How to show that $\bar\psi\gamma^\mu\psi$ of a Dirac spinor $\psi$ transforms as a vector? This is part 2 of exercise II.1.1 of Zee's QFT in a Nutshell (here's part 1).
This is what I have got:
\begin{align}
\bar\psi\gamma^\lambda\psi \mapsto \bar\psi^{\,\prime}\gamma^\lambda\psi^{\,\prime} & = {\psi^{\,\prime}}^\dagger{\gamma^0}^\dagger\gamma^\lambda\psi^{\,\prime} = (S(\Lambda)\psi)^\dagger\gamma^0\gamma^\lambda(S(\Lambda)\psi) \\
& = \psi^\dagger S(\Lambda)^\dagger\gamma^0\gamma^\lambda S(\Lambda)\psi\\
& = \tag{1}\psi^\dagger e^{\frac i4\omega_{\mu\nu}{\sigma^{\mu\nu}}^\dagger}\gamma^0\gamma^\lambda e^{-\frac i4\omega_{\mu\nu}\sigma^{\mu\nu}}\psi\\
& = \psi^\dagger\left(1+\frac i4\omega_{\mu\nu}(\sigma^{\mu\nu})^\dagger+O({\omega_{\mu\nu}}^2)\right)\gamma^0\gamma^\lambda\left(1-\frac i4\omega_{\mu\nu}\sigma^{\mu\nu}+O({\omega_{\mu\nu}}^2)\right)\psi\\
& = \psi^\dagger\gamma^0\left(\gamma^\lambda+\frac i4\omega_{\mu\nu}[[\gamma^\mu,\gamma^\nu],\gamma^\lambda]+O({\omega_{\mu\nu}}^2)\right)\psi\\
& = \psi^\dagger\gamma^0\left(\gamma^\lambda+\frac 12\omega_{\mu\nu}[\sigma^{\mu\nu},\gamma^\lambda]+O({\omega_{\mu\nu}}^2)\right)\psi
\end{align}
Two questions:


*

*How exactly does the last line prove that $\bar\psi\gamma^\lambda\psi$ transforms as a vector under Lorentz transformations? It certainly looks like a vector transformation to me, because of the commutator between the Lorentz generators and the "vector components" $\gamma^\mu$, but how can I prove this quantitatively?

*Then, a more general question: How can I get rid of the $O({\omega_{\mu\nu}}^2)$? I know that it can be ignored since we consider infinitesimal transformations only, so the transformation behavior is governed by first-order terms only. But what's the mathematical rigorous way to get rid of them? Do I just write a $\simeq$ sign and leave them out at the right-hand side of the $\simeq$? That would not seem right to me, because when dealing with expansions, a $\simeq$ does not imply strict equality, it just implies "equality up to a certain order".

 A: It gets easier if you use the result from part 1. Then you also don't have to deal with the $\mathcal O(\omega^2)$ (see my answer to your other question).
In your calculation, you transformed $\bar\psi$ and $\psi$, but not $\gamma^\lambda$.
This is correct, as I will show in the end, but I will take another point of view which is really helpful here:
$\gamma^\lambda$ is an object with one Lorentz index $\lambda$ and two fermionic indices, hence it should transform as
$$ {\gamma'}^\lambda = {\Lambda^\lambda}_\nu (S \gamma^\nu S^{-1}) $$
(according to the general rules how tensors transform).
Because we already know from part 1 that $\bar\psi \to \bar\psi S^{-1}$, we immediately get
$$ \bar\psi \gamma^\lambda \psi \to {\Lambda^\lambda}_\nu \bar\psi S^{-1} S \gamma^\nu S^{-1} S \psi = {\Lambda^\lambda}_\nu \bar\psi \gamma^\nu \psi \;, $$
this is the expected transformation behavior of a Lorentz vector.

What you did is also correct, of course, but missing one ingredient.
Since
$$ {\Lambda^\lambda}_\nu (S \gamma^\nu S^{-1}) = \gamma^\lambda \;, $$
the $\gamma$ matrices actually do not transform.
Proving this is the trickier bit (also not too hard, though).
I haven't read Zee's book, but I would guess he proves it somewhere?
(You could start by writing $\bar\psi \gamma^\lambda \psi \to \bar\psi S^{-1} \gamma^\lambda S \psi$ and then use this identity to get to the same result.)
