Chapter 2 Special Orthogonal Group SO(N)

Transcription

1 Chapter 2 Special Orthogonal Group SO(N) 1 Introduction Since the exactly solvable higher-dimensional quantum systems with certain central potentials are usually related to the real orthogonal group O(N) deined by orthogonal n n matrices, we shall give a brie review o some basic properties o group O(N) based on the monographs and textbooks [ ]. Beore proceeding to do so, we irst outline the development in order to make the reader recognize its importance in physics. We oten apply groups throughout mathematics and the sciences to capture the internal symmetry o other structures in the orm o automorphism groups. It is well-known that the internal symmetry o the structure is usually related to an invariant mathematical property, and a set o transormations that preserve this kind o property together with the operation o composition o transormations orm a group named a symmetry group. It should be noted that Galois theory is the historical origin o the group concept. He used groups to describe the symmetries o the equations satisied by the solutions o a polynomial equation. The solvable groups are thus named due to their prominent role in this theory. The concept o the Lie group named or mathematician Sophus Lie plays a very important role in the study o dierential equations and maniolds; they combine analysis and group theory and are thereore the proper objects or describing symmetries o analytical structures. An understanding o group theory is o importance in physics. For example, groups describe the symmetries which the physical laws seem to obey. On the other hand, physicists are very interested in group representations, especially o the Lie groups, since these representations oten point the way to the possible physical theories and they play an essential role in the algebraic method or solving quantum mechanics problems. As a common knowledge, the study o the groups is always related to the corresponding algebraic method. Up to now, the algebraic method has become the subject o interest in various ields o physics. The elegant algebraic method was irst S.-H. Dong, Wave Equations in Higher Dimensions, DOI / _2, Springer Science+Business Media B.V

2 14 2 Special Orthogonal Group SO(N) introduced in the context o the new matrix mechanics around Since the introduction o the angular momentum in quantum mechanics, which was intimately connected with the representations o the rotation group SO(3) associated with the rotational invariance o central potentials, its importance was soon recognized and the necessary ormalism was developed principally by a number o pioneering scientists including Weyl, Racah, Wigner and others [136, ]. Until now, the algebraic method to treat the angular momentum theory can be ound in almost all textbooks o quantum mechanics. On the other hand, it oten runs parallel to the dierential equation approach due to the great scientist Schrödinger. Pauli employed algebraic method to deal with the hydrogen atom in 1926 [145] and Schrödinger also solved the same problem almost at the same time [146], but their ates were quiet dierent. This is because the standard dierential equation approach was more accessible to the physicists than the algebraic method. As a result, the algebraic approach to determine the energy levels o the hydrogen atom was largely orgotten and the algebraic techniques went into abeyance or several decades. Until the middle o 1950s, the algebraic techniques revived with the development o theories or the elementary particles since the explicit orms o the Hamiltonian or those elementary particle systems are unknown and the physicists have to make certain assumptions on their internal symmetries. Among various attempts to solve this diicult problem, the particle physicists examined some non-compact Lie algebras and hoped that they would provide a clue to the classiication o the elementary particles. Unortunately, this hope did not materialize. Nevertheless, it is ound that the Lie algebras o the compact Lie groups enable such a classiication or the elementary particles [147] and the non-compact groups are relevant or the dynamic groups in atomic physics [148] and the non-classical properties o quantum optical systems involving coherent and squeezed states as well as the beam splitting and linear directional coupling devices [ ]. It is worth pointing out that one o the reasons why the algebraic techniques were accepted very slowly and the original group theoretical and algebraic methods proposed by Pauli [145] were neglected is undoubtedly related to the abstract character and inherent complexity o group theory. Even though the proper understanding o group theory requires an intimate knowledge o the standard theory o inite groups and o the topology and maniold theory, the basic concepts o group theory are quite simple, specially when we present them in the context o physical applications. Basically, we attempt to introduce them as simple as possible so that the common reader can master the basic ideas and essence o group theory. The detailed inormation on group theory can be ound in the textbooks [ , 154]. On the other hand, during the development o algebraic method, Racah algebra techniques played an important role in physics since it enables us to treat the integration over the angular coordinates o a complex many-particle system analytically and leads to the ormulas expressed in terms o the generalized CGCs, Wigner n-j symbols, tensor spherical harmonics and/or rotation matrices. With the development o algebraic method in the late 1950s and early 1960s, the algebraic method proposed by Pauli was systematized and simpliied greatly by using the concepts o the Lie algebras. Up to now, the algebraic method has been widely applied to

3 2 Abstract Groups 15 various ields o physics such as nuclear physics [155], ield theory and particle physics [156], atomic and molecular physics [ ], quantum chemistry [161], solid state physics [162], quantum optics [149, 151, ] and others. 2 Abstract Groups We now give some basic deinitions about the abstract groups 1 based on textbooks by Weyl, Wybourne, Miller, Ma and others [136, 137, 139, 140, 169]. Deinition A group G is a set o elements {e,,g,h,k,...} together with a binary operation. This binary operation named a group multiplication is subject to the ollowing our requirements: Closure:i,g G, then g G too, Identity element: there exists an identity element e in G (a unit) such that e = e= or any G, Inverses: or every G there exists an inverse element 1 G such that 1 = 1 = e, Associative law: the identity (hk) = ( h)k is satisied or all elements,h,k G. Subgroup: a subgroup o G is a subset S G, which is itsel a group under the group multiplication deined in G, i.e.,,h S h S. Homomorphism: a homomorphism o groups G and H is a mapping rom a group G into a group H, which transorms products into products, i.e., G H. Isomorphism: an isomorphism is a homomorphism which is one-to-one and onto [169]. From the viewpoint o the abstract group theory, isomorphic groups can be identiied. In particular, isomorphic groups have identical multiplication tables. Representation: a representation o a group G is a homomorphism o the group into the group o invertible operators on a certain (most oten complex) Hilbert space V (called representation space). I the representation is to be initedimensional, it is suicient to consider homomorphisms G GL(n).TheGL(n) represents a general linear group o non-singular matrices o dimension n. Usually, the image o the group in this homomorphism is called a representation as well. Irreducible representation: an irreducible representation is a representation whose representation space contains no proper subspace invariant under the operators o the representation. Commutation relation: since a Lie algebra has an underlying vector space structure we may choose a basis set {L i } (i = 1, 2, 3,...,N) or the Lie algebra. In 1 There exist two kinds o dierent meanings o the terminology abstract group during the irst hal o the 20th century. The irst meaning was that o a group deined by our axioms given above, but the second one was that o a group deined by generators and commutation relations.

