Kappa Snyder deformations of Minkowski spacetime, realizations, and Hopf algebra

Abstract

We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and κ-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from κdeformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the Hopf structure and star product are found. Special cases, particularly Snyder and κ-Minkowski in Maggiore-type realizations are discussed. The same generalized Hopf algebraic structures are as well considered in the case of an arbitrary allowable kind of realisation and results are given perturbatively up to second order in deformation parameters.

Scalar field theory on noncommutative Snyder spacetime

“…It can be obtained by a straightforward implementation of the Campbell-BakerHausdorff formula or by the more elegant method developed in [24,49]. The star product between two plane waves is defined by F ðk; qÞ as e iðkxÞ ?…”

“…Consider now a noncommutative plane wave e iðkxÞ , in whichx refers to a given realization (6), and k are the eigenvalues of p ¼ Ài@=@x . It is then possible to show that [24,49] e iðkxÞ xI ¼ e iðKxÞ ;…”

“…In particular, the Snyder space-time is not a special case of Minkowski [22,23]. In fact, as clarified in [24], spaces are based on Lie algebra while Snyder space is grounded on trilinear commutations relations.…”

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

“…5.3), the star product is non-associative and the twist does not satisfy the cocycle condition. Field theories defined on spaces with non-associative star products are constructed; see for example versions on κ-Snyder space [44,45] and on Snyder space [42,[58][59][60]. The properties of field theories on non-associative star products are currently under investigation.…”

Plane waves in noncommutative fluids

“…The noncommutative fluid was constructed as a noncommutative field theory [13,14] in the realization method approach [15,16,17,18,19,20,21,22,23,24,25,26,27] by generalizing the first order action functional of the commutative perfect relativistic fluid [28,29,30,31]. In this approach one obtains the fluid dynamics of the long wavelength degrees of freedom of the system bypassing the statistical analysis of the microscopic degrees of freedom which in the noncommutative spaces is not well understood yet (see for tentative approaches [32,33,34]).…”