Field theory for a finite temperature normal fluid Can normal-fluid (not superfluid) hydrodynamics be derived from some classical field theory? Here I mean conservation (continuity) equations for mass (density), momentum, and energy (entropy, temperature).
On a related note, can the speed of sound be seen as a pole in the Green's function of such a system? 
If yes, could you point me to some relevant textbook or a review article? If not, then why?
Thanks!
 A: Of course, conservation laws for particle number, momentum, and energy follow directly from the classical equations of motion. However, by fluid dynamics we mean more than that. We mean that the conservation laws can be expressed in terms of coarse grained, hydrodynamic variables, such as the density, the energy density, and the fluid velocity. For example, the off-diagonal components of the stress tensor in a non-relativistic field theory are $T_{xy}\sim m^{-1}\psi^\dagger \nabla_x\nabla_y\psi$. How do we know that this can be written as $T_{xy}=\rho u_xu_y+\ldots$?
Fluid dynamics is a general effective theory for a (quantum) many body system at finite temperature. This means that it should always be possible to derive fluid dynamics from an underlying field theory, and hydrodynamic modes (like sound) should appear as poles of retarded Green functions. However, the classical limit of a thermal field theory is a very special limit, and is typically not hydrodynamic.
1) The easiest way to derive fluid dynamics from a microscopic theory is to start from kinetic theory and the Boltzmann equation. Taking moments of the Boltzmann equation gives the conservation laws. Expanding the distribution function around the stationary (equilibrium) distribution gives the conservation laws in their fluid dynamic form. Parameters in the Boltzmann equation (collision terms, quasi-particle energies, etc) can be computed from an underlying field theory. 
2) The kinetic theory itself can be derived from quantum field theory in certain limits (typically, perturbative interactions). The Boltzmann equation follows from the equations of motion for non-equilibrium Green functions, as shown by Kadanoff and Baym (and others), and described in their text book.
3) To derive fluid dynamics from field theory beyond weak coupling is a difficult problem. A new class of theories where this has been achieved is the strong coupling limit of field theories that have a dual holographic description (the AdS/CFT correspondence). In that case, fluid dynamics in d+1 dimensions follows from classical field theory in a d+2 dimensional space time containing a horizon. 
4) The classical limit of a thermal field theory is the limit of large occupation numbers. For a normal fluid this is not close to equilibrium (the equilibrium distribution has $f=O(1)$, and fluid dynamics does not emerge. There are cases where the classical description can be shown to be equivalent to the Boltzmann equation.
5) Retarded correlation functions have hydro poles, but to see this from summing Feynman diagrams is not entirely trivial, even in weak coupling. Typically, a large class of Feynman diagrams has to be summed (again, this is easier for a superfluid than for a normal fluid).  
6) Finally, could it be that fluid dynamics itself is some kind of field theory (can I write down a lagrangian in terms of fluid dynamical variables)? Not in the usual sense, because the Navier-Stokes equation is irreversible -- it cannot be the equation of motion of a unitary field theory. However, I can take the equations of motion and derive them from a suitable effective action. This action is indeed useful for deriving fluctuation corrections to the Navier-Stokes equation. See the review by Hohenberg and Halperin. 
A: I'm not quite familiar with this subject. Nevertheless, I suggest you to read E. M. Lifthitz, and L. P. Pitaevskii, Statistical Physics, part 2, translated by J. B. Sykes, and M. J. Kearsley (Pergamon Press Ltd., Oxford, 1980) section 24.
