Simple generalization of the Feynman rules for QFT to thermal QFT?

Question

Assuming that one knows Feynman rules for QFT, what is the simplest way to generalize them for $T \neq 0$ case? What is the main difference? Can we just read them off from Lagrangian the same way as in QFT?

Since the partition function $Z$ in thermal quantum field theory is just $Z_{QFT}$ with substitution $t \rightarrow i\tau$, with $\tau$ being integrated from $0$ to $\beta$, as opposed to time $t$ being integrated from $-\infty$ to $\infty$, and with different boundary conditions for the fields (periodic/anti-periodic for bosons/fermions), can we use the same analogy for the generalization of the Feynman rules?

Answer 1

Finite temperature Feynman rules are simply zero temperature Feynman rules for Euclidean ($t\to i\tau$) QFT in periodic imaginary time. So instead of continuous values for the momenta, you will have a discrete spectrum for the timelike moments (such as in the infinite potential well in basic quantum mechanics). It's called the Matsubara formalism, if you want to google for more.