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

  • #1
12
0
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

[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?
 

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
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.
 

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