4 16 2 Special Orthogonal Group SO(N) general, the Lie algebra can be completely deined by speciying the commutators o these basis elements: [L i,l j ]= c ij k L k, i,j,k= 1, 2, 3,...,N, (2.1) k in which the coeicients c ij k and the elements L i are the structure constants and the generators o the Lie algebra, respectively. It is worth noting that the set o operators, which commute with all elements o the Lie algebra, are called Casimir operators. We shall constraint ourselves in the ollowing parts to study some basic properties o the compact group SO(N) alongside the well-known compact so(n) Lie algebra o the generalized angular momentum theory since it shall be helpul in successive Chapters. We suggest the reader reer to the textbooks on group theory [ , 144, 154, 169] or Appendices A C or more inormation. 3 Orthogonal Group SO(N) For every positive integer N, the orthogonal group O(N) is the group o N N orthogonal matrices A satisying AA T = 1, A = A. (2.2) Because the determinant o an orthogonal matrix is either 1 or 1, and so the orthogonal group has two components. The component containing the identity 1 is the special orthogonal group SO(N). AnN-dimensional real matrix contains N 2 real parameters. The column matrices o a real orthogonal matrix are normal and orthogonal to each other. There exist N real matrix constraints or the normalization and N(N 1)/2 real constraints or the orthogonality. Thus, the number o independent real parameters or characterizing the elements o the groups SO(N) is equal to N 2 [N + N(N 1)/2]=N(N 1)/2. The group space is a doubly-connected closed region so that the SO(N) is a compact Lie group with rank N(N 1)/2. 4 Tensor Representations o the Orthogonal Group SO(N) In this section we are going to study the reduction o a tensor space o the SO(N) and calculation o the orthonormal irreducible basis tensors [139, 140]. 4.1 Tensors o the Orthogonal Group SO(N) We begin by studying the tensors o the SO(N). For a given rank n o the SO(N), we know that there are N n components with a ollowing transorm, R T c1 c n OR T c1 c n = R c1 d 1 R cn d n T d1 d n, R SO(N). (2.3) d 1 d n

5 4 Tensor Representations o the Orthogonal Group SO(N) 17 It is noted that only one nonvanishing component o a basis tensor is equal to 1, i.e., (θ a1 a n ) b1 b n = δ a1 b 1 δ an b n = (θ a1 ) b1 (θ an ) bn, (2.4) O R (θ a1 a n ) = c 1 c n (θ c1 c n )R c1 a 1 R cn a n, (2.5) rom which one may expand any tensor in such a way T b1 b n = a 1 a n T a1 a n (θ a1 a n ) b1 b n. (2.6) The tensor space is an invariant linear space both in the SO(N) and in the permutation group S n. Since the SO(N) transormation commutes with the permutation so that one can reduce the tensor space in the orthogonal groups SO(N) by the projection o the Young operators, which are conveniently used to deal with the permutation group S n. Note that there are several important characteristics or the tensors o the SO(N) group: The real and imaginary parts o a tensor o the SO(N) transorm independently in Eq. (2.3). As a result, we need only study their real tensors. There is no any dierence between a covariant tensor and a contra-variant tensor or the SO(N) transormations. The contraction o a tensor can be achieved between any two indices. Thereore, beore projecting a Young operator, the tensor space must be decomposed into a series o traceless tensor subspaces, which remain invariant in the SO(N). Denote by T the traceless tensor space o rank n. Ater projecting a Young operator, T [λ] μ = y μ [λ] T is a traceless tensor subspace with a given permutation symmetry. T [λ] μ will become a null space i the summation o the numbers o boxes in the irst two columns o the Young pattern 2 [λ] is larger than the dimension N. I the row number m o the Young pattern [λ] is larger than N/2, then the basis tensor y [λ] μ θ b1 b m c can be changed to a dual basis tensor by a totally antisymmetric tensor ɛ a1 a N, [y [λ] θ] a1 a N m c = 1 m! whose inverse transormation is given by a N m+1 a N ɛ a1 a N m a N m+1 a N y [λ] θ an a N m+1 c, (2.7) 2 A Young pattern [λ] has n boxes lined up on the top and on the let, where the jth row contains λ j boxes. For instance, the Young pattern [2, 1] is It should be noted that the number o boxes in the upper row is not less than in the lower row, and the number o boxes in the let column is not less than that in the right column. We suggest the reader reer to the permutation group S n in Appendix A or more inormation..

6 18 2 Special Orthogonal Group SO(N) 1 (N m)! = a m+1 a N ɛ b1 b m a m+1 a N 1 m!(n m)! [y [λ] θ] an a m+1 c a 1 a N ɛ b1 b m a m+1 a N ɛ an a m+1 a m a 1 y [λ] θ a1 a m c = ( 1) N(N 1)/2 y [λ] θ b1 b m c. (2.8) Ater some algebraic manipulations, it is ound that the correspondence between two sets o basis tensors is one-to-one and the dierence between them is only in the arranged order. Thus, a traceless tensor subspace T [λ] μ is equivalent to a traceless tensor subspace T [λ ] ν, where the row number o the Young pattern [λ ] is (N m) < N/2, { [λ ] [λ], λ i = λi, i (N m), (2.9) 0, i >(N m), where m (N/2,N]. I N = 2l, i.e., the row number l o [λ] is equal to N/2, then the Young pattern [λ] is the same as its dual Young pattern, called the sel-dual Young pattern. To remove the phase actor ( 1) N(N 1)/2 = ( 1) l appearing in Eq. (2.8), we introduce a actor ( i) l in Eq. (2.7), Deine [y [λ] θ] a1 a l c = ( i)l l! y [λ] θ a1 a l c = ( i)l l! ɛ a1 a l a l+1 a 2l y [λ] θ a2l a l+1 c, (2.10) a l+1 a 2l [y [λ] θ] a2l a l+1 c. (2.11) a l+1 a 2l ɛ a1 a l a l+1 a 2l ψ ± a 1 a l c = 1 2{ y [λ] θ a1 a l c ± [y [λ] θ] a1 a l c }. (2.12) We observe that ψ a + 1 a l c keeps invariant in the dual transormation so that we call it sel-dual basis tensor. On the contrary, we call ψa 1 a l c the anti-sel-dual basis tensor because it changes its sign in dual transormation. For example, or even N = 2l we may construct the sel-dual and anti-sel-dual basis tensors as ollows: ψ ± 1 l = 1 2{ y [1 l] θ 1 l ± ( i) l y [1l] θ (2l) (l+1) }. (2.13) Thereore, when l = N/2 the representation space T [λ] μ can be divided to the sel-dual and the anti-sel-dual tensor subspaces with the same dimension. Notice that the combinations by the Young operators and the dual transormations (2.7) and (2.13) are all real except that the dual transormation (2.13) with N = 4l + 2is complex. In conclusion, the traceless tensor subspace T [λ] μ corresponds to a representation [λ] o the SO(N), where the row number l o Young pattern [λ] is less than N/2. When l = N/2 the traceless tensor subspace T [λ] μ can be decomposed into the sel- corresponding dual tensor subspace T [+λ] μ and anti-sel-dual tensor subspace T [ λ] μ

