How to write a matrix from $SU(2)$ in terms of one angle and one complex number $z$ , where $z$ is from sphere $S^{2}$

Question

For given a matrix from $SU(2)$ , how can represent it in terms of two parameters: one angle and one complex number $z$ from the sphere $S^{2}$ ? Does this have any links with : $\mathrm{SU}(2)$ axis and angle representation ? Can someone help me do this? Thank you in advance!

Answer 1

A matrix in $\mathrm{SU}(2)$ is of the form $$ \left( \begin{matrix} a & -\overline{b}\\ b & \overline{a} \end{matrix} \right), \qquad \lvert a\rvert^2 + \lvert b\rvert^2 = 1, \quad a,b \in \mathbb C. $$ Write $a=x+yi, b=z+wi$. Note that the equation $\lvert a\rvert^2 + \lvert b\rvert^2 = 1$ says that $$x^2+y^2+z^2+w^2=1,$$ which is the equation of a three-sphere $\mathbb S^3$ in $\mathbb R^4$. Identify $\mathbb{R}^4$ with the space of quaternions $\mathbb H$, so that $(x,y,z,w)$ corresponds to $x+yi+zj+wij \in \mathbb H$ (this is only a correspondence of vector spaces). Now, since $i,j,ij$ span an $\mathbb R^3$ in $\mathbb H$, define $u$ as a unit vector in their span which is proportional to $yi+zj+wij$, so that $u \in \mathbb{S}^2 \subset \mathbb R^3$ (there are two choices for this vector). You can write the quaternion $x+yi+zj+wij$ as $$\cos \vartheta 1+ \sin \vartheta u, \qquad \text{ for some } \vartheta \in \mathbb S^1.$$ Obviously $x^2 = \cos^2 \vartheta$ and $y^2+z^2+w^2=\sin^2 \vartheta$. Therefore, $(\vartheta,u) \in \mathbb S^1 \times \mathbb S^2$ is one of the possible pairs you are looking for.