# Confusion about signs in the Dirac equation for an external electromagnetic field

I'm working through Maggiore's A Modern Introduction to Quantum Field Theory, and I'm studying the Dirac equation in an external electromagnetic field given by:

$$\left[\gamma^{\mu} \left(i\partial_{\mu} - A_{\mu} \right) -m \right] \begin{pmatrix} \phi \\ \chi \end{pmatrix} = 0.$$

This is found in Equation (3.178) at the beginning of Section 3.6. He then expands the above into two equations by first defining $$\phi' = \exp\left[ imt\right] \phi$$ and $$\chi' = \exp\left[ imt\right] \chi$$. Substituting these into the above equation and grinding the gears of algebra, he comes up with the following two equations:

$$\left[i\partial_0 - eA_0 \right] \phi' = - \textbf{\sigma} . \left(i \nabla + e\textbf{A} \right) \chi',$$

$$\left[i\partial_0 - eA_0 + 2m \right] \chi' = - \textbf{\sigma} . \left(i \nabla + e\textbf{A} \right) \phi'.$$

I can get these two equations without too much trouble, but the problem I'm having is that the $$+ e\textbf{A}$$ on the right-hand side of the equations has a negative sign for me.

When I explicitly worked out the matrix on the left-hand side of the first equation in this post, I got the following:

$$\begin{pmatrix} \left[ i\partial_0 - eA_0 - m \right] & \left[ \textbf{\sigma} . \left(i \nabla - e\textbf{A} \right) \right] \\ \left[ - \textbf{\sigma} . \left(i \nabla - e\textbf{A} \right) \right] & \left[ -i\partial_0 + eA_0 - m \right] \end{pmatrix} \begin{pmatrix} \phi \\ \chi \end{pmatrix} = 0.$$

I don't think there's a mistake in the book, but I can't seem to figure out where my sign issue is. There should be a $$+e$$ in the matrix, but I seem to expanding the terms incorrectly. Does anyone have some insight into where the extra negative sign comes from?

The gamma matrices I'm using are:

$$\gamma^0 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$

$$\gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^1 & 0 \end{pmatrix}.$$

I just found the answer to my own question. It has to do with the definition for the gradient. It turns out that $$i\partial^i = -i\partial_i$$ which brings the equations back to what Maggiore has.