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Let $R_z(\theta)$ be a rotation matrix around $z$ axis, where $R_z(\theta)\in\operatorname{SO}(3)$. I know that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so $R_z(\theta)$ has to pre-images in $\operatorname{SU}(2)$. But how can I found them?

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  • $\begingroup$ Use the map here. $\endgroup$ Commented Aug 5, 2022 at 11:35
  • $\begingroup$ @DietrichBurde I'm familiar with this map. But I didn't succeed to get a lift to rotations. I thought maybe there is a more simple way or answer to this... $\endgroup$ Commented Aug 5, 2022 at 11:37

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We have $\mathfrak{so}(3)\cong\mathfrak{su}(2)$, see here or Masahito Hayashi, Group Representation for Quantum Theory, 2017 (See here, equation (3.27) on page 82). The Pauli matrices $\sigma_1$, $\sigma_2$ and $\sigma_3$ are a basis for $\mathfrak{su}(2)$ with: $$[\sigma_i,\sigma_j]=2i\varepsilon_{ijk}\sigma_k \Leftrightarrow \left[\frac{\sigma_i}{2i},\frac{\sigma_j}{2i}\right]=\varepsilon_{ijk}\frac{\sigma_k}{2i}.$$ The matrices: $$L_1=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix},\; L_2=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},\; L_3=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ known in quantum mechanics to describe angular momentum for the angular momentum quantum number $l=1$ with $m_l=-1;0;+1$ are a basis of $\mathfrak{so}(3)$ with: $$[L_i,L_j]=\varepsilon_{ijk}L_k.$$ Therefore the isomorphism is given by the identification $\frac{\sigma_j}{2i}\leftrightarrow L_j$.

Write $\vec\sigma$ and $\vec{L}$ for the collection of the three respective matrices into a single vector, then using a positive scalar $\theta$ and a unit vector $\vec{n}$, we have (See here and here): $$e^{i\theta\vec{n}\cdot\vec\sigma} =1\cos(\theta)+i\vec{n}\cdot\sigma\sin(\theta) \in\operatorname{SU}(2) \tag{1}$$ $$e^{i\theta\vec{n}\cdot\vec{L}} =1+\underbrace{2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)}_{=\sin(\theta)}(\vec{n}\cdot\vec{L}) +2\sin\left(\frac{\theta}{2}\right)^2(\vec{n}\cdot\vec{L})^2 \in\operatorname{SO}(3) \tag{2}$$ The matrix in equation (2) is exactly the matrix describing the rotation by the angle $\theta$ around the unit vector $\vec{n}$ (which in your case is $e_z$ and simplifies the formula). Given a matrix in $\operatorname{SO}(3)$, $\vec{n}$ can therefore be obtained as its eigenvector and $\theta$ by putting a vector perpendicular to $\vec{n}$ into the matrix. If we have $\theta$ and $\vec{n}$, we can put them into equation (1) to compute a preimage in $\operatorname{SU}(2)$ unter the covering and simply by just swapping the sign, getting the other.

It should be noted that this results in $\operatorname{PSU}(2)=\operatorname{SU}(2)/\{\pm 1\}$ being homeomorphic to $\operatorname{SO}(3)$ (See here).

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