13 Contraction Contents 13.1 Preliminaries 233 13.2 In¨on¨u–Wigner Contractions 233 13.3 Simple Examples of In¨on¨u–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 parame- ter 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 sim- ple class of contractions — the In¨on¨u–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 lim- iting 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 In¨on¨u–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 In¨on¨u–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 In¨on¨u–Wigner Contractions In this section we illustrate several facets of In¨on¨u–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- direc- tions, 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 In¨on¨u–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 be- come 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 rota- tion through an angle θ1 = d2/R about the x-axis producing a displace- ment 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 displace- ments 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 an- gular 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 inhomo- geneous 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.