Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group

{{main article|Special unitary group}}

The special unitary group SU is the group of unitary matrices whose determinant is equal to 1.P. Carruthers (1966) Introduction to Unitary symmetry, Interscience. [https://books.google.com/books?id=t33soAEACAAJ online]. This set is closed under matrix multiplication. All transformations characterized by the special unitary group leave norms unchanged. The {{math|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 {{math|SU(3)}} is a subgroup of group {{math|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 {{math|U(3)}} group is 9. Furthermore, multiplying a U by a phase, {{math|e^iφ}} leaves the norm invariant. Thus {{math|U(3)}} can be decomposed into a direct product {{math|U(1) × SU(3)/Z₃}}. Because of this additional constraint, {{math|SU(3)}} has dimension 8.

Every unitary matrix {{mvar|U}} can be written in the form

: $U=e^\{iH\}\; \backslash ,$

where H is hermitian. The elements of {{math|SU(3)}} can be expressed as

: $U=e^\{i\backslash sum\{a\_k\backslash lambda\_k\}\}$

where $\backslash lambda\_k$ are the 8 linearly independent matrices forming the basis of the Lie algebra of {{math|SU(3)}}, in the triplet representation. The unit determinant condition requires the $\backslash lambda\_k$ matrices to be traceless, since

: $\backslash det(e^A)=e^\{\backslash operatorname\{tr\}\; (A)\}$.

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,

: $$

\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 {{math|SU(3)}} group in the triplet representation, and they are normalized as

: $\backslash operatorname\{tr\}(\backslash lambda\_j\backslash lambda\_k)=2\backslash delta\_\{jk\}\; .$

The Lie algebra structure constants of the group are given by the commutators of $\backslash lambda\_k$

: $[\backslash lambda\_j,\backslash lambda\_k]=2if\_\{jkl\}\backslash lambda\_l\; ~,$

where $f\_\{jkl\}$ are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol $\backslash epsilon\_\{jkl\}$ of {{math|SU(2)}}.

In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric {{mvar|λ}}s. Note $f\_\{ljk\}=\backslash frac\{-i\}\{4\}\backslash mathrm\{tr\}([\backslash lambda\_l\; ,\; \backslash lambda\_j]\; \backslash lambda\_k)$.

Moreover,

: $\backslash \{\backslash lambda\_j,\backslash lambda\_k\backslash \}=\backslash frac\{4\}\{3\}\backslash delta\_\{jk\}\; +2d\_\{jkl\}\backslash lambda\_l$

where $d\_\{jkl\}$ are the completely symmetric coefficient constants. They vanish if the number of indices from the set {{mset|2, 5, 7}} is odd. In terms of the matrices,

: $d\_\{jkl\}\; =\; \backslash frac\{1\}\{4\}\backslash operatorname\{tr\}(\backslash \{\backslash lambda\_j,\backslash lambda\_k\backslash \}\backslash lambda\_l)\; =\; \backslash frac\{1\}\{4\}\backslash operatorname\{tr\}(\backslash \{\backslash lambda\_k,\backslash lambda\_l\backslash \}\backslash lambda\_j)\; =\; d\_\{klj\}\; =\; d\_\{kjl\}$

File:Root_system_A2_with_labels.png of {{math|SU(3)}}. The 6 roots are mutually inclined by {{math|π/3}} to form a hexagonal lattice: {{mvar|α}} corresponds to isospin; {{mvar|β}} to U-spin; and {{math|α + β}} to V-spin.]]

A slightly differently normalized standard basis consists of the F-spin operators, which are defined as $\backslash hat\{F\_i\}=\backslash frac\{1\}\{2\}\backslash lambda\_i$ for the 3, and are utilized to apply to any representation of this algebra.

The Cartan–Weyl basis of the Lie algebra of {{math|SU(3)}} is obtained by another change of basis, where one defines,Introduction to Elementary Particles- David J. Griffiths, {{ISBN|978-3527406012}}, Chapter-1, Page33-38

: $\backslash hat\{I\}\_\{\backslash pm\}=\backslash hat\{F\}\_1\backslash pm\; i\backslash hat\{F\}\_2$

: $\backslash hat\{I\}\_3=\backslash hat\{F\_3\}$

: $\backslash hat\{V\}\_\{\backslash pm\}=\backslash hat\{F\}\_4\backslash pm\; i\backslash hat\{F\}\_5$

: $\backslash hat\{U\}\_\{\backslash pm\}=\backslash hat\{F\}\_6\backslash pm\; i\backslash hat\{F\}\_7$

: $\backslash hat\{Y\}=\backslash frac\{2\}\{\backslash sqrt\{3\}\}\backslash hat\{F\}\_8~.$

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.{{harvnb|Hall|2015}} Section 6.2

The standard form of generators of the {{math|SU(3)}} group satisfies the commutation relations given below,

{{colbegin}}

: $[\backslash hat\{Y\},\backslash hat\{I\}\_3]=0,$

: $[\backslash hat\{Y\},\backslash hat\{I\}\_\backslash pm]=0,$

: $[\backslash hat\{Y\},\backslash hat\{U\}\_\backslash pm]=\backslash pm\; \backslash hat\{U\}\_\backslash pm,$

: $[\backslash hat\{Y\},\backslash hat\{V\}\_\backslash pm]=\backslash pm\; \backslash hat\{V\}\_\backslash pm,$

: $[\backslash hat\{I\}\_3,\backslash hat\{I\}\_\backslash pm]=\backslash pm\; \backslash hat\{I\}\_\backslash pm,$

: $[\backslash hat\{I\}\_3,\backslash hat\{U\}\_\backslash pm]=\backslash mp\backslash frac\{1\}\{2\}\backslash hat\{U\}\_\backslash pm,$

: $[\backslash hat\{I\}\_3,\backslash hat\{V\}\_\backslash pm]=\backslash pm\; \backslash frac\{1\}\{2\}\backslash hat\{V\}\_\backslash pm,$

: $[\backslash hat\{I\}\_+,\backslash hat\{I\}\_-]=\; 2\backslash hat\; I\_3,$

: $[\backslash hat\{U\}\_+,\backslash hat\{U\}\_-]=\; \backslash frac\{3\}\{2\}\backslash hat\{Y\}-\backslash hat\{I\}\_3,$

: $[\backslash hat\{V\}\_+,\backslash hat\{V\}\_-]=\; \backslash frac\{3\}\{2\}\backslash hat\{Y\}+\backslash hat\{I\}\_3,$

: $[\backslash hat\{I\}\_+,\backslash hat\{V\}\_-]=\; -\backslash hat\; U\_-,$

: $[\backslash hat\{I\}\_+,\backslash hat\{U\}\_+]=\; \backslash hat\; V\_+,$

: $[\backslash hat\{U\}\_+,\backslash hat\{V\}\_-]=\; \backslash hat\; I\_-,$

: $[\backslash hat\{I\}\_+,\backslash hat\{V\}\_+]=\; 0,$

: $[\backslash hat\{I\}\_+,\backslash hat\{U\}\_-]=\; 0,$

: $[\backslash hat\{U\}\_+,\backslash hat\{V\}\_+]=\; 0.$

{{colend}}

All other commutation relations follow from hermitian conjugation of these operators.

