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I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps an example derivation of the well known expression of spherical harmonics would be helpful)? I know it has something to do with being able to compute the multiplicities of the tensor product of the irreducible representations, but I don't know how one would do that. |
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The Clebsch-Gordan coefficients appear in the representation theory of the [Lie] group of rotations $SO(3)$ [and its fundamental cover $SU(2)$]. When expressing the tensor product of two irreducible representations of this group [itself being a reducible representation] as a direct sum of irreducible representations, the normalized coefficients of the expansion are the Clebsch-Gordan coefficients. They express the multiplicity of each irreducible representation in the decomposition. The Clebsch-Gordan coefficients are themselves orthonormal, with orthonormality relation $\sum_{|m_1|\leq j_1,|m_2| \leq j_2} C(j_1,j_2,m_1,m_2|j,m) C(j_1,j_2,m_1,m_2|j',m')= \delta_{j,j'} \delta_{m,m'}$ $\sum_{j=|j_1-j_2|}^{j_1+j_2} \sum_{|m|\leq j} C(j_1,j_2,m_1,m_2|j,m) C(j_1,j_2,m_1',m_2'|j,m)= \delta_{m_1,m_1'} \delta_{m_2,m_2'}$ and as exposed above appear when decomposing reducible representations into sums of irreducible representation. In terms of angular momentum states $|j_1,m_1 \rangle \bigotimes|j_2,m_2\rangle=\sum_{j,m} C(j_1,j_2,m_1,m_2|j,m)|j,m\rangle$ where $C(j_1,j_2,m_1,m_2|j,m)=\langle j,m|j_1,j_2,m_1,m_2\rangle$ The Clebsch-Gordan coefficients appears also, in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. The derivation of the formula is a bit cumbersome and the result looks like this $Y_{l_1}^{m_1}(\theta,\varphi)Y_{l_2}^{m_2}(\theta,\varphi)=\sum_{l,m} \ \sqrt{\dfrac{(2l_1+1)(2l_2+1)}{4 \pi(2l+1)}} \\ \times C(l_1,l_2,m_1,m_2|l,m)C(l_1,l_2,0,0|l,m)Y_{l}^m(\theta,\varphi)$ They are also related to other more complicated structures like the Wigner 3-j symbols or the Racah coefficients. In addition, I may add that there exists a closed formula for them in $3$ dimensions (derived by Racah) and that this formula is not known for arbitrary dimensions. |
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