In any textbook, hermitian conjugate of spinor is defined like $ \psi_{\alpha}^{+}=\bar\psi_{\dot{\alpha}} $ and $(\psi^{\alpha})^{+}=\bar{\psi}^{\dot\alpha}$.
We have Pauli matrices $\sigma^{\mu}_{\alpha\dot\beta}$ and $(\bar\sigma^{\mu})^{\dot\alpha\beta}$ they are hermitian matrices i.e.
$(\sigma^{\mu}_{\alpha\dot\beta})^{+}=\sigma^{\mu}_{\alpha\dot\beta}$ and $((\bar\sigma^{\mu})^{\dot\alpha\beta})^{+}=(\bar\sigma^{\mu})^{\dot\alpha\beta}$.

Now consider the known expression $$(\xi\sigma^{\mu}\bar\psi)^{+}=\psi\sigma^{\mu}\bar\xi$$

that is $(\xi^{\alpha}\sigma^{\mu}_{\alpha\dot\beta}\bar\psi^{\dot\beta})^{+}=(\bar\psi^{\dot\beta})^{+}(\sigma^{\mu}_{\alpha\dot\beta})^{+}(\xi^{\alpha})^{+}=\psi^{\beta}\sigma^{\mu}_{\alpha\dot\beta}\bar\xi^{\dot\alpha}=???$

there is no more possible to sum over indices at all. Where is the mistake?

And one more question: By definition hermitian conjugate of two spinors$(\psi\xi)^{+}=\xi^{+}\psi^{+}$. But how is defined $(P^{\mu}Q_{\alpha})^{+}$ where $P^{\mu}$ is momentum operator and $Q_{\alpha}$ is spinor. On which space ${"+"}$ acts?

This post imported from StackExchange Physics at 2015-02-11 11:52 (UTC), posted by SE-user Paramore