These commutation relations can be used to construct the irreducible representations of the {{math|SU(3)}} group.

The representations of the group lie in the 2-dimensional {{math|I₃−Y}} plane. Here, $\backslash hat\{I\}\_3$ stands for the z-component of Isospin and $\backslash hat\{Y\}$ 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: {{math|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.

{{Main article|Casimir operator}}

The Casimir operator is an operator that commutes with all the generators of the Lie group. In the case of {{math|SU(2)}}, the quadratic operator {{math|J²}} is the only independent such operator.

In the case of {{math|SU(3)}} group, by contrast, two independent Casimir operators can be constructed, a quadratic and a cubic: they are,{{Cite journal | doi = 10.1016/0029-5582(61)90253-X| title = Group theory of harmonic oscillators (II). The integrals of Motion for the quadrupole-quadrupole interaction| journal = Nuclear Physics| volume = 23| pages = 177–199| year = 1961| last1 = Bargmann | first1 = V.| last2 = Moshinsky | first2 = M.|bibcode = 1961NucPh..23..177B }}

: $\backslash hat\{C\_1\}=\backslash sum\_k\; \backslash hat\{F\_k\}\; \backslash hat\{F\_k\}\; \backslash qquad\; \backslash qquad\; \backslash hat\{C\_2\}=\backslash sum\_\{jkl\}d\_\{jkl\}\; \backslash hat\{F\_j\}\; \backslash hat\{F\_k\}\; \backslash hat\{F\_l\}\; ~.$

These Casimir operators serve to label the irreducible representations of the Lie group algebra {{math|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, {{math|D(1,0)}}, the eigenvalue of {{tmath|\hat{C}_1}} is 4/3, and of {{tmath|\hat{C}_2}}, 10/9.

More generally, from Freudenthal's formula, for generic {{math|D(p,q)}}, the eigenvalueSee eq. 3.65 in {{Cite journal | doi = 10.1103/RevModPhys.38.215| title = Dynamical Symmetry in Particle Physics| journal = Reviews of Modern Physics| volume = 38| issue = 2| pages = 215–255| year = 1966| last1 = Pais | first1 = A.|bibcode = 1966RvMP...38..215P}} of {{tmath|\hat{C}_1}} is

: $(p^2+q^2+3p+3q+pq)/3$.

The eigenvalue ("anomaly coefficient") of {{tmath|\hat{C}_2}} isPais, ibid. (3.66)

: $(p-q)(3+p+2q)(3+q+2p)/18\; .$

It is an odd function under the interchange {{math|p ↔ q}}. Consequently, it vanishes for real representations {{math|1=p = q}}, such as the adjoint, {{math|D(1,1)}}, i.e. both {{tmath|\hat{C}_2}} and anomalies vanish for it.

{{see also|Representation theory of semisimple Lie algebras#The case of sl(3,C)}}

The irreducible representations of SU(3) are analyzed in various places, including Hall's book.{{harvnb|Hall|2015}} Chapter 6 Since the SU(3) group is simply connected,{{harvnb|Hall|2015}} Proposition 13.11 the representations are in one-to-one correspondence with the representations of its Lie algebra{{harvnb|Hall|2015}} Theorem 5.6 su(3), or the complexification{{harvnb|Hall|2015}} Section 3.6 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 the proof of Proposition 6.17 in {{harvnb|Hall|2015}} (See also the section of Young tableaux below: {{mvar|p}} is the number of single-box columns, "quarks", and {{mvar|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

: .

(The elements $H\_1$ and $H\_2$ are linear combinations of the elements $\backslash hat\{I\}\_3$ and $\backslash hat\{Y\}$, but normalized so that the eigenvalues of $H\_1$ and $H\_2$ 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{{harvnb|Hall|2015}} Theorem 6.27 and Example 10.23 File:Baryon-decuplet.svg

: $d(p,q)=\backslash frac\{1\}\{2\}(p+1)(q+1)(p+q+2),$

their irreducible characters are given by{{harvnb|Greiner|Müller|2012|loc=Ch. 10.15}} Note: There is a typo in the final quoting of the result - in Equation 10.121 the first $\backslash varphi$ should instead be a $\backslash psi$.

: $\backslash chi^\{p,q\}(\backslash theta,\; \backslash phi)\; =\; e^\{i\backslash frac\{\backslash theta\}\{3\}\; (2p+q)\}\; \backslash sum\backslash limits\_\{k=0\}^\{p\}\backslash sum\backslash limits\_\{l=0\}^q\; e^\{-i(k+l)\; \backslash theta\}\; \backslash left(\backslash frac\{\backslash sin((k-l+q+1)\backslash phi/2)\}\{\backslash sin\; (\backslash phi/2)\}\; \backslash right),$

and the corresponding Haar measure is

$\backslash mu(SU(3))=64\; \backslash sin\backslash left(\backslash frac\; \backslash phi\; 2\backslash right)^2$

\sin\left(\frac 12 (\theta+\phi/2)\right)^2

\sin\left(\frac 12 (\theta-\phi/2)\right)^2

such that $-2\backslash pi\backslash leq\; \backslash phi\backslash leq\; 2\backslash pi$ and $-3\backslash pi\backslash leq\backslash theta\backslash leq\; 3\backslash pi$,

: $V(SU(3))=\backslash int\_\{-\backslash pi\}^\{\backslash pi\}\backslash !\backslash int\_\{-\backslash pi\}^\backslash pi\; d\backslash !\backslash left(\backslash frac\; \backslash phi\; 2\backslash right)d\backslash !\backslash left(\backslash frac\; \backslash theta\; 3\backslash right)\; \backslash mu(SU(3))=\backslash int\_\{-2\backslash pi\}^\{2\backslash pi\}\backslash frac\; \{d\backslash phi\}\{2\}\backslash int\_\{-3\backslash pi\}^\{3\backslash pi\}\; \backslash frac\; \{d\backslash theta\}\{3\}\; \backslash mu(SU(3))=24\backslash pi^2.$

An {{math|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 {{math|SU(2)}} multiplets, namely the eigenvalues of its quadratic Casimir and of {{mvar|I}}₃.

