Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries

TL;DR

This paper reviews Finsler and Hamilton geometries as frameworks for describing particle kinematics affected by Planck-scale effects, analyzing their properties and implications through a $q$-de Sitter-inspired example.

Contribution

It provides a comparative analysis of Finsler and Hamilton geometries in modeling quantum configuration and phase spaces, highlighting their advantages and limitations.

Findings

Finsler and Hamilton geometries offer distinct perspectives on quantum phase spaces.

The $q$-de Sitter-inspired relation serves as a test case for these geometrical approaches.

Both approaches have specific strengths and weaknesses in describing Planck-scale physics.

Abstract

In this paper, we review two approaches that can describe, in a geometrical way, the kinematics of particles that are affected by Planck-scale departures, named Finsler and Hamilton geometries. By relying on maps that connect the spaces of velocities and momenta, we discuss the properties of configuration and phase spaces induced by these two distinct geometries. In particular, we exemplify this approach by considering the so-called $q$-de Sitter-inspired modified dispersion relation as a laboratory for this study. We finalize with some points that we consider as positive and negative ones of each approach for the description of quantum configuration and phases spaces.