Editing Pauli matrices (section)

==== Anticommutation relations====
They also satisfy the [[anticommutator|anticommutation]] relations:
:<math>\{\sigma_j, \sigma_k\} = 2 \delta_{j k}\,I,</math>

where <math>\{\sigma_j, \sigma_k\}</math> is defined as <math>\sigma_j \sigma_k + \sigma_k \sigma_j,</math> and {{math|''δ{{sub|jk}}''}} is the [[Kronecker delta]]. {{mvar|I}} denotes the {{math|2 × 2}} identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the [[Clifford algebra]] for <math>\mathbb{R}^3,</math> denoted <math>\mathrm{Cl}_3(\mathbb{R}) .</math>

The usual construction of generators <math>\sigma_{jk} = \tfrac{1}{4} [\sigma_j, \sigma_k]</math> of <math>\mathfrak{so}(3)</math> using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.

A few explicit commutators and anti-commutators are given below as examples:
{| style="text-align:left;"
! Commutators
! Anticommutators
|-
| <math>\begin{align}
\left[\sigma_1, \sigma_1\right] &= 0 \\
\left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \\
\left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \\
\left[\sigma_3, \sigma_1\right] &= 2i\sigma_2 
\end{align}</math>{{quad}}
| <math>\begin{align}
\left\{\sigma_1, \sigma_1\right\} &= 2I \\
\left\{\sigma_1, \sigma_2\right\} &= 0  \\
\left\{\sigma_2, \sigma_3\right\} &= 0  \\
\left\{\sigma_3, \sigma_1\right\} &= 0
\end{align}</math>
|}