Spin-weighted spherical harmonics

Spin-weighted spherical harmonics (SWSHs) generalize the standard spherical-harmonic functions. In particular, there is a spin weight $s$ associated with each class of SWSHs ${}_{s}Y_{\ell,m}$, and $s=0$ corresponds to the standard spherical harmonics $Y_{\ell,m}$. They can be thought of as special cases of the Wigner $\mathfrak{D}$ matrices. This allows us to use the representation property of the $\mathfrak{D}$ matrices to understand how SWSHs rotate. More importantly, if a spin-weighted function is expanded in a basis of SWSHs, we can derive a formula for the coefficients of that expansion with respect to a rotated frame. These properties will all be derived and discussed below.

This discussion is essentially an updated version of the one found in my paper of a couple years ago. For more references, see the citations therein.

Spin-weighted functions

Spin-weighted functions on the sphere are unusual objects. They take values in the complex numbers $\mathbb{C}$. One would be forgiven, therefore, for thinking that the field’s value at a point is a scalar. The key property of a scalar is its trivial transformation law, which spin-weighted functions do not obey.¹ To be more precise, imagine we have some function ${}_{s}f(\boldsymbol{n})$, where $\boldsymbol{n}$ is the vector from the origin to the corresponding point on the sphere, and $s$ is the spin weight. Now, if we rotate our basis, we have a new function ${}_{s}f’$; the same rotation gives us a new field point $\boldsymbol{n}’$. A scalar function obeys \begin{equation} \label{eq:ScalarTransformation} {}_{0}f’(\boldsymbol{n}’) = {}_{0}f(\boldsymbol{n}) \end{equation} On the other hand, a spin-weighted function obeys the somewhat stranger transformation law \begin{equation} \label{eq:SpinWeightedTransformation} {}_{s}f’(\boldsymbol{n}’) = {}_{s}f(\boldsymbol{n})\, e^{-i\, s\, \gamma} \end{equation} for some angle $\gamma$. The most important example of this transformation is a rotation about the $\boldsymbol{n}$ vector (in the positive, right-handed sense) through an angle $\gamma$.²

It may be surprising that a transformation for which $\boldsymbol{n}’ = \boldsymbol{n}$ actually changes the value of a spin-weighted function. The reason for this strange behavior is that spin-weighted functions are actually contractions of tensors with various terms of the form $(\boldsymbol{\vartheta} + i\, \boldsymbol{\varphi})/\sqrt{2}$, or the conjugate of that term. In this sense, the spin-weighted functions are actually tensor components with respect to some coordinate basis. So it makes sense that the components should transform under coordinate transformations.

Different ways of thinking about spin-weighted functions

It seems a little inconsistent that we should write spin-weighted functions as functions of $\boldsymbol{n}$ alone; strictly speaking, it seems like it would make more sense to write ${}_{s}f(\boldsymbol{n}, \boldsymbol{\vartheta}, \boldsymbol{\varphi})$. In fact, we can do a little better. Those three vectors are a little redundant; all the information provided by them is actually carried in a single rotation operator: \begin{align*} \boldsymbol{\vartheta} &= \mathcal{R} \{\basis{x}\}, \\\
\boldsymbol{\varphi} &= \mathcal{R} \{\basis{y}\}, \\\
\boldsymbol{n} &= \mathcal{R} \{\basis{z}\}. \end{align*} And so we can consider the spin-weighted functions not to be functions of points on the sphere, but functions of rotation operators. Given that the group of unit quaternions (rotors) is so vastly preferable as a way of representing rotations, we will frequently write spin-weighted functions as functions of a rotor ${}_{s}f(\quat{R})$.