Since $[\backslash hat\{I\}\_3,\backslash hat\{Y\}]=0$, we can

label different states by the eigenvalues of $\backslash hat\{I\}\_3$ and $\backslash hat\{Y\}$ operators, $|t,y\backslash rangle$, for a given eigenvalue of the isospin Casimir. The action of operators on this states are,[http://www.ks.uiuc.edu/Services/Class/PHYS480/qm_PDF/chp12.pdf Senner & Schulten]

: $\backslash hat\{I\}\_3|t,y\backslash rangle=t|t,y\backslash rangle$File:Representation of SU(3).jpeg

: $\backslash hat\{Y\}|t,y\backslash rangle=y|t,y\backslash rangle$

: $\backslash hat\{U\}\_0|t,y\backslash rangle=\backslash Bigl\; (\backslash frac\{3\}\{4\}y-\backslash frac\{1\}\{2\}t\backslash Bigr)|t,y\backslash rangle$

: $\backslash hat\{V\}\_0|t,y\backslash rangle=\backslash Bigl(\backslash frac\{3\}\{4\}y+\backslash frac\{1\}\{2\}t\backslash Bigr)|t,y\backslash rangle$

: $\backslash hat\{I\}\_\backslash pm|t,y\backslash rangle=\backslash alpha|t\backslash pm1,y\backslash rangle$

: $\backslash hat\{U\}\_\backslash pm|t,y\backslash rangle=\backslash beta|t\backslash mp\backslash frac\{1\}\{2\},y\backslash pm1\backslash rangle$

: $\backslash hat\{V\}\_\backslash pm|t,y\backslash rangle=\backslash gamma|t\backslash pm\backslash frac\{1\}\{2\},y\backslash pm1\backslash rangle$

Here,

: $\backslash hat\{U\}\_0\backslash equiv\; \backslash frac\{1\}\{2\}[\backslash hat\{U\}\_+,\backslash hat\{U\}\_-]=\backslash frac\{3\}\{4\}\backslash hat\{Y\}-\backslash frac\{1\}\{2\}\backslash hat\{I\}\_3$

and

: $\backslash hat\{V\}\_0\backslash equiv\; \backslash frac\{1\}\{2\}[\backslash hat\{V\}\_+,\backslash hat\{V\}\_-]=\backslash frac\{3\}\{4\}\backslash hat\{Y\}+\backslash frac\{1\}\{2\}\backslash hat\{I\}\_3.$

File:A2example.pdf

All the other states of the representation can be constructed by the successive application of the ladder operators $\backslash hat\{I\}\_\backslash pm,\; \backslash hat\{U\}\_\backslash pm$ and $\backslash hat\{V\}\_\backslash pm$ and by identifying the base states which are annihilated by the action of the lowering operators. These operators can be pictured as arrows whose endpoints form the vertices of a hexagon (picture for generators above).

{{see also|Quark model}}

The product representation of two irreducible representations $D(p\_1,q\_1)$ and $D(p\_2,q\_2)$ is generally reducible. Symbolically,

: $D(p\_1,q\_1)\backslash otimes\; D(p\_2,q\_2)=\backslash sum\_\{P,Q\}\backslash oplus\backslash sigma(P,Q)D(P,Q)~,$

where {{tmath|\sigma(P,Q)}} is an integer.

For example, two octets (adjoints) compose to

: $D(1,1)\backslash otimes\; D(1,1)=\; D(2,2)\backslash oplus\; D(3,0)\backslash oplus\; D(1,1)\; \backslash oplus\; D(1,1)\backslash oplus\; D(0,3)\; \backslash oplus\; D(0,0)\; ~,$

that is, their product reduces to an icosaseptet (27), decuplet, two octets, an antidecuplet, and a singlet, 64 states in all.

The right-hand series is called the Clebsch–Gordan series. It implies that the representation {{tmath|D(P,Q)}} appears {{tmath|\sigma(P,Q)}} times in the reduction of this direct product of $D(p\_1,q\_1)$ with $D(p\_2,q\_2)$.

Now a complete set of operators is needed to specify uniquely the states of each irreducible representation inside the one just reduced.

The complete set of commuting operators in the case of the irreducible representation {{tmath|D(P,Q)}} is

: $\backslash \{\backslash hat\{C\}\_1,\; \backslash hat\{C\}\_2,\; \backslash hat\{I\}\_3,\; \backslash hat\{I\}^2,\; \backslash hat\{Y\}\backslash \}~,$

where

: $I^2\backslash equiv\{I\_1\}^2+\{I\_2\}^2+\{I\_3\}^2$.

The states of the above direct product representation are thus completely represented by the set of operators

: $\backslash \{\backslash hat\{C\}\_1(1),\; \backslash hat\{C\}\_2(1),\; \backslash hat\{I\}\_3(1),\; \backslash hat\{I\}^2(1),\; \backslash hat\{Y\}(1),\; \backslash hat\{C\}\_1(2),\; \backslash hat\{C\}\_2(2),\; \backslash hat\{I\}\_3(2),\; \backslash hat\{I\}^2(2),\; \backslash hat\{Y\}(2)\backslash \},$

where the number in the parentheses designates the representation on which the operator acts.

An alternate set of commuting operators can be found for the direct product representation, if one considers the following set of operators,

: $\backslash begin\{align\}$

\hat{\mathbb{C}}_1&=\hat{C}_1(1)+\hat{C}_1(2)\\

\hat{\mathbb{C}}_2&=\hat{C}_2(1)+\hat{C}_2(2)\\

\hat{\mathbb{I}}^2&=\hat{I}^2(1)+\hat{I}^2(2)\\

\hat{\mathbb{Y}} &=\hat{Y}(1)+\hat{Y}(2)\\

\hat{\mathbb{I}}_3&=\hat{I}_3(1)+\hat{I}_3(2).

