Quadratic algebra contractions and 2nd order ...mille003/VinetTalk.pdfQuadratic algebras are...

Quadratic algebra contractions and 2nd ordersuperintegrable systems

Willard Miller, [Joint with E.G. Kalnins (Waikato), Sarah Post (Hawaii),Eyal Subag (Technion) and Robin Heinonen (Minnesota]

University of Minnesota

W. Miller (University of Minnesota) Quadratic algebra contractions Vinet 60th 1 / 44

Abstract

Quadratic algebras are generalizations of Lie algebras; they include thesymmetry algebras of 2nd order superintegrable systems in 2 dimensions asspecial cases. The superintegrable systems are exactly solvable physicalsystems in classical and quantum mechanics. We describe a contractiontheory for quadratic algebras and show that for constant curvaturesuperintegrable systems, ordinary Lie algebra contractions inducecontractions of the quadratic algebras of the superintegrable systems thatcorrespond to geometrical pointwise limits of the physical systems. Oneconsequence is that by contracting function space realizations ofrepresentations of the generic superintegrable quantum system on the2-sphere (which give the structure equations for Racah/Wilson polynomials)to the other superintegrable systems one obtains the full Askey scheme oforthogonal hypergeometric polynomials.

Outline

1 Introduction

2 Representatives of Nondegenerate Systems

3 Representatives of Degenerate Systems

4 Models of Superintegrable Systems

5 The S9 Difference Operator Model

6 Lie algebra contractions

7 Contractions of Superintegrable SystemsD2 contractions

8 Discussion and Conclusions

Introduction

Superintegrable Systems: HΨ = EΨ

A quantum superintegrable system is an integrable Hamiltonian systemon an n-dimensional Riemannian/pseudo-Riemannian manifold withpotential:

H = ∆n + V

that admits 2n − 1 algebraically independent partial differential operatorscommuting with H, the maximum possible.

[H,Lj ] = 0, L2n−1 = H, n = 1,2, · · · ,2n − 1.

Superintegrability captures the properties of quantum Hamiltoniansystems that allow the Schrödinger eigenvalue problem HΨ = EΨ to besolved exactly, analytically and algebraically.A system is of order K if the maximum order of the symmetry operators,other than H, is K . For n = 2, K = 1,2 all systems are known.

Introduction

H = ∆n + V

[H,Lj ] = 0, L2n−1 = H, n = 1,2, · · · ,2n − 1.

Introduction

H = ∆n + V

[H,Lj ] = 0, L2n−1 = H, n = 1,2, · · · ,2n − 1.

Introduction

1st order systems

These are the (zero-potential) Laplace-Beltrami eigenvalue equations onconstant curvature spaces.

Simplest examples: Euclidean Helmholtz equation (P21 + P2

2 )Φ = −λ2Φ (orthe Klein-Gordon equation (P2

1 − P22 )Φ = −λ2Φ), and Laplace -Beltrami

eigenvalue equation on the 2-sphere (J21 + J2

2 + J23 )Ψ = −j(j + 1)Ψ.

Here the symmetry algebras close under commutation to form the Liealgebras e(2,R), e(1,1) or o(3,R).

One can find 2-variable differential operator models of the irreduciblerepresentations of these Lie algebras in which the basis eigenfunctions arethe spherical harmonics (o(3,R)) or Bessel functions (e(2,R)).

Introduction

Free 2nd order superintegrable systems, (no potential,K = 2)We apply these ideas to 2nd order systems in 2D (2n − 1 = 3). The complexspaces with Laplace-Beltrami operators admitting at least three 2nd ordersymmetries were classified by Koenigs (1896). They are:

The two constant curvature spaces, six linearly independent 2nd ordersymmetries and three 1st order symmetries,The four Darboux spaces, four 2nd order symmetries and one 1st ordersymmetry,

ds2 = 4x(dx2 + dy2), ds2 =x2 + 1

x2 (dx2 + dy2),

ds2 =ex + 1

e2x (dx2 + dy2), ds2 =2 cos 2x + b

sin2 2x(dx2 + dy2),

Eleven 4-parameter Koenigs spaces. No 1st order symmetries. Example

ds2 = (c1

x2 + y2 +c2

x2 +c3

y2 + c4)(dx2 + dy2).

