Gamma matrices invariant under lorentz transformation

I know this has been asked before but I just can't seem to get my head around it based on the answers I've read.

So the idea is that we have the gamma matrices $\gamma^{\mu}$. Now from my understanding, this $\mu$ corresponds to a space time index, whilst for each fixed value of $\mu$, $\gamma^{\mu}$ defines a $4 \times 4$ matrix which is defined in spinor space, thus more explicitly $\gamma^{\mu}_{ab}$ where $a,b$ denote the spinor indices.

Now usually when showing covariance of the Dirac equation, we take the gamma matrices to be scalars which thus don't transform. We then find that they must satisfy $$\gamma^{\mu}=S[\Lambda]\Lambda_{\nu}^{\mu} \gamma^{\nu} S[\Lambda]^{-1}$$ in order for covariance to be satisfied.

I just can't for the life of me understand why we would start by treating $\gamma^{\mu}$ as a scalar. To me it seems clear that it would transform as a field with two spinor indices and a vector index, that is $$\gamma^{\mu} \to \gamma^{\mu'}=S[\Lambda] \Lambda_{\nu}^{\mu} \gamma^{\nu} S[\Lambda]^{-1}$$

Why are we able to just ignore the spacetime and spinior indices associated to the gamma matrices whilst treating them like scalars?

Edit: I feel like this paper https://arxiv.org/pdf/1309.7070.pdf talks exactly about what I'm not really understanding on page 5, however it doesn't really give an answer to any of the problems it poses using that method, but instead switches to a different method/picture. "In addition, the mathematical origin of $\gamma^{\mu}$ matrices is explained, from which it becomes clear why they are fixed matrices which do not transform", something that explains this line would be super helpful!

• Actually the $\psi\dagger \gamma^0 \gamma^\mu \psi$ transform as a vector under Lorentz transformations. – my2cts Apr 25 '18 at 22:10
• I deleted my answer because the paper you mentioned in the edit is saying the same thing that I have said in a more elegant and comprehensive way. I think the answer to your question lies on the expressed fact that gamma matrices can be treated separately as a vector and a spinor matrix. However these separated transformations are related by the equation (14). – Oktay Doğangün Apr 25 '18 at 22:36
• Actually, this issue become clearer when you work on the Dirac equation in curved spacetime (but to learn tetrad/vierbein formalism of gravity is the cost). – Oktay Doğangün Apr 25 '18 at 22:53
• In order to understand why this is the case you have to know the isomorphism theorem $\text{SO}^{\uparrow}(3,1)\cong \text{SL}(2,\mathbb{C})/\mathbb{Z}_2$. Then use the fact that the (reducible) dirac representation is the direct sum of two fundamental representations of $\text{SL}(2,\mathbb{C}$. The Pauli matrices (which have one vector, one dotted spinor and one undotted spinor indice) transform in a very similar way. They are the more fundamental object. – NormalsNotFar Apr 25 '18 at 23:27