How would you evaluate 
\begin{equation}|iD\!\!\!\!/-m|\end{equation} Where $D\_{\mu}=\partial\_{\mu}-ieA\_{\mu}$.
I have an idea of how to do this without the gauge field, because it's essentially 
\begin{equation}|i\partial\!\!\!/-m|=|\partial^{2}-m^{2}|^{1/2}\end{equation}
 and A. Zee's book covers this is some detail. From what I can tell, you use $\ln|A|=\mathrm{tr}\ln A$. The trace is in momentum space 
\begin{equation}\frac{1}{2}\int \frac{d^{4}k}{(2\pi)^{4}} \ln (k^{2}-m^{2}-i\epsilon)+C\end{equation}
How do you do this in the presence of a gauge field?
I've gotten this far
\begin{equation}|iD\!\!\!\!/-m|=e^{\frac{1}{2}\mathrm{tr}\ln \left(D^{2}-(e/2)\sigma^{\mu\nu}F\_{\mu\nu}+m^{2}\right)}\end{equation}
With $\sigma^{\mu\nu}=(i/2)[\gamma^{\mu},\gamma^{\nu}]$. But I have no idea how to proceed.