13 Contraction

Contents

13.1 Preliminaries 233 13.2 Inönü–Wigner Contractions 233 13.3 Simple Examples of Inönü–Wigner Contractions 234 13.3.1 The Contraction SO(3) → ISO(2) 234 13.3.2 The Contraction SO(4) → ISO(3) 235 13.3.3 The Contraction SO(4, 1) → ISO(3, 1)237 13.4 The Contraction U(2) → H4 239 13.4.1 Contraction of the Algebra 239 13.4.2 Contraction of the Casimir Operators 240 13.4.3 Contraction of the Parameter Space 240 13.4.4 Contraction of Representations 241 13.4.5 Contraction of Basis States 241 13.4.6 Contraction of Matrix Elements 242 13.4.7 Contraction of BCH Formulas 242 13.4.8 Contraction of Special Functions 243 13.5 Conclusion 244 13.6 Problems 245

New Lie groups can be constructed from old by a process called group contraction. Contraction involves reparameterization of the Lie group’s parameter space in such a way that the group multiplication properties, or commutation relations in the Lie algebra, remain well defined even in a singular limit. In general, the properties of the original Lie group have well-defined limits in the contracted Lie group. For example, the parameter space for the contracted group is well-defined and noncompact. Other properties with well-defined limits include: Casimir operators; basis states of representations; matrix elements of operators; and Baker–Campbell– Hausdorff formulas. Contraction provides limiting relations among the special functions of mathematical physics. We describe a particularly simple class of contractions — the Inönü–Wigner contractions — and treat one example of a contraction not in this class.

232 13.1 Preliminaries 233 13.1 Preliminaries It is possible to construct new Lie algebras from old by a certain limiting process called contraction. In this process a new set of basis vectors Yr is related to the initial set of basis vectors Xi through a i parameter-dependent change of basis: Yr = Mr (ǫ)Xi. The structure t constants have the transformation properties of a tensor: Crs (ǫ) = i j k 1 t Mr (ǫ)Ms (ǫ)Cij (M(ǫ)− )k (cf. Eq. 4.22). As long as the change of basis transformation is nonsingular the Lie algebra is unchanged. t If the transformation becomes singular, the structure constants Crs (ǫ) may still converge to a well-defined limit. It is often the case that the structure constants t t Crs (0) = lim Crs (ǫ) (13.1) ǫ 0 → exist and define a Lie algebra that is different from the original Lie algebra.

13.2 Inönü–Wigner Contractions If a Lie algebra g has a subalgebra h and a complementary subspace p with commutation relations of the form g = h + p [h, h] h subalgebra ⊆ (13.2) [h, p] p this is important ⊆ [p, p] h + p this is always true ⊆ then the Inönü–Wigner contraction of g g′ involves the following change → of basis transformation h I 0 h ′ = dim(h) (13.3) p 0 ǫ I p ′ dim(p) where dim(h) is the dimension of the subalgebra h. The commutation relations of g′ are well-defined for all values of ǫ, including the singular limit ǫ 0: → [h′, h′] = [h, h] h ⊆ [h′, p′] = [h,ǫp] = ǫ [h, p] limǫ 0 ǫp p′ 2 → 2 → [p′, p′] = [ǫp,ǫp] = ǫ [p, p] limǫ 0 ǫ (h + p) 0 → → (13.4) In the limit ǫ 0 the contracted algebra g′ is the semidirect sum of the → original subalgebra h and the subalgebra p′, where p′ is commutative and [h, p′] p′: ⊆ 234 Contraction

g = h + p p p′ = ǫp g′ = h + p′ → [h, h] h [h, h] h ⊆ −→ ⊆ (13.5) [h, p] p [h, p′] p′ ⊆ ⊆ [p, p] h + p [p′, p′] = 0 ⊆

13.3 Simple Examples of Inönü–Wigner Contractions In this section we illustrate several facets of Inönü–Wigner contractions by contracting three different orthogonal groups.

