[Physics] Hermitian Adjoint of Spinor

Question

Say we have a four component spinor $\psi$:
$$
\psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}
$$
Is the Hermitian adjoint of this:
$$
\psi^\dagger =\begin{pmatrix}\psi_L^\dagger \psi_R^\dagger\end{pmatrix}
$$
OR
$$
\psi^\dagger =\begin{pmatrix}\psi_L^* \psi_R^*\end{pmatrix}~?
$$

Answer 1

Its the first one. This is exactly what the "dagger" does. It transposes the spinor, converting it from a column spinor to a row spinor, and takes every entry to its complex conjugate, i.e:

$$ \psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix} \xrightarrow{\dagger} \begin{pmatrix}(\psi^T_L)^* (\psi^T_R)^*\end{pmatrix} = \begin{pmatrix}\psi_L^\dagger \psi_R^\dagger\end{pmatrix} $$