spinor vs vector indices of Dirac gamma matrices

Gamma matrices are defined by the Clifford algebra

$$ \{\gamma^\mu, \gamma^\nu\}= 2g^{\mu\nu}\mathbb I_n \,. $$

So, you see the index $\mu$ in $\gamma^\mu$ runs from $0$ upto $D-1$ where $D$ is the number of spacetime dimensions. It does not mean $\gamma^\mu$ is a vector. The $\mu$ index here only tells you how many gamma matrices are there. The dimensionality of the matrices themselves is $n= 2^{[D/2]}$ where $[\cdot]$ gives you the integer part of a number. For example, in $(1+2)-$dimensions, $D=3$ and hence the Dirac matrices are $2^{[1.5]}= 2$ dimensional, which you recognize are the Pauli matrices. The labels of the entries of the gamma matrices are known as spinor indices. So, in 3 dimensions, for example, the $a,b$ in $\gamma^\mu_{ab}$ would run from $1$ to $2$.

What is a $4$-vector? It is something that transforms like a vector under Lorentz transformations $\Lambda$. Namely, $X^\mu$ is a vector if it transforms like

$$ X^\mu\to {\Lambda^\mu}_\nu X^\nu \,. $$

That's the definition! Just having a $4$-dimensional column vector with Greek indices labelling its entries does not make it a Lorentz vector. It needs to transform the right way.

Okay, so what is a spinor? A spinor is something that transforms like a spinor. Namely, $\psi$ is a spinor if it transforms, under a Lorentz transformation parametrized by $\omega_{\mu\nu}$, like

$$ \psi \to \Lambda_{1/2} \psi\, \qquad (\Rightarrow \overline\psi \to \overline\psi\ \Lambda_{1/2}^{-1}\ ) \,, $$

where $\Lambda_{1/2} = \exp{(-\frac i2 \omega_{\mu\nu} S^{\mu\nu})}$ and $S^{\mu\nu} = \frac i4 [\gamma^\mu, \gamma^\nu]$ generates an $n-$dimensional representation of the Lorentz algebra.

Let's make a remark on why we use something like $\overline \psi = \psi^\dagger \gamma^0$. Well, because we want to construct bilinear Lorentz scalars like $\psi^\dagger \psi$, but $\psi^\dagger \psi$ is not a Lorentz scalar precisely because the matrix $\Lambda_{1/2}$ is not unitary. Under a Loretz transformation,

$$ \psi^\dagger \to \psi^\dagger \Lambda_{1/2}^\dagger \ne \psi^\dagger \Lambda_{1/2}^{-1}\,.$$

However, we notice an interesting property of the gamma matrix $\gamma^0$.

$$ \boxed{ \Lambda_{1/2}^\dagger \gamma^0 = \gamma^0 \Lambda_{1/2}^{-1} }$$

This immediately tells us that defining something like $\overline \psi \equiv \psi^\dagger \gamma^0$ will do the job.

$$ \overline \psi \to (\psi^\dagger \Lambda_{1/2}^\dagger)\gamma^0 = \psi^\dagger \gamma^0 \Lambda_{1/2}^{-1} = \overline\psi \Lambda_{1/2}^{-1} $$

Because of this special property of $\gamma^0$, now we have that $\overline\psi\psi\to \overline\psi\psi$.

You can check that the gamma matrices also satisfy the relation

$$ \Lambda_{1/2}^{-1} \gamma^\mu_{ab} \Lambda_{1/2} = {\Lambda^\mu}_\nu \gamma^\nu_{ab}\,. $$

Understand that this is not a transformation of the gamma matrices under a Lorentz transformation. Gamma matrices are fixed constant matrices that form the basis of an algebra. They do not transform. The above is just a property of the gamma matrices due to them being generators of a particular representation of the Lorentz algebra.

However, this relation allows you to take the $\mu$ index in $\gamma^\mu$ "seriously". Because, due to this you can immediately see that under a Lorentz transformation, the current $J^\mu := \overline\psi \gamma^\mu \psi= \overline\psi^a \gamma^\mu_{ab} \psi^b$ indeed transforms like a vector.

$$ J^\mu \to {\Lambda^\mu}_\nu J^\nu \,.$$