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      Physical Constraints on Quantum Deformations of Spacetime Symmetries

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          Abstract

          In this work we study the deformations into Lie bialgebras of the three relativistic Lie algebras: de Sitter, Anti-de Sitter and Poincar\'e, which describe the symmetries of the three maximally symmetric spacetimes. These algebras represent the centrepiece of the kinematics of special relativity (and its analogue in (Anti-)de Sitter spacetime), and provide the simplest framework to build physical models in which inertial observers are equivalent. Such a property can be expected to be preserved by Quantum Gravity, a theory which should build a length/energy scale into the microscopic structure of spacetime. Quantum groups, and their infinitesimal version `Lie bialgebras', allow to encode such a scale into a noncommutativity of the algebra of functions over the group (and over spacetime, when the group acts on a homogeneous space). In 2+1 dimensions we have evidence that the vacuum state of Quantum Gravity is one such `noncommutative spacetime' whose symmetries are described by a Lie bialgebra. It is then of great interest to study the possible Lie bialgebra deformations of the relativistic Lie algebras. In this paper, we develop a classification of such deformations in 2, 3 and 4 spacetime dimensions, based on physical requirements based on dimensional analysis, on various degrees of `manifest isotropy' (which implies that certain symmetries, i.e. Lorentz transformations or rotations, are `more classical'), and on discrete symmetries like P and T. On top of a series of new results in 3 and 4 dimensions, we find a no-go theorem for the Lie bialgebras in 4 dimensions, which singles out the well-known `\(\kappa\)-deformation' as the only one that depends on the first power of the Planck length, or, alternatively, that possesses `manifest' spatial isotropy.

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          Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale

          I show that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of space-times whose short-distance structure is governed by an observer-independent length scale. The consistency of these postulates proves incorrect the expectation that modifications of the rules of kinematics involving the Planck length would necessarily require the introduction of a preferred class of inertial observers. In particular, it is possible for every inertial observer to agree on physical laws supporting deformed dispersion relations of the type \(E^2- c^2 p^2- c^4 m^2 + f(E,p,m;L_p)=0\), at least for certain types of \(f\).
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            Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity

            , (2009)
            We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. By adding boundary terms to the first order Einstein Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2+1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL(2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL(2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein Gordon equation.
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              Quantum \(\kappa\)-Poincare in Any Dimensions

              The \(\kappa\)-deformation of the D-dimensional Poincar\'e algebra \((D\geq 2)\) with any signature is given. Further the quadratic Poisson brackets, determined by the classical \(r\)-matrix are calculated, and the quantum Poincar\'e group "with noncommuting parameters" is obtained.
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                Author and article information

                Journal
                26 February 2018
                Article
                1802.09483
                fdf5a069-3966-4439-896f-37404511d077

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                12 pages, 1 table
                hep-th

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