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?

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?

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 apparantly 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?

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?