13.3.1 The Contraction SO(3) ISO(2) → The inﬁnitesimal generators of the Lie group SO(3) may be chosen as L = X = x ∂ x ∂ = ǫ x ∂ , with L and L deﬁned by cyclic 1 23 2 3 − 3 2 1jk j k 2 3 permutation. The commutation relations are

[L ,L ] = L 1 2 − 3 [L ,L ] = L (13.6) 2 3 − 1 [L ,L ] = L 3 1 − 2 Under contraction with respect to the subalgebra of rotations about the z-axis (inﬁnitesimal generator L3) the operators L1 and L2 go to

ǫL ǫ=1/R (1/R)L P 1 1 − 2 (13.7) ǫL −→ (1/R)L → +P 2 2 1 The commutation relations of the contracted algebra, ISO(2) = E(2), are

[L , P ] = P 3 1 − 2 [L3, P2] = +P1 (13.8)

[P1, P2] = 0

The three operators L3, P1, P2 generate the group of Euclidean motions of the plane, E(2), or inhomogeneous orthogonal transformations in the plane R2, ISO(2). This group consists of rotations about the z-axis, generated by L3, and displacements of the origin in the x- and y- directions, generated by P1 = ∂1 and P2 = ∂2. To verify this interpretation we can imagine the group SO(3) acting on the sphere x2 + y2 + z2 = R2 in the neighborhood of the north pole 13.3 Simple Examples of Inönü–Wigner Contractions 235 (0, 0, R), as shown in Fig. 13.1. An element in the Lie algebra so(3) can be written in the form L L θ L + θ L + θ L ( d ) 1 + (+d ) 2 + θ L (13.9) 1 1 2 2 3 3 −→ − 2 R 1 R 3 3 In the limit R we find → ∞ 1 L = 1 (x2∂ x3∂ )= 1 (y∂/∂z R∂/∂y) ∂/∂y = ∂ = P R 1 R 3 − 2 R − → − − 2 − 2 1 L = 1 (x3∂ x1∂ )= 1 (R∂/∂x x∂/∂z) +∂/∂x =+∂ =+P R 2 R 1 − 3 R − → 1 1 (13.10) The contracted limits of the operators L1 and L2 in the limit of a sphere of very large radius are operators P , +P describing displacements in − 2 1 the y and +x directions. In addition, the parameters θ ,θ and d , d − 1 2 1 2 are related by

d1 = +Rθ2 d = Rθ (13.11) 2 − 1 As the radius of the sphere becomes very large, the two angles θ1,θ2 become small with the product Rθi (i = 1, 2) approaching a well-deﬁned limit. This corresponds to a rotation through an angle θ2 = d1/R about the y-axis producing a displacement of d1 in the x-direction, and a rotation through an angle θ1 = d2/R about the x-axis producing a displacement of d in the y-direction. − 2 The Casimir operator for the group SO(3) contracts to an invariant operator as follows: 2[SO(3)] = L2 + L2 + L3 C 1 2 3 2[ISO(2)] = lim(1/R2) 2[SO(3)] C C 2 2 2 = lim[(L1/R) + (L2/R) + (L3/R) ] ∂2 ∂2 = ( P )2 + (+P )2 +0= + (13.12) − 2 1 ∂y2 ∂x2 This is just the Laplacian operator on the plane R2.

13.3.2 The Contraction SO(4) ISO(3) → This group is similar to SO(3) and can be treated similarly. The six generators are

Li = ǫijkxj ∂k V = x ∂ x ∂ 1 i = j = k 3 (13.13) i i 4 − 4 i ≤ 6 6 ≤ 236 Contraction

Fig. 13.1. Rotations on the surface of a sphere of radius R approach displacements in the plane as R →∞.

