PlanetMath: Pauli matrices

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Pauli matrices

(Definition)

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:

$\displaystyle \left[ \sigma_i, \sigma_j \right]$	$\displaystyle = 2i\epsilon_{ijk} \sigma_k$ where $\epsilon_{ijk}$ is the Levi-Civita symbol
$\displaystyle \lbrace \sigma_i, \sigma_j \rbrace$	$\displaystyle =2 \mathbf{I} \delta_{ij}$ where $\mathbf{I}$ is the identity matrix and $\delta_{ij}$ is the Kronecker delta

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:

$\displaystyle \delta_0$	$\displaystyle = \begin{pmatrix}1 & 0\\ 0 & 1 \end{pmatrix}$
$\displaystyle \delta_1$	$\displaystyle = \begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}$
$\displaystyle \delta_2$	$\displaystyle = \begin{pmatrix}0 & -i\\ i & 0 \end{pmatrix}$
$\displaystyle \delta_3$	$\displaystyle = \begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}$

This choice is useful when writing the Dirac matrices.

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Other names:

sigma matrices

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Cross-references: Dirac matrices, group, identity matrix, identities, theory, unitary matrices, Hermitian
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This is version 6 of Pauli matrices, born on 2008-03-28, modified 2008-04-17.
Object id is 10449, canonical name is PauliMatrices.
Accessed 416 times total.

Classification:

AMS MSC:

15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

Pending Errata and Addenda

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