What do the Pauli matrices say about the Threefold Way?

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!

• They reflect the existence of a sequence of embeddings $\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset M_2(\mathbb{C})$, and various properties of composites of those embeddings, I guess. Feb 1, 2023 at 20:27
• TeX note: please use, e.g., $\sin \theta$ \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. Feb 1, 2023 at 20:56
• @LSpice Thank you for editing and explaining. Feb 1, 2023 at 20:57
• @QiaochuYuan Thank you! Yes, that is helpful. Still, I am hoping for something more. Feb 1, 2023 at 20:58

1 Answer

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.

• Thank you @Carlo_Beenakker for your enlightening, clear and helpful answer! I see that the article en.wikipedia.org/wiki/Random_matrix further explains that the ensembles are invariant, respectively, under orthogonal conjugation, unitary conjugation and conjugation by the symplectic group. They model Hamiltonians with time reversal symmetry (GOE), without time reversal symmetry (GUE) and with time reversal symmetry but no rotational symmetry. There is a lot for me to explore. Thank you. Feb 2, 2023 at 17:33
• Your answer concerns the Pauli matrices themselves, as Hermitian matrices, but my question, as I elaborated, is actually about the three rotation matrices: "Do these distinctions reflect any deep insights into the relationship between the real numbers, the complex numbers and the quaternions?" The rotation matrices are not Hermitian. They are special unitary. I wait for more answers. Or maybe, some day, in a few years, I can answer this question myself. Feb 2, 2023 at 17:49
• thank you for the feedback; I'm afraid I cannot say anything sensible about the connection between $U_p=e^{i\theta\sigma_p}$ and "rotations expressed, respectively, in quaternions, real numbers and complex numbers"; the usual way to think about this object is as a rotation of a spinor on the Bloch sphere; $U_1,U_2,U_3$ rotate about, respectively, the $x$-axis, the $y$-axis, and the $z$-axis. Feb 2, 2023 at 22:03
• @Carlo_Beenakker Yes, that's very helpful. Thank you! I wasn't aware of the Block sphere but I see it relates to a qubit. en.wikipedia.org/wiki/Bloch_sphere The article mentions, as you say, that these are the rotation operators about the Bloch basis. Thus you answered my question. I will study this, wait for more answers, but yours may very well be the best answer I will receive. Thank you! Feb 3, 2023 at 14:56