Galileo Symmetries in Polymer Particle Representation

TL;DR

This paper explores how Galileo symmetries are represented in a discrete quantum geometry model, showing that they are approximately recovered at low energies despite the underlying discreteness.

Contribution

It demonstrates the implementation and near-recovery of Galileo symmetries within the polymer particle framework, a simple model for discrete quantum geometry.

Findings

Galileo transformations are well-defined in the polymer Hilbert space.

Galileo symmetries are approximately recovered at low energies.

Deviations from classical symmetries are highly suppressed in the low-energy limit.

Abstract

To illustrate the conceptual problems for the low-energy symmetries in the continuum of spacetime emerging from the discrete quantum geometry, Galileo symmetries are investigated in the polymer particle representation of a non-relativistic particle as a simple toy model. The complete Galileo transformations (translation, rotation and Galileo boost) are naturally defined in the polymer particle Hilbert space and Galileo symmetries are recovered with highly suppressed deviations in the low-energy regime from the underlying polymer particle description.