Details

This page carefully works through all the conventions used in this package, starting from first principles to motivate the choices and ensure that each step is on firm footing. The previous page collects the results in a more concise form.

Note that we will use Euler angles and spherical coordinates here, but they are not used internally in this package — though conversion functions are available. It is almost always a bad idea to use Euler angles in computing; quaternions are clearly the preferred representation for numerous reasons. However, Euler angles are important for (a) comparing to other sources, and (b) performing analytic integrations. These are the only two uses we will make of Euler angles.

Throughout, $ℓ$, $m'$, $m$, and $s$ are either all integers or all half-integers; every statement below holds in both cases unless noted otherwise.

Three-dimensional space

The space we are working in is naturally three-dimensional Euclidean space, so we start with a right-handed Cartesian coordinate system $(x, y, z)$. These also give us the unit basis vectors $(𝐱, 𝐲, 𝐳)$. Note that these basis vectors are assumed to have unit norm, but we omit the hats just to keep the notation simple. Any vector in this space can be written as

\[𝐯 = v_x 𝐱 + v_y 𝐲 + v_z 𝐳,\]

in which case the Euclidean norm is given by

\[\| 𝐯 \| = \sqrt{v_x^2 + v_y^2 + v_z^2}.\]

Equivalently, we can write the components of the Euclidean metric as

\[g_{ij} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)_{ij}.\]

Note that, because the points of the space are in one-to-one correspondence with the vectors, we will frequently use a vector to label a point in space.

Spherical coordinates

We will be working on the sphere, so it will be very convenient to use spherical coordinates $(r, θ, ϕ)$. We choose the standard "physics" conventions for these, in which we relate to the Cartesian coordinates by

\[\begin{aligned} r &= \sqrt{x^2 + y^2 + z^2} &&\in [0, \infty), \\ θ &= \arccos\left(\frac{z}{r}\right) &&\in [0, π], \\ ϕ &= \arctan\left(\frac{y}{x}\right) &&\in [0, 2π), \end{aligned}\]

where we assume the $\arctan$ in the expression for $ϕ$ is really the two-argument form that gives the correct quadrant. The inverse transformation is given by

\[\begin{aligned} x &= r \sin θ \cos ϕ, \\ y &= r \sin θ \sin ϕ, \\ z &= r \cos θ. \end{aligned}\]

We can use this to find the components of the metric in spherical coordinates:

\[g_{i'j'} = \sum_{i,j} \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^j}{\partial x^{j'}} g_{ij} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2θ \end{array} \right)_{i'j'}.\]

The unit coordinate vectors in spherical coordinates are given by differentiating the vector to the point in question, $𝐩=[x,y,z]$, with respect to the coordinate in question, producing

\[\begin{aligned} 𝐧 &= \sin θ \cos ϕ 𝐱 + \sin θ \sin ϕ 𝐲 + \cos θ 𝐳, \\ \boldsymbol{θ} &= \cos θ \cos ϕ 𝐱 + \cos θ \sin ϕ 𝐲 - \sin θ 𝐳, \\ \boldsymbol{ϕ} &= -\sin ϕ 𝐱 + \cos ϕ 𝐲, \end{aligned}\]

where, again, we omit the hats on the unit vectors to keep the notation simple. Conversely, we can express the Cartesian basis vectors in terms of the spherical basis vectors as

\[\begin{aligned} 𝐱 &= \sin θ \cos ϕ 𝐧 + \cos θ \cos ϕ \boldsymbol{θ} - \sin ϕ \boldsymbol{ϕ}, \\ 𝐲 &= \sin θ \sin ϕ 𝐧 + \cos θ \sin ϕ \boldsymbol{θ} + \cos ϕ \boldsymbol{ϕ}, \\ 𝐳 &= \cos θ 𝐧 - \sin θ \boldsymbol{θ}. \end{aligned}\]

One seemingly obvious — but extremely important — fact is that the unit basis frame $(𝐱, 𝐲, 𝐳)$ can be rotated onto $(\boldsymbol{θ}, \boldsymbol{ϕ}, 𝐧)$ by first rotating through the "polar" angle $θ$ about the $𝐲$ axis, and then through the "azimuthal" angle $ϕ$ about the $𝐳$ axis. This becomes important when we consider spin-weighted functions.

Integration in Cartesian coordinates is, of course, trivial as

\[\int_{\mathbb{R}^3} f\, d^3𝐫 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f\, dx\, dy\, dz.\]

In spherical coordinates, the integrand involves the square-root of the determinant of the metric, so we have

\[\int_{\mathbb{R}^3} f\, d^3𝐫 = \int_0^\infty \int_0^π \int_0^{2π} f\, r^2 \sin θ\, dr\, dθ\, dϕ.\]

Restricting to the unit sphere, we obtain the usual surface element

\[\int_{𝕊²} f\, d^2\Omega = \int_0^π \int_0^{2π} f\, \sin θ\, dθ\, dϕ.\]

Note that $\int_{𝕊²} d^2\Omega = 4π$.

Four-dimensional space: Quaternions and rotations

Geometric algebra

Given the basis vectors $(𝐱, 𝐲, 𝐳)$ and the Euclidean norm, we can define the geometric algebra. The key feature is the geometric product, which we could define for any pair of vectors as $𝐯$ and $𝐰$ as

\[𝐯 𝐰 = 𝐯 ⋅ 𝐰 + 𝐯 ∧ 𝐰,\]

where the dot product is the usual scalar product and the wedge product is the antisymmetric part of the tensor product — acting just like the standard exterior product from the algebra of differential forms. The geometric product is linear, associative, distributive, and has the property that

\[𝐯𝐯 = \| 𝐯 \|^2.\]

The most useful properties of the geometric product are that parallel vectors commute with each other, while orthogonal vectors anticommute. Since the geometric product is linear, the product of any two vectors can be decomposed into parallel and orthogonal parts.

The basis for this entire space is then the set

\[\begin{gather} 𝟏, \\ 𝐱, 𝐲, 𝐳,\\ 𝐱𝐲, 𝐱𝐳, 𝐲𝐳, \\ 𝐱𝐲𝐳. \end{gather}\]

The standard presentation of quaternions (including the confused historical development) uses different symbols for these last four basis elements:

\[\begin{gather} 𝐢 = 𝐳𝐲 = -𝐲𝐳, \\ 𝐣 = 𝐱𝐳 = -𝐳𝐱, \\ 𝐤 = 𝐲𝐱 = -𝐱𝐲, \\ 𝐈 = 𝐱𝐲𝐳. \end{gather}\]

Note that each of these squares to -1. For example, recalling that orthogonal vectors anticommute, the product is associative, and the product of a vector with itself is just its squared norm, we have

\[𝐱𝐲𝐱𝐲 = -𝐱𝐲𝐲𝐱 = -𝐱(𝐲𝐲)𝐱 = -𝐱𝐱 = -1.\]

Any of these could act like the unit imaginary; $𝐱𝐲$ is probably the canonical choice.

$𝐈$ is sometimes called the pseudoscalar. Its inverse is $𝐈^{-1} = 𝐳𝐲𝐱 = -𝐱𝐲𝐳$, which can also serve as something very much like the Hodge star operator,^[1] mapping elements to their "dual" elements. In particular, we have

\[\begin{aligned} 𝐢 &= 𝐈^{-1}𝐱, \\ 𝐣 &= 𝐈^{-1}𝐲, \\ 𝐤 &= 𝐈^{-1}𝐳. \end{aligned}\]

We will see that $𝐢$ generates right-handed rotations in the positive sense about $𝐱$, $𝐣$ about $𝐲$, and $𝐤$ about $𝐳$. Moreover, this mapping between $(𝐱, 𝐲, 𝐳)$ and $(𝐢, 𝐣, 𝐤)$ is a vector-space isomorphism. In fact, the reader who is not familiar with geometric algebra but is familiar with quaternions may be able to read an expression like $𝐣 𝐱 𝐣⁻¹$ as if it is just an abuse of notation, and mentally replace $𝐱$ with $𝐢$ to read those symbols as a valid quaternion expression; both viewpoints are equally correct by the isomorphism.

Quaternions and Euler angles

Note that there are different conventions for the signs of the $(𝐢, 𝐣, 𝐤)$ basis. Everyone agrees that $𝐢² = 𝐣² = 𝐤² = -1$, but we could easily flip the sign of any basis element, and these would still be satisfied. The identifications we chose above are made to ensure that $𝐢$ generates rotations about $𝐱$, and so on, but even that depends on how we define quaternions as acting on vectors (to be discussed below). A different choice of the latter would result in all flipping the sign of all three basis elements, which is a convention that is commonly used — though almost exclusively in aerospace. The key expressions that eliminate ambiguity are the multiplications

\[\begin{aligned} 𝐢 𝐣 &= 𝐤, \\ 𝐣 𝐤 &= 𝐢, \\ 𝐤 𝐢 &= 𝐣. \end{aligned}\]

We can also use these rules above to determine $𝐢𝐣𝐤 = -𝟏$. All four of these equations have flipped signs in other conventions. See Sommer et al. for a discussion of the different conventions.

We use coordinates $(W, X, Y, Z)$ on the space of quaternions, so that a quaternion would be written as

\[𝐐 = W𝟏 + X𝐢 + Y𝐣 + Z𝐤,\]

though we usually omit the $𝟏$. The space of all quaternions is thus four dimensional. The norm is just the standard Euclidean norm, so that the norm of a quaternion is

\[\| 𝐐 \| = \sqrt{W^2 + X^2 + Y^2 + Z^2}.\]

An important operation is the conjugate, which is defined as

\[\overline{𝐐} = W - X𝐢 - Y𝐣 - Z𝐤.\]

Note that the squared norm can be written as the quaternion times its conjugate. Any nonzero quaternion has an inverse, which is just the conjugate divided by the squared norm:

\[𝐐^{-1} = \frac{\overline{𝐐}}{𝐐\overline{𝐐}} = \frac{\overline{𝐐}}{\| 𝐐 \|^2}.\]

The other important operation is exponentiation. Since a scalar commutes with any quaternion, including a nonzero scalar component in the quaternion will simply multiply the result by the exponential of that scalar component. Moreover, we will not have any use for such an exponential, so we assume that the argument to the exponential function is a "pure" quaternion — that is, one with zero scalar component. Moreover, we write it as a unit quaternion $𝐮$ times some real number $\sigma$. In particular, note that $𝐮^2 = -1$, so that it acts like the imaginary unit, which means we already know how to exponentiate it:

\[\exp(𝐮\, \sigma) = \cos\sigma + 𝐮\, \sin\sigma.\]

Note that the inverse of the result can be obtained simply by negating the argument, as usual.

