Effective field theory for non-relativistic hydrodynamics

Abstract

We write down a Schwinger-Keldysh effective field theory for non-relativistic (Galilean) hydrodynamics. We use the null background construction to covariantly couple Galilean field theories to a set of background sources. In this language, Galilean hydrodynamics gets recast as relativistic hydrodynamics formulated on a one dimension higher spacetime admitting a null Killing vector. This allows us to import the existing field theoretic techniques for relativistic hydrodynamics into the Galilean setting, with minor modifications to include the additional background vector field. We use this formulation to work out an interacting field theory describing stochastic fluctuations of energy, momentum, and density modes around thermal equilibrium. We also present a translation of our results to the more conventional Newton-Cartan language, and discuss how the same can be derived via a non-relativistic limit of the effective field theory for relativistic hydrodynamics.

Effective field theory for hydrodynamics without boosts

“…10,42-50], the EFT description of boost-agnostic hydrodynamics needs to treat time and space directions on independent footing. A similar discussion for Galilean hydrodynamics appeared recently in [11], where the time and space directions were indeed treated independently, but nonetheless had to be tied down to respect the underlying Milne boost symmetry. As noted there, Milne boosts actually make things quite hard for an effective field theorist; to make this symmetry manifest, one needs to pass to a higher-dimensional "null-background" representation followed by a null reduction to obtain the final results.…”

“…Since boost-agnostic hydrodynamics does not worry about boosts altogether, the ensuing EFT is formally simpler than its Galilean cousin. In fact, the following discussion is mostly a reproduction of section 5 of [11], but with the Milne boost symmetry revoked. The lack of a symmetry does mean that many more terms can now enter the effective action at a given derivative order that were previously not allowed, making the boost-agnostic case structurally more richer; we will see an example of this for one-derivative fluids in section 4.…”

“…Here, β 0 = (k B T 0 ) −1 is the (constant) inverse temperature and µ 0 the (constant) chemical potential of the global thermal state. Given these choices, we are left with residual spatial reparametrisation freedom τ → τ + f ( σ) and σ i → σ i ( σ) as well as with U(1) phase shifts ϕ s → ϕ s + λ( σ) (see [11]).…”

“…This form of the action makes all the spacetime and worldvolume symmetries of the effective theory manifest. However, the action is also required to obey a set of Schwinger-Keldysh requirements on the account of describing generic thermal field theories [8,11]. These are…”

“…Treating hydrodynamics as a bona fide thermal field theory, a more fundamental approach has been developed in the context of relativistic fluids in the past few years [8][9][10], and has recently been adapted to Galilean-invariant fluids as well [11]. This formulation is based on the Schwinger-Keldysh effective field theory (EFT) framework for non-equilibrium thermal systems; see [12] for a review.…”