Introduction

2nd order systems with potential, K = 2

The symmetry operators of each system close under commutation togenerate a quadratic algebra, and the irreducible representations of thisalgebra determine the eigenvalues of H and their multiplicityAll the 2nd order superintegrable systems are limiting cases of a singlesystem: the generic 3-parameter potential on the 2-sphere, S9 in ourlisting. Analogously all quadratic symmetry algebras of these systemsare contractions of S9.

S9 : H = ∆2 +a1

s23, s2

1 + s22 + s2

3 = 1,

L1 = (s2∂s3 − s3∂s2 )2 +a3s2

s22, L2, L3,

H = L1 + L2 + L3 + a1 + a2 + a3

Introduction

2nd order systems with potential, K = 2

The symmetry operators of each system close under commutation togenerate a quadratic algebra, and the irreducible representations of thisalgebra determine the eigenvalues of H and their multiplicityAll the 2nd order superintegrable systems are limiting cases of a singlesystem: the generic 3-parameter potential on the 2-sphere, S9 in ourlisting. Analogously all quadratic symmetry algebras of these systemsare contractions of S9.

S9 : H = ∆2 +a1

s23, s2

1 + s22 + s2

3 = 1,

L1 = (s2∂s3 − s3∂s2 )2 +a3s2

s22, L2, L3,

H = L1 + L2 + L3 + a1 + a2 + a3

Introduction

3 types of 2nd order superintegrable systems:

1 Nondegenerate:

V (x) = a1V(1)(x) + a2V(2)(x) + a3V(3)(x) + a4

2 Degenerate:V (x) = a1V(1)(x) + a2

3 Free:V = a1, no potential

Introduction

Nondegenerate systems (2n − 1 = 3 generators)

The symmetry algebra generated by H,L1,L2 always closes undercommutation. Define 3rd order commutator R by R = [L1,L2]. Then

[Lj ,R] = A(j)1 L2

1 + A(j)2 L2

2 + A(j)3 H2 + A(j)

4 {L1, L2}+ A(j)5 HL1 + A(j)

+A(j)7 L1 + A(j)

8 L2 + A(j)9 H + A(j)

10, {L1, L2} = L1L2 + L2L1,

R2 = b1L31 + b2L3

2 + b3H3 + b4{L21, L2}+ b5{L1, L2

2}+ b6L1L2L1 + b7L2L1L2

+b8H{L1, L2}+ b9HL21 + b10HL2

2 + b11H2L1 + b12H2L2 + b13L21 + b14L2

2 + b15{L1, L2}

+b16HL1 + b17HL2 + b18H2 + b19L1 + b20L2 + b21H + b22,

This structure is an example of a quadratic algebra. If we know the expansionfor R2 we can compute the other structure relations.

Introduction

Nondegenerate systems (2n − 1 = 3 generators)

The symmetry algebra generated by H,L1,L2 always closes undercommutation. Define 3rd order commutator R by R = [L1,L2]. Then

[Lj ,R] = A(j)1 L2

1 + A(j)2 L2

2 + A(j)3 H2 + A(j)

4 {L1, L2}+ A(j)5 HL1 + A(j)

+A(j)7 L1 + A(j)

8 L2 + A(j)9 H + A(j)

10, {L1, L2} = L1L2 + L2L1,

R2 = b1L31 + b2L3

2 + b3H3 + b4{L21, L2}+ b5{L1, L2

2}+ b6L1L2L1 + b7L2L1L2

+b8H{L1, L2}+ b9HL21 + b10HL2

2 + b11H2L1 + b12H2L2 + b13L21 + b14L2

2 + b15{L1, L2}

+b16HL1 + b17HL2 + b18H2 + b19L1 + b20L2 + b21H + b22,

This structure is an example of a quadratic algebra. If we know the expansionfor R2 we can compute the other structure relations.

Introduction

Degenerate systems (2n − 1 = 3)

There are 4 generators: one 1st order X and 3 second order H,L1,L2.

