In the following subsections, we work 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. First, we can just list the most important conventions. Note that we will use Euler angles and spherical coordinates here. 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.
Right-handed Cartesian coordinates $(x, y, z)$ and unit basis vectors $(๐ฑ, ๐ฒ, ๐ณ)$.
Spherical coordinates $(r, \theta, \phi)$ and unit basis vectors $(๐ซ, \boldsymbol{\theta}, \boldsymbol{\phi})$. The "polar angle" $\theta \in [0, \pi]$ measures the angle between the specified direction and the positive $๐ณ$ axis. The "azimuthal angle" $\phi \in [0, 2\pi)$ measures the angle between the projection of the specified direction onto the $๐ฑ$-$๐ฒ$ plane and the positive $๐ฑ$ axis, with the positive $๐ฒ$ axis corresponding to the positive angle $\phi = \pi/2$.
Quaternions $๐ = W + X๐ข + Y๐ฃ + Z๐ค$, where $๐ข๐ฃ๐ค = -1$. In software, this quaternion is represented by $(W, X, Y, Z)$. We will depict a three-dimensional vector $๐ฏ = v_x ๐ฑ + v_y ๐ฒ + v_z ๐ณ$ interchangeably as a quaternion $v_x ๐ข + v_y ๐ฃ + v_z ๐ค$.
A rotation represented by the unit quaternion $๐$ acts on a vector $๐ฏ$ as $๐\, ๐ฏ\, ๐^{-1}$.
Where relevant, rotations will be assumed to be right-handed, so that a quaternion characterizing the rotation through an angle $\vartheta$ about a unit vector $๐ฎ$ can be expressed as $๐ = \exp(\vartheta ๐ฎ/2)$. Note that $-๐$ would deliver the same rotation, which is why the group of unit quaternions $\mathrm{Spin}(3) = \mathrm{SU}(2)$ is a double cover of the group of rotations $\mathrm{SO}(3)$.
Euler angles parametrize a unit quaternion as $๐ = \exp(\alpha ๐ค/2)\, \exp(\beta ๐ฃ/2)\, \exp(\gamma ๐ค/2)$. The angles $\alpha$ and $\beta$ take values in $[0, 2\pi)$. The angle $\beta$ takes values in $[0, 2\pi]$ to parametrize the group of unit quaternions $\mathrm{Spin}(3) = \mathrm{SU}(2)$, or in $[0, \pi]$ to parametrize the group of rotations $\mathrm{SO}(3)$.
A point on the unit sphere with spherical coordinates $(\theta, \phi)$ can be represented by Euler angles $(\alpha, \beta, \gamma) = (\phi, \theta, 0)$. The rotation with these Euler angles takes the positive $๐ณ$ axis to the specified direction. In particular, any function of spherical coordinates can be promoted to a function on Euler angles using this identification.
For a complex-valued function $f(๐)$, we define two operators, the left and right Lie derivatives:
\[L_๐ฎ f(๐) = \left.-i \frac{d}{d\epsilon}\right|_{\epsilon=0}
f\left(e^{\epsilon ๐ฎ/2}\, ๐\right)
\qquad \text{and} \qquad
R_๐ฎ f(๐) = \left.-i \frac{d}{d\epsilon}\right|_{\epsilon=0}
-f\left(๐\, e^{\epsilon ๐ฎ/2}\right),\]
where $๐ฎ$ can be any pure-vector quaternion. In particular, $L$ represents the standard angular-momentum operators, and we can compute the expressions in Euler angles for the basis vectors:\[\begin{aligned}
-L_x = L_๐ข &= -i \left( \sin\alpha \frac{\partial}{\partial\beta} + \cos\alpha \cot\beta \frac{\partial}{\partial\alpha} \right), \\
-L_y = L_๐ฃ &= -i \left( \cos\alpha \frac{\partial}{\partial\beta} - \sin\alpha \cot\beta \frac{\partial}{\partial\alpha} \right), \\
-L_z = L_๐ค &= -i \frac{\partial}{\partial\alpha}.
