On the superintegrability of the rational Ruijsenaars-Schneider-van Diejen models

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On the superintegrability of the rational Ruijsenaars-Schneider-van Diejen models

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Viktor Ayadi

Department of Theoretical Physics, University of Szeged, Hungary

June 13, 2012

1Based on joint work with L. Feh´er.

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Content

1 (Super)integrability

2 Wojciechowski’s Poisson algebra

3 Rational Ruijsenaars-Schneider model

4 BC_n generalization

5 Conclusion

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5 Conclusion

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Let us consider a Hamiltonian system (M, ω,h), where (M, ω) is a 2n dimensional symplectic manifold and h is the Hamiltonian.

Liouville integrability

The system is Liouville integrableif there exist n independent functions,h_i ∈C^∞(M) (i = 1, . . . ,n); that mutually Poisson commute and the Hamiltonianh is equal to one of the h_i. Maximal superintegrability

A Liouville integrable system is called maximally

superintegrableif it has (n−1) additional constants of motion (denote these asfj ∈C^∞(M)) that are time independent, globally smooth and the (2n−1) functions

h1, . . . ,hn,f1, . . . ,fn−1

are independent (their differentials are linearly independent) on a dense submanifold ofM.

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Examples of superintegrable systems Isotropic harmonic oscillator

The Kepler problem and some magnetic analogues^a Rational Calogero model^b

Rational Ruijsenaars-Schneider model^c

aA. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco,Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles, Ann. Phys. 324(2009)

bS. Wojciechowski,Superintegrability of the Calogero-Moser system, Phys.

Lett.95A(1983) 279-281

cV. Ayadi and L. Feh´er,On the superintegrability of the rational Ruijsenaars-Schneider model, Phys.Let.A374(2010)

Remark: Systems with repulsive interactions are expected to be superintegrable in general, but even for such systems it is interesting to exhibit extra constants of motion in explicit form.

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Superintegrability of Kepler problem

Let us consider the Hamiltonian system (T^∗R³,P

idp_i ∧dq_i,H), where the Hamiltonian is given by

H(q,p) = 1

2mp²−k q .

The system is Liouville integrable: H,J_z,J² are the independent Poisson commuting constants of motions (J =q×p).

Additional constants of motions are the components of the Laplace-Runge-Lenz vector:

A=p×J−mkˆq.

Only two components ofAare independent, sinceA·J= 0.

With the constants of motionsH,Jz,J²,Az,A² the Kepler problem ismaximally superintegrable.

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5 Conclusion

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Let us focus on then particle rational Calogero model defined by H(x,p) = 1

2

n

X

j=1

p_j²+X

j6=k

χ² (x_j −x_k)².

Following Wojciechowski², consider the functions

Ij = tr(L^j) I_k¹ = tr(XL^k+1), j,k ∈Z,j ≥1,k ≥ −1,

with then×n matrices

Xij =δijxi and Lij =δijpi+ (1−δij)i χ (xi−xj).

2S. Wojciechowski,Superintegrability of the Calogero-Moser system, Phys.

Lett.95A(1983) 279-281

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The functions on the previous slide satisfy the Poisson bracket algebra

{I_k,Ij}_M = 0, {I_k¹,Ij}_M =jIj+k,{I_k¹,I_j¹}_M = (j −k)I_k+j¹ .

Wojciechowski used this algebra to show the superintegrability of the Calogero Hamiltonian given in terms of the Lax matrix as H=I2/2.

Extra constants of motions

In fact, he found the following independent extra constants of motion

Kj =I_j¹₋₂I1−I₋₁¹ Ij, j ∈ {2, . . . ,n}.

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We shall see later that the rational Calogero and RS models share the same ”Wojciechowski algebra”

{I_j,I_k}_M = 0 {I_k¹,I_j}_M =jI_j+k {I_k¹,I_j¹}_M = (j−k)I_k+j¹ . One difference is that in the RS case the functionsI_j and I_k¹ are globally smoothfor all integersj and k. In particular, in this case the functionsI_k¹ realize the centerless Virasoro algebra over the full phase space.

