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 $\kappa$-Poincar\'e 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.
Highlights
Unravel some aspects of quantum gravity, those that emerge in the low energy limit and relate them to observable physical phenomena
Among many approaches in this direction we chose the approach of noncommutative (NC) gravity and field theory based on deformations of symmetry [1,2,3,4,5,6,7,8]
In a similar way the symmetry of the field theory, which is the Poincaré symmetry, is replaced by its twist deformed version. Both symmetry deformations can be interpreted as quantum symmetries and the mathematical framework for their study is the theory of quantum groups and Hopf algebras [9,10,11,12,13,14]
Summary
We want to construct the most general algebra of differential one-forms ξμ ≡ dxμ ∈ Ω 1 compatible with κ-Minkowski spacetime that is bicovariant, i.e. closed in differential forms (the differential calculus is of classical dimension). Solutions C1, C2, C3 are new and each of them corresponds to time-like, light-like and space-like deformation parameter aμ. In [48, 49], we have constructed universal κ-Poincaré covariant differential calculus over κ-Minkowski space This universal algebra has been generated by xμ, Mμν, η, ξμ (where Mμν are Lorentz generators) for time-like, lightlike and space-like deformation parameter aμ. In case of light-like deformation a2 = 0, the one-forms have classical dimension, i.e. the additional one-form does not appear. Among all the solutions for light-like deformation (a2 = 0), only solution C4 coincides with light-like case constructed in [48, 49] It follows that solution C4 is compatible with κ-Poincare Hopf algebra, which is demonstrated in sections 5.1 and 6.3
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