The commutation relations are

[L ,L ] = ǫ L i j − ijk k [L , V ] = ǫ V (13.14) i j − ijk k [V , V ] = ǫ L i j − ijk k We contract with respect to the subgroup SO(3) generated by the angular momentum operators Li, deﬁning 1 1 Pi = lim Vi = lim (xi∂4 x4∂i)= ∂i (13.15) R R R R − →∞ →∞ − − The commutation relations of the contracted algebra are

[L ,L ] = ǫ L i j − ijk k [L , P ] = ǫ P (13.16) i j − ijk k [Pi, Pj ] = 0

The operators Pi describe displacements in the x-, y-, and z- (i =1, 2, 3) directions. The contracted group is ISO(3), the Euclidean, or inhomogeneous orthogonal group, on R3. 13.3 Simple Examples of In¨on¨u–Wigner Contractions 237 As in the case SO(3) ISO(2), we can contract the second order → Casimir operator of SO(4) to that of ISO(3)

2 2 1 [ISO(3)] = lim (1/R )(L L + V V) R C →∞ · · = lim [(L/R) (L/R) + (V/R) (V/R)] (13.17) R · · →∞ ∂2 ∂2 ∂2 = 0+ P P = 2 = + + (13.18) · ∇ ∂x2 ∂y2 ∂z2 As before, this is no surprise. The contracted operator is the Laplacian on R3. What is a surprise is that there is a second nontrivial invariant operator. For SO(4) this is [cf. Eq. (9.24)] 2[SO(4)] = ǫijklX X 8L V (13.19) C2 ij kl → · The contracted limit of this operator is

2 2 [ISO(3)]/8 = lim (1/R)(L V) R C →∞ · = lim [L (V/R)] = L P (13.20) R · − · →∞ The two invariant operators P P = 2 and L P = L form a · ∇ · − ·∇ complete set of invariant operators for the group ISO(3).

13.3.3 The Contraction SO(4, 1) ISO(3, 1) → The group ISO(3, 1) consists of proper Lorentz transformations [SO(3, 1)] that leave invariant the quadratic form x2 + y2 + z2 (ct)2 (13.21) − as well as displacements of the origin in the three spacelike directions and one timelike direction. The inhomogeneous Lorentz group, or Poincar´e group, leaves invariant space-time intervals

2 2 2 2 (x x′) + (y y′) + (z z′) (ct ct′) (13.22) − − − − − This group can be contracted from either SO(4, 1) or SO(3, 2). We choose as inﬁnitesimal generators for the group SO(4, 1) the operators X = x ∂ x ∂ = ǫ L 1 i, j, k 3 rotations ij i j − j i ijk k ≤ ≤ B = x ∂ + x ∂ 1 i 3 boosts i4 i 4 4 i ≤ ≤ T = x ∂ x ∂ i =1, 2, 3 sign space displacements i5 i 5 ± 5 i − i = 4 +sign time displacements (13.23) 238 Contraction This set of generators is contracted with respect to the subgroup SO(3, 1) generated by rotations and boosts. The second order Casimir invariant for SO(4, 1) and its contraction to the second order Casimir invariant for the Poincar´egroup are

2 2 [SO(4, 1)] = L L B B + T T T45 C · − · · − 1 ∂2 2[ISO(3, 1)] = 0 0 + C − ∇·∇ − c2 ∂t2 However, SO(4, 1) has a second Casimir operator, since it is a real form for the rank-two root space B2. This is a fourth-degree operator that is derived by analytic continuation from the fourth-order Casimir operastor of SO(5) [cf. Eq. (9.22) and (9.23)]

4[SO(5)] = W αW C α α αβγµν W = ǫ XβγXµν (13.24) where ǫαβγµν is the Levi-Civita symbol (antisymmetric tensor) on ﬁve α symbols, and Wα is similarly deﬁned. The contracted limit of W is nonzero only if one of the four remaining symbols (e.g. ν) is 5:

α αβγµ5 lim (1/R)W lim ǫ Xβγ[(1/R)Xµ5] R R →∞ → →∞ αβγµ5 µ = ǫ Xβγ(∂/∂x ) (13.25)

