How can I construct the 4D real representation of SU(2)?

[Dilatino]

An element of $SU(2)$, such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as

$$U(x) = e^{ixT_1} = \left( \begin{array}{cc} \cos\frac{x}{2} & i\sin\frac{x}{2} \\ i\sin\frac{x}{2} & \cos\frac{x}{2} \\ \end{array} \right) = \left( \begin{array}{cccc} c & 0 & 0 & -s \\ 0 & c & s & 0 \\ 0 & -s & c & 0 \\ s & 0 & 0 & c \\ \end{array} \right)$$

I assume that here $c = \cos\frac{x}{2}$ and $s = \sin\frac{x}{2}$.The last 4 by 4 matrice is said to be constructed by treating the real and complex parts of each complex number as two real numbers. However, when doing this I would rather have expected that each complex number in the 2 by 2 matrice is expanded into its own 2 by 2 matrice, such that the resulting 4 by 4 matrice would schematically look like

$$U(x) = \left( \begin{array}{cccc} Re(U_{11}) & Im(U_{11}) & Re(U_{12}) & Im(U_{12}) \\ Im(U_{11}) & Re(U_{11}) & Im(U_{12}) & Re(U_{12}) \\ Re(U_{21}) & Im(U_{21}) & Re(U_{22}) & Im(U_{22}) \\ Im(U_{21}) & Re(U_{21}) & Im(U_{11}) & Re(U_{22}) \\ \end{array} \right)$$

But this is obviously not how the 4 by 4 matrice is constructed. What am I missing or misunderstanding?

 

[fzero]

I think you want to let $U$ act on
$$ U \begin{pmatrix} a+ib \\ c+ id \end{pmatrix} = \begin{pmatrix} a'+ib' \\ c'+ id' \end{pmatrix}$$
and then rewrite this as a 4x4 matrix acting on $(a~b~c~d)^T$. The reason that $U_{4\times 4}$ does not take the 2nd form you wrote is because of the factors of $i$ and $i^2$.

Note that this doesn't actually give what we would call a representation of the group, because, if we do the same mapping on the generators (Pauli matrices), the image matrices don't satisfy the SU(2) algebra. If you really want the 4-dimensional representation, then a good way to work it out is by noting that it is the spin 3/2 representation. Then we know how the $J_\pm, J_z$ act on the states $|3/2,m\rangle$ and we can work out appropriate matrices for them.