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A Representations of SU(2) A Representations of SU(2) In this appendix we provide details of the parameterization of the group SU(2) and differential forms on the group space. An arbitrary representation of the group SU(2) is given by the set of three generators Tk, which satisfy the Lie algebra [Ti,Tj ]=iεijkTk, with ε123 =1. The element of the group is given by the matrix U =exp{iT · ω} , (A.1) 1 where in the fundamental representation Tk = 2 σk, k =1, 2, 3, with 01 0 −i 10 σ = ,σ= ,σ= , 1 10 2 i 0 3 0 −1 standard Pauli matrices, which satisfy the relation σiσj = δij + iεijkσk . The vector ω has components ωk in a given coordinate frame. Geometrically, the matrices U are generators of spinor rotations in three- 3 dimensional space R and the parameters ωk are the corresponding angles of rotation. The Euler parameterization of an arbitrary matrix of SU(2) transformation is defined in terms of three angles θ, ϕ and ψ,as ϕ θ ψ iσ3 iσ2 iσ3 U(ϕ, θ, ψ)=Uz(ϕ)Uy(θ)Uz(ψ)=e 2 e 2 e 2 ϕ ψ ei 2 0 cos θ sin θ ei 2 0 = 2 2 −i ϕ θ θ − ψ 0 e 2 − sin cos 0 e i 2 2 2 i i θ 2 (ψ+ϕ) θ − 2 (ψ−ϕ) cos 2 e sin 2 e = i − − i . (A.2) − θ 2 (ψ ϕ) θ 2 (ψ+ϕ) sin 2 e cos 2 e Thus, the SU(2) group manifold is isomorphic to three-sphere S3. The Euler angles θ, ϕ and ψ take values within the intervals 0 ≤ θ ≤ π,0≤ ϕ ≤ 2π and 0 ≤ ψ ≤ 4π. Note that the reduction to the parametric space of the 502 A Representations of SU(2) orthogonal group SO(3) can be achieved, if we fix the range of values of the angle ψ to be restricted to the interval 0 ≤ ψ ≤ 2π and make the identification ψ ∼ ψ +2π. Using the matrices (A.2), we can define so-called canonical left and right one-forms on the group SU(2), which are also called Maurer–Cartan one- forms (note that dU −1 U = −U −1 dU) i i R = U −1 dU = σ R ,L= dU U −1 = σ L . (A.3) 2 k k 2 k k Since det U = 1, we have the condition on these forms d det U = detr ln U =trL =trR =0. The components of the Maurer–Cartan forms in the basis given by the Pauli matrices are written as R1 = − sin ψdθ +cosψ sin θdϕ, L1 =sinϕdθ − cos ϕ sin θdψ, R2 =cosψdθ +sinψ sin θdϕ, L2 =cosϕdθ +sinϕ sin θdψ, R3 = dψ +cosθdϕ, L3 = dϕ +cosθdψ . (A.4) Clearly, they satisfy the Maurer–Cartan equations 1 1 dR = ε R ∧ R ,dL= − ε L ∧ L . n 2 nmk m k n 2 nmk m k In the same way, we can define the set of angular coordinates ψ,˜ θ,˜ ϕ˜,that parameterizes the sphere SO(3) and the one-forms on the space of parameters of this group. The left and right forms on the group SU(2) are dual to the vector field ξk, components of which form the standard basis of the Lie algebra on the group SU(2): (R) (L) ξk ,Rm = δkm, ξk ,Lm = δkm . Here the right and left Killing vectors are related with generators of rotations about the corresponding axis of Cartesian coordinates. They can be written in terms of the Euler parameterization as ∂ ∂ cos ψ ∂ ξ(R) = − cot θ cos ψ − sin ψ + , 1 ∂ψ ∂θ sin θ ∂ϕ ∂ ∂ sin ψ ∂ ξ(R) = − cot θ sin ψ +cosψ + , 2 ∂ψ ∂θ sin θ ∂ϕ ∂ ξ(R) = , (A.5) 3 ∂ψ and A Representations of SU(2) 503 cos ϕ ∂ ∂ ∂ ξ(L) = − +sinϕ +cotθ cos ϕ , 1 sin θ ∂ψ ∂θ ∂ϕ sin ϕ ∂ ∂ ∂ ξ(L) = +cosϕ − cot θ sin ϕ , 2 sin θ ∂ψ ∂θ ∂ϕ ∂ ξ(L) = , (A.6) 3 ∂ϕ for the left and right Killing vector field, respectively. The vector fields on the parameter space of the SO(3) group can be constructed in the same way. Note that the generators of the left and right rotations commute, while left and right Killing vectors satisfy the SU(2) Lie algebra (R) (R) − (R) (L) (L) (L) (L) (R) [ξm ,ξn ]= εmnkξk , [ξm ,ξn ]=εmnkξk , [ξm ,ξn ]=0. Thus, the right one-form Rn is invariant with respect to the left action of the SU(2) group while the left one-form Ln is invariant with respect to the right action of the group SU(2), i.e., the corresponding Lie derivative with (L) (R) respect to ξn and ξn vanishes. The metric on the group manifold, which is constructed using the one-forms Rn, by definition is left-invariant with the (L) Killing vectors