αβγµ5 µ α The four vector ǫ Xβγ(∂/∂x ) is fairly complicated. Since W Wα is invariant, it is convenient to compute it for a particle of mass m in a frame in which the particle is at rest

Pµ = (0, 0, 0,mc) (13.26)

In this frame

α αβγµ W = ǫ Xβγmc = Lαmc (13.27)

Therefore the invariant is

W αW = (L L)(P P) (13.28) α · · with P P = P P µ = (mc)2. · µ − It should be emphasized that if an operator is an invariant and its P spectrum or interpretation is desired, the operator should be viewed from the coordinate system which most simplifies its determination (principle of maximum laziness). 13.4 The Contraction U(2) H 239 → 4 13.4 The Contraction U(2) H → 4 In this section we consider a group contraction that is not of Inönü- Wigner type. This is the contraction of the compact unitary group U(2) to the solvable group H4. This contraction relates the angular momentum operators to the single mode photon operators. These are the infinitesimal generators of the groups U(2) and H4, respectively. This contraction leads to a number of useful relations that are explored in successive sections.

13.4.1 Contraction of the Algebra

The Lie algebra u(2) is spanned by inﬁnitesimal generators J3, J , J0 ± with commutation relations

[J3, J ] = J ± ± ± [J+, J ] = 2J3 (13.29) − [J0, J] = 0

The operators h3,h ,h0 are related to J3, J , J0 by the following change ± ± of basis

h+ c J+ h c J  −  =  1   −  (13.30) h3 1 2c2 J3        h0   1   J0        These operators satisfy the following commutation relations

[h3,h ] = h ± ± ± 2 [h+,h ] = 2c h3 h0 (13.31) − − [h0, h] = 0

In the limit c 0 the change of basis transformation become singular → but the commutation relations [Eq (13.31)] converge to a well-deﬁned limit satisﬁed by the single mode photon operators

1 1 h3 nˆ + 2 I = 2 a,a† h+ c 0 a†   →   (13.32) h −→ a  −     h0   I      240 Contraction 13.4.2 Contraction of the Casimir Operators The group U(2) has rank two. Its two Casimir operators are of ﬁrst and second order

1 = J C 0 2 2 1 = J3 + (J+J + J J+) (13.33) C 2 − − Under contraction J h but the second Casimir operator has a more 0 → 0 interesting limit

2 2 2 1 2 1 lim c = lim c (h3 h0) + [(cJ+)(cJ ) + (cJ )(cJ+)] c 0 c 0 2 − − → C → − 2c 2 2 2 1 2 1 2 = lim c h3 (h3h0 + h0h3)+ c ( h0) + (13.34) c 0 2 → − 2 −2c 1 [(h+)(h ) + (h )(h+)] 2 − −

2 The operator (h0/2c) is proportional to the square of the ﬁrst Casimir operator. It therefore commutes with all elements in the Lie algebra. Therefore the remaining set of operators on the right hand side of (13.34) must also commute with all operators in the Lie algebra. In the limit c 0, (ch )2 0 and the remaining operators go to a well-deﬁned limit → 3 →

2 2 2 2 1 1 1 1 lim c [U(2)] (h0/2c) [H4]= (ˆn + I)I + I(ˆn + I) + (aa†+a†a) c 0 C − →C −2 2 2 2 → (13.35) 1 This is a quadratic operator in the generatorsn ˆ + 2 I,a†,a, and I of H4. The value of this operator in the standard Fock space spanned by the photon number states 0 , 1 , 2 , is zero. It is the other “invisible | i | i | i · · · invariant” for H4.