\end{align}

Thus, the set of commuting operators includes

: $\backslash \{\backslash hat\{\backslash mathbb\{C\}\}\_1,\; \backslash hat\{\backslash mathbb\{C\}\}\_2,\; \backslash hat\{\backslash mathbb\{Y\}\},\; \backslash hat\{\backslash mathbb\{I\}\}\_3,\; \backslash hat\{\backslash mathbb\{I\}\}^2,\; \backslash hat\{C\}\_1(1),\; \backslash hat\{C\}\_1(2),\; \backslash hat\{C\}\_2(1),\; \backslash hat\{C\}\_2(2)\backslash \}.$

This is a set of nine operators only. But the set must contain ten operators to define all the states of the direct product representation uniquely. To find the last operator {{mvar|Γ}}, one must look outside the group. It is necessary to distinguish different {{tmath|D(P,Q)}} for similar values of {{mvar|P}} and {{mvar|Q}}.

: $\backslash begin\{array\}\{cc|cc|cc|cc\}$

\text{Operator} & \text{Eigenvalue} & \text{Operator} & \text{Eigenvalue} & \text{Operator} & \text{Eigenvalue} & \text{Operator} & \text{Eigenvalue}

\\

\hline

\hat{C}_1(1) & {c^1}_1 & \hat{C}_1(2) & {c^1}_2 & \hat{C}_2(1) & {c^2}_1 & \hat{C}_2(2) & {c^2}_2

\\

\hat{\mathbb{I}}^2 & {i^2} & \hat{\mathbb{I}}_3 & {i^z} & \hat{\mathbb{Y}} & {y} & \hat{\Gamma} & \gamma

\\

\hat{\mathbb{C}}_1 & {c^1} & \hat{\mathbb{C}}_2 & {c^1} & \hat{Y}_1 & {y}_1 & \hat{Y}_2 & {y}_2

\\

\hat{I}^2(1) & {i^2}_1 & \hat{I}^2(2) & {i^2}_2 & \hat{I}_3(1) & {i^z}_1 & \hat{I}_3(2) & {i^z}_2

\end{array}

Thus, any state in the direct product representation can be represented by the ket,

: $|\{c^1\}\_1,\; \{c^1\}\_2,\; \{c^2\}\_1,\; \{c^2\}\_2,\; y\_1,\; y\_2,\; \{i^2\}\_1,\; \{i^2\}\_2,\; \{i^z\}\_1,\; \{i^z\}\_2\backslash rangle$

also using the second complete set of commuting operator, we can define the states in the direct product representation as

: $|\{c^1\}\_1,\; \{c^1\}\_2,\; \{c^2\}\_1,\; \{c^2\}\_2,\; y,\; \backslash gamma,\; \{i^2\},\; \{i^z\},\; c^1,\; c^2\backslash rangle$

We can drop the $\{c^1\}\_1,\; \{c^1\}\_2,\; \{c^2\}\_1,\; \{c^2\}\_2$ from the state and label the states as

: $|y\_1,\; y\_2,\; \{i^2\}\_1,\; \{i^2\}\_2,\; \{i^z\}\_1,\; \{i^z\}\_2\backslash rangle$

using the operators from the first set, and,

: $|y,\; \backslash gamma,\; \{i^2\},\; \{i^z\},\; c^1,\; c^2\backslash rangle\; ~,$

using the operators from the second set.

Both these states span the direct product representation and any states in the representation can be labeled by suitable choice of the eigenvalues.

Using the completeness relation,

{{Equation box 1

|indent =:

|equation = $|y,\; \backslash gamma,\; \{i^2\},\; \{i^z\},\; c^1,\; c^2\backslash rangle\; =\backslash sum\_\{y\_1,\; y\_2\}\backslash sum\_\{\{i^2\}\_1,\; \{i^2\}\_2\}\backslash sum\_\{\{i^z\}\_1,\; \{i^z\}\_2\}\backslash langle\; y\_1,\; y\_2,\; \{i^2\}\_1,\; \{i^2\}\_2,\; \{i^z\}\_1,\; \{i^z\}\_2|y,\; \backslash gamma,\; \{i^2\},\; \{i^z\},\; c^1,\; c^2\backslash rangle\; |y\_1,\; y\_2,\; \{i^2\}\_1,\; \{i^2\}\_2,\; \{i^z\}\_1,\; \{i^z\}\_2\backslash rangle$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4}}

Here, the coefficients

{{Equation box 1

|indent =:

|equation = $\backslash langle\; y\_1,\; y\_2,\; \{i^2\}\_1,\; \{i^2\}\_2,\; \{i^z\}\_1,\; \{i^z\}\_2|y,\; \backslash gamma,\; \{i^2\},\; \{i^z\},\; c^1,\; c^2\backslash rangle$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4

}}

are the Clebsch–Gordan coefficients.

To avoid confusion, the eigenvalues $c^1,\; c^2$ can be simultaneously denoted by {{mvar|μ}} and the eigenvalues $i^2,\; i^z,\; y$ are simultaneously denoted by {{mvar|ν}}. Then the eigenstate of the direct product representation {{tmath|D(P,Q)}} can be denoted by{{Cite journal | doi = 10.1103/RevModPhys.35.916| title = The Octet Model and its Clebsch-Gordan Coefficients| journal = Reviews of Modern Physics| volume = 35| issue = 4| pages = 916–939| year = 1963| last1 = De Swart | first1 = J. J.|bibcode = 1963RvMP...35..916D | url =https://cds.cern.ch/record/345262/files/RevModPhys.35.916.pdf}} {{erratum|{{cite journal|doi=10.1103/RevModPhys.37.326|title=none|journal=Reviews of Modern Physics|volume=37|issue=2|pages=326|year=1965|last1=De Swart|first1=J. J.|bibcode=1965RvMP...37..326D|doi-access=free}}|checked=yes}}

: $\backslash psi$

\begin{pmatrix}

\mu_1 & \mu_2 & \gamma\\

& & \nu

\end{pmatrix},

where $\backslash mu\_1$ is the eigenvalues of $\{c^1\}\_1,\; \{c^2\}\_1$ and $\backslash mu\_2$ is the eigenvalues of $\{c^1\}\_2,\; \{c^2\}\_2$ denoted simultaneously. Here, the quantity expressed by the parenthesis is the Wigner 3-j symbol.

