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

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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.

This post has been migrated from (A51.SE)
asked Feb 28, 2012
retagged Mar 18, 2014
disclaimer: I also asked this at [MathOverflow](http://mathoverflow.net/questions/89955/a-nice-overview-and-maybe-derivation-of-the-poincare-transformations-of-the-vec)

This post has been migrated from (A51.SE)

## 1 Answer

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The problem with Poincaré group is in the fact that it is not compact. That's why this question is non-trivial. Though, properly formulated search gives few papers on this topic. Try to find the answer in this paper http://arxiv.org/abs/math-ph/0507056 . The paper itself may be not that interesting, but there is a nice introduction with a number of useful references.

This post has been migrated from (A51.SE)
answered Feb 28, 2012 by (340 points)

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