Partial derivatives in the derivation of a Dirac Spinor

Question

As per JF132's answer to Conservation of the axial current using Dirac equations of motion,

"since the gamma matrices $\gamma^\mu$ are $4\times 4$ matrices, and the conjugate Dirac spinors $\bar{\psi}$ are a $1\times 4$ row vectors, the following are not equivalent because of the rules of matrix multiplication,"

$$ \begin{align} i\partial_\mu\bar{\psi}\gamma^\mu+m\bar{\psi} &=0 \\ i\gamma^\mu\partial_\mu\bar{\psi}+m\bar{\psi} &=0 \\ (i\gamma^\mu \partial_\mu + m)\bar{\psi} &= 0 \end{align} $$

My confusion is regarding how the partial derivative works here. It seems to me that $\partial_\mu \bar{\psi}$ is a scalar, and so I'm free to multiply $\gamma^\mu$ on the right instead of on the left if I want.

What is the flaw in my understanding?

Answer 1

Working out all the indices, the partial derivative has one Lorentz index $\mu$ and a Dirac index (spinor index) $a$ so that $$\partial_\mu \bar{\psi} \equiv \partial_\mu\bar{\psi}_a$$ So this quantity is not a Lorentz scalar, neither a Dirac scalar. The gamma matrices have also two Dirac indices $$\gamma^\mu \equiv \left(\gamma^\mu\right)^a_{\;b}$$ so you need to be careful on how you move things.