Angular Momentum Matrices
\(\forall \ell, \exists (2\ell + 1) \times
(2\ell + 1)\) matrices
\(\mathbf{J}_x^\ell\), \(\mathbf{J}_y^\ell\), \(\mathbf{J}_z^\ell\) satisfying commutation
relations
\(\left[\mathbf{J}_i, \mathbf{J}_j\right] =
-i\epsilon_{ijk}\mathbf{J}_k\), with
\(\mathbf{J}_z \hat{=} \mathrm{diag}(-\ell,
-\ell + 1 \ldots \ell - 1, \ell).\)
Example: \(\ell =
1\)
\[\small\mathbf{J}_x \hat{=} {1 \over
\sqrt{2}}\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0
& 1 & 0\end{bmatrix};
\mathbf{J}_y \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & i & 0
\\ -i & 0 & i \\ 0 & -i & 0\end{bmatrix};
\mathbf{J}_z \hat{=} \begin{bmatrix}-1 & 0 & 0 \\ 0 &0
&0 \\ 0& 0 &1\end{bmatrix}.\]
Rotation as an Eigenvalue Problem
For fixed \(m\) (to leading
order):
\[\left(-\mathbf{\Omega}_0^2 + 2 \omega m
\mathbf{R} + \omega^2 \mathbf{\Delta}\right)\mathbf{c} = 0\]
For fixed \(n\) (to leading
order):
\[(-\omega_0^2 \mathbb{1}_{2\ell+1} + 2
\omega \mathbf{J}_z R_{n,n} + \omega^2 \mathbb{1}_{2\ell+1})\mathbf{y} =
0\]
Combined Angular Momentum Operator
(Aligned Case)
\[\vec{\xi}_{n\ell m}(r, \theta, \varphi)
= \vec{\xi}_{n\ell}(r) Y_\ell^m(\theta, \phi) \iff
\underbrace{\tilde{R}_{n, n', m, m'} = R_{n,n'}
J_{m,m'}}_{\text{this is a }\textbf{tensor product}!}\]
\[\implies \left(-\mathbf{\Omega_0}^2
\otimes \mathbb{1}_{2\ell+1} + 2 \omega \underbrace{\mathbf{R} \otimes
\mathbf{J}_z}_{\tilde{\mathbf{R}}} + \omega^2 \mathbf{\Delta} \otimes
\mathbb{1}_{2\ell+1}\right)\mathbf{x} = 0\]
The Misaligned Angular Momentum Operator
Let’s associate with each mass shell at \(r\) both \(\Omega(r)\)
(as is customary), and also an axis \(\hat{\mathbf{n}}(r) =\sum_i n_i
\mathbf{e}_i\).
\[\small
\begin{aligned}
\mathbf{R}_{n\ell, n\ell} &= b_{n\ell}\int
{\mathbf{d}^\ell}^\dagger(\beta(r)) \Omega(r) \mathbf{J}_z
{\mathbf{d}^\ell}(\beta(r))\ K(r)\ \mathrm{d} r\\
&= b_{n\ell}\int \Omega(r) (\hat{\mathbf{n}} \cdot
\vec{\mathbf{J}})\ K(r)\ \mathrm{d} r\\
&= \boxed{b_{n\ell}\left(\int \vec{\mathbf{\Omega}}(r) K(r)\
\mathrm{d} r\right) \cdot \vec{\mathbf{J}}}.
\end{aligned}
\]
\[
\boxed{\tilde{\mathbf{R}} = \int
\mathbf{K}\otimes\left(\Omega_\text{rot}(r)\ \hat{\mathbf{n}}\cdot
\vec{\mathbf{J}}\right)\ \mathrm{d}r}
\]
\(\implies\) For each mode, AM
matrix is
specified by usual vector addition.
Mixed Modes
We only assume that the pure p- and g-mode
solutions
are separately amenable to separation of
variables;
the mixed-mode eigenfunctions need not be.
\[
\small
\left(\begin{bmatrix}
{\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\
\mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}}
\end{bmatrix} \otimes \mathbb{1}_{2\ell+1}
+ 2 \omega \begin{bmatrix} {\color{forestgreen}\tilde{\mathbf{R}}_\pi}
& 0 \\ 0 &
{\color{forestgreen}\tilde{\mathbf{R}}_\gamma}\end{bmatrix}
+ \omega^2 \begin{bmatrix}
\mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1}
\end{bmatrix} \otimes \mathbb{1}_{2\ell+1}
\right)\mathbf{x} = 0.
\]
Mode mixing yields avoided crossings
between multiplet components of identical \(m\)
(cf. Mosser+ 2012, Ouazzani+ 2013, Deheuvels+ 2017)
\(\beta = 0\)
\(\beta = {\pi\over10}\)
\(\beta = {\pi\over2}\)
\(\beta = \pi\)
