# Does finite temperature QFT partition function capture both quantum and statistical fluctuations?

I've been working on understanding finite temperature field theory and am stuck with the following concepts: For our thermal system in equilibrium we have the usual partition function $$$$Z_\text{Thermal}=\text{tr}(e^{-\beta \hat{H}}),$$$$ which we can write as the path integral $$$$Z_\text{Thermal}=\int\mathcal{D}\phi \, \text{exp}\left[ i\int d^dx \mathcal{L}_E \right],$$$$ where we've introduced imaginary time $$\tau=it$$ and our (say real scalar) fields are periodic $$\phi(\vec{x},\tau)=\phi(\vec{x},\tau+i\beta)$$. From here we can construct a generating functional $$Z_\text{Thermal}[J]$$ and compute all thermodynamic properties of interest in a similar manner to $$T=0$$ QFT. I am stuck with

1) I haven't really understood the significance of the imaginary vs. real time formalism. Why would one pick one over the other in a given problem?

2) Looking at the partition function above, intuitively I can see the statistical fluctuations of the system being captured by the path integral as we would see in any statistical field theory. But does this partition function also capture quantum fluctuations? Is the only difference between a classical statistical field theory and a QFT at finite temperature the imaginary time with periodicity? I would've intuitively thought we would expect something like

$$$$Z_{\text{QFT}+\text{Statistical}} \sim Z_{\text{QFT}} Z_{\text{Thermal}},$$$$ so that the "total" partition function captures both quantum and thermal fluctuations?

3)When we calculate correlation functions from our generator $$Z_\text{Thermal}[J]$$, what are these objects capturing? Say the 2 point function, does this only represent thermal fluctuations? Where is the QFT bit?

Sorry for the long post, hopefully it is clear enough. Thanks in advance!

• In principle, the thermal path integral captures both quantum and thermal fluctuations. In such instances, it is impossible to separate "quantum" and "thermal" fluctuations into two different parts; see the answers and discussion on this previous question (I especially like the quotes from Landau & Lifschitz): physics.stackexchange.com/questions/8049/… – Seth Whitsitt Mar 11 at 22:44