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Confined dynamical systems with Carroll or Galilei symmetries

Andrea Barducci, Roberto Casalbuoni, Joaquim Gomis

TL;DR

This paper develops a general, configuration-space framework to construct dynamical systems invariant under generalized Carroll or Galilei symmetries realized as k-contractions of the Poincaré group. By partitioning space-time into a Minkowski subspace and a Euclidean sector and introducing a confining sector with Lagrange multipliers, the authors build discrete particle models and continuous brane theories that are invariant under the corresponding contracted algebras with zero central charge. A central insight is that boosts mixing the two sectors can be compensated by suitably transforming the confinement multipliers, yielding a robust mechanism for confinement and symmetry realization. The approach yields both Carroll dynamical systems and dual Galilei (instantonic) pictures, and is illustrated through explicit examples including Carroll and Galilei particles, light-like variants, a two-particle system, and Carroll strings, with potential relevance to non-relativistic holography and flat-space symmetry structures.

Abstract

We introduce a general method to construct classes of dynamical systems invariant under generalizations of the Carroll and of the Galilei groups. The method consists in starting from a space-time in $D+1$ dimensions and partitioning it in two parts, the first Minkowskian and the second Euclidean. Tha action consist of two terms that are separately invariant the Minkwoskian and Euclidean partitioning. One of those contains a system of lagrangian multiplies that confine the system to a subspace. The other term defines the dynamics of the system. The total lagrangian is invariant under the Carroll or the Galilei groups with zero central charge.

Confined dynamical systems with Carroll or Galilei symmetries

TL;DR

This paper develops a general, configuration-space framework to construct dynamical systems invariant under generalized Carroll or Galilei symmetries realized as k-contractions of the Poincaré group. By partitioning space-time into a Minkowski subspace and a Euclidean sector and introducing a confining sector with Lagrange multipliers, the authors build discrete particle models and continuous brane theories that are invariant under the corresponding contracted algebras with zero central charge. A central insight is that boosts mixing the two sectors can be compensated by suitably transforming the confinement multipliers, yielding a robust mechanism for confinement and symmetry realization. The approach yields both Carroll dynamical systems and dual Galilei (instantonic) pictures, and is illustrated through explicit examples including Carroll and Galilei particles, light-like variants, a two-particle system, and Carroll strings, with potential relevance to non-relativistic holography and flat-space symmetry structures.

Abstract

We introduce a general method to construct classes of dynamical systems invariant under generalizations of the Carroll and of the Galilei groups. The method consists in starting from a space-time in dimensions and partitioning it in two parts, the first Minkowskian and the second Euclidean. Tha action consist of two terms that are separately invariant the Minkwoskian and Euclidean partitioning. One of those contains a system of lagrangian multiplies that confine the system to a subspace. The other term defines the dynamics of the system. The total lagrangian is invariant under the Carroll or the Galilei groups with zero central charge.

Paper Structure

This paper contains 14 sections, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $k$-contractions of the Poincaré group
  3. $k$-contractions in configuration space
  4. Building Dynamical Models
  5. Discrete models
  6. Continuous models
  7. Examples
  8. The Carroll massive particle
  9. The Galilei particle
  10. A light-like particle of Galilei type
  11. A light-like particle of Carroll type
  12. A model of two particles
  13. 2-contraction for a Carroll string
  14. Conclusions and outlook

Figures (3)

  • Figure 1: The two types of contractions considered in the text. In the left panel the Carroll type. In the right panel the Galilei type. The rescaled variables are underlined. The arrows denote the directions in which the boosts act in the two cases.
  • Figure 2: The two types of point-like systems considered in the text. In the left panel the Carroll type. In the right panel the Galilei type. The lines are a sketch of the world-lines of particles living in the $M(1,k-1)$ Minkowski subspace (left panel) and of the world-lines of the instantons living in the euclidean subspace $E(D+1-k)$ (right panel) .
  • Figure 3: The shaded figures are a sketch of the extended objects living in the $M(1,k-1)$ Minkowski subspace (left panel) and of the ones living in the euclidean subspace $E(D+1-k)$(right panel) .The variables $(\tau,\sigma_1,\cdots,\sigma_m)$ describe the world-sheet of the extended objects. We assume $m\le k-1$ in the Carroll case and $m\le D-k$ in the Galilei case.