Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

TL;DR

This paper constructs quantum inhomogeneous conformal symmetries and twistorial DSR algebras using noncommutative twistors, introducing new quantum deformations and phase spaces relevant for quantum gravity models.

Contribution

It introduces palatial NC twistors from quantum conformal Hopf algebras and develops twistorial DSR algebras linked with Planck-scale deformations.

Findings

Construction of palatial NC twistors as quantum-covariant modules.

Introduction of twistorial DSR algebra with Planck-scale deformation.

Development of generalized quantum twistorial phase space.

Abstract

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomegeneous conformal Hopf algebras $\mathcal{U}_{\theta }(su(2,2)\ltimes T^{4}$) and $\mathcal{U}_{\bar{\theta}}(su(2,2)\ltimes\bar{T}^{4}$), where $T^{4}$ describe complex twistor coordinatesand $\bar{T}^{4}$ 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) $su(2,2)\ltimes H^{4,4}_\hslash$ ($H^{4,4}_\hslash=\bar{T}^4 \ltimes_\hslash T_4$ 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 $\lambda_p$ will be called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We shall describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and by the quantization map in $H_\hslash^{4,4}$. We introduce as well generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra $\mathcal{U}_\theta(su(2,2)\ltimes T^4).$