Non-relativistic conformal symmetries and Newton-Cartan structures

TL;DR

This paper classifies the conformal symmetries of non-relativistic Newton-Cartan spacetime, revealing a family of infinite-dimensional Lie algebras parameterized by a rational dynamical exponent, with applications to various physical systems.

Contribution

It provides a unifying classification of non-relativistic conformal symmetries using Newton-Cartan structures and introduces a family of infinite-dimensional Lie algebras labeled by a rational dynamical exponent.

Findings

Lie algebras form a family labeled by a rational dynamical exponent z

Schr"odinger-Virasoro algebra corresponds to z=2

Symmetries apply to classical particles, hydrodynamics, and Galilean electromagnetism

Abstract

This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational "dynamical exponent", $z$. The Schr\"odinger-Virasoro algebra of Henkel et al. corresponds to $z=2$. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schr\"odinger Lie algebra, for which z=2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) and Lukierski, Stichel and Zakrzewski [alias "$\alt$" of Henkel], with $z=1$. Physical systems realizing these symmetries include, e.g., classical systems of massive, and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.