Since low-energy, non-relativistic thermal field theories are defined in Euclidean spacetime, while high-energy relativistic theories are define in Minkowski spacetime, I was wondering if there are renormalization methods that can show such a change in metric signature.


1 Answer 1


The time contour really has nothing to do with renormalization. Rather it is something you choose at the outset for the purpose of the calculation you want to do. With any choice of time contour the renormalization theory is pretty much the same. What renormalization does (understood in terms of Kadanoff/Wilsonian renormalization group) is generate higher dimension effective operators in the Lagrangian. The addition of operators to the Lagrangian has no effect on what is your choice of time contour to integrate them on!

The reason for the choice of time contour is a little more subtle, and you've probably only seen the two most common special cases. Exposure to the general case may clarify what's going on with the imaginary time thing, even if you never use the most general case. The general correlation function (simplifying to a single scalar field) can be written

$$ \langle \phi(x_1,t_1)\cdots\phi(x_n,t_n) \rangle = \mathrm{Tr}\left\{ \rho(t_0) U(t_0,t_1) \phi(x_1,t_1) U(t_1,t_2)\cdots U(t_{n-1},t_n)\phi(x_n,t_n)U(t_n,t_0) \right\}$$

where the time evolution operators $U(t_i,t_j)$ come from working in the Heisenberg (or interaction) picture and $\rho$ is an arbitrary initial density matrix describing the system at the initial time $t_0$. This is all standard stuff similar to what you'll see in any QFT course.

Here comes a trick (part 1): you can write any density matrix you like as $\mathrm{e}^{-\beta H^M}$. Completely general. $H^M$ is not necessarily the Hamiltonian of your system, though if it is you have a thermal equilibrium state at temperature $\beta^{-1}$. Now the trick (part 2): notice that $\mathrm{e}^{-\beta H^M} = \mathrm{e}^{-i (-i\beta) H^M} = U(t_0 - i\beta, t_0; H^M)$. This is just a trick: imaginary time evolution with "Hamiltonian" $H^M$ gives you a density matrix. If $H^M = H$ this is just a thermal state. If not it's not. The general formalism can cope with the real time dynamics of an arbitrary non-equilibrium state.

Now have a look at page 107 of Stefanucci & van Leeuwen. I reproduce the relevant figure below (I believe it's fair use, but I heartily recommend you read the whole book if you get the chance):

time contours from Fig. 4.5 of Stefanucci and van Leeuwen

The first figure shows the general situation I've described: the time evolution starts at $t_0$, runs up the real axis to catch any $\phi(x,t)$ operators that are there, then back down to $t_0$ to "meet" the initial density matrix, which we make by evolving down the imaginary axis with $H^M$ which may or may not be $H$.

Now we can make approximations. If all you care about are thermal equilibrium properties and not non-equilibrium time evolution, you can measure all thermal correlations by taking all times at the initial time and $H^M=H$. The real time part of the contour collapses and you are just left with the imaginary time contour you know. It's not so much that thermal field theory is defined on an imaginary time contour. It's just that that is what's left when you don't care about anything else.

On the other hand you can start with some non-interacting state at $t_0\to -\infty$ and slowly (adiabatically) turn on an interaction and watch what happens. This gives the second set of contours (Fig. b), known as the Schwinger-Keldysh contours and often used for studying nonequilibrium situations like electric currents in nanostructures etc.

Finally if you take the density matrix to be an equilibrium density matrix at zero temperature then you can use the Gell-Mann-Low theorem to remove the backwards time contour completely. This gives you the usual one way real time contour that you probably know from ordinary QFT (Fig. c). This works because a vacuum state at $t\to -\infty$ adiabatically turns into a vacuum state at $t\to +\infty$. In a non-equilibrium situation you can't rely on this and you need the full contour.

  • $\begingroup$ Thanks a lot for bringing my attention to such a generalized formalism! To rephrase my earlier question in this new language: Consider a process that evolves dynamically before settling in a thermal equilibrium state (the reverse of your first diagram). Is there a way in which the transition point (from real-time direction to imaginary-time direction) can be dynamically determined from the system? $\endgroup$
    – GuSuku
    Nov 5, 2013 at 19:48

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