Here, the relation to the standard way of writing functions on the sphere is clear. ${}_{s}f(\vartheta, \varphi)$ is the function’s value at the point given by coordinates $(\vartheta, \varphi)$. But for spin-weighted functions, there is an implicit dependence on the standard tangent basis at that point, $(\boldsymbol{\vartheta}, \boldsymbol{\varphi})$. By writing the spin-weighted function as a function of the rotor $\quat{R}$, we are simultaneously

1. encoding the point $(\vartheta, \varphi)$ as $\quat{R}\, \basis{z}\, \quat{R}^{-1}$, and
2. encoding the dependence on the tangent basis as $\quat{R}\, \basis{x}\, \quat{R}^{-1}$ and $\quat{R}\, \basis{y}\, \quat{R}^{-1}$.

Moreover, this will make the relationship between SWSHs and the Wigner $\mathfrak{D}$ matrices very simple, and remove apparent singularities associated with the coordinates $(\vartheta, \varphi)$. (In fact, this approach will also be very helpful with the more general Newman-Penrose formalism and Lorentz transformations.)

As a side note, a common mathematical device for simultaneously recording a position on the sphere and the orientation of the tangent basis is called the “Hopf fibration”. This is a fiber bundle, where the base space is the sphere $S^2$ (representing positions on the sphere), and the fibers are circles $S^1$ (corresponding to orientations of the tangent basis at that point). The total space is simply $S^3$. This, not coincidentally, is also the space of unit quaternions. So we see that the Hopf fibration is just another way of looking at rotors.

SWSHs in terms of Wigner’s $\mathfrak{D}$ matrices

It is important to make contact with other sources in the literature, to ensure that we use the same conventions—or at least understand how our conventions differ. First, we note that as shown before, the standard spherical coordinates $(\vartheta, \varphi)$ and the corresponding standard tangent basis vectors can be expressed simply in the form of the rotor \begin{equation*} \rthetaphi = e^{\varphi \basis{z}/2}\, e^{\vartheta \basis{y}/2}. \end{equation*} Or, to put this in a form more useful for us, we have \begin{equation*} \quat{R}_a = e^{i\,\varphi/2}\, \cos\frac{\beta}{2}, \qquad \quat{R}_b = e^{-i\,\varphi/2}\, \sin\frac{\beta}{2}. \end{equation*} Plugging these expressions into the expression for $\mathfrak{D}$ given here, for example, and comparing to the expressions in the data-formats paper, we see that SWSHs can be written as \begin{equation} \label{eq:SWSHFromWignerD} {}_{s}Y_{\ell,m} (\vartheta, \varphi) = {}_{s}Y_{\ell,m} (\quat{R}_{(\vartheta, \varphi)}) = (-1)^s\, \sqrt{\frac{2\ell+1} {4\pi}}\, \mathfrak{D}^{(\ell)}_{m,-s} \left(\quat{R}_{(\vartheta, \varphi)} \right). \end{equation}

Rotating SWSHs

Now, using Eq. \eqref{eq:SWSHFromWignerD} and the representation property of Wigner’s $\mathfrak{D}$ matrices, we can show how to rotate the SWSH functions, and show the transformation of modes of a decomposition of a general spin-weighted function by SWSHs.

We start of with the fact that the $\mathfrak{D}$ matrices form a representation of the rotation group. That is, given any two rotors $\quat{R}_1$ and $\quat{R}_2$, we have \begin{equation} \mathfrak{D}^{(\ell)}_{m’,m} (\quat{R}_1\, \quat{R}_2) = \sum_{m’’} \mathfrak{D}^{(\ell)}_{m’,m’’} (\quat{R}_1)\, \mathfrak{D}^{(\ell)}_{m’‘,m} (\quat{R}_2). \end{equation} From Eq. \eqref{eq:SWSHFromWignerD}, we see that it is possible to rewrite this as \begin{equation} \label{eq:SWSHTransformation1} {}_{s}Y_{\ell,m} (\quat{R}_1\, \quat{R}_2) = \sum_{m’} \mathfrak{D}^{(\ell)}_{m,m’} (\quat{R}_1)\, {}_{s}Y_{\ell,m’}(\quat{R}_2). \end{equation} In principle, this is the rotation law needed for SWSHs.

