Functional Determinant of a system of differential operators?

[Yellotherephysics]

TL;DR Summary

When applying the stationary phase approximation for path integrals we need to calculate the determinant of an operator; but what can we do if this operator is itself a matrix (made out of operators!)

So in particular, how could the determinant of some general "operator" like

$$ \begin{pmatrix}
f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x)
\end{pmatrix} $$

with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary phase approximation context to divide by some reference operator such as

$$ \begin{pmatrix}
0 & \frac{d}{dx} \\ \frac{d}{dx} & 0
\end{pmatrix} $$

and use this value, for the quantum corrections to the classical path?

 

[vanhees71]

Maybe, my lecture notes on many-body QFT helps. I use the heat-kernel method:

https://itp.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

 

[Yellotherephysics]

Thanks for linking, your notes look really good vanhees71!

Would this be a fair summary for the Heat-Kernel method (starting on page 93 of the linked notes)?
If we call the operator we're interested in H.

1. Work out the propogator $\langle x | exp(- H \theta) | x' \rangle$
2. Then get some kind of expression for $\langle x | exp(- H \theta) | x \rangle$
3. Find the Heat-Kernel (a function of $\theta$) by integrating over the coordinates
4. Find the other Heat-Kernel thing, denoted by $\tilde{H}$ (a function of the complex parameter $\alpha$)
5. Take the complex parameter $\alpha$ to zero, getting the trace of the log of H, which we can then relate to the determinant

Is this algorithm still safe to use if H is something like
\begin{pmatrix}

a(x) & \frac{d}{dx} & b(x) \\ \frac{d}{dx} & c(x) & d(x) \\ e(x) & f(x) & g(x)

\end{pmatrix}
In the notes $\langle x | exp(- H \theta) | x' \rangle$ is turned into something concrete by integrating over the momentum identity, would I be able to do something similar with my matrix example?