Confusion with finite temperature and lattice formulation

[diegzumillo]

I think the title sums up pretty well my doubts. I learned QFT from Peskin and Schroeder and other common sources, all implicitly defined QFT at zero temperature. Then I started learning about lattice QCD, how to define the action, how to find continuum limits, the importance of the dependence of coupling constant with lattice spacing etc. So far so good. We want to do our calculations on the lattice but keeping in mind the continuum limit is where the original model is, so we want to lower g as much as possible, or increase beta, or whatever you call these variables.

But as I read articles on the subject I kept encountering this statistical mechanical terminology all over the place. In particular, finite-temperature phase transitions. So I kept digging and learned that there is this whole world of physics of finite temperature QFT. This is new to me but I think I understand the gist of it, but what bothers me is that it seems to be related to lattice formulation but not in an obvious way. Like a phase transition far from the continuum limit interpreted as a finite temperature phase transition. It seems to imply temperature and coupling constant are the same thing, which doesn't sound right at all. On the other hand, I've seen people talk about lattice formulation of finite temperature QCD, which sounds like confirmation that these are completely different things.

I'm a little confused and overwhelmed. Any light on this subject would be greatly appreciated.

 

[king vitamin]

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
$$\langle \Omega | \phi(t_1,x_1) \phi(t_2,x_2) \cdots \phi(t_n,x_n) | \Omega \rangle = \int \mathcal{D}\phi(t,x) e^{i \mathcal{S}[\phi]}$$
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,
$$\mathcal{Z}_C = \sum_{\{\sigma_i \}} e^{- \beta H[\sigma_i]}$$
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:
$$\langle \phi(X_1) \phi(X_2) \cdots \phi(X_n) \rangle = \int \mathcal{D}\phi(X) e^{- \beta H[\phi]}$$
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
$$i\mathcal{S} \rightarrow -\mathcal{S}_E = -\int d \tau \, d^d x \left[ (\partial_{\tau} \phi)^2 + (\nabla \phi)^2 + V(\phi) \right]$$
[/tex]
Now, just re-label some things:
$$\mathcal{S}_E \sim \beta H, \qquad (\tau,x) \sim (X)$$
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:
$$\mathcal{Z}_Q = \mathrm{Tr}\left( e^{- \beta H} \right) = \sum_n \langle n| e^{- \beta H} | n \rangle$$
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:
$$\langle \{\phi_2\} | e^{- i H t} | \{\phi_1\} \rangle = \int_{\{\phi_1\}}^{\{\phi_2\}} \mathcal{D} \phi \, e^{i \int_0^t d t' \, L[\phi]}$$
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:
$$\mathcal{Z}_Q = \int_{\mathrm{PBC}} \mathcal{D}\phi(\tau,x) e^{- \int_0^{\beta} L_E[\phi]}$$
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.

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:
$$\mathcal{S} = \frac{1}{g} \int d^{d+1}x \, F^2$$
or the nonlinear-sigma model:
$$\mathcal{S} = \frac{1}{g} \int d^{d+1}x \, (\partial_{\mu} \phi)^2$$
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
$$(\partial_\mu \phi)^2 + \frac{m^2}{2} \phi^2 + u \phi^4 \longrightarrow \frac{1}{\sqrt{u}}(\partial_\mu \phi)^2 + \frac{m^2}{2\sqrt{u}} \phi^2 + \phi^4$$
and now $T \propto \sqrt{u}$ roughly.

EDIT: Fixed some typos.

 

[diegzumillo]

All hail King Vitamin! The amount of effort you put in explaining this makes me ashamed of the amount of effort I put in my first post. Seriously though, thanks, I really appreciate it.

So the thing that confuses me is that I'm used to seeing statistical mechanics as the macroscopic behavior of some fundamental theory, but it seems the connection we are making here between the statistical mechanics (classical/quantum) and QFT (ground state/finite temperature) is almost purely mathematical, with thermodynamic quantities gaining new interpretations.

It's not the first time I realize my confusion stems from my underdeveloped statistical mechanics, so some of these things sound like they should be old news to me. For example, things got a little confusing at the finite temperature part, where you considered $t = - i \beta$. Sure, this goes back to what I said before about the relation being mathematical, but the intuitive notion of temperature still seems to be somewhat preserved there, as that time scale (periodic, disregarding UV effects etc) suggests an energy scale, kind of like the usual interpretation temperature. So... not purely mathematical then? Also, maybe this is another connection between coupling constant and temperature?

Am I misinterpreting anything? Honestly I feel a lot less confused!

 

[king vitamin]

Hmm, I'll try to break down what I consider "physical" and "mathematical" about these relations, but this will necessarily be subjective.

I think the main physical relation between the QFT-classical field theory correspondence is the fact that they apply to some system where the physically measured length/time scales are extremely large compared with any microscopic length/time scales (I'm talking about UV cutoffs on the QFT side), and in addition one has contributions from all length scales from the physically accessible scales to the UV cutoff/lattice spacing.

To unpack this last paragraph using an example I learned from Ken Wilson, what I mean is that physical divergences in both QFT and classical critical phenomena involve integrals of the type
$$\int_{\mu}^{\Lambda} \frac{dk}{k}$$
Even though you are led to this integral thinking about physical* momentum scales $\mu$ and $\Lambda$, this integral has no characteristic scale. By this, I mean that all momenta between $\mu$ and $\Lambda$ contribute equally to the final answer. As a result, you have a sensitivity of your answer to the scale $\mu$ that you are probing the theory at. This is the essence of the renormalization group.

So from a certain physical point of view, one may expect QFT and classical criticality to take the same form because they are both physical problems with approximate scale invariance, and systems like these are naturally treated with the renormalization group (which appears naturally in field theory). For more on this point of view, I recommend reading Ken Wilson's Nobel speech, as well as the beginning of his review article on the Kondo problem.

On the other hand, I have to admit that the imaginary time formalism for finite temperature QFT has always been a little mysterious to me. I might be a bit ashamed to admit that if I hadn't heard some pretty smart/well-known physicists say the same. At the end of the day you are interested in some dynamic information of correlation functions, but this is an analytic continuation where you consider an imaginary time lying on a circle of radius $\beta$ and then analytically continuing back and considering all values of time. It's not clear to me what this has to do with temperature per-se.

So I have to admit that I can only interpret the imaginary time QFT as just a mathematical relation. It's a current goal of mine to learn the Schwinger-Keldysh formulation for finite-temperature QFT, of which I understand that the above formalism (the "Matsubara formalism") is a special case.

* I will consider the cutoff energy $\Lambda$ physical in a QFT, e.g. I am thinking of effective QFTs. So for a particle physics example, you can use four-Fermi theory for the weak force if you work at scales much less than the W/Z masses, which are like the cutoff $\Lambda$.