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Non-relativistic conformal symmetries and Newton-Cartan structures

Christian Duval, Péter A. Horvathy

TL;DR

This work provides a geometric, Newton-Cartan–based classification of non-relativistic conformal symmetries, revealing an infinite family of conformal Galilei algebras $ rak{cgal}_z(d)$ parameterized by a rational dynamical exponent $z$. It further develops conformal Newton-Cartan structures, introducing the expanded Schrödinger algebra $ ilde{ rak{sch}}$ and its $z$-dependent subalgebras $ rak{sch}_z(d)$, as well as lightlike-geodesic extensions $ rak{cnc}(d)$ and $ rak{cmil}(d)$ with finite-dimensional subalgebras like $ rak{sch}_2(d)$ and $ rak{cmil}_1(d)$. The authors connect these algebras to physical systems—massive and massless Galilean particles, hydrodynamics, and Galilean electromagnetism—via explicit transformation laws and conserved quantities, showing how geometry dictates possible symmetries. Overall, the paper provides a unifying, geometry-driven framework that links non-relativistic conformal symmetries to NC structures and their geodesic content, with a clear path to identifying relevant symmetries in concrete non-relativistic models.

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 Schrödinger-Virasoro algebra of Henkel et al. corresponds to $z=2$. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger 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.

Non-relativistic conformal symmetries and Newton-Cartan structures

TL;DR

This work provides a geometric, Newton-Cartan–based classification of non-relativistic conformal symmetries, revealing an infinite family of conformal Galilei algebras parameterized by a rational dynamical exponent . It further develops conformal Newton-Cartan structures, introducing the expanded Schrödinger algebra and its -dependent subalgebras , as well as lightlike-geodesic extensions and with finite-dimensional subalgebras like and . The authors connect these algebras to physical systems—massive and massless Galilean particles, hydrodynamics, and Galilean electromagnetism—via explicit transformation laws and conserved quantities, showing how geometry dictates possible symmetries. Overall, the paper provides a unifying, geometry-driven framework that links non-relativistic conformal symmetries to NC structures and their geodesic content, with a clear path to identifying relevant symmetries in concrete non-relativistic models.

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", . The Schrödinger-Virasoro algebra of Henkel et al. corresponds to . Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which z=2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) and Lukierski, Stichel and Zakrzewski [alias "" of Henkel], with . 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.

Paper Structure

This paper contains 25 sections, 156 equations.

Table of Contents

  1. Introduction
  2. Newton-Cartan structures
  3. Galilei structures and Newton-Cartan connections
  4. NC-gauge transformations, and NC-Milne structures
  5. Gauge transformations
  6. NC-Milne structure
  7. Conformal Galilei transformations, Schrö-dinger-Virasoro Lie algebra
  8. The Lie algebra, $\mathfrak{cgal}$, of conformal Galilei transformations
  9. Conformal Galilei transformations, $\mathfrak{cgal}_z$, with dynamical ex-ponent $z$
  10. Conformal Newton-Cartan transformations
  11. The Schrödinger Lie algebra
  12. The expanded Schrödinger Lie algebra, $\widetilde{\mathfrak{sch}}$, of projective Galilei conformal transformations
  13. The Schrödinger Lie algebras, $\mathfrak{sch}_z$, with dynamical exponent $z$
  14. Transformation law of NC-connections under conformal Galilei rescalings
  15. Conformal NC transformations: lightlike geodesics
  16. ...and 10 more sections