93. Second order hydrodynamic coefficients in some field theories (like QCD)

By definition, we call an IR mode of a quantum field theory hydrodynamical, if its relaxation time goes to infinity, while its wave length . If this mode is hydrodynamical, does it mean that its evolution is determined by one of the equations we study in hydrodynamics – like, say, Euler equation or Navier-Stokes equation?

For example, when people are talking about viscosity of the quark-gluon plasma, does it mean that we can describe its behaviour by the Navier-Stokes equation with effective viscosity term added by hands? The answer is negative. It turns out that relativistic Navier-Stokes equations are acausal, and we do need relativistic equations to describe QG plasma in the regime under interest. Indeed, note that Navier-Stokes equations are Euler equations plus a momentum-diffusion term: the viscosity is the momentum-diffusion coefficient. Since diffusion equations are acasual (information can propagate with infinite velocity in the system described by the diffusion equation), we conclude that Navier-Stokes equations are also acasual.

Relativistic Navier-Stokes equations are also unstable. However, Israel and Stewart showed long time ago that the instability problem can be cured if one takes into account second-order terms into the gradient expansion of corrections to ideal Eulerian hydrodynamics (viscosity is the first order term in such an expansion).

Currently, a large part of the RHIC community involved in building SYM models of ion collisions is focused on calculating these second order terms by mapping the strong coupling regime to the gravity on AdS background via AdS/CFT.

The paper by York and Moore introduces another approach to the problem. First order hydrodynamic coefficients are essentially calculated by means of linear responce approach – the latter is nothing but a linear approximation in the corresponding problem of quantum kinetics. Therefore, we can

• write down Schwinger-Keldysh Green functions for a given field theory (well, the authors do not actually do it),
• derive kinetic equation from them,
• in the limit of large relaxation times derive second (third etc.) coefficients from the developed kinetic theory.

The authors pursue this program for several field theories – QCD (note that they do not need conformal SYM!), QED and scalar field theory with quartic interaction .

I actually have a couple of questions remaining fter reading their paper, that I would like to ask them in case they give a seminar at HIP. Here they are:

1. Kinetic equation is actually first one in the BBGKY chain, derived under the rigid gas approximation, i.e., weak coupling and small occupation numbers. In the regime under interest for the QG plasma, both conditions actually break down. That is why AdS/CFT approach is so valuable. Using gauge/gravity duality, we can at least get some clue on physics in the regime of strong coupling.

2. If we derive kinetic theory from the Schwinger-Keldysh, we actually get two equations – kinetic equation governing occupation numbers and mass shell equation showing what is the spectrum of elementary excitations. In the second order w.r.t. the coupling the mass shell equation is non-trivial, and I don’t honestly see where the fluctuations of mass shell are taken into account in the paper.

Anyway, the paper is actually fascinating to read, and reading it, one starts to think at some point: “it is so simple, why I did not do it myself?”