# Parametrizing $SU(2)$ with Hermitian matrices

There is something that is not clear to me

Here is what I know:

• Pauli matrices are $$\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$$, $$\sigma_2 = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$$, $$\sigma_3 = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$$
• Each matrix in $$A=SU(2)$$ can be represented as $$\begin{pmatrix}\alpha & -\bar \beta \\ \beta & \bar\alpha\end{pmatrix},\, |\alpha|^2 + |\beta|^2 = 1$$ and decomposed as $$A=wI-ix\sigma_1-iy\sigma_2-iz\sigma_3$$ with $$x^2+y^2+z^2+w^2=1$$. That is $$I,-i\sigma_1,-i\sigma_2, -i\sigma_3$$ are sort of a basis for $$SU(2)$$, but values have to be normalized
• Each matrix in $$A=SU(2)$$ can be obtained as $$A=\exp (H)$$ where $$H$$ is skew Hermitian. Now a basis for skew Hermitian matrices is also $$-i\sigma_1,-i\sigma_2, -i\sigma_3$$ and $$H=-\alpha i\sigma_1-\beta i\sigma_2 -i\gamma \sigma_3$$

My question is: is there a simple relation between the parametrization $$w,x,y,z\in S^3$$ for $$A\in SU(2), A=wI-ix\sigma_1-iy\sigma_2-iz\sigma_3$$ and the parametrization $$\alpha,\beta,\gamma\in \mathbb{R}^3$$ for $$A\in SU(2), A=\exp (H),H=-\alpha i\sigma_1-\beta i\sigma_2 -i\gamma \sigma_3$$

The question arises because I see often the first parametrization in use (for example as a way to represent unit quaternions) but never the second one

The exponentiating a Pauli vector is straightforward, $$A\in SU(2),~~ A=\exp (H), \qquad H=-\alpha i\sigma_1-\beta i\sigma_2 -i\gamma \sigma_3 .$$ Define $$\theta ^2\equiv \alpha^2+\beta^2+\gamma^2 , \qquad n_1=\alpha /\theta;\qquad n_2=\beta /\theta;\qquad n_3=\gamma /\theta ,$$ so that the antihermitean logarithm of the group element is but $$H= -i\theta~ \vec n \cdot \vec \sigma , ~~ \Longrightarrow ~\exp (H)= \cos \theta -i\sin\theta ~~\vec n \cdot \vec \sigma ~.$$
That is to say $$w=\cos\theta; \qquad x= \frac{\sin\theta}{\theta} \alpha ; \qquad y= \frac{\sin\theta}{\theta} \beta ; \qquad z= \frac{\sin\theta}{\theta} \gamma ~$$ a euclidean unit four-vector, alright.