13.4.3 Contraction of the Parameter Space An arbitrary element in the Lie algebra u(2) and its counterpart in the algebra h4 with basis h3,h ,h0 is [1, 31] ±

1 iφ 1 +iφ iθµJµ = θe− J+ θe J + iθ3J3 + iθ0J0 = 2 − 2 −

θ iφ θ +iφ θ3 e− h+ e h + iθ3h3 + i(θ0 )h0 (13.36) 2c − 2c − − 2c2 13.4 The Contraction U(2) H 241 → 4 In the limit c 0 the parameter θ must approach zero so that the limits → θ iφ lim + e− +α c 0 → 2c → θ +iφ lim e α∗ (13.37) c 0 → −2c → −

θ3 exist. In addition, θ should diverge so that θ 2 remains well deﬁned. 0 0 − 2c

13.4.4 Contraction of Representations The action of the operators h on the angular momentum state J, M 3 | i is J 1 J 1 J h = (J + J ) = (M + ) (13.38) 3| M i 3 2c2 0 | M i 2c2 | M i

It is useful to measure states from the “lowest” state J, J in the | − i angular momentum multiplet. The state with the quantum number M is the ground state if M = J, and the nth state when − n = J + M (13.39)

In order for the action of h on J, M to have a well-defined limit, we 3 | i insist that 1 1 lim(M + ) = lim(n J + ) (13.40) c 0 2 c 0 2 → 2c → − 2c be well defined. This is the case when we go through a sequence of larger and larger representations J of dimension (2J +1) as c becomes smaller and smaller. Specifically, we require c and J to be related by [1,31] 1 lim( J + ) = 0 implies 2Jc2 = 1 (13.41) c 0 2 → − 2c In this case J lim h3 = n ∞ (13.42) c → 0 | M i | n i J →∞

13.4.5 Contraction of Basis States The basis states J, M for an angular momentum multiplet are con- | i structed by applying the angular momentum shift up operator n = J+M 242 Contraction times to the ground state J, J . These states are contracted to the | − i harmonic oscillator states as follows J (J )n J = + | M = J + n i [(2J)!n!/(2J n)!]1/2 | J i − − − n (cJ+) ∞ = lim ∞ (13.43) n J [(2Jc2)nn!]1/2 0 | i →∞ | i n (a†) = ∞ √n! | 0 i

13.4.6 Contraction of Matrix Elements The matrix elements of the angular momentum operators on the angular momentum basis states contract readily to the matrix elements of the photon operators on the Fock states

1 J a†a ∞ = limc 0(J3 + 2 ) = | n i → 2c | M i (13.44) 1 J limc 0[J + M + ( 2 J)] (n + 0) ∞ → 2c − | M = n J i → | n i − J a† ∞ = limc 0 cJ+ = | n i → | M i

J 2 limc 0 (J M)(J + M + 1)c √n +1 ∞ → | M +1 i − → | n +1 i p (13.45) J a ∞ = limc 0 cJ = | n i → −| M i (13.46) J 2 limc 0 (J + M)(J M + 1)c √n ∞ → | M 1 i − → | n 1 i − p −

13.4.7 Contraction of BCH Formulas Baker–Campbell–Hausdorﬀ formulas, which can easily be derived for U(2) in its faithful 2 2 matrix representation, can readily be contracted × to BCH formulas for H4, which can be derived with only a little more diﬃculty in its faithful 3 3 matrix representation [cf. Eq. (7.36)]. For × example, the following BCH formula for U(2)

∗ ∗ ∗ (ζJ+ ζ J−) τJ+ ln(1+τ τ)J3 τ J− ζ e − = e e e− tan ζ = τ (13.47) ζ | | | | 13.4 The Contraction U(2) H 243 → 4 contracts under limc 0 ζ/c α to the BCH formula for H4 → → (αa† α∗a) αa† 1 α∗αI α∗a e − = e e− 2 e− α = lim ζ/c (13.48) c 0 →

