An element of [itex]SU(2)[/itex], such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

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)

[/tex]

I assume that here [itex]c = \cos\frac{x}{2}[/itex] and [itex]s = \sin\frac{x}{2}[/itex].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

[tex]

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)

[/tex]

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

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

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