-\end{aligned}\]
Angular-momentum operators defined in Lie terms, translated to Euler angles and spherical coordinates.- Spherical harmonics
- Wigner D-matrices
- Spin-weighted spherical harmonics
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 $(๐ฑ, ๐ฒ, ๐ณ)$. 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
\[\mathbf{v} = v_x \mathbf{๐ฑ} + v_y \mathbf{๐ฒ} + v_z \mathbf{๐ณ},\]
in which case the Euclidean norm is given by
\[\| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2 + v_z^2}.\]
Equivalently, we can write the components of the Euclidean metric as
\[g_{ij} = \left( \begin{array}{ccc}
+f\left(๐\, e^{\epsilon ๐ฎ/2}\right),\]
where $๐ฎ$ can be any pure-vector quaternion. In particular, $L$ represents the standard angular-momentum operators, and we can compute the expressions in Euler angles for the basis vectors:
\[\begin{aligned}
+ L_x = L_๐ข &= -i \left\{
+ -\frac{\cos\alpha}{\tan\beta} \frac{\partial} {\partial \alpha}
+ - \sin\alpha \frac{\partial} {\partial \beta}
+ +\frac{\cos\alpha}{\sin\beta} \frac{\partial} {\partial \gamma}
+ \right\} \\
+ L_y = L_๐ฃ &= -i \left\{
+ -\frac{\sin\alpha}{\tan\beta} \frac{\partial} {\partial \alpha}
+ + \cos\alpha \frac{\partial} {\partial \beta}
+ +\frac{\sin\alpha}{\sin\beta} \frac{\partial} {\partial \gamma}
+ \right\} \\
+ L_z = L_๐ค &= -i \frac{\partial} {\partial \alpha} \\
+ K_x = K_๐ข &= -i \left\{
+ -\frac{\cos\gamma}{\sin\beta} \frac{\partial} {\partial \alpha}
+ +\sin\gamma \frac{\partial} {\partial \beta}
+ +\frac{\cos\gamma}{\tan\beta} \frac{\partial} {\partial \gamma}
+ \right\} \\
+ K_y = K_๐ฃ &= -i \left\{
+ \frac{\sin\gamma}{\sin\beta} \frac{\partial} {\partial \alpha}
+ +\cos\gamma \frac{\partial} {\partial \beta}
+ -\frac{\sin\gamma}{\tan\beta} \frac{\partial} {\partial \gamma}
+ \right\} \\
+ K_z = K_๐ค &= -i \frac{\partial} {\partial \gamma}
+\end{aligned}\]
We can lift any function on $S^2$ to a function on $S^3$ โ or more precisely any function on spherical coordinates to a function on the space of Euler angles โ by the correspondence $(\theta, \phi) \mapsto (\alpha, \beta, \gamma) = (\phi, \theta, 0)$. We can then express the angular-momentum operators in their more common form, in terms of spherical coordinates:
\[\begin{aligned}
+ L_x &= -i \left\{
+ -\frac{\cos\phi}{\tan\theta} \frac{\partial} {\partial \phi}
+ - \sin\phi \frac{\partial} {\partial \theta}
+ \right\} \\
+ L_y &= -i \left\{
+ -\frac{\sin\phi}{\tan\theta} \frac{\partial} {\partial \phi}
+ + \cos\phi \frac{\partial} {\partial \theta}
+ \right\} \\
+ L_z &= -i \frac{\partial} {\partial \phi}
+\end{aligned}\]
(The $R$ operators make less sense for a function of spherical coordinates.)
Spherical harmonics
Wigner D-matrices
Spin-weighted spherical harmonics
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 $(๐ฑ, ๐ฒ, ๐ณ)$. 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
\[\mathbf{v} = v_x \mathbf{๐ฑ} + v_y \mathbf{๐ฒ} + v_z \mathbf{๐ณ},\]
in which case the Euclidean norm is given by
\[\| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2 + v_z^2}.\]
Equivalently, we can write the components of the Euclidean metric as
\[g_{ij} = \left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
@@ -88,4 +117,18 @@
+ \cos\rho\, ๐ฏ_โ + \sin\rho\, \hat{\mathfrak{r}}\times ๐ฏ_โ
\end{aligned}\]
The final expression shows that this is precisely what we expect when rotating $๐ฏ$ through an angle $\rho$ (in a positive, right-handed sense) about the axis $\hat{\mathfrak{r}}$.
Note that the presence of two factors of $๐$ 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 $๐$ 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 $๐$ results in the same rotation. Thus, for any rotation, there are two (precisely opposite) quaternions that represent it.
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 $\gamma$ about the $๐ณ$ axis, followed by a rotation through $\beta$ about the $๐ฒ$ axis, and finally a rotation through $\alpha$ about the $๐ณ$ axis. Note that the axes are fixed, and not subject to any preceding rotations. More precisely, we can write the unit quaternion as
\[๐ = \exp\left(\frac{\alpha}{2} ๐ค\right)
\exp\left(\frac{\beta}{2} ๐ฃ\right)
- \exp\left(\frac{\gamma}{2} ๐ค\right).\]