ξn . The group space of SU(2) group is isomorphic to one of the “remark- able” spheres S0,S1,S3 and S7, which are characterized by the left × right parallelism. The vector fields on the sphere S3 are related with the angular momentum operator as (R) − (R) (L) (L) Ln = iξn ,Ln = iξn . It follows from the relation (A.7) that the components of the operator of angular momentum satisfy the usual commutation relation, which does not distinguish between left and right rotations: [Ln,Lm]=iεnmkLk . Eigenfunctions of the operator of angular momentum are known as Wigner functions l ≡ imϕ l iµψ Dmµ(ϕ, θ, ψ) e dmµ(θ)e , (A.7) l where dmµ(θ) are defined as [10]: 1 2 (l − m)!(l + m)! m+µ − m−µ dl (θ)= (1 − x) 2 (1 + x) 2 mµ (l − µ)!(l + µ)! × (−m−µ,−m+µ) Pl+m (x) , (A.8) (a,b) x =cosθ and Pn (x) is a Jacobi polynomial 504 A Representations of SU(2) (−1)n dn P (a,b)(x)= (1 − x)−a(1 + x)−b (1 − x)a+n(1 + x)b+n . n 2nn! dxn The Wigner function is related to the generalized spherical harmonics as l − Yµlm(θ, ϕ)=Dµm( ϕ, θ, ϕ) . The matrices (A.2), which correspond to the fundamental representation of 1/2 the group SU(2), are particular cases of the Wigner functions: Dµm(ϕ, θ, ψ)= U(ϕ, θ, ψ). However, the difference between left and right rotations on the sphere S3, which is hidden behind the general definition of the operator of angular momentum, reappears if we consider the ladder operators L± = L1 ± L2. Then the Wigner functions satisfy the equations (R) l l (R) l l L± D = l(l +1)− µ(µ ± 1)D ± ,LD = µD ; mµ mµ 1 3 mµ mµ (L) l − − ∓ l (L) l − l L± Dmµ = l(l +1) m(m 1)Dm∓1µ ,L3 Dmµ = mDmµ . (A.9) B Quaternions Four-dimensional Euclidean space R4 is quite special, since it admits a nat- ural multiplicative structure. This becomes very important in clarifying the description of the moduli spaces of the monopoles. In this Appendix, we briefly give addition material to that used in Sect. 6.5.1. Let us consider a set of 2 × 2 complex matrices R4. It is closed under matrix addition and multiplication by real scalars and, therefore, may be considered as a real vector space. The bases of the space R4 are given by the set of matrices 10 0 −i e = = I ,e= = −iσ , 1 01 2 2 −i 0 1 0 −1 −i 0 e = = −iσ ,e= = −iσ , (B.1) 3 10 2 4 0 i 3 which satisfy the algebra e4eµ = eµe4 = eµ,enem = −δnm + εnmkek (n, m, k =1, 2, 3) . (B.2) 4 4 The basis {eµ} provides a natural isomorphism from R to R given by the mapping X = e1x1 + e2x2 + e3x3 + e4x4 → xµ =(x1,x2,x3,x4) . Since the basis {eµ} is orthonormal, this mapping does not change the norm 2 2 2 2 2 X = x1 + x2 + x3 + x4 and such an isomorphism is an isometry. Note that a matrix X of the space R4 can be written as x − ix − ix − x X = 1 4 2 3 , (B.3) −ix2 + x3 x1 + ix4 and then the norm X 2=detX. The commutation relations (B.2) is a particular case of the so-called al- gebra of quaternions. The space of quaternions H can be viewed as the set of complex matrices R4 equipped with a standard set of matrix operations, or as the vector space R4 with multiplicative structure. Since e1 is a multiplicative identity, we can drop it and write an arbitrary quaternion as X = x0 + xnen. The operation of the quaternionic conjugation is defined as 506 B Quaternions X =→ X¯ = xµe¯µ = x0 − xnen . Thus, if a quaternion X is considered as a matrix in R4, its conjugated X¯ is the conjugated transpose matrix. The product of two quaternions can be computed using the relations (B.2). In particular, we have XX¯ = XX¯ = X 2 and XY = Y¯ X¯.Thereal and imaginary parts of a quaternion are 1 1 Re X = (X + X¯)=X , Im X = (X − X¯)=X e . 2 4 2 n n Quaternions whose imaginary part is equal to zero are called real quaternions. A unit quaternion satisfies the relation X 2= 1. Clearly, these quaternions correspond to the elements of R4 with unit determinant, that is, the group of unit quaternions is actually the group SU(2). Its group space, a sphere S3 naturally arises as a subspace of R4.
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