7 4 Tensor Representations o the Orthogonal Group SO(N) 19 to the representation [±λ], respectively. All irreducible representations [λ] and [±λ] are real except or [±λ] with N = 4l + 2. As ar as the orthonormal irreducible basis tensors o the SO(N), we are going to address two problems. The irst is how to decompose the standard tensor Young tableaux into a sum o the traceless basis tensors. The second is how to combine the basis tensors such that they are the common eigenunctions o H j and orthonormal to each other. The advantage o the method based on the standard tensor Young tableaux is that the basis tensors are known explicitly and the multiplicity o any weight is equivalent to the number o the standard tensor Young tableaux with the weight. For group SO(N), the key issue or inding the orthonormal irreducible basis is to ind the common eigenstates o H i and the highest weight state in an irreducible representation. For odd and even N, i.e., the groups SO(2l + 1) and SO(2l), the generators T ab o the sel-representation satisy [T ab ] cd = i(δ ac δ bd δ ad δ bc ), [T ab,t cd ]= i(δ bc T ad + δ ad T bc δ bd T ac δ ac T bd ). (2.14) The bases H i in the Cartan subalgebra can be written as H i = T (2i 1)(2i), i [1,N/2]. (2.15) As what ollows, we are going to study the irreducible basis tensors o the SO(2l + 1) and SO(2l), respectively. 4.2 Irreducible Basis Tensors o the SO(2l + 1) It is known that the Lie algebra o the SO(2l + 1) is B l. The simple roots o the SO(2l + 1) are given by [139, 140] r ν = e ν e ν+1, ν [1,l 1], r l = e l, (2.16) where r ν are the longer roots with d ν = 1 and r l is the shorter root with d l = 1/2. Based on the deinition o the Chevalley bases, which include 3l bases E ν,f ν, and H ν or the generators, E rν dν E ν, E rν dν F ν, 1 d ν l (r ν ) i H i 1 r ν H H ν, (2.17) d ν i=1

8 20 2 Special Orthogonal Group SO(N) one is able to calculate the Chevalley bases o the SO(2l + 1) in the selrepresentation as ollows: H ν = T (2ν 1)(2ν) T (2ν+1)(2ν+2), E ν = 1 2 [T (2ν)(2ν+1) it (2ν 1)(2ν+1) it (2ν)(2ν+2) T (2ν 1)(2ν+2) ], F ν = 1 2 [T (2ν)(2ν+1) + it (2ν 1)(2ν+1) + it (2ν)(2ν+2) T (2ν 1)(2ν+2) ], H l = 2T (2l 1)(2l), E l = T (2l)(2l+1) it (2l 1)(2l+1), F l = T (2l)(2l+1) + it (2l 1)(2l+1). (2.18) Note that θ a are not the common eigenvectors o H ν. By generalizing the spherical harmonic basis vectors or the SO(3) group, we may deine the spherical harmonic basis vectors or the sel-representation o the SO(2l + 1) as ollows: ( 1) l β (θ 2β 1 + iθ 2β ), β [1,l], φ β = θ 2l+1, β = l + 1, (2.19) 1 2 (θ 4l 2β+3 iθ 4l 2β+4 ), β [l + 2, 2l + 1], which are orthonormal and complete. In the spherical harmonic basis vectors φ β, the nonvanishing matrix entries in the Chevalley bases are given by H ν φ ν = φ ν, H ν φ ν+1 = φ ν+1, H ν φ 2l ν+1 = φ 2l ν+1, H ν φ 2l ν+2 = φ 2l ν+2, H l φ l = 2φ l, H l φ l+2 = 2φ l+2, E ν φ ν+1 = φ ν, E ν φ 2l ν+2 = φ 2l ν+1, (2.20) E l φ l+1 = 2φ l, E l φ l+2 = 2φ l+1, F ν φ ν = φ ν+1, F ν φ 2l ν+1 = φ 2l ν+2, F l φ l = 2φ l+1, F l φ l+1 = 2φ l+2, where ν [1,l 1]. That is to say, the diagonal matrices o H ν and H l in the spherical harmonic basis vectors φ β are expressed as ollows: H ν = diag{0,...,0, 1, 1, 0,...,0, 1, 1, 0,...,0}, }{{}}{{}}{{} ν 1 2l 2ν 1 ν 1 H l = diag{0,...,0, 2, 0, 2, 0,...,0}. }{{}}{{} l 1 l 1 (2.21) The spherical harmonic basis tensor φ β1 β n o rank n or the SO(2l +1) becomes the direct product o n spherical harmonic basis vectors φ β1 φ βn. The standard tensor Young tableaux y [λ] ν φ β1 β n are the common eigenstates o the H ν, but generally neither orthonormal nor traceless. The eigenvalue o H ν in the standard tensor

9 4 Tensor Representations o the Orthogonal Group SO(N) 21 Young tableaux y [λ] ν φ β1 β n is equal to the number o the digits ν and (2l ν + 1) in the tableau, minus the number o (ν + 1) and (2l ν + 2). The eigenvalue o H l in the standard tensor Young tableau is equal to the number o l in the tableau, minus the number o (l + 2), and then multiplied with a actor 2. The action F ν on the standard tensor Young tableau is equal to the sum o all possible tensor Young tableaux, each o which can be obtained rom the original one through replacing one illed digit ν by the digit (ν + 1), or through replacing one illed digit (2l ν + 1) by the digit (2l ν + 2). But the action o the F l on the standard tensor Young tableau is equal to the sum, multiplied with a actor 2, o all possible tensor Young tableaux, each o which can be obtained rom the original one through replacing one illed digit l by (l + 1) or through replacing one illed (l + 1) by (l + 2). However, the actions o E ν and E l on the standard tensor Young tableau are opposite to those o F ν and F l. Even though the obtained tensor Young tableaux may be not standard, they can be transormed into the sum o the standard tensor Young tableaux by symmetry. Two standard tensor Young tableaux with dierent sets o illed digits are orthogonal to each other. For a given irreducible representation [λ] o the SO(2l + 1), where the row number o Young pattern [λ] is not larger than l, the highest weight state corresponds to the standard tensor Young tableau, in which each box in the βth row is illed with the digit β because each raising operator E ν annihilates it. The highest weight M = ν ω νm ν can be calculated rom (2.20) as ollows: M ν = λ ν λ ν+1, ν [1,l), M l = 2λ l. (2.22) The tensor representation [λ] o the SO(2l + 1) with even M l is a single-valued representation, while the representation with odd M l becomes a double-valued (spinor) representation. The standard tensor Young tableaux y [λ] ν φ β1 β n are generally not traceless, but the standard tensor Young tableau with the highest weight is traceless because it only contains φ β with β<l+ 1asshowninEq.(2.19). For example, the tensor basis θ 1 θ 1 is not traceless, but φ 1 φ 1 is traceless. Since the highest weight is simple, the highest weight state is orthogonal to any other standard tensor Young tableau in the irreducible representation. Thereore, one is able to obtain the remaining orthonormal and traceless basis tensors in the representation [λ] o the SO(2l + 1) rom the highest weight state by the lowering operators F ν based on the method o the block weight diagram. 4.3 Irreducible Basis Tensors o the SO(2l) The Lie algebra o the SO(2l) is D l and its simple roots are given by r ν = e ν e ν+1, ν [1,l 1], r l = e l 1 + e l. (2.23)

