Editing Pauli matrices (section)

===Relation to dot and cross product===
Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives
:<math>
\begin{align} 
  \left[ \sigma_j, \sigma_k\right] + \{\sigma_j, \sigma_k\}  &= (\sigma_j \sigma_k - \sigma_k \sigma_j ) + (\sigma_j \sigma_k + \sigma_k \sigma_j) \\
    2i\varepsilon_{j k \ell}\,\sigma_\ell + 2 \delta_{j k}I &= 2\sigma_j \sigma_k 
\end{align}
</math>
so that,  
{{Equation box 1
 |indent =:
 |equation =  <math>~~ \sigma_j \sigma_k = \delta_{j k}I + i\varepsilon_{j k \ell}\,\sigma_\ell ~ .~</math>
 |cellpadding= 6
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 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

[[tensor contraction|Contracting]] each side of the equation with components of two {{math|3}}-vectors {{math|''a{{sub|p}}''}} and {{math|''b{{sub|q}}''}} (which commute with the Pauli matrices, i.e., {{math|1=''a{{sub|p}}σ{{sub|q}}'' = ''σ{{sub|q}}a{{sub|p}}'')}} for each matrix {{math|''σ{{sub|q}}''}} and vector component {{math|''a{{sub|p}}''}} (and likewise with {{math|''b{{sub|q}}''}}) yields
:<math>~~ \begin{align} 
  a_j b_k \sigma_j \sigma_k & = a_j b_k \left(i\varepsilon_{jk\ell}\,\sigma_\ell + \delta_{jk}I\right) \\
  a_j \sigma_j b_k \sigma_k & = i\varepsilon_{jk\ell}\,a_j b_k \sigma_\ell + a_j b_k \delta_{jk}I 
\end{align}.~</math>

Finally, translating the index notation for the [[dot product]] and [[cross product#Index notation for tensors|cross product]] results in 
{{NumBlk||{{Equation box 1
  |indent =:
  |equation = <math>~~\Bigl(\vec{a} \cdot \vec{\sigma}\Bigr)\Bigl(\vec{b} \cdot \vec{\sigma}\Bigr) = \Bigl(\vec{a} \cdot \vec{b}\Bigr) \, I + i \Bigl(\vec{a} \times \vec{b}\Bigr) \cdot \vec{\sigma}~~</math>
  |cellpadding= 6
  |border
  |border colour = #0073CF
  |bgcolor=#F9FFF7
 }}
 |{{EquationRef|1}}
}}

If {{mvar|i}} is identified with the pseudoscalar {{math|''σ{{sub|x}}σ{{sub|y}}σ{{sub|z}}''}} then the right hand side becomes <math> a \cdot b + a \wedge b </math>, which is also the definition for the product of two vectors in geometric algebra.

If we define the spin operator as {{math|1='''''J''''' = {{sfrac|''ħ''|2}}'''''σ'''''}}, then {{math|1='''''J'''''}} satisfies the commutation relation:<math display="block">\mathbf{J} \times \mathbf{J} = i\hbar \mathbf{J}</math>Or equivalently, the Pauli vector satisfies:<math display="block">\frac{\vec{\sigma}}{2} \times \frac{\vec{\sigma}}{2} = i\frac{\vec{\sigma}}{2}</math>