T-Minkowski noncommutative spacetimes I: Poincaré groups, differential calculi and braiding

Abstract

This paper introduces and investigates a class of noncommutative spacetimes that I will call ``T-Minkowski,'' whose quantum Poincaré group of isometries exhibits unique and physically motivated characteristics. Notably, the coordinates on the Lorentz subgroup remain commutative, while the deformation is confined to the translations (hence the T in the name), which act like an integrable set of vector fields on the Lorentz group. This is similar to Majid's bicrossproduct construction, although my approach allows the description of spacetimes with commutators that include a constant matrix as well as terms that are linear in the coordinates (the resulting structure is that of a centrally-extended Lie algebra). Moreover, I require that one can define a covariant braided tensor product representation of the quantum Poincaré group, describing the algebra of N-points. This also implies that a 4-dimensional bicovariant differential calculus exists on the noncommutative spacetime. The resulting models can all be described in terms of a numerical triangular R-matrix through RTT relations (as well as RXX, RXY and RXdX relations for the homogeneous spacetime, the braiding and the differential calculus). The R-matrices I find are in one-to-one correspondence with the triangular r-matrices on the Poincaré group without quadratic terms in the Lorentz generators. These have been classified, up to automorphisms, by Zakrzewski, and amount to 16 inequivalent models. This paper is the first of a series, focusing on the identification of all the quantum Poincaré groups that are allowed by my assumptions, as well as the associated quantum homogeneous spacetimes, differential calculi and braiding constructions.

arXiv comment25 pages, no figures. Matches the version accepted for publication by Progress of Theoretical and Experimental Physics

References in corpus (43)

• String Theory and Noncommutative Geometry
• Noncommutative Field Theory
• Quantum Field Theory on Noncommutative Spaces
• The quantum structure of spacetime at the Planck scale and quantum fields
• Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale
• Modern tests of Lorentz invariance
• Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry
• Minimal Length Scale Scenarios for Quantum Gravity
• The Spectral Action Principle
• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• Minimal Length Uncertainty Relation and Ultraviolet Regularisation
• Noncommutative Geometry and Gravity
• 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory
• Non-commutative superspace from string theory
• Quantum $\kappa$-Poincare in Any Dimensions
• Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity
• Untwisting Noncommutative R^d and the Equivalence of Quantum Field Theories
• Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
• New Lie-Algebraic and Quadratic Deformations of Minkowski Space from Twisted Poincare Symmetries
• Poisson structures on the Poincare group
• On "full" twisted Poincare' symmetry and QFT on Moyal-Weyl spaces
• Real Forms of Complex Quantum Anti de Sitter Algebra $U_q(Sp(4,C))$ and their Contraction Schemes
• A New Null-Plane Quantum Poincare Algebra
• Construction of $\theta$-Poincaré Algebras and their Invariants on ${\mathcal{M}}_{\theta }$
• 4D Quantum Affine Algebras and Space--Time q-Symmetries
• Braided Quantum Field Theory
• Twisted Classical Poincaré Algebras
• Noncommutative Lorentz Symmetry and the Origin of the Seiberg-Witten Map
• A non-commutative Minkowskian spacetime from a quantum AdS algebra
• Non-Commutative Geometry and Twisted Conformal Symmetry
• Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces
• Null-plane Quantum Universal $R$-matrix
• First Combined Study on LIV from Observations of Energy-dependent Time Delays from Multiple-type Gamma-ray Sources -- Part I. Motivation, Method Description and Validation through Simulations of H.E.S.S., MAGIC and VERITAS Datasets
• Phase spaces of Doubly Special Relativity
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• The Momentum Spaces of $\kappa$-Minkowski noncommutative spacetime
• Braided Symmetries in Noncommutative Field Theory
• An Interpretation of Noncommutative Field Theory in Terms of a Quantum Shift
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• Localization and observers in $\varrho$-Minkowski spacetime
• Quantum particles in non-commutative space-time: an identity crisis
• On the consequences of twisted Poincare' symmetry upon QFT on Moyal noncommutative spaces
• Multiparticle states in braided lightlike $\kappa$-Minkowski noncommutative QFT

Cited by in corpus (1)

• T-Minkowski noncommutative spacetimes II: classical field theory