[X , Lj ] = C(j)1 L1 + C(j)

2 L2 + C(j)3 H + C(j)

4 X 2 + C(j)5 , j = 1, 2,

[L1, L2] = E1{L1,X}+ E2{L2,X}+ E3HX + E4X 3 + E5X ,

Since 2n − 1 = 3 there must be an identity satisfied by the 4 generators. It isof 4th order:

c1L21 + c2L2

2 + c3H2 + c4{L1, L2}+ c5HL1 + c6HL2 + c7X 4 + c8{X 2, L1}+ c9{X 2, L2}

+c10HX 2 + c11XL1X + c12XL2X + c13L1 + c14L2 + c15H + c16X 2 + c17 = 0

If we know the 4th order identity, we can compute the other structure relationsto within an overall scale factor.

Introduction

Stäckel Equivalence Classes

All 2nd order 2d superintegrable systems with potential are known. There are59 types, on a variety of manifolds, but under the Stäckel transform, aninvertible structure preserving mapping, they divide into 12 equivalenceclasses with representatives on flat space and the 2-sphere, 6 withnondegenerate 3-parameter potentials

{S9,E1,E2,E3′,E8,E10}

and 6 with degenerate 1-parameter potentials

{S3,E3,E4,E5,E6,E14}

Representatives of Nondegenerate Systems

Example: S9

H = J21 + J2

2 + J23 +

where J3 = s1∂s2 − s2∂s1 and J2, J3 are obtained by cyclic permutations of indices.

Basis symmetries: (J3 = s2∂s1 − s1∂s2 , · · · )

L1 = J21 +

, L2 = J22 +

, L3 = J23 +

Structure equations:

[Li ,R] = 4{Li , Lk} − 4{Li , Lj} − (8 + 16aj)Lj + (8 + 16ak )Lk + 8(aj − ak ),

R2 =83{L1, L2, L3} − (16a1 + 12)L2

1 − (16a2 + 12)L22 − (16a3 + 12)L2

523({L1, L2}+{L2, L3}+{L3, L1})+

13(16+176a1)L1+

13(16+176a2)L2+

13(16+176a3)L3

+323(a1 + a2 + a3) + 48(a1a2 + a2a3 + a3a1) + 64a1a2a3, R = [L1, L2].

Interesting 2 variable O.P.’s: Prorial-Karlin-MacGregor, (right triangle)W. Miller (University of Minnesota) Quadratic algebra contractions Vinet 60th 12 / 44

Example: S9

H = J21 + J2

2 + J23 +

where J3 = s1∂s2 − s2∂s1 and J2, J3 are obtained by cyclic permutations of indices.

Basis symmetries: (J3 = s2∂s1 − s1∂s2 , · · · )

L1 = J21 +

, L2 = J22 +

, L3 = J23 +

[Li ,R] = 4{Li , Lk} − 4{Li , Lj} − (8 + 16aj)Lj + (8 + 16ak )Lk + 8(aj − ak ),

R2 =83{L1, L2, L3} − (16a1 + 12)L2

1 − (16a2 + 12)L22 − (16a3 + 12)L2

523({L1, L2}+{L2, L3}+{L3, L1})+

13(16+176a1)L1+

13(16+176a2)L2+

13(16+176a3)L3

+323(a1 + a2 + a3) + 48(a1a2 + a2a3 + a3a1) + 64a1a2a3, R = [L1, L2].

Interesting 2 variable O.P.’s: Prorial-Karlin-MacGregor, (right triangle)W. Miller (University of Minnesota) Quadratic algebra contractions Vinet 60th 12 / 44

Example: E1

H = ∂2x + ∂2

y − ω2(x2 + y2) +b1

x2 +b2

Generators:

