JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 41, NUMBER 9
SEPTEMBER 2000
Clebsch–Gordan coefficients of SU„3… in SU„2… and SO„3… bases D. J. Rowe and C. Bahria) Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
共Received 4 Feburary 2000; accepted for publication 3 April 2000兲 New algorithms are developed for the purpose of optimizing the efficient calculation of SU共3兲 Clebsch–Gordan coefficients in both SU共2兲- and SO共3兲-coupled bases. The new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods. The methods are developed for SU共3兲 but apply to other compact semi-simple Lie groups. © 2000 American Institute of Physics. 关S0022-2488共00兲00109-2兴 I. INTRODUCTION
The group SU共3兲 and its Lie algebra su共3兲 play central roles both in nuclear1 and elementary particle physics.2 Applications of SU共3兲 and other unitary groups are also useful in the analysis of quantum interferometers.3 Moreover, SU共3兲 occurs as a physically significant subgroup of many groups needed in physical theory. For example, the subgroup chain U共6兲傻SU共3兲傻SO共3兲 is used in the interacting boson model4 of nuclear physics and the chain Sp(3,R)傻SU共3兲傻SO共3兲 is used in the microscopic theory of nuclear collective models.5,6 In such applications, one needs the matrices of SU共3兲 irreps 共irreducible representations兲 and CG 共Clebsch–Gordan兲 coefficients for reducing the tensor products of irreps. Algorithms and computer codes for computing SU共3兲 irreps and CG coefficients have long existed in both SU共2兲- and SO共3兲-coupled bases 共cf., for example, the papers of Moshinsky,7 Hecht,8 Resnikoff,9 Vergados,10 Sharp et al.,11 Draayer et al.,12 Millener,13 Alis˘auskas,14 Pluhar,15 Klink,16 LeBlanc, Hecht, and Biedenharn,17 and Rowe and Repka18兲. However, because the dimensions of some applications, e.g., the nuclear symplectic model, can be huge, it is important that the algorithms be as fast and efficient as they can be. Thus it is worthwhile to see if some of the new techniques of VCS 共vector coherent state兲 theory,19 developed for the purpose of computing Lie group and Lie algebra representations and coupling coefficients, result in improved efficiencies. The SU共3兲 algorithms of Draayer et al.,12 LeBlanc et al.,17 and Rowe and Repka,18 for example, rely on the shift tensor methods of Biedenharn et al.20 As shown in Refs. 17, 21, VCS theory gives complete sets of shift tensors that can be used to compute the CG coefficients of a wide range of semi-simple Lie groups. However, in this paper, we develop simpler and more efficient methods. The most important advance is the use of VCS representation theory to identify the highest grade states in the tensor product of two irreps. This enables one to infer seed coefficients from which all other CG coefficients are derived, both in SU共2兲- and SO共3兲-coupled bases. The methods given are systematic and well-suited for the development of corresponding algorithms for other groups 共algebras兲 and subgroup 共subalgebra兲 chains. II. THE CANONICAL SU„2…-COUPLED BASIS A. Basis states
The su共3兲 Lie algebra 共more precisely its complex extension兲 is spanned by operators ˆ ij , C
i⬍ j, raising operators,
a兲
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0022-2488/2000/41(9)/6544/22/$17.00
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© 2000 American Institute of Physics
J. Math. Phys., Vol. 41, No. 9, September 2000
ˆ ij , C ˆ 11⫺C ˆ 22 , hˆ 1 ⫽C
Clebsch-Gordan coefficients of SU(3)
共1兲
i⬎ j, lowering operators, hˆ 2 ⫽Cˆ 22⫺Cˆ 33 ,
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Cartan operators,
and has commutation relations ˆ i j ,Cˆ kl 兴 ⫽ ␦ jk Cˆ il ⫺ ␦ il Cˆ k j . 关C
共2兲
A highest weight state 兩 典 for an irrep of highest weight (, ) satisfies the equations ˆ i j 兩 典 ⫽0, C hˆ 1 兩 典 ⫽ 兩 典 ,
i⬍ j,
hˆ 2 兩 典 ⫽ 兩 典 .
共3兲
ˆ 33 Without loss of generality, we may suppose that 兩 典 is also an eigenstate of the operator C with zero eigenvalue so that ˆ 11兩 典 ⫽ 共 ⫹ 兲 兩 典 , C
Cˆ 22兩 典 ⫽ 兩 典 ,
Cˆ 33兩 典 ⫽0.
共4兲
The Hilbert space, H( ) , for the SU共3兲 irrep with highest weight (, ) thereby becomes a Hilbert space for a U共3兲 irrep of highest weight (⫹ , ,0). Let SU共2兲23 denote the SU共2兲傺SU共3兲 subgroup whose Lie algebra is spanned by the I-spin operators, ˆ 23 , Iˆ ⫹ ⫽C
Iˆ ⫺ ⫽Cˆ 32 ,
Iˆ 0 ⫽ 21 共 Cˆ 22⫺Cˆ 33兲 .
共5兲
The highest weight state 兩 典 is then the m⫽s⫽ /2 state of a SU共2兲23 multiplet of states 兵 兩 ( )0sm 典 ;m⫽⫺s,...,s 其 which satisfy the conditions ˆ 13兩 共 兲 0sm 典 ⫽0, ˆ 12兩 共 兲 0sm 典 ⫽C C
ˆ 11兩 共 兲 0sm 典 ⫽ 共 ⫹ 兲 兩 共 兲 0sm 典 , C
Iˆ 0 兩 共 兲 0sm 典 ⫽m 兩 共 兲 0sm 典 .
共6兲
ˆ 11 in the space and are referred to as The states 兵 兩 ( )0sm 典 其 have the largest eigenvalues of C highest grade states. Lower grade states are generated by repeated application of lowering operators to the highest grade states. Since the lowering operators are components of a SU共2兲23 spin- 21 tensor ˜P 1/2 with components 1/2 ˆ ˆ 21 , ˜P 1/2 共 f 兲 ⫽ ˆf 2 ⫽C
1/2 ˆ 31 , ˜P ⫺1/2 共 ˆf 兲 ⫽ ˆf 3 ⫽C
共7兲
we define a spin-j lowering tensor ˜P j ( ˆf ) with components ˜P mj 共 ˆf 兲 ⫽
ˆf (2j⫹m) ˆf (3j⫺m)
冑共 j⫹m 兲 ! 共 j⫺m 兲 !
,
m⫽⫺ j,...,⫹ j.
共8兲
An orthonormal SU共2兲23-coupled basis for a ( ) irrep is then given, to within norm factors, by 兩 共 兲 jIN 典 ⫽
1
1
兺 共 sn, jm 兩 IN 兲 ˜P mj共 ˆf 兲 兩 共 兲 0sn 典 ⫽ K (jI ) 关 ˜P j 共 ˆf 兲 丢 兩 共 兲 0s 典 ] NI ,
) K ( mn jI
共9兲
where the coupling is right-to-left SU共2兲 coupling. The label j defines the grade of a state. If we define a grading operator,
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D. J. Rowe and C. Bahri
ˆ 11⫺Cˆ 22⫺Cˆ 33 , Xˆ ⫽2C
共10兲
Xˆ 兩 共 兲 jIN 典 ⫽ 共 2⫹ ⫺6 j 兲 兩 共 兲 jIN 典 ,
共11兲
then
) is derived by and the state 兩 ( ) jIN 典 is said to have grade (2⫹ ⫺6 j). The norm factor K ( jI 18 VCS methods and given below 关cf. Eq. 共17兲兴.
B. Vector-coherent-state representation
In a VCS representation 共details are given in Ref. 18兲, the highest grade ( j⫽0) states are ( ) ⬅ 兩 ( )0sm 典 其 for an irrep ( ) of the direct product group identified with a basis 兵 m U共1兲1⫻SU(2) 23 for which ( ) ( ) ⫽ 共 ⫹ 兲 m , Cˆ 11 m
( ) ( ) sˆ ⫾ m ⫽ 冑共 s⫿m 兲共 s⫾m⫹1 兲 m⫾1 ,
( ) ( ) sˆ 0 m ⫽m m ,
共12兲
ˆ 11), sˆ k ⫽ ( ) (Iˆ k ) and s⫽ /2. A state 兩 典 苸H( ) is then assigned a VCS where Cˆ 11⫽ ( ) (C wave function with vector values,
共 z 兲 ⫽ 兺 m( ) 具 共 兲 0sm 兩 e zˆ 兩 典 ,
共13兲
zˆ ⫽z 2 Cˆ 12⫹z 3 Cˆ 13 .
共14兲
m
where
It is then determined18 that the wave functions for the 共orthonormal兲 basis states 兵 兩 ( ) jIN 典 其 are given by ( ) ˜ j ) ( ) I ( 兴N , jIN 共 z 兲 ⫽K jI 关 P 共 z 兲 丢
共15兲
with ˜P mj 共 z 兲 ⫽
z (2j⫹m) z (3j⫺m)
冑共 j⫹m 兲 ! 共 j⫺m 兲 !
,
m⫽⫺ j,...,⫹ j,
共16兲
and that the normalization factors have values 共with s⫽ /2兲 ) K ( jI ⫽
⌫
冑
共 ⫹ ⫹1 兲 !! . 共 ⫹s⫹I⫺ j⫹1 兲 ! 共 ⫹s⫺I⫺ j 兲 !
