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In Giuliani & Vignale's Quantum Theory of the Electron Liquid, in page 126, they point out that the absorption and emission spectra are related by $$S_{AA^\dagger}(-\omega)=e^{-\beta\hbar\omega}S_{A^\dagger A}(\omega)$$ and that to obtain such relation, you should just start off from the definition $$S_{AA^\dagger}(\omega)=\sum_{nm}P_m|A_{mn}|^2\delta(\omega-\omega_{mn})$$ and take $\omega\rightarrow-\omega$ and interchange $n$ and $m$. I know that $\omega_{mn}=-\omega_{nm}$ and $\hat A$ is an hermitian operator. But so far I haven't managed to understand how taking such considerations can lead one to the first relation. Any help would be welcome, thank you!

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The averaging here is a thermal trace: $$S_{AA^\dagger}(t)= Tr[e^{-\beta H}A(t)A^\dagger(0)].$$ You need to permute it in order to change the order of the operators.

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