Covariant dynamics on the momentum space

TL;DR

This paper presents a covariant formulation of quantum dynamics in momentum space, exploring deformations like Snyder models, and showing their equivalence or triviality in certain geometrical settings.

Contribution

It introduces a geometrical, covariant approach to momentum space dynamics, unifying different deformed models and simplifying their analysis.

Findings

Deformation of flat momentum space results in trivial dynamics.

Different Snyder models are shown to be dynamically equivalent.

The approach provides a unified geometric framework for deformed momentum spaces.

Abstract

A geometrical interpretation of Schr\"odinger's kinetic and potential energy operators is proposed, allowing for a covariant momentum space formulation of the dynamics that is relevant for the theories with the deformation of the momentum space structure. Some specific examples are discussed in the context of flat space deformations and the Euclidean Snyder (spherical momentum space) model. In this formulation the dynamics for the deformations of the flat momentum space becomes trivial, while different versions of the Snyder model turn out to be dynamically equivalent.