As we saw above, spin-weighted functions should be regarded as functions on the rotation group, rather than as functions on points of the sphere. Unfortunately, this is not the standard treatment; a canonical tangent basis is assumed from the coordinate system. So we need to write the arguments of the SWSHs in terms of spherical coordinates. But there’s a wrinkle here. If we assume that the argument of the SWSH on the left-hand side above can be written as spherical coordinates $\quat{R}_{(\vartheta’, \varphi’)}$, the argument of the SWSH on the right-hand side cannot be written purely as spherical coordinates; there is some additional rotation needed. In general, then, we must write the transformation in the form \begin{equation} \quat{R}_{(\vartheta’, \varphi’)} = \quat{R}_{\text{phys}}\, \quat{R}_{(\vartheta, \varphi)}\, e^{\gamma\, \basis{z}/2}. \end{equation} The final term in this equation represents an initial rotation about the $\basis{z}$ basis vector, which is equivalent to a final rotation about the vector joining the origin to the point $(\vartheta’, \varphi’)$. This term must be included because that is the only way the equation can be true for general sets of angles and overall rotation $\quat{R}_{\text{phys}}$. We have placed a subscript “$\text{phys}$” on this overall rotor because, as will be discussed below, this represents a transformation of the physical system.

This $\gamma$ factor at first appears troubling. It is a function of both $\quat{R}_{\text{phys}}$ and $(\vartheta, \varphi)$. In fact, if we insert our rotations into Eq. \eqref{eq:SWSHTransformation1}, we obtain a surprising result. Using the known behavior of spin-weighted functions we obtain \begin{align} {}_{s}Y_{\ell,m} (\quat{R}_{(\vartheta’, \varphi’)}) &= \sum_{m’} \mathfrak{D}^{(\ell)}_{m,m’} (\quat{R}_{\text{phys}})\, {}_{s}Y_{\ell,m’}(\quat{R}_{(\vartheta, \varphi)}\, e^{\gamma\, \basis{z}/2}) \nonumber \\\\ \label{eq:SWSHRotation} &= \sum_{m’} \mathfrak{D}^{(\ell)}_{m,m’} (\quat{R}_{\text{phys}})\, {}_{s}Y_{\ell,m’}(\quat{R}_{(\vartheta, \varphi)})\, e^{-i\, s\, \gamma}. \end{align} The spin-weighted spherical harmonics, regarded as functions of the spherical coordinates alone, do not transform among themselves under a representation of the rotation group—except in the familiar case of $s=0$!

This curious factor of $e^{-i\, s\, \gamma}$, however, is precisely what is needed to ensure that modes of an expansion in SWSHs do transform among themselves under rotations. To be more precise, let us take a function ${}_{s}f$, with a well defined spin weight $s$. This can be expanded in the basis of SWSHs as \begin{equation*} {}_{s}f(\quat{R}_{(\vartheta, \varphi)}) = \sum_m f^{\ell,m} {}_{s}Y_{\ell,m}(\quat{R}_{(\vartheta, \varphi)}). \end{equation*} Now, there is another field ${}_{s}f’$ representing the physically rotated field having the same magnitude at $(\vartheta’, \varphi’)$ as ${}_{s}f$ has at $(\vartheta, \varphi)$. This has a similar expansion: \begin{equation*} {}_{s}f’(\quat{R}_{(\vartheta’, \varphi’)}) = \sum_m f’^{\ell,m} {}_{s}Y_{\ell,m}(\quat{R}_{(\vartheta’, \varphi’)}). \end{equation*} Now, recalling from Eq. \eqref{eq:SpinWeightedTransformation} the relation ${}_{s}f’(\quat{R}_{(\vartheta’, \varphi’)}) = {}_{s}f(\quat{R}_{(\vartheta, \varphi)})\, e^{-i\, s\, \gamma}$, we can combine the last three equations. Using orthogonality of the SWSHs, we can pull out the transformation law for the modes: \begin{equation} f^{\ell,m} = \sum_{m’} f’^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m} (\quat{R}_{\text{phys}}). \end{equation} Note that the spin dependence $e^{-i\, s\, \gamma}$ has canceled out. We can also rewrite this as \begin{equation} f’^{\ell,m} = \sum_{m’} f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m} (\quat{R}^{-1}_{\text{phys}}). \end{equation} This is the more interesting formula, because we assume that we’ve started with the field ${}_{s}f$, and wish to derive the modes of the rotated field ${}_{s}f’$.