10 22 2 Special Orthogonal Group SO(N) The lengths o all simple roots are same, d ν = 1. Similarly, based on the deinition o the Chevalley bases (2.17), we ind that its Chevalley bases in the sel-representation are same as those o the SO(2l + 1) except or ν = l, H l = T (2l 3)(2l 2) + T (2l 1)(2l), E l = 1 2 [T (2l 2)(2l 1) it (2l 3)(2l 1) + it (2l 2)(2l) + T (2l 3)(2l) ], (2.24) F l = 1 2 [T (2l 2)(2l 1) + it (2l 3)(2l 1) it (2l 2)(2l) + T (2l 3)(2l) ]. Likewise, θ a are not the common eigenvectors o the H ν. By generalizing the spherical harmonic basis vectors or the SO(4) group, we deine the spherical harmonic basis vectors or the sel-representation o the SO(2l) as ollows: φ β = ( 1) l β 1 2 (θ 2β 1 + iθ 2β ), β [1,l], 1 2 (θ 4l 2β+1 iθ 4l 2β+2 ), β [l + 1, 2l], (2.25) which are orthonormal and complete. In these basis vectors, the nonvanishing matrix entries o the Chevalley bases are given by H ν φ ν = φ ν, H ν φ ν+1 = φ ν+1, H ν φ 2l ν = φ 2l ν, H ν φ 2l ν+1 = φ 2l ν+1, H l φ l 1 = φ l 1, H l φ l = φ l, H l φ l+1 = φ l+1, H l φ l+2 = φ l+2, E ν φ ν+1 = φ ν, E ν φ 2l ν+1 = φ 2l ν, E l φ l+1 = φ l 1, E l φ l+2 = φ l, F ν φ ν = φ ν+1, F ν φ 2l ν = φ 2l ν+1, F l φ l 1 = φ l+1, F l φ l = φ l+2, (2.26) where ν [1,l 1]. As a result, the diagonal matrices o the H ν and H l in the spherical harmonic basis vectors φ β are calculated as: H ν = diag{0,...,0, 1, 1, 0,...,0, 1, 1, 0,...,0}, }{{}}{{}}{{} ν 1 2l 2ν 2 H l = diag{0,...,0, 1, 1, 1, 1, 0,...,0}. }{{}}{{} l 2 l 2 ν 1 (2.27) The spherical harmonic basis tensor φ β1 β n o rank n or the SO(2l) is the direct product o n spherical harmonic basis vectors φ β1 φ βn. The standard tensor Young tableaux y ν [λ] φ β1 β n are the common eigenstates o the H ν, but in general neither orthonormal nor traceless. The eigenvalue o H ν in the standard tensor Young tableaux y [λ] ν φ β1 β n is equal to the number o the digits ν and (2l ν) in the tableau,

11 4 Tensor Representations o the Orthogonal Group SO(N) 23 minus the number o (ν + 1) and (2l ν + 1). The eigenvalue o H l in the standard tensor Young tableau is equal to the number o the digits (l 1) and l in the tableau, minus the number o (l + 1) and (l + 2). The eigenvalues orm the weight m o standard tensor Young tableau. The action o F ν on the standard tensor Young tableau is equal to the sum o all possible tensor Young tableaux, each o which can be obtained rom the original one through replacing one illed digit (2l ν) by the digit (2l ν + 1). The action o F l on the standard tensor Young tableau is equal to the sum o all possible tensor Young tableaux, each o which is obtained rom the original one through replacing one illed digit (l 1) by the digit (l + 1) or through replacing one illed l by the digit (l + 2). However, the actions o E ν and E l are opposite to those o F ν and F l. The obtained tensor Young tableaux may be not standard, but they can be transormed into the sum o the standard tensor Young tableaux by symmetry. Two standard tensor Young tableaux with dierent weights are orthogonal to each other. For a given irreducible representation [λ] or [+λ] o the SO(2l), where the row number o Young pattern [λ] is not larger than l, the highest weight state corresponds to the standard tensor Young tableau where each box in the βth row is illed with the digit β because every raising operator E ν annihilates it. In the standard tensor Young tableau with the highest weight o the representation [ λ], the box in the βth row is illed with the digit β, but the box in the lth row with the digit (l + 1). The highest weight M = ν ω νm ν is calculated rom (2.20) as M ν = λ ν λ ν+1, ν [1,l 1), M l 1 = M l = λ l 1, λ l = 0, M l 1 = λ l 1 λ l, M l = λ l 1 + λ l, or [+λ], M l 1 = λ l 1 + λ l, M l = λ l 1 λ l, or [ λ]. (2.28) The tensor representation [λ] o the SO(2l) with even (M l 1 + M l ) is a singlevalued representation. However, the representation with odd (M l 1 + M l ) is a double-valued (spinor) representation. The standard tensor Young tableaux are generally not traceless, but the standard tensor Young tableau with the highest weight is traceless because it only contains φ β with β<l+ 2. Furthermore, l and l + 1 do not appear in the tableau simultaneously as illustrated in Eq. (2.25). Since the highest weight is simple, the highest weight state is orthogonal to any other standard tensor Young tableau in the irreducible representation. Hence, we can obtain the remaining orthonormal and traceless basis tensors in the irreducible representation o the SO(2l) rom the highest weight state by the lowering operators F ν in light o the method o the block weight diagram. The multiplicity o a weight in the representation can be easily obtained by counting the number o the traceless tensor Young tableaux with this weight.

12 24 2 Special Orthogonal Group SO(N) 4.4 Dimensions o Irreducible Tensor Representations The dimension d [λ] o the representation [λ] o the SO(N) can be calculated by hook rule [139, 140]. The dimension is expressed as a quotient, where the numerator and the denominator are denoted by the symbols Y [λ] T and Y [λ] h, respectively: d [±λ] [SO(2l)]= Y [λ] T 2Y [λ] h d [λ] [SO(N)]= Y [λ] T 2Y [λ] h, when λ l 0,, others. (2.29) The irst ormula in Eq. (2.29) corresponds to the case where the row number o the Young pattern [λ] is equal to N/2. The hook path (i, j) in the Young pattern [λ] is deined as a path which enters the Young pattern at the rightmost o the ith row, goes letward in the i row, turns downward at the j column, goes downward in the j column, and leaves rom the Young pattern at the bottom o the j column. The inverse hook path denoted by (i, j) is the same path as the hook path (i, j),but with opposite direction. The number o boxes contained in the path (i, j), aswell as in its inverse, is the hook number h ij.they [λ] h represents a tableau o the Young pattern [λ] where the box in the jth column o the ith row is illed with the hook number H ij. However, the Y [λ] T is a tableau o the Young pattern [λ] where each box is illed with the sum o the digits which are respectively illed in the same box o each tableau Y [λ] T b in the series. The notation Y [λ] T means the product o the illed digits in it, so does the notation Y [λ] [λ] h. Here, the tableaux Y T b can be obtained by the ollowing rules: Y [λ] T 0 is a tableau o the Young pattern [λ], where the box in the jth column o the ith row is illed with the digit (N + j i). Let [λ (1) ]=[λ]. Starting with [λ (1) ], deine recursively the Young pattern [λ (b) ] by removing the irst row and the irst column o the Young pattern [λ (b 1) ] until [λ (b) ] contains less two columns. I [λ (b) ] contains more than one column, deine Y [λ] T b as a tableau o the Young pattern [λ] where the boxes in the irst (b 1) row and in the irst (b 1) column are illed with 0, and the remaining part o the Young pattern is [λ (b) ]. Let [λ (b) ] have r rows. Fill the irst r boxes along the hook path (1, 1) o the Young pattern [λ (b) ], starting with the box on the rightmost, with the digits (λ (b) 1 1), (λ (b) 2 1),...,(λ (b) r 1), box by box, and ill the irst (λ (b) i 1) boxes in each inverse hook path (i, 1) o the Young pattern [λ (b) ],i [1,r] with 1. The remaining boxes are illed with 0. I several 1 are illed in the same box, the digits are summed. The sum o all illed digits in the pattern Y [λ] T b with b>0 is equal to 0.

13 4 Tensor Representations o the Orthogonal Group SO(N) Adjoint Representation o the SO(N) We are going to study the adjoint representation o the SO(N) by replacing the tensors. The N(N 1)/2 generators T ab in the sel-representation o the SO(N) construct the complete bases o N-dimensional antisymmetric matrices. Denote T cd by T A or convenience, A [1,N(N 1)/2]. Then we have Tr(T A T B ) = 2δ AB. (2.30) Based on the adjoint representation D ad (G) satisying D(R)I B D(R) 1 = I D DDB ad (R), D R SO(N), (2.31) where R is an ininitesimal element, we have N(N 1)/2 RT A R 1 = T B DBA ad (R). (2.32) B=1 The antisymmetric tensor T ab o rank 2 o the SO(N) satisies a similar relation in the SO(N) transormation R (O R T) cd = ij R ci T ij (R 1 ) jd = (RT R 1 ) cd, (2.33) where T ab like an antisymmetric matrix can be expanded by (T A ) ab as ollows: N(N 1)/2 T cd = (T A ) cd F A, F A = 1 (T A ) dc T cd, (2.34) 2 A=1 where the coeicient F A is a tensor that transorms in the SO(N) transormation R as ollows: (O R T) cd = (RT R 1 ) cd cd = A = B (O R T) cd = B (RT A R 1 ) cd F A { (T B ) cd A (T B ) cd O R F B. D ad BA (R)F A }, (2.35) Thus, in terms o the adjoint representation o the SO(N) we can transorm F A in such a way (O R F) B = A D ad BA (R)F A. (2.36) The adjoint representation o the SO(N) is equal to the antisymmetric tensor representation [1, 1] o rank 2. The adjoint representation o the SO(N) or N = 3 or N>4 is irreducible. Except or N = 2, 4, the SO(N) is a simple Lie group.

14 26 2 Special Orthogonal Group SO(N) 4.6 Tensor Representations o the Groups O(N) It is known that the group O(N) is a mixed Lie group with two disjoint regions corresponding to det R =±1. Its invariant subgroup SO(N) has a connected group space corresponding to det R = 1. The set o elements related to the det R = 1is the coset o SO(N). The property o the O(N) can be characterized completely by the SO(N) and a representative element in the coset [139, 140]. For odd N = 2l + 1, we may choose ε = 1 as the representative element in the coset since ε is sel-inverse and commutes with every element in O(2l + 1). Thus, the representation matrix D(ε) in the irreducible representation o O(2l + 1) is a constant matrix D(ε) = c1, D(ε) 2 = 1, c=±1. (2.37) Denote by R the element in SO(2l + 1) and by R = εr the element in the coset. From each irreducible representation D [λ] (SO(2l + 1)) one obtains two induced irreducible representations D [λ]± (O(2l + 1)), D [λ]± (R) = D [λ] (R), D [λ]± (εr) =±D [λ] (R). (2.38) Two representations D [λ]± (O(2l + 1)) are inequivalent because o dierent characters o the ε in two representations. For even N = 2l,ε = 1 belongs to SO(2l). We may choose the representative element in the coset to be a diagonal matrix σ, in which the diagonal entries are 1 except or σ NN = 1. Even though σ 2 = 1, σ does not commute with some elements in O(2l). Any tensor Young tableau y [λ] ν θ β1 β n is an eigentensor o the σ with the eigenvalue 1 or 1 depending on whether the number o illed digits N in the tableau is even or odd. In the spherical harmonic basis tensors, σ interchanges the illed digits l and l + 1 in the tensor Young tableau y [λ] ν φ β1 β n. Thereore, the representation matrix D [λ] (σ ) is known. Denote by R the element in the SO(2l) and by R = σr the element in the coset. From each irreducible representation D [λ] (SO(2l)), where the row number o [λ] is less than l, we obtain two induced irreducible representations D [λ]± (O(2l)), D [λ]± (R) = D [λ] (R), D [λ]± (σ R) =±D [λ] (σ )D [λ] (R). (2.39) Likewise, two representations D [λ]± (O(2l)) are inequivalent due to the dierent characters o the σ in two representations. When l = N/2 there are two inequivalent irreducible representations D [(±)λ] o the SO(2l). Their basis tensors are given in Eq. (2.12). Since two terms in Eq. (2.12) contain dierent numbers o the subscripts N, then σ changes the tensor Young tableau in [±λ] to that in [ λ], i.e., the representation spaces o both D [±λ] (SO(2l)) correspond to an irreducible representation D [λ] o the O(2l), D [λ] (R) = D [+λ] (R) D [ λ] (R), D [λ] (σ R) = D [λ] (σ )D [λ] (R), (2.40) where the representation matrix D [λ] (σ ) is calculated by interchanging the illed digits l and (l +1) in the tensor Young tableau y [λ] ν φ β1 β n. Two representations with

15 5 Ɣ Matrix Groups 27 dierent signs o D [λ] (σ ) are equivalent since they might be related by a similarity transormation ( ) 1 0 X =. (2.41) Ɣ Matrix Groups Dirac generalized the Pauli matrices to our γ matrices, which satisy the anticommutation relations. In terms o the γ matrices, Dirac established the Dirac equation to describe the relativistic particle with spin 1/2. In the language o group theory, Dirac ound the spinor representation o the Lorentz group. In this section we irst generalize the γ matrices and ind that the set o products o the γ matrices orms the matrix group Ɣ. 5.1 Fundamental Property o Ɣ Matrix Groups First, let us review the property o the Ɣ matrix groups [88 90]. We deine N matrices γ a, which satisy the ollowing anticommutation relations {γ a,γ b }=γ a γ b + γ b γ a = 2δ ab 1, a,b [1,N]. (2.42) That is, γ 2 a = 1 and γ aγ b = γ b γ a or a b. The set o all products o the γ a matrices, in the multiplication rule o matrices, orms a group, denoted by Ɣ N.Ina product o γ a matrices, two γ b with the same subscript can be moved together and eliminated by Eq. (2.42) so that Ɣ N is a inite matrix group. We choose a aithul irreducible unitary representation o the Ɣ N as its selrepresentation. It is known rom Eq. (2.42) that γ a is unitary and hermitian, whose eigenvalue is 1 or 1. Let γ (N) = γ 1 γ 2 γ N, γ a = γ 1 a = γ a, (2.43) ( (N)) 2 γ = ( 1) N(N 1)/2 1. (2.44) For odd N, since γ (N) commutes with every γ a matrix, then it is a constant matrix according to the Schur theorem (see Appendix B): { γ (N) ±1, or N = 4l + 1, = (2.45) ±i1, or N = 4l 1. Two groups with dierent γ (4l+1) are isomorphic through a one-to-one correspondence, say γ a γ a, a [1, 4l], γ 4l+1 γ 4l+1. (2.46)

