Physical Constraints on Quantum Deformations of Spacetime Symmetries

Abstract

In this work we study the deformations into Lie bialgebras of the three relativistic Lie algebras: de Sitter, Anti-de Sitter and Poincaré, 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.

arXiv comment13 pages, 1 table, matches version published on Nuclear Physics B

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Cited by in corpus (11)

• Localization and Reference Frames in $\kappa$-Minkowski Spacetime
• Pauli-Jordan Function and Scalar Field Quantization in $\kappa$-Minkowski Noncommutative Spacetime
• The Momentum Spaces of $\kappa$-Minkowski noncommutative spacetime
• Noncommutative spaces of worldlines
• Light Cone in a Quantum Spacetime
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• The Poincaré group as a Drinfel'd double
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Multiparticle states in braided lightlike $\kappa$-Minkowski noncommutative QFT
• The Weyl-Mellin quantization map for $\kappa$-Minkowski Noncommutative Spacetime
• Fuzzy worldlines with $\kappa$-Poincaré symmetries