In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory. Generalization to SU(3) of Clebsch–Gordan coefficients is useful because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists (the eightfold way) that connects the three light quarks: up, down, and strange.
SU(3) group
The special unitary group SU is the group of unitary matrices whose determinant is equal to 1. This set is closed under matrix multiplication. All transformations characterized by the special unitary group leave norms unchanged. The SU(3) symmetry appears in the light quark flavour symmetry (among up, down, and strange quarks) dubbed the Eightfold Way (physics). The same group acts in quantum chromodynamics on the colour quantum numbers of the quarks that form the fundamental (triplet) representation of the group. The group SU(3) is a subgroup of group U(3), the group of all 3×3 unitary matrices. The unitarity condition imposes nine constraint relations on the total 18 degrees of freedom of a 3×3 complex matrix. Thus, the dimension of the U(3) group is 9. Furthermore, multiplying a U by a phase, eiφ leaves the norm invariant. Thus U(3) can be decomposed into a direct product U(1) × SU(3)/Z3. Because of this additional constraint, SU(3) has dimension 8.
Generators of the Lie algebra Every unitary matrix U can be written in the form
U = e i H {\displaystyle U=e^{iH}\,}
where H is hermitian. The elements of SU(3) can be expressed as
U = e i ∑ a k λ k {\displaystyle U=e^{i\sum {a_{k}\lambda _{k}}}}
where λ k {\displaystyle \lambda _{k}} are the 8 linearly independent matrices forming the basis of the Lie algebra of SU(3), in the triplet representation. The unit determinant condition requires the λ k {\displaystyle \lambda _{k}} matrices to be traceless, since
det ( e A ) = e tr ( A ) {\displaystyle \det(e^{A})=e^{\operatorname {tr} (A)}} . An explicit basis in the fundamental, 3, representation can be constructed in analogy to the Pauli matrix algebra of the spin operators. It consists of the Gell-Mann matrices,
λ 1 = ( 0 1 0 1 0 0 0 0 0 ) λ 2 = ( 0 − i 0 i 0 0 0 0 0 ) λ 3 = ( 1 0 0 0 − 1 0 0 0 0 ) λ 4 = ( 0 0 1 0 0 0 1 0 0 ) λ 5 = ( 0 0 − i 0 0 0 i 0 0 ) λ 6 = ( 0 0 0 0 0 1 0 1 0 ) λ 7 = ( 0 0 0 0 0 − i 0 i 0 ) λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 − 2 ) . {\displaystyle {\begin{array}{ccc}\lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}&\lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}&\lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}\\\\\lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}&\lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}\\\\\lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}&\lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}&\lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.\end{array}}}
These are the generators of the SU(3) group in the triplet representation, and they are normalized as
tr ( λ j λ k ) = 2 δ j k . {\displaystyle \operatorname {tr} (\lambda _{j}\lambda _{k})=2\delta _{jk}.}
The Lie algebra structure constants of the group are given by the commutators of λ k {\displaystyle \lambda _{k}}
[ λ j , λ k ] = 2 i ∑ l f j k l λ l , {\displaystyle [\lambda _{j},\lambda _{k}]=2i\sum _{l}f_{jkl}\lambda _{l}~,}
where f j k l {\displaystyle f_{jkl}} are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol ϵ j k l {\displaystyle \epsilon _{jkl}} of SU(2). In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric λs. Note f l j k = − i 4 t r ( [ λ l , λ j ] λ k ) {\displaystyle f_{ljk}={\frac {-i}{4}}\mathrm {tr} ([\lambda _{l},\lambda _{j}]\lambda _{k})} . Moreover,
{ λ j , λ k } = 4 3 δ j k + 2 d j k l λ l {\displaystyle \{\lambda _{j},\lambda _{k}\}={\frac {4}{3}}\delta _{jk}+2d_{jkl}\lambda _{l}}
where d j k l {\displaystyle d_{jkl}} are the completely symmetric coefficient constants. They vanish if the number of indices from the set {2, 5, 7} is odd. In terms of the matrices,
d j k l = 1 4 tr ( { λ j , λ k } λ l ) = 1 4 tr ( { λ k , λ l } λ j ) = d k l j = d k j l {\displaystyle d_{jkl}={\frac {1}{4}}\operatorname {tr} (\{\lambda _{j},\lambda _{k}\}\lambda _{l})={\frac {1}{4}}\operatorname {tr} (\{\lambda _{k},\lambda _{l}\}\lambda _{j})=d_{klj}=d_{kjl}}
Standard basis
A slightly differently normalized standard basis consists of the F-spin operators, which are defined as F i ^ = 1 2 λ i {\displaystyle {\hat {F_{i}}}={\frac {1}{2}}\lambda _{i}} for the 3, and are utilized to apply to any representation of this algebra. The Cartan–Weyl basis of the Lie algebra of SU(3) is obtained by another change of basis, where one defines,
I ^ ± = F ^ 1 ± i F ^ 2 {\displaystyle {\hat {I}}_{\pm }={\hat {F}}_{1}\pm i{\hat {F}}_{2}}
I ^ 3 = F 3 ^ {\displaystyle {\hat {I}}_{3}={\hat {F_{3}}}}
V ^ ± = F ^ 4 ± i F ^ 5 {\displaystyle {\hat {V}}_{\pm }={\hat {F}}_{4}\pm i{\hat {F}}_{5}}
U ^ ± = F ^ 6 ± i F ^ 7 {\displaystyle {\hat {U}}_{\pm }={\hat {F}}_{6}\pm i{\hat {F}}_{7}}
Y ^ = 2 3 F ^ 8 . {\displaystyle {\hat {Y}}={\frac {2}{\sqrt {3}}}{\hat {F}}_{8}~.}
Because of the factors of i in these formulas, this is technically a basis for the complexification of the su(3) Lie algebra, namely sl(3,C). The preceding basis is then essentially the same one used in Hall's book.
