How do I show that Thermal Field Theory works?

Question

Of course it does work. What I'm asking is:
What is the easiest-to-understand, most direct example that shows that Thermal Field Theory is a predictive theory that actually describes how temperature affects reality?

Anyone who has taken a couple of courses in modern QFT has encountered the idea that if you take the time variable $t$ in your partition function and make it complex ($\tau=it$) and compact ($0\le\tau\le\beta$), then you're describing a theory with temperature $T=1/\beta$. I know where this idea comes from, that the exponential of the partition function is $-\beta H$ and the one of the (left) evolution operator is $-itH$, and now that I've used all of this for some time it makes natural sense to me. But when I tried to explain this to a mathematician friend of mine he pointed out that it's kind of a big leap. I wanted to reply "It is, but it works!" but I couldn't find an example to prove my point.

Answer 1

First of all, there is more logic connecting thermal field theory and quantum field theory than what you wrote. In particular, there is a Wick rotation that relates the path integral integrand $\sim e^{i S/\hbar}$ (where $S$ is the action) to $\sim e^{- \beta H}$. Then, if you started from the definition of the partition function in statistical mechanics, you would get a sum over states with a weight $\sim e^{- \beta H}$. So there is a legitimate (at least at a physics-level-of-rigor) mathematical step that relates the natural definition of a QFT partition function to the natural definition of a statistical mechanics partition function. It's more than noticing a formal similarity between two formulas.

Second, if you are looking for an example where a path integral is useful in statistical mechanics, some classic examples are Landau-Ginzburg theory and effective field theory, for example applied to the Ising model.

However, using a path integral is kind of a technical detail, that has more to do with the state space than the fundamental underlying Boltzmann distribution $\sim e^{-\beta H}$ that underlies statistical mechanics. The Boltzmann distribution is used in many contexts, maybe the simplest is the ideal gas which is used to derive bounds on the efficiency of heat engines, among other things. Quantum mechanically, the Boltzmann distribution also underlies the Fermi-Dirac and Bose-Einstein distributions, which (among other things) are used to describe semi-conductor physics and the Planck spectrum of radiation, respectively.