Much as with standard three-dimensional space, we could introduce a generalization of spherical coordinates, though we use a slight variant: extended Euler coordinates. We will see below how to interpret these as a series of rotations. For now, we simply state the relation:

\[\begin{aligned} R &= \sqrt{W^2 + X^2 + Y^2 + Z^2} &&\in [0, \infty), \\ α &= \arctan\frac{Z}{W} + \arctan\frac{-X}{Y} &&\in [0, 2π), \\ β &= 2\arccos\sqrt{\frac{W^2+Z^2}{W^2+X^2+Y^2+Z^2}} &&\in [0, π], \\ γ &= \arctan\frac{Z}{W} - \arctan\frac{-X}{Y} &&\in [0, 4π), \end{aligned}\]

where we again assume the $\arctan$ in the expressions for $α$ and $γ$ is really the two-argument form that gives the correct quadrant, and if relevant, we use mod to limit the values on output. Note that here, $γ$ ranges up to $4π$ rather than just $2π$, as in the standard Euler angles. This is because we are describing the space of quaternions, rather than just the space of rotations. If we restrict to quaternions with magnitude $R=1$, we have exactly the group of unit quaternions $\mathrm{Spin}(3)=\mathrm{SU}(2)$, which is a double cover of the rotation group $\mathrm{SO}(3)$. This extended range for $γ$ is necessary to cover the entire space of quaternions; if we further restrict to $[0, 2π)$, we would only cover the space of rotations. This and the inclusion of $R$ identify precisely how this coordinate system extends the standard Euler angles.

Note that it would also be reasonable to limit $γ$ to $2π$, while allowing $β$ to range up to $2π$ to cover the entire space of quaternions. This is just somewhat more delicate to compute, and is simply not conventional. Also, using $γ ∈ [0,4π)$ integrates nicely with our framework of a telescope with $γ$ representing the rotation about its line of sight; a full $4π$ rotation is required for the polarizer to explore the full range of states of half-integer spin fields.

The inverse transformation is given by

\[\begin{aligned} W &= R\, \cos\frac{β}{2} \cos\frac{α+γ}{2}, \\ X &= -R\, \sin\frac{β}{2} \sin\frac{α-γ}{2}, \\ Y &= R\, \sin\frac{β}{2} \cos\frac{α-γ}{2}, \\ Z &= R\, \cos\frac{β}{2} \sin\frac{α+γ}{2}. \end{aligned}\]

As with the spherical coordinates, we can use this to find the components of the metric in our extended Euler coordinates:

\[g_{i'j'} = \sum_{i,j} \frac{\partial X^i}{\partial X^{i'}} \frac{\partial X^j}{\partial X^{j'}} g_{ij} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{R^2}{4} & 0 & \frac{R^2 \cos β}{4} \\ 0 & 0 & \frac{R^2}{4} & 0 \\ 0 & \frac{R^2 \cos β}{4} & 0 & \frac{R^2}{4} \end{array} \right)_{i'j'}.\]

Again, we can compute the unit basis vectors in extended Euler coordinates in terms of the unit basis vectors in quaternion coordinates by differentiating $𝐐$ (using the expressions for $W, X, Y, Z$ given above) with respect to the extended-Euler coordinate in question, then normalizing — which divides out a factor of $1$ for $𝐑$ and $R/2$ for each of $α$, $β$ and $γ$, in accordance with the metric above:

\[\begin{aligned} 𝐑 &= \left( \cos \frac{β}{2} \cos \frac{α+γ}{2} 𝟏 - \sin \frac{β}{2} \sin \frac{α-γ}{2} 𝐢 + \sin \frac{β}{2} \cos \frac{α-γ}{2} 𝐣 + \cos \frac{β}{2} \sin \frac{α+γ}{2} 𝐤 \right), \\ \boldsymbol{α} &= \left( -\cos \frac{β}{2} \sin \frac{α+γ}{2} 𝟏 - \sin \frac{β}{2} \cos \frac{α-γ}{2} 𝐢 - \sin \frac{β}{2} \sin \frac{α-γ}{2} 𝐣 + \cos \frac{β}{2} \cos \frac{α+γ}{2} 𝐤 \right), \\ \boldsymbol{β} &= \left( -\sin \frac{β}{2} \cos \frac{α+γ}{2} 𝟏 - \cos \frac{β}{2} \sin \frac{α-γ}{2} 𝐢 + \cos \frac{β}{2} \cos \frac{α-γ}{2} 𝐣 - \sin \frac{β}{2} \sin \frac{α+γ}{2} 𝐤 \right), \\ \boldsymbol{γ} &= \left( -\cos \frac{β}{2} \sin \frac{α+γ}{2} 𝟏 + \sin \frac{β}{2} \cos \frac{α-γ}{2} 𝐢 + \sin \frac{β}{2} \sin \frac{α-γ}{2} 𝐣 + \cos \frac{β}{2} \cos \frac{α+γ}{2} 𝐤 \right). \end{aligned}\]

Invariant measure

Again, integration involves a square-root of the determinant of the metric, which reduces to $R^3 \sin β / 8$. The integral over the entire space of quaternions is then

\[\int_{\mathbb{R}^4} f\, d^4𝐐 = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty f\, dW\, dX\, dY\, dZ = \int_0^\infty \int_0^{2π} \int_0^{π} \int_0^{4π} f\, \frac{R^3}{8} \sin β\, dR\, dα\, dβ\, dγ.\]

Restricting to the unit sphere $R=1$, we obtain the measure on $\mathrm{Spin}(3)$,

\[\int_{\mathrm{Spin}(3)} f\, d^3\Omega = \frac{1}{8} \int_0^{2π} \int_0^{π} \int_0^{4π} f\, \sin β\, dα\, dβ\, dγ,\]

where $\int_{\mathrm{Spin}(3)} d^3\Omega = 2π^2$ is the volume of the unit 3-sphere. This measure is inherited from the Euclidean metric of $\mathbb{R}^4$, which is invariant under multiplication by unit quaternions on either side; it is therefore the (bi-invariant) Haar measure of the group, and it is the measure we use whenever we integrate over $\mathrm{Spin}(3)$ — in particular for the orthogonality of the Wigner $𝔇$ matrices below. Finally, restricting to the space of rotations by limiting $γ$ to $[0, 2π)$, we can further simplify this to

\[\int_{\mathrm{SO}(3)} f\, d^3\Omega = \frac{1}{8} \int_0^{2π} \int_0^{π} \int_0^{2π} f\, \sin β\, dα\, dβ\, dγ,\]

where $\int_{\mathrm{SO}(3)} d^3\Omega = π^2$. (Many references instead normalize the measure on $\mathrm{SO}(3)$ to $8π^2$ by omitting the factor of $1/8$, or to $1$; only the relative normalizations matter below.) These volume factors are verified symbolically on the metrics and integration page.

Rotations

We restrict to a unit quaternion $𝐑$, for which $W^2 + X^2 + Y^2 + Z^2 = 1$. Given this constraint we can, without loss of generality, write the quaternion as

\[𝐑 = \exp\left(\frac{\rho}{2} \hat{𝔯}\right) = \cos\frac{\rho}{2} + \sin\frac{\rho}{2}\, \hat{𝔯},\]

where $\rho$ is an angle of rotation and $\hat{𝔯}$ is a unit "pure-vector" quaternion. We can multiply a vector $𝐯$ as

\[𝐑\, 𝐯\, 𝐑^{-1}.\]

Splitting $𝐯 = 𝐯_⟂ + 𝐯_∥$ into components perpendicular and parallel to $\hat{𝔯}$, we see that $𝐯_∥$ commutes with $𝐑$ and $𝐑^{-1}$, while $𝐯_⟂$ anticommutes with $\hat{𝔯}$. To find the full rotation, we expand the product:

\[\begin{aligned} 𝐑\, 𝐯\, 𝐑^{-1} &= 𝐯_∥ + \left(\cos\frac{\rho}{2} + \sin\frac{\rho}{2}\, \hat{𝔯}\right) 𝐯_⟂ \left(\cos\frac{\rho}{2} - \sin\frac{\rho}{2}\, \hat{𝔯}\right) \\ &= 𝐯_∥ + \left(\cos\frac{\rho}{2}\, 𝐯_⟂ + \sin\frac{\rho}{2}\, \hat{𝔯}\, 𝐯_⟂\right) \left(\cos\frac{\rho}{2} - \sin\frac{\rho}{2}\, \hat{𝔯}\right) \\ &= 𝐯_∥ + \cos^2\frac{\rho}{2}\, 𝐯_⟂ + \sin\frac{\rho}{2}\, \cos\frac{\rho}{2}\, \hat{𝔯}\, 𝐯_⟂ - \sin\frac{\rho}{2}\, \cos\frac{\rho}{2}\, 𝐯_⟂ \, \hat{𝔯} - \sin^2\frac{\rho}{2}\, \hat{𝔯}\, 𝐯_⟂\, \hat{𝔯} \\ &= 𝐯_∥ + \cos^2\frac{\rho}{2}\, 𝐯_⟂ + \sin\frac{\rho}{2}\, \cos\frac{\rho}{2}\, [\hat{𝔯}, 𝐯_⟂] - \sin^2\frac{\rho}{2}\, 𝐯_⟂ \\ &= 𝐯_∥ + \cos\rho\, 𝐯_⟂ + \sin\rho\, \hat{𝔯}\times 𝐯_⟂ \end{aligned}\]

The final expression shows that this is precisely what we expect when rotating $𝐯$ through an angle $\rho$ (in a positive, right-handed sense) about the axis $\hat{𝔯}$.