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Additional constants of motion

We can immediately see from the ”Wojciechowski algebra” that I_j¹{I_k¹,I}_M −I_k¹{I_j¹,I}_M

have vanishing Poisson brackets withI; for anyI =I(I₁, . . . ,I_n).

In our cases the 2n functions

I₁, . . . ,I_n,I₁¹, . . . ,I_n¹

are independent. This means that the Jacobian J= det ∂(I1, . . . ,In,I₁¹, . . . ,I_n¹)

∂(p1, . . . ,pn,q1, . . . ,qn) is non-vanishing generically.

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Consequence: Superintegrability ofI_j for all j=1, . . . ,n For any fixedj the functions

C_k,j :=I_k¹I_2j −I_j¹I_k_+j, k ∈ {1,2, . . . ,n} \ {j}

Poisson commute withI_j. Using the functions I1, . . .In,I₁¹, . . .I_n¹ as coordinates we can determine the Jacobian

Jj := det∂(I_a,C_b,j)

∂(I_α,I_β¹) , where a, α∈ {1, . . . ,n}

b, β∈ {1, . . . ,n} \ {j} . The value of the determinant isJj = (I2j)ⁿ⁻¹.

SinceJ_j is generically non-zero, it follows that I_j is superintegrable with the (2n−1) independent functions

I₁, . . . ,I_n,C_k_,j ,where k ∈ {1, . . .n} \ {j}.

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5 Conclusion

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The rational Ruijsenaars-Schneider model The phase space of the model is

M =C_n×Rⁿ ={(q,p)|q ∈ C_n,p ∈Rⁿ}, where

C_n:={q ∈Rⁿ |q₁>q₂ >· · ·>q_n}.

The symplectic structure ω=Pn

k=1dp_k ∧dq_k corresponds to the fundamental Poisson brackets

{q_i,pj}_M =δi,j, {q_i,qj}_M ={p_i,pj}_M = 0.

The rational RS Hamiltonian is given by H_RS =

n

X

k=1

cosh(p_k)Y

j6=k

1 + χ² (qk −qj)²

¹₂ .

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The rational RS model can be considered as a relativistic generalization of the rational Calogero model. To see this³, introduce the functions:

P_RS =

n

X

k=1

sinh (p_k)Y

j6=k

1 + χ² (q_k −q_j)²

¹₂ ,

and

BRS =−

n

X

i=1

qi.

(H_RS,P_RS,B_RS) are the generators of the Poincar´e algebra in (1 + 1) dimension

{H_RS,B_RS}_M =P_RS, {P_RS,B_RS}_M =H_RS, {H_RS,P_RS}_M = 0.

3S.N.M. Ruijsenaars and H. Schneider,A new class of integrable models and their relation to solitons, Ann. Phys. 170(1986)

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The commuting HamiltoniansI_k are traces of thekth powers of the Lax matrixL. The Lax matrix (Hermitian, positive definite)

L(q,p)_j,k =uj(q,p)

iχ iχ+ (q_j −q_k)

u_k(q,p),

with theR+ valued functions uj(q,p) :=e^p^j Y

m6=j

1 + χ² (q_j −q_m)²

¹₄ .

We define the functionsI_j,I_k¹ ∈C^∞(M) by Ij(q,p) := tr L(q,p)^j

, I_k¹(q,p) := tr

L(q,p)^kq

∀j,k∈Z, using the diagonal matrixq:= diag(q1, . . . ,qn).

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The functionsIj,I_k¹ satisfy the same ”Wojciechowski Poisson bracket relations” that were mentioned previously.

The RS Hamiltonian’s expression in terms ofI_k-s H_RS = (1/2)(I₁+I−1).

Extra constants of motion

Examining the expressionI_j¹{I₁¹,H_RS}_M −I₁¹{I_j¹,H_RS}_M, we get extra constants of motion for the superintegrability ofH_RS. These are given by

Kj :=I_j¹(I2−n)−I₁¹(Ij+1−Ij−1), j = 2, . . . ,n.

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We can show thatI₁, . . . ,I_n,I₁¹, . . . ,I_n¹ generically form coordinates on the phase space, so they are functionally independent.

