The $\kappa$-Newtonian and $\kappa$-Carrollian algebras and their noncommutative spacetimes

Abstract

We derive the non-relativistic $c\to \infty$ and ultra-relativistic $c\to 0$ limits of the $\kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $\kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $\kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $\kappa$-Newtonian and $\kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $\kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $\kappa$, the curvature parameter $\eta$ and the speed of light parameter $c$.

arXiv comment19 pages. v2 matches version accepted for publication by Phys. Lett. B; some references added

References in corpus (39)

• Quantum gravity and minimum length
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Minimal Length Scale Scenarios for Quantum Gravity
• Quantum Spacetime Phenomenology
• Gauge Theory on Noncommutative Spaces
• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Doubly Special Relativity theories as different bases of $\kappa$--Poincaré algebra
• Gauging the Carroll Algebra and Ultra-Relativistic Gravity
• Deformed Boost Transformations That Saturate at the Planck Scale
• Carroll versus Galilei Gravity
• Spin and Statistics on the Groenewold-Moyal Plane: Pauli-Forbidden Levels and Transitions
• Dynamics of Carroll Particles
• Waves on Noncommutative Spacetime and Gamma-Ray Bursts
• Real Forms of Complex Quantum Anti de Sitter Algebra $U_q(Sp(4,C))$ and their Contraction Schemes
• Relative-locality distant observers and the phenomenology of momentum-space geometry
• Planck-scale-modified dispersion relations in FRW spacetime
• Dynamics of Carroll Strings
• Quantum evolution in spacetime foam
• Non-Relativistic Spacetimes with Cosmological Constant
• Covariant Non-Commutative Space-Time
• Conformal compactification and cycle-preserving symmetries of spacetimes
• Must Quantum Spacetimes Be Euclidean?
• Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras
• A non-commutative Minkowskian spacetime from a quantum AdS algebra
• Casimir invariants for the complete family of quasi-simple orthogonal algebras
• Kinematical Lie algebras via deformation theory
• The kappa-(A)dS quantum algebra in (3+1) dimensions
• The $\kappa$-(A)dS noncommutative spacetime
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• Classical dynamics on curved Snyder space
• On the Ultrarelativistic Limit of General Relativity
• Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups
• Kinematics of particles with quantum de Sitter symmetries
• Noncommutative spaces of worldlines
• Central Extensions of the Quasi-orthogonal Lie algebras
• Deformed Carroll particle from 2+1 gravity
• Relativistic compatibility of the interacting $\kappa$-Poincaré model and implications for the relative locality framework
• Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras
• (Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach

Cited by in corpus (14)

• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Double Quantization
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Deformed classical-quantum mechanics transition
• Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei
• $\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines
• The noncommutative space of light-like worldlines
• Fate of $\kappa$-Minkowski space-time in non-relativistic (Galilean) and ultra-relativistic (Carrollian) regimes
• Quantum Galilei group as quantum reference frame transformations
• Fuzzy worldlines with $\kappa$-Poincaré symmetries
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs
• On Carrollian and Galilean contractions of BMS algebra in 3 and 4 dimensions