Furthermore, $\{\backslash phi^\{\backslash mu\_1\}\}\_\{\backslash nu\_1\}$ are considered to be the basis states of $D(p\_1,q\_1)$ and $\{\backslash phi^\{\backslash mu\_2\}\}\_\{\backslash nu\_2\}$ are the basis states of $D(p\_2,q\_2)$. Also $\{\backslash phi^\{\backslash mu\_1\}\}\_\{\backslash nu\_1\},\{\backslash phi^\{\backslash mu\_2\}\}\_\{\backslash nu\_2\}$ are the basis states of the product representation. Here $\backslash nu\_1,\; \backslash nu\_2$ represents the combined eigenvalues $(\{i^2\}\_1,\; \{i^z\}\_1,\; y\_1)$ and $(\{i^2\}\_2,\; \{i\; ^z\}\_2,\; y\_2)$ respectively.

Thus the unitary transformations that connects the two bases are

: $\backslash psi$

\begin{pmatrix}

\mu_1 & \mu_2 & \gamma\\

& & \nu

\end{pmatrix}

=\sum_{\nu_1,\nu_2}\begin{pmatrix}

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

{\phi^{\mu_1}}_{\nu_1}{\phi^{\mu_2}}_{\nu_2}

This is a comparatively compact notation. Here,

: $\backslash begin\{pmatrix\}$

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

are the Clebsch–Gordan coefficients.

The Clebsch–Gordan coefficients form a real orthogonal matrix. Therefore,

: $\{\backslash phi^\{\backslash mu\_1\}\}\_\{\backslash nu\_1\}\{\backslash phi^\{\backslash mu\_2\}\}\_\{\backslash nu\_2\}=\backslash sum\_\{\backslash mu,\backslash nu,\backslash gamma\}\backslash begin\{pmatrix\}$

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

\psi

\begin{pmatrix}

\mu_1 & \mu_2 & \gamma\\

& & \nu

\end{pmatrix}.

Also, they follow the following orthogonality relations,

: $\backslash sum\_\{\backslash nu\_1,\backslash nu\_2\}\backslash begin\{pmatrix\}$

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

\begin{pmatrix}

\mu_1 & \mu_2 & \gamma'\\

\nu_1 & \nu_2 & \nu'

\end{pmatrix}=\delta_{\nu \nu'}\delta_{\gamma, \gamma'}

: $\backslash sum\_\{\backslash mu\; \backslash nu\; \backslash gamma\}\backslash begin\{pmatrix\}$

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

\begin{pmatrix}

\mu_1 & \mu_2 & \gamma\\

\nu_1' & \nu_2' & \nu

\end{pmatrix}=\delta_{\nu_1 \nu_1'}\delta_{\nu_2, \nu_2'}

If an irreducible representation $\{\backslash scriptstyle\{\backslash mu\}\_\{\backslash gamma\}\}$ appears in the Clebsch–Gordan series of $\{\backslash scriptstyle\{\backslash mu\}\_1\backslash otimes\{\backslash mu\}\_2\}$, then it must appear in the Clebsch–Gordan series of $\{\backslash scriptstyle\{\backslash mu\}\_2\backslash otimes\{\backslash mu\}\_1\}$. Which implies,

: $\backslash begin\{pmatrix\}$

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

=\xi_1\begin{pmatrix}

\mu_2 & \mu_1 & \gamma\\

\nu_2 & \nu_1 & \nu

\end{pmatrix}

Where $\backslash xi\_1=\backslash xi\_1(\backslash mu\_1,\backslash mu\_2,\backslash gamma)=\backslash pm1$

Since the Clebsch–Gordan coefficients are all real, the following symmetry property can be deduced,

: $\backslash begin\{pmatrix\}$

\mu_1 & \mu_2 & \gamma\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}=\xi_2\begin{pmatrix}

{\mu_1}^* & {\mu_2}^* & {\gamma}^*\\

\nu_1 & \nu_2 & \nu

\end{pmatrix}

Where $\backslash xi\_2=\backslash xi\_2(\backslash mu\_1,\; \backslash mu\_2,\; \backslash gamma)=\backslash pm\; 1$.

A three-dimensional harmonic oscillator is described by the Hamiltonian

: $\backslash hat\{H\}=-\backslash tfrac\{1\}\{2\}\backslash nabla^2+\backslash tfrac\{1\}\{2\}(x^2+y^2+z^2),$

where the spring constant, the mass and the Planck constant have been absorbed into the definition of the variables, {{math|1=ħ = m =1}}.

It is seen that this Hamiltonian is symmetric under coordinate transformations that preserve the value of $V=x^2+y^2+z^2$. Thus, any operators in the group {{math|SO(3)}} keep this Hamiltonian invariant.

More significantly, since the Hamiltonian is Hermitian, it further remains invariant under operation by elements of the much larger {{math|SU(3)}} group.

{{Math proof|drop=hidden|title=Proof that the symmetry group of linear isotropic 3D harmonic oscillator is {{math|SU(3)}}{{cite journal |doi=10.1119/1.1971373|title=Three-Dimensional Isotropic Harmonic Oscillator and SU3|journal=American Journal of Physics|volume=33|issue=3|pages=207–211|year=1965|last1=Fradkin|first1=D. M.|bibcode=1965AmJPh..33..207F}}

|proof=

A symmetric (dyadic) tensor operator analogous to the Laplace–Runge–Lenz vector for the Kepler problem may be defined,

: $\backslash hat\{A\}\_\{ij\}=\backslash frac\{1\}\{2\}p\_ip\_j+\backslash omega^2r\_ir\_j$

which commutes with the Hamiltonian,

: $[\backslash hat\{A\}\_\{ij\},\backslash hat\{H\}]=0.$

Since it commutes with the Hamiltonian (its trace), it represents 6−1=5 constants of motion.

