Abstract
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).
Highlights
One can distinguish two basic levels in quantization procedure of physical models describing contemporary fundamental interactions: (i) The first level can be called quantum-mechanical with canonically quantized phase space coordinates and possible presence of classical gravity only as a static background
Quantum gravity (QG) remains a subject of rather hypothetical models, it is mostly agreed that QG effects require at ultra-short distances the replacement of classical Einsteinian space-time by quantum noncommutative space-time geometry
The first parameter, c, appears in the physical basis of the relativistic classical algebra A and the second parameter, λ, determines the QG-induced modification of the algebraic structure (2). It was argued already in the 1930s [18] that QG models should at the basic level depend on three fundamental nonvanishing constants, c, } and G, where G can be replaced by λ p or m p (see (1)); if the cosmological constant or de Sitter radius of the Universe is finite, it introduces additional geometric parameter
Summary
One can distinguish two basic levels in quantization procedure of physical models describing contemporary fundamental interactions:. The first parameter, c, appears in the physical basis of the relativistic classical algebra A and the second parameter, λ, determines the QG-induced modification of the algebraic structure (2) It was argued already in the 1930s [18] that QG models should at the basic level depend on three fundamental nonvanishing constants, c, } and G, where G can be replaced by λ p or m p (see (1)); if the cosmological constant or de Sitter radius of the Universe is finite, it introduces additional geometric parameter. The model of quantum space-time symmetries, which was an inspiration for introducing DSR algebras, is provided by the κ-deformed Poincaré–Hopf algebra [20,21] with semi-direct product structure presented in [22] in so-called bicrossproduct basis. It should be recognized here that there are already several interesting papers dealing with quantum deformations of twistors and their geometries (see, e.g., [23–29])
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