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Physical constraints on quantum deformations of spacetime symmetries. (English) Zbl 1395.83031

Summary: 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 centerpiece 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 characterization of such deformations in 2, 3 and 4 spacetime dimensions motivated by 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.

MSC:

83C45 Quantization of the gravitational field
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B62 Lie bialgebras; Lie coalgebras
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83A05 Special relativity
83C65 Methods of noncommutative geometry in general relativity
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