Hamilton geometry: Phase space geometry from modified dispersion relations

TL;DR

This paper develops a Hamiltonian geometric framework for phase space that incorporates curved spacetime and momentum space, applying it to models with Planck-scale modified dispersion relations inspired by quantum groups.

Contribution

It introduces a unified Hamilton geometry approach to phase space that captures both spacetime and momentum space curvature, extending the geometric understanding of modified dispersion relations.

Findings

Derived explicit curvature expressions for Planck-scale models

Identified cases with curved spacetime and momentum space

Showed flat momentum space in metric-induced dispersion relations

Abstract

We describe the Hamilton geometry of the phase space of particles whose motion is characterised by general dispersion relations. In this framework spacetime and momentum space are naturally curved and intertwined, allowing for a simultaneous description of both spacetime curvature and non-trivial momentum space geometry. We consider as explicit examples two models for Planck-scale modified dispersion relations, inspired from the $q$-de Sitter and $\kappa$-Poincar\'e quantum groups. In the first case we find the expressions for the momentum and position dependent curvature of spacetime and momentum space, while for the second case the manifold is flat and only the momentum space possesses a nonzero, momentum dependent curvature. In contrast, for a dispersion relation that is induced by a spacetime metric, as in General Relativity, the Hamilton geometry yields a flat momentum space and the usual curved spacetime geometry with only position dependent geometric objects.