Pauli matrices

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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}$

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.

Synonym:

sigma matrices

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