16 28 2 Special Orthogonal Group SO(N) On the other hand, or a given γ (4l+1),theγ (4l+1) 4l+1 can be expressed as a product o other γ a matrices. As a result, all elements both in Ɣ 4l and in Ɣ 4l+1 can be expressed as the products o matrices γ a,a [1, 4l] so that they are isomorphic. In addition, since γ (4l 1) is equal to either i1 or i1, Ɣ 4l 1 is isomorphic onto a group composed o the Ɣ 4l 2 and iɣ 4l 2, Ɣ 4l+1 Ɣ 4l, Ɣ 4l 1 {Ɣ 4l 2,iƔ 4l 2 }. (2.47) 5.2 Case N = 2l Let us calculate the order g (2l) o the Ɣ 2l. Obviously, i R Ɣ 2l, then R Ɣ 2l, too. I we choose one element in each pair o elements ±R, then we obtain a set Ɣ 2l containing g(2l) /2 elements. Denote by S n a product o n dierent γ a. Since the number o dierent S n contained in the set Ɣ 2l is equal to the combinatorics o n among 2l, then we have 2l ( ) 2l g (2l) = 2 = 2(1 + 1) 2l = 2 2l+1. (2.48) n n=0 For any element S n Ɣ 2l except or ±1, we may ind a matrix γ a which is anticommutable with S n. In act, when n is even and γ appears in the product S n, one has γ a S n = S n γ a. However, when n is odd there exists at least one γ a which does not appear in the product S n so that γ a S n = S n γ a. Thereore, we ind that Tr S n = Tr(γ 2 a S n) = Tr(γ a S n γ a ) = Tr S n = 0. (2.49) That is to say, the character o the element S in the sel-representation o the Ɣ 2l is { ±d (S) = (2l), when S =±1, (2.50) 0, when S ±1, where d (2l) is the dimension o the γ a. Since the sel-representation o the Ɣ 2l is irreducible, we have 2 ( d (2l)) 2 = (S) 2 = g (2l) = 2 2l+1, d (2l) = 2 l. (2.51) S Ɣ 2l BasedonEqs.(2.43) and (2.50), we have det γ a = 1orl>1. Since γ (2l) γ (2l) is anticommutable with every γ a, one may deine γ (2l) satisies Eq. (2.42), i.e., with a actor such that γ (2l) γ (2l) = ( i) l γ (2l) = ( i) l γ 1 γ 2 γ 2l, Actually, γ (2l) can also be deined as the matrix γ 2l+1 in Ɣ 2l+1. by multiplying ( (2l)) 2 γ = 1. (2.52)

17 5 Ɣ Matrix Groups 29 The matrices in the set Ɣ 2l are linearly independent. Otherwise, there exists a linear relation S C(S)S = 0, S Ɣ 2l. By multiplying it with R 1 /d (2l) and taking the trace, one obtains any coeicient C(R) = 0. Thus, the set Ɣ 2l contains 2 2l linear independent matrices o dimension d (2l) = 2 l so that they orm a complete set o basis matrices. Any matrix M o dimension d (2l) can be expanded by S Ɣ 2l as ollows: M = S Ɣ 2l C(S)S, C(S) = 1 d (2l) Tr(S 1 M). (2.53) According to Eq. (2.42), the ±S orm a class, while 1 and 1 orm two classes, respectively. The Ɣ 2l group contains (2 2l + 1) classes. Their representation is one-dimensional. Arbitrary chosen n matrices γ a correspond to 1 and the remaining matrices γ b correspond to 1. The number o the one-dimensional nonequivalent representations is calculated as 2l n=0 ( ) 2l = 2 2l. (2.54) n The remaining irreducible representation o the Ɣ 2l must be d (2l) -dimensional, which is aithul. The γ a matrices in the representation are called the irreducible γ a matrices, which may be written as: γ 2n 1 = 1 } 1 {{} σ 1 σ 3 σ 3, }{{} n 1 l n γ 2n = 1 } 1 {{} σ 2 σ 3 σ 3, }{{} n 1 l n γ (2l) = σ 3 σ }{{} 3. l (2.55) Since γ (2l) is diagonal, the orms o Eq. (2.55) are called the reduced spinor representations. Remember that the eigenvalues ±1 are arranged in the diagonal line o the γ (2l) in mixed way. Let us mention an equivalent theorem or the γ a matrices. Theorem 2.1 Two sets o d (2l) -dimensional matrices γ a and γ a satisying the anticommutation relation (2.42), where N = 2l, are equivalent γ a = X 1 γ a X, a [1, 2l]. (2.56) The similarity transormation matrix X is determined up to a constant actor. I the determinant o the matrix X is constrained to be 1, there are d (2l) choices or the actor: exp[ i2nπ/d (2l) ], n [0,d (2l) ). (2.57)

18 30 2 Special Orthogonal Group SO(N) 5.3 Case N = 2l + 1 Since γ (2l) and (2l) matrices γ a in Ɣ 2l, a [1, 2l], satisy the antisymmetric relation (2.42), then they can be deined to be the (2l + 1) matrices γ a in Ɣ 2l+1.Inthis deinition, γ (2l+1) in Ɣ 2l+1 is chosen as γ 2l+1 = γ (2l), γ (2l+1) = γ 1 γ 2l+1 = i l 1. (2.58) Obviously, the dimension d (2l+1) o the matrices in Ɣ 2l+1 is the same as d (2l) in Ɣ 2l, d (2l+1) = d (2l) = 2 l. (2.59) For odd N, the equivalent theorem must be modiied because the multiplication rule o elements in Ɣ 2l+1 includes Eq. (2.45). A similarity transormation cannot change the sign o γ (2l+1), i.e., the equivalent condition or two sets o γ a and γ a has to include a new condition γ = γ, in addition to those given in Theorem 2.1. I we take γ a = (γ a ) T, then we have γ (2l+1) = γ 1 γ 2l+1 = {γ 2l+1 γ 1 } T } T = ( 1) l+1{ γ (2l+1) = ( 1) l+1 γ (2l+1). (2.60) 6 Spinor Representations o the SO(N) 6.1 Covering Groups o the SO(N) BasedonasetoN irreducible unitary matrices γ a satisying the anticommutation relation (2.42), we deine γ a = N R ai γ i, R SO(N). (2.61) i=1 Since R is a real orthogonal matrix, then γ a satisy γ a γ b + γ b γ a = ij R ai R bj {γ i γ j + γ j γ i } = 2 i R ai R bi 1 Due to Eq. (2.42) and a R 1aR 2a = 0, we have = 2δ ab 1. (2.62)

