nLab Clebsch-Gordan coefficient

Idea

Given a tensor product $\rho_{j_1} \otimes \rho_{j_2}$ of two representations of some Lie group $G$ (by default often the special orthogonal group $SO(3)$) and given its decomposition into a direct sum of irreducible representations $\{\rho_{t_{tot}}\}$ by an isomorphism

$$\rho_{j_1} \otimes \rho_{j_2} \stackrel{\simeq}{\longrightarrow} \underset{j_{tot}}{\oplus} C_{j_1 j_2}{}^{j_{tot}} \rho_{j_{tot}}$$

then the matrix elements of this linear map in some standard basis are called – in the physics literature – the Clebsch-Gordan coefficients or equivalently (up to a constant) the Wigner 3j symbols .

Specifically for $G = SO(3)$ the rotation group in 3-dimensional Cartesian space, then the standard basis elements of the representation $\rho_{j}$ of total angular momentum $j \in \mathbb{N}$ are traditionally denoted

$$|j,m\rangle \in \rho_{j} \;\;\,,\;\; -j \leq m \leq j$$

and their inner product is traditionally denoted by

$$\langle j_1, m_1 | j_2, m_2\rangle \in \mathbb{C} \,.$$

In these basis elements that above matrix then has components given by

$$\langle (j_1, m_1)\otimes(j_2,m_2) | j_{tot}, m_{tot}\rangle \in \mathbb{C} \,.$$

These expressions are specifically the Clebsch-Gordan coefficients as they appear in the physics literature.

Related concepts

• Wigner 6-j symbol

• Wigner 9-j symbol

• Fierz identity

• fusion ring

References

Lecture notes include for instance

• M. Tuckerman, Quantum mechanics and dynamics – Addition of angular momenta – The general problem)