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Ravendark

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**1./2. Homework Statement**

In my QFT lecture we were introduced to the 1PI effective action ##\Gamma[\varphi]## for a scalar theory (in Euclidean space-time). In one-loop approximation we've found [tex] \Gamma^{(\text{1-loop})}[\varphi] = S[\varphi] + \frac{1}{2} \operatorname{Tr} \log D^{-1}[/tex] where ##D^{-1} = \delta^2 S[\varphi] / \delta \varphi \delta \varphi## denotes the inverse classical propagator.

Now I'm asked to compute the derivative of ##\Gamma^{(\text{1-loop})}## with respect to ##D##. And this is is point where I got stuck.

## Homework Equations

We had the functional derivative in the exercises and we derived formulas like [tex]\frac{\delta}{\delta f(y)} f(x) = \delta(x-y) \; , \qquad \frac{\delta}{\delta f(y)} \int dx \, f(x)^n = n f(y)^{n-1} \quad (n \in \mathbb{N})\; .[/tex] Furthermore, I know that the functional derivative obeys the product and chain rule.

## The Attempt at a Solution

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Since the action does not depend on ##D##, I have: [tex] \frac{\delta}{\delta D(x,y)} \Gamma^{(\text{1-loop})}[\varphi] = \frac{1}{2} \frac{\delta}{\delta D(x,y)} \operatorname{Tr} \log D^{-1}[/tex] Now I'm not sure how do deal with the trace...I think it is meant in the functional sense, thus [tex]\frac{\delta}{\delta D(x,y)} \Gamma^{(\text{1-loop})}[\varphi] = \frac{1}{2} \frac{\delta}{\delta D(x,y)} \int d^4z \, \operatorname{tr} \log D^{-1}(z,z)[/tex] where the lowercase tr runs over internal indices (colour, flavour...). Now I have no idea how to proceed...there is this trace, a log and furthermore the inverse of ##D## instead ##D## itself. Can someone give me a hint, please?