nLab
Clebsch-Gordan coefficient
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Physics
Contents
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.

References
Lecture notes include for instance

M. Tuckerman, Quantum mechanics and dynamics – Addition of angular momenta – The general problem )
Last revised on January 6, 2017 at 11:24:05.
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