I'm trying to find how does this quantity $$\psi^\dagger\psi$$ transforms under a Lorentz transformation. Where $\psi$ is a Dirac spinor.
What I've tried so far:
It is known that a Dirac spinor transforms as $$\psi' = S\psi.$$ The matrix $S$ satisfy certain properties. I calculated $\psi'^\dagger=\psi^\dagger S^\dagger$, then: $$\psi'^\dagger\psi'=\psi^\dagger S^\dagger S\psi.$$
I don't known how to continue from here.
Another way I have tried was rewrite $\psi'^\dagger$ this way: $$\psi'^\dagger = \bar{\psi'}\gamma^0$$ Where $\bar{\psi'}=\psi'^\dagger \gamma^0$. Then, using $\bar{\psi'}=\bar{\psi}S^{-1}$ $$\psi'^\dagger\psi'=\bar{\psi}S^{-1}\gamma^0 S\psi.$$ Using this property of the $S$ matrix $(S^{-1})_{\alpha\beta}(\gamma^\lambda)_{\rho\sigma}(S)_{\sigma\beta}=a^{\lambda}_{\mu}(\gamma^{\mu})_{\alpha\beta}$ with $\lambda=0$: $$\psi'^\dagger\psi' = a^{0}_{\mu}\bar{\psi}\gamma^\mu\psi.$$
The coefficients $a^{\mu}_{\nu}$ are a general Lorentz transformation (proper Lorentz transformation, rotations,etc). I don't know if this is the correct way of approaching this problem. I know that $\psi^\dagger\psi$ is not a Lorentz scalar.
Any help is appreciated.