We should note the meaning of $\quat{R}_{\text{phys}}$, now that we understand how it fits into the transformations. It is a rotation taking a physical system $f$ with some value at the point $(\vartheta, \varphi)$ into a physical system $f’$ with that value at the point $(\vartheta’, \varphi’)$. We can think of the rotor $\quat{R}^{-1}_{\text{phys}}$ as the same rotation, but applied to the coordinate basis itself, while leaving the physical system in place. We will therefore write it as $\quat{R}_{\text{basis}}$—the rotation being applied to the basis itself, while leaving the physical system fixed. The field ${}_{s}f$ is measured with respect to the original basis, and ${}_{s}f’$ is the field measured with respect to this rotated basis. Then we have \begin{equation} f’^{\ell,m} = \sum_{m’} f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m} (\quat{R}_{\text{basis}}). \end{equation}

Composing rotations

The final detail we need to understand is the composition of rotations. We can write down a basic rule:

• rotations of the physical system compose by left-multiplication;
• rotations of the coordinate basis compose by right-multiplication.

Thus, for example, if we have an initial rotation of the physical system $\quat{R}_{\text{p}1}$ and then apply a second rotation of the physical system $\quat{R}_{\text{p}2}$, this is equivalent to applying a single rotation $\quat{R}_{\text{p}2}\, \quat{R}_{\text{p}1}$ (note the ordering). Similarly, if we have an initial rotation of the coordinate basis $\quat{R}_{\text{b}1}$ and then apply a second rotation of the coordinate basis $\quat{R}_{\text{b}2}$, this is equivalent to applying a single rotation $\quat{R}_{\text{b}1}\, \quat{R}_{\text{b}2}$.

It is instructive to see explicitly how these rotations play out in terms of the rotation of modes. We start out with modes $f^{\ell,m}$ in the initial system; we then apply the first rotation (of either the physical system or the basis) to obtain the modes $f’^{\ell,m}$; and finally apply the second rotation to obtain the modes $f’’^{\ell,m}$.

First, for rotations of the physical system: \begin{align*} f’’^{\ell,m} &= \sum_{m’’} \left( f’^{\ell,m’’} \right)\, \mathfrak{D}^{(\ell)}_{m’‘,m} (\quat{R}^{-1}_{\text{p}2}) \\\
&= \sum_{m’‘,m’} \left( f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m’’} (\quat{R}^{-1}_{\text{p}1}) \right)\, \mathfrak{D}^{(\ell)}_{m’‘,m} (\quat{R}^{-1}_{\text{p}2}) \\\
&= \sum_{m’} f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m} (\quat{R}^{-1}_{\text{p}1}\, \quat{R}^{-1}_{\text{p}2}) \\\
&= \sum_{m’} f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m} \left( (\quat{R}_{\text{p}2}\, \quat{R}_{\text{p}1})^{-1} \right) \end{align*} Next, for rotations of the basis with respect to which the modes are measured: \begin{align*} f’’^{\ell,m} &= \sum_{m’’} \left( f’^{\ell,m’’} \right)\, \mathfrak{D}^{(\ell)}_{m’‘,m} (\quat{R}_{\text{b}2}) \\\
&= \sum_{m’‘,m’} \left( f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m’’} (\quat{R}_{\text{b}1}) \right)\, \mathfrak{D}^{(\ell)}_{m’‘,m} (\quat{R}_{\text{b}2}) \\\
&= \sum_{m’} f^{\ell,m’}\, \mathfrak{D}^{(\ell)}_{m’,m} (\quat{R}_{\text{b}1}\, \quat{R}_{\text{b}2}) \end{align*} Thus, we see that applying the rotations in the natural order is fine, but if we wish to keep track of what those rotations were, we need to remember that rotations do not commute.