# Observables and Dispersion Relations in k-Minkowski Spacetime

## TL;DR

This paper explores the structure of quantum symmetries and wave equations in k-Minkowski noncommutative spacetime, deriving deformed dispersion relations and connecting them to physical observables and phenomenology.

## Contribution

It introduces a quantum Lie algebra framework for noncommutative spacetime symmetries and derives deformed wave operators and dispersion relations in k-Minkowski space.

## Key findings

- Quantum Poincare'-Weyl algebra is obtained for k-Minkowski space.
- The d'Alembert operator is a quadratic Casimir of quantum translations.
- Deformed energy-momentum relations are consistent with Deformed Special Relativity.

## Abstract

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.