Gauge fixing, BRS invariance and Ward identities for randomly stirred flows

TL;DR

This paper demonstrates that Galilean invariance in the Navier-Stokes equation can be viewed as a gauge symmetry, and uses gauge fixing and BRS symmetry to derive Ward identities, clarifying the role of symmetries in fluid dynamics.

Contribution

It introduces a gauge fixing approach to handle Galilean invariance and extended Galilean invariance in randomly stirred flows, deriving associated Ward identities and clarifying their effects.

Findings

Galilean invariance acts as a gauge symmetry in fluid dynamics.

Gauge fixing resolves multiple counting of inertial frames in path integrals.

Derived Ward identities relate response and vertex functions, constraining only the zero mode.

Abstract

The Galilean invariance of the Navier-Stokes equation is shown to be akin to a global gauge symmetry familiar from quantum field theory. This symmetry leads to a multiple counting of infinitely many inertial reference frames in the path integral approach to randomly stirred fluids. This problem is solved by fixing the gauge, i.e., singling out one reference frame. The gauge fixed theory has an underlying Becchi-Rouet-Stora (BRS) symmetry which leads to the Ward identity relating the exact inverse response and vertex functions. This identification of Galilean invariance as a gauge symmetry is explored in detail, for different gauge choices and by performing a rigorous examination of a discretized version of the theory. The Navier-Stokes equation is also invariant under arbitrary rectilinear frame accelerations, known as extended Galilean invariance (EGI). We gauge fix this extended symmetry and derive the generalized Ward identity that follows from the BRS invariance of the gauge-fixed theory. This new Ward identity reduces to the standard one in the limit of zero acceleration. This gauge-fixing approach unambiguously shows that Galilean invariance and EGI constrain only the zero mode of the vertex but none of the higher wavenumber modes.