Question

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.

Answer 1

Unfortunately, I'm not familiar with the optics literature on this kind of scattering, so I don't know where this result would be available in the literature. It does seem like the papers of Stein and of Cruzan are not online. I can only make a suggestion for how to rederive the transformation formulas independently.

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 $\psi^{(1)}_{l'm'}(s,\chi,\eta)$.

Lorentz transformations could be handled similarly.