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.