3.29)The operators ˜ L , ˜ L ± and ˜ b f j ; f j − l are closely related to the Dunkl operators [6, 15].The annihilation-creation operators provide an algebraic solution method of the Calogerosystems. The entire Hilbert space is generated by the multiple application of creation oper-ators on the ground state wavefunction φ ( q ) = e W ( q ) : r Y j =1 Y 1) = (2 i ) r Ts( Q r (sp) ) = (2 i ) r η ( r ) = r X l =0 ( − l b (sp) r ; r − l , (3.35) η ( r ) = (2 i ) − r r X l =0 ( − l b (sp) r ; r − l ∝ q q · · · q r , (3.36) η ( r ) ( t ) def = e i H t η ( r ) e − i H t = (2 i ) − r r X l =0 ( − l b (sp) r ; r − l e i ( r − l ) t . (3.37)For D even , the situation is slightly more complicated, partly because of the existence ofanother sinusoidal coordinate η ( r/ , which is a Coxeter invariant polynomial of { q j } of degree r , too. Here we prepare the Lax matrices based on both the spinor L ± (sp) and the anti-spinorweights L ± (as) , since η ( r ) ∝ Ts( Q r (sp) ) − Ts( Q r (as) ) ∝ q q · · · q r . (3.38)It is quite elementary to verify (3.36) and (3.38). Let us introduce two ‘generating functions’ G ( r )(sp) ( s ) def = Ts (cid:0) ( L +(sp) + s L − (sp) ) r (cid:1) = r X l =0 b (sp) r ; r − l s l , (3.39) G ( r )(as) ( s ) def = Ts (cid:0) ( L +(as) + s L − (as) ) r (cid:1) = r X l =0 b (as) r ; r − l s l . (3.40)13ext we introduce the sinusoidal coordinate η ( r ) as their difference at s = − η ( r ) def = (2 i ) − r (cid:0) G ( r )(sp) ( − − G ( r )(as) ( − (cid:1) = Ts( Q r (sp) ) − Ts( Q r (as) )= (2 i ) − r r X l =0 ( − l (cid:0) b (sp) r ; r − l − b (as) r ; r − l (cid:1) . (3.41)We obtain the corresponding exact Heisenberg operator solution η ( r ) ( t ) def = e i H t η ( r ) e − i H t = (2 i ) − r r X l =0 ( − l (cid:0) b (sp) r ; r − l − b (as) r ; r − l (cid:1) e i ( r − l ) t . (3.42)This completes the derivation of the exact Heisenberg operator solutions for the D -typeCalogero systems. A complete set of exact Heisenberg operator solutions, as many as the degree of freedom, isconstructed for the Calogero systems based on any root system, including the exceptional andnon-crystallographic ones. Based on the complete set, one can write down the Heisenbergoperator solution e i H t A e − i H t for any operator A expressible as a polynomial in the sinusoidalcoordinates { η ( j ) } . This is the first demonstration of the exact solvability of multi-particlequantum mechanics in the Heisenberg picture . At the same time, these Heisenberg operatorsolutions provide the explicit forms of various annihilation-creation operators , as the positive and negative energy parts . Their commutation relations are, in general, quite involved. As inthe simplest case of degree one quantum mechanics [1], these sinusoidal coordinates and theirexpansion into the annihilation-creation operators provide the explicit forms of the multi-variable generalisation of the three term recursion relations for the orthogonal polynomialsconstituting the multi-variable eigenfunctions. The derivation of the Heisenberg operatorsolutions is a simple consequence of the universal Lax pair, which manifests the quantumintegrability of Calogero systems based on any root system .Let us conclude this paper with a few comments on possible future directions of thepresent research. For better understanding of multi-particle quantum mechanics in general, itis desirable to enlarge the list of exact Heisenberg operator solutions. The obvious candidatesare: the Sutherland systems [7] having trigonometric potentials, various super-symmetricgeneralisations of the Calogero-Sutherland systems [5, 11], the Ruijsenaars-Schneider-van-14iejen systems [16] which are ‘discrete’ counterparts of the Calogero-Sutherland systems,and the (affine) Toda molecules.The newly found annihilation-creation operators suggest an interesting possibility of in-troducing multi-particle coherent states as common eigenstates of certain annihilation oper-ators. It is a good challenge to construct explicit examples of such multi-particle coherentstates having mathematically elegant structure and/or practical use.A completely integrable system, including the Calogero-Sutherland-Moser systems, hasthe so-called hierarchy structure . It is characterised by the existence of mutually involutiveconserved quantities H , H , . . . , H r , [ H j , H k ] = 0 , j, k = 1 , . . . , r, (4.1)which could be adopted as independent Hamiltonians generating different but compatibletime-flows; t , t ,. . . , t r , as many as the degree of freedom. It is a good challenge to constructcommon Heisenberg operator solutions to all the flows of the hierarchy e i P rj =1 H j t j ˜ η ( k ) e − i P rj =1 H j t j , k = 1 , . . . , r. (4.2) Acknowledgements This work is supported in part by Grants-in-Aid for Scientific Research from the Ministryof Education, Culture, Sports, Science and Technology, No.18340061 and No.19540179. Appendix: The Preferred Choice of R Here we list, for the exceptional and the non-crystallographic root systems, the set R to beused for the explicit evaluation of the Heisenberg operator solutions. They are of the lowestdimensionality.1. E : The weights of (or ) dimensional representation of the Lie algebra. 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