It has the following properties,

: $\backslash sum\_j\backslash hat\{A\}\_\{ij\}\backslash hat\{L\}\_j=\backslash sum\_i\backslash hat\{L\}\_\{i\}\backslash hat\{A\}\_\{ij\}=0$

: $\backslash sum\_j\backslash hat\{A\}\_\{ij\}\backslash hat\{A\}\_\{jk\}=\backslash hat\{H\}\backslash hat\{A\}\_\{ik\}+\backslash frac\{1\}\{4\}\backslash omega^2\backslash \{\backslash hat\{L\}\_i\backslash hat\{L\}\_k-\backslash delta\_\{ik\}\backslash hat\{L\}^2+2[\backslash hat\{L\}\_i,\backslash hat\{L\}\_k]-2\backslash hbar^2\backslash delta\_\{ik\}\backslash \}$

: $Tr[\backslash hat\{A\}\_\{ij\}]=\backslash sum\_i\{\backslash hat\{A\}\_\{ii\}\}=\backslash hat\{H\}.$

Apart from the tensorial trace of the operator$\backslash hat\{A\}\_\{ij\},$, which is the Hamiltonian, the remaining 5 operators can be rearranged into their spherical component form as

: $\backslash hat\{A\}\_0=\backslash omega^\{-1\}(2\backslash hat\{A\}\_\{33\}-\backslash hat\{A\}\_\{11\}-\backslash hat\{A\}\_\{22\})$

: $\backslash hat\{A\}\_\{\backslash pm\}=\backslash mp\backslash omega^\{-1\}(\backslash hat\{A\}\_\{13\}\backslash pm\; i\backslash hat\{A\}\_\{23\})$

: $\backslash hat\{A\text{'}\}\_\{\backslash pm\}=\backslash omega^\{-1\}(\backslash hat\{A\}\_\{11\}-\backslash hat\{A\}\_\{22\}\backslash pm\; 2i\backslash hat\{A\}\_\{22\})$

Further, the angular momentum operators are written in spherical component form as

: $\backslash hat\{L\}\_3=\backslash hat\{L\}\_3$

: $\backslash hat\{L\}\_\{\backslash pm\}=(\backslash hat\{L\}\_1\backslash pm\; i\backslash hat\{L\}\_2).$

They obey the following commutation relations,

: $[\backslash hat\{L\}\_\{3\},\backslash hat\{A\}\_\{0\}]=[\backslash hat\{A\}\_\{0\},\backslash hat\{A\text{'}\}\_\{\backslash pm\}]=[\backslash hat\{A\}\_\{\backslash pm\},\backslash hat\{A\text{'}\}\_\{\backslash pm\}]=[\backslash hat\{L\}\_\{\backslash pm\},\backslash hat\{A\text{'}\}\_\{\backslash pm\}]\; =0$

: $[\backslash hat\{L\}\_\{\backslash pm\},\backslash hat\{L\}\_\{\backslash mp\}]=-4[\backslash hat\{A\}\_\{\backslash pm\},\backslash hat\{A\}\_\{\backslash mp\}]=\backslash frac\{1\}\{2\}[\backslash hat\{A\text{'}\}\_\{\backslash pm\},\backslash hat\{A\text{'}\}\_\{\backslash mp\}]=\backslash pm2\backslash hbar\backslash hat\{L\}\_3$

: $[\backslash hat\{L\}\_\{\backslash pm\},\backslash hat\{A\}\_\{\backslash mp\}]=\backslash hbar\backslash hat\{A\}\_0$

: $\backslash pm[\backslash hat\{L\}\_\{3\},\backslash hat\{L\}\_\{\backslash pm\}]=-\backslash frac\{2\}\{3\}[\backslash hat\{A\}\_\{0\},\backslash hat\{A\}\_\{\backslash pm\}]=[\backslash hat\{A\}\_\{\backslash mp\},\backslash hat\{A\text{'}\}\_\{\backslash pm\}]=\backslash hbar\backslash hat\{L\}\_\{\backslash pm\}$

: $\backslash pm[\backslash hat\{L\}\_\{3\},\backslash hat\{A\}\_\{\backslash pm\}]=-\backslash frac\{1\}\{6\}[\backslash hat\{A\}\_\{0\},\backslash hat\{L\}\_\{\backslash pm\}]=\backslash frac\{1\}\{4\}[\backslash hat\{L\}\_\{\backslash mp\},\backslash hat\{A\text{'}\}\_\{\backslash pm\}]=\backslash hbar\backslash hat\{A\}\_\{\backslash pm\}$

: $\backslash pm[\backslash hat\{L\}\_\{3\},\backslash hat\{A\text{'}\}\_\{\backslash pm\}]=2[\backslash hat\{L\}\_\{\backslash pm\},\backslash hat\{A\}\_\{\backslash pm\}]=2\backslash hbar\backslash hat\{A\text{'}\}\_\{\backslash pm\}.$

The eight operators (consisting of the 5 operators derived from the traceless symmetric tensor operator {{mvar|Â_ij}} and the three independent components of the angular momentum vector) obey the same commutation relations as the infinitesimal generators of the {{math|SU(3)}} group, detailed above.

As such, the symmetry group of Hamiltonian for a linear isotropic 3D Harmonic oscillator is isomorphic to {{math|SU(3)}} group.

}}

More systematically, operators such as the Ladder operators

: $\backslash sqrt\{2\}\backslash hat\{a\}\_\{i\}=\backslash hat\{X\}\_\{i\}+i\backslash hat\{P\}\_\{i\}~~$ and $~~\backslash sqrt\{2\}\backslash hat\{a\}\_\{i\}^\backslash dagger=\backslash hat\{X\}\_\{i\}-i\backslash hat\{P\}\_\{i\}$

can be constructed which raise and lower the eigenvalue of the Hamiltonian operator by 1.

The operators $\backslash hat\; \{a\}\_\{i\}\backslash neq\; \backslash hat\; \{a\}\_\{i\}^\backslash dagger$ are not hermitian, but hermitian operators can be constructed from combinations of $\backslash hat\{a\}\_i\backslash hat\{a\}\_\{j\}^\backslash dagger$,

: $\backslash hat\{a\}\_i\backslash hat\{a\}\_\{j\}^\backslash dagger\; +\; \backslash hat\{a\}\_j\backslash hat\{a\}\_\{i\}^\backslash dagger~~$ and $~~\; i\backslash left(\backslash hat\{a\}\_i\backslash hat\{a\}\_\{j\}^\backslash dagger\; -\; \backslash hat\{a\}\_j\backslash hat\{a\}\_\{i\}^\backslash dagger\backslash right)$.

