Group Theory 3.8 Direct Product Representations, Clebsch-Gordan Coefficients

Direct Product Vector Space: Let U and V be inner product spaces and \{\mathbf{u}_i; i=1,\dots, n_u\} and \{\mathbf{v}_j; j=1,\dots, n_v\} be orthonormal basis in the two spaces respectively. Then the direct product space W=U\times V consists of all linear combinations of the orthonormal basis vector \{\mathbf{w}_k; k=(i, j); i=1,\dots, n_u; j=1,\dots, n_v\}, which can be regarded as \mathbf{w}_k=\mathbf{u}_i\times \mathbf{v}_j, such that
(i) Inner product: \langle w_{k'}|w_k\rangle=\delta_{k'k};
(ii) W=\{\mathbf{x}; |x\rangle=|w_k\rangle x_k\} where the complex number x_k are the components of x;
(iii) \langle x|y\rangle=x^{\ast}_k y_k.

Direct Product Representation: Let G be a symmetry group of a physical system, and W be the direct product space of physical solutions consisting of two sets of degrees of freedom U, V. Suppose D^{\mu}(G) and D^{\nu}(G) are the representations of G on U and V respectively. The the operators D^{\mu \times \nu}(G)=D^{\mu}(G)\times D^{\nu}(G) on W forms the direct product representation of D^{\mu}(G) (on U) and D^{\nu}(G) (on V).

The group characters of the direct product representation D^{\mu\times\nu} are equal to the product of the characters of the two representation D^{\mu} and D^{\nu}: i.e.

(1)   \begin{equation*} \chi^{\mu\times\nu} = \chi^{\mu}\chi^{\nu} \end{equation*}

Clebsch-Gordan Coefficients: The vector space W can be decomposed into a direct sum of invariant subspaces with basis \{|k\rangle\}, which are related to basis \{|i,j\rangle\} with a unitary transformation,

(2)   \begin{equation*} |k\rangle = \sum_{i,j}|i,j\rangle\langle i,j|k\rangle \end{equation*}

the matrix elements \langle i,j|k\rangle are called Clebsch-Gordan Coefficients.

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