19 6 Spinor Representations o the SO(N) 31 R 1c1 R 2c2 γ c1 γ c2 = 1 R 1c1 R 2c2 (γ c1 γ c2 γ c2 γ c1 ), (2.63) 2 c 1 c 2 c 1 c 2 γ 1 γ 2 γ N = R 1 c1 R NcN γ c1 γ cn c 1 c N = R 1 c1 R NcN ɛ c1 c N γ 1 γ 2 γ N c 1 c N = (det R)γ 1 γ 2 γ N = γ 1 γ 2 γ N. (2.64) From Theorem 2.1, we know that γ a and γ a are related by a unitary similarity transormation D(R) with determinant 1, D(R) 1 γ a D(R) = N R ai γ i, det D(R) = 1, (2.65) i=1 where D(R) is determined up to a constant exp[ i2nπ/d (N) ],n [0,d (N) ).In terms o the deinition o the group, the set o D(R) deined in Eq. (2.65) and operated in the multiplication rule o matrices, orms a Lie group G N. There exists a d (N) -to-one correspondence between the elements in G N and those in SO(N), and the correspondence keeps invariant in the multiplication o elements. Thereore, the G N is homomorphic to SO(N). Because the group space o the SO(N) is doublyconnected, its covering group is homomorphic to it by a two-to-one correspondence. As a result, the group space o the G N must all into several disjoint pieces, where the piece containing the identity element E orms an invariant subgroup G N o the G N.TheG N is a connected Lie group and becomes the covering group o the SO(N). Since the group space o G N is connected, based on the property o the ininitesimal elements, a discontinuous condition can be ound to pick up G N rom the G N. Let R be an ininitesimal element. We may expand R and D(R) with respect to the ininitesimal parameters ω αβ as ollows R ab = δ ab i α<β ω αβ (T αβ ) ab = δ ab ω ab, D(R) = 1 i α<β ω αβ S αβ, (2.66) where T αβ are the generators in the sel-representation o the SO(N) as given in Eq. (2.14). The S αβ are the generators in G N.FromEq.(2.65) one has [γ c,s αβ ]= d (T αβ ) cd γ d = i{δ αc γ β δ βc γ α }, (2.67) rom which we obtain S αβ = 1 4i (γ αγ β γ β γ α ). (2.68) It is easy to prove that S αβ is hermitian since D(R) is unitary.

20 32 2 Special Orthogonal Group SO(N) Deine 3 Basedonthis,wehave { B C = (N), when N = 4l + 1, C (N), when N 4l + 1. C 1 S αβ C = (S αβ ) T = S αβ, C 1 D(R)C ={D(R 1 )} T = D(R). (2.69) (2.70) This discontinuous condition restricts the actor in D(R) such that there is a twoto-one correspondence between ±D(R) in G N and R in SO(N) through relations (2.65) and (2.70). That is to say, the G N is the covering group o SO(N), SO(N) G N, (2.71) where the G N is the undamental spinor representation denoted by D [s] (SO(N)). Thereore, the S αβ represent the spinor angular momentum operators [88 90]. The irreducible tensor representation [λ] is a single-valued representation o the SO(N), but a non-aithul representation o G N because its aithul representation is a double-valued representation o the SO(N). Since the products S n span a complete set o the d (N) -dimensional matrices, this can be decided by checking the commutation relations o the S n with the generators S αβ whether there exists a non-constant matrix commutable with all S αβ. It is ound that only γ (N) is commutable with all S αβ.theγ (2l+1) is a constant matrix so that the undamental spinor representation D [s] (SO(2l + 1)) is irreducible and selconjugate. On the contrary, since γ (2l) is not a constant matrix so that the undamental spinor representation D [s] (SO(2l)) is reducible. By a similarity transormation X,theγ (2l) can be transerred to σ 3 1 and D [s] (SO(2l)) is reduced to the direct sum o two irreducible representations X 1 D [s] (R)X = ( D [+s] ) (R) 0 0 D [ s]. (2.72) (R) Two representations D [±s] (SO(2l)) are proved inequivalent by leading to an absurdity. In act, i Z 1 D [ s] (R)Z = D [+s] (R) and Y = 1 Z, then all generators (XY ) 1 S αβ XY are commutable with σ 1 1, but their product is not commutable with it, 2 l (XY ) 1 (S 12 S 34 S (2l 1)(2l) )XY = Y 1 [X 1 γ (2l) X]Y = σ 3 1, (2.73) which results in a contradiction. 3 B N is the strong space-time relection matrix and C N is the charge conjugation matrix, which are usually used in particle physics.

21 6 Spinor Representations o the SO(N) 33 Introduce two project operators P ± [139, 140], P ± = 1 ( (2l)) 1 ± γ, P± D [s] (R) = D [s] P ±, 2 ( ) ( X P + X =, X 1 P D [s] D [+s] ) (R) 0 (R)X =, 0 0 ( ) ( ) X P X =, X 1 P 0 1 D [s] 0 0 (R)X = 0 D [ s]. (R) From the ollowing relation (2.74) (C (2l) ) 1 γ (2l) C (2l) = ( i) l (γ 1 ) T (γ 2 ) T (γ 2l ) T = ( 1) l (γ (2l) ) T, (2.75) where C (2l) and T denote the charge conjugation matrix and the transpose o the matrix, respectively, one has { C 1 D [s] D (R)P ± C = [s] (R) P ±, when N = 4l, D [s] (R) (2.76) P, when N = 4l + 2. Two non-equivalent representations D [±s] (R) are conjugate to each other when N = 4l + 2, while they are sel-conjugate when N = 4l. The dimension o the irreducible spinor representations o the SO(N) is calculated as d [s] [SO(2l + 1)]=2 l, d [±s] [SO(2l)]=2 (l 1). (2.77) 6.2 Fundamental Spinors o the SO(N) For an SO(N) transormation R, is called the undamental spinor o the SO(N) i it transorms through the undamental spinor representation D [s] (R): (O R ) ν = D νμ [s] (R) μ, O R = D [s] (R), (2.78) μ where is a column matrix with d [s] components. The Chevalley bases H ν (S), E ν (S) and F ν (S) with respect to the spinor angular momentum can be obtained rom Eqs. (2.18) and (2.24) through replacing T ab by S ab. In the chosen orms o γ a given in Eq. (2.55), the Chevalley bases or the SO(2l + 1) group are given by H ν (S) = 1 } 1 {{} 1 2 {σ σ 3 } 1 } 1 {{}, ν 1 l ν 1 H l (S) = 1 } 1 {{} σ 3, l 1 E ν (S) = 1 } 1 {{} {σ + σ } 1 } 1 {{} = F ν (S) T, ν 1 l ν 1 E l (S) = σ 3 σ }{{} 3 σ + = F l (S) T, l 1 (2.79)