There are nine such operators for $i,\backslash ,j=1,\backslash ,2,\backslash ,3$.

The nine hermitian operators formed by the bilinear forms {{math|â_i^†â_j}} are controlled by the fundamental commutators

: $[\backslash hat\{a\}\_i,\backslash hat\{a\}\_\{j\}^\backslash dagger]=\backslash delta\_\{ij\},$

: $[\backslash hat\{a\}\_i,\backslash hat\{a\}\_j]=[\backslash hat\{a\}\_\{i\}^\backslash dagger,\backslash hat\{a\}\_\{j\}^\backslash dagger]=0,$

and seen to not commute among themselves. As a result, this complete set of operators don't share their eigenvectors in common, and they cannot be diagonalized simultaneously. The group is thus non-Abelian and degeneracies may be present in the Hamiltonian, as indicated.

The Hamiltonian of the 3D isotropic harmonic oscillator, when written in terms of the operator $\backslash hat\{N\_i\}=\backslash hat\{a\}\_\{i\}^\backslash dagger\; \backslash hat\{a\}\_\{i\}$ amounts to

: $\backslash hat\{H\}\; =\; \backslash omega\backslash bigl[\; \backslash tfrac\{3\}\{2\}\; +\; \backslash hat\{N\}\_\{1\}\; +\; \backslash hat\{N\}\_2\; +\; \backslash hat\{N\}\_\{3\}\; \backslash bigr]$.

The Hamiltonian has 8-fold degeneracy. A successive application of {{mvar|â_i}} and {{mvar|â_j}}^† on the left preserves the Hamiltonian invariant, since it increases {{mvar|N_i}} by 1 and decrease {{mvar|N_j}} by 1, thereby keeping the total

: $N=\backslash sum\{N\_i\}$   constant. (cf. quantum harmonic oscillator)

{{see also|Jordan map}}

Since the operators belonging to the symmetry group of Hamiltonian do not always form an Abelian group, a common eigenbasis cannot be found that diagonalizes all of them simultaneously. Instead, we take the maximally commuting set of operators from the symmetry group of the Hamiltonian, and try to reduce the matrix representations of the group into irreducible representations.

{{see also|Complete set of commuting observables}}

The Hilbert space of two particles is the tensor product of the two Hilbert spaces of the two individual particles,

: $\backslash mathbb\{H\}=\backslash mathbb\{H\}\_1\backslash otimes\; \backslash mathbb\{I\}+\backslash mathbb\{I\}\backslash otimes\backslash mathbb\{H\}\_2~,$

where $\backslash mathbb\{H\}\_1$ and $\backslash mathbb\{H\}\_2$ are the Hilbert space of the first and second particles, respectively.

The operators in each of the Hilbert spaces have their own commutation relations, and an operator of one Hilbert space commutes with an operator from the other Hilbert space. Thus the symmetry group of the two particle Hamiltonian operator is the superset of the symmetry groups of the Hamiltonian operators of individual particles. If the individual Hilbert spaces are {{mvar|N}}-dimensional, the combined Hilbert space is {{math|{{var|N}}²}}-dimensional.

The symmetry group of the Hamiltonian is {{math|SU(3)}}. As a result, the Clebsch–Gordan coefficients can be found by expanding the uncoupled basis vectors of the symmetry group of the Hamiltonian into its coupled basis. The Clebsch–Gordan series is obtained by block-diagonalizing the Hamiltonian through the unitary transformation constructed from the eigenstates which diagonalizes the maximal set of commuting operators.

{{Main article|Young tableau}}

A Young tableau (plural tableaux) is a method for decomposing products of an SU(N) group representation into a sum of irreducible representations. It provides the dimension and symmetry types of the irreducible representations, which is known as the Clebsch–Gordan series. Each irreducible representation corresponds to a single-particle state and a product of more than one irreducible representation indicates a multiparticle state.

Since the particles are mostly indistinguishable in quantum mechanics, this approximately relates to several permutable particles. The permutations of {{mvar|n}} identical particles constitute the symmetric group S_{{mvar|n}}. Every {{mvar|n}}-particle state of S_{{mvar|n}} that is made up of single-particle states of the fundamental {{mvar|N}}-dimensional SU(N) multiplet belongs to an irreducible SU(N) representation. Thus, it can be used to determine the Clebsch–Gordan series for any unitary group.{{cite book |first1=George B. |last1=Arfken |first2=Hans J. |last2=Weber |chapter=4. Group Theory |title=Mathematical Methods For Physicists International Student Edition |chapter-url=https://books.google.com/books?id=tNtijk2iBSMC&pg=PA241 |date=2005 |edition=6th |publisher=Elsevier |isbn=978-0-08-047069-6 |pages=241–320}}

Any two particle wavefunction $\backslash psi\_\{1,2\}$, where the indices 1,2 represents the state of particle 1 and 2, can be used to generate states of explicit symmetry using the symmetrizing and the anti-symmetrizing operators.http://hepwww.rl.ac.uk/Haywood/Group_Theory_Lectures/Lecture_4.pdf {{Bare URL PDF|date=March 2022}}

: $\backslash mathbf\{S\_\{12\}\}=\backslash mathbf\{I\}+\backslash mathbf\{P\_\{12\}\}$

: $\backslash mathbf\{A\_\{12\}\}=\backslash mathbf\{I\}-\backslash mathbf\{P\_\{12\}\}$

where the $\backslash mathbf\{P\_\{12\}\}$ are the operator that interchanges the particles (Exchange operator).

The following relation follows:-

: $\backslash mathbf\{P\_\{12\}P\_\{12\}\}=\backslash mathbf\{I\}$

: $\backslash mathbf\{P\_\{12\}S\_\{12\}\}=\backslash mathbf\{P\_\{12\}\}+\backslash mathbf\{I\}=\backslash mathbf\{S\_\{12\}\}$

: $\backslash mathbf\{P\_\{12\}A\_\{12\}\}=\backslash mathbf\{P\_\{12\}\}-\backslash mathbf\{I\}=-\backslash mathbf\{A\_\{12\}\}$

: $\backslash mathbf\{P\_\{12\}\}\backslash mathbf\{S\_\{12\}\}\backslash psi\_\{12\}=+\backslash mathbf\{S\_\{12\}\}\backslash psi\_\{12\}$

thus,

: $\backslash mathbf\{P\_\{12\}\}\backslash mathbf\{A\_\{12\}\}\backslash psi\_\{12\}=-\backslash mathbf\{A\_\{12\}\}\backslash psi\_\{12\}.$

Starting from a multiparticle state, we can apply $\backslash mathbf\{S\_\{12\}\}$ and $\backslash mathbf\{A\_\{12\}\}$ repeatedly to construct states that are:

1. Symmetric with respect to all particles.
2. Antisymmetric with respect to all particles.
3. Mixed symmetries, i.e. symmetric or antisymmetric with respect to some particles.