L1 = ∂2x − ω2x2 +

x2 , L2 = ∂2y − ω2y2 +

y2 , L3 = (x∂y − y∂x)2 + y2 b1

x2 + x2 b2

Structure relations:[R, L1] = 8L2

1 − 8HL1 − 16ω2L3 + 8ω2,

[R, L3] = 8HL3 − 8{L1, L3}+ (16b1 + 8)H − 16(b1 + b2 + 1)L1,

R2 +83{L1, L1, L3}−8H{L1, L3}+(16b1 +16b2 +

)L21−16ω2L2

3− (32b1 +176

+(16b1 + 12)H2 +1763

ω2L3 + 16ω2(3b1 + 3b2 + 4b1b2 +23) = 0

Example: E1

H = ∂2x + ∂2

y − ω2(x2 + y2) +b1

x2 +b2

Generators:

L1 = ∂2x − ω2x2 +

x2 , L2 = ∂2y − ω2y2 +

y2 , L3 = (x∂y − y∂x)2 + y2 b1

x2 + x2 b2

Structure relations:[R, L1] = 8L2

1 − 8HL1 − 16ω2L3 + 8ω2,

[R, L3] = 8HL3 − 8{L1, L3}+ (16b1 + 8)H − 16(b1 + b2 + 1)L1,

R2 +83{L1, L1, L3}−8H{L1, L3}+(16b1 +16b2 +

)L21−16ω2L2

3− (32b1 +176

+(16b1 + 12)H2 +1763

ω2L3 + 16ω2(3b1 + 3b2 + 4b1b2 +23) = 0

Representatives of Degenerate Systems

Relations between nondegenerate/degeneratesystems

Every 1-parameter potential can be obtained from some 3-parameterpotential by parameter restriction.It is not simply a restriction, however, because the structure of thesymmetry algebra changes.A formally skew-adjoint 1st order symmetry appears and this induces anew 2nd order symmetry.Thus the restricted potential has a strictly larger symmetry algebra than isinitially apparent.

Example: S3 (Higgs oscillator)

H = J21 + J2

2 + J23 +

The system is the same as S9 with a1 = a2 = 0, a3 = a with the former L2 replaced by

L2 =12(J1J2 + J2J1)−

andX = J3 = s2∂s3 − s3∂s2 .

Structure relations:

[L1,X ] = 2L2, [L2,X ] = −X 2 − 2L1 + H − a, [L1, L2] = −(L1X + XL1)− (12+ 2a)X ,

(X 2L1 + XL1X + L1X 2

1 + L22 −HL1 + (a +

)X 2 − 16

H + (a− 23)L1 −

Interesting 2-variable O.P.’s: Koschmieder, Zerneke, (disk)

Example: E3 (Harmonic oscillator)

H = ∂2x + ∂2

y − ω2(x2 + y2)

Basis symmetries:

L1 = ∂2x − ω2x2, L3 = ∂xy − ω2xy , X = x∂y − y∂x .

Also we set L2 = ∂2y − ω2y2 = H − L1.

[L1,X ] = 2L3, [L3,X ] = H − 2L1, [L1, L3] = 2ω2X ,

L21 + L2

3 − L1H − ω2X 2 + ω2 = 0

Free triplets

We say that the 2D system without potential,

H0 = ∆2

and with 3 algebraically independent second-order symmetries is a 2nd orderfree triplet. The possible spaces admitting free triplets are just those classifiedby Koenigs: 2 constant curvature, 4 Darboux, and 11 Koenigs spaces.

Note that every nondegenerate or degenerate superintegrable system definesa free triplet, simply by setting the parameters aj = 0 in the potential.Similarly, this free triplet defines a free quadratic algebra, i.e., a quadraticalgebra with all aj = 0.

In general, a free triplet cannot be obtained as a restriction of asuperintegrable system and its associated algebra does not close to a freequadratic algebra.

Closure Theorems (Kalnins-Miller, 2014)

TheoremA free triplet extends to a superintegrable system if and only if it generates afree quadratic algebra Q.

TheoremA superintegrable system, degenerate or nondegenerate, with quadraticalgebra Q, is uniquely determined by its free quadratic algebra.Q.

Remark 1: These theorems are true for constant curvature spaces where thefree triplets are contained in the enveloping algebras of the Lie symmetryalgebras of the spaces as well as for Darboux spaces (1-dimensional Liesymmetry algebra) and Koenig spaces (no symmetry Lie algebra) where thisis no longer true.