We immediately obtain the following equation det∂(Ia,K_b)

∂(Iα,I_β¹) = (I₂−n)ⁿ⁻¹, a, α∈ {1, . . . ,n}

b, β∈ {2, . . . ,n} , which is generically non-vanishing. This implies that the (2n−1) functionIa,K_b are independent.

Thus we’ve proved the superintegrability of the Hamiltonian H_RS.

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We’ve established the ”Wojciechowski Poisson algebra” using the derivation of rational RS model by Hamiltonian reduction^a. Next we describe the basic idea of this procedure.

aL. Feh´er and C. Klimˇc´ık,On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models, J. Phys. A: Math. Theor. 42 (2009)

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The unreduced phase space

T^∗GL(n,C)× O_χ≡GL(n,C)×gl(n,C)× O_χ ={(g,J^R, ξ)}. HereO_χ is a minimal coadjoint orbit of the groupU(n)

O_χ:={iχ(1n−vv^†)|v ∈Cⁿ,|v|²=n}.

The symplectic form is

Ω =dhJ^R,g⁻¹dgi+ω^O^χ. The corresponding Poisson-brackets are

{g,hX,J^Ri}=gX, {hX,J^Ri,hY,J^Ri}=−h[X,Y],J^Ri, {hX₊, ξi,hY₊, ξi}=h[X₊,Y₊], ξi, whereX₊,Y₊ are the anti-hermitian parts of matrices X,Y ∈gl(n,C).

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Symmetry group

We reduce using the symmetry groupK :=U(n)×U(n); for an element (η_L, η_R)∈K the action is given by

Ψ_(η_L_,η_R₎: (g,J^R, ξ)7→(η_Lgη⁻¹_R , η_RJ^Rη⁻¹_R , η_Lξη⁻¹_L ).

Momentum constraint

We set the momentum map corresponding to the action to zero, in other words we introduce first class constraints

J₊^R = 0 and (gJ^Rg⁻¹)₊+ξ = 0.

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Reduced phase space

The resulting reduced phase space can be identified with the Ruijsenaars-Schneider phase space (M, ω). In fact the following manifold is a global gauge-slice

S :={(L(q,p)¹²,−2q, ξ(q,p))|(q,p)∈ C_n×Rⁿ}.

Hereξ is anO_χ valued function onM:

ξ(q,p) :=iχ(1n−v(q,p)v(q,p)^†), wherev(q,p) :=L(q,p)⁻¹²u(q,p), with the Lax matrixL.

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We can realize the functionsI_j,I_k¹ ∈C^∞(M) as restrictions of K-invariant functionsI_j,I¹_k to S. These functions are given by

I_j(g,J^R, ξ) := tr (g^†g)^j

,I¹_k(g,J^R, ξ) :=−1

2<tr (g^†g)^kJ^R .

TheK-invariant functions survive the reduction. We can easily verify the relations

{I_j,I_k}= 0, {I¹_k,I_j}=jI_j+k, {I¹_k,I¹_j}= (j−k)I¹_k+j

for allj,k∈Z. These relations imply

{I_j¹,I_k}_M = 0, {I_k¹,I_j}_M =jI_j+k, {I_k¹,I_j¹}_M = (j−k)I_k+j¹ .

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5 Conclusion

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The rational Ruijsenaars-Schneider-van Diejen model

TheBC(n) generalization of the rational RS model is due to van Diejen.^a

The Hamiltonian is given by H_RSvD =

n

X

c=1

cosh(2θ_c)

1 +ν² λ²_c

¹₂ 1 + κ²

λ²_c ¹₂

n

Y

(d6=c)d=1

1 + 4µ² (λc−λd)²

¹₂

1 + 4µ² (λc+λd)²

¹₂

+ νκ 4µ²

n

Y

c=1

1 +4µ² λ²_c

− νκ 4µ²,

with κ, µ, ν real parameters that satisfyµ6= 06=ν andκν≥0.

aJ.F. van Diejen, Deformations of Calogero–Moser systems and finite Toda chains, Theor. Math. Phys. 99(1994)

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The RSvD phase space

The phase space of the rational RSvD model is given by P =C_n×Rⁿ={(λ, θ)|λ∈ C_n, θ∈Rⁿ}, where we’ve introduced the notation

C_n:={λ∈Rⁿ |λ1 > λ2 >· · ·> λn>0}.