13.4.8 Contraction of Special Functions Special functions that are associated with the group SU(2) include Ja- cobi polynomials, the associated Legendre polynomials and spherical harmonics, and the Legendre polynomials. The special functions associated with the “harmonic oscillator” group H4 are the Hermite polynomials and the harmonic oscillator wavefunctions. One might reasonably expect that the Hermite polynomials and harmonic oscillator wavefunctions are related to the Jacobi or associated Legendre polynomials in some contraction limit. This is so. l The spherical harmonics Ym(θ, φ) and associated Legendre polynomi- l als Pm(cos θ) are related by [1,31]

imφ l e l l m l Ym(θ, φ)= Pm(cos θ) Y m(θ, φ) = ( ) Y+m(θ, φ)∗ (13.49) √2π − − The associated legendre polynomials are deﬁned by

l+m l l+m 1 2l +1 (l m)! 2 +m/2 d 2 l Pm(u) = ( ) l − (1 u ) l+m (1 u ) − 2 l!r 2 s(l + m)! − du − (13.50) These polynomials are contracted to harmonic oscillator wavefunctions under u x/√l and l + m = n: → 1/4 l √ lim l− Pm(u = x/ l)= c 0 →

1/2 n n (2l)!l 1 2 2 ( 1/2c2)/2 d 2 2 1/2c2 lim( ) [1 2c x ] − [1 2c x ] c 0 (2l) 2nn!(2lc2)n dxn → − r 2 l!l! s − − (13.51) The limit is taken as c 0,l ,l + m = n, 2lc2 = 1. The limit inside → → ∞ n 1 the ﬁrst square root is 1/√π, that within the second is (2 n!)− . The result of this contraction is n 1/4 l √ 1 x2/2 d x2 lim l− Pm(u = x/ l)= e e− = ψn(x) c 0 n −dx → 2 n!√π (13.52) p 244 Contraction where ψn(x) is the appropriately normalized harmonic oscillator eigenfunction

1 x2/2 ψn(x)= Hn(x)e− (13.53) 2nn!√π and Hn(x) is the nth Hermitep polynomial. Under contraction the orthogonality relations obeyed by the associated Legendre functions go over to the orthogonality relations for the harmonic oscillator eigenfunctions

+1 l l δmm′ = Pm(u)Pm′ (u)du 1 → Z−

+√l 1 l √ 1 l √ √ lim 1/4 Pm(x/ l) 1/4 Pm′ (x/ l) d(u l) l √l l l → →∞ Z−

+ ∞ ψn(x)ψn′ (x)dx = δnn′ (13.54) Z−∞ Unfortunately, it is not possible to derive the completeness relations for the harmonic oscillator eigenfunctions from the completeness relations for the Jacobi or associated Legendre polynomials. However, there is a very simple and beautiful proof of the completeness relations for all special functions associated with compact Lie groups. It is due to Wigner and Stone.

13.5 Conclusion Contraction of groups to form inequivalent groups can be carried out whenever a singular change of basis can be constructed under which the structure constants have a well-defined limit. Contraction is a particularly useful way to construct nonsemisimple Lie groups from simple and semisimple Lie groups. The contracted group is always noncompact. Contraction of groups provides many useful relations between the original group and its contracted limit. These involve the commutation relations in the Lie algebra, the range of values in the parameter spaces that map onto the groups, the Casimir operators, the basis states of representations, operator matrix elements, Baker–Campbell–Hausdorff formulas, and limiting relations among special functions. These relations have all been illustrated by example. 13.6 Problems 245 13.6 Problems 1. Under the contraction SO(3) ISO(2) the representations of SO(3) → contract to representations of ISO(2). Since ISO(2) is a noncompact group it has no faithful finite-dimensional unitary representations. We therefore consider the following limit

lim a 0 aJ P a2l(l + 1) p2 ﬁnite ↓ ± → ± → l p l J P ↑ ∞ 3 → 3 | m i → | m i

(p/a)β = lβ = x ﬁnite

a. Compute the matrix elements of the operators P in the algebra ± iso(2) and show

l l lim p p aJ P h m′ | ±| m i −→ h m′ | ±| m i

lim a (l m)(l m + 1) δm′,m 1 p δm′,m 1 ∓ ± ± −→ ± b. Computep the contracted limit of the Jacobi polynomials and show that l m n lim Pmn (cos(x/l)) = ( ) − Jm n(x) − − where Jk(x) is the kth Bessel function [1, 31]. c. Contract the spherical harmonics and show that