Note that the presence of two factors of $𝐑$ in the expression for rotating a vector explains two things. First, it explains why the angle of rotation is twice the angle of the quaternion: one factor of $𝐑$ either commutes and cancels or anti-commutes and combines with the the other factor. Second, it explains why the quaternion group is a double cover of the rotation group: negating $𝐑$ results in the same rotation. Thus, for any rotation, there are two (precisely opposite) quaternions that represent it.

Euler angles and spherical coordinates

Now that we understand how rotations work, we can provide geometric intuition for the expressions given above for Euler angles. The Euler angles in our convention represent an initial rotation through $γ$ about the $𝐳$ axis, followed by a rotation through $β$ about the $𝐲$ axis, and finally a rotation through $α$ about the $𝐳$ axis. Note that the axes are fixed, and not subject to any preceding rotations. More precisely, we can write the unit quaternion as

\[𝐑 = \exp\left(\frac{α}{2} 𝐤\right) \exp\left(\frac{β}{2} 𝐣\right) \exp\left(\frac{γ}{2} 𝐤\right).\]

One of the more important interpretations of a rotor is considering what it does to the basis triad $(𝐱, 𝐲, 𝐳)$. In particular, the vector $𝐳$ is rotated onto the point given by spherical coordinates $(θ, ϕ) = (β, α)$, while $𝐱$ and $𝐲$ are rotated into the plane spanned by the unit basis vectors $\boldsymbol{θ}$ and $\boldsymbol{ϕ}$ corresponding to that point. If $γ = 0$ the rotation is precise, meaning that $𝐱$ is rotated onto $\boldsymbol{θ}$ and $𝐲$ onto $\boldsymbol{ϕ}$; if $γ ≠ 0$ then they are rotated within that plane by the angle $γ$ about the $𝐧$ axis. Thus, we identify the spherical coordinates $(θ, ϕ)$ with the Euler angles $(α, β, γ) = (ϕ, θ, 0)$.

Rotation and angular-momentum operators

Complex-valued functions

Starting with Cartesian coordinates and the Euclidean norm on $\mathbb{R}^3$, we have constructed the geometric algebra over that space, as well as the spaces $\mathrm{Spin}(3) = \mathrm{SU}(2)$ (topologically $𝕊³$), $\mathrm{SO}(3)$ (topologically $\mathbb{RP}^3$), and $𝕊²$. We will be defining complex-valued functions on these spaces, and defining operators to construct and classify them. In particular, because we have constructed the spaces, they are naturally supplied with coordinates that are effectively inherited from the original Cartesian system. We will be using these coordinate systems to construct both the operators and functions. However, it is important to note that the coordinate systems may have singularities, which means that the spaces of coordinates may have different topologies than the spaces they represent. For example, Euler angles have topology $𝕊¹ \times I \times 𝕊¹$ instead of the $𝕊³$ and $\mathbb{RP}^3$ topologies of the spaces they represent; spherical coordinates have topology $𝕊¹ \times I$ instead of $𝕊²$.

Defining functions on the coordinate system of a space is subtly different from defining functions on the space itself. For example, spin-weighted functions are generally written as functions of ($𝕊²$) spherical coordinates. However, they cannot be defined as functions on $𝕊²$ itself; some notion of a reference tangent direction is needed at each point. The difference is that spherical coordinates supply a natural choice for the reference tangent direction: the unit vector in the $\boldsymbol{θ}$ direction. This supplies just enough information to define the spin-weighted functions — though this ends up not being a useful form when more general transformations or deeper understanding are needed.

Because of this variety of spaces, we will need to use function composition in several ways; functions defined on one space can be "lifted" or "lowered" to another via maps between the spaces. In the diagram below, the function $F$ can be used to define the function $f$ via the mapping $m$ as $f = m \circ F$.

For example, $A$ could be the space of spherical coordinates, $B$ could be $\mathrm{Spin}(3)$, and $F$ could be a spin-weighted function. There are many maps from spherical coordinates into $\mathrm{Spin}(3)$; we expect that all such maps will be related by rotations from $\mathrm{SO}(3)$, and in some sense equivalent via some universality relation. However, for singular maps — such as coordinate singularities where multiple coordinate values correspond to a single "physical" point — we find exceptions to the universality. These compositions will be useful, in that we can define functions on the "largest" available space, and extend them to any space that maps into the first.

In principle, our functions should be defined on $\mathrm{Spin}(3)$ or even the quaternions in general, though in practice we will define them on the space of coordinates on those spaces. In any case, we will classify the functions by their behavior with respect to actions of $\mathrm{Spin}(3)$ on the argument to the function. Therefore, we need to consider the general behavior of functions under such actions.

Finite rotations

We work with functions $f: A \to \mathbb{C}$, where $A$ is either the group of unit quaternions, or the full algebra of quaternions. Any non-zero quaternion can be expressed as $e^𝔤$ for some finite quaternion $𝔤$, which is referred to as the "generator" of the action of $e^𝔤$. This can act on a function $f$ by multiplying the argument by $e^𝔤$. However, there is an ambiguity: we could multiply either on the left or the right:^[2]

\[f\left(𝐐\right) \mapsto f\left(e^𝔤 𝐐\right) \qquad \text{or} \qquad f\left(𝐐\right) \mapsto f\left(𝐐 e^𝔤\right).\]

There is an additional ambiguity, in that this action rotates the argument of the function, whereas we will often prefer to think in terms of rotating the function itself. For example, our function may describe the measurement of some field in a particular coordinate system. Here, the argument $𝐐$ describes a particular value of the coordinates, and $e^𝔤$ changes the point under consideration. If, on the other hand $e^𝔤$ describes how the field itself is rotated, then we can write the rotated field as a function $f'$ which is related to the original function $f$ by

\[f'\left(𝐐\right) = f\left(e^{-𝔤} 𝐐\right) \qquad \text{or} \qquad f'\left(𝐐\right) = f\left(𝐐 e^{-𝔤}\right).\]

Note that the exponent is negated, because the action of $e^𝔤$ on the argument is the inverse of the action of $e^{-𝔤}$ on the function. This is a general property of the action of a group on a space, and is a consequence of the group action being a homomorphism.

To validate the signs here, it may be helpful to work through a simple example involving the sphere $𝕊²$. We define the function $f$ on spherical coordinates as

\[f(θ, ϕ) = \sin θ \sin ϕ.\]

Recall that we can map the spherical coordinates into the Euler angles, and the Euler angles into the quaternion

\[(θ, ϕ) \mapsto (ϕ, θ, 0) \mapsto 𝐐 = \exp\left(\frac{ϕ}{2} 𝐤\right) \exp\left(\frac{θ}{2} 𝐣\right).\]

It is straightforward to see that we can write $f$ as a function of $𝐐$ as

\[f(𝐐) = \left\langle 𝐐\, 𝐤\, 𝐐^{-1} \right\rangle_{𝐣},\]

where the angle brackets and subscript indicate that we are taking the $𝐣$ component. That is, $f$ is the $y$ component of the vector $𝐳$ rotated by $𝐐$.

Now, we imagine rotating the field by an angle $α$ in the positive sense about the $z$ axis. Visualizing the situation, we can see that the rotated field should be represented by

\[f'(θ, ϕ) = \sin θ \sin(ϕ - α).\]

For example, the rotated field evaluated at the point $(θ, ϕ) = (π/2, 0)$ along the positive $x$ axis should correspond to the original field evaluated at the point $(θ, ϕ) = (π/2, -α)$. This rotation is generated by $𝔤 = α 𝐤 / 2$, which allows us to immediately calculate

\[\begin{aligned} f(e^𝔤 𝐐) &= \sin θ \sin(ϕ + α) &&& f(𝐐 e^𝔤) &= \sin θ \sin ϕ \\ f(e^{-𝔤} 𝐐) &= \sin θ \sin(ϕ - α) &&& f(𝐐 e^{-𝔤}) &= \sin θ \sin ϕ. \end{aligned}\]

Thus, we see that left-multiplication by $e^{-𝔤}$ corresponds to rotation of the field while leaving the coordinates fixed; left-multiplication by $e^𝔤$ corresponds to rotation of the coordinates while leaving the field fixed; and right-multiplication by either doesn't affect this function at all.

Of course, right-multiplication using other choices for $𝔤$ could certainly have some effect on this function, and this choice of $𝔤$ could have an effect on other functions. Note that right-multiplication can also be interpreted as left-multiplication, where the generator itself is rotated by the argument to the function. That is,

\[\begin{aligned} f(𝐐 e^𝔤) &= f(𝐐 e^{𝔤} 𝐐^{-1} 𝐐) = f(e^{𝔤'} 𝐐) \\ f(𝐐 e^{-𝔤}) &= f(𝐐 e^{-𝔤} 𝐐^{-1} 𝐐) = f(e^{-𝔤'} 𝐐), \end{aligned}\]

where $𝔤' = 𝐐 𝔤 𝐐^{-1}$. In this example, $𝔤'$ generates a rotation by an angle $α$ about the point in question, which leaves that point fixed, and since this is a scalar function it has no effect on the value. Of course, we will see below that changing by a phase proportional to $α$ is the defining feature of a spin-weighted function.

Differential rotations

We now define a pair of operators that differentiate a function with respect to infinitesimal rotations we apply to the functions themselves:

\[\begin{aligned} L_{𝔤} f(𝐐) &= \lambda \left. \frac{\partial} {\partial θ} f \left( e^{-θ 𝔤 / 2} 𝐐 \right) \right|_{θ=0}, \\ R_{𝔤} f(𝐐) &= \rho \left. \frac{\partial} {\partial θ} f \left( 𝐐 e^{-θ 𝔤 / 2} \right) \right|_{θ=0}. \end{aligned}\]

Here, we have introduced the constants $\lambda$ and $\rho$ because we will actually be able to derive their values — up to signs — based on the requirement that raising and lowering operators exist for each. Finally, we will choose the signs based on demands that these operators correspond as naturally as possible to the standard canonical angular-momentum operators.

