# $(1/2,1/2)$ Representation transformation laws in Schwichtenberg has extra transpose

I'm reading Schwichtenberg's Physics from Symmetry. I have a question about the derivation of the $$(1/2,1/2)$$ representation of the lorentz group.

The issue is that Schwichtenberg adds an extra traspose to his 'lower dot' trasnformation law.

The problem is with the following equation (3.225):

$$v \rightarrow v' =v'_{a\dot{b}} =(e^{i\theta\sigma/2+\phi\sigma/2})_a^cv_{c\dot{d}}((e^{-i\theta\sigma^*/2+\phi\sigma^*/2})^\dot{d}_\dot{b})^T$$

It seems to me that the equation should not include the transpose on the right hand side, and that the equation should instead look like this:

$$v \rightarrow v' =v'_{a\dot{b}} =(e^{i\theta\sigma/2+\phi\sigma/2})_a^cv_{c\dot{d}}(e^{-i\theta\sigma^*/2+\phi\sigma^*/2})^\dot{d}_\dot{b}$$

I don't see why Schwichtenberg adds the transpose, the transformation law on 3.210 doesn't include a transpose, so it seems to come out of nowhere.

$$X_\dot{a}= (e^{-i\theta\sigma^*/2+\phi\sigma^*/2})^\dot{b}_\dot{a} {X_\dot{b}}$$

I'm sure I'm missing something, thank you.

• Which page? Which eqs? Commented Jul 23 at 17:25
• @Qmechanic page 82 equation 3.225, the transformation law used is on page 78 and equation 3.210. Commented Jul 23 at 17:36