The symplectic structureω =−2Pn

k=1dθk ∧dλk corresponds to the fundamental Poisson brackets

2{θ_i, λj}_P =δi,j, {λ_i, λj}_P ={θ_i, θj}_P = 0.

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First we need to introduce some matrix valued functions on the phase space on the next few slides. We’ll rely on Pusztai’s article^a. He derived the RSvD model by Hamiltonian reduction.

aB. G. Pusztai,The hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars-Schneider-van Diejen models: Lax matrices and duality, Nucl. Phys.

B856(2012)

Then we’ll explicitly construct the ”Wojciechowski algebra” and the extra constants of motions.

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A(λ, θ) is a 2n×2n Hermitian matrix, where the matrix entries lying in the diagonaln×n blocks are given by the formulae

A_a,b(λ, θ) =e^θ^a^+θ^b|z_a(λ)z_b(λ)|¹² 2iµ 2iµ+λ_a−λ_b, A_n+a,n+b(λ, θ) =e^−θ^a^−θ^b z_a(λ)z_b(λ)

|z_a(λ)zb(λ)|¹²

2iµ 2iµ−λ_a+λ_b, and the matrix entries of then×n off-diagonal blocks have the form

A_a,n+b(λ, θ) =A_n+b,a(λ, θ)

=e^θ^a^−θ^bz_b(λ)|z_a(λ)z_b(λ)⁻¹|¹² 2iµ

2iµ+λ_a+λ_b +i(µ−ν) iµ+λ_aδ_a,b, for anya,b ∈ {1, . . . ,n}. Here we use the functions

C_n3λ→za(λ) =−

1 +iν λa

n

Y

d=1d6=a

1 + 2iµ

λa−λ_d 1 + 2iµ λa+λ_d

.

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Let us define then×n diagonal matrixλ= diag(λ₁, . . . , λ_n). Now the Hermitian 2n×2n matrixh(λ) is provided by

h(λ) =

α(λ) β(λ)

−β(λ) α(λ)

,

with the functions α(x) =

px+√

x²+κ²

√2x and β(x) =iκ 1

√2x

1 px+√

x²+κ² ,

wherex∈(0,∞). Introduce also the diagonal matrix Λ := diag(λ1, . . . , λn,−λ₁,· · · −λn).

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Let us define the functionsI_j,I_k¹ ∈C^∞(P) by Ij(λ, θ) = tr[(h⁻¹Ah⁻¹)^j]

I_k¹(λ, θ) = (−1/4) tr[(hΛh⁻¹+h⁻¹Λh)(h⁻¹Ah⁻¹)^k],

for all j,k ∈Z.

Using Pusztai’s work, the Poisson bracket relations of the functions I_j,I_k¹ can be determined in a similar way as in the previous model.

In this case we obtain the same ”Wojciechowski algebra”

again.

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The principal Hamiltonian can be expressed according to HRSvD = (1/2)I1.

Then (n−1) independent extra constants of motions are given by K_j =I_j¹I₂−I₁¹I_j₊₁, j ∈ {2, . . . ,n}.

Thus we’ve proved the superintegrability of the RSvD model.

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5 Conclusion

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Result

We’ve shown explicitly the superintegrability of the rational A_n Ruijsenaars-Schneider model and the rational BC_n RSvD model

Remark

The functionsI_k¹ have linear time dependence under the

Hamiltonian flow ofI =I(I1, . . . ,In). Thus we have obtained an explicit linearisation of the dynamics.

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Thank you for your attention!

Viktor Ayadi

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Acknowledgement

The presentation is supported by the European Union and

co-funded by the European Social Fund. Project title: ”Broadening the knowledge base and supporting the long term professional sustainability of the Research University Centre of Excellence at the University of Szeged by ensuring the rising generation of excellent scientists.” Project number: T ´AMOP-4.2.2/B-10/1-2010-0012