Commutation algebra of the generators The standard form of generators of the SU(3) group satisfies the commutation relations given below,
All other commutation relations follow from hermitian conjugation of these operators. These commutation relations can be used to construct the irreducible representations of the SU(3) group. The representations of the group lie in the 2-dimensional I3−Y plane. Here, I ^ 3 {\displaystyle {\hat {I}}_{3}} stands for the z-component of Isospin and Y ^ {\displaystyle {\hat {Y}}} is the Hypercharge, and they comprise the (abelian) Cartan subalgebra of the full Lie algebra. The maximum number of mutually commuting generators of a Lie algebra is called its rank: SU(3) has rank 2. The remaining 6 generators, the ± ladder operators, correspond to the 6 roots arranged on the 2-dimensional hexagonal lattice of the figure.
Casimir operators
The Casimir operator is an operator that commutes with all the generators of the Lie group. In the case of SU(2), the quadratic operator J2 is the only independent such operator. In the case of SU(3) group, by contrast, two independent Casimir operators can be constructed, a quadratic and a cubic: they are,
C 1 ^ = ∑ k F k ^ F k ^ C 2 ^ = ∑ j k l d j k l F j ^ F k ^ F l ^ . {\displaystyle {\hat {C_{1}}}=\sum _{k}{\hat {F_{k}}}{\hat {F_{k}}}\qquad \qquad {\hat {C_{2}}}=\sum _{jkl}d_{jkl}{\hat {F_{j}}}{\hat {F_{k}}}{\hat {F_{l}}}~.}
These Casimir operators serve to label the irreducible representations of the Lie group algebra SU(3), because all states in a given representation assume the same value for each Casimir operator, which serves as the identity in a space with the dimension of that representation. This is because states in a given representation are connected by the action of the generators of the Lie algebra, and all generators commute with the Casimir operators. For example, for the triplet representation, D(1,0), the eigenvalue of C ^ 1 {\displaystyle {\hat {C}}_{1}} is 4/3, and of C ^ 2 {\displaystyle {\hat {C}}_{2}} , 10/9. More generally, from Freudenthal's formula, for generic D(p,q), the eigenvalue of C ^ 1 {\displaystyle {\hat {C}}_{1}} is
( p 2 + q 2 + 3 p + 3 q + p q ) / 3 {\displaystyle (p^{2}+q^{2}+3p+3q+pq)/3} . The eigenvalue ("anomaly coefficient") of C ^ 2 {\displaystyle {\hat {C}}_{2}} is
( p − q ) ( 3 + p + 2 q ) ( 3 + q + 2 p ) / 18. {\displaystyle (p-q)(3+p+2q)(3+q+2p)/18.}
It is an odd function under the interchange p ↔ q. Consequently, it vanishes for real representations p = q, such as the adjoint, D(1,1), i.e. both C ^ 2 {\displaystyle {\hat {C}}_{2}} and anomalies vanish for it.
