Observables and Dispersion Relations in k-Minkowski Spacetime

Abstract

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.

30 pages. Revised version further motivating Poincare'-Weyl (conformal) symmetry. Added explicit proof of covariance of massive scalar fields wave equation under quantum translations (eq. (76)). To appear in JHEP

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• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Poisson gauge theory
• Braided Symmetries in Noncommutative Field Theory
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Physical Constraints on Quantum Deformations of Spacetime Symmetries
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Dispersion Relations in $\kappa$-Noncommutative Cosmology
• Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Interpolations between Jordanian Twists Induced by Coboundary Twists
• Accelerated horizons and Planck-scale kinematics
• Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces
• $\kappa$-deformed BMS symmetry
• Twist for Snyder space
• Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries
• BMS algebras in 4 and 3 dimensions, their quantum deformations and duals
• On Light-like Deformations of the Poincaré Algebra
• Two ${\theta }_{\mu \nu }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids
• Interpolations between Jordanian twists, the Poincaré-Weyl algebra and dispersion relations
• Electromagnetic Theory with Quantum Principal Bundles