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Suppose I have a QFT action describing interaction of two objects and I can perturbative write the action as $S=S_0+S_i$ where $S_0$ is the non-interacting, unperturbed action and $S_i$ is the interaction part (to appropriate order in perturbation). Then the interaction energy could be written as $E_i=\langle 1-e^{-S_i}\rangle_{C,0}$ where we are taking connected correlation function in the unperturbed background. How to derive this? I guess in terms of partition function it is $Z=Z_0e^{\langle iS_i-1\rangle_{C,0}}$. Many thanks in advance!

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