Effective field theory for non-relativistic hydrodynamics

TL;DR

This paper develops a covariant effective field theory for non-relativistic (Galilean) hydrodynamics by embedding it into a higher-dimensional relativistic framework, enabling the use of relativistic techniques to analyze fluctuations and dynamics.

Contribution

It introduces a novel null background construction to formulate Galilean hydrodynamics as a relativistic theory in higher dimensions, facilitating new analytical approaches.

Findings

Formulation of Galilean hydrodynamics as relativistic on null backgrounds.

Derivation of an interacting field theory for stochastic fluctuations.

Translation of results to Newton-Cartan geometry and non-relativistic limits.

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.