共17兲
In the VCS representation an element Aˆ in the u共3兲 Lie algebra is represented by the operator (Aˆ ), where
( )
关 ⌫ ( ) 共 Aˆ 兲 兴共 z 兲 ⫽
兺m m( ) 具 共 兲 0sm 兩 e zˆ Aˆ 兩 典 .
共18兲
From this definition, it is inferred18 that raising and lowering operators of the su共3兲 algebra have VCS representations, ⌫ ( ) 共 eˆ i 兲 ⫽ⵜ i , where
ˆ ,z i 兴 , ⌫ ( ) 共 ˆf i 兲 ⫽ 关 ⌳
共19兲
J. Math. Phys., Vol. 41, No. 9, September 2000
Clebsch-Gordan coefficients of SU(3)
eˆ i ⫽Cˆ i1 , ⵜ i⫽ / z i ,
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ˆf i ⫽Cˆ 1i ,
Cˆ i j ⫽ ( ) 共 Cˆ i j 兲 ,
i, j⫽2,3,
共20兲
) ˆ is an operator that is diagonal in the 兵 ( and ⌳ jIN 其 basis with eigenvalues given by ( ) ( ) ) ˆ ( ⌳ jIN ⫽⍀ jI jIN ,
) ⍀ ( jI ⫽ 共 2⫹ ⫹3 兲 j⫺ j 共 j⫹1 兲 ⫺I 共 I⫹1 兲 .
共21兲
) It will be noted that the subset of wave functions of highest grade in the set 兵 ( jIN 其 are those ( ) that are annihilated by the raising operators ⌫ (eˆ i )⫽ⵜ i ; these are the z-independent functions ( ) ( ) (z)⫽ m 兵 0sm 其.
C. Highest grade states in a tensor product space
Basis states, 兵 兩 ( 1 1 ) j 1 I 1 N 1 典 丢 兩 ( 2 2 ) j 2 I 2 N 2 典 其 , for the tensor product of two SU共3兲 irreps of highest weights ( 1 1 ) and ( 2 2 ) have VCS wave functions in two sets of complex variables, z 1 ⫽(z 12 ,z 13 ) and z 2 ⫽(z 22 ,z 23 ), with values ( 1 1 ) 1 共z 兲 1I1N1
j
( 2 2 ) 2 共 z 兲. 2I2N2
j
共22兲
The highest grade states for irreducible subrepresentations are the linear combinations of these products which are annihilated by the raising operators eˆ 2 and eˆ 3 . Thus, the identification of highest grade states in the tensor product space is easy in the VCS representation for which the raising operators take the simple form ⌫ ( 1 1 ) 丢 ( 2 2 ) 共 eˆ i 兲 ⫽ⵜ 1i ⫹ⵜ 2i .
共23兲
Observation: A complete set of 共unnormalized兲 highest grade states for the subrepresentation of highest weight ( ) in the tensor product ⌫ ( 1 1 ) 丢 ⌫ ( 2 2 ) are given for s in the range s ⫽ 12 兩 1 ⫺ 2 兩 ,..., 21 ( 1 ⫹ 2 ) by linear combinations ⌽ Ni( ) ⫽
兺s ⌽ Ns( ) a si
共24兲
of wave functions 兵 ⌽ Ns( ) ; N⫽⫺ /2, . . . , /2其 whose values are expressible in the form ⌽ Ns( ) 共 z 1 ,z 2 兲 ⫽ 关 ˜P j 共 z 2 ⫺z 1 兲 丢 共 ( 2 , 2 ) 丢 ( 1 , 1 ) 兲 s 兴 N /2
共25兲
6 j⫽2 1 ⫹ 1 ⫹2 2 ⫹ 2 ⫺2⫺ .
共26兲
with
Proof: From the definition of the grading operator, given by Eq. 共10兲, one determines that ⌽ Ns( ) has grade 2 1 ⫹ 1 ⫹2 2 ⫹ 2 ⫺6 j. The state ⌽ Ns( ) is evidently annihilated by the raising operators and so is a highest grade state for some irreducible subrepresentation of the tensor product representation. A state of highest weight ( ) has grade 2⫹ . Thus, the observation follows. The running multiplicity index i serves to distinguish different subirreps of the same highest weight.
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D. Clebsch–Gordan coefficients
A complete set of SU共3兲 Clebsch–Gordan coefficients can be generated from a set of seed coefficients. These are SU共2兲-reduced CG coefficients 兵 „( 1 1 ) j 1 I 1 ,( 2 2 ) j 2 I 2 储 ( )0 /2…其 for coupling to highest grade states. From these coefficients, one obtains, for example, „共 1 1 兲 j 1 I 1 N 1 , 共 2 2 兲 j 2 I 2 N 2 兩 共 兲 0, /2,N… ⫽ 共 I 1 N 1 ,I 2 N 2 兩 /2,N 兲 „共 1 1 兲 j 1 I 1 , 共 2 2 兲 j 2 I 2 储 共 兲 0, /2…,
共27兲
where (I 1 N 1 ,I 2 N 2 兩 /2,N) is a SU共2兲 CG coefficient and indexes the multiplicity of different couplings to irreps of the same highest weight ( ). ( ) If 兵 兩 ⌿ 0, /2,N 典 其 is an orthornormal set of highest grade states within the tensor product space of two irreps of highest weights ( 1 , 1 ) and ( 2 , 2 ), respectively, then the seed coefficients are the overlaps, ( 2 2 ) ( ) /2 ( ) 丢 j I1 1 兴 N 兩 ⌿ 0, /2,N 典 . 2I2 1 1
„共 1 1 兲 j 1 I 1 , 共 2 2 兲 j 2 I 2 储 共 兲 0, /2…⫽ 具 关 j
共28兲
The other SU共3兲 CG coefficients are obtained by stepping down from the highest grade states with the lowering operators. In this way, one obtains 共as shown in Ref. 18兲 the expressions „共 1 1 兲 j 1 I 1 ; 共 2 2 兲 j 2 I 2 储 共 兲 jI… ⫽
( 1 1 ) ( 2 2 ) Kj I 1I1 2 2
Kj
) K ( jI
⫻
⫻W
兺
j 1⬘ I ⬘1 j ⬘2 I ⬘2
冉
⫻
冑共 2 j⫹1 兲共 ⫹1 兲共 2I 1 ⫹1 兲共 2I 2 ⫹1 兲共 2 j 1 ⫹1 兲 ! 共 2 j 2 ⫹1 兲 !
共 ⫺1 兲 I 1 ⫹I 2 ⫺I 1⬘ ⫺I 2⬘ ⫺ j
冊 冉 冉
冑
共 2I 1⬘ ⫹1 兲共 2I 2⬘ ⫹1 兲
共 2 j ⬘1 兲 ! 共 2 j 1 ⫺2 j ⬘1 兲 ! 共 2 j 2⬘ 兲 ! 共 2 j 2 ⫺2 j 2⬘ 兲 !
1 2 j 1 I 1⬘ j 1 ⫺ j 1⬘ ;I 1 j 1⬘ W j I ⬘ j ⫺ j ⬘ ;I j ⬘ 2 2 2 2 2 2 2 2
冦
I ⬘1
2
I 2⬘
j 1 ⫺ j 1⬘
j 2 ⫺ j 2⬘
I1
I2
j I
冧
冊冉
2j 2 j 1 ⫺2 j 1⬘
冊
1/2
共 1 1 兲 j 1⬘ I ⬘1 , 共 2 2 兲 j ⬘2 I 2⬘ 储 共 兲 0 ( 1 1 )
Kj
I1 1⬘ ⬘
2
( 2 2 )
Kj
冊
共29兲
.
⬘2 I ⬘2
Claim 1: Coupling coefficients for the nonorthonormal highest grade states are given by
( ) ( ) 具 关 j 2 I22 2 丢 j 1 I11 1 兴 N /2兩 ⌽ Ni( ) 典 ⫽
兺s
共 ⫺1 兲 2 j 1 ( 1 1 ) ( 2 2 ) Kj I 1I1 2 2
Kj
冑
冋
s1
s2
s
j1
j2
j
I1
I2
/2
s1
s2
s
j1
j2
j
I1
I2
/2
共 2 j 兲! 共 2 j 1兲!共 2 j 2兲!
册
a si , 共30兲
where j⫽ j 1 ⫹ j 2 and
冋
s1
s2
s
j1
j2
j
I1
I2
/2
册
⫽ 冑共 2s⫹1 兲共 2 j⫹1 兲共 2I 1 ⫹1 兲共 2I 2 ⫹1 兲
再
冎
共31兲
J. Math. Phys., Vol. 41, No. 9, September 2000
Clebsch-Gordan coefficients of SU(3)
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is a unitary SU共2兲 9 j-symbol; 兵 a si 其 is a set of solutions to the equations
兺s
冋
s1
s2
s
j1
j2
j
I1
I2
/2
册
a si ⫽0
共32兲 ( 1 1 ) 1I1
for all values of j 1 I 1 and j 2 I 2 for which the corresponding K-factors, K j
( 2 2 ) , 2I2
and K j
vanish.
Proof: Making use of the expression 共 j 1 m 1 , j 2 m 2 兩 j⫽ j 1 ⫹ j 2 ,m⫽m 1 ⫹m 2 兲 ⫽
冑
共 2 j 1 兲 ! 共 2 j 2 兲 ! 共 j⫹m 兲 ! 共 j⫺m 兲 ! , 共 2 j 兲 ! 共 j 1 ⫹m 1 兲 ! 共 j 1 ⫺m 1 兲 ! 共 j 2 ⫹m 2 兲共 j 2 ⫺m 2 兲 !
