0
$\begingroup$

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?

$\endgroup$

1 Answer 1

1
$\begingroup$

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 $\Psi^\dagger$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.