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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:

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

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

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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.

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