共33兲
for a stretched SU共2兲 CG coefficient, one determines that ˜P mj (z 2 ⫺z 1 ) has the expansion ˜P mj 共 z 2 ⫺z 1 兲 ⫽
兺 共 ⫺1 兲 2 j 冑共 2 j 1 兲 ! 共 2 j 2 兲 ! 关 ˜P j 共 z 2 兲 丢 ˜P j 共 z 1 兲兴 mj . j ⫹j ⫽j 1
1
共 2 j 兲!
The claim then follows by making the recoupling
关关 ˜P 共 z 兲 丢 ˜P 共 z 兲兴 丢 关 j2
2
j1
1
j
2
1
共34兲
2
( 2 2 )
/2 丢 ( 1 1 ) 兴 s 兴 N ⫽
兺
I1I2
冋
s1
s2
s
j1
j2
j
I1
I2
/2
册
关关 ˜P j 2 共 z 2 兲 丢 ( 2 2 ) 兴 I 2 /2
丢 关˜ P j 1 共 z 1 兲 丢 ( 1 1 ) 兴 I 1 兴 N
.
共35兲
The linear combinations ⌽ Ni( ) ⫽ 兺 s ⌽ Ns( ) a si must be chosen such that they only involve states that appear in the space of the tensor product ⌫ ( 1 1 ) 丢 ⌫ ( 2 2 ) . To simplify the notation, let us temporarily denote by ␣ the set of quantum numbers of a product state
␣ N ⫽†关 ˜P j 2 共 z 2 兲 丢 ( 2 2 ) 兴 J 2 丢 关 ˜P j 1 共 z 1 兲 丢 ( 1 1 ) 兴 I 1 ‡N /2
共36兲
that appears on the rhs of Eq. 共33兲 and define the dot product
␣ N •  M ⫽ ␦ ␣ ␦ NM .
共37兲
The desired 兵 a si 其 coefficients are then solutions of the constraint equations
兺s ␣ N •⌽ Ns( ) a si ⫽0
共38兲
for all ␣ for which ␣ N is a disallowed state. 共Note that Eq. 共38兲 is independent of N.兲 Using Eqs. 共25兲, 共34兲, 共35兲, and 共37兲, we obtain Eq. 共32兲. If none of the K-factors vanishes for any j 1 I 1 and j 2 I 2 appearing in Eq. 共33兲, the constraint equations become redundant; one can set a s j i ⫽ ␦ i j , where s j runs over the values of s. Using the claim, we can evaluate the expression
具 ⌽ Ni⬘ ( ) 兩 ⌽ Ni( ) 典 ⫽
兺 j I j I
1 1 2 2
( 2 ) ( ) /2 ( ) ( ) i ( ) 丢 j I1 1 兴 N 兩 ⌽ N⬘ 典具 关 j 2 I22 2 丢 j 1 I11 1 兴 N /2兩 ⌽ Ni( ) 典 1 1
具 关 j 2 I22
共39兲
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D. J. Rowe and C. Bahri
for the overlaps of the nonorthonormal highest grade states with one another and find orthonormal combinations, ( ) ⌿ 0, /2,N ⫽
兺i ⌽ Ni( ) B i ( ) ⫽ 兺s ⌽ Ns( ) C s ( ) ,
冉
C s ( ) ⫽
兺i a si B i ( )
冊
.
共40兲
Thus, one determines the seed coefficients
„共 1 1 兲 j 1 I 1 , 共 2 2 兲 j 2 I 2 储 共 兲 0, /2…⫽
兺s 具 关 (j I ) 丢 (j I ) 兴 N /2兩 ⌽ Ns( ) 典 C s ( ) . 2 2
2 2
1 1
1 1
共41兲
E. A multiplicity-free example
The seed coefficients for multiplicity-free couplings are particularly easy to determine. Consider, for example, the CG coefficients for the decomposition,
共 2 ,0兲 丢 共 1 ,0兲 →
兺 共 ⫽ 1 ⫹ 2 ⫺2 , 兲 ,
共42兲
of a tensor product of two ⫽0 irreps. Since there is no multiplicity, the recoupling involved in deriving Eq. 共30兲 is trivial and the expression for the overlaps reduces to
( ,0)
具关 I2 2
( 1 ,0) /2 兴 N 兩 ⌽ N( ) 典 ⫽ 共 ⫺1 兲 2I 1 1
丢 I
冑
! 共 1 ⫺2I 1 兲 ! 共 2 ⫺2I 2 兲 ! ␦ I 1 ⫹I 2 , /2 . 1 ! 2 ! 共 2I 1 兲 ! 共 2I 2 兲 !
共43兲
Thus, the squared norm of the state ⌽ N( ) is simply
具 ⌽ N( ) 兩 ⌽ N( ) 典 ⫽
! 共 1 ⫺m 兲 ! 共 2 ⫺n 兲 ! 共 ⫹ ⫹1 兲 ! 共 1 ⫺ 兲 ! 共 2 ⫺ 兲 ! ⫽ , 兺 1 ! 2 ! m⫹n⫽ m!n! 1 ! 2 ! 共 ⫹1 兲 !
共44兲
where we have used the identity
兺 m ⫹m ⫽ j 1
2
3
共 j 1 ⫹m 1 兲 ! 共 j 2 ⫹m 2 兲 ! 共 j 1 ⫹ j 2 ⫹ j 3 ⫹1 兲 ! 共 j 2 ⫹ j 3 ⫺ j 1 兲 ! 共 j 1 ⫹ j 3 ⫺ j 2 兲 ! ⫽ , 共 j 1 ⫺m 1 兲 ! 共 j 2 ⫺m 2 兲 ! 共 2 j 3 ⫹1 兲 ! 共 j 1 ⫹ j 2 ⫺ j 3 兲 !
共45兲
derived by Sharp.22 Thus, we obtain the seed coefficients, 共共 1 ,0兲 I 1 , 共 2 ,0兲 I 2 储共 兲 0, /2兲
⫽ 共 ⫺1 兲 2I 1
冑
共 ⫹1 兲 ! ! 共 1 ⫺2I 1 兲 ! 共 2 ⫺2I 2 兲 ! ␦ , 共 ⫹ ⫹1 兲 ! 共 1 ⫺ 兲 ! 共 2 ⫺ 兲 ! 共 2I 1 兲 ! 共 2I 2 兲 ! I 1 ⫹I 2 , /2
共46兲
in agreement with a known result.18 As remarked in Ref. 18, the general ( 2 ,0) 丢 ( 1 ,0) →(, ) CG coefficients are then determined, by Eq. 共29兲 to be given by
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Clebsch-Gordan coefficients of SU(3)
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共共 1 0 兲 I 1 , 共 2 0 兲 I 2 储共 兲 jI 兲
⫽
冑 冑
共 ⫹1 兲共 ⫹1 兲 ! 共 ⫹ ⫹I⫺I 1 ⫺I 2 ⫹1 兲 ! 共 ⫹ ⫺I⫺I 1 ⫺I 2 兲 ! 共 2I 1 ⫹2I 2 ⫺ ⫹1 兲 ! 共 1 ⫺ 兲 ! 共 2 ⫺ 兲 ! 共 1 ⫺2I 1 兲 ! 共 2 ⫺2I 2 兲 ! 共 1 ⫺2I 1⬘ 兲 ! 共 2 ⫺2I ⬘2 兲 ! 共 2I 1 ⫹1 兲 ! 共 2I 2 ⫹1 兲 ! 共 ⫺1 兲 2I 2⬘ 共 ⫹ ⫹1 兲 ! 共 2I 1⬘ 兲 ! 共 2I 1 ⫺2I 1⬘ 兲 ! 共 2I 2⬘ 兲 ! 共 2I 2 ⫺2I ⬘2 兲 ! I ⬘ ⫹I ⬘ ⫽ /2
⫻
兺
1
⫻
冦
I ⬘1
I 2⬘
2
I 1 ⫺I 1⬘
I 2 ⫺I 2⬘
j
I1
I2
I
冧
2
共47兲
.
Expressions for these multiplicity-free coefficients were derived previously by VCS methods and given in terms of ordinary angular–momentum recoupling coefficients in Ref. 17.
III. SU„3… CLEBSCH–GORDAN COEFFICIENTS IN A SO„3…-COUPLED BASIS A. Construction of orthonormal basis states
Claim 2: Let H( ) denote the Hilbert space for a SU共3兲 irrep of highest weight ( ) and let F( ) denote the space of complex functions over SO共3兲 spanned by
KLM ⫽
冑
2L⫹1 L D KM , 82
共48兲
L is a Wigner 共rotation matrix coefficient兲 function with K, L, and M in the ranges where D KM
K⫽ , L⫽
再
⫺2,...,0 or 1,
⫹K,⫹K⫺1,...,K,
for K⫽0,
,⫺2,...,0 or 1,
for K⫽0.
共49兲
M ⫽⫺L, . . . ,⫹L . Then there is a map23 F( ) →H( ) in which
哫兩典⫽
冕
R 共 ⍀ ⫺1 兲 兩 典 共 ⍀ 兲 d⍀,
共50兲
where 兩 典 苸H( ) is the highest weight state and d⍀( ␣ ,  , ␥ )⫽d␣ sin  d d␥ is the invariant SO共3兲 volume element. The space F( ) is a Hilbert space, isomorphic to H( ) , with the inner product given by 共 , ⬘ 兲⬅ 具 兩 ⬘典 ⫽
冕 冕 *
共 ⍀ ⬘ 兲 具 兩 R 共 ⍀ ⬘ ⍀ ⫺1 兲 兩 典 ⬘ 共 ⍀ 兲 d⍀ ⬘ d⍀,
共51兲
with kernel 具 兩 R(⍀ ⬘ ⍀ ⫺1 ) 兩 典 , given24 in terms of Euler angles by
具 兩 R 共 ␣ ,  , ␥ 兲 兩 典 ⫽ 共 cos  兲 共 cos ␣ cos ␥ ⫺sin ␣ cos  sin ␥ 兲 .
