What do the Pauli matrices say about the Threefold Way?

Question

The Pauli matrices $$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

are closely related to rotations expressed, respectively, in quaternions, real numbers and complex numbers, as follows.

For $\theta\in\mathbb{R}$ we can define \begin{gather*} g_1(\theta)=e^{i\theta\sigma_1}=\begin{pmatrix} \cos \theta & i\ \sin \theta \\ i\ \sin \theta & \cos \theta \end{pmatrix}, \\ g_2(\theta)=e^{i\theta\sigma_2}=\begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}, \\ g_3(\theta)=e^{i\theta\sigma_3}=\begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix}. \end{gather*}

$g_3(\theta)$ expresses a rotation as a complex number $e^{i\theta}$ (and its inverse $e^{-i\theta}$). Geometrically, I suppose this is nothing more than an angle.

$g_2(\theta)$ expresses a rotation as a $2\times 2$ matrix in the real numbers. Geometrically, this may express rotation as a distance, such as an arc or a spread $\sin^2\theta$.

$g_1(\theta)$ expresses a rotation in terms of quaternions. For any unit quaternion can be written as $\textrm{cos}\ \theta + u\ \textrm{sin}\ \theta$ where $u$ is a unit vector in $\mathbb{R}i + \mathbb{R}j + \mathbb{R}k$. We can identify the vector $u$ with the complex number $i$. For any $u$, $g_1(\theta)g_1(\psi)=g_1(\theta+\psi)$ defines a subalgebra within the quaternions. Geometrically, the rotation can be identified with an oriented area.

Do these distinctions reflect any deep insights into the relationship between the real numbers, the complex numbers and the quaternions?

I am aware of John Baez's posts about Dyson's threefold way but I don't have access to Dyson's 1962 paper The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics. I have found the transcript of Vladimir Arnold's 1997 lecture Symplectization Complexification and Mathematical Trinities and the video of his 1988 lecture Polymathematics: complexification, symplectization and all that. Thank you for any expositions that would help make sense of this!

Answer 1

Q: What do the Pauli matrices say about the Threefold Way?

In the context of Dyson's threefold way, the Pauli matrices produce two of the three ensembles of random Hamiltonians.

A Hermitian matrix $H$ with normally distributed matrix elements belongs to the Gaussian Orthogonal Ensemble (GOE) if the matrix elements are real, to the Gaussian Unitary Ensemble (GUE) if the matrix elements are complex, and to the Gaussian Symplectic Ensemble (GSE) if the matrix elements are linear combinations of Pauli matrices of the form $$H_{nm}=a^{(0)}_{nm}I_2 + i\sum_{p=1}^3 a^{(p)}_{nm}\sigma_p,\quad a^{(0)},a^{(1)},a^{(2)},a^{(3)}\in\mathbb{R}.$$ The restriction to real coefficients is essential, without it the Hamiltonian ensemble is the GUE instead of the GSE. The GOE cannot be obtained from Pauli matrices.