The classical basis for the κ-Poincaré Hopf algebra and doubly special relativity theories

Abstract

Several issues concerning quantum κ−Poincaré algebra are discussed and reconsidered here. We propose two different formulations of κ−Poincaré quantum algebra. Firstly we present a complete Hopf algebra formulae of κ−Poincaré in classical Poincaré basis. Further by adding one extra generator, which modifies the classical structure of Poincaré algebra, we eliminate non polynomial functions in the κ− parameter. Hilbert space representations of such algebras make Doubly Special Relativity (DSR) similar to the Stueckelberg's version of (proper-time) relativistic Quantum Mechanics.

Unified description for $$\kappa $$ κ -deformations of orthogonal groups

“…The above reminds one of some formulas from [39]. This enables us to introduce the new system of generators…”

“…In order to preserve a compact form for the formulas (5)- (6) we have also introduced the following notation (extending our previous notation from [39]):…”

“…Drinfeld-Jimbo-type) deformation one can always switch to the so-called q-analog version with all infinite series hidden in the one additional generator. In the case of κ-Poincaré it is −1 τ which solves a specialization problem for κ, for details see [39].…”

On Hopf algebroid structure of κ-deformed Heisenberg algebra

“…In particular one can choose F µ ( p) in a way leading to classical Poincaré algebra in algebraic sector of κ-deformed Poincaré algebra [33]- [35]. In such a way we get more complicated nonabelian formulae for the composition law of fourmomenta, which unfortunately are not endowed with physical interpretation.…”

Дубли Гейзенберга Квантованных Алгебр Пуанкаре

“…Позднее мы на-помним θ-твистованную алгебру Пуанкаре [19], [20]. В завершающей части мы также рассмотрим κ-деформированный дубль Гейзенберга алгебры Пуанкаре [2], но в клас-сическом базисе [21], а не в используемом обычно базисе би-смеш-произведения.…”