Clebsch–Gordan coefficients for SU(3)

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.

All transformations characterized by the special unitary group leave norms unchanged.

The unitarity condition imposes nine constraint relations on the total 18 degrees of freedom of a 3×3 complex matrix.

are the 8 linearly independent matrices forming the basis of the Lie algebra of SU(3), in the triplet representation.

matrices to be traceless, since An explicit basis in the fundamental, 3, representation can be constructed in analogy to the Pauli matrix algebra of the spin operators.

are the structure constants completely antisymmetric and are analogous to the Levi-Civita symbol

In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric λs.

In terms of the matrices, A slightly differently normalized standard basis consists of the F-spin operators, which are defined as

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

is the Hypercharge, and they comprise the (abelian) Cartan subalgebra of the full Lie algebra.

The remaining 6 generators, the ± ladder operators, correspond to the 6 roots arranged on the 2-dimensional hexagonal lattice of the figure.

In the case of SU(3) group, by contrast, two independent Casimir operators can be constructed, a quadratic and a cubic: they are,[4] 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.

⁠ is[6] 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 ⁠

The irreducible representations of SU(3) are analyzed in various places, including Hall's book.

The representations have dimension[12] their irreducible characters are given by[13] and the corresponding Haar measure is[13]

This generalizes the mere two labels for SU(2) multiplets, namely the eigenvalues of its quadratic Casimir and of I3.

and by identifying the base states which are annihilated by the action of the lowering operators.

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 ⁠

But the set must contain ten operators to define all the states of the direct product representation uniquely.

Thus, any state in the direct product representation can be represented by the ket, also using the second complete set of commuting operator, we can define the states in the direct product representation as We can drop the

Thus the unitary transformations that connects the two bases are This is a comparatively compact notation.

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

A symmetric (dyadic) tensor operator analogous to the Laplace–Runge–Lenz vector for the Kepler problem may be defined, which commutes with the Hamiltonian, Since it commutes with the Hamiltonian (its trace), it represents 6−1=5 constants of motion.

As a result, this complete set of operators don't share their eigenvectors in common, and they cannot be diagonalized simultaneously.

The Hamiltonian of the 3D isotropic harmonic oscillator, when written in terms of the operator

A successive application of âi and âj† on the left preserves the Hamiltonian invariant, since it increases Ni by 1 and decrease Nj by 1, thereby keeping the total 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.

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.

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.