Noncommutative differential forms on the kappa-deformed space

Abstract

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity.We also consider the star-product of classical differential forms. The starproduct is well-defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.

Relativistic motion enhanced quantum estimation of $$\kappa $$ κ -deformation of spacetime

“…In the limit of κ → ∞, the Poincaré algebra is recovered. To construct the κ-Poincaré algebra, we require the realizations of the noncommutative coordinatesx μ in terms of ordinary commutative coordinates x μ and their derivatives ∂ μ , where [24][25][26][27][28][29][30][31]. A family of realizations for noncommu-tative coordinatesx μ satisfying the algebra in Eq.…”

Scalar field theory on noncommutative Snyder spacetime

“…Consider two momentap and p in two distinct realizations. Since momenta transform as vectors under the Lorentz symmetry, see (3), the relatioñ p ¼ p fðAÞ (14) holds. The function fðAÞ, such that fð0Þ ¼ 1, depends on the realization ' 1 and can be obtained as follows.…”

“…Because different momenta are related by (14), the antipode Sðp Þ is exactly (not only at the first order) trivial in all realizations. On the other hand, the co-unit "ðgÞ is also trivial for any g 2 P S .…”

“…For example Minkowski, a particular case of Lie algebra-type space-time, has been developed at different levels specifying star products [11,12], differential calculus [13,14], scalar field theory [15][16][17][18], and conserved charges [19][20][21]. The key difference between Snyder and Minkowski is that in the latter the momentum space has the structure of a non-Abelian Lie group and thus the coproduct is noncommutative, but still associative.…”

Κ-Deformed DIRAC EQUATION

“…It has been shown in [8], that the symmetry algebra of the κ-deformed space-time is defined by the κ-Poincare algebra, which is a Hopf algebra. The symmetry algebra of the κ-deformed space-time can also be realised using undeformed κ-Poincare algebra, where the defining relations are same as that of the usual Poincare algebra, but then the explicit form of the generators are deformed [9][10][11][12][13].Various studies have been carried out analysing field theory models defined in κ-deformed spacetime [14][15][16][17][18][19][20][21][22][23][24]. In most of these studies, field equations, which are invariant under the symmetry algebra, defined in κ-deformed space-times are set up from the κ-deformed quadratic Casimir [9] of the algebra.…”