Conventions

Defined exclusively in terms of quaternions, the Wigner \(\mathfrak{D}\) matrices and spherical-harmonic functions (spin-weighted and scalar) are pretty simple, and easy to make internally consistent. However, it is important to establish which conventions are in use — especially in comparison to other sources. Here, I carefully examine all the assumptions built in to the conventions for the spherical package, and relate these choices to those made by other authors.

On this page, we simply introduce the basic conventions and notation for rotations in terms of the beautiful, elegant, efficient, and highly intuitive presentation with quaternions. We then compare to uglier, old-fashioned presentations in terms of spherical coordinates and the profoundly hideous Euler angles. On another page, we will see how to get directly from quaternions to highly efficient and accurate formulas for the Wigner \(\mathfrak{D}\) matrices, with no use of Euler angles whatsoever. Similarly, we will be able to express spin-weighted spherical harmonics directly in terms of quaternions, though with a simple translation to and from standard spherical coordinates. This will allow us to derive simple rotation laws for the SWSHs and modes of a general decomposition in terms of SWSHs.