共52兲
Proof: The claim follows from the observations, proved by Elliott,1 that the set of states 兵 R(⍀) 兩 典 ;⍀苸SO(3) 其 spans the Hilbert space H( ) and the states
6552
J. Math. Phys., Vol. 41, No. 9, September 2000
兩 KLM 典 ⫽
冑
2L⫹1 82
冕
D. J. Rowe and C. Bahri
L R 共 ⍀ ⫺1 兲 兩 典 D KM 共 ⍀ 兲 d⍀,
共53兲
with K, L and M in the ranges specified by Eq. 共49兲, are a basis for H( ) . The claim yields the known overlaps of the Elliott basis states24
具 KLM 兩 K ⬘ L ⬘ M ⬘ 典 ⫽ ␦ LL ⬘ ␦ M M ⬘ S KK ⬘ , L
共54兲
with L
S KK ⬘ ⫽
冕
* D KK 共 ⍀ 兲 具 兩 R 共 ⍀ 兲 兩 典 d⍀. ⬘ L
共55兲
These overlap integrals are evaluated in Sec. IV C. Given the positive Hermitian matrix S L , one can determine the expansion coefficients of an orthonormal basis, 兩 ␣ LM 典 ⫽
兺K 兩 KLM 典 K¯ K ␣共 L 兲 ,
共56兲
兩 KLM 典 ⫽
兺␣ KK*␣共 L 兲 兩 ␣ LM 典 ,
共57兲
and the inverse expansion
where
兺K K¯ K ␣共 L 兲 KK*共 L 兲 ⫽ ␦ ␣ .
共58兲
It follows that the K matrices are related to the S L matrices by L
S KK ⬘ ⫽
兺␣ KK ␣共 L 兲 KK*⬘␣共 L 兲 .
共59兲
For example, one can choose K(L) to be the Hermitian square root of S L .
B. Clebsch–Gordan coefficient
If 兩 典 is a highest weight state in the tensor product space of two irreps of highest weights ( 1 , 1 ) and ( 2 , 2 ), then the seed coefficients for a SO共3兲-coupled basis are the coefficients in the expansion 兩典⫽
兺
共共 1 1 兲 ␣ 1 L 1 M 1 , 共 2 2 兲 ␣ 2 L 2 M 2 兩 兲 关 兩 共 2 2 兲 ␣ 2 L 2 M 2 典 丢 兩 共 1 1 兲 ␣ 1 L 1 M 1 典 ].
共60兲
These coefficients are given in terms of the SU共2兲 seeds and overlaps by „共 1 1 兲 ␣ 1 L 1 M 1 , 共 2 2 兲 ␣ 2 L 2 M 2 兩 … ⫽
兺
具 共 1 1 兲 ␣ 1L 1 M 1兩共 1 1 兲 j 1I 1N 1典 具 共 2 2 兲 ␣ 2L 2 M 2兩共 2 2 兲 j 2I 2N 2典
⫻ 共 I 1 N 1 ,I 2 N 2 兩 /2, /2兲 „共 1 1 兲 j 1 I 1 , 共 2 2 兲 j 2 I 2 储 共 兲 0, /2…,
共61兲
J. Math. Phys., Vol. 41, No. 9, September 2000
Clebsch-Gordan coefficients of SU(3)
6553
where the sum is over all repeated indices. Given the seed coefficients, one can construct a highest weight state 兩 ( ) 典 and from it, by Eqs. 共53兲 and 共56兲, determine the correponding SO共3兲-coupled basis states; 兩 共 兲 ␣ LM 典 ⫽
⫽
冑
2L⫹1 ¯ ( ) 共 L 兲 K 8 2 K⭓0 K ␣
兺
兺
M ⬘1 M ⬘2 M 1 M 2
⫽
冑
兺
„共 1 1 兲 ␣ 1 L 1 M 1 , 共 2 2 兲 ␣ 2 L 2 M 2 兩 …
2L⫹1 ¯ ( ) 共 L 兲 K 8 2 K⭓0 K ␣
兺
冕
D
L 2*
M 2 M ⬘2
L*
L 共 ⍀ 兲 D M1 M 共 ⍀ 兲 D KM 共 ⍀ 兲 d⍀ 1
1⬘
关 兩共 2 2 兲␣ 2L 2典
M 1M 2
丢 兩共 1 1 兲␣ 1L 1典 ] M L
⫻
L R 共 ⍀ ⫺1 兲 兩 共 兲 典 D KM 共 ⍀ 兲 d⍀
兩 共 2 2 兲 ␣ 2 L 2 M 2⬘ 典
丢 兩 共 1 1 兲 ␣ 1 L 1 M 1⬘ 典
⫻
冕
冑
„共 1 1 兲 ␣ 1 L 1 M 1 , 共 2 2 兲 ␣ 2 L 2 M 2 兩 …
82 ¯ ( ) 共 L 兲 共 L 1 M 1 ,L 2 M 2 兩 LK 兲 . K 2L⫹1 K⭓0 K ␣
兺
共62兲
This leads to the 共reduced兲 Clebsch–Gordan coefficients, „共 1 1 兲 ␣ 1 L 1 , 共 2 2 兲 ␣ 2 L 2 储 共 兲 ␣ L… ⫽
冑 ⫻
82 „共 1 1 兲 ␣ 1 L 1 M 1 , 共 2 2 兲 ␣ 2 L 2 M 2 兩 … 2L⫹1 M 1 M 2
兺
兺
K⭓0
¯ K(␣ ) 共 L 兲 . 共 L 1 M 1 ,L 2 M 2 兩 LK 兲 K
共63兲
C. The overlap integrals
To evaluate the seed coefficients 兵 (( 1 1 ) ␣ 1 L 1 M 1 ,( 2 2 ) ␣ 2 L 2 M 2 兩 ) 其 , we need the overlaps ¯ K*␣ 共 L 兲 . 具 共 兲 ␣ LM 兩 共 兲 jIN 典 ⫽ 兺 具 共 兲 KLM 兩 共 兲 jIN 典 K K⭓0
共64兲
The first factor in this expression is given by
具 共 兲 KLM 兩 共 兲 jIN 典 ⫽
冑
2L⫹1 82
冕
L* ) D KM 共 ⍀ 兲 ( jIN 共 ⍀ 兲 d⍀,
共65兲
) where ( jIN is the coherent state wave function for the state 兩 ( ) jIN 典 ; i.e., ) ( jIN 共 ⍀ 兲 ⫽ 具 兩 R 共 ⍀ 兲 兩 共 兲 jIN 典 .
共66兲
An element ⍀ of SO共3兲 is parametrized in terms of Euler angles in the usual way by ˆ
ˆ
ˆ
⍀ 共 ␣ ,  , ␥ 兲 ⫽e ⫺i␣ L z e ⫺i L y e ⫺i␥ L z , with
共67兲
6554
J. Math. Phys., Vol. 41, No. 9, September 2000
ˆ 23⫺Cˆ 32兲 ⫽2Iˆ y , Lˆ z ⫽⫺i共 C
D. J. Rowe and C. Bahri
Lˆ y ⫽⫺i共 Cˆ 12⫺Cˆ 21兲 ⫽2Tˆ y ,
共68兲
where ˆI y and Tˆ y are infinitesimal generators of the SU共2兲23 and SU共2兲12 subgroups of SU共3兲, respectively. Thus, ⍀( ␣ ,  , ␥ ) is expressed as ⍀ 共 ␣ ,  , ␥ 兲 ⫽R 23共 0,2␣ ,0兲 R 12共 0,2 ,0兲 R 23共 0,2␥ ,0兲
共69兲
s I ) ( jIN 共 ⍀ 兲 ⫽ 兺 d sm 共 2 ␣ 兲 具 共 兲 0sm 兩 R 12共 0,2 ,0 兲 兩 共 兲 jIM 典 d M N 共 2 ␥ 兲 .
共70兲
and
mM
As shown in Ref. 25, matrix elements of R 12(0,2 ,0) are evaluated by writing R 12共 0,2 ,0兲 ⫽ P 132R 23共 0,2 ,0兲 P 123 ,
共71兲
where P i jk is a Weyl permutation of the 共1,2,3兲 axes. Matrix elements of the Weyl permutations are given in Refs. 25, 26 共cf. also the Appendix兲. The relevant matrix elements for the present purposes are given by
具 共 兲 0sm 兩 P 132兩 共 兲 kJJ 典 ⫽ ␦ 2k,⫹s⫹m ␦ 2J,⫹s⫺m 共 ⫺1 兲 ,
共72兲
and, with the understanding that 2k⫽⫹s⫹m,
2J⫽⫹s⫺m,
s⫽ /2,
共73兲
by
具 共 兲 kJK 兩 P 123兩 共 兲 jIM 典 ⫽ ␦ M , j⫹m ␦ K,J⫺2 j 共 ⫺1 兲 I⫹⫺ j⫺s 冑共 2J⫹1 兲共 2I⫹1 兲
⫻
再
⫹ ⫺2 j 2
2 j⫹s⫹m 2
J
s⫺m 2
2
I
冎
共74兲
.
