= Weyl integration formula for SU(2)
{c}
{title2=$\int_{\mathrm{SU}(2)}F(g)\,dg=\frac2\pi\int_0^\pi F(\operatorname{diag}(e^{i\theta},e^{-i\theta}))\sin^2\theta\,d\theta$}
= Weyl integration formula for SU2
{c}
{synonym}
For a <class function> on <SU(2)> and normalized <Haar measure>, the displayed formula reduces a group integral to its conjugacy-class angle. The weight follows by identifying <SU(2)> with the unit sphere in four real dimensions: writing a unit quaternion as $\cos\theta+\mathbf n\sin\theta$ gives spherical volume proportional to $\sin^2\theta\,d\theta\,d\mathbf n$. Since $\int_0^\pi\sin^2\theta\,d\theta=\pi/2$, normalization gives $2/\pi$.
Back to article page