Note that when composing operators, it is critical to keep track of the order of operations, which may look slightly unnatural:

\[\begin{aligned} L_𝔤 L_𝔥 f(𝐐) % &= \left. \lambda \frac{\partial} {\partial γ} f'\left(e^{-γ 𝔤 / 2} 𝐐 \right) \right|_{γ=0}, \\ &= \left. \lambda^2 \frac{\partial} {\partial γ} \frac{\partial} {\partial \eta} f\left(e^{-\eta 𝔥 / 2} e^{-γ 𝔤 / 2} 𝐐 \right) \right|_{γ=\eta=0}, \\ R_𝔤 R_𝔥 f(𝐐) % &= \rho \left. \frac{\partial} {\partial γ} f' \left( 𝐐 e^{-γ 𝔤 / 2} \right) \right|_{γ=0} \\ &= \left. \rho^2 \frac{\partial} {\partial γ} \frac{\partial} {\partial \eta} f\left( 𝐐 e^{-γ 𝔤 / 2} e^{-\eta 𝔥 / 2} \right) \right|_{γ=\eta=0}. \end{aligned}\]

We can prove the first of these, for example, by defining $f'(𝐐) = L_𝔥 f(𝐐)$, then applying the definition of $L_𝔤$ to $f'(𝐐)$, and finally substituting the definition of $f'$ back in. If we failed to use the correct order of operations, we would get sign errors when trying to evaluate the commutators.

These operators have some nice properties. For any scalar $s$, we have

\[\begin{aligned} L_{s 𝔤} &= s L_{𝔤}, \\ R_{s 𝔤} &= s R_{𝔤}. \end{aligned}\]

Given any basis $𝐞_n$ for the quaternions, we can use the multivariable chain rule to expand the operators in terms of components:

\[\begin{aligned} L_{𝔤} &= \sum_n g_n\, L_{𝐞_n}, \\ R_{𝔤} &= \sum_n g_n\, R_{𝐞_n}. \end{aligned}\]

This implies that vector addition holds more generally:

\[\begin{aligned} L_{𝔤 + 𝔥} &= L_{𝔤} + L_{𝔥} \\ R_{𝔤 + 𝔥} &= R_{𝔤} + R_{𝔥}. \end{aligned}\]

Moreover, we can show that these operators form a Lie algebra with the commutator as the Lie bracket. That is, we have

\[\begin{aligned} [L_{𝔤}, L_{𝔥}] &= \frac{\lambda}{2} L_{[𝔤, 𝔥]}, \\ [R_{𝔤}, R_{𝔥}] &= -\frac{\rho}{2} R_{[𝔤, 𝔥]}, \\ [L_{𝔤}, R_{𝔥}] &= 0. \end{aligned}\]

Conventionally, we single out the $𝐳$ axis — or equivalently the generator $𝐤 = 𝐲𝐱$ — as a sort of fiducial axis, and $L_z = L_𝐤$ and $R_z = R_𝐤$ as the fiducial operators. Then, by definition, their raising operators $L_+$ and $R_+$ and lowering operators $L_-$ and $R_-$ satisfy

\[\begin{aligned} [L_z, L_\pm] &= \pm L_\pm, \\ [R_z, R_\pm] &= \pm R_\pm. \end{aligned}\]

Assuming that the raising and lowering operators can be written as linear combinations of the basis operators, these equations imply that they have no component proportional $L_𝐳$, and that both of the remaining components must be nonzero. This actually allows us to deduce that $\lambda^2 = \rho^2 = -1$. This, in turn, allows us to deduce the values of the raising and lowering operators up to an overall factor. Conventionally the factor is chosen so that

\[\begin{aligned} L_\pm &= L_𝐱 \pm i L_𝐲, \\ R_\pm &= R_𝐱 \pm i R_𝐲. \end{aligned}\]

To pin down $\lambda$ and $\rho$, suppose the raising operator for $L_z$ is some linear combination $L_+ = a L_𝐱 + b L_𝐲 + c L_𝐳$. Using the commutator relation above together with $[𝐤, 𝐢] = 2𝐣$ and $[𝐤, 𝐣] = -2𝐢$, we find

\[[L_z, L_+] = \lambda \left( a L_𝐲 - b L_𝐱 \right).\]

Setting this equal to $L_+$ requires $c = 0$, $a = -\lambda b$, and $b = \lambda a$, which together imply

\[\lambda^2 = -1 \qquad \text{and} \qquad L_+ \propto L_𝐱 + \lambda L_𝐲.\]

Thus the mere existence of ladder operators forces $\lambda = \pm i$, and the sign of $\lambda$ is tied to which of the two combinations we choose to call the raising operator. Adopting the universal convention

\[L_\pm = L_𝐱 \pm i L_𝐲\]

therefore fixes $\lambda = i$. The same argument for the right operators, with the extra minus sign in their commutator relation, gives $[R_z, R_+] = -\rho\, (a R_𝐲 - b R_𝐱)$, hence $\rho^2 = -1$ and $R_+ \propto R_𝐱 - \rho R_𝐲$. Adopting the analogous convention

\[R_\pm = R_𝐱 \pm i R_𝐲\]

fixes $\rho = -i$. We therefore have

\[\begin{aligned} \lambda &= i, \\ \rho &= -i, \end{aligned}\]

which are the values used in the definitions given in the Summary. Two consequences deserve emphasis, both of which will be borne out below. First, $L$ then coincides with the standard angular-momentum operator of quantum mechanics ($L_z = -i\partial_\phi$ on the sphere), with the standard commutation relations $[L_a, L_b] = i\epsilon_{abc} L_c$. Second, $R$ obeys exactly the same commutation relations, and its fiducial component is $R_z = i\partial_γ$, which we will find to have eigenvalue $+s$ on a function of spin weight $s$. The opposite sign for $\rho$ — as used, for example, in the body-fixed operators of Wikipedia and in earlier versions of this package — leads to "anomalous" commutation relations with an extra minus sign and to an eigenvalue of $-s$.

Once the constants are fixed, the commutator relations read

\[[L_𝔤, L_𝔥] = \frac{i}{2} L_{[𝔤, 𝔥]}, \qquad [R_𝔤, R_𝔥] = \frac{i}{2} R_{[𝔤, 𝔥]}, \qquad [L_𝔤, R_𝔥] = 0,\]

and the Casimir operators $L^2 = L_𝐱^2 + L_𝐲^2 + L_𝐳^2$ and $R^2$ commute with everything. In fact $L^2 = R^2$, because both equal the Laplacian on $\mathrm{Spin}(3)$ up to a constant factor — see the section on Laplacians.

Angular-momentum operators in Euler angles

Having defined $L$ and $R$ in terms of the group structure, we can express them in any coordinate system over the group $\mathrm{Spin}(3)$ — most obviously, in the system of Euler angles, which will help us make contact with the standard expressions in the literature. The procedure is not geometrically insightful, but it is algebraically mechanical: write $e^{-ϵ𝐮/2}\, 𝐑_{α, β, γ}$ (or $𝐑_{α, β, γ}\, e^{-ϵ𝐮/2}$) in terms of its quaternion components, equate this to a new rotor $𝐑_{α', β', γ'}$ and extract the new Euler angles $(α', β', γ')$ as functions of $ϵ$, then differentiate at $ϵ=0$, and apply the chain rule. This is done symbolically on the $L_j$ and $R_j$ with Euler angles page, which also verifies the commutation relations claimed above. The results are collected in the Summary; the essential ones are

\[L_𝐤 = -i \frac{\partial}{\partial α} \qquad \text{and} \qquad R_𝐤 = i \frac{\partial}{\partial γ}.\]

Restricting to spin-weight-0 functions via $(α, β, γ) = (ϕ, θ, 0)$ reproduces the textbook expressions for $L_x$, $L_y$, $L_z$, and $L_\pm$ in spherical coordinates, as shown there. The $R$ operators have no analog on the 2-sphere: their expressions retain derivatives with respect to $γ$, which is the first sign that spin-weighted functions cannot be defined on $𝕊²$.

Wigner's 𝔇 matrices

Rotation operators

Sakurai [19] says that

Because rotations affect physical systems, the state ket corresponding to a rotated system is expected to look different from the state ket corresponding to the original unrotated system. Given a rotation operation $R$, characterized by a $3×3$ orthogonal matrix $R$, we associate an operator $𝒟(R)$ in the appropriate ket space such that

\[|α\rangle_R = 𝒟(R) |α\rangle,\]

where $|α\rangle_R$ and $|α\rangle$ stand for the kets of the rotated and original system, respectively.

In our setting the "kets" are complex-valued functions on $\mathrm{Spin}(3)$, and we saw above that rotating the field by $𝐑$ while leaving the coordinates fixed means rotating the argument by $𝐑^{-1}$. We therefore define the rotation operator $U(𝐑)$ by

\[\left[U(𝐑) f\right](𝐐) = f\left(𝐑^{-1}\, 𝐐\right).\]

A short calculation shows that $U(𝐑_1)\, U(𝐑_2) = U(𝐑_1 𝐑_2)$, so $U$ is a representation of $\mathrm{Spin}(3)$ on the space of functions.^[3] It is unitary with respect to the inner product defined by the invariant measure, and it commutes with every right operator $R_𝐮$ (left and right multiplication commute). For an infinitesimal rotation we have

\[\begin{aligned} \left[U\left(e^{ϵ 𝐮/2}\right) f\right](𝐑) &= f\left(e^{-ϵ 𝐮/2}𝐑\right) \\ &\approx f\left(𝐑\right) + ϵ \left. \frac{d}{dϵ} \right|_{ϵ=0} f\left(e^{-ϵ 𝐮/2}𝐑\right) \\ &= f\left(𝐑\right) - i ϵ L_𝐮 f\left(𝐑\right), \end{aligned}\]

which is precisely Sakurai's Eq. (3.1.15),

\[𝒟\left(\hat{𝐧}, dϕ \right) = 1 - i \left( 𝐉 \cdot \hat{𝐧} \right) dϕ,\]

with $𝐉 \to 𝐋$. Exponentiating, $U(e^{ϑ 𝐮/2}) = \exp(-i ϑ L_𝐮)$, and in particular for Euler angles

\[U(𝐑_{α, β, γ}) = \exp[-iα L_z]\, \exp[-iβ L_y]\, \exp[-iγ L_z].\]

This is the operator whose matrix elements every source in our priority list (LALSuite, Wikipedia, Sakurai, Shankar, Zettili, Varshalovich et al.) uses to define Wigner's $𝔇$ matrices; we do the same.