Thus, we obtain ( ) s J I ) ( jIN 共 ⍀ 兲 ⫽ 兺 C m 共 jI 兲 d sm 共 2 ␣ 兲 d J,J⫺2 j 共 2  兲 d j⫹m,N 共 2 ␥ 兲 ,
共75兲
m
with
( ) Cm 共 jI 兲 ⫽ 共 ⫺1 兲 I⫹⫹s⫺ j
冑共 2J⫹1 兲共 2I⫹1 兲
再
⫹ ⫺2 j 2
2 j⫹s⫹m 2
s⫺m 2
2
J I
冎
.
共76兲
( ) ( jI)→C (0) (I)⫽1 and For example, if ⫽0, then m⫽s⫽0, j⫽I, C m (0) /2 I IN 共 ␣ ,  , ␥ 兲 ⫽d /2,/2⫺2I 共 2  兲 d I,N 共 2␥ 兲.
The needed overlaps are now given by
共77兲
J. Math. Phys., Vol. 41, No. 9, September 2000
具 共 兲 KLM 兩 共 兲 jIN 典 ⫽
Clebsch-Gordan coefficients of SU(3)
冑 冕 冕
2L⫹1 82
⫻
2
s d sm 共 2 ␣ 兲 e iK ␣ d␣
J L d J,J⫺2 j 共 2  兲 d KM 共  兲 sin  d
0
⫻
兺m C m( )共 jI 兲 冕0
6555
2
0
d Ij⫹m,N 共 2 ␥ 兲 e iM ␥ d␥ ,
共78兲
and it remains to perform the integrals in this expansion. The reduced Wigner functions have the well-known expansion J⫺M
d JM N 共 2  兲 ⫽
兺
⫽0
d 共 M ,J,N, 兲 共 cos  兲 2J⫹N⫺M ⫺2 共 sin  兲 M ⫺N⫹2 ,
共79兲
with d 共 M ,J,N, 兲 ⫽ 共 ⫺1 兲 M ⫺N⫹
冑共 J⫹M 兲 ! 共 J⫺M 兲 ! 共 J⫹N 兲 ! 共 J⫺N 兲 ! 共 J⫹N⫺ 兲 ! ! 共 J⫺M ⫺ 兲 ! 共 M ⫺N⫹ 兲 !
共80兲
.
Let I(p,q,K) denote the integral I 共 p,q,K 兲 ⫽
冕
2
0
共 cos ␣ 兲 p 共 sin ␣ 兲 q e iK ␣ d␣ .
共81兲
Then, as shown by Draayer and Williams,12 I 共 p,q,K 兲 ⫽
1 2 共 2i兲 q p
冕
2
0
共 e i␣ ⫹e ⫺i␣ 兲 p 共 e i␣ ⫺e ⫺i␣ 兲 q e iK ␣ d␣ ⫽2
共 ⫺i兲 q 2 p⫹q
兺n 共 ⫺1 兲 n
冉 冊冉 冊 p
q
⫺n
n
,
共82兲 where ⫽ (p⫹q⫹K) 关note that I(p,q,K) vanishes unless is an integer in the range 0⭐ ⭐p⫹q]. Thus, we obtain 1 2
冕
2
0
冕
2
0
s d sm 共 2 ␣ 兲 e i␣ d␣ ⫽d 共 s,s,m,0兲 I 共 s⫹m,s⫺m, 兲 ,
共83兲
I⫺ j⫺m
d Ij⫹m,N 共 2 ␥ 兲 e iM ␥ d␥ ⫽
兺
⫽0
d 共 j⫹m,I,N, 兲 I 共 2I⫹N⫺ j⫺m⫺2 , j⫹m⫺N⫹2 ,M 兲 . 共84兲
We now consider the integral
冕
0
J L d J,J⫺2 j 共 2  兲 d KM 共  兲 sin  d ⫽
兺 d 共 J,J,J⫺2 j,0兲 d 共 K,L,M , 兲 冕0 共 cos  兲 2J⫺2 j 共 sin  兲 2 j⫹1
⫻ 共 cos  /2兲 2L⫹M ⫺K⫺2 共 sin  /2兲 K⫺M ⫹2 d . Standard tables of integrals 共e.g., Gradshteyn and Ryzhik27兲 give
冕
/2
0
冉
冊
冉
共 p⫹q⫺2 兲 /2 1 p⫹1 q⫹1 1 , ⫽ 共 cos  兲 p 共 sin  兲 q d ⫽ B 2 2 2 p⫹q 共 p⫺1 兲 /2
共85兲
冊
⫺1
where B is the beta function and the binomial coefficient is defined, generally, by
,
共86兲
6556
J. Math. Phys., Vol. 41, No. 9, September 2000
D. J. Rowe and C. Bahri
冉冊
⌫ 共 ⫹1 兲 ⫽ . ⌫ 共 ⫺ ⫹1 兲 ⌫ 共 ⫹1 兲
共87兲
It follows that
冕
0
J d J,J⫺2 j共 2  兲
冉 冊
共 ⫺2 兲 2 j⫹1 2J d ⫽⫺ 2J⫹L⫹1 2 j
L d KM 共  兲 sin 
⫻
冉
1/2 L⫺K
2(J⫺ j)
兺 d 共 K,L,M , 兲 r⫽0 兺 ⫽0
2J⫹L 共 K⫺M ⫹2r⫹2 ⫹2 j 兲 /2
冊
共 ⫺1 兲 r
冉
2J⫺2 j r
⫺1
冊
共88兲
.
¯ „ L … and S„ L … matrices D. The K
¯ (L) matrices, for an irrep of highest weight (, ), are obtained by solution of Eqs. 共58兲 The K and 共59兲 with L
S KK ⬘ ⫽
冕
* D KK 共 ⍀ 兲 共 cos  兲 关 cos ␣ cos ␥ ⫺sin ␣ cos  sin ␥ 兴 d⍀ ⬘ L
⫽
兺 共 ⫺1 兲 ⫽0 ⫻
冕
冉 冊冕
共 cos ␣ 兲 ⫺ 共 sin ␣ 兲 e iK ␣ d␣
冕
共 cos ␥ 兲 ⫺ 共 sin ␥ 兲 e iK ⬘ ␥ d␥
d KK ⬘ 共  兲共 cos  兲 ⫹ sin  d . L
共89兲
With the substitution (⫹ )/2 共 cos  兲 ⫹ ⫽d (⫹ )/2,(⫹ )/2共 2  兲 ,
共90兲
L
all the integrals in this expression for S KK ⬘ are special cases of the integrals given above. Thus, we obtain
L S KK ⬘ ⫽2
兺
⫽0 ⫹
⫻
冉冊
1 I 共 ⫺ , ,K 兲 I 共 ⫺ , ,K ⬘ 兲 共 ⫺1 兲 ⫹ ⫹L⫹1
兺 共 ⫺1 兲 r r⫽0
冉 冊冉 ⫹ r
⫹ ⫹L 共 K⫺K ⬘ ⫹2n⫹2r 兲 /2
冊
L⫺K
兺
n⫽0
d 共 K,L,K ⬘ ,n 兲
⫺1
共91兲
.
The S L overlap integrals simplify when there are no multiplicities. For example, for a multiplicity-free (, ⫽0) irrep, the overlaps are given by L ⫽ 具 共 0 兲 0LM 兩 共 0 兲 0LM 典 S 00
⫽4 2
冕
0
L d 00 共  兲 共 cos  兲 sin  d ⫽4 2
冕
1
⫺1
P L 共 z 兲 z dz.
共92兲
From Eq. 共8.14.15兲 of Abramowitz and Stegun,28 we have
冕
1
0
and, hence,
P L 共 z 兲 z dz⫽
! , 共 ⫺L 兲 !! 共 ⫹L⫹1 兲 !!
共93兲
J. Math. Phys., Vol. 41, No. 9, September 2000
L S 00 ⫽ 兩 K (0) 共 L 兲 兩 2 ⫽4 2
Clebsch-Gordan coefficients of SU(3)
! 关 1⫹ 共 ⫺1 兲 ⫹L 兴 . ⫺L !! 兲 共 共 ⫹L⫹1 兲 !!
6557
共94兲
For a multiplicity-free (⫽0, ) irrep,
L ⫽2 S LL
兺
⫽0
共 ⫺1 兲
冉冊
共 兲 ! 共 ⫹L⫺r 兲 ! . 关 I 共 ⫺ , ,L 兲兴 2 兺 共 ⫺1 兲 r 共 ⫺r 兲 ! 共 ⫹L⫹1 兲 ! r⫽0
共95兲
For an arbitrary irrep, the states of lowest angular momentum are always multiplicity free. For example, if and are both even, there is single state of angular momentum L⫽0 whereas, if ⫹ is odd, the state of angular momentum L⫽1 is multiplicity free. Suppose and are both even. Then
0 ⫽ S 00
兺 共 ⫺1 兲 ⫽0
冉 冊冋 冕
2
0
共 cos ␣ 兲 ⫺ 共 sin ␣ 兲 d␣
册冕 2
0
共 cos  兲 ⫹ sin  d
⫽
4 2 共 ⫺1 兲 !! 共 ⫺1 兲 !! 共 ⫺ ⫺1 兲 !! , 共 1⫹ 共 ⫺1 兲 兲 !! 共 ⫹ ⫹1 兲 !! 共 ⫺ 兲 !! ⫽0
兺
共96兲
with the understanding that 共 ⫺1 兲 !!⫽1.
