Pauli matrices

The matrices

σ_{x} = [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}], σ_{y} = [\begin{array}{cc} 0 & - i \\ i & 0 \end{array}], σ_{z} = [\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}]

See this.
They satisfy:

$σ_{j}^{2} = I d$
$σ_{i} σ_{j} = - σ_{j} σ_{i}$ for $i \neq j$ .
Both are summarized in $σ_{i} σ_{j} + σ_{j} σ_{i} = 2 I δ_{i j}$ (anti-commutator ${σ_{i}, σ_{j}} = σ_{i} σ_{j} + σ_{j} σ_{i}$ )

The Pauli matrices are Hermitian, and they span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, $σ_{k}$ represents the observable corresponding to spin along the $k^{t h}$ coordinate axis in three-dimensional Euclidean space $R^{3}$ .

Pauli vectors

Given a vector $v = (x, y, z) \in R^{3}$ , the corresponding matrix

σ_{v} = x σ_{x} + y σ_{y} + z σ_{z} =

[\begin{array}{cc} z & x - i y \\ x + i y & - z \end{array}]

is called a Pauli vector. This injection

R^{3} \to M_{2} (C)

let us understand the vector space $R^{3}$ as an algebra. Some of the operations have a nice geometric interpretation:

The square of a vector is the length of the vector times $I d$ .
Perpendicular vectors anti-commute: if $v \cdot w = 0$ then $σ_{v} σ_{w} = - σ_{w} σ_{v}$ .
Negative conjugation by a unit Pauli vector is a reflection in the plane perpendicular to the vector. For example, negative conjugation by $σ_{z}$ is a reflection in the $x y$ -plane:

\begin{aligned} σ_{x} \to - σ_{z} σ_{x} σ_{z} = σ_{x} \\ σ_{y} \to - σ_{z} σ_{y} σ_{z} = σ_{y} \\ σ_{z} \to - σ_{z} σ_{z} σ_{z} = - σ_{z} \end{aligned}

We are entering the world of Geometric Algebras.

Rotations: are the composition of two reflections. They correspond to

V \to A V A^{†}, A \in S U (2)

Important facts

Pauli matrices multiplied by $i$ give rise to the Lie algebra $su (2)$ .
On the other hand, they constitute a representation of the Clifford algebra $C l_{3}$ .

Relation to spinors

(See this video)
This is related, to my knowledge, with spinors $S = C^{2}$ , with basis $s_{b}$ , in the following way. Consider the dual space $S^{d u a l}$ , with dual basis $s^{a}$ . Then we can think of a map $σ : R^{3} \to S \otimes S^{d u a l}$ :

\begin{aligned} e_{x} & \to σ_{x} = s_{2} \otimes s^{1} + s_{1} \otimes s^{2} \\ e_{y} & \to σ_{y} = i s_{2} \otimes s^{1} - i s_{1} \otimes s^{2} \\ e_{z} & \to σ_{z} = s_{1} \otimes s^{1} - s_{2} \otimes s^{2} \end{aligned}

This map has three indices: $σ_{i b}^{a}$ and acts on a general $v = v^{i} e_{i} \in R^{3}$ sending it to the element $V_{b}^{a} s_{b} \otimes s^{a}$ , where

σ_{i b}^{a} v_{i} = V_{b}^{a}

It can be thought as if one vector index $i$ is equivalent to two spinorial indices $a, b$ .