# $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations

## TL;DR

This paper explores different implementations of the $$-deformed relativistic phase space, focusing on Jordanian twists and their impact on the Lorentz-Weyl algebra and physical phenomena like the hydrogen atom spectrum.

## Contribution

It introduces a framework for $$-deformed phase spaces using Jordanian twists and analyzes how different realizations affect the Lorentz-Weyl algebra and physical predictions.

## Key findings

- Spectrum of the relativistic hydrogen atom varies with algebra realization.
- Different implementations lead to distinct physical interpretations.
- The Jordanian twist provides a specific realization of noncommutative coordinates.

## Abstract

We consider $\kappa$-deformed relativistic quantum phase space and possible implementations of the Lorentz algebra. There are two ways of performing such implementations. One is a simple extension where the Poincar\'e algebra is unaltered, while the other is a general extension where the Poincar\'e algebra is deformed. As an example we fix the Jordanian twist and the corresponding realization of noncommutative coordinates, coproduct of momenta and addition of momenta. An extension with a one-parameter family of realizations of the Lorentz generators, dilatation and momenta closing the Poincar\'e-Weyl algebra is considered. The corresponding physical interpretation depends on the way the Lorentz algebra is implemented in phase space. We show how the spectrum of the relativistic hydrogen atom depends on the realization of the generators of the Poincar\'e-Weyl algebra.