# A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics

With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining $$\psi_{lm}^{(j)}(r,\vartheta,\varphi)=z_l^{(j)}(r)Y_{lm}(\vartheta,\varphi),$$ what are representations of the Poincaré transformations applied to the Vector Spherical Harmonics

$$\vec L_{lm}^{(j)} = \vec\nabla \psi_{lm}^{(j)},\\ \vec M_{lm}^{(j)} = \vec\nabla\times\vec r \psi_{lm}^{(j)},\\ \vec N_{lm}^{(j)} = \vec\nabla\times\vec M_{lm}^{(j)}$$

? Does any publication cover all Poincaré-transformations, i.e. not only translations and rotations but also Lorentz boosts? I'd prefer one publication covering all transformations at once due to the different normalizations sometimes used.

• disclaimer: I also asked this at theoreticalphysics.stackexchange.com/q/981/19 Mar 1 '12 at 12:18
• Can you say in what context spherical harmonics would be used in conjunction with spatial translations? Since spherical harmonics are adapted to transform nicely only under rotations, you might not be able to find this information anywhere other than the literature on some specialized application. Mar 1 '12 at 13:41
• @Igor: For example in the Generalized Lorentz-Mie Theory, i.e. electromagnetic scattering by multiple small particles (see e.g. opticsinfobase.org/ao/abstract.cfm?id=45782, opticsinfobase.org/josaa/abstract.cfm?id=856, iopscience.iop.org/1464-4258/1/6/309 - I can unfortunately not find the 1960s references by Stein and Cruzan, but I'm hoping for something including Lorentz transformations as well anyway) Mar 1 '12 at 14:15
• arxiv.org/abs/1403.6117
– user114668
Oct 30 '20 at 9:51

A promising starting point is the expansion of a plane wave in spherical harmonics which can be found as formula (10.43) in section 10.3 of the 3rd edition of Jackson's textbook on Electrodynamics. Multiplying it by $e^{-i\omega t}$ on both sides gives $$e^{-i\omega t + i\mathbf{k}\cdot\mathbf{r}} = 4\pi e^{-i\omega t} \sum_{l=0}^\infty i^l j_l(kr) \sum_{m=-l}^l Y^*_{lm}(\theta,\phi) Y_{lm}(\theta',\phi') ,$$ where $(r,\theta,\phi)$ and $(k,\theta',\phi')$ are the spherical coordinates of the vectors $\mathbf{r}$ and $\mathbf{k}$ respectively. The left hand side essentially gives the functions $\psi^{(1)}_{lm}(r,\theta,\phi)$ that you are interested in, when expanded in the basis of $Y_{lm}(\theta',\phi')$. I'm not sure which expressions would generate $\psi^{(j)}_{lm}(r,\theta,\phi)$ for your other $j$-cases.
The same left hand side also transforms nicely with respect to Poincaré transformations. I think you should be able to use that property to get the transformation formulas that you need. For example, if you consider the translation by $\mathbf{r}=\mathbf{s}+\mathbf{d}$, with $(s,\chi,\eta)$ the spherical coordinates of $\mathbf{s}$, you get $$e^{i\mathbf{k}\cdot\mathbf{r}} = e^{i\mathbf{k}\cdot \mathbf{d}} e^{i\mathbf{k}\cdot\mathbf{s}} .$$ Expanding each of the exponentials in the basis of $Y_{lm}(\theta',\phi')$ and using the Clebsch-Gordan formula for expanding products of spherical harmonics will give a formula for $\psi^{(1)}_{lm}(r,\theta,\phi)$ in terms of $\psi^{(1)}_{l'm'}(s,\chi,\eta)$.