Interpolations between Jordanian twists, the Poincar\'e-Weyl algebra and dispersion relations

TL;DR

This paper explores a family of Jordanian twists and their interpolations, analyzing their algebraic structures, and applies these to construct deformations of spacetime and dispersion relations relevant for quantum physics.

Contribution

It introduces a new two-parameter family of Jordanian twists, analyzes their relations, and applies similarity transformations to derive novel dispersion relations in quantum spacetime models.

Findings

Existence of a one-parameter family of twists identical to a simple Jordanian twist

Construction of deformed $$-Minkowski spacetime with star product and coordinate realizations

Derivation of two types of dispersion relations from Poincare9-Weyl algebra deformations

Abstract

We consider a two parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and discussed. It is proved that there exists a one parameter family of twists identical to a simple Jordanian twist. The twisted coalgebra, star product and coordinate realizations of the $\kappa$-Minkowski noncommutative space time are presented. Real forms of Jordanian deformations are also discussed. The method of similarity transformations is applied to the Poincar\'e-Weyl Hopf algebra and two types of one parameter families of dispersion relations are constructed. Mathematically equivalent deformations, that are related to nonlinear changes of symmetry generators and linked with similarity maps, may lead to differences in the description of physical phenomena.