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!

`\sin \theta`

, not $\textrm{sin}\ \theta$`\textrm{sin}\ \theta`

. Of course, not every operator name you want to use comes with a pre-defined shortcut; for those that don't, you can use`\DeclareMathOperator`

. So, if`\sin`

didn't already exist, you could do`\DeclareMathOperator\sin{sin}`

, and then it would (or you can use $\operatorname{sin} \theta$`\operatorname{sin} \theta`

for one-off operators). I have edited accordingly. $\endgroup$