On systems having Poincar\'e and Galileo symmetry

TL;DR

This paper explores how certain physical systems can exhibit both Poincaré and Galileo symmetries simultaneously, revealing new structures and symmetries in wave equations, fluid models, and Maxwell's equations across different dimensions.

Contribution

It introduces a method to obtain dynamics-dependent symmetry representations and uncovers hidden symmetries in the Chaplygin gas and Maxwell's equations, linking relativistic and non-relativistic frameworks.

Findings

Revealed non-relativistic Chaplygin gas features using Lorentz structure.

Discovered field-dependent symmetries and Noether charges in the Chaplygin system.

Showed Maxwell's equations can be exactly Galileo covariant with field-dependent transformations.

Abstract

Using the wave equation in d > or = 1 space dimensions it is illustrated how dynamical equations may be simultaneously Poincar\'e and Galileo covariant with respect to different sets of independent variables. This provides a method to obtain dynamics-dependent representations of the kinematical symmetries. When the field is a displacement function both symmetries have a physical interpretation. For d = 1 the Lorentz structure is utilized to reveal hitherto unnoticed features of the non-relativistic Chaplygin gas, including a relativistic structure with a limiting case that exhibits the Carroll group, and field-dependent symmetries and associated Noether charges. The Lorentz transformations of the potentials naturally associated with the Chaplygin system are given. These results prompt the search for further symmetries and it is shown that the Chaplygin equations support a nonlinear superposition principle. A known spacetime mixing symmetry is shown to decompose into label-time and superposition symmetries. It is shown that a quantum mechanical system in a stationary state behaves as a Chaplygin gas. The extension to d > 1 is used to illustrate how the physical significance of the dual symmetries is contingent on the context by showing that Maxwells equations exhibit an exact Galileo covariant formulation where Lorentz and gauge transformations are represented by field-dependent symmetries. A natural conceptual and formal framework is provided by the Lagrangian and Eulerian pictures of continuum mechanics.