Representations of the SU(3) group
The irreducible representations of SU(3) are analyzed in various places, including Hall's book. Since the SU(3) group is simply connected, the representations are in one-to-one correspondence with the representations of its Lie algebra su(3), or the complexification of its Lie algebra, sl(3,C). We label the representations as D(p,q), with p and q being non-negative integers, where in physical terms, p is the number of quarks and q is the number of antiquarks. Mathematically, the representation D(p,q) may be constructed by tensoring together p copies of the standard 3-dimensional representation and q copies of the dual of the standard representation, and then extracting an irreducible invariant subspace. (See also the section of Young tableaux below: p is the number of single-box columns, "quarks", and q the number of double-box columns, "antiquarks"). Still another way to think about the parameters p and q is as the maximum eigenvalues of the diagonal matrices
H 1 = ( 1 0 0 0 − 1 0 0 0 0 ) , H 2 = ( 0 0 0 0 1 0 0 0 − 1 ) {\displaystyle H_{1}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}},\quad H_{2}={\begin{pmatrix}0&0&0\\0&1&0\\0&0&-1\end{pmatrix}}} . (The elements H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} are linear combinations of the elements I ^ 3 {\displaystyle {\hat {I}}_{3}} and Y ^ {\displaystyle {\hat {Y}}} , but normalized so that the eigenvalues of H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} are integers.) This is to be compared to the representation theory of SU(2), where the irreducible representations are labeled by the maximum eigenvalue of a single element, h.
The representations have dimension
d ( p , q ) = 1 2 ( p + 1 ) ( q + 1 ) ( p + q + 2 ) , {\displaystyle d(p,q)={\frac {1}{2}}(p+1)(q+1)(p+q+2),}
their irreducible characters are given by
χ p , q ( θ , ϕ ) = e i θ 3 ( 2 p + q ) ∑ k = 0 p ∑ l = 0 q e − i ( k + l ) θ ( sin ( ( k − l + q + 1 ) ϕ / 2 ) sin ( ϕ / 2 ) ) , {\displaystyle \chi ^{p,q}(\theta ,\phi )=e^{i{\frac {\theta }{3}}(2p+q)}\sum \limits _{k=0}^{p}\sum \limits _{l=0}^{q}e^{-i(k+l)\theta }\left({\frac {\sin((k-l+q+1)\phi /2)}{\sin(\phi /2)}}\right),}
and the corresponding Haar measure is
μ ( S U ( 3 ) ) = 64 sin ( ϕ 2 ) 2 sin ( 1 2 ( θ + ϕ / 2 ) ) 2 sin ( 1 2 ( θ − ϕ / 2 ) ) 2 {\displaystyle \mu (SU(3))=64\sin \left({\frac {\phi }{2}}\right)^{2}\sin \left({\frac {1}{2}}(\theta +\phi /2)\right)^{2}\sin \left({\frac {1}{2}}(\theta -\phi /2)\right)^{2}}
such that − 2 π ≤ ϕ ≤ 2 π {\displaystyle -2\pi \leq \phi \leq 2\pi } and − 3 π ≤ θ ≤ 3 π {\displaystyle -3\pi \leq \theta \leq 3\pi } ,
V ( S U ( 3 ) ) = ∫ − π π ∫ − π π d ( ϕ 2 ) d ( θ 3 ) μ ( S U ( 3 ) ) = ∫ − 2 π 2 π d ϕ 2 ∫ − 3 π 3 π d θ 3 μ ( S U ( 3 ) ) = 24 π 2 . {\displaystyle V(SU(3))=\int _{-\pi }^{\pi }\!\int _{-\pi }^{\pi }d\!\left({\frac {\phi }{2}}\right)d\!\left({\frac {\theta }{3}}\right)\mu (SU(3))=\int _{-2\pi }^{2\pi }{\frac {d\phi }{2}}\int _{-3\pi }^{3\pi }{\frac {d\theta }{3}}\mu (SU(3))=24\pi ^{2}.}
An SU(3) multiplet may be completely specified by five labels, two of which, the eigenvalues of the two Casimirs, are common to all members of the multiplet. This generalizes the mere two labels for SU(2) multiplets, namely the eigenvalues of its quadratic Casimir and of I3. Since [ I ^ 3 , Y ^ ] = 0 {\displaystyle [{\hat {I}}_{3},{\hat {Y}}]=0} , we can label different states by the eigenvalues of I ^ 3 {\displaystyle {\hat {I}}_{3}} and Y ^ {\displaystyle {\hat {Y}}} operators, | t , y ⟩ {\displaystyle |t,y\rangle } , for a given eigenvalue of the isospin Casimir. The action of the operators on these states is given by
I ^ 3 | t , y ⟩ = t | t , y ⟩ {\displaystyle {\hat {I}}_{3}|t,y\rangle =t|t,y\rangle }
Y ^ | t , y ⟩ = y | t , y ⟩