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?
Best 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 $\Psi^\dagger$.