In David McMohans QFT Demystified book on p66 it is claimed that the correspondance between $SO(3)$ which rotates space vectors $\vec{x} = (x,y,z)^T$ and the group $SU(2)$ which acts on two-component spinors $\psi = (\alpha, \beta)^T$ can be seen by making use of the maping between space vectors and two-component spinors given by

\[x = \frac{1}{2}(\beta^2-\alpha^2); \quad y= -\frac{i}{2} (\beta^2+\alpha^2); \quad z = \alpha\beta \]

Where does this specific mapping come from? Is it unique or are there other maps between space vectors and spinors?

Then, the author goes forward and writes that the 3 parameters of $SU(2)$ can be associated with angles of rotation of $SO(3)$ such that for an arbitrary angle $\alpha$ (not a spinor component now) a 3D rotation around the x axis can be written as 2x2 or 3x3 matrix as

\[U = \left( \begin{array}{cc} \cos(\alpha/2) & i\sin(\alpha/2) \\ i\sin(\alpha/2) & \cos(\alpha/2) \\ \end{array} \right) \quad \mathrm{or} \quad R_x(\omega) \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos\omega & \sin\omega \\ 0 & - \sin\omega & \cos\omega \\ \end{array} \right) \]

a 3D rotation around the y-axis as by the angle $\beta$ as

\[U = \left( \begin{array}{cc} \cos(\beta/2) & \sin(\beta/2) \\ -\sin(\beta/2) & \cos(\beta/2) \\ \end{array} \right) \quad \mathrm{or} \quad R_y(\phi) \left( \begin{array}{ccc} \cos\phi & 0 & \sin\phi \\ 0 & 1 & 0 \\ -\sin\phi & 0 & \cos\phi \\ \end{array} \right) \]

and a 3D rotation around the z-axis by the angle $\gamma$ as

\[U = \left( \begin{array}{cc} e^{i\gamma/2} & 0 \\ 0 & e^{-i\gamma/2} \\ \end{array} \right) \quad \mathrm{or} \quad R_z(\theta) \left( \begin{array}{ccc} \cos\theta & \cos\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \end{array} \right) \]

First of all, do the angles of the two representations correspond to each other too, such that $\alpha = \omega$, $\beta = \phi$, and $\gamma = \theta$?

And how can it be proven or seen that the specific 2x2 matrix of $SU(2)$ represent exactly the same 3D rotation as the 3x3 matrix of $SO(3)$ in each case?