共97兲
If is odd and is even,
1 ⫽ S 00
兺 共 ⫺1 兲 ⫽0
冉 冊冋 冕
2
0
共 cos ␣ 兲 ⫺ 共 sin ␣ 兲 d␣
册冕 2
0
1 d 00 共  兲 共 cos  兲 ⫹ sin  d
⫽
4 2 共 ⫺1 兲 !! 共 ⫺1 兲 !! 共 ⫺ ⫺1 兲 !! . 共 1⫹ 共 ⫺1 兲 兲 !! 共 ⫹ ⫹2 兲 !! 共 ⫺ 兲 !! ⫽0
兺
共98兲
If is even and is odd,
1 ⫽ S 11
兺 共 ⫺1 兲 ⫽0
冉 冊冋 冕
2
0
共 cos ␣ 兲 ⫺ 共 sin ␣ 兲 e i ␣ d␣
冋
册冕 2
0
1 d 11 共  兲 共 cos  兲 ⫹ sin  d
册
1⫹ 共 ⫺1 兲 共 ⫺1 兲 !! 共 ⫺ 兲 !! 1⫺ 共 ⫺1 兲 !! 共 ⫺ ⫺1 兲 !! 2 2 !! ⫹ ⫽ . ⫹ ⫹2 共 ⫺1 兲 !! 共 ⫺ 兲 !! 共 ⫹1 兲 共 ⫹1 兲 !! ⫽0 ⫹ ⫹1 !! 共 ⫺ ⫺1 兲 !!
兺
共99兲 E. CG coefficients for some multiplicity-free couplings
According to Eq. 共63兲, CG coefficients for the multiplicity-free ( 2 ,0) 丢 ( 1 ,0)→( 1 ⫹ 2 ⫺2 , ) couplings are given by 共共 1 0 兲 L 1 , 共 2 0 兲 L 2 储共 兲 ␣ L 典 ⫽
冑
82 共共 1 0 兲 L 1 M 1 , 共 2 0 兲 L 2 M 2 兩 兲 2L⫹1 M 1 M 2 K
兺
¯ K(␣ ) 共 L 兲 , ⫻共 L 1 M 1 ,L 2 M 2 兩 LK 兲 K
共100兲
with SO共3兲 seed coefficients given in terms of the SU共2兲 seeds and overlaps by Eq. 共61兲. Inserting the explicit expression for the SU共2兲 seeds, given by Eq. 共46兲, we obtain
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J. Math. Phys., Vol. 41, No. 9, September 2000
„共 1 0 兲 L 1 M 1 , 共 2 0 兲 L 2 M 2 兩 …⫽
兺
I 1 ,I 2
D. J. Rowe and C. Bahri
␦ I 1 ⫹I 2 , /2具 共 1 0 兲 L 1 M 1 兩 共 1 0 兲 I 1 I 1 典
⫻ 具 共 20 兲 L 2 M 2兩共 20 兲 I 2I 2典 ⫻共 ⫺1 兲 2I 1
冑
共 ⫹1 兲 ! ! 共 1 ⫺2I 1 兲 ! 共 2 ⫺2I 2 兲 ! . 共 ⫹ ⫹1 兲 ! 共 1 ⫺ 兲 ! 共 2 ⫺ 兲 ! 共 2I 1 兲 ! 共 2I 2 兲 !
共101兲 The overlaps 具 (0)LM 兩 (0)II 典 , given by Eqs. 共64兲 and 共78兲, are
具 共 0 兲 LM 兩 共 0 兲 II 典 ⫽
冑 冕
2L⫹1 (0) ¯K 共 L 兲 2
⫻
0
/2 d /2,/2⫺2I 共 2  兲 d L0M 共  兲 sin  d
冕
2
0
I d I,I 共 2 ␥ 兲 e iM ␥ d␥ .
共102兲
The second integral in this equation, evaluated by means of Eq. 共83兲, is given by
冕
2
0
I d I,I 共 2 ␥ 兲 e iM ␥ d␥ ⫽I 共 2I,0,M 兲 ⫽
冉
冊
2 2I 2I I⫹M /2 , 2
共103兲
and is seen to vanish unless I⫹M /2 is a positive integer less than 2I. With the substitutions /2 d /2,/2⫺2I 共 2  兲 ⫽ 共 ⫺1 兲 2I
d L0M 共  兲 ⫽ 共 ⫺1 兲 M
冋
冉 冊 2I
1/2
共 cos  兲 ⫺2I 共 sin  兲 2I ,
共 L⫺M 兲 ! 共 L⫹M 兲 !
册
共104兲
1/2
P LM 共 cos  兲 ,
共105兲
where P LM (cos ) is an associated Legendre polynomial, the first integral is expressed in terms of the standard integral27
冕
1
0
z ⫺2I 共 1⫺z 2 兲 I P LM 共 z 兲 dz⫽
2 M 共 2I⫺M 兲 !! 共 ⫺2I⫺1 兲 !! F 共共 L⫺M ⫹1 兲 /2, 共 ⫺M 兲 ! 共 ⫺M ⫹1 兲 !! 3 2 ⫺ 共 M ⫹L 兲 /2,I⫹1⫺M /2;1⫺M , 共 3⫹⫺M 兲 /2;1 兲 ,
共106兲
where 3 F 2 (...) is a hypergeometric function. Because I⫹M /2 is an integer for nonvanishing overlaps, the parity of the integrand in Eq. 共106兲 is determined to be (⫺1) ⫹L . So the overlaps ¯ (0) (L), from Eq. 共94兲, vanish for ⫹L odd. Finally, using the explicit expression of K
具 共 0 兲 LM 兩 共 0 兲 II 典 ⫽2 M ⫺2I ⫻
冉 冊冋 2I
冋 册
1⫹ 共 ⫺1 兲 ⫹L 共 2L⫹1 兲共 ⫺L 兲 !! 共 ⫹L⫹1 兲 !! 2 ! 1/2
共 L⫺M 兲 ! 共 L⫹M 兲 !
1/2
册冉 1/2
2I I⫹M /2
冊
共 2I⫺M 兲 !! 共 ⫺2I⫺1 兲 !! F „共 L⫺M ⫹1 兲 /2, 共 ⫺M 兲 ! 共 ⫺M ⫹1 兲 !! 3 2
⫺ 共 M ⫹L 兲 /2,I⫹1⫺M /2;1⫺M , 共 3⫹⫺M 兲 /2;1….
共107兲
With this expression for the overlaps, Eqs. 共100兲 and 共101兲 give an explicit, albeit complicated, expression for the CG coefficient (( 1 0)L 1 ,( 2 0)L 2 储 ( ) ␣ L) to within multiplication by the
J. Math. Phys., Vol. 41, No. 9, September 2000
Clebsch-Gordan coefficients of SU(3)
6559
¯ ( ) (L) matrix. If the final irrep ( ) is generic, there is an inner multiplicity and one must K resort to numerical methods to take the square root of the S L matrix to determine the K ( ) (L) ¯ ( ) (L). matrix and its inverse K For the special case of ⫽0, there is neither outer nor inner multiplicity and complete expressions are obtained. Equation 共63兲 gives „共 1 0 兲 L 1 , 共 2 0 兲 L 2 储共 0 兲 L…⫽
冑
82 „共 1 0 兲 L 1 0,共 2 0 兲 L 2 兩 0… 2L⫹1
¯ (0) 共 L 兲 , ⫻ 共 L 1 0,L 2 0 兩 L0 兲 K
共108兲
where, from Eq. 共61兲, „共 1 0 兲 L 1 0,共 2 0 兲 L 2 0 兩 0…⫽
兺
具 共 1 0 兲 L 1 0 兩 1 0 典 具 共 2 0 兲 L 2 0 兩 2 0 典 共 I 1 N 1 ,I 2 N 2 兩 00兲
⫻„共 1 0 兲 I 1 , 共 2 0 兲 I 2 储 0….
共109兲
„共 1 0 兲 I 1 , 共 2 0 兲 I 2 储 0…⫽ ␦ I 1 ,0␦ I 2 ,0 ,
共110兲
From Eq. 共46兲, we have
and, from Eqs. 共56兲 and 共65兲,
具 共 0 兲 LM 兩 共 0 兲 0 典 ⫽ 具 共 0 兲 LM 兩 0 典 ⫽
冑
2L⫹1 (0) ¯K 共 L 兲 82
⫽4 2 ␦ M 0
冑
冕
* 共 ⍀ 兲 0 兩 R 共 ⍀ 兲 兩 0 d⍀ D L0M 具 典
2L⫹1 (0) ¯K 共 L 兲 82
冕
L d 00 共  兲 共 cos  兲 sin  d .
共111兲
Evaluating this expression with the results of Eqs. 共92兲–共94兲 gives 1 2
具 共 0 兲 LM 兩 共 0 兲 0 典 ⫽ ␦ M 0 关 1⫹ 共 ⫺1 兲 ⫹L 兴
冑
共 2L⫹1 兲 ! . 共 ⫺L 兲 !! 共 ⫹L⫹1 兲 !!
共112兲
Putting these results together, we obtain „共 1 0 兲 L 1 , 共 2 0 兲 L 2 储共 0 兲 L… 1 ⫽ 关 1⫹ 共 ⫺1 兲 1 ⫹L 1 兴关 1⫹ 共 ⫺1 兲 2 ⫹L 2 兴共 L 1 0,L 2 0 兩 L0 兲 4 ⫻
冑
共 2L 1 ⫹1 兲共 2L 2 ⫹1 兲 1 ! 2 ! 共 ⫺L 兲 !! 共 ⫹L⫹1 兲 !! . 共 2L⫹1 兲 ! 共 1 ⫺L 1 兲 !! 共 2 ⫺L 2 兲 !! 共 1 ⫹L 1 ⫹1 兲 !! 共 2 ⫹L 2 ⫹1 兲 !!
共113兲
IV. DISCUSSION
The above algorithms have been programmed to give computer codes for the computation of any SU共3兲 Clebsch–Gordan coefficient in either a SU共2兲 or a SO共3兲 basis. For comparison, we have also programmed the algorithm of Rowe and Repka18 for the canonical SU共2兲 basis. The results show that the Rowe–Repka algorithm leads to little or no increase in speed over the Draayer–Akiyama code.12 This is perhaps not surprising because both algorithms resolve the outer multiplicity problems with recursive methods based on similar shift-tensor concepts. In contrast,
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D. J. Rowe and C. Bahri
the new algorithms presented in this paper increase the speed of computation by ⬃15% for the SU共2兲 basis and a factor of ⬃2 for the SO共3兲 basis. The latter is a particularly significant improvement. The new algorithms have their foundations in recent developments in vector-coherent-state theory. Most applications of VCS theory in the past have been to determine the explicit matrices of Lie algebra representations. For such applications, only the ratios of the important K-matrices, which give the norms of states, are needed. However, for the calculation of Clebsch-Gordan coefficients, absolute values of the K-matrices are needed. It is important to be assured that the K-matrices are the same, even if derived differently. When there are missing labels 共i.e., multiplicity indices are needed to distinguish basis states兲, there is arbitrariness in the choice of basis states and a corresponding arbitrariness in the choice of K matrices. For example, the SU共3兲傻SO共3兲 multiplicity has a natural resolution in which the K matrices are taken to be the Hermitian square roots of the corresponding 共uniquely defined兲 S matrices 关cf. Eq. 共59兲兴. However, for some purposes it may be convenient to choose a canonical ˆ 丢 Lˆ ) 0 operator.29 Such a basis has the propSU共3兲傻SO共3兲 basis that diagonalizes Racah’s (Lˆ 丢 Q erty that the reduced matrix elements of the quadrupole operator between states of the same angular momentum are diagonal in the multiplicity indices, i.e., LL . 具 共 兲 ␣ L 储 Qˆ 储共 兲  L 典 ⫽ ␦ ␣ M ␣␣
共114兲
The matrix element of the SU共3兲 quadrupole operator in a SU共3兲傻SO共3兲 basis have been evaluated and determined30,6 to be given by
具 共 兲  L ⬘ 储 Qˆ 储共 兲 ␣ L 典 ⫽ 冑2L⫹1
兺 兺
K ⬘ ⭓0 K⭓0
¯ K  共 L ⬘ 兲 M L ⬘ L KK ␣ 共 L 兲 , K K⬘K ⬘
共115兲
with
冋
册
1 1 L L M K⬘⬘ K ⫽ 共 2⫹ ⫹3 兲 ⫹ L 共 L⫹1 兲 ⫺ L ⬘ 共 L ⬘ ⫹1 兲 共 LK,20兩 L ⬘ K ⬘ 兲 ⫺ ␦ K,1共 ⫺1 兲 ⫹L 2 2 ⫻共 L,⫺1,22兩 L ⬘ K ⬘ 兲 ⫹
冑
3 共 ⫺K 兲共 LK,22兩 L ⬘ K ⬘ 兲 ⫹ 2
冑
3 共 ⫹1 兲 2
冑
3 共 ⫹K 兲共 LK,2,⫺2 兩 L ⬘ K ⬘ 兲 其 . 2 共116兲
Thus, a basis that diagonalizes the Racah operator corresponds to a choice of K matrices that bring the matrices M LL to diagonal form. Since the S-matrices commute with the matrix representations of the Racah operator,31 it follows that this construction gives K-matrices which satisfy Eq. 共59兲. Note also that, since all K-matrices that satisfy Eq. 共59兲 are the same to within unitary transforLL mations, K-matrices that diagonalize M K ⬘ K are unitarily equivalent to others which are the Hermitian square roots of the S-matrices. Although the algorithms reported in this paper are given explicitly for SU共3兲, they also apply, with suitable adjustment, to other groups of importance in physics. For example, they readily adapt to SO共4兲 and to SO共5兲⬃Sp共2兲. As a useful extension of the present program, we plan to derive the asymptotic expressions to which the SU共3兲 Clebsch–Gordan coefficients approach in the limit of large-dimensional irreps. It is our belief that such limiting values become very accurate for the kinds of large-dimensional irreps that occur, for example, in applications of the SU共3兲 model and its various symplectic model extensions to heavy deformed nuclei.
J. Math. Phys., Vol. 41, No. 9, September 2000
Clebsch-Gordan coefficients of SU(3)
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APPENDIX A: MATRIX ELEMENTS OF THE SU„3… WEYL OPERATORS
Matrix elements of the SU共3兲 Weyl operators have been derived in a Gel’fand–Tsetlin basis32 by Chaco´n and Moshinsky26 and by Rowe, Sanders, and de Guise.25 In particular, the latter article gave
具 共 兲 ⬘ I ⬘ 兩 P 123兩 共 兲 I 典 ⫽ ␦ 1⬘ , 3 ␦ 2⬘ , 1 ␦ ⬘3 , 2 共 ⫺1 兲 ( 3 ⫹2I ⬘ ⫹2 ⫹)/2冑共 2I⫹1 兲共 2I ⬘ ⫹1 兲 ⫻
再
1 /2 2 /2 I ⬘ 3 /2
/2
I
冎
共A1兲
and
具 共 兲 ⬘ I ⬘ 兩 P 132兩 共 兲 I 典 ⫽ 具 共 兲 I 兩 P 123兩 共 兲 ⬘ I ⬘ 典 * .
共A2兲
As shown below, the basis states 兵 兩 ( ) I 典 其 of Ref. 25 are related to the VCS basis used in this paper by 兩 共 兲 jIN 典 ⫽ 共 ⫺1 兲 I⫺ j⫺s 兩 共 兲 I 典 ,
共A3兲
with
1 ⫽⫹2s⫺2 j,
共 2s⫽ 兲 ,
2 ⫽s⫹ j⫹N,
共A4兲
3 ⫽s⫹ j⫺N. Thus, both bases are Gel’fand–Tsetlin bases, albeit with different phase conventions. Generic irreps of SU共3兲 can be constructed in a space of two copies of a three-dimensional harmonic oscillator. Let 兵 a i†␣ ;i⫽1,2,3, ␣ ⫽1,2其 denote the raising operators of two threedimensional harmonic oscillators. These operators transform as bases for two three-dimensional irreps of U共3兲, under U共3兲 transformation of the i-index, and as three two-dimensional 共spin- 21兲 irreps of U共2兲 under U共2兲 transformation of the ␣-index; the bar is used to distinguish this U共2兲 from subgroups of U共3兲. Simultaneous highest weight states for U共3兲 and U共2兲 are given 共to within a proportionality constant兲 by † † † † † a 21兴 兩 0 典 . 兲 关 a 11a 22⫺a 12 共 a 11
共A5兲
This state has a U共3兲 weight (⫹ , ,0) and a U共2兲 weight (⫹ , ). The two commuting groups U共3兲 and U共2兲 are said to form a dual pair. † † The doublet (a i1 ,a i2 ) can be regarded 共for each i兲 as the two components of a spin- 21 U共2兲 † tensor a i . It is then inferred that (a †i ) 2 j / 冑(2 j)! is a spin-j tensor with components 关共 a †i 兲 2 j 兴 mj
冑共 2 j 兲 !
兩0典⫽
† j⫹m † j⫺m 兲 共 a i1 共 a i2 兲
冑共 j⫹m 兲 ! 共 j⫺m 兲 !
0 ,
m⫽⫺ j, . . . ,⫹ j.
共A6兲
The states 兵 兩 ( ) I 典 其 can now be defined as U共2兲-coupled states, 兩共 兲 I 典 ⫽
1
冑 1 ! 2 ! 3 !
/2 兩0典. 关共 a †1 兲 1 丢 关共 a †2 兲 2 丢 共 a †3 兲 3 兴 I 兴 /2
共A7兲
It is seen by inspection that 兩 ( ) I 典 is a U共2兲 highest weight state of SU共2兲 spin /2. Also it is seen that the total number of oscillator quanta in the state is 1 ⫹ 2 ⫹ 3 . Thus, if we set
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J. Math. Phys., Vol. 41, No. 9, September 2000
D. J. Rowe and C. Bahri
1 ⫹ 2 ⫹ 3 ⫽2⫹ , in accord with Eq. 共A4兲, the state 兩 ( ) I 典 is determined to have U共2兲 共highest兲 weight (⫹ , ). It follows by duality that the set of states 兵 兩 ( ) I 典 其 belong to a U共3兲 irrep of highest weight (⫹ , ,0). In fact, since the U共3兲 operators are realized by 2
ˆ i j⫽ C
兺
␣ ⫽1
共A8兲
a i†␣ a j ␣ ,
it is seen that the components of ⫽( 1 , 2 , 3 ) are eigenvalues of the weight operators, ˆ ii 兩 共 兲 jIN 典 ⫽ i 兩 共 兲 jIN 典 , C
共A9兲
i⫽1,2,3.
