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This is a fairly basic QFT question but it's bothering me. And Peskin/Schroeder fails at explaining basic stuff, so here I am.

After calculating Z for a particular theory I know this can be used to calculate all kinds of correlation functions. Itself, however, is the probability amplitude of vacuum-vacuum, correct? So when a problem asks me to calculate some vacuum diagrams I simply get a few terms of Z.

My difficulty is that I'm supposed to compare with the ground state energy that I calculated using a different method (for reference, it's the perturbed oscillator, or anharmonic oscillator). And when I realized I had exactly the same expansion for both ground energy and Z my whole understanding of it fell apart.

Considering the representation of Z using the time evolution operator (exponential of Hamiltonian etc) being sandwiched between vacuum states, I was expecting to be able to relate Z and E through an exponential/logarithm. Which goes well with its relation to statistical mechanics. But I'm not seeing that happen.

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# Partition function and vacuum energy (basic stuff)

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