# $SU(2)$ in two complex dimensions

I am just a beginner of group theory. I saw an $$SU(2)$$ example (example 4.16) in the book by Nadir Jeevanjee, An introduction to Tensors and group theory for Physicists. For $$SU(2)$$ elements, they satisfy:

$$A^{\dagger}=A^{-1}.$$

The generic element is:

$$\begin{pmatrix} \alpha\ \ \beta\\ -\bar{\beta}\ \bar{\alpha} \end{pmatrix} ,\ \alpha,\beta\in\mathbb{C} ,\ |\alpha|^2+|\beta|^2=1.$$

The author just directly write one kind of parametrization: $$\begin{pmatrix} e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\ ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}} \end{pmatrix}.$$

The three angles $$\theta,\psi,\phi$$ are Euler angles. From $$|\alpha|^2+|\beta|^2=1$$, I can get a 3-sphere.

If $$\alpha=u+iv,\ \beta=x+iy$$, I get: $$u^2+v^2+x^2+y^2=1.$$ I can use a parametrization:

$$u=\cos{\phi_1}\\ v=\sin{\phi_1}\cos{\phi_2}\\ x=\sin{\phi_1}\sin{\phi_2}\cos{\phi_3}\\ y=\sin{\phi_1}\sin{\phi_2}\sin{\phi_3}.$$ But apparently this is not the case which is related to Euler angles. How can I get the parametrization by the author and relate to the Euler angles?

To get the “usual” Euleurian expression as the product $$R_z(\zeta)R_y(\eta)R_z(\gamma)$$, with $$R_k(\varphi)=e^{i\varphi \sigma_k}$$ you need to set \begin{align} \zeta=\psi+\pi/4\, ,\qquad \gamma=\phi-\pi/4\, ,\qquad \eta=\theta\, . \end{align} The rest you can read off: $$\alpha=e^{i(\psi+\phi)}\cos\theta/2 = \cos\left((\psi+\phi)/2\right)\cos(\theta/2) + i \sin\left((\psi+\phi)/2\right)\cos(\theta/2)$$ so $$u=\cos\left((\psi+\phi)/2\right)\cos\theta/2$$ etc.