Pauli matrices

The Pauli matrices are a set of three Hermitian, unitary matrices used by Wolfgang Pauli in his theory of quantum-mechanical spin. They are given by:

	$\displaystyle\sigma_{1}$	$\displaystyle=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$
	$\displaystyle\sigma_{2}$	$\displaystyle=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}$
	$\displaystyle\sigma_{3}$	$\displaystyle=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$

They satisfy the following commutation and anticommutation identities:

	$\displaystyle\left[\sigma_{i},\sigma_{j}\right]$	$\displaystyle=2i\epsilon_{ijk}\sigma_{k}\text{where $\epsilon_{ijk}$ is the % Levi-Civita symbol}$
	$\displaystyle\{\sigma_{i},\sigma_{j}\}$	$\displaystyle=2\mathbf{I}\delta_{ij}\text{where $\mathbf{I}$ is the identity % matrix and $\delta_{ij}$ is the Kronecker delta}$

0.1 Delta notation

With the identity matrix I, the Pauli matrices form a group. When combined in this way, they are often given the symbols $\delta_{i}$ , as follows:

	$\displaystyle\delta_{0}$	$\displaystyle=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$
	$\displaystyle\delta_{1}$	$\displaystyle=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$
	$\displaystyle\delta_{2}$	$\displaystyle=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}$
	$\displaystyle\delta_{3}$	$\displaystyle=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$

This choice is useful when writing the Dirac matrices.