# Doubt on dotted indices notation for Weyl spinors

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?

## 1 Answer

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 $$\Psi^\dagger$$.