Pauli matricesPauliMatrices2008-03-28 12:01:552008-04-17 10:21:50Definition% this is the default PlanetMath preamble. as your knowledge
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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:
\begin{align*}
\sigma_1 &= \begin{pmatrix} 0 & 1\\
1 & 0
\end{pmatrix}\\
\sigma_2 &= \begin{pmatrix} 0 & -i\\
i & 0
\end{pmatrix}\\
\sigma_3 &= \begin{pmatrix} 1 & 0\\
0 & -1
\end{pmatrix}\\
\end{align*}
They satisfy the following commutation and anticommutation identities:
\begin{align*}
\left[ \sigma_i, \sigma_j \right] &= 2i\epsilon_{ijk} \sigma_k\text{where $\epsilon_{ijk}$ is the Levi-Civita symbol}\\
\lbrace \sigma_i, \sigma_j \rbrace &=2 \mathbf{I} \delta_{ij} \text{where $\mathbf{I}$ is the identity matrix and $\delta_{ij}$ is the Kronecker delta}
\end{align*}
\subsection{Delta notation}
With the identity matrix $\textbf{I}$, the Pauli matrices form a group. When combined in this way, they are often given the symbols $\delta_i$, as follows:
\begin{align*}
\delta_0 &= \begin{pmatrix} 1 & 0\\
0 & 1
\end{pmatrix}\\
\delta_1 &= \begin{pmatrix} 0 & 1\\
1 & 0
\end{pmatrix}\\
\delta_2 &= \begin{pmatrix} 0 & -i\\
i & 0
\end{pmatrix}\\
\delta_3 &= \begin{pmatrix} 1 & 0\\
0 & -1
\end{pmatrix}\\
\end{align*}
This choice is useful when writing the Dirac matrices.