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

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

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

<br /> U(x) =<br /> \left(<br /> \begin{array}{cccc}<br /> Re(U_{11}) &amp; Im(U_{11}) &amp; Re(U_{12}) &amp; Im(U_{12}) \\<br /> Im(U_{11}) &amp; Re(U_{11}) &amp; Im(U_{12}) &amp; Re(U_{12}) \\<br /> Re(U_{21}) &amp; Im(U_{21}) &amp; Re(U_{22}) &amp; Im(U_{22}) \\<br /> Im(U_{21}) &amp; Re(U_{21}) &amp; Im(U_{11}) &amp; Re(U_{22}) \\<br /> \end{array}<br /> \right)<br />

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.