Definition

Let $|ℓ, m\rangle$ denote an orthonormal set of simultaneous eigenfunctions of $L^2,$ $L_z,$ and $R_z,$ with eigenvalues $ℓ(ℓ+1),$ $m,$ and some fixed $s$. (The value of $s$ is irrelevant here because $U(𝐑)$ commutes with $R_z$; for integer $ℓ$ we may take $s=0$, so that the $|ℓ, m\rangle$ are just the ordinary spherical harmonics defined below.) Because $U(𝐑)$ commutes with $L^2$, it maps the $(2ℓ+1)$-dimensional eigenspace of $L^2$ to itself, and we define Wigner's $𝔇$ matrix as the matrix of $U(𝐑)$ in this basis — Sakurai's Eq. (3.5.42):

\[𝔇^{(ℓ)}_{m',m}(𝐑) = \langle ℓ, m' | U(𝐑) | ℓ, m \rangle, \qquad \text{equivalently} \qquad U(𝐑)\, |ℓ, m\rangle = \sum_{m'} |ℓ, m'\rangle\, 𝔇^{(ℓ)}_{m',m}(𝐑).\]

Here $ℓ \in \{0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots\}$ and $m', m \in \{-ℓ, -ℓ+1, \ldots, ℓ\}$, so that $ℓ$, $m'$, and $m$ are either all integers or all half-integers. In particular, we note three points about this definition, because they are the main points on which the literature disagrees:

1. Argument. $𝔇$ is a function of the rotation $𝐑$ itself — the rotation applied to the field — not of its inverse.
2. Index order. The first index $m'$ is attached to the bra, and will be found below to pair with the first Euler angle $α$ and with the left operators $L$; the second index $m$ is attached to the ket, and pairs with the last Euler angle $γ$ and the right operators $R$.
3. Conjugation. With $U$ as defined above, $𝔇$ involves the phases $e^{-i m' α}$ and $e^{-i m γ}$ — see below. Several sources (including Wigner, Edmonds, Goldberg et al., and Boyle (2016), whose convention was used by versions of this package before 3.0) define a matrix that is the complex conjugate of this one, possibly with additional signs $(-1)^{m'-m}$ or transposition. The comparisons pages record the precise relationship for each source.

Basic properties

Because $U$ is a unitary representation, $𝔇$ is too — as summarized by Sakurai's Eq. (3.5.46):

\[𝔇^{(ℓ)}(𝐑_1\, 𝐑_2) = 𝔇^{(ℓ)}(𝐑_1)\, 𝔇^{(ℓ)}(𝐑_2), \qquad 𝔇^{(ℓ)}(𝟏) = 𝟏, \qquad 𝔇^{(ℓ)}(𝐑^{-1}) = 𝔇^{(ℓ)}(𝐑)^{-1} = 𝔇^{(ℓ)}(𝐑)^\dagger,\]

where the products are matrix products in the indices $(m', m)$. Since $e^{π 𝐮} = -𝟏$ for any unit vector $𝐮$, and $U(e^{π𝐮}) = \exp(-i2π L_𝐮)$ has eigenvalues $e^{-i2π m}$ on the eigenspace of $L^2$, we also have

\[𝔇^{(ℓ)}(-𝐑) = (-1)^{2ℓ}\, 𝔇^{(ℓ)}(𝐑).\]

The step from the eigenvalues to that equation deserves a word. The eigenvalues of $L_𝐮$ on the $ℓ$ eigenspace of $L^2$ are the $2ℓ+1$ values $m = -ℓ, \ldots, ℓ$, so a priori $U(-𝟏)$ could act differently on each. But $m$ and $ℓ$ differ by an integer, so $e^{-i2π m} = (-1)^{2m} = (-1)^{2ℓ}$ for every $m$ in that range: $U(-𝟏)$ is the scalar $(-1)^{2ℓ}$ times the identity there. Hence $𝔇^{(ℓ)}(-𝐑) = 𝔇^{(ℓ)}(-𝟏)\, 𝔇^{(ℓ)}(𝐑) = (-1)^{2ℓ}\, 𝔇^{(ℓ)}(𝐑)$, independently of $m$ or $m'$.

That is, for integer $ℓ$ the two quaternions representing a given rotation give the same matrix, so $𝔇^{(ℓ)}$ is a representation of $\mathrm{SO}(3)$; for half-integer $ℓ$ it is a representation of $\mathrm{Spin}(3)$ only, and changes sign rather than returning to the same value under a rotation through $2π$. This is the group-theoretic reason that the Euler angle $γ$ must range over $[0, 4π)$ to cover $\mathrm{Spin}(3)$.

The other basic symmetry follows from the reality of the $d$ matrix introduced below and the structure of the phases:

\[\overline{𝔇^{(ℓ)}_{m',m}(𝐑)} = (-1)^{m'-m}\, 𝔇^{(ℓ)}_{-m',-m}(𝐑).\]

Note that $m'-m$ is always an integer, so this holds for half-integer $ℓ$ as well.

Euler-angle form and the $d$ matrix

Using the Euler-angle factorization of $U$ and the eigenvalue property of $L_z$ on the basis kets, we find — Sakurai's Eq. (3.5.50) —

\[\begin{aligned} 𝔇^{(ℓ)}_{m',m}(α, β, γ) &= \langle ℓ, m' | \exp[-iL_z α]\exp[-iL_y β]\exp[-iL_z γ] | ℓ, m \rangle \\ &= e^{-i m' α}\, d^{(ℓ)}_{m',m}(β)\, e^{-i m γ}, \end{aligned}\]

where the "small" $d$ matrix is

\[d^{(ℓ)}_{m',m}(β) = \langle ℓ, m' | \exp[-iL_y β] | ℓ, m \rangle.\]

We derive the phases very explicitly. Recall that in general

\[\left(\left\langle ψ | A\, B\, C | \chi \right\rangle\right)^\ast = \left\langle \chi | C^\dag\, B^\dag\, A^\dag | ψ \right\rangle, \qquad \left( e^{-i ϵ L_u} \right)^\dag = e^{i ϵ L_u^\dag} = e^{i ϵ L_u},\]

so that $\langle ℓ, m' | e^{-iαL_z} = \left(e^{iαL_z} |ℓ, m'\rangle \right)^\dagger = e^{-i m' α} \langle ℓ, m'|$, and similarly on the right. Because $L_y = (L_+ - L_-)/2i$ has purely imaginary matrix elements in the standard basis (whose ladder coefficients are real and positive — the Condon–Shortley convention discussed below), $d$ is real. Wigner's explicit formula for it is

\[d^{(ℓ)}_{m',m}(β) = \sum_{k} (-1)^{k - m + m'} \frac{\sqrt{(ℓ+m)!\,(ℓ-m)!\,(ℓ+m')!\,(ℓ-m')!}} {(ℓ+m-k)!\,k!\,(ℓ-m'-k)!\,(k-m+m')!} \left(\cos\frac{β}{2}\right)^{2ℓ+m-m'-2k} \left(\sin\frac{β}{2}\right)^{2k-m+m'},\]

where the sum runs over all integers $k$ for which the factorials have non-negative arguments, namely $\max(0, m-m') \leq k \leq \min(ℓ+m, ℓ-m')$. This is the form given by Sakurai (Eq. 3.8.33), Zettili (Eq. 7.56), and Wikipedia, and it is the form implemented by LALSuite. The simplest cases are

\[d^{(1/2)}(β) = \begin{pmatrix} \cos\frac{β}{2} & -\sin\frac{β}{2} \\ \sin\frac{β}{2} & \phantom{-}\cos\frac{β}{2} \end{pmatrix}, \qquad d^{(1)}(β) = \begin{pmatrix} \frac{1+\cos β}{2} & -\frac{\sin β}{\sqrt{2}} & \frac{1-\cos β}{2} \\ \frac{\sin β}{\sqrt{2}} & \cos β & -\frac{\sin β}{\sqrt{2}} \\ \frac{1-\cos β}{2} & \frac{\sin β}{\sqrt{2}} & \frac{1+\cos β}{2} \end{pmatrix},\]

with rows indexed by $m'$ and columns by $m$, both decreasing from $ℓ$ to $-ℓ$ (the usual textbook layout). The $d$ matrix satisfies

\[\begin{aligned} d^{(ℓ)}_{m',m}(β) &= (-1)^{m'-m}\, d^{(ℓ)}_{m,m'}(β) = d^{(ℓ)}_{-m,-m'}(β), \\ d^{(ℓ)}_{m',m}(-β) &= d^{(ℓ)}_{m,m'}(β), \\ d^{(ℓ)}_{m',m}(π - β) &= (-1)^{ℓ+m'}\, d^{(ℓ)}_{m',-m}(β), \\ d^{(ℓ)}_{m',m}(π) &= (-1)^{ℓ-m}\, δ_{m',-m}, \\ d^{(ℓ)}_{m',m}(2π + β) &= (-1)^{2ℓ}\, d^{(ℓ)}_{m',m}(β), \end{aligned}\]

which are the standard relations found, e.g., on Wikipedia. Note that this package does not compute $d$ from the formula above, which is numerically disastrous for large $ℓ$; the formula is the definition against which the recursive algorithms (documented in the Notes) are tested.