22 34 2 Special Orthogonal Group SO(N) where ν [1,l). The Chevalley bases or the SO(2l) group are the same as those or the SO(2l + 1) except or ν = l, H l (S) = } 1 1 {{} 1 2 {σ σ 3 }, l 2 E l (S) = 1 } 1 {{} {σ + σ + }=F l (S) T. l 2 (2.80) The basis spinor [m] o the SO(N) can also be expressed as a direct product o l two-dimensional basis spinors (β), [m]=(β 1,β 2,...,β l ) = (β 1 )(β 2 )...(β l ), (2.81) ( ) ( ) 1 0 (+) =, ( ) =. (2.82) 0 1 For even N, the undamental spinor space can be decomposed into two subspaces by the project operators P ±, ± = P ±, corresponding to irreducible spinor representations D [±s]. The basis spinor in the representation space o D [+s] contains even number o actors ( ), and that o D [ s] contains odd number o actors ( ).The highest weight states [M] and their highest weights M are given by (+)...(+) (+), }{{} M =[0,...,0, 1], }{{} [s] o the SO(2l + 1), l 1 l 1 (+)...(+) (+), }{{} M =[0,...,0, 0, 1], }{{} [+s] o the SO(2l), (2.83) l 1 l 2 (+)...(+) ( ), }{{} M =[0,...,0, 1, 0], }{{} [ s] o the SO(2l). l 1 l 2 The remaining basis states are calculated by the applications o lowering operators F ν (S). 6.3 Direct Products o Spinor Representations Since the spinor representation is unitary so that we have O R = D [s] (R) 1, (2.84) = μ μ μ = μν μ δ μν ν, (2.85) O R ( ) = D [s] (R) 1 D [s] (R) =, which means that keeps invariant in the SO(N) transormations and is a scalar o the SO(N). In other words, the products o μ and ν span an invariant linear space, corresponding to the direct product representation D [s] D [s] o the SO(N). In the reduction o D [s] D [s] there is an identical representation where the CGCs are δ μν. In general, one has

23 6 Spinor Representations o the SO(N) 35 O R ( γ a1 γ an ) = D [s] (R) 1 γ a1 γ an D [s] (R) = R a1 c 1 R an c n γ c1 γ cn, (2.86) c 1...c n where γ a1 γ an is an antisymmetric tensor o rank n o the SO(N) corresponding to the Young pattern [1 n ] with n N. Otherwise, the respective γ a can be moved together and eliminated. When N = 2l + 1, the γ (2l+1) is a constant matrix so that the product o (N n) matrices γ a can be changed to a product o n matrices γ a. Thus, the rank n o the tensor (2.86) is less than N/2, and the Clebsch-Gordan series is given by [s] [s] [s] [s] [0] [1] [1 2 ] [1 l ], or SO(2l + 1). (2.87) The matrix entries o product o γ a are the CGCs. The highest weight in product space is given by M =[0,...,0, 2] corresponding to representation [1 l ]. When N = 2l, according to the property o the project operators P ±, P + P = P P + = 0, P ± P ± = P ±, γ (2l) P ± =±P ±, (2.88) P γ b1 γ b2n P ± = 0, P ± γ b1 γ b2n+1 P ± = 0, the product o the (N n) matrices γ b can still be changed to a product o n matrices γ b.in = l,wehave γ 1 γ 2 γ l = ( i) l γ 2l γ 2l 1 γ l+1 γ (2l), γ 1 γ 2 γ l P ± = 1 (2.89) 2 {γ 1γ 2 γ l ± ( i) l γ 2l γ 2l 1 γ l+1 }P ±. I N = 4l,wehave [±s] [±s] [±s] [±s] [0] [1 2 ] [1 4 ] [(±1)1 2l ], (2.90) [ s] [±s] [ s] [±s] [1] [1 3 ] [1 5 ] [1 2l 1 ]. I N = 4l + 2, one has [±s] [±s] [ s] [±s] [0] [1 2 ] [1 4 ] [1 2l ], (2.91) [ ] [±s] [±s] [±s] [1] [1 3 ] [1 5 ] [(±)1 2l+1 ]. The sel-dual and anti-sel-dual representations occur in the reduction o the direct product [±s] [±s], but not in the reduction o [+s] [ s]. The highest weights are M =[0,...,0, 0, 2] in the product space [+s] [+s], M = [0,..., 0, 2, 0] in [ s] [ s], and M =[0,...,0, 1, 1] in [+s] [ s]. 6.4 Spinor Representations o Higher Ranks In the SO(3) group, D 1/2 is a undamental spinor representation. The spinor representations D j o higher ranks can be obtained by reducing the direct product o the undamental spinor representation and a tensor representation, D 1/2 D l D l+1/2 D l 1/2. (2.92)

24 36 2 Special Orthogonal Group SO(N) The spinor representations o higher ranks o the SO(N) can also be obtained in a similar way. A spinor a1...a n with the tensor indices is called a spin-tensor i it transorms in R SO(N) as ollows: (O R ) a1 a n = R a1 c 1 R an c n D [s] (R) c1 c n. (2.93) c 1 c n The tensor part o the spin-tensor can be decomposed into a direct sum o the traceless tensors with dierent ranks. Each traceless tensor subspace can be reduced by the projection o the Young operators. Thus, the reduced subspace o the traceless tensor part o the spin-tensor is denoted by a Young pattern [λ] or [±λ] where the row number o [λ] is not larger than N/2. However, this subspace o the spin-tensor corresponds to the direct product o the undamental spinor representation [s] and the irreducible tensor representation [λ] or [±λ], and it is still reducible. It is required to ind a new restriction to pick up the irreducible subspace like the subspace o D l+1/2 in Eq. (2.92) ortheso(3) group. The restriction is rom the so-called trace o the second kind o the spin-tensor which keeps invariant in the SO(N) transormations: and a1 a i 1 a i+1 a n = N γ c a1 a i 1 ca i+1 a n, (2.94) c=1 (O R ) a1 a i 1 a i+1 a n = [ ] R a1 c 1 R an c n γ c R cc D [s] (R) c1 c i 1 c c i+1 c n c 1 c n c c = [ ] R a1 c 1 R an c n D [s] (R) γ c c1 c i 1 c c i+1 c n c 1 c n c = R a1 c 1 R an c n D [s] (R) c1 c i 1 c i+1 c n. (2.95) c 1 c n The irreducible subspace o the SO(N) contained in the spin-tensor space, in addition to the projection o a Young operator, satisies the usual traceless conditions o tensors and the traceless conditions o the second kind ψ a d d c = 0, γ d ψ a d c = 0. (2.96) d The highest weight M o the irreducible representation is the highest weight in the direct product space. The irreducible representation is denoted by [s,λ] or the SO(2l + 1) { [s] [λ] [s,λ], (2.97) M =[(λ 1 λ 2 ),...,(λ l 1 λ l ), (2λ l + 1)], d