2π l lim Ym(β = x/l) Jm(x) r l → d. Contract the Legendre polynomials and show that

lim P l (cos(β = x/l)) J (x) → 0 e. In the generating function expression

αJ+ l l l l e Ym(θ, φ)= AkYm+k(θ, φ)= Ym(θ′, φ′) k 0 X≥ l compute the coeﬃcients Ak and the arguments θ′, φ′ explicitly. Contract these results to construct the classical generating functions for Bessel functions. f. Show that the operator L L contracts to 2 in the plane. · ∇ 246 Contraction g. Show that the Casimir invariant operator for SO(3) becomes the Laplace-Beltrami operator on S2 = SO(3)/SO(2) when restricted to the sphere surface, and this operator contracts to the Bessel equation. 2. Under the contraction u(2) h the representations of the unitary → 4 group U(2) contract to representations of the noncompact Heisenberg group H4. Since H4 is noncompact it has no faithful ﬁnite dimensional unitary irreducible representations. We therefore contract through a series of representations of U(2) of ever increasing dimensions, as follows:

lim ǫ ǫJ h 2jǫ2 1 → ∞ ± → ± → 1 j j + ,m J3 + 2 h3 ∞ → ∞ → −∞ 2ǫ → | m i → | n i j + m = n (ﬁnite) θ π √2ǫx → 2 − a. Compute the matrix elements

j j lim ǫJ ∞ h ∞ h m′ | ±| m i −→ h n′ | ±| n i

lim √n +1 δn′,n+1 ǫ (j m)(j m + 1) δm′,m 1 ∓ ± ± −→ √n δn′,n 1 p − b. Contract the spherical harmonics and show

1/4 l π lim x2/2 l Pn l,0 √2ǫx ψn(x)= NnHn(x)e− − 2 − −→ where ψn(x) is the nth excited state wavefunction for the harmonic oscillator, Hn(x) is the nth Hermite polynomial, and Nn 1 is the usual normalization coeﬃcient, Nn = . √2nn!√π c. Carry out steps c. - f. of the previous problem. The results are obtained by making the following replacements: Bessel function harmonic oscillator eigenfunction → Bessel equation Schr¨odinger equation for harmonic oscilator →

3. Contract the Lie algebra su(2) spanned by J3, J ([J3, J ] = ± ± J , [J+, J ]=2J3) with respect to the subalgebra J . Use a simple ± ± − − In¨on¨u-Wigner contraction to show

limǫ 0 ǫ(2J3) P P ′ = ∂x → → limǫ 0 ǫ(J+) TT ′ = ∂t → → limǫ 0 (J ) V V ′ = t∂x → − → 13.6 Problems 247 Construct the commutation relations of the contracted operators and show that the operators on the right (P ′,T ′, V ′) satisfy an isomorphic set of commutation relations. The operators ∂x, ∂t, t∂x generate the Galilean group in one dimension. Conclude that if the Lie algebra a1 is contracted with respect to one of its shift operators the galilean algebra gal(1) results. 4. Contract SO(n + 1) with respect to the subgroup SO(n) and show how the invariant metric and measure on the sphere Sn = SO(n + 1)/SO(n) reduce to the familiar metric and measure on Rn = ISO(n)/SO(n). 5. Disentangling formulas can also be contracted. a. Use the deﬁning 2 2 matrix representation for su(2) to construct × the disentangling theorem

∗ ∗ ∗ ζJ+ ζ J− τJ+ log(1+τ τ)J3 τ J− e − = e e e−

and show τ = (ζ/ ζ )tan( ζ ). | | | | b. Use a faithful matrix representation of the Lie algebra h4 to construct the disentangling theorem

