Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories

Abstract

Classification of differential forms on $\kappa$-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the $\kappa$-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to $\kappa$-Poincaré Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

37 pages, major revision, new section added, accepted for publication in JHEP

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Cited by in corpus (28)

• The $\kappa$-(A)dS noncommutative spacetime
• Observables and Dispersion Relations in k-Minkowski Spacetime
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebras
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Minimal and maximal lengths from position-dependent noncommutativity
• Effects of quantum deformation on the integer quantum Hall effect
• $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations
• Twisted statistics and the structure of Lie-deformed Minkowski spaces
• Associative realizations of $\kappa$-deformed extended Snyder model
• Vacuum energy and the cosmological constant problem in $\kappa$-Poincaré invariant field theories
• Noncommutative correction to the entropy of charged BTZ black hole
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Twisted conformal algebra related to $\kappa$-Minkowski space
• The Weyl realizations of Lie algebras and left-right duality
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Symmetries of $\kappa$ Minkowski space-time: A possibility of exotic momentum space geometry?
• Twist for Snyder space
• Central tetrads and quantum spacetimes
• Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
• Propagation of spinors on a noncommutative spacetime: equivalence of the formal and the effective approach
• Two-particle system in Coulomb potential for twist-deformed space-time
• A Stochastic Origin of Spacetime Non-Commutativity
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications
• Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries
• Twist deformations leading to kappa-Poincare Hopf algebra and their application to physics
• Generalization of Weyl realization to a class of Lie superalgebras
• Maxwell's equations and Lorentz force in doubly special relativity