Differential relations

Viewed as functions on $\mathrm{Spin}(3)$, the matrix elements of $𝔇$ are eigenfunctions of our differential operators. The representation property gives $𝔇_{m',m}(𝐑\, e^{-ϵ𝐮/2}) = \sum_k 𝔇_{m',k}(𝐑)\, 𝔇_{k,m}(e^{-ϵ𝐮/2})$, and $𝔇_{k,m}(e^{-ϵ𝐮/2}) = \langle ℓ, k| e^{iϵL_𝐮} |ℓ, m\rangle$, so differentiating the definitions of $R_𝐮$ and (similarly) $L_𝐮$ yields

\[R_𝐮\, 𝔇^{(ℓ)}_{m',m} = \sum_k 𝔇^{(ℓ)}_{m',k}\, (J_𝐮)_{k,m}, \qquad L_𝐮\, 𝔇^{(ℓ)}_{m',m} = -\sum_k (J_𝐮)_{m',k}\, 𝔇^{(ℓ)}_{k,m},\]

where $(J_𝐮)_{k,m} = \langle ℓ, k | L_𝐮 | ℓ, m \rangle$ are the standard $(2ℓ+1)$-dimensional angular-momentum matrices. Inserting the standard matrix elements of $J_z$ and $J_\pm$ gives

\[\begin{aligned} L^2\, 𝔇^{(ℓ)}_{m',m} &= R^2\, 𝔇^{(ℓ)}_{m',m} = ℓ(ℓ+1)\, 𝔇^{(ℓ)}_{m',m}, \\ L_z\, 𝔇^{(ℓ)}_{m',m} &= -m'\, 𝔇^{(ℓ)}_{m',m}, & R_z\, 𝔇^{(ℓ)}_{m',m} &= m\, 𝔇^{(ℓ)}_{m',m}, \\ L_\pm\, 𝔇^{(ℓ)}_{m',m} &= -\sqrt{(ℓ \pm m')(ℓ \mp m' + 1)}\, 𝔇^{(ℓ)}_{m' \mp 1, m}, & R_\pm\, 𝔇^{(ℓ)}_{m',m} &= \sqrt{(ℓ \mp m)(ℓ \pm m + 1)}\, 𝔇^{(ℓ)}_{m', m \pm 1}. \end{aligned}\]

These can be checked directly on the Euler-angle form: $L_z = -i\partial_α$ acting on $e^{-im'α}$ gives $-m'$, while $R_z = i\partial_γ$ acting on $e^{-imγ}$ gives $m$. The minus sign and the reversed direction of the $L$ ladder are unavoidable consequences of defining $𝔇$ through matrix elements of $U(𝐑)$: as a function of its argument, $𝔇_{m',m}$ behaves like a row of expansion coefficients rather than like a basis function. It is the complex conjugate $\overline{𝔇_{m',m}}$ that behaves like a wavefunction, with $L_z$ eigenvalue $+m'$ and $R_z$ eigenvalue $-m$ — which is exactly why the spin-weighted spherical harmonics below are proportional to $\overline{𝔇_{m,-s}}$. (Wikipedia states the same fact in the language of the symmetric top: it is $D^{j\ast}_{m'm}$ that is the eigenfunction of the space-fixed $𝒥_z$ with eigenvalue $m'$ and of the body-fixed $𝒫_z = -R_z$ with eigenvalue $m$.)

Orthogonality and completeness

By the Peter–Weyl theorem (Schur orthogonality), the matrix elements of the irreducible unitary representations form a complete orthogonal basis of $L^2(\mathrm{Spin}(3))$. With the invariant measure normalized to total volume $2π^2$, the orthogonality relation reads

\[\int_{\mathrm{Spin}(3)} \overline{𝔇^{(ℓ_1)}_{m'_1, m_1}(𝐑)}\, 𝔇^{(ℓ)}_{m', m}(𝐑)\, d^3Ω = \frac{2π^2}{2ℓ+1}\, δ_{ℓ_1, ℓ}\, δ_{m'_1, m'}\, δ_{m_1, m}.\]

Restricting the sum over $ℓ$ to integers gives the corresponding statement for $\mathrm{SO}(3)$, whose volume in this normalization is $π^2$. This is the sense in which the $𝔇$ matrices generalize Fourier analysis from the circle to the rotation group, and it is the basis of the spin-weighted spherical-harmonic transforms (SSHTs) implemented in this package.

Spherical harmonics and spin-weighted spherical harmonics

Spherical harmonics

Fortunately, there does not seem to be any disagreement in the physics literature about the definition of the spherical harmonics $Y_{ℓ, m}(θ, ϕ)$; everyone uses the Condon–Shortley convention, and our $Y_{ℓ,m}$ is the standard one. The only place where care is needed is that many sources define $Y_{ℓ,m}$ in terms of associated Legendre functions, for which there is a sign ambiguity $(-1)^m$ that is sometimes absorbed into the Legendre function and sometimes not. To be unambiguous, we go back to the original.

The Condon–Shortley phase convention is a choice of phase factors in the definition of the spherical harmonics that requires the coefficients in

\[L_{\pm} |ℓ,m\rangle = α^{\pm}_{ℓ,m} |ℓ, m \pm 1\rangle\]

to be real and positive, together with the requirement that $Y_{ℓ,0}$ be real and positive on the positive $𝐳$ axis. The reasoning behind this choice is explained more fully in Section 2 of Ufford and Shortley (1932). As a practical matter, Condon and Shortley's Section 4³, Eq. (15) (page 52) gives

\[\Theta(ℓ, m) = (-1)^ℓ \sqrt{\frac{2ℓ+1}{2} \frac{(ℓ+m)!}{(ℓ-m)!}} \frac{1}{2^ℓ ℓ!} \frac{1}{\sin^mθ} \frac{d^{ℓ-m}}{d(\cos θ)^{ℓ-m}} \sin^{2ℓ}θ,\]

and the spherical harmonic is $Y_{ℓ,m}(θ, ϕ) = \Theta(ℓ, m)\, \Phi(m)$ with $\Phi(m) = e^{imϕ} / \sqrt{2π}$ from their Eq. (5). This expression is tested directly against this package (see the Condon–Shortley comparison page), so we can definitively say that the spherical-harmonic functions provided by this package obey the Condon–Shortley phase convention. For example,

\[Y_{0,0} = \frac{1}{\sqrt{4π}}, \qquad Y_{1,0} = \sqrt{\frac{3}{4π}} \cos θ, \qquad Y_{1,\pm 1} = \mp \sqrt{\frac{3}{8π}} \sin θ\, e^{\pm iϕ}.\]

An equivalent closed form that avoids Legendre functions altogether — and is the $s=0$ case of the general expression for spin-weighted spherical harmonics given below — is

\[\begin{aligned} Y_{ℓ,m}(θ, ϕ) &= \sqrt{\frac{2ℓ+1}{4π}}\, e^{imϕ} \sum_{k = k_1}^{k_2} \frac{(-1)^k\, ℓ!\, [(ℓ+m)!(ℓ-m)!]^{1/2}} {(ℓ+m-k)!\,(ℓ-k)!\,k!\,(k-m)!} \left(\cos\frac{θ}{2}\right)^{2ℓ+m-2k} \left(\sin\frac{θ}{2}\right)^{2k-m}, \end{aligned}\]

where $k_1 = \max(0, m)$ and $k_2 = \min(ℓ+m, ℓ)$.

Now, $Y_{ℓ,m}$ is a function on $𝕊²$, but we have argued that the natural domain for everything in this package is $\mathrm{Spin}(3)$. We lift it in the obvious way: for a unit quaternion $𝐑$ that rotates $𝐳$ onto the direction with spherical coordinates $(θ, ϕ)$ — that is, for $𝐑 = 𝐑_{θ, ϕ}\, e^{γ 𝐤 / 2}$ with any $γ$ — we define $Y_{ℓ,m}(𝐑) = Y_{ℓ,m}(θ, ϕ)$. The lifted function is independent of $γ$, so $R_z Y_{ℓ,m} = 0$: the spherical harmonics have spin weight zero. Comparing the closed form above with Wigner's formula for $d$, we find

\[Y_{ℓ,m}(𝐑) = \sqrt{\frac{2ℓ+1}{4π}}\, \overline{𝔇^{(ℓ)}_{m,0}(𝐑)},\]

which is the standard relation (e.g., on Wikipedia), and — recalling the differential relations — is consistent with $L_z Y_{ℓ,m} = m Y_{ℓ,m}$ and $R_z Y_{ℓ,m} = 0$. The complex conjugate is essential; without it the sign of $m$ would be wrong.

Rotation of spherical harmonics

The definition of $𝔇$ as the matrix of $U(𝐑)$ in the basis $|ℓ, m\rangle$ is precisely the statement of how spherical harmonics transform under rotation:

\[\left[U(𝐑)\, Y_{ℓ,m}\right](𝐐) = Y_{ℓ,m}\left(𝐑^{-1}\, 𝐐\right) = \sum_{m'} 𝔇^{(ℓ)}_{m',m}(𝐑)\, Y_{ℓ,m'}(𝐐).\]

In words: the field $Y_{ℓ,m}$ rotated by $𝐑$ is a combination of the unrotated $Y_{ℓ,m'}$ with coefficients given by the column $m$ of $𝔇(𝐑)$. Using unitarity, the same law can be written with the rotation in the other position,

\[Y_{ℓ,m}\left(𝐑\, 𝐐\right) = \sum_{m'} \overline{𝔇^{(ℓ)}_{m,m'}(𝐑)}\, Y_{ℓ,m'}(𝐐),\]

which is the form given by Wigner (his Eq. A.8) and is the one that arises when evaluating the harmonics at a rotated point. The first form — with $𝐑^{-1}$ in the argument and no conjugate — is the one used by Sakurai, Le Bellac, Torres del Castillo, and others, and we take it as canonical. Exactly the same law holds for the spin-weighted spherical harmonics defined below, with $Y$ replaced by ${}_sY$, because $U(𝐑)$ commutes with $R_z$.

