Kappa-Minkowski spacetime: mathematical formalism and applications in Planck scale physics

TL;DR

This paper explores the mathematical formalism of kappa-Minkowski spacetime and its application to Doubly Special Relativity, highlighting how deformed symmetries and dispersion relations can model Planck scale physics.

Contribution

It introduces a formalism connecting noncommutative spacetime with deformed relativistic symmetries using Hopf algebras, and demonstrates how DSR algebra can be derived from undeformed algebra.

Findings

Deformed relativistic symmetries described by kappa-Poincare Hopf algebra.

DSR algebra constructed as a nonlinear transformation of undeformed algebra.

Realizations of quantum spacetime lead to deformed dispersion relations.

Abstract

The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, which unifies noncommutative coordinates with generators of the symmetry algebra. It contains the deformed Heisenberg-Weyl subalgebra. It was proved that DSR algebra can be obtained by nonlinear change of generators from undeformed algebra. We show that the possibility of applications in Planck scale physics is connected with certain realizations of quantum spacetime, which in turn leads to deformed dispersion relations.