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# 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.

Related:

Spinor, SchrodingersWaveEquation, UnitaryGroup, HermitianMatrix, DiracMatrices, DiracEquation

Synonym:

sigma matrices

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15A57*no label found*

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