Rotation of mode weights

The law above is stated for the harmonics; the consequence for the weights of a function expanded in them is generally more useful. We start with $f(𝐐) = \sum_{ℓ,m} f_{ℓ,m}\, Y_{ℓ,m}(𝐐)$. The actively rotated field $f'$ — meaning that the field is rotated, rather than the coordinates of its argument — obeys $f'(𝐐) = f(𝐑^{-1}\, 𝐐)$. Substituting the law and relabelling the summation index gives

\[\begin{aligned} f'(𝐐) &= \sum_{ℓ,m'} f'_{ℓ,m'}\, Y_{ℓ,m'}(𝐐) \\ &= \sum_{ℓ,m} f_{ℓ,m}\, Y_{ℓ,m}(𝐑^{-1}\, 𝐐) \\ &= \sum_{ℓ,m} f_{ℓ,m}\, \sum_{m'} \overline{𝔇^{(ℓ)}_{m,m'}(𝐑^{-1})}\, Y_{ℓ,m'}(𝐐) \\ &= \sum_{ℓ,m',m} f_{ℓ,m}\, 𝔇^{(ℓ)}_{m',m}(𝐑)\, Y_{ℓ,m'}(𝐐) \\ &= \sum_{ℓ,m'} \left[ \sum_m 𝔇^{(ℓ)}_{m',m}(𝐑)\, f_{ℓ,m}\right] Y_{ℓ,m'}(𝐐). \end{aligned}\]

The first line is effectively the definition of $f'_{ℓ,m'}$; the second uses the relation between $f'$ and $f$, and the expansion of $f$; the third uses the rotation law for $Y_{ℓ,m}$ from above; the fourth just combines the sums, then uses the conjugate-transpose property of $𝔇$; and the fifth just groups the terms in a nice way. Comparing the first and last lines (or using orthonormality of the harmonics) gives the law for the weights of a rotated field:

\[f'_{ℓ,m'} = \sum_m 𝔇^{(ℓ)}_{m',m}(𝐑)\, f_{ℓ,m}.\]

Note that $𝔇^{(ℓ)}$ is seemingly transposed relative to the expression for the rotation of the harmonics themselves (the primes are in different places). This is closely analogous to the "opposite" transformation law that we saw for the differential operators.

Spin-weighted functions

Newman and Penrose [4] define the spherical tangent basis vectors as

\[m^\mu = \frac{1}{\sqrt{2}} \left( \boldsymbol{θ} + i \boldsymbol{ϕ} \right)^\mu\]

and discuss spin weight in terms of the rotation

\[(m^\mu)' = e^{iψ} m^\mu,\]

where the tangent basis rotates but we are "keeping the coordinates fixed". They then define a function $\eta$ to have spin weight $s$ if it transforms as

\[\eta' = e^{isψ} \eta.\]

Such functions are generally the result of contracting a tensor field with some number of $m^\mu$ and some number of $\bar{m}^\mu$ vectors (though spinor extensions resulting in half-integer spin weights are also possible).

This definition shows that it is impossible to define spin-weighted functions on the 2-sphere alone; the 2-sphere includes no information about the directions of basis vectors in its tangent space. Instead, spin-weighted functions live on the "unit tangent bundle" of the 2-sphere, which is homeomorphic to the 3-sphere — the space of unit quaternions. Thus, we think of spin-weighted functions as functions on $\mathrm{Spin}(3)$, and frequently discuss them in terms of Euler angles.

As we saw above, the basis $m^\mu$ corresponds to the Euler angles $(ϕ, θ, 0)$, while $(m^\mu)'$ corresponds to $(ϕ, θ, -ψ)$. Written as a function of Euler angles, the defining transformation becomes

\[\eta(ϕ, θ, -ψ) = e^{isψ} \eta(ϕ, θ, 0), \qquad \text{or} \qquad \eta(α, β, γ) = e^{-isγ} \eta(α, β, 0),\]

and, independently of the choice of Euler angles,

\[\eta\left(𝐐\, e^{γ 𝐤/2}\right) = e^{-isγ}\, \eta(𝐐).\]

This is the crucial definition giving us the behavior of spin-weighted functions: applying $R_z = i\partial_γ$ we find

\[R_z\, \eta = s\, \eta,\]

so spin-weighted functions are eigenfunctions of $R_z$ with eigenvalue equal to the spin weight. It was to make this sign come out positive that we chose $\rho = -i$ in the definition of $R$.

The spin-raising and -lowering operators — canonically denoted $\eth$ and $\bar{\eth}$ — were introduced by Newman and Penrose in their Eq. (3.8) as

\[\begin{aligned} \eth \eta &= -\sin^s θ \left\{ \frac{\partial}{\partial θ} + \frac{i}{\sin θ} \frac{\partial}{\partial ϕ} \right\} \left(\eta \sin^{-s} θ\right), \\ \bar{\eth} \eta &= -\sin^{-s} θ \left\{ \frac{\partial}{\partial θ} - \frac{i}{\sin θ} \frac{\partial}{\partial ϕ} \right\} \left(\eta \sin^{s} θ\right). \end{aligned}\]

These expressions look like operators on the 2-sphere, but that appearance is misleading: they depend on $s$, which encodes the missing information about the tangent basis. Writing out the Euler-angle expressions for $R_𝐱$ and $R_𝐲$, replacing $\partial_γ$ by $-is$ (as is legitimate when acting on a function of definite spin weight), and converting the remaining angles to spherical coordinates — see the calculation page — we find that these are just the ladder operators of $R_z$:

\[\eth = R_+ = R_𝐱 + i R_𝐲, \qquad \bar{\eth} = -R_- = -\left(R_𝐱 - i R_𝐲\right).\]

The minus sign in the second relation is real and worth remembering. It arises because $\bar{\eth}$ is defined as the complex-conjugate operator of $\eth$ ($\bar{\eth}\eta = \overline{\eth\bar{\eta}}$), and $R_\pm$ contain an explicit factor of $i$, so that $\overline{R_+ \bar{\eta}} = -R_- \eta$. As a result $R_-$ has the Condon–Shortley-positive coefficient $R_- {}_sY_{ℓ,m} = +\sqrt{(ℓ+s)(ℓ-s+1)}\, {}_{s-1}Y_{ℓ,m}$, whereas Newman–Penrose's operator has $\bar{\eth}\, {}_sY_{ℓ,m} = -\sqrt{(ℓ+s)(ℓ-s+1)}\, {}_{s-1}Y_{ℓ,m}$, exactly as in Eq. (2.7b) of Goldberg et al., and $\bar{\eth}\eth\, {}_sY_{ℓ,m} = -(ℓ-s)(ℓ+s+1)\, {}_sY_{ℓ,m}$. Since the literature on spin-weighted functions universally uses Newman–Penrose's $\bar{\eth}$, we keep their sign and simply note that $\bar{\eth} \neq R_-$. Either way, $[R_z, \eth] = \eth$ and $[R_z, \bar{\eth}] = -\bar{\eth}$.

Spin-weighted spherical harmonics

We now have everything needed to define the spin-weighted spherical harmonics as functions on $\mathrm{Spin}(3)$. We require ${}_sY_{ℓ,m}$ to be a simultaneous eigenfunction of $L^2$, $L_z$, and $R_z$ with eigenvalues $ℓ(ℓ+1)$, $m$, and $s$, normalized in the standard way, with phases fixed by the Condon–Shortley requirement that both sets of ladder operators have real, positive coefficients:

\[\begin{aligned} L_\pm\, {}_sY_{ℓ,m} &= \sqrt{(ℓ \mp m)(ℓ \pm m + 1)}\, {}_sY_{ℓ,m \pm 1}, \\ R_\pm\, {}_sY_{ℓ,m} &= \sqrt{(ℓ \mp s)(ℓ \pm s + 1)}\, {}_{s \pm 1}Y_{ℓ,m}. \end{aligned}\]

The differential relations show that $\overline{𝔇^{(ℓ)}_{m,-s}}$ has exactly the required eigenvalues, so ${}_sY_{ℓ,m} = c_{ℓ,s}\, \overline{𝔇^{(ℓ)}_{m,-s}}$ for some constants $c_{ℓ,s}$, which the ladder conditions relate to one another. Because $R_\pm$ contain an explicit factor of $i$, they do not commute with complex conjugation; rather $R_+ \bar{g} = -\overline{R_- g}$, and therefore

\[R_+ \overline{𝔇^{(ℓ)}_{m,-s}} = -\overline{R_- 𝔇^{(ℓ)}_{m,-s}} = -\sqrt{(ℓ-s)(ℓ+s+1)}\; \overline{𝔇^{(ℓ)}_{m,-s-1}},\]

while $L_+ \overline{𝔇^{(ℓ)}_{m,-s}} = +\sqrt{(ℓ-m)(ℓ+m+1)}\, \overline{𝔇^{(ℓ)}_{m+1,-s}}$. Positivity of the $L_+$ coefficient is thus automatic, but positivity of the $R_+$ coefficient requires $c_{ℓ,s+1} = -c_{ℓ,s}$: the phase must alternate with $s$. Identifying the $s=0$ case with the standard spherical harmonics fixes $c_{ℓ,0} = \sqrt{(2ℓ+1)/4π}$, and we obtain the definition

\[\boxed{ {}_sY_{ℓ,m}(𝐑) = (-1)^s \sqrt{\frac{2ℓ+1}{4π}}\; \overline{𝔇^{(ℓ)}_{m,-s}(𝐑)} }\]

The factor $(-1)^s$ is therefore not an arbitrary phase: it is the Condon–Shortley convention applied to $R_\pm$, and it is exactly the factor that appears in the standard expressions of the gravitational-wave literature — LALSuite, the NINJA paper [26], and (up to their differing $𝔇$ convention) Goldberg et al.. In Euler angles, $\overline{𝔇_{m,-s}(ϕ, θ, γ)} = e^{imϕ}\, d_{m,-s}(θ)\, e^{-isγ}$, so restricting to $γ=0$ gives the familiar

\[{}_sY_{ℓ,m}(θ, ϕ) = (-1)^s \sqrt{\frac{2ℓ+1}{4π}}\; d^{(ℓ)}_{m,-s}(θ)\, e^{imϕ},\]

and inserting Wigner's formula for $d$ gives the explicit expression

\[\begin{aligned} {}_{s}Y_{ℓ,m}(θ, ϕ) &= (-1)^s\sqrt{\frac{2ℓ+1}{4π}}\, e^{imϕ} \sum_{k = k_1}^{k_2} \frac{(-1)^k[(ℓ+m)!(ℓ-m)!(ℓ-s)!(ℓ+s)!]^{1/2}} {(ℓ+m-k)!\,(ℓ+s-k)!\,k!\,(k-s-m)!} \left(\cos\frac{θ}{2}\right)^{2ℓ+m+s-2k} \left(\sin\frac{θ}{2}\right)^{2k-s-m}, \end{aligned}\]

where $k_1 = \max(0, m+s)$ and $k_2 = \min(ℓ+m, ℓ+s)$. This is Eq. (II.7)–(II.8) of the NINJA paper and the expression coded in LALSuite; both are tested against this package on the comparisons pages. Again, this package does not use this form; it is shown here to make comparison with other sources easy.

Summarizing the properties of ${}_sY_{ℓ,m}$ that follow directly from those of $𝔇$:

\[\begin{aligned} L^2\, {}_sY_{ℓ,m} &= ℓ(ℓ+1)\, {}_sY_{ℓ,m}, \qquad L_z\, {}_sY_{ℓ,m} = m\, {}_sY_{ℓ,m}, \qquad R_z\, {}_sY_{ℓ,m} = s\, {}_sY_{ℓ,m}, \\ {}_sY_{ℓ,m}\left(𝐑\, e^{γ𝐤/2}\right) &= e^{-isγ}\, {}_sY_{ℓ,m}(𝐑), \qquad {}_sY_{ℓ,m}\left(𝐑^{-1}\, 𝐐\right) = \sum_{m'} 𝔇^{(ℓ)}_{m',m}(𝐑)\, {}_sY_{ℓ,m'}(𝐐), \\ \overline{{}_sY_{ℓ,m}} &= (-1)^{m+s}\, {}_{-s}Y_{ℓ,-m}, \qquad {}_sY_{ℓ,m}(-𝐑) = (-1)^{2ℓ}\, {}_sY_{ℓ,m}(𝐑), \\ \int_{\mathrm{Spin}(3)} \overline{{}_{s'}Y_{ℓ',m'}}\; {}_sY_{ℓ,m}\; d^3Ω &= \frac{π}{2}\, δ_{ℓ',ℓ}\, δ_{m',m}\, δ_{s',s}, \qquad \int_{𝕊^2} \overline{{}_{s}Y_{ℓ',m'}(θ,ϕ)}\; {}_sY_{ℓ,m}(θ,ϕ)\; \sin θ\, dθ\, dϕ = δ_{ℓ',ℓ}\, δ_{m',m}. \end{aligned}\]

The first orthogonality relation is the natural one on $\mathrm{Spin}(3)$; the second is the conventional one, which integrates over the 2-sphere at fixed $γ$ and is valid only when both functions have the same spin weight (functions of different spin weight are not orthogonal on $𝕊²$, and the integral is not even coordinate-independent). The conjugation relation is the one given by Goldberg et al. (their Eq. 2.6); note that the exponent $m+s$ is always an integer.

Rotation of spin-weighted spherical harmonics and modes

Using the relationship between ${}_sY_{ℓ,m}$ and $𝔇$, the rotation law for the spin-weighted spherical harmonics is

\[\begin{aligned} \left[U(𝐑)\, {}_sY_{ℓ,m}\right](𝐐) &= {}_sY_{ℓ,m}\left(𝐑^{-1}\, 𝐐\right) \\ &= (-1)^s \sqrt{\frac{2ℓ+1}{4π}}\; \overline{𝔇^{(ℓ)}_{m,-s}(𝐑^{-1}\, 𝐐)} \\ &= (-1)^s \sqrt{\frac{2ℓ+1}{4π}}\; \sum_{m'} \overline{𝔇^{(ℓ)}_{m,m'}(𝐑^{-1})}\; \overline{𝔇^{(ℓ)}_{m',-s}(𝐐)} \\ &= \sum_{m'} \overline{𝔇^{(ℓ)}_{m,m'}(𝐑^{-1})}\; {}_sY_{ℓ,m'}(𝐐) \\ &= \sum_{m'} {}_sY_{ℓ,m'}(𝐐)\, 𝔇^{(ℓ)}_{m',m}(𝐑). \end{aligned}\]

This is precisely the same law as for the ordinary spherical harmonics, with extra $s$ indices added. And exactly the same reasoning as for the rotation of scalar mode weights gives the law for the weights of a rotated spin-weighted function:

\[{}_sf'_{ℓ,m'} = \sum_m 𝔇^{(ℓ)}_{m',m}(𝐑)\, {}_sf_{ℓ,m}.\]

Again, $f'$ is the actively rotated field, with the coordinates of its argument unchanged.

Half-integer indices

Everything above is valid when $ℓ$, $m$, and $s$ are all half-integers, with one exception: the factor $(-1)^s$ is then $\pm i$, and a branch must be chosen. It is worth being precise about what is and is not forced here. The ladder conditions determine all the ${}_sY_{ℓ,m}$ for a given $ℓ$ up to one overall phase $c_ℓ$, and there is no $s=0$ member to anchor it. Requiring the conjugation relation $\overline{{}_sY_{ℓ,m}} = (-1)^{m+s}\, {}_{-s}Y_{ℓ,-m}$ to hold as written forces $c_ℓ$ to be real, so $c_ℓ = \pm 1$; requiring the standard product formula (with real Clebsch–Gordan coefficients) to hold across integer and half-integer $ℓ$ forces the same sign for every half-integer $ℓ$. Nothing forces that one remaining global sign, because $(-1)^{2s}$ is itself multiplicative. This is the exact analog of the arbitrary choice "$Y_{ℓ,0}$ is positive at the north pole" in the integer case. We choose the principal branch,

\[(-1)^s \equiv e^{iπ s} = i^{2s},\]

so that the definition above is well defined for all $s$, and the anchor condition can be stated uniformly as

\[{}_sY_{ℓ,-s}(𝟏) = i^{2s} \sqrt{\frac{2ℓ+1}{4π}}\]

(recall that $𝔇^{(ℓ)}(𝟏)$ is the identity, so only $m=-s$ survives at $𝐑=𝟏$). The same branch is used for the prefactor of the explicit formula above.

A related trap: using the conjugation symmetry of $𝔇$, the definition can be rewritten without a complex conjugate as

\[{}_sY_{ℓ,m}(𝐑) = (-1)^s (-1)^{m+s} \sqrt{\frac{2ℓ+1}{4π}}\; 𝔇^{(ℓ)}_{-m,s}(𝐑) = e^{iπ(m+2s)} \sqrt{\frac{2ℓ+1}{4π}}\; 𝔇^{(ℓ)}_{-m,s}(𝐑).\]

For integer indices this is simply $(-1)^m \sqrt{(2ℓ+1)/4π}\, 𝔇^{(ℓ)}_{-m,s}(𝐑)$, which is a perfectly good alternative form. For half-integer $s$, however, the naive replacement $e^{iπ(m+2s)} \to (-1)^m = e^{iπm}$ is off by $(-1)^{2s} = -1$. We therefore treat the conjugate form as the definition and the $𝔇_{-m,s}$ form as an integer-index corollary.

Laplacians

Bander and Itzykson [27] show that Wigner's D matrices (extended to the full space of quaternions with arbitrary norm) are harmonic with respect to the Laplacian of the full 4-D space. We also know that

\[\Delta_{𝕊^{n-1}} f(x) = \Delta_{\mathbb{R}^n} f(x/|x|),\]

and

\[\Delta_{\mathbb{R}^n} f(x) = \frac{1}{r^{n-1}} \frac{\partial}{\partial r} \left( r^{n-1} \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \Delta_{𝕊^{n-1}} f.\]

The matrix elements of $𝔇^{(ℓ)}$ are homogeneous polynomials, so we are interested in the case where $f$ is homogeneous with degree $d$:

\[f(x) = r^d f(x/|x|),\]

and we can easily compute

\[\frac{1}{r^{n-1}} \frac{\partial}{\partial r} \left( r^{n-1} \frac{\partial f(x)}{\partial r} \right) % = % d\frac{1}{r^{n-1}} \frac{\partial}{\partial r} \left( r^{d+n-2} \right)f % = % d(d+n-2) r^{d-2} f(x/|x|) % = % d(d+n-2) r^{d-2} f(x/|x|) % = % \frac{d(d+n-2)}{r^2} r^{d} f(x/|x|) = \frac{d(d+n-2)}{r^2} f(x).\]

For any element of $𝔇^{(ℓ)}$ we have $d=2ℓ$ and $n=4$, so the restriction to the space of unit quaternions is not harmonic with respect to the Laplacian on the 3-sphere, but is an eigenfunction with eigenvalue $-2ℓ(2ℓ+2) = -4ℓ(ℓ+1)$. That is,

\[\Delta_{\mathrm{Spin}(3)}\, 𝔇^{(ℓ)}_{m',m} = -4ℓ(ℓ+1)\, 𝔇^{(ℓ)}_{m',m}.\]

Comparing with the eigenvalue $ℓ(ℓ+1)$ of the Casimir operators, we see that

\[\Delta_{\mathrm{Spin}(3)} = -4 L^2 = -4 R^2,\]

which is the group-theoretic reason that $L^2 = R^2$: both are the Laplacian of the bi-invariant metric, and the Laplacian commutes with both left and right multiplication.

Note that Lee [28] points out that there is a sign ambiguity in the Laplacian. As I see it, the geometry community skews toward including a negative sign (which means that all eigenvalues are non-negative), while the physics community skews toward excluding it (which means that all eigenvalues are non-positive). It's also easy to prove that on a closed and connected manifold, eigenfunctions with distinct eigenvalues are orthogonal, since

\[(\lambda_u - \lambda_v) \int f_u f_v = \int (\lambda_u f_u) f_v - \int f_u (\lambda_v f_v) = \int (\Delta f_u) f_v - \int f_u (\Delta f_v) = 0\]

(the last equality by Green's theorem). Since the eigenvalues are distinct, equating the first and last expressions implies that this can only be true if $\int f_u f_v=0$.

Boyd and Petschek [29] produced an interesting discussion with numerous little insights into the use of special functions on different spaces. In particular, they show why associated Legendre functions are preferred to Chebyshev polynomials for the spherical harmonics. They also mention that since the Laplacian measures curvature, and spherical harmonics of a given degree have the same Laplacian eigenvalue, the spherical harmonics with a given $ℓ$ all have the same "amount" of curvature. So, for example, the $m = ±ℓ$ modes vary most rapidly with longitude and least rapidly with latitude, while the $m = 0$ mode varies most rapidly with latitude but not at all with longitude.

Vasil et al. [30] use spin-weighted spherical harmonics to do tensor calculus in the 3-ball, and have a lot of formulas for derivatives, as a result.