Quantum Deformations of Einstein's Relativistic Symmetries

TL;DR

This paper explores two approaches to modifying Einstein's Poincaré symmetries using quantum group structures, introducing deformations characterized by fundamental constants, and discusses their implications for quantum spacetime models.

Contribution

It presents two types of quantum Poincaré symmetries, one with constant noncommutativity and another with Lie-algebraic structure, incorporating fundamental constants into relativistic symmetries.

Findings

Describes $ heta_{ ueta}$-deformation with constant noncommutativity.

Introduces $oldsymbol{ ext{kappa}}$-deformation with Lie-algebraic structure.

Highlights the role of fundamental mass or length in quantum symmetries.

Abstract

We shall outline two ways of introducing the modification of Einstein's relativistic symmetries of special relativity theory - the Poincar\'{e} symmetries. The most complete way of introducing the modifications is via the noncocommutative Hopf-algebraic structure describing quantum symmetries. Two types of quantum relativistic symmetries are described, one with constant commutator of quantum Minkowski space coordinates ($\theta_{\mu\nu}$-deformation) and second with Lie-algebraic structure of quantum space-time, introducing so-called $\kappa$-deformation. The third fundamental constant of Nature - fundamental mass $\kappa$ or length $\lambda$ - appears naturally in proposed quantum relativistic symmetry scheme. The deformed Minkowski space is described as the representation space (Hopf-module) of deformed Poincar\'{e} algebra. Some possible perspectives of quantum-deformed relativistic symmetries will be outlined.