# A Triple Product Integral Identity for Vector Spherical Harmonics

Last updated 2021-05-21 01:08:20 SGT

As everyone knows, the product of three spherical harmonics can be related to the Wigner $3j$ symbols (and therefore Clebsch-Gordan coefficients) in the form

Lately, however, I have been extensively using vector spherical harmonics, for which I have not been able to find equivalent identities to this scalar version. In particular I have been confronted with situations where I have needed to perform integrals of the form

where here I am schematically using 1, 2, 3 to stand in for the relevant multi-indices.

To proceed, let us first consider how one recovers the orthogonality relation of the poloidal vector spherical harmonics $\boldsymbol{\Psi}_l^m = r \nabla Y_l^m$. Limiting our attention to the unit sphere, we have $\nabla_3 = \partial_r + {1 \over r} \nabla_2$ (although $Y_l^m$ has no $r$ dependence in any case). Therefore

where the derivative here is understood to be purely horizontal (acting only on angles). We now integrate by parts to obtain

The first term vanishes by Stokes' Theorem as it is the integral of a 2-divergence over the sphere (which has no boundary). For the second term, we recall the definition of the spherical harmonics is that they are the eigenfunctions of the Laplacian: $\left(\nabla^2 + l(l+1)\right)Y_{l}^{m} = 0$. Thus we obtain

as expected.

Let us now turn our attention to the triple product integral, eq. 2. Again integrating by parts, we obtain

Again, the integral of the 2-divergence vanishes, and we are left with 2 terms. The first is an integral of similar form to eq. 2, but with the multi-indices having undergone a cyclic permutation (and with a sign change). The second is the scalar triple integral in eq. 1, which does not depend on the ordering of the multi-indices. Effecting this cyclic permutation twice more, we arrive at

In full: