Field-dependent symmetries of a non-relativistic fluid model

TL;DR

This work studies a 1+1D non-relativistic fluid model with action ${\cal L}_{0}=-R\partial_{t}\Theta-\tfrac{1}{2}R(\partial_{x}\Theta)^2-V(R)$, uncovering a hidden dynamical symmetry that, in the free case, elevates to the ${\rm O}(3,2)$ conformal group. By promoting the phase $\Theta$ to a vertical coordinate $s$ on an extended space $M$, the authors linearize the action of the conformal group and show how extended-space symmetries project to a field-dependent action on ordinary space, with the membrane potential $V(R)=c/R$ singled out as the unique potential compatible with these extensions. When $V=0$, the full conformal symmetry is recovered on $M$, while projection to ordinary space yields the Schrödinger or Poincaré subgroups depending on the potential, and the associated Noether charges close the ${\rm o}(3,2)$ algebra. The article further constructs conserved currents from the Bargmann energy–momentum tensor, analyzes the role of curvature terms to preserve conformal invariance, and connects these results to the Schrödinger equation symmetries discussed by Jevicki and others, offering a geometric, extended-space perspective on nonrelativistic dynamical symmetries.

Abstract

As found by Bordemann and Hoppe and by Jevicki, a certain non-relativistic model of an irrotational and isentropic fluid, related to membranes and to partons, admits a Poincaré symmetry. Bazeia and Jackiw associate this dynamical symmetry to a novel type of ``field dependent'' action on space-time. The ``Kaluza-Klein type'' framework of Duval et al. is used to explain the origin of these symmetries and to derive the associated conserved quantities. In the non-interacting case, the symmetry extends to the entire conformal group.