Doubt on dotted indices notation for Weyl spinors

Question

I'm starting Bailin and Love "Supersymmetric gauge field theory and string theory" and trying to get used to dotted indices.

Let's consider a Dirac spinor written in terms of left and right Weyl spinors

$$\Psi=\begin{pmatrix}\psi_\alpha\\ \bar{\chi}^\dot{\alpha}\end{pmatrix}~~~~,~~~~\alpha,\beta=1,2\tag{1.83}$$

$$(\psi_\alpha)^*=\bar{\psi}_\dot{\alpha}~~~~,~~~~(\bar{\chi}^\dot{\alpha})^*=\chi^\alpha\tag{1.77}$$

Now, consider the $\gamma$ matrices in the Weyl representation $$\gamma^\mu=\begin{pmatrix}0&\sigma^\mu\\\bar\sigma^\mu&0\end{pmatrix},$$ where $$(\sigma^\mu)_{\alpha\dot\beta}=(\mathbb{I}_2,\vec{\sigma})_{\alpha\dot\beta}~~~~,~~~~(\bar\sigma^\mu)^{\dot\alpha\beta}=(\mathbb{I}_2,-\vec{\sigma})^{\dot\alpha\beta}\tag{1.58}$$

What I see is that $\Psi^\dagger\gamma^\mu$ (which includes $\bar\Psi=\Psi^\dagger\gamma^0$) doesn't seem to match indices: $$\Psi^\dagger\gamma^\mu=\begin{pmatrix}\bar{\psi}_\dot{\alpha}&\chi^\alpha\end{pmatrix}\begin{pmatrix}0&(\sigma^\mu)_{\alpha\dot\beta}\\(\bar\sigma^\mu)^{\dot\alpha\beta}&0\end{pmatrix}``="\begin{pmatrix}\chi^\alpha(\bar\sigma^\mu)^{\dot\alpha\beta}&\bar{\psi}_\dot{\alpha}(\sigma^\mu)_{\alpha\dot\beta}\end{pmatrix}$$

Am I doing something wrong?

Answer 1

Indeed $$ \Psi^\dagger \gamma^\mu\Psi $$ is not a Lorentz invariant bilinear. You have to write $$ \Psi^\dagger \gamma^0\gamma^\mu\Psi = \Psi^\dagger \left( \begin{array}{cc}0&\mathbb{I}\\\mathbb{I}&0 \end{array} \right)\gamma^\mu\Psi = \chi^\alpha(\sigma^\mu)_{\alpha\dot\beta}\bar{\chi}^{\dot\beta} + \bar{\psi}_{\dot{\alpha}}(\bar{\sigma}^\mu)^{\dot\alpha\beta}\psi_\beta\,. $$ The $\gamma^0$ is kind of a "cheat." You don't have to think about where to place the indices but use it just to swap the two entries of the $\Psi^\dagger$.