Remark 2: These theorems are constructive. Given a free quadratic algebraQ one can compute the potential V and the symmetries of the quadraticalgebra Q.

What are models?A representation of a quadratic algebra Q is a homomorphism of Q intothe associative algebra of linear operators on some vector space:products go to products, commutators to commutators, etc.A model M is a faithful representation of Q in which the vector space is aspace of polynomials in one complex variable and the action is viadifferential/difference operators acting on that space.The dimensions of the finite dimensional irreducible representationsdetermine the multiplicities of the quantum energy eigenspaces and therepresentations determine the eigenvalues.

What is the significance of models?

They are easier to work with than the original abstract algebras.Eigenspaces in the models can be interpreted as expansion coefficientsof one separable eigenbasis of the original quantum system with potentialin terms of another eigenbasis.Racah and Wilson polynomials are related to S9 in this way.

Models of Superintegrable Systems

The big picture: Special functions and contractions

Connection with special functions

Special functions arise in two distinct ways:As separable eigenfunctions of the quantum Hamiltonian.As eigenfunctions in the model. Often solutions of difference equations.

Connection with contractionsTaking coordinate limits starting from quantum system S9 we can obtainother superintegrable systems.These limits induce limit relations between the special functionsassociated with the superintegrable systems.The limits induce contractions of the associated quadratic algebras, andvia the models, limit relations between the associated special functions.For constant curvature systems the required limits are all induced byWigner-type Lie algebra contractions of o(3.C) and e(2,C).The Askey scheme for orthogonal functions of hypergeometric type fitsnicely into this picture. (Kalnins-Miller-Post, 2014)

Model interplay

Figure :

The S9 Difference Operator Model

The S9 difference operator model. 1

L2 fn,m = (−4t2 −1

1 + B23 )fn,m,

L3 fn,m = (−4τ∗τ − 2[B1 + 1][B2 + 1] +1

2)fn,m,

H = L1 + L2 + L3 +3

4− (B2

1 + B22 + B2

3 ) = −4(m + 1)(B1 + B2 + B3 + m + 1) − 2(B1B2 + B1B3 + B2B3) +3

4− (B2

1 + B22 + B2

Here n = 0, 1, · · · ,m if m is a nonnegative integer and n = 0, 1, · · · otherwise. Also

4− B2

j , α = −(B1 + B3 + 1)/2 − m, β = (B1 + B3 + 1)/2, γ = (B1 − B3 + 1)/2, δ = (B1 + B3 − 1)/2 + B2 + m + 2,

EAF (t) = F (t + A), τ =1

2t(E1/2 − E−1/2), τ

∗ =1

[(α + t)(β + t)(γ + t)(δ + t)E1/2 − (α − t)(β − t)(γ − t)(δ − t)E−1/2

wn(t2) = (α + β)n(α + γ)n(α + δ)n4F3

(−n, α + β + γ + δ + n − 1, α − t, α + tα + β, α + γ, α + δ

= (α + β)n(α + γ)n(α + δ)nΦ(α,β,γ,δ)n (t2), Φn ≡ fn,m,

τ∗τΦn = n(n + α + β + γ + δ − 1)Φn,

where (a)n is the Pochhammer symbol and 4F3(1) is a hypergeometric function of unit argument. The polynomial wn(t2) is symmetric in α, β, γ, δ. For

the finite dimensional representations the spectrum of t2 is {(α + k)2, k = 0, 1, · · · ,m} and the orthogonal basis eigenfunctions are Racah

polynomials. In the infinite dimensional case they are Wilson polynomials.

