Quantum Deformations of D = 4 Poincaré Algebra

Recent work showed that κ-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter Λ, the Planck scale quantum deformation parameter κ and the speed of light parameter c to move to the well-studied κ-Poincaré algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these algebras of relativistic symmetries and their associated noncommutative spacetimes, emphasizing the nontrivial effects of interplay between curvature, quantum deformation and speed of light parameters.

Download Full-text

Colourful Poincaré symmetry, gravity and particle actions

Journal of High Energy Physics ◽

10.1007/jhep08(2021)047 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Joaquim Gomis ◽

Euihun Joung ◽

Axel Kleinschmidt ◽

Karapet Mkrtchyan

Keyword(s):

Field Theory ◽

Free Particle ◽

Three Dimensional ◽

Background Field ◽

Quantum Field ◽

Symmetry Factor ◽

Starting Point ◽

Poincaré Algebra ◽

Poincaré Symmetry ◽

Three Space

Abstract We construct a generalisation of the three-dimensional Poincaré algebra that also includes a colour symmetry factor. This algebra can be used to define coloured Poincaré gravity in three space-time dimensions as well as to study generalisations of massive and massless free particle models. We present various such generalised particle models that differ in which orbits of the coloured Poincaré symmetry are described. Our approach can be seen as a stepping stone towards the description of particles interacting with a non-abelian background field or as a starting point for a worldline formulation of an associated quantum field theory.

Download Full-text

Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

Symmetry ◽

10.3390/sym13081309 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1309

Author(s):

Jerzy Lukierski

Keyword(s):

Phase Space ◽

Hopf Algebras ◽

Deformation Parameter ◽

Heisenberg Algebra ◽

Planck Length ◽

Covariant Quantum ◽

Drinfeld Twist ◽

Quantum Deformations ◽

Heisenberg Double ◽

Conformal Symmetries

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

Download Full-text