From Quantum Deformations of Relativistic Symmetries to Modified Kinematics and Dynamics

TL;DR

This paper reviews how noncommutative space-time and quantum deformations of symmetries influence classical and quantum theories, including mechanics, field theory, and gravity, highlighting mathematical tools like Hopf algebras.

Contribution

It provides a concise overview of noncommutative space-time applications in quantum-deformed theories and discusses specific examples like canonical and κ-deformed geometries.

Findings

Noncommutative space-time modifies classical and quantum dynamics.

Hopf algebras are essential for describing quantum-deformed symmetries.

Noncommutative geometry leads to modifications in Einstein gravity.

Abstract

We present a short review describing the use of noncommutative space-time in quantum-deformed dynamical theories: classical and quantum mechanics as well as classical and quantum field theory. We expose the role of Hopf algebras and their realizations (noncommutative modules) as important mathematical tool describing quantum-deformed symmetries: quantum Lie groups and quantum Lie algebras. We consider in some detail the most studied examples of noncommutative space-time geometry: the canonical and $\kappa$-deformed cases. Finally we briefly describe the modifications of Einstein gravity obtained by introduction of noncommutative space-time coordinates.