Pauli matrices

The matrices

$$ \sigma_x=\left[\begin{array}{cc} {0} & {1} \\ {1} & {0} \end{array}\right], \sigma_y=\left[\begin{array}{cc} {0} & {-i} \\ {i} & {0} \end{array}\right], \sigma_z=\left[\begin{array}{cc} {1} & {0} \\ {0} & {-1} \end{array}\right] $$

See this.

They satisfy:

$\sigma_j^2=Id$

$\sigma_i \sigma_j=-\sigma_j \sigma_i$ for $i\neq j$.

Both are summarized in $\sigma_i \sigma_j+\sigma_j \sigma_i=2I\delta_{ij}$ (anti-commutator $\{\sigma_i,\sigma_j\}=\sigma_i \sigma_j+\sigma_j \sigma_i$)

Pauli vectors

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

$$ \sigma_v=x\sigma_x+y\sigma_y+z\sigma_z= $$ $$ \left[\begin{array}{cc} {z} & {x-i y} \\ {x+i y} & {-z} \end{array}\right] $$

is called a Pauli vector. This injection

$$ \mathbb R^3\to M_2(\mathbb C) $$

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

The square of a vector is the length of the vector times $Id$.

Perpendicular vectors anti-commute: if $v\cdot w=0$ then $\sigma_v \sigma_w=-\sigma_w \sigma_v$.

Negative conjugation by a unit Pauli vector is a reflection in the plane perpendicular to the vector. For example, negative conjugation by $\sigma_z$ is a reflection in the $xy$-plane:

$$ \begin{aligned} & \sigma_x \rightarrow -\sigma_z \sigma_x \sigma_z=\sigma_x \\ & \sigma_y \rightarrow -\sigma_z \sigma_y \sigma_z=\sigma_y \\ & \sigma_z \rightarrow -\sigma_z \sigma_z \sigma_z=-\sigma_z \end{aligned} $$

We are entering the world of Geometric Algebras.

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

$$ V\to AVA^{\dagger}, \quad A\in SU(2) $$

Important facts

Pauli matrices multiplied by $i$ give rise to the Lie algebra $\mathfrak{su}(2)$.

On the other hand, they constitute a representation of the Clifford algebra $Cl_3$.

Relation to spinors

(See this video

This is related, to my knowledge, with spinors $S=\mathbb C^2$, with basis $s_b$, in the following way. Consider the dual space $S^{dual}$, with dual basis $\mathfrak{s}^a$. Then we can think of a map $\sigma:\mathbb R^3 \to S \otimes S^{dual}$:

$$ \begin{aligned} e_x&\to \sigma_x=s_2\otimes \mathfrak{s}^1+s_1\otimes \mathfrak{s}^2\\ e_y&\to \sigma_y=is_2\otimes \mathfrak{s}^1-is_1\otimes \mathfrak{s}^2\\ e_z&\to \sigma_z=s_1\otimes \mathfrak{s}^1-s_2\otimes \mathfrak{s}^2\\ \end{aligned} $$

This map has three indices: $\sigma^{\,a}_{i\, b}$ and acts on a general $v=v^ie_i \in \mathbb R^3$ sending it to $V^a_{\quad b}s_b\otimes \mathfrak{s}^a$, ie.,

$$ \sigma^{\,a}_{i\, b} v_i=V^a_{\quad b} $$

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

________________________________________

Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

INDEX: