There are a few different relations here, and I agree that it can be confusing. It is best to explain everything using the path integral formalism for QFT. Let's consider some field theory which just involves a single scalar field, \phi(t,x),
Ground state (zero-temperature) quantum field theory
Consider a quantum field theory which involves a single scalar field, \phi(t,x) in d spatial dimensions, and assume it is described by an action of the form
\mathcal{S}[\phi] = \int dt \, d^dx \left[ (\partial_t \phi)^2 - (\nabla \phi)^2 - V(\phi) \right].
Here, we calculate correlation functions as
<br />
\langle \Omega | \phi(t_1,x_1) \phi(t_2,x_2) \cdots \phi(t_n,x_n) | \Omega \rangle = \int \mathcal{D}\phi(t,x) e^{i \mathcal{S}[\phi]}<br />
where |\Omega \rangle is the ground state, and one normalizes the path integral such that \langle \Omega |\Omega \rangle = 1. This is the kind of theory you're used to dealing with in a QFT course, which are usually based around computing S matrix amplitudes in particle physics.
Next:
Classical statistical field theory
The idea here is that if you have a statistical system, say like an Ising model with spin variables \sigma_i on a D-dimensional spatial lattice, you have a partition function,
<br />
\mathcal{Z}_C = \sum_{\{\sigma_i \}} e^{- \beta H[\sigma_i]}<br />
where H[\sigma_i] is the Hamiltonian describing the interactions of these spins. Then you can compute correlation functions in the usual way, \langle \sigma_j \sigma_k \rangle = \frac{1}{\mathcal{Z}_C} \sum_{\{\sigma_i\}} \sigma_j \sigma_k e^{- \beta H[\sigma_i]} etc.
Now, probably the most interesting phenomenon in stat mech is a continuous phase transition (aka a critical point/line/phase), where the correlations of the spins become
much larger than the lattice spacing. Near these transitions, you can describe the physics in terms of coarse-grained fields, call them \phi(X) (where X now denotes a D-dimensional vector). These fields vary extremely slowly on the lattice scale, and you can effectively take a continuum limit, obtaining a continuum statistical field theory:
<br />
\langle \phi(X_1) \phi(X_2) \cdots \phi(X_n) \rangle = \int \mathcal{D}\phi(X) e^{- \beta H[\phi]}<br />
Now the Hamiltonian in an integral over some local Hamiltonian density, say
\beta H[\phi] = \int d^D X \left[ (\nabla \phi)^2 + V(\phi) \right]
just to give an example suggestive of the quantum example above.
Note that there is no time-dependence here! Near classical critical point, the time-dependence decouples from the statics, and needs to be determined from other means. There is also nothing "quantum" about these theories. They describe classical physics, and these field theories can be used to obtain, for example, properties of an iron ferromagnet at its Curie point (1000 Kelvin!). In fact, it is crucial that these transitions occur at a finite temperature, where quantum effects are unimportant (thermal fluctuations dominate over quantum fluctuations).
There is one last class of theories, for precisely when thermal fluctuations become similar to or less important than quantum fluctuations. But before getting to that, I want to discuss the relation between these last two theories, since it will introduce Wick rotations.
The relation between ground state QFT in (d+1)-dimensional spacetime and classical SFT in D = (d+1)-dimensional space
Hopefully I set this up in a way where you anticipate this next step. We take the specific QFT above, and Wick rotate
t \rightarrow - i \tau
Then the action I wrote above becomes
<br />
i\mathcal{S} \rightarrow -\mathcal{S}_E = -\int d \tau \, d^d x \left[ (\partial_{\tau} \phi)^2 + (\nabla \phi)^2 + V(\phi) \right]
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Now, just re-label some things:
<br />
\mathcal{S}_E \sim \beta H, \qquad (\tau,x) \sim (X)<br />
and you'll notice that this is precisely the classical statistical field theory above, but the SFT lives in (d+1) spatial dimensions while the QFT lives in d spatial dimensions. Here we need to associate the Euclidean action of the QFT with the Hamiltonian of the field theory. (I'll answer your question about temperature as the coupling below.)
For a lot of purposes, these two theories have a ton of similar properties. For example, the renormalization group properties are the same (so you get the same scaling dimensions, which are among the most important observables in either a QFT or an SFT). However, you do need to Wick rotate back to real-time to get dynamical properties of the quantum theory. For example, in a gapped phase the correlation functions of the SFT all decay like e^{- m X} in all spatial directions, but after Wick-rotating one finds that the unequal-time correlations of the QFT do not decay: they oscillate like e^{i m t} (as should be expected for a coherent quantum state).
Finite-temperature quantum statistical field theory
Ok, now let's consider the case where we need finite temperature, and quantum field theory. The starting point here is the partition function you learned in quantum stat mech:
<br />
\mathcal{Z}_Q = \mathrm{Tr}\left( e^{- \beta H} \right) = \sum_n \langle n| e^{- \beta H} | n \rangle<br />
The usual way to proceed here is to recall the path-integral representation of the matrix elements of the evolution operator evaluated at some time t:
<br />
\langle \{\phi_2\} | e^{- i H t} | \{\phi_1\} \rangle = \int_{\{\phi_1\}}^{\{\phi_2\}} \mathcal{D} \phi \, e^{i \int_0^t d t' \, L[\phi]}<br />
Hopefully my notation is clear: I'm considering a field theory in terms of \phi which evolves from field configurations \{\phi_1\} to \{\phi_2\} over the course of some time t. The limits on the path integral mean I take all field configurations which satisfy the proper boundary conditions at times 0 and t.
Now we apply this to the quantum partition function by taking t = - i \beta . The form of the quantum partition functions also implies that we sum over all field configurations which the same at \tau = 0,\beta. So we find:
<br />
\mathcal{Z}_Q = \int_{\mathrm{PBC}} \mathcal{D}\phi(\tau,x) e^{- \int_0^{\beta} L_E[\phi]}<br />
Here, the Lagrangian L_E is the same which appears in the Wick-rotated zero-temperature action above, but now our imaginary-time direction is finite, with radius \beta.
So now we can alternatively interpret this theory as a classical statistical field theory on a space where one of the spatial dimensions is a finite circle with radius \beta. Because adding a finite dimension is an IR property, this still doesn't change the RG properties. But when you need to get dynamical information about this theory by Wick rotating back, you now get both coherent quantum parts and thermal parts.
diegzumillo said:
It seems to imply temperature and coupling constant are the same thing, which doesn't sound right at all.
I think this is referring to the T=0 QFT to T>0 SFT relation (the first two I mentioned). If you have, say, a gauge theory:
<br />
\mathcal{S} = \frac{1}{g} \int d^{d+1}x \, F^2<br />
or the nonlinear-sigma model:
<br />
\mathcal{S} = \frac{1}{g} \int d^{d+1}x \, (\partial_{\mu} \phi)^2<br />
then after Wick-rotation you very clearly find g \propto T for the corresponding classical theories. And even with the scalar theories I gave above, you can imagine redefining your fields to get something like
<br />
(\partial_\mu \phi)^2 + \frac{m^2}{2} \phi^2 + u \phi^4 \longrightarrow \frac{1}{\sqrt{u}}(\partial_\mu \phi)^2 + \frac{m^2}{2\sqrt{u}} \phi^2 + \phi^4<br />
and now T \propto \sqrt{u} roughly.
EDIT: Fixed some typos.