Instead of using ψ, in Young tableaux, we use square boxes (□) to denote particles and i to denote the state of the particles. File:Young tableaux 1.jpeg

The complete set of $n\_p$ particles are denoted by arrangements of $n\_p$ □s, each with its own quantum number label (i).

The tableaux is formed by stacking boxes side by side and up-down such that the states symmetrised with respect to all particles are given in a row and the states anti-symmetrised with respect to all particles lies in a single column. Following rules are followed while constructing the tableaux:

1. A row must not be longer than the one before it.
2. The quantum labels (numbers in the □) should not decrease while going left to right in a row.
3. The quantum labels must strictly increase while going down in a column.

For N = 3 that is in the case of SU(3), the following situation arises. In SU(3) there are three labels, they are generally designated by (u,d,s) corresponding to up, down and strange quarks which follows the SU(3) algebra. They can also be designated generically as (1,2,3). For a two-particle system, we have the following six symmetry states:

:

\hline

1 & 1 \\

\hline

\end{array}

\atop

uu

}

|| || $$

{

\begin{array}

\hline

1 & 2 \\

\hline

\end{array}

\atop

\frac 1 \sqrt 2 (ud+du)

}

|| || $$

{

\begin{array}

\hline

1 & 3 \\

\hline

\end{array}

\atop

\frac 1 \sqrt 2 (us+su)

}

|| || $$

{

\begin{array}

\hline

2 & 2 \\

\hline

\end{array}

\atop

dd

}

|| || $$

{

\begin{array}

\hline

2 & 3 \\

\hline

\end{array}

\atop

\frac 1 \sqrt 2 (ds+sd)

}

|| || $$

{

\begin{array}

\hline

3 & 3 \\

\hline

\end{array}

\atop

ss

}

|}

and the following three antisymmetric states:

: $$

{

\begin{array}

\hline

1 \\

\hline

2 \\

\hline

\end{array}

\atop

\frac 1 \sqrt 2 (ud-du)

}

\qquad

{

\begin{array}

\hline

1 \\

\hline

3 \\

\hline

\end{array}

\atop

\frac 1 \sqrt 2 (us-su)

}

\qquad

{

\begin{array}

\hline

2 \\

\hline

3 \\

\hline

\end{array}

\atop

\frac 1 \sqrt 2 (ds-sd)

}

The 1-column, 3-row tableau is the singlet, and so all tableaux of nontrivial irreps of SU(3) cannot have more than two rows. The representation {{math|D(p,q)}} has

{{math|p+q}} boxes on the top row and {{mvar|q}} boxes on the second row.

Clebsch–Gordan series is the expansion of the tensor product of two irreducible representation into direct sum of irreducible representations.

$D(p\_1,q\_1)\backslash otimes\; D(p\_2,q\_2)=\backslash sum\_\{P,Q\}\backslash oplus\; D(P,Q)$. This can be easily found out from the Young tableaux.

:

The tensor product of a triplet with an octet reducing to a deciquintuplet (15), a sextet, and a triplet

: $D(1,0)\backslash otimes\; D(1,1)=\; D(2,1)\backslash oplus\; D(0,2)\; \backslash oplus\; D(1,0)$

appears diagrammatically as-

center

center

a total of 24 states.

Using the same procedure, any direct product representation is easily reduced.

{{See also|Hook length formula}}

• Wigner D-matrix
• Tensor operator
• Wigner–Eckart theorem
• Representation theory
• Racah W-coefficient
• Gell-Mann–Okubo mass formula

{{reflist}}

• {{cite book|last=Lichtenberg|first=D.B.| title=Unitary Symmetry and Elementary Particles|year=2012|edition=2nd|isbn=978-0123941992|publisher= Academic Press}}
• {{cite book|last1=Greiner|first=W.|last2=Müller|first2=B.|author-link1=Walter Greiner|title=Quantum Mechanics: Symmetries|year=2012|edition=2nd|isbn=978-3540580805|publisher=Springer|url-access=registration|url=https://archive.org/details/quantummechanics0001grei}}
• {{citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
• {{cite journal | doi = 10.1103/RevModPhys.36.1005| title = Tables of Clebsch–Gordan Coefficients of SU3| journal = Reviews of Modern Physics| volume = 36| issue = 4| pages = 1005| year = 1964| last1 = McNamee | first1 = P.| last2 = j. | first2 = S.| last3 = Chilton | first3 = F. |bibcode = 1964RvMP...36.1005M }}
• {{cite journal|title=Irreducible representations of the SU3 group|last=Mandel'tsveig|first=V. B.|year=1965|journal=Sov Phys JETP|volume=20|number=5|pages=1237–1243}} [http://jetp.ac.ru/cgi-bin/dn/e_020_05_1237.pdf online] {{Webarchive|url=https://web.archive.org/web/20160311171345/http://jetp.ac.ru/cgi-bin/dn/e_020_05_1237.pdf |date=2016-03-11 }}
• {{cite web |last1=Coleman |first1=Sidney|year=1965|publisher=IAEA |title=Fun with SU(3) |url=https://inspirehep.net/literature/1720680 |website=INSPIREHep}}
• {{cite journal |doi=10.1088/0305-4470/19/1/007|title=Clebsch-Gordan coefficients of SU(3) with simple symmetry properties|journal=Journal of Physics A: Mathematical and General|volume=19|issue=1|pages=21–28|year=1986|last1=Pluhar|first1=Z.|last2=Smirnov|first2=Yu F.|last3=Tolstoy|first3=V. N.|bibcode=1986JPhA...19...21P}}

{{DEFAULTSORT:Clebsch-Gordan coefficient for SU(3)}}

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