αa† α∗a αa† 1 α∗α I α∗a e − = e e− 2 e− c. Use the contraction relation Eq. (13.30) for u(2) h to show that → 4 the u(2) disentangling theorem contracts to the h4 disentangling theorem in the limit α = limc 0 ζ/c. →

6. Thermal expectation values of the operator X are constructed by β β taking the trace: X = tr Xe− H/tr e− H, and a generating function h i αX αX β β for expectation values is e = tr e e− H/tr e− H. When the h i operators X and are elements in a ﬁnite dimensional Lie algebra H these expectation values can often be computed rather simply. a. Assume = ǫJ and X is in the Lie algebra su(2). Show that in the H 3 2 2 deﬁning matrix representation × θ J cosh(θ/2)+(θz/θ) sinh(θ/2) (θx iθy)/θ sinh(θ/2) e · − → (θ + iθ )/θ sinh(θ/2) cosh(θ/2) (θ /θ) sinh(θ/2) x y − z

βǫ/2 β e− 0 e− H → 0 e+βǫ/2 248 Contraction b. Show that the trace of this product is 2 cosh(θ/2) cosh(βǫ/2) 2(θ /θ) sinh(θ/2) sinh(βǫ/2) (= 2cosh(ψ/2)) − z c. Show that in the 2 2 matrix representation with j = 1 and 2j+1=2 × 2 θ J e · = (sinh ψ/ sinh(ψ/2)) / (sinh βǫ/ sinh(βǫ/2)) h i d. Show that in the (2j + 1) (2j + 1) dimensional representation × θ J sinh((2j + 1)ψ/2)/ sinh(ψ/2) e · = h i sinh((2j + 1)βǫ/2)/ sinh(βǫ/2) e. As j becomes large, show that this ratio simpliﬁes to

θ J j e · →∞ sinh(jψ)/ sinh(jβǫ) h i −→ f. Contract this generating function to the Heisenberg algebra.

7. One real form of D3 is the conformal group SO(4, 2). a. Write down the quadratic, cubic, and quartic Casimir operators for SO(4, 2). These are analytic continuations of 2 = X2 , 3 = C ij ij C ǫabcdef X X X , and 4 = Y 2, where Y = ǫijcdef X X . ab cd ef C ij ij ij Pcd ef b. Contract SO(4, 2) with respect to the subgroup SO(4) SO(2). P ⊗ c. Construct the quadratic, cubic, and quartic Casimir operators of the contracted group. These are analytic continuations of the contractions of the three operators of part a. If we deﬁne Ai = limǫ 0 ǫXi5 and → Bi = limǫ 0 ǫXi6, then show that the Casimirs contract to →

2 A A + B B C → · · 3 ǫijklX A B C → ij k l 4 (ǫijklA B )2 C → ij k l In these expressions the indices rangeP from 1 to 4. d. Write down the Laplace-Beltrami operators in the eight-dimensional spaces SO(4, 2)/ [SO(4) SO(2)] and I [SO(4) SO(2)] / [SO(4) SO(2)]. ⊗ ⊗ ⊗ 8. Riemannian symmetric spaces have been classiﬁed using the Cartan decomposition of simple Lie algebras:

[h, h] h ⊆ g = h + p [h, p] = p [p, p] h ⊆ Operators Xi span h and Xα span p. a. Show that the metric on p is 13.6 Problems 249

i γ γ i gα,β = Cα,γ Cβ,i + Cα,i Cβ,γ

b. Show that in the contracted limit Yα = limǫ 0 ǫXα a metric tensor → on p is well deﬁned by

2 g(p′)α,β = lim(Yα, Yβ)/ǫ = (Xα,Xβ) ǫ 0 → Use the structure constants to show this. c. Show that this metric is unchanged on the contracted space P ′ = G′/H, as opposed to the metric on P = G/H, which varies from place to place on the space.