We say that the state 兩 ( ) I 典 is a weight state of weight . Claim 3: A VCS state 兩 ( ) jIN 典 is also a weight state with weight given by Eq. 共A4兲. Moreover, the states 兵 兩 ( ) jIN 典 其 form a Gel’fand–Tsetlin basis for a U共3兲 irrep. Proof: The highest weight state 兩 典 ⬅ 兩 ( )0ss 典 is, by definition 关cf. Eq. 共4兲兴, an eigenstate of the U共3兲 weight operators, Cˆ 11兩 共 兲 0ss 典 ⫽ 共 ⫹ 兲 兩 共 兲 0ss 典 ,
Cˆ 22兩 共 兲 0ss 典 ⫽ 兩 共 兲 0ss 典 ,
Cˆ 33兩 共 兲 0ss 典 ⫽0, 共A10兲
of weight (⫹ , ,0)⫽(⫹2s,2s,0). The commutation relations, ˆ 11 ,Iˆ ⫺ 兴 ⫽0, 关C
ˆ 22 ,Iˆ ⫺ 兴 ⫽⫺Iˆ ⫺ , 关C
ˆ 33 ,Iˆ ⫺ 兴 ⫽Iˆ ⫺ , 关C
共A11兲
then imply that Cˆ 11兩 共 兲 0sm 典 ⫽ 共 ⫹2s 兲 兩 共 兲 0sm 典 ,
Cˆ 22兩 共 兲 0sm 典 ⫽ 共 s⫹m 兲 兩 共 兲 0sm 典 ,
ˆ 33兩 共 兲 0sm 典 ⫽ 共 s⫺m 兲 兩 共 兲 0sm 典 . C
共A12兲
And the commutation relations, ˆ 11 , ˜P nj 共 ˆf 兲兴 ⫽⫺2 j ˜P nj 共 ˆf 兲 , 关C
ˆ 22 , ˜P nj 共 ˆf 兲兴 ⫽ 共 j⫹n 兲 ˜P nj 共 ˆf 兲 , 关C
ˆ 33 , ˜P nj 共 ˆf 兲兴 ⫽ 共 j⫺n 兲 ˜P nj 共 ˆf 兲 , 关C 共A13兲
imply that ˆ 11兩 共 兲 jIN 典 ⫽ 共 ⫹ ⫺2 j 兲 兩 共 兲 jIN 典 , C
Cˆ 22兩 共 兲 jIN 典 ⫽ 共 s⫹ j⫹N 兲 兩 共 兲 jIN 典 ,
Cˆ 33兩 共 兲 jIN 典 ⫽ 共 s⫹ j⫺N 兲 兩 共 兲 jIN 典 .
共A14兲
Since the states 兵 兩 ( ) jIN 典 其 reduce the subgroup chain U共3兲傻U共2兲傻U共1兲, they are, by definition, a canonical Gel’fand–Tsetlin basis, albeit expressed in an unfamiliar manner. Claim 4: If is given by Eq. 共A4兲, then 兩 共 兲 jIN 典 ⫽ 共 ⫺1 兲 I⫺ j⫺s 兩 共 兲 I 典 .
共A15兲
Proof: It has been shown that the states 兩 ( ) jIN 典 and 兩 ( ) I 典 are both eigenstates of the Cˆ ii operators and that, with j and N related to the components of by Eq. 共A4兲, they have the same eigenvalues. That they also have the same SU共2兲23 angular momentum I is seen by observing that, if ˆ 23兩 共 2 3 兲 II 典 ⫽0, C
共A16兲
J. Math. Phys., Vol. 41, No. 9, September 2000
Clebsch-Gordan coefficients of SU(3)
6563
then the state 兩 共 2 3 兲 II 典 ⫽ 关共 a †2 兲 2 丢 共 a †3 兲 3 兴 II 兩 0 典
共A17兲
is simultaneously of SU共2兲23 and U共2兲 highest weight. The condition 共A16兲 is satisfied when 2 ⫺ 3 ⫽2I. It follows, by duality, that the states 兵 兩 ( 2 3 )IM 典 其 belong to SU共2兲23 and U共2兲 irreps of angular momentum I. Thus, with given by Eq. 共A4兲, the states 兩 ( ) jIN 典 and 兩 ( ) I 典 have the same weight and the same SU共2兲23 angular momentum. Hence they are the same to within a phase factor. It remains to derive the phase factor. The highest weight state 兩 ( )0ss 典 can be set equal to 兩 共 兲 0ss 典 ⫽ 兩 共 兲共 ⫹2s,2s,0兲 s 典 ,
共A18兲
thereby fixing the phase of this state to accord with the claim. Comparing the actions of the ˆ 32 on the states 兩 ( )0sm 典 and 兩 ( )(⫹2s,s⫹m,s⫺m)s 典 , SU共2兲23 lowering operator C Cˆ 32兩 共 兲 0sm 典 ⫽ 冑共 s⫹m 兲共 s⫺m⫹1 兲 兩 共 兲 0sm⫺1 典 ,
共A19兲 Cˆ 32兩 共 兲共 ⫹2s,s⫹m,s⫺m 兲 s 典 ⫽ 冑共 s⫹m 兲共 s⫺m⫹1 兲 兩 共 兲共 ⫹2s,s⫹m⫺1,s⫺m⫹1 兲 s 典 , then implies the identity 兩 共 兲 0sm 典 ⫽ 兩 共 兲共 ⫹2s,s⫹m,s⫺m 兲 s 典 ,
共A20兲
for all highest-grade states, consistent with the claim. ˆ i j of Eq. 共A8兲 as the SU共2兲 scalars: Writing the su共3兲 operators C Cˆ i j ⫽& 关 a †i ⫻a j 兴 0 ,
共A21兲
one can show that ˜P mj 共 ˆf 兲 ⫽
冑
共 2 j⫹1 兲 关共 a †3 兲 j⫺m 丢 共 a †2 兲 j⫹m 丢 共 a 1 兲 2 j 兴 0 共 j⫹m 兲 ! 共 j⫺m 兲 !
共A22兲
and, hence, that 兩 共 兲 jIN 典 ⫽
⫽
1
兺 共 sm, jn 兩 IN 兲 ˜P nj共 ˆf 兲 兩 共 兲 0sm 典
) K ( mn jI
1
兺 共 sm, jn 兩 IN 兲 冑共 j⫹n 兲 ! 共 j⫺n 兲 ! 共 s⫹m 兲 ! 共 s⫺m 兲 ! 关关共 a †3 兲 j⫺m 丢 共 a †2 兲 j⫹m 共 2 j⫹1 兲
) K ( mn jI
/2
丢 共 a 1 兲 2 j 兴 0 关共 a 1 兲 ⫹ 丢 关共 a 2 兲 兴 s 兴 /2兩 0 典 . †
†
共A23兲
It remains to combine and recouple the operators in this expression to prove the claim. The a 1 and a †1 operators can be combined using the identity 兩 0 典 ⫽ 共 ⫺1 兲 m⫺n 关共 a 1 兲 n 丢 共 a †1 兲 m 兴 (m⫺n)/2
m! 共 m⫺n 兲 !
冑
m⫹1 兩0典. 关共 a †1 兲 m⫺n 兴 (m⫺n)/2 m⫺n⫹1
共A24兲
Then, after some tedious but straightforward recoupling, one obtains 兩 共 兲 jIN 典 ⫽
兺I C I(⬘ )共 jIN 兲 兩 共 兲 I ⬘ 典 ; ⬘
共A25兲
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D. J. Rowe and C. Bahri
with the i values given by Eq. 共A4兲 and ( )
C I ⬘ 共 jIN 兲 ⫽
共 sm, jn 兩 IN 兲 冑 兺 共 j⫹n 兲 ! 共 j⫺n 兲 ! 共 s⫹m 兲 ! 共 s⫺m 兲 ! mn
⫻
冋 再
共 s⫹ j⫹N 兲 ! 共 s⫹ j⫺N 兲 !
s⫺m 2
s⫹m 2
s
j⫺n 2
j⫹n 2
j
s⫹ j⫺N 2
s⫹ j⫹N 2
I⬘
s
⫻ ⫹2s⫺2 j 2
I⬘
j 2
册
冑
共 2I ⬘ ⫹1 兲共 ⫹s⫹I⫺ j⫹1 兲 ! 共 ⫹s⫺I⫺ j 兲 ! ! 共 ⫹2s⫺2 j 兲 !
冎
⫹2s ; 2
共A26兲
the 9j-symbol with square brackets is the unitary 9j-coefficient 共i.e., with the square root factors included兲. The 9j-and 6j-symbols used here are of a special type, i.e., some arguments are the sum of some others. Consequently, Eq. 共A26兲 reduces to ( )
C I ⬘ 共 jIN 兲 ⫽
冑
共 ⫹s⫹I⫺ j⫹1 兲 ! 共 ⫹s⫺I⫺ j 兲 ! 共 ⫹s⫹I ⬘ ⫺ j⫹1 兲 ! 共 ⫹s⫺I ⬘ ⫺ j 兲 !
⫻
共 sm, jn 兩 IN 兲 兺 mn
冉
冏
冊
s⫹ j⫺N s⫺m⫺ j⫹n s⫹ j⫹N s⫹m⫺ j⫺n , I ⬘ ,s⫺ j . 2 2 2 2 共A27兲
Using the properties of Regge symmetries of the SU共2兲 Clebsch–Gordan coefficients, ( )
C I ⬘ 共 jIN 兲 ⫽ 共 ⫺1 兲 I⫺s⫺ j ␦ II ⬘ .
共A28兲
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