Editing Pauli matrices (section)

====The group composition law of {{math|SU(2)}}====
A straightforward application of  formula {{EquationNote|(2)}}  provides a parameterization of the composition law of  the group {{math|SU(2)}}.{{efn|The relation among {{math|''a, b, c,'' ''' n, m, k '''}} derived here in the {{math|2 × 2}} representation holds for ''all representations'' of {{math|SU(2)}}, being a ''group identity''. Note that, by virtue of the standard normalization of that group's generators as ''half'' the Pauli matrices, the parameters ''a'',''b'',''c'' correspond to ''half'' the rotation angles of the rotation group. That is, the Gibbs formula linked amounts to <math>\hat k \tan c/2=  (\hat n \tan a/2+ \hat m \tan b/2 -\hat m \times \hat n \tan a/2 ~ \tan b/2 )/(1-\hat m\cdot \hat n \tan a/2 ~\tan b/2 )</math>.}} One may directly solve for {{mvar|c}} in 
<math display=block>\begin{align}
  e^{ia\left(\hat{n} \cdot \vec{\sigma}\right)} e^{ib\left(\hat{m} \cdot \vec{\sigma}\right)}
    &= I\left(\cos a \cos b - \hat{n} \cdot \hat{m} \sin a \sin b\right) + i\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n} \times \hat{m} ~ \sin a \sin b \right) \cdot \vec{\sigma} \\
    &= I\cos{c} + i \left(\hat{k} \cdot \vec{\sigma}\right) \sin c \\
    &= e^{ic \left(\hat{k} \cdot \vec{\sigma}\right)},
\end{align}</math>

which specifies the generic group multiplication, where, manifestly,  
<math display=block>\cos c = \cos a \cos b - \hat{n} \cdot \hat{m} \sin a \sin b~,</math>
the [[spherical law of cosines]].  Given {{mvar|c}}, then, 
<math display=block>\hat{k} = \frac{1}{\sin c}\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} \sin a \sin b\right).</math>

Consequently, the composite rotation parameters in this group element (a closed form of the respective [[Baker–Campbell–Hausdorff formula|BCH expansion]] in this case) simply amount to<ref>{{cite book |first=J.W. |last=Gibbs |year=1884 |title=Elements of Vector Analysis |place=New Haven, CT |page=67 |author-link=J. W. Gibbs |chapter=4. Concerning the differential and integral calculus of vectors |chapter-url={{GBurl|VurzAAAAMAAJ|p=67}} |publisher=Tuttle, Moorehouse & Taylor }} In fact, however, the formula goes back to [[Olinde Rodrigues]] (1840), replete with half-angle: {{cite journal |first=Olinde |last=Rodrigues |author-link=Olinde Rodrigues |year=1840 |title=Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire |journal=[[J. Math. Pures Appl.]] |volume=5 |pages=380–440 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1840_1_5_A39_0.pdf}}</ref>

<math display=block> e^{ic \hat{k} \cdot \vec{\sigma}} =
  \exp \left( i\frac{c}{\sin c} \left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b\right) \cdot \vec{\sigma}\right). </math>

(Of course, when <math>\hat{n}</math> is parallel to <math>\hat{m}</math>, so is <math>\hat{k}</math>, and {{math|1=''c'' = ''a + b''}}.)
{{see also|Rotation formalisms in three dimensions#Rodrigues vector|Spinor#Three dimensions}}