Partition function in Statistical mechanics is given by $$ Z = \sum_ne^{-\beta E_n} $$ For QFT, it is defined in terms of a path integral: $$ Z = \int D\phi e^{-S[\phi]} $$ How can we see the relation between these definitions. I.e, can we show that: $$ \int D\phi e^{-S[\phi]}=\sum_ne^{-TE_n} $$ For time $T$ and $E_n$ the energy of the eigenstates of the quantum theory?
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$\begingroup$ You want to use the appropriate partition function. In your case, that would be the partition function for a continous quantum mechanical system which looks different from what you wrote, which is for a discrete classical system. You can then probably use the derivation of the path integral formalism from the Hamiltonian to derive one partition function from the other. Also, why do you replace $\beta$ with time on the last line? $\endgroup$– paulinaCommented Jun 21 at 16:05
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$\begingroup$ Yes, this is how you do thermal QFT. The whole trick is to view temperature as cyclic, imaginary time. Zee gives a nice overview in QFT in a Nutshell $\endgroup$– LPZCommented Jun 21 at 21:59
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