L2 fn,m = (−4t2 −1

1 + B23 )fn,m,

L3 fn,m = (−4τ∗τ − 2[B1 + 1][B2 + 1] +1

2)fn,m,

H = L1 + L2 + L3 +3

4− (B2

1 + B22 + B2

3 ) = −4(m + 1)(B1 + B2 + B3 + m + 1) − 2(B1B2 + B1B3 + B2B3) +3

4− (B2

1 + B22 + B2

Here n = 0, 1, · · · ,m if m is a nonnegative integer and n = 0, 1, · · · otherwise. Also

4− B2

j , α = −(B1 + B3 + 1)/2 − m, β = (B1 + B3 + 1)/2, γ = (B1 − B3 + 1)/2, δ = (B1 + B3 − 1)/2 + B2 + m + 2,

EAF (t) = F (t + A), τ =1

2t(E1/2 − E−1/2), τ

∗ =1

[(α + t)(β + t)(γ + t)(δ + t)E1/2 − (α − t)(β − t)(γ − t)(δ − t)E−1/2

wn(t2) = (α + β)n(α + γ)n(α + δ)n4F3

(−n, α + β + γ + δ + n − 1, α − t, α + tα + β, α + γ, α + δ

= (α + β)n(α + γ)n(α + δ)nΦ(α,β,γ,δ)n (t2), Φn ≡ fn,m,

τ∗τΦn = n(n + α + β + γ + δ − 1)Φn,

where (a)n is the Pochhammer symbol and 4F3(1) is a hypergeometric function of unit argument. The polynomial wn(t2) is symmetric in α, β, γ, δ. For

the finite dimensional representations the spectrum of t2 is {(α + k)2, k = 0, 1, · · · ,m} and the orthogonal basis eigenfunctions are Racah

polynomials. In the infinite dimensional case they are Wilson polynomials.

The action of L2 and L3 on an L3 eigenbasis is

L2fn,m = −4K (n + 1, n)fn+1,m − 4K (n, n)fn,m − 4K (n − 1, n)fn−1,m + (B21 + B2

3 −12)fn,m,

L3fn,m = −(4n2 + 4n[B1 + B2 + 1] + 2[B1 + 1][B2 + 1]− 12)fn,m,

K (n + 1, n) =(B1 + B2 + n + 1)(n −m)(−B3 −m + n)(B2 + n + 1)

(B1 + B2 + 2n + 1)(B1 + B2 + 2n + 2),

K (n − 1, n) =n(B1 + n)(B1 + B2 + B3 + m + n + 1)(B1 + B2 + m + n + 1)

(B1 + B2 + 2n)(B1 + B2 + 2n + 1),

K (n, n) = [B1 + B2 + 2m + 1

2]2 − K (n + 1, n)− K (n − 1, n),

Lie algebra contractions

Let (A; [; ]A), (B; [; ]B) be two complex Lie algebras. We say B is a contractionof A if for every ε ∈ (0; 1] there exists a linear invertible map tε : B → A suchthat for every X ,Y ∈ B,

limε→0

t−1ε [tεX , tεY ]A = [X ,Y ]B.

Thus, as ε→ 0 the 1-parameter family of basis transformations can becomenonsingular but the structure constants go to a finite limit.

Lie algebra contractions

Contractions of e(2,C) and o(3,C)

These are the symmetry algebras of free systems on constant curvaturespaces. Their contractions have long since been classified. The are 6nontrivial contractions of e(2,C) and 4 of o(3,C).

Example: A Wigner-Inönü contraction of o(3,C). We use the classicalrealization for o(3,C) acting on the 2-sphere, with basisJ1 = s2p3 − s3p2, J2 = s3p1 − s1p3, J3 = s1p2 − s2p1, commutation relations

[J2, J1] = J3, [J3, J2] = J1, [J1, J3] = J2,

and Hamiltonian H = J21 + J2

2 + J23 . Here s2

1 + s22 + s2

3 = 1.

Basis change : {J ′1, J ′2, J ′3} = {εJ1, εJ2, J3}, o < ε ≤ 1

coordinate implementation x =s1

ε, y =

ε, s3 ≈ 1, J = J3

New structure relations : [J ′2, J′1] = ε2J ′3, [J ′3, J

′2] = J ′1, [J ′1, J

′3] = J ′2,

Let ε→ 0 : [J ′2, J′1] = 0, [J ′3, J

′2] = J ′1, [J ′1, J

′3] = J ′2, get e(2,C)

Contractions of Superintegrable Systems

Contractions of nondegenerate systems. 1

Suppose we have a nondegenerate superintegrable system with generatorsH,L1,L2, R = [L1,L2] and the usual structure equations, defining a quadraticalgebra Q. If we make a change of basis to new generators H, L1, L2 andparameters a1, a2, a3 such that L1

A1,1 A1,2 A1,3

A2,1 A2,2 A2,3

0 0 A3,3

B1,1 B1,2 B1,3

B2,1 B2,2 B2,3

B3,1 B3,2 B3,3

C1,1 C1,2 C1,3

C2,1 C2,2 C2,3

C3,1 C3,2 C3,3

for some 3× 3 constant matrices A = (Ai,j ),B,C such that det A · det C 6= 0,we will have the same system with new structure equations of the same formfor R = [L1, L2], [Lj , R], R2, but with transformed structure constants.

Choose a continuous 1-parameter family of basis transformation matricesA(ε),B(ε),C(ε, 0 < ε ≤ 1 such that A(1) = C(1) is the identity matrix,B(1) = 0 and det A(ε) 6= 0, det C(ε) 6= 0.Now suppose as ε→ 0 the basis change becomes singular, (i.e., thelimits of A,B,C either do not exist or, if they exist do not satisfydet A(0) det C(0) 6= 0) but the structure equations involvingA(ε),B(ε),C(ε), go to a limit, defining a new quadratic algebra Q′.We call Q′ a contraction of Q in analogy with Lie algebra contractions.

There is a similar definition of a contraction of a degenerate superintegrablesystem.

Lie algebra contractions⇒ quadratic algebracontractions

Constant curvature spaces:

Theorem

(Kalnins-Miller, 2014) Every Lie algebra contraction of A = e(2,C) orA = o(3,C) induces uniquely a contraction of a free quadratic algebra Qbased on A, which in turn induces uniquely a contraction of the quadraticalgebra Q with potential. This is true for both classical and quantum algebras.

A complication: Darboux spaces: The Lie symmetry algebra is only1-dimensional so Wigner contractions don’t apply and geometrical freequadratic algebra contractions must be determined on a case-by-case basis.

Figure : The Askey scheme and contractions of superintegrable systems

Figure : The Askey contraction scheme

Example: free nondegenerate Darboux 2 systemsThe space Darboux 2 (D2) has free degenerate Hamiltonian

x2 + 1(p2

x + p2y )

Killing vector: K = py . Basis, {H,K2,X1,X2} for the 4-dimensional space of2nd order Killing tensors.

X1 = 2xpxpy +2y

x2 + 1(p2

y−x2p2x ), X2 = 2xypxpy +

(y2 − x4)p2y + x2(1− y2)p2

x2 + 1.

Functional relation

4HX2 + X 21 − 4K2X2 − 4H2 = 0.

Possible free systems:1 ~D2C: L1 = X2, L2 = X1, R2 = 4L1L2

2 + 16L21H− 16L1H2.

2 ~D2B: L1 = X2, L2 = K2, R2 = 16L1L22 − 16L1L2H+ 16L2H2.

3 ~D2A: L1 = X1, L2 = K2, R2 = 16L32 − 32L2

2H+ 16L2H2.W. Miller (University of Minnesota) Quadratic algebra contractions Vinet 60th 31 / 44

Extension of D2C to a nondegenerate superintegrablesystem.

x2 + 1(p2

x + p2y ) + V (x , y),

V (x , y) =x2

x2 + y2(x2 + 1)

y +√

x2 + y2+

−y +√

x2 + y2

)+ c4.

The Casimir is

R2−4L1L22−16L2

1H+16L1H2−4(c2 +c3)H2 +16c4L21 +(c2 +c3)L2

2−32c4HL1

+(8c2c4 − c21 + 4c2c3 + 8c3c4)H+ (16c2

4 + c21)L1 + c1(−c3 + c2)L2

c21c2 +

c21c3 + c2

1c4 − 4c3c24 − 4c2c3c4 − 4c2c2

Example: D2 contractions

1 Letx = x ′ +

1ε, y = y ′, H′ = H.

As ε→ 0 we have

H′ = p2x ′ + p2

y ′ , εX1 ≈ 2px ′py ′ , ε2X2 ≈ −p2

y ′ .

Contractions constructed from such limits have flat space as the targetmanifold.

2 Letx = εx ′, y = εy ′, H′ = H.

As ε→ 0 we have H′ = x ′2(p2x ′ + p2

y ′) and

εX1 ≈ 2x ′px ′py ′ + 2y ′p2y ′ , εX2 = 2x ′y ′px ′py ′ + y ′2p2

y ′ + x ′2p2x ′ , εK = py ′ .

Contractions constructed from such limits have the complex 2-sphere asthe target manifold.

Example: Contraction 2 applied to D2C

In the limit, and in Cartesian coordinates s1, s2, s3 on the complex 2-sphere,where s2

1 + s22 + s2

3 = 1 and J1, J2, J3 are defined by J3 = s1p2 − s2p3 andcyclic permutations, we have

H = J 21 + J 2

2 + J 23 +

(s1 − is2)2 +k2s3√s2

1 + s22

(s1 − is2)√

s21 + s2

This system is real on a real hyperboloid.

In general, constant curvature superintegrable systems cannot contract toDarboux or Koenig systems but contractions in the reverse order occur.

Table of Darboux 2 contractions

Darboux 2 to flat space Darboux 2 to 2-sphereD2A E3′ S1D2B E2 S2D2C E3′ S4D2D∗ E5∗ S5∗

The full contractions hierarchy is too complicated to be shown on a singleslide. We break it up into connected parts.

Table of Darboux 2 contractions

Darboux 2 to flat space Darboux 2 to 2-sphereD2A E3′ S1D2B E2 S2D2C E3′ S4D2D∗ E5∗ S5∗

The full contractions hierarchy is too complicated to be shown on a singleslide. We break it up into connected parts.

Contraction hierarchy 1

Figure : Second part of nondegenerate contraction diagram. This can in principle bejoined with the previous.

Figure : Diagram showing hierarchy of contractions of degenerate systems.W. Miller (University of Minnesota) Quadratic algebra contractions Vinet 60th 38 / 44

Abstract contractions

We are in the process of classifying all abstract contractions of freedegenerate and nondegenerate quadratic algebras. Every Lie algebrainduced contraction is also an abstract contraction, but the converse is false.There are many abstract contractions that cannot be obtained by takingpointwise coordinate limits of quantum superintegrable systems. For example:E7 to E2 and S7 to E15. The significance of these non-physical limits isunclear to us at present.

Abstract contractions between nondegenerate freesystems

Abstract contractions induced by Lie algebra contractions

Discussion and Conclusions

Wrap-up. 1

Free quadratic algebras uniquely determine associated superintegrablesystems with potential.A contraction of a free quadratic algebra to another uniquely determinesa contraction of the associated superintegrable systems.For a 2D superintegrable systems on a constant curvature space thesecontractions can be induced by Lie algebra contractions of the underlyingLie symmetry algebra.Every 2D superintegrable system is obtained either as a sequence ofcontractions from S9 or is Stäckel equivalent to a system that is soobtained.

Wrap-up. 2

Taking contractions step-by-step from the S9 model we can recover theAskey Scheme. However, the contraction method is more general. Itapplies to all special functions that arise from the quantum systems viaseparation of variables, not just polynomials of hypergeometric type, andit extends to higher dimensions.The special functions arising from the models can be described as thecoefficients in the expansion of one separable eigenbasis for the originalquantum system in terms of another separable eigenbasis.The functions in the Askey Scheme are just those hypergeometricpolynomials that arise as the expansion coefficients relating twoseparable eigenbases that are both of hypergeometric type. Thus, thereare some contractions which do not fit in the Askey scheme since thephysical system fails to have such a pair of separable eigenbases.

Wrap-up. 3

Even though 2nd order 2D nondegenerate superintegrable systemsadmit no group symmetry, their structure is determined completely by theunderlying symmetry of constant curvature spaces.

Quadratic algebra contractions and 2nd order ...mille003/VinetTalk.pdfQuadratic algebras are...

Documents

Transcript of Quadratic algebra contractions and 2nd order ...mille003/VinetTalk.pdfQuadratic algebras are...