Noncommutative Differential Calculus on the Kappa-Minkowski Space

Abstract

Following the construction of the $\kappa$-Minkowski space from the bicrossproduct structure of the $\kappa$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $\kappa$-Minkowski space and and we briefly discuss the main properties of its differential calculi.

LaTeX 10 pages

References in corpus (4)

• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Quantum $\kappa$-Poincare in Any Dimensions
• The Quantum Deformed Dirac Equation from the k-Poincare` Algebra
• The Dirac-Coulomb Problem for the $\kappa$-Poincare Quantum Group

Cited by in corpus (80)

• Quantum Spacetime Phenomenology
• Fractal properties of quantum spacetime
• Deformed Field Theory on kappa-spacetime
• Kappa-Minkowski space-time and the star product realizations
• Doubly Special Relativity and de Sitter space
• Covariant realizations of kappa-deformed space
• Field theory on $\kappa$--Minkowski space revisited: Noether charges and breaking of Lorentz symmetry
• Scalar Field Theory on Non-commutative Snyder Space-Time
• From noncommutative kappa-Minkowski to Minkowski space-time
• Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
• Speed of particles and a relativity of locality in $\kappa$-Minkowski quantum spacetime
• Velocity of particles in Doubly Special Relativity
• Dirac spinors for Doubly Special Relativity and $\kappa$-Minkowski noncommutative spacetime
• Gravity induced from quantum spacetime
• Scalar field theory in Snyder space-time: alternatives
• Generalized kappa-deformed spaces, star-products, and their realizations
• Scalar field theory on $\kappa$-Minkowski space-time and Doubly Special Relativity
• Derivatives, forms and vector fields on the kappa-deformed Euclidean space
• Planck-scale-modified dispersion relations in homogeneous and isotropic spacetimes
• Noncommutative Differential Forms on the kappa-deformed Space
• A twisted look on kappa-Minkowski: U(1) gauge theory
• The $\kappa$-(A)dS noncommutative spacetime
• Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space
• $\kappa$-Poincaré invariant quantum field theories with KMS weight
• Differential structure on kappa-Minkowski space, and kappa-Poincare algebra
• Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces
• Gauge theories on quantum spaces
• K-Poincare-Hopf algebra and Hopf algebroid structure of phase space from twist
• Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories
• Discreteness of area in noncommutative space
• Gravitational Collapse of a Homogeneous Scalar Field in Deformed Phase Space
• Relativistic and Newtonian kappa spacetimes
• Unified description for kappa-deformations of orthogonal groups
• Differential algebras on kappa-Minkowski space and action of the Lorentz algebra
• Differential structure on the $\kappa$-Minkowski spacetime from twist
• Deformed and extended Galilei group Hopf algebras
• Relative locality in a quantum spacetime and the pregeometry of $\kappa$-Minkowski
• Differential Structure on kappa-Minkowski Spacetime Realized as Module of Twisted Weyl Algebra
• Noncommutative fields and the short-scale structure of spacetime
• Gauge theories on $\kappa$-Minkowski spaces: Twist and modular operators
• Differential calculi on quantum Minkowski space
• $\kappa$-deformed complex fields and discrete symmetries
• Almost commutative Riemannian geometry: wave operators
• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Signal propagation on $\kappa$-Minkowski spacetime and non-local two-point functions
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• $\kappa -$Minkowski spacetime and a uniformly accelerating observer
• Scalar Field theory in $\kappa$-Minkowski spacetime from twist
• Quantum groups and deformed special relativity
• Quantum kappa-deformed differential geometry and field theory
• Differential forms and k-Minkowski spacetime from extended twist
• Universal $\kappa$-Poincaré covariant differential calculus over $\kappa$-Minkowski space
• Quasitriangular structure and twisting of the 2+1 bicrossproduct model
• Spectral dimensions and dimension spectra of quantum spacetimes
• Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras
• Noncommutative spherically symmetric spacetimes at semiclassical order
• Quantum geodesics on $\lambda$-Minkowski spacetime
• Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
• On the 5D differential calculus and translation transformations in 4D kappa-Minkowski noncommutative spacetime
• Single Extra Dimension from $\kappa$-Poincaré and Gauge Invariance
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Blackbody radiation in κ-Minkowski spacetime
• Gauge fields on noncommutative geometries with curvature
• A group theoretic description of the $\kappa$-Poincaré Hopf algebra
• Extending general covariance: Moyal-type noncommutative manifolds
• A short introduction to $\kappa$-deformation
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Compact $\kappa$-deformation and spectral triples
• Quantum Koszul formula on quantum spacetime
• Quantum instability of gauge theories on $\kappa$-Minkowski space
• Central tetrads and quantum spacetimes
• Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
• Scaling Limit of the Noncommutative Black Hole
• Fate of $\kappa$-Minkowski space-time in non-relativistic (Galilean) and ultra-relativistic (Carrollian) regimes
• Covariant formulation of Noether's Theorem for translations on kappa-Minkowski spacetime
• $\kappa$-General-Relativity I: a Non-Commutative GR Theory with the $\kappa$-Minkowski Spacetime as its Flat Limit
• Generalization of Weyl realization to a class of Lie superalgebras
• Time dependent transitions with time-space noncommutativity & its implications in Quantum Optics
• Asymmetry in momentum space: restoring $\mathcal